pdf2htmlEX/demo/cheat5.page

<div class="pd w0 h0"><div id="pf5" class="pf" data-page-no="5"><div class="pc pc5"><img class="bi x0 y0 w1 h1" alt="" src="data:image/png;base64,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"/><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">Theoretical<span class="_ _0"> </span>Computer<span class="_ _0"> </span>Science<span class="_ _0"> </span>Cheat<span class="_ _0"> </span>Sheet</div><div class="t m0 xe3 h3 y2 ff2 fs0 fc0 sc0 ls0 ws0">Num<span class="_ _5"></span>b<span class="_ _3"></span>er Theory<span class="_ _57"> </span>Graph Theory</div><div class="t m0 x80 h3 y3 ff2 fs0 fc0 sc0 ls0 ws0">The<span class="_ _8"> </span>Chinese<span class="_ _8"> </span>remainder<span class="_ _8"> </span>theorem:<span class="_ _0"> </span>There<span class="_ _8"> </span>ex-</div><div class="t m0 x80 h3 y5 ff2 fs0 fc0 sc0 ls0 ws0">ists a n<span class="_ _5"></span>umber <span class="ff3">C<span class="_ _34"> </span></span>suc<span class="_ _5"></span>h that:</div><div class="t m0 xe3 h4 y3ab ff3 fs0 fc0 sc0 ls0 ws0">C <span class="ff4">≡<span class="_ _7"> </span></span>r</div><div class="t m0 x148 h5 y3ac ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x6 h3 y3ad ff2 fs0 fc0 sc0 ls0 ws0">mo<span class="_ _3"></span>d<span class="_ _7"> </span><span class="ff3">m</span></div><div class="t m0 x1a0 h5 y3ac ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x107 h3 y2f1 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x107 h3 y19 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x107 h3 y15 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x18e h3 y2f1 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x18e h3 y19 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x18e h3 y15 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x6 h3 y2f1 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x6 h3 y19 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x6 h3 y15 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 xe3 h4 y242 ff3 fs0 fc0 sc0 ls0 ws0">C <span class="ff4">≡<span class="_ _7"> </span></span>r</div><div class="t m0 x148 h5 y3ae ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x6 h3 y2f4 ff2 fs0 fc0 sc0 ls0 ws0">mo<span class="_ _3"></span>d<span class="_ _7"> </span><span class="ff3">m</span></div><div class="t m0 xd5 h5 y3ae ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x80 h3 y130 ff2 fs0 fc0 sc0 ls0 ws0">if<span class="_ _7"> </span><span class="ff3">m</span></div><div class="t m0 x98 h5 y3af ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x8f h3 y130 ff2 fs0 fc0 sc0 ls0 ws0">and<span class="_ _7"> </span><span class="ff3">m</span></div><div class="t m0 x100 h5 y3af ff6 fs1 fc0 sc0 ls0 ws0">j</div><div class="t m0 x143 h4 y130 ff2 fs0 fc0 sc0 ls0 ws0">are<span class="_ _7"> </span>relatively<span class="_ _7"> </span>prime<span class="_ _7"> </span>for<span class="_ _7"> </span><span class="ff3">i<span class="_ _7"> </span><span class="ff4"></span></span>=<span class="_ _7"> </span><span class="ff3">j<span class="_ _15"></span></span>.</div><div class="t m0 x80 h3 y3b0 ff2 fs0 fc0 sc0 ls0 ws0">Euler’s<span class="_ _11"> </span>function:<span class="_ _17"> </span><span class="ff3">φ</span>(<span class="ff3">x</span>)<span class="_ _11"> </span>i<span class="_ _5"></span>s<span class="_ _11"> </span>the<span class="_ _1e"> </span>num<span class="_ _5"></span>b<span class="_ _3"></span>er<span class="_ _11"> </span>of</div><div class="t m0 x80 h3 y351 ff2 fs0 fc0 sc0 ls0 ws0">p<span class="_ _3"></span>ositiv<span class="_ _5"></span>e<span class="_ _e"> </span>in<span class="_ _5"></span>tegers<span class="_ _e"> </span>less<span class="_ _58"> </span>than<span class="_ _e"> </span><span class="ff3">x<span class="_ _58"> </span></span>relativ<span class="_ _5"></span>ely</div><div class="t m0 x80 h3 y2b ff2 fs0 fc0 sc0 ls0 ws0">prime<span class="_ _34"> </span>to<span class="_ _34"> </span><span class="ff3">x</span><span class="ls1d">.I<span class="_ _27"></span>f</span></div><div class="t m0 x120 h6 y330 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x18e h5 y3b1 ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x18e h5 y354 ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">=1</span></div><div class="t m0 x94 h3 y2b ff3 fs0 fc0 sc0 ls0 ws0">p</div><div class="t m0 x8b h5 y5a ff6 fs1 fc0 sc0 ls0 ws0">e</div><div class="t m0 x10a h7 y13a ffa fs2 fc0 sc0 ls0 ws0">i</div><div class="t m0 x8b h5 y3b2 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x10b h3 y2b ff2 fs0 fc0 sc0 ls0 ws0">is<span class="_ _34"> </span>the<span class="_ _34"> </span>prime<span class="_ _1e"> </span>fac-</div><div class="t m0 x80 h3 y334 ff2 fs0 fc0 sc0 ls0 ws0">torization of <span class="ff3">x </span>then</div><div class="t m0 xfc h3 y3b3 ff3 fs0 fc0 sc0 ls0 ws0">φ<span class="ff2">(</span>x<span class="ff2 ls1">)=</span></div><div class="t m0 x13f h5 y3b4 ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x1a1 h6 y3b5 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x1a1 h5 y3b6 ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">=1</span></div><div class="t m0 x149 h3 y3b7 ff3 fs0 fc0 sc0 ls0 ws0">p</div><div class="t m0 x6 h5 y3b8 ff6 fs1 fc0 sc0 ls0 ws0">e</div><div class="t m0 x1a2 h7 y5e ffa fs2 fc0 sc0 ls0 ws0">i</div><div class="t m0 x8b h5 y3b8 ff8 fs1 fc0 sc0 ls0 ws0">−<span class="ff5">1</span></div><div class="t m0 x6 h5 y142 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x10b h3 y3b7 ff2 fs0 fc0 sc0 ls0 ws0">(<span class="ff3">p</span></div><div class="t m0 x124 h5 y3b9 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x14a h4 y3b7 ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff2">1)<span class="ff3">.</span></span></div><div class="t m0 x80 h3 y25e ff2 fs0 fc0 sc0 ls0 ws0">Euler’s<span class="_ _0"> </span>theorem:<span class="_ _12"> </span>If<span class="_ _34"> </span><span class="ff3">a<span class="_ _0"> </span></span>and<span class="_ _34"> </span><span class="ff3">b<span class="_ _34"> </span></span>are<span class="_ _0"> </span>relatively</div><div class="t m0 x80 h3 y359 ff2 fs0 fc0 sc0 ls0 ws0">prime then</div><div class="t m0 xe3 h4 y40 ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _7"> </span><span class="ff4">≡<span class="_ _7"> </span><span class="ff3">a</span></span></div><div class="t m0 x115 h5 y284 ff6 fs1 fc0 sc0 ls0 ws0">φ<span class="ff5">(</span>b<span class="ff5">)</span></div><div class="t m0 x8b h3 y40 ff2 fs0 fc0 sc0 ls0 ws0">mo<span class="_ _3"></span>d<span class="_ _7"> </span><span class="ff3">b.</span></div><div class="t m0 x80 h3 y3ba ff2 fs0 fc0 sc0 ls0 ws0">F<span class="_ _5"></span>ermat’s<span class="_ _7"> </span>theorem:</div><div class="t m0 xe3 h4 y44 ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _7"> </span><span class="ff4">≡<span class="_ _7"> </span><span class="ff3">a</span></span></div><div class="t m0 x108 h5 y2cc ff6 fs1 fc0 sc0 ls0 ws0">p<span class="ff8">−<span class="ff5">1</span></span></div><div class="t m0 x8b h3 y44 ff2 fs0 fc0 sc0 ls0 ws0">mo<span class="_ _3"></span>d<span class="_ _7"> </span><span class="ff3">p.</span></div><div class="t m0 x80 h3 y3bb ff2 fs0 fc0 sc0 ls0 ws0">The<span class="_ _0"> </span>Euclidean<span class="_ _0"> </span>algorithm:<span class="_ _11"> </span>if<span class="_ _0"> </span><span class="ff3 ls1e">a&gt;b</span>are<span class="_ _0"> </span>in-</div><div class="t m0 x80 h3 y73 ff2 fs0 fc0 sc0 ls0 ws0">tegers then</div><div class="t m0 x1a3 h3 y1f5 ff2 fs0 fc0 sc0 ls0 ws0">gcd(<span class="ff3">a,<span class="_ _6"> </span>b</span>)<span class="_ _7"> </span>=<span class="_ _7"> </span>gcd(<span class="ff3">a<span class="_ _7"> </span></span>mo<span class="_ _3"></span>d<span class="_ _7"> </span><span class="ff3">b,<span class="_ _8"> </span>b<span class="_ _5"></span><span class="ff2">)<span class="ff3">.</span></span></span></div><div class="t m0 x80 h3 y3bc ff2 fs0 fc0 sc0 ls0 ws0">If</div><div class="t m0 x1a4 h6 y3bd ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xc7 h5 y79 ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 xc7 h5 y1f8 ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">=1</span></div><div class="t m0 xc6 h3 y3be ff3 fs0 fc0 sc0 ls0 ws0">p</div><div class="t m0 x1a5 h5 y3bf ff6 fs1 fc0 sc0 ls0 ws0">e</div><div class="t m0 x18c h7 y38e ffa fs2 fc0 sc0 ls0 ws0">i</div><div class="t m0 x1a5 h5 y1f8 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x100 h3 y3be ff2 fs0 fc0 sc0 ls0 ws0">is<span class="_ _0"> </span>the<span class="_ _0"> </span>prime<span class="_ _0"> </span>factorization<span class="_ _0"> </span>of<span class="_ _34"> </span><span class="ff3">x</span></div><div class="t m0 x80 h3 y7a ff2 fs0 fc0 sc0 ls0 ws0">then</div><div class="t m0 xe1 h3 yab ff3 fs0 fc0 sc0 ls0 ws0">S<span class="_ _15"></span><span class="ff2">(</span>x<span class="ff2 ls1">)=</span></div><div class="t m0 x130 h6 y3c0 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x107 h5 y3c1 ff6 fs1 fc0 sc0 ls0 ws0">d<span class="ff8">|</span>x</div><div class="t m0 x18e h3 yab ff3 fs0 fc0 sc0 ls0 ws0">d<span class="_ _7"> </span><span class="ff2">=</span></div><div class="t m0 xe6 h5 y3c2 ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x13e h6 y3c3 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x13e h5 y3c4 ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">=1</span></div><div class="t m0 x144 h3 y1b5 ff3 fs0 fc0 sc0 ls0 ws0">p</div><div class="t m0 x82 h5 y3c5 ff6 fs1 fc0 sc0 ls0 ws0">e</div><div class="t m0 x1a0 h7 y3c6 ffa fs2 fc0 sc0 ls0 ws0">i</div><div class="t m0 x121 h5 y3c5 ff5 fs1 fc0 sc0 ls0 ws0">+1</div><div class="t m0 x82 h5 y3c7 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x118 h4 y1b5 ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff2">1</span></div><div class="t m0 x145 h3 y3c8 ff3 fs0 fc0 sc0 ls0 ws0">p</div><div class="t m0 x121 h5 y1bf ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 xfd h4 y3c8 ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff2">1</span></div><div class="t m0 x119 h3 yab ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x80 h3 yaf ff2 fs0 fc0 sc0 ls0 ws0">P<span class="_ _5"></span>erfect<span class="_ _8"> </span>Numbers:<span class="_ _0"> </span><span class="ff3">x<span class="_ _8"> </span></span>is<span class="_ _8"> </span>an<span class="_ _8"> </span>even<span class="_ _8"> </span>perfect<span class="_ _8"> </span>num-</div><div class="t m0 x80 h3 y8d ff2 fs0 fc0 sc0 ls10 ws0">be<span class="_ _5"></span>r<span class="_ _8"> </span>iff<span class="_ _8"> </span><span class="ff3 ls0">x<span class="_ _7"> </span></span><span class="ls1">=2</span></div><div class="t m0 x100 h5 y270 ff6 fs1 fc0 sc0 ls0 ws0">n<span class="ff8">−<span class="ff5">1</span></span></div><div class="t m0 x123 h3 y8d ff2 fs0 fc0 sc0 ls0 ws0">(2</div><div class="t m0 x13f h5 y270 ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x148 h4 y8d ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _15"></span><span class="ff2">1)<span class="_ _7"> </span>and<span class="_ _8"> </span>2</span></div><div class="t m0 x121 h5 y270 ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 xd6 h4 y8d ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _15"></span><span class="ff2">1<span class="_ _7"> </span>is<span class="_ _8"> </span>prime.</span></div><div class="t m0 x80 h3 y90 ff2 fs0 fc0 sc0 ls0 ws0">Wilson’s theorem:<span class="_ _1e"> </span><span class="ff3">n<span class="_ _7"> </span></span>is a prime iff</div><div class="t m0 x88 h4 y1c4 ff2 fs0 fc0 sc0 ls0 ws0">(<span class="ff3">n<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span></span>1)!<span class="_ _7"> </span><span class="ff4 ls1">≡−<span class="_ _b"></span><span class="ff2">1m<span class="_ _b"></span>o<span class="_ _9"></span>d<span class="_ _5"></span><span class="ff3 ls0">n.</span></span></span></div><div class="t m0 x80 h3 y205 ff2 fs0 fc0 sc0 ls0 ws0">M¨<span class="_ _2b"></span>obius in<span class="_ _5"></span>version:</div><div class="t m0 x16c h3 y210 ff3 fs0 fc0 sc0 ls0 ws0">µ<span class="ff2">(</span>i<span class="ff2 ls1">)=</span></div><div class="t m0 x140 h6 y1cf ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x140 h6 y3c9 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x140 h6 y30e ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x140 h6 y3ca ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x140 h6 y376 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x88 h3 ya0 ff2 fs0 fc0 sc0 ls1f ws0">1i<span class="_ _59"></span>f<span class="_ _5a"></span><span class="ff3 ls0">i<span class="_ _7"> </span><span class="ff2 ls1">=1<span class="_ _b"></span>.</span></span></div><div class="t m0 x88 h3 y2d3 ff2 fs0 fc0 sc0 ls1f ws0">0i<span class="_ _59"></span>f<span class="_ _5a"></span><span class="ff3 ls0">i <span class="ff2">is not square-free.</span></span></div><div class="t m0 x88 h4 y3cb ff2 fs0 fc0 sc0 ls0 ws0">(<span class="ff4">−</span>1)</div><div class="t m0 xe5 h5 y3cc ff6 fs1 fc0 sc0 ls0 ws0">r</div><div class="t m0 x108 h3 y3cd ff2 fs0 fc0 sc0 ls0 ws0">if <span class="ff3">i </span>is the pro<span class="_ _3"></span>duct of</div><div class="t m0 x108 h3 y3ce ff3 fs0 fc0 sc0 ls0 ws0">r<span class="_ _0"> </span><span class="ff2">distinct primes.</span></div><div class="t m0 x80 h3 y3cf ff2 fs0 fc0 sc0 ls0 ws0">If</div><div class="t m0 x18f h3 yc8 ff3 fs0 fc0 sc0 ls0 ws0">G<span class="ff2">(</span>a<span class="ff2 ls1">)=</span></div><div class="t m0 x142 h6 y3d0 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x6 h5 yce ff6 fs1 fc0 sc0 ls0 ws0">d<span class="ff8">|</span>a</div><div class="t m0 x8c h3 yc8 ff3 fs0 fc0 sc0 ls0 ws0">F<span class="_ _6"> </span><span class="ff2">(</span>d<span class="ff2">)</span>,</div><div class="t m0 x80 h3 y3d1 ff2 fs0 fc0 sc0 ls0 ws0">then</div><div class="t m0 xc8 h3 y3d2 ff3 fs0 fc0 sc0 ls0 ws0">F<span class="_ _6"> </span><span class="ff2">(</span>a<span class="ff2 ls1">)=</span></div><div class="t m0 xf3 h6 y1da ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x18e h5 yd1 ff6 fs1 fc0 sc0 ls0 ws0">d<span class="ff8">|</span>a</div><div class="t m0 x101 h3 y3d3 ff3 fs0 fc0 sc0 ls0 ws0">µ<span class="ff2">(</span>d<span class="ff2">)</span>G</div><div class="t m0 x145 h6 y192 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x10c h3 y2de ff3 fs0 fc0 sc0 ls0 ws0">a</div><div class="t m0 x10c h3 y3d4 ff3 fs0 fc0 sc0 ls0 ws0">d</div><div class="t m0 x132 h6 y192 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xd7 h3 y3d5 ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x80 h3 y199 ff2 fs0 fc0 sc0 ls0 ws0">Prime n<span class="_ _5"></span>umbers:</div><div class="t m0 x92 h3 y3d6 ff3 fs0 fc0 sc0 ls0 ws0">p</div><div class="t m0 x8f h5 y3d7 ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x18d h4 y3d8 ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff3">n<span class="_ _6"> </span></span>ln<span class="_ _8"> </span><span class="ff3">n<span class="_ _8"> </span></span>+<span class="_ _8"> </span><span class="ff3">n<span class="_ _2"></span></span>ln<span class="_ _8"> </span>ln<span class="_ _2"> </span><span class="ff3">n<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>n<span class="_ _8"> </span></span>+<span class="_ _8"> </span><span class="ff3">n</span></div><div class="t m0 x129 h3 y2e0 ff2 fs0 fc0 sc0 ls0 ws0">ln<span class="_ _6"> </span>ln<span class="_ _8"> </span><span class="ff3">n</span></div><div class="t m0 x119 h3 y2ae ff2 fs0 fc0 sc0 ls0 ws0">ln<span class="_ _6"> </span><span class="ff3">n</span></div><div class="t m0 x146 h3 y3d9 ff2 fs0 fc0 sc0 ls0 ws0">+<span class="_ _8"> </span><span class="ff3">O</span></div><div class="t m0 x120 h6 y3da ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x148 h3 y2b2 ff3 fs0 fc0 sc0 ls0 ws0">n</div><div class="t m0 xf3 h3 y3db ff2 fs0 fc0 sc0 ls0 ws0">ln<span class="_ _6"> </span><span class="ff3">n</span></div><div class="t m0 x94 h6 y3dc ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x10a h3 y3dd ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x85 h3 y3de ff3 fs0 fc0 sc0 ls0 ws0">π<span class="_ _3"></span><span class="ff2">(</span>n<span class="ff2 ls1">)=</span></div><div class="t m0 x12f h3 y103 ff3 fs0 fc0 sc0 ls0 ws0">n</div><div class="t m0 xc8 h3 y3df ff2 fs0 fc0 sc0 ls0 ws0">ln<span class="_ _6"> </span><span class="ff3">n</span></div><div class="t m0 x130 h3 y3e0 ff2 fs0 fc0 sc0 ls0 ws0">+</div><div class="t m0 x2 h3 y103 ff3 fs0 fc0 sc0 ls0 ws0">n</div><div class="t m0 x131 h3 y3df ff2 fs0 fc0 sc0 ls0 ws0">(ln<span class="_ _6"> </span><span class="ff3">n</span>)</div><div class="t m0 x13e h5 yf8 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x1a6 h3 y3e0 ff2 fs0 fc0 sc0 ls0 ws0">+</div><div class="t m0 x14a h3 y103 ff2 fs0 fc0 sc0 ls0 ws0">2!<span class="ff3">n</span></div><div class="t m0 x82 h3 y3df ff2 fs0 fc0 sc0 ls0 ws0">(ln<span class="_ _6"> </span><span class="ff3">n</span>)</div><div class="t m0 x83 h5 yf8 ff5 fs1 fc0 sc0 ls0 ws0">3</div><div class="t m0 x146 h3 y109 ff2 fs0 fc0 sc0 ls0 ws0">+<span class="_ _8"> </span><span class="ff3">O</span></div><div class="t m0 x120 h6 y3a4 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x116 h3 y3e1 ff3 fs0 fc0 sc0 ls0 ws0">n</div><div class="t m0 xf3 h3 y107 ff2 fs0 fc0 sc0 ls0 ws0">(ln<span class="_ _6"> </span><span class="ff3">n</span>)</div><div class="t m0 xe6 h5 y3e2 ff5 fs1 fc0 sc0 ls0 ws0">4</div><div class="t m0 x8c h6 y3a4 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x95 h3 y109 ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x179 h3 y327 ff2 fs0 fc0 sc0 ls0 ws0">Definitions:</div><div class="t m0 x179 h3 y3e3 fff fs0 fc0 sc0 lsc ws0">Loo<span class="_ _3"></span>p<span class="_ _49"> </span><span class="ff2 ls0">An edge<span class="_ _0"> </span>connecting<span class="_ _0"> </span>a ver-</span></div><div class="t m0 x137 h3 y11d ff2 fs0 fc0 sc0 ls0 ws0">tex to itself.</div><div class="t m0 x179 h3 y3e4 fff fs0 fc0 sc0 ls0 ws0">Dir<span class="_ _5"></span>e<span class="_ _5"></span>cte<span class="_ _5"></span>d<span class="_ _d"> </span><span class="ff2">Each edge has a direction.</span></div><div class="t m0 x179 h3 y19 fff fs0 fc0 sc0 ls0 ws0">Simple<span class="_ _5b"> </span><span class="ff2">Graph<span class="_ _21"> </span>with<span class="_ _36"> </span>no<span class="_ _21"> </span>lo<span class="_ _15"></span>ops<span class="_ _21"> </span>or</span></div><div class="t m0 x137 h3 y3e5 ff2 fs0 fc0 sc0 ls0 ws0">m<span class="_ _5"></span>ulti-edges.</div><div class="t m0 x179 h3 y3e6 fff fs0 fc0 sc0 ls0 ws0">Walk<span class="_ _5c"> </span><span class="ff2">A sequence <span class="ff3">v</span></span></div><div class="t m0 xad h5 y249 ff5 fs1 fc0 sc0 ls0 ws0">0</div><div class="t m0 x15a h3 y3e7 ff3 fs0 fc0 sc0 ls0 ws0">e</div><div class="t m0 x10 h5 y249 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 xaf h3 y3e7 ff3 fs0 fc0 sc0 ls0 ws0">v</div><div class="t m0 x2e h5 y249 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x14 h3 y3e7 ff3 fs0 fc0 sc0 ls7 ws0">...e</div><div class="t m0 x11c h5 y249 ff6 fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 x4a h3 y3e7 ff3 fs0 fc0 sc0 ls0 ws0">v</div><div class="t m0 xce h5 y249 ff6 fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 x13a h3 y3e7 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x179 h3 y14f fff fs0 fc0 sc0 ls0 ws0">T<span class="_ _5"></span>r<span class="_ _3d"></span>ail<span class="_ _5d"> </span><span class="ff2">A<span class="_ _8"> </span>walk<span class="_ _8"> </span>with<span class="_ _7"> </span>distinct<span class="_ _8"> </span>edges.</span></div><div class="t m0 x179 h3 y3e8 fff fs0 fc0 sc0 ls0 ws0">Path<span class="_ _5e"> </span><span class="ff2">A<span class="_ _31"> </span>trail<span class="_ _5f"> </span>with<span class="_ _31"> </span>distinct</span></div><div class="t m0 x137 h3 y32e ff2 fs0 fc0 sc0 ls0 ws0">v<span class="_ _5"></span>ertices.</div><div class="t m0 x179 h3 y3e9 fff fs0 fc0 sc0 ls0 ws0">Conne<span class="_ _5"></span>cte<span class="_ _5"></span>d<span class="_ _24"> </span><span class="ff2">A<span class="_ _7"> </span>graph<span class="_ _7"> </span>where<span class="_ _7"> </span>there<span class="_ _7"> </span>exists</span></div><div class="t m0 x137 h3 y27b ff2 fs0 fc0 sc0 ls0 ws0">a<span class="_ _12"> </span>path<span class="_ _12"> </span>b<span class="_ _3"></span>et<span class="_ _5"></span>ween<span class="_ _12"> </span>an<span class="_ _5"></span>y<span class="_ _12"> </span>t<span class="_ _5"></span>w<span class="_ _5"></span>o</div><div class="t m0 x137 h3 y3ea ff2 fs0 fc0 sc0 ls0 ws0">v<span class="_ _5"></span>ertices.</div><div class="t m0 x179 h3 y30 fff fs0 fc0 sc0 ls0 ws0">Comp<span class="_ _5"></span>onent<span class="_ _22"> </span><span class="ff2">A<span class="_ _60"> </span>maximal<span class="_ _60"> </span>connected</span></div><div class="t m0 x137 h3 y3eb ff2 fs0 fc0 sc0 ls0 ws0">subgraph.</div><div class="t m0 x179 h3 y3ec fff fs0 fc0 sc0 lsc ws0">T ree<span class="_ _61"> </span><span class="ff2 ls0">A<span class="_ _7"> </span>connected<span class="_ _8"> </span>acyclic<span class="_ _7"> </span>graph.</span></div><div class="t m0 x179 h3 y3ed fff fs0 fc0 sc0 lsc ws0">F ree<span class="_ _34"> </span>t<span class="_ _15"></span>ree<span class="_ _32"> </span><span class="ff2 ls0">A tree with<span class="_ _0"> </span>no root.</span></div><div class="t m0 x179 h3 y261 fff fs0 fc0 sc0 ls0 ws0">DAG<span class="_ _62"> </span><span class="ff2">Directed acyclic graph.</span></div><div class="t m0 x179 h3 y3ee fff fs0 fc0 sc0 ls0 ws0">Eulerian<span class="_ _32"> </span><span class="ff2">Graph<span class="_ _7"> </span>with a trail visiting</span></div><div class="t m0 x137 h3 y48 ff2 fs0 fc0 sc0 ls0 ws0">eac<span class="_ _5"></span>h edge exactly once.</div><div class="t m0 x179 h3 y6a fff fs0 fc0 sc0 ls0 ws0">Hamiltonian<span class="_ _12"> </span><span class="ff2">Graph<span class="_ _8"> </span>with<span class="_ _7"> </span>a<span class="_ _8"> </span>cycle<span class="_ _7"> </span>visiting</span></div><div class="t m0 x137 h3 y6b ff2 fs0 fc0 sc0 ls0 ws0">eac<span class="_ _5"></span>h vertex exactly once.</div><div class="t m0 x179 h3 y3ef fff fs0 fc0 sc0 ls0 ws0">Cut<span class="_ _63"> </span><span class="ff2">A<span class="_ _11"> </span>set<span class="_ _1e"> </span>of<span class="_ _11"> </span>edges<span class="_ _11"> </span>whose<span class="_ _1e"> </span>re-</span></div><div class="t m0 x137 h3 y72 ff2 fs0 fc0 sc0 ls0 ws0">mo<span class="_ _5"></span>v<span class="_ _5"></span>al<span class="_ _1e"> </span>increases<span class="_ _11"> </span>the<span class="_ _1e"> </span>n<span class="_ _5"></span>um-</div><div class="t m0 x137 h3 y79 ff2 fs0 fc0 sc0 ls0 ws0">b<span class="_ _3"></span>er of comp<span class="_ _3"></span>onen<span class="_ _5"></span>ts.</div><div class="t m0 x179 h3 y3f0 fff fs0 fc0 sc0 ls0 ws0">Cut-set<span class="_ _4d"> </span><span class="ff2">A minimal cut.</span></div><div class="t m0 x179 h3 y3c0 fff fs0 fc0 sc0 ls0 ws0">Cut<span class="_ _0"> </span>e<span class="_ _5"></span>dge<span class="_ _4"> </span><span class="ff2">A size 1 cut.</span></div><div class="t m0 x179 h3 y3f1 fff fs0 fc0 sc0 ls0 ws0">k-Conne<span class="_ _5"></span>cte<span class="_ _5"></span>d<span class="_ _58"> </span><span class="ff2">A<span class="_ _36"> </span>graph<span class="_ _36"> </span>connected<span class="_ _36"> </span>with</span></div><div class="t m0 x137 h4 y308 ff2 fs0 fc0 sc0 ls0 ws0">the<span class="_ _1e"> </span>remo<span class="_ _5"></span>v<span class="_ _5"></span>al<span class="_ _1e"> </span>of<span class="_ _1e"> </span>an<span class="_ _5"></span>y<span class="_ _1e"> </span><span class="ff3">k <span class="ff4">−<span class="_ _0"> </span></span></span>1</div><div class="t m0 x137 h3 y3f2 ff2 fs0 fc0 sc0 ls0 ws0">v<span class="_ _5"></span>ertices.</div><div class="t m0 x179 h4 y1c1 fff fs0 fc0 sc0 ls0 ws0">k-T<span class="_ _5"></span>ough<span class="_ _64"> </span><span class="ff4">∀<span class="ff3">S<span class="_ _21"> </span></span>⊆<span class="_ _12"> </span><span class="ff3 ls20">V,<span class="_ _2"></span>S<span class="_ _12"> </span></span><span class="ff2">=<span class="_ _12"> </span></span>∅<span class="_ _1e"> </span><span class="ff2 lsc">we<span class="_ _11"> </span>h<span class="_ _3"></span>ave</span></span></div><div class="t m0 x137 h4 y1ca ff3 fs0 fc0 sc0 ls0 ws0">k<span class="_ _7"> </span><span class="ff4">·<span class="_ _8"></span></span>c<span class="_ _5"></span><span class="ff2">(<span class="ff3">G<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>S<span class="_ _15"></span></span>)<span class="_ _7"> </span><span class="ff4 ls1">≤|<span class="_ _b"></span><span class="ff3 ls0">S<span class="_ _15"></span><span class="ff4">|<span class="ff2">.</span></span></span></span></span></div><div class="t m0 x179 h3 y3f3 fff fs0 fc0 sc0 ls0 ws0">k-R<span class="_ _5"></span>e<span class="_ _5"></span>gular<span class="_ _65"> </span><span class="ff2">A graph<span class="_ _0"> </span>where all<span class="_ _0"> </span>v<span class="_ _5"></span>ertices</span></div><div class="t m0 x137 h3 yb3 ff2 fs0 fc0 sc0 ls0 ws0">ha<span class="_ _5"></span>ve degree <span class="ff3">k</span>.</div><div class="t m0 x179 h3 yb5 fff fs0 fc0 sc0 ls0 ws0">k-F<span class="_ _5"></span>actor<span class="_ _55"> </span><span class="ff2">A<span class="_ _24"> </span><span class="ff3">k</span>-regular<span class="_ _24"> </span>spanning</span></div><div class="t m0 x137 h3 yb7 ff2 fs0 fc0 sc0 ls0 ws0">subgraph.</div><div class="t m0 x179 h3 y3f4 fff fs0 fc0 sc0 ls0 ws0">Matching<span class="_ _54"> </span><span class="ff2">A<span class="_ _34"> </span>set<span class="_ _1e"> </span>of<span class="_ _34"> </span>edges,<span class="_ _1e"> </span>no<span class="_ _1e"> </span>t<span class="_ _5"></span>w<span class="_ _5"></span>o<span class="_ _34"> </span>of</span></div><div class="t m0 x137 h3 y3f5 ff2 fs0 fc0 sc0 ls0 ws0">whic<span class="_ _5"></span>h are adjacent.</div><div class="t m0 x179 h3 y2da fff fs0 fc0 sc0 ls0 ws0">Clique<span class="_ _48"> </span><span class="ff2">A<span class="_ _36"> </span>set<span class="_ _21"> </span>of<span class="_ _36"> </span>v<span class="_ _5"></span>ertices,<span class="_ _e"> </span>all<span class="_ _21"> </span>of</span></div><div class="t m0 x137 h3 y18a ff2 fs0 fc0 sc0 ls0 ws0">whic<span class="_ _5"></span>h are adjacent.</div><div class="t m0 x179 h3 y3f6 fff fs0 fc0 sc0 ls0 ws0">Ind.<span class="_ _1e"> </span>set<span class="_ _66"> </span><span class="ff2">A<span class="_ _1e"> </span>set<span class="_ _34"> </span>of<span class="_ _1e"> </span>v<span class="_ _5"></span>ertices,<span class="_ _1e"> </span>none<span class="_ _1e"> </span>of</span></div><div class="t m0 x137 h3 y2eb ff2 fs0 fc0 sc0 ls0 ws0">whic<span class="_ _5"></span>h are adjacent.</div><div class="t m0 x179 h3 y3f7 fff fs0 fc0 sc0 ls0 ws0">V<span class="_ _5"></span>ertex<span class="_ _14"> </span>c<span class="_ _5"></span>over<span class="_ _12"> </span><span class="ff2">A<span class="_ _58"> </span>set<span class="_ _e"> </span>of<span class="_ _e"> </span>v<span class="_ _5"></span>ertices<span class="_ _e"> </span>whic<span class="_ _5"></span>h</span></div><div class="t m0 x137 h3 y3f8 ff2 fs0 fc0 sc0 ls0 ws0">co<span class="_ _5"></span>ver all edges.</div><div class="t m0 x179 h3 y3f9 fff fs0 fc0 sc0 ls0 ws0">Planar<span class="_ _0"> </span>gr<span class="_ _5"></span>aph<span class="_ _14"> </span><span class="ff2">A graph which<span class="_ _14"> </span>can be em-</span></div><div class="t m0 x137 h3 y219 ff2 fs0 fc0 sc0 ls0 ws0">b<span class="_ _3"></span>eded in the plane.</div><div class="t m0 x179 h3 y383 fff fs0 fc0 sc0 ls0 ws0">Plane<span class="_ _0"> </span>gr<span class="_ _5"></span>aph<span class="_ _f"> </span><span class="ff2">An<span class="_ _0"> </span>em<span class="_ _5"></span>bedding<span class="_ _0"> </span>of<span class="_ _0"> </span>a planar</span></div><div class="t m0 x137 h3 y2ae ff2 fs0 fc0 sc0 ls0 ws0">graph.</div><div class="t m0 x9c h6 yea ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x45 h5 y3fa ff6 fs1 fc0 sc0 ls0 ws0">v<span class="_ _3"></span><span class="ff8">∈</span>V</div><div class="t m0 x9d h3 y3fb ff2 fs0 fc0 sc0 ls0 ws0">deg(<span class="ff3">v<span class="_ _3"></span></span><span class="ls1">)=2<span class="_ _b"></span><span class="ff3 ls0">m.</span></span></div><div class="t m0 x179 h4 y3de ff2 fs0 fc0 sc0 ls0 ws0">If <span class="ff3">G </span>is planar then <span class="ff3">n<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>m<span class="_ _8"> </span></span>+<span class="_ _8"> </span><span class="ff3">f<span class="_ _0"> </span></span><span class="ls1">=2<span class="_ _b"></span>,<span class="_ _15"></span>s<span class="_ _b"></span>o</span></div><div class="t m0 x25 h4 y3fc ff3 fs0 fc0 sc0 ls0 ws0">f<span class="_ _0"> </span><span class="ff4">≤<span class="_ _7"> </span><span class="ff2">2</span></span>n<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span><span class="ff2">4</span></span><span class="lsa">,m<span class="_ _a"></span><span class="ff4 ls0">≤<span class="_ _7"> </span><span class="ff2">3<span class="ff3">n<span class="_ _8"> </span></span></span>−<span class="_ _8"> </span><span class="ff2">6<span class="ff3">.</span></span></span></span></div><div class="t m0 x179 h3 y112 ff2 fs0 fc0 sc0 ls0 ws0">An<span class="_ _5"></span>y planar graph has a vertex with de-</div><div class="t m0 x179 h4 y1a0 ff2 fs0 fc0 sc0 ls0 ws0">gree <span class="ff4">≤<span class="_ _7"> </span></span>5.</div><div class="t m0 x162 h3 y34d ff2 fs0 fc0 sc0 ls0 ws0">Notation:</div><div class="t m0 x162 h3 y6 ff3 fs0 fc0 sc0 ls0 ws0">E<span class="_ _15"></span><span class="ff2">(</span>G<span class="ff2">)<span class="_ _5f"> </span>Edge set</span></div><div class="t m0 x162 h3 ya ff3 fs0 fc0 sc0 ls0 ws0">V<span class="_ _8"> </span><span class="ff2">(</span>G<span class="ff2">)<span class="_ _5f"> </span>V<span class="_ _3d"></span>ertex set</span></div><div class="t m0 x162 h3 y10 ff3 fs0 fc0 sc0 ls0 ws0">c<span class="ff2">(</span>G<span class="ff2">)<span class="_ _67"> </span>Num<span class="_ _5"></span>b<span class="_ _3"></span>er of comp<span class="_ _3"></span>onents</span></div><div class="t m0 x162 h3 y3fd ff3 fs0 fc0 sc0 ls0 ws0">G<span class="ff2">[</span>S<span class="_ _15"></span><span class="ff2">]<span class="_ _67"> </span>Induced subgraph</span></div><div class="t m0 x162 h3 y32a ff2 fs0 fc0 sc0 ls0 ws0">deg(<span class="ff3">v<span class="_ _3"></span></span>)<span class="_ _f"> </span>Degree of <span class="ff3">v</span></div><div class="t m0 x162 h3 y128 ff2 fs0 fc0 sc0 ls0 ws0">∆(<span class="ff3">G</span>)<span class="_ _c"> </span>Maximum degree</div><div class="t m0 x162 h3 y32c ff3 fs0 fc0 sc0 ls0 ws0">δ<span class="_ _3"></span><span class="ff2">(</span>G<span class="ff2">)<span class="_ _67"> </span>Minim<span class="_ _5"></span>um<span class="_ _14"> </span>degree</span></div><div class="t m0 x162 h3 y3fe ff3 fs0 fc0 sc0 ls0 ws0">χ<span class="ff2">(</span>G<span class="ff2">)<span class="_ _60"> </span>Chromatic n<span class="_ _5"></span>umber</span></div><div class="t m0 x162 h3 y3ff ff3 fs0 fc0 sc0 ls0 ws0">χ</div><div class="t m0 x34 h5 y13b ff6 fs1 fc0 sc0 ls0 ws0">E</div><div class="t m0 x6d h3 y400 ff2 fs0 fc0 sc0 ls0 ws0">(<span class="ff3">G</span>)<span class="_ _58"> </span>Edge chromatic n<span class="_ _5"></span>um<span class="_ _5"></span>b<span class="_ _3"></span>er</div><div class="t m0 x162 h3 y401 ff3 fs0 fc0 sc0 ls0 ws0">G</div><div class="t m0 x43 h5 y58 ff6 fs1 fc0 sc0 ls0 ws0">c</div><div class="t m0 x35 h3 y402 ff2 fs0 fc0 sc0 ls0 ws0">Complemen<span class="_ _5"></span>t graph</div><div class="t m0 x162 h3 y403 ff3 fs0 fc0 sc0 ls0 ws0">K</div><div class="t m0 x43 h5 y29 ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x35 h3 y404 ff2 fs0 fc0 sc0 ls0 ws0">Complete graph</div><div class="t m0 x162 h3 y3b4 ff3 fs0 fc0 sc0 ls0 ws0">K</div><div class="t m0 x43 h5 y141 ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x11d h7 y24f ffc fs2 fc0 sc0 ls0 ws0">1</div><div class="t m0 xec h5 y141 ff6 fs1 fc0 sc0 ls0 ws0">,n</div><div class="t m0 x7b h7 y24f ffc fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x35 h3 y3b4 ff2 fs0 fc0 sc0 ls0 ws0">Complete bipartite graph</div><div class="t m0 x162 h3 y405 ff2 fs0 fc0 sc0 ls0 ws0">r</div><div class="t m0 x16b h3 y405 ff2 fs0 fc0 sc0 ls0 ws0">(<span class="ff3 ls10">k,<span class="_ _6"> </span><span class="_ _5"></span><span class="ff2 ls0">)<span class="_ _28"> </span>Ramsey n<span class="_ _5"></span>um<span class="_ _5"></span>b<span class="_ _3"></span>er</span></span></div><div class="t m0 xb8 h3 y406 ff2 fs0 fc0 sc0 ls0 ws0">Geometry</div><div class="t m0 x162 h3 y407 ff2 fs0 fc0 sc0 ls0 ws0">Pro<span class="_ _15"></span>jectiv<span class="_ _5"></span>e<span class="_ _e"> </span>co<span class="_ _3"></span>ordinates:<span class="_ _31"> </span>triples</div><div class="t m0 x162 h3 y408 ff2 fs0 fc0 sc0 ls0 ws0">(<span class="ff3">x,<span class="_ _6"> </span>y<span class="_ _15"></span>,<span class="_ _6"> </span>z<span class="_ _15"></span></span>), not all <span class="ff3">x</span>, <span class="ff3">y<span class="_ _0"> </span></span>and <span class="ff3">z<span class="_ _0"> </span></span>zero.</div><div class="t m0 x186 h4 y409 ff2 fs0 fc0 sc0 ls0 ws0">(<span class="ff3">x,<span class="_ _6"> </span>y<span class="_ _15"></span>,<span class="_ _6"> </span>z<span class="_ _15"></span></span><span class="ls1">)=(<span class="_ _b"></span><span class="ff3 ls0">cx,<span class="_ _6"> </span>cy<span class="_ _15"></span>,<span class="_ _6"> </span>cz<span class="_ _15"></span><span class="ff2">)<span class="_ _28"> </span><span class="ff4">∀</span></span>c<span class="_ _7"> </span><span class="ff4"><span class="ff2 ls1">=0<span class="_ _b"></span><span class="ff3 ls0">.</span></span></span></span></span></div><div class="t m0 x162 h3 y40a ff2 fs0 fc0 sc0 ls0 ws0">Cartesian<span class="_ _55"> </span>Pro<span class="_ _3"></span>jective</div><div class="t m0 x162 h3 y40b ff2 fs0 fc0 sc0 ls0 ws0">(<span class="ff3">x,<span class="_ _6"> </span>y<span class="_ _15"></span></span><span class="ls21">)(<span class="_ _68"></span><span class="ff3 ls0">x,<span class="_ _6"> </span>y<span class="_ _15"></span>,<span class="_ _6"> </span><span class="ff2">1)</span></span></span></div><div class="t m0 x162 h4 y1ad ff3 fs0 fc0 sc0 ls0 ws0">y <span class="ff2">=<span class="_ _7"> </span></span>mx<span class="_ _8"> </span><span class="ff2">+<span class="_ _8"> </span></span>b<span class="_ _31"> </span><span class="ff2">(</span>m,<span class="_ _8"> </span><span class="ff4">−<span class="_ _5"></span><span class="ff2">1<span class="ff3 ls4">,b<span class="_ _1f"></span><span class="ff2 ls0">)</span></span></span></span></div><div class="t m0 x162 h4 y40c ff3 fs0 fc0 sc0 ls0 ws0">x<span class="_ _7"> </span><span class="ff2">=<span class="_ _7"> </span></span>c<span class="_ _5d"> </span><span class="ff2">(1</span>,<span class="_ _6"> </span><span class="ff2">0</span>,<span class="_ _8"> </span><span class="ff4">−<span class="_ _5"></span><span class="ff3">c<span class="ff2">)</span></span></span></div><div class="t m0 x162 h3 y40d ff2 fs0 fc0 sc0 ls0 ws0">Distance<span class="_ _36"> </span>form<span class="_ _5"></span>ula,<span class="_ _e"> </span><span class="ff3">L</span></div><div class="t m0 xf0 h5 y165 ff6 fs1 fc0 sc0 ls0 ws0">p</div><div class="t m0 xa4 h3 y40e ff2 fs0 fc0 sc0 ls0 ws0">and<span class="_ _36"> </span><span class="ff3">L</span></div><div class="t m0 x38 h5 y165 ff8 fs1 fc0 sc0 ls0 ws0">∞</div><div class="t m0 x162 h3 y38e ff2 fs0 fc0 sc0 ls0 ws0">metric:</div><div class="t m0 xec h6 y40f ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x7b h3 y410 ff2 fs0 fc0 sc0 ls0 ws0">(<span class="ff3">x</span></div><div class="t m0 x35 h5 ya8 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x4f h4 y410 ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff3">x</span></div><div class="t m0 x51 h5 ya8 ff5 fs1 fc0 sc0 ls0 ws0">0</div><div class="t m0 x70 h3 y410 ff2 fs0 fc0 sc0 ls0 ws0">)</div><div class="t m0 xbb h5 y168 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x64 h3 y410 ff2 fs0 fc0 sc0 ls2 ws0">+(<span class="_ _9"></span><span class="ff3 ls0">y</span></div><div class="t m0 xdf h5 ya8 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x55 h4 y410 ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff3">y</span></div><div class="t m0 xa6 h5 ya8 ff5 fs1 fc0 sc0 ls0 ws0">0</div><div class="t m0 xe0 h3 y410 ff2 fs0 fc0 sc0 ls0 ws0">)</div><div class="t m0 x1a7 h5 y168 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xd2 h3 y410 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x6d h6 y411 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xec h4 yab ff4 fs0 fc0 sc0 ls0 ws0">|<span class="ff3">x</span></div><div class="t m0 x7b h5 y412 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x60 h4 yab ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff3">x</span></div><div class="t m0 x62 h5 y412 ff5 fs1 fc0 sc0 ls0 ws0">0</div><div class="t m0 x1a h4 yab ff4 fs0 fc0 sc0 ls0 ws0">|</div><div class="t m0 x36 h5 y413 ff6 fs1 fc0 sc0 ls0 ws0">p</div><div class="t m0 x63 h4 yab ff2 fs0 fc0 sc0 ls0 ws0">+<span class="_ _8"> </span><span class="ff4">|<span class="ff3">y</span></span></div><div class="t m0 x1e h5 y412 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 xa3 h4 yab ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff3">y</span></div><div class="t m0 x21 h5 y412 ff5 fs1 fc0 sc0 ls0 ws0">0</div><div class="t m0 x7e h4 yab ff4 fs0 fc0 sc0 ls0 ws0">|</div><div class="t m0 x66 h5 y413 ff6 fs1 fc0 sc0 ls0 ws0">p</div><div class="t m0 xbe h6 y411 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xe0 h5 y414 ff5 fs1 fc0 sc0 ls0 ws0">1<span class="ff6">/p</span></div><div class="t m0 xc0 h3 yab ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x181 h3 y85 ff2 fs0 fc0 sc0 ls0 ws0">lim</div><div class="t m0 x33 h5 y2cf ff6 fs1 fc0 sc0 ls0 ws0">p<span class="ff8">→∞</span></div><div class="t m0 x4e h6 y415 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x7a h4 y85 ff4 fs0 fc0 sc0 ls0 ws0">|<span class="ff3">x</span></div><div class="t m0 x35 h5 y416 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x6e h4 y85 ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff3">x</span></div><div class="t m0 x51 h5 y416 ff5 fs1 fc0 sc0 ls0 ws0">0</div><div class="t m0 x70 h4 y85 ff4 fs0 fc0 sc0 ls0 ws0">|</div><div class="t m0 x180 h5 y1bb ff6 fs1 fc0 sc0 ls0 ws0">p</div><div class="t m0 x1d h4 y85 ff2 fs0 fc0 sc0 ls0 ws0">+<span class="_ _8"> </span><span class="ff4">|<span class="ff3">y</span></span></div><div class="t m0 x199 h5 y416 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x20 h4 y85 ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff3">y</span></div><div class="t m0 x23 h5 y416 ff5 fs1 fc0 sc0 ls0 ws0">0</div><div class="t m0 x7d h4 y85 ff4 fs0 fc0 sc0 ls0 ws0">|</div><div class="t m0 xbf h5 y1bb ff6 fs1 fc0 sc0 ls0 ws0">p</div><div class="t m0 xba h6 y415 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x165 h5 y3c1 ff5 fs1 fc0 sc0 ls0 ws0">1<span class="ff6">/p</span></div><div class="t m0 x185 h3 y85 ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x162 h3 y270 ff2 fs0 fc0 sc0 ls0 ws0">Area<span class="_ _0"> </span>of<span class="_ _0"> </span>triangle<span class="_ _34"> </span>(<span class="ff3">x</span></div><div class="t m0 xd0 h5 y417 ff5 fs1 fc0 sc0 ls0 ws0">0</div><div class="t m0 x37 h3 y270 ff3 fs0 fc0 sc0 ls4 ws0">,y</div><div class="t m0 x20 h5 y417 ff5 fs1 fc0 sc0 ls0 ws0">0</div><div class="t m0 x21 h3 y270 ff2 fs0 fc0 sc0 ls0 ws0">),<span class="_ _0"> </span>(<span class="ff3">x</span></div><div class="t m0 xa5 h5 y417 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x105 h3 y270 ff3 fs0 fc0 sc0 ls4 ws0">,y</div><div class="t m0 x38 h5 y417 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x185 h3 y270 ff2 fs0 fc0 sc0 ls0 ws0">)</div><div class="t m0 x162 h3 y203 ff2 fs0 fc0 sc0 ls0 ws0">and (<span class="ff3">x</span></div><div class="t m0 xa1 h5 y1ca ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x169 h3 y203 ff3 fs0 fc0 sc0 ls4 ws0">,y</div><div class="t m0 x6e h5 y1ca ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x62 h3 y203 ff2 fs0 fc0 sc0 ls0 ws0">):</div><div class="t m0 xec h5 y418 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 xec h5 y419 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x4e h3 y41a ff2 fs0 fc0 sc0 ls0 ws0">abs</div><div class="t m0 x188 h6 y96 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x188 h6 y41b ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x188 h6 yb2 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x188 h6 y419 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x113 h3 y41c ff3 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 x6f h5 y99 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x51 h4 y41c ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff3">x</span></div><div class="t m0 x64 h5 y99 ff5 fs1 fc0 sc0 ls0 ws0">0</div><div class="t m0 x37 h3 y41c ff3 fs0 fc0 sc0 ls0 ws0">y</div><div class="t m0 x7c h5 y99 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x55 h4 y41c ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff3">y</span></div><div class="t m0 xa6 h5 y99 ff5 fs1 fc0 sc0 ls0 ws0">0</div><div class="t m0 x113 h3 y30d ff3 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 x6f h5 y177 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x51 h4 y30d ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff3">x</span></div><div class="t m0 x64 h5 y177 ff5 fs1 fc0 sc0 ls0 ws0">0</div><div class="t m0 x37 h3 y30d ff3 fs0 fc0 sc0 ls0 ws0">y</div><div class="t m0 x7c h5 y177 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x55 h4 y30d ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff3">y</span></div><div class="t m0 xa6 h5 y177 ff5 fs1 fc0 sc0 ls0 ws0">0</div><div class="t m0 x182 h6 y96 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x182 h6 y41b ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x182 h6 yb2 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x182 h6 y419 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xd2 h3 y41d ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x162 h3 y1ce ff2 fs0 fc0 sc0 ls0 ws0">Angle formed b<span class="_ _5"></span>y three p<span class="_ _3"></span>oints:</div><div class="t m0 x19 h3 y41e ff3 fs0 fc0 sc0 ls0 ws0">θ</div><div class="t m0 x35 h3 y41f ff2 fs0 fc0 sc0 ls0 ws0">(0<span class="ff3">,<span class="_ _6"> </span></span>0)</div><div class="t m0 x55 h3 y420 ff2 fs0 fc0 sc0 ls0 ws0">(<span class="ff3">x</span></div><div class="t m0 x66 h5 y29c ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 xbe h3 y420 ff3 fs0 fc0 sc0 ls4 ws0">,y</div><div class="t m0 x1a7 h5 y29c ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 xd2 h3 y420 ff2 fs0 fc0 sc0 ls0 ws0">)</div><div class="t m0 xdf h3 y3cb ff2 fs0 fc0 sc0 ls0 ws0">(<span class="ff3">x</span></div><div class="t m0 x21 h5 y2d2 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xa4 h3 y3cb ff3 fs0 fc0 sc0 ls4 ws0">,y</div><div class="t m0 x7d h5 y2d2 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x182 h3 y3cb ff2 fs0 fc0 sc0 ls0 ws0">)</div><div class="t m0 x1d h3 y421 ff3 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x54 h5 y422 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x1a8 h3 y420 ff3 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x1e h5 y29c ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x79 h3 y1dd ff2 fs0 fc0 sc0 ls0 ws0">cos<span class="_ _6"> </span><span class="ff3">θ </span>=</div><div class="t m0 x1a h3 y423 ff2 fs0 fc0 sc0 ls0 ws0">(<span class="ff3">x</span></div><div class="t m0 x63 h5 y424 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 xbb h3 y425 ff3 fs0 fc0 sc0 ls4 ws0">,y</div><div class="t m0 x54 h5 y424 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 xa2 h4 y425 ff2 fs0 fc0 sc0 ls0 ws0">)<span class="_ _8"> </span><span class="ff4">·<span class="_ _8"> </span></span>(<span class="ff3">x</span></div><div class="t m0 x21 h5 y424 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x7e h3 y425 ff3 fs0 fc0 sc0 ls4 ws0">,y</div><div class="t m0 xa6 h5 y424 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xbf h3 y425 ff2 fs0 fc0 sc0 ls0 ws0">)</div><div class="t m0 x1e h3 y426 ff3 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xbc h5 y427 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x37 h3 y426 ff3 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xdf h5 y427 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xba h3 y1dd ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x162 h3 yd5 ff2 fs0 fc0 sc0 ls0 ws0">Line<span class="_ _34"> </span>through<span class="_ _34"> </span>tw<span class="_ _5"></span>o<span class="_ _34"> </span>points<span class="_ _34"> </span>(<span class="ff3">x</span></div><div class="t m0 xa5 h5 y1de ff5 fs1 fc0 sc0 ls0 ws0">0</div><div class="t m0 x105 h3 yd5 ff3 fs0 fc0 sc0 ls4 ws0">,y</div><div class="t m0 x38 h5 y1de ff5 fs1 fc0 sc0 ls0 ws0">0</div><div class="t m0 x185 h3 yd5 ff2 fs0 fc0 sc0 ls0 ws0">)</div><div class="t m0 x162 h3 y37e ff2 fs0 fc0 sc0 ls0 ws0">and (<span class="ff3">x</span></div><div class="t m0 xa1 h5 y197 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x169 h3 y37e ff3 fs0 fc0 sc0 ls4 ws0">,y</div><div class="t m0 x6e h5 y197 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x62 h3 y37e ff2 fs0 fc0 sc0 ls0 ws0">):</div><div class="t m0 x35 h6 y1df ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x35 h6 y31d ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x35 h6 y2ac ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x35 h6 y428 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x35 h6 y382 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x35 h6 y429 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x4f h3 ydf ff3 fs0 fc0 sc0 ls22 ws0">xy<span class="_ _1f"></span><span class="ff2 ls0">1</span></div><div class="t m0 x50 h3 ydc ff3 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 x113 h5 y42a ff5 fs1 fc0 sc0 ls0 ws0">0</div><div class="t m0 x17e h3 ydc ff3 fs0 fc0 sc0 ls0 ws0">y</div><div class="t m0 x52 h5 y42a ff5 fs1 fc0 sc0 ls0 ws0">0</div><div class="t m0 x1e h3 ydc ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x50 h3 ye1 ff3 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 x113 h5 y42b ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x17e h3 ye1 ff3 fs0 fc0 sc0 ls0 ws0">y</div><div class="t m0 x52 h5 y42b ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x1e h3 ye1 ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x53 h6 y1df ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x53 h6 y31d ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x53 h6 y2ac ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x53 h6 y42c ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x53 h6 y382 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x53 h6 y42d ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xdf h3 ydc ff2 fs0 fc0 sc0 ls1 ws0">=0<span class="_ _b"></span><span class="ff3 ls0">.</span></div><div class="t m0 x162 h3 yf3 ff2 fs0 fc0 sc0 ls0 ws0">Area of circle, v<span class="_ _5"></span>olume of sphere:</div><div class="t m0 x13b h3 y2ba ff3 fs0 fc0 sc0 ls0 ws0">A<span class="_ _7"> </span><span class="ff2">=<span class="_ _7"> </span></span><span class="ls10">πr</span></div><div class="t m0 x6f h5 y3db ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x36 h3 y2ba ff3 fs0 fc0 sc0 lsd ws0">,V<span class="_ _69"></span><span class="ff2 ls0">=</span></div><div class="t m0 x65 h5 yeb ff5 fs1 fc0 sc0 ls0 ws0">4</div><div class="t m0 x65 h5 y19d ff5 fs1 fc0 sc0 ls0 ws0">3</div><div class="t m0 x66 h3 y2ba ff3 fs0 fc0 sc0 ls10 ws0">πr</div><div class="t m0 x167 h5 y3db ff5 fs1 fc0 sc0 ls0 ws0">3</div><div class="t m0 xa5 h3 y2ba ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x17c h3 y3df ff2 fs0 fc0 sc0 ls0 ws0">If<span class="_ _7"> </span>I<span class="_ _7"> </span>hav<span class="_ _5"></span>e<span class="_ _7"> </span>seen<span class="_ _7"> </span>farther<span class="_ _7"> </span>than<span class="_ _7"> </span>others,</div><div class="t m0 x17c h3 y42e ff2 fs0 fc0 sc0 ls0 ws0">it<span class="_ _0"> </span>is b<span class="_ _3"></span>ecause<span class="_ _0"> </span>I<span class="_ _0"> </span>ha<span class="_ _5"></span>v<span class="_ _5"></span>e<span class="_ _0"> </span>sto<span class="_ _3"></span>o<span class="_ _3"></span>d<span class="_ _0"> </span>on the</div><div class="t m0 x17c h3 y42f ff2 fs0 fc0 sc0 ls0 ws0">shoulders of gian<span class="_ _5"></span>ts.</div><div class="t m0 x17c h3 y430 ff2 fs0 fc0 sc0 ls0 ws0">– Issac Newton</div></div><div class="pi" data-data='{"ctm":[1.673203,0.000000,0.000000,1.673203,0.000000,0.000000]}'></div></div></div>