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1110.3870

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present study deals with the model calculations of co cameron band and @xmath4doublet ultraviolet emissions in martian dayglow and the impact of solar euv flux on the calculated intensities . photoelectrons generated due to photoionization in the martian atmosphere have been degraded in the atmosphere using monte carlo model based analytical yield spectrum technique . emission rates of cameron and @xmath4 uv doublet bands due to photon and electron impact on @xmath19 have been calculated using euvac and s2k solar euv flux models . densities of prominent ions and co(a@xmath1 ) in martian upper atmosphere are calculated under steady state photochemical equilibrium condition . production rates of cameron and @xmath4 uv doublet bands are height - integrated to calculate overhead intensity and along the line of sight to obtain limb intensities . limb intensities are compared with the spicam / mars express and uv spectrometer / mariner observed intensities . due to higher euv fluxes at longer ( 7001050 ) wavelengths in the s2k model , the contribution of photodissociation of @xmath19 in producing cameron band is about 50% higher in low as well as in high solar activity conditions . variations in euv fluxes at longer wavelengths from solar minimum to solar maximum are less prominent in the euvac solar euv flux model compared to the s2k model . for low solar activity condition , limb intensities of cameron and @xmath4 uv doublet bands around peak brightness calculated using s2k model are @xmath23040% higher than those calculated using euvac model . comparison of calculated intensities has been made with the spicam - observed values for condition similar to the viking . intensities calculated using both s2k and euvac models are higher than the observed values . calculated altitude of emission peak of co cameron and @xmath4 uv doublet bands is also higher by @xmath25 km than the observed value . a reduction in the e-@xmath19 cross section forming cameron band by a factor of 2 and the density of @xmath19 in model atmosphere by a factor of 1.5 brings the calculated intensity ( using euvac model ) of cameron band in close agreement with the spicam observation . while modelling the recent observations made by spicam on - board mars express , we have taken two set of conditions with different model atmospheres and solar longitudes . in the first case , ls @xmath54 130@xmath22 ( ls = 100 - 130@xmath22 ) , atmosphere is taken from mars thermospheric general circulation model @xcite and calculations are made for the day 24 october 2004 with moderated solar activity flux ( f10.7 = 88 ) . total intensities of co cameron and @xmath4 uv doublet bands calculated using s2k model are around @xmath2615% higher than those calculated using euvac model . contribution of @xmath19 photodissociative excitation in cameron band production is 50% higher when s2k model is used . dissociative recombination of @xmath4 is an important source of cameron band in this case ( cf . [ fig : ver - cam - shem ] ) due to higher densities of @xmath4 ion ( fig . [ fig : ionden ] ) compared to those calculated for low solar activity condition ( viking condition ) . calculated intensities of cameron and uv doublet bands have been compared with the spicam - observed limb intensities . intensities calculated using s2k and euvac solar flux models are higher than the observed values by a factor of 1.7 to 2 for cameron and a factor of 1.4 for uv doublet bands ( see fig . [ fig : int - min ] ) . we found that altitude of peak emission of both cameron and uv doublet bands are 2 to 3 km higher than observed profiles . this discrepancy in observed and calculated intensities and altitudes could be due to the fact that observed values are averaged over several days of observations while calculation are carried out for a particular day . due to the dust storm during ls @xmath45 130@xmath22 , observed emission peak is around 10 km higher for both cameron and uv doublet compared with the spicam - observed values for mex orbit 1426 on 26 feb . 2005 @xcite . to model the emission during ls @xmath53 130@xmath22 , we have taken atmospheric model for solar maximum condition . intensities calculated using the s2k solar flux model are @xmath2818% higher than those calculated using the euvac model . these calculated intensities are higher than the observed - averaged values by a factor of @xmath132 for cameron band and @xmath1350% for uv doublet band . the calculated intensity of cameron band ( after reducing the e-@xmath19 cross sections by a factor of 2 ) is in agreement with the observed values ( fig . [ fig : int - max ] ) . in all three conditions discussed above , i.e. , low solar activity ( viking ) , and first ( ls @xmath32 130@xmath22 ) and second ( ls @xmath45 130@xmath22 ) cases , calculated intensities of both cameron and uv doublet bands are higher than observations . on an average , model values of @xcite for cameron and @xmath4 uv doublet bands are around a factor of 1.74 and 1.41 , respectively , higher than the spicam observations . @xcite also found that their calculated intensities of cameron and uv doublet bands are around 25% higher than the spicam - observed values . this shows that these discrepancies in the model and observed values are due the uncertainties in the input physical parameters in the model . uncertainties in cross sections , namely , e-@xmath19 cross section producing cameron band and photoionization of @xmath19 forming uv doublet band can be one of the causes of discrepancies in the model and observations . calculations are also made for the high solar ( mariner 6 and 7 observations ) activity condition . though the contribution of @xmath19 photodissociative excitation in cameron band production is higher when s2k model is used , but the total intensity of co cameron band calculated using the euvac model is slightly higher than that calculated using the s2k model . this is because of the higher photoelectron flux when euvac model is used ( fig . [ fig : pef ] ) . the calculated intensities of both cameron and uv doublet band are lower than the observed values ( fig . [ fig : int - mariner ] ) . 1 . generally , solar euv fluxes in bands are higher in s2k model except at few bands at shorter wavelength range ( @xmath32 250 ) . solar euv fluxes at longer wavelengths are higher in s2k model , specially in the 1000 - 1050 bin , where flux is around an order of magnitude higher than the corresponding flux in euvac model . solar euv flux at lines is smaller in the s2k model compared to that in the euvac model . 2 . due to higher euv flux at lines in the euvac model , peaks at 2030 ev range in the photoelectron flux are more prominent when euvac model is used . 3 . during high solar activity condition , calculated photoelectron fluxes are higher for euvac model due to higher euv fluxes below 250 in the euvac model . hence , intensities calculated using euvac model are higher by 510% than those calculated using s2k model . 4 . during solar minimum condition , the cameron and @xmath4 uv doublet intensities calculated using s2k solar flux model are @xmath23040% higher than those calculated using the euvac model . 5 . during both , solar minimum as well as maximum conditions , cameron band production due to photodissociative excitation of @xmath19 is about 50% higher when s2k solar euv flux model is used . altitude of peak production rate of cameron and @xmath4 uv doublet bands is independent of solar euv flux model used in the calculations . however , for the cameron band the altitude where photodissociation of @xmath19 takes over electron impact dissociation is higher in the euvac model compared to that in the s2k model . 7 . reduction in the e-@xmath19 cross section producing cameron band and photoionization cross section producing @xmath4 uv doublet band is required to bring the model calculations in agreement with the observations . 8 . for a given set of observation , and accounting for the uncertainties in the cross sections , intensities calculated using the euvac model are in better agreement with the observation than those calculated using the s2k model . simultaneous observation of solar euv flux with dayglow measurements would be very helpful in improving our understanding of the processes that governs the dayglow emissions on mars . more accurate measurements of cross sections for electron impact dissociation of @xmath19 producing cameron band and photoionization of @xmath19 in @xmath29 state are required for the better modelling of co cameron band and @xmath4 uv doublet band in martian atmosphere , as well as in other @xmath19containing atmospheres , like venus and comets . avakyan , s. v. , iiin , r. n. , lavrov , v. m. , ogurtsov , g. n. , 1998 . in : avakyan , s. v. ( ed . ) , collision processes and excitation of uv emission from planetary atmospheric gases : a handbook of cross sections . gordon and breach science publishers . barth , c. a. , hord , c. w. , pearce , j. b. , kelly , k. k. , anderson , g. p. , stewart , a. i. , 1971 . mariner 6 and 7 ultraviolet spectrometer experiment : upper atmosphere data . j. geophys . 76 , 2213 2227 . . bhardwaj , a. , 1999 . on the role of solar euv , photoelectrons , and auroral electrons in the chemistry of c(@xmath72d ) and the production of ci 1931 in the inner cometary coma : a case for comet p / halley . j. geophys . 104 , 1929 1942 . . bhardwaj , a. , singhal , r. p. , 1993 . optically thin h lyman alpha production on outer planets : low - energy proton acceleration in parallel electric fields and neutral h atom precipitation from ring current . j. geophys . 98 ( a6 ) , 9473 9481 . . bougher , s. w. , engel , s. , hinson , d. p. , murphy , j. r. , 2004 . mgs radio science electron density profiles : interannual variability and implications for the martian neutral atmosphere . j. geophys . res . 109 . . bougher , s. w. , engel , s. , roble , r. g. , foster , b. , 1999 . comparative terrestrial planet thermospheres : 2 . solar cycle variation of global structure and winds at equinox . j. geophys . 104 , 16591 16611 . . bougher , s. w. , engel , s. , roble , r. g. , foster , b. , 2000 . comparative terrestrial planet thermospheres : 3 . solar cycle variation of global structure and winds at solstices . 105 , 17669 17692 . . buonsanto , m. j. , richards , p. g. , tobiska , w. k. , solomon , s. c. , tung , y. .-k . , fennelly , j. a. , 1995 . ionospheric electron densities calculated using different euv flux models and cross sections : comparison with radar data . j. geophys . 100 ( a8 ) , 14569 14580 . . forget , f. , montmessin , f. , bertaux , j. l. , galindo , f. g. , lebonnois , s. , qumerais , e. , reberac , a. , dimarellis , e. , valverde , m. a. l. , 2009 . density and temperatures of the upper martian atmosphere measured by stellar occultations with mars express spicam . j. geophys . res . 114 . . gilijamse , j. j. , hoekstra , s. , meek , s. a. , metsl , m. , van de meerakker , s. y. t. , t , s. y. , meijer , g. , groenenboom , g. c. , c. , g. , 2007 . the radiative lifetime of metastable co ( a@xmath76=0 ) . j. chem . 127 , 2211024 . . haider , s. a. , bhardwaj , a. , 2005 . radial distribution of production rates , loss rates and densities corresponding to ion masses @xmath7740 amu in the inner coma of comet halley : composition and chemistry . icarus 177 , 196 216 . . halmann , m , laulicht , i , february 1966 . isotope effects on vibrational transition probabilities.iv . electronic transitions of isotopic c@xmath3 , co , cn , h@xmath3 , and ch molecules . astrophysical journal supplement 12 , 307 321 . . lean , j. l. , woods , t. n. , eparvier , f. g. , meier , r. r. , strickland , d. j. , correira , j. t. , evans , j. s. , 2011 . solar extreme ultraviolet irradiance : present , past , and future . j. geophys . res . 116 . . rosati , r. e. , johnsen , r. , golde , m. f. , 2003 . absolute yields of co ( @xmath79 , @xmath80 , @xmath81 ) + o from the dissociative recombination of co@xmath0 ions with electrons . j. chem . 119 , 11630 11635 . . seiersen , k. , al - khalili , a. , heber , o. , jensen , m. j. , nielsen , i. b. , pedersen , h. b. , safvan , c. p. , andersen , l. h. , aug 2003 . dissociative recombination of the cation and dication of co@xmath3 . a 68 ( 2 ) , 022708 . . shematovich , v. i. , bisikalo , d. v. , grard , j .- c . , cox , c. , bougher , s. w. , leblanc , f. , 2008 . monte carlo model of electron transport for the calculation of mars dayglow emissions . j. geophys . res . 113 . . , m. p. , gougousi , t. , johnsen , r. , golde , m. f. , may 1998 . measurement of the absolute yield of co(a@xmath82@xmath83)+o products in the dissociative recombination of co@xmath84 ions with electrons . j. 108 , 8400 8407 . . tobiska , w. k. , woods , t. , eparvier , f. , viereck , r. , floyd , l. , bouwer , d. , rottman , g. , white , o. r. , 2000 . the solar2000 empirical solar irradiance model and forecast tool . 62 , 1233 1250 . . lccccccccc band & band origin & & & + ( @xmath86 ) & & euvac & s2k & & euvac & s2k + 0 - 0 & 2063.0 & 716 & 845 & & 20 & 24 + 0 - 1 & 2158.4 & 713 & 840 & & 20 & 23.2 + 0 - 2 & 2261.0 & 311 & 367 & & 9 & 10 + 0 - 3 & 2374.0 & 79 & 93 & & 2 & 2.6 + 1 - 0 & 1992.5 & 1151 & 1358 & & 32 & 38 + 2 - 0 & 1927.5 & 603 & 711 & & 17 & 20 + 3 - 0 & 1868.1 & 181 & 213 & & 5 & 6 + 4 - 0 & 1813.0 & 38 & 45 & & 1 & 1.3 + . the ratio of the photoelectron flux at 130 km calculated using the two solar flux models is also shown with magnitude on right side y - axis . thin dotted horizontal line depicts the s2k / euvac ratio = 1 . ] and o@xmath41 ions and co(a@xmath1 ) molecule for solar minimum condition calculated using euvac ( solid curve ) and s2k ( dashed curve ) solar euv flux models . density of co(a@xmath1 ) molecule is plotted after multiplying by a factor of 100 . dotted curves show the densities of @xmath4 and o@xmath41 ions for first case ( ls @xmath54 130@xmath22 ) using euvac solar flux . ] uv doublet bands ( left panel ) and co cameron ( right panel ) for low solar activity condition . solid squares with error bars represents the spicam - observed values taken from @xcite . dashed curves show the calculated intensity ( using euvac model ) after reducing the density of @xmath19by a factor of 1.5 . dash - dotted curve shows the calculated intensity ( using euvac ) of cameron band with reduced density ( by a factor of 1.5 ) and reduced ( by a factor of 2 ) e-@xmath19 cameron band production cross section . ] uv doublet band ( left panel ) and co cameron band ( right panel ) for ls @xmath32 130@xmath22 . symbols represent the spicam - observed values taken from @xcite . dash - dotted curve shows the calculated intensity of cameron band with reduced ( by a factor of 2 ) e-@xmath19 cross section . ] uv doublet band ( left panel ) and co cameron band ( right panel ) for ls @xmath45 130@xmath22 . dash dotted curve shows the cameron band intensity for euvac model with e-@xmath19 cross section forming co(a@xmath1 ) reduced by a factor of 2 . open circles with error bars represent the spicam - observed intensity taken from @xcite . ] . solid curve with symbols shows the limb intensity calculated using s2k model at sza = 45@xmath22 . dashed curve shows the calculated intensity ( using euvac model ) at sza = 0@xmath22 . dash dotted curve shows the calculated intensity ( using euvac model ) at sza = 0@xmath22 and after reducing the e-@xmath19 cross sections forming co(a@xmath1 ) by a factor of 2 . symbol represents the observed intensity of cameron band and uv doublet band measured by mariner 6 and 7 @xcite . observed values are shown for 2 orbits each of mariner 6 ( for sza = 27 and 0@xmath22 ; open and solid triangle , respectively ) and mariner 7 ( for sza = 44 and 0@xmath22 ; open and solid circle , respectively ) . ]

calculated limb intensities profiles are compared with spicam / mars express and mariner observations . analytical yield spectrum densities of prominent ions and co molecule in excited triplet a state are calculated using major ion - neutral reactions . volume emission rates of co cameron and co uv doublet bands have been calculated for different observations ( viking condition , mariner and mars express spicam observations ) on mars . altitude of peak limb brightness of co cameron and co uv doublet band is found to be independent of solar euv flux models . calculated

this study is aimed at making a calculation about the impact of the two most commonly used solar euv flux models solar2000 ( s2k ) of and euvac model of on photoelectron fluxes , volume emission rates , ion densities and co cameron and co uv doublet band dayglow emissions on mars in three solar activity conditions : minimum , moderate , and maximum . calculated limb intensities profiles are compared with spicam / mars express and mariner observations . analytical yield spectrum ( ays ) approach has been used to calculate photoelectron fluxes in martian upper atmosphere . densities of prominent ions and co molecule in excited triplet a state are calculated using major ion - neutral reactions . volume emission rates of co cameron and co uv doublet bands have been calculated for different observations ( viking condition , mariner and mars express spicam observations ) on mars . for the low solar activity condition , dayglow intensities calculated using the s2k model are% higher than those calculated using the euvac model . during high solar activity , due to the higher euv fluxes at wavelengths below 250 in the euvac model , intensities calculated using euvac model are slightly higher (% ) than those calculated using s2k model . irrespective of the solar activity condition , production of cameron band due to photodissociative excitation of co is around 50% higher when s2k model is used . altitude of peak limb brightness of co cameron and co uv doublet band is found to be independent of solar euv flux models . calculated limb intensities of co cameron and co uv doublet bands are on an average a factor of and.5 , respectively , higher than the spicam mars express observation , while they are consistent with the mariner observations . 2

1702.08003

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one of the most elementary system configuation models in multi - user information theory is the broadcast channel ( bc ) . it has been introduced in the early seventies of the twentieth century by cover @xcite , and since then , a vast amount of papers and books , studying different topics of the broadcast problem , have been published . generally speaking , the bc is a communication model , where a single transmitter wishes to communicate different messages to two or more receivers . the various messages may be private ( i.e. , aimed to one receiver only ) or common ( i.e. , aimed to two or more receivers ) . although the characterization of the capacity region of the general bc is still an open problem , some special cases have been solved , most notably , the degraded bc ( dbc ) , first presented in @xcite . the capacity region of the dbc , conjectured by cover , was first proved to be achievable by bergmans @xcite , and the converse was established by bergmans @xcite and gallager @xcite . another special case , which is somewhat more general than the dbc and which was first introduced and solved by krner and marton @xcite , is the broadcast channel with degraded message sets , also known as the asymmetric broadcast channel ( abc ) . the direct part of their coding theorem relys on bergmans scheme , which suggested the use of an hierarchical random code : first generate cloud centers " , which designate messages intended to both the receiver with the relatively high channel quality , henceforth referred to as the _ strong user _ , and the receiver with the relatively low channel quality , henceforth referred to as the _ weak user_. then , in the second step , around " each cloud center , generate a codeword for each message that is intended to the strong user only . the transmitter sends a codeword pertaining to one of the clouds . the strong decoder fully decodes both the common message and his private message , whereas the weak decoder decodes the common message only . other channels in which one receiver is superior to another and channels with nested information were studied by csiszr and krner @xcite and by el gamal @xcite , to name a few . multi user information theory is , first and foremost , driven by the quest to characterize capacity regions , i.e. , the region of all sets of rates that allow reliable communication ( a.k.a . achievable rates ) . a somewhat sharper performance metric concerns the exponential decay rate ( the error exponent ) of the probability of error for each user , as a function of the coding rates within the interior of the capacity region . on top of that , an interesting question concerns the trade off between the error exponent of the strong user and the one of the weak user , or equivalently , the achievable region in the plane of error exponents for a given set of coding rates . while the capacity regions of the dbc and the abc have been known for many years , only little has been known about their reliability functions . earlier works on error exponents for the general dbc and abc include those of gallager @xcite and krner and sgarro @xcite , respectively . in both works , the coding scheme of @xcite was adopted , but the decoder was sub optimal . more recently , kaspi and merhav @xcite have derived some tighter lower bounds to the reliability functions of both users by analyzing random coding error exponents of their optimal decoders . while their derivation was exponentially tight at most of the steps , there were still some steps in @xcite where exponential tightness might have been compromised . moreover , kaspi and merhav have analyzed ensembles of i.i.d.codes , which are not as good as ensembles of fixed composition codes ( * ? ? ? * section 7.3 ) . these two points give rise to the thought that there is room for improvement upon the results of @xcite , and indeed , such an improvement is one of the contributions of this work . in fact , the exponential error bounds , derived in this paper , both for the strong user and the weak one , are tight in the sense that they provide the exact random coding exponents for the ensemble of fixed composition codes . moreover , the resulting expressions are much simpler and easier to calculate than those of the best exponential bounds of kaspi and merhav ( see , in particular , the second part of @xcite ) . interestingly , one of the ingredients that contributes significantly to this simplification in the error exponent expressions , is the derivation of _ universal decoders _ for both users , and this simplification is achieved thanks to a simple sandwich argument , asserting that a lower bound to the error exponent of the universal decoder can not be larger than an upper bound to the error exponent of the optimal decoder , but on the other hand , the latter turns out to be mathematically smaller than or equal to the former , and so , by contrasting the two exponential error bounds , which must therefore be equivalent , the expressions are considerably simplified . in other words , beyond this simplification of the error exponent bounds , there is an additional bonus , which is in obtaining universal decoders for both users . these decoders achieve the same random coding error exponents as the corresponding optimal decoders of the two users . both universal decoders are certain variants of the maximum mutual information ( mmi ) decoder ( * ? ? ? * theorem 5.2 ) , but they are different from the earlier proposed mmi - like universal decoders for the abc , due to krner and sgarro @xcite . for one thing , our universal decoder for the weak user depends explicitly on the entire code , unlike the one in @xcite , which depends on the cloud centers only . since we rely heavily on the method of types , our exponential error bounds have the flavor of those of csiszr and krner @xcite . while exponentially tight , their shortcoming is that they are not easy to calculate since they involve minimizations over auxiliary channels , and these might be computationally painful especially for large alphabets . to alleviate this difficulty , we also propose gallager style bounds @xcite , which require optimizations over very few ( one or two ) parameters , but the caveat is that exponential tightness might be sacrificed . moreover , the gallager style bounds lend themselves to better intuitive understanding on the behavior of the error exponents for both of the users . specifically , we derive a _ phase diagram _ for the weak user , which fully describes the functional behavior of the bound in different regions of the plane of rates . we also demonstrate our results numerically for an example of the binary symmetric bc , and compare our results to those in earlier works , showing explicitly the improvement . the remaining part of the paper is organized as follows . in section 2 , we establish notation conventions , formalize the model and the problem , and finally , review some preliminaries . in section 3 , we summarize the main theoretical results of this paper , and give some numerical results for the binary symmetric bc . section 4 provides the proofs concerning the strong user in the abc ( the exact random coding error exponent and the universal decoder ) , and section 5 contains a similar treatment for the weak user . in section 6 , we derive lower bounds on the exact random coding error exponents , and in section 7 we study them .

this work contains two main contributions concerning the asymmetric broadcast channel . the first is an analysis of the exact random coding error exponents for both users , and the second is the derivation of universal decoders for both users . these universal decoders are certain variants of the maximum mutual information ( mmi ) universal decoder , which achieve the corresponding random coding exponents of optimal decoding . in addition , we introduce some lower bounds , which involve optimization over very few parameters , unlike the original , exact exponents , which involve minimizations over auxiliary probability distributions . numerical results for the binary symmetric broadcast channel show improvements over previously derived error exponents for the same model . + * index terms : * error exponent , asymmetric broadcast channel , universal decoding , mmi . the andrew & erna viterbi faculty of electrical engineering + technion - israel institute of technology + technion city , haifa 3200004 , israel + \{rans@campus , merhav@ee}.technion.ac.il

this work contains two main contributions concerning the asymmetric broadcast channel . the first is an analysis of the exact random coding error exponents for both users , and the second is the derivation of universal decoders for both users . these universal decoders are certain variants of the maximum mutual information ( mmi ) universal decoder , which achieve the corresponding random coding exponents of optimal decoding . in addition , we introduce some lower bounds , which involve optimization over very few parameters , unlike the original , exact exponents , which involve minimizations over auxiliary probability distributions . numerical results for the binary symmetric broadcast channel show improvements over previously derived error exponents for the same model . + * index terms : * error exponent , asymmetric broadcast channel , universal decoding , mmi . the andrew & erna viterbi faculty of electrical engineering + technion - israel institute of technology + technion city , haifa 3200004 , israel + \{rans@campus , merhav@ee}.technion.ac.il

0805.3731

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new @xmath9-dimensional @xmath0- and @xmath1- and @xmath2-transforms were recently described in @xcite . each transform is based on a compact semisimple lie group of rank @xmath9 and comes in three versions : analogs of fourier series , fourier integrals , and fourier transforms on an @xmath9-dimensional lattice . they are named @xmath0- and @xmath1- and @xmath2-transforms @xcite in recognition of the fact that they can be understood as generalizations of the one dimensional cosine , sine , and exponential fourier transform . the aim of this paper is to set the grounds for the 3-dimensional exploitation of transforms described here as continuous transforms in a finite region @xmath3 of the 3-dimensional euclidean space @xmath10 , and also as discrete transforms of functions given on a lattice grid of points @xmath4 of any density in ready - to - use form . a positive integer @xmath5 specifies the density . in some cases , the density of the grid is more flexible as it is dictated by not one , but two or three positive intergers . the grid could thus be made denser along certain axes . the symmetry of the lattice is dictated by the shape of @xmath3 , or equivalently , by the choice of the lie group . there are seven compact semisimple lie groups of rank 3 : @xmath11 throughout the paper we identify these cases by symbols that are often used for their respective lie algebras : @xmath12 the immediate motivation for this paper is our anticipation of the extensive use of the transforms given the need for processing the rapidly increasing amount of 3d digital data gathered today . in 2d , our group transforms offered only in some cases more than a marginal advantage , having emerged when satisfactory practical methods had already been developed and adequately implemented . so far , practical use of the functions in 2d rested on the fact that the continuous extension of the transformed lattice data displayed remarkably smooth interpolation between lattice points @xcite ( see also references therein ) . special functions , which serve as the kernel of our transform ( we call them @xmath0- , @xmath1- , and @xmath2-functions or orbit functions ) , have simple symmetry property under the action of the corresponding affine weyl group . the affine group contains as a subgroup the group of translations in @xmath13 , which underlies the common fourier transform . this is the primary reason for the superior performance of our transforms , although detailed comparisons , rather than examples , will have to provide quantitative content to substantiate such a claim . other properties of the @xmath0- , @xmath1- , and @xmath2-functions are not less important . within each family , functions are described in a uniform way for semisimple lie groups of any type and rank . in this work , we illustrate this uniformity by considering all seven rank 3 group cases in parallel . the price to pay for the uniformity of methods is having to work with non - orthogonal bases which are not normalized . the functions are defined in @xmath13 and have continuous derivatives of all degrees . their orthogonality , when integrated over the finite region @xmath3 appropriate for each lie group , was shown in @xcite . the discrete orthogonality of @xmath0-functions in @xmath6 has already been described in @xcite and extensively used ( see for example @xcite and references therein ) . the completeness of these systems of functions directly follows from the completeness of the system of exponential functions . a laplace operator for each lie group is given in a different set of coordinates . the @xmath0- and @xmath1-functions are its eigenfunctions with known eigenvalues . on the boundary of @xmath3 , the @xmath0-functions have a vanishing normal derivative , while @xmath1-functions reach zero at the boundary . the functions have a number of other useful properties , which can be found in @xcite . for example , the decomposition of their products into sums , the splitting of functions into as many mutually exclusive congruence classes as is the order of the center of the lie group , etc@xmath14 . a different but valid viewpoint on some of the special functions presented here , namely , functions symmetrized by the summation of constituent functions over a finite group @xcite , may turn out to be rather useful . the finite group , in the case of @xmath0- and @xmath1-functions , is the weyl group of the corresponding semisimple lie group . in the case of @xmath2-functions , it is the even subgroup of the weyl group . the weyl group of @xmath15 is isomorphic to the group @xmath16 of the permutation of @xmath9 elements . this led to the recent implementations in @xcite , where instead of the weyl group of @xmath15 , the @xmath16 group is used , and variables are given relative to an orthonormal system of coordinates . furthermore , the even subgroup of @xmath16 is the alternating group . related transforms were introduced most recently in @xcite . the paper is organized as follows . in section [ sec_properties_definitions ] , necessary definitions and properties of lie groups and algebras are given and discussed . semisimple lie groups of rank 3 are considered in detail in section [ sec_algebras ] . for each of these groups , we lay down the information necessary to construct and use their orbit functions for 3d continuous and discrete transforms . section [ sec_orbit - funcs ] is devoted to @xmath0- , @xmath1- and @xmath2- orbit functions and to their pertinent properties . continuous and discrete orbit - function transforms are presented in section [ sec_c - s - e - transforms ] . some problems and possible applications arising in connection with orbit functions are formulated in the conclusion . an example of the application of orbit function transforms in the case of the group @xmath17 is given at the end of the paper .

three dimensional continuous and discrete fourier - like transforms , based on the three simple and four semisimple compact lie groups of rank 3 , are presented . for each simple lie group , pertinent properties of the functions are described in detail , such as their orthogonality within each family , when integrated over a finite region of the 3-dimensional euclidean space ( continuous orthogonality ) , as well as when summed up over a lattice grid ( discrete orthogonality ) . the positive integer sets up the density of the lattice containing .

three dimensional continuous and discrete fourier - like transforms , based on the three simple and four semisimple compact lie groups of rank 3 , are presented . for each simple lie group , there are three families of special functions (- ,- , and-functions ) on which the transforms are built . pertinent properties of the functions are described in detail , such as their orthogonality within each family , when integrated over a finite region of the 3-dimensional euclidean space ( continuous orthogonality ) , as well as when summed up over a lattice grid ( discrete orthogonality ) . the positive integer sets up the density of the lattice containing . the expansion of functions given either on or on is the paper s main focus . centre de recherches mathmatiques , universit de montral , c.p.6128-centre ville , montral , h3c3j7 , qubec , canada ; patera@crm.umontreal.ca + institute of mathematics of nas of ukraine , 3 , tereshchenkivska str , kyiv-4 , 04216 , ukraine ; maryna@imath.kiev.ua

1003.2341

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tio@xmath0 has been one of the most studied oxides over the past few years . this is due to the broad range of applications it offers in strategic fields of scientific , technological , environmental , and commercial relevance . in particular , tio@xmath0 surfaces and nanocrystals provide a rich variety of suitably tunable properties from structure to opto - electronics . special attention has been paid to tio@xmath0 s optical properties . this is because tio@xmath0 is regarded as one of the best candidate materials to efficiently produce hydrogen via photocatalysis @xcite . tio@xmath0 nanostructures are also widely used in dye - sensitized solar cells , one of the most promising applications in the field of hybrid solar cells @xcite . since the first experimental formation of hydrogen by photocatalysis in the early 1980s , @xcite tio@xmath0 has been the catalyst of choice . reasons for this include the position of tio@xmath0 s conduction band above the energy of hydrogen formation , the relatively long lifetime of excited electrons which allows them to reach the surface from the bulk , tio@xmath0 s high corrosion resistance compared to other metal oxides , and its relatively low cost @xcite . however , the large optical band gap of bulk tio@xmath1 ( @xmath2 3 ev ) means that only high energy uv light may excite its electrons . this effectively blocks most of the photons which pierce the atmosphere , typically in the visible range , from participating in any bulk tio@xmath1 based photocatalytic reaction . on the other hand , the difference in energy between excited electrons and holes , i.e. the band gap , must be large enough ( @xmath3 1.23 ev ) to dissociate water into hydrogen and oxygen . for these reasons it is of great interest to adjust the band gap @xmath4 of tio@xmath1 into the range 1.23 @xmath5 2.5 ev , while maintaining the useful properties mentioned above @xcite . with this aim , much research has been done on the influence of nanostructure @xcite and dopants @xcite on tio@xmath1 photocatalytic activity . for low dimensional nanostructured materials , electrons and holes have to travel shorter distances to reach the surface , allowing for a shorter quasi - particle lifetime . however , due to quantum confinement effects , lower dimensional tio@xmath6 nanostructures tend to have _ larger _ band gaps @xcite . on the other hand , although doping may introduce mid - gap states , recent experimental studies have shown that boron and nitrogen doping of bulk tio@xmath1 yields band gaps _ smaller _ than the threshold for water splitting @xcite . this suggests that low dimensional structures with band gaps larger than about @xmath7 ev may be a better starting point for doping . recently , several promising new candidate structures have been proposed @xcite . these small ( @xmath8 5 ) tio@xmath1 nanotubes , with a hexagonal abc pto@xmath1 structure ( hexabc ) , were found to be surprisingly stable , even in the boron and nitrogen doped forms . this stability may be attributed to their structural similarity to bulk rutile tio@xmath9 , with the smallest nanotube having the same structure as a rutile nanorod . a further difficulty for any photocatalytic system is controlling how electrons and holes travel through the system @xcite . for this reason , methods for reliably producing both @xmath10-type and @xmath11-type tio@xmath1 semiconducting materials are highly desirable . so far , doped tio@xmath1 tends to yield only @xmath10-type semiconductors . however , it has recently been proposed that @xmath11-type tio@xmath0 semiconducting materials may be obtained by nitrogen doping surface sites of low dimensional materials @xcite . in this chapter we will discuss in detail the effects of quantum confinement and doping on the optical properties of tio@xmath0 . tio@xmath0 nanostructures are also one of the main components of hybrid solar cells . in a typical grtzel cell @xcite , tio@xmath0 nanoparticles with average diameters around 20 nm collect the photoelectron transferred from a dye molecule adsorbed on the surface @xcite . such processes are favoured by a proper energy level alignment between solid and organic materials , although the dynamic part of the process also plays an important role in the charge transfer . clearly , tio@xmath0 s characteristics of long quasi - particle lifetimes , high corrosion resistance , and relatively low cost , must be balanced with control of its energy level alignment with molecular states , and a fast electron injection at the interface . despite all the engineering efforts , the main scientific goal remains to optimize the efficiency of solar energy conversion into readily available electricity . different research approaches have been devoted to benefit from quantum size properties emerging at the nanoscale @xcite , find an optimal donor - acceptor complex @xcite , mix nanoparticles and one - dimensional structures , such as nanotubes or nanowires @xcite , and control the geometry of the tio@xmath0 nano - assembly @xcite . a clear theoretical understanding of tio@xmath0 s optoelectronic properties is necessary to help unravel many fundamental questions concerning the experimental results . in particular , the properties of excitons , photo - injected electrons , and surface configuration in tio@xmath0 nanomaterials may play a critical role in determining their overall behaviour in solar cells . for tio@xmath0 at both the nanoscale and macroscale regime , it is necessary to first have a complete picture of the optical properties in order to clarify the contribution of excitons . octahedrons ( right ) of rutile ( top ) and anatase ( bottom ) . the lattice parameters in are denoted @xmath12 , @xmath13 , and @xmath14 , while @xmath15 and @xmath16 are the distances in between a ti@xmath17 ion and its nearest and next - nearest neighbour o@xmath18 ions , respectively . in the case of rutile the interstitial ti impurity site is shown with a green circle ( see left top ) . , scaledwidth=80.0% ] despite the clear importance of its surfaces and nanostructures , investigations of tio@xmath0 bulk ( see [ lfig1 ] ) electronic and optical properties have not provided , so far , a comprehensive description of the material . important characteristics , such as the electronic band gap , are still undetermined . most of the experimental and computational work has been focused on synthesis and analysis of systems with reduced dimensionality . the experimental synthesis and characterization of nanostructured materials is in general a costly and difficult task . however , predictions of a dye or nanostructure s properties from simulations can prove a great boon to experimentalists . modern large - scale electronic structure calculations have become important tools by providing realistic descriptions and predictions of structure and electronic properties for systems of technological interest . although it will not be treated in this chapter , it is important to mention the problem of electron localization in reduced titania @xcite . this will provide a glimpse of the complexity faced , from the theoretical point of view , when studying transition metal atoms . the localization of @xmath19 electrons makes the accurate description of their exchange correlation interaction a difficult task @xcite . the electron localization in defective titania has been an open question from both experimental and theoretical points of view , and caused much controversy during the past few years @xcite . oxygen vacancies are quite common in tio@xmath0 , and their presence and behaviour can significantly affect the properties of nanostructures . when an oxygen vacancy is created in tio@xmath0 , i.e. when tio@xmath0 is reduced , the two electrons coming from the removed o@xmath18 ion must be redistributed within the structure . one possibility is that these two extra electrons remain localized onto two ti ions close to the o@xmath18 vacancy . in this way a pair of ti@xmath17 ions become ti@xmath20 ions . another option is for the two extra electrons to delocalize along the whole structure , i.e. they do not localize on any particular ti ion . finally , an intermediate situation , with one electron localized and the other spread , is also possible . concerning the tio@xmath0 bulk , conventional density functional theory ( dft ) calculations using either local density approximations ( lda ) or generalized gradient approximations ( gga ) for the exchange - correlation ( xc)-functionals show a scenario with both electrons delocalized . on the other hand , hybrid functionals and hartree - fock calculations give rise to a situation with both electrons localized . for gga+u calculations , the results are very sensitive to the value of the u parameter . for certain values of u both electrons remain localized , while for others there is an intermediate situation @xcite . experimentally , there are electron paramagnetic resonance ( epr ) measurements suggesting that the extra electrons are mainly localized on interstitial ti@xmath20 ions @xcite . these interstitial ti@xmath20 ions are impurities placed at the natural interstices of the rutile lattice ( see [ lfig1 ] ) and , similarly to the ti@xmath17 ions of the pure lattice , they also form tio@xmath21 octahedrons . recent stm and pes experiments have shown that the interstitial ti@xmath20 ions play a key role in the localization of the electrons when a bridge oxygen is removed from the tio@xmath0 ( 110 ) surface @xcite . these experiments concluded the controversial discussion about the localization of electrons in the bridge oxygen defective tio@xmath0 ( 110 ) surface ( see refs . @xcite for more details ) . however , the problem remains unresolved for the bulk case . in summary , in this chapter we first analyze the full _ ab initio _ treatment of electronic and optical properties in [ dft ] and [ mbpt ] , before applying it to the two most stable bulk phases , rutile and anatase in [ bulk ] . these are also the phases most easily found when nanostructures are synthesized . we will focus on their optical properties and excitonic behaviour . we then explore the possibility of tuning the oxide band gap using quantum confinement effects and dimensionality , by analyzing atomic clusters , nanowires and nanotubes in [ nanostructure ] . a further component whose effect has to be evaluated is that of doping , which may further tune the optical behaviour by introducing electronic states in the gap , as presented in [ doping].combining the effects of quantum confinement and doping is hoped to produce a refined properties control . in [ cells ] , we report some details on modeling for dye - sensitized solar cells , before providing a summary and our concluding remarks .

titanium dioxide is one of the most widely investigated oxides . this is due to its broad range of applications , from catalysis to photocatalysis to photovoltaics . despite this large interest , many of its bulk properties further , some of tio s most important properties , such as its electronic band gap , the localized character of excitons , and the localized nature of states induced by oxygen vacancies , are still under debate . indeed , we address one of the main challenges to tio-photocatalysis , namely band gap narrowing , by showing how to combine nanostructural changes with doping . with this aim while quantum confinement effects lead to a widening of the energy gap , it has been shown that substitutional doping with boron or nitrogen gives rise to ( meta-)stable structures and the introduction of dopant and mid - gap states which effectively reduce the band gap .

titanium dioxide is one of the most widely investigated oxides . this is due to its broad range of applications , from catalysis to photocatalysis to photovoltaics . despite this large interest , many of its bulk properties have been sparsely investigated using either experimental techniques or _ ab initio _ theory . further , some of tio s most important properties , such as its electronic band gap , the localized character of excitons , and the localized nature of states induced by oxygen vacancies , are still under debate . we present a unified description of the properties of rutile and anatase phases , obtained from _ ab initio _ state of the art methods , ranging from density functional theory ( dft ) to many body perturbation theory ( mbpt ) derived techniques . in so doing , we show how advanced computational techniques can be used to quantitatively describe the structural , electronic , and optical properties of tio nanostructures , an area of fundamental importance in applied research . indeed , we address one of the main challenges to tio-photocatalysis , namely band gap narrowing , by showing how to combine nanostructural changes with doping . with this aim we compare tio s electronic properties for 0d clusters , 1d nanorods , 2d layers , and 3d bulks using different approximations within dft and mbpt calculations . while quantum confinement effects lead to a widening of the energy gap , it has been shown that substitutional doping with boron or nitrogen gives rise to ( meta-)stable structures and the introduction of dopant and mid - gap states which effectively reduce the band gap . finally , we report how _ ab initio _ methods can be applied to understand the important role of tio as electron - acceptor in dye - sensitized solar cells . this task is made more difficult by the hybrid organic - oxide structure of the involved systems .

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determining the physical parameters of large samples of galaxies is crucial to constrain our knowledge of the complicated physical processes that govern the formation and evolution of these systems . ultimately , we seek methods that quickly and effectively map observed data from @xmath0 galaxies to sets of parameters that describe the star formation and chemical histories of these galaxies . due to the complexity of the mechanisms of galaxy evolution , it is critical that we adopt a model that uses few assumptions about how galaxies evolve . moreover , full utilization of the available theoretical stellar models , avoiding ad hoc simplifications , is critical to obtaining accurate parameter estimates . there has been a large amount of work dedicated to estimating the physical parameters of databases of galaxies using different types of data . for example , spectral indices ( @xcite ) , emission features ( @xcite ) , and full high - resolution , broad - band spectra ( @xcite ) have all been utilized to estimate the star formation histories ( sfhs ) and metallicities of galaxies . recently , due to the abundance of high - quality , homogeneous databases of galaxy spectra such as the sloan digital sky survey ( sdss , @xcite ) , which provides high - resolution spectra for hundreds of thousands of galaxies , the approach of full high - resolution spectral fitting is now possible to infer the properties of large populations of galaxies . recent work has shown promise in the application of these methods to sdss galaxy spectra ( see @xcite and @xcite for reviews of results achieved by two such fitting methods ) . however , large - scale analyses , while possible , are not necessarily accurate , and can be computationally challenging . there is much room for improvement in accuracy of sfh parameter estimation . it is critical that we understand the shortcomings of the current models and improve their accuracy . at the same time , to achieve our goal of obtaining accurate sfh estimates for a large sample of galaxies , we can not sacrifice the computational efficiency of current methods . a common technique in the literature , called empirical population synthesis , is to model galaxies as mixtures of simple stellar populations ( ssps ) with known physical parameters . recent studies that have used this method are , e.g. @xcite , @xcite , @xcite , and @xcite . under this model , galaxy data are treated as linear combinations of the observed properties of ssps . historically , ssp parameters and observables were derived from observations of well - understood stellar systems . more recent studies have instead used model - produced ssps , such as those from the evolutionary population synthesis models of @xcite . the starlight spectral fitting code , introduced by @xcite , fits observed spectra with linear combinations of ssps from the models of @xcite . in @xcite , it was shown that sfh parameters of simulated galaxy spectra could be recovered by starlight in the absence of noise . however , their simulated spectra were generated and fit with the _ same basis _ of 45 ssps , rendering the results difficult to generalize to the expected performance on real galaxies , which can be composed of ssps on infinitely fine grids of age and metallicity . results of the starlight fitting code are highly dependent on the choice of basis of ssp spectra . the database of ssps available to us could theoretically be infinitely large , encompassing all combinations of age , metallicity , initial mass function , evolutionary prescription , etc . including too many ssps in our basis causes the code to be prohibitively slow and the solution to be degenerate due to the inclusion of ssps with almost identical spectra . if , however , we simply hand - select a few ssps on a coarse grid of age and metallicity with which to model our data , then we effectively ignore large subsets of our ssp parameter space , leading to suboptimal estimates for fits to databases of galaxies . using a hand - selected basis of ssps may also lead to the inclusion of multiple ssps whose spectra are essentially identical , which can lead to degeneracies . there is much room for improvement in the accuracy of sfh estimates by exploring numerical methods of selecting small bases of ssps . in this paper , we propose a method of choosing a small set of _ prototype _ spectra from a large database of ssp spectra based on their observable quantities . our method attempts to capture much of the variability of a large set of ssps in a few numerically chosen prototype ssps . the method is based on the diffusion map of @xcite and @xcite , a non - linear technique that seeks a simple , natural representation of data that are complex and high - dimensional , such as high - resolution astronomical spectra . we utilize the _ diffusion @xmath1-means _ method to numerically find a basis of prototype ssps with which we can fit galaxies to accurately estimate their sfhs . in a future study , we will consider the problem of computational speed in methods such as starlight , which takes @xmath2 minutes to fit each sdss galaxy spectrum for a basis of size 150 , and show how to quickly extend the sfh estimates obtained by using the methods in this paper to a large database of galaxies by taking advantage of the geometry of the manifold on which the data lie . the applicability of the methods introduced here extend beyond the problem of sfh estimation using high - resolution spectra . any astronomical task where observed data ( i.e. spectra , photometric data , spectral indices , etc . ) are fit as linear combinations of observable data from simpler systems can benefit from intelligent numerical selection of prototypes . the outline of the paper is as follows . in 2 we describe the starlight spectral synthesis code , discuss the drawbacks of ssp bases employed in the literature , and introduce the diffusion @xmath1-means method of prototype selection . we directly compare the basis of prototypes selected by our method to those used in the literature and those found using other methods . in 3 we fit several sets of realistic simulated galaxy spectra using the starlight code with different bases to directly compare the ability of each basis to accurately estimate galaxy parameters . in 4 we fit a set of sdss galaxy spectra and analyse physical parameter estimates obtained by different bases . we conclude with some remarks and discussion of future directions in 5 .

to further our knowledge of the complex physical process of galaxy formation , it is essential that we characterize the formation and evolution of large databases of galaxies . the spectral synthesis starlight code of cid fernandes et al . ( 2004 ) was designed for this purpose . results of starlight are highly dependent on the choice of input basis of simple stellar population ( ssp ) spectra . we analyze a sample of 3046 galaxies in sdss dr6 and compare the parameter estimates obtained from different basis choices . [ firstpage ] methods : data analysis methods : statistical methods : numerical galaxies : evolution galaxies : formation galaxies : stellar content

to further our knowledge of the complex physical process of galaxy formation , it is essential that we characterize the formation and evolution of large databases of galaxies . the spectral synthesis starlight code of cid fernandes et al . ( 2004 ) was designed for this purpose . results of starlight are highly dependent on the choice of input basis of simple stellar population ( ssp ) spectra . speed of the code , which uses random walks through the parameter space , scales as the square of the number of basis spectra , making it computationally necessary to choose a small number of ssps that are coarsely sampled in age and metallicity . in this paper , we develop methods based on diffusion map ( lafon & lee , 2006 ) that , for the first time , choose appropriate bases of prototype ssp spectra from a large set of ssp spectra designed to approximate the continuous grid of age and metallicity of ssps of which galaxies are truly composed . we show that our techniques achieve better accuracy of physical parameter estimation for simulated galaxies . specifically , we show that our methods significantly decrease the age - metallicity degeneracy that is common in galaxy population synthesis methods . we analyze a sample of 3046 galaxies in sdss dr6 and compare the parameter estimates obtained from different basis choices . [ firstpage ] methods : data analysis methods : statistical methods : numerical galaxies : evolution galaxies : formation galaxies : stellar content

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we adopt the model of galaxy spectra as a mixture of simple systems introduced in @xcite : @xmath3 where @xmath4 is the model flux in wavelength bin @xmath5 . the component @xmath6 is the @xmath7th basis spectrum normalized at wavelength @xmath8 . the basis of ssp spectra @xmath9 is chosen before performing the analysis . the scalar @xmath10 $ ] is the proportion of flux contribution of the @xmath7th component at @xmath8 , where @xmath11 ; the vector @xmath12 with components @xmath13 is the _ population vector _ of a galaxy . these flux fractions can be converted to mass fractions , @xmath14 , using the light - to - mass ratios of each basis spectrum @xmath15 at @xmath8 . to describe the sfh of a galaxy , we will use time - binned versions of @xmath12 and @xmath16 , denoted @xmath17 and @xmath18 , respectively . using binned population vectors lets us avoid degeneracies that can occur in the estimation of the original population vectors . other components of the model in ( [ cfmodel ] ) are the scaling parameter @xmath19 , the reddening term , @xmath20 , and the gaussian convolution , @xmath21 . the reddening term describes distortion in the observed spectrum due to foreground dust , and is modeled by the extinction law of @xcite with @xmath22 . the gaussian convolution , @xmath21 , accounts for movement of stars within the observed galaxy with respect to our line - of - sight , and is parametrized by a central velocity @xmath23 and dispersion @xmath24 . to fit individual galaxies , we use the starlight fitting routine introduced by @xcite . the code uses a metropolis algorithm plus simulated annealing to find the minimum of @xmath25 ^ 2,\ ] ] where @xmath26 is the observed flux in the wavelength bin @xmath5 , @xmath4 is the model flux in ( [ cfmodel ] ) , and @xmath27 is the inverse of the noise in the measurement @xmath26 . the summation is over the @xmath28 wavelength bins in the observed spectrum . the minimization of ( [ chisq ] ) is performed over @xmath29 parameters : @xmath30 , and @xmath24 , where @xmath1 is the number of basis spectra in @xmath31 . the speed of the algorithm scales as @xmath32 . for the analysis of , e.g. @xmath33 galaxy spectra in the sdss database , it is crucial to pick a basis with a small number of spectra . in , for example , @xcite and @xcite , the starlight model ( [ cfmodel ] ) and fitting algorithm are tested using simulations . the simulated spectra are generated from the model in ( [ cfmodel ] ) using @xmath34 ssps from @xcite and @xmath35 ssps from @xcite , respectively . these authors fit the simulations using the same basis of ssps that was used to generate the simulations . from their analyses , they conclude that in the absence of noise , the algorithm accurately recovers the input parameters . in the presence of noise , the population vector @xmath12 is not recovered , and it is advised that a time - binned description of the sfh be adopted . although their use of the same basis for both generating and fitting simulated spectra is an appropriate test that the algorithm works , it is not a fair assessment of the expected performance of the methods for a database of real galaxy spectra . in [ models ] we discuss our concerns with their simulation assessments . in [ basis ] we present a new suite of methods that we test with several sets of simulated galaxy spectra . these simulations , which are described in [ simulations ] , are designed to be more representative of a large database of galaxy spectra , such as sdss . in reality , the population of all observed galaxies is not constrained to be mixtures of a small number of ssps on a discrete grid of age and metallicity . a more physically accurate representation of galaxies is as mixtures of an infinitely large basis of ssps on a continuous grid of age and metallicity and , depending on the complexity of the underlying physics , a possibly infinite grid of prescriptions for initial mass function and evolutionary track . in our simulations in [ simulations ] we simulate galaxies from a large database of @xmath36 ssps on a fine grid of age and metallicity to determine whether we can accurately describe their sfhs , metallicities , and kinematic parameters using an appropriately chosen , computationally tractable basis of @xmath35 or 150 spectra . we adopt a ssp database containing 1278 spectra . this set of spectra is used to both choose an appropriate basis ( [ basis ] ) and generate simulated spectra ( [ simulations ] ) . the database is meant to represent the large population of ssps from which observed galaxies can be composed . we use the spectra from the models of @xcite , computed for a @xcite imf , ` padova 1994 ' evolutionary tracks ( @xcite ) , and stelib library ( @xcite ) . theoretically , we could also vary the imf and evolutionary tracks across spectra and use the same methods developed in [ basis ] , but for simplicity decide to fix them in this study . the ssps in our database are generated on a grid of 213 approximately evenly log - spaced time bins from 0 to 18 gyrs and 6 different metallicities : @xmath37 = 0.0001 , 0.0004 , 0.004 , 0.008 , 0.02 and 0.05 , where @xmath37 is the fraction of the mass of a star composed of metals ( @xmath38 ) . this grid is designed to approximate a continuous variation across age and @xmath37 . the coarseness in the current @xmath37 grid is due to limitations in the current stellar population models . again , a finer grid of age and @xmath37 can be implemented with the techniques that we develop in [ basis ] . in previous studies using the starlight code , researchers hand - selected sets of ssp spectra to use in the model fitting . according to @xcite , `` the elements of the base should span the range of spectral properties observed in the sample galaxies and provide enough resolution in age and metallicity to address the desired scientific questions . '' in this same paper , they claim that their basis of @xmath34 ssps adequately recovers parameters of age and metallicity in their simulations . however , as discussed in [ specsyn ] , these simulations are unrealistic since they employ the same basis to both generate and fit the simulated spectra . in this section we present methods to numerically determine a basis @xmath31 of a small number of _ prototype spectra _ , @xmath15 , from a large set of ssp spectra . the resultant basis of prototype spectra will be designed to capture a large proportion of the variation of the set of ssps , which is chosen to span the range of observed spectral properties of a data set of galaxies at a high resolution in age and metallicity , such as the database of 1278 ssps described in [ models ] . the numerically chosen basis of prototype spectra , @xmath31 , should recover physical parameters better than any hand - chosen basis of the same size because hand - selected bases ignore subsets of the parameter space and may cause the solution to be degenerate by including multiple ssps with nearly identical spectra . in [ simulations ] we show that the bases computed from our numerical methods outperform hand - selected bases used in the literature in parameter estimation for simulated galaxies . diffusion map is a popular new technique for data parametrization and dimensionality reduction that has been developed by @xcite and @xcite , and was recently used by @xcite to analyse sdss spectra and @xcite to estimate photometric redshifts . as shown in these latter works , diffusion map has the powerful ability to uncover simple structure in complicated , high - dimensional data . these analyses demonstrate that this simple structure can be used to make precise statistical inferences about the data . in the problem at hand , we begin with a set @xmath39 of high - resolution broad - band ssp spectra @xmath40 , where @xmath41 can be very large and each @xmath42 has flux measurements in @xmath43 wavelength bins . from this set , our goal is to derive a small basis of @xmath1 prototype spectra , @xmath44 , each of length @xmath45 , that captures most of the variability in @xmath39 . in what follows , we briefly review the basics of the diffusion map technique and then show how it can be used to find an appropriate basis @xmath31 . the main idea of the diffusion map is that it finds a parametrization of a data set that preserves the connectivity of data points in a way that is dependent only on the relationship of each datum to points in a local neighborhood . by preserving the local interactions in a data set , diffusion map is able to learn , e.g. a natural parametrization of the of the spiral in fig . 1 of @xcite where other methods , such as principal components analysis , fail . starting with our set of ssp spectra , @xmath39 , in units of @xmath46/ per initial @xmath47 , we normalize the spectra at @xmath8 to ensure that we are not confused by absolute scale differences between individual spectra . we retain the scaling factors @xmath48 for later use to convert back to the original units . next , we select a discrepancy measure between ssps , i.e. @xmath49 which is the euclidean distance between two ssp spectra . the specific choice of @xmath50 is often not crucial to the diffusion map ; any reasonable measure of discrepancy will return similar results . it is important to note here the flexibility of the diffusion map method . for each problem , we can define a different discrepancy measure based on the type of data available to us and the types of information we want to incorporate . for example , we could define our measure @xmath51 by weighting more heavily those regions around absorption lines which are highly correlated with stellar age and metallicity . contrast this flexibility with the rigidity of principal components analysis , which is only based on the correlation structure of the data . from our discrepancy measure in ( [ distance ] ) , we construct a weighted graph on our data set @xmath39 , where the nodes are the individual ssp spectra and the weights on the edges of the graph reflect the amount of similarity between pairwise spectra . we define the weights as @xmath52 where @xmath53 should be small enough such that @xmath54 unless @xmath55 and @xmath56 are similar , but large enough so that the entire graph is connected ( i.e. every ssp has at least one non - zero weight to another ssp ) . in practice , we choose @xmath53 that produces a basis that achieves the best fits to galaxy spectra ( see [ simulations ] ) . by normalizing the rows of the weight matrix , @xmath57 , to sum to unity , we can define a markov random walk on the graph where the probability of going from @xmath55 to @xmath56 in one step is @xmath58 . we store all of these one - step probabilities in a matrix @xmath59 . by the theory of markov chains , the @xmath60-step transition probabilities are defined by @xmath61 . in the previous analyses in @xcite and @xcite , the authors fixed @xmath60 . in these previous papers , the choice of @xmath60 was unimportant because the task was linear regression on the diffusion coordinates , in which the multipliers @xmath62 of the diffusion map were absorbed into the linear regression coefficients @xmath63 . in the present work , our task is to determine a set of @xmath1 prototype spectra . the tool we use is @xmath1-means clustering in diffusion space , where results are dependent upon intra - point euclidean distances of the diffusion map , which in turn are dependent on @xmath60 . in this paper we introduce a diffusion operator that considers _ all _ scales @xmath64 simultaneously . we define the _ multi - scale _ diffusion map as @xmath65.\ ] ] where @xmath66 and @xmath67 are the right eigenvectors and eigenvalues of @xmath59 , respectively , in a bi - orthogonal decomposition . euclidean distance in the @xmath68-dimensional space described by equation ( [ dmap2 ] ) approximates the multi - scale diffusion distance , a distance measure that utilizes the geometry of the data set by simultaneously considering all possible paths between any two data points at all scales @xmath60 , in the markov random walk constructed above . an advantage to using the multi - scale diffusion map , besides elimination of the tuning parameter @xmath60 , is that it gives a more robust description of the structure of the data by considering the propagation of local interactions of data points through all scales . we also have three other tuning parameters : @xmath53 , @xmath68 ( dimensionality of the diffusion map ) , and @xmath1 ( number of prototypes ) over which to optimize the fits to galaxy spectra . for simplicity , we fix @xmath68 by choosing the dimensionality of the diffusion map to coincide with a 95% drop - off in the eigenvalue multipliers in the multi - scale diffusion map . this is reasonable because most of the information about the structure of the data will be in the few dimensions with large eigenvalue multipliers . simulations show that this choice of @xmath68 produces results comparable to optimizing over a large grid of @xmath68 . for the parameter @xmath1 , fits should improve as the basis gets larger . however , we want to keep @xmath1 small both for computational reasons and to avoid the occurrence of nearly identical spectra in our basis , which can cause degeneracies in our fits . in practice , we choose @xmath1=45 or 150 to directly compare our results with the bases used by @xcite ( cf05 ) and @xcite ( asa07 ) . this leaves us with only one tuning parameter , @xmath53 , over which to optimize the diffusion @xmath1-means basis in fits to galaxy spectra . finally , we determine a set of @xmath1 prototype spectra by performing @xmath1-means clustering in the @xmath68-dimensional diffusion map representation ( [ dmap2 ] ) of the ssp spectra , @xmath39 . the @xmath1-means algorithm is a standard machine learning method used to cluster @xmath41 data points into @xmath1 groups . it works by minimizing the total intra - cluster variance , @xmath69 where @xmath70 is the set of points in the @xmath7th cluster and @xmath71 is the @xmath68-dimensional geometric centroid of the set @xmath70 . as described in 3.2 of @xcite , for diffusion maps @xmath72 where @xmath73 is the trivial left eigenvector of @xmath59 . the @xmath1-means algorithm begins with an initial partition of the data and alternately computes cluster centroids and reallocates points to the cluster with the nearest centroid . the algorithm stops when no points are allocated differently in two consecutive iterations . the final centroids define the @xmath1 prototypes to be used in our basis . in fig . [ sspdmap ] , we plot the mapping of 1278 ssp spectra into the @xmath74 dimensional diffusion space described by ( [ dmap2 ] ) . the large black dots denote the @xmath1-means centroids for @xmath35 . individual ssps are coloured by cluster membership . the @xmath1 prototypes capture the variability of the ssp spectra along a low - dimensional manifold in diffusion space . note that the density of prototypes varies along different parts of the manifold . this is due both to the local complexity of the manifold and the sampling of the original base of ssps . this latter effect , that we tend to obtain more prototypes in areas where there are more ssps in our basis , can easily be corrected for by using a weighted @xmath1-means method , described in 2.2.3 , that accounts for varying density of ssps along the manifold . the last step of the diffusion @xmath1-means algorithm is to determine the @xmath1 protospectra and their properties . first , the normalization constant for the @xmath7th prototype spectrum is @xmath75 and the @xmath7th protospectrum is @xmath76 similarly , the age , metallicity , and `` stellar - mass fraction''the percent of the initial stellar mass still in stars of prototype @xmath7 are naturally defined as @xmath77 where @xmath78 , and @xmath79 are the age , metallicity and stellar - mass fraction of the @xmath80th ssp , respectively . equations ( [ eqn : protot])-([eqn : protosmf ] ) are natural definitions of the prototype parameters in the sense that the fit of this basis to a galaxy would yield exactly the same parameter estimates as the equivalent mixture of the original set of ssps , @xmath39 . in fig . [ protot ] we plot the 45 ssp spectra in cf05 and the 45 prototype spectra found by diffusion @xmath1-means . all spectra are normalized to 1 at @xmath8=4020 and are coloured by log age . the diffusion @xmath1-means prototypes spread themselves evenly over the range of spectral profiles , capturing a gradual trend from young to old spectra . on the other hand , the cf05 basis includes many similar spectra from younger populations and sparsely covers the range of spectral profiles of older populations . see algorithm 1 for a summary of the diffusion @xmath1-means algorithm . sample matlab and r code is available on the web at http://www.stat.cmu.edu/~annlee/software.htm . the diffusionmap r package is available at http://cran.r - project.org/. normalize ssp spectra , @xmath39 , at @xmath8 , fix @xmath81 compute similarity matrix , @xmath82 ( [ distance ] ) and weight matrix @xmath57 ( [ weights ] ) compute @xmath59 : @xmath58 decompose @xmath59 : @xmath83 project @xmath39 to the @xmath68-dimensional diffusion map , @xmath84 ( [ dmap2 ] ) set @xmath85=1 . fix @xmath1 . randomly partition data into sets @xmath86 @xmath87 for some @xmath7 @xmath88 compute cluster centroids , @xmath89 ( [ centroid ] ) partition the data so that @xmath90 @xmath1 protospectra ( [ eqn : protospec ] ) and corresponding parameters ( [ eqn : protot]-[eqn : protosmf ] ) in [ simulations ] , we compare the fits obtained by the diffusion @xmath1-means basis to those found by two other numerical methods : principal components ( pc ) @xmath1-means and standard @xmath1-means . each of these two methods is similar to diffusion @xmath1-means , but both have critical drawbacks . _ pc @xmath1-means _ works similarly to diffusion @xmath1-means ( algorithm 1 ) except that @xmath1-means is performed on the projection of the normalized ssp spectra into _ principal components _ ( pc ) space , not diffusion space . for an example of the application of principal components analysis to ssp spectra see @xcite . the main drawback to pc @xmath1-means is its assumption that the ssp spectra lie on a linear subspace of the original @xmath91 dimensional space . if the ssps actually lie on a non - linear manifold , then the @xmath1 prototypes may poorly capture the intrinsic variation of the original ssps because the non - linear structure will have been inappropriately collapsed on to a linear space by the principal components projection . _ standard @xmath1-means _ is also similar to algorithm 1 , except that @xmath1-means is performed in the original @xmath92 dimensional space ; i.e. , no reduction in dimensionality is done before running @xmath1-means . there are two obvious drawbacks to this procedure . first , the algorithm generally is slow because distance computations are cumbersome in high dimensions and @xmath1-means usually takes more iterations to converge . for comparison , the dimensionality of the spaces used by diffusion and pc @xmath1-means are each @xmath93 , a factor of 100 smaller than @xmath45 . second , and more importantly , appropriate prototype spectra are difficult to find by standard @xmath1-means because euclidean distances , used by @xmath1-means to define clusters , are only physically meaningful over short distances . diffusion @xmath1-means avoids this problem by clustering in diffusion space , in which euclidean distance approximates diffusion distance , a measure that has physical meaning on all scales . the result is that standard @xmath1-means inappropriately relates ssps that are not physically similar . in fig . [ prototz ] , we plot @xmath94 versus @xmath95 for @xmath1=150 prototypes in asa07 and diffusion , standard , and pc @xmath1-means . notably , diffusion @xmath1-means finds a much higher density of prototypes with high @xmath94 or high @xmath95 , reflecting the complicated manner in which ssps with those properties vary with respect to @xmath37 and @xmath60 . at the other extreme , the asa07 prototypes reside on a regular grid , and thus include many prototype spectra that are essentially identical and also exclude prototypes that have unique spectral properties . the standard and pc @xmath1-means prototypes also estimate more high @xmath94 and high @xmath95 prototypes than a regular grid . the methods introduced above use only the observable properties of ssps to choose representative prototypes . it may be the case that we want to incorporate other _ a priori _ information that we have about the ssps and their relationship with the galaxies we are fitting . for instance , we might know that a ssp with a particular age and metallicity is generally found in the types of galaxies we are trying to fit , and hence will want to include in our basis a prototype with characteristics closely matching those of this ssp . this information can easily be incorporated with the framework introduced above by defining an _ a priori _ weight , @xmath96 for each ssp in our database , where higher weights signify more importance of the ssp . by modifying the definition of the diffusion map geometric centroid ( [ centroid ] ) to be @xmath97 and altering the @xmath1-means algorithm to minimize @xmath98 instead of @xmath99 in ( [ kmeans ] ) , we choose a basis that reflects both the numerical observable properties of the ssps and our _ a priori _ knowledge . this weighted @xmath1-means algorithm can be used to correct for the tendency of the prototype ssps to depend on the specific set of ssps , @xmath39 , chosen . if , for example , we define @xmath100 as the inverse of the local density of ssp @xmath80 , then the prototype ssps in fig . [ sspdmap ] would more uniformly cover the manifold of ssps in diffusion space , and not tend to more heavily sample denser regions of the ssp manifold . in this manner , the prototype spectra will only depend on the spectral properties , and not the particular sampling of the set of ssps .

speed of the code , which uses random walks through the parameter space , scales as the square of the number of basis spectra , making it computationally necessary to choose a small number of ssps that are coarsely sampled in age and metallicity . in this paper , we develop methods based on diffusion map ( lafon & lee , 2006 ) that , for the first time , choose appropriate bases of prototype ssp spectra from a large set of ssp spectra designed to approximate the continuous grid of age and metallicity of ssps of which galaxies are truly composed . we show that our techniques achieve better accuracy of physical parameter estimation for simulated galaxies . specifically , we show that our methods significantly decrease the age - metallicity degeneracy that is common in galaxy population synthesis methods .

to further our knowledge of the complex physical process of galaxy formation , it is essential that we characterize the formation and evolution of large databases of galaxies . the spectral synthesis starlight code of cid fernandes et al . ( 2004 ) was designed for this purpose . results of starlight are highly dependent on the choice of input basis of simple stellar population ( ssp ) spectra . speed of the code , which uses random walks through the parameter space , scales as the square of the number of basis spectra , making it computationally necessary to choose a small number of ssps that are coarsely sampled in age and metallicity . in this paper , we develop methods based on diffusion map ( lafon & lee , 2006 ) that , for the first time , choose appropriate bases of prototype ssp spectra from a large set of ssp spectra designed to approximate the continuous grid of age and metallicity of ssps of which galaxies are truly composed . we show that our techniques achieve better accuracy of physical parameter estimation for simulated galaxies . specifically , we show that our methods significantly decrease the age - metallicity degeneracy that is common in galaxy population synthesis methods . we analyze a sample of 3046 galaxies in sdss dr6 and compare the parameter estimates obtained from different basis choices . [ firstpage ] methods : data analysis methods : statistical methods : numerical galaxies : evolution galaxies : formation galaxies : stellar content

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galaxies do not live isolated lives , but exist in the tidal fields of their environment . arp ( 1966 ) , in his atlas of peculiar galaxies , lay the observational groundwork for the modern study of interacting galaxy systems by identifying many `` peculiar '' systems , later interpreted as various stages of major galaxy mergers . strong galaxy - galaxy interactions may dramatically alter the stellar populations ( e.g. larson & tinsley 1978 ; kennicutt _ et al . _ 1987 ; turner 1998 ; kennicutt 1998 ) , morphology ( e.g. toomre & toomre 1972 ; hernquist , heyl & spergel 1993 ) and kinematics of galaxies ( e.g. toomre & toomre 1972 ; barnes & hernquist 1992 ) driving evolution along the hubble sequence . massive mergers are also capable of funneling gas into the center of galaxies causing nuclear starbursts ( barnes & hernquist 1991 ; mihos , richstone & bothun 1992 ; barnes & hernquist 1996 ) and qso activity ( e.g. sanders _ 1988 ) . at the present epoch , however , major mergers are fairly rare events ( e.g. kennicutt _ et al . _ 1987 ) and their broad evolutionary importance is unclear . minor mergers and , in general , weak tidal interactions between galaxies occur with much higher frequency than major ones ( e.g. lacey & cole 1993 ) . by weak interactions we mean those which do not destroy the disk of the `` target '' spiral . hierarchical structure formation models ( e.g. cold dark matter ) predict that the merging histories for high mass objects today contained multiple low mass accretion events in their past ( e.g. lacey & cole 1993 ) . the specific roles which weak interactions play in the evolution of galaxies , however , is uncertain . weak interactions may cause disk heating ( e.g. toth & ostriker 1992 ; quinn , hernquist & fullagar 1993 ) and satellite remnants may build up the stellar halo ( e.g. searle & zinn 1978 ; johnston , hernquist & bolte 1996 ) . ( 1987 ) studied the relation between interaction strength and star formation by making a comparison between isolated galaxies , close pairs , and galaxies from the arp atlas . they found that close pairs have larger values of @xmath15 , i.e. higher star formation rates ( sfr ) than isolated galaxies . while pair spacing is weakly correlated with the sfr , they could not determine the specific role of interaction strength on the sfr . ( 1998 ) and allam _ ( 1999 ) both studied the hubble type specific effects of environment on the sfr in galaxies . they found that the sfr/_mass of existing stars _ was inversely proportional to the local galaxy density . they postulate that the anti - correlation is due partly to gas stripping and due partly to the anti - correlation of the merger cross - section with the galaxy - galaxy velocity dispersion . there is also evidence that interactions excite nuclear activity . in their close pair and strongly interacting sample kennicutt _ ( 1987 ) found a strong correlation between @xmath16 emission in the disk and that in the nucleus . such a correlation between disk and nuclear emission is supported by theoretical work ; mihos & hernquist ( 1994 ) and hernquist & mihos ( 1995 ) demonstrated that minor interactions form bar instabilities in the disk which in turn funnel large amounts of gas into the nucleus . the effectiveness of this process is suppressed by the presence of a dense bulge , which prevents bar formation . due to the numerical expense in computing high resolution n - body / sph ( collisionless particle / smoothed particle hydrodynamics ) models , the exact interaction parameters which result in such activity are uncertain . weak interactions may also manifest themselves as kinematic or structural irregularities . roughly @xmath17 of all spiral galaxies have asymmetric hi profiles and rotation curves ( baldwin , lynden - bell & sancisi 1980 ; richter & sancisi 1994 ; haynes _ et al . ( 1980 ) postulated that these asymmetries are caused by weak interactions in the galaxy s past or by lopsided orbits . ( 1999 ) examined the optical rotation curves of a set of observed and simulated interacting disk galaxies . they showed that interactions can cause large scale , time dependent asymmetries in the rotation curves of their sample galaxies . ( 1999 ) studied the kinematic asymmetries present in two galaxies lopsided in their optical and hi distributions . they qualitatively reproduced the kinematic asymmetries by placing closed orbits in mildly lopsided potential . a dynamical indicator of weak interactions may be `` lopsidedness . '' in the context of this paper ( following rudnick & rix 1998 ; hereafter rr98 ) , lopsidedness is defined as a bulk asymmetry in the _ mass _ distribution of a galactic disk . surveys for lopsidedness in the stellar light of galaxies were first carried out by rix & zaritsky ( 1995 ; hereafter rz95 ) and zaritsky & rix ( 1997 ; hereafter zr97 ) . using near - ir surface photometry of face - on spiral galaxies ( spanning all hubble types ) they examined the magnitude of the @xmath2 azimuthal fourier component of the i and k - band surface brightness , thus characterizing the global asymmetry of the stellar light . rz95 and zr97 found that a quarter of the galaxies in their sample were significantly lopsided . using a larger , magnitude limited sample restricted to early type disks ( s0 to sab ) and imaged in the r - band , rr98 found that the fraction of significantly lopsided early type disks is identical to that for late - type disks . rr98 convincingly demonstrated that lopsidedness is not an effect of dust , but is in fact the asymmetric distribution of the light from old stars and hence from the _ stellar mass _ in the disk . some theoretical work has been done in examining long lived @xmath2 modes ( syer & tremaine 1996 ; zang & hohl 1978 ; sellwood & merritt 1994 ) , little convincing evidence however has been put forth to show that isolated galaxies will form stable @xmath2 modes without external perturbations or significant counter - rotating populations . without invoking the special cases above , long lived lopsidedness is possible if the disk resides in a lopsided potential . the question remains however : how is a lopsided potential created / maintained ? numerical simulations of hyperbolic encounters between disk galaxies fail to produce @xmath2 modes of amplitude @xmath18 without destroying the pre - existing stellar disk ( naab , t. ; private communication ) . minor mergers and possibly some weak interactions therefore remain as the most probable cause of lopsidedness ( rr98 ) . recent work has shown that perturbations in the outer halo of a galaxy may be amplified and even transmitted down into the disk ( weinberg 1994 ) . work by walker , mihos & hernquist ( 1996 ) and zr97 showed that the type and magnitude of lopsidedness seen in rz95 , zr97 and rr98 is comparable to the result of the accretion of a small satellite , if the mass ratio with the main galaxy is @xmath19 . in a preliminary study ( i.e. a rigid halo with no dynamical friction ) levine and sparke ( 1998 ) showed that lopsided galaxies may be formed by disks orbiting off center and retrograde in a flat - cored , dark matter dominated halo . they postulated that a galaxy may be pushed off center by a satellite accretion . using phase mixing arguments ( baldwin _ et al . _ 1980 ; rz95 ) and analysis of n - body simulations ( walker _ et al . _ 1996 ; zr97 ) the lifetime of lopsided features has been estimated at @xmath20 gyr . that lopsidedness is transient ( @xmath21 ) yet common , requires that it must be recurring and therefore lopsidedness may have significant evolutionary consequences . the current paper focuses on the impact that minor mergers ( observed as lopsidedness ) may have on boosting the sfr and the recent star formation history ( sfh ) of disk galaxies . for the purpose of this discussion , we will assume that lopsidedness is caused by minor mergers . regardless of what causes lopsidedness however , the perturbation in the gravitational potential manifestly exists and therefore may affect the gas in the galaxy to such a degree as to boost the sfr . indeed , zr97 find that lopsidedness is correlated ( at @xmath22 confidence ) with the `` excess '' of blue luminosity ( over what is predicted by the tully - fisher relation ) . modeling the integrated spectral evolution of starbursts using evolutionary population synthesis ( eps ) codes has been been well studied ( e.g. couch and sharples 1987 ; barger _ _ 1996 ; turner 1998 ) and despite its limitations , is a useful tool in determining the relative sfh over the past @xmath23 gyr . the same techniques used to probe the sfh in massive starbursts should also work to probe the recent sfh in the putative mini - bursts which we seek to study . by comparing measured indicators of recent sf ( e.g. @xmath4 , @xmath3 break strength , a star content ) , to the same indicators derived from the eps models , we will place limits on the mini - burst mass and duration . we have obtained spatially integrated spectra of a sample of 40 late type spiral galaxies ( sab - sbc ) of varying degrees of lopsidedness with the intent of using their relative stellar populations ( as determined from stellar template fitting and eps models ) to determine their recent sf histories . unlike the mass - normalized blue light excess , @xmath24 used in zr97 , our method operates independently of assumptions about a galaxy s mass , inclination or luminosity . in addition to probing the recent ( @xmath25 gyr ) sfh with studies of the stellar continuum we probe the current sfr by measuring the integrated balmer line emission strengths ( e.g. kennicutt _ et al . _ 1994 ) . the layout of the paper is as follows . in 2 we discuss the sample selection , observations , data reduction and determination of galaxy lopsidedness ; in 3 we examine our methods for determining the current sfr and recent sfh via the measurement of emission and stellar continuum properties as a function of lopsidedness . the discussion of the significance of these results , including the correlation of the boost parameters with other galaxy characteristics and the impact of our results on previous works ( i.e. rz95,zr97 & rr98 ) is contained in 4 . in 5 we present a summary and possible directions for future work .

to investigate the link between weak tidal interactions in disk galaxies and the boosting of their recent star formation , we obtain images and spatially integrated spectra ( ) for 40 late - type spiral galaxies ( sab - sbc ) with varying degrees of lopsidedness ( a dynamical indicator of weak interactions ) . we quantify lopsidedness as the amplitude , of the fourier component of the azimuthal surface brightness distribution , averaged over a range of radii . we compare the young stellar content , quantified by and the strength of the break ( ) , with lopsidedness and find a correlation between the two .

to investigate the link between weak tidal interactions in disk galaxies and the boosting of their recent star formation , we obtain images and spatially integrated spectra ( ) for 40 late - type spiral galaxies ( sab - sbc ) with varying degrees of lopsidedness ( a dynamical indicator of weak interactions ) . we quantify lopsidedness as the amplitude , of the fourier component of the azimuthal surface brightness distribution , averaged over a range of radii . the median spectrum of the most lopsided galaxies shows strong evidence for a more prominent young stellar population ( i.e. strong balmer absorption , strong nebular emission , a weak break and a blue continuum ) when compared to the median spectrum of the most symmetric galaxies . we compare the young stellar content , quantified by and the strength of the break ( ) , with lopsidedness and find a correlation between the two . we also find a correlation between and lopsidedness . using the evolutionary population synthesis code of bruzual & charlot we model the spectra as an `` underlying population '' and a superimposed `` boost population '' with the aim of constraining the fractional boost in the sfr averaged over the past gyr ( the characteristic lifetime of lopsidedness ) . from the difference in both and the strength of the break ( ) between the most and least symmetric thirds of our sample , we infer that of stars are formed over the duration of a lopsided event in addition to the `` underlying '' sfh ( assuming a final galactic stellar mass of ) . this corresponds to a factor of increase in the sfr over the past years . for the nuclear spectra , all of the above correlations except vs. are weaker than for the disk , indicating that in lopsided galaxies , the sf boost is not dominated by the nucleus .

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to quantify the correlation between the recent sf histories of present - day spiral galaxies and their global asymmetry , we compare the integrated spectral properties of late - type spirals of varying lopsidedness . we find that the recent ( @xmath159 gyr ) sfh and current ( @xmath160 years ) sfr are both strongly correlated with @xmath150 although there is appreciable scatter in the individual galaxy - to - galaxy properties . for @xmath74 , reflecting the current sfr , we find a @xmath7 spearman - rank correlation with @xmath150 . we fit a combination of a0v and g0iii stellar spectra to our galaxy spectra to quantify the relative abundance of a - stars in the disk ( which traces the sfr within 0.5 gyrs ) . from these best fit model spectra , @xmath92 , we measure a number of spectral indices , and find that @xmath99 , @xmath5 , and @xmath82 are correlated with @xmath150 at the @xmath161 , @xmath162 , and @xmath163 levels , respectively . we measure the same spectral indices in the nucleus , and find them less correlated with @xmath150 ( except @xmath5 ) . unless a nuclear starburst is obscured , the disk and not the nucleus is the primary site of the sf increase we see in lopsided galaxies . this is in contrast to numerical simulations where minor mergers funnel gas into the nucleus of galaxies , causing intense starbursts ( mihos & hernquist 1994 ; hernquist & mihos 1995 ) . only by the presence of a dense bulge can the formation of a bar , and the subsequent funneling of gas , be prevented . to quantify the mass of additional stars formed in lopsided galaxies , we defined a boost vector in @xmath99 vs. @xmath5 space , by comparing the median values of these properties for the most symmetric third and the most lopsided third of our sample . we find @xmath164 and @xmath165 . we fit this vector with an `` underlying population @xmath88 boost '' eps model corresponding to a progenitor galaxy with @xmath134 , @xmath135 myr , and boost age of @xmath9 gyr . using this best fit eps model , we find that @xmath10 is formed in the boost in addition to the `` underlying '' sfh ( assuming a final stellar mass of @xmath166 ) . this is a considerable fraction ( @xmath167 ) of the final stellar mass of the galaxy and corresponds to a factor of 8 increase in the sfr over the past @xmath13 years . given the increasing merger rates and increasing gas fractions towards higher redshifts , minor merger induced sf boosts of short duration played an important role in assembling the present day stellar content of galaxies . finally , we address by how much the frequency of lopsidedness from a magnitude limited sample is increased by the corresponding luminosity boost . our best fit eps boost model corresponds to a @xmath158 magnitude brightening when galaxies becomes lopsided , increasing their presence four - fold in magnitude limited samples . we lack the statistics however , to examine any hubble type dependent differences in the luminosity boost . it is obvious that more work needs to be done to fully understand the cause of lopsidedness as well as the sfh of lopsided galaxies . to quantify the hubble type specific boost in the recent sfh , a large sample should be obtained with significant numbers of galaxies in each hubble type bin . since imaging and spectroscopy will be needed for this project , a volume limited sample may be constructed which bypasses many of the problems encountered when selecting galaxies according to an apparent magnitude limit . companion searches to sufficiently faint magnitudes will help to study the possible link between environment and lopsidedness ( as caused by weak tidal interactions ) . with the recent commissioning of large area imaging and spectroscopy surveys such as sloan digital sky survey , constructing such a sample will become relatively straightforward . numerical simulations have shown to be a useful tool in studying the evolution of the stellar and gas distributions in minor mergers . high resolution simulations with a live halo are crucial for studying the detailed response of the disk to the merger ( walker _ et al . a thorough exploration of interaction parameter space is needed to quantify the structural and kinematic response in the stellar and gas components . high resolution n - body studies are also needed to explore the global stability of isolated galactic disks . this work was completed with partial support from nsf grants ast9870151 , ast9421145 and ast9900789 . greg rudnick and h .- w . rix would like to thank nelson caldwell for many valuable discussions on measuring sfhs from integrated spectra . greg rudnick would like to thank craig kulesa and christopher gottbrath for many useful discussions in the early hours of the morning . greg rudnick would also like to thank megan sosey and chris gottbrath for assisting with our observing program . finally , we would like to thank the steward observatory 2.3-m telescope operators for their assistance in the 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l. 1996 , , 460 , 121 weinberg , m. d. 1994 , apj , 421 , 481 zang , t. a. , & hohl , f. 1978 , , 226 , 521 zaritsky , d. , & rix , h .- w . 1997 , , 477 , 118 llllll a 1219 + 41 & 3.0 & 1.19 & 6927 & 0.85 & 7.52@xmath1681.03 ic 0520 & 2.0 & 1.95 & 3528 & 0.79 & 6.06@xmath1680.90 ic 0900 & 4.0 & 1.62 & 7067 & 0.65 & 9.30@xmath1680.89 ic 1269 & 4.0 & 1.70 & 6116 & 0.74 & 8.21@xmath1681.57 ngc 2347 & 3.0 & 1.78 & 4422 & 0.71 & 5.32@xmath1680.79 ngc 2582 & 2.0 & 1.23 & 4439 & 1.00 & 2.79@xmath1680.35 ngc 2744 & 2.0 & 1.66 & 3428 & 0.65 & 1.62@xmath1680.33 ngc 2916 & 3.0 & 2.45 & 3730 & 0.68 & 5.50@xmath1680.76 ngc 3066 & 4.0 & 1.10 & 2049 & 0.89 & 0.78@xmath1680.12 ngc 3162 & 4.0 & 3.02 & 1298 & 0.83 & 1.41@xmath1680.18 ngc 3177 & 3.0 & 1.44 & 1302 & 0.81 & 0.46@xmath1680.06 ngc 3310 & 4.0 & 3.09 & 0980 & 0.78 & 1.84@xmath1680.18 ngc 3353 & 3.0 & 1.35 & 0944 & 0.71 & 0.22@xmath1680.03 ngc 3681 & 4.0 & 2.51 & 1239 & 0.79 & 1.49@xmath1680.48 ngc 3684 & 4.0 & 3.09 & 1163 & 0.69 & 1.57@xmath1680.50 ngc 3897 & 4.0 & 1.95 & 6411 & 1.00 & 7.93@xmath1681.17 ngc 3928 & 3.0 & 1.51 & 0982 & 1.00 & 0.24@xmath1680.03 ngc 3963 & 4.0 & 2.75 & 3186 & 0.91 & 5.23@xmath1681.67 ngc 4017 & 4.0 & 1.78 & 3454 & 0.78 & 3.72@xmath1681.18 ngc 4041 & 4.0 & 2.69 & 1234 & 0.93 & 1.27@xmath1680.16 ngc 4351 & 2.0 & 2.00 & 2310 & 0.68 & 1.79@xmath1680.26 ngc 4412 & 3.0 & 1.41 & 2294 & 0.89 & 1.43@xmath1680.21 ngc 4430 & 3.0 & 2.29 & 1443 & 0.89 & 0.85@xmath1680.13 ngc 4595 & 3.0 & 1.74 & 0633 & 0.65 & 0.12@xmath1680.18 ngc 4639 & 4.0 & 2.75 & 1010 & 0.68 & 0.59@xmath1680.05 ngc 4814 & 3.0 & 3.09 & 2513 & 0.74 & 2.48@xmath1680.50 ngc 4911 & 4.0 & 1.45 & 7970 & 0.91 & 12.04@xmath1681.53 ngc 5218 & 3.0 & 1.82 & 2807 & 0.69 & 2.31@xmath1681.02 ngc 5614 & 2.0 & 2.45 & 3892 & 0.83 & 7.50@xmath1680.95 ngc 5653 & 3.0 & 1.74 & 3567 & 0.74 & 4.63@xmath1680.59 ngc 5713 & 4.0 & 2.75 & 1883 & 0.89 & 3.32@xmath1680.49 ngc 5915 & 4.0 & 1.82 & 5570 & 1.00 & 2.04@xmath1680.26 ngc 5923 & 2.0 & 1.74 & 2291 & 0.72 & 5.20@xmath1680.88 ngc 5936 & 3.0 & 1.45 & 4004 & 0.89 & 4.76@xmath1680.60 ngc 6012 & 2.0 & 2.09 & 1854 & 0.72 & 1.69@xmath1680.21 ngc 6195 & 3.0 & 1.55 & 9000 & 0.69 & 12.89@xmath1682.19 ngc 6574 & 4.0 & 1.41 & 2282 & 0.78 & 2.02@xmath1680.26 ngc 6711 & 4.0 & 1.35 & 4671 & 0.93 & 3.63@xmath1680.50 ngc 6824 & 3.0 & 1.70 & 3337 & 0.69 & 3.58@xmath1680.72 ngc 7177 & 3.0 & 3.10 & 1150 & 0.64 & 0.99@xmath1680.12 llllllll a 1219 + 41 & 19.7 & 3.15 & 0.083@xmath1680.057 & -3.7@xmath1680.3 & 8.1@xmath1680.6 & 0.34 & 1.18@xmath1680.01 ic 0520 & 19.8 & 3.82 & 0.031@xmath1680.039 & -2.3@xmath1680.2 & 4.4@xmath1680.7 & 0.16 & 1.72@xmath1680.01 ic 0900 & 11.6 & 6.54 & 0.156@xmath1680.068 & -6.1@xmath1680.3 & 8.0@xmath1680.7 & 0.38 & 1.32@xmath1680.01 ic 1269 & 13.0 & 2.93 & 0.239@xmath1680.108 & -6.3@xmath1680.5 & 6.9@xmath1680.9 & 0.32 & 1.34@xmath1680.02 ngc 2347 & 10.6 & 5.34 & 0.083@xmath1680.044 & -4.9@xmath1680.3 & 6.5@xmath1680.7 & 0.32 & 1.12@xmath1680.01 ngc 2582 & 20.0 & 2.82 & 0.061@xmath1680.042 & -2.7@xmath1680.3 & 5.6@xmath1680.9 & 0.24 & 1.59@xmath1680.02 ngc 2744 & 10.8 & 4.28 & 0.233@xmath1680.047 & -5.4@xmath1680.4 & 9.6@xmath1680.8 & 0.60 & 1.32@xmath1680.01 ngc 2916 & 23.0 & 3.29 & 0.177@xmath1680.062 & -4.3@xmath1680.2 & 7.0@xmath1680.6 & 0.36 & 1.47@xmath1680.01 ngc 3066 & 13.7 & 3.40 & 0.356@xmath1680.132 & -6.4@xmath1680.2 & 7.0@xmath1680.6 & 0.37 & 1.05@xmath1680.01 ngc 3162 & 26.0 & 2.91 & 0.181@xmath1680.070 & -6.8@xmath1680.3 & 8.6@xmath1680.7 & 0.52 & 1.42@xmath1680.01 ngc 3177 & 8.6 & 6.58 & 0.183@xmath1680.050 & -6.6@xmath1680.2 & 7.2@xmath1680.5 & 0.37 & 1.213@xmath1680.005 ngc 3310 & 9.92 & 7.63 & 0.334@xmath1680.137 & -20.1@xmath1680.2 & 8.0@xmath1680.4 & 0.45 & 0.963@xmath1680.002 ngc 3353 & 12.1 & 4.23 & 0.643@xmath1680.059 & -22.2@xmath1680.2 & 6.8@xmath1680.6 & 0.38 & 0.883@xmath1680.004 ngc 3681 & 24.9 & 3.03 & 0.027@xmath1680.038 & -2.6@xmath1680.2 & 3.7@xmath1680.7 & 0.12 & 1.18@xmath1680.01 ngc 3684 & 16.8 & 4.51 & 0.042@xmath1680.046 & -5.1@xmath1680.3 & 6.6@xmath1680.7 & 0.34 & 1.23@xmath1680.01 ngc 3897 & 12.6 & 4.49 & 0.099@xmath1680.054 & -3.8@xmath1680.3 & 5.6@xmath1680.9 & 0.25 & 1.12@xmath1680.01 ngc 3928 & 18.7 & 4.03 & 0.009@xmath1680.036 & -6.6@xmath1680.3 & 6.7@xmath1680.7 & 0.35 & 1.42@xmath1680.01 ngc 3963 & 25.4 & 2.98 & 0.135@xmath1680.047 & -3.4@xmath1680.2 & 8.0@xmath1680.6 & 0.34 & 1.19@xmath1680.01 ngc 4017 & 16.0 & 3.89 & 0.083@xmath1680.087 & -7.0@xmath1680.3 & 8.3@xmath1680.7 & 0.48 & 1.15@xmath1680.01 ngc 4041 & 10.8 & 6.97 & 0.041@xmath1680.041 & -6.4@xmath1680.2 & 6.8@xmath1680.5 & 0.35 & 1.20@xmath1680.005 ngc 4351 & 18.1 & 4.18 & 0.153@xmath1680.047 & -4.2@xmath1680.3 & 7.3@xmath1680.7 & 0.42 & 0.98@xmath1680.01 ngc 4412 & 23.7 & 2.90 & 0.396@xmath1680.098 & -6.0@xmath1680.2 & 7.8@xmath1680.6 & 0.40 & 1.26@xmath1680.01 ngc 4430 & 28.3 & 2.67 & 0.245@xmath1680.089 & -5.1@xmath1680.3 & 7.4@xmath1680.8 & 0.35 & 1.29@xmath1680.01 ngc 4595 & 16.3 & 3.82 & 0.170@xmath1680.097 & -4.2@xmath1680.2 & 7.9@xmath1680.6 & 0.46 & 1.003@xmath1680.005 ngc 4639 & 23.4 & 3.24 & 0.072@xmath1680.043 & -2.9@xmath1680.2 & 4.9@xmath1680.6 & 0.15 & 1.52@xmath1680.01 ngc 4814 & 13.5 & 5.58 & 0.068@xmath1680.041 & -3.2@xmath1680.3 & 5.2@xmath1680.7 & 0.17 & 1.50@xmath1680.01 ngc 4911 & 14.5 & 3.54 & 0.129@xmath1680.055 & -3.2@xmath1680.3 & 6.5@xmath1680.8 & 0.27 & 1.38@xmath1680.01 ngc 5218 & 8.7 & 4.86 & 0.182@xmath1680.046 & -1.6@xmath1680.2 & 7.6@xmath1680.6 & 0.29 & 1.37@xmath1680.01 ngc 5614 & 11.9 & 4.76 & 0.074@xmath1680.080 & -1.7@xmath1680.3 & 3.9@xmath1680.8 & 0.10 & 1.71@xmath1680.01 ngc 5653 & 6.8 & 5.63 & 0.129@xmath1680.131 & -5.2@xmath1680.2 & 8.0@xmath1680.5 & 0.36 & 1.343@xmath1680.005 ngc 5713 & 14.6 & 4.27 & 0.349@xmath1680.106 & -6.2@xmath1680.2 & 9.6@xmath1680.5 & 0.48 & 1.113@xmath1680.003 ngc 5915 & 5.0 & 3.48 & 0.204@xmath1680.109 & -11.5@xmath1680.2 & 8.7@xmath1680.6 & 0.51 & 1.146@xmath1680.005 ngc 5923 & 15.2 & 2.52 & 0.115@xmath1680.055 & -2.6@xmath1680.4 & 6.2@xmath1680.9 & 0.23 & 1.44@xmath1680.02 ngc 5936 & 11.7 & 3.61 & 0.114@xmath1680.065 & -7.6@xmath1680.2 & 8.3@xmath1680.6 & 0.44 & 1.218@xmath1680.005 ngc 6012 & 12.2 & 3.46 & 0.028@xmath1680.042 & -3.5@xmath1680.3 & 4.6@xmath1680.9 & 0.15 & 1.45@xmath1680.02 ngc 6195 & 11.2 & 4.17 & 0.098@xmath1680.048 & -3.6@xmath1680.4 & 6.1@xmath1680.9 & 0.25 & 1.30@xmath1680.02 ngc 6574 & 8.6 & 4.46 & 0.111@xmath1680.050 & -5.6@xmath1680.2 & 7.2@xmath1680.5 & 0.31 & 1.43@xmath1680.01 ngc 6711 & 9.2 & 3.13 & 0.166@xmath1680.050 & -6.2@xmath1680.3 & 7.5@xmath1680.8 & 0.38 & 1.25@xmath1680.01 ngc 6824 & 14.5 & 4.64 & 0.068@xmath1680.1 & -2.4@xmath1680.2 & 6.5@xmath1680.5 & 0.24 & 1.50@xmath1680.01 ngc 7177 & 10.9 & 5.66 & 0.140@xmath1680.1 & -1.7@xmath1680.2 & 4.1@xmath1680.6 & 0.12 & 1.59@xmath1680.01 [ tbl : table2.7.23.99.in ]

the median spectrum of the most lopsided galaxies shows strong evidence for a more prominent young stellar population ( i.e. strong balmer absorption , strong nebular emission , a weak break and a blue continuum ) when compared to the median spectrum of the most symmetric galaxies . using the evolutionary population synthesis code of bruzual & charlot we model the spectra as an `` underlying population '' and a superimposed `` boost population '' with the aim of constraining the fractional boost in the sfr averaged over the past gyr ( the characteristic lifetime of lopsidedness ) . from the difference in both and the strength of the break ( ) between the most and least symmetric thirds of our sample , we infer that of stars are formed over the duration of a lopsided event in addition to the `` underlying '' sfh ( assuming a final galactic stellar mass of ) . this corresponds to a factor of increase in the sfr over the past years . for the nuclear spectra , all of the above correlations except vs. are weaker than for the disk , indicating that in lopsided galaxies , the sf boost is not dominated by the nucleus .

to investigate the link between weak tidal interactions in disk galaxies and the boosting of their recent star formation , we obtain images and spatially integrated spectra ( ) for 40 late - type spiral galaxies ( sab - sbc ) with varying degrees of lopsidedness ( a dynamical indicator of weak interactions ) . we quantify lopsidedness as the amplitude , of the fourier component of the azimuthal surface brightness distribution , averaged over a range of radii . the median spectrum of the most lopsided galaxies shows strong evidence for a more prominent young stellar population ( i.e. strong balmer absorption , strong nebular emission , a weak break and a blue continuum ) when compared to the median spectrum of the most symmetric galaxies . we compare the young stellar content , quantified by and the strength of the break ( ) , with lopsidedness and find a correlation between the two . we also find a correlation between and lopsidedness . using the evolutionary population synthesis code of bruzual & charlot we model the spectra as an `` underlying population '' and a superimposed `` boost population '' with the aim of constraining the fractional boost in the sfr averaged over the past gyr ( the characteristic lifetime of lopsidedness ) . from the difference in both and the strength of the break ( ) between the most and least symmetric thirds of our sample , we infer that of stars are formed over the duration of a lopsided event in addition to the `` underlying '' sfh ( assuming a final galactic stellar mass of ) . this corresponds to a factor of increase in the sfr over the past years . for the nuclear spectra , all of the above correlations except vs. are weaker than for the disk , indicating that in lopsided galaxies , the sf boost is not dominated by the nucleus .

1105.1482

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the relationship between the transmitted symbol and the received signal vector in many communication systems can be expressed in the form @xmath1 where @xmath2 is the @xmath3 transmitted vector whose entries are chosen from a finite symbol alphabet , @xmath4 and @xmath5 are the @xmath6 received signal and noise vectors , respectively , and @xmath7 is @xmath8 channel matrix . as a practical decoding scheme when a code constraint is imposed , _ iterative detection and decoding _ ( idd ) has been applied to various digital communication systems including channel equalization @xcite , multi - input multi - output ( mimo ) detection @xcite , and multi - user detection @xcite . motivated by the _ turbo principle _ @xcite , an idd receiver exchanges soft information between a symbol detector and a channel decoder to achieve performance close to the channel capacity . the symbol detector computes _ a posteriori _ probabilities ( app ) on the bits comprising @xmath2 , using _ a priori _ probabilities provided by the channel decoder and the observation @xmath4 . then , the detector exchanges this soft information ( so called _ extrinsic _ information ) with a soft - input soft - output decoder , such as the max - log - map decoder @xcite . in the sequel , we refer to such a detector as an _ app detector_. direct computation of the app involves marginalization over all configurations of the vector @xmath2 , leading to exponential complexity in the system size ( e.g. , number of antenna elements in mimo systems ) . as a means of approximately performing the app detection at reduced complexity , _ tree detection techniques _ have received much attention recently @xcite . ( refer to @xcite for an overview of tree detection techniques . ) the essence of these approaches is to produce a set of promising symbol candidates via a tree search for estimating the app over this reduced set . thus far , a variety of tree detection algorithms have been proposed . in @xcite , the _ sphere decoding algorithm _ ( sda ) @xcite with a fixed radius was used to find symbol candidates . in @xcite , _ a priori _ information obtained from the channel decoder was exploited to improve the search efficiency of the sda . in @xcite , a hard sphere decoder was employed to find a single _ maximum a posteriori _ probability ( map ) symbol estimate maximizing @xmath9 and a candidate list was generated by flipping bits in the map estimate . in @xcite , the apps of all bits in @xmath2 are obtained simultaneously by modifying a bound tightening rule of a single sphere search . additionally , a more sophisticated extension of this idea was introduced in @xcite . the computational complexity of these tree detection algorithms varies depending on the channel and noise realizations , and in the worst - case the search complexity is the same as that of exhaustive search . in order to limit the worst case complexity of the tree detection approach , _ fixed - complexity tree search _ techniques @xcite have been proposed . for example , an @xmath0-algorithm was extended to soft - input soft - output detection in @xcite and an intelligent candidate adding algorithm for improving efficiency of the @xmath0-algorithm was proposed in @xcite . the stack algorithm was also exploited for list generation in combination with soft augmentation of tail bits of stack elements @xcite . other fixed - complexity soft - input soft - output detection algorithms can be found in @xcite . the @xmath0-algorithm @xcite , also known as _ @xmath10-best algorithm _ in the mimo detection literature @xcite , selects only a finite set of the @xmath0 best candidates for each layer of the detection tree . the @xmath0-algorithm is a practical candidate for soft - input soft - output detection due to its inherent nature to facilitate parallel and pipelined processing @xcite . in spite of this benefit , the @xmath0-algorithm suffers from a poor performance - complexity trade - off due to the greedy nature of the algorithm . to be specific , the algorithm checks the validity of paths in the forward direction and never traverses back for reconsideration . once the correct path is rejected , it will never be selected again in subsequent selections , resulting in wasteful search effort . moreover , these erroneous decisions often occur in early candidate selection stages where the accumulated path metric considers only a few symbol spans . one way to alleviate such error propagation is symbol detection ordering @xcite . by processing each layer in an appropriate order , the chances of errors propagating to the next stage can be reduced . nevertheless , error propagation severely limits the performance of the @xmath0-algorithm especially when the system size is large . in this paper , we pursue an improvement of the performance - complexity trade - off of soft - input soft - output @xmath0-algorithms . towards this end , we propose a new path metric capturing the contribution of the entire symbol path . while the conventional path metric accounts for the contributions of symbols along the visited path only , the new path metric looks ahead to the unvisited paths and estimates their contributions through a soft unconstrained linear symbol estimate . in fact , a _ bias term _ reflecting the information from as - yet undecided symbols is incorporated into the conventional path metric for this purpose . in order to distinguish this improved path metric from the conventional path metric and other look - ahead metrics , we henceforth refer to it as a _ linear estimate - based look - ahead ( le - la ) path metric_. we apply the le - la path metric to the soft - input soft - output @xmath0-algorithm , introducing an _ improved soft - in soft - out @xmath0-algorithm _ ( iss - ma ) . by sorting paths based on the le - la path metric , the iss - ma lessens the chance of rejecting the correct path from the candidate list and eventually improves the detection performance especially for systems of large dimension . indeed , from an analysis of the probability of correct path loss ( cpl ) , we show that the le - la path metric benefits the candidate selection process of the @xmath0-algorithm . the idea of using a look - ahead path metric has been explored in artificial intelligence search problems @xcite and can also be found in soft decoding of linear block codes @xcite . in @xcite , computationally efficient methods to obtain the bias term were investigated using semi - definite programming and @xmath11 estimation techniques . while these approaches search for a _ deterministic _ bias term ( lower - bound of future cost ) to guarantee the optimality of the sequential or depth - first search , our approach uses linear estimation to derive a bias term designed to improve candidate selection of the breadth - first search . the key advantage of using a linear estimator is that _ a priori _ information can be easily incorporated into the bias term so that the look - ahead operation benefits from the decoder output in each iteration . it is also worth emphasizing the difference between the proposed path metric and fano matric @xcite . the fano metric exploits the _ a posteriori _ probability of each path as its path metric . for a binary symmetric channel , the fano metric introduces a bias term proportional to the path length to penalize paths of short length . the extension of the fano metric to channels with memory or mimo channels is not straightforward , since it involves marginalization over the distribution of the undecided symbols . modification of the fano metric is considered for equalization of intersymbol interference ( isi ) channels in @xcite and for multi - input multi - output detection in @xcite . as a means to improve path metric of the sda , the idea of probabilistic pruning was introduced in @xcite . in @xcite , the probability density of an observed signal estimated from a separate tree search is used as a bias term . while these approaches assign an equal bias term for paths of the same length , the iss - ma provides a distinct bias term for each path in the tree , allowing for the application of a breadth - first search such as the @xmath0-algorithm . as such , our path metric can be readily combined with any type of tree - based soft - input soft - output detector . the rest of this paper is organized as follows . in section [ sec : problem ] , we briefly review the idd system and the tree detection algorithm . in section [ sec : iss - ma ] , we present the le - la path metric along with its efficient computation . we also describe the application of the le - la path metric to the soft - input soft - output @xmath0-algorithm . in section [ sec : performance ] , we present the performance analysis of the iss - ma . in section [ sec : simulation ] , we provide simulation results and conclude in section [ sec : conclusion ] . we briefly summarize the notation used in this paper . uppercase and lowercase letters written in boldface denote matrices and vectors , respectively . the superscripts @xmath12 and @xmath13 denote transpose and conjugate transpose ( hermitian operator ) , respectively . @xmath14 denotes an @xmath15-norm square of a vector and @xmath16 is a diagonal matrix that has elements on the main diagonal . @xmath17 and @xmath18 are @xmath19 matrix whose entries are all ones or zeros , respectively . the subscript is omitted if there is no risk of confusion . @xmath20 denotes a circular symmetric complex gaussian density with mean @xmath21 and variance @xmath22 . @xmath23 $ ] denotes expectation over the random variable @xmath24 . @xmath25 denotes @xmath26 - e[\mathbf{x}]e[\mathbf{y}^{h}]$ ] . for a hermitian matrix @xmath27 , @xmath28 ( or @xmath29 ) means that @xmath27 is semi - positive definite ( or positive definite ) . @xmath30 means probability of the event @xmath31 . @xmath32 denotes a joint probability density function ( pdf ) for the random variables @xmath33 .

tree detection techniques are often used to reduce the complexity of _ a posteriori probability _ ( app ) detection in high dimensional multi - antenna wireless communication systems . in this paper , we introduce an efficient soft - input soft - output tree detection algorithm that employs a new type of look - ahead path metric in the computation of its branch pruning ( or sorting ) . while conventional path metrics depend only on symbols on a visited path , the new path metric accounts for unvisited parts of the tree in advance through an unconstrained linear estimator and adds a bias term that reflects the contribution of as - yet undecided symbols . by applying the linear estimate - based look - ahead path metric to an-algorithm that selects the best paths for each level of the tree we develop a new soft - input soft - output tree detector , called an _ improved soft - input soft - output-algorithm _ ( iss - ma ) . based on an analysis of the probability of correct path loss , we show that the improved path metric offers substantial performance gain over the conventional path metric .

tree detection techniques are often used to reduce the complexity of _ a posteriori probability _ ( app ) detection in high dimensional multi - antenna wireless communication systems . in this paper , we introduce an efficient soft - input soft - output tree detection algorithm that employs a new type of look - ahead path metric in the computation of its branch pruning ( or sorting ) . while conventional path metrics depend only on symbols on a visited path , the new path metric accounts for unvisited parts of the tree in advance through an unconstrained linear estimator and adds a bias term that reflects the contribution of as - yet undecided symbols . by applying the linear estimate - based look - ahead path metric to an-algorithm that selects the best paths for each level of the tree we develop a new soft - input soft - output tree detector , called an _ improved soft - input soft - output-algorithm _ ( iss - ma ) . based on an analysis of the probability of correct path loss , we show that the improved path metric offers substantial performance gain over the conventional path metric . we also demonstrate through simulations that the iss - ma provides a better performance - complexity trade - off than existing soft - input soft - output detection algorithms .

1312.6129

i

turbulence is a common phenomenon in astrophysical fluids ( see , for example , elmegreen & scalo 2004 ) and it is obvious that most astrophysical fluids are permeated by magnetic fields ( brandenburg & subramanian 2005 ) . such magnetized fluids can be investigated by numerical simulations of driven magnetohydrodynamic ( mhd ) turbulence ( e.g. biskamp 2003 ) . to maintain turbulence in fluids , energy must be injected into the fluids . many turbulence simulations have been performed with either solenoidal ( @xmath8@xmath9@xmath10@xmath110 ) or compressive ( @xmath8@xmath12@xmath10@xmath110 ) forcing ( e.g. , meneguzzi , frisch , & pouquet 1981 ; cho & vishniac 2000 ; brandenburg 2001 ; ostriker , stone , & gammie 2001 ; federrath et al . most of these studies have adopted energy injection on a single scale in fourier space ( wavenumber space ) . however , multiple driving scales should be considered in simulations in order to better imitate real astrophysical fluids , such as the interstellar medium ( ism ) and the intracluster medium ( icm ) , because multiple astrophysical driving mechanisms may act on different scales simultaneously in those systems . there are many possible energy sources for ism turbulence . mac low ( 2004 ) examined available driving mechanisms for turbulence in the ism . magnetorotational instabilities , gravitational instabilities , protostellar outflows , expansion of h ii regions , stellar winds from massive stars , and supernova explosions have been considered as candidates for energy sources of ism turbulence . haverkorn et al . ( 2008 ) observed faraday rotation of extragalatic radio sources through the galactic plane and determined the outer scale of turbulence in the galactic ism . they suggested that stellar sources , such as stellar winds and protostellar outflows , drive turbulence on parsec scales in the spiral arms and supernova and superbubble explosions on @xmath13100 parsec scales in the interarm regions . han et al . ( 2004 ) showed the large - scale magnetic energy spectrum in our galaxy and suggested that ism turbulence is driven by stellar winds and supernova explosions on scales from 10 parsecs to 100 parsecs . it is also clear that the icm is in a turbulent state with multiple driving scales . schuecker et al . ( 2004 ) obtained a pressure map of the coma cluster using xmm - newton data and derived properties of turbulence from the map . according to their result , the largest eddy size and the smallest eddy size in the central region of the cluster are 145 kpc and 20 kpc , respectively . many possible driving mechanisms exist for icm turbulence . first , there are mechanisms that can provide energy injection on large scales . for example , cosmological shocks ( ryu et al . 2003 ; pfrommer et al . 2006 ) or mergers ( de young 1992 ; tribble 1993 ; norman & bryan 1999 ; roettiger et al . 1999 ; ricker & sarazin 2001 ) can produce turbulence in which the outer scale is similar to the size of the entire cluster . in fact , the outer scale of turbulence observed in some simulations during the formation of galaxy clusters is up to @xmath13 several hundred kpc ( norman & bryan 1999 ; ricker & sarazin 2001 ) , which is a few times smaller than the cluster size of @xmath13mpc . second , there are also mechanisms that can provide energy injection on small scales . for example , infall of small structures ( takizawa 2005 ) , agn jets ( see , for example , scannapieco & brggen 2008 ) , or galaxy wakes ( roland 1981 ; bregman & david 1989 ; kim 2007 ) can produce turbulence in which the outer scale is much smaller than the size of a cluster . due to variety of turbulence driving scales in the ism and the icm , it is necessary to inject energy on several different scales to simulate turbulence in those systems . however , there has been no rigorous research in this direction . in this paper , we study mhd turbulence driven at two scales . we mainly focus on the behavior of kinetic , magnetic , and density spectra in the presence of the driving at two scales . the outline of this study is as follows . we start by explaining numerical methods , initial conditions and forcing used for this work in section 2 . theoretical expectations are given in section 3 . then , results for incompressible mhd turbulence are shown in section 4 . results for compressible mhd turbulence , especially density , rotation measure ( rm ) , and velocity centroid ( vc ) spectra , are provided in section 5 . we discuss astrophysical implications in section 6 and give summary in section 7 .

turbulence is ubiquitous in astrophysical fluids such as the interstellar medium ( ism ) and the intracluster medium ( icm ) . in turbulence studies , it is customary to assume that fluid is driven on a single scale . however , in astrophysical fluids , there can be many different driving mechanisms that act on different scales .

turbulence is ubiquitous in astrophysical fluids such as the interstellar medium ( ism ) and the intracluster medium ( icm ) . in turbulence studies , it is customary to assume that fluid is driven on a single scale . however , in astrophysical fluids , there can be many different driving mechanisms that act on different scales . if there are multiple energy - injection scales , the process of energy cascade and turbulence dynamo will be different compared with the case of single energy - injection scale . in this work , we perform three - dimensional incompressible / compressible magnetohydrodynamic ( mhd ) turbulence simulations . we drive turbulence in fourier space in two wavenumber ranges , 2k ( large - scale ) and 15k 26 ( small - scale ) . we inject different amount of energy in each range by changing the amplitude of forcing in the range . we present the time evolution of the kinetic and magnetic energy densities and discuss the turbulence dynamo in the presence of energy injections at two scales . we show how kinetic , magnetic and density spectra are affected by the two - scale energy injections and we discuss the observational implications . in the case , where and are energy - injection rates at the large and small scales , respectively , our results show that even a tiny amount of large - scale energy injection can significantly change the properties of turbulence . on the other hand , when , the small - scale driving does not influence the turbulence statistics much unless .

0704.3889

c

we have compared the predictions for the liquid - vapour coexistence curve of a long - range yukawa fluid obtained from advanced theoretical approaches with gibbs ensemble monte carlo simulations . concerning the simulations care has to be taken when the range of the potential exceeds the length of the simulation box . this was done by performing an ewald sum in the case of a cubic simulation box and by using hyperspherical boundary conditions . the theoretical approaches that we applied comprised the self - consistent ornstein - zernike approximation ( scoza ) , the hierarchical reference theory ( hrt ) , as well as an optimised mean field theory ( omf ) . while the omf yields the exact result in the limit of infinite range of the potential , it deteriorates with decreasing interaction range . on the other hand , hrt and scoza turn out to be in perfect agreement with simulation results over the whole interaction range considered . this study complements a recent comparison @xcite for the case of intermediate and short range yukawa fluid .

two liquid state theories , the self - consistent ornstein - zernike equation ( scoza ) and the hierarchical reference theory ( hrt ) are shown , by comparison with monte carlo simulations , to perform extremely well in predicting the liquid - vapour coexistence of the hard core yukawa ( hcy ) fluid when the interaction is long range . in addition , we present an analytical optimised mean field theory which is exact in the limit of an infinitely long range interaction .

two liquid state theories , the self - consistent ornstein - zernike equation ( scoza ) and the hierarchical reference theory ( hrt ) are shown , by comparison with monte carlo simulations , to perform extremely well in predicting the liquid - vapour coexistence of the hard core yukawa ( hcy ) fluid when the interaction is long range . the long range of the potential is treated in the simulations using both an ewald sum and hyperspherical boundary conditions . in addition , we present an analytical optimised mean field theory which is exact in the limit of an infinitely long range interaction . the work extends a previous one by caccamo _ et al _ [ _ phys . rev . e , _ * 60 * , 5533 ( 1999 ) ] for short range interactions . * keywords * : yukawa potential , critical phenomena , monte carlo simulations , scoza , hrt .

0709.1114

i

in the past several years h@xmath0 has been detected in diffuse interstellar clouds @xcite where it had been expected to exist in abundances below observable limits . this surprising result raised various questions about the diffuse cloud environment . the rather simple chemistry of h@xmath0 allows for only three variable parameters in determining its abundance when the steady state approximation is used : the h@xmath0-electron recombination rate , the electron to hydrogen ratio , and the cosmic - ray ionization rate . previous work @xcite has shown that the first two of these are relatively well constrained . this leaves the cosmic - ray ionization rate as an unconstrained parameter . because the low energy cosmic - rays responsible for most of the ionization in diffuse clouds can not be directly measured in the solar system , we must rely on molecules to act as tracers of the ionization rate . using h@xmath0 , @xcite found the cosmic - ray ionization rate of molecular hydrogen , @xmath10 , to be much larger along the sightline to @xmath11 per than the canonical value of @xmath12 s@xmath6 . prior to the detection of h@xmath0 in diffuse clouds , oh and hd were the molecules of choice for estimating the cosmic - ray ionization rate there . estimates using these molecules required determining rate constants and modeling various reactions on the pathways to forming oh and hd @xcite . the derived values of the ionization rate tended to agree with the canonical value of @xmath4 , the primary cosmic - ray ionization rate , but differ greatly from the value derived from the recent h@xmath0 measurement toward @xmath11 per ( the relation between @xmath4 and @xmath10 is explained in 4.2 and quantified by equation ( [ eq10 ] ) ) . of the three molecules , the simple chemistry of h@xmath0 provides the most direct determination of @xmath4 @xcite , suggesting that measurements of h@xmath0 should produce more accurate results and be a more reliable tracer of the cosmic - ray ionization rate than oh or hd . the higher ionization rate found by @xcite towards @xmath11 per implies the production of more h@xmath0 , and if generally applicable , could account for the higher than expected column densities found in several diffuse clouds @xcite . however , prior to the present work the enhanced ionization rate was known to exist for certain only along one line of sight , and thus could have been considered an anomaly . to test if an enhanced ionization rate is a general property of the diffuse interstellar medium ( ism ) , we have performed a survey of h@xmath0 in nineteen diffuse cloud sightlines . h@xmath0 is detected in eight of the clouds and the overall results , including analysis of previous observations by our group , support a higher ionization rate . when coupled with further arguments , this strongly suggests that a greatly enhanced ionization rate is a typical property of the diffuse ism .

h is detected in eight diffuse cloud sightlines with column densities varying from to . because of the simple chemistry associated with h production and destruction , these column density measurements can be used in concert with various other data to infer the primary cosmic - ray ionization rate , .

using high resolution infrared spectroscopy we have surveyed twenty sightlines for h absorption . h is detected in eight diffuse cloud sightlines with column densities varying from to . this brings to fourteen the total number of diffuse cloud sightlines where h has been detected . these detections are mostly along sightlines concentrated in the galactic plane , but well dispersed in galactic longitude . the results imply that abundant h is common in the diffuse interstellar medium . because of the simple chemistry associated with h production and destruction , these column density measurements can be used in concert with various other data to infer the primary cosmic - ray ionization rate , . values range from s to s with an average of s . where h is not detected the upper limits on the ionization rate are consistent with this range . the average value of is about an order of magnitude larger than both the canonical rate and rates previously reported by other groups using measurements of oh and hd . the discrepancy is most likely due to inaccurate measurements of rate constants and the omission of effects which were unknown when those studies were performed . we believe that the observed column density of h is the most direct tracer for the cosmic - ray ionization rate due to its simple chemistry . recent models of diffuse cloud chemistry require cosmic - ray ionization rates on the order of 10 s to reproduce observed abundances of various atomic and molecular species , in rough accord with our observational findings .

0709.1114

c

values of the cosmic - ray ionization rate for diffuse clouds observed here as well as those determined for other diffuse clouds observed previously by us are given in the right hand column of table [ tbl4 ] . the detected values in the lines of sight to fourteen sources cover the range 0.53.2@xmath71 s@xmath6 . upper limits , which are given for fifteen diffuse clouds , are consistent with this range of ionization rates , with the possible exception of hd 168607 . while most of the detections of h@xmath0 are confined to the galactic plane , they are widely dispersed in galactic longitude . we therefore conclude that the values of the cosmic - ray ionization rate listed in table [ tbl4 ] are typical for galactic diffuse interstellar clouds . a few of the sightlines we investigated have been studied previously to derive cosmic - ray ionization rates . all of these studies used column densities of either oh , hd , or both in their calculations . because the formation pathways of oh and hd include the ionization of atomic hydrogen , they can be used to determine the h ionization rate . most of these studies @xcite then derived the primary cosmic - ray ionization rate from the h ionization rate , but @xcite did not because they still considered ionization via x - rays to be important . our values of the primary ionization rate for @xmath11 per , @xmath21 per , @xmath23 per , @xmath22 per , and @xmath11 oph are shown in table [ tbl5 ] along with the rates derived from oh and hd measurements as well as cloud modeling . for @xmath11 per our value is over an order of magnitude larger than those reported by @xcite and @xcite . while the rest of our new measurements in table [ tbl5 ] are only upper limits , these are also typically orders of magnitude larger than previously published values . the only exception is @xmath21 per where both papers cite values of @xmath4 about one fourth to one half our upper limit . various model calculations were performed by @xcite to investigate three of the sightlines that we study here : @xmath11 oph , @xmath11 per , and @xmath21 per . in creating these models they used the most recent measurements of rate constants and the column densities of diagnostic species such as h and h@xmath28 as input parameters . by varying a few uncertain parameters , they would then generate lists of predicted column densities for many atomic and molecular species under slightly different conditions . when their paper was written , it was believed that the h@xmath0-electron recombination rate constant was much lower than the currently accepted value . @xcite reported an upper limit correspnding to @xmath72 @xmath73 s@xmath6 at @xmath74 k , and @xcite lowered the upper limit to @xmath75 @xmath73 s@xmath6 at @xmath76 k. due to the wide range of possible recombination rate constants , @xcite performed calculations using both @xmath72 and @xmath77 @xmath73 s@xmath6 . the cosmic - ray ionization rates from their paper listed in table [ tbl5 ] were computed by determining @xmath4 necessary to reproduce observed oh column densities when @xmath78 @xmath73 s@xmath6 . we choose to compare these ionization rates to ours because we obtain @xmath79 @xmath73 s@xmath6 when @xmath80 k is used as the input temperature in equation ( [ eq8 ] ) . the value of @xmath4 inferred by @xcite is about the same as ours for @xmath11 per , but the lower limits they derived for @xmath11 oph and @xmath21 per are larger than our upper limits for both of those sightlines . for their models that used @xmath81 @xmath73 s@xmath6 , @xcite obtained cosmic - ray ionization rates that are about a factor of 1 to 5 times smaller than ours . from these models they also predicted the column density of h@xmath0 along each sightline . their results are all on the order of @xmath51(h@xmath0 ) @xmath82 @xmath2 , which is a few times larger than the observed column densities or upper limits in any of these sightlines . because @xcite use only a slightly smaller cosmic - ray ionization rate ( corresponding to the formation rate ) but a much smaller recombination rate ( corresponding to the destruction rate ) , their prediction of an h@xmath0 column density similar to observed values seems somewhat serendipitous . in addition to the overestimate of the h@xmath0 column density , a small h@xmath0-electron recombination rate constant may have further consequences . @xcite noted that a small value of @xmath33 may have been responsible for underestimates of the primary cosmic - ray ionization rate in the past . this is because a slower destruction rate requires a slower formation rate to produce a given abundance . in addition to the slow recombination rate , there are some other possible explanations for differences between the cosmic - ray ionization rate inferred from h@xmath0 and those inferred from oh and hd . @xcite pointed out that the rate constant associated with the endothermic charge transfer from h@xmath47 to o varies over the temperatures typically associated with diffuse clouds . this means that the oh production rate is temperature dependent . the ionization rates towards @xmath11 per and @xmath11 oph quoted in @xcite were derived using temperatures of 120 k and 110 k , respectively , for the warm components of the cloud models along each sightline @xcite . as these temperatures are about twice as large as the values determined from h@xmath28 , their oh production is much more efficient . the result is a smaller cosmic - ray ionization rate needed to produce the observed oh column density than if a lower temperature had been used . this problem was addressed by the later models of @xcite where tempertaure and density were varied as functions of cloud depth . @xcite went on to make a comprehensive chemical model of the cloud towards @xmath11 per . they determined the value of @xmath4 that would best reproduce all observed atomic and molecular column densities to be @xmath83 s@xmath6 , which is in good agreement with our estimate of @xmath84 s@xmath6 . the difference in these values may arise because we assume a uniform distribution of gas while @xcite invoke a three phase model which includes diffuse gas , dense gas , and magnetohydrodynamic shocks . two different effects may lead to underestimates of @xmath4 from measurements of hd . the first has to do with an overestimate of the total deuterium to hydrogen ratio @xmath85/@xmath39 . this ratio can be used to estimate the molecular deuterium fraction , @xmath86 . however , the observed values of @xmath86 are about an order of magnitude smaller than those predicted by @xmath85/@xmath39 . to explain this discrepancy , @xcite argued that the atomic deuterium fraction must be larger than the total deuterium fraction . this means that approximating @xmath85/@xmath39 with @xmath87 overestimates the total deuterium to hydrogen ratio . @xcite showed that the cosmic - ray ionization rate is inversely related to the deuterium fraction , so an overestimate of @xmath85/@xmath39 will underestimate @xmath4 . secondly , @xcite emphasized the importance of grain neutralization proposed by @xcite . this process reduces the number of h@xmath47 ions in the gas through charge transfer with small grains . by lowering the abundance of h@xmath47 , the production rate of hd will decrease . this is because hd formation is dependent upon the reaction involving the charge transfer from h@xmath47 to d. since neutralization slows down hd production , a larger value of @xmath4 is needed to create a given abundance than if the effect were not taken into account . @xcite used a model which includes grain neutralization and showed that both h@xmath0 and hd column densities can be reproduced with a single ionization rate of @xmath88 s@xmath6 . since oh formation is dependent on a similar charge transfer reaction , grain neutralization and thus a larger cosmic - ray ionization rate may be necessary in its analysis as well . @xcite studied h@xmath0 in the sightline towards @xmath11 per . using nearly the same analysis as this paper , they inferred a value of @xmath89 s@xmath6 which is equivalent to @xmath90 s@xmath6 shown in table [ tbl5 ] . this higher ionization rate is due to various differences in input parameters . in terms of the parameters in this paper , @xcite used 1.5@xmath33 , 1.2@xmath51(h@xmath0 ) , 1.2@xmath39 , and 0.8@xmath63 for the following reasons . the h@xmath0-electron recombination rate constant differs because they approximated the electron temperature with the h@xmath0 temperature instead of the h@xmath28 temperature in equation ( [ eq8 ] ) . further observations have more than doubled the total integration time so that the spectrum and h@xmath0 column density change slightly between papers . the value of @xmath39 used by @xcite was an average number density computed from various measurements , whereas the value used in this paper comes from the c@xmath28 analysis of @xcite . finally , we have adopted a single value of @xmath63 to be used in all calculations while they used h@xmath28 and c@xmath47 column densities measured towards @xmath11 per . while all of the observations and models above are viable methods for finding the cosmic - ray ionization rate , we believe that the use of h@xmath0 should produce the best results due to its relatively simple chemistry . using either oh or hd to calculate @xmath4 requires more measurements , more assumptions , and more variable parameters than using h@xmath0 . more parameters give the opportunity for a greater uncertainty to accumulate during the calculation . fewer uncertainties coupled with advances in instrumentation lead us to speculate that the cosmic - ray ionization rates inferred from h@xmath0 may be the most accurate to date for diffuse clouds . however , improved estimates of @xmath50 and @xmath39 , the two most uncertain values in our calculations , would make h@xmath0 an even better probe of the cosmic - ray ionization rate . several theoretical calculations of @xmath4 have been performed in the last half - century @xcite . in these papers the authors derived a cosmic - ray ionization rate starting from the observed flux of cosmic - rays in our solar system . unfortunately , there are large uncertainties associated with this method . the cosmic - ray spectrum is well measured above about 1 gev , but lower energy particles are deflected from the inner solar system by the magnetic field coupled to the solar wind . the particles which are most important for ionizing species in diffuse clouds are likely those with energies from about 2 to 10 mev . since this portion of the spectrum can not be directly measured , the flux at low energies must somehow be extrapolated from existing data . @xcite assumed that the power law which applies to the flux of high energy cosmic - rays continues down to 10 mev where the spectrum peaks and then decreases linearly with energy . from these assumptions they derived an ionization rate of @xmath91 s@xmath6 . @xcite , however , fit a curve to measurements of cosmic - rays with energies near 100 mev that also matches the high energy spectrum power law . with this method , their spectrum peaks around 100 mev and falls off for lower energies . the result of using their fit is a lower limit of @xmath92 s@xmath6 . in the same paper they derived an upper limit of @xmath93 s@xmath6 via arguments that low energy cosmic - ray protons are accelerated in type i supernova shells . @xcite used data from the _ pioneer _ and _ voyager _ spacecraft as they travelled outward in the solar system where the weaker solar wind allows for the detection of lower - energy cosmic - rays . these data were then combined with previous observations to infer the interstellar proton spectrum . using this proton spectrum and a heavy nuclei spectrum both with low energy cut - offs at 10 mev and an electron spectrum cut - off below 2 mev , @xcite calculated the primary cosmic - ray ionization rate to be @xmath94 s@xmath6 . our ionization rates fall neatly within the bounds formed by these studies and so are not inconsistent with constraints based on direct cosmic - ray measurements and theoretical particle physics . in contrast to our findings in diffuse clouds , the cosmic - ray ionization rate in dense clouds does seem to agree with the canonical value . observations of h@xmath0 towards dense clouds have found column densities roughly the same as those seen in diffuse clouds @xcite . these measurements have been used to calculate the product @xmath95 . when @xmath10 is taken to be the canonical value of @xmath12 s@xmath6 , the resulting pathlength is on the order of a parsec . this is a typical size for dense clouds as measured by other methods such as extinction mapping . since h@xmath0should be a reliable tracer for the cosmic - ray ionization rate in both environments , there must be some mechanism causing the difference between dense and diffuse clouds . one possibility examined by both @xcite and @xcite is cosmic - ray self - confinement . in this process cosmic - rays generate alfvn waves which can effectively confine the lower energy particles ( @xmath96 mev ) to diffuse material , thus preventing them from entering dense clouds . because cosmic - rays in the 1100 mev range are the most efficient at ionization , self - confinement naturally leads to a higher ionization rate in diffuse clouds than in dense clouds . another possibility is that there is a previously unrecognized high flux of low energy cosmic - rays that can penetrate diffuse but not dense clouds . assuming that typical column densities of diffuse clouds are of order @xmath97 @xmath2 and those of dense clouds are of order @xmath98 @xmath2 , cosmic - rays with energies @xmath99220 mev @xcite would contribute to the ionization rate only in diffuse clouds . as we foresee no observational techniques that would distinguish between these two possibilities , a resolution to this question will depend on more sophisticated theoretical treatments .

these detections are mostly along sightlines concentrated in the galactic plane , but well dispersed in galactic longitude . values range from s to s with an average of s . where h is not detected the upper limits on the ionization rate are consistent with this range . the average value of is about an order of magnitude larger than both the canonical rate and rates previously reported by other groups using measurements of oh and hd . the discrepancy is most likely due to inaccurate measurements of rate constants and the omission of effects which were unknown when those studies were performed . we believe that the observed column density of h is the most direct tracer for the cosmic - ray ionization rate due to its simple chemistry . recent models of diffuse cloud chemistry require cosmic - ray ionization rates on the order of 10 s to reproduce observed abundances of various atomic and molecular species , in rough accord with our observational findings .

using high resolution infrared spectroscopy we have surveyed twenty sightlines for h absorption . h is detected in eight diffuse cloud sightlines with column densities varying from to . this brings to fourteen the total number of diffuse cloud sightlines where h has been detected . these detections are mostly along sightlines concentrated in the galactic plane , but well dispersed in galactic longitude . the results imply that abundant h is common in the diffuse interstellar medium . because of the simple chemistry associated with h production and destruction , these column density measurements can be used in concert with various other data to infer the primary cosmic - ray ionization rate , . values range from s to s with an average of s . where h is not detected the upper limits on the ionization rate are consistent with this range . the average value of is about an order of magnitude larger than both the canonical rate and rates previously reported by other groups using measurements of oh and hd . the discrepancy is most likely due to inaccurate measurements of rate constants and the omission of effects which were unknown when those studies were performed . we believe that the observed column density of h is the most direct tracer for the cosmic - ray ionization rate due to its simple chemistry . recent models of diffuse cloud chemistry require cosmic - ray ionization rates on the order of 10 s to reproduce observed abundances of various atomic and molecular species , in rough accord with our observational findings .

1210.6353

i

both theoretically and observationally , core - collapse supernova ( ccsn ) explosions are linked with the death of massive stars . however , because precise measurements of ccsn progenitor masses are scarce , the mapping between explosion scenario and the progenitor mass distribution is less clear . the most common method of progenitor mass determination is direct imaging , in which one identifies the progenitor star in multi - band pre - explosion imaging , fits a spectral energy distribution to the photometry , and verifies the star is gone once the sn has faded . this methodology is ideal given appropriate data , but it is limited to contemporary sne that have pre - explosion hubble space telescope ( hst ) or deep ground - based imaging . as a result , only @xmath125 sne have any constraint on their progenitor masses , and half of these are only upper limits @xcite . a significant increase in the number of progenitor masses could allow us a powerful window into further understanding supernovae . direct imaging has created a prototypical picture of massive star death . type ii - p sne are assumed to be created by red supergiant ( rsg ) stars with intact hydrogen envelopes , while more exotic sne may be created by higher mass stars . however , a number of questions remain to be answered regarding supernova physics . @xcite identify what they term to be the ` red supergiant problem , ' an observed lack of progenitors of @xmath0@xmath116 - 30 m@xmath7that we would expect to explode as type ii - p sn . a variety of channels have been proposed to account for these missing explosions , including direct black hole formation or the stars exploding as different sn types . binarity of the progenitor system can also play a significant role in sn explosions . @xcite show that numbers of type ibc and iib explosions are significantly underestimated if single - star explosions are assumed to be the only sn channel . these issues are difficult to address without further progenitors to analyze . unfortunately , while direct - imaging of the progenitor star would be ideal , there are constraints that limit the frequency with which it may be applied . first , direct imaging is limited by the sn rate : roughly one per century in a large spiral galaxy @xcite . second , direct imaging requires that the host galaxy be close enough to resolve the progenitor star , an effective limit of @xmath120 mpc @xcite . third , preexisting sub - arcsecond images of the site must exist , necessitating archival hst images with either the wide field planetary camera 2 ( wfpc-2 ) or the advanced camera for surveys ( acs ) . finally , even with the existence of appropriate images , there is no guarantee that the precursor will be identified . for example , @xcite summarized 20 progenitor detections in type ii - p explosions . only five progenitors were directly observed , and an additional two which fell on compact star clusters were well constrained . the remainder had no detection , but were given upper bounds based on the maximum luminosity with which they could escape detection ( see also @xcite ) . in short , while direct imaging is ideal given appropriate data , it suffers from prerequisites that limit the number of opportunities in which it may be applied . it is clear that development of an independent and complementary technique would be of interest . in this paper we make use of an alternative technique , stellar population analysis , to infer progenitor masses . the technique involves examining a color - magnitude diagram ( cmd ) of the surrounding population of stars to infer an age and mass for the progenitor star . the advantage of our methodology is that we are not reliant on individual identification and photometry of the specific progenitor star , leaving us free to apply our method to cataloged supernova remnants ( snrs ) in addition to directly observed sne . this ability drastically increases the number of progenitor masses we may find , allowing us to make a more complete measurement of the underlying distribution of progenitor masses . our method offers no direct way to probe the binarity of the system , which is relevant given the role binarity likely plays in certain ccsn scenarios . however , our estimate of the age of the surrounding stellar population is not affected by the particular details of a given progenitor system . our method also offers no way to determine the type of ccsn explosion we are observing , so for snr analysis we are not able to provide a direct link between sn explosion type and progenitor mass . however , we will be able to comment on the ranges of zams masses that are producing ccsn explosions . stellar population analysis has been used many times historically to analyze the characteristics of progenitors . many groups have carried out age dating of stellar clusters coincident with supernovae and derived corresponding stellar masses @xcite . @xcite examined the sfh map of the lmc published by @xcite . they estimate ages and masses for the four most recent snr produced by ccsne in the lmc . in @xcite , we employed a technique identical to the one used in this paper to estimate the age of the progenitor of ngc 300 ot2008 - 1 . we found that the star was likely to have formed 8 - 13 myr ago , corresponding to a @xmath0 of 12 - 17 m@xmath7 . finally , in @xcite we applied the same technique as this paper to archival images of sn 2011dh , finding a most likely age of 17@xmath8 myr and a @xmath0 of 13@xmath9 m@xmath7 . clearly there is a long list of successful applications of stellar population analysis leading to constraints on progenitor physical properties . in this paper we examine 59 snrs in the spiral galaxy m31 , listed in table [ tab_snr ] . in 2 we outline the methods used to determine star formation histories ( sfhs ) , ages , and masses for the progenitor stars . in 3 we present the results of the analysis . we find that 53 snr display recent star formation , and we derive masses for these . we also present four example cases that are representative of the overall sample . finally , in 4 we examine the distribution of progenitor masses . we search for indications of a minimum mass , which we find to be between 7.0 and 7.8 m@xmath7 , and note that the distribution is more bottom - heavy than a salpeter imf distribution .

application of stellar evolution models allows us to infer thefrom this age . because our technique is not contingent on identification or precise location of the progenitor star , it can be applied to the location of any known snr . if we assume a single power law distribution , , we find a distribution that is steeper than a salpeter imf ( ) . the result is preliminary and requires more snrs and further analysis .

using hubble space telescope ( hst ) photometry , we age - date 59 supernova remnants ( snrs ) in the spiral galaxy m31 and use these ages to estimate zero - age main sequence masses ( ) for their progenitors . to accomplish this , we create color - magnitude diagrams ( cmds ) and employ cmd fitting to measure the recent star formation history ( sfh ) of the regions surrounding cataloged snr sites . we identify any young coeval population that likely produced the progenitor star , then assign an age and uncertainty to that population . application of stellar evolution models allows us to infer thefrom this age . because our technique is not contingent on identification or precise location of the progenitor star , it can be applied to the location of any known snr . we identify significant young star formation around 53 of the 59 snrs and assign progenitor masses to these , representing a factor of increase over currently measured progenitor masses . we consider the remaining 6 snrs as either probable type ia candidates or the result of core - collapse progenitors that have escaped their birth sites . in general , the distribution of recovered progenitor masses is bottom heavy , showing a paucity of the most massive stars . if we assume a single power law distribution , , we find a distribution that is steeper than a salpeter imf ( ) . in particular , we find values of outside the range to be inconsistent with our measured distribution at 95% confidence . if instead we assume a distribution that follows a salpeter imf up to some maximum mass , we find that values of are inconsistent with the measured distribution at 95% confidence . in either scenario , the data suggest that some fraction of massive stars may not explode . the result is preliminary and requires more snrs and further analysis . in addition , we use our distribution to estimate a minimum mass for core collapse between 7.0 and 7.8 m .

1210.6353

m

the basic procedure for deriving a mass estimate for each progenitor is as follows . we first use coordinates from three snr catalogs as explained in 2.2 , which we assume are accurate to within a few arcseconds and unbiased to any one type of sn . we then search for any appropriate hst fields that contain the snr in question . we perform photometry on the stars in each field , and create cmds of the region within @xmath150 pc of the snr coordinates , assuming a distance modulus of @xmath10 @xcite . we use cmd fitting to measure the sfh of each region . for each region that displays recent sf , we use the age of this recent sf to associate an age with the progenitor star . finally , we apply stellar evolution models to convert the age of the progenitor to the @xmath0 of the progenitor . our method requires several assumptions to produce results . above all , we must assume that some of the stars surrounding the snr are in fact coeval with the progenitor star . there are several pieces of evidence suggesting this assumption is reasonable . first , 90% of stars form in clusters with sizes of @xmath11 pc @xcite . these stars are expected to stay spatially associated for timescales greater than the lifetimes of ccsne progenitors , even for dissolving , unbound clusters . second , theoretical predictions are that stars will stay spatially associated on scales of @xmath1100 pc for 100 myr @xcite . observational constraints support this @xcite , as do simulations by @xcite who found that @xmath185% of ccsne will explode within 100 pc of their birth site . in addition , even if a progenitor would travel farther , we would still expect to see a fraction of the coeval young stellar population which is sufficient for age dating . the precise number of interest is the fraction of young stars that will stay within a certain distance in the lifetime of a core collapse supernova progenitor . the question is a complex one , and we know of no research that quantifies it precisely at this time . we conclude that it is a reasonable assumption that most of the young stellar population around our observed snrs is coeval with their progenitors . based on the analysis in @xcite , we adopt a value of @xmath150 pc for the radius of our star selection annulus . our method does not assume any information about the type of sn that exploded . we assume that snrs associated with recent star formation are ccsne , but do not distinguish individual subtypes . for ccsne , the particular type of supernova has no effect on our mass determination process . however , the progenitors of thermonuclear type ia sne are likely to arise from older stellar populations which may vary considerably more in age than the coeval populations of massive stars . while there exists considerably discussion in the literature as to the precise nature of type ia progenitor systems and their distribution in age ( see @xcite and references therein ) , we would expect to see type ia progenitors ranging from ages of a few hundred myr to several gyr . since these timescales are comparable to or significantly greater than the dynamical timescale of m31 , our methodology of identifying and analyzing coeval stars is ineffective at measuring the precise ages of type ia progenitors . our method does have the risk of misinterpreting a type ia snr as a ccsnr . in their volume limited sample ( d @xmath11 60 mpc ) , @xcite find that type ia sn compose 24% of observed sn . however , there are several qualitative reasons as to why one would expect a smaller ia / ccsn fraction in our survey . first , there is some discussion as to whether such a volume - limited survey may underestimate the fraction of ccsn due to their faintness compared to type ia sn ( see discussions in @xcite ) . this potential bias suggests that 24% could be considered an upper limit . in addition , the areas of m31 with greater hst coverage are primarily star forming regions , where we would naively expect ccsn to be more common , increasing the relative fraction of ccsn . thus the actual fraction of type ia snr we expect in our sample is probably somewhat less than 24% . moreover , it is likely that some type ia snr can be identified by their lack of an associated young stellar population . @xcite examined four ia sites in the lmc and found that three displayed a significant lack of recent star formation . out of the 59 examined , we found 6 snr displayed no recent sf in their surrounding stellar populations . finally , we have examined what effect additional hidden ia contamination will have on our data . the procedure we used is explained in 4.3 . we found that additional ia contamination up to a total fraction of @xmath125% has very little effect on the overall distribution observed in the sample . to summarize , we argue that there are several qualitative reasons to assume that the effective ia fraction in our sample is smaller than 24% . in addition , our method offers a means to identify and remove some snr that are likely ia progenitors . finally , our overall results are largely insensitive to inclusion of additional ia contamination up to the 24% observed in the volume - limited survey from @xcite . another possibility for snrs with no coincident young sf is that they are the result of high - velocity progenitor stars that have left their birth sites . as a test of this possibility , we examined any site with no observed recent sf in galex fuv data . we found that all 6 sites with no recent sf are still relatively close ( @xmath150 to 100 pc ) to galex fuv sources . if we assume that the progenitor star resulted from the most nearby fuv source for each zero - sf snr , then a 50 myr old star would only require velocities of a few km / s to reach the snr site from the fuv source site , well within reasonable velocities for runaway stars . higher mass stars would of course require higher velocities to leave the fuv source in their shorter lifetimes . this analysis suggests that our study is unable to differentiate between type ia snr and snr resulting from high - velocity stars . we note , however , that a null - result for progenitor mass will not affect the distribution of progenitor masses we measure . it will simply reduce our effective sample size . finally , our method is highly contingent on the accuracy of the stellar evolution models used to model our observed cmds . because the models are theoretical , they generally do not have easily quantifiable uncertainties . the wide array of available model sets tend to systematically differ in bolometric luminosity and temperature , but are consistent within about 0.2 mag in bolometric luminosity and 0.02 dex in log temperature . we therefore quantify the uncertainty of our results due to the models used by including random shifts in the bolometric luminosity and temperature of the models as part of our monte carlo ( mc ) tests ( see also @xcite ) . we detail the process further in 2.5 . there are only a few extensive catalogs of snrs in m31 . we use the catalogs of @xcite , @xcite , and @xcite . all three make identifications based on [ sii]-to - h@xmath4 ratios , and @xcite make additional use of morphology and ob star associations . @xcite identify three confidence levels based on approximate levels of contamination , although they note that the confidence levels are somewhat subjective . we only perform analysis on their first and second confidence candidates . the catalogs are not mutually exclusive , and the coverage areas of the three catalogs overlap in several areas , causing several remnants to be identified in both catalogs . to avoid treating a double - identified remnant as two separate remnants , we identify all remnants from two separate catalogs in which the 50 pc areas overlap and only select coordinates from one catalog . in these duplicate cases , we adopted the coordinates first from @xcite , then from @xcite if @xcite did not identify a remnant in that position . on occasion , a single catalog identified two remnants in which these 50 pc regions overlapped , but we still performed analysis on both candidates in such a case . in addition , we cross - referenced the snr catalogs with @xcite and eliminated two , k567 and k884 , which were found to not be snr based on their optical / x - ray properties . for our analysis , we require that the stellar population surrounding the snr be imaged by either the advanced camera for surveys ( acs ) or the wide - field planetary camera 2 ( wfpc2 ) instruments on hst in at least two broadband filters . we also require our 50% completeness limits ( see 2.3 ) for a given cmd to be at least 24.5 magnitude in f475w , f555w , or f606w , whichever represents the blue filter ( all qualifying fields of sufficient depth were either f475w , f555w , or f606w vs. f814w ) . the location of the ms turnoff is very similar in color in these filter sets , so adopting a similar magnitude cut in all three is reasonable . our comparisons between snrs imaged multiple times at varying depths indicate that simply reaching the ms turnoff for a given cmd is insufficient to correctly determine the age . cmd fitting is highly sensitive to densities of stars at given ages , requiring a well - sampled stellar population at the age of interest . in general , we found that our methodology applied to shallower data had a tendency to miss older bursts of sf which still may have resulted in ccsn . the available 2-filter broadband data for m31 happens to be distributed in two groups , including many shallow fields ( blue depth @xmath12 ) and many deep fields ( blue depth @xmath13 ) , with very few in between . the choice of our depth cut causes us to reject five snr that have only been imaged with shallow wfpc-2 data . in table [ tab_snr ] we list the snr for which our criteria is met , along with the corresponding hst fields . in cases where the regions in question were imaged multiple times , we selected whichever set of images had the greatest number of stellar detections . in fig . [ spatial ] we chart the locations of the examined remnants on a star - subtracted image of m31 in h@xmath4 . those remnants colored red represent probable ia candidates , where no young sf was found . there is a clear grouping of ccsn candidates along the star forming arms of m31 . we performed resolved stellar photometry using the photometry pipeline developed for the acs nearby galaxy treasury program @xcite . this pipeline uses the dolphot stellar photometry package @xcite to fit the well - characterized acs point spread function to all of the point sources in the images . we then converted fluxes to vega magnitudes using the standard zero - points and aperture corrections from the acs handbook . we assess photometric errors and completeness using fake star tests . at least @xmath14 tests are performed by inserting fake stars of known color and magnitude into the data one at a time and blindly attempting to recover them with the same software . both fake star tests and photometry were performed on the full hst fields . cmd fitting is a powerful tool for measuring star formation histories @xcite . to estimate star formation histories of the regions surrounding the snr , we used the software package match @xcite . match works by creating many model cmds based on theoretical isochrones for a variety of ages and metallicities . a linear combination of these model cmds are then fit to the observed cmd . we use the models of @xcite and @xcite both for cmd fitting and estimation of @xmath0 . for each field , we selected stars from our photometry catalog within a 15 radius around the snr coordinates , which equates to a physical size of @xmath150 pc at an assumed distance modulus of @xmath10 @xcite . this is consistent with the spatial correlation discussed in 2.1 . we account for photometric errors through the use of fake star tests . we selected fake stars in a region @xmath12.5 times the radius of the real star annulus to ensure that at least a few thousand recovered fake stars are included for each snr . we defined lower magnitude limits as the point at which fake star completeness dipped below 50% . match requires a variety of parameters to generate and fit cmds . we assumed a salpeter imf , @xmath15 @xcite , and a binary fraction of 0.35 . @xcite demonstrated that varying the imf value from -2.0 to -2.7 or varying the binary fraction from 0.2 to 0.5 had no effect on the epoch assigned to a recent burst . we also varied the imf from -1.3 to -3.3 for select regions and found no significant difference in relative fractions of sf for various epochs . although absolute amplitudes of sf did change , only relative amplitudes have an effect on our method of determining ages . note that the imf and binary fraction are used by match purely for purposes of populating the models . we assume nothing about either value with regards to a potential ccsn progenitor system or the overall progenitor mass distribution . match produces fits in logarithmic age bins . we adopted 71 age bins increasing in .05 increments from 6.60 ( 4 myr ) up to 10.10 ( 12.5 gyr ) . we are unable to fit for ages younger than this due to the lack of isochrones at younger ages . match will interpret any sf from a population younger than the 6.60 to 6.65 ( 4 to 4.5 myr ) bin as being included in this bin . thus our youngest bin actually includes all sf from present times back to 4.5 myr . as a result , we may only quote an upper age limit for any sf found in this youngest bin . in fig . [ isochrones ] we plot the isochrones for every - other age bin from 6.60 through 8.00 in f555w vs. f555w - f814w . fitting metallicity with match is not viable given the very weak dependence of the optical colors of the upper main sequence on metallicity . we therefore constrained the metallicity to a spread of @xmath10.15 dex and to increase with the lifetime of the galaxy , which consistently produced best - fit metallicities of solar . this value is consistent with the known gas - phase metallicity of m31 @xcite . finally , we binned our cmds in units of 0.3 in magnitude and 0.15 in color . this binning accounted for the fact that some fields had few upper ms stars , allowing us to reduce the impact of this paucity on our fits . however , we still maintain a number of bins significantly larger than the number of free parameters used in the fitting . we experimented with using finer binning and found no change in our age determinations within our uncertainties . recent sfhs are very sensitive to the treatment of reddening . our most simplistic treatment of reddening is to assume that all reddening is due to a galactic foreground , search over a specified range of reddening values for a best fit , and apply this value to the cmd as a whole . however , we frequently found significant differential reddening ( @xmath16 ) across our small snr - centered regions , which manifests as an increased width to the ms and red clump . star forming regions will typically have extensive amounts of dust and gas , often not uniformly distributed , and as such there is no reason to believe that a single reddening value would correctly describe the region as a whole . match by default allows for 0.5 mag of differential reddening for populations below 40 myr , after which it falls linearly down to 0.0 mag at 100 myr . in addition , the user has the ability to add additional full - field differential reddening across the cmd . we almost always found that the default treatment of reddening was inadequate to account for all the reddening present in the regions . as an example of how ignoring differential reddening can lead to an erroneous result , we consider the snr bw-102 . the photometry is quite deep , extending to magnitudes of @xmath17 and @xmath18 . in the left panel of fig . [ bw102_nodav ] we plot the observed cmd of the region , as well as the best - fit model generated by match in the background in greyscale . in the right panel , we plot the cumulative sfh as a fraction of total sfh in the past 50 myr , assuming a fixed differential reddening of @xmath19 mags affecting only the young stars . we found a best - fit reddening value of @xmath20 . all the star formation is concentrated in the youngest time bin ( @xmath11 4.5 myr ) , corresponding to a lower limit @xmath0 of 52 m@xmath7 . however , several qualities of the cmd would cause us to be suspicious of this result . the main sequence appears very dim , and appears to contain lack the young , bright , blue stars that the model predicts at @xmath21 . we would not expect this given a result of very extensive young star formation . looking beyond the main sequence , we see that the red clump is extended over nearly 2 magnitudes in f475w . indeed , the model very poorly models the stars at @xmath22 , @xmath23thus not only was the default value of differential reddening insufficient to model the entire cmd , the treatment of differential reddening as something unique to the young stellar population is clearly not reflected in the reality of the cmd . for regions such as this , it is necessary to apply differential reddening across the cmd . we must then define the amount of differential reddening to add . to understand our technique for determining the amount of @xmath24 to use , we must expand on the differential reddening model used by match . match applies differential reddening to the model cmds by applying a top - hat distribution along the reddening line . the low end of the top - hat is defined by a foreground value of @xmath25 for which match fits . the width of the distribution is then defined by the specified value provided by the user . when applying additional differential reddening , fit values tended to improve until we reached unphysical values of @xmath16 the only true constraint we may apply is that this minimum value of @xmath25 must be at least equivalent to the milky way foreground value . @xcite find a foreground reddening value to m31 of @xmath26 . while this value is likely to change slightly over the angular size of m31 , the changes will clearly be small compared to the magnitude of differential reddening internal to m31 . our solution to cmds such as bw-102 is to increase the width of the differential reddening distribution applied to all ages until the best - fit value of @xmath25 match finds reaches the foreground value . we then use this differential reddening distribution to measure the sfh and derive the age of the sn progenitor . in fig . [ bw102dav ] we plot the sfh results for a variety of differential reddening values , increasing from @xmath27 up to @xmath28 magnitudes , in increments of 0.1 . note that this differential reddening is in addition to the default young @xmath19 , leading to up to 2.0 magnitudes of differential reddening on the very youngest stellar populations . we found that a value of @xmath28 was the point at which the overall reddening value dropped to the expected foreground value for bw-102 . for this value of differential reddening , the sfh corresponded to an age of 36 myr and a mass of @xmath19 m@xmath7 , a much more reasonable result given the observed cmd . note that the sfh reaches this result ( within uncertainty ) before the best - fit @xmath29 value reaches foreground , suggesting that our result is not highly contingent on the precise value of foreground reddening assumed . in the left panel of fig [ bw102 ] we plot the observed cmd with the best - fit model plotted in greyscale in the background , created using the above @xmath24 procedure . in the right panel , we plot the cumulative sfh of the region for the past 50 myr , with error bars from the monte carlo analysis described in 2.5 we found essentially all fields suffered from differential reddening in excess of the default values used by match . when no extra differential reddening was included in the models , many regions without obvious young star formation were best fit by high overall reddening values and included star formation in the youngest few time bin . the issue arises from faint main sequence stars revealed by deep photometry . without inclusion of differential reddening , these stars are fit by the uniform foreground reddening value for the region , which is typically a high value ( e.g. @xmath30 ) . using this erroneously high value , the stars are corrected to be brighter and bluer than they actually are , leading match to mistake them as evidence of young star formation . by allowing for differential reddening , we can incorporate the full range of different reddening values across the field . for our final procedure , we increase the amount of full - cmd differential reddening until we find the best fit @xmath25 reaches the foreground value , @xmath31 . we then adopt this value of differential reddening for subsequent analysis of that particular snr . our ultimate goal is to estimate the mass of the progenitor . we do this by identifying the burst of recent star formation from which the progenitor star most likely originated , allowing us to assign an age to each progenitor star . ideally , each sfh would have a single isolated burst , making it easy to associate a single age to the young stellar population . in many cases , however , multiple bursts of varying rates and durations appear in the young star formation history . as a result , we examine the cumulative star formation history of each cmd as a fraction of the total recent star formation . when examined as such , the fraction of star formation in each age bin corresponds to the probability that the progenitor will be of that age . because we are only interested in recent sf , we examine only the most recent 50 myr of sf ( see 4.1 for an explanation of this limit ) . our results suffer from both random uncertainties due to the sampling of the cmd and systematic errors due to differences between the theoretical isochrones and observations . for example , if the models were consistently redder than the observed stars , we would generally find younger ages when these models are applied to real data . the random errors are generally highly dependent on the number of upper main sequence stars in the field , with higher numbers providing a tighter constraint on the data ( although more upper main sequence stars may also simply indicate a younger population ) . see @xcite for further discussion of this point . to analyze our uncertainties , we performed a series of monte carlo ( mc ) realizations of the data ( see also @xcite ) . in each mc run , we re - sampled the observed cmd to account for the poisson errors of the stars . in addition , to estimate how systematic model differences can impact the results , we add in random shifts to the models for each run . we used random shifts of @xmath32 in temperature and @xmath33 in bolometric luminosity . these systematic offsets are much larger than uncertainty in the distance modulus , thereby incorporating both effects into our overall uncertainty analysis . uncertainty in the age of the recent star formation manifests itself in two distinct ways . the first is uncertainty measured by the mc analysis performed above , while the second is due to the width of the intrinsic spread of the burst across multiple age bins . we estimate the latter of these as the difference between the median age and the ages where 16% and 84% of star formation has occurred . we estimate the former of these as the rms difference between the median age of the best fit and the median ages of sf from the mc tests . we then add these differences in quadrature to assign a confidence interval to each result . thus for each progenitor we determine a median age , as well as ( potentially asymmetric ) uncertainties about this median . finally , we make the assumption that we are unable to determine the age of sf to a greater precision than the age bins in which we have measured this sf . we thus round the age range to the age of the next isochrone out from the median . we perform this step for both younger and older star formation , always rounding away from the median . the ages of these isochrones determine our final uncertainties . in order to use this method of analysis , we must define the maximum age of star formation which may produce core - collapse progenitors . as explained in 4.1 , we use our distribution to estimate a minimum mass for core - collapse between 7.8 and 7.0 m@xmath7 . as a result , we adopt 50 myr as the maximum age of a core - collapse progenitor and only perform the above mass estimation analysis over the most recent 50 myr of sf . the @xcite and @xcite models specify a maximum mass for an isochrone at a given age . more massive stars will have already died off . thus the isochrones we have identified as bounding our confidence interval can be linked directly to values for @xmath0 . our value for the median progenitor mass comes from interpolating the final isochrone masses between isochrones to the median age value . this is necessary because our median age wo nt line up exactly with a defined isochrone . in fig . [ massplot ] we plot the @xcite and @xcite isochrones for final isochrone mass vs. age for metallicities of z = 0.004 , 0.008 , 0.019 , and 0.030 . the models produce very similar age to mass conversions regardless of the assumed metallicity . the vast majority of our regions produce best fit metallicities of approximately solar . as a result , we adopt the z = 0.019 isochrone for mass determinations . note that masses change very quickly for younger populations , while masses for older populations change at a much slower rate . this means that our results for less massive progenitors will naturally be more precise . note that we have neglected systematic uncertainties in the age - to - mass conversion process for our individual progenitor results . this leads to very small error bars on some progenitor masses , especially those at older ages , where mass does nt change significantly as a function of age . our mass distributions are not sensitive to this model - dependent systematic uncertainty , as shown by similarity between the median progenitor distribution and the probability distribution ( see 4.2 ) . however , uncertainties are almost certainly underestimated for some progenitors , especially at the low - mass end . to estimate the magnitude of this systematic uncertainty , we compared the solar metallicity isochrones of @xcite to the @xcite isochrones used for our analysis . we found that for a @xmath150 myr lifetime star , the @xcite isochrones find a maximum mass of 6.7 m@xmath7 , compared to the 7.3 m@xmath7predicted for the @xcite isochrones . at a lifetime of @xmath122 myr , the youngest stars tested by @xcite , the maximum masses for the different isochrone sets are 9.9 m@xmath7and 10.9 m@xmath7respectively . these suggest systematic uncertainties of around 0.5 - 1.0 m@xmath7for the age - mass conversion process . again , we do not include these systematic uncertainties in the reported values in table 2 ; the reader should keep this in mind when interpreting the results of any individual progenitor star in our study .

, we create color - magnitude diagrams ( cmds ) and employ cmd fitting to measure the recent star formation history ( sfh ) of the regions surrounding cataloged snr sites . we identify any young coeval population that likely produced the progenitor star , then assign an age and uncertainty to that population . we identify significant young star formation around 53 of the 59 snrs and assign progenitor masses to these , representing a factor of increase over currently measured progenitor masses . in addition , we use our distribution to estimate a minimum mass for core collapse between 7.0 and 7.8 m .

using hubble space telescope ( hst ) photometry , we age - date 59 supernova remnants ( snrs ) in the spiral galaxy m31 and use these ages to estimate zero - age main sequence masses ( ) for their progenitors . to accomplish this , we create color - magnitude diagrams ( cmds ) and employ cmd fitting to measure the recent star formation history ( sfh ) of the regions surrounding cataloged snr sites . we identify any young coeval population that likely produced the progenitor star , then assign an age and uncertainty to that population . application of stellar evolution models allows us to infer thefrom this age . because our technique is not contingent on identification or precise location of the progenitor star , it can be applied to the location of any known snr . we identify significant young star formation around 53 of the 59 snrs and assign progenitor masses to these , representing a factor of increase over currently measured progenitor masses . we consider the remaining 6 snrs as either probable type ia candidates or the result of core - collapse progenitors that have escaped their birth sites . in general , the distribution of recovered progenitor masses is bottom heavy , showing a paucity of the most massive stars . if we assume a single power law distribution , , we find a distribution that is steeper than a salpeter imf ( ) . in particular , we find values of outside the range to be inconsistent with our measured distribution at 95% confidence . if instead we assume a distribution that follows a salpeter imf up to some maximum mass , we find that values of are inconsistent with the measured distribution at 95% confidence . in either scenario , the data suggest that some fraction of massive stars may not explode . the result is preliminary and requires more snrs and further analysis . in addition , we use our distribution to estimate a minimum mass for core collapse between 7.0 and 7.8 m .

1210.6353

c

our method measures the prominence of a burst as a fraction of the total recent sf . this approach requires that we define the period of time that we consider as `` recent '' sf . specifically , we must identify the age range where star formation can produce stars massive enough to result in ccsne . theoretical arguments and observational evidence point to a minimum mass necessary for progenitors to undergo core - collapse to a neutron star . stars below this mass are generally assumed to leave behind white dwarf stars , producing no sn explosion . thus one may constrain this cross - over mass by either measuring the maximum mass from which a star may create a white dwarf , or the minimum mass necessary for a star to explode . measurements of white dwarfs have defined a lower limit on this minimum mass of 6.3 - 7.1 m@xmath7@xcite , corresponding to an age of between @xmath155 and 63 myr , and direct progenitor mass measurements has have converged on a value of 8@xmath381 m@xmath7@xcite , corresponding to an age of between @xmath133 and 55 myr . in our observed progenitor mass distribution , we would expect to see the following behavior : above the minimum mass for core - collapse , we expect the distribution of progenitor masses to follow the imf , assuming that the recent sfr is approximately constant . below the minimum mass , the inferred progenitor mass should have no physical connection to the ccsne process and should reflect random sampling of the sfr at @xmath4250 myr , producing an essentially flat distribution in inferred progenitor mass . in the left panel of fig . [ minmass ] , we plot the distribution of progenitor masses for a variety of assumed minimum masses . we vary the minimum mass from 9.6 m@xmath7to 6.0 m@xmath7(our chosen masses are mapped using the @xmath0from @xcite and @xcite isochrones ) . we find that for assumed minimum mass greater than 8.1 m@xmath7 , the distribution increases until the assumed minimum mass is reached . for assumed minimum mass values below 8.1 m@xmath7 , there is a peak between 7.5 and 8.5 m@xmath7 , below which the number of progenitors drops , suggesting a minimum mass in this range . to find the actual minimum mass , we lower the assumed minimum mass until the measured minimum mass no longer reflects this assumed value . we note the amplitude of the peak in the distribution is greatest for an assumed minimum of 7.3 m@xmath7 . to quantify the location of this peak , we calculate the derivative of the number of progenitors as a function of mass . we assume that the maximum value of this derivative occurs at the minimum mass a star undergoes core - collapse , as this value identifies the beginning of the peak . in the right panel of fig . [ minmass ] we plot the location of this maximum against our assumed minimum mass . we note that above an assumed minimum mass of 7.3 m@xmath7 , the peak value traces the assumed minimum mass . for 7.3 m@xmath7 and below , however , the peak value is always around @xmath17.5 m@xmath7 . if we assume our uncertainties are at least the width of the mass bins , the we find a minimum mass for core - collapse between 7.0 and 7.8 m@xmath7 . this range is consistent with the observational measurements of @xcite and @xcite . we therefore have adopted the 44.7 to 50 myr ( 7.7 to 7.3 m@xmath7 ) bin as the oldest included for our final results . in the left panel of fig . [ diffhist ] we plot the histogram of median progenitor masses , restricted to 7 m@xmath7 and above . in the right panel of fig . [ diffhist ] we plot the cumulative fraction of progenitor masses . unless otherwise noted , we assume a maximum mass of 120 m@xmath7 , although the choice of this value at high masses is essentially irrelevant to the overall distributions given the rarity of extremely massive stars . qualitatively , the observed distribution shows a lack of the most massive stars when compared to a salpeter imf ( @xmath2 , where @xmath3 ) . we performed a kolomogorov - smirnov ( ks ) test , assuming a single power law distribution . we found values of @xmath4 outside the range @xmath5 inconsistent with the measured distribution at 95% confidence . alternatively , we may consider a model distribution that is a salpeter imf ( @xmath3 ) up to some maximum mass , which we may vary . we found that this model was inconsistent with the data at 95% confidence at assumed maximum masses @xmath43 m@xmath7 . however , precise determination of this value is difficult in our survey due to the intrinsic rarity of massive stars and the mass spacing in our isochrines ( see fig . [ massplot ] ) . rather , this value represents the sort of mass range in which one must consider ccsn possible in order to maintain a salpeter imf . in either scenario , the full distribution of measured masses suggests that some fraction of massive stars are not exploding as ccsn . this result has interesting implications for ccsn physics . a wide variety of sn channels have been explored both theoretically and observationally in the literature . theoretical predictions have explored the possibility of direct black hole formation beyond a certain mass threshold somewhere around @xmath125 m@xmath7 . the manifestation of such events in an overall mass distribution would be an observed lack of progenitors beyond the mass threshold , essentially a more bottom - heavy imf than that of all massive stars . while the reality is likely something more complicated than a well - defined threshold between ccsn and black hole formation ( many different scenarios likely combine to produce a complicated mass distribution ) , the qualitative effect will be that which we observe in our distribution . @xcite first identified the red supergiant problem , an observed lack of type iip progenitors between 16 and 30 m@xmath7 . many solutions have been proposed to explain the problem ( see @xcite and references therein ) . in addition , the recent sn 2012aw may fall in this mass range @xcite , suggesting the possibility that type iip progenitors do exist in this range and have simply not yet been observed in sufficient number . we identify six progenitors with median progenitor masses between 16 and 30 m@xmath7 , although the uncertainties on many of these are large . if our progenitors were sampled uniformly from a salpeter imf , we would expect to find @xmath110 progenitors ( 20% ) in the mass range from 16 to 30 m@xmath7 . thus while we do nt observe a complete lack of progenitors in the specified mass range , we do observe fewer than we would expect given a salpeter imf distribution . finally , while we assumed our snr catalogs constituted a complete , unbiased sample , the possibility exists that selection effects in the catalogs lead to the lower end of the progenitor mass distribution being sampled more heavily . in particular , extremely massive progenitors are likely associated with strong h ii regions , where identification of snrs is a more difficult observational task . it is possible snrs of this type are systematically undersampled in the survey . we attempted to quantify both the imf slope and minimum mass using markov chain monte carlo methods , but found that the data could not produce meaningful constraints beyond those determined by our more simplistic techniques . we believe the chief reason for this is the size of errors due to differential reddening of the fields . part of the problem is treatment of differential reddening as a top - hat distribution . in addition , the inclination of m31 contributes to these high differential extinction values . the application of this technique to a less inclined galaxy would be of benefit in this respect . finally , the simple addition of more progenitor mass estimates would allow us to better constrain our analysis . we are currently performing identical analysis on an additional @xmath165 snr in m33 in pursuit of these final two points . type ia snrs coincident with star - forming regions could in principle affect our mass distribution . to test the possibility of additional type ia contamination beyond the 11% observed , we examined galex fuv fluxes at the sites of all snr in our sample . we assumed that sites with the lowest fuv flux corresponded to possible older type ia sites which happened to be coincident with a small amount of recent sf . the eight lowest flux snr ( not including the 6 with no recent sf ) were bw-18 , bw-69 , 1 - 006 , 1 - 010 , 2 - 024 , 2 - 050 , k891 , and k956a . from table [ tab_results ] , these additional snrs in general have fewer ms stars and less total sf then most snrs in the sample . we found that after removing these snr , our observed distribution now ranged from @xmath44 , which is consistent with our earlier measurement . this suggests that additional type ia contamination has little effect on our overall result , and still results in an imf that is steeper than salpeter ( -2.35 ) . using resolved hst photometry , we have analyzed the stars surrounding 59 snr in m31 . using cmd fitting , we calculate a sfh within a 50 pc radius of each snr . we find that 53 of the snr regions display significant evidence of recent star formation , which we use to age - date the progenitor star . the remaining six regions display no recent star formation , and we consider them either possible type ia candidates or the result of massive runaway progenitor stars . we examine the distribution of progenitor masses for our ccsn candidates and find a lack of massive stars compared to a standard salpeter imf ( @xmath2 , where @xmath3 ) . if a uniform single imf is assumed , we find values for @xmath4 outside the range @xmath45 inconsistent with the measured distribution at 95% confidence . alternatively , if we consider a distribution that is a salpeter imf up to some maximum mass , we place an upper limit on the maximum mass allowed at @xmath46 m@xmath7 . we also estimate a minimum mass for core collapse of between 7.0 and 7.8 m@xmath7 , which is both greater than the maximum mass for white dwarf collapse @xcite and consistent with direct progenitor measurements @xcite . is supported in part by funding from the mary gates endowment . z.g.j . , b.f.w . , and j.j.d . are supported in part by go-12055 . is supported in part by an nsf astronomy and astrophysics postdoctoral fellowship under award ast-0802315 . this work is based on observations made with the nasa / esa hubble space telescope , obtained from the data archive at the space telescope science institute . support for this work was provided by nasa through hubble fellowship grant 51273.01 awarded to k.m.g . by the space telescope science institute . stsci is operated by the association of universities for research in astronomy , inc . under nasa contract nas 5 - 26555 . map of m31 . snrs in red correspond to those with no young sf detected . we consider these as possible type ia locations , or as the results of runaway stars . we note that a large amount of our ccsn candidates fall along the star forming arms of m31 . this is clearly expected ( ccsn candidates will tend to be found in regions of recent sf ) , but it also indicates the constraints imposed on our sample by the locations of archival hst data ( star - forming regions are targeted significantly more than other areas).,scaledwidth=80.0% ] + 1 - 006 & 10.6318 & 41.1005 & pos-33 & 10273 & wfpc-2 & f555w=24.5 , f814w=23.4 + 1 - 008 & 10.7675 & 41.6031 & pos-21 & 10273 & acs & f555w=26.6 , f814w=26.0 + 1 - 009 & 10.7975 & 41.6256 & pos-23 & 10273 & acs & f555w=25.6 , f814w=25.3 + 1 - 010 & 10.7979 & 41.4853 & pos-18 & 10273 & wfpc-2 & f555w=24.9 , f814w=23.7 + 2 - 016 & 10.3196 & 40.9554 & g-87 & 6671 & wfpc-2 & f555w=25.2 , f814w=23.9 + 2 - 020 & 10.4508 & 41.1138 & g-104 & 10260 & acs & f606w=25.6 , f814w=23.5 + 2 - 021 & 10.4773 & 40.7866 & g-119 & 6671 & wfpc-2 & f555w=26.2 , f814w=25.0 + 2 - 024 & 10.5958 & 41.0036 & pos-29 & 10273 & acs & f555w=25.8 , f814w=25.1 + 2 - 025 & 10.6448 & 40.9688 & pos-29 & 10273 & acs & f555w=25.9 , f814w=25.1 + 2 - 026 & 10.6698 & 41.0447 & pos-30 & 10273 & acs & f555w=26.0 , f814w=25.2 + 2 - 028 & 10.7376 & 40.9698 & pos-41 & 10273 & acs & f555w=25.7 , f814w=25.4 + 2 - 044 & 11.1989 & 41.4654 & b08-f10 & 12075 & acs & f475w=27.3 , f814w=26.0 + 2 - 046 & 11.2928 & 41.5993 & b12-f17 & 12071 & acs & f475w=27.5 , f814w=26.1 + 2 - 048 & 11.3094 & 41.6033 & b12-f17 & 12071 & acs & f475w=27.5 , f814w=26.1 + 2 - 049 & 11.3254 & 41.8683 & b15-f08 & 12056 & acs & f475w=27.4 , f814w=26.0 + 2 - 050 & 11.3662 & 41.8698 & b15-f07 & 12056 & acs & f475w=27.4 , f814w=26.1 + + bw-11 & 10.2467 & 40.6081 & g-76 & 11081 & wfpc-2 & f606w=26.2 , f814w=24.9 + bw-18 & 10.3988 & 41.1155 & g-104 & 10260 & acs & f606w=26.1 , f814w=24.9 + bw-19 & 10.5408 & 40.9472 & pos-27 & 10273 & acs & f555w=26.1 , f814w=25.5 + bw-20 & 10.5433 & 40.8644 & pos-26 & 10273 & acs & f555w=26.0 , f814w=25.0 + bw-31 & 10.7317 & 40.9956 & pos-41 & 10273 & acs & f555w=26.4 , f814w=25.4 + bw-32 & 10.7329 & 40.9717 & pos-41 & 10273 & acs & f555w=25.7 , f814w=24.6 + bw-36 & 10.7725 & 41.3750 & g-205 & 10260 & acs & f606w=25.5 , f814w=24.5 + bw-39 & 10.7933 & 41.6282 & pos-23 & 10273 & acs & f555w=25.8 , f814w=25.4 + bw-44 & 10.8779 & 41.6882 & pos-24 & 10273 & acs & f555w=26.5 , f814w=25.0 + bw-60 & 11.0883 & 41.9018 & b15-f12 & 12056 & acs & f475w=28.0 , f814w=26.8 + bw-61 & 11.1054 & 41.3501 & b06-f10 & 12105 & acs & f475w=27.3 , f814w=25.9 + bw-65 & 11.1488 & 41.4227 & b06-f04 & 12105 & acs & f475w=27.3 , f814w=25.9 + bw-66 & 11.1550 & 41.8666 & b15-f17 & 12056 & acs & f475w=27.5 , f814w=26.2 + bw-69 & 11.1825 & 41.9645 & b17-f18 & 12059 & acs & f475w=27.6 , f814w=26.5 + bw-71 & 11.1958 & 41.4886 & b08-f04 & 12075 & acs & f475w=27.5 , f814w=26.0 + bw-74 & 11.2129 & 41.4847 & b08-f04 & 12075 & acs & f475w=27.1 , f814w=25.8 + bw-76 & 11.2267 & 41.5121 & b08-f04 & 12075 & acs & f475w=27.3 , f814w=25.9 + bw-77 & 11.2269 & 41.5306 & b08-f04 & 12075 & acs & f475w=27.3 , f814w=25.9 + bw-81 & 11.2858 & 41.6101 & b12-f17 & 12071 & acs & f475w=27.4 , f814w=26.1 + bw-82 & 11.2892 & 41.8523 & b15-f08 & 12056 & acs & f475w=27.6 , f814w=26.3 + bw-84 & 11.3162 & 41.6561 & b12-f11 & 12071 & acs & f475w=27.4 , f814w=26.0 + bw-86 & 11.3387 & 41.6668 & b12-f11 & 12071 & acs & f475w=27.3 , f814w=26.0 + bw-89 & 11.3650 & 41.9036 & b15-f01 & 12056 & acs & f475w=27.5 , f814w=26.3 + bw-102 & 11.4675 & 42.1618 & b21-f11 & 12055 & acs & f475w=27.6 , f814w=26.8 + bw-105 & 11.6296 & 41.9886 & b18-f03 & 12108 & acs & f475w=27.6 , f814w=26.6 + bw-106 & 11.6417 & 42.1804 & b21-f08 & 12055 & acs & f475w=28.0 , f814w=27.0 + bw-110 & 11.6896 & 42.2183 & b21-f01 & 12055 & acs & f475w=27.7 , f814w=27.0 + + k180 & 10.9186 & 41.1814 & b02-f11 & 12073 & acs & f475w=27.2 , f814w=25.9 + k376 & 11.0850 & 41.5804 & b09-f14 & 12057 & acs & f475w=27.1 , f814w=25.8 + k446 & 11.1298 & 41.3572 & b06-f04 & 12105 & acs & f475w=27.3 , f814w=26.0 + k497 & 11.1562 & 41.4133 & b06-f10 & 12105 & acs & f475w=27.3 , f814w=26.0 + k516/bw-67 & 11.1699 & 41.4145 & b06-f04 & 12105 & acs & f475w=27.4 , f814w=26.0 + k525a & 11.1820 & 41.4372 & b08-f10 & 12075 & acs & f475w=27.3 , f814w=26.0 + k526a & 11.1715 & 41.4653 & b08-f10 & 12075 & acs & f475w=27.3 , f814w=26.0 + k527a & 11.1834 & 41.4465 & b08-f10 & 12075 & acs & f475w=27.3 , f814w=26.0 + k574 & 11.2101 & 41.4649 & b08-f10 & 12075 & acs & f475w=27.4 , f814w=26.1 + k594 & 11.2201 & 41.9161 & b15-f10 & 12056 & acs & f475w=27.2 , f814w=26.0 + k856a & 11.4313 & 41.9313 & b16-f05 & 12106 & acs & f475w=27.6 , f814w=26.4 + k891 & 11.5405 & 42.2198 & b21-f04 & 12055 & acs & f475w=27.6 , f814w=26.6 + k908 & 11.6231 & 41.9685 & b18-f03 & 12108 & acs & f475w=28.0 , f814w=27.0 + k934/bw-107 & 11.6467 & 42.2266 & b21-f01 & 12055 & acs & f475w=27.6 , f814w=26.9 + k947/2 - 047 & 11.6689 & 42.1911 & b21-f07 & 12055 & acs & f475w=27.6 , f814w=26.8 + k956a & 11.6792 & 42.2171 & b21-f01 & 12055 & acs & f475w=27.7 , f814w=27.0 + [ tab_snr ] bw-74 & 17@xmath48 & 11@xmath49 & 825 & 4572 & 169 & 0.3 + bw-44 & @xmath50 & @xmath51 & 644 & 2085 & 3 & 1.5 + bw-86 & 9.6@xmath52 & 28@xmath53 & 575 & 4344 & 48 & 0.3 + bw-84 & 8.7@xmath54 & 34@xmath55 & 572 & 5158 & 49 & 0.3 + k527a & 17@xmath56 & 11@xmath57 & 531 & 4925 & 62 & 0.1 + 2 - 049 & 7.6@xmath58 & 47@xmath59 & 517 & 4364 & 60 & 0.6 + bw-81 & 12@xmath60 & 19@xmath61 & 502 & 5212 & 16 & 0.4 + k525a & 20@xmath62 & 9.4@xmath63 & 501 & 4677 & 121 & 0.2 + bw-65 & 8.1@xmath64 & 40@xmath65 & 500 & 5263 & 57 & 0.5 + bw-77 & 7.6@xmath66 & 47@xmath67 & 473 & 4968 & 31 & 0.2 + k934 & @xmath6813 & @xmath6816 & 455 & 1934 & 75 & 0.7 + k908 & 8.6@xmath69 & 36@xmath70 & 418 & 3360 & 47 & 0.3 + bw-66 & 36@xmath71 & 5.5@xmath72 & 411 & 3310 & 63 & 1.6 + bw-71 & 10@xmath73 & 25@xmath74 & 401 & 5281 & 37 & 0.6 + bw-31 & 9.9@xmath75 & 27@xmath76 & 397 & 2288 & 23 & 0.3 + k446 & 9.0@xmath77 & 32@xmath78 & 389 & 4921 & 26 & 0.6 + k594 & 8.4@xmath79 & 38@xmath76 & 379 & 2537 & 26 & 0.5 + bw-106 & 19@xmath80 & 10@xmath81 & 372 & 2642 & 64 & 1.3 + k497 & 10@xmath82 & 26@xmath83 & 360 & 5106 & 26 & 0.2 + k856a & 27@xmath84 & 7@xmath85 & 356 & 2125 & 54 & 0.3 + k376 & 16@xmath86 & 12@xmath86 & 350 & 6874 & 55 & 0.7 + bw-76 & 8.5@xmath87 & 37@xmath88 & 345 & 5013 & 35 & 0.3 + 1 - 008 & 7.6@xmath89 & 46@xmath90 & 336 & 1895 & 61 & 1.7 + bw-61 & 15@xmath91 & 13@xmath92 & 303 & 4835 & 106 & 0.8 + 2 - 050 & 7.6@xmath93 & 47@xmath94 & 289 & 4343 & 11 & 0.5 + k526a & 8.4@xmath79 & 38@xmath76 & 288 & 4842 & 3 & 0.3 + bw-82 & 8.1@xmath95 & 40@xmath96 & 281 & 4017 & 8 & 0.3 + bw-89 & 8.1@xmath97 & 40@xmath98 & 265 & 4296 & 43 & 0.9 + k947 & 13@xmath99 & 16@xmath100 & 245 & 1874 & 37 & 0.9 + bw-60 & 14@xmath60 & 15@xmath101 & 243 & 3277 & 39 & 1.5 + bw-105 & @xmath688 & @xmath10244 & 239 & 1945 & 13 & 0.3 + bw-11 & 10@xmath103 & 26@xmath104 & 235 & 1199 & 105 & 0.9 + 1 - 009 & 8.9@xmath105 & 33@xmath106 & 219 & 739 & 36 & 1.2 + bw-20 & 7.9@xmath75 & 42@xmath107 & 192 & 1339 & 22 & 1.8 + 2 - 044 & 8.4@xmath108 & 37@xmath109 & 186 & 4368 & 13 & 0.2 + 2 - 046 & 11@xmath86 & 21@xmath76 & 185 & 4445 & 16 & 0.4 + 2 - 024 & 16@xmath86 & 12@xmath86 & 164 & 1934 & 5 & 0.5 + bw-39 & @xmath6811 & @xmath10220 & 163 & 844 & 104 & 2.1 + k574 & 8.1@xmath110 & 40@xmath111 & 162 & 4207 & 18 & 0.5 + k180 & 8.8@xmath79 & 33@xmath76 & 162 & 3950 & 23 & 0.6 + 2 - 048 & 7.7@xmath112 & 45@xmath113 & 160 & 3581 & 33 & 0.7 + k891 & @xmath689 & @xmath10236 & 149 & 1441 & 23 & 1.1 + 2 - 025 & 8.2@xmath114 & 39@xmath115 & 140 & 1287 & 13 & 0.3 + bw-18 & 9.9@xmath116 & 27@xmath76 & 117 & 2165 & 25 & 2.0 + k516 & 8.5@xmath117 & 36@xmath118 & 109 & 2210 & 10 & 0.4 + k956a & 8.1@xmath119 & 40@xmath120 & 103 & 1457 & 22 & 1.1 + bw-69 & 24@xmath121 & 7.5@xmath122 & 100 & 1467 & 8 & 1.5 + bw-32 & 9.3@xmath75 & 30@xmath76 & 91 & 562 & 6 & 0.5 + bw-110 & 12@xmath60 & 20@xmath123 & 86 & 1391 & 12 & 0.8 + bw-102 & 8.5@xmath124 & 36@xmath53 & 73 & 1568 & 31 & 1.5 + 2 - 020 & 18@xmath60 & 11@xmath9 & 38 & 1137 & 20 & 1.4 + 1 - 006 & 11@xmath86 & 24@xmath125 & 10 & 601 & 6 & 0.0 + 1 - 010 & 12@xmath86 & 19@xmath9 & 10 & 335 & 9 & 0.2 + + bw-19 & - & - & 291 & 2520 & 0 & 0.5 + bw-36 & - & - & 10 & 4162 & 0 & 1.3 + 2 - 026 & - & - & 191 & 2421 & 0 & 0.1 + 2 - 021 & - & - & 122 & 1268 & 0 & 0.0 + 2 - 028 & - & - & 53 & 374 & 0 & 0.5 + 2 - 016 & - & - & 5 & 466 & 0 & 0.6 + [ tab_results ]

using hubble space telescope ( hst ) photometry , we age - date 59 supernova remnants ( snrs ) in the spiral galaxy m31 and use these ages to estimate zero - age main sequence masses ( ) for their progenitors . to accomplish this we consider the remaining 6 snrs as either probable type ia candidates or the result of core - collapse progenitors that have escaped their birth sites . in general , the distribution of recovered progenitor masses is bottom heavy , showing a paucity of the most massive stars . in particular , we find values of outside the range to be inconsistent with our measured distribution at 95% confidence . if instead we assume a distribution that follows a salpeter imf up to some maximum mass , we find that values of are inconsistent with the measured distribution at 95% confidence . in either scenario , the data suggest that some fraction of massive stars may not explode .

using hubble space telescope ( hst ) photometry , we age - date 59 supernova remnants ( snrs ) in the spiral galaxy m31 and use these ages to estimate zero - age main sequence masses ( ) for their progenitors . to accomplish this , we create color - magnitude diagrams ( cmds ) and employ cmd fitting to measure the recent star formation history ( sfh ) of the regions surrounding cataloged snr sites . we identify any young coeval population that likely produced the progenitor star , then assign an age and uncertainty to that population . application of stellar evolution models allows us to infer thefrom this age . because our technique is not contingent on identification or precise location of the progenitor star , it can be applied to the location of any known snr . we identify significant young star formation around 53 of the 59 snrs and assign progenitor masses to these , representing a factor of increase over currently measured progenitor masses . we consider the remaining 6 snrs as either probable type ia candidates or the result of core - collapse progenitors that have escaped their birth sites . in general , the distribution of recovered progenitor masses is bottom heavy , showing a paucity of the most massive stars . if we assume a single power law distribution , , we find a distribution that is steeper than a salpeter imf ( ) . in particular , we find values of outside the range to be inconsistent with our measured distribution at 95% confidence . if instead we assume a distribution that follows a salpeter imf up to some maximum mass , we find that values of are inconsistent with the measured distribution at 95% confidence . in either scenario , the data suggest that some fraction of massive stars may not explode . the result is preliminary and requires more snrs and further analysis . in addition , we use our distribution to estimate a minimum mass for core collapse between 7.0 and 7.8 m .

1502.03583

r

in bulk a fluid of hard rectangles of aspect ratio @xmath1 undergoes a phase transition from an isotropic to a nematic phase upon increasing the density @xcite . the bulk transition is continuous , presumably of the kosterlitz - thouless type @xcite . the aim of the present work is to study confinement properties , and hence we have not analysed in detail the bulk properties . however we have run bulk simulations using a square box of side length @xmath52 with periodic boundary conditions and found a continuous isotropic - nematic transition at @xmath53 , in agreement with the predictions of the scale particle theory @xcite . in what follows we show the states that occur under confinement . for each geometry we group the states in a diagram as a function of the system size and the packing fraction . we use the local fields ( density , uniaxial order parameter and tilt angle ) to distinguish between distinct states . however , one should bear in mind that given the confined geometries analyzed here and the dimensionality of the system , one expects gradual transitions between the different states . in addition , the distinction between states , such as nematic , smectic , or isotropic , in confined geometry is not clearcut . and packing fraction @xmath21 for hard rectangles ( @xmath1 ) confined in a homeotropic cell . the color indicates the average uniaxial order parameter @xmath54 . white circles show the state points where @xmath55 ( the white - dashed line is a guide to the eye ) . the black lines delimit approximate boundaries between different states ( no simulation data is available for the regions depicted in white at high packing fractions ) . ] we first consider hard rectangles ( @xmath1 ) confined between two parallel planar walls inducing homeotropic anchoring ( see fig . [ fig1]a ) . the state diagram in the plane of packing fraction as a function of pore width is depicted in fig . we have calculated about @xmath56 state points with @xmath16 varying between @xmath57 and @xmath58 . the color map shows the average of the uniaxial order parameter inside the pore @xmath59 * isotropic , nematic and smectic states . * first we focus on the larger pores that we have investigated , @xmath60 . we identified three distinct states : isotropic ( i ) , nematic ( n ) , and smectic ( s@xmath61 , with @xmath3 the number of smectic layers inside the pore ) . examples of the particle configurations and the density profiles for each state in a pore with @xmath62 are shown in figs . [ fig3 ] and [ fig4 ] , respectively . at low densities the particles form an isotropic state ( fig . [ fig3]a ) . the density profile ( fig . [ fig4 ] , top panel ) is rather constant with a small amount of adsorption of particles close to the walls . the uniaxial order parameter ( middle panel ) is zero except in a small region near the walls where it shows incipient orientational order due to the walls . in this region the particles are ( slightly ) aligned with their long axes perpendicular to the walls , as the tilt angle ( bottom panel ) shows . the maximum in density occurs at contact with the surfaces . however , the maximum of the uniaxial order parameter is shifted @xmath63 away from the walls . this is a general feature of a hard center - wall that has been previously reported in three- @xcite and two- @xcite dimensional systems on the basis of mc simulation and dft . by increasing the density , the capillary nematization ( i.e. the formation of a nematic state inside the pore ) occurs in a continuous fashion , as expected . in the nematic state all the particles are oriented , on average , perpendicular to the walls ( fig . [ fig3]b , and fig . [ fig4 ] bottom panel ) . the uniaxial order parameter is positive in the whole capillary ( fig . [ fig4 ] middle ) and there is incipient positional order that propagates into the pore from the walls ( fig . [ fig4 ] top ) . by further increasing the density the particles form a smectic state ( fig . [ fig3]c ) with well - defined layers ( fig . [ fig4]a ) , the number of which is the result of commensuration between the size of the pore and the smectic period . in the range of pore sizes investigated here , we have found smectic states with @xmath64 to @xmath65 layers . the smectic layers are slightly tilted , especially those in the center of the pore ( see the tilt angle profile in fig . [ fig4]c ) . the reason is that the commensuration is not perfect , i.e. the ratio between the pore size and the smectic period is not an integer number . by reducing the size of the pore , the nematization occurs at lower packing fractions . in order to visualize this effect we have depicted a line of constant average uniaxial order parameter , @xmath66 , in the state diagram ( fig . [ fig2 ] ) . this line monotonically increases with @xmath16 and asymptotically tends to the bulk packing fraction at which @xmath66 . therefore , confinement promotes nematic order . this is an expected behaviour because even at low densities the walls induce some homeotropic anchoring . similarly , confinement promotes smectic order as oscillations in the density profile start to appear at lower packing fractions in smaller pores . of the states in a homeotropic cell with @xmath62 . ( top ) scaled density profile @xmath67 , ( middle ) uniaxial order parameter profile @xmath68 and ( bottom ) tilt angle profile @xmath69 . dotted line : isotropic state , @xmath70 . dashed line : nematic state , @xmath71 . solid line : smectic state ( s@xmath72 ) , @xmath73 . snapshots of the particle configurations corresponding to these profiles are depicted in fig . [ fig3 ] . ] the most interesting phenomenology arises when the particles are strongly confined in narrow pores and at high packing fractions . in this regime new states with symmetries different than those of the stable bulk phases appear . * smectic c*. for pores with @xmath74 the particles form a nematic state at intermediate densities . by further increasing the density , the rods align into two well - defined layers as already present in the s@xmath75 state . here , however , the particles are strongly tilted with respect to the direction perpendicular to the layers . we call this the smectic c state , s@xmath76 , where @xmath64 indicates the number of layers . the particle configurations and the order parameter profiles of the s@xmath76 state are depicted in fig . [ fig5 ] panels ( a ) and ( b ) , respectively . the tilt profile ( bottom of panel b ) presents two minima shifted from the location of the the maximum density . the smectic c state appears because the size of the pore is not commensurate with the smectic period and the number of layers is reduced . as a result the particles tilt in order to fill efficiently the available space . by increasing the density , the tilt angle decreases . this is consistent with the fact that the smectic period monotonically decreases with the density . in fig . [ fig5]c ) we plot the tilt angle at contact with the wall , @xmath77 , as a function of @xmath16 . for each packing fraction , above a certain threshold , there is a critical pore size below which the particles start to tilt . the smaller the pore is the more tilted the particles are . increasing the number of particles : @xmath78 , nematic state ( left ) ; @xmath79 , smectic c state ( middle ) ; @xmath80 , smectic c state ( right ) . ( b ) local fields as a function of @xmath12 : density profile ( top ) , uniaxial order parameter ( middle ) , and tilt angle ( bottom ) . the different sets correspond to the states showed in panel ( a ) : @xmath78 , nematic state ( dotted line ) ; @xmath79 , smectic c state ( dashed line ) ; @xmath80 , smectic c state ( solid line ) . ( c ) smectic c tilt angle ( absolute value ) as a function of the pore width for different packing fractions , as indicated . ] we have also found a smectic c state with three layers , s@xmath81 , in a small region of the state diagram around @xmath82 ( see fig . [ fig2 ] ) . this region is significantly smaller than the stability region of the s@xmath83 state , and it appears at higher packing fractions than @xmath84 does . at very high packing fractions the smectic period is sufficiently small such that three non - tilted layers fit inside the capillary and the s@xmath81 state is replaced by a s@xmath85 state . we have not found smectic c states with more than three layers although we can not rule out their existence in regions of the state diagram at very high packing fractions . nevertheless , we are confident that those regions shrink rapidly by increasing the pore size and eventually might cease to exist . this can be understood as follows . there is a minimum @xmath86 and a maximum @xmath87 layer spacing between which the formation of non - tilted smectic layers is stable . consider a capillary with @xmath88 smectic layers inside . as a rough estimate the layers will tilt if there is no sufficient space to accommodate @xmath88 non - tilted layers , i.e. if the condition @xmath89 is satisfied . in addition @xmath90 should hold as well , because otherwise an s@xmath91 state , and not an s@xmath92 state , would be stable . both conditions together roughly set the limits in pore size for a smectic c phase with @xmath88 layers as @xmath93 the above equation also shows that the range in pore size in which the smectic c is stable decreases with increasing @xmath88 . indeed , there is a maximum pore size above which no tilted smectic is expected : @xmath94 which results from taking the equality on both sides of eq . ( [ minmax ] ) . for hard rectangles @xmath95 ( because the particles can not overlap ) . we can express @xmath96 , with @xmath97 being the maximum expansion of the layer spacing for the smectic state in units of @xmath86 . then , using eq . ( [ pinpanpum ] ) , @xmath98 . we did not find sm@xmath99 states for @xmath100 . it implies @xmath101 , which seems to be a reasonable value . . ( a ) snapshot of characteristic configurations : nematic state , @xmath102 ( left ) ; nematic - brush , @xmath103 ( second ) ; brush , @xmath104 ( third ) ; smectic - brush , @xmath105 ( right ) . ( b ) local fields as a function of @xmath106 of the states showed in panel ( a ) : density profile ( top ) , uniaxial order parameter ( middle ) , and tilt angle ( bottom ) . the different sets are : nematic ( dotted line ) , nematic - brush ( red dotted - dashed line ) , brush ( dashed line ) , smectic brush ( solid line ) . ( c ) average of the tilt angle profile in a pore with @xmath107 as a function of the packing fraction . results obtained by increasing and decreasing the number of particles . simulation parameters : @xmath108 , @xmath14 ( black - dashed line ) ; @xmath109 , @xmath14 ( blue - dotted line ) ; @xmath108 , @xmath110 ( red - solid line ) . ] * brush states . * the remaining states in the diagram depicted in fig . [ fig2 ] are the brush nematic b@xmath61 and the brush smectic b@xmath111 , both with @xmath3 homeotropic layers . for pores with size in the vicinity of @xmath112 and high packing fractions there is a region where b@xmath75 and b@xmath113 are stable . in fig . [ fig6 ] we show the corresponding order parameter profiles and characteristic particle configurations . by increasing the density from a stable nematic state , some of the particles located in the middle of the pore rotate by ninety degrees , placing their long axis parallel to the walls ( nematic - brush state ) . a further increase in the number of particles results in the pure brush nematic state b@xmath75 , with one layer of particles with homeotropic anchoring adjacent to each wall . the particles at the center of the cavity are aligned parallel to the walls . to rule out the possibility that this state is an artefact of our method of increasing the number of particles , we have initialized a system with @xmath114 and @xmath112 in a nematic state with all the particles perpendicular to the walls . after an equilibration stage of about @xmath115 mcs the particles formed the brush state . hence we are confident that the brush state is indeed stable . the extent of the central region where the particles are oriented parallel to the walls grows by increasing the density ( see e.g. the tilt angle profile in fig.[fig6 ] , bottom of panel b ) . in the b@xmath75 state those particles in the central region posses orientational but no positional order . however at sufficiently high packing fractions smectic layers occur also at the center . this additional order constitutes the smectic brush state , b@xmath113 . the transitions between the different states is gradual and hence no differences should appear if we e.g. fix the size of the pore and track the order parameters by first increasing and then decreasing the density . this is actually what we have found for all the states discussed throughout of the paper except for the nematic - brush transition : we plot in fig . [ fig6]c the average of the tilt angle profile , @xmath116 as a function of the packing fraction , as obtained by either increasing or decreasing the number of particles . @xmath117 is approximately zero in the nematic state and is different than zero in the brush - nematic state due to the particles aligning perpendicular to the walls . we show in the figure three sets of data corresponding to simulations with different numbers of mcs and different lateral pore sizes . in the three cases there is strong hysteresis , most likely related to a finite size effect because our system can be effectively considered as a one - dimensional system with short range interactions where no first order transitions are expected according to the so - called van hove theorem . nevertheless , the van hove theorem does not apply in the presence of external fields ( see @xcite for a discussion about the exceptions to the van hove theorem ) . ( a , b ) and @xmath118 ( c ) at an average packing fraction @xmath105 . ] brush nematic and smectic states with three layers of particles that are perpendicular to the walls also appear in the region of the state diagram where @xmath119 ( see the labels b@xmath120 and b@xmath121 in fig . [ fig2 ] ) . an example of the b@xmath121 state is shown in fig . the @xmath122 and @xmath123 states are in general not symmetric in @xmath12 . the region of particles aligned parallel to the walls is not located in the middle of the cell . in order to test the stability of the unexpected symmetry breaking of these states , we have initialized the system in a symmetric configuration where two small regions of rods parallel to the walls are placed between layers of particles with homeotropic configuration . the resulting configurations after running more than @xmath124 mcs are shown in panels ( b ) and ( c ) of fig . [ fig7 ] . as we had to initialize the system at very high packing fractions we were not able to recover the asymmetric brush profile shown in ( a ) . however , the states in ( b ) and ( c ) are again asymmetric . they resemble coexistence states between the state represented in ( a ) and its mirror image . hence , we conclude that the symmetric b@xmath120 and b@xmath121 phases are not stable . this could have been anticipated because the symmetric state has four interfaces between parallel and perpendicular rods , whereas the asymmetric one contains only two . asymmetric profiles in symmetric pores have been previously found in three - dimensional mixtures of hard rods @xcite and monocomponent and binary mixtures with soft interactions , see e.g. @xcite . the states shown in ( b ) and ( c ) are probably not stable because they have larger interfaces than the one obtained by gradually increasing @xmath5 ( a ) . in addition in both cases the central layer is distorted , which increases the elastic energy of the system . nevertheless , due to the finite - time simulations and the finite lateral pore size , the system may show a bimodal behaviour oscillating between states ( b ) and ( c ) as if it were a genuine phase transition . as for the smectic c states , the regions in the state diagram where the brush states are stable move to high packing fractions and shrink with the size of the pore . we could not find brush states above b@xmath120 but their existence in narrow regions and high packing fractions can not be ruled out . instead of forming a brush state , the particles could tilt and form a smectic c state reaching very high packing fractions , similar to those in the brush state . hence , an interesting question is : why are there regions of the state diagram where the brush states are more stable than the smectic c ? to answer this question we compare the excess in free energy of both states with respect to non - distorted nematic or smectic states . in the brush state the excess in free energy is dominated by the two interfaces of perpendicularly aligned particles . in the smectic c state there are two important contributions : the violation of the anchoring imposed by the surfaces and the formation of tilted layers . both contributions increase by increasing the tilt angle . the tilt angle in the smectic c state , and hence the excess in free energy , increases by reducing the size of the pore . as a consequence the brush states appear replacing the smectic c when the size of pore is reduced . this simple argument explains not only the appearance of the brush states but also their relative position to the s@xmath99 in the state diagram , cf . [ fig2 ] . the regions in the state diagram ( fig . [ fig2 ] ) where smectic c and brush states become stable possess a smaller average uniaxial order parameter than the surrounding regions . when these states start to form there are regions in the pore where the particles align according to the incipient state and other regions where the particles remain in the nematic state . in addition , the order parameter profiles of the brush states depend on both the vertical and the horizontal coordinates . however we calculate them only as a function of the horizontal coordinate . both effects result in an artificially reduced uniaxial order parameter that , on the other hand , is useful to distinguish the boundaries between states in the state diagram . next we investigate the behaviour of hard rectangles with the same aspect ratio @xmath1 as before , but confined between two parallel walls that promote antagonistic anchoring , the so - called hybrid cell . the left wall induces homeotropic anchoring and the right wall promotes planar alignment of the particles ( see a schematic of the geometry in fig . [ the state diagram is depicted in fig . [ fig8 ] in the plane of packing @xmath21 fraction and scaled - effective pore width @xmath125 . as in the previous case the color map indicates the value of the averaged uniaxial order parameter inside the pore . we have run more than @xmath126 simulations with pore widths @xmath127 to generate the diagram . and the packing fraction @xmath21 for hard rectangles ( @xmath1 ) confined in a hybrid planar cell . the color map represents the value of the averaged uniaxial order parameter @xmath54 . the empty circles connected with a dashed white line show the state points for which @xmath66 . the empty squares connected via a solid white line show the approximate boundary between the linear and the uniform states . ] for any pore width the isotropic state is stable at low densities . in this state there is a small layer of particles oriented perpendicular ( parallel ) to the left ( right ) wall . the remaining particles do not show orientational order . as discussed the anchoring imposed by the planar wall is stronger than that of the center - hard wall . a manifestation of this is the the value of uniaxial order parameter in the isotropic state ( not shown ) , which is higher close to the planar wall than close to the homeotropic one . confinement in a hybrid cell promotes , as in the homeotropic cell , orientational order of the particles ( see e.g. the line of constant uniaxial order parameter in the state diagram ) . first we focus on the regime of large pore sizes . by increasing the number of particles the following sequence of states appears : isotropic ( i ) , step ( st ) , linear ( l ) , uniform nematic ( u ) , and uniform smectic ( u@xmath128 ) . examples of the configuration of the particles and the order parameter profiles in the intermediate and high density states for a pore with @xmath62 are shown in figs . [ fig9 ] and [ fig10 ] , respectively . * step state . * also known as director - exchange phase or biaxial phase , the step phase was proposed by schopohl and sluckin @xcite and by palffy - muhoray et al . it has been study in three - dimensional systems with landau - de gennes theory @xcite , simulation @xcite , and density functional theory @xcite . in the st state there are two nematic regions with uniform and opposite directors following the anchoring imposed by the surfaces ( see fig . [ fig9]a ) . the interface between both regions is sharp ; the director rotates by ninety degrees in a region of about two molecular lengths ( see the tilt profile in fig . [ fig10 ] dotted line ) . at the interface the uniaxial order parameter drops to zero . for large pores the st state is stable in a very narrow region of packing fractions contiguous to the isotropic state . actually , as suggested in @xcite , the st state could be a manifestation of the isotropic state at densities close to the capillary nematization in sufficiently narrow pores . of the states in a hybrid cell with @xmath62 . ( top ) density profile , ( middle ) uniaxial order parameter profile , ( bottom ) tilt angle profile . dotted line : step state , @xmath129 . red dotted - dashed line : linear nematic state , @xmath130 . dashed line : uniform nematic state , @xmath131 . solid line : uniform smectic state , @xmath132 . snapshots of the particle configurations corresponding to these profiles are depicted in fig . [ fig9 ] . ] * linear state . * increasing the packing fraction from the st state in large pores gives rise to the formation of the linear state ( see fig . [ fig9]b ) . here the director rotates continuously from homeotropic to planar anchoring ( see fig . [ fig10 ] , red dotted - dashed line ) . far enough from both substrates the tilt profile varies linearly with the distance across the pore . in this way the elastic free energy is minimized . the formation of the l state is the analogue to the capillary nematization in a symmetric pore . the l state is compatible with the anchoring imposed by both substrates and at the same time minimizes the elastic free energy . * uniform nematic state . * by further increasing the packing fraction there is a configurational change from the linear state to the uniform nematic state . the u state is a nematic with uniform director parallel to the wall except for the first layer of particles adsorbed at the homeotropic wall , where the particles are perpendicular to the substrate ( see figs . [ fig9]c and [ fig10 ] ) . this first layer is most likely a consequence of the peculiarities of the hard center - wall , which allows for a high packing fraction of particles only in the case that rods are aligned perpendicular to the wall . the linear - uniform transition is a consequence of the stronger anchoring induced by the planar wall in comparison to the hard - center wall . although it occurs gradually as we increase the packing fraction , the range in @xmath21 at which the transition occurs is small , enabling us to draw a line in the state diagram that approximately indicates its location ( see fig . [ fig2 ] ) . the density at the l - u configurational change and , therefore , the range in packing fractions at which the l state is stable , increases with the pore width . actually , the l state may replace the uniform nematic state in the regime of very large pores . to understand this , consider the excess in free energy of the l and u states over a bulk undistorted nematic , @xmath133 . in the l state @xmath134 , where @xmath135 and @xmath136 are due to the anchoring imposed by the right and the left walls , respectively , and @xmath137 is the elastic energy due to the deformations of the director field . for very large pores the director varies linearly and rotates by ninety degrees in the pore . hence , the divergence of the director is @xmath138 and the elastic energy @xmath139 , with @xmath140 the splay elastic constant . in the uniform state both anchoring constraints are satisfied and contribute to the excess in free energy as in the linear state . the director is not distorted ( @xmath141 ) but there is an interface generated by the first layer of particles with homeotropic alignment . hence , @xmath142 , where @xmath143 is the free energy of the nematic - nematic interface . the elastic contribution in the l state decreases with @xmath15 but @xmath143 does not . therefore we expect the l state to replace the u state for sufficiently wide pores . note that the same argument is valid if instead of an interface between two nematics with opposite directors in the u state there is a violation of the anchoring imposed by one of the substrates . in that case the anchoring energy in the u state would be higher than that in the l state and would not decrease with the size of the pore . we have performed simulations in a pore with @xmath144 , and the l - u transition occurs at @xmath145 , considerably higher than e.g. the case @xmath62 ( @xmath146 ) . this scenario , in which the l state replaces the u state in very wide pores , is therefore plausible . nevertheless , we can not rule out another scenario in which the u state is stable for any pore width , as it has been found in @xcite . in @xcite a system of spherocylinders confined in a hybrid cell is analyzed with dft , and the lu transition persists at any pore length due to an anchoring transition at one of the substrates , i.e. the type of anchoring induced by the wall changes by varying the density . in our case , however , such an anchoring transition is not expected as we deal with hard core potentials . note , nevertheless , that the first layer of particles adsorbed on the hard - center wall could effectively act as a hard wall for the second layer if the density is sufficiently high , which in practice could be viewed as an anchoring transition . in this second scenario the u state would be stable even for very wide pores . simulations for pores wider than those considered here could help to elucidate this point . * uniform smectic state . * finally at very high packing fractions the particles parallel to the walls form smectic layers . the resulting state is similar to the uniform nematic but with positional ordering . an example of the particle configurations is presented in fig . [ fig9]d . the corresponding order parameter profiles are shown in fig . [ fig10 ] ( solid lines ) . the formation of layers by increasing the density from the u state takes place very gradually and we could not identify the packing fraction of the u - u@xmath128 transition in the state diagram . we find that the direction of the layers is not perpendicular to the walls . the director is tilted with respect to the direction perpendicular to the layers , like it is in a smectic c. we did not find any relation between the tilt angle and the size of the pore . the fact that the layers are tilted could be a finite size effect related to the vertical size of the pore , or it could also be related to high fluctuations in the tilt angle . in contrast to the homeotropic case , the state diagram of the hybrid cell does not show additional states in the regime of small pores . the only significant difference in the region of small pores with respect to the region of large pores is that the linear state disappears . we could not find the linear state in pores with @xmath147 . ) confined in a square cavity with planar anchoring in the packing fraction - side length plane . the color map represents the average of the uniaxial order parameter inside the cavity , @xmath54 . empty circles indicate the position where @xmath66 . empty squares roughly show the boundary between the elastic and the bridge smectic states . lines are guides to the eye . ] we next consider confinement of the rods in a square cavity favouring planar alignment of the particles ( see a sketch of the geometry in fig.[fig1]c ) . confinement in all spatial directions adds additional constraints on the orientational ordering of the particles that might result in e.g. the formation of topological defects . the state diagram in the plane of packing fraction and side length is depicted in fig . [ fig11 ] . we found three distinct states : isotropic ( i ) , elastic ( e ) , and bridge smectic ( br@xmath128 ) . representative results of these states are shown in fig . [ fig12 ] for a cavity with side length @xmath148 . at low densities the isotropic state is stable , here the fluid is disordered . only a thin layer of particles close to the walls shows some degree of orientational order ( see the uniaxial order parameter in fig . [ fig12]a ) . the density ( not shown ) is rather uniform in the whole cavity , showing only a small desorption of particles close to walls , especially near the corners of the cavity . the uniaxial order parameter is also smaller in the vicinity of the corners . the nematization occurs by increasing the number of particles . the result is a gradual transition from the isotropic to the elastic state ( see fig . [ fig12]b ) . in the e state the nematic can not adopt a uniform configuration due to the surfaces and six disclinations arise in the cavity . four disclinations are located in the corners of the cavity . in the middle of the cavity the rods align along one of the diagonals . this leads to the formation of two further disclinations with topological charge @xmath149 located along the other diagonal , at a distance of about @xmath150 from the corners . the disclinations are clearly visible as a drop of the uniaxial order parameter ( see fig . [ fig12]b ) . the density profile ( not shown ) also reveals a depletion of particles close the defect cores . the position of the cores of the @xmath149 defects fluctuates during the simulation but they always stay away from each other . the inner @xmath149 defects are connected with the adjacent corner defects , see the uniaxial order parameter in fig . [ fig12]b ) . the smaller the cavity becomes the stronger this effect is . the packing fraction at which the capillary nematization occurs increases monotonically with the size of the cavity and tends asymptotically to the bulk value ( see for example the line of constant average uniaxial order parameter depicted in the state diagram , fig . [ fig11 ] ) . the average order parameter @xmath54 depends nontrivially on @xmath16 and @xmath21 . in the i region , e.g. , @xmath151 , @xmath54 decreases by increasing @xmath16 because the walls induce order in a small region close to them and the ratio between this region and the whole cavity decreases as @xmath16 is increased . once the nematic is formed the trend is reversed . for instance , at @xmath152 the smaller the cavity becomes the lower @xmath54 is . here the whole cavity is in a nematic state , except in those regions where the disclinations appear , and the ratio between the surface occupied by the disclinations and the whole cavity decreases with @xmath16 . next we focus on the regime of high packing fraction . by increasing @xmath21 from the e state the particles show incipient positional order , forming smectic layers without changing their director field ( not shown ) . then , at higher packing fractions there is a complete structural change to the bridge smectic state ( see fig . [ fig12]c ) . in the br@xmath128 state the particles that were oriented along one diagonal in the e state rotate by @xmath153 generating three domains where the director is almost uniform . the domains are separated by domain walls where the director rotates by @xmath154 ( the uniaxial order parameter vanishes at the domain walls , see fig . [ fig12]c ) . the domain walls connect two corners and divide the cavity in three regions with uniform director . the size of the domains fluctuates but the central domain is always bigger than the others . the domain walls become more rigid as the density is increased . the same state has been predicted recently using density functional theory in a system of rectangles with restricted orientations ( zwanzig approximation ) confined in the same geometry @xcite . the authors of @xcite classify the br@xmath128 state according to the number of smectic layers in the central domain . such a criterion is not applicable in our case due to the large fluctuations of the domain walls , but obviously the number of smectic layers in the cavity varies with the side length . in order to estimate the packing fraction of the e - br@xmath128 transition we have made a histogram of the global tilt angle @xmath155 inside the cavity , i.e. the tilt angle resulting of a diagonalization of the tensorial order parameter formed by all the particles . in the isotropic state @xmath155 fluctuates between @xmath156 and @xmath157 . as soon as the the elastic state arises , @xmath155 fluctuates between the values for both diagonal directions ; the histogram shows two peaks at @xmath158 and @xmath159 . at high densities , but still in the e state , the system stops fluctuating between the diagonals ( during the available simulation time ) and the histogram shows only one peak either at @xmath158 or @xmath159 . finally , in the br@xmath128 state there is a single peak centered at @xmath160 or @xmath161 governed by the particles of the main domain . again at these packing fractions the particles can not fluctuate between both equivalent states with @xmath160 or @xmath161 during the available simulation time . the behaviour of @xmath155 allows us to estimate the e - br@xmath128 transition as the packing fraction at which @xmath155 changes from @xmath158 or @xmath159 to @xmath156 or @xmath161 . the result is plotted in the state diagram , fig . [ fig11 ] . the bigger the cavity is the higher the packing fraction at the e - br@xmath128 transition is . we can rationalize the transition as follows . let @xmath162 be the excess in free energy of the confined system over a bulk undistorted state . in the br@xmath128 state @xmath163 with @xmath164 being the anchoring free energy due to the interaction with the walls and @xmath165 the contribution due to the domain walls . in the e state @xmath166 , with @xmath167 accounting for the elastic deformations of the director field , and @xmath168 for the disclination cores . @xmath164 is similar in both cases because the anchoring is satisfied in both states . @xmath165 is proportional to the length of the domain walls and hence to @xmath16 . @xmath168 does not depend on the size of the cavity and , finally , the elastic energy is @xcite @xmath169,\ ] ] where @xmath170 is the director field and @xmath171 and @xmath172 are the splay and bend elastic constants , respectively . for rods confined in a circular cavity , the elastic energy grows logarithmically with the radius of the cavity @xcite . here , we have computed numerically the divergence and the rotational of the director in the e state and we have found that the dependence of the elastic energy with the cavity size is also weak , increasing slower than linear in @xmath16 . on the other hand , in the br@xmath128 state , the size of the domain walls is proportional to the size of the cavity , and hence @xmath173 . therefore , for a fixed @xmath21 we expect the bridge state to be replaced by the elastic state at sufficiently big cavity sizes due to different dependence of @xmath162 with @xmath16 in both states . the increase of the packing fraction at the e - br@xmath128 can be understood given the behaviour of the elastic constants with the packing fraction ; both @xmath171 and @xmath172 monotonically increase with @xmath21 . in a very recent study @xcite garlea et al . have simulated a quasi - monolayer of hard spherocylinders confined in a square prism as well as the two - dimensional limit of discorectangles in a square cavity . the authors observe a state very similar to the elastic state in which the inner @xmath149 defect and its adjacent corner defect form a kind of line defect . actually , in our case , for the smaller cavities it is difficult to say whether those defects are actually two independent defects or whether they form a single structure . garlea et al . have also found smectic ordering in their simulations , but in contrast to our findings they did not observe domain walls at high packing fractions , although they state in @xcite `` ... we do sometimes observe particles trapped perpendicularly to the smectic layers , invariably next to the wall '' . the differences at high packing fractions between both systems are probably due to the slightly different geometry of the particles ( spherocylinders vs rectangles ) . in contrast to hard spherocylinders ( or discorectangles in two dimensions ) , hard rectangles posses degenerate close packing states and have a higher tendency to cluster @xcite . this may explain the presence of domain walls in a system of hard rectangles and its absence in a system of hard spherocylinders . it is unlikely that the dimensionality plays a dominant role because garlea et al . have studied both the quasi two dimensional system and the strict two - dimensional limit , and found no differences between them . confined in a square cavity with homeotropic anchoring : packing fraction - side length plane . the color map indicates the average of the uniaxial order parameter @xmath54 . empty circles mark the packing fraction at which @xmath66 . empty squares roughly indicate the elastic - bridge nematic transition . lines are guides to the eye . ] finally we investigate the confinement of rods in a square cavity that promotes homeotropic anchoring ( a schematic of the geometry is shown in fig . [ fig1]d ) . the state diagram and representative states for @xmath148 are shown in figs . [ fig13 ] and [ fig14 ] , respectively . here , as in the case of the planar cavity , the elastic state consists of particles aligned along one diagonal ( see fig . [ fig14]b ) . however , in contrast to the planar cell , the alignment of the particles leads to the formation of only two disclinations with topological charge @xmath174 ( see the drop of the uniaxial order parameter in panel b of fig . [ fig14 ] ) . the fluctuations in the position of the disclinations is high , much higher than in the planar cavity . this is most likely related to the dominant elastic deformations of the director involved in each disclination : splay - like deformations in the case of @xmath174 disclinations and bend - like in @xmath149 disclinations . as @xmath175 ( see e.g. , @xcite ) we expect more fluctuations in the positions of @xmath174 disclinations than in @xmath149 disclinations . as an example of the high fluctuations of the @xmath174 disclination cores , we show in fig . [ fig14]a ) a state where both disclinations have merged forming a single @xmath176 disclination . this state is a variation of the elastic state that we observe sometimes , especially at low densities . this configuration is metastable because it involves higher elastic deformations and the energy of one @xmath176 disclination core is higher than that of two @xmath174 disclination cores ( the energy of a disclination core increases with the square of its topological charge ) . by further increasing the density , we find a gradual transition from the elastic to the bridge nematic state ( fig . [ fig14]c ) . in the bridge nematic state there are three domains of particles with uniform director . in contrast to the planar cavity , here the bridge state is not accompanied by positional order because it appears at lower density ( compare the position of the elastic - bridge transition in the states diagrams of figs . [ fig11 ] and [ fig13 ] ) and the particles remain in a nematic state . another difference involves the domain walls that stay always at a distance of about one molecular length from the ( effective ) walls . the position of the domain walls fluctuates less than in the case of a planar cavity . at higher packing fractions the rods in the main domain form well - defined layers . we call this the bridge smectic state , b@xmath111 , where @xmath3 indicates the number of layers of the main domain . the number of smectic layers is well defined due to the stable positions of domain walls in the cell . the number of layers is the result of commensuration between the side length of the cavity and the smectic layer spacing . the approximate regions of the distinct b@xmath111 states are indicated in the state diagram of fig . [ fig12 ] .

we present results for the state diagram as a function of the packing fraction and the degree of confinement . under extreme confinement , unexpected states appear with lower symmetries than those of the corresponding stable states in bulk , such as the formation of states that break the anchoring constraints or the symmetry imposed by the surfaces . in both types of square cavities the particles form disclinations at intermediate densities . at high densities , however , the elastic stress is relaxed via the formation of domain walls where the director rotates abruptly by ninety degrees .

using monte carlo simulation , we analyse the behaviour of two - dimensional hard rods in four different types of geometric confinement : ( i ) a slit pore where the particles are confined between two parallel walls with homeotropic anchoring ; ( ii ) a hybrid slit pore formed by a planar and a homeotropic wall ; square cavities that frustrate the orientational order by imposing either ( iii ) homeotropic or ( iv ) planar wall anchoring . we present results for the state diagram as a function of the packing fraction and the degree of confinement . under extreme confinement , unexpected states appear with lower symmetries than those of the corresponding stable states in bulk , such as the formation of states that break the anchoring constraints or the symmetry imposed by the surfaces . in both types of square cavities the particles form disclinations at intermediate densities . at high densities , however , the elastic stress is relaxed via the formation of domain walls where the director rotates abruptly by ninety degrees .

1502.03583

c

in summary , we have performed a systematic analysis of the behaviour of two - dimensional hard rods confined in slit pores and in square cavities . in the case of slit pores we have shown that our simple hard core model contains much of the phenomenology observed in corresponding confined three dimensional systems . examples are the capillary nematization and smectization in homeotropic pores , and the formation of linear and step states that occurs in hybrid planar cells . in addition we have found new states that have not been experimentally observed or theoretically predicted . an example is the smectic c and the brush state that we have observed in homeotropic cells . both states break the anchoring imposed by the surfaces . the asymmetric brush state breaks also the symmetry of the cell . in all cases we have rationalized the stability by comparing the excess in free energy to the corresponding undistorted bulk phase . in recent experiments on vertically vibrated monolayers of rods confined in a circular cavity @xcite the same textures were found as mc studies predict for equilibrium rods @xcite . actually , the elastic state we have found in the square planar cavity has been observed in vibrated granular rods @xcite . granular materials flow and diffuse anomalously @xcite . although being non - thermal fluids , under certain circumstances such systems form steady states with the textures of thermal fluids . a comparison between mc simulation of confined rods ( thermal fluid ) and vibrated granular rods ( non - thermal fluid ) would help to find the analogies between both systems . the elastic state of the planar square cell has been also observed experimentally in confined actin filaments @xcite , colloidal particles @xcite , and predicted using onsager - like density functional theory @xcite . the authors of @xcite found , using experiments on confined colloids and oseen frank elastic theory , that the elastic state is metastable with respect to another state that contains two corner disclinations and is free of bulk disclinations ( diagonal state ) . in the diagonal state the total deformation of the director is higher than in the elastic state , but on the other hand , the diagonal state has no bulk disclinations . the total elastic energy decreases with the size of the cavity , and the energetic cost associated to a disclination is independent of cavity size . hence , we expect the diagonal state to replace the elastic state in our system for cavities much bigger than the ones studied here . the rate between the splay and bend elastic constant also plays a role determining the relative stability of the confined states . in @xcite the case of equal elastic constants is analysed , whereas in our system we expect the bend elastic constant to be much higher than the splay one . although we have analysed a two - dimensional model , our results may be of relevance to gaining a better understanding of three - dimensional systems where similar phenomenology has been already found . for example , capillary nematization @xcite and smectization @xcite have been studied in confined rods and platelets between two parallel walls . the hybrid cell has been also analysed in three dimensions @xcite , and phases with the same symmetry as those found here appear . our results indicate that other states , not observed yet in three - dimensional systems , can arise under extreme confinement . for example , states that break the anchoring , like the smectic c or the brush states found here , or states with symmetry breaking ( i.e. , asymmetric states in confined symmetric pores ) such as the asymmetric brush state . those states might be difficult to find in e.g. density functional studies where one typically assumes that the symmetry of the order parameter profiles is the same as the one imposed by the surfaces . it is interesting to compare the confinement of rods in square cavities , secs . [ sp ] and [ sh ] , with the recent study of confined rods in circular cavities @xcite . in both cases at high densities the system form domain walls in an attempt to reduce the elastic distortions of the director field . although the domain walls will probably disappear in larger cavities , they might be a general mechanism to reduce elastic stresses under extreme confinement . some of the states found in the slit - pore geometry show lateral ordering , such as for example the brush smectic states . for selected pore sizes , we have performed simulations varying the lateral size of the cell , @xmath177 from @xmath178 to @xmath179 and no differences have been found . we are , therefore , confident that the lateral ordering is not induced by the applied boundary conditions . nevertheless , monte carlo simulations in the isothermal isobaric ensemble ( npt ) might help to elucidate the role that the lateral size of the pore plays in the stability of such states . we have restricted the analysis to hard rectangles with length - to - width ratio of @xmath180 . we expect a similar phenomenology for particles with aspect ratio higher than @xmath181 because the bulk behaviour is qualitatively the same . however , for particles with shorter aspect ratios completely new phenomenology will presumably appear because states with tetratic correlations are stable in bulk @xcite and might modify the phase behaviour presented here .

using monte carlo simulation , we analyse the behaviour of two - dimensional hard rods in four different types of geometric confinement : ( i ) a slit pore where the particles are confined between two parallel walls with homeotropic anchoring ; ( ii ) a hybrid slit pore formed by a planar and a homeotropic wall ; square cavities that frustrate the orientational order by imposing either ( iii ) homeotropic or ( iv ) planar wall anchoring .

using monte carlo simulation , we analyse the behaviour of two - dimensional hard rods in four different types of geometric confinement : ( i ) a slit pore where the particles are confined between two parallel walls with homeotropic anchoring ; ( ii ) a hybrid slit pore formed by a planar and a homeotropic wall ; square cavities that frustrate the orientational order by imposing either ( iii ) homeotropic or ( iv ) planar wall anchoring . we present results for the state diagram as a function of the packing fraction and the degree of confinement . under extreme confinement , unexpected states appear with lower symmetries than those of the corresponding stable states in bulk , such as the formation of states that break the anchoring constraints or the symmetry imposed by the surfaces . in both types of square cavities the particles form disclinations at intermediate densities . at high densities , however , the elastic stress is relaxed via the formation of domain walls where the director rotates abruptly by ninety degrees .

1009.2140

i

the magnetic field is an essential ingredient in present - day star formation . observations have shown that molecular cloud cores have magnetic energy comparable to gravitational energy @xcite . the cloud cores also have angular momentum @xcite . stars are born in such cloud cores . as the cloud collapses , the central region of the collapsing cloud rotates rapidly , and this rapid rotation suppresses further collapse and subsequent protostar formation . in the collapsing cloud , however , the magnetic field can effectively transfer the excess angular momentum from the center of the cloud by magnetic braking ( and protostellar outflow ) , and promote further collapse and protostar formation . moreover , after the protostar formation , the magnetic field greatly contributes to the evolution of a protostar and a circumstellar disk . because the angular momentum in the circumstellar disk is transferred by magnetic effects , the gas in the circumstellar disk can fall onto the protostar , increasing the protostellar mass in the main accretion phase . thus , it may be expected that only a small - size ( or no ) disk appears around the protostar if the angular momentum of the disk is effectively transferred by magnetic effects . in various star - forming regions , however , many circumstellar disks with a size of several hundred au can be observed ( e.g. , @xcite ) , indicating that some of the angular momentum survives the magnetic braking catastrophe and contributes to the growth of the circumstellar disk . the formation and evolution of the circumstellar disk are important not only for star formation but also for planet formation . the size and mass of the circumstellar disk determine the mode of formation of a planet ( i.e. , core accretion or gravitational instability mode ; see @xcite ) . the angular velocity of the molecular cloud core that is the host cloud for star formation creates a circumstellar disk in the star formation process . thus , to understand the formation of the circumstellar disk , we need to investigate the evolution of the molecular cloud core from the prestellar stage . numerical simulations can reproduce the disk formation in the molecular cloud core . however , because there is a large difference in both spatial and time scales between the molecular cloud core and the circumstellar disk , we need special numerical techniques such as adaptive mesh refinement or nested grid and smoothed particle hydrodynamics . using these techniques , several studies have investigated the disk formation in the molecular cloud core from the prestellar stage . @xcite and @xcite investigated the formation of the protostar and circumstellar disk in an unmagnetized cloud ; they showed that the first core that forms before protostar formation in the gas - collapsing phase directly evolves into a circumstellar disk after the protostar formation . @xcite and @xcite showed that the circumstellar disk , which originates from the first core , is more massive than or comparable to the protostar in the main accretion phase and has a size of several hundred au when the host cloud is unmagnetized . using numerical simulations , the disk formation in magnetized clouds has also been investigated . these studies reported that the calculated size of the circumstellar disk appearing in a magnetized cloud is much smaller than the observed sizes . using an ideal mhd calculation , @xcite showed that the circumstellar disk formed in a magnetized cloud has a size of @xmath3au at maximum . @xcite also showed that large - size disk formation is suppressed in a magnetized cloud . this is because the angular momentum of the circumstellar disk ( or infalling gas onto the circumstellar disk ) is excessively transferred outward by magnetic braking , and the gas in the disk effectively falls onto the protostar . in reality , however , the magnetic field dissipates in a high - density gas region by ohmic dissipation and ambipolar diffusion , because the degree of ionization is considerably low ( e.g. , @xcite ) . such a high - density gas region , where the neutral gas is decoupled from the magnetic field , corresponds well to the disk - forming region in the early main accretion phase ( @xmath4au , @xcite ) . thus , if the magnetic field can sufficiently dissipate around the protostar , magnetic braking becomes ineffective and a large - size circumstellar disk comparable to observations may appear even in strongly magnetized clouds . in contrast , with non ideal mhd calculations , @xcite and @xcite reported that the disk size is considerably smaller than observations even when the magnetic field dissipates in the high - density gas region , although the disk in non ideal mhd calculations is somewhat larger than that in ideal mhd calculations . these studies , however , overlooked the effect of the finite mass of the host cloud for star and circumstellar disk formation . in the early main accretion phase , a massive gas envelope ( or remnant of the molecular cloud ) remains around the circumstellar disk . the massive envelope rotates much slower than the circumstellar disk , and thus it can brake the circumstellar disk through the magnetic field lines that connect the less massive rapidly rotating disk to the more massive slowly rotating envelope . in other words , the angular momentum of the circumstellar disk is transferred into the infalling envelope by torsional alfv@xmath5n waves . this is the magnetic braking mechanism for the circumstellar disk formation . thus , even with very strong magnetic fields , the circumstellar disk rotation velocity does not decrease when no massive envelope exists around the disk . the mass of the infalling envelope decreases with time in the main accretion phase when an isolated molecular cloud core is assumed . the infalling envelope disappears by the end of the main accretion phase : a part of the envelope is blown away from the cloud by the protostellar outflow @xcite , and the remainder falls onto either the circumstellar disk or the protostar . thus , we can expect that magnetic braking becomes gradually ineffective as the mass of the infalling envelope decreases , and it rarely works when the mass of the circumstellar disk is much larger than that of the infalling envelope . note that , in reality , magnetic braking can work even after the infalling envelope disappears , because the low - density interstellar medium can receive angular momentum from the circumstellar disk . also note that because the efficiency of magnetic braking decreases as the ambient density decreases , the angular momentum of the circumstellar disk is not effectively transferred into the low - density interstellar medium ( see , [ sec : diss ] ) . in addition , in an isolated cloud core , although magnetic braking can redistribute the angular momentum inside the cloud , it can not effectively transfer the angular momentum outside the cloud ( or into the interstellar medium ) . the infalling envelope receives the angular momentum from the circumstellar disk by magnetic braking , while it finally falls onto the circumstellar disk with the angular momentum by the end of the main accretion phase . however , when the infalling envelope is depleted and magnetic braking becomes ineffective , no sufficiently large disk appears if insufficient mass to form the disk remains in the infalling envelope . in summary , to investigate the real size and mass of the circumstellar disk , we have to investigate the disk evolution in the collapsing cloud core , at least till the end of the main accretion phase ( or until the infalling gas disappears ) . in previous studies , the calculation was stopped much before the infalling envelope disappears . thus , it is natural that magnetic braking is effective and no large - size disk appears during this phase ( i.e. , early main accretion phase ) . in this study , however , we calculated the evolution of the circumstellar disk until almost all gas of the infalling envelope disappears and found that a large - size disk comparable to observations appears when the infalling envelope becomes less massive than the circumstellar disk . the structure of this paper is as follows . the framework of our models and the numerical method are given in 2 . the numerical results are presented in 3 . we discuss the magnetic braking timescale and the effect of the infalling envelope on the circumstellar disk formation in 4 .

, however , the circumstellar disk grows rapidly and exceedsau by the end of the main accretion phase . this rapid growth of the circumstellar disk is caused by the depletion of the infalling envelope , while magnetic braking is effective when the infalling envelope is more massive than the circumstellar disk . the infalling envelope can not brake the circumstellar disk when the latter is more massive than the former .

using resistive magnetohydrodynamics simulation , we investigate circumstellar disk formation in a strongly magnetized cloud . as the initial state , an isolated cloud core embedded in a low - density interstellar medium with a uniform magnetic field is adopted . the cloud evolution is calculated until almost all gas inside the initial cloud falls onto either the circumstellar disk or a protostar , and a part of the gas is ejected into the interstellar medium by the protostellar outflow driven by the circumstellar disk . in the early main accretion phase , the disk size is limited toau because the angular momentum of the circumstellar disk is effectively transferred by both magnetic braking and the protostellar outflow . in the later main accretion phase , however , the circumstellar disk grows rapidly and exceedsau by the end of the main accretion phase . this rapid growth of the circumstellar disk is caused by the depletion of the infalling envelope , while magnetic braking is effective when the infalling envelope is more massive than the circumstellar disk . the infalling envelope can not brake the circumstellar disk when the latter is more massive than the former . in addition , the protostellar outflow weakens and disappears in the later main accretion phase , because the outflow is powered by gas accretion onto the circumstellar disk . although the circumstellar disk formed in a magnetized cloud is considerably smaller than that in an unmagnetized cloud , a circumstellar disk exceedingau can form even in a strongly magnetized cloud .

1009.2140

c

in this paper , to investigate the formation of the circumstellar disk in a realistic setting , we set a dense molecular cloud core embedded in a low - density interstellar medium in a large simulation box with size 16 times the radius of the molecular cloud core , in which a uniform magnetic field parallel to the rotation axis is adopted . in the gravitationally collapsing cloud , it is expected that the disk formation is suppressed by magnetic braking which effectively transfers the angular momentum from the circumstellar disk , as described in past studies @xcite . in reality , our results indicate that the circumstellar disk formed in a magnetized cloud is much smaller than that in an unmagnetized cloud ( see [ sec : mags ] ) . however , we also showed that the circumstellar disk exceeding @xmath240au can form even in a strongly magnetized cloud ( @xmath253 ) , which contradicts previous studies . the past studies predicted that magnetic braking suppresses the disk formation and limits the size of the circumstellar disk within @xmath3au . this inconsistency is attributed to the initial setting of the host cloud and the calculation time after the circumstellar disk formation . in this study , we calculated the cloud evolution until almost all gas inside the initial cloud falls onto either the circumstellar disk or the protostar . in other words , we execute the calculation until the freefall timescale at the cloud boundary is exceeded . as described in [ sec : intro ] , as the cloud mass is depleted , magnetic braking is expected to be ineffective because the less massive infalling envelope can not brake the circumstellar disk . in [ sec : results ] , we numerically showed that magnetic braking weakens with time and a large - size circumstellar disk can form . in this section , we analytically discuss the timescale of magnetic braking and disk formation . the angular momentum of the cloud or disk is transferred into the external medium through a torsional alfv@xmath5n wave . magnetic braking becomes effective when the angular momentum transferred into the external medium is comparable to that of the rotator ( e.g. , the molecular cloud or disk ) . in other words , magnetic braking becomes effective when the moment of inertia of the gas through which the alfv@xmath5n waves have propagated becomes comparable to the moment of inertia of the cloud or disk . with this condition , the magnetic braking timescale can be described as @xmath254 where @xmath255 , @xmath256 , @xmath257 and @xmath258 are the half thickness of the cloud ( or the disk ) , alfv@xmath5n velocity in the external medium , cloud ( or disk ) density , and density of the external medium , respectively @xcite . thus , magnetic braking becomes effective when a typical timescale of the system ( @xmath259 ) is longer than the magnetic braking timescale @xmath260 . with equation ( [ eq : mb ] ) , the condition for magnetic braking becoming effective is described as @xmath261 the left - hand side of equation ( [ eq : mb2 ] ) corresponds to the line density of the external medium in the range at which the alfv@xmath5n wave can reach in @xmath259 , while the right - hand side corresponds to the line density of the cloud or disk . equation ( [ eq : mb2 ] ) indicates that magnetic braking is effective only when the mass of the external medium is greater than the mass of the rotator . in other words , magnetic braking does not take place when sufficient mass does not exit around the rotator . this is natural because a less massive external medium can not brake a massive object . this condition can be also understood from equation ( [ eq : mb ] ) . when an uniform magnetic field is assumed , equation ( [ eq : mb ] ) indicates that the magnetic braking timescale is proportional to @xmath262 . note that the alfv@xmath5n speed is proportional to @xmath263 . thus , the magnetic braking timescale becomes longer as the external density decreases . for example , when the external density is extremely low , magnetic braking does not take place . we estimate the magnetic braking timescale for model 1 ( initially the strongest magnetic field model ) . first , we estimate the magnetic - braking timescale of the molecular cloud core embedded in the interstellar medium . for model 1 , the density contrast between the center of the cloud core and the external medium is 82 at the initial state , and alfv@xmath5n speed in the external mediums is @xmath264kms@xmath15 . thus , with equation ( [ eq : mb ] ) , the magnetic braking timescale becomes @xmath265yr , which is much longer than the freefall timescale of the cloud ( @xmath266yr ) or our calculation time ( @xmath267yr ) . thus , it is expected that the cloud rarely loses its angular momentum by magnetic braking of the interstellar medium . in [ sec : typical ] , we showed that the total angular momentum of the cloud core is rarely decreased through the calculation . next , we estimate the magnetic - braking timescale of the circumstellar disk . first , we assume that the circumstellar disk exists only in the low - density interstellar medium ( thus , not including the infalling envelope ) to simply estimate the effect of the low - density interstellar medium outside the gravitational sphere . the density contrast between the circumstellar disk ( @xmath268 ) and interstellar medium ( @xmath269 ) is about @xmath270 , and the circumstellar disk has a scale height of @xmath271au . with these parameters , the magnetic braking timescale of the circumstellar disk is @xmath272yr . thus , the interstellar medium can not brake the circumstellar disk in our setting . note that , in this study , we adopted a somewhat higher density of the interstellar medium ( @xmath269 ) than the observed value of the interstellar medium to limit the alfv@xmath5n speed in the interstellar medium , because it is difficult to calculate the cloud evolution over a long time with a large alfv@xmath5n speed . thus , the magnetic braking timescale in the actual interstellar medium is longer than in our setting with a lower external density , because it is proportional to @xmath263 . in summary , magnetic braking from the low - density interstellar medium is completely ignored in the formation and evolution of the circumstellar disk . on the other hand , the infalling envelope in the region of @xmath273 can brake the circumstellar disk in the main accretion phase . the torsional alfv@xmath5n wave generated by the center of the cloud can reach the cloud boundary ( @xmath274au ) in @xmath275yr ( @xmath276 ) , where the alfv@xmath5n speed of the external medium @xmath264kms@xmath15 is adopted . note that , in reality , because the magnetic field is amplified , the torsional alfv@xmath5n wave can reach the cloud boundary in several freefall timescales . thus , when the infalling envelope is more massive than the circumstellar disk , the angular momentum of the circumstellar disk is effectively transferred by magnetic braking . in the main accretion phase , however , the mass of the infalling envelope decreases with time . thus , it is expected that magnetic braking gradually becomes ineffective with time . however , it is difficult to estimate the magnetic braking timescale of the infalling envelope quantitatively , because the infalling envelope drastically changes its density distribution in a several free - fall timescale that corresponds to the timescale for the circumstellar disk formation . thus , the disk formation process and final size of the circumstellar disk can not be investigated using a simple analytical framework . only a less - massive and small - size circumstellar disk may appear in a magnetized cloud when magnetic braking significantly transfers the angular momentum by the end of the main accretion phase . in this study , however , we calculated the cloud evolution by the end of the main accretion phase and showed that the circumstellar disk exceeding @xmath240au can form even in a strongly magnetized cloud . in a magnetized cloud , magnetic braking becomes ineffective and the disk size increases rapidly in the later main accretion phase . it is natural that magnetic braking only redistributes the angular momentum from the circumstellar disk into the infalling envelope . because the infalling envelope finally falls onto the circumstellar disk , the excess angular momentum gives rise to a rotation supporting the disk in the later main accretion phase . instead of magnetic braking , only the protostellar outflow can transfer the angular momentum to the interstellar medium , because the gas with a certain amount of angular momentum is directly ejected into the interstellar medium from the circumstellar disk by the protostellar outflow . because the protostellar outflow is powered by mass accretion onto the circumstellar disk , the outflow disappears in the later main accretion phase , which also corresponds to rapid growth of the circumstellar disk . thus , the circumstellar disk begins to grow after the protostellar outflow sufficiently weakens or disappears . in addition , in the later main accretion phase , the spiral structure appearing in the circumstellar disk can contribute to the redistribution of the angular momentum in the circumstellar disk . thus , it is difficult to specify which mechanism is effective for the angular momentum transfer in a magnetized collapsing cloud . however , the large - size disk comparable to observations can form in a strongly magnetized cloud . in this study , we mainly investigated the evolution of the collapsing cloud with a mass of @xmath277 in the initial state and found that the circumstellar disk with a size of several hundred au appears in the later main accretion phase . on the other hand , we can not calculate the cloud evolution for an initially massive cloud until the infalling envelope disappears . however , a massive cloud has a larger angular momentum ( or larger centrifugal radius ) . thus , a circumstellar disk much larger than @xmath240au may form in an initially massive cloud . finally , we comment on the geometry of the initial magnetic field . in this study , we assumed the magnetic field lines to be parallel to the rotation axis . with ideal mhd calculation , @xcite already reported the formation of the circumstellar disk when the magnetic field lines are inclined from the rotation axis . thus , we need to investigate the inclined field case with non - ideal mhd effects in the future . numerical computations were carried out on nec sx-9 at center for computational astrophysics , cfca , of national astronomical observatory of japan , and nec sx-8 at the yukawa institute computer facility . this work was supported by the grants - in - aid from mext ( 20540238 , 21740136 ) . ) for models with different initial magnetic field strengths @xmath56 or different cloud masses are plotted against the initial cloud mass ( left panel ) and cloud radius ( right panel ) in the upper panels . the centrifugal radii ( @xmath279 ) for different angular velocities @xmath57 or different cloud masses are plotted in the lower panels . , width=529 ] but on the @xmath186 plane . in each panel , the black line corresponds to the circumstellar disk . the boundary between outflowing ( @xmath280 ) and inflowing ( @xmath281 ) gas is described by the white line ; inside this boundary , the gas is outflowing from the circumstellar disk . , width=566 ] on the @xmath186 plane at the same epochs as figs . [ fig:3 ] and [ fig:5 ] except for panel _ a _ ( initial state ) . the white broken line is the boundary between the gravitational sphere and interstellar medium ; gravity ( gas self - gravity and the gravity of the protostar ) is imposed inside this boundary . , width=566 ] on the @xmath186 plane at the same epochs as figures [ fig:3 ] , [ fig:5 ] and [ fig:6 ] . the toroidal component ( @xmath282 ) is greater than the poloidal component ( @xmath283 ) inside the black line . the white line corresponds to the contour of @xmath284 . the arrows indicate the magnetic field direction on the @xmath186 plane . , width=566 ] against the cloud mass . the dashed line is the distribution of the angular momentum for model 5 at @xmath285yr . the initial distribution of the angular momentum is plotted by a thin solid line . the angular momentum and mass are integrated in the region of @xmath286 . the sum of the protostellar mass @xmath134 and circumstellar disk mass @xmath135 of each epoch is plotted by the diamond symbol ( @xmath287 ) on each line . the region on the left - hand side of the diamond symbol corresponds to the circumstellar disk region . , width=566 ] plane for the resistive ( model 1 ; upper panel ) and ideal ( model 4 ; lower panel ) mhd models . the thick white line is the contour of @xmath288 . the arrows indicate the magnetic field direction on the @xmath186 plane . , width=529 ]

using resistive magnetohydrodynamics simulation , we investigate circumstellar disk formation in a strongly magnetized cloud . as the initial state , an isolated cloud core embedded in a low - density interstellar medium with a uniform magnetic field the cloud evolution is calculated until almost all gas inside the initial cloud falls onto either the circumstellar disk or a protostar , and a part of the gas is ejected into the interstellar medium by the protostellar outflow driven by the circumstellar disk . in the early main accretion phase , the disk size is limited toau because the angular momentum of the circumstellar disk is effectively transferred by both magnetic braking and the protostellar outflow . in the later main accretion phase in addition , the protostellar outflow weakens and disappears in the later main accretion phase , because the outflow is powered by gas accretion onto the circumstellar disk . although the circumstellar disk formed in a magnetized cloud is considerably smaller than that in an unmagnetized cloud , a circumstellar disk exceedingau can form even in a strongly magnetized cloud .

using resistive magnetohydrodynamics simulation , we investigate circumstellar disk formation in a strongly magnetized cloud . as the initial state , an isolated cloud core embedded in a low - density interstellar medium with a uniform magnetic field is adopted . the cloud evolution is calculated until almost all gas inside the initial cloud falls onto either the circumstellar disk or a protostar , and a part of the gas is ejected into the interstellar medium by the protostellar outflow driven by the circumstellar disk . in the early main accretion phase , the disk size is limited toau because the angular momentum of the circumstellar disk is effectively transferred by both magnetic braking and the protostellar outflow . in the later main accretion phase , however , the circumstellar disk grows rapidly and exceedsau by the end of the main accretion phase . this rapid growth of the circumstellar disk is caused by the depletion of the infalling envelope , while magnetic braking is effective when the infalling envelope is more massive than the circumstellar disk . the infalling envelope can not brake the circumstellar disk when the latter is more massive than the former . in addition , the protostellar outflow weakens and disappears in the later main accretion phase , because the outflow is powered by gas accretion onto the circumstellar disk . although the circumstellar disk formed in a magnetized cloud is considerably smaller than that in an unmagnetized cloud , a circumstellar disk exceedingau can form even in a strongly magnetized cloud .

0706.2473

c

we use the simple population synthesis to model the stellar contributions in double - peaked sdss agns . the reliable stellar velocity dispersions are obtained for 52 medium - luminous double - peaked sdss agns with obvious stellar features . we find that : 1 ) the black hole mass is from @xmath173 to @xmath174 and the eddington ratio is from about 0.01 to about 1 ; 2 ) the factor @xmath10 far deviates from the virialized value 0.75 , suggesting the non - virial dynamics of blrs ; 3 ) the peak separation is mildly correlated with the eddington ratio and smbh mass with almost the same correlation coefficients , which can be interpreted in the doppler shift of thin annulus of blrs created by gravitational instability ; 4 ) based on the line - emitting accretion disk model , we need external illumination of the accretion disk to produce the observed strength of h@xmath11 line . in the future , using different models , we would fit the double - peaked profiles to constrain the nature of double - peaked agns . we can also use the double - peaked agns to constrain the blrs origin ( nicastro 2000 ; laor 2003 ; bian & gu 2007 ) .

the stellar velocity dispersions ( ) are obtained for 52 double - peaked agns with obvious stellar absorption features , ranging from 106 to 284 ^-1 km s . the black hole masses are calculated to range from to , and the eddington ratio from about 0.01 to about 1 . we find that far deviates from 0.75 , suggesting the non - virial dynamics of broad line regions . the peak separation is mildly correlated with the eddington ratio and smbh mass with almost the same correlation coefficients . it implies that it is difficult to detect obvious double - peak agns with higher eddington ratios . using the monochromatic luminosity at 5100 to trace the bolometric luminosity , we find that the external illumination of the accretion disk is needed to produce the observed strength of h emission line .

using the stellar population synthesis , we model the stellar contribution for a sample of 110 double - peaked broad - lines agns from the sloan digital sky survey ( sdss ) . the stellar velocity dispersions ( ) are obtained for 52 double - peaked agns with obvious stellar absorption features , ranging from 106 to 284 ^-1 km s . we also use multi - component profiles to fit$] and h emission lines . using the well - established relation , the black hole masses are calculated to range from to , and the eddington ratio from about 0.01 to about 1 . comparing with the known relation , we can get the factor , which indicates blrs geometry , inclination and kinematics . we find that far deviates from 0.75 , suggesting the non - virial dynamics of broad line regions . the peak separation is mildly correlated with the eddington ratio and smbh mass with almost the same correlation coefficients . it implies that it is difficult to detect obvious double - peak agns with higher eddington ratios . using the monochromatic luminosity at 5100 to trace the bolometric luminosity , we find that the external illumination of the accretion disk is needed to produce the observed strength of h emission line .

1511.04705

c

we have investigated , by means of numerical transfer - matrix approach in the angular - momentum space , the effects of the skew - interlayer hopping integrals ( the trigonal warping ) on selected transport characteristics of bilayer - graphene ( blg ) corbino disks . additionally , the analytical mode - matching for an artificial ( nanotube - like ) system , formed of a blg strip upon applying the periodic boundary conditions , was briefly presented and the analogies between these two systems were put forward . if the fermi energy is close to the charge - neutrality point , both the scaling behavior at zero magnetic field ( which would require a comparison between devices of different sizes in an experimental study ) and the single - device magnetotransport discussion unveils several phenomena , in which transport characteristics , such as the conductivity , the fano factor , and the third charge - transfer cumulant , are noticeably affected by the trigonal warping . in the pseudodiffusive transport regime , corresponding to the disk radii ratios @xmath276 , the conductivity shows a one - parameter scaling , in agreement with predictions of ref . @xcite for a rectangular sample . in the corbino geometry , however , the role a crystallographic orientation is eliminated , and the zero - field minimal conductivity can be approximated by @xmath277,\ ] ] where @xmath278 depends only on the skew - interlayer hopping @xmath6 , and varies from @xmath279 nm for @xmath90ev to @xmath280 nm for @xmath113ev . in the uniform magnetic field @xmath42 , the conductivity increases reaching the maximal value @xmath281 near the so - called lifshitz field @xmath253 , for which the magnetic length follows the relation @xmath282 where @xmath283 is defined by @xmath6 and other microscopic parameters : the lattice spacing @xmath284 as well as the nearest - neighbor intra- and interlayer hoppings @xmath4 and @xmath5 . above @xmath253 , the conductivity gradually decreases showing the asymptotic behavior @xmath285 [ see eq . ( [ eq : przewbc ] ) ] . the second and third charge transfer cumulants stay close to their pseudodiffusive values ( @xmath266 , @xmath267 ) when varying the system size or the magnetic field . in the opposite , quantum - tunneling regime ( corresponding to @xmath286 ) , the charge - transfer characteristics are also sensitive to @xmath6 . at zero field and @xmath14 , the transport is governed by two quantum channels with angular momenta @xmath2 . for @xmath15 , the backscattering is enhanced for these channels , as the related wavefunctions no longer match the symmetry of the low - energy hamiltonian . at high magnetic fields , all the charge - transfer characteristics show quasiperiodic beating patterns , with the envelope period @xmath287 [ see eq . ( [ eq : okrcor ] ) ] . most remarkably , the beating patterns , triggered by the trigonal - warping ( @xmath15 ) , remain well - pronounced in the landau quantization regime . ( unlike the average conductivity enhancement , which is usually eliminated by a few tesla field . ) it seems this finite - system version of the lifshitz transition can be related to numerous phenomena appearing in different branches of physics , starting from semiconducting heterostructures @xcite , via strongly - correlated electron systems @xcite , to neutrino physics @xcite , in which scattering the particles between quantum states with different effective masses leads to oscillations in relevant counting statistics , although this time the interference occurs between the evanescent waves . we stress here that finding the lifshitz field @xmath253 , via the asymptotic behavior of the conductivity , or via the beating period , may allow one at least in principle to determine the value of @xmath6 from a single - device magnetotransport measurement . apart from the possible verification of tight - binding parameters in blg hamiltonian , we believe the effects we describe , when confirmed experimentally , will provide a thorough insight into the interplay between massless- and massive - chiral states ruling the quantum transport through blg devices near the charge - neutrality point . for instance , the conductivity enhancement for @xmath288 may dominate the signatures of interaction - related magnetic catalysis phenomenon @xcite ( particularly in finite - size systems ) , and one should precisely distinguish single- and many - body aspects when searching for this intriguing phenomenon in blg . as we have focused on clean ballistic systems , several factors which may modify the transport properties of graphene - based devices , including the disorder @xcite , lattice defects @xcite , or magnetic impurities @xcite , are beyond the scope of this work . some experimental @xcite and numerical @xcite findings suggest that charge - transfer characteristics in the pseudodiffusive transport regime are quite robust against such factors . for the opposite , quantum - tunneling regime , we put forward the following reasoning : in the presence of trigonal warping , the rotational symmetry supposed in earlier studies of mlg @xcite or blg @xcite disks , no longer applies . in spite of this fact , the basic oscillation period [ @xmath289 , in terms of magnetic flux piercing the disk area ] remains unaltered , allowing one to believe that oscillations and beating patterns would appear in a more general situation as well .

using the transfer matrix in the angular - momentum space we investigate the impact of trigonal warping on magnetotransport and scaling properties of a ballistic bilayer graphene in the corbino geometry . although the conductivity at the charge - neutrality point and zero magnetic field exhibits a one - parameter scaling , the shot - noise characteristics , quantified by the fano factor and the third charge - transfer cumulant , remain pseudodiffusive . this shows that the pseudodiffusive transport regime in bilayer graphene is not related to the universal value of the conductivity but can be identified by higher charge - transfer cumulants . for corbino disks with larger radii ratios the conductivity is suppressed by the trigonal warping , mainly because the symmetry reduction amplifies backscattering for normal modes corresponding to angular - momentum eigenvalues . weak magnetic fields enhance the conductivity , reaching the maximal value near the crossover field^{-1}$ ] , where ( ) is the nearest - neighbor intra- ( inter- ) layer hopping integral , is the skew - interlayer hopping integral , and ( ) is the outer ( inner ) disk radius . for magnetic fields we observe quasiperiodic conductance oscillations characterized by the decreasing mean value , where . the conductivity , as well as higher charge - transfer cumulants , show beating patterns with an envelope period proportional to .

using the transfer matrix in the angular - momentum space we investigate the impact of trigonal warping on magnetotransport and scaling properties of a ballistic bilayer graphene in the corbino geometry . although the conductivity at the charge - neutrality point and zero magnetic field exhibits a one - parameter scaling , the shot - noise characteristics , quantified by the fano factor and the third charge - transfer cumulant , remain pseudodiffusive . this shows that the pseudodiffusive transport regime in bilayer graphene is not related to the universal value of the conductivity but can be identified by higher charge - transfer cumulants . for corbino disks with larger radii ratios the conductivity is suppressed by the trigonal warping , mainly because the symmetry reduction amplifies backscattering for normal modes corresponding to angular - momentum eigenvalues . weak magnetic fields enhance the conductivity , reaching the maximal value near the crossover field^{-1}$ ] , where ( ) is the nearest - neighbor intra- ( inter- ) layer hopping integral , is the skew - interlayer hopping integral , and ( ) is the outer ( inner ) disk radius . for magnetic fields we observe quasiperiodic conductance oscillations characterized by the decreasing mean value , where . the conductivity , as well as higher charge - transfer cumulants , show beating patterns with an envelope period proportional to . this constitutes a qualitative difference between the high - field ( ) magnetotransport in the case ( earlier discussed in ref . ) and in the case , providing a finite - system analog of the lifshitz transition . = 1

1010.4673

i

this article is about the properties of a quantum statistical ensemble , which is , in a sense , opposite to the conventional micro - canonical ensemble . it is an extension of the previous work of one of us@xcite . the present work contains a discussion of underlying conceptual issues and detailed analytical and numerical investigations of finite quantum systems . these two aspects of the article are complementary but otherwise quite different in scope and style . sections [ narrow ] and [ prospects ] are dealing with conceptual issues . they motivate and outline a broader agenda . the remaining sections ( [ qmc]-[spins ] plus appendices ) are more specialized . we hope that the conceptual part would motivate the readers unfamiliar with the above agenda to look into the rest of the article .

usual approach to the foundations of quantum statistical physics is based on conventional micro - canonical ensemble as a starting point for deriving boltzmann - gibbs ( bg ) equilibrium . it leaves , however , a number of conceptual and practical questions unanswered . here we discuss these questions , thereby motivating the study of a natural alternative known as quantum micro - canonical ( qmc ) ensemble .

usual approach to the foundations of quantum statistical physics is based on conventional micro - canonical ensemble as a starting point for deriving boltzmann - gibbs ( bg ) equilibrium . it leaves , however , a number of conceptual and practical questions unanswered . here we discuss these questions , thereby motivating the study of a natural alternative known as quantum micro - canonical ( qmc ) ensemble . we present a detailed numerical study of the properties of the qmc ensemble for finite quantum systems revealing a good agreement with the existing analytical results for large quantum systems . we also propose the way to introduce analytical corrections accounting for finite - size effects . with the above corrections , the agreement between the analytical and the numerical results becomes very accurate . the qmc ensemble leads to an unconventional kind of equilibrium , which may be realizable after strong perturbations in small isolated quantum systems having large number of levels . we demonstrate that the variance of energy fluctuations can be used to discriminate the qmc equilibrium from the bg equilibrium . we further suggest that the reason , why bg equilibrium commonly occurs in nature rather than the qmc - type equilibrium , has something to do with the notion of quantum collapse .

1010.4673

c

in conclusion , we have demonstrated that the large-@xmath29 analytical description of the qmc ensemble developed in ref.@xcite allows one to make good predictions for the monte - carlo sampling of the qmc ensemble for finite-@xmath29 systems . the description of ref.@xcite is amenable to finite-@xmath29 corrections , and once these corrections are introduced , the theory produces accurate quantitative agreement with the monte - carlo results for systems with @xmath217 . we have also studied numerically the implications of the qmc ensemble for a subsystem of a small spin system . our results indicate that , already in this rather artificial case , the behavior of subsystem s density matrix qualitatively follows the analytical result for the large-@xmath29 case . we have further suggested that the subsystem energy variance , as well as the average value of subsystem s energy itself , can be used to discriminate experimentally between the qmc and the bg statistics . in the context of foundations of quantum statistical physics , the apparent impossibility of obtaining the boltzmann - gibbs equilibrium from the qmc ensemble indicates that one should be looking closer at the basis and the applicability limit of the narrow energy window condition of the conventional micro - canonical ensemble . in this respect , quantum collapse may play a fundamental role . _ note added : _ as we were finishing editing this manuscript for resubmission , we discovered a series of papers by fresch and moro@xcite very closely related to the scope of the present work and ref.@xcite . in particular , in ref.@xcite , the above authors have pursued similar monte - carlo sampling of the qmc ensemble in finite dimensional hilbert spaces . their results appear to be consistent with and , in some aspects , complementary to ours . in particular , they have used metropoilis - hastings algorithm , as opposed to the direct sampling routine of our paper , and thereby were able to explore significantly larger hilbert spaces . on the other hand , the present paper advances further the subject of finite-@xmath29 corrections to the large-@xmath29 qmc results . we have also learned from ref.@xcite that , already in 1990 , wootters have considered the qmc ensemble and arrived to what we call the small-@xmath46 approximation@xcite .

we demonstrate that the variance of energy fluctuations can be used to discriminate the qmc equilibrium from the bg equilibrium . we further suggest that the reason , why bg equilibrium commonly occurs in nature rather than the qmc - type equilibrium , has something to do with the notion of quantum collapse .

usual approach to the foundations of quantum statistical physics is based on conventional micro - canonical ensemble as a starting point for deriving boltzmann - gibbs ( bg ) equilibrium . it leaves , however , a number of conceptual and practical questions unanswered . here we discuss these questions , thereby motivating the study of a natural alternative known as quantum micro - canonical ( qmc ) ensemble . we present a detailed numerical study of the properties of the qmc ensemble for finite quantum systems revealing a good agreement with the existing analytical results for large quantum systems . we also propose the way to introduce analytical corrections accounting for finite - size effects . with the above corrections , the agreement between the analytical and the numerical results becomes very accurate . the qmc ensemble leads to an unconventional kind of equilibrium , which may be realizable after strong perturbations in small isolated quantum systems having large number of levels . we demonstrate that the variance of energy fluctuations can be used to discriminate the qmc equilibrium from the bg equilibrium . we further suggest that the reason , why bg equilibrium commonly occurs in nature rather than the qmc - type equilibrium , has something to do with the notion of quantum collapse .

1412.1086

i

several recent experiments @xcite have provided strong evidence for a dramatic change in the nature of the low temperature electronic state of the hole - doped cuprate superconductors near optimal doping ( @xmath0 ) . moreover , zero field photoemission experiments carried out in the normal state have seen evidence for a ` large ' fermi - surface for @xmath1 , consistent with the overall luttinger count @xcite , and disconnected fermi ` arcs ' near the nodal regions for @xmath2 @xcite . at high fields , quantum oscillations also reveal a ` large ' fermi - surface for @xmath1 @xcite , but a closed electron - like fermi - surface with an area that constitutes a small fraction of the entire brillouin - zone for @xmath2 @xcite . it is therefore quite natural to associate the transition with decreasing @xmath3 at @xmath0 with the loss of a ` large ' fermi - surface and the simultaneous opening of a pseudogap . there has also been significant experimental progress @xcite in understanding the structure of the density - wave ordering at lower doping , which is likely responsible for the reconstructed electron - like fermi - surface seen in quantum oscillation experiments @xcite . in this paper we will use these advances to motivate and develop a previously proposed model @xcite for the physics of the strange metal near optimal doping . we argue that the rich phenomenology observed in the underdoped cuprates is primarily driven by a transition between non - fermi liquid metals with large and small fermi surfaces which does not directly involve any broken global symmetry . all states with broken symmetry observed at low temperatures and low doping are not part of the critical field theory @xcite , but are derived as low energy instabilities of the parent small fermi surface phase . this diminished role for broken symmetries is consistent with absence of any observed order with a significant correlation length at higher temperatures . we will also construct a global phase diagram to describe the many phases and crossovers around the strange metal . a quantum phase transition which does not involve broken symmetries is necessarily associated with a _ topological _ change in the character of the ground state wavefunction . emergent gauge fields are a powerful method of describing this topological structure , and they remain applicable also to the gapless metallic phases of interest to us here . given the fundamental connection between emergent gauge fields and the size of the fermi surface , which was established in ref . using oshikawa s method @xcite , we are naturally led to a quantum phase transition in which there is a change in the structure of the deconfined gauge excitations . indeed , this describes a higgs transition in a metal , such as that discussed in ref . . this argument is a general motivation for higgs criticality near optimal doping in the cuprates , which applies beyond the specific model considered here . we emphasize that we are using the traditional particle - physics terminology in which a `` higgs transition '' describes the breaking of a local gauge invariance . we are not referring to the longitudinal mode of a broken global symmetry , which has also been labeled `` higgs '' in condensed matter contexts @xcite . the primary new motivation for the model of ref . arises from our recent work @xcite analyzing the @xmath4-form factor density waves observed in scanning tunnelling microscopy @xcite and x - ray experiments @xcite . in this work @xcite , we argued that such density waves arise most naturally as an instability of a metallic higher temperature pseudogap state with small fermi surfaces described as a @xcite ` fractionalized fermi liquid ' ( fl * ) ; other works with related ideas on the pseudogap are refs . . specifically , we used a theory of the fl * involving a background u(1 ) spin liquid with bosonic spinons @xcite : it is therefore convenient to dub this metallic state for the pseudogap as a u(1)-fl*. these results are also easily extended to a @xmath5 spin liquid , and we will consider this case in appendix [ app : spiral ] . the presence of a small fermi surface without symmetry breaking requires topological order and emergent gauge fields @xcite , and so also a higgs transition to the large fermi surfaces at larger doping : here we provide a natural embedding of a fl * theory into such a transition , and we expect similar approaches are possible for other possible topological orders in the underdoped regime . we now consider the evolution of the u(1)-fl * , and its small electronic fermi surfaces , to the conventional ` large ' fermi surface fermi liquid state at large doping . there is an existing conventional theory of the transformation from small to large fermi surfaces driven by the disappearance of antiferromagnetic order . this is a transition between two fermi liquids , and the vicinity of the transition is described by the hertz - millis theory @xcite and its field - theoretic extensions @xcite , as shown in fig . [ fig : phase1 ] . . note that the u(1)-fl * has a ` small ' fermi surface of electrons due to the presence of topological order , while phase a above has a ` small ' fermi surface of electrons because of translational symmetry breaking.,width=432 ] here , we describe a detour from this direct route @xcite in which two new non - fermi liquid phases appear between the conventional phases of hertz - millis theory . the detour is described by a su(2 ) gauge theory , and the transition from small to large fermi surfaces is now a higgs transition without any local order parameter , in which the emergent gauge structure describing the topological order in the ground state changes from u(1 ) to su(2 ) . the higgs field of this transition is a measure of the local antiferromagnetic correlations in a rotating reference frame to be introduced below in eq . ( [ r ] ) . note that the higgs transition in fig . [ fig : phase1 ] is between metallic states which we denote as ` algebraic charge liquids ' ( acl ) . the small and large fermi surfaces in the acls are those of spinless fermions which carry the electromagnetic charge of the electron . for the u(1 ) acl , a bound state forms between the spinless fermions and a spin @xmath6 boson @xcite , leading to small fermi surfaces of fermionic quasiparticles carrying the same quantum numbers as the electron in the u(1)-fl * : so photoemission will detect a small fermi surface of electrons in the u(1)-fl*. we anticipate that similar effects are also present in the su(2 ) acl metal : there is a large density of states of thermally excited @xmath6 bosons at low energy , so that the photoemission spectral function reflects the large fermi surface of the spinless fermions . we also note that although the higgs field plays a central role in our phase diagram , its direct experimental detection will be difficult . it is overdamped via its coupling to the fermi surfaces , and gauge invariance prohibits any experimental probe from coupling linearly to it . nevertheless , we will see below that it has significant experimental consequences via its strong effect on the fermionic spectrum . we will present details of this theory starting from a microscopic model in section [ su2 ] , but first , in section [ sec : prelim ] , we shall describe some key aspects using our proposed phase diagram in fig . [ phasecup ] .

the theory contains a quantum phase transition between metals with large and small fermi surfaces of spinless fermions carrying the electromagnetic charge of the electron , but the transition does not directly involve any broken global symmetries . this higgs field measures the local antiferromagnetic correlations in a ` rotating reference frame ' .

we analyze a candidate theory for the strange metal near optimal hole - doping in the cuprate superconductors . the theory contains a quantum phase transition between metals with large and small fermi surfaces of spinless fermions carrying the electromagnetic charge of the electron , but the transition does not directly involve any broken global symmetries . the two metals have emergent su(2 ) and u(1 ) gauge fields respectively , and the transition is driven by the condensation of a real higgs field , carrying a finite lattice momentum and an adjoint su(2 ) gauge charge . this higgs field measures the local antiferromagnetic correlations in a ` rotating reference frame ' . we propose a global phase diagram around this higgs transition , and describe its relationship to a variety of recent experiments on the cuprate superconductors .

0801.2159

i

the galactic bulge underwent an intense burst of star formation early in the formation of the galaxy , leading to a very different stellar population and chemical evolution history than found in the milky way disk or halo ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . in particular , massive stars may dominate the pollution at almost all metallicities , leading to unique abundance patterns ( e.g. , * ? ? ? * ) . similar events are thought to mark the formation of other galactic spheroids , making the bulge stellar population a template for interpreting extragalactic observations . as a result of its unique formation history in the galaxy , the bulge has been the subject of intensive study . the detection of rr lyrae stars @xcite first indicated that the bulge contained old stars . with deeper photometry , the main sequence turnoff ( msto ) of the bulge was detected . @xcite found a mean age of 11 - 14 gyr for stars in baade s window , with a negligible fraction of stars with ages @xmath2 5 gyr . photometry reaching more than 2 magnitudes below the msto with _ hubble space telescope _ confirmed the generally old nature of the bulge @xcite , although @xcite included a reminder that a young metal - rich population has similar msto colors and luminosities to an older , more metal - poor population . therefore , deriving ages reliably from photometry of the msto requires adequate knowledge of the bulge metallicity distribution function ( mdf ) , specifically for the msto stars . however , because of the faintness of those stars , the measurement of the bulge mdf has historically relied on giants . since the discovery of both m giants and rr lyr stars , it has been known that the bulge giants span a wide range in metallicity . low - dispersion spectra provided the first quantitative measure of the mdf @xcite . @xcite measured indices from low - dispersion spectra of 268 k giants ( both red clump stars and first ascent giants ) to derive a mean metallicity @xmath3 log@xmath4(n@xmath5/n@xmath6)@xmath7 log@xmath4(n@xmath5/n@xmath6)@xmath8 ) @xmath9}\rangle = -0.11 $ ] , with a dispersion of 0.46 dex . recalibration by @xcite based on high - resolution spectra of 15 stars in common with the @xcite sample reduced the mean metallicity to @xmath10 . @xcite measured @xmath9}\rangle = -0.21 $ ] from low - dispersion near - infrared spectra for 72 m giants in the inner bulge . the good agreement between @xcite result and the recalibrated @xcite result is somewhat surprising . at the bright end of the giant branch , only metal - rich first ascent giants become m giants . however , both k giants and m giants become red clump stars , and lower luminosity metal - rich giants are k stars as well . therefore , the inclusion of red clump stars and fainter giants in the @xcite sample make the biases in their sample more similar to those of the @xcite study . @xcite measured both the mdf of the bulge and the age of the stars using deep photometry of the bulge in the optical and near - infrared wavelengths . they constructed the m@xmath11 and ( v - k)@xmath12 color magnitude diagram for a low - reddening window at ( @xmath13)=(0.277 , @xmath146.167 ) . because the slope of the red giant branch ( rgb ) in these colors depends on the metallicity , rgb stars with different metallicities fall on different parts of the color - magnitude diagram . they could therefore use their photometry to derive the mdf of 503 giants by comparing the positions of bright ( m@xmath15 ) rgb stars with globular cluster fiducials of known metallicity . after correcting for small biases in their mdf caused by their magnitude cutoff , they find an mdf with a peak at [ m / h]@xmath16 , a sharp cutoff at [ m / h]@xmath17 and few stars with [ m / h]@xmath18 . adopting the metallicities derived from the giants for the msto dwarfs , they estimated that the bulge is coeval with the halo and argued that the lack of stars above the prominent msto of the bulge ruled out a significantly younger population . these measurements of the bulge giant mdf can be improved by metallicity measurements from high - dispersion measurements of many stars in several fields throughout the bulge . recently , @xcite and @xcite have obtained a total of @xmath19 k giant spectra at ( r@xmath2020,000 ) in 4 bulge windows . they confirm the metal - rich nature of the bulge . with such a metal - rich population having been reached so quickly after star formation began , the bulge is expected to have a distinct chemical evolution compared to other galactic populations , such as the halo or the disk because , for example , the contributions of longer - lived polluters , such as type ia supernovae ( sne ) or low - mass agb stars , should be small . indeed , @xcite measured high [ @xmath1/fe ] ratios in giants , in particular , high [ mg / fe ] for [ fe / h ] values up to solar , arguing for little type ia sn contribution of fe compared to the thick or thin disks . @xcite confirmed the overall enhancement in [ mg / fe ] and strengthened the conclusion of @xcite that the other @xmath1 abundance ratios do not track [ mg / fe ] exactly . [ o / fe ] , [ si / fe ] , [ ca / fe ] and [ ti / fe ] begin to decrease around [ fe / h]=0 , while [ mg / fe ] does so at supersolar [ fe / h ] . @xcite suggested that metallicity - dependent type ii sn yields could explain the different behaviors of the @xmath1 elements . @xcite explained the low [ o / mg ] ratios in metal - rich bulge giants through a different metallicity - dependent mechanism : wolf - rayet winds leading to less effective o production in metal - rich massive stars . by [ fe / h]@xmath200.2 , all the [ @xmath1/fe ] ratios have begun to decline , probably indicating the introduction of large amounts of fe from type ia sne ( e.g. , * ? ? ? several studies have looked at the abundances of the light elements na and al in the bulge , two elements whose production should depend on the metallicity of the massive stars that exploded as type ii sne . @xcite measured na in 7 k and m giants and found the predicted increase in [ na / fe ] and [ na / o ] in the most metal - rich stars . @xcite found supersolar [ al / fe ] at all metallicities and , for [ fe / h ] @xmath210 , enhanced [ na / fe ] compared to the ratios in disk stars . interestingly , at higher metallicities , the scatter in [ al / fe ] and [ na / fe ] increased and became larger than could be explained by observational errors . interpreting the na and al abundances as the result of type ii sn production may be problematic . the surface abundances of al and na have been shown to be increased by large amounts of internal mixing in metal - poor globular cluster stars ( e.g. * ? ? ? * ) , where the products of proton - capture reactions deep inside the star are mixed up to the surface , leading to enhancements in these two elements . however , @xcite argued that the c and n abundances in the giants they studied were consistent with only mild mixing and , therefore , that the high na and al had to be due to the overall chemical evolution of the bulge . @xcite also found evidence for mild mixing in giants , affecting c and n , but not o , na , or al . finally , there is little information on the neutron - capture elements in the bulge . the absorption lines for these elements are concentrated in the blue part of the optical spectrum , where the crowding from fe , cn , and other lines is severe . near - ir spectra have essentially no lines of these elements . while there are a few lines of ba in the red , these lines in metal - rich giants are so saturated that reliable measurements are very difficult . as a result , only the neutron - capture element eu has published results so far . @xcite found [ eu / fe]@xmath210 in bulge giants , which is likely because of the production of eu in the r - process . measuring additional neutron - capture element abundances would test this idea , because the r - process is better at making some elements ( e.g. , eu ) than others ( e.g. , ba ) . our knowledge of the metallicity and abundance ratios of bulge stars has generally been confined to the bright giants , which are usually the only ones accessible to high - resolution observations . but studying the dwarfs has several advantages . their abundances of the elements such as c and n are unaffected by dredge - up processes on the giant branch . we can measure elements , such as s and zn , that not measured in giants , because the hotter temperatures of the dwarfs decrease the strength of cn and increase the strength of certain atomic lines . in addition , it is critical to know the metallicity of stars at the msto to accurately measure the ages from their color and luminosity . finally , individual ages can be assigned to dwarf stars near the turnoff . the advent of large surveys to identify and follow - up microlensing events , such as the microlensing observations in astrophsycis ( moa ) collaborationiabond / alert / alert.html ] , the optical gravitational lens experiment ogle / ogle3/ews / ews.html ] ( ogle ) , the microlensing follow up networkmicrofun/ ] ( @xmath22fun ) and the probing lensing anomalies network ( planet ) , provides an opportunity to study otherwise unobservable bulge dwarfs . during high - magnification microlensing events , it is possible to obtain high - resolution , high signal - to - noise ratio spectra of faint stars with a huge savings in observing time : a factor @xmath23 where @xmath24 is the magnification and @xmath25 is the number of magnitudes below sky of the unmagnified star . in @xcite , we reported the detailed abundances for a highly - magnified bulge dwarf , , which from a 15 minute exposure at magnification @xmath26 , was shown to be one of the most metal - rich stars ever observed . it also provided the first measurements of s and cu in the bulge . here we present a high - resolution spectrum of the bulge g - dwarf moa-2006-blg-99s , taken at magnification @xmath27 . finally , we respond the challenge : `` ask not what microlensing can do for stellar spectroscopy ask what stellar spectroscopy can do for microlensing . '' there is one important way that the spectroscopic study of bulge dwarfs can benefit microlensing . whenever a source approaches or transits a `` caustic '' ( line of infinite magnification ) caused by the lens , one can measure @xmath28 , the ratio of the angular source radius to angular einstein radius @xmath29 , from the microlens lightcurve . then @xmath30 is inferred from the dereddened color and magnitude of the source to yield @xmath31 , which in turn provides important constraints on the lens properties . because spectroscopy is not normally available for these microlensed sources , the dereddened color and magnitude are estimated by comparing the source position on an instrumental color - magnitude diagram with that of the clump and then assuming that the bulge clump is similar to the local clump as measured by _ hipparcos _ @xcite . this procedure undoubtedly suffers some statistical errors and could suffer systematic errors as well . for example , the bulge clump may have a different color from the local one . high - resolution spectra of an ensemble of microlensed bulge sources will test this procedure for both statistical and systematic errors . for ogle-2006-blg-265s , the standard microlensing procedure yielded @xmath32 , whereas @xcite obtained @xmath33 from high - resolution spectroscopy . this difference hints at a possible discrepancy , but only by repeating this procedure on a number of dwarfs can this be confirmed .

we analyze a high - resolution spectrum of a microlensed g - dwarf in the galactic bulge , acquired when the star was magnified by a factor of 110 . we find that the abundance ratios of alpha and iron - peak elements are similar to those of bulge giants with the same metallicity .

we analyze a high - resolution spectrum of a microlensed g - dwarf in the galactic bulge , acquired when the star was magnified by a factor of 110 . we measure a spectroscopic temperature , derived from the wings of the balmer lines , that is the same as the photometric temperature , derived using the color determined by standard microlensing techniques . we measure [ fe / h]= , which places this star at the upper end of the bulge giant metallicity distribution . in particular , this star is more metal - rich than any bulge m giant with high - resolution abundances . we find that the abundance ratios of alpha and iron - peak elements are similar to those of bulge giants with the same metallicity . for the first time , we measure the abundances of k and zn for a star in the bulge . the [ k / mg ] ratio is similar to the value measured in the halo and the disk , suggesting that k production closely tracks production . the [ cu / fe ] and [ zn / fe ] ratios support the theory that those elements are produced in type ii sne , rather than type ia sne . we also measured the first c and n abundances in the bulge that have not been affected by first dredge - up . the [ c / fe ] and [ n / fe ] ratios are close to solar , in agreement with the hypothesis that giants experience only canonical mixing .

0801.2159

r

in table 2 , we summarize the abundances measured for 17 elements in . we include both log@xmath54 and its error , as well as [ x / fe ] and its error . to give an idea of the uncertainty due to scatter from the lines , rather than from atmospheric parameters , we give @xmath57 , the rms of abundances derived from individual lines as well as the number of lines . we also give our measurements of the solar abundances , which we will use to calculate ratios . for reference , we include the @xcite solar abundances in the final column . we measure [ fe / h]@xmath59 for . in @xcite , we measured [ fe / h]@xmath60 for the dwarf . the stars that are microlensed are unbiased in metallicity . the criterion for spectroscopic follow - up is that the unmagnified source be faint enough to be a bulge dwarf , regardless of color . therefore , especially considering the large and variable reddening toward the bulge , we are not biased in our high - resolution follow - up toward high metallicity sources . the high metallicities of these two dwarfs is surprising given that work on giants has indicated an average metallicity near solar . in figure [ fig : mdf1 ] , we compare the metallicity distribution function ( mdf ) of the two dwarfs with several mdfs based on studies of giant stars . the mdf of @xcite , which is based on high - resolution analysis of m giants and should be biased _ toward _ the highest metallicity objects , lacks any giants as metal - rich as these dwarfs . this is particularly notable , since the work of @xcite on k giants with low - dispersion spectra shows some extremely metal - rich stars . a complicating factor is the possible presence of a metallicity gradient in the bulge . in figure [ fig : mdf2 ] , we compare the metallicities of the dwarfs with mdfs derived by @xcite from high - resolution spectra for giants in three bulge fields : 4 , 6 , and 12away from the galactic center . the inner field is more metal - rich than the outer . however , these two dwarfs are located 6.5 ( ) and 4.9 ( ) away from the galactic center , and therefore gradients can not explain their anomalously high metallicities . these results hint that the mdfs of the bulge giants and dwarfs may be different . whether this is true can be established by more observations of bulge dwarfs and by resolution of the discrepancies among the mdfs derived for giants , particularily between the low - dispersion and high - dispersion studies . because the microlensed dwarfs are found at a range of distances from the galactic center , comparison of the giant and dwarf mdfs also depends on measuring the metallicity gradient ( and the size of deviations from that gradient ) in the bulge . ideally , the mdf for giants in the same field as the microlensed dwarf would be measured . with our measurements of t@xmath43 , log @xmath48 , and [ fe / h ] from spectroscopy , we can compare the position of on the hertzsprung - russell diagram with theoretical isochrones ( fig . [ fig : iso ] ) . we use the yonsei - yale isochrones @xcite with [ fe / h]=0.385 , [ @xmath1/fe]=0 for comparison . the best fit age for this star is @xmath205 gyr . instead of log g , we can also use the i - band magnitude to plot the star on the h - r diagram . we adopt a distance of 8.5 kpc , placing this dwarf on the far side of the bulge , its most likely position because the optical depth for lensing is larger there . figure [ fig : imag ] shows that a similar age is obtained . indeed , shifting the position of in the vertical direction , by changing its distance or luminosity has little effect on the young age that we derive for this star , because , at this metallicity , there should be no stars this hot with ages @xmath61 6 gyr . however , the uncertainty in the temperature combined with the effect that changing the temperature has on the derived metallicity produce large uncertainties in the age . if we instead adopt the t@xmath43 on the lower edge of our range ( 5600k ) , the metallicity calculated from the lines drops to [ fe / h]=0.16 dex . using isochrones of this metallicity , the new t@xmath43 gives an age of @xmath209 gyrs . improvements in the accuracy of temperatures are needed to get better age constraints for bulge dwarfs , but in principal , ages can be measured for individual stars near the main - sequence turnoff . -t@xmath43 ) . m@xmath62 was calculated using the i@xmath12 magnitude and assuming a distance of 8.5 kpc . the t@xmath43 is the temperature from the balmer lines . we also show isochrones from @xcite the solid lines show isochrones for [ fe / h]=0.385 and for ages 2 , 3 , 4 , 5 and 7 gyr . the dashed line shows a 5 gyr isochrone for a solar metallicity.,width=604 ] as stars move up the giant branch , they pass through first dredge - up , which brings up material that has been processed in the cn cycle . the c and n abundances measured in giants no longer represent the original c and n endowments of the stars , although c+n will remain constant as long as only material processed in the cn cycle , and not the on cycle , is mixed to the surface . @xcite and @xcite measured c and n in giants in the bulge . they found that the giants lie to the n - rich side of the line defined by the c / n ratio of the sun ( figure [ fig : mix ] ) . @xcite concluded that a small amount of mixing had occurred in the giants . this conclusion is only valid if the original abundances in the giants lie close to the line . otherwise , if the bulge dwarfs have non - solar c / n ratios , the c / n ratios measured in bulge giants could imply either no mixing ( and a n - rich original composition ) or substantial mixing ( and a c - rich original composition ) . the abundances of c and n for are also plotted on figure [ fig : cn ] and shows that the assumption of @xcite is justified and that the expected amount of mixing has occurred in the giants . li has been created since the big bang by stellar nucleosynthesis and by cosmic ray spallation . a li abundance for a star in the bulge , measuring how fast li was made in the early galaxy , would be very interesting . however , most stars no longer have the same amount of li on their surfaces as was present in their natal gas clouds . li is easily burned during pre - main sequence and main - sequence phases of stars and is either destroyed throughout the convective envelope during the rgb phase or ( for a brief time ) created in the star itself and dredged up . we have no detection of li in this star , only a 3-@xmath57 upper limit of log@xmath54(li)=1.84 dex based on a @xmath63 fit to the data ( see @xcite for more details ) . in figure [ fig : li ] we show this upper limit compared with li measurements in open cluster stars having a range of ages as well as field stars from @xcite . we also include the upper limit from . lower values can be expected in field stars because of astration on the main - sequence and because they are often older than the clusters featured in figure [ fig : li ] and were formed out of gas that had not been polluted by as much li . figure [ fig : li ] shows that the li upper limits in the bulge dwarfs are consistent with the upper limits in field dwarfs . a dwarf with t@xmath43@xmath64k is probably needed to measure the amount of li produced by spallation in the bulge . vs. log@xmath54(li ) for the hyades and m67 from @xcite and for field stars from @xcite compared with the upper limit for and . the bulge dwarf li limits are consistent with disk stars , which is not surprising given the age and temperature of these stars.,width=604 ] the bulge has a different star formation history than the halo / disk . the ratios of type ii / type ia pollution or type ii / agb star pollution at a given [ fe / h ] are therefore different as well , and the abundance ratios reflect this . we compare the abundances for both and with bulge giants and field stars from the thick / thin disk and halo from literature sources . in table 3 , we summarize the literature sources we use for each element . for many elements , observing red giant stars in the bulge is an effective method of measuring their abundances . however , as shown in 5.2.1 , internal mixing on the rgb alters the abundances of c and n. the abundances of c and n that we measure in therefore represent the first observations of the primordial c and n produced by the chemical evolution of the bulge . has [ c / fe]=0.04@xmath65 and [ n / fe]=@xmath140.06@xmath66 ( fig . [ fig : cn ] ) . the solar values of [ c / fe ] and n / fe ] show that c and n production kept pace with the fe production in the bulge . there are many sources of c in the universe ( * ? ? ? type ii sne and agb stars certainly contribute substantial amounts of c and n ; the roles of novae and wolf - rayet stars are less clear . these contributions , whatever they are , track the production of fe in the chemical evolution of the bulge . finally , we attempted to measure the @xmath67c/@xmath68c ratio , which is sensitive to the source of c , being low for low - mass agb stars and high for type ii sne . we could only set an uninteresting limit ( @xmath67c/@xmath68c > 1 ) on this very interesting number . and [ n / fe ] for ( filled black square ) compared to bulge giants ( filled blue circles ) and disk dwarfs ( open red triangles ) . the low [ c / fe ] values for the bulge giants are the result of internal mixing . the [ c / fe ] and [ n / fe ] values in , on the other hand , are the result of the pollution of the gas of the bulge by previous generations of stars . the solar ratios for these elements impose constraints on the inefficiency of c and n production in the bulge.,width=604 ] the abundances of na and al are elevated in bulge giants @xcite compared to disk stars . metal - rich type ii sne are predicted to produce more of the odd - z elements such as na and al than metal - poor type ii sne and could potentially be the explanation of the difference between the bulge and the disk . in figure [ fig : naal ] , we show the [ na / fe ] and [ al / fe ] value for and . these two unmixed stars are on the lower end of the scatter seen in the giants . this could be a hint that the larger [ na / fe ] and [ al / fe ] values seen in giants are due to internal mixing , but because the dwarf values fall within the scatter outlined in the scatter indicates that more measurements in dwarf stars are needed before any differences in the distribution of [ na / fe ] and [ al / fe ] in dwarfs and giants can be seen . and [ al / fe ] for and ( filled black squares ) compared to bulge giants ( filled blue circles ) and field stars ( open red triangles ) . the dwarfs fall within the distribution of [ na / fe ] and [ al / fe ] values seen in the giants , but at the lower edge of that distribution . , width=604 ] figure [ fig : alpha ] shows the [ o / fe ] , [ mg / fe ] , [ si / fe ] , and [ ca / fe ] for compared with halo stars , thin and thick disk stars , and bulge giants . the metallicity of is in the range of metallicities measured for bulge giants , allowing direct comparison of [ @xmath1/fe ] ratios between dwarfs and giants . the agreement is good , as both the giants and the dwarf have [ @xmath1/fe ] below the high [ @xmath1/fe ] values of the more metal - poor ( [ fe / h]@xmath690 ) stars . this decline in all [ @xmath1/fe ] ratios suggests that fe from type ia sne is being added and that the more metal - rich stars formed sufficiently later to have this ejectum in their gas . , [ mg / fe ] , [ si / fe ] and [ ca / fe ] for and ( filled black squares ) compared to bulge giants ( filled blue circles ) and field stars ( open red triangles ) . the [ @xmath1/fe ] ratios in agree well with the values measured in giants of similar metallicity . , width=453 ] k is an odd - z element and is predicted in nucleosynthesis models to be underproduced relative to the @xmath1 elements in metal - poor sne . there are few measurements in the literature , and those that exist show the opposite trend of increasing [ k / fe ] with decreasing [ fe / h ] @xcite . however , the only k line available for study in most stars , the resonance line at 7698 , is affected by non - lte effects , and these corrections have not yet been applied to large samples . @xcite derived nlte corrections for each star in their sample . the [ k / fe ] values were still supersolar at low metallicities , with the thin disk stars showing a drop in [ k / fe ] for[fe / h]@xmath70 . the [ k / fe ] ratios in the thick disk stars remain high . however , the [ k / mg ] ratios showed much smaller variations among the different galactic populations . they argued that the constant [ k / mg ] ratio ( [ k / mg]=@xmath71 ) in the stars indicated that the nucleosynthesis of k is closely coupled to that of the @xmath1-elements , which is somewhat surprising given the theoretical predictions . we measured the 7698 line in and the sun . we also measured the k abundance in using turbospectrum and the model atmosphere described in @xcite . the [ k / fe ] we measure for is @xmath72 figure [ fig : k ] compares the results for the bulge to the @xcite results . we applied no nlte correction , but assumed that the lte abundances in the sun and would be affected by the same amount . the [ k / mg ] values are 0.07@xmath73 for and 0.12@xmath74 for ; the [ k / mg ] abundance is still within a narrow range , even in this very different chemical evolution history . for and ( filled black squares ) compared to field stars ( open red triangles ) . the [ k / fe ] values in the dwarfs fall in the range seen in the disk stars.,width=604 ] the abundances for ti , sc , mn and ni are shown in figure [ fig : iron ] . @xcite found that [ ti / fe ] behaves like [ o / fe ] in the bulge , with supersolar [ ti / fe ] ratios for many stars , followed by [ ti / fe ] decreasing to solar for [ fe / h @xmath75 . the abundance of ti is also enhanced in halo stars , leading it to be classified as an `` @xmath1-element '' for observational purposes . in , [ ti / fe ] is close to solar , in line with the @xmath1-elements discussed above . the ratios of [ sc / fe ] and [ ni / fe ] are observed to be close to solar for a wide range of populations : halo , thick and thin disk and bulge . the data for show that abundances in this bulge dwarf agree with this picture . for [ fe / h ] @xmath76 in the galactic halo / disk , there is a plateau at [ mn / fe]@xmath77 @xcite . because type ia sne have not polluted the most metal - poor stars in the galaxy , we can derive the ratio of mn / fe produced in ( metal - poor ) type ii sne from this plateau . [ mn / o ] starts to rise before [ o / fe ] starts to drop in the disk . because the drop in [ o / fe ] signals the onset of substantial type ia sn contribution , the rise in mn relative to o can not be due to type ia sne , but rather to increased production of mn by more metal - rich type ii sne @xcite . @xcite also measured mn in 13 stars in the sagittarius dwarf galaxy . these stars have substantial type ia sne contributions to their gas , but [ mn / fe ] values about 0.2 dex below the trend seen in the galactic disk , providing additional evidence that metal - rich type ii sne are responsible for mn production . the near solar [ mn / fe ] values for and ogle-99 are also in support of this idea , because they were seen in an environment that had many type ii sne occur , unlike the sagittarius stars . , [ ti / fe ] , [ mn / fe ] , and [ ni / fe ] for and ( filled black squares ) compared to bulge giants ( filled blue circles ) and field stars ( open red triangles).,width=453 ] the [ ti / fe ] value in agrees well with [ ti / fe ] ratios measured in bulge giants of similar metallicity . within the error bars , the other [ iron - peak / fe ] ratios follow the trends seen the disk stars . [ fig : iron ] and [ zn / fe ] for and ( filled black squares ) compared to field stars ( open red triangles ) . the @xcite models predict [ zn / fe]@xmath78 for high - metallicity stars in the bulge . therefore , the low [ zn / fe ] disagrees with the theory that zn is produced in large amounts of type ia sne . the solar and supersolar [ cu / fe ] values in the bulge are consistent with either type ia sn production or metal - rich type ii sn production.,width=604 ] the percent of cu and zn production to be ascribed to different nucleosynthesis sites ( e.g. , type ii sne , agb stars , type ia sne ) is uncertain . the observations show that [ cu / fe ] @xmath79 at the lowest metallicities and then rises to solar by [ fe / h]@xmath80 ( e.g. * ? ? ? [ zn / fe ] , on the other hand , has supersolar values at the lowest metallicities and then decreases to closer to solar ( e.g. * ? ? ? @xcite used new weak s - process calculations and available sn models to argue that approximately two - thirds of the zn and cu production in the universe is due to type ia sne . the rest of the zn is from a primary process in massive stars , while the cu comes from a secondary ( = metallicity - dependent ) process in massive stars . using this model and a chemical evolution model for the bulge , @xcite predicted that both [ cu / fe ] and [ zn / fe ] would be @xmath20 0.2 at [ fe / h]=0.3 . the sn models available to @xcite did not include important effects , such as detailed calculation of neutron - capture elements beyond fe . @xcite used updated results and considered zn and cu production by neutron - capture in the o - rich parts of type ii sne ( `` weak sr - process '' ) . in their analysis , cu is mostly produced in this weak sr - process , a secondary process . a small amount of primary cu is made as radioactive zn in the inner regions of type ii sne . zn production is also due to massive star nucleosynthesis , but here there is a large primary production in the @xmath1-rich freeze out in type ii sne , which is supplemented at higher metallicities by a secondary contribution from the weak - sr process . @xcite find no need for contributions to cu and zn from type ia sne or agb stars . the measurements of [ cu / fe ] and [ zn / fe ] in support the @xcite model and argue against the production of large amounts of zn in type ia sne . because the abundance of cu is dominated by a secondary process , the solar [ cu / fe ] at [ fe / h]=0.36 and the supersolar value at [ fe / h]=0.56 are the result of copious cu production in metal - rich type ii sne . however , the primary production of zn and the smaller secondary contribution is not sufficient to keep up with the fe from both type ii sne and type ia sne . the observations also show again that the chemical evolution of zn is separate from fe . for and ( filled black squares ) compared to field stars ( open red triangles ) . [ ba / fe ] shows the largest deviation from the trends seen in the disk stars of any element studied in this paper . this low ba value is consistent with the idea that the r - process is the dominate producer of heavy elements in the bulge . it also puts constraints on s - process contributions to ba ( and the accompanying c and n contributions ) from agb stars . , width=604 ] the [ ba / fe ] for falls considerably below the solar value ( [ ba / fe]=@xmath81 ( fig . [ fig : ba ] ) . it falls in the range not seen in other parts of galaxy , except for the metal - poor halo . we note that we could only measure 1 line of ba in , and the solar value we measure is the most discrepant from the solar value in @xcite . however , even if the grevesse & sauval value is used , the [ ba / fe ] for is still subsolar ( [ ba / fe]=@xmath140.24 ) . if we use the solar value of ba ( log(@xmath54(ba)=2.39 ) reported in @xcite , has a [ ba / fe]@xmath82 . the paucity of ba in the halo stars is explained by the fact that the r - process is the only available channel for producing the heavy elements in the early univere , and the r - process is not an efficient producer of ba . it is tempting to ascribe the low ba in to the same cause , especially in light of the high [ eu / fe ] measurements in bulge giants by @xcite . stellar populations dominated by type ii sne and r - process production should have high [ eu / fe ] and [ eu / ba ] before type ia sne and agb stars eventually add fe and ba to the ism . whether the ba deficiency in can be explained by a lack of contributions from agb stars depends on the relative timescales of type ia sn pollution and agb pollution and whether the low [ @xmath1/fe ] values in are the result of type ia pollution . if we assume that type ia sne have contributed significantly to the abundances of , which is reasonable , then agb pollution must trail type ia sn production . the evidence on this point is mixed . @xcite saw , in addition to a wide range at any given metallicity , a rise in the [ la / fe ] ratios at [ fe / h ] @xmath83 . because la , like ba , is mostly due to the s - process , this would indicates the s - process from agb stars is added before fe from type ia sne causes the [ @xmath1/fe ] ratios to turn over . on the other hand , @xcite argued based on the isotope ratios of mg that 3 - 6 m@xmath84 stars did not start contributing to the halo until [ fe / h]@xmath85 , at the same time or later than the type ia sne . the [ ba / fe ] measured in is closer to the solar value , and suggests that by [ fe / h]=0.5 the bulge had reached the same point in chemical evolution as the solar neighborhood , with substantial amounts of ba supplied by the s - process in agb stars balancing the iron supplied by type ia sne . ba abundances for bulge stars with a wide range of metallicity would help clarify the origin of the ba abundance in the bulge .

for the first time , we measure the abundances of k and zn for a star in the bulge . the [ k / mg ] ratio is similar to the value measured in the halo and the disk , suggesting that k production closely tracks production . the [ cu / fe ] and [ zn / fe ] ratios support the theory that those elements are produced in type ii sne , rather than type ia sne . we also measured the first c and n abundances in the bulge that have not been affected by first dredge - up . the [ c / fe ] and [ n / fe ] ratios are close to solar , in agreement with the hypothesis that giants experience only canonical mixing .

we analyze a high - resolution spectrum of a microlensed g - dwarf in the galactic bulge , acquired when the star was magnified by a factor of 110 . we measure a spectroscopic temperature , derived from the wings of the balmer lines , that is the same as the photometric temperature , derived using the color determined by standard microlensing techniques . we measure [ fe / h]= , which places this star at the upper end of the bulge giant metallicity distribution . in particular , this star is more metal - rich than any bulge m giant with high - resolution abundances . we find that the abundance ratios of alpha and iron - peak elements are similar to those of bulge giants with the same metallicity . for the first time , we measure the abundances of k and zn for a star in the bulge . the [ k / mg ] ratio is similar to the value measured in the halo and the disk , suggesting that k production closely tracks production . the [ cu / fe ] and [ zn / fe ] ratios support the theory that those elements are produced in type ii sne , rather than type ia sne . we also measured the first c and n abundances in the bulge that have not been affected by first dredge - up . the [ c / fe ] and [ n / fe ] ratios are close to solar , in agreement with the hypothesis that giants experience only canonical mixing .

astro-ph0110235

c

the previous section shows that a simple analysis , not taking into account morphology or activity level ( h@xmath0 luminosity ) of the galaxies , gives that the percentage of liners , transition and absorption line galaxies with nearby companions is significantly higher than that of seyferts and hii galaxies . this result contradicts all previous works in this field , which either found that seyferts and hii galaxies have the same percentage of companions as other galaxies , or that they have a higher percentage of companions . however , we show that when we take into account the morphological types of the galaxies and split them into subgroups containing only ellipticals and only spirals , the situation is different . there is no difference in the percentage of galaxies with companions among different activity types if we consider only galaxies of similar morphological types . this result is consistent with the one found by fuentes - williams & stocke ( 1988 ) , bushouse ( 1986 , 1987 ) , de robertis et al . ( 1998 ) , and also with more recent results on clustering of low luminosity agns at higher redshifts ( brown et al . 2001 ; schreier et al . 2001 ) . the percentage of elliptical galaxies with companions is approximately two times higher than that of spirals . this explains why liners , transition and absorption line galaxies have higher percentages of companions when all galaxies are considered , since a higher percentage of these galaxies is found in ellipticals when compared to seyferts and hii galaxies . given the fact that most of the hii galaxies in the palomar sample have only small quantities of recent star formation , most of them should be considered as normal , quiescent galaxies . we found that a there is a higher percentage of galaxies with companions among hii galaxies with l(h@xmath0)@xmath1 erg s@xmath2 then in lower h@xmath0 luminosity hii galaxies . the percentage of companions increases even more for hii galaxies with higher h@xmath0 luminosities , which confirms previous results ( kennicutt et al . 1997 , keel et al . 1985 , bushouse 1986 ) . the results we obtained separating the galaxies by morphological types is somewhat expected . it was shown by dressler ( 1980 ) and whitmore , gilmore & jones ( 1993 ) in the study of clusters of galaxies that the percentage of ellipticals increases in higher density environments . although this result was based on clusters , postman & geller ( 1984 ) found that the results also apply for groups of galaxies . furthermore , whitmore et al . ( 1993 ) found that the percentage of ellipticals in close pairs is higher than that of spirals , supporting the results we found . charlton , whitmore & gilmore ( 1995 ) showed that ellipticals are found more often in pairs than spirals in clusters and high density environments , like large groups . we found that @xmath430% of the ellipticals in our sample are in the virgo cluster , while only 20% of the spirals are found in virgo . another @xmath430% of the ellipticals are found in groups of galaxies with 10 or more galaxies garcia ( 1993 ) , suggesting that the higher percentage of ellipticals with companions is in fact due to the morphology - density relation . the study of the local galaxy densities ( section 4 ) shows that , in most of the cases , the distribution of this quantity does not differ for different activity types . in the cases where we find significant differences , they can be explained if we exclude the galaxies which belong to the virgo cluster , or if we separate the galaxies into different morphological types . these results agree with the ones obtained in the comparison between the percentage of companions in galaxies with different activity types . it can be argued that the results found for seyfert galaxies is somewhat questionable , since the palomar survey has a large portion of low luminosity seyfert galaxies , which are usually not observed in other surveys . in this way , the results presented here would be biased towards low luminosity seyferts , which , in a way similar to hii galaxies , may have a higher percentage of companions as the luminosity increases . evidence of this effect is given by heckman , carty & bothun ( 1985 ) , heckman et al . ( 1984 ) and hutchings ( 1983 ) , who showed that radio galaxies and quasars are found in higher density environments . we believe that this is not the case . we compare the results presented here with the ones from schmitt et al . ( 2001 ) for warm infrared seyfert galaxies . the median [ oiii ] luminosity of their seyfert galaxies , which is believed to be an isotropic indicator of the intrinsic luminosity of these galaxies , is l([oiii])@xmath50 erg s@xmath2 , calculated using de grijp et al . ( 1992 ) values . this value corresponds to 70 times the median value of the [ oiii ] luminosity of seyfert galaxies in the palomar survey ( l([oiii])@xmath51 erg s@xmath2 ) , obtained from ho et al . ( 1997a ) . schmitt et al . ( 2001 ) observed that between 19% and 28% of their seyfert galaxies have companions . their criteria were a little different from ours , assuming that galaxies were companions if the distance between the primary and the secondary was smaller than 3 times the diameter of the primary , and the brightness difference between them smaller than 3 mag . since it was not possible for them to find radial velocities for all possible companion galaxies , they could only put limits on the percentage of galaxies with companions . the lower limit is 19% , which corresponds to those galaxies where it was possible to confirm that the companion has a radial velocity difference smaller than 1000 km s@xmath2 , while the upper limit is 28% , corresponding to all galaxies , including those without information about the velocity of the secondary . the uncertainty in these measurements is @xmath45% , given by poisson statistics . the number of seyfert galaxies with companions at distances smaller than 3 diameters in the palomar survey is 8 in a sample of 46 galaxies , a total of 17%@xmath52% . given the uncertainties involved in these measurements , there is no significant difference in the percentage of seyfert galaxies with companions in the palomar and schmitt et al . ( 2001 ) samples , which shows that low and high luminosity seyfert galaxies similar environments . the contingency table analysis gives that there is a 16% probability that the two samples have the same number of companions . the explanation of why other papers found that there is a higher percentage of seyferts with companions than normal galaxies with companions is not very clear . one possibility would be that they mixed galaxies of different morphological types . however , the percentage of seyferts in ellipticals is small and this effect could be smeared out in a larger sample . another explanation is related to the way the previous works selected their samples of seyferts . in the case of laurikainen & salo ( 1995 ) and dultzin - hacyan et al . 1994 ) , their samples included a large portion of seyferts selected from ultraviolet surveys . in seyfert 2 galaxies , the unified model predicts that the ultraviolet emission is either nuclear radiation reflected towards the observer ( antonucci 1993 ) , or a circumnuclear starburst ( cid fernandes & terlevich 1995 ) . if the seyfert 1 s and seyfert 2 s have the same amount of ultraviolet excess , this means that the seyfert 2 s were selected from two orders of magnitude higher in the luminosity function , since only 1% of the nuclear light is believed to be reflected , or that their circumnuclear region is dominated by a luminous starburst which is responsible for most of the ultraviolet emission ( see schmitt et al . 2001 for a discussion about this ) . in the latter case , the fact that these papers observed a higher percentage of seyfert 2 s with companions , but not seyfert 1 s , would be due to the fact that the percentage of starbursts with companions goes up as the starburst luminosity goes up , as shown above . the result that seyfert galaxies have the same percentage of companion galaxies as galaxies with other activity types is intriguing and may have important consequences for the theories of how the gas is moved from kiloparsec scales to the inner parsec region of the galaxy and feeds the nuclear black hole . taken at face value , this result indicates that interactions are not important in this process . however , this result can be interpreted in a different way . since we expect a delay between the time when the interaction occurs and when the gas reaches the nucleus , we may be seeing different stages of this process . it has been shown by several simulations ( e.g. byrd et al . 1986 , 1987 , byrd & valtonen 2001 , lin et al . 1988 , hernquist & mihos 1995 ) that interactions move the gas to the nuclear region of the galaxy , where its density and temperature increases , a starburst is formed and later the gas feeds the black hole . also , taking into account the evidence that many galaxies have black holes in their nuclei , suggesting that this may be the case in all galaxies ( magorrian et al . 1998 , ho 1999b , gebhardt et al . 2000 , ferrarese & merritt 2000 ) , it may be possible that all galaxies pass through a period of activity . the galaxies may also pass through different activity types , where they first present a star formation period , when the gas has just moved into the nuclear region , later they present a period of seyfert activity , when the gas is being accreted by the black hole , and finally they go into a period where they turn into a liner or transition galaxy , when the amount of gas available to fuel the nucleus is small . we would like to thank jim ulvestad , luis ho and the referee for comments and suggestions which significantly improved this manuscript . this research made use of the nasa / ipac extragalactic database ( ned ) , which is operated by the jet propulsion laboratory , caltech , under contract with nasa . we also used the digitized sky survey , which was produced at the space telescope science institute under u.s . government grant nagw-2166 . the national radio astronomy observatory is a facility of the national science foundation operated under cooperative agreement by associated universities , inc . lrrrlrrrl ngc0507 & -1.70 & 0.46 & 602 & ngc0508 & & s0 & & ngc1023 & -2.87 & 0.33 & 111 & ngc1023a & & s0 & & ngc2300 & 0.02 & 2.19 & 454 & ngc2276 & & s0 & & ngc2950 & -2.83 & 3.15 & 21 & ugc5179 & & s0 & & ngc3384 & 0.57 & 1.46 & 154 & ngc3379 & & s0 & & ngc3613 & -0.69 & 4.97 & 512 & ngc3619 & & e & & ngc3640 & -2.78 & 0.64 & 441 & ngc3641 & & e & & ngc4262 & -2.09 & 4.92 & 29 & ic0781 & & s0 & yes & ngc4291 & -0.30 & 3.41 & 399 & ngc4319 & & e & & ngc4340 & 0.08 & 1.68 & 326 & ngc4350 & & s0 & yes & ngc4365 & -2.91 & 1.60 & 444 & ngc4370 & & e & yes & ngc4382 & -1.52 & 1.12 & 200 & ngc4394 & & s0 & yes & ngc4406 & -1.96 & 1.21 & 480 & ngc4402 & & e & yes & ngc4417 & 0.07 & 3.76 & 422 & ngc4424 & & s0 & yes & ngc4473 & -0.27 & 3.37 & 884 & ngc4477 & & e & yes & ngc4478 & -0.67 & 2.47 & 596 & ngc4476 & & e & yes & ngc4503 & -2.56 & 2.12 & 136 & ic3470 & & l & yes & ngc4564 & 1.67 & 10.5@xmath53 & 379 & ngc4579 & & e & yes & ngc4638 & -2.76 & 0.81 & 16 & ngc4637 & & s0 & yes & ngc4649 & -2.11 & 0.35 & 302 & ngc4647 & & e & yes & ngc4754 & 0.32 & 2.57 & 412 & ngc4762 & & s0 & yes & ngc5576 & -1.42 & 0.91 & 171 & ngc5574 & & e & & ngc5638 & -1.53 & 0.76 & 68 & ngc5636 & & e & & ngc7332 & -0.15 & 1.50 & 44 & ngc7339 & & s0 & & ngc7457 & -2.30 & 1.87 & 96 & ugc12311 & & s0 & & ngc7619 & -2.60 & 1.06 & 369 & ngc7617 & & e & & ngc0315 & -1.72 & 1.81 & 130 & ngc0311 & 39.55 & e & & ngc0474 & -0.04 & 0.79 & 11 & ngc0470 & 38.52 & s0 & & ngc1961 & -2.76 & 2.71 & 42 & ugc3342 & 39.81 & l & & ngc2336 & -1.61 & 2.73 & 158 & ic0467 & 38.39 & l & & ngc3166 & -2.87 & 1.00 & 4 & ngc3165 & 39.10 & l & & ngc3169 & -0.05 & 1.76 & 110 & ngc3166 & 39.02 & l & & ngc3190 & -1.67 & 1.07 & 253 & ngc3187 & 38.82 & l & & ngc3193 & 0.29 & 1.92 & 108 & ngc3190 & 38.20 & e & & ngc3226 & 1.27 & 0.73 & 165 & ngc3227 & 38.93 & e & & pair ngc3379 & -0.57 & 1.39 & 185 & ngc3384 & 37.94 & e & & ngc3414 & -2.41 & 2.24 & 220 & ugc5958 & 39.23 & s0 & & ngc3507 & -0.65 & 3.78 & 152 & ngc3501 & 39.39 & l & & ngc3607 & -2.21 & 0.64 & 266 & ngc3605 & 38.93 & s0 & & ngc3608 & 0.90 & 1.79 & 189 & ngc3607 & 38.28 & e & & ngc3623 & 0.52 & 2.07 & 80 & ngc3627 & 37.77 & l & & ngc3718 & -0.72 & 1.44 & 31 & ngc3729 & 38.46 & l & & ngc3998 & -1.97 & 1.13 & 353 & ngc3990 & 40.00 & s0 & & ngc4036 & -0.27 & 4.05 & 164 & ngc4041 & 39.35 & s0 & & ngc4111 & -2.43 & 2.40 & 136 & ngc4117 & 39.40 & s0 & & ngc4261 & -2.47 & 0.87 & 322 & ngc4264 & 39.35 & e & yes & ngc4278 & -1.88 & 0.84 & 410 & ngc4283 & 39.17 & e & & ngc4374 & -2.87 & 1.59 & 395 & ngc4387 & 38.89 & e & yes & ngc4394 & 1.52 & 2.04 & 198 & ngc4382 & 38.33 & l & yes & ngc4438 & -1.12 & 0.53 & 717 & ngc4435 & 39.37 & l & yes & ngc4486 & -2.72 & 1.10 & 67 & ngc4478 & 39.44 & e & yes & ngc4550 & -0.41 & 1.10 & 785 & ngc4551 & 38.41 & s0 & yes & ngc4762 & -0.32 & 1.58 & 416 & ngc4754 & 37.49 & s0 & yes & ngc5077 & -0.91 & 1.61 & 39 & ngc5079 & 39.67 & e & yes & ngc5195 & 1.71 & 0.78 & 22 & ngc5194 & 37.94 & s0 & & pair ngc5353 & -0.36 & 0.62 & 352 & ngc5354 & 39.12 & s0 & & ngc5363 & 0.08 & 3.56 & 103 & ngc5364 & 39.42 & s0 & & pair ngc5485 & -1.26 & 2.79 & 594 & ngc5486 & 38.35 & s0 & & ngc5566 & -2.93 & 0.64 & 267 & ngc5569 & 38.66 & e & & ngc5813 & -2.81 & 2.99 & 395 & ngc5811 & 38.56 & e & & ngc5850 & 0.48 & 2.36 & 770 & ngc5846 & 38.66 & l & & ngc5970 & -2.97 & 2.81 & 52 & ic1131 & 38.06 & l & & ngc5982 & 0.67 & 3.08 & 387 & ngc5985 & 38.46 & e & & pair ngc5985 & -0.67 & 1.35 & 386 & ngc5982 & 38.94 & l & & pair ngc6340 & -2.56 & 1.89 & 39 & ic1251 & 38.50 & l & & ngc7626 & 0.13 & 2.57 & 381 & ngc7619 & 38.81 & e & & ngc0520 & 0.00 & 0.02 & 9 & pair & 39.33 & l & & merging ngc0672 & -0.83 & 1.07 & 83 & ic1727 & 37.81 & l & & pair ngc0697 & -2.03 & 4.61 & 167 & ngc0694 & 39.01 & l & & ngc0812 & -2.99 & 3.64 & 647 & ugc1585 & 40.08 & l & & ngc0877 & -0.99 & 4.59 & 173 & ngc0871 & 39.18 & l & & ngc2146 & -2.01 & 2.94 & 604 & ngc2146a&39.76 & l & & ngc2342 & -0.92 & 1.66 & 44 & ngc2341 & 40.93 & l & & ngc2276 & -0.02 & 2.14 & 454 & ngc2300 & 39.87 & l & & ngc2750 & -0.61 & 0.36 & 2 & mcg+04 - 22 - 012&40.60 & l & & ngc2964 & -1.21 & 2.10 & 246 & ngc2968 & 40.01 & l & & ngc3034 & 1.47 & 3.29 & 250 & ngc3031 & 39.71 & l & & ngc3338 & -2.42 & 3.30 & 85 & ugc5832 & 38.15 & l & & ngc3389 & 1.08 & 2.33 & 595 & ngc3384 & 38.60 & l & & ngc3395 & -0.23 & 0.57 & 7 & ngc3396 & 39.40 & l & & pair ngc3430 & -0.66 & 1.56 & 90 & ngc3424 & 39.04 & l & & ngc3504 & -1.36 & 4.46 & 162 & ngc3512 & 40.81 & l & & ngc3646 & -2.84 & 2.01 & 731 & ngc3649 & 39.42 & l & & ngc3684 & 0.19 & 4.53 & 9 & ngc3686 & 38.67 & l & & ngc3686 & -0.19 & 4.32 & 9 & ngc3684 & 39.80 & l & & ngc3690 & 0.00 & 0.05 & 31 & pair & 40.62 & l & & merging ngc3729 & 0.72 & 4.15 & 30 & ngc3718 & 39.41 & l & & ngc3963 & -0.98 & 3.02 & 181 & ngc3958 & 39.61 & l & & ngc4041 & 0.27 & 5.72@xmath53 & 164 & ngc4036 & 39.46 & l & & ngc4088 & -1.58 & 1.97 & 7 & ngc4085 & 39.00 & l & & ngc4123 & -0.33 & 3.20 & 19 & ngc4116 & 40.35 & l & & ngc4274 & -2.27 & 2.80 & 129 & ngc4283 & 38.47 & l & & ngc4273 & -2.36 & 0.81 & 114 & ngc4277 & 40.26 & l & & ngc4298 & 0.26 & 0.73 & 9 & ngc4302 & 39.00 & l & yes & ngc4380 & -2.58 & 2.62 & 2 & ic3328 & 38.02 & l & yes & ngc4424 & -0.07 & 3.06 & 412 & ngc4417 & 39.02 & l & yes & ngc4469 & -2.54 & 4.20 & 15 & ugc7596 & 38.16 & l & yes & ngc4477 & -2.08 & 1.40 & 472 & ngc4479 & 38.84 & s0 & yes & ngc4485 & 2.41 & 1.53 & 83 & ngc4490 & 37.23 & l & & ngc4490 & -2.41 & 0.55 & 83 & ngc4485 & 37.78 & l & & ngc4496a&0.00 & 0.23 & 6 & ngc4496 & 38.47 & l & yes & merging ngc4517 & -2.70 & 1.63 & 410 & ngc4517a&37.37 & l & & ngc4532 & -2.66 & 4.26 & 31 & holmberg vii&39.69 & l & yes & ngc4535 & -1.83 & 4.25 & 736 & ngc4519 & 39.72 & l & yes & ngc4536 & -2.86 & 1.11 & 51 & ngc4533 & 39.38 & l & yes & ngc4567 & 0.61 & 0.44 & 11 & ngc4568 & 38.78 & l & yes & pair ngc4568 & -0.61 & 0.28 & 11 & ngc4567 & 38.95 & l & yes & pair ngc4618 & -1.72 & 1.99 & 66 & ngc4625 & 38.16 & l & & ngc4631 & -1.49 & 2.08 & 38 & ngc4656 & 37.42 & l & & ngc4647 & 2.11 & 0.87 & 302 & ngc4649 & 38.52 & l & yes & ngc4654 & -1.10 & 3.57 & 29 & ngc4639 & 39.07 & l & yes & ngc4656 & 0.00 & 0.24 & 27 & ngc4657 & 37.95 & l & & merging ngc5112 & -0.94 & 3.34 & 24 & ngc5107 & 38.19 & l & & ngc5364 & -0.08 & 2.10 & 103 & ngc5363 & 38.14 & l & & ngc5775 & -1.25 & 1.01 & 111 & ngc5774 & 38.76 & l & & ngc5905 & -0.07 & 3.27 & 82 & ngc5908 & 40.25 & l & & ngc5962 & -2.50 & 3.34 & 42 & ugc9925 & 39.21 & l & & ngc0777 & -2.05 & 2.86 & 448 & ngc0778 & 38.73 & e & & ngc1068 & -1.25 & 4.34 & 140 & ngc1055 & 41.55 & l & & ngc3031 & -1.47 & 1.34 & 240 & ngc3034 & 37.64 & l & & ngc3227 & -1.27 & 0.40 & 165 & ngc3226 & 40.38 & l & & pair ngc3735 & -2.60 & 3.96 & 87 & ugc6532 & 39.82 & l & & ngc4168 & -2.07 & 1.06 & 424 & ngc4165 & 37.60 & e & yes & ngc4258 & -2.79 & 1.83 & 578 & ngc4217 & 38.35 & l & & ngc4565 & -1.45 & 4.28 & 126 & ngc4494 & 37.97 & l & & ngc4579 & -1.67 & 4.89 & 379 & ngc4564 & 39.44 & l & yes & ngc4725 & -2.37 & 2.26 & 16 & ngc4747 & 38.19 & l & & ngc5194 & -1.71 & 0.40 & 22 & ngc5195 & 38.88 & l & & pair ngc5395 & -1.27 & 0.66 & 15 & ngc5394 & 38.87 & l & & ngc0410 & -1.31 & 2.24 & 352 & ngc0407 & 39.43 & e & & ngc0521 & -1.30 & 2.53 & 411 & ic1694 & 39.16 & l & & ngc0524 & -2.80 & 3.48 & 11 & ngc0516 & 38.57 & s0 & & ngc0660 & -1.47 & 2.55 & 78 & ugc1195 & 38.89 & l & & ic1727 & 0.83 & 1.12 & 83 & ngc0672 & 37.69 & l & & pair ngc1055 & 1.25 & 3.96 & 142 & ngc1068 & 37.92 & l & & ngc1161 & -0.71 & 1.00 & 588 & ngc1160 & 38.70 & s0 & & ngc3245 & -1.59 & 2.91 & 26 & ngc3245a&39.59 & s0 & & ngc3627 & -0.52 & 2.17 & 80 & ngc3623 & 38.50 & l & & ngc3628 & -0.34 & 2.41 & 39 & ngc3623 & 36.87 & l & & ngc3917 & -2.77 & 2.22 & 133 & ngc3917a&37.29 & l & & ngc4145 & -2.70 & 2.16 & 154 & ugc7175 & 37.72 & l & & ngc4216 & -2.75 & 1.45 & 101 & ngc4222 & 38.53 & l & yes & ngc4220 & -0.99 & 4.68 & 250 & ngc4218 & 38.26 & l & & ngc4281 & -2.25 & 1.93 & 186 & ngc4277 & 38.61 & s0 & & ngc4350 & -0.08 & 2.07 & 319 & ngc4340 & 38.26 & s0 & yes & ngc4435 & 1.12 & 1.64 & 710 & ngc4438 & 38.98 & s0 & yes & ngc4459 & -2.64 & 2.51 & 293 & ngc4468 & 38.75 & s0 & yes & ngc4527 & -2.78 & 3.15 & 19 & ngc4533 & 38.86 & l & yes & ngc4552 & -2.20 & 3.33 & 862 & ngc4551 & 38.52 & e & yes & ngc4569 & -2.53 & 3.87 & 430 & ngc4531 & 39.91 & l & yes & ngc5354 & 0.36 & 0.85 & 352 & ngc5353 & 38.71 & s0 & & ngc5746 & -1.87 & 2.36 & 147 & ngc5740 & 38.48 & l & & ngc5846 & -2.58 & 1.75 & 324 & ngc5845 & 38.81 & e & & lrrrrrrrrrr all galaxies & 46&12 ( 26%@xmath548%)&193&51 ( 26%@xmath544% ) & 88&40 ( 45%@xmath547%)&63&24 ( 38%@xmath548%)&61&26 ( 43%@xmath548%)excluding virgo & 37&10 ( 27%@xmath549%)&168&38 ( 23%@xmath544% ) & 73&33 ( 45%@xmath548%)&49&17 ( 35%@xmath548%)&35&13 ( 37%@xmath5410%)ellipticals & 3 & 2 ( 67%@xmath5447% ) & 0 & 0 & 19&13 ( 68%@xmath5419% ) & 5 & 3 ( 60%@xmath5435%)&22&12 ( 55%@xmath5416%)s0 s & 10 & 0 & 9 & 1 ( 11%@xmath5411% ) & 20&12 ( 60%@xmath5417%)&18 & 8 ( 44%@xmath5416%)&34&13 ( 38%@xmath5411%)late types & 33&10 ( 29%@xmath5410%)&184&50 ( 27%@xmath544% ) & 49&15 ( 31%@xmath548%)&40&13 ( 33%@xmath549%)&5&1 ( 20%@xmath5420%)l@xmath55 & 22 & 8 ( 36%@xmath5413%)&112&23 ( 21%@xmath544% ) & 57&24 ( 42%@xmath549%)&52&20 ( 38%@xmath549% ) & 0 & 0 l@xmath43 & 24 & 4 ( 17%@xmath548% ) & 81&28 ( 35%@xmath547% ) & 31&16 ( 52%@xmath5413%)&11&4 ( 36%@xmath5418% ) & 0 & 0 lrrrrrrrrrr all galaxies&44.5%&3.3%&0.2%&77.8%&16.9%&6.5%&65.9%&96.3%&7.7%&1.6%all galaxies distance @xmath563 diameters&13.6%&0.6%&@xmath560.1%&22.5%&16.2%&2.4%&79.8%&82.6%&10.5%&1.0%excluding virgo&24.7%&6.5%&0.04%&42.8%&44.8%&8.7%&81.7%&56.7%&35.8%&7.2%ellipticals&72.2%&&&36.4%&&&82.5%&&&ellipticals @xmath47 s0s&21.0%&&&6.2%&&&79.6%&&&late types&73.5%&97.6%&66.4%&14.5%&84.0%&52.3%&13.0%&73.8%&15.1%&17.0%late types @xmath47 s0s&73.5%&8.2%&4.8%&73.9%&16.3%&14.8%&97.5%&66.8%&20.9%&23.0%l@xmath55&69.8%&64.1%&0.3%&&86.5%&1.5%&&10.8%&&l@xmath43&87.2%&5.9%&62.8%&&19.7%&90.6%&&9.4%&&

the comparison of the percentage of galaxies with nearby companions showed that there is a higher percentage of liners , transition , and absorption line galaxies with companions than seyferts and hii galaxies . however , we find that when we consider only galaxies of similar morphological types ( ellipticals or spirals ) , there is no difference in the percentage of galaxies with companions among different activity types , indicating that the former result was due to the morphology - density effect . also , only small differences are found when we consider galaxies with similar h luminosities . the fact that we find that galaxies of different activity types have the same percentage of companions , suggests that interactions between galaxies is not a necessary condition to trigger the nuclear activity in agns . we compare our results with previous ones and discuss their implications .

we analyze the idea that nuclear activity , either agn or star formation , can be triggered by interactions , studying the percentage of active , hii and quiescent galaxies with companions . our sample was selected from the palomar survey , and avoids selection biases faced by previous studies . this sample was split into 5 different groups , seyfert galaxies , liners , transition galaxies , hii galaxies and absorption line galaxies . the comparison between the local galaxy density distributions of the different groups showed that in most cases there is no statistically significant difference among galaxies of different activity types , with the exception that absorption line galaxies are seen in higher density environments , since most of them are in the virgo cluster . the comparison of the percentage of galaxies with nearby companions showed that there is a higher percentage of liners , transition , and absorption line galaxies with companions than seyferts and hii galaxies . however , we find that when we consider only galaxies of similar morphological types ( ellipticals or spirals ) , there is no difference in the percentage of galaxies with companions among different activity types , indicating that the former result was due to the morphology - density effect . also , only small differences are found when we consider galaxies with similar h luminosities . the comparison between hii galaxies of different h luminosities shows that there is a significantly higher percentage of galaxies with companions among hii galaxies with l(h) erg s , than among those with l(h) erg s , indicating that interactions increase the amount of circumnuclear star formation , in agreement with previous results . the fact that we find that galaxies of different activity types have the same percentage of companions , suggests that interactions between galaxies is not a necessary condition to trigger the nuclear activity in agns . we compare our results with previous ones and discuss their implications .

astro-ph0703350

i

fluorescent fe k lines are common in the x - ray spectra of accreting black holes , ranging from x - ray binaries to active galactic nuclei ( agns ) , and provide one of the best probes for studying accretion disks . if the fe k line is emitted from the inner part of the accretion disk , it becomes broad and asymmetric due to both doppler shift and gravitational redshift ( fabian et al . 1989 ) . the best example of a broad fe k line is seen in the seyfert 1 galaxy mcg63015 ( tanaka et al . 1995 ) . broad fe k lines have been observed in many other sources and are thought to be very common in the x - ray spectra of seyfert 1 nuclei ( nandra et al . 1997a ) . recent observations of agns with _ xmm - newton _ show that the broad fe k line is not as common as previously believed . while a few seyfert 1 galaxies ( e.g. , mcg63015 , mrk 205 , mrk 509 ) indeed show an unambiguous broad line , fe lines in other galaxies are dominated by a relatively narrow feature . from an analysis of 53 type 1 agns observed with _ xmm - newton _ , page et al . ( 2004a ) showed that a broad fe k line ( @xmath4kev ) is seen in only 13 sources , while the remaining 40 sources have a narrow line ( @xmath5kev ) or no lines . some of the 13 sources show a narrow line as well as the broad line . in a similar study , yaqoob & padmanabhan ( 2004 ) reported the results of 18 observations of 15 seyfert 1 galaxies observed with the _ chandra _ high energy grating . they measured the width of the line core and obtained a weighted mean of fwhm = 2380@xmath6 760 kms@xmath7 , which is slightly larger than the instrument resolution ( fwhm @xmath8 1860 kms@xmath7 ) . evidence of underlying broad - line emission was also found in at least four sources . based on these recent studies , it is still controversial whether or not relativistically broadened fe k emission is truly common in nearby agns . streblyanska et al . ( 2005 ) derived an average rest - frame spectrum of agns detected in a 770 ksec _ xmm - newton _ observation of the lockman hole field . they used a sample of 104 x - ray sources with optical redshifts measured by lehmann et al . ( 2001 ) , analyzing separately the type 1 and type 2 subsamples defined by optical spectra ( schmidt et al . 1998 ; lehman et al . 2001 ) . from composite rest - frame spectra generated for the type 1 and type 2 sources , they found evidence for a broad line peaking at a rest - frame energy @xmath96.4kev with an equivalent width ( ew ) @xmath9560ev and @xmath9460ev , respectively . however , it should be noted that there are some systematic uncertainties in modelling the continuum , since it is virtually impossible to distinguish a very broad line clearly from the continuum for sources as faint as those analyzed by streblyanska et al . in addition , making composite spectra of faint sources may introduce spurious spectral features ( yaqoob 2005 ) . brusa et al . ( 2005 ) and comastri et al . ( 2007 ) studied average spectra of agns detected in the _ chandra _ deep field north and south . they stacked the x - ray counts in the observed frame from spectroscopically identified agns , using a large number of source spectra within sufficiently narrow redshift ranges such that the energy ( redshift ) spread is negligible . a broad fe line is seen in some of the stacked spectra and is fitted with a relativistic disk line model peaking at a rest - energy of 6.4 kev . since the fluxes of the objects in the sample are low , this study may suffer from the same uncertainties affecting the streblyanska et al . analysis of the lockman hole . what parameters of the agn affect the strength or profile of the fe k line ? an inverse correlation between the equivalent width of fe k line and x - ray luminosity , known as the x - ray baldwin effect , was first pointed out by iwasawa & taniguchi ( 1993 ) . this correlation has been confirmed with _ asca _ ( nandra et al . 1997b ) and _ xmm - newton _ data ( page et al . 2004a ; see also jimnez - bailn et al . the dependence of the iron line profile with luminosity was investigated by nandra et al . ( 1997b ) using 18 seyfert 1s and 21 quasars observed with _ asca_. they divided the sample into five luminosity bins and examined average fe k line profiles . the line shows a very similar profile , which is composed of a narrow core and a broad red wing for groups with a luminosity below @xmath10 . the intensity of the narrow core , however , becomes weak above this luminosity . the red wing becomes weak and the peak energy shifts to higher energy for the luminosity range of @xmath11 ; no evidence for line emission is observed above @xmath12 . jimnez - bailn et al . ( 2005 ) , analyzing a sample of 39 palomar - green ( pg ; schmidt & green 1983 ) quasars observed with _ xmm - newton _ , found a similar result as that reported by nandra et al . ( 1997b ) . the luminosity dependence of the fe k line may be related to the ionization level of the accretion disk . this paper examines the behavior of the fe k line as a function of accretion rate , as parameterized by the eddington ratio , taking advantage of the availability of black hole mass and bolometric luminosity estimates . the eddington ratio is a fundamental parameter that may control both the ionization state and geometric structure of the accretion disk . if so , we expect the profile of the fe k line , in particular its width and central energy , to vary systematically with eddington ratio . the goal of this study is to investigate this issue . this paper is organized as follows . section 2 describes the observations , and section 3 shows the results of the spectral analysis of average spectra and its eddington ratio dependence . we discuss the physical origin of the eddington ratio dependence in section 4 . our findings are summarized in section 5 .

we analyze x - ray spectra of 43 palomar - green quasars observed with _ a feature resembling an emission line at 6.4 kev , identified with an fe k line , is detected in 33 objects . the fe line is relatively narrow (kev ) , with a center energy of 6.48kev and a mean equivalent width ( ew ) of 248ev . by combining black hole masses estimated from the virial method and bolometric luminosities derived from full spectral energy distributions , this result suggests that the accretion rate onto the black hole directly influences the geometrical structure and ionization state of the accretion disk .

we analyze x - ray spectra of 43 palomar - green quasars observed with _ xmm - newton _ in order to investigate their mean fe k line profile and its dependence on physical properties . the continuum spectra of 39 objects are well reproduced by a model consisting of a power law and a blackbody modified by galactic absorption . the spectra of the remaining four objects require an additional power - law component absorbed with a column density of . a feature resembling an emission line at 6.4 kev , identified with an fe k line , is detected in 33 objects . approximately half of the sample show an absorption feature around 0.650.95 kev , which is due to absorption lines and edges of and . we fit the entire sample simultaneously to derive average fe line parameters by assuming a common fe line shape . the fe line is relatively narrow (kev ) , with a center energy of 6.48kev and a mean equivalent width ( ew ) of 248ev . by combining black hole masses estimated from the virial method and bolometric luminosities derived from full spectral energy distributions , we examine the dependence of the fe k line profile on eddington ratio . as the eddington ratio increases , the line becomes systematically stronger ( ew = 130 to 280 ev ) , broader ( to 0.7kev ) , and peaks at higher energies ( 6.4 to 6.8kev ) . this result suggests that the accretion rate onto the black hole directly influences the geometrical structure and ionization state of the accretion disk . we also examine a two - component model consisting of a gaussian and a diskline to constrain the intensity of the broad line . the mean equivalent widths areev for the four eddington ratio groups , although the standard deviations in each group are very large . this suggests that the broad line is not ubiquitous .

astro-ph0703350

r

the spectra of most objects are not well represented by a simple power law ; soft excess emission and an emission line - like feature around 6 kev are also often seen . therefore , we considered a model consisting of a power law , a blackbody component to represent the soft excess emission below @xmath9 2kev , and a gaussian emission line around 6 kev , all modified by galactic absorption . we used a two blackbody model for the soft excess if the single blackbody did not represent the data well . the hydrogen column densities of the galactic absorption are left free . some objects show absorption features around 0.650.95 kev . we multiplied one or two edges to the above model for such cases . the spectra of 39 objects are well reproduced by this model . we obtained hydrogen column densities of @xmath27 @xmath8 ( 0.003@xmath240.26)@xmath28 , photon indices @xmath29 1.5 to 2.5 , and temperatures of the blackbody @xmath30kev . the column densities are consistent with the galactic value derived from the hi map by dickey & lockman ( 1990 ) . except for a few cases ( see appendix ) , our results for individual objects are in good agreement with the independent analysis published in the references given in table 1 . the spectra of the remaining four objects ( pg 1004 + 130 , 1411 + 442 , 1535 + 547 , and 2214 + 139 ) show broad excess emission at energies above @xmath9 23 kev . we added an absorbed power - law component to the model used above to represent this component . this model provides a good description of the spectra . the hydrogen column densities of the absorbed power law are @xmath31 for all the cases . details of the spectral fits to these four quasars are given in appendix[appe : pg qso ] . the spectral parameters for the other components are similar to those obtained for the 39 objects without an additional absorbed power law . an absorption feature around 0.650.95 kev , which can be well fitted by one or two edges , is seen in 20 out of the 43 objects . the majority of the sources ( 18/20 ) require only a single edge , with a mean edge energy of @xmath32kev and an optical depth of @xmath33 . the absorption features in the two remaining objects are represented by two edges : for pg 1211 + 143 , the edge energies are @xmath34/@xmath35kev and the optical depths are @xmath36/@xmath37 ; for pg 1404 + 226 , the prameters are @xmath38/@xmath39kev and @xmath40/@xmath41 ) . similar features are often seen in seyfert 1s and quasars observed with the energy resolution of ccd . high - resolution spectroscopy of such features with grating spectrometers onboard _ xmm - newton _ and _ chandra _ have shown that they consist of many absorption lines of ionized species such as he - like and hydrogenic o , ne , and mg . the observed edge energies are in agreement with such ionized absorbers observed with ccd resolution ( e.g. , george et al . 1998 , 2000 ) . the fluxes and luminosities obtained for the best - fit model , as well as the detected count rates , are summarized in table 2 . neither the fluxes nor the luminosities are corrected for absorption . we derived a co - added rest - frame spectrum of the 43 quasars in order to examine the average shape of the fe k emission line . since each object has a different redshift and thus a different observed line energy , the detector response , which is energy dependent , is different for each object . as the co - added rest - frame spectrum is a summation of spectra taken with different energy resolution , this situation complicates quantitative analysis . therefore , the spectrum derived here is for presentation purpose only , and quantitative analysis by simultaneous fitting is done in the next subsection . the co - added spectrum was constructed as follows . first , ratios of the data to the best - fit continuum model were calculated . the ratios were then multiplied by the unfolded best - fit continuum model . the resulting spectrum is composed of an unfolded continuum and a folded emission - line component . the energy scales of both the unfolded continuum spectrum and the spectrum consisting of the unfolded continuum plus the folded line were shifted to the source redshift . we selected bin widths of 0.25kev for @xmath428kev and 2.5kev for @xmath438kev in this process , and used a monte carlo method to redistribute the observed events into the new spectral bins . the 43 spectra with the folded line are then co - added . the unfolded continuum model spectra were also co - added . finally , the ratio of the two spectra is calculated , as shown in figure[fig : 43pg ratio ] . a prominent narrow core at @xmath96.4kev and a very weak low - energy wing are seen . in contrast to previous studies , we employ simultaneous fits rather than fits to stacked spectra . since the fe k line in each quasar has a different peak energy in the observed frame according to their redshifts , a technique that properly treats the energy - dependent detector response is necessary . we fitted all spectra simultaneously in order to determine mean fe k line parameters . the spectral parameters of the best - fit continuum models for each object were used as initial parameters in the spectral fits . all the continuum parameters were left free , while the parameters for the edge model are fixed at the best - fit values for individual fits . a gaussian model with common peak energy and width is added to the continuum model . the normalizations of the gaussian component were left free individually . the fitting results are shown in table[tab : fitting 43pg ] . the best - fit peak energy and the dispersion of the gaussian line are @xmath44kev and @xmath45kev , respectively . the mean of the distribution of the equivalent widths , shown in figure[fig : ew distribution all ] , is 248ev , with a standard deviation of 168ev . a relativistic line model ( diskline model in xspec ) instead of a gaussian was also examined . we fixed several parameters ; the peak energy at 6.4kev , the outer radius @xmath46 at 500@xmath47 , where @xmath48 is the schwarzschild radius , the emissivity index @xmath14 at @xmath242 , and the inclination angle of the disk @xmath49 at 30 degrees . the free parameters are the inner radius @xmath50 and the intensity of the line . again , the parameters in the continuum model were initially set to the best - fit values in the individual fits and left free except for the edge component . the results are summarized in table[tab : fitting 43pg ] . the best - fit inner radius @xmath51 is @xmath52 . the lower boundary is fixed at @xmath53 , which is the minimum value allowed in the diskline model assuming a schwarzschild black hole . the mean and standard deviation of the equivalent widths are 210ev and 133ev , respectively . the line profile in figure[fig : 43pg ratio ] shows a prominent narrow core and a very weak low - energy wing . such a profile may be indicative of the presence of multiple line components , and a single gaussian or a diskline profile may not be a good representation . therefore , we examined a two - component line model consisting of either two gaussians or a combination of a gaussian and a diskline component . with regards to the double - gaussian model , although the @xmath26 improved by 103 for 3 additional parameters , not all the line parameters were tightly constrained . thus , we first fixed the continuum component to obtain a rough value of the gaussian widths , and then the widths were fixed at these values to constrain the equivalent widths of the lines . all the parameters except for the widths were left free in the latter fit . note that the normalizations of the lines were determined for each object . the resulting peak energies of the broad and narrow lines are 5.06kev and 6.52kev , respectively ( see table[tab : fitting 43pg ] ) . the mean equivalent width for the narrow component is 261 ev , and for the broad component it is 150 ev , although the standard deviations of the distributions are large ( 171 and 181 ev , respectively ) . the distributions suggest that the narrow component is commonly seen and that the broad line is present in a fraction of the sample . the gaussian plus diskline model was then examined . we fixed the peak energy and the dispersion of the gaussian to the best - fit parameters of the narrow component in the double - gaussian fit . the fixed parameters for the diskline component were the same as those in the single diskline fit presented above . the free parameters are the inner radius and the normalization of the diskline , and the parameters for the gaussian and continuum . as summarized in table[tab : fitting 43pg ] , the best - fit inner radius is @xmath50 = @xmath54 , and the mean equivalent width of the diskline is ew=102ev . the large standard deviation ( 181 ev ) again suggests that the diskline is required in only a fraction of the sample . ccccccccccc & & + model & energy & @xmath55 & ew & energy & @xmath50 & @xmath46 & @xmath14 & @xmath49 & ew & @xmath26/d.o.f . + & ( kev ) & ( kev ) & ( ev ) & ( kev ) & ( @xmath48 ) & ( @xmath48 ) & & ( deg ) & ( ev ) & + gaussian & @xmath56 & @xmath57 & @xmath58 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & 8958/8129 + diskline & @xmath24 & @xmath24 & @xmath24 & 6.4f & @xmath59 & 500f & @xmath242f & 30f & @xmath60 & 8975/8130 + gaussian@xmath61gaussian & @xmath62 & 0.61f & 150(181 ) & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & + & @xmath63 & 0.34f & 261(171 ) & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & 8853/8086 + gaussian@xmath61diskline & 6.52f & 0.34f & 155(181 ) & 6.4f & @xmath64 & 500f & @xmath242f & 30f & @xmath65 & 8912/8087 + we now turn to the eddington ratio dependence of the fe line profile . the 43 quasars were divided into four groups according to the eddington ratio as follows : ( 1)@xmath66 , ( 2)@xmath67 , ( 3)@xmath68 , and ( 4)@xmath69 ( see table[tab : the sample ] ) . the spectra in each group were co - added as in section [ subsec : fitting 43pg ] for the purposes of presentation ; figure[fig : 10pg ratio ] shows the resulting data - to - continuum model ratios . as the eddington ratio increases , the width of the fe k line appear to becomes broader and its peak energy rises . following the procedure outlined in section [ subsec : fitting 43pg ] , we fitted the spectra in each group simultaneously to characterize the fe k line quantitatively . the following four models were examined for the fe k line component : single gaussian , single diskline , double gaussians , and a gaussian plus diskline model . the results of the spectral fits are shown in table[tab : fitting 10pg ] , and the eddington ratio dependence of the peak energy and the width of the line obtained from the single gaussian model fits are shown in figure[fig : dependence of the param ] . we again find that the width and peak energy of the line become broader and higher as the eddington ratio increases . the peak energy gradually increases from 6.37 kev in group 4 to 6.77 kev in group 1 . the width is relatively narrow ( @xmath55 = 0.15kev ) in group 4 , but definitely resolved ( @xmath70 = 0.07kev ) , and becomes wider ( @xmath55 = 0.68 kev in group 1 ) as @xmath71 increases . the distributions of the equivalent widths for each group obtained from the gaussian fits are shown in figure[fig : ew distribution ] ; the mean equivalent widths in each group are 280 , 276 , 287 , and 131ev , respectively . a diskline model was examined next . we fixed several diskline parameters . the line center energy was chosen to be 6.4 kev for groups 3 and 4 and 6.7 kev for groups 1 and 2 , for which the gaussian fits suggest that the line is from ionized fe . the outer radius @xmath46 , the emissivity index @xmath14 , and the inclination angle @xmath49 were fixed at 500@xmath72 , @xmath242 , and 30 degrees , respectively . the free parameters are the inner radius @xmath50 and the normalization . the parameters for the continuum model were also left free , except for those of the edge component . the results are shown in table[tab : fitting 10pg ] . the resulting values of @xmath26 are similar to or slightly worse than those for the gaussian fits . the best - fit inner radius is @xmath73 , which is the last stable orbit of a schwartzschild black hole . we also examined a two - component line model since a strong narrow core and a broad wing are visible in groups 3 and 4 , and possibly in group 2 . a double - gaussian model was first tested in the same way as in section[subsec : fitting 43pg ] ; the results are shown in table[tab : fitting 10pg ] . the resulting @xmath26 values are similar to or only slightly better than those of the single - gaussian fit , indicating that the additional broad line is statistically not required . the distributions of the ew are broad and indicate that some objects may have a broad component . a gaussian plus diskline model was then used in order to constrain the equivalent width of the diskline , in which the inner radius is fixed at 3@xmath72 . the peak energy and the dispersion of the gaussian were fixed at the best - fit parameters for the narrow component in the double - gaussian fit . the inner radius of the disk was also fixed at 3@xmath72 , and the other parameters in the diskline model were fixed to the same values as in the single - diskline fits described above . only the normalizations of the gaussian and diskline were left free . the parameters for the continuum model were treated in a similar way as in section 3.3 . the results are summarized in table[tab : fitting 10pg ] . the resulting values of @xmath26 are similar to or only slightly better than those for the single - gaussian or diskline fits . the mean equivalent widths of the diskline in each group are 99 , 176 , 132 , and 67ev , respectively , although the standard deviations as large . the large standard deviations again suggest that the broad diskline is not ubiquitous . cccccccccccc & & & + group & model & energy & @xmath55 & ew & energy & @xmath50 & @xmath46 & @xmath14 & @xmath49 & ew & @xmath26/d.o.f . + & & ( kev ) & ( kev ) & ( ev ) & ( kev ) & ( @xmath48 ) & ( @xmath48 ) & & ( deg ) & ( ev ) & + 1 & gaussian & @xmath74 & @xmath75 & @xmath76 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & 2308/2237 + & diskline & @xmath24 & @xmath24 & @xmath24 & 6.7f & @xmath77 & 500f & @xmath242f & 30f & @xmath78 & 2316/2238 + & gaussian@xmath61gaussian & @xmath79 & 0.42f & @xmath80 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & + & & @xmath81 & 0.48f & @xmath82 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & 2305/2248 + & gaussian@xmath61diskline & 7.47f & 0.48f & @xmath83 & 6.7f & 3.0f & 500f & @xmath242f & 30f & @xmath84 & 2304/2250 + 2 & gaussian & @xmath85 & @xmath86 & @xmath87 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & 2636/2197 + & diskline & @xmath24 & @xmath24 & @xmath24 & 6.7f & @xmath88 & 500f & @xmath242f & 30f & @xmath89 & 2662/2198 + & gaussian@xmath61gaussian & @xmath90 & 0.38f & @xmath91 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & + & & @xmath92 & 0.24f & @xmath93 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & 2634/2218 + & gaussian@xmath61diskline & 6.52f & 0.24f & @xmath94 & 6.7f & 3.0f & 500f & @xmath242f & 30f & @xmath95 & 2646/2220 + 3 & gaussian & @xmath96 & @xmath97 & @xmath98 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & 1796/1774 + & diskline & @xmath24 & @xmath24 & @xmath24 & 6.4f & @xmath99 & 500f & @xmath242f & 30f & @xmath100 & 1801/1775 + & gaussian@xmath61gaussian & @xmath101 & 0.48f & @xmath102 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & + & & @xmath103 & 0.25f & @xmath104 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & 1800/1798 + & gaussian@xmath61diskline & 6.49f & 0.25f & @xmath105 & 6.4f & 3.0f & 500f & @xmath242f & 30f & 132 ( 132 ) & 1803/1800 + 4 & gaussian & @xmath106 & @xmath107 & @xmath108 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & 2184/1935 + & diskline & @xmath24 & @xmath24 & @xmath24 & 6.4f & @xmath109 & 500f & @xmath242f & 30f & @xmath110 & 2186/1936 + & gaussian@xmath61gaussian & @xmath111 & 0.70f & @xmath112 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & + & & @xmath113 & 0.09f & 82 ( 70 ) & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & @xmath24 & 2173/1957 + & gaussian@xmath61diskline & 6.37f & 0.09f & @xmath114 & 6.4f & 3.0f & 500f & @xmath242f & 30f & @xmath115 & 2180/1959 +

xmm - newton _ in order to investigate their mean fe k line profile and its dependence on physical properties . approximately half of the sample show an absorption feature around 0.650.95 kev , which is due to absorption lines and edges of and . we fit the entire sample simultaneously to derive average fe line parameters by assuming a common fe line shape . we also examine a two - component model consisting of a gaussian and a diskline to constrain the intensity of the broad line . the mean equivalent widths areev for the four eddington ratio groups , although the standard deviations in each group are very large . this suggests that the broad line is not ubiquitous .

we analyze x - ray spectra of 43 palomar - green quasars observed with _ xmm - newton _ in order to investigate their mean fe k line profile and its dependence on physical properties . the continuum spectra of 39 objects are well reproduced by a model consisting of a power law and a blackbody modified by galactic absorption . the spectra of the remaining four objects require an additional power - law component absorbed with a column density of . a feature resembling an emission line at 6.4 kev , identified with an fe k line , is detected in 33 objects . approximately half of the sample show an absorption feature around 0.650.95 kev , which is due to absorption lines and edges of and . we fit the entire sample simultaneously to derive average fe line parameters by assuming a common fe line shape . the fe line is relatively narrow (kev ) , with a center energy of 6.48kev and a mean equivalent width ( ew ) of 248ev . by combining black hole masses estimated from the virial method and bolometric luminosities derived from full spectral energy distributions , we examine the dependence of the fe k line profile on eddington ratio . as the eddington ratio increases , the line becomes systematically stronger ( ew = 130 to 280 ev ) , broader ( to 0.7kev ) , and peaks at higher energies ( 6.4 to 6.8kev ) . this result suggests that the accretion rate onto the black hole directly influences the geometrical structure and ionization state of the accretion disk . we also examine a two - component model consisting of a gaussian and a diskline to constrain the intensity of the broad line . the mean equivalent widths areev for the four eddington ratio groups , although the standard deviations in each group are very large . this suggests that the broad line is not ubiquitous .

astro-ph0703350

c

we examined mean fe k line shapes for the four groups sorted by the eddington ratios , and found that the peak energy , width , and equivalent width become higher ( 6.4 to 6.8kev ) , broader ( 0.1 to 0.7kev ) , and larger ( 130 to 280ev ) , respectively , as the eddington ratio increases . in this section , we discuss possible reasons of the observed eddington ratio dependence of the fe line profile . the line center energy of the fe k line is governed by a combination of the doppler effect caused by the orbital motion , gravitational redshift , and the ionization state of the emitter . an fe line from a rotating disk around a black hole shows a double - peaked profile with a peak energy that depends on the inclination angle of the disk . since each of the four groups we used contains from 10 to 11 objects , it is unlikely that any one of the groups has a mean inclination angle considerably different from those of other groups . therefore , the different line center energies obtained for the four groups probably reflect different ionization states rather than different mean disk inclination angles . the eddington ratio dependence of the peak energy thus strongly suggests that the ionization state of the accretion disk depends on the eddington ratio . many aspects of an ionized disk have been studied theoretically over the last decade . calculations have been performed for constant - density atmospheres ( e.g. , matt et al . 1993 ; ross & fabian 1993 ; @xmath116ycki et al . 1994 ) , as well as for atmospheres in hydrostatic equilibrium ( nayakshin et al . 2000 ; ballantyne et al . 2001 ) . these authors calculated reflection spectra from an ionized disk for different conditions . in the lowest ionization regime , the most prominent feature is the cold iron line at 6.4kev . as the mass accretion rate goes up , the ionized stage becomes higher , and the fe k line is dominated by fe xxv and xxvi , and k lines from the lighter elements emerge in the @xmath117kev band . at very high accretion rates , the surface of the disk is highly ionized so that the only noticeable line is a compton - broadened fe k line peaking at @xmath97kev . the eddington ratio dependence of the peak energy and the width of the fe k line we observed in the pg quasar sample is in good agreement with the predictions of ionized disk models . the large ew in the high - eddington ratio groups is also likely to be related to the ionization state of the accretion disk . calculations of the reflected x - rays from an ionized disk have shown that the ew of the fe k line can be higher than in the case of a disk in a low - ionization state . the resonant trapping effect plays a role in the intermediate - ionization state higher than . the ew increases again from , reaching a maximum value of @xmath118ev ( matt et al . 1993 ; ross & fabian 1993 ; @xmath116ycki & czerny 1994 ) . the observed line center energies and ews for the three groups with a largest eddington ratios are consistent with the latter expectation , although the scatter of the actual ew distributions is large . the reduction of ew caused by resonant trapping is not clearly seen in our data . this may be due to the coarse binning of the eddington ratio we have adopted . future analysis of a larger sample may be able to see the ionized disk in an intermediate - ionization state . the eddington ratio dependence of the line width also suggests a change of the geometrical structure of the disk , if the velocity width indeed reflects emission from the inner part of an accretion disk broadened by doppler effects and gravitational redshift . although the line widths obtained from the single - gaussian fits are consistent with systematic variations in the inner disk radius @xmath50 , such a trend is not clear in the fits obtained using the diskline model . it is possible that the broader line widths at high eddington ratios may in part be accounted for by a blend of lines from fe in different ionization states . streblyanska et al . ( 2005 ) derived an average fe line profile of 53 type 1 agn detected in a 770 ks observation of the lockman hole with _ xmm - newton_. their average spectrum shows a broad line - like feature ( @xmath119kev ) peaking at 6.4kev with an equivalent width of ew = 420ev . the profile is asymmetric , with a significant wing seen toward low energies . the line width ( @xmath55 = 0.36kev ) we obtained for the pg quasar sample is significantly narrower , and the equivalent width ( ew@xmath8250ev ) smaller , than that of streblyanska et al.s profile . we discuss possible causes for the difference . appropriate modeling of the continuum is essential to derive a correct fe line profile . a single power - law model was assumed in the analysis of the agns in the lockman hole . since the spectra of many agns show a signature of absorption by ionized and/or cold matter , a simple power law may not be a good approximation of the underlying continuum . indeed , four quasars in our sample are absorbed by a large column density ( @xmath120 @xmath121 ) . if there are some highly absorbed objects in the lockman hole sample , using a single power - law model for the continuum may mimic a broad - line profile . the flux of a highly absorbed power law with a column density of @xmath122 at 5kev is about 50% of that of an unabsorbed power law . if the fraction of highly absorbed quasars is 1/10 , their contribution to the low - energy side of the fe k line profile around 5 kev is less than 5% . the contribution of the absorbed continuum depends on the fraction of absorbed quasars . mateos et al . ( 2005 ) measured the absorbed fraction for their sample of lockman hole agns with sufficient x - ray counts . they found that seven out of 46 type 1 agns show a highly absorbed spectrum . this fraction is similar to that in our pg quasar sample . note , however , that streblyanska et al.s sample may contain more highly absorbed objects because their flux limit is lower than that used by mateos et al . ( 2005 ) and x - ray surveys have shown that objects with lower flux tend to have a harder spectrum . if the absorbed fraction is 10% or higher , rather than 5% , in the lockman hole sample , a considerable fraction of the red wing might be explained by the contribution of highly absorbed emission . the physical state of an accretion disk may depend on luminosity . if the mean luminosity of our sample is fairly different from that of the lockman hole agns , it may introduce a difference in the fe k line profile . in order to examine this possibility , we compared the distributions of the @xmath123kev luminosity between the two samples ( fig.[fig : lumin distri ] ) . the luminosities of the lockman hole agns were taken from mainieri et al . they analyzed 39 type 1 agns in the _ xmm - newton _ observation of the lockman hole field . most of the 53 type 1 agns in streblyanska et al . ( 2005 ) have been analyzed by mainieri et al . the two distributions are very similar , and their mean luminosities are essentially identical : the mean luminosity of our sample is @xmath124ergss@xmath7 , while that of the lockman hole agns is @xmath125ergss@xmath7 . we conclude that luminosity differences is not the main cause of the difference of the line profiles . yaqoob ( 2005 ) pointed out a spurious effect in adding spectra of faint sources . he showed that conventional methods for averaging low signal - to - noise ratio x - ray spectra result in a weakening of emission lines , a broad dip in the continuum above an emission line , and a spectral hardening at the highest energies , if sources at different redshifts are stacked . yaqoob ( 2005 ) simulated 200 spectra from sources with redshifts 0.52.5 and 210 kev fluxes 660@xmath126 ergs @xmath121 s@xmath7 and found an artifact wing at energies below the peak of the fe k line . this feature occurs at the level of about 5%10% at 46 kev . this kind of spectral distortion may affect the fe line shape in the analysis of streblyanska et al . ( 2005 ) since their sample has a relatively low average flux ( @xmath127 ) , compared to our pg quasar sample ( @xmath128 ) .

we examine the dependence of the fe k line profile on eddington ratio . as the eddington ratio increases , the line becomes systematically stronger ( ew = 130 to 280 ev ) , broader ( to 0.7kev ) , and peaks at higher energies ( 6.4 to 6.8kev ) .

we analyze x - ray spectra of 43 palomar - green quasars observed with _ xmm - newton _ in order to investigate their mean fe k line profile and its dependence on physical properties . the continuum spectra of 39 objects are well reproduced by a model consisting of a power law and a blackbody modified by galactic absorption . the spectra of the remaining four objects require an additional power - law component absorbed with a column density of . a feature resembling an emission line at 6.4 kev , identified with an fe k line , is detected in 33 objects . approximately half of the sample show an absorption feature around 0.650.95 kev , which is due to absorption lines and edges of and . we fit the entire sample simultaneously to derive average fe line parameters by assuming a common fe line shape . the fe line is relatively narrow (kev ) , with a center energy of 6.48kev and a mean equivalent width ( ew ) of 248ev . by combining black hole masses estimated from the virial method and bolometric luminosities derived from full spectral energy distributions , we examine the dependence of the fe k line profile on eddington ratio . as the eddington ratio increases , the line becomes systematically stronger ( ew = 130 to 280 ev ) , broader ( to 0.7kev ) , and peaks at higher energies ( 6.4 to 6.8kev ) . this result suggests that the accretion rate onto the black hole directly influences the geometrical structure and ionization state of the accretion disk . we also examine a two - component model consisting of a gaussian and a diskline to constrain the intensity of the broad line . the mean equivalent widths areev for the four eddington ratio groups , although the standard deviations in each group are very large . this suggests that the broad line is not ubiquitous .

1101.5939

i

in this paper , we have investigated s - wave proton deuteron scattering in pionless effective field theory . in the quartet channel , we have calculated the elastic scattering phase shift up to using the power counting for coulomb contributions suggested by rupak and kong @xcite . the coulomb effects are included at accuracy in our calculation . using an optimised integration mesh we were able to extend their calculation into the threshold region were the coulomb interaction becomes highly non - perturbative . we found good agreement both with available phase shift analyses and with the results of rupak and kong at momenta @xmath160 . moreover , we extended the power counting to the doublet channel and performed a complete calculation of the phase shifts to in agreement with the available phase shift data . we also carried out a partial calculation that neglected the contribution of the subleading three - body force entering at this order . the results of this calculation are stable under variations of the cutoff . furthermore , there is good agreement with the phase shift data at low momenta and room for a small contribution of the neglected three - body force at larger momenta . overall , however , the doublet channel phase shifts are only weakly sensitive to the subleading three - body force entering at . although we were mainly interested in @xmath4@xmath5 scattering , we have also calculated the coulomb contribution to the @xmath0he@xmath0h binding energy difference @xmath161 . this observable has previously been calculated in the pionless theory by treating the coulomb interaction non - perturbatively @xcite . here , we treat the coulomb potential between proton and deuteron in first order perturbation theory using trinucleon wave functions . higher order corrections to this quantity are expected to be small . our result is in reasonable agreement with the experimental value and other evaluations . we find @xmath161 to be more sensitive to the subleading three - body force . the partial result is about 10% too large , thus leaving room for a contribution from the omitted three - body force . we also observe steps in the calculated value of @xmath161 as the cutoff is increased beyond its natural range . whenever the leading three - nucleon force has gone through a pole , a drop in the calculated binding energy occurs . we interpret this as an artifact of the theory related to the efimov effect . at higher cutoffs , spurious deep three - body bound states appear and the triton is not the true ground state anymore . it appears that an additional short - distance counterterm is required to cancel these contributions if one wants to go to cutoffs much larger than the pion mass . a further study of this issue would be interesting . in the future , a full calculation including the subleading three - body force and the electromagnetic interaction terms generated from gauging its momentum dependence should be carried out . such an accuracy will , _ e.g. _ , be required for high - precision calculations of low - energy astrophysical processes in pionless effective field theory and the effective field theory for halo nuclei . we thank a. rusetsky , b. metsch , m. hoferichter and p. hagen for discussions and s .- ando , s. coon , d. r. phillips and u. van kolck for comments on the manuscript . this research was supported in part by the dfg through sfb / tr 16 `` subnuclear structure of matter '' and the bmbf under contract no . 06bn9006 . s.k . was supported by the `` studienstiftung des deutschen volkes '' and by the bonn - cologne graduate school of physics and astronomy .

deuteron scattering in the framework of pionless effective field theory . in the quartet channel , we calculate the elastic scattering phase shift up to next - to - next - to - leading order in the power counting . in the doublet channel , we perform a next - to - leading order calculation . we obtain good agreement with the available phase shift analyses down to the scattering threshold . the phase shifts in the region of non - perturbative coulomb interactions are calculated by using an optimised integration mesh . moreover ,

we calculate low - energy proton deuteron scattering in the framework of pionless effective field theory . in the quartet channel , we calculate the elastic scattering phase shift up to next - to - next - to - leading order in the power counting . in the doublet channel , we perform a next - to - leading order calculation . we obtain good agreement with the available phase shift analyses down to the scattering threshold . the phase shifts in the region of non - perturbative coulomb interactions are calculated by using an optimised integration mesh . moreover , the coulomb contribution to theheh binding energy difference is evaluated in first order perturbation theory . we comment on the implications of our results for the power counting of subleading three - body forces .

1112.5070

i

let @xmath0 be a standard one - dimensional brownian motion , @xmath1 be an integer , and let @xmath2 be a symmetric element of @xmath3 . denote by @xmath4 the @xmath5-tuple wiener - it integral of @xmath2 with respect to @xmath6 . it is well known that multiple wiener - it integrals of different orders are uncorrelated but not necessarily independent . in an important paper @xcite , stnel and zakai gave the following characterization of the independence of multiple wiener - it integrals . [ uzthmintro ] let @xmath7 be integers and let @xmath8 and @xmath9 be symmetric . then , random variables @xmath10 and @xmath11 are independent if and only if @xmath12 rosiski and samorodnitsky @xcite observed that multiple wiener - it integrals are independent if and only if their squares are uncorrelated : @xmath13 this condition can be viewed as a generalization of the usual covariance criterion for the independence of jointly gaussian random variables ( the case of @xmath14 ) . in the seminal paper @xcite , nualart and peccati discovered the following surprising central limit theorem . [ npthmintro ] let @xmath15 , where @xmath16 is fixed and @xmath17 are symmetric . assume also that @xmath18= 1 $ ] for all @xmath19 . then convergence in distribution of @xmath20 to the standard normal law is equivalent to convergence of the fourth moment . that is , as @xmath21 , @xmath22\to 3.\ ] ] shortly afterwards , peccati and tudor @xcite established a multidimensional extension of theorem [ npthmintro ] . since the publication of these two important papers , many improvements and developments on this theme have been considered . in particular , nourdin and peccati @xcite extended theorem [ npthmintro ] to the case when the limit of @xmath23 s is a centered gamma distributed random variable . we refer the reader to the book @xcite for further information and details of the above results . heuristic argument linking theorem [ uzthmintro ] and theorem [ npthmintro ] was given by rosiski @xcite , while addressing a question of albert shiryaev . namely , let @xmath24 and @xmath25 be two i.i.d . centered random variables with fourth moment and unit variance . the link comes via a simple formula @xmath26 - 3,\ ] ] criterion , as well as the celebrated bernstein s theorem that asserts that @xmath24 and @xmath25 are gaussian if and only if @xmath27 and @xmath28 are independent . a rigorous argument to carry through this idea is based on a characterization of the asymptotic independence of multiple wiener - it integrals , which is much more difficult to handle than the plain independence , and may also be of an independent interest . the covariance between the squares of multiple wiener - it integrals plays the pivotal role in this characterization . at this point we should also mention an extension of to the multivariate setting . let @xmath29 be a finite set and @xmath30 be a sequence of non - negative integers . let @xmath31 be a multiple wiener - it integral of order @xmath32 , @xmath33 . consider a partition of @xmath29 into disjoint blocks @xmath34 , so that @xmath35 , and the resulting random vectors @xmath36 , @xmath37 . then @xmath38 the proof of this criterion is similar to the proof of in @xcite . in this paper in theorem [ mainblock ] we establish an asymptotic version of characterizing the asymptotic moment - independence between blocks of multiple wiener - it integrals . as a consequence of this result , we deduce the fourth moment theorem of nualart and peccati @xcite in theorem [ nupec ] , its multidimensional extension due to peccati and tudor @xcite in theorem [ pectud ] , and some neat estimates on the speed of convergence in theorem [ pectud - bd ] . furthermore , we obtain new multidimensional extension of a theorem of nourdin and peccati @xcite in theorem [ t : noupec ] , and give another new result on the bivariate convergence of vectors consisting of multiple wiener - it integrals in theorem [ t : bivariateblock ] . proposition [ bm - dmt ] applies theorem [ t : bivariateblock ] to establish the limit process for functions of short and long range dependent stationary gaussian time series in the spirit of the celebrated breuer - major @xcite and dobrushin - major - taqqu @xcite theorems . in theorem [ thmmoo ] we establish the asymptotic moment - independence for discrete non - gaussian chaoses using some techniques of mossel , odonnel and oleszkiewicz @xcite . the paper is organized as follows . in section [ s : pre ] we list some basic facts from gaussian analysis and prove some lemmas needed in the present work . in particular , we establish lemma [ l : cs ] , which a version of the cauchy - schwarz inequality well suited to deal with contractions of functions , see . it is used in the proof of the main result , theorem [ mainblock ] . section [ s : main ] is devoted to the main results on the asymptotic independence . section [ s : apps ] gives some immediate consequences and related applications of the main result . section [ s : f - apps ] provides further applications to the study of short and long range dependent stochastic processes and multilinear random forms in non - gaussian random variables .

we characterize the asymptotic independence between blocks consisting of multiple wiener - it integrals . as a consequence of this characterization , we derive the celebrated fourth moment theorem of nualart and peccati , its multidimensional extension , and other related results on the multivariate convergence of multiple wiener - it integrals , that involve gaussian and non gaussian limits . we give applications to the study of the asymptotic behavior of functions of short and long range dependent stationary gaussian time series and establish the asymptotic independence for discrete non - gaussian chaoses .

we characterize the asymptotic independence between blocks consisting of multiple wiener - it integrals . as a consequence of this characterization , we derive the celebrated fourth moment theorem of nualart and peccati , its multidimensional extension , and other related results on the multivariate convergence of multiple wiener - it integrals , that involve gaussian and non gaussian limits . we give applications to the study of the asymptotic behavior of functions of short and long range dependent stationary gaussian time series and establish the asymptotic independence for discrete non - gaussian chaoses .

1602.06556

c

we have investigated pair creation by the schwinger mechanism in @xmath7 . specifically , we considered a charged massive scalar field coupled to a constant background electric field in @xmath7 . after the canonical quantization , bogoliubov coefficients were obtained , and then the decay rate and the density of created pairs were computed ; see these main results in eqs . ( [ rate ] ) and ( [ density ] ) . also , using a semiclassical method the decay rate and the density were computed ; see appendix [ app : sem ] . both methods agree to say that in the semiclassical approximation , the screening orientation stays and the antiscreening ordination is suppressed . the density of created pairs is constant with respect to time . it signals that the pair creation in @xmath7 from electric and gravitational fields exactly balances the dilution from the expansion of the universe . under the semiclassical condition we computed the conduction current of the created particles in any dimension . we find that in the strong electric field regime , @xmath129 , the semiclassical current becomes independent of the scalar field mass and responds as @xmath1 , and in the heavy scalar field regime , @xmath130 , due to the presence of a boltzmann mass suppression factor it exponentially damped . our main goal has been to study the induced quantum vacuum expectation value of the conduction current of the created pairs . thus , in the case of a @xmath2 dimensional ds , the expectation value of the spacelike component of the current operator has been computed in the in - vacuum state . as expected , a linear uv divergence appeared . applying an adiabatic subtraction regularization scheme the divergent term was removed and a finite expression was obtained for the current and the corresponding conductivity . they have been plotted in figs . [ fig1 ] and [ fig2 ] , respectively . the current and conductivity have been also analytically investigated . we find that in the strong electric field regime , @xmath129 , the current responds as @xmath3 and becomes independent of scalar field mass parameter @xmath212 ( [ mds ] ) . in the weak electric field regime , @xmath203 , the current has a linear response in @xmath4 and is inversely proportional to @xmath212 . for the case of a massless minimally coupled scalar field , i.e. , @xmath204 , for @xmath297 , the current varies as @xmath5 . consequently , in this regime , the current and conductivity are increasing unbounded for decreasing electric field , which leads to the phenomenon of ir - hc . the regime of ir - hc has been extensively discussed in sec . [ sec : irhc ] from both numerical and analytical points of view . it has been shown that ir - hc happens for @xmath298 with @xmath299 for @xmath300 and @xmath233 positive but unbounded in @xmath139 . this difference comes from the renormalization scheme used in @xmath139 . the behavior of the current has also been derived in the ir - hc regime for any dimension in eq . ( [ eq : approx ] ) up to the renormalization factors . a proposed relation of ir - hc with conformality and tachyonicity remains also to be further explored but is beyond the scope of this paper . until sec . [ sec : gravity ] , the gravitational and electric fields were treated as an external field , and one important next step is to take into account backreactions of the created pairs to those two fields . indeed , as soon as the energy of the population of the schwinger created pairs becomes of the order of the energy carried by the constant electric field or of the gravitational energy , backreaction effects become unavoidable . investigating these effects could be used to find specific forms of electric fields or specific classes of spacetimes which favor or disfavor pair creation . furthermore , it could also be a fruitful way to make cosmological statements about magnetogenesis , matter - antimatter asymmetry , primordial gravitational waves , or the way inflation is driven and ends . those issues are currently under investigation @xcite . our first results on gravitational backreaction effects were depicted in sec . [ sec : gravity ] . using a semiclassical approach the energy - momentum tensor of the schwinger pairs has been computed in the heavy scalar field regime ; see eq . ( [ emt ] ) . we showed that creation of particles leads to a decay of the hubble constant . in the limit of zero electric field , our result is consistent with a previous study @xcite up to a factor of 2 in the exponent but disagrees with a very recent work @xcite . a more consistent calculation of this effect must dynamically study the evolution of the hubble constant @xmath16 through einstein equations , and this will explicitly break de sitter invariance by introducing a preferred time slicing . we argue that it should be possible to compute it together with the corrections from the schwinger effect and the presence of an electric field to the vacuum fluctuation during an inflationary phase . this could in turn affect the power spectrum at the end of inflation , as it was already suggested in the conclusion of @xcite . after the evolution of the primordial power spectrum through the reheating and the radiation dominated era , in principle it could be measured by cosmic microwave background experiments .

we consider a charged scalar field in a-dimensional de sitter spacetime and investigate pair creation by a schwinger mechanism in a constant electric field background . using a semiclassical approximation we find that the semiclassical current of the created pairs in the strong electric field limit responds as . going further but restricting to dimensional de sitter spacetime , the quantum expectation value of the spacelike component of the induced current has been computed in the in - vacuum state by applying an adiabatic subtraction scheme . we find that , in the strong electric field limit , the current responds as . in the weak electric field

we consider a charged scalar field in a-dimensional de sitter spacetime and investigate pair creation by a schwinger mechanism in a constant electric field background . using a semiclassical approximation the current of the created pairs has been estimated . we find that the semiclassical current of the created pairs in the strong electric field limit responds as . going further but restricting to dimensional de sitter spacetime , the quantum expectation value of the spacelike component of the induced current has been computed in the in - vacuum state by applying an adiabatic subtraction scheme . we find that , in the strong electric field limit , the current responds as . in the weak electric field limit the current has a linear response in and an inverse dependence on the mass of the scalar field . in the case of a massless scalar field , the current varies with which leads to a phenomenon of infrared hyperconductivity . a new relation between infrared hyperconductivity , tachyons , and conformality is discussed , and a scheme to avoid an infrared hyperconductivity regime is proposed . in dimension , we eventually presented some first estimates of the backreaction of the schwinger pairs to the gravitational field , and we find a decrease of the hubble constant due to the pair creation .

0709.4581

i

for more than two decades milgrom s modified newtonian dynamics ( mond ) mond has been able to explain galaxy rotation curves which are conventionally considered as an evidence of cold dark matter ( cdm ) on galactic scales . mond modifies newton s second law of motion to @xmath1 where @xmath2 and @xmath3 are the acceleration and newtonian gravitational potential , respectively ; @xmath4 is an effectively free function tending to unity in the limit @xmath5 , with @xmath6 being a new fundamental constant , which must have a numerical value of @xmath7 in order to match observations on a galactic scale . this theory looks like newton s when accelerations are large but is significantly different when accelerations are small . on galactic scales , @xmath8 , so the newtonian dynamics is modified , but in a way that can fit spiral galaxy rotation curves . subsequently , bekenstein @xcite built a relativistic theory which has mond as a non - relativistic , weak - field limit , thus making the study of cosmology possible . in addition to the conventional tensor gravitational field , bekenstein s theory involves a vector and a scalar field , and is therefore dubbed teves . interestingly , it has been argued that teves could also explain the large - scale structure formation of the universe without recurring to cdm @xcite , thanks to the presence of the vector field @xcite . recently , the authors of ref . @xcite showed that teves is equivalent to a vector - tensor theory of gravitation where the vector field has a non - fixed norm . they also showed that the correct mondian limit could be realized with a single vector field having non - canonical kinetic terms @xcite . these results indicate that a vector field in the gravity sector might be an interesting component of the universe ( indeed the model in @xcite and its generalized version we shall presented below in eq . ( [ eq : ouraction ] ) could be used to explain dark energy and dark matter in background cosmology ) and merits more detailed investigations . the idea of a vector field coupled to gravity has a long worldline ( see for example @xcite for a review and @xcite for further references ) , but in this work we will focus on the model described in @xcite , which is the most well - studied one , and investigate its cosmological implications . this particular theory is based on a dynamical vector field coupled to gravitation that picks up a preferred frame and preserves general covariance . this vector field is unit - norm , timelike , and violates local lorentz invariance . it is called the ther field ( or simply - field ) and we will refer to the associated einstein - ther theory as - theory defined by the - lagrangian . the - lagrangian considered in @xcite is a special case of our general model introduced in eq . ( [ eq : ouraction ] ) below for a unit - norm vector field that includes terms up to second order in derivatives , and it has been extensively studied in various contexts @xcite . in refs . @xcite , the background cosmology and primordial power spectra of perturbations from inflation of a slight different model were also considered . here , we investigate these for the model presented in ref . @xcite , and also study the evolution of linear perturbation to the - field during the radiation and matter - dominated epochs . as we will show below , if we restrict the parameter space of the underlying theory so as to satisfy the local experimental gravity constraints , @xcite , this perturbation becomes sourceless and decays during the epoch of inflation and late - matter domination . however , it is sourced by the evolutions of the photon and neutrino anisotropic stresses during the radiation era and early matter era , which have some imprints on the cosmological observables . our presentation is organized as follows . in sec . [ sect : equations ] , we briefly introduce the general - theory and derive perturbation equations for the background friedmann - like cosmology in the covariant and gauge invariant ( cgi ) formalism ( see @xcite for a derivation of the perturbation equations in conformal newtonian gauge ) . in sec . [ sect : cosmology ] we shall use these equations to discuss the perturbation dynamics for the cosmological models of ref . @xcite . first , we summarize the existing constraints on the model in sec . [ sect : parameterspace ] ; then , in sec . [ sect : perturbationevolution ] , we present the evolution equations for the perturbation variables and then use them to show how the primordial spectra of scalar and tensor perturbations in this theory are unmodified and modified , respectively , on comparing them with the predictions of general relativity ( gr ) . the late - time evolution of the - field perturbation and its effects on cosmological observables are also studied there . finally , our discussion and conclusions are presented in sec . [ sect : conclusion ] . throughout this work our convention is @xmath9u^{c } = r^{\ \ c}_{ab\ d}u^{d } , r_{ab } = r_{acb}^{\ \ \ c}$ ] , where @xmath10 are respectively the riemann tensor and ricci tensor ; the metric signature is @xmath11 and the universe is assumed to be spatially flat , filled with photons , baryons , cdm , 3 species of neutrinos and a cosmological constant .

the spacetime splitting approach is used to derive covariant and gauge invariant perturbation equations which are valid for a general class of lagrangians . restricting attention to the parameter space of these theories we also study the implications for late - time cosmology and find that the evolution of photon and neutrino anisotropic stresses can source the field perturbation during the radiation and matter dominated epochs , and as a result the cmb and matter power spectra are modified

we consider cosmology in the einstein - ther theory ( the generally covariant theory of gravitation coupled to a dynamical timelike lorentz - violating vector field ) with a linear - lagrangian . the spacetime splitting approach is used to derive covariant and gauge invariant perturbation equations which are valid for a general class of lagrangians . restricting attention to the parameter space of these theories which is consistent with local gravity experiments , we show that there are tracking behaviors for the field , both in the background cosmology and at linear perturbation level . the primordial power - spectrum of scalar perturbations in this model is shown to be the same that predicted by standard general relativity . however , the power - spectrum of tensor perturbation is different from that in general relativity , but has a smaller amplitude and so can not be detected at present . we also study the implications for late - time cosmology and find that the evolution of photon and neutrino anisotropic stresses can source the field perturbation during the radiation and matter dominated epochs , and as a result the cmb and matter power spectra are modified . however these effects are degenerate with respect to other cosmological parameters , such as neutrino masses and the bias parameter in the observed galaxy spectrum .

0709.4581

c

in this paper we have studied the cosmology of the einstein- ther theory . after presenting the general field equations for such theories in the cgi formalism , we focussed on a specific class of models described in eq . ( [ eq : aelagrangian ] ) , and confined ourselves to the parameter space of models described by eq . ( [ eq : parameterspace2 ] ) which pass the ppn and @xmath118erenkov constraints . this parameter space is known to have the same locally- and cosmologically - felt gravitational constants ( @xmath270 , which are different from the true bare @xmath94 ) and this tracking behavior indicates that we can consider its background expansion just by using the measured value @xmath271 and ignoring the presence of the -field ( since the only effect of the ther field is to track other matter species , the model in eq . ( [ eq : aelagrangian ] ) clearly can not explain dark energy . more general models , such as the @xmath266 proposed in @xcite and our sec . [ sect : equations ] , may serve this purpose ) . we find that it is a general feature that the tracking behavior not only occurs at the background level but also at the linear order in perturbation theory . for example , the quantity @xmath272 tracks that of the other matter species on super - horizon scales . this indicates that , whatever type of perturbation is generated during inflation , the presence of -field will not alter it . in particular , in the single - field inflation model we consider , no isocurvature perturbation is produced . this is an important characteristic , and it would be interesting to see whether similar behavior occurs for general , higher - order choices of @xmath266 . we generalized the analysis of primordial power spectra for the -theory @xcite to our model . for the parameter ranges which satisfy local gravity bounds , we find that the evolution of the large - scale gravitational potential , @xmath158 , is unmodified as compared with that in gr , and show that the primordial power spectrum of @xmath158 also has the same form as in gr [ c.f . ( [ eq : pps ] ) ] . if we assume that the bare gravitational constants in gr and in the -model are the same , then the magnitudes of the spectra are different in these two models . however , contrary to the discussion in @xcite , we argue that we do not know the true bare @xmath273 , but only know the measured value , @xmath274 . in both gr and our -model , @xmath275 is equal to the cosmological value @xmath276 while in the latter @xmath277 . as a result , we show that with the same inflationary potential the primordial power spectra of @xmath158 in these two models should have the same shape and the same magnitude . meanwhile , we also find that the power spectrum of tensor perturbations in our model is smaller in magnitude than is predicted in gr , and so currently we can not use it to distinguish between gr and the -model . for the late - time evolution of the perturbations , it is shown that , the -field perturbation is driven by the evolution of the anisotropic photon and neutrino stresses the @xmath278^{\prime}$ ] term in its propagation equation although it is sourceless during inflation . therefore , it decays away exponentially during the inflationary era , grows again in the radiation - dominated epoch when @xmath279^{\prime}$ ] is significant , and finally diminishes again , oscillating when @xmath279^{\prime}$ ] eventually becomes negligible . as a result , the cmb and matter power spectra are modified by the existence of the -field . we also remark that , depending on the nature of the dark energy , it is possible that the -field perturbation will have non - trivial dynamics driven by the @xmath280 term , where @xmath281 is the possible dark - energy anisotropic stress @xcite . we also note that there recently appeared a later paper @xcite which investigated in details the structure formation of the model proposed in @xcite . in their sec . v the authors considered a simple power - law model @xmath282 in which @xmath283 corresponds to our lagrangian eq . ( [ eq : aelagrangian ] ) ( except for the @xmath23 term ) . they obtained criteria for there to be a growing mode in the evolution of the ( scalar - mode ) - field perturbation . however their criteria can not be applied directly to the model we consider here . to see why , note that in eq . ( 32 ) of @xcite the quantities @xmath284 also contain @xmath285 ( their @xmath285 is equivalently our @xmath286 ) . however , by their argument the @xmath285 terms in @xmath287 are suppressed by a small quantity @xmath288 ( which in _ our _ notation is just @xmath289 ) and are ultimately neglected from the - field equation of motion , their eq . ( 43 ) . in our model , there is no argument that @xmath290 ( its magnitude actually _ is _ 1 ) and so we can no longer neglect the @xmath286 terms appearing in @xmath291 . when these are taken into account we obtain a different equation of motion , _ i.e. _ , the @xmath292s in eq . ( 43 ) of @xcite are different in our work , and consequently the criteria for the existence of growing modes are different as well . in addition , are the facts that we have a @xmath23 term in our lagrangian and use a different parameter space , eqs . ( [ eq : parameterspace ] , [ eq : parameterspace2 ] ) , which also contribute to the differences between our results and those in @xcite . as an aside , we stress that the key relation in our work , @xmath293 , is a consequence of eq . ( [ eq : parameterspace ] ) and has nothing to do with whether or not the - gravity waves propagate superluminally . in fact , we could do our calculation dropping the constraint that these waves propagate superluminally , and in this case the perturbation dynamics might place some constraints on the parameter space ; for example , in some portion of the parameter space the growth of @xmath286 may become unstable . in our work we choose the parameters leading to superluminal gravity waves simply to avoid constraints from @xmath294erenkov radiation . there are still some debates about this kind of choice , see however @xcite for a different and conservative point of view . there are also other general differences between the ea model we consider here and the gea model of @xcite . first , for the static and weak field limit , the @xmath23 term and the choice of parameter space eq . ( [ eq : parameterspace ] ) are crucial for the ea model to evade ppn tests @xcite ; in @xcite there is no @xmath23 term , but the nonlinearity in the ther field lagrangian guarantees that the local gravitational tests are not a problem because at high densities the modification is negligible . second , as we have seen above , our conclusion that the primordial power spectrum of density perturbation is unmodified compared with gr relies on the fact @xmath293 so that the background expansion during inflation in our model is the same as in gr ( again the @xmath23 term is crucial here ) . if the @xmath23 term is not included , the local and cosmological gravitational constants are different , meaning that the background evolution during inflation is different from gr ; as a result the background quantities at horizon crossing , which determine the observables in inflationary models through the slow - roll parameters , may be different from those in gr this means that the ea model will predict different shape of primordial density spectrum other than gr @xcite . for the gea model considered in @xcite , again the nonlinear @xmath266 makes the modifications to gr during early times small enough to be neglected , and as such the primordial power spectrum is also the same as in gr . note however that in the latter case @xcite the late time evolution of vector field perturbation is also modified in a complex way , making the corresponding cosmological behaviors different from ours . so , in conclusion , although the background expansion of our -model is exactly the same as that in gr , the cosmological data on cmb and matter power spectra can be used to constrain the -model . however , the constraint is not expected to be very stringent because the linear perturbation spectra depend weakly on the model parameters . other considerations of the behaviour in strong gravitational fields , such as those studies of the compact stars or black holes @xcite would enable better constraints on the present model to be obtained . alao , as general interests for cosmology , studies in more complicated - field lagrangian as those performed in @xcite need to be explored thoroughly . we thank ted jacobson , eugene lim , constantinos skordis , xiaoting wang , hongsheng zhao , tom zlosnik and the referee for helpful discussions . the numerical calculation in this work uses a modified version of the public camb code @xcite . bl is supported by the overseas research studentship , cambridge overseas trust and the damtp at university of cambridge . dfm acknowledges the humboldt foundation . arianto , f. p. zen , bobby , e. gunara , triyanta and supardi ( 2007 ) , arxiv:0709.3688 [ hep - th ] ; s. kanno and j. soda , phys . d * 74 * , 063505 ( 2006 ) ; a. tartaglia and n. radicella ( 2007 ) , arxiv:0708.0675 [ gr - qc ] ; m. libanov , v. rubakov , e. papantonopoulos , m. sami and s. tsujikawa , jcap * 0708 * , 010 ( 2007 ) ; p. g. ferreira , b. m. gripaios , r. saffari and t. g. zlosnik , phys . d * 75 * , 044014 ( 2007 ) . and , * * , ( ) ; and , in _ theoretial and observational cosmology , _ ed . m. lachize - rey ( springer , new york , 1998 ) ; and , * * , ( ) ; , phd dissertation , queens college and astrophysics group , cavendish laboratory , cambridge university , 2000 . and , * * , ( ) [ arxiv : astro - ph/0610486 ] ; and , * * , ( ) [ arxiv : gr - qc/0701111 ] ; , and , * * , ( ) [ arxiv : astro - ph/0610794 ] ; , and , * * , ( ) [ arxiv:0705.3795 [ gr - qc ] ] ; t. koivisto and d. f. mota , phys . lett . b * 644 * , 104 ( 2007 ) ; t. koivisto and d. f. mota , phys . rev . d * 75 * , 023518 ( 2007 ) ; , and , phys . d * 76 * , 104047 ( 2007 ) ( arxiv:0707.2664 [ gr - qc ] ) . we want to elaborate this point as it will also be used below . suppose we have a universe described by either standard gr or the -theory eq . ( [ eq : aelagrangian ] ) and we have a measured gravitational constant @xmath295 . if we assume gr , then we have @xmath296 , while otherwise we have @xmath297 , so that whenever we need @xmath298 we could relate it to the textbook - value @xmath295 by @xmath299 ( in the gr case ) , or @xmath300 ( in the -case ) . obviously , when @xmath301 , @xmath302 the background cosmologies are the same in gr and our -model . t. clifton , d. f. mota and j. d. barrow , mon . not . * 358 * ( 2005 ) 601 [ arxiv : gr - qc/0406001 ] . , _ theory and experiment in gravitational physics _ ( cambridge university press , cambridge , 1993 ) . t. koivisto and d. f. mota , phys . d * 73 * , 083502 ( 2006 ) [ arxiv : astro - ph/0512135 ] . t. koivisto and d. f. mota , arxiv:0707.0279 [ astro - ph ] . d. f. mota , j. r. kristiansen , t. koivisto and n. e. groeneboom , arxiv:0708.0830 [ astro - ph ] . with general choices of parameters @xmath303 , eq . ( eq : propagationphils ) is only valid if @xmath304 . thus , in the radiation - dominated era it can not apply even for superhorizon scales , and its solution eq . ( [ eq : lsphi ] ) should be modified as well .

we consider cosmology in the einstein - ther theory ( the generally covariant theory of gravitation coupled to a dynamical timelike lorentz - violating vector field ) with a linear - lagrangian . the primordial power - spectrum of scalar perturbations in this model is shown to be the same that predicted by standard general relativity . however , the power - spectrum of tensor perturbation is different from that in general relativity , but has a smaller amplitude and so can not be detected at present .

we consider cosmology in the einstein - ther theory ( the generally covariant theory of gravitation coupled to a dynamical timelike lorentz - violating vector field ) with a linear - lagrangian . the spacetime splitting approach is used to derive covariant and gauge invariant perturbation equations which are valid for a general class of lagrangians . restricting attention to the parameter space of these theories which is consistent with local gravity experiments , we show that there are tracking behaviors for the field , both in the background cosmology and at linear perturbation level . the primordial power - spectrum of scalar perturbations in this model is shown to be the same that predicted by standard general relativity . however , the power - spectrum of tensor perturbation is different from that in general relativity , but has a smaller amplitude and so can not be detected at present . we also study the implications for late - time cosmology and find that the evolution of photon and neutrino anisotropic stresses can source the field perturbation during the radiation and matter dominated epochs , and as a result the cmb and matter power spectra are modified . however these effects are degenerate with respect to other cosmological parameters , such as neutrino masses and the bias parameter in the observed galaxy spectrum .

0805.4309

i

the kerr family , discovered in 1963 @xcite , comprises perhaps the most important family of exact solutions to the einstein vacuum equations @xmath18 the governing equations of general relativity . for parameter values @xmath19 ( here @xmath1 denotes the mass and @xmath0 angular momentum per unit mass ) , the kerr solutions represent black hole spacetimes : i.e. asymptotically flat spacetimes which possess a region which can not communicate with future null infinity . the celebrated schwarzschild family sits as the one - parameter subfamily of kerr corresponding to @xmath20 . much of current theoretical astrophysics is based on the hypothesis that isolated systems described by kerr metrics are ubiquitous in the observable universe . despite the centrality of the kerr family to the general relativistic world picture , the most basic questions about the behaviour of linear waves on kerr backgrounds have remained to this day unanswered . this behaviour is in turn intimately connected to the stability properties of the kerr metrics themselves as solutions of @xmath21 , and thus , with the very physical tenability of the notion of black hole . in particular , even the question of the uniform boundedness ( pointwise , or in energy ) of solutions @xmath7 to the linear wave equation @xmath22 in the domain of outer communications has not been previously resolved , except for the schwarzschild subfamily . the main theorems of this paper give the resolution of the boundedness problem for @xmath23 , for the case @xmath2 . solutions to @xmath23 arising from regular initial data remain uniformly bounded in the domain of outer communications . the bound is quantitative , i.e. it is computable in terms of the initial supremum and initial energy - type quantities on initial data . in fact , the results of this paper apply to a much more general setting than the specific kerr metric : boundedness is proven for solutions of @xmath23 on the exterior region of any stationary axisymmetric spacetime sufficiently close to a schwarzschild spacetime with mass @xmath5 . thus , the methods may be of relevance in the ultimate goal of this analysis : understanding the dynamics of the einstein equations @xmath21 in a neighborhood of a kerr metric . we first give a statement of the main results for the special case of kerr and the related kerr - newman family ( this is a family of solutions to the coupled einstein - maxwell system ) . we refer the reader to @xcite . let @xmath4 denote the kerr solution with parameters @xmath24 or more generally the kerr - newman solution with parameters @xmath25 , with @xmath26 and let @xmath27 denote the closure of a domain of outer communications . ( the parameter @xmath28 is known as the charge . ) let @xmath9 be a cauchy hypersurface in @xmath4 crossing the event horizon to the future of the sphere of bifurcation , and such that @xmath29 coincides with a constant-@xmath30 hypersurface , for large @xmath31 , where @xmath30 and @xmath31 denote here the standard boyer - lindquist coordinates on @xmath32 . recall that in such coordinates , the stationary killing field @xmath33 is given by @xmath34 . the kerr - newman solutions are moreover axisymmetric . the penrose diagram , say along the axis of symmetry ( where the axisymmetric killing field vanishes ) , is depicted below : @xmath35 note that @xmath29 is a past cauchy hypersurface for @xmath36 . denotes causal future , not to be confused with currents @xmath37 to be defined later . ] we have that @xmath38 is foliated by @xmath39 for @xmath40 , where @xmath39 is the future translation of @xmath41 by the flow generated by the stationary killing field @xmath42 for time @xmath43 . let @xmath44 denote the unit future normal of @xmath39 . let @xmath45 denote a translation invariant null generator for @xmath46 , and give @xmath47 the induced volume from @xmath48 and @xmath49 . let @xmath50 denote the standard energy momentum tensor associated to a solution @xmath7 of the wave equation @xmath23 @xmath51 define @xmath52 by @xmath53 and @xmath54 by @xmath55 note that the former current is positive definite when contracted with a future - timelike vector field , but is not conserved , whereas the latter current is conserved , but not positive definite when so contracted . [ prwto ] let @xmath4 , @xmath27 , @xmath39 be as above . there exists a universal positive constant @xmath56 , and a constant @xmath12 depending on @xmath1 and the choice of @xmath57 such that if @xmath58 then the following statement holds . let @xmath7 be a solution of @xmath23 on @xmath4 such that @xmath59 . then @xmath60 @xmath61 @xmath62 here the integrals are with respect to the induced volume forms . the integral on the left hand side of @xmath63 can be defined via a limiting procedure . [ deutero ] under the assumptions of the previous theorem , the following holds . let @xmath7 be a solution of the wave equation @xmath23 on @xmath4 such that @xmath64 then @xmath65 in @xmath66 . the hypothesis of theorem [ prwto ] can be re - expressed as the statement that local energy as measured by a local observer be finite , i.e. that @xmath67 , @xmath68 be in @xmath69 , together with the global assumption that @xmath70 the latter in turn is certainly satisfied if @xmath71 vanishes in a neighborhood of @xmath72 . similarly , the hypothesis of theorem [ deutero ] is satisfied for @xmath73 , @xmath68 in @xmath74 , if @xmath71 vanishes in a neighborhood of @xmath72 . finally , note that given an arbitrary cauchy surface @xmath75 for kerr , sufficiently well behaved at @xmath72 , it follows that the right hand side of @xmath76 is bounded by @xmath77 thus the above regularity assumptions could be imposed on an arbitrary cauchy surface . there are no unphysical restrictions on the support of the solution in a neighborhood of @xmath78 . the results of theorems [ prwto ] and [ deutero ] remain true when the kerr or kerr - newman metric is replaced by an arbitrary stationary axisymmetric black hole exterior metric suitably close to schwarzschild , and with suitable assumptions on the geometry of the killing fields . in particular , in addition to smallness , it is required that as in the kerr solution the null generator of the horizon is in the span of the killing fields . the precise assumptions are outlined in section [ ta3n ] . the elusiveness of the results of theorems [ prwto ] and [ deutero ] stems from the well - known phenomenon of _ this is related to the fact that the killing field @xmath33 ( with respect to which the kerr solution is stationary ) is not everywhere - timelike in the domain of outer communications . in particular , there is a region of spacetime where @xmath33 is spacelike , the so - called _ ergoregion_. the boundary of this region is called the _ ergosphere_. the presence of the ergoregion means that the energy current @xmath79 is not positive definite when integrated over spacelike hypersurfaces . thus , the conservation of @xmath79 does not imply _ a priori _ bounds on an @xmath80-based quantity . in particular , the local energy of the solution can be greater than the initial total energy , even if the energy is initially supported where @xmath79 is positive definite . a test - particle version of this fact , where a particle coming in from infinity splits into one of negative energy entering the black hole and one of greater positive energy returning to infinity , is known as the _ penrose process_. the pioneering study by christodoulou @xcite of the `` black hole transformations '' obtainable via a penrose process led to a subject known as `` black hole thermodynamics '' . in the physics literature , where discussion of these issues is inextricably linked to the separability of @xmath23 and decomposition of @xmath7 into modes , the problem of the ergoregion appears as a formidable and perhaps intractable obstacle . it turns out , however , that there are other physical mechanisms at play which have an important role but are not necessarily well reflected from the point of view of separability . in particular , the tendency of waves to eventually disperse ( true in any asymptotically flat spacetime ) coupled with the powerful red - shift effect at the horizon . indeed , these properties , which depend only loosely on the stationarity , tend to make solutions not only stay bounded but decay to a constant in time , even if the local energy increases for a short time . unfortunately , the dispersive properties of waves on black hole backgrounds are severely complicated by the presence of trapped null geodesics . ( the presence of these can easily be inferred by a continuity argument in view of the fact that there exist both null geodesics crossing the horizon and going to null infinity . ) it is only very recently that the role of trapping has been sufficiently well understood in the special case of the schwarzschild family to allow for the first proofs of decay for general solutions of @xmath23 on such backgrounds . see the results described in section [ prevsch ] . in the case of kerr , the techniques introduced for controlling trapping on schwarzschild can not be readily perturbed . this has to do with the fact that these techniques seem to exploit the special property that the trapping concentrates asymptotically on a set of codimension @xmath81 in physical space , the so - called photon sphere . in contrast , in kerr the codimensionality of the space of trapped geodesics can only be properly understood in phase space . this indicates that controlling trapping requires a far more delicate analysis . it would appear from the above that the problem of superradiance could in principle be overcome , but at the expense of a very delicate analysis of trapping . a closer look , however , reveals that the situtation is considerably more favourable . at a heuristic level , the reason for this is the following remark : if one could separate out the `` superradiant '' part of the solution from the `` non - superradiant '' part , then one only has to exploit dispersion for the superradiant part . this latter problem turns out to be much easier than understanding dispersion for the whole solution . to decompose the solution , we must first cut off the solution @xmath7 in the `` time''-interval of interest to obtain @xmath82 and then decompose into two pieces @xmath83 where @xmath84 is to be supported in frequency space ( real frequencies @xmath85 and integer @xmath86 here defined with respect to coordinates @xmath30 and @xmath87 ) only in the range @xmath88 , whereas @xmath89 is to be supported in frequency space only in the range @xmath90 . for spacetimes sufficiently close to schwarzschild , for a suitable choice of the parameter @xmath91 , one can view @xmath89 as essentially non - superradiant , and @xmath84 as the superradiant part . if one can show boundedness for @xmath89 and dispersion for @xmath84 , then one will have proven the uniform boundedness of the sum @xmath7 . for spacetimes sufficiently close to schwarzschild , one can choose @xmath91 sufficiently small so that trapping essentially does not occur for @xmath84 , and the dispersive mechanism of schwarzschild is stable . this relies on the stability of the red - shift effect for considerations close to the horizon . in complete contrast to the standard picture , it is the superradiant part of the solution which would be the better behaved one . in practice , the analysis is of course not as simple as what has been portrayed above , and here again , the stabilising effect of the red - shift acting near the horizon plays an important role . in view of the cutoffs in time , the equations satisfied by @xmath92 and @xmath89 are coupled . moreover , the statement that @xmath89 is non - superradiant while @xmath84 is dispersive must also be understood modulo error terms . it turns out that to control these error terms , one of necessity must have at their disposal an energy quantity which does not degenerate on the horizon , that is to say , the @xmath80-based quantity for which one shows uniform boundedness must be the one of theorem [ prwto ] , and not a quantity analogous to @xmath79 in schwarzschild . in particular , one must understand the red - shift mechanism even for the `` non - superradiant '' part @xmath89 , for which one does not understand dispersion . such stable estimates at the horizon ( corresponding to the energy measured by local observers ) exploiting the red - shift effect were first attained for schwarzschild in our previous @xcite . it is interesting to note , however , that in @xcite , understanding of the red - shift mechanism was always coupled with understanding dispersion , i.e. controlling the trapping phenomenon . in particular , one had to appeal to an understanding of dispersion even to obtain the result of theorems [ prwto ] for schwarzschild . in this paper , we show how understanding the red - shift can be decoupled from understanding dispersion in the non - superradiant case . in addition , we show that the red - shift effect allows us to commute the wave equation with a vector field transverse to the horizon , yielding a new route to higher order estimates and pointwise estimates . an extra side - benefit of our results is thus a new , simpler and more robust proof of theorems [ prwto ] and [ deutero ] even for the case of schwarzschild . see section [ further1 ] . we review in detail previous work on this and related problems . results of the type of theorems [ prwto ] and [ deutero ] for static perturbations of minkowski space pose little difficulty . ( indeed , the analogue of theorem [ prwto ] is immediate , and theorem [ deutero ] can be proven with the help of sobolev inequalities after commuting the equation with the static killing field . ) thus , we shall pass directly to the black hole case . the analogue of theorem [ deutero ] for schwarzschild is a celebrated result of kay and wald @xcite , building on previous work of wald @xcite where the theorem had been proven for the restricted class of data whose support was assumed not to contain the bifurcation sphere @xmath93 . in view of the positive definiteness of @xmath79 in the domain of outer communications , the only essential difficulty is obtaining bounds for @xmath7 up to the horizon ( where @xmath33 becomes null ) , as bounds away from the horizon can be obtained essentially as described immediately above for static perturbations of minkowski space . the arguments of kay and wald to prove the analogue of theorem [ deutero ] relied on the staticity to realize a solution @xmath7 as @xmath94 where @xmath95 is again a solution of @xmath23 constructed by inverting an elliptic operator acting on initial data . in addition , a pretty geometric construction exploiting the discrete symmetries of maximal schwarzschild was used to remove the unphysical restriction on the support near @xmath93 necessary for constructing @xmath96 in the original @xcite . unfortunately , neither of these methods is particularly robust to perturbation . the reason the authors had to resort to such techniques was that theorem [ deutero ] was proven _ without _ proving the analogue of theorem [ prwto ] , rather , using only the conserved flux @xmath79 whose control degenerates as @xmath46 is approached . theorem [ prwto ] for schwarzschild was only proven as part of the decay results of @xcite to be discussed below . turning now to the issue of decay , the first non - quantitative decay result for @xmath23 on schwarzschild is contained in the thesis of twainy @xcite . the first quantitative decay results for solutions of @xmath23 on schwarzschild ( and more generally , reissner - nordstrm ) were proven in @xcite , but were restricted to spherically symmetric solutions , or alternatively , the @xmath97th spherical harmonic @xmath98 of a general solution @xmath7 . ( in fact , this was a byproduct of the main result of @xcite , which concerns decay rates for spherically symmetric solutions to the coupled einstein-(maxwell)-scalar field system . ) quantitative decay results for the whole solution @xmath7 , both pointwise and in energy , were proven in @xcite , in particular , the uniform decay result @xmath99 is an eddington - finkelstein advanced time coordinate and @xmath100 is an appropriate quantity computable on initial data , and @xmath101 denotes say @xmath102 . inequality @xmath103 is sharp as a uniform decay rate in @xmath104 . the results of @xcite exploit both the red - shift effect near the horizon and the dispersive properties . the estimates are derived using a variety of vector field multipliers , in particular , a vector field multiplier @xmath105 such that the flux of @xmath106 gives the local energy at the horizon . the energy identity of @xmath105 quantifies the red - shift effect . weaker decay results were proven independently by blue and sterbenz @xcite for initial data vanishing on @xmath93 , but with control which degenerates on the horizon . in particular , the estimates of @xcite are unstable to perturbation near the horizon . the stability of the estimates of @xcite near the horizon will be of critical importance here . both @xcite and @xcite control trapping effects with the help of vector field multipliers which must be carefully chosen for each spherical harmonic separately . these were inspired by a series of papers by soffer and collaborators , for instance @xcite ; see , however @xcite . the first proof of decay for @xmath7 not relying on spherical harmonic decomposition for the construction of these multipliers is provided by our more recent @xcite . since uniform boundedness is the most basic question which can be asked about @xmath23 on kerr , previous results in this setting are of necessity of a partial nature . in particular , essentially all previous work on @xmath23 is restricted to the projection of @xmath7 to a single azimuthal frequency , or equivalently , to the case where the data are of the form @xmath107 solutions arising from @xmath108 are then of the form @xmath109 . let us call such solutions azimuthal modes . in principle , one could attempt to deduce properties of general @xmath7 by summing relations deduced for each individual azimuthal mode . as we shall see , however , due to the non - quantitative nature of the results described below , in of themselves they unfortunately yield no information about general @xmath7 . nonetheless , even the study of such @xmath110 without regard to uniform control in @xmath86 turns out to be a non - trivial problem . indeed , even for such individual azimuthal modes , the most basic questions had not been previously answered , in particular , the analogue of theorems [ prwto ] or [ deutero ] . this being said , there are interesting partial results concerning @xmath108 that had been previously obtained . in particular , most recently finster _ et al . _ @xcite had been able to show for smooth @xmath110 that for fixed @xmath111 and @xmath112 , @xmath113 under the assumption that the support of @xmath110 does not contain @xmath114 . see however @xcite . in particular , one can deduce @xmath115 for each fixed @xmath111 and @xmath112 , without however a bound on the @xmath116 . the results rest on an explicit integral representation of the solution which is derived using the remarkable ( but all too fragile ) separability properties of the kerr metric . the arguments contain many pretty applications of contour integral methods of classical complex analysis . since these techniques are essentially algebraic , no restriction on the size of @xmath117 need be made provided @xmath118 . in @xcite , under the same assumption on the initial support , the authors deduce that for each @xmath119 , @xmath120 thus , the energy of each mode in the region @xmath121 remains finite but again , no quantitative bound in terms of data can be produced . moreover , from the results of @xcite , one can not deduce that the @xmath116 of @xmath122 and @xmath123 commute with taking @xmath124 or @xmath125 , i.e. @xmath122 is _ a priori _ compatible with @xmath110 blowing up along the horizon : @xmath126 and @xmath123 is compatible with infinite energy concentration near the horizon : @xmath127 as explained before , no statement could be inferred for the general solution @xmath7 from the above statements on individual azimuthal modes , not even a weak statement like @xmath122 or @xmath123 . this is because the @xmath128 and @xmath116 of @xmath129 , @xmath122 and @xmath123 do not _ a priori _ commute with summation over @xmath86 . must be understood as the projection @xmath110 , to agree with what is proven in the body of the paper . ] of course , in view of theorems [ prwto ] and [ deutero ] , one can now infer from @xmath129 corollary [ nqc ] of section [ nqd ] . the somewhat unsatisfying nature of the above previous results deduced with the help of separability are indicative of how difficult it is to obtain quantitative statements about solutions of the wave equation @xmath23 even in the algebraically special case where one has explicit representations of the solution . perhaps this is for the best , however . remarkable though they are , the separability properties of the kerr metric are unstable to perturbation . just as in the case of stability of minkowski space @xcite , understanding the stability properties of the einstein equations near the kerr solution will undoubtedly require robust methods . we hope that the techniques employed here will have further applications in this direction . a related problem to the wave equation is that of the klein - gordon equation @xmath130 with @xmath131 . there is a well - developed scattering theory on schwarzschild for the class of solutions of @xmath132 with finite energy associated to the killing @xmath33 . in particular , an asymptotic completeness statement has been proven in @xcite . this analysis in of itself , however , when specialised to @xmath74 solutions in the geometric sense , only gives very weak information about the solution . in particular , it does not give @xmath80 control of @xmath7 or its angular derivatives on @xmath46 . in the case of kerr , there are again certain partial results for @xmath132 in the direction of scattering for a `` non - superradiant '' subspace of initial data @xcite . these interesting results do not , however , address the characteristic difficulties of superradiance . see also @xcite . finally , we mention that there has been a series of interesting papers concerning the dirac equation on kerr and kerr - newman . see @xcite . for dirac , considerations turn out to be much easier as this equation does not exhibit the phenomenon of superradiance . we shall not comment more about this here but refer the reader to the nice article @xcite . we can not do justice here to the vast work on this subject in the physics literature . see @xcite for a nice survey . the authors thank g. holzegel for comments on the manuscript . the authors also thank the niels bohr institute , copenhagen , for hospitality while this work was being completed . m. d. is support in part by a grant from the european research council . i. r. is support in part by nsf grant dms-0702270 .

we consider kerr spacetimes with parameters and such that , kerr - newman spacetimes with parameters , , and more generally , stationary axisymmetric black hole exterior spacetimes which are sufficiently close to a schwarzschild metric with parameter , with appropriate geometric assumptions on the plane spanned by the killing fields . we show uniform boundedness on the exterior for sufficiently regular solutions to the wave equation , i.e. we show that solutions arising from smooth initial data prescribed on an arbitrary cauchy surface satisfy in the domain of outer communications . no unphysical restrictions are imposed on the behaviour of near the bifurcation surface of the event horizon . the norm is finite if , and is well - behaved at spatial infinity , in particular , it is sufficient to assume is supported away from spatial infinity . the pointwise estimate derives in fact from the uniform boundedness of a positive definite energy flux . note that in view of the very general assumptions , the separability properties of the wave equation on the kerr background are not used .

we consider kerr spacetimes with parameters and such that , kerr - newman spacetimes with parameters , , and more generally , stationary axisymmetric black hole exterior spacetimes which are sufficiently close to a schwarzschild metric with parameter , with appropriate geometric assumptions on the plane spanned by the killing fields . we show uniform boundedness on the exterior for sufficiently regular solutions to the wave equation , i.e. we show that solutions arising from smooth initial data prescribed on an arbitrary cauchy surface satisfy in the domain of outer communications . in particular , the bound holds up to and including the event horizon . here , is a norm on initial data and depends only on the parameters of the nearby schwarzschild metric . no unphysical restrictions are imposed on the behaviour of near the bifurcation surface of the event horizon . the norm is finite if , and is well - behaved at spatial infinity , in particular , it is sufficient to assume is supported away from spatial infinity . the pointwise estimate derives in fact from the uniform boundedness of a positive definite energy flux . note that in view of the very general assumptions , the separability properties of the wave equation on the kerr background are not used .

cond-mat0210211

i

the problem of electron motion in a disordered conductor in a periodic potential and strong magnetic field displays a rich combination of phenomena of both classical and quantum origin . the complexity arises from the appearance of two independent types of periodicities , of the potential and of the cyclotron orbits , which may interplay in complicated ways . perhaps the most striking effect is known as weiss oscillations@xcite , whereby very large oscillations in the resistivity are induced by even a weak periodic potential at moderate magnetic fields . at different magnetic fields or temperatures , other types of contributions to the resistivity become significant . for example , at very small fields , a positive magnetoresistance results from a classical mechanism of channeling of orbits@xcite . at larger magnetic fields , the resistivity develops shubnikov de haas oscillations , which originate from the onset of landau quantization and may still be affected by the periodic potential@xcite . at even higher fields , the integer quantum hall effect@xcite ( iqhe ) becomes operative due to the contribution of quantum interference processes . much of the above phenomenology has been thoroughly analyzed theoretically , by a variety of techniques including quantum mechanical approaches@xcite involving diagrammatics@xcite or solution of a quantum boltzmann equation@xcite , and quasiclassical approaches@xcite . the experimental realization of these systems is also well advanced@xcite with precise confirmation of the theoretical predictions already possible . so far however a theory for the influence of quantum interference processes in a periodic potential and high magnetic field has been missing . such processes lead to weak localization corrections to the conductivity which become significant at low temperatures ( see e.g. refs.@xcite for reviews ) . at high magnetic fields , they are also responsible for the operation of the iqhe@xcite , whereby the hall conductivity becomes quantized at low temperatures . at the same time , the observation of the iqhe in systems with a weak potential modulation is well within current experimental capability@xcite . this paper aims to fill this gap by showing how the standard theory for weak localization and the iqhe may be generalized so as to incorporate the presence of a periodic potential . we employ a field - theory approach based on a nonlinear @xmath0 model@xcite which is well established in the study of mesoscopic disordered conductors . in deriving the appropriate field theory , we are able to show how previous theoretical calculations of the conductivity , within both quasiclassical@xcite and quantum mechanical approaches@xcite , may be recovered in a natural way within the field theory formalism . this enables us to extend previous theoretical results in a consistent way so as to include the influence of quantum interference processes . the experiments of weiss _ et al_.@xcite employed weakly modulated two - dimensional ( 2d ) electron systems of high mobility with a well - known period , @xmath1 , much less than the mean free path , @xmath2 . such samples were engineered using a holographic modulation technique , based on the persistent photoconductivity effect in gaas / al@xmath3ga@xmath4as heterostructures at low temperatures @xmath5 . the weiss oscillations appear in only one component of the resistivity tensor , @xmath6 , when the modulation is in the @xmath7 direction . furthermore they appear at magnetic fields @xmath8 such that the cyclotron radius @xmath9 ( where @xmath10 is the cyclotron frequency , @xmath11 is the electron charge and @xmath12 is the fermi velocity ) satisfies the commensurability condition @xmath13 , for integer @xmath14 . hence the oscillations are periodic in @xmath15 . in addition they are relatively stable with respect to temperature , suggesting a quasiclassical origin . at even higher magnetic fields , such that the cyclotron radius is much less than the period of modulation ( @xmath16 ) , the quasiclassical theory predicts that the @xmath6 component shows a large , nonoscillatory increase proportional to @xmath17 , leading to a strong positive magnetoresistance . this result has been confirmed in experiments@xcite for which the temperature was kept deliberately high so as to avoid the intervention of the iqhe . a limitation of the quasiclassical approach is that it fails to account for the renormalization of single - particle properties , such as the density of states and scattering lifetime , by the strong magnetic field according to landau quantization . even in unmodulated samples , quantization leads to oscillations in the density of states with respect to magnetic fields , and hence to shubnikov de haas oscillations in the resistivity . in samples with the periodic potential , the shubnikov de haas oscillations start to appear at higher magnetic fields than the weiss oscillations ( for @xmath18 an integer multiple of the fermi wavelength @xmath19 ) . a quantum - mechanical approach that does allow for such quantization effects has been provided by zhang and gerhardts@xcite ( see also peeters and vasilopoulos@xcite ) ; it is a diagrammatic treatment that generalizes the approach of ando and co workers@xcite to modulated samples . the quantum - mechanical approach is then capable of describing both weiss oscillations and the shubnikov de haas oscillations ( also affected by the modulation ) at higher magnetic fields . in principle , the calculation of weak localization corrections in a strong magnetic field ( unitary ensemble ) is possible even in the presence of a periodic potential , by a generalization of the diagrammatic approach of zhang and gerhardts@xcite , but the procedure would be complicated ( although the simpler orthogonal case has been examined diagrammatically for a periodic magnetic field@xcite ) . instead , the calculation of high - order diagrams is more convenient in the field - theory formalism and , furthermore , with this method the possibility exists of calculating contributions of diagrams to all orders by the renormalization - group technique . we show how the field - theory takes the form of a nonlinear @xmath0 model with a topological term @xcite . the effective lagrangian is slightly nonstandard since it contains an anisotropy in the coefficients , corresponding to the difference between the longitudinal conductivities in the @xmath7 and @xmath20 directions , due to the periodic potential . the effect of weak localization corrections to the conductivity for unmodulated samples is accounted for by a scaling ( one - parameter scaling ) of the conductivity with the system size . as a first step we derive the analog of the one - parameter scaling for the conductivity@xcite in the modulated system by means of a perturbative rg analysis of the effective lagrangian . we then turn to the study of the iqhe , implementing a generalization of a two - parameter scaling , which has been conjectured@xcite as a model for the iqhe in unmodulated systems . we examine how the resistivity tensor should be affected at low temperatures by the iqhe in modulated samples for parameters that are realizable in actual experiments . we see , for example , how the hall conductivity becomes quantized under scaling at low temperatures , while in the regions between the plateaus the longitudinal conductivities develop peaks of differing heights according to the anisotropy in the @xmath7 and @xmath20 directions . _ in the following , we employ the hamiltonian for the disordered conductor in a magnetic field and periodic potential in two dimensions : @xmath21 here @xmath22 is the vector potential , so that @xmath23 , where @xmath8 is the perpendicular , uniform magnetic field . also @xmath24 is the periodic potential which is taken to be weak ( @xmath25 , where @xmath26 is the fermi energy ) and with modulation period @xmath27 . @xmath28 is the disorder potential , which we assume to be @xmath29-correlated in space , with the associated scattering time @xmath30 . we assume that the mean free path @xmath31 greatly exceeds the modulation period @xmath32 . the plan for the remainder of the paper is as follows . in sec . [ sec : sigma ] we describe the field theory approach and derive the effective lagrangian for the system . in sec . [ sec : scaling ] we show how the field theory provides the scaling of the conductivity tensor under changes of length scale due to the contribution of quantum interference processes , and hence a description of the iqhe in these samples . section [ sec : summary ] concludes with a summary and discussion .

we consider magnetotransport in a disordered two - dimensional electron gas in the presence of a periodic modulation in one direction . existing quasiclassical and quantum approaches to this problem account for weiss oscillations in the resistivity tensor at moderate magnetic fields , as well as a strong modulation - induced modification of the shubnikov de haas oscillations at higher magnetic fields . they do not account , however , for the operation at even higher magnetic fields of the integer quantum hall effect , for which quantum interference processes are responsible . we introduce a field theory approach , based on a nonlinear model , which encompasses naturally both the quasiclassical and quantum mechanical approaches , as well as providing a consistent means of extending them to include quantum interference corrections . a perturbative renormalization - group analysis of the field theory shows how weak localization corrections to the conductivity tensor may be described by a modification of the usual one - parameter scaling , such as to accommodate the anisotropy of the bare conductivity tensor . we also show how the two - parameter scaling , conjectured as a model for the quantum hall effect in unmodulated systems , may be generalized similarly for the modulated system . within this model

we consider magnetotransport in a disordered two - dimensional electron gas in the presence of a periodic modulation in one direction . existing quasiclassical and quantum approaches to this problem account for weiss oscillations in the resistivity tensor at moderate magnetic fields , as well as a strong modulation - induced modification of the shubnikov de haas oscillations at higher magnetic fields . they do not account , however , for the operation at even higher magnetic fields of the integer quantum hall effect , for which quantum interference processes are responsible . we introduce a field theory approach , based on a nonlinear model , which encompasses naturally both the quasiclassical and quantum mechanical approaches , as well as providing a consistent means of extending them to include quantum interference corrections . a perturbative renormalization - group analysis of the field theory shows how weak localization corrections to the conductivity tensor may be described by a modification of the usual one - parameter scaling , such as to accommodate the anisotropy of the bare conductivity tensor . we also show how the two - parameter scaling , conjectured as a model for the quantum hall effect in unmodulated systems , may be generalized similarly for the modulated system . within this model we illustrate the operation of the quantum hall effect in modulated systems for parameters that are realistic for current experiments .

1210.3841

i

let @xmath8 be an algebraically closed field and @xmath9 be integers with @xmath10 . let @xmath11 denote the affine variety in @xmath12 defined by the vanishing of all @xmath13 minors of an @xmath3 matrix whose entries are independent indeterminates over @xmath8 . equivalently , @xmath11 is the locus of @xmath3 matrices over @xmath8 of rank @xmath14 . this is a classical and well - studied object and a number of its properties are known . for example , we know that @xmath11 is irreducible , rational , arithmetically cohen - macaulay and projectively normal . moreover the multiplicity of @xmath11 ( at its vertex , since @xmath11 is evidently a cone ) or equivalently , the degree of the corresponding projective subvariety of @xmath15 is given by the following elegant formula ( cf . 20.18 and 20.19 ) or ( * ? ? 6.2 ) ; see also @xcite for an alternative proof and ( * ? ? ? * ch . 2 , 4 ) or @xcite for an alternative approach and a different formula ) : @xmath16 more generally , the hilbert series of @xmath11 ( or more precisely , of the corresponding projective subvariety of @xmath15 ) is also known and is explicitly given by @xmath17 where @xmath18 is the dimension of @xmath11 ( as an affine variety ) , and the coefficients @xmath19 are given by sums of binomial determinants as follows : @xmath20 for a proof of this formula , we refer to @xcite ( see also @xcite and @xcite ) . using this ( cf . @xcite ) , or otherwise ( cf . @xcite ) , it can be shown that @xmath11 is gorenstein if and only if @xmath7 . moreover , one can also show that the @xmath6-invariant of the ( homogeneous ) coordinate ring of @xmath11 [ which , by definition , is the least degree of a generator of its graded canonical module ] is @xmath21 ; see , e.g. , @xcite or ( * ? ? ? we now turn to jet schemes , which have been of much recent interest due in large part to nash s suggestion @xcite that jet schemes should give information about singularities of the base ; see , e.g. , @xcite . if @xmath22 is a scheme of finite type over @xmath8 and @xmath23 a positive integer , then a @xmath24-jet on @xmath22 is a morphism @xmath25/(t^k ) \to { \mathcal{z}}$ ] . the set of @xmath24-jets on @xmath22 forms a scheme of finite type over @xmath8 , denoted @xmath26 and called the @xmath27 jet scheme of @xmath22 . a little more concretely , suppose @xmath22 is the affine scheme @xmath28 defined by the ideal @xmath29 in the polynomial ring @xmath30 $ ] . consider independent indeterminates @xmath31 and @xmath32 ( @xmath33 and @xmath34 ) over @xmath8 and the corresponding polynomial ring @xmath35 in the @xmath36 variables @xmath32 . for each @xmath37 , the polynomial @xmath38 is of the form @xmath39 modulo @xmath40 for unique @xmath41 ( @xmath42 ) . @xmath26 is then the affine scheme @xmath43 , where @xmath44 is the ideal generated by all @xmath45 , @xmath46 , @xmath42 . ( often in the literature , authors conflate the algebraic set in @xmath47 consisting of the zeros of the polynomials @xmath45 with @xmath26 itself . this is generally harmless , especially when considering topological properties such as components , since the points of this algebraic set correspond bijectively with the set of closed points of @xmath26 as @xmath8 is algebraically closed , and the set of closed points of an affine scheme is dense in the scheme . see ( * ? ? ? 3.49 ) for instance . ) when @xmath22 is smooth of dimension @xmath48 , the jet scheme @xmath26 is known to be smooth of dimension @xmath49 . in general , @xmath26 can have multiple irreducible components , and these include a principal component that corresponds to the closure of the set of jets supported over the smooth points of the base scheme @xmath22 . these components are usually quite complicated and interesting . in fact , very little seems to be known about the structure of these components and their numerical invariants such as multiplicities . for example , even when @xmath22 is a monomial scheme such as the one given by @xmath50 , where @xmath51 , determining the irreducible components and the multiplicity of @xmath26 appears to require some effort ; see , e.g. , @xcite and @xcite . irreducible components of jet schemes of toric surfaces are discussed in @xcite , while the irreducibility of jet schemes of the commuting matrix pairs scheme is discussed in @xcite . in a more recent work @xcite , the hilbert series of arc spaces ( that are , in a sense , limits of @xmath52 jet schemes as @xmath53 ) of seemingly simple objects such as the double line @xmath54 are shown to have connections with the rogers - ramanujan identities . now determinantal varieties such as @xmath11 above are natural examples of singular algebraic varieties and it is not surprising that the study of their jet schemes has been of considerable interest . this was done first by koir and sethuraman in @xcite and @xcite ( see also yuen @xcite ) . to describe the related results , henceforth we fix positive integers @xmath55 with @xmath56 , and let @xmath57 denote the @xmath27 jet scheme on @xmath11 . it was shown in @xcite that @xmath57 is irreducible of codimension @xmath58 when @xmath59 , and if @xmath60 , then it can have @xmath61 irreducible components with equality when @xmath62 or @xmath63 . a more unified result has recently been obtained by docampo @xcite who shows that @xmath57 has exactly @xmath64 irreducible components . at any rate , the best understood case with multiple components is @xmath0 , where @xmath65 . in this case @xmath66 , where @xmath67 is isomorphic to @xmath68 while @xmath69 is the principal component which is the closure of the jets supported over the smooth points of the base variety @xmath1 . here it will be convenient to consider @xmath70 indeterminates denoted @xmath71 for @xmath72 , @xmath73 , and the corresponding polynomial ring @xmath74 $ ] . also let @xmath75 and @xmath76 denote , respectively , the ideals of @xmath77 corresponding to the jet scheme @xmath0 and its principal component @xmath69 . in @xcite , it was shown that both @xmath78 and @xmath76 are homogeneous radical ideals of @xmath77 ( so that @xmath76 is prime ) and moreover , their grbner bases were explicitly determined . the leading term ideal @xmath79 of @xmath76 with respect to this grbner basis is generated by squarefree monomials and hence @xmath80 is the stanley - reisner ring of a simplicial complex @xmath81 . jonov @xcite subsequently studied this simplicial complex . he showed that @xmath81 is shellable and thus deduced that @xmath82 is cohen - macaulay . ( this last result was independently obtained by smith and weyman as well in @xcite , using their geometric technique for computing syzygies . ) jonov also found a formula for the multiplicity of @xmath82 , namely , @xmath83 equation ( [ mult ] ) above is the starting point of the present paper . we first show that the right side of this equation simplifies remarkably to yield the pretty result @xmath84 ( this was already mentioned in ( * ? ? ? 2.8 ) by way of a remark ) . next , we proceed to determine the hilbert series of @xmath82 or of the principal component @xmath69 . we use the well - known connections between the hilbert series of @xmath82 , that of @xmath80 , and the shelling of the facets of the simplicial complex @xmath81 obtained in @xcite . with some effort we are led to an initial formula for the hilbert series of @xmath82 , which is enormously complicated and involves multiple sums of products of binomials in the same vein as the right side of . but we persist with the combinatorics and are eventually rewarded with the main result of this paper . namely , just like the multiplicity , the hilbert series of @xmath82 is precisely the square of the hilbert series of the base determinantal variety @xmath1 . as a corollary of this , we are able to determine the @xmath6-invariant of @xmath82 and the hilbert series of its graded canonical module . moreover , we show that , as in the case of classical determinantal varieties , @xmath69 is gorenstein if and only if @xmath7 . the proofs given here are completely elementary , but highly combinatorial and rather intricate . heuristically , it appears to us that up to some flat deformation ( such as the grbner deformation of @xmath76 to @xmath79 , which preserves the hilbert series ) , the coordinate ring of the principal component ( suitably deformed ) should look like the tensor product of the coordinate ring of the base ( similarly deformed ) with itself . ( this would reflect the fact that at the smooth points , the base variety locally looks like its tangent space . ) it would follow then that the hilbert series of the principal component is the square of that of @xmath1 . we emphasize that this is only heuristics ( with all of its ever - present dangers ) ; nevertheless , we suspect that analogous results relating the hilbert series of the principal component to that of the base scheme should hold more generally for all @xmath57 , and possibly also for jet schemes over a wider class of affine base schemes . we do not know how to prove this , and leave it open for investigation .

when , this jet scheme has two irreducible components : a trivial component , isomorphic to an affine space , and a nontrivial component that is the closure of the jets supported over the smooth locus of . this second component is referred to as the _ principal component _ of ; it is , in fact , a cone and can also be regarded as a projective subvariety of .

we consider the affine variety of first order jets over , where is the classical determinantal variety given by the vanishing of all minors of a generic matrix . when , this jet scheme has two irreducible components : a trivial component , isomorphic to an affine space , and a nontrivial component that is the closure of the jets supported over the smooth locus of . this second component is referred to as the _ principal component _ of ; it is , in fact , a cone and can also be regarded as a projective subvariety of . we prove that the degree of the principal component of is the square of the degree of and more generally , the hilbert series of the principal component of is the square of the hilbert series of . as an application , we compute the-invariant of the principal component of and show that the principal component of is gorenstein if and only if .

1502.05655

i

we consider a discrete - time complex valued branching random walk . the system starts with an initial particle , called the root , at time @xmath0 . at time @xmath1 , the particle dies and gives birth to @xmath2 particles , which form the particles at generation @xmath3 . at time @xmath4 , each of these particles dies and gives birth to @xmath2 new particles , and so on ... for all @xmath5 , we denote @xmath6 the genealogical tree associated to the @xmath7-th generation : its elements have length @xmath7 which we denote by @xmath8 . then , we set @xmath9 . we consider a family of independent complex gaussian random variables @xmath10 indexed by the nodes of this tree and identically distributed with common law @xmath11 , whose real part is independent of the imaginary part ( see figure [ tree ] ) . = [ ->,very thick , rounded corners=4pt ] ; = [ circle , draw , fill = yellow , scale=0.6 ] = [ minimum width=2 cm , minimum height=0.8 cm , rectangle , rounded corners=10pt , draw , fill = yellow!30,text = red , font= * * ] ( r ) at ( 0,10)root ; ( r0 ) at ( -5,8 ) ; ( r1 ) at ( 5,8 ) ; ( r00 ) at ( -7,6 ) ; ( r01 ) at ( -3,6 ) ; ( r10 ) at ( 3,6 ) ; ( r11 ) at ( 7,6 ) ; ( r000 ) at ( -8,4 ) ; ( r001 ) at ( -6,4 ) ; ( r010 ) at ( -4,4 ) ; ( r011 ) at ( -2,4 ) ; ( r100 ) at ( 2,4 ) ; ( r101 ) at ( 4,4 ) ; ( r110 ) at ( 6,4 ) ; ( r111 ) at ( 8,4 ) ; ( r0000 ) at ( -8.5,3 ) ; ( r0001 ) at ( -7.5,3 ) ; ( r0010 ) at ( -6.5,3 ) ; ( r0011 ) at ( -5.5,3 ) ; ( r0100 ) at ( -4.5,3 ) ; ( r0101 ) at ( -3.5,3 ) ; ( r0110 ) at ( -2.5,3 ) ; ( r0111 ) at ( -1.5,3 ) ; ( r1000 ) at ( 1.5,3 ) ; ( r1001 ) at ( 2.5,3 ) ; ( r1010 ) at ( 3.5,3 ) ; ( r1011 ) at ( 4.5,3 ) ; ( r1100 ) at ( 5.5,3 ) ; ( r1101 ) at ( 6.5,3 ) ; ( r1110 ) at ( 7.5,3 ) ; ( r1111 ) at ( 8.5,3 ) ; ( time1 ) at ( -13,8)1st generation ; ( time2 ) at ( -13,6)2nd generation ; ( time3 ) at ( -13,4)3rd generation ; ( time4 ) at ( -13,3)@xmath12 ; ( r ) ( r0 ) node [ left ] @xmath13 ( r00 ) node [ left ] @xmath14 ( r000 ) node[left ] @xmath15 ; ( r ) ( r0 ) ( r00 ) ( r001 ) node[right ] @xmath16 ; ( r ) ( r0 ) ( r01 ) node[right ] @xmath17 ( r010 ) node[right ] @xmath18 ; ( r ) ( r0 ) ( r01 ) ( r011 ) node[right ] @xmath19 ; ( r ) ( r1 ) ( r10 ) ( r100 ) node[left ] @xmath20 ; ( r ) ( r1 ) node[right ] @xmath21 ( r10 ) node[left ] @xmath22 ( r101)node[right ] @xmath23 ; ( r ) ( r1 ) ( r11 ) node[right ] @xmath24 ( r110)node[right ] @xmath25 ; ( r ) ( r1 ) ( r11 ) ( r111)node[right ] @xmath26 ; ( r ) ( r0 ) ( r01 ) ( r011 ) ; ( theta ) at ( 0,2 ) @xmath27 ; ( theta ) to[bend left ] ( r011 ) ; ( r000)(r0000 ) ; ( r000)(r0001 ) ; ( r001)(r0010 ) ; ( r001)(r0011 ) ; ( r010)(r0100 ) ; ( r010)(r0101 ) ; ( r011)(r0110 ) ; ( r011)(r0111 ) ; ( r100)(r1000 ) ; ( r100)(r1001 ) ; ( r101)(r1010 ) ; ( r101)(r1011 ) ; ( r110)(r1100 ) ; ( r110)(r1101 ) ; ( r111)(r1110 ) ; ( r111)(r1111 ) ; * * if @xmath28 and @xmath29 or @xmath30 , we set @xmath31 . each particle @xmath32 is given a complex weight @xmath33 corresponding to the sum of the weights @xmath34 encountered along the shortest path joining @xmath35 to the root ( see figure [ tree ] ) , i.e. @xmath36 we define the real and imaginary part of @xmath37 @xmath38 which are independent . we assume that the real part of the branching random walk is critical @xmath39 put in other words , the real part of @xmath11 has law @xmath40 where @xmath41 is a standard gaussian random variable . the imaginary part of @xmath11 has law @xmath41 and is independent of its real part . for any @xmath42 , we define : @xmath43 one gets a martingale out of @xmath44 by renormalizing by the mean , namely by considering @xmath45)_n$ ] . it is natural to wonder for which values of the parameters @xmath46 this renormalization by the mean gives a martingale converging almost surely towards a non trivial limit @xmath47 . the real case @xmath48 has given rise to an extensive literature ranging from the study of mandelbrot s multiplicative cascades ( see @xcite among many others ) to some extensions like the study of branching random walks ( see @xcite and references therein ) . the complex case has been answered in @xcite ( actually , @xcite is concerned with gaussian multiplicative chaos , which is a different context , but the methods apply to our context ) . the reader may have in mind the resulting phase diagram in figure [ diagram ] . ( 3,6 ) ; ( 3.5,3.5 ) ( 12,3.5 ) ( 0,7 ) ; ( 5,0 ) arc ( 0:45:5 ) ( 0,0 ) ( 5,0 ) ; ( 3.53,3.5 ) ( 12,3.5 ) ; ( 3.6,3.5 ) ( 0,7 ) ; ( 5,0 ) arc ( 0:45:5 ) ; ( 0,0 ) ( 0,9 ) node[left]@xmath49 ; ( 0,0 ) ( 12.2,0)node[below]@xmath50 ; ( 7,3.5 ) node[above]@xmath51 ; ( 2,5.2 ) node[above , rotate=-45,line width=2pt]@xmath52 ; ( 4.8,1.5 ) node[above , rotate=-70,line width=2pt]@xmath53 ; ( 1,2 ) node[right ] phase i ; ( 5,7 ) node[right ] phase ii ; ( 7,2 ) node[right ] phase iii ; it is proved in @xcite that we have almost sure convergence towards a non trivial limit in the so - called phase i , i.e. @xmath54\frac{1}{2},1[\text { and } \gamma+\beta<1\big)\ ] ] and the boundary case @xmath55\frac{1}{2},1[\text { and } \gamma+\beta=1\ ] ] is treated in @xcite on a related model , gaussian multiplicative chaos . in all other cases , a renormalization by the mean is a martingale that does not converge to something non trivial . a mathematical understanding of the limiting objects we get is therefore a natural question . in particular , one may look at more generally as a complex measure on @xmath56 $ ] . the canonical way to do so is to use the dyadic decomposition of the reals that belong to @xmath56 $ ] . more specifically , if @xmath57 then we set @xmath58 $ ] ; the complex measure @xmath59 is then defined on @xmath56 $ ] by @xmath60t_z , t_z+2^{-n}]}(x)\,2^n\,dx.\ ] ] in the case when @xmath48 , these complex measures turn out to be positive random measures and an important question is to study the diffusivity properties of the scaling limit . it is known nowadays that the limiting measures are diffuse in phase i @xcite or its boundary ( i.e. @xmath61 , see @xcite ) : they are atom free . in the complex case ( i.e. @xmath62 ) , this question makes sense when asking whether the family of random functions @xmath63\mapsto m_n^{\gamma,\beta}[0,t]\ ] ] converge uniformly towards a continuous limit after renormalization by the mean of the total mass @xmath64)$ ] . in the inner phase i , it is proved in @xcite that we have uniform convergence of this family of functions ( even in the hlder sense ) as an application of the kolmogorov criterion . in this paper , we aim at answering this question in the boundary case , i.e. the frontier of phases i / ii excluding the extremal points . in this case , the kolmogorov criterion ( or refined versions ) breaks down . hence establishing uniform convergence is more difficult as one can no longer rely on general machinery on convergence or tightness of stochastic processes . let us also mention that we have chosen a normalization of our parameters @xmath46 so that the mean in the boundary case is exactly one , that is for any @xmath65 with @xmath66 and any borel set @xmath67 of @xmath56 $ ] , the family @xmath68 is a complex valued martingale with mean @xmath3 . now , we can state the main results of this paper : [ th : main ] let @xmath46 belong to the boundary of phases i / ii , i.e. . the sequence of functions @xmath69 $ ] converges almost surely in the space of continuous functions towards some random continuous function @xmath47 . there exists some ( non explicit ) constant @xmath70 ( given by @xmath71 where @xmath72 appears in proposition [ boundness ] ) such that for all @xmath73 the function @xmath47 satisfies almost surely the following modulus estimate @xmath74 is some random constant . we could in principle give an explicit formula for @xmath75 ( as a function of @xmath49 ) but since our method is not optimal , we did not try to keep track of this information . as mentioned in the introduction , it turns out that the space of continuous functions is the right space to study convergence of @xmath76 . indeed , though for each @xmath77 @xmath76 is a complex measure , one can show that the limiting function @xmath47 is not of finite variation ( see subsection [ infinitevar ] ) [ coro : main ] let @xmath46 belong to the boundary of phases i / ii , i.e. . the mapping @xmath78 $ ] is not of finite variation almost surely . the authors of @xcite studied complex gaussian multiplicative chaos ( gmc ) in dimension @xmath79 , i.e. the renormalization theory of @xmath80 where @xmath81 is the lebesgue measure , @xmath82 , @xmath83 and @xmath84 are two independent log - correlated fields on some domain @xmath85 ( of course , we could absorb the @xmath86 in the parameters @xmath49 and @xmath50 but we will not do so in order to compare complex gmc with the complex branching random walk studied in this paper ) . in particular , the work @xcite enables to define @xmath87 where @xmath83 and @xmath84 are two independent log - correlated fields on @xmath88 with covariance given by @xmath89= \ln \frac{1}{|y - x|}$ ] and when @xmath90 satisfies the condition @xmath91 condition i / ii refers to the so - called frontier between phase i and phase ii : see @xcite . more precisely , if @xmath92 are appropriate cut - off approximations of @xmath93 ( as @xmath94 goes to @xmath95 ) then , if @xmath46 are in phase i / ii , the random distribution @xmath96 } dx\ ] ] converges almost surely ( in the space of distributions ) towards a distribution @xmath97 . in this case , the operator @xmath97 is a conformally invariant boundary operator in the framework of @xmath98 string theory on the upper half plane ( see @xcite ) . the work @xcite did not establish but conjectured that convergence holds in the space of continuous functions . now , one expects that the complex branching random walk of this paper and complex gmc have a similar behaviour . one justification for this is that the branching random walk also has logarithmic correlations but with respect to the underlying ultrametric distance on the tree @xmath99 . hence , theorem [ th : main ] gives additional support to the conjecture of @xcite . another important question on this topic is the following . when the renormalized martingale defined by converges towards a non trivial limit , it is readily seen that the limit satisfies a distributional equation of the type @xmath100 where @xmath101 are complex random variables independent of @xmath102 , which are i.i.d . random variables with law @xmath47 . such an equation is known under the name of smoothing transform and has been extensively studied in the case when @xmath101 are positive ( see @xcite ) , real valued @xcite . we would like to emphasize the fact that understanding this equation in the complex case is an important point . let us mention some work to appear @xcite in this perspective . the authors would like to thank matthias meiners for interesting discussions .

@addtoreset p l [ section ] [ theorem]definition [ theorem]lemma [ theorem]proposition [ theorem]remark [ theorem]corollary [ theorem]conjecture * key words or phrases : * branching random walk , complex weights , boundary case , diffusivity . * msc 2000 subject classifications : 60j80 , 60g57 , 60g50 , 60g17 . * = 2truecm = 2truecm

we consider the complex branching random walk on a dyadic tree with gaussian weights on the boundary between the diffuse phase and the glassy phase . we study the branching random walk in the space of continuous functions and establish convergence in this space . the main difficulty here is that the expected modulus of continuity of the limit is too weak in order to show tightness in the space of continuous functions by means of standard tools from the theory of stochastic processes . @addtoreset p l [ section ] [ theorem]definition [ theorem]lemma [ theorem]proposition [ theorem]remark [ theorem]corollary [ theorem]conjecture * key words or phrases : * branching random walk , complex weights , boundary case , diffusivity . * msc 2000 subject classifications : 60j80 , 60g57 , 60g50 , 60g17 . * = 2truecm = 2truecm

1502.05655

r

in this section , we first introduce some further notations and then give the proof of theorem [ th : main ] based on some auxiliary results , the proofs of which are postponed to the following sections in the paper . in the sequel , we will denote by @xmath103 a generic positive constant which can change from line to line ; we will also denote @xmath72 a generic constant belonging to @xmath104 whose value can also change from line to line . usually , it will be clear from the context that @xmath103 or @xmath72 can depend on other constants but to lighten notations , we will make this dependence implicit . for any @xmath105 , we let @xmath106 be the left child of @xmath107 and @xmath108 be the right child of @xmath107 ( see figure [ rootened ] ) . notice that if @xmath109 then @xmath110 . furthermore , to simplify the notations , we mention that we will implicitly assume in the following that the notation @xmath111 means @xmath112 ( note that it is not clear otherwise whether one must take the child after or before restricting to the k - th node of @xmath35 ) . the same convention holds for the right child . for any @xmath113 , we further consider the sub - tree @xmath114 of @xmath99 rootened in @xmath35 ( see figure [ rootened ] ) , namely @xmath115 finally , we define an order on @xmath116 . we write @xmath117 if @xmath35 is a descendant of @xmath107 in the tree @xmath99 , i.e. if @xmath118 and @xmath119 ( we adopt similar conventions for @xmath120 ) . = [ ->,very thick , rounded corners=3pt ] ; = [ circle , draw , fill = yellow , scale=0.6,minimum height=1 cm ] = [ minimum width=2 cm , minimum height=0.8 cm , rectangle , rounded corners=10pt , draw , fill = yellow!30,text = red , font= * * ] ( r ) at ( 3,9 ) ; ( r1 ) at ( 0,8)z ; ( r10 ) at ( -2,6)@xmath106 ; ( r11 ) at ( 2,6)@xmath108 ; ( r100 ) at ( -3,4 ) ; ( r101 ) at ( -1,4 ) ; ( r110 ) at ( 1,4 ) ; ( r111 ) at ( 3,4 ) ; ( r ) ( r1 ) ( r10 ) ( r100 ) ; ( r ) ( r1 ) ( r10 ) ( r101 ) ; ( r1 ) ( r10 ) ; ( r1 ) ( r11 ) ; ( r ) ( r1 ) ( r11 ) ( r110 ) ; ( r ) ( r1 ) ( r11) ( r111 ) ; * * = [ ->,very thick , rounded corners=4pt ] ; = [ circle , draw , fill = yellow , scale=0.6 ] = [ minimum width=2 cm , minimum height=0.8 cm , rectangle , rounded corners=10pt , draw , fill = yellow!30,text = red , font= * * ] ( r ) at ( 0,10)root ; ( r0 ) at ( -5,8)0 ; ( r1 ) at ( 5,8)1 ; ( r00 ) at ( -7,6)00 ; ( r01 ) at ( -3,6)01 ; ( r10 ) at ( 3,6)10 ; ( r11 ) at ( 7,6)11 ; ( r000 ) at ( -8,4 ) ; ( r001 ) at ( -6,4 ) ; ( r010 ) at ( -4,4 ) ; ( r011 ) at ( -2,4 ) ; ( r100 ) at ( 2,4 ) ; ( r101 ) at ( 4,4 ) ; ( r110 ) at ( 6,4 ) ; ( r111 ) at ( 8,4 ) ; ( r0000 ) at ( -8.5,3 ) ; ( r0001 ) at ( -7.5,3 ) ; ( r0010 ) at ( -6.5,3 ) ; ( r0011 ) at ( -5.5,3 ) ; ( r0100 ) at ( -4.5,3 ) ; ( r0101 ) at ( -3.5,3 ) ; ( r0110 ) at ( -2.5,3 ) ; ( r0111 ) at ( -1.5,3 ) ; ( r1000 ) at ( 1.5,3 ) ; ( r1001 ) at ( 2.5,3 ) ; ( r1010 ) at ( 3.5,3 ) ; ( r1011 ) at ( 4.5,3 ) ; ( r1100 ) at ( 5.5,3 ) ; ( r1101 ) at ( 6.5,3 ) ; ( r1110 ) at ( 7.5,3 ) ; ( r1111 ) at ( 8.5,3 ) ; ( r ) ( r0 ) ( r00 ) ( r000 ) ; ( r ) ( r0 ) ( r00 ) ( r001 ) ; ( r ) ( r0 ) ( r01 ) ( r010 ) ; ( r ) ( r0 ) ( r01 ) ( r011 ) ; ( r ) ( r1 ) ( r10 ) ( r100 ) ; ( r ) ( r1 ) ( r10) ( r101 ) ; ( r ) ( r1 ) ( r11 ) ( r110 ) ; ( r ) ( r1 ) ( r11 ) ( r111 ) ; ( r ) ( r0 ) ( r01 ) ( r011 ) ; ( r ) ( r0 ) ( r01 ) ( r010 ) ; ( r000)(r0000 ) ; ( r000)(r0001 ) ; ( r001)(r0010 ) ; ( r001)(r0011 ) ; ( r010)(r0100 ) ; ( r010)(r0101 ) ; ( r011)(r0110 ) ; ( r011)(r0111 ) ; ( r100)(r1000 ) ; ( r100)(r1001 ) ; ( r101)(r1010 ) ; ( r101)(r1011 ) ; ( r110)(r1100 ) ; ( r110)(r1101 ) ; ( r111)(r1110 ) ; ( r111)(r1111 ) ; * * we set @xmath121 . for any @xmath122 and @xmath123 with @xmath29 , we consider the mass of the subtree rootened at @xmath107 up to generation @xmath7 ( see figure [ mnz ] ) @xmath124\\ = & \sum_{|z|=n,\ , z_{|l}=u } \ee^{-\gamma [ v(z)-v(u)]+ i\beta\sqrt{2\ln 2}[x(z)-x(u)]}.\nonumber\end{aligned}\ ] ] = [ ->,very thick , rounded corners=4pt ] ; = [ circle , draw , fill = yellow , scale=0.7 ] = [ minimum width=2 cm , minimum height=0.8 cm , rectangle , rounded corners=10pt , draw , fill = yellow!30,text = red , font= * * ] ( 1r ) at ( 5,12 ) ; ( r ) at ( 0,10 ) ; ( r0 ) at ( -5,8)z ; ( r1 ) at ( 5,8 ) ; ( r00 ) at ( -7,6 ) ; ( r01 ) at ( -3,6 ) ; ( r10 ) at ( 3,6 ) ; ( r11 ) at ( 7,6 ) ; ( r000 ) at ( -8,4 ) ; ( r001 ) at ( -6,4 ) ; ( r010 ) at ( -4,4 ) ; ( r011 ) at ( -2,4 ) ; ( r100 ) at ( 2,4 ) ; ( r101 ) at ( 4,4 ) ; ( r110 ) at ( 6,4 ) ; ( r111 ) at ( 8,4 ) ; ( time1 ) at ( -13,8)k - th generation ; ( time2 ) at ( -13,6)@xmath12 ; ( time3 ) at ( -13,4)n - th generation ; ( r ) ( r0 ) ( r00 ) ( r000 ) ; ( r ) ( r0 ) ( r00 ) ( r001 ) ; ( r ) ( r0 ) ( r01 ) node[right ] @xmath17 ( r010 ) node[below ] @xmath18 ; ( r ) ( r0 ) ( r01 ) ( r011 ) node[below ] @xmath19 ; ( r ) ( r1 ) ( r10 ) ( r100 ) ; ( r ) ( r1 ) ( r10 ) ( r101 ) ; ( r ) ( r1 ) ( r11 ) ( r110 ) ; ( r ) ( r1 ) ( r11 ) ( r111 ) ; ( r0 ) ( r01 ) ( r011 ) ; ( r01 ) ( r010 ) ; ( 1r)(r ) ; * * with these notations , we have for @xmath8 @xmath125= \sum_{k=0}^{n-1 } \ee^{-\gamma v(u_{|k}^{(l ) } ) + i\beta\sqrt{2\ln 2}x(u_{|k}^{(l ) } ) } m_{n}^{\gamma,\beta}(u_{|k}^{(l ) } ) 1_{\ { u_{| k+1}\neq u_{|k}^{(l ) } \}}.\ ] ] this can be seen by summing over all the subtrees located on the left - hand side of the path joining the root to the particle @xmath35 ( see figure [ lefttree ] ) . = [ ->,very thick , rounded corners=4pt ] ; = [ circle , draw , fill = yellow , scale=0.7 ] = [ minimum width=2 cm , minimum height=0.8 cm , rectangle , rounded corners=10pt , draw , fill = yellow!30,text = red , font= * * ] ( 1r ) at ( 5,12 ) ; ( r ) at ( 0,10 ) ; ( r0 ) at ( -5,8 ) ; ( r1 ) at ( 5,8 ) ; ( r00 ) at ( -7,6 ) ; ( r01 ) at ( -3,6 ) ; ( r10 ) at ( 3,6 ) ; ( r11 ) at ( 7,6 ) ; ( r000 ) at ( -8,4 ) ; ( r001 ) at ( -6,4 ) ; ( r010 ) at ( -4,4 ) ; ( r011 ) at ( -2,4 ) ; ( r100 ) at ( 2,4 ) ; ( r101 ) at ( 4,4)u ; ( r110 ) at ( 6,4 ) ; ( r111 ) at ( 8,4 ) ; ( time1 ) at ( -13,8)k - th generation ; ( time2 ) at ( -13,6)@xmath12 ; ( time3 ) at ( -13,4)n - th generation ; ( r ) ( r0 ) ( r00 ) ( r000 ) ; ( r ) ( r0 ) ( r00 ) ( r001 ) ; ( r ) ( r0 ) ( r01 ) ( r010 ) ; ( r ) ( r0 ) ( r01 ) ( r011 ) ; ( r ) ( r1 ) ( r10 ) ( r100 ) ; ( r ) ( r1 ) ( r10 ) ( r101 ) ; ( r ) ( r1 ) ( r11 ) ( r110 ) ; ( r ) ( r1 ) ( r11 ) ( r111 ) ; ( r ) ( r1 ) ( r10 ) ( r101 ) ; ( 1r)(r ) ; ( r ) ( -6,8.5 ) ( -9,3.5 ) ( -1,3.5 ) ( -3.5,8) ( r ) ; ( r10 ) ( 1,3.5 ) ( 2.5,3.5 ) ( r10 ) ; * * in fact , we can extend this decomposition : if @xmath29 and @xmath8 @xmath126= & \ee^{-\gamma v(u_{|l } ) + i\beta\sqrt{2\ln 2}x(u_{|l } ) } \\ & \times\sum_{k=0}^{n - l-1 } \ee^{-\gamma ( v(u_{|k+l}^{(l)})-v(u_{|l } ) ) + i\beta\sqrt{2\ln 2 } ( x ( u_{|k+l}^{(l ) } ) -x(u_{|l } ) ) } m_{n}^{\gamma,\beta}(u_{|k+l}^{(l ) } ) 1_{\ { u_{|k+l+1}\neq u_{|k+l}^{(l ) } \ } } .\end{aligned}\ ] ] finally , we introduce the following quantity for @xmath127 , @xmath29 and @xmath128 @xmath129 and @xmath130 . note that by the triangle inequality we have @xmath131| \leq || m^{\gamma,\beta}_{n , p } ||_{\infty}\ ] ] hence upper bounds on @xmath132 lead to upper bounds on @xmath133|$ ] ( which of course is equal to @xmath134|$ ] ) . finally , by the recursive structure on the tree , i.e. the subtree starting from any vertex @xmath35 has same distribution as the original tree , the variable @xmath135 has the same distribution as @xmath136 for @xmath137 . we introduce the centered standard gaussian walk @xmath138 . when the walk starts from a point @xmath139 , we denote the associated probability measure @xmath140 and the corresponding measure @xmath141 . when @xmath142 , we will omit the subscript . we will denote @xmath143 the infimum of the walk on the set @xmath144 . finally , we recall the many to one lemma which is very useful in the context of branching random walks ; for all function @xmath145 @xmath146\ ] ] recall also that it is proven in @xcite that : -(see lemma 2.4 ) there exists @xmath147 such that for any @xmath148 , @xmath149 and @xmath150 , @xmath151\right)\leq \frac{c(1+x)(1+b)(1+(b - a))}{n^{\frac{3}{2}}}.\ ] ] -(see lemma b.2 ) for any @xmath152 there exists @xmath153 such that for any @xmath149 @xmath154 \leq c(\kappa)\ ] ] in fact , lemma 2.4 and lemma b.2 of @xcite are much more general and concerns more general walks than the gaussian one . we first explain the main idea behind the proof of theorem [ th : main ] . essentially , the proof relies on the fact that one can extract from a uniform ( in @xmath7 ) bound on the supremum of @xmath155 $ ] a uniform ( in @xmath7 ) bound on the increments of @xmath155 $ ] . more precisely , the main estimate of this paper is given by proposition [ boundness ] below which gives a uniform bound on @xmath132 for all @xmath156 . recall that , by , one should see inequality below as an estimate on the supremum of @xmath157 $ ] . in fact , we will only use the estimate for @xmath158 . now , by the recursive structure on the tree , i.e. the subtree starting from any vertex @xmath35 has same distribution as the original tree , one can transfer such global estimates to estimates on the modulus of continuity of @xmath159 : this is essentially the content of inequality ( recall that @xmath160 is defined in and has same distribution as @xmath161 with @xmath162 when @xmath127 ) . this is sufficient to get tightness in the space of continuous functions and then almost sure convergence is a consequence of a non trivial theorem on banach valued martingales in @xcite ( the main theorem of @xcite states that an @xmath163 bounded banach valued martingale converges almost surely if and only if it is tight : here we will work in the banach space of continuous functions ) . now , let us fix @xmath164 and @xmath165 such that @xmath166 . since @xmath167 for @xmath127 we have : @xmath168 } |m_r^{\gamma,\beta}[s , t]| \leq 2 e^{-\gamma v(u ) } || m^{\gamma,\beta}_r(u ) ||_{\infty}.\ ] ] also , for @xmath127 and @xmath169 the right neighboor on the tree ( i.e. @xmath170 ) , we have : @xmath171 } |m_r^{\gamma,\beta}[s , t]| \leq 2 e^{-\gamma v(u ) } || m^{\gamma,\beta}_r(u ) ||_{\infty}+ e^{-\gamma v(u ' ) } || m^{\gamma,\beta}_r(u ' ) ||_{\infty}.\ ] ] if @xmath172 then there are two possibilities : _ first case _ : @xmath173 and @xmath174 lie in the same dyadic interval of the form @xmath175 $ ] in which case we have @xmath176| \leq |m_r^{\gamma,\beta}[\frac{k}{2^l},s]|+ |m_r^{\gamma,\beta}[\frac{k}{2^l},t]| $ ] _ second case _ : @xmath173 and @xmath174 lies in some dyadic interval of the form @xmath175 $ ] and @xmath174 in the dyadic interval @xmath177 $ ] in which case we have @xmath176| \leq |m_r^{\gamma,\beta}[\frac{k}{2^l},s]|+ |m_r^{\gamma,\beta}[\frac{k}{2^l},\frac{k+1}{2^l}]| + |m_r^{\gamma,\beta}[\frac{k+1}{2^l},t]| $ ] therefore , we get the bound : @xmath178| \leq 3 \sup_{|u|=l } e^{-\gamma v(u ) } || m^{\gamma,\beta}_r(u ) ||_{\infty}\ ] ] recall that the recursive structure of the tree entails that @xmath179 above is distributed for each @xmath35 like @xmath180 with @xmath162 . let @xmath73 . we set @xmath181 where @xmath182 is such that the bound page holds . finally , we introduce @xmath183 recall that for all @xmath184 , there exists @xmath147 such that ( see @xcite lemma 2.3 for example ) @xmath185= \e\left [ l^{\frac{1}{2 } } \frac{(1+s_l^\eta)}{l^{\eta \over 2 } } 1 _ { \min_{j\leq l } s_j\geq -\log l } \right ] \leq c(\log l).\ ] ] therefore , we have for all @xmath186 \\ & \leq \frac{c\log l } { l^{\frac{\epsilon}{2 } } } .\end{aligned}\ ] ] then it leads to the estimate @xmath187 thus , by the bound , we have for all @xmath73 and @xmath188 @xmath189| \geq ( \delta_{l,\epsilon})^\gamma \big ) & \leq \p ( \mathbb{a}_{l,\epsilon}^c ) + \p\left ( \sup_{|u|=l } e^{-\gamma v(u ) } || m^{\gamma,\beta}_r(u ) ||_{\infty } \geq \frac{(\delta_{l,\epsilon})^\gamma}{3 } \ , | \ , \mathbb{a}_{l,\epsilon } \right ) \\ & \leq \frac{c}{l^{\frac{\epsilon}{4}}}+ 1- \p\left ( \max_{|u|=l } \ee^{-\gamma v(u ) } || m^{\gamma,\beta}_r(u ) ||_{\infty } \leq \frac{(\delta_{l,\epsilon})^\gamma}{3 } \ , | \ , \mathbb{a}_{l,\epsilon } \right ) \\ & \leq \frac{c}{l^{\frac{\epsilon}{4}}}+ 1- \e\left ( \prod_{|u|=l } \p\left ( \ee^{-\gamma v(u ) } || m^{\gamma,\beta}_r(u ) ||_{\infty } \leq \frac{(\delta_{l,\epsilon})^\gamma}{3 } \ : \big | v(u ) \right ) \ , | \ , \mathbb{a}_{l,\epsilon } \right).\end{aligned}\ ] ] in the last line , we have conditioned on @xmath190 et used the independence of the mass of the trees rooted at @xmath35 . hence @xmath189| \geq \delta_{l,\epsilon } \big ) & \leq \frac{c}{l^{\frac{\epsilon}{4}}}+ 1- \e\left ( \prod_{|u|=l}\left\ { 1- \p\left ( || m^{\gamma,\beta}_r(u ) ||_{\infty } \geq \frac{(\delta_{l,\epsilon})^\gamma}{3 } \ee^{\gamma v(u ) } \big | v(u ) \right ) \ , | \ , \mathbb{a}_{l,\epsilon } \right\ } \right ) \\ & \leq \frac{c}{l^{\frac{\epsilon}{4}}}+ 1- \e\left ( \prod_{|u|=l } \left\{1- \frac{3c}{\delta_{l,\epsilon}}(1 + v(u)^\eta 1_{\min_{j\leq l } v(u_{|j } ) \geq -\log l})\ee^{-v(u ) } \right\ } \ , | \ , \mathbb{a}_{l,\epsilon } \right ) \\ & \leq \frac{c}{l^{\frac{\epsilon}{4}}}+ \frac{c}{l^{\frac{\epsilon}{2 } } } \\ & \leq \frac{c}{l^{\frac{\epsilon}{4 } } } .\end{aligned}\ ] ] therefore , we get @xmath191| \geq ( \delta_{l,\epsilon})^\gamma \right ) \leq \frac{c}{l^{\frac{\epsilon}{4}}}\ ] ] and then @xmath192| \geq ( \delta_{l,\epsilon})^\gamma \right ) = 0,\ ] ] which implies that @xmath193 is tight in the space of continuous functions . now , by using theorem 3 of @xcite where we view @xmath193 as a martingale which takes values in the banach space of continuous functions , we conclude that @xmath193 converges almost surely in the space of continuous functions towards some continuous function @xmath194 . finally , the estimate implies that @xmath195| \geq \delta_{l,\epsilon } \right ) \leq \frac{c}{l^{\frac{\epsilon}{4}}}\ ] ] which gives the stated modulus of continuity estimate .

we consider the complex branching random walk on a dyadic tree with gaussian weights on the boundary between the diffuse phase and the glassy phase . we study the branching random walk in the space of continuous functions and establish convergence in this space . the main difficulty here is that the expected modulus of continuity of the limit is too weak in order to show tightness in the space of continuous functions by means of standard tools from the theory of stochastic processes .

we consider the complex branching random walk on a dyadic tree with gaussian weights on the boundary between the diffuse phase and the glassy phase . we study the branching random walk in the space of continuous functions and establish convergence in this space . the main difficulty here is that the expected modulus of continuity of the limit is too weak in order to show tightness in the space of continuous functions by means of standard tools from the theory of stochastic processes . @addtoreset p l [ section ] [ theorem]definition [ theorem]lemma [ theorem]proposition [ theorem]remark [ theorem]corollary [ theorem]conjecture * key words or phrases : * branching random walk , complex weights , boundary case , diffusivity . * msc 2000 subject classifications : 60j80 , 60g57 , 60g50 , 60g17 . * = 2truecm = 2truecm

1108.4720

i

critical phenomena of partial differential equations ( pdes ) associated with the critical parameter can be found in various physical applications . for example , in @xcite the gravitational collapse of the scalar field was considered . the critical phenomenon of the gravitational collapse was discovered ; there exists a critical initial mass such that if the initial mass is less than the critical mass , the scalar field eventually disperses . conversely , it forms a black hole if the value exceeds the critical mass . if the initial parameter is arbitrarily close to the critical value , scale - invariant behavior of the solution occurs @xcite . for pdes from nonlinear optics , a similar phenomenon has been found and studied , e.g. the sine - gordon equation or nonlinear schrdinger equation with a point - like singular potential term @xcite . for the sine - gordon equation , there exists a kink soliton solution ; this yields a critical behavior once a self - interacting singular potential is placed . this phenomenon is known as the wave phenomenon in disordered media with a defect . if the initial velocity of the kink soliton solution exceeds the critical velocity , the kink soliton eventually passes through the potential , which is known as the _ particle - pass_. if the initial velocity is less than the critical velocity , the soliton is either captured by or reflected by the potential , known as the _ particle - capture _ or _ particle - reflection _ , since the sine - gordon equation is not integrable when the singular potential term is present , no exact solution is available ; no exact value of the critical parameter is available either . there are many unknown properties of the solution behaviors in this case . for more detailed study , we refer the readers to @xcite . a similar phenomenon has been also found with nonlinear schrdinger equations @xcite . for example , in @xcite the singular potential term perturbs the soliton propagation . similar phenomena of particle - pass , particle - capture , and particle - reflection were observed for some critical parameters . in this paper , we consider the critical phenomenon that is induced by the point - like singular source term in the pdes , such as the sine - gordon equation mentioned above . in particular , we consider the klein - gordon and sine - gordon equations to be perturbed by the singular potential term . for both cases , the solution behaviors are highly sensitive to the critical value of the parameter . determination of the critical value and the mean solution is an important task because the wave or soliton interactions with the local defect should be well understood for better optical communications . in either particle - capture or particle - reflection , the optical signal is not transmitted with the associated parametric value if it is less than the critical value . the signal should be transmitted with some values of the parameter that exceed the critical value for the communication without blocking or trapping @xcite . finding the critical value is doable but challenging . there are two reasons . first , the singular potential is given by the dirac @xmath0-function , so the given pdes are not well - defined . for the sine - gordon equation , no exact solution is known with the presence of the singular potential term , which means that no analytic form of the critical value is available only the numerical or real laboratory experimental approach can render the critical value measurable . the numerical determination of the critical parameter value relies on expensive monte - carlo(mc ) simulations based on the bi - section method . moreover , it is highly expensive to determine the critical value up to an arbitrary accuracy ( for the sine - gordon case , it is not even known whether the critical value is rational or irrational ) . furthermore , the numerically obtained critical value depends on the numerical method , considered with given numerical settings such as the truncation number , domain size , final time etc . due to the singularity in the potential term , the gibbs phenomenon also appears in the finite truncated solution , resulting in a slow convergence . thus when we talk about finding the critical value , we mean the critical value defined by the numerical method used , not the _ exact _ value from the pdes . this will be explained in detail , using the klein - gordon equation in the paper . second , although the singular potential term is used to mimic the effect of the local defect , such a model is not perfect . associated parameters , such as the strength and the location of the singular potential terms , have a great degree of uncertainty . thus , determining the critical value needs to be understood within the context of the uncertainty analysis . in this paper , as the first step toward thoroughly analyzing these two issues , we propose to use the generalized polynomial chaos ( gpc ) method @xcite for finding the critical value of both the klein - gordon and sine - gordon equations and the associated statistical quantities such as mean solutions around the critical value . the gpc method has been successfully implemented to analyze the uncertainty quantification for problems such as fluid flows with uncertain initial and boundary conditions , sensitivity analysis , steady - state problems , etc . @xcite . in our proposed method , we consider the associated parameter that yields the critical phenomenon of the system as a random variable , and assume that we will try to find a critical parameter value with the given numerical setting . for the klein - gordon equation , the associated parameter is the strength or amplitude of the singular potential term ; for the sine - gordon equation , the associated parameter is the initial velocity of the soliton solution in the presence of the singular potential term that has a fixed amplitude . the original equations are then parameterized by the random variable and the solution of the parameterized equations is obtained by the gpc method . that is , the solution is expanded by the orthogonal polynomials in the random space . the associated orthogonal polynomials are determined by the probabilistic nature of the random variable , which is the critical parameter in our case . for example , if the random variable has the uniform probability density function ( pdf ) , the associated polynomial is the legendre polynomial ; if the pdf is normal distribution , the associated polynomial is the hermite polynomial @xcite . we first consider the klein - gordon equation with the singular potential and find some conditions for the critical phenomenon . the klein - gordon equation with the singular source term has been considered in various studies . for example , in @xcite , the singular source term is considered in adopting the model , the meson theory , that the particle mass varies with space and time . in @xcite , two @xmath0-functions are used as the singular potentials for schrdinger equation and the corresponding eigenfunctions are derived using the weyl - titchmarsh - kodaira ( wtk ) spectral theorem . in @xcite , the klein - gordon equation with the singular defect was considered and the chaotic behavior of the solution is studied . however , not every singular potential term yields the critical phenomenon . we consider the self - interacting singular potential term and show the existence of the critical phenomenon with a certain type of boundary condition . that is , the energy norm increases , vanishes , or decreases if the parameter value exceeds , is equal to , or is less than the critical value . the critical value of the parameter is obtained analytically in this case , which makes it possible to do an error analysis . for the critical value , we assume the uniform pdf and use the legendre polynomial for the expansion . the resulting deterministic equations are solved using the chebyshev collocation method with the consistent formulation for the dirac @xmath0-function @xcite . once the expansion coefficients are obtained , the solution is reconstructed for various values of the parameter . using the fact that the energy norm remains constant for the critical value when the time is large , the invariant point is found as the critical value . the mean solution is also found by the first gpc mode . we compare the gpc approach and the mc approach . then we apply the same method for the sine - gordon equation . the sine - gordon equation with singular potential terms is also found in much of the existing literature . the singularly perturbed solution of the sine - gordon equation shows many interesting mathematical structures , such as the fractality , self - similarity , etc . . moreover , such an equation yields the critical phenomenon associated with the critical parameter , as explained above briefly . in our work , we use the initial velocity of the soliton solution as the random variable , and try to find the critical initial velocity around which the particle - pass , particle - capture , and particle - reflection occur . for the random variable , we consider both the hermite and legendre polynomials that is , the normal and the uniform distributions as the pdf . unlike the klein - gordon equation , the reconstruction of the solution using the gpc method is not obtained directly . this is because the receiving soliton signal has two extreme cases , i.e. _ transmitted _ or _ not - transmitted _ , and because of the nonlinearity . the extreme solution behavior renders the pdf as the two @xmath0-function - like pdfs . for the particle - capture , the amplitude of the tail at the domain exit is @xmath1 ; it vanishes eventually for the particle - pass . such strong discontinuity in the solution behavior causes the gibbs phenomenon in the reconstruction . these two extreme cases are not separable in the reconstruction . in the paper , we show such non - separable reconstruction for both the hermite and legendre cases . thus instead of using the reconstruction by the gpc method , we use the gpc mean for the determination of the critical value . in this case , the legendre gpc method yields much faster convergence than the hermite gpc method . numerical results show that the legendre gpc method determines the critical value very efficiently and accurately . the determined critical value agrees well with the value found by the many direct mc simulations . to the best of our knowledge , gpc analysis for the singularly perturbed pdes by the dirac @xmath0-functions has not been thoroughly studied in existing literature . in particular , for the nonlinear optics equations with a singular defect in the disordered media , the uncertainty quantification is necessary . this is because the defect is defined in the highly localized regime and thus contains a high degree of intrinsic uncertainty . in our previous work @xcite , some linear wave equations with the singular source term were considered with the legendre pc method . our current work takes the first step to investigate the uncertainty quantification of nonlinear optics equations , with the source term confined in the highly localized regime . the paper is composed of the following sections . in section 2 , the klein - gordon equation with the singular source term is considered . in this section , we show that there exists a critical phenomenon and find the exact value of the critical parameter value . in section 3 , we briefly explain the gpc method and derive the gpc expansion of the klein - gordon equation using the legendre polynomials . in section 4 , numerical methods used for the numerical approximation of the deterministic equations derived in section 3 are explained including the consistent method for the approximation of the dirac @xmath0-function . in section 5 , numerical results of the gpc method for the klein - gordon equation are given . comparison between the mc method and the gpc method is given . in section 6 , the critical phenomenon for the sine - gordon equation is explained . the method of finding the critical initial velocity is explained using the pdf of the receiving signals and the gpc mean value . the hermite gpc method is presented . in section 7 , the legendre gpc is used for the sine - gordon equation and the deterministic equations are derived . numerical results and remark on the error are presented . in section 8 , we briefly outlined the gpc method for the case that the randomess is in the amplitue of the singular potential function and explain that the same gpc method can be applied to this case . in section 9 , concluding remark and future works are presented .

the singular potential term yields a critical phenomenon that is , the solution behavior around the critical parameter value bifurcates into two extreme cases . pinpointing the critical value with arbitrary accuracy is even more challenging . in this work , we adopt the generalized polynomial chaos ( gpc ) method to determine the critical values and the mean solutions around such values . we show the existence of a critical behavior with certain boundary conditions . the obtained partial differential equations are solved using the chebyshev collocation method . due to the existence of the singularity along with the consistent chebyshev method , determines the critical value and the mean solution highly efficiently . we then consider the sine - gordon equation , for which the critical value is associated with the initial velocity of the kink soliton solution . the critical behavior in this case is that the solution _ passes _ through ( particle - pass ) , is _ trapped _ by ( particle - capture ) , or is _ reflected _ by ( particle - reflection ) the singular potential if the initial velocity of the soliton solution is greater than , equal to , or less than the critical value , respectively . due to the nonlinearity of the equation numerical results show that the critical value can be determined efficiently and accurately by using the proposed method . critical phenomenon , klein - gordon equation , sine - gordon equation , galerkin methods , generalized polynomial chaos , spectral method , legendre polynomials , hermite polynomials , uncertainty quantification , gibbs phenomenon

we consider the klein - gordon and sine - gordon type equations with a point - like potential , which describes the wave phenomenon in disordered media with a defect . the singular potential term yields a critical phenomenon that is , the solution behavior around the critical parameter value bifurcates into two extreme cases . finding such critical parameter values and the associated statistical quantities demands a large number of individual simulations with different parameter values . pinpointing the critical value with arbitrary accuracy is even more challenging . in this work , we adopt the generalized polynomial chaos ( gpc ) method to determine the critical values and the mean solutions around such values . first , we consider the critical value associated with the strength of the singular potential for the klein - gordon equation . we show the existence of a critical behavior with certain boundary conditions . then we expand the solution in the random variable associated with the parameter . the obtained partial differential equations are solved using the chebyshev collocation method . due to the existence of the singularity , the gibbs phenomenon appears in the solution , yielding a slow convergence of the numerically computed critical value . to deal with the singularity , we adopt the consistent spectral collocation method . the gpc method , along with the consistent chebyshev method , determines the critical value and the mean solution highly efficiently . we then consider the sine - gordon equation , for which the critical value is associated with the initial velocity of the kink soliton solution . the critical behavior in this case is that the solution _ passes _ through ( particle - pass ) , is _ trapped _ by ( particle - capture ) , or is _ reflected _ by ( particle - reflection ) the singular potential if the initial velocity of the soliton solution is greater than , equal to , or less than the critical value , respectively . due to the nonlinearity of the equation , we use the gpc mean value rather than reconstructing the solution to find the critical parameter . numerical results show that the critical value can be determined efficiently and accurately by using the proposed method . the results are also compared with the results using the monte - carlo method . critical phenomenon , klein - gordon equation , sine - gordon equation , galerkin methods , generalized polynomial chaos , spectral method , legendre polynomials , hermite polynomials , uncertainty quantification , gibbs phenomenon

1108.4720

c

in this paper , we proposed an efficient method using the gpc method to find the critical parameter values and the related statistical quantities for the klein - gordon and sine - gordon equations with a point - like singular potential function . by assuming the unknown critical parameter values as a random variable , and expanding the solution in the orthogonal polynomials associated with the random variable , we computed the critical parameter with much less computation than the mc approach . for the klein - gordon equation , the critical parameter is associated with the amplitude of the singular potential function . using the gpc reconstruction of the solution , the invariant point is found to be the critical parameter value . for the sine - gordon equation , the initial velocity of the soliton solution is used as the random variable . for this case , both the hermite and legendre gpc methods are used . due to the nonlinearity and the @xmath0-function like pdf of the receiving signals , the reconstruction could not be used ; the gpc mean was used instead . the legrendre gpc mean converges quickly , and it efficiently and accurately determines the critical parameter value . as our numerical results show , the gpc method offers several advantages in finding the critical value as compared to the mc method . first of all , one can avoid the large number of individual simulations for different parametric values . our proposed method suggests that , with a small number of mc simulations , it is possible to make the first guess about the critical value . once the first guess is made , one can apply the gpc method to narrow down or pinpoint the critical value accurately and efficiently . moreover , regarding the gpc method for the sine - gordon equation , the stochastic variable appears only in the initial and boundary conditions , and does not appear in the main pde . to perform the integration related to @xmath29 , one can use the legendre - gauss quadrature for the legendre chaos and gauss - hermite quadrature for the hermite chaos and obtain a fast convergence . in this paper , only @xmath453 quadrature points are used . adopting the quadrature rule , one can easily speed up the whole process . during our study , we also found a limitation of the gpc method regarding the determination of the initial critical velocity or the critical amplitude of the singular potential term for the sine - gordon equation . that is , we could not use the reconstruction constructed by the gpc modes to find the pdf of the receiving solutions due to the non - linearity and non - separability related to the gibbs phenomenon of the spectral method . despite this limitation , we find that the gpc method works well for finding the critical values for the singularly perturbed klein - gordon and sine - gordon equations . since no significant gpc analysis exists for the singularly perturbed pdes by the dirac @xmath0-function , our study could be a good resource for this type of research . in our future work , we will apply a similar gpc method for different pdes , such as the nonlinear schrdinger equations with singular potential terms . we will also consider more general types of uncertainties associated with the singular potential term , for which more than one random variables are involved and the gpc expansion should be in the multi - dimensional random space . to reduce the complexity due to the dimensionality , we will attempt to use the gpc collocation method with the fast sparse grid methods @xcite . .1 in * acknowledgement : * we are grateful to bruce pitman , gino biondini , and emmanuel lorin for their useful discussions and references . some important properties of the legendre polynomials are used in the paper . legendre polynomials are defined by the solutions @xmath459 of the following sturm - liouville equation : @xmath460+n(n+1)p_{n}=0 , \quad x \in [ -1,1],\ ] ] where @xmath459 is the legendre polynomial of degree @xmath461 . first few legendre polynomials are @xmath462 @xmath463 @xmath464 @xmath465 @xmath466 . also @xmath467 , @xmath468 if @xmath461 is even and @xmath469 if @xmath461 is odd , also @xmath470 if @xmath461 is odd with the orthogonality @xmath471 from eq . ( .2 ) we have , @xmath473 multiplying both sides by @xmath474 and integrating from @xmath475 to @xmath476 , we get , @xmath477 by using the orthogonality of the legendre polynomials we get , @xmath478 by using eq . ( .3 ) we get the followings . + for @xmath461 is odd number : @xmath479 for @xmath461 is even number : @xmath480 the derivative of legendre polynomials as the finite linear sum of legendre polynomials are used in the paper and some examples are as follows : for @xmath481 , @xmath482 and for @xmath483 , @xmath484 by orthogonal property @xmath485 . so in general if @xmath165 is even and @xmath461 is odd or @xmath461 is even and @xmath165 is odd , @xmath486 + if both @xmath165 and @xmath461 are even numbers and @xmath487 then by using the orthogonal property of legendre polynomials and from eq . ( [ even_derivatives ] ) we have , @xmath488=n(n+1).\nonumber \end{aligned}\ ] ] if both @xmath165 and @xmath461 are odd numbers and @xmath487 then the orthogonal property and eq . ( [ odd_derivatives ] ) yield , @xmath489=n(n+1).\ ] ] + + k. dalbey , a. k. patra , e. b. pitman , m. i. bursik , and m. f. sheridan , input uncertainty propagation methods and hazard mapping of geophysical mass flows , _ j. geophy . research _ , 113 , 2008 , pp . b05203 : 116 . w. c. k. mak , b. a. malomed , and p. l. chu , interaction of a soliton with a local defect in a bragg grating , _ j. opt b _ 20 , 2003 , pp . . l. morales - molina and r. a. vicencio , trapping of discrete solitons by defects in nonlinear waveguides arrays , _ optics letters _ 31 , 2006 , pp . 966968 . a. trombettoni , a. smerzi , and a. r. bishop , discrete nonlinaer schrodinger equation with defects , _ phys . e _ 67 , 2003 , pp . 016607 : 111 . d. wang , j .- h . jung , and g. biondini , detailed study of numerical methods for the perturbed sine - gordon equation with impulse forcing , _ in preparation _ , 2011 .

we consider the klein - gordon and sine - gordon type equations with a point - like potential , which describes the wave phenomenon in disordered media with a defect . finding such critical parameter values and the associated statistical quantities demands a large number of individual simulations with different parameter values . then we expand the solution in the random variable associated with the parameter . , the gibbs phenomenon appears in the solution , yielding a slow convergence of the numerically computed critical value . to deal with the singularity , we adopt the consistent spectral collocation method . the gpc method , , we use the gpc mean value rather than reconstructing the solution to find the critical parameter .

we consider the klein - gordon and sine - gordon type equations with a point - like potential , which describes the wave phenomenon in disordered media with a defect . the singular potential term yields a critical phenomenon that is , the solution behavior around the critical parameter value bifurcates into two extreme cases . finding such critical parameter values and the associated statistical quantities demands a large number of individual simulations with different parameter values . pinpointing the critical value with arbitrary accuracy is even more challenging . in this work , we adopt the generalized polynomial chaos ( gpc ) method to determine the critical values and the mean solutions around such values . first , we consider the critical value associated with the strength of the singular potential for the klein - gordon equation . we show the existence of a critical behavior with certain boundary conditions . then we expand the solution in the random variable associated with the parameter . the obtained partial differential equations are solved using the chebyshev collocation method . due to the existence of the singularity , the gibbs phenomenon appears in the solution , yielding a slow convergence of the numerically computed critical value . to deal with the singularity , we adopt the consistent spectral collocation method . the gpc method , along with the consistent chebyshev method , determines the critical value and the mean solution highly efficiently . we then consider the sine - gordon equation , for which the critical value is associated with the initial velocity of the kink soliton solution . the critical behavior in this case is that the solution _ passes _ through ( particle - pass ) , is _ trapped _ by ( particle - capture ) , or is _ reflected _ by ( particle - reflection ) the singular potential if the initial velocity of the soliton solution is greater than , equal to , or less than the critical value , respectively . due to the nonlinearity of the equation , we use the gpc mean value rather than reconstructing the solution to find the critical parameter . numerical results show that the critical value can be determined efficiently and accurately by using the proposed method . the results are also compared with the results using the monte - carlo method . critical phenomenon , klein - gordon equation , sine - gordon equation , galerkin methods , generalized polynomial chaos , spectral method , legendre polynomials , hermite polynomials , uncertainty quantification , gibbs phenomenon

1511.00031

i

the study of hadronic properties at finite temperature @xmath2 is one of the theoretical ingredients needed to understand the behaviour of matter created in relativistic heavy ion collision experiments , such as those in rhic and lhc ( alice ) . in particular , the qcd transition involving chiral symmetry restoration and deconfinement plays a crucial role , as it is clear from the many recent advances of lattice groups in the study of the phase diagram and other thermodynamical properties @xcite . for vanishing baryon chemical potential , the qcd transition is a crossover for 2 + 1 flavours with physical quark masses , the transition temperature being about @xmath5 150 -160 mev . in the chiral limit ( vanishing light quark masses for fixed strange mass ) it is believed to become a second - order phase transition belonging to the universality class of the @xmath6 model @xcite . lattice simulations also support this fact . actually , in @xcite it is shown that the lattice results are compatible with a @xmath7-like restoration pattern in the chiral limit and for physical masses , by studying the scaling of different thermodynamical observables near @xmath8 . the expected reduction in the transition temperature from the physical mass case to the chiral limit one based on those analysis is about 15 - 20@xmath9 , although subject to many lattice uncertainties @xcite . from the theoretical side , it is important to provide solid analysis of this chiral restoration pattern based on effective theories , given the limitations of perturbative qcd at those temperature scales . a simple model realization was historically the linear sigma model ( lsm ) based on @xmath10 spontaneous symmetry breaking @xcite , where the @xmath11-component of the @xmath6 field acquires a thermal vacuum expectation value and mass , both of them vanish at the transition in the chiral limit , and @xmath12 mesons degenerate as chiral partners . however , such a simple description is nowadays in conflict with observations : on the one hand , the @xmath13 broad resonance produced in pion - pion scattering and listed in the pdg @xcite is not compatible with a particle - like state ( see @xcite for a recent review ) . on the other hand , to reproduce consistently pion data , the lsm requires working in a strong coupling regime , invalidating the perturbative description . nevertheless , it is clear that the @xmath14 state must play an important role in chiral restoration , since it shares the quantum numbers of the qcd vacuum . chiral symmetry restoration has also been studied within qcd inspired models like the nambu - jona - lasinio or gross - neveu ones @xcite . a systematic and model - independent framework that takes into account the relevant light meson degrees of freedom and their interactions is chiral perturbation theory ( chpt ) @xcite . the effective chpt lagrangian is constructed as a derivative and mass expansion @xmath15 , where @xmath16 denotes generically a meson energy scale compared to the chiral scale @xmath17 1 gev . the lowest order lagrangian @xmath18 is the non - linear sigma model ( nlsm ) . the use of energy expansions in chiral effective theories is also justified at finite temperature to describe heavy ion physics . pions are actually the most copiously produced particles after a heavy ion collision and most of their properties from hadronization to thermal freeze - out can be reasonably described within the temperature range where these theories are applicable . thus , the chiral restoring behaviour in terms of the quark condensate is qualitatively obtained within chpt @xcite . moreover , the introduction of realistic pion interactions by demanding unitarity through the inverse amplitude method ( iam ) @xcite extended at finite @xmath2 @xcite improves chpt , providing a more accurate description of several effects of interest in a heavy - ion environment , such as thermal resonances , transport coefficients and electromagnetic corrections @xcite . this approach also provides a novel understanding of the role of the @xmath13 broad resonant state in chiral symmetry restoration , without having to deal with the typical lsm drawbacks . thus , the unitarized @xmath19 scattering amplitude within chpt at finite temperature develops a @xmath20 thermal pole at @xmath21 ^ 2 $ ] , which for @xmath3 corresponds to the pdg state , and whose trajectory in the complex plane as @xmath2 varies shows some interesting features : the sudden drop of @xmath22 towards the two - pion threshold can be interpreted in terms of chiral symmetry restoration , as opposed for instance to the @xmath23 @xmath24-channel where the mass drop is much softer . in addition , it has been recently shown @xcite that the scalar susceptibility saturated with this @xmath11-like state , with squared mass @xmath25 , develops a maximum near @xmath26 compatible with lattice data , unlike the pure chpt prediction which is monotonically increasing . moreover , chiral partners in the scalar - pseudoscalar sector are understood through degeneration of correlators and susceptibilities @xcite , something which is also directly seen in lattice data @xcite . the role of the @xmath4 state for chiral restoration could become more complicated if its possible tetraquark component is also considered at finite temperature @xcite . a crucial step in the unitarized approach is the condition of exact thermal unitarity for the partial waves , with a thermal space factor modified by the bose - einstein distribution function . this condition holds perturbatively in chpt @xcite and the unitarized amplitude is constructed by requiring thermal unitarity to all orders , based on the physical collision processes occurring in the thermal bath @xcite . however , it is important to emphasize that thermal unitarity for the full amplitude was not formally proven in those works ; in fact , that will be one of the relevant issues discussed in the present work . although the approaches based on effective theories in terms of the lightest mesons provide a good description of the physics involved , especially in what concerns the effect of the lightest resonances ( as discussed above ) , a more accurate treatment near @xmath26 would require including heavier degrees of freedom . that is for instance the framework of the hadron resonance gas , which describes the system through the statistical ensemble of all free states thermally available , and where corrections due to interactions and lattice masses can be also accounted for @xcite . effective chiral models including explicitly vector and axial - vector resonances have also been successfully used to depict several hadron thermal properties relevant for observables such as the dilepton and photon spectra and @xmath27 mixing / degeneration at the chiral transition @xcite . in this work , we will consider an alternative approach to the thermal pion scattering amplitude , namely the limit of large number of nambu - goldstone bosons @xmath1 , or in other words , large number of light flavours with no strangeness , as treated before at @xmath3 in @xcite . previous large-@xmath1 analysis at @xmath28 can be found in @xcite . within this framework , the lowest order chiral effective lagrangian for low - energy qcd will be the @xmath0 nlsm , whose corresponding symmetry breaking pattern is @xmath29 . as we have just commented , the latter is believed to take place in chiral symmetry restoration for @xmath30 , since @xmath6 and @xmath31 are respectively isomorphic to the isospin groups @xmath32 and @xmath33 . this technique has the advantage of allowing for a partial resummation of the scattering amplitude preserving many physical properties such as unitarity and the dynamical generation of the @xmath4 pole , which will help us to shed more light on the chiral restoring issues discussed before . we will work in the chiral limit , since it simplifies considerably the analysis , besides enhancing chiral - restoring effects , as explained above . at this point , it is important to remark that massless pions remain massless at finite temperature @xcite , unlike many other instances in thermal field theory where elementary massless excitations acquire mass in the thermal bath , like high-@xmath2 fermions @xcite , gauge fields @xcite including large-@xmath34 analysis @xcite or when electromagnetic corrections are switched on @xcite . the study of the large-@xmath1 approach in low - energy qcd implies a simplification of the pion dynamics @xcite without changing essential features such as analyticity , unitarity and the low - energy behaviour for pion scattering . this is fully accomplished when working in the functional formalism of the theory @xcite so that the lagrangian is built as @xmath35 covariant and @xmath36 invariant ( in the chiral limit ) . furthermore , as we will see in detail here , this approach will allow to describe consistently the @xmath4 state through its pole in the second riemann sheet , where the parameters of the model are fitted to pion - pion scattering phase shift data . thanks to the fact that the model is exactly unitary , we will be granted to go beyond the standard perturbative chpt description for the scattering @xcite , as a complementary description of unitarization methods such as the iam . an additional observation that makes this approach suitable for studying chiral restoring effects is that in order to reproduce correctly the @xmath37 pole ( linked to chiral restoration as mentioned before ) , the dominant contributions to @xmath19 scattering are the loop diagrams from the leading order chiral lagrangian , rather than the particular form of higher order terms needed to renormalize the amplitude @xcite . thus , the large-@xmath1 limit framework provides a resummation of the dominant loop contributions needed to maintain exact unitarity , so that the scalar pole can be correctly described . with the above motivations kept in mind , we will analyze in this work elastic pion - pion scattering at finite temperature within the large-@xmath1 @xmath0 model in the chiral limit . we will show that at @xmath3 one gets reasonable values for the @xmath37 pole from a fit to experimental data of a two - parameter partial wave . the extension to @xmath28 includes a formal discussion of the renormalizability of the model , which as expected can be carried out in terms of @xmath3 counterterms , although with important subtleties to be taken into account . the important feature of exact unitarity is demonstrated , including thermal corrections , something that allows us to define the second - sheet pole . having fixed the @xmath3 pole position , its @xmath2 dependence is obtained and it is shown that the results are compatible with a second - order chiral restoring phase transition , consistently with previous determinations and lattice data . the paper is organized as follows : in section [ sec : scattering ] we introduce our large @xmath1 approach within the framework of the massless nlsm and work out the diagrammatic expansion for pion scattering , both at zero and at finite temperature ; section [ sec : ren ] is devoted to explain the renormalization procedure , for which technical details are relegated to appendix [ app : ren ] . in section [ sec : pw ] we perform the analysis of the @xmath20 partial wave , providing a fit to @xmath3 data and showing that the large-@xmath1 amplitude satisfies exactly unitarity at zero and finite temperature . the latter grants us to define the riemann second - sheet pole corresponding to the @xmath4 state , which we study in detail in section [ sec : pole ] , paying special attention to its thermal evolution and the connection with chiral symmetry restoration . our conclusions are presented in section [ sec : conc ] .

this formulation provides a good description of scattering data in the scalar channel and generates dynamically the pole , the pole position lying within experimental determinations . previous results with this model are updated using newer analysis of pion scattering data . next , we analyze the behaviour of the pole at finite , which is consistent with chiral symmetry restoration when the scalar susceptibility is saturated by the state , in a second - order transition scenario and in accordance with lattice and theoretical analysis .

we consider the non - linear sigma model for large as an effective theory for low - energy qcd at finite temperature , in the chiral limit . at this formulation provides a good description of scattering data in the scalar channel and generates dynamically the pole , the pole position lying within experimental determinations . previous results with this model are updated using newer analysis of pion scattering data . we calculate the pion scattering amplitude at finite and show that it satisfies exactly thermal unitarity , which had been assumed but not formally proven in previous works . we discuss the main differences with the result and we show that one can define a proper renormalization scheme with counterterms such that the renormalized amplitude can be chosen to depend only on a few parameters . next , we analyze the behaviour of the pole at finite , which is consistent with chiral symmetry restoration when the scalar susceptibility is saturated by the state , in a second - order transition scenario and in accordance with lattice and theoretical analysis .

1511.00031

c

we have studied pion scattering in the large-@xmath1 @xmath0 model at finite temperature in the chiral limit and its consequences regarding the @xmath4 pole and chiral symmetry restoration . our analysis gives rise to interesting theoretical and phenomenological results , consistent with previous analysis and lattice data . after calculating the relevant feynman diagrams , which include an effective thermal vertex from tadpole resummation , an important part of our work has been devoted to show that it is possible to find a renormalization scheme rendering the thermal amplitude finite with a @xmath3 renormalization of the corresponding vertices this is a nontrivial extension of the @xmath3 renormalization of the scattering amplitude , since the breaking of lorentz covariance in the thermal bath induces crossed terms between tadpole - like and @xmath221 loop functions . in the low - energy expansion of the model , up to @xmath167 , we have checked explicitly this renormalization scheme , providing a diagrammatic and lagrangian interpretation . another relevant result is that the large-@xmath1 thermal amplitude satisfies exactly the thermal unitarity relation , imposed in previous works as a physical condition for the exact amplitude . its low - energy properties are also preserved , being consistent for instance with the thermal dependence of the pion decay constant . by a suitable choice of the low - energy constants , similarly to the @xmath3 case , compatible with the scale evolution of the renormalized couplings , we end up with a phenomenological unitary amplitude depending only on two parameters , @xmath180 and @xmath99 . by fitting those parameters to experimental data in the @xmath20 channel , which is more reliable for data not very close to threshold in the elastic region , we reproduce the pole position of the @xmath4 in the second riemann sheet fairly consistently with pdg values and recent determinations . the chiral limit character of our approach implies a larger value for @xmath180 than phenomenologically expected , but it allows to obtain pole position parameters @xmath222 closer to the physical case . the fits to data are actually very good in the chosen region , precisely the most relevant energy range concerning this resonant state . once the @xmath3 pole has been fixed to physical values , we have studied its evolution with temperature . the @xmath4 pole remains a wide state for all the temperature range of interest , the real and imaginary parts @xmath223 and @xmath224 behaving similarly to the iam analysis , showing the signature of chiral restoration . in order to explore this further , we define a scalar susceptibility @xmath225 saturated by the inverse of @xmath226 , corresponding to the real part of the scalar state self - energy at zero four - momentum , which diverges at a given @xmath8 with a power law , as it corresponds to a continuous second - order phase transition in the chiral limit . the values obtained for @xmath8 , as well as the critical exponent of @xmath227 , are consistent with those obtained with other analytical approaches , such as the iam , and with lattice analysis , being compatible with a @xmath6 scaling . the combination of the large-@xmath1 framework with the phenomenological features of the @xmath4 pole allows to improve the predictions of previous approaches based on the partition function . thus , we obtain a very reasonable description of the chiral restoration transition within this approach , given the different uncertainties involved , such as possible @xmath85 corrections near the physical @xmath30 case or the absence of heavier degrees of freedom , which should play an important role near the transition and improve our simple pion gas scenario .

we consider the non - linear sigma model for large as an effective theory for low - energy qcd at finite temperature , in the chiral limit . at we calculate the pion scattering amplitude at finite and show that it satisfies exactly thermal unitarity , which had been assumed but not formally proven in previous works .

we consider the non - linear sigma model for large as an effective theory for low - energy qcd at finite temperature , in the chiral limit . at this formulation provides a good description of scattering data in the scalar channel and generates dynamically the pole , the pole position lying within experimental determinations . previous results with this model are updated using newer analysis of pion scattering data . we calculate the pion scattering amplitude at finite and show that it satisfies exactly thermal unitarity , which had been assumed but not formally proven in previous works . we discuss the main differences with the result and we show that one can define a proper renormalization scheme with counterterms such that the renormalized amplitude can be chosen to depend only on a few parameters . next , we analyze the behaviour of the pole at finite , which is consistent with chiral symmetry restoration when the scalar susceptibility is saturated by the state , in a second - order transition scenario and in accordance with lattice and theoretical analysis .

1504.05002

i

consider the minimization problem @xmath0 where @xmath1 is a compact polyhedral set , @xmath2 and @xmath3 is strongly convex and continuously differentiable over @xmath4 . note that for a general matrix @xmath5 , the function @xmath6 is not necessarily strongly convex . when the problem at hand is large - scale , first order methods , which have relatively low computational cost per iteration , are usually utilized . these methods include , for example , the class of projected ( proximal ) gradient methods . a drawback of these methods is that under general convexity assumptions , they posses only a sublinear rate of convergence @xcite , while linear rate of convergence can be established only under additional conditions such as strong convexity of the objective function @xcite . luo and tseng @xcite showed that the strong convexity assumption can be relaxed and replaced by an assumption on the existence of a local error bound , and under this assumption , certain classes algorithms , which they referred to as feasible descent methods " , converge in an asymptotic linear time . the model ( [ eq : problem ] ) with assumptions on strong convexity of @xmath7 , compactness and polyhedrality of @xmath8 was shown in @xcite to satisfy the error bound . in @xcite wang and lin extended the work @xcite and showed that there exists a _ global _ error bound for problem ( [ eq : problem ] ) with the additional assumption of compactness of @xmath8 ; and derived the exact linear rate for this case . we note that the family of feasible descent methods " include the block alternating minimization algorithm ( under the assumption of block strong convexity ) , as well as gradient projection methods , and therefore are usually at least as complex as evaluating the orthogonal projection operator onto the feasible set @xmath8 at each iteration . an alternative to algorithms which are based on projection ( or proximal ) operators are _ linear - oracle_-based algorithms such as the conditional gradient ( cg ) method . the cg algorithm was presented by frank and wolfe in 1956 @xcite , for minimizing a convex function over a compact polyhedral set . at each iteration , the algorithm requires a solution to the problem of minimizing a linear objective function over the feasible set . it is assumed that this solution is obtained by a call to a linear - oracle , i.e. , a black box which , given a linear function , returns an optimal solution of this linear function over the feasible set ( see an exact definition in section [ sec : cg ] ) . in some instances , and specifically for certain types of polyhedral sets , obtaining such a linear - oracle can be done more efficiently than computing the orthogonal projection onto the feasible set ( see examples in @xcite ) , and therefore the cg algorithm has an advantage over projection - based algorithms . the original paper of frank and wolfe also contained a proof of an @xmath9 rate of convergence of the function values to the optimal value . levitin and polyak showed in @xcite that this @xmath9 rate can also be extended to the case where the feasible set is a general compact convex set . cannon and culum proved in @xcite that this rate is in fact _ tight_. however , if in addition to strong convexity of the objective function , the optimal solution is in the interior of the feasible set , then linear rate of convergence of the cg method can be established @xcite . epelman and freund @xcite , as well as beck and teboulle @xcite showed a linear rate of convergence of the conditional gradient with a special stepsize choice in the context of finding a point in the intersection of an affine space and a closed and convex set under a slater - type assumption . another setting in which linear rate of convergence can be derived is when the feasible set is uniformly ( strongly ) convex and the norm of the gradient of the objective function is bounded away from zero @xcite . + another approach for deriving a linear rate of convergence is to modify the algorithm . for example , hazan and garber used _ local _ linear - oracles in @xcite in order to show linear rate of convergence of a localized " version of the conditional gradient method . a different modification , which is viable when the feasible set is a compact polyhedral , is to use a variation of the conditional gradient method that incorporates away steps . this version of the conditional gradient method , which we refer to as _ away steps conditional gradient _ ( ascg ) , was initially suggested by wolfe in @xcite and then studied by guelat and marcotte @xcite , where a linear rate of convergence was established under the assumption that the objective function is strongly convex , as well as an assumption on the location of the optimal solution . in @xcite jaggi and lacoste - julien were able to extend this result for the more general model ( [ eq : problem ] ) for the case where @xmath10 , without restrictions on the location of the solution . we note that the ascg requires that the linear - oracle will produce an optimal solution of the associated problem which is an extreme point . we will call such an oracle a _ vertex linear - oracle _ ( see the discussion in section [ sec : vertexlinearoracles ] ) . * contribution . * in this work , our starting point and main motivation are the results of jaggi and lacoste - julien @xcite . our contribution is threefold : * we extend the results given in @xcite and show that the ascg algorithm converges linearly for the general case of problem , that is , for any value of @xmath5 and @xmath11 . + the additional linear term @xmath12 enables us to consider much more general models . for example , consider the @xmath13-regularized least squares problem @xmath14 where @xmath15 is a compact polyhedral , @xmath16 and @xmath17 . since @xmath18 is compact , we can find a constant @xmath19 for which @xmath20 for any @xmath21 . we can now rewrite the model as @xmath22 } { \left\|{{{\bf b}}{{\bf x}}-{{\bf c}}}\right\|}^2+\lambda y,\ ] ] which obviously fits the general model ( [ eq : problem ] ) * the analysis in @xcite assumes the existence of a _ vertex _ linear - oracle on the set @xmath23 , rather than an oracle for the set @xmath8 . this fact is not significant for the pure " cg algorithm , since it only requires a linear - oracle and not a _ vertex _ linear - oracle . this means that for the cg algorithm , a linear - oracle on @xmath23 can be easily obtained by applying @xmath5 on the output of the linear - oracle on @xmath8 . on the other hand , this argument fails for the ascg algorithm that specifically requires the oracle to return an extreme point of the feasible set , and finding such a vertex linear - oracle on @xmath23 might be a complex task , see section [ sec : vertexlinearoracles ] for more details . our analysis only requires a vertex linear - oracle on the original set @xmath8 . * we present an analysis based on simple duality arguments , which are completely different than the geometric arguments in @xcite . consequently , we obtain a computable constant for the rate of convergence , which is explicitly expressed as a function of the problem s parameters and the geometry of the feasible set . this constant , which we call the vertex - facet distance constant " , replaces the so - called _ pyramidal width _ constant from @xcite , which reflects the geometry of the feasible set and is obtained as the optimal value of a very complex mixed integer saddle point optimization problem whose exact value is unknown even for simple polyhedral sets . * paper layout . * the paper is organized as follows . section [ sec : preliminaries ] presents some preliminary results and definitions needed for the analysis . in particular , it provides a brief introduction to the classical cg algorithm and linear oracles . section [ sec : awaystepconditionalgradient ] presents the ascg algorithm and the convergence analysis , and is divided into four subsections . in section [ sec : vertexlinearoracles ] the concept of vertex linear - oracle , needed for the implementation of ascg , is presented , and the difficulties of obtaining a vertex linear - oracle on a linear transformation of the feasible set are discussed . in section [ sec : ascgmethod ] we present the ascg method with different possible stepsize choices . in section [ sec : rateconvergenceanalysis ] , we provide the rate of convergence analysis of the ascg for problem , and present the new _ vertex - facet distance _ constant used in the analysis . finally , in section [ sec : findingomegaforpolyhedrons ] , we demonstrate how to compute this new constant for a few examples of simple polyhedral sets . * notations . * we denote the cardinality of set @xmath24 by @xmath25 . the difference , union and intersection of two given sets @xmath24 and @xmath26 are denoted by @xmath27 , @xmath28 and @xmath29 respectively . subscript indices represent elements of a vector , while superscript indices represent iterates of the vector , i.e. , @xmath30 is the @xmath31th element of vector @xmath32 , @xmath33 is a vector at iteration @xmath34 , and @xmath35 is the @xmath31th element of @xmath33 . the vector @xmath36 is the @xmath31th vector of the standard basis of @xmath37 , @xmath38 is the all - zeros vector , and @xmath39 is the vector of all ones . given two vectors @xmath40 , their dot product is denoted by @xmath41 . given a matrix @xmath42 and vector @xmath43 , @xmath44 denotes the spectral norm of @xmath45 , and @xmath46 denotes the @xmath47 norm of @xmath32 , unless stated otherwise . @xmath48 , @xmath49 and @xmath50 represent the transpose , rank and image of @xmath45 respectively . we denote the @xmath31th row of a given matrix @xmath45 by @xmath51 , and given a set @xmath52 , @xmath53 is the submatrix of @xmath45 such that @xmath54 for any @xmath55 . if @xmath45 is a symmetric matrix , then @xmath56 is its minimal eigenvalue . if a matrix @xmath45 is also invertible , we denote its inverse by @xmath57 . given matrices @xmath58 and @xmath59 , the matrix @xmath60\in { \mathbb{r}}^{n\times { ( m+k)}}$ ] is their horizontal concatenation . given a point @xmath32 and a closed convex set @xmath8 , the distance between @xmath32 and @xmath8 is denoted by @xmath61 . the standard unit simplex in @xmath37 is denoted by @xmath62 and its relative interior by @xmath63 . given a set @xmath64 , its convex hull is denoted by @xmath65 . given a convex set @xmath66 , the set of all its extreme points is denoted by @xmath67 .

jaggi and lacoste - julien showed that the conditional gradient method with away steps employed on the aforementioned problem without the additional linear term has linear rate of convergence , depending on the so - called pyramidal width of the feasible set . this constant replaces the pyramidal width , which is difficult to evaluate .

we consider the problem of minimizing a function , which is the sum of a linear function and a composition of a strongly convex function with a linear transformation , over a compact polyhedral set . jaggi and lacoste - julien showed that the conditional gradient method with away steps employed on the aforementioned problem without the additional linear term has linear rate of convergence , depending on the so - called pyramidal width of the feasible set . we revisit this result and provide a variant of the algorithm and an analysis that is based on simple duality arguments , as well as corresponding error bounds . this new analysis ( a ) enables the incorporation of the additional linear term , ( b ) does not require a linear - oracle that outputs an extreme point of the linear mapping of the feasible set and ( c ) depends on a new constant , termed the vertex - facet distance constant " , which is explicitly expressed in terms of the problem s parameters and the geometry of the feasible set . this constant replaces the pyramidal width , which is difficult to evaluate .

1203.0447

i

let @xmath10 be a probability space and let @xmath11 be a sequence of i.i.d . random variables on @xmath10 taking values in @xmath12 , @xmath13 . let us define @xmath14 and @xmath15 . the sequence @xmath16 is called a _ random walk _ with jumps @xmath11 . the random walk @xmath16 is said to be _ recurrent _ if @xmath17 and _ transient _ if @xmath18 it is well known that every random walk is either recurrent or transient ( see @xcite , theorem 4.2.1 ) . recall that a random walk @xmath16 in @xmath12 is called _ truly @xmath19-dimensional _ if @xmath20 holds for all @xmath21 it is also well known that every truly @xmath19-dimensional random walk is transient if @xmath22 ( see @xcite , theorem 4.2.13 ) . an @xmath12-valued random variable @xmath23 is said to have _ stable distribution _ if , for any @xmath24 , there are @xmath25 and @xmath26 , such that @xmath27 where @xmath28 are independent copies of @xmath23 and @xmath29 denotes equality in distribution . it turns out that @xmath30 for some @xmath31 $ ] which is called the index of stability ( see @xcite , definition 1.1.4 and corollary 2.1.3 ) . the case @xmath32 corresponds to the gaussian random variable . a random walk @xmath16 is said to be stable if the random variable @xmath33 has stable distribution . in the class of truly two - dimensional stable random walks in @xmath34 , by @xcite , theorem 4.2.9 , the only recurrent case is the case when @xmath16 is a truly two - dimensional random walk with zero mean gaussian jumps . in the case @xmath35 , every stable distribution is characterized by four parameters : the stability parameter @xmath31,$ ] the skewness parameter @xmath36 $ ] , the scale parameter @xmath37 and the shift parameter @xmath38 ( see @xcite , definition 1.1.6 ) . using the notation from @xcite , we denote one - dimensional stable distributions by @xmath39 for symmetric stable distributions , that is , for @xmath40 ( see @xcite , property 1.2.5 ) , we write s@xmath7s . a s@xmath7s random walk is recurrent if and only if @xmath41 ( see the discussion after @xcite , lemma 4.2.12 ) . in this paper , we generalize the s@xmath7s random walk in the way that the index of stability of the jump distribution depends on the current position and study the transience and recurrence property of the generalization . actually , we will not need the stability property of transition jumps . all we will need is a tail behavior of transition jumps . let us introduce the notation @xmath42 when @xmath43 , for @xmath44 where @xmath45 $ ] . recall that if @xmath46 is the density function of a s@xmath7s distribution with @xmath47 and @xmath37 ( for the existence of densities of @xmath48 distributions see @xcite , definition 1.1.6 and @xcite , theorem 3.3.5 ) , then @xmath49 when @xmath50 where @xmath51 and @xmath52 for @xmath53 see @xcite , property 1.2.15 . now , let @xmath54 and @xmath55 be arbitrary functions and let @xmath56 be a family of density functions on @xmath8 such that 1 . @xmath57 is a borel measurable function for all @xmath58 and 2 . @xmath59 when @xmath50 for all @xmath60 . let us define a markov chain @xmath16 on @xmath8 by the following transition kernel @xmath61 the chain @xmath16 jumps from the state @xmath62 with transition density @xmath63 , with the power - law decay with exponent @xmath3 , and this jump distribution depends only on the current state @xmath62 . transition densities @xmath56 are asymptotically equivalent to the densities of s@xmath7s distributions , and we call such chain a _ stable - like chain_. the aim of this paper is to find conditions for the recurrence and transience property of the stable - like chain @xmath16 in terms of the function @xmath64 to the best of our knowledge , all methods used in establishing conditions for recurrence and transience in the random walk case are based on the i.i.d . property of random walk jumps , that is , laws of large numbers ( chung fuchs theorem ) , central limit theorems , characteristic functions approach ( stone ornstein formula ) etc . ( see @xcite , theorems 4.2.7 , 4.2.8 and 4.2.9 ) . although we deal with distributions similar to s@xmath7s distributions , it is not clear if these methods can be used in the case of the non - constant function @xmath65 . special cases of this problem have been considered in @xcite and @xcite . in @xcite and @xcite , the authors consider the countable state space @xmath66 and the function @xmath65 is a two - valued step function which takes one value on negative integers and the other one on nonnegative integers . the processes considered in @xcite and @xcite run in continuous time . the function @xmath65 considered in @xcite is a two - valued step function which takes one value on negative reals and the other one on nonnegative reals , while in @xcite the author considers the case when the function @xmath65 is periodic and continuously differentiable . the methods used in @xcite and @xcite , actually reduce the process to random walks and lvy processes . also , it is not clear if these methods can be used in the general case , that is , when the function @xmath65 is an arbitrary function . in this paper , under certain assumptions on the functions @xmath65 , @xmath67 and on the family of density functions @xmath56 , we give sufficient conditions for the recurrence and transience property of the stable - like chain @xmath16 in terms of the function @xmath64 let us denote by @xmath68 the borel @xmath69-algebra on @xmath8 , by @xmath70 the lebesgue measure on @xmath68 and for arbitrary @xmath71 and @xmath60 we define @xmath72 . assume that the family of probability densities @xmath56 satisfies additional three conditions : 1 . there exists @xmath73 such that @xmath74^{c}}\biggl{\vert}f_x(y ) \frac { |y|^{\alpha(x)+1}}{c(x)}-1\biggr{\vert}=0;\ ] ] 2 . @xmath75 for every compact set @xmath76^{c}$ ] ; 3 . there exists @xmath77 such that for every compact set @xmath78^{c}$ ] with @xmath79 , we have @xmath80}\int_{c - x}f_x(y)\,\mathrm{d}y>0.\ ] ] condition ( c3 ) ensures that out of some compact set all jump densities of the stable - like chain @xmath16 can be replaced by their tail behavior uniformly . this condition is crucial in proving certain structural properties of the chain @xmath16 and in finding sufficient conditions for the recurrence and transience . another essential property of the chain @xmath16 is that every compact set is a petite set . a petite set is a set which assumes a role of a singleton for markov chains on general state space ( for the exact definition of the petite set see definition [ d22 ] ) . this is the reason why compact sets are important in conditions ( c3 ) , ( c4 ) and ( c5 ) . besides ensuring that all compact sets are petite sets ( singletons ) , conditions ( c4 ) and ( c5 ) ensure also that the chain is irreducible . condition ( c4 ) ensures that the scaling function @xmath67 does not vanish on petite sets , and condition ( c5 ) ensures that the petite set @xmath81 $ ] communicates with the rest of the state space . note that condition ( c3 ) implies @xmath82^{c}}c(x)<\infty.\ ] ] indeed , let @xmath83 be arbitrary . then there exists @xmath84 such that for all @xmath85 we have @xmath86 for all @xmath87^{c}$ ] . therefore , upon integrating over @xmath88 we get @xmath89 for every @xmath87^{c}$ ] . an example of a stable - like chain which satisfies conditions ( c3)(c5 ) is the chain which has exactly @xmath90 jumps at each location @xmath62 , where the functions @xmath65 , @xmath91 and @xmath92 are borel measurable and take finitely many values ( see proposition [ p55 ] for details ) . before stating the main results of this paper we recall relevant definitions of recurrence and transience . let @xmath93 be a markov chain on @xmath94 . a. the chain @xmath93 is _ @xmath95-irreducible _ if there exists a probability measure @xmath96 on @xmath68 such that for every @xmath60 there exists @xmath24 such that @xmath97 implies @xmath98 . b. the chain @xmath93 is _ recurrent _ if it is @xmath96-irreducible and if @xmath99 holds for all @xmath100 and all @xmath71 , such that @xmath97 . c. the chain @xmath93 is _ transient _ if it is @xmath96-irreducible and if there exists a countable cover of @xmath8 with sets @xmath101 , such that for each @xmath102 there is a finite constant @xmath103 such that @xmath104 holds for all @xmath60 . the following two constants will appear in the statements of the main results : for @xmath105 , let @xmath106 and for @xmath107 and @xmath108 let @xmath109 where @xmath110 is the digamma function , @xmath111 is the gauss hypergeometric function and @xmath112 is the incomplete beta function ( see section [ sec3 ] for the definition of these functions ) . the constants @xmath113 and @xmath114 are strictly positive ( see proofs of theorems [ tm13 ] and [ tm14 ] ) . furthermore , it is not hard to see that the constant @xmath113 , as a function of @xmath105 , is strictly increasing , @xmath115 and @xmath116 the constant @xmath114 , as a function of @xmath108 for fixed @xmath117 , is strictly positive and @xmath118 while considered as a function of @xmath119 for fixed @xmath120 , it is strictly decreasing , @xmath121 and @xmath122 [ tm13 ] let @xmath123 be an arbitrary function such that @xmath124 furthermore , let @xmath55 be an arbitrary function and let @xmath56 be a family of density functions on @xmath8 which satisfies conditions _ ( c1)(c5 ) _ and such that @xmath125<r(\alpha)\ ] ] when @xmath126 , and the left - hand side in ( [ eq13 ] ) is finite when @xmath32 . then the stable - like markov chain @xmath16 given by the transition kernel @xmath127 is recurrent . [ tm14 ] let @xmath128 be an arbitrary function such that @xmath129 and let @xmath108 be arbitrary . furthermore , let @xmath55 be an arbitrary function and let @xmath56 be a family of density functions which satisfies conditions _ ( c1)(c5 ) _ and there exists @xmath130 , such that @xmath131 for all @xmath132 then the stable - like markov chain @xmath16 given by the transition kernel @xmath127 is transient . actually , instead of condition ( [ eq13 ] ) , in the proof of theorem [ tm13 ] , we use the following more technical but equivalent condition @xmath133 ( see section [ sec5 ] for details ) . conditions ( [ eq13 ] ) ( i.e. , ( [ eq15 ] ) ) and ( [ eq14 ] ) are needed to control the behavior of the family of density functions @xmath56 on sets symmetric around the origin . condition ( [ eq13 ] ) actually says that when the chain @xmath16 has moved far away from the origin , since @xmath134 , it can not have strong tendency to move further from the origin . since @xmath134 , it is clear that condition ( [ eq13 ] ) is satisfied if @xmath135 and if @xmath136 holds for all @xmath58 and for all @xmath137 large enough . for a non - symmetric example , one can take @xmath1 to be the density function of a @xmath138 distribution , when @xmath139 , and the density function of a @xmath140 distribution , when @xmath141 , where @xmath142 , @xmath143 @xmath144 and @xmath145 . using the concavity property of the function @xmath146 , for @xmath108 , condition ( [ eq14 ] ) follows from the condition @xmath147 ( see section [ sec5 ] for details ) . note that condition ( [ eq17 ] ) actually says that the function @xmath67 can not decrease too fast . since @xmath148 and @xmath149 , a simple example which satisfies condition ( [ eq17 ] ) is the case when @xmath150 , for some @xmath151 and for all @xmath137 large enough , where @xmath152 is arbitrary . furthermore , one can prove that the function @xmath153 is strictly decreasing on @xmath154 . hence , according to the condition ( [ eq17 ] ) , we choose @xmath155 close to @xmath156 . in the random walk case , that is , when the family of density functions @xmath56 is reduced to a single density function @xmath46 such that @xmath157 , when @xmath158 , where @xmath47 and @xmath159 , conditions ( c1)(c5 ) are trivially satisfied . hence , by theorem [ tm13 ] and the condition ( [ eq13 ] ) , if @xmath160 and if @xmath161 the random walk with the jump density @xmath46 is recurrent , and if @xmath162 , by theorem [ tm14 ] and the condition ( [ eq17 ] ) , the random walk with the jump density @xmath46 is transient . this result can be strengthened . if we assume that @xmath163 for all @xmath58 , from the discussion in @xcite , page 88 , the random walk with the jump density @xmath46 is recurrent if and only if @xmath164 as a simple consequence of theorems [ tm13 ] and [ tm14 ] , we get the following well - known recurrence and transience conditions for the s@xmath7s random walk case . [ c15 ] a s@xmath7s , @xmath165 , random walk is recurrent . a @xmath166 , @xmath167 , random walk with arbitrary shift is transient . the previous corollary can be generalized . if the functions @xmath168 @xmath91 and @xmath92 are borel measurable and take finitely many values , then the stable - like chain with s@xmath65s jumps is recurrent if @xmath135 for all @xmath60 . if @xmath149 for all @xmath60 , then the stable - like chain with @xmath90 jumps is transient . conditions in theorems [ tm13 ] and [ tm14 ] are only sufficient conditions for recurrence and transience of the stable - like chain @xmath16 . on the countable state space @xmath66 , when @xmath169 for @xmath170 , in @xcite it is proved that if @xmath171 , the associated chain is recurrent , and if @xmath172 , the associated chain is transient . a similar result , with @xmath173 for @xmath170 , is proved in the continuous time case in @xcite , that is , a stable - like process with the symbol @xmath174 is recurrent if and only if @xmath175 in @xcite , in the case when the function @xmath65 is periodic and continuously differentiable function , it is proved that all that matters is the minimum of the function @xmath65 . if @xmath176 , then a stable - like process with the symbol @xmath174 is recurrent if and only if @xmath177 now we explain our strategy of proving the main results . the proof of theorems [ tm13 ] and [ tm14 ] is based on the _ foster lyapunov drift criterion _ for recurrence and transience of markov chains ( see @xcite , theorems 8.4.2 and 8.4.3 ) . this criterion is based on finding an appropriate test function @xmath178 ( positive and unbounded in the recurrence case and positive and bounded in the transience case ) , and an appropriate set @xmath179 ( petite set ) such that @xmath180 , in the recurrence case , and @xmath181 , in the transience case , for every @xmath182 the idea is to find test functions @xmath178 such that the associated level sets @xmath183 are compact sets , that is , petite sets , and that @xmath184 , when @xmath185 , in the case of recurrence and @xmath184 , when @xmath186 , in the case of transience . in the recurrence case for the test function , we take @xmath187 , and in the transience case we take @xmath188 where @xmath189 ( recall that @xmath190 ) . now , by proving that @xmath191 in the recurrence case , and @xmath192 in the transience case , since compact sets are petite sets , the proofs of theorems [ tm13 ] and [ tm14 ] are accomplished . a similar approach , by using similar test functions @xmath178 , can be found in @xcite and @xcite . in @xcite , the author considers a markov chain on the nonnegative real line with uniformly bounded transition jumps , while in @xcite the authors generalize this result to the case of uniformly bounded @xmath193-moments of transition jumps , for some @xmath194 . if we allow that @xmath195 and assume the following additional assumption : @xmath196 , for every compact set @xmath76^{c}$ ] ( recall that the constant @xmath197 is defined in condition ( c3 ) ) , one can prove all nice structural properties of the chain @xmath16 , given by ( [ eq11 ] ) , proved in section [ sec2 ] . hence , since the chain @xmath16 is recurrent if and only if the chain @xmath198 is recurrent , @xcite covers the case when @xmath199 the paper is organized as follows . in section [ sec2 ] , we give several structural properties of the stable - like chain @xmath16 which will be crucial in finding sufficient conditions for the recurrence and transience property . in sections [ sec3 ] and [ sec4 ] , using foster lyapunov drift criterion for recurrence and transience of markov chains , we prove theorems [ tm13 ] and [ tm14 ] . in section [ sec5 ] , we extend our model from the model of asymptotically symmetric transition jumps to the model of asymptotically non - symmetric transition jumps . further , we prove that the change of the chain @xmath16 on bounded sets will not affect the recurrence and transience property . throughout the paper , we use the following notation . we write @xmath200 and @xmath201 for nonnegative and nonpositive integers , respectively . for @xmath202 let @xmath203 and @xmath204 . furthermore , @xmath16 will denote the stable - like markov chain on @xmath8 given by ( [ eq11 ] ) with transition densities satisfying conditions ( c1)(c5 ) , while @xmath93 will denote an arbitrary markov chain on @xmath94 given by the transition kernel @xmath205 for @xmath60 and @xmath71 . for @xmath60 , @xmath206 and @xmath24 let @xmath207 and @xmath208 .

we consider the recurrence and transience problem for a time - homogeneous markov chain on the real line with transition kernel , where the density functions , for large , have a power - law decay with exponent , where . in this paper , under a uniformity condition on the density functions and an additional mild drift condition , we prove that when , the chain is recurrent . similarly , under the same uniformity condition on the density functions and some mild technical conditions , we prove that when , the chain is transient . as a special case of these results , we give a new proof for the recurrence and transience property of a symmetric-stable random walk on with the index of stability

we consider the recurrence and transience problem for a time - homogeneous markov chain on the real line with transition kernel , where the density functions , for large , have a power - law decay with exponent , where . in this paper , under a uniformity condition on the density functions and an additional mild drift condition , we prove that when , the chain is recurrent . similarly , under the same uniformity condition on the density functions and some mild technical conditions , we prove that when , the chain is transient . as a special case of these results , we give a new proof for the recurrence and transience property of a symmetric-stable random walk on with the index of stability

1109.3336

i

the statistical analysis of functional data , commonly referred to as functional data analysis ( fda ) , is an established area of statistics with a great number of practical applications ; see the books @xcite and references therein for various examples . when the data is available as finely sampled curves , say in time , it is common to treat it as a collection of continuous - time curves or functions , each being observed in totality . these datasets are then termed `` functional , '' and various statistical procedures applicable in finite dimensions can be extended to this functional setting . among such procedures is principal component analysis ( pca ) , which is the focus of present work . if one thinks of continuity as a mathematical abstraction of reality , then treating functional data as continuous curves is arguably a valid modeling device . however , in practice , one is faced with finite computational resources and is forced to implement a ( finite - dimensional ) approximation of true functional procedures by some sort of truncation procedure , for instance , in the frequency domain . it is then important to understand the effects of this truncation on the statistical performance of the procedure . in other situations , such as in longitudinal data analysis @xcite , a continuous curve model is justified as a hidden underlying generating process to which one has access only through sparsely sampled measurements in time , possibly corrupted by noise . studying how the time - sampling affects the estimation of the underlying functions in the presence of noise shares various common elements with the frequency - domain problem described above . the aim of this paper is to study effects of `` sampling''in a fairly general sense on functional principal component analysis in smooth function spaces . in order to do so , we adopt a functional - theoretic approach by treating the sampling procedure as a ( continuous ) linear operator . this set - up provides us with a notion of sampling general enough to treat both the frequency - truncation and time - sampling within a unified framework . we take as our smooth function space a hilbert subspace @xmath0 of @xmath6 $ ] and denote the sampling operator by @xmath7 . we assume that there are functions @xmath8 , @xmath9 $ ] , in @xmath0 for @xmath10 , generated i.i.d . from a probabilistic model ( to be discussed ) . we then observe the collection @xmath11 in noise . we refer to the index @xmath5 as the number of _ statistical samples _ , and to the index @xmath3 as the number of _ functional samples_. we analyze a natural @xmath4-estimator which takes the form of a regularized pca in @xmath1 , and provide nonasymptotic bounds on the estimation error in terms of @xmath5 and @xmath3 . the eigen - decay of two operators govern the rates , the product of the sampling operator @xmath12 and its adjoint , and the product of the map embedding @xmath0 in @xmath2 and its adjoint . our focus will be on the setting where @xmath0 is a reproducing kernel hilbert space ( rkhs ) , in which case the two eigen - decays are intimately related through the kernel function @xmath13 . in such cases , the two components of the rate interact and give rise to optimal values for the number of functional samples ( @xmath3 ) in terms of the number of statistical samples ( @xmath5 ) or vice versa . this has practical appeal in cases where obtaining either type of samples is costly . our model for the functions @xmath14 is an extension to function spaces of the _ spiked covariance model _ introduced by johnstone and his collaborators @xcite , and studied by various authors ( e.g. , @xcite ) . we consider such models with @xmath15 components , each lying within the hilbert ball @xmath16 of radius @xmath17 , with the goal of recovering the @xmath15-dimensional subspace spanned by the spiked components in this functional model . we analyze our @xmath4-estimators within a high - dimensional framework that allows both the number of statistical samples @xmath5 and the number of functional samples @xmath3 to diverge together . our main theoretical contributions are to derive nonasymptotic bounds on the estimation error as a function of the pair @xmath18 , which are shown to be sharp ( minimax - optimal ) . although our rates also explicitly track the number of components @xmath15 and the smoothness parameter @xmath17 , we do not make any effort to obtain optimal dependence on these parameters . the general asymptotic properties of pca in function spaces have been investigated by various authors ( e.g. , @xcite ) . accounting for smoothness of functions by introducing various roughness / smoothness penalties is a standard approach , used in the papers @xcite , among others . the problem of principal component analysis for sampled functions , with a similar functional - theoretic perspective , is discussed by besse and ramsey @xcite for the noiseless case . a more recent line of work is devoted to the case of functional pca with noisy sampled functions @xcite . cardot @xcite considers estimation via spline - based approximation , and derives mise rates in terms of various parameters of the model . hall et al . @xcite study estimation via local linear smoothing , and establish minimax - optimality in certain settings that involve a fixed number of functional samples . both papers @xcite demonstrate trade - offs between the numbers of statistical and functional samples ; we refer the reader to hall et al . @xcite for an illuminating discussion of connections between fda and lda approaches ( i.e. , having full versus sampled functions ) , which inspired much of the present work . we note that the regularization present in our @xmath4-estimator is closely related to classical roughness penalties @xcite in the special case of spline kernels , although the discussion there applies to fully - observed functions , as opposed to the sampled models considered here . after initial posting of this work , we became aware of more recent work on sampled functional pca . working within the framework of hall et al . @xcite , the analysis of li and hsing @xcite allows for more flexible sample sizes per curve ; they derive optimal uniform ( i.e. , @xmath19 ) rates of convergence for local linear smoothing estimators of covariance function and the resulting eigenfunctions . another line of work @xcite has analyzed sampled forms of silverman s criterion @xcite , with some variations . huang et al . @xcite derive a criterion based on rank - one approximation coupled with scale invariance considerations , combined with an extra weighting of the covariance matrix . xi and zhao @xcite also show the consistency of their estimator for both regular and irregular sampling . the regular ( time ) sampling setup in both papers have an overlap with our work ; the eigenfunctions are assumed to lie in a second order sobolev space , corresponding to a special case of a rkhs . however , even in this particular case , our estimator is different , and it is an interesting question whether a version of the results presented here can be used to show the minimax optimality of these silverman - type criteria . there has also been recent work with emphasis on sampled functional covariance estimation , including the work of cai and yuan @xcite , who analyze an estimator which can be described as regularized least - squares with penalty being the norm of tensor product of rkhs with itself . they provide rates of convergence for the covariance function , from which certain rates ( argued to be optimal within logarithmic factors ) for eigenfunctions follow . as mentioned above , our sampled model resembles very much that of spiked covariance model for high - dimensional principal component analysis . a line of work on this model has treated various types of sparsity conditions on the eigenfunctions @xcite ; in contrast , here the smoothness condition on functional components translates into an ellipsoid condition on the vector principal components . perhaps an even more significant difference is that in this paper , the effective scaling of noise in @xmath1 is substantially smaller in some cases ( e.g. , the case of time sampling ) . this difference could explain why the difficulty of `` high - dimensional '' setting is not observed in such cases as one lets @xmath20 . on the other hand , a difficulty particular to our sampled model is the lack of orthonormality between components after sampling . it not only leads to identifiability issues , but also makes recovering individual components difficult . in order to derive nonasymptotic bounds on our @xmath4-estimator , we exploit various techniques from empirical process theory ( e.g. , @xcite ) , as well as the concentration of measure ( e.g. , @xcite ) . we also exploit recent work @xcite on the localized rademacher complexities of unit balls in a reproducing kernel hilbert space , as well as techniques from nonasymptotic random matrix theory , as discussed in davidson and szarek @xcite , in order to control various norms of random matrices . these techniques allow us to obtain finite - sample bounds that hold with high probability , and are specified explicitly in terms of the pair @xmath18 , and the underlying smoothness of the hilbert space . the remainder of this paper is organized as follows . section [ secbackground ] is devoted to background material on reproducing kernel hilbert spaces , adjoints of operators , as well as the class of sampled functional models that we study in this paper . in section [ secmest ] , we describe @xmath4-estimators for sampled functional pca , and discuss various implementation details . section [ secmain ] is devoted to the statements of our main results , and discussion of their consequences for particular sampling models . in subsequent sections , we provide the proofs of our results , with some more technical aspects deferred to the supplementary material @xcite . section [ secproofsubspace ] is devoted to bounds on the subspace - based error . we conclude with a discussion in section [ secdiscuss ] . in the supplementary material @xcite , section 7 is devoted to proofs of bounds on error in the function space , whereas section 8 provides proofs of matching lower bounds on the minimax error , showing that our analysis is sharp . _ notation_. we will use to denote the hilbert schmidt norm of an operator or a matrix . the corresponding inner product is denoted as @xmath21 . if @xmath22 is an operator on a hilbert space @xmath0 with an orthonormal basis @xmath23 , then @xmath24 . for a matrix @xmath25 , we have @xmath26 , the range as @xmath27 and the kernel as @xmath28 .

this model includes time and frequency sampling as special cases . in contrast to classical approach in fpca in which access to entire functions is assumed , having a limited number of functional samples places limitations on the performance of statistical procedures . we study these effects by analyzing the rate of convergence of an-estimator for the subspace spanned by the leading components in a multi - spiked covariance model . the estimator takes the form of regularized pca , and hence is computationally attractive . we analyze the behavior of this estimator within a nonasymptotic framework , and provide bounds that hold with high probability as a function of the number of statistical samples and the number of functional samples .

we consider the sampling problem for functional pca ( fpca ) , where the simplest example is the case of taking time samples of the underlying functional components . more generally , we model the sampling operation as a continuous linear map from to , where the functional components to lie in some hilbert subspace of , such as a reproducing kernel hilbert space of smooth functions . this model includes time and frequency sampling as special cases . in contrast to classical approach in fpca in which access to entire functions is assumed , having a limited number of functional samples places limitations on the performance of statistical procedures . we study these effects by analyzing the rate of convergence of an-estimator for the subspace spanned by the leading components in a multi - spiked covariance model . the estimator takes the form of regularized pca , and hence is computationally attractive . we analyze the behavior of this estimator within a nonasymptotic framework , and provide bounds that hold with high probability as a function of the number of statistical samples and the number of functional samples . we also derive lower bounds showing that the rates obtained are minimax optimal .

1109.3336

i

in this section , we begin by introducing background on reproducing kernel hilbert spaces , as well as linear operators and their adjoints . we then introduce the functional and observation model that we study in this paper , and conclude with discussion of some approximation - theoretic issues that play an important role in parts of our analysis . we begin with a quick overview of some standard properties of reproducing kernel hilbert spaces ; we refer the reader to the books @xcite and references therein for more details . a reproducing kernel hilbert space ( or rkhs for short ) is a hilbert space @xmath0 of functions @xmath29 that is equipped with a symmetric positive semidefinite function @xmath30 , known as the kernel function . we assume the kernel to be continuous , and the set @xmath31 to be compact . for concreteness , we think of @xmath32 $ ] throughout this paper , but any compact set of @xmath33 suffices . for each @xmath34 , the function @xmath35 belongs to the hilbert space @xmath0 and it acts as the _ representer of evaluation _ , meaning that @xmath36 for all @xmath37 . the kernel @xmath38 defines an integral operator @xmath39 on @xmath40 , mapping the function @xmath41 to the function @xmath42 . by the spectral theorem in hilbert spaces , this operator can be associated with a sequence of eigenfunctions @xmath43 , in @xmath0 , orthogonal in @xmath0 and orthonormal in @xmath40 , and a sequence of nonnegative eigenvalues @xmath44 . most useful for this paper is the fact that any function @xmath37 has an expansion in terms of these eigenfunctions and eigenvalues , namely @xmath45 for some @xmath46 . in terms of this expansion , we have the representations @xmath47 and @xmath48 . many of our results involve the decay rate of these eigenvalues : in particular , for some parameter @xmath49 , we say that the kernel operator has eigenvalues with _ polynomial-@xmath50 decay _ if there is a constant @xmath51 such that @xmath52 let us consider an example to illustrate . in the case @xmath32 $ ] and @xmath53 , we can consider the kernel function @xmath54 . as discussed in appendix a of the supplementary material @xcite , this kernel generates the class of functions @xmath55\bigr ) \mid f(0 ) = 0 , f \mbox { absolutely continuous and } f ' \in l^2\bigl([0,1]\bigr ) \bigr\}.\ ] ] the class @xmath0 is an rkhs with inner product @xmath56 , and the ball @xmath16 corresponds to a sobolev space with smoothness @xmath53 . the eigen - decomposition of the kernel integral operator is @xmath57^{-2},\qquad \psi_k(t ) = \sqrt{2 } \sin\bigl ( \mu_k^{-1/2 } t \bigr),\qquad k=1,2,\ldots.\ ] ] consequently , this class has polynomial decay with parameter @xmath53 . we note that there are natural generalizations of this example to @xmath58 , corresponding to the sobolev classes of @xmath50-times differentiable functions ; for example , see the books @xcite . in this paper , the operation of generalized sampling is defined in terms of a bounded linear operator @xmath59 on the hilbert space . its adjoint is a mapping @xmath60 , defined by the relation @xmath61 for all @xmath37 and @xmath62 . in order to compute a representation of the adjoint , we note that by the riesz representation theorem , the @xmath63th coordinate of this mapping namely , @xmath64_j$]can be represented as an inner product @xmath65 , for some element , and we can write @xmath66^t.\ ] ] consequently , we have @xmath67 , so that for any @xmath62 , @xmath68 this adjoint operator plays an important role in our analysis . let @xmath69 be a fixed sequence of positive numbers , and let @xmath70 be a fixed sequence of functions orthonormal in @xmath6 $ ] . consider a collection of @xmath5 i.i.d . random functions @xmath71 , generated according to the model @xmath72 where @xmath73 are i.i.d . @xmath74 across all pairs @xmath75 . this model corresponds to a finite - rank instantiation of functional pca , in which the goal is to estimate the span of the unknown eigenfunctions @xmath76 . typically , these eigenfunctions are assumed to satisfy certain smoothness conditions ; in this paper , we model such conditions by assuming that the eigenfunctions belong to a reproducing kernel hilbert space @xmath0 embedded within @xmath6 $ ] ; more specifically , they lie in some ball in @xmath0 , @xmath77 for statistical problems involving estimation of functions , the random functions might only be observed at certain times @xmath78 , such as in longitudinal data analysis , or we might collect only projections of each @xmath79 in certain directions , such as in tomographic reconstruction . more concretely , in a _ time - sampling model _ , we observe @xmath3-dimensional vectors of the form @xmath80^t + \sigma_0 w_i\qquad\mbox{for $ i = 1 , 2,\ldots , n$},\ ] ] where @xmath81 is a fixed collection of design points , and @xmath82 is a noise vector . another observation model is the _ basis truncation model _ in which we observe the projections of @xmath41 onto the first @xmath3 basis functions @xmath83 of the kernel operator namely , @xmath84^t + \sigma_0 w_i\nonumber\\[-8pt]\\[-8pt ] & & \eqntext{\mbox{for $ i = 1 , 2,\ldots , n$},}\end{aligned}\ ] ] where @xmath85 represents the inner product in @xmath6 $ ] . in order to model these and other scenarios in a unified manner , we introduce a linear operator @xmath86 that maps any function @xmath87 in the hilbert space to a vector @xmath88 of @xmath3 samples , and then consider the linear observation model @xmath89 this model ( [ eqnlinobs ] ) can be viewed as a functional analog of the spiked covariance models introduced by johnstone @xcite as an analytically - convenient model for studying high - dimensional effects in classical pca . both the time - sampling ( [ eqntimesamp ] ) and frequency truncation ( [ eqnbasistrun ] ) models can be represented in this way , for appropriate choices of the operator @xmath86 . recall representation ( [ eqnlinoprep ] ) of @xmath86 in terms of the functions @xmath90 . * for the time sampling model ( [ eqntimesamp ] ) , we set @xmath91 , so that by the reproducing property of the kernel , we have @xmath92 for all @xmath37 , and @xmath93 . with these choices , the operator @xmath86 maps each @xmath37 to the @xmath3-vector of rescaled samples @xmath94^t.\ ] ] defining the rescaled noise @xmath95 yields an instantiation of model ( [ eqnlinobs ] ) which is equivalent to time - sampling ( [ eqntimesamp ] ) . * for the basis truncation model ( [ eqnbasistrun ] ) , we set @xmath96 so that the operator @xmath12 maps each function @xmath37 to the vector of basis coefficients @xmath97^t$ ] . setting @xmath98 then yields another instantiation of model ( [ eqnlinobs ] ) , this one equivalent to basis truncation ( [ eqnbasistrun ] ) . a remark on notation before proceeding : in the remainder of the paper , we use @xmath99 as shorthand notation for @xmath100 , since the index @xmath3 should be implicitly understood throughout our analysis . in this paper , we provide and analyze estimators for the @xmath15-dimensional eigen - subspace spanned by @xmath101 , in both the sampled domain @xmath1 and in the functional domain . to be more specific , for @xmath102 , define the vectors @xmath103 , and the subspaces @xmath104 and let @xmath105 and @xmath106 denote the corresponding estimators . in order to measure the performance of the estimators , we will use projection - based distances between subspaces . in particular , let @xmath107 and @xmath108 be orthogonal projection operators into @xmath109 and @xmath105 , respectively , considered as subspaces of @xmath110 . similarly , let @xmath111 and @xmath112 be orthogonal projection operators into @xmath113 and @xmath106 , respectively , considered as subspaces of @xmath114 . we are interested in bounding the deviations @xmath115 where @xmath116 is the hilbert schmidt norm of an operator ( or matrix ) . one object that plays an important role in our analysis is the matrix @xmath117 . from the form of the adjoint , it can be seen that @xmath118_{ij } = \langle\phi_i , \phi_j \rangle_{\mathcal{h}}$ ] . for future reference , let us compute this matrix for the two special cases of linear operators considered thus far : * for the time sampling model ( [ eqntimesamp ] ) , we have @xmath91 for all @xmath119 , and hence @xmath118_{ij } = \frac{1}{m}\langle{\mathbb{k}}(\cdot , t_i),{\mathbb{k}}(\cdot , t_j)\rangle_{{\mathcal{h } } } = \frac{1}{m } { \mathbb{k}}(t_i , t_j)$ ] , using the reproducing property of the kernel . * for the basis truncation model ( [ eqnbasistrun ] ) , we have @xmath120 , and hence @xmath118_{ij } = \langle\mu_i \psi_i,\mu_j\psi_j\rangle_{{\mathcal{h } } } = \mu_i\delta_{ij}$ ] . thus , in this special case , we have @xmath121 . in general , the matrix @xmath122 is a type of gram matrix , and so is symmetric and positive semidefinite . we assume throughout this paper that the functions @xmath90 are linearly independent in @xmath0 , which implies that @xmath122 is strictly positive definite . consequently , it has a set of eigenvalues which can be ordered as @xmath123 under this condition , we may use @xmath122 to define a norm on @xmath1 via @xmath124 . moreover , we have the following interpolation lemma , which is proved in appendix b.1 of the supplementary material @xcite : [ leminterpolate ] for any @xmath37 , we have @xmath125 , with equality if and only if @xmath126 . moreover , for any @xmath127 , the function @xmath128 has smallest hilbert norm of all functions satisfying @xmath129 , and is the unique function with this property . this lemma is useful in constructing a function - based estimator , as will be clarified in section [ secmest ] . in our analysis of the functional error @xmath130 , a number of approximation - theoretic quantities play an important role . as a mapping from an infinite - dimensional space @xmath0 to @xmath1 , the operator @xmath12 has a nontrivial nullspace . given the observation model ( [ eqnlinobs ] ) , we receive no information about any component of a function @xmath131 that lies within this nullspace . for this reason , we define the width of the nullspace in the @xmath2-norm , namely the quantity @xmath132 in addition , the observation operator @xmath12 induces a semi - norm on the space @xmath0 , defined by @xmath133_j^2.\ ] ] it is of interest to assess how well this semi - norm approximates the @xmath2-norm . accordingly , we define the quantity @xmath134 which measures the worst - case gap between these two ( semi)-norms , uniformly over the hilbert ball of radius one , restricted to the subspace of interest @xmath135 . given knowledge of the linear operator @xmath12 , the quantity @xmath136 can be computed in a relatively straightforward manner . in particular , recall the definition of the matrix @xmath122 , and let us define a second matrix @xmath137 with entries @xmath138 . [ lemdefect ] we have the equivalence @xmath139 where denotes the @xmath140-operator norm . see appendix b.2 of the supplementary material @xcite for the proof of this claim .

we consider the sampling problem for functional pca ( fpca ) , where the simplest example is the case of taking time samples of the underlying functional components . more generally , we model the sampling operation as a continuous linear map from to , where the functional components to lie in some hilbert subspace of , such as a reproducing kernel hilbert space of smooth functions .

we consider the sampling problem for functional pca ( fpca ) , where the simplest example is the case of taking time samples of the underlying functional components . more generally , we model the sampling operation as a continuous linear map from to , where the functional components to lie in some hilbert subspace of , such as a reproducing kernel hilbert space of smooth functions . this model includes time and frequency sampling as special cases . in contrast to classical approach in fpca in which access to entire functions is assumed , having a limited number of functional samples places limitations on the performance of statistical procedures . we study these effects by analyzing the rate of convergence of an-estimator for the subspace spanned by the leading components in a multi - spiked covariance model . the estimator takes the form of regularized pca , and hence is computationally attractive . we analyze the behavior of this estimator within a nonasymptotic framework , and provide bounds that hold with high probability as a function of the number of statistical samples and the number of functional samples . we also derive lower bounds showing that the rates obtained are minimax optimal .

0906.2597

i

although the gamma - ray burst ( grb ) was discovered about 50 years ago first through its prompt @xmath1-ray emission , large uncertainties still remain in understanding the prompt emission site , namely , the distance of the emission site from the explosion centre @xmath0 , with controversial evidence . there are three possible sites discussed in the literature . one is the standard internal - shock site which depends on the fluctuation time scale @xmath7 seen in grb light curves ( e.g. , rees & mszros 1994 , see piran 2005 , mszros 2006 for reviews ) . it can have a large range of @xmath8 cm because @xmath7 and @xmath2 vary largely from burst to burst . the second is the photospheric radius at @xmath9 cm at which the prompt emission arises as a combination of the photosphere thermal emission and a comptonized component above it , the latter being induced by some energy dissipation process below and above the photosphere ( e.g. rees & mszros 2005 ; ryde et al . 2006 ; thompson et al . 2007 ) . the third one is a large radius ( @xmath10 cm ) as is supported by the swift xrt data ( lazzati & begelman 2005 ; lyutikov 2006 ; kumar et al . 2007 ) and fermi data of grb 080916c ( abdo et al . 2009 ; zhang & peer 2009 ) , possibly due to magnetic dissipation ( e.g. , lyutikov & blandford 2003 ) . the rapidly responding ability of a few grb - dedicated ground or space based optical telescopes , e.g. , rotse ( akerlof et al . 2003 ) , raptor ( vestrand et al . 2002 ) , tortora ( racusin et al . 2008 ) and the uvot ( roming et al . 2005 ) on aboard the swift satellite , has enabled the time - resolved detection of bright prompt optical emission before the @xmath1-rays die off , for about a dozen of grbs . five of these grbs , i.e. , 041219a ( vestrand et al . 2005 ) , 050820a ( vestrand et al . 2006 ) , 051111 ( yost et al . 2007a ) , 061121 ( page et al . 2007 ) and 080319b ( racusin et al . 2008 ) , show a temporal correlation between the strongly variable optical flux and the @xmath1-ray pulses , which suggests that the optical emission most likely shares the same dynamical process that is responsible for the highly variable @xmath1-ray emission . while the other four bursts have optical flux densities below or marginally consistent with the extrapolations from the low - energy power law of the @xmath1-ray spectra , the optical flux density in grb 080319b exceeds the @xmath1-ray extrapolation by 4 orders of magnitude ( racusin et al . 2008 ; kumar & panaitescu 2008 ) , suggesting that for this burst alone the optical emission has a spectral origin different from that of the @xmath1-rays . in this paper , for the four grbs - 041219a , 050820a , 051111 and 061121 - we assume that the prompt optical and the @xmath1-ray emissions are components belonging to the same synchrotron radiation continuum of a group of hot electrons . based on this assumption , the self - absorption frequency of the synchrotron electrons , @xmath11 , which causes a break in the long - wavelength part of the continuum , can be determined or constrained by studying the optical - to-@xmath1-ray spectral energy distribution ( sed ) to interpret the diversity in the prompt optical / @xmath1-ray temporal correspondence . ] . since @xmath11 is dependent on the properties of the prompt emission source , such as the distance of the emission site from the explosion centre @xmath0 , the bulk lorentz factor ( lf ) @xmath2 and the magnetic field @xmath3 of the source , from @xmath11 we can determine or make constraints on @xmath0 for these bursts , using information on @xmath2 and @xmath3 obtained in other ways . this is the main goal of this paper . since the prompt optical and @xmath1-ray components in grb 080319b are most likely of different spectral origins because of its peculiar sed shape , our approach is not applicable to this burst . on the other hand , for some other long grbs the rapid response of the dedicated rotse telescope has returned only upper limits of the optical flux density during the prompt phase ( yost et al . another goal of this paper is to get constraints on @xmath0 for these optically `` dark '' bursts and to study whether the prompt optical non - detection is caused by a heavier self - absorption due to a closer emission site to the explosion centre . in this paper , we first derive analytically @xmath11 in terms of @xmath0 , @xmath2 , @xmath3 and the emission properties in sec . the arguments that support our assumption of one synchrotron continuum component for both optical and @xmath1/x - ray are given in sec . we derive in sec . 4 the constraints on @xmath0 through @xmath11 explicitly , by determining the location of @xmath11 in the optical - to-@xmath1-ray sed and considering all possible spectral regimes . we apply this method to a prompt optical detection grb sample and a prompt optical non - detection sample which are described in sec . the results are presented in sec . 6 . finally the conclusion and discussions are given in sec .

we constrain the distance of the gamma - ray burst ( grb ) prompt emission site from the explosion centre , , by determining the location of the electron s self absorption frequency in the grb prompt optical - to - x/-ray spectral energy distribution , assuming that the optical and the-ray emissions are among the same synchrotron radiation continuum of a group of hot electrons . all possible spectral regimes are considered in our analysis . we identify a small sample of 4 bursts that satisfy the following three criteria : ( 1 ) they all have simultaneous optical and-ray detections in multiple observational time intervals ; ( 2 ) they all show temporal correlations between the optical and-ray light curves ; and ( 3 ) the optical emission is consistent with belonging to the same spectral component as the-ray emission . for all the time intervals of these 4 bursts , it is inferred that cm . for a small fraction of the sample , the constraint can be pinned down to cm for . for a second sample of bursts with prompt optical non - detections , gamma - rays : bursts - gamma - rays : theory - radiation mechanisms : non - thermal - radiative transfer

we constrain the distance of the gamma - ray burst ( grb ) prompt emission site from the explosion centre , , by determining the location of the electron s self absorption frequency in the grb prompt optical - to - x/-ray spectral energy distribution , assuming that the optical and the-ray emissions are among the same synchrotron radiation continuum of a group of hot electrons . all possible spectral regimes are considered in our analysis . the method has only two assumed parameters , namely , the bulk lorentz factor of the emitting source , and the magnetic field strength in the emission region ( with a weak dependence ) . we identify a small sample of 4 bursts that satisfy the following three criteria : ( 1 ) they all have simultaneous optical and-ray detections in multiple observational time intervals ; ( 2 ) they all show temporal correlations between the optical and-ray light curves ; and ( 3 ) the optical emission is consistent with belonging to the same spectral component as the-ray emission . for all the time intervals of these 4 bursts , it is inferred that cm . for a small fraction of the sample , the constraint can be pinned down to cm for . for a second sample of bursts with prompt optical non - detections , only upper limits on can be obtained . we find no inconsistency between the-constraints for this non - detection sample and those for the detection sample . gamma - rays : bursts - gamma - rays : theory - radiation mechanisms : non - thermal - radiative transfer

1607.03630

i

we constructed the new godunov type relativistic hydrodynamic code in milne coordinates based on the algorithm in cartesian coordinates @xcite . we evaluated the flux terms , using the numerical solution of the riemann problem with the initial condition at the constant proper time @xmath11 . we checked correctness of our algorithm from the comparison between numerical calculations and analytical solutions of shock tube , expansion of matter into the vacuum , the landau - khalatnikov solution , propagation of fluctuation around bjorken flow and the gubser flow . we investigated the energy and momentum conservation of our code from calculation of longitudinal hydrodynamic expansion with an initial condition for high - energy heavy - ion collisions . in particular , we focused on the effects of the source terms in relativistic numerical hydrodynamics in milne coordinates on stability and numerical viscosity . we analyzed those effects in the test problems of expansion into the vacuum and the conservation property . in expansion of matter into the vacuum , we showed that numerical results from the code without the source terms is closer to the analytical solution compared with that with source terms . besides , the code without the source terms is more stable and has less numerical viscosity than the code with the source terms . in addition , we observed that the code written in the conservative form keeps the conservation property with high accuracy in the expansion from the fluctuating initial longitudinal profile for high - energy heavy - ion collisions even on the coarse grid . our algorithm is easily extended to the code with the qcd equation of state and finite viscosities @xcite . after that , we shall employ our hydrodynamic code to investigate experimental results at rhic and lhc and understand the detailed qgp bulk property using a reliable 3d relativistic viscous hydrodynamic expansion with small numerical viscosity .

we check the correctness of the numerical algorithm by comparing numerical calculations and analytical solutions in various problems , such as shock tubes , expansion of matter into the vacuum , landau - khalatnikov solution , propagation of fluctuations around bjorken flow and gubser flow . we investigate the energy and momentum conservation property of our code in a test problem of longitudinal hydrodynamic expansion with an initial condition for high - energy heavy - ion collisions . furthermore , we discuss how the numerical stability is affected by the source terms of relativistic numerical hydrodynamics in milne coordinates .

we construct a new godunov type relativistic hydrodynamics code in milne coordinates , using a riemann solver based on the two - shock approximation which is stable under existence of large shock waves . we check the correctness of the numerical algorithm by comparing numerical calculations and analytical solutions in various problems , such as shock tubes , expansion of matter into the vacuum , landau - khalatnikov solution , propagation of fluctuations around bjorken flow and gubser flow . we investigate the energy and momentum conservation property of our code in a test problem of longitudinal hydrodynamic expansion with an initial condition for high - energy heavy - ion collisions . we also discuss numerical viscosity in the test problems of expansion of matter into the vacuum and conservation properties . furthermore , we discuss how the numerical stability is affected by the source terms of relativistic numerical hydrodynamics in milne coordinates .

1311.6465

c

in this work , we have constructed a new kinematical variable that gives , under certain circumstances , the cm angular distribution , @xmath184 , of a decaying particle @xmath2 when one of its daughter particles @xmath7 is not detected . among the requirements for this method are that the masses of the mother particle @xmath2 as well as both daughter particles @xmath1 and @xmath7 must be known . additionally , we have shown that the magnitude of @xmath184 only depends on the component of the momentum of the measured particle @xmath1 which is transverse to the momentum of @xmath2 , which we call @xmath12 , as can be seen in eq . . we have also shown that @xmath12 is uniquely related to the energy of the missed particle @xmath7 , @xmath21 , as seen in eqs . and . therefore , if either @xmath12 or @xmath21 can be determined event by event , then the magnitude of @xmath184 is known , event by event , unambiguously . the sign of our observable , on the other hand , has an inherent two - fold ambiguity . we find that it can be consistently split into two solutions which we call the large and small solutions , based on the size of the component of @xmath185 which is parallel to @xmath30 . we find that the large solution always has a negative sign , as in eq . while the small solution s sign depends on the energy of the measured particle @xmath1 , as in eq . . although this sign can not be determined event by event , the distributions of the large and small cm angular distributions , eq . and eq . , respectively , contain the information required to reconstruct the true @xmath184 in many situations . the large solution gives the negative absolute value of the distribution of @xmath184 , from which it is easy to reconstruct the full symmetrized version of the @xmath184 distribution by dividing it in half and taking the mirror image on the @xmath186 side . although this distribution can not tell us anything about the parity violation in the true @xmath184 distribution , it can give us the full symmetrized dependence of the differential cross - section on @xmath184 and therefore the spin - combinations of the mother and daughter particles which can produce that distribution . in many cases , this is already sufficient to determine the spin of @xmath2 and @xmath7 , where the spin of @xmath1 is already known . although the full structure of our kinematical variable was first described in the present work , a special case of this kinematical variable was applied to d - y production of a charged lepton and a neutrino where the spin of the intermediate resonance was shown to be unambiguously determined by the large solution @xcite . on the other hand , the small solution contains nontrivial information about the sign of @xmath184 . although it does not agree with the true sign event by event , its distribution contains the clear signatures of the parity violation present in the true distribution . in @xcite , it was shown that in the special case of d - y production of a charged lepton and a neutrino , a simple reconstruction technique could be applied to the small solution to fully reconstruct the true cm angular distribution , including its parity violating features , almost exactly . it was , further , shown that this reconstruction technique was universal and did not depend on the spin of @xmath2 or the parity - violation . in other words , the reconstruction technique could be applied blindly and give the correct results in all spin cases . moreover , it was also shown that the parity violation could be determined directly from the small solution without applying the reconstruction technique . in @xcite , it was also shown that acceptance cuts , and even the rather large @xmath187gev cut only affected the @xmath188 edges of the distributions , but left the majority of the distribution unaffected , preserving its power . in the present work , we extended this to consider the effect of a finite width . we found that an approximately @xmath125% width and smaller widths only affected the @xmath189 edges and the @xmath190 central region , but left the rest of the distribution unchanged , as seen in fig . [ fig : dylarge ] . this shows that this variable works quite well for realistic d - y charged resonance production at the lhc . in the present work , we applied our kinematical variables , for the first time , to the antler process @xmath181 at the ilc where @xmath7 was taken to be a self - charge - conjugate dark matter particle . we showed that @xmath21 is known in this case , therefore @xmath183 is known , event by event , unambiguously , as described above . we focused , in this article , on only parity - symmetric operators , therefore all our angular distributions were inherently symmetric . we calculated analytically the dependence of the differential cross - section on @xmath184 and included it in table [ tab : analytic cm angular distributions ] . we found agreement of the true cm angular distribution with these formulas in all cases . the cm angular distributions split up into four classes . the first class ( denoted by i in table [ tab : analytic cm angular distributions ] and figs . [ fig : classes plots ] through [ fig : isr plots ] ) is given by a flat distribution and , unfortunately , includes both the cases where @xmath2 is spin-@xmath108 and spin-@xmath124 . the second class ( denoted by ii in table [ tab : analytic cm angular distributions ] and figs . [ fig : classes plots ] through [ fig : isr plots ] ) is given by a concave negative parabola and includes only the case where @xmath2 is spin-@xmath125 and @xmath7 is spin-@xmath124 . the third class ( denoted by iii in table [ tab : analytic cm angular distributions ] and figs . [ fig : classes plots ] through [ fig : isr plots ] ) is given by a concave positive parabola and includes the case where @xmath2 is spin-@xmath125 and @xmath7 is spin-@xmath129 and the cases where @xmath2 is spin-@xmath129 and @xmath7 is spin-@xmath108 , @xmath125 or @xmath94 . although their distributions are all concave positive parabolas , we show that with appropriate masses and collision energy , they can be separated and distinguished as in fig . [ fig : iii splitting]iii . the last class ( denoted by iv in table [ tab : analytic cm angular distributions ] and figs . [ fig : classes plots ] through [ fig : isr plots ] ) is given by a m " shape and includes the cases where @xmath2 is spin-@xmath94 and @xmath7 is spin-@xmath124 or @xmath129 . again , we show that for appropriate parameters , these can be distinguished as in fig . [ fig : iii splitting]iv . we then showed that the large cm angular distribution after a simple reconstruction , gives exact agreement with the true distribution in the narrow width limit and in the absence of isr and beamstrahlung , as shown in fig . [ fig : classes plots ] . if the width is not infinitesimal , on the other hand , we found a small reduction on the edges and , to a lesser extent , a small enhancement in the center of the distribution occured . however , for a @xmath125% width , we found the effect to be practically negligible while a @xmath169% width made a noticeable effect but left the shape of the distributions largely intact for most spin combinations , as seen in fig . [ fig : fw plots ] . the effect of isr and beamstrahlung , on the other hand , was much more pronounced . it also appeared as a reduction on the edges and , to a lesser extent , an enhancement in the center . however , as can be seen in fig . [ fig : isr plots ] , the shape of the true distribution is still clearly visible for most of the spin combinations . furthermore , the effect of isr and beamstrahlung can be well modeled . from these results , we see that our method should work quite well in this process and that as long as @xmath2 is spin-@xmath125 or higher and the masses do nt conspire to make the distributions within a class identical , it should be possible to determine the spin of both @xmath2 and the dark matter particle @xmath7 using our kinematical variables in this process at the ilc . _ acknowledgements _ n.d.c . was supported in part by pitt pacc and the u.s . department of energy under grant no . de - fg02 - 95er40896 . d.s . was supported in part by pitt pacc , the pennsylvania space grant consortium research scholarship and the dietrich school of arts and sciences summer undergraduate research award for independent research . we would like to thank tao han , adam leibovich and ayres freitas for their encouragement , helpful discussions and support during the completion of this project . we would also like to thank the university of granada high energy theory group for their hospitality during our visit where part of this research was completed .

this enables us to determine the spin of the mother particle and the dark matter particle in certain cases . , we give a brief summary of other methods to measure the spin of dark matter . [ sec : dylhc ] , we summarize the application of these methods to charged drell - yan production and consider the effects of a finite width . in section [ sec : antlers ] , we describe how the application of our kinematical variables to antler processes at the ilc can be used to determine the spin of dark matter . in section [ sec : conclusion ] , we conclude .

we construct a new kinematical variable that is able to fully reconstruct the absolute value , and partially reconstruct the sign , of the angular distribution in the center of momentum system of a decaying particle in certain cases where the center of momentum system is only known up to a two - fold ambiguity . after making contact with drell - yan production at the large hadron collider , we apply this method to the pair - production of dark matter in association with two charged leptons at the international linear collider and show that for a small intermediate width , perfect agreement is found with the true angular distribution in the absence of initial state radiation . in the presence of initial state radiation , we find that the modification to the angular distributions is small for most angles and that different spin combination classes should still be distinguishable . this enables us to determine the spin of the mother particle and the dark matter particle in certain cases . the existence of dark matter has been well established through a combination of galactic rotation curves , weak and strong gravitational lensing , big bang nucleosynthesis , the cosmic microwave background and the bullet cluster . from these observations , we know that dark matter is electrically neutral , non - baryonic and composes roughly 83% of the matter and 23% of the energy of the universe . however , these observations do not tell us the detailed properties of dark matter such as its mass , spin and how it interacts with visible matter . for that , we need to observe a dark matter particle ( dmp ) in the laboratory . because the standard model ( sm ) of particle physics does not contain dark - matter ( among other things ) it is a low - energy effective theory that fits inside a larger , more complete theory . two prominent examples of these theories are the minimal supersymmetric extension of the sm ( mssm ) and the universal extra - dimension ( ued ) model . in the present context , one of the most important features of these models is the presence of a new parity symmetry with the consequence that the lightest parity - odd particle ( lpp ) is stable and ( if neutral ) a dark - matter candidate . in these theories , the lpp is a weakly interacting massive particle ( wimp ) and , so , can be pair produced at particle colliders , such as the large hadron collider ( lhc ) and the international linear collider ( ilc ) . to determine the spin of a dmp at a collider , ideally , we would like to boost into the center of momentum ( cm ) frame of its parent particle and histogram the angle of its decay with respect to the boost direction ( see figure [ fig : bldsystem ] ) . we will call this the cm angular distribution , where is the angle of the decay product with respect to the boost direction in the cm system . if the width of the parent particle is narrow , this distribution will correspond with linear combinations of squares of the wigner-functions where and correspond with the spin and spin - component along the boost direction of the parent particle and corresponds with the difference of the helicities of the final state particles and ( see appendix [ sec : wigner d - functions ] for a brief discussion . ) the challenge for dark - matter particles is that they do not interact with particle detectors and are , thus , not measured . therefore , since we do not know their momentum , we often can not reconstruct the cm system . decay of into and where is an observed sm - particle , is the missing dark - matter particle and is the parent of this decay . ] in this paper , we introduce a new kinematical variable that is able to fully reconstruct the absolute value of the cm angular distribution unambiguously and its sign up to a two - fold ambiguity even in some cases where the cm system is not known . this method is a generalization of that used to reconstruct the spin of a new charged resonance in drell - yan processes at the lhc . our result is the following . the absolute value of the cosine of this angle is given by where , and are the masses of the parent particle ( ) , the observed particle ( ) and the dmp ( ) , is the component of the observed momentum of the visible particle ( ) that is transverse to the momentum of in the lab - frame , and the two possible signs are given by and which have been labeled by and , to be explained below and is the energy of the observed particle ( ) in the lab frame . furthermore , can be expressed purely in terms of known quantities and the energy of in the lab frame as where the requirements for this method are that : the masses of , and must be known ; the full momentum of must be known ; the width of must be narrow ; , and must all be on - shell ; and either or must be known . under these circumstances , even if the b cm system can not be reconstructed , the cm angular distribution can be calculated , up to the sign ambiguity outlined above . in this paper , we will describe two scenarios where these requirements are satisfied . the first is in the discovery of a new resonance in charged drell - yan production of a charged lepton and a neutrino at the lhc . we will summarize this scenario and refer to for further details . the second is in the antler production of two charged leptons and two dark matter particles at the ilc , which we will describe in detail in this paper . before moving on , we give a brief summary of other methods to measure the spin of dark matter . an analysis of the spin - correlation in various cascade decay chains has shown that in many cases the resulting distributions were sufficient to determine the spin . it has also been found that in certain cases the production cross - section varies with spin . additionally , the shapes of some other distributions have a dependence on the spin . in addition to these methods , our method has the benefit of reconstructing the actual cm angular distribution even when the cm system can not be reconstructed , in many cases . the rest of this paper is organized as follows . in section [ sec : derivation ] , we derive these kinematical variables in detail . in section [ sec : dylhc ] , we summarize the application of these methods to charged drell - yan production and consider the effects of a finite width . in section [ sec : antlers ] , we describe how the application of our kinematical variables to antler processes at the ilc can be used to determine the spin of dark matter . in section [ sec : conclusion ] , we conclude .

hep-th0210246

i

following earlier work on maximally supersymmetric solutions of eleven - dimensional supergravity @xcite , a new maximally supersymmetric solution of type iib supergravity @xcite was found . this paved the way for quantisation in light - cone gauge of superstring theory in a constant flux @xcite . the spectrum consists of a unique massless groundstate on which a tower of massive states is constructed @xcite using creation operators whose masses are of order @xmath2 where @xmath3 @xmath4 is momentum along the @xmath5 light - cone direction and @xmath6 is the flux . this new string background merits further investigation . since the string spectrum in the pp - wave background is now known , string interactions are the next step in the study of the pp - wave background . in this background ten - dimensional lorentz invariance is broken by the non - zero flux , and hence it is no longer possible to set @xmath4 to zero in general scattering amplitudes . this obstruction significantly hinders the vertex operator approach to string interactions . there is only one other known way of studying string interactions in light - cone gauge pioneered by mandelstam @xcite for the bosonic string . in this approach an interaction vertex for the scattering of three strings can be constructed by requiring continuity of string fields on the worldsheet depicted in figure [ fig1 ] . this continuity is enforced by a delta functional on string fields @xmath7 for computational purposes it is essential to express the delta functional in fourier modes @xcite . the interaction vertex may then be written as an exponential of creation operators which enforce the delta functional conditions mode by mode . the functional approach @xcite can be extended to the superstring @xcite . the cubic interaction vertex is a first order in string coupling @xmath8 correction to the free hamiltonian . in order to satisfy the superalgebra with this new hamiltonian , dynamical supercharges also receive corrections . this essential difference from the bosonic string modifies the form of the vertex @xcite . in particular @xcite , in the oscillator basis , the superstring interaction vertex as well as the dynamical supercharges not only have a part exponential in creation operators , but also have a prefactor which is polynomial in creation operators . the exact form of the prefactors is determined by two requirements . firstly , they should not destroy worldsheet continuity enforced by the exponential part of the vertex . secondly , the superalgebra should be satisfied order by order in string coupling . in @xcite an oscillator expression for the flat spacetime superstring vertex and dynamical supercharges was constructed and shown to satisfy all said consistency conditions . in this paper we apply the formalism of @xcite to the pp - wave background in order to construct an interaction vertex and dynamical supercharges in the oscillator basis . we prove explicitly that our expressions satisfy all the aforementioned consistency conditions . we use this vertex to compute various three - point functions in string theory . the pp - wave interaction vertex has also been recently investigated in @xcite . our results differ from the expressions presented in @xcite in several ways . firstly , our bosonic prefactor contains a piece , not present in @xcite , proportional to @xmath9 ( _ cf_. equation ) . this piece is crucial in proving ( see section [ sec4 ] ) that the dynamically generated supercharges and interaction hamiltonian satisfy the pp - wave superalgebra at first order in string coupling . further , without it , three - point functions of states containing only fermionic creation operators would vanish ( see section [ sec52 ] ) . secondly , we also present the oscillator - basis expression for the fermionic prefactor ; in particular , the @xmath10-dependent normalisation ( _ cf_. equation ) is essential in proving that the dynamical superalgebra is satisfied . the exponential part of our vertex also contains a piece linear in fermionic zero - modes @xcite ( _ cf_. equation ) and our expression for the coefficients of the negatively moded bosonic creation operators in the prefactor ( _ cf_. equation ) differs by a factor of @xmath11 @xcite from the original one in @xcite . discrepancy . ] since we prove that our expressions for the dynamical supercharges and hamiltonian satisfy the superalgebra , we believe that the vertex constructed here is the correct one . the pp - wave background can be obtained as a penrose limit @xcite of @xmath12 @xcite . through the ads / cft correspondence @xcite , this has led to a conjectured duality between string theory in the pp - wave background and a double scaling limit of su(@xmath13 ) , @xmath14 super yang - mills theory @xcite . explicitly one considers a sector of ( bmn ) operators with @xmath15 r - charge @xmath16 and conformal dimension @xmath17 such that the difference @xmath18 remains fixed in the limit @xmath16 , @xmath19 , @xmath20 and @xmath21 fixed @xcite . the bmn operators are field theory duals of perturbative string states in the pp - wave background . the sym theory is believed to be expandable in a double series in the effective genus counting parameter @xmath22 and effective coupling @xmath23 @xcite @xmath24 moreover , the finite quantity @xmath18 is a function of @xmath23 @xcite and @xmath22 @xcite . the expansion parameters @xmath25 and @xmath23 are related to string theory parameters by @xcite @xmath26 one of the most appealing aspects of this duality is that both string theory and the bmn sector of gauge theory are simultaneously perturbatively accessible . while some further investigation seems necessary to clarify the exact form of this correspondence , at the planar ( interacting ) field theory two - point function / free string level , there is increasing evidence @xcite that the bmn operators do correspond to the free string theory . the extension of the duality beyond this limit , by considering string interactions and the non - planar sector of ( interacting ) gauge theory respectively , needs further clarification . indeed , the exact notion of the penrose limit in a cft needs to be better understood @xcite . by focusing on a certain subclass of the bmn operators , several proposals for such an extension of the bmn correspondence have been made and were studied in @xcite . in this paper we use our interaction vertex to compute three - point functions for various string states . in particular , for the class of amplitudes studied so far in the literature the term in the cubic vertex proportional to @xmath27 does not contribute and we recover the results of @xcite . we show that so - called impurity preserving three - point functions of states dual to a class of protected operators vanish in string theory , as would be expected by the extension of the duality recently proposed in @xcite . further , we compute three - point functions for string states having only fermionic creation operators . here only the @xmath27 term contributes and these amplitudes may provide a check of any extension of the string - bit dynamical supercharges to terms bilinear in fermions , which are not known at present @xcite . we hope our results can shed some light on the bmn correspondence and that the three - string vertex can be used to investigate the duality further . finally if the bmn duality is to be believed , d - branes in the pp - wave background @xcite should play an important role in gauge theory . this paper is organized as follows . in section [ sec2 ] we briefly review the free string in the pp - wave background and set our notation . in sections [ sec3 ] and [ sec4 ] we construct the superstring vertex in the pp - wave background . section [ sec3 ] focuses on the exponential part of the vertex . most of the results of this section have appeared in @xcite and are included here for completeness . in section [ sec4 ] we construct the prefactors and show that the superalgebra is satisfied with these modified generators . using the vertex , we compute some three - point functions and compare them to field theory in the bmn limit in section [ sec5 ] . several appendices , containing computational details are also included .

we show that these satisfy the pp - wave superalgebra at first order in string coupling . the cubic interaction vertex and dynamical supercharges presented here differ from the expressions previously given in the literature . using this vertex we compute various string theory three - point functions and comment on their relation to gauge theory in the bmn limit . hep - th/0210246 + aei-2002 - 086 + spin-2002/33 + itp-2002/54 0.5 cm 1.0 cm a. pankiewicz and b. stefaski , jr

we construct the cubic interaction vertex and dynamically generated supercharges in light - cone superstring field theory in the pp - wave background . we show that these satisfy the pp - wave superalgebra at first order in string coupling . the cubic interaction vertex and dynamical supercharges presented here differ from the expressions previously given in the literature . using this vertex we compute various string theory three - point functions and comment on their relation to gauge theory in the bmn limit . hep - th/0210246 + aei-2002 - 086 + spin-2002/33 + itp-2002/54 0.5 cm 1.0 cm a. pankiewicz and b. stefaski , jr . 0.5 cm _ max - planck - institut fr gravitationsphysik , albert - einstein institut + am mhlenberg 1 , d-14476 golm , germany _ 0.5 cm _ spinoza institute , university of utrecht + postbus 80.195 , 3508 td utrecht , the netherlands _ 1.0 cm october 2002 1.0 cm

1611.02932

i

the search for a theory of quantum gravity will only be successful if one eventually finds a way to test the candidate theories by experiment or observation @xcite . because of the extremely high energies at which quantum - gravitational effects are expected to be strong , many researchers have looked for features of quantum - gravitational origin in the anisotropies of the cosmic microwave background ( cmb ) ; see , for example , @xcite . these anisotropies are thought to originate during the inflationary phase of the primordial universe from quantum fluctuations of the metric and the scalar inflaton field . hence , they are in a sense already a consequence of the quantum nature of space - time and thus an effect of quantum gravity . what we are going to study in the following is to derive the corrections that arise from quantizing the universe as a whole in the canonical formalism that leads to the wheeler - dewitt equation . one way to obtain corrections to the power spectra of the inflationary scalar and tensor perturbations , which lead to the cmb anisotropies , is to perform a semiclassical approximation of the wheeler - dewitt equation @xcite . this procedure leads to a schrdinger equation with quantum - gravitational correction terms , which can be used to calculate the corrected power spectra . this was analyzed for a simple model containing only non - gauge - invariant perturbations of a scalar field in @xcite . an alternative semiclassical approximation was presented in @xcite and was used to calculate cmb corrections in @xcite . the results of both approaches were essentially the same , with differences only in the numerical factors . we have previously studied the corrections to the power spectra of gauge - invariant scalar and tensor perturbations and have made explicit calculations for the de sitter case @xcite . here , we shall extend our analysis and use a _ generic slow - roll approximation _ that is compatible with all observational data obtained so far and which encompasses a wide range of inflaton potentials . our paper is organized as follows . in sec . ii , we summarize the quantization procedure and the semiclassical approximation . this leads to the equations we use in sec . iii to obtain general expressions for the power spectra without and with quantum gravity corrections . in sec . iv , we introduce the slow - roll approximation , and in sec . v we calculate the corresponding uncorrected power spectra . vi represents the main part of the paper ; here , we calculate the slow - roll power spectra with the quantum - gravitational corrections . in sec . vii , we derive from this the corrections to the spectral index , its running , and to the tensor - to - scalar ratio . we also comment on the observability of the calculated effects and compute the correction for the cmb temperature anisotropies . finally , in sec . viii , we summarize and perform an outlook . we also compare our results with the results obtained in @xcite .

we continue our study on corrections from canonical quantum gravity to the power spectra of gauge - invariant inflationary scalar and tensor perturbations . a semiclassical approximation is applied in order to obtain a schrdinger equation with quantum - gravitational correction terms , from which we calculate the corrections to the power spectra . finally , the effects for the temperature anisotropies in the cosmic microwave background are qualitatively obtained .

we continue our study on corrections from canonical quantum gravity to the power spectra of gauge - invariant inflationary scalar and tensor perturbations . a direct canonical quantization of a perturbed inflationary universe model is implemented , which leads to a wheeler - dewitt equation . for this equation , a semiclassical approximation is applied in order to obtain a schrdinger equation with quantum - gravitational correction terms , from which we calculate the corrections to the power spectra . we go beyond the de sitter case discussed earlier and analyze our model in the first slow - roll approximation , considering terms linear in the slow - roll parameters . we find that the dominant correction term from the de sitter case , which leads to an enhancement of power on the largest scales , gets modified by terms proportional to the slow - roll parameters . a correction to the tensor - to - scalar ratio is also found at second order in the slow - roll parameters . making use of the available experimental data , the magnitude of these quantum - gravitational corrections is estimated . finally , the effects for the temperature anisotropies in the cosmic microwave background are qualitatively obtained .

1611.02932

c

in this paper quantum - gravitational corrections for the power spectra of the gauge - invariant scalar and tensor perturbations have been obtained in the slow - roll regime . in particular , the corrected form for the scalar and tensor power spectrum is given , respectively , by and , where @xmath144 is given in and @xmath151 in in terms of the numerical coefficients @xmath145 , @xmath181 and @xmath182 . these coefficients have been computed by solving the linearized evolution equation for the gaussian width @xmath130 with natural initial data ( in the sense that the initial state , which is constructed as a small deformation of the usual bunch davies vacuum , best describes in this context the expected properties of a freely - evolving mode ) . their values are given in , , and , respectively . the above results generalize the results for the de sitter case obtained earlier in @xcite to a more realistic scenario of slow - roll inflation . in particular , and as one would naively expect , the main part of the correction is due to the de sitter contribution ( which introduces an enhancement of the spectrum ) , whereas the slow - roll part slightly modifies it . let us at this point briefly comment on the results of @xcite , obtained from an alternative expansion of the wheeler - dewitt equation . as in our treatment , the authors find quantum - gravitational correction terms proportional to @xmath242 . nonetheless , their result for the slow - roll approximation is not just a small perturbation of the de sitter case , but can give a comparable contribution for large scales , which can even lead to a power - loss instead of an enhancement of the power spectra . moreover , let us stress that the kind of correction that has been obtained in this analysis being proportional to the factor @xmath243 has appeared in several different approaches in the context of quantum geometrodynamics @xcite . the form of this correction is not completely unexpected . in fact , it is possible to argue , already on dimensional grounds , that @xmath244 is the only non - dimensional parameter that one could use to include perturbatively ( as a power series expansion ) quantum - gravity corrections . furthermore since , due to the background homogeneity , one needs to introduce explicitly a volume ( @xmath245 ) in order to regularize the spatial integral in the action , another dimensionless quantity @xmath246 enters the game . nevertheless , in principle , the power of this latter quantity might have been different and thus it is very interesting to see how the same correction is explicitly realized in different specific models . in the last section we have also analyzed the magnitude of the obtained corrections and the possibility of observing them experimentally . the most difficult issue in order to give a precise estimation is that , due to the regularization of the spatial integral in the action , a length scale needs to be considered . the power spectrum then depends on that length scale and there seems to be nothing physical to fix it . as we have commented , in the main part of the paper , the most reasonable choice is to take it as an infrared cut - off , relating it to the largest observable scale in the cmb . in our case , just to give an approximated estimate , we have chosen it as the length scale of a typical mode that affects the cmb . in particular we have chosen the pivot scale selected by the planck mission . in this way , we have obtained that the corrections for all different parameters of the power spectra ( spectral indices and runnings ) are well inside the current experimental error bars . finally , we have also obtained the qualitative form of the correction induced in the cmb temperature anisotropies by this quantum - gravity effect . the analysis we have performed is valid for large scales ( small @xmath218 ) , for which quantum - gravity effects are expected to be more relevant . in particular , it shows that a correction of the form @xmath226 which , as commented above , seems very generic in this context , leads to a relative correction of the order @xmath230 for the anisotropies , which thus quickly declines with increasing @xmath218 . with this paper we conclude our investigations on quantum - gravitational corrections arising from a canonical quantization of a perturbed universe model using the wheeler - dewitt equation . the effects on large scales we have obtained are , for a reasonable choice of @xmath198 , not observable in the cmb data and since we have used a generic slow - roll model that encompasses a wide range of inflationary models , using more refined models that obey the slow - roll approximation would not enhance the corrections . nonetheless , it is still an open question whether such corrections can be observed in situations where cosmic variance is not present ; for example , in galaxy - galaxy correlation functions .

a direct canonical quantization of a perturbed inflationary universe model is implemented , which leads to a wheeler - dewitt equation . for this equation , we find that the dominant correction term from the de sitter case , which leads to an enhancement of power on the largest scales , gets modified by terms proportional to the slow - roll parameters .

we continue our study on corrections from canonical quantum gravity to the power spectra of gauge - invariant inflationary scalar and tensor perturbations . a direct canonical quantization of a perturbed inflationary universe model is implemented , which leads to a wheeler - dewitt equation . for this equation , a semiclassical approximation is applied in order to obtain a schrdinger equation with quantum - gravitational correction terms , from which we calculate the corrections to the power spectra . we go beyond the de sitter case discussed earlier and analyze our model in the first slow - roll approximation , considering terms linear in the slow - roll parameters . we find that the dominant correction term from the de sitter case , which leads to an enhancement of power on the largest scales , gets modified by terms proportional to the slow - roll parameters . a correction to the tensor - to - scalar ratio is also found at second order in the slow - roll parameters . making use of the available experimental data , the magnitude of these quantum - gravitational corrections is estimated . finally , the effects for the temperature anisotropies in the cosmic microwave background are qualitatively obtained .

0911.0109

i

the aim of this paper is to define , analyze and possibly accustom new distributions in @xmath0 . they are defined with a help of two one - dimensional distributions that first appeared recently , partially in noncommutative context and are defined through infinite products . that is why it is difficult to analyze them straightforwardly using ordinary calculus . one has to refer to some extent to notations and results of so called @xmath1series theory . however the distributions we are going to define and examine have _ purely commutative , classical probabilistic meaning_. they appeared first in an excellent paper of boejko et al . @xcite as a by product of analysis of some non - commutative model . later they also appeared in purely classical context of so called one - dimensional random fields first analyzed by w. bryc at al . in @xcite and @xcite . from these papers we can deduce much information on these distributions . in particular we are able to indicate sets of polynomials that are orthogonal with respect to measures defined by these distributions . those are so called @xmath1hermite and al - salam - chihara polynomials - a generalizations of well known sets of polynomials . thus in particular we know all moments of the discussed one - dimensional distributions . what is interesting about distributions discussed in this paper is that many of their properties resemble similar properties of normal distribution . as stated in the title we consider three families of distributions , however properties of one , called multidimensional @xmath1normal , are main subject of the paper . the properties of the remaining two are in fact only sketched . all distributions considered in this paper have densities . the distributions in this paper are parametrized by several parameters . one of this parameters , called @xmath4 belongs to @xmath5 $ ] and for @xmath6 the distributions considered in this paper become ordinary normal . two out of three families of distributions defined in this paper have the property that all their marginals belong to the same class as the joint , hence one of the important properties of normal distribution . conditional distributions considered in this paper have the property that conditional expectation of a polynomial is also a polynomial of the same order - one of the basic properties of normal distributions . distributions considered in this paper satisfy gebelein inequality -property discovered first in the normal distribution context . furthermore as in the normal case lack of correlation between components of a random vectors considered in the paper lead to independence of these components . finally conditional distribution @xmath7 considered in this paper can be expanded in series of the form @xmath8 where @xmath9 is a marginal density , @xmath10 are orthogonal polynomials of @xmath9 and @xmath11 are also polynomials . in particular if @xmath12 that is when instead of conditional distribution of @xmath13 we consider only distribution of @xmath14 then @xmath15 . in this case such expansion formula it is a so called poisson - mehler formula , a generalization of a formula with @xmath16 being ordinary hermite polynomials and @xmath17 that appeared first in the normal distribution context . on the other hand one of the conditional distributions that can be obtained with the help of distributions considered in this paper is in fact a re - scaled and normalized ( that is multiplied by a constant so its integral is equal to @xmath18 ) askey - wilson weight function . hence we are able to prove some properties of this askey - wilson density . in particular we will obtain a generalization of poisson - mehler expansion formula for this density . to define briefly and swiftly these one - dimensional distributions that will be later used to construct multidimensional generalizations of normal distributions , let us define the following sets @xmath19 & if & \left\vert q\right\vert < 1 \\ \left\ { -1,1\right\ } & if & q=-1% \end{array}% \right . .\]]let us set also @xmath20 and @xmath21 if @xmath22 . sometimes to simplify notation we will use so called indicator functions @xmath23 the two one - dimensional distributions ( in fact families of distributions ) are given by their densities . the first one has density:@xmath24defined for @xmath25 @xmath26 . we will set also @xmath27for @xmath28 considered distribution does not have density , is discrete with two equal mass points at @xmath29 . since this case leads to non - continuous distributions we will not analyze it in the sequel . the fact that such definition is reasonable i.e. that distribution defined by @xmath30 tends to normal @xmath31 as @xmath32 will be justified in the sequel . the distribution defined by @xmath33 @xmath34 will be referred to as @xmath35normal distribution . the second distribution has density : @xmath36 defined for @xmath25 @xmath37 , @xmath38 @xmath39 . it will be referred to as @xmath40conditional normal , distribution . for @xmath6 we set @xmath41 ( in the sequel we will justify this fact ) . notice that we have @xmath42 for all @xmath43 . the simplest example of multidimensional density that can be constructed from these two distribution is two dimensional density @xmath44that will be referred to in the sequel as @xmath45 and two parameters . one playing similar rle to parameter @xmath46 in two - dimensional normal distribution . the other parameter @xmath47 has a different rle . in particular it is responsible for modality of the distribution and of course it defines its support . as stated above , distribution defined by @xmath30 appeared in 1997 in @xcite in basically non - commutative context . it turns out to be important both for classical and noncommutative probabilists as well as for physicists . this distribution has been accustomed i.e. equivalent form of the density and methods of simulation of i.i.d . sequences drawn from it are e.g. presented in @xcite . distribution @xmath48 although known earlier in nonprobabilistic context , appeared ( as an important probability distribution ) in the paper of w. bryc @xcite in a classical context as a conditional distribution of certain markov sequence . in the following section we will briefly recall basic properties of these distributions as well as of so called @xmath1hermite polynomials ( a generalization of ordinary hermite polynomials ) . to do this we have to refer to notation and some of the results of @xmath1series theory . the paper is organized as follows . in section 2 after recall some of the results of @xmath1series theory we present definition of multivariate @xmath1normal distribution . the following section presents main result . the last section contains lengthy proofs of the results from previous section .

we define and study distributions in that we callnormal . for they are really multidimensional normal , for they have densities , compact support and many properties that resemble properties of ordinary multidimensional normal distribution . we also consider some generalizations of these distributions and indicate close relationship of these distributions to askey - wilson weight function i.e. weight with respect to which askey - wilson polynomials are orthogonal and prove some properties of this weight function . in particular we prove a generalization of poisson - mehler expansion formula .

we define and study distributions in that we callnormal . for they are really multidimensional normal , for they have densities , compact support and many properties that resemble properties of ordinary multidimensional normal distribution . we also consider some generalizations of these distributions and indicate close relationship of these distributions to askey - wilson weight function i.e. weight with respect to which askey - wilson polynomials are orthogonal and prove some properties of this weight function . in particular we prove a generalization of poisson - mehler expansion formula .

1101.5783

i

supervised classification , also known as pattern recognition , is a fundamental problem in statistics , as it represents an abstraction of the decision - making problem faced by many applied practitioners . examples include a doctor making a medical diagnosis , a handwriting expert performing an authorship analysis , or an email filter deciding whether or not a message is genuine . classifiers based on nearest neighbours are perhaps the simplest and most intuitively appealing of all nonparametric classifiers . the @xmath0-nearest neighbour classifier was originally studied in the seminal works of @xcite ( later republished as @xcite ) and @xcite , but it retains its popularity today . surprisingly , it is only recently that detailed understanding of the nature of the error probabilities has emerged @xcite . arguably the most obvious defect with the @xmath0-nearest neighbour classifier is that it places equal weight on the class labels of each of the @xmath0 nearest neighbours to the point @xmath5 being classified . intuitively , one would expect improvements in terms of the misclassification rate to be possible by putting decreasing weights on the class labels of the successively more distant neighbours . the first purpose of this paper is to describe the asymptotic structure of the difference between the misclassification rate ( risk ) of a weighted nearest neighbour classifier and that of the optimal bayes classifier for classification problems with feature vectors in @xmath6 . theorem [ thm : main ] in section [ sec : main ] below shows that , subject to certain regularity conditions on the underlying distributions of each class and the weights , this excess risk ( or _ regret _ ) asymptotically decomposes as a sum of two dominant terms , one representing bias and the other representing variance . for simplicity of exposition , we will deal initially with binary classification problems , though we also indicate the appropriate extension to general multicategory problems . our second contribution , following on from the first , is to derive the vector of non - negative weights that is asymptotically optimal in the sense of minimising the misclassification rate ( cf . theorem [ thm : optweights ] ) . in fact this asymptotically optimal weight vector has a relatively simple form : let @xmath7 denote the sample size and let @xmath8 denote the weight assigned to the @xmath9th nearest neighbour ( normalised so that @xmath10 . then the optimal choice is to set @xmath11 ( an explicit expression for @xmath12 is given in ( [ eq : kstar ] ) below ) and then let @xmath13 & \mbox{for $ i=1,\ldots , k^*$ } \\ 0 & \mbox{for $ i = k^*+1,\ldots , n$. } \end{array } \right.\ ] ] thus , in the asymptotically optimal weighting scheme , only a proportion @xmath14 of the weights are positive . the maximal weight is almost @xmath15 times the average positive weight , and the discrete distribution on @xmath16 defined by the asymptotically optimal weights decreases in a concave fashion when @xmath4 , in a linear fashion when @xmath17 and in a convex fashion when @xmath18 ; see figure [ fig : decweights ] . when @xmath1 is large , about @xmath19 of the weights are above the average positive weight . [ t][t]0 [ t][t]0.002 [ t][t]0.004 [ t][t]0.006 [ t][t]0.008 [ t][t]0.01 [ t][t]0 [ t][t]20 [ t][t]40 [ t][t]60 [ t][t]80 [ t][t]100 [ t][t]neighbour order [ t][t]optimal weight another consequence of theorem [ thm : optweights ] is that @xmath20 is bigger by a factor of @xmath21 than the asymptotically optimal choice of @xmath0 for traditional , unweighted @xmath0-nearest neighbour classification . it is notable that this factor , which is around 1.27 when @xmath4 and increases towards 2 for large @xmath1 , does not depend on the underlying populations . this means that there is a natural correspondence between any unweighted @xmath0-nearest neighbour classifier and one of optimally weighted form , obtained by multiplying @xmath0 by this dimension - dependent factor to obtain the number @xmath22 of positive weights for the weighted classifier , and then using the weights given in ( [ eq : optweights ] ) with @xmath22 replacing @xmath20 . in corollary 3 we describe the asymptotic improvement in the excess risk that is attainable using the procedure described in the previous paragraph . since the rate of convergence to zero of the excess risk is @xmath23 in both cases , the improvement is in the leading constant , and again it is notable that the asymptotic improvement does not depend on the underlying populations . the improvement is relatively modest , which goes some way to explaining the continued popularity of the ( unweighted ) @xmath0-nearest neighbour classifier . nevertheless , for @xmath24 , the improvement in regret is at least 5% , though it is negligible as @xmath3 ; the greatest improvement occurs when @xmath2 , and here it is just over 8% . see figure [ fig : asympimp ] . [ t][t]0.90 [ t][t]0.92 [ t][t]0.94 [ t][t]0.96 [ t][t]0.98 [ t][t]1.00 [ t][t]1.02 [ t][t]0 [ t][t]10 [ t][t]20 [ t][t]30 [ t][t]40 [ t][t]50 [ t][t]@xmath1 [ t][t]regret ratio another popular way of improving the performance of a classifier is by _ bagging _ @xcite . short for ` bootstrap aggregating ' , bagging involves combining the results of many empirically simulated predictions . empirical analyses , e.g. @xcite , have reported that bagging can result in improvements over unweighted @xmath0-nearest neighbour classification . moreover , as explained by @xcite , understanding the properties of the bagged nearest neighbour classifier is also of interest because they provide insight into _ random forests _ @xcite . random forest algorithms have been some of the most successful ensemble methods for regression and classification problems , but their theoretical properties remain relatively poorly understood . when bagging the nearest neighbour classifier , we can draw resamples from the data either with- or without - replacement . we treat the ` infinite simulation ' case , where both versions take the form of a weighted nearest neighbour classifier with weights decaying approximately exponentially on successively more distant observations from the point being classified @xcite . the crucial choice is that of the resample size , or equivalently the sampling fraction , i.e. the ratio of the resample size to the original sample size . in section [ sec : bagged ] , we describe the asymptotically optimal resample fraction ( showing in particular that it is the same for both with- and without - replacement sampling ) and compare its regret with those of the weighted and unweighted @xmath0-nearest neighbour classifiers . finally , in section [ sec : negweights ] , we consider the problem of choosing optimal weights without the restriction that they should be non - negative . the situation here is somewhat analogous to the use of higher order kernels for classifiers based on kernel density estimates of each of the population densities . in particular , subject to additional smoothness assumptions on the population densities , we find that powers of @xmath7 arbitrarily close to the ` parametric rate ' of @xmath25 for the excess risk are attainable . all proofs are deferred to the appendix . classification has been the subject of several book - length treatments , including @xcite , @xcite and @xcite . in particular , classifiers based on nearest neighbours form a central theme of @xcite . the review paper by @xcite contains 243 references and provides a thorough survey of the classification literature up to 2005 . more recently , @xcite have discussed the relative merits of _ plug - in _ classifiers ( a family to which weighted nearest neighbour classifiers belong ) and classifiers based on _ empirical risk minimisation _ , such as support vector machines @xcite . weighted nearest neighbour classifiers were first studied by @xcite ; see also @xcite . @xcite proved that if @xmath26 as @xmath27 and @xmath28 for some @xmath29 with @xmath30 as @xmath27 , then risk of the weighted nearest neighbour classifier converges to the risk of the bayes classifier ; see also @xcite . as mentioned above , this work attempts to study the difference between these risks more closely . weighted nearest neighbour classifiers are also related to classifiers based on kernel estimates of each of the class densities ; see for example the review by @xcite , as well as @xcite . the @xmath23 rates of convergence obtained in this paper for non - negative weights are the same as those obtained by @xcite under similar twice - differentiable conditions with second - order kernel estimators of the class densities . @xcite proved that in a certain sense this is the minimax optimal rate , though his assumptions and context are slightly different from what is studied here . further related work includes the literature on _ highest density region _ or _ level set _ estimation @xcite . @xcite and @xcite proved an analogous result for the bagged nearest neighbour classifier to the @xcite result described in the previous paragraph . more precisely , if the resample size @xmath31 used for the bagging diverges to infinity , and @xmath32 as @xmath27 , then the risk of the bagged nearest neighbour classifier converges to the bayes risk . note that this result does not depend on whether the resamples are taken with or without replacement from the training data . @xcite have recently proved a striking rate of convergence result for the bagged nearest neighbour estimate ; this is described in greater detail in section [ sec : bagged ] .

we derive an asymptotic expansion for the excess risk ( regret ) of a weighted nearest - neighbour classifier . this allows us to find the asymptotically optimal vector of non - negative weights , which has a rather simple form . the improvement is greatest when , but thereafter decreases as . finally , we argue that improvements in the rate of convergence are possible under stronger smoothness assumptions , provided we allow negative weights .

we derive an asymptotic expansion for the excess risk ( regret ) of a weighted nearest - neighbour classifier . this allows us to find the asymptotically optimal vector of non - negative weights , which has a rather simple form . we show that the ratio of the regret of this classifier to that of an unweighted-nearest neighbour classifier depends asymptotically only on the dimension of the feature vectors , and not on the underlying populations . the improvement is greatest when , but thereafter decreases as . the popular bagged nearest neighbour classifier can also be regarded as a weighted nearest neighbour classifier , and we show that its corresponding weights are somewhat suboptimal when is small ( in particular , worse than those of the unweighted-nearest neighbour classifier when ) , but are close to optimal when is large . finally , we argue that improvements in the rate of convergence are possible under stronger smoothness assumptions , provided we allow negative weights . * key words : * bagging , classification , nearest neighbours , weighted nearest neighbour classifiers .

1301.4391

i

in this paper we focus on the a posteriori error control and adaptivity for fully discrete crank - nicolson finite element ( cnfe ) schemes for the general form of linear schrdinger equation : @xmath1 $ , } & \\ & u=0 & & \quad\mbox{on $ \partial \varomega\times ( 0,t]$ , } & \\ & u(\cdot,0)=u_0 & & \quad\mbox{in $ \varomega$ } , & \end{aligned } \right.\ ] ] where @xmath2 is a convex `` polygonal '' domain in @xmath3 , @xmath4 , with boundary @xmath5 , and @xmath6 in , @xmath7 is a positive constant , @xmath8\to\mathbb{r}$ ] and @xmath9\to\mathbb{c}$ ] are given functions and @xmath10 is a given initial value . a special case of is the so - called linear schrdinger equation in the semiclassical regime : @xmath11 with high frequency initial data . it is clear that can be obtained from by setting @xmath12 , @xmath13 and @xmath14 . in , @xmath15 ( @xmath16 ) is the scaled planck constant , @xmath17 is an @xmath18 time - dependent potential and @xmath19 is the wave function . the wave function @xmath19 is used to define primary physical quantities , called _ observables _ ( @xcite ) , such as the _ position density _ , @xmath20 and the _ current density _ , @xmath21 problems related to are of great interest in physics and engineering . however , the solution of is complicated from the theoretical as well as the numerical analysis point of view . it is well known that for @xmath15 small ( close to zero ) , the solution of oscillates with wavelength @xmath22 , preventing @xmath19 to converge strongly as @xmath23 . because of this , standard numerical methods fail to correctly approximate @xmath19 and the observables , unless very fine mesh sizes and time steps are used . in particular , previous works ( cf . , e.g. , @xcite ) suggested that for standard finite element ( fe ) methods there is a very restrictive dispersive relation connecting the mesh sizes ( space and time ) with parameter @xmath15 ; cf . , e.g. , below . this restrictive dispersive relation can be relaxed using the so - called time - splitting spectral methods , introduced earlier by bao , jin & markowich in @xcite , for the approximation of the solution of . in this paper , our goal is to show that constructing adaptive algorithms based on rigorous a posteriori error control leads to cnfe schemes which are competitive to the best available methods for the approximation of the solution ( and the observables ) of the semiclassical schrdinger equation , and in general of linear schrdinger equations of the form . it also permits , for the first time , realistic computations for rough potentials for the linear schrdinger equation in the semiclassical regime.to achieve our goal , in the current work we : 1 . provide rigorous a posteriori error analysis for for cnfe approximations using fe spaces that are allowed to change in time ; 2 . study the advantages of adaptivity through the obtained estimators for the efficient error control of . optimal order a posteriori error estimates for the heat equation for cnfe schemes with fe spaces that are allowed to change in time have been derived very recently by bnsch , karakatsani & makridakis in @xcite . however the extension of those ideas from the simple heat equation to the linear schrdinger equation is of increased difficulty due to the complex - value and multiscale nature of the problem . because of this , novel ideas and techniques are introduced . more precisely , our main contributions are : 1 . derivation of optimal order a posteriori error bounds in the @xmath0norm for cnfe schemes for . the fact that the analysis includes time - dependent potentials , makes the problem more challenging since there are no rigorous results for schrdinger equations for such potentials . in addition the existing literature on a posteriori error analysis for problems with time - dependent operators of the form @xmath24 is quite limited . to the best of our knowledge , only in @xcite the authors consider similar operators . moreover the derived estimates hold for @xmath25type potentials as well , in contrast to the existing literature . in particular , existing results require smooth @xmath26type potentials . however , this regularity requirement on the potential is rather restrictive from applications point of view . including @xmath18 time - dependent potentials in the analysis is important for another reason : it can be considered as the first step for the a posteriori error control of nonlinear schrdinger ( nls ) equations . more precisely , the relaxation scheme introduced by besse in @xcite suggests that a posteriori error bounds for linear schrdinger equations with @xmath25type time - dependent potentials is essential for the efficient approximation of the solution of certain nls equations . introduction of a _ novel elliptic reconstruction _ leading to upper bounds that do not involve the global @xmath25norm of @xmath27 , and thus , to bounds that do reflect the physical properties of the problem.the elliptic reconstruction was developed by makridakis & nochetto in @xcite to derive optimal order @xmath28 a posteriori error bounds for fe spatial discrete schemes for the heat equation using energy techniques . a straightforward generalization of this notion of the elliptic reconstruction to schrdinger equations leads to estimates that involve the @xmath25norm of the potential . consequently , the obtained estimates are practically useless and adaptivity is inefficient , even in the simplest case of constant potentials . therefore , proposing a modified elliptic reconstruction based on the physical properties of the problem under consideration is crucial for the efficient error control of ( and so ) . additionally , the new ideas developed for this purpose might be useful for other problems as well , such as convection - diffusion or reaction - diffusion problems a detailed numerical study on the reliability and robustness of the a posteriori estimators through a _ time - space adaptive algorithm_. our starting point is the adaptive algorithm proposed in @xcite , adapted to the linear schrdinger equation , . the a posteriori estimators derived in this work are on the solution @xmath19 of . however , in many applications observables like the position density , or the current density are far more important than the solution itself . thus , we introduce an appropriate modification of the a posteriori estimators and the adaptive algorithm . this modification is based on a heuristic idea and the results concerning the observables are very encouraging . overall , the adaptive algorithm reduces the computational cost substantially and provides efficient error control of @xmath19 and the observables for small values of the planck constant @xmath15 . it is very difficult to obtain such results via standard techniques and without adaptivity . we point out that our purpose is not to prove convergence and optimality of the considered time - space adaptive algorithm , but rather to show that adaptivity based on rigorous a posteriori error control can be proven beneficial for the approximation of the solution ( and the observables ) of the linear semiclassical schrdinger equation . in addition , it is to be emphasized that as long as the adaptive algorithm converges , we can guarantee rigorously , based on the a posteriori error analysis , that _ total error remains below a given tolerance_. for parabolic problems , a number of adaptive algorithms exists in the literature ; cf . , e.g. , @xcite and the references therein . however , convergence and optimality of time - space adaptive algorithms are very delicate and difficult issues . in the literature exists only one proven convergent time - space adaptive algorithm for evolution problems and can be found in @xcite . this algorithm is appropriate for the heat equation and backward euler fe schemes and it is not clear how to generalize it to other problems and higher order in time methods . despite the fact that problem ( and thus ) is linear , a posteriori error bounds and adaptive algorithms for linear schrdinger equations are very limited in the literature . in particular , a posteriori error estimates in the @xmath0norm for fully discrete cnfe schemes have been proven earlier by drfler in @xcite ; these estimates are first order accurate in time , thus not optimal . using these estimates , drfler also proposes an adaptive algorithm in @xcite . in @xcite ( see also @xcite ) , we considered only time - discrete approximations and we managed to prove optimal order a posteriori error estimates for in the @xmath28 and @xmath29norms . this was achieved using the crank - nicolson reconstruction proposed by akrivis , makridakis & nochetto in @xcite . similar estimates for , using an alternative reconstruction , proposed by lozinski , picasso & prachittham in @xcite , can be found in @xcite . to the best of our knowledge , optimal order a posteriori error estimates for fully discrete cnfe schemes do not exist in the literature . some preliminary results to that direction can be found in @xcite . however , the a posteriori estimators derived in @xcite are scaled by the global @xmath25norm of @xmath27 . hence , as already mentioned , the derived estimators do not reflect the physical properties of the problem , which makes adaptivity through these estimates not reliable . a posteriori error estimates in the @xmath0norm have been proven earlier in @xcite for uniform partitions and the time - splitting spectral methods for the linear schrdinger equation in the semiclassical regime . in @xcite , only the one - dimensional case in space is studied and the analysis , as in @xcite , permits only time - independent potentials , without being obvious how the theory can be extended to time - dependent potentials . in addition , the time - spectral methods require smooth potentials ; the particular analysis is not applicable for @xmath25type potentials . the analysis of the current paper is based on the introduction of appropriate space - time reconstructions . such reconstructions for cnfe methods and fe spaces that are allowed to change in time were introduced , for the first time , very recently , by bnsch , karakatsani & makridakis in @xcite , for the proof of optimal order a posteriori estimates in the @xmath0norm for the heat equation . to define those time - space reconstructions , the authors combined the idea of the elliptic reconstruction in @xcite with the crank - nicolson reconstruction of @xcite . the notion of the elliptic reconstruction has also been used earlier in @xcite and @xcite for the derivation of optimal order a posteriori error estimates for backward euler fe schemes for the heat and the wave equation , respectively . the reconstruction technique is a useful tool for deriving optimal order a posteriori error bounds ; usually , this is not feasible via a direct comparison of the exact and the numerical solution ; cf . , e.g. , @xcite . in our context , time - space reconstructions can be defined through the novel elliptic reconstruction we introduce and the crank - nicolson reconstruction of @xcite . more precisely , the paper is organized as follows . in section [ prelim ] , we introduce notation , the variational formulation of problem and the fully discrete scheme . we propose the novel elliptic reconstruction and discuss its properties . with the aim of this new elliptic reconstruction , we then define appropriate time - space reconstructions . the main theoretical results are stated in section [ apost ] , where the a posteriori analysis is developed and optimal order error bounds are derived using energy techniques , residual - type error estimators and the properties of the reconstructions . the two last sections are devoted to the numerical investigation of the efficiency of the estimators . in particular , in section [ unif ] , we validate numerically the optimal order of convergence of the estimators using uniform partitions . for the linear schrdinger equation in the semiclassical regime , we verify numerically that the estimators have the expected behavior with respect to the scaled parameter @xmath15 . finally , in section [ adapt ] , we appropriately modify and apply to the one - dimensional semiclassical schrdinger equation a time - space adaptive algorithm described in @xcite ( see also @xcite ) . we further develop the algorithm and we make it applicable for the approximation not only of the exact solution @xmath19 but also for the observables , and we discuss in detail the benefits of adaptivity for equations of the form .

the derivation of the estimators is based on a novel elliptic reconstruction that leads to estimates which reflect the physical properties of schrdinger equations . the final estimates are obtained using energy techniques and residual - type estimators . , we further develop and analyze an existing time - space adaptive algorithm to the cases of schrdinger equations . the adaptive algorithm reduces the computational cost substantially and provides efficient error control for the solution and the observables of the problem , especially for small values of the planck constant .

we derive optimal order a posteriori error estimates for fully discrete approximations of linear schrdinger - type equations , in thenorm . for the discretization in time we use the crank - nicolson method , while for the space discretization we use finite element spaces that are allowed to change in time . the derivation of the estimators is based on a novel elliptic reconstruction that leads to estimates which reflect the physical properties of schrdinger equations . the final estimates are obtained using energy techniques and residual - type estimators . various numerical experiments for the one - dimensional linear schrdinger equation in the semiclassical regime , verify and complement our theoretical results . the numerical implementations are performed with both uniform partitions and adaptivity in time and space . for adaptivity , we further develop and analyze an existing time - space adaptive algorithm to the cases of schrdinger equations . the adaptive algorithm reduces the computational cost substantially and provides efficient error control for the solution and the observables of the problem , especially for small values of the planck constant .

cond-mat9705307

i

the low temperature behavior of the strongly interacting quantum liquid @xmath0he has been a subject of experimental and theoretical research for decades . in 1908 helium was first liquified by kamerlingh onnes and in 1911 he discovered a sharp maximum in the density at what is now commonly called the @xmath1-point.@xcite after that , a number of macroscopic quantum phenomena like superfluid flow , second sound , the fountain effect and quantized vortices were observed . phenomenological theories were developed and justified from a microscopic point of view.@xcite also , the famous kosterlitz - thouless transition was first observed in thin superfluid helium films.@xcite in the solid phase of @xmath0he , which is reached only at low temperature and high pressure ( cf . [ fig1 ] ) , one also expects to observe macroscopic quantum phenomena because of the large zero point vibration of the atoms about their equilibrium position.@xcite because of this , solid helium has been termed a quantum solid . in such a solid , the interstitials and vacancies are effectively delocalized due to their ability to tunnel through the potential barriers . at low temperatures these point defects then form a weakly interacting bose gas . furthermore , the large zero point motion results in an unusually rapid exchange rate of nearest neighbour atoms , which may lead to large ring exchanges between the helium atoms.@xcite bose - einsein condensation of the defects or exchange processes of the lattice atoms may then open two routes to a new phase of matter at low temperature in which long - range crystalline and superfluid order coexist . this is called the supersolid phase . @xcite theoretically the existence of such a phase has since long been anticipated.@xcite however , it was only recently claimed to have been observed experimentally that three dimensional solid @xmath0he is a spatially ordered superfluid , or supersolid , at sufficiently low temperatures and densities . the experiments leading to this claim were performed by lengua and goodkind , who measured the attenuation and velocity of sound in solid @xmath0he for relatively high purities and low atomic densities of the quantum crystal.@xcite the temperature dependence of the attenuation revealed a coupling to thermally activated excitations , consistent with the existence of a propagating mode in the gas of point defects that is expected to be present in a quantum crystal.@xcite furthermore , assuming the speed of sound of the defect mode to depend on the density of defects in the same way as in a dilute bose gas , they found a relation between the temperature dependence of the phase velocity and that of the defect density . to consistently interpret their data they then had to assume a macroscopic population of the zero momentum state of the point defects , i.e. a bose - einstein condensation of the point defects . thus the phase diagram of @xmath0he in three dimensions would be qualitatively given by fig . [ fig1 ] . following a certain trajectory in this phase diagram , @xmath0he may undergo a transition from the normal phase to the superfluid phase at some temperature @xmath2 and subsequently from the superfluid to the supersolid phase at a temperature @xmath3 . as mentioned above , the possibility of superfluid flow in a solid has since long been anticipated theoretically . andreev and lifschitz were the first to attempt to derive the hydrodynamics of a supersolid by including the effect of bose - einstein condensation of the defects on the hydrodynamics of an ideal crystal.@xcite in addition to this pioneering work , liu has more recently presented a thorough discussion of the andreev and lifschitz hydrodynamics.@xcite however , it was pointed out by martin _ et.al . _ that the conventional treatment of the hydrodynamic equations for a classical crystal , which does nt include defects , is neccesarily incomplete since it yields the wrong number of modes.@xcite they identified the missing mode as a mode in the defect density . this implies that the hydrodynamics of andreev and lifshits is also incomplete , because it does not include the non - condensed defects and as a result does not lead to the required defect mode in the normal state of the crystal . in addition , martin _ et.al . _ assume diffusive dynamics for their defect mode . this seems to be appropriate for a classical but not for a quantum crystal , where the defect mode is expected to be a propagating mode , as is confirmed by the experiments of lengua and goodkind . recently stoof _ et.al._ , in respons to experiments with submonolayer superfluid helium film,@xcite derived the hydrodynamic equations for an isotropic supersolid in two dimensions which did include propagating behavior of the crucial defect mode.@xcite moreover , the longitudinal part of the solid hydrodynamics derived by these authors turns out to be identical to the system of two coupled wave equations that lengua and goodkind used to accurately model their data . however , to apply these promising results to the experiments with solid helium , we have to extend them in two ways . first of all we have to consider a three dimensional system , and second of all we have to take into account the anisotropy of solid @xmath0he , which is a hexogonally closed packed ( hcp ) crystal . thus we hope to justify from a microscopic point of view the phenomenological equations that succesfully explained the propagation of sound in solid @xmath0he and led to the first claim of a supersolid phase in this system . the paper is organized as follows . in sec . [ section2 ] the hydrodynamic equations describing a normal solid with point defects will be derived . this is achieved by deriving an action describing a solid with dislocations , using methods developed by kleinert.@xcite from this action we obtain the interaction between phonons and a point defect , by seeing the point defect as a limiting case of a dislocation . also , a more microscopic point of view is presented and dissipation is included . in sec . [ section3 ] we then add a superfluid degree of freedom to our hydrodynamic equations in the usual way and in sec . [ section4 ] we discuss the experiment by lengua and goodkind in the light of our results . it should be noted that in order to understand this experiment it is not neccessary to include temperature flucuations into our considerations and we will neglect them in the rest of this article . we conclude with a discussion and outlook in sec . [ section5 ] .

in particular , the usual hydrodynamics is modified in such a way that it leads to the presence of a propagating instead of a diffusive defect mode . the former is appropriate for a quantum crystal and observed in recent experiments . the observation of these additional modes is a clear experimental signature of the supersolid phase .

we derive the hydrodynamic equations of motion of solid and supersolidhe , that describe the collective modes of these phases . in particular , the usual hydrodynamics is modified in such a way that it leads to the presence of a propagating instead of a diffusive defect mode . the former is appropriate for a quantum crystal and observed in recent experiments . furthermore , we find that in supersolid helium there are two additional modes associated with the superfluid degrees of freedom . the observation of these additional modes is a clear experimental signature of the supersolid phase .

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glassy systems of extremely diverse types exist in nature . they all share several common features like a very slow , non - equilibrium dynamics . the development of a full theoretical description of the glassy phase is one of the most important challenges in condensed matter physics . a variety of techniques , that range from scaling arguments to mean - field approaches have been , and are still used , with the aim of attempting a satisfactory description of the glassy properties . one of these techniques is due to thouless , anderson and palmer ( tap ) @xcite , who introduced an approach to classical disordered systems based on the study of a free - energy landscape . the key object is the legendre transform of the free - energy @xmath1 with respect to a number of order parameters that are sufficient to describe the transition and the different phases in the system . this function behaves as an effective potential whose minima represent different possible phases . in a classical fully - connected ising model only one order parameter is needed , the global magnetization @xmath2 . the two possible minima of @xmath3 correspond to the two possible states of positive and negative magnetization , @xmath4 . focusing on the sherrington - kirkpatrick ( sk ) mean - field model for spin - glasses , tap showed that _ all _ the local magnetizations @xmath5 , @xmath6 , have to be included in order to derive the relevant free - energy landscape . the extremization condition of the tap free - energy on the @xmath7s leads to the tap equations . it was soon after realized by bray and moore @xcite that the number of solutions to the tap equations for the sk model is exponential in the number of spins in the system , for temperatures below the spin - glass transition @xcite . a very useful alternative derivation of the tap equations was given by plefka @xcite who showed that these equations can also be obtained from a power expansion of the gibbs potential up to second order in the exchange couplings . the advantage of this derivation is twofold : it allows to show convergence of the power expansion for all temperatures and it is easily applicable to other mean - field glassy models . moreover , georges and yedidia @xcite showed that the high temperature series , at fixed order parameter , of the free - energy can be used to derive tap - like equations , and its corrections , for models in finite dimensions or , equivalently , with finite range interactions . the connection between the tap approach and the more standard analysis of the partition function of a disordered model has been exhibited by de dominicis and young @xcite who showed that , for the sk model , one recovers the equilibrium results of the replica or the cavity method @xcite via weighted boltzmann averages over solutions of the tap equations . more recently , the tap approach has been applied to other classical disordered models . in particular , two models that we shall discuss in the following , the spherical and ising @xmath0 spin - glass models @xcite and the ghatak - sherrington ( gs ) model @xcite have been analyzed with this method @xcite . glassy systems , and in particular disordered ones , are characterized by having a very slow dynamics with non equilibrium effects at low temperatures @xcite . mean - field models , like the the spherical @xmath0 spin - glass model @xcite or the sk spin - glass @xcite , capture this phenomenology . the dynamic solution for the evolution starting from random initial conditions , that represent a quench from high temperatures analytically , is intimately connected to the structure and organization of tap solutions . one of the most striking results of the dynamic analysis of @xmath0 spin - glass - like models is that the energy density ( and other one time - quantities ) converges asymptotically to the energy density of high lying solutions of the tap equations . this level has been called _ threshold_. the energy - density in equilibrium is different . this and other related results suggest that an interpretation of the dynamics in terms of a motion in a tap free - energy landscape can be given @xcite . the generalization of the tap approach to dynamics that has been developed in @xcite allows one to make this statement precise : the evolution is determined by a gradient - descent in the tap free - energy landscape with the most important addition of non - markovian terms . usually , glasses can be analyzed with a fully classical approach since their transition temperatures are rather high . nevertheless , in many cases of great interest the critical temperature can be lowered by tuning another external parameter and quantum fluctuations become very important . this is the case for the insulating magnetic compound liho@xmath8y@xmath9f@xmath10 , that is an experimental realization of a quantum spin - glass , and presently receives much attention @xcite . other examples where glassy properties in the presence of quantum fluctuations have been observed are mixed hydrogen bonded ferro - antiferro electric crystals @xcite , interacting electron systems @xcite , cuprates like la@xmath11sr@xmath8cuo@xmath10 @xcite , amorphous insulators @xcite , etc . the quantum fluctuations in liho@xmath8y@xmath9f@xmath10 can be controlled by tuning the strength of an external field that is transverse to the preferred direction of the randomly located magnetic impurities . after a series of experiments presented in @xcite the authors conclusions are : ( 1 ) the samples undergo a paramagnetic to spin - glass transition in the @xmath12 plane , where @xmath13 and @xmath14 is the strength of the transverse field . ( 2 ) the transition is of second order ( in the thermodynamic sense ) close to the classical critical point @xmath15 but crosses over to first order close to the quantum critical point @xmath16 . ( 3 ) the system undergoes out of equilibrium dynamics in the glassy phase as demonstrated by the fact that the dynamics strongly depends on the preparation of the sample for all subsequent times explored experimentally . the theoretical study of quantum spin - glasses started with bray and moore s analysis of the equilibrium properties of the fully connected heisenberg model @xcite . in this article , bray and moore introduced a path - integral representation in imaginary time of the partition function that they analyzed with the replica trick . many articles on the equilibrium of this , and related , mean - field models have been published since @xcite . the static properties of low dimensional models have been studied and it has been shown that , in finite dimensions , griffiths - mccoy singularities are very important close to the quantum critical point @xcite . in all these models , the transition from the paramagnetic to the spin - glass phase has been reported to be of second order throughout . in most classical disordered models studied so far the transition from the disordered to the ordered phase is of second order in the thermodynamic sense . in the exact solution of the sk model , the spin - glass order parameter @xmath17 is continuous at the transition which is of second order in the thermodynamic sense @xcite . in other classical glassy models like the potts glass @xcite or the spherical @xcite and ising @xcite @xmath0 spin - glasses , the order parameter jumps at the transition which , however , is still of second - order in the thermodynamic sense since there is neither a jump in the susceptibility nor a latent heat . a classical model that exhibits a first - order transition is the anisotropic @xmath0 spin - glass , @xmath18 , in which the spins take integer values between @xmath19 and @xmath20 and there is an extra term in the hamiltonian @xmath21 , proportional to a coupling constant @xmath22 , that controls the crystalline tendency . in this case , a crossover from a second - order to a first - order thermodynamic transition in the plane @xmath23 has been exhibited in the exact solution @xcite . the classical ghatak - sherrington ( gs ) model @xcite is another candidate to exhibit a second to first order crossover in the thermodynamic transition . it is the anisotropic extension of the sk model , or the @xmath24 limit of the previous model . in this case , a crossover from a second order to a first order transition in the plane @xmath23 has been exhibited in an _ approximate _ solution ( one step replica symmetry breaking ) @xcite . the exact solution has not been derived yet and it is then not well established if it has a true first - order thermodynamic transition . in quantum problems , first order transitions have been reported in three models . the first one is the so - called `` fermionic ising spin - glass '' analyzed by oppermann and collaborators @xcite . this model , however , is thermodynamically equivalent to the classical gs model discussed above @xcite . the other two models are very similar indeed and they are different ways of extending the classical spherical @xmath0 spin - glass model @xcite to include quantum fluctuations . in one case , the continuous spins are generalized to @xmath25 component vectors and a global spherical constraint as well as commutation relations are imposed @xcite . the other one uses the fact that the spherical @xmath0 spin - glass model can be interpreted as a particle moving in an infinite dimensional hyper - sphere with a random potential . quantization is then done by imposing commutation relations between coordinates and momenta @xcite . the latter can also be interpreted as an extension of the a quantum rotor model @xcite that includes @xmath0 interactions . the relation between the critical properties of the quantum versions of @xmath0 spin - glass models and the experiments in @xcite has been put forward in @xcite . in addition , the connection between the static calculation supplemented by the marginality condition and the analysis of the out of equilibrium dynamics in contact with an environment developed in @xcite was also discussed in @xcite . however , the reason why the transition changes from second to first order close to the quantum critical point was not clear from this analysis . it is one of the aims of this article to clarify this point , and study to what extent one can claim it to be general , with the use of the tap approach . quantum tap equations for the sk model in a transverse field have been presented by ishii and yamamoto @xcite and cesare _ et al _ @xcite . the former use a perturbative expansion of the free - energy in the strength of the transverse field , and then follow closely tap s techniques ; the latter implement a cavity method . the tap equations derived by rehker and oppermann @xcite for the fermionic spin - glass model coincide with the ones presented by yokota @xcite for the classical gs model since these two models are thermodynamically equivalent @xcite . hence , our aim is twofold . on the one hand we want to present a quantum extension of the tap approach to the statistical properties of disordered systems . thus , after a short revision of the classical tap approach in section [ introtap ] , we discuss in section [ sec1:formalism ] the derivation of the quantum tap free - energy and tap equations using a general approach that extends the ones developed by plefka @xcite and georges and yedidia @xcite . the advantage with respect to previous derivations of quantum tap equations is that this method can be applied to any quantum disordered model and it allows to obtain the tap equations as well as the tap free energy . in section [ p - spin ] we present , as an example , the tap free - energy and tap - equations for the quantum extension of the @xmath0 spin spherical spin - glass model studied in @xcite . we show that the tap equations can be easily related to the equations for the order parameter in the matsubara replica approach and also to some of the equations appearing in the real - time dynamic approach . the tap analysis of this model furnishes a benchmark to study the generalization to the quantum case of the methods and interpretations developed for classical systems . section [ transition ] is devoted to the second aim of this article . via the tap approach we show that the same type of phase diagram naturally emerges for all systems having a discontinuous phase transition in their classical limit ( these are models solved by a one - step replica symmetry breaking ansatz within the replica analysis ) . in particular we relate the first - order transition close to the quantum critical point to the structure of metastable states . finally , we present our conclusions in section [ conclu ] .

in particular , we argue that a crossover from a second order thermodynamic transition close to the classical critical point to a first order thermodynamic transition close to the quantum critical point is to be expected in a large class of systems . lpt - ens/0033 , lpthe/0034 .

we derive thouless - anderson - palmer ( tap ) equations for quantum disordered systems . we apply them to the study of the paramagnetic and glassy phases in the quantum version of the spherical spin - glass model . we generalize several useful quantities ( complexity , threshold level , etc . ) and various ideas ( configurational entropy crisis , etc ) , that have been developed within the classical tap approach , to quantum systems . the analysis of the quantum tap equations allows us to show that the phase diagram ( temperature - quantum parameter ) of the spin - glass model should be generic . in particular , we argue that a crossover from a second order thermodynamic transition close to the classical critical point to a first order thermodynamic transition close to the quantum critical point is to be expected in a large class of systems . lpt - ens/0033 , lpthe/0034 .

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now that we have become accustomed to referring to the hot gas in groups and clusters of galaxies as `` cooling flows '' , no cooling gas has been observed . although energy is lost by x - ray emission , the absence of spectral evidence for gas cooling to low temperatures has been well - documented in many observational papers ( e.g. peterson et al . 2001 ; xu et al . the hot gas in groups and clusters of galaxies may cool , but at a much lower rate than previously thought . this has led to numerous theoretical models in which the gas is assumed to be reheated in some fashion by powerful radio jets , by rising bubbles , by shocks , by magic etc . most or all of these models have been unsatisfactory at some level . the most common difficulty of heated flows is not reproducing the observed radial temperature and density profiles ( e.g. brighenti & mathews 2002 , 2003 ) , which are smooth and approximately similar for galaxy clusters of all scales . less theoretical effort has been expended in understanding the radial abundance gradients in the hot gas , but it is fair to say that little progress has been made so far on this important problem . in this paper we show that a combination of heating and radial mass redistribution can satisfy these stringent observational constraints and simultaneously explain the observed radial metallicity profiles . while the flows described here retain some useful features of traditional cooling flows , we refer to them as `` circulation flows '' since gas flows in both directions simultaneously . cooling to low temperatures is greatly or entirely eliminated . the primary source of heat that drives the circulation is assumed to be moderate agn activity in the cores of luminous elliptical galaxies that reside at or near the center of the diffuse x - ray emission in groups and poor clusters . our theoretical and observational studies of the viralized hot gas in galaxy clusters have been motivated in several ways . first , we have maintained that hot gas in galaxy groups is more likely to reveal the physical nature and dynamical evolution of the gas than hotter gas in rich clusters . we assume that some of the groups are old and , if cosmically isolated , are less likely to have been upset and mixed by recent mergers or powerful radio sources . second , we have argued that the radial variation of the gaseous iron abundance in these groups retains an imprint of the origin and dynamical evolution of the gas since it received its iron . finally , at the relatively low temperatures of gas in galaxy groups , @xmath3 kev , the influence of thermal conductivity @xmath4 on the scale of the flow can be ignored . in many galaxy groups and clusters the gas - phase iron abundance has a noticeable peak centered on the central elliptical galaxy that extends out to @xmath1 kpc . the iron abundance increases to @xmath5solar at the center . xmm spectra indicate that 70 - 80 percent of this iron was formed in type ia supernovae . of particular interest for our discussion here is the realization that the total amount of gas - phase iron in this central region is comparable to the total amount of iron that could be created by all type ia supernovae in the central galaxy since its stars were formed . this is only possible if little or none of the iron - enriched gas has cooled . it is also significant that the region of enhanced iron emission around the central galaxy is much larger in extent than the half - light radius of the optical galaxy . to explain these important features , we describe a simple time - dependent model for the long term evolution of hot gas in galaxy groups in which gas enriched by type ia supernovae inside the central galaxy is heated and buoyantly transported far into the surrounding gas where it is stored over time . meanwhile , most of the hot x - ray emitting gas loses energy and flows inward , as in a standard cooling flow . the cooling inflow receives both iron - enriched gas and energy from clouds or bubbles of heated gas that are moving outward . to avoid flows with catastrophic cooling or discordant temperature profiles , it is essential that the inflowing gas receive both energy and mass from the central regions . the physical nature of this heating @xmath6 compression , sound wave dissipation , weak shocks , etc , is left unspecified for the time being . for successful flows the mass and energy deposition must be spatially broad , not concentrated near a single radius in the flow . this can be accomplished if inflowing gas arriving near the central galaxy is heated to a variety of entropy levels and ultimately floats upward approximately to that radius where its entropy matches that of the surrounding , inflowing gas . the discussion below is a time - dependent generalization of the steady state circulation flows that we discussed in a previous paper ( mathews et al . 2003 ) . flows in which gas moves in both radial directions at each radius are notoriously difficult to reproduce faithfully even with the most sophisticated multidimensional numerical codes because numerical diffusion across the eulerian grid blurs the distinction between the counter - streaming flows . for this reason , in this initial study of the time - dependent evolution of radially circulating gas , we study the evolution of the inward flowing gas with a standard eulerian code , but simplify our treatment of the outflowing gas that is difficult to resolve on computational grids . the physical processes by which gas near the central elliptical galaxy is heated are left unspecified . some of the less important aspects of bubble dynamics that we discussed in our study of steady state circulation flows such as momentum exchange by the drag interaction of rising bubbles , expansion of individual bubbles , etc are either not treated in detail here or are represented by source terms in the equations for the inflowing gas . since the dynamical time for buoyant gas is much less than the radiative cooling time , we assume that heated outflowing clouds move rapidly to their final destination . nevertheless , we do not expect that the detailed inclusion of these complications will alter the basic character or the success of the circulation flows described here . we show that time - dependent circulation flows can quite naturally produce all of the major observed radial profiles in the hot gas : its temperature , density and metallicity . in addition to these important attributes , our circulation flows are compatible with the x - ray observation that very little if any gas cools to low temperatures .

x - ray spectra indicate that little or no gas is cooling to low temperatures . this gaseous iron has been circulating for many gyrs , unable to cool . as dense inflowing this gas dominates the local x - ray spectrum but shares the total available volume with centrally heated , outflowing gas . the rapid radial recirculation of gas within kpc results in a flat core in the gas iron abundance , similar to many group and cluster observations .

we describe a new type of dynamical model for hot gas in galaxy groups and clusters in which gas moves simultaneously in both radial directions . the observational motivations for this type of flow are compelling . x - ray spectra indicate that little or no gas is cooling to low temperatures . bubbles of hot gas typically appear in chandra x - ray images and xmm x - ray spectra within kpc of the central elliptical galaxy . these bubbles must be buoyant . furthermore , the elemental composition and total mass of gas - phase iron observed within kpc of the center can be understood as the accumulated outflow of most or all of the iron produced by type ia supernovae in the central galaxy over time . this gaseous iron has been circulating for many gyrs , unable to cool . as dense inflowing gas cools , it produces a positive central temperature gradient , a characteristic feature of normal cooling flows . this gas dominates the local x - ray spectrum but shares the total available volume with centrally heated , outflowing gas . circulating flows eventually cool catastrophically if the outward flowing gas transports mass but no heat ; to maintain the circulation both mass and energy must be supplied to the inflowing gas over a large volume , extending to the cooling radius . the rapid radial recirculation of gas within kpc results in a flat core in the gas iron abundance , similar to many group and cluster observations . we believe the circulation flows described here are the first gasdynamic , long - term evolutionary models that are in good agreement with all essential features observed in the hot gas : little or no gas cools as required by xmm spectra , the gas temperature increases outward near the center , and the gaseous iron abundance is about solar near the center and decreases outward . .2 in

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in this paper we describe a new model for the dynamical evolution of hot gas in groups and clusters of galaxies in which gas flows in both radial directions simultaneously . the incoming gas resembles a traditional cooling flow and dominates the x - ray emission . the outgoing gas consists of an ensemble of heated buoyant bubbles that rise to distant regions where they merge with the inflowing gas . we have not calculated the dynamics of the rising bubbles of heated gas in detail , and this must be an objective for the future . instead , we constructed a schematic time - dependent model guided in part by observations and by the detailed bubble - flow interactions described in our previous study of steady state circulation flows ( mathews et al . it is noteworthy and fortunate that the density and temperature profiles of the cooling , inflowing gas are not sensitive to the filling factor @xmath53 , which is easier to determine reliably from the observations than from dynamical models for the bubbles . this insensitivity may explain the approximate similarity of @xmath156 and @xmath238 profiles in clusters of all sizes . we have shown here that the steady state circulation flows we proposed earlier ( mathews et al . 2003 ) can be generalized to include time variation . an essential feature of successful circulation flows is that both mass and energy must be distributed from the center throughout a large volume of the flow within the cooling radius . the outward increase in gas temperature near the center of the flow , a characteristic feature of most galaxy clusters , results largely from a conventional cooling inflow , modified somewhat by additional energy received from locally outflowing bubbles . after evolving for many gyrs , circulation flows are in accord with the important observation that most or all of the iron produced by type ia supernovae in the central elliptical galaxy has been stored in a region typically extending to @xmath1 kpc . from the perspective of the circulation flow hypothesis , a record of the enrichment - heating history of groups is retained in currently observed iron and temperature profiles of the hot gas and it may be possible to retrieve this information by comparison with more accurate circulation flows . the central peak in the radial iron distribution within @xmath0 kpc is very sensitive to the ( negative ) cooling inflow velocity at small radii . if this velocity is high , the iron abundance remains low and flat - topped . iron abundance profiles that peak sharply toward the center , such as that observed in ngc 5044 , can occur only if the flow velocity is reduced to a very low level by ( recent ) large heating near the center . we have shown that the central iron peak is more pronounced when the mass redistribution and heating are more strongly peaked at smaller radii . concentrated peaks in the iron abundance within @xmath1 kpc are also typically observed in many rich clusters : sersic 159 - 03 ( kaastra et al 2001 ) , cygnus a ( smith et al . 2002 ) , abell 1795 ( ettori et al . 2002 ) and abell 2029 ( lewis , stocke & buote 2002 ) . we speculate that these regions are also mostly enriched by the circulation of iron from type ia supernovae in the central galaxies and there is some observational support for this ( e.g. ettori et al . 2002 ) . if so , this may indicate a characteristic heating luminosity @xmath239 ergs s@xmath194 for all clusters . however , if there is an upper limit on the heating luminosity @xmath111 , perhaps set by the physical conditions near the central black hole , it is possible that the mass inflow rate @xmath240 in rich clusters is too large for the inflowing gas to be heated in the way we describe here . this may give rise to an additional more violent energy releases in the form of powerful radio jets which are rare in galaxy groups that have more modest mass cooling rates @xmath240 . we believe the circulation flows described here are the first gasdynamic , long - term evolutionary models that are in good agreement with all essential features of the hot gas : the gas temperature maximum at several @xmath44 , the approximately linear rise of entropy with radius , the dominance of snia products in the x - ray spectra , the total iron mass within 100 kpc and its radial distribution . the relative constancy of these observed profiles during many gyrs of our calculated flows is consistent with the similarity of the observed x - ray properties among galaxy groups and clusters . finally , little or no gas cools in circulation models , and this is consistent with the much decreased or absence of cooling gas in x - ray spectra of both galaxy groups and clusters . .4 in studies of the evolution of hot gas in elliptical galaxies at uc santa cruz are supported by nasa grants nag 5 - 8409 & atp02 - 0122 - 0079 and nsf grants ast-9802994 & ast-0098351 for which we are very grateful . blanton , e. l. , sarazin , c. l. & mcnamara , b. r. 2003 , apj , 585 , 227 brighenti , f. & mathews , w. g. 2003 , apj 587 , 580 brighenti , f. & mathews , w. g. 2002 , apj 573 , 542 buote , d. a. 2000a , apj 539 , 172 buote , d. a. 2000b , mnras , 311 , 176 buote , d. a. et al . 2003a , apj , 594 , 741 buote , d. a. et al . 2003b , apj , 595 , 151 buote , d. a. , brighenti , f. & mathews , w. g. 2004 , apj ( in press ) ( astro - ph/0404430 ) cappellaro , e. , evans , r. & turatto , m. 1999 , a&a , 351 , 459 de grandi , s. , ettori , s. , monghetti , m. & molendi , s. 2004 , a&a , 419 , 7 de grandi , s. & molendi , s. 2001 , apj , 551 , 153 dupke , r. a. & white , r. e. iii 2003 , apj , 583 , l13 ettori , s. , fabian , a. c. , allen , s. w. & johnstone , r. m. 2002 , mnras , 331 , 635 conference , august 2000 ( astro - ph/0103392 ) fabian , a. c. et al . 2000 , mnras , 318 , l65 gal - yam , a. & maoz , d. 2004 , mnras , 347 , 942 gastaldello , f. & molendi , s. 2002 , apj , 572 , 160 kim , d .- w . & fabbiano , g. 2004 , apj , submitted ( astro - ph/0403105 ) johnstone , r. m. , allen , s. w. , fabian , a. c. & sanders , j. s. 2002 , mnras , 336 , 299 kaastra , j. s. et al . 2001 , a&a , 365 , l99 lewis , a. d. , stocke , j. t. & buote , d. a. 2002 , apj , 573 , l13 loewenstein , m. , mushotzky , r. f. , angelini , l. , arnaud , k. a. & quataert , e. 2001 , apj 555 , l21 loken , c. , norman , m. l. , nelson , e. , bryan , g. l. & motl , p. 2002 , apj , 579 , 571 mathews , w. g. 1989 , aj , 97 , 42 mathews , w. g. & brighenti , f. 2003 , araa , 41 , 191 mathews , w. g. 1997 , aj , 113 , 755 mathews , w. g. , brighenti , f. , buote , d. a. & lewis , a. d. 2003 , apj , 596 , 159 mathews , w. g. & brighenti , f. 2003 , apj 590 , 5 morris , r. g. & fabian , a. c. 2003 , mnras , 338 , 824 navarro , j. , frenk , c. & white , s. 1997 , apj 490 , 493 osullivan , e. , vrtilek , j. m. , read , a. m. , david , l. p. & ponman , t. j. 2003 , mnras , 346 , 525 pain , r. et al . 2002 , apj , 577 , 120 peterson , j. r. , paerels , f. b. s. , kaastra , j. s. , arnaud , m. , reiprich , t. h. , fabian , a. c. , mushotzky , r. f. , jernigan , j. g. & sakelliou , i. 2001 , a&a 365 , l104 rickes , m. g. , pastoriza , m. g. & bonatto , ch . 2004 , a&a , 419 , 449 sadler , e. m. , slee , o. b. , reynolds , j. e. & ekers , r. d. 1994 , _ the first stromlo symposium : the physics of active galaxies _ , asp conf . 54 , 335 sanders , j. s. & fabian , a. c. 2002 , mnras , 331 , 273 schmidt , r. w. , fabian a. c. , & sanders , j. s. 2002 , mnras , 337 , 71 smith , d. a. , wilson , a. s. , arnaud , k. a. , terashima , y. & young , a. j. 2002 , apj 565 , 195 stone , j. m. & norman , m. l. 1992 , apjs , 88 , 253 sun , m. et al . 2003 , apj , 598 , 250 sutherland , r. s. , & dopita , m. a. 1993 , apjs , 88 , 253 xu , h. et al . 2002 , apj 579 , 600 xue , y .- j . , bhringer , h. & matsushita , k. 2004 , a&a , ( in press ) ( astro - ph/0403026 ) . the dotted lines show the density and temperature profiles observed in ngc 5044 that serve as initial conditions for the computed profiles ; the dotted line pressure and entropy profiles are derived from the observations . the filling factors correspond to @xmath153 , 0.5 and 1.0 , all with @xmath56 kpc . in descending order from the top the panels show profiles for the gas entropy @xmath242 , the gas density @xmath243 ( solid lines ) and pressure @xmath244 ( dashed lines ) in units of @xmath245 dyne , the gas temperature @xmath246 , the gas velocity @xmath247 . and the flow filling factor @xmath52 . the low amplitude velocity profiles with @xmath153 and @xmath160 are fortuitously identical . , title="fig : " ] .7 in is concentrated near @xmath248 kpc , ( @xmath166 ) = ( 2,18,10,0 ) . the flow began at cosmic time @xmath162 gyrs and is shown at @xmath249 gyrs when severe cooling occurs . the dashed lines show the gas pressure and the iron abundance in solar units . the dotted lines show the observed @xmath238 and @xmath156 profiles for ngc 5044 . , title="fig : " ] .7 in is a broad function extending out to several hundred kpc , ( @xmath166 ) = ( 1,1.5,30,0 ) . the flow is shown at time @xmath250 gyrs when catastrophic cooling occurred at the inner radius . the dashed lines show the gas pressure and the iron abundance in solar units . the dotted lines show the observed @xmath238 and @xmath156 profiles for ngc 5044 . , title="fig : " ] .7 in that give good agreement at time @xmath221 gyrs with the observed iron abundance and temperature profiles observed in ngc 5044 . the dashed lines show the gas pressure and the iron abundance in solar units . heavy and light lines correspond to probabilities @xmath70 defined by ( @xmath166 ) = ( 1,1.1,41,1.9 ) and ( @xmath166 ) = ( 1,1.5,45,1.6 ) respectively . the dotted lines show the observed @xmath238 and @xmath156 profiles for ngc 5044 . , title="fig : " ] .7 in ) = ( 1,1.1,41,1.9 ) ( light lines ) and ( 1,1.5,45,1.6 ) ( heavy lines ) . shown are the mass inflow rate at @xmath51 @xmath191 in @xmath13 yr@xmath194 ( long dashed lines ) , the total stellar mass loss rate @xmath192 in @xmath13 yr@xmath194 in the central e galaxy ( short dashed lines ) , and the heating luminosity @xmath251 in units of @xmath193 ergs s@xmath194 ( solid lines ) . , title="fig : " ] .7 in

we describe a new type of dynamical model for hot gas in galaxy groups and clusters in which gas moves simultaneously in both radial directions . furthermore , the elemental composition and total mass of gas - phase iron observed within kpc of the center can be understood as the accumulated outflow of most or all of the iron produced by type ia supernovae in the central galaxy over time . gas cools , it produces a positive central temperature gradient , a characteristic feature of normal cooling flows . circulating flows eventually cool catastrophically if the outward flowing gas transports mass but no heat ; to maintain the circulation both mass and energy must be supplied to the inflowing gas over a large volume , extending to the cooling radius . we believe the circulation flows described here are the first gasdynamic , long - term evolutionary models that are in good agreement with all essential features observed in the hot gas : little or no gas cools as required by xmm spectra , the gas temperature increases outward near the center , and the gaseous iron abundance is about solar near the center and decreases outward .

we describe a new type of dynamical model for hot gas in galaxy groups and clusters in which gas moves simultaneously in both radial directions . the observational motivations for this type of flow are compelling . x - ray spectra indicate that little or no gas is cooling to low temperatures . bubbles of hot gas typically appear in chandra x - ray images and xmm x - ray spectra within kpc of the central elliptical galaxy . these bubbles must be buoyant . furthermore , the elemental composition and total mass of gas - phase iron observed within kpc of the center can be understood as the accumulated outflow of most or all of the iron produced by type ia supernovae in the central galaxy over time . this gaseous iron has been circulating for many gyrs , unable to cool . as dense inflowing gas cools , it produces a positive central temperature gradient , a characteristic feature of normal cooling flows . this gas dominates the local x - ray spectrum but shares the total available volume with centrally heated , outflowing gas . circulating flows eventually cool catastrophically if the outward flowing gas transports mass but no heat ; to maintain the circulation both mass and energy must be supplied to the inflowing gas over a large volume , extending to the cooling radius . the rapid radial recirculation of gas within kpc results in a flat core in the gas iron abundance , similar to many group and cluster observations . we believe the circulation flows described here are the first gasdynamic , long - term evolutionary models that are in good agreement with all essential features observed in the hot gas : little or no gas cools as required by xmm spectra , the gas temperature increases outward near the center , and the gaseous iron abundance is about solar near the center and decreases outward . .2 in

1602.04685

i

in classical electrodynamics , the dynamics of @xmath0 charges and their corresponding electromagnetic fields is governed by the lorentz equations @xmath1,\end{aligned}\ ] ] the maxwell equations @xmath2 and the maxwell constraints @xmath3 for @xmath4 . in our notation , @xmath5 denote the position and momentum of the @xmath6th charge at time @xmath7 . for simplicity we give all charges the same mass @xmath8 and rigid electric charge density @xmath9 and use units such that the speed of light equals one and the vacuum permittivity equals @xmath10 . note that by virtue of , the constraints at @xmath11 imply that they hold for all times @xmath12 . contrary to the text - book presentation , see , e.g. , @xcite , in which one employs only one total electric and magnetic field , it will be convenient for our discussion to associate with each charge @xmath6 an individual electric and magnetic field @xmath13 . thanks to the linearity of the maxwell equations in the field degrees of freedom , the equations of motion - coincide with the one given in text - books when setting @xmath14 . other choices of @xmath15 allow to switch on or off the interaction of the @xmath16-th field on the @xmath6-th charge . for arbitrary @xmath15 and smooth and compactly supported @xmath17 it has been proven that the coupled system of equations - has a well - posed initial problem for any initial data @xmath18 with reasonably regular fields @xmath19 fulfilling the constraints ; see @xcite . spinning charges were discussed in @xcite and the semi - relativistic system was considered in @xcite . very early , however , it was observed , e.g. , in @xcite , that replacing the charge density @xmath20 by a dirac delta distribution @xmath21 ( for simplicity , setting the total electric charge equal one ) renders the self - interaction summand @xmath22 on the right - hand side of the lorentz equation , and thereby , also the coupled system of equations - , ill - defined . the reason for this is that , in the point - charge case @xmath23 , the maxwell fields @xmath24 are not entirely smooth anymore but have a second order pole at @xmath25 which is exactly where they would have to be evaluated in @xmath22 . in order to distinguish the case of general @xmath20 from the point - charge case of @xmath23 , we use the convention that lower - case fields @xmath26 solve the equations - for @xmath23 , which then implies the relation @xmath27 , where @xmath28 denotes the convolution and @xmath29 is a solution to the free maxwell equations , i.e. - for @xmath30 . to see the divergent behavior of @xmath26 , thanks to the linearity , it suffices to regard a special solution to - for a fixed charge trajectory @xmath31 . in the following we drop the index @xmath6 to keep the notation slim . two well - known solutions of - are the advanced and retarded linard - wiechert fields @xmath32=({\boldsymbol{e}}^\pm_t,{\boldsymbol{b}}^\pm_t)$ ] , where the square bracket notation emphasizes the functional dependence on the charge trajectory @xmath33 . they are given by @xmath34}{|{\boldsymbol{x } } -{\boldsymbol{q}}|(1\pm { { \boldsymbol{n}}}\cdot{\boldsymbol{v}})^3}\bigg|^\pm , \nonumber \\ { \boldsymbol{b}}^{\pm}_t({\boldsymbol{x } } ) & : = \mp { \boldsymbol{n}}^{\pm}\wedge{\boldsymbol{e}}^{\pm}_t({\boldsymbol{x } } ) , \label{eq : lw}\end{aligned}\ ] ] where we have used the abbreviations @xmath35 cf . all other solutions @xmath36 to - for the same trajectory @xmath33 can then be represented as @xmath37+(1-\lambda){\boldsymbol{f}}^+_t[{\boldsymbol{q}},{\boldsymbol{p}}]+{\boldsymbol{f}}^0_t\end{aligned}\ ] ] for @xmath38 $ ] , where @xmath39 is a solution to the corresponding homogeneous equations , i.e. , - for @xmath40 . for smooth @xmath39 , the explicit expressions in imply that all corresponding fields @xmath36 are smooth on @xmath41 where they admit the discussed singular behavior that renders the term @xmath22 in ill - defined for @xmath23 . to still make sense out of this ill - defined self - interaction , an informal mass renormalization argument is usually employed , see @xcite , which effectively replaces the problematic term @xmath22 with the finite abraham - lorentz - dirac back reaction @xmath42 . in the non - relativistic regime , the latter may be approximated by @xmath43 , with @xmath44 denoting the electric charge . this procedure cures the original problem , however , introduces a dynamical instability as for almost all but very special initial accelerations , which now must be provided along with initial positions and momenta , the corresponding charge trajectories approach the speed of light exponentially fast . nevertheless , it was shown that the subset of physically sensible solutions can be well approximated in certain regimes by a dynamically stable version that was suggested by landau and lifschitz ; see @xcite . after replacing the ill - defined term @xmath22 appropriately or simply omitting it by setting @xmath45 , which often can be justified as its renormalized version is usually small ( e.g. , for small acceleration , jerk , and electric charge ) , one might hope that there are no further obstacles in arriving at a solution theory for the maxwell - lorentz system - in the point - charge limit @xmath46 . a general proof of the well - posedness of the corresponding initial value problem , however , is difficult and remains open . the first two difficulties are obvious : 1 ) the charges must not collide , otherwise @xmath47 in blows up ; and 2 ) the charges must not approach the speed of light too fast , otherwise the factors @xmath48 in may blow up . mathematically , difficulty 1 ) poses a similarly delicate problem as in the @xmath0-particle problem of gravitation , only now with the additional complication that the coulomb potentials in are lorentz - boosted and to be evaluated at _ delayed _ or _ advanced _ times @xmath49 as given in . difficulty 2 ) is due to the accumulation of the escaping fields along the light cone and must be excluded with an a priori bound on the charge velocities . when handled with care , it is reasonable to expect that at most only very few initial values @xmath50 lead to catastrophic events due to these two difficulties . however , there is a third difficulty which is more subtle and , to our knowledge , has not received attention yet . given a charge trajectory @xmath51 , only rather special initial fields @xmath52 give rise to solutions @xmath26 to - that are sufficiently regular outside a neighborhood of @xmath25 in order to be evaluated in the terms @xmath53 in for all times . generic initial fields will generate singular fronts in the fields traveling at the speed of light , and another charge @xmath16 having velocities below the speed of light is bound to traverse such fronts in finite time . + in section [ sec : steps ] we explain the mathematical origin of this questionable artifact and discuss how solutions with singular light fronts can be ruled out by appropriate restrictions on the initial values . in section [ sec : ml - system ] , for the point - charge case @xmath23 , we give necessary conditions for global existence of piecewise as well as globally smooth solutions to the maxwell - lorentz system - . we discuss that this point - charge phenomenon has a straight - forward analogue in quantum field theory , and furthermore , implications on the case of extended charges @xmath20 . in the latter , the singular light fronts qualitatively persist in @xmath24 , however , in a smoothened version as can be seen from the convolution relation @xmath27 . due to this additional smoothness , the singular light fronts cause no trouble concerning the solution theory anymore . nevertheless , as illustrated in a quantitative example in the end of section [ sec : admissible ] , they can cause sharp , though smooth , steps on the length scale of the diameter of @xmath20 , and therefore , in principal observable radiation . moreover , in section [ sec : admissible ] , we demonstrate that the initial value problem is ill - posed when demanding smooth global solutions , as the necessary restriction on the initial fields @xmath52 requires information about the , at @xmath11 , unknown charge trajectories @xmath54 . we introduce a mathematical procedure for finding admissible initial fields despite this fact . the latter , however , introduces an unwanted arbitrariness which , as we suggest in section [ sec : conclusion ] , can be eliminated by physical reasoning . the resulting restrictions on the initial values naturally turn the equations of motion of the maxwell - lorentz system - into a class of delay differential equations that include the fokker - schwarzschild - tetrode equations of motion of wheeler - feynman electrodynamics @xcite and the synge equations @xcite as prime examples .

we provide explicit formulas for the corresponding fields , demonstrate how this phenomenon renders the initial value problem ill - posed , and show how such bad initial data can be ruled out by extra conditions in addition to the maxwell constraints . these extra conditions , however , require knowledge of the history of the solution and , as we discuss , effectively turn the maxwell - lorentz system into a system of delay equations much like the fokker - schwarzschild - tetrode equations . for extended charges such singular light fronts persist in a smoothened form and , as we argue , yield physically doubtful solutions .

we describe a seemingly unnoticed feature of the text - book maxwell - lorentz system of classical electrodynamics which challenges its formulation in terms of an initial value problem . for point - charges , even after appropriate renormalization , we demonstrate that most of the generic initial data evolves to develop singularities in the electromagnetic fields along the light cones of the initial charge positions . we provide explicit formulas for the corresponding fields , demonstrate how this phenomenon renders the initial value problem ill - posed , and show how such bad initial data can be ruled out by extra conditions in addition to the maxwell constraints . these extra conditions , however , require knowledge of the history of the solution and , as we discuss , effectively turn the maxwell - lorentz system into a system of delay equations much like the fokker - schwarzschild - tetrode equations . for extended charges such singular light fronts persist in a smoothened form and , as we argue , yield physically doubtful solutions . our results also apply to some extent to expectation values of field operators in quantum field theory .

astro-ph9904115

i

recent discoveries of slow - moving objects beyond the orbit of neptune have radically changed our understanding of the outer solar system . these observations have revealed a large population of kuiper belt objects ( kbos ) in orbits with semi - major axes of 3945 au ( @xcite , 1995 ; @xcite ; @xcite ; @xcite ; @xcite ; @xcite ; @xcite ) . kbos with reliable orbits have a cumulative size distribution that follows @xmath14 , with @xmath15 ( @xcite ; @xcite ) . the estimated population of @xmath13 kbos with radii @xmath16 50 km indicates a total mass , @xmath17 g ] , for reasonable assumptions about the albedo and distance distribution . the small mass in kbos is a problem for current planet formation theories . in most theories , small planetesimals in the solar nebula grow by collisional accumulation ( e.g. , @xcite ; @xcite , 1984 ; @xcite , 1993 ; @xcite ) . this growth eventually produces one or more ` cores ' that accumulate most , if not all , of the solid mass in an annular ` feeding zone ' defined by balancing the gravity of the growing planetesimal with the gravity of the sun and the rest of the disk . large cores with masses of 110 @xmath18 can also accrete gas from the feeding zone ( @xcite ) . applied to the inner solar system , this model generally accounts for the masses of the terrestrial and gas giant planets ( e.g. , @xcite , @xcite ; but see @xcite ) . at 3050 au , however , the timescale to produce planet - sized , @xmath19 1000 km , objects exceeds the disk lifetime , @xmath20 100 myr , unless the mass of the outer disk is @xmath21@xmath22 ( e.g. , @xcite , 1984 ; @xcite ; @xcite , 1996 ; @xcite ) . the presence of large kbos in a small mass kuiper belt is thus a mystery . in kenyon & luu ( 1998 , hereafter kl98 ) , we began to address this issue by considering planetesimal growth in a single annulus at 35 au . we showed that calculations including accretion and velocity evolution naturally produce several `` plutos '' with radii exceeding 1000 km and numerous 50 km radius kbos on timescales of @xmath23 20200 myr for a wide range of initial conditions in plausible solar nebulae . these timescales indicate that pluto can form in the outer solar system in parallel with the condensation of the outermost large planets . in this paper , we extend kl98 by adding fragmentation to our planetesimal evolution code . the code generally matches published calculations at 1 au ( wetherill & stewart 1993 , hereafter ws93 ) and at 40 au ( davis & farinella 1997 ) . our numerical results demonstrate that fragmentation and velocity evolution damp runaway growth to provide a self - limiting mechanism for the formation of kbos and pluto in the kuiper belt . these calculations produce several plutos and at least @xmath24 50 km radius kbos on timescales of 2100 myr in annuli with modest surface densities , 2.00.14 g @xmath25 , of solid material at 35 au . our analysis sets a lower limit on the intrinsic strength of kbos , @xmath19 300 erg g@xmath8 , and indicates that the initial size distribution , the initial eccentricity of the planetesimal swarm , and the details of the fragmentation algorithm have little impact on the resulting size distribution of kbos . our results appear to resolve the mystery of large kbos in a small mass kuiper belt . planetesimal evolution at 3550 au in a minimum mass solar nebula ( @xcite ) naturally produces large kbos in numbers close to those currently observed . most of the disk mass ends up in smaller objects , 0.110 km , that can be collisionally depleted over the age of the solar system . this depletion rate depends on the intrinsic strength and eccentricity distribution of kbos ( @xcite ; @xcite ; @xcite , 1997b ) . future observations can place better constraints on these physical parameters and provide additional tests of our interpretation of kbo formation . we outline the fragmentation model and tests in sec . 2 , describe calculations for the kuiper belt in sec . 3 , and conclude with a discussion and summary in sec . the appendix contains a complete description of the fragmentation algorithms and updates of the coagulation code from kl98 .

the power law indices are nearly independent of the initial mass in the annulus , ; the initial eccentricity of the planetesimal swarm , ; and the initial size distribution of the planetesimal swarm . the maximum size of objects depends on their intrinsic tensile strength , ; pluto formation requires 300 erg g . our models yield 3040 myr for planetesimals with in a minimum mass solar nebula . the production of several ` plutos ' and 50 km radius kuiper belt objects leaves most of the initial mass in 0.110 km radius objects that can be collisionally depleted over the age of the solar system . these results resolve the puzzle of large kuiper belt objects in a small mass kuiper belt . # 1 # 1 # 1 = = = 1=1=0pt = 2=2=0pt 2ex 4ex scott j. kenyon harvard - smithsonian center for astrophysics 60 garden street , cambridge , ma 02138 e - mail : skenyon@cfa.harvard.edu 1ex and 1ex jane x. luu leiden observatory po box 9513 , 2300 ra leiden , the netherlands e - mail : luu@strw.leidenuniv.nl 3ex to appear in _ the astronomical journal _

we describe new planetesimal accretion calculations in the kuiper belt that include fragmentation and velocity evolution . all models produce two power law cumulative size distributions , for radii 0.33 km and for radii 13 km . the power law indices are nearly independent of the initial mass in the annulus , ; the initial eccentricity of the planetesimal swarm , ; and the initial size distribution of the planetesimal swarm . the transition between the two power laws moves to larger radii as increases . the maximum size of objects depends on their intrinsic tensile strength , ; pluto formation requires 300 erg g . the timescale to produce pluto - sized objects , , is roughly proportional to and , and is less sensitive to other input parameters . our models yield 3040 myr for planetesimals with in a minimum mass solar nebula . the production of several ` plutos ' and 50 km radius kuiper belt objects leaves most of the initial mass in 0.110 km radius objects that can be collisionally depleted over the age of the solar system . these results resolve the puzzle of large kuiper belt objects in a small mass kuiper belt . # 1 # 1 # 1 = = = 1=1=0pt = 2=2=0pt 2ex 4ex scott j. kenyon harvard - smithsonian center for astrophysics 60 garden street , cambridge , ma 02138 e - mail : skenyon@cfa.harvard.edu 1ex and 1ex jane x. luu leiden observatory po box 9513 , 2300 ra leiden , the netherlands e - mail : luu@strw.leidenuniv.nl 3ex to appear in _ the astronomical journal _ july 1999

1603.03962

c

in this paper we presented the mathematical underpinnings of the pvb ( biorthogonal von neumann ) method for quantum mechanical simulations . the pvb method exploits the phase phase space localization of the von neumann basis to provide a sparse representation of quantum mechanical states that spans only the part of phase space where there is significant amplitude . this in turn can lead to significant computational savings in both cpu and memory . a detailed analysis was given of the subtle issues of projection onto subspaces of biorthogonal bases . two complementary ways of understanding this projection were provided . the first focuses on the basis functions : it was shown that under projection one of the biorthogonal bases remains unchanged while the other becomes significantly distorted . we showed that the distortions may be viewed as arising from subtraction of the @xmath156 gaussians that span the complementary subspace . this explains why gaussians near the boundary of the reduced phase space boundary are significantly distorted , while gaussians far from the boundary are essentially unperturbed . the second way to understand the effect of non - orthogonal projection focuses on the coefficients : in the @xmath156 basis all @xmath310 basis vectors contribute to the coefficients , since all basis vectors overlap one another due to the non - orthogonality of the basis . we then analyzed the various representations of the schrdinger equation in the reduced basis and approximations thereto . we concluded that for high - accuracy applications @xmath311 ( eq . [ eq : h1_def ] ) is the preferred form , although it comes which a relatively high computational cost . for medium to low accuracy applications , an approximate form , @xmath312 ( eq . [ eq : h2 ] ) may be used . several numerical examples were brought , showing the relative merits of @xmath238 and @xmath282 . a more challenging application of pvb , the double ionization of helium , is presented in @xcite . despite the significant methodological progress further development is possible . specifically , one may further reduce the representation by decomposing multi - dimensional objects into a sum - of - products , and truncating the sum when the correlation is sufficiently low . this strategy is used by the potfit algorithm to decompose the potential , but a similar approach could be used for the wavefunction and the reduced hamiltonian . this is a challenging problem , however , as the dynamics continuously modify the reduced basis , which generally is not easily decomposable . further areas of research include the correspondence between pvb and other phase space representations , including the discrete husimi and wigner representation . we also plan to explore the treatment of particle symmetries ( bosonic , fermionic ) in multi - particle implementations of the pvb method . beyond method development , there is a wide range of problems which are amenable to the pvb methodology , including high - harmonic generation , multi - electron ionization , photodissociation and chemical reactions . we intend to explore these applications in the near future . to conclude , pvb is an accurate , scalable and efficient method for quantum dynamics simulations , and it is our hope that it will find its place as part of the standard quantum numerics toolbox .

we describe the mathematical underpinnings of the biorthogonal von neumann method for quantum mechanical simulations ( pvb ) . in particular , we present a detailed discussion of the important issue of non - orthogonal projection onto subspaces of biorthogonal bases , and how this differs from orthogonal projection . we present various representations of the schrdinger equation in the reduced basis and discuss their relative merits .

we describe the mathematical underpinnings of the biorthogonal von neumann method for quantum mechanical simulations ( pvb ) . in particular , we present a detailed discussion of the important issue of non - orthogonal projection onto subspaces of biorthogonal bases , and how this differs from orthogonal projection . we present various representations of the schrdinger equation in the reduced basis and discuss their relative merits . we conclude with illustrative examples and a discussion of the outlook and challenges ahead for the pvb representation .

1608.00465

c

we have used a general and robust method to determine the mid - infrared and optical luminosity evolutions and luminosity functions simultaneously for quasars using a sdss @xmath16 _ wise _ dataset , which combines 22 @xmath0 m infrared and @xmath25-band optical data for over 20,000 quasars ranging in redshifts from 0.08 to 4.97 . as discussed in [ intro ] quite different strategies can be used to assemble an infrared agn data sample for determination of infrared population characteristics . these strategies have advantages and disadvantages . here we have chosen to assemble a large sample of tens of thousands of objects with definite spectroscopic redshifts and known and straightforward flux truncations for inclusion , from which the true intrinsic population characteristics of interest can be determined directly and non - parametrically with limited modeling and assumptions . here we find , as discussed in [ method ] , that quasars have undergone significant luminosity evolution with redshift in the mid - infrared , but less than in the optical band , and , comparing to previous results ( e.g. * ? ? ? * ; * ? ? ? * ) , both of these evolutions are less dramatic than in the radio band . this provides an important input to constrain models of jet , accretion disk , and torus emission and their evolution over the history of the universe . for example , in the basic models of agn where the spin energy of the black hole is tapped to create the jets ( e.g * ? ? ? * ; * ? ? ? * ) , faster radio evolution than optical would indicate that the spin parameters of black holes were higher in the past since radio emission overwhelmingly results from jets . since mid - infrared emission in agn is some combination of emission from the dusty tori , the jets , and the host galaxies , the significantly less rapid evolution of infrared emission in comparison with radio would confirm that jet emission is a sub - dominant source of infrared emission . we also show in [ localinf ] that the local 22 @xmath0 m mid - infrared luminosity function of quasars @xmath152 shows a dramatic flattening at luminosities below @xmath175erg s@xmath2hz@xmath2 . a flattening of this sort is also seen in the 15@xmath0 m luminosity function of type-1 and type-2 agn by @xcite , and at 24 @xmath0 m by @xcite . such dramatic flattening of the luminosity function is not seen in optical wavebands at the luminosities probed by this analysis . however a similar flattening _ is _ seen in the local radio luminosity function at 1.4 ghz luminosities below @xmath176erg s@xmath2hz@xmath2 by @xcite . do these mid - infrared and radio luminosity function flattenings result from the same process ? a simple scaling of the radio emission to the mid - infrared with a synchrotron - like power law spectral index of @xmath177 would put the mid - infrared luminosity equivalent to the 1.4 ghz break at @xmath178erg s@xmath2hz@xmath2 . as this is not where the infrared break is observed , the two flattenings seemingly have different physical causes . this again points to jet emission being a sub - dominant component of the infrared emission . rather , it is some phenomenon of the tori or host galaxies that causes the relative scarcity of quasars with mid - infrared luminosities below @xmath175erg s@xmath2hz@xmath2 . given that that a differential luminosity function which is flat at the faint end corresponds to a cumulative luminosity function which has a power law slope of -1 at the faint end , we can conclude that the contribution of quasars to the integrated mid - infrared light output in the universe peaks at 22 @xmath0 m luminosities of around @xmath165erg s@xmath2hz@xmath2 . could the inferred flattening of @xmath152 result from selection effects , in particular the flux limit for optically - identified quasars in the data set used in this analysis ? in principle the analysis of this work accesses the true intrinsic distributions of local luminosities ( among other quantities ) and corrects for the effects of survey truncations . it is always possible that the population differs significantly _ intrinsically _ at combinations of @xmath136 , @xmath137 , and @xmath15 outside of those present in the data set from how it is at combinations represented in the data set . however , the data set spans two decades in local optical luminosity and local infrared luminosity with significant spread in the @xmath136 , @xmath137 plane . we believe it is most likely that the flattening in @xmath137 is intrinsic in the population of quasars , although can not rule out from this analysis alone an additional population of low infrared luminosity objects which almost universally have a low optical luminosity . [ [ integrated - emission - from - quasars - and - contribution - to - the - infrared - background - light ] ] integrated emission from quasars and contribution to the infrared background light ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ given the distributions calculated here , we can calculate the total integrated output of quasars in the unverse at 22 @xmath0 m . one should integrate the overall mid - infrared luminosity function @xmath179 times the flux corresponding to each luminosity and redshift over all redshifts and luminosities : @xmath180 in terms of the density evolution function @xmath46 and the mid - infrared luminosity portion @xmath181 this is @xmath182 in terms of the local mid - infrared luminosity function @xmath152 we would need the value of this function for the local luminosity corresponding to each luminosity and redshift combination : @xmath183 carrying out this integration results in a calculated value of @xmath184 w m@xmath185 sr@xmath2 hz@xmath2 . in @xmath186 units this is @xmath187 w m@xmath185 sr@xmath2 , which can be compared to e.g. the results obtained by @xcite at 15 @xmath0 m who report a value of ( 4.2 12.1 ) @xmath188 w m@xmath185 sr@xmath2 for type-1 agn and ( 5.5 14.6 ) @xmath188 w m@xmath185 sr@xmath2 for type-2 agn . it would be enlightening to compare the number obtained here for the integrated output of quasars at 22 @xmath0 m to the total cosmic infrared background light ( cib ) level at this wavelength . unfortunately a gap exists in reported measurement of the cib between 3.5 @xmath0 m and 60 @xmath0 m , with reported values of @xmath189 w m@xmath185 sr@xmath2 at the former @xcite and @xmath190 w m@xmath185 sr@xmath2 at the latter @xcite . taking an intermediate value between these two as an estimate of the cib at 22 @xmath0 m would indicate that quasars contribute less than one percent of the total integrated mid - infrared light output in the universe . alternately , one could also compare the value obtained here for the output of quasars to the total output calculated from source counts for all sources at this wavelength . @xcite calculate this output from all sources to be @xmath191 w m@xmath185 sr@xmath2 at 24 @xmath0 m , which , ignoring any spectral shape between 24 @xmath0 m and 22 @xmath0 m and given the value obtained here , would make quasars responsible for between 1% and 5% of the total output from 22 @xmath0 m sources . as discussed in [ roevsec ] , the mid - infrared and optical luminosities are highly correlated , but the power - law correlation index between the mid - infrared and optical luminosities ( @xmath300.8 ) is less than that found previously for the radio and optical luminosities ( @xmath301.0 ) . as discussed in [ ifcorr ] , the subject of luminosity - luminosity correlations is complicated and it is not straightforward to determine how much of these correlations are intrinsic to the waveband emissions in the population and how much are induced by similar redshift evolutions and the truncations of the data set . we will explore this issue in a future work . for the present , we can speculate that _ if _ these correlations are intrinsic ( or if the induced portion is roughly the same in the infrared - optical case as the radio - optical case ) , the radio - optical correlation being more powerful than the infrared - optical correlation could support the idea that the mass and/or spin of the black hole affect the size and/or temperature of the accretion disk and power of the jets more than they affect the size and/or temperature of the torus . a full understanding of the true nature of luminosity - luminosity correlations in agn , and an extension of these considerations to the x - ray band , will be useful in exploring these and other questions . funding for the sdss and sdss - ii has been provided by the alfred p. sloan foundation , the participating institutions , the national science foundation , the u.s . department of energy , the national aeronautics and space administration , the japanese monbukagakusho , the max planck society , and the higher education funding council for england . the sdss web site is http://www.sdss.org/. this publication makes use of data products from the wide - field infrared survey explorer , which is a joint project of the university of california , los angeles , and the jet propulsion laboratory / california institute of technology , and _ neowise _ , which is a project of the jet propulsion laboratory / california institute of technology . _ wise _ and _ neowise _ are funded by the national aeronautics and space administration . we thank the referee for very insightful comments . abazajian , k. , adelman - mccarthy , k. , ageros , m. et al . 2009 , , 182 , 543 babbedge , t. , rowan - robinson , m. , vaccari , m. et al . 2006 , , 370 , 1159 blandford , r. & znajek , r. 1977 , , 179 , 433 broderick , j. w. , & fender , r. p. 2011 , , 417 , 184 boyle , b. , shanks , s. , croom , r. , smith , l. , loaring , n. & heymans , c. 2000 , , 317 , 1014 brown , m. , brand , k. , dey , a. et al . 2006 , , 638 , 88 dermer , c. 2007 , , 659 , 958 eddington , a. 1940 , , 100 , 35 efron , b. & petrosian , v. 1992 , , 399 , 345 efron , b. & petrosian , v. 1999 , jasa , 94 , 447 , mill valley , ca : university science books 1989 finkbeiner , d. , davis , m. , & schlegel , d. 2000 , , 544 , 81 gallagher , s. , richards , g. , lacy , m. , hines , d. , elitzur , m. , & storrie - lombardi , l. 2008 , , 661 , 30 gorjian , v. , wright , e. , & chary , r. 2000 , , 536 , 550 hopkins , p. , richards , g. & hernquist , l. 2007 , , 654 , 731 kimball , a. , kellerman , k. , condon , j. , ivezic , z. , & perley , r. 2011 , , 739 , l29 lacy , m. , ridgway , s. , sajina , a. , petric , a. , gates , e. , urrutia , t. , & storrie - lombardi , l. 2015 , , 802 , 102 lafranca , f. , melini , g. , & fiore , f. 2010 , , 718 , 368 lonsdale , c. , polletta , m. , surace , j. et al . 2004 , , 154 , 54 lynden - bell , b. 1971 , , 155 , 95 mainzer , a. , bauer , j. , grav , t. et al . 2011 , , 731 , 53 maloney , a. & petrosian , v. 1999 , , 518 , 32 mateos , s. , alonso - herrero , f. , carrera , f. , blain , a. , severgnini , p. , caccianiga , a. , & ruiz , a. 2012 , , 426 , 3271 mateos , s. , alonso - herrero , f. , carrera , f. , blain , a. , severgnini , p. , caccianiga , a. , & ruiz , a. 2012 , , 434 , 941 matute , i. , lafranca , f. , pozzi , f. , gruppioni , c. , lari , c. , & zamorani , g. 2006 , , 451 , 443 miller , l. , peacock , j. , & mead , a. 1990 , , 244 , 207 papovich , c. , dole , h. , egami , e. et al . 2004 , , 154 , 70 pris , i. , petitjean , p. , aubourg , . 2014 , , 563 , a54 petrosian , v. 1992 , in statistical challenges in modern astronomy , ed . feigelson & g.h . babu ( new york : springer ) , 173 petrosian , v. 1973 , , 183 , 359 petrosian , v. , & singal , j. 2015 , in proc . iau s313 , extragalactic jets from every angle , eds . f. massaro , c. cheung , e. lopez , & a siemiginowska ( cambridge , uk : cambridge university press ) richards , g. , strauss , m. , fan , x. et al . 2006a , , 131 , 2766 richards , g. , strauss , m. , fan , x. et al . 2006b , , 131 , 2766 schmidt , m. 1972 , , 176 , 273 schneider , d. , richards , g , hall , p. et al . 2010 , , 166 , 470 shaver , p. , wall , j. , kellermann , k. , jackson , c. , & hawkins , m. 1996 , , 384 , 439 shupe , d. , rowan - robinson , m. , lonsdale , c. et al . 2008 , , 135 , 1050 singal , j. , petrosian , v. , lawrence , a. , & stawarz , . , 2011 , , 743 , 104 singal , j. , petrosian , v. , & ajello , m. 2012 , , 753 , 45 singal , j. , petrosian , v. , stawarz , . , & lawrence , a. 2013 , , 764 , 43 singal , ko , a. , & petrosian , v. 2014 , apj , 786 , 109 teerikorpi , p. 2004 , , 424 , 73 xu , c. , carol , j. , lonsdale , j. , shupe , d. , olinger , j. , & masci , f. 2001 , , 562 , 179 wright , n. , eisenhardt , p. , mainzer , a. et al . 2010 , , 140 , 1868

we determine the 22 m luminosity evolution and luminosity function for quasars from a data set of over 20,000 objects obtained by combining flux - limited sloan digital sky survey optical and wide field infrared survey explorer mid - infrared data . we find that the population of quasars exhibits positive luminosity evolution with redshift in the mid - infrared , but with considerably less mid - infrared evolution than in the optical or radio bands . with the luminosity evolutions accounted for the latter displays a sharp flattening at local luminosities belowerg shz , which has been reported previously at 15 m for agn classified as both type-1 and type-2 . we calculate the integrated total emission from quasars at 22 m and find it to be a small fraction of both the cosmic infrared background light and the integrated emission from all sources at this wavelength .

we determine the 22 m luminosity evolution and luminosity function for quasars from a data set of over 20,000 objects obtained by combining flux - limited sloan digital sky survey optical and wide field infrared survey explorer mid - infrared data . we apply methods developed in previous works to access the intrinsic population distributions non - parametrically , taking into account the truncations and correlations inherent in the data . we find that the population of quasars exhibits positive luminosity evolution with redshift in the mid - infrared , but with considerably less mid - infrared evolution than in the optical or radio bands . with the luminosity evolutions accounted for , we determine the density evolution and local mid - infrared luminosity function . the latter displays a sharp flattening at local luminosities belowerg shz , which has been reported previously at 15 m for agn classified as both type-1 and type-2 . we calculate the integrated total emission from quasars at 22 m and find it to be a small fraction of both the cosmic infrared background light and the integrated emission from all sources at this wavelength .

1507.04107

i

the rapid progress in experimentally observing and even controlling electron dynamics in atoms and molecules demands powerful theoretical approaches ; see , e.g. , @xcite for reviews on this subject . one of the most challenging and , therefore , interesting tasks is the accurate description and understanding of the ultrafast and complex behavior arising from the electron - electron interaction . it can be expected that mean field , i.e. , hartree - fock - type approaches are insufficient , and that electronic correlations are important . these are especially difficult to treat in time - dependent theories with more than two active electrons , due to the complexity of the multi - electron wave function and even more so if the continuum is included for photoionization ; see @xcite for an overview . of particular interest in the context of strong field physics are molecular systems due to their much more complex dynamics and degrees of freedom owing to their geometrical structure . with the development of alignment and even orientation techniques @xcite measurements in the molecular - fixed frame of reference become accessible , which allows for an investigation beyond orientation - averaged quantities . one question is the preferred direction of electron emission with respect to the electrical field direction of a linearly polarized laser and the influence of correlation effects in strong - field excitation scenarios of heteronuclear molecules . in a first approximation , the tunnel ionization maps the highest - occupied molecular orbital ( homo ) to the continuum @xcite , and by recollision strong - field ionization was even used to illustrate the homo experimentally @xcite . however , experimental evidence using co molecules @xcite showed that this simplified one - electron picture needs to be adjusted , as effects such as inner - shell polarizations @xcite , stark shifts and orbital distortions @xcite have impact on the ionization dynamics . the question to what extent electronic correlations are important remains open . in ref . @xcite this topic has been addressed within a one - dimensional model of the four - electron lih molecule , and a shift of the preferred direction of emission is observed when electronic correlations are included . the immediate question of whether these effects are present in a full three - dimensional analysis shall be answered by this work and completed by angle - resolved investigations . all of these above - discussed issues call for a general , time - dependent theory including external ( possibly strong ) fields beyond a perturbative approach . the fundamental equation describing the physics of these quantum systems is the ( non - relativistic ) time - dependent schrdinger equation ( tdse ) . however , its direct numerical solution , even by means of supercomputers , is limited to systems consisting of only one or two electrons , e.g. helium @xcite or molecular hydrogen @xcite . semi - analytical theories , such as the strong - field approximation and tunneling theories @xcite provide physical insight but often draw on a simplified picture of the electron - electron interactions . in order to solve the time - dependent schrdinger equation for more than two active electrons including the electrons interactions , approximate numerical techniques need to be employed . these include the time - dependent configuration interaction singles ( td - cis ) method @xcite , multi - configuration time - dependent hartree - fock ( mc - tdhf ) @xcite or its generalizations time - dependent restricted or complete active space self - consistent - field ( td - ras / cas - scf ) @xcite and the state - specific - expansion approach @xcite ( see also ref . @xcite for an overview ) . further , time - dependent density - functional theory ( td - dft ) @xcite and time - dependent close - coupling solutions of the tdse by using pseudo - potentials for the description of more than two electrons @xcite have been applied to photoionization of molecules . especially the mctdhf family suffers from complicated non - linear numerics , and its applicability to photoionization is not yet fully understood . td - dft and the pseudo - potential approaches , on the other hand , rely strongly on the chosen functionals or potentials with unknown accuracy and lack tunable parameters to achieve convergence to the fully correlated solution . one of the most successful methods which bears some similarities to our present approach , is the time - dependent r - matrix method @xcite . the aims of the present work are ( i ) to provide a fully _ ab - initio _ time - dependent approach to electron dynamics in diatomic molecules exposed to strong laser fields including a systematic ( i.e. controllable ) approach to electron correlation without relying on pseudo - potentials and ( ii ) to demonstrate the method by shining light onto the question of whether electronic correlation decides from which end an electron leaves a heteronuclear molecule which is exposed to a strong electric single - cycle pulse . our approach is based on the time - dependent generalized - active - space configuration interaction ( td - gas - ci ) formalism which we apply within a prolate spheroidal single - particle basis set in combination with the well - established partition - in - space concept to tackle the scattering part of the hamiltonian . the paper is organized as follows . after a brief introduction into the theory of td - gas - ci , we give a detailed overview on the used basis set and details of our implementation , in section [ sec : theory ] . technical aspects and the explicit formulas and strategies of their efficient numerical handling are presented in the corresponding appendices . in sections [ sec : results ] and [ sec : sfi - lih ] , we show illustrative numerical examples and demonstrate the abilities of the present approach . we focus on the lih molecule in strong single - cycle infrared ( ir ) pulses and explore the influence of electronic correlations on the molecular photoelectron angular distributions ( pads ) and the preferred direction of electron emission as a function of the geometrical set - up . the paper closes with conclusions and a discussion of future applications of the present theory .

we develop a time - dependent theory to investigate electron dynamics and photoionization processes of diatomic molecules interacting with strong laser fields including electron - electron correlation effects . we combine the recently formulated time - dependent generalized - active - space configuration interaction theory [ d. hochstuhl and m. bonitz , phys . rev . rev . are investigated : , with the electrical field pointing from h to li , and the opposite case of .

we develop a time - dependent theory to investigate electron dynamics and photoionization processes of diatomic molecules interacting with strong laser fields including electron - electron correlation effects . we combine the recently formulated time - dependent generalized - active - space configuration interaction theory [ d. hochstuhl and m. bonitz , phys . rev . a * 86 * , 053424 ( 2012 ) ; s. bauch et al . , phys . rev . a * 90 * , 062508 ( 2014 ) ] with a prolate spheroidal basis set including localized orbitals and continuum states to describe the bound electrons and the outgoing photoelectron . as an example , we study the strong - field ionization of the two - center four - electron lithium hydride molecule in different intensity regimes . by using single - cycle pulses , two orientations of the asymmetric heteronuclear molecule are investigated : , with the electrical field pointing from h to li , and the opposite case of . the preferred orientation for ionization is determined and we find a transition from , for low intensity , to , for high intensity . the influence of electron correlations is studied at different levels of approximation , and we find a significant change in the preferred orientation . for certain intensity regimes , even an interchange of the preferred configuration is observed , relative to the uncorrelated simulations . further insight is provided by detailed comparisons of photoelectron angular distributions with and without correlation effects taken into account .

1507.04107

c

in this paper we presented a time - dependent approach to correlated electron dynamics following the excitation of diatomic molecules with strong electromagnetic fields . the method is based on the td - gas - ci approach using a prolate - spheroidal representation of the single - particle orbitals within a partition - in - space concept to allow for good convergence of the truncated ci expansion . thereby , parts of the multi - particle wave function close - by the nuclei are represented within a hartree - fock - like orbital basis and the ejected part is represented in a grid - like fe - dvr basis set . we illustrated the method by its application to the calculation of angle - resolved photoelectron spectra of the four - electron heteronuclear lih molecule with and without taking electron - electron correlation contributions into account . to demonstrate the capabilities of the present approach , we then concentrated on the strong - field ionization of lih using single - cycle pulses . the ionization yield for the two opposite orientations of the molecule along the linearly polarized electric field was calculated and an intensity - dependent shift of the preferred configuration was observed : while for low intensities in the tunneling regime , ionization for is larger , for high intensities well above the barrier , shows higher yields . in between both regimes , a smooth transition is found . by turning on electronic correlations in the simulation , we find that especially yields in the tunneling regime are affected whereas the high - intensity regime is well described using the sae or cis approximations . correlations shift the preferred configuration from to for low intensities and vice versa for high intensities . in a certain intermediate intensity regime even an interchange of the preferred configuration in comparison to uncorrelated calculations is observed . additionally , angle - resolved photoionization distributions were presented and discussed , and the correlation effects were singled out by comparison to sae calculations . our results demonstrate the importance of electron - electron correlations in strong - field excitation scenarios of diatomic molecules . we expect the td - gas - ci approach in combination with the prolate spheroidal basis set to be applicable to larger systems such as the co molecule and to arbitrary polarization of the exciting pulse in the near future , where experimental data is available @xcite . further , the application to two - color excitation scenarios , such as streaking and xuv - xuv pump - probe , and the exploration of correlation effects in molecular systems on ultrashort time scales , e.g. , post - collision interaction effects @xcite or the time - delay in photoemission is within reach . the authors thank c. hinz for indispensable optimizations of the td - gas - ci code . the authors gratefully acknowledge discussions with l. b. madsen . h. r. larsson acknowledges financial support by the `` studienstiftung des deutschen volkes '' and the `` fonds der chemischen industrie '' . this work was supported by the bmbf in the frame of the `` verbundprojekt fsp 302 '' and computing time at the hlrn via grants ` shp00006 ` and ` shp00013 ` .

a * 90 * , 062508 ( 2014 ) ] with a prolate spheroidal basis set including localized orbitals and continuum states to describe the bound electrons and the outgoing photoelectron . as an example the preferred orientation for ionization is determined and we find a transition from , for low intensity , to , for high intensity . the influence of electron correlations is studied at different levels of approximation , and we find a significant change in the preferred orientation . for certain intensity regimes , even an interchange of the preferred configuration is observed , relative to the uncorrelated simulations . further insight is provided by detailed comparisons of photoelectron angular distributions with and without correlation effects taken into account .

we develop a time - dependent theory to investigate electron dynamics and photoionization processes of diatomic molecules interacting with strong laser fields including electron - electron correlation effects . we combine the recently formulated time - dependent generalized - active - space configuration interaction theory [ d. hochstuhl and m. bonitz , phys . rev . a * 86 * , 053424 ( 2012 ) ; s. bauch et al . , phys . rev . a * 90 * , 062508 ( 2014 ) ] with a prolate spheroidal basis set including localized orbitals and continuum states to describe the bound electrons and the outgoing photoelectron . as an example , we study the strong - field ionization of the two - center four - electron lithium hydride molecule in different intensity regimes . by using single - cycle pulses , two orientations of the asymmetric heteronuclear molecule are investigated : , with the electrical field pointing from h to li , and the opposite case of . the preferred orientation for ionization is determined and we find a transition from , for low intensity , to , for high intensity . the influence of electron correlations is studied at different levels of approximation , and we find a significant change in the preferred orientation . for certain intensity regimes , even an interchange of the preferred configuration is observed , relative to the uncorrelated simulations . further insight is provided by detailed comparisons of photoelectron angular distributions with and without correlation effects taken into account .

nucl-th0312021

i

for over forty years now the three - nucleon problem has , with considerable success , been used as a testing ground for the nucleon - nucleon interaction . usually the three - body equations used are based on the schrdinger equation or its implementation for scattering in the form of the faddeev @xcite equations with the two - body interaction being one of finite range . however , discussions of the case in which the range of the interaction is significantly less than the wavelengths of interest , i.e. @xmath0 , also have a long history . there has been renewed interest in this case with the advent of effective field theory ( eft ) descriptions of few - nucleon systems at low energy @xcite . in an eft treatment of the problem of two- and three - body scattering at energies such that @xmath1 the two - body scattering problem requires renormalization since the leading - order two - body potential is a three - dimensional delta function . but in low - energy @xmath2 scattering @xmath3 and @xmath4 are not the only scales in the problem . the presence of a low - energy bound state in the @xmath2 system the deuteron means that we must also account for the deuteron binding momentum @xmath5with @xmath6 the unnaturally - large @xmath2 scattering length when we do an eft analysis of this problem . in technical terms the presence of an enhanced two - body scattering length or equivalently a near - zero - energy bound state means that there is a non - trivial fixed point in the renormalization - group evolution of this leading - order potential . as long as this is accounted for , and a power counting built around the scale hierarchy : @xmath7 a systematic eft can be established and renormalized using a variety of regularization schemes @xcite . a similar scale hierarchy ( and hence a similar eft ) governs the low - energy interactions of helium-4 atoms @xcite . it is also relevant to the physics of bose - einstein condensates , if the external magnetic field is adjusted such that the atoms ( e.g. @xmath8rb ) are near a feshbach resonance @xcite . as a first step in extending this eft to heavier nuclei , the three - nucleon system was considered , and the faddeev equations for the particular case of a zero - range interaction were solved . it was soon discovered that the leading - order ( lo ) eft equation for the quartet ( total angular momentum @xmath9 ) channel yielded a unique solution @xcite , while for the doublet ( total angular momentum @xmath10 ) channel the corresponding equation did not yield a unique solution at least in the absence of three - body forces @xcite . this could be simply understood on the grounds that in the quartet channel the effective interaction between the neutron and the deuteron is repulsive as a result of the pauli principle , and this ultimately means that the neutron and deuteron do not experience a zero - range interaction . in contrast , in the doublet channel the effective neutron - deuteron interaction is attractive and the full difficulties of the zero - range interaction manifest themselves . these difficulties were first elucidated by thomas , who pointed out that if two - body forces alone are employed the nuclear force must have a finite range if the binding energy of nuclei is to be finite @xcite . the three - body scattering problem for zero - range interactions considered in the seminal work of bedaque and collaborators @xcite was first considered in research that antedates faddeev s landmark 1961 paper : by skorniakov and ter - martirosian @xcite and by danilov @xcite . these authors found similar difficulties to bedaque _ et al . _ , and traced the non - uniqueness to the fact that in the asymptotic region this three - body equation for scattering reduces to a homogeneous equation whose solution can be added to the solution of the inhomogeneous equation with an arbitrary weighting a point recently reiterated by blankleider and gegelia @xcite . in their 1999 papers @xcite , bedaque _ et al . _ introduced a three - body force into the leading - order three - body eft equation , so as to obtain a unique solution for 1 + 2 phase shifts . they adjusted this force in order to reproduce the experimental 1 + 2 scattering length . the energy dependence of the 1 + 2 phase shift was then predicted @xcite . the introduction of this three - body force is unexpected if naive dimensional analysis is used to estimate the size of various effects in the eft , but it is apparently necessary if the equations are to yield sensible , unique predictions for physical observables . this also accords with the 1995 paper of adhikhari , frederico , and goldman , who pointed out that the divergences in the kernel of the faddeev equations for a zero - range interaction may necessitate the introduction of a piece of three - body data so that these divergences can be renormalized away @xcite . ( but see refs . @xcite for a conflicting view . ) in an attempt to get some insight into alternative ways to establish a unique solution to the three - body scattering problem at leading order in the effective field theory , we try to bridge the gap between the faddeev approach in which the interaction has a finite range and the eft formulation of this problem . in sec . [ sec2 ] , we examine the amado model @xcite for the case of three spinless bosons . here we look at scattering in which the interaction of an incident particle on a composite system of the other two is considered within the framework of the lagrangian for the lee model @xcite . if three - body forces are neglected then the only difference between this approach and those at lo in the eft of refs . @xcite is that in the lee model lagrangian one may introduce a form factor that plays the role of a cut - off in the theory . in this way we can connect the lo eft equations ( without a three - body force ) to those found in the amado model , by taking the limit as the range of the interaction goes to zero . the resulting equation has a non - compact kernel unless a cutoff is imposed on the momentum integration . we then reproduce and reiterate the results of refs . @xcite , demonstrating that the low - energy solution of the equation changes radically as the cutoff is varied . using a renormalization - group analysis we trace this unreasonable cutoff dependence to the presence of eigenvalues equal to 1 in the kernel of the integral equation . in section [ sec3 ] we use a subtraction originally developed by hammer and mehen @xcite to remove these eigenvalues . our analysis of sec . [ sec2 ] then allows us to demonstrate that the subtracted three - body equation is renormalization - group - invariant . the subtraction of ref . @xcite was employed at a specific energy , and used experimental data from the three - body system to determine the half - off - shell behaviour of the 1 + 2-amplitude . here we go further , and show that using low - energy two - body data plus just one piece of experimental data for the three - body system the 1 + 2 scattering length we can predict the low - energy three - body phase shifts . we do this by first solving the subtracted integral equation for the half - off - shell threshold 1 + 2-amplitude . we then use this result to derive unique predictions for the full off - shell behaviour of the 1 + 2-threshold - amplitude , and thence for the 1 + 2-amplitude at any energy . the equations derived in this way are equivalent to those of bedaque _ et al . _ , but represent a reformulation of the problem in which only physical , renormalized quantities appear . in consequence the leading - order three - body force of refs . @xcite does not appear in our equations . our single subtraction ultimately allows us to generate predictions for the energy - dependence of the 1 + 2 phase shifts at leading order in the eft without the presence of an explicit three - body force . the subtraction does , though , require data from the three - body system ( namely the 1 + 2 scattering length ) before other three - body observables can be predicted . in sec . [ sec4 ] we apply the formalism of secs . [ sec2][sec3 ] to the conceptually identical but technically more complicated case of the doublet channel in nd scattering . here , we compare the numerical solution to our once - subtracted equation with phase - shift data . in sec . [ sec5 ] we consider higher - order corrections to the lo eft and illustrate that the results from the eft are in good agreement with the nd data below three - nucleon breakup threshold if the sub - leading ( two - body ) terms in the eft expansion are adjusted so as to reproduce the asymptotic s - state normalization of deuterium . the resulting description of the doublet phase shifts is very good up to the deuteron breakup threshold . finally in sec . [ sec6 ] we present some concluding remarks regarding the limitations of this method and discuss the convergence and usefulness of the eft .

we discuss effective field theory treatments of the problem of three particles interacting via short - range forces . one case of such a system is neutron - deuteron scattering at low energies . we also show that these eigenvalues can be removed from the kernel by one subtraction , resulting in an equation which is renormalization - group invariant . a unique solution for 1 + 2 scattering phase shifts we give an explicit demonstration of our procedure for both the case of three spinless bosons and the case of the doublet channel in nd scattering .

we discuss effective field theory treatments of the problem of three particles interacting via short - range forces . one case of such a system is neutron - deuteron scattering at low energies . we demonstrate that in attractive channels the renormalization - group evolution of the 1 + 2 scattering amplitude may be complicated by the presence of eigenvalues greater than unity in the kernel . we also show that these eigenvalues can be removed from the kernel by one subtraction , resulting in an equation which is renormalization - group invariant . a unique solution for 1 + 2 scattering phase shifts is then obtained . we give an explicit demonstration of our procedure for both the case of three spinless bosons and the case of the doublet channel in nd scattering . after the contribution of the two - body effective range is included in the effective field theory , it gives a good description of the nd doublet phase shifts below deuteron breakup threshold .

1503.06028

i

let @xmath0 be a non - empty bounded open subset of @xmath1 for @xmath2 and let @xmath3 be the dirichlet laplacian on @xmath0 . then the spectrum @xmath4 of @xmath5 is discrete and forms a positive increasing sequence @xmath6 where the eigenvalues are repeated according to their multiplicity . interest in the geometric information about @xmath0 encoded by @xmath4 started a little over 100 years ago and was crystallised by kac in his paper @xcite entitled ` can one hear the shape of a drum ? ' or more precisely , does @xmath4 determine @xmath0 up to isometry ? the answer to that question is no in general , as shown in @xcite ; see also @xcite for a concise presentation of a family of counterexamples . however some geometric information about @xmath0 can be recovered . weyl s theorem shows that the eigenvalue counting function @xmath7 defined by @xmath8 has asymptotic expansion @xmath9 as @xmath10 , for some constant @xmath11 depending only on @xmath12 , where @xmath13 denotes the @xmath12-dimensional lebesgue measure . aside from prompting kac s question this result has led to a large body of work on the behaviour of the eigenvalue counting function and we now give a very brief description of the results that have motivated the work we will present here . as a first extension it is natural to ask about the second order term in this expansion . if @xmath14 is smooth , then under some assumptions , that there are not too many periodic geodesics , the expansion has a second order term @xmath15 as @xmath10 , for some other constant @xmath16 depending only on @xmath12 . the reader is referred to @xcite and references therein for more information . this means that , under some regularity conditions , we can recover the size of the domain and that of the boundary from the spectral asymptotics ; in particular , using the isoperimetric inequality , we can determine whether or not @xmath0 is an open ball . interest in the second term of the expansion of @xmath7 grew further when berry studied the spectral asymptotics of domains with a fractal boundary in @xcite . he conjectured that the hausdorff dimension of @xmath14 should drive the second order term . brossard and carmona in @xcite studied the associated partition function , a smoothed version of the eigenvalue counting function , and showed that the minkowski dimension , @xmath17 , was the relevant notion of dimension for the second order term in the short time expansion of this function . for the counting function itself a general result of lapidus @xcite shows that , if @xmath18 , the second order term is of order @xmath19 provided the minkowski content of the boundary is finite . in general it is difficult to determine the precise order of growth for the second order term for arbitrary boundaries , however for one - dimensional domains @xcite it was shown that the minkowski dimension captures the order of growth of the second term in the asymptotics and the minkowski content , the constant , when they exist . the problem of determining the spectral asymptotics has also been considered for sets which are themselves fractal . for some classes of fractal , such as the sierpinski gasket , or more generally p.c.f . self - similar sets @xcite or generalised sierpinski carpets @xcite , a laplacian can be defined and shown to have a discrete spectrum . the exponent for the leading order growth rate in the eigenvalue counting function is called the spectral dimension and differs from the hausdorff or minkowski dimension of the set . if the fractal has enough symmetry , such as for instance the sierpinski gasket , then a weyl type theorem is no longer true @xcite , @xcite in that the rescaled limit of the eigenvalue counting function does not converge . however the weyl limit does exist for ` generic ' deterministic p.c.f . self - similar sets @xcite and also for random sierpinski gaskets @xcite and it is natural to ask about the growth of the second order term in these settings . our aim is to consider some random fractals where we anticipate more generic behaviour of the counting function . we will consider both domains with fractal boundaries and fractal sets here . firstly we will consider the case of open subsets with fractal boundaries in the one - dimensional case of a so called fractal string . our second case will be an example where the set itself is a fractal , the continuum random tree . in both cases the first order terms in the spectral asymptotics due to the fractal structure are understood and we will focus on the behaviour of the second order terms . a fractal string is a set obtained as the complement of a cantor set in the unit interval , so can be thought of as a sequence of intervals of decreasing length @xcite . the dirichlet laplacian is then the union of the dirichlet laplacians on each interval . some discussion of the spectral asymptotics of random fractal strings can be found in @xcite where it is shown that for cantor sets constructed via random iterated function systems , the second order term due to the boundary exists almost surely . we will consider a suitable subset of these random fractal strings and determine when the order of the fluctuations about the boundary term is given by a central limit theorem ( clt ) . this turns out to be a subtle question and the existence of a clt is determined by the rate of convergence in an associated renewal theorem . we will give a precise statement after introducing all the terminology in theorem [ thm::spectralasympstring ] . we will then show that when the fractal is generated using a dirichlet distribution , the existence of a central limit theorem depends on the particular dirichlet distribution considered . an example of what we are able to show is the following . let @xmath20 , for @xmath21 , be the random fractal string obtained as the complement of the random cantor set generated by subdividing any interval of length @xmath22 into three , retaining two intervals of size @xmath23 , and removing one of length @xmath24 , where the pair @xmath25 is independent for each interval and distributed as dirichlet(@xmath26 ) ( that is a beta(@xmath26 ) distribution in this simple case ) and @xmath27 . we write @xmath28 for the probability law for the random fractal string and @xmath29 for expectation with respect to @xmath28 . we note that @xmath30 will be the minkowski dimension of the random cantor set @xmath28-almost surely , that is the dimension of the boundary of the string . we write @xmath31 for the associated eigenvalue counting function . [ thm : ex1 ] ( i ) for all @xmath32 and @xmath33 there is a strictly positive deterministic constant @xmath34 such that as @xmath35 @xmath36 ( ii ) if @xmath37 , then there exists a strictly positive deterministic constant @xmath38 such that as @xmath35 @xmath39 where @xmath40 is normally distributed with mean 0 and variance @xmath41 . + ( iii ) there exists an @xmath42 and a @xmath33 such that : if @xmath43 , then there exists a not - identically - zero periodic function @xmath44 such that @xmath45 where @xmath46 , @xmath47 and , for this range of @xmath48 we have @xmath49 . in particular @xmath50 does not converge in distribution as @xmath35 . \(1 ) the first result gives the almost sure behaviour of the second term in the counting function asymptotics and is true for random fractal strings constructed using a wide class of distributions on the simplex . + ( 2 ) in part ( iii ) we conjecture that it is possible to take @xmath51 and any @xmath52 . indeed , towards proving the above result , we first provide conditions under which a clt holds ( see theorem [ thm::spectralasympstring ] and section [ subsec::numericalexample ] ) , and explain when one will not ( see remark [ rmk::sharpspeedconvergence ] ) . this distinction is determined by the rate of convergence in a related renewal theorem and depends on the values of the roots of @xmath53 , which we solve numerically ( we can also solve this equation analytically for small values of @xmath48 ) . these computations demonstrate that we can take @xmath54 to be at least @xmath55 . furthermore , although we are not able to prove it rigorously , the monotonicity of the results suggests that @xmath54 can be taken arbitrarily large . + ( 3 ) we also conjecture that , in the case where there is no clt , i.e. @xmath56 , the size of the second order term is determined by @xmath57 , in that , @xmath28 almost surely for @xmath58 , @xmath59 where @xmath60 and @xmath61 as @xmath62 . + ( 4 ) the proof of the above result shows that the period of @xmath63 is given by @xmath64 , where @xmath65 is one of the complex conjugate pair of roots whose real part gives @xmath57 . observe that , as @xmath48 increases , the beta@xmath66 distribution becomes closer to the distribution given by a delta measure at the point ( 1/2,1/2 ) . if we take @xmath67 , then we anticipate that our random fractal string should converge ( in a suitable sense ) to the cantor string ( the string formed as the complement of the classical ternary cantor set ) as @xmath48 goes to infinity . it is known that for the cantor string there is a non - constant periodic function that appears in the second order term in the counting function asymptotics @xcite . thus our result suggests that there is a non - trivial transition in the parameter space from the case where there is ` enough randomness ' for a clt about the second order term , to the case where there is not , through to the limit , where there is not even a strong law of large numbers for this term . we will also consider the case of the brownian continuum random tree , a random self - similar fractal . it was shown in @xcite that there was a weyl limit for the counting function in this case . it was also shown that the second order term for this fractal set was of order 1 in mean which would be anticipated as the boundary of the tree is just two points , a 0-dimensional set . in this paper we show that there is a clt about the almost sure asymptotics . however at this point we have not shown strict positivity of the variance due to the complexity of the correlation structure in the variance of the limit of the rescaled counting function . we conjecture that there will be a non - trivial clt for this counting function . this will show that the randomness in the construction means the second order term in the spectral asymptotics is determined by the fluctuations about the leading order term , as these are much greater than the effects due to the boundary of the set . the main technical tool we develop is a central limit theorem for the general crump - mode - jagers branching process . in our setting the random fractal sets , the random cantor set boundary of the string , or the continuum random tree , can be encoded as general branching processes . we are able to use a characteristic associated with these processes to determine the behaviour of the counting function . in this case there may be dependence on the offspring of an individual and we obtain a clt in this more general setting , extending the work of @xcite . we also remark that the techniques used here can easily be applied to geometric counting functions or other functions associated with heat flow , such as the partition function or heat content of the set . we anticipate similar behaviour in the fluctuations of these quantities about their almost sure limits . the paper is organised as follows . in section [ bpcltsec ] , we recall the definition of the general branching process and some laws of large numbers for such processes . we then prove our central limit theorem for the general branching process using a taylor expansion proof . in section [ sec::dirichletweights ] , we restrict ourselves to general branching processes where a suitable function of the birth times is chosen to lie on an @xmath68-dimensional simplex , which will ensure that the limit of the usual branching process martingale is a constant . we will call such processes @xmath69-gbps and discuss extensively how to establish the conditions required for the central limit theorem in this setting as this will allow us to illustrate when we do and do not have a central limit theorem for the associated general branching process . in section [ sec::cantorstrings ] , we define a family of open subsets @xmath70 of @xmath71 $ ] whose random boundary is a statistically self - similar cantor set built using scale factors on the simplex . we are then able to show our main result which gives conditions for the existence of a central limit theorem . in section 5 we consider some examples where the law of the @xmath69-gbp is given by a dirichlet distribution . we show that , for some dirichlet weights , the eigenvalue counting function of the set @xmath70 satisfies a central limit theorem . as a consequence we will be able to establish theorem [ thm : ex1 ] . in section 6 we turn to the continuum random tree . we recall that this tree can be viewed as a random self - similar set and how to construct a laplace operator on it . we then show that the conditions for the general branching process central limit theorem hold and hence there is a clt in the spectral asymptotics . for convenience , we will use the shorthand notation @xmath72 with @xmath73 to mean some positive constant whose value is fixed for the length of a proof or a subsection .

we will show examples from a class of random fractals generated from dirichlet distributions as this is a relatively simple setting in which there are sets where there will and will not be a central limit theorem . the brownian continuum random tree can also be viewed as a random fractal generated by a dirichlet distribution . the first order term in the spectral asymptotics is known almost surely and here we show that there is a central limit theorem describing the fluctuations about this , though the positivity of the variance arising in the central limit theorem is left open . in both cases these fractals can be described through a general crump - mode - jagers branching process and we exploit this connection to establish our central limit theorems for the higher order terms in the spectral asymptotics . our main tool is a central limit theorem for such general branching processes which we prove under conditions which are weaker than those previously known . + * msc : * 28a80 ( primary ) , 60j80 , 35p20 ( secondary ) .

we discuss the spectral asymptotics of some open subsets of the real line with random fractal boundary and of a random fractal , the continuum random tree . in the case of open subsets with random fractal boundary we establish the existence of the second order term in the asymptotics almost surely and then determine when there will be a central limit theorem which captures the fluctuations around this limit . we will show examples from a class of random fractals generated from dirichlet distributions as this is a relatively simple setting in which there are sets where there will and will not be a central limit theorem . the brownian continuum random tree can also be viewed as a random fractal generated by a dirichlet distribution . the first order term in the spectral asymptotics is known almost surely and here we show that there is a central limit theorem describing the fluctuations about this , though the positivity of the variance arising in the central limit theorem is left open . in both cases these fractals can be described through a general crump - mode - jagers branching process and we exploit this connection to establish our central limit theorems for the higher order terms in the spectral asymptotics . our main tool is a central limit theorem for such general branching processes which we prove under conditions which are weaker than those previously known . + * msc : * 28a80 ( primary ) , 60j80 , 35p20 ( secondary ) .

astro-ph9708231

i

the classification of an active galactic nucleus ( agn ) depends upon the wavelength at which one observes . historically , optical observations yielded the categories of seyfert types 1 and 2 , classifying most of the sources which are now `` famous '' . seyfert-2 galaxies differ from type-1 in that the former show only narrow emission lines in their optical spectra . it was soon realized that some seyferts showed weak broad components along with the narrow emission lines and consequently the subclasses seyfert-1.5 , 1.8 and 1.9 were introduced to quantify the differences in strength of the broad - line components relative to the narrow lines . narrow emission line galaxies ( nelgs ) are bright and variable x - ray sources , discovered in early x - ray sky surveys ( marshall 1979 ) . the narrow optical emission lines often have weak , broad h@xmath0 and p@xmath1 emission ( ward 1978 , veron 1980 , shuder , 1980 ) making the optical spectra similar to seyfert-1.9 galaxies . thus the nelg classification is indicative that the source was discovered in an x - ray survey , but many nelgs are otherwise indistinguishable from seyfert-1.9 galaxies . optical spectroscopy and spectropolarimetry , infrared spectroscopy , x - ray spectroscopy and temporal studies plus @xmath2-ray spectra are all important in the determination of the fundamental nature of obscured nuclei . unification models for agn postulate that large amounts of dense , molecular material exists between the broad - emission - line region ( blr ) and the narrow - emission - line region ( nlr ) , in some cases within parsecs of the active nucleus ( see antonucci 1993 for a review of unified models for agn ) . the simplest geometry consistent with observations is a torus , and consequently it has been suggested that one of the primary factors in seyfert classification is the orientation of the absorbing torus to our line - of - sight . this hypothesis is consistent with the existence of circumnuclear molecular gas suggested by absorption measurements in a number of wavebands ( e.g. braatz 1993 , greenhill 1996 ) . in unified models , sources observed within the opening angle of the torus correspond to those classified optically as type-1 agn , while sources with lines - of - sight intersecting the torus correspond to type-2 agn . in the latter case , the nuclear light can nevertheless be observed via scattering or transmission . antonucci & miller ( 1985 ) provided compelling support for this model when they detected broad , seyfert-1 type emission lines in the polarized optical spectrum of the seyfert-2 galaxy ngc 1068 , and similar results were later obtained for a number of seyfert-2 galaxies ( miller and goodrich 1990 ; tran , miller and kay 1992 ) . recently , veilleux ( 1997 ) provided further support for the model with their infrared observations of pa@xmath1 , br@xmath2 and br@xmath0 lines in many seyfert 2 galaxies , revealing hidden blrs which were not always detectable in scattered optical light . optically - thick torii would be expected to result in collimation of nuclear continuum radiation , and imaging of optical emission lines has shown preferential elongation of some narrow - line regions along the radio axis of the agn ( haniff , wilson & ward 1988 ) . these regions must be ionized by a more intense radiation field than is directly observed , again , supporting the unified model ( e.g. pogge 1988 , tadhunter and tsvetanov 1989 ) . further support came from x - ray spectra , which showed large absorbing columns and iron k@xmath0 lines of high equivalent width ( ew ) ( e.g. awaki 1991 ) . the _ asca _ satellite ( makishima et al . 1996 ) consists of four co - aligned grazing - incidence x - ray telescopes ( xrts ; serlemitsos et al . the focal - plane instruments are two solid - state imaging spectrometers ( siss ) , each consisting of four ccd chips , providing an effective bandpass @xmath30.410 kev ( burke et al . 1994 ) , and two gas imaging spectrometers ( giss ) at the focus of the other two xrts , providing coverage over @xmath30.810 kev ( ohashi et al . 1996 and references therein ) . the sources presented here were systematically analyzed in the same way as the broad - line seyfert galaxies presented in nandra 1997 ( hereafter n97 ) . the analysis method is also described in paper i. in a previous paper we presented the basic data - analysis results from a sample of _ asca _ observations of type-2 seyfert galaxies ( turner 1997a , hereafter paper i ) , i.e. those having predominantly narrow optical emission lines . the original sample of 26 observations of 25 narrow - line agn comprised 17 seyfert-2 galaxies and 8 nelgs drawn from the _ asca _ public archive . in paper i we found the 0.6 - 10 kev spectra of a sample of seyfert-2 galaxies and nelgs to be complex , often containing a heavily - absorbed continuum component , a soft x - ray component and numerous x - ray emission lines . we found the 6 - 7 kev regime to be dominated by line flux from gas with ionization - state @xmath4 fe xvi . several sites are expected to produce significant x - ray line emission in agn including the line - of - sight absorber , optically - thick material out of the line - of - sight ( both the putative accretion disk and other larger - scale systems such as the torus ) , ionized ( scattering ) gas and regions of starburst emission . the absence of a strong 6.4 kev iron line component in starburst galaxies ( ptak 1997 ) , indicates that the presence of such a line is likely to be an indication of nuclear activity . iron k@xmath0 yields the strongest observed x - ray emission line in seyfert galaxies , and thus provides an important probe of conditions in the reprocessing material . a discussion of sources dominated by scattered and compton - reflected x - rays was presented in a second paper ( turner 1997b , hereafter paper ii ) . fig . 1 ( repeated from paper ii , for ease of reference ) shows the equivalent width of the iron k - shell line plotted against neutral x - ray absorbing column , @xmath5 , for the sample sources . equivalent widths were measured against the continuum component dominating the 6 - 8 kev range , and based upon the fit to a narrow gaussian profile ( see paper i for details ) . the dot - dashed line in fig 1 denotes the equivalent width of iron k@xmath0 predicted to be produced by transmission through a uniform shell of neutral material ( with solar abundances subtending 4@xmath6 to a continuum source of photon index @xmath7 , leahy & creighton 1993 ) , where the photon flux @xmath8 . the dashed line shows the equivalent width predicted via compton - reflection from optically - thick material , as a function of , assuming that only the power - law is absorbed . in this case the reflection is assumed to be produced from the accretion disk with an equivalent width @xmath9 ev ( n97 ) , as typically observed in seyfert-1 galaxies . coincidently , 230 ev represents both the maximum equivalent width observed when iron k@xmath0 was parameterized as a narrow gaussian line , and the mean equivalent width assuming a relativistic line profile ( n97 ) . sources can lie significantly above both of these model lines if the direct continuum is hidden but the reprocessed emission is observed , as the line equivalent width is then measured against a suppressed continuum . consideration of the iron k@xmath0 , ` [ ` oiii ` ] ` @xmath10 line and x - ray variability together suggested that ngc 1068 , ngc 4945 , ngc 2992 , mrk 3 , mrk 463e and mrk 273 are dominated by reprocessed x - rays ( paper ii ) . these sources were denoted `` group c '' ( marked with squares on fig . 1 ) . sources lying on the `` leahy and creighton line '' were denoted `` group a '' ( marked as circles on fig . 1 ) . thus group a is composed of ngc 1808 , ngc 4507 , ngc 5252 , ngc 6240 , eso 103-g35 , ic 5063 , ngc 7172 and ngc 7582 . ( table 1 shows the group designation of the sources based upon the original sample ) . sources with iron k@xmath0 equivalent widths lying between that line and the 230 ev line are consistent with seyfert-1 spectra transmitted through a high absorbing column and are denoted `` group b '' ( marked as stars on fig . 1 ) . group b is composed of ngc 526a , ngc 2110 , mcg-5 - 23 - 16 and ngc 7314 , which are all nelgs . this classification implies we see the nuclear component directly in group b , and this is supported by the observation of rapid x - ray variability in their flux ( paper i and hayashi 1996 ) and rapid variability of the iron line profile in ngc 7314 ( yaqoob 1996 ) . this division into groups a , b and c leaves mcg-01 - 01 - 043 , ngc 4968 , ngc 6251 , e 0449 , ngc 5135 and ngc 1667 unclassified , i.e. those with the lowest signal - to - noise ratio in the data . as we will demonstrate , this crude classification of sources yields a useful insight into the relative importance of the regions contributing to the x - ray spectra in several different cases . the distribution of sample sources is shown in table 2 . to summarize , in paper ii we showed that in group - c sources the iron k@xmath0 complex contains significant contributions from neutral and high - ionization species of iron , thus compton - reflection , hot gas and starburst emission all could make significant contributions to the observed x - ray spectra . mrk 3 appeared to be the only source which had little contamination by starburst activity and in this case the _ asca _ spectrum below 3 kev is dominated by gas with an x - ray ionization parameter @xmath11 ( as defined by netzer 1996 ) and effective column density @xmath12 . this material is more highly ionized than the zone of material comprising the warm absorber seen in seyfert-1 galaxies ( george 1997 , hereafter g97 ) , but may contain a contribution from shock - heated gas associated with the jet . in this paper the x - ray properties of sources in groups a and b are examined in the context of unified models for agn .

we discuss the spectral properties of a sample of type-2 seyfert galaxies based upon the analysis of data . in this paper we consider the sources for which the x - ray spectra appear to be dominated by the nuclear continuum , transmitted through a large column of absorbing material . we find that both seyfert-2 galaxies and nelgs show iron k line profiles indicative of reprocessing of nuclear x - rays in a face - on accretion disk .

we discuss the spectral properties of a sample of type-2 seyfert galaxies based upon the analysis of data . in this paper we consider the sources for which the x - ray spectra appear to be dominated by the nuclear continuum , transmitted through a large column of absorbing material . we find that both seyfert-2 galaxies and nelgs show iron k line profiles indicative of reprocessing of nuclear x - rays in a face - on accretion disk . such line profiles are also observed in seyfert-1 galaxies . this result is contrary to unification models , which would predict the inner regions of seyfert-2 galaxies to be observed edge - on . this raises some questions as to the orientation of the circumnuclear absorber . if the observed differences between seyfert type-1 and type-2 galaxies , and nelgs are not due to differences in the orientation of the absorbing material , then we suggest that differences in dust composition and grain size , and in the density of the circumnuclear gas could be of primary importance . = = = 1=1=0pt = 2=2=0pt = 2=2=0pt

1207.3810

i

in its most general form , the dynamical state of the interior of a star is one of differential rotation and entropy stratification . if isobaric and isochoric surfaces do not coincide , the angular velocity need not be constant on cylinders . a noteable example is the sun , for which helioseismology studies have fashioned a remarkably detailed and rich portrait . where the sun is stably stratified in entropy , in the bulk of the radiative zone , it tends not to be differentially rotating . however , surrounding the radius of vanishing entropy gradient , in both the convective _ and _ radiative layers , there is significant differential rotation generally dominated by the radial component of the angular velocity gradient . higher in the convection zone , the rotation contours show an abrupt change in morphology , with the sudden emergence of a distinctly conical pattern ( e.g. miesch & toomre 2009 ) . finally , approaching the sun s surface , there is once again an abrupt shift in contour morphology , apparently associated with the onset of high - velocity convection . in previous work ( balbus , latter , & weiss 2012 [ blw ] and references therein ) , it has been shown that the pattern of coaxial cones in the bulk of the solar convective zone ( scz ) can be understood as an elementary solution of the vorticity equation ( in the limit of thermal wind balance ) under certain well - posed assumptions . one of these assumptions is that a small angular entropy gradient is present , for without this there can be no axial component of the angular velocity gradient . where does this all - important entropy gradient come from ? is it , as is often argued , an ineluctable consequence of convection and the coriolis force , or is something more or something else involved ? for that matter , is the entropy gradient more or less fundamental than the concomitant angular velocity gradient ? to address these questions , we begin with a very general study of the linear behavior of three - dimensional fluid displacements in a shearing and stratified background medium . the background angular velocity and entropy profiles may depend upon both poloidal coordinates . it is demonstrated that for a medium in uniform rotation , the most unstable displacements do _ not _ deflect from spherically radial paths , despite the presence of coriolis forces . when a axial component of the angular velocity gradient is _ already _ present however , it is shown that there is a significant polar deflection of higher entropy fluid elements . more precisely , the sense of this deflection is poleward for outward - moving displacements if the axial angular velocity gradient is negative ( as in the sun ) , and _ equatorial _ if this gradient is positive . thus , a axial angular angular velocity gradient is self - reinforcing , and may thus be reshaped , in a convective fluid . angular velocity gradients in cylindrical radius are much less effective in this regard : they have no first order effect on convective displacements . because a background axial angular velocity gradient requires a vorticity source , this has far reaching consequences , and we use this finding as a thin edge of wedge to pry further into the origins of the sun s baroclinic differential rotation . we argue in particular that it is likely that the rotation pattern of the scz has emerged by responding to a preexisting angular entropy gradient , rather than generating such a gradient internally . this is entirely consistent with the experience of numerical simulations , in which rotation on cylinders stubbornly persists unless latitudinal entropy boundary conditions are present , in which case solar - like profiles emerge relatively easily ( e.g. miesch , brun , toomre 2006 ) . indeed , if the conclusions of this paper are well - founded , the direct imposition of such boundary conditions is the `` correct '' procedure ! in the second part of this paper , we put forth the case that vorticity generation is all but inevitable near the outer edge of the radiative zone where the entropy gradient vanishes . the combination of diffusive heating and centrifugal distortion of equipotential surfaces is incompatible with radiative and dynamical equilibrium in a uniformly rotating medium . the classic remedy of introducing a tiny amount of meridional circulation ( e.g. , schwarzschild 1958 ) breaks down at a surface of zero entropy gradient . instead , radiative equilibrium is re - established with very slightly different isothermal and isochoric surfaces . if , as one would expect from radiative considerations , the isotherms are more spherical than the isochoric surfaces , an incipient tachocline is generated , bearing many of the features observed of the true solar tachocline : a negative axial gradient of the angular velocity everywhere , a dominant ( spherical ) radial component of this gradient , and an increasingly dominant cylindrical disposition of the isorotation contours toward the equator . the generation of vorticity at a level stemming from the centrifugal distortion of the equipotential surfaces has not been hitherto viewed as an important component of the solar differential rotation profile . however , not only is the centrifugal distortion of precisely the correct order - of - magnitude for this problem , we are argue that it is the principal causal agent . if this is correct , numerical simulations whose goal is to reproduce the sun s internal rotation accurately from first principles will ultimately have to accomodate this @xmath0 effect . the organization of the paper is as follows . in 2 , we present the governing equations for three - dimensional fluid displacements in an axisymmetric but otherwise fully general entropy - stratified , shearing background . this is an interesting gasdynamical problem in its own right , and particularly relevant for the sun . general solutions are presented in 3 , but we focus on the most rapidly growing modes , for which a simple analysis is possible . the solution shows explicitly the relationship between shear , coriolis forces , and the deflection of convective trajectories . in 4 , we integrate our findings with standard solar models , arguing that the seed angular entropy gradient is a result of centrifugal distortion of equipotential surfaces in the radiative zone together with the disappearance of the entropy gradient at the scz boundary . finally , 5 summarizes our results .

we examine the linear behavior of three - dimensional lagrangian displacements in a stratified , shearing background . the isentropic and iso - rotation surfaces of the equilibrium flow are assumed to be axisymmetric , but otherwise fully two - dimensional . the model , in principle very general , is used to study the behavior of fluid displacements in an environment resembling the solar convection zone . the tendency for the angular velocity to remain constant with depth in the bulk of the convective zone , together with other critical features of the rotation profile , emerge from little more than a visual inspection of the governing equation . in the absence of a background axial angular velocity gradient , displacements exhibit no poleward bias , suggesting that solar convection `` plays - off '' of prexisting shear rather than creates it . we argue that baroclinic vorticity of precisely the right order is generated at the radiative / convective zone boundary due to centrifugal distortion of equipotential surfaces that is not precisely followed by isothermal surfaces .

we examine the linear behavior of three - dimensional lagrangian displacements in a stratified , shearing background . the isentropic and iso - rotation surfaces of the equilibrium flow are assumed to be axisymmetric , but otherwise fully two - dimensional . three - dimensional magnetic fields are included in the perturbation equations ; however the equilibrium is assumed to be well - described by purely hydrodynamic forces . the model , in principle very general , is used to study the behavior of fluid displacements in an environment resembling the solar convection zone . some very suggestive results emerge . all but high - latitude displacements align themselves with the observed surfaces of constant angular velocity . the tendency for the angular velocity to remain constant with depth in the bulk of the convective zone , together with other critical features of the rotation profile , emerge from little more than a visual inspection of the governing equation . in the absence of a background axial angular velocity gradient , displacements exhibit no poleward bias , suggesting that solar convection `` plays - off '' of prexisting shear rather than creates it . we argue that baroclinic vorticity of precisely the right order is generated at the radiative / convective zone boundary due to centrifugal distortion of equipotential surfaces that is not precisely followed by isothermal surfaces . if so , many features of the sun s internal rotation become more clear , including : i ) the general appearance of the tachocline ; ii ) the extension of differential rotation well into the radiative zone ; iii ) the abrupt change of morphology of convective zone isorotation surfaces ; and iv ) the inability of current numerical simulations to reproduce the solar rotation profile without imposed entropy boundary conditions .

1207.3810

c

the current work combines two very different types of calculation , linking the linear dynamics of the convective zone to the origins of solar differential rotation and vorticity . we begin with the second part first . we have argued that the centrifugal distortion of equipotential surfaces combined with demands of thermal equilibrium requires the cleaving of isobaric and isochoric surfaces , and is likely to be the underlying cause of the sun s differential rotation . current numerical simulations are not designed to capture a process in which an order @xmath172 radiative effect is turned into relative angular velocity gradients of order @xmath173 . it may well be possible , however , to devise other computational schemes tailored to working with this point - of - view . an important technical point is that the tachocline pressure and density ( and therefore entropy ) must have both @xmath174 and @xmath133 angular structure . although this in itself is not a particularly new result , neither has it been widely appreciated , and we have exploited it in a rather novel manner . with @xmath145 and @xmath129 structure in place , not only may one understand the simple form of the vorticity equation in the tachocline , with reasonable approximations one may explicitly solve the equation . an important parameter of this equation , namely the precise angle at which @xmath175 changes sign , is probably determined by minimizing the torque on the radiative interior . finally , it is interesting to note that roxburgh ( 2001 ) calculated the sun s multipole moments using models based on helioseismic inversions . he found that the size of the octopole term @xmath176 was comparable to the _ change _ in the quadrupole term @xmath177 when differential rotation was included . this is what one would expect if differential rotation were _ modifying _ @xmath129 and _ creating _ @xmath145 angular structures . in the first part of this paper , we have carried out a very general lagrangian linear calculation of the fluid displacements in an arbitrary , magnetized , two - dimensional background , stratified in both @xmath1 and @xmath3 ( or @xmath4 and @xmath5 ) . applied to conditions appropriate to the scz , in the absence of a background @xmath178 gradient , we find no poleward deflection of hot convective fluid elements . such an effect is sometimes invoked to produce an angular entropy gradient which would in turn lead to differential rotation via thermal wind balance . although there is nothing in our calculations that would prohibit the emergence of the required gradients in @xmath7 and @xmath3 at nonlinear order in a turbulent fluid , it is some significance that the dominant leading order linear response is completely different depending upon whether @xmath178 is present or absent . when a finite @xmath178 is _ a priori _ present , there are reinforcing convective deflections : a negative axial @xmath7 gradient , in particular , engenders poleward deflections of warmer fluid elements , which via thermal wind balance strengthen and maintain this same @xmath7 gradient . the presence or absence of background baroclinic vorticity is thus mirrored in the leading behavior of convective displacements . the actual generation of baroclinic vorticity may well lie outside the realm of convection dynamics . we propose that a vorticity source will inevitably appear at the location of a vanishing entropy gradient . in general , because of the effects centrifugal flattening , the constraints of thermal and dynamical equilibrium will force different iso - surfaces for density , temperature , and pressure . this must lead to an axial angular velocity gradient . by making some simplifying but plausible approximations , one may calculate a time - steady solution of the vorticity equation for the angular velocity . this explicit solution yields a tachocline structure that certainly resembles the observations , and bears comparison with an earlier , more accurate , but also more phenomenological , calculation ( blw ) . with the onset of fully developed convection , surfaces of constant angular velocity and residual entropy coincide , and the character of the isorotation contours takes on the classical conical form well - known from helioseismology . the strength of this view of the origin of solar differential is that it emerges from just a few rather simple and largely inevitable processes : the demands of thermal energy balance in a rotating system ( which creates a baroclinic axial gradient of @xmath7 at the radiative / convective boundary ) , the kinematics of shear ( which , via embedded wavenumbers , incorporates @xmath7 gradients into the response of the fluid displacements ) , and the linear dynamics of convection ( which causes poleward displacements of entropy bearing fluid elements ) . while direct numerical simulation of this scenario is likely to be very challenging , it may well be possible to design a test - of - principle proxy system .

all but high - latitude displacements align themselves with the observed surfaces of constant angular velocity . if so , many features of the sun s internal rotation become more clear , including : i ) the general appearance of the tachocline ; ii ) the extension of differential rotation well into the radiative zone ; iii ) the abrupt change of morphology of convective zone isorotation surfaces ; and iv ) the inability of current numerical simulations to reproduce the solar rotation profile without imposed entropy boundary conditions .

we examine the linear behavior of three - dimensional lagrangian displacements in a stratified , shearing background . the isentropic and iso - rotation surfaces of the equilibrium flow are assumed to be axisymmetric , but otherwise fully two - dimensional . three - dimensional magnetic fields are included in the perturbation equations ; however the equilibrium is assumed to be well - described by purely hydrodynamic forces . the model , in principle very general , is used to study the behavior of fluid displacements in an environment resembling the solar convection zone . some very suggestive results emerge . all but high - latitude displacements align themselves with the observed surfaces of constant angular velocity . the tendency for the angular velocity to remain constant with depth in the bulk of the convective zone , together with other critical features of the rotation profile , emerge from little more than a visual inspection of the governing equation . in the absence of a background axial angular velocity gradient , displacements exhibit no poleward bias , suggesting that solar convection `` plays - off '' of prexisting shear rather than creates it . we argue that baroclinic vorticity of precisely the right order is generated at the radiative / convective zone boundary due to centrifugal distortion of equipotential surfaces that is not precisely followed by isothermal surfaces . if so , many features of the sun s internal rotation become more clear , including : i ) the general appearance of the tachocline ; ii ) the extension of differential rotation well into the radiative zone ; iii ) the abrupt change of morphology of convective zone isorotation surfaces ; and iv ) the inability of current numerical simulations to reproduce the solar rotation profile without imposed entropy boundary conditions .

hep-ph9906487

i

in recent years solar - neutrino and atmospheric - neutrino measurements have provided a growing body of evidence for the existence of neutrino oscillations @xcite . the observed solar neutrino deficit @xcite can be interpreted as evidence for the oscillation of electron neutrinos ( @xmath8 ) into neutrinos of a different flavor . the recent atmospheric - neutrino results from the super - kamiokande collaboration @xcite , along with other experiments @xcite suggest the oscillation of muon neutrinos ( @xmath9 ) into neutrinos of a different flavor , which are dominantly either tau neutrinos ( @xmath10 ) or sterile neutrinos ( @xmath11 ) . taken together , these results suggest the mixing of at least three different neutrino types @xcite . in addition , the lsnd collaboration has reported @xcite results from @xmath12 decays at rest that could be interpreted as the first evidence for @xmath13 oscillations in an accelerator based experiment . lsnd has also reported @xcite results from measurements of @xmath14 decays in flight that could be the first evidence for @xmath2 oscillations . if all of the above reported effects survive , then oscillations to a sterile neutrino are required , since there would then be three distinct mass - squared difference ( @xmath15 ) values @xcite . the solar - neutrino , atmospheric - neutrino , and lsnd results have generated much interest in future accelerator - based neutrino oscillation experiments . we can anticipate that an extensive experimental program will be needed to firmly establish the existence of neutrino oscillations and to precisely determine all the parameters relevant to the phenomenon . the up - coming accelerator - based experiments @xcite will firmly ground the existence of neutrino oscillations , and measure some of the associated parameters . also , reactor experiments will tightly constrain @xmath8 disappearance @xcite ; a sensitivity down to @xmath16 ev@xmath6 is expected in kamland . however , precise measurements of neutrino mass - squared differences and the mixing matrix that relates neutrino - flavor eigenstates to neutrino - mass eigenstates will almost certainly require a further generation of experiments exploiting higher intensity and/or higher quality neutrino beams than currently available @xcite . other goals of such experiments might be ( i ) to determine whether the atmospheric neutrino oscillations are @xmath17 , @xmath1 , or a mixture of both @xcite , ( ii ) to detect earth matter effects on neutrino oscillations @xcite , ( iii ) when matter effects are present , to determine the ordering of the neutrino mass eigenstates responsible for the oscillation being measured @xcite , and ( iv ) to detect @xmath4 and @xmath18 violation if it exists in the lepton sector @xcite . it has been suggested @xcite that higher intensity and higher quality neutrino beams could be made by exploiting the very intense muon sources that are currently being developed as a part of ongoing muon collider @xcite feasibility studies . the muons would be accelerated up to the desired energy , and injected into a storage ring consisting of two long straight sections joined together by two arcs . muons that decay in the straight sections would form neutrino beams consisting of 50% @xmath9 and 50% @xmath19 if negative muons are stored , and 50% @xmath8 and 50% @xmath20 if positive muons are stored . a compact muon storage ring neutrino source could be tilted downwards at a large angle to enable neutrino beams to be directed through the earth . if the muons from a muon collider muon source are accelerated to energies of @xmath21 gev and injected into a suitable storage ring , it has been shown that the neutrino fluxes are sufficient to detect hundreds of neutrino charged current ( cc ) interactions per year in a reasonable size detector on the other side of the earth @xcite . the muon storage ring neutrino source idea has led to several recent workshops @xcite . a number of papers @xcite have discussed the physics potential of this new type of neutrino facility . in addition , a preliminary muon storage ring neutrino source design study has been made @xcite , and further studies are planned . the evolution of the existing accelerator complexes at fermilab @xcite and cern @xcite towards a muon collider with a muon storage ring neutrino source has also been considered . much further work will be required to develop a realistic design before the first muon storage ring neutrino source can be proposed . in this paper we consider the physics potential of muon storage ring neutrino sources that are being considered in explicit upgrade scenarios for the fermilab accelerator complex . two different geometries are considered : ( i ) a neutrino source at fermilab pointing toward the soudan mine in minnesota ( @xmath22 km ) , and ( ii ) a neutrino source at fermilab pointing toward the gran sasso underground laboratory in italy ( baseline length @xmath23 km ) . the paper is organized as follows . in section ii the characteristics of neutrino beams from muon storage ring sources are discussed and the basic oscillation formulas are presented . the role of muon storage ring neutrino sources in exploring the parameters associated with @xmath0 and @xmath1 oscillations is considered in section iii . section iv discusses @xmath24 oscillations , including matter effects and the possibility of detecting @xmath4 violation . in section v we discuss @xmath25 oscillations . finally , a summary is given in section vi .

we examine the physics capabilities of known flavor neutrino beams from intense muon sources . furthermore , they can test whether the dominant atmospheric neutrino oscillations are and/or , determine the content of atmospheric neutrino oscillations , and measure appearance . depending on the oscillation parameters , they may be able to detect earth matter and violation effects and to determine the ordering of some of the mass eigenstates . to v. barger , s. geer , and k.

we examine the physics capabilities of known flavor neutrino beams from intense muon sources . we find that long - baseline neutrino experiments based on such beams can provide precise measurements of neutrino oscillation mass and mixing parameters . furthermore , they can test whether the dominant atmospheric neutrino oscillations are and/or , determine the content of atmospheric neutrino oscillations , and measure appearance . depending on the oscillation parameters , they may be able to detect earth matter and violation effects and to determine the ordering of some of the mass eigenstates . to v. barger , s. geer , and k. whisnant _department of physics , university of wisconsin , madison , wi 53706 , usa +fermi national accelerator laboratory , p.o . box 500 , batavia , il 60510 , usa +department of physics and astronomy , iowa state university , ames , ia 50011 , usa _ 14.60.pq , 13.15.+g , 13.35.bv , 07.77ka

1101.1139

i

quantum optics is governed by rules imposed by commutation relations which have to be kept during time evolution . optical amplification is no exception to this story . typically , the amplified output is suffered from inevitable excess noise . this limitation is quantum - mechanically imposed , thus does not depend on the specific realization methods . caves classified general linear amplification into phase - insensitive amplification ( pia ) and phase - sensitive amplification ( psa ) @xcite . he also systematically derived the quantum limit of excess noise for such general linear amplification with arbitrary gain from the requirement to preserve commutation relations . this excess noise originates from quantum fluctuations in the auxiliary system required to keep energy conservation . we concentrate on pia , supposing the target of amplification to be optical wave amplitude of a single mode , which is denoted by the term `` signal '' . classical counterpart of pia is a conversion of arbitrary complex wave amplitude @xmath1 into @xmath2 , where @xmath3 is the gain of amplification . as is found in ordinary textbooks , annihilation operators in quantum optics correspond to complex amplitudes in classical optics . therefore , we describe the amplifying process by the transformation of annihilation operators . quantum - mechanically optimal pia in the sense that the excess noise is minimized can be achieved by the following transformation @xcite : @xmath4 where @xmath5 and @xmath6 are the signal mode s annihilation operators before and after the amplification , respectively . there is an extra term @xmath7 which is introduced in order to meet the commutation relation of @xmath8=1 $ ] for both the input and output signal modes . here , @xmath9 is an arbitrary phase factor , and @xmath10 is another mode s annihilation operator in the auxiliary system . throughout this paper , the ancilla mode represented by @xmath10 is denoted by the term `` idler '' and distinguished from other ancilla modes . . becomes the input - output relation of optimal pia when the idler is in a vacuum state . the quantum fluctuation of the idler contaminates the amplified signal . this is the inevitable excess noise of pia . note that this penalty prevents amplification from being a loophole of the uncertainty relation in joint measurements @xcite . at the limit of high - gain amplification , we can see the famous @xmath11 db cost of the noise figure for pia of coherent states . in addition to this intrinsic excess noise , further nonintrinsic excess noise may be caused by other ancilla modes in nonoptimal pia . there are numerous practical realizations of optical amplification . doped fiber amplifiers ( dfas ) and semiconductor optical amplifiers ( soas ) utilize stimulated emissions @xcite , and raman amplifiers ( ras ) and optical parametric amplifiers ( opas ) utilize nonlinear optical effects . in principle , there is no quantum - mechanical reason to prevent these realizations from achieving the optimal pia in the form of eq . . however , the real devices with current technology are accompanied by further excess noises . recently , pia operating almost at the optimal level is experimentally demonstrated by josse _ et al . _ by utilizing feedforward @xcite . the reason for the high efficiency of josse s pia is that it does not require inefficient nonclassical operations or nonclassical ancillas . it uses a vacuum state as an ancilla which is present everywhere , and linear optics and homodyne measurements followed by feedforward which are highly efficient . although josse s pia is a good attainment , it is not the end of the story . the signal transformation in eq . is an irreversible thermalizing process . complete pia should have _ unitary _ realization on an expanded hilbert space . in order to unitarize pia , two - mode description is sufficient . the full input - output relation becomes as follows : [ eq : piaunitary ] @xmath12 note that the roles of the signal and idler are symmetric in this relation . the significance of unitarization must be the reversibility . the inverse transformation is easily derived when we take notice of the fact that eq . is equivalent to two - mode squeezing operation . a two - mode squeezing operation parametrized by @xmath13 is canceled by another two - mode squeezing operation where the squeezing direction is opposite , i.e. , @xmath14 . nonetheless , in many amplification schemes including josse s experimental demonstration @xcite , the idler output is lost in the inextractable environment , making the process irreversible . in order to realize idler - preserving and close - to - optimal pia , we require some nonclassicality for the amplifier . this is contrastive to josse s idler - nonpreserving pia which does not require any nonclassicality . a typical strategy to introduce nonclassicality into feedforward - based quantum circuits is to use nonclassical states as ancillas . continuous - variable ( cv ) quantum teleportation @xcite and cv error correction @xcite are good examples . in these examples , squeezed states are used as ancillas that support the performance beyond the classical limit , and the complex operations after the state preparation stage are efficiently implemented by linear optics . in this paper , by employing the feedforward - based scheme proposed in ref . @xcite , we demonstrate pia of coherent states which preserve the idler output . the scheme basically relies on linear optics including homodyne measurements and feedforward . squeezed vacuum states are used as ancillas , which inject nonclassicality into our pia . only for generating the nonclassical ancilla states , we resort to nonlinear optical effects . our demonstration is for the amplification gain of @xmath0 , which is tuned via passive optical devices and feedforward electric circuits . combining pia for @xmath0 with a half beamsplitter , we also demonstrate @xmath15 approximate cloning of coherent states , where an `` anticlone '' remains at the output . ( anticlone will be explained in sec . [ sec : clone ] . ) in principle , our amplifier and cloner becomes quantum - mechanically optimum at the limit of infinite squeezing of the ancillas . for the case of finite squeezing , as is the real situation in experiments , further excess noise invades in accordance to the level of the squeezing . however , the degradation is small enough to retain nonclassical features . the behaviors of our amplifier and cloner are fully characterized by using several coherent states as inputs . furthermore , we also pay much attention to the output correlations , because nonclassical properties clearly appear in them . for the pia experiment , we check the einstein - podolsky - rosen ( epr ) correlation between the signal and idler outputs . for the cloning experiment , we check bipartite entanglement between each clone and the anticlone , which as a whole proves tripartite entanglement of class i @xcite . moreover , for both experiments , the reversibility is investigated from the output correlations . our idler - preserving pia is significant in several respects . first of all , the reversibility will pave the way to new schemes . recently , there is a proposal of a cv quantum interface that enables in principle a unit fidelity of transfer using such reversible pia @xcite . moreover , the reversibility in cloning is also advantageous . cloning of unknown states is distribution of information , and its reversibility reserves the option to recover the distributed fragments of the information . this will be further discussed in sec . [ sec : clone ] . secondly , our pia would have some applications as two - mode squeezing operation . note that one - mode squeezing operation is already demonstrated successfully in ref . @xcite with similar approach . in this introduction , pia has been described together with a brief historical review . especially , the nonclassical property of pia is discussed , which is obscure in many amplification processes because the idler output is lost in the inextractable environment . the subsequent contents of this paper are as follows . in sec . [ sec : ffpia ] , feedforward - based pia is described , explicitly showing the excess noise due to finite squeezing of ancillas . in sec . [ sec : clone ] , cv quantum state cloning and its connection with pia are described . in sec . [ sec : setup ] , the experimental setup is described . in sec . [ sec : resultspia ] , the experimental results for pia of coherent states with @xmath0 are shown . in sec . [ sec : resultsclone ] , the experimental results for @xmath15 approximate cloning of coherent states are shown . in sec . [ sec : summary ] , our experimental achievements are summarized .

we experimentally demonstrate phase - insensitive linear optical amplification which preserves the idler at the output . the amplification gain of is demonstrated . in addition , combining this amplifier with a beamsplitter , we also demonstrate approximate cloning of coherent states where an anticlone is present . we investigate the reversibility by reconstructing the initial state from the output correlations , and the results furthermore , full characterization of the amplifier and cloner is given by using coherent states with several different mean values as inputs . our amplifier is based on linear optics , offline preparation of nonclassical ancillas , and homodyne measurements followed by feedforward . squeezed states are used as the ancillas , and nonlinear optical effects are exploited only for their generation .

we experimentally demonstrate phase - insensitive linear optical amplification which preserves the idler at the output . since our amplification operation is unitary up to small excess noise , it is reversible beyond the classical limit . the entanglement between the two output modes is the resource for the reversibility . the amplification gain of is demonstrated . in addition , combining this amplifier with a beamsplitter , we also demonstrate approximate cloning of coherent states where an anticlone is present . we investigate the reversibility by reconstructing the initial state from the output correlations , and the results are slightly beyond the cloning limit . furthermore , full characterization of the amplifier and cloner is given by using coherent states with several different mean values as inputs . our amplifier is based on linear optics , offline preparation of nonclassical ancillas , and homodyne measurements followed by feedforward . squeezed states are used as the ancillas , and nonlinear optical effects are exploited only for their generation . the ancillas introduce nonclassicality into the amplifying operation , making entanglement at the output .

astro-ph0507530

i

faint blue objects discovered in deep hubble space telescope images have been the subject of discussion in recent years . the extreme depth of the hubble deep field ( hdf ) north ( williams et al . 1996 ) and south ( casertano et al . 2000 ) and the hubble ultra deep field ( hudf ; beckwith et al . 2004 ) enables us to study faint stellar objects in the regions of the color - magnitude diagram that are devoid of standard galactic stars . supported by the observed microlensing events toward the large magellanic cloud ( alcock et al . 2000 ) , several investigators proposed that the faint blue objects observed in the hdf images can explain part of the dark matter in the galaxy and a significant population of the halo of the galaxy may be in the form of low - luminosity white dwarfs . claims by mendez & minniti ( 2000 ; hereafter m&m ) that the faint blue sources in the hdf north and south are galactic stars seemed to be consistent with earlier findings of ibata et al . ( 1999 ) who found 5 faint halo white dwarf candidates with detectable proper motions in the hdf north . however , further analysis by richer et al . ( 2001 ) and kilic et al . ( 2004 ) showed that the faint blue objects in the hdf north do not show any significant proper motion . a detailed analysis of the point sources in the hdf north ( kilic et al . 2004 ) and the hudf ( pirzkal et al . 2005 ) showed that blue extra - galactic sources may be confused with white dwarfs . the ( 10@xmath3 ) limiting ab magnitudes of the @xmath4 band images for these two fields are 27.6 and 29.0 , respectively . using a 7 year baseline , kilic et al . ( 2004 ) obtained proper motion measurements for the point sources in the hdf north including the 5 faint blue objects ; they identified two possible disk white dwarfs , one of which now also appears spectroscopically to be a white dwarf ( d. stern , private communication ) . using low resolution spectroscopy and proper motion measurements , pirzkal et al . ( 2005 ) identified 20 late type stars , 2 quasars , and 4 possible white dwarf candidates in the hudf . kilic , von hippel & winget ( 2005 ) showed that only two of these candidates ( hudf 4839 and 9020 ) are firm white dwarf candidates with hudf 4839 possibly being a thick disk object , and hudf 9020 being either a disk or a halo object . none of these white dwarf candidates show significant proper motion ( pirzkal et al . non - detection of high velocity white dwarfs in the hudf is consistent with non - detection of halo white dwarfs in the hdf north . lack of detection of high velocity white dwarfs in these two fields implies that white dwarfs account for less than 10% of the galactic dark matter ( pirzkal et al . 2005 ; kilic , von hippel & winget 2005 ) . ibata et al.s ( 2000 ) and oppenheimer et al.s ( 2001 ) discoveries of apparent halo white dwarfs from kinematic surveys were enough to explain 2% of the dark matter in the solar neighbourhood . on the other hand , further analysis by several investigators showed that most of these white dwarfs are associated with the thick disk population of the galaxy ( reid et al . 2001 ; reyle et al . 2001 ; bergeron et al . nevertheless , old halo white dwarfs are observed in the globular clusters m4 ( hansen et al . 2004 ) and ngc 6397 ( mendez 2002 ) and in the field toward m4 ( kalirai et al . reid ( 2004 ; see for a complete review on high velocity white dwarfs ) found that these observations do not require additions to the standard galactic populations . the hdf south data provides another opportunity to test whether the faint blue objects in deep hubble images can be old halo white dwarfs and if they can explain part of the galactic dark matter . m&m found 22 galactic stars and 10 faint blue objects in the hdf south . if these 10 faint blue objects are halo white dwarfs , then they would explain 3050% of the dark matter in the solar neighborhood . we extend kilic et al.s ( 2004 ) work to the hdf south by using the original hdf south data and images of the same field taken 3 years later for the go-9267 proposal ( wfpc2 supernova search , pi : s. beckwith ) to measure the proper motions of the point sources in the hdf south . section 2 describes our first epoch data and the classification of the point sources , while data reduction procedures for our second epoch images are discussed in 3 . 4 describes the proper motion measurements for the point sources . we present our spectral energy distribution fitting results in 5 , and various implications of these results are then discussed in 6 .

we explore the nature of the faint blue objects in the hubble deep field south . we have derived proper motions for the point sources in the hubble deep field south using a 3-year baseline . combining our proper motion measurements with spectral energy distribution the other faint blue objects analyzed by mendez & minniti do not show any significant proper motion and are inconsistent with being halo white dwarfs ; they do not contribute to the galactic dark matter .

we explore the nature of the faint blue objects in the hubble deep field south . we have derived proper motions for the point sources in the hubble deep field south using a 3-year baseline . combining our proper motion measurements with spectral energy distribution fitting enabled us to identify 4 quasars and 42 stars , including 3 white dwarf candidates . two of these white dwarf candidates , hdfs 1444 and 895 , are found to display significant proper motion , 21.1 7.9 mas yr and 34.9 8.0 mas yr , and are consistent with being thick disk or halo white dwarfs located at kpc . the other faint blue objects analyzed by mendez & minniti do not show any significant proper motion and are inconsistent with being halo white dwarfs ; they do not contribute to the galactic dark matter . the observed population of stars and white dwarfs is consistent with standard galactic models .

astro-ph0507530

c

we identified 4 quasars and 42 stars , including 36 k0 or later type stars , with our sed fitting technique . we adopt the absolute magnitude ( @xmath54 ) for each spectral type from pickles ( 1998 ; table 2 ) . we use the @xmath54 for each object to simulate the absolute magnitudes in the wfpc2 f606w filter ( @xmath55 ) . we calculated photometric distances for all of the objects in our sample using the simulated @xmath55 and the observed @xmath56 magnitudes . combining our proper motion measurements with photometric distances , we are able to derive tangential velocities for the stars . the photometric errors for the point sources in the hdf south are small , hence they cause only relatively small errors in the estimated distances . on the other hand , if the assigned spectral types are wrong by 1 index ( for example m3 instead of m2 ) , then the absolute magnitudes could be wrong by as much as 1 magnitude . we also note that our stellar templates have approximately solar metallicity . therefore , the distances to the metal poor halo objects , which will be intrinsically fainter for the same spectral type , may be over - estimated by our sed fitting method . for example , a metal poor g0 dwarf ( @xmath57=-0.8 $ ] ) would be 0.5 magnitude fainter than a solar metallicity g0 star ( pickles 1998 ) . hence , the distance to a metal poor g0 type star would be over - estimated by 26% . in order to determine the effect of different metallicites , we use synthetic spectra from a phoenix model atmosphere grid ( brott & hauschildt 2005 ) for stars with 2000 k @xmath58 10000 k , [ z / h]= 0.0 , -0.5,-2.0 , and log(g)=4.5 to simulate photometric colors in the hst bands . we found that using different metallicities changes the best fit @xmath59 by 50 450 k for g0 and later type stars , and the spectral types obtained from metal poor atmosphere models are usually 1 - 2 later in spectral types , e.g. m2 instead of m0 , than the ones obtained from the models with solar metallicity . we also used kurucz model atmospheres ( kurucz 1995 ) with 3500 k @xmath58 6000 k to test the metallicity effect , and found similar results . using girardi et al . ( 2002 ) theoretical isochrones , we estimate @xmath54 and @xmath55 for different metallicities for the point sources in our sample . spectral types from the sed fitting procedure ( using the pickles library ) , estimated absolute magnitudes , derived distances , and tangential velocities are given in table 3 . the ranges of absolute magnitudes , distances , and tangential velocities show the effect of using models with different metallicities . despite these potential systematic errors , the total information we have available for the point sources ( the morphological information , proper motion measurements , and the sed fitting results ) is sufficient to determine their nature . figure 9 shows the histogram of the number of stars observed at a given distance in the hdf south along with the predictions from the star count models ( reid & majewski 1993 ; dashed line ) and the observed distribution of stars in the hudf ( pirzkal et al . 2005 ; solid line ) . the top panel shows the distribution of stars if we use pickles main sequence library , and the bottom panel shows the same distribution if we use synthetic spectra with [ z / h]=-2.0 ( halo - like metallicity ) . both panels show that there is a deficit of stars in the 515 kpc range in the hdf south compared to the star count models . if we assume that all of the stars in the hdf south have solar metallicity , then we see an excess number of stars at @xmath235 kpc , and 4 more stars ( hdfs 441 , 1020 , 2596 , and 261 ) with estimated distances larger than 50 kpc , which is not expected from the star count models , nor would the distances be consistent with the metallicities for galactic stars . most of the objects observed at @xmath235 kpc are k type stars , whereas the four objects with distances @xmath60 kpc fit late type star seds fairly well , and they have essentially the same spectral type ( k7-m0 ) . if we assume that their classification as point sources is reliable , our sed fitting procedure works well to identify quasars and stars , and that they have solar metallicity , then we could claim that we discovered two new populations of stars ; a cluster of stars at 35 kpc and several other stars at @xmath12 50 kpc . hdf south and hudf have similar galactic latitudes ( -49.21 and -54.39 , respectively ) , therefore they should have similar distributions of galactic objects . we do not find any stars with distances larger than 50 kpc in the hdf north or the hudf . even though hdf south ( @xmath61328.25 , @xmath62 - 49.21 ) is about 25@xmath63 away from the center of the small magellanic cloud ( smc ; @xmath61302.80 , @xmath62 - 44.31 ) , the magellanic stream has over dense regions near the hdf south ( 1@xmath64 10@xmath65 atoms @xmath66 ; see the hi surface density maps of mathewson & ford 1984 ) . the average distance to the 4 objects with @xmath67 kpc is 66.8 @xmath0 13.4 kpc ( assuming solar metallicity ) , whereas the distance to the smc is measured to be 60.6 @xmath0 3.8 kpc ( hilditch , howarth , & harries 2005 ) . can these objects be members of the smc ? assuming that they have metallicities similar to the smc ( @xmath57=-0.7 $ ] ; lennon 1999 ) , we estimate their average distance to be 17.8 @xmath0 3.4 kpc , therefore they are not consistent with being in the smc . we expect the majority of the stars in the hdf south to be halo stars ( from the star count models and the velocity distribution of figure 7 ) . therefore , the bottom panel of figure 9 is likely to represent the real distribution of stars better than the top panel . a comparison of the distances , tangential velocities , and photometric colors show that all of the 42 stars except three ( hdfs 1444 , 895 , and 2488 ) are consistent with being dwarf / subdwarf stars in the galaxy . seven of these 42 stars ( hdfs 191 , 1576 , 1922 , 2072 , 10081 , 10326 , and 10617 ) show significant proper motion ( @xmath68 ) and have spectral types m2m5 , therefore are probable halo m dwarfs . large errors in our proper motion measurements prevent us from classifying the kinematic properties of the other stars , nevertheless , they are most likely to be g0 and later type dwarfs in the thick disk or halo of the galaxy . the three stars ( hdfs 1444 , 895 , and 2488 ) with estimated distances larger than 90 kpc ( both for the metal rich and the metal poor case ) are discussed in the next section . hdfs 1444 , 895 , and 2488 would have to be at very large distances ( @xmath69 kpc ) if they were main - sequence stars of any metallicity . on the other hand , as white dwarfs , they would be at more reasonable distances . in order to find the temperatures of these objects , we simulated the colors for blackbody seds with temperatures in the range 3000 80000 k. figure 10 shows the best fitting blackbody seds ( solid lines ) for hdfs 1444 ( top panel ) , 895 ( middle panel ) , and hdfs 2488 ( bottom panel ) . we calculate the blackbody temperatures for these objects to be 10547 k , 6096 k , and 10463 k , respectively . we have also used da white dwarf models ( d. saumon and d. koester , private communication ) to simulate colors for white dwarfs with @xmath70 g = 8 and 3000 k @xmath71 20000 k. cool white dwarfs show depressed infrared fluxes due to the effects of collision induced absorption ( cia ) due to molecular hydrogen ( hansen 1998 , saumon & jacobson 1999 ) . the pure h white dwarf models that we used include the cia opacities , therefore , we are able to compare the spectral energy distributions of young and old white dwarfs simultaneously and find the best - fit solution for our white dwarf candidates . assuming that hdfs 1444 , 895 , and 2488 are pure - h atmosphere white dwarfs , we estimate the temperatures of these objects to be 10681 k , 5882 k , and 11000 k , respectively . our best fitting da white dwarf seds are shown as dotted lines in figure 10 . using our best fit da white dwarf atmosphere solutions , we estimate the absolute magnitudes for our white dwarf candidates using the tables from bergeron et al . we use the white dwarf models to predict @xmath55 , and therefore to calculate the distances and tangential velocities for these three objects . estimated distances and kinematic properties of our white dwarf candidates are given in table 3 . hdfs 1444 displays a significant proper motion , 21.07 @xmath0 7.93 mas yr @xmath1 , and is consistent with being a thick disk or halo object at 1.5 kpc with @xmath38 148 @xmath0 56 km s@xmath1 . likewise , hdfs 895 displays a proper motion of 34.9 @xmath0 8.0 mas yr @xmath1 , and is more likely to be a halo white dwarf at 2.1 kpc with @xmath38 346 @xmath0 79 km s@xmath1 . hdfs 2488 does not display any significant proper motion , and its classification as a point source is questionable ( see 2.1 ) , nevertheless , if it is a star , then it would have to be a halo white dwarf at 9 kpc . figure 11 shows the @xmath47 vs. @xmath49 velocity diagram for the stars ( triangles ; assuming [ m / h]=-2.0 ) and the likely white dwarfs ( circles ) in the hdf south . we use the results of chiba & beers ( 2000 ) for the expected velocity distribution of halo ( solid line ) and thick disk ( dashed line ) objects . it is apparent from this figure that we have several thick disk stars in our sample . nevertheless , the majority of the stars are more likely to be in the galactic halo . one of the white dwarf candidates , hdfs 1444 , may well be a thick disk white dwarf , whereas the other two candidates , hdfs 895 and 2488 , are more likely to be halo white dwarfs . m&m identified 10 faint blue objects in the hdf south . we classified two of these 10 objects ( hdfs 1812 and 1827 ) as resolved ( see figure 2 ) . we measure proper motions of 2.26 @xmath0 7.93 and 3.25 @xmath0 6.98 mas yr@xmath1 for these two objects , respectively . one of the faint blue objects ( hdfs 1332 ) is fainter than @xmath72 ( therefore not included in our analysis ) , and has a proper motion of 9.31 @xmath0 7.93 mas yr@xmath1 . six of the faint blue objects ( hdfs 1945 , 2007 , 2178 , 441 , 1020 , 261 ) have proper motions in the range 2.56 to 10.27 mas yr@xmath1 . our sed fitting analysis showed that hdfs 1945 , 2007 , and 2178 have colors more consistent with being quasars than stars . in addition , we classify hdfs 441 , 1020 , and 261 as metal poor stars in the halo of the galaxy . therefore , only one of the faint blue objects identified by m&m , hdfs 1444 , plus two more white dwarf candidates identified in our analysis ( hdfs 895 and 2488 ) are consistent with being white dwarfs . we use reid & majewski ( 1993 ) star count models and our own calculations based on gilmore et al . ( 1989 ; see kilic et al . 2004 for a complete discussion ) to predict the number of stars and white dwarfs expected in the hdf south . we expect to find 4552 stars and 0.662.31 white dwarfs , including 0.240.50 disk white dwarfs in the hdf south . the star count models mildly over - predict the observed number of stars . the observed population of 23 white dwarfs is consistent with the standard galactic models .

fitting enabled us to identify 4 quasars and 42 stars , including 3 white dwarf candidates . two of these white dwarf candidates , hdfs 1444 and 895 , are found to display significant proper motion , 21.1 7.9 mas yr and 34.9 8.0 mas yr , and are consistent with being thick disk or halo white dwarfs located at kpc . the observed population of stars and white dwarfs is consistent with standard galactic models .

we explore the nature of the faint blue objects in the hubble deep field south . we have derived proper motions for the point sources in the hubble deep field south using a 3-year baseline . combining our proper motion measurements with spectral energy distribution fitting enabled us to identify 4 quasars and 42 stars , including 3 white dwarf candidates . two of these white dwarf candidates , hdfs 1444 and 895 , are found to display significant proper motion , 21.1 7.9 mas yr and 34.9 8.0 mas yr , and are consistent with being thick disk or halo white dwarfs located at kpc . the other faint blue objects analyzed by mendez & minniti do not show any significant proper motion and are inconsistent with being halo white dwarfs ; they do not contribute to the galactic dark matter . the observed population of stars and white dwarfs is consistent with standard galactic models .

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the subject of this paper is the coulomb sum rule ( csr ) for inelastic electron scattering @xcite , in a reformulation which extends its validity to relativistic momentum transfers @xmath2 , where @xmath3 is the three - momentum transfer to the target nucleus , and @xmath4 is the nucleon mass . the purpose of this work is to provide practical methods for the analysis of the longitudinal @xmath5 response of nuclear targets , at the higher energies now available at the continuous electron beam accelerator facility ( cebaf ) and other electron accelerators . in a previous paper @xcite , we derived a relativistic coulomb sum rule ( rcsr ) which incorporates the relativistic effects of nucleon recoil at large @xmath1 , as well as of fermi motion . in the present work we extend this theoretical approach to include the effects of nuclear interactions on the rcsr . in our recent article , we discussed in detail the assumptions under which a rcsr could be derived . the basic approximation is that only nucleon , i.e. , as opposed to antinucleon , degrees of freedom enter the coulomb response in the spacelike regime accessible by @xmath5 experiments . we call this the `` nucleons - only '' approximation ; it ignores effects of antinucleons , but includes fully the relativity of the nucleons . following conventional treatments , we adopt an impulse approximation which ignores explicit contributions from the exchange of charged mesons . we include nucleon anomalous moments and elastic form factors , which are important at higher energies . the final assumption is less conventional , but absolutely necessary . in order to derive a _ non - energy - weighted _ sum rule , it must be possible to factor _ all _ dependence on the photon energy @xmath6 from the current matrix element , otherwise dynamical effects enter into the coulomb response function and complicate the isolation of correlation effects . in this paper , we restrict our attention to form factor models which have factorable dependence on @xmath6 . we define the nuclear coulomb sum in terms of the coulomb response function @xmath7 and the proton electric form factor @xmath8 : @xmath9 where the lower limit @xmath10 excludes the quasielastic peak , and the upper limit @xmath1 restricts the integration to spacelike four - momenta . under the assumptions of ref . @xcite mentioned above , we obtain a rcsr which can be expressed in terms of one- and two - body contributions : @xmath11 where the one - body contribution is of the form @xmath12 @xmath13 includes two - body correlation information in momentum space , and @xmath14 is the uncorrelated two - body part which is related to the square of the nuclear elastic form factor , as discussed in ref . @xcite . in the one - body term ( [ ac ] ) , @xmath15 is the nucleon momentum distribution function for isospin projection @xmath16 and one spin projection , and @xmath17 is a kinematic factor which arises due to relativistic nucleon recoil and fermi motion in the target . in the nonrelativistic limit ( @xmath18 ) , we have @xmath19 and @xmath20 , which leads to @xmath21 ; then the sum rule ( [ ab ] ) is the result of ref . @xcite . the relativistic effects are all in the functions @xmath17 , representing recoil of the struck nucleon and fermi motion in the target ground state . in ref . @xcite we further showed that ( [ ac ] ) does not depend strongly on the details of @xmath15 , but only on the lowest momentum moments , e.g. , @xmath22 . this leads to a method of evaluating ( [ ac ] ) accurately in a weakly model - dependent manner , and in principle permits the extraction of the correlation function @xmath13 from the the experimentally measured coulomb sum ( [ aa ] ) , using ( [ ab ] ) . in the present paper , we investigate what changes are required in the sum rule when the coulomb sum @xmath23 is modified by relativistic two - body interactions . we study the effect of the average interaction in the nucleus using the mean field approach of quantum hadrodynamics , a relativistic field theory for nuclear physics @xcite . in particular , we consider qhd - i , which includes vector and scalar isoscalar mesons only . we then reformulate the coulomb sum rule of ref . @xcite so that results ( [ aa])([ac ] ) have a similar structure , but with modifications reflecting the mean - field effects of these relativistic interactions . these modifications have two main effects on the rcsr : first , there are kinematical effects resulting from mean - field interactions of the nucleons , which are represented in qhd - i by a reduction in the effective nucleon mass @xmath0 in the medium . consequences of a reduced effective nucleon mass @xmath0 for the coulomb sum have been considered previously in a fermi gas model @xcite . these calculations include interactions of both initial and final plane - wave nucleon states with the mean fields , through the effective mass @xmath0 , which can be interpreted as binding in the initial state , and final state interactions of the ejected nucleon with the nucleus . chinn , picklesimer and van orden @xcite have studied the effects of final state interactions on the coulomb response of a fermi gas , using more realistic interactions , and have seen similar effects to those seen due to @xmath24 . second , the electromagnetic coupling of the nucleon in medium may be modified by the mean fields , entering through the off - shell behavior of the nucleon elastic form factors . both modifications introduce a degree of model - dependence in the rcsr which is not present in the nonrelativistic formulation , nor in the relativistic formulation of ref . @xcite . we show how these features can be incorporated into the theory to allow the evaluation of the one - body contribution @xmath25 , and the subsequent extraction of the two - body correlation function @xmath13 from the measured coulomb response . this paper is organized as follows : in section 2 we introduce the basic formalism to include relativistic mean - field effects in the coulomb sum rule ( rcsr ) . we introduce two models ( f and g ) for the electromagnetic charge operator , and investigate the resulting behavior connected to different off - shell assumptions for the nucleon elastic form factors . in section 3 , we derive a modified version of the rcsr ( [ aa])([ac ] ) , concentrating on the explicit changes to the one - body term @xmath25 in off - shell models f and g. in section 4 , we illustrate the operation of the sum rule in a simple nuclear system : uniform nuclear matter treated in the mean - field approximation , with nuclear binding effects incorporated using qhd - i . we examine the sensitivity of the rcsr to @xmath0 and to the choice of off - shell models ( f and g ) , focussing on the convergence of the moment expansion in each case . we further demonstrate that the particular form of the rcsr given here is applicable to _ both _ models , and argue that the same form should also be valid for a broad class of form factor models . in section 5 , we draw conclusions , give guidelines for the application of the rcsr to data , and indicate important directions for future work .

relativistic interactions are included by using a dirac representation adapted from a vector - scalar field theory . we consider two models for the off - shell behavior of the nuclear electromagnetic current , and demonstrate that the sum rule is accurate for applications to data over the interesting range of and three momentum .

we extend the formulation of relativistic coulomb sum rules to account for the average effects of nuclear binding on the initial and final states of ejected nucleons . relativistic interactions are included by using a dirac representation adapted from a vector - scalar field theory . the scalar field reduces the effective nucleon mass and increases the relativistic effects of recoil and fermi motion . we consider two models for the off - shell behavior of the nuclear electromagnetic current , and demonstrate that the sum rule is accurate for applications to data over the interesting range of and three momentum . we further indicate that the form of the sum rule is sufficiently general to accommodate a broad class of off - shell form factor models.pacs numbers : 25.30.fj , 11.55.hx , 24.10.jv @=11 @=12 0.5 true in 0.5 true in submitted to _ physical review c _

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the main results of this paper can be summarized as follows : the effects of nuclear binding in the mean - field approximation enter the coulomb response function @xmath7 and the coulomb sum @xmath23 in a way that can be characterized by a reduced effective mass @xmath0 . these effects depend on the behavior of the electromagnetic current for the off - shell kinematics in the nuclear medium . the sensitivity of the coulomb sum @xmath23 to the choice of off - shell model has been discussed previously @xcite . we consider two illustrative models : f and g. we demonstrate in these models that the rcsr of ref . @xcite can be extended to account for relativistic binding effects , in addition to the purely kinematic effects of recoil and fermi motion treated in ref . the resulting rcsr is no longer model - independent , since one must make some assumptions about @xmath0 and the off - shell behavior of the form factors . however , to the extent that the dominant effects of the nuclear medium can be characterized by the simple parameter @xmath0 , the coulomb sum rule analysis can be applied to data , with the goal of looking for nuclear structure effects beyond the mean field , e.g. , two - body correlations . we emphasize that the form of the sum rule is sufficiently general to accommodate a broad class of off - shell form factor models , i.e. , those for which the recoil function @xmath17 can be expressed in the form ( 3.9 ) . the real change in transforming to the @xmath0 basis enters through the nucleons - only approximation , under which the sum rule is derived . strong potentials in relativistic models mix free @xmath57 components into the initial and final interacting nuclear states , and modify the spacelike response function . chinn , picklesimer and van orden @xcite have isolated the effects of this mixing of @xmath57 components , and show results qualitatively similar to those seen in fig . 4 . ( their form factors are closer to our model f than g. ) the transformation to an @xmath0 basis automatically incorporates these potential effects in a convenient representation , although the use of an effective mass independent of position or momentum may only approximate the physical situation . how might one apply this rcsr to experimental data ? first , the coulomb sum @xmath23 must be calculated from the measured coulomb response function , as in ( [ aa ] ) . then , one must make some specific assumptions about the off - shell behavior of the current operator , as discussed in section 3 . this should be cast in the @xmath222 form as in ( 3.10 ) and ( 3.11 ) for model f ( or @xmath223 for model g ) , with the substitution ( 3.12 ) to remove ( approximately ) any residual @xmath6-dependence from @xmath182 and @xmath183 . the choice of model is not restricted to those presented in sections 3.1 and 3.2 . with the factored moment expansion , the relativistic recoil function ( 3.1 ) takes the form ( 3.13 ) and the ratios @xmath224 are evaluated at @xmath128 , as in ( 3.14 ) . the recoil factor @xmath225 is then evaluated in a moment expansion , e.g. , to @xmath168 as in ( 4.4 ) , with @xmath226 fixed by other experimental information , or by a model as in ( 4.5 ) , or as a free parameter . the expansion coefficients @xmath108 are given in appendix b , and are functions of the parameter @xmath0 . the modified coulomb sum @xmath158 is obtained by forming the ratio ( 4.2 ) of the experimentally determined numerator @xmath23 , and the ( model dependent ) recoil denominator @xmath227 . the expected behavior of @xmath23 with increasing @xmath1 is that it will approach the one - body term @xmath25 of ( 1.3 ) , assuming that both @xmath13 and @xmath228 in ( 1.2 ) , as @xmath194 . the modified sum @xmath158 will then approach unity _ if _ the assumed form of the current is correct , _ and if _ the effective mass @xmath0 is appropriately chosen . should that be the case for a given set of data , it would be reasonable to assume that the recoil functions have been correctly chosen . the sum rule ( 1.3 ) can then be used to investigate the two - nucleon correlation function in the ratio form is related to the standard nonrelativistic correlation function . ( see eqs . ( 5.8 ) and ( 5.16 ) in ref . an interesting application of the rcsr is the constraint of the viable off - shell form factor models by analyzing the experimentally determined coulomb sum @xmath23 in different off - shell models and comparing to @xmath231 beyond the expected range of correlations . a further remark about the analysis of data with the rcsr seems appropriate . it has become customary for the experimental coulomb response data to be integrated in a modified form of ( 1.1 ) , as suggested by deforest @xcite , in which the proton electric form factor @xmath119 is replaced by @xmath232 , with @xmath78 ( see , e.g. , refs . we explained in ref . @xcite why this procedure will not lead to a _ non - energy - weighted _ sum rule : to obtain a sum rule of the form ( 1.2 ) or ( 1.5 ) , the extra @xmath6-dependence should not be introduced into the definition of the coulomb sum . the kinematic effects included in the recoil factors @xmath225 depend on the 3-momentum transfer * q * , rather than on the invariant @xmath71 . it is this property which preserves the sum rule in passing from ( 1.2 ) to ( 5.1 ) . the use of the @xmath0-basis for both initial and final states , with the same value of @xmath0 , implicitly assumes that the nucleus is large enough that the nucleon kinematics in the final state are sensitive to the scalar field . for smaller nuclei , it may be necessary to account for the effects of finite size . this could possibly be accomplished through a rederivation of the rcsr in a hartree representation based on bound , localized nuclear states . another issue is the momentum - dependence of the mean - field potentials , as discussed in refs . @xcite , for example . since the rcsr has been derived in momentum space , such a modification is relatively straightforward . as a first approximation , which is consistent with our conclusion that fermi motion effects tend to be small , it seems reasonable to use @xmath233 for initial states and @xmath234 for final states . this prescription preserves the basic structure of the rcsr , in that no further dependence on @xmath47 is introduced . this would require the initial- and final - state masses to be treated distinctly , however , and would complicate the form of the coefficients in appendix b. we are currently investigating the effects of virtual @xmath55 pairs ( of mass @xmath0 ) on the rcsr , as they enter intermediate excited states in the random phase approximation . these were considered previously by horowitz @xcite , and appear to be significant . we are also interested in how energy dependent terms , which can enter in off - shell models where a substitution of the form ( 3.12 ) is not appropriate , affect the rcsr . this research was supported in part by the u.s . department of energy under grant no . de - fg02 - 88er40425 with the university of rochester . the authors would also like to thank the high energy physics group for use of the vax computer . in model g , we use the on - shell forms of @xmath85 and @xmath86 , given in ( [ aae ] ) . in model f , we use @xmath186 and @xmath187 , given in ( 3.11 ) . expression ( a5 ) makes clear the origin of the functional forms in ( 3.6 ) and ( 3.7 ) . in this appendix , we give the coefficients @xmath245 , for even powers through @xmath171 in the moment expansion . these are written as they appear in model g , i.e. , in terms of the usual nucleon charge ( @xmath116 ) and magnetic moment ( @xmath117 ) : @xmath248\nonumber\\ & & \label{abc}\\ & & + \,\mu_\sigma^2\ \bigl[-64{e_{\bf q}^*}^9 - 24{e_{\bf q}^*}^8m^ * + 16{e_{\bf q}^*}^7{m^*}^2\nonumber\\ & & \nonumber\\ & & \qquad-16{e_{\bf q}^*}^6{m^*}^3 -48{e_{\bf q}^*}^5{m^*}^4 - 24{e_{\bf q}^*}^4{m^*}^5\bigr ] . \biggr]\nonumber\end{aligned}\ ] ] k. w. mcvoy and l. van hove , phys . rev . * 125 * , 1034 ( 1962 ) . fetter and j.d . walecka , _ quantum theory of many - particle systems , _ ( mcgraw - hill , new york , 1971 ) . ferre and d.s . koltun , phys . c * 49 * , 1961 ( 1994 ) . serot and j.d . walecka , adv . * 16 * , 1 ( 1986 ) . t. matsui , phys . lett . b * 132 * , 260 ( 1983 ) . g. do dang , m. lhuillier , nguyen van giai and j. w. van orden , phys . c * 35 * , 1637 ( 1987 ) . chinn , a. picklesimer and j.w . van orden , phys . c * 40 * , 790 ( 1989 ) . chinn , a. picklesimer and j.w . van orden , phys . c * 40 * , 1159 ( 1989 ) . r. schiavilla , r.b . wiringa and j. carlson , phys . * 70 * , 3856 ( 1993 ) . t. de forest , jr . , * a392 * , 232 ( 1983 ) . chinn and a. picklesimer , _ nuovo cim . a _ * 105 * , 1149 ( 1992 ) . s.j . wallace , ann . nucl . & part . , * 37 * , 267 ( 1987 ) . t. de forest , jr . * a414 * , 347 ( 1984 ) . chen _ et . al . , _ phys . * 66 * , 1283 ( 1991 ) . meziani _ et . al . , _ phys . lett . * 69 * , 41 ( 1992 ) . g. do dang and pham van thieu , phys . c * 28 * , 1845 ( 1983 ) . h. kim , c.j . horowitz and m.r . frank , indiana university nuclear theory center , preprint no . 94 - 11 , submitted to phys . c ( 1994 ) . c. j. horowitz , phys . b * 208 * , 8 ( 1988 ) . barnes , phys . * 1 * , 166 ( 1962 ) . hand , d.g . miller and r. wilson , rev . mod . phys . * 35 * , 335 ( 1963 ) . scadron , _ advanced quantum theory , _ ( springer - verlag , new york , 1979 ) . ( see problem 5.8c , pg . 352 . ) 0.3 true in * figure captions * 0.2 true in fig . 1 coulomb sum @xmath23 for a uniform system of dirac protons , at two values of @xmath0 . two different evaluations of the lowest - order rcsr are shown as indicated.fig . 2 . rcsr evaluated for a uniform system of nucleons with anomalous magnetic moments , for @xmath162 . the coulomb sum @xmath23 in ( 4.1 ) and @xmath158 in ( 4.2 ) are shown as indicated.fig . same as fig . 2 , but for @xmath24 in model g. results are shown for ( a ) @xmath143 and ( b ) @xmath142 . same as fig . 3 , but in model f. note that @xmath169 is calculated in two ways : the `` factored '' ( dashed ) and `` consistent '' ( similar dot - dashed ) moment expansions.fig . 5 . electric and magnetic recoil functions , @xmath129 and @xmath130 , in a convenient separation . results are shown for ( a ) @xmath162 and ( b ) @xmath142 . the functions @xmath190 ( solid ) , @xmath196 ( dashed ) and @xmath197 ( dot - dashed ) are shown ; @xmath191 . ( note the scale changes for @xmath198 and @xmath199.)fig . effective form factors ( squared ) in model f , for @xmath142 . solid curves are obtained by averaging ( 3.11 ) with ( 3.12 ) over the fermi sphere , and include fermi corrections to all orders . dashed curves are @xmath202 and @xmath203 of ( 3.14 ) , as used in the `` factored '' moment expansion .

we further indicate that the form of the sum rule is sufficiently general to accommodate a broad class of off - shell form factor models.pacs numbers : 25.30.fj , 11.55.hx , 24.10.jv @=11 @=12 0.5 true in 0.5 true in submitted to _ physical review c _

we extend the formulation of relativistic coulomb sum rules to account for the average effects of nuclear binding on the initial and final states of ejected nucleons . relativistic interactions are included by using a dirac representation adapted from a vector - scalar field theory . the scalar field reduces the effective nucleon mass and increases the relativistic effects of recoil and fermi motion . we consider two models for the off - shell behavior of the nuclear electromagnetic current , and demonstrate that the sum rule is accurate for applications to data over the interesting range of and three momentum . we further indicate that the form of the sum rule is sufficiently general to accommodate a broad class of off - shell form factor models.pacs numbers : 25.30.fj , 11.55.hx , 24.10.jv @=11 @=12 0.5 true in 0.5 true in submitted to _ physical review c _

1512.03705

c

we have developed a generalised langevin equation ( gle ) approach to treat non - equilibrium conditions when a central classical region is connected to two realistic thermal baths at two different temperatures . the method is called gle-2b for generalised langevin equation with two baths . following the original gle approach @xcite , the extended langevin dynamics scheme is modified to take into account two sets of auxiliary degrees of freedom , each of which characterises the vibrational properties of the baths . these auxiliary degrees of freedom are then used to solve the non - markovian dissipative dynamics of the central region . we have developed the corresponding algorithm for md simulations and implemented it within the md code lammps . as a first application , we have studied the heat transport properties of a short al nanowire , that connects the left and the right al baths , in the steady - state regime . we have mostly considered the establishment of a local temperature profile within the system when the two bath temperatures are different . our results are interpreted in terms of the properties of harmonic versus non - harmonic systems , and the presence or the absence of defects . in agreement with earlier studies , we found that in a purely harmonic ( ballistic ) thermal conductor ( with spatially extended normal modes ) , there is no temperature gradient across the central part of the system . whenever the system presents some form of thermal resistance ( finite conductance ) due to anharmonic effects , disorder , or extra random processes , a temperature gradient is present in the system . furthermore a concrete example of such effects in a model of a one - dimension al chain is provided in appendix [ app:1dmodel ] . we have also compared the results of the simulations using the gle-2b approach to the results of other simulations that were carried out using standard thermostatting approaches ( based on markovian langevin and nose - hoover thermostats , see fig . [ fig : gle_and_otherthermostats ] ) . in the latter cases , either a flat temperature profile or a temperature gradient across the central system can be obtained depending on the value used for the damping parameter . upon the choice of this parameter , two different physical results can be obtained . such a dilemma does not exist in the gle-2b approach as it does not contain any adjustable parameters . furthermore , we have shown that the gle-2b is able to treat , within the same scheme , two widely different transport regimes , i.e. systems which have ballistic ( with no temperature gradient ) or diffusive ( with temperature gradient ) thermal transport properties . this is a crucial point since the crossover between ballistic and diffusive transport regimes has been observed experimentally @xcite in organic molecules of different lengths connecting two electrodes , after having been predicted theoretically @xcite . penultimately we would like to add that the gle-2b has also another advantage over the more commonly used thermostatting approaches . this method has been derived explicitly in order to be able to treat inherently non - equilibrium properties which can not be simulated ( in principle ) by the nh thermostats . furthermore , we have already shown in appendix d of ref . [ ] that we can derive the gle dynamics with a coloured noise which is not simply proportional to the memory kernel ( as is the case for the classical limit of the equilibrium fluctuation - dissipation theorem ) . this means that quantum effects of the baths can be incorporated in the gle dynamics . the importance of such effects has been considered in refs . [ , ] . finally , it should be noticed that our gle-2b approach is also perfectly appropriate to study time - dependent phenomena . such interesting phenomena , which involve proper dynamical behaviour of systems , will be the subject of future studies . ) is performed only on the atoms of the central region . the @xmath105 and @xmath104 baths ( displaced further to the left and right for clarity ) enter into the gle calculations via the sets of fitting parameters @xmath94 and the @xmath139 quantities . in such calculations , the positions of the bath atoms are fixed at their equilibrium positions shown in fig . [ fig : system_gle ] . ( _ lower panel _ ) the atoms in the central region follow newton s eom and the atoms in the @xmath105 and @xmath104 baths follow a dissipative lg or nh dynamics . the atoms with a black cross are kept fixed to ensure the overall stability of the system . the two approaches , gle-2b and lg / nh , represent two different kinds of stochastic processes . , title="fig:",width=340 ] + ) is performed only on the atoms of the central region . the @xmath105 and @xmath104 baths ( displaced further to the left and right for clarity ) enter into the gle calculations via the sets of fitting parameters @xmath94 and the @xmath139 quantities . in such calculations , the positions of the bath atoms are fixed at their equilibrium positions shown in fig . [ fig : system_gle ] . ( _ lower panel _ ) the atoms in the central region follow newton s eom and the atoms in the @xmath105 and @xmath104 baths follow a dissipative lg or nh dynamics . the atoms with a black cross are kept fixed to ensure the overall stability of the system . the two approaches , gle-2b and lg / nh , represent two different kinds of stochastic processes . , title="fig:",width=226 ] we acknowledge financial support from the uk epsrc , under grant no . ep / j019259/1 . hn , cdl and lk acknowledge the stimulating research environment provided by the epsrc centre for doctoral training in cross - disciplinary approaches to non - equilibrium systems ( canes , ep / l015854/1 ) . finally , ag would like to acknowledge the department of physics at king s college london for funding the summer internship which resulted in her contribution to this project . finally the authors thank one of the referees for a careful and critical analysis of our results and for suggestions that strengthened the value of the present work .

b * 89 * , 134303 ( 2014 ) ] to model a central classical region connected to two realistic thermal baths at two different temperatures . in such nonequilibrium conditions a heat flow following the original gle approach , the extended langevin dynamics scheme is modified to take into account two sets of auxiliary degrees of freedom corresponding to the mapping of the vibrational properties of each bath . these auxiliary variables are then used to solve the non - markovian dissipative dynamics of the central region . the results of the simulations using the gle-2b approach are compared to the results of other simulations that were carried out using standard thermostatting approaches ( based on markovian langevin and nose - hoover thermostats ) . we concentrate on the steady state regime and study the establishment of a local temperature profile within the system . the results show that the gle-2b approach is able to treat , within a single scheme , two widely different thermal transport regimes , i.e. ballistic systems , with no temperature gradient , and diffusive systems with a temperature gradient .

we extend the generalised langevin equation ( gle ) method [ phys . rev . b * 89 * , 134303 ( 2014 ) ] to model a central classical region connected to two realistic thermal baths at two different temperatures . in such nonequilibrium conditions a heat flow is established , via the central system , in between the two baths . the gle-2b ( gle two baths ) scheme permits us to have a realistic description of both the dissipative central system and its surrounding baths . following the original gle approach , the extended langevin dynamics scheme is modified to take into account two sets of auxiliary degrees of freedom corresponding to the mapping of the vibrational properties of each bath . these auxiliary variables are then used to solve the non - markovian dissipative dynamics of the central region . the resulting algorithm is used to study a model of a short al nanowire connected to two baths . the results of the simulations using the gle-2b approach are compared to the results of other simulations that were carried out using standard thermostatting approaches ( based on markovian langevin and nose - hoover thermostats ) . we concentrate on the steady state regime and study the establishment of a local temperature profile within the system . the conditions for obtaining a flat profile or a temperature gradient are examined in detail , in agreement with earlier studies . the results show that the gle-2b approach is able to treat , within a single scheme , two widely different thermal transport regimes , i.e. ballistic systems , with no temperature gradient , and diffusive systems with a temperature gradient .

nucl-th9909010

i

we have explored whether the parameter set tm1 @xcite can be improved by adding new couplings that stem from the modern effective field theory approach to relativistic nuclear phenomenology . we have been concerned with analyzing the possibilities of the new couplings to ensure a reasonable agreement with the density dependence of the scalar and vector components of the dbhf self - energies , while performing well for finite nuclei . the extended parameter set has been called tm1*. it is able to reproduce ground - state properties of spherical nuclei for @xmath17 with a quality similar to conventional sets like nl1 or nl3 , and with the appealing feature of having a positive quartic scalar self - coupling . this could not be achieved with the set tm1 which had to be restricted to @xmath15 larger than 20 in order to keep @xmath39 positive @xcite . it is important to note that this limitation seems to be common to any set of parameters containing only a quartic vector self - interaction on top of the standard non - linear @xmath0 model . to check this point we have performed calculations with the recently proposed nl - sv1 and nl - sv2 parameter sets @xcite that include a quartic vector self - coupling ( like tm1 ) . for light nuclei we find a good agreement with experiment when we use the nl - sv1 set which has a negative @xmath39 coupling , whereas this is not the case with the nl - sv2 set where @xmath39 is positive . in comparison with the dbhf results in nuclear matter the extended set tm1 * shows a significant improvement over tm1 due to the addition of the @xmath40 and @xmath41 couplings . the latter couplings ( at least @xmath40 ) are very helpful to bring the vector and scalar potentials closer towards the dbhf calculations as the density grows . to the end of computing finite nuclei we have introduced the @xmath61 , @xmath25 , @xmath26 , @xmath27 and @xmath62 parameters on top of the set that describes nuclear matter . we remark that the new parameters have a minor influence on the investigated properties of finite nuclei . however , they allow the full tm1 * force to improve the agreement with experiment for double - closed shell nuclei compared with the starting tm1 parameters and to obtain better results for light - mass nuclei , which was a shortcoming of the tm1 set . we also have tested the tm1 * force for isotopic energy differences , isotopic changes in charge radii and two - neutron and two - proton separation energies . nuclei near the drip lines have been explored for some particular cases by taking into account quasi - bound states in the bcs calculation following the method of ref . it should be mentioned that by including all of the relevant couplings in the energy density expansion compatible with the eft approach to qhd , as developed in refs . @xcite , the tm1 * model is more consistent with our current understanding of effective field theories . nevertheless , we have seen that some of the new couplings of the eft model remain underdetermined in spite of the information taken into account about the equation of state and the self - energies at higher densities . in conclusion , the relativistic mean field approach extended by the new non - linear meson self - interactions and tensor couplings based upon effective field theory , allows one to reproduce at the same time the trends of microscopic dbhf calculations up to relatively high densities and various finite nuclei properties . in the low - density domain ( that corresponds to the finite nuclei region ) the main properties are almost fixed by the nuclear matter properties around saturation , and then the new parameters have only a small contribution . however , as the density increases the vector - vector and scalar - vector meson interactions play an important role in providing enough flexibility to the model to be able to follow the tendency of the dbhf calculations . extended sets like tm1 * may be more useful for systems having relatively higher density and temperature , whereas they will serve the same purpose for normal systems as the conventional parameter sets . to further constrain the new eft parameters additional observables will be required . nuclear phenomena involving currents could prove helpful for couplings such as @xmath25 and @xmath26 that imply the derivatives of the fields . on the side of the isovector channel , information from many - body dbhf calculations of asymmetric and neutron matter as well as data on neutron radii and the neutron skin thickness should be relevant .

we extend the relativistic mean field theory model of sugahara and toki ( tm1 ) by adding new couplings suggested by modern effective field theories . an improved set of parameters ( tm1 * ) is developed with the goal to test the ability of the models based on effective field theory to describe the properties of finite nuclei and , at the same time , to be consistent with the trends of dirac brueckner hartree fock calculations at densities away from the saturation region . * effects of new non - linear couplings in relativistic effective field theory * + m. del estal , m. centelles , x. vias and s.k . departament destructura i constituents de la matria , facultat de fsica , + universitat de barcelona , diagonal _ 647 _ , e-_08028 _ barcelona , spain _ _ pacs : _ 21.60.-n , 21.10.dr , 21.65.+f , 21.30.fe relativistic mean field approach , effective field theory , non - linear self - interactions , dirac brueckner hartree fock , nuclear structure

we extend the relativistic mean field theory model of sugahara and toki ( tm1 ) by adding new couplings suggested by modern effective field theories . an improved set of parameters ( tm1 * ) is developed with the goal to test the ability of the models based on effective field theory to describe the properties of finite nuclei and , at the same time , to be consistent with the trends of dirac brueckner hartree fock calculations at densities away from the saturation region . we compare our calculations with other relativistic nuclear force parameters for various nuclear phenomena . * effects of new non - linear couplings in relativistic effective field theory * + m. del estal , m. centelles , x. vias and s.k . patra + _ departament destructura i constituents de la matria , facultat de fsica , + universitat de barcelona , diagonal _ 647 _ , e-_08028 _ barcelona , spain _ _ pacs : _ 21.60.-n , 21.10.dr , 21.65.+f , 21.30.fe relativistic mean field approach , effective field theory , non - linear self - interactions , dirac brueckner hartree fock , nuclear structure

hep-lat0404014

i

recent years have witnessed steady progress in the lattice qcd calculation of the light hadron spectrum @xcite . in the quenched approximation ignoring quark vacuum polarization effects , well - controlled chiral and continuum extrapolations enabled a calculation of hadron masses with an accuracy of 0.53% @xcite . at the same time the study established a systematic deviation of the quenched light hadron spectrum from experiment by approximately 10% . we then have made an attempt of full qcd calculation that allows chiral and continuum extrapolations within a consistent set of simulations @xcite . the deviations from experiment in the light hadron spectrum are significantly reduced and the light quark mass decreases by about 25% with the inclusion of dynamical @xmath8 and @xmath9 quarks . with currently available computer power and simulation algorithms , however , the sea quark mass that can be explored is far from the physical value and a long chiral extrapolation is involved to get to the physical @xmath8 and @xmath9 quark mass . an attempt has been made to push down the simulation to a small quark mass corresponding to @xmath10 in full qcd with the kogut - susskind(staggered)-type quark action @xcite . the staggered action , however , poses a problem of flavor mixing , which would modify the hadron spectrum and its quark mass dependence near the chiral limit . the staggered action also suffers from ambiguities in hadron operators and has a potential problem of non - locality . the wilson - type quark actions have the advantage of simplicity : they are local and respect flavor symmetry , but a larger computational cost limits the simulations to relatively large quark masses corresponding to @xmath11 @xcite . an important problem is to examine whether chiral extrapolations from such a quark mass range lead to results viable in the chiral limit . chiral extrapolations are usually made with polynomials in the quark mass . the problem is that they are not consistent with the logarithmic singularity expected in the chiral limit . in reality , the physical quarks are not exactly massless and hence the polynomial extrapolation should in principle work . however , increasingly higher orders are needed should one wish to increase the accuracy of the extrapolation . it is compelling to estimate the systematic errors due to higher order contributions when the data are extrapolated using a low - order polynomial . an alternative choice for chiral extrapolations is to incorporate chiral perturbation theory ( chpt ) @xcite . the present lattice data , however , are not quite consistent with the chpt predictions . the high - statistics jlqcd simulation of two - flavor full qcd , using the plaquette gauge action and the @xmath12-improved wilson quark action at @xmath13 ( @xmath14 fm ; the spatial size @xmath151.77 fm ) , shows no signature for the logarithmic singularity in the pion mass and pion decay constant @xcite . a possible reason for the failure to find the chiral logarithm is that sea quark masses , corresponding to @xmath160.6 , are too large . higher order corrections of chpt may have to be included to describe the data , as suggested from a partially quenched analysis , which shows that @xmath170.3 is required for the convergence of one - loop formula @xcite . another possibility is explicit chiral symmetry breaking of the wilson quark actions that may invalidate the chpt formulae . modifications due to finite lattice spacings may be needed for an analysis of data obtained on a coarse lattice . recently studies were made to adapt chpt to the wilson - type fermion at finite lattice spacings ( wchpt ) @xcite , with subtle differences in the order counting , and hence the resulting formulae for observables , among the authors . the work @xcite assumes the @xmath12 chiral symmetry breaking effects being smaller than those from the quark mass , and only the effects linear in lattice spacing are retained in the chiral lagrangian . this contrasts to the authors of refs . @xcite who include the @xmath18 effects in the chiral lagrangian , however , with different order countings . in ref . @xcite the @xmath12 terms are treated as being comparable to the quark mass term while the @xmath18 terms are assumed to be subleading : in this case , @xmath12 effects are essentially absorbed into the redefinition of the quark mass in the one - loop formulae and the @xmath18 terms provide additional counter terms . in ref . @xcite , on the other hand , the terms of @xmath18 are kept at the leading order , because the existence of parity - broken phase and vanishing of pion mass depend on them in a critical way @xcite . the coefficients of chiral logarithm terms receive @xmath12 contributions , and hence the logarithmic chiral behavior is modified at a finite lattice spacing . similar attempts to include the @xmath18 flavor mixing for the staggered - type quark action were made in refs . @xcite . the qq+q collaboration @xcite applied the one - loop chpt and wchpt with the prescription of refs . @xcite to their data obtained at @xmath190.5 . their simulations were made at coarse lattices of @xmath20 fm @xmath21 and @xmath22 fm @xmath23 using the plaquette gauge action and the unimproved wilson quark action ( @xmath24 fm ) . they reported that their data are described by these formulae . however , their sea quark masses are not quite small , and , since large scaling violation is suspected with unimproved actions at coarse lattice spacings and lattice artifacts are suggested at strong couplings @xcite , it should be demonstrated at weaker couplings in order that the discretization effects are actually under control . the ukqcd collaboration reported a result at @xmath25 obtained with the actions and the lattice spacing the same as those of jlqcd , with @xmath26 fm @xcite . they indicated the pion decay constant to bend slightly downward at this quark mass , but further work is required for quantitative comparison with the chpt predictions . in this paper , we follow up on our previous two - flavor full qcd work @xcite with an rg - improved gauge action and tadpole - improved @xmath12-improved wilson - clover quark action at @xmath270.55 and attempt to lower the quark mass to give @xmath28 down to 0.35 . since the computational costs grows rapidly toward the chiral limit , roughly proportional to @xmath29 @xcite , we concentrate our effort on the coarsest lattice of @xmath30 fm at @xmath1 , while using improved actions . generation of configurations below @xmath31 demands technical improvements . the bicgstab algorithm sometimes fails to converge , which we overcome by an improvement called bicgstab(ds-@xmath32 ) @xcite . another problem is the emergence of instabilities in the hmc molecular dynamics evolution @xcite . this seems to be caused by very small eigenvalues of the dirac operator , leading to the change of the molecular dynamics orbit from elliptic to hyperbolic . the only resolution at present is to reduce the time step size . in this manner , we generated 4000 trajectories at @xmath33 , 0.5 and 0.4 and 1400 trajectories at the smallest quark mass of @xmath34 on a @xmath35 lattice with @xmath36 fm . to examine the finite - size effect , we also generated 2000 trajectories at @xmath33 and 0.5 on a @xmath37 lattice with @xmath38 fm . we calculate the light hadron spectrum and the quark mass on these configurations , and examine the validity of the quadratic chiral extrapolations by comparing the extrapolations made in the previous work with our new data at smaller quark masses . it turns out that the new data are increasingly lower than the extrapolation toward a smaller sea quark mass . we then examine how our data compare with the wchpt formulae , and whether wchpt fits using only the previous data at large quark masses predict correctly the new small quark mass data . this serves as a test to verify the viability of wchpt and of chiral extrapolations . computing for the present work was made on the vpp5000/80 at the information processing center of university of tsukuba . we used 4 or 8 nodes , each node having the peak speed of 9.6 gflops . the present simulation costed 0.119 tflops@xmath39years of computing time measured in terms of the peak speed . this paper is organized as follows . we describe configuration generations in sec . [ section : simulation ] . the method of measurement of hadron masses , decay constants , quark masses and the static quark potential is explained in sec . [ section : measurement ] . the finite - size effects on hadron masses are also discussed in the same section . [ section : chiral_extrapolation_polynomials ] discusses chiral extrapolations with conventional polynomials , and those based on chpt are presented in sec . [ section : chiral_extrapolation_chpt ] . our conclusion is given in sec . [ section : conclusion ] . preliminary results of these calculations were reported in ref .

we extend the study of the light hadron spectrum and the quark mass in two - flavor qcd to smaller sea quark mass , corresponding to.35 . numerical simulations are carried out using the rg - improved gauge action and the meanfield - improved clover quark action at ( fm from meson mass ) . we observe that the light hadron spectrum for small sea quark mass does not follow the expectation from chiral extrapolations with quadratic functions made from the region of.55 . , we find the mean up and down quark mass being smaller than the previous result from quadratic chiral extrapolation by approximately 10% , [ mev ] in the continuum limit .

we extend the study of the light hadron spectrum and the quark mass in two - flavor qcd to smaller sea quark mass , corresponding to.35 . numerical simulations are carried out using the rg - improved gauge action and the meanfield - improved clover quark action at ( fm from meson mass ) . we observe that the light hadron spectrum for small sea quark mass does not follow the expectation from chiral extrapolations with quadratic functions made from the region of.55 . whereas fits with either polynomial or continuum chiral perturbation theory ( chpt ) fails , the wilson chpt ( wchpt ) that includes effects associated with explicit chiral symmetry breaking successfully fits the whole data : in particular , wchpt correctly predicts the light quark mass spectrum from simulations for medium heavy quark mass , such as . reanalyzing the previous data with the use of wchpt , we find the mean up and down quark mass being smaller than the previous result from quadratic chiral extrapolation by approximately 10% , [ mev ] in the continuum limit .

1103.0520

i

the construction of microstate geometries for black holes and black rings is now a fairly well developed subject . the defining idea of such geometries is that they have the same asymptotics at infinity as a given black hole or black ring and yet the solutions are completely smooth and , as a consequence , they resolve the black - hole or black - ring singularity into smooth `` bubbled geometries . '' more generally , it is interesting to see what classes of smooth , multi - centered geometries can be inserted into a black - hole , or black - ring throat since such additional structure may be thought of as `` hair '' on the black object . since the throat geometries are typically asymptotic to an anti de sitter space , such geometric `` hair in the back of a throat '' can be studied quite precisely using holography . one of the primary purposes in finding such backgrounds is not only to elucidate the possible microstate geometries but also to obtain a better semi - classical description of black - hole microstates in general . most of the progress on microstate geometries has been for bps solutions @xcite because such solutions are generically far simpler and are , in fact , governed by a linear system of differential equations @xcite . however , recent work has shown that there are also linear systems of equations that govern very large families of extremal , non - bps solutions @xcite . these families not only include most of the known extremal , non - bps black holes and black rings but also greatly generalize these solutions by including more charges and extending the results to completely new non - bps , multi - centered black - ring solutions . the construction of such solutions is a very important first step in understanding the structure and properties of non - bps solutions in general , particularly since such solutions have played a pivotal role in the analysis of attractor flows @xcite , black - hole bound states , deconstruction @xcite , wall crossing and entropy enigmas @xcite . ultimately one would like to find the non - bps analogs of bubbled solutions in which the singularity of the underlying black object has been resolved and the black - hole , or black - ring , throat is rounded off in some form of smooth geometry . such solutions have come to be known as _ microstate geometries _ and the requisite geometric transition to the bubbled geometries that lies at the heart of such objects has been found and extensively studied for bps solutions @xcite . however , there are very few examples of the corresponding smooth , non - bps geometries @xcite . indeed , there are rather few examples of non - bps solutions with more than one non - trivial bubble in a black - hole background . one of the purposes of this paper is to investigate an interesting new class of bubbled geometries and to find new non - bps bubbled solutions . a crucial first step in finding bubbled geometries is to find smooth , four - dimensional `` base geometries '' with non - trivial homology ( the `` bubbles '' ) that supports the cohomological fluxes that give rise to the charges of the solution . in bps solutions ( and in some non - bps solutions ) these base geometries are necessarily hyper - khler manifolds and are usually chosen to be asymptotic to @xmath4 or @xmath5 so that the final , five - dimensional geometry is asymptotic to five - dimensional , or four - dimensional , minkowski space . there are a rich variety of such admissible base geometries because the metrics are allowed to be ambi - polar , that is , the base metric is allowed to reverse sign from regions with signature @xmath6 to regions with signature @xmath7 . in spite of this apparent peculiarity , the presence of very non - trivial warp factors means that such base metrics can give rise to perfectly viable , smooth , lorentzian five - dimensional geometries . the ambi - polar generalizations of gibbons - hawking ( gh ) metrics @xcite have proven extremely valuable in that they have given rise to extensive and highly computable classes of bps microstate geometries . it was one of the important realizations of @xcite that one could relax the hyper - khler condition on the base geometry and still obtain non - bps solutions from linear systems of equations . in particular , it was shown in @xcite how this could be achieved by starting from a four - dimensional euclidean solution to einstein s equations coupled to an electromagnetic field . this led to some interesting examples based upon israel - wilson metrics but it turns out that many of these naturally arise as spectral flows of more standard non - bps solutions based upon gibbons - hawking metrics . one can also construct a regular non - bps solution with an euclidean kerr - newman base @xcite , however the base has only one topological two - cycle and there is no obvious way to generalize it easily . a natural follow - up question is whether there are any other classes of euclidean electrovac solutions that contain appropriate non - trivial homology and that can be used to generate non - bps microstate geometries . it turns out that there is a relatively straightforward generalization of the gibbons - hawking metrics to khler electrovac solutions and these geometries were studied extensively by lebrun in @xcite . the central result in @xcite is to find the explicit local form of all euclidean , four - dimensional khler metrics that have a @xmath0 isometry and a vanishing ricci scalar . it is then shown in a follow - up paper , @xcite , that these solutions are necessarily electrovac solutions whose electromagnetic field is related to the khler form . this generalizes earlier work on the classification of hyper - khler metrics with @xmath0 isometry @xcite in which it is shown that the metric is determined by a single function that is required to satisfy the affine toda equation . the gibbons - hawking metrics then emerge as precisely the metrics for which the @xmath0 action preserves all three complex structures ( that is , the @xmath0 is tri - holomorphic ) . the `` lebrun metrics '' in @xcite are defined by two functions , one of which must satisfy the affine toda equation and the other of which must essentially be harmonic in a background defined by the affine toda solution . this family of solutions collapses to either the general class of @xmath0-invariant hyper - khler metrics , or to the gh family if one makes an essentially trivial choice for one of the two functions . amongst the general class of lebrun metrics is what we will call the `` lebrun - burns metrics , '' which provide a simple , explicit class of khler metrics on @xmath8 with @xmath9 points blown up s . ] . the end result is rather similar to the gibbons - hawking metrics except that the @xmath10 sections and the harmonic functions on @xmath10 are now replaced by the hyperbolic space , @xmath11 , and its harmonic functions . the lebrun - burns metrics are also asymptotic to @xmath4 and , interestingly enough , the @xmath0 isometry on this @xmath4 does _ not _ act in a manner that matches the tri - holomorphic @xmath0 action on a gibbons - hawking metric that is similarly asymptotic to @xmath4 . thus the solutions obtained from the lebrun - burns metrics will be intrinsically different from those obtained from gh solutions . in this paper we explicitly solve the equations of motion of five - dimensional supergravity on an axisymmetric lebrun - burns base and thus provide an infinite class of non - supersymmetric multi - centered solution . we find that , due to the maxwell flux on the four - dimensional khler base , the five - dimensional backgrounds are not asymptotically flat and have the asymptotics of a warped , rotating @xmath12 space . for certain choices of parameters this becomes the near horizon limit of a bmpv black hole @xcite . we would like to emphasize that although our approach to finding these solutions was inspired by the construction of bps microstate geometries we find the most general axisymmetric solutions within the floating brane ansatz @xcite and with an axisymmetric lebrun - burns base . in particular our solutions include superpositions of non - supersymmetric concentric black rings and other potentially interesting solutions with horizons . in section 2 of this paper we review the process through which one can construct five - dimensional non - bps solutions using the floating - brane ansatz @xcite . in section 3 we introduce the general lebrun metrics and begin solving the linear system in this background , and in section 4 we specialize to the lebrun - burns metrics , discuss their properties and further reduce the linear system and exhibit all the necessary green functions . we will also show , in section 4 , that because the maxwell field on the four - dimensional base involves the khler form , the energy - momentum tensor does not fall off at infinity and hence the lebrun - burns base metrics do not naturally lead to five - dimensional solutions that are asymptotic to flat space . in section 5 we start with the simplest possible lebrun - burns metric , @xmath13 , and use the linear system and green functions of section 4 to construct very simple , explicit examples of non - bps solutions . we find that the natural asymptotic geometries that arise from lebrun - burns base metrics are a warped , rotating @xmath12 space . in section 6 we consider more general lebrun - burns base metrics with non - trivial topology and find the general axisymmetric solution to the system of non - bps equations on such a base . five - dimensional regularity and the absence of closed time - like curves ( ctcs ) puts very stringent conditions on our solutions but in spite of this we find a family of regular solutions with non - trivial bubbles that are asymptotic to the near - horizon region of a bmpv black hole . in section 7 we present our conclusions and suggest some directions for further work . some of the technical details of the construction of the solutions are relegated to the appendix .

we find a class of non - supersymmetric multi - center solutions of the stu model of five - dimensional ungauged supergravity . we show that there is an infinite number of regular multi - center solutions with non - trivial topology that are asymptotic to the near - horizon limit of a bmpv black hole . = 10000 * hair in the back of a throat : * * non - supersymmetric multi - center solutions from khler manifolds * * nikolay bobev , ben niehoff and nicholas p. warner + * simons center for geometry and physics + stony brook university + stony brook , ny 11794 , usa + department of physics and astronomy + university of southern california + los angeles , ca 90089 , usa + nbobev@scgp.stonybrook.edu , bniehoff@usc.edu , warner@usc.edu +

we find a class of non - supersymmetric multi - center solutions of the stu model of five - dimensional ungauged supergravity . the solutions are determined by a system of linear equations defined on a four - dimensional khler manifold with vanishing ricci scalar and a isometry . the most general class of such khler manifolds was studied by lebrun and they have non - trivial-cycles that can support the topological fluxes characteristic of bubbled geometries . after imposing an additional symmetry on the base we find explicit multi - center supergravity solutions . we show that there is an infinite number of regular multi - center solutions with non - trivial topology that are asymptotic to the near - horizon limit of a bmpv black hole . = 10000 * hair in the back of a throat : * * non - supersymmetric multi - center solutions from khler manifolds * * nikolay bobev , ben niehoff and nicholas p. warner + * simons center for geometry and physics + stony brook university + stony brook , ny 11794 , usa + department of physics and astronomy + university of southern california + los angeles , ca 90089 , usa + nbobev@scgp.stonybrook.edu , bniehoff@usc.edu , warner@usc.edu +

0909.0938

i

in this paper we study the ricci flow on three - dimensional , unimodular metric lie algebras . metric lie algebras are in one - to - one correspondence with left - invariant riemannian metrics on simply - connected lie groups , and ricci flow on such metrics has been studied by a number of authors ( e.g. , @xcite @xcite @xcite @xcite @xcite @xcite @xcite @xcite ) . the major advances in this paper are ( 1 ) a unification of the trajectories for the ricci flow , previously viewed individually in case - by - case studies of bianchi classes , into a single global topological picture , and ( 2 ) use of a new technique of flowing the lie structure constants , which highlights different features of the system than the usual evolution of metric coefficients . the space of metric lie algebras has been studied by a number of authors ( e.g. , @xcite @xcite @xcite ) . understanding ricci flow on the space of metric lie algebras is important for studying both homogeneous spaces and ricci flow of general manifolds . a number of ricci soliton metrics ( fixed points of the ricci flow up to diffeomorphism invariance and rescaling ) have been found on homogeneous spaces ( see , e.g. , @xcite @xcite @xcite @xcite @xcite @xcite ) , and it has been suggested that finding ricci solitons may be a promising way to attack alekseevskii s conjecture ( see @xcite and @xcite ) . lott has shown that three - dimensional , type iii solutions to ricci flow converge to the known homogeneous expanding solitons as they collapse in the limit ( see @xcite ) . ricci flow on homogeneous spaces is also useful in constructing self - dual solutions of euclidean vacuum einstein s equations ( see @xcite ) . we will consider the set @xmath0 of three - dimensional , nonabelian , unimodular metric lie algebras modulo isometry and scaling . milnor gives an excellent description of such metric lie algebras in @xcite , in particular showing that there exists a special orthonormal basis @xmath1 which diagonalizes both the ricci endomorphism and the lie bracket ( we say that the lie bracket is diagonalized if @xmath2 $ ] is a scalar multiple of @xmath3 ) . thus the set of three - dimensional , unimodular metric lie algebras depends only on three parameters . in fact , there are two natural choices of those three parameters , and the ricci flow through these parameter spaces takes one of the following forms : 1 . fix the lie algebra and let the metric vary . 2 . evolve the frame to keep it orthonormal and let the structure constants vary . in both cases , the lie bracket and inner product remain diagonal with respect to the frame . however , in the first case the lie bracket coefficients are fixed and the lengths of basis elements change . in the second case , the lie bracket coefficients change but the lengths of basis elements do not change ( since the basis evolves to stay orthonormal ) . it is extremely important that the frame remains orthogonal under the flow , which follows from the fact that both the structure constants and the ricci curvature can always be diagonalized at the same time as the metric . this is true for three - dimensional , unimodular metric lie algebras , but not in general . the lack of such a frame is the major obstacle for classifying ricci flow on four - dimensional , simply - connected homogeneous spaces ; isenberg - jackson - lu @xcite classify ricci flow for some riemannian homogeneous spaces which do admit such a frame . since @xmath0 is a three - dimensional space considered modulo rescaling , we have a two - dimensional system of odes , which is reasonable to analyze as a dynamical system in the plane . most previous work on ricci flow on homogeneous spaces takes the first parametrization ( e.g. , @xcite , @xcite , @xcite ) . in contrast , we will take the second parametrization , and consider @xmath0 as a quotient of the space of structure constants . this method has previously been used by the second author to study ricci flow on nilmanifolds @xcite ( see also @xcite and @xcite ) . let @xmath4 denote the flow on @xmath0 determined by the ricci flow . theorem a describes the topological dynamics of the flow @xmath5 [ theorem a]the phase space @xmath0 is the disjoint union of the following invariant sets ( see figures [ figure1 ] and [ figure2 ] ) : * four points @xmath6 and @xmath7 * six one - dimensional trajectories @xmath8 and @xmath9 ; and * three connected two - dimensional open sets @xmath10 and @xmath11 ; such that * the points @xmath6 and @xmath12 are fixed by @xmath13 * the orbit of a point @xmath14 in a @xmath15 or @xmath16 has @xmath17 and @xmath18 and * the orbit of a point @xmath14 in @xmath19 or @xmath20 has @xmath17 and @xmath21 theorem b interprets theorem a geometrically . in the sequel , we will say a point in the phase space represents a particular metric , although we actually mean that it represents the homothety class of the metric , i.e. , the equivalence class of the metric up to isometry and scaling . the decomposition of the phase space @xmath0 in theorem a corresponds geometrically as follows . 1 . each of the four fixed points @xmath6 and @xmath12 represents a soliton metric : * @xmath22 represents the soliton metric on the three - dimensional heisenberg group @xmath23 * @xmath24 represents the soliton metric on the three - dimensional solvable group @xmath25 * @xmath26 represents the flat metric on the three - dimensional euclidean group @xmath27 * @xmath12 represents the round metric on the group @xmath28 . 2 . the five trajectories @xmath29 @xmath30 @xmath31 @xmath32 and @xmath9 have riemannian submersion structures : * @xmath33 consists of left - invariant metrics on @xmath34 ( often denoted @xmath35 ) . these metrics fiber as riemannian submersions over @xmath36 . * @xmath37 consists of left - invariant metrics on @xmath38 . these metrics fiber as riemannian submersions over @xmath36 . * @xmath39 consists of left - invariant metrics on @xmath40 which fiber as riemannian submersions over the hyperbolic plane @xmath41 * @xmath42 and @xmath9 consist of left - invariant metrics on @xmath43 which fiber as riemannian submersions over the round sphere @xmath44 ( these riemannian manifolds are often called berger spheres ) . the trajectory @xmath42 corresponds to submersions whose fibers are larger than those of the round 3-sphere ( corresponding to the point @xmath12 ) and the trajectory @xmath9 corresponds to submersions whose fibers are smaller than those of the round 3-sphere . 3 . the three connected open sets @xmath10 and @xmath11 have the structures : * @xmath45 consists of left - invariant metrics on @xmath28 . * @xmath46 and @xmath11 consist of left - invariant metrics on @xmath47 . note that the trajectory @xmath48 is still somewhat mysterious . this trajectory was discovered independently by cao , guckenheimer , and saloff - coste @xcite , and evidence for it was present in @xcite . preliminary evidence suggests that this trajectory is not invariant under cross curvature flow , which may indicate it does not arise from extra symmetries of the riemannian metric , as the other special orbits do ( see remark [ remark symmetry ] ) . the organization of this paper is as follows . in section [ section space of unimod ] , we introduce notation and discuss the space of three - dimensional , unimodular metric lie algebras and their curvatures . in section [ section rf ] we derive the ricci flow equations on the space of structure constants . in section [ dynamics ] we analyze the dynamics of the ricci flow equations on a natural phase space and then on @xmath0 , completing the proof of theorem a. in section [ geometry section ] we analyze the dynamics geometrically , relating fixed points and special trajectories to ricci solitons and riemannian submersions , proving theorem b. in section [ section convergence ] we discuss how our convergence results relate to the current literature on ricci flows on three - dimensional , unimodular metric lie algebras and lie groups . finally , we include an appendix which give the details of characterizing the space @xmath0 . ( 40,60)(0,0 ) ( 10,0)(0,1)30 ( 30,30)(0,1)20 ( 10,30)(1,0)20 ( 10,0)(2,3)20 ( 10,30)(1,1)20 ( 10,15)(4,3)20 ( 10,0 ) ( 10,15 ) ( 10,30 ) ( 30,30 ) ( 30,50 ) ( 10,8)(0,1)0.8 ( 10,22)(0,-1)0.8 ( 30,40)(0,1)0.8 ( 20,30)(1,0)0.8 ( 22,24)(4,3)0.8 ( 20,15)(2,3)0.8 ( 20,40)(1,1)0.8 ( 6,0)@xmath49 ( 6,15)@xmath50 ( 6,30)@xmath49 ( 32,30)@xmath51 ( 32,50)@xmath52 ( 4,22)@xmath33 ( 4,8)@xmath33 ( 20,12)@xmath53 ( 14,22)@xmath48 ( 20,31)@xmath37 ( 16,42)@xmath42 ( 31,40)@xmath9 ( 22,37)@xmath45 ( 16,27)@xmath46 ( 12,13)@xmath54 with ricci flow lines . ]

this system is amenable to direct phase plane analysis , and we find that the fixed points and special trajectories in the phase plane correspond to special metric lie algebras , including ricci solitons and special riemannian submersions . these results are one way to unify the study of ricci flow on left invariant metrics on three - dimensional , simply - connected , unimodular lie groups , which had previously been studied by a case - by - case analysis of the different bianchi classes . in an appendix = 1

we give a global picture of the ricci flow on the space of three - dimensional , unimodular , nonabelian metric lie algebras considered up to isometry and scaling . the ricci flow is viewed as a two - dimensional dynamical system for the evolution of structure constants of the metric lie algebra with respect to an evolving orthonormal frame . this system is amenable to direct phase plane analysis , and we find that the fixed points and special trajectories in the phase plane correspond to special metric lie algebras , including ricci solitons and special riemannian submersions . these results are one way to unify the study of ricci flow on left invariant metrics on three - dimensional , simply - connected , unimodular lie groups , which had previously been studied by a case - by - case analysis of the different bianchi classes . in an appendix , we prove a characterization of the space of three - dimensional , unimodular , nonabelian metric lie algebras modulo isometry and scaling . = 1

1305.6569

i

consider the stochastic dynamics @xmath0 on @xmath1 satisfying @xmath2 called _ brownian dynamics _ or _ overdamped langevin dynamics_. here @xmath3 is a smooth function , @xmath4 is a positive constant , and @xmath5 is a standard @xmath6-dimensional brownian motion @xcite . the dynamics is used to model the evolution of the position vector @xmath0 of @xmath7 particles ( in which case @xmath8 ) in an energy landscape defined by the potential energy @xmath9 . this is the so - called _ molecular dynamics_. typically this energy landscape has many metastable states , and in applications it is of interest to understand how @xmath0 moves between them . temperature accelerated dynamics ( tad ) is an algorithm for computing this _ metastable dynamics _ efficiently . ( see @xcite for the original algorithm , @xcite for some modifications , and @xcite for an overview of tad and other similar methods for accelerating dynamics . ) each metastable state corresponds to a basin of attraction @xmath10 for the gradient dynamics @xmath11 of a local minimum of the potential @xmath9 . in tad , temperature is raised to force @xmath0 to leave each basin more quickly . what would have happened at the original low temperature is then extrapolated . to generate metastable dynamics of @xmath12 at low temperature , this procedure is repeated in each basin . this requires the assumptions : * @xmath0 immediately reaches local equilibrium upon entering a given basin @xmath10 ; and * an arrhenius law may be used to extrapolate the exit event at low temperature . the arrhenius ( or eyring - kramers ) law states that , in the small temperature regime , the time it takes to transition between neighboring basins @xmath10 and @xmath13 is @xmath14,\ ] ] where @xmath15 is the difference in potential energy between the local minimum in @xmath10 and the lowest saddle point along a path joining @xmath10 to @xmath13 . here @xmath16 is a constant ( called a _ prefactor _ ) depending on the eigenvalues of the hessian of @xmath9 at the local minimum and at the saddle point , but not on the temperature . in practice the arrhenius law is used when @xmath17 . we refer to @xcite for details . tad is a very popular technique , in particular for applications in material sciences ; see for example @xcite . in this article we provide a mathematical framework for tad , and in particular a mathematical formalism for ( h1)-(h2 ) . our analysis will actually concern a slightly modified version of tad . in this modified version , which we call _ modified tad _ , the dynamics is allowed to reach local equilibrium after entering a basin , thus circumventing assumption ( h1 ) . the assumption ( h1 ) is closely related to the no recrossings assumption in transition state theory ; in particular one can see the local equilibration steps ( modifications ( m1 ) and ( m2 ) below ) in modified tad as a way to account for recrossings . we note that modified tad can be used in practice and , since it does not require the assumption ( h1 ) , may reduce some of the numerical error in ( the original ) tad . to analyze modified tad , we first make the notion of local equilibration precise by using _ quasistationary distributions _ , in the spirit of @xcite , and then we circumvent ( h2 ) by introducing an idealized extrapolation procedure which is _ exact_. the result , which we call _ idealized tad _ , yields exact metastable dynamics ; see theorem [ mainthm ] below . idealized tad is not a practical algorithm because it depends on quantities related to quasistationary distributions which can not be efficiently computed . however , we show that idealized tad agrees with modified tad at low temperature . in particular we justify ( h2 ) in modified tad by showing that at low temperature , the extrapolation procedure of idealized tad agrees with that of modified tad ( and of tad ) , which is based on the arrhenius law ; see theorem [ theorem2 ] below . in this article , we focus on the overdamped langevin dynamics for simplicity . the algorithm is more commonly used in practice with the langevin dynamics @xmath18 the notion of quasistationary distributions still makes sense for the langevin dynamics @xcite , so an extension of our analysis to that dynamics is in principle possible , though the mathematics there are much more difficult due to the degeneracy of the infinitesimal generator of . in particular , some results on the low temperature asymptotics of the principal eigenvalue and eigenvector for hypoelliptic diffusions are still missing . the paper is organized as follows . in section [ sec : tad ] , we recall tad and present modified tad . in section [ sec : idealtad ] , we introduce idealized tad and prove it is exact in terms of metastable dynamics . finally , in section [ sec : theta ] , we show that idealized tad and modified tad are essentially equivalent in the low temperature regime . our analysis in section [ sec : theta ] is restricted to a one - dimensional setting . the extension of this to higher dimensions will be the purpose of another work . throughout the paper it will be convenient to refer to various objects related to the dynamics at a high and low temperature , @xmath19 and @xmath20 , as well as at a generic temperature , @xmath21 . to do so , we use superscripts @xmath22 and @xmath23 to indicate that we are looking at the relevant object at @xmath24 or @xmath25 , respectively . we drop the superscripts to consider objects at a generic temperature @xmath21 .

we give a mathematical framework for temperature accelerated dynamics ( tad ) , an algorithm proposed by m.r . voter in to efficiently generate metastable stochastic dynamics . using the notion of _ quasistationary distributions _ , we propose some modifications to tad . then considering the modified algorithm in an idealized setting , . accelerated molecular dynamics , temperature accelerated dynamics , langevin dynamics , stochastic dynamics , metastability , quasi - stationary distributions , kinetic monte carlo 82c21 , 82c80

we give a mathematical framework for temperature accelerated dynamics ( tad ) , an algorithm proposed by m.r . srensen and a.f . voter in to efficiently generate metastable stochastic dynamics . using the notion of _ quasistationary distributions _ , we propose some modifications to tad . then considering the modified algorithm in an idealized setting , we show how tad can be made mathematically rigorous . accelerated molecular dynamics , temperature accelerated dynamics , langevin dynamics , stochastic dynamics , metastability , quasi - stationary distributions , kinetic monte carlo 82c21 , 82c80

astro-ph9712046

c

we have analyzed the _ asca _ and _ rosat _ pspc x - ray observations of the rich cluster abell 2029 . the _ asca _ gis and _ rosat _ pspc are in good agreement on the global x - ray spectrum , and give an average ambient gas temperature of @xmath70 kev ( including the effects of a cooling flow ) . if the gas temperature is assumed to be constant in the analysis of the spatially resolved _ asca _ sis and gis , the average temperature is found to be 8.6 kev . the iron abundance in the gas is @xmath7 of the solar value . there is no significant evidence for any variation in the abundance with position in the cluster . the global x - ray spectra , central x - ray spectra , and _ rosat _ surface brightness all require a cooling flow at the cluster center . the global x - ray spectrum implies that the total cooling rate is @xmath8 yr@xmath3 . the global x - ray spectra are consistent with the galactic value for the soft x - ray absorption toward the cluster . the _ rosat _ pspc spectra of the central regions of the cluster of completely inconsistent with the large value of foreground excess absorption found by white et al . ( 1991 ) based on the _ einstein _ sss spectrum . the upper limit on excess foreground absorption is @xmath9 @xmath10 . however , the spectra do not rule out a significant amount of intrinsic absorbing gas located within the cooling flow region . one concern with modeling the cooling flow spectrum and absorption in a2029 is that the _ asca _ sis and _ rosat _ pspc spectra do not appear to be consistent . this may indicate that there are calibration problems with the _ asca _ sis spectra at low energies . the pspc image shows that the cluster is elliptical , but is very regular and smooth . this agrees with previous analyses of the optical and x - ray distribution of the cluster ( slezak et al . 1994 ; david et al . 1995 ; buote & tsai 1996 ) . we also find that there is no significant evidence for any irregularities in the temperature distribution in the cluster , as would be produced by a subcluster merger . structure in the x - ray surface brightness and particularly the gas temperature distribution has been interpreted as evidence for mergers and other hydrodynamic activity in many other clusters ( e.g. , henry & briel 1995 ; markevitch et al . 1996,1997 ) . the lack of structure in the x - ray properties of a2029 suggests that the cluster is relaxed and that the gas is in hydrostatic equilibrium . we use the assumption of hydrostatic equilibrium to determine the gravitational mass of the cluster as a function of radius . within a spherical radius of @xmath0 ( @xmath1 mpc ) , the total gravitational mass is @xmath4 , while the mass of gas is @xmath5 . the gas fraction of the cluster at this radius is @xmath169 , and the gas fraction is is increasing with radius at the largest radii . thus , a2029 is a particularly strong example of the so - called `` baryon catastrophe '' in clusters ( e.g. , david et al . 1995 ) . we thank the referee , joel bregman , for a number of helpful suggestions . c. l. s. was supported in part by nasa rosat grants nag 51891 , nag 53308 , nasa asca grant nag 5 - 2526 , and nasa astrophysical theory program grant 5 - 3057 . m. l. m. was supported by nasa grant nag5 - 2611 .

we have analyzed _ asca _ and _ rosat _ pspc spectra and images of the galaxy cluster abell 2029 . the pspc image shows that the cluster is very regular and smooth . also , there is no significant evidence for any irregularities in the temperature distribution in the cluster , as would be produced by a subcluster merger . these results suggest that a2029 is a relaxed cluster , and that the gas is in hydrostatic equilibrium . we use the assumption of equilibrium to determine the gravitational mass of the cluster as a function of radius . at a radius of ( mpc ; km s mpc ) , the gravitational mass is , while the mass of gas is . there is no significant evidence for any variation in the abundance with position in the cluster . the global x - ray spectra , central x - ray spectra , and _ rosat _ surface brightness all require a cooling flow at the cluster center . the global x - ray spectrum implies that the total cooling rate is yr . the global x - ray spectra are consistent with the galactic value for the soft x - ray absorption toward the cluster . the _ rosat _ pspc spectra of the central regions of the cluster are inconsistent with the large value of foreground excess absorption found by white et al . ( 1991 ) based on the _ einstein _ sss spectrum . the upper limit on excess foreground absorption is . however , the spectra do not rule a significant amount of intrinsic absorbing gas located within the cooling flow region .

we have analyzed _ asca _ and _ rosat _ pspc spectra and images of the galaxy cluster abell 2029 . the _ asca _ spectra of the cluster indicate that the gas temperature declines with radius . the pspc image shows that the cluster is very regular and smooth . also , there is no significant evidence for any irregularities in the temperature distribution in the cluster , as would be produced by a subcluster merger . these results suggest that a2029 is a relaxed cluster , and that the gas is in hydrostatic equilibrium . we use the assumption of equilibrium to determine the gravitational mass of the cluster as a function of radius . at a radius of ( mpc ; km s mpc ) , the gravitational mass is , while the mass of gas is . the gas fraction is found to increase with radius ; within a spherical radius of , the fraction is . the iron abundance in the gas is found to be of solar . there is no significant evidence for any variation in the abundance with position in the cluster . the global x - ray spectra , central x - ray spectra , and _ rosat _ surface brightness all require a cooling flow at the cluster center . the global x - ray spectrum implies that the total cooling rate is yr . the global x - ray spectra are consistent with the galactic value for the soft x - ray absorption toward the cluster . the _ rosat _ pspc spectra of the central regions of the cluster are inconsistent with the large value of foreground excess absorption found by white et al . ( 1991 ) based on the _ einstein _ sss spectrum . the upper limit on excess foreground absorption is . however , the spectra do not rule a significant amount of intrinsic absorbing gas located within the cooling flow region .

0809.1443

i

there is now overwhelming evidence that the vast majority of massive early - type galaxies harbor a supermassive black hole ( bh ) at their centers , and that bhs play an important role during the assembly and evolution of their host galaxies ( see reviews in @xcite ) . the existence of the @xmath8 correlation between bh mass and central stellar velocity dispersion links the growth of the bh with the formation of the host bulge @xcite , perhaps through feedback mechanisms associated with an active galactic nucleus ( agn ) . the detection of bhs can be achieved by several methods , including direct kinematics of surrounding gas and stars ( e.g. , @xcite ) , and optical spectroscopic identification of nuclear emission lines originating from an agn ( e.g. , @xcite ) . the first method is limited to nearby , inactive or mildly active galaxies , however , while optical spectroscopic identification is generally limited to moderately bright nuclei that are not strongly confused by circumnuclear star formation or heavily obscured by dust . an extensive spectroscopic survey of nearby galaxies has revealed that approximately @xmath9% of galaxies of hubble type e sb contain an agn @xcite , broadly consistent with the idea that bhs are ubiquitous in these systems @xcite . for types later than sc , where classical bulges disappear @xcite , only @xmath10% of galaxies show optical agn activity . this leads to the notion that classical bulges or ellipticals are necessary for bh formation . there are , however , a few striking counter examples . ngc4395 , an essentially bulgeless sm galaxy , exhibits all the hallmarks of a seyfert 1 nucleus @xcite , and it has a bh mass of @xmath11 @xcite . pox52 shares an almost identical optical spectrum to ngc4395 , but is hosted in a spheroidal galaxy @xcite , a visually similar but distinct morphological type compared to elliptical galaxies or classical bulges . more recently , an agn was discovered in ngc3621 , a late - type sd galaxy , using infrared spectroscopic observations @xcite . on the other hand , the nearby scd galaxy m33 has a firm upper limit on the mass of any central dark object of @xmath12 @xcite , far below the typical @xmath13 bhs found in galaxies such as ngc4395 . thus bhs can indeed exist in bulgeless systems , but exactly how common such systems are remains unknown . from a spectral analysis of sdss data , @xcite have found a population of low - mass bhs ( @xmath14@xmath15 ) accreting at appreciable eddington ratios . the host galaxies from the initial sample of 19 sources @xcite , using _ hst _ imaging , are found to be mid- to late - type spirals or spheroidals similar to pox52 @xcite . none are as late - type as ngc4395 , however , and so the incidence of bhs in very late - type systems remains uncertain . for such systems , there is also the possibility that bhs are replaced by nuclear star clusters ( ncs ) . both @xcite and @xcite suggest that bhs in the @xmath8 relation are replaced by compact massive objects at low mass , with a generalized correlation @xmath16 . such compact objects could include ncs instead of bhs , commonly found in late - type spirals @xcite . it is therefore not immediately obvious that such late - type spirals _ should _ contain bhs . establishing a bh census for late - type spirals is important to determine if bhs are indeed formed in earlier - type galaxies only ( ellipticals or early - type spirals ) , or if they also exist in very late - type galaxies . one complication in late - type spirals is the abundance of gas and dust , with strong levels of star formation . if these systems contain bhs , they are likely to be of relatively low mass , and hence the agn , even when fully active , will be of quite low luminosity . while the mere presence of an agn indicates a bh , energetically weak or heavily embedded agns are extraordinarily difficult to observe , particularly at optical wavelengths where most of these observations have historically occurred . dust obscuration severely attenuates optical and ultraviolet flux , and central star formation can dominate the optical / infrared emission . x - ray observations allow us to peer through the obscuring gas and dust column , since high column densities are required to suppress x - ray flux ( particularly energetic x - rays ) . x - ray detections are therefore a useful indicator of agn activity , and since the launch of the _ chandra x - ray observatory _ , the sensitivity with which to carry out such studies has been greatly improved . in particular , the advanced ccd imaging spectrometer ( acis ; @xcite ) provides better than 05 resolution with very low background contamination ( @xmath17 counts pix@xmath6 sec@xmath6 between 0.5 and 8 kev ) . this allows low flux limits to be achieved with very short ( few ks ) exposures , enabling surveys of non - x - ray selected samples @xcite . @xcite have used x - ray detections successfully to identify bhs in low - mass bulges , and have begun to establish statistics on agn x - ray activity as a function of host bulge mass . x - ray observations have also been useful in searching for nuclear activity in quiescent ( i.e. optically faint ) bhs residing in massive elliptical galaxies @xcite , and in star - forming early - type spiral galaxies @xcite . in this paper we investigate the nuclear activity in a sample of late - type spiral galaxies using archival _ chandra _ data , which provide a clearer method of detecting agn activity in such star formation - dominated , potentially heavily obscured environments . systems with nuclear x - ray detections coincident with the optical nucleus are identified as agn candidates , and after considering the possible contamination of our results from x - ray binaries , we estimate the prevalence of agns in late - type spiral galaxies . we discuss our sample selection in [ sec : sample ] and our results in [ sec : results ] . we conclude with a discussion of possible non - agn sources and directions for future work in [ sec : disc ] .

these x - ray sources range in luminosity from(210 kev ) = to ergs s . considering possible contamination from low - mass x - ray binaries ( lmxbs ) , we estimate that detections are possible lmxbs instead of true agns , based on the probability of observing a lmxb in a nuclear star cluster typically found in these late - type spiral galaxies . the incidence of agn activity in such late - type spiral galaxies also suggests that nuclear massive black holes can form and grow in galaxies with little or no evidence for bulges .

we have assembled a sample of 64 late - type spiral galaxies ( types 6.09.0 , corresponding to hubble types scd sm ) with archival _ chandra _ data . at a signal - to - noise ( s / n ) threshold of 3 , we find 12 objects with x - ray point - source detections in close proximity with the optical or near - infrared position of the nucleus ( median offset = 16 ) , suggestive of possible low - luminosity active galactic nuclei ( agns ) . including measurements with 3 s / n 1.5 , our detections increase to 18 . these x - ray sources range in luminosity from(210 kev ) = to ergs s . considering possible contamination from low - mass x - ray binaries ( lmxbs ) , we estimate that detections are possible lmxbs instead of true agns , based on the probability of observing a lmxb in a nuclear star cluster typically found in these late - type spiral galaxies . given the typical ages of nuclear star clusters , contamination by high - mass x - ray binaries is unlikely . this agn fraction is higher than that observed in optical surveys , indicating that active nuclei , and hence central black holes , are more common than previously suggested . the incidence of agn activity in such late - type spiral galaxies also suggests that nuclear massive black holes can form and grow in galaxies with little or no evidence for bulges . follow - up multiwavelength observations will be necessary to confirm the true nature of these sources .

astro-ph9605112

i

the information that is available about galaxies primarily relates to their morphology , internal kinematics , and spectral energy distribution . the further away one goes , the less important become the first two , compared to the third . clearly , if one wants to study galaxy evolution , study of the electromagnetic spectrum is of maximum importance . since the spectrum of a galaxy generally consists of a combination of stellar spectra , emission from gas , and possibly non - thermal radiation , partly extinguished by dust , and since also stellar spectra exist in a very large number of varieties , it is clear that understanding the spectrum of a galaxy is very difficult . to study complicated spectra it is necessary to first understand the spectral energy distributions of relatively simple objects . for this reason we will discuss here a model that analyzes the spectra of early - type galaxies . these objects appear to contain relatively little dust extinction and gaseous interstellar medium , and to have little recent star formation . they have , for these reasons , been the most studied objects in the current literature on population synthesis . there have been a number of accepted ways to attack the problem of understanding the spectrum of an elliptical . to understand why we adopt our current method , we will summarize shortly some of the population synthesis methods most often used in the literature . a stellar population synthesis program tries to find a combination of stars for which the integrated spectrum agrees with the observed spectrum of the object under study . in practise the problem is often underconstrained , i.e. a number of combinations of stars can be found which are able to fit the spectrum . to overcome this problem one generally forces the solution to obey certain constraints . these range from simple continuity requirements ( e.g. the luminosity function should decrease monotonically ) to the requirement that the distribution of stars is determined completely by stellar evolution calculations . models with very few physical constraints are generally called _ empirical _ population synthesis models , as opposed to _ evolutionary _ models . empirical models have been used with some success by spinrad & taylor ( 1971 ) , faber ( 1972 ) , oconnell ( 1976 , 1980 ) and pickles ( 1985 ) . these papers often make use of linear programming to obtain their results . some workers , notably bica ( 1988 ) , have attempted to take into account evolutionary effects by using , as units of population , distributions of stars observed in clusters of our galaxy , instead of individual stars . evolutionary models use a theoretical isochrone or hr diagram , convert isochrone parameters to observed spectra in some way and , finally , integrate along the isochrone . they all need to make an assumption about which _ initial mass function _ imf to use . also , the models need a recipe prescribing when the stars have been formed . since the imf is not very well known at the present time , its treatment is not very different from one model to another . however , as far as the _ star formation rate _ ( sfr ) is concerned , some models assume that all stars are formed at the same time , others prescribe that the sfr has to decrease exponentially with time , while still others explicitly try to describe the whole formation of a galaxy from a gas cloud and form stars when the physical conditions in the gas are adequate . examples of this kind of evolutionary models can be found in tinsley ( 1968,1972,1978a,1978b,1980 ) , searle _ et al . _ ( 1973 ) , tinsley & gunn ( 1976 ) , turnrose ( 1976 ) , whitford ( 1978 ) , larson & tinsley ( 1978 ) , wu _ et al . _ ( 1980 ) , vandenberg ( 1983 ) , bruzual ( 1983,1992 ) , stetson & harris ( 1988 ) , renzini & buzzoni ( 1986 ) , rocca - volmerange & guiderdoni ( 1987,1988,1990 ) , guiderdoni & rocca - volmerange ( 1987,1990,1991 ) , yoshii & takahara ( 1988 ) , rocca - volmerange ( 1989 ) , buzzoni ( 1989 ) , charlot & bruzual ( 1991 ) , lacey _ et al . _ ( 1993 ) and bruzual & charlot ( 1993 ) . some of these models not only predict colors but also line - strengths notably those of peletier ( 1989 ) and worthey ( 1994 ) . there are also models which combine evolutionary population predictions with considerations of chemical evolution . these models follow the evolution of the gas and make use of isochrones of more than one metallicity ( solar ) . examples of these _ chemo - evolutionary _ population synthesis models are arimoto & yoshii ( 1986,1987 ) , casuso ( 1991 ) , bressan _ et al . _ ( 1994 ) as well as the model presented in this paper . note that for this type of models only the global metallicity , z , is normally taken into account to determine the stellar populations . however it is in principle possible to follow the abundance distribution separately for each of several important elements in the gas . examples of these so called chemical evolution models can be found in larson ( 1972 ) , matteucci & tornambe ( 1987 ) , tosi ( 1988 ) and there are many more . however , calculating colors and especially absorption line - strengths in such models has not been attempted up to now . finally , among this kind of models there are a few which combine chemical evolution with dynamics , e.g. theis , burkert & hensler ( 1992 ) . one can not say that one type of model is better than another . in general , more observables and physical parameters can be calculated if one makes more assumptions . if no assumptions are made about the physics , as in the empirical models , one may end up with solutions that are unphysical . if however wrong assumptions are made , one will not learn anything about the stellar evolution history either . we show later in this paper that our results can be reproduced using the _ simple _ analytical model and possibly with infall . this means that we could have replaced the part of the model that deals with the evolution of the gas by some analytical calculations . however , we have preferred to build up the numerical machinery , since it offers much more flexibility , and even improved insight . it is clear that our understanding of all these aspects is improving with time , which means that the models that are to be applied can legitemately be more and more complicated , and in this context we have developed the evolutionary population synthesis code presented in this paper . for the reasons mentioned above it should never be used as a static black box out from which a theoretical fit to the data is to be taken , but as an evolving tool , which might help in disentangling stellar populations in a composite system . if one wants to calculate the final spectrum of a galaxy that has evolved from a gas cloud , one has to integrate over time the spectra of all stars that are still _ living _ at the current time . the number of stars formed at a certain epoch is determined by the star formation rate . little is known about this , so many models give it a prescribed form , and let it decrease e.g. exponentially . in this paper we assume that the sfr is proportional to the gas density ( schmidt 1959 ) . the gas density itself and the chemical evolution of the gas is calculated taking into account the original gas , and the metal - enriched gas that is ejected by stars . the yields for the various elements needed for this calculation are not especially well known , and better knowledge would significantly improve the model . other factors that may affect the sfr , such as inflow or outflow , are not considered in the context of the present models . the stars living at the current epoch contain stellar populations with a mixture of ages and metallicities . to calculate the final spectrum we decompose the stars into single stellar populations ( ssp ) each of a single age and metallicity , and calculate their spectra . to do this , one needs in the first place theoretical isochrones . the parameters of the isochrones depend , amongst others , on opacities of ions and molecules , and are reasonably well known for the early stages of stellar evolution . however for later phases such as the agb and the post - agb , evolutionary calculations are very complicated , and could still benefit from significant improvement . the isochrones are much better for solar and sub - solar metallicity than for metal rich stars , because the former can be , and have been , tested observationally on globular clusters . in general , the relative composition of the elements heavier than helium in the models is close to solar ; only recently have people included for example oxygen - enhancement ( vandenberg 1992 ) or @xmath0-element enhancement ( weiss _ et al . _ 1995 ) . the following step is to obtain a spectrum for a star with physical parameters given by the isochrones . model atmospheres needed for this ( e.g. kurucz 1992 ) appear to be reasonably reliable in the blue . in the red , molecular opacities make them , for the time being , much less reliable . for that reason several authors ( faber _ et al . _ 1985 , gorgas _ et al . _ 1993 and worthey _ et al . _ 1994 ) have developed a method that depends much more on observations . they use a grid of observed stars with various theoretical parameters to determine fitting functions that can be used to calculate an absorption line index for any combination of theoretical isochrone parameters . the problem however is that these fitting functions at the moment are available only for a limited number of absorption lines , since one needs many stars of various types and metallicities to make them . if one wants to carry out population synthesis , one needs to cover as large a wavelength range as possible , and also at as high a spectral resolution . this is to cover many colors and line indices , since in principle every color and absorption line is affected in a different way by the parameters above : the sfr , the imf , the metallicity and abundance ratios etc . in practice a model with complete coverage from uv to near - ir is hard to implement , because of lack of fitting functions , color calibrations etc . , and the observations are hard to obtain . in this paper we are concentrating on early - type galaxies , for which high spectral resolution is less important , due to their large velocity dispersions . we have produced model output for colors and lines that are relatively easy to obtain and reproduce , and that allow us to separate effects due to various relevant physical parameters . to some extent we are limited by the availability of libraries of stars , especially for the fitting functions , so this aspect too , can be significantly improved in the future . the layout of the paper is as follows . in section 2 we will describe how we obtain colors and absorption line indices for a single stellar population , emphasizing differences from previous studies , and especially any improvements . in section 3 we introduce our chemical evolutionary model . in section 4 we apply our model to a limited set of data from three standard galaxies , for which the data is presented in an accompanying paper ( vazdekis _ et al . _ 1996 , * paper ii * ) . we perform the fits for an ssp ( the _ static _ option ) and for a full evolutionary case . in section 5 the conclusions are presented . finally , in paper ii , apart from presenting the observational basis of the data - set used here , we carry out a comprehensive fit of the whole set of indices and colors on the basis of the scenarios suggested in this paper . in particular , we compare the fits obtained assuming a single - age single - metallicity stellar population or using the full chemical evolution model . we also discuss the relations between elements .

it can synthesize single age , single metallicity stellar populations or follow the galaxy through its evolution from an initial gas cloud to the present time . we have applied our model to new data for a set of three early - type galaxies , to find out whether these can be fitted using single - age old metal - rich stellar populations , as is normal practice when one uses other stellar models of this kind . however the data , as well as our fits , will be discussed in much more detail in a second paper ( vazdekis _ et al .

we have developed a new stellar population synthesis model designed to study early - type galaxies . it provides optical and near - infrared colors , and line indices for 25 absorption lines . it can synthesize single age , single metallicity stellar populations or follow the galaxy through its evolution from an initial gas cloud to the present time . the model incorporates the new isochrones of the padova group and the latest stellar spectral libraries . we have applied our model to new data for a set of three early - type galaxies , to find out whether these can be fitted using single - age old metal - rich stellar populations , as is normal practice when one uses other stellar models of this kind . the model is extensively compared with previous ones in the literature to establish its accuracy as well as the accuracy of this kind of models in general . using the evolutionary version of the model we find that we can not fit the most metal - rich elliptical galaxies if we keep the imf constant and do not allow infall of gas . we do however reproduce the results of arimoto & yoshii ( 1986 ) for the evolution of the gas , and produce colors , and , for the first time with this type of models , absorption line - strengths . it is in fact possible to fit the data for the elliptical galaxies by varying the imf with time . our numerical model is in good broad agreement with the analytical _ simple model_. we prefer however to calculate the evolution of the gas numerically instead of using the _ simple model _ , since it offers more flexibility , and even improved insight , when comparing with observations . in the present paper we describe the model , and compare a few key observables with new data for three early - type _ standard _ galaxies . however the data , as well as our fits , will be discussed in much more detail in a second paper ( vazdekis _ et al . _ 1996 ) , where some conclusions will be drawn about elliptical galaxies on the basis of this model . 23truecm

astro-ph9605112

c

we have developed a stellar population model to apply to early - type galaxies . it produces optical and near - infrared colors , and absorption line strengths in lines from 4100 - 8800 on the lick system , is applicable to systems of intermediate or old age , and metallicity larger than 0.1 z@xmath94 . the model is chemo - evolutionary , i.e. it calculates the properties of a stellar system , starting from a primordial gas cloud . however , it can easily be used to predict the properties of systems with a single age and metallicity . the model colors and line strengths are determined by integrating stellar observables along theoretical isochrones , in this way obtaining single stellar populations ( ssps ) , and then integrating these ssps over time . the model uses isochrones with solar metal abundance ratios . as far as possible empirical calibrations have been used to convert theoretical isochrone values to observables for individual stars . some properties of the models are : * extensive comparisons with models from other authors show that there is broad overall agreement in the colors and line strengths . we have also made independent estimates of the errors in our observables . * our chemical evolutionary model is in good agreement with arimoto & yoshii ( 1986 ) . we present optical and ir colors which are likely to be improved as a result of new calibration relations . also , for the first time , we give integrated line strengths for an evolutionary model . * we find that for a closed box approximation the most metal rich elliptical galaxies can not be fitted with a single imf that is constant in time . to solve this problem , we propose a scenario invoking an imf skewed towards high - mass stars during a short , initial period ( smaller than 1 gyr ) , followed by preferential low - mass star formation in the remaining time . * we have briefly tested the model here by fitting it to a few key colors and line strength indices , using a new data set for 3 standard early - type galaxies . in general , satisfactory results are reported . a much more comprehensive comparison of theory with observations is given in paper ii . to conclude , we need to make a few statements about the applicability of this model . given the nature of this type studies , the numbers given in the text will soon cease to be the most accurate possible , because better isochrones become available , or better calibrations linking one parameter to another . therefore , we will try to update the model as time goes on , and interested people can always obtain the most recent version electronically from the authors . there are however a few areas in which we think that further effort by the astronomical community is needed to improve models of this kind . these are : * we show that absorption line strengths are generally more accurate than integrated colors . to change this situation better color - color relations are needed , especially for very high and very low metallicities . * inclusion of more absorption line observations will make this kind of models more useful . especially in the near - uv or near - ir very little work has been done , except in the region of the ca ii triplet . more libraries like the one of worthey _ ( 1994 ) are urgently needed . * it looks as if a substantial part of the uncertainties are due to incorrect treatment of the later stages of stellar evolution , such as the agb . theoretical work in this area would be very valuable . we are grateful to franco fagotto for discussions and for his valuable comments and suggestions on the manuscript , which have greatly improved the final version . we also would like to thank the referee , guy worthey , for improving substantially the quality of this paper . we are grateful to the padova group for providing us with the set of theoretical isochrones . we are indebted to onno pols who kindly gave us the calculations for the very low - mass stars , and to angel alonso , santiago arribas and carlos martnez who have made available their observational results before publication . we also thank gustavo bruzual and stphane charlot who provided us with their latest calculations of integrated colors and absorption line indices . we also thank nobuo arimoto , marijn franx and bianca poggianti for useful comments on the manuscript . this work was partially supported by grants pb91 - 0510 and pb94 - 1107 of the spanish dgicyt . thanks the kapteyn institute in groningen for financial assistance during a working visit . alexander , d.r . & ferguson , j.w . , 1994 , , 473 , 879 alongi , m. , bertelli , g. , bressan , a. , chiosi , c. , fagotto , f. , greggio , l. , nasi , e. , 1993 , , 97 , 851 alonso , a. , arribas , s. & martnez - roger , c. , 1994 , , 107 , 365 alonso , a. , arribas , s. & martnez - roger , c. , 1995 , , 297 , 197 alonso , a. , arribas , s. & martnez - roger , c. , 1996 , , in press arimoto , n. , & yoshii , y. , 1986 , , 164 , 260 ( * ay86 * ) arimoto , n. , & yoshii , y. , 1987 , , 173 , 23 arnaud , m. , rothenflug , r. , boulade , o. , vigroux , l. & vangioni - flam , e. , , 254 , 49 audouze , j. & tinsley , b.m . , 1976 , , 14 , 43 bertelli , g. , bressan , a. , chiosi , c. , fagotto , f. & nasi , e. , 1994 , , 106 , 275 ( * bbcfn * ) bessell , m.s . , 1983 , , 95 , 480 bessell , m.s . , 1990 , , 102 , 1181 bessell , m.s . , wood , p. r. , 1984 , , 96 , 247 bessell , m.s . & brett , j.m . , 1988 , , 100 , 1134 bessell , m.s . , brett , j.m . , scholz , m. & wood , p.r . , 1989 , , 77 , 1 bessell , m.s . , brett , j.m . , scholz , m. & wood , p.r . , 1991 , , 89 , 335 bica , e. , 1988 , , 195 , 76 bica , e. , pastoriza , m. , maia , m. , da silva , l. & dottori , h. , 1991 , , 102 , 1702 bica , e. , clara , j.j & dottori , h. , 1992 , , 103 , 1859 bressan , a. , chiosi , c. , fagotto , f. , 1994 , , 94 , 63 bressan , a. , chiosi , c. & tantalo , r. , 1996 , , submitted bruzual , g. , 1983 , , 273 , 205 bruzual , g. , 1992 , in the stellar populations of galaxies , ed . b. barbuy & a. renzini ( dordrecht : kluwer ) , 311 bruzual , g. & charlot , s. , 1993 , , 405 , 538 bruzual , g. & charlot , s. , 1996 , in preparation burstein , d. , faber , s. m. , gaskell , c. m. & krumm , n. , 1984 , , 287 , 586 buser , r. & kurucz , r. , 1978 , 70 , 555 buzzoni , a. , 1989 , , 71 , 817 caldwell , n. , kennicutt , r. & schommer , r. , 1994 , , 108 , 1186 casuso , e. , 1991 , ph . d. thesis , univ . of la laguna casuso , e. , vazdekis , a. , peletier , r. & beckman , j.e . , 1996 , , 458 , 533 charlot , s. & bruzual , g. , 1991 , , 367 , 126 charlot , s. , worthey , g. & bressan , a. , 1996 , , 457 , 625 clayton , d.d . , 1985 , in nucleosynthesis , univ . chicago press , w. d. arnett & truran , eds . , p. 65 clayton , d.d . , 1986 , , 98 , 968 code , a.d . , davis , j. , bless , r.c . , & hanbury brown , r. , 1976 , , 203,417 covino , s. , galletti , s. & pasinetti , l.e . , 1995 , , 303 , 79 daz , a. , terlevich , e. & terlevich , r. , 1989 , , 239 , 325 eggleton , p.p . , 1973 , , 163 , 279 elbaz , d. , arnaud , m & vangioni - flam , e. , 1995 , , 303 , 345 faber , s.m . , 1972 , , 20 , 361 . faber , s.m . , friel , e.d . , burstein , d. , gaskell , c.m . , 1985 , , 57 , 711 fluks , m.a . , plez , b. , th , p.s . , de winter , d. , westerlund , b.e . & steenman , h.c . , 1994 , , 105 , 311 frogel , j.a . , persson , s.e . & cohen , j.g . , 1981 , 246 , 842 gibson , b.k . , 1996 , , submitted girardi , l. , chiosi , c. , bertelli , g. & bressan , a. , 1995 , , 298 , 87 gorgas , j. , faber , s.m . , burstein , d. , gonzalez , j.j . , courteau , s. & prosser , c. , 1993 , , 86 , 153 guiderdoni , b. , rocca - volmerange , b. , 1987 , , 186 , 1 guiderdoni , b. , rocca - volmerange , b. , 1990 , , 227 , 362 guiderdoni , b. , rocca - volmerange , b. , 1991 , , 252 , 435 han , z. , podsiadlowski , p. & eggleton , p.p . , 1994 , , 270 , 121 he s , r. & peletier , r. , , 268 , 539 huebner , w.f . , merts , a.l . , magee , n.h . & argo , m.f . , 1977 , los alamos sci . lab . la-6760-m houdashelt , m.l . , frogel , j.a . & cohen , j.g . , 1992 , 103 , 163 iglesias , c.a . , rogers , f.j . , wilson , b.g . , 1992 , , 397 , 717 johnson , h.l . , 1966 , , 4 , 193 kennicutt , r.c . , 1989 , , 344 , 685 kroupa , p. , tout , c.a . & gilmore , g. , 1993 , , 262 , 545 kurucz , r.l . , 1992 , in the stellar populations of galaxies , ed . b. barbuy & a. renzini ( dordrecht : kluwer ) , 225 lacey , g. , fall , s.m . , 1985 , , 290 , 154 lacey , c.g . , guiderdoni , b. , rocca - 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volmerange , b. , 1989 , , 236 , 7 rocca - volmerange , b. & guiderdoni , b. , 1987 , , 175 , 15 rocca - volmerange , b. & guiderdoni , b. , 1988 , , 75 , 93 rocca - volmerange , b. & guiderdoni , b. , 1990 , , 247 , 166 scalo , j.m . , 1986 , fund . cosmic phys . , 11 , 1 . schmidt , m. , 1959 , , 129 , 243 schmidt , m. , 1963 , , 137 , 758 searle , l. , sargent , w.l.w . & bagnuolo , w.g . , 1973 , , 179 , 427 searle , l. , wilkinson , a. & bagnuolo , w.g . , 1980 , , 239 , 803 spinrad , h. & taylor , b.j . , 1971 , , 22 , 445 stetson , p.b . & harris , w.e . , 1988 , , 96 , 909 tantalo , r. , chiosi , c. bressan , a. & fagotto , f. , 1995 , , submitted terndrup , d.m . , frogel , j.a . & withford , a.e . , 1990 , , 357 , 453 terndrup , d.m . , frogel , j.a . & withford , a.e . , 1991 , 378 , 742 tinsley , b.m . , 1968 , , 151 , 547 tinsley , b.m . , 1972 , , 20 , 383 tinsley , b.m & gunn , j.e.,1976 , , 203 , 52 tinsley , b.m . , 1978a , , 220 , 816 tinsley , b.m . , 1978b , , 222 , 14 tinsley , b.m . , 1980 , fund . cosmic phys . , 5 , 287 theis , c. , burkert , a. , hensler , g. , 1992 , , 265 , 465 tosi , m. 1988 , , 197 , 33 turnrose , b.e . , 1976 , , 210 , 33 twarog , b.a . , 1980 , , 242 , 242 vandenberg , d.a . , 1983 , , 51,29 vandenberg , d.a . , 1992 , , 391 , 685 vazdekis , a. , peletier , r. f. , beckman , j. e. & casuso , e. , 1996 , , submitted ( * paper ii * ) weiss , a. , peletier , r.f . & matteucci , f. , 1995 , , 296 , 73 whitford , a.e . , 1978 , , 226 , 777 worthey , g. , faber , s.m . & gonzalez , j.j . , 1992 , , 398 , 69 worthey , g. , faber , s. , gonzlez , j. & burstein , d. , 1994 , , 94 , 687 worthey , g. , 1994 , , 95 , 107 ( * w94 * ) wu , c.c . , faber , s.m . , gallagher , j.s . , peck , m. & tinsley , b.m . , 1980 , , 237 , 290 yoshii , y. & takahara , f. , 1988 , , 299 , 593

we have developed a new stellar population synthesis model designed to study early - type galaxies . it provides optical and near - infrared colors , and line indices for 25 absorption lines . our numerical model is in good broad agreement with the analytical _ , since it offers more flexibility , and even improved insight , when comparing with observations . in the present paper we describe the model , and compare a few key observables with new data for three early - type _ standard _ galaxies .

we have developed a new stellar population synthesis model designed to study early - type galaxies . it provides optical and near - infrared colors , and line indices for 25 absorption lines . it can synthesize single age , single metallicity stellar populations or follow the galaxy through its evolution from an initial gas cloud to the present time . the model incorporates the new isochrones of the padova group and the latest stellar spectral libraries . we have applied our model to new data for a set of three early - type galaxies , to find out whether these can be fitted using single - age old metal - rich stellar populations , as is normal practice when one uses other stellar models of this kind . the model is extensively compared with previous ones in the literature to establish its accuracy as well as the accuracy of this kind of models in general . using the evolutionary version of the model we find that we can not fit the most metal - rich elliptical galaxies if we keep the imf constant and do not allow infall of gas . we do however reproduce the results of arimoto & yoshii ( 1986 ) for the evolution of the gas , and produce colors , and , for the first time with this type of models , absorption line - strengths . it is in fact possible to fit the data for the elliptical galaxies by varying the imf with time . our numerical model is in good broad agreement with the analytical _ simple model_. we prefer however to calculate the evolution of the gas numerically instead of using the _ simple model _ , since it offers more flexibility , and even improved insight , when comparing with observations . in the present paper we describe the model , and compare a few key observables with new data for three early - type _ standard _ galaxies . however the data , as well as our fits , will be discussed in much more detail in a second paper ( vazdekis _ et al . _ 1996 ) , where some conclusions will be drawn about elliptical galaxies on the basis of this model . 23truecm

astro-ph0404256

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we modeled the emission - line luminosity and profile from the debris released by the tidal disruption of a star by a black hole in the early phase of evolution . our model predicts prompt optical evolution of post - disruption debris and profile shapes different from circular and elliptical disk model profiles . since line profiles observed so far in liners look more disk - like and evolve slowly , the observations are likely to have caught the event at late times ( @xmath226 6 months after the initial disruption ) , after the debris has settled into a quasi - stable configuration . the line profiles can take a variety of shapes for different orientations of the debris tail relative to the observer . due to the very diverse morphology of the debris , it is almost impossible to uniquely match the multi - peaked profile with the exact emission geometry . nevertheless , the profile widths and shifts are strongly indicative of the velocity distribution and the location of matter emitting the bulk of the h@xmath0 light . profile shapes do not depend sensitively on the shape of the light curve of the x - rays illuminating the debris . they strongly depend on the distance of the emitting material from the central ionizing source , which is a consequence of the finite propagation time of the ionization front and the redistribution of the debris in phase space . it may be possible to distinguish between the two effects observationally , based on their different characteristic time scales . the onset of the optically thick spheroidal halo should cause the disappearance of the broad h@xmath0 emission line on the time scale of months , and give rise to the emission of narrower , strong , blueshifted or redshifted emission line , arising from the portion of the tidal tail unobscured by the halo . if x - ray flares and the predicted variable profiles could be observed from the same object they could be used to identify the tidal disruption event in its early phase . the x - ray flares can be promptly detected by all - sky synoptic x - ray surveys and high energy burst alert missions such as _ swift_. the evolution of the tidal event may then be followed with optical telescopes from the ground on longer time scales and give an insight in the next stage of development of the debris . thus , simulations of the tidal disruption process on longer time scales ( of order several months to a few years ) are sorely needed . calculations of the long - term evolution of a tidal disruption event can predict the type of structure that the debris finally settles into and whether its emission - line signature resembles the transient double - peaked lines observed in liners . this study would provide an important insight into the evolution of liners . finally , the observed rate of tidally disrupted solar type stars can constrain the rate of captured compact objects ( which are important gravitational wave sources ) , and the capture rate of main sequence stars in our galaxy , which are expected to emit the peak of the gravitational radiation in the lisa frequency band and can be detected in the local universe . we are indebted to j. charlton for her help with cloudy . t.b . also thanks m. falanga for his valuable comments . we are also grateful to the anonymous referee for very insightful and helpful comments and suggestions . we acknowledge the support of the center for gravitational wave physics funded by the nsf under cooperative agreement phy-0114375 , nsf grants phy-9800973 and phy-0244788 , the zaccheus daniel fellowship , and the eberly college of science . lccccccc tail & @xmath229@xmath230 & @xmath231@xmath232 & @xmath233 & @xmath234 & @xmath235 & @xmath1740.1 & @xmath236 + disk & @xmath237@xmath238 & @xmath239@xmath240 & @xmath241@xmath242 & @xmath243@xmath244 & @xmath245 & 20 & @xmath246 + halo & @xmath238 & @xmath247@xmath240 & @xmath248 & @xmath233 & @xmath249 & 27 & @xmath250 +

we have modeled the time - variable profiles of the h emission line from the non - axisymmetric disk and debris tail created in the tidal disruption of a solar - type star by a black hole . we model the emission line profiles in the period immediately after the accretion rate onto the black hole became significant .

we have modeled the time - variable profiles of the h emission line from the non - axisymmetric disk and debris tail created in the tidal disruption of a solar - type star by a black hole . two tidal disruption event simulations were carried out using a three dimensional relativistic smooth - particle hydrodynamic code , to describe the early evolution of the debris during the first fifty to ninety days . we have calculated the physical conditions and radiative processes in the debris using the photoionization code cloudy . we model the emission line profiles in the period immediately after the accretion rate onto the black hole became significant . we find that the line profiles at these very early stages of the evolution of the post - disruption debris do not resemble the double peaked profiles expected from a rotating disk since the debris has not yet settled into such a stable structure . as a result of the uneven distribution of the debris and the existence of a `` tidal tail '' ( the stream of returning debris ) , the line profiles depend sensitively on the orientation of the tail relative to the line of sight . moreover , the predicted line profiles vary on fairly short time scales ( of order hours to days ) . given the accretion rate onto the black hole we also model the h light curve from the debris and the evolution of the h line profiles in time .

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we know of @xmath220 x - ray binaries with dynamically confirmed black hole ( bh ) accretors ; these include 15 low mass x - ray binaries ( lmxbs ) and 3 high mass x - ray binaries ( hmxbs ) in the milky way and magellanic clouds ( see e.g. * ? ? ? * and references within ) , as well as bh + wolf - rayet binaries in ic10 @xcite and ngc300 @xcite . these systems were identified using x - ray and optical observations ; the mass function is calculated from periodic radial velocity shifts in emission lines from the optical counterpart . all of the bh lmxbs identified this way are necessarily transient , because the optical spectra of bright lmxbs are dominated by the accretion disc ( see e.g. * ? ? ? we have established a method for identifying bhcs from their x - ray properties alone . this makes use of the `` low / hard '' emission state seen in bh and neutron star ( ns ) xbs @xcite , that is only seen at 0.011000 kev luminosities @xmath010% eddington in ns xbs @xcite ; @xcite recently found that the low / hard state is limited to luminosities @xmath010% eddington in bh xbs also . bh xbs may exhibit low / hard emission states at considerably higher luminosities than ns xbs , due to the higher accretor mass . however , it is necessary to differentiate between our bhcs and distant active galactic nuclei ( agn ) , since agn and xb emission spectra are often similar . we have identified 10 bhcs from their high luminosity low states to date . of these , 9 are associated with m31 gcs @xcite , and are therefore likely lmxbs . only one of these is transient ; however , persistently bright gc bh xbs are consistent with tidal capture of a main sequence donor ( * ? ? ? * although the donor may be disrupted in the process ) , or with an ultra - compact system with a degenerate donor @xcite . we identified our first bhc outside of a gc from its x - ray spectra , long term ( @xmath212 year ) behaviour , and a serendipitous hst observation @xcite ; the faint optical counterpart ( @xmath5 @xmath6 @xmath70.4 ) suggests a low mass donor for this system also . we cover 8 of the 9 gc bhcs in our survey : bhcs 1 , 2 , 20 , 25 , 28 , 31 , 32 , and 34 . the field bhc described in @xcite is bhc3 . we obtained accurate positions for our x - ray sources by registering 27 x - ray sources associated with m31 gcs to the m31 field 5 b band image provided by @xcite . the r.m.s . offsets were 0.11@xmath8 in r.a . and 0.09@xmath8 in dec @xcite ; this would be extremely unlikely unless the 27 x - ray sources used for calibration were indeed associated with the gcs . the x - ray spectra of bh binaries are usually described with two components : a thermal component ( often modelled as a multi - temperature disk blackbody ) , and a power law component to represent unsaturated , inverse - compton scattering of cool photons on hot electrons @xcite . the hard state is classified by a power law component with photon index ( @xmath9 ) = 1.42.1 , and a thermal component that contributes @xmath10 of the 220 kev flux @xcite . our very first gc bhc ( xb045 , xbo 45 in * ? ? ? * ) was associated with the m31 gc b045 , named following the revised bologna catalogue v.3.4 ( rbc , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? its @xmath217,000 count xmm - newton / pn spectrum was well described by an absorbed power law with line - of - sight absorption ( @xmath11 ) = 1.41@xmath120.11@xmath13 atom @xmath14 , and @xmath9 = 1.45@xmath120.04 ; @xmath15/dof = 517/487 ( good fit probability , gfp , = 0.17 ) . adding a blackbody component improved the fit somewhat , but the thermal contribution to the flux was too small to be constrained ; we therefore considered xbo 45 to be in its hard state . since its unabsorbed 0.310 kev luminosity was 2.5@xmath120.2@xmath16 erg s@xmath17 , 140% eddington for a 1.4 @xmath18 ns , we considered it a bhc . the 8 remaining gcs bhcs are included in this survey , unlike xbo 45 , so we do nt describe each spectrum in detail here . however , we note that fitting a two component model to xb144 ( xbo 144 in * ? ? ? * ) yielded k@xmath19 = 0.0082@xmath120.0016 kev , indicating a complete lack of thermal component in the spectrum . we also note that even though xb082 was best described by @xmath9 = 1.20@xmath120.09 , the @xmath15/dof for that fit was @xmath20.9 , and we were able to obtain fits where @xmath9=1.4 and @xmath15/dof @xmath201 @xcite . the unabsorbed 0.310 kev luminosities of our gc bhcs exhibiting hard state spectra range over @xmath2545@xmath21 erg s@xmath17 @xcite . comparison with the 0.510 kev agn flux distribution obtained by @xcite yields a 2.4@xmath22 probability that our gc bhcs are coincident agn . the probability that our brightest gc bhc , xb135 , is a coincident agn is 1.2@xmath23 . we will present evidence in a separate paper that xb135 may contain the most massive stellar mass bh known to date ; it may have been formed by direct collapse of a high mass , metal poor star ( r. barnard et al . , 2013 , in prep ) . we found that the gcs hosting these very bright x - ray sources were significantly more massive and/or metal rich than the other gcs in m31 , agreeing with previous work . however , two gcs had particularly low metalicities ; one of these is b135 , consistent with the direct collapse formation scenario for xb135 ( see * ? ? ? * and references within ) . @xcite have shown that while bh masses are limited to @xmath215 @xmath18 for solar metalicties , they could theoretically reach @xmath280 @xmath18 for metalicities @xmath20.01 solar . high accretion rate ns xbs exhibit multi - component emission that may appear to be hard state spectra in extragalactic x - ray sources , where the spectra have relatively few photons . to compare the emission spectra of our bhcs with ns xrb , we use the double thermal ( disk blackbody + blackbody ) model used by lin et al . ( 2007 , 2009 , 2012 ) to describe hundreds of rxte observations of ns xbs in all their varied emission states . we note that the physical interpretation of their model is contradicted for persistent xbs by a substantial body of work ; however , lin et al . ( 2007 , 2009 , 2012 ) have sampled spectra from the full range of ns lmxb emission states in a consistent manner , allowing us to compare our bhcs with ns xbs in a single parameter space . for a long time , ns xbs were divided into two types , z - sources ( high lumonisity ) and atoll sources ( low luminosity ) , based on their luminosities and color - color diagrams ( cds ) ; the cds of z - sources had three branches ( horizontal , normal , and flaring ) , and evolved along these branches without ever jumping from one branch to the other ; the cds of the atoll sources were more fragmented @xcite . furthermore , the z - sources were split into those like cygnus x-2 , and those like scorpius x-1 @xcite . it was believed that the differences were due to more than just the accretion rate , because the z - sources varied by a factor of a few when tracing their z - shaped cds , while atoll source intensities varied by 12 orders of magnitude @xcite . however , we now know of two transient ns systems that exhibited both types of z - source behavior before evolving to atoll source behavior during decay @xcite . @xcite examined the spectral evolution of two galactic x - ray transients , aql x-1 and 4u1608@xmath752 , over many rxte observations covering @xmath620 outbursts . they devised a new double - thermal model ( disk blackbody + blackbody ) to describe a ns transient soft state that is analagous to the bh soft state described by @xcite . they have since applied their model to rxte observations of xtej1701@xmath7462 , one of the transients that exhibits cyg - like and sco - like z - source behavior as well as atoll behavior @xcite , and also to the sco - like z - source gc 17 + 2 @xcite . they have applied their model to hundreds of rxte spectra from galactic ns binaries including the full gamut of ns spectral behavior . they found that their disk blackbody + blackbody was unsuccessful in two situations . firstly , they found the hard state spectra to be power law dominated , as expected @xcite . we therefore expect our bhcs to inhabit a separate parameter space to the ns xbs because fitting the double thermal model to hard state spectra will yield unphysical results . secondly , they found that z - sources required a three - component spectrum ( disk blackbody + blackbody + power law ) on the horizontal branch @xcite . @xcite also performed broadband analysis of suzaku and bepposax observations of the persistently bright galactic ns xb 4u1705@xmath744 , with energy ranges 0.1600 kev and 0.1300 kev respectively . when using a disk blackbody + blackbody model to the soft state spectra , they found temperatures that were similar to those they observed in the rxte observations @xcite . fitting an additional comptonization component , via either a power law or simpl convolution model ( following * ? ? ? * ) resulted only in small changes in the thermal components ; this is because the comptonized component only contributed @xmath010% of the total flux in the soft state ( see fig . 6 in * ? ? ? hence the parametric differences between our bhcs and the ns xbs should not be due to differences in energy bands used in the observations , or the lack of a third ( power law ) component . evidence against the double thermal model indicates an extended corona that contributes a substantial portion of the x - ray flux . this evidence includes the ingress times of periodic absorption dips in the x - ray lightcurves of the high inclination xbs ( the dipping sources , see e.g. * ? ? ? * ; * ? ? ? * and references within ) , and also broadened emission lines in a chandra grating spectroscopy of cyg x-2 @xcite . furthermore , compact corona models where the inner disk temperature is tied to the seed photon energy for comptonization are rejected for ulxs in ngc253 , and the confirmed bh+wolf - rayet binary ic10 x-1 @xcite , as well as bhc3 in the steep power law state @xcite . in @xcite , we applied structure function analysis to xbs for the first time , following @xcite , who created an ensemble structure function for agn . they used a structure function ( sf ) to estimate the mean intensity deviation for data separated by time @xmath24 : @xmath25 where @xmath26 is the photon noise and @xmath27 is the x - ray flux . they grouped the sf into logarithmic bins with width 0.5 ; each bin in the range log(@xmath24 ) = 0.03.0 contained more than 100 measurements . our sample consisted of 37 x - ray sources associated with objects in the rbc ; these were classified by @xcite as 30 confirmed gcs , 4 gc candidates , 1 star , and 2 agn . the sfs of gc xbs with 0.310 kev luminosities @xmath2250@xmath28 erg s@xmath17 tended to show significantly more variability than agn over a wide range of time - scales . the sfs of brighter xbs generally showed comparable or less variability than agn , despite their high signal to noise ; however , their high fluxes made them unlikely agn . sfs provide an effective mechanism for distinguishing between xbs and agn @xcite . in this work , we combine these techniques to search for bhcs in the whole region covered by 152 chandra observations from our monitoring programme ; the roll angle is unrestricted , resulting an approximately circular field of view with radius @xmath29 . we surveyed 530 x - ray sources in this region . we also obtain spectra from archival xmm - newton observations , to strengthen the cases for several bhcs . the 100@xmath3 region surrounding m31 * is particularly interesting , because @xcite found an excess of x - ray binaries over the radial distribution expected from k band light ( tracing stellar mass ) ; this excess had the distribution expected of dynamically formed xbs @xcite . dynamical xb formation requires stellar densities rarely seen outside gcs , and @xcite proposed that the m31 bulge is sufficiently large and dense to form a significant number of xbs dynamically . however , since the stellar velocities in the m31 bulge are @xmath2510 times higher than in gcs , @xcite expect only short period binaries to survive , with most of these containing bh accretors . evidence for bhcs in this region would therefore provide strong support for their theory . in the next section , we discuss the observations and data analysis ; this is followed by our results , then by our discussion .

they exhibit `` hard state '' emission spectra that are seen at luminosities% eddington in x - ray binaries ( xbs ) containing a neutron star ( ns ) or black hole ( bh ) , at luminosities that significantly exceed the ns threshold . nine of these are associated with globular clusters ( gcs ) ; hence , these are most likely low mass x - ray binaries ( lmxbs ) ; eight are included in this survey .

we have previously identified 10 m31 black hole candidates ( bhcs ) in m31 , from their x - ray properties alone . they exhibit `` hard state '' emission spectra that are seen at luminosities% eddington in x - ray binaries ( xbs ) containing a neutron star ( ns ) or black hole ( bh ) , at luminosities that significantly exceed the ns threshold . nine of these are associated with globular clusters ( gcs ) ; hence , these are most likely low mass x - ray binaries ( lmxbs ) ; eight are included in this survey . we have recently discovered that analysis of the long term 0.54.5 kev variability of xbs via structure functions allows us to separate xbs from agn , even though the emission spectra are often similar ; this has enabled us to search for bhcs outside of gcs . we have identified 26 new bhcs ( 12 strong , 14 plausible ) within 20 of the m31 nucleus ( m31 * ) , using 152 chandra observations spaced over years ; some of our classifications were enhanced with xmm - newton observations . of these , 7 appear within 100 of m31 * ; this supports the theory suggesting that this region experiences enhanced xb production via dynamical processes similar to those seen in gcs . we have found a parameter space where our black hole candidates are separated from galactic neutron star binaries : we show that modelling a simulated hard state spectrum with a disk blackbody + blackbody model yields parameters that lie outside the space occupied by neutron star binaries that are modeled this way . the probability that our bhcs all lie within the ns parameter space is .

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we have identified 26 new black hole candidates in the central region of m31 , using their structure functions or luminosities to identify them as x - ray binaries , and their high luminosity hard state spectra to classify them as bhcs . of these , 12 are strong candidates , and 14 are plausible candidates that may benefit from further observations . we were previously limited to identifying bhcs in globular clusters , due to the similarities between xb and agn spectra . this brings the total number of bhcs within 20@xmath1 of m31@xmath56 identified by their high luminosity hard states to 35 . ccccc bhc & @xmath11 / 10@xmath60 @xmath14 & k@xmath72 / kev & k@xmath74 / kev + 1 & 0.35 f & 1.99@xmath126 & 1@xmath103 + 3 & 0.07 f & 1.07@xmath127 & 0.42@xmath128 + 16 & 0.07 f & 1.44@xmath129 & 0.8@xmath106 + 20 & 0.197@xmath120.011 & 1.29@xmath107 & 0.73@xmath130 + 21 & 0.11@xmath120.07 & 1.6@xmath131 & 0.7@xmath132 + 31 & 0.07 f & 1.27@xmath133 & 0.43@xmath108 + the structure functions of most of our bhcs reveal them to be substantially more variable than typical agn ( as measured by vagnetti et al . , 2011 ) over a wide range of time scales . those bhcs with comparable or less variability than the agn have 0.510 kev luminosities matched by @xmath20.6 agn within the observed region , according to @xcite . it is therefore unlikely that our new bhcs are agn . we have found that our bhc spectra exist in a separate parameter space to galactic neutron star systems when we compare our best fits for absorbed blackbody + disk blackbody emission models with the systems studied by lin et al . ( 2007 , 2009 , 2012 ) . this is expected because this double thermal model fails to fit hard state spectra ; the disk blackbody component is forced low temperatures in order to provide the low energy flux . indeed , the probability that our bhcs all lie in the ns parameter space when fitted with the double - thermal model is just @xmath23@xmath71 . seven of our bhcs ( 20% of our total sample ) were found within 100@xmath3 of the m31 nucleus , lending support to the hypothesis that the m31 bulge is sufficiently dense to form a significant number of xbs dynamically , as seen in globular clusters ; since the stellar velocities in the m31 bulge are considerably higher than in gcs , surviving xbs are expected to have short periods and are likely to contain black hole accretors @xcite .

we have previously identified 10 m31 black hole candidates ( bhcs ) in m31 , from their x - ray properties alone . we have recently discovered that analysis of the long term 0.54.5 kev variability of xbs via structure functions allows us to separate xbs from agn , even though the emission spectra are often similar ; this has enabled us to search for bhcs outside of gcs . we have identified 26 new bhcs ( 12 strong , 14 plausible ) within 20 of the m31 nucleus ( m31 * ) , using 152 chandra observations spaced over years ; some of our classifications were enhanced with xmm - newton observations . of these , 7 appear within 100 of m31 * ; this supports the theory suggesting that this region experiences enhanced xb production via dynamical processes similar to those seen in gcs . we have found a parameter space where our black hole candidates are separated from galactic neutron star binaries : we show that modelling a simulated hard state spectrum with a disk blackbody + blackbody model yields parameters that lie outside the space occupied by neutron star binaries that are modeled this way . the probability that our bhcs all lie within the ns parameter space is .

we have previously identified 10 m31 black hole candidates ( bhcs ) in m31 , from their x - ray properties alone . they exhibit `` hard state '' emission spectra that are seen at luminosities% eddington in x - ray binaries ( xbs ) containing a neutron star ( ns ) or black hole ( bh ) , at luminosities that significantly exceed the ns threshold . nine of these are associated with globular clusters ( gcs ) ; hence , these are most likely low mass x - ray binaries ( lmxbs ) ; eight are included in this survey . we have recently discovered that analysis of the long term 0.54.5 kev variability of xbs via structure functions allows us to separate xbs from agn , even though the emission spectra are often similar ; this has enabled us to search for bhcs outside of gcs . we have identified 26 new bhcs ( 12 strong , 14 plausible ) within 20 of the m31 nucleus ( m31 * ) , using 152 chandra observations spaced over years ; some of our classifications were enhanced with xmm - newton observations . of these , 7 appear within 100 of m31 * ; this supports the theory suggesting that this region experiences enhanced xb production via dynamical processes similar to those seen in gcs . we have found a parameter space where our black hole candidates are separated from galactic neutron star binaries : we show that modelling a simulated hard state spectrum with a disk blackbody + blackbody model yields parameters that lie outside the space occupied by neutron star binaries that are modeled this way . the probability that our bhcs all lie within the ns parameter space is .

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the mechanism responsible for high @xmath1 superconductivity is still highly controversial @xcite . while a large part of the theoretical effort is based on the hypothesis that cooper pairs are formed in high @xmath1 compounds , there is no agreement on the physical origin of the pairing interaction nor on the symmetry of the pair wavefunction . these two questions are actually intimately related . indeed repulsive interactions , such as produced by spin fluctuations @xcite , require an order parameter which changes sign on the fermi surface in order to produce pairs . this leads , in the simplest hypothesis , to pairing with d - wave symmetry if we assume singlet pairing . on the other hand purely attractive interactions lead to pairing with s - wave symmetry since any change of sign is unfavorable in this case . therefore an experimental determination of the order parameter symmetry should greatly help to identify the physical interaction responsible for pair formation : although it would not be enough to provide a unique identification , it would strongly narrow the remaining possibilities . for this reason a large part of the recent experimental work has been aimed toward providing a clear signature for the symmetry of the order parameter . quite surprisingly recent experiments on @xmath3 designed with this purpose of identifying the symmetry have given clear cut , but contradictory answers @xcite . indeed the observation of a sizeable josephson current @xcite in a c - axis tunnelling junction between @xmath3 and pb is quite difficult to reconcile with a pure d - wave symmetry while it is in full agreement with s - wave symmetry . similarly the fact that there is no angular dependence in the critical current of ybco - ybco grain boundary junctions in the a - b plane @xcite goes clearly in the direction of an s - wave interpretation . on the other hand a number of experiments are in favor of a d - wave symmetry . many experiments , including tunnelling , nmr , raman scattering , photoemission , penetration depth @xcite , have shown the existence of low energy excited states . actually these experiments are compatible with a strongly anisotropic s - wave order parameter . or more simply one may look for extrinsic effects and wonder if these states do not arise from surface effects or defects , which would provide quasi - normal regions . however the existence of a linear t dependence of the penetration depth over a large range of temperature , simultaneously in cristals and films , with the same slope @xcite , makes an extrinsic interpretation for all these experiments unlikely ( while this kind of explanation may very well be valid for some of them ) . moreover some experiments specifically designed to check if the order parameter changes sign over the fermi surface have given positive answers . these are the corner squid experiments @xcite which give a clear indication for a change of sign of the order parameter between the a and the b axis , and the observation of a spontaneous magnetization corresponding to a half magnetic flux quantum in 3 grain - boundary josephson junctions @xcite which implies a @xmath4 shift , in clear agreement with d - wave symmetry . on the other hand the d - wave interpretation is not free of problems . for example recent experiments show that some thermodynamical superconducting properties are markedly anisotropic ( the a and b - axis results are different ) . indeed the penetration depth in good ybco cristals display a strong anisotropy of the penetration depth @xcite , the specific heat anomaly at @xmath1 of the parent compound @xmath5 ( with same @xmath1 as ybco ) has a marked anisotropy as a function of the orientation of an applied magnetic field @xcite . it is difficult to ascribe these anisotropies to the weak orthorhombic distorsion , and it is more likely that the cuo chains play a significant role in the superconducting properties . one of the most conspicuous problem of the d - wave interpretation is the weak sensitivity of the critical temperature of @xmath3 to the presence of impurities . indeed any kind of impurities , whether magnetic or not , produces in d - wave superconductors @xcite an effect analogous to pair - breaking by magnetic impurities in standard s - wave superconductors @xcite . in particular the critical temperature @xcite decreases rapidly with increasing impurity concentration following the abrikosov - gorkov law @xcite , and superconductivity disappears at a critical concentration . in contrast all samples of @xmath3 seem to have a @xmath1 around 90 k. it is difficult to believe that all samples ( including the earlier ones ) are clean enough to affect only weakly the critical temperature , whereas all microscopic studies show that there are always more structural defects than what is generally admitted . moreover ion @xcite or electron @xcite irradiation experiments have shown a rather weak sensitivity of the critical temperature of @xmath3 on the inverse lifetime deduced from resistivity measurements @xcite . actually zn impurities are known @xcite to have a depressing effect on @xmath6 but this can be interpreted as a standard pair - breaking effect since the environment of a zn impurity , located in the @xmath7 planes , is known to be magnetic @xcite . in order to solve these contradictions we have proposed recently @xcite for @xmath3 a model which mixes s - wave and d - wave features . in our model , in addition to the @xmath7 planes , the cuo chains play an essential role . the pairing interaction within the planes is attractive ( it can be for example produced by phonons ) . on the other hand the pairing interaction between planes and chains is repulsive ( it can be produced by coulomb interaction ) . in this way the order parameter has opposite signs on the planes and on the chains . moreover we include the hybridization between planes and chains , which corresponds physically to take into account the possibility for an electron to jump from planes to chains or vice - versa . naturally the coupling responsible for this hybridization is fairly small , but it is a well known feature of all band structure calculations @xcite . it is of importance only when the plane and the chain band intersect . in this case it leads to an anticrossing in the dispersion relations , and similarly to an anticrossing of the fermi surfaces , wherever the ( uncoupled ) pieces of the fermi surface related to plane and chain cross . as a result , when we move on a given sheet of the fermi surface , we go from a part which corresponds physically to a plane electron , to a part which corresponds physically to a chain electron . since the order parameter has opposite signs for plane and chain electrons , this implies that the order parameter changes sign on a given sheet of the fermi surface and therefore has nodes on this sheet by continuity . therefore our model provides an order parameter which is quite analogous to a d - wave order parameter and it can in this way explain @xcite all the experiments in favor of d - wave symmetry . on the other hand it does not have a d - wave symmetry since the nodes , which occur at the intersection between plane and chain bands , have no reason to satisfy @xmath8 = @xmath9 , and the average of the order parameter has no symmetry reason to be zero . hence there is for example no problem with the nonzero josephson current in @xmath3 - pb junctions along the c - axis @xcite . moreover since the attractive in - plane interaction and the repulsive interaction between plane and chain help each other , there is no problem to explain the high value of the critical temperature of @xmath3 @xcite . naturally our model is specific of @xmath3 , but we can think that it can be generalized to other compounds where the role of the chains can be played by other parts of the structure , such as the bio planes in bssco . on the other hand there is no possibility of this kind in lsco and accordingly we do not expect experiments to display in this compound the same physical features as in ybco . our model has common features with many other models . as a two band model it is quite similar to the two band models introduced by suhl , matthias and walker and others @xcite to describe superconductivity in transition metals . the possibility of an interband repulsive interaction in two band models was already introduced by kondo @xcite . more recently a two band s - n model has been introduced by abrikosov and klemm @xcite to account for raman scattering data . the idea of an interband repulsion has been put forward by various groups @xcite recently in the context of high @xmath1 superconductivity in order to show that experiments displaying a change of sign of the order parameter did not necessarily imply a spin fluctuation mechanism . our model is also similar to the one proposed by abrikosov @xcite , where there are attractive and repulsive interactions within a single band , leading to a change of sign of the order parameter within this band . finally , as noted above , chain - plane hybridization is a standard feature of band structure calculations @xcite and there have been various suggestions of the importance of the chains in the physics of @xmath10 whether for intrinsic or for extrinsic reasons @xcite , these models allowing for the electron to jump between planes and chains . in this paper we consider the effect of impurities on the critical temperature in our model . we will actually restrict ourselves to non magnetic impurities . indeed magnetic impurities are easily included , but they will naturally lead to pair - breaking and produce a @xmath1 following the ag law as in other models . hence magnetic impurities can not discriminate between various models and we ignore them for simplicity . the conclusion of our study is that , in our model , for generic parameters , the critical temperature is much less sensitive to impurities than in standard d - wave models . therefore we might say that , with respect to the sensitivity of the critical temperature to non magnetic impurities , our model behaves as a weak d - wave model . this happens for two independent reasons . first the reduction of @xmath1 is due to plane - chain scattering , which is weak compared to plane - plane scattering . next the fact that we have a two band model provides further possibilities for a weak impurity sensitivity . in section ii we present our model and we calculate the critical temperature of the clean superconductor as a first step toward the calculation of the impurity dependence , which is dealt with in section iii . finally section iv is devoted to a comparison with experimental results and to our conclusion .

pairing in the planes is due to phonons , while coulomb repulsion induces in the chains an order parameter with opposite sign . due to the anticrossing produced by hybridization between planes and chains , the order parameter changes sign on a single sheet of the fermi surface resulting in nodes in the gap . we find that , in our model , the critical temperature is much less sensitive to impurities than in standard d - wave models .

we have studied the effect of impurities on the critical temperature for a model of ybco involving pairing both in the cuo planes and in the cuo chains . in this model pairing in the planes is due to phonons , while coulomb repulsion induces in the chains an order parameter with opposite sign . due to the anticrossing produced by hybridization between planes and chains , the order parameter changes sign on a single sheet of the fermi surface resulting in nodes in the gap . we find that , in our model , the critical temperature is much less sensitive to impurities than in standard d - wave models . one reason is that impurities produce essentially plane - plane and chain - chain scattering , which does not affect the critical temperature . is reduced by the scattering between parts of the fermi surface which have opposite signs for the order parameter , just as in standard d - wave . in our model this is due to plane - chain scattering . we have found that this scattering , whatever its origin , will be smaller by a factor of order t ( that is hybridization coupling over fermi energy ) compared to plane - plane and chain - chain scattering . accordingly the sensitivity of to impurities in our model is reduced by a similar factor , compared to the d - wave situation . in the specific case which we have studied in details and which reduces to the two - band model , we have found a further reduction of the sensitivity of to impurities with a behaviour which can vary continuously from s - wave like to d - wave like depending on the parameters . we expect a similar behaviour and reduction to occur in the general case .

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the conclusion of our study is that , in our model , the critical temperature is much less sensitive to impurities than it is in standard d - wave models . one basic reason is that we expect physically impurities to produce essentially plane - plane and chain - chain scattering . just as in s - wave superconductors , this scattering does not affect the critical temperature . on the other hand @xmath1 is reduced by scattering between parts of the fermi surface which have opposite signs for the order parameter , just as in standard d - wave . in our model this is due to plane - chain scattering . we have found that this scattering , whatever its origin , will be smaller by a factor of order t / @xmath306 ( that is hybridization coupling over fermi energy ) compared to plane - plane and chain - chain scattering . from the band structure calculations we expect this factor to be typically somewhere between 0.1 and 0.3 . therefore the sensitivity of @xmath1 to impurities in our model is reduced by a similar factor , compared to the d - wave situation . in the specific case which we have studied in details and which reduces to the two - band model , we have found a further reduction of the sensitivity of @xmath1 to impurities with a behaviour which can vary continuously from s - wave like to d - wave like depending on the parameters . from our discussion of its physical origin , we expect a similar behaviour and reduction to occur in the general case . it is unfortunately not possible to make quantitative comparison with experiments . indeed it seems surprisingly quite difficult @xcite to avoid a magnetic character for substitutional impurities in ybco . naturally this occurs when the isolated impurity atom itself has a magnetic moment . but this happens also when a non magnetic impurity acquires a magnetic moment due to its interaction with the environment , as it is the case for zn for example @xcite . naturally an impurity with a magnetic character will produce pair - breaking leading to an abrikosov - gorkov like law in any model , in agreement with what is observed experimentally . hence these kind of experiments can not be used directly to eliminate theoretical models . nevertheless , after these words of caution with respect to a simple - minded interpretation of impurities experimental results , we note that the reduction of @xmath1 down to 13 k for 8% zn is obtained by an increase of the residual resistivity @xcite by a factor of order 10 . the corresponding @xmath307 / @xmath308 should be then of order 20 @xmath1 ( see below ) . this is much more than what is necessary to destroy superconductivity within a d - wave model according to the abrikosov and gorkov law . moreover it is useful to plot the results of ref.17 and ref.18 for the critical temperature as a function of the residual resistivity . this is done on fig.2 . it can be seen that the behaviour of some of our results found in fig.1 is quite similar to the experimental results ( we assume naturally that the residual resistivity is proportional to the scattering rate due to impurities ) . we have naturally enough adjustable parameters to fit them nicely . however such a fit would be rather meaningless because , in addition to the magnetic problem , the scatter in the data is rather important at low impurity content and the interpretation of the data for larger concentration is uncertain ( localisation effects which are not taken into account in our theory might play a significant role ) . if we turn to irradiation experiments , whether by electrons or by light ions , the interpretation is also not an easy one . it is known that high @xmath1 compounds have a critical temperature much more sensitive to irradiation than standard superconducting materials . however , although it is likely that most of the created defects do not have a magnetic character , we can not eliminate , from our knowledge on substitutional impurities , the possibility that some are magnetic . the experimental evidence is controversial in this respect @xcite . then there is some evidence that localisation effects might be important since a metal - insulator transition is observed in ybco under light ion irradiation , and there is no intermediate normal phase between the superconducting phase and the insulating one @xcite . this sensitivity to localisation is easy to understand because of the two - dimensional nature of the @xmath7 planes . localisation effects are not included in our theoretical study . it is also not clear at all that the created disorder can be considered as homogeneous @xcite since the resistivity measurements do not always display a sharp drop at the critical temperature . similarly one might expect that the defects created by irradiation are randomly distributed at the microscopic scale , but it is actually quite likely that the chains are more sensitive to irradiation than the planes . there is finally the obvious problem of having an experimental determination of the quasiparticle lifetime produced by disorder . measuring the increase in resistivity appears the best way to do it @xcite although it is far from perfect since , for example , it measures at best transport lifetime which we expect to be somewhat larger than quasiparticle lifetime . notwithstanding the above problems let us try to interpret the irradiation experiments with our theoretical model , just to see what comes out . from drude s law , with a typical resistivity of 100 @xmath309 cm at @xmath1 and a plasma frequency of 1.1 ev @xcite , we have a typical inverse lifetime @xmath310 / @xmath308 @xmath53 2 @xmath311 since superconductivity disappears in d - wave when the inverse lifetime due to impurities is of order 2 @xmath1 we would expect that an increase of resistivity of 100 @xmath309 cm leads to the suppression of @xmath1 @xcite . this corresponds roughly to a decrease of @xmath1 of 1k per @xmath309 cm ( the ag law is essentially linear ) . the experimental results of ref.15 give a linear decrease of @xmath1 with respect to resistivity with a slope 0.3 k / @xmath309 cm . in ref.14 there is an upward curvature at low resistivity with a maximal initial slope of 0.1 k/ @xmath309 cm . since in our model a reduction by a typical factor 1/10 with respect to the d - wave result corresponds to a typical choice of our parameters , we see that this last experimental result agrees with our expectation . but the result of ref.14 could easily be explained for example by a larger value of the hybridization energy or by a suitable choice of the other parameters of our two - band model . quite generally we have enough parameters to vary in our model so we can get easily agreement with these various experimental results . however it must also be kept in mind that , as discussed above , there are other possible physical processes which we have not taken into account and which will add up to produce a faster decrease of the critical temperature with the resistivity . a clear example of this is found in ref.14 where a more rapid decrease is found for a set of samples and attributed to extrinsic effects , while good samples show an average decrease of 0.03 k/ @xmath309 cm . therefore we can consider an experimental result as an upper bound for our theoretical result , but it may quite well be larger than what we find . clearly it is more difficult to explain in a d - wave model the slow dependence of @xmath1 on resistivity found in ref.14 than it is in our model to explain the somewhat stronger dependence found in ref.15 . in conclusion we have seen that our model is quite coherent with the present experimental evidence . naturally , with respect to this problem of the effect of impurities on the critical temperature , it would be much better to have experiments providing stronger constraints on theoretical models , but this might prove difficult to achieve . let us vary eq.(9 ) with respect to the phonon frequencies @xmath45 and @xmath69 . we assume for example @xmath312 = - @xmath313 / @xmath45 = - @xmath314 @xmath295 @xmath69 @xmath116 @xmath315 the variation of eq.(9 ) gives : where @xmath317 . the isotope effect is reduced if we show that @xmath318 @xmath116 0 . we set @xmath319 = ( @xmath320 ) ( @xmath321 ) - @xmath322 . we have @xmath323 ( @xmath324 ) = @xmath325 @xmath326 / d , and @xmath327 = @xmath65 @xmath326 / d + d ( @xmath61 - kk ) @xmath312 @xmath47 @xmath65 @xmath326 / d ( we assume @xmath61 - kk @xmath47 0 , which implies @xmath328 - @xmath329 ) . since we have y @xmath47 @xmath62 and y @xmath47 @xmath330 and @xmath326 @xmath47 0 it is enough to prove that : @xmath335 = \nonumber \\ - k^*k'^ * [ \lambda ^*-\lambda ^{\prime * } + \mu ^*-\mu ^{\prime * } ] ^{2 } - ( \mu ^*\mu ^{\prime * } - k^*k'^ * ) [ ( \lambda ^*-\lambda ^{\prime * } ) ^{2 } + 4 k^*k'^ * ] < 0 \label{eqa3}\end{aligned}\ ] ] where @xmath336 = 1/ @xmath272 and @xmath333 = 1/ @xmath273 . since we have y @xmath47 @xmath336 @xmath47 y , eq.(48 ) is satisfied and the isotope effect is indeed reduced . there is no reduction only in the limiting case @xmath61 = kk and @xmath337 for a recent review of the controversy about the order parameter symmetry , see r. c. dynes , sol . 92 , 53 ( 1994 ) and j. r. schrieffer , sol . 92 , 129 ( 1994 ) . d. j. scalapino , e. loh and j. e. hirsch , phys . rev . b * 34 * , 8190 ( 1986 ) ; a. kampf and j. r. schrieffer , phys . rev . b*42 * , 7967 ( 1990 ) ; p. monthoux , a. v. balatsky and d. pines , phys.rev.lett.*67 * , 3448 ( 1991 ) . a. g. sun , d. a. gajewski , m. b. maple and r. c. dynes , phys.rev.lett.*72 * , 2267 ( 1994 ) . p. chaudhari and shawn - yu lin , phys.rev.lett.*72 * , 1084 ( 1994 ) . w. n. hardy , d. a. bonn , d. c. morgan , r. liang and k. zhang , phys . rev . lett . * 70 * , 3999 ( 1993 ) ; l. a. de vaulchier , j. p. vieren , y. guldner , n. bontemps , r. combescot , y. lemaitre and j. c. mage , europhys * 33 * , 153 ( 1996 ) . d. a. wollman , d. j. van harlingen , w. c. lee , d. m. ginsberg and a. j. leggett , phys.rev.lett.*71 * , 2134 ( 1993 ) ; d. a. brawner and h. r. ott , phys.rev.b * 50 * , 6530 ( 1994 ) . c. c. tsuei , j. r. kirtley , c. c. chi , l. s. yu - jahnes , a. gupta , t. shaw , j. z. sun and m. b. ketchen , phys.rev.lett . * 73 * , 593 ( 1994 ) d. n. basov _ _ , phys.rev.lett.*74 * , 598 ( 1995 ) . j. buan , b. zhou , c. c. huang , j. z. liu and r. n. shelton , phys.rev . b * 49 * , 12220 ( 1994 ) . p. hirschfeld , p. wlfle and d. einzel , phys.rev.b * 37 * , 83 ( 1988 ) . r. j. radtke , k. levin , h. b. schttler and m. r. norman , phys.rev.b * 48 * , 653 ( 1993 ) . p. muzikar , d. rainer and j. a. sauls , fermi liquid theory of non s - wave superconductivity in lecture notes for the nato advanced study institute : vortices in superfluids to be published by kluwer ( 1994 ) . a. a. abrikosov and l. p. gorkov , zh . . fiz . * 39 * , 1781 ( 1960 ) [ sov . jetp * 12 * , 1243 ( 1961 ) ] . j. lesueur , l. dumoulin , s. quillet and j. radcliffe , j. alloys and compounds * 195 * , 527 ( 1993 ) . j. giapintzakis , d. m. ginsberg , m. a. kirk and s. ockers , phys.rev . b * 50 * , 15967 ( 1994 ) . r. liang , t. nakamura , h. kawaji and m. itoh , physica c * 170 * , 307 ( 1990 ) . t. r. chien , z. z. wang and n. p. ong , phys.rev.lett . * 67 * , 2088 ( 1991 ) . s. zagoulaev , p. monod and j. jegoudez , phys.rev . b * 52 * , 10474 ( 1995 ) . r. combescot and x. leyronas , phys.rev.lett.*75 * , 3732 ( 1995 ) . j. yu , s. massida , a. j. freeman and d. d. koeling , phys . lett . a * 122 * , 203 ( 1987 ) ; w.e . pickett et al.,science * 255 * , 46 ( 1992 ) . b. ashauer , w. lee and j. rammer , z. phys . b * 67 * , 147 ( 1987 ) ; l.n . bulaevskii , o.v . dolgov , m.o . ptitsyn , phys.rev . b * 38 * , 11290 ( 1988 ) . h. suhl , b. t. matthias and l. r. walker , phys.rev.lett . * 3 * , 552 ( 1959 ) . v. a. moskalenko , fiz . * 8 * , 503 ( 1959 ) . j. kondo , prog . * 29 * , 1 ( 1963 ) . t. soda and y. wada , prog . phys.*36 * , 1111 ( 1966 ) . v. a. moskalenko and m. e. palistrant , zh . * 49 * , 770 ( 1965 ) [ sov . jetp * 22 * , 536 ( 1966 ) ] ; c. c. sung and v. k. wong , j. phys . * 28 * , 1933 ( 1967 ) ; w. s. chow , phys.rev . * 172 * , 467 ( 1968 ) ; t. kusakabe , prog . phys.*43 * , 907 ( 1970 ) . a. a. abrikosov , physica c * 182 * , 191 ( 1991 ) ; a. a. abrikosov and r. a. klemm , physica c * 191 * , 224 ( 1992 ) . a. a. golubov , o. v. dolgov , e. g. maksimov , i. i. mazin and s. v. shulga , physica c * 235 - 240 * , 2383 ( 1994 ) . a. a. golubov and i. i. mazin physica c * 243 * , 153 ( 1995 ) . a. i. liechtenstein , i. i. mazin and o. k. andersen , phys . rev . lett.*74 * , 2303(1995 ) . d. z. liu , k. levin and j. maly , phys.rev.b * 51 * , 8680 ( 1995 ) . a. a. abrikosov , physica c * 244 * , 243 ( 1995 ) . z. tesanovic , a. r. bishop and r. l. martin , sol . comm , * 68 * , 337 ( 1988 ) . v. z. kresin and s. a. wolf , phys.rev . b * 46 * , 6458 ( 1992 ) . r. a. klemm and s. h. liu , phys.rev.lett.*74 * , 2343 ( 1995 ) . d. rainer , sol . comm * 6 * , 111 ( 1968 ) . r. combescot , europh.lett.10 , 177 ( 1989 ) ; r.combescot , phys . b * 42 * , 7810 ( 1990 ) . a. a. abrikosov , l. p. gorkov and i. e. dzyaloshinski , methods of quantum field theory in statistical physics ( dover , n.y . , 1975 ) . see for example k. maki in superconductivity , ed.by r. d. parks ( dekker , n. y. 1969 ) ; p.b.allen and b. mitrovic , sol . state phys . 37 , ed.f . seitz , d. turnbull and h. ehrenreich ( academic , n. y. 1982 ) . d. markowitz and l.p . kadanoff , phys.rev . * 131 * , 563 ( 1963 ) . the symmetric case @xmath156 = @xmath158 and k = k has @xmath277 = @xmath273 = @xmath156 - k , and is solved explicitely into two branches @xmath338 + @xmath265 and @xmath339 with the physical solution corresponding to the lower value for x. this degenerate situation is interesting to keep in mind , since the general case can be considered as resulting from the coupling of these two branches when one moves away from this particular case .

one reason is that impurities produce essentially plane - plane and chain - chain scattering , which does not affect the critical temperature . is reduced by the scattering between parts of the fermi surface which have opposite signs for the order parameter , just as in standard d - wave . in our model this is due to plane - chain scattering . we have found that this scattering , whatever its origin , will be smaller by a factor of order t ( that is hybridization coupling over fermi energy ) compared to plane - plane and chain - chain scattering . accordingly the sensitivity of to impurities in our model is reduced by a similar factor , compared to the d - wave situation . in the specific case which we have studied in details and which reduces to the two - band model , we have found a further reduction of the sensitivity of to impurities with a behaviour which can vary continuously from s - wave like to d - wave like depending on the parameters . we expect a similar behaviour and reduction to occur in the general case .

we have studied the effect of impurities on the critical temperature for a model of ybco involving pairing both in the cuo planes and in the cuo chains . in this model pairing in the planes is due to phonons , while coulomb repulsion induces in the chains an order parameter with opposite sign . due to the anticrossing produced by hybridization between planes and chains , the order parameter changes sign on a single sheet of the fermi surface resulting in nodes in the gap . we find that , in our model , the critical temperature is much less sensitive to impurities than in standard d - wave models . one reason is that impurities produce essentially plane - plane and chain - chain scattering , which does not affect the critical temperature . is reduced by the scattering between parts of the fermi surface which have opposite signs for the order parameter , just as in standard d - wave . in our model this is due to plane - chain scattering . we have found that this scattering , whatever its origin , will be smaller by a factor of order t ( that is hybridization coupling over fermi energy ) compared to plane - plane and chain - chain scattering . accordingly the sensitivity of to impurities in our model is reduced by a similar factor , compared to the d - wave situation . in the specific case which we have studied in details and which reduces to the two - band model , we have found a further reduction of the sensitivity of to impurities with a behaviour which can vary continuously from s - wave like to d - wave like depending on the parameters . we expect a similar behaviour and reduction to occur in the general case .

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the local structure of , with an orthorhombic phase and = 60 k , has been studied at the cu k@xmath142 , ba k@xmath142 , and tl l@xmath110edges using the xafs technique . several new approaches have been implemented , _ e.g. _ , direct structural simulations up to 7 using the feff5 code , constrained fits , and comparisons of ppdfs for different models . both the qualitative and quantitative data analyses clearly show that the cu , o(1 ) , o(2 ) , and ba atoms are at their ideal sites in the unit cell as given by the diffraction studies , while the tl and o(3 ) atoms are displaced from the sites suggested by the average crystal structure . the tl - tl distance at r=3.5 between the tlo layers remains , but the tl - tl distance at 3.9 in the tlo layer is not observed and the tl - ba and ba - tl peaks are very broad . the shorter tl - o(3 ) distance in the tlo layer is about 2.33 , significantly shorter than the distance calculated with both the tl and o(3 ) atoms at their ideal @xmath0 sites ( @xmath10 or @xmath2 ) . an excellent fit to the xafs data can be achieved with a correlated displacement model shown in fig . the fitting results show that the tl atom is displaced along the @xmath3 direction from its ideal site by about 0.11 and the o(3 ) atom is shifted from the @xmath0 site by about 0.53 roughly along the @xmath4 direction . this model also fits very well to the xafs data collected from two tetragonal samples with different t@xmath143s . the xafs data do not support the uncorrelated displacement models proposed by diffraction investigations@xcite . our model is similar to that proposed by the pdf analyses of the neutron scattering data on tl-2212@xcite , but is more specific about the relative shift of the tl atoms between the two consecutive tlo layers . however , the tl displacement , 0.11 , obtained in our study , is only one third of the value ( 0.32 ) suggested by the pdf . the estimated cu and tl valences imply a charge transfer between the and tlo layers which is consistent with other experimental results@xcite . the estimated tl valence also supports the distorted structure of the tlo layer obtained from our xafs analysis . the estimated tl valence ( 2.95 + ) in the distorted tlo layer is much closer to its formal valence ( 3 + ) than that ( 2.77 + ) estimated from the ideal tlo layer . a comparison of the xafs data at the tl l@xmath110 edge from three samples clearly shows a correlation between the transition temperature and the local o environment around the tl atoms . this implies that the hole concentration in the layer is mainly controlled by the o arrangement about the tl in those samples , and not by cu substitution for the tl . we thank corwin booth for help in the data collection . the experiments were performed at the stanford synchrotron radiation laboratory , which is operated by the u.s . department of energy , division of chemical sciences , and by the nih , biomedical resource technology program , division of research resources . the experiment is partially carried out on uc / national laboratories prt beam time . the work is supported in part by nsf grant dmr-92 - 05204 . b. h. toby , w. dmowksi , t. egami , j. d. jorgensen , m. a. subramanian , j. gopalakrishnan , and a. w. sleight , in _ high temperature superconductors : relationships between properties , structure , and solid - state chemistry _ , edited by j. d. jorgensen _ et al _ ( materials research society , pittsburgh , pennsylvinia , 1989 ) , p.309 . in this paper , we have used @xmath52=0.9 for the tl - tl pair . in ref . 9 , @xmath52 was set to be about 0.5 which seems too small according to our recent detailed comparison of feff5 calculation with experimental xafs@xcite . .cu local structure obtained from xafs and diffraction . nbrs " indicates the weighted number of neighbors . the estimated errors are : number of neighbors , @xmath144 15 % ; distances , @xmath144 0.02 . [ cols="^,^,^,^,^,^,^ " , ]

we have used the xafs ( x - ray - absorption fine structure ) technique to investigate the local structure about the cu , ba , and tl atoms in orthorhombic with a superconducting transition temperature = 60 k. our results clearly show that the o(1 ) , o(2 ) , cu , and ba atoms are at their ideal sites as given by the diffraction measurements , while the tl and o(3 ) atoms are more disordered than suggested by the average crystal structure . the tl - tl distance at 3.5 between the tlo layers does not change , but the tl - tl distance at 3.9 within the tlo layer is not observed and the tl - ba and ba - tl peaks are very broad . the shorter tl - o(3 ) distance in the tlo layer is about 2.33 , significantly shorter than the distance calculated with both the tl and o(3 ) atoms at their ideal sites ( or ) . a model based on these results shows that the tl atom is displaced along the directions from its ideal site by about 0.11 ; the displacements of neighboring tl atoms are correlated . the o(3 ) atom is shifted from the site by about 0.53 roughly along the directions . a comparison of the tl l-edge xafs spectra from three samples , with = 60 k , 76 k , and 89 k , shows that the o environment around the tl atom is sensitive to while the tl local displacement is insensitive to and the structural symmetry .

we have used the xafs ( x - ray - absorption fine structure ) technique to investigate the local structure about the cu , ba , and tl atoms in orthorhombic with a superconducting transition temperature = 60 k. our results clearly show that the o(1 ) , o(2 ) , cu , and ba atoms are at their ideal sites as given by the diffraction measurements , while the tl and o(3 ) atoms are more disordered than suggested by the average crystal structure . the tl - tl distance at 3.5 between the tlo layers does not change , but the tl - tl distance at 3.9 within the tlo layer is not observed and the tl - ba and ba - tl peaks are very broad . the shorter tl - o(3 ) distance in the tlo layer is about 2.33 , significantly shorter than the distance calculated with both the tl and o(3 ) atoms at their ideal sites ( or ) . a model based on these results shows that the tl atom is displaced along the directions from its ideal site by about 0.11 ; the displacements of neighboring tl atoms are correlated . the o(3 ) atom is shifted from the site by about 0.53 roughly along the directions . a comparison of the tl l-edge xafs spectra from three samples , with = 60 k , 76 k , and 89 k , shows that the o environment around the tl atom is sensitive to while the tl local displacement is insensitive to and the structural symmetry . these conclusions are compared with other experimental results and the implications for charge transfer and superconductivity are discussed .

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generalizations of the ising model including further than nearest - neighbor interactions may give rise to complicated spatial orderings and produce complex and rich critical behavior @xcite . the transitions between ordered and disordered phases may be continuous or of first - order with a tricritical point between them . the exactly soluble ising model in @xmath5 @xcite with the addition of next - nearest - neighbor interactions becomes an analytically intractable problem and an understanding of the effects of adding such further couplings on the critical behavior of the system is an open important problem of great theoretical and experimental interest . we will be particularly interested is cases involving competing interactions with ground - state arrangements that mimic a sublattice order or superantiferromagnetic ( saf ) order in which ferromagnetic lines along the directions of the lattice alternate with lines of opposite oriented spins . such models are of great theoretical and experimental interest . they simulate various types of antiferromagnets @xcite , but also important models of alloy orderings @xcite . due to the well - known correspondence between spin systems and lattice gas , they provide an understanding of the behavior of adsorbed gases on crystal surfaces @xcite . it should be noted here that , remarkable efforts have been made to predict the order of the transition according to the landau - lifshitz rules , which , on the basis of group theory arguments , select the ordered configurations that can be reached from the disordered phase by a continuous transition @xcite . k. binder @xcite has reviewed these rules and has also pointed out well - known counter - examples , emphasizing the experimental difficulties in distinguishing a weak first - order transition from a second - order transition . second - order transitions are more special than first - order transitions , but they are theoretically much better understood . at a first - order transition there is no diverging correlation length , and in general one can not restrict attention to long wavelength phenomena , thus no universality , as in critical phenomena , is to be expected . furthermore , it is well known that antiferromagnetic arrangements , with several order - parameter components , may be produced by the competition of the interactions @xcite . however , no general theoretical agreement exists connecting the symmetry of spin structures , the number of order - parameter components , and the range of interaction with the expected critical behavior and in particular the order of the phase transition . furthermore , many authors have demonstrated the difficulties in properly identifying the order of the transition on a ground of high - temperature expansion , scaling , renormalization group transformations , and monte carlo simulations ( see ref . @xcite and references therein ) . roomany and wyld @xcite pointed out that the occurrence of second - order type effects at the weakly first - order transitions can be explained by a comparison of the correlation length @xmath6 with the lattice size @xmath7 . in the case of the @xmath8 potts model s weak first - order transition , peczak and landau @xcite have observed pseudocritical exponents close to the conjectured values of the critical indices in the @xmath9 potts model . this paper is concerned with the analysis of numerical data , obtained via an accurate entropic monte carlo scheme , on triangular ising finite systems with nearest - neighbor ( @xmath0 ) and next - nearest - neighbor ( @xmath1 ) antiferromagnetic interactions in ratio @xmath2 . rastelli et al . @xcite have recently presented numerical evidence ( for @xmath10 , @xmath11 , and @xmath12 ) that , in the thermodynamic limit , this model undergoes a first - order phase transition , from a layered ordered phase ( saf phase ) to a high temperature paramagnetic phase . here , we will present a more detailed identification of the nature of this transition by looking again at the size dependence of the traditional thermodynamic quantities but also by implementing the lee - kostelirlitz method @xcite and detecting the first - order character of the transition via the size - dependence of the free energy barrier . furthermore , we will improve the original estimates of rastelli et al . @xcite and try to verify the predictions of the finite - size scaling ( fss ) theory of first - order transitions , including transition point shifts , thermal , and magnetic anomalies . the theory of fss near second - order transitions has a rich and longstanding literature @xcite , starting with the pioneering works of fisher @xcite and fisher and barber @xcite . this theory has been extended to first - order transitions @xcite and fss and renormalization group methods have proven to be very useful tools for the study of first- and second - order phase transitions @xcite . our attempt here to elucidate the distinctive first - order features of the present model , by an extensive numerical study , will closely follow previous analogous studies carried out on the @xmath8 , @xmath13 , and @xmath14 potts model @xcite . in these studies the existing theories of first - order transitions have been tested and verified but several important aspects have not been thoroughly clarified , especially in the cases of weak first - order transitions . not only the demonstration of a weak first - order transition is more difficult but also strong finite - size effects may obscure the true asymptotic behavior @xcite and some theoretical predictions may not be met even at quite large lattices ( since the correlation length may be very large ) . in such cases , it is very difficult to discriminate between wrong phenomenological expectations and correct theoretical predictions . therefore , comparisons of different models and alternative studies may provide useful information and a better understanding of finite - size effects , something that is necessary for the correct interpretation of the numerical data and the verification of the theories . the present model , showing a weak first - order phase transition , offers the opportunity of such a contrasting test with the well - studied cases of the potts model . our interest in the present model stems from our recent studies on the analogous square saf model @xcite , which is again an ising model on the square lattice with nearest - neighbor ( @xmath0 ) and next - nearest - neighbor ( @xmath1 ) antiferromagnetic interactions . the ground state of the square saf model is four - fold degenerate and consists of the arrangements in which ferromagnetic rows ( columns ) alternate with opposite oriented spins . in the square model the saf order can be obtained in both cases of a ferromagnetic or an antiferromagnetic nearest - neighbor coupling ( for its @xmath15 phase diagram see refs . similarly , the present triangular saf model , with antiferromagnetic interactions in ratio @xmath2 , has a six - fold degenerate ground state and consists of the six arrangements in which ferromagnetic lines alternate with opposite oriented spins in the three lattice directions ( @xmath15 phase diagrams are given in refs . the hamiltonian that governs these systems , in zero - field , is @xmath16 where here both nearest - neighbor ( @xmath0 ) and next - nearest - neighbor ( @xmath1 ) interactions will be assumed to be positive ( antiferromagnetic ) . this hamiltonian has been studied also in @xmath17 fcc lattices and evidence for both first- and second - order phase transitions have been presented @xcite , although the distinction of the order of the transition was difficult in that case too . the square model , governed by eq . ( [ eq:1 ] ) , develops at low temperatures saf order for @xmath18 and several early studies @xcite , mainly based on importance sampling @xcite , have suggested that this system possess anomalous exponents and a non - universal critical behavior with exponents depending on the coupling ratio @xmath19 . however , despite this early accepted scenario , there has been recently renewed interest and some attempts to re - examine the behavior of this model appeared . in several papers lopez et al . @xcite have used the cluster variation method ( cvm ) to study this model , concluding that the system undergoes a first - order transition for a particular range of the coupling ratio @xmath19 ( @xmath20 ) . on the other hand , this different scenario - predicting first - order transitions between ordered and disordered phases followed by continuous transitions outside the first - order region - has now been seriously questioned by the present authors , who have presented very strong evidence @xcite for the case @xmath21 that points out a well - obeyed second - order scaling behavior . the buzano and pretti @xcite attempt to verify the scenario of lopez et al . @xcite , by including an additional @xmath22-body coupling , and using again the cvm , has revealed a further peculiar inadequacy of the cvm . the limiting case ( @xmath23 ) , where the exact solution of the baxter model @xcite applies , was also predicted by the cvm to undergo a first - order transition , in contradiction to the exact solution . the above introduction describes a rather controversial situation on the class of models with antiferromagnetic ground - state arrangements . therefore , it is of interest to follow closely the previous fss analysis applied to the potts models and study in more detail the triangular saf model . in this case , we will insist that this system undergoes a genuine , but weak , first - order transition , as originally predicted by rastelli et al . @xcite . having a six - fold degenerate ground state arrangement , this model will be shown to produce first - order characteristics that lie between the @xmath8 and @xmath24 potts model . the rest of the paper is organized as follows . in the next section we outline an extensive entropic sampling program . this program goes beyond our previous practice in other applications @xcite and is based on ( i ) the wang - landau ( wl ) method @xcite , ( ii ) our dominant energy restriction ( crmes ) scheme @xcite , and ( iii ) a second - stage improvement that combines the wl @xcite , the lee entropic @xcite , and the broad histogram ( bh ) @xcite or transition matrix @xcite methods . in section [ sec:3 ] we shall review all necessary theoretical aspects of first - order transitions that are then used for the analysis of our numerical data . the free energy barrier method of lee and kosterlitz @xcite , used in the literature for the identification of a first - order transition , is discussed in subsection [ sec:3aa ] , while the size - dependence of thermal properties derived from the well - known double gaussian approximation is summarized in subsection [ sec:3b ] . finally , some new transition points derived from the number - of - phases parameter @xcite are presented in subsection [ sec:3c ] . section [ sec:4 ] presents the analysis of our numerical data . in subsection [ sec:4a ] we make use of several alternative methods for the estimation of the transition temperature presenting a detailed analysis of the new transition points derived from the number - of - phases parameter and a discussion on their conjectured exponentially small shift behavior . in subsection [ sec:4b ] we explore our data for the energy cumulants and the magnetic susceptibility , emphasizing on the examination of the higher - order corrections predicted by the double gaussian approximation . the characteristic values of the energy cumulants , i.e. the coefficient of the dominant diverging term of the specific heat and the limiting values of the second- and fourth - order energy cumulants are determined and the corresponding theoretical predictions are critically discussed . this serves as a useful self - consistency test of our numerical data but also of the theoretical predictions . finally , our conclusions are summarized in section [ sec:5 ] .

we implement a new and accurate numerical entropic scheme to investigate the first - order transition features of the triangular ising model with nearest - neighbor ( ) and next - nearest - neighbor ( ) antiferromagnetic interactions in ratio . our results improve the original estimates of rastelli et al . and verify all the generally accepted predictions of the finite - size scaling theory of first - order transitions , including transition point shifts , thermal , and magnetic anomalies . the behavior of transition points , derived from the number - of - phases parameter , is not in accordance with the theoretically conjectured exponentially small shift behavior and the well - known double gaussian approximation does not correctly describe higher correction terms of the energy cumulants .

we implement a new and accurate numerical entropic scheme to investigate the first - order transition features of the triangular ising model with nearest - neighbor ( ) and next - nearest - neighbor ( ) antiferromagnetic interactions in ratio . important aspects of the existing theories of first - order transitions are briefly reviewed , tested on this model , and compared with previous work on the potts model . using lattices with linear sizes and we estimate the thermal characteristics of the present weak first - order transition . our results improve the original estimates of rastelli et al . and verify all the generally accepted predictions of the finite - size scaling theory of first - order transitions , including transition point shifts , thermal , and magnetic anomalies . however , two of our findings are not compatible with current phenomenological expectations . the behavior of transition points , derived from the number - of - phases parameter , is not in accordance with the theoretically conjectured exponentially small shift behavior and the well - known double gaussian approximation does not correctly describe higher correction terms of the energy cumulants . it is argued that this discrepancy has its origin in the commonly neglected contributions from domain wall corrections . , and first - order transitions , triangular ising model- superantiferromagnetism , entropic sampling 05.50+q , 75.10.hk , 05.10.ln , 64.60.fr

0809.2537

c

the triangular ising model with nearest- and next - nearest - neighbor antiferromagnetic interactions has been studied and its first - order transition features have been clarified , when the interaction ratio is @xmath21 . we have outlined the most important aspects of the existing theories of first - order transitions and we have tested on this model some basic hypothesis from these theories , comparing our results and findings with previous work on the potts model . our numerical data have been used to obtain accurate estimates for all the thermal characteristics of the present weak first - order transition . all the generally accepted predictions of the finite - size scaling theory for first - order transitions , concerning transition point shifts , thermal , and magnetic anomalies , have been well verified for the present model . however , two of our findings for this model are not compatible with some theoretical or phenomenological expectations . the first of these concerns the behavior of the new transition point observables introduced by borgs and janke @xcite and also some similar transition points introduced in this paper . these finite - size transition points are suitable finite - size approximations of the fundamental number - of - phases parameter and it has been theoretically argued @xcite that should be expected to obey exponentially small shifts . this expectation is not verified for the present weak first - order transition . finally , we have shown that the well - known double gaussian approximation does not describe correctly the higher correction terms for all energy cumulants of the present model . it appears that the first correction term in the expansions of energy cumulants is of the form @xmath214 and not of the form @xmath215 , expected from the double gaussian approximation . lee and kosterlitz @xcite have pointed out the inadequacy of the gaussian approximation to produce shifts in the locations of the energies of the equal - height peaks in agreement with those described by eqs . ( [ eq:10 ] ) and ( [ eq:11 ] ) , which are rather well observed in simulations ( see ref . @xcite and fig . [ fig:2](b ) ) . in fact the shortcomings of the gaussian behavior has been critically discussed in the early work of challa et al . our analysis is therefore suggesting again the need for a more realistic theory . it is tempting to assume that the attempted here and well obeyed expansions for the energy cumulants , starting with the correction term @xmath205 , may have their origin in the neglected domain wall corrections and could have the same explanation with the existing puzzling situation concerning the shift behavior of the free energy minima pointed out by lee and kosterlitz @xcite . this research was supported by the special account for research grants of the university of athens under grant nos . fytas would like to thank the the alexander s. onassis public benefit foundation for financial support . l. onsager , phys . 65 ( 1944 ) 117 ; r.j . baxter , exactly solved models in statistical mechanics , academic press , london , 1982 ; b. mccoy , t. wu , the two dimensional ising model , harvard university press , cambridge , 1972 . c. yamaguchi , y. okabe , j. phys . a 34 ( 2001 ) 8781 ; p.n . vorontsov - velyaminov , n.a . volkov , a.a . yurchenko , j. phys . a 37 ( 2004 ) 1573 ; n. rathore , j.j . de pablo , j. chem . ( 2002 ) 7225 ; m.s . shell , p.g . debenedetti , a.z . panagiotopoulos , phys . e 66 ( 2003 ) 056703 ; p. poulain , f. calvo , r. antoine , m. broyer , ph . dugourd , _ ibid . _ 73 ( 2006 ) 056704 ; c. zhou , t.c . schulthess , s. torbrgge , d.p . landau , phys . 96 ( 2006 ) 120201 . ( a ) plot of the barrier height @xmath216 . the dotted line is an extrapolation to @xmath103 , giving a nonzero value for the surface tension of the order of @xmath217 . the inset is used as a guide for the reader . ( b ) the energy minima @xmath218 and @xmath219 and their difference as a function of the inverse linear size . the dotted lines are linear fits indicating the values of the bulk energies @xmath94 and @xmath95 , and that of the latent heat @xmath220 at @xmath221 . ] temperature - dependence of the ratios @xmath222 ( a ) and @xmath223 ( b ) ( eqs . ( [ eq:24 ] ) and ( [ eq:30 ] ) , respectively ) . in panel ( a ) all pairs are shown , i.e. ( @xmath224 , @xmath225)=(@xmath148 , @xmath149 ) , ( @xmath150 , @xmath151 ) , ( @xmath152 , @xmath153 ) , ( @xmath154 , @xmath155 ) , ( @xmath149 , @xmath156 ) , ( @xmath151 , @xmath157 ) , and ( @xmath153 , @xmath158 ) , while in panel ( b ) only the pairs ( @xmath224 , @xmath225)=(@xmath149 , @xmath156 ) , ( @xmath151 , @xmath157 ) , and ( @xmath153 , @xmath158 ) are shown . ] ( a ) illustration of the crossing point corresponding to the peak of the number - of - phases parameter . the behavior of the most probable energies of the pair of systems is also illustrated . ( b ) behavior of the differences , @xmath226 , of the two characteristic energies for each system of the chosen pair . for the larger lattice the graph corresponding to the double of the difference @xmath226 is also shown in order to facilitate the illustration of the crossing relationship ( [ eq:31 ] ) for @xmath166 ( @xmath159 ) . ] ( a ) illustration of the behavior of the three peaks of the reduced number - of - phases parameter defined in in eq . ( [ eq:30 ] ) for @xmath166 . ( b ) behavior of six finite - size transition points and comparison with our estimate ( solid line ) for the transition point and also with the estimate @xmath191 of ref . @xcite ( dashed line ) . ] ( a ) fss behavior of the specific heat data at the five pseudocritical temperatures @xmath121 using an @xmath188 correction . ( b ) the same with panel ( a ) but with an @xmath187 correction . note the large difference in the @xmath200 values . ] ( a ) fss behavior of @xmath228 . two fitting attempts are applied as shown in the figure . @xmath200 is much smaller for the case of an @xmath188 correction . ( b ) fss of @xmath228 , now against @xmath188 . the line corresponds to the fitting formulae shown in panel ( b ) and gives an estimate @xmath229 . ] ( a ) fss behavior of the susceptibility data at the five pseudocritical temperatures @xmath121 . the solid lines are simultaneous fitting attempts , using an @xmath188 correction . ( b ) the same with panel ( a ) but with a correction of the order of @xmath187 . ]

important aspects of the existing theories of first - order transitions are briefly reviewed , tested on this model , and compared with previous work on the potts model . using lattices with linear sizes and we estimate the thermal characteristics of the present weak first - order transition . however , two of our findings are not compatible with current phenomenological expectations . it is argued that this discrepancy has its origin in the commonly neglected contributions from domain wall corrections . , and first - order transitions , triangular ising model- superantiferromagnetism , entropic sampling 05.50+q , 75.10.hk , 05.10.ln , 64.60.fr

we implement a new and accurate numerical entropic scheme to investigate the first - order transition features of the triangular ising model with nearest - neighbor ( ) and next - nearest - neighbor ( ) antiferromagnetic interactions in ratio . important aspects of the existing theories of first - order transitions are briefly reviewed , tested on this model , and compared with previous work on the potts model . using lattices with linear sizes and we estimate the thermal characteristics of the present weak first - order transition . our results improve the original estimates of rastelli et al . and verify all the generally accepted predictions of the finite - size scaling theory of first - order transitions , including transition point shifts , thermal , and magnetic anomalies . however , two of our findings are not compatible with current phenomenological expectations . the behavior of transition points , derived from the number - of - phases parameter , is not in accordance with the theoretically conjectured exponentially small shift behavior and the well - known double gaussian approximation does not correctly describe higher correction terms of the energy cumulants . it is argued that this discrepancy has its origin in the commonly neglected contributions from domain wall corrections . , and first - order transitions , triangular ising model- superantiferromagnetism , entropic sampling 05.50+q , 75.10.hk , 05.10.ln , 64.60.fr

1003.0676

c

we have studied the effects of self - annihilating dark matter on the collapse of the gas structures harboring the formation of the first stars , known as pop iii . for the first time in the literature we follow self - consistently the evolution of the dark matter profile as a consequence of the gravitational drag of the collapsing gas and include the feedback of energy injection by dm annihilations on the chemistry of the gas , in the yet unexplored regime between the virialization of the halo and the formation of a hydrostatic core . we have explored the effects of dmas by spanning a range of masses and annihilation cross sections around the values of the vanilla wimp scenario , namely those able to reproduce the relic abundance of dm with a thermal decoupling , finding similar results but with variations in the details and onset times of the different phases that we are to summarize . in the following , quoted numbers refer to our fiducial case ( @xmath152 , @xmath153 , @xmath154 ) . as expected , heavier wimps ( or smaller self - annihilation rates , or lower opacity since the energy injection term is proportional to @xmath21/m@xmath155 and depends also on @xmath35 ) effects become relevant at later times during the collapse , the same dma rate being achieved at higher dm densities . lighter particles ( or higher @xmath21 , or higher @xmath35 ) produce the same effects at earlier times ( and therefore smaller gas densities ) . this general scenario is robust with respect to parameter variations within physically acceptable ranges . a few key points of our study are worth some emphasis : _ ( i ) _ as known from existing literature ( rmf07 ) , before the virialization of the halo , at gas density @xmath156 ( and therefore extremely small gas opacity ) dmas do not sensibly affect any gas process ; _ ( ii ) _ between halo virialization and a gas density @xmath157 dmas contribute mainly through indirect feedback effects : the free electron floor created by the ionizations induced by dmas catalyzes formation . in turn , molecular provides more cooling to the cloud than in the standard case ( without dmas ) and the temperature of the cloud _ decreases _ as a consequence of dmas ; _ ( iii ) _ finally , at @xmath158 the dma heating rate becomes equal to the gas cooling rate . to a first approximation this leads to a balance between losses and gain , which we dubbed as the `` critical point '' . our results generally agree with previous ones : as the semi - analytical estimates of sfg08 and the analytical study ( based on simulation data ) from natarajan , tan and o shea ( 2009 ) , we confirm the existence of a `` critical point '' . we also find that the equality of cooling vs dma heating terms takes place ( in the innermost shell ) at central densities of approximately @xmath85 , details depending on parameters ( dm mass , opacity coefficient etc . see previous section , and following discussion ) , in agreement with the above mentioned studies . this is particularly relevant as our analysis is the first fully numerical , self - consistent study including a reasonably accurate treatment of radiative transfer allowing to reach the critical point and beyond it in at least three cases . it had been previously suggested ( sfg08 ) that , after reaching of the `` critical point '' , the collapse would stop and the whole structure stall , thus generating a new type of celestial object . with our numerical simulations we have accessed this stage of the collapse , finding that the system does _ not _ halt its collapse in three cases out of four ; in the only case where the collapse stops , this happens far after the critical point was reached , and its duration is very short ( @xmath159 ) after which the gas restarts contracting . moreover , the dm parameters for which the astrophysical system finds important changes after the critical point are actually strongly disfavored by dm constraints based on local and primordial universe observations . by changing the dm mass or self - annihilation rate , the scenario we have described does not change qualitatively , within the range of values studied and the physical regime we have accessed with our simulations . while stressing again that when our objects reach the critical point the structure does _ not _ halt the collapse , and instead it continues its evolution toward the formation of a protostellar core , it is also useful to comment on the alteration of the dynamics of the collapse induced by dmas . the duration of the gas collapse from halo virialization down to densities of @xmath160 , in presence of dmas appears to be _ shorter _ ( especially in the first phases ) than in the standard case ( by about 1 per cent ) of the total collapse time . however we point out that the change is a small fraction of the total , and that the effects of dmas on collapse time are very difficult to quantify . in fact , the dmas effect can indeed accelerate the collapse in some shells , and decelerate it in others , thus making it difficult to obtain a homogeneous definition of time delay . for example , the reduction of the infall velocity of shells enclosing @xmath132 implies that the collapse time of these shells is longer ( by 1020 per cent ) than in the control ( nodm ) case . we feel confident in stressing , however , that the shells which are mostly affected by the infall time decrease ( @xmath7020% of the infall velocity ) are placed between 10@xmath118m@xmath117 and 10@xmath145m@xmath117 , and would become part of a hydrostatic protostar only in the final stages of the collapse . it is not clear whether these shells will eventually end up in the hydrostatic core ( omukai & palla 2003 ) , however it is reasonable to expect that such modification will not alter the _ total _ time for the formation of the star by more than @xmath7020 per cent . even if our runs could not reach the regime when a hydrostatic core forms ( with the partial exception of the maximal run , where we probably got close ) , it is worth to examine the likely consequences of our results for the further evolution of the protostar in presence of dmas . we start reminding that in the standard case ( without dmas ) the simulations of r02 show that when the protostar becomes opaque to continuum radiation ( at @xmath161 ) , the formation of the hydrostatic core is delayed by the thermostatic effects of dissociations , so that the core actually forms when @xmath162 . however , at the end of our runs the central abundance in the maximal and sub - maximal runs is @xmath163 ; even in the fiducial run , the abundance , while still high , is decreasing much earlier than in the standard case ; this is very different from the results of r02 , where @xmath164 up to @xmath165 . because of the lower amount of h@xmath34 , it is reasonable to expect that in the cases with dmas the delaying effects of dissociation are absent , or smaller than in the standard case . then , the hydrostatic core would form at lower densities ( e.g. , @xmath166 , if the protostar becomes optically thick to continuum radiation at the same density as in the r02 runs ) , and its initial mass would be larger ( probably in the 0.010.1 @xmath167 range , rather than @xmath168 ) . however , we point out that r02 found that in the case with no dmas the mass of the hydrostatic core grows very fast , reaching @xmath169 in less than 3 years ( see e.g. their fig . 6 ) : then , the difference in the initial size is relatively unimportant . it is probably more relevant to note that the lower temperatures and infall velocities of the layers outside the hydrostatic core imply that in the cases with dmas the accretion rate might be slower than what was found by r02 .

we include an energy term based on dark matter ( dm ) self - annihilation during the cooling and subsequent collapse of the metal - free gas , in halos hosting the formation of the first stars in the universe . we have found that the feedback induced on the chemistry of the cloud _ does _ modify the properties of the gas throughout the collapse . we have also found that when the rate of energy produced by the dm annihilations and absorbed by the gas equals the chemical cooling ( at densities yet far from the actual formation of a proto stellar core ) the structure does _ not _ halt its collapse , although that proceeds more slowly by a factor smaller than few per cent of the total collapse time .

we include an energy term based on dark matter ( dm ) self - annihilation during the cooling and subsequent collapse of the metal - free gas , in halos hosting the formation of the first stars in the universe . we have found that the feedback induced on the chemistry of the cloud _ does _ modify the properties of the gas throughout the collapse . however , the modifications are not dramatic , and the typical jeans mass within the halo is conserved throughout the collapse , for all the dm parameters we have considered . this result implies that the presence of dark matter annihilations does not substantially modify the initial mass function of the first stars , with respect to the standard case in which such additional energy term is not taken into account . we have also found that when the rate of energy produced by the dm annihilations and absorbed by the gas equals the chemical cooling ( at densities yet far from the actual formation of a proto stellar core ) the structure does _ not _ halt its collapse , although that proceeds more slowly by a factor smaller than few per cent of the total collapse time . stars : formation stars : populationiii dark ages , reionization , first stars dark matter

math0610735

i

we begin with a brief review of standard notation . we write @xmath4 ( respectively , @xmath5 ) to indicate that @xmath1 is a partition ( composition ) of @xmath6 , and denote by @xmath7 the number of parts of @xmath1 . the partition with @xmath8 parts equal to @xmath9 is denoted by @xmath10 $ ] . the _ cycle type _ of a permutation @xmath11 in the symmetric group @xmath12 is the partition of @xmath6 determined by the lengths of the disjoint cycles comprising @xmath11 . the conjugacy class of @xmath12 consisting of all permutations of cycle type @xmath13 is denoted by @xmath14 . this notation is extended to allow @xmath15 to be a composition , in which case the ordering of the parts of @xmath15 is simply ignored . members of @xmath16}$ ] are called _ cycles of length @xmath17 _ , or _ @xmath17-cycles _ , while elements of @xmath18}$ ] are _ full cycles _ in @xmath12 . for vectors @xmath19 and @xmath20 we use the abbreviations @xmath21 and @xmath22 . finally , if @xmath23 $ ] is a formal power series , then we write @xmath24\,f(\mathbf{x})$ ] for the coefficient of the monomial @xmath25 in @xmath26 . a * transitive factorization * of a permutation @xmath27 is a tuple @xmath28 of permutations @xmath29 such that ( 1 ) @xmath30 , and ( 2 ) the group @xmath31 generated by the factors acts transitively on @xmath12 . if @xmath32 for @xmath33 and @xmath34 , then one can show @xcite that @xmath35 in the case of equality above , @xmath36 is said to be * minimal transitive*. for example , since @xmath37 the tuple @xmath38 is a factorization of @xmath39 . ( fixed points have been suppressed in the factors . ) this factorization is easily verified to be minimal transitive . minimal transitive factorizations have been very well studied , with much of this attention stemming from the fact that they serve geometers as combinatorial models for branched coverings of the sphere by the sphere . in this context , transitivity guarantees connectedness of the associated covering , while minimality implies the covering surface is the sphere . for further information on these connections see @xcite and the references therein . the focus of this paper is the class of * cycle factorizations * , by which we mean minimal transitive factorizations , such as above , whose factors are all cycles of length at least two . in particular , given a composition @xmath1 and a sequence @xmath40 of nonnegative integers ( called the * cycle index * ) , we wish to determine the number @xmath41 of cycle factorizations of any fixed permutation @xmath34 into exactly @xmath2 2-cycles , @xmath3 3-cycles , _ etc_. our results will be formulated in terms of the generating series @xmath42 here , and throughout , @xmath43 denotes the total number of factors in any factorization counted by @xmath44 , and @xmath45 is a vector of indeterminates . the structure of generic cycle factorizations is not well understood and , aside from explicit evaluations of @xmath46}({\mathbf{{\mathbf{i}}}})}$ ] ( see @xcite and theorem [ thm : springer ] of this paper ) , little work has been done on their enumeration . however , a significant effort has been directed toward a natural specialization of this problem , which is to count what we call * @xmath17-cycle factorizations*. these are cycle factorizations whose factors are all @xmath17-cycles for some fixed @xmath17 . the case @xmath47 ( transposition factors ) is particularly important geometrically . counting factorizations of permutations is known as the _ hurwitz problem _ , and dates back to hurwitz s original investigations into the classification of almost simple ramified coverings of the sphere by the sphere @xcite . the following formula , suggested but not completely proved by hurwitz himself , gives the number of factorizations of any permutation of cycle type @xmath48 : @xmath49 although it has been extensively studied from various points of view , ranging from analytic to combinatorial ( see @xcite , for example ) , no purely bijective proof of this striking enumerative formula is known . such a proof would be of tremendous interest as it could provide further insight into the underlying geometry . this is particularly true in light of a recent celebrated result of ekedahl , lando , shapiro , and vainshtein @xcite that identifies the enumeration of factorizations ( _ i.e. _ almost simple coverings ) with the evaluation of certain hodge integrals , objects of great interest in the intersection theory of the moduli space of curves . see @xcite for further details on these fascinating connections . indeed , the ultimate goal of our approach to factorization problems is a full combinatorialization of hodge integrals . for arbitrary @xmath50 , counting factorizations has not been as thoroughly examined and appears quite difficult . substantial progress was made in @xcite , where generating series formulations for the number of such factorizations of permutations with up to three cycles are given . in particular , letting @xmath51 denote the series obtained by specializing @xmath52 at @xmath53 and @xmath54 for @xmath55 , it is shown there that @xmath56 and @xmath57 where @xmath58 $ ] ( which depends on @xmath17 ) is the unique series satisfying @xmath59 a more complicated expression for @xmath60 is also given in terms of @xmath61 , but the calculations necessary to evaluate @xmath51 for @xmath62 become intractable . there is a natural equivalence relation on cycle factorizations induced by permitting commutations of disjoint factors . that is , we say two cycle factorizations are * equivalent * if one can be obtained from the other by repeatedly exchanging adjacent factors that are disjoint in the sense that no symbol is moved by both . for example , the following factorizations are equivalent : @xmath63 let @xmath64 denote the number of inequivalent cycle factorizations of @xmath34 with cycle index @xmath65 . we shall study these numbers through the series @xmath66 as before , let @xmath67 be the restricted series counting inequivalent factorizations obtained from @xmath68 by setting @xmath53 and @xmath54 for @xmath55 in @xmath68 . the problem of counting factorizations up to commutation can apparently be traced back to stanley , who originally posed it in the context of factorizations . the first result along these lines came from eidswick @xcite and longyear @xcite , who proved ( independently ) that the number of inequivalent factorizations of the full cycle @xmath69 is the generalized catalan number @xmath70 longyear s approach involved commutation of factorizations into a canonical form . this led to the following cubic functional equation for the generating series @xmath71 , from which is easily deduced : @xmath72 springer @xcite later generalized longyear s argument to obtain an explicit formula for @xmath73}({\mathbf{i}})}}$ ] . his result is recovered here as theorem [ thm : ispringer ] . also see @xcite for an alternative derivation of the number of inequivalent factorizations of a full cycle . more recently , goulden , jackson , and latour @xcite counted inequivalent factorizations of any permutation @xmath74}$ ] , proving that @xmath75 where @xmath76 is defined by . their method again relies on commutation to canonical form , with the additional aid of a clever combinatorial construction and an intricate inclusion - exclusion argument . the primary goal of this paper is to introduce a new technique in the enumeration of both cycle factorizations and their equivalence classes under commutation . we believe our approach to be of interest for two principal reasons : first , it conveniently allows one to ignore the fine detail of factorizations ( _ i.e. _ element - wise analysis ) and focus on the grander structure , and second , it makes clearer the structural parallels between the enumeration of factorizations and their equivalence classes . what follows is a brief overview of the paper highlighting our main results . in section [ sec : preliminaries ] the reader is introduced to various constructs and conventions that are used extensively throughout the paper . our analysis of cycle factorizations then begins in section [ sec : cyclefacts ] , where we describe a graphical representation that allows to be viewed as a generating series for a special class of labelled planar maps . cycle factorizations of full cycles are then seen to correspond with particularly simple maps , namely _ cacti _ , which are a natural generalization of trees . this leads to the recovery of a known explicit formula for the number @xmath46}({\mathbf{i}})}$ ] of cycle factorizations of a full cycle ( see theorem [ thm : springer ] ) . it also marks our first encounter with the series @xmath77[[x]]$ ] defined as the unique solution of the functional equation @xmath78 where @xmath79[[z]]$ ] is given by @xmath80 clearly @xmath81 is a generalization of the series @xmath61 given by , making it no surprise that it plays a central role in our analysis . in particular , we shall present a graphical decomposition of maps ( called _ pruning _ ) that identifies an algebraic dependence of @xmath52 on @xmath81 . this is the content of theorem [ thm : cactuspruning ] , and is the centerpiece of our method . by exploiting the pruning decomposition we deduce following extension of to factorizations of arbitrary cycle index . [ thm : mainthm1 ] let @xmath81 and @xmath82 be defined as above and , for @xmath83 , set @xmath84 . then @xmath85 in the end , our proof of theorem [ thm : mainthm1 ] still rests on an _ ad hoc _ enumeration that we have not been able to generalize . thus a formulation of @xmath52 for arbitrary @xmath86 remains out of reach . we believe that it will be more tedious than difficult to extend our methods to arrive at an expression for @xmath87 , but this has only yet been done in a special case . we comment further on these developments in [ ssec : furtherresults ] . in section [ sec : inequiv ] we turn to the enumeration of cycle factorizations up to the equivalence defined in [ ssec : equivalence ] . by modifying our graphical representation of cycle factorizations to allow for commutations of adjacent factors , we are led to springer s formula @xcite for the number of inequivalent cycle factorizations of a full cycle ( theorem [ thm : ispringer ] ) . we discover that the unique solution @xmath88[[x]]$ ] of the functional equation @xmath89 plays a role directly analogous to that of @xmath81 in the enumeration of ordered cycle factorizations . again we develop a pruning decomposition ( theorem [ thm : inequivpruning ] ) from which we deduce the following generalization of to factorizations of arbitrary cycle index . [ thm : mainthm2 ] let @xmath90 and @xmath82 be defined as above and , for @xmath83 , set @xmath91 . then @xmath92 upon setting @xmath93 and @xmath94 for @xmath95 , the defining equation of @xmath90 transforms to @xmath96 , thus identifying @xmath90 it with longyear s series @xmath97 ( see ) . under these same specializations , it is easy to check that theorem [ thm : mainthm2 ] reduces to . again , we have been unable to extend theorem [ thm : mainthm2 ] to give a general expression for @xmath68 . however , we have used our method to deduce a raw form of @xmath98 , thus extending the goulden - jackson - latour result in a different direction . see [ ssec : ifurtherresults ] for further comments on this and related matters .

we introduce a new approach to an enumerative problem closely linked with the geometry of branched coverings ; that is , we study the number of ways a given permutation ( with cycles described by the partition ) can be decomposed into a product of exactly 2-cycles , 3-cycles , _ etc . _ , with certain minimality and transitivity conditions imposed on the factors . the method is to encode such factorizations as planar maps with certain _ descent structure _ and apply a new combinatorial decomposition to make their enumeration more manageable . we apply our technique to determine when has one or two parts , extending earlier work of goulden and jackson . we also show how these methods are readily modified to count _ inequivalent _ factorizations , where equivalence is defined by permitting commutations of adjacent disjoint factors . our technique permits a substantial generalization of recent work of goulden , jackson , and latour , while allowing for a considerable simplification of their analysis .

we introduce a new approach to an enumerative problem closely linked with the geometry of branched coverings ; that is , we study the number of ways a given permutation ( with cycles described by the partition ) can be decomposed into a product of exactly 2-cycles , 3-cycles , _ etc . _ , with certain minimality and transitivity conditions imposed on the factors . the method is to encode such factorizations as planar maps with certain _ descent structure _ and apply a new combinatorial decomposition to make their enumeration more manageable . we apply our technique to determine when has one or two parts , extending earlier work of goulden and jackson . we also show how these methods are readily modified to count _ inequivalent _ factorizations , where equivalence is defined by permitting commutations of adjacent disjoint factors . our technique permits a substantial generalization of recent work of goulden , jackson , and latour , while allowing for a considerable simplification of their analysis .

cond-mat0206571

c

we conclude by summarizing the strong points as well as the limitations of the new quantum impurity solver introduced in this paper , as well as possible extensions and applications . on the positive side , the dsr method provides an interpolating scheme between the weak coupling and atomic limits ( at half - filling ) . it is also free of some of the pathologies encountered in the simplest finite-@xmath0 extensions of nca ( negative lifetimes at low temperature ) . when applied in the context of dmft , it is able to reproduce many of the qualitatively important features associated with the mott transition , such as coexisting insulating and metallic solutions and the existence of two energy scales in the dmft description of a correlated metal ( the quasi - particle coherence bandwith and the `` preformed gap '' ) . hence the dsr solver is quite useful in the dmft context , at a low computational cost , and might be applicable to electronic structure calculations for systems close to half - filling when the orbital degeneracy becomes large . to incorporate more realistic modelling , one can introduce different energy levels for each correlated orbital , while the extension to non - symmetric coulomb interactions ( such as the hund s coupling ) may require some additional work . the dsr method does not reproduce fermi - liquid behaviour at low energy however , which makes it inadequate to address physical properties in the very low - energy regime ( as is also the case with nca ) . the main limitation however is encountered when departing from half - filling ( i.e from @xmath228 electrons in an @xmath5-fold degenerate orbital ) . while the dsr approximation can be used at small dopings , it fails to reproduce the correct atomic limit when the occupancy differs significantly from @xmath228 ( and in particular can not deal with the mott transition at other integer fillings in the multi - orbital case ) . we would like to emphasize however that this results from extending the slave rotor variable to a field with a large number of components . it is possible to improve this feature of the dsr method by dealing directly with an @xmath3 phase variable , which does reproduce accurately the atomic limit even when the constraint is treated at the mean - field level . we intend to address this issue in a future work . another possible direction is to examine systematic corrections beyond the saddle - point approximation in the large @xmath125 expansion . finally , we would like to outline some other possible applications of the slave rotor representation introduced in this paper ( sec . [ sec : rotor ] ) . this representation is both physically natural and economical . in systems with strong coulomb interactions , the phase variable dual to the local charge is an important collective field . promoting this single field to the status of a slave particle avoids the redundancies of usual slave - boson representations . in forthcoming publications , we intend to use this representation for : i ) constructing impurity solvers in the context of _ extended _ dmft @xcite , in which the frequency - dependent charge correlation function must be calculated @xcite ii ) constructing mean - field theories of _ lattice _ models of correlated electrons ( e.g the hubbard model ) @xcite and iii ) dealing with quantum effects on the coulomb blockade in mesoscopic systems .

we demonstrate that this impurity solver can be applied in the context of dynamical mean - field theory , at or close to half - filling . good agreement with established results on the mott transition is found , and large values of the orbital degeneracy can be investigated at low computational cost .

we introduce a representation of electron operators as a product of a spin - carrying fermion and of a phase variable dual to the total charge ( slave quantum rotor ) . based on this representation , a new method is proposed for solving multi - orbital anderson quantum impurity models at finite interaction strength . it consists in a set of coupled integral equations for the auxiliary field green s functions , which can be derived from a controlled saddle - point in the limit of a large number of field components . in contrast to some finite- extensions of the non - crossing approximation , the new method provides a smooth interpolation between the atomic limit and the weak - coupling limit , and does not display violation of causality at low - frequency . we demonstrate that this impurity solver can be applied in the context of dynamical mean - field theory , at or close to half - filling . good agreement with established results on the mott transition is found , and large values of the orbital degeneracy can be investigated at low computational cost .