%Paper: 
%From: Jacques Goldberg <goldberg@techunix.technion.ac.il>
%Date: Tue, 25 Jan 1994 13:10:12 +0200

%  ABOUT THE DRAWING:
%  There is ONE figure in this article
%  Select the right documentstyle below, to enable inclusion of the drawing.
%  If the drawing is wanted, also delete the % in the \epsfig command near
%  the end of the present file.
%  The drawing is supplied as Encapsulated PostScript file  jjgnufig.eps
%  which is appended, after a ---- cut here ------- line, behind the TeX stuff
%  The extra style file epsfig.sty is from CERN and I will gladly send it to
%  you if necessary.
%  An alternate regular PostScript copy of the drawing is file jjgnufig.ps,
%  which any PostScript printer should print as is, available on request
%  from  goldberg@tx.technion.ac.il
%
\documentstyle[prl,preprint,aps]{revtex}
%\documentstyle[epsfig,prl,preprint,aps]{revtex}
\begin{document}
% \draft command makes pacs numbers print
\tighten
\draft
\preprint{TECHNION-PH-94-2}
% repeat the \author\address pair as needed
\title{What is the statistical significance\\ of the solar
neutrino flux problem?}
\author{Jacques Goldberg}
\address{Department of Physics, Technion, Haifa, Israel}
\date{\today}
\maketitle
\begin{abstract}
The discrepancy between measured and predicted neutrino fluxes on
earth may be less significant than generally believed. Indeed,
a technically incorrect method has so far been used to propagate the
errors on the standard solar model input parameters through the
calculation of the predicted neutrino flux. Input parameters should have
been drawn from uniform not Gaussian distributions. The uncertainty on the
flux may therefore
have been substantially underestimated and the significance of
the observed discrepancy correspondingly overestimated. Results from
different experiments which cannot be directly
compared without reference to a model, may also be less incompatible
than currently believed.

\begin{center}
(Submitted to Physical Review Letters)
\end{center}
\end{abstract}
% insert suggested PACS numbers in braces on next line

\pacs{PACS numbers: 96.40.Tv, 96.60.Kx, 95.30.Cq, 02.50.Ng, 02.70.Lq}

%\twocolum
Uncomfortably, an almost trivial remark must be made about the technique
employed to compare the measured flux on Earth of solar neutrini,
with the prediction of the standard solar model.
For a clear statement of the problem,
the reader is referred to John Bahcall's concise talk at the XXVI International
Conference on High Energy Physics\cite{JBDAL}, with apologies to all other
scientists who contributed so much to this subject.

Very briefly, the flux of solar neutrini on Earth
is measured by several experiments using
various techniques\cite{DAVIS,GALLEX,SAGE,KAMIO}. This flux is independently
predicted using the standard solar model\cite{JB88,CT,JB92}. The prediction
depends on a collection of input parameters derived from prior experiments with
substantial uncertainties. These uncertainties are propagated in the neutrino
flux prediction, which is found incompatible with the experiments,
and further renders
incompatible such experiments that cannot be directly compared
but are nevertheless related through the model\cite{JBDAL}.

The complexity of the standard solar model and the subsequent neutrino
flux derivation precludes the straightforward analytical computation of the
covariance matrix which, together with the errors on the input parameters,
would supply the uncertainty associated with a predicted value. Standard
solar model authors\cite{JB88,CT} estimate this uncertainty by repeating
the calculation of the model with several, typically one thousand\cite{JBDAL},
sets of randomly selected input parameters, each drawn from a Gaussian
distribution defined by the experimental
value and error for the input parameter.

This technique is of course perfectly legitimate {\em except that the
sets of input parameters should be drawn from uniform, not Gaussian,
distributions.}

The rest of this note explains why, and what the consequences may be.

The source of this widely spread wrong practice
is the confusion between quantities whose
nature is random and quantities which are fixed even if unknown.

When an experimentalist measures some physical parameter, the
outcome of the measurement is a random variable, but the parameter is
a fixed constant of perfectly well defined value even if nothing is known
yet about that value. Usually, many independent random sources (of finite
variances to be absolutely correct) contribute to the probability distribution
of the outcome of the measurement which is Gaussian by virtue of the central
limit theorem; this Gaussian probability has a very well defined but unknown
expectation value (the actual value of the parameter being measured), and its
variance is the sum of the contributing variances.

When the value of the parameter is known, it is legitimate to
simulate the possible outcome of a measurement by drawing random numbers
from a Gaussian distribution of known expectation value and variance.
This is common practice, for example, in High Energy Physics
detector simulation, where
particle momenta are originally produced by an event generator with strict
momentum conservation, and are then smeared, and a detector momentum
dependent response
guessed, by adding random errors.

Because a physical parameter is not a random number, it is
completely meaningless to talk about the probability that such a parameter
has some given value\cite{SLM}. Usual intuitive speculations about the value
of an unknown parameter
begin by wrongly considering it as a probabilistic quantity. What an
experimental result stated as $P=\hat P\pm\sigma$ teaches us about the
actual value of some parameter $P$ which is being measured, is simply
this\cite{RPP}:

``There is no more than 31.73\% probability to be wrong when asserting that
the interval $[\hat P-\sigma,\hat P+\sigma]$ covers the actual value
of the parameter $P$.''

This is the common
definition of a confidence interval. Nothing is gained, no
statement is made more conservative, by making the interval twice as large
and correspondingly limiting the risk for a wrong statement to 4.55\%, or
any other combination, as long as the error distribution is Gaussian.
There is no additional information about where the actual value of $P$ is
located; it is even not necessarily covered by the confidence interval.
There is therefore no
reason to give preference to any value. After having taken a calculated risk
to be wrong when taking for granted that the actual value is indeed covered
by the confidence interval, there still is no reason to give preference
to any value within that interval.
Therefore, values to be tried for the parameter should be selected uniformly
over the arbitrarily wide confidence interval (without forgetting the risk
of systematic wrong choice since the interval may fail to cover
the actual value of the parameter), rather than following a
Gaussian distribution which very strongly prefers points close to the center
of the interval. The common intuition that values close to the center are
more likely, simply fails to recognize that the unknown but fixed central
value (the parameter) of a Gaussian distribution and one random but known
outcome (the measurement) drawn from this distribution are not functionally
symmetrical and may therefore not be exchanged.

By taking an infinitely wide confidence interval, the probability to make
a wrong statement when saying that this interval covers the actual value of
the parameter vanishes. This is the uninteresting situation in which
all models fit all data. Remembering that $\sigma$ is the initial
experimental error on the input parameter, the width
$2k\sigma$ of the interval to be used can however
be selected {\em after} an arbitrary (reasonable) threshold for wrong
statements $\epsilon$ has been set. For example, if the author of this note
would always
have set $\epsilon$ to the one standard deviation level, 31.73\% of
the papers carrying his name would contain false results. Before invoking
the need for new Physics following a discrepancy between an experiment and
a model, one might probably at least want a safer 1\% level, still
 one or two wrong
papers in a life time, which corresponds to $k=2.58$ ( 2.58
standard deviations).

The extension to more than one parameter is straightforward even if, again,
seemingly anti--intuitive. For a point in $n$-dimensional space to be located
outside of a given hypercube, it is necessary and sufficient that at least
one coordinate be located outside the corresponding confidence interval; more
coordinates out of their confidence intervals are not relevant. The probability
that at least one confidence interval does
not cover the corresponding parameter, which is
simply the complement of the probability that all intervals
cover their own parameter, is $1-(1-\epsilon)^n\approx n\epsilon$. The common
reduced width of uniform distributions for $n$ parameters is therefore the
number of standard deviations at which the Gaussian tail is equal to
$\epsilon/n$ where, we remind, $\epsilon$ is the threshold for the
acceptation of publishing a wrong result.

Predictions for the solar neutrino flux have been compared with
experiment\cite{JBDAL} by generating 1,000 solar models with 1,000
sets of input parameters drawn from Gaussian distributions. Although
finite width uniform distributions will clearly not let the guessed
parameters vary as far as normal distributions, the uniform density in
the hypercube is very different from the clustering around the central
values that Gaussian distributions provide.
Thus, the
distribution of predicted fluxes will almost certainly become
very substantially wider using uniform distributions.
We will however have to wait until the correct
method for guessing parameters is applied
to existing computer programs\cite{JB88,CT},
to see how the significance of
the observed discrepancies is affected.

Discussions with colleagues have shown that the very mundane facts
about statistics quoted above are not well perceived.
A very simple example is therefore presented to illustrate the
issue. Consider a simple pendulum, made of a point mass at the free end
of a thin string of length $l$. The model to be checked says that the
period $T$ is given by the formula $T=2\pi\sqrt{l\over g}$ with
$g={GM\over R^2}$ where $G$ is the gravitational constant, $M$ the mass of
a spherical Earth and $R$ its radius. The length $l$ has supposedly
been directly measured, and
a prediction for $l$ derived from measurements of $\pi,G,M,R,T$. The
measured value of $l$ is compared with the prediction, possibly
requiring ``new Physics''
such as massive strings or non spherical earth or highly viscous air etc...

Let us begin with the physical constant $\pi$, which as everybody knows
has a very well defined value. Hopefully, nobody will ever ask the question
``what is the probability that $\pi=3.17$ ?'', since $\pi$ is not a random
variable. Assume that we get the value of $\pi$ from a very cheap three
digits display pocket calculator, showing $\pi=3.14$. We know nothing about the
next digit (a fresh student might perhaps
write $\pi=3.145\pm0.005$: this however is not fair, because we do not take
31.73\% chances to make a wrong statement when stating that the interval
$[3.14,3.15]$ covers the true value of $\pi$, we take exactly 0\% such risk).
When estimating the effect on the predicted length of the
pendulum of the uncertainty on $\pi$, would it come to anybody's mind to
draw values for $\pi$ from a normal distribution centered around $3.145$
and of $\sigma=0.002$ ($\sigma$ set such that the random number
almost always remains in the $[3.14,3.15]$ interval)? Obviously, one will
rather try each of $0,1,\ldots,9$ in turn as the next digit; random numbers
from an {\em uniform} distribution do the same: the order of the trial is of
no importance, nor the exact value if enough trials are made.

Let us now turn to any of the four other parameters in our trivial problem.
Is there any reason to handle them differently, just because their
100\% confidence
intervals are not finite?

This trivial example has been simulated. Figure \ref{f1}
 shows the expected Gaussian
distribution of a direct measurement of a string of known length, and
simulated model predictions using Gaussian and uniform errors for the input
parameters.
We have artificially
considered $\pi$ as a parameter to enhance the
effect of having many parameters. Values and errors for $G,M,R$ have
been taken from litterature\cite{RPP,HP}, and errors of
the same order of magnitude have been
set for $\pi$ and the ``measured'' length and period. The period was set
to two seconds, the well known beat of a one meter pendulum.
The nominal true length was slightly shifted as shown
in the figure so that Gaussian simulated predictions come out incompatible
within several standard deviations while uniformly simulated ones ``agree''
quite often, always at the 1\% risk of wrong publication level. This exercise
has no other purpose than illustrating the broadening of the distribution
of the model prediction when uniform distributions are used. It does not imply
anything about the solar neutrino flux. At best, it may
hopefully trigger authors
of solar models to rerun their programs with uniform errors.

In conclusion, a more reliable estimate of the significance
of the discrepancy between experiments and standard solar model neutrino
fluxes could easily be obtained by rerunning computer simulations of the
standard solar model with input parameters drawn from uniform rather than
Gaussian distributions, of widths controlled by the priorily set
accepted risk
of publishing a wrong result.

Thanks are due to Arnon Dar and Asher Peres for several illuminating
discussions, and to the students of Physics 114103 whose endless questions
have helped me begin to learn a little bit of statistics.
This work was supported in part by the Technion Fund for the Promotion of
Research.


\begin{references}
\bibitem{JBDAL} J. Bahcall,
{\em XXVI International Conference
on High Energy Physics}, edited by J.R. Sanford (AIP, New York, 1993),
p. 1111.
\bibitem{DAVIS} R. Davis {\em et al., Proc. 21st ICRC},
edited by R.J. Protheroe, {\bf 12}, 143 (1990).
\bibitem{GALLEX} D. Vignaud,
{\em XXVI International Conference
on High Energy Physics}, edited by J.R. Sanford (AIP, New York, 1993),
p. 1093.
\bibitem{SAGE} V.N. Gavrin {\em et al.},
{\em XXVI International Conference
on High Energy Physics}, edited by J.R. Sanford (AIP, New York, 1993),
p 1101.
\bibitem{KAMIO} K. Nakamura, Nucl. Phys. B (Suppl.) {\bf 31} 105 (1993).
\bibitem{JB88} J.N. Bahcall and R.K. Ulrich, Rev. Mod. Phys. {\bf 60}, 297
(1988).
\bibitem{CT} S. Turck-Chi\`eze {\em et al.}, Astrophys. J.
{\bf 335}, 415 (1988).; S.~Turck-Chi\`eze {\em et al.}, Phys. Rep. {\bf 230},
57 (1993).
\bibitem{JB92} J.N. Bahcall and M.H. Pinsonneault, Rev. Mod. Phys.
{\bf 64}, 885 (1992).
\bibitem{SLM} S.L. Meyer, {\em Data analysis for scientists and engineers},
p. 294 (John Wiley \& Sons, New York, 1975).
\bibitem{RPP} Particle Data Group, Phys. Rev. D1 {\bf 45}, III.38  (1992).
\bibitem{HP} {\em Handbook of Physics and Chemistry}, 55th Edition,
p. F164, (CRC Press, Cleveland, 1974).
\end{references}


\begin{figure}
%
% Delete % at the beginning of next line to include the drawing.
%\epsfig{file=jjgnufig.eps,width=16cm}
\caption{The
expected distributions of direct length measurements (solid line), and
of the length predicted from a measurement
of the period using Gaussian random input parameters (broken line), or
uniform
random input parameters (dotted line). A cross shows the nominal length and
its one standard deviation error.}
\label{f1}
\end{figure}
\end{document}
% Behind this TeX input and not including the cut line is an Encapsulated
% PostScript file. See the header before the TeX stuff.
------------------------ cut here -----------------------------
%!PS-Adobe-2.0 EPSF-2.0
%%BoundingBox: 0 0 567 567
%%Title: JJGNUFIG.EPS
%%Creator: HIGZ Version 1.17/10D (FineSoft,JINR, fine@main1.jinr.dubna.su)
%%CreationDate: 21/01/94   12.18
%%EndComments
80 dict begin
/s {stroke} def /l {lineto} def /m {moveto} def /t { translate} def
/sw {stringwidth} def /r {rotate} def /rl {roll} def
/d {rlineto} def /rm {rmoveto} def /gr {grestore} def /f {eofill} def
/c {setrgbcolor} def /lw {setlinewidth} def /sd {setdash} def
/cl {closepath} def /sf {scalefont setfont} def
/box {m dup 0 exch d exch 0 d 0 exch neg d cl} def
/bl {box s} def /bf {box f} def
/mp {newpath /y exch def /x exch def} def
/side {[w .77 mul w .23 mul] .385 w mul sd w 0 l currentpoint t -144 r} def
/mr {mp x y w2 0 360 arc} def /m24 {mr s} def /m20 {mr f} def
/mb {mp x y w2 add m w2 neg 0 d 0 w neg d w 0 d 0 w d cl} def
/mt {mp x y w2 add m w2 neg w neg d w 0 d cl} def
/m21 {mb f} def /m25 {mb s} def /m22 {mt f} def /m26 {mt s} def
/m23 {mp x y w2 sub m w2 w d w neg 0 d cl f} def
 /m27 {mp x y w2 add m w3 neg w2 neg d w3 w2 neg d w3 w2 d cl s} def
 /m28 {mp x w2 sub y w2 sub w3 add m w3 0 d 0 w3 neg d w3 0 d 0 w3 d w3 0 d
 0 w3 d w3 neg 0 d 0 w3 d w3 neg 0 d 0 w3 neg d w3 neg 0 d cl s } def
 /m29 {mp gsave x w2 sub y w2 add w3 sub m currentpoint t
 4 {side} repeat cl fill gr} def
 /m30 {mp gsave x w2 sub y w2 add w3 sub m currentpoint t
 5 {side} repeat s gr} def /m31 {mp x y w2 sub m 0 w d x w2 sub y m w 0 d
 x w2 sub y w2 add m w w neg d x w2 sub y w2
 sub m w w d s} def
/m2 {mp x y w2 sub m 0 w d x w2 sub y m w 0 d s} def
/m5 {mp x w2 sub y w2 sub m w w d x w2 sub y w2 add m w w neg d s} def
/reencdict 24 dict def /ReEncode {reencdict begin /nco&na exch def
/nfnam exch def /basefontname exch def /basefontdict basefontname findfont def
/newfont basefontdict maxlength dict def basefontdict {exch dup /FID ne
{dup /Encoding eq {exch dup length array copy newfont 3 1 roll put} {exch
newfont 3 1 roll put} ifelse} {pop pop} ifelse } forall newfont
/FontName nfnam put nco&na aload pop nco&na length 2 idiv {newfont
/Encoding get 3 1 roll put} repeat nfnam newfont definefont pop end } def
/accvec [ 176 /agrave 181 /Agrave 190 /acircumflex 192 /Acircumflex
201 /adieresis 204 /Adieresis 209 /ccedilla 210 /Ccedilla 211 /eacute
212 /Eacute 213 /egrave 214 /Egrave 215 /ecircumflex 216 /Ecircumflex
217 /edieresis 218 /Edieresis 219 /icircumflex 220 /Icircumflex
221 /idieresis 222 /Idieresis 223 /ntilde 224 /Ntilde 226 /ocircumflex
228 /Ocircumflex 229 /odieresis 230 /Odieresis 231 /ucircumflex 236
/Ucircumflex
237 /udieresis 238 /Udieresis 239 /aring 242 /Aring 243 /ydieresis
244 /Ydieresis 246 /aacute 247 /Aacute 252 /ugrave 253 /Ugrave] def
/Times-Roman /Times-Roman accvec ReEncode
/Times-Italic /Times-Italic accvec ReEncode
/Times-Bold /Times-Bold accvec ReEncode
/Times-BoldItalic /Times-BoldItalic accvec ReEncode
/Helvetica /Helvetica accvec ReEncode
/Helvetica-Oblique /Helvetica-Oblique accvec ReEncode
/Helvetica-Bold /Helvetica-Bold accvec ReEncode
/Helvetica-BoldOblique /Helvetica-BoldOblique  accvec ReEncode
/Courier /Courier accvec ReEncode
/Courier-Oblique /Courier-Oblique accvec ReEncode
/Courier-Bold /Courier-Bold accvec ReEncode
/Courier-BoldOblique /Courier-BoldOblique accvec ReEncode
/oshow {gsave [] 0 sd true charpath stroke gr} def
/stwn { /fs exch def /fn exch def /text exch def fn findfont fs sf
 text sw pop xs add /xs exch def} def
/stwb { /fs exch def /fn exch def /nbas exch def /textf exch def
textf length /tlen exch def nbas tlen gt {/nbas tlen def} if
fn findfont fs sf textf dup length nbas sub nbas getinterval sw
pop neg xs add /xs exch def} def
/accspe [ 65 /plusminus 66 /bar 67 /existential 68 /universal
69 /exclam 70 /numbersign 71 /greater 72 /question 73 /integral
74 /colon 75 /semicolon 76 /less 77 /bracketleft 78 /bracketright
79 /greaterequal 80 /braceleft 81 /braceright 82 /radical
83 /spade 84 /heart 85 /diamond 86 /club 87 /lessequal
88 /multiply 89 /percent 90 /infinity 48 /circlemultiply 49 /circleplus
50 /emptyset 51 /lozenge 52 /bullet 53 /arrowright 54 /arrowup
55 /arrowleft 56 /arrowdown 57 /arrowboth 48 /degree 44 /comma 43 /plus
 45 /angle 42 /angleleft 47 /divide 61 /notequal 40 /equivalence 41 /second
 97 /approxequal 98 /congruent 99 /perpendicular 100 /partialdiff 101 /florin
 102 /intersection 103 /union 104 /propersuperset 105 /reflexsuperset
 106 /notsubset 107 /propersubset 108 /reflexsubset 109 /element 110
/notelement
 111 /gradient 112 /logicaland 113 /logicalor 114 /arrowdblboth
 115 /arrowdblleft 116 /arrowdblup 117 /arrowdblright 118 /arrowdbldown
 119 /ampersand 120 /omega1 121 /similar 122 /aleph ] def
/Symbol /Special accspe ReEncode
gsave .25 .25 scale
%%EndProlog
 gsave 0 0 t 0 0 0 c [] 0 sd 1 lw 1814 1814 227 227 bl 227 227 m 227 227 l 257
 227 l 287 227 l 318 227 l 348 227 l 378 227 l 408 227 l 438 227 l 469 227 l
499
 227 l 529 227 l 559 227 l 590 227 l 620 227 l 650 227 l 680 227 l 711 227 l
741
 227 l 771 227 l 801 227 l 832 227 l 862 227 l 892 227 l 922 227 l 953 227 l
983
 227 l 983 227 l 1013 227 l 1043 227 l 1074 227 l 1104 227 l 1134 227 l 1164
227
 l 1194 227 l 1225 227 l 1255 227 l 1285 227 l 1315 227 l 1346 227 l 1346 326 l
 1376 326 l 1376 1006 l 1406 1006 l 1406 1761 l 1436 1761 l 1436 1860 l 1467
 1860 l 1467 831 l 1497 831 l 1497 283 l 1527 283 l 1527 241 l 1557 241 l 1557
 227 l 1588 227 l 1618 227 l 1648 227 l 1678 227 l 1709 227 l 1739 227 l 1739
 227 l 1769 227 l 1799 227 l 1830 227 l 1860 227 l 1890 227 l 1920 227 l 1950
 227 l 1981 227 l 2011 227 l 2041 227 l s 227 227 m 227 2041 l s 261 227 m 227
 227 l s 244 274 m 227 274 l s 244 321 m 227 321 l s 244 368 m 227 368 l s 244
 416 m 227 416 l s 261 463 m 227 463 l s 244 510 m 227 510 l s 244 557 m 227
557
 l s 244 604 m 227 604 l s 244 652 m 227 652 l s 261 699 m 227 699 l s 244 746
m
 227 746 l s 244 793 m 227 793 l s 244 840 m 227 840 l s 244 888 m 227 888 l s
 261 935 m 227 935 l s 244 982 m 227 982 l s 244 1029 m 227 1029 l s 244 1076 m
 227 1076 l s 244 1124 m 227 1124 l s 261 1171 m 227 1171 l s 244 1218 m 227
 1218 l s 244 1265 m 227 1265 l s 244 1312 m 227 1312 l s 244 1359 m 227 1359 l
 s 261 1407 m 227 1407 l s 244 1454 m 227 1454 l s 244 1501 m 227 1501 l s 244
 1548 m 227 1548 l s 244 1595 m 227 1595 l s 261 1643 m 227 1643 l s 244 1690 m
 227 1690 l s 244 1737 m 227 1737 l s 244 1784 m 227 1784 l s 244 1831 m 227
 1831 l s 261 1879 m 227 1879 l s 261 1879 m 227 1879 l s 244 1926 m 227 1926 l
 s 244 1973 m 227 1973 l s 244 2020 m 227 2020 l s
 /xs 0 def
(0)
 /Times-Italic   48 stwn
 gsave 181 211
 t   0 r  xs neg 0 t 0 0 m
 /Times-Italic findfont   48 sf 0    0 m
(0)
 show
 gr
 /xs 0 def
(50)
 /Times-Italic   48 stwn
 gsave 181 447
 t   0 r  xs neg 0 t 0 0 m
 /Times-Italic findfont   48 sf 0    0 m
(50)
 show
 gr
 /xs 0 def
(100)
 /Times-Italic   48 stwn
 gsave 181 683
 t   0 r  xs neg 0 t 0 0 m
 /Times-Italic findfont   48 sf 0    0 m
(100)
 show
 gr
 /xs 0 def
(150)
 /Times-Italic   48 stwn
 gsave 181 919
 t   0 r  xs neg 0 t 0 0 m
 /Times-Italic findfont   48 sf 0    0 m
(150)
 show
 gr
 /xs 0 def
(200)
 /Times-Italic   48 stwn
 gsave 181 1155
 t   0 r  xs neg 0 t 0 0 m
 /Times-Italic findfont   48 sf 0    0 m
(200)
 show
 gr
 /xs 0 def
(250)
 /Times-Italic   48 stwn
 gsave 181 1391
 t   0 r  xs neg 0 t 0 0 m
 /Times-Italic findfont   48 sf 0    0 m
(250)
 show
 gr
 /xs 0 def
(300)
 /Times-Italic   48 stwn
 gsave 181 1627
 t   0 r  xs neg 0 t 0 0 m
 /Times-Italic findfont   48 sf 0    0 m
(300)
 show
 gr
 /xs 0 def
(350)
 /Times-Italic   48 stwn
 gsave 181 1863
 t   0 r  xs neg 0 t 0 0 m
 /Times-Italic findfont   48 sf 0    0 m
(350)
 show
 gr
 227 227 m 2041 227 l s 227 261 m 227 227 l s 287 244 m 287 227 l s 348 244 m
 348 227 l s 408 244 m 408 227 l s 469 244 m 469 227 l s 529 261 m 529 227 l s
 590 244 m 590 227 l s 650 244 m 650 227 l s 711 244 m 711 227 l s 771 244 m
771
 227 l s 832 261 m 832 227 l s 892 244 m 892 227 l s 953 244 m 953 227 l s 1013
 244 m 1013 227 l s 1074 244 m 1074 227 l s 1134 261 m 1134 227 l s 1194 244 m
 1194 227 l s 1255 244 m 1255 227 l s 1315 244 m 1315 227 l s 1376 244 m 1376
 227 l s 1436 261 m 1436 227 l s 1497 244 m 1497 227 l s 1557 244 m 1557 227 l
s
 1618 244 m 1618 227 l s 1678 244 m 1678 227 l s 1739 261 m 1739 227 l s 1799
 244 m 1799 227 l s 1860 244 m 1860 227 l s 1920 244 m 1920 227 l s 1981 244 m
 1981 227 l s 2041 261 m 2041 227 l s
 /xs 0 def
(0.97)
 /Times-Italic   48 stwn
 gsave 227 172
 t   0 r  xs 2 div neg 0 t 0 0 m
 /Times-Italic findfont   48 sf 0    0 m
(0.97)
 show
 gr
 /xs 0 def
(0.98)
 /Times-Italic   48 stwn
 gsave 529 172
 t   0 r  xs 2 div neg 0 t 0 0 m
 /Times-Italic findfont   48 sf 0    0 m
(0.98)
 show
 gr
 /xs 0 def
(0.99)
 /Times-Italic   48 stwn
 gsave 832 172
 t   0 r  xs 2 div neg 0 t 0 0 m
 /Times-Italic findfont   48 sf 0    0 m
(0.99)
 show
 gr
 /xs 0 def
(1)
 /Times-Italic   48 stwn
 gsave 1134 172
 t   0 r  xs 2 div neg 0 t 0 0 m
 /Times-Italic findfont   48 sf 0    0 m
(1)
 show
 gr
 /xs 0 def
(1.01)
 /Times-Italic   48 stwn
 gsave 1436 172
 t   0 r  xs 2 div neg 0 t 0 0 m
 /Times-Italic findfont   48 sf 0    0 m
(1.01)
 show
 gr
 /xs 0 def
(1.02)
 /Times-Italic   48 stwn
 gsave 1739 172
 t   0 r  xs 2 div neg 0 t 0 0 m
 /Times-Italic findfont   48 sf 0    0 m
(1.02)
 show
 gr
 /xs 0 def
(1.03)
 /Times-Italic   48 stwn
 gsave 2041 172
 t   0 r  xs 2 div neg 0 t 0 0 m
 /Times-Italic findfont   48 sf 0    0 m
(1.03)
 show
 gr
 [12 12] 0 sd 227 227 m 227 227 l 257 227 l 287 227 l 318 227 l 348 227 l 378
 227 l 408 227 l 438 227 l 469 227 l 499 227 l 529 227 l 559 227 l 590 227 l
590
 241 l 620 241 l 620 236 l 650 236 l 650 260 l 680 260 l 680 283 l 711 283 l
711
 312 l 741 312 l 741 387 l 771 387 l 771 378 l 801 378 l 801 472 l 832 472 l
832
 534 l 862 534 l 862 675 l 892 675 l 892 670 l 922 670 l 922 807 l 953 807 l
953
 670 l 983 670 l 983 661 l 1013 661 l 1013 694 l 1043 694 l 1043 472 l 1074 472
 l 1074 491 l 1104 491 l 1104 345 l 1134 345 l 1134 354 l 1164 354 l 1164 255 l
 1194 255 l 1194 250 l 1225 250 l 1225 246 l 1255 246 l 1255 236 l 1285 236 l
 1285 232 l 1315 232 l 1315 227 l 1346 227 l 1376 227 l 1406 227 l 1436 227 l
 1467 227 l 1497 227 l 1527 227 l 1557 227 l 1588 227 l 1618 227 l 1648 227 l
 1678 227 l 1709 227 l 1739 227 l 1739 227 l 1769 227 l 1799 227 l 1830 227 l
 1860 227 l 1890 227 l 1920 227 l 1950 227 l 1981 227 l 2011 227 l 2041 227 l s
 [4 8] 0 sd 227 227 m 227 227 l 257 227 l 287 227 l 318 227 l 348 227 l 378 227
 l 408 227 l 438 227 l 438 236 l 469 236 l 469 232 l 499 232 l 499 274 l 529
274
 l 529 283 l 559 283 l 559 316 l 590 316 l 590 312 l 620 312 l 620 274 l 650
274
 l 650 345 l 680 345 l 680 364 l 711 364 l 711 383 l 741 383 l 741 411 l 771
411
 l 771 425 l 801 425 l 801 477 l 832 477 l 832 496 l 862 496 l 862 458 l 892
458
 l 892 515 l 922 515 l 922 449 l 953 449 l 953 510 l 983 510 l 983 430 l 1013
 430 l 1013 449 l 1043 449 l 1043 463 l 1074 463 l 1074 453 l 1104 453 l 1104
 434 l 1134 434 l 1134 425 l 1164 425 l 1164 401 l 1194 401 l 1225 401 l 1225
 359 l 1255 359 l 1255 326 l 1285 326 l 1285 288 l 1315 288 l 1315 269 l 1346
 269 l 1346 236 l 1376 236 l 1376 265 l 1406 265 l 1406 241 l 1436 241 l 1436
 232 l 1467 232 l 1467 227 l 1497 227 l 1527 227 l 1557 227 l 1588 227 l 1618
 227 l 1648 227 l 1678 227 l 1709 227 l 1739 227 l 1739 227 l 1769 227 l 1799
 227 l 1830 227 l 1860 227 l 1890 227 l 1920 227 l 1950 227 l 1981 227 l 2011
 227 l 2041 227 l s [] 0 sd 1406 345 m 1467 345 l s 1436 335 m 1436 354 l s
gr gr
end
%%EOF

