\input rmacro.sty
%\hsize16383.99999pt %uncomment to display all hyphens in the log file
%\parfillskip 0pt plus0pt minus0pt
%\showboxdepth-1
%\tolerance-1
%\hbadness0
%\hyphenpenalty=-1000 %just have lots of hyphenation in the dvi file
%\pretolerance-1
\tolerance=1000
%\doublehyphendemerits=-100000
%\finalhyphendemerits=-100000
\uchyph=1
\defaulthyphenchar=`\-
\hyphenchar\tenrm=-1

%Load the fonts
\font\cyr=xncyr10\relax %upright
\font\bcyr=xncyrb10\relax %bold
\font\icyr=xncyri10\relax %italic
\font\ccyr=xncyrc10\relax %small caps
\font\hcyr=xncyrh10\relax %sans serif
\font\sf=cmss10\relax %Latin sans serif
% AMSTeX ripoff
\font\teneuf=eufm10
\font\tenmsy=msym10
\newfam\euffam
\newfam\msyfam
\textfont\euffam=\teneuf
\textfont\msyfam=\tenmsy
\def\frak#1{{\fam\euffam#1}}
\def\Bbb#1{{\fam\msyfam#1}}
%\let\Bbb=\relax %If you don't have these
%\let\frak=\relax %If you don't have these

\def\acircle{\mathaccent"7017 } %circle over math accent
\def\qed{\ifhmode\unskip\nobreak\fi\ifmmode\ifinner\else\hskip5pt\fi\fi
 \hbox{\ \vrule width4pt height6pt depth1.5pt\hskip0pt}}

\cdefTranslitX

\noindent{\rm Russian alphabet, transliterated one way:}

\hbox{0123456789:;<=>?@ABCDEFGHIJKLMNO}

\hbox{PQRSTUVWXYZ[\]^_`abcdefghijklmno}

\cdefTranslitY

\noindent{\rm Russian alphabet, transliterated another way:}

\hbox{0123456789:;<=>?@ABCDEFGHIJKLMNO}

\hbox{PQRSTUVWXYZ[\]^_`abcdefghijklmno}

\cdefRussian

\cyr
\noindent{\rm Russian alphabet, native:}

\hbox{0123456789:;<=>?@ABCDEFGHIJKLMNO}

\hbox{PQRSTUVWXYZ[\]^_`abcdefghijklmno}

{\icyr
\noindent{\rm Russian alphabet,} Zc`aXR:

\hbox{0123456789:;<=>?@ABCDEFGHIJKLMNO}

\hbox{PQRSTUVWXYZ[\]^_`abcdefghijklmno}

}

\bigskip\noindent{\sf From} {\hcyr 4.0. Ac_`c]U]Z^, <3`c__k \Pb`Xf>}

<PZaX\P[l]kU ]U_`XR^TX\kU [^ZP[l]^ S`c__k $GL(n,{\Bbb R})$ ]PT _^[U\
RUiUabRU]]ke gXaU[ $\Bbb R$ ^_XaP]]k Ac_`c]U]Z^ R abPblU (9), STU cabP]^R[U]k
a[UTcniXU bU^`U\k:

1. 5a[X ]UgUb]^U gXa[^ $n>1$, b^ R $GL(n,{\Bbb R})$ ]Ub ]U_`XR^TX\ke [^ZP[l]^
]X[l_^bU]b]ke _^TS`c__.

2. 5a[X $n$---gUb]^U gXa[^, ]U oR[oniUUao abU_U]ln TR^YZX, b^ R $GL(n,{\Bbb
R})$ a b^g]^abln T^ a^_`oVU]]^abX X\UUbao b^[lZ^ ^T]P \PZaX\P[l]Po ]U_`XR^TX\Po
[^ZP[l]^ ]X[l_^bU]b]Po _^TS`c__P.

3. 5a[X $n=2^\alpha\quad(\alpha\ge1)$, b^ \]^VUabR^ RaUe \PZaX\P[l]ke
]U_`XR^TX\ke [^ZP[l]^ ]X[l_^bU]b]ke _^TS`c__ S`c__ $GL(n,{\Bbb R})$ `PWQXRPUbao
]P TRP Z[PaaP a^_`oVU]]ke _^TS`c__.

4. ;nQPo ]U_`XR^TX\Po [^ZP[l]^ ]X[l_^bU]b]Po _^TS`c__P R $GL(n,{\Bbb R})$ _`X
$n>2$ X\_`X\XbXR]P.

\bigskip\noindent{\sf From} {\hcyr  =.2. 3^S^[l, <BP`Pa 1c[lQP>}

:^[UQ]c[Pal Rao b^[_P. A]PgP[P ]P \XS _`^]Ua[^al _^ RaU\c QU`USc \^[gP]XU,
Z^b^`^U cabP]PR[XRPUbao _U`UT aRX`U_^n Qc`UY, X _^b^\ RT`cS _^T]o[Xal `UgX, X
RUal WPS^R^`X[ QU`US.

---:PZ, gb^Qk VXTk TU`VP[X ]P P`U]TU e`XabXP]aZXU fU`ZRX! Gb^Qk ZaU]TWk
WP_`oSP[X R ^S[^Q[X _`PR^a[PR]ke e`XabXP]! :PZ, gb^Qk _^_cabXbl bPZXU \cgU]Xo
]P `caaZ^Y WU\[U ^b _`^Z[obke ]UT^RU`Z^R! Gb^Qk R^b bPZ _^abc_P[X a
_^[Z^R]XZP\X X SUbl\P]^\! 4P ]U QcTUb VU aUS^, ]U QcTUb!

BPZXU a[^RP _U`U[UbP[X _^ RaU\ Z^]fP\. 7Phc\U[X WP_^`^Vfk X _^gco[X aR^n aX[k.
Bcb cVU ]U Qk[^ R^[]U]XY [USZ^\ka[U]]^S^ ]P`^TP; R^[]^RP[Xal RaU eP`PZbU`k
boVU[kU X Z`U_ZXU, Z^b^`kU ]U aZ^`^ ]PZP[o[Xal, ]^, ]PZP[XRhXal, c_^`]^ X T^[S^
e`P]X[X R aUQU R]cb`U]]XY VP`.

---?U`URUhPbl Ran VXT^Rc!---`PWTP[^al XW b^[_k.---?cabl VU ]U hlnb XW _^_^RaZXe
`XW nQ^Z aR^X\ VXT^RZP\! ?cabl VU ]U abPRob W]PgZ^R ]P aRobke _PaePe!
?U`Ub^_Xbl Xe RaUe, _^SP]fUR, R 4]U_`U.---A[^RP mbX, _`^XW]UaU]]kU ZU\-b^ XW
b^[_k, _`^[UbU[X \^[]XUY _^ RaU\ S^[^RP\, X b^[_P `X]c[Pal ]P _`UT\UablU, a
VU[P]XU\ _U`U`UWPbl RaUe VXT^R.

1UT]kU ak]k 8W`PX[o, `PabU`oRhX RaU _`XacbabRXU aR^US^ X QUW b^S^ \U[Z^S^ TceP,
_`obP[Xal R _cabke S^`U[^g]ke Q^gZPe, R _UgZPe X TPVU WP_^[WkRP[X _^T nQZX
aR^Xe VXT^R^Z; ]^ Z^WPZX RUWTU Xe ]Pe^TX[X.

---Oa]^RU[l\^V]kU _P]k!---Z`XgP[ ^TX], Rka^ZXY X T[X]]kY, ZPZ _P[ZP, VXT,
Rkac]cRhX XW ZcgX aR^Xe b^RP`XiUY VP[Zcn aR^n `^Vc, XaZ^RU`ZP]]cn
ab`Pe^\.---Oa]^RU[l\^V]kU _P]k! A[^R^ b^[lZ^ TPYbU ]P\ aZPWPbl, ^T]^ a[^R^! <k
bPZ^U ^QloRX\ RP\, gUS^ UiU ]XZ^STP ]U a[khP[X, bPZ^U RPV]^U, gb^ ]U \^V]^
aZPWPbl ZPZ^U RPV]^U!

---=c, _cabl aZPVcb,---aZPWP[ 1c[lQP, Z^b^`kY RaUSTP [nQX[ Rka[chPbl
^QRX]oU\^S^.

---Oa]kU _P]k!---_`^XW]Ua VXT.---BPZXe _P]^R UiU ]XZ^STP ]U RXTkRP]^. 5Y Q^Sc,
]XZ^STP! BPZXe T^Q`ke, e^`^hXe X e`PQ`ke ]U Qk[^ UiU ]P aRUbU!..---3^[^a US^
c\X`P[ X T`^VP[ ^b ab`PeP.---:PZ \^V]^, gb^Qk \k Tc\P[X _`^ WP_^`^VfUR
gb^-]XQcTl ]Ue^`^hUU! BU a^RaU\ ]U ]PhX, bU, gb^ P`U]TPbc`abRcnb ]P CZ`PY]U! 5Y
Q^Sc, ]U ]PhX! B^ a^RaU\ ]U VXTk; b^ gU`b W]PUb gb^. B^ bPZ^U, gb^ b^[lZ^
_^_[URPbl ]P ]US^, TP X Q`^aXbl! 2^b X ^]X aZPVcb b^ VU. =U _`PRTP [X, H[U\P,
X[X bk, H\c[l?

---5Y Q^Sc, _`PRTP!---^bRUgP[X XW b^[_k H[U\P X H\c[l R XW^T`P]]ke o[^\ZPe, ^QP
QU[kU, ZPZ S[X]P.

---<k ]XZ^STP UiU,---_`^T^[VP[ T[X]]ke VXT,---]U a^S[PhP[Xal a ]U_`XobU[o\X. 0
ZPb^[XZ^R \k X W]Pbl ]U e^bX\; _cabl X\ gU`b _`Xa]Xbao! <k a WP_^`^VfP\X ZPZ
Q`Pblo `^T]kU...

---:PZ? Gb^Qk WP_^`^Vfk Qk[X a RP\X Q`Pblo?---_`^XW]Ua ^TX] XW b^[_k.---=U
T^VTUbUal, _`^Z[obkU VXTk! 2 4]U_` Xe, _P]^RU! 2aUe _^b^_Xbl, _^SP]fUR!

MbX a[^RP Qk[X aXS]P[^\. 6XT^R `PaeRPbP[X _^ `cZP\ X ]PgP[X hRk`obl R R^[]k.
6P[ZXY Z`XZ `PWTP[ao a^ RaUe ab^`^], ]^ ac`^RkU WP_^`^Vfk b^[lZ^ a\Uo[Xal,
RXTo, ZPZ VXT^RaZXU ]^SX R QPh\PZPe X gc[ZPe Q^[bP[Xal ]P R^WTceU. 1UT]kY
^`Pb^`, ]PZ[XZPRhXY aP\ ]P aR^n hUn QUTc, RkaZ^gX[ XW ZPdbP]P, WP Z^b^`kY Qk[^
US^ ceRPbX[X, R ^T]^\ _US^\ X cWZ^\ ZP\W^[U, aeRPbX[ WP ]^SX 1c[lQc X VP[ZX\
S^[^a^\ \^[X[:

---2U[XZXY S^a_^TX], oa]^RU[l\^V]kY _P]! O W]P[ X Q`PbP RPhUS^, _^Z^Y]^S^
4^`^hP! 1k[ R^X] ]P cZ`PhU]XU RaU\c `kfP`abRc. O U\c R^aU\la^b fUZX]^R TP[,
Z^STP ]cV]^ Qk[^ RkZc_Xblao XW _[U]P c bc`Z^R.

---Bk W]P[ Q`PbP?---a_`^aX[ BP`Pa.

---5Y Q^Sc, W]P[! 2U[XZ^Tch]kY Qk[ _P].

---0 ZPZ bUQo W^Rcb?

---O]ZU[l.

---E^`^h^,---aZPWP[ BP`Pa X _^b^\, _^Tc\PR, ^Q`PbX[ao Z Z^WPZP\ X _`^S^R^`X[
bPZ:---6XTP QcTUb RaUSTP R`U\o _^RUaXbl, Z^STP QcTUb ]cV]^, P ]P aUS^T]o
^bTPYbU US^ \]U.---AZPWPRhX mb^, BP`Pa _^RU[ US^ Z aR^U\c ^Q^Wc, R^W[U Z^b^`^S^
ab^o[X Z^WPZX US^.---=c, _^[UWPY _^T bU[USc, [UVX bP\ X ]U _^hURU[Xal; P Rk,
Q`Pbfk, ]U Rk_caZPYbU VXTP.

\bigskip\noindent{\sf From} {\hcyr  D.8. :P`_U[URXg, ;.5. APT^RaZXY, <M[U\U]bk
[X]UY]^Y P[SUQ`k X [X]UY]^S^ _`^S`P\\X`^RP]Xo>}

\centerline{\bcyr \S1.9 @P]S \Pb`Xfk}

1. ?`UT_^[^VX\, gb^ ]P\ TP]P \Pb`XfP $U$ XW $m$ ab`^Z X $n$ ab^[Qf^R:
$$U = \left(\matrix{
a_{11} & a_{12} & \ldots & a_{1n} \cr
a_{21} & a_{22} & \ldots & a_{2n} \cr
\ldots & \ldots & \ldots & \ldots \cr
a_{m1} & a_{m2} & \ldots & a_{mn} \cr
}\right).$$
2kTU[X\ R ]UY _`^XWR^[l]^ $k$ ab`^Z X $k$ ab^[Qf^R. M[U\U]bk, ab^oiXU ]P
_U`UaUgU]XX RkTU[U]]ke ab`^Z X ab^[Qf^R, ^Q`PWcnb ZRPT`Pb]cn \Pb`Xfc $k$-S^
_^`oTZP. >_`UTU[XbU[l mb^Y \Pb`Xfk ]PWkRPUbao {\icyr \X]^`^\} $k$-S^ _^`oTZP
\Pb`Xfk $U$. 2kQX`Po RaUR^W\^V]k\X a_^a^QP\X _^ $k$ ab`^Z X $k$ ab^[Qf^R,
_^[cgPU\ RaUR^W\^V]kU \X]^`k $k$-S^ _^`oTZP \Pb`Xfk $U$. >gURXT]^, gb^ ^QiUU
gXa[^ bPZXe \X]^`^R a^abPR[oUb $C_m^k C_n^k$.

<Pb`XfP $U$ ^Q[PTPUb, bPZX\ ^Q`PW^\, \X]^`P\X `PW[Xg]ke _^`oTZ^R. <X]^`P\X
_U`R^S^ _^`oTZP oR[onbao aP\X m[U\U]bk \Pb`Xfk $U$. <PZaX\P[l]^ R^W\^V]kY
_^`oT^Z \X]^`^R `PRU], ^gURXT]^, ]PX\U]lhU\c XW gXaU[ $m$ X $n$.

2. @Paa\^b`X\ R \Pb`XfU $U$ RaU UU \X]^`k RaUe R^W\^V]ke _^`oTZ^R. 2kTU[X\ bU
XW ]Xe, Z^b^`kU ^b[Xg]k ^b ]c[o. 2RUTU\ a[UTcniUU ^_`UTU[U]XU.

{\ccyr >_`UTU[U]XU 4.} =PXQ^[lhXY XW _^`oTZ^R \X]^`^R, ^b[Xg]ke ^b ]c[o,
]PWkRPUbao {\icyr `P]S^\ \Pb`Xfk.}

{\ccyr @PWjoa]U]XU.} 5a[X `P]S \Pb`Xfk $U$ `PRU] $r$, b^ mb^ ^W]PgPUb, gb^ R
\Pb`XfU $U$ X\UUbao ^b[Xg]kY ^b ]c[o \X]^` _^`oTZP $r$, ]^ RaoZXY \X]^` _^`oTZP
Q^[lhUS^, gU\ $r$, `PRU] ]c[n.

5a[X a`UTX m[U\U]b^R \Pb`Xfk X\UUbao e^bo Qk ^TX] ^b[Xg]kY ^b ]c[o, b^ `P]S
\Pb`Xfk WPRUT^\^ ]U \U]lhU UTX]Xfk $(r\ge1)$, bPZ ZPZ \X]^`P\X _U`R^S^ _^`oTZP
oR[onbao aP\X m[U\U]bk \Pb`Xfk. 5a[X VU RaU m[U\U]bk \Pb`Xfk `PR]k ]c[n, b^ X
`P]S UU _`X]X\Pnb `PR]k\ ]c[n.

@P]S \Pb`Xfk $U$ ^Q^W]PgPUbao gU`UW $r(U)$. 2 TP[l]UYhU\ _^[UW]^ X\Ubl R RXTc
a[UTcnicn [U\\c.

{\ccyr ;U\\P 1.} {\icyr 5a[X RaU \X]^`k $l$-S^ _^`oTZP \Pb`Xfk $U$ `PR]k ]c[n,
b^ RaU UU \X]^`k _^`oTZ^R Q^[lhXe, gU\ $l$, bPZVU `PR]k ]c[n.}

{\ccyr 4^ZPWPbU[labR^.} ?cabl, a^S[Pa]^ ca[^RXn [U\\k, RaU \X]^`k $l$-S^
_^`oTZP WPTP]]^Y \Pb`Xfk `PR]k ]c[n. 4^ZPVU\ a]PgP[P, gb^ `PR]k ]c[n RaU UU
\X]^`k $l+1$-S^ _^`oTZP. 2 aP\^\ TU[U, `Paa\^b`X\ [nQ^Y \X]^` $l+1$-S^ _^`oTZP.
Mb^b \X]^`, ZPZ ^_`UTU[XbU[l $l+1$-S^ _^`oTZP, `PRU] ac\\U _`^XWRUTU]XY
m[U\U]b^R ZPZ^S^-[XQ^ US^ ab^[QfP ]P Xe P[SUQ`PXgUaZXU T^_^[]U]Xo.
0[SUQ`PXgUaZ^U T^_^[]U]XU [nQ^S^ m[U\U]bP \X]^`P $M$ oR[oUbao, ^gURXT]^,
]UZ^b^`k\ \X]^`^\ $l$-S^ _^`oTZP \Pb`Xfk $U$, RWobk\ a a^^bRUbabRcniX\ W]PZ^\.
?^mb^\c P[SUQ`PXgUaZ^U T^_^[]U]XU [nQ^S^ m[U\U]bP \X]^`P $M$ `PR]^ ]c[n, P
W]PgXb, `PRU] ]c[n X aP\ \X]^` $M$.

0]P[^SXg]k\X `PaacVTU]Xo\X, Xa_^[lWco cVU T^ZPWP]]kY ]P\X dPZb `PRU]abRP ]c[n
RaUe \X]^`^R $l+1$-S^ _^`oTZP, T^ZPWkRPU\ `PRU]abR^ ]c[n RaUe \X]^`^R $l+2$-S^
_^`oTZP X b.T. ;U\\P T^ZPWP]P. \qed

\bigskip\noindent{\sf From} {\hcyr DUT^` <XePY[^RXg 4^ab^URaZXY, <?`Uabc_[U]XU
X ]PZPWP]XU>}

2 ]PgP[U Xn[o, R g`UWRkgPY]^ VP`Z^U R`U\o, _^T RUgU`, ^TX] \^[^T^Y gU[^RUZ
RkhU[ XW aR^UY ZP\^`ZX, Z^b^`cn ]P]X\P[ ^b VX[lf^R R Ab^[o`]^\ _U`Uc[ZU, ]P
c[Xfc, X \UT[U]]^, ZPZ Qk R ]U`UhX\^abX, ^b_`PRX[ao Z :^ZchZX]c \^abc.

>] Q[PS^_^[cg]^ XWQUS]c[ Rab`UgX a^ aR^UY e^WoYZ^Y ]P [Uab]XfU. :P\^`ZP US^
_`Xe^TX[Pal _^T aP\^n Z`^R[UY Rka^Z^S^ _obXmbPV]^S^ T^\P X _^e^TX[P Q^[UU ]P
hZPd, gU\ ]P ZRP`bX`c. :RP`bX`]Po VU e^WoYZP US^, c Z^b^`^Y ^] ]P]X\P[ mbc
ZP\^`Zc a ^QUT^\ X _`Xa[cS^Y, _^\UiP[Pal ^T]^Y [Uab]XfUY ]XVU, R ^bTU[l]^Y
ZRP`bX`U, X ZPVTkY `PW, _`X Rke^TU ]P c[Xfc, U\c ]U_`U\U]]^ ]PT^ Qk[^ _`^e^TXbl
\X\^ e^WoYZX]^Y Zce]X, _^gbX RaUSTP ]PabUVl ^bR^`U]]^Y ]P [Uab]Xfc. 8 ZPVTkY
`PW \^[^T^Y gU[^RUZ, _`^e^To \X\^, gcRabR^RP[ ZPZ^U-b^ Q^[UW]U]]^U X b`ca[XR^U
^iciU]XU, Z^b^`^S^ abkTX[ao X ^b Z^b^`^S^ \^`iX[ao. >] Qk[ T^[VU] Z`cS^\
e^WoYZU X Q^o[ao a ]UY Rab`UbXblao.

=U b^ gb^Q ^] Qk[ bPZ b`ca[XR X WPQXb, a^RaU\ TPVU ]P_`^bXR; ]^ a ]UZ^b^`^S^
R`U\U]X ^] Qk[ R `PWT`PVXbU[l]^\ X ]P_`oVU]]^\ a^ab^o]XX, _^e^VU\ ]P
X_^e^]T`Xn. >] T^ b^S^ cS[cQX[ao R aUQo X cUTX]X[ao ^b RaUe, gb^ Q^o[ao TPVU
RaoZ^Y Rab`UgX, ]U b^[lZ^ Rab`UgX a e^WoYZ^Y. >] Qk[ WPTPR[U] QUT]^abln; ]^
TPVU abUa]U]]^U _^[^VU]XU _U`UabP[^ R _^a[UT]UU R`U\o boS^bXbl US^. =Paci]k\X
TU[P\X aR^X\X ^] a^RaU\ _U`UabP[ X ]U e^bU[ WP]X\Pblao. =XZPZ^Y e^WoYZX R
aci]^abX ^] ]U Q^o[ao, gb^ Qk bP ]X WP\kh[o[P _`^bXR ]US^. =^ ^abP]PR[XRPblao
]P [Uab]XfU, a[chPbl RaoZXY RWT^` _`^ Ran mbc ^QkTU]]cn T`UQUTU]l, T^ Z^b^`^Y
U\c ]Ub ]XZPZ^S^ TU[P, RaU mbX _`XabPRP]Xo ^ _[PbUVU, cS`^Wk, VP[^Qk, X _`X
mb^\ aP\^\c XWR^`PgXRPblao, XWRX]oblao, [SPbl---]Ub, cV [cghU _`^aZ^[lW]cbl
ZPZ-]XQcTl Z^hZ^Y _^ [Uab]XfU X c[XW]cbl, gb^Qk ]XZb^ ]U RXTP[.

2_`^gU\, ]P mb^b `PW ab`Pe Rab`UgX a^ aR^UY Z`UTXb^`hUY TPVU US^ aP\^S^ _^`PWX[
_^ Rke^TU ]P c[Xfc.

\bigskip\noindent{\sf From} {\hcyr  5.A. 2U]fU[l ({\sf aka} 8. 3`UZ^RP),
<BU^`Xo RU`^ob]^abUY>}

\centerline{\bcyr 11.4. Cb^g]U]XU `UWc[lbPb^R, _^[cgU]]ke \Ub^T^\ [X]UP`XWPfXX}

2 ]UZ^b^`ke WPTPgPe _`PZbXZX R^W]XZPUb a^\]U]XU R _`X\U]X\^abX \Ub^TP
[X]UP`XWPfXX R aRoWX a bU\, gb^ TXP_PW^] a[cgPY]ke P`Sc\U]b^R ]U ]Pab^[lZ^ \P[,
gb^Qk R US^ _`UTU[Pe dc]ZfXo \^S[P Qkbl a T^abPb^g]^Y b^g]^abln [X]UP`XW^RP]P.

2 mbXe a[cgPoe T[o _`^RU`ZX _`X\U]X\^abX \Ub^TP [X]UP`XWPfXX X T[o cb^g]U]Xo
_^[cgU]]ke `UWc[lbPb^R \^VUb Qkbl _`X\U]U] \Ub^T, ^a]^RP]]kY ]P a^e`P]U]XX R
`PW[^VU]XX dc]ZfXX ]U b^[lZ^ [X]UY]ke g[U]^R, ]^ X ]UZ^b^`ke _^a[UTcniXe g[U]^R
Q^[UU Rka^ZXe _^`oTZ^R X ^fU]ZU _^S`Uh]^abUY, aRoWP]]ke a mbX\X g[U]P\X.

4[o b^S^ gb^Qk _^oa]Xbl mb^b \Ub^T, `Paa\^b`X\ a]PgP[P ]PXQ^[UU _`^ab^Y a[cgPY
dc]ZfXX ^T]^S^ a[cgPY]^S^ P`Sc\U]bP. A[cgPY]Po RU[XgX]P $Y$ Uabl dc]ZfXo
a[cgPY]^S^ P`Sc\U]bP $X$:
$$ Y=\phi(X), \eqno(11.4.1) $$
_`XgU\ dc]ZfXo $\phi$ a`PR]XbU[l]^ \P[^ ^b[XgPUbao ^b [X]UY]^Y ]P
cgPabZU _`PZbXgUaZX R^W\^V]ke W]PgU]XY P`Sc\U]bP $X$, ]^ RaU VU ^b[XgPUbao
]Pab^[lZ^, gb^ R^W]XZPUb a^\]U]XU R _`X\U]X\^abX \Ub^TP [X]UP`XWPfXX. 4[o
_`^RU`ZX mb^S^ ^Qab^obU[labRP _`X\U]X\ Q^[UU b^g]kY \Ub^T, P X\U]]^: `PW[^VX\
dc]ZfXn $\phi$ R `oT BUY[^`P R ^Z`Uab]^abX b^gZX $m_x$ X a^e`P]X\ R `PW[^VU]XX
_U`RkU b`X g[U]P:
$$ y=\phi(x)\approx \phi(m_x) + \phi^\prime (m_x) (x-m_x) + {1\over2}
\phi^{\prime\prime} (m_x) (x-m_x)^2. \eqno(11.4.2) $$
BP VU d^`\c[P QcTUb, ^gURXT]^, _`XQ[XVU]]^ aRoWkRPbl a[cgPY]kU
RU[XgX]k $Y$ X $X$:
$$ Y=\phi(m_x) + \phi^\prime (m_x) (X-m_x) + {1\over2} \phi^{\prime\prime}
(m_x) (X-m_x)^2= \phi(m_x) + \phi^\prime (m_x) \acircle X + {1\over2}
\phi^{\prime\prime} (m_x) \acircle X^2. \eqno(11.4.3) $$

?^[lWcoal Rk`PVU]XU\ (11.4.3), ]PYTU\ \PbU\PbXgUaZ^U ^VXTP]XU X TXa_U`aXn
RU[XgX]k $Y$. ?`X\U]oo bU^`U\k ^ gXa[^Rke eP`PZbU`XabXZPe, X\UU\:

$$ m_y=\phi(m_x)+{1\over2} \phi^{\prime\prime} (m_x) M[\acircle X^2]
= \phi(m_x) +{1\over2} \phi^{\prime\prime} (m_x) D_x.
\eqno(11.4.4) $$

?^ d^`\c[U (11.4.4) \^V]^ ]PYbX cb^g]U]]^U W]PgU]XU \PbU\PbXgUaZ^S^ ^VXTP]Xo X
a`PR]Xbl US^ a bU\ W]PgU]XU\ $\phi(m_x)$, Z^b^`^U _^[cgPUbao \Ub^T^\
[X]UP`XWPfXX; _^_`PRZ^Y, cgXbkRPniUY ]U[X]UY]^abl dc]ZfXX, oR[oUbao Rb^`^Y g[U]
d^`\c[k (11.4.4).

>_`UTU[oo TXa_U`aXn _`PR^Y X [UR^Y gPabX d^`\c[k (11.4.3), X\UU\:

$$ D_y=[\phi^\prime(m_x)]^2D_x+{1\over4} [\phi^{\prime\prime} (m_x)]^2
D[\acircle X^2] + \phi^\prime (m_x) \phi^{\prime\prime} (m_x) K[\acircle
X,\acircle X^2], \eqno(11.4.5) $$
STU $K[\acircle X,\acircle X^2]$---Z^``U[ofX^]]kY \^\U]b RU[XgX]
$\acircle X,\acircle X^2$.

2k`PWX\ Re^ToiXU R d^`\c[c (11.4.5) RU[XgX]k gU`UW fU]b`P[l]kU \^\U]bk RU[XgX]k
$X$:

$$
D[\acircle X^2] = M[\acircle X^4] - \{ M[\acircle X^2] \}^2 = \mu_4[X]-D_x^2,
$$
$$
K[\acircle X,\acircle X^2] = M[\acircle X\{\acircle X^2-M[\acircle X^2] \}].
$$

>Z^]gPbU[l]^ X\UU\:

$$ D_y=[\phi^\prime(m_x)]^2D_x+ {1\over4} [\phi^{\prime\prime} (m_x)]^2
( \mu_4 [X] -D_x^2)+ \phi^\prime (m_x) \phi^{\prime\prime} (m_x) \mu_3 [X].
\eqno(11.4.6) $$

D^`\c[P (11.4.6) TPUb cb^g]U]]^U W]PgU]XU TXa_U`aXX _^ a`PR]U]Xn a \Ub^T^\
[X]UP`XWPfXX; UU Rb^`^Y X b`UbXY g[U]k _`UTabPR[onb a^Q^Y _^_`PRZc ]P
]U[X]UY]^abl dc]ZfXX. 2 d^`\c[c, Z`^\U TXa_U`aXX P`Sc\U]bP $D_x$, Re^Tob UiU
b`UbXY X gUbRU`bkY fU]b`P[l]kU \^\U]bk $\mu_3[X]$, $\mu_4[X]$. 5a[X mbX \^\U]bk
XWRUab]k, b^ _^_`PRZP Z TXa_U`aXX \^VUb Qkbl ]PYTU]P ]U_^a`UTabRU]]^ _^ d^`\c[U
(11.4.6). >T]PZ^ WPgPabcn ]Ub ]U^Qe^TX\^abX R UU b^g]^\ ^_`UTU[U]XX; T^abPb^g]^
[Xhl W]Pbl UU _^`oT^Z. =P _`PZbXZU gPab^ Rab`UgPnbao a[cgPY]kU RU[XgX]k,
`Pa_`UTU[U]]kU _`XQ[XWXbU[l]^ _^ a[cgPY]^\c WPZ^]c. 4[o a[cgPY]^Y RU[XgX]k,
_^TgX]U]]^Y ]^`\P[l]^\c WPZ^]c,

$$ \mu_3[X]=0; \qquad \mu_4[X]=3\sigma_x^4=3D_x^2, \eqno(11.4.7) $$
X d^`\c[P (11.4.6) _`X]X\PUb RXT
$$ D_y=[\phi^\prime(m_x)]^2 D_x+{1\over2} [\phi^{\prime\prime}(m_x)]^2 D^2_x.
\eqno(11.4.8) $$

D^`\c[^Y (11.4.8) \^V]^ _^[lW^RPblao T[o _`XQ[XVU]]^Y ^fU]ZX _^S`Uh]^abX \Ub^TP
[X]UP`XWPfXX R a[cgPU, Z^STP P`Sc\U]b `Pa_`UTU[U] _^ WPZ^]c, Q[XWZ^\c Z
]^`\P[l]^\c.

\bigskip\noindent{\sf From} {\hcyr <0[S^`Xb\XWPfXo X ^a]^Rk _`^S`P\\X`^RP]Xo>,
`UT. 0.O. APRU[lUR, Z]XSP 3}

\centerline{\bcyr 3. 1^[UU a[^V]kU m[U\U]bk oWkZP 1mYaXZ}

\centerline{\bcyr \S4.1 <PaaXRk}

?U`U\U]]kU, Z^b^`kU \k Xa_^[lW^RP[X _`X a^abPR[U]XX _`UTkTciXe _`^S`P\\,
]PWkRPnbao _`^abk\X _U`U\U]]k\X. :PVTPo bPZPo _U`U\U]]Po X\UUb X\o X _^T ZPVTcn
XW ]Xe RkTU[oUbao ogUYZP _P\obX, ^Q`PiU]XU Z Z^b^`^Y ^aciUabR[oUbao _^ mb^\c
X\U]X. 8a_^[lW^RP]XU b^[lZ^ _`^abke _U`U\U]]ke WPb`cT]oUb X[X TU[PUb R^^QiU
]UR^W\^V]k\ `UhU]XU \]^SXe RPV]ke WPTPg.

<PaaXR^\ ]PWkRPUbao c_^`oT^gU]]Po _^a[UT^RPbU[l]^abl RU[XgX], ^Q^W]PgPU\Po
^T]X\ X\U]U\. C_^`oT^gU]]^abl WPZ[ngPUbao R b^\, gb^ m[U\U]bk \PaaXRP
`Pa_^[^SPnbao R _^a[UT^RPbU[l]ke ogUYZPe _P\obX.

<PaaXR \^V]^ _`UTabPRXbl aUQU R RXTU ]UaZ^[lZXe ^TX]PZ^Rke oiXZ^R, aZ`U_[U]]ke
R\UabU. =P Ran Z^]ab`cZfXn ZPZ fU[^U RUhPUbao ^TX] o`[kZ a X\U]U\ \PaaXRP
(]P_`X\U` {\tt a}). OiXZX R ^T]^Y bPZ^Y Z^]ab`cZfXX ]c\U`cnbao, ]PgX]Po a 1.
=^\U` oiXZP ]PWkRPUbao X]TUZa^\ m[U\U]bP \PaaXRP, _^\UiU]]^S^ R mb^b oiXZ.
Gb^Qk _^[cgXbl T^abc_ Z oiXZc, ]cV]^ cZPWPbl X\o \PaaXRP X US^ X]TUZa. 8]TUZa
WP_XakRPUbao R Z`cS[ke aZ^QZPe _^a[U X\U]X \PaaXRP. 5a[X, ]P_`X\U`, W]PgU]XU 5
]cV]^ _`XaR^Xbl b`UblU\c U[U\U]bc \PaaXRP {\tt a}, a[UTcUb ]P_XaPbl {\tt
a(3)=5}. ?`X Rk_^[]U]XX mb^S^ ^_U`Pb^`P ]Pe^TXbao \PaaXR {\tt a}, ^bagXbkRPUbao
b`UbXY oiXZ (ogUYZP _P\obX) X R ]US^ Z[PTUbao W]PgU]XU 5. 5a[X ]cV]^ RaU\ hUabX
m[U\U]bP\ \PaaXRP {\tt a} _`XaR^Xbl W]PgU]XU 5, a[UTcUb ]P_XaPbl:

\bigskip\noindent{\sf From} {\hcyr  0[UZaP]T` AU`SUURXg ?chZX] (?)}

%Quoted in Barbara Beeton, Mathematical Symbols and Cyrillic Fonts Ready for
%Distribution, TUGboat 6 \#2.

<UiP]aZXY c]XRU`aXbUb, ]Pe^ToiXYao ]P _cbX Z =UaZcg]^\c, _`PWT]^RP[ ]P T]oe
aR^Y _obXTUaobX[Ub]XY nQX[UY. :^S^ R^WX[X R BXbk X[X S^`^TaZcn Q^[l]Xfc, b^b,
Z^]Ug]^, _^\]Xb WT^`^RU]]UYhXY, b`UembPV]kY T^\XiU _^ _`PRcn `cZc a RkRUaZ^Y
<1^SPTU[l]Po X <UiP]aZXU cgX[XiP> X b^\c ]PRU`]^U Rab`UgP[Xal ]P _cbX RU`U]Xfk
cgU]XgUaZXe _P`, a^[XT]^ _`^Sc[XRPU\ke ]PTWX`PbU[o\X. 2 <UiP]aZ^\ cgX[XiU
_`^aRUiPnb a[PRo] XaZ[ngXbU[l]^ \UiP]aZ^S^ X Zc_UgUaZ^S^ _`^Xae^VTU]Xo. 3`Pdk,
QP`^]k, \P`ZXWk X RXZ^]bk ]U _`X]X\Pnbao. ?`U_^TPnb R ]U\ a[UTcniXU ]PcZX:
Qc[SPZbU`Xn, iU[ZP]lU ]P agUbPe, P[SUQ`c, SU^\Ub`Xn, Q[PS^]`PRXU, ^b\U]]kY
_^gU`Z X SP[P]bU`UY]kU \P]U`k. 2^a_XbkRPnb R ^b`^ZU Rka^ZXU gcRabRP X
XaZ^`U]onb R ]U\, ZPZ aZPWP]^ R ^bgUbU, aP\^\]U]XU: RaoZ^S^, R^^Q`PWXRhUS^ aUQo
HUZa_X`^\ X[X ZXbPYaZX\ \P]TP`X]^\, ^abPR[onb QUW ZPhX. :^]gXRhXY Zc`a c\UUb ]U
b^[lZ^ gXbPbl, _XaPbl X agXbPbl, ]^ \^VUb bPZVU QUW WP_X]ZX, ]P aP\^\ gXab^\
_P`XVaZ^\ PZfU]bU RkS^R^`Xbl <\U`aX-a> X ]P_XaPbl <_`^QP _U`P> bPZX\
WP[XeRPbaZX\ _^gU`Z^\, gb^ Rk ]U R^WTU`VXbUal, gb^Qk ]U R^aZ[XZ]cbl:

$${\bf S^{\rm-1}TS=dg}(\omega_1,\ldots,\omega_n)=\bf\Lambda$$
$$
A=\pmatrix{
a_{11}&\ldots&a_{1n}\cr
\vdots&\ddots&\vdots\cr
a_{m1}&\ldots&a_{mn}\cr},\quad
\biggl(\int_{-\infty}^\infty e^{-x^2}\,dx\biggr)^2=\pi,
$$

\bigskip\noindent{\sf From} {\hcyr 4P]XU[l 4. <PZ-:`PZU], CX[lo\ A. 4^`],
<GXa[U]]kU \Ub^Tk X _`^S`P\\X`^RP]XU ]P D>@B@0=U>}

\centerline{\bcyr 3[PRP 1. >a]^Rk _`^S`P\\X`^RP]Xo ]P D>@B@0=U}

\centerline{\bcyr \S1.1 ?`X\U]U]XU fXd`^Rke RkgXa[XbU[l]ke \PhX]}

M[UZb`^]]kU fXd`^RkU RkgXa[XbU[l]kU \PhX]k (MF2<) hX`^Z^ _`X\U]onbao T[o
`UhU]Xo ]Pcg]ke, bUe]XgUaZXe X mZ^]^\XgUaZXe WPTPg. >]X a_^a^Q]k _`^XWR^TXbl
RkgXa[U]Xo ^gU]l Qkab`^, RkTPRPbl ^gU]l b^g]kU `UWc[lbPbk, WP_^\X]Pbl Q^[lhXU
\PaaXRk X]d^`\PfXX X _`^XWR^TXbl T[X]]kU X a[^V]kU _^a[UT^RPbU[l]^abX
RkgXa[U]XY QUW R\UhPbU[labRP gU[^RUZP.

2 mb^Y Z]XSU QcTcb R ^a]^R]^\ `Paa\^b`U]k \Ub^Tk `UhU]Xo ]P MF2< ]Pcg]ke X
bUe]XgUaZXe WPTPg. =UZ^b^`^U _`UTabPR[U]XU ^Q mb^\ Z`cSU WPTPg \^V]^ _^[cgXbl
XW a[UTcniXe _`X\U`^R. :^]ab`cX`^RP]XU ]^R^S^ aP\^[UbP b`UQcUb WPb`Pbk bkaog
gPa^R \PhX]]^S^ R`U\U]X T[o Xaa[UT^RP]Xo RWPX\^aRoWP]]ke b`UQ^RP]XY Z
Z^]abcZfX^]]k\ \PbU`XP[P\, PU`^TX]P\XZU, TRXSPbU[o\ X aXabU\P\ c_`PR[U]Xo _`X
`PW[Xg]ke ca[^RXoe _^[UbP. ?`^UZbX`^RP]XU eX\XgUaZ^S^ WPR^TP WPRXaXb ^b
`PagUb^R _`^TcZbXR]^abX, ca[^RXY _`^XWR^TabRP X ^b \]^SXe T`cSXe ^Qab^obU[labR.
?`X _`^UZbX`^RP]XX [X]XX m[UZb`^_U`UTPgX ]U^Qe^TX\^ XWcgU]XU ]PS`cW^Z, Z^b^`kU
QcTcb _`X[^VU]k Z `PW[Xg]k\ cgPabZP\ [X]XX _`X XW\U]U]XX _^b`UQ[U]Xo
m[UZb`^m]U`SXX X[X _`X ]U^Qkg]ke aXbcPfXoe.

2 mb^\ RRUTU]XX ]U [Xh]X\ QcTUb ^b\UbXbl, gb^ MF2< ]U <`UhPUb WPTPgc>. >]P
_^\^SPUb ]P\ [Xhl Xaa[UT^RPbl `PW[Xg]kU RP`XP]bk. <k ]U WPTPU\ \PhX]U R^_`^a:
<ZPZ ]cV]^ _^ab`^Xbl mb^ cab`^YabR^?>, \k WPTPU\ R^_`^a: <ZPZ QcTUb mb^
cab`^YabR^ `PQ^bPbl _`X TP]]^Y aXabU\U ca[^RXY, Ua[X ^]^ _^ab`^U]^ ^_`UTU[U]]k\
^Q`PW^\?> AciUabRcUb \]^VUabR^ a_^a^Q^R _^ab`^U]Xo X]VU]U`]^Y aXabU\k, Q^[lh^U
`PW]^^Q`PWXU ca[^RXY, _`X Z^b^`ke mb^Y aXabU\U _`XTUbao `PQ^bPbl, X `PW[Xg]kU,
gPab^ _`^bXR^`UgXRkU Z`XbU`XX, Z^b^`k\ mbP aXabU\P T^[V]P cT^R[UbR^`obl. MF2<
]U \^VUb WPTPblao Xae^T]k\X TP]]k\X T[o _`^UZbX`^RP]Xo, _U`UgXa[Xbl ca[^RXo,
T[o Z^b^`ke ]PT^ Xaa[UT^RPbl `PQ^bc aXabU\k, ^_`UTU[Xbl X[X ]PYbX `PWc\]cn
abU_U]l Z^\_`^\XaaP \UVTc _`^bXR^`UgXRk\X Z`XbU`Xo\X. >Qkg]^ ^]P [Xhl \^VUb
^ZPWPbl ]P\ Q^[lhcn _^\^il, `PaagXbkRPo X _`UTaZPWkRPo `UWc[lbPbk ]PhUS^ RkQ^`P
R mbXe R^_`^aPe.

\centerline{\bcyr 1.3 ?`^S`P\\P ]P D>@B@0=U}

0[S^`Xb\ `UhU]Xo WPTPgX, WP_XaP]]kY a _^\^iln D>@B@0=P, a^ab^Xb XW
_^a[UT^RPbU[l]^abX ^_U`Pb^`^R. MbX ^_U`Pb^`k \^Scb _`X]PT[UVPbl Z ]UaZ^[lZX\
`PW[Xg]k\ bX_P\. >T]X XW ]Xe ^_`UTU[onb P`Xd\UbXgUaZXU ^_U`PfXX, oR[oniXUao
^a]^R]k\ a^TU`VP]XU\ P[S^`Xb\P, T`cSXU ^_U`UTU[onb _^`oT^Z RR^TP X RkR^TP
X]d^`\PfXX, ZPZ, ]P_`X\U`, RR^T gXaU[ a _U`d^ZP`bk [XQ^ RkR^T gXaU[ ]P _UgPbl
X[X ]P _U`d^`PfXn. >_U`Pb^`k mbXe TRce _U`Rke bX_^R Xa_^[]onbao R b^\ _^`oTZU,
R Z^b^`^\ ^]X ]P_XaP]k. B`UbXY bX_ ^_U`Pb^`^R XW\U]oUb _^`oT^Z Rk_^[]U]Xo
^_U`PfXY, bPZ gb^ S`c__k ^_U`PgXY \^Scb Rk_^[]oblao _^Rb^`]^ X[X R ZPZ^\-[XQ^
T`cS^\ _^`oTZU, ^b[Xg]^\ ^b b^S^, R Z^b^`^\ ^]X ]P_XaP]k. >_U`Pb^`k gUbRU`b^S^
bX_P a^TU`VPb X]d^`\PfXn ^Q P[S^`Xb\U, ]^ ]XZPZXe TUYabRXY ]U ^_`UTU[onb.

2\UabU RWobkU, RaU ^_U`Pb^`k, ^_`UTU[oniXU P[S^`Xb\ `UhU]Xo WPTPgX, a^abPR[onb
Xae^T]cn _`^S`P\\c. ?^a[U b^S^ ZPZ Xae^T]Po _`^S`P\\P ]P_XaP]P X
^b_U`d^`X`^RP]P ]P _U`d^ZP`bPe, ^]P _`U^Q`PWcUbao a _^\^iln b`P]a[ob^`P
D>@B@0=P R `PQ^gcn _`^S`P\\c. @PQ^gPo _`^S`P\\P _`UTabPR[oUb a^Q^Y
_^a[UT^RPbU[l]^abl m[U\U]bP`]ke Z^\\P]T T[o MF2<, bPZXe, ZPZ a[^VU]XU,
a`PR]U]XU TRce gXaU[, WP_^\X]P]XU gXa[P R _P\obX \PhX]k, X b.T.\ _`U^Q`PW^RP]XU
Xae^T]^Y _`^S`P\\k R `PQ^gcn _`^S`P\\c ]U^Qe^TX\^ _^b^\c, gb^ oWkZ D>@B@0=
oR[oUbao S^`PWT^ Q^[UU a[^V]k\, ]UVU[X oWkZ Z^\\P]T aP\^Y MF2<. @PagUbk ]P MF2<
_`^XWR^Tobao X\U]]^ _^ `PQ^gUY _`^S`P\\U, Z^b^`Po X _`XR^TXb Z gXa[U]]k\
`UWc[lbPbP\.

A[^R^ D>@B@0=, bPZX\ ^Q`PW^\, ^b]^aXbao ZPZ Z oWkZc T[o ^_XaP]Xo RkgXa[XbU[l]ke
P[S^`Xb\^R, bPZ X Z b`P]a[ob^`c. B`P]a[ob^` D>@B@0=, ]PWkRPU\kY bPZVU X]^STP
_`^fUaa^`^\ X[X Z^\_X[ob^`^\, oR[oUbao aP\ Q^[lh^Y _`^S`P\\^Y ]P oWkZU \PhX]k.
(?^]obXU b`P]a[ofXX _^R[Xo[^ ]P aP\^ ]PWRP]XU oWkZP D>@B@0=: D>@\c[P X
B@0=a[ofXo.)

BU_U`l \k ^Q`PiPU\ao Z XWcgU]Xn m[U\U]b^R, XW Z^b^`ke a^abPR[ocbao ^_U`Pb^`k:
Z^]abP]b, _U`U\U]]ke, ^_U`PfXY, Rk`PVU]XY X dc]ZfXY. ?^a[U XWcgU]Xo mbXe
^a]^R]ke m[U\U]b^R oWkZP \k ]PcgX\ao a^abPR[obl XW ]Xe ^_U`Pb^`k X _XaPbl
]UZ^b^`kU _`^abkU _`^S`P\\k.

\centerline{\bcyr 1.4 :^]abP]bk}

<k ]PgX]PU\ a `Paa\^b`U]Xo TRce bX_^R gXaU[, _`X\U]oU\ke R D>@B@0=U,---fU[ke X
TUYabRXbU[l]ke.

FU[^U gXa[^ \^VUb Qkbl `PR]^ ]c[n, \^VUb Qkbl _^[^VXbU[l]k\ X[X ^b`XfPbU[l]k\,
]^ _^ PQa^[nb]^Y RU[XgX]U ]U T^[V]^ _`UR^ae^TXbl 32768. :PZ QcTUb _^ZPWP]^
]XVU, fU[kU gXa[P Xa_^[lWcnbao b^[lZ^ R ]UZ^b^`ke a_UfXP[l]ke aXbcPfXoe.
1^[lhX]abR^ gXaU[, Xa_^[lWcU\ke R D>@B@0=U, oR[onbao TUYabRXbU[l]k\X gXa[P\X.
D^`\P _`UTabPR[U]Xo bPZXe gXaU[, ]PWkRPU\Po d^`\^Y _`UTabPR[U]Xo a _[PRPniUY
WP_ob^Y, STU gXa[^ WP_XakRPUbao R RXTU T`^QX RU[XgX]^Y \UVTc 0.1 X 1,
c\]^VU]]^Y ]P fU[cn abU_U]l 10, X\UUb ae^TabR^ a d^`\^Y WP_XaX gXaU[ R ]Pcg]^Y
[XbU`Pbc`U. 0Qa^[nb]Po RU[XgX]P gXa[P, _`UTabPR[U]]^S^ bPZX\ ^Q`PW^\, T^[V]P
[UVPbl _`XQ[XWXbU[l]^ R _`UTU[Pe ^b $10^{-38}$ T^ $10^{38}$.

:PZ [USZ^ RXTUbl, TUYabRXbU[l]kU gXa[P \^Scb Qkbl fU[k\X X[X X\Ubl T`^Q]cn
gPabl. 1^[lhX]abR^ MF2< _`^XWR^TXb ^_U`PfXX a TUYabRXbU[l]k\X gXa[P\X bPZX\
^Q`PW^\, gb^ _`^S`P\\Xab \^VUb ]U WPQ^bXblao ^ _^[^VU]XX TUaobXg]^Y WP_ob^Y.
2k`PR]XRP]XU _^`oTZ^R gXaU[, ]P_`X\U` _`X a[^VU]XX X[X RkgXbP]XX, _`^XWR^TXbao
R MF2< PRb^\PbXgUaZX (^bZcTP, a^QabRU]]^, X _`^XW^hU[ bU`\X] <_[PRPniPo
WP_obPo>).

2aU TUYabRXbU[l]kY gXa[P, _`X\U]oU\kU R D>@B@0=U, oR[ocbao `PfX^]P[l]k\X;
_`X\U]U]XU X``PfX^]P[l]ke X X]^STP Z^\_[UZa]ke gXaU[ WP_`UiU]^.

>T]PZ^ RkgXa[U]Xo a Z^\_[UZa]k\X gXa[P\X \^V]^ _`^XWR^TXbl, Rk_^[]oo
P`Xd\UbXgUaZXU TUYabRXo a Xe TUYabRXbU[l]k\X X \]X\k\X gPabo\X.

;nQ^U gXa[^, Z^b^`^U _^oR[oUbao R _`^S`P\\U R oR]^\ RXTU, ]PWkRPUbao
Z^]abP]b^Y, R b^ R`U\o ZPZ RU[XgX]P, Z^b^`^Y _`XaRPXRPUbao ]PX\U]^RP]XU X
Z^b^`Po \^VUb _`X]X\Pbl R _`^fUaaU RkgXa[U]XY `PW[Xg]kU gXa[^RkU W]PgU]Xo,
]PWkRPUbao _U`U\U]]^Y. =P_`X\U`, ]XVU QcTUb _^ZPWP]^, gb^ a[UTcniXU Z^\QX]PfXX
aX\R^[^R oR[onbao P`Xd\UbXgUaZX\X ^_U`Pb^`P\X:

{\tt i-2}

{\tt x-P+12.7}

7TUal 2 X 12.7 oR[onbao Z^]abP]bP\X; {\tt i}, {\tt x}, X {\tt a} oR[onbao
_U`U\U]]k\X.

2 D>@B@0=U fU[kU X TUYabRXbU[l]kU Z^]abP]bk `PW[XgPnbao _^ ^bacbabRXn X[X
_`XacbabRXn TUaobXg]^Y b^gZX. BPZ, 3---fU[Po Z^]abP]bP, R b^ R`U\o ZPZ 3.0 (P
bPZVU 3. X[X 3.0000 X b.T.)---TUYabRXbU[l]Po Z^]abP]bP. 4RP mbX bX_P gXaU[ ]U
RWPX\^WP\U]oU\k, _^b^\c gb^ a_^a^Qk WP_XaX Xe R _P\obX MF2< X Rk_^[]U]Xo a ]X\X
P`Xd\UbXgUaZXe ^_U`PfXY a^RU`hU]]^ `PW[Xg]k.

\bigskip\noindent{\sf From} {\hcyr  1XQ[Xo, :]XSP _Ua]X _Ua]UY A^[^\^]P}

(1) 1 4P [^QWPUb ^] \U]o [^QWP]XU\ cab aR^Xe! 8Q^ [PaZX bR^X [cghU RX]P. 2 >b
Q[PS^R^]Xo \PabUY bR^Xe X\o bR^U, ZPZ `PW[Xb^U \X`^; _^mb^\c TURXfk [nQob bUQo.
3 2[UZX \U]o, \k _^QUVX\ WP b^Q^n;---fP`l RRU[ \U]o R gU`b^SX aR^X,---QcTU\
R^aeXiPblao X `PT^RPblao b^Q^n, _`UR^W]^aXbl [PaZX bR^X Q^[lhU ]UVU[X RX]^;
T^ab^Y]^ [nQPb bUQo! 4 4iU`X XU`caP[X\aZXU! gU`]P o, ]^ Z`PaXRP, ZPZ hPb`k
:XTP`aZXU, ZPZ WPRUak A^[^\^]^Rk. 5 =U a\^b`XbU ]P \U]o, gb^ o a\cS[P; XQ^
a^[]fU ^_P[X[^ \U]o: ak]^Rlo \PbU`X \^UY `PWS]URP[Xal ]P \U]o, _^abPRX[X \U]o
abU`Ugl RX]^S`PT]XZX,---\^US^ a^QabRU]]^S^ RX]^S`PT]XZP o ]U abU`US[P. 6 AZPVX
\]U, bk, Z^b^`^S^ [nQXb TchP \^o: STU _PaUhl bk? STU ^bTkePUhl R _^[TU]l? Z
gU\c \]U Qkbl aZXbP[XfUn R^W[U abPT b^RP`XiUY bR^Xe? 7 5a[X bk ]U W]PUhl mb^S^,
_`UZ`Pa]UYhPo XW VU]iX], b^ XTX aUQU _^ a[UTP\ ^RUf, X _PaX Z^W[ob bR^Xe _^T[U
hPb`^R _PabchUaZXe. 8 :^Qk[XfU \^UY R Z^[Ua]XfU dP`P^]^R^Y o c_^T^QX[ bUQo,
R^W[nQ[U]]Po \^o. 9 ?`UZ`Pa]k [P]Xbk bR^X _^T _^TRUaZP\X, hUo bR^o R ^VU`U[loe;
10 7^[^bkU _^TRUaZX \k aTU[PU\ bUQU a aU`UQ`o]]k\X Q[UaZP\X. 11 4^Z^[U fP`l Qk[
WP ab^[^\ aR^X\, ]P`T \^Y XWTPRP[ Q[PS^R^]XU aR^U. 12 <X``^RkY
_cg^Z---R^W[nQ[U]]kY \^Y c \U]o; c S`cTUY \^Xe _`UQkRPUb. 13 :PZ ZXabl ZX_U`P,
R^W[nQ[U]]kY \^Y c \U]o R RX]^S`PT]XZPe 5]SUTaZXe. 14 > bk _`UZ`Pa]P,
R^W[nQ[U]]Po \^o, bk _`UZ`Pa]P! S[PWP bR^X S^[cQX]kU. 15 >, bk _`UZ`PaU],
R^W[nQ[U]]kY \^Y, X [nQUWU]! X [^VU c ]Pa---WU[U]l; 16 :`^R[X T^\^R
]PhXe---ZUT`k, _^b^[ZX ]PhX---ZX_P`Xak.

(2) 1 O ]P`fXaa AP`^]aZXY, [X[Xo T^[X]! 2 Gb^ [X[Xo \UVTc bU`]P\X, b^
R^W[nQ[U]]Po \^o \UVTc TURXfP\X. 3 Gb^ oQ[^]l \UVTc [Ua]k\X TU`URlo\X, b^
R^W[nQ[U]]kY \^Y \UVTc n]^hP\X. 2 bU]X UU [nQ[n o aXTUbl, P _[^Tk UU a[PTZX T[o
S^`bP]X \^UY. 4 >] RRU[ \U]o R T^\ _X`P, X W]P\o US^ ]PT^ \]^n---[nQ^Rl. 5
?^TZ`U_XbU \U]o RX]^\, ^aRUVXbU \U]o oQ[^ZP\X, XQ^ o XW]U\^SPn ^b [nQRX. 6
;URPo `cZP US^ c \U]o _^T S^[^R^n, P _`PRPo ^Q]X\PUb \U]o. 7 7PZ[X]Pn RPa,
TiU`X 8U`caP[X\aZXU, aU`]P\X X[X _^[URk\X [P]o\X: ]U QcTXbU X ]U b`UR^VlbU
R^W[nQ[U]]^Y, T^Z^[U UY cS^T]^. 8 3^[^a R^W[nQ[U]]^S^ \^US^! R^b, ^] XTUb,
aZPgUb _^ S^`P\, _`kSPUb _^ e^[\P\. 9 4`cS \^Y _^e^V ]P aU`]c X[X ]P \^[^T^S^
^[U]o. 2^b, ^] ab^Xb c ]Pa WP abU]^n, WPS[oTkRPUb R ^Z]^, \U[lZPUb aZR^Wl
`UhUbZc. 10 2^W[nQ[U]]kY \^Y ]PgP[ S^R^`Xbl \]U: RabP]l, R^W[nQ[U]]Po \^o,
_`UZ`Pa]Po \^o, RkYTX! 11 2^b, WX\P cVU _`^h[P; T^VTl \X]^RP[, _U`UabP[; 12
fRUbk _^ZPWP[Xal ]P WU\[U; R`U\o _U]Xo ]PabP[^, X S^[^a S^`[Xfk a[khU] R ab`P]U
]PhUY; 13 a\^Z^R]Xfk `Pa_cabX[X aR^X _^gZX, X RX]^S`PT]kU [^Wk, `PafRUbPo,
XWTPnb Q[PS^R^]XU. 2abP]l, R^W[nQ[U]]Po \^o, _`UZ`Pa]Po \^o, RkYTX! 14 3^[cQXfP
\^o R ciU[XX aZP[k _^T Z`^R^\ cbUaP! _^ZPVX \]U [Xf^ aR^U, TPY \]U ca[khPbl
S^[^a bR^Y; _^b^\c, gb^ S^[^a bR^Y a[PT^Z X [Xf^ bR^U _`Xob]^. 15 ;^RXbU ]P\
[XaXf, [XaU]ob, Z^b^`kU _^`bob RX]^S`PT]XZX, P RX]^S`PT]XZX ]PhX R fRUbU. 16
2^W[nQ[U]]kY \^Y _`X]PT[UVXb \]U, P o U\c; ^] _PaUb \UVTc [X[Xo\X. 17 4^Z^[U
TU]l TkhUb _`^e[PT^n, X cQUSPnb bU]X, R^WR`PbXal, QcTl _^T^QU] aU`]U X[X
\^[^T^\c ^[U]n ]P `PaaU[X]Pe S^`.

(3) 1 =P [^VU \^U\ ]^gln XaZP[P o b^S^, Z^b^`^S^ [nQXb TchP \^o, XaZP[P US^ X
]U ]Ph[P US^. 2 2abP]c VU o, _^YTc _^ S^`^Tc, _^ c[XfP\ X _[^iPTo\, X QcTc
XaZPbl b^S^, Z^b^`^S^ [nQXb TchP \^o; XaZP[P US^ X ]U ]Ph[P US^. 3 2ab`UbX[X
\U]o ab`PVX, ^Qe^ToiXU S^`^T: <]U RXTP[X [X Rk b^S^, Z^b^`^S^ [nQXb TchP \^o?>
4 =^ UTRP o ^b^h[P ^b ]Xe, ZPZ ]Ph[P b^S^, Z^b^`^S^ [nQXb TchP \^o; ceRPbX[Pal
WP ]US^, X ]U ^b_cabX[P US^, T^Z^[U ]U _`XRU[P US^ R T^\ \PbU`X \^UY X R^
R]cb`U]]XU Z^\]Pbk `^TXbU[l]Xfk \^UY. 5 7PZ[X]Pn RPa, TiU`X 8U`caP[X\aZXU,
aU`]P\X X[X _^[URk\X [P]o\X: ]U QcTXbU X ]U b`UR^VlbU R^W[nQ[U]]^Y, T^Z^[U UY
cS^T]^. 6 :b^ mb^, R^ae^ToiPo ^b _cabk]X ZPZ Qk ab^[Qk Tk\P, ^Zc`XRPU\Po \X``^n
X dX\XP\^\, RaoZX\X _^`^hZP\X \X`^RP`]XZP? 7 2^b ^T` US^---A^[^\^]P: hUablTUaob
aX[l]ke R^Z`cS ]US^, XW aX[l]ke 8W`PX[URke. 8 2aU ^]X TU`VPb _^ \Ugc, ^_kb]k R
Q^n, c ZPVT^S^ \Ug US^ _`X QUT`U US^ `PTX ab`PeP ]^g]^S^. 9 =^aX[l]ke ^T`
aTU[P[ aUQU fP`l A^[^\^] XW TU`UR ;XRP]aZXe; 10 ab^[_fk US^ aTU[P[ XW aU`UQ`P,
[^Z^b]XZX US^ XW W^[^bP, aUTP[XiU US^ XW _c`_c`^R^Y bZP]X; R]cb`U]]^abl US^
cQ`P]P a [nQ^RXn TiU`o\X 8U`caP[X\aZX\X. 11 ?^YTXbU X _^a\^b`XbU, TiU`X
AX^]aZXU, ]P fP`o A^[^\^]P R RU]fU, Z^b^`k\ cRU]gP[P US^ \Pbl US^ R TU]l
Q`PZ^a^gUbP]Xo US^, R TU]l `PT^ab]kY T[o aU`TfP US^.

\bigskip\noindent{\sf From} {\hcyr  1^`Xa ?PabU`]PZ, <4^Zb^` 6XRPS^>, 11.4}

?^ \UVTc]P`^T]^Y Z^]RU]fXX ^ :`Pa]^\ :`UabU R^U]]kU R`PgX X a[cVPiXU aP]XbP`]ke
gPabUY ]U X\Unb _`PRP R^^`cVU]]^ cgPabR^RPbl R Q^URke TUYabRXoe R^nniXe. =^
^T]PVTk T^Zb^`c _`^bXR R^[X _`Xh[^al ]P`chXbl mb^ _`PRX[^. 7PRoWPRhPoao abkgZP
WPabP[P US^ ]P _^[U X WPabPRX[P `PWTU[Xbl acTlQc a`PVPniXeao X ^bab`U[XRPblao.

?P`bXWP]aZPo fU_l, R Z^b^`^Y WPabXS]cbkY ^S]U\ T^Zb^` WP[US `oT^\ a
bU[US`PdXab^\ ^b`oTP, WP]X\P[P [Ua]cn ^_chZc. 7P a_X]^n _P`bXWP] Qk[P bPYSP,
R_U`UTX---^bZ`kbPo _^[o]P, ^S^[U]]^U, ]UWPiXiU]]^U _`^ab`P]abR^, _^ Z^b^`^\c
h[X QU[kU, ]Pabc_Po.

>]X _`XQ[XVP[Xal X Qk[X cVU Q[XWZ^. 4^Zb^` e^`^h^ Xe RXTU[, ZPVT^S^ R [Xf^. Mb^
Qk[X \P[lgXZX X n]^hX XW ]UR^U]]ke a[^UR ab^[Xg]^S^ ^QiUabRP X [nTX Q^[UU
_^VX[kU, \^QX[XW^RP]]kU XW WP_PaP. =^ b^] WPTPRP[X _U`RkU, \^[^TUVl, abcTU]bk
_U`R^Zc`a]XZX X SX\]PWXabk, R^al\XZ[Paa]XZX, ]UTPR]^ WP_XaPRhXUao R
T^Q`^R^[lfk.

4^Zb^` ]U W]P[ ]XZ^S^ XW ]Xe, ]^ [XfP _^[^RX]k ZPWP[Xal U\c _`XRkg]k\X,
RXTU]]k\X, W]PZ^\k\X. >T]X ]P_^\X]P[X U\c Qk[ke hZ^[l]ke b^RP`XiUY. <^VUb
abPblao, mb^ Qk[X Xe \[PThXU Q`Pblo? 4`cSXe ^] a[^R]^ Rab`UgP[ R bUPb`P[l]^Y
X[X c[Xg]^Y b^[_U R Qk[kU S^Tk. 8e Rk`PWXbU[l]kU, _`XR[UZPbU[l]kU dXWX^]^\XX
ZPWP[Xal Q[XWZX\X, aR^X\X.

A[cVU]XU T^[Sc, ZPZ ^]X US^ _^]X\P[X, ^TchUR[o[^ Xe R^ab^`VU]]k\ \^[^TgUabR^\,
]U]cV]k\, RkWkRPniX\. >]X h[X `Paak_]k\ `UTZX\ ab`^U\, Rk_`o\XRhXal R^ RUal
`^ab, _`UR^ae^To Rk_`PRZ^Y ZPT`^Rke SRP`TUYfUR X, Q`PRX`co ^_Pa]^abln, ]U
_`XQUSP[X Z _U`UQUVZU X WP[USP]ln ]P _^[U, e^bo ]P _^[o]U Qk[X ]U`^R]^abX,
QcS^`ZX X Z^gZX, WP Z^b^`k\X \^V]^ Qk[^ cZ`kblao. ?c[X _P`bXWP] _^gbX _^S^[^R]^
RkZPhXRP[X Xe.

?^a`UTX hX`^Z^S^ S^[^S^ _^[o, _^ Z^b^`^\c TRXSP[Xal R_U`UT QU[kU, ab^o[^
\U`bR^U ^QS^`U[^U TU`UR^. >]^ Qk[^ ^QcS[U]^ \^[]XUY X[X _[P\U]U\ Z^ab`P, X[X
`PaiU_[U]^ X ^_P[U]^ _`UThUabRcniX\X a`PVU]Xo\X. :PVTkY ]Pabc_PRhXY
T^Q`^R^[lgUaZXY ab`U[^Z Q`^aP[ ]P ]US^ RWS[oTk, Q^`oal a XaZchU]XU\ WPYbX WP
US^ abR^[ T[o Q^[UU QUW^_Pa]^S^ X RkRU`U]]^S^ _`XfU[P, ]^ _`U]UQ`USP[ a^Q[PW]^\
X hU[ TP[lhU.

C _P`bXWP] Qk[^ ^S`P]XgU]]^U gXa[^ _Pb`^]^R. 8e a[UT^RP[^ QU`Ugl. 8\U[ao
_`XZPW, _^TTU`VP]]kY Z`cS^Rk\ cS^R^`^\, ab`U[obl a Z^`^bZXe TXabP]fXY, X
RX]b^R^Z, `PR]ke gXa[c RXTX\ke \XhU]UY.

4^Zb^` [UVP[ QUW ^`cVXo R b`PRU X ]PQ[nTP[ WP e^T^\ Q^o. 2aU US^ a^gcRabRXU
Qk[^ ]P ab^`^]U SU`^XgUaZX SXQ]cRhXe TUbUY. >] ^b TchX VU[P[ X\ cTPgX. Mb^ Qk[X
^b_`kaZX aU\UYabR, RU`^ob]^, Q[XWZXe U\c _^ Tcec, US^ R^a_XbP]Xo, US^
]`PRabRU]]^S^ aZ[PTP, US^ _^]obXY.

HURU[l]c[Pal c ]US^ \ka[l RkQUVPbl Z ]X\ ]P _^[o]c X aTPblao, X bPZX\ ^Q`PW^\
^Q`UabX XWQPR[U]XU. =^ hPS Qk[ `XaZ^RP]]kY, a^_`oVU]]kY a ^_Pa]^abln.

?^ZP ^] T^QUVP[ Qk T^ aU`UTX]k _^[o]k, _^T]oR RRU`e `cZX, US^ \^S[X Qk c[^VXbl
a ^QUXe ab^`^], _^`PVU]]k\ R S`cTl X a_X]c, aR^X---R ]PZPWP]XU WP a^RU`hU]]cn
XW\U]c, gcVXU---]U `PW^Q`PR US^ ]P\U`U]XY. >] RUTl ]U `PW QkRP[ R _^T^Q]ke
_^[^VU]Xoe, _`^Tc\P[ RaU R^W\^V]^abX X TPR]^ _`XW]P[ mbX _[P]k a_PaU]Xo
]U_`XS^T]k\X. 8, \X`oal a TR^YabRU]]^abln gcRabR, T^Zb^` _`^T^[VP[ [UVPbl ]P
VXR^bU, [Xf^\ Z _^[o]U X QUW ^`cVXo a[UTXbl XW b`PRk WP e^T^\ Q^o.

>T]PZ^, a^WU`fPbl X _`UQkRPbl R QUWTUYabRXX a`UTX ZX_URhUY Z`cS^\ Q^`lQk ]U ]P
VXR^b, P ]P a\U`bl, Qk[^ ]U\ka[X\^ X RkhU gU[^RUgUaZXe aX[. 8 TU[^ Qk[^ ]U R
US^ a^QabRU]]^Y aP\^WPiXbU, P R a[UT^RP]XX _^`oTZc a^RU`hPRhUS^ao, R _^TgX]U]XX
WPZ^]P\ b^S^, gb^ `PWkS`kRP[^al _U`UT ]X\ X R^Z`cS ]US^. 1k[^ _`^bXR _`PRX[
^abPRPblao Z mb^\c R QUWcgPabXX. =PT^ Qk[^ TU[Pbl b^ VU, gb^ TU[P[X T`cSXU. HU[
Q^Y. 2 ]US^ X b^RP`XiUY ab`U[o[X. =PT^ Qk[^ ^bab`U[XRPblao.

8 Z^STP bU[Ud^]Xab `oT^\ a ]X\ R fU_X WPQX[ao R acT^`^SPe X _^b^\ WP\U` X
Rkbo]c[ao, WPabkR R ]U_^TRXV]^abX, N`XY 0]T`UURXg _^[WZ^\ _^Tbo]c[ao Z ]U\c,
a]o[ a ]US^ ac\Zc, RWo[ US^ RX]b^RZc X RU`]cRhXal ]P _`UV]UU \Uab^, abP[
`PW`oVPbl UU Rkab`U[ WP Rkab`U[^\.

=^ VP[^abl ]U _^WR^[o[P U\c fU[Xblao R \^[^Tke [nTUY, Z^b^`k\X ^] [nQ^RP[ao X
Z^b^`k\ a^gcRabR^RP[. 0 ab`U[obl aTc`c R R^WTce Qk[^ a[XhZ^\ S[c_k\ X _`PWT]k\
WP]obXU\, _`^bXRX`UgPiX\ US^ ]P\U`U]Xo\. 8 RkQX`Po \X]cbk, Z^STP \UVTc ]X\ X
US^ \XhU]ln ]U abP]^RX[ao ]XZb^ XW ]P_PTPniXe, ^] abP[ ab`U[obl R fU[l _^
^QS^`U[^\c TU`URc. C ]US^ Qk[X bcb aR^X _`XU\k.

FU[oal X _^ \U`U RaU cb^g]oniUYao ]PR^TZX ]UWP\Ub]^ X ]U T^ Z^]fP caX[XRPo
]PVX\ a^QPgZX, ZPZ Qk QUW `PagUbP Z^STP-]XQcTl Rkab`U[Xbl, _^ZP a_caZ Zc`ZP X
Rkab`U[ ]U a[UT^RP[X aP\X a^Q^Y ZPZ Qk aRU`e ^VXTP]Xo, T^Zb^` abP[ a _`XRkg]^Y
\UbZ^abln `PWQ`PakRPbl R^Z`cS _^\U`bRU[^S^ TU`URP aQXbkU a ]US^ ]XV]XU ^ba^ehXU
acglo.

=^, ^ cVPa! :PZ ]X ^abU`USP[ao T^Zb^`, ZPZ Qk ]U _^_Pabl R Z^S^-]XQcTl, b^
^TX], b^ T`cS^Y ]Pabc_PniXY RTRXSP[Xal R `UhPniXY \XS \UVTc ]X\ X TU`UR^\, X
_U`UaUZP[X _`XfU[l]cn [X]Xn R \^\U]b `cVUY]^S^ `PW`oTP. 4Rce ^] WPTU[ X `P]X[,
P b`UblU\c ]UagPab[XRfc, aRP[XRhU\cao ]UTP[UZ^ ^b TU`URP, mb^ ab^X[^ VXW]X.

\bigskip\noindent{\sf From} {\hcyr 1XQ[Xo, A^Q^`]^U _^a[P]XU aRob^S^ P_^ab^[P
8cTk}

1 8cTP, `PQ 8XacaP E`XabP, Q`Pb 8PZ^RP, _`XWRP]]k\, Z^b^`kU ^aRoiU]k 1^S^\
>bf^\ X a^e`P]U]k 8Xaca^\ E`Xab^\: 2 \X[^abl RP\ X \X` X [nQ^Rl TP c\]^VPbao. 3
2^W[nQ[U]]kU! X\Uo RaU caU`TXU _XaPbl RP\ ^Q ^QiU\ a_PaU]XX, o _^gU[ WP ]cV]^U
]P_XaPbl RP\ cRUiURP]XU---_^TRXWPblao WP RU`c, ^T]PVTk _`UTP]]cn aRobk\. 4 8Q^
RZ`P[Xal ]UZ^b^`kU [nTX, XWT`UR[U _`UT]PW]PgU]]kU Z aU\c ^acVTU]Xn, ]UgUabXRkU,
^Q`PiPniXU Q[PS^TPbl 1^SP ]PhUS^ R _^R^T Z `Pa_cbabRc X ^bRU`SPniXUao UTX]^S^
2[PTkZX 1^SP X 3^a_^TP ]PhUS^ 8XacaP E`XabP. 5 O e^gc ]P_^\]Xbl RP\, cVU
W]PniX\ mb^, gb^ 3^a_^Tl, XWQPRXR ]P`^T XW WU\[X 5SX_UbaZ^Y, _^b^\ ]URU`^RPRhXe
_^ScQX[, 6 X P]SU[^R, ]U a^e`P]XRhXe aR^US^ T^ab^X]abRP, ]^ ^abPRXRhXe aR^U
VX[XiU, a^Q[nTPUb R RUg]ke cWPe, _^T \`PZ^\, ]P acT RU[XZ^S^ T]o. 7 :PZ A^T^\ X
3^\^``P X ^Z`Uab]kU S^`^TP, _^T^Q]^ X\ Q[cT^TUYabR^RPhXU X e^TXRhXU WP X]^n
_[^bXn, _^TRU`ShXal ZPW]X ^S]o RUg]^S^, _^abPR[U]k R _`X\U`,---8 bPZ b^g]^
QcTUb X a aX\X \UgbPbU[o\X, Z^b^`kU ^aZRU`]onb _[^bl, ^bRU`SPnb ]PgP[labRP X
W[^a[^Rob Rka^ZXU R[PabX. 9 <XePX[ 0`eP]SU[, Z^STP S^R^`X[ a TXPR^[^\, a_^`o ^
<^XaUUR^\ bU[U, ]U a\U[ _`^XW]Uabl cZ^`XW]U]]^S^ acTP, ]^ aZPWP[: <TP WP_`UbXb
bUQU 3^a_^Tl>. 10 0 aXX W[^a[^Rob b^, gUS^ ]U W]Pnb; gb^ VU _^ _`X`^TU, ZPZ
QUaa[^RUa]kU VXR^b]kU W]Pnb, bU\ `Pab[URPnb aUQo. 11 3^`U X\, _^b^\c gb^ XTcb
_cbU\ :PX]^Rk\, _`UTPnbao ^Q^[liU]Xn \WTk, ZPZ 2P[PP\, X R c_^`abRU _^SXQPnb,
ZPZ :^`UY. 12 BPZ^RkU QkRPnb a^Q[PW]^\ ]P RPhXe RUgU`oe [nQRX: _X`hUabRco a
RP\X, QUW ab`PeP cbcg]onb aUQo; mb^---QUWR^T]kU ^Q[PZP, ]^aX\kU RUb`^\; ^aU]]XU
TU`URlo, QUa_[^T]kU, TRPVTk c\U`hXU, Xab^`S]cbkU; 13 aRX`U_kU \^`aZXU R^[]k,
_U]oiXUao a`P\^bP\X aR^X\X; WRUWTk Q[cVTPniXU, Z^b^`k\ Q[nTUbao \`PZ bl\k ]P
RUZX. 14 > ]Xe _`^`^gUabR^RP[ X 5]^e, aUTl\kY ^b 0TP\P, S^R^`o: <aU, XTUb
3^a_^Tl a^ bl\P\X aRobke (0]SU[^R) aR^Xe---15 a^bR^`Xbl acT ]PT RaU\X X
^Q[XgXbl RaUe \UVTc ]X\X ]UgUabXRke R^ RaUe TU[Pe, Z^b^`kU _`^XWRU[^ Xe
]UgUabXU, X R^ RaUe VUab^ZXe (a[^RPe), Z^b^`kU _`^XW]^aX[X ]P =US^ ]UgUabXRkU
S`Uh]XZX>. 16 Mb^ `^_^b]XZX, ]XgU\ ]UT^R^[l]kU, _^abc_PniXU _^ aR^X\ _^e^bo\
(]UgUabXR^ X QUWWPZ^]]^); cabP Xe _`^XW]^aob ]PTcbkU a[^RP; ^]X ^ZPWkRPnb
[XfU_`XobXU T[o Z^`kabX. 17 =^ Rk, R^W[nQ[U]]kU, _^\]XbU _`UTaZPWP]]^U
0_^ab^[P\X 3^a_^TP ]PhUS^ 8XacaP E`XabP; 18 ^]X S^R^`X[X RP\, gb^ R _^a[UT]UU
R`U\o _^oRobao `cSPbU[X, _^abc_PniXU _^ aR^X\ ]UgUabXRk\ _^e^bo\. 19
Mb^---[nTX, ^bTU[oniXU aUQo (^b UTX]abRP RU`k), TchUR]kU, ]U X\UniXU TceP. 20 0
Rk, R^W[nQ[U]]kU, ]PWXTPo aUQo ]P aRobUYhUY RU`U RPhUY, \^[oal 4ce^\ ARobk\, 21
a^e`P]oYbU aUQo R [nQRX 1^VXUY, ^VXTPo \X[^abX ^b 3^a_^TP ]PhUS^ 8XacaP E`XabP,
T[o RUg]^Y VXW]X. 22 8 Z ^T]X\ QcTlbU \X[^abXRk, a `Paa\^b`U]XU\; 23 P T`cSXe
ab`Pe^\ a_PaPYbU, Xab^`SPo XW ^S]o, ^Q[XgPYbU VU a^ ab`Pe^\, S]chPoal TPVU
^TUVT^n, Z^b^`Po ^aZRU`]U]P _[^bln. 24 <^SciU\c VU a^Q[nabX RPa ^b _PTU]Xo X
_^abPRXbl _`UT a[PR^n AR^Un ]U_^`^g]k\ R `PT^abX, 25 UTX]^\c _`U\cT`^\c 1^Sc,
a_PaXbU[n ]PhU\c g`UW 8XacaP E`XabP 3^a_^TP ]PhUS^, a[PRP X RU[XgXU, aX[P X
R[Pabl _`UVTU RaUe RUZ^R, ]k]U X R^ RaU RUZX. 0\X]l.

\bigskip\noindent{\sf From} {\hcyr  0.?.\ GUe^R, <B^[abkY X b^]ZXY>}

=P R^ZWP[U =XZ^[PURaZ^Y VU[UW]^Y T^`^SX Rab`UbX[Xal TRP _`XobU[o: ^TX] b^[abkY,
T`cS^Y b^]ZXY. B^[abkY b^[lZ^ gb^ _^^QUTP[ ]P R^ZWP[U, X ScQk US^, _^TU`]cbkU
\Pa[^\, [^a]X[Xal ZPZ a_U[kU RXh]X. ?Pe[^ ^b ]US^ eU`Ua^\ X d[U`T^`P]VU\.
B^]ZXY VU b^[lZ^ gb^ RkhU[ XW RPS^]P X Qk[ ]PRlngU] gU\^TP]P\X, cW[P\X X
ZP`b^]ZP\X. ?Pe[^ ^b ]US^ RUbgX]^Y X Z^dUY]^Y SciUY. 8W-WP US^ a_X]k
RkS[oTkRP[P ecTU]lZPo VU]iX]P a T[X]]k\ _^TQ^`^TZ^\---US^ VU]P, X Rka^ZXY
SX\]PWXab a _`Xic`U]]k\ S[PW^\---US^ ak].

---?^`dX`XY!---R^aZ[XZ]c[ b^[abkY, cRXTUR b^]Z^S^. ---Bk [X mb^? 3^[cQgXZ \^Y!
AZ^[lZ^ WX\, aZ^[lZ^ [Ub!

---1PbnhZX!---XWc\X[ao b^]ZXY.---<XhP! 4`cS TUbabRP! >bZcTP bk RWo[ao?

?`XobU[X b`^UZ`Pb]^ ^Q[^QkWP[Xal X cab`U\X[X T`cS ]P T`cSP S[PWP, _^[]kU a[UW.
>QP Qk[X _`Xob]^ ^hU[^\[U]k.

---<X[kY \^Y!---]PgP[ b^]ZXY _^a[U [^QkWP]Xo.---2^b ]U ^VXTP[! 2^b an`_`XW! =c,
TP _^S[oTX VU ]P \U]o e^`^hU]lZ^! BPZ^Y VU Z`PaPRUf, ZPZ X Qk[! BPZ^Y VU
Tch^]^Z X iUS^[l! 0e bk, S^a_^TX! =c, gb^ VU bk? 1^SPb? 6U]Pb? O cVU VU]Pb, ZPZ
RXTXhl... Mb^ R^b \^o VU]P, ;cXWP, c`^VTU]]Po 2P]fU]QPe... [nbU`P]ZP... P mb^
ak] \^Y, =PdP]PX[, cgU]XZ b`UblUS^ Z[PaaP. Mb^, =PdP]o, T`cS \^US^ TUbabRP! 2
SX\]PWXX R\UabU cgX[Xal!

=PdP]PX[ ]U\]^S^ _^Tc\P[ X a]o[ hP_Zc.

---2 SX\]PWXX R\UabU cgX[Xal!---_`^T^[VP[ b^]ZXY. ---?^\]Xhl, ZPZ bUQo
T`PW]X[X? BUQo T`PW]X[X 3U`^ab`Pb^\ WP b^, gb^ bk ZPWU]]cn Z]XVZc _P_X`^aZ^Y
_`^VUS, P \U]o MdXP[lb^\ WP b^, gb^ o oQUT]XgPbl [nQX[. E^-e^... 4Ubl\X Qk[X!
=U Q^Yao, =PdP]o! ?^T^YTX Z ]U\c _^Q[XVU... 0 mb^ R^b \^o VU]P, c`^VTU]]Po
2P]fU]QPe... [nbU`P]ZP...

=PdP]PX[ ]U\]^S^ _^Tc\P[ X a_`obP[ao WP a_X]c ^bfP.

---=c, ZPZ VXRUhl, T`cS?---a_`^aX[ b^[abkY, R^ab^`VU]]^ S[oTo ]P
T`cSP.---A[cVXhl STU? 4^a[cVX[ao?

---A[cVc, \X[kY \^Y! :^[[UVaZX\ PaUaa^`^\ cVU Rb^`^Y S^T X AbP]Xa[PRP X\Un.
6P[^RP]XU _[^e^U... =c, TP Q^S a ]X\! 6U]P c`^ZX TPUb, o _^`baXSP`k _`XRPb]^ XW
TU`URP TU[Pn. >b[Xg]kU _^`baXSP`k! ?^ `cQ[n WP hbcZc _`^TPn. 5a[X Zb^ QU`Ub
TUaobl hbcZ X Q^[UU, b^\c, _^]X\PUhl, cabc_ZP. ?`^QPR[oU\ao Z^U-ZPZ. A[cVX[,
W]PUhl, R TU_P`bP\U]bU, P bU_U`l anTP _U`URUTU] ab^[^]PgP[l]XZ^\ _^ b^\c VU
RUT^\abRc... 7TUal QcTc a[cVXbl. =c, P bk ZPZ? =UQ^al cVU abPbaZXY? 0?

---=Ub, \X[kY \^Y, _^Tk\PY _^RkhU,---aZPWP[ b^[abkY.---O cVU T^ bPY]^S^
T^a[cVX[ao... 4RU WRUWTk X\Un.

B^]ZXY RT`cS _^Q[UT]U[, ^ZP\U]U[, ]^ aZ^`^ [Xf^ US^ XaZ`XRX[^al R^ RaU ab^`^]k
hX`^gPYhUY c[kQZ^Y; ZPWP[^al, gb^ ^b [XfP X S[PW US^ _^ak_P[Xal XaZ`k. AP\ ^]
ajUVX[ao, aS^`QX[ao, acWX[ao... 5S^ gU\^TP]k, cW[k X ZP`b^]ZX ajUVX[Xal,
_^\^`iX[Xal... 4[X]]kY _^TQ^`^T^Z VU]k abP[ UiU T[X]]UU; =PdP]PX[ Rkbo]c[ao R^
d`c]b X WPabUS]c[ RaU _cS^RZX aR^US^ \c]TX`P...

---O, RPhU _`UR^ae^TXbU[labR^... >gU]l _`Xob]^! 4`cS, \^V]^ aZPWPbl, TUbabRP X
RT`cS Rkh[X R bPZXU RU[l\^VX! EX-eX-a.

---=c, _^[]^!---_^\^`iX[ao b^[abkY.---4[o gUS^ mb^b b^]? <k a b^Q^Y T`cWlo
TUbabRP---X Z gU\c bcb mb^ gX]^_^gXbP]XU!

---?^\X[cYbU... Gb^ Rk-a...---WPeXeXZP[ b^]ZXY, UiU Q^[UU
ajUVXRPoal.---<X[^abXR^U R]X\P]XU RPhUS^ _`UR^ae^TXbU[labRP... 2`^TU ZPZ Qk
VXRXbU[l]^Y R[PSX... Mb^ R^b, RPhU _`UR^ae^TXbU[labR^, ak] \^Y =PdP]PX[... 6U]P
;cXWP, [nbU`P]ZP, ]UZ^b^`k\ ^Q`PW^\...

B^[abkY e^bU[ Qk[^ R^W`PWXbl gb^-b^, ]^ ]P [XfU c b^]Z^S^ Qk[^ ]P_XaP]^ ab^[lZ^
Q[PS^S^RU]Xo, a[PT^abX X _^gbXbU[l]^Y ZXa[^bk, gb^ bPY]^S^ a^RUb]XZP ab^h]X[^.
>] ^bRU`]c[ao ^b b^]Z^S^ X _^TP[ U\c ]P _`^iP]lU `cZc.

B^]ZXY _^VP[ b`X _P[lfP, _^Z[^]X[ao RaU\ bc[^RXiU\ X WPeXeXZP[, ZPZ ZXbPUf:
<eX-eX-eX>. 6U]P c[kQ]c[Pal. =PdP]PX[ hP`Z]c[ ]^S^Y X c`^]X[ dc`PVZc. 2aU b`^U
Qk[X _`Xob]^ ^hU[^\[U]k.

\bigskip\noindent{\sf From} {\hcyr 2.>. :[ngURaZXY, <:c`a `caaZ^Y Xab^`XX>}

2k _`^a[chP[X cVU ]UaZ^[lZ^ Zc`a^R _^ RaU^QiUY Xab^`XX, _^W]PZ^\X[Xal a
WPTPgP\X X _`XU\P\X c]XRU`aXbUbaZ^S^ XWcgU]XP mb^Y ]PcZX. =PgX]Po Zc`a `caaZ^Y
Xab^`XX, o _`UT_^h[n U\c ]UaZ^[lZ^ aP\ke ^QiXe U[U\U]bP`]ke a^^Q`PVU]XY, fU[l
Z^b^`ke---aRoWPbl aTU[P]]kU RP\X ]PQ[nTU]Xo X Rk]UaU]]kU R_UgPb[U]Xo _^
RaU^QiUY Xab^`XX a WPTPgUY X _`XU\P\X ^bTU[l]^S^ XWcgU]Xo Xab^`XX @^aaXX.

{\bcyr =Pcg]Po WPTPgP XWcgU]Xo \Uab]^Y Xab^`XX}\quad ?^]obU] _`PZbXgUaZXY
X]bU`Ua, _^QcVTPniXY ]Pa XWcgPbl Xab^`Xn @^aaXX ^a^Q^, RkTU[oo UU XW a^abPRP
RaU^QiUY Xab^`XX: RUTl mb^ Xab^`Xo ]PhUS^ ^bUgUabRP. =^ mb^b R^a_XbPbU[l]kY,
b.U.\ _`PZbXgUaZXY, X]bU`Ua ]U XaZ[ngPUb ]Pcg]^S^, ]P_`^bXR, T^[VU] b^[lZ^
_`XTPRPbl U\c Q^[UU TXTPZbXgUaZ^Y aX[k. 8bPZ, ]PgX]Po ^a^QkY Zc`a `caaZ^Y
Xab^`XX, \^V]^ _^abPRXbl bPZ^Y ^QiXY R^_`^a: ZPZcn ]Pcg]cn fU[l \^VUb X\Ubl
a_UfXP[l]^U XWcgU]XU Xab^`XX ^T]^Y ZPZ^Y-[XQ^ ab`P]k, ZPZ^S^-[XQ^ ^bTU[l]^S^
]P`^TP? MbP fU[l T^[V]P Qkbl RkRUTU]P XW ^QiXe WPTPg Xab^`XgUaZ^S^ XWcgU]Xo,
b.U.\ XW WPTPg XWcgU]Xo ^QiUY Xab^`XX gU[^RUgUabRP.

{\bcyr 8ab^`XgUaZXY _`^fUaa}\quad =P ]Pcg]^\ oWkZU a[^R^ <Xab^`Xo>
c_^b`UQ[oUbao R TR^oZ^\ a\ka[U: 1) ZPZ TRXVU]XU R^ R`U\U]X, _`^fUaa, X 2) ZPZ
_^W]P]XU _`^fUaaP. ?^mb^\c RaU, gb^ a^RU`hPUbao R^ R`U\U]X X\UUb aR^n Xab^`Xn.
A^TU`VP]XU\ Xab^`XX, ZPZ ^bTU[l]^Y ]PcZX, a_UfXP[l]^Y ^b`Pa[X ]Pcg]^S^ W]P]Xo,
a[cVXb Xab^`XgUaZXY _`^fUaa, b.U.\ e^T, ca[^RXo X ca_UeX gU[^RUgUaZ^S^
^QiUVXbXo X[X VXW]l gU[^RUgUabRP R UU `PWRXbXX X `UWc[lbPbPe. GU[^RUgUaZ^U
^QiUVXbXU---bPZ^Y VU dPZb \X`^R^S^ QkbXo, ZPZ X VXW]l ^Z`cVPniUY ]Pa _`X`^Tk, X
]Pcg]^U _^W]P]XU mb^S^ dPZbP---bPZPo VU ]Ucab`P]X\Po _^b`UQ]^abl gU[^RUgUaZ^S^
c\P, ZPZ X XWcgU]XU VXW]X mb^Y _`X`^Tk. GU[^RUgUaZ^U ^QiUVXbXU Rk`PVPUbao R
`PW]^^Q`PW]ke [nTaZXe a^nWPe, Z^b^`kU \^Scb Qkbl ]PWRP]k Xab^`XgUaZX\X bU[P\X X
Z^b^`kU R^W]XZPnb, `Pabcb X `PW\]^VPnbao _^T^Q]^ ^`SP]XgUaZX\ bU[P\ _`X`^Tk.
2^W]XZ]^RU]XU, `^ab X a\U]P mbXe a^nW^R a^ RaU\X ca[^RXo\X X _^a[UTabRXo\X Xe
VXW]X X Uabl b^, gb^ \k ]PWkRPU\ Xab^`XgUaZX\ _`^fUaa^\.

\bigskip\noindent{\sf From} {\hcyr ;X_\P] 1U`a, <<PbU\PbXgUaZXU R^_`^ak
T^WRcZ^R^Y X ^Z^[^WRcZ^R^Y SPW^R^Y TX]P\XZX>}

\centerline{\bcyr ?`UTXa[^RXU PRb^`P Z `caaZ^\c XWTP]Xn}

O ^gU]l `PT, gb^ \^o \^]^S`PdXo ^ ]UZ^b^`ke \PbU\PbXgUaZXe WPTPgPe SPW^R^Y
TX]P\XZX _^oR[oUbao R `caaZ^\ _U`UR^TU. O Q[PS^TP`U] _U`UR^TgXZc X `UTPZb^`c, P
bPZVU X bU\ Z^[[USP\, Z^b^`kU cZPWP[X ^hXQZX X ^_UgPbZX, T^_ciU]]kU R
P]S[XYaZ^\ XWTP]XX.

:^STP `cZ^_Xal Qk[P WPZ^]gU]P, o X\U[ a\U[^abl ]PTUoblao, gb^ ^]P _`UTabPR[oUb
a^Q^Y T^abPb^g]^ _^[]kY ^QW^` a^ab^o]Xo bU^`XX R bUe cWZXe _`UTU[Pe, Z^b^`kU
Qk[X ]P\UgU]k. AUS^T]o mb^b ^QW^` _^ ]U^Qe^TX\^abX ]UaZ^[lZ^ cabP`U[. O c_^\o]c
b^[lZ^ ^ b`Ue R^_`^aPe, Z^b^`kU ]U_`U\U]]^ a[UT^RP[^ Qk RZ[ngXbl R a[cgPU
_U`Ua\^b`P bUZabP. Mb^ _`UVTU RaUS^ `UWc[lbPbk 4U-4V^`TVX (1) X =PhP (1),
Z^b^`k\ cTP[^al aciUabRU]]^ _`^]XZ]cbl R bU^`Xn ZRPWX[X]UY]ke c`PR]U]XY \]^SXe
_U`U\U]]ke, P bPZVU ]UTPR]UU c_`^iU]XU T^ZPWUbU[labRP, _`UT[^VU]]^U <^WU`^\
(1). 7PbU\ ]U^Qe^TX\^ Qk[^ Qk ^b`PWXbl Q^[lhcn `PQ^bc, _`^TU[P]]cn _^ gXa[U]]k\
`PagUbP\ bUgU]Xo WP aX[l]k\ ^ba^UTX]U]]k\ aZPgZ^\ (R gPab]^abX, \Ub^Tk
4^`^T]Xfk]P (1,2), 3P`PQUTo]P (1), @Xeb\PYU`P (1) X 2P] 4PYZP (1,2)), X,
]PZ^]Uf, TP]]^U 6U`\U]^\ ^_XaP]XU bUgU]Xo RQ[XWX _U`UaUgU]Xo aZPgZP a WRcZ^R^Y
(`UhPniUU WPTPgc, _^abPR[U]]cn ]P ab`.~134--136).

A T`cS^Y ab^`^]k, \^V]^ ^b\UbXbl, gb^ Q^[lhPo gPabl WPTPg, ad^`\c[X`^RP]]ke R
mb^Y Z]XSU, RaU UiU ^abPUbao ]U`UhU]]^Y.

;X_\P] 1U`a, 30 \Po 1960 S.

\centerline{\bcyr 3[PRP 1. 4XddU`U]fXP[l]kU c`PR]U]Xo _^bU]fXP[l]^S^ bUgU]Xo
SPWP}

\centerline{\bcyr \S2. 4XddU`U]fXP[l]kU c`PR]U]Xo X Z`PURkU ca[^RXo}

<k Z`PbZ^ _^Rb^`X\ RkR^T c`PR]U]XY T[o _^bU]fXP[P bUgU]Xo aVX\PU\^Y VXTZ^abX.
1^[UU TUbP[l]^U ^_XaP]XU \^V]^ ]PYbX R^ \]^SXe Z]XSPe (a\., ]P_`X\U` :c`P]b X
D`XT`Xea (1)).

>a]^R]kU c`PR]U]Xo. BUgU]XU a^RU`hU]]^Y VXTZ^abX ^_XakRPUbao R bPZ ]PWkRPU\^\
mY[U`^R^\ _`UTabPR[U]XX WPTP]XU\ _[^b]^abX $\rho$ X Z^\_^]U]b^R aZ^`^abX $u_1$,
$u_2$, $u_3$ ZPZ dc]ZfXY TUZP`b^Rke Z^^`TX]Pb $x_1$, $x_2$, $x_3$ X R`U\U]X
$t$. ?^[]^U ^_XaP]XU b`UQcUb bPZVU W]P]Xo TRce T`cSXe bU`\^TX]P\XgUaZXe
RU[XgX], aZPVU\ TPR[U]Xo $p$ X bU\_U`Pbc`k X[X TPR[U]Xo X m]b`^_XX. >T]PZ^ \k
X\UU\ TU[^ a PTXPQPbXgUaZX\ X XW^m]b`^_XgUaZX\ bUgU]XU\. 2 mb^\ a[cgPU TPR[U]XU
Uabl XWRUab]Po dc]ZfXo _[^b]^abX. 4[o XTUP[l]^S^ SPWP
$$
p=c \rho^\tau,
\eqno(2.1)
$$
STU $\tau>1$---_^ab^o]]Po (^b]^hU]XU cTU[l]ke bU_[^U\Z^abUY). AbP]TP`b]^U
W]PgU]XU $\tau$ T[o R^WTceP `PR]^ $1{,}4$. =P\ QcTUb cT^Q]^ ]U ^S`P]XgXRPblao
mbX\ a^^b]^hU]XU\ \UVTc TPR[U]XU\ X _[^b]^abln, P `Paa\Pb`XRPbl bPZVU ^QiXY
a[cgPY QP`^b`^_]^Y VXTZ^abX, b.U.\ VXTZ^abX R Z^b^`^Y
$$
p=p(\rho),
\eqno(2.2)
$$
STU $p(\rho)$---]UZ^b^`Po T^abPb^g]^ S[PTZPo R^W`PabPniPo dc]ZfXo.

:^\_^]U]bk aZ^`^abX X _[^b]^abl aRoWP]k c`PR]U]XU\ ]U`PW`kR]^abX, Rk`PVPniX\
WPZ^] a^e`P]U]Xo \Paak,
$$
{\partial\rho\over\partial t} +
\sum_{i=1}^3 {\partial(\rho u_i)\over\partial x_i}=0.
\eqno(2.3)
$$

A T`cS^Y ab^`^]k, Z^\_^]U]bk aZ^`^abX T^[V]k cT^R[UbR^`obl mY[U`^Rk\ c`PR]U]Xo\
TRXVU]Xo, oR[oniX\ao Rk`PVU]XU\ Rb^`^S^ WPZ^]P =lnb^]P. 5a[X _`U]UQ`Ugl
\Paa^Rk\X aX[P\X, R gPab]^abX boVUabln, b^ mbX c`PR]U]Xo X\Unb RXT
$$
{\partial u_i\over\partial  t}
\sum_{j=1}^3 u_i
{\partial u_i\over\partial x_j} + {1\over\rho}
{\partial p\over\partial x_i}=0
\quad (i=1,2,3).
\eqno(2.4)
$$

7P\UbX\, gb^ a _^\^iln a^^b]^hU]Xo (2.2) TPR[U]XU \^VUb Qkbl XaZ[ngU]^ XW
c`PR]U]XY MY[U`P. C`PR]U]Xo TRXVU]Xo X c`PR]U]XU ]U`PW`kR]^abX cT^R[UbR^`onb
_`X]fX_c ^b]^aXbU[l]^abX Z[PaaXgUaZ^Y \UeP]XZX, b.U.\ ^]X a_`PRUT[XRk R [nQ^Y
X]U`fXP[l]^Y aXabU\U Z^^`TX]Pb, Ua[X ^]X a_`PRUT[XRk R ^T]^Y XW bPZXe aXabU\.

\bigskip\noindent {\sf And naturally$\ldots$}

\bigskip\noindent{\sf From} {\hcyr AU`SUY 8RP]^RXg 0To], <> d`PS\U]bPe a[^RP
$\Delta$ R S`c__U Z^a>}

3`c__P Z^a ${\frak B}_{n+1}$ WPTPUbao ^Q`PWcniX\X
$$
a_1, a_2, \ldots, a_n \eqno(1)
$$
X ^_`UTU[oniX\X a^^b]^hU]Xo\X
$$
a_i a_j = a_j a_i,\quad \left|i-j\right|>1 \eqno(2)
$$
$$
a_i a_{i+1} a_i = a_{i+1} a_i a_{i+1} \quad (1\le i\le n). \eqno(3)
$$

1. 2 `PQ^bU [1] 3P`aPYT _^[cgX[ `UhU]XU _^abPR[U]]^Y 0`bX]^\ R `PQ^bU [2]
_`^Q[U\k a^_`oVU]]^abX T[o S`c__ ${\frak B}_{n+1}$. 2 ^a]^RU T^ZPWPbU[labRP
[UVPb cabP]^R[U]]kU R [1] aR^YabP bPZ ]PWkRPU\^S^ dc]TP\U]bP[l]^S^ a[^RP
$$
\Delta \rightleftharpoons a_1 a_2 \cdots a_n a_1 a_2 \cdots a_{n-1} \cdots
a_2 a_2 a_3 a_1 a_2 a_1.
$$
2 gPab]^abX, $\Delta^2$ _^`^VTPUb fU]b` S`c__k ${\frak B}_{n+1}$.

?cabl $\Pi_{n+1}$ Uabl _^[cS`c__P, WPTP]]Po ^Q`PWcniX\X (1) X ^_`UTU[oniX\X
^b]^hU]Xo\X (2) X (3). 2 [1] T^ZPWP]^, gb^ $\Pi_{n+1}$ RZ[PTkRPUbao R S`c__c
${\frak B}_{n+1}$. GU`UW $\doteq$ ^Q^W]PgPUbao `PRU]abR^ a[^R R $\Pi_{n+1}$.

5a[X $X\doteq PQ$, b^ S^R^`ob, gb^ a[^R^ $X$ TU[Xbao R _^[cS`c__U $\Pi_{n+1}$
a[URP ]P $P$ (a_`PRP ]P $Q$). 8W `UWc[lbPb^R `PQ^bk [1] [USZ^ a[UTcUb, gb^
dc]TP\U]bP[l]^U a[^R^ $\Delta$ a b^g]^abln T^ `PRU]abRP R $\Pi_{n+1}$ \^V]^
^_`TU[Xbl ZPZ Z`PbgPYhUU a[^R^ R P[dPRXbU (1), Z^b^`^U TU[Xbao R $\Pi_{n+1}$
a[URP (a_`PRP) ]P RaU QcZRk (1).

\bigskip\noindent{\sf From}
{\hcyr 0[UZaUY 8RP]^RXg :^ab`XZX], <2^Z`cS 1U`]aPYTP>}

\centerline{\bcyr \S2.2 A_caZ Z b^]ZX\ am]TRXgP\ (a[cgPY $p \gg n$)}

5a[X ^a]^R]^U _^[U $F$ X\UUb ]c[URcn X[X T^abPb^g]^ Q^[lhcn eP`PZbU`XabXZc (_^
a`PR]U]Xn a $n$), b^ T^ZPWPbU[labR^ aciUabR^RP]Xo am]TRXgP R P[SUQ`U ;X $L$ a
ca[^RXU\ $E_n$ aX[l]^ c_`^iPUbao. ?`UVTU gU\ \PbU`XP[XW^RPbl mb^ cbRU`VTU]XU,
_^a\^b`X\ ]P XTUP[ $E_{n-1}(L)$ R $L$, _^`^VTU]]kY RaU\X m[U\U]bP\X
$[uv^{n-1}]$ a $u,v\in L$. 8W aZPWP]]^S^ R S[.\ 1 T^R^[l]^ ^gURXT]^, gb^
$$
E_{n-1}(L)= \left\langle [ uv^{n-1}] \mid u,v \in L \right\rangle_F
$$
---[X]UY]^U _`^ab`P]abR^ ]PT $F$, ]Pbo]cb^U ]P m[U\U]bk $[ uv^{n-1}]$. =PS[oT]^
mb^ X[[nab`X`cUbao d^`\c[^Y
$$
[uv^{n-1}w]=[uwv^{n-1}] + \sum_i \alpha_i [uz_i^{n-1}],
$$
[USZ^ RkbUZPniUY XW d^`\c[k (1.2):
$$
v^{n-1}w-wv^{n-1}=\sum_{i=0}^{n-2} v^i [vw] v^{n-2-i} =
\left\{ \matrix{ [vw] & v \cr 1 & n-2 \cr} \right\} =
\sum_i \alpha_i z_i^{n-1},\quad z \in L.
$$
>ZPWkRPUbao, gb^ _`X $p \gg n$ XTUP[ $E_{n-1}(L)$ am]TRXgUR. B^g]UU,
a_`PRUT[XRP

{\ccyr BU^`U\P 3.1.} {\icyr 4[o [nQ^Y _P`k m[U\U]b^R $u,v$ XW $n-$m]SU[UR^Y
P[SUQ`k ;X $L$ ]PT _^[U\ $F$ eP`PZbU`XabXZX $p>n+[n/2]$ X\UUb \Uab^ b^VTUabR^}
$$
[uv^{n-1}]^2=0.
$$

\bigskip\noindent{\sf From} {\hcyr =.=. @U_X], <?^[X[X]UY]kU b^VTUabRP R
Z^[lfPe ;X, Paa^fXX`^RP]]ke a _U`X^TXgUaZX\X S`c__P\X>}

?cabl $B(\infty,p)$---aR^Q^T]Po _U`X^TXgUaZPo S`c__P agUb]^S^ `P]SP _`^ab^S^
_U`X^TP $p$, P $L(B(\infty,p))$---P[SUQ`P ;X, Paa^fXX`^RP]]Po a mb^Y S`c__^Y. 2
`PQ^bU [1] 2^m]-;X cZPWP[ aU\UYabR^ _^[X[X]UY]ke b^VTUabR, Rk_^[]U]]ke R
P[SUQ`U $L(B(\infty,p))$, bPZ^U, gb^ RaoZ^U _^[X[X]UY]^U b^VUTabR^ P[SUQ`k
$L(B(\infty,p))$ oR[oUbao a[UTabRXU\ b^VTUabR mb^S^ aU\UYabRP.

2 ]Pab^oiUY `PQ^bU _^[cgU] P]P[^SXg]kY `UWc[lbPb T[o S`c__ $B(\infty,q)$
_`^XWR^[l]^S^ _U`X^TP $q\in {\Bbb N}$.

\centerline{\S1. ?^[X[X]UY]kU b^VTUabRP $K_n (x_1, \ldots, x_n) \equiv 0$}

?cabl ${\frak L}$---aR^Q^T]^U Z^[lf^ ;X ]P ^Q`PWcniXe $x_1,x_2,x_3,\ldots$ a
fU[k\X Z^UddXfXU]bP\X. ?cabl $S$---Z^]Ug]^U \]^VUabR^ ]Pbc`P[l]ke gXaU[,
$S=\{i,j,\ldots,k\}$, $i<j<\ldots,k$. :PZ X R `PQ^bU [1], T[o m[U\U]bP
$u\in{\frak L}$ gU`UW $[u,x_s]$ \k ^Q^W]PgPU\ Z^\\cbPb^` $[u,x_i,
x_j,\ldots,x_k]$, P gU`UW $[u,\bar x_s]$ m[U\U]b $(-1)^{\mathop{\rm Card} (S)}
[u,x_k,\ldots, x_j,x_i]$. BPZ^U VU ^Q^W]PgU]XU $[u,x_s]$ \k Xa_^[lWcU\ T[o
S`c__^R^S^ Z^\\cbPb^`P $[u,x_i, x_j,\ldots,x_k]$ R aR^Q^T]^Y S`c__U $\langle
x_1, x_2, x_3, \ldots ; \emptyset \rangle$. =U[X]UY]kY _^ ZPZ^Y-[XQ^ ^Q`PWcniUY
$x_i$ Z^\\cbPb^` ]P _^`^VTPniXe $x_1, x_2, x_3, \ldots $ QcTU\ ]PWkRPbl {\icyr
$x_i^2$-Z^\\cbPb^`^\.}

\bye