Create app.py

app.py

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+LONG_ARTICLE = """"for about 20 years the problem of properties of
+short - term changes of solar activity has been
+considered extensively . many investigators
+studied the short - term periodicities of the
+various indices of solar activity . several
+periodicities were detected , but the
+periodicities about 155 days and from the interval
+of @xmath3 $ ] days ( @xmath4 $ ] years ) are
+mentioned most often . first of them was
+discovered by @xcite in the occurence rate of
+gamma - ray flares detected by the gamma - ray
+spectrometer aboard the _ solar maximum mission (
+smm ) . this periodicity was confirmed for other
+solar flares data and for the same time period
+@xcite . it was also found in proton flares during
+solar cycles 19 and 20 @xcite , but it was not
+found in the solar flares data during solar cycles
+22 @xcite . _    several autors confirmed above
+results for the daily sunspot area data . @xcite
+studied the sunspot data from 18741984 . she found
+the 155-day periodicity in data records from 31
+years . this periodicity is always characteristic
+for one of the solar hemispheres ( the southern
+hemisphere for cycles 1215 and the northern
+hemisphere for cycles 1621 ) . moreover , it is
+only present during epochs of maximum activity (
+in episodes of 13 years ) .
+similarinvestigationswerecarriedoutby + @xcite .
+they applied the same power spectrum method as
+lean , but the daily sunspot area data ( cycles
+1221 ) were divided into 10 shorter time series .
+the periodicities were searched for the frequency
+interval 57115 nhz ( 100200 days ) and for each of
+10 time series . the authors showed that the
+periodicity between 150160 days is statistically
+significant during all cycles from 16 to 21 . the
+considered peaks were remained unaltered after
+removing the 11-year cycle and applying the power
+spectrum analysis . @xcite used the wavelet
+technique for the daily sunspot areas between 1874
+and 1993 . they determined the epochs of
+appearance of this periodicity and concluded that
+it presents around the maximum activity period in
+cycles 16 to 21 . moreover , the power of this
+periodicity started growing at cycle 19 ,
+decreased in cycles 20 and 21 and disappered after
+cycle 21 . similaranalyseswerepresentedby + @xcite
+, but for sunspot number , solar wind plasma ,
+interplanetary magnetic field and geomagnetic
+activity index @xmath5 . during 1964 - 2000 the
+sunspot number wavelet power of periods less than
+one year shows a cyclic evolution with the phase
+of the solar cycle.the 154-day period is prominent
+and its strenth is stronger around the 1982 - 1984
+interval in almost all solar wind parameters . the
+existence of the 156-day periodicity in sunspot
+data were confirmed by @xcite . they considered
+the possible relation between the 475-day (
+1.3-year ) and 156-day periodicities . the 475-day
+( 1.3-year ) periodicity was also detected in
+variations of the interplanetary magnetic field ,
+geomagnetic activity helioseismic data and in the
+solar wind speed @xcite . @xcite concluded that
+the region of larger wavelet power shifts from
+475-day ( 1.3-year ) period to 620-day ( 1.7-year
+) period and then back to 475-day ( 1.3-year ) .
+the periodicities from the interval @xmath6 $ ]
+days ( @xmath4 $ ] years ) have been considered
+from 1968 . @xcite mentioned a 16.3-month (
+490-day ) periodicity in the sunspot numbers and
+in the geomagnetic data . @xcite analysed the
+occurrence rate of major flares during solar
+cycles 19 . they found a 18-month ( 540-day )
+periodicity in flare rate of the norhern
+hemisphere . @xcite confirmed this result for the
+@xmath7 flare data for solar cycles 20 and 21 and
+found a peak in the power spectra near 510540 days
+. @xcite found a 17-month ( 510-day ) periodicity
+of sunspot groups and their areas from 1969 to
+1986 . these authors concluded that the length of
+this period is variable and the reason of this
+periodicity is still not understood . @xcite and +
+@xcite obtained statistically significant peaks of
+power at around 158 days for daily sunspot data
+from 1923 - 1933 ( cycle 16 ) . in this paper the
+problem of the existence of this periodicity for
+sunspot data from cycle 16 is considered . the
+daily sunspot areas , the mean sunspot areas per
+carrington rotation , the monthly sunspot numbers
+and their fluctuations , which are obtained after
+removing the 11-year cycle are analysed . in
+section 2 the properties of the power spectrum
+methods are described . in section 3 a new
+approach to the problem of aliases in the power
+spectrum analysis is presented . in section 4
+numerical results of the new method of the
+diagnosis of an echo - effect for sunspot area
+data are discussed . in section 5 the problem of
+the existence of the periodicity of about 155 days
+during the maximum activity period for sunspot
+data from the whole solar disk and from each solar
+hemisphere separately is considered . to find
+periodicities in a given time series the power
+spectrum analysis is applied . in this paper two
+methods are used : the fast fourier transformation
+algorithm with the hamming window function ( fft )
+and the blackman - tukey ( bt ) power spectrum
+method @xcite . the bt method is used for the
+diagnosis of the reasons of the existence of peaks
+, which are obtained by the fft method . the bt
+method consists in the smoothing of a cosine
+transform of an autocorrelation function using a
+3-point weighting average . such an estimator is
+consistent and unbiased . moreover , the peaks are
+uncorrelated and their sum is a variance of a
+considered time series . the main disadvantage of
+this method is a weak resolution of the
+periodogram points , particularly for low
+frequences . for example , if the autocorrelation
+function is evaluated for @xmath8 , then the
+distribution points in the time domain are :
+@xmath9 thus , it is obvious that this method
+should not be used for detecting low frequency
+periodicities with a fairly good resolution .
+however , because of an application of the
+autocorrelation function , the bt method can be
+used to verify a reality of peaks which are
+computed using a method giving the better
+resolution ( for example the fft method ) . it is
+valuable to remember that the power spectrum
+methods should be applied very carefully . the
+difficulties in the interpretation of significant
+peaks could be caused by at least four effects : a
+sampling of a continuos function , an echo -
+effect , a contribution of long - term
+periodicities and a random noise . first effect
+exists because periodicities , which are shorter
+than the sampling interval , may mix with longer
+periodicities . in result , this effect can be
+reduced by an decrease of the sampling interval
+between observations . the echo - effect occurs
+when there is a latent harmonic of frequency
+@xmath10 in the time series , giving a spectral
+peak at @xmath10 , and also periodic terms of
+frequency @xmath11 etc . this may be detected by
+the autocorrelation function for time series with
+a large variance . time series often contain long
+- term periodicities , that influence short - term
+peaks . they could rise periodogram s peaks at
+lower frequencies . however , it is also easy to
+notice the influence of the long - term
+periodicities on short - term peaks in the graphs
+of the autocorrelation functions . this effect is
+observed for the time series of solar activity
+indexes which are limited by the 11-year cycle .
+to find statistically significant periodicities it
+is reasonable to use the autocorrelation function
+and the power spectrum method with a high
+resolution . in the case of a stationary time
+series they give similar results . moreover , for
+a stationary time series with the mean zero the
+fourier transform is equivalent to the cosine
+transform of an autocorrelation function @xcite .
+thus , after a comparison of a periodogram with an
+appropriate autocorrelation function one can
+detect peaks which are in the graph of the first
+function and do not exist in the graph of the
+second function . the reasons of their existence
+could be explained by the long - term
+periodicities and the echo - effect . below method
+enables one to detect these effects . ( solid line
+) and the 95% confidence level basing on thered
+noise ( dotted line ) . the periodogram values are
+presented on the left axis . the lower curve
+illustrates the autocorrelation function of the
+same time series ( solid line ) . the dotted lines
+represent two standard errors of the
+autocorrelation function . the dashed horizontal
+line shows the zero level . the autocorrelation
+values are shown in the right axis . ]     because
+the statistical tests indicate that the time
+series is a white noise the confidence level is
+not marked . ]    . ] the method of the diagnosis
+of an echo - effect in the power spectrum ( de )
+consists in an analysis of a periodogram of a
+given time series computed using the bt method .
+the bt method bases on the cosine transform of the
+autocorrelation function which creates peaks which
+are in the periodogram , but not in the
+autocorrelation function . the de method is used
+for peaks which are computed by the fft method (
+with high resolution ) and are statistically
+significant . the time series of sunspot activity
+indexes with the spacing interval one rotation or
+one month contain a markov - type persistence ,
+which means a tendency for the successive values
+of the time series to remember their antecendent
+values . thus , i use a confidence level basing on
+the red noise of markov @xcite for the choice of
+the significant peaks of the periodogram computed
+by the fft method . when a time series does not
+contain the markov - type persistence i apply the
+fisher test and the kolmogorov - smirnov test at
+the significance level @xmath12 @xcite to verify a
+statistically significance of periodograms peaks .
+the fisher test checks the null hypothesis that
+the time series is white noise agains the
+alternative hypothesis that the time series
+contains an added deterministic periodic component
+of unspecified frequency . because the fisher test
+tends to be severe in rejecting peaks as
+insignificant the kolmogorov - smirnov test is
+also used . the de method analyses raw estimators
+of the power spectrum . they are given as follows
+@xmath13    for @xmath14 + where @xmath15 for
+@xmath16 + @xmath17 is the length of the time
+series @xmath18 and @xmath19 is the mean value .
+the first term of the estimator @xmath20 is
+constant . the second term takes two values (
+depending on odd or even @xmath21 ) which are not
+significant because @xmath22 for large m. thus ,
+the third term of ( 1 ) should be analysed .
+looking for intervals of @xmath23 for which
+@xmath24 has the same sign and different signs one
+can find such parts of the function @xmath25 which
+create the value @xmath20 . let the set of values
+of the independent variable of the autocorrelation
+function be called @xmath26 and it can be divided
+into the sums of disjoint sets : @xmath27 where +
+@xmath28 + @xmath29 @xmath30 @xmath31 + @xmath32 +
+@xmath33 @xmath34 @xmath35 @xmath36 @xmath37
+@xmath38 @xmath39 @xmath40    well , the set
+@xmath41 contains all integer values of @xmath23
+from the interval of @xmath42 for which the
+autocorrelation function and the cosinus function
+with the period @xmath43 $ ] are positive . the
+index @xmath44 indicates successive parts of the
+cosinus function for which the cosinuses of
+successive values of @xmath23 have the same sign .
+however , sometimes the set @xmath41 can be empty
+. for example , for @xmath45 and @xmath46 the set
+@xmath47 should contain all @xmath48 $ ] for which
+@xmath49 and @xmath50 , but for such values of
+@xmath23 the values of @xmath51 are negative .
+thus , the set @xmath47 is empty .    . the
+periodogram values are presented on the left axis
+. the lower curve illustrates the autocorrelation
+function of the same time series . the
+autocorrelation values are shown in the right axis
+. ] let us take into consideration all sets
+\{@xmath52 } , \{@xmath53 } and \{@xmath41 } which
+are not empty . because numberings and power of
+these sets depend on the form of the
+autocorrelation function of the given time series
+, it is impossible to establish them arbitrary .
+thus , the sets of appropriate indexes of the sets
+\{@xmath52 } , \{@xmath53 } and \{@xmath41 } are
+called @xmath54 , @xmath55 and @xmath56
+respectively . for example the set @xmath56
+contains all @xmath44 from the set @xmath57 for
+which the sets @xmath41 are not empty . to
+separate quantitatively in the estimator @xmath20
+the positive contributions which are originated by
+the cases described by the formula ( 5 ) from the
+cases which are described by the formula ( 3 ) the
+following indexes are introduced : @xmath58
+@xmath59 @xmath60 @xmath61 where @xmath62 @xmath63
+@xmath64 taking for the empty sets \{@xmath53 }
+and \{@xmath41 } the indices @xmath65 and @xmath66
+equal zero . the index @xmath65 describes a
+percentage of the contribution of the case when
+@xmath25 and @xmath51 are positive to the positive
+part of the third term of the sum ( 1 ) . the
+index @xmath66 describes a similar contribution ,
+but for the case when the both @xmath25 and
+@xmath51 are simultaneously negative . thanks to
+these one can decide which the positive or the
+negative values of the autocorrelation function
+have a larger contribution to the positive values
+of the estimator @xmath20 . when the difference
+@xmath67 is positive , the statement the
+@xmath21-th peak really exists can not be rejected
+. thus , the following formula should be satisfied
+: @xmath68    because the @xmath21-th peak could
+exist as a result of the echo - effect , it is
+necessary to verify the second condition :
+@xmath69\in c_m.\ ] ]    . the periodogram values
+are presented on the left axis . the lower curve
+illustrates the autocorrelation function of the
+same time series ( solid line ) . the dotted lines
+represent two standard errors of the
+autocorrelation function . the dashed horizontal
+line shows the zero level . the autocorrelation
+values are shown in the right axis . ]    to
+verify the implication ( 8) firstly it is
+necessary to evaluate the sets @xmath41 for
+@xmath70 of the values of @xmath23 for which the
+autocorrelation function and the cosine function
+with the period @xmath71 $ ] are positive and the
+sets @xmath72 of values of @xmath23 for which the
+autocorrelation function and the cosine function
+with the period @xmath43 $ ] are negative .
+secondly , a percentage of the contribution of the
+sum of products of positive values of @xmath25 and
+@xmath51 to the sum of positive products of the
+values of @xmath25 and @xmath51 should be
+evaluated . as a result the indexes @xmath65 for
+each set @xmath41 where @xmath44 is the index from
+the set @xmath56 are obtained . thirdly , from all
+sets @xmath41 such that @xmath70 the set @xmath73
+for which the index @xmath65 is the greatest
+should be chosen .    the implication ( 8) is true
+when the set @xmath73 includes the considered
+period @xmath43 $ ] . this means that the greatest
+contribution of positive values of the
+autocorrelation function and positive cosines with
+the period @xmath43 $ ] to the periodogram value
+@xmath20 is caused by the sum of positive products
+of @xmath74 for each @xmath75-\frac{m}{2k},[\frac{
+2m}{k}]+\frac{m}{2k})$ ] .    when the implication
+( 8) is false , the peak @xmath20 is mainly
+created by the sum of positive products of
+@xmath74 for each @xmath76-\frac{m}{2k},\big [
+\frac{2m}{n}\big ] + \frac{m}{2k } \big ) $ ] ,
+where @xmath77 is a multiple or a divisor of
+@xmath21 . it is necessary to add , that the de
+method should be applied to the periodograms peaks
+, which probably exist because of the echo -
+effect . it enables one to find such parts of the
+autocorrelation function , which have the
+significant contribution to the considered peak .
+the fact , that the conditions ( 7 ) and ( 8) are
+satisfied , can unambiguously decide about the
+existence of the considered periodicity in the
+given time series , but if at least one of them is
+not satisfied , one can doubt about the existence
+of the considered periodicity . thus , in such
+cases the sentence the peak can not be treated as
+true should be used .    using the de method it is
+necessary to remember about the power of the set
+@xmath78 . if @xmath79 is too large , errors of an
+autocorrelation function estimation appear . they
+are caused by the finite length of the given time
+series and as a result additional peaks of the
+periodogram occur . if @xmath79 is too small ,
+there are less peaks because of a low resolution
+of the periodogram . in applications @xmath80 is
+used . in order to evaluate the value @xmath79 the
+fft method is used . the periodograms computed by
+the bt and the fft method are compared . the
+conformity of them enables one to obtain the value
+@xmath79 .    . the fft periodogram values are
+presented on the left axis . the lower curve
+illustrates the bt periodogram of the same time
+series ( solid line and large black circles ) .
+the bt periodogram values are shown in the right
+axis . ] in this paper the sunspot activity data (
+august 1923 - october 1933 ) provided by the
+greenwich photoheliographic results ( gpr ) are
+analysed . firstly , i consider the monthly
+sunspot number data . to eliminate the 11-year
+trend from these data , the consecutively smoothed
+monthly sunspot number @xmath81 is subtracted from
+the monthly sunspot number @xmath82 where the
+consecutive mean @xmath83 is given by @xmath84 the
+values @xmath83 for @xmath85 and @xmath86 are
+calculated using additional data from last six
+months of cycle 15 and first six months of cycle
+17 .    because of the north - south asymmetry of
+various solar indices @xcite , the sunspot
+activity is considered for each solar hemisphere
+separately . analogously to the monthly sunspot
+numbers , the time series of sunspot areas in the
+northern and southern hemispheres with the spacing
+interval @xmath87 rotation are denoted . in order
+to find periodicities , the following time series
+are used : + @xmath88   + @xmath89    + @xmath90
++ in the lower part of figure [ f1 ] the
+autocorrelation function of the time series for
+the northern hemisphere @xmath88 is shown . it is
+easy to notice that the prominent peak falls at 17
+rotations interval ( 459 days ) and @xmath25 for
+@xmath91 $ ] rotations ( [ 81 , 162 ] days ) are
+significantly negative . the periodogram of the
+time series @xmath88 ( see the upper curve in
+figures [ f1 ] ) does not show the significant
+peaks at @xmath92 rotations ( 135 , 162 days ) ,
+but there is the significant peak at @xmath93 (
+243 days ) . the peaks at @xmath94 are close to
+the peaks of the autocorrelation function . thus ,
+the result obtained for the periodicity at about
+@xmath0 days are contradict to the results
+obtained for the time series of daily sunspot
+areas @xcite .    for the southern hemisphere (
+the lower curve in figure [ f2 ] ) @xmath25 for
+@xmath95 $ ] rotations ( [ 54 , 189 ] days ) is
+not positive except @xmath96 ( 135 days ) for
+which @xmath97 is not statistically significant .
+the upper curve in figures [ f2 ] presents the
+periodogram of the time series @xmath89 . this
+time series does not contain a markov - type
+persistence . moreover , the kolmogorov - smirnov
+test and the fisher test do not reject a null
+hypothesis that the time series is a white noise
+only . this means that the time series do not
+contain an added deterministic periodic component
+of unspecified frequency . the autocorrelation
+function of the time series @xmath90 ( the lower
+curve in figure [ f3 ] ) has only one
+statistically significant peak for @xmath98 months
+( 480 days ) and negative values for @xmath99 $ ]
+months ( [ 90 , 390 ] days ) . however , the
+periodogram of this time series ( the upper curve
+in figure [ f3 ] ) has two significant peaks the
+first at 15.2 and the second at 5.3 months ( 456 ,
+159 days ) . thus , the periodogram contains the
+significant peak , although the autocorrelation
+function has the negative value at @xmath100
+months .    to explain these problems two
+following time series of daily sunspot areas are
+considered : + @xmath101   + @xmath102     + where
+@xmath103    the values @xmath104 for @xmath105
+and @xmath106 are calculated using additional
+daily data from the solar cycles 15 and 17 .
+and the cosine function for @xmath45 ( the period
+at about 154 days ) . the horizontal line ( dotted
+line ) shows the zero level . the vertical dotted
+lines evaluate the intervals where the sets
+@xmath107 ( for @xmath108 ) are searched . the
+percentage values show the index @xmath65 for each
+@xmath41 for the time series @xmath102 ( in
+parentheses for the time series @xmath101 ) . in
+the right bottom corner the values of @xmath65 for
+the time series @xmath102 , for @xmath109 are
+written . ] ( the 500-day period ) ]    the
+comparison of the functions @xmath25 of the time
+series @xmath101 ( the lower curve in figure [ f4
+] ) and @xmath102 ( the lower curve in figure [ f5
+] ) suggests that the positive values of the
+function @xmath110 of the time series @xmath101 in
+the interval of @xmath111 $ ] days could be caused
+by the 11-year cycle . this effect is not visible
+in the case of periodograms of the both time
+series computed using the fft method ( see the
+upper curves in figures [ f4 ] and [ f5 ] ) or the
+bt method ( see the lower curve in figure [ f6 ] )
+. moreover , the periodogram of the time series
+@xmath102 has the significant values at @xmath112
+days , but the autocorrelation function is
+negative at these points . @xcite showed that the
+lomb - scargle periodograms for the both time
+series ( see @xcite , figures 7 a - c ) have a
+peak at 158.8 days which stands over the fap level
+by a significant amount . using the de method the
+above discrepancies are obvious . to establish the
+@xmath79 value the periodograms computed by the
+fft and the bt methods are shown in figure [ f6 ]
+( the upper and the lower curve respectively ) .
+for @xmath46 and for periods less than 166 days
+there is a good comformity of the both
+periodograms ( but for periods greater than 166
+days the points of the bt periodogram are not
+linked because the bt periodogram has much worse
+resolution than the fft periodogram ( no one know
+how to do it ) ) . for @xmath46 and @xmath113 the
+value of @xmath21 is 13 ( @xmath71=153 $ ] ) . the
+inequality ( 7 ) is satisfied because @xmath114 .
+this means that the value of @xmath115 is mainly
+created by positive values of the autocorrelation
+function . the implication ( 8) needs an
+evaluation of the greatest value of the index
+@xmath65 where @xmath70 , but the solar data
+contain the most prominent period for @xmath116
+days because of the solar rotation . thus ,
+although @xmath117 for each @xmath118 , all sets
+@xmath41 ( see ( 5 ) and ( 6 ) ) without the set
+@xmath119 ( see ( 4 ) ) , which contains @xmath120
+$ ] , are considered . this situation is presented
+in figure [ f7 ] . in this figure two curves
+@xmath121 and @xmath122 are plotted . the vertical
+dotted lines evaluate the intervals where the sets
+@xmath107 ( for @xmath123 ) are searched . for
+such @xmath41 two numbers are written : in
+parentheses the value of @xmath65 for the time
+series @xmath101 and above it the value of
+@xmath65 for the time series @xmath102 . to make
+this figure clear the curves are plotted for the
+set @xmath124 only . ( in the right bottom corner
+information about the values of @xmath65 for the
+time series @xmath102 , for @xmath109 are written
+. ) the implication ( 8) is not true , because
+@xmath125 for @xmath126 . therefore ,
+@xmath43=153\notin c_6=[423,500]$ ] . moreover ,
+the autocorrelation function for @xmath127 $ ] is
+negative and the set @xmath128 is empty . thus ,
+@xmath129 . on the basis of these information one
+can state , that the periodogram peak at @xmath130
+days of the time series @xmath102 exists because
+of positive @xmath25 , but for @xmath23 from the
+intervals which do not contain this period .
+looking at the values of @xmath65 of the time
+series @xmath101 , one can notice that they
+decrease when @xmath23 increases until @xmath131 .
+this indicates , that when @xmath23 increases ,
+the contribution of the 11-year cycle to the peaks
+of the periodogram decreases . an increase of the
+value of @xmath65 is for @xmath132 for the both
+time series , although the contribution of the
+11-year cycle for the time series @xmath101 is
+insignificant . thus , this part of the
+autocorrelation function ( @xmath133 for the time
+series @xmath102 ) influences the @xmath21-th peak
+of the periodogram . this suggests that the
+periodicity at about 155 days is a harmonic of the
+periodicity from the interval of @xmath1 $ ] days
+. ( solid line ) and consecutively smoothed
+sunspot areas of the one rotation time interval
+@xmath134 ( dotted line ) . both indexes are
+presented on the left axis . the lower curve
+illustrates fluctuations of the sunspot areas
+@xmath135 . the dotted and dashed horizontal lines
+represent levels zero and @xmath136 respectively .
+the fluctuations are shown on the right axis . ]
+the described reasoning can be carried out for
+other values of the periodogram . for example ,
+the condition ( 8) is not satisfied for @xmath137
+( 250 , 222 , 200 days ) . moreover , the
+autocorrelation function at these points is
+negative . these suggest that there are not a true
+periodicity in the interval of [ 200 , 250 ] days
+. it is difficult to decide about the existence of
+the periodicities for @xmath138 ( 333 days ) and
+@xmath139 ( 286 days ) on the basis of above
+analysis . the implication ( 8) is not satisfied
+for @xmath139 and the condition ( 7 ) is not
+satisfied for @xmath138 , although the function
+@xmath25 of the time series @xmath102 is
+significantly positive for @xmath140 . the
+conditions ( 7 ) and ( 8) are satisfied for
+@xmath141 ( figure [ f8 ] ) and @xmath142 .
+therefore , it is possible to exist the
+periodicity from the interval of @xmath1 $ ] days
+. similar results were also obtained by @xcite for
+daily sunspot numbers and daily sunspot areas .
+she considered the means of three periodograms of
+these indexes for data from @xmath143 years and
+found statistically significant peaks from the
+interval of @xmath1 $ ] ( see @xcite , figure 2 )
+. @xcite studied sunspot areas from 1876 - 1999
+and sunspot numbers from 1749 - 2001 with the help
+of the wavelet transform . they pointed out that
+the 154 - 158-day period could be the third
+harmonic of the 1.3-year ( 475-day ) period .
+moreover , the both periods fluctuate considerably
+with time , being stronger during stronger sunspot
+cycles . therefore , the wavelet analysis suggests
+a common origin of the both periodicities . this
+conclusion confirms the de method result which
+indicates that the periodogram peak at @xmath144
+days is an alias of the periodicity from the
+interval of @xmath1 $ ] in order to verify the
+existence of the periodicity at about 155 days i
+consider the following time series : + @xmath145
++ @xmath146    + @xmath147   + the value @xmath134
+is calculated analogously to @xmath83 ( see sect .
+the values @xmath148 and @xmath149 are evaluated
+from the formula ( 9 ) . in the upper part of
+figure [ f9 ] the time series of sunspot areas
+@xmath150 of the one rotation time interval from
+the whole solar disk and the time series of
+consecutively smoothed sunspot areas @xmath151 are
+showed . in the lower part of figure [ f9 ] the
+time series of sunspot area fluctuations @xmath145
+is presented . on the basis of these data the
+maximum activity period of cycle 16 is evaluated .
+it is an interval between two strongest
+fluctuations e.a . @xmath152 $ ] rotations . the
+length of the time interval @xmath153 is 54
+rotations . if the about @xmath0-day ( 6 solar
+rotations ) periodicity existed in this time
+interval and it was characteristic for strong
+fluctuations from this time interval , 10 local
+maxima in the set of @xmath154 would be seen .
+then it should be necessary to find such a value
+of p for which @xmath155 for @xmath156 and the
+number of the local maxima of these values is 10 .
+as it can be seen in the lower part of figure [ f9
+] this is for the case of @xmath157 ( in this
+figure the dashed horizontal line is the level of
+@xmath158 ) . figure [ f10 ] presents nine time
+distances among the successive fluctuation local
+maxima and the horizontal line represents the
+6-rotation periodicity . it is immediately
+apparent that the dispersion of these points is 10
+and it is difficult to find even few points which
+oscillate around the value of 6 . such an analysis
+was carried out for smaller and larger @xmath136
+and the results were similar . therefore , the
+fact , that the about @xmath0-day periodicity
+exists in the time series of sunspot area
+fluctuations during the maximum activity period is
+questionable .    . the horizontal line represents
+the 6-rotation ( 162-day ) period . ]    ]    ]
+to verify again the existence of the about
+@xmath0-day periodicity during the maximum
+activity period in each solar hemisphere
+separately , the time series @xmath88 and @xmath89
+were also cut down to the maximum activity period
+( january 1925december 1930 ) . the comparison of
+the autocorrelation functions of these time series
+with the appriopriate autocorrelation functions of
+the time series @xmath88 and @xmath89 , which are
+computed for the whole 11-year cycle ( the lower
+curves of figures [ f1 ] and [ f2 ] ) , indicates
+that there are not significant differences between
+them especially for @xmath23=5 and 6 rotations (
+135 and 162 days ) ) . this conclusion is
+confirmed by the analysis of the time series
+@xmath146 for the maximum activity period . the
+autocorrelation function ( the lower curve of
+figure [ f11 ] ) is negative for the interval of [
+57 , 173 ] days , but the resolution of the
+periodogram is too low to find the significant
+peak at @xmath159 days . the autocorrelation
+function gives the same result as for daily
+sunspot area fluctuations from the whole solar
+disk ( @xmath160 ) ( see also the lower curve of
+figures [ f5 ] ) . in the case of the time series
+@xmath89 @xmath161 is zero for the fluctuations
+from the whole solar cycle and it is almost zero (
+@xmath162 ) for the fluctuations from the maximum
+activity period . the value @xmath163 is negative
+. similarly to the case of the northern hemisphere
+the autocorrelation function and the periodogram
+of southern hemisphere daily sunspot area
+fluctuations from the maximum activity period
+@xmath147 are computed ( see figure [ f12 ] ) .
+the autocorrelation function has the statistically
+significant positive peak in the interval of [ 155
+, 165 ] days , but the periodogram has too low
+resolution to decide about the possible
+periodicities . the correlative analysis indicates
+that there are positive fluctuations with time
+distances about @xmath0 days in the maximum
+activity period . the results of the analyses of
+the time series of sunspot area fluctuations from
+the maximum activity period are contradict with
+the conclusions of @xcite . she uses the power
+spectrum analysis only . the periodogram of daily
+sunspot fluctuations contains peaks , which could
+be harmonics or subharmonics of the true
+periodicities . they could be treated as real
+periodicities . this effect is not visible for
+sunspot data of the one rotation time interval ,
+but averaging could lose true periodicities . this
+is observed for data from the southern hemisphere
+. there is the about @xmath0-day peak in the
+autocorrelation function of daily fluctuations ,
+but the correlation for data of the one rotation
+interval is almost zero or negative at the points
+@xmath164 and 6 rotations . thus , it is
+reasonable to research both time series together
+using the correlative and the power spectrum
+analyses . the following results are obtained :
+1 . a new method of the detection of statistically
+significant peaks of the periodograms enables one
+to identify aliases in the periodogram . 2 .   two
+effects cause the existence of the peak of the
+periodogram of the time series of sunspot area
+fluctuations at about @xmath0 days : the first is
+caused by the 27-day periodicity , which probably
+creates the 162-day periodicity ( it is a
+subharmonic frequency of the 27-day periodicity )
+and the second is caused by statistically
+significant positive values of the autocorrelation
+function from the intervals of @xmath165 $ ] and
+@xmath166 $ ] days . the existence of the
+periodicity of about @xmath0 days of the time
+series of sunspot area fluctuations and sunspot
+area fluctuations from the northern hemisphere
+during the maximum activity period is questionable
+. the autocorrelation analysis of the time series
+of sunspot area fluctuations from the southern
+hemisphere indicates that the periodicity of about
+155 days exists during the maximum activity period
+. i appreciate valuable comments from professor j.
+jakimiec ."""
+
+from transformers import LEDForConditionalGeneration, LEDTokenizer
+import torch
+
+tokenizer = LEDTokenizer.from_pretrained("allenai/led-large-16384-arxiv")
+
+input_ids = tokenizer(LONG_ARTICLE, return_tensors="pt").input_ids.to("cuda")
+global_attention_mask = torch.zeros_like(input_ids)
+# set global_attention_mask on first token
+global_attention_mask[:, 0] = 1
+
+model = LEDForConditionalGeneration.from_pretrained("allenai/led-large-16384-arxiv", return_dict_in_generate=True).to("cuda")
+
+sequences = model.generate(input_ids, global_attention_mask=global_attention_mask).sequences
+
+summary = tokenizer.batch_decode(sequences)