for several isotopes beta decay is energetically forbidden but the simultaneous occurrence of two beta decays ( @xmath6 ) is allowed . this process has been observed in eleven nuclei with half - lives in the range of @xmath7  yr  @xcite . extensions of the standard model predict that also neutrinoless double beta ( @xmath8 ) decay should exist : ( a , z)@xmath9(a , z+2)@xmath10 . in this process lepton number is violated by two units and the observation would have far - reaching consequences  @xcite . it would prove that neutrinos have a majorana mass component . assuming the exchange of light majorana neutrinos , an effective neutrino mass can be evaluated by using predictions for the nuclear matrix element ( nme ) . the experimental signature of @xmath8  decay is a peak at the @xmath11-value of the decay . the two most sensitive experiments with the candidate nucleus @xmath0ge  ( @xmath12  kev  @xcite ) were heidelberg - moscow ( hdm )  @xcite and the international germanium experiment ( igex )  @xcite . they found no evidence for the @xmath8  decay of @xmath0ge  and set lower limits on the half - life @xmath3 @xmath13  yr and @xmath14  yr at 90% c.l . , respectively . part of hdm  published a claim to have observed @xmath15 @xmath8  decays  @xcite and reported @xmath3@xmath16  yr . later , pulse shape information was used to strengthen the claim  @xcite . because of inconsistencies in the latter reference pointed out recently  @xcite , the present comparison is restricted to the result of ref . @xcite .    until recently , the claim has not been scrutinized . the currently most sensitive experiments are kamland - zen  @xcite and exo-200  @xcite looking for @xmath8  decay of @xmath17xe and  @xcite employing @xmath0ge . nuclear matrix element calculations are needed to relate the different isotopes . thus the experiments using @xmath17xe can not refute the claim in a model - independent way . is able to perform a direct test using the same isotope and also using mostly the same detectors as hdm  and igex . this paper reports the @xmath8  results of phase  i of . the  experiment  @xcite is located at the laboratori nazionali del gran sasso ( lngs ) of infn in italy . high - purity germanium ( hpge ) detectors made from isotopically modified material with @xmath0ge  enriched to @xmath1886% ( @xmath19ge ) are mounted in low - mass copper supports and immersed in a 64  m@xmath20 cryostat filled with liquid argon ( lar ) . the lar serves as cooling medium and shield against external backgrounds . the shielding is complemented by 3  m of water which is instrumented with photo multipliers to detect cherenkov light generated by muons . the hpge detector signals are read out with custom - made charge sensitive amplifiers optimized for low radioactivity which are operated close to the detectors in lar . the analog signals are digitized with 100  mhz flash adcs and analyzed offline . if one of the detectors has an energy deposition above the trigger threshold ( 40 - 100  kev ) , all channels are analyzed for possible coincidences .    reprocessed @xmath21-type semi - coaxial detectors from the hdm  and  experiments were operated together with newly produced  phase  ii detectors . the latter are of bege type manufactured by canberra  @xcite . the active volume fraction  of the detectors was determined beforehand amounting to 0.87 ( 0.92 ) for the semi - coaxial ( bege ) detectors  @xcite . data acquisition started in november 2011 with eight @xmath19ge  detectors ( ang  1 - 5 from hdm  and rg  1 - 3 from igex ) , totaling a weight of 17.67  kg . five enriched  phase  ii detectors of 3.63  kg in total were deployed in july 2012 . ang  1 and rg  3 started to draw leakage current soon after their deployment , and are omitted in this analysis . one bege detector showed an unstable behavior and is omitted as well . since march 2013 , rg  2 is no longer used since it is operated below its full depletion voltage . a fraction of 5% of the data was discarded because of temperature - related instabilities . results from the data collected until may 2013 ( 492.3   live days ) are reported here . the total exposure considered for the analysis amounts to 21.6  kg@xmath1yr  of @xmath19ge  detector mass , yielding @xmath22  mol@xmath1yr of @xmath0ge  within the active volume . the offline analysis of the digitized charge pulses is performed with the software tool gelatio  @xcite and the procedure described in ref . the deposited energy is reconstructed by a digital filter with semi - gaussian shaping . events generated by discharges or due to electromagnetic noise are rejected by a set of quality cuts . the energy scale of the individual detectors is determined with @xmath23th sources once every one or two weeks . the differences between the reconstructed peak positions and the ones from the calibration curves are smaller than 0.3  kev . the energy resolution was stable over the entire data acquisition period . the gain variation between consecutive calibrations is less than 0.05%  @xcite , which corresponds to @xmath24 of the expected energy resolution ( full width half maximum , fwhm ) at @xmath25 . between calibrations , the stability is monitored by regularly injecting charge pulses into the input of the amplifiers . the energy spectrum and its decomposition into individual sources is discussed in ref . peaks from @xmath26k , @xmath27k , @xmath28bi , @xmath28pb and @xmath29tl @xmath30 rays can be identified as well as @xmath31 decays from the @xmath32ra decay chain , and @xmath33 events from @xmath34ar . all @xmath30-ray peaks are reconstructed at the correct energy within their statistical uncertainty . the energy resolution ( fwhm ) of the strongest line ( 1524.6  kev from @xmath27k ) is 4.5 ( 3.1 )  kev for the semi - coaxial ( bege ) detectors . these values are about 10% larger than the resolutions obtained from calibrations . the broadening is due to fluctuations of the energy scale between calibrations . the interpolated fwhm at @xmath25  for physics data is detector dependent and varies between 4.2 and 5.7  kev for the semi - coaxial detectors , and between 2.6 and 4.0  kev for the bege detectors . the exposure - averaged values are @xmath35  kev and @xmath36  kev , respectively . for the first time in the field of @xmath8  decay search , a blind analysis was performed in order to avoid bias in the event selection criteria . events with energies within @xmath25@xmath37  kev were not processed . after the energy calibration and the background model were finalized the window was opened except for @xmath38  kev ( @xmath39  kev ) around @xmath25  for the semi - coaxial ( bege ) detectors . after all selections discussed below had been frozen , the data in the @xmath25  region were analyzed . the validity of the offline energy reconstruction and of the event selection procedures have been cross - checked with a fully independent analysis . the signature for @xmath8  decay is a single peak at @xmath25 . furthermore , events from @xmath8  decays have a distinct topology , which allows to distinguish them from @xmath30-induced background . for @xmath8  events , energy is deposited by the two electrons , which have a short range in germanium : more than 90% of @xmath8  events are expected to deposit all energy localized within few mm@xmath20 ( single - site events , sse ) . on the other hand , most background events from @xmath30-ray interactions have energy depositions in many detectors or at different , well separated , positions ( multi - site events , mse ) . only events with an energy deposition in a single detector are accepted resulting in a background reduction by about 15% around @xmath25 , with no efficiency loss for @xmath8  decays . events in the hpge detectors are rejected if they are in coincidence within 8  @xmath40s  with a signal from the muon veto . this leads to a further background reduction by about 7% . events which are preceded or followed by another event in the same detector within 1  ms are excluded . this allows to reject background events from the @xmath28bi-@xmath28po cascade ( bipo ) in the @xmath41rn decay chain . less than 1% of the events at @xmath25  are affected by this cut . due to the low counting rate in gerda  and due to the low muon flux at lngs , the dead time due to the muon veto and bipo cuts is negligible . the detector signals are different for sse and mse , and also surface events from @xmath33 or @xmath31 decays exhibit a characteristic shape . thus , pulse shape discrimination ( psd ) techniques can improve the sensitivity .    for bege detectors , a simple and effective psd is based on the ratio of the maximum of the current pulse ( called @xmath42 ) over the energy @xmath43 the @xmath44 cut efficiency is determined from calibration data using events in the double escape peak ( dep ) of the 2615  kev @xmath30 ray from @xmath29tl . it is cross - checked with @xmath6  decays of @xmath0ge . the acceptance of signal events at @xmath25  is = @xmath45 , while only 20% of the background events at this energy survive .    for the semi - coaxial detectors , a psd method based on an artificial neural network ( ann )  @xcite is used . the signal acceptance = @xmath46 is adjusted with dep events and the uncertainty is derived from the @xmath6  spectrum and from events at the compton edge . about 55% of the background events around @xmath25  are classified as sse - like and considered for the analysis . two alternative psd methods were developed based on a likelihood ratio and on a combination of @xmath44 and the asymmetry of the current pulse ; they are used for cross - checks . the three psd methods use very different training samples and selection criteria but more than 90% of the events rejected by ann are also rejected by the two other algorithms . the half - life on @xmath8  decay is calculated as @xmath47 with @xmath48 being avogadro s constant ,  the total exposure ( detector mass @xmath1 live time ) , and @xmath49  g the molar mass of the enriched material . @xmath50 is the observed signal strength or the corresponding upper limit . the efficiency @xmath51 accounts for the fraction of @xmath0ge  atoms ( ) , the active volume fraction ( ) , the signal acceptance by psd ( @xmath52 ) , and @xmath53 . the latter is the probability that a @xmath8  decay taking place in the active volume of a detector releases its entire energy in it , contributing to the full energy peak at @xmath25 . energy losses are due to bremsstrahlung photons , fluorescence x - rays , or electrons escaping the detector active volume . monte carlo simulations yield @xmath54 ( 0.90 ) for semi - coaxial ( bege ) detectors . the gerda  background model  @xcite predicts approximately a flat energy distribution between 1930 and 2190  kev from compton events of @xmath30 rays of @xmath29tl and @xmath28bi decays , degraded @xmath31 events , and @xmath33 rays from @xmath27k and @xmath28bi . the signal region ( @xmath55 )  kev and the intervals ( @xmath56 )  kev and ( @xmath57 )  kev , which contain known @xmath30-ray peaks from @xmath29tl and @xmath28bi , respectively , are excluded in the background calculation . the net width of the window used for the evaluation of the constant background is hence 230  kev . data are grouped into three subsets with similar characteristics : ( @xmath58 ) data from the bege detectors form one set , ( @xmath59 ) the _ golden _ data set contains the major part of the data from the semi - coaxial detectors except ( @xmath60 ) two short periods with higher background levels when the bege detectors were inserted ( _ silver _ data set ) . table  [ tab : results ] lists the observed number of events in the interval @xmath61  kev for the three data sets , the number of background events in the 230  kev window and the exposure - weighted average efficiency @xmath62 over all detectors . table  [ tab : results2 ] reports the details of these events including the results from the psd analysis . the combined energy spectrum around @xmath25 , with and without the psd selection , is displayed in fig . [ fig : spectrum ] . crlrrr data set & [ kg@xmath1yr ] & @xmath62 & bkg & bi @xmath63 & cts +   + _ golden _ & 17.9 & @xmath64 & 76 & 18@xmath652 & 5 + _ silver _ & 1.3 & @xmath64 & 19 & 63@xmath66 & 1 + _ bege _ & 2.4 & @xmath67 & 23 & 42@xmath68 & 1 +   + _ golden _ & 17.9 & @xmath69 & 45 & 11@xmath652 & 2 + _ silver _ & 1.3 & @xmath69 & 9 & 30@xmath70 & 1 + _ bege _ & 2.4 & @xmath71 & 3 & 5@xmath72 & 0 +    @xmath63 in units of @xmath73  cts/(kev@xmath1kg@xmath1yr ) . seven events are observed in the range @xmath61  kev before the psd , to be compared to @xmath74 expected background counts . no excess of events beyond the expected background is observed in any of the three data sets . this interpretation is strengthened by the pulse shape analysis . of the six events from the semi - coaxial detectors , three are classified as sse by ann , consistent with the expectation . five of the six events have the same classification by at least one other psd method . the event in the bege data set is rejected by the a / e cut . no events remain within @xmath75 after psd . all results quoted in the following are obtained with psd . .[tab : results2 ] list of all events within @xmath76  kev [ cols="<,<,>,<,^ " , ]      the combined energy spectrum from all @xmath19ge  detectors without ( with ) psd is shown by the open ( filled ) histogram . the lower panel shows the region used for the background interpolation . in the upper panel , the spectrum zoomed to @xmath25  is superimposed with the expectations ( with psd selection ) based on the central value of ref . @xcite , @xmath77  yr ( red dashed ) and with the 90% upper limit derived in this work , corresponding to @xmath78  yr ( blue solid ) . ]    to derive the signal strength @xmath50 and a frequentist coverage interval , a profile likelihood fit of the three data sets is performed . the fitted function consists of a constant term for the background and a gaussian peak for the signal with mean at @xmath25  and standard deviation @xmath79 according to the expected resolution . the fit has four free parameters : the backgrounds of the three data sets and @xmath80 , which relates to the peak integral by eq . [ eq : counts ] . the likelihood ratio is only evaluated for the physically allowed region @xmath81 . it was verified that the method has always sufficient coverage . the systematic uncertainties due to the detector parameters , selection efficiency , energy resolution and energy scale are folded in with a monte carlo approach which takes correlations into account . the best fit value is @xmath82 , namely no excess of signal events above the background . the limit on the half - life is @xmath83 including the systematic uncertainty . the limit on the half - life corresponds to @xmath84  counts . the systematic uncertainties weaken the limit by about 1.5% . given the background levels and the efficiencies of table  [ tab : results ] , the median sensitivity for the @xmath85c.l . limit is @xmath86 . a bayesian calculation  @xcite was also performed with the same fit described above . a flat prior distribution is taken for @xmath80 between 0 and @xmath87  yr@xmath88 . the toolkit bat  @xcite is used to perform the combined analysis on the data sets and to extract the posterior distribution for @xmath3  after marginalization over all nuisance parameters . the best fit is again @xmath82 and the 90% credible interval is @xmath89  yr ( with folded systematic uncertainties ) . the corresponding median sensitivity is @xmath90  yr . the gerda  data show no indication of a peak at @xmath25 , i.e.  the claim for the observation of @xmath8  decay in @xmath0ge  is not supported . taking @xmath3  from ref . @xcite , @xmath91 decays are expected  ( see note   from ref . @xcite for the active exposure ( @xmath1 ) . all other efficiency factors of eq . ( [ eq : eff ] ) approximately cancel with the exception of . the expected number of events after the psd selection is 6.8 ( 6.5 in @xmath92 ) . the difference with respect to the value calculated from @xmath3  is due to the efficiency factor @xmath53 , which is taken to be 100% in ref .  @xcite . ] ) in @xmath93 and @xmath94 background events after the psd cuts , as shown in fig . [ fig : spectrum ] . this can be compared with three events detected , none of them within @xmath95 . the model ( @xmath96 ) , which includes the claimed @xmath8  signal from ref . @xcite , gives in fact a worse fit to the data than the background - only model ( @xmath97 ) : the bayes factor , namely the ratio of the probabilities of the two models , is @xmath98 . assuming the model @xmath96 , the probability to obtain @xmath82 as the best fit from the profile likelihood analysis is @xmath99=0.01 . the gerda  result is consistent with the limits by hdm  and igex . the profile likelihood fit is extended to include the energy spectra from hdm  ( interval 2000 - 2080  kev ; fig .  4 of ref . @xcite ) and igex  ( interval 2020 - 2060  kev ; table  ii of ref . constant backgrounds for each of the five data sets and gaussian peaks for the signal with common @xmath80 are assumed . experimental parameters ( exposure , energy resolution , efficiency factors ) are obtained from the original references or , when not available , extrapolated from the values used in gerda . the best fit yields @xmath82 and a limit of @xmath100 the bayes factor is @xmath101 ; the claim is hence strongly disfavored . limits ( 90% c.l . ) on @xmath3  of @xmath0ge  ( this work ) and @xmath17xe  @xcite compared with the signal claim for @xmath0ge  of ref . @xcite ( 68% c.l the lines in the shaded gray band are the predictions for the correlation of the half - lives in @xmath17xe and in @xmath0ge  according to different nme calculations  @xcite . the selection of calculations and the labels are taken from ref . @xcite . ]    whereas only @xmath0ge  experiments can test the claimed signal in a model - independent way , nme calculations can be used to compare the present @xmath0ge  result to the recent limits on the @xmath17xe half - life from kamland - zen  @xcite and exo-200  @xcite . fig .  [ fig : comparison ] shows the experimental results , the claimed signal ( labeled `` claim ( 2004 ) '' ) and the correlations for different predictions , assuming that the exchange of light majorana neutrinos is the leading mechanism . within this assumption , the present result can be also combined with the @xmath17xe experiments to scrutinize ref . the most conservative exclusion is obtained by taking the smallest ratio @xmath102(@xmath17xe)/@xmath102(@xmath0ge)@xmath103  @xcite of the calculations listed in ref . this leads to an expected signal count of 23.6@xmath655.6 ( 3.6@xmath650.9 ) for kamland - zen ( exo-200 ) . the comparison with the corresponding background - only models  21.5 counts . the equivalent observed counts are -1.17@xmath104 lower , i.e.  about 435 events  @xcite . exo-200 expects @xmath105 counts in the interval @xmath106 and observes 5 events . ] yields a bayes factor @xmath107 of 0.40 for kamland - zen and 0.23 for exo-200 . including the  result , the bayes factor becomes 0.0022 . also in this case the claim is strongly excluded ; for a larger ratio of nmes the exclusion becomes even stronger . note , however , that other theoretical approximations might lead to even smaller ratios and thus weaker exclusions . the range for the upper limit on the effective electron neutrino mass @xmath108 is 0.2 - 0.4  ev . this limit is obtained by using the combined @xmath0ge  limit of eq . [ eq : allge ] , the recently re - evaluated phase space factors of ref . @xcite and the nme calculations mentioned above  @xcite . scaling due to different parameters @xmath109 and @xmath110 for nme is obeyed as discussed in ref . @xcite .    in conclusion , due to the unprecedented low background counting rate and the good energy resolution intrinsic to hpge detectors , establishes after only 21.6  kg@xmath1yr  exposure the most stringent @xmath8  half - life limit for @xmath0ge . the long - standing claim for a @xmath8  signal in @xmath0ge  is strongly disfavored , which calls for a further exploration of the degenerate majorana mass scale . this will be pursued by gerda  phase  ii aiming for a sensitivity increased by a factor of about 10 . the gerda  experiment is supported financially by the german federal ministry for education and research ( bmbf ) , the german research foundation ( dfg ) via the excellence cluster universe , the italian istituto nazionale di fisica nucleare ( infn ) , the max planck society ( mpg ) , the polish national science centre ( ncn ) , the foundation for polish science ( mpd programme ) , the russian foundation for basic research ( rfbr ) , and the swiss national science foundation ( snf ) . the institutions acknowledge also internal financial support . 38ifxundefined [ 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 @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * , ( ) link:\doibase 10.1103/physrevc.81.032501 [ * * ,   ( ) ] @noop * * , ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop @noop ( ) ,  ,   @noop * * ,   ( ) @noop * * , ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop ( ) ,  ,   @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop ( ) ,   @noop ( ) ,   @noop * * ,   ( ) link:\doibase 10.1103/physrevc.81.0285021 [ * * ,   ( ) ] @noop * * ,   ( )

neutrinoless double beta decay is a process that violates lepton number conservation .
 it is predicted to occur in extensions of the standard model of particle physics .
 this letter reports the results from phase  i of the germanium detector array ( gerda ) experiment at the gran sasso laboratory ( italy ) searching for neutrinoless double beta decay of the isotope @xmath0ge .
 data considered in the present analysis have been collected between november 2011 and may 2013 with a total exposure of 21.6  kg@xmath1yr .
 a blind analysis is performed .
 the background index is about @xmath2  cts/(kev@xmath1kg@xmath1yr )  after pulse shape discrimination .
 no signal is observed and a lower limit is derived for the half - life of neutrinoless double beta decay of @xmath0ge , @xmath3@xmath4  yr ( 90  % c.l . ) . the combination with the results from the previous experiments with @xmath0ge  yields @xmath3@xmath5  yr ( 90  % c.l . ) .


many structural properties of _ a_-si , such as defect concentration and variation in mean bond angle , are difficult to determine experimentally . this is because it is impossible to measure directly the coordinates of the atoms in _ a_-si . however , important information on the structure of _ a_-si can be obtained indirectly , through a number of experimental techniques . these techniques include neutron , x - ray and raman scattering , electron - spin resonance , and x - ray photo - absorption . compared to other methods , raman scattering is more sensitive to small changes in the short - range order of _ a_-si . for this reason , raman measurements on _ a_-si are frequently used to obtain structural information  @xcite . the experimental raman spectra of _ a_-si show two distinct peaks , at about 150 @xmath2 and 480 @xmath2 , associated with the transverse acoustic ( ta ) and the transverse optic ( to ) vibrational modes , respectively . certain features in the raman spectrum are highly sensitive to the structural properties of the _ a_-si sample . for example , the width of the to peak is related to the root - mean - square bond - angle variation @xmath1 in the amorphous network@xcite .    in several computational studies@xcite , the relation between @xmath0 and @xmath1 was quantified . all studies indicate a broadening of the to peak with increasing @xmath1 , but there is no quantitative agreement . beeman s linear relation , @xmath3 , which dates back to 1985 , is often used by experimentalists to determine @xmath1 from raman measurements . here , @xmath0 is in @xmath2 and @xmath1 in degrees . beeman derived his relation using nine structural models of _ a_-si . of these models , five were generated from the same 238-atom , hand - built model by connell and temkin@xcite , which contains only even - membered rings . in contrast , all simulations on _ a_-si find an abundance of five- and seven - fold rings . moreover , these five connell - temkin models are statistically dependent and not periodic , consequently containing a large fraction of surface atoms . experimental values of @xmath1 , based on the radial distribution function of _ a_-si obtained in neutron - diffraction studies , range from 9.9 to 11.0 degrees@xcite . of the nine structural models used by beeman , only three exhibit values of @xmath1 in this range . new techniques to generate _ a_-si structures , such as art  @xcite , as well as more powerful computers , have made it possible to generate larger and more realistic _ a_-si systems via computer simulation . also the description of the raman scattering process has improved . beeman used the bond polarizibility model proposed by alben _ et al._@xcite , which dates back to 1975 . characteristic for this model is the inclusion of three weighting parameters , whose values must be set somewhat arbitrarily . several studies have indicated that the values originally proposed by alben yield an incorrect value for the depolarization ratio  @xcite . these studies therefore propose different weights . since then , other polarizibility models have been proposed , for example by marinov and zotov  @xcite .    in this manuscript , we will re - investigate the relation between @xmath0 and @xmath1 by computer simulation . this simulation is based on a large number of 1000-atom , periodic configurations , with structural properties ( radial distribution function , spread in mean bond angle ) that are in excellent agreement with experiment . furthermore , recent advances in neutron scattering techniques have made it possible to directly compare the bond polarizibility models to experiment  @xcite . we therefore also include a detailed comparison of the model of alben and the model of marinov and zotov to experiment .    additionally , we present two other methods to obtain structural information from the raman spectrum . the ta / to intensity ratio  @xcite and the location of the to - peak  @xcite are believed to be directly related to @xmath1 ; these relations will be quantified . the outline of this paper is as follows . in section  [ sect : method ] , we explain the generation of the _ a_-si configurations used in this study . we then discuss how the raman spectrum is obtained from these configurations . the results and conclusions are presented in sections [ sect : results ] and [ sect : conclusions ] , respectively . to calculate raman spectra , three ingredients are required : ( 1 ) a potential describing the atomic interactions in the sample , ( 2 ) a continuous random network representing a realistic sample of _ a_-si , and ( 3 ) a model assigning raman activities to the vibrational eigenmodes of the sample .    in the present work , we use a modified version of the stillinger - weber potential for all calculations . this potential has the same functional form as the original sw potential  @xcite , but with different parameters . the parameters were chosen specifically to describe _ a_-si , see ref . @xcite .      to study the effect of @xmath1 on the raman spectrum , we require a number of _ a_-si configurations with varying values of @xmath1 . these configurations were generated using the activation - relaxation technique ( art )  @xcite . as was shown in previous studies  @xcite , art yields structures in good agreement with experiment . they display a low density of coordination defects , a narrow bond - angle distribution and an excellent overlap with the experimental radial distribution function ( rdf ) . the method is outlined below :    1 . initially , 1000 atoms are placed at random in a periodic cubic cell ; the configuration is then relaxed at zero pressure . 2 .   the configuration is annealed using art . one art move consists of two steps : ( 1 ) the sample is brought from a local energy minimum to a nearby saddle - point ( activation ) , and ( 2 ) then relaxed to a new minimum with a local energy minimization scheme including volume optimization , at zero pressure . the new minimum is accepted with a metropolis probability at temperature @xmath4 ev . 3 .   every 50 art moves , up to approximately five art moves per atom when the energy has reached a plateau , the configuration is stored . for 1000-atom samples , this procedure yields a set containing 100 samples . this procedure is repeated nine times , generating nine statistically independent sets of 100 correlated configurations each . for each set , we found that @xmath1 ranges from approximately 10@xmath5 for the well - annealed configurations , to approximately 14@xmath5 for the poorly annealed configurations . we focus on the reduced raman spectrum @xmath6 , with thermal and harmonic oscillator factors removed . this spectrum is a function of frequency @xmath7 and of the form @xmath8 where @xmath9 is the vibrational density of states ( vdos ) and @xmath10 a coupling parameter , which depends on frequency and on the polarization ( hh or hv ) of the incident light used in the raman experiment .    to calculate the vdos , the hessian is calculated . diagonalization of the hessian gives the frequencies of the vibrational modes , from which the vdos is obtained . the function @xmath10 is obtained from the polarizibility tensor @xmath11 . in terms of @xmath11 , the coupling parameter for hh and hv raman scattering becomes @xmath12 and @xmath13 , respectively , with @xmath14 and @xmath15 given by the tensor invariants @xmath16   \\\ ] ] and @xmath17 + \\        & \frac{1}{2 } & \left [ ( \alpha_{11}-\alpha_{22})^2 +                              ( \alpha_{22}-\alpha_{33})^2 +                             ( \alpha_{33}-\alpha_{11})^2 \right ] ; \nonumber\end{aligned}\ ] ] see for instance ref . @xcite . the form of the polarizibility tensor @xmath11 still needs to be specified ; this is the most uncertain part of the calculation . several models have been proposed , amongst which the commonly used model of alben _ et al . _ @xcite and the more recent model of marinov and zotov  @xcite .    in the model of alben , a cylindrical symmetry of the individual bonds is assumed and each bond is treated independently as a homopolar , diatomic molecule . three different forms for the bond polarizibility tensor are introduced : @xmath18 , \\ \alpha_2(\omega_p ) & = & \sum_{l,\delta } \vec{u}_l \cdot \vec{r}_\delta      \left [ \frac{1}{2 } \left ( \vec{r}_\delta\vec{u}_l + \vec{u}_l\vec{r}_\delta      \right ) - \frac{1}{3 } { \bf i } \right ] , \\ \alpha_3(\omega_p ) & = & \sum_{l,\delta } ( \vec{u}_l \cdot \vec{r}_\delta ) { \bf i}. \end{aligned}\ ] ] here , the summation runs over all atoms @xmath19 in the sample and their nearest neighbors @xmath20 , @xmath21 is the unit vector from the equilibrium position of atom @xmath19 to the nearest neighbor @xmath20 , @xmath22 is the displacement vector of atom @xmath19 when it is vibrating in mode @xmath23 and @xmath24 is the unit dyadic . the total polarizibility tensor @xmath25 is a weighted sum of the three terms , i.e. @xmath26 . as was stated in the introduction , the precise choice of the weights @xmath27 is somewhat arbitrary and this is the major shortcoming of the model . several studies have indicated that mechanisms 1 and 3 provide the main contribution to the raman scattering process ; these propose to use @xmath28 proportional to @xmath29 , respectively  @xcite . in this paper , we will use this set of weights . the model of marinov and zotov ( mz ) has no free parameters . in this model , the bond polarizibility is expressed as a sum of three components ; two components parallel to the bond arising from bonding and non - bonding electrons and a third component perpendicular to the bond , see ref  @xcite . under these assumptions , the polarizibility tensor takes the form : @xmath30 . \end{aligned}\ ] ] here , the summation runs over all bonds @xmath31 in the sample , @xmath32 is a unit vector parallel to the bond , @xmath33 is the bond length and @xmath34 is defined as @xmath35 ; where @xmath36 and @xmath37 are the displacement vectors of atoms @xmath38 and @xmath39 constituting the @xmath31th bond , when vibrating in mode @xmath23 . the coupling parameter for _ a_-si has also been determined experimentally  @xcite through neutron scattering methods . according to this experiment , the coupling parameter is a slowly increasing function of frequency . we will use the experimental result of ref . @xcite to test the validity of both the model of alben and the mz model . first , in section  [ sect : coupling ] , we compare the polarizibility models of alben and marinov and zotov to experiment . in the subsequent sections , we investigate the relation between the spread in the bond angle @xmath1 and :    1 . raman to peak width , 2 . raman to peak position , and 3 . raman to / ta intensity ratio . we show results for hv polarized light only ; this is the usual experimental situation . results for hh polarized light have also been obtained and are available upon request . the solid curves in fig .  [ fig : coupling ] show the hv raman coupling parameter for _ a_-si calculated using the model of alben ( top ) and the mz model ( bottom ) for the bond polarizibility . the experimental result of ref . @xcite is also shown ( dashed ) . for this calculation , we used a well - annealed , 1000-atom configuration with @xmath40 , since this will most closely resemble the experimental sample . we have checked that the general features of the curves in fig . [ fig : coupling ] do not depend on the details of the configuration used : a number of other , well - annealed , configurations , with @xmath1 ranging from 10.0@xmath41 to 11.0@xmath41 , gave similar results . [ fig : coupling ] shows that both models yield an increasing coupling parameter for frequencies up to around 500 @xmath2 . this is in qualitative agreement with experiment . for higher frequencies , the model calculations predict a sharp decrease in the coupling parameter . this is not confirmed by experiment . the quantitative agreement with experiment is rather poor , especially in the low - frequency regime ; both models provide substantially less activity in this regime than observed in experiment . for this reason , raman spectra calculated using either of the two models yield ta peak amplitudes far below experimental values . this , in our opinion , is their major shortcoming . this point is further illustrated in the top graph of fig . [ fig : spectra ] , where we show the raman spectrum calculated using the mz model ( solid ) and an experimental spectrum ( dashed ) taken from ref .  @xcite . the experimental spectrum was obtained from ion - implanted _ a_-si which had been annealed at 500@xmath5c for two hours . agreement between model and experiment , particularly in the low - frequency regime , is poor . given the overall poor performance of both the model of alben and the mz model , it may be feasible to follow a semi - experimental approach in which a computer generated _ a_-si sample is used to calculate the vdos and experimental data is used to describe the coupling parameter . this approach is justified because several studies have indicated that changes in the raman spectrum are due to changes in the vdos and not to changes in the coupling parameter  @xcite . the results of this approach are illustrated in the bottom graph of fig . [ fig : spectra ] . here , we show the raman spectrum obtained by multiplying a computer - generated vdos with experimental coupling parameter data ( solid ) . to calculate the vdos , we used a well - annealed _ a_-si sample with @xmath40 ; the experimental coupling parameter was taken from ref . the dashed line shows again the experimental raman spectrum taken from ref . agreement with experiment has improved substantially . of the two models considered here , the mz model provides slightly more activity in the low - frequency regime than the model of alben ; comparison with experiment would therefore favor the mz model . however , given the overall poor performance of both models , we will also show in the following sections results obtained using the semi - experimental approach . the intensity of the raman to peak decreases abruptly on the high frequency side , but not on the low frequency side . beeman therefore defines @xmath0 as twice the half - width at half the maximum height on the high frequency side of the to peak , as a meaningful parameter to specify the to peak width  @xcite . in this paper , we use the same definition .    the solid lines in fig . [ fig : pwhv ] show the relation between @xmath0 and @xmath1 for hv polarized light , derived using the model of alben ( top ) , the mz model ( middle ) and the semi - experimental approach ( bottom ) . also shown is the result obtained by beeman ( dashed ) . both the model of alben and the mz model produce similar results ; linear least square fits yield the equations @xmath42 and @xmath43 , respectively . compared to the result of beeman , @xmath44 , we see agreement on the sensitivity ( i.e.  slope of the lines ) of @xmath0 to @xmath1 , but not on the overall offset ( i.e.  intercepts of the lines ) . the same holds for the result obtained in the semi - experimental approach ; least square fitting yields @xmath45 in that case .      as another way to obtain structural information on _ a_-si from its raman spectrum , we investigate the relation between the to peak frequency ( @xmath46 ) and @xmath1 . fig . [ fig : to ] shows the relation between @xmath46 and @xmath1 for hv polarized light , derived using the model of alben ( top ) , the mz model ( middle ) and the semi - experimental approach ( bottom ) .    according to fig . [ fig : to ] , @xmath46 shifts to higher frequency as @xmath1 decreases . however , agreement with experiment for both the model of alben and the mz model is poor . experimental raman spectra of well - annealed _ a_-si samples yield @xmath46 in the order of 480 @xmath2  @xcite . for well - annealed configurations , for which @xmath1 ranges from 9.9 to 11.0 degrees , the models exceed the experimental value by around 20 @xmath2 . the semi - experimental approach is in much better agreement with experiment ; a linear fit yields the equation @xmath47 which for @xmath40 leads to @xmath48 @xmath2 . the cause of this is that in the model of alben and the mz model , @xmath46 is determined by the peak in the coupling parameter , whereas in the semi - experimental approach , it is determined by the vdos . next , we confirm that the ta / to intensity ratio is directly related to @xmath1 . fig . [ fig : ta ] shows the relation between reduced raman ta / to intensity and @xmath1 for hv polarized light for the model of alben ( top ) , the mz model ( middle ) and the semi - experimental approach ( bottom )    from fig . [ fig : ta ] , we see that the ta / to intensity ratio increases with increasing @xmath1 . the increase is approximately linear . hv raman experiments on well - annealed _ a_-si samples yield a reduced ta / to intensity ratio around 0.11  @xcite . for both the model of alben and the mz model , the hv ta / to ratio is of order @xmath49 , i.e.  one order of magnitude below the experimental value . this is consistent with the earlier finding that both models underestimate the raman activity in the low - frequency regime of the spectrum , see section  [ sect : coupling ] . the semi - experimental approach , on the other hand , yields a ta / to ratio of 0.14 for @xmath40 , in better agreement with experiment . we have generated nine independent sets of 1000-atom samples of _ a_-si that display a variety of short - range order ; the spread in nearest - neighbor bond angles ranges from 10 to 14 degrees . for these samples , the hv raman spectra are calculated . to describe the raman scattering process , we have used the earlier bond polarizibility model of alben _ et al . _ , the more recent model of marinov and zotov as well as experimental data taken from ref . @xcite .    comparison to experiment shows that both the model of alben and the mz model greatly underestimate the raman activity in the low - frequency regime of the spectrum . this makes these models less suitable to describe low - frequency features of the raman spectrum , for instance the ta peak . of the two models considered here , the mz model is closer to experiment . however , for a more accurate calculation of raman spectra , we propose a semi - experimental approach . in this approach , the vdos is obtained in computer simulation and experimental data is used to describe the coupling parameter .    as ways to obtain structural information on _ a_-si from its raman spectrum , we have investigated the relation between the to peak - width @xmath0 and @xmath1 , as well as the relations between the to peak frequency and the ta / to intensity ratio as functions of @xmath1 .    according to our results , where we used the semi - experimental approach , @xmath0 and @xmath1 are related by @xmath45 for hv polarized light . here , @xmath0 is in @xmath2 and @xmath1 in degrees . comparing this to the result of beeman ( @xmath50 ) , we find that our result is similar . our results also show a shift of the raman to peak frequency ( @xmath46 ) towards higher frequency , as @xmath1 decreases . in the semi - experimental approach , a linear least - square fit yields @xmath47 for hv polarized light . here , @xmath46 is in @xmath2 and @xmath1 in degrees . according to this equation , the shift of @xmath46 is approximately 7.5 @xmath2 , going from unannealed _ a_-si @xmath51 to annealed _ a_-si @xmath52 . this is in quantitative agreement with experiment  @xcite . finally , we have shown that the reduced raman ta / to intensity ratio @xmath53 is directly related to @xmath1 ; @xmath54 decreases linearly with decreasing @xmath1 . using the semi - experimental approach , we obtain the relation @xmath55 , where @xmath1 is in degrees .

in 1985 , beeman , tsu and thorpe established an almost linear relation between the raman transverse - optic ( to ) peak width @xmath0 and the spread in mean bond angle @xmath1 in _ a_-si .
 this relation is often used to estimate the latter quantity in experiments . in the last decade
 , there has been significant progress in the computer generation of sample networks of amorphous silicon . exploiting this progress
 , this manuscript presents a more accurate determination of the relation between @xmath0 and @xmath1 using 1000-atom configurations . also investigated and quantified
 are the relations between the to peak frequency and the ratio of the intensities of the transverse - acoustic ( ta ) and to peak , both as functions of @xmath1 .
 as @xmath1 decreases , the ta / to intensity ratio decreases and the to peak frequency increases .
 these relations offer additional ways to obtain structural information on _
 a_-si from raman measurements .


the paradigm of observables defined on an infinite hilbert space being mutually incompatible in quantum mechanics is provided by the heisenberg commutation relations for the position and momentum operators . the associated heisenberg group in connection with the corresponding weyl algebra of phase - space translations is still relevant for systems with a finite number of orthogonal states , providing a basis of the space @xmath1 . as first studied by schwinger , for each dimension @xmath5 there is a set of unitary operators which give rise to a discrete equivalent of the heisenberg - weyl group @xcite . we may somehow a priori expect that the kinematics of composite quantum systems will not depend on the dimensions of their building blocks . that is , that composite systems let us say consisting in the tensor product of two different hilbert spaces ( differing only in the corresponding dimensions ) will undergo a similar evolution since they are structurally identical . mathematically , the previous fact would imply the state spaces @xmath1 to be structurally identical , at least with respect to properties closely related to the heisenberg - weyl group . however , it is surprising that the aforementioned group allows one to construct @xmath6 so - called mutually unbiased ( mu ) bases of the space @xmath1 if @xmath7 is the power of a prime number @xcite , whereas the construction fails in all other dimensions . in point of fact , no other successful method to construct @xmath6 mu bases in all dimensions is known @xcite .    given @xmath8 orthonormal bases in the space @xmath9 , they are _ mutually unbiased _ if the moduli of the scalar products among the @xmath10 basis vectors take these values :    @xmath11    where @xmath12 . mu bases have useful applications in many quantum information processing . such ( complete ) sets of mu bases are ideally suited to reconstruct quantum states @xcite while sets of up to @xmath6 mu bases have applications in quantum cryptography @xcite and in the solution of the mean king s problem @xcite . even for @xmath4 , we do not know whether there exist four mu bases or not @xcite . hence the research on the maximum number of bases for @xmath4 and construction of mu bases in @xmath13 is of great importance . the issue of mu bases constitutes another part in the field of quantum information theory that is involved in pure mathematics , such as number theory , abstract algebra and projective algebra . the methods to construct complete sets of mu bases typically deal with all prime or prime - power dimensions . they are constructive methods and effectively lead to the same bases . two ( or more ) mu bases thus correspond to two ( or more ) unitary matrices , one of which can always be mapped to the identity @xmath14 of the space @xmath9 , using an overall unitary transformation . it then follows from the conditions ( [ mub ] ) that the remaining unitary matrices must be complex hadamard matrices : the moduli of all their matrix elements equal @xmath15 . this representation of mu bases links their classification to the classification of complex hadamard matrices @xcite .    in this paper , we choose a different method to study mu bases in dimension six or any other dimension @xmath7 . we will approach the problem by directly exploring the unitary matrices  randomly distributed , but according to the haar measure  whose columns vectors constitute the bases elements , which must fulfill a series of requirements concerning their concomitant bases being unbiased . the overall scenario reduces to a simple though a bit involved optimization procedure . in point of fact , we shall perform a two - fold search employing i ) an amoeba optimization procedure , where the optimal value is obtained at the risk of falling into a local minimum and ii ) the so called simulated annealing @xcite well - known search method , a monte carlo method , inspired by the cooling processes of molten metals . the advantage of this duplicity of computations is that we can be absolutely confident about the final result reached . indeed , the second recipe contains a mechanism that allows a local search that eventually can escape from local optima . this paper is organized as follows . in section ii we describe the generation of unitary matrices according to their natural haar measure . section iii explains how the optimization is performed and the concomitant results are shown in section iv . finally , some conclusions are drawn in section v. the applications that have appeared so far in quantum information theory , in the form of dense coding , teleportation , quantum cryptography and specially in algorithms for quantum computing ( quantum error correction codes for instance ) , deal with finite numbers of qubits . a quantum gate which acts upon these qubits or even the evolution of that system is represented by a unitary matrix @xmath16 , with @xmath17 being the dimension of the associated hilbert space @xmath18 . the state @xmath19 describing a system of @xmath20 qubits is given by a hermitian , positive - semidefinite ( @xmath21 ) matrix , with unit trace . in view of these facts , it is natural to think that an interest has appeared in the _ quantification _ of certain properties of these systems , most of the times in the form of the characterization of a certain state @xmath19 , described by @xmath21 matrices of finite size . natural applications arise when one tries to simulate certain processes through random matrices , whose probability distribution ought to be described accordingly . this enterprise requires a quantitative measure @xmath22 on a given set of matrices . there is one natural candidate measure , the * haar measure * on the group @xmath23 of unitary matrices . in mathematical analysis , the haar measure @xcite is known to assign an  invariant volume " to what is known as subsets of locally compact topological groups . here we present the formal definition @xcite : given a locally compact topological group @xmath24 ( multiplication is the group operation ) , consider a @xmath25-algebra @xmath26 generated by all compact subsets of @xmath24 . if @xmath27 is an element of @xmath24 and @xmath28 is a set in @xmath26 , then the set @xmath29 @xmath30 @xmath31 @xmath32 also belongs to @xmath26 . a measure @xmath22 on @xmath26 will be letf - invariant if @xmath33 for all @xmath27 and @xmath28 . such an invariant measure is the haar measure @xmath22 on @xmath24 ( it happens to be both left and right invariant ) . in other words @xcite , the haar measure defines the unique invariant integration measure for lie groups . it implies that a volume element d@xmath34 is identified by defining the integral of a function @xmath35 over @xmath24 as @xmath36 , being left and right invariant    @xmath37    the invariance of the integral follows from the concomitant invariance of the volume element d@xmath34 . it is plain , then , that once d@xmath34 is fixed at a given point , say the unit element @xmath38 , we can move performing a left or right translation . we do not gain much physical insight with these definitions of the haar measure and its invariance , unless we identify @xmath24 with the group of unitary matrices @xmath23 , the element @xmath27 with a unitary matrix @xmath39 and @xmath28 with subsets of the group of unitary matrices @xmath23 , so that given a reference state @xmath40 and a unitary matrix @xmath41 , we can associate a state @xmath42 to @xmath40 . physically what is required is a probability measure @xmath22 invariant under unitary changes of basis in the space of pure states , that is ,    @xmath43    these requirements can only be met by the haar measure , which is rotationally invariant .    now that we have justified what measure we need , we should be able to generate random matrices according to such a measure in arbitrary dimensions . the theory of random matrices @xcite specifies different _ ensembles _ of matrices , classified according to their different properties . in particular , the circular unitary ensemble ( cue ) consists of all matrices with the ( normalized ) haar measure on the unitary group @xmath23 . the circular orthogonal ensemble ( coe ) is described in similar terms using orthogonal matrices , and it was useful in order to describe the entanglement features of two-_rebits _ systems . given a @xmath21 unitary matrix @xmath39 , the minimum number of independent entries is @xmath44 . this number should match those elements that need to describe the haar measure on @xmath23 . this is best seen from the following reasoning . suppose that a matrix @xmath39 is decomposed as a product of two ( also unitary ) matrices @xmath45 . in the vicinity of @xmath26 , we have @xcite @xmath46 , where @xmath47 is a hermitian matrix with elements @xmath48 . then the probability measure nearby @xmath49 is @xmath50 , which accounts for the number of independent variables . such measure for cue is invariant @xcite and therefore proportional to the haar measure . yet , the aforementioned description is not useful for practical purposes . we need to parameterize the unitary matrices according to the haar measure . according to the parameterization for cue dating back to hurwitz @xcite using euler angles , the basic assumption is that an arbitrary unitary matrix can be decomposed into elementary two - dimensional transformations , denoted by @xmath51 :    @xmath52    using these elementary rotations we define the composite transformations    @xmath53    we finally form the matrix    @xmath54    with the angles parameterizing the rotations    @xmath55    the ensuing ( normalized ) haar measure @xcite    @xmath56 d\psi_{rs } \cr & & \prod_{1 < s \le n } d\chi_{1s}\end{aligned}\ ] ]    provides us with a random matrix belonging to cue . finally , we randomly generate the angles ( [ angles ] ) _ uniformly _ and obtain the desired random matrix @xmath39 ( [ u ] ) . let us formulate the problem of having @xmath0 orthonormal bases @xmath57 in terms of the elements of a unitary matrix . all basis elements or vectors are obtained from a unitary matrix by identifying them with the corresponding columns . unitarity guarantees that all vectors will therefore be orthonormal . now we have to cope with the bases being unbiased amidst them . since each basis is represented by a unitary matrix , we then have @xmath58 . this condition can be addressed by imposing that matrix elements    @xmath59    where @xmath60 , have to be equal to @xmath15 . in other words , @xmath61 has to be proportional to a hadamard - like matrix . the aforementioned conditions has to be applied to all possible @xmath62 bipartite combinations of bases @xmath63 . let us define the following quantities as the _ residuals _ @xmath64    thus , the problem of finding a set of unbiased orthonormal bases is translated into the optimization procedure of finding the minimum of @xmath65 being equal to zero . if the minimum is different from zero , given @xmath7 and @xmath0 , we definitely do not have a set of unbiased bases . in addition , our function resembles very much the quantity used in @xcite to define the notion of `` unbiasedness '' between two orthonormal bases . to whether or not the aforementioned quantity represents a metric is something not checked . now that the we have translated the problem of finding mu bases into an operational one , one has to be able to explore all possible bases . this fact means that we have to be able to survey the set space of unitary matrices . since we described in the previous section how to generate random unitary matrices properly , we will have to numerically explore all unitary matrices . the way to pursue that is to consider the angles ( [ angles ] ) given @xmath7 and @xmath0 in all cases in ( [ ucdot ] ) as the variables of the function @xmath65 to be minimized . provided the concomitant optimal value ( the sum of all residuals @xmath66 ) is equal to zero , we may then have found a set of mu bases . otherwise , that may not be possible given the constraints on @xmath7 and @xmath0 . now that we have the tools to perform a numerical survey over the set of unitary matrices , we carry out the optimization described in the previous section .      the @xmath67 case with three bases is know to exist , so our numerical procedure must return a minimal value of zero . the results are depicted in fig . ( [ figa ] ) . as can be appreciated , convergence is reached very fast after each monte carlo step ( formed by 15000 different configurations each ) . therefore , we are quite confident that we have found a set of mu bases in the @xmath68case . case with three basis vs. the number of monte carlo steps . as can be clearly appreciated , a global zero minimum is reached . see text for details.,width=332 ]    however , we must bear in mind an important issue regarding numerical surveys . not all our simulations lead to a zero minimum , so the lack of convergence is in favor of the argument that some sets of mu bases can not be extended to further number of bases . in @xmath4 there are some sets of 2 mu bases that can not be extended to 3 mu bases ( see ref . @xcite and references therein ) . from numerical simulations it is knows that null measure sets can not be reached . in @xcite , a subset of the karlson s family of complex hadamard matrices can not be extended to 3 mu bases . additionally , the karlson s family has dimension 2 and the maximal set of complex hadamard matrices in dimension 6 has dimension 4 , so it is a null measure set . therefore , one could never achieve a unitary matrix from random simulations such that it belongs to the karlsson s family . moreover , there are 1 dimensional families and even more , isolated complex hadamard matrices in dimension six . when considering the extension of the number of mu bases @xmath69 , @xmath70 being a hadamard matrix , provided by a certain number , we then know that that function lacks the property of continuity @xcite . the absence of continuity together with an incomplete knowledge about the number discontinuities in the number of mu bases makes the overall problem a difficult one . however , in our approach , we succeed in finding at least a few cases where @xmath71 holds . the study of the case with three bases confirms that our approach to the problem is a good one . as a matter of fact , we could study the problem for any @xmath72case , but the overall optimization procedure as it is indeed the case for any simulation of a quantum system becomes intractable at some point . with the numerical tools being a valid one , we can now tackle the problem of whether @xmath67 can sustain @xmath3 mu bases . now that we have implemented the tools for performing a search in the space of unitary matrices of a given dimension @xmath21 , we are in a position a bit closer to ascertain whether it is possible to have four mu bases in the @xmath73case . we start the numerical search and the outcome of if is shown in fig . ( [ figb ] ) . the evolution is such that the total function to be minimized rapidly decreases , and attains a value that is not zero . several repetitions of the same optimization procedure lead to the same conclusion : the value which is optimized is of @xmath74 . thus , we have more evidence that four mutually unbiased bases can not occur in @xmath67 . however , in the light of the previous discussion on continuity , it still remains doubts that our numerical procedure may not arrive at the minimum of 0 because we are trying to explore a set of zero measure . this fact implies that our numerical approach to the problem may have ( still ) some loopholes as far as reaching a conclusive answer . all facts points towards that @xmath3 is incompatible with @xmath4 , but we have no theorem that ascertains whether the function which is optimized reaches may ever reach a minimum of zero . in addition , we are left with an intriguing question : what is the meaning of having a set of four almost mu bases ? ( let us call them @xmath75mu bases from now on ) . definitely , if we have found one such @xmath75mu bases set , it may not be unique . in point of fact , there may exist as many as different vales for the function to be optimized are reached . however , what is the physics that entails that one family of these @xmath75mu bases reaches a minimum minimorum ? in operational terms , what role could these @xmath75mu bases play in practice ? it may be the case , for instance , that a subset of the four bases is mutually unbiased . case with four basis vs. the number of monte carlo steps . this typical evolution of the function to be optimized the sum of residuals in our case does not reach a minimum of zero . it approaches zero but the corresponding value is aways of @xmath74 . see text for details.,width=332 ] we have translated the problem of the existence of @xmath76bases in @xmath67 into an optimization procedure . as expected , the concomitant numerical optimization has provided a satisfactory answer for known cases such as @xmath77 in @xmath67 . this new approach to the problem of to whether or not there exist a set of @xmath3 mu bases for @xmath4 has provided more evidence in favor that this is not case , although no theorem guarantees this argument . in addition , we are left with the interesting question on the limitations that pose the use of sets of imperfect mu bases in quantum information tasks , an issue that is certainly of interest for in experiments one has to deal with imperfections . also , our procedure is capable to explore more dimensions and bases in a straightforward manner , although taking into account that a computational limitation is reached , and therefore opens the door to similar studies in the future @xcite . j. batle acknowledges partial support from the physics department , uib . j. batle acknowledges fruitful discussions with d. goyeneche , j. rossell and maria del mar batle .

there has been great interest in finding sets of @xmath0 mutually unbiased bases which are compatible with a given space @xmath1 , specially in physics due to their interesting applications in quantum information theory .
 several general results have been obtained so far , but surprising results may occur for definite @xmath2-values .
 one such case that has remained an open question ( the simplest case ) is the one regarding the existence of @xmath3 mutually orthogonal bases for @xmath4 . in the present work we introduce a new approach to the problem by translating it into an optimization procedure for a given pair @xmath2 .


the association between meteoroid streams and comets is well - established , though the idea that some originate from asteroids is quite old ( see williams , 2011 ) . the formation mechanism of a meteor stream from a comet is well - understood . ices sublimate generating a gas outflow that carries away small dust grains . inter - asteroid collisions , as was seen in 2010a2 ( linear ) ( jewitt _ et al . _ , 2010 ) is the most obvious formation process , though there are other possibilities . in all cases , the ejection velocity is small compared to the orbital speed so that a stream is formed on an orbit that is similar to that of the parent ( the physics and mathematics involved are given in williams , 2004 ) . individual meteoroid orbits can evolve in a significantly different way from the parent . the obvious mechanism causing this is a close planetary encounter ( hughes williams & fox , 1981 ) but collisions between stream meteoroids , the parent or interplanetary dust particles can also play a part . these processes feed the sporadic population , making it hard to associate such meteoroids with a given parent . the major tool for determining a pairing of parent and meteoroid stream has been orbit similarity . to quantify this , southworth & hawkins ( 1963 ) and drummond ( 1981 ) proposed orbital similarity d - criteria , both being widely used . using such criteria , many pairings between comets and meteor streams have been established and also between minor showers and near - earth asteroids ( neas ) . because of the vast increase in the number of known asteroids , the probability of a chance similarity of orbits is high . for this reason , poruban , korno & williams ( 2004 ) proposed that orbits need to remain similar for 5000 years before any association can be claimed . three major showers , the geminids , the quadrantids and the taurids all have asteroids associated with them . whipple ( 1983 ) pointed out the orbital similarity between the geminid stream and ( 3200 ) phaethon , while fox , williams & hughes ( 1983 ) showed that phaethon could be the parent of the geminid meteoroid stream assuming the ejection mechanism was cometary . battams & watson ( 2009 ) reported that phaethon brightened by at least 2 magnitudes in 2009 . the asteroids 2005 ud and 1999 yc have orbits that are similar to that of phaethon , all suggesting that a fragmenting large comet may be the actual parent .    jenniskens ( 2004 ) noted the similarity between the orbits of the quadrantids and asteroid 2003 eh1(now 196256 ) . in the activity curve of the quadrantids there is a broad background and a sharp narrow central peak suggesting that an old comet ( probably 96p / machholz ) , produced the background while a recent outburst produced the sharp peak . this could the fragmentation of c/1490 y1 a few hundred years ago ( williams _ et al _ 2004 ) , producing asteroid 2003eh1 . the taurid stream is a complex of several smaller streams and filaments . there are many associated asteroids and lists are given in poruban , korno & williams ( 2006 ) . the taurid complex has an active comet associated with it , 2p / encke . asher , clube & steel ( 1993 ) suggested that the family of meteoroid streams , comet 2p / encke and associated apollo asteroids could all have formed by the fragmentation of a giant comet 20 - 30 ky ago . the sporadic meteors have lost the identity of their parent s orbit . in order to discriminate between the orbits of comets and asteroids , several one parameter criteria have been proposed , they are listed in jopek & williams ( 2013 ) . they have been used by several authors : whipple , 1954 , found that cometary types dominated within photographic meteors , jones & sarma , 1985 ; steel , 1996 , found that the proportion of cometary and asteroidal were about equal within tv meteors ; according to steel , 1996 asteroidal meteoroids dominated among kharkov radio meteors , while voloshchuk _ et al . _ , 1997 , claimed the opposite .    applying several criteria to 780 comets and 7830 neas , jopek & williams ( 2013 ) found that the q- and e - criterion were the most reliable . they also classified @xmath0 sporadic meteoroids collected from several sources . using one parameter q- ( or e- ) criterion the authors found that amongst all sporadic meteoroids @xmath1-@xmath2% can not be classified correctly . the orbits of such objects have asteroidal aphelia @xmath3 ( or semi - major axes @xmath4 ) and cometary inclinations @xmath5 deg . the ecliptic radiants of the meteors corresponding to these objects concentrate around the position of the apex of the earth motion . such meteoroids as was shown by jones et al . 2001 , nesvorny et al . 2011 can be of cometary origin . therefore to classify the sporadic meteoroids jopek & williams proposed that two parameter criteria such as @xmath6-@xmath7 are used . for cometary orbits we have : @xmath8 } , \hspace{0.3 cm }   \rm{or } \hspace{0.3 cm }   i > 75\deg \label{qicriterion}\ ] ] where : @xmath9  semi - major axis ; @xmath10  eccentricity ; @xmath7 inclination of the meteoroid orbit . asher d.j clube s.v.m . steel d.i . , 1993 , mnras , 264 , 93 battams k.watson a. 2009 , iau circ . , 9054 drummond j.d . , 1981 , icarus , 45 , 453 fox k. williams i.p . hughes d. w. , 1983 , mnras , 205 , 1155 hughes d. w. williams i. p. fox k. , 1981 , mnras , 195 , 625 jenniskens p. , 2004 , aj , 127 , 3018 jewitt d. weaver h. agarwal j. mutchler m. drahus m. , 2010 , nature , 467 , 817 jones j. sarma t. , 1985 , bull . czechosl . , 36 , 103 jones j. , campbell - brown m. , nikolova s. , 2001 , in meteors 2001 , ed warmbeim b. , esa - sp495 575 jopek t.j . williams i.p . , 2013 , mnras , 430 , 2377 nesvorny d. , vokrouhlicky d. , pokorny p. , janches d. , 2011 , apj , 747 , 1 poruban v. korno l. williams i.p . , 2004 , em&p . 95 , 697 poruban v. korno l. williams i.p . , 2006 , contr . skalnate pleso , 36 , 103 southworth r.b . hawkins g.s . , 1963 , smiths . astroph.,7 , 261 steel d.i . , 1996 , space sci . , 78 , 507 voloshchuk yu . i. vorgul a.  v. kashcheev b. l. , 1997 . astronomicheskii vestnik , 31 , 345 whipple f.l . , 1954 , aj , 59 , 201 whipple f.l . , 1983 , iau circ 3881 . williams i.p . , 2004 , wgn j. imo , 32 , 11 williams i.p . , 2011 , a & g , 52 , 2.20 williams i.p hughes d. w. mcbride n. wu z. , 1993 , mnras , 260 , 43 williams i.p . ryabova g.o . baturin a. p. chernitsov a. m. , 2004 , mnras , 355 , 1171

neo s come close to the earth s orbit so that any dust ejected from them , might be seen as a meteor shower .
 orbits evolve rapidly , so that a similarity of orbits at one given time is not sufficient to prove a relationship , orbital evolution over a long time interval also has to be similar .
 sporadic meteoroids can not be associated with a single parent body , they can only be classified as cometary or asteroidal .
 however , by considering one parameter criteria , many sporadics are not classified properly therefore two parameter approach was proposed .


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Premise: Gays y lesbianas.

Hypothesis: heterosexuales.


Technical Field
The present invention relates generally to seals for rotary machines and, more particularly, to a seal assembly and rotary machine containing such seal.

Related Art
In many rotary machines, such as a gas turbine or jet engine, a gas is compressed in a compressor and mixed with a fuel source in a combustor.

The combination of gas and fuel is then ignited for generating combustion gases that are directed to turbine stage(s) that derive energy therefrom.

Both turbine stage(s) and compressor have stationary or non-rotating components, e.g., vanes, that cooperate with rotating components, e.g., blades, for compressing and expanding the operational gases.

The operational gases change in pressure through the machine and a variety of seals are provided to preserve the differential pressures where necessary to maximize machine efficiency and performance.

An exemplary seal may be provided between a turbine rotor and a cooperating stator or stator body so the rotor may be pressurized to provide thrust balance relative to the rearwardly directed force generated by the engine and the forward direction of the engine.

In the above-described settings, turbine components and seals exceed the operating temperature range of flexible organic compound elastomer seals used in lower temperature applications.

Accordingly, seals used must be capable of operation in a high temperature environment.

In addition, the seals used must address the close operating clearances required in machinery of this type.

Rotary machine seal design also requires consideration of the relative motion between components produced by the differential thermal expansion that occurs throughout the machinery operating cycle compared to cold clearance at assembly.

One structure commonly provided to control leakage flow along a turbine shaft or other rotating surface is a labyrinth seal.

In this setting, a variety of blocking seal strips and obstructions are used between stationary turbine components.

Solid labyrinth seals typically have a relatively large clearance to avoid rub damage.

Labyrinth seals, therefore, do not maximize machine performance.

Another commonly used seal is a brush seal, which include a pack of metal bristles that contact a rotor at free ends thereof to maintain a seal with the rotor.

The bristles may be inclined relative to the rotor and may be supported by plates.

Brush seals have been aggressively pursued in recent years to provide tighter clearances in rotating machinery seal designs because they have some resilience to accommodate rubbing against the rotating component.

5,090,710, issued to Flower, a brush seal is comprised of closely packed fine wires or filaments that are weld assembled in a carrier assembly that is then inserted in a machine with the bristles wiping the rotating surface.

The bristles and assembly are fabricated of materials suitable for the fluid temperature and, compared to a labyrinth seal, leakage is reduced through and past the bristles in close contact with the rotating surface.

Brush seals, however, pose a number of deficiencies.

First, the multistep brush seal manufacturing process is costly.

Second, brush seal bristles do not always maintain a close running clearance because of their inherent inability to withstand long term wear.

Third, brush seals exposed to solid particles are subject to erosion or other deterioration.

Finally, brush seals are also subject to vibration due to movement of the pressurized fluid being sealed.

Therefore, brush seals oftentimes require dampening features.

Another type seal is disclosed in U.S. Pat.

5,042,823 and 5,071,138, both issued to Mackay et al.

These disclosures reveal a laminated finger seal providing a planar array of radially and circumferentially extending fingers separated by gaps.

This structure suffers from a number of disadvantages.

For instance, each stacked lamination is a solid ring (not segmented) and, therefore, is limited in application to large diameter machines that require installation/replacement without rotor removal.

In addition, the finger geometry provided is provided in a substantially radial plane, which may prevent adequate flexure of the fingers.

In view of the foregoing, there is a need in the art for a seal assembly having low cost manufacture and capable of withstanding the operational sensitivities described above.

In accordance with the invention a seal assembly is provided that has a number of seal members or xe2x80x9cleafxe2x80x9d seals.

The seal assembly may be manufactured from rolled shim stock using wire electro-discharge manufacturing (EDM) to make narrow, precision slots to produce the desired seal member geometry.

The seal members may be angled between their free ends and their fixed ends and may include a support for supporting the angle.

The invention provides similar benefits as brush seals and finger seals in rotary machine applications but at lower cost and with more robust attributes.

Seal member geometry is engineered with respect to thickness, width, length, and number of members to meet specific application requirements of differential pressure and anticipated differential motion.

The support serves to limit member movement in one direction and withstand differential pressure, while force imposed by a rub engagement on a rotating component is reduced with the elastic flexure of the seal assembly.

Seal member end geometry may be shaped to provide a precision diameter and may also incorporate geometry for aerodynamic lift that would minimize wear in those rotor seal applications that anticipate a heavy transient rub.

In a first aspect of the invention is provided a seal assembly for sealing against a rotatable component, the seal assembly comprising: a first element including a plurality of spaced seal members having slots therebetween; a second element including a plurality of spaced seal members having slots therebetween; and wherein each seal member is angled between a fixed end and a free end thereof, and the free ends slidably engage the rotatable component to seal thereagainst.

A second aspect of the invention provides a rotary machine comprising: a rotatable component and a non-rotatable component, the components lying about a common axis; a seal assembly between the components, the seal assembly including: a plurality of juxtaposed elements, each element including a plurality of spaced seal members defining slots therebetween, each seal member including an angle disposed between a fixed end and a free end thereof, wherein the free ends slidably engage the rotatable component to seal thereagainst; and a support for supporting the angle of each seal member.

In a third aspect of the invention is provided a method of inhibiting fluid flow through an annular slot defined between a stator body and a rotor received in the stator body, the rotor having a longitudinal axis, the method including the steps of: disposing on the stator body a plurality of arcuate elements each having a band portion and an integral plurality of circumferentially disposed seal members having slots therebetween, wherein the seal members include an angle therein and extend inwardly from the stator body at an angle relative to the longitudinal axis to slidably contact the rotor along a circumference thereof; circumferentially aligning and axially juxtaposing the elements; employing the cooperatively disposed elements to define an annular seal extending between the stator body and the rotor; and inhibiting fluid flow through the annular slot with the annular seal.

In a fourth aspect of the invention is provided a method of fabricating a seal assembly for sealing pressurized chambers of a rotary machine having a stator body and a rotor, the method comprising the steps of: (a) cutting slots axially into an edge of a material to form a plurality of seal members; (b) layering the material; (c) securing the layers of material together; (d) forming an angle in the seal members; and (e) coupling the layers of material to a holder that is adapted to be coupled to a stator body of the rotary machine, whereby the seal members slidably engage the rotor of the rotary machine, when in operation, to seal the pressurized chambers.

The foregoing and other features and advantages of the invention will be apparent from the following more particular description of preferred embodiments of the invention.


The role of hand calculations in ground water flow modeling.

Most ground water modeling courses focus on the use of computer models and pay little or no attention to traditional analytic solutions to ground water flow problems.

Why waste time to learn about the method of images, or why study analytic solutions to one-dimensional or radial flow problems?

Computer models solve much more realistic problems and offer sophisticated graphical output, such as contour plots of potentiometric levels and ground water path lines.

However, analytic solutions to elementary ground water flow problems do have something to offer over computer models: insight.

For instance, an analytic one-dimensional or radial flow solution, in terms of a mathematical expression, may reveal which parameters affect the success of calibrating a computer model and what to expect when changing parameter values.

Similarly, solutions for periodic forcing of one-dimensional or radial flow systems have resulted in a simple decision criterion to assess whether or not transient flow modeling is needed.

Basic water balance calculations may offer a useful check on computer-generated capture zones for wellhead protection or aquifer remediation.

An easily calculated "characteristic leakage length" provides critical insight into surface water and ground water interactions and flow in multi-aquifer systems.

Familiarity with elementary analytic solutions and the capability of performing some simple hand calculations can promote appropriate (computer) modeling techniques, avoids unnecessary complexity, improves reliability, and is likely to save time and money.

Training in basic hand calculations should be an important part of the curriculum of ground water modeling courses.


Today we announce the departure of Philip 'aizy' Aistrup from our Counter-Strike: Global Offensive team.

He is transferring to G2 Esports for an undisclosed fee in agreement with all parties involved.

aizy has been instrumental in our team's success over the past 10 months he's been with us and we have nothing but gratitude for his time with us and the work he's put into improving himself and his team mates.


I'm currently working off a query from this post.

That query is written in Legacy SQL and will not work in my environment.

I've modified the query to use the modern SQL functions and updated the SELECT date as date to use timestamp_micros.

I should also mention that the rows I'm trying to select are coming in from Firebase Analytics.

My Query:
SELECT
    FORMAT_TIMESTAMP('%Y-%m-%d', TIMESTAMP_MICROS(event.timestamp_micros)) as date,
    SUM(CASE WHEN period = 7  THEN users END) as days_07,
    SUM(CASE WHEN period = 14 THEN users END) as days_14,
    SUM(CASE WHEN period = 30 THEN users END) as days_30
FROM (
    SELECT
        FORMAT_TIMESTAMP('%Y-%m-%d', TIMESTAMP_MICROS(event.timestamp_micros)) as date,
        periods.period as period,
        COUNT(DISTINCT user_dim.app_info.app_instance_id) as users
    FROM `com_sidearm_fanapp_uiowa_IOS.*

` as activity
    CROSS JOIN
        UNNEST(event_dim) as event
    CROSS JOIN (
        SELECT
            FORMAT_TIMESTAMP('%Y-%m-%d', TIMESTAMP_MICROS(event.timestamp_micros)) as date
        FROM `com_sidearm_fanapp_uiowa_IOS.*

`
        CROSS JOIN
            UNNEST(event_dim) as event
        GROUP BY event.timestamp_micros
    ) as dates
    CROSS JOIN (
        SELECT
            period
        FROM
            (
                SELECT 7 as period
                UNION ALL
                SELECT 14 as period
                UNION ALL
                SELECT 30 as period
            )
    ) as periods
    WHERE
        dates.date >= activity.date
    AND
        SAFE_CAST(FLOOR(TIMESTAMP_DIFF(dates.date, activity.date, DAY)/periods.period) AS INT64) = 0
    GROUP BY 1,2
)
CROSS JOIN
    UNNEST(event_dim) as event
GROUP BY date
ORDER BY date DESC

A:

Column name period is ambiguous at [24:13] error.

to fix this particular error - you should fix below
    CROSS JOIN (
        SELECT
            period
        FROM
            (SELECT 7 as period),
            (SELECT 14 as period),
            (SELECT 30 as period)
    ) as periods

so it should look like:
    CROSS JOIN (
        SELECT
            period
        FROM
            (SELECT 7 as period UNION ALL
            SELECT 14 as period UNION ALL
            SELECT 30 as period)
    ) as periods

Answer on your updated question

Try below.

I didn't have chance to test it but hope it can help you fix your query
SELECT
    date,
    SUM(CASE WHEN period = 7  THEN users END) as days_07,
    SUM(CASE WHEN period = 14 THEN users END) as days_14,
    SUM(CASE WHEN period = 30 THEN users END) as days_30
FROM (
    SELECT
        activity.date as date,
        periods.period as period,
        COUNT(DISTINCT user) as users
    FROM (
      SELECT
          event.timestamp_micros as date,
           user_dim.app_info.app_instance_id as user
      FROM `yourTable` CROSS JOIN UNNEST(event_dim) as event
    ) as activity
    CROSS JOIN (
        SELECT
            event.timestamp_micros  as date
        FROM `yourTable` CROSS JOIN UNNEST(event_dim) as event
        GROUP BY event.timestamp_micros
    ) as dates
    CROSS JOIN (
        SELECT period
        FROM
            (SELECT 7 as period UNION ALL
            SELECT 14 as period UNION ALL
            SELECT 30 as period)
    ) as periods
    WHERE dates.date >= activity.date
    AND SAFE_CAST(FLOOR(TIMESTAMP_DIFF(TIMESTAMP_MICROS(dates.date), TIMESTAMP_MICROS(activity.date), DAY)/periods.period) AS INT64) = 0
    GROUP BY 1,2
)
GROUP BY date
ORDER BY date DESC


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Context: The normal force is due to repulsive forces of interaction between atoms at close contact. When their electron clouds overlap, Pauli repulsion (due to fermionic nature of electrons) follows resulting in the force that acts in a direction normal to the surface interface between two objects.:93 The normal force, for example, is responsible for the structural integrity of tables and floors as well as being the force that responds whenever an external force pushes on a solid object. An example of the normal force in action is the impact force on an object crashing into an immobile surface.

Question: What is the repulsive force of close range atom interaction?

normal force
normal force
normal force
normal force


Context: Overactive immune responses comprise the other end of immune dysfunction, particularly the autoimmune disorders. Here, the immune system fails to properly distinguish between self and non-self, and attacks part of the body. Under normal circumstances, many T cells and antibodies react with "self" peptides. One of the functions of specialized cells (located in the thymus and bone marrow) is to present young lymphocytes with self antigens produced throughout the body and to eliminate those cells that recognize self-antigens, preventing autoimmunity.

Question: Where are the specialized cells that eliminate cells that recognize self-antigens located?

thymus and bone marrow
thymus and bone marrow
thymus and bone marrow


Context: French irregular forces (Canadian scouts and Indians) harassed Fort William Henry throughout the first half of 1757. In January they ambushed British rangers near Ticonderoga. In February they launched a daring raid against the position across the frozen Lake George, destroying storehouses and buildings outside the main fortification. In early August, Montcalm and 7,000 troops besieged the fort, which capitulated with an agreement to withdraw under parole. When the withdrawal began, some of Montcalm's Indian allies, angered at the lost opportunity for loot, attacked the British column, killing and capturing several hundred men, women, children, and slaves. The aftermath of the siege may have contributed to the transmission of smallpox into remote Indian populations; as some Indians were reported to have traveled from beyond the Mississippi to participate in the campaign and returned afterward having been exposed to European carriers.

Question: During withdrawal from Fort William Henry, what did some Indian enemies of French do?


Context: Consultant pharmacy practice focuses more on medication regimen review (i.e. "cognitive services") than on actual dispensing of drugs. Consultant pharmacists most typically work in nursing homes, but are increasingly branching into other institutions and non-institutional settings. Traditionally consultant pharmacists were usually independent business owners, though in the United States many now work for several large pharmacy management companies (primarily Omnicare, Kindred Healthcare and PharMerica). This trend may be gradually reversing as consultant pharmacists begin to work directly with patients, primarily because many elderly people are now taking numerous medications but continue to live outside of institutional settings. Some community pharmacies employ consultant pharmacists and/or provide consulting services.

Question: What is consultant pharmacy not concerned with?


Context: Notable faculty in physics have included the speed of light calculator A. A. Michelson, elementary charge calculator Robert A. Millikan, discoverer of the Compton Effect Arthur H. Compton, the creator of the first nuclear reactor Enrico Fermi, "the father of the hydrogen bomb" Edward Teller, "one of the most brilliant and productive experimental physicists of the twentieth century" Luis Walter Alvarez, Murray Gell-Mann who introduced the quark, second female Nobel laureate Maria Goeppert-Mayer, the youngest American winner of the Nobel Prize Tsung-Dao Lee, and astrophysicist Subrahmanyan Chandrasekhar.

Question: Who is also known at the father of the hydrogen bomb?

Edward Teller
Edward Teller
Edward Teller


Context: News of the two battles reached England in August. After several months of negotiations, the government of the Duke of Newcastle decided to send an army expedition the following year to dislodge the French. They chose Major General Edward Braddock to lead the expedition. Word of the British military plans leaked to France well before Braddock's departure for North America. In response, King Louis XV dispatched six regiments to New France under the command of Baron Dieskau in 1755. The British, intending to blockade French ports, sent out their fleet in February 1755, but the French fleet had already sailed. Admiral Edward Hawke detached a fast squadron to North America in an attempt to intercept the French.

Question: What was the goal of Braddock's expedition?

dislodge the French
dislodge the French
dislodge the French
dislodge the French


Context: For a precise definition of what it means to solve a problem using a given amount of time and space, a computational model such as the deterministic Turing machine is used. The time required by a deterministic Turing machine M on input x is the total number of state transitions, or steps, the machine makes before it halts and outputs the answer ("yes" or "no"). A Turing machine M is said to operate within time f(n), if the time required by M on each input of length n is at most f(n). A decision problem A can be solved in time f(n) if there exists a Turing machine operating in time f(n) that solves the problem. Since complexity theory is interested in classifying problems based on their difficulty, one defines sets of problems based on some criteria. For instance, the set of problems solvable within time f(n) on a deterministic Turing machine is then denoted by DTIME(f(n)).

Question: What is not used for a precise definition of what it means to solve a problem using a given amount of time and space?


Context: In the United Kingdom and several other Commonwealth countries including Australia and Canada, the use of the term is generally restricted to primary and secondary educational levels; it is almost never used of universities and other tertiary institutions. Private education in North America covers the whole gamut of educational activity, ranging from pre-school to tertiary level institutions. Annual tuition fees at K-12 schools range from nothing at so called 'tuition-free' schools to more than $45,000 at several New England preparatory schools.

Question: What term does the UK, Australia and Canada always use when referring to universities?


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What is the best off-road motorcycle trail ?

i hear that the mojave road is amazing!<br />\nsearch for it online.


Trans fats occur in manufactured foods during the process of partial hydrogenation, when hydrogen gas is bubbled through vegetable oil to increase shelf life and stabilize the original polyunsatured oil. The resulting fat is similar to saturated fat, which raises "bad" LDL cholesterol and can lead to clogged arteries and heart disease. \nUntil very recently, food labels were not required to list trans fats, and this health risk remained hidden to consumers. In early July, FDA regulations changed, and food labels will soon begin identifying trans fat content in processed foods.


according to the www.fedex.com web site:\nAir Fleet
\n
\n670 aircraft, including:
\n47 Airbus A300-600s  17 Boeing DC10-30s
\n62 Airbus A310-200/300s  36 Boeing MD10-10s
\n2 ATR 72s  5 Boeing MD10-30s
\n29 ATR 42s  57 Boeing MD11s
\n18 Boeing 727-100s  10 Cessna 208As
\n94 Boeing 727-200s  246 Cessna 208Bs
\n30 Boeing DC10-10s  17 Fokker F-27s


In the san francisco bay area, does it make sense to rent or buy ?

renting vs buying depends on your goals. <br />\ngenerally thinking is that buying is better b/c the payments that would go into the rent start building equity in your home. the govt also incentivizes you to buy by making your property tax payments and mortgage interest payments tax deductible.\nhaving said that current housing status in the bay area is such that housing cost to purchase is relatively high and rental prices (compared to ownership cost) are relatively low (relative to the rest of the country). it makes lese sense to buy vs. other places.\nbottom line you should base your decision on whether you think the market will keep going up or not. the other numbers tend to even out, the main gain or loss in buying comes from appreciation/depreciation.


What's the best way to clean a keyboard?

There are commercial kits available, but a can of compressed air, a lint-free cloth or wipes, mild dishwashing liquid, and a vacuum cleaner with a soft brush are all you really need for a basic cleaning. \nAfter turning off your computer and unplugging your keyboard, gently shake the keyboard upside down over some newspaper to dislodge loose crumbs and particles. \nUse the can of compressed air to blow a stream of air between the keys. \nDampen a lint-free cloth with a diluted solution of dishwashing liquid and water to wipe down the keys. \nUse the vacuum cleaner brush attachment to suck away any remaining dirt or debris. <br />\nIf you have a membrane-type keyboard (it doesn't have a spring under each key) and the keys are truly grungy, you can remove the keys with a small screwdriver and use cotton swabs and 90% isopropyl alcohol to get it sparkling clean. If you are still feeling industrious, next you can tackle cleaning your mouse. \nOf course, if cleaning your keyboard seems a truly undesirable task, you can always buy a new one for around $20.