article
stringlengths 0
10k
| abstract
stringlengths 21
108k
|
|---|---|
for fixed integers @xmath0 and @xmath1 , we consider the admissible sequences of @xmath2 lattice paths in a colored @xmath3 square given in @xcite . each admissible sequence of paths can be associated with a partition @xmath10 of @xmath4 . in section
[ paths ] , we show that the number of self - conjugate admissible sequences of paths associated with @xmath10 is equal to the number of standard young tableaux of shape @xmath10 , and thus can be calculated using the hook length formula .
we extend this result to include the non - self - conjugate admissible sequences of paths and show that the number of all such admissible sequences of paths is equal to the sum of squares of the number of standard young tableaux of partitions of @xmath4 with height less than or equal to @xmath11 . using the rsk correspondence in @xcite ,
it is shown in ( @xcite , corollary 7.23.12 ) that the sum of squares of the number of standard young tableaux of partitions of @xmath4 with height less than or equal to @xmath11 is equal to the number of @xmath6-avoiding permutations of @xmath7 . in section [ multiplicities ]
, we apply our results to the representation theory of the affine kac - moody algebra @xmath8 .
let @xmath12 , @xmath13 and @xmath14 denote the simple roots , simple coroots , and fundamental weights respectively .
note that @xmath15
. for @xmath16 , set @xmath17 and @xmath18 . as shown in @xcite , @xmath19
are maximal dominant weights of the irreducible @xmath8-module @xmath9 .
we show that the multiplicity of the weight @xmath19 in @xmath9 is the number of @xmath6-avoiding permutations of @xmath7 , which proves conjecture 4.13 in @xcite .
for fixed integers @xmath0 and @xmath1 , consider the @xmath3 square containing @xmath20 unit boxes in the fourth quadrant so that the top left corner of the square is at the origin .
we assign color @xmath21 to a box if its upper left corner has coordinates @xmath22 .
this gives the following @xmath3 colored square @xmath23 : a lattice path @xmath25 on @xmath23 is a path joining the lower left corner @xmath26 to the upper right corner @xmath27 moving unit lengths up or right . for two lattice paths @xmath28 on @xmath23
we say that @xmath29 if the boxes above @xmath30 are also above @xmath25 .
now , we draw @xmath2 lattice paths , @xmath31 on @xmath23 such that @xmath32 . for integers
@xmath33 , where @xmath34 , @xmath35 , we define @xmath36 to be the number of @xmath37-colored boxes between @xmath38 and @xmath39 .
we define @xmath40 to be the number of @xmath37-colored boxes below @xmath41 and @xmath42 to be the number of @xmath37-colored boxes above @xmath43 .
denote by @xmath49 the set of all admissible sequences of @xmath2 paths .
notice that there are @xmath4 0-colored boxes in @xmath23 and hence for any admissible sequence of paths , @xmath50 .
in addition , it follows from definition [ pathsdef](2 ) that @xmath51 for any admissible sequence of paths .
thus , we can and do associate an admissible sequence of paths @xmath44 on @xmath23 with a partition @xmath52 of @xmath4 . in this case , we say that this admissible sequence of paths is of type @xmath10 and often draw @xmath10 as a young diagram .
figure [ adseq](a ) is an element of @xmath53 , where @xmath54 and @xmath55 are shown in figures [ adseq](b ) , [ adseq](c ) , and [ adseq](d ) , respectively .
notice that this admissible sequence of paths is of type @xmath56 . | for @xmath0 and @xmath1 , we consider certain admissible sequences of @xmath2 lattice paths in a colored @xmath3 square .
we show that the number of such admissible sequences of lattice paths is given by the sum of squares of the number of standard young tableaux of partitions of @xmath4 with height @xmath5 , which is also the number of @xmath6-avoiding permutations of @xmath7 .
finally , we apply this result to the representation theory of the affine lie algebra @xmath8 and show that this quantity gives the multiplicity of certain maximal dominant weights in the irreducible module @xmath9 . |
a magnitude limited complete census of variable stars in nearby dwarf galaxies allows important contributions to the star formation history of these systems . measurements of some variable stars can supply improved distance determinations for the host galaxies , others will provide important constraints for the population analysis .
different classes of variables can further improve the understanding of the star formation history of these system , functioning as tracers of star formation during different epochs .
we expect the data set of our long term monitoring program to be especially well suited to study the contents of red long - period variables and to re - investigate the paucity of cepheids with @xmath1 days as reported by sandage & carlson ( 1985 ) .
we selected a sample of six local group dwarf irregular galaxies which are visible with the 0.8 m telescope of our institute at mt .
the names and additional data from the literature compilation by mateo ( 1998 ) are shown in table 1 .
.names , variable star counts , absolute @xmath2-band brightness in mag , and current distance estimation in kpc for the dwarf galaxies observed in our project .
the data are taken from the literature compilation by mateo ( 1995 ) .
for leo a the data are from the work of dolphin et .
al ( 2002 ) and from this work . [ cols="<,<,^,^,^,^,^ " , ] @xmath3 this work the observations so far were carried out in @xmath4 and @xmath2-band , sparsely sampling a three year period starting with test observations in 1999 .
this part of the data set should be sensitive for long period variable stars with periods up to @xmath5 days .
additional observations in @xmath4 , @xmath2 and @xmath6-band were obtained during 3 observing campaigns at the 1.23 m telescope on calar alto densely sampling three two week long periods .
these observations should provide a ground for a search for variable stars with shorter periods ranging from @xmath7 days up to @xmath8 days .
the acquired data were bias subtracted , flat - fielded and cosmic ray rejected .
then , the images from one night were astrometrically aligned to a common reference frame and combined with individual weights proportional to their @xmath9 . for each epoch , consisting of all the stacked images of a single night , a difference image against a common deep reference frame was created using an implementation ( gssl & riffeser , 2002 , 2003 ) of the alard algorithm ( alard & lupton , 1998 ) .
finally , these difference images were convolved with a stellar psf . to extract lightcurves from the reduced data ,
first all pixels deviating significantly ( @xmath10 ) from the reference image in a minimum number of epochs @xmath11 were flagged , utilizing the complete per - pixel error propagation of our data reduction pipeline .
then , using these coordinates as input , values and associated errors are read from the difference images and the lightcurve data are assembled . to search for periodic signals in the extracted difference fluxes , a lomb ( 1976 ) algorithm using the interpretation from scargle ( 1982 )
is applied .
the photometric calibration was conducted using the hst data published by schulte - ladbeck et al .
for the galaxies leo a , and ugca 92 , we have a very good monitoring and a large fraction of the data passed already the pipeline .
the leo a data set serves as test case : a total of 26 variable star candidates were detected . among them
, we identified 16 secure long period variables ( typical average values @xmath12 , and @xmath13 period [ days ] @xmath14 ) , and we have 8 further candidates for lpvs .
in addition we were able to identify two good candidates for @xmath0 cephei stars with best fitting periods of 6.4 and 1.69 days .
the later candidate was previously described by dolphin et al .
( 2002 ) as c2-v58 with a period of 1.4 days . the dolphin et al .
period solution fails in deriving a reliable lightcurve with our data , yet , applying our period value to their data set yields reasonable results .
the phase convolved lightcurves for the two @xmath0 cephei variables are shown in figure 1 .
the color magnitude diagram shown in the left panel of figure 2 is based upon the hst data published by tolstoy et al .
( 1996 ) and schulte - ladbeck et al . flagged by bigger symbols
are those variables from our sample that lie inside the hst field of view , two @xmath0 cephei variables in the instability strip ( crosses ) and the candidates for long term variability ( triangles ) in the regime of the red giants .
tolstoy et al .
( 1996 ) based on ground - based data found a distance modulus for leo a of 24.2 and a resulting distance of 690 kpc ( see also schulte - ladbeck et al . ) .
this result got further support by the search for short periodic variables with the wiyn telescope within 3 consecutive days in dec . 2000 ( dolphin et al .
our data complement this dataset for longer periods .
the right hand panel of figure 2 shows the period - luminosity ( pl ) relation of the smc shifted to the distance determined by tolstoy et al .
the short period variables measured by dolphin coincide with the shown pl relation .
the overplotted values for the two cepheids from our survey ( crosses ) support this relation also in the regime of longer periods .
we presented preliminary results for our survey for variable stars in a sample of irregular local group dwarf galaxies . for the leo a dwarf galaxy , the best analysed case so far
, we already identified a total of 26 candidates for variability , 16 of these as long period variables and 2 @xmath0 cephei stars .
we compared the later with the period - luminosity relation and the short period variables discussed by dolphin et al .
we found , that our cepheids fully support their findings and the resulting distance estimate for leo a. this result is further in good agreement with the trgb distance ( tolstoy et al .
, schulte - ladbeck et al . ) .
the location of the lpvs in the color - magnitude diagram indicate that most of them are early asymptotic giant branch stars .
while a complete census of these intermediate age stars is missing for most of the local group members , a proper statistic of their appearance can guide the reconstruction of the star formation history at the age of several gyr by - passing the age metalicity degeneracy inherent to color magnitude diagram studies .
we like to thank drs .
i. drozdovsky , c. maraston , r.e .
schulte - ladbeck , and e. tolstoy for helpful discussion .
we acknowledge the support of the calar alto and wendelstein staff .
j. fliri and a. riffeser carried out some of our observations .
the project is supported by the deutsche forschungsgemeinschaft grant ho 1812/3 - 1 and ho 1812/3 - 2 .
alard , c. & lupton , r. h. , , 503 , 325 dolphin , a. e. et al .
2002 , , 123 , 3154 gssl c. a. & riffeser a. 2002 , , 381 , 1095 gssl , c. a. & riffeser , a. 2003 , asp conf .
295 , 229 lomb n. r. 1976 , , 39 , 447 mateo m. l. 1998 , , 36 , 435 sandage , a. & carlson , g. 1985 , , 90 , 1464 scargle j. d. 1982 , , 263 , 835 schulte - ladbeck r. et al .
2002 , , 124 , 896 tolstoy e. et al .
1996 , , 116 , 1244 | dwarf galaxies in the local group provide a unique astrophysical laboratory . despite their proximity some of these systems still lack a reliable distance determination as well as studies of their stellar content and star formation history .
we present first results of our survey of variable stars in a sample of six local group dwarf irregular galaxies . taking the leo a dwarf galaxy as an example we describe observational strategies and data reduction .
we discuss the lightcurves of two newly found @xmath0 cephei stars and place them into the context of a previously derived p - l relation .
finally we discuss the lpv content of leo a. |
there are reasons to believe that cosmic rays ( crs ) around the ankle at @xmath0 gev are dominated by extragalactic protons @xcite .
scattering processes in the cosmic microwave background ( cmb ) limit the propagation of ultra high energy ( uhe ) charged particles in our universe .
a continuation of a power - like cr spectrum above the greisen - zatsepin - kuzmin ( gzk ) cutoff @xcite at about @xmath1 gev is only consistent with the proton dominance if the sources lie within the proton attenuation length of about 50 mpc .
very few astrophysical accelerators can generate crs with energies above the gzk cutoff ( see e.g. @xcite for a review ) and so far none of the candidate sources have been confirmed in our local environment .
it has been speculated that decaying superheavy particles , possibly some new form of dark matter or remnants of topological defects , could be a source of uhe crs , but also these proposals are not fully consistent with the cr spectrum at lower energies @xcite .
the observation of gzk excesses has led to speculations about a different origin of uhe crs .
berezinsky and zatsepin @xcite proposed that _
cosmogenic _ neutrinos produced in the decay of the gzk photopions could explain these events assuming a strong neutrino nucleon interaction .
we have followed this idea in ref .
@xcite and investigated the statistical goodness of scenarios with strongly interacting neutrinos from optically thin sources using cr data from agasa @xcite and hires @xcite ( see fig . [ cr ] ) and limits from horizontal events at agasa @xcite and contained events at rice @xcite .
-branes , and string excitations ( see ref .
@xcite ) . ]
the flux of uhe extragalactic protons from distant sources is redshifted and also subject to @xmath2 pair production and photopion - production in the cmb which can be taken into account by means of propagation functions .
the resonantly produced photopions provide a _ guaranteed _ source of cosmogenic uhe neutrinos observed at earth . in astrophysical accelerators inelastic scattering of the beam protons off the ambient photon gas in the source will also produce photopions which provide an additional source of uhe neutrinos . the corresponding spectrum will in general depend on the details of the source such as the densities of the target photons and the ambient gas @xcite .
we have used the flux of crs from _ optically thin _ sources using the luminosities given in ref .
@xcite in the goodness - of - fit test . for a reasonable and consistent contribution of extragalactic neutrinos in vertical crs
one has to assume a strong and rapid enhancement of the neutrino nucleon interaction .
the realization of such a behavior has been proposed in scenarios beyond the ( perturbative ) sm ( see ref .
@xcite ) . for convenience
, we have approximated the strong neutrino nucleon cross section in our analysis by a @xmath3-behavior shown in fig .
[ fig ] , parameterized by the energy scale and width of the transition , and the amplification compared to the standard model predictions .
our analysis showed that uhe crs measured at agasa and hires can be interpreted to the 90% cl as a composition of extragalactic protons and strongly interacting neutrinos from optically thin sources in agreement with experimental results from horizontal events at agasa and contained events at rice ( see fig .
[ fig ] ) .
the pierre auger observatory combines the experimental techniques of agasa and hires as a hybrid detector . with a better energy resolution , much higher statistics and also stronger bounds on horizontal showers
it will certainly help to clarify our picture of uhe crs in the future .
the author would like to thank the organizers of the erice school on nuclear physics 2005 _ `` neutrinos in cosmology , in astro , particle and nuclear physic '' _ for the inspiring workshop and vihkos ( _ `` virtuelles institut fr hochenergiestrahlungen aus dem kosmos '' _ ) for support .
m. ahlers , a. ringwald , and h. tu , _ astropart .
( to appear ) , preprint astro - ph/0506698 .
v. berezinsky , a. z. gazizov and s. i. grigorieva , preprint hep - ph/0204357 ; v. berezinsky , a. z. gazizov and s. i. grigorieva , .
m. ahlers _
et al . _ , .
k. greisen , ; g. t. zatsepin and v. a. kuzmin , .
d. f. torres and l. a. anchordoqui , .
d. v. semikoz and g. sigl , .
v. s. beresinsky and g. t. zatsepin , .
m. takeda _
et al . _ [ agasa ] , .
d. j. bird _
et al . _
[ hires ] , ; r. u. abbasi _ et al . _ [ hires ] , ; r. u. abbasi _ et al . _ [ hires ] , . s. yoshida _
_ [ agasa ] , . | the origin and chemical composition of ultra high energy cosmic rays is still an open question in astroparticle physics .
the observed large - scale isotropy and also direct composition measurements can be interpreted as an extragalactic proton dominance above the _ ankle _ at about @xmath0 gev .
photopion production of extragalactic protons in the cosmic microwave background predicts a cutoff at about @xmath1 gev in conflict with excesses reported by some experiments . in this report
we will outline a recent statistical analysis @xcite of cosmic ray data using strongly interacting neutrinos as primaries for these excesses . |
in solid - core photonic crystal fibers ( pcf ) the air - silica microstructured cladding ( see fig . [ fig1 ] ) gives rise to a variety of novel phenomena @xcite including large - mode area ( lma ) endlessly - single mode operation @xcite .
though pcfs typically have optical properties very different from that of standard fibers they of course share some of the overall properties such as the susceptibility of the attenuation to macro - bending .
macrobending - induced attenuation in pcfs has been addressed both experimentally as well as theoretically / numerically in a number of papers @xcite . however , predicting bending - loss is no simple task and typically involves a full numerical solution of maxwell s equations as well as use of a phenomenological free parameter , _
e.g. _ an effective core radius . in this paper
we revisit the problem and show how macro - bending loss measurements on high - quality pcfs can be predicted with high accuracy using easy - to - evaluate empirical relations .
predictions of macro - bending induced attenuation in photonic crystal fibers have been made using various approaches including antenna - theory for bent standard fibers @xcite , coupling - length criteria @xcite , and phenomenological models within the tilted - index representation @xcite . here
, we also apply the antenna - theory of sakai and kimura @xcite , but contrary to refs .
@xcite we make a full transformation of standard - fiber parameters such as @xmath1 , @xmath2 , and @xmath0 @xcite to fiber parameters appropriate to high - index contrast pcfs with a triangular arrangement of air holes . in the large - mode area
limit we get ( see appendix ) @xmath3 for the power - decay , @xmath4 , along the fiber . for a conversion to a db - scale @xmath5 should be multiplied by @xmath6 . in eq .
( [ alpha_lma ] ) , @xmath7 is the bending radius , @xmath8 is the effective area @xcite , @xmath9 is the index of silica , and @xmath10 is the recently introduced effective v - parameter of a pcf @xcite .
the strength of our formulation is that it contains no free parameters ( such as an arbitrary core radius ) and furthermore empirical expressions , depending only on @xmath11 and @xmath12 , have been given recently for both @xmath8 and @xmath13 @xcite . from the function
@xmath14 we may derive the parametric dependence of the critical bending radius @xmath15 .
the function increases dramatically when the argument is less than unity and thus we may define a critical bending radius from @xmath16 where @xmath17 . typically the pcf is operated close to cut - off where @xmath18 @xcite so that the argument may be written as @xmath19 this dependence was first reported and experimentally confirmed by birks
_ et al . _
@xcite and recently a pre - factor of order unity was also found experimentally in ref .
we have fabricated three lma fibers by the stack - and - pull method and characterized them using the conventional cut - back technique .
all three fibers have a triangular air - hole array and a solid core formed by a single missing air - hole in the center of the structure , see fig .
[ fig1 ] . for the lma-20 macro - bending loss has been measured for bending radii of r=8 cm and r=16 cm and
the results are shown in fig .
the predictions of eq .
( [ alpha_lma ] ) are also included .
it is emphasized that the predictions are based on the empirical relations for @xmath8 and @xmath13 provided in refs . @xcite and @xcite respectively and therefore do not require any numerical calculations .
similar results are shown in figs .
[ fig3 ] and [ fig4 ] for the lma-25 and lma-35 fibers , respectively .
the pcf , in theory , exhibits both a short and long - wavelength bend - edge .
however , the results presented here only indicate a short - wavelength bend - edge .
the reason for this is that the long - wavelength bend - edge occurs for @xmath20 @xcite . for typical lma - pcfs
it is therefor located in the non - transparent wavelength regime of silica .
in conclusion we have demonstrated that macro - bending loss measurements on high - quality pcfs can be predicted with good accuracy using easy - to - evaluate empirical relations with only @xmath21 and @xmath22 as input parameters .
since macro - bending attenuation for many purposes and applications is the limiting factor we believe that the present results will be useful in practical designs of optical systems employing photonic crystal fibers .
the starting point is the bending - loss formula for a gaussian mode in a standard - fiber @xcite @xmath23 where @xmath8 is the effective area , @xmath24 is the core radius , @xmath7 is the bending radius , and the standard - fiber parameters are given by @xcite @xmath25 substituting these parameters into eq .
( [ alpha1 ] ) we get @xmath26 in the relevant limit where @xmath27 . here ,
@xmath28 and @xmath29 in eqs . ( [ alpha_lma ] ) and ( [ v_pcf ] ) have been introduced . for large - mode area fibers we make a further simplification for the isolated propagation constant ; using that @xmath30 we arrive at eq .
( [ alpha_lma ] ) .
m. d. nielsen acknowledges financial support by the danish academy of technical sciences . | we report on an easy - to - evaluate expression for the prediction of the bend - loss for a large mode area photonic crystal fiber ( pcf ) with a triangular air - hole lattice .
the expression is based on a recently proposed formulation of the v - parameter for a pcf and contains no free parameters .
the validity of the expression is verified experimentally for varying fiber parameters as well as bend radius . the typical deviation between the position of the measured and the predicted bend loss edge is within measurement uncertainty . 10 url # 1`#1`urlprefix[2][]#2 j. c. knight , `` photonic crystal fibres , '' nature * 424 * , 847851 ( 2003 ) .
t. a. birks , j. c. knight , and p. s. j. russell , `` endlessly single mode photonic crystal fibre , '' opt . lett .
* 22 * , 961963 ( 1997 ) .
t. srensen , j. broeng , a. bjarklev , e. knudsen , and s. e. b. libori , `` macro - bending loss properties of photonic crystal fibre , '' electron . lett . * 37 * , 287289 ( 2001 ) .
t. srensen , j. broeng , a. bjarklev , t. p. hansen , e. knudsen , s. e. b. libori , h. r. simonsen , and j. r. jensen , `` spectral macro - bending loss considerations for photonic crystal fibres , '' iee proc .- opt .
* 149 * , 206 ( 2002 ) .
n. a. mortensen and j. r. folkenberg , `` low - loss criterion and effective area considerations for photonic crystal fibers , '' j. opt . a : pure appl . opt . * 5 * , 163167 ( 2003 ) .
j. c. baggett , t. m. monro , k. furusawa , v. finazzi , and d. j. richardson , `` understanding bending losses in holey optical fibers , '' opt .
commun .
* 227 * , 317335 ( 2003 ) .
j. sakai and t. kimura , `` bending loss of propagation modes in arbitrary - index profile optical fibers , '' appl . opt . *
17 * , 14991506 ( 1978 ) .
j. sakai , `` simplified bending loss formula for single - mode optical fibers , '' appl . opt .
* 18 * , 951952 ( 1979 ) .
a. w. snyder and j. d. love , _ optical waveguide theory _
( chapman & hall , new york , 1983 ) .
n. a. mortensen , `` effective area of photonic crystal fibers , '' opt .
express * 10 * , 341348 ( 2002 ) .
http://www.opticsexpress.org/abstract.cfm?uri=opex-10-7-341 .
n. a. mortensen , j. r. folkenberg , m. d. nielsen , and k. p. hansen , `` modal cut - off and the @xmath0parameter in photonic crystal fibers , '' opt . lett . * 28 * , 18791881 ( 2003 ) .
m. d. nielsen , n. a. mortensen , j. r. folkenberg , and a. bjarklev , `` mode - field radius of photonic crystal fibers expressed by the @xmath0parameter , '' opt
. lett .
* 28 * , 23092311 ( 2003 ) . m. d. nielsen and n. a. mortensen , `` photonic crystal fiber design based on the @xmath0parameter , '' opt .
express * 11 * , 27622768 ( 2003 ) .
http://www.opticsexpress.org / abstract.cfm?uri = opex-11 - 21 - 2762% [ http://www.opticsexpress.org / abstract.cfm?uri = opex-11 - 21 - 2762% ] . |
in @xcite a database containing a solution of the 3d incompressible navier - stokes ( ns ) equations is presented .
the equations were solved numerically with a standard pseudo - spectral simulation in a periodic domain , using a real space grid of @xmath0 grid points .
a large - scale body force drives a turbulent flow with a taylor microscale based reynolds number @xmath1 . out of this solution ,
@xmath2 snapshots were stored , spread out evenly over a large eddy turnover time .
more on the simulation and on accessing the data can be found at http://turbulence.pha.jhu.edu . in practical terms
, we have easy access to the turbulent velocity field and pressure at every point in space and time .
one usual way of visualising a turbulent velocity field is to plot vorticity isosurfaces see for instance the plots from @xcite .
the resulting pictures are usually very `` crowded '' , in the sense that there are many intertwined thin vortex tubes , generating an extremely complex structure .
in fact , the picture of the entire dataset from @xcite looks extremely noisy and it is arguably not very informative about the turbulent dynamics . in this work ,
we follow a different approach .
first of all , we use the alternate quantity @xmath3 first introduced in @xcite .
secondly , the tool being used has the option of displaying data only inside clearly defined domains of 3d space .
we can exploit this facility to investigate the multiscale character of the turbulent cascade . because vorticity is dominated by the smallest available scales in the velocity
, we can visualize vorticity at scale @xmath4 by the curl of the velocity box - filtered at scale @xmath4 .
we follow a simple procedure : * we filter the velocity field , using a box filter of size @xmath5 , and we generate semitransparent surfaces delimitating the domains @xmath6 where @xmath7 ; * we filter the velocity field , using a box filter of size @xmath8 , and we generate surfaces delimitating the domains @xmath9 where @xmath10 , but only if these domains are contained in one of the domains from @xmath6 ; and this procedure can be used iteratively with several scales ( we use at most 3 scales , since the images become too complex for more levels ) .
additionally , we wish sometimes to keep track of the relative orientation of the vorticity vectors at the different scales . for this purpose
we employ a special coloring scheme for the @xmath11 isosurfaces : for each point of the surface , we compute the cosine of the angle @xmath12 between the @xmath13 filtered vorticity and the @xmath5 filtered vorticity : @xmath14 the surface is green for @xmath15 , yellow for @xmath16 and red for @xmath17 , following a continuous gradient between these three for intermediate values .
the opening montage of vortex tubes is very similar to the traditional visualisation : a writhing mess of vortices . upon coarse - graining
, additional structure is revealed .
the large - scale vorticity , which appears as transparent gray , is also arranged in tubes . as a next step ,
we remove all the fine - scale vorticity outside the large - scale tubes . the color scheme for the small - scale vorticity
is that described earlier , with green representing alignment with the large - scale vorticity and red representing anti - alignment .
clearly , most of the small - scale vorticity is aligned with the vorticity of the large - scale tube that contains it .
we then remove the fine - grained vorticity and pan out to see that the coarse - grained vortex tubes are also intricately tangled and intertwined . introducing a yet larger scale , we repeat the previous operations
. the relative orientation properties of the vorticity at these two scales is similar to that observed earlier .
next we visualize the vortex structures at all three scales simultaneously , one inside the other .
it is clear that the small vortex tubes are transported by the larger tubes that contain them .
however , this is not just a passive advection .
the small - scale vortices are as well being distorted by the large - scale motions . to focus on this more clearly
, we now render just the two smallest scales .
one can observe the small - scale vortex tubes being both stretched and twisted by the large - scale motions .
the stretching of small vortex tubes by large ones was suggested by orszag and borue @xcite as being the basic mechanism of the turbulent energy cascade .
as the small - scale tubes are stretched out , they are `` spun up '' and gain kinetic energy . here , this phenomenon is clearly revealed .
the twisting of small - scale vortices by large - scale screw motions has likewise been associated to helicity cascade @xcite .
the video thus allows us to view the turbulent cascade in progress .
next we consider the corresponding view with three levels of vorticity simultaneously .
since the ratio of scales is here 1:15:49 we are observing less than two decades of the turbulent cascade .
one must imagine the complexity of a very extended inertial range with many scales of motion .
not all of the turbulent dynamics is tube within tube . in our last scene
we visualize in the right half domain all the small - scale vortices , and in the left domain only the small - scale vortices inside the larger scale ones . in the right half ,
the viewer can observe stretching of the small - scale vortex structures taking place externally to the large - scale tubes .
the spin - up of these vortices must contribute likewise to the turbulent energy cascade .
6ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo
secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty link:\doibase 10.1080/14685240802376389 [ * * ( ) , 10.1080/14685240802376389 ] @noop * * ( ) , in http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1592886[__ ] ( ) p. @noop _ _ , ( ) link:\doibase 10.1017/s0022112097008306 [ * * , ( ) ] http://journals.cambridge.org / production / action / cjogetfulltext?fulltextid=4% 00523 [ * * , ( ) ] | the jhu turbulence database @xcite can be used with a state of the art visualisation tool @xcite to generate high quality link : anc / dfdsubmissionquarterres.mpg[fluid dynamics videos ] . in this work
we investigate the classical idea that smaller structures in turbulent flows , while engaged in their own internal dynamics , are advected by the larger structures .
they are not advected undistorted , however .
we see instead that the small scale structures are sheared and twisted by the larger scales .
this illuminates the basic mechanisms of the turbulent cascade . |
in recent years electron transfer ( et ) between molecular adsorbates and semiconductor nanomaterials and surfaces has been subject of much research @xcite .
the injection of an electron into the conduction band is a prototype reaction for a lot of electrochemical and photoelectrochemical interfacial processes such as photography , solar energy conversion , quantum dot devices , etc .
interfacial et between discrete molecular levels and a conducting surface is the simplest of all surface reactions : it involves only the exchange of an electron , and so no bonds are broken @xcite .
the ultrafast nature of the charge injection from adsorbed molecules to the conduction band of semiconductor surfaces was shown in recent experiments @xcite . the theoretical description of such experiments demands an adequate treatment of the et dynamics to be able to describe short time - scale phenomena such as coherences .
this can be done within the reduced density matrix ( rdm ) description used in the present contribution .
recently @xcite the electron injection from a chromophore to a semiconductor conduction band was described using the time - dependent schrdinger equation , thus neglecting relaxation processes .
the neglect of relaxation processes was motivated by the experimental finding that injected electrons relax only within 150 fs in the perylene - tio@xmath0 system . here
we include relaxation to be able to treat a larger class of experiments where , for example , the adsorbed molecule is surrounded by a liquid environment , and longer times .
in the rdm theory the full system is divided into a relevant system part and a heat bath .
therefore the total hamiltonian consists of three terms the system part @xmath1 , the bath part @xmath2 , and the system - bath interaction @xmath3 : @xmath4 the rdm @xmath5 is obtained from the density matrix of the full system by tracing out the degrees of freedom of the environment .
this reduction together with a second - order perturbative treatment of @xmath3 and the markov approximation leads to the redfield equation @xcite : @xmath6 + { \mathcal r } \rho = { \mathcal
l } \rho .
\label{eq : redfield}\ ] ] in this equation @xmath7 denotes the redfield tensor .
if one assumes bilinear system - bath coupling with system part @xmath8 and bath part @xmath9 @xmath10 one can take advantage of the following decomposition @xcite : @xmath11 + [ \lambda\rho , k]+ [ k,\rho\lambda^{\dagger } ] .
\label{eq : pf - form}\ ] ] the @xmath12 operator can be written in the form @xmath13 where @xmath14 is the operator @xmath8 in the interaction representation .
the system bath interaction is taken to be linear in the reaction coordinate as well as in the bath coordinates .
neither the rotating wave nor the secular approximation have been invoked .
the so - called diabatic damping approximation which has numerical advantages @xcite is not used because it could lead to wrong results in the present system studied @xcite . in the following
we direct our attention to et between an excited molecular state and a conduction band .
the hamiltonian modeling this system consists of the ground and one excited state of the molecule and a quasi - continuum describing the conduction band together with one vibrational coordinate @xmath15 here @xmath16 can be equal to @xmath17 for the ground state , @xmath18 for the excited state , and @xmath19 for the quasi - continuum . as in ref .
@xcite we choose the frequency of the vibrational mode to be @xmath20 .
the coupling between the excited state and the continuum states is assumed to be constant : @xmath21 .
a box - shaped uniform density of states is used . instead of modeling the excitation from the ground state
explicitly we assume a @xmath22-pulse .
the excited state potential energy surface is shifted 0.1 along the reaction coordinate with respect to the ground state potential energy surface .
this results in an initial vibrational wave packet on the excited state with significant population in the lowest 4 - 5 vibrational states .
the shift between the excited state energy surface and the continuum parabola is 0.2 .
the thermal bath is characterized by its spectral density @xmath23 . because all system oscillators have the same frequency the coupling to the bath
can be given by one parameter @xmath24 in the diabatic damping approximation . denoting the effective mass of the harmonic oscillator by @xmath25 the strength of the damping
is chosen as @xmath26 . to be able to study the effects of dissipation
we do not model the quasi - continuum with such a large number of electronic states as in ref .
@xcite . in that work
a band of width 2 ev was described using an energy difference of 2.5 mev leading to 801 electronic surfaces .
these calculations are already demanding using wave packet propagation but almost impossible using direct density matrix propagation .
for doing such a large system one would have to use the monte carlo wave function scheme @xcite .
we use a much simpler model and describe only that part of the conduction band which really takes part in the injection process .
the total width of the conduction band may be significantly larger . in the following ,
a band of width 0.75 ev is treated with 31 electronic surfaces . in each of these electronic states
five vibrational states are taken into account .
we are aware that this is only a minimal model but hope that it catches the effects of dissipation on the electron injection process .
here we look at two different populations arising in the process of electron injection .
the time - dependent population of the electronic states in the conduction band is calculated as the sum over the vibrational levels of each electronic surface @xmath27 . as a second quantity
we look at the time - dependent population of the vibrational levels of the excited molecular state @xmath28 .
these two probability distributions give some hints on the effect of dissipation .
figure 1 shows the electronic population for the quasi - continuum , i.e. the probability distribution of the injected electron , versus the energy of the conduction band .
as described above , the four lowest vibrational states are populated significantly at @xmath29 .
the structure arising in the upper panel of fig . 1
was already explained by ramakrishna et al .
it can be estimated using the golden rule .
the electronic probabilities in the quasi - continuum are given as @xmath30 where @xmath31 is the initial vibronic distribution in the excited state and @xmath32 and @xmath33 are the vibronic parts of the wave packet in the excited and quasi - continuum states , respectively .
the energy @xmath34 denotes the middle of the band . turning on dissipation two
effects can be seen .
first , the vibrational populations in the excited state of the molecule no longer only decay into the quasi - continuum states but also relax within the excited state ( see fig . 2 ) .
second , the vibrational populations also relax within the quasi - continuum states .
the recurrences back into the excited state become much smaller .
only those parts of the wave packet which are still high enough in energy can go back to the molecule . in summary , we extended the work by ramakrishna , willig , and may @xcite by including relaxation processes into the description of electron injection into the conduction band of a semiconductor .
this will , at least , become important for modeling electron injection in the presence of a fluid surrounding the attached molecule . | electron injection from an adsorbed molecule to the substrate ( heterogeneous electron transfer ) is studied .
one reaction coordinate is used to model this process .
the surface phonons and/or the electron - hole pairs together with the internal degrees of freedom of the adsorbed molecule as well as possibly a liquid surrounding the molecule provide a dissipative environment , which may lead to dephasing , relaxation , and sometimes excitation of the relevant system . in the process studied the adsorbed molecule is excited by a light pulse .
this is followed by an electron transfer from the excited donor state to the quasi - continuum of the substrate .
it is assumed that the substrate is a semiconductor .
the effects of dissipation on electron injection are investigated . electron transfer , density matrix theory , molecules at surfaces |
the open connectome project ( located at http://openconnecto.me ) aims to annotate all the features in a 3d volume of neural em data , connect these features , and compute a high resolution wiring diagram of the brain , known as a connectome .
it is hoped that such work will help elucidate the structure and function of the human brain .
the aim of this work is to automatically annotate axoplasmic reticula , since it is extremely time consuming to hand - annotate them .
specifically , the objective is to achieve an operating point with high precision , to enable robust contextual inference .
there has been very little previous work towards this end @xcite .
axoplasmic reticula are present only in axons , indicating the identity of the surrounding process and informing automatic segmentation .
the brain data we are working with was color corrected using gradient - domain image - stitching techniques @xcite to adjust contrast through the slices .
we use this data as the testbed for running our filters and annotating axoplasmic reticula .
the bilateral filter @xcite is a non - linear filter consisting of one 2d gaussian kernel @xmath0 , which decays with spatial distance , and one 1d gaussian kernel @xmath1 , which decays with pixel intensity : @xmath2_p = \frac{1}{w_p}\sum_{q\in s}g_{\sigma_{s}}(||p - q||)g_{\sigma_{r}}(i_p - i_q)i_q,\\ & \hspace{4mm}\textrm{where } w_p = \sum_{q\in s}g_{\sigma_{s}}(||p - q||)g_{\sigma_{r}}(i_p - i_q ) \end{split}\ ] ] is the normalization factor .
this filter smooths the data by averaging over neighboring pixels while preserving edges , and consequently important detail , by not averaging over pixels with large intensity difference . applying this filter
accentuates features like axoplasmic reticula in our data .
even with a narrow gaussian in the intensity domain , the bilateral filter causes some color bleeding across edges .
we try to undo this effect through laplacian sharpening .
the laplacian filter computes the difference between the intensity at a pixel and the average intensity of its neighbors .
therefore , adding a laplacian filtered image to the original image results in an increase in intensity where the average intensity of the surrounding pixels is less than that of the center pixel , an intensity drop where the average is greater , and no change in areas of constant intensity .
hence , we use the 3x3 laplacian filter to highlight edges around dark features such as axoplasmic reticula .
we use a morphological region growing algorithm on our filtered data to locate and annotate axoplasmic 26.5 mm @xmath3 26.5 mm @xmath3 26.5 mm @xmath3 26.5 mm @xmath3 26.5 mm 26.5 mm 2 reticula .
we implement this by iterating over the filtered image and looking for dark pixels , where a dark pixel is defined as a pixel with value less than a certain specified threshold .
when a dark pixel is found , we check its 8-neighborhood to determine if the surrounding pixels are also below the threshold .
then , we check the pixels surrounding these , and we do this until we find only high intensity pixels , or until we grow larger than the diameter of an axoplasmic reticula .
the thresholds we use in our algorithm are biologically motivated and tuned empirically .
finally , we track our annotations through the volume to verify their correctness and identify axoplasmic reticula that were missed initially . for each slice , we traverse the annotations and check if an axoplasmic reticulum is present in the corresponding xy - location ( with some tolerance ) in either of the adjacent slices .
if a previously annotated axoplasmic reticulum object is present , we confirm the existing annotation .
otherwise , the adjacent slice locations are checked for axoplasmic reticula with a less restrictive growing algorithm , and new annotations are added in the corresponding slice . if no axoplasmic reticulum object is found in either of the adjacent slices , then we assume the annotation in the current slice to be incorrect , and delete it .
we qualitatively evaluated our algorithm on 20 slices from the kasthuri11 dataset , and quantitatively compared our results against ground truth from a neurobiologist .
our algorithm annotates axoplasmic reticulum objects with 87% precision , and 52% recall .
these numbers are approximate since there is inherent ambiguity even among expert annotators .
our current algorithm is designed to detect transverally sliced axoplasmic reticula . in future work , we plan to extend our morphological region growing algorithm to also find dilated axoplasmic reticula , and to incorporate a more robust tracking method such as kalman or particle filtering .
additionally , our algorithm can be adapted to annotate other features in neural em data , such as mitochondria , by modifying the morphological region growing algorithm . | * _ abstract _ : * in this paper , we present a new pipeline which automatically identifies and annotates axoplasmic reticula , which are small subcellular structures present only in axons .
we run our algorithm on the kasthuri11 dataset , which was color corrected using gradient - domain techniques to adjust contrast .
we use a bilateral filter to smooth out the noise in this data while preserving edges , which highlights axoplasmic reticula .
these axoplasmic reticula are then annotated using a morphological region growing algorithm . additionally , we perform laplacian sharpening on the bilaterally filtered data to enhance edges , and repeat the morphological region growing algorithm to annotate more axoplasmic reticula .
we track our annotations through the slices to improve precision , and to create long objects to aid in segment merging .
this method annotates axoplasmic reticula with high precision .
our algorithm can easily be adapted to annotate axoplasmic reticula in different sets of brain data by changing a few thresholds .
the contribution of this work is the introduction of a straightforward and robust pipeline which annotates axoplasmic reticula with high precision , contributing towards advancements in automatic feature annotations in neural em data . + 2 |
nuclei in interaction with external fields display a wide variety of collective vibrations known as giant resonances , associated with various degrees of freedom and multipolarities .
the giant isovector dipole resonance and the giant isoscalar quadrupole resonance are the most studied examples in this class of phenomena .
a particular mode , that is associated with vibrations in the number of particles , has been predicted in the 70s@xcite and discussed , under the name of giant pairing resonance , in the middle of the 80 s in a number of papers@xcite .
this phenomenon , despite some early efforts aimed to resolve some broad bump in the high - lying spectrum in ( p , t ) reactions@xcite , is still without any conclusive experimental confirmation . for a discussion , in particluar in connection with two - particle transfer reactions , on many aspects of pairing correlations in nuclei
we refer to a recent review@xcite .
we have studied the problem of collective pairing modes at high excitation energy in two neutron transfer reactions with the aim to prove the advantage of using unstable beam as a new tool to enhance the excitation of such modes @xcite .
the main point is that with standard available beams one is faced with a large energy mismatch that strongly hinders the excitation of high - lying states and favours the transition to the ground state of the final system .
instead the optimum q - value condition in the ( @xmath3he,@xmath4he ) stripping reaction suppresses the ground state and should allow the transition to 10 - 15 mev energy region .
we have performed particle - particle rpa calculations on lead and bcs+rpa on tin , as paradigmatic examples of normal and superfluid systems , evaluating the response to the pairing operator .
subsequently the two - neutron transfer form factors have been constructed in the framework of the macroscopic model@xcite and used in dwba computer codes .
we have estimated cross - sections of the order of some millibarns , dominating over the mismatched transition to the ground state .
recently we added similar calculations on other much studied targets to give some guide for experimental work .
the formal analogy between particle - hole and particle - particle excitations is very well established both from the theoretical side@xcite and from the experimental side for what concern low - lying pairing vibrations around closed shell nuclei and pairing rotations in open shells .
the predicted concentration of strength of a @xmath5 character in the high - energy region ( 8 - 15 mev for most nuclei ) is understood microscopically as the coherent superposition of 2p ( or 2h ) states in the next major shell above the fermi level .
we have roughly depicted the situation in fig .
( [ fig1 ] ) . in closed shell nuclei the addition of a pair of particles ( or holes ) to the next major shell , with a total energy @xmath6 , is expected to have a high degree of collectivity .
also in the case of open shell nuclei the same is expected for the excitation of a pair of particles with @xmath7 energies .
for normal nuclei the hamiltonian with a monopole strength interaction reads : @xmath8 where @xmath9 annihilates a pair of particles coupled to @xmath10 total angular momentum . getting rid of all the technicalities of the solution of the pp - rpa equations ( that may be found in the already cited work by the author ) we merely state that the pairing phonon may be expressed as a superposition of 2p ( or 2h ) states with proper forward and backward amplitudes ( @xmath11 and @xmath12 ) .
the pair transfer strength , that is a measure of the amount of collectivity of a each state @xmath13 , is given by : @xmath14 .
\label{p5}\ ] ] this quantity is plotted in the first column of fig .
( [ fig2 ] ) for the removal ( upper panel ) and addition mode ( lower panel ) . in the same figure
are reported the pairing strength parameters for the states of @xmath1sn . to obtain these last quantities for superfluid spherical
nuclei one has to rewrite the hamiltonian according to the bcs transformation and has to solve more complex rpa equations . in this case
the pairing strength for the addition of two particles is given , for each state @xmath13 , by : @xmath15_{00}|0\rangle = \sum_{j } \sqrt{2j+1 } [ u^{2}_{j } x_{n}(j ) + v^{2}_{j}y_{n}(j)]\ ] ] where the @xmath16 and @xmath17 are the usual occupation probabilities .
the amount of collectivity is a clear signal of the structural existence of giant pairing vibrations in the high - lying energy region .
we also report here a number of analogous results for other commonly studied targets = 9.4pc = 9.4pc = 9.4pc with the aim of giving some indications to experimentalists on the reasons why we think that lead and tin are some of the most promising candidates .
we have studied two isotopes of calcium with closed shells .
even if the absolute magnitudes of the @xmath18 is lower , it is worthwhile to notice that some enhancement is seen in the more neutron - rich @xmath19ca with respect to @xmath20ca .
an important role in this change is certainly due to the different shell structure of the two nuclei as well as to the scheme that we implemented to obtain the set of single particle levels .
the latter is responsible for the collectivity of the removal modes in both ca isotopes and also for the difficulty in finding out a collective state in the addition modes .
we display also results for @xmath21zr where the strength is much more fragmented and the identification of the gpv is more difficult . in the work of broglia and bes estimates for the energy of the pairing resonance are given as @xmath22 mev and @xmath23 mev for normal and superfluid systems respectively .
our figures follow roughly these prescriptions based on simple arguments ( and much more grounded in the case of normal nuclei ) as evident from table [ ta1 ] .
.comparison of position of gpv between our calculation and the broglia and bes estimate .
[ cols="^,^,^",options="header " , ] = 13.8pc = 13.8pc these cross - sections have been derived for sharp states , and we refer to the numbers in the last table when speaking of order of magnitude estimates . obviously cross - section in the high - lying energy region have a finite ( and large ) width that should be inserted for a more realistic description of the spectrum .
we have chosen a simple scheme that gives a lorentzian distribution with a width that grows quadratically with the excitation energy , @xmath24 , with @xmath25 adjusted to give a width of 4 mev for the gpv .
this could seem rather arbitrary since there is no reason for an _ a priori _ assignment of this quantity .
we have been brought to this simple prescription because other collective states ( of different nature ) lying in the same energy region display similar values for their width , and it is reasonable to assume some rule to narrow the low - energy states and to broaden the high - energy ones .
the final achievements for the four reactions studied in detail are presented in figure [ fig4 ] where the areas corresponding to the cross - sections given above have been shaded to give a feeling of the relative magnitudes of the transition to the ground states and to the gpv s . it is worthwhile to note that in the case of pb there is a considerable gain in using unstable beams , while in sn is much less evident .
one sees the need for unstable helium when compares the magnitude for the pairing resonance in the right a ) and b ) panels with the peak at zero energy : in the first panel the transition to the ground state is extremely hindered .
a @xmath3he beam is currently available ( or it will be available in the very near future ) in many radioactive ion beams facilities around the world and the calculations that we have presented could allow a planning for future experiments aimed to study the not yet completely unraveled role of pairing interaction in common nuclei , using exotic weakly bound nuclei as useful tools .
the author wishes to gratefully acknowledge discussions with andrea vitturi , hugo sofia and wolfram von oertzen on various aspects of theoretical and experimental nuclear physics .
the participation at the _ vii international school - seminar on heavy ion physics , dubna , russia _ 2002 has been supported by the infn .
xxxx r.a.broglia and d.r.bes , plb691291977 .
m.w.herzog , r.j.liotta and l.j.sibanda , _ phys .
_ c * 31 * , 259 , ( 1985 ) .
et al _ , prl 3914511977 .
w.von oertzen and a.vitturi , _ rep .
phys . _ * 64 * , 1247 - 1337 , ( 2001 ) .
l.fortunato , w.von oertzen , h.m.sofia and a. vitturi , _ eur .
_ a * 14 * , ( 2002 ) , in press . c.h.dasso and a.vitturi ( editors ) ,
_ collective aspects in pair transfer phenomena _ , sif proc .
18 , ( editrice compositori bologna , 1987 ) .
d.r.bes and r.a.broglia , _ phys .
_ c * 3 * , 2349 , ( 1971 ) .
c.h.dasso and g.pollarolo , plb 1552231985 .
c.h.dasso and a.vitturi , prl 596341987 . | we investigate the possible signature of the presence of giant pairing states at excitation energy of about 10 mev via two - particle transfer reactions induced by neutron - rich weakly - bound projectiles . performing particle - particle rpa calculations on @xmath0pb and bcs+rpa calculations on @xmath1sn
, we obtain the pairing strength distribution for two particles addition and removal modes .
estimates of two - particle transfer cross sections can be obtained in the framework of the macroscopic model. the weak - binding nature of the projectile kinematically favours transitions to high - lying states . in the case of @xmath2
reaction we predict a population of the giant pairing vibration with cross sections of the order of a millibarn , dominating over the mismatched transition to the ground state . |
the schwinger - dyson ( sd ) equation is one of the most popular approaches to investigate the non - perturbative features of quantum field theory .
the analyses by making use of the sd equation for quark propagator are well - known .
recently , the coupled sd equations for the gluon and ghost propagators in yang - mills theory have been studied mainly in the lorentz ( landau ) gauge.@xcite in this paper , we derive the sd equations for the @xmath0 yang - mills theory in the maximal abelian ( ma ) gauge and solve them analytically in the infrared ( ir ) asymptotic region .
the ma gauge is useful to investigate the yang - mills theory from the view point of the dual superconductivity . in the ma gauge , in contrast to the ordinary lorentz gauge
, we must explicitly distinguish the diagonal components of the fields from the off - diagonal components .
this is indeed the case even in the perturbative analysis in the uv region.@xcite therefore , we must take account of the four propagators for the diagonal gluon , off - diagonal gluon , diagonal ghost and off - diagonal ghost
. numerical behaviors of gluon propagators in the ma gauge are also investigated on a lattice simulation.@xcite
first , we derive the sd equations from the @xmath0 yang - mills action in the ma gauge@xcite . the graphical representation of sd equations are shown in figure [ fig : sde ] .
= .001 in ( 6000,1800 ) ( 0,-200)(0,500)(0,150)(450,300)(600,160)(800,200)(1250,300)(1400,160)(1600,0)(2000,350)(2200,160)(2400,160)(3600,160)(3800,160)(0,1000)(0,150)(450,300)(600,160)(800,200)(1250,300)(1400,160)(1600,100)(2000,350)(2200,160)(2400,160)(3600,160)(3800,160)(0,1500)(0,150)(0,250)(450,300)(600,160)(800,200)(1000,250)(1250,300)(1400,160)(1600,0)(1570,230)(2200,160)(2400,0)(2370,230)(3000,160)(3200,160)(4400,160)(4600,160)(0,0)(0,150)(0,250)(450,300)(600,160)(800,200)(1000,250)(1250,300 ) for the diagonal gluon propagator , we adopt the landau gauge so that the diagonal gluon propagator @xmath1 has only the transverse part @xmath2 where we defined the form factor @xmath3 . while , the off - diagonal gluon propagator @xmath4 has both the transverse and longitudinal parts @xmath5\delta^{ab},\ ] ] where we defined the form factors @xmath6 and @xmath7 .
the form factor @xmath8 for the off - diagonal ghost propagator @xmath9 is defined @xmath10 the diagonal ghost propagator is decoupled from the other fields so that we omit it hereafter .
now , we write down the sd equations : @xmath11 @xmath12 and @xmath13 here the contributions from the two - loop graphs have been omitted .
the full form of sd equations will be given in a separate paper@xcite .
@xmath14 is the full vertex function for the diagonal gluon , off - diagonal ghost and off - diagonal antighost interaction , while @xmath15 is the full vertex function for an interaction of the diagonal gluon and two off - diagonal gluons , and the superscript `` @xmath16 '' means a _ bare _ propagator or vertex function . in the ma gauge
, we obtain the slavnov - taylor ( st ) identities @xmath17 @xmath18
in order to solve the sd equations analytically , we employ the following approximations .
@xmath19 we neglect the two - loop contributions . instead of the full vertex functions ,
we adopt modified vertex functions which are compatible with the st identities .
we adopt approximations for vertex functions as @xmath20 and @xmath21 here , we adopt the feynman gauge for the off - diagonal gluon for simplicity , that is , @xmath22 and @xmath23 . substituting the bare form factors , which are @xmath24 , into the right hand side of the ansatz ( [ eq : acc ] ) and ( [ eq : aaa ] )
, we obtain the bare vertex functions .
moreover , these ansatz are compatible with the st identities ( [ eq : sti - c ] ) and ( [ eq : sti - a ] ) in the limit of @xmath25 . in the momentum integration
, we use the higashijima - miransky approximation@xcite as @xmath26
now we adopt the ansatz for the form factors in the ir region : @xmath27 g(p^2 ) = b(p^2)^v+\cdots,\\[1 mm ] f_{\rm t}(p^2 ) = c(p^2)^w+\cdots . \end{array } \label{eq : ir solutions}\ ] ] substituting the ansatz ( [ eq : ir solutions ] ) for the form factors , and the ansatz ( [ eq : acc ] ) and ( [ eq : aaa ] ) for vertex functions into the sd equations ( [ eq : diagonal gluon ] ) , ( [ eq : off - diagonal ghost ] ) and ( [ eq : off - diagonal gluon ] ) , and comparing the leading term in the both sides of each equation , we obtain the following results for @xmath22 . from eqs .
( [ eq : off - diagonal ghost ] ) and ( [ eq : off - diagonal gluon ] ) , we obtain the relations @xmath28 and @xmath29 . in the case of @xmath30 and @xmath31 , from the eq .
( [ eq : diagonal gluon ] ) , we obtain the relation @xmath32 so that @xmath33 is less than @xmath34 . in the case of @xmath35 and @xmath31 , we need redefine the form factor @xmath8 as @xmath36 with @xmath37 since contributions from the leading term of @xmath8 are canceled each other in the ansatz ( [ eq : acc ] ) .
therefore we need the information of next leading term of the form factor @xmath8 . in this case
we obtain the relation @xmath38 from the eq .
( [ eq : diagonal gluon ] ) so that @xmath33 is also less than @xmath34 .
next , we consider the case of @xmath30 and @xmath39 . as well as the above case , we need redefine the form factor @xmath6 as @xmath40 with @xmath41 and we obtain the relation @xmath42 ( @xmath43 ) .
similarly , in the case of @xmath44 , we obtain the relation @xmath45 ( @xmath43 ) .
the results are summarized in table [ tbl : feynman gauge ] .
@xmath32 & @xmath42 @xmath35 & @xmath38 & @xmath45 [ tbl : feynman gauge ] in the gauge other than the feynman gauge , that is , @xmath46 , the calculation and discussion are very tedious
. however , the qualitative results are identical to the above case except for the following one point . in this case , even if @xmath39 , there occurs no cancellation as in the above two cases 2c and 2d .
this is because the off - diagonal gluon propagator has the momentum dependent tensor structure for @xmath46 , while it is proportional to @xmath47 for @xmath22 .
therefore , we obtain the relation @xmath48 in the case of @xmath39 .
( see table [ tbl : not feynman gauge ] . )
@xmath30 & @xmath32 & @xmath48 @xmath35 & @xmath38 & @xmath48 [ tbl : not feynman gauge ]
in the ir limit , the form factors of each propagator behave as @xmath49 @xmath50 @xmath51 therefore the solution shows that the diagonal gluon propagator is enhanced in the ir limit , while the off - diagonal gluon and off - diagonal ghost propagators are suppressed in the ir region .
our results are compatible with a hypothesis of abelian dominance@xcite . | we derive the schwinger - dyson equations for the @xmath0 yang - mills theory in the maximal abelian gauge and solve them in the infrared asymptotic region .
we find that the infrared asymptotic solutions for the gluon and ghost propagators are consistent with the hypothesis of abelian dominance . |
it is common accepted that braking of pulsars is caused by the magneto - dipole radiation of the rotating magnetic star . in this case
the rate of losses of the neutron star rotation energy can be equated to the power of its magneto - dipole radiation : @xmath1 + where _ i _ is the moment of inertia of the neutron star , @xmath2 - the angular speed of its rotation , @xmath3 - its magnetic moment , @xmath0 - the angle between the rotation axis and the magnetic moment , _ c _ - speed of light . for standard parameters of neutron stars : masses of order of the solar mass ( @xmath4 ) and radii _ r _ of order of @xmath5 cm we can put _ i _ = @xmath6 . for the magnetic moment
we have @xmath7 + here @xmath8 is the magnetic induction at the magnetic pole , @xmath9 ?
the induction at the magnetic equator . instead of @xmath2 the rotation period @xmath10
is usually measured and we can obtain from ( 1 ) and ( 2 ) : @xmath11 + this equality is used usually to calculate magnetic inductions of pulsars assuming that @xmath12 for all objects . the known catalogs ( see , for example manchester et al
. , 2005 ) contain as a rule @xmath9 instead of @xmath8 .
here we propose to decline the assumption on the constancy of @xmath13 and use some estimations of this parameter to calculate more accurate values of pulsar magnetic inductions .
in a number of our works ( malov & nikitina , 2011a , b , 2013 ) some methods for calculations of the angle @xmath0 have been put forward and applied to some catalogs of pulsars ( keith et al . , 2010 ; van ommen et al . , 1997 ; weltevrede & johnston , 2008 ) at approximately 10 , 20 and 30 cm .
basic equations for this aim are ( manchester & taylor , 1977 ) : @xmath14 @xmath15 + here @xmath16 is the angle between the line of sight and the rotation axis , @xmath17 - the angular radius of the emission cone , @xmath18 - a half of the angular width of the observed pulse , @xmath19 - the position angle of the linear polarization , @xmath20 - longitude . the simplest case for the calculations of the angle @xmath0 is realized when the line of sight passes through the center of the emission cone , i.e. @xmath21 + in this case we can use the dependence of the observed pulse width @xmath22 at the @xmath23 level on the rotation period and determine the lower boundary in the corresponding diagram to obtain @xmath24 + as the result we have from ( 4 ) , ( 5 ) and ( 7 ) ( malov & nikitina , 2011a ) : @xmath25 + the values of angles calculated by this method are denoted as @xmath26 and given in the table 1 .
usually polarization measurements are made inside the pulse longitudes only . in this case
we can use the maximal derivative of the position angle . from ( 5 ) we have @xmath27 we can obtain from the dependence of @xmath22 on _ p _ by the least squares method @xmath28 + the third equation for the calculations of the angle @xmath0 is ( 4 ) . from these three equations we obtain @xmath29y^2 + 2c(d - b^2)y+c^2d^2-b^2(1+c^2)=0.\\\ ] ] + here @xmath30
+ we can transform the equation ( 9 ) to the following form @xmath31 + then finding the value of y from the equation ( 11 ) we can calculate @xmath0 from ( 13 ) .
we have calculated values of @xmath0 by this method and list them in the table 1 as @xmath32 . here
we correct the misprint in the equation ( 11 ) made in our papers ( malov & nikitina , 2011a , b , 2013 ) .
there is an additional way to calculate angles @xmath0 .
this way uses observable values of position angles and shapes of average profiles for individual pulsars . in this case , original equations form the closed system for calculations of the angles @xmath17 , @xmath16 and @xmath0 : @xmath33 as the observed pulsar profiles have various forms , the coefficient _
n _ has a different value depending on a profile structure .
we put arbitrary the following values of _ n _ ( fig.1 ) . if the ratio of the intensity @xmath34 in the center of the pulse to the maximal intensity @xmath35 is zero then @xmath36 . for @xmath37 @xmath38 , @xmath39 @xmath40 , @xmath41 @xmath42 , and for @xmath43 @xmath44 .
it is worth noting that the solution of the system ( 14 ) can be obtained numerically for any value of _
n_. for example , if @xmath45 , the solution for @xmath46 can be obtained from the equation : @xmath47 at n = 2 : @xmath48 y^4 + 2c \left [ c^2 ( 1 + d - 2d^2 ) - 2 - d \right ] y^3 + \left [ 2dc^4 ( 1 - d ) - \right .
. - c^2 ( 2d^2 - 6d + 7 ) + 5 \right ] y^2 + 2c \left [ c^2 d^2 + d(1 + c^2 ) - 2 ( c^2 - 1 ) \right ] y + c^2 d^2 ( 1 + c^2 ) - ( c^2 - 1)^2 = 0;\\ \end{array}\ ] ] at n = 3/2 : @xmath49 \sqrt{\frac{1 + \frac{c + y}{\sqrt{c^2 + 2cy + 1}}}{2 } } - c y^2 ( 1 - d ) - y - cd = 0;\ ] ] at n = 5/4 : @xmath50 this method gives angles @xmath51 ( see the table 1 ) . for some pulsars calculations
were made by one method only . when it was possible we used two or all three methods . in these cases ,
the mean value of the angle @xmath0 has been calculated .
the resulting values @xmath52 are listed in the table 1 .
some other authors ( for example , kuzmin & dagkesamanskaya , 1983 ; kuzmin et al . , 1984 ;
lyne & manchester , 1988 ) carried out calculations of the angle @xmath0 earlier for the shorter samples of pulsars using some additional assumptions .
we will use further our estimations to calculate magnetic inductions at the surface of the neutron stars .
the distribution of the angles @xmath0 from the table 1 ( fig.2 ) shows that the majority of pulsars have rather small inclinations of the magnetic moments .
these pulsars are old enough , and we can conclude that they evolve to the aligned geometry .
the average characteristic age for our sample of pulsars is @xmath53 years .
we must note however that the angles calculated by the method * _ 1 ) _ * are the lower limits of this parameter .
this explains partly the predominance of the small values of @xmath0 . from the table 1.,width=453 ]
.values of the angle @xmath0 ( deg ) . [ cols="^,^,^,^,^,^,^,^,^,^,^ " , ]
1 . some methods for calculations of the angle @xmath0 between rotation and magnetic axes were applied to obtain the values of @xmath0 for 376 radio pulsars .
the distribution of these values shows the predominance of small inclinations of the magnetic axes . 2 .
magnetic inductions at the surface of 375 pulsars considered were calculated .
there is no the measured derivative @xmath54 for the pulsar j1713 - 3949 and it is excluded from the consideration .
the distribution of the calculated magnetic inductions can be described by the gaussian with the maximal value of @xmath55 and the width in the logarithmic scale nearly 1 .
the calculated inductions are higher than the catalog equatorial inductions with the mean value of the ratio of these quantities of 5 . for the pulsar j1410 - 7404 @xmath56 . the maximal value of the ratio @xmath57 for the pulsar j2007 + 0809 .
this work has been carried out with the financial support of basic research program of the presidium of the russian academy of sciences * _ transitional and explosive processes in astrophysics _ * ( p-41 ) .
we thank a.v.biryukov for very useful comments and discussions .
99 keith m.j . , johnston s. , weltevrede p. and kramer m. , 2010 , mnras , 402 , 745 kuzmin a.d . ,
dagkesamanskaya i.m . , 1983
, soviet astron .
letters , 9 , 80 kuzmin a.d . ,
dagkesamanskaya i.m . ,
pugachev v.d . ,
1984 , soviet astron . letters , 10 , 357 lyne a.g . ,
manchester r.n . , 1988 ,
mnras , 243 , 477 manchester r.n .
, taylor j.h . , 1977 , pulsars .
w.h.freeman and company , san francisco manchester r.n .
et al . , 2005 ,
j. , 129 , 1993 .
malov i.f . ,
nikitina e.b . , 2011a , astron.rep . , 55
, 19 malov i.f . ,
nikitina e.b . , 2011b , astron.rep . , 55 , 878 malov i.f . , nikitina e.b
, 2013 , astron.rep . , 57 , 833 van ommen t.d .
et al . , 1997 ,
mnras , 287 , 1210 weltevrede p. , johnston s. , 2008 , mnras , 391 , 1210 | we used the magneto - dipole radiation mechanism for the braking of radio pulsars to calculate the new values of magnetic inductions at the surfaces of neutron stars . for this aim
we estimated the angles @xmath0 between the rotation axis and the magnetic moment of the neutron star for 376 radio pulsars using three different methods .
it was shown that there was the predominance of small inclinations of the magnetic axes . using the obtained values of the angle @xmath0 we calculated the equatorial magnetic inductions for pulsars considered .
these inductions are several times higher as a rule than corresponding values in the known catalogs .
* keywords * magnetic fields ; methods : data analysis ; methods : statistical ; _ ( stars : ) _ pulsars : general |
one of the most promising solutions to the hierarchy problem is the randall - sundrum ( rs ) model @xcite . in this model
there is a single extra dimension compactified on @xmath1 with the non - factorizable metric of @xmath2 @xmath3 here @xmath4 is the extra - dimensional coordinate , and @xmath5 is the inverse of the @xmath6 curvature , @xmath7 .
two branes define the boundaries of the extra dimension .
one , at @xmath8 , is called the uv , or planck , brane .
the other , at @xmath9 , is the ir , or tev , brane . picking @xmath10 , which is natural in realistic stabilization mechanisms ,
solves the hierarchy problem @xcite . in the original rs model
the sm was confined to the ir brane , and only gravity propagated in the bulk @xcite .
it has since been realized that both gauge and fermion fields can live in the bulk in a realistic model @xcite .
these models are realistic , but the parameter space can be strongly reduced by precision electroweak constraints .
much of this problem can be traced to the fact that the massive gauge fields receive a contribution to their mass from the bulk geometry which does not respect the custodial @xmath11 .
this can be fixed by expanding the gauge group to @xmath12 , which dramatically improves the electroweak fit @xcite .
the breaking of this extended electroweak symmetry proceeds in two stages : on the uv brane @xmath13 ; on the ir brane @xmath14 , where @xmath15 is the diagonal of the @xmath16 groups .
this paper investigates the properties of the higgs sector that accomplishes this breaking .
we now ask what drives the breaking on each brane . on the planck brane
all degrees of freedom will have planck scale masses , so we can ignore them .
we can then implement the breaking with boundary conditions to good approximation .
this leads to the boundary conditions at @xmath8 @xmath17 here @xmath18 and @xmath19 are ratios of 5d gauge couplings : @xmath20 , and @xmath21 . on the tev brane
, the masses will be tev scale , so we should look at the higgs sector in detail . the simplest structure that will create
the breaking pattern is a real higgs that is a bidoublet under @xmath22 .
this leads to the boundary conditions at @xmath9 @xmath23 note that in the @xmath24 limit we obtain the usual higgsless boundary conditions , and this model reduces to the higgsless model in @xcite
. we will use this parameter , @xmath25 to interpolate between the sm limit ( @xmath26 ) , and the higgsless limit .
[ fig : wroot ] to write down the effective 4d theory , we expand the 5d fields into kaluza klein ( kk ) fields , @xmath27 we can now obtain the gauge boson wavefunctions by solving the equation of motion subject to the boundary conditions ( [ eq : gaugebcr ] ) and ( [ eq : gaugebcrp ] ) .
this produces a spectrum of eigenvalues corresponding to the excitations of the gauge fields .
the lowest masses in each of the charged and neutral sectors will correspond to the @xmath28 and @xmath29 bosons .
the neutral sector also contains a zero mode , corresponding to the photon .
fig [ fig : wroot ] shows the eigenvalue for the @xmath28 as a function of the parameter @xmath25 .
[ fig : wcoup ]
one interesting feature of this model is that the @xmath28 and @xmath29 wavefunctions are suppressed near the ir brane , as can be seen by inspecting the boundary conditions
. this suppression increases for increasing @xmath25 .
this means that the coupling of massive gauge bosons to the higgs will generically be suppressed .
[ fig : wcoup ] shows the coupling of the @xmath28 to the higgs . for values of @xmath25 near unity
the lep bounds on the higgs mass can be dramatically reduced .
( for larger values of @xmath25 the model is effectively higgsless . )
the fermion sector of this model is more complicated .
again , the higgs vev induces mixed boundary conditions that link left and right handed fields to give the fermions masses .
however , there are two new degrees of freedom . first
, since 5d fermions are achiral , ther can always be a mass term in the bulk @xmath30 .
the main effect of this term is to shift the location of the fermion zero mode in the bulk . by changing this parameter
we can cause the zero mode to be localized either near the uv or ir brane , and also can change the degree of this localization .
this allows us to control the overlap of the zero mode with the ir brane , and consequently the strength with which the fermion interacts with the higgs . in this way
the hierarchy of fermion masses can be generated by order one changes in the 5d masses .
the second complication arises from the @xmath31 symmetry which enforces that , for example , @xmath32 if unbroken .
this mass relation can be modified by mixing with new fermions localized to the planck brane , where the @xmath31 is broken . for full details , see @xcite . note that there are tree - level corrections to precision electroweak observables , coming largely from the kk excitations of the gauge bosons .
unfortunately , the magnitude of these corrections is highly sensitive to the configuration of the fermion sector . for the specific configuration studied in @xcite
we find the constraint @xmath33 .
there are , however , special points in the fermion parameter space where the constraint becomes trivial , so a wide range of @xmath25 should be considered .
the final interesting shift in higgs properties is in the couplings to massless gauge bosons , _
i.e. _ gluons and photons .
the coupling of the higgs to gluon pairs is induced through top loops . however , in this model the kk excitations also couple to the higgs .
furthermore , note that we have arranged small 4d yukawa couplings for the other fermions by small wavefunction overlaps with the would - be zero modes .
the 5d yukawa couplings are all order 1 , and the excited states have _ no _ wavefunction suppression .
hence there are large contributions to the higgs - glue - glue coupling from the kk excitations of _ all _ colored fermions .
this leads to an enhancement in that coupling , as seen in fig .
[ fig : hgg ] .
there are similar corrections to the higgs - gamma - gamma coupling .
the situation there is more complicated , however , since there are also contributions from @xmath28 boson loops , which are dominant in the sm , and the higgs coupling to @xmath28s is suppressed .
we can now look at the behavior of the higgs branching ratios , as shown in fig .
[ fig : widthbr ] . note in particular the dominance of @xmath34 over a wide range , and the late onset of @xmath35 .
this is driven by the @xmath31 symmetry , which gives a large enhancement of the @xmath36-quark yukawa , and the suppression of the gauge boson couplings .
note also the reduction in the @xmath37 mode .
this will make discovery at the lhc difficult .
the suppression of the coupling to gauge bosons also means that production at the ilc will be reduced , making this higgs a particularly difficult one to find .
however , it will be essential to measure the higgs couplings with precision to identify a warped extra dimension as the correct theory of new physics , if indeed this is what is realized in nature .
9 b. lillie , arxiv : hep - ph/0505074 .
l. randall and r. sundrum , phys .
lett . * 83 * , 3370 ( 1999 ) [ arxiv : hep - ph/9905221 ] .
w. d. goldberger and m. b. wise , phys .
* 83 * , 4922 ( 1999 ) [ arxiv : hep - ph/9907447 ] .
h. davoudiasl , j. l. hewett and t. g. rizzo , phys .
lett . * 84 * , 2080 ( 2000 ) [ arxiv : hep - ph/9909255 ] . h. davoudiasl , j. l. hewett and t. g. rizzo , phys .
b * 473 * , 43 ( 2000 ) [ arxiv : hep - ph/9911262 ] and phys .
d * 63 * , 075004 ( 2001 ) [ arxiv : hep - ph/0006041 ] ; a. pomarol , phys .
b * 486 * , 153 ( 2000 ) [ arxiv : hep - ph/9911294 ] ; y. grossman and m. neubert , phys .
b * 474 * , 361 ( 2000 ) [ arxiv : hep - ph/9912408 ] ; t. gherghetta and a. pomarol , nucl .
b * 586 * , 141 ( 2000 ) [ arxiv : hep - ph/0003129 ] .
k. agashe , a. delgado , m. j. may and r. sundrum , jhep * 0308 * , 050 ( 2003 ) [ arxiv : hep - ph/0308036 ] . c. csaki , c. grojean , h. murayama , l. pilo and j. terning , phys .
d * 69 * , 055006 ( 2004 ) [ arxiv : hep - ph/0305237 ] . c. csaki , c. grojean , l. pilo and j. terning , phys .
* 92 * , 101802 ( 2004 ) [ arxiv : hep - ph/0308038 ] .
g. cacciapaglia , c. csaki , c. grojean and j. terning , phys .
d * 70 * , 075014 ( 2004 ) [ arxiv : hep - ph/0401160 ] and phys .
d * 71 * , 035015 ( 2005 ) [ arxiv : hep - ph/0409126 ] ; g. cacciapaglia , c. csaki , c. grojean , m. reece and j. terning , arxiv : hep - ph/0505001 . y. nomura , jhep * 0311 * , 050 ( 2003 ) [ arxiv : hep - ph/0309189 ] .
r. barbieri , a. pomarol and r. rattazzi , phys .
b * 591 * , 141 ( 2004 ) [ arxiv : hep - ph/0310285 ] .
h. davoudiasl , j. l. hewett , b. lillie and t. g. rizzo , phys .
d * 70 * , 015006 ( 2004 ) [ arxiv : hep - ph/0312193 ] and jhep * 0405 * , 015 ( 2004 ) [ arxiv : hep - ph/0403300 ] ; j. l. hewett , b. lillie and t. g. rizzo , jhep * 0410 * , 014 ( 2004 ) [ arxiv : hep - ph/0407059 ] ; r. s. chivukula , e. h. simmons , h. j. he , m. kurachi and m. tanabashi , phys .
d * 70 * , 075008 ( 2004 ) [ arxiv : hep - ph/0406077 ] and phys .
b * 603 * , 210 ( 2004 ) [ arxiv : hep - ph/0408262 ] .
a. birkedal , k. matchev and m. perelstein , phys .
* 94 * , 191803 ( 2005 ) [ arxiv : hep - ph/0412278 ] . c. csaki , c. grojean , j. hubisz , y. shirman and j. terning , phys .
d * 70 * , 015012 ( 2004 ) [ arxiv : hep - ph/0310355 ] . | we study the corrections to higgs physics in a model of a single warped extra dimension with all fields except the higgs in the bulk , and a gauge symmetry extended to @xmath0 .
we find that generically the higgs coupling to electroweak gauge boson pairs is suppressed , the coupling to gluons is enhanced , and the coupling to photons is often suppressed , but can be enhanced . |
the proper - motion observations of pulsars show that the pulsars had the kick velocity in the formation stage .
the young pulsars have proper velocity of @xmath4 @xcite .
the physical mechanism of such kick velocity may be due to the harrison
tademaru mechanism @xcite , anisotropic emission of neutrinos , anisotropic explosion and so on ( see lorimer @xcite for the review ) .
therefore , it is also reasonable to assume the existence of the proper motion of the pulsars in the formation process of pop iii nss , although there is no direct evidence since no pop iii star or pulsar is observed . while , repetto et al .
@xcite suggest that bhs also have a natal kick velocity comparable to pulsars from the galactic latitude distribution of the low mass x - ray binaries in our galaxy .
but , first , this is not the direct observation of proper motion of bhs , and second , since the mass of pop iii bhs is larger than pop i and pop ii bhs , their kick velocity might be so small that it can be neglected .
therefore , we take into account the natal kick for pop iii nss but not for pop iii bhs in this paper .
the kick speed @xmath5 obeys a maxwellian distribution as @xmath6 \,,\ ] ] where @xmath7 is the dispersion .
the details of the method how to calculate the natal kick are shown in ref .
@xcite . in this paper
, we perform population synthesis monte carlo simulations of pop iii binary stars .
we calculate the pop iii ns - bh and pop i and ii ns - bh for comparison .
pop i and pop ii stars mean solar metal stars and metal poor stars whose metallicity is less than 10% of solar metallicity , respectively . in this paper , we consider five metallicity cases of @xmath8 ( pop iii ) , @xmath9 and @xmath10 ( pop i ) .
there are important differences between pop iii and pop i and ii .
pop iii stars are ( 1 ) more massive , @xmath11 , ( 2 ) smaller stellar radius compared with that of pop i and ii , and ( 3 ) no stellar wind mass loss . these properties play key roles in binary interactions . in order to estimate the event rate of ns - bh mergers and the properties of ns - bh , we use the binary population synthesis method @xcite which is the monte calro simulation of binary evolution .
first , we choose the binary initial conditions such as the primary mass @xmath12 , the mass ratio @xmath13 , the separation @xmath14 , and the eccentricity @xmath15 when the binary is born .
these binary initial conditions are chosen by the monte calro method and the initial distribution functions such as the initial mass function ( imf ) , the initial mass ratio function ( imrf ) , the initial separation function ( isf ) , and the initial eccentricity distribution function ( ief ) .
we adopt these distribution functions for pop iii stars and pop i and ii stars as table [ idf ] . [ cols="^,^,^",options="header " , ]
this work was supported by mext grant - in - aid for scientific research on innovative areas , `` new developments in astrophysics through multi - messenger observations of gravitational wave sources '' , no .
24103006 ( tn , hn ) , by the grant - in - aid from the ministry of education , culture , sports , science and technology ( mext ) of japan no .
15h02087 ( tn ) , and jsps grant - in - aid for scientific research ( c ) , no .
16k05347 ( hn ) .
a. g. lyne and d. r. lorimer , nature * 369 * , 127 ( 1994 ) .
b. m. s. hansen and e. s. phinney , mon . not .
soc . * 291 * , 569 ( 1997 ) [ astro - ph/9708071 ] .
e. r. harrison and e. p. tademaru , astrophys .
j. * 201 * , 447 ( 1975 ) .
d. r. lorimer , living rev .
* 11 * , 8 ( 2008 ) [ arxiv:0811.0762 [ astro - ph ] ] .
s. repetto , m. b. davies and s. sigurdsson , mon . not .
425 * , 2799 ( 2012 ) [ arxiv:1203.3077 [ astro-ph.ga ] ] .
j. r. hurley , c. a. tout and o. r. pols , mon . not .
roy . astron .
soc . * 329 * , 897 ( 2002 ) [ astro - ph/0201220 ] .
t. kinugawa , k. inayoshi , k. hotokezaka , d. nakauchi and t. nakamura , mon . not .
soc . * 442 * , 2963 ( 2014 ) [ arxiv:1402.6672 [ astro-ph.he ] ] .
t. kinugawa , a. miyamoto , n. kanda and t. nakamura , mon . not .
soc . * 456 * , 1093 ( 2016 ) [ arxiv:1505.06962 [ astro-ph.sr ] ] . c. e. rhoades , jr . and r. ruffini , phys . rev .
lett . * 32 * , 324 ( 1974 ) .
j. b. hartle , phys .
rep , * 46 * , 201 ( 1978 ) r. m. kulsrud , r. cen , j. p. ostriker and d. ryu , astrophys . j. * 480 * , 481 ( 1997 ) [ astro - ph/9607141 ] . l. m. widrow , rev .
phys . * 74 * , 775 ( 2002 ) [ astro - ph/0207240 ] .
m. langer , j. l. puget and n. aghanim , phys .
d * 67 * , 043505 ( 2003 ) [ astro - ph/0212108 ] .
k. doi and h. susa , astrophys .
j. * 741 * , 93 ( 2011 ) [ arxiv:1108.4504 [ astro-ph.co ] ] .
h. nieuwenhuijzen and c. de jager , astron .
astrophys . * 231 * , 134 ( 1990 ) .
e. vassiliadis and p. r. wood , astrophys .
j. * 413 * , 641 ( 1993 ) .
r. m. humphreys and k. davidson , publ .
. soc . pac .
* 106 * , 1025 ( 1989 ) .
n. smith , ann .
astrophys .
* 52 * , 487 ( 2014 ) [ arxiv:1402.1237 [ astro-ph.sr ] ] . k. belczynski , t. bulik , c. l. fryer , a. ruiter , j. s. vink and j. r. hurley , astrophys . j. * 714 * , 1217 ( 2010 ) [ arxiv:0904.2784 [ astro-ph.sr ] ] .
j. s. vink and a. de koter , astron .
astrophys . *
442 * , 587 ( 2005 ) [ astro - ph/0507352 ] .
r. s. de souza , n. yoshida and k. ioka , astron .
astrophys . * 533 * , a32 ( 2011 ) [ arxiv:1105.2395 [ astro-ph.co ] ] .
e. visbal , z. haiman and g. l. bryan , mon . not .
* 453 * , 4456 ( 2015 ) [ arxiv:1505.06359 [ astro-ph.co ] ] .
t. hartwig , m. volonteri , v. bromm , r. s. klessen , e. barausse , m. magg and a. stacy , mon . not .
* 460 * , l74 ( 2016 ) [ arxiv:1603.05655 [ astro-ph.ga ] ] .
k. inayoshi , k. kashiyama , e. visbal and z. haiman , mon . not .
soc . * 461 * , 2722 ( 2016 ) [ arxiv:1603.06921 [ astro-ph.ga ] ] .
p. madau and m. dickinson , ann .
rev . astron .
astrophys . * 52 * , 415 ( 2014 ) [ arxiv:1403.0007 [ astro-ph.co ] ] .
x. ma , p. f. hopkins , c. a. faucher - giguere , n. zolman , a. l. muratov , d. keres and e. quataert , mon . not .
soc . * 456 * , 2140 ( 2016 ) [ arxiv:1504.02097 [ astro-ph.ga ] ] .
a. fontana _ et al .
_ , astron .
astrophys .
* 459 * , 745 ( 2006 ) [ astro - ph/0609068 ] .
k. kawaguchi , k. kyutoku , h. nakano , h. okawa , m. shibata and k. taniguchi , phys .
d * 92 * , 024014 ( 2015 ) [ arxiv:1506.05473 [ astro-ph.he ] ] .
d. keppel and p. ajith , phys .
d * 82 * , 122001 ( 2010 ) [ arxiv:1004.0284 [ gr - qc ] ] .
j. aasi _ et al .
_ [ ligo scientific and virgo collaborations ] , living rev . rel . *
19 * , 1 ( 2016 ) [ arxiv:1304.0670 [ gr - qc ] ] .
t. nakamura _ et al .
_ , arxiv:1607.00897 [ astro-ph.he ] . n. dalal , d. e. holz , s. a. hughes and b. jain , phys . rev .
d * 74 * , 063006 ( 2006 ) [ astro - ph/0601275 ] .
p. ajith _ et al .
. rev .
lett . * 106 * , 241101 ( 2011 ) [ arxiv:0909.2867 [ gr - qc ] ] . | in the population synthesis simulations of pop iii stars , many bh ( black hole)-bh binaries with merger time less than the age of the universe @xmath0 are formed , while ns ( neutron star)-bh binaries are not .
the reason is that pop iii stars have no metal so that no mass loss is expected .
then , in the final supernova explosion to ns , much mass is lost so that the semi major axis becomes too large for pop iii ns - bh binaries to merge within @xmath1 .
however it is almost established that the kick velocity of the order of @xmath2 exists for ns from the observation of the proper motion of the pulsar .
therefore , the semi major axis of the half of ns - bh binaries can be smaller than that of the previous argument for pop iii ns - bh binaries to decrease the merging time .
we perform population synthesis monte carlo simulations of pop iii ns - bh binaries including the kick of ns and find that the event rate of pop iii ns - bh merger rate is @xmath3 .
this suggests that there is a good chance of the detection of pop iii ns - bh mergers in o2 of advanced ligo and advanced virgo from this autumn . |
i am grateful to alekos kechris for informing me of t.dyck/ ; the proof given seems to be due to alain louveau .
i thank norm levenberg for references .
hough , j.b .
, krishnapur , m. , peres , y. , and virg , b. , _ zeros of gaussian analytic functions and determinantal point processes_. university lecture series , * 51*. american mathematical society , providence , ri , 2009 .
mester , p. , invariant monotone coupling need not exist .
* 41 * ( 2013 ) , 3a , 11801190 .
morris , b. , the components of the wired spanning forest are recurrent . _
probab . theory related fields _ * 125 * ( 2003 ) , 259265 . | we describe the fundamental constructions and properties of determinantal probability measures and point processes , giving streamlined proofs .
we illustrate these with some important examples . we pose several general questions and conjectures .
primary 60k99 , 60g55 ; secondary 42c30 , 37a15 , 37a35 , 37a50 , 68u99 .
random matrices , eigenvalues , orthogonal projections , positive contractions , exterior algebra , stochastic domination , negative association , point processes , mixtures , spanning trees , orthogonal polynomials , completeness , bernoulli processes .
determinantal point processes were originally defined by macchi @xcite in physics . starting in the 1990s ,
determinantal probability began to flourish as examples appeared in numerous parts of mathematics @xcite .
recently , applications to machine learning have appeared @xcite .
a discrete determinantal probability measure is one whose elementary cylinder probabilities are given by determinants . more specifically ,
suppose that @xmath0 is a finite or countable set and that @xmath1 is an @xmath2 matrix .
for a subset @xmath3 , let @xmath4 denote the submatrix of @xmath1 whose rows and columns are indexed by @xmath5 .
if @xmath6 is a random subset of @xmath0 with the property that for all finite @xmath7 , we have e.dpm = ( qa ) , then we call @xmath8 a . the inclusion - exclusion principle in combination with yields the probability of each elementary cylinder event .
therefore , for every @xmath1 , there is at most one probability measure , to be denoted @xmath9 , on subsets of @xmath0 that satisfies .
conversely , it is known ( see , e.g. , b.lyons:det/ ) that there is a determinantal probability measure corresponding to @xmath1 if @xmath1 is the matrix of a positive contraction on @xmath10 ( in the standard orthonormal basis ) .
technicalities are required even to define the corresponding concept of determinantal point process for @xmath0 being euclidean space or a more general space .
we present a virtually complete development of their basic properties in a way that minimizes such technicalities by adapting the approach of b.lyons:det/ from the discrete case .
in addition , we use an idea of goldman b.goldman/ to deduce properties of the general case from corresponding properties in the discrete case .
space limitations prevent mention of most of what is known in determinantal probability theory , which pertains largely to the analysis of specific examples .
we focus instead on some of the basic properties that hold for all determinantal processes and on some intriguing open questions .
let @xmath0 be a denumerable set .
we identify a subset of @xmath0 with an element of @xmath11 in the usual way .
there are several approaches to prove the basic existence results and identities for determinantal probability measures .
we sketch the one used by b.lyons : det/. this depends on understanding first the case where @xmath1 is the matrix of an orthogonal projection .
it also relies on exterior algebra so that the existence becomes immediate .
any unit vector @xmath12 in a hilbert space with orthonormal basis @xmath0 gives a probability measure @xmath13 on @xmath0 , namely , @xmath14 associated to orthogonal projections @xmath15 .
we refer to b.lyons:det/ for details not given here .
identify @xmath0 with the standard orthonormal basis of the real or complex hilbert space @xmath10 . for @xmath16 ,
let @xmath17 denote a collection of ordered @xmath18-element subsets of @xmath0 such that each @xmath18-element subset of @xmath0 appears exactly once in @xmath17 in some ordering .
define @xmath19 if @xmath20 , then @xmath21 and @xmath22 .
we also define @xmath23 to be the scalar field , @xmath24 or @xmath25 .
the elements of @xmath26 are called of @xmath18 , or for short .
we then define the ( or ) of multivectors in the usual alternating multilinear way : @xmath27 for any permutation @xmath28 , and @xmath29 for any scalars @xmath30 ( @xmath31,\ ; e \in e'$ ] ) and any finite @xmath32 .
( thus , @xmath33 unless all @xmath34 are distinct . ) the inner product on @xmath26 satisfies e.ipdet = _ i , j when @xmath35 and @xmath36 are 1-vectors .
( this also shows that the inner product on @xmath26 does not depend on the choice of orthonormal basis of @xmath37 . )
we then define the ( or ) @xmath38 , where the summands are declared orthogonal , making it into a hilbert space .
( throughout the paper , @xmath39 is used to indicate the sum of orthogonal summands , or , if there are an infinite number of orthogonal summands , the closure of their sum . )
vectors @xmath40 are linearly independent iff @xmath41 . for a @xmath18-element subset @xmath3 with ordering @xmath42 in @xmath17 , write @xmath43 .
we also write @xmath44 for any function @xmath45 .
although there is an isometric isomorphism @xmath46 for @xmath47 , this does not simplify matters in the discrete case
. it will be very useful in the continuous case later , however .
if @xmath48 is a closed linear subspace of @xmath37 , written @xmath49 , then we identify @xmath50 with its inclusion in @xmath51 .
that is , @xmath52 is the closure of the linear span of the @xmath18-vectors @xmath53 . in particular ,
if @xmath54 , then @xmath55 is a 1-dimensional subspace of @xmath51 ; denote by @xmath56 a unit multivector in this subspace .
note that @xmath56 is unique up to a scalar factor of modulus 1 ; which scalar is chosen will not affect the definitions below .
we denote by @xmath15 the orthogonal projection onto @xmath48 for any @xmath49 or , more generally , @xmath57 .
l.projection for every closed subspace @xmath49 , every @xmath16 , and every @xmath58 , we have @xmath59 write @xmath60 and expand the product .
all terms but @xmath61 have a factor of @xmath62 in them , making them orthogonal to @xmath50 by e.ipdet/. a multivector is called or if it is the wedge product of 1-vectors .
b.whitney:book/ , p. 49 , shows that e.whitney .
we shall use the defined by duality : @xmath63 in particular , if @xmath64 and @xmath65 is a multivector that does not contain any term with @xmath66 in it ( that is , @xmath67 ) , then @xmath68 and @xmath69 . more generally , if @xmath70 with @xmath71 and @xmath72 , then @xmath73 and @xmath74 .
note that the interior product is sesquilinear , not bilinear , over @xmath25 .
for @xmath70 , write @xmath75 $ ] for the subspace of scalar multiples of @xmath12 in @xmath37 . if @xmath48 is a finite - dimensional subspace of @xmath37 and @xmath76 , then e.hwedge _ h
e = p_h^e_h + [ e ] ( up to signum ) . to see this ,
let @xmath77 be an orthonormal basis of @xmath48 , where @xmath78 .
put @xmath79 .
then @xmath80 is an orthonormal basis of @xmath81 $ ] , whence @xmath82 } = u_1 \wedge u_2 \wedge \cdots \wedge u_r \wedge v = \mv_h \wedge v = \mv_h \wedge e/\|p_h^\perp e\|\ ] ] since @xmath83 .
this shows e.hwedge/. similarly , if @xmath84 , then e.hvee _ h e = p_h e_h e^ ( up to signum ) .
indeed , put @xmath85 .
let @xmath86 be an orthonormal basis of @xmath48 with @xmath87 .
then @xmath88 ( up to signum ) , as desired .
finally , we claim that e.reverse = .
indeed , @xmath89 , so this is equivalent to @xmath90 thus , it suffices to show that @xmath91 by sesquilinearity , it suffices to show this for @xmath92 members of an orthonormal basis of @xmath48 .
but then it is obvious . for a more detailed presentation of exterior algebra ,
see b.whitney : book/. let @xmath48 be a subspace of @xmath37 of dimension @xmath93 . define the probability measure @xmath94 on subsets @xmath95 by e.xihpr ^h(\{b } ) : = ||^2 .
note that this is non-0 only for @xmath96 .
also , by l.projection/ , @xmath97 for @xmath96 , which is non-0 iff @xmath98 are linearly independent .
that is , @xmath99 iff the projections of the elements of @xmath100 form a basis of @xmath48
. let @xmath101 be any basis of @xmath48 .
if we use e.ipdet/ and the fact that @xmath102 for some scalar @xmath103 , then we obtain another formula for @xmath94 : we use @xmath104 to denote a random subset of @xmath0 arising from a probability measure @xmath94 . to see that e.dpm/ holds for the matrix of @xmath15 , observe that for @xmath96 , @xmath105 = \bigip{p_{\ext(h ) } \theta_b , \theta_b } = \bigip{\bigwedge_{e \in b } p_h e , \bigwedge_{e \in b } e } = \det [ \ip{p_h e , f}]_{e , f \in b}\ ] ] by e.ipdet/.
this shows that e.dpm/ holds for @xmath106 since @xmath107 @xmath94-a.s .
the general case is a consequence of multilinearity , which gives the following extension of e.dpm/. we use the convention that @xmath108 and @xmath109 for any multivector @xmath65 .
t.genprs if @xmath110 and @xmath111 are ( possibly empty ) subsets of a finite set @xmath0 , then e.genprs ^h[a_1 , a_2 = ] = . in particular , for every @xmath3
, we have e.included ^h[a ] = p_(h ) _ a^2 .
c.dualrep if @xmath0 is finite , then for every subspace @xmath49 , we have e.dualrep ^h^(\{e b } ) = ^h(\{b } ) .
these extend to infinite @xmath0 . in order to define @xmath94
when @xmath48 is infinite dimensional , we proceed by finite approximation .
let @xmath112 be infinite .
consider first a finite - dimensional subspace @xmath48 of @xmath37 .
define @xmath113 as the image of the orthogonal projection of @xmath48 onto the span of @xmath114 . by considering a basis of @xmath48 , we see that @xmath115 in the weak operator topology ( wot ) , i.e. , matrix - entrywise , as @xmath116 .
it is also easy to see that if @xmath117 , then @xmath118 for all large @xmath18 and , in fact , @xmath119 in the usual norm topology .
it follows that e.genprs/ holds for this subspace @xmath48 and for every finite @xmath120 .
now let @xmath48 be an infinite - dimensional closed subspace of @xmath37 .
choose finite - dimensional subspaces @xmath121 .
it is well known that @xmath115 ( wot ) . then e.detgenprs a ( p_h_k a ) ( p_h a ) , whence @xmath122 has a weak@xmath123 limit that we denote @xmath94 and that satisfies e.genprs/. we also note that for _ any _ sequence of subspaces @xmath113 , if @xmath124 ( wot ) , then @xmath125 weak@xmath123 because e.detgenprs/ then holds .
we call @xmath1 a if @xmath1 is a self - adjoint operator on @xmath37 such that for all @xmath126 , we have @xmath127 . a of @xmath1 is an orthogonal projection @xmath15 onto a closed subspace @xmath128 for some @xmath129 such that for all @xmath126 , we have @xmath130 , where we regard @xmath131 as the orthogonal sum @xmath132 . in this case , @xmath1 is also called the of @xmath15 to @xmath37 .
choose such a dilation ( see e.vecdilate/ or e.dilate/ ) and define @xmath9 as the law of @xmath133 when @xmath104 has the law @xmath94 .
then e.dpm/ for @xmath1 is a special case of e.dpm/ for @xmath15 .
of course , when @xmath1 is the orthogonal projection onto a subspace @xmath48 , then @xmath134 .
basic properties of @xmath9 follow from those for orthogonal projections , such as : t.q if @xmath1 is a positive contraction , then for all finite @xmath135 , e.qgenprs ^q= .
if e.dpm/ is given , then e.qgenprs/ can be deduced from e.dpm/ without using our general theory and , in fact , without assuming that the matrix @xmath1 is self - adjoint . indeed , suppose that @xmath136 is any diagonal matrix .
denote its @xmath137-entry by @xmath138 . comparing coefficients of @xmath138
shows that e.dpm/ implies , for finite @xmath3 , e.xe = ( ( q + x ) a ) . replacing @xmath5 by @xmath139 and
choosing @xmath140 gives e.qgenprs/. on the other hand , if we substitute @xmath141 , then we may rewrite e.xe/ as e.ze = ( ( q z + i - q ) a ) , where @xmath142 is the diagonal matrix of the variables @xmath143 .
let @xmath0 be finite .
write @xmath144 for @xmath3
. then e.ze/ is equivalent to e.affine _ a e ^q[= a ] z^a = ( i - q+qz ) .
this is the same as the laplace transform of @xmath9 after a trivial change of variables .
when @xmath145 , we can write @xmath146 with @xmath147 . thus , for all @xmath3 , we have a probability measure @xmath8 on @xmath148 is called if its generating polynomial @xmath149 z^a$ ] satisfies the inequality for all @xmath150 and all real @xmath151 .
this property is satisfied by every determinantal probability measure , as was shown by b.bbl:rayleigh/ , who demonstrated its usefulness in showing other properties , such as negative associations and preservation under symmetric exclusion processes . for a set @xmath152 ,
denote by @xmath153 the @xmath154-field of events that are measurable with respect to the events @xmath155 for @xmath156 .
define the @xmath154-field to be the intersection of @xmath157 over all finite @xmath158 .
we say that a measure @xmath8 on @xmath148 has if every event in the tail @xmath154-field has measure either 0 or 1 .
t.tail b.lyons:det/ if @xmath1 is a positive contraction , then @xmath9 has trivial tail . for finite @xmath0 and a positive contraction @xmath1 ,
define the of @xmath9 to be @xmath159 numerical calculation supports the following conjecture b.lyons:det/ : g.concave for all positive contractions @xmath160 and
@xmath161 , we have e.concave ( ( q_1+q_2)/2 ) ( ( q_1 ) + ( q_2))/2 . let @xmath0 be denumerable .
a function @xmath162 is called if for all @xmath163 and all @xmath164 , we have @xmath165 . an event is called increasing or if its indicator is increasing . given two probability measures @xmath166 , @xmath167 on @xmath148 , we say that and write @xmath168 if for all increasing events @xmath169 , we have @xmath170 .
this is equivalent to @xmath171 for all bounded increasing @xmath172 .
a of two probability measures @xmath166 , @xmath167 on @xmath148 is a probability measure @xmath173 on @xmath174 whose coordinate projections are @xmath166 , @xmath167 ; it is if @xmath175 by strassen s theorem @xcite , stochastic domination @xmath176 is equivalent to the existence of a monotone coupling of @xmath166 and @xmath167 .
t.dominate-infinite b.lyons:det/ if @xmath177 , then @xmath178
. it would be very interesting to find a natural or explicit monotone coupling .
a coupling @xmath173 has @xmath8 if for all events @xmath179 , we have @xmath180 .
q.unioncoupling @xcite given @xmath181 , is there a coupling of @xmath182 and @xmath183 with union marginal @xmath94 ? a positive answer is supported by some numerical calculation .
it is easily seen to hold when @xmath184 by c.dualrep/. in the sequel , we write @xmath185 if @xmath186 for all @xmath126 .
t.dominate @xcite if @xmath187 , then @xmath188 . by t.dominate-infinite/ , it suffices that there exist orthogonal projections @xmath189 and @xmath190 that are dilations of @xmath160 and @xmath161 such that @xmath191 .
this follows from namark s dilation theorem @xcite , which says that any measure whose values are positive operators , whose total mass is @xmath192 , and which is countably additive in the weak operator topology dilates to a spectral measure .
the measure in our case is defined on a 3-point space , with masses @xmath160 , @xmath193 , and @xmath194 , respectively . if we denote the respective dilations by @xmath195 , @xmath196 , and @xmath197 , then we set @xmath198 and @xmath199
. a positive answer in general to q.unioncoupling/ would give the following more general result by compression : if @xmath160 , @xmath161 and @xmath200 are positive contractions on @xmath37 , then there is a coupling of @xmath201 and @xmath202 with union marginal @xmath203
. it would be very useful to have additional sufficient conditions for stochastic domination : see the end of s.orthogpoly/ and g.fkdom/. for examples where more is known , see t.gmdom/. we shall say that the events in @xmath153 are @xmath158 and likewise for functions that are measurable with respect to @xmath153 .
we say that @xmath8 has if for every pair @xmath204 , @xmath205 of increasing functions that are measurable with respect to complementary subsets of @xmath0 , e.negass .
@xcite if @xmath206 , then @xmath9 has negative associations .
the details for finite @xmath0 were given in b.lyons : det/. for infinite @xmath0 , let @xmath204 and @xmath205 be increasing bounded functions measurable with respect to @xmath207 and @xmath208 , respectively . choose finite @xmath209 .
the conditional expectations @xmath210 $ ] and @xmath211 $ ] are increasing functions to which e.negass/ applies ( because restriction to @xmath212 corresponds to a compression of @xmath1 , which is a positive contraction ) and which , being martingales , converge to @xmath204 and @xmath205 in @xmath213 .
write @xmath214 for the distribution of a bernoulli random variable with expectation @xmath215 . for @xmath216 $ ] ,
let @xmath217 be the distribution of a sum of independent @xmath218 random variables . recall that @xmath75 $ ] is the set of scalar multiples of @xmath12 .
t.eigmix @xcite ; lemma 3.4 of @xcite ; ( 2.38 ) of @xcite ; @xcite let @xmath1 be a positive contraction with spectral decomposition @xmath219}$ ] , where @xmath220 are orthonormal .
let @xmath221 be independent .
let @xmath222 $ ] ; thus , @xmath223 . then @xmath224 .
hence , if @xmath225 , then @xmath226 . by t.dominate/
, it suffices to prove it when only finitely many @xmath227 . then by t.q/
, we have @xmath228 = \bigip{\bigwedge_{e \in a } q e , \theta_a } $ ] for all @xmath3 . now
@xmath229 } e & = \sum_{j \colon a \to \bbn } \prod_{e \in a } \lambda_{j(e ) } \bigwedge_{e \in a } p_{[v_{j(e ) } ] } e \\ & = \sum_{j \colon a \rightarrowtail \bbn } \prod_{e \in a } \lambda_{j(e ) } \bigwedge_{e \in a } p_{[v_{j(e ) } ] } e\end{aligned}\ ] ] because @xmath230 and @xmath231 } e$ ] is a multiple of @xmath12 , so none of the terms where @xmath232 is not injective contribute .
thus , @xmath233 } e = \ebig{\sum_{j \colon a \rightarrowtail \bbn } \prod_{e \in a } i_{j(e ) } \bigwedge_{e \in a } p_{[v_{j(e ) } ] } e } \\ & = \ebig{\sum_{j \colon a \to \bbn } \prod_{e \in a } i_{j(e ) } \bigwedge_{e \in a } p_{[v_{j(e ) } ] } e } = \be\bigwedge_{e \in a } \sum_k i_k p_{[v_k ] } e = \be \bigwedge_{e \in a } p_{\rh } e \,.\end{aligned}\ ] ] we conclude that @xmath234 = \be \leftip{\bigwedge_{e \in a } p_{\rh } e , \theta_a } = \ebig { \bp^{\rh}\left [ a \subseteq \ba \right ] } $ ] by e.qgenprs/. we sketch another proof : let @xmath235 be disjoint from @xmath0 with the same cardinality .
choose an orthonormal sequence @xmath236 in @xmath131 .
define then @xmath1 is the compression of @xmath15 to @xmath37 . expanding @xmath237 in the obvious way into orthogonal pieces and restricting to @xmath0 , we obtain the desired equation from e.xihpr/. the first proof shows more generally the following : let @xmath238 be a positive contraction .
let @xmath220 be ( not necessarily orthogonal ) vectors such that @xmath239 } \lloew i$ ] .
let @xmath240 be independent bernoulli random variables with @xmath241 .
write @xmath242}$ ]
. then @xmath243 .
this was observed by ghosh and krishnapur ( personal communication , 2014 )
. note that in the mixture of t.eigmix/ , the distribution of @xmath244 is determinantal corresponding to the diagonal matrix with diagonal @xmath245 .
thus , it is natural to wonder whether @xmath246 can be taken to be a general determinantal measure .
if such a mixture is not necessarily determinantal , must it be strongly rayleigh or at least have negative correlations ? here
, we say that a probability measure @xmath8 on @xmath148 has if for every pair @xmath5 , @xmath100 of finite disjoint subsets of @xmath0 , we have @xmath247 \le \bp [ a \subseteq \qba ] \bp [ b \subseteq \qba ] $ ] .
note that negative associations is stronger than negative correlations .
the most well - known example of a ( nontrivial discrete ) determinantal probability measure is that where @xmath6 is a uniformly chosen random spanning tree of a finite connected graph @xmath248 with @xmath249 . here ,
we regard a spanning tree as a set of edges .
the fact that holds for the uniform spanning tree is due to b.burpem/ and is called the transfer current theorem .
the case with @xmath250 was shown much earlier by b.kirchhoff/ , while the case with @xmath251 was first shown by b.bsst/. write @xmath252 for the uniform spanning tree measure on @xmath253 . to see that @xmath252 is indeed determinantal , consider the vertex - edge incidence matrix @xmath254 of @xmath253 , where each edge is oriented ( arbitrarily ) and the @xmath255-entry of @xmath254 equals 1 if @xmath256 is the head of @xmath66 , @xmath257 if @xmath256 is the tail of @xmath66 , and 0 otherwise .
identifying an edge with its corresponding column of @xmath254 , we find that a spanning tree is the same as a basis of the column space of @xmath254 .
given @xmath258 , define the at @xmath256 to be the @xmath256-row of @xmath254 , regarded as a vector @xmath259 in the row space , @xmath260 .
it is easy that the row - rank of @xmath254 is @xmath261 .
let @xmath262 and let @xmath65 be the wedge product ( in some order ) of the stars at all the vertices other than @xmath263 .
thus , @xmath264 for some @xmath265 .
since spanning trees are bases of the column space of @xmath254 , we have @xmath266 iff @xmath5 is a spanning tree .
that is , the only non - zero coefficients of @xmath65 are those in which choosing one edge in each @xmath259 for @xmath267 yields a spanning tree ; moreover , each spanning tree occurs exactly once since there is exactly one way to choose an edge incident to each @xmath267 to get a given spanning tree .
this means that its coefficient is @xmath268 .
hence , @xmath269 is indeed uniform on spanning trees .
simultaneously , this proves the matrix tree theorem that the number of spanning trees equals @xmath270_{x , y \ne x_0}$ ] , since this determinant is @xmath271
. one can define analogues of @xmath252 on infinite connected graphs @xcite by weak limits . for brevity ,
we simply define them here as determinantal probability measures .
again , all edges of @xmath253 are oriented arbitrarily .
we define @xmath272 as the closure of the linear span of the stars .
an element of @xmath273 that is finitely supported and orthogonal to @xmath272 is called a ; the closed linear span of the cycles is @xmath274 .
the is @xmath275 , while the is @xmath276 .
our discussion of the continuous " case includes the discrete case , but the discrete case has the more elementary formulations given earlier .
let @xmath0 be a measurable space .
as before , @xmath0 will play the role of the underlying set on which a point process forms a counting measure . while before we implicitly used counting measure on @xmath0 itself , now we shall have an arbitrary measure @xmath173 ; it need not be a probability measure . the case of lebesgue measure on euclidean space is a common one .
the hilbert spaces of interest will be @xmath277
. there may be no natural order in @xmath0 , so to define , e.g. , a probability measure on @xmath278 points of @xmath0 , it is natural to use a probability measure on @xmath279 that is symmetric under coordinate changes and that vanishes on the diagonal @xmath280 .
likewise , for exterior algebra , it is more convenient to identify @xmath281 with @xmath282 for @xmath283 .
thus , @xmath284 is identified with the function @xmath285_{i , j \in \{1 , \ldots , n\}}/\sqrt{n ! } $ ] .
note that @xmath286 \det [ { v_i(x_j ) } ] = \det [ u_i(x_j ) ] \det [ { v_i(x_j)}]^t \nonumber \\ & = \det [ u_i(x_j)][{v_i(x_j)}]^t = \det [ k(x_i , x_j)]_{i , j \in \{1 , \ldots , n\}}\end{aligned}\ ] ] with @xmath287 .
here , @xmath288 denotes transpose .
suppose from now on that @xmath0 is a locally compact polish space ( equivalently , a locally compact second countable hausdorff space ) .
let @xmath173 be a radon measure on @xmath0 , i.e. , a borel measure that is finite on compact sets .
let @xmath289 be the set of radon measures on @xmath0 with values in @xmath290 .
we give @xmath289 the vague topology generated by the maps @xmath291 for continuous @xmath172 with compact support ; then @xmath289 is polish .
the corresponding borel @xmath154-field of @xmath289 is generated by the maps @xmath292 for borel @xmath3 .
let @xmath293 be a simple point process on @xmath0 , i.e. , a random variable with values in @xmath289 such that @xmath294 for all @xmath295 .
the power @xmath296 lies in @xmath297 .
thus , @xmath298 $ ] is a borel measure on @xmath299 ; the part of it that is concentrated on @xmath300 is called the of @xmath293 .
if the intensity measure is absolutely continuous with respect to @xmath301 , then its radon - nikodym derivative @xmath302 is called the or the : since the intensity measure vanishes on the diagonal @xmath303 , we take @xmath302 to vanish on @xmath303
. we also take @xmath302 to be symmetric under permutations of coordinates .
intensity functions are the continuous analogue of the elementary probabilities e.dpm/. since the sets @xmath304 generate the @xmath154-field on @xmath300 for pairwise disjoint borel @xmath305 , a measurable function @xmath306 is the " @xmath18-point intensity function iff since @xmath293 is simple , @xmath307 , where @xmath308 . since @xmath302 vanishes on the diagonal , it follows from e.rn/ that for disjoint @xmath309 and non - negative @xmath310 summing to @xmath18 , again , this characterizes @xmath302 , even if we use only @xmath311 . in the special case that @xmath312 a.s .
for some @xmath313 , then the definition e.rn/ shows that a random ordering of the @xmath278 points of @xmath293 has density @xmath314 .
more generally , e.rn/ shows that for all @xmath315 , whence in this case , we call @xmath293 if for some measurable @xmath316 and all @xmath16 , @xmath317 @xmath301-a.e . here
, @xmath318 is the matrix @xmath319_{i , j \le k}$ ] . in this case
, we denote the law of @xmath293 by @xmath320 .
we consider only @xmath158 that are locally square integrable ( i.e. , @xmath321 is radon ) , are hermitian ( i.e. , @xmath322 for all @xmath323 ) , and are positive semidefinite ( i.e. , @xmath324 is positive semidefinite for all finite @xmath325 , written @xmath326 ) . in this case , @xmath158 defines a positive semidefinite integral operator @xmath327 on functions @xmath328 with compact support . for every borel @xmath3 , we denote by @xmath329 the measure @xmath173 restricted to borel subsets of @xmath5 and by @xmath330 the compression of @xmath158 to @xmath5 , i.e. , @xmath331 for @xmath332 .
the operator @xmath158 is locally trace - class , i.e. , for every compact @xmath3 , the compression @xmath330 is trace class , having a spectral decomposition @xmath333 , where @xmath334 are orthonormal eigenfunctions of @xmath330 with positive summable eigenvalues @xmath335 . if @xmath110 is the set where @xmath336 , then @xmath337 and @xmath338 converges on @xmath339 , with sum @xmath340-a.e .
equal to @xmath158 .
we normally redefine @xmath158 on a set of measure 0 to equal this sum .
such a @xmath158 defines a determinantal point process iff the integral operator @xmath158 extends to all of @xmath341 as a positive contraction @xcite .
the joint intensities determine uniquely the law of the point process ( * ? ? ?
* lemma 4.2.6 ) .
poisson processes are not determinantal processes , but when @xmath173 is continuous , they are distributional limits of determinantal processes . to see that a positive contraction defines a determinantal point process , we first consider @xmath158 that defines an orthogonal projection onto a finite - dimensional subspace , @xmath48
. then @xmath342 for every orthonormal basis @xmath343 of @xmath48 and @xmath344 is a unit multivector in the notation of s.ext/. because of e.prodtensor/ , we have i.e. , @xmath345/n!$ ] is a density with respect to @xmath346 .
although in the discrete case , the absolute squared coefficients of @xmath347 give the elementary probabilities , now coefficients are replaced by a function whose absolute square gives a probability density .
as noted already , e.firstdensity/ means that @xmath348 is the @xmath278-point intensity function . in order to show that this density gives a determinantal process with kernel @xmath158
, we use the cauchy - binet formula , which may be stated as follows : for @xmath349 matrices @xmath350 $ ] and @xmath351 $ ] with @xmath352_{\substack{i \le k \\ j \in j}}$ ] , we have @xmath353 [ b_{i , j}]^t\big ) = \sum_{|j| = k } \det a^j \cdot \det b^j = \sum _ { \substack{\sigma , \tau \in \sym(k , n ) \\
\operatorname{im}(\sigma ) = \operatorname{im}(\tau ) } } ( -1)^\sigma ( -1)^\tau \prod_{i=1}^k a_{i , \sigma(i ) } b_{i , \tau(i ) } \,,\ ] ] where @xmath354 denotes the image of @xmath154 and the sums extend over all pairs of injections @xmath355 here , the sign @xmath356 of @xmath154 is defined in the usual way by the parity of the number of pairs @xmath357 for which @xmath358 .
we have @xmath359 \,d\mu^{n - k}(x_{k+1 } , \ldots , x_n ) \nonumber \\ & = \frac{1}{(n - k ) ! }
\int_{e^{n - k } } \sum_{\sigma \in \sym(n ) } ( -1)^\sigma \prod_{i=1}^n \phi_{\sigma(i)}(x_i ) \cdot { } \nonumber \\
\noalign{$\displaystyle \hfill \cdot \sum_{\tau \in \sym(n ) } ( -1)^\tau \prod_{i=1}^n \overline{\phi_{\tau(i)}(x_i ) } \,d\mu^{n - k}(x_{k+1 } , \ldots , x_n ) $ } & = \sum_{\substack{\sigma , \tau \in \sym(k , n ) \\
\operatorname{im}(\sigma ) = \operatorname{im}(\tau ) } } ( -1)^\sigma ( -1)^\tau \prod_{i=1}^k \phi_{\sigma(i)}(x_i ) \overline{\phi_{\tau(i)}(x_i ) } \nonumber \\ & = \det \big(k \restrict ( x_1 , \ldots , x_k)\big ) \,.\end{aligned}\ ] ] here , the first equality uses e.integrate/ , the second equality uses e.prodtensor/ , the third equality uses the fact that @xmath360 is 1 or 0 according as @xmath361 or not , and the fourth equality uses cauchy - binet .
note that a factor of @xmath362 arises because for every pair of injections @xmath363 with equal image , there are @xmath362 extensions of them to permutations @xmath364 with @xmath361 for all @xmath365 ; in this case , @xmath366 .
we write @xmath94 for the law of the associated point process on @xmath0 .
l.weaklimit let @xmath367 with @xmath368 for some @xmath369 .
then @xmath370 is tight and every weak limit point of @xmath371 is simple . by using the kernel @xmath372 with respect to the measure @xmath373 , we may assume that @xmath374 .
tightness follows from @xmath375 \le \be[\sx_n(a ) ] = \int_a k_n(x , x ) \,d\mu(x)\,.\ ] ] for the rest , we may assume that @xmath0 is compact and @xmath376 .
let @xmath293 be a limit point of @xmath371 .
let @xmath377 be the atomic part of @xmath173 and @xmath378 .
choose @xmath379 and partition @xmath0 into sets @xmath380 with @xmath381 .
let @xmath5 be such that @xmath382 and @xmath383 .
let @xmath384 be open such that @xmath385 and @xmath386 .
then @xmath387 & \le \limsup_n \big(\bp[\sx_n(u \setminus a ) \ge 1 ] + \bp[\texists i \sx_n(a_i ) \ge 2]\big ) \\ & \le \limsup_n \big(\be[\sx_n(u \setminus a ) ] + \sum_i \be[(\sx_n(a_i))_2]\big ) \\ & \le \muc(u ) + \sum_i \mu(a_i)^2 < 2/m \ , .
\tag*{\qedhere}\end{aligned}\ ] ] now , given any locally trace - class orthogonal projection @xmath158 onto @xmath48 , choose finite - dimensional subspaces @xmath388 with corresponding projections @xmath389 .
clearly @xmath390 @xmath391-a.e.and @xmath392 @xmath173-a.e .
thus , the joint intensity functions converge a.e . by dominated convergence , if @xmath393 is relatively compact and borel , then @xmath394 \to \int_a \det ( k \restrict f ) \,d\mu^k(f)$ ] . by uniform exponential moments of @xmath395 (
* ? ? ?
* proof of lemma 4.2.6 ) , it follows that all weak limit points of @xmath396 are equal , and hence , by l.weaklimit/ , define @xmath94 with kernel @xmath158 .
( in s.cinequalities/ , we shall see that @xmath397 is stochastically increasing . ) finally , let @xmath158 be any locally trace - class positive contraction . define the orthogonal projection on @xmath398 whose block matrix is take an isometric isomorphism of @xmath277 to @xmath131 for some denumerable set @xmath235 and interpret the above as an orthogonal projection @xmath399 on @xmath400 .
then @xmath399 is clearly locally trace - class and @xmath158 is the compression of @xmath399 to @xmath0 .
thus , we define @xmath320 by intersecting samples of @xmath401 with @xmath0 .
we remark that by writing @xmath399 as a limit of increasing finite - rank projections that we then compress , we see that @xmath320 may be defined as a limit of determinantal processes corresponding to increasing finite - rank positive contractions .
g.ctail if @xmath158 is a locally trace - class positive contraction , then @xmath320 has trivial tail in that every event in @xmath402 is trivial . rather than using compressions as in the last paragraph above , an alternative approach to defining @xmath320 uses mixtures and starts from finite - rank projections , as in s.mix/. this approach is due to b.hkpv : survey/. consider first a finite - rank @xmath403 .
let @xmath404 be independent .
let @xmath405 $ ] ; thus , @xmath406 . we claim that @xmath407 is determinantal with kernel @xmath158 .
indeed , it is clearly a simple point process .
write @xmath408 , @xmath409 , and @xmath410 .
let @xmath411 .
combining cauchy - binet with e.prodtensor/ yields @xmath412 .
similarly , the joint intensities of @xmath413 are the expectations of the joint intensities of @xmath414 , which equal @xmath415 essentially the same works for trace - class @xmath416 ; we need merely take , in the last step , a limit in the above equation as @xmath417 for @xmath418 , since all terms are non - negative and @xmath419 a.e .
given this construction of @xmath320 for trace - class @xmath158 , one can then construct @xmath320 for a general locally trace - class positive contraction by defining its restriction to each relatively compact set @xmath5 via the trace - class compression @xmath330 . as noted by b.hkpv:survey/ ,
a consequence of the mixture representation is a clt due originally to b.soshnikov:gauss/ : t.clt let @xmath389 be trace - class positive contractions on spaces @xmath420 .
let @xmath367 and write @xmath421 . if @xmath422 as @xmath417 , then @xmath423 obeys a clt .
in order to simulate @xmath320 when @xmath158 is a trace - class positive contraction , it suffices , by taking a mixture as above , to see how to simulate @xmath424 when @xmath425 . the following algorithm ( * ? ? ? * algo .
18 ) gives a uniform random ordering of @xmath293 as @xmath426 . since @xmath427 , the measure @xmath428/n = n^{-1 } k(x , x)\,d\mu(x)$ ] is a probability measure on @xmath0 .
select a point @xmath429 at random from that measure .
if @xmath430 , then we are done .
if not , then let @xmath431 be the orthogonal complement in @xmath48 of the function @xmath432 , where @xmath343 is an orthonormal basis for @xmath48 .
then @xmath433 and we may repeat the above for @xmath431 to get the next point , @xmath434 , then @xmath435 , etc .
the conditional density of @xmath436 given @xmath437 is @xmath438 by e.densityktuple/ , i.e. , @xmath439 times the squared distance from @xmath440 to the linear span of @xmath441 .
it can help for rejection sampling to note that this is at most @xmath442 .
one can also sample faster by noting that the conditional distribution of @xmath436 is the same as that of @xmath443 , where @xmath444 is a uniformly random vector on the unit sphere of @xmath113 .
note that if @xmath445 are bounded @xmath446-valued random variables , then the function @xmath447 determines the joint distribution of @xmath448 since it gives the derivatives at @xmath449 of the probability generating function @xmath450 .
let us re - examine e.falling/ in the context of a finite - rank @xmath451 .
given disjoint @xmath452 and non - negative @xmath310 summing to @xmath18 , it will be convenient to write @xmath453 for @xmath454 .
we have by cauchy - binet @xmath455 = \int_{\prod_{\ell=1}^r a_\ell^{k_\ell } } \rho_k \,d\mu^k = \int_{\prod_{\ell=1}^r a_\ell^{k_\ell } } \det ( k \restrict ( x_1 , \ldots , x_k ) ) \,\prod_{j=1}^k d\mu(x_j ) \\ & = \int_{\prod_{\ell=1}^r a_\ell^{k_\ell } } \sum_{\substack{\sigma , \tau \in \sym(k , n ) \\
\operatorname{im}(\sigma ) = \operatorname{im}(\tau ) } } ( -1)^\sigma ( -1)^\tau \prod_{j=1}^k \lambda_{\sigma(j ) } \phi_{\sigma(j)}(x_j ) \overline{\phi_{\tau(j)}(x_j ) } \,\prod_{j=1}^k d\mu(x_j ) \\ & = \sum_{\substack{\sigma , \tau \in \sym(k , n ) \\
\operatorname{im}(\sigma ) = \operatorname{im}(\tau ) } } ( -1)^\sigma ( -1)^\tau \prod_{j=1}^k \int_{a_{\kappa(j ) } } \lambda_{\sigma(j ) } \phi_{\sigma(j)}(x_j ) \overline{\phi_{\tau(j)}(x_j ) } \,d\mu(x_j ) \\ & = \sum_{\substack{\sigma , \tau \in \sym(k , n ) \\
\operatorname{im}(\sigma ) = \operatorname{im}(\tau ) } } ( -1)^\sigma ( -1)^\tau \lambda^{\operatorname{im}(\sigma ) } \prod_{j=1}^k \bigip{\boi{a_{\kappa(j ) } } \phi_{\sigma(j ) } , \overline{\phi_{\tau(j ) } } } \\ & = \sum_{\sigma \in \sym(k , n ) } ( -1)^\sigma \lambda^{\operatorname{im}(\sigma ) } \det \big [ \bigip{\boi{a_{\kappa(j ) } } \phi_{\sigma(j ) } , \overline{\phi_\ell } } \big]_{\substack{j \le k \hfill \\ \ell
\in \operatorname{im}(\sigma ) } } \,.\end{aligned}\ ] ] as an immediate consequence of this formula , we obtain the following important principle of goldman ( * ? ? ?
* proposition 12 ) that allows one to infer properties of continuous determinantal point processes from corresponding properties of discrete determinantal probability measures : t.transfer let @xmath456 and @xmath457 be two radon measure spaces on locally compact polish sets .
let @xmath458 be pairwise disjoint borel subsets of @xmath0 and @xmath459 be pairwise disjoint borel subsets of @xmath325 .
let @xmath460 $ ] with @xmath461 .
let @xmath462 be orthonormal in @xmath277 and @xmath463 be orthonormal in @xmath464 .
let @xmath465 and @xmath466 . if @xmath467 for all @xmath468 , then the @xmath320-distribution of @xmath469 equals the @xmath470-distribution of @xmath471 .
when only finitely many @xmath227 , this follows from our previous calculation .
the general case follows from weak convergence of the processes corresponding to the partial sums , as in the paragraph following l.weaklimit/. this permits us to compare to discrete measures via ( * ? ? ?
* lemma 16 ) : l.compare let @xmath173 be a radon measure on a locally compact polish space , @xmath0 .
let @xmath458 be pairwise disjoint borel subsets of @xmath0 .
let @xmath472 for @xmath16 .
then there exists a denumerable set @xmath325 , pairwise disjoint subsets @xmath459 of @xmath325 , and @xmath473 such that @xmath474 and @xmath475 for all @xmath468 . without loss of generality
, we may assume that @xmath476 .
for each @xmath477 , fix an orthonormal basis @xmath478 for the subspace of @xmath277 spanned by @xmath479 . here ,
@xmath480 .
define @xmath481 and @xmath482 .
let @xmath483 be the isometric isomorphism from the span of @xmath484 to @xmath485 that sends @xmath486 to @xmath487 .
defining @xmath488 yields the desired vectors .
we now show how the discrete models of s.transf/ allow us to obtain the analogues of the stochastic inequalities known to hold for discrete determinantal probability measures . for a borel set @xmath7 , let @xmath207 denote the @xmath154-field on @xmath289 generated by the functions @xmath489 for borel @xmath490 .
we say that a function that is measurable with respect to @xmath207 is , more simply , measurable with respect to @xmath5 .
the obvious partial order on @xmath289 allows us to define what it means for a function @xmath491 to be . as in the discrete case
, we say that @xmath8 has if @xmath492 \le \be[f_1 ] \be[f_2 ] $ ] for every pair @xmath204 , @xmath205 of bounded increasing functions that are measurable with respect to complementary subsets of @xmath0 .
an event is increasing if its indicator is increasing .
then @xmath8 has negative associations iff for every pair @xmath493 , @xmath494 of increasing events that are measurable with respect to complementary subsets of @xmath0 .
we also say that and write @xmath495 if @xmath496 for every increasing event @xmath169 .
call an event if it has the form @xmath497 , where @xmath100 is a relatively compact borel set and @xmath498 .
write @xmath499 for the closure under finite unions and intersections of the collection of elementary increasing events with @xmath500 ; the notation @xmath501 is chosen for upwardly closed " .
note that every event in @xmath499 is measurable with respect to some finite collection of functions @xmath502 for pairwise _ disjoint _ relatively compact borel @xmath503 .
write @xmath504 for the closure of @xmath499 under monotone limits , i.e. , under unions of increasing sequences and under intersections of decreasing sequences ; these events are also increasing .
this is the same as the closure of @xmath499 under countable unions and intersections .
l.approxincr let @xmath5 be a borel subset of a locally compact polish space , @xmath0 .
then @xmath504 is exactly the class of increasing borel sets in @xmath207 .
we give a proof at the end of this subsection .
first , we derive two consequences . a weaker version ( negative correlations of elementary increasing events ) of the initial one is due to b.ghosh/. t.cfm let @xmath173 be a radon measure on a locally compact polish space , @xmath0 .
let @xmath158 be a locally trace - class positive contraction on @xmath277 .
then @xmath320 has negative associations .
let @xmath505 be borel .
let @xmath506 and @xmath507 .
then @xmath508 for some compact @xmath100 by definition of @xmath509 .
we claim that e.cnegass/ holds for @xmath493 , @xmath494 , and @xmath510 , i.e. , for @xmath511 .
now @xmath493 is measurable with respect to a finite number of count functions @xmath502 for some disjoint @xmath512 ( @xmath513 ) and likewise @xmath494 is measurable with respect to a finite number of functions @xmath514 for some disjoint @xmath515 ( @xmath513 ) .
thus , there are functions @xmath516 and @xmath517 such that @xmath518 and @xmath519 . by t.transfer/ and l.compare/
, there is some discrete determinantal probability measure @xmath9 on some denumerable set @xmath325 and pairwise disjoint sets @xmath520 such that the joint @xmath521-distribution of all @xmath522 and @xmath523 is equal to the joint @xmath9-distribution of all @xmath524 and @xmath525 .
define the corresponding events @xmath526 by @xmath527 and @xmath528 .
since @xmath526 depend on disjoint subsets of @xmath325 , t.fm/ gives that @xmath529 .
this is the same as e.cnegass/ by t.transfer/. the same e.cnegass/ clearly then holds in the less restrictive setting @xmath530 by taking monotone limits .
l.approxincr/ completes the proof .
t.cdom theorem 3 of b.goldman/ suppose that @xmath531 and @xmath532 are two locally trace - class positive contractions such that @xmath533
. then @xmath534 .
it suffices to show that @xmath535 for every @xmath536 .
again , it suffices to assume that @xmath537 are trace class .
l.compare/ applied to all eigenfunctions of @xmath531 and @xmath532 yields a denumerable @xmath325 and two positive contractions @xmath538 on @xmath485 , together with an event @xmath539 , such that @xmath540 for @xmath541 .
furthermore , by construction , every function in @xmath485 is the image of a function in @xmath542 under the isometric isomorphism @xmath483 used to prove l.compare/ , whence @xmath543 .
therefore t.dominate/ yields @xmath544 , as desired .
again , it would be very interesting to have a natural monotone coupling of @xmath545 with @xmath546 .
for some examples where this would be desirable , see s.orthogpoly/. l.approxincr/ will follow from this folklore variant of a theorem of dyck b.dyck/ : t.dyck let @xmath136 be a polish space on which @xmath547 is a partial ordering that is closed in @xmath548 . let @xmath501 be a collection of open increasing sets that generates the borel subsets of @xmath136 .
let @xmath549 be the closure of @xmath501 under countable intersections and countable unions .
suppose that for all @xmath550 , either @xmath551 or there is @xmath552 and an open set @xmath553 such that @xmath554 , @xmath555 , and @xmath556 . then @xmath549 equals the class of increasing borel sets .
obviously every set in @xmath549 is borel and increasing . to show the converse
, we prove a variant of lusin s separation theorem .
namely , we show that if @xmath557 is increasing and analytic ( with respect to the paving of closed sets , as usual ) and if @xmath558 is analytic with @xmath559 , then there exists @xmath560 such that @xmath561 and @xmath562 . taking @xmath563 to be borel and @xmath564 forces @xmath565 and gives the desired conclusion . to prove this separation property , we first show a stronger conclusion in a special case
: suppose that @xmath566 are compact such that @xmath110 is contained in an increasing set @xmath563 that is disjoint from @xmath111 ; then there exists an open @xmath560 and an open @xmath567 such that @xmath568 , @xmath569 , and @xmath556 .
indeed , since @xmath563 is increasing , for every @xmath570 , we do _ not _ have that @xmath551 , whence by hypothesis , there exist @xmath571 and an open @xmath572 with @xmath573 , @xmath574 , and @xmath575 . because @xmath111 is compact , for each @xmath576 , we may choose @xmath577 such that @xmath578 .
define @xmath579 .
then @xmath580 is open , contains @xmath256 , and is disjoint from @xmath581 , whence compactness of @xmath110 ensures the existence of @xmath582 with @xmath583 .
then @xmath584 is open , contains @xmath111 , and is disjoint from @xmath384 , as desired . to prove the general case ,
let @xmath585 and @xmath586 be the two coordinate projections on @xmath587 .
define @xmath588 for @xmath589 to be 0 if there exists @xmath560 such that @xmath590 and @xmath591 ; and to be 1 otherwise .
we claim that @xmath192 is a capacity in the sense of ( * ? ? ?
* ( 30.1 ) ) .
it is obvious that @xmath592 if @xmath593 and it is simple to check that if @xmath594 , then @xmath595 .
suppose for the final property that @xmath5 is compact and @xmath596 ; we must find an open @xmath597 for which @xmath598 .
there exists some @xmath599 with @xmath600 and @xmath601 .
then the result of the second paragraph yields sets @xmath384 and @xmath567 that give @xmath602 as desired
. now let @xmath563 and @xmath603 be as in the first paragraph .
if @xmath604 is compact , then setting @xmath605 and applying the second paragraph shows that @xmath596 .
thus , by the choquet capacitability theorem ( * ? ? ?
* ( 30.13 ) ) , @xmath606 .
l.approxincr/ clearly every set in @xmath504 is increasing and in @xmath207 .
for the converse , endow @xmath5 with a metric so that it becomes locally compact polish while preserving its class of relatively compact sets and its borel @xmath154-field : choose a denumerable partition of @xmath5 into relatively compact sets @xmath607 and make each one compact and of diameter at most 1 ; make the distance between @xmath256 and @xmath608 be 1 if @xmath256 and @xmath608 belong to different @xmath607 .
let @xmath609 with the vague topology and let @xmath501 be the class of elementary increasing events defined with respect to ( relatively compact ) sets @xmath500 that are open for this new metric .
apply t.dyck/. since @xmath610 , the result follows .
natural examples of determinantal point processes arise from orthogonal polynomials with respect to a probability measure @xmath173 on @xmath25 .
assume that @xmath173 has infinite support and finite moments of all orders .
let @xmath389 denote the orthogonal projection of @xmath611 onto the linear span @xmath612 of the functions @xmath613 .
there exist unique ( up to signum ) polynomials @xmath614 of degree @xmath18 such that for every @xmath278 , @xmath615 is an orthonormal basis of @xmath612 . by elementary row operations
, we see that for variables @xmath616 , the map @xmath617_{i , j \le n}$ ] is a vandermonde polynomial up to a constant factor , whence @xmath618 [ \phi_i(z_j)]^ * = c_n \prod_{1 \le i < j \le n } |z_i - z_j|^2\ ] ] for some constant @xmath619 . therefore , the density of @xmath620 ( with points randomly ordered ) with respect to @xmath346 is given by @xmath621 times the square of a vandermonde determinant .
classical examples include the following : 1
. if @xmath173 is gaussian measure on @xmath24 , i.e. , @xmath622 , then @xmath614 are the hermite polynomials , @xmath623 , and @xmath620 is the law of the , which is the set of eigenvalues of @xmath624 , where @xmath625 is an @xmath626 matrix whose entries are independent standard complex gaussian .
( a standard complex gaussian random variable is the same as a standard gaussian vector in @xmath627 divided by @xmath628 in order that the complex variance equal 1 .
its density is @xmath629 with respect to lebesgue measure on @xmath25 . )
this is due to wigner ; see b.mehta/. 2 .
if @xmath173 is unit lebesgue measure on the unit circle @xmath630 , then @xmath631 , so @xmath632 , and @xmath620 is the law of the , which is the set of eigenvalues of a random matrix whose distribution is haar measure on the set of @xmath626 unitary matrices .
this ensemble was introduced by dyson , but the law of the eigenvalues is due to weyl ; see b.hkpv : book/. 3 .
if @xmath173 is standard gaussian measure on @xmath25 , then @xmath633 , @xmath623 , and @xmath620 is the law of the , which is the set of eigenvalues of an @xmath626 matrix whose entries are independent standard complex gaussian .
this is due to ginibre ; see b.hkpv : book/. 4 .
if @xmath173 is unit lebesgue measure on the unit disk @xmath634 , then @xmath635 , so @xmath636 , and the limit of @xmath620 is the law of the zero set of the random power series whose coefficients are independent standard complex gaussian , which converges in the unit disk a.s .
this is due to peres and virg b.peresvirag/. 5 .
if @xmath173 has density @xmath637 with respect to lebesgue measure on @xmath25 , then @xmath638 for @xmath315 , so @xmath639 , and @xmath620 is the law of the , which is the set of eigenvalues of @xmath640 when @xmath641 are independent @xmath626 matrices whose entries are independent standard complex gaussian .
( here , we are limited to @xmath612 since the larger spaces do not lie in @xmath341 . )
this is due to krishnapur b.krishnapur:thesis/ ; see b.hkpv : book/. the process was studied earlier by b.caillol/ and b.fjm/ , but without observing the connection to eigenvalues .
inverting stereographic projection , we identify this process with one whose density with respect to lebesgue measure on the unit sphere in @xmath642 is proportional to @xmath643 . for additional information on such processes ,
see @xcite . for an extension to complex manifolds ,
see @xcite . by t.cdom/ ,
the processes @xmath620 stochastically increase in @xmath278 for each of the examples above except the last
. it would be interesting to see natural monotone couplings .
perhaps the last example also increases stochastically in @xmath278 .
the is the limit of the @xmath278th ginibre processes as @xmath417 ; it has the kernel @xmath644 with respect to standard gaussian measure on @xmath25 .
this process is invariant under all isometries of @xmath25 . for each of the plane , sphere , and hyperbolic disk ,
there is only a 1-parameter family of determinantal point processes having a kernel @xmath645 that is holomorphic in @xmath646 and in @xmath647 and whose law is isometry invariant ( * ? ? ?
* theorem 3.0.5 ) . for the sphere , that family
has already been given above ; the parameter is a positive integer . for the other two families ,
the parameter is a positive real number , @xmath648 . in the case of the plane ,
the processes are related simply by homotheties , @xmath649 .
the push - forward of the ginibre process with respect to @xmath650 has kernel @xmath651 with respect to the measure @xmath652 , where @xmath173 is lebesgue measure on @xmath25 .
do these processes increase stochastically in @xmath648 , like poisson processes do ? in the hyperbolic disk , the processes have kernel @xmath653 with respect to the measure @xmath654 , where @xmath173 is lebesgue measure on @xmath655 .
( we fix a branch of @xmath656 for @xmath657 . )
these give orthogonal projections onto the generalized bergman spaces .
the case @xmath658 is that of the limiting ope4 above .
do these processes stochastically increase in @xmath648 ?
recall that when @xmath48 is a finite - dimensional subspace of @xmath37 , the measure @xmath94 is supported by those subsets @xmath95 that project to a basis of @xmath48 under @xmath15 .
similarly , when @xmath158 is the kernel of a finite - rank orthogonal projection onto @xmath659 , define the functions @xmath660 .
then the measure @xmath320 is supported by those @xmath661 such that @xmath662 is a basis of @xmath48 , since @xmath663 .
here , @xmath664 means that @xmath665 .
the question of extending this to infinite - dimensional @xmath48 turns out to be very interesting .
a basis of a finite - dimensional vector space is a minimal spanning set .
although @xmath666 is @xmath94-a.s .
linearly independent , minimality does not hold in general , even for the wired spanning forest of a tree , as shown by the examples in b.heicklenlyons/. see also c.ell2min/. however , the other half of being a basis does hold in the discrete case and is open in the continuous case .
let @xmath667 $ ] be the closed linear span of @xmath668 .
t.basis b.lyons:det/ for every @xmath49 , we have @xmath669 = h$ ] @xmath94-a.s .
we give an application of t.basis/ for @xmath670 , but it has an analogous statement for every countable abelian group . let @xmath671 be the unit circle equipped with unit lebesgue measure
. for a measurable function @xmath672 and @xmath673 , the of @xmath172 at @xmath278 is @xmath674 .
let @xmath675 denote the restriction of @xmath676 to @xmath677 .
if @xmath678 is measurable , we say @xmath679 is @xmath5 if the set @xmath680 is dense in @xmath681 , where we identify @xmath681 with the set of functions in @xmath682 that vanish outside @xmath5 .
the case where @xmath5 is an interval is quite classical ; see b.redheffer/ for a review . a crucial role in that case is played by the following notion of density of @xmath677 .
d.bm for an interval @xmath683 \subset \bbr \setminus \ { 0 \}$ ] , define its @xmath684\big ) : = \max \big\ { |a| , |b| \big\}/ \min \big\ { |a| , |b| \big\ } \,.\ ] ] for a discrete @xmath685 , the of @xmath677 , denoted @xmath686 , is the supremum of those @xmath687 for which there exist disjoint nonempty intervals @xmath688 with @xmath689 for all @xmath278 and @xmath690 ^ 2 = \infty$ ] .
a simpler form of the beurling - malliavin density was provided by b.red:two/ , who showed that e.bmred ( s ) = \ { c : s _ k s | - | < } .
c.seqdual b.lyons:det/ let @xmath691 be lebesgue measurable with measure @xmath692 .
then there is a set of beurling - malliavin density @xmath692 in @xmath693 that is complete for @xmath5 .
indeed , let @xmath694 be the determinantal probability measure on @xmath695 corresponding to the toeplitz matrix @xmath696
. then @xmath694-a.e .
@xmath697 is complete for @xmath5 and has @xmath698 .
when @xmath5 is an interval , the celebrated theorem of beurling and malliavin b.bm/ says that if @xmath677 is complete for @xmath5 , then @xmath699 , and that if @xmath700 , then @xmath677 is complete for @xmath5 .
( this holds for @xmath677 that are not necessarily sets of integers , but we are concerned in this subsection only with @xmath679 . )
c.seqdual/ can be compared ( take @xmath701 and @xmath702 ) to a theorem of bourgain and tzafriri b.btz/ , according to which there is a set @xmath697 of ( schnirelman ) density at least @xmath703 such that if @xmath704 and @xmath676 vanishes off @xmath677 , then @xmath705 it would be interesting to find a quantitative strengthening of c.seqdual/ that would encompass this theorem of @xcite .
the following theorem is equivalent to t.basis/ by duality : t.morris b.lyons:det/ for every @xmath49 , we have @xmath706 } h } = [ \ba]$ ] @xmath94-a.s . as an example , consider the wired spanning forest of a graph , @xmath253 . here , @xmath707 .
in this case , @xmath708 } \star(g ) } = \star(b)$ ] for @xmath709 .
thus , the conclusion of t.morris/ is that @xmath710 , which equals @xmath711 , is concentrated on the singleton @xmath712 for @xmath713-a.e .
@xmath714 .
this was a conjecture of , established by b.morris/. c.ell2min for every @xmath49 , @xmath94-a.s . the maps @xmath715 \to h$ ] and @xmath716 } \colon h \to [ \ba]$ ] are injective with dense image .
both statements are equivalent to @xmath717 \cap h^\perp = \{0\ } = h \cap \ba^\perp$ ] , and these are the contents of theorems [ thm : basis ] and [ thm : morris ] . proved that on any network @xmath718 ( where @xmath253 is the underlying graph and @xmath719 is the function assigning conductances , or weights , to the edges ) , for @xmath720-a.e .
forest @xmath714 and for every component tree @xmath483 of @xmath714 , the @xmath721 of @xmath722 equals @xmath483 a.s .
this suggested b.lyons:det/ the following extension .
given a subspace @xmath48 of @xmath37 and a set @xmath95 , the subspace of @xmath723 $ ] most like " or closest to " @xmath48 is the closure of the image of @xmath48 under the orthogonal projection @xmath724}$ ] ; we denote this subspace by @xmath725 .
for example , if @xmath726 , then @xmath727 since for each @xmath728 , we have @xmath724 } ( \star_x^g ) = \star_x^b$ ] .
to say that @xmath729 is concentrated on @xmath730 is the same as to say that @xmath731 $ ] .
this motivated the following theorem and shows how it is an extension of morris s theorem .
if @xmath158 is a locally trace - class orthogonal projection onto @xmath48 , then for @xmath732 , we have @xmath733 in other words , @xmath158 is a reproducing kernel for @xmath48 .
a subset @xmath677 of @xmath48 is called if the closed linear span of @xmath677 equals @xmath48 ; equivalently , the only element of @xmath48 that is orthogonal to @xmath677 is 0 .
an analogue of t.basis/ was conjectured by lyons and peres in 2010 : g.cbasis if @xmath158 is a locally trace - class orthogonal projection onto @xmath48 , then for @xmath320-a.e .
@xmath293 , @xmath734 = h$ ] , i.e. , if @xmath735 and @xmath736 , then @xmath737 .
just as in the discrete case , this appears to be on the critical border for many special instances , as we illustrate for several processes where @xmath738 : 1 .
let @xmath173 be lebesgue measure on @xmath24 and @xmath739 , the .
denote the fourier transform on @xmath24 by @xmath740 for @xmath741 , and , by isometric extension , for @xmath742 .
write @xmath743}$ ] . since @xmath744 , we have @xmath745 , where @xmath746 is the inverse fourier transform of @xmath172 .
therefore , the induced operator @xmath158 arises from the orthogonal projection onto the paley - wiener space @xmath747 .
the sine - kernel process arises frequently ; e.g. , it is various scaling limits of the @xmath278th gaussian unitary ensemble in the bulk " as @xmath417 .
( a related scaling limit of the gue is wigner s semicircle distribution . )
we may more easily interpret g.cbasis/ for fourier transforms of functions in @xmath748 $ ] : it says that for @xmath320-a.e .
@xmath293 , the only @xmath749 $ ] such that @xmath750 is @xmath751 .
although the beurling - malliavin theorem applies , no information can be deduced because @xmath752 a.s .
however , ghosh b.ghosh/ has proved this case .
2 . let @xmath173 be standard gaussian measure on @xmath25 and @xmath753 .
this is the ginibre process .
it corresponds to orthogonal projection onto the @xmath754 consisting of the entire functions that lie in @xmath611 ; this is the space of power series @xmath755 such that @xmath756 .
completeness of a set of elements @xmath757 in @xmath754 is equivalent to completeness in @xmath758 ( with lebesgue measure ) of the gabor system of windowed complex exponentials @xmath759 \st \lambda \in \sqrt{2}\lambda\big\ } \,,\ ] ] which is used in time - frequency analysis of non - band - limited signals
. the equivalence is proved using the bargmann transform @xmath760 \,dt \big ) \,,\ ] ] which is an isometry from @xmath758 to @xmath754 . that the critical density is 1 was shown in various senses going back to von neumann ; see b.clp/. this case has also been proved by ghosh b.ghosh/. 3 .
let @xmath173 be unit lebesgue measure on the unit disk @xmath761 and @xmath762 .
this process is the limiting ope4 in s.orthogpoly/. it corresponds to orthogonal projection onto the @xmath763 consisting of the analytic functions that lie in @xmath764 .
what is known about the zero sets of functions in the bergman space b.duren/ is insufficient to settle g.cbasis/ in this case and it remains open .
the two instances above that have been proved by ghosh b.ghosh/ follow from his more general result that g.cbasis/ holds whenever @xmath173 is continuous and @xmath320 is , which means that @xmath765 is measurable with respect to the @xmath320-completion of @xmath766 for every ball @xmath767 .
the limiting process ope4 is not rigid b.hs : tolerance/. ghosh and krishnapur ( personal communication , 2014 ) have shown that @xmath320 is rigid only if @xmath158 is an orthogonal projection .
it is not sufficient that @xmath158 be a projection , as the example of the bergman space shows .
a necessary and sufficient condition to be rigid is not known .
let @xmath158 be a locally trace - class orthogonal projection onto @xmath768 .
for a function @xmath172 , write @xmath769 for the function @xmath770
. let @xmath771 .
clearly @xmath772 for a.e .
@xmath293 . also , for @xmath773
, the function @xmath774 is bounded .
a conjecture analogous to c.ell2min/ is that @xmath293 is a sort of set of interpolation for @xmath48 in the sense that given any countable dense set @xmath775 , for a.e .
@xmath293 , the set @xmath776 is dense in @xmath777 .
one may also ask about completeness for appropriate poisson point processes .
suppose @xmath778 is a group that acts on @xmath0 and that @xmath158 is @xmath778-invariant , i.e. , @xmath779 for all @xmath780 , @xmath295 , and @xmath781 .
( this is equivalent to the operator @xmath158 being @xmath778-equivariant . )
then the probability measure @xmath320 is @xmath778-invariant .
this contact with ergodic theory and other areas of mathematics suggests many interesting questions .
lack of space prevents us from considering more than just a few aspects of the case where @xmath0 is discrete and from giving all definitions .
let @xmath782 . in this case
, @xmath158 is invariant iff @xmath783 for some @xmath784 $ ] , where @xmath785 .
we write @xmath786 in place of @xmath320 . some results and questions from b.ls:dyn/
follow .
t.bern for all @xmath172 , the process @xmath786 is isomorphic to a bernoulli process .
this was shown in dimension 1 by b.shitak:ii/ for those @xmath172 such that @xmath787 by showing that those @xmath786 are weak bernoulli ( wb ) , also called @xmath788-mixing " and absolutely regular " . despite its name
, it is known that wb is strictly stronger than bernoullicity . the precise class of @xmath172 for which @xmath786 is wb
is not known . as usual ,
the of a nonnegative function @xmath172 is @xmath789 .
t.gmdom for all @xmath172 , the process @xmath786 stochastically dominates product measure @xmath790 and is stochastically dominated by product measure @xmath791 .
these bounds are optimal .
we conjecture that ( kolmogorov - sinai ) entropy is concave , as would follow from g.concave/. g.invconcave for all @xmath172 and @xmath792 , we have @xmath793 .
q.block let @xmath794 $ ] be a trigonometric polynomial of degree @xmath795 .
then @xmath786 is @xmath795-dependent , as are all @xmath796-block factors of independent processes .
is @xmath797 an @xmath796-block factor of an i.i.d .
process ?
this is known when @xmath798 b.broman/. let @xmath778 be a sofic group , a class of groups that includes all finitely generated amenable groups and all finitely generated residually amenable groups .
no finitely generated group is known not to be sofic .
let @xmath0 be @xmath778 or , more generally , a set acted on by @xmath778 with finitely many orbits , such as the edges of a cayley graph of @xmath778 .
the following theorems are from b.lyonsthom/. t.sbern for every @xmath778-equivariant positive contraction @xmath1 on @xmath37 , the process @xmath9 is a @xmath799-limit of finitely dependent ( invariant ) processes . if @xmath778 is amenable and @xmath800 , then @xmath9 is isomorphic to a bernoulli process . even if @xmath166 and @xmath167 are @xmath778-invariant probability measures on @xmath801 with @xmath168 , there need not be a @xmath778-invariant monotone coupling of @xmath166 and @xmath167 b.mester : mono/. the proof of the preceding theorem depends on the next one : t.monojoin if @xmath160 and @xmath161 are two @xmath778-equivariant positive contractions on @xmath37 with @xmath185 , then there exists a @xmath778-invariant monotone coupling of @xmath201 and @xmath202 .
the proof of t.sbern/ also uses the inequality @xmath802 for equivariant positive contractions , @xmath1 and @xmath803 , where @xmath804 is the schatten 1-norm .
when @xmath1 and @xmath803 commute , one can improve this bound to @xmath805 we do not know whether this inequality always holds .
write @xmath806 for the fuglede - kadison determinant of @xmath1 when @xmath1 is a @xmath778-equivariant operator
. the following would extend t.gmdom/. it is open even for finite groups .
g.fkdom for all @xmath778-equivariant positive contractions @xmath1 on @xmath807 , the process @xmath9 stochastically dominates product measure @xmath808 and is stochastically dominated by product measure @xmath809 , and these bounds are optimal .
it turns out that the expected degree of a vertex in the free uniform spanning forest of a cayley graph depends only on the group , via its first @xmath810-betti number , @xmath811 , and not on the generating set used to define the cayley graph b.lyons:betti/ : t.betti in every cayley graph @xmath253 of a group @xmath778 , we have @xmath812 = 2 \beta_1(\gp ) + 2 \,.\ ] ] this is proved using the representation of @xmath813 as a determinantal probability measure .
it can be used to give a uniform bound on expansion constants b.lpv/ : t.lpv for every finite symmetric generating set @xmath677 of a group @xmath778 , we have @xmath814 for all finite non - empty @xmath815 .
there are extensions of these results to higher - dimensional cw - complexes and higher @xmath810-betti numbers b.lyons : betti/. in unpublished work with d. gaboriau @xcite , we have shown the following : t.damien let @xmath253 be a cayley graph of a finitely generated group @xmath778 and @xmath816 . then there exists a @xmath778-invariant finitely dependent determinantal probability measure @xmath9 on @xmath817 that stochastically dominates @xmath818 and such that @xmath819 \le \be_\fsf\big[\deg_\fo(\bp)\big ] + \epsilon \,.\ ] ] in addition , if @xmath778 is sofic , then @xmath820 .
if it could be shown that @xmath9 , or indeed every invariant finitely dependent probability measure that dominates @xmath813 , yields a connected subgraph a.s . , then it would follow that @xmath821 is equal to the cost of @xmath778 , a major open problem of b.gaboriau : invar/. |
the study of the high - redshift progenitors of today s massive galaxies can provide us with invaluable insights into the key mechanisms that shape the evolution of galaxies in the high - mass regime .
the latest generation of galaxy formation models are now able to explain the number densities and ages of massive galaxies at high redshift . however , this is only part of the challenge , as recent studies have posed new questions about how the morphologies of massive galaxies evolve with redshift .
in addition to the basic question of how high - redshift galaxies evolve in size , there is also still much debate about how these massive galaxies evolve in terms of their fundamental morphological type .
extensive studies of the local universe have revealed a bimodality in the colour - morphology plane , with spheroidal galaxies typically inhabiting the red sequence and disk galaxies making up the blue cloud ( e.g. ( * ? ? ?
* baldry et al . 2004 ) ) .
however , recent studies at both low ( e.g. ( * ? ? ?
* bamford et al . 2009 ) ) and high redshift ( e.g. ( * ? ? ? * van der wel et al . 2011 ) ) have uncovered a significant population of passive disk - dominated galaxies , providing evidence that the physical processes which quench star - formation may be distinct from those responsible for driving morphological transformations .
this result is particularly interesting in light of the latest morphological studies of high - redshift massive galaxies by and ( * ? ? ?
* van der wel et al . ( 2011 ) ) who find that , in contrast to the local population of massive galaxies ( which is dominated by bulge morphologies ) , by @xmath6 massive galaxies are predominantly disk - dominated systems . in this work
we attempt to provide significantly improved clarity on these issues .
the candels ( ( * ? ? ?
* grogin et al . 2011 ) , ( * ? ? ?
* koekemoer et al . 2011 ) ) near - infrared f160w data provides the necessary combination of depth , angular resolution , and area to enable the most detailed study to date of the rest - frame optical morphologies of massive ( @xmath1 ) galaxies at @xmath2 in the ukidss ultra deep survey ( ( * ? ? ?
* lawrence et al . 2007 ) ) .
for this study we have constructed a sample based on photometric redshifts and stellar mass estimates which were determined using the stellar population synthesis models of ( * ? ? ?
* bruzual & charlot ( 2003 ) ) assuming a chabrier initial mass function ( see ( * ? ? ?
* bruce et al .
2012 ) for full details ) .
this provides us with a total mass - complete sample of @xmath7 galaxies .
we have employed the galfit ( ( * ? ? ?
* peng et al . 2002 ) ) morphology fitting code to determine the morphological properties for all the objects in our sample . to conduct the double component fitting we define three components : a srsic index fixed at @xmath8 bulge , an @xmath9 fixed disk and a centrally concentrated psf component to account for any agn or nuclear starbursts within our galaxies .
these three components are combined to generate six alternative multiple component model fits , of varying complexity , for every object in the sample .
these models are formally nested , and thus @xmath10 statistics can be used to determine the `` best '' model given the appropriate number of model parameters .
armed with this unparalleled morphological information on massive galaxies at high redshift we can consider how the relative number density of galaxies of different morphological type changes during the key epoch in cosmic history probed here . in fig .
1 we illustrate this by binning our sample into four redshift bins of width @xmath11 , and consider three alternative cuts in morphological classification as measured by @xmath12 from our bulge - disk decompositions . in the left - hand panel of fig . 1 we have simply split the sample into two categories : bulge - dominated ( @xmath13 ) and disk - dominated ( @xmath14 ) . in the central panel we have separated the sample into three categories , with any object for which @xmath15 classed as `` intermediate '' .
finally , in the right - hand panel we have expanded this intermediate category to encompass all objects for which @xmath16 . and using three alternative cuts in morphological classification ( both to try to provide a complete picture , and to facilitate comparison with different categorisations in the literature).,width=528 ] from these panels it can be seen that @xmath6 marks a key transition phase , above which massive galaxies are predominantly disk - dominated systems and below which they become increasingly mixed bulge+disk systems .
we also note that at the lowest redshifts probed by this study ( @xmath17 ) it is seen that , while bulge - dominated objects are on the rise , pure - bulge galaxies ( i.e. objects comparable to present - day giant ellipticals ) have yet to emerge in significant numbers , with @xmath18% of these high - mass galaxies still retaining a significant disk component .
this is compared with @xmath19 of the local @xmath1 galaxy population , which would be classified as pure - bulges from our definition ( @xmath20 , corresponding to @xmath21 ) from the sample of ( * ? ? ?
* buitrago et al . ( 2013 ) ) .
thus , our results further challenge theoretical models of galaxy formation to account for the relatively rapid demise of massive star - forming disks , but the relatively gradual emergence of genuinely bulge - dominated morphologies .
in addition to our morphological decompositions we also make use of the sed fitting already employed in the sample selection to explore the relationship between star - formation activity and morphological type .
2 shows specific star - formation rate ( @xmath22 ) versus morphological type for the massive galaxies in our sample , where morphology is quantified by single srsic index in the left - hand panel , and by bulge - to - total @xmath23-band flux ratio ( @xmath12 ) in the right - hand panel .
the values of @xmath22 plotted are derived from the original optical - infrared sed fits employed in the sample selection , and include correction for dust extinction as assessed from the best fitting value of @xmath24 derived during the sed fitting . as a check of the potential failure of this approach to correctly identify reddened dusty star - forming galaxies , we have also searched for 24@xmath25 m counterparts in the _ spitzer _ spuds mips imaging of the uds , and have highlighted in blue stars those objects which yielded a mips counterpart within a search radius of @xmath26arcsec . to first order ,
our results show that the well - documented bimodality in the colour - morphology plane seen at low redshift , where spheroidal galaxies inhabit the red sequence , while disk galaxies occupy the blue cloud is at least partly already in place by @xmath6 .
nonetheless , the sample also undoubtedly contains star - forming bulge - dominated galaxies and , perhaps more interestingly , a significant population of apparently quiescent disk - dominated objects . to highlight and quantify this population
we have indicated by a box on both the panels the region occupied by objects with disk - dominated morphologies and @xmath27 . in the left - hand panel ,
disk - dominated is defined as @xmath28 , and @xmath29% of the quiescent galaxies lie within this box ( if we exclude the 24@xmath25 m detections ) , while in the right - hand panel , disk - dominated is defined by @xmath14 , in which case @xmath30% of the quiescent objects lie within this region .
the presence of a significant population of passive disks among the massive galaxy population at these redshifts indicates that star - formation activity can cease without a disk galaxy being turned directly into a disk - free spheroid , as generally previously expected if the process that quenches star formation is a major merger .
one possible mechanism for this arises from the latest generation of hydrodynamical simulations ( e.g. ( * ? ? ?
* kere et al . 2005 ) , ( * ? ? ?
* dekel et al . 2009a ) ) and analytic theories ( e.g. ( * ? ? ?
* birnboim & dekel 2003 ) ) , which suggest a formation scenario whereby at high redshift star - formation is fed through inflows of cold gas .
another scenario which can account for star - formation quenching , whilst still being consistent with the existence of passive disks , is the model of violent disk instabilities ( e.g. ( * ? ? ?
* dekel et al . 2009b ) ) , coupled with
morphology quenching " ( ( * ? ? ?
* martig et al . 2009 ) ) . | we have used high - resolution , hst wfc3/ir , near - infrared imaging to conduct a detailed bulge - disk decomposition of the morphologies of @xmath0 of the most massive ( @xmath1 ) galaxies at @xmath2 in the candels - uds field .
we find that , while such massive galaxies at low redshift are generally bulge - dominated , at redshifts @xmath3 they are predominantly mixed bulge+disk systems , and by @xmath4 they are mostly disk - dominated .
interestingly , we find that while most of the quiescent galaxies are bulge - dominated , a significant fraction ( @xmath5% ) of the most quiescent galaxies , have disk - dominated morphologies
. thus , our results suggest that the physical mechanisms which quench star - formation activity are not simply connected to those responsible for the morphological transformation of massive galaxies . |
natural hamiltonian systems are the mathematical models of those physical systems for which the energy is constant , for example harmonic oscillators or the kepler system .
often , as in the previous two examples , more quantities are constants of the motion ( or _ first integrals _ ) : angular momentum , laplace - runge - lentz vector , etc . usually , these constants are expressed by quadratic polynomials in the momenta or , for quantum systems , by second - order differential operators .
hamiltonian systems with constants of the motion of degree higher than two are less common , nevertheless , some of them are of great interest , as for instance the three - body jacobi - calogero and wolfes systems .
these systems represent the dynamics of three point - masses on a line under forces determined by the potential functions @xmath4 respectively ( we do not consider here the harmonic oscillator terms ) and they have essentially the same dynamics @xcite . both the resulting natural hamiltonians in @xmath5 admit one linear and one quadratic in the momenta constants of the motion , making the systems liouville - integrable and solvable by separation of variables ( see @xcite and references therein ) .
other two independent constants of the motion do exist , one quadratic , due to the multiseparability of the hamiltonian , and one cubic .
the systems are then maximally superintegrable ( ms ) , having a number of functionally independent constants of the motion equal to twice the degrees of freedom , minus one ( for quantum systems , the same number of algebraically independent symmetry operators ) .
ms systems are of the greatest importance in mathematical physics , harmonic oscillators and kepler are ms and this makes them to satisfy bertrand s theorem . indeed , maximal superintegrability manifests itself , for classical systems , in the fact that all finite orbits of ms systems are closed while , for quantum systems , in the fact that the energy levels are totally degenerate @xcite .
in recent years , several techniques made possible the construction of classical and quantum hamiltonian systems , ms and not , with first integrals of arbitrarily high degree @xcite whose study , still in development , produced remarkable results in special functions , quantum algebras , canonical quantization theories @xcite . in this note it is shortly introduced the work on the topic done by claudia chanu , luca degiovanni and the author ( in short cdr ) in several joint articles . in few words ,
the extension procedure ( theorem [ teo0 ] ) adds one degree of freedom to some suitable hamiltonian @xmath6 in such a way an extra non trivial first integral , polynomial of degree @xmath7 , of the new hamiltonian do exist .
the following theorem , stated in @xcite , defines and characterizes what we intend for `` extensions '' in the particular case of natural hamiltonians on cotangent bundles of riemannian manifolds ; for a more general definition , see @xcite .
given an @xmath8-dimensional natural hamiltonian @xmath6 on the cotangent bundle of a ( pseudo)-riemannian manifold @xmath9 , let be @xmath10 and @xmath11 where @xmath12 is the hamiltonian vector field of @xmath6 , then [ teo0 ] let @xmath9 be a @xmath8-dimensional ( pseudo-)riemannian manifold with metric tensor @xmath13 . the natural hamiltonian @xmath14 on @xmath15 with canonical coordinates @xmath16 admits an extension @xmath17 in the form ( [ hamext ] ) with a first integral @xmath18 with @xmath19 given by ( [ u ] ) and @xmath20 , if and only if the following conditions hold : 1 . the functions @xmath21 and @xmath22 satisfy @xmath23 @xmath24 where @xmath25 is the hessian tensor of @xmath21 .
2 . for @xmath26 the extended hamiltonian @xmath17 and
the first integral @xmath27 are @xmath28 for @xmath29 the extended hamiltonian @xmath17 and the first integral @xmath27 are @xmath30 with @xmath31 , @xmath32 , @xmath33 and @xmath34 in @xcite it is proved that @xmath35 is functionally independent from @xmath17 , @xmath6 and any other first integral of @xmath6 in @xmath36 .
the integrability conditions of ( [ hessteo ] ) are discussed in @xcite and it is found that their complete integrability requires @xmath37 , where @xmath31 is the constant curvature of @xmath9 . then , the function @xmath20 can depend linearly on up to @xmath38 parameters and the maximal number of parameters is attained on constant curvature manifolds only .
however , non complete solutions can be found in non - constant curvature manifolds ( cdr to appear ) . from equation ( [ vteo ] ) , the expressions of the admissible potentials @xmath22 can be computed .
several examples are given in @xcite .
the particular form of @xmath19 makes possible to explicit any @xmath35 by expanding the @xmath39-th power of a binomial , obtaining @xcite @xmath40 with @xmath41}{\left ( \begin{matrix } m \cr 2k \end{matrix } \right ) \gamma^{2k}p_u^{m-2k}\left(-2m(cl+l_0)\right)^k},\ ] ] @xmath42}{\left ( \begin{matrix } m \cr 2k+1 \end{matrix } \right ) \gamma^{2k+1}p_u^{m-2k-1}\left(-2m(cl+l_0)\right)^k } , \quad m>1,\ ] ] where @xmath43 $ ] denotes the integer part and @xmath44 .
we remark that first integrals of high degree obtained in other ways than by the extension procedure @xcite can be explicitly expressed only thanks to the fact that the dynamical equations are in these cases separated in some coordinate system . as a first example of the extension procedure we consider the one - dimensional hamiltonian @xcite @xmath45 the geodesic term of the extended hamiltonian @xmath17 is @xmath46 where @xmath31 is here the constant curvature of the extended configuration manifold .
the solutions of equations ( [ hessteo ] ) and ( [ vteo ] ) are @xmath47 where @xmath48 . when @xmath49 and @xmath50 , the configuration manifold of @xmath17 is the euclidean plane , the sphere @xmath51 and the pseudosphere @xmath52 respectively , while for @xmath53 and @xmath50 , the minkowski plane , the desitter and anti - desitter manifolds , respectively . after a rescaling of the coordinate @xmath54 , the parameter @xmath39 in @xmath17 passes into @xmath6 and @xmath55
this makes evident that the extension procedure introduces some discrete symmetry into @xmath17 , in this case a dihedral symmetry of order @xmath56 , somehow connected with the extra first integral @xmath35 . in the euclidean plane ( i.e. @xmath57 ) with @xmath58 and @xmath59 ,
@xmath22 is associated with the jacobi - calogero potential or , equivalently , with the wolfes potential @xcite . indeed , in cylindrical coordinates of @xmath5 , @xmath60 , with axis @xmath61 parallel to @xmath62 w.r .
to cartesian coordinates @xmath63 , we have @xmath64 the procedure of extension provides two new functionally independent first integrals to the extended hamiltonian @xmath17 : @xmath17 itself and @xmath27 .
this fact is particularly relevant when the hamiltonian @xmath6 is ms . in this case , @xmath17 is ms too , admitting @xmath65 functionally independent first integrals . in @xcite this property of extended hamiltonians
is studied in several cases .
for example , let us consider @xmath66 that is a particular case of the generalized tremblay - turbiner - winternitz system ( ttw ) @xcite for @xmath67 and @xmath68 defined on constant - curvature manifolds of curvature @xmath69 .
this system is ms for any rational parameter @xmath70 and admits polynomial first integrals of degree related to @xmath70 @xcite . in @xcite
it is shown that @xmath6 always admits extensions of the form @xmath71 with @xmath72 , creating in this way new ms systems .
similarly , harmonic oscillators in @xmath73 , isotropic or not , can be extended into harmonic oscillators in @xmath74 @xcite .
the assumption @xmath20 can be generalized to @xmath75 .
this leads to important generalizations , introduced in @xcite , that will be developed in a paper in preparation ( cdr ) .
the extension procedure can be in this case applied with @xmath39 substituted by any positive rational @xmath76 after a suitable definition of @xmath77 , so that the generalized ttw system of above , with @xmath78 , can be written as an extension for any rational @xmath70 . to classical extended hamiltonians and their first integrals
can be associated quantum hamiltonians and symmetry operators by some procedure of quantization , usually in form of laplace - beltrami operators .
when the curvature of @xmath9 is not constant , the quantization requires additional terms ( quantum corrections ) in order to keep integrability or superintegrability . the quantum correction
is then determined by the scalar curvature and by the weyl tensor @xcite . at least in the case
@xmath79 , the simultaneous quantization of @xmath17 and @xmath80 is possible , allowing the preservation of maximal superintegrability of ms classical extended systems to the quantum limit .
this will be shown in a paper in preparation ( cdr ) . | given an n - dimensional natural hamiltonian l on a riemannian or pseudo - riemannian manifold , we call `` extension '' of l the n+1 dimensional hamiltonian @xmath0 with new canonically conjugated coordinates @xmath1 . for suitable l ,
the functions @xmath2 and @xmath3 can be chosen depending on any natural number m such that h admits an extra polynomial first integral in the momenta of degree m , explicitly determined in the form of the m - th power of a differential operator applied to a certain function of coordinates and momenta .
in particular , if l is maximally superintegrable ( ms ) then h is ms also .
therefore , the extension procedure allows the creation of new superintegrable systems from old ones . for m=2 , the extra first integral generated by the extension procedure
determines a second - order symmetry operator of a laplace - beltrami quantization of h , modified by taking in account the curvature of the configuration manifold .
the extension procedure can be applied to several hamiltonian systems , including the three - body calogero and wolfes systems ( without harmonic term ) , the tremblay - turbiner - winternitz system and n - dimensional anisotropic harmonic oscillators .
we propose here a short review of the known results of the theory and some previews of new ones . |
the effect of temperature and angular momentum on pairing properties is an interesting subject in the study of nuclear structure . because of its simplicity
, the bcs theory is often used , which offers a good description of pairing correlation in the macroscopic systems such as metallic superconductors .
it predicts a collapse of the pairing gap at @xmath0 , which signals the sharp superfluid - normal ( sn ) phase transition at finite temperature .
the bcs theory , however , ignores quantal and thermal fluctuations , which are significant in finite small systems .
therefore , it needs to be corrected for the application to finite nuclei .
various theoretical approaches have been proposed to study the effects of fluctuations on nuclear pairing @xcite .
their results show that , at zero angular momentum , thermal fluctuations smear out the sharp sn phase transition , resulting in a pairing gap , which does not collapse at finite temperature . in rotating nuclei ,
a phenomenon of temperature induced pair correlations , which reflects the strong fluctuations of the order parameter in small systems , has also been predicted @xcite .
the recent microscopic approach , called the modified bcs ( mbcs ) theory @xcite has shown , for the fist time , that the microscopic source causing the non - collapsing pairing gap is the quasiparticle - number fluctuation ( qnf ) .
recently , we proposed the self - consistent quasiparticle random - phase approximation ( scqrpa ) @xcite , which includes the qnf as well as the quantal fluctuations due to dynamic coupling to pair vibrations .
the purpose of present work is to extend this approach to finite temperature and finite angular momentum .
the pairing hamiltonian is considered , which describes a system of @xmath1 particles interacting via a pairing force with the parameter @xmath2 and rotating with angular velocity @xmath3 and a fixed angular momentum projection @xmath4 on the laboratory ( or body ) fixed @xmath5 axis : @xmath6 where @xmath7 ( @xmath8 ) is the operator that creates ( annihilates ) a particle with angular momentum @xmath9 , spin projection @xmath10 or @xmath11 , and energy @xmath12 . for simplicity , the subscripts @xmath9 label the single - particle states @xmath13 with @xmath14 0 ,
whereas @xmath15 denote the time - reversal states @xmath16 .
the particle number operator @xmath17 is defined as @xmath18 , whereas @xmath19 is the @xmath20-projection of total angular momentum .
the variational procedure is applied to minimize the expectation value of this hamiltonian in the grand canonical ensemble .
the result yields the final equations for the pairing gap , particle number and total angular momentum , which include the effect of qnf in the form @xmath21 @xmath22~ , \hspace{5 mm } m = \sum_k m_k(n_k^{+ } - n_k^{-})~ , \label{nm}\ ] ] where the quasiparticle energy @xmath23 and renormalized single - particle energy @xmath24 are given as @xmath25 @xmath26 with @xmath27 , and @xmath28 .
the expectation values @xmath29 and @xmath30 are evaluated by solving a set of coupled equations , which contain the scqrpa @xmath31 and @xmath32 amplitudes .
the qnf is given as @xmath33 , where the quasiparticle occupation numbers @xmath34 are found from the integral equations @xmath35^{2}+[\gamma_{k}^{\pm}(\omega)]^2}d\omega~ , \label{nkcoupling}\ ] ] with the mass operators @xmath36 obtained by solving the set of equations for double - time quasiparticle green s functions and those of a quasiparticle coupled with scqrpa pair vibrations .
the quasiparticle dampings are given as @xmath37 $ ] .
the proposed approach is called the ftbcs1+scqrpa theory . neglecting the coupling to scqrpa , i.e. the factors @xmath29 and @xmath30
, it becomes the ftbcs1 theory , which is different from the conventional ftbcs theory by the presence of the qnf .
the violation of particle number at zero angular momentum is approximately removed by applying the lipkin - nogami ( ln ) method .
the corresponding approaches are called the ftln1+scqrpa and ftln1 .
the numerical calculations are carried out within the @xmath38 doubly degenerate equidistant model with the number @xmath38 of levels equal to that of particles , @xmath1 , as well as for @xmath39o , @xmath40ca , @xmath41fe , and @xmath42sn .
the results obtained show that , at zero angular momentum , under the effect of qnf within the ftbcs1 ( ftln1 ) , the sharp sn phase transition predicted by the ftbcs theory is smoothed out . as the result ,
the pairing gap does not collapse at @xmath43 , but has a tail , which extends to high @xmath44 .
the dynamic coupling to the scqrpa vibrations significantly improves the agreement with the exact results for the total energies and heat capacities obtained for @xmath45 as well as those obtained @xmath41fe within the finite - temperature quantum monte carlo method @xcite [ figs .
[ fig ] ( a ) [ fig ] ( c ) ] .
however , for heavy nuclei such as @xmath42sn , the scqrpa corrections are found to be negligible in comparison with the ftbcs1 ( ftln1 ) results . for @xmath39o and @xmath40ca , the ftbcs1 pairing gaps , obtained at different @xmath4 , decreases as @xmath44 increases and do not collapses at high @xmath44 .
at @xmath4 higher than the critical value @xmath46 , where the ftbcs gap for @xmath47 disappears , there appear thermally assisted pairing correlations , in which the ftbcs1 gap reappears at a given @xmath48 , and remains finite at @xmath49 [ fig .
[ fig ] ( d ) ] .
this phenomenon is caused by the qnf within the ftbcs1 theory . at @xmath47 ,
the qnf is zero , so the ftbcs and ftbcs1 gaps are the same as functions of @xmath4 ( or @xmath3 ) , and both collapse at @xmath50 . however , with increasing @xmath44 , the ftbcs1 gaps , which are obtained at different @xmath44 , collapse at @xmath51 , and remain finite even at very high @xmath44 , whereas those given by the conventional ftbcs theory vanish at @xmath52 and @xmath53 [ figs .
( e ) and [ fig ] ( f ) ] .
0 l. g. moretto , phys .
b * 35 * , 397 ( 1971 ) ; nucl . phys .
a * 185 * , 145 ( 1971 ) .
l. g. moretto , phys .
b * 40 * , 1 ( 1972 ) .
a. l. goodman , nucl .
a * 352 * , 30 ( 1981 ) ; phys .
c * 29 * , 1887 ( 1984 ) .
r. rossignoli , p. ring , and n.d .
dang , phys .
b * 297 * , 9 ( 1992 ) .
s. frauendorf , _ et .
b * 68 * , 024518 ( 2003 ) .
n.d . dang and v. zelevinsky , phys .
c * 64 * , 064319 ( 2001 ) ; n.d .
dang and a. arima , phys . rev .
c * 67 * , 014304 ( 2003 ) ; n. d. dang , nucl .
a * 784 * , 147 ( 2007 ) .
n.q . hung and n.d .
dang , phys .
c * 76 * , 054302 ( 2007 ) , ibid . * 77 * , 029905(e ) ( 2008 ) .
s. rombouts , k. heyde , and n. jachowicz , phys .
c * 58 * , 3295 ( 1998 ) . | an approach is proposed to nuclear pairing at finite temperature and angular momentum , which includes the effects of the quasiparticle - number fluctuation and dynamic coupling to pair vibrations within the self - consistent quasiparticle random - phase approximation .
the numerical calculations of pairing gaps , total energies , and heat capacities are carried out within a doubly folded multilevel model as well as several realistic nuclei .
the results obtained show that , in the region of moderate and strong couplings , the sharp transition between the superconducting and normal phases is smoothed out , causing a thermal pairing gap , which does not collapse at a critical temperature predicted by the conventional bardeen - cooper - schrieffer s ( bcs ) theory , but has a tail extended to high temperatures .
the theory also predicts the appearance of a thermally assisted pairing in hot rotating nuclei . |
* the anomalous x - ray pulsars ( axps ) * are a group of x - ray pulsars whose spin periods fall in a narrow range ( @xmath0 s ) , whose x - ray spectra are very soft , and which show no evidence that they accrete from a binary companion ( see mereghetti 1999 for a recent review ) . these objects may be isolated neutron stars with extremely strong ( @xmath1 g ) surface magnetic fields , or they may be accreting from a `` fallback '' accretion disk .
optical measurements could potentially help discriminate between these models .
an optical counterpart to one axp , 4u 0142 + 61 , has recently been identified and shown to have peculiar optical colors ( hulleman et al .
* the radio - quiet neutron stars ( rqnss ) * are a group of compact x - ray sources found near the center of young supernova remnants .
their x - ray spectra are roughly consistent with young , cooling neutron stars , but they show no evidence for the non - thermal emission associated with `` classical '' young pulsars like the crab ( see brazier & johnston 1999 for a review )
. the x - ray spectral properties of the rqnss and the axps are similar ( see , e.g. , chakrabarty et al .
below in table 1 , the general properties of the three rqnss as our targets in the southern sky are listed .
clcccc & & @xmath2 & age & @xmath3 & + source & snr & ( kpc ) & ( @xmath4 yr ) & ( kev ) & refs + 1e 08204247 & pup a & 2.0 & 3.7 & 0.28 & 1 - 3 + 1e 16145055 & rcw 103 & 3.3 & 1 - 3 & 0.56 & 4 - 6 + 1e 12075209 & pks 120952 & 1.5 & 7 & 0.25 & 7 - 9 + + + +
our observations were made using the magellan instant camera ( magic ) on the magellan-1/walter baade 6.5-meter telescope at las campanas observatory , chile .
magic is a ccd filter photometer built by mit and cfa for the @xmath5 focus of the baade telescope .
the current detector is a 2048@xmath62048 site ccd with a 69 mas / pixel scale and a 142@xmath6142 arcsec field of view .
we used the sloan filter set , which have the following central wavelengths ( fukugita et al .
1996 ) : @xmath7=3540 ; @xmath8=4770 ; @xmath9=6230 ; @xmath10=7620 ; and @xmath11=9130 .
brazier , k.t.s .
, & johnston , s. 1999 , mnras , 303 , l1 bignami , g.f . , caraveo , g.a . , & mereghetti , s. 1992 , apj , 389 , l67 chakrabarty , d. et al .
2001 , apj , 548 , 800 fukugita , m. et al .
1996 , aj , 111 , 1748 garmire , g.p . ,
pavlov , g.g .
, & garmire , a.b .
2000 , iauc , 7350 , 2 gotthelf , e.v . ,
petre , r. , & hwang , u. 1997 , 487 , l175 helfand , d.j . , & becker , r.h .
1984 , nature , 307 , 215 hulleman , f. , kerkwijk , m.h . , & kulkarni , s.r .
2000 , nature , 408 , 689 mereghetti , s. 1999 , in the neutron star black hole connection , ed . c. kouveliotou et al . , ( dordrecht : kluwer ) mereghetti , s. , caraveo , p. , & bignami , g.f .
1992 , a & a , 263 , 172 mereghetti , s. , bignami , g.f .
, & caraveo , p.a .
1996 , apj , 464 , 842 pavlov , g. g. , zavlin , v.e . , & trmper , j. 1999 , apj , 511 , l45 petre , r. , becker , c.m . , &
winkler , p.f .
1996 , apj , 465 , l43 petre , et al . 1982 , apj , 258 , 22 seward , f.d .
1990 , apjss , 73 , 781 tuohy , i. , & garmire , g. 1980 , apj , 239 , 107 | we report on our search for the optical counterparts of the southern hemisphere anomalous x - ray pulsar 1e1048.1 - 5937 and the radio - quiet neutron stars in supernova remnants puppis a , rcw 103 , and pks 1209 - 52 .
the observations were carried out with the new mit / cfa magic camera on the magellan - i 6.5 m telescope in chile .
we present deep multiband optical images of the x - ray error circles for each of these targets and discuss the resulting candidates and limits .
# 1_#1 _ # 1_#1 _ = # 1 1.25 in .125 in .25 in |
End of preview. Expand
in Dataset Viewer.
- Downloads last month
- 20