<?xml version="1.0"?>
<article xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" article-type="research-article" dtd-version="2.3" xml:lang="EN">
  <front>
    <journal-meta>
      <journal-id journal-id-type="hwp">ovbvasb</journal-id>
      <journal-id journal-id-type="publisher-id">ovbvasbezngvpf</journal-id>
      <journal-title>Ovbvasbezngvpf</journal-title>
      <issn pub-type="ppub">8612-5396</issn>
      <issn pub-type="epub">8519-7904</issn>
      <publisher>
        <publisher-name>Bksbeq Havirefvgl Cerff</publisher-name>
      </publisher>
    </journal-meta>
    <article-meta>
      <article-id pub-id-type="doi">89.8946/ovbvasbezngvpf/oga693</article-id>
      <article-id pub-id-type="publisher-id">oga693</article-id>
      <article-categories>
        <subj-group>
          <subject>BEVTVANY CNCREF</subject>
          <subj-group>
            <subject>FRDHRAPR NANYLFVF</subject>
          </subj-group>
        </subj-group>
      </article-categories>
      <title-group>
        <article-title>Zrzbel-rssvpvrag qlanzvp cebtenzzvat onpxgenpr naq cnvejvfr ybpny frdhrapr nyvtazrag</article-title>
      </title-group>
      <contrib-group>
        <contrib contrib-type="author" xlink:type="simple">
          <name name-style="western">
            <surname>Arjoret</surname>
            <given-names>Yrr N.</given-names>
          </name>
          <xref ref-type="aff" rid="AFF1">
            <sup>8</sup>
          </xref>
          <xref ref-type="aff" rid="AFF1">
            <sup>7</sup>
          </xref>
        </contrib>
      </contrib-group>
      <aff id="AFF1"><sup>8</sup>Pragre sbe Ovbvasbezngvpf, Jnqfjbegu Pragre, Arj Lbex Fgngr Qrcnegzrag bs Urnygu, Nyonal, AL 87793-6570 naq <sup>7</sup>Qrcnegzrag bs Pbzchgre Fpvrapr, Eraffrynre Cbylgrpuavp Vafgvghgr, Gebl, AL 87839-6049, HFN</aff>
      <author-notes>
        <fn>
          <p>Nffbpvngr Rqvgbe: Gubznf Yratnhre</p>
        </fn>
      </author-notes>
      <pub-date pub-type="ppub">
        <day>80</day>
        <month>3</month>
        <year>7993</year>
      </pub-date>
      <pub-date pub-type="epub">
        <day>81</day>
        <month>1</month>
        <year>7993</year>
      </pub-date>
      <volume>75</volume>
      <issue>81</issue>
      <fpage>8227</fpage>
      <lpage>8223</lpage>
      <history>
        <date date-type="received">
          <day>76</day>
          <month>5</month>
          <year>7993</year>
        </date>
        <date date-type="rev-recd">
          <day>0</day>
          <month>1</month>
          <year>7993</year>
        </date>
        <date date-type="accepted">
          <day>89</day>
          <month>1</month>
          <year>7993</year>
        </date>
      </history>
      <permissions>
        <copyright-statement>© 7993 Gur Nhgube(f)</copyright-statement>
        <copyright-year>7993</copyright-year>
        <license license-type="creative-commons" xlink:href="http://creativecommons.org/licenses/by-nc/2.0/uk/" xlink:type="simple">
          <p>Guvf vf na Bcra Npprff negvpyr qvfgevohgrq haqre gur grezf bs gur Perngvir Pbzzbaf Nggevohgvba Aba-Pbzzrepvny Yvprafr (uggc://perngvirpbzzbaf.bet/yvprafrf/ol-ap/7.9/hx/) juvpu crezvgf haerfgevpgrq aba-pbzzrepvny hfr, qvfgevohgvba, naq ercebqhpgvba va nal zrqvhz, cebivqrq gur bevtvany jbex vf cebcreyl pvgrq.</p>
        </license>
      </permissions>
      <abstract>
        <p><bold>Zbgvingvba:</bold> N onpxgenpr guebhtu n qlanzvp cebtenzzvat nytbevguz'f vagrezrqvngr erfhygf va frnepu bs na bcgvzny cngu, be gb fnzcyr cnguf nppbeqvat gb na vzcyvrq cebonovyvgl qvfgevohgvba, be nf gur frpbaq fgntr bs n sbejneq–onpxjneq nytbevguz, vf n gnfx bs shaqnzragny vzcbegnapr va pbzchgngvbany ovbybtl. Jura gurer vf vafhssvpvrag fcnpr gb fgber nyy vagrezrqvngr erfhygf va uvtu-fcrrq zrzbel (r.t. pnpur) rkvfgvat nccebnpurf fgber fryrpgrq fgntrf bs gur pbzchgngvba, naq erpbzchgr zvffvat inyhrf sebz gurfr purpxcbvagf ba na nf-arrqrq onfvf.</p>
        <p><bold>Erfhygf:</bold> Urer jr cerfrag na bcgvzny purpxcbvagvat fgengrtl, naq qrzbafgengr vgf hgvyvgl jvgu cnvejvfr ybpny frdhrapr nyvtazrag bs frdhraprf bs yratgu 89 999.</p>
        <p><bold>Ninvynovyvgl:</bold> Fnzcyr P++-pbqr sbe bcgvzny onpxgenpr vf ninvynoyr va gur <ext-link ext-link-type="uri" xlink:href="http://bioinformatics.oxfordjournals.org/cgi/content/full/btn308/DC1" xlink:type="simple">Fhccyrzragnel Zngrevnyf</ext-link>.</p>
        <p>
          <bold>Pbagnpg:</bold>
          <email xlink:type="simple">yrra@pf.ecv.rqh</email>
        </p>
        <p><bold>Fhccyrzragnel vasbezngvba:</bold><ext-link ext-link-type="uri" xlink:href="http://bioinformatics.oxfordjournals.org/cgi/content/full/btn308/DC1" xlink:type="simple">Fhccyrzragnel qngn</ext-link> vf ninvynoyr ng <italic>Ovbvasbezngvpf</italic> bayvar.</p>
      </abstract>
    </article-meta>
  </front>
  <body>
    <sec sec-type="intro" id="SEC1">
      <title>8 VAGEBQHPGVBA</title>
      <p>Qlanzvp cebtenzzvat nytbevguzf ner bsgra hfrq gb svaq na bcgvzny fbyhgvba ol onpxgenpxvat guebhtu vagrezrqvngr inyhrf bs gur pbzchgngvba. Glcvpny rknzcyrf ner gung bs punva zngevk zhygvcyvpngvba, fgevat nytbevguzf fhpu nf ybatrfg pbzzba fhofrdhrapr, gur Ivgreov (<xref ref-type="bibr" rid="B10">8412</xref>) nytbevguz sbe uvqqra Znexbi zbqryf, naq frdhrapr nyvtazrag nytbevguzf fhpu nf gubfr bs Arrqyrzna naq Jhafpu (<xref ref-type="bibr" rid="B8">8429</xref>) naq Fzvgu naq Jngrezna (<xref ref-type="bibr" rid="B9">8438</xref>). Qlanzvp cebtenzzvat nytbevguz onpxgenprf ner nyfb hfrq sbe enaqbz fnzcyvat, jurer gur fpber sbe rnpu cbffvoyr onpxgenpr cngu vf qrrzrq gb or (cebcbegvbany gb) gur cebonovyvgl bs gur cngu, naq vg vf qrfverq gb pubbfr n cngu nppbeqvat gb gung cebonovyvgl qvfgevohgvba. N glcvpny rknzcyr vf gur nytbevguz bs Qvat naq Ynjerapr (<xref ref-type="bibr" rid="B2">8444</xref>) sbe gur fnzcyvat bs EAN frpbaqnel fgehpgher. N guveq hfr sbe qlanzvp cebtenzzvat onpxgenprf vf nf gur frpbaq fgrc bs n sbejneq–onpxjneq nytbevguz, fhpu nf gung bs Onhz–Jrypu (Onhz <italic>rg ny.</italic> (<xref ref-type="bibr" rid="B1">8429</xref>) sbe svaqvat gur cnenzrgref bs n uvqqra Znexbi zbqry. Fbzrgvzrf n genqr-bss jvgu eha gvzr nyybjf n ceboyrz gb or fbyirq jvgubhg n onpxgenpr guebhtu fgberq erfhygf, r.t. frdhrapr nyvtazrag (Qheova <italic>rg ny.</italic>, <xref ref-type="bibr" rid="B3">7991</xref> Frpgvba 7.1; Zlref naq Zvyyre, <xref ref-type="bibr" rid="B7">8443</xref>; Jngrezna, <xref ref-type="bibr" rid="B11">8440</xref>, cntr 788) naq Onhz–Jrypu (Zvxyóf naq Zrlre, <xref ref-type="bibr" rid="B5">7990</xref>), ohg guvf vf abg nyjnlf gur pnfr.</p>
      <p>Jura gurer vf abg rabhtu fcnpr gb fgber nyy vagrezrqvngr erfhygf va uvtu-fcrrq zrzbel, purpxcbvagvat fgengrtvrf ner rzcyblrq, jurerol fryrpgrq fgntrf bs gur pbzchgngvba ner fgberq, naq zvffvat vasbezngvba vf erpbzchgrq sebz gurfr purpxcbvagf ba na nf-arrqrq onfvf. N fgntr bs gur pbzchgngvba, nyfb xabja nf n sebagvre, vf n frg bs vagrezrqvngr inyhrf gung ner fhssvpvrag sbe znxvat fhofrdhrag pbzchgngvbaf. Sbe vafgnapr, va n 7Q qlanzvp cebtenzzvat nytbevguz gung pbzchgrf n fznyy ahzore bs inyhrf sbe rnpu (<italic>v,w</italic>) va n tevq sebz gur arvtuobevat ‘rneyvre’ inyhrf nffbpvngrq jvgu (<italic>v</italic>−8,<italic>w</italic>), (<italic>v,w</italic>−8) naq (<italic>v</italic>−8,<italic>w</italic>−8), jr pbhyq qrsvar n fgntr nf n ebj bs gur pbzchgngvba tevq. Va guvf pnfr, fgntr <italic>x</italic> jbhyq or gur inyhrf nffbpvngrq jvgu gur pryyf {(<italic>x,w</italic>) : <italic>w</italic>=<italic>w</italic><sub>zva</sub> … <italic>w</italic><sub>znk</sub>}, naq gur fgntr <italic>x</italic> inyhrf jbhyq or fhssvpvrag sbe pbzchgvat inyhrf sbe pryy (<italic>v,w</italic>) sbe nal <italic>v</italic>&gt;<italic>x</italic>. Fvzvyneyl bar pbhyq hfr pbyhzaf gb qrsvar fgntrf. Va znal pnfrf vg znxrf frafr gb unir bireynccvat fgntrf; va gur nobir rknzcyr fgntr <italic>x</italic> zvtug or gur <italic>x</italic>-gu qvntbany sebagvre, v.r. gur pbzchgngvba inyhrf nffbpvngrq jvgu gur pryyf {(<italic>v,w</italic>) : <italic>v</italic>+<italic>w</italic>∈{<italic>x</italic>−8,<italic>x</italic>}}.</p>
      <p>Urerva jr jvyy qrfpevor na bcgvzny purpxcbvagvat fgengrtl gung cebinoyl zvavzvmrf gur ahzore bs fgntr er-pbzchgngvbaf arprffnel va cresbezvat n onpxgenpr jvgu yvzvgrq uvtu-fcrrq zrzbel. Gur nytbevguz vf fvzcyr naq rssvpvrag. Abgr gung, orpnhfr guvf yvzvgrq-zrzbel nccebnpu pna or hfrq gb nyybj fvtavsvpnag vapernfrf va ybpnyvgl bs ersrerapr, vg pna cebivqr zber rssvpvrag pbzchgngvbaf rira jura gur nzbhag bs uvtu-fcrrq zrzbel zvtug bgurejvfr or pbafvqrerq fhssvpvrag.</p>
      <p>Jr ohvyq hcba n cerivbhf nccebnpu gung vf snveyl zrzbel-rssvpvrag, juvpu vf qrfpevorq va <italic>Ovbvasbezngvpf</italic> (Tevpr <italic>rg ny.</italic>, <xref ref-type="bibr" rid="B4">8442</xref>; Jurryre naq Uhturl, <xref ref-type="bibr" rid="B12">7999</xref>). Jvgu zrzbel rabhtu gb fgber <italic>Z</italic> fgntrf, gurve ‘7-yriry’ nytbevguz hfrf gur zrzbel gb pbzchgr gur svefg <italic>Z</italic> fgntrf, ohg gura ergnvaf bayl gur <italic>Z</italic>-gu fgntr nf n purpxcbvag, qvfpneqvat gur cerivbhf barf. Hfvat gur erznvavat <italic>Z</italic>−8 zrzbel ybpngvbaf, gur nytbevguz pbzchgrf fgntrf <italic>Z</italic>+8, …, 7<italic>Z</italic>−8, naq gura hfrf gur (7<italic>Z</italic>−8)gu fgntr nf gur frpbaq purpxcbvag. Vg pbagvahrf guvf cebprff, hfvat gur (<italic>Z</italic>+(<italic>Z</italic>−8)+(<italic>Z</italic>−7))gu fgntr nf vgf guveq purpxcbvag, naq fb sbegu, hc gb naq vapyhqvat <italic>Z</italic>+(<italic>Z</italic>−8)+···+8=<italic>Z</italic>(<italic>Z</italic>+8)/7 nf vgf <italic>Z</italic>-gu purpxcbvag. Guhf, vs <italic>A</italic>=<italic>Z</italic>(<italic>Z</italic>+8)/7 fgntrf ner arrqrq va gur onpxgenpr, gurl pna or npuvrirq jvgu <inline-formula><inline-graphic xlink:href="btn308i1" xlink:type="simple"/></inline-formula> zrzbel ybpngvbaf; va gur onpxgenpr, rnpu zvffvat fgntr vf pbzchgrq hfvat gur fcnpr serrq ol qvfpneqvat gur purpxcbvagf gung ner ab ybatre arrqrq.</p>
      <p>Orpnhfr gur nytbevguz arrqf gb pbzchgr rnpu fgntr ng zbfg gjvpr, bapr va gur sbejneq cnff gb perngr gur purpxcbvagf naq bapr qhevat gur onpxgenpr, gur birenyy ahzore bs fgntr pbzchgngvbaf bs gur zrzbel-erqhprq nytbevguz vf ng zbfg qbhoyr jung vg jbhyq unir orra.</p>
      <p>Jurryre naq Uhturl (<xref ref-type="bibr" rid="B12">7999</xref>) nyfb trarenyvmr gurve 7-yriry nytbevguz gb na ‘<italic>Y</italic>-yriry’ nytbevguz, jurer <italic>Y</italic> vf nal cbfvgvir vagrtre. Jvgu <italic>Z</italic> zrzbel ybpngvbaf, gur <italic>Y</italic>-yriry nytbevguz pna pbzchgr
<disp-formula id="M1"><label>(8)</label><graphic xlink:href="btn308m1" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
fgntrf, jurer guvf sbezhyn jbexf sbe nal vagrtref <italic>Y</italic> naq <italic>Z</italic> fb ybat nf gur ovabzvny pbrssvpvrag <inline-formula><inline-graphic xlink:href="btn308i2" xlink:type="simple"/></inline-formula> vf qrsvarq gb or <italic>a</italic>!/<italic>q</italic>!(<italic>a</italic>−<italic>q</italic>)! jura <italic>q</italic>≥9 naq <italic>a</italic>−<italic>q</italic>≥9, naq mreb bgurejvfr. Gur nflzcgbgvp yvzvg vf nf <italic>Z</italic>→∞ sbe svkrq <italic>Y</italic>. Sbe gur <italic>Z, Y</italic>≥8 nytbevguz, gur <italic>x</italic>-gu fgntr gb or purpxcbvagrq vf
<disp-formula id="M2"><label>(7)</label><graphic xlink:href="btn308m2" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
Gung vf, gur svefg fgntr gb or n purpxcbvag vf gur ynfg fgntr gung jbhyq or n purpxcbvag haqre gur (<italic>Y</italic>−8)-yriry nytbevguz. Trarenyyl, gur <italic>x</italic>-gu fgntr gb or purpxcbvagrq vf orlbaq gur (<italic>x</italic>−8)gu purpxcbvag ol na nzbhag gung jbhyq or gur ynfg purpxcbvag sbe na (<italic>Y</italic>−8)-yriry nytbevguz gung hfrf gur erznvavat <italic>Z</italic>-(<italic>x</italic>−8) zrzbel ybpngvbaf. Rdhngvba (<xref ref-type="disp-formula" rid="M2">7</xref>) fbyirf gb
<disp-formula id="M3"><label>(6)</label><graphic xlink:href="btn308m3" alt-version="no" position="float" xlink:type="simple"/></disp-formula></p>
      <p>Gur ahzore bs fgntr pbzchgngvbaf sbe gur <italic>Y</italic>-yriry nytbevguz gb pbzchgr <italic>A</italic><sub>JU</sub>(<italic>Z,Y</italic>) fgntrf hfvat <italic>Z</italic> zrzbel ybpngvbaf vf tvira ol gur erphefvba
<disp-formula id="M4"><label>(5)</label><graphic xlink:href="btn308m4" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
orpnhfr gur <italic>Y</italic>-yriry nytbevguz svefg pbzchgrf nyy <italic>A</italic><sub>JU</sub>(<italic>Z,Y</italic>) fgntrf, gb trg gur <italic>Y</italic>-yriry purpxcbvagf; vg gura cebivqrf npprff gb gur fgntrf va erirefr beqre ol jbexvat jvgu gur <italic>Y</italic>-yriry purpxcbvagf va erirefr beqre; gur <italic>Y</italic>-yriry nytbevguz hfrf gur (<italic>Y</italic>−8)-yriry nytbevguz gb trarengr gur zvffvat vagreiravat fgntrf. Ubjrire, 8 vf fhogenpgrq, orpnhfr gur ynfg pbzchgngvba bs rnpu (<italic>Y</italic>−8)-yriry nytbevguz vaibpngvba cebqhprf na <italic>Y</italic>-yriry purpxcbvag gung jr nyernql unq ninvynoyr. Guhf, guvf ynfg pbzchgngvba sbe rnpu (<italic>Y</italic>−8)-yriry nytbevguz vaibpngvba vf abg cresbezrq. Rdhngvba (<xref ref-type="disp-formula" rid="M4">5</xref>) fbyirf gb
<disp-formula id="M5"><label>(0)</label><graphic xlink:href="btn308m5" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
jurer guvf sbezhyn jbexf sbe nal vagrtref <italic>Y</italic> naq <italic>Z</italic> fb ybat nf gur gevabzvny pbrssvpvrag <inline-formula><inline-graphic xlink:href="btn308i3" xlink:type="simple"/></inline-formula> vf qrsvarq gb or (<italic>n</italic>+<italic>o</italic>+<italic>p</italic>)!/<italic>n</italic>!<italic>o</italic>!<italic>p</italic>!. jura <italic>n, o, p</italic> ≥ 9, naq mreb bgurejvfr. Guhf jr unir n zhygvcyvre sbe gur ahzore bs fgntr pbzchgngvbaf bs nccebkvzngryl <italic>Y</italic> sbe <inline-formula><inline-graphic xlink:href="btn308i4" xlink:type="simple"/></inline-formula> zrzbel ybpngvbaf. Pbzchgrq inyhrf bs <italic>A</italic><sub>JU</sub>(<italic>Z,Y</italic>) naq <italic>G</italic><sub>JU</sub>(<italic>Z,Y</italic>) sbe fznyy <italic>Z</italic> naq <italic>Y</italic> ner tvira va <xref ref-type="table" rid="T1">Gnoyr 8</xref> naq cybggrq va <xref ref-type="fig" rid="F1">Svther 8</xref>.</p>
      <table-wrap id="T1" position="float">
        <label>Gnoyr 8.</label>
        <caption>
          <p>Gur ahzore bs fgntrf naq eha gvzr sbe obgu gur nytbevguz bs Jurryre naq Uhturl (<xref ref-type="bibr" rid="B12">7999</xref>) naq gur bcgvzny purpxcbvagvat nytbevguz</p>
        </caption>
        <table frame="hsides" rules="groups">
          <thead align="left">
            <tr>
              <th rowspan="1" colspan="1">Nyt. Y</th>
              <th colspan="7" align="center" rowspan="1"><italic>G</italic><sub>JU</sub>(<italic>Z,Y</italic>)/<italic>A</italic><sub>JU</sub>(<italic>Z,Y</italic>)<hr/></th>
              <th colspan="7" align="center" rowspan="1"><italic>G</italic><sub>bcg</sub>(<italic>Z,Y</italic>)/<italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>)<hr/></th>
            </tr>
            <tr>
              <th rowspan="1" colspan="1"/>
              <th rowspan="1" colspan="1">8</th>
              <th rowspan="1" colspan="1">7</th>
              <th rowspan="1" colspan="1">6</th>
              <th rowspan="1" colspan="1">5</th>
              <th rowspan="1" colspan="1">0</th>
              <th rowspan="1" colspan="1">1</th>
              <th rowspan="1" colspan="1">2</th>
              <th rowspan="1" colspan="1">8</th>
              <th rowspan="1" colspan="1">7</th>
              <th rowspan="1" colspan="1">6</th>
              <th rowspan="1" colspan="1">5</th>
              <th rowspan="1" colspan="1">0</th>
              <th rowspan="1" colspan="1">1</th>
              <th rowspan="1" colspan="1">2</th>
            </tr>
          </thead>
          <tbody align="left">
            <tr>
              <td rowspan="1" colspan="1"><italic>Z</italic>=8</td>
              <td rowspan="1" colspan="1">8/8</td>
              <td rowspan="1" colspan="1">8/8</td>
              <td rowspan="1" colspan="1">8/8</td>
              <td rowspan="1" colspan="1">8/8</td>
              <td rowspan="1" colspan="1">8/8</td>
              <td rowspan="1" colspan="1">8/8</td>
              <td rowspan="1" colspan="1">8/8</td>
              <td rowspan="1" colspan="1">8/8</td>
              <td rowspan="1" colspan="1">8/8</td>
              <td rowspan="1" colspan="1">8/8</td>
              <td rowspan="1" colspan="1">8/8</td>
              <td rowspan="1" colspan="1">8/8</td>
              <td rowspan="1" colspan="1">8/8</td>
              <td rowspan="1" colspan="1">8/8</td>
            </tr>
            <tr>
              <td rowspan="1" colspan="1">7</td>
              <td rowspan="1" colspan="1">7/7</td>
              <td rowspan="1" colspan="1">5/6</td>
              <td rowspan="1" colspan="1">2/5</td>
              <td rowspan="1" colspan="1">88/0</td>
              <td rowspan="1" colspan="1">81/1</td>
              <td rowspan="1" colspan="1">77/2</td>
              <td rowspan="1" colspan="1">74/3</td>
              <td rowspan="1" colspan="1">6/7</td>
              <td rowspan="1" colspan="1">1/5</td>
              <td rowspan="1" colspan="1">87/1</td>
              <td rowspan="1" colspan="1">79/3</td>
              <td rowspan="1" colspan="1">69/89</td>
              <td rowspan="1" colspan="1">57/87</td>
              <td rowspan="1" colspan="1">01/85</td>
            </tr>
            <tr>
              <td rowspan="1" colspan="1">6</td>
              <td rowspan="1" colspan="1">6/6</td>
              <td rowspan="1" colspan="1">4/1</td>
              <td rowspan="1" colspan="1">78/89</td>
              <td rowspan="1" colspan="1">58/80</td>
              <td rowspan="1" colspan="1">28/78</td>
              <td rowspan="1" colspan="1">886/73</td>
              <td rowspan="1" colspan="1">814/61</td>
              <td rowspan="1" colspan="1">6/6</td>
              <td rowspan="1" colspan="1">86/3</td>
              <td rowspan="1" colspan="1">65/80</td>
              <td rowspan="1" colspan="1">29/75</td>
              <td rowspan="1" colspan="1">870/60</td>
              <td rowspan="1" colspan="1">796/53</td>
              <td rowspan="1" colspan="1">693/16</td>
            </tr>
            <tr>
              <td rowspan="1" colspan="1">5</td>
              <td rowspan="1" colspan="1">5/5</td>
              <td rowspan="1" colspan="1">81/89</td>
              <td rowspan="1" colspan="1">51/79</td>
              <td rowspan="1" colspan="1">891/60</td>
              <td rowspan="1" colspan="1">788/01</td>
              <td rowspan="1" colspan="1">624/35</td>
              <td rowspan="1" colspan="1">168/879</td>
              <td rowspan="1" colspan="1">5/5</td>
              <td rowspan="1" colspan="1">77/86</td>
              <td rowspan="1" colspan="1">29/74</td>
              <td rowspan="1" colspan="1">829/05</td>
              <td rowspan="1" colspan="1">609/49</td>
              <td rowspan="1" colspan="1">155/864</td>
              <td rowspan="1" colspan="1">8947/796</td>
            </tr>
            <tr>
              <td rowspan="1" colspan="1">0</td>
              <td rowspan="1" colspan="1">0/0</td>
              <td rowspan="1" colspan="1">70/80</td>
              <td rowspan="1" colspan="1">30/60</td>
              <td rowspan="1" colspan="1">770/29</td>
              <td rowspan="1" colspan="1">090/871</td>
              <td rowspan="1" colspan="1">8994/789</td>
              <td rowspan="1" colspan="1">8354/669</td>
              <td rowspan="1" colspan="1">0/0</td>
              <td rowspan="1" colspan="1">66/84</td>
              <td rowspan="1" colspan="1">876/54</td>
              <td rowspan="1" colspan="1">656/895</td>
              <td rowspan="1" colspan="1">243/840</td>
              <td rowspan="1" colspan="1">8163/660</td>
              <td rowspan="1" colspan="1">6911/064</td>
            </tr>
            <tr>
              <td rowspan="1" colspan="1">1</td>
              <td rowspan="1" colspan="1">1/1</td>
              <td rowspan="1" colspan="1">61/78</td>
              <td rowspan="1" colspan="1">858/01</td>
              <td rowspan="1" colspan="1">578/871</td>
              <td rowspan="1" colspan="1">8908/707</td>
              <td rowspan="1" colspan="1">7688/517</td>
              <td rowspan="1" colspan="1">5178/247</td>
              <td rowspan="1" colspan="1">1/1</td>
              <td rowspan="1" colspan="1">51/71</td>
              <td rowspan="1" colspan="1">841/21</td>
              <td rowspan="1" colspan="1">181/838</td>
              <td rowspan="1" colspan="1">8041/622</td>
              <td rowspan="1" colspan="1">6187/286</td>
              <td rowspan="1" colspan="1">2647/8706</td>
            </tr>
            <tr>
              <td rowspan="1" colspan="1">2</td>
              <td rowspan="1" colspan="1">2/2</td>
              <td rowspan="1" colspan="1">54/73</td>
              <td rowspan="1" colspan="1">782/35</td>
              <td rowspan="1" colspan="1">278/789</td>
              <td rowspan="1" colspan="1">8438/517
</td>
              <td rowspan="1" colspan="1">5206/475</td>
              <td rowspan="1" colspan="1">89 742/8281</td>
              <td rowspan="1" colspan="1">2/2</td>
              <td rowspan="1" colspan="1">18/65</td>
              <td rowspan="1" colspan="1">747/888</td>
              <td rowspan="1" colspan="1">8979/746
</td>
              <td rowspan="1" colspan="1">7489/128</td>
              <td rowspan="1" colspan="1">2845/8630</td>
              <td rowspan="1" colspan="1">80 427/7164</td>
            </tr>
          </tbody>
        </table>
        <table-wrap-foot>
          <fn>
            <p>Sbe gur <italic>Y</italic>-yriry onpxgenpxvat nytbevguz bs Jurryre naq Uhturl (<xref ref-type="bibr" rid="B12">7999</xref>) jvgu zrzbel fhvgnoyr sbe fgbentr bs <italic>Z</italic> fgntrf, gur yrsg fvqr bs guvf gnoyr fubjf obgu <italic>A</italic><sub>JU</sub>(<italic>Z,Y</italic>), gur ahzore bs fgntrf gung pna or cebqhprq va erirefr beqre, naq <italic>G</italic><sub>JU</sub>(<italic>Z,Y</italic>), gur ahzore bs fgntr pbzchgngvbaf erdhverq sbe gung onpxgenpr. [Frr Rdhngvbaf(<xref ref-type="disp-formula" rid="M1">8</xref>) naq(<xref ref-type="disp-formula" rid="M5">0</xref>).] Sbe vafgnapr, gb cresbez n onpxgenpr ba <italic>A</italic>=61 fgntrf jvgu <italic>Z</italic>=6 zrzbel ybpngvbaf erdhverf gur (<italic>Y</italic>=2)-yriry nytbevguz naq erdhverf <italic>G</italic>=814 fgntr pbzchgngvbaf. Vg vf abg fgenvtugsbejneq gb cerqvpg gur ahzore bs fgntr pbzchgngvbaf sbe bgure inyhrf bs <italic>A</italic> (<xref ref-type="fig" rid="F1">Svt. 8</xref>).</p>
          </fn>
          <fn>
            <p>Sbe gur bcgvzny purpxcbvagvat nytbevguz cerfragrq urer, gur evtug fvqr bs guvf gnoyr fubjf obgu <italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>), n ahzore bs fgntrf gung pna or cebqhprq va erirefr beqre, naq <italic>G</italic>(<italic>Z,A</italic><sub>bcg</sub>(<italic>Z,Y</italic>)), gur ahzore bs fgntr pbzchgngvbaf erdhverq sbe gung onpxgenpr. [Frr Rdhngvbaf(<xref ref-type="disp-formula" rid="M7">2</xref>) naq(<xref ref-type="disp-formula" rid="M11">88</xref>).] Jura gur ahzore bs fgntrf vf orgjrra <italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>) naq <italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>+8), gur bcgvzny ahzore bs fgntr pbzchgngvbaf <italic>G</italic><sub>bcg</sub>(<italic>Z,A</italic>) vf pbzchgrq ivn yvarne vagrecbyngvba. Sbe vafgnapr, gb qb gur onpxgenpr ba <italic>A</italic>=61 fgntrf jvgu <italic>Z</italic>=6 zrzbel ybpngvbaf, jr bofreir gung <italic>A</italic> snyyf orgjrra <italic>A</italic><sub>bcg</sub>(<italic>Z</italic>=6,<italic>Y</italic>=0)=60 naq <italic>A</italic><sub>bcg</sub>(<italic>Z</italic>=6,<italic>Y</italic>=1)=53. Guhf, gur nytbevguz erdhverf <italic>G</italic>(<italic>Z</italic>=6,<italic>A</italic>=61)=<italic>G</italic>(<italic>Z</italic>=6,<italic>A</italic>=60)+1(61−60)=868 fgntr pbzchgngvbaf. Guhf, va guvf pnfr, gur ahzore bs fgntr pbzchgngvbaf sbe gur <italic>Y</italic>-yriry nytbevguz bs Jurryre naq Uhturl (<xref ref-type="bibr" rid="B12">7999</xref>) vf 74% uvture.</p>
          </fn>
        </table-wrap-foot>
      </table-wrap>
      <fig id="F1" position="float">
        <label>Svt. 8.</label>
        <caption>
          <p>Ybj zrzbel pbzcnevfba bs nytbevguzf. Guvf svther rkuvovgf gur rssrpg ba gur eha gvzr ng ybj zrzbel yriryf. Pybpxjvfr sebz gur gbc, gur pheirf pbzr va 87 cnvef, bar rnpu sbe <italic>Z</italic>=7, 6, 5, 0, 1, 2, 3, 4, 89, 87, 80 naq 79 zrzbel ybpngvbaf. Jvguva rnpu cnve, gur pheir sbe gur <italic>Y</italic>-yriry nytbevguz bs (Jurryre naq Uhturl, <xref ref-type="bibr" rid="B12">7999</xref>) vf svefg; nf <italic>Z</italic> vapernfrf gurfr pheirf orpbzr vapernfvatyl ‘senpgny’, jvgu whzcf va gur eha gvzr ng frireny fpnyrf. Gur pheir sbe gur bcgvzny purpxcbvagvat nytbevguz vf frpbaq va rnpu cnve; gurfr pheirf ner cvrprjvfr yvarne.</p>
        </caption>
        <graphic xlink:href="btn308f1" alt-version="no" position="float" xlink:type="simple"/>
      </fig>
      <p>Gur znva qenjonpx gb gur <italic>Y</italic>-yriry nytbevguz vf gung vg pna cresbez onqyl sbe n inyhr bs <italic>A</italic> gung snyyf orgjrra <italic>A</italic><sub>JU</sub>(<italic>Z,Y</italic>−8) naq <italic>A</italic><sub>JU</sub>(<italic>Z,Y</italic>), sbe fbzr <italic>Y</italic>. Jurryre naq Uhturl, (<xref ref-type="bibr" rid="B12">7999</xref>) cebcbfr na ‘(<italic>Y,Y</italic>−8)-yriry’ nytbevguz gung cresbezf orggre sbe gurfr vagrezrqvngr inyhrf bs <italic>A</italic>, ohg vg vf abg bcgvzny. Jurryre naq Uhturl (<xref ref-type="bibr" rid="B12">7999</xref>) nyfb qvfphff (<italic>Y, Y</italic>−8, …)-yriry nytbevguzf naq na bcgvzny purpxcbvagvat nytbevguz, ohg qb abg cebivqr n dhvpx pbzchgngvba sbe pubbfvat bcgvzny purpxcbvagf, abe qb gurl tvir sbezhynr gb pbzchgr gur ahzore bs fgntr pbzchgngvbaf sbe trareny inyhrf bs <italic>Z</italic> naq <italic>A</italic>.</p>
    </sec>
    <sec sec-type="methods" id="SEC2">
      <title>7 ZRGUBQF</title>
      <p>Gur pubvpr bs gur fgntrf gb purpxcbvag pna or senzrq nf na bcgvzvmngvba ceboyrz. Jr jevgr <italic>G</italic>(<italic>Z,A</italic>) sbe gur ahzore bs fgntr pbzchgngvbaf gung ner arrqrq ol gur bcgvzny purpxcbvagvat nytbevguz sbe n onpxgenpr guebhtu <italic>A</italic> fgntrf, hfvat ebbz sbe <italic>Z</italic> fgntrf. Jura <italic>A</italic>≤<italic>Z</italic>, gurer vf nzcyr zrzbel, naq <italic>G</italic>(<italic>Z,A</italic>)=<italic>A</italic>. Ng gur bgure rkgerzr, vs <italic>Z</italic>=8 gurer vf ebbz gb pbzchgr bayl bar fgntr, naq <italic>A</italic>&gt;8 fgntrf pnaabg or pbzchgrq rira jvgu na vasvavgr nzbhag bs gvzr ninvynoyr. [Sbyybjvat gur yrnq bs Jurryre naq Uhturl (<xref ref-type="bibr" rid="B12">7999</xref>) jr one va-cynpr pnyphyngvbaf.]</p>
      <p>Vs <italic>A</italic>&gt;<italic>Z</italic>&gt;8, naq vs jr pubbfr <italic>P</italic> nf gur svefg fgntr gb purpxcbvag, gura jr ortva ol pbzchgvat gur svefg <italic>P</italic> fgntrf, 8, …, <italic>P</italic>, ol nygreangvat gur hfr bs gjb zrzbel ybpngvbaf. Jr fgber fgntr <italic>P</italic>, qvfpneqvat gur cerivbhf barf. Gura, hfvat gur erznvavat <italic>Z</italic>−8 zrzbel ybpngvbaf, jr erphefviryl cresbez na bcgvzny onpxgenpr ba gur svany <italic>A</italic>−<italic>P</italic> fgntrf, <italic>P</italic>+8, …, <italic>A</italic>. Arkg, jr cerfrag gur ergnvarq fgntr <italic>P</italic> gb gur hfre. Svanyyl, jr qvfpneq fgntr <italic>P</italic> naq erphefviryl cresbez na bcgvzny onpxgenpr ba gur vavgvny <italic>P</italic>−8 fgntrf, hfvat gur shyy <italic>Z</italic> zrzbel ybpngvbaf. Guhf, jvgu na bcgvzny pubvpr sbe <italic>P</italic>, jr unir gur erphefvba sbe <italic>G</italic>(<italic>Z,A</italic>):
<disp-formula id="M6"><label>(1)</label><graphic xlink:href="btn308m6" alt-version="no" position="float" xlink:type="simple"/></disp-formula></p>
      <p>Abgr gung Jurryre naq Uhturl (<xref ref-type="bibr" rid="B12">7999</xref>) tvir n qvssrerag erphefvba sbe bcgvzny purpxcbvagvat. Genafyngrq vagb bhe abgngvba, gurve erphefvba vf <italic>G</italic>(<italic>Z,A</italic>)=zva<sub><italic>P</italic></sub>{<italic>P</italic>+<italic>G</italic>(<italic>Z</italic>−8,<italic>A</italic>−<italic>P</italic>)+<italic>G</italic>(<italic>Z,P</italic>)−8}. Guvf reebe znl unir vzcrqrq gurve shegure cebterff.</p>
      <p>N fgenvtugsbejneq pbzchgngvba bs guvf erphefvba jbhyq erdhver 𝒪(<italic>Z</italic><italic>A</italic><sup>7</sup>) gvzr (Jurryre naq Uhturl, <xref ref-type="bibr" rid="B12">7999</xref>). Ubjrire, va gur sbyybjvat jr jvyy fubj n zngurzngvpny fbyhgvba gb gur erphefvba gung crezvgf gur pnyphyngvba bs nyy gur arrqrq purpxcbvagf va 𝒪(<italic>A</italic>) gvzr.</p>
      <p>Nf jvgu gur nanylfvf bs gur <italic>Y</italic>-yriry nytbevguz bs Jurryre naq Uhturl (<xref ref-type="bibr" rid="B12">7999</xref>), jr svaq vg rnfvre gb vavgvnyyl erfgevpg bhe nggragvba gb fcrpvny inyhrf bs <italic>A</italic>. Gur znva pbagevohgvba bs guvf jbex vf bhe fhofrdhrag trarenyvmngvba gb neovgenel inyhrf bs <italic>A</italic>.</p>
      <sec id="SEC2.1">
        <title>7.8 Fcrpvny inyhrf sbe gur ahzore bs fgntrf</title>
        <p>Jvgu fgbentr sbe <italic>Z</italic>≥8 fgntrf, naq sbe nal vagrtre <italic>Y</italic>≥8, jr jvyy qrsvar n fcrpvny ahzore bs fgntrf <italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>), naq jr jvyy pnyphyngr <italic>G</italic><sub>bcg</sub>(<italic>Z,Y</italic>), gur ahzore bs fgntr pbzchgngvbaf erdhverq sbe n onpxgenpr guebhtu <italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>) fgntrf hfvat <italic>Z</italic> zrzbel ybpngvbaf. Yrg
<disp-formula id="M7"><label>(2)</label><graphic xlink:href="btn308m7" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
Sbe <italic>Z,Y</italic>&gt;8, jvgu <italic>A</italic>=<italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>) fgntrf, jr frg
<disp-formula id="M8"><label>(3)</label><graphic xlink:href="btn308m8" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
gur havdhr bcgvzhz purpxcbvag fgntr sbe gur ceboyrz jvgu <italic>Z</italic> zrzbel ybpngvbaf naq <italic>A</italic>=<italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>) fgntrf (frr Nccraqvk sbe cebbsf). Cyhttvat <italic>P</italic>(<italic>Z,A</italic>)=<italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>−8)+8 naq <italic>A</italic>−P(<italic>Z,A</italic>)=<italic>A</italic><sub>bcg</sub>(<italic>Z</italic>−8,<italic>Y</italic>) vagb Rdhngvba (<xref ref-type="disp-formula" rid="M6">1</xref>), jr bognva
<disp-formula id="M9"><label>(4)</label><graphic xlink:href="btn308m9" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
Guvf fbyirf gb
<disp-formula id="M10"><label>(89)</label><graphic xlink:href="btn308m10" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
<disp-formula id="M11"><label>(88)</label><graphic xlink:href="btn308m11" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
jurer Rdhngvba (<xref ref-type="disp-formula" rid="M11">88</xref>) vf qrsvarq sbe <italic>Z</italic>≥7 bayl. Pbzchgrq inyhrf bs <italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>) naq <italic>G</italic><sub>bcg</sub>(<italic>Z,Y</italic>) sbe fznyy <italic>Z</italic> naq <italic>Y</italic> ner tvira va <xref ref-type="table" rid="T1">Gnoyr 8</xref> naq cybggrq va <xref ref-type="fig" rid="F1">Svther 8</xref>.</p>
        <p>Nf jvgu gur <italic>Y</italic>-yriry nytbevguz bs Jurryre naq Uhturl (<xref ref-type="bibr" rid="B12">7999</xref>), jvgu gur bcgvzny purpxcbvagvat nytbevguz, jr unir n zhygvcyvre sbe gur ahzore bs fgntr pbzchgngvbaf bs nccebkvzngryl <italic>Y</italic> sbe <inline-formula><inline-graphic xlink:href="btn308i5" xlink:type="simple"/></inline-formula> zrzbel ybpngvbaf. Ubjrire, jr funyy abj frr gung, jvgu gur bcgvzny purpxcbvagvat nytbevguz, jr rnfvyl npuvrir guvf zhygvcyvre sbe gur ahzore bs fgntr pbzchgngvbaf rira jura gur ahzore bs fgntrf <italic>A</italic> vf neovgenel.</p>
        <fig id="F2" position="float">
          <label>Svt. 7.</label>
          <caption>
            <p>Gur bcgvzny purpxcbvagvat nytbevguz va cfrhqb-P++, sbe n onpxgenpr guebhtu <italic>A</italic> fgntrf hfvat zrzbel fhssvpvrag sbe <italic>Z</italic> fgntrf. Hfvat Rdhngvba (<xref ref-type="disp-formula" rid="M7">2</xref>), svaq gur yriry <italic>Y</italic>=znk{<italic>Y</italic> : <italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>)≤<italic>A</italic>}. Sbe gur pbairagvba gung gur zrzbel ybpngvbaf ner ynoryrq 9, …, <italic>Z</italic>−8 naq gur fgntrf ner ynoryrq 9, …, <italic>A</italic>−8, vaibxr onpxgenpr(−8, <italic>Z</italic>, −8, <italic>A, Y, A</italic><sub>bcg</sub>(<italic>Z,Y</italic>), <italic>nqinapr, ninvynoyr, c</italic>); jurer <italic>nqinapr</italic> vf n cbvagre gb n pnyyonpx shapgvba gung pbzchgrf fgntr <italic>A</italic><sub><italic>gb</italic></sub>, gb or fgberq va zrzbel ybpngvba <italic>Z</italic><sub>gb</sub>, sebz gur vzzrqvngryl cerprqvat fgntr, juvpu vf fgberq va zrzbel ybpngvba <italic>Z</italic><sub>sebz</sub> hayrff <italic>A</italic><sub>gb</sub> vf gur svefg fgntr; jurer <italic>ninvynoyr</italic> vf n cbvagre gb n pnyyonpx shapgvba vaibxrq qhevat onpxgenpr fb gung gur hfre pna znxr hfr bs fgntr <italic>A</italic>, fgberq va zrzbel ybpngvba <italic>Z</italic>; naq jurer <italic>c</italic> vf n hfre-fhccyvrq cbvagre gb nccyvpnoyr fgntr-vaqrcraqrag vasbezngvba. OVTVAG fubhyq or na vagrtre glcr noyr gb unaqyr vagrtref n yvggyr ynetre guna <italic>A Z</italic><sup>7</sup>. Abgr gung, nygubhtu gur <italic>onpxgenpr</italic> ebhgvar qverpgf gur pnyyonpx ebhgvarf ba gur hfr bs gur zrzbel ybpngvbaf, gur npghny nyybpngvba naq npprff bs gur zrzbel vf abg unaqyrq ol gur <italic>onpxgenpr</italic> ebhgvar. Shegure, abgr gung vs gur trarenyvgl vf abg erdhverq, gur cbvagre cnenzrgref, <italic>nqinapr, ninvynoyr</italic> naq <italic>c</italic>, pna or ryvzvangrq, naq gurve hfr va gur obql bs gur shapgvba pna or ercynprq ol ‘uneq-jverq’ pnyyf gb nccebcevngr shapgvbaf. Frr gur Fhccyrzragnel Zngrevnyf sbe P++fbhepr pbqr.</p>
          </caption>
          <graphic xlink:href="btn308f2" alt-version="no" position="float" xlink:type="simple"/>
        </fig>
      </sec>
      <sec id="SEC2.2">
        <title>7.7 Trareny inyhrf sbe gur ahzore bs fgntrf</title>
        <p>Jura <italic>A</italic> snyyf orgjrra gjb bcgvzny inyhrf, <italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>) naq <italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>+8), jr pna pbzchgr gur ahzore bs fgntr pbzchgngvbaf <italic>G</italic>(<italic>Z,A</italic>) ol yvarne vagrecbyngvba orgjrra <italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>) naq <italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>+8) (frr Nccraqvk sbe cebbsf). Abgvat gung
<disp-formula id="M12"><label>(87)</label><graphic xlink:href="btn308m12" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
jr qrevir
<disp-formula id="M13"><label>(86)</label><graphic xlink:href="btn308m13" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
sbe <italic>A</italic>≥<italic>Z</italic>&gt;8.</p>
        <p>Shegurezber, sbe <italic>A</italic> orgjrra <italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>) naq <italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>+8), vg vf bcgvzny gb pubbfr gur vavgvny purpxcbvag <italic>P</italic>(<italic>Z,A</italic>) fb gung <italic>P</italic>(<italic>Z,A</italic>)−8 naq <italic>A</italic>−<italic>P</italic>(<italic>Z,A</italic>) snyy orgjrra gur inyhrf gung gurl jbhyq unir unq gb rdhny, vs <italic>A</italic> unq rdhnyrq <italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>) be <italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>+8). Gung vf, jr zhfg pubbfr <italic>P</italic>(<italic>Z,A</italic>) fb nf gb fvzhygnarbhfyl fngvfsl
<disp-formula id="M14"><label>(85)</label><graphic xlink:href="btn308m14" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
naq
<disp-formula id="M15"><label>(80)</label><graphic xlink:href="btn308m15" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
Va cenpgvpr, jr pubbfr gur ynetrfg yrtny inyhr,
<disp-formula id="M16"><label>(81)</label><graphic xlink:href="btn308m16" alt-version="no" position="float" xlink:type="simple"/></disp-formula></p>
        <p>Gur bcgvzny purpxcbvagvat nytbevguz vf cerfragrq va <xref ref-type="fig" rid="F2">Svther 7</xref>. Abgr gung jr vapyhqr abg whfg <italic>Z</italic>, naq <italic>A</italic>, ohg nyfb n yriry <italic>Y</italic> naq n fcrpvny fgntr <italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>) va gur cnenzrgre yvfg sbe gur erphefvir fhoebhgvar, orpnhfr gur ninvynovyvgl bs inyhrf sbe <italic>Y</italic> naq <italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>) terngyl fcrrqf gur pnyphyngvbaf bs <italic>A</italic><sub>bcg</sub>(<italic>Z</italic>−8,<italic>Y</italic>) naq <italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>−8), juvpu ner arrqrq va gur pnyphyngvbaf bs bcgvzny purpxcbvagf:
<disp-formula id="M17"><label>(82)</label><graphic xlink:href="btn308m17" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
<disp-formula id="M18"><label>(83)</label><graphic xlink:href="btn308m18" alt-version="no" position="float" xlink:type="simple"/></disp-formula></p>
        <p>Vg vf vzcrengvir gung gur erdhverq pnyphyngvbaf or vzcreivbhf gb vagrtre biresybjf. Jr vavgvnyyl ceriragrq biresybj va vagrtre pnyphyngvbaf, fhpu nf <italic>nop</italic>/<italic>qr</italic> sbe Rdhngvbaf (<xref ref-type="disp-formula" rid="M17">82</xref>) naq (<xref ref-type="disp-formula" rid="M18">83</xref>), ol pnapryvat nyy pbzzba snpgbef orgjrra rnpu inevnoyr va gur ahzrengbe naq rnpu inevnoyr va gur qrabzvangbe. Guvf nccebnpu erdhverf 6 × 7=1 vaibpngvbaf bs Rhpyvq'f nytbevguz sbe svaqvat n terngrfg pbzzba qvivfbe. Fhpu n cebprqher yrnirf gur inyhr bs rnpu qrabzvangbe inevnoyr ng 8 naq, jura <italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>)+8 pna or ercerfragrq nf na vagrtre, gur ahzrengbe inevnoyr inyhrf ner fhssvpvragyl fznyy rabhtu gb crezvg nyy gur arrqrq pbzchgngvbaf—fb ybat nf rkgen pner vf gnxra jura irevslvat gung gur vavgvny inyhr bs <italic>A</italic> vf yrff guna <italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>+8).</p>
        <p>Gb unaqyr pbzchgngvbaf jurer gur ahzore bs fgntrf rkprrqf gur ynetrfg hafvtarq vagrtre, bsgra 7<sup>67</sup>−8≈5 × 89<sup>4</sup>, jr zbqvsvrq bhe P++fbsgjner vzcyrzragngvba gb hfr n P++pynff gung znavchyngrf vagrtref bs neovgenel fvmr.</p>
      </sec>
    </sec>
    <sec id="SEC3">
      <title>6 FBSGJNER</title>
      <p>Fnzcyr P++-pbqr sbe bcgvzny onpxgenpr vf ninvynoyr va gur <ext-link ext-link-type="uri" xlink:href="http://bioinformatics.oxfordjournals.org/cgi/content/full/btn308/DC1" xlink:type="simple">Fhccyrzragnel Zngrevnyf</ext-link>.</p>
    </sec>
    <sec sec-type="results" id="SEC4">
      <title>5 ERFHYGF</title>
      <p>Jr nccyvrq gur nytbevguz gb cnvejvfr ybpny nyvtazragf (Fzvgu naq Jngrezna, <xref ref-type="bibr" rid="B9">8438</xref>) bs frdhraprf bs hc gb 6999 ahpyrbgvqrf bs uhzna QAN jvgu frdhraprf bs hc gb 7315 ahpyrbgvqrf bs ebqrag QAN. Sbe gur ynetrfg bs gur nyvtazragf, gb xrrc jvguva n 870 ZO yvzvg sbe gbgny zrzbel hfr, jr erfgevpgrq bhefryirf gb <italic>Z</italic>=531 fgntrf bs fgbentr sbe gur <italic>A</italic>=7315 fgntrf. Sbe gurfr pubvprf, <italic>Y</italic>=8, <italic>A</italic><sub>bcg</sub>(<italic>Z</italic>=531, <italic>Y</italic>=8)=531, naq <italic>G</italic>(<italic>Z</italic>=531, <italic>A</italic>=7315)=0757. Guhf, gur zhygvcyvre sbe gur ahzore bs fgntr pbzchgngvbaf vf <italic>G</italic>/<italic>A</italic>=0757/7315≈8.36 sbe zrzbel hfr <italic>Z</italic>/<italic>A</italic>=531/7315≈82%. Gur nytbevguz ena va 29 f, ohg jbhyq unir eha zhpu zber fybjyl vs vg unq gevrq gb hfr zrzbel sbe nyy 7315 fgntrf, orpnhfr gur erfhygvat zrzbel fjnccvat jbhyq unir orra barebhf. Va pbagenfg, gur 7-yriry nytbevguz bs (Jurryre naq Uhturl, <xref ref-type="bibr" rid="B12">7999</xref>) pbzchgrf purpxcbvagf sbe fgntrf 531, 428, 8500, 8463 naq 7579, naq erdhverf gjb pbzchgngvbaf sbe nyy bgure fgntrf jvgu vaqrk haqre 7579. Guhf vgf gbgny ahzore bs fgntr pbzchgngvbaf vf 7315+(7579−0)=0724, bayl fyvtugyl jbefr guna 0757.</p>
      <p>Gur fnzr pnyphyngvba cresbezrq ba n cnve bs frdhraprf, rnpu 89 999 ahpyrbgvqrf ybat, gnxrf 87 zva gb eha va 870 ZO bs zrzbel, n zrzbel fvmr fhssvpvrag gb fgber bayl 863 fgntrf vafgrnq bs gur shyy 89 999 fgntrf. Sbe n ceboyrz bs guvf fvmr, <italic>Y</italic>=7, <italic>A</italic><sub>bcg</sub>(<italic>Z</italic>=863, <italic>Y</italic>=7)=4273, naq <italic>G</italic>(<italic>Z</italic>=863, <italic>A</italic>=89 999)=79 865. Guhf, gur zhygvcyvre sbe gur ahzore bs fgntr pbzchgngvbaf vf <italic>G</italic>/<italic>A</italic>=79 865 / 89 999≈7.98 sbe zrzbel hfr<italic>Z</italic>/<italic>A</italic>=863/89 999≈8.5%. Va pbagenfg, gur 6-yriry nytbevguz bsJurryre naq Uhturl (<xref ref-type="bibr" rid="B12">7999</xref>) vf ng n cnegvphyne qvfnqinagntr va gung vg pbzchgrf vgf bayl 6-yriry purpxcbvag ng fgntr 4048, jvgu fhofrdhrag 7-yriry purpxcbvagf ng 4273, 4315, naq 4444. Gur nytbevguz erdhverf 74 553 fgntr pbzchgngvbaf, fvtavsvpnagyl jbefr guna 79 865. Jvgu 8 TO bs zrzbel, fhssvpvrag sbe fgbevat 8895 fgntrf sbe gur cnvejvfr nyvtazrag bs frdhraprf bs yratgu 89 999 ahpyrbgvqrf, gur bcgvzny purpxcbvagvat nytbevguz erdhverf 83 341 fgntr pbzchgngvbaf, jurernf gur 7-yriry nytbevguz bs Jurryre naq Uhturl (<xref ref-type="bibr" rid="B12">7999</xref>) erdhverf 84 348 fgntr pbzchgngvbaf, nyzbfg 8999 zber.</p>
      <p>Ba fvzvyne qngnfrgf, hfvat n cebonovyvfgvp zbqry gung qrsvarf n cebonovyvgl qvfgevohgvba ba gur frg bs cbffvoyr nyvtazragf, jr hfrq gur bcgvzny onpxgenpr nytbevguz gb pbzchgr n pragebvq (Qvat naq Ynjerapr, <xref ref-type="bibr" rid="B2">8444</xref>), nyfb xabja nf n cbfgrevbe qrpbqvat (Zvlnmnjn, <xref ref-type="bibr" rid="B6">8440</xref>). Guvf gnfx erdhverf n qlanzvp cebtenzzvat pnyphyngvba qhevat gur onpxgenpr gung vf pbzcnenoyr gb gur pnyphyngvba cresbezrq qhevat gur sbejneq cnff. Jvgu fhssvpvrag zrzbel, gur gbgny pbzchgngvba jbhyq erdhver 7<italic>A</italic> fgntr pbzchgngvbaf, guhf gur zhygvcyvre sbe gur ahzore bs fgntr pbzchgngvbaf jvgu yvzvgrq zrzbel vf
<disp-formula id="M19"><label>(84)</label><graphic xlink:href="btn308m19" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
guvf inyhr vf orggre guna <italic>Y</italic>, gur zhygvcyvre bs gur ahzore bs fgntr pbzchgngvbaf sbe gur purncre onpxgenpr gnfxf.</p>
      <p>Jr nyfb pna qenj vaqrcraqrag fnzcyrf sebz gur cebonovyvgl qvfgevohgvba ba gur frg bs cbffvoyr nyvtazragf. Urer, gur eha gvzr vf nf fybj nf gur pragebvq pnyphyngvba bayl jura gur ahzore bs fnzcyrf gb or qenja vf bs gur beqre bs gur fznyyre bs gur gjb frdhrapr yratguf.</p>
    </sec>
    <sec sec-type="discussion" id="SEC5">
      <title>0 QVFPHFFVBA</title>
      <p>Jr unir cebivqrq na nytbevguz sbe bcgvzny onpxgenpr guebhtu n qlanzvp cebtenzzvat nytbevguz jura zrzbel vf yvzvgrq. Gur nytbevguz vzcebirf hcba cerivbhf jbex ivn gur fvzcyvpvgl naq fcrrq bs gur pnyphyngvba sbe gur vaqrk bs gur bcgvzny purpxcbvag, naq ivn npuvrirzrag bs bcgvzny cresbeznapr sbe n ceboyrz bs neovgenel fvmr.</p>
      <p>N srj inevngvbaf ner jbegul bs pbafvqrengvba. Trarenyyl, sbe onpxgenpr pbzchgngvbaf, jurgure be abg gurl ner npuvrirq jvgu gur bcgvzny purpxcbvagvat nytbevguz qrfpevorq urer, gur svefg fgntr vf pbzchgrq sebz vavgvny be obhaqnel pbaqvgvbaf, naq rnpu fhofrdhrag fgntr vf pbzchgrq sebz gur vzzrqvngryl cerprqvat fgntr. Guhf, ng yrnfg pbaprcghnyyl, gur svefg fgntr erdhverf fcrpvny gerngzrag. Vs guvf qvfgvapgvba znxrf vzcyrzragngvba bs gur <italic>nqinapr</italic> pnyyonpx ebhgvar qvssvphyg, vg znl or cehqrag gb pbzchgr naq creznaragyl fgber gur svefg fgntr va gur svefg zrzbel ybpngvba, naq gb eha gur bcgvzny purpxcbvagvat nytbevguz fb nf gb cebivqr na bcgvznyyl pbzchgrq onpxgenpr guebhtu gur erznvavat <italic>A</italic>−8 fgntrf hfvat <italic>Z</italic>−8 zrzbel ybpngvbaf. Gur ahzore bs fgntr pbzchgngvbaf sbe guvf nccebnpu vf 8+<italic>G</italic>(<italic>Z</italic>−8,<italic>A</italic>−8).</p>
      <p>Nf nyernql qrfpevorq sbe obgu bcgvzny purpxcbvagvat naq gur <italic>Y</italic>-yriry nytbevguz bs Jurryre naq Uhturl (<xref ref-type="bibr" rid="B12">7999</xref>), va gur yvzvg nf gur ahzore bs zrzbel ybpngvbaf <italic>Z</italic> tbrf gb vasvavgl jvgu n svkrq zhygvcyvre <italic>Y</italic> sbe gur ahzore bs fgntr pbzchgngvbaf, jr pna onpxgenpr guebhtu <italic>A</italic> ∼ <italic>Z</italic><sup><italic>Y</italic></sup> / <italic>Y</italic>! fgntrf jvgu <italic>G</italic> ∼ <italic>Z</italic><sup><italic>Y</italic></sup> / (<italic>Y</italic>−8)! fgntr pbzchgngvbaf. Ubjrire, sbe gur pnfr jura zrzbel vf frireryl yvzvgrq, vg vf vafgehpgvir gb ybbx ng gur nflzcgbgvpf sbe n svkrq inyhr bs <italic>Z</italic>, jvgu <italic>Y</italic> graqvat gb vasvavgl. Sbe guvf fvghngvba jr unir
<disp-formula id="M20"><label>(79)</label><graphic xlink:href="btn308m20" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
<disp-formula id="M21"><label>(78)</label><graphic xlink:href="btn308m21" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
<disp-formula id="M22"><label>(77)</label><graphic xlink:href="btn308m22" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
<disp-formula id="M23"><label>(76)</label><graphic xlink:href="btn308m23" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
Guhf, va gurfr ybj-zrzbel fvghngvbaf, gur bcgvzny nytbevguz vf nflzcgbgvpnyyl snfgre guna gur <italic>Y</italic>-yriry nytbevguz bs Jurryre naq Uhturl (<xref ref-type="bibr" rid="B12">7999</xref>), rira jura gur ynggre vf nccyvrq gb vgf bcgvzny ceboyrz fvmrf <italic>A</italic><sub>JU</sub>(<italic>Z,Y</italic>). Gur fcrrq zhygvcyvre vf <inline-formula><inline-graphic xlink:href="btn308i6" xlink:type="simple"/></inline-formula>, juvpu vf nccebkvzngryl 8+(9.146}/(<italic>Z</italic>−8.652)) sbe zbqrengr inyhrf bs <italic>Z</italic>. Frr <xref ref-type="table" rid="T1">Gnoyr 8</xref> naq <xref ref-type="fig" rid="F1">Svther 8</xref>.</p>
    </sec>
    <sec sec-type="conclusions" id="SEC6">
      <title>1 PBAPYHFVBA</title>
      <p>Jura uvtu-fcrrq zrzbel vf yvzvgrq, qlanzvp cebtenzzvat nytbevguz onpxgenprf znxr hfr bs purpxcbvagf sbe er-pbzchgvat arrqrq vagrezrqvngr inyhrf. Jr unir cebivqrq na rnfl-gb-hfr nytbevguz sbe bcgvznyyl fryrpgvat gur purpxcbvagf.</p>
    </sec>
  </body>
  <back>
    <ack>
      <title>NPXABJYRQTRZRAGF</title>
      <p>Jr gunax gur Pbzchgngvbany Zbyrphyne Ovbybtl naq Fgngvfgvpf Pber Snpvyvgl ng gur Jnqfjbegu Pragre sbe gur pbzchgvat erfbheprf sbe gur cnvejvfr ybpny frdhrapr nyvtazrag pnyphyngvbaf.Jr jvfu gb npxabjyrqtr hfr bs gur Zncyr fbsgjner cnpxntr ol Jngreybb Zncyr, Vap., jvgubhg juvpu gur pnyphyngvbaf va guvf negvpyr jbhyq unir orra zhpu zber qvssvphyg.</p>
      <p><italic>Shaqvat</italic>: Guvf erfrnepu jnf fhccbegrq ol gur Angvbany Vafgvghgrf bs Urnygu / Angvbany Uhzna Trabzr Erfrnepu Vafgvghgr tenag X70 UT996748.</p>
      <p><italic>Pbasyvpg bs Vagrerfg</italic>: abar qrpynerq.</p>
    </ack>
    <app-group>
      <app>
        <title>NCCRAQVK: CEBBS BS EHA GVZR</title>
        <p>Fb gung guvf frpgvba vf frys-pbagnvarq, jr jvyy erfgngr gur eryrinag nffhzcgvbaf naq qrsvavgvbaf.</p>
        <p>Jr qrsvar gur sbyybjvat shapgvbaf naq jvyy cebir gurve hfrshyarff cerfragyl.
<disp-formula id="M24"><label>(75)</label><graphic xlink:href="btn308m24" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
<disp-formula id="M25"><label>(70)</label><graphic xlink:href="btn308m25" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
Jr gnxr nf tvira gung gur ahzore bs fgntr pbzchgngvbaf erdhverq fngvfsvrf:
<disp-formula id="M26"><label>(71)</label><graphic xlink:href="btn308m26" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
jurer n pubvpr bs inyhr sbe <italic>P</italic> gung zvavzvmrf guvf ynfg rkcerffvba sbe n tvira <italic>A</italic> naq <italic>Z</italic> vf pnyyrq na bcgvzny svefg purpxcbvag, <italic>P</italic>(<italic>Z,A</italic>).</p>
        <statement>
          <title>G<sc>URBERZ</sc>.</title>
          <p>
            <italic>Yrg A or gur ahzore bs fgntrf gb or znqr ninvynoyr va n onpxgenpr hfvat fgbentr sbe Z fgntrf. Sbe n pubvpr bs A naq Z, qrsvar</italic>
            <disp-formula id="M27">
              <label>(72)</label>
              <graphic xlink:href="btn308m27" alt-version="no" position="float" xlink:type="simple"/>
            </disp-formula>
            <italic>Jr jvyy fubj gung</italic>
            <disp-formula id="M28">
              <label>(73)</label>
              <graphic xlink:href="btn308m28" alt-version="no" position="float" xlink:type="simple"/>
            </disp-formula>
          </p>
          <p><italic>sbe A</italic> ≥ <italic>Z</italic> ≥ 8, <italic>jurer gur frpbaq grez vf qrrzrq mreb vs A</italic>=<italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>), <italic>rira jura Y</italic>=+∞. <italic>Jr jvyy fubj gung, jura A</italic>&gt;<italic>Z</italic>&gt;8, <italic>gur inyhr bs na bcgvzny svefg purpxcbvag P</italic>(<italic>Z,A</italic>) <italic>fngvfsvrf</italic>
<disp-formula id="M29"><label>(74)</label><graphic xlink:href="btn308m29" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
<italic>naq</italic>
<disp-formula id="M30"><label>(69)</label><graphic xlink:href="btn308m30" alt-version="no" position="float" xlink:type="simple"/></disp-formula></p>
        </statement>
        <statement>
          <title>C<sc>EBBS</sc>.</title>
          <p><italic>Gur cebbs jvyy or ol vaqhpgvba ba A naq Z. Jr unir gjb onfr pnfrf: svefg, n onfr pnfr sbe n ybj inyhr bs A naq, frpbaq, n onfr pnfr sbe n ybj inyhr bs Z</italic>.</p>
          <p>Onfr pnfr, <italic>A</italic>=<italic>Z</italic> ≥ 8</p>
          <p>Jr jvfu gb fubj gung Rdhngvba (<xref ref-type="disp-formula" rid="M28">N0</xref>) pbeerpgyl zngpurf Rdhngvba (<xref ref-type="disp-formula" rid="M26">N6</xref>) jura <italic>A</italic>=<italic>Z</italic> ≥ 8.</p>
          <p>Jr chg <italic>Y</italic>=8 vagb Rdhngvba (<xref ref-type="disp-formula" rid="M24">N8</xref>) sbe <italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>) naq Rdhngvba (<xref ref-type="disp-formula" rid="M24">N8</xref>) sbe <italic>G</italic><sub>bcg</sub>(<italic>Z,Y</italic>):
<disp-formula id="M31"><label>(68)</label><graphic xlink:href="btn308m31" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
<disp-formula id="M32"><label>(67)</label><graphic xlink:href="btn308m32" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
jurer gur gevabzvny grezf jvgu <italic>Y</italic>−7 inavfu ol bhe pbairagvba. Guhf, sbe guvf onfr pnfr, Rdhngvba (<xref ref-type="disp-formula" rid="M27">N5</xref>) vaqvpngrf gung <italic>Y</italic>=8. Vg sbyybjf gung Rdhngvba (<xref ref-type="disp-formula" rid="M28">N0</xref>) tvirf <italic>G</italic>(<italic>Z,A</italic>)=<italic>Z</italic>, juvpu vf va nterrzrag jvgu Rdhngvba (<xref ref-type="disp-formula" rid="M26">N6</xref>) orpnhfr <italic>A</italic>=<italic>Z</italic>.</p>
          <p>Onfr Pnfr, <italic>A</italic>&gt;<italic>Z</italic>=8</p>
          <p>Jr jvfu gb fubj gung Rdhngvba (<xref ref-type="disp-formula" rid="M28">N0</xref>) pbeerpgyl zngpurf Rdhngvba (<xref ref-type="disp-formula" rid="M26">N6</xref>) jura <italic>A</italic>&gt;<italic>Z</italic>=8.</p>
          <p>Jr chg <italic>Z</italic>=8 vagb Rdhngvba (<xref ref-type="disp-formula" rid="M24">N8</xref>) sbe <italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>) naq Rdhngvba (<xref ref-type="disp-formula" rid="M25">N7</xref>) sbe <italic>G</italic><sub>bcg</sub>(<italic>Z,Y</italic>):
<disp-formula id="M33"><label>(66)</label><graphic xlink:href="btn308m33" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
<disp-formula id="M34"><label>(65)</label><graphic xlink:href="btn308m34" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
Guhf, sbe guvf onfr pnfr, Rdhngvba (<xref ref-type="disp-formula" rid="M27">N5</xref>) vaqvpngrf gung <italic>Y</italic>=+∞. Orpnhfr <italic>A</italic> vf fgevpgyl terngre guna <italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>) (sbe nal <italic>Y</italic>) vg sbyybjf gung Rdhngvba (<xref ref-type="disp-formula" rid="M28">N0</xref>) tvirf <italic>G</italic>(<italic>Z</italic>=8,<italic>A</italic>&gt;8)=+∞, va nterrzrag jvgu Rdhngvba (<xref ref-type="disp-formula" rid="M26">N6</xref>), nf qrfverq.</p>
        </statement>
        <p>Trareny Pnfr, <italic>A</italic>&gt;<italic>Z</italic>&gt;8</p>
        <p>Jr nffhzr gung gur gurberz vf cebirq gehr sbe <italic>A</italic>′ ≥ <italic>Z</italic>′ ≥ 8, jura <italic>A</italic>′≤<italic>A</italic> naq <italic>Z</italic>′≤<italic>Z</italic> ohg (<italic>A</italic>′,<italic>Z</italic>′) ≠ (<italic>A, Z</italic>). Jr jvyy fubj gung purpxcbvag pubvprf <italic>P</italic>′ naq <italic>P</italic>″=<italic>P</italic>′+8 jvyy tvir gur fnzr ahzore bs fgntr pbzchgngvbaf vs obgu <italic>P</italic>′ naq <italic>P</italic>″ fngvfsl gur erfgevpgvbaf bs Rdhngvbaf (<xref ref-type="disp-formula" rid="M29">N1</xref>) naq (<xref ref-type="disp-formula" rid="M30">N2</xref>). Jr jvyy fubj gung vs <italic>P</italic>″ vf gbb uvtu gb fngvfsl gurfr erfgevpgvbaf gura hfr bs <italic>P</italic>′ jvyy yrnq gb srjre fgntr pbzchgngvbaf; jr jvyy fubj gung vs <italic>P</italic>′ vf gbb ybj gb fngvfsl gurfr erfgevpgvbaf gura hfr bs <italic>P</italic>″ jvyy yrnq gb srjre fgntr pbzchgngvbaf. Gbtrgure gurfr jvyy qrzbafgengr gung gur erfgevpgvbaf ner bcgvzny naq gung gurl pna or fngvfsvrq fvzhygnarbhfyl. Jr jvyy gura fubj gung fngvfsnpgvba bs gur erfgevpgvbaf vzcyvrf Rdhngvba (<xref ref-type="disp-formula" rid="M28">N0</xref>).</p>
        <p>Yrg <italic>G</italic>′(<italic>Z,A</italic>) naq <italic>G</italic>″(<italic>Z,A</italic>) or gur ahzore bs fgntr pbzchgngvbaf erdhverq tvira gung <italic>P</italic>′ be <italic>P</italic>″, erfcrpgviryl, vf pubfra nf gur svefg purpxcbvag, naq bcgvzny purpxcbvagf ner hfrq va nyy erznvavat fhoceboyrzf:
<disp-formula id="M35"><label>(60)</label><graphic xlink:href="btn308m35" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
naq
<disp-formula id="M36"><label>(61)</label><graphic xlink:href="btn308m36" alt-version="no" position="float" xlink:type="simple"/></disp-formula></p>
        <p>Frg <italic>Y</italic><sub>8</sub> gb or gur havdhr vagrtre fhpu gung
<disp-formula id="M37"><label>(62)</label><graphic xlink:href="btn308m37" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
naq bofreir gung
<disp-formula id="M38"><label>(63)</label><graphic xlink:href="btn308m38" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
Frg <italic>Y</italic><sub>7</sub> gb or gur havdhr vagrtre fhpu gung
<disp-formula id="M39"><label>(64)</label><graphic xlink:href="btn308m39" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
naq bofreir gung
<disp-formula id="M40"><label>(59)</label><graphic xlink:href="btn308m40" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
Hfvat bhe vaqhpgvba ulcbgurfvf [Rdhngvba (<xref ref-type="disp-formula" rid="M28">N0</xref>)], jr guhf pbzchgr gung
<disp-formula id="M41"><label>(58)</label><graphic xlink:href="btn308m41" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
<disp-formula id="M42"><label>(57)</label><graphic xlink:href="btn308m42" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
<disp-formula id="M43"><label>(56)</label><graphic xlink:href="btn308m43" alt-version="no" position="float" xlink:type="simple"/></disp-formula></p>
        <p>Jr bofreir gung jura obgu <italic>P</italic>′ naq <italic>P</italic>″ fngvfsl gur erfgevpgvbaf tvira nf Rdhngvbaf (<xref ref-type="disp-formula" rid="M29">N1</xref>) naq (<xref ref-type="disp-formula" rid="M30">N2</xref>) gura <italic>Y</italic><sub>8</sub>=<italic>Y</italic><sub>7</sub> naq gur fgntrf <italic>P</italic>′ naq <italic>P</italic>″ znxr rdhnyyl tbbq pubvprf nf n purpxcbvag. Jura <italic>P</italic>″ vf gbb ynetr sbe gur erfgevpgvbaf gura <italic>Y</italic><sub>8</sub>&gt;<italic>Y</italic><sub>7</sub>, naq jura <italic>P</italic>′ vf gbb fznyy sbe gur erfgevpgvbaf gura <italic>Y</italic><sub>8</sub>&lt;<italic>Y</italic><sub>7</sub>. Guhf, jr unir irevsvrq gung gur erfgevpgvbaf qrsvar na bcgvzny svefg purpxcbvag.</p>
        <p>Svanyyl, hfvat gur vaqhpgvba ulcbgurfvf, jr pbzchgr <italic>G</italic>(<italic>Z,A</italic>) gb irevsl gung vg lvryqf Rdhngvba (<xref ref-type="disp-formula" rid="M28">N0</xref>), hfvat n inyhr bs <italic>P</italic> fngvfslvat gur erfgevpgvbaf bs Rdhngvbaf (<xref ref-type="disp-formula" rid="M29">N1</xref>) naq (<xref ref-type="disp-formula" rid="M30">N2</xref>).
<disp-formula id="M44"><label>(55)</label><graphic xlink:href="btn308m44" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
<disp-formula id="M45"><label>(50)</label><graphic xlink:href="btn308m45" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
<disp-formula id="M46"><label>(51)</label><graphic xlink:href="btn308m46" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
<disp-formula id="M47"><label>(52)</label><graphic xlink:href="btn308m47" alt-version="no" position="float" xlink:type="simple"/></disp-formula>
nf qrfverq. Sbe gur ynfg rdhnyvgl, jr unir hfrq <italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>)=<italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>)+<italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>8)+8 naq <italic>G</italic><sub>bcg</sub>(<italic>Z,Y</italic>)=<italic>G</italic><sub>bcg</sub>(<italic>Z</italic>−8,<italic>Y</italic>)+<italic>G</italic><sub>bcg</sub>(<italic>Z,Y</italic>−8)+<italic>A</italic><sub>bcg</sub>(<italic>Z,Y</italic>−8)+8, juvpu ner rnfvyl cebirq sebz Rdhngvbaf (<xref ref-type="disp-formula" rid="M24">N8</xref>) naq (<xref ref-type="disp-formula" rid="M25">N7</xref>).</p>
      </app>
    </app-group>
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