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"# Π‘Ρ€Π΅Π΄Π½Π΅Π΅ количСство Π²Π΅Ρ€ΡˆΠΈΠ½ Π² Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠ΅\n", + "\n", + "Алгоритм ДТарвиса ΠΈΠΌΠ΅Π΅Ρ‚ ΡΠ»ΠΎΠΆΠ½ΠΎΡΡ‚ΡŒ $O(X N)$, Π³Π΄Π΅ $X$ -- число Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ, Π° $N$ -- ΠΎΠ±Ρ‰Π΅Π΅ число Ρ‚ΠΎΡ‡Π΅ΠΊ. Π§Ρ‚ΠΎΠ±Ρ‹ ΠΎΡ†Π΅Π½ΠΈΡ‚ΡŒ ΡΠ»ΠΎΠΆΠ½ΠΎΡΡ‚ΡŒ этого Π°Π»Π³ΠΎΡ€ΠΈΡ‚ΠΌΠ° Π² срСднСм случаС, Π½Π΅ΠΎΠ±Ρ…ΠΎΠ΄ΠΈΠΌΠΎ Π²Ρ‹Ρ‡ΠΈΡΠ»ΠΈΡ‚ΡŒ $E(X)$ -- матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ Π²Π΅Π»ΠΈΡ‡ΠΈΠ½Ρ‹ $X$. Для Π΅Π³ΠΎ Π°Π½Π°Π»ΠΈΠ·Π° ΠΌΡ‹ Π΄ΠΎΠ»ΠΆΠ½Ρ‹ ΠΎΠΏΡ€Π΅Π΄Π΅Π»ΠΈΡ‚ΡŒ ΠΎΠ³Ρ€Π°Π½ΠΈΡ‡Π΅Π½Π½ΡƒΡŽ ΠΎΠ±Π»Π°ΡΡ‚ΡŒ, ΠΈΠ· ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ Ρ‚ΠΎΡ‡ΠΊΠΈ, Ρ‚Π°ΠΊ ΠΊΠ°ΠΊ ΠΎΠ½ΠΈ ΠΌΠΎΠ³ΡƒΡ‚ Π±Ρ‹Ρ‚ΡŒ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны лишь Π² мноТСствС с ΠΊΠΎΠ½Π΅Ρ‡Π½ΠΎΠΉ ΠΌΠ΅Ρ€ΠΎΠΉ Π›Π΅Π±Π΅Π³Π° [[Kendall, Moran (1963)](https://doi.org/10.1002/bimj.19650070322)]. \n", + "\n", + "НиТС ΠΏΡ€ΠΈΠ²Π΅Π΄Ρ‘Π½ ряд Ρ‚Π΅ΠΎΡ€Π΅ΠΌ, ΠΏΡ€Π΅Π΄ΡΡ‚Π°Π²Π»ΡΡŽΡ‰ΠΈΡ… ΠΎΡ†Π΅Π½ΠΊΡƒ срСднСго количСства Π²Π΅Ρ€ΡˆΠΈΠ½ Π² Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠ΅ ΠΏΡ€ΠΈ Ρ€Π°Π·Π»ΠΈΡ‡Π½Ρ‹Ρ… распрСдСлСниях Ρ‚ΠΎΡ‡Π΅ΠΊ.\n", + "\n", + "**Π’Π΅ΠΎΡ€Π΅ΠΌΠ° 1** [[RΓ©nyi, Sulanke (1963)](https://doi.org/10.1007/BF00535300)]. *Если $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ ΠΈ нСзависимо ΠΈΠ· Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° Π½Π° плоскости, Ρ‚ΠΎ ΠΏΡ€ΠΈ $N\\to\\infty$ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ этих Ρ‚ΠΎΡ‡Π΅ΠΊ $E(X)=\\frac23 r(\\log N + \\gamma) + O(1)$, Π³Π΄Π΅ $\\gamma$ - константа Π­ΠΉΠ»Π΅Ρ€Π°.*\n", + "\n", + "**Π’Π΅ΠΎΡ€Π΅ΠΌΠ° 2** [[Raynaud (1970)](https://doi.org/10.1017/S0021900200026917)]. *Если $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ ΠΈ нСзависимо ΠΈΠ· внутрСнности $k$-ΠΌΠ΅Ρ€Π½ΠΎΠΉ гипСрсфСры, Ρ‚ΠΎ ΠΏΡ€ΠΈ $N\\to\\infty$ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π³ΠΈΠΏΠ΅Ρ€Π³Ρ€Π°Π½Π΅ΠΉ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ этих Ρ‚ΠΎΡ‡Π΅ΠΊ асимптотичСски стрСмится ΠΊ $E(f) = O(N^{(k-1)/(k+1)})$*\n", + "\n", + "**Π’Π΅ΠΎΡ€Π΅ΠΌΠ° 3** [[Raynaud (1970)](https://doi.org/10.1017/S0021900200026917)]. *Если $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ нСзависимо Π² соотвСтствии с $k$-ΠΌΠ΅Ρ€Π½Ρ‹ΠΌ Π½ΠΎΡ€ΠΌΠ°Π»ΡŒΠ½Ρ‹ΠΌ распрСдСлСниСм, Ρ‚ΠΎ ΠΏΡ€ΠΈ $N\\to\\infty$ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ этих Ρ‚ΠΎΡ‡Π΅ΠΊ асимптотичСски стрСмится ΠΊ $E(f) = O((\\log N)^{(k-1)/2})$*\n", + "\n", + "**Π’Π΅ΠΎΡ€Π΅ΠΌΠ° 4** [[Bentley, Kung, Schkolnick, Thompson (1978)](https://doi.org/10.1145/322092.322095)]. *Если ΠΊΠΎΠΎΡ€Π΄ΠΈΠ½Π°Ρ‚Ρ‹ $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ Π² $k$-ΠΌΠ΅Ρ€Π½ΠΎΠΌ пространствС Π²Ρ‹Π±Ρ€Π°Π½Ρ‹ нСзависимо Π² соотвСтствии с ΠΏΡ€ΠΎΠΈΠ·Π²ΠΎΠ»ΡŒΠ½Ρ‹ΠΌ Π½Π΅ΠΏΡ€Π΅Ρ€Ρ‹Π²Π½Ρ‹ΠΌ распрСдСлСниСм (Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΠΎ, для ΠΊΠ°ΠΆΠ΄ΠΎΠΉ ΠΊΠΎΠΎΡ€Π΄ΠΈΠ½Π°Ρ‚Ρ‹ ΠΈΡΠΏΠΎΠ»ΡŒΠ·ΡƒΠ΅Ρ‚ΡΡ своё распрСдСлСниС), Ρ‚ΠΎ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ этих Ρ‚ΠΎΡ‡Π΅ΠΊ асимптотичСски стрСмится ΠΊ $E(X) = O((\\log N)^{k-1})$*. \n", + "\n", + "Условиям этой Ρ‚Π΅ΠΎΡ€Π΅ΠΌΡ‹ ΡƒΠ΄ΠΎΠ²Π»Π΅Ρ‚Π²ΠΎΡ€ΡΡŽΡ‚ ΠΌΠ½ΠΎΠ³ΠΈΠ΅ распрСдСлСния, Π²ΠΊΠ»ΡŽΡ‡Π°Ρ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎΠ΅ распрСдСлСниС Π² Π³ΠΈΠΏΠ΅Ρ€ΠΊΡƒΠ±Π΅.\n", + "\n", + "Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ,\n", + "- $E(X) = O(\\log N)$ для $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ, Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Ρ… ΠΈΠ· Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ°\n", + "- $E(X) = O(N^{1/3})$ для $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ, Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Ρ… ΠΈΠ· ΠΊΡ€ΡƒΠ³Π°\n", + "- $E(X) = O(N^{1/2})$ для $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ, Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Ρ… ΠΈΠ· сфСры\n", + "- $E(X) = O((\\log N)^2)$ для $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ, Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Ρ… ΠΈΠ· ΠΊΡƒΠ±Π°\n", + "- $E(X) = O(\\sqrt{\\log N})$ для $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ, Π½ΠΎΡ€ΠΌΠ°Π»ΡŒΠ½ΠΎ распрСдСлённых Π½Π° плоскости\n", + "\n", + "Π—Π΄Π΅ΡΡŒ прСдставлСн ΠΏΠ΅Ρ€Π΅Π²ΠΎΠ΄ Ρ€Π°Π±ΠΎΡ‚Ρ‹ A. RΓ©nyi ΠΈ R. Sulanke, Π³Π΄Π΅ приводится Π΄ΠΎΠΊΠ°Π·Π°Ρ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ ΠΎΡ†Π΅Π½ΠΊΠΈ матСматичСского оТидания числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ Π² Π΄Π²ΡƒΠΌΠ΅Ρ€Π½ΠΎΠΌ случаС." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# О Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠ΅ N случайно Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Ρ… Ρ‚ΠΎΡ‡Π΅ΠΊ\n", + "> [A. Renyi ΠΈ R. Sulanke (1963 Π³.)](https://1drv.ms/b/s!AsLTB1ON1LwNjMhGXq0xv1LW6Wx0-w)\n", + "\n", + "ΠŸΡƒΡΡ‚ΡŒ $P_i\\:(i=1,\\:2,\\:\\ldots,\\:n)$ -- Ρ‚ΠΎΡ‡ΠΊΠΈ Π½Π° плоскости, Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Π΅ случайно ΠΈ нСзависимо Π² соотвСтствии с Π½Π΅ΠΊΠΎΡ‚ΠΎΡ€Ρ‹ΠΌ распрСдСлСниСм, Π° $H_n$ - выпуклая ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠ° этих Ρ‚ΠΎΡ‡Π΅ΠΊ, которая, ΠΊΠ°ΠΊ извСстно, являСтся Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΌ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΎΠΌ. Число $X_n$ Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ -- случайная Π²Π΅Π»ΠΈΡ‡ΠΈΠ½Π°, ΠΈ Π² Π΄Π°Π½Π½ΠΎΠΉ Ρ€Π°Π±ΠΎΡ‚Π΅ Π±ΡƒΠ΄Π΅Ρ‚ исслСдовано Π΅Ρ‘ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ $E_n = E(X_n)$ ΠΏΡ€ΠΈ $n\\to\\infty$ Π² Ρ€Π°Π·Π»ΠΈΡ‡Π½Ρ‹Ρ… прСдполоТСниях ΠΎ распрСдСлСнии Ρ‚ΠΎΡ‡Π΅ΠΊ $P_i$.\n", + "\n", + "* Π’ $\\S1$ рассмотрим случай, ΠΊΠΎΠ³Π΄Π° Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i$ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны Π²Π½ΡƒΡ‚Ρ€ΠΈ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ r-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$. Как Π±ΡƒΠ΄Π΅Ρ‚ ΠΏΠΎΠΊΠ°Π·Π°Π½ΠΎ, $E_n = (2r/3)(\\log n + \\gamma) + T + O(1)$, Π³Π΄Π΅ $\\gamma$ - постоянная Π­ΠΉΠ»Π΅Ρ€Π°:\n", + " \n", + " $$\\gamma = \\lim_{n\\to\\infty} \\left( \\sum_{k=1}^{n} \\frac1k - \\log n \\right)=\\lim_{n\\to\\infty} \\left( 1+\\frac12+\\frac13+\\ldots+\\frac1n - \\log n \\right) \\notag$$\n", + " \n", + " ΠŸΡ€ΠΈ этом Π²Π΅Π»ΠΈΡ‡ΠΈΠ½Π° $T = T(K)$ зависит ΠΎΡ‚ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° K.\n", + "\n", + "* Π’ $\\S2$ Π΄ΠΎΠΊΠ°ΠΆΠ΅ΠΌ ΡΠ»Π΅ΠΌΠ΅Π½Ρ‚Π°Ρ€Π½ΡƒΡŽ Π³Π΅ΠΎΠΌΠ΅Ρ‚Ρ€ΠΈΡ‡Π΅ΡΠΊΡƒΡŽ Ρ‚Π΅ΠΎΡ€Π΅ΠΌΡƒ ΠΎ Ρ‚ΠΎΠΌ, Ρ‡Ρ‚ΠΎ $T(K)$ ΠΏΡ€ΠΈΠ½ΠΈΠΌΠ°Π΅Ρ‚ ΠΌΠ°ΠΊΡΠΈΠΌΠ°Π»ΡŒΠ½Ρ‹Π΅ значСния для ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΎΠ², ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Π΅ Π°Ρ„Ρ„ΠΈΠ½Π½Ρ‹ΠΌΠΈ прСобразованиями ΠΌΠΎΠΆΠ½ΠΎ привСсти ΠΊ ΠΏΡ€Π°Π²ΠΈΠ»ΡŒΠ½ΠΎΠΌΡƒ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΡƒ, ΠΈ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ для Π½ΠΈΡ….\n", + "\n", + "* Π’ $\\S3$ исслСдуСм асимптотичСскоС ΠΏΠΎΠ²Π΅Π΄Π΅Π½ΠΈΠ΅ $E_n$, ΠΊΠΎΠ³Π΄Π° Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i$ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны Π² Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ области с Π³Π»Π°Π΄ΠΊΠΎΠΉ Π³Ρ€Π°Π½ΠΈΡ†Π΅ΠΉ. ΠžΠΊΠ°Π·Ρ‹Π²Π°Π΅Ρ‚ΡΡ, Ρ‡Ρ‚ΠΎ Π² Ρ‚Π°ΠΊΠΎΠΌ случаС $E_n$ ΠΈΠΌΠ΅Π΅Ρ‚ порядок $\\sqrt[3]n$.\n", + "\n", + "* НаконСц, Π² $\\S4$ ΠΌΡ‹ Π΄ΠΎΠΊΠ°ΠΆΠ΅ΠΌ, Ρ‡Ρ‚ΠΎ $E_n$ Π²Π΅Π΄Ρ‘Ρ‚ сСбя ΠΊΠ°ΠΊ $\\sqrt{\\log n}$, Ссли Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i$ ΠΈΠΌΠ΅ΡŽΡ‚ Π½ΠΎΡ€ΠΌΠ°Π»ΡŒΠ½ΠΎΠ΅ распрСдСлСниС." + ] + }, + { + "attachments": { + "1.png": { + "image/png": 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xzpqRjuiYGFz51xs0qIyIiLQ0YOBQnH/BZKSnfwaX6kJC80QsXbIA540ch4mTzA2+TklJMf79UhoGDR6Oy6+4LuSHfEMy7KOjY9CsWfMaz87v/n0HspYtxkOPPOPd8Y6IiMKHEALX33Abpk+/Evv27YGiKLj9zn94J9k1RHGxHa++8iwmTJiEMWMvAFAR/gX5J9CufUd/le5XIRn2QghcfsV1+OjDN/HjDwsxZOhI/LH3d3z55cd48KGnvAsrEBFReGqekOhdUe9M2O02vPP2K0hNnVFlTti2rb/C4Shh2Afa2HEXonefftiQvQbfL5yHtm3b4amn/4OIiEitSyMiohBksxXh3y8+ia5de2DX779h1+8Vs/BLHQ5s/jUb191wm8YVNl7Ihj0AtGzZ+ozGYIiIiOry0gtPYN++PbWuqiqEQPsQvasHQjzsm0rl1ZKIiCg83XBT3TPuFUUJ6e3SwzLsDx7ch5ycwzhZWICiUyfxxx+7sWL5D4iMisJZPc6p8XwmERHpXyg/R386YRn2ZWVlUFUVzRMScdsdD3iPO8vL4XSF/laGRERElYVl2Hfv3hPdu/fUugwiIqKACMkV9IiIiKjhGPZEREQ6x7AnIiLSOYY9ERGRzjHsiYiIdI5hT0REpHMMeyIiIp1j2BMREekcw56IiEjnGPZEREQ6x7AnIiLSOYY9ERGRzjHsiYiIdI5hT0REpHMMeyIiIp1j2BMREekcw56IiEjnGPZEREQ6Z9S6ACIianpSSpw8WdigtrGxcTCZTH6uiLTEsCci0qHfd+3AfffdhpOFhdi+fQtUVUVERAT69R+EuNg4AEBZeRmOH8/D/v1/oOdZ5+CaGTfixptug9HIaNAb/hclItKhnmf3wvwFWQCACeOGYNOmbNxx5/144sl/1Wh7/Hgennj8H3jowbuxYP53+HpOJiIjIwNcMfkTx+yJiHSspKQY27dvAQCcf8HEWtu0bNkK//fGB+jYsTOWL/8Rb7/9WiBLpABg2BMR6diaNatRVlaGyKgoDBkyos52BoMBnTt3BQAszMwIVHkUIAx7IiIdW7XyJwDAsKHnISoqqt62Bw/uD0RJpAGGPRGRjq1cmQUAGD1mfL3tDhzYh/37/wAAjB17vp+rokBj2BMR6VRJSTE2ZK8FAIwePb7etp99+gEAoG1yO/zttnv8XRoFGMOeiEinPOP1UdHRGDxkeJ3tNm/eiDfe+A+Sklrg89nfoUWLlgGskgKBYU9EpFOe8frhw0bW+ihdYWEBXn/tJVw0cTRGDB+FpT+uwaBBQwNdJgUAn7MnItIpz3j94cMHccN1l3uPO0odOHXqJOw2G4YNH4l5GUsxdNh5GlVJgcCwJyLSoZKSYmzcsA4A8PQz/8bkKZYGv/bHHxfDOu8bREVHAwC2bd2MSRel4Na/3c3V9UIU/6sREenQmjWrUVpaCkVRMOK80Q1+3YoVy/D8v55E5sLl3vXyd+/ehRHDemPLlk1453+f+qtk8iOO2RMR6ZBnvL5Pn35ITExq8Ou+X2jF+nW/IGvZEu+xHj16okuXbvj6q1lwOBxNXiv5H+/siYh0yDNeP2r0uDN63XXX3YJmzZpXeZ2qqjh6NBfJye1PuzAPBSeGPRGRzlQerx816szCvufZvfDQw09WOTYnfTbsdhteefXtJquRAothT0SkM57xeiEEho8Y1ahrHM3NwaZN2diyZRPmW+di1udzMWXqtCaulAKFY/ZERDqzYvkyAMDZ5/RGq1atG3UNe7EdJY4SdOzYGW3aJmPJ4kwUFhY0ZZkUQLyzJyLSgT17fsfBg/tx8MB+fPrJewAAl8uFhZkZiG/WDMOHj/LOrm+Ibt16oFu3HgCA6ZdcgbFjBmLihecha3k2YmJi/fI7kP8w7Ik09MH7b+HQwQP1tlEMBiQmJqF797MwZuwExMXFB6g6CiU/ZS3FwQMVu9Zde90tiIqKQrHdjrVrVqPcWY7evfsiKalFo64dERGB6dMvxwvPp+GjD/+HO+68rylLpwBg2BNpqEuXboiMjML8+XOx6Pv5MJlMeObZl9GmbbK3TUFBPvb9sQdvvvkK7DYbHn4kDbffca+GVVMwuvGm23y+RmFhAcaOHojBg4fho0++rnLO8yVz167ffH4fCjyGPZGGLrjwIgAVXbCLvp+P884bg1v/dnetbe+97xFcNGk0Hv3nfVBVFXfedX8gS6UwUFCQj4MH93u77yvb/ftOAMCAAYMDXRY1AU7QIwoCq1ZVLIAyZuyEOtskJCTiuutuAQC89OLTKC0tDUhtFD66du2OK66cgdtu/3uV44cOHcA333yJYcNH4uprbtSoOvIF7+yJNFZcbMemjesBnH7PcSEEAKCo6BQOHtyPHj16+rs8CjNvvPkh3nn7dXzzzZeIi40DhMAvv6zErX+7G/ff/09ERERoXSI1AsOeSGO//LIK5eXliI6OwaDBw+ptu3XrrwAAg8GAtpXG9YmaitFo9A4RlZeXo7TUwUmhOsCwJ9LYyhVZAIDhw0fWe9dUXl6OpUu/BwCYzZfwA5j8zmQyndHjehS8OGZPpDHPGuajx4yvt90Xsz/B0dwctG7dBs+/8Jr/CyMi3WDYE2mo8nj9mDF1T877Yen3+Ocj96Jr1+6YvyALbZPbBapEItIBduMTacgzXm8ymaAoCjZtyvaeczqd+H3XDsyd+zU2ZK/F7Xfci3vvexjR0TEaVkxEoYhhT6Qhz3h9u3YdsGD+d1VPCoFWrVrjzrvux7BhI2tsLZqffwJZWUtRUlyMTp27YMiQ4fwiQD45dbIQv+3YgqJTJxEbG4ezzzkXSUkttS6LmgDDnkhDnvF6y7RL8eRTLzT4dV9+8Sl++ukHTLooBREREXjn7deRnb0WaWkv4Mq/XuunakmvpJT4YekCLPtxIZxOp/e4oigYcd44pJhTYTAYNKyQfMWwJ9JISUkxfnV325/JnuPr1v6MDRvW4e13PvEeS738atz2t+tw+23XIympBSZOmtrk9ZJ+LV6UgR9/yKxxXFVVrF61DGVlpUi9/DoNKqOmwgl6RBpZvXoFysrK3HdPoxv8ujlzvoDV+i12795V5fgdd94HKSU+/PCdpi6VdCw//ziyln1fb5v161Zj/749AaqI/IFhT6SRVSsrlsg999z+aN48ocGva948AYWFBSgqOlXleOtWbQBUjOUTNdTWLRuhqupp223enH3aNhS82I1PpBHPeP2o0Q3vwgeAfz76NP7x4OM1Fjv59dcNAIBBg4Y2SX2kI1JCKSyAkncU4vgxKMePQck7BuVEHmx5OQ26REEBv0SGMoY9kQYaO17vUT3oVVXF66+/hISERNxzz4NNUiOFEFWFUpgP5VilMHcHujheEeqoNPEOBgPUpJaQLVsjpkUr4ETead8iJibWj78A+RvDnkgDS5YsRFlZGQBg6LDzfL7ey/9+Ftu2bkb6nEwkt2vv8/Uo+Ai7DcqxXIi8o1CO5cJwNBfKsVygML/ijt399wkAYDRCTWoJNSERsnM3OIeOhKtNW6it20K2agO1RSvAPbu+2/69wBsvnvb9e57dx1+/GgUAw54oQNauWY133n4d+fknsG7dL97jV16egk6duuCy1KuQYp5+xtf9+ON3MSd9Npb88At3wQtVLheUUychCvNrhLnIO1pxZ+5y/dneHeayVRuoXXvANXBY1TBv2RpQGjYlq1Pnbujdpz+2b/u1zjYdO3bBuecO9PW3JA0x7IkCpG+/AXgi7XkAQGREJKJjYlBeXg673QYAaNmy1Rlf88MP3oY141ssWrIaiYlJAIB9+/aiS5duTVc4+c7phJJ/HEpBPnCyoGaYHz8GVJ4kZzJBTWwB2aoNXH36ozwxCWpi0p9h3qoN4N7uuClcceUN+OyTd7B7944a59q374Rrr7sNSgO/PFBwYtgTBUh0dEytIdyqVetGXe+9d9/Ajh3b8fWcTO8YvtPpxFNPPoyPPvnap1rpDJWXQyk44b0rVwryK/6pI8xlRARkQpI3zMvatIVsngg1qYVfwvx0oqKicfPMv2Pr1o3YseBbnDyWi+iBw3BOr74YOGg4g14HGPZEIeidt1/H9wutuP7GW7FgQcUyu8V2O3bs2IaOnTprXJ0OVQtzw9FciMKKQBd5R6HkHQWk9DavLczV1m2B5okVd+gBDvOGEEKgb99BGLp+DYzHjsE2Y6bWJVETYtgThZiPPvofHnn47wCAn376ocb5N978MNAlhb6yMii1jJejMN/7yFqVMI+Ng5qQCCQkVQlz2aoNZEJFl3uoUmxFkLFxWpdBTYxhTxRihg07D3PnLanzfL9+nEhVnSgrq3Pymyg4AaWwoEp7GRtXEd4JiXC171QlzNU2yZA6fgxN2G2Ajn+/cMWwJwoxffr007qEoCPstronvx3NgSi2V2nvDfNWbeDq2avq5Le27SDDefdAuw2ydVutq6AmxrAnoqDnfca8IB/iZEHVMM89AlFSXKV95TAv79O/IswTK8bQ1eT2kFHRGv0mwU/YbVDZja87DHsi0lyVBWOqz2RvQJi73LPZZWIS1HYdICOjNPpNQp8otjPsdYhhT0R+V331N0+gi7yjUI4cgih1VGlfW5h7Z7O3bgMZEanRb6JzUkIU2zlBT4cY9kQhRkoJu60IpogIRAbJHWxtS7mKwnyIwoKaYW4wQI1vBuleNKZymFdfypUCS5QUA6oKGcsJenrDsCcKEQ5HCZYumY/s9T+j2D3hrG1ye5x//mT0H+DHne7qWcr1tOuy1xbmZ7CUKwWWcK/myDt7/WHYE4WA4mI73nn7ZRzNPVLleG7OYcz+/H0cOXIQk6dc0riLu1xQTuTVPZu9+rrslZZy9XVddgouDHv9YtgThYDvvp1dI+gry1q2CN269cTZ55xb86R7XXaRdxRK/omas9lPsy671ku5UgC5wx5x8drWQU2OYU8U5IqKTmHz5uzTtlv1/Tycm3O45mx2HSzlSoHBO3v9YtgTBbkjhw9AVgrruhw+uA/R69dAxsRCbdUGsmUruAYOheruWpetWkNt0QqSd21UB2/YcwU93WHYE+mEGhOLove+DO/V38gnwm4HFIV/h3SIs2iIglxycgeIBnSrt+/YhR/S5BPFVlRxV89hHN1h2BMFMcMfu9H2vy+gf0nJaduOHDUhABWRngm7jeP1OsVufKIgpOSfQOTcL2HKWgw1sQUumXYlDmWvxvHjx2ptP3LUBPTu0z/AVZLeMOz1i2FPFEREWSlMi+cj8ruvASlROv1KlKZcisiICNwxciwWLfwOGzasQVlZKQAgKaklzr9gCoYOG6Vx5aQHwm4DGPa6xLAnCgZSwrhuNaJmfwTlRB7KR41H6ZXXQ01I9DaJiYnF9EuvhnnaFTh5sgAmownNmidoWDTpTrEdsk2y1lWQHzDsiTRm2Ps7Ime9D+Ou3+Dq1Rf2e/8JV+dudbY3Go1o0aJVACukcCFsRXB1O0vrMsgPGPZEGlHyjyPy689gWpUFtXVbFN/9EJzsjicNCbuNz9jrFMOeKMC84/JzvwKAinF582WAyaRxZRTWpIQoKeYEPZ1i2BMFimdc/vMPoeQfR/mo8XD89QZIjrtTEBDFdkBKrrCoUwx7ogAw7NlVMS7/+w64eveD/YHH4erYReuyiLy4Lr6+MeyJ/KjKuHybZI7LU9Dyhn0cw16PGPZEfiBKHYhYMBcR878BjCY4rrgWZZMsHJen4OXZ3pYT9HSJYU/UlKSEaeUyRH71CZSiUygfeyEcqddANmuudWVE9WI3vr4x7ImaiHH3TkTMeh/G3Tvh6tMf9mtuhqtjZ63LImqQP7vxOUFPjxj2RD6qMi7fth3H5SkkKXYbYDBARkZpXQr5AcOeqJGqjMubIirG5S+aBhj5vxWFHu+COtzeVpf4qUR0pjguTzrEHe/0jWFPdAYMv21B1KwPYNi/l+PypCvCxrDXM4Y9UQMoR3MQ9dWnMK5dBTW5PYofeALOAUO0LouoyfDOXt8Y9kT18IzLR1rnQEZGwTHjFpRdOAUwGLQujahp2W2Q7TpoXQX5CcOeqDbucfmoLz+GsBVVjMtfPgMyvpnWlRH55aBs1gAAIABJREFUBe/s9Y1hT1SNYftmRH32PgwH98HVpz8cM26Bq0Mnrcsi8iuGvb4x7InclNwjiPr6MxjXroKrXQeOy1P4UFUIRwnDXscY9hT2hN2GSOs3iFiUwXF5Ckve7W0Z9rrFsKfw5XLBtHwpotJnQdhtKLtwCkovuxoyOkbryogCyrtULjfB0S2GPYUlw7ZfETXrfRgO7odz4FA4ZtwCtXVbrcsi0oQn7MF18XWLYa8zjpISOEodp21nNBoRF4b/Yys5hxGVPqtiXL59RxQ/mAZnv0Fal0WkLe5lr3sMe515+eXnsOzHxTh0+CCOHc0FAHTo0Amdu3SFyVixl3phYQGOHDkEp9OJ8y+YhPvuewS9ep+rZdl+5x2X/34eZFR0xbj8X6YCiqJ1aUSaU7i9re4x7HXmscefxWOPP4slizNxeepUREREYNXPm9Gs2rrtUkosW7YEN9/4V1gzvsHHn6Tjoslmjar2o8rj8sV2lF0wmePyRNVwL3v9422NTmVnrwUADB4yvEbQA4AQAuefPxHPPPsySktLccftN6C42B7oMv3KsO1XxD72d0R/8CZc3XvC9tJbcMy4hUFPVI3wbG8bEal1KeQnvLPXqZUrswAAY8ZMqLddt249AAD5+Sfw8+oVuODCi/xdmt8pOYcR9fkHMG5aD1eX7ih+9F9w9tL3MAWRL7wL6nB7W91i2OuQw+HA+vVrAACjx4yvt+2hQwf+/JcQ/x9d2IoQOfdLRCzNhBrfDCU33YHycX/huDzRaXD1PP1j2OvQunU/o9ThQGRkJIYOGVFv2xUrlgEAYmJiMXjwsECU1/Q84/JffwZRUszn5YnOkLDZoDLsdY1hr0OrVv4EABgydASioqPrbFdUdArfzU0HANzz9weRkJAYkPqakmHbr4j67D0YDh2oeF7+2plQW7XRuiyi0GK3AQx7XWPY65BnvH706PF1tlFVFQ/cfweKik7h8iuuwQP/eCwwxTURw+GDiPzio4px+a49UPzY83Ce00frsohCkrDboCZysyc9Y9jrTEPG69euWY1nn3kMGzeux/MvvIZb/3Y3RIiM13vH5ZcsgNo8oWJcfvzEkJ9vQKQljtnrH8NeZzzj9UIIvP3Wa/jgvbe8506dOom8vGNo3aYtpkydhs8+/xbNmydoWO0ZcLkQsTQTkXM+B1xOlE2ZjtKLL4eMqnuYgogahmGvfwx7nfGM1/fp0w+fz/7ujF+fm3MEr776Ag4e3I/ZX8xr6vIaxbhhLaJmvQ8l7yicQ0fCcdUNUFu21rosIn1wuSBKHQx7nWPY64xnvH7U6HFn9Lq33nwVP/6wCJ06d8V867foP2CwH6o7M4bDBxH5+Qcwbt5QMS5/69/hPLu31mUR6Yqw27i9bRhg2OtI5fH6UaPOLOxvv+Ne3H7HvQCA5T/90OS1nYkq4/IJiRyXJ/IjLpUbHhj2OlJ5vH7EeaO1LufMOZ2I+GGhe1zeVTEuP/0KyMgorSsj0i3hWSabYa9rDHsd8YzXn31Ob7RqFVpj2hyXJ9IG7+zDA8NeR7zj9WfYha8lw749iJz1Pow7tsHV7SwU33YfnGedo3VZRGGDYR8eGPY6UVxs947Xjxw1VuNqTk8pLEDkN7Nh+mkJ1IQklNz6d5SPnsBxeaIA84Z9HMNezxj2IczhcGBD9lrY7TZ8++1XKHU4AADbt21BUlILdO7cFV27dte4ymqqjcuXXnwFylIuhYyI0LoyorAkbEWA0cjtbXWOYR/Cjh8/hoyMbxAdFY22bZLx4ENPoKy0FC6nEz/+sAh9+vQLqrA3bliLqM/eg3L8WMW4/NU3Qm3RSuuyiMKaKLazCz8MMOxDWIcOnfDCi69rWkNZWSlWr8rCjh1bcOpkIZonJKJ37/4Ycd5YmEwVd+uGP3ZXPC+/Yxtc3Xui+Pb7OS5PFCSEnTvehQOGPTVafv5xfPDe6zh+/Jj32IkTedi7ZxfWrl2JmanXofXSTJiyFkNNbMFxeaIgxKVywwPDnqo4duwoCgsLcDQ3B0VFpxAf36zWdqqq4uOP3qwS9FWuczQHs157Dv84dQql069EacqlAMfliYIPt7cNCwx7AgDMSZ+NBfMr1tK/4MKLoKoq7r7zZgDADTf9DWPHnl+l/ZYtG3A090i919xnNCD7tntx9pDz/FM0EflM2GxQk1pqXQb5GcOeAACXpV6Fy1KvanD7vbt3Nqjd7tzDOLuxRRE1MZfLhYMH9zeobatWrREbBne8otgOGROrdRnkZwx7ahRHqaNB7TyPAxIFg82bN+Jfzz2BU6dOInv9GrhcLsTFxWPw4GEQleaS5OefwM5dv6FLl264/vqZuPmWO2A06vPjkmP24UGff3vJ71o2cCnbFlzyloLIwIFDkD4nEwDQt09nHDp0AI8/8Rxm3npXjbb5+Sfw4D/uwiMP/x1Ll36PL7+y6i/wXS4IRwlkXLzWlZCfKVoXQKFpwIChUJT6//oYjUb066/9VrlE1eUcOYxDhw4AACZM+EutbZKSWuCttz9GYmISflj6PT777INAlhgQXCo3fDDsqVFatW6LCedPrrfNxEnTkJjYIkAVETWcZx+JNm2TcVbPutd8iIiIQJu2yQCARd/PD0RpAcWwDx8665OiQJo4yYKYmFgsXpSB0kpj+NHRMZg8ZTqGjwj+NfopPHnCfvTo8fW2c7lcOHzoIADAZDL5uarAU2xFFX9g2Osew558MnrMBRg6bBSOvPgkTkqJqMtnoFv3nojgOtsUxDzbQY8eM77edhs3rkdR0SkAdXf3hzTe2YcNhj35LDIyCn0UBYiOQXGvvlqXQ1SvnCOHsWfP7wBOf2f/ycfvAgDO6dUHV111vZ8rCzx244cPjtlTkxDl5YCB3x0p+Hm68JPbtUePHj3rbLdkcSZmf/4xOnfuii++zEBUdHRgCgwgUWwHwLAPBwx7ahpOJ6TeHksiXTrdeP2+fXvx2KP34+qrLsZlqVdh6Y9r0KVLt8AVGEDCboOMiOAW02GAn87UNFQXoMMJTKQ/nvH6DdlrMX3an+PwTqcTZeVlMJlMGDlyLFb/srXeO3+n04l9+/bW2ybYCbuNq+eFCYY9NQnhdAIGg9ZlENUrN+eId7z+f+/NwuDBwxp9rWefeRRbt27GnG8WNlV5AcfV88IHu/GpaZSXQ3LMnoLc8uU/AgBiY+PQr9/ARl9n2bIleOvNVwEpm6o0TQhbEcM+TDDsqWm4nADH7CnIrVpV0YU/YsSoRj83f/x4Huakz8bAQUObsjRtFNv5jH2YYNhTkxAujtlT8Fu5IgsAMGr0uEa9XkqJp9IexhNP/Ou0y0WHAmG3cV38MBH6f1spOJSXQyocs6fglZtzBHv37gYAjBrVuLB/5+3XkZIy3buEbqjjBL3wwbCnpsFufApynvH66OgYDBg45Ixfv33bFpw4cRyTLkpp6tI0I2ycoBcu+OlMTUI4GfYU3FasWAYAGDp0BCLO8Lny4mI7Xn31Bbz51kf+KE0b5eUQZaUM+zDBT2fynZSAqnI2PgWdjRvXY//+P3Do4AF8M+dLAEBBQT6+/fYrxMfF4/wLJsHQgEdG0558GA8+9MQZf0kIZgqXyg0r/HQm35WXV/zknT0FmQ3Za5GXdwxR0dF48OEnvMf3/bEHDocDY8aef9qw/+67dJhMJtjtNmzalO09brfZ4HK5sGlTNiIjItGr97l++z38gevihxd+OpPPhMtZ8QeGPQWZm26+3edrRJgiEGGKwLy56VWO7927Gy1btsK8uelISEwKvbB3r4vPR+/CAz+dyXfOirBnNz7p0ZSp0zBl6rQax7/48lP06NETTz71ggZVNQHe2YcVfjqTz4TLVfEH3tlTGCgsLEBBQT5sRUXIzc3B0dwcxMTGIj6+mdalnRF244cXfjqT75wVY/bc9Y7CwQP334H4uHikpl4FAHjh+TQ0a94cTz39ksaVnRlhKwIAqHzOPizw05l85+7G5372FA7e/2C21iU0Cc/2ttDREwZUNy6qQz4TTk7QIwo13PEuvDDsyXfu2fjsxicKHaLYznXxwwjDnnzHO3uikMM7+/DCsCefebrxhZG73hGFDLsN4OS8sMFbMfKZZDc+6diunduwZcsGFOSfQHR0DM7q2RuDBo+AMcT/vgtbEdTWbbUugwIktP+2UlDwTtBrwBrjRKGivLwcX87+AFu3bqxyfPPmbKxYsRQ33HgnkpJaalSd79iNH17YjU++46N3pEPzvvuiRtB7HDuag48+fANOz9/9ECSK7Qz7MMKwJ995uvFNHLMnfcjLO4r161bX2+bY0RysX19/m6BVVgZRVsbZ+GGEYU8+43P2pDe/79oOKWWD2oUixb0JjuQEvbDBsCffecJe4Zg96UOxZ0e407C715cPNZ6lctmNHz4Y9uQ7J2fjk74kJCQ1abtg49kEh9vbhg+GPfnMu589x+xJJ87p1Rcm0+nXjO/Xb3AAqvED7ngXdhj25DvuZ086ExcXj0kX1dzDvrJevfuhV+9+AaqoaXnu7FWGfdjgpzP5LkDP2R84sA833XBlgx53atGiJfr2HYArrpyBc3r18WtdpE9jxl4IAFj0/TyUl5d5jwshMGjwCEy/5CoIIbQqzyeCE/TCDsOefCac5RVB7+cPvk6dumCe9QfYbDb89QozNmxYh79MnIK33/kEhkpfNE6eLMSGDevwysv/wn//+2888I/H8Mg/n/JrbaRPY8ZeiIGDhuO37ZuRn38csbFx6NmzN1q3Sda6NJ8Iuw0yMopDb2GEYU++c7kgA7QufkxMLGJiYnEk5zAA4NJLr0SLFlVXMUtISETnzl0xbtwFGDGsN1568Wn06NETqZdfHZAaSV/i4uIxdNgorctoUlw9L/xwzJ5853QG9Bn73bt3ITfnCABg9JjxdbZLSmqBgYOGAgA++uh/gSiNKCQIWxFkLLvwwwnDnnwmXM6Arou/cmUWAKBr1+5o375jvW0LCwsAAMeP5/m7LKKQwTv78MOwJ985nQHrxgeAlSuyAABjxk6ot11paSk2b65Y27zvuf39XRZR6LDbAC6VG1YY9uQzEeBu/FWrfgIAjB49vt52CxZ8B0dJCQwGA267494AVEYUGrgJTvhp0Cf01KlTxyiKMgqA4nK5Ps/MzNzv57oolLicAVs9r8p4fT1hX1xsxwvPpwEAnkx7AUOGDA9AdUShgd344afeT+iLL764i8vl+kRKWSiEeFxKWWYwGN6yWCxvSSk/BHCX1Wr9OkC1UrByBm7M3jNe36NHTyS3a19rm99/34nbb7seuTlH8H9vfIBrZtwYkNqIQoVit/EZ+zBTZ9hPmTKlp8vl+gnAiiFDhlyZlpamuo/fbjAYdgCIEkKcflso0r2KbvzAjNmvWlnRhR8ZGYXXXnvxzxNSIi/vGDZtysbhQwcxffrl+Oyzb9A2uV1A6iIKFaKsFCgv5519mKk17CdOnBhrMBgyABjLysr+5gl6AMjMzNxvNpv3ATgbwPKGvMmUKVN6ZmZm7mqKgikIBbAb33NnP37ChRgwoOq65HFx8bjzzvtrveP/5yP3olevczFg4GDExMTiaG4Otm3bjKioaMy49qZAlE4UFISN6+JrzWw2nwPgXillM0VRXFLKFVar9V2z2fyXqKioNenp6Seb+j1r/YSOiIh4FBVh/uKiRYvyK5+bOHFiLIBuALZlZGQcPd0bpKSk3KIoygPu65EeBWiCXuXx+quvvgG9ep/b4NfOSZ+NvLxjVY5NTbkYb7z5YZPWSBTsBDfB0ZTFYpkppfy3EOJmq9WaDgBms/lZi8XysZTyWofDcQeAt5r6fWt8Qlsslngp5Z0AIIT4qvp5k8k0GkCElPKH01186tSpvYQQ/5FS5jZJtRSUhNMJBGATHM9dfWJiEs4+p/cZvbZ7j564864HUJB/As0TE3HBBZPQt+8AP1RJFNy8Yc9H7wIuJSXlFinlOwBuycjISPccLyoqejY+Pj4XAFRV3eKP967tE/ovAOIB5GRkZGyqflIIcb7757L6Ljx58uRIRVH+KaVcJoTo1STVUlCSLicQGeX39/GM148aPQ6KcmZPjSYltcDd9/zDH2URhRZP2HOCXkBZLJZ+Usr/A7DearVW6VLMyspymM3mkwBMMTExa2p7fUpKylmKooxUVdWhqurOzMzMGvlcnxqfmFJKz0DoSgA1JuC5w14tKytbUd+FTSbTM4qivCSEKDmTgigEBagb3/N8/ahR4/z+XkR6xW58bUgpXwAQCeB/qJatU6ZM6QygE4CV6enpZZXPjR8/3mg2m18VQtwqpTwphGhjMBi+TklJWTVt2rS+DX3/2j6hEwBACLGx+ompU6cmAhgohNjsGcs3m83XW63Wjyu3M5vNk1VV/X3+/PlbzGZzQ2uhECWcTr/vZb979y7kHKnY/GbU6DMPe0dJCV54Pg1FRaegqir27t2NsWPPx+133Buy25QSNQbDPvAsFksPKeVkAKrRaLRWP68oyvnuP/5Y/VyzZs0eU1X1x/nz53tfN3ny5I+NRuNWVVWXTJo0qXf1uXW1qfEJLYTYL6WEqqqHazk3EYBBSrkcANLS0pTs7OxpAD6u9Eu1AWCxWq23ne7NSSec/p+N7xmvb948AX369Dvj15eVleGKK2ega9fuACoW3Rk/bgjy8o4i7akXT/NqIv0QdhtkVHRA97MgXOT++evcuXOPVT/pGR6XUtYYHpdSzgAwCoA37BcuXHjKYrG8J6V8OjIy8jIA756ugNoGPn8EIIUQLSofnDx5cgchxLPuN98EABs2bBgnpfR256elpSlSyhcNBsNjp3tj0hGXf7vxjx/Pw+zPPwYAdO7cFaqq1tu+NtYFy7xBD1RslXvVVdfhzTdewc4d25uoUqLgx9XzAk9V1R4AIKVcVUeTCQCKbDbbhlrO2YQQ3dPS0qrktaqque5rtqzlNTXUCPuMjIz1AD4FcN348eOjAMBsNg8yGo3/kVJeDcAhhIhzB/s9qqrO9rw2Ozv7HwA+nTt37omGvDnpg79m4z/z9KMYMqgnUi+djPKyMgwYMBiKomDCuCEYPbI/9u//o0HXKSsrg9PprHG8S9fucDqdWLw4s6lLJwpaXBc/8MSfY4U7qp9zP3PfHsCqrKwsp/vYx57zUVFRg4qKinpWXu/GfU3P40TrG1JDrZ/QycnJt+Tk5NzTrFmz2WazOVcIcTAyMvL69PT0ErPZPFUIcdv69ev7KoryemZmZq67uKFSyu5SyvSpU6d2q3S5OAAmz7EFCxbsbUhhFEL81I3/+BPP4fEnnvP5OqNH9UdcbBx+zFpX5bjR/QWlrKzU5/cgChW8s9fETvfPwlrOzXD/XAEAqampzR0Oh/dGPD093VX9BRaLpZ27e3+Z1Wpd0pACav2Efvfdd8sBvFzbOavV+iNqmUSgqmqUoij5QoiZ1U4NABCjKIrn+COoZZY/hS7hLA/orndnSgiB4SNG1Th+4MA+AKj1HJFu2YqAxCStqwgrTqczw2g0vgygT+XjFovlSlVVxwshIKXcDwAlJSV/FUJ8W9e1Zs6cacrJyflKCLHL5XJdigbmaZN9Qi9YsGAF3N9MKktJSRkjhGhltVofbqr3oiDjcvl9Nr4vrrxyRo0FdBwOBz799H1cNNl82q1yifREsdvgat9R6zLCysKFCw+ZzeabhRCvmc3m1Yqi7He5XNcAOFFWVjYxMjJyjaIoo81m8x4AU4uKiqbXdp3x48cbc3JyZgE45nA4rl28eLG9oTX4bT/7tLQ05ZJLLkkWQrQGkDht2rSOqampnP6pRwHez/5M3X3Pg1i18iekPfkQlizOxNy5X+PiaRdixPBR+ODDL7QujyighN3G1fM0YLVaZ7tcrqEAOqmqOl5K+b+MjIyX3IE9TFXVn4UQ/YQQV3nG7iubOXOmqVmzZl8A2D148ODUxYsX28ePHx9lsVgatNuX3z6h169f/7oQIg7uLn9VVZ8qLS01ALjOX+9J2hDO8qB+jMdgMODJp17A/v1/IDt7LSJMEfjkk3S0aZusdWlEAccJetrJzMzcj1rWvbdarcWomBhfq9TU1IgjR458oijKPKvVOstqrXgKr3nz5qNVVe1Z2zWr81vYz58//y5/XZuCiJSAqkIGaItbX3Tu3BWdO3fVugwizYhSR8WEWoZ9yEhNTY1wOBxfCyH2Aii1WCypACCljFVVNRXASw25TvD2vVJocLkqAj+Iu/GJqAJXzws9DofjawDTAEBKeW/1806n8/qGXIef0OQT4XIPLTHsiYKeJ+xVboITSj4TQnxe2wkppbpw4cK8hlyEn9Dkm/JyAIAM4jF7IqogbBVhD97Zhwyr1fpNU1yHYU++cbnXewjiMfuTJwtQVHQKcXHxSEjg88UUxtiNH7YY9uSTYO7G37tnF+Zb03H48AHvseTkDphqvgxnndVLw8qItOEds+ejd2HHb8/ZU5hwrznv713vztTWLRvx3ruvVgl6AMjJOYQP3/8vNm1cq1FlRNoRdhsgBGR0jNalUIAx7Mknwj1m74+NcBqruNiO9K8/qXN3PFVV8c2cWbAVnQpwZUTaEsV2bm8bphj25Jsg7Mb/ddM6OBwl9bYpKyvFhg1rAlQRUXAQtiKO14cphj35xj1BL5i68XNyDjew3SE/V0IUXLjjXfhi2JNPhHvMXgRRN76hgV2UisK//hRehN3Gx+7CFD/tyCfS6X7OPoju7Dt1atiSuJ06d/NzJUTBhXf24YthT75xBt+Yfd++g077PH18s+YYOHBYgCoiCg6i2A4Zx7APRwx78onwLKoTRN34RpMJ11x7K2LqWBI0MjIKM2bcioiIyABXRqQtTtALX8HzCU2hqTz4uvEBoGPHLrj7749iyWIrtm3dBIejBJGRUejdpz/+MtGMFi1aaV0iUcBxe9vwFVyf0BR6gvDRO4/ExBa4/IrrgSsqHrXjnTyFM+EoAVwuSG6CE5bYjU8+Ed4x++BdGx8Ag57CnnfHOy6VG5YY9uQbz3K5XJGLKKhxL/vwxrAnnwTzRjhEVImd29uGM4Y9+cZ7Z8+wJwpmnr3seWcfnhj25BvPmL0puMfsicKdtxufE/TCEsOefCJcTkBRACG0LoWI6uHd3pZhH5YY9uQbpzPonrEnopqE3Vaxjz33hAhL/K9OvnG5gmr1PCKqHdfFD2/8lCafCJeT4/VhwFFSgk2bsuEoddTbLjoqGkktWqJbtx4N3n2QAoNhH94Y9uSb8nI+Yx8GDh85hNmzP0ZBQT6WLv0ejpISJCYm4aLJZkS6Fywqd5bj+PE8bNy4Ho6SElx19fV45J9PoVmz5hpXTwCgcHvbsMawJ984nezGDwPdu5+F//7f+wCA80acix2/bcPf730Yd9/zjxpty8vL8dKLT+Plfz+Ln376AYsWr0J8fLNAl0zV2W2QLVtrXQVphGP25BN244eX/PwT2LXzNwDAuPEX1NrGZDLh0ceeQbduPfDb9q34v/++HMgSqQ7sxg9vDHvyjcvF2fhhZPWq5VBVFQkJiejbd0C9bVu2qriLzMpaGojS6DQY9uGNYU++KS8HFI7Zh4uVK7MAACNHjYVSzyNcUkr8sXc3ACCOAaM9KSFKiqHG8b9FuGLYk0+E08l18cPIyhVZAIDRo8fX227r1l+Rl3cMADDpohQ/V0Wn493ell+8whbDnnzjckJyzD4s5OefwG+/bQUAjB4zvt62H7z3FgCgd5++uO66W/xdGp0Gd7wj3pKRTwQX1QkbnvH6xMQk9OnTr852C+Z/h88++wBnnXU2vvzKiqjo6ABWSbVh2BPv7Mk3fM4+bNQ3Xu90OrFx43rcdedNuOnGK3HjTbdh6Y9r0LFjZw0qpRo829tyXfywxVsy8ol0OQHeuYUFz3j9zh3bMX3aX6qcc7lcSEpqgfNGjsFjjz2LNm2Ta73GqVMncfJkIdq2bQcTh38CxntnHxevcSWkFYY9+UQ4nZBGfmjrXUFBvne8/tXX/3faCXrVrV+/Bq++8jy6dOmGzp27YseO7eg/YBDH8wNE2IoAsBs/nDHsyTd8zj4srFyRBVVVERkZiSGDh5/Ra+fNm4PHH30A86w/oGvX7gCA/7z8HB5+6B5ccfk1HNMPAGG3V2xvGx2jdSmkEX5Kk2+c5Xz0LgysWvUTAGDQ4GFnFM67d+/Cbbdehw8/+tIb9AAQERGJgQOHwBQR0eS1Uk3CbqvYx14IrUshjXCCHvlEOJ1cVCcMeMbrR40ad0ave+U//0J8fDwmTppa5fhddz+AzIXLuTNegChcPS/s8ZaMfOPkc/Z6V3m8ftTohoe90+lE5oJ5GDx4GMrKyvDFF59g9+870bFjZ1yWehVatmzlr5KpGi6VS7yzJ9+4uIKe3nnG600mE4YNO6/Brzt06ABOnixEXFw8HnnoHgwfNhJPPPk82rXvgDGjB3DN/AAS3N427PFTmnwiuMWt7i1f/iMAoP+AwYg5g+e0T54sBAAsXfo9Vv28GV26dAMAWCyXYvOvG3HtNZdi/YZdaN26TdMXTVXZbZCt22pdBWmId/bkG6eTs/F1aPnyH/HRh+/gqScfxqxZHwIAjubm4M03XsGnn7yP4mL7aa/h+WJw9tm9vEHvMXLUWBQVncJXX37W9MVTDezGJ35Kk0+EywlwkpXuOEpKIBQFnbt2w//93/swmkwotttRVl4GVaoQDZjVnZzcDoqieLe6rcyzE96e3buavHaqSdhtUBn2YY1hT77hrne6VH32fGPExcWjV69zvd35lZWWlQJArV8EqIm5t7flnX14Yzc+NZ7LBUgJyTF7qsPV19yALZs3wVFSUuX43r27IYSAZdqlGlUWPkRJMaCqDPswx7CnRhMuZ8Uf+Ogd1eHmW+5A334D8Oyzj3mPHTp0AK/+53k8+NAT6NdvoIbVhYc/18Vn2Icz3pJR4zkrwp4T9KguJpMJ385dhJdfehYzrrkBYkd8AAAgAElEQVQESUktUJCfj+dfeA2Tp1i0Li8scF18Ahj25At32HMFPapPbGwcnnzqBa3LCF+eJycY9mGN3fjUaMIT9ryzJwpa3m58hn1YY9hT47nH7LlcLlHwYtgTwLAnX3ju7DkbnyhoCbsNUBTIKG4lHM4Y9tRo3m58LqpDFLQUG7e3JU7QI1+wG5/qcWD/Xmzbtsm7GU6v3v3QvfvZWpcVdoStiF34xLCnxvPc2Qt241Ml5eXlmJP+KTZtXFvl+IrlS9Grdz/89aqbEBkZpVF14UcU2xn2xG588gFn41Mtvv1mVo2g9/ht+2Z8Puu9AFcU3ri9LQEMe/KBdHFRHarq8OED2JD9S71tdu7Yil27tgeoIgI3wSEw7MkHory84g/sxie337ZvbtJ25Dtub0sAw5584XIB4J09/amo6FTD2p066edKyINhTwDDnnzh4pg9VdWsWfOGtWue4OdKCACgqtzelgAw7MkH7Man6nr36Q/RgOe5e/fpH4BqSBTbK7ahZtiHPYY9NR53vaNqkpM7YOiwUfW2ObfvQPTocU6AKgpvwr0JDsOe+ClNjadWjNmzG58qm3bxXwEA69augpSyyrn+/Yfgssuv1aKssMS97MmDn9LUaNz1jmpjNBpx6WUzcN7I8di2dROKTp1EbFw8evXqi06du2ldXnhxhz2fsyd+SlPjucfsJcfsqRbt2nVEu3YdtS4jrAlbEQB24xPH7MkHwuWs2FyDG+EQBSWF29uSG8OeGs/lYhc+URATxXbAYIDkXgRhj2FPjed0sgufKIh5F9Th9rZhj2FPjSacTt7ZEwUxrp5HHgx7ajyXk8/YEwUxYbNBxsRqXQYFAYY9NZ7TydXziIIY7+zJg2FPjSbKy9mNTxTMGPbkxrCnxlNd7MYnCmK8sycPhj01ntMJGE1aV0FEdRB2G2RcvNZlUBBg2FOjCaeTC+oQBStVhXCUcIIeAWDYky+c5ezGJwpSwm7j9rbkxbCnxnO5IDgbnygoCS6VS5Uw7KnRhNMJmDhmTxSMBHe8o0oY9tRo0uWE5Jg9UXDiXvZUCcOeGk2Ul3NRHaIgpRTbAbAbnyow7KnxnFwulyhYccyeKmPYU+OpLo7ZEwUpYSvi9rbkxbCnRuNz9kTBi6vnUWUMe2q88nLuZ08UpBj2VBnDnhrPxf3siYKVKLZD5VK55Mawp0YTLo7ZEwUtm43P2JMXw54aj934REGL3fhUGcOeGs/FCXpEwUrYbdwEh7wY9tRowskxe6JgJWxFvLMnL4Y9NY6UgKqyG58oGLlcEGWlDHvyYthT45SXV/zknT1R0OH2tlQdw54aRbicFX9g2BMFHe9SuXz0jtwY9tQ4zoqwZzc+UfAR7k1wwAl65MZP6hD3xexPcPRYbr1tjAYjkpJaoEePnhg0eBiMTXA3Llwu98X5V4go2AhbEQBugkN/4p19iFMUBQIC3y+04qknH8bzzz2BUocDCQmJ3n8cjhKsXJmFS6ZPQv++XfHtN1/6/sbOijF77npHFHy44x1Vx0/qEHfFlTMAAEeP5mDNL6swZuz5eOjhJ2tt++hjz+AvF56Hm2+6Ci6XC6mXX934N3Z343M/e6Lg8+eYPcOeKvDOXidWr1oOABgzZkKdbdq374hrr70ZUko88cSDUFW10e8nPGHP5XKJgo6w2wCTCTIiUutSKEgw7HXg5MlCbN36KwBgzNi6wx4AIiMr/ufPzTmCo7k5jX9Tl2eCHlfQIwo2otgOlV34VAnDXgd+Xr0CLpcL8fHN0K/fwHrbbt++FQBgNBqR1KJl49/UyUfviIIV18Wn6hj2OrByZRYA4LyRY+qdaV9aWoqsZUsAABbLpd67/MbwdOMLI7vxiYKNsBVxXXyqgmGvAytXZAEARo8eX2+7L774BMeP56Fly1Z45rmXfXpP6enG5509UfCxc3tbqophH+IaOl6/cmUWHn/0AbRr1wHzMn5Au3YdfHpf7wQ9jtkTBR1241N1vC0LcZ7x+ri4eHTs2BmFhQUAAKfTicLCAuzcsR3ffvMllixZiBnX3oSHH0lDs2bNfX9jPnpH/9/enYdHVd19AP+eOzNZICGELEAg7CAurEELAjVYAWNyB0QHKlq1Vql7W7CllfqKWmuX14UutoL1VbG1bazITCIiCCiggKwiQQIhQIAQCAmQbTIz9573j8ykSSbLZJlMZvh+nscncO65Z37zqPnde1bqskRFBbfKpXr4mzrIecbrY2N74VfPLvG6Hp+QiNm3zsXLy17zSvK6ruPAga9w/Fg+EhN7Y+y4CQgLC/Ptgz3d+Fx6R9Tl8M2eGmKyD3Ke8fq58+7CL5/6lc/35RzYjxXL/4QpU1ORnDwQR47k4vlfPYVHHl2IGTPTW7xfcDY+UdfkckFU25nsqR7+pg5ily5drB2vnzzlBp/vq6yswKJFD2PVB+sQEREBALjuW9dj5s0ZmDplLD5c8xkGDRrSfCM8CIeoS6rdPY+z8akOTtALYlu3fApN02A0GnHttRN9vm/btq2oqqysTfQecXHxGD16HPbs2dlyI5ygR9QlKdwXnxrBZB/Etm79FAAwdtwERLViMk7fPknYt283nlj0CCo9R2ECKC8vw4Gvv8KYMeNbbKP2PHuO2RN1KZ43ey69o7qY7IPY1i01yX7KZN+78AFg5JVX46bpafjb669iQsoVePcfb+HkyRO4924LnlzyLIYMGdZyI+zGJ+qaag/B4Wx8+i8m+yB16dJF7N+/F0DrxusBQAiBt1e+hwU/fAxnCk/j4YfuxbgxQzH/zntxx/x7fGuE3fhEXRKPt6XGMNkHqfXrP4KmaRBC4LpvXd/q+4/lH8Xx4/n41fMvYurUaXC5XHjg/jux9OnFkFK2eL9wOWsSvRBtCZ+I/MST7HVO0KM62AcbRHZs/xz/fPdtXLhQik8+WQsAkFLigfvvRFLffrjdMt+nt/yDOV9j/h2zsGr1OgwaNAQPP/IT2Kzv45dLFmHZK79DQkJvPPLowuYb0TRI7otP1OWIinLIsDDA1z0z6LLAZB9Ehg4bgbvvfQAA8JNFv0B0dA9UVlTA4XRAStnycjm3JU8uxIIfPlavvmqegxtSv4N5czPw1pvLW072LhfX2BN1QaKinMvuyAt/WweRuLh4xLXnWFq3o0ePYPjwK7zKe/SIwaJFT+Lxxx9osY3zleUoDg+DfjQX/fsPRFhY20/QI6KOIyrKOTmPvDDZX4bGj78W69d/hJump3ldy8s7jLQ0c5P3ni0qxH/+8w6O5R8BwsOBv7wIo8mEKZNvxIybZ8HACXtEAcWtcqkxnKB3GXr+1y9h06b1+N1vn0WFezKPw+HAP/7+JtZ9/CGeXvpCo/cVnTmNV//8u5pEX4fL6cSmTWvxzsrXfJrcR0R+VFnBNfbkhW/2l6G+Sf2wZes+fLDq31jyi5qx+YjISEyZkorM/6yBaGKG/XvvrURVVWWT7eYc2Ifdu7YhZcIkv8RNRC0TFeXQE3oHOgzqYpjsL1NGoxG3W+bjdst8n+qfLSrEieNHW6y3c+fnTPZEASTKyzhBj7ww2ZNPzp4941u9kydgzD0IvVcc9JhYbqdL1MlEZWWrxuydTmftcF5TFEXxOiKbgguTPfnE5GPSDqusQLdnF9f+XfaIgd6zF2SvOMievWoeAmLjIGN7QcbFQ4+JheQvEaKO4XS2+njb226diTNnTuPEiWOorq4GAAwZMqxeci8tLcHZs0UYOGgwVHUOHn1sEZN/kGGyJ58kDxgMg8EATdOarTdwdAoqbpgOUVoCcbEUSmlJzT9nz0AcPghjyXmIhuP+RiP0qGjI2DjInrE1DwWx7n8S+wAxsdDj4iEju/nxGxIFP6UN++JbszYAAJ595km8/NILuOaaMfhsyx6vuTvl5WV4fcWf8ewzT+Jf/1yJ7DWfon//AR0XPPkVkz35pFu37rjuW1PxxeebmqxjMBjw7RkZ0JKSgcFNtyUcDogLNQ8BuFgK5eyZ2ocCXCiB4VQBTOfPAQ0eLGRYWM2DQM9YoGfNg4DnoQAxsTV/jkvgfv102WrPvvh73Udb33TTzY1O0o2KisaPf/JzfPNNDv71z5VY+JOH8O/M7PYFTJ2GyZ58lpFxO4rPFeHw4YNe1xRFwZzb7kJSUnKL7ciwMMjEPjVv7c0QFeWNPhCICyUQ54pgzM2BcqHUu/3uUdAT+3j1EsiY2JrhA3cZUahp6/G2DocD27d/DgCYMjW12bojRowEAHyy/iNUVJSjO5f5BQUme/KZ0WTCffc/jh3bN2PXzi9QVFSIsPBwDBk8HDdMm4l+/Tq2S092j4LWPQroPwDa1WMar+RwQKnbS1DngUApLYFyYJ9vvQR1Hgj0XnGQCb3ZS0DBx/Nm38rZ+Ht2f4nKygqYTCZMnDSl2bqnTp2s+Qwpua9GEGGyp1ZRFAUTJ92AiZNad6yu34SF1XTnt6aXoOQ8xMVSGIrO1O8luHgBaPDLS3aPqvdAIHv2gta7T/1egp6xPP2PuoS2duNv3fopAGDsuAktvql76o4ZMx5R3JY3aDDZ02WhXi9BU5WcTijlZTUPAO4Jhp4HAqW0BEr+ESjnixFmr6p/n8kEvXtUvQmGngcCvVdczXyC+ATIiEh/f026zNUeb9vKZL9l8yYAwJQpqc3W2/nlNhz6JgdCCCz55XNtCZEChMmeyMNkqhnLj+1VO8HQ2Ug1zwRDca6otpfAa4Lhp+sAXa93n2foQCb0ru0l8JpgGJ8IKNzFmtqm5njb8Fbtb+F0On0ar7dXVWHRwochhMDSZ37b6Nka1HUx2RO1kmeCIRL7NN1L4HJBKbtUr5eg3nyCUwVQdu+AqKyofx+XIVI71Jx417q3+nrj9RMne13XdR0bN3yMJUsW4cKFUry98j/IUG/tqJCpkzDZE/mD0ejVS9AYr2WIdfYlaNUyxIYPBFyGeFlqy4l3W7ZsAgAYjEbcecfsetc0TcO54rMYOHAwHnlkISyW+YiIrBmOyss7jKFDh3dI3OR/TPZEAdShyxBbmmBYZ18CLkMMTW1K9u7x+vnz78WLL73q0z2lpSWYPGkUpkxJRWLvPggPC/eqM33GLbglfVarYiH/YbInCgIdsgwxNwfG4nMQ1fb6bbewDBExsdATEiHDIzrhm1K7VJQD0T18rl53vH7yFN9X2OQeOojq6mp88snaRq/HxPTEo48/4XN75H9M9kShoh3LEGsfDFq5DLH2wSChN5chdgGiohx6nySf6+/etQOV7nkj10+a6vN9hw4dxG9+uwxz593ltdve4p89jttvv4Nd/F0Mkz3RZabDliE2ds4BlyEGVGu78T1r5ocNG4E+fX1/SCgoOI777nsQsQ2GgN5+63WMGDES02fc4nNb1DmY7InIG5chBiVRWdG6ZL+lJtlPnty6TbIaW2O/d+8ufPbZBqx4/e+taos6B5M9EbWZT8sQ4R46OHumdcsQ0fI5B3rvvq3eGjZUCYcDwuHwOdk7nU5s27YVQOvG6xvjcrnw3DNP4vU33m30EB0KPCZ7IvI72T0K2uBhXIboR6KV++Lv2rn9v+P1k7/drs9+7a9/wOgx47269anrYLInoi6hTcsQ23rOQVPLEIN4gqEv++KXlV3CkSO5OHe2CH/644sAao6m3rtnF4qKzmDIkGHo2TO2VZ/rcDjwysu/wTt/X9X24MnvmOyJKKj4NMHQ4YBSUf7f+QStWYbYrXvNBEP3fIKuvgyxvLwMO7ZvxokDX0Hv1QuJ+3ZifGLvRo+b3rlzO6wfvAcAGDN2PFJSrgMAbNjwMXRNg2Xuna1+y1+zxoqSkvO4ZlQTS0KpS2CyJ6LQExYGPcwzwXBYk9WCfRni4cMH8c7br8HuOZwpPByHvt6DLQf24js3pWP6DLVe/WnTpmPatOkdGkN21geIju7Bc+27OCZ7Irps+dRL0Mw5B8rZM00vQ/TzOQfnzhXh7Tf/Aoej2vt7SYn167LQs2cvXHud9373HWnP7i951G0QYLInImpOW885qLOtcasmGNaZT9DcBMNP1mc3mujr+uijD5AyYRIUPy5hLDh5AgOSB/qtfeoYTPZERB3ApwmGmgbl4gUoJcVAaQmU0vMQ7p9KyXng1AmY9u3ymksAgwG6uycAPWOhx8bh8Df7W4ypvOwSCgtPol+/Ae38dk2Lj0vAgAGD/NY+dQwmeyKizmIwQO8VVzPZrxmi2g6l+FztXIK6DwW4UApj3mFUhhl9mhNQ4Z6l7y/bv8yByRTm18+g9mOyJyLqYmR4BLR+yUC/5CbnEvR8YQlKSopbbKtXr/iODa4BTswLDtyLkogoCI0eM6HFOklJyYiPT+yEaKirY7InIgpCqdNmNpvIjUYjZt96RydGRF0Zkz0RURCKjOyGBQ8uxKBG9hGIiYnFvfc9ioGDhgYgMuqKOGZPRBSkYmJi8dDDP8XxY3k4diwPTqcDffv2x4grrobJZAp0eNSFMNkTEQW5gYOG8i2emsVufCIiohDHZE9ERBTimOyJiIhCHJM9ERFRiGOyJyIiCnFM9kRERCGOyZ6IiCjEMdkTERGFOCZ7IiKiEMdkT0REFOKY7ImIiEIckz0REVGIY7InIiIKcUz2REREIY7JnoiIKMQx2RMREYU4JnsiIqIQx2RPREQU4pjsiYiIQhyTPRERUYhjsiciIgpxTPZEREQhjsmeiIgoxDHZExERhTgmeyIiohDHZE9ERBTimOyJiIhCHJM9ERFRiGOyJyIiCnFM9kRERCGOyZ6IiCjEMdkTERGFOCZ7IiKiEMdkT0REFOKMgQ6gM3z99V5s37YZmqYhJWUiUiZMDHRIRETUSfKOHMK6ddkwmUwoLy/DPfc8iJ6xvQIdVqcK+WS/4ZM1KC4+h3u//zBcLid+sfhRREVF44qRVwc6NCIi8rMjh7/B2o+sePDhRTAYDPhozWq89teXsfgXzwU6tE4V0t3458+fQ15eLubOuxsGgwHh4REYNWocsrPfD3RoRETkZ1JKZGe/jwd++GMYDAYAwJVXjcKBA/tQVFQY4Og6V0gn++ys96GaLfXKYnrG4nDuQUgpAxQVERF1hpycrzBy5DUICwurLYuO7gGgpmv/chLSyf5CaQn69EmqVyalREVFOUpLzwcoKiIi6gz5Rw9j+Igr65U5HA4AQGHhqUCEFDAhnew1XfMqu3ChBABQUVHe2eEQEVEnqq6uru2+9/C86NntVYEIKWBCOtmHh4Xj0qWL9cpKS2r+RRuNpkCEREREnWT06PHY9sVn9cpKzhcDAMLDIwIRUsCERLKXUmLjxrX4x9//hpycr2rLZ82ehzffeBUOR3Vt2fET+RBCIPYyW3ZBRBTKtmzegBXLl+H1FX9ASUlNQh8+4ko4nU5s2byhtl5+/hEAQK+4+IDEGSghkew/zF6F8LBw3PidNPzm17/Ejh1bAQD9+g+AefZc/OXVF7Hzyy9QXHwWFy+UIjY2DhERkQGOmoiIOkJeXi5OnTqBBxb8CPaqKrz4+2drr9151/1wOKrx6p9+j9xDOTh69DAAoH+/AYEKNyCCPtk7HA7kHz2M6yenIiGhNwwGQ72nuEGDhuLRxxajsrICr/75fwHUPNFVVlYEKmQiIupAn6z/EOkZtwEAjCYTCgqO1Q7ZCiFw43fScM/3H0Jubg6OHP4GAHDmzGlUVVUGLObOFvTJfu/eL3HNqHEAgKIzp6FpGkwNxuMNBgO+fcNNSErqDwBISuqPdR9nNduulLK2u4eIiLqm6mo7KirKERUVDQA4WXAcBoMB3d1/9+jePQoDBgwGAAwddgXi4hNQXl5Wr46UEqdPFWD//j04fvxo7cz9UBD0yf7c2SKMHTsBAHDEvW5y8NDhjdb9ev9eADVj+bNmz2u23XUfZ2HZy7/uwEiJiKijlZeXITV1BgCgqqoSJ08eR3LyoHpr6z32798DALjuusm4+uoxSEjoXXstLy8Xy197BQcP7gekxP6vduOnT/wQGzeu7Zwv4mdBv11uesac2j8fOnQAADDyCu+tcE+fKkBJSTGGDR9Z719wY06cyMd7me+gW7fuHRssERF1qLi4BMTFJQAAcg/lQNd1XHnVqEbr7t+/BwaDAZMmfbteua7reOXl5zFv3j2YMvVGAMCo0eNx8eIFvPnGq4iPT8Qodw9ysAr6N/u6DubsR3h4BAYP8X6z37ZtMwDgxhtvbrYNh8OB7Kz3a3sLiIgoOHzzzdcAgJEjr/G6lp9/BKdPFSAlZaLXITjl5WUou3QRn2/dVK+8b99+kFLi+PGjfou5swT9m73H+fPnUFx8FqNGjfPaREHTNHz26XrExyfiWxOnNtvOe5krMWvWXO6fT0QUZL45+DWEEBhxxVVe1zZtWAshBMyz53pd69EjBr96fhl6xPSsV15QcBxA473FwSZk3uwPu2dYDhs+0uvavr07UVJSjDm3zYfJ1PRmOju//AJJSclI6pfstziJiKjj6bqOY8fy0KdvP3TvHlXvWmVlBb744jNcPzm1dpJeQ/2TB6JHj5javxcWnsLmzz5B2i2zG80rwSZk3uwvlNZsg9vYXvirP/gXrr56DCZPmdbk/SUlxfhq3y7cd/+jfo2TiIg6nt1eBU3TGp2TlWV9D+Hh4Zh/5w+abUPTNPwn8x2cO1eEgwf3497vP1Q7hh/sQibZe57IhBD1ytevy8bFixewcNFTXtc8pJTI/PdK3HnX/X6Pk4iIOl5kZDdERUV77Xmfl5eL9es/xM8WP1N74l1TDAYD5n73HmiahqN5ufjb3/6EY8fy8N07vg+jMbjTZch0448bfx3i4xOxaePHcDqdAIDt27dg/foPsfgXzyGmZ2yT91pX/xvfvuGm2nWaREQUXIQQuH3u95B35BC2bN4Au70KOTlfYcXyZVj0xP+0qiveYDBg+Igrcc+9D+LjtTb8893/82PknaPxV93O13P06HGHPt28O7E9jVy6dBEbN3yEkwXHERXdA3Fx8Zg+I6PZAw/y8nLx8Vobbk6bVa981fvv4mheLhb99GkANTvxNdUzQEREXUNh4Sns2rUNZZcuonfvvrh+cmqL26M7HNU4daoAgwcPq1eu6zruv+92CEXB8hX/8pr83dD4scOK8/PzEgHI9n6Pjhbc/RIN9OgR0+JmOQ0ZjUYkDxiEAwf21Ss/WXAcTqeztnzQoKEdFicREflH3779kOHeOtdXy//6Cnbs2IqFi57C2HHX1pZrmgZd1yE1DdXV9qDeeyWkkn1bDBw4BAMHDvEq37d3J3Rdb/V/NEREFFwiIiORnDwIAwbWn6l/4kQ+dF3HVVeNDupED4TQmH1HqaqqxOlTBSgtOY/KygqcOJHvtX8yERGFDovle4hPSERRUSGkrOmBv3TpIt5ZuQKJiX1w/4LHAxxh+3WVQegOGbPvCKvefxcVFeUIM9Xsq+x0OaHrOr5394IAR0ZERP7idDiwZetGHD92FLquobKiAiOuuAo3pE5vdt5XXRyzDyK3zrkj0CEQEVEnM4WFYdq0mYEOw2/YjU9ERBTimOyJiIhCHJM9ERFRiGOyJyIiCnFM9kRERCGOyZ6IiCjEdZmld9WOakN+fl6gwyAiImoTp9PRZV+gu8qmOt3i4xPei4jo5tvOBURERF1MZWVZZUlJSUag4yAiIiIiIiIiIiIiIiIiIiIiIiIiIqK2sFgsYRaLJSbQcRBR19FVlt4RUTupqjoFwP9IKc8DKFYUpUd4ePgjdrv9PgBbbDbb7gCHSEQB0qZNdVRVjQcwzGazbevgeIioDcxm88+klE8pinLr6tWr1wNAenr6VLvdvg7ARCHESwCY7IkuU23d7ectAMs6MhAiahuz2fyQlPK3AJ7wJHoAyM7O3gxAAoCUcnug4iOi9ktLSwtvz/2tfrNXVXUWgFuEECfb88FE1H7p6elXSilfAnAoJSVlhc1ma1hFAJAul2tjI7eLjIyMGxVFmQogWtf1U0KIVTabLd/fcROR72bPnj1I07T9ZrN5mtVq3dmWNlr1Zm+xWCIBvAwAUsreS5cu7bL7ABNdDhRFWQogQgjxxtKlS/W618xmczSAFAD716xZc67uNYvFEqaq6ltCiN6api2zWq1PKIpSAOCgqqo/7rQvQEQt0jTtZQBRuq4nt7WNViVru93+cwA73H81bd++Pa6tH0xE7TNz5sxeAOYAgBAiq+F1XdenAjAB2NDwmt1uTwMwF4ArOzu7FIC0Wq2ZAD4F8KKqqgP8GTsR+UZV1RkAZgOAoih92tqOz8l+9uzZQ4UQ06SUL3jKTCZT37Z+MBG1j8lkugE1Q3GFq1evzml4XQhxo/unVxe+ruslAMoAOOqWSyntABQhRKRfgiYin7nH6Z8FcBoApJRtzrk+j9m7XK6XhBCLhBCnPGWapvUF8FVbP5yI2k5RlAFSSgBoavLdNAC6pmmbASAtLS1hzZo1xQCke/JeQt3KM2bM6C6EmARgp9VqzfVj6ETkA6PR+ISU8k0hxDwASQDanOx9erM3m81mRVEKbDbblzab7TzcbwMGg6HNXQpE1D66rle6/3ik4bVbb701DsBYIcRX7m56GI3GFy0WS6PHSFsslsjw8PBlAAoVRZkD9yx+IgqMWbNmJQOYERkZuQJAobvYf2/2Fosl0m63/9JoNKa5iySAIgDJ7elSIKL2URQlR0oJIYS94TVN0x4DoEgpPwdqJuTZ7faIzMzMqrr1MjIyhiuKstButw8FkKwoymOrV68u6JxvQERN0XX9ZQCLMzMzNbPZXOj+f73NL9gtJvvq6urFUsq/rFq16nyd4kIAyWjHUwYRtc/48eO/2LVr1z4p5aS65aqq3q/repIQAgDOA4Ddbr9dCPFRwzaysrIOA3gIANLS0vobjcbNqqp+UVZWdvemTZtcnfA1iKgBs9k8XUp53rNxnZSy0P3TP934t9xyy0Ap5Q0TJkx4q8ElT5cCu/GJAmTp0qW6rutzAM1/kL8AAAXiSURBVCSqqvoHs9n8XVVVXweArKysBe5d89IyMjLuBnBbnz59VjbX3po1a04C+D2AO6Kjo7n8jigALBZLmJTyGQBLPGWeZA8g0WKxGNrSbrPJ3mAwLBNCLGy4flcIUej+6fNTRnp6+pCMjIw72xIkETUuOzv7aEpKylgAb+q6fqa6uvpHNpvtdQCwWq2LhBB3KIqSV1ZWNm/58uVOz31ms/kus9n8UCNNHnD/nNUJ4RNRA3a7/QkhxJs2m63YU+bJuQCM1dXV8W1pt8lu/PT09JlSygKbzban4bXWdiksWLDAVFhY+A8AZwH8vS2BElHj3A/jje57b7Vaj6CRCXxSytcBmCwWy8rMzMxyT7kQoq97bLDCbwETUaNmzZqVrOv6jPHjx99otVpry3VdL1SUmndzl8vVFzXz5poiZs6cGbt27dpS1Jlo22iyT01NjVAU5RUA3VRVvaWRKtHunz514xcWFj4NYDhqkj0RBd42AB/VTfQAIKWcgZpfEH8MSFRElzFd119UFGVxw950k8lUqGkaAMBgMPQFsLfhvenp6VcaDIbHpZQOAGdVVZ0EwAVgic1mO9Boso+KiloshPiN1WptOFYPAMjIyFCFEFYAURaLJarhL4wGdVOFEOeklJzhS9RFGI3GuS6X64+qqv5YCLFB0zSnEGIugNsBPGC1WrMDHSPR5cQ9Ke8mXdehqmq9a55EDwBSSq+XbFVVuwHI1jTNkp2dvctdLMxm86dSyk0ZGRljvZK9e5vMqVar9dmmglIUpdC9mQeqqqr6AjjcWL20tLQEIcTdVqv1B6qqfr/Fb0tEnWLVqlVnAcxTVfVaABMVRekF4LCu6wM96/KJqHO4l8Y+LaVMd7+ZezEYDJOklP0bmyunKMooXdcHK4ryYwDfcxdLXde3CCGmAkht7M1+mZTySTSzqYau64XuZT1QFKWpZC+MRuOvNU1rti0iChybzfYlgC8DHQfR5cxuty+UUr6VlZX1RVN1VFUtANC/sblyYWFhO6uqqhbouv5p3fI6u2x+XS/Zq6p6G4DirKysHWhGUlLS2cLCQh2Aout6o5P0VFV9VAjxwYcffnimubaIiIguV2lpaf0BpEVGRt7YQtUmV8FlZmZqAFbULXMvnZ8DYHlWVta+2mTvHi94Qwjxw5aCy83NNURHR1cDiFQUpX/D6+np6dcAiLdarZzkQ0RE1ASj0bhMSvlzd8JukhDCM3ze5MT4GTNmdA8PDzcDGAJgHoAnUlJS/mqz2WBUVfVlANdLKa8DACnlq6qqzrTZbF5j7Onp6TMNBsP9UsoUAJHu+j83m80TAey2Wq0vuGfyL4yIiHiwrV+eiIgo1Kmq+oQQYqDNZmuy+74Oz7r7Jpe8x8TEOCsrK3OFEMcACCHED3bv3n0cQLaYNWvWKACay+WyA4CUsocQojo7O/tgw4bMZnNvTdP6G43GKk99g8GgaJoWA6A0Ozv7qNlsfk7X9d5CiLwGt/8IwAUAbwkhzlmt1jd8+HJEREQhY/bs2YN0XZ8hpbwVwM0AqoQQLyiKsvKDDz441rC+2WyeI6UcLoR4VErZH4AmhHhK1/WjTqdz3dq1a0ua+ixVVX8O4NcA5hhXr16939cgrVZrEZpfzA+Xy/W6ECLeM1vfQ1GUp4UQRzRNW280GquauJ2IiChkuVyuvkKIWClllqIoqwBESymNTqczqmFdi8USVl1dPQwApJS/E0JU67quAIgRQgzq1q2bsW7dzMzMejP5DQZDpqZpLwB4Vvj5e9VSVbUcwEabzaa2WJmIiIh8YjabM6WUNwMYbbPZ8j3lGRkZ/YQQJwFUtnjqXXssXbpU2bVr1zwAgwF0AzA+IyPjEUVRcqxW60Z/fjYREdHlQEo5EECVyWSqd9y1oihXuXvZP/NrsgcAIYQLwG4hxAwA0DQt0t+fSUREdBl5HsCsqqqq2jMtUlNTI6SUSwCc1TTtR53WjU9ERET+kZ6enqIoygIhhC6lNAIYAeBrIcTzVqv19P8DmITywrfyjBsAAAAASUVORK5CYII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" 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" + } + }, + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## 1. Π‘Π»ΡƒΡ‡Π°ΠΉΠ½ΠΎ Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Π΅ Ρ‚ΠΎΡ‡ΠΊΠΈ Π² Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΌ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ΅\n", + "\n", + "ΠŸΡƒΡΡ‚ΡŒ $K$ - Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΉ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ с Π²Π΅Ρ€ΡˆΠΈΠ½Π°ΠΌΠΈ $A_1,\\:A_2,\\:\\ldots,\\:A_r$ ΠΈ ΡΠΎΠΎΡ‚Π²Π΅Ρ‚ΡΡ‚Π²ΡƒΡŽΡ‰ΠΈΠΌΠΈ ΡƒΠ³Π»Π°ΠΌΠΈ $\\vartheta_1,\\:\\vartheta_2,\\:\\ldots,\\:\\vartheta_r$. ΠžΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Π·Π° $a_k$ Π΄Π»ΠΈΠ½Ρƒ стороны $A_k A_{k+1}$, Π³Π΄Π΅ $A_{r+1} = A_1$ $(k = 1,\\:2,\\:\\ldots,\\:r)$.\n", + "\n", + "![1.png](attachment:1.png)\n", + "\n", + "ΠŸΡƒΡΡ‚ΡŒ $\\varepsilon_{ij} = 1$, Ссли ΠΎΡ‚Ρ€Π΅Π·ΠΎΠΊ $P_i P_j$ ΠΏΡ€ΠΈ $i \\neq j$ ΠΏΡ€ΠΈΠ½Π°Π΄Π»Π΅ΠΆΠΈΡ‚ Π³Ρ€Π°Π½ΠΈΡ†Π΅ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ $H_n$, ΠΈΠ½Π°Ρ‡Π΅ $\\varepsilon_{ij} = 0$. Π’ΠΎΠ³Π΄Π° ΠΎΡ‡Π΅Π²ΠΈΠ΄Π½ΠΎ, Ρ‡Ρ‚ΠΎ\n", + "\n", + "$$X_n = \\sum_{1 \\le i \\lt j \\le n}\\varepsilon_{ij} \\label{eq:1} \\tag{1}$$\n", + "\n", + "Π’ силу случайного распрСдСлСния Π²Π΅Ρ€ΠΎΡΡ‚Π½ΠΎΡΡ‚ΡŒ $W_n = P(\\varepsilon_{ij} = 1)$ ΠΎΠ΄ΠΈΠ½Π°ΠΊΠΎΠ²Π° для всСх ΠΏΠ°Ρ€ $(i, j)$, Ρ‚Π°ΠΊΠΈΡ…, Ρ‡Ρ‚ΠΎ $1 \\le i \\lt j \\leq n$, ΠΈ Ρ‚ΠΎΠ³Π΄Π°\n", + "\n", + "$$E_n = E(X_n) = \\sum_{1 \\le i \\lt j \\le n} E(\\varepsilon_{ij}) = \\sum_{1 \\le i \\lt j \\le n} \\left( 0 \\cdot (1 - W_n) + 1 \\cdot W_n \\right) = C_n^2 \\cdot W_n \\label{eq:2} \\tag{2}$$\n", + "\n", + "Π§Ρ‚ΠΎΠ±Ρ‹ ΠΎΠΏΡ€Π΅Π΄Π΅Π»ΠΈΡ‚ΡŒ асимптотичСскоС ΠΏΠΎΠ²Π΅Π΄Π΅Π½ΠΈΠ΅ $E_n$, достаточно Π²Ρ‹Ρ‡ΠΈΡΠ»ΠΈΡ‚ΡŒ Π²Π΅Ρ€ΠΎΡΡ‚Π½ΠΎΡΡ‚ΡŒ $W_n$. МоТСм Ρ€Π°ΡΡΠΌΠΎΡ‚Ρ€Π΅Ρ‚ΡŒ $W_n$ ΠΊΠ°ΠΊ Π²Π΅Ρ€ΠΎΡΡ‚Π½ΠΎΡΡ‚ΡŒ Ρ‚ΠΎΠ³ΠΎ, Ρ‡Ρ‚ΠΎ всС Ρ‚ΠΎΡ‡ΠΊΠΈ $P_h \\: (h = 3,\\:4,\\:\\ldots,\\:n)$ Π»Π΅ΠΆΠ°Ρ‚ ΠΏΠΎ ΠΎΠ΄Π½Ρƒ сторону ΠΎΡ‚ прямой $P_1 P_2$. \n", + "\n", + "![2.png](attachment:2.png)\n", + "\n", + "ΠŸΡƒΡΡ‚ΡŒ $F$ ΠΎΠ±ΠΎΠ·Π½Π°Ρ‡Π°Π΅Ρ‚ ΠΏΠ»ΠΎΡ‰Π°Π΄ΡŒ $K$. Если Ρ‡Π΅Ρ€Π΅Π· $F_1 \\le \\frac12 F$ ΠΈ $F_2 = F - F_1$ ΠΎΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΡ‚ΡŒ части, Π½Π° ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Π΅ прямая $P_1 P_2$ Π΄Π΅Π»ΠΈΡ‚ ΠΎΠ±Π»Π°ΡΡ‚ΡŒ $K$, Ρ‚ΠΎ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ\n", + "\n", + "$$W_n = \\frac1{F^2} \\int\\limits_K \\int\\limits_K \\left[ \\left(1 - \\frac{F_1}{F}\\right)^{n-2} + \\left(\\frac{F_1}{F}\\right)^{n-2}\\right] dP_1 dP_2 \\label{eq:3} \\tag{3}$$\n", + "\n", + "ΠΊΠ°ΠΊ Π²Π΅Ρ€ΠΎΡΡ‚Π½ΠΎΡΡ‚ΡŒ Ρ‚ΠΎΠ³ΠΎ, Ρ‡Ρ‚ΠΎ всС Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i\\:(i=3,\\:4,\\:\\ldots,\\:n)$ ΠΏΡ€ΠΈΠ½Π°Π΄Π»Π΅ΠΆΠ°Ρ‚ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ ΠΎΠ΄Π½ΠΎΠΉ ΠΈΠ· частСй $F$.\n", + "\n", + "Π’Π°ΠΊ ΠΊΠ°ΠΊ $F_1/F \\le \\frac12$, ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π» Π²Ρ‚ΠΎΡ€ΠΎΠ³ΠΎ слагаСмого ΠΎΠ³Ρ€Π°Π½ΠΈΡ‡Π΅Π½ константой:\n", + "\n", + "$$\\frac1{F^2} \\int\\limits_K \\int\\limits_K \\left(\\frac{F_1}{F}\\right)^{n-2} dP_1 dP_2 \\le \\frac1{2^{n - 2}} \\label{eq:4} \\tag{4}$$\n", + "\n", + "А ΡΠ»Π΅Π΄ΠΎΠ²Π°Ρ‚Π΅Π»ΡŒΠ½ΠΎ,\n", + "\n", + "$$E_n \\sim C_n^2 \\cdot \\frac1{F^2} \\int\\limits_K \\int\\limits_K \\left(1 - \\frac{F_1}{F}\\right)^{n-2} dP_1 dP_2 \\label{eq:5} \\tag{5}$$\n", + "\n", + "$$\\text{Π³Π΄Π΅} \\: A \\sim B \\Leftrightarrow \\lim_{n\\to+\\infty} \\frac{A_n}{B_n} = 1 \\notag$$\n", + "\n", + "\n", + "ΠžΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Ρ‡Π΅Ρ€Π΅Π· $f_i$ ΠΏΠ»ΠΎΡ‰Π°Π΄ΡŒ Ρ‚Ρ€Π΅ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $A_{i-1}A_iA_{i+1}$ $(A_{r+j} = A_j)$, ΠΈ ΠΏΡƒΡΡ‚ΡŒ $f = \\min\\limits_{1 \\le i \\le r} f_i$\n", + "\n", + "![3.png](attachment:3.png)\n", + "\n", + "ΠžΡ‡Π΅Π²ΠΈΠ΄Π½ΠΎ, Ρ‡Ρ‚ΠΎ ΠΏΡ€ΠΈ ΠΈΠ·ΡƒΡ‡Π΅Π½ΠΈΠΈ асимптотичСского повСдСния $E_n$ ΠΌΠΎΠΆΠ½ΠΎ ΠΎΠ³Ρ€Π°Π½ΠΈΡ‡ΠΈΡ‚ΡŒΡΡ ΠΏΠ°Ρ€Π°ΠΌΠΈ Ρ‚ΠΎΡ‡Π΅ΠΊ $(P_1,\\:P_2)$, для ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… прямая $P_1 P_2$ пСрСсСкаСтся Π»ΠΈΠ±ΠΎ с двумя сосСдними сторонами $K$, Π»ΠΈΠ±ΠΎ с двумя сторонами $K$, Ρ€Π°Π·Π΄Π΅Π»Ρ‘Π½Π½Ρ‹ΠΌΠΈ Ρ‚Ρ€Π΅Ρ‚ΡŒΠ΅ΠΉ стороной. Для всСх ΠΎΡΡ‚Π°Π»ΡŒΠ½Ρ‹Ρ… прямых выполняСтся ΠΎΠ³Ρ€Π°Π½ΠΈΡ‡Π΅Π½ΠΈΠ΅\n", + "\n", + "$$1 - \\frac{F_1}{F} \\leqq 1 - \\frac{f}{F} \\notag$$\n", + "\n", + "ΠŸΠΎΡΡ‚ΠΎΠΌΡƒ ΠΌΡ‹ ΠΌΠΎΠΆΠ΅ΠΌ ΠΏΡ€Π΅Π½Π΅Π±Ρ€Π΅Ρ‡ΡŒ ΡΠΎΠΎΡ‚Π²Π΅Ρ‚ΡΡ‚Π²ΡƒΡŽΡ‰Π΅ΠΉ Ρ‡Π°ΡΡ‚ΡŒΡŽ ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π»Π° ΠΈ Π·Π°ΠΏΠΈΡΠ°Ρ‚ΡŒ \n", + "\n", + "$$E_n \\sim C_n^2 \\cdot \\frac1{F^2} \\left( \\sum\\limits_{i=1}^r (I_i + J_i) \\right) \\label{eq:6} \\tag{6}$$ \n", + "\n", + "$$I_i = \\int\\limits_{C_i} \\left(1 - \\frac{F_1}{F}\\right)^{n-2} dP_1 dP_2 \\label{eq:7} \\tag{7}$$ \n", + "\n", + "$$J_i = \\int\\limits_{D_i} \\left(1 - \\frac{F_1}{F}\\right)^{n-2} dP_1 dP_2 \\label{eq:8} \\tag{8}$$\n", + "\n", + "Π—Π΄Π΅ΡΡŒ $C_i$ ΠΎΠ·Π½Π°Ρ‡Π°Π΅Ρ‚ мноТСство ΠΏΠ°Ρ€ Ρ‚ΠΎΡ‡Π΅ΠΊ $(P_1,\\:P_2)$, для ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… прямая $P_1P_2$ пСрСсСкаСт смСТныС стороны $A_{i-1}A_i$ ΠΈ $A_iA_{i+1}$ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$, a $D_i$ - мноТСство ΠΏΠ°Ρ€ $(P_1,\\:P_2)$, для ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… прямая $P_1P_2$ пСрСсСкаСт стороны $A_{i-1}A_i$ ΠΈ $A_{i+1}A_{i+2}$ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$, Ρ€Π°Π·Π΄Π΅Π»Ρ‘Π½Π½Ρ‹Π΅ Π΄Ρ€ΡƒΠ³ΠΎΠΉ стороной.\n", + "\n", + "Π‘Π½Π°Ρ‡Π°Π»Π° рассмотрим ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π» $I_i$. ΠžΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Ρ‡Π΅Ρ€Π΅Π· $G_{ab}$ мноТСство ΠΏΠ°Ρ€ $(P_1,\\:P_2)$, для ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… Π²Ρ‹ΠΏΠΎΠ»Π½Π΅Π½Ρ‹ нСравСнства $|A_iQ_1| < a$ ΠΈ $|A_iQ_2| < b$, Π³Π΄Π΅ $Q_1$ - Ρ‚ΠΎΡ‡ΠΊΠ° пСрСсСчСния прямой $P_1P_2$ со стороной $A_{i-1}A_i$, Π° $Q_2$ - Ρ‚ΠΎΡ‡ΠΊΠ° пСрСсСчСния $P_1P_2$ со стороной $A_iA_{i+1}$. Π—Π΄Π΅ΡΡŒ ΠΈ Π΄Π°Π»Π΅Π΅ $|AB|$ -- Π΄Π»ΠΈΠ½Π° ΠΎΡ‚Ρ€Π΅Π·ΠΊΠ° $AB$.\n", + "\n", + "По элСмСнтарным расчётам получаСтся \n", + "\n", + "$$\\int\\limits_{G_{ab}} dP_1dP_2 = \\frac{a^2b^2\\sin^2\\vartheta_i}{12} \\label{eq:9} \\tag{9}$$\n", + "\n", + "ΠŸΠ»ΠΎΡ‰Π°Π΄ΡŒ Ρ‚Ρ€Π΅ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $Q_1A_iQ_2$, отсСкаСмого прямой $P_1P_2$, Ρ€Π°Π²Π½Π° $\\frac{1}{2} a b \\sin \\vartheta_i$. Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, взяв $a_{i-1} = |A_{i-1}A_i|$, $a_i = |A_iA_{i+1}|$, ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ \n", + "\n", + "$$I_i = \\int\\limits_0^{a_{i-1}} \\int\\limits_0^{a_i} \\left(1 - \\frac{ab \\sin \\vartheta_i}{2F}\\right)^{n-2} \\sin^2 \\vartheta_i \\frac{ab}{3} \\,da\\,db \\label{eq:10} \\tag{10}$$ \n", + "\n", + "Π’Ρ‹ΠΏΠΎΠ»Π½ΠΈΠΌ Π·Π°ΠΌΠ΅Π½Ρƒ \n", + "\n", + "$$X = a \\sqrt\\frac{\\sin \\vartheta_i}{2F}, \\; Y = b \\sqrt\\frac{\\sin \\vartheta_i}{2F} \\label{eq:11} \\tag{11}$$ \n", + "\n", + "ΠΈ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ\n", + "\n", + "$$I_i = \\frac{4F^2}{3} \\int\\limits_0^{X_i} \\int\\limits_0^{Y_i} (1-XY)^{n-2} XY \\: dX dY \\label{eq:12} \\tag{12}$$\n", + "\n", + "Π³Π΄Π΅\n", + "\n", + "$$X_i = a_{i-1} \\sqrt\\frac{\\sin \\vartheta_i}{2F}, \\; Y_i = a_i \\sqrt\\frac{\\sin \\vartheta_i}{2F} \\notag$$\n", + "\n", + "ΠΈ Ρ‚Π°ΠΊΠΆΠ΅ Π·Π°ΠΌΠ΅Ρ‚ΠΈΠΌ, Ρ‡Ρ‚ΠΎ\n", + "\n", + "$$\\varrho_i = X_iY_i = \\frac{a_{i-1}a_i\\sin \\vartheta_i}{2F} = \\frac{f_i}{F} < 1 \\label{eq:13} \\tag{13}$$\n", + "\n", + "Π’Π΅ΠΏΠ΅Ρ€ΡŒ ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΡƒΠ΅ΠΌ ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π»:\n", + "\n", + "$$\\int\\limits_0^{X_i} \\int\\limits_0^{Y_i} (1-XY)^{n-2} XY \\: dX dY = \\int\\limits_0^{X_i} \\int\\limits_0^{Y_i} (1-XY)^{n-2} \\: dX dY - \\int\\limits_0^{X_i} \\int\\limits_0^{Y_i} (1-XY)^{n-1} \\: dX dY \\notag$$\n", + "\n", + "Π”Π°Π»Π΅Π΅ ΠΏΡ€ΠΈΠΌΠ΅Π½ΠΈΠΌ \n", + "\n", + "$$\\int\\limits_0^{X_i} \\int\\limits_0^{Y_i} (1-XY)^{n-1} \\: dX dY = \\int\\limits_0^{X_i} \\frac{1 - (1 - XY_i)^n}{nX} \\: dX = \\\\ = \\frac1n \\left( 1 + \\frac12 + \\ldots + \\frac1n - \\sum\\limits_{k=1}^n \\frac{(1-\\varrho_i)^k}{k} \\right) \\notag$$\n", + "\n", + "Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, ΠΌΡ‹ ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ Π·Π½Π°Ρ‡Π΅Π½ΠΈΠ΅ ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π»Π° $I_i$:\n", + "\n", + "$$I_i = \\frac{4F^2}{3} \\left[ \\frac{\\cfrac12 + \\cfrac13 + \\ldots + \\cfrac1n}{n(n-1)} - \\frac{\\sum\\limits_{k=1}^{n-1} \\cfrac{(1 - \\varrho_i)^k}{k}}{n(n-1)} + \\frac{(1-\\varrho_i)^n}{n^2} \\right] \\label{eq:14} \\tag{14}$$\n", + "\n", + "Если ΠΏΠΎΠ»ΠΎΠΆΠΈΠΌ \n", + "\n", + "$$S_i = \\sum\\limits_{k=1}^\\infty \\frac{(1 - \\varrho_i)^k}{k} = \\log \\frac1{\\varrho_i} \\label{eq:15} \\tag{15}$$\n", + "\n", + "Ρ‚ΠΎ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ Π² ΠΊΠΎΠ½Π΅Ρ‡Π½ΠΎΠΌ ΠΈΡ‚ΠΎΠ³Π΅\n", + "\n", + "$$E_n = \\frac23 (\\gamma - 1 + \\log n) r - \\frac23 \\sum\\limits_{i=1}^r S_i + o(1) + \\frac{C_n^2}{F^2} \\sum\\limits_{i=1}^r J_i \\label{eq:16} \\tag{16}$$ \n", + "\n", + "Π³Π΄Π΅ $\\gamma$ - постоянная Π­ΠΉΠ»Π΅Ρ€Π°.\n", + "\n", + "Π’Π΅ΠΏΠ΅Ρ€ΡŒ вычислим ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π» $J_i$. ΠŸΡ€ΠΎΠ΄Π»ΠΈΠΌ стороны $A_{i-1}A_i$ ΠΈ $A_{i+1}A_{i+2}$ Π΄ΠΎ Ρ‚ΠΎΡ‡ΠΊΠΈ ΠΈΡ… пСрСсСчСния $A_i^*$; случай, ΠΊΠΎΠ³Π΄Π° эти стороны ΠΏΠ°Ρ€Π°Π»Π»Π΅Π»ΡŒΠ½Ρ‹, ΠΌΠΎΠΆΠ½ΠΎ Ρ€Π°ΡΡΠΌΠ°Ρ‚Ρ€ΠΈΠ²Π°Ρ‚ΡŒ Π°Π½Π°Π»ΠΎΠ³ΠΈΡ‡Π½ΠΎ.\n", + "\n", + "ΠŸΡƒΡΡ‚ΡŒ снова $Q_1$ - Ρ‚ΠΎΡ‡ΠΊΠ° пСрСсСчСния прямой $P_1P_2$ со стороной $A_{i-1}A_i$, Π° $Q_2$ - Ρ‚ΠΎΡ‡ΠΊΠ° пСрСсСчСния $P_1P_2$ со стороной $A_{i+1}A_{i+2}$, ΠΈ ΠΏΡƒΡΡ‚ΡŒ\n", + "\n", + "$$a' = |A_iA_i^*|, \\;\\; b' = |A_{i+1}A_i^*|, \\;\\; a = |A_iQ_1|, \\;\\; b = |A_{i+1}Q_2| \\notag$$\n", + "\n", + "$G_{ab}$ Π±Ρ‹Π»ΠΎ ΠΎΠΏΡ€Π΅Π΄Π΅Π»Π΅Π½ΠΎ Π²Ρ‹ΡˆΠ΅. ΠŸΠΎΠ»ΡƒΡ‡ΠΈΠΌ ΠΈΠ· $(9)$:\n", + "\n", + "$$\\int\\limits_{G_{ab}} dP_1 dP_2 = \\frac{\\sin^2\\beta_i}{12} (a^2 + 2aa')(b^2 + 2bb') \\label{eq:17} \\tag{17}$$ \n", + "\n", + "Π³Π΄Π΅ $\\beta_i$ - ΡƒΠ³ΠΎΠ» ΠΏΡ€ΠΈ $A_i^*$. Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ,\n", + "\n", + "$$J_i = \\int\\limits_{D_i} \\left(1 - \\frac{(ab+a'b+ab')\\sin \\beta_i}{2F} \\right)^{n-2} \\frac{\\sin^2 \\beta_i}{3} (a+a')(b+b')\\,da\\,db \\label{eq:18} \\tag{18}$$\n", + "\n", + "ΠΈ этот ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π» асимптотичСски эквивалСнтСн\n", + "\n", + "$$J_i \\sim \\int\\limits_{D_i} \\left(1 - \\frac{(a'b+ab')\\sin \\beta_i}{2F} \\right)^{n-2} \\frac{\\sin^2 \\beta_i}{3} a'b'\\,da\\,db \\label{eq:19} \\tag{19}$$\n", + "\n", + "Π”Π°Π»Π΅Π΅ сдСлаСм Π·Π°ΠΌΠ΅Π½Ρƒ\n", + "\n", + "$$a = \\frac{F}{b' \\sin \\beta_i} X, \\;\\; b = \\frac{F}{a' \\sin \\beta_i} Y \\notag$$\n", + "\n", + "ΠΈ Π½Π°ΠΉΠ΄Ρ‘ΠΌ Π·Π½Π°Ρ‡Π΅Π½ΠΈΠ΅ ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π»Π°\n", + "\n", + "$$J_i \\sim \\frac{F^2}{3} \\int\\limits_0^{a_{i-1}'} \\int\\limits_0^{a_{i+1}'} \\left( 1 - \\frac{X+Y}{2} \\right)^{n-2} dX\\,dY = \\\\ = \\frac{4F^2}{3n(n-1)} \\left[ 1 - \\left( 1 - \\frac{a_{i-1}'}{2} \\right)^n - \\left( 1 - \\frac{a_{i+1}'}{2} \\right)^n + \\left( 1 - \\frac{a_{i-1}' + a_{i+1}'}{2} \\right)^n \\right] \\notag$$\n", + "\n", + "Π³Π΄Π΅ \n", + "\n", + "$$a_{i-1}' = \\frac{a_ib'\\sin\\beta_i}{F}, \\;\\; a_{i+1}' = \\frac{a_{i+2}a'\\sin\\beta_i}{F} \\notag$$\n", + "\n", + "Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ ΠΈΡ‚ΠΎΠ³ΠΎΠ²Ρ‹ΠΉ Ρ€Π΅Π·ΡƒΠ»ΡŒΡ‚Π°Ρ‚:\n", + "\n", + "$$E_n = \\frac{2r}{3} (\\log n + \\gamma) - \\frac23 \\sum\\limits_{i=1}^{r} S_i + o(1) \\notag$$\n", + "\n", + "**Π£Ρ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 1.** ΠŸΡƒΡΡ‚ΡŒ $K$ - Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΉ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ с $r$ сторонами. ΠœΠ°Ρ‚Π΅ΠΌΠ°Ρ‚ΠΈΡ‡Π΅ΡΠΊΠΎΠ΅ ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ $E_n$ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ $H_n$, построСнной Π½Π° $n$ случайных Ρ‚ΠΎΡ‡ΠΊΠ°Ρ… ΠΈΠ· $K$, Ρ€Π°Π²Π½ΠΎ\n", + "\n", + "$$E_n = \\frac{2r}{3} (\\log n + \\gamma) + \\frac23 \\log \\frac{\\prod\\limits_1^r f_i}{F^r} + o(1) \\label{eq:20} \\tag{20}$$\n", + "\n", + "Π—Π΄Π΅ΡΡŒ $F$ ΠΎΠ±ΠΎΠ·Π½Π°Ρ‡Π°Π΅Ρ‚ ΠΏΠ»ΠΎΡ‰Π°Π΄ΡŒ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$, Π° $f_i$ - ΠΏΠ»ΠΎΡ‰Π°Π΄ΡŒ Ρ‚Ρ€Π΅ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $A_{i-1}A_iA_{i+1}$.\n", + "\n", + "ΠŸΡƒΡΡ‚ΡŒ, Π½Π°ΠΏΡ€ΠΈΠΌΠ΅Ρ€, $K$ - Ρ‚Ρ€Π΅ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ, Ρ‚ΠΎΠ³Π΄Π° $f_1 = f_2 = f_3 = F$ ΠΈ \n", + "\n", + "$$E_n = 2(\\log n + \\gamma) + o(1) \\label{eq:21} \\tag{21}$$\n", + "\n", + "Π—Π°ΠΌΠ΅Ρ‚ΠΈΠΌ, Ρ‡Ρ‚ΠΎ Π² Ρ‚Π°ΠΊΠΎΠΌ случаС Ρ„ΠΎΡ€ΠΌΡƒΠ»Π° для $E_n$ Π½Π΅ зависит ΠΎΡ‚ Ρ„ΠΎΡ€ΠΌΡ‹ Ρ‚Ρ€Π΅ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ°, Ρ‚Π°ΠΊ ΠΊΠ°ΠΊ $E_n$, ΠΎΡ‡Π΅Π²ΠΈΠ΄Π½ΠΎ, Π΄ΠΎΠ»ΠΆΠ½ΠΎ Π±Ρ‹Ρ‚ΡŒ ΠΈΠ½Π²Π°Ρ€ΠΈΠ°Π½Ρ‚Π½ΠΎ ΠΊ Π°Ρ„Ρ„ΠΈΠ½Π½Ρ‹ΠΌ прСобразованиям. Π’ $\\S2$ ΠΌΡ‹ ΠΏΠΎΠΊΠ°ΠΆΠ΅ΠΌ, Ρ‡Ρ‚ΠΎ константа \n", + "\n", + "$$Z(K) = \\frac{\\prod\\limits_{i=1}^r f_i}{F^r} \\label{eq:22} \\tag{22}$$\n", + "\n", + "которая зависит ΠΎΡ‚ Ρ„ΠΎΡ€ΠΌΡ‹ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$ ΠΈ появляСтся Π² Π²Ρ‹Ρ€Π°ΠΆΠ΅Π½ΠΈΠΈ $(\\ref{eq:20})$, становится максимальной для ΠΏΡ€Π°Π²ΠΈΠ»ΡŒΠ½ΠΎΠ³ΠΎ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° ΠΈ Π°Ρ„Ρ„ΠΈΠ½Π½Ρ‹Ρ… ΠΊ Π½Π΅ΠΌΡƒ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΎΠ²." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## 2. Π­ΠΊΡΡ‚Ρ€Π΅ΠΌΠ°Π»ΡŒΠ½Π°Ρ Π·Π°Π΄Π°Ρ‡Π° для Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹Ρ… ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΎΠ²\n", + "\n", + "ΠŸΡ€Π΅ΠΆΠ΄Π΅ Ρ‡Π΅ΠΌ ΠΏΠ΅Ρ€Π΅ΠΉΡ‚ΠΈ ΠΊ дальнСйшим асимптотичСским ΠΎΡ†Π΅Π½ΠΊΠ°ΠΌ для $E_n$ ΠΏΡ€ΠΈ Π΄Ρ€ΡƒΠ³ΠΈΡ… прСдполоТСниях ΠΎ распрСдСлСнии Ρ‚ΠΎΡ‡Π΅ΠΊ $P_i$, Π΄ΠΎΠΊΠ°ΠΆΠ΅ΠΌ ΡΠ»Π΅Π΄ΡƒΡŽΡ‰Π΅Π΅ ΡƒΡ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅:\n", + "\n", + "**Π£Ρ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 2.** ΠŸΡƒΡΡ‚ΡŒ $K$ - Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΉ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ с $r$ сторонами. Ѐункция \n", + "\n", + "$$Z(K) = \\frac1{F^r} \\prod\\limits_{i=1}^r f_i \\label{eq:23} \\tag{23}$$\n", + "\n", + "ΠΈΠ· ΠΏΡ€Π΅Π΄Ρ‹Π΄ΡƒΡ‰Π΅Π³ΠΎ ΠΏΠ°Ρ€Π°Π³Ρ€Π°Ρ„Π° максимальна Ρ‚ΠΎΠ³Π΄Π° ΠΈ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ Ρ‚ΠΎΠ³Π΄Π°, ΠΊΠΎΠ³Π΄Π° $K$ ΠΌΠΎΠΆΠ½ΠΎ ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΠΎΠ²Π°Ρ‚ΡŒ Π² ΠΏΡ€Π°Π²ΠΈΠ»ΡŒΠ½Ρ‹ΠΉ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ Π°Ρ„Ρ„ΠΈΠ½Π½Ρ‹ΠΌΠΈ прСобразованиями.\n", + "\n", + "**Π”ΠΎΠΊΠ°Π·Π°Ρ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ:**\n", + "ΠŸΡƒΡΡ‚ΡŒ $\\alpha_i = \\overrightarrow{A_{i-1}A_i}$, Π³Π΄Π΅ $A_{r+i} = A_i$. Π’ΠΎΠ³Π΄Π°\n", + "\n", + "$$f_i = \\frac12 [\\alpha_i, \\: \\alpha_{i+1}] \\\\ F = \\frac12 \\sum\\limits_{i=1}^{r-2} [\\alpha_1 + \\alpha_2 + \\ldots + \\alpha_i, \\: \\alpha_{i+1}] \\label{eq:24} \\tag{24}$$\n", + "\n", + "Π³Π΄Π΅ $[\\alpha, \\: \\beta]$ - ΠΌΠΎΠ΄ΡƒΠ»ΡŒ Π²Π΅ΠΊΡ‚ΠΎΡ€Π½ΠΎΠ³ΠΎ произвСдСния $\\alpha$ ΠΈ $\\beta$. Ѐункция $Z(K)$ Ρ‚Π΅ΠΏΠ΅Ρ€ΡŒ являСтся Ρ„ΡƒΠ½ΠΊΡ†ΠΈΠ΅ΠΉ ΠΎΡ‚ $r$ Π²Π΅ΠΊΡ‚ΠΎΡ€ΠΎΠ² $\\alpha_i \\: (i=1, \\: \\ldots, \\: r)$, ΡƒΠ΄ΠΎΠ²Π»Π΅Ρ‚Π²ΠΎΡ€ΡΡŽΡ‰ΠΈΡ… ΡƒΡΠ»ΠΎΠ²ΠΈΡŽ замкнутости:\n", + "\n", + "$$\\sum\\limits_{i=1}^r \\alpha_i = 0 \\label{eq:25} \\tag{25}$$\n", + "\n", + "Π’Π΅ΠΏΠ΅Ρ€ΡŒ допустим, Ρ‡Ρ‚ΠΎ Π²Π΅Ρ€ΡˆΠΈΠ½Ρ‹ $A_i$ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$ ΠΏΠΎΠ΄Π²Π΅Ρ€Π³Π°ΡŽΡ‚ΡΡ достаточно ΠΌΠ°Π»Ρ‹ΠΌ смСщСниям, ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Π΅ ΠΎΠΏΠΈΡΡ‹Π²Π°ΡŽΡ‚ΡΡ Π½Π΅ΠΏΡ€Π΅Ρ€Ρ‹Π²Π½ΠΎ Π΄ΠΈΡ„Ρ„Π΅Ρ€Π΅Π½Ρ†ΠΈΡ€ΡƒΠ΅ΠΌΠΎΠΉ Ρ„ΡƒΠ½ΠΊΡ†ΠΈΠ΅ΠΉ ΠΎΡ‚ ΠΏΠ°Ρ€Π°ΠΌΠ΅Ρ‚Ρ€Π° $t$. Π’Π°ΠΊ ΠΊΠ°ΠΊ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ $K$, ΠΊΠΎΡ‚ΠΎΡ€Ρ‹ΠΉ Π΄ΠΎΠ»ΠΆΠ΅Π½ ΡΠΎΠΎΡ‚Π²Π΅Ρ‚ΡΡ‚Π²ΠΎΠ²Π°Ρ‚ΡŒ Π·Π½Π°Ρ‡Π΅Π½ΠΈΡŽ ΠΏΠ°Ρ€Π°ΠΌΠ΅Ρ‚Ρ€Π° $t = 0$, являСтся Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΌ r-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΎΠΌ, Ρ‚ΠΎ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΈ, ΡΠ²Π»ΡΡŽΡ‰ΠΈΠ΅ΡΡ Ρ€Π΅Π·ΡƒΠ»ΡŒΡ‚Π°Ρ‚ΠΎΠΌ достаточно ΠΌΠ°Π»ΠΎΠ³ΠΎ измСнСния $t$, Ρ‚Π°ΠΊΠΆΠ΅ Π±ΡƒΠ΄ΡƒΡ‚ Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΌΠΈ r-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ°ΠΌΠΈ. Ѐункция $Z$, рассматриваСмая ΠΊΠ°ΠΊ функция ΠΎΡ‚ $t$, Π΄ΠΎΠ»ΠΆΠ½Π° Π΄ΠΎΡΡ‚ΠΈΠ³Π°Ρ‚ΡŒ максимума ΠΏΡ€ΠΈ $t = 0$, Ссли $K$ - ΠΌΠ°ΠΊΡΠΈΠΌΠ°Π»ΡŒΠ½Ρ‹ΠΉ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ. НСобходимоС условиС Ρ‚ΠΎΠ³ΠΎ, Ρ‡Ρ‚ΠΎ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ $K$ являСтся ΠΌΠ°ΠΊΡΠΈΠΌΠ°Π»ΡŒΠ½Ρ‹ΠΌ, ΠΌΡ‹ ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ ΠΈΠ· нСслоТного расчёта: \n", + "\n", + "$$\\sum\\limits_{i=1}^r \\frac{F}{f_i} ([\\dot \\alpha_i, \\: \\alpha_{i+1}] + [\\alpha_i, \\: \\dot \\alpha_{i+1}]) - \\\\ - r \\sum\\limits_{i=1}^{r-2}([\\dot \\alpha_1 + \\ldots + \\dot \\alpha_i, \\: \\alpha_{i+1}] + [\\alpha_1 + \\ldots + \\alpha_i, \\: \\dot \\alpha_{i+1}]) = 0 \\label{eq:26} \\tag{26}$$ \n", + "\n", + "Π“Π΄Π΅ $\\dot \\alpha_i = \\left.\\frac{d}{dt} \\alpha_i \\right|_{t=0}$ - ΠΏΡ€ΠΎΠΈΠ·Π²ΠΎΠ»ΡŒΠ½Ρ‹Π΅ Π²Π΅ΠΊΡ‚ΠΎΡ€Ρ‹, ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Π΅ Ρ‚ΠΎΠΆΠ΅ Π΄ΠΎΠ»ΠΆΠ½Ρ‹ ΡƒΠ΄ΠΎΠ²Π»Π΅Ρ‚Π²ΠΎΡ€ΡΡ‚ΡŒ ΡƒΡΠ»ΠΎΠ²ΠΈΡŽ замкнутости, ΡΠ»Π΅Π΄ΡƒΡŽΡ‰Π΅ΠΌΡƒ ΠΈΠ· $(25)$:\n", + "\n", + "$$\\sum\\limits_{i=1}^r \\dot\\alpha_i = 0 \\label{eq:27} \\tag{27}$$\n", + "\n", + "Если ΠΌΡ‹ ΠΏΠΎΠ»ΠΎΠΆΠΈΠΌ ΠΏΠΎΡΠ»Π΅Π΄ΠΎΠ²Π°Ρ‚Π΅Π»ΡŒΠ½ΠΎ\n", + "\n", + "$$\\dot \\alpha_j = \\beta, \\;\\; \\dot \\alpha_{j+1} = -\\beta, \\;\\; \\dot \\alpha_i = 0 \\: \\text{для} \\: i \\neq j, \\;\\; i \\neq j+1 \\label{eq:28} \\tag{28}$$\n", + "\n", + "Π³Π΄Π΅ $\\beta$ - ΠΈΠ·Π½Π°Ρ‡Π°Π»ΡŒΠ½ΠΎ ΠΏΡ€ΠΎΠΈΠ·Π²ΠΎΠ»ΡŒΠ½Ρ‹ΠΉ Π²Π΅ΠΊΡ‚ΠΎΡ€, Ρ‚ΠΎ условиС замкнутости $(\\ref{eq:27})$, ΠΎΡ‡Π΅Π²ΠΈΠ΄Π½ΠΎ, выполняСтся, ΠΈ ΠΌΡ‹ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ для $j = 1, \\: \\ldots, \\: r$:\n", + "\n", + "$$\\frac{F}{r} \\left\\{ \\frac{[\\alpha_{j-1}, \\: \\beta]}{f_{j-1}} - \\frac{[\\beta, \\: \\alpha_{j+2}]}{f_{j+1}} + \\frac1{f_j} [\\beta, \\: \\alpha_j + \\alpha_{j+1}] \\right\\} = [\\beta, \\: \\alpha_j + \\alpha_{j+1}] \\label{eq:29} \\tag{29}$$\n", + "\n", + "Если ΠΌΡ‹ Ρ‚Π΅ΠΏΠ΅Ρ€ΡŒ Π½Π°ΠΌΠ΅Ρ€Π΅Π½Π½ΠΎ Π²Ρ‹Π±Π΅Ρ€Π΅ΠΌ $\\beta = \\alpha_j$, Ρ„ΠΎΡ€ΠΌΡƒΠ»Π° ΠΏΡ€ΠΈΠΌΠ΅Ρ‚ Π²ΠΈΠ΄\n", + "\n", + "$$\\frac{F}{r} \\left\\{ 4 - \\frac{[\\alpha_j, \\: \\alpha_{j+2}]}{f_{j+1}} \\right\\} = 2 f_j \\label{eq:30} \\tag{30}$$\n", + "\n", + "Π° для $\\beta = -\\alpha_{j+1}$\n", + "\n", + "$$\\frac{F}{r} \\left\\{ 4 - \\frac{[\\alpha_{j-1}, \\: \\alpha_{j+1}]}{f_{j-1}} \\right\\} = 2 f_j \\label{eq:31} \\tag{31}$$\n", + "\n", + "Π‘Ρ€Π°Π²Π½ΠΈΠΌ эти Π΄Π²Π΅ Ρ„ΠΎΡ€ΠΌΡƒΠ»Ρ‹ ΠΈ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ\n", + "\n", + "$$[\\alpha_j, \\: \\alpha_{j+2}] = \\frac{f_{j+1}}{f_{j-1}} [\\alpha_{j-1}, \\: \\alpha_{j-1}] \\label{eq:32} \\tag{32}$$\n", + "\n", + "Π’Π΅ΠΏΠ΅Ρ€ΡŒ ΠΌΡ‹ запишСм ΡƒΡ€Π°Π²Π½Π΅Π½ΠΈΠ΅ $(\\ref{eq:31})$ Π² Ρ„ΠΎΡ€ΠΌΠ΅\n", + "\n", + "$$2 \\left( 2 - \\frac{r f_j}{F} \\right) = \\frac{[\\alpha_{j-1}, \\: \\alpha_{j+1}]}{f_{j-1}} \\notag$$\n", + "\n", + "ΠΈ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ ΠΏΡƒΡ‚Ρ‘ΠΌ ΠΏΠΎΠ²Ρ‚ΠΎΡ€Π½ΠΎΠ³ΠΎ примСнСния $(\\ref{eq:32})$:\n", + "\n", + "$$2 \\left( 2 - \\frac{r f_j}{F} \\right) = \\frac{f_j}{f_1 \\cdot f_2} [\\alpha_1, \\: \\alpha_3] \\notag$$\n", + "\n", + "ΠΈΠ»ΠΈ\n", + "\n", + "$$f_j = \\frac2{\\cfrac{r}{F} + \\cfrac{[\\alpha_1, \\: \\alpha_3]}{2 f_1 \\cdot f_2}} \\label{eq:33} \\tag{33}$$\n", + "\n", + "Для $r = 3$ наша ΡΠΊΡΡ‚Ρ€Π΅ΠΌΠ°Π»ΡŒΠ½Π°Ρ Π·Π°Π΄Π°Ρ‡Π° Ρ‚Ρ€ΠΈΠ²ΠΈΠ°Π»ΡŒΠ½Π°. Π’ случаС $r \\ge 4$ получаСтся, Ρ‡Ρ‚ΠΎ $f_3 = f_4 = \\ldots = f_r$. ΠŸΡ€ΠΈ цикличСской пСрСстановкС Π½ΠΎΠΌΠ΅Ρ€ΠΎΠ² Π²Π΅Ρ€ΡˆΠΈΠ½ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ\n", + "\n", + "$$f_1 = f_2 = \\ldots = f_r \\label{eq:34} \\tag{34}$$\n", + "\n", + "Π° послС $(\\ref{eq:32})$ Ρ‚Π°ΠΊΠΆΠ΅ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ\n", + "\n", + "$$[\\alpha_i, \\: \\alpha_{i+2}] = [\\alpha_1, \\: \\alpha_3] = \\beta \\;\\; (i = 1, \\: 2, \\: \\ldots, \\: r) \\label{eq:35} \\tag{35}$$\n", + "\n", + "Π³Π΄Π΅ $\\beta$ - нСкоторая константа.\n", + "\n", + "**ΠŸΡ€ΠΈΠΌΠ΅Ρ‡Π°Π½ΠΈΠ΅.** Π’ случаС $\\beta = [\\alpha_i, \\: \\alpha_{i+2}] = 0$ получаСтся, Ρ‡Ρ‚ΠΎ $f_j = 2F/r$. Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, Π²Π΅ΠΊΡ‚ΠΎΡ€ $\\alpha_{i+2}$ всСгда ΠΏΠ°Ρ€Π°Π»Π»Π΅Π»Π΅Π½ $\\alpha_i$. Π­Ρ‚ΠΎ Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΠΎ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ Π² Ρ‚ΠΎΠΌ случаС, Ссли $r = 4$, Π° $K$ - ΠΏΠ°Ρ€Π°Π»Π»Π΅Π»ΠΎΠ³Ρ€Π°ΠΌΠΌ.\n", + "\n", + "Π’Π΅ΠΏΠ΅Ρ€ΡŒ вСрнёмся ΠΊ ΠΎΠ±Ρ‰Π΅ΠΌΡƒ ΡΠ»ΡƒΡ‡Π°ΡŽ. Π‘ ΠΏΠΎΠΌΠΎΡ‰ΡŒΡŽ Π°Ρ„Ρ„ΠΈΠ½Π½Ρ‹Ρ… ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΠΎΠ²Π°Π½ΠΈΠΉ ΠΌΡ‹ всСгда ΠΌΠΎΠΆΠ΅ΠΌ Π΄ΠΎΠ±ΠΈΡ‚ΡŒΡΡ Ρ‚ΠΎΠ³ΠΎ, Ρ‡Ρ‚ΠΎ $f_j = 1/2$, Ρ‚ΠΎ Π΅ΡΡ‚ΡŒ\n", + "\n", + "$$[\\alpha_i, \\: \\alpha_{i+1}] = 1 \\label{eq:36} \\tag{36}$$\n", + "\n", + "Из $(\\ref{eq:33})$ сразу слСдуСт\n", + "\n", + "$$F = \\frac{r}{2(2-\\beta)} \\label{eq:37} \\tag{37}$$\n", + "\n", + "ΠΈ, ΡΠ»Π΅Π΄ΠΎΠ²Π°Ρ‚Π΅Π»ΡŒΠ½ΠΎ, $\\beta \\lt 2$.\n", + "\n", + "Π’Π°ΠΊ ΠΊΠ°ΠΊ Π²Π΅ΠΊΡ‚ΠΎΡ€Ρ‹ $\\alpha_{i-2}$ ΠΈ $\\alpha_{i-1}$ Π»ΠΈΠ½Π΅ΠΉΠ½ΠΎ нСзависимы, ΠΈΠ· $\\alpha_i = x \\alpha_{i-2} + y \\alpha_{i-1}$, учитывая $(\\ref{eq:35})$ ΠΈ $(\\ref{eq:36})$, слСдуСт:\n", + "\n", + "$$\\alpha_i = -\\alpha_{i-2} + b \\alpha_{i-1} \\label{eq:38} \\tag{38}$$\n", + "\n", + "Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, ΠΌΡ‹ ΠΏΠΎΠΊΠ°Π·Π°Π»ΠΈ, Ρ‡Ρ‚ΠΎ Π²Π΅ΠΊΡ‚ΠΎΡ€Ρ‹ $\\alpha_i$ сторон максимального Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° Π΄ΠΎΠ»ΠΆΠ½Ρ‹ ΡƒΠ΄ΠΎΠ²Π»Π΅Ρ‚Π²ΠΎΡ€ΡΡ‚ΡŒ рСкурсивной Ρ„ΠΎΡ€ΠΌΡƒΠ»Π΅ $(\\ref{eq:38})$. Π—Π°Π²Π΅Ρ€ΡˆΠΈΠΌ Π΄ΠΎΠΊΠ°Π·Π°Ρ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ, ΠΏΠΎΠΊΠ°Π·Π°Π², Ρ‡Ρ‚ΠΎ Π² этом случаС ΠΌΠΎΠΆΠ½ΠΎ ΠΎΠΏΡ€Π΅Π΄Π΅Π»ΠΈΡ‚ΡŒ Π΅Π²ΠΊΠ»ΠΈΠ΄ΠΎΠ²Ρƒ ΠΌΠ΅Ρ‚Ρ€ΠΈΠΊΡƒ Π² Π°Ρ„Ρ„ΠΈΠ½Π½ΠΎΠΉ плоскости, ΠΎΡ‚Π½ΠΎΡΠΈΡ‚Π΅Π»ΡŒΠ½ΠΎ ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ $K$ являСтся ΠΏΡ€Π°Π²ΠΈΠ»ΡŒΠ½Ρ‹ΠΌ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΎΠΌ. \n", + "\n", + "Π’ΠΎ-ΠΏΠ΅Ρ€Π²Ρ‹Ρ…, Π·Π°ΠΌΠ΅Ρ‚ΠΈΠΌ, Ρ‡Ρ‚ΠΎ всСгда Π΄ΠΎΠ»ΠΆΠ½ΠΎ Π±Ρ‹Ρ‚ΡŒ\n", + "\n", + "$$-1 \\leq b \\lt 2 \\label{eq:39} \\tag{39}$$\n", + "\n", + "Π”Π΅ΠΉΡΡ‚Π²ΠΈΡ‚Π΅Π»ΡŒΠ½ΠΎ, Ссли $b \\lt -1$, Ρ‚ΠΎ сторона $\\alpha_3$ Π±ΡƒΠ΄Π΅Ρ‚ ΠΏΠ΅Ρ€Π΅ΡΠ΅ΠΊΠ°Ρ‚ΡŒ сторону $\\alpha_1$, Ρ‡Ρ‚ΠΎ ΠΏΡ€ΠΎΡ‚ΠΈΠ²ΠΎΡ€Π΅Ρ‡ΠΈΡ‚ выпуклости $K$. ВСрхняя ΠΎΡ†Π΅Π½ΠΊΠ° $b \\lt 2$ ΡƒΠΆΠ΅ Π΄ΠΎΠΊΠ°Π·Π°Π½Π° Π² $(\\ref{eq:37})$.\n", + "Π•Π²ΠΊΠ»ΠΈΠ΄ΠΎΠ²Π° ΠΌΠ΅Ρ‚Ρ€ΠΈΠΊΠ° ΠΎΠ΄Π½ΠΎΠ·Π½Π°Ρ‡Π½ΠΎ устанавливаСтся, Ссли ΠΌΡ‹ ΠΎΠΏΡ€Π΅Π΄Π΅Π»ΠΈΠΌ скалярноС ΠΏΡ€ΠΎΠΈΠ·Π²Π΅Π΄Π΅Π½ΠΈΠ΅ Π²Π΅ΠΊΡ‚ΠΎΡ€ΠΎΠ² $\\alpha_1$ ΠΈ $\\alpha_2$, ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Π΅ ΠΌΡ‹ рассматриваСм ΠΊΠ°ΠΊ базис:\n", + "\n", + "$$\\alpha_1^2 = \\alpha_2^2 = \\frac{2}{\\sqrt{4-b^2}}, \\;\\; \\alpha_1\\alpha_2 = \\frac{b}{\\sqrt{4-b^2}} \\label{eq:40} \\tag{40}$$\n", + "\n", + "Π—Π°ΠΌΠ΅Ρ‚ΠΈΠΌ, Ρ‡Ρ‚ΠΎ\n", + "\n", + "$$[\\alpha_1, \\alpha_2]^2 = \\begin{vmatrix} \\alpha_1^2 & \\alpha_1\\alpha_2 \\\\ \\alpha_1\\alpha_2 & \\alpha_2^2 \\end{vmatrix} = 1 \\notag$$\n", + "\n", + "Π‘Π»Π΅Π΄ΠΎΠ²Π°Ρ‚Π΅Π»ΡŒΠ½ΠΎ, эта ΠΌΠ΅Ρ‚Ρ€ΠΈΠΊΠ° ΠΏΠΎΠ»ΠΎΠΆΠΈΡ‚Π΅Π»ΡŒΠ½ΠΎ ΠΎΠΏΡ€Π΅Π΄Π΅Π»Π΅Π½Π°, Ρ‚.Π΅. являСтся Π΅Π²ΠΊΠ»ΠΈΠ΄ΠΎΠ²ΠΎΠΉ ΠΌΠ΅Ρ‚Ρ€ΠΈΠΊΠΎΠΉ. Π’Π΅ΠΏΠ΅Ρ€ΡŒ, ΠΈΡΠΏΠΎΠ»ΡŒΠ·ΡƒΡ Ρ€Π΅ΠΊΡƒΡ€ΡΠΈΠ²Π½ΡƒΡŽ Ρ„ΠΎΡ€ΠΌΡƒΠ»Ρƒ ΠΈΠ· $(\\ref{eq:38})$, Π»Π΅Π³ΠΊΠΎ Π΄ΠΎΠΊΠ°Π·Π°Ρ‚ΡŒ ΠΏΠΎ ΠΈΠ½Π΄ΡƒΠΊΡ†ΠΈΠΈ, Ρ‡Ρ‚ΠΎ для всСх $j$ Π΄ΠΎΠ»ΠΆΠ½ΠΎ Π²Ρ‹ΠΏΠΎΠ»Π½ΡΡ‚ΡŒΡΡ\n", + "\n", + "$$\\alpha_j^2 = \\frac{2}{\\sqrt{4-b^2}}, \\;\\; \\alpha_j\\alpha_{j+1} = \\frac{b}{\\sqrt{4-b^2}}, \\;\\; [\\alpha_j, \\:\\alpha_{j+1}] = 1 \\notag$$\n", + "\n", + "ΠžΡ‚ΡΡŽΠ΄Π° слСдуСт, Ρ‡Ρ‚ΠΎ всС стороны нашСго $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$ ΠΈΠΌΠ΅ΡŽΡ‚ ΠΎΠ΄ΠΈΠ½Π°ΠΊΠΎΠ²ΡƒΡŽ Π΄Π»ΠΈΠ½Ρƒ ΠΈ Ρ‡Ρ‚ΠΎ Π»ΡŽΠ±Ρ‹Π΅ Π΄Π²Π΅ смСТныС стороны ΠΏΠ΅Ρ€Π΅ΡΠ΅ΠΊΠ°ΡŽΡ‚ΡΡ ΠΏΠΎΠ΄ ΠΎΠ΄Π½ΠΈΠΌ ΡƒΠ³Π»ΠΎΠΌ. Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ Π΄ΠΎΠ»ΠΆΠ΅Π½ Π±Ρ‹Ρ‚ΡŒ ΠΏΡ€Π°Π²ΠΈΠ»ΡŒΠ½Ρ‹ΠΌ ΠΎΡ‚Π½ΠΎΡΠΈΡ‚Π΅Π»ΡŒΠ½ΠΎ этой ΠΌΠ΅Ρ‚Ρ€ΠΈΠΊΠΈ.\n", + "\n", + "Π­Ρ‚ΠΎ Π΄ΠΎΠΊΠ°Π·Ρ‹Π²Π°Π΅Ρ‚ Π΅Π΄ΠΈΠ½ΡΡ‚Π²Π΅Π½Π½ΠΎΡΡ‚ΡŒ максимального ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ°. Π•Π³ΠΎ сущСствованиС слСдуСт ΠΈΠ· ограничСнности $Z(K)$ ΠΈ нСпрСрывности $F$ ΠΈ $Z$. Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, ΡƒΡ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 2 ΠΏΠΎΠ»Π½ΠΎΡΡ‚ΡŒΡŽ Π΄ΠΎΠΊΠ°Π·Π°Π½ΠΎ." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## 3. Π‘Π»ΡƒΡ‡Π°ΠΉΠ½ΠΎ Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Π΅ Ρ‚ΠΎΡ‡ΠΊΠΈ Π² Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ области с Π³Π»Π°Π΄ΠΊΠΎΠΉ Π³Ρ€Π°Π½ΠΈΡ†Π΅ΠΉ\n", + "\n", + "ΠŸΡƒΡΡ‚ΡŒ Ρ‚Π΅ΠΏΠ΅Ρ€ΡŒ $K$ - выпуклая ΠΎΠ±Π»Π°ΡΡ‚ΡŒ, Π³Ρ€Π°Π½ΠΈΡ†Π° ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ ΠΈΠΌΠ΅Π΅Ρ‚ Π½Π΅ΠΏΡ€Π΅Ρ€Ρ‹Π²Π½ΡƒΡŽ ΠΊΡ€ΠΈΠ²ΠΈΠ·Π½Ρƒ ΠΊΠ°ΠΊ Ρ„ΡƒΠ½ΠΊΡ†ΠΈΡŽ ΠΎΡ‚ Π΄Π»ΠΈΠ½Ρ‹ Π΄ΡƒΠ³ΠΈ. Π’ обозначСниях ΠΏΡ€Π΅Π΄Ρ‹Π΄ΡƒΡ‰ΠΈΡ… ΠΏΠ°Ρ€Π°Π³Ρ€Π°Ρ„ΠΎΠ² ΠΌΡ‹ снова ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ\n", + "\n", + "$$E_n = C_n^2 \\cdot W_n \\notag$$\n", + "\n", + "Богласно ΡƒΡ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΡŽ ΠΎ ΠΌΠ΅Ρ€Π΅ мноТСства ΠΏΠ°Ρ€ Ρ‚ΠΎΡ‡Π΅ΠΊ, извСстному ΠΈΠ· ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π»ΡŒΠ½ΠΎΠΉ Π³Π΅ΠΎΠΌΠ΅Ρ‚Ρ€ΠΈΠΈ (см. Blascke [2], S 17, (86)):\n", + "\n", + "$$dP_1 dP_2 = |t_1 - t_2| \\, dt_1 \\, dt_2 \\, d\\varphi \\, dp \\label{eq:41} \\tag{41}$$\n", + "\n", + "Π³Π΄Π΅ $p$ -- Π΄Π»ΠΈΠ½Π° пСрпСндикуляра ΠΊ прямой $P_1 P_2$ ΠΈΠ· Π½Π°Ρ‡Π°Π»Π° ΠΊΠΎΠΎΡ€Π΄ΠΈΠ½Π°Ρ‚, $\\varphi$ -- ΡƒΠ³ΠΎΠ» Π½Π°ΠΊΠ»ΠΎΠ½Π° этого пСрпСндикуляра, $t_1, \\: t_2$ -- расстояния ΠΏΠΎ этой прямой ΠΎΡ‚ $P_1$ ΠΈ $P_2$ Π΄ΠΎ Ρ‚ΠΎΡ‡ΠΊΠΈ пСрСсСчСния $P_1 P_2$ с пСрпСндикуляром. ΠžΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Ρ‡Π΅Ρ€Π΅Π· $f$ ΠΏΠ»ΠΎΡ‰Π°Π΄ΡŒ участка, отсСкаСмого ΠΎΡ‚ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$ прямой $P_1 P_2$. Π’ΠΎΠ³Π΄Π° ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ\n", + "\n", + "$$E_n = \\cfrac{C_n^2}{F^2} \\int\\limits_0^{2\\pi} \\left\\{ \\int\\limits_0^{p(\\varphi)} \\left(1-\\frac{f}{F} \\right)^{n-2} ( \\int \\int |t_1 - t_2| \\, dt_1 \\, dt_2) \\, dp \\right\\} d\\varphi \\label{eq:42} \\tag{42}$$\n", + "\n", + "Π§Π΅Ρ€Π΅Π· $l(p, \\phi)$ ΠΎΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Π΄Π»ΠΈΠ½Ρƒ Ρ…ΠΎΡ€Π΄Ρ‹ области $K$, Π»Π΅ΠΆΠ°Ρ‰Π΅ΠΉ Π½Π° прямой с ΠΊΠΎΠΎΡ€Π΄ΠΈΠ½Π°Ρ‚Π°ΠΌΠΈ $p, \\varphi$. Π’ΠΎΠ³Π΄Π°\n", + "\n", + "$$\\int \\int |t_1 - t_2| \\, dt_1 \\, dt_2 = \\frac{l^3(p, \\varphi)}{3} \\label{eq:43} \\tag{43}$$\n", + "\n", + "Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ,\n", + "\n", + "$$E_n = \\cfrac{C_n^2}{3F^2} \\int\\limits_0^{2\\pi} \\left\\{ \\int\\limits_0^{p(\\varphi)} \\left(1-\\frac{f}{F} \\right)^{n-2} l^3(p, \\varphi) \\, dp \\right\\} \\, d\\varphi \\label{eq:44} \\tag{44}$$\n", + "\n", + "ΠŸΡ€ΠΈ расчСтС асимптотичСского значСния $E_n$ ΠΌΡ‹ ΠΌΠΎΠΆΠ΅ΠΌ, ΠΎΡ‡Π΅Π²ΠΈΠ΄Π½ΠΎ, ΠΎΠ³Ρ€Π°Π½ΠΈΡ‡ΠΈΡ‚ΡŒΡΡ ΠΏΠ°Ρ€Π°ΠΌΠΈ Π·Π½Π°Ρ‡Π΅Π½ΠΈΠΉ $(p, \\: \\varphi)$, для ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… выполняСтся $f/F < \\varepsilon$; Π΄Ρ€ΡƒΠ³ΠΈΠ΅ ΠΏΠ°Ρ€Ρ‹ Π·Π½Π°Ρ‡Π΅Π½ΠΈΠΉ Π΄Π°ΡŽΡ‚ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ остаточный Ρ‡Π»Π΅Π½ порядка $n^2(1-\\varepsilon)^n = o(1)$. Рассмотрим Ρ‚Π΅ΠΏΠ΅Ρ€ΡŒ фиксированноС Π·Π½Π°Ρ‡Π΅Π½ΠΈΠ΅ $\\varphi$. ΠŸΡƒΡΡ‚ΡŒ $P(\\varphi)$ - Ρ‚ΠΎΡ‡ΠΊΠ°, Π² ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ ΠΊΠ°ΡΠ°Ρ‚Π΅Π»ΡŒΠ½Π°Ρ ΠΊ $K$ ΠΈΠΌΠ΅Π΅Ρ‚ ΡƒΠ³ΠΎΠ» Π½Π°ΠΊΠ»ΠΎΠ½Π° $\\varphi$. Как становится ясно ΠΈΠ· Π½ΠΈΠΆΠ΅ΡΠ»Π΅Π΄ΡƒΡŽΡ‰ΠΈΡ… ΠΎΡ†Π΅Π½ΠΎΠΊ, ΠΌΠΎΠΆΠ½ΠΎ Π·Π°ΠΌΠ΅Π½ΠΈΡ‚ΡŒ $f$ ΠΈ $l$ Π½Π° ΡΠΎΠΎΡ‚Π²Π΅Ρ‚ΡΡ‚Π²ΡƒΡŽΡ‰ΠΈΠ΅ Π²Π΅Π»ΠΈΡ‡ΠΈΠ½Ρ‹ $\\bar f$ ΠΈ $\\bar l$ окруТности ΠΊΡ€ΠΈΠ²ΠΈΠ·Π½Ρ‹ Π² Ρ‚ΠΎΡ‡ΠΊΠ΅ $P(\\varphi)$. ΠŸΡƒΡΡ‚ΡŒ $\\alpha$ - Ρ†Π΅Π½Ρ‚Ρ€Π°Π»ΡŒΠ½Ρ‹ΠΉ ΡƒΠ³ΠΎΠ», ΡΠΎΠΎΡ‚Π²Π΅Ρ‚ΡΡ‚Π²ΡƒΡŽΡ‰ΠΈΠΉ Ρ…ΠΎΡ€Π΄Π΅ $l$, ΠΈ $r(\\varphi)$ - радиус окруТности ΠΊΡ€ΠΈΠ²ΠΈΠ·Π½Ρ‹ Π² Ρ‚ΠΎΡ‡ΠΊΠ΅ $P(\\varphi)$. Π’ΠΎΠ³Π΄Π°\n", + "\n", + "$$\\bar l = 2 \\, r(\\varphi) \\, \\sin(\\alpha/2) \\quad \\text{ΠΈ} \\quad \\bar f = \\frac12 (\\alpha - \\sin\\alpha) \\, r^2(\\varphi) \\label{eq:45} \\tag{45}$$\n", + "\n", + "ΠšΡ€ΠΎΠΌΠ΅ Ρ‚ΠΎΠ³ΠΎ,\n", + "\n", + "$$|dp| = \\frac12 r(\\varphi) \\sin(\\alpha/2) |d\\alpha| \\label{eq:46} \\tag{46}$$\n", + "\n", + "НаконСц,\n", + "\n", + "$$l - \\bar l = o(\\alpha^2) \\quad \\text{ΠΈ} \\quad f - \\bar f = o(\\alpha^4) \\label{eq:47} \\tag{47}$$\n", + "\n", + "Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ Ρ€Π΅Π·ΡƒΠ»ΡŒΡ‚Π°Ρ‚ для $E_n$:\n", + "\n", + "$$E_n = \\cfrac{C_n^2}{12F^2} \\int\\limits_0^{2\\pi} r^4 \\int\\limits_0^{\\alpha(\\varphi)} \\left( 1 - \\frac{\\alpha^3r^2}{12F^2} + o(\\alpha^3) \\right)^{n-2} (\\alpha^4 + o(\\alpha^5)) \\, d\\alpha \\, d\\varphi \\label{eq:48} \\tag{48}$$\n", + "\n", + "ΠŸΠΎΠ΄ΡΡ‚Π°Π²ΠΈΠ²\n", + "\n", + "$$\\frac{\\alpha^3r^2}{12F} = \\frac{X}{n} \\label{eq:49} \\tag{49}$$\n", + "\n", + "ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ\n", + "\n", + "$$S(\\varphi) = \\int\\limits_0^{\\alpha(\\varphi)} \\left(1 - \\frac{\\alpha^3r^2}{12F} + o(\\alpha^3)\\right)^{n-2} (\\alpha^4 + o(\\alpha^5)) \\, d\\alpha \\label{eq:50} \\tag{50}$$\n", + "$$S(\\varphi) = \\frac13 \\left(\\frac{12F}{r^2 n}\\right)^{5/3} \\left(\\int\\limits_0^\\infty \\left(1 - \\frac{X}{n}\\right)^{n-2} x^{2/3} \\, dx + o(1)\\right) = \\\\ = \\frac13 \\, \\Gamma \\left( \\frac{5}{3} \\right) \\left(\\frac{12F}{r^2 n}\\right)^{5/3} (1 + o(1)) \\notag$$\n", + "\n", + "Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ\n", + "\n", + "$$E_n \\sim \\Gamma \\left(\\frac{5}{3}\\right) \\, \\sqrt[3]{\\frac{2n}{3F}} \\, \\int\\limits_0^{2\\pi} r^{2/3} \\, d\\varphi \\label{eq:51} \\tag{51}$$\n", + "\n", + "ΠžΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Π·Π° $s$ Π΄Π»ΠΈΠ½Ρƒ Π΄ΡƒΠ³ΠΈ Π³Ρ€Π°Π½ΠΈΡ†Ρ‹ $K$ ΠΈ, считая $d\\varphi/ds = 1/r$, ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ ΡΠ»Π΅Π΄ΡƒΡŽΡ‰Π΅Π΅ ΡƒΡ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅:\n", + "\n", + "**Π£Ρ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 3.** Если $K$ - выпуклая ΠΎΠ±Π»Π°ΡΡ‚ΡŒ, граничная кривая ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ ΠΈΠΌΠ΅Π΅Ρ‚ Π½Π΅ΠΏΡ€Π΅Ρ€Ρ‹Π²Π½ΡƒΡŽ ΠΊΡ€ΠΈΠ²ΠΈΠ·Π½Ρƒ $x = x(s)$, Ρ‚ΠΎ\n", + "\n", + "$$E_n \\sim \\Gamma \\left(\\frac{5}{3}\\right) \\, \\sqrt[3]{\\frac23} \\, (F^{-1/3} \\int\\limits_L x^{1/3} \\, ds) \\sqrt[3]{n} \\label{eq:52} \\tag{52}$$\n", + "\n", + "Π˜Π½Ρ‚Π΅Π³Ρ€Π°Π» $\\int\\limits_L x^{1/3} \\, ds$, входящий Π² $(\\ref{eq:52})$, являСтся Π°Ρ„Ρ„ΠΈΠ½Π½ΠΎΠΉ Π΄Π»ΠΈΠ½ΠΎΠΉ ΠΊΡ€ΠΈΠ²ΠΎΠΉ $L$. Π›Π΅Π³ΠΊΠΎ Π²ΠΈΠ΄Π΅Ρ‚ΡŒ, Ρ‡Ρ‚ΠΎ ΠΌΠ½ΠΎΠΆΠΈΡ‚Π΅Π»ΡŒ $\\sqrt[3]{n}$ являСтся Π°Ρ„Ρ„ΠΈΠ½Π½ΠΎ-ΠΈΠ½Π²Π°Ρ€ΠΈΠ°Π½Ρ‚Π½Ρ‹ΠΌ. Из Π°Ρ„Ρ„ΠΈΠ½Π½ΠΎΠ³ΠΎ изопСримСтричСского свойства эллипсов (см. [3]) нСпосрСдствСнно слСдуСт, Ρ‡Ρ‚ΠΎ этот ΠΌΠ½ΠΎΠΆΠΈΡ‚Π΅Π»ΡŒ максималСн для эллипсов ΠΈ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ для эллипсов." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## 4. ΠΠΎΡ€ΠΌΠ°Π»ΡŒΠ½ΠΎΠ΅ распрСдСлСниС случайных Ρ‚ΠΎΡ‡Π΅ΠΊ\n", + "\n", + "Π’ этом ΠΏΠ°Ρ€Π°Π³Ρ€Π°Ρ„Π΅ ΠΌΡ‹ Π΄ΠΎΠΊΠ°ΠΆΠ΅ΠΌ ΡΠ»Π΅Π΄ΡƒΡŽΡ‰Π΅Π΅ ΡƒΡ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅:\n", + "\n", + "**Π£Ρ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 4.** ΠŸΡƒΡΡ‚ΡŒ Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i \\:(i = 1, \\:2, \\:\\ldots, \\:n)$ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ Π½Π° плоскости нСзависимо Π΄Ρ€ΡƒΠ³ ΠΎΡ‚ Π΄Ρ€ΡƒΠ³Π° Π² соотвСтствии с Π½ΠΎΡ€ΠΌΠ°Π»ΡŒΠ½Ρ‹ΠΌ распрСдСлСниСм. Π’ΠΎΠ³Π΄Π°\n", + "\n", + "$$E_n \\sim 2 \\sqrt{2 \\pi \\log n} \\label{eq:53} \\tag{53}$$\n", + "\n", + "**Π”ΠΎΠΊΠ°Π·Π°Ρ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ:**\n", + "Π’Π°ΠΊ ΠΊΠ°ΠΊ $E_n$ ΠΈΠ½Π²Π°Ρ€ΠΈΠ°Π½Ρ‚Π½ΠΎ ΠΎΡ‚Π½ΠΎΡΠΈΡ‚Π΅Π»ΡŒΠ½ΠΎ Π°Ρ„Ρ„ΠΈΠ½Π½Ρ‹Ρ… ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΠΎΠ²Π°Π½ΠΈΠΉ плоскости, ΠΌΠΎΠΆΠ½ΠΎ ΡΡ‡ΠΈΡ‚Π°Ρ‚ΡŒ, Ρ‡Ρ‚ΠΎ функция плотности нашСго Π½ΠΎΡ€ΠΌΠ°Π»ΡŒΠ½ΠΎΠ³ΠΎ распрСдСлСния ΠΈΠΌΠ΅Π΅Ρ‚ Π²ΠΈΠ΄ $\\frac{1}{2\\pi} \\, exp \\left(-\\frac{x^2 + y^2}{2}\\right) \\notag$. Как ΠΈ Π² ΠΏΡ€Π΅Π΄Ρ‹Π΄ΡƒΡ‰ΠΈΡ… ΠΏΠ°Ρ€Π°Π³Ρ€Π°Ρ„Π°Ρ…, \n", + "\n", + "$$E_n = C_n^2 \\cdot W_n \\notag$$\n", + "\n", + "Π³Π΄Π΅ $W_n$ - Π²Π΅Ρ€ΠΎΡΡ‚Π½ΠΎΡΡ‚ΡŒ Ρ‚ΠΎΠ³ΠΎ, Ρ‡Ρ‚ΠΎ всС Ρ‚ΠΎΡ‡ΠΊΠΈ $P_3, \\:\\ldots, \\:P_n$ Π»Π΅ΠΆΠ°Ρ‚ ΠΏΠΎ ΠΎΠ΄Π½Ρƒ сторону ΠΎΡ‚ прямой $P_1 P_2$.\n", + "Из Ρ„ΠΎΡ€ΠΌΡƒΠ»Ρ‹ ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π»ΡŒΠ½ΠΎΠΉ Π³Π΅ΠΎΠΌΠ΅Ρ‚Ρ€ΠΈΠΈ \n", + "\n", + "$$dP_1dP_2 = |t_1 - t_2| \\, dt_1 \\, dt_2 \\, dp \\, d\\varphi \\notag$$ \n", + "\n", + "ΠΈ ΠΈΠ·-Π·Π° ΠΊΡ€ΡƒΠ³ΠΎΠ²ΠΎΠΉ симмСтрии Π½ΠΎΡ€ΠΌΠ°Π»ΡŒΠ½ΠΎΠ³ΠΎ распрСдСлСния ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ\n", + "\n", + "$$W_n = \\frac{1}{2\\pi} \\iiint (g(p)^{n-2} + (1 - g(p))^{n-2}) \\, |t_1 - t_2| \\, exp \\left(-p^2 - \\frac{t_1^2 + t_2^2}{2}\\right) \\, dt_1 \\, dt_2 \\, dp \\label{eq:54} \\tag{54}$$\n", + "\n", + "$$\\text{Π³Π΄Π΅} \\: g(p) = \\int\\limits_{-\\infty}^{+\\infty} \\int\\limits_{P}^{+\\infty} \\frac{1}{2\\pi} \\, exp\\left(-\\frac{x^2+y^2}{2}\\right) \\, dx \\, dy = 1 - \\varPhi(p) \\label{eq:55} \\tag{55}$$\n", + "\n", + "$$\\varPhi(p) = \\frac{1}{\\sqrt{2\\pi}} \\int\\limits_{-\\infty}^p exp\\left(-\\frac{u^2}{2}\\right) \\, du \\label{eq:56} \\tag{56}$$\n", + "\n", + "Π§Π΅Ρ€Π΅Π· Π·Π°ΠΌΠ΅Π½Ρƒ\n", + "\n", + "$$u = \\frac{t_1 - t_2}{\\sqrt 2}, \\: v = \\frac{t_1 + t_2}{\\sqrt 2}, \\: \\frac{\\partial(t_1, t_2)}{\\partial(u, v)} = 1 \\label{eq:57} \\tag{57}$$\n", + "\n", + "ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ\n", + "\n", + "$$\\frac{1}{2\\pi} \\int\\int |t_1 - t_2| \\, exp\\left(-\\frac{t_1^2 + t_2^2}{2}\\right) \\, dt_1 \\, dt_2 = \\frac{2}{\\sqrt \\pi} \\label{eq:58} \\tag{58}$$\n", + "\n", + "Π‘Π»Π΅Π΄ΠΎΠ²Π°Ρ‚Π΅Π»ΡŒΠ½ΠΎ,\n", + "\n", + "$$W_n = \\frac{2}{\\sqrt \\pi} \\int\\limits_0^\\infty [\\varPhi^{n-2}(p) + (1 - \\varPhi(p))^{n-2}] e^{-p^2} dp \\label{eq:59} \\tag{59}$$\n", + "\n", + "ΠΈ, ΠΎΡ‡Π΅Π²ΠΈΠ΄Π½ΠΎ, $1 - \\varPhi(p) \\leqq 1/2$ для $p \\geqq 0$. А Π·Π½Π°Ρ‡ΠΈΡ‚,\n", + "\n", + "$$W_n \\sim \\frac{2}{\\sqrt \\pi} \\int\\limits_0^\\infty \\varPhi^{n-2}(p) e^{-p^2} dp \\label{eq:60} \\tag{60}$$\n", + "\n", + "Π’Π΅ΠΏΠ΅Ρ€ΡŒ для $p > 1$ \n", + "\n", + "$$\\varPhi(p) = 1 - \\frac{exp \\left(-\\frac{p^2}{2}\\right)}{\\sqrt{2\\pi}p\\left(1 + \\frac{\\theta p}{p^2}\\right)} \\label{eq:61} \\tag{61}$$\n", + "\n", + "Π³Π΄Π΅ $0 < \\theta_p < 1$ (см. [4], стр. 136, Π·Π°Π΄Π°Ρ‡Π° 18).\n", + "\n", + "Π’Ρ‹ΠΏΠΎΠ»Π½ΠΈΠΌ подстановку \n", + "\n", + "$$p = \\sqrt{2\\log n - \\log(2\\log n)} + \\frac{u - \\log \\sqrt{2\\pi}}{\\sqrt{2 \\log n}} \\label{eq:62} \\tag{62}$$\n", + "\n", + "ΠΈ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ \n", + "\n", + "$$W_n \\sim \\frac{4\\sqrt{2\\pi\\log n}}{n^2}$$\n", + "\n", + "Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, $E_n \\sim 2\\sqrt{2\\pi \\log n}$, Ρ‡Ρ‚ΠΎ ΠΈ Ρ‚Ρ€Π΅Π±ΠΎΠ²Π°Π»ΠΎΡΡŒ Π΄ΠΎΠΊΠ°Π·Π°Ρ‚ΡŒ." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Π›ΠΈΡ‚Π΅Ρ€Π°Ρ‚ΡƒΡ€Π°\n", + "\n", + "1) A. RΓ©nyi, R. Sulanke: Über die konvexe HΓΌlle von n zufΓ€llig gewΓ€hlten Punkten. Z. Wahrscheinlichkeitstheorie 2 (1963), 75--84 \n", + "2) W. Blaschke: Integralgeometrie. 3. Aufl. Berlin: VEB Deutscher Verlag der Wissenschaften 1955, 1--130. \n", + "3) W. Blaschke, K. Reidemeister: Differentialgeometrie II, Berlin: Springer 1923, 60. \n", + "4) A. RΓ©nyi: Wahrscheinlichkeitsrechnung, mit einem Anhang ΓΌber Informationstheorie. Berlin: VEB Deutscher Verlag der Wissenschaften 1962, 1--547." + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.6.3" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} From 3fa831dc51e309d87e88ec6459e9999617a0de52 Mon Sep 17 00:00:00 2001 From: Egor Makarenko Date: Tue, 3 Apr 2018 04:21:38 +0300 Subject: [PATCH 2/5] Small markdown & latex fixes --- ..._expectation_of_convex_hull_vertices.ipynb | 82 +++++++++++++------ 1 file changed, 56 insertions(+), 26 deletions(-) diff --git a/33_expectation_of_convex_hull_vertices/33_expectation_of_convex_hull_vertices.ipynb b/33_expectation_of_convex_hull_vertices/33_expectation_of_convex_hull_vertices.ipynb index 85403b4..0fe9035 100644 --- a/33_expectation_of_convex_hull_vertices/33_expectation_of_convex_hull_vertices.ipynb +++ b/33_expectation_of_convex_hull_vertices/33_expectation_of_convex_hull_vertices.ipynb @@ -2,15 +2,19 @@ "cells": [ { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "slideshow": { + "slide_type": "-" + } + }, "source": [ "# Π‘Ρ€Π΅Π΄Π½Π΅Π΅ количСство Π²Π΅Ρ€ΡˆΠΈΠ½ Π² Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠ΅\n", "\n", - "Алгоритм ДТарвиса ΠΈΠΌΠ΅Π΅Ρ‚ ΡΠ»ΠΎΠΆΠ½ΠΎΡΡ‚ΡŒ $O(X N)$, Π³Π΄Π΅ $X$ -- число Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ, Π° $N$ -- ΠΎΠ±Ρ‰Π΅Π΅ число Ρ‚ΠΎΡ‡Π΅ΠΊ. Π§Ρ‚ΠΎΠ±Ρ‹ ΠΎΡ†Π΅Π½ΠΈΡ‚ΡŒ ΡΠ»ΠΎΠΆΠ½ΠΎΡΡ‚ΡŒ этого Π°Π»Π³ΠΎΡ€ΠΈΡ‚ΠΌΠ° Π² срСднСм случаС, Π½Π΅ΠΎΠ±Ρ…ΠΎΠ΄ΠΈΠΌΠΎ Π²Ρ‹Ρ‡ΠΈΡΠ»ΠΈΡ‚ΡŒ $E(X)$ -- матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ Π²Π΅Π»ΠΈΡ‡ΠΈΠ½Ρ‹ $X$. Для Π΅Π³ΠΎ Π°Π½Π°Π»ΠΈΠ·Π° ΠΌΡ‹ Π΄ΠΎΠ»ΠΆΠ½Ρ‹ ΠΎΠΏΡ€Π΅Π΄Π΅Π»ΠΈΡ‚ΡŒ ΠΎΠ³Ρ€Π°Π½ΠΈΡ‡Π΅Π½Π½ΡƒΡŽ ΠΎΠ±Π»Π°ΡΡ‚ΡŒ, ΠΈΠ· ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ Ρ‚ΠΎΡ‡ΠΊΠΈ, Ρ‚Π°ΠΊ ΠΊΠ°ΠΊ ΠΎΠ½ΠΈ ΠΌΠΎΠ³ΡƒΡ‚ Π±Ρ‹Ρ‚ΡŒ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны лишь Π² мноТСствС с ΠΊΠΎΠ½Π΅Ρ‡Π½ΠΎΠΉ ΠΌΠ΅Ρ€ΠΎΠΉ Π›Π΅Π±Π΅Π³Π° [[Kendall, Moran (1963)](https://doi.org/10.1002/bimj.19650070322)]. \n", + "Алгоритм ДТарвиса ΠΈΠΌΠ΅Π΅Ρ‚ ΡΠ»ΠΎΠΆΠ½ΠΎΡΡ‚ΡŒ $O(X N)$, Π³Π΄Π΅ $X$ β€” число Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ, Π° $N$ β€” ΠΎΠ±Ρ‰Π΅Π΅ число Ρ‚ΠΎΡ‡Π΅ΠΊ. Π§Ρ‚ΠΎΠ±Ρ‹ ΠΎΡ†Π΅Π½ΠΈΡ‚ΡŒ ΡΠ»ΠΎΠΆΠ½ΠΎΡΡ‚ΡŒ этого Π°Π»Π³ΠΎΡ€ΠΈΡ‚ΠΌΠ° Π² срСднСм случаС, Π½Π΅ΠΎΠ±Ρ…ΠΎΠ΄ΠΈΠΌΠΎ Π²Ρ‹Ρ‡ΠΈΡΠ»ΠΈΡ‚ΡŒ $E(X)$ β€” матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ Π²Π΅Π»ΠΈΡ‡ΠΈΠ½Ρ‹ $X$. Для Π΅Π³ΠΎ Π°Π½Π°Π»ΠΈΠ·Π° ΠΌΡ‹ Π΄ΠΎΠ»ΠΆΠ½Ρ‹ ΠΎΠΏΡ€Π΅Π΄Π΅Π»ΠΈΡ‚ΡŒ ΠΎΠ³Ρ€Π°Π½ΠΈΡ‡Π΅Π½Π½ΡƒΡŽ ΠΎΠ±Π»Π°ΡΡ‚ΡŒ, ΠΈΠ· ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ Ρ‚ΠΎΡ‡ΠΊΠΈ, Ρ‚Π°ΠΊ ΠΊΠ°ΠΊ ΠΎΠ½ΠΈ ΠΌΠΎΠ³ΡƒΡ‚ Π±Ρ‹Ρ‚ΡŒ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны лишь Π² мноТСствС с ΠΊΠΎΠ½Π΅Ρ‡Π½ΠΎΠΉ ΠΌΠ΅Ρ€ΠΎΠΉ Π›Π΅Π±Π΅Π³Π° [[Kendall, Moran (1963)](https://doi.org/10.1002/bimj.19650070322)]. \n", "\n", "НиТС ΠΏΡ€ΠΈΠ²Π΅Π΄Ρ‘Π½ ряд Ρ‚Π΅ΠΎΡ€Π΅ΠΌ, ΠΏΡ€Π΅Π΄ΡΡ‚Π°Π²Π»ΡΡŽΡ‰ΠΈΡ… ΠΎΡ†Π΅Π½ΠΊΡƒ срСднСго количСства Π²Π΅Ρ€ΡˆΠΈΠ½ Π² Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠ΅ ΠΏΡ€ΠΈ Ρ€Π°Π·Π»ΠΈΡ‡Π½Ρ‹Ρ… распрСдСлСниях Ρ‚ΠΎΡ‡Π΅ΠΊ.\n", "\n", - "**Π’Π΅ΠΎΡ€Π΅ΠΌΠ° 1** [[RΓ©nyi, Sulanke (1963)](https://doi.org/10.1007/BF00535300)]. *Если $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ ΠΈ нСзависимо ΠΈΠ· Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° Π½Π° плоскости, Ρ‚ΠΎ ΠΏΡ€ΠΈ $N\\to\\infty$ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ этих Ρ‚ΠΎΡ‡Π΅ΠΊ $E(X)=\\frac23 r(\\log N + \\gamma) + O(1)$, Π³Π΄Π΅ $\\gamma$ - константа Π­ΠΉΠ»Π΅Ρ€Π°.*\n", + "**Π’Π΅ΠΎΡ€Π΅ΠΌΠ° 1** [[RΓ©nyi, Sulanke (1963)](https://doi.org/10.1007/BF00535300)]. *Если $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ ΠΈ нСзависимо ΠΈΠ· Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° Π½Π° плоскости, Ρ‚ΠΎ ΠΏΡ€ΠΈ $N\\to\\infty$ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ этих Ρ‚ΠΎΡ‡Π΅ΠΊ $E(X)=\\frac23 r \\, (\\log N + \\gamma) + O(1)$, Π³Π΄Π΅ $\\gamma$ - константа Π­ΠΉΠ»Π΅Ρ€Π°.*\n", "\n", "**Π’Π΅ΠΎΡ€Π΅ΠΌΠ° 2** [[Raynaud (1970)](https://doi.org/10.1017/S0021900200026917)]. *Если $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ ΠΈ нСзависимо ΠΈΠ· внутрСнности $k$-ΠΌΠ΅Ρ€Π½ΠΎΠΉ гипСрсфСры, Ρ‚ΠΎ ΠΏΡ€ΠΈ $N\\to\\infty$ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π³ΠΈΠΏΠ΅Ρ€Π³Ρ€Π°Π½Π΅ΠΉ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ этих Ρ‚ΠΎΡ‡Π΅ΠΊ асимптотичСски стрСмится ΠΊ $E(f) = O(N^{(k-1)/(k+1)})$*\n", "\n", @@ -35,15 +39,15 @@ "metadata": {}, "source": [ "# О Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠ΅ N случайно Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Ρ… Ρ‚ΠΎΡ‡Π΅ΠΊ\n", - "> [A. Renyi ΠΈ R. Sulanke (1963 Π³.)](https://1drv.ms/b/s!AsLTB1ON1LwNjMhGXq0xv1LW6Wx0-w)\n", + "[[RΓ©nyi, Sulanke (1963)](https://doi.org/10.1007/BF00535300)]\n", "\n", - "ΠŸΡƒΡΡ‚ΡŒ $P_i\\:(i=1,\\:2,\\:\\ldots,\\:n)$ -- Ρ‚ΠΎΡ‡ΠΊΠΈ Π½Π° плоскости, Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Π΅ случайно ΠΈ нСзависимо Π² соотвСтствии с Π½Π΅ΠΊΠΎΡ‚ΠΎΡ€Ρ‹ΠΌ распрСдСлСниСм, Π° $H_n$ - выпуклая ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠ° этих Ρ‚ΠΎΡ‡Π΅ΠΊ, которая, ΠΊΠ°ΠΊ извСстно, являСтся Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΌ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΎΠΌ. Число $X_n$ Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ -- случайная Π²Π΅Π»ΠΈΡ‡ΠΈΠ½Π°, ΠΈ Π² Π΄Π°Π½Π½ΠΎΠΉ Ρ€Π°Π±ΠΎΡ‚Π΅ Π±ΡƒΠ΄Π΅Ρ‚ исслСдовано Π΅Ρ‘ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ $E_n = E(X_n)$ ΠΏΡ€ΠΈ $n\\to\\infty$ Π² Ρ€Π°Π·Π»ΠΈΡ‡Π½Ρ‹Ρ… прСдполоТСниях ΠΎ распрСдСлСнии Ρ‚ΠΎΡ‡Π΅ΠΊ $P_i$.\n", + "ΠŸΡƒΡΡ‚ΡŒ $P_i\\:(i=1,\\:2,\\:\\ldots,\\:n)$ β€” Ρ‚ΠΎΡ‡ΠΊΠΈ Π½Π° плоскости, Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Π΅ случайно ΠΈ нСзависимо Π² соотвСтствии с Π½Π΅ΠΊΠΎΡ‚ΠΎΡ€Ρ‹ΠΌ распрСдСлСниСм, Π° $H_n$ - выпуклая ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠ° этих Ρ‚ΠΎΡ‡Π΅ΠΊ, которая, ΠΊΠ°ΠΊ извСстно, являСтся Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΌ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΎΠΌ. Число $X_n$ Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ β€” случайная Π²Π΅Π»ΠΈΡ‡ΠΈΠ½Π°, ΠΈ Π² Π΄Π°Π½Π½ΠΎΠΉ Ρ€Π°Π±ΠΎΡ‚Π΅ Π±ΡƒΠ΄Π΅Ρ‚ исслСдовано Π΅Ρ‘ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ $E_n = E(X_n)$ ΠΏΡ€ΠΈ $n\\to\\infty$ Π² Ρ€Π°Π·Π»ΠΈΡ‡Π½Ρ‹Ρ… прСдполоТСниях ΠΎ распрСдСлСнии Ρ‚ΠΎΡ‡Π΅ΠΊ $P_i$.\n", "\n", - "* Π’ $\\S1$ рассмотрим случай, ΠΊΠΎΠ³Π΄Π° Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i$ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны Π²Π½ΡƒΡ‚Ρ€ΠΈ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ r-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$. Как Π±ΡƒΠ΄Π΅Ρ‚ ΠΏΠΎΠΊΠ°Π·Π°Π½ΠΎ, $E_n = (2r/3)(\\log n + \\gamma) + T + O(1)$, Π³Π΄Π΅ $\\gamma$ - постоянная Π­ΠΉΠ»Π΅Ρ€Π°:\n", + "* Π’ $\\S1$ рассмотрим случай, ΠΊΠΎΠ³Π΄Π° Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i$ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны Π²Π½ΡƒΡ‚Ρ€ΠΈ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ r-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$. Как Π±ΡƒΠ΄Π΅Ρ‚ ΠΏΠΎΠΊΠ°Π·Π°Π½ΠΎ, $E_n = (2r\\,/\\,3)(\\log n + \\gamma) + T + O(1)$, Π³Π΄Π΅ $\\gamma$ - постоянная Π­ΠΉΠ»Π΅Ρ€Π°:\n", " \n", " $$\\gamma = \\lim_{n\\to\\infty} \\left( \\sum_{k=1}^{n} \\frac1k - \\log n \\right)=\\lim_{n\\to\\infty} \\left( 1+\\frac12+\\frac13+\\ldots+\\frac1n - \\log n \\right) \\notag$$\n", " \n", - " ΠŸΡ€ΠΈ этом Π²Π΅Π»ΠΈΡ‡ΠΈΠ½Π° $T = T(K)$ зависит ΠΎΡ‚ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° K.\n", + " ΠŸΡ€ΠΈ этом Π²Π΅Π»ΠΈΡ‡ΠΈΠ½Π° $T = T(K)$ зависит ΠΎΡ‚ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$.\n", "\n", "* Π’ $\\S2$ Π΄ΠΎΠΊΠ°ΠΆΠ΅ΠΌ ΡΠ»Π΅ΠΌΠ΅Π½Ρ‚Π°Ρ€Π½ΡƒΡŽ Π³Π΅ΠΎΠΌΠ΅Ρ‚Ρ€ΠΈΡ‡Π΅ΡΠΊΡƒΡŽ Ρ‚Π΅ΠΎΡ€Π΅ΠΌΡƒ ΠΎ Ρ‚ΠΎΠΌ, Ρ‡Ρ‚ΠΎ $T(K)$ ΠΏΡ€ΠΈΠ½ΠΈΠΌΠ°Π΅Ρ‚ ΠΌΠ°ΠΊΡΠΈΠΌΠ°Π»ΡŒΠ½Ρ‹Π΅ значСния для ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΎΠ², ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Π΅ Π°Ρ„Ρ„ΠΈΠ½Π½Ρ‹ΠΌΠΈ прСобразованиями ΠΌΠΎΠΆΠ½ΠΎ привСсти ΠΊ ΠΏΡ€Π°Π²ΠΈΠ»ΡŒΠ½ΠΎΠΌΡƒ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΡƒ, ΠΈ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ для Π½ΠΈΡ….\n", "\n", @@ -69,6 +73,12 @@ "source": [ "## 1. Π‘Π»ΡƒΡ‡Π°ΠΉΠ½ΠΎ Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Π΅ Ρ‚ΠΎΡ‡ΠΊΠΈ Π² Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΌ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ΅\n", "\n", + "**Π£Ρ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 1.** Если Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i$ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны Π²Π½ΡƒΡ‚Ρ€ΠΈ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ r-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$, матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ $E_n = (2r\\,/\\,3)(\\log n + \\gamma) + T + O(1)$, Π³Π΄Π΅ $\\gamma$ - постоянная Π­ΠΉΠ»Π΅Ρ€Π°.\n", + "\n", + "**Π”ΠΎΠΊΠ°Π·Π°Ρ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ:**\n", + "\n", + "$\\triangleright$\n", + "
\n", "ΠŸΡƒΡΡ‚ΡŒ $K$ - Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΉ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ с Π²Π΅Ρ€ΡˆΠΈΠ½Π°ΠΌΠΈ $A_1,\\:A_2,\\:\\ldots,\\:A_r$ ΠΈ ΡΠΎΠΎΡ‚Π²Π΅Ρ‚ΡΡ‚Π²ΡƒΡŽΡ‰ΠΈΠΌΠΈ ΡƒΠ³Π»Π°ΠΌΠΈ $\\vartheta_1,\\:\\vartheta_2,\\:\\ldots,\\:\\vartheta_r$. ΠžΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Π·Π° $a_k$ Π΄Π»ΠΈΠ½Ρƒ стороны $A_k A_{k+1}$, Π³Π΄Π΅ $A_{r+1} = A_1$ $(k = 1,\\:2,\\:\\ldots,\\:r)$.\n", "\n", "![1.png](attachment:1.png)\n", @@ -120,13 +130,13 @@ "\n", "Π—Π΄Π΅ΡΡŒ $C_i$ ΠΎΠ·Π½Π°Ρ‡Π°Π΅Ρ‚ мноТСство ΠΏΠ°Ρ€ Ρ‚ΠΎΡ‡Π΅ΠΊ $(P_1,\\:P_2)$, для ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… прямая $P_1P_2$ пСрСсСкаСт смСТныС стороны $A_{i-1}A_i$ ΠΈ $A_iA_{i+1}$ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$, a $D_i$ - мноТСство ΠΏΠ°Ρ€ $(P_1,\\:P_2)$, для ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… прямая $P_1P_2$ пСрСсСкаСт стороны $A_{i-1}A_i$ ΠΈ $A_{i+1}A_{i+2}$ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$, Ρ€Π°Π·Π΄Π΅Π»Ρ‘Π½Π½Ρ‹Π΅ Π΄Ρ€ΡƒΠ³ΠΎΠΉ стороной.\n", "\n", - "Π‘Π½Π°Ρ‡Π°Π»Π° рассмотрим ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π» $I_i$. ΠžΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Ρ‡Π΅Ρ€Π΅Π· $G_{ab}$ мноТСство ΠΏΠ°Ρ€ $(P_1,\\:P_2)$, для ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… Π²Ρ‹ΠΏΠΎΠ»Π½Π΅Π½Ρ‹ нСравСнства $|A_iQ_1| < a$ ΠΈ $|A_iQ_2| < b$, Π³Π΄Π΅ $Q_1$ - Ρ‚ΠΎΡ‡ΠΊΠ° пСрСсСчСния прямой $P_1P_2$ со стороной $A_{i-1}A_i$, Π° $Q_2$ - Ρ‚ΠΎΡ‡ΠΊΠ° пСрСсСчСния $P_1P_2$ со стороной $A_iA_{i+1}$. Π—Π΄Π΅ΡΡŒ ΠΈ Π΄Π°Π»Π΅Π΅ $|AB|$ -- Π΄Π»ΠΈΠ½Π° ΠΎΡ‚Ρ€Π΅Π·ΠΊΠ° $AB$.\n", + "Π‘Π½Π°Ρ‡Π°Π»Π° рассмотрим ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π» $I_i$. ΠžΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Ρ‡Π΅Ρ€Π΅Π· $G_{ab}$ мноТСство ΠΏΠ°Ρ€ $(P_1,\\:P_2)$, для ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… Π²Ρ‹ΠΏΠΎΠ»Π½Π΅Π½Ρ‹ нСравСнства $\\left|A_iQ_1\\right| < a$ ΠΈ $\\left|A_iQ_2\\right| < b$, Π³Π΄Π΅ $Q_1$ - Ρ‚ΠΎΡ‡ΠΊΠ° пСрСсСчСния прямой $P_1P_2$ со стороной $A_{i-1}A_i$, Π° $Q_2$ - Ρ‚ΠΎΡ‡ΠΊΠ° пСрСсСчСния $P_1P_2$ со стороной $A_iA_{i+1}$. Π—Π΄Π΅ΡΡŒ ΠΈ Π΄Π°Π»Π΅Π΅ $\\left|AB\\right|$ β€” Π΄Π»ΠΈΠ½Π° ΠΎΡ‚Ρ€Π΅Π·ΠΊΠ° $AB$.\n", "\n", "По элСмСнтарным расчётам получаСтся \n", "\n", "$$\\int\\limits_{G_{ab}} dP_1dP_2 = \\frac{a^2b^2\\sin^2\\vartheta_i}{12} \\label{eq:9} \\tag{9}$$\n", "\n", - "ΠŸΠ»ΠΎΡ‰Π°Π΄ΡŒ Ρ‚Ρ€Π΅ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $Q_1A_iQ_2$, отсСкаСмого прямой $P_1P_2$, Ρ€Π°Π²Π½Π° $\\frac{1}{2} a b \\sin \\vartheta_i$. Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, взяв $a_{i-1} = |A_{i-1}A_i|$, $a_i = |A_iA_{i+1}|$, ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ \n", + "ΠŸΠ»ΠΎΡ‰Π°Π΄ΡŒ Ρ‚Ρ€Π΅ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $Q_1A_iQ_2$, отсСкаСмого прямой $P_1P_2$, Ρ€Π°Π²Π½Π° $\\frac{1}{2} a b \\sin \\vartheta_i$. Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, взяв $a_{i-1} = \\left|A_{i-1}A_i\\right|$, $a_i = \\left|A_iA_{i+1}\\right|$, ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ \n", "\n", "$$I_i = \\int\\limits_0^{a_{i-1}} \\int\\limits_0^{a_i} \\left(1 - \\frac{ab \\sin \\vartheta_i}{2F}\\right)^{n-2} \\sin^2 \\vartheta_i \\frac{ab}{3} \\,da\\,db \\label{eq:10} \\tag{10}$$ \n", "\n", @@ -172,7 +182,7 @@ "\n", "ΠŸΡƒΡΡ‚ΡŒ снова $Q_1$ - Ρ‚ΠΎΡ‡ΠΊΠ° пСрСсСчСния прямой $P_1P_2$ со стороной $A_{i-1}A_i$, Π° $Q_2$ - Ρ‚ΠΎΡ‡ΠΊΠ° пСрСсСчСния $P_1P_2$ со стороной $A_{i+1}A_{i+2}$, ΠΈ ΠΏΡƒΡΡ‚ΡŒ\n", "\n", - "$$a' = |A_iA_i^*|, \\;\\; b' = |A_{i+1}A_i^*|, \\;\\; a = |A_iQ_1|, \\;\\; b = |A_{i+1}Q_2| \\notag$$\n", + "$$a' = \\left|A_iA_i^*\\right|, \\;\\; b' = \\left|A_{i+1}A_i^*\\right|, \\;\\; a = \\left|A_iQ_1\\right|, \\;\\; b = \\left|A_{i+1}Q_2\\right| \\notag$$\n", "\n", "$G_{ab}$ Π±Ρ‹Π»ΠΎ ΠΎΠΏΡ€Π΅Π΄Π΅Π»Π΅Π½ΠΎ Π²Ρ‹ΡˆΠ΅. ΠŸΠΎΠ»ΡƒΡ‡ΠΈΠΌ ΠΈΠ· $(9)$:\n", "\n", @@ -202,7 +212,10 @@ "\n", "$$E_n = \\frac{2r}{3} (\\log n + \\gamma) - \\frac23 \\sum\\limits_{i=1}^{r} S_i + o(1) \\notag$$\n", "\n", - "**Π£Ρ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 1.** ΠŸΡƒΡΡ‚ΡŒ $K$ - Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΉ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ с $r$ сторонами. ΠœΠ°Ρ‚Π΅ΠΌΠ°Ρ‚ΠΈΡ‡Π΅ΡΠΊΠΎΠ΅ ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ $E_n$ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ $H_n$, построСнной Π½Π° $n$ случайных Ρ‚ΠΎΡ‡ΠΊΠ°Ρ… ΠΈΠ· $K$, Ρ€Π°Π²Π½ΠΎ\n", + "
\n", + "$\\triangleleft$\n", + "\n", + "ΠŸΡƒΡΡ‚ΡŒ $K$ - Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΉ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ с $r$ сторонами. ΠœΠ°Ρ‚Π΅ΠΌΠ°Ρ‚ΠΈΡ‡Π΅ΡΠΊΠΎΠ΅ ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ $E_n$ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ $H_n$, построСнной Π½Π° $n$ случайных Ρ‚ΠΎΡ‡ΠΊΠ°Ρ… ΠΈΠ· $K$, Ρ€Π°Π²Π½ΠΎ\n", "\n", "$$E_n = \\frac{2r}{3} (\\log n + \\gamma) + \\frac23 \\log \\frac{\\prod\\limits_1^r f_i}{F^r} + o(1) \\label{eq:20} \\tag{20}$$\n", "\n", @@ -234,6 +247,9 @@ "ΠΈΠ· ΠΏΡ€Π΅Π΄Ρ‹Π΄ΡƒΡ‰Π΅Π³ΠΎ ΠΏΠ°Ρ€Π°Π³Ρ€Π°Ρ„Π° максимальна Ρ‚ΠΎΠ³Π΄Π° ΠΈ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ Ρ‚ΠΎΠ³Π΄Π°, ΠΊΠΎΠ³Π΄Π° $K$ ΠΌΠΎΠΆΠ½ΠΎ ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΠΎΠ²Π°Ρ‚ΡŒ Π² ΠΏΡ€Π°Π²ΠΈΠ»ΡŒΠ½Ρ‹ΠΉ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ Π°Ρ„Ρ„ΠΈΠ½Π½Ρ‹ΠΌΠΈ прСобразованиями.\n", "\n", "**Π”ΠΎΠΊΠ°Π·Π°Ρ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ:**\n", + "\n", + "$\\triangleright$\n", + "
\n", "ΠŸΡƒΡΡ‚ΡŒ $\\alpha_i = \\overrightarrow{A_{i-1}A_i}$, Π³Π΄Π΅ $A_{r+i} = A_i$. Π’ΠΎΠ³Π΄Π°\n", "\n", "$$f_i = \\frac12 [\\alpha_i, \\: \\alpha_{i+1}] \\\\ F = \\frac12 \\sum\\limits_{i=1}^{r-2} [\\alpha_1 + \\alpha_2 + \\ldots + \\alpha_i, \\: \\alpha_{i+1}] \\label{eq:24} \\tag{24}$$\n", @@ -329,7 +345,9 @@ "\n", "ΠžΡ‚ΡΡŽΠ΄Π° слСдуСт, Ρ‡Ρ‚ΠΎ всС стороны нашСго $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$ ΠΈΠΌΠ΅ΡŽΡ‚ ΠΎΠ΄ΠΈΠ½Π°ΠΊΠΎΠ²ΡƒΡŽ Π΄Π»ΠΈΠ½Ρƒ ΠΈ Ρ‡Ρ‚ΠΎ Π»ΡŽΠ±Ρ‹Π΅ Π΄Π²Π΅ смСТныС стороны ΠΏΠ΅Ρ€Π΅ΡΠ΅ΠΊΠ°ΡŽΡ‚ΡΡ ΠΏΠΎΠ΄ ΠΎΠ΄Π½ΠΈΠΌ ΡƒΠ³Π»ΠΎΠΌ. Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ Π΄ΠΎΠ»ΠΆΠ΅Π½ Π±Ρ‹Ρ‚ΡŒ ΠΏΡ€Π°Π²ΠΈΠ»ΡŒΠ½Ρ‹ΠΌ ΠΎΡ‚Π½ΠΎΡΠΈΡ‚Π΅Π»ΡŒΠ½ΠΎ этой ΠΌΠ΅Ρ‚Ρ€ΠΈΠΊΠΈ.\n", "\n", - "Π­Ρ‚ΠΎ Π΄ΠΎΠΊΠ°Π·Ρ‹Π²Π°Π΅Ρ‚ Π΅Π΄ΠΈΠ½ΡΡ‚Π²Π΅Π½Π½ΠΎΡΡ‚ΡŒ максимального ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ°. Π•Π³ΠΎ сущСствованиС слСдуСт ΠΈΠ· ограничСнности $Z(K)$ ΠΈ нСпрСрывности $F$ ΠΈ $Z$. Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, ΡƒΡ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 2 ΠΏΠΎΠ»Π½ΠΎΡΡ‚ΡŒΡŽ Π΄ΠΎΠΊΠ°Π·Π°Π½ΠΎ." + "Π­Ρ‚ΠΎ Π΄ΠΎΠΊΠ°Π·Ρ‹Π²Π°Π΅Ρ‚ Π΅Π΄ΠΈΠ½ΡΡ‚Π²Π΅Π½Π½ΠΎΡΡ‚ΡŒ максимального ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ°. Π•Π³ΠΎ сущСствованиС слСдуСт ΠΈΠ· ограничСнности $Z(K)$ ΠΈ нСпрСрывности $F$ ΠΈ $Z$. Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, ΡƒΡ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 2 ΠΏΠΎΠ»Π½ΠΎΡΡ‚ΡŒΡŽ Π΄ΠΎΠΊΠ°Π·Π°Π½ΠΎ.\n", + "
\n", + "$\\triangleleft$" ] }, { @@ -338,21 +356,27 @@ "source": [ "## 3. Π‘Π»ΡƒΡ‡Π°ΠΉΠ½ΠΎ Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Π΅ Ρ‚ΠΎΡ‡ΠΊΠΈ Π² Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ области с Π³Π»Π°Π΄ΠΊΠΎΠΉ Π³Ρ€Π°Π½ΠΈΡ†Π΅ΠΉ\n", "\n", + "**Π£Ρ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 3.** Если Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i$ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны Π² Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ области с Π³Π»Π°Π΄ΠΊΠΎΠΉ Π³Ρ€Π°Π½ΠΈΡ†Π΅ΠΉ Π½Π° плоскости, матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ $E_n = O(\\sqrt[3]n)$.\n", + "\n", + "**Π”ΠΎΠΊΠ°Π·Π°Ρ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ:** \n", + "\n", + "$\\triangleright$\n", + "
\n", "ΠŸΡƒΡΡ‚ΡŒ Ρ‚Π΅ΠΏΠ΅Ρ€ΡŒ $K$ - выпуклая ΠΎΠ±Π»Π°ΡΡ‚ΡŒ, Π³Ρ€Π°Π½ΠΈΡ†Π° ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ ΠΈΠΌΠ΅Π΅Ρ‚ Π½Π΅ΠΏΡ€Π΅Ρ€Ρ‹Π²Π½ΡƒΡŽ ΠΊΡ€ΠΈΠ²ΠΈΠ·Π½Ρƒ ΠΊΠ°ΠΊ Ρ„ΡƒΠ½ΠΊΡ†ΠΈΡŽ ΠΎΡ‚ Π΄Π»ΠΈΠ½Ρ‹ Π΄ΡƒΠ³ΠΈ. Π’ обозначСниях ΠΏΡ€Π΅Π΄Ρ‹Π΄ΡƒΡ‰ΠΈΡ… ΠΏΠ°Ρ€Π°Π³Ρ€Π°Ρ„ΠΎΠ² ΠΌΡ‹ снова ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ\n", "\n", "$$E_n = C_n^2 \\cdot W_n \\notag$$\n", "\n", "Богласно ΡƒΡ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΡŽ ΠΎ ΠΌΠ΅Ρ€Π΅ мноТСства ΠΏΠ°Ρ€ Ρ‚ΠΎΡ‡Π΅ΠΊ, извСстному ΠΈΠ· ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π»ΡŒΠ½ΠΎΠΉ Π³Π΅ΠΎΠΌΠ΅Ρ‚Ρ€ΠΈΠΈ (см. Blascke [2], S 17, (86)):\n", "\n", - "$$dP_1 dP_2 = |t_1 - t_2| \\, dt_1 \\, dt_2 \\, d\\varphi \\, dp \\label{eq:41} \\tag{41}$$\n", + "$$dP_1 dP_2 = \\left|t_1 - t_2\\right| \\, dt_1 \\, dt_2 \\, d\\varphi \\, dp \\label{eq:41} \\tag{41}$$\n", "\n", - "Π³Π΄Π΅ $p$ -- Π΄Π»ΠΈΠ½Π° пСрпСндикуляра ΠΊ прямой $P_1 P_2$ ΠΈΠ· Π½Π°Ρ‡Π°Π»Π° ΠΊΠΎΠΎΡ€Π΄ΠΈΠ½Π°Ρ‚, $\\varphi$ -- ΡƒΠ³ΠΎΠ» Π½Π°ΠΊΠ»ΠΎΠ½Π° этого пСрпСндикуляра, $t_1, \\: t_2$ -- расстояния ΠΏΠΎ этой прямой ΠΎΡ‚ $P_1$ ΠΈ $P_2$ Π΄ΠΎ Ρ‚ΠΎΡ‡ΠΊΠΈ пСрСсСчСния $P_1 P_2$ с пСрпСндикуляром. ΠžΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Ρ‡Π΅Ρ€Π΅Π· $f$ ΠΏΠ»ΠΎΡ‰Π°Π΄ΡŒ участка, отсСкаСмого ΠΎΡ‚ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$ прямой $P_1 P_2$. Π’ΠΎΠ³Π΄Π° ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ\n", + "Π³Π΄Π΅ $p$ β€” Π΄Π»ΠΈΠ½Π° пСрпСндикуляра ΠΊ прямой $P_1 P_2$ ΠΈΠ· Π½Π°Ρ‡Π°Π»Π° ΠΊΠΎΠΎΡ€Π΄ΠΈΠ½Π°Ρ‚, $\\varphi$ β€” ΡƒΠ³ΠΎΠ» Π½Π°ΠΊΠ»ΠΎΠ½Π° этого пСрпСндикуляра, $t_1, \\: t_2$ β€” расстояния ΠΏΠΎ этой прямой ΠΎΡ‚ $P_1$ ΠΈ $P_2$ Π΄ΠΎ Ρ‚ΠΎΡ‡ΠΊΠΈ пСрСсСчСния $P_1 P_2$ с пСрпСндикуляром. ΠžΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Ρ‡Π΅Ρ€Π΅Π· $f$ ΠΏΠ»ΠΎΡ‰Π°Π΄ΡŒ участка, отсСкаСмого ΠΎΡ‚ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$ прямой $P_1 P_2$. Π’ΠΎΠ³Π΄Π° ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ\n", "\n", - "$$E_n = \\cfrac{C_n^2}{F^2} \\int\\limits_0^{2\\pi} \\left\\{ \\int\\limits_0^{p(\\varphi)} \\left(1-\\frac{f}{F} \\right)^{n-2} ( \\int \\int |t_1 - t_2| \\, dt_1 \\, dt_2) \\, dp \\right\\} d\\varphi \\label{eq:42} \\tag{42}$$\n", + "$$E_n = \\cfrac{C_n^2}{F^2} \\int\\limits_0^{2\\pi} \\left\\{ \\int\\limits_0^{p(\\varphi)} \\left(1-\\frac{f}{F} \\right)^{n-2} ( \\iint \\left|t_1 - t_2\\right| \\, dt_1 \\, dt_2) \\, dp \\right\\} d\\varphi \\label{eq:42} \\tag{42}$$\n", "\n", "Π§Π΅Ρ€Π΅Π· $l(p, \\phi)$ ΠΎΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Π΄Π»ΠΈΠ½Ρƒ Ρ…ΠΎΡ€Π΄Ρ‹ области $K$, Π»Π΅ΠΆΠ°Ρ‰Π΅ΠΉ Π½Π° прямой с ΠΊΠΎΠΎΡ€Π΄ΠΈΠ½Π°Ρ‚Π°ΠΌΠΈ $p, \\varphi$. Π’ΠΎΠ³Π΄Π°\n", "\n", - "$$\\int \\int |t_1 - t_2| \\, dt_1 \\, dt_2 = \\frac{l^3(p, \\varphi)}{3} \\label{eq:43} \\tag{43}$$\n", + "$$\\iint \\left|t_1 - t_2\\right| \\, dt_1 \\, dt_2 = \\frac{l^3(p, \\varphi)}{3} \\label{eq:43} \\tag{43}$$\n", "\n", "Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ,\n", "\n", @@ -364,7 +388,7 @@ "\n", "ΠšΡ€ΠΎΠΌΠ΅ Ρ‚ΠΎΠ³ΠΎ,\n", "\n", - "$$|dp| = \\frac12 r(\\varphi) \\sin(\\alpha/2) |d\\alpha| \\label{eq:46} \\tag{46}$$\n", + "$$\\left| \\, dp \\, \\right| = \\frac12 r(\\varphi) \\sin(\\alpha/2) \\left| \\, d\\alpha \\, \\right| \\label{eq:46} \\tag{46}$$\n", "\n", "НаконСц,\n", "\n", @@ -389,11 +413,13 @@ "\n", "ΠžΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Π·Π° $s$ Π΄Π»ΠΈΠ½Ρƒ Π΄ΡƒΠ³ΠΈ Π³Ρ€Π°Π½ΠΈΡ†Ρ‹ $K$ ΠΈ, считая $d\\varphi/ds = 1/r$, ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ ΡΠ»Π΅Π΄ΡƒΡŽΡ‰Π΅Π΅ ΡƒΡ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅:\n", "\n", - "**Π£Ρ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 3.** Если $K$ - выпуклая ΠΎΠ±Π»Π°ΡΡ‚ΡŒ, граничная кривая ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ ΠΈΠΌΠ΅Π΅Ρ‚ Π½Π΅ΠΏΡ€Π΅Ρ€Ρ‹Π²Π½ΡƒΡŽ ΠΊΡ€ΠΈΠ²ΠΈΠ·Π½Ρƒ $x = x(s)$, Ρ‚ΠΎ\n", + "Если $K$ - выпуклая ΠΎΠ±Π»Π°ΡΡ‚ΡŒ, граничная кривая ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ ΠΈΠΌΠ΅Π΅Ρ‚ Π½Π΅ΠΏΡ€Π΅Ρ€Ρ‹Π²Π½ΡƒΡŽ ΠΊΡ€ΠΈΠ²ΠΈΠ·Π½Ρƒ $x = x(s)$, Ρ‚ΠΎ\n", "\n", "$$E_n \\sim \\Gamma \\left(\\frac{5}{3}\\right) \\, \\sqrt[3]{\\frac23} \\, (F^{-1/3} \\int\\limits_L x^{1/3} \\, ds) \\sqrt[3]{n} \\label{eq:52} \\tag{52}$$\n", "\n", - "Π˜Π½Ρ‚Π΅Π³Ρ€Π°Π» $\\int\\limits_L x^{1/3} \\, ds$, входящий Π² $(\\ref{eq:52})$, являСтся Π°Ρ„Ρ„ΠΈΠ½Π½ΠΎΠΉ Π΄Π»ΠΈΠ½ΠΎΠΉ ΠΊΡ€ΠΈΠ²ΠΎΠΉ $L$. Π›Π΅Π³ΠΊΠΎ Π²ΠΈΠ΄Π΅Ρ‚ΡŒ, Ρ‡Ρ‚ΠΎ ΠΌΠ½ΠΎΠΆΠΈΡ‚Π΅Π»ΡŒ $\\sqrt[3]{n}$ являСтся Π°Ρ„Ρ„ΠΈΠ½Π½ΠΎ-ΠΈΠ½Π²Π°Ρ€ΠΈΠ°Π½Ρ‚Π½Ρ‹ΠΌ. Из Π°Ρ„Ρ„ΠΈΠ½Π½ΠΎΠ³ΠΎ изопСримСтричСского свойства эллипсов (см. [3]) нСпосрСдствСнно слСдуСт, Ρ‡Ρ‚ΠΎ этот ΠΌΠ½ΠΎΠΆΠΈΡ‚Π΅Π»ΡŒ максималСн для эллипсов ΠΈ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ для эллипсов." + "Π˜Π½Ρ‚Π΅Π³Ρ€Π°Π» $\\int\\limits_L x^{1/3} \\, ds$, входящий Π² $(\\ref{eq:52})$, являСтся Π°Ρ„Ρ„ΠΈΠ½Π½ΠΎΠΉ Π΄Π»ΠΈΠ½ΠΎΠΉ ΠΊΡ€ΠΈΠ²ΠΎΠΉ $L$. Π›Π΅Π³ΠΊΠΎ Π²ΠΈΠ΄Π΅Ρ‚ΡŒ, Ρ‡Ρ‚ΠΎ ΠΌΠ½ΠΎΠΆΠΈΡ‚Π΅Π»ΡŒ $\\sqrt[3]{n}$ являСтся Π°Ρ„Ρ„ΠΈΠ½Π½ΠΎ-ΠΈΠ½Π²Π°Ρ€ΠΈΠ°Π½Ρ‚Π½Ρ‹ΠΌ. Из Π°Ρ„Ρ„ΠΈΠ½Π½ΠΎΠ³ΠΎ изопСримСтричСского свойства эллипсов (см. [3]) нСпосрСдствСнно слСдуСт, Ρ‡Ρ‚ΠΎ этот ΠΌΠ½ΠΎΠΆΠΈΡ‚Π΅Π»ΡŒ максималСн для эллипсов ΠΈ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ для эллипсов.\n", + "
\n", + "$\\triangleleft$" ] }, { @@ -402,13 +428,14 @@ "source": [ "## 4. ΠΠΎΡ€ΠΌΠ°Π»ΡŒΠ½ΠΎΠ΅ распрСдСлСниС случайных Ρ‚ΠΎΡ‡Π΅ΠΊ\n", "\n", - "Π’ этом ΠΏΠ°Ρ€Π°Π³Ρ€Π°Ρ„Π΅ ΠΌΡ‹ Π΄ΠΎΠΊΠ°ΠΆΠ΅ΠΌ ΡΠ»Π΅Π΄ΡƒΡŽΡ‰Π΅Π΅ ΡƒΡ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅:\n", - "\n", "**Π£Ρ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 4.** ΠŸΡƒΡΡ‚ΡŒ Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i \\:(i = 1, \\:2, \\:\\ldots, \\:n)$ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ Π½Π° плоскости нСзависимо Π΄Ρ€ΡƒΠ³ ΠΎΡ‚ Π΄Ρ€ΡƒΠ³Π° Π² соотвСтствии с Π½ΠΎΡ€ΠΌΠ°Π»ΡŒΠ½Ρ‹ΠΌ распрСдСлСниСм. Π’ΠΎΠ³Π΄Π°\n", "\n", "$$E_n \\sim 2 \\sqrt{2 \\pi \\log n} \\label{eq:53} \\tag{53}$$\n", "\n", - "**Π”ΠΎΠΊΠ°Π·Π°Ρ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ:**\n", + "**Π”ΠΎΠΊΠ°Π·Π°Ρ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ:** \n", + "\n", + "$\\triangleright$\n", + "
\n", "Π’Π°ΠΊ ΠΊΠ°ΠΊ $E_n$ ΠΈΠ½Π²Π°Ρ€ΠΈΠ°Π½Ρ‚Π½ΠΎ ΠΎΡ‚Π½ΠΎΡΠΈΡ‚Π΅Π»ΡŒΠ½ΠΎ Π°Ρ„Ρ„ΠΈΠ½Π½Ρ‹Ρ… ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΠΎΠ²Π°Π½ΠΈΠΉ плоскости, ΠΌΠΎΠΆΠ½ΠΎ ΡΡ‡ΠΈΡ‚Π°Ρ‚ΡŒ, Ρ‡Ρ‚ΠΎ функция плотности нашСго Π½ΠΎΡ€ΠΌΠ°Π»ΡŒΠ½ΠΎΠ³ΠΎ распрСдСлСния ΠΈΠΌΠ΅Π΅Ρ‚ Π²ΠΈΠ΄ $\\frac{1}{2\\pi} \\, exp \\left(-\\frac{x^2 + y^2}{2}\\right) \\notag$. Как ΠΈ Π² ΠΏΡ€Π΅Π΄Ρ‹Π΄ΡƒΡ‰ΠΈΡ… ΠΏΠ°Ρ€Π°Π³Ρ€Π°Ρ„Π°Ρ…, \n", "\n", "$$E_n = C_n^2 \\cdot W_n \\notag$$\n", @@ -456,7 +483,9 @@ "\n", "$$W_n \\sim \\frac{4\\sqrt{2\\pi\\log n}}{n^2}$$\n", "\n", - "Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, $E_n \\sim 2\\sqrt{2\\pi \\log n}$, Ρ‡Ρ‚ΠΎ ΠΈ Ρ‚Ρ€Π΅Π±ΠΎΠ²Π°Π»ΠΎΡΡŒ Π΄ΠΎΠΊΠ°Π·Π°Ρ‚ΡŒ." + "Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, $E_n \\sim 2\\sqrt{2\\pi \\log n}$, Ρ‡Ρ‚ΠΎ ΠΈ Ρ‚Ρ€Π΅Π±ΠΎΠ²Π°Π»ΠΎΡΡŒ Π΄ΠΎΠΊΠ°Π·Π°Ρ‚ΡŒ.\n", + "
\n", + "$\\triangleleft$" ] }, { @@ -465,14 +494,15 @@ "source": [ "## Π›ΠΈΡ‚Π΅Ρ€Π°Ρ‚ΡƒΡ€Π°\n", "\n", - "1) A. RΓ©nyi, R. Sulanke: Über die konvexe HΓΌlle von n zufΓ€llig gewΓ€hlten Punkten. Z. Wahrscheinlichkeitstheorie 2 (1963), 75--84 \n", - "2) W. Blaschke: Integralgeometrie. 3. Aufl. Berlin: VEB Deutscher Verlag der Wissenschaften 1955, 1--130. \n", + "1) A. RΓ©nyi, R. Sulanke: Über die konvexe HΓΌlle von n zufΓ€llig gewΓ€hlten Punkten. Z. Wahrscheinlichkeitstheorie 2 (1963), 75β€”84 \n", + "2) W. Blaschke: Integralgeometrie. 3. Aufl. Berlin: VEB Deutscher Verlag der Wissenschaften 1955, 1β€”130. \n", "3) W. Blaschke, K. Reidemeister: Differentialgeometrie II, Berlin: Springer 1923, 60. \n", - "4) A. RΓ©nyi: Wahrscheinlichkeitsrechnung, mit einem Anhang ΓΌber Informationstheorie. Berlin: VEB Deutscher Verlag der Wissenschaften 1962, 1--547." + "4) A. RΓ©nyi: Wahrscheinlichkeitsrechnung, mit einem Anhang ΓΌber Informationstheorie. Berlin: VEB Deutscher Verlag der Wissenschaften 1962, 1β€”547." ] } ], "metadata": { + "celltoolbar": "Slideshow", "kernelspec": { "display_name": "Python 3", "language": "python", From b3c49be1baffafec8ac5e94a73e6d514a4ec6f1f Mon Sep 17 00:00:00 2001 From: Egor Makarenko Date: Tue, 3 Apr 2018 05:30:27 +0300 Subject: [PATCH 3/5] Added slideshow mode --- ..._expectation_of_convex_hull_vertices.ipynb | 812 +- ...tation_of_convex_hull_vertices.slides.html | 12714 ++++++++++++++++ 2 files changed, 13392 insertions(+), 134 deletions(-) create mode 100644 33_expectation_of_convex_hull_vertices/33_expectation_of_convex_hull_vertices.slides.html diff --git a/33_expectation_of_convex_hull_vertices/33_expectation_of_convex_hull_vertices.ipynb b/33_expectation_of_convex_hull_vertices/33_expectation_of_convex_hull_vertices.ipynb index 0fe9035..bda7249 100644 --- a/33_expectation_of_convex_hull_vertices/33_expectation_of_convex_hull_vertices.ipynb +++ b/33_expectation_of_convex_hull_vertices/33_expectation_of_convex_hull_vertices.ipynb @@ -4,45 +4,140 @@ "cell_type": "markdown", "metadata": { "slideshow": { - "slide_type": "-" + "slide_type": "slide" } }, "source": [ "# Π‘Ρ€Π΅Π΄Π½Π΅Π΅ количСство Π²Π΅Ρ€ΡˆΠΈΠ½ Π² Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠ΅\n", "\n", - "Алгоритм ДТарвиса ΠΈΠΌΠ΅Π΅Ρ‚ ΡΠ»ΠΎΠΆΠ½ΠΎΡΡ‚ΡŒ $O(X N)$, Π³Π΄Π΅ $X$ β€” число Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ, Π° $N$ β€” ΠΎΠ±Ρ‰Π΅Π΅ число Ρ‚ΠΎΡ‡Π΅ΠΊ. Π§Ρ‚ΠΎΠ±Ρ‹ ΠΎΡ†Π΅Π½ΠΈΡ‚ΡŒ ΡΠ»ΠΎΠΆΠ½ΠΎΡΡ‚ΡŒ этого Π°Π»Π³ΠΎΡ€ΠΈΡ‚ΠΌΠ° Π² срСднСм случаС, Π½Π΅ΠΎΠ±Ρ…ΠΎΠ΄ΠΈΠΌΠΎ Π²Ρ‹Ρ‡ΠΈΡΠ»ΠΈΡ‚ΡŒ $E(X)$ β€” матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ Π²Π΅Π»ΠΈΡ‡ΠΈΠ½Ρ‹ $X$. Для Π΅Π³ΠΎ Π°Π½Π°Π»ΠΈΠ·Π° ΠΌΡ‹ Π΄ΠΎΠ»ΠΆΠ½Ρ‹ ΠΎΠΏΡ€Π΅Π΄Π΅Π»ΠΈΡ‚ΡŒ ΠΎΠ³Ρ€Π°Π½ΠΈΡ‡Π΅Π½Π½ΡƒΡŽ ΠΎΠ±Π»Π°ΡΡ‚ΡŒ, ΠΈΠ· ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ Ρ‚ΠΎΡ‡ΠΊΠΈ, Ρ‚Π°ΠΊ ΠΊΠ°ΠΊ ΠΎΠ½ΠΈ ΠΌΠΎΠ³ΡƒΡ‚ Π±Ρ‹Ρ‚ΡŒ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны лишь Π² мноТСствС с ΠΊΠΎΠ½Π΅Ρ‡Π½ΠΎΠΉ ΠΌΠ΅Ρ€ΠΎΠΉ Π›Π΅Π±Π΅Π³Π° [[Kendall, Moran (1963)](https://doi.org/10.1002/bimj.19650070322)]. \n", - "\n", - "НиТС ΠΏΡ€ΠΈΠ²Π΅Π΄Ρ‘Π½ ряд Ρ‚Π΅ΠΎΡ€Π΅ΠΌ, ΠΏΡ€Π΅Π΄ΡΡ‚Π°Π²Π»ΡΡŽΡ‰ΠΈΡ… ΠΎΡ†Π΅Π½ΠΊΡƒ срСднСго количСства Π²Π΅Ρ€ΡˆΠΈΠ½ Π² Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠ΅ ΠΏΡ€ΠΈ Ρ€Π°Π·Π»ΠΈΡ‡Π½Ρ‹Ρ… распрСдСлСниях Ρ‚ΠΎΡ‡Π΅ΠΊ.\n", - "\n", - "**Π’Π΅ΠΎΡ€Π΅ΠΌΠ° 1** [[RΓ©nyi, Sulanke (1963)](https://doi.org/10.1007/BF00535300)]. *Если $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ ΠΈ нСзависимо ΠΈΠ· Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° Π½Π° плоскости, Ρ‚ΠΎ ΠΏΡ€ΠΈ $N\\to\\infty$ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ этих Ρ‚ΠΎΡ‡Π΅ΠΊ $E(X)=\\frac23 r \\, (\\log N + \\gamma) + O(1)$, Π³Π΄Π΅ $\\gamma$ - константа Π­ΠΉΠ»Π΅Ρ€Π°.*\n", - "\n", - "**Π’Π΅ΠΎΡ€Π΅ΠΌΠ° 2** [[Raynaud (1970)](https://doi.org/10.1017/S0021900200026917)]. *Если $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ ΠΈ нСзависимо ΠΈΠ· внутрСнности $k$-ΠΌΠ΅Ρ€Π½ΠΎΠΉ гипСрсфСры, Ρ‚ΠΎ ΠΏΡ€ΠΈ $N\\to\\infty$ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π³ΠΈΠΏΠ΅Ρ€Π³Ρ€Π°Π½Π΅ΠΉ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ этих Ρ‚ΠΎΡ‡Π΅ΠΊ асимптотичСски стрСмится ΠΊ $E(f) = O(N^{(k-1)/(k+1)})$*\n", - "\n", - "**Π’Π΅ΠΎΡ€Π΅ΠΌΠ° 3** [[Raynaud (1970)](https://doi.org/10.1017/S0021900200026917)]. *Если $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ нСзависимо Π² соотвСтствии с $k$-ΠΌΠ΅Ρ€Π½Ρ‹ΠΌ Π½ΠΎΡ€ΠΌΠ°Π»ΡŒΠ½Ρ‹ΠΌ распрСдСлСниСм, Ρ‚ΠΎ ΠΏΡ€ΠΈ $N\\to\\infty$ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ этих Ρ‚ΠΎΡ‡Π΅ΠΊ асимптотичСски стрСмится ΠΊ $E(f) = O((\\log N)^{(k-1)/2})$*\n", - "\n", - "**Π’Π΅ΠΎΡ€Π΅ΠΌΠ° 4** [[Bentley, Kung, Schkolnick, Thompson (1978)](https://doi.org/10.1145/322092.322095)]. *Если ΠΊΠΎΠΎΡ€Π΄ΠΈΠ½Π°Ρ‚Ρ‹ $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ Π² $k$-ΠΌΠ΅Ρ€Π½ΠΎΠΌ пространствС Π²Ρ‹Π±Ρ€Π°Π½Ρ‹ нСзависимо Π² соотвСтствии с ΠΏΡ€ΠΎΠΈΠ·Π²ΠΎΠ»ΡŒΠ½Ρ‹ΠΌ Π½Π΅ΠΏΡ€Π΅Ρ€Ρ‹Π²Π½Ρ‹ΠΌ распрСдСлСниСм (Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΠΎ, для ΠΊΠ°ΠΆΠ΄ΠΎΠΉ ΠΊΠΎΠΎΡ€Π΄ΠΈΠ½Π°Ρ‚Ρ‹ ΠΈΡΠΏΠΎΠ»ΡŒΠ·ΡƒΠ΅Ρ‚ΡΡ своё распрСдСлСниС), Ρ‚ΠΎ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ этих Ρ‚ΠΎΡ‡Π΅ΠΊ асимптотичСски стрСмится ΠΊ $E(X) = O((\\log N)^{k-1})$*. \n", - "\n", - "Условиям этой Ρ‚Π΅ΠΎΡ€Π΅ΠΌΡ‹ ΡƒΠ΄ΠΎΠ²Π»Π΅Ρ‚Π²ΠΎΡ€ΡΡŽΡ‚ ΠΌΠ½ΠΎΠ³ΠΈΠ΅ распрСдСлСния, Π²ΠΊΠ»ΡŽΡ‡Π°Ρ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎΠ΅ распрСдСлСниС Π² Π³ΠΈΠΏΠ΅Ρ€ΠΊΡƒΠ±Π΅.\n", - "\n", - "Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ,\n", + "Алгоритм ДТарвиса ΠΈΠΌΠ΅Π΅Ρ‚ ΡΠ»ΠΎΠΆΠ½ΠΎΡΡ‚ΡŒ $O(X N)$, Π³Π΄Π΅ $X$ β€” число Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ, Π° $N$ β€” ΠΎΠ±Ρ‰Π΅Π΅ число Ρ‚ΠΎΡ‡Π΅ΠΊ. Π§Ρ‚ΠΎΠ±Ρ‹ ΠΎΡ†Π΅Π½ΠΈΡ‚ΡŒ ΡΠ»ΠΎΠΆΠ½ΠΎΡΡ‚ΡŒ этого Π°Π»Π³ΠΎΡ€ΠΈΡ‚ΠΌΠ° Π² срСднСм случаС, Π½Π΅ΠΎΠ±Ρ…ΠΎΠ΄ΠΈΠΌΠΎ Π²Ρ‹Ρ‡ΠΈΡΠ»ΠΈΡ‚ΡŒ $E(X)$ β€” матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ Π²Π΅Π»ΠΈΡ‡ΠΈΠ½Ρ‹ $X$. Для Π΅Π³ΠΎ Π°Π½Π°Π»ΠΈΠ·Π° ΠΌΡ‹ Π΄ΠΎΠ»ΠΆΠ½Ρ‹ ΠΎΠΏΡ€Π΅Π΄Π΅Π»ΠΈΡ‚ΡŒ ΠΎΠ³Ρ€Π°Π½ΠΈΡ‡Π΅Π½Π½ΡƒΡŽ ΠΎΠ±Π»Π°ΡΡ‚ΡŒ, ΠΈΠ· ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ Ρ‚ΠΎΡ‡ΠΊΠΈ, Ρ‚Π°ΠΊ ΠΊΠ°ΠΊ ΠΎΠ½ΠΈ ΠΌΠΎΠ³ΡƒΡ‚ Π±Ρ‹Ρ‚ΡŒ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны лишь Π² мноТСствС с ΠΊΠΎΠ½Π΅Ρ‡Π½ΠΎΠΉ ΠΌΠ΅Ρ€ΠΎΠΉ Π›Π΅Π±Π΅Π³Π° [[Kendall, Moran (1963)](https://doi.org/10.1002/bimj.19650070322)]. " + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "notes" + } + }, + "source": [ + "Π”Π°Π»Π΅Π΅ ΠΏΡ€ΠΈΠ²Π΅Π΄Ρ‘Π½ ряд Ρ‚Π΅ΠΎΡ€Π΅ΠΌ, ΠΏΡ€Π΅Π΄ΡΡ‚Π°Π²Π»ΡΡŽΡ‰ΠΈΡ… ΠΎΡ†Π΅Π½ΠΊΡƒ срСднСго количСства Π²Π΅Ρ€ΡˆΠΈΠ½ Π² Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠ΅ ΠΏΡ€ΠΈ Ρ€Π°Π·Π»ΠΈΡ‡Π½Ρ‹Ρ… распрСдСлСниях Ρ‚ΠΎΡ‡Π΅ΠΊ." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ + "**Π’Π΅ΠΎΡ€Π΅ΠΌΠ° 1** [[RΓ©nyi, Sulanke (1963)](https://doi.org/10.1007/BF00535300)]. *Если $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ ΠΈ нСзависимо ΠΈΠ· Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° Π½Π° плоскости, Ρ‚ΠΎ ΠΏΡ€ΠΈ $N\\to\\infty$ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ этих Ρ‚ΠΎΡ‡Π΅ΠΊ $E(X)=\\frac23 r \\, (\\log N + \\gamma) + O(1)$, Π³Π΄Π΅ $\\gamma$ - константа Π­ΠΉΠ»Π΅Ρ€Π°.*" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "**Π’Π΅ΠΎΡ€Π΅ΠΌΠ° 2** [[Raynaud (1970)](https://doi.org/10.1017/S0021900200026917)]. *Если $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ ΠΈ нСзависимо ΠΈΠ· внутрСнности $k$-ΠΌΠ΅Ρ€Π½ΠΎΠΉ гипСрсфСры, Ρ‚ΠΎ ΠΏΡ€ΠΈ $N\\to\\infty$ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π³ΠΈΠΏΠ΅Ρ€Π³Ρ€Π°Π½Π΅ΠΉ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ этих Ρ‚ΠΎΡ‡Π΅ΠΊ асимптотичСски стрСмится ΠΊ $E(X) = O(N^{(k-1)/(k+1)})$*" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ + "**Π’Π΅ΠΎΡ€Π΅ΠΌΠ° 3** [[Raynaud (1970)](https://doi.org/10.1017/S0021900200026917)]. *Если $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ нСзависимо Π² соотвСтствии с $k$-ΠΌΠ΅Ρ€Π½Ρ‹ΠΌ Π½ΠΎΡ€ΠΌΠ°Π»ΡŒΠ½Ρ‹ΠΌ распрСдСлСниСм, Ρ‚ΠΎ ΠΏΡ€ΠΈ $N\\to\\infty$ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ этих Ρ‚ΠΎΡ‡Π΅ΠΊ асимптотичСски стрСмится ΠΊ $E(X) = O((\\log N)^{(k-1)/2})$*" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "**Π’Π΅ΠΎΡ€Π΅ΠΌΠ° 4** [[Bentley, Kung, Schkolnick, Thompson (1978)](https://doi.org/10.1145/322092.322095)]. *Если ΠΊΠΎΠΎΡ€Π΄ΠΈΠ½Π°Ρ‚Ρ‹ $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ Π² $k$-ΠΌΠ΅Ρ€Π½ΠΎΠΌ пространствС Π²Ρ‹Π±Ρ€Π°Π½Ρ‹ нСзависимо Π² соотвСтствии с ΠΏΡ€ΠΎΠΈΠ·Π²ΠΎΠ»ΡŒΠ½Ρ‹ΠΌ Π½Π΅ΠΏΡ€Π΅Ρ€Ρ‹Π²Π½Ρ‹ΠΌ распрСдСлСниСм (Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΠΎ, для ΠΊΠ°ΠΆΠ΄ΠΎΠΉ ΠΊΠΎΠΎΡ€Π΄ΠΈΠ½Π°Ρ‚Ρ‹ ΠΈΡΠΏΠΎΠ»ΡŒΠ·ΡƒΠ΅Ρ‚ΡΡ своё распрСдСлСниС), Ρ‚ΠΎ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ этих Ρ‚ΠΎΡ‡Π΅ΠΊ асимптотичСски стрСмится ΠΊ $E(X) = O((\\log N)^{k-1})$*. " + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "notes" + } + }, + "source": [ + "Условиям этой Ρ‚Π΅ΠΎΡ€Π΅ΠΌΡ‹ ΡƒΠ΄ΠΎΠ²Π»Π΅Ρ‚Π²ΠΎΡ€ΡΡŽΡ‚ ΠΌΠ½ΠΎΠ³ΠΈΠ΅ распрСдСлСния, Π²ΠΊΠ»ΡŽΡ‡Π°Ρ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎΠ΅ распрСдСлСниС Π² Π³ΠΈΠΏΠ΅Ρ€ΠΊΡƒΠ±Π΅." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "skip" + } + }, + "source": [ + "Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ," + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ "- $E(X) = O(\\log N)$ для $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ, Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Ρ… ΠΈΠ· Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ°\n", "- $E(X) = O(N^{1/3})$ для $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ, Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Ρ… ΠΈΠ· ΠΊΡ€ΡƒΠ³Π°\n", "- $E(X) = O(N^{1/2})$ для $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ, Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Ρ… ΠΈΠ· сфСры\n", "- $E(X) = O((\\log N)^2)$ для $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ, Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Ρ… ΠΈΠ· ΠΊΡƒΠ±Π°\n", - "- $E(X) = O(\\sqrt{\\log N})$ для $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ, Π½ΠΎΡ€ΠΌΠ°Π»ΡŒΠ½ΠΎ распрСдСлённых Π½Π° плоскости\n", - "\n", + "- $E(X) = O(\\sqrt{\\log N})$ для $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ, Π½ΠΎΡ€ΠΌΠ°Π»ΡŒΠ½ΠΎ распрСдСлённых Π½Π° плоскости" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "skip" + } + }, + "source": [ "Π—Π΄Π΅ΡΡŒ прСдставлСн ΠΏΠ΅Ρ€Π΅Π²ΠΎΠ΄ Ρ€Π°Π±ΠΎΡ‚Ρ‹ A. RΓ©nyi ΠΈ R. Sulanke, Π³Π΄Π΅ приводится Π΄ΠΎΠΊΠ°Π·Π°Ρ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ ΠΎΡ†Π΅Π½ΠΊΠΈ матСматичСского оТидания числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ Π² Π΄Π²ΡƒΠΌΠ΅Ρ€Π½ΠΎΠΌ случаС." ] }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, "source": [ "# О Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠ΅ N случайно Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Ρ… Ρ‚ΠΎΡ‡Π΅ΠΊ\n", "[[RΓ©nyi, Sulanke (1963)](https://doi.org/10.1007/BF00535300)]\n", "\n", - "ΠŸΡƒΡΡ‚ΡŒ $P_i\\:(i=1,\\:2,\\:\\ldots,\\:n)$ β€” Ρ‚ΠΎΡ‡ΠΊΠΈ Π½Π° плоскости, Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Π΅ случайно ΠΈ нСзависимо Π² соотвСтствии с Π½Π΅ΠΊΠΎΡ‚ΠΎΡ€Ρ‹ΠΌ распрСдСлСниСм, Π° $H_n$ - выпуклая ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠ° этих Ρ‚ΠΎΡ‡Π΅ΠΊ, которая, ΠΊΠ°ΠΊ извСстно, являСтся Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΌ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΎΠΌ. Число $X_n$ Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ β€” случайная Π²Π΅Π»ΠΈΡ‡ΠΈΠ½Π°, ΠΈ Π² Π΄Π°Π½Π½ΠΎΠΉ Ρ€Π°Π±ΠΎΡ‚Π΅ Π±ΡƒΠ΄Π΅Ρ‚ исслСдовано Π΅Ρ‘ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ $E_n = E(X_n)$ ΠΏΡ€ΠΈ $n\\to\\infty$ Π² Ρ€Π°Π·Π»ΠΈΡ‡Π½Ρ‹Ρ… прСдполоТСниях ΠΎ распрСдСлСнии Ρ‚ΠΎΡ‡Π΅ΠΊ $P_i$.\n", - "\n", + "ΠŸΡƒΡΡ‚ΡŒ $P_i\\:(i=1,\\:2,\\:\\ldots,\\:n)$ β€” Ρ‚ΠΎΡ‡ΠΊΠΈ Π½Π° плоскости, Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Π΅ случайно ΠΈ нСзависимо Π² соотвСтствии с Π½Π΅ΠΊΠΎΡ‚ΠΎΡ€Ρ‹ΠΌ распрСдСлСниСм, Π° $H_n$ - выпуклая ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠ° этих Ρ‚ΠΎΡ‡Π΅ΠΊ, которая, ΠΊΠ°ΠΊ извСстно, являСтся Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΌ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΎΠΌ. Число $X_n$ Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ β€” случайная Π²Π΅Π»ΠΈΡ‡ΠΈΠ½Π°, ΠΈ Π² Π΄Π°Π½Π½ΠΎΠΉ Ρ€Π°Π±ΠΎΡ‚Π΅ Π±ΡƒΠ΄Π΅Ρ‚ исслСдовано Π΅Ρ‘ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ $E_n = E(X_n)$ ΠΏΡ€ΠΈ $n\\to\\infty$ Π² Ρ€Π°Π·Π»ΠΈΡ‡Π½Ρ‹Ρ… прСдполоТСниях ΠΎ распрСдСлСнии Ρ‚ΠΎΡ‡Π΅ΠΊ $P_i$." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ "* Π’ $\\S1$ рассмотрим случай, ΠΊΠΎΠ³Π΄Π° Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i$ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны Π²Π½ΡƒΡ‚Ρ€ΠΈ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ r-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$. Как Π±ΡƒΠ΄Π΅Ρ‚ ΠΏΠΎΠΊΠ°Π·Π°Π½ΠΎ, $E_n = (2r\\,/\\,3)(\\log n + \\gamma) + T + O(1)$, Π³Π΄Π΅ $\\gamma$ - постоянная Π­ΠΉΠ»Π΅Ρ€Π°:\n", " \n", " $$\\gamma = \\lim_{n\\to\\infty} \\left( \\sum_{k=1}^{n} \\frac1k - \\log n \\right)=\\lim_{n\\to\\infty} \\left( 1+\\frac12+\\frac13+\\ldots+\\frac1n - \\log n \\right) \\notag$$\n", @@ -57,50 +152,97 @@ ] }, { - "attachments": { - "1.png": { - "image/png": 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" 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" + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" } }, - "cell_type": "markdown", - "metadata": {}, "source": [ "## 1. Π‘Π»ΡƒΡ‡Π°ΠΉΠ½ΠΎ Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Π΅ Ρ‚ΠΎΡ‡ΠΊΠΈ Π² Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΌ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ΅\n", "\n", - "**Π£Ρ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 1.** Если Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i$ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны Π²Π½ΡƒΡ‚Ρ€ΠΈ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ r-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$, матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ $E_n = (2r\\,/\\,3)(\\log n + \\gamma) + T + O(1)$, Π³Π΄Π΅ $\\gamma$ - постоянная Π­ΠΉΠ»Π΅Ρ€Π°.\n", - "\n", + "**Π£Ρ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 1.** Если Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i$ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны Π²Π½ΡƒΡ‚Ρ€ΠΈ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ r-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$, матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ $E_n = (2r\\,/\\,3)(\\log n + \\gamma) + T + O(1)$, Π³Π΄Π΅ $\\gamma$ - постоянная Π­ΠΉΠ»Π΅Ρ€Π°." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "**Π”ΠΎΠΊΠ°Π·Π°Ρ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ:**\n", "\n", "$\\triangleright$\n", - "
\n", - "ΠŸΡƒΡΡ‚ΡŒ $K$ - Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΉ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ с Π²Π΅Ρ€ΡˆΠΈΠ½Π°ΠΌΠΈ $A_1,\\:A_2,\\:\\ldots,\\:A_r$ ΠΈ ΡΠΎΠΎΡ‚Π²Π΅Ρ‚ΡΡ‚Π²ΡƒΡŽΡ‰ΠΈΠΌΠΈ ΡƒΠ³Π»Π°ΠΌΠΈ $\\vartheta_1,\\:\\vartheta_2,\\:\\ldots,\\:\\vartheta_r$. ΠžΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Π·Π° $a_k$ Π΄Π»ΠΈΠ½Ρƒ стороны $A_k A_{k+1}$, Π³Π΄Π΅ $A_{r+1} = A_1$ $(k = 1,\\:2,\\:\\ldots,\\:r)$.\n", - "\n", - "![1.png](attachment:1.png)\n", "\n", + "ΠŸΡƒΡΡ‚ΡŒ $K$ - Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΉ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ с Π²Π΅Ρ€ΡˆΠΈΠ½Π°ΠΌΠΈ $A_1,\\:A_2,\\:\\ldots,\\:A_r$ ΠΈ ΡΠΎΠΎΡ‚Π²Π΅Ρ‚ΡΡ‚Π²ΡƒΡŽΡ‰ΠΈΠΌΠΈ ΡƒΠ³Π»Π°ΠΌΠΈ $\\vartheta_1,\\:\\vartheta_2,\\:\\ldots,\\:\\vartheta_r$. ΠžΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Π·Π° $a_k$ Π΄Π»ΠΈΠ½Ρƒ стороны $A_k A_{k+1}$, Π³Π΄Π΅ $A_{r+1} = A_1$ $(k = 1,\\:2,\\:\\ldots,\\:r)$." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ + "![1.png](1.png)" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "ΠŸΡƒΡΡ‚ΡŒ $\\varepsilon_{ij} = 1$, Ссли ΠΎΡ‚Ρ€Π΅Π·ΠΎΠΊ $P_i P_j$ ΠΏΡ€ΠΈ $i \\neq j$ ΠΏΡ€ΠΈΠ½Π°Π΄Π»Π΅ΠΆΠΈΡ‚ Π³Ρ€Π°Π½ΠΈΡ†Π΅ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ $H_n$, ΠΈΠ½Π°Ρ‡Π΅ $\\varepsilon_{ij} = 0$. Π’ΠΎΠ³Π΄Π° ΠΎΡ‡Π΅Π²ΠΈΠ΄Π½ΠΎ, Ρ‡Ρ‚ΠΎ\n", "\n", "$$X_n = \\sum_{1 \\le i \\lt j \\le n}\\varepsilon_{ij} \\label{eq:1} \\tag{1}$$\n", "\n", "Π’ силу случайного распрСдСлСния Π²Π΅Ρ€ΠΎΡΡ‚Π½ΠΎΡΡ‚ΡŒ $W_n = P(\\varepsilon_{ij} = 1)$ ΠΎΠ΄ΠΈΠ½Π°ΠΊΠΎΠ²Π° для всСх ΠΏΠ°Ρ€ $(i, j)$, Ρ‚Π°ΠΊΠΈΡ…, Ρ‡Ρ‚ΠΎ $1 \\le i \\lt j \\leq n$, ΠΈ Ρ‚ΠΎΠ³Π΄Π°\n", "\n", - "$$E_n = E(X_n) = \\sum_{1 \\le i \\lt j \\le n} E(\\varepsilon_{ij}) = \\sum_{1 \\le i \\lt j \\le n} \\left( 0 \\cdot (1 - W_n) + 1 \\cdot W_n \\right) = C_n^2 \\cdot W_n \\label{eq:2} \\tag{2}$$\n", - "\n", + "$$E_n = E(X_n) = \\sum_{1 \\le i \\lt j \\le n} E(\\varepsilon_{ij}) = \\sum_{1 \\le i \\lt j \\le n} \\left( 0 \\cdot (1 - W_n) + 1 \\cdot W_n \\right) = C_n^2 \\cdot W_n \\label{eq:2} \\tag{2}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "Π§Ρ‚ΠΎΠ±Ρ‹ ΠΎΠΏΡ€Π΅Π΄Π΅Π»ΠΈΡ‚ΡŒ асимптотичСскоС ΠΏΠΎΠ²Π΅Π΄Π΅Π½ΠΈΠ΅ $E_n$, достаточно Π²Ρ‹Ρ‡ΠΈΡΠ»ΠΈΡ‚ΡŒ Π²Π΅Ρ€ΠΎΡΡ‚Π½ΠΎΡΡ‚ΡŒ $W_n$. МоТСм Ρ€Π°ΡΡΠΌΠΎΡ‚Ρ€Π΅Ρ‚ΡŒ $W_n$ ΠΊΠ°ΠΊ Π²Π΅Ρ€ΠΎΡΡ‚Π½ΠΎΡΡ‚ΡŒ Ρ‚ΠΎΠ³ΠΎ, Ρ‡Ρ‚ΠΎ всС Ρ‚ΠΎΡ‡ΠΊΠΈ $P_h \\: (h = 3,\\:4,\\:\\ldots,\\:n)$ Π»Π΅ΠΆΠ°Ρ‚ ΠΏΠΎ ΠΎΠ΄Π½Ρƒ сторону ΠΎΡ‚ прямой $P_1 P_2$. \n", "\n", - "![2.png](attachment:2.png)\n", - "\n", + "![2.png](2.png)" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "ΠŸΡƒΡΡ‚ΡŒ $F$ ΠΎΠ±ΠΎΠ·Π½Π°Ρ‡Π°Π΅Ρ‚ ΠΏΠ»ΠΎΡ‰Π°Π΄ΡŒ $K$. Если Ρ‡Π΅Ρ€Π΅Π· $F_1 \\le \\frac12 F$ ΠΈ $F_2 = F - F_1$ ΠΎΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΡ‚ΡŒ части, Π½Π° ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Π΅ прямая $P_1 P_2$ Π΄Π΅Π»ΠΈΡ‚ ΠΎΠ±Π»Π°ΡΡ‚ΡŒ $K$, Ρ‚ΠΎ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ\n", "\n", "$$W_n = \\frac1{F^2} \\int\\limits_K \\int\\limits_K \\left[ \\left(1 - \\frac{F_1}{F}\\right)^{n-2} + \\left(\\frac{F_1}{F}\\right)^{n-2}\\right] dP_1 dP_2 \\label{eq:3} \\tag{3}$$\n", "\n", - "ΠΊΠ°ΠΊ Π²Π΅Ρ€ΠΎΡΡ‚Π½ΠΎΡΡ‚ΡŒ Ρ‚ΠΎΠ³ΠΎ, Ρ‡Ρ‚ΠΎ всС Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i\\:(i=3,\\:4,\\:\\ldots,\\:n)$ ΠΏΡ€ΠΈΠ½Π°Π΄Π»Π΅ΠΆΠ°Ρ‚ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ ΠΎΠ΄Π½ΠΎΠΉ ΠΈΠ· частСй $F$.\n", - "\n", + "ΠΊΠ°ΠΊ Π²Π΅Ρ€ΠΎΡΡ‚Π½ΠΎΡΡ‚ΡŒ Ρ‚ΠΎΠ³ΠΎ, Ρ‡Ρ‚ΠΎ всС Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i\\:(i=3,\\:4,\\:\\ldots,\\:n)$ ΠΏΡ€ΠΈΠ½Π°Π΄Π»Π΅ΠΆΠ°Ρ‚ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ ΠΎΠ΄Π½ΠΎΠΉ ΠΈΠ· частСй $F$." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "Π’Π°ΠΊ ΠΊΠ°ΠΊ $F_1/F \\le \\frac12$, ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π» Π²Ρ‚ΠΎΡ€ΠΎΠ³ΠΎ слагаСмого ΠΎΠ³Ρ€Π°Π½ΠΈΡ‡Π΅Π½ константой:\n", "\n", "$$\\frac1{F^2} \\int\\limits_K \\int\\limits_K \\left(\\frac{F_1}{F}\\right)^{n-2} dP_1 dP_2 \\le \\frac1{2^{n - 2}} \\label{eq:4} \\tag{4}$$\n", @@ -109,17 +251,43 @@ "\n", "$$E_n \\sim C_n^2 \\cdot \\frac1{F^2} \\int\\limits_K \\int\\limits_K \\left(1 - \\frac{F_1}{F}\\right)^{n-2} dP_1 dP_2 \\label{eq:5} \\tag{5}$$\n", "\n", - "$$\\text{Π³Π΄Π΅} \\: A \\sim B \\Leftrightarrow \\lim_{n\\to+\\infty} \\frac{A_n}{B_n} = 1 \\notag$$\n", - "\n", - "\n", + "$$\\text{Π³Π΄Π΅} \\: A \\sim B \\Leftrightarrow \\lim_{n\\to+\\infty} \\frac{A_n}{B_n} = 1 \\notag$$" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "ΠžΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Ρ‡Π΅Ρ€Π΅Π· $f_i$ ΠΏΠ»ΠΎΡ‰Π°Π΄ΡŒ Ρ‚Ρ€Π΅ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $A_{i-1}A_iA_{i+1}$ $(A_{r+j} = A_j)$, ΠΈ ΠΏΡƒΡΡ‚ΡŒ $f = \\min\\limits_{1 \\le i \\le r} f_i$\n", "\n", - "![3.png](attachment:3.png)\n", - "\n", + "![3.png](3.png)" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "ΠžΡ‡Π΅Π²ΠΈΠ΄Π½ΠΎ, Ρ‡Ρ‚ΠΎ ΠΏΡ€ΠΈ ΠΈΠ·ΡƒΡ‡Π΅Π½ΠΈΠΈ асимптотичСского повСдСния $E_n$ ΠΌΠΎΠΆΠ½ΠΎ ΠΎΠ³Ρ€Π°Π½ΠΈΡ‡ΠΈΡ‚ΡŒΡΡ ΠΏΠ°Ρ€Π°ΠΌΠΈ Ρ‚ΠΎΡ‡Π΅ΠΊ $(P_1,\\:P_2)$, для ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… прямая $P_1 P_2$ пСрСсСкаСтся Π»ΠΈΠ±ΠΎ с двумя сосСдними сторонами $K$, Π»ΠΈΠ±ΠΎ с двумя сторонами $K$, Ρ€Π°Π·Π΄Π΅Π»Ρ‘Π½Π½Ρ‹ΠΌΠΈ Ρ‚Ρ€Π΅Ρ‚ΡŒΠ΅ΠΉ стороной. Для всСх ΠΎΡΡ‚Π°Π»ΡŒΠ½Ρ‹Ρ… прямых выполняСтся ΠΎΠ³Ρ€Π°Π½ΠΈΡ‡Π΅Π½ΠΈΠ΅\n", "\n", - "$$1 - \\frac{F_1}{F} \\leqq 1 - \\frac{f}{F} \\notag$$\n", - "\n", + "$$1 - \\frac{F_1}{F} \\leqq 1 - \\frac{f}{F} \\notag$$" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "ΠŸΠΎΡΡ‚ΠΎΠΌΡƒ ΠΌΡ‹ ΠΌΠΎΠΆΠ΅ΠΌ ΠΏΡ€Π΅Π½Π΅Π±Ρ€Π΅Ρ‡ΡŒ ΡΠΎΠΎΡ‚Π²Π΅Ρ‚ΡΡ‚Π²ΡƒΡŽΡ‰Π΅ΠΉ Ρ‡Π°ΡΡ‚ΡŒΡŽ ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π»Π° ΠΈ Π·Π°ΠΏΠΈΡΠ°Ρ‚ΡŒ \n", "\n", "$$E_n \\sim C_n^2 \\cdot \\frac1{F^2} \\left( \\sum\\limits_{i=1}^r (I_i + J_i) \\right) \\label{eq:6} \\tag{6}$$ \n", @@ -128,8 +296,17 @@ "\n", "$$J_i = \\int\\limits_{D_i} \\left(1 - \\frac{F_1}{F}\\right)^{n-2} dP_1 dP_2 \\label{eq:8} \\tag{8}$$\n", "\n", - "Π—Π΄Π΅ΡΡŒ $C_i$ ΠΎΠ·Π½Π°Ρ‡Π°Π΅Ρ‚ мноТСство ΠΏΠ°Ρ€ Ρ‚ΠΎΡ‡Π΅ΠΊ $(P_1,\\:P_2)$, для ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… прямая $P_1P_2$ пСрСсСкаСт смСТныС стороны $A_{i-1}A_i$ ΠΈ $A_iA_{i+1}$ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$, a $D_i$ - мноТСство ΠΏΠ°Ρ€ $(P_1,\\:P_2)$, для ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… прямая $P_1P_2$ пСрСсСкаСт стороны $A_{i-1}A_i$ ΠΈ $A_{i+1}A_{i+2}$ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$, Ρ€Π°Π·Π΄Π΅Π»Ρ‘Π½Π½Ρ‹Π΅ Π΄Ρ€ΡƒΠ³ΠΎΠΉ стороной.\n", - "\n", + "Π—Π΄Π΅ΡΡŒ $C_i$ ΠΎΠ·Π½Π°Ρ‡Π°Π΅Ρ‚ мноТСство ΠΏΠ°Ρ€ Ρ‚ΠΎΡ‡Π΅ΠΊ $(P_1,\\:P_2)$, для ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… прямая $P_1P_2$ пСрСсСкаСт смСТныС стороны $A_{i-1}A_i$ ΠΈ $A_iA_{i+1}$ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$, a $D_i$ - мноТСство ΠΏΠ°Ρ€ $(P_1,\\:P_2)$, для ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… прямая $P_1P_2$ пСрСсСкаСт стороны $A_{i-1}A_i$ ΠΈ $A_{i+1}A_{i+2}$ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$, Ρ€Π°Π·Π΄Π΅Π»Ρ‘Π½Π½Ρ‹Π΅ Π΄Ρ€ΡƒΠ³ΠΎΠΉ стороной." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "Π‘Π½Π°Ρ‡Π°Π»Π° рассмотрим ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π» $I_i$. ΠžΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Ρ‡Π΅Ρ€Π΅Π· $G_{ab}$ мноТСство ΠΏΠ°Ρ€ $(P_1,\\:P_2)$, для ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… Π²Ρ‹ΠΏΠΎΠ»Π½Π΅Π½Ρ‹ нСравСнства $\\left|A_iQ_1\\right| < a$ ΠΈ $\\left|A_iQ_2\\right| < b$, Π³Π΄Π΅ $Q_1$ - Ρ‚ΠΎΡ‡ΠΊΠ° пСрСсСчСния прямой $P_1P_2$ со стороной $A_{i-1}A_i$, Π° $Q_2$ - Ρ‚ΠΎΡ‡ΠΊΠ° пСрСсСчСния $P_1P_2$ со стороной $A_iA_{i+1}$. Π—Π΄Π΅ΡΡŒ ΠΈ Π΄Π°Π»Π΅Π΅ $\\left|AB\\right|$ β€” Π΄Π»ΠΈΠ½Π° ΠΎΡ‚Ρ€Π΅Π·ΠΊΠ° $AB$.\n", "\n", "По элСмСнтарным расчётам получаСтся \n", @@ -138,8 +315,17 @@ "\n", "ΠŸΠ»ΠΎΡ‰Π°Π΄ΡŒ Ρ‚Ρ€Π΅ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $Q_1A_iQ_2$, отсСкаСмого прямой $P_1P_2$, Ρ€Π°Π²Π½Π° $\\frac{1}{2} a b \\sin \\vartheta_i$. Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, взяв $a_{i-1} = \\left|A_{i-1}A_i\\right|$, $a_i = \\left|A_iA_{i+1}\\right|$, ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ \n", "\n", - "$$I_i = \\int\\limits_0^{a_{i-1}} \\int\\limits_0^{a_i} \\left(1 - \\frac{ab \\sin \\vartheta_i}{2F}\\right)^{n-2} \\sin^2 \\vartheta_i \\frac{ab}{3} \\,da\\,db \\label{eq:10} \\tag{10}$$ \n", - "\n", + "$$I_i = \\int\\limits_0^{a_{i-1}} \\int\\limits_0^{a_i} \\left(1 - \\frac{ab \\sin \\vartheta_i}{2F}\\right)^{n-2} \\sin^2 \\vartheta_i \\frac{ab}{3} \\,da\\,db \\label{eq:10} \\tag{10}$$ " + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "Π’Ρ‹ΠΏΠΎΠ»Π½ΠΈΠΌ Π·Π°ΠΌΠ΅Π½Ρƒ \n", "\n", "$$X = a \\sqrt\\frac{\\sin \\vartheta_i}{2F}, \\; Y = b \\sqrt\\frac{\\sin \\vartheta_i}{2F} \\label{eq:11} \\tag{11}$$ \n", @@ -154,20 +340,47 @@ "\n", "ΠΈ Ρ‚Π°ΠΊΠΆΠ΅ Π·Π°ΠΌΠ΅Ρ‚ΠΈΠΌ, Ρ‡Ρ‚ΠΎ\n", "\n", - "$$\\varrho_i = X_iY_i = \\frac{a_{i-1}a_i\\sin \\vartheta_i}{2F} = \\frac{f_i}{F} < 1 \\label{eq:13} \\tag{13}$$\n", - "\n", + "$$\\varrho_i = X_iY_i = \\frac{a_{i-1}a_i\\sin \\vartheta_i}{2F} = \\frac{f_i}{F} < 1 \\label{eq:13} \\tag{13}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "Π’Π΅ΠΏΠ΅Ρ€ΡŒ ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΡƒΠ΅ΠΌ ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π»:\n", "\n", "$$\\int\\limits_0^{X_i} \\int\\limits_0^{Y_i} (1-XY)^{n-2} XY \\: dX dY = \\int\\limits_0^{X_i} \\int\\limits_0^{Y_i} (1-XY)^{n-2} \\: dX dY - \\int\\limits_0^{X_i} \\int\\limits_0^{Y_i} (1-XY)^{n-1} \\: dX dY \\notag$$\n", "\n", "Π”Π°Π»Π΅Π΅ ΠΏΡ€ΠΈΠΌΠ΅Π½ΠΈΠΌ \n", "\n", - "$$\\int\\limits_0^{X_i} \\int\\limits_0^{Y_i} (1-XY)^{n-1} \\: dX dY = \\int\\limits_0^{X_i} \\frac{1 - (1 - XY_i)^n}{nX} \\: dX = \\\\ = \\frac1n \\left( 1 + \\frac12 + \\ldots + \\frac1n - \\sum\\limits_{k=1}^n \\frac{(1-\\varrho_i)^k}{k} \\right) \\notag$$\n", - "\n", + "$$\\int\\limits_0^{X_i} \\int\\limits_0^{Y_i} (1-XY)^{n-1} \\: dX dY = \\int\\limits_0^{X_i} \\frac{1 - (1 - XY_i)^n}{nX} \\: dX = \\\\ = \\frac1n \\left( 1 + \\frac12 + \\ldots + \\frac1n - \\sum\\limits_{k=1}^n \\frac{(1-\\varrho_i)^k}{k} \\right) \\notag$$" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, ΠΌΡ‹ ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ Π·Π½Π°Ρ‡Π΅Π½ΠΈΠ΅ ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π»Π° $I_i$:\n", "\n", - "$$I_i = \\frac{4F^2}{3} \\left[ \\frac{\\cfrac12 + \\cfrac13 + \\ldots + \\cfrac1n}{n(n-1)} - \\frac{\\sum\\limits_{k=1}^{n-1} \\cfrac{(1 - \\varrho_i)^k}{k}}{n(n-1)} + \\frac{(1-\\varrho_i)^n}{n^2} \\right] \\label{eq:14} \\tag{14}$$\n", - "\n", + "$$I_i = \\frac{4F^2}{3} \\left[ \\frac{\\cfrac12 + \\cfrac13 + \\ldots + \\cfrac1n}{n(n-1)} - \\frac{\\sum\\limits_{k=1}^{n-1} \\cfrac{(1 - \\varrho_i)^k}{k}}{n(n-1)} + \\frac{(1-\\varrho_i)^n}{n^2} \\right] \\label{eq:14} \\tag{14}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "Если ΠΏΠΎΠ»ΠΎΠΆΠΈΠΌ \n", "\n", "$$S_i = \\sum\\limits_{k=1}^\\infty \\frac{(1 - \\varrho_i)^k}{k} = \\log \\frac1{\\varrho_i} \\label{eq:15} \\tag{15}$$\n", @@ -176,15 +389,35 @@ "\n", "$$E_n = \\frac23 (\\gamma - 1 + \\log n) r - \\frac23 \\sum\\limits_{i=1}^r S_i + o(1) + \\frac{C_n^2}{F^2} \\sum\\limits_{i=1}^r J_i \\label{eq:16} \\tag{16}$$ \n", "\n", - "Π³Π΄Π΅ $\\gamma$ - постоянная Π­ΠΉΠ»Π΅Ρ€Π°.\n", - "\n", + "Π³Π΄Π΅ $\\gamma$ - постоянная Π­ΠΉΠ»Π΅Ρ€Π°." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "Π’Π΅ΠΏΠ΅Ρ€ΡŒ вычислим ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π» $J_i$. ΠŸΡ€ΠΎΠ΄Π»ΠΈΠΌ стороны $A_{i-1}A_i$ ΠΈ $A_{i+1}A_{i+2}$ Π΄ΠΎ Ρ‚ΠΎΡ‡ΠΊΠΈ ΠΈΡ… пСрСсСчСния $A_i^*$; случай, ΠΊΠΎΠ³Π΄Π° эти стороны ΠΏΠ°Ρ€Π°Π»Π»Π΅Π»ΡŒΠ½Ρ‹, ΠΌΠΎΠΆΠ½ΠΎ Ρ€Π°ΡΡΠΌΠ°Ρ‚Ρ€ΠΈΠ²Π°Ρ‚ΡŒ Π°Π½Π°Π»ΠΎΠ³ΠΈΡ‡Π½ΠΎ.\n", "\n", "ΠŸΡƒΡΡ‚ΡŒ снова $Q_1$ - Ρ‚ΠΎΡ‡ΠΊΠ° пСрСсСчСния прямой $P_1P_2$ со стороной $A_{i-1}A_i$, Π° $Q_2$ - Ρ‚ΠΎΡ‡ΠΊΠ° пСрСсСчСния $P_1P_2$ со стороной $A_{i+1}A_{i+2}$, ΠΈ ΠΏΡƒΡΡ‚ΡŒ\n", "\n", "$$a' = \\left|A_iA_i^*\\right|, \\;\\; b' = \\left|A_{i+1}A_i^*\\right|, \\;\\; a = \\left|A_iQ_1\\right|, \\;\\; b = \\left|A_{i+1}Q_2\\right| \\notag$$\n", "\n", - "$G_{ab}$ Π±Ρ‹Π»ΠΎ ΠΎΠΏΡ€Π΅Π΄Π΅Π»Π΅Π½ΠΎ Π²Ρ‹ΡˆΠ΅. ΠŸΠΎΠ»ΡƒΡ‡ΠΈΠΌ ΠΈΠ· $(9)$:\n", + "$G_{ab}$ Π±Ρ‹Π»ΠΎ ΠΎΠΏΡ€Π΅Π΄Π΅Π»Π΅Π½ΠΎ Π²Ρ‹ΡˆΠ΅. " + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ + "ΠŸΠΎΠ»ΡƒΡ‡ΠΈΠΌ ΠΈΠ· $(\\ref{eq:9})$:\n", "\n", "$$\\int\\limits_{G_{ab}} dP_1 dP_2 = \\frac{\\sin^2\\beta_i}{12} (a^2 + 2aa')(b^2 + 2bb') \\label{eq:17} \\tag{17}$$ \n", "\n", @@ -194,8 +427,17 @@ "\n", "ΠΈ этот ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π» асимптотичСски эквивалСнтСн\n", "\n", - "$$J_i \\sim \\int\\limits_{D_i} \\left(1 - \\frac{(a'b+ab')\\sin \\beta_i}{2F} \\right)^{n-2} \\frac{\\sin^2 \\beta_i}{3} a'b'\\,da\\,db \\label{eq:19} \\tag{19}$$\n", - "\n", + "$$J_i \\sim \\int\\limits_{D_i} \\left(1 - \\frac{(a'b+ab')\\sin \\beta_i}{2F} \\right)^{n-2} \\frac{\\sin^2 \\beta_i}{3} a'b'\\,da\\,db \\label{eq:19} \\tag{19}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "Π”Π°Π»Π΅Π΅ сдСлаСм Π·Π°ΠΌΠ΅Π½Ρƒ\n", "\n", "$$a = \\frac{F}{b' \\sin \\beta_i} X, \\;\\; b = \\frac{F}{a' \\sin \\beta_i} Y \\notag$$\n", @@ -206,21 +448,48 @@ "\n", "Π³Π΄Π΅ \n", "\n", - "$$a_{i-1}' = \\frac{a_ib'\\sin\\beta_i}{F}, \\;\\; a_{i+1}' = \\frac{a_{i+2}a'\\sin\\beta_i}{F} \\notag$$\n", - "\n", + "$$a_{i-1}' = \\frac{a_ib'\\sin\\beta_i}{F}, \\;\\; a_{i+1}' = \\frac{a_{i+2}a'\\sin\\beta_i}{F} \\notag$$" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ ΠΈΡ‚ΠΎΠ³ΠΎΠ²Ρ‹ΠΉ Ρ€Π΅Π·ΡƒΠ»ΡŒΡ‚Π°Ρ‚:\n", "\n", "$$E_n = \\frac{2r}{3} (\\log n + \\gamma) - \\frac23 \\sum\\limits_{i=1}^{r} S_i + o(1) \\notag$$\n", "\n", - "
\n", - "$\\triangleleft$\n", "\n", + "$\\triangleleft$" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "ΠŸΡƒΡΡ‚ΡŒ $K$ - Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΉ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ с $r$ сторонами. ΠœΠ°Ρ‚Π΅ΠΌΠ°Ρ‚ΠΈΡ‡Π΅ΡΠΊΠΎΠ΅ ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ $E_n$ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ $H_n$, построСнной Π½Π° $n$ случайных Ρ‚ΠΎΡ‡ΠΊΠ°Ρ… ΠΈΠ· $K$, Ρ€Π°Π²Π½ΠΎ\n", "\n", "$$E_n = \\frac{2r}{3} (\\log n + \\gamma) + \\frac23 \\log \\frac{\\prod\\limits_1^r f_i}{F^r} + o(1) \\label{eq:20} \\tag{20}$$\n", "\n", - "Π—Π΄Π΅ΡΡŒ $F$ ΠΎΠ±ΠΎΠ·Π½Π°Ρ‡Π°Π΅Ρ‚ ΠΏΠ»ΠΎΡ‰Π°Π΄ΡŒ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$, Π° $f_i$ - ΠΏΠ»ΠΎΡ‰Π°Π΄ΡŒ Ρ‚Ρ€Π΅ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $A_{i-1}A_iA_{i+1}$.\n", - "\n", + "Π—Π΄Π΅ΡΡŒ $F$ ΠΎΠ±ΠΎΠ·Π½Π°Ρ‡Π°Π΅Ρ‚ ΠΏΠ»ΠΎΡ‰Π°Π΄ΡŒ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$, Π° $f_i$ - ΠΏΠ»ΠΎΡ‰Π°Π΄ΡŒ Ρ‚Ρ€Π΅ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $A_{i-1}A_iA_{i+1}$." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "ΠŸΡƒΡΡ‚ΡŒ, Π½Π°ΠΏΡ€ΠΈΠΌΠ΅Ρ€, $K$ - Ρ‚Ρ€Π΅ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ, Ρ‚ΠΎΠ³Π΄Π° $f_1 = f_2 = f_3 = F$ ΠΈ \n", "\n", "$$E_n = 2(\\log n + \\gamma) + o(1) \\label{eq:21} \\tag{21}$$\n", @@ -234,7 +503,11 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, "source": [ "## 2. Π­ΠΊΡΡ‚Ρ€Π΅ΠΌΠ°Π»ΡŒΠ½Π°Ρ Π·Π°Π΄Π°Ρ‡Π° для Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹Ρ… ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΎΠ²\n", "\n", @@ -244,36 +517,83 @@ "\n", "$$Z(K) = \\frac1{F^r} \\prod\\limits_{i=1}^r f_i \\label{eq:23} \\tag{23}$$\n", "\n", - "ΠΈΠ· ΠΏΡ€Π΅Π΄Ρ‹Π΄ΡƒΡ‰Π΅Π³ΠΎ ΠΏΠ°Ρ€Π°Π³Ρ€Π°Ρ„Π° максимальна Ρ‚ΠΎΠ³Π΄Π° ΠΈ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ Ρ‚ΠΎΠ³Π΄Π°, ΠΊΠΎΠ³Π΄Π° $K$ ΠΌΠΎΠΆΠ½ΠΎ ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΠΎΠ²Π°Ρ‚ΡŒ Π² ΠΏΡ€Π°Π²ΠΈΠ»ΡŒΠ½Ρ‹ΠΉ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ Π°Ρ„Ρ„ΠΈΠ½Π½Ρ‹ΠΌΠΈ прСобразованиями.\n", - "\n", + "ΠΈΠ· ΠΏΡ€Π΅Π΄Ρ‹Π΄ΡƒΡ‰Π΅Π³ΠΎ ΠΏΠ°Ρ€Π°Π³Ρ€Π°Ρ„Π° максимальна Ρ‚ΠΎΠ³Π΄Π° ΠΈ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ Ρ‚ΠΎΠ³Π΄Π°, ΠΊΠΎΠ³Π΄Π° $K$ ΠΌΠΎΠΆΠ½ΠΎ ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΠΎΠ²Π°Ρ‚ΡŒ Π² ΠΏΡ€Π°Π²ΠΈΠ»ΡŒΠ½Ρ‹ΠΉ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ Π°Ρ„Ρ„ΠΈΠ½Π½Ρ‹ΠΌΠΈ прСобразованиями." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "**Π”ΠΎΠΊΠ°Π·Π°Ρ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ:**\n", "\n", "$\\triangleright$\n", - "
\n", + "\n", "ΠŸΡƒΡΡ‚ΡŒ $\\alpha_i = \\overrightarrow{A_{i-1}A_i}$, Π³Π΄Π΅ $A_{r+i} = A_i$. Π’ΠΎΠ³Π΄Π°\n", "\n", "$$f_i = \\frac12 [\\alpha_i, \\: \\alpha_{i+1}] \\\\ F = \\frac12 \\sum\\limits_{i=1}^{r-2} [\\alpha_1 + \\alpha_2 + \\ldots + \\alpha_i, \\: \\alpha_{i+1}] \\label{eq:24} \\tag{24}$$\n", "\n", "Π³Π΄Π΅ $[\\alpha, \\: \\beta]$ - ΠΌΠΎΠ΄ΡƒΠ»ΡŒ Π²Π΅ΠΊΡ‚ΠΎΡ€Π½ΠΎΠ³ΠΎ произвСдСния $\\alpha$ ΠΈ $\\beta$. Ѐункция $Z(K)$ Ρ‚Π΅ΠΏΠ΅Ρ€ΡŒ являСтся Ρ„ΡƒΠ½ΠΊΡ†ΠΈΠ΅ΠΉ ΠΎΡ‚ $r$ Π²Π΅ΠΊΡ‚ΠΎΡ€ΠΎΠ² $\\alpha_i \\: (i=1, \\: \\ldots, \\: r)$, ΡƒΠ΄ΠΎΠ²Π»Π΅Ρ‚Π²ΠΎΡ€ΡΡŽΡ‰ΠΈΡ… ΡƒΡΠ»ΠΎΠ²ΠΈΡŽ замкнутости:\n", "\n", - "$$\\sum\\limits_{i=1}^r \\alpha_i = 0 \\label{eq:25} \\tag{25}$$\n", - "\n", - "Π’Π΅ΠΏΠ΅Ρ€ΡŒ допустим, Ρ‡Ρ‚ΠΎ Π²Π΅Ρ€ΡˆΠΈΠ½Ρ‹ $A_i$ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$ ΠΏΠΎΠ΄Π²Π΅Ρ€Π³Π°ΡŽΡ‚ΡΡ достаточно ΠΌΠ°Π»Ρ‹ΠΌ смСщСниям, ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Π΅ ΠΎΠΏΠΈΡΡ‹Π²Π°ΡŽΡ‚ΡΡ Π½Π΅ΠΏΡ€Π΅Ρ€Ρ‹Π²Π½ΠΎ Π΄ΠΈΡ„Ρ„Π΅Ρ€Π΅Π½Ρ†ΠΈΡ€ΡƒΠ΅ΠΌΠΎΠΉ Ρ„ΡƒΠ½ΠΊΡ†ΠΈΠ΅ΠΉ ΠΎΡ‚ ΠΏΠ°Ρ€Π°ΠΌΠ΅Ρ‚Ρ€Π° $t$. Π’Π°ΠΊ ΠΊΠ°ΠΊ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ $K$, ΠΊΠΎΡ‚ΠΎΡ€Ρ‹ΠΉ Π΄ΠΎΠ»ΠΆΠ΅Π½ ΡΠΎΠΎΡ‚Π²Π΅Ρ‚ΡΡ‚Π²ΠΎΠ²Π°Ρ‚ΡŒ Π·Π½Π°Ρ‡Π΅Π½ΠΈΡŽ ΠΏΠ°Ρ€Π°ΠΌΠ΅Ρ‚Ρ€Π° $t = 0$, являСтся Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΌ r-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΎΠΌ, Ρ‚ΠΎ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΈ, ΡΠ²Π»ΡΡŽΡ‰ΠΈΠ΅ΡΡ Ρ€Π΅Π·ΡƒΠ»ΡŒΡ‚Π°Ρ‚ΠΎΠΌ достаточно ΠΌΠ°Π»ΠΎΠ³ΠΎ измСнСния $t$, Ρ‚Π°ΠΊΠΆΠ΅ Π±ΡƒΠ΄ΡƒΡ‚ Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΌΠΈ r-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ°ΠΌΠΈ. Ѐункция $Z$, рассматриваСмая ΠΊΠ°ΠΊ функция ΠΎΡ‚ $t$, Π΄ΠΎΠ»ΠΆΠ½Π° Π΄ΠΎΡΡ‚ΠΈΠ³Π°Ρ‚ΡŒ максимума ΠΏΡ€ΠΈ $t = 0$, Ссли $K$ - ΠΌΠ°ΠΊΡΠΈΠΌΠ°Π»ΡŒΠ½Ρ‹ΠΉ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ. НСобходимоС условиС Ρ‚ΠΎΠ³ΠΎ, Ρ‡Ρ‚ΠΎ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ $K$ являСтся ΠΌΠ°ΠΊΡΠΈΠΌΠ°Π»ΡŒΠ½Ρ‹ΠΌ, ΠΌΡ‹ ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ ΠΈΠ· нСслоТного расчёта: \n", + "$$\\sum\\limits_{i=1}^r \\alpha_i = 0 \\label{eq:25} \\tag{25}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ + "Π’Π΅ΠΏΠ΅Ρ€ΡŒ допустим, Ρ‡Ρ‚ΠΎ Π²Π΅Ρ€ΡˆΠΈΠ½Ρ‹ $A_i$ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$ ΠΏΠΎΠ΄Π²Π΅Ρ€Π³Π°ΡŽΡ‚ΡΡ достаточно ΠΌΠ°Π»Ρ‹ΠΌ смСщСниям, ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Π΅ ΠΎΠΏΠΈΡΡ‹Π²Π°ΡŽΡ‚ΡΡ Π½Π΅ΠΏΡ€Π΅Ρ€Ρ‹Π²Π½ΠΎ Π΄ΠΈΡ„Ρ„Π΅Ρ€Π΅Π½Ρ†ΠΈΡ€ΡƒΠ΅ΠΌΠΎΠΉ Ρ„ΡƒΠ½ΠΊΡ†ΠΈΠ΅ΠΉ ΠΎΡ‚ ΠΏΠ°Ρ€Π°ΠΌΠ΅Ρ‚Ρ€Π° $t$. Π’Π°ΠΊ ΠΊΠ°ΠΊ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ $K$, ΠΊΠΎΡ‚ΠΎΡ€Ρ‹ΠΉ Π΄ΠΎΠ»ΠΆΠ΅Π½ ΡΠΎΠΎΡ‚Π²Π΅Ρ‚ΡΡ‚Π²ΠΎΠ²Π°Ρ‚ΡŒ Π·Π½Π°Ρ‡Π΅Π½ΠΈΡŽ ΠΏΠ°Ρ€Π°ΠΌΠ΅Ρ‚Ρ€Π° $t = 0$, являСтся Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΌ r-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΎΠΌ, Ρ‚ΠΎ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΈ, ΡΠ²Π»ΡΡŽΡ‰ΠΈΠ΅ΡΡ Ρ€Π΅Π·ΡƒΠ»ΡŒΡ‚Π°Ρ‚ΠΎΠΌ достаточно ΠΌΠ°Π»ΠΎΠ³ΠΎ измСнСния $t$, Ρ‚Π°ΠΊΠΆΠ΅ Π±ΡƒΠ΄ΡƒΡ‚ Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΌΠΈ r-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ°ΠΌΠΈ. Ѐункция $Z$, рассматриваСмая ΠΊΠ°ΠΊ функция ΠΎΡ‚ $t$, Π΄ΠΎΠ»ΠΆΠ½Π° Π΄ΠΎΡΡ‚ΠΈΠ³Π°Ρ‚ΡŒ максимума ΠΏΡ€ΠΈ $t = 0$, Ссли $K$ - ΠΌΠ°ΠΊΡΠΈΠΌΠ°Π»ΡŒΠ½Ρ‹ΠΉ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ. " + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ + "НСобходимоС условиС Ρ‚ΠΎΠ³ΠΎ, Ρ‡Ρ‚ΠΎ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ $K$ являСтся ΠΌΠ°ΠΊΡΠΈΠΌΠ°Π»ΡŒΠ½Ρ‹ΠΌ, ΠΌΡ‹ ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ ΠΈΠ· нСслоТного расчёта: \n", "\n", "$$\\sum\\limits_{i=1}^r \\frac{F}{f_i} ([\\dot \\alpha_i, \\: \\alpha_{i+1}] + [\\alpha_i, \\: \\dot \\alpha_{i+1}]) - \\\\ - r \\sum\\limits_{i=1}^{r-2}([\\dot \\alpha_1 + \\ldots + \\dot \\alpha_i, \\: \\alpha_{i+1}] + [\\alpha_1 + \\ldots + \\alpha_i, \\: \\dot \\alpha_{i+1}]) = 0 \\label{eq:26} \\tag{26}$$ \n", "\n", "Π“Π΄Π΅ $\\dot \\alpha_i = \\left.\\frac{d}{dt} \\alpha_i \\right|_{t=0}$ - ΠΏΡ€ΠΎΠΈΠ·Π²ΠΎΠ»ΡŒΠ½Ρ‹Π΅ Π²Π΅ΠΊΡ‚ΠΎΡ€Ρ‹, ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Π΅ Ρ‚ΠΎΠΆΠ΅ Π΄ΠΎΠ»ΠΆΠ½Ρ‹ ΡƒΠ΄ΠΎΠ²Π»Π΅Ρ‚Π²ΠΎΡ€ΡΡ‚ΡŒ ΡƒΡΠ»ΠΎΠ²ΠΈΡŽ замкнутости, ΡΠ»Π΅Π΄ΡƒΡŽΡ‰Π΅ΠΌΡƒ ΠΈΠ· $(25)$:\n", "\n", - "$$\\sum\\limits_{i=1}^r \\dot\\alpha_i = 0 \\label{eq:27} \\tag{27}$$\n", - "\n", + "$$\\sum\\limits_{i=1}^r \\dot\\alpha_i = 0 \\label{eq:27} \\tag{27}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "Если ΠΌΡ‹ ΠΏΠΎΠ»ΠΎΠΆΠΈΠΌ ΠΏΠΎΡΠ»Π΅Π΄ΠΎΠ²Π°Ρ‚Π΅Π»ΡŒΠ½ΠΎ\n", "\n", "$$\\dot \\alpha_j = \\beta, \\;\\; \\dot \\alpha_{j+1} = -\\beta, \\;\\; \\dot \\alpha_i = 0 \\: \\text{для} \\: i \\neq j, \\;\\; i \\neq j+1 \\label{eq:28} \\tag{28}$$\n", "\n", "Π³Π΄Π΅ $\\beta$ - ΠΈΠ·Π½Π°Ρ‡Π°Π»ΡŒΠ½ΠΎ ΠΏΡ€ΠΎΠΈΠ·Π²ΠΎΠ»ΡŒΠ½Ρ‹ΠΉ Π²Π΅ΠΊΡ‚ΠΎΡ€, Ρ‚ΠΎ условиС замкнутости $(\\ref{eq:27})$, ΠΎΡ‡Π΅Π²ΠΈΠ΄Π½ΠΎ, выполняСтся, ΠΈ ΠΌΡ‹ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ для $j = 1, \\: \\ldots, \\: r$:\n", "\n", - "$$\\frac{F}{r} \\left\\{ \\frac{[\\alpha_{j-1}, \\: \\beta]}{f_{j-1}} - \\frac{[\\beta, \\: \\alpha_{j+2}]}{f_{j+1}} + \\frac1{f_j} [\\beta, \\: \\alpha_j + \\alpha_{j+1}] \\right\\} = [\\beta, \\: \\alpha_j + \\alpha_{j+1}] \\label{eq:29} \\tag{29}$$\n", - "\n", + "$$\\frac{F}{r} \\left\\{ \\frac{[\\alpha_{j-1}, \\: \\beta]}{f_{j-1}} - \\frac{[\\beta, \\: \\alpha_{j+2}]}{f_{j+1}} + \\frac1{f_j} [\\beta, \\: \\alpha_j + \\alpha_{j+1}] \\right\\} = [\\beta, \\: \\alpha_j + \\alpha_{j+1}] \\label{eq:29} \\tag{29}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "Если ΠΌΡ‹ Ρ‚Π΅ΠΏΠ΅Ρ€ΡŒ Π½Π°ΠΌΠ΅Ρ€Π΅Π½Π½ΠΎ Π²Ρ‹Π±Π΅Ρ€Π΅ΠΌ $\\beta = \\alpha_j$, Ρ„ΠΎΡ€ΠΌΡƒΠ»Π° ΠΏΡ€ΠΈΠΌΠ΅Ρ‚ Π²ΠΈΠ΄\n", "\n", "$$\\frac{F}{r} \\left\\{ 4 - \\frac{[\\alpha_j, \\: \\alpha_{j+2}]}{f_{j+1}} \\right\\} = 2 f_j \\label{eq:30} \\tag{30}$$\n", @@ -284,8 +604,17 @@ "\n", "Π‘Ρ€Π°Π²Π½ΠΈΠΌ эти Π΄Π²Π΅ Ρ„ΠΎΡ€ΠΌΡƒΠ»Ρ‹ ΠΈ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ\n", "\n", - "$$[\\alpha_j, \\: \\alpha_{j+2}] = \\frac{f_{j+1}}{f_{j-1}} [\\alpha_{j-1}, \\: \\alpha_{j-1}] \\label{eq:32} \\tag{32}$$\n", - "\n", + "$$[\\alpha_j, \\: \\alpha_{j+2}] = \\frac{f_{j+1}}{f_{j-1}} [\\alpha_{j-1}, \\: \\alpha_{j-1}] \\label{eq:32} \\tag{32}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "Π’Π΅ΠΏΠ΅Ρ€ΡŒ ΠΌΡ‹ запишСм ΡƒΡ€Π°Π²Π½Π΅Π½ΠΈΠ΅ $(\\ref{eq:31})$ Π² Ρ„ΠΎΡ€ΠΌΠ΅\n", "\n", "$$2 \\left( 2 - \\frac{r f_j}{F} \\right) = \\frac{[\\alpha_{j-1}, \\: \\alpha_{j+1}]}{f_{j-1}} \\notag$$\n", @@ -296,8 +625,17 @@ "\n", "ΠΈΠ»ΠΈ\n", "\n", - "$$f_j = \\frac2{\\cfrac{r}{F} + \\cfrac{[\\alpha_1, \\: \\alpha_3]}{2 f_1 \\cdot f_2}} \\label{eq:33} \\tag{33}$$\n", - "\n", + "$$f_j = \\frac2{\\cfrac{r}{F} + \\cfrac{[\\alpha_1, \\: \\alpha_3]}{2 f_1 \\cdot f_2}} \\label{eq:33} \\tag{33}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "Для $r = 3$ наша ΡΠΊΡΡ‚Ρ€Π΅ΠΌΠ°Π»ΡŒΠ½Π°Ρ Π·Π°Π΄Π°Ρ‡Π° Ρ‚Ρ€ΠΈΠ²ΠΈΠ°Π»ΡŒΠ½Π°. Π’ случаС $r \\ge 4$ получаСтся, Ρ‡Ρ‚ΠΎ $f_3 = f_4 = \\ldots = f_r$. ΠŸΡ€ΠΈ цикличСской пСрСстановкС Π½ΠΎΠΌΠ΅Ρ€ΠΎΠ² Π²Π΅Ρ€ΡˆΠΈΠ½ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ\n", "\n", "$$f_1 = f_2 = \\ldots = f_r \\label{eq:34} \\tag{34}$$\n", @@ -306,10 +644,28 @@ "\n", "$$[\\alpha_i, \\: \\alpha_{i+2}] = [\\alpha_1, \\: \\alpha_3] = \\beta \\;\\; (i = 1, \\: 2, \\: \\ldots, \\: r) \\label{eq:35} \\tag{35}$$\n", "\n", - "Π³Π΄Π΅ $\\beta$ - нСкоторая константа.\n", - "\n", - "**ΠŸΡ€ΠΈΠΌΠ΅Ρ‡Π°Π½ΠΈΠ΅.** Π’ случаС $\\beta = [\\alpha_i, \\: \\alpha_{i+2}] = 0$ получаСтся, Ρ‡Ρ‚ΠΎ $f_j = 2F/r$. Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, Π²Π΅ΠΊΡ‚ΠΎΡ€ $\\alpha_{i+2}$ всСгда ΠΏΠ°Ρ€Π°Π»Π»Π΅Π»Π΅Π½ $\\alpha_i$. Π­Ρ‚ΠΎ Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΠΎ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ Π² Ρ‚ΠΎΠΌ случаС, Ссли $r = 4$, Π° $K$ - ΠΏΠ°Ρ€Π°Π»Π»Π΅Π»ΠΎΠ³Ρ€Π°ΠΌΠΌ.\n", - "\n", + "Π³Π΄Π΅ $\\beta$ - нСкоторая константа." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ + "**ΠŸΡ€ΠΈΠΌΠ΅Ρ‡Π°Π½ΠΈΠ΅.** Π’ случаС $\\beta = [\\alpha_i, \\: \\alpha_{i+2}] = 0$ получаСтся, Ρ‡Ρ‚ΠΎ $f_j = 2F/r$. Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, Π²Π΅ΠΊΡ‚ΠΎΡ€ $\\alpha_{i+2}$ всСгда ΠΏΠ°Ρ€Π°Π»Π»Π΅Π»Π΅Π½ $\\alpha_i$. Π­Ρ‚ΠΎ Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΠΎ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ Π² Ρ‚ΠΎΠΌ случаС, Ссли $r = 4$, Π° $K$ - ΠΏΠ°Ρ€Π°Π»Π»Π΅Π»ΠΎΠ³Ρ€Π°ΠΌΠΌ." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "Π’Π΅ΠΏΠ΅Ρ€ΡŒ вСрнёмся ΠΊ ΠΎΠ±Ρ‰Π΅ΠΌΡƒ ΡΠ»ΡƒΡ‡Π°ΡŽ. Π‘ ΠΏΠΎΠΌΠΎΡ‰ΡŒΡŽ Π°Ρ„Ρ„ΠΈΠ½Π½Ρ‹Ρ… ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΠΎΠ²Π°Π½ΠΈΠΉ ΠΌΡ‹ всСгда ΠΌΠΎΠΆΠ΅ΠΌ Π΄ΠΎΠ±ΠΈΡ‚ΡŒΡΡ Ρ‚ΠΎΠ³ΠΎ, Ρ‡Ρ‚ΠΎ $f_j = 1/2$, Ρ‚ΠΎ Π΅ΡΡ‚ΡŒ\n", "\n", "$$[\\alpha_i, \\: \\alpha_{i+1}] = 1 \\label{eq:36} \\tag{36}$$\n", @@ -318,19 +674,47 @@ "\n", "$$F = \\frac{r}{2(2-\\beta)} \\label{eq:37} \\tag{37}$$\n", "\n", - "ΠΈ, ΡΠ»Π΅Π΄ΠΎΠ²Π°Ρ‚Π΅Π»ΡŒΠ½ΠΎ, $\\beta \\lt 2$.\n", - "\n", + "ΠΈ, ΡΠ»Π΅Π΄ΠΎΠ²Π°Ρ‚Π΅Π»ΡŒΠ½ΠΎ, $\\beta \\lt 2$." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "Π’Π°ΠΊ ΠΊΠ°ΠΊ Π²Π΅ΠΊΡ‚ΠΎΡ€Ρ‹ $\\alpha_{i-2}$ ΠΈ $\\alpha_{i-1}$ Π»ΠΈΠ½Π΅ΠΉΠ½ΠΎ нСзависимы, ΠΈΠ· $\\alpha_i = x \\alpha_{i-2} + y \\alpha_{i-1}$, учитывая $(\\ref{eq:35})$ ΠΈ $(\\ref{eq:36})$, слСдуСт:\n", "\n", - "$$\\alpha_i = -\\alpha_{i-2} + b \\alpha_{i-1} \\label{eq:38} \\tag{38}$$\n", - "\n", + "$$\\alpha_i = -\\alpha_{i-2} + b \\alpha_{i-1} \\label{eq:38} \\tag{38}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, ΠΌΡ‹ ΠΏΠΎΠΊΠ°Π·Π°Π»ΠΈ, Ρ‡Ρ‚ΠΎ Π²Π΅ΠΊΡ‚ΠΎΡ€Ρ‹ $\\alpha_i$ сторон максимального Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° Π΄ΠΎΠ»ΠΆΠ½Ρ‹ ΡƒΠ΄ΠΎΠ²Π»Π΅Ρ‚Π²ΠΎΡ€ΡΡ‚ΡŒ рСкурсивной Ρ„ΠΎΡ€ΠΌΡƒΠ»Π΅ $(\\ref{eq:38})$. Π—Π°Π²Π΅Ρ€ΡˆΠΈΠΌ Π΄ΠΎΠΊΠ°Π·Π°Ρ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ, ΠΏΠΎΠΊΠ°Π·Π°Π², Ρ‡Ρ‚ΠΎ Π² этом случаС ΠΌΠΎΠΆΠ½ΠΎ ΠΎΠΏΡ€Π΅Π΄Π΅Π»ΠΈΡ‚ΡŒ Π΅Π²ΠΊΠ»ΠΈΠ΄ΠΎΠ²Ρƒ ΠΌΠ΅Ρ‚Ρ€ΠΈΠΊΡƒ Π² Π°Ρ„Ρ„ΠΈΠ½Π½ΠΎΠΉ плоскости, ΠΎΡ‚Π½ΠΎΡΠΈΡ‚Π΅Π»ΡŒΠ½ΠΎ ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ $K$ являСтся ΠΏΡ€Π°Π²ΠΈΠ»ΡŒΠ½Ρ‹ΠΌ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΎΠΌ. \n", "\n", "Π’ΠΎ-ΠΏΠ΅Ρ€Π²Ρ‹Ρ…, Π·Π°ΠΌΠ΅Ρ‚ΠΈΠΌ, Ρ‡Ρ‚ΠΎ всСгда Π΄ΠΎΠ»ΠΆΠ½ΠΎ Π±Ρ‹Ρ‚ΡŒ\n", "\n", "$$-1 \\leq b \\lt 2 \\label{eq:39} \\tag{39}$$\n", "\n", - "Π”Π΅ΠΉΡΡ‚Π²ΠΈΡ‚Π΅Π»ΡŒΠ½ΠΎ, Ссли $b \\lt -1$, Ρ‚ΠΎ сторона $\\alpha_3$ Π±ΡƒΠ΄Π΅Ρ‚ ΠΏΠ΅Ρ€Π΅ΡΠ΅ΠΊΠ°Ρ‚ΡŒ сторону $\\alpha_1$, Ρ‡Ρ‚ΠΎ ΠΏΡ€ΠΎΡ‚ΠΈΠ²ΠΎΡ€Π΅Ρ‡ΠΈΡ‚ выпуклости $K$. ВСрхняя ΠΎΡ†Π΅Π½ΠΊΠ° $b \\lt 2$ ΡƒΠΆΠ΅ Π΄ΠΎΠΊΠ°Π·Π°Π½Π° Π² $(\\ref{eq:37})$.\n", + "Π”Π΅ΠΉΡΡ‚Π²ΠΈΡ‚Π΅Π»ΡŒΠ½ΠΎ, Ссли $b \\lt -1$, Ρ‚ΠΎ сторона $\\alpha_3$ Π±ΡƒΠ΄Π΅Ρ‚ ΠΏΠ΅Ρ€Π΅ΡΠ΅ΠΊΠ°Ρ‚ΡŒ сторону $\\alpha_1$, Ρ‡Ρ‚ΠΎ ΠΏΡ€ΠΎΡ‚ΠΈΠ²ΠΎΡ€Π΅Ρ‡ΠΈΡ‚ выпуклости $K$. ВСрхняя ΠΎΡ†Π΅Π½ΠΊΠ° $b \\lt 2$ ΡƒΠΆΠ΅ Π΄ΠΎΠΊΠ°Π·Π°Π½Π° Π² $(\\ref{eq:37})$." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "Π•Π²ΠΊΠ»ΠΈΠ΄ΠΎΠ²Π° ΠΌΠ΅Ρ‚Ρ€ΠΈΠΊΠ° ΠΎΠ΄Π½ΠΎΠ·Π½Π°Ρ‡Π½ΠΎ устанавливаСтся, Ссли ΠΌΡ‹ ΠΎΠΏΡ€Π΅Π΄Π΅Π»ΠΈΠΌ скалярноС ΠΏΡ€ΠΎΠΈΠ·Π²Π΅Π΄Π΅Π½ΠΈΠ΅ Π²Π΅ΠΊΡ‚ΠΎΡ€ΠΎΠ² $\\alpha_1$ ΠΈ $\\alpha_2$, ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Π΅ ΠΌΡ‹ рассматриваСм ΠΊΠ°ΠΊ базис:\n", "\n", "$$\\alpha_1^2 = \\alpha_2^2 = \\frac{2}{\\sqrt{4-b^2}}, \\;\\; \\alpha_1\\alpha_2 = \\frac{b}{\\sqrt{4-b^2}} \\label{eq:40} \\tag{40}$$\n", @@ -339,50 +723,112 @@ "\n", "$$[\\alpha_1, \\alpha_2]^2 = \\begin{vmatrix} \\alpha_1^2 & \\alpha_1\\alpha_2 \\\\ \\alpha_1\\alpha_2 & \\alpha_2^2 \\end{vmatrix} = 1 \\notag$$\n", "\n", - "Π‘Π»Π΅Π΄ΠΎΠ²Π°Ρ‚Π΅Π»ΡŒΠ½ΠΎ, эта ΠΌΠ΅Ρ‚Ρ€ΠΈΠΊΠ° ΠΏΠΎΠ»ΠΎΠΆΠΈΡ‚Π΅Π»ΡŒΠ½ΠΎ ΠΎΠΏΡ€Π΅Π΄Π΅Π»Π΅Π½Π°, Ρ‚.Π΅. являСтся Π΅Π²ΠΊΠ»ΠΈΠ΄ΠΎΠ²ΠΎΠΉ ΠΌΠ΅Ρ‚Ρ€ΠΈΠΊΠΎΠΉ. Π’Π΅ΠΏΠ΅Ρ€ΡŒ, ΠΈΡΠΏΠΎΠ»ΡŒΠ·ΡƒΡ Ρ€Π΅ΠΊΡƒΡ€ΡΠΈΠ²Π½ΡƒΡŽ Ρ„ΠΎΡ€ΠΌΡƒΠ»Ρƒ ΠΈΠ· $(\\ref{eq:38})$, Π»Π΅Π³ΠΊΠΎ Π΄ΠΎΠΊΠ°Π·Π°Ρ‚ΡŒ ΠΏΠΎ ΠΈΠ½Π΄ΡƒΠΊΡ†ΠΈΠΈ, Ρ‡Ρ‚ΠΎ для всСх $j$ Π΄ΠΎΠ»ΠΆΠ½ΠΎ Π²Ρ‹ΠΏΠΎΠ»Π½ΡΡ‚ΡŒΡΡ\n", + "Π‘Π»Π΅Π΄ΠΎΠ²Π°Ρ‚Π΅Π»ΡŒΠ½ΠΎ, эта ΠΌΠ΅Ρ‚Ρ€ΠΈΠΊΠ° ΠΏΠΎΠ»ΠΎΠΆΠΈΡ‚Π΅Π»ΡŒΠ½ΠΎ ΠΎΠΏΡ€Π΅Π΄Π΅Π»Π΅Π½Π°, Ρ‚.Π΅. являСтся Π΅Π²ΠΊΠ»ΠΈΠ΄ΠΎΠ²ΠΎΠΉ ΠΌΠ΅Ρ‚Ρ€ΠΈΠΊΠΎΠΉ. " + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ + "Π’Π΅ΠΏΠ΅Ρ€ΡŒ, ΠΈΡΠΏΠΎΠ»ΡŒΠ·ΡƒΡ Ρ€Π΅ΠΊΡƒΡ€ΡΠΈΠ²Π½ΡƒΡŽ Ρ„ΠΎΡ€ΠΌΡƒΠ»Ρƒ ΠΈΠ· $(\\ref{eq:38})$, Π»Π΅Π³ΠΊΠΎ Π΄ΠΎΠΊΠ°Π·Π°Ρ‚ΡŒ ΠΏΠΎ ΠΈΠ½Π΄ΡƒΠΊΡ†ΠΈΠΈ, Ρ‡Ρ‚ΠΎ для всСх $j$ Π΄ΠΎΠ»ΠΆΠ½ΠΎ Π²Ρ‹ΠΏΠΎΠ»Π½ΡΡ‚ΡŒΡΡ\n", "\n", "$$\\alpha_j^2 = \\frac{2}{\\sqrt{4-b^2}}, \\;\\; \\alpha_j\\alpha_{j+1} = \\frac{b}{\\sqrt{4-b^2}}, \\;\\; [\\alpha_j, \\:\\alpha_{j+1}] = 1 \\notag$$\n", "\n", "ΠžΡ‚ΡΡŽΠ΄Π° слСдуСт, Ρ‡Ρ‚ΠΎ всС стороны нашСго $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$ ΠΈΠΌΠ΅ΡŽΡ‚ ΠΎΠ΄ΠΈΠ½Π°ΠΊΠΎΠ²ΡƒΡŽ Π΄Π»ΠΈΠ½Ρƒ ΠΈ Ρ‡Ρ‚ΠΎ Π»ΡŽΠ±Ρ‹Π΅ Π΄Π²Π΅ смСТныС стороны ΠΏΠ΅Ρ€Π΅ΡΠ΅ΠΊΠ°ΡŽΡ‚ΡΡ ΠΏΠΎΠ΄ ΠΎΠ΄Π½ΠΈΠΌ ΡƒΠ³Π»ΠΎΠΌ. Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ Π΄ΠΎΠ»ΠΆΠ΅Π½ Π±Ρ‹Ρ‚ΡŒ ΠΏΡ€Π°Π²ΠΈΠ»ΡŒΠ½Ρ‹ΠΌ ΠΎΡ‚Π½ΠΎΡΠΈΡ‚Π΅Π»ΡŒΠ½ΠΎ этой ΠΌΠ΅Ρ‚Ρ€ΠΈΠΊΠΈ.\n", "\n", "Π­Ρ‚ΠΎ Π΄ΠΎΠΊΠ°Π·Ρ‹Π²Π°Π΅Ρ‚ Π΅Π΄ΠΈΠ½ΡΡ‚Π²Π΅Π½Π½ΠΎΡΡ‚ΡŒ максимального ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ°. Π•Π³ΠΎ сущСствованиС слСдуСт ΠΈΠ· ограничСнности $Z(K)$ ΠΈ нСпрСрывности $F$ ΠΈ $Z$. Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, ΡƒΡ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 2 ΠΏΠΎΠ»Π½ΠΎΡΡ‚ΡŒΡŽ Π΄ΠΎΠΊΠ°Π·Π°Π½ΠΎ.\n", - "
\n", + "\n", "$\\triangleleft$" ] }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, "source": [ "## 3. Π‘Π»ΡƒΡ‡Π°ΠΉΠ½ΠΎ Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Π΅ Ρ‚ΠΎΡ‡ΠΊΠΈ Π² Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ области с Π³Π»Π°Π΄ΠΊΠΎΠΉ Π³Ρ€Π°Π½ΠΈΡ†Π΅ΠΉ\n", "\n", - "**Π£Ρ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 3.** Если Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i$ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны Π² Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ области с Π³Π»Π°Π΄ΠΊΠΎΠΉ Π³Ρ€Π°Π½ΠΈΡ†Π΅ΠΉ Π½Π° плоскости, матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ $E_n = O(\\sqrt[3]n)$.\n", - "\n", + "**Π£Ρ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 3.** Если Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i$ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны Π² Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ области с Π³Π»Π°Π΄ΠΊΠΎΠΉ Π³Ρ€Π°Π½ΠΈΡ†Π΅ΠΉ Π½Π° плоскости, матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ $E_n = O(\\sqrt[3]n)$." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "**Π”ΠΎΠΊΠ°Π·Π°Ρ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ:** \n", "\n", "$\\triangleright$\n", - "
\n", - "ΠŸΡƒΡΡ‚ΡŒ Ρ‚Π΅ΠΏΠ΅Ρ€ΡŒ $K$ - выпуклая ΠΎΠ±Π»Π°ΡΡ‚ΡŒ, Π³Ρ€Π°Π½ΠΈΡ†Π° ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ ΠΈΠΌΠ΅Π΅Ρ‚ Π½Π΅ΠΏΡ€Π΅Ρ€Ρ‹Π²Π½ΡƒΡŽ ΠΊΡ€ΠΈΠ²ΠΈΠ·Π½Ρƒ ΠΊΠ°ΠΊ Ρ„ΡƒΠ½ΠΊΡ†ΠΈΡŽ ΠΎΡ‚ Π΄Π»ΠΈΠ½Ρ‹ Π΄ΡƒΠ³ΠΈ. Π’ обозначСниях ΠΏΡ€Π΅Π΄Ρ‹Π΄ΡƒΡ‰ΠΈΡ… ΠΏΠ°Ρ€Π°Π³Ρ€Π°Ρ„ΠΎΠ² ΠΌΡ‹ снова ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ\n", "\n", - "$$E_n = C_n^2 \\cdot W_n \\notag$$\n", + "ΠŸΡƒΡΡ‚ΡŒ Ρ‚Π΅ΠΏΠ΅Ρ€ΡŒ $K$ - выпуклая ΠΎΠ±Π»Π°ΡΡ‚ΡŒ, Π³Ρ€Π°Π½ΠΈΡ†Π° ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ ΠΈΠΌΠ΅Π΅Ρ‚ Π½Π΅ΠΏΡ€Π΅Ρ€Ρ‹Π²Π½ΡƒΡŽ ΠΊΡ€ΠΈΠ²ΠΈΠ·Π½Ρƒ ΠΊΠ°ΠΊ Ρ„ΡƒΠ½ΠΊΡ†ΠΈΡŽ ΠΎΡ‚ Π΄Π»ΠΈΠ½Ρ‹ Π΄ΡƒΠ³ΠΈ. Π’ обозначСниях ΠΏΡ€Π΅Π΄Ρ‹Π΄ΡƒΡ‰ΠΈΡ… ΠΏΠ°Ρ€Π°Π³Ρ€Π°Ρ„ΠΎΠ² ΠΌΡ‹ снова ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ\n", "\n", + "$$E_n = C_n^2 \\cdot W_n \\notag$$" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "Богласно ΡƒΡ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΡŽ ΠΎ ΠΌΠ΅Ρ€Π΅ мноТСства ΠΏΠ°Ρ€ Ρ‚ΠΎΡ‡Π΅ΠΊ, извСстному ΠΈΠ· ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π»ΡŒΠ½ΠΎΠΉ Π³Π΅ΠΎΠΌΠ΅Ρ‚Ρ€ΠΈΠΈ (см. Blascke [2], S 17, (86)):\n", "\n", "$$dP_1 dP_2 = \\left|t_1 - t_2\\right| \\, dt_1 \\, dt_2 \\, d\\varphi \\, dp \\label{eq:41} \\tag{41}$$\n", "\n", "Π³Π΄Π΅ $p$ β€” Π΄Π»ΠΈΠ½Π° пСрпСндикуляра ΠΊ прямой $P_1 P_2$ ΠΈΠ· Π½Π°Ρ‡Π°Π»Π° ΠΊΠΎΠΎΡ€Π΄ΠΈΠ½Π°Ρ‚, $\\varphi$ β€” ΡƒΠ³ΠΎΠ» Π½Π°ΠΊΠ»ΠΎΠ½Π° этого пСрпСндикуляра, $t_1, \\: t_2$ β€” расстояния ΠΏΠΎ этой прямой ΠΎΡ‚ $P_1$ ΠΈ $P_2$ Π΄ΠΎ Ρ‚ΠΎΡ‡ΠΊΠΈ пСрСсСчСния $P_1 P_2$ с пСрпСндикуляром. ΠžΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Ρ‡Π΅Ρ€Π΅Π· $f$ ΠΏΠ»ΠΎΡ‰Π°Π΄ΡŒ участка, отсСкаСмого ΠΎΡ‚ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$ прямой $P_1 P_2$. Π’ΠΎΠ³Π΄Π° ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ\n", "\n", - "$$E_n = \\cfrac{C_n^2}{F^2} \\int\\limits_0^{2\\pi} \\left\\{ \\int\\limits_0^{p(\\varphi)} \\left(1-\\frac{f}{F} \\right)^{n-2} ( \\iint \\left|t_1 - t_2\\right| \\, dt_1 \\, dt_2) \\, dp \\right\\} d\\varphi \\label{eq:42} \\tag{42}$$\n", - "\n", + "$$E_n = \\cfrac{C_n^2}{F^2} \\int\\limits_0^{2\\pi} \\left\\{ \\int\\limits_0^{p(\\varphi)} \\left(1-\\frac{f}{F} \\right)^{n-2} ( \\iint \\left|t_1 - t_2\\right| \\, dt_1 \\, dt_2) \\, dp \\right\\} d\\varphi \\label{eq:42} \\tag{42}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "Π§Π΅Ρ€Π΅Π· $l(p, \\phi)$ ΠΎΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Π΄Π»ΠΈΠ½Ρƒ Ρ…ΠΎΡ€Π΄Ρ‹ области $K$, Π»Π΅ΠΆΠ°Ρ‰Π΅ΠΉ Π½Π° прямой с ΠΊΠΎΠΎΡ€Π΄ΠΈΠ½Π°Ρ‚Π°ΠΌΠΈ $p, \\varphi$. Π’ΠΎΠ³Π΄Π°\n", "\n", "$$\\iint \\left|t_1 - t_2\\right| \\, dt_1 \\, dt_2 = \\frac{l^3(p, \\varphi)}{3} \\label{eq:43} \\tag{43}$$\n", "\n", "Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ,\n", "\n", - "$$E_n = \\cfrac{C_n^2}{3F^2} \\int\\limits_0^{2\\pi} \\left\\{ \\int\\limits_0^{p(\\varphi)} \\left(1-\\frac{f}{F} \\right)^{n-2} l^3(p, \\varphi) \\, dp \\right\\} \\, d\\varphi \\label{eq:44} \\tag{44}$$\n", - "\n", - "ΠŸΡ€ΠΈ расчСтС асимптотичСского значСния $E_n$ ΠΌΡ‹ ΠΌΠΎΠΆΠ΅ΠΌ, ΠΎΡ‡Π΅Π²ΠΈΠ΄Π½ΠΎ, ΠΎΠ³Ρ€Π°Π½ΠΈΡ‡ΠΈΡ‚ΡŒΡΡ ΠΏΠ°Ρ€Π°ΠΌΠΈ Π·Π½Π°Ρ‡Π΅Π½ΠΈΠΉ $(p, \\: \\varphi)$, для ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… выполняСтся $f/F < \\varepsilon$; Π΄Ρ€ΡƒΠ³ΠΈΠ΅ ΠΏΠ°Ρ€Ρ‹ Π·Π½Π°Ρ‡Π΅Π½ΠΈΠΉ Π΄Π°ΡŽΡ‚ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ остаточный Ρ‡Π»Π΅Π½ порядка $n^2(1-\\varepsilon)^n = o(1)$. Рассмотрим Ρ‚Π΅ΠΏΠ΅Ρ€ΡŒ фиксированноС Π·Π½Π°Ρ‡Π΅Π½ΠΈΠ΅ $\\varphi$. ΠŸΡƒΡΡ‚ΡŒ $P(\\varphi)$ - Ρ‚ΠΎΡ‡ΠΊΠ°, Π² ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ ΠΊΠ°ΡΠ°Ρ‚Π΅Π»ΡŒΠ½Π°Ρ ΠΊ $K$ ΠΈΠΌΠ΅Π΅Ρ‚ ΡƒΠ³ΠΎΠ» Π½Π°ΠΊΠ»ΠΎΠ½Π° $\\varphi$. Как становится ясно ΠΈΠ· Π½ΠΈΠΆΠ΅ΡΠ»Π΅Π΄ΡƒΡŽΡ‰ΠΈΡ… ΠΎΡ†Π΅Π½ΠΎΠΊ, ΠΌΠΎΠΆΠ½ΠΎ Π·Π°ΠΌΠ΅Π½ΠΈΡ‚ΡŒ $f$ ΠΈ $l$ Π½Π° ΡΠΎΠΎΡ‚Π²Π΅Ρ‚ΡΡ‚Π²ΡƒΡŽΡ‰ΠΈΠ΅ Π²Π΅Π»ΠΈΡ‡ΠΈΠ½Ρ‹ $\\bar f$ ΠΈ $\\bar l$ окруТности ΠΊΡ€ΠΈΠ²ΠΈΠ·Π½Ρ‹ Π² Ρ‚ΠΎΡ‡ΠΊΠ΅ $P(\\varphi)$. ΠŸΡƒΡΡ‚ΡŒ $\\alpha$ - Ρ†Π΅Π½Ρ‚Ρ€Π°Π»ΡŒΠ½Ρ‹ΠΉ ΡƒΠ³ΠΎΠ», ΡΠΎΠΎΡ‚Π²Π΅Ρ‚ΡΡ‚Π²ΡƒΡŽΡ‰ΠΈΠΉ Ρ…ΠΎΡ€Π΄Π΅ $l$, ΠΈ $r(\\varphi)$ - радиус окруТности ΠΊΡ€ΠΈΠ²ΠΈΠ·Π½Ρ‹ Π² Ρ‚ΠΎΡ‡ΠΊΠ΅ $P(\\varphi)$. Π’ΠΎΠ³Π΄Π°\n", + "$$E_n = \\cfrac{C_n^2}{3F^2} \\int\\limits_0^{2\\pi} \\left\\{ \\int\\limits_0^{p(\\varphi)} \\left(1-\\frac{f}{F} \\right)^{n-2} l^3(p, \\varphi) \\, dp \\right\\} \\, d\\varphi \\label{eq:44} \\tag{44}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ + "ΠŸΡ€ΠΈ расчСтС асимптотичСского значСния $E_n$ ΠΌΡ‹ ΠΌΠΎΠΆΠ΅ΠΌ, ΠΎΡ‡Π΅Π²ΠΈΠ΄Π½ΠΎ, ΠΎΠ³Ρ€Π°Π½ΠΈΡ‡ΠΈΡ‚ΡŒΡΡ ΠΏΠ°Ρ€Π°ΠΌΠΈ Π·Π½Π°Ρ‡Π΅Π½ΠΈΠΉ $(p, \\: \\varphi)$, для ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… выполняСтся $f/F < \\varepsilon$; Π΄Ρ€ΡƒΠ³ΠΈΠ΅ ΠΏΠ°Ρ€Ρ‹ Π·Π½Π°Ρ‡Π΅Π½ΠΈΠΉ Π΄Π°ΡŽΡ‚ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ остаточный Ρ‡Π»Π΅Π½ порядка $n^2(1-\\varepsilon)^n = o(1)$. Рассмотрим Ρ‚Π΅ΠΏΠ΅Ρ€ΡŒ фиксированноС Π·Π½Π°Ρ‡Π΅Π½ΠΈΠ΅ $\\varphi$. ΠŸΡƒΡΡ‚ΡŒ $P(\\varphi)$ - Ρ‚ΠΎΡ‡ΠΊΠ°, Π² ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ ΠΊΠ°ΡΠ°Ρ‚Π΅Π»ΡŒΠ½Π°Ρ ΠΊ $K$ ΠΈΠΌΠ΅Π΅Ρ‚ ΡƒΠ³ΠΎΠ» Π½Π°ΠΊΠ»ΠΎΠ½Π° $\\varphi$. Как становится ясно ΠΈΠ· Π½ΠΈΠΆΠ΅ΡΠ»Π΅Π΄ΡƒΡŽΡ‰ΠΈΡ… ΠΎΡ†Π΅Π½ΠΎΠΊ, ΠΌΠΎΠΆΠ½ΠΎ Π·Π°ΠΌΠ΅Π½ΠΈΡ‚ΡŒ $f$ ΠΈ $l$ Π½Π° ΡΠΎΠΎΡ‚Π²Π΅Ρ‚ΡΡ‚Π²ΡƒΡŽΡ‰ΠΈΠ΅ Π²Π΅Π»ΠΈΡ‡ΠΈΠ½Ρ‹ $\\bar f$ ΠΈ $\\bar l$ окруТности ΠΊΡ€ΠΈΠ²ΠΈΠ·Π½Ρ‹ Π² Ρ‚ΠΎΡ‡ΠΊΠ΅ $P(\\varphi)$. " + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ + "ΠŸΡƒΡΡ‚ΡŒ $\\alpha$ - Ρ†Π΅Π½Ρ‚Ρ€Π°Π»ΡŒΠ½Ρ‹ΠΉ ΡƒΠ³ΠΎΠ», ΡΠΎΠΎΡ‚Π²Π΅Ρ‚ΡΡ‚Π²ΡƒΡŽΡ‰ΠΈΠΉ Ρ…ΠΎΡ€Π΄Π΅ $l$, ΠΈ $r(\\varphi)$ - радиус окруТности ΠΊΡ€ΠΈΠ²ΠΈΠ·Π½Ρ‹ Π² Ρ‚ΠΎΡ‡ΠΊΠ΅ $P(\\varphi)$. Π’ΠΎΠ³Π΄Π°\n", "\n", "$$\\bar l = 2 \\, r(\\varphi) \\, \\sin(\\alpha/2) \\quad \\text{ΠΈ} \\quad \\bar f = \\frac12 (\\alpha - \\sin\\alpha) \\, r^2(\\varphi) \\label{eq:45} \\tag{45}$$\n", "\n", @@ -392,12 +838,30 @@ "\n", "НаконСц,\n", "\n", - "$$l - \\bar l = o(\\alpha^2) \\quad \\text{ΠΈ} \\quad f - \\bar f = o(\\alpha^4) \\label{eq:47} \\tag{47}$$\n", - "\n", + "$$l - \\bar l = o(\\alpha^2) \\quad \\text{ΠΈ} \\quad f - \\bar f = o(\\alpha^4) \\label{eq:47} \\tag{47}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ Ρ€Π΅Π·ΡƒΠ»ΡŒΡ‚Π°Ρ‚ для $E_n$:\n", "\n", - "$$E_n = \\cfrac{C_n^2}{12F^2} \\int\\limits_0^{2\\pi} r^4 \\int\\limits_0^{\\alpha(\\varphi)} \\left( 1 - \\frac{\\alpha^3r^2}{12F^2} + o(\\alpha^3) \\right)^{n-2} (\\alpha^4 + o(\\alpha^5)) \\, d\\alpha \\, d\\varphi \\label{eq:48} \\tag{48}$$\n", - "\n", + "$$E_n = \\cfrac{C_n^2}{12F^2} \\int\\limits_0^{2\\pi} r^4 \\int\\limits_0^{\\alpha(\\varphi)} \\left( 1 - \\frac{\\alpha^3r^2}{12F^2} + o(\\alpha^3) \\right)^{n-2} (\\alpha^4 + o(\\alpha^5)) \\, d\\alpha \\, d\\varphi \\label{eq:48} \\tag{48}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "ΠŸΠΎΠ΄ΡΡ‚Π°Π²ΠΈΠ²\n", "\n", "$$\\frac{\\alpha^3r^2}{12F} = \\frac{X}{n} \\label{eq:49} \\tag{49}$$\n", @@ -405,42 +869,83 @@ "ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ\n", "\n", "$$S(\\varphi) = \\int\\limits_0^{\\alpha(\\varphi)} \\left(1 - \\frac{\\alpha^3r^2}{12F} + o(\\alpha^3)\\right)^{n-2} (\\alpha^4 + o(\\alpha^5)) \\, d\\alpha \\label{eq:50} \\tag{50}$$\n", - "$$S(\\varphi) = \\frac13 \\left(\\frac{12F}{r^2 n}\\right)^{5/3} \\left(\\int\\limits_0^\\infty \\left(1 - \\frac{X}{n}\\right)^{n-2} x^{2/3} \\, dx + o(1)\\right) = \\\\ = \\frac13 \\, \\Gamma \\left( \\frac{5}{3} \\right) \\left(\\frac{12F}{r^2 n}\\right)^{5/3} (1 + o(1)) \\notag$$\n", - "\n", + "$$S(\\varphi) = \\frac13 \\left(\\frac{12F}{r^2 n}\\right)^{5/3} \\left(\\int\\limits_0^\\infty \\left(1 - \\frac{X}{n}\\right)^{n-2} x^{2/3} \\, dx + o(1)\\right) = \\\\ = \\frac13 \\, \\Gamma \\left( \\frac{5}{3} \\right) \\left(\\frac{12F}{r^2 n}\\right)^{5/3} (1 + o(1)) \\notag$$" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ\n", "\n", "$$E_n \\sim \\Gamma \\left(\\frac{5}{3}\\right) \\, \\sqrt[3]{\\frac{2n}{3F}} \\, \\int\\limits_0^{2\\pi} r^{2/3} \\, d\\varphi \\label{eq:51} \\tag{51}$$\n", "\n", - "ΠžΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Π·Π° $s$ Π΄Π»ΠΈΠ½Ρƒ Π΄ΡƒΠ³ΠΈ Π³Ρ€Π°Π½ΠΈΡ†Ρ‹ $K$ ΠΈ, считая $d\\varphi/ds = 1/r$, ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ ΡΠ»Π΅Π΄ΡƒΡŽΡ‰Π΅Π΅ ΡƒΡ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅:\n", - "\n", + "ΠžΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Π·Π° $s$ Π΄Π»ΠΈΠ½Ρƒ Π΄ΡƒΠ³ΠΈ Π³Ρ€Π°Π½ΠΈΡ†Ρ‹ $K$ ΠΈ, считая $d\\varphi/ds = 1/r$, ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ ΡΠ»Π΅Π΄ΡƒΡŽΡ‰Π΅Π΅ ΡƒΡ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅:" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "Если $K$ - выпуклая ΠΎΠ±Π»Π°ΡΡ‚ΡŒ, граничная кривая ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ ΠΈΠΌΠ΅Π΅Ρ‚ Π½Π΅ΠΏΡ€Π΅Ρ€Ρ‹Π²Π½ΡƒΡŽ ΠΊΡ€ΠΈΠ²ΠΈΠ·Π½Ρƒ $x = x(s)$, Ρ‚ΠΎ\n", "\n", "$$E_n \\sim \\Gamma \\left(\\frac{5}{3}\\right) \\, \\sqrt[3]{\\frac23} \\, (F^{-1/3} \\int\\limits_L x^{1/3} \\, ds) \\sqrt[3]{n} \\label{eq:52} \\tag{52}$$\n", "\n", "Π˜Π½Ρ‚Π΅Π³Ρ€Π°Π» $\\int\\limits_L x^{1/3} \\, ds$, входящий Π² $(\\ref{eq:52})$, являСтся Π°Ρ„Ρ„ΠΈΠ½Π½ΠΎΠΉ Π΄Π»ΠΈΠ½ΠΎΠΉ ΠΊΡ€ΠΈΠ²ΠΎΠΉ $L$. Π›Π΅Π³ΠΊΠΎ Π²ΠΈΠ΄Π΅Ρ‚ΡŒ, Ρ‡Ρ‚ΠΎ ΠΌΠ½ΠΎΠΆΠΈΡ‚Π΅Π»ΡŒ $\\sqrt[3]{n}$ являСтся Π°Ρ„Ρ„ΠΈΠ½Π½ΠΎ-ΠΈΠ½Π²Π°Ρ€ΠΈΠ°Π½Ρ‚Π½Ρ‹ΠΌ. Из Π°Ρ„Ρ„ΠΈΠ½Π½ΠΎΠ³ΠΎ изопСримСтричСского свойства эллипсов (см. [3]) нСпосрСдствСнно слСдуСт, Ρ‡Ρ‚ΠΎ этот ΠΌΠ½ΠΎΠΆΠΈΡ‚Π΅Π»ΡŒ максималСн для эллипсов ΠΈ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ для эллипсов.\n", - "
\n", + "\n", "$\\triangleleft$" ] }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, "source": [ "## 4. ΠΠΎΡ€ΠΌΠ°Π»ΡŒΠ½ΠΎΠ΅ распрСдСлСниС случайных Ρ‚ΠΎΡ‡Π΅ΠΊ\n", "\n", "**Π£Ρ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 4.** ΠŸΡƒΡΡ‚ΡŒ Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i \\:(i = 1, \\:2, \\:\\ldots, \\:n)$ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ Π½Π° плоскости нСзависимо Π΄Ρ€ΡƒΠ³ ΠΎΡ‚ Π΄Ρ€ΡƒΠ³Π° Π² соотвСтствии с Π½ΠΎΡ€ΠΌΠ°Π»ΡŒΠ½Ρ‹ΠΌ распрСдСлСниСм. Π’ΠΎΠ³Π΄Π°\n", "\n", - "$$E_n \\sim 2 \\sqrt{2 \\pi \\log n} \\label{eq:53} \\tag{53}$$\n", - "\n", + "$$E_n \\sim 2 \\sqrt{2 \\pi \\log n} \\label{eq:53} \\tag{53}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "**Π”ΠΎΠΊΠ°Π·Π°Ρ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ:** \n", "\n", "$\\triangleright$\n", - "
\n", + "\n", "Π’Π°ΠΊ ΠΊΠ°ΠΊ $E_n$ ΠΈΠ½Π²Π°Ρ€ΠΈΠ°Π½Ρ‚Π½ΠΎ ΠΎΡ‚Π½ΠΎΡΠΈΡ‚Π΅Π»ΡŒΠ½ΠΎ Π°Ρ„Ρ„ΠΈΠ½Π½Ρ‹Ρ… ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΠΎΠ²Π°Π½ΠΈΠΉ плоскости, ΠΌΠΎΠΆΠ½ΠΎ ΡΡ‡ΠΈΡ‚Π°Ρ‚ΡŒ, Ρ‡Ρ‚ΠΎ функция плотности нашСго Π½ΠΎΡ€ΠΌΠ°Π»ΡŒΠ½ΠΎΠ³ΠΎ распрСдСлСния ΠΈΠΌΠ΅Π΅Ρ‚ Π²ΠΈΠ΄ $\\frac{1}{2\\pi} \\, exp \\left(-\\frac{x^2 + y^2}{2}\\right) \\notag$. Как ΠΈ Π² ΠΏΡ€Π΅Π΄Ρ‹Π΄ΡƒΡ‰ΠΈΡ… ΠΏΠ°Ρ€Π°Π³Ρ€Π°Ρ„Π°Ρ…, \n", "\n", "$$E_n = C_n^2 \\cdot W_n \\notag$$\n", "\n", - "Π³Π΄Π΅ $W_n$ - Π²Π΅Ρ€ΠΎΡΡ‚Π½ΠΎΡΡ‚ΡŒ Ρ‚ΠΎΠ³ΠΎ, Ρ‡Ρ‚ΠΎ всС Ρ‚ΠΎΡ‡ΠΊΠΈ $P_3, \\:\\ldots, \\:P_n$ Π»Π΅ΠΆΠ°Ρ‚ ΠΏΠΎ ΠΎΠ΄Π½Ρƒ сторону ΠΎΡ‚ прямой $P_1 P_2$.\n", + "Π³Π΄Π΅ $W_n$ - Π²Π΅Ρ€ΠΎΡΡ‚Π½ΠΎΡΡ‚ΡŒ Ρ‚ΠΎΠ³ΠΎ, Ρ‡Ρ‚ΠΎ всС Ρ‚ΠΎΡ‡ΠΊΠΈ $P_3, \\:\\ldots, \\:P_n$ Π»Π΅ΠΆΠ°Ρ‚ ΠΏΠΎ ΠΎΠ΄Π½Ρƒ сторону ΠΎΡ‚ прямой $P_1 P_2$." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "Из Ρ„ΠΎΡ€ΠΌΡƒΠ»Ρ‹ ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π»ΡŒΠ½ΠΎΠΉ Π³Π΅ΠΎΠΌΠ΅Ρ‚Ρ€ΠΈΠΈ \n", "\n", "$$dP_1dP_2 = |t_1 - t_2| \\, dt_1 \\, dt_2 \\, dp \\, d\\varphi \\notag$$ \n", @@ -451,30 +956,66 @@ "\n", "$$\\text{Π³Π΄Π΅} \\: g(p) = \\int\\limits_{-\\infty}^{+\\infty} \\int\\limits_{P}^{+\\infty} \\frac{1}{2\\pi} \\, exp\\left(-\\frac{x^2+y^2}{2}\\right) \\, dx \\, dy = 1 - \\varPhi(p) \\label{eq:55} \\tag{55}$$\n", "\n", - "$$\\varPhi(p) = \\frac{1}{\\sqrt{2\\pi}} \\int\\limits_{-\\infty}^p exp\\left(-\\frac{u^2}{2}\\right) \\, du \\label{eq:56} \\tag{56}$$\n", - "\n", + "$$\\varPhi(p) = \\frac{1}{\\sqrt{2\\pi}} \\int\\limits_{-\\infty}^p exp\\left(-\\frac{u^2}{2}\\right) \\, du \\label{eq:56} \\tag{56}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "Π§Π΅Ρ€Π΅Π· Π·Π°ΠΌΠ΅Π½Ρƒ\n", "\n", "$$u = \\frac{t_1 - t_2}{\\sqrt 2}, \\: v = \\frac{t_1 + t_2}{\\sqrt 2}, \\: \\frac{\\partial(t_1, t_2)}{\\partial(u, v)} = 1 \\label{eq:57} \\tag{57}$$\n", "\n", "ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ\n", "\n", - "$$\\frac{1}{2\\pi} \\int\\int |t_1 - t_2| \\, exp\\left(-\\frac{t_1^2 + t_2^2}{2}\\right) \\, dt_1 \\, dt_2 = \\frac{2}{\\sqrt \\pi} \\label{eq:58} \\tag{58}$$\n", - "\n", + "$$\\frac{1}{2\\pi} \\int\\int |t_1 - t_2| \\, exp\\left(-\\frac{t_1^2 + t_2^2}{2}\\right) \\, dt_1 \\, dt_2 = \\frac{2}{\\sqrt \\pi} \\label{eq:58} \\tag{58}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "Π‘Π»Π΅Π΄ΠΎΠ²Π°Ρ‚Π΅Π»ΡŒΠ½ΠΎ,\n", "\n", "$$W_n = \\frac{2}{\\sqrt \\pi} \\int\\limits_0^\\infty [\\varPhi^{n-2}(p) + (1 - \\varPhi(p))^{n-2}] e^{-p^2} dp \\label{eq:59} \\tag{59}$$\n", "\n", "ΠΈ, ΠΎΡ‡Π΅Π²ΠΈΠ΄Π½ΠΎ, $1 - \\varPhi(p) \\leqq 1/2$ для $p \\geqq 0$. А Π·Π½Π°Ρ‡ΠΈΡ‚,\n", "\n", - "$$W_n \\sim \\frac{2}{\\sqrt \\pi} \\int\\limits_0^\\infty \\varPhi^{n-2}(p) e^{-p^2} dp \\label{eq:60} \\tag{60}$$\n", - "\n", + "$$W_n \\sim \\frac{2}{\\sqrt \\pi} \\int\\limits_0^\\infty \\varPhi^{n-2}(p) e^{-p^2} dp \\label{eq:60} \\tag{60}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "Π’Π΅ΠΏΠ΅Ρ€ΡŒ для $p > 1$ \n", "\n", "$$\\varPhi(p) = 1 - \\frac{exp \\left(-\\frac{p^2}{2}\\right)}{\\sqrt{2\\pi}p\\left(1 + \\frac{\\theta p}{p^2}\\right)} \\label{eq:61} \\tag{61}$$\n", "\n", - "Π³Π΄Π΅ $0 < \\theta_p < 1$ (см. [4], стр. 136, Π·Π°Π΄Π°Ρ‡Π° 18).\n", - "\n", + "Π³Π΄Π΅ $0 < \\theta_p < 1$ (см. [4], стр. 136, Π·Π°Π΄Π°Ρ‡Π° 18)." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ "Π’Ρ‹ΠΏΠΎΠ»Π½ΠΈΠΌ подстановку \n", "\n", "$$p = \\sqrt{2\\log n - \\log(2\\log n)} + \\frac{u - \\log \\sqrt{2\\pi}}{\\sqrt{2 \\log n}} \\label{eq:62} \\tag{62}$$\n", @@ -484,13 +1025,17 @@ "$$W_n \\sim \\frac{4\\sqrt{2\\pi\\log n}}{n^2}$$\n", "\n", "Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, $E_n \\sim 2\\sqrt{2\\pi \\log n}$, Ρ‡Ρ‚ΠΎ ΠΈ Ρ‚Ρ€Π΅Π±ΠΎΠ²Π°Π»ΠΎΡΡŒ Π΄ΠΎΠΊΠ°Π·Π°Ρ‚ΡŒ.\n", - "
\n", + "\n", "$\\triangleleft$" ] }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, "source": [ "## Π›ΠΈΡ‚Π΅Ρ€Π°Ρ‚ΡƒΡ€Π°\n", "\n", @@ -502,7 +1047,6 @@ } ], "metadata": { - "celltoolbar": "Slideshow", "kernelspec": { "display_name": "Python 3", "language": "python", diff --git a/33_expectation_of_convex_hull_vertices/33_expectation_of_convex_hull_vertices.slides.html b/33_expectation_of_convex_hull_vertices/33_expectation_of_convex_hull_vertices.slides.html new file mode 100644 index 0000000..e7f281f --- /dev/null +++ b/33_expectation_of_convex_hull_vertices/33_expectation_of_convex_hull_vertices.slides.html @@ -0,0 +1,12714 @@ + + + + + + + + + + + +33_expectation_of_convex_hull_vertices slides + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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Среднее количество вершин в выпуклой оболочке

Алгоритм ДТарвиса ΠΈΠΌΠ΅Π΅Ρ‚ ΡΠ»ΠΎΠΆΠ½ΠΎΡΡ‚ΡŒ $O(X N)$, Π³Π΄Π΅ $X$ β€” число Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ, Π° $N$ β€” ΠΎΠ±Ρ‰Π΅Π΅ число Ρ‚ΠΎΡ‡Π΅ΠΊ. Π§Ρ‚ΠΎΠ±Ρ‹ ΠΎΡ†Π΅Π½ΠΈΡ‚ΡŒ ΡΠ»ΠΎΠΆΠ½ΠΎΡΡ‚ΡŒ этого Π°Π»Π³ΠΎΡ€ΠΈΡ‚ΠΌΠ° Π² срСднСм случаС, Π½Π΅ΠΎΠ±Ρ…ΠΎΠ΄ΠΈΠΌΠΎ Π²Ρ‹Ρ‡ΠΈΡΠ»ΠΈΡ‚ΡŒ $E(X)$ β€” матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ Π²Π΅Π»ΠΈΡ‡ΠΈΠ½Ρ‹ $X$. Для Π΅Π³ΠΎ Π°Π½Π°Π»ΠΈΠ·Π° ΠΌΡ‹ Π΄ΠΎΠ»ΠΆΠ½Ρ‹ ΠΎΠΏΡ€Π΅Π΄Π΅Π»ΠΈΡ‚ΡŒ ΠΎΠ³Ρ€Π°Π½ΠΈΡ‡Π΅Π½Π½ΡƒΡŽ ΠΎΠ±Π»Π°ΡΡ‚ΡŒ, ΠΈΠ· ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ Ρ‚ΠΎΡ‡ΠΊΠΈ, Ρ‚Π°ΠΊ ΠΊΠ°ΠΊ ΠΎΠ½ΠΈ ΠΌΠΎΠ³ΡƒΡ‚ Π±Ρ‹Ρ‚ΡŒ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны лишь Π² мноТСствС с ΠΊΠΎΠ½Π΅Ρ‡Π½ΠΎΠΉ ΠΌΠ΅Ρ€ΠΎΠΉ Π›Π΅Π±Π΅Π³Π° [Kendall, Moran (1963)].

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Π’Π΅ΠΎΡ€Π΅ΠΌΠ° 1 [RΓ©nyi, Sulanke (1963)]. Если $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ ΠΈ нСзависимо ΠΈΠ· Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° Π½Π° плоскости, Ρ‚ΠΎ ΠΏΡ€ΠΈ $N\to\infty$ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ этих Ρ‚ΠΎΡ‡Π΅ΠΊ $E(X)=\frac23 r \, (\log N + \gamma) + O(1)$, Π³Π΄Π΅ $\gamma$ - константа Π­ΠΉΠ»Π΅Ρ€Π°.

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Π’Π΅ΠΎΡ€Π΅ΠΌΠ° 2 [Raynaud (1970)]. Если $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ ΠΈ нСзависимо ΠΈΠ· внутрСнности $k$-ΠΌΠ΅Ρ€Π½ΠΎΠΉ гипСрсфСры, Ρ‚ΠΎ ΠΏΡ€ΠΈ $N\to\infty$ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π³ΠΈΠΏΠ΅Ρ€Π³Ρ€Π°Π½Π΅ΠΉ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ этих Ρ‚ΠΎΡ‡Π΅ΠΊ асимптотичСски стрСмится ΠΊ $E(X) = O(N^{(k-1)/(k+1)})$

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Π’Π΅ΠΎΡ€Π΅ΠΌΠ° 3 [Raynaud (1970)]. Если $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ нСзависимо Π² соотвСтствии с $k$-ΠΌΠ΅Ρ€Π½Ρ‹ΠΌ Π½ΠΎΡ€ΠΌΠ°Π»ΡŒΠ½Ρ‹ΠΌ распрСдСлСниСм, Ρ‚ΠΎ ΠΏΡ€ΠΈ $N\to\infty$ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ этих Ρ‚ΠΎΡ‡Π΅ΠΊ асимптотичСски стрСмится ΠΊ $E(X) = O((\log N)^{(k-1)/2})$

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Π’Π΅ΠΎΡ€Π΅ΠΌΠ° 4 [Bentley, Kung, Schkolnick, Thompson (1978)]. Если ΠΊΠΎΠΎΡ€Π΄ΠΈΠ½Π°Ρ‚Ρ‹ $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ Π² $k$-ΠΌΠ΅Ρ€Π½ΠΎΠΌ пространствС Π²Ρ‹Π±Ρ€Π°Π½Ρ‹ нСзависимо Π² соотвСтствии с ΠΏΡ€ΠΎΠΈΠ·Π²ΠΎΠ»ΡŒΠ½Ρ‹ΠΌ Π½Π΅ΠΏΡ€Π΅Ρ€Ρ‹Π²Π½Ρ‹ΠΌ распрСдСлСниСм (Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΠΎ, для ΠΊΠ°ΠΆΠ΄ΠΎΠΉ ΠΊΠΎΠΎΡ€Π΄ΠΈΠ½Π°Ρ‚Ρ‹ ΠΈΡΠΏΠΎΠ»ΡŒΠ·ΡƒΠ΅Ρ‚ΡΡ своё распрСдСлСниС), Ρ‚ΠΎ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ этих Ρ‚ΠΎΡ‡Π΅ΠΊ асимптотичСски стрСмится ΠΊ $E(X) = O((\log N)^{k-1})$.

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  • $E(X) = O(\log N)$ для $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ, Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Ρ… ΠΈΠ· Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ°
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  • $E(X) = O(N^{1/3})$ для $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ, Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Ρ… ΠΈΠ· ΠΊΡ€ΡƒΠ³Π°
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  • $E(X) = O(N^{1/2})$ для $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ, Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Ρ… ΠΈΠ· сфСры
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  • $E(X) = O((\log N)^2)$ для $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ, Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Ρ… ΠΈΠ· ΠΊΡƒΠ±Π°
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  • $E(X) = O(\sqrt{\log N})$ для $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ, Π½ΠΎΡ€ΠΌΠ°Π»ΡŒΠ½ΠΎ распрСдСлённых Π½Π° плоскости
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О выпуклой оболочке N случайно выбранных точек

[RΓ©nyi, Sulanke (1963)]

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ΠŸΡƒΡΡ‚ΡŒ $P_i\:(i=1,\:2,\:\ldots,\:n)$ β€” Ρ‚ΠΎΡ‡ΠΊΠΈ Π½Π° плоскости, Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Π΅ случайно ΠΈ нСзависимо Π² соотвСтствии с Π½Π΅ΠΊΠΎΡ‚ΠΎΡ€Ρ‹ΠΌ распрСдСлСниСм, Π° $H_n$ - выпуклая ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠ° этих Ρ‚ΠΎΡ‡Π΅ΠΊ, которая, ΠΊΠ°ΠΊ извСстно, являСтся Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΌ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΎΠΌ. Число $X_n$ Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ β€” случайная Π²Π΅Π»ΠΈΡ‡ΠΈΠ½Π°, ΠΈ Π² Π΄Π°Π½Π½ΠΎΠΉ Ρ€Π°Π±ΠΎΡ‚Π΅ Π±ΡƒΠ΄Π΅Ρ‚ исслСдовано Π΅Ρ‘ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ $E_n = E(X_n)$ ΠΏΡ€ΠΈ $n\to\infty$ Π² Ρ€Π°Π·Π»ΠΈΡ‡Π½Ρ‹Ρ… прСдполоТСниях ΠΎ распрСдСлСнии Ρ‚ΠΎΡ‡Π΅ΠΊ $P_i$.

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  • Π’ $\S1$ рассмотрим случай, ΠΊΠΎΠ³Π΄Π° Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i$ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны Π²Π½ΡƒΡ‚Ρ€ΠΈ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ r-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$. Как Π±ΡƒΠ΄Π΅Ρ‚ ΠΏΠΎΠΊΠ°Π·Π°Π½ΠΎ, $E_n = (2r\,/\,3)(\log n + \gamma) + T + O(1)$, Π³Π΄Π΅ $\gamma$ - постоянная Π­ΠΉΠ»Π΅Ρ€Π°:

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    $$\gamma = \lim_{n\to\infty} \left( \sum_{k=1}^{n} \frac1k - \log n \right)=\lim_{n\to\infty} \left( 1+\frac12+\frac13+\ldots+\frac1n - \log n \right) \notag$$

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    ΠŸΡ€ΠΈ этом Π²Π΅Π»ΠΈΡ‡ΠΈΠ½Π° $T = T(K)$ зависит ΠΎΡ‚ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$.

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  • Π’ $\S2$ Π΄ΠΎΠΊΠ°ΠΆΠ΅ΠΌ ΡΠ»Π΅ΠΌΠ΅Π½Ρ‚Π°Ρ€Π½ΡƒΡŽ Π³Π΅ΠΎΠΌΠ΅Ρ‚Ρ€ΠΈΡ‡Π΅ΡΠΊΡƒΡŽ Ρ‚Π΅ΠΎΡ€Π΅ΠΌΡƒ ΠΎ Ρ‚ΠΎΠΌ, Ρ‡Ρ‚ΠΎ $T(K)$ ΠΏΡ€ΠΈΠ½ΠΈΠΌΠ°Π΅Ρ‚ ΠΌΠ°ΠΊΡΠΈΠΌΠ°Π»ΡŒΠ½Ρ‹Π΅ значСния для ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΎΠ², ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Π΅ Π°Ρ„Ρ„ΠΈΠ½Π½Ρ‹ΠΌΠΈ прСобразованиями ΠΌΠΎΠΆΠ½ΠΎ привСсти ΠΊ ΠΏΡ€Π°Π²ΠΈΠ»ΡŒΠ½ΠΎΠΌΡƒ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΡƒ, ΠΈ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ для Π½ΠΈΡ….

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  • Π’ $\S3$ исслСдуСм асимптотичСскоС ΠΏΠΎΠ²Π΅Π΄Π΅Π½ΠΈΠ΅ $E_n$, ΠΊΠΎΠ³Π΄Π° Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i$ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны Π² Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ области с Π³Π»Π°Π΄ΠΊΠΎΠΉ Π³Ρ€Π°Π½ΠΈΡ†Π΅ΠΉ. ΠžΠΊΠ°Π·Ρ‹Π²Π°Π΅Ρ‚ΡΡ, Ρ‡Ρ‚ΠΎ Π² Ρ‚Π°ΠΊΠΎΠΌ случаС $E_n$ ΠΈΠΌΠ΅Π΅Ρ‚ порядок $\sqrt[3]n$.

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  • НаконСц, Π² $\S4$ ΠΌΡ‹ Π΄ΠΎΠΊΠ°ΠΆΠ΅ΠΌ, Ρ‡Ρ‚ΠΎ $E_n$ Π²Π΅Π΄Ρ‘Ρ‚ сСбя ΠΊΠ°ΠΊ $\sqrt{\log n}$, Ссли Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i$ ΠΈΠΌΠ΅ΡŽΡ‚ Π½ΠΎΡ€ΠΌΠ°Π»ΡŒΠ½ΠΎΠ΅ распрСдСлСниС.

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1. Случайно выбранные точки в выпуклом многоугольнике

Π£Ρ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 1. Если Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i$ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны Π²Π½ΡƒΡ‚Ρ€ΠΈ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ r-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$, матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ $E_n = (2r\,/\,3)(\log n + \gamma) + T + O(1)$, Π³Π΄Π΅ $\gamma$ - постоянная Π­ΠΉΠ»Π΅Ρ€Π°.

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Π”ΠΎΠΊΠ°Π·Π°Ρ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ:

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$\triangleright$

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ΠŸΡƒΡΡ‚ΡŒ $K$ - Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΉ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ с Π²Π΅Ρ€ΡˆΠΈΠ½Π°ΠΌΠΈ $A_1,\:A_2,\:\ldots,\:A_r$ ΠΈ ΡΠΎΠΎΡ‚Π²Π΅Ρ‚ΡΡ‚Π²ΡƒΡŽΡ‰ΠΈΠΌΠΈ ΡƒΠ³Π»Π°ΠΌΠΈ $\vartheta_1,\:\vartheta_2,\:\ldots,\:\vartheta_r$. ΠžΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Π·Π° $a_k$ Π΄Π»ΠΈΠ½Ρƒ стороны $A_k A_{k+1}$, Π³Π΄Π΅ $A_{r+1} = A_1$ $(k = 1,\:2,\:\ldots,\:r)$.

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1.png

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ΠŸΡƒΡΡ‚ΡŒ $\varepsilon_{ij} = 1$, Ссли ΠΎΡ‚Ρ€Π΅Π·ΠΎΠΊ $P_i P_j$ ΠΏΡ€ΠΈ $i \neq j$ ΠΏΡ€ΠΈΠ½Π°Π΄Π»Π΅ΠΆΠΈΡ‚ Π³Ρ€Π°Π½ΠΈΡ†Π΅ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ $H_n$, ΠΈΠ½Π°Ρ‡Π΅ $\varepsilon_{ij} = 0$. Π’ΠΎΠ³Π΄Π° ΠΎΡ‡Π΅Π²ΠΈΠ΄Π½ΠΎ, Ρ‡Ρ‚ΠΎ

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$$X_n = \sum_{1 \le i \lt j \le n}\varepsilon_{ij} \label{eq:1} \tag{1}$$

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Π’ силу случайного распрСдСлСния Π²Π΅Ρ€ΠΎΡΡ‚Π½ΠΎΡΡ‚ΡŒ $W_n = P(\varepsilon_{ij} = 1)$ ΠΎΠ΄ΠΈΠ½Π°ΠΊΠΎΠ²Π° для всСх ΠΏΠ°Ρ€ $(i, j)$, Ρ‚Π°ΠΊΠΈΡ…, Ρ‡Ρ‚ΠΎ $1 \le i \lt j \leq n$, ΠΈ Ρ‚ΠΎΠ³Π΄Π°

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$$E_n = E(X_n) = \sum_{1 \le i \lt j \le n} E(\varepsilon_{ij}) = \sum_{1 \le i \lt j \le n} \left( 0 \cdot (1 - W_n) + 1 \cdot W_n \right) = C_n^2 \cdot W_n \label{eq:2} \tag{2}$$

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Π§Ρ‚ΠΎΠ±Ρ‹ ΠΎΠΏΡ€Π΅Π΄Π΅Π»ΠΈΡ‚ΡŒ асимптотичСскоС ΠΏΠΎΠ²Π΅Π΄Π΅Π½ΠΈΠ΅ $E_n$, достаточно Π²Ρ‹Ρ‡ΠΈΡΠ»ΠΈΡ‚ΡŒ Π²Π΅Ρ€ΠΎΡΡ‚Π½ΠΎΡΡ‚ΡŒ $W_n$. МоТСм Ρ€Π°ΡΡΠΌΠΎΡ‚Ρ€Π΅Ρ‚ΡŒ $W_n$ ΠΊΠ°ΠΊ Π²Π΅Ρ€ΠΎΡΡ‚Π½ΠΎΡΡ‚ΡŒ Ρ‚ΠΎΠ³ΠΎ, Ρ‡Ρ‚ΠΎ всС Ρ‚ΠΎΡ‡ΠΊΠΈ $P_h \: (h = 3,\:4,\:\ldots,\:n)$ Π»Π΅ΠΆΠ°Ρ‚ ΠΏΠΎ ΠΎΠ΄Π½Ρƒ сторону ΠΎΡ‚ прямой $P_1 P_2$.

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ΠŸΡƒΡΡ‚ΡŒ $F$ ΠΎΠ±ΠΎΠ·Π½Π°Ρ‡Π°Π΅Ρ‚ ΠΏΠ»ΠΎΡ‰Π°Π΄ΡŒ $K$. Если Ρ‡Π΅Ρ€Π΅Π· $F_1 \le \frac12 F$ ΠΈ $F_2 = F - F_1$ ΠΎΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΡ‚ΡŒ части, Π½Π° ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Π΅ прямая $P_1 P_2$ Π΄Π΅Π»ΠΈΡ‚ ΠΎΠ±Π»Π°ΡΡ‚ΡŒ $K$, Ρ‚ΠΎ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ

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$$W_n = \frac1{F^2} \int\limits_K \int\limits_K \left[ \left(1 - \frac{F_1}{F}\right)^{n-2} + \left(\frac{F_1}{F}\right)^{n-2}\right] dP_1 dP_2 \label{eq:3} \tag{3}$$

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ΠΊΠ°ΠΊ Π²Π΅Ρ€ΠΎΡΡ‚Π½ΠΎΡΡ‚ΡŒ Ρ‚ΠΎΠ³ΠΎ, Ρ‡Ρ‚ΠΎ всС Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i\:(i=3,\:4,\:\ldots,\:n)$ ΠΏΡ€ΠΈΠ½Π°Π΄Π»Π΅ΠΆΠ°Ρ‚ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ ΠΎΠ΄Π½ΠΎΠΉ ΠΈΠ· частСй $F$.

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Π’Π°ΠΊ ΠΊΠ°ΠΊ $F_1/F \le \frac12$, ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π» Π²Ρ‚ΠΎΡ€ΠΎΠ³ΠΎ слагаСмого ΠΎΠ³Ρ€Π°Π½ΠΈΡ‡Π΅Π½ константой:

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$$\frac1{F^2} \int\limits_K \int\limits_K \left(\frac{F_1}{F}\right)^{n-2} dP_1 dP_2 \le \frac1{2^{n - 2}} \label{eq:4} \tag{4}$$

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А ΡΠ»Π΅Π΄ΠΎΠ²Π°Ρ‚Π΅Π»ΡŒΠ½ΠΎ,

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$$E_n \sim C_n^2 \cdot \frac1{F^2} \int\limits_K \int\limits_K \left(1 - \frac{F_1}{F}\right)^{n-2} dP_1 dP_2 \label{eq:5} \tag{5}$$

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$$\text{Π³Π΄Π΅} \: A \sim B \Leftrightarrow \lim_{n\to+\infty} \frac{A_n}{B_n} = 1 \notag$$

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ΠžΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Ρ‡Π΅Ρ€Π΅Π· $f_i$ ΠΏΠ»ΠΎΡ‰Π°Π΄ΡŒ Ρ‚Ρ€Π΅ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $A_{i-1}A_iA_{i+1}$ $(A_{r+j} = A_j)$, ΠΈ ΠΏΡƒΡΡ‚ΡŒ $f = \min\limits_{1 \le i \le r} f_i$

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ΠžΡ‡Π΅Π²ΠΈΠ΄Π½ΠΎ, Ρ‡Ρ‚ΠΎ ΠΏΡ€ΠΈ ΠΈΠ·ΡƒΡ‡Π΅Π½ΠΈΠΈ асимптотичСского повСдСния $E_n$ ΠΌΠΎΠΆΠ½ΠΎ ΠΎΠ³Ρ€Π°Π½ΠΈΡ‡ΠΈΡ‚ΡŒΡΡ ΠΏΠ°Ρ€Π°ΠΌΠΈ Ρ‚ΠΎΡ‡Π΅ΠΊ $(P_1,\:P_2)$, для ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… прямая $P_1 P_2$ пСрСсСкаСтся Π»ΠΈΠ±ΠΎ с двумя сосСдними сторонами $K$, Π»ΠΈΠ±ΠΎ с двумя сторонами $K$, Ρ€Π°Π·Π΄Π΅Π»Ρ‘Π½Π½Ρ‹ΠΌΠΈ Ρ‚Ρ€Π΅Ρ‚ΡŒΠ΅ΠΉ стороной. Для всСх ΠΎΡΡ‚Π°Π»ΡŒΠ½Ρ‹Ρ… прямых выполняСтся ΠΎΠ³Ρ€Π°Π½ΠΈΡ‡Π΅Π½ΠΈΠ΅

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$$1 - \frac{F_1}{F} \leqq 1 - \frac{f}{F} \notag$$

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ΠŸΠΎΡΡ‚ΠΎΠΌΡƒ ΠΌΡ‹ ΠΌΠΎΠΆΠ΅ΠΌ ΠΏΡ€Π΅Π½Π΅Π±Ρ€Π΅Ρ‡ΡŒ ΡΠΎΠΎΡ‚Π²Π΅Ρ‚ΡΡ‚Π²ΡƒΡŽΡ‰Π΅ΠΉ Ρ‡Π°ΡΡ‚ΡŒΡŽ ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π»Π° ΠΈ Π·Π°ΠΏΠΈΡΠ°Ρ‚ΡŒ

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$$E_n \sim C_n^2 \cdot \frac1{F^2} \left( \sum\limits_{i=1}^r (I_i + J_i) \right) \label{eq:6} \tag{6}$$

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$$I_i = \int\limits_{C_i} \left(1 - \frac{F_1}{F}\right)^{n-2} dP_1 dP_2 \label{eq:7} \tag{7}$$

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$$J_i = \int\limits_{D_i} \left(1 - \frac{F_1}{F}\right)^{n-2} dP_1 dP_2 \label{eq:8} \tag{8}$$

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Π—Π΄Π΅ΡΡŒ $C_i$ ΠΎΠ·Π½Π°Ρ‡Π°Π΅Ρ‚ мноТСство ΠΏΠ°Ρ€ Ρ‚ΠΎΡ‡Π΅ΠΊ $(P_1,\:P_2)$, для ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… прямая $P_1P_2$ пСрСсСкаСт смСТныС стороны $A_{i-1}A_i$ ΠΈ $A_iA_{i+1}$ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$, a $D_i$ - мноТСство ΠΏΠ°Ρ€ $(P_1,\:P_2)$, для ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… прямая $P_1P_2$ пСрСсСкаСт стороны $A_{i-1}A_i$ ΠΈ $A_{i+1}A_{i+2}$ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$, Ρ€Π°Π·Π΄Π΅Π»Ρ‘Π½Π½Ρ‹Π΅ Π΄Ρ€ΡƒΠ³ΠΎΠΉ стороной.

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Π‘Π½Π°Ρ‡Π°Π»Π° рассмотрим ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π» $I_i$. ΠžΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Ρ‡Π΅Ρ€Π΅Π· $G_{ab}$ мноТСство ΠΏΠ°Ρ€ $(P_1,\:P_2)$, для ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… Π²Ρ‹ΠΏΠΎΠ»Π½Π΅Π½Ρ‹ нСравСнства $\left|A_iQ_1\right| < a$ ΠΈ $\left|A_iQ_2\right| < b$, Π³Π΄Π΅ $Q_1$ - Ρ‚ΠΎΡ‡ΠΊΠ° пСрСсСчСния прямой $P_1P_2$ со стороной $A_{i-1}A_i$, Π° $Q_2$ - Ρ‚ΠΎΡ‡ΠΊΠ° пСрСсСчСния $P_1P_2$ со стороной $A_iA_{i+1}$. Π—Π΄Π΅ΡΡŒ ΠΈ Π΄Π°Π»Π΅Π΅ $\left|AB\right|$ β€” Π΄Π»ΠΈΠ½Π° ΠΎΡ‚Ρ€Π΅Π·ΠΊΠ° $AB$.

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По элСмСнтарным расчётам получаСтся

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$$\int\limits_{G_{ab}} dP_1dP_2 = \frac{a^2b^2\sin^2\vartheta_i}{12} \label{eq:9} \tag{9}$$

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ΠŸΠ»ΠΎΡ‰Π°Π΄ΡŒ Ρ‚Ρ€Π΅ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $Q_1A_iQ_2$, отсСкаСмого прямой $P_1P_2$, Ρ€Π°Π²Π½Π° $\frac{1}{2} a b \sin \vartheta_i$. Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, взяв $a_{i-1} = \left|A_{i-1}A_i\right|$, $a_i = \left|A_iA_{i+1}\right|$, ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ

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$$I_i = \int\limits_0^{a_{i-1}} \int\limits_0^{a_i} \left(1 - \frac{ab \sin \vartheta_i}{2F}\right)^{n-2} \sin^2 \vartheta_i \frac{ab}{3} \,da\,db \label{eq:10} \tag{10}$$

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Π’Ρ‹ΠΏΠΎΠ»Π½ΠΈΠΌ Π·Π°ΠΌΠ΅Π½Ρƒ

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$$X = a \sqrt\frac{\sin \vartheta_i}{2F}, \; Y = b \sqrt\frac{\sin \vartheta_i}{2F} \label{eq:11} \tag{11}$$

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ΠΈ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ

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$$I_i = \frac{4F^2}{3} \int\limits_0^{X_i} \int\limits_0^{Y_i} (1-XY)^{n-2} XY \: dX dY \label{eq:12} \tag{12}$$

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Π³Π΄Π΅

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$$X_i = a_{i-1} \sqrt\frac{\sin \vartheta_i}{2F}, \; Y_i = a_i \sqrt\frac{\sin \vartheta_i}{2F} \notag$$

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ΠΈ Ρ‚Π°ΠΊΠΆΠ΅ Π·Π°ΠΌΠ΅Ρ‚ΠΈΠΌ, Ρ‡Ρ‚ΠΎ

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$$\varrho_i = X_iY_i = \frac{a_{i-1}a_i\sin \vartheta_i}{2F} = \frac{f_i}{F} < 1 \label{eq:13} \tag{13}$$

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Π’Π΅ΠΏΠ΅Ρ€ΡŒ ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΡƒΠ΅ΠΌ ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π»:

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$$\int\limits_0^{X_i} \int\limits_0^{Y_i} (1-XY)^{n-2} XY \: dX dY = \int\limits_0^{X_i} \int\limits_0^{Y_i} (1-XY)^{n-2} \: dX dY - \int\limits_0^{X_i} \int\limits_0^{Y_i} (1-XY)^{n-1} \: dX dY \notag$$

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Π”Π°Π»Π΅Π΅ ΠΏΡ€ΠΈΠΌΠ΅Π½ΠΈΠΌ

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$$\int\limits_0^{X_i} \int\limits_0^{Y_i} (1-XY)^{n-1} \: dX dY = \int\limits_0^{X_i} \frac{1 - (1 - XY_i)^n}{nX} \: dX = \\ = \frac1n \left( 1 + \frac12 + \ldots + \frac1n - \sum\limits_{k=1}^n \frac{(1-\varrho_i)^k}{k} \right) \notag$$

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Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, ΠΌΡ‹ ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ Π·Π½Π°Ρ‡Π΅Π½ΠΈΠ΅ ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π»Π° $I_i$:

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$$I_i = \frac{4F^2}{3} \left[ \frac{\cfrac12 + \cfrac13 + \ldots + \cfrac1n}{n(n-1)} - \frac{\sum\limits_{k=1}^{n-1} \cfrac{(1 - \varrho_i)^k}{k}}{n(n-1)} + \frac{(1-\varrho_i)^n}{n^2} \right] \label{eq:14} \tag{14}$$

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Если ΠΏΠΎΠ»ΠΎΠΆΠΈΠΌ

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$$S_i = \sum\limits_{k=1}^\infty \frac{(1 - \varrho_i)^k}{k} = \log \frac1{\varrho_i} \label{eq:15} \tag{15}$$

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Ρ‚ΠΎ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ Π² ΠΊΠΎΠ½Π΅Ρ‡Π½ΠΎΠΌ ΠΈΡ‚ΠΎΠ³Π΅

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$$E_n = \frac23 (\gamma - 1 + \log n) r - \frac23 \sum\limits_{i=1}^r S_i + o(1) + \frac{C_n^2}{F^2} \sum\limits_{i=1}^r J_i \label{eq:16} \tag{16}$$

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Π³Π΄Π΅ $\gamma$ - постоянная Π­ΠΉΠ»Π΅Ρ€Π°.

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Π’Π΅ΠΏΠ΅Ρ€ΡŒ вычислим ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π» $J_i$. ΠŸΡ€ΠΎΠ΄Π»ΠΈΠΌ стороны $A_{i-1}A_i$ ΠΈ $A_{i+1}A_{i+2}$ Π΄ΠΎ Ρ‚ΠΎΡ‡ΠΊΠΈ ΠΈΡ… пСрСсСчСния $A_i^*$; случай, ΠΊΠΎΠ³Π΄Π° эти стороны ΠΏΠ°Ρ€Π°Π»Π»Π΅Π»ΡŒΠ½Ρ‹, ΠΌΠΎΠΆΠ½ΠΎ Ρ€Π°ΡΡΠΌΠ°Ρ‚Ρ€ΠΈΠ²Π°Ρ‚ΡŒ Π°Π½Π°Π»ΠΎΠ³ΠΈΡ‡Π½ΠΎ.

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ΠŸΡƒΡΡ‚ΡŒ снова $Q_1$ - Ρ‚ΠΎΡ‡ΠΊΠ° пСрСсСчСния прямой $P_1P_2$ со стороной $A_{i-1}A_i$, Π° $Q_2$ - Ρ‚ΠΎΡ‡ΠΊΠ° пСрСсСчСния $P_1P_2$ со стороной $A_{i+1}A_{i+2}$, ΠΈ ΠΏΡƒΡΡ‚ΡŒ

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$$a' = \left|A_iA_i^*\right|, \;\; b' = \left|A_{i+1}A_i^*\right|, \;\; a = \left|A_iQ_1\right|, \;\; b = \left|A_{i+1}Q_2\right| \notag$$

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$G_{ab}$ Π±Ρ‹Π»ΠΎ ΠΎΠΏΡ€Π΅Π΄Π΅Π»Π΅Π½ΠΎ Π²Ρ‹ΡˆΠ΅.

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ΠŸΠΎΠ»ΡƒΡ‡ΠΈΠΌ ΠΈΠ· $(\ref{eq:9})$:

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$$\int\limits_{G_{ab}} dP_1 dP_2 = \frac{\sin^2\beta_i}{12} (a^2 + 2aa')(b^2 + 2bb') \label{eq:17} \tag{17}$$

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Π³Π΄Π΅ $\beta_i$ - ΡƒΠ³ΠΎΠ» ΠΏΡ€ΠΈ $A_i^*$. Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ,

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$$J_i = \int\limits_{D_i} \left(1 - \frac{(ab+a'b+ab')\sin \beta_i}{2F} \right)^{n-2} \frac{\sin^2 \beta_i}{3} (a+a')(b+b')\,da\,db \label{eq:18} \tag{18}$$

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ΠΈ этот ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π» асимптотичСски эквивалСнтСн

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$$J_i \sim \int\limits_{D_i} \left(1 - \frac{(a'b+ab')\sin \beta_i}{2F} \right)^{n-2} \frac{\sin^2 \beta_i}{3} a'b'\,da\,db \label{eq:19} \tag{19}$$

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Π”Π°Π»Π΅Π΅ сдСлаСм Π·Π°ΠΌΠ΅Π½Ρƒ

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$$a = \frac{F}{b' \sin \beta_i} X, \;\; b = \frac{F}{a' \sin \beta_i} Y \notag$$

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ΠΈ Π½Π°ΠΉΠ΄Ρ‘ΠΌ Π·Π½Π°Ρ‡Π΅Π½ΠΈΠ΅ ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π»Π°

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$$J_i \sim \frac{F^2}{3} \int\limits_0^{a_{i-1}'} \int\limits_0^{a_{i+1}'} \left( 1 - \frac{X+Y}{2} \right)^{n-2} dX\,dY = \\ = \frac{4F^2}{3n(n-1)} \left[ 1 - \left( 1 - \frac{a_{i-1}'}{2} \right)^n - \left( 1 - \frac{a_{i+1}'}{2} \right)^n + \left( 1 - \frac{a_{i-1}' + a_{i+1}'}{2} \right)^n \right] \notag$$

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Π³Π΄Π΅

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$$a_{i-1}' = \frac{a_ib'\sin\beta_i}{F}, \;\; a_{i+1}' = \frac{a_{i+2}a'\sin\beta_i}{F} \notag$$

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Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ ΠΈΡ‚ΠΎΠ³ΠΎΠ²Ρ‹ΠΉ Ρ€Π΅Π·ΡƒΠ»ΡŒΡ‚Π°Ρ‚:

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$$E_n = \frac{2r}{3} (\log n + \gamma) - \frac23 \sum\limits_{i=1}^{r} S_i + o(1) \notag$$

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$\triangleleft$

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ΠŸΡƒΡΡ‚ΡŒ $K$ - Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΉ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ с $r$ сторонами. ΠœΠ°Ρ‚Π΅ΠΌΠ°Ρ‚ΠΈΡ‡Π΅ΡΠΊΠΎΠ΅ ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ $E_n$ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ $H_n$, построСнной Π½Π° $n$ случайных Ρ‚ΠΎΡ‡ΠΊΠ°Ρ… ΠΈΠ· $K$, Ρ€Π°Π²Π½ΠΎ

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$$E_n = \frac{2r}{3} (\log n + \gamma) + \frac23 \log \frac{\prod\limits_1^r f_i}{F^r} + o(1) \label{eq:20} \tag{20}$$

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Π—Π΄Π΅ΡΡŒ $F$ ΠΎΠ±ΠΎΠ·Π½Π°Ρ‡Π°Π΅Ρ‚ ΠΏΠ»ΠΎΡ‰Π°Π΄ΡŒ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$, Π° $f_i$ - ΠΏΠ»ΠΎΡ‰Π°Π΄ΡŒ Ρ‚Ρ€Π΅ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $A_{i-1}A_iA_{i+1}$.

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ΠŸΡƒΡΡ‚ΡŒ, Π½Π°ΠΏΡ€ΠΈΠΌΠ΅Ρ€, $K$ - Ρ‚Ρ€Π΅ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ, Ρ‚ΠΎΠ³Π΄Π° $f_1 = f_2 = f_3 = F$ ΠΈ

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$$E_n = 2(\log n + \gamma) + o(1) \label{eq:21} \tag{21}$$

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Π—Π°ΠΌΠ΅Ρ‚ΠΈΠΌ, Ρ‡Ρ‚ΠΎ Π² Ρ‚Π°ΠΊΠΎΠΌ случаС Ρ„ΠΎΡ€ΠΌΡƒΠ»Π° для $E_n$ Π½Π΅ зависит ΠΎΡ‚ Ρ„ΠΎΡ€ΠΌΡ‹ Ρ‚Ρ€Π΅ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ°, Ρ‚Π°ΠΊ ΠΊΠ°ΠΊ $E_n$, ΠΎΡ‡Π΅Π²ΠΈΠ΄Π½ΠΎ, Π΄ΠΎΠ»ΠΆΠ½ΠΎ Π±Ρ‹Ρ‚ΡŒ ΠΈΠ½Π²Π°Ρ€ΠΈΠ°Π½Ρ‚Π½ΠΎ ΠΊ Π°Ρ„Ρ„ΠΈΠ½Π½Ρ‹ΠΌ прСобразованиям. Π’ $\S2$ ΠΌΡ‹ ΠΏΠΎΠΊΠ°ΠΆΠ΅ΠΌ, Ρ‡Ρ‚ΠΎ константа

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$$Z(K) = \frac{\prod\limits_{i=1}^r f_i}{F^r} \label{eq:22} \tag{22}$$

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которая зависит ΠΎΡ‚ Ρ„ΠΎΡ€ΠΌΡ‹ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$ ΠΈ появляСтся Π² Π²Ρ‹Ρ€Π°ΠΆΠ΅Π½ΠΈΠΈ $(\ref{eq:20})$, становится максимальной для ΠΏΡ€Π°Π²ΠΈΠ»ΡŒΠ½ΠΎΠ³ΠΎ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° ΠΈ Π°Ρ„Ρ„ΠΈΠ½Π½Ρ‹Ρ… ΠΊ Π½Π΅ΠΌΡƒ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΎΠ².

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2. Экстремальная задача для выпуклых многоугольников

ΠŸΡ€Π΅ΠΆΠ΄Π΅ Ρ‡Π΅ΠΌ ΠΏΠ΅Ρ€Π΅ΠΉΡ‚ΠΈ ΠΊ дальнСйшим асимптотичСским ΠΎΡ†Π΅Π½ΠΊΠ°ΠΌ для $E_n$ ΠΏΡ€ΠΈ Π΄Ρ€ΡƒΠ³ΠΈΡ… прСдполоТСниях ΠΎ распрСдСлСнии Ρ‚ΠΎΡ‡Π΅ΠΊ $P_i$, Π΄ΠΎΠΊΠ°ΠΆΠ΅ΠΌ ΡΠ»Π΅Π΄ΡƒΡŽΡ‰Π΅Π΅ ΡƒΡ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅:

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Π£Ρ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 2. ΠŸΡƒΡΡ‚ΡŒ $K$ - Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΉ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ с $r$ сторонами. Ѐункция

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$$Z(K) = \frac1{F^r} \prod\limits_{i=1}^r f_i \label{eq:23} \tag{23}$$

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ΠΈΠ· ΠΏΡ€Π΅Π΄Ρ‹Π΄ΡƒΡ‰Π΅Π³ΠΎ ΠΏΠ°Ρ€Π°Π³Ρ€Π°Ρ„Π° максимальна Ρ‚ΠΎΠ³Π΄Π° ΠΈ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ Ρ‚ΠΎΠ³Π΄Π°, ΠΊΠΎΠ³Π΄Π° $K$ ΠΌΠΎΠΆΠ½ΠΎ ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΠΎΠ²Π°Ρ‚ΡŒ Π² ΠΏΡ€Π°Π²ΠΈΠ»ΡŒΠ½Ρ‹ΠΉ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ Π°Ρ„Ρ„ΠΈΠ½Π½Ρ‹ΠΌΠΈ прСобразованиями.

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Π”ΠΎΠΊΠ°Π·Π°Ρ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ:

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$\triangleright$

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ΠŸΡƒΡΡ‚ΡŒ $\alpha_i = \overrightarrow{A_{i-1}A_i}$, Π³Π΄Π΅ $A_{r+i} = A_i$. Π’ΠΎΠ³Π΄Π°

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$$f_i = \frac12 [\alpha_i, \: \alpha_{i+1}] \\ F = \frac12 \sum\limits_{i=1}^{r-2} [\alpha_1 + \alpha_2 + \ldots + \alpha_i, \: \alpha_{i+1}] \label{eq:24} \tag{24}$$

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Π³Π΄Π΅ $[\alpha, \: \beta]$ - ΠΌΠΎΠ΄ΡƒΠ»ΡŒ Π²Π΅ΠΊΡ‚ΠΎΡ€Π½ΠΎΠ³ΠΎ произвСдСния $\alpha$ ΠΈ $\beta$. Ѐункция $Z(K)$ Ρ‚Π΅ΠΏΠ΅Ρ€ΡŒ являСтся Ρ„ΡƒΠ½ΠΊΡ†ΠΈΠ΅ΠΉ ΠΎΡ‚ $r$ Π²Π΅ΠΊΡ‚ΠΎΡ€ΠΎΠ² $\alpha_i \: (i=1, \: \ldots, \: r)$, ΡƒΠ΄ΠΎΠ²Π»Π΅Ρ‚Π²ΠΎΡ€ΡΡŽΡ‰ΠΈΡ… ΡƒΡΠ»ΠΎΠ²ΠΈΡŽ замкнутости:

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$$\sum\limits_{i=1}^r \alpha_i = 0 \label{eq:25} \tag{25}$$

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Π’Π΅ΠΏΠ΅Ρ€ΡŒ допустим, Ρ‡Ρ‚ΠΎ Π²Π΅Ρ€ΡˆΠΈΠ½Ρ‹ $A_i$ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$ ΠΏΠΎΠ΄Π²Π΅Ρ€Π³Π°ΡŽΡ‚ΡΡ достаточно ΠΌΠ°Π»Ρ‹ΠΌ смСщСниям, ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Π΅ ΠΎΠΏΠΈΡΡ‹Π²Π°ΡŽΡ‚ΡΡ Π½Π΅ΠΏΡ€Π΅Ρ€Ρ‹Π²Π½ΠΎ Π΄ΠΈΡ„Ρ„Π΅Ρ€Π΅Π½Ρ†ΠΈΡ€ΡƒΠ΅ΠΌΠΎΠΉ Ρ„ΡƒΠ½ΠΊΡ†ΠΈΠ΅ΠΉ ΠΎΡ‚ ΠΏΠ°Ρ€Π°ΠΌΠ΅Ρ‚Ρ€Π° $t$. Π’Π°ΠΊ ΠΊΠ°ΠΊ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ $K$, ΠΊΠΎΡ‚ΠΎΡ€Ρ‹ΠΉ Π΄ΠΎΠ»ΠΆΠ΅Π½ ΡΠΎΠΎΡ‚Π²Π΅Ρ‚ΡΡ‚Π²ΠΎΠ²Π°Ρ‚ΡŒ Π·Π½Π°Ρ‡Π΅Π½ΠΈΡŽ ΠΏΠ°Ρ€Π°ΠΌΠ΅Ρ‚Ρ€Π° $t = 0$, являСтся Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΌ r-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΎΠΌ, Ρ‚ΠΎ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΈ, ΡΠ²Π»ΡΡŽΡ‰ΠΈΠ΅ΡΡ Ρ€Π΅Π·ΡƒΠ»ΡŒΡ‚Π°Ρ‚ΠΎΠΌ достаточно ΠΌΠ°Π»ΠΎΠ³ΠΎ измСнСния $t$, Ρ‚Π°ΠΊΠΆΠ΅ Π±ΡƒΠ΄ΡƒΡ‚ Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΌΠΈ r-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ°ΠΌΠΈ. Ѐункция $Z$, рассматриваСмая ΠΊΠ°ΠΊ функция ΠΎΡ‚ $t$, Π΄ΠΎΠ»ΠΆΠ½Π° Π΄ΠΎΡΡ‚ΠΈΠ³Π°Ρ‚ΡŒ максимума ΠΏΡ€ΠΈ $t = 0$, Ссли $K$ - ΠΌΠ°ΠΊΡΠΈΠΌΠ°Π»ΡŒΠ½Ρ‹ΠΉ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ.

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НСобходимоС условиС Ρ‚ΠΎΠ³ΠΎ, Ρ‡Ρ‚ΠΎ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ $K$ являСтся ΠΌΠ°ΠΊΡΠΈΠΌΠ°Π»ΡŒΠ½Ρ‹ΠΌ, ΠΌΡ‹ ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ ΠΈΠ· нСслоТного расчёта:

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$$\sum\limits_{i=1}^r \frac{F}{f_i} ([\dot \alpha_i, \: \alpha_{i+1}] + [\alpha_i, \: \dot \alpha_{i+1}]) - \\ - r \sum\limits_{i=1}^{r-2}([\dot \alpha_1 + \ldots + \dot \alpha_i, \: \alpha_{i+1}] + [\alpha_1 + \ldots + \alpha_i, \: \dot \alpha_{i+1}]) = 0 \label{eq:26} \tag{26}$$

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Π“Π΄Π΅ $\dot \alpha_i = \left.\frac{d}{dt} \alpha_i \right|_{t=0}$ - ΠΏΡ€ΠΎΠΈΠ·Π²ΠΎΠ»ΡŒΠ½Ρ‹Π΅ Π²Π΅ΠΊΡ‚ΠΎΡ€Ρ‹, ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Π΅ Ρ‚ΠΎΠΆΠ΅ Π΄ΠΎΠ»ΠΆΠ½Ρ‹ ΡƒΠ΄ΠΎΠ²Π»Π΅Ρ‚Π²ΠΎΡ€ΡΡ‚ΡŒ ΡƒΡΠ»ΠΎΠ²ΠΈΡŽ замкнутости, ΡΠ»Π΅Π΄ΡƒΡŽΡ‰Π΅ΠΌΡƒ ΠΈΠ· $(25)$:

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$$\sum\limits_{i=1}^r \dot\alpha_i = 0 \label{eq:27} \tag{27}$$

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Если ΠΌΡ‹ ΠΏΠΎΠ»ΠΎΠΆΠΈΠΌ ΠΏΠΎΡΠ»Π΅Π΄ΠΎΠ²Π°Ρ‚Π΅Π»ΡŒΠ½ΠΎ

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$$\dot \alpha_j = \beta, \;\; \dot \alpha_{j+1} = -\beta, \;\; \dot \alpha_i = 0 \: \text{для} \: i \neq j, \;\; i \neq j+1 \label{eq:28} \tag{28}$$

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Π³Π΄Π΅ $\beta$ - ΠΈΠ·Π½Π°Ρ‡Π°Π»ΡŒΠ½ΠΎ ΠΏΡ€ΠΎΠΈΠ·Π²ΠΎΠ»ΡŒΠ½Ρ‹ΠΉ Π²Π΅ΠΊΡ‚ΠΎΡ€, Ρ‚ΠΎ условиС замкнутости $(\ref{eq:27})$, ΠΎΡ‡Π΅Π²ΠΈΠ΄Π½ΠΎ, выполняСтся, ΠΈ ΠΌΡ‹ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ для $j = 1, \: \ldots, \: r$:

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$$\frac{F}{r} \left\{ \frac{[\alpha_{j-1}, \: \beta]}{f_{j-1}} - \frac{[\beta, \: \alpha_{j+2}]}{f_{j+1}} + \frac1{f_j} [\beta, \: \alpha_j + \alpha_{j+1}] \right\} = [\beta, \: \alpha_j + \alpha_{j+1}] \label{eq:29} \tag{29}$$

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Если ΠΌΡ‹ Ρ‚Π΅ΠΏΠ΅Ρ€ΡŒ Π½Π°ΠΌΠ΅Ρ€Π΅Π½Π½ΠΎ Π²Ρ‹Π±Π΅Ρ€Π΅ΠΌ $\beta = \alpha_j$, Ρ„ΠΎΡ€ΠΌΡƒΠ»Π° ΠΏΡ€ΠΈΠΌΠ΅Ρ‚ Π²ΠΈΠ΄

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$$\frac{F}{r} \left\{ 4 - \frac{[\alpha_j, \: \alpha_{j+2}]}{f_{j+1}} \right\} = 2 f_j \label{eq:30} \tag{30}$$

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а для $\beta = -\alpha_{j+1}$

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$$\frac{F}{r} \left\{ 4 - \frac{[\alpha_{j-1}, \: \alpha_{j+1}]}{f_{j-1}} \right\} = 2 f_j \label{eq:31} \tag{31}$$

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Π‘Ρ€Π°Π²Π½ΠΈΠΌ эти Π΄Π²Π΅ Ρ„ΠΎΡ€ΠΌΡƒΠ»Ρ‹ ΠΈ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ

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$$[\alpha_j, \: \alpha_{j+2}] = \frac{f_{j+1}}{f_{j-1}} [\alpha_{j-1}, \: \alpha_{j-1}] \label{eq:32} \tag{32}$$

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Π’Π΅ΠΏΠ΅Ρ€ΡŒ ΠΌΡ‹ запишСм ΡƒΡ€Π°Π²Π½Π΅Π½ΠΈΠ΅ $(\ref{eq:31})$ Π² Ρ„ΠΎΡ€ΠΌΠ΅

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$$2 \left( 2 - \frac{r f_j}{F} \right) = \frac{[\alpha_{j-1}, \: \alpha_{j+1}]}{f_{j-1}} \notag$$

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ΠΈ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ ΠΏΡƒΡ‚Ρ‘ΠΌ ΠΏΠΎΠ²Ρ‚ΠΎΡ€Π½ΠΎΠ³ΠΎ примСнСния $(\ref{eq:32})$:

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$$2 \left( 2 - \frac{r f_j}{F} \right) = \frac{f_j}{f_1 \cdot f_2} [\alpha_1, \: \alpha_3] \notag$$

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ΠΈΠ»ΠΈ

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$$f_j = \frac2{\cfrac{r}{F} + \cfrac{[\alpha_1, \: \alpha_3]}{2 f_1 \cdot f_2}} \label{eq:33} \tag{33}$$

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Для $r = 3$ наша ΡΠΊΡΡ‚Ρ€Π΅ΠΌΠ°Π»ΡŒΠ½Π°Ρ Π·Π°Π΄Π°Ρ‡Π° Ρ‚Ρ€ΠΈΠ²ΠΈΠ°Π»ΡŒΠ½Π°. Π’ случаС $r \ge 4$ получаСтся, Ρ‡Ρ‚ΠΎ $f_3 = f_4 = \ldots = f_r$. ΠŸΡ€ΠΈ цикличСской пСрСстановкС Π½ΠΎΠΌΠ΅Ρ€ΠΎΠ² Π²Π΅Ρ€ΡˆΠΈΠ½ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ

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$$f_1 = f_2 = \ldots = f_r \label{eq:34} \tag{34}$$

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Π° послС $(\ref{eq:32})$ Ρ‚Π°ΠΊΠΆΠ΅ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ

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$$[\alpha_i, \: \alpha_{i+2}] = [\alpha_1, \: \alpha_3] = \beta \;\; (i = 1, \: 2, \: \ldots, \: r) \label{eq:35} \tag{35}$$

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Π³Π΄Π΅ $\beta$ - нСкоторая константа.

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ΠŸΡ€ΠΈΠΌΠ΅Ρ‡Π°Π½ΠΈΠ΅. Π’ случаС $\beta = [\alpha_i, \: \alpha_{i+2}] = 0$ получаСтся, Ρ‡Ρ‚ΠΎ $f_j = 2F/r$. Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, Π²Π΅ΠΊΡ‚ΠΎΡ€ $\alpha_{i+2}$ всСгда ΠΏΠ°Ρ€Π°Π»Π»Π΅Π»Π΅Π½ $\alpha_i$. Π­Ρ‚ΠΎ Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΠΎ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ Π² Ρ‚ΠΎΠΌ случаС, Ссли $r = 4$, Π° $K$ - ΠΏΠ°Ρ€Π°Π»Π»Π΅Π»ΠΎΠ³Ρ€Π°ΠΌΠΌ.

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Π’Π΅ΠΏΠ΅Ρ€ΡŒ вСрнёмся ΠΊ ΠΎΠ±Ρ‰Π΅ΠΌΡƒ ΡΠ»ΡƒΡ‡Π°ΡŽ. Π‘ ΠΏΠΎΠΌΠΎΡ‰ΡŒΡŽ Π°Ρ„Ρ„ΠΈΠ½Π½Ρ‹Ρ… ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΠΎΠ²Π°Π½ΠΈΠΉ ΠΌΡ‹ всСгда ΠΌΠΎΠΆΠ΅ΠΌ Π΄ΠΎΠ±ΠΈΡ‚ΡŒΡΡ Ρ‚ΠΎΠ³ΠΎ, Ρ‡Ρ‚ΠΎ $f_j = 1/2$, Ρ‚ΠΎ Π΅ΡΡ‚ΡŒ

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$$[\alpha_i, \: \alpha_{i+1}] = 1 \label{eq:36} \tag{36}$$

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Из $(\ref{eq:33})$ сразу слСдуСт

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$$F = \frac{r}{2(2-\beta)} \label{eq:37} \tag{37}$$

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ΠΈ, ΡΠ»Π΅Π΄ΠΎΠ²Π°Ρ‚Π΅Π»ΡŒΠ½ΠΎ, $\beta \lt 2$.

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Π’Π°ΠΊ ΠΊΠ°ΠΊ Π²Π΅ΠΊΡ‚ΠΎΡ€Ρ‹ $\alpha_{i-2}$ ΠΈ $\alpha_{i-1}$ Π»ΠΈΠ½Π΅ΠΉΠ½ΠΎ нСзависимы, ΠΈΠ· $\alpha_i = x \alpha_{i-2} + y \alpha_{i-1}$, учитывая $(\ref{eq:35})$ ΠΈ $(\ref{eq:36})$, слСдуСт:

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$$\alpha_i = -\alpha_{i-2} + b \alpha_{i-1} \label{eq:38} \tag{38}$$

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Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, ΠΌΡ‹ ΠΏΠΎΠΊΠ°Π·Π°Π»ΠΈ, Ρ‡Ρ‚ΠΎ Π²Π΅ΠΊΡ‚ΠΎΡ€Ρ‹ $\alpha_i$ сторон максимального Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° Π΄ΠΎΠ»ΠΆΠ½Ρ‹ ΡƒΠ΄ΠΎΠ²Π»Π΅Ρ‚Π²ΠΎΡ€ΡΡ‚ΡŒ рСкурсивной Ρ„ΠΎΡ€ΠΌΡƒΠ»Π΅ $(\ref{eq:38})$. Π—Π°Π²Π΅Ρ€ΡˆΠΈΠΌ Π΄ΠΎΠΊΠ°Π·Π°Ρ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ, ΠΏΠΎΠΊΠ°Π·Π°Π², Ρ‡Ρ‚ΠΎ Π² этом случаС ΠΌΠΎΠΆΠ½ΠΎ ΠΎΠΏΡ€Π΅Π΄Π΅Π»ΠΈΡ‚ΡŒ Π΅Π²ΠΊΠ»ΠΈΠ΄ΠΎΠ²Ρƒ ΠΌΠ΅Ρ‚Ρ€ΠΈΠΊΡƒ Π² Π°Ρ„Ρ„ΠΈΠ½Π½ΠΎΠΉ плоскости, ΠΎΡ‚Π½ΠΎΡΠΈΡ‚Π΅Π»ΡŒΠ½ΠΎ ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ $K$ являСтся ΠΏΡ€Π°Π²ΠΈΠ»ΡŒΠ½Ρ‹ΠΌ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΎΠΌ.

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Π’ΠΎ-ΠΏΠ΅Ρ€Π²Ρ‹Ρ…, Π·Π°ΠΌΠ΅Ρ‚ΠΈΠΌ, Ρ‡Ρ‚ΠΎ всСгда Π΄ΠΎΠ»ΠΆΠ½ΠΎ Π±Ρ‹Ρ‚ΡŒ

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$$-1 \leq b \lt 2 \label{eq:39} \tag{39}$$

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Π”Π΅ΠΉΡΡ‚Π²ΠΈΡ‚Π΅Π»ΡŒΠ½ΠΎ, Ссли $b \lt -1$, Ρ‚ΠΎ сторона $\alpha_3$ Π±ΡƒΠ΄Π΅Ρ‚ ΠΏΠ΅Ρ€Π΅ΡΠ΅ΠΊΠ°Ρ‚ΡŒ сторону $\alpha_1$, Ρ‡Ρ‚ΠΎ ΠΏΡ€ΠΎΡ‚ΠΈΠ²ΠΎΡ€Π΅Ρ‡ΠΈΡ‚ выпуклости $K$. ВСрхняя ΠΎΡ†Π΅Π½ΠΊΠ° $b \lt 2$ ΡƒΠΆΠ΅ Π΄ΠΎΠΊΠ°Π·Π°Π½Π° Π² $(\ref{eq:37})$.

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Π•Π²ΠΊΠ»ΠΈΠ΄ΠΎΠ²Π° ΠΌΠ΅Ρ‚Ρ€ΠΈΠΊΠ° ΠΎΠ΄Π½ΠΎΠ·Π½Π°Ρ‡Π½ΠΎ устанавливаСтся, Ссли ΠΌΡ‹ ΠΎΠΏΡ€Π΅Π΄Π΅Π»ΠΈΠΌ скалярноС ΠΏΡ€ΠΎΠΈΠ·Π²Π΅Π΄Π΅Π½ΠΈΠ΅ Π²Π΅ΠΊΡ‚ΠΎΡ€ΠΎΠ² $\alpha_1$ ΠΈ $\alpha_2$, ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Π΅ ΠΌΡ‹ рассматриваСм ΠΊΠ°ΠΊ базис:

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$$\alpha_1^2 = \alpha_2^2 = \frac{2}{\sqrt{4-b^2}}, \;\; \alpha_1\alpha_2 = \frac{b}{\sqrt{4-b^2}} \label{eq:40} \tag{40}$$

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Π—Π°ΠΌΠ΅Ρ‚ΠΈΠΌ, Ρ‡Ρ‚ΠΎ

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$$[\alpha_1, \alpha_2]^2 = \begin{vmatrix} \alpha_1^2 & \alpha_1\alpha_2 \\ \alpha_1\alpha_2 & \alpha_2^2 \end{vmatrix} = 1 \notag$$

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Π‘Π»Π΅Π΄ΠΎΠ²Π°Ρ‚Π΅Π»ΡŒΠ½ΠΎ, эта ΠΌΠ΅Ρ‚Ρ€ΠΈΠΊΠ° ΠΏΠΎΠ»ΠΎΠΆΠΈΡ‚Π΅Π»ΡŒΠ½ΠΎ ΠΎΠΏΡ€Π΅Π΄Π΅Π»Π΅Π½Π°, Ρ‚.Π΅. являСтся Π΅Π²ΠΊΠ»ΠΈΠ΄ΠΎΠ²ΠΎΠΉ ΠΌΠ΅Ρ‚Ρ€ΠΈΠΊΠΎΠΉ.

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Π’Π΅ΠΏΠ΅Ρ€ΡŒ, ΠΈΡΠΏΠΎΠ»ΡŒΠ·ΡƒΡ Ρ€Π΅ΠΊΡƒΡ€ΡΠΈΠ²Π½ΡƒΡŽ Ρ„ΠΎΡ€ΠΌΡƒΠ»Ρƒ ΠΈΠ· $(\ref{eq:38})$, Π»Π΅Π³ΠΊΠΎ Π΄ΠΎΠΊΠ°Π·Π°Ρ‚ΡŒ ΠΏΠΎ ΠΈΠ½Π΄ΡƒΠΊΡ†ΠΈΠΈ, Ρ‡Ρ‚ΠΎ для всСх $j$ Π΄ΠΎΠ»ΠΆΠ½ΠΎ Π²Ρ‹ΠΏΠΎΠ»Π½ΡΡ‚ΡŒΡΡ

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$$\alpha_j^2 = \frac{2}{\sqrt{4-b^2}}, \;\; \alpha_j\alpha_{j+1} = \frac{b}{\sqrt{4-b^2}}, \;\; [\alpha_j, \:\alpha_{j+1}] = 1 \notag$$

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ΠžΡ‚ΡΡŽΠ΄Π° слСдуСт, Ρ‡Ρ‚ΠΎ всС стороны нашСго $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$ ΠΈΠΌΠ΅ΡŽΡ‚ ΠΎΠ΄ΠΈΠ½Π°ΠΊΠΎΠ²ΡƒΡŽ Π΄Π»ΠΈΠ½Ρƒ ΠΈ Ρ‡Ρ‚ΠΎ Π»ΡŽΠ±Ρ‹Π΅ Π΄Π²Π΅ смСТныС стороны ΠΏΠ΅Ρ€Π΅ΡΠ΅ΠΊΠ°ΡŽΡ‚ΡΡ ΠΏΠΎΠ΄ ΠΎΠ΄Π½ΠΈΠΌ ΡƒΠ³Π»ΠΎΠΌ. Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ Π΄ΠΎΠ»ΠΆΠ΅Π½ Π±Ρ‹Ρ‚ΡŒ ΠΏΡ€Π°Π²ΠΈΠ»ΡŒΠ½Ρ‹ΠΌ ΠΎΡ‚Π½ΠΎΡΠΈΡ‚Π΅Π»ΡŒΠ½ΠΎ этой ΠΌΠ΅Ρ‚Ρ€ΠΈΠΊΠΈ.

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Π­Ρ‚ΠΎ Π΄ΠΎΠΊΠ°Π·Ρ‹Π²Π°Π΅Ρ‚ Π΅Π΄ΠΈΠ½ΡΡ‚Π²Π΅Π½Π½ΠΎΡΡ‚ΡŒ максимального ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ°. Π•Π³ΠΎ сущСствованиС слСдуСт ΠΈΠ· ограничСнности $Z(K)$ ΠΈ нСпрСрывности $F$ ΠΈ $Z$. Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, ΡƒΡ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 2 ΠΏΠΎΠ»Π½ΠΎΡΡ‚ΡŒΡŽ Π΄ΠΎΠΊΠ°Π·Π°Π½ΠΎ.

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$\triangleleft$

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3. Случайно выбранные точки в выпуклой области с гладкой границей

Π£Ρ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 3. Если Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i$ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны Π² Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ области с Π³Π»Π°Π΄ΠΊΠΎΠΉ Π³Ρ€Π°Π½ΠΈΡ†Π΅ΠΉ Π½Π° плоскости, матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ $E_n = O(\sqrt[3]n)$.

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Π”ΠΎΠΊΠ°Π·Π°Ρ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ:

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$\triangleright$

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ΠŸΡƒΡΡ‚ΡŒ Ρ‚Π΅ΠΏΠ΅Ρ€ΡŒ $K$ - выпуклая ΠΎΠ±Π»Π°ΡΡ‚ΡŒ, Π³Ρ€Π°Π½ΠΈΡ†Π° ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ ΠΈΠΌΠ΅Π΅Ρ‚ Π½Π΅ΠΏΡ€Π΅Ρ€Ρ‹Π²Π½ΡƒΡŽ ΠΊΡ€ΠΈΠ²ΠΈΠ·Π½Ρƒ ΠΊΠ°ΠΊ Ρ„ΡƒΠ½ΠΊΡ†ΠΈΡŽ ΠΎΡ‚ Π΄Π»ΠΈΠ½Ρ‹ Π΄ΡƒΠ³ΠΈ. Π’ обозначСниях ΠΏΡ€Π΅Π΄Ρ‹Π΄ΡƒΡ‰ΠΈΡ… ΠΏΠ°Ρ€Π°Π³Ρ€Π°Ρ„ΠΎΠ² ΠΌΡ‹ снова ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ

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$$E_n = C_n^2 \cdot W_n \notag$$

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Богласно ΡƒΡ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΡŽ ΠΎ ΠΌΠ΅Ρ€Π΅ мноТСства ΠΏΠ°Ρ€ Ρ‚ΠΎΡ‡Π΅ΠΊ, извСстному ΠΈΠ· ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π»ΡŒΠ½ΠΎΠΉ Π³Π΅ΠΎΠΌΠ΅Ρ‚Ρ€ΠΈΠΈ (см. Blascke [2], S 17, (86)):

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$$dP_1 dP_2 = \left|t_1 - t_2\right| \, dt_1 \, dt_2 \, d\varphi \, dp \label{eq:41} \tag{41}$$

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Π³Π΄Π΅ $p$ β€” Π΄Π»ΠΈΠ½Π° пСрпСндикуляра ΠΊ прямой $P_1 P_2$ ΠΈΠ· Π½Π°Ρ‡Π°Π»Π° ΠΊΠΎΠΎΡ€Π΄ΠΈΠ½Π°Ρ‚, $\varphi$ β€” ΡƒΠ³ΠΎΠ» Π½Π°ΠΊΠ»ΠΎΠ½Π° этого пСрпСндикуляра, $t_1, \: t_2$ β€” расстояния ΠΏΠΎ этой прямой ΠΎΡ‚ $P_1$ ΠΈ $P_2$ Π΄ΠΎ Ρ‚ΠΎΡ‡ΠΊΠΈ пСрСсСчСния $P_1 P_2$ с пСрпСндикуляром. ΠžΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Ρ‡Π΅Ρ€Π΅Π· $f$ ΠΏΠ»ΠΎΡ‰Π°Π΄ΡŒ участка, отсСкаСмого ΠΎΡ‚ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$ прямой $P_1 P_2$. Π’ΠΎΠ³Π΄Π° ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ

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$$E_n = \cfrac{C_n^2}{F^2} \int\limits_0^{2\pi} \left\{ \int\limits_0^{p(\varphi)} \left(1-\frac{f}{F} \right)^{n-2} ( \iint \left|t_1 - t_2\right| \, dt_1 \, dt_2) \, dp \right\} d\varphi \label{eq:42} \tag{42}$$

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Π§Π΅Ρ€Π΅Π· $l(p, \phi)$ ΠΎΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Π΄Π»ΠΈΠ½Ρƒ Ρ…ΠΎΡ€Π΄Ρ‹ области $K$, Π»Π΅ΠΆΠ°Ρ‰Π΅ΠΉ Π½Π° прямой с ΠΊΠΎΠΎΡ€Π΄ΠΈΠ½Π°Ρ‚Π°ΠΌΠΈ $p, \varphi$. Π’ΠΎΠ³Π΄Π°

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$$\iint \left|t_1 - t_2\right| \, dt_1 \, dt_2 = \frac{l^3(p, \varphi)}{3} \label{eq:43} \tag{43}$$

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Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ,

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$$E_n = \cfrac{C_n^2}{3F^2} \int\limits_0^{2\pi} \left\{ \int\limits_0^{p(\varphi)} \left(1-\frac{f}{F} \right)^{n-2} l^3(p, \varphi) \, dp \right\} \, d\varphi \label{eq:44} \tag{44}$$

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ΠŸΡ€ΠΈ расчСтС асимптотичСского значСния $E_n$ ΠΌΡ‹ ΠΌΠΎΠΆΠ΅ΠΌ, ΠΎΡ‡Π΅Π²ΠΈΠ΄Π½ΠΎ, ΠΎΠ³Ρ€Π°Π½ΠΈΡ‡ΠΈΡ‚ΡŒΡΡ ΠΏΠ°Ρ€Π°ΠΌΠΈ Π·Π½Π°Ρ‡Π΅Π½ΠΈΠΉ $(p, \: \varphi)$, для ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… выполняСтся $f/F < \varepsilon$; Π΄Ρ€ΡƒΠ³ΠΈΠ΅ ΠΏΠ°Ρ€Ρ‹ Π·Π½Π°Ρ‡Π΅Π½ΠΈΠΉ Π΄Π°ΡŽΡ‚ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ остаточный Ρ‡Π»Π΅Π½ порядка $n^2(1-\varepsilon)^n = o(1)$. Рассмотрим Ρ‚Π΅ΠΏΠ΅Ρ€ΡŒ фиксированноС Π·Π½Π°Ρ‡Π΅Π½ΠΈΠ΅ $\varphi$. ΠŸΡƒΡΡ‚ΡŒ $P(\varphi)$ - Ρ‚ΠΎΡ‡ΠΊΠ°, Π² ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ ΠΊΠ°ΡΠ°Ρ‚Π΅Π»ΡŒΠ½Π°Ρ ΠΊ $K$ ΠΈΠΌΠ΅Π΅Ρ‚ ΡƒΠ³ΠΎΠ» Π½Π°ΠΊΠ»ΠΎΠ½Π° $\varphi$. Как становится ясно ΠΈΠ· Π½ΠΈΠΆΠ΅ΡΠ»Π΅Π΄ΡƒΡŽΡ‰ΠΈΡ… ΠΎΡ†Π΅Π½ΠΎΠΊ, ΠΌΠΎΠΆΠ½ΠΎ Π·Π°ΠΌΠ΅Π½ΠΈΡ‚ΡŒ $f$ ΠΈ $l$ Π½Π° ΡΠΎΠΎΡ‚Π²Π΅Ρ‚ΡΡ‚Π²ΡƒΡŽΡ‰ΠΈΠ΅ Π²Π΅Π»ΠΈΡ‡ΠΈΠ½Ρ‹ $\bar f$ ΠΈ $\bar l$ окруТности ΠΊΡ€ΠΈΠ²ΠΈΠ·Π½Ρ‹ Π² Ρ‚ΠΎΡ‡ΠΊΠ΅ $P(\varphi)$.

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ΠŸΡƒΡΡ‚ΡŒ $\alpha$ - Ρ†Π΅Π½Ρ‚Ρ€Π°Π»ΡŒΠ½Ρ‹ΠΉ ΡƒΠ³ΠΎΠ», ΡΠΎΠΎΡ‚Π²Π΅Ρ‚ΡΡ‚Π²ΡƒΡŽΡ‰ΠΈΠΉ Ρ…ΠΎΡ€Π΄Π΅ $l$, ΠΈ $r(\varphi)$ - радиус окруТности ΠΊΡ€ΠΈΠ²ΠΈΠ·Π½Ρ‹ Π² Ρ‚ΠΎΡ‡ΠΊΠ΅ $P(\varphi)$. Π’ΠΎΠ³Π΄Π°

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$$\bar l = 2 \, r(\varphi) \, \sin(\alpha/2) \quad \text{ΠΈ} \quad \bar f = \frac12 (\alpha - \sin\alpha) \, r^2(\varphi) \label{eq:45} \tag{45}$$

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ΠšΡ€ΠΎΠΌΠ΅ Ρ‚ΠΎΠ³ΠΎ,

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$$\left| \, dp \, \right| = \frac12 r(\varphi) \sin(\alpha/2) \left| \, d\alpha \, \right| \label{eq:46} \tag{46}$$

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НаконСц,

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$$l - \bar l = o(\alpha^2) \quad \text{ΠΈ} \quad f - \bar f = o(\alpha^4) \label{eq:47} \tag{47}$$

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Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ Ρ€Π΅Π·ΡƒΠ»ΡŒΡ‚Π°Ρ‚ для $E_n$:

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$$E_n = \cfrac{C_n^2}{12F^2} \int\limits_0^{2\pi} r^4 \int\limits_0^{\alpha(\varphi)} \left( 1 - \frac{\alpha^3r^2}{12F^2} + o(\alpha^3) \right)^{n-2} (\alpha^4 + o(\alpha^5)) \, d\alpha \, d\varphi \label{eq:48} \tag{48}$$

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ΠŸΠΎΠ΄ΡΡ‚Π°Π²ΠΈΠ²

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$$\frac{\alpha^3r^2}{12F} = \frac{X}{n} \label{eq:49} \tag{49}$$

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ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ

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$$S(\varphi) = \int\limits_0^{\alpha(\varphi)} \left(1 - \frac{\alpha^3r^2}{12F} + o(\alpha^3)\right)^{n-2} (\alpha^4 + o(\alpha^5)) \, d\alpha \label{eq:50} \tag{50}$$ +$$S(\varphi) = \frac13 \left(\frac{12F}{r^2 n}\right)^{5/3} \left(\int\limits_0^\infty \left(1 - \frac{X}{n}\right)^{n-2} x^{2/3} \, dx + o(1)\right) = \\ = \frac13 \, \Gamma \left( \frac{5}{3} \right) \left(\frac{12F}{r^2 n}\right)^{5/3} (1 + o(1)) \notag$$

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Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ

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$$E_n \sim \Gamma \left(\frac{5}{3}\right) \, \sqrt[3]{\frac{2n}{3F}} \, \int\limits_0^{2\pi} r^{2/3} \, d\varphi \label{eq:51} \tag{51}$$

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ΠžΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Π·Π° $s$ Π΄Π»ΠΈΠ½Ρƒ Π΄ΡƒΠ³ΠΈ Π³Ρ€Π°Π½ΠΈΡ†Ρ‹ $K$ ΠΈ, считая $d\varphi/ds = 1/r$, ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ ΡΠ»Π΅Π΄ΡƒΡŽΡ‰Π΅Π΅ ΡƒΡ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅:

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Если $K$ - выпуклая ΠΎΠ±Π»Π°ΡΡ‚ΡŒ, граничная кривая ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ ΠΈΠΌΠ΅Π΅Ρ‚ Π½Π΅ΠΏΡ€Π΅Ρ€Ρ‹Π²Π½ΡƒΡŽ ΠΊΡ€ΠΈΠ²ΠΈΠ·Π½Ρƒ $x = x(s)$, Ρ‚ΠΎ

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$$E_n \sim \Gamma \left(\frac{5}{3}\right) \, \sqrt[3]{\frac23} \, (F^{-1/3} \int\limits_L x^{1/3} \, ds) \sqrt[3]{n} \label{eq:52} \tag{52}$$

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Π˜Π½Ρ‚Π΅Π³Ρ€Π°Π» $\int\limits_L x^{1/3} \, ds$, входящий Π² $(\ref{eq:52})$, являСтся Π°Ρ„Ρ„ΠΈΠ½Π½ΠΎΠΉ Π΄Π»ΠΈΠ½ΠΎΠΉ ΠΊΡ€ΠΈΠ²ΠΎΠΉ $L$. Π›Π΅Π³ΠΊΠΎ Π²ΠΈΠ΄Π΅Ρ‚ΡŒ, Ρ‡Ρ‚ΠΎ ΠΌΠ½ΠΎΠΆΠΈΡ‚Π΅Π»ΡŒ $\sqrt[3]{n}$ являСтся Π°Ρ„Ρ„ΠΈΠ½Π½ΠΎ-ΠΈΠ½Π²Π°Ρ€ΠΈΠ°Π½Ρ‚Π½Ρ‹ΠΌ. Из Π°Ρ„Ρ„ΠΈΠ½Π½ΠΎΠ³ΠΎ изопСримСтричСского свойства эллипсов (см. [3]) нСпосрСдствСнно слСдуСт, Ρ‡Ρ‚ΠΎ этот ΠΌΠ½ΠΎΠΆΠΈΡ‚Π΅Π»ΡŒ максималСн для эллипсов ΠΈ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ для эллипсов.

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$\triangleleft$

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4. Нормальное распределение случайных точек

Π£Ρ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 4. ΠŸΡƒΡΡ‚ΡŒ Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i \:(i = 1, \:2, \:\ldots, \:n)$ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ Π½Π° плоскости нСзависимо Π΄Ρ€ΡƒΠ³ ΠΎΡ‚ Π΄Ρ€ΡƒΠ³Π° Π² соотвСтствии с Π½ΠΎΡ€ΠΌΠ°Π»ΡŒΠ½Ρ‹ΠΌ распрСдСлСниСм. Π’ΠΎΠ³Π΄Π°

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$$E_n \sim 2 \sqrt{2 \pi \log n} \label{eq:53} \tag{53}$$

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Π”ΠΎΠΊΠ°Π·Π°Ρ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ:

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$\triangleright$

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Π’Π°ΠΊ ΠΊΠ°ΠΊ $E_n$ ΠΈΠ½Π²Π°Ρ€ΠΈΠ°Π½Ρ‚Π½ΠΎ ΠΎΡ‚Π½ΠΎΡΠΈΡ‚Π΅Π»ΡŒΠ½ΠΎ Π°Ρ„Ρ„ΠΈΠ½Π½Ρ‹Ρ… ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΠΎΠ²Π°Π½ΠΈΠΉ плоскости, ΠΌΠΎΠΆΠ½ΠΎ ΡΡ‡ΠΈΡ‚Π°Ρ‚ΡŒ, Ρ‡Ρ‚ΠΎ функция плотности нашСго Π½ΠΎΡ€ΠΌΠ°Π»ΡŒΠ½ΠΎΠ³ΠΎ распрСдСлСния ΠΈΠΌΠ΅Π΅Ρ‚ Π²ΠΈΠ΄ $\frac{1}{2\pi} \, exp \left(-\frac{x^2 + y^2}{2}\right) \notag$. Как ΠΈ Π² ΠΏΡ€Π΅Π΄Ρ‹Π΄ΡƒΡ‰ΠΈΡ… ΠΏΠ°Ρ€Π°Π³Ρ€Π°Ρ„Π°Ρ…,

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$$E_n = C_n^2 \cdot W_n \notag$$

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Π³Π΄Π΅ $W_n$ - Π²Π΅Ρ€ΠΎΡΡ‚Π½ΠΎΡΡ‚ΡŒ Ρ‚ΠΎΠ³ΠΎ, Ρ‡Ρ‚ΠΎ всС Ρ‚ΠΎΡ‡ΠΊΠΈ $P_3, \:\ldots, \:P_n$ Π»Π΅ΠΆΠ°Ρ‚ ΠΏΠΎ ΠΎΠ΄Π½Ρƒ сторону ΠΎΡ‚ прямой $P_1 P_2$.

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Из Ρ„ΠΎΡ€ΠΌΡƒΠ»Ρ‹ ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π»ΡŒΠ½ΠΎΠΉ Π³Π΅ΠΎΠΌΠ΅Ρ‚Ρ€ΠΈΠΈ

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$$dP_1dP_2 = |t_1 - t_2| \, dt_1 \, dt_2 \, dp \, d\varphi \notag$$

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ΠΈ ΠΈΠ·-Π·Π° ΠΊΡ€ΡƒΠ³ΠΎΠ²ΠΎΠΉ симмСтрии Π½ΠΎΡ€ΠΌΠ°Π»ΡŒΠ½ΠΎΠ³ΠΎ распрСдСлСния ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ

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$$W_n = \frac{1}{2\pi} \iiint (g(p)^{n-2} + (1 - g(p))^{n-2}) \, |t_1 - t_2| \, exp \left(-p^2 - \frac{t_1^2 + t_2^2}{2}\right) \, dt_1 \, dt_2 \, dp \label{eq:54} \tag{54}$$

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$$\text{Π³Π΄Π΅} \: g(p) = \int\limits_{-\infty}^{+\infty} \int\limits_{P}^{+\infty} \frac{1}{2\pi} \, exp\left(-\frac{x^2+y^2}{2}\right) \, dx \, dy = 1 - \varPhi(p) \label{eq:55} \tag{55}$$

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$$\varPhi(p) = \frac{1}{\sqrt{2\pi}} \int\limits_{-\infty}^p exp\left(-\frac{u^2}{2}\right) \, du \label{eq:56} \tag{56}$$

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Π§Π΅Ρ€Π΅Π· Π·Π°ΠΌΠ΅Π½Ρƒ

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$$u = \frac{t_1 - t_2}{\sqrt 2}, \: v = \frac{t_1 + t_2}{\sqrt 2}, \: \frac{\partial(t_1, t_2)}{\partial(u, v)} = 1 \label{eq:57} \tag{57}$$

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ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ

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$$\frac{1}{2\pi} \int\int |t_1 - t_2| \, exp\left(-\frac{t_1^2 + t_2^2}{2}\right) \, dt_1 \, dt_2 = \frac{2}{\sqrt \pi} \label{eq:58} \tag{58}$$

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Π‘Π»Π΅Π΄ΠΎΠ²Π°Ρ‚Π΅Π»ΡŒΠ½ΠΎ,

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$$W_n = \frac{2}{\sqrt \pi} \int\limits_0^\infty [\varPhi^{n-2}(p) + (1 - \varPhi(p))^{n-2}] e^{-p^2} dp \label{eq:59} \tag{59}$$

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ΠΈ, ΠΎΡ‡Π΅Π²ΠΈΠ΄Π½ΠΎ, $1 - \varPhi(p) \leqq 1/2$ для $p \geqq 0$. А Π·Π½Π°Ρ‡ΠΈΡ‚,

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$$W_n \sim \frac{2}{\sqrt \pi} \int\limits_0^\infty \varPhi^{n-2}(p) e^{-p^2} dp \label{eq:60} \tag{60}$$

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Π’Π΅ΠΏΠ΅Ρ€ΡŒ для $p > 1$

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$$\varPhi(p) = 1 - \frac{exp \left(-\frac{p^2}{2}\right)}{\sqrt{2\pi}p\left(1 + \frac{\theta p}{p^2}\right)} \label{eq:61} \tag{61}$$

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Π³Π΄Π΅ $0 < \theta_p < 1$ (см. [4], стр. 136, Π·Π°Π΄Π°Ρ‡Π° 18).

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Π’Ρ‹ΠΏΠΎΠ»Π½ΠΈΠΌ подстановку

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$$p = \sqrt{2\log n - \log(2\log n)} + \frac{u - \log \sqrt{2\pi}}{\sqrt{2 \log n}} \label{eq:62} \tag{62}$$

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ΠΈ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ

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$$W_n \sim \frac{4\sqrt{2\pi\log n}}{n^2}$$

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Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, $E_n \sim 2\sqrt{2\pi \log n}$, Ρ‡Ρ‚ΠΎ ΠΈ Ρ‚Ρ€Π΅Π±ΠΎΠ²Π°Π»ΠΎΡΡŒ Π΄ΠΎΠΊΠ°Π·Π°Ρ‚ΡŒ.

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$\triangleleft$

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Литература

1) A. RΓ©nyi, R. Sulanke: Über die konvexe HΓΌlle von n zufΓ€llig gewΓ€hlten Punkten. Z. Wahrscheinlichkeitstheorie 2 (1963), 75β€”84
+2) W. Blaschke: Integralgeometrie. 3. Aufl. Berlin: VEB Deutscher Verlag der Wissenschaften 1955, 1β€”130.
+3) W. Blaschke, K. Reidemeister: Differentialgeometrie II, Berlin: Springer 1923, 60.
+4) A. RΓ©nyi: Wahrscheinlichkeitsrechnung, mit einem Anhang ΓΌber Informationstheorie. Berlin: VEB Deutscher Verlag der Wissenschaften 1962, 1β€”547.

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+ + + + + + + From 49aeaceaaecd8a987055be95f6c51a5605aa3c0e Mon Sep 17 00:00:00 2001 From: Egor Makarenko Date: Sat, 7 Apr 2018 11:30:32 +0300 Subject: [PATCH 4/5] More fixes --- .../33_expectation_of_convex_hull_vertices.ipynb | 4 ++-- .../33_expectation_of_convex_hull_vertices.slides.html | 4 ++-- 2 files changed, 4 insertions(+), 4 deletions(-) diff --git a/33_expectation_of_convex_hull_vertices/33_expectation_of_convex_hull_vertices.ipynb b/33_expectation_of_convex_hull_vertices/33_expectation_of_convex_hull_vertices.ipynb index bda7249..d23354e 100644 --- a/33_expectation_of_convex_hull_vertices/33_expectation_of_convex_hull_vertices.ipynb +++ b/33_expectation_of_convex_hull_vertices/33_expectation_of_convex_hull_vertices.ipynb @@ -138,7 +138,7 @@ } }, "source": [ - "* Π’ $\\S1$ рассмотрим случай, ΠΊΠΎΠ³Π΄Π° Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i$ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны Π²Π½ΡƒΡ‚Ρ€ΠΈ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ r-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$. Как Π±ΡƒΠ΄Π΅Ρ‚ ΠΏΠΎΠΊΠ°Π·Π°Π½ΠΎ, $E_n = (2r\\,/\\,3)(\\log n + \\gamma) + T + O(1)$, Π³Π΄Π΅ $\\gamma$ - постоянная Π­ΠΉΠ»Π΅Ρ€Π°:\n", + "* Π’ $\\S1$ рассмотрим случай, ΠΊΠΎΠ³Π΄Π° Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i$ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны Π²Π½ΡƒΡ‚Ρ€ΠΈ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$. Как Π±ΡƒΠ΄Π΅Ρ‚ ΠΏΠΎΠΊΠ°Π·Π°Π½ΠΎ, $E_n = (2r\\,/\\,3)(\\log n + \\gamma) + T + O(1)$, Π³Π΄Π΅ $\\gamma$ - постоянная Π­ΠΉΠ»Π΅Ρ€Π°:\n", " \n", " $$\\gamma = \\lim_{n\\to\\infty} \\left( \\sum_{k=1}^{n} \\frac1k - \\log n \\right)=\\lim_{n\\to\\infty} \\left( 1+\\frac12+\\frac13+\\ldots+\\frac1n - \\log n \\right) \\notag$$\n", " \n", @@ -161,7 +161,7 @@ "source": [ "## 1. Π‘Π»ΡƒΡ‡Π°ΠΉΠ½ΠΎ Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Π΅ Ρ‚ΠΎΡ‡ΠΊΠΈ Π² Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΌ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ΅\n", "\n", - "**Π£Ρ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 1.** Если Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i$ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны Π²Π½ΡƒΡ‚Ρ€ΠΈ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ r-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$, матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ $E_n = (2r\\,/\\,3)(\\log n + \\gamma) + T + O(1)$, Π³Π΄Π΅ $\\gamma$ - постоянная Π­ΠΉΠ»Π΅Ρ€Π°." + "**Π£Ρ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 1.** Если Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i$ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны Π²Π½ΡƒΡ‚Ρ€ΠΈ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$, матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ $E_n = (2r\\,/\\,3)(\\log n + \\gamma) + T + O(1)$, Π³Π΄Π΅ $\\gamma$ - постоянная Π­ΠΉΠ»Π΅Ρ€Π°." ] }, { diff --git a/33_expectation_of_convex_hull_vertices/33_expectation_of_convex_hull_vertices.slides.html b/33_expectation_of_convex_hull_vertices/33_expectation_of_convex_hull_vertices.slides.html index e7f281f..38231d0 100644 --- a/33_expectation_of_convex_hull_vertices/33_expectation_of_convex_hull_vertices.slides.html +++ b/33_expectation_of_convex_hull_vertices/33_expectation_of_convex_hull_vertices.slides.html @@ -11998,7 +11998,7 @@

    -

  • Π’ $\S1$ рассмотрим случай, ΠΊΠΎΠ³Π΄Π° Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i$ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны Π²Π½ΡƒΡ‚Ρ€ΠΈ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ r-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$. Как Π±ΡƒΠ΄Π΅Ρ‚ ΠΏΠΎΠΊΠ°Π·Π°Π½ΠΎ, $E_n = (2r\,/\,3)(\log n + \gamma) + T + O(1)$, Π³Π΄Π΅ $\gamma$ - постоянная Π­ΠΉΠ»Π΅Ρ€Π°:

    +

  • Π’ $\S1$ рассмотрим случай, ΠΊΠΎΠ³Π΄Π° Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i$ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны Π²Π½ΡƒΡ‚Ρ€ΠΈ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$. Как Π±ΡƒΠ΄Π΅Ρ‚ ΠΏΠΎΠΊΠ°Π·Π°Π½ΠΎ, $E_n = (2r\,/\,3)(\log n + \gamma) + T + O(1)$, Π³Π΄Π΅ $\gamma$ - постоянная Π­ΠΉΠ»Π΅Ρ€Π°:

    $$\gamma = \lim_{n\to\infty} \left( \sum_{k=1}^{n} \frac1k - \log n \right)=\lim_{n\to\infty} \left( 1+\frac12+\frac13+\ldots+\frac1n - \log n \right) \notag$$

    ΠŸΡ€ΠΈ этом Π²Π΅Π»ΠΈΡ‡ΠΈΠ½Π° $T = T(K)$ зависит ΠΎΡ‚ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$.

    • @@ -12017,7 +12017,7 @@

    -

    1. Случайно выбранные точки в выпуклом многоугольнике

    Π£Ρ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 1. Если Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i$ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны Π²Π½ΡƒΡ‚Ρ€ΠΈ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ r-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$, матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ $E_n = (2r\,/\,3)(\log n + \gamma) + T + O(1)$, Π³Π΄Π΅ $\gamma$ - постоянная Π­ΠΉΠ»Π΅Ρ€Π°.

    +

    1. Случайно выбранные точки в выпуклом многоугольнике

    Π£Ρ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 1. Если Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i$ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны Π²Π½ΡƒΡ‚Ρ€ΠΈ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$, матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ $E_n = (2r\,/\,3)(\log n + \gamma) + T + O(1)$, Π³Π΄Π΅ $\gamma$ - постоянная Π­ΠΉΠ»Π΅Ρ€Π°.

From 1539b56af411214def735a8cf243763581ae5710 Mon Sep 17 00:00:00 2001 From: Egor Makarenko Date: Sun, 6 Oct 2019 13:58:42 +0500 Subject: [PATCH 5/5] Add markdown file from hackmd This article was originally written in markdown. I think that it will be useful to save the original file too. --- .../33_expectation_of_convex_hull_vertices.md | 426 ++++++++++++++++++ 1 file changed, 426 insertions(+) create mode 100644 33_expectation_of_convex_hull_vertices/33_expectation_of_convex_hull_vertices.md diff --git a/33_expectation_of_convex_hull_vertices/33_expectation_of_convex_hull_vertices.md b/33_expectation_of_convex_hull_vertices/33_expectation_of_convex_hull_vertices.md new file mode 100644 index 0000000..634aa0f --- /dev/null +++ b/33_expectation_of_convex_hull_vertices/33_expectation_of_convex_hull_vertices.md @@ -0,0 +1,426 @@ +# Π‘Ρ€Π΅Π΄Π½Π΅Π΅ количСство Π²Π΅Ρ€ΡˆΠΈΠ½ Π² Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠ΅ + +Алгоритм ДТарвиса ΠΈΠΌΠ΅Π΅Ρ‚ ΡΠ»ΠΎΠΆΠ½ΠΎΡΡ‚ΡŒ $O(X N)$, Π³Π΄Π΅ $X$ -- число Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ, Π° $N$ -- ΠΎΠ±Ρ‰Π΅Π΅ число Ρ‚ΠΎΡ‡Π΅ΠΊ. Π§Ρ‚ΠΎΠ±Ρ‹ ΠΎΡ†Π΅Π½ΠΈΡ‚ΡŒ ΡΠ»ΠΎΠΆΠ½ΠΎΡΡ‚ΡŒ этого Π°Π»Π³ΠΎΡ€ΠΈΡ‚ΠΌΠ° Π² срСднСм случаС, Π½Π΅ΠΎΠ±Ρ…ΠΎΠ΄ΠΈΠΌΠΎ Π²Ρ‹Ρ‡ΠΈΡΠ»ΠΈΡ‚ΡŒ $E(X)$ -- матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ Π²Π΅Π»ΠΈΡ‡ΠΈΠ½Ρ‹ $X$. Для Π΅Π³ΠΎ Π°Π½Π°Π»ΠΈΠ·Π° ΠΌΡ‹ Π΄ΠΎΠ»ΠΆΠ½Ρ‹ ΠΎΠΏΡ€Π΅Π΄Π΅Π»ΠΈΡ‚ΡŒ ΠΎΠ³Ρ€Π°Π½ΠΈΡ‡Π΅Π½Π½ΡƒΡŽ ΠΎΠ±Π»Π°ΡΡ‚ΡŒ, ΠΈΠ· ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ Ρ‚ΠΎΡ‡ΠΊΠΈ, Ρ‚Π°ΠΊ ΠΊΠ°ΠΊ ΠΎΠ½ΠΈ ΠΌΠΎΠ³ΡƒΡ‚ Π±Ρ‹Ρ‚ΡŒ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны лишь Π² мноТСствС с ΠΊΠΎΠ½Π΅Ρ‡Π½ΠΎΠΉ ΠΌΠ΅Ρ€ΠΎΠΉ Π›Π΅Π±Π΅Π³Π° [[Kendall, Moran (1963)](https://doi.org/10.1002/bimj.19650070322)]. + +НиТС ΠΏΡ€ΠΈΠ²Π΅Π΄Ρ‘Π½ ряд Ρ‚Π΅ΠΎΡ€Π΅ΠΌ, ΠΏΡ€Π΅Π΄ΡΡ‚Π°Π²Π»ΡΡŽΡ‰ΠΈΡ… ΠΎΡ†Π΅Π½ΠΊΡƒ срСднСго количСства Π²Π΅Ρ€ΡˆΠΈΠ½ Π² Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠ΅ ΠΏΡ€ΠΈ Ρ€Π°Π·Π»ΠΈΡ‡Π½Ρ‹Ρ… распрСдСлСниях Ρ‚ΠΎΡ‡Π΅ΠΊ. + +**Π’Π΅ΠΎΡ€Π΅ΠΌΠ° 1** [[RΓ©nyi, Sulanke (1963)](https://doi.org/10.1007/BF00535300)]. *Если $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ ΠΈ нСзависимо ΠΈΠ· Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° Π½Π° плоскости, Ρ‚ΠΎ ΠΏΡ€ΠΈ $N\to\infty$ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ этих Ρ‚ΠΎΡ‡Π΅ΠΊ $E(X)=\frac23 r(\log N + \gamma) + O(1)$, Π³Π΄Π΅ $\gamma$ - константа Π­ΠΉΠ»Π΅Ρ€Π°.* + +**Π’Π΅ΠΎΡ€Π΅ΠΌΠ° 2** [[Raynaud (1970)](https://doi.org/10.1017/S0021900200026917)]. *Если $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ ΠΈ нСзависимо ΠΈΠ· внутрСнности $k$-ΠΌΠ΅Ρ€Π½ΠΎΠΉ гипСрсфСры, Ρ‚ΠΎ ΠΏΡ€ΠΈ $N\to\infty$ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π³ΠΈΠΏΠ΅Ρ€Π³Ρ€Π°Π½Π΅ΠΉ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ этих Ρ‚ΠΎΡ‡Π΅ΠΊ асимптотичСски стрСмится ΠΊ $E(f) = O(N^{(k-1)/(k+1)})$* + +**Π’Π΅ΠΎΡ€Π΅ΠΌΠ° 3** [[Raynaud (1970)](https://doi.org/10.1017/S0021900200026917)]. *Если $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ нСзависимо Π² соотвСтствии с $k$-ΠΌΠ΅Ρ€Π½Ρ‹ΠΌ Π½ΠΎΡ€ΠΌΠ°Π»ΡŒΠ½Ρ‹ΠΌ распрСдСлСниСм, Ρ‚ΠΎ ΠΏΡ€ΠΈ $N\to\infty$ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ этих Ρ‚ΠΎΡ‡Π΅ΠΊ асимптотичСски стрСмится ΠΊ $E(f) = O((\log N)^{(k-1)/2})$* + +**Π’Π΅ΠΎΡ€Π΅ΠΌΠ° 4** [[Bentley, Kung, Schkolnick, Thompson (1978)](https://doi.org/10.1145/322092.322095)]. *Если ΠΊΠΎΠΎΡ€Π΄ΠΈΠ½Π°Ρ‚Ρ‹ $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ Π² $k$-ΠΌΠ΅Ρ€Π½ΠΎΠΌ пространствС Π²Ρ‹Π±Ρ€Π°Π½Ρ‹ нСзависимо Π² соотвСтствии с ΠΏΡ€ΠΎΠΈΠ·Π²ΠΎΠ»ΡŒΠ½Ρ‹ΠΌ Π½Π΅ΠΏΡ€Π΅Ρ€Ρ‹Π²Π½Ρ‹ΠΌ распрСдСлСниСм (Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΠΎ, для ΠΊΠ°ΠΆΠ΄ΠΎΠΉ ΠΊΠΎΠΎΡ€Π΄ΠΈΠ½Π°Ρ‚Ρ‹ ΠΈΡΠΏΠΎΠ»ΡŒΠ·ΡƒΠ΅Ρ‚ΡΡ своё распрСдСлСниС), Ρ‚ΠΎ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ этих Ρ‚ΠΎΡ‡Π΅ΠΊ асимптотичСски стрСмится ΠΊ $E(X) = O((\log N)^{k-1})$*. + +Условиям этой Ρ‚Π΅ΠΎΡ€Π΅ΠΌΡ‹ ΡƒΠ΄ΠΎΠ²Π»Π΅Ρ‚Π²ΠΎΡ€ΡΡŽΡ‚ ΠΌΠ½ΠΎΠ³ΠΈΠ΅ распрСдСлСния, Π²ΠΊΠ»ΡŽΡ‡Π°Ρ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎΠ΅ распрСдСлСниС Π² Π³ΠΈΠΏΠ΅Ρ€ΠΊΡƒΠ±Π΅. + +Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, +- $E(X) = O(\log N)$ для $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ, Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Ρ… ΠΈΠ· Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° +- $E(X) = O(N^{1/3})$ для $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ, Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Ρ… ΠΈΠ· ΠΊΡ€ΡƒΠ³Π° +- $E(X) = O(N^{1/2})$ для $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ, Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Ρ… ΠΈΠ· сфСры +- $E(X) = O((\log N)^2)$ для $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ, Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Ρ… ΠΈΠ· ΠΊΡƒΠ±Π° +- $E(X) = O(\sqrt{\log N})$ для $N$ Ρ‚ΠΎΡ‡Π΅ΠΊ, Π½ΠΎΡ€ΠΌΠ°Π»ΡŒΠ½ΠΎ распрСдСлённых Π½Π° плоскости + +Π—Π΄Π΅ΡΡŒ прСдставлСн ΠΏΠ΅Ρ€Π΅Π²ΠΎΠ΄ Ρ€Π°Π±ΠΎΡ‚Ρ‹ A. RΓ©nyi ΠΈ R. Sulanke, Π³Π΄Π΅ приводится Π΄ΠΎΠΊΠ°Π·Π°Ρ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ ΠΎΡ†Π΅Π½ΠΊΠΈ матСматичСского оТидания числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ Π² Π΄Π²ΡƒΠΌΠ΅Ρ€Π½ΠΎΠΌ случаС. + +--- + +# О Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠ΅ N случайно Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Ρ… Ρ‚ΠΎΡ‡Π΅ΠΊ +> [A. Renyi ΠΈ R. Sulanke (1963 Π³.)](https://1drv.ms/b/s!AsLTB1ON1LwNjMhGXq0xv1LW6Wx0-w) + +ΠŸΡƒΡΡ‚ΡŒ $P_i\:(i=1,\:2,\:\ldots,\:n)$ -- Ρ‚ΠΎΡ‡ΠΊΠΈ Π½Π° плоскости, Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Π΅ случайно ΠΈ нСзависимо Π² соотвСтствии с Π½Π΅ΠΊΠΎΡ‚ΠΎΡ€Ρ‹ΠΌ распрСдСлСниСм, Π° $H_n$ - выпуклая ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠ° этих Ρ‚ΠΎΡ‡Π΅ΠΊ, которая, ΠΊΠ°ΠΊ извСстно, являСтся Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΌ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΎΠΌ. Число $X_n$ Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ -- случайная Π²Π΅Π»ΠΈΡ‡ΠΈΠ½Π°, ΠΈ Π² Π΄Π°Π½Π½ΠΎΠΉ Ρ€Π°Π±ΠΎΡ‚Π΅ Π±ΡƒΠ΄Π΅Ρ‚ исслСдовано Π΅Ρ‘ матСматичСскоС ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ $E_n = E(X_n)$ ΠΏΡ€ΠΈ $n\to\infty$ Π² Ρ€Π°Π·Π»ΠΈΡ‡Π½Ρ‹Ρ… прСдполоТСниях ΠΎ распрСдСлСнии Ρ‚ΠΎΡ‡Π΅ΠΊ $P_i$. + +* Π’ $\S1$ рассмотрим случай, ΠΊΠΎΠ³Π΄Π° Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i$ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны Π²Π½ΡƒΡ‚Ρ€ΠΈ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ r-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$. Как Π±ΡƒΠ΄Π΅Ρ‚ ΠΏΠΎΠΊΠ°Π·Π°Π½ΠΎ, $E_n = (2r/3)(\log n + \gamma) + T + O(1)$, Π³Π΄Π΅ $\gamma$ - постоянная Π­ΠΉΠ»Π΅Ρ€Π°: + + $$\gamma = \lim_{n\to\infty} \left( \sum_{k=1}^{n} \frac1k - \log n \right)=\lim_{n\to\infty} \left( 1+\frac12+\frac13+\ldots+\frac1n - \log n \right) \notag$$ + + ΠŸΡ€ΠΈ этом Π²Π΅Π»ΠΈΡ‡ΠΈΠ½Π° $T = T(K)$ зависит ΠΎΡ‚ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° K. + +* Π’ $\S2$ Π΄ΠΎΠΊΠ°ΠΆΠ΅ΠΌ ΡΠ»Π΅ΠΌΠ΅Π½Ρ‚Π°Ρ€Π½ΡƒΡŽ Π³Π΅ΠΎΠΌΠ΅Ρ‚Ρ€ΠΈΡ‡Π΅ΡΠΊΡƒΡŽ Ρ‚Π΅ΠΎΡ€Π΅ΠΌΡƒ ΠΎ Ρ‚ΠΎΠΌ, Ρ‡Ρ‚ΠΎ $T(K)$ ΠΏΡ€ΠΈΠ½ΠΈΠΌΠ°Π΅Ρ‚ ΠΌΠ°ΠΊΡΠΈΠΌΠ°Π»ΡŒΠ½Ρ‹Π΅ значСния для ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΎΠ², ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Π΅ Π°Ρ„Ρ„ΠΈΠ½Π½Ρ‹ΠΌΠΈ прСобразованиями ΠΌΠΎΠΆΠ½ΠΎ привСсти ΠΊ ΠΏΡ€Π°Π²ΠΈΠ»ΡŒΠ½ΠΎΠΌΡƒ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΡƒ, ΠΈ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ для Π½ΠΈΡ…. + +* Π’ $\S3$ исслСдуСм асимптотичСскоС ΠΏΠΎΠ²Π΅Π΄Π΅Π½ΠΈΠ΅ $E_n$, ΠΊΠΎΠ³Π΄Π° Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i$ Ρ€Π°Π²Π½ΠΎΠΌΠ΅Ρ€Π½ΠΎ распрСдСлСны Π² Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ области с Π³Π»Π°Π΄ΠΊΠΎΠΉ Π³Ρ€Π°Π½ΠΈΡ†Π΅ΠΉ. ΠžΠΊΠ°Π·Ρ‹Π²Π°Π΅Ρ‚ΡΡ, Ρ‡Ρ‚ΠΎ Π² Ρ‚Π°ΠΊΠΎΠΌ случаС $E_n$ ΠΈΠΌΠ΅Π΅Ρ‚ порядок $\sqrt[3]n$. + +* НаконСц, Π² $\S4$ ΠΌΡ‹ Π΄ΠΎΠΊΠ°ΠΆΠ΅ΠΌ, Ρ‡Ρ‚ΠΎ $E_n$ Π²Π΅Π΄Ρ‘Ρ‚ сСбя ΠΊΠ°ΠΊ $\sqrt{\log n}$, Ссли Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i$ ΠΈΠΌΠ΅ΡŽΡ‚ Π½ΠΎΡ€ΠΌΠ°Π»ΡŒΠ½ΠΎΠ΅ распрСдСлСниС. + +## 1. Π‘Π»ΡƒΡ‡Π°ΠΉΠ½ΠΎ Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Π΅ Ρ‚ΠΎΡ‡ΠΊΠΈ Π² Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΌ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ΅ + +ΠŸΡƒΡΡ‚ΡŒ $K$ - Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΉ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ с Π²Π΅Ρ€ΡˆΠΈΠ½Π°ΠΌΠΈ $A_1,\:A_2,\:\ldots,\:A_r$ ΠΈ ΡΠΎΠΎΡ‚Π²Π΅Ρ‚ΡΡ‚Π²ΡƒΡŽΡ‰ΠΈΠΌΠΈ ΡƒΠ³Π»Π°ΠΌΠΈ $\vartheta_1,\:\vartheta_2,\:\ldots,\:\vartheta_r$. ΠžΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Π·Π° $a_k$ Π΄Π»ΠΈΠ½Ρƒ стороны $A_k A_{k+1}$, Π³Π΄Π΅ $A_{r+1} = A_1$ $(k = 1,\:2,\:\ldots,\:r)$. + +![](https://i.imgur.com/qDXplcM.png) + +ΠŸΡƒΡΡ‚ΡŒ $\varepsilon_{ij} = 1$, Ссли ΠΎΡ‚Ρ€Π΅Π·ΠΎΠΊ $P_i P_j$ ΠΏΡ€ΠΈ $i \neq j$ ΠΏΡ€ΠΈΠ½Π°Π΄Π»Π΅ΠΆΠΈΡ‚ Π³Ρ€Π°Π½ΠΈΡ†Π΅ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ $H_n$, ΠΈΠ½Π°Ρ‡Π΅ $\varepsilon_{ij} = 0$. Π’ΠΎΠ³Π΄Π° ΠΎΡ‡Π΅Π²ΠΈΠ΄Π½ΠΎ, Ρ‡Ρ‚ΠΎ + +$$X_n = \sum_{1 \le i \lt j \le n}\varepsilon_{ij} \label{eq:1} \tag{1}$$ + +Π’ силу случайного распрСдСлСния Π²Π΅Ρ€ΠΎΡΡ‚Π½ΠΎΡΡ‚ΡŒ $W_n = P(\varepsilon_{ij} = 1)$ ΠΎΠ΄ΠΈΠ½Π°ΠΊΠΎΠ²Π° для всСх ΠΏΠ°Ρ€ $(i, j)$, Ρ‚Π°ΠΊΠΈΡ…, Ρ‡Ρ‚ΠΎ $1 \le i \lt j \leq n$, ΠΈ Ρ‚ΠΎΠ³Π΄Π° + +$$E_n = E(X_n) = \sum_{1 \le i \lt j \le n} E(\varepsilon_{ij}) = \sum_{1 \le i \lt j \le n} \left( 0 \cdot (1 - W_n) + 1 \cdot W_n \right) = C_n^2 \cdot W_n \label{eq:2} \tag{2}$$ + +Π§Ρ‚ΠΎΠ±Ρ‹ ΠΎΠΏΡ€Π΅Π΄Π΅Π»ΠΈΡ‚ΡŒ асимптотичСскоС ΠΏΠΎΠ²Π΅Π΄Π΅Π½ΠΈΠ΅ $E_n$, достаточно Π²Ρ‹Ρ‡ΠΈΡΠ»ΠΈΡ‚ΡŒ Π²Π΅Ρ€ΠΎΡΡ‚Π½ΠΎΡΡ‚ΡŒ $W_n$. МоТСм Ρ€Π°ΡΡΠΌΠΎΡ‚Ρ€Π΅Ρ‚ΡŒ $W_n$ ΠΊΠ°ΠΊ Π²Π΅Ρ€ΠΎΡΡ‚Π½ΠΎΡΡ‚ΡŒ Ρ‚ΠΎΠ³ΠΎ, Ρ‡Ρ‚ΠΎ всС Ρ‚ΠΎΡ‡ΠΊΠΈ $P_h \: (h = 3,\:4,\:\ldots,\:n)$ Π»Π΅ΠΆΠ°Ρ‚ ΠΏΠΎ ΠΎΠ΄Π½Ρƒ сторону ΠΎΡ‚ прямой $P_1 P_2$. + +![](https://i.imgur.com/g0S8ogT.png) + +ΠŸΡƒΡΡ‚ΡŒ $F$ ΠΎΠ±ΠΎΠ·Π½Π°Ρ‡Π°Π΅Ρ‚ ΠΏΠ»ΠΎΡ‰Π°Π΄ΡŒ $K$. Если Ρ‡Π΅Ρ€Π΅Π· $F_1 \le \frac12 F$ ΠΈ $F_2 = F - F_1$ ΠΎΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΡ‚ΡŒ части, Π½Π° ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Π΅ прямая $P_1 P_2$ Π΄Π΅Π»ΠΈΡ‚ ΠΎΠ±Π»Π°ΡΡ‚ΡŒ $K$, Ρ‚ΠΎ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ + +$$W_n = \frac1{F^2} \int\limits_K \int\limits_K \left[ \left(1 - \frac{F_1}{F}\right)^{n-2} + \left(\frac{F_1}{F}\right)^{n-2}\right] dP_1 dP_2 \label{eq:3} \tag{3}$$ + +ΠΊΠ°ΠΊ Π²Π΅Ρ€ΠΎΡΡ‚Π½ΠΎΡΡ‚ΡŒ Ρ‚ΠΎΠ³ΠΎ, Ρ‡Ρ‚ΠΎ всС Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i\:(i=3,\:4,\:\ldots,\:n)$ ΠΏΡ€ΠΈΠ½Π°Π΄Π»Π΅ΠΆΠ°Ρ‚ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ ΠΎΠ΄Π½ΠΎΠΉ ΠΈΠ· частСй $F$. + +Π’Π°ΠΊ ΠΊΠ°ΠΊ $F_1/F \le \frac12$, ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π» Π²Ρ‚ΠΎΡ€ΠΎΠ³ΠΎ слагаСмого ΠΎΠ³Ρ€Π°Π½ΠΈΡ‡Π΅Π½ константой: + +$$\frac1{F^2} \int\limits_K \int\limits_K \left(\frac{F_1}{F}\right)^{n-2} dP_1 dP_2 \le \frac1{2^{n - 2}} \label{eq:4} \tag{4}$$ + +А ΡΠ»Π΅Π΄ΠΎΠ²Π°Ρ‚Π΅Π»ΡŒΠ½ΠΎ, + +$$E_n \sim C_n^2 \cdot \frac1{F^2} \int\limits_K \int\limits_K \left(1 - \frac{F_1}{F}\right)^{n-2} dP_1 dP_2 \label{eq:5} \tag{5}$$ + +$$\text{Π³Π΄Π΅} \: A \sim B \Leftrightarrow \lim_{n\to+\infty} \frac{A_n}{B_n} = 1 \notag$$ + + +ΠžΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Ρ‡Π΅Ρ€Π΅Π· $f_i$ ΠΏΠ»ΠΎΡ‰Π°Π΄ΡŒ Ρ‚Ρ€Π΅ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $A_{i-1}A_iA_{i+1}$ $(A_{r+j} = A_j)$, ΠΈ ΠΏΡƒΡΡ‚ΡŒ $f = \min\limits_{1 \le i \le r} f_i$ + +![](https://i.imgur.com/JxZ8lff.png) + +ΠžΡ‡Π΅Π²ΠΈΠ΄Π½ΠΎ, Ρ‡Ρ‚ΠΎ ΠΏΡ€ΠΈ ΠΈΠ·ΡƒΡ‡Π΅Π½ΠΈΠΈ асимптотичСского повСдСния $E_n$ ΠΌΠΎΠΆΠ½ΠΎ ΠΎΠ³Ρ€Π°Π½ΠΈΡ‡ΠΈΡ‚ΡŒΡΡ ΠΏΠ°Ρ€Π°ΠΌΠΈ Ρ‚ΠΎΡ‡Π΅ΠΊ $(P_1,\:P_2)$, для ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… прямая $P_1 P_2$ пСрСсСкаСтся Π»ΠΈΠ±ΠΎ с двумя сосСдними сторонами $K$, Π»ΠΈΠ±ΠΎ с двумя сторонами $K$, Ρ€Π°Π·Π΄Π΅Π»Ρ‘Π½Π½Ρ‹ΠΌΠΈ Ρ‚Ρ€Π΅Ρ‚ΡŒΠ΅ΠΉ стороной. Для всСх ΠΎΡΡ‚Π°Π»ΡŒΠ½Ρ‹Ρ… прямых выполняСтся ΠΎΠ³Ρ€Π°Π½ΠΈΡ‡Π΅Π½ΠΈΠ΅ + +$$1 - \frac{F_1}{F} \leqq 1 - \frac{f}{F} \notag$$ + +ΠŸΠΎΡΡ‚ΠΎΠΌΡƒ ΠΌΡ‹ ΠΌΠΎΠΆΠ΅ΠΌ ΠΏΡ€Π΅Π½Π΅Π±Ρ€Π΅Ρ‡ΡŒ ΡΠΎΠΎΡ‚Π²Π΅Ρ‚ΡΡ‚Π²ΡƒΡŽΡ‰Π΅ΠΉ Ρ‡Π°ΡΡ‚ΡŒΡŽ ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π»Π° ΠΈ Π·Π°ΠΏΠΈΡΠ°Ρ‚ΡŒ + +$$E_n \sim C_n^2 \cdot \frac1{F^2} \left( \sum\limits_{i=1}^r (I_i + J_i) \right) \label{eq:6} \tag{6}$$ + +$$I_i = \int\limits_{C_i} \left(1 - \frac{F_1}{F}\right)^{n-2} dP_1 dP_2 \label{eq:7} \tag{7}$$ + +$$J_i = \int\limits_{D_i} \left(1 - \frac{F_1}{F}\right)^{n-2} dP_1 dP_2 \label{eq:8} \tag{8}$$ + +Π—Π΄Π΅ΡΡŒ $C_i$ ΠΎΠ·Π½Π°Ρ‡Π°Π΅Ρ‚ мноТСство ΠΏΠ°Ρ€ Ρ‚ΠΎΡ‡Π΅ΠΊ $(P_1,\:P_2)$, для ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… прямая $P_1P_2$ пСрСсСкаСт смСТныС стороны $A_{i-1}A_i$ ΠΈ $A_iA_{i+1}$ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$, a $D_i$ - мноТСство ΠΏΠ°Ρ€ $(P_1,\:P_2)$, для ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… прямая $P_1P_2$ пСрСсСкаСт стороны $A_{i-1}A_i$ ΠΈ $A_{i+1}A_{i+2}$ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$, Ρ€Π°Π·Π΄Π΅Π»Ρ‘Π½Π½Ρ‹Π΅ Π΄Ρ€ΡƒΠ³ΠΎΠΉ стороной. + +Π‘Π½Π°Ρ‡Π°Π»Π° рассмотрим ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π» $I_i$. ΠžΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Ρ‡Π΅Ρ€Π΅Π· $G_{ab}$ мноТСство ΠΏΠ°Ρ€ $(P_1,\:P_2)$, для ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… Π²Ρ‹ΠΏΠΎΠ»Π½Π΅Π½Ρ‹ нСравСнства $|A_iQ_1| < a$ ΠΈ $|A_iQ_2| < b$, Π³Π΄Π΅ $Q_1$ - Ρ‚ΠΎΡ‡ΠΊΠ° пСрСсСчСния прямой $P_1P_2$ со стороной $A_{i-1}A_i$, Π° $Q_2$ - Ρ‚ΠΎΡ‡ΠΊΠ° пСрСсСчСния $P_1P_2$ со стороной $A_iA_{i+1}$. Π—Π΄Π΅ΡΡŒ ΠΈ Π΄Π°Π»Π΅Π΅ $|AB|$ -- Π΄Π»ΠΈΠ½Π° ΠΎΡ‚Ρ€Π΅Π·ΠΊΠ° $AB$. + +По элСмСнтарным расчётам получаСтся + +$$\int\limits_{G_{ab}} dP_1dP_2 = \frac{a^2b^2\sin^2\vartheta_i}{12} \label{eq:9} \tag{9}$$ + +ΠŸΠ»ΠΎΡ‰Π°Π΄ΡŒ Ρ‚Ρ€Π΅ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $Q_1A_iQ_2$, отсСкаСмого прямой $P_1P_2$, Ρ€Π°Π²Π½Π° $\frac{1}{2} a b \sin \vartheta_i$. Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, взяв $a_{i-1} = |A_{i-1}A_i|$, $a_i = |A_iA_{i+1}|$, ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ + +$$I_i = \int\limits_0^{a_{i-1}} \int\limits_0^{a_i} \left(1 - \frac{ab \sin \vartheta_i}{2F}\right)^{n-2} \sin^2 \vartheta_i \frac{ab}{3} \,da\,db \label{eq:10} \tag{10}$$ + +Π’Ρ‹ΠΏΠΎΠ»Π½ΠΈΠΌ Π·Π°ΠΌΠ΅Π½Ρƒ + +$$X = a \sqrt\frac{\sin \vartheta_i}{2F}, \; Y = b \sqrt\frac{\sin \vartheta_i}{2F} \label{eq:11} \tag{11}$$ + +ΠΈ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ + +$$I_i = \frac{4F^2}{3} \int\limits_0^{X_i} \int\limits_0^{Y_i} (1-XY)^{n-2} XY \: dX dY \label{eq:12} \tag{12}$$ + +Π³Π΄Π΅ + +$$X_i = a_{i-1} \sqrt\frac{\sin \vartheta_i}{2F}, \; Y_i = a_i \sqrt\frac{\sin \vartheta_i}{2F} \notag$$ + +ΠΈ Ρ‚Π°ΠΊΠΆΠ΅ Π·Π°ΠΌΠ΅Ρ‚ΠΈΠΌ, Ρ‡Ρ‚ΠΎ + +$$\varrho_i = X_iY_i = \frac{a_{i-1}a_i\sin \vartheta_i}{2F} = \frac{f_i}{F} < 1 \label{eq:13} \tag{13}$$ + +Π’Π΅ΠΏΠ΅Ρ€ΡŒ ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΡƒΠ΅ΠΌ ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π»: + +$$\int\limits_0^{X_i} \int\limits_0^{Y_i} (1-XY)^{n-2} XY \: dX dY = \int\limits_0^{X_i} \int\limits_0^{Y_i} (1-XY)^{n-2} \: dX dY - \int\limits_0^{X_i} \int\limits_0^{Y_i} (1-XY)^{n-1} \: dX dY \notag$$ + +Π”Π°Π»Π΅Π΅ ΠΏΡ€ΠΈΠΌΠ΅Π½ΠΈΠΌ + +$$\int\limits_0^{X_i} \int\limits_0^{Y_i} (1-XY)^{n-1} \: dX dY = \int\limits_0^{X_i} \frac{1 - (1 - XY_i)^n}{nX} \: dX = \\ = \frac1n \left( 1 + \frac12 + \ldots + \frac1n - \sum\limits_{k=1}^n \frac{(1-\varrho_i)^k}{k} \right) \notag$$ + +Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, ΠΌΡ‹ ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ Π·Π½Π°Ρ‡Π΅Π½ΠΈΠ΅ ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π»Π° $I_i$: + +$$I_i = \frac{4F^2}{3} \left[ \frac{\cfrac12 + \cfrac13 + \ldots + \cfrac1n}{n(n-1)} - \frac{\sum\limits_{k=1}^{n-1} \cfrac{(1 - \varrho_i)^k}{k}}{n(n-1)} + \frac{(1-\varrho_i)^n}{n^2} \right] \label{eq:14} \tag{14}$$ + +Если ΠΏΠΎΠ»ΠΎΠΆΠΈΠΌ + +$$S_i = \sum\limits_{k=1}^\infty \frac{(1 - \varrho_i)^k}{k} = \log \frac1{\varrho_i} \label{eq:15} \tag{15}$$ + +Ρ‚ΠΎ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ Π² ΠΊΠΎΠ½Π΅Ρ‡Π½ΠΎΠΌ ΠΈΡ‚ΠΎΠ³Π΅ + +$$E_n = \frac23 (\gamma - 1 + \log n) r - \frac23 \sum\limits_{i=1}^r S_i + o(1) + \frac{C_n^2}{F^2} \sum\limits_{i=1}^r J_i \label{eq:16} \tag{16}$$ + +Π³Π΄Π΅ $\gamma$ - постоянная Π­ΠΉΠ»Π΅Ρ€Π°. + +Π’Π΅ΠΏΠ΅Ρ€ΡŒ вычислим ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π» $J_i$. ΠŸΡ€ΠΎΠ΄Π»ΠΈΠΌ стороны $A_{i-1}A_i$ ΠΈ $A_{i+1}A_{i+2}$ Π΄ΠΎ Ρ‚ΠΎΡ‡ΠΊΠΈ ΠΈΡ… пСрСсСчСния $A_i^*$; случай, ΠΊΠΎΠ³Π΄Π° эти стороны ΠΏΠ°Ρ€Π°Π»Π»Π΅Π»ΡŒΠ½Ρ‹, ΠΌΠΎΠΆΠ½ΠΎ Ρ€Π°ΡΡΠΌΠ°Ρ‚Ρ€ΠΈΠ²Π°Ρ‚ΡŒ Π°Π½Π°Π»ΠΎΠ³ΠΈΡ‡Π½ΠΎ. + +ΠŸΡƒΡΡ‚ΡŒ снова $Q_1$ - Ρ‚ΠΎΡ‡ΠΊΠ° пСрСсСчСния прямой $P_1P_2$ со стороной $A_{i-1}A_i$, Π° $Q_2$ - Ρ‚ΠΎΡ‡ΠΊΠ° пСрСсСчСния $P_1P_2$ со стороной $A_{i+1}A_{i+2}$, ΠΈ ΠΏΡƒΡΡ‚ΡŒ + +$$a' = |A_iA_i^*|, \;\; b' = |A_{i+1}A_i^*|, \;\; a = |A_iQ_1|, \;\; b = |A_{i+1}Q_2| \notag$$ + +$G_{ab}$ Π±Ρ‹Π»ΠΎ ΠΎΠΏΡ€Π΅Π΄Π΅Π»Π΅Π½ΠΎ Π²Ρ‹ΡˆΠ΅. ΠŸΠΎΠ»ΡƒΡ‡ΠΈΠΌ ΠΈΠ· $(9)$: + +$$\int\limits_{G_{ab}} dP_1 dP_2 = \frac{\sin^2\beta_i}{12} (a^2 + 2aa')(b^2 + 2bb') \label{eq:17} \tag{17}$$ + +Π³Π΄Π΅ $\beta_i$ - ΡƒΠ³ΠΎΠ» ΠΏΡ€ΠΈ $A_i^*$. Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, + +$$J_i = \int\limits_{D_i} \left(1 - \frac{(ab+a'b+ab')\sin \beta_i}{2F} \right)^{n-2} \frac{\sin^2 \beta_i}{3} (a+a')(b+b')\,da\,db \label{eq:18} \tag{18}$$ + +ΠΈ этот ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π» асимптотичСски эквивалСнтСн + +$$J_i \sim \int\limits_{D_i} \left(1 - \frac{(a'b+ab')\sin \beta_i}{2F} \right)^{n-2} \frac{\sin^2 \beta_i}{3} a'b'\,da\,db \label{eq:19} \tag{19}$$ + +Π”Π°Π»Π΅Π΅ сдСлаСм Π·Π°ΠΌΠ΅Π½Ρƒ + +$$a = \frac{F}{b' \sin \beta_i} X, \;\; b = \frac{F}{a' \sin \beta_i} Y \notag$$ + +ΠΈ Π½Π°ΠΉΠ΄Ρ‘ΠΌ Π·Π½Π°Ρ‡Π΅Π½ΠΈΠ΅ ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π»Π° + +$$J_i \sim \frac{F^2}{3} \int\limits_0^{a_{i-1}'} \int\limits_0^{a_{i+1}'} \left( 1 - \frac{X+Y}{2} \right)^{n-2} dX\,dY = \\ = \frac{4F^2}{3n(n-1)} \left[ 1 - \left( 1 - \frac{a_{i-1}'}{2} \right)^n - \left( 1 - \frac{a_{i+1}'}{2} \right)^n + \left( 1 - \frac{a_{i-1}' + a_{i+1}'}{2} \right)^n \right] \notag$$ + +Π³Π΄Π΅ + +$$a_{i-1}' = \frac{a_ib'\sin\beta_i}{F}, \;\; a_{i+1}' = \frac{a_{i+2}a'\sin\beta_i}{F} \notag$$ + +Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ ΠΈΡ‚ΠΎΠ³ΠΎΠ²Ρ‹ΠΉ Ρ€Π΅Π·ΡƒΠ»ΡŒΡ‚Π°Ρ‚: + +$$E_n = \frac{2r}{3} (\log n + \gamma) - \frac23 \sum\limits_{i=1}^{r} S_i + o(1) \notag$$ + +**Π£Ρ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 1.** ΠŸΡƒΡΡ‚ΡŒ $K$ - Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΉ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ с $r$ сторонами. ΠœΠ°Ρ‚Π΅ΠΌΠ°Ρ‚ΠΈΡ‡Π΅ΡΠΊΠΎΠ΅ ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΠ΅ $E_n$ числа Π²Π΅Ρ€ΡˆΠΈΠ½ Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡ‡ΠΊΠΈ $H_n$, построСнной Π½Π° $n$ случайных Ρ‚ΠΎΡ‡ΠΊΠ°Ρ… ΠΈΠ· $K$, Ρ€Π°Π²Π½ΠΎ + +$$E_n = \frac{2r}{3} (\log n + \gamma) + \frac23 \log \frac{\prod\limits_1^r f_i}{F^r} + o(1) \label{eq:20} \tag{20}$$ + +Π—Π΄Π΅ΡΡŒ $F$ ΠΎΠ±ΠΎΠ·Π½Π°Ρ‡Π°Π΅Ρ‚ ΠΏΠ»ΠΎΡ‰Π°Π΄ΡŒ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$, Π° $f_i$ - ΠΏΠ»ΠΎΡ‰Π°Π΄ΡŒ Ρ‚Ρ€Π΅ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $A_{i-1}A_iA_{i+1}$. + +ΠŸΡƒΡΡ‚ΡŒ, Π½Π°ΠΏΡ€ΠΈΠΌΠ΅Ρ€, $K$ - Ρ‚Ρ€Π΅ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ, Ρ‚ΠΎΠ³Π΄Π° $f_1 = f_2 = f_3 = F$ ΠΈ + +$$E_n = 2(\log n + \gamma) + o(1) \label{eq:21} \tag{21}$$ + +Π—Π°ΠΌΠ΅Ρ‚ΠΈΠΌ, Ρ‡Ρ‚ΠΎ Π² Ρ‚Π°ΠΊΠΎΠΌ случаС Ρ„ΠΎΡ€ΠΌΡƒΠ»Π° для $E_n$ Π½Π΅ зависит ΠΎΡ‚ Ρ„ΠΎΡ€ΠΌΡ‹ Ρ‚Ρ€Π΅ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ°, Ρ‚Π°ΠΊ ΠΊΠ°ΠΊ $E_n$, ΠΎΡ‡Π΅Π²ΠΈΠ΄Π½ΠΎ, Π΄ΠΎΠ»ΠΆΠ½ΠΎ Π±Ρ‹Ρ‚ΡŒ ΠΈΠ½Π²Π°Ρ€ΠΈΠ°Π½Ρ‚Π½ΠΎ ΠΊ Π°Ρ„Ρ„ΠΈΠ½Π½Ρ‹ΠΌ прСобразованиям. Π’ $\S2$ ΠΌΡ‹ ΠΏΠΎΠΊΠ°ΠΆΠ΅ΠΌ, Ρ‡Ρ‚ΠΎ константа + +$$Z(K) = \frac{\prod\limits_{i=1}^r f_i}{F^r} \label{eq:22} \tag{22}$$ + +которая зависит ΠΎΡ‚ Ρ„ΠΎΡ€ΠΌΡ‹ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$ ΠΈ появляСтся Π² Π²Ρ‹Ρ€Π°ΠΆΠ΅Π½ΠΈΠΈ $(\ref{eq:20})$, становится максимальной для ΠΏΡ€Π°Π²ΠΈΠ»ΡŒΠ½ΠΎΠ³ΠΎ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° ΠΈ Π°Ρ„Ρ„ΠΈΠ½Π½Ρ‹Ρ… ΠΊ Π½Π΅ΠΌΡƒ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΎΠ². + +## 2. Π­ΠΊΡΡ‚Ρ€Π΅ΠΌΠ°Π»ΡŒΠ½Π°Ρ Π·Π°Π΄Π°Ρ‡Π° для Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹Ρ… ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΎΠ² + +ΠŸΡ€Π΅ΠΆΠ΄Π΅ Ρ‡Π΅ΠΌ ΠΏΠ΅Ρ€Π΅ΠΉΡ‚ΠΈ ΠΊ дальнСйшим асимптотичСским ΠΎΡ†Π΅Π½ΠΊΠ°ΠΌ для $E_n$ ΠΏΡ€ΠΈ Π΄Ρ€ΡƒΠ³ΠΈΡ… прСдполоТСниях ΠΎ распрСдСлСнии Ρ‚ΠΎΡ‡Π΅ΠΊ $P_i$, Π΄ΠΎΠΊΠ°ΠΆΠ΅ΠΌ ΡΠ»Π΅Π΄ΡƒΡŽΡ‰Π΅Π΅ ΡƒΡ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅: + +**Π£Ρ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 2.** ΠŸΡƒΡΡ‚ΡŒ $K$ - Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΉ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ с $r$ сторонами. Ѐункция + +$$Z(K) = \frac1{F^r} \prod\limits_{i=1}^r f_i \label{eq:23} \tag{23}$$ + +ΠΈΠ· ΠΏΡ€Π΅Π΄Ρ‹Π΄ΡƒΡ‰Π΅Π³ΠΎ ΠΏΠ°Ρ€Π°Π³Ρ€Π°Ρ„Π° максимальна Ρ‚ΠΎΠ³Π΄Π° ΠΈ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ Ρ‚ΠΎΠ³Π΄Π°, ΠΊΠΎΠ³Π΄Π° $K$ ΠΌΠΎΠΆΠ½ΠΎ ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΠΎΠ²Π°Ρ‚ΡŒ Π² ΠΏΡ€Π°Π²ΠΈΠ»ΡŒΠ½Ρ‹ΠΉ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ Π°Ρ„Ρ„ΠΈΠ½Π½Ρ‹ΠΌΠΈ прСобразованиями. + +**Π”ΠΎΠΊΠ°Π·Π°Ρ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ:** +ΠŸΡƒΡΡ‚ΡŒ $\alpha_i = \overrightarrow{A_{i-1}A_i}$, Π³Π΄Π΅ $A_{r+i} = A_i$. Π’ΠΎΠ³Π΄Π° + +$$f_i = \frac12 [\alpha_i, \: \alpha_{i+1}] \\ F = \frac12 \sum\limits_{i=1}^{r-2} [\alpha_1 + \alpha_2 + \ldots + \alpha_i, \: \alpha_{i+1}] \label{eq:24} \tag{24}$$ + +Π³Π΄Π΅ $[\alpha, \: \beta]$ - ΠΌΠΎΠ΄ΡƒΠ»ΡŒ Π²Π΅ΠΊΡ‚ΠΎΡ€Π½ΠΎΠ³ΠΎ произвСдСния $\alpha$ ΠΈ $\beta$. Ѐункция $Z(K)$ Ρ‚Π΅ΠΏΠ΅Ρ€ΡŒ являСтся Ρ„ΡƒΠ½ΠΊΡ†ΠΈΠ΅ΠΉ ΠΎΡ‚ $r$ Π²Π΅ΠΊΡ‚ΠΎΡ€ΠΎΠ² $\alpha_i \: (i=1, \: \ldots, \: r)$, ΡƒΠ΄ΠΎΠ²Π»Π΅Ρ‚Π²ΠΎΡ€ΡΡŽΡ‰ΠΈΡ… ΡƒΡΠ»ΠΎΠ²ΠΈΡŽ замкнутости: + +$$\sum\limits_{i=1}^r \alpha_i = 0 \label{eq:25} \tag{25}$$ + +Π’Π΅ΠΏΠ΅Ρ€ΡŒ допустим, Ρ‡Ρ‚ΠΎ Π²Π΅Ρ€ΡˆΠΈΠ½Ρ‹ $A_i$ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$ ΠΏΠΎΠ΄Π²Π΅Ρ€Π³Π°ΡŽΡ‚ΡΡ достаточно ΠΌΠ°Π»Ρ‹ΠΌ смСщСниям, ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Π΅ ΠΎΠΏΠΈΡΡ‹Π²Π°ΡŽΡ‚ΡΡ Π½Π΅ΠΏΡ€Π΅Ρ€Ρ‹Π²Π½ΠΎ Π΄ΠΈΡ„Ρ„Π΅Ρ€Π΅Π½Ρ†ΠΈΡ€ΡƒΠ΅ΠΌΠΎΠΉ Ρ„ΡƒΠ½ΠΊΡ†ΠΈΠ΅ΠΉ ΠΎΡ‚ ΠΏΠ°Ρ€Π°ΠΌΠ΅Ρ‚Ρ€Π° $t$. Π’Π°ΠΊ ΠΊΠ°ΠΊ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ $K$, ΠΊΠΎΡ‚ΠΎΡ€Ρ‹ΠΉ Π΄ΠΎΠ»ΠΆΠ΅Π½ ΡΠΎΠΎΡ‚Π²Π΅Ρ‚ΡΡ‚Π²ΠΎΠ²Π°Ρ‚ΡŒ Π·Π½Π°Ρ‡Π΅Π½ΠΈΡŽ ΠΏΠ°Ρ€Π°ΠΌΠ΅Ρ‚Ρ€Π° $t = 0$, являСтся Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΌ r-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΎΠΌ, Ρ‚ΠΎ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΈ, ΡΠ²Π»ΡΡŽΡ‰ΠΈΠ΅ΡΡ Ρ€Π΅Π·ΡƒΠ»ΡŒΡ‚Π°Ρ‚ΠΎΠΌ достаточно ΠΌΠ°Π»ΠΎΠ³ΠΎ измСнСния $t$, Ρ‚Π°ΠΊΠΆΠ΅ Π±ΡƒΠ΄ΡƒΡ‚ Π²Ρ‹ΠΏΡƒΠΊΠ»Ρ‹ΠΌΠΈ r-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ°ΠΌΠΈ. Ѐункция $Z$, рассматриваСмая ΠΊΠ°ΠΊ функция ΠΎΡ‚ $t$, Π΄ΠΎΠ»ΠΆΠ½Π° Π΄ΠΎΡΡ‚ΠΈΠ³Π°Ρ‚ΡŒ максимума ΠΏΡ€ΠΈ $t = 0$, Ссли $K$ - ΠΌΠ°ΠΊΡΠΈΠΌΠ°Π»ΡŒΠ½Ρ‹ΠΉ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ. НСобходимоС условиС Ρ‚ΠΎΠ³ΠΎ, Ρ‡Ρ‚ΠΎ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ $K$ являСтся ΠΌΠ°ΠΊΡΠΈΠΌΠ°Π»ΡŒΠ½Ρ‹ΠΌ, ΠΌΡ‹ ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ ΠΈΠ· нСслоТного расчёта: + +$$\sum\limits_{i=1}^r \frac{F}{f_i} ([\dot \alpha_i, \: \alpha_{i+1}] + [\alpha_i, \: \dot \alpha_{i+1}]) - \\ - r \sum\limits_{i=1}^{r-2}([\dot \alpha_1 + \ldots + \dot \alpha_i, \: \alpha_{i+1}] + [\alpha_1 + \ldots + \alpha_i, \: \dot \alpha_{i+1}]) = 0 \label{eq:26} \tag{26}$$ + +Π“Π΄Π΅ $\dot \alpha_i = \left.\frac{d}{dt} \alpha_i \right|_{t=0}$ - ΠΏΡ€ΠΎΠΈΠ·Π²ΠΎΠ»ΡŒΠ½Ρ‹Π΅ Π²Π΅ΠΊΡ‚ΠΎΡ€Ρ‹, ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Π΅ Ρ‚ΠΎΠΆΠ΅ Π΄ΠΎΠ»ΠΆΠ½Ρ‹ ΡƒΠ΄ΠΎΠ²Π»Π΅Ρ‚Π²ΠΎΡ€ΡΡ‚ΡŒ ΡƒΡΠ»ΠΎΠ²ΠΈΡŽ замкнутости, ΡΠ»Π΅Π΄ΡƒΡŽΡ‰Π΅ΠΌΡƒ ΠΈΠ· $(25)$: + +$$\sum\limits_{i=1}^r \dot\alpha_i = 0 \label{eq:27} \tag{27}$$ + +Если ΠΌΡ‹ ΠΏΠΎΠ»ΠΎΠΆΠΈΠΌ ΠΏΠΎΡΠ»Π΅Π΄ΠΎΠ²Π°Ρ‚Π΅Π»ΡŒΠ½ΠΎ + +$$\dot \alpha_j = \beta, \;\; \dot \alpha_{j+1} = -\beta, \;\; \dot \alpha_i = 0 \: \text{для} \: i \neq j, \;\; i \neq j+1 \label{eq:28} \tag{28}$$ + +Π³Π΄Π΅ $\beta$ - ΠΈΠ·Π½Π°Ρ‡Π°Π»ΡŒΠ½ΠΎ ΠΏΡ€ΠΎΠΈΠ·Π²ΠΎΠ»ΡŒΠ½Ρ‹ΠΉ Π²Π΅ΠΊΡ‚ΠΎΡ€, Ρ‚ΠΎ условиС замкнутости $(\ref{eq:27})$, ΠΎΡ‡Π΅Π²ΠΈΠ΄Π½ΠΎ, выполняСтся, ΠΈ ΠΌΡ‹ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ для $j = 1, \: \ldots, \: r$: + +$$\frac{F}{r} \left\{ \frac{[\alpha_{j-1}, \: \beta]}{f_{j-1}} - \frac{[\beta, \: \alpha_{j+2}]}{f_{j+1}} + \frac1{f_j} [\beta, \: \alpha_j + \alpha_{j+1}] \right\} = [\beta, \: \alpha_j + \alpha_{j+1}] \label{eq:29} \tag{29}$$ + +Если ΠΌΡ‹ Ρ‚Π΅ΠΏΠ΅Ρ€ΡŒ Π½Π°ΠΌΠ΅Ρ€Π΅Π½Π½ΠΎ Π²Ρ‹Π±Π΅Ρ€Π΅ΠΌ $\beta = \alpha_j$, Ρ„ΠΎΡ€ΠΌΡƒΠ»Π° ΠΏΡ€ΠΈΠΌΠ΅Ρ‚ Π²ΠΈΠ΄ + +$$\frac{F}{r} \left\{ 4 - \frac{[\alpha_j, \: \alpha_{j+2}]}{f_{j+1}} \right\} = 2 f_j \label{eq:30} \tag{30}$$ + +Π° для $\beta = -\alpha_{j+1}$ + +$$\frac{F}{r} \left\{ 4 - \frac{[\alpha_{j-1}, \: \alpha_{j+1}]}{f_{j-1}} \right\} = 2 f_j \label{eq:31} \tag{31}$$ + +Π‘Ρ€Π°Π²Π½ΠΈΠΌ эти Π΄Π²Π΅ Ρ„ΠΎΡ€ΠΌΡƒΠ»Ρ‹ ΠΈ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ + +$$[\alpha_j, \: \alpha_{j+2}] = \frac{f_{j+1}}{f_{j-1}} [\alpha_{j-1}, \: \alpha_{j-1}] \label{eq:32} \tag{32}$$ + +Π’Π΅ΠΏΠ΅Ρ€ΡŒ ΠΌΡ‹ запишСм ΡƒΡ€Π°Π²Π½Π΅Π½ΠΈΠ΅ $(\ref{eq:31})$ Π² Ρ„ΠΎΡ€ΠΌΠ΅ + +$$2 \left( 2 - \frac{r f_j}{F} \right) = \frac{[\alpha_{j-1}, \: \alpha_{j+1}]}{f_{j-1}} \notag$$ + +ΠΈ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ ΠΏΡƒΡ‚Ρ‘ΠΌ ΠΏΠΎΠ²Ρ‚ΠΎΡ€Π½ΠΎΠ³ΠΎ примСнСния $(\ref{eq:32})$: + +$$2 \left( 2 - \frac{r f_j}{F} \right) = \frac{f_j}{f_1 \cdot f_2} [\alpha_1, \: \alpha_3] \notag$$ + +ΠΈΠ»ΠΈ + +$$f_j = \frac2{\cfrac{r}{F} + \cfrac{[\alpha_1, \: \alpha_3]}{2 f_1 \cdot f_2}} \label{eq:33} \tag{33}$$ + +Для $r = 3$ наша ΡΠΊΡΡ‚Ρ€Π΅ΠΌΠ°Π»ΡŒΠ½Π°Ρ Π·Π°Π΄Π°Ρ‡Π° Ρ‚Ρ€ΠΈΠ²ΠΈΠ°Π»ΡŒΠ½Π°. Π’ случаС $r \ge 4$ получаСтся, Ρ‡Ρ‚ΠΎ $f_3 = f_4 = \ldots = f_r$. ΠŸΡ€ΠΈ цикличСской пСрСстановкС Π½ΠΎΠΌΠ΅Ρ€ΠΎΠ² Π²Π΅Ρ€ΡˆΠΈΠ½ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ + +$$f_1 = f_2 = \ldots = f_r \label{eq:34} \tag{34}$$ + +Π° послС $(\ref{eq:32})$ Ρ‚Π°ΠΊΠΆΠ΅ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ + +$$[\alpha_i, \: \alpha_{i+2}] = [\alpha_1, \: \alpha_3] = \beta \;\; (i = 1, \: 2, \: \ldots, \: r) \label{eq:35} \tag{35}$$ + +Π³Π΄Π΅ $\beta$ - нСкоторая константа. + +**ΠŸΡ€ΠΈΠΌΠ΅Ρ‡Π°Π½ΠΈΠ΅.** Π’ случаС $\beta = [\alpha_i, \: \alpha_{i+2}] = 0$ получаСтся, Ρ‡Ρ‚ΠΎ $f_j = 2F/r$. Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, Π²Π΅ΠΊΡ‚ΠΎΡ€ $\alpha_{i+2}$ всСгда ΠΏΠ°Ρ€Π°Π»Π»Π΅Π»Π΅Π½ $\alpha_i$. Π­Ρ‚ΠΎ Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΠΎ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ Π² Ρ‚ΠΎΠΌ случаС, Ссли $r = 4$, Π° $K$ - ΠΏΠ°Ρ€Π°Π»Π»Π΅Π»ΠΎΠ³Ρ€Π°ΠΌΠΌ. + +Π’Π΅ΠΏΠ΅Ρ€ΡŒ вСрнёмся ΠΊ ΠΎΠ±Ρ‰Π΅ΠΌΡƒ ΡΠ»ΡƒΡ‡Π°ΡŽ. Π‘ ΠΏΠΎΠΌΠΎΡ‰ΡŒΡŽ Π°Ρ„Ρ„ΠΈΠ½Π½Ρ‹Ρ… ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΠΎΠ²Π°Π½ΠΈΠΉ ΠΌΡ‹ всСгда ΠΌΠΎΠΆΠ΅ΠΌ Π΄ΠΎΠ±ΠΈΡ‚ΡŒΡΡ Ρ‚ΠΎΠ³ΠΎ, Ρ‡Ρ‚ΠΎ $f_j = 1/2$, Ρ‚ΠΎ Π΅ΡΡ‚ΡŒ + +$$[\alpha_i, \: \alpha_{i+1}] = 1 \label{eq:36} \tag{36}$$ + +Из $(\ref{eq:33})$ сразу слСдуСт + +$$F = \frac{r}{2(2-\beta)} \label{eq:37} \tag{37}$$ + +ΠΈ, ΡΠ»Π΅Π΄ΠΎΠ²Π°Ρ‚Π΅Π»ΡŒΠ½ΠΎ, $\beta \lt 2$. + +Π’Π°ΠΊ ΠΊΠ°ΠΊ Π²Π΅ΠΊΡ‚ΠΎΡ€Ρ‹ $\alpha_{i-2}$ ΠΈ $\alpha_{i-1}$ Π»ΠΈΠ½Π΅ΠΉΠ½ΠΎ нСзависимы, ΠΈΠ· $\alpha_i = x \alpha_{i-2} + y \alpha_{i-1}$, учитывая $(\ref{eq:35})$ ΠΈ $(\ref{eq:36})$, слСдуСт: + +$$\alpha_i = -\alpha_{i-2} + b \alpha_{i-1} \label{eq:38} \tag{38}$$ + +Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, ΠΌΡ‹ ΠΏΠΎΠΊΠ°Π·Π°Π»ΠΈ, Ρ‡Ρ‚ΠΎ Π²Π΅ΠΊΡ‚ΠΎΡ€Ρ‹ $\alpha_i$ сторон максимального Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠ³ΠΎ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° Π΄ΠΎΠ»ΠΆΠ½Ρ‹ ΡƒΠ΄ΠΎΠ²Π»Π΅Ρ‚Π²ΠΎΡ€ΡΡ‚ΡŒ рСкурсивной Ρ„ΠΎΡ€ΠΌΡƒΠ»Π΅ $(\ref{eq:38})$. Π—Π°Π²Π΅Ρ€ΡˆΠΈΠΌ Π΄ΠΎΠΊΠ°Π·Π°Ρ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ, ΠΏΠΎΠΊΠ°Π·Π°Π², Ρ‡Ρ‚ΠΎ Π² этом случаС ΠΌΠΎΠΆΠ½ΠΎ ΠΎΠΏΡ€Π΅Π΄Π΅Π»ΠΈΡ‚ΡŒ Π΅Π²ΠΊΠ»ΠΈΠ΄ΠΎΠ²Ρƒ ΠΌΠ΅Ρ‚Ρ€ΠΈΠΊΡƒ Π² Π°Ρ„Ρ„ΠΈΠ½Π½ΠΎΠΉ плоскости, ΠΎΡ‚Π½ΠΎΡΠΈΡ‚Π΅Π»ΡŒΠ½ΠΎ ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ $K$ являСтся ΠΏΡ€Π°Π²ΠΈΠ»ΡŒΠ½Ρ‹ΠΌ $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠΎΠΌ. + +Π’ΠΎ-ΠΏΠ΅Ρ€Π²Ρ‹Ρ…, Π·Π°ΠΌΠ΅Ρ‚ΠΈΠΌ, Ρ‡Ρ‚ΠΎ всСгда Π΄ΠΎΠ»ΠΆΠ½ΠΎ Π±Ρ‹Ρ‚ΡŒ + +$$-1 \leq b \lt 2 \label{eq:39} \tag{39}$$ + +Π”Π΅ΠΉΡΡ‚Π²ΠΈΡ‚Π΅Π»ΡŒΠ½ΠΎ, Ссли $b \lt -1$, Ρ‚ΠΎ сторона $\alpha_3$ Π±ΡƒΠ΄Π΅Ρ‚ ΠΏΠ΅Ρ€Π΅ΡΠ΅ΠΊΠ°Ρ‚ΡŒ сторону $\alpha_1$, Ρ‡Ρ‚ΠΎ ΠΏΡ€ΠΎΡ‚ΠΈΠ²ΠΎΡ€Π΅Ρ‡ΠΈΡ‚ выпуклости $K$. ВСрхняя ΠΎΡ†Π΅Π½ΠΊΠ° $b \lt 2$ ΡƒΠΆΠ΅ Π΄ΠΎΠΊΠ°Π·Π°Π½Π° Π² $(\ref{eq:37})$. +Π•Π²ΠΊΠ»ΠΈΠ΄ΠΎΠ²Π° ΠΌΠ΅Ρ‚Ρ€ΠΈΠΊΠ° ΠΎΠ΄Π½ΠΎΠ·Π½Π°Ρ‡Π½ΠΎ устанавливаСтся, Ссли ΠΌΡ‹ ΠΎΠΏΡ€Π΅Π΄Π΅Π»ΠΈΠΌ скалярноС ΠΏΡ€ΠΎΠΈΠ·Π²Π΅Π΄Π΅Π½ΠΈΠ΅ Π²Π΅ΠΊΡ‚ΠΎΡ€ΠΎΠ² $\alpha_1$ ΠΈ $\alpha_2$, ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Π΅ ΠΌΡ‹ рассматриваСм ΠΊΠ°ΠΊ базис: + +$$\alpha_1^2 = \alpha_2^2 = \frac{2}{\sqrt{4-b^2}}, \;\; \alpha_1\alpha_2 = \frac{b}{\sqrt{4-b^2}} \label{eq:40} \tag{40}$$ + +Π—Π°ΠΌΠ΅Ρ‚ΠΈΠΌ, Ρ‡Ρ‚ΠΎ + +$$[\alpha_1, \alpha_2]^2 = \begin{vmatrix} \alpha_1^2 & \alpha_1\alpha_2 \\ \alpha_1\alpha_2 & \alpha_2^2 \end{vmatrix} = 1 \notag$$ + +Π‘Π»Π΅Π΄ΠΎΠ²Π°Ρ‚Π΅Π»ΡŒΠ½ΠΎ, эта ΠΌΠ΅Ρ‚Ρ€ΠΈΠΊΠ° ΠΏΠΎΠ»ΠΎΠΆΠΈΡ‚Π΅Π»ΡŒΠ½ΠΎ ΠΎΠΏΡ€Π΅Π΄Π΅Π»Π΅Π½Π°, Ρ‚.Π΅. являСтся Π΅Π²ΠΊΠ»ΠΈΠ΄ΠΎΠ²ΠΎΠΉ ΠΌΠ΅Ρ‚Ρ€ΠΈΠΊΠΎΠΉ. Π’Π΅ΠΏΠ΅Ρ€ΡŒ, ΠΈΡΠΏΠΎΠ»ΡŒΠ·ΡƒΡ Ρ€Π΅ΠΊΡƒΡ€ΡΠΈΠ²Π½ΡƒΡŽ Ρ„ΠΎΡ€ΠΌΡƒΠ»Ρƒ ΠΈΠ· $(\ref{eq:38})$, Π»Π΅Π³ΠΊΠΎ Π΄ΠΎΠΊΠ°Π·Π°Ρ‚ΡŒ ΠΏΠΎ ΠΈΠ½Π΄ΡƒΠΊΡ†ΠΈΠΈ, Ρ‡Ρ‚ΠΎ для всСх $j$ Π΄ΠΎΠ»ΠΆΠ½ΠΎ Π²Ρ‹ΠΏΠΎΠ»Π½ΡΡ‚ΡŒΡΡ + +$$\alpha_j^2 = \frac{2}{\sqrt{4-b^2}}, \;\; \alpha_j\alpha_{j+1} = \frac{b}{\sqrt{4-b^2}}, \;\; [\alpha_j, \:\alpha_{j+1}] = 1 \notag$$ + +ΠžΡ‚ΡΡŽΠ΄Π° слСдуСт, Ρ‡Ρ‚ΠΎ всС стороны нашСго $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$ ΠΈΠΌΠ΅ΡŽΡ‚ ΠΎΠ΄ΠΈΠ½Π°ΠΊΠΎΠ²ΡƒΡŽ Π΄Π»ΠΈΠ½Ρƒ ΠΈ Ρ‡Ρ‚ΠΎ Π»ΡŽΠ±Ρ‹Π΅ Π΄Π²Π΅ смСТныС стороны ΠΏΠ΅Ρ€Π΅ΡΠ΅ΠΊΠ°ΡŽΡ‚ΡΡ ΠΏΠΎΠ΄ ΠΎΠ΄Π½ΠΈΠΌ ΡƒΠ³Π»ΠΎΠΌ. Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, $r$-ΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊ Π΄ΠΎΠ»ΠΆΠ΅Π½ Π±Ρ‹Ρ‚ΡŒ ΠΏΡ€Π°Π²ΠΈΠ»ΡŒΠ½Ρ‹ΠΌ ΠΎΡ‚Π½ΠΎΡΠΈΡ‚Π΅Π»ΡŒΠ½ΠΎ этой ΠΌΠ΅Ρ‚Ρ€ΠΈΠΊΠΈ. + +Π­Ρ‚ΠΎ Π΄ΠΎΠΊΠ°Π·Ρ‹Π²Π°Π΅Ρ‚ Π΅Π΄ΠΈΠ½ΡΡ‚Π²Π΅Π½Π½ΠΎΡΡ‚ΡŒ максимального ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ°. Π•Π³ΠΎ сущСствованиС слСдуСт ΠΈΠ· ограничСнности $Z(K)$ ΠΈ нСпрСрывности $F$ ΠΈ $Z$. Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, ΡƒΡ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 2 ΠΏΠΎΠ»Π½ΠΎΡΡ‚ΡŒΡŽ Π΄ΠΎΠΊΠ°Π·Π°Π½ΠΎ. + +## 3. Π‘Π»ΡƒΡ‡Π°ΠΉΠ½ΠΎ Π²Ρ‹Π±Ρ€Π°Π½Π½Ρ‹Π΅ Ρ‚ΠΎΡ‡ΠΊΠΈ Π² Π²Ρ‹ΠΏΡƒΠΊΠ»ΠΎΠΉ области с Π³Π»Π°Π΄ΠΊΠΎΠΉ Π³Ρ€Π°Π½ΠΈΡ†Π΅ΠΉ + +ΠŸΡƒΡΡ‚ΡŒ Ρ‚Π΅ΠΏΠ΅Ρ€ΡŒ $K$ - выпуклая ΠΎΠ±Π»Π°ΡΡ‚ΡŒ, Π³Ρ€Π°Π½ΠΈΡ†Π° ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ ΠΈΠΌΠ΅Π΅Ρ‚ Π½Π΅ΠΏΡ€Π΅Ρ€Ρ‹Π²Π½ΡƒΡŽ ΠΊΡ€ΠΈΠ²ΠΈΠ·Π½Ρƒ ΠΊΠ°ΠΊ Ρ„ΡƒΠ½ΠΊΡ†ΠΈΡŽ ΠΎΡ‚ Π΄Π»ΠΈΠ½Ρ‹ Π΄ΡƒΠ³ΠΈ. Π’ обозначСниях ΠΏΡ€Π΅Π΄Ρ‹Π΄ΡƒΡ‰ΠΈΡ… ΠΏΠ°Ρ€Π°Π³Ρ€Π°Ρ„ΠΎΠ² ΠΌΡ‹ снова ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ + +$$E_n = C_n^2 \cdot W_n \notag$$ + +Богласно ΡƒΡ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΡŽ ΠΎ ΠΌΠ΅Ρ€Π΅ мноТСства ΠΏΠ°Ρ€ Ρ‚ΠΎΡ‡Π΅ΠΊ, извСстному ΠΈΠ· ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π»ΡŒΠ½ΠΎΠΉ Π³Π΅ΠΎΠΌΠ΅Ρ‚Ρ€ΠΈΠΈ (см. Blascke [2], S 17, (86)): + +$$dP_1 dP_2 = |t_1 - t_2| \, dt_1 \, dt_2 \, d\varphi \, dp \label{eq:41} \tag{41}$$ + +Π³Π΄Π΅ $p$ -- Π΄Π»ΠΈΠ½Π° пСрпСндикуляра ΠΊ прямой $P_1 P_2$ ΠΈΠ· Π½Π°Ρ‡Π°Π»Π° ΠΊΠΎΠΎΡ€Π΄ΠΈΠ½Π°Ρ‚, $\varphi$ -- ΡƒΠ³ΠΎΠ» Π½Π°ΠΊΠ»ΠΎΠ½Π° этого пСрпСндикуляра, $t_1, \: t_2$ -- расстояния ΠΏΠΎ этой прямой ΠΎΡ‚ $P_1$ ΠΈ $P_2$ Π΄ΠΎ Ρ‚ΠΎΡ‡ΠΊΠΈ пСрСсСчСния $P_1 P_2$ с пСрпСндикуляром. ΠžΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Ρ‡Π΅Ρ€Π΅Π· $f$ ΠΏΠ»ΠΎΡ‰Π°Π΄ΡŒ участка, отсСкаСмого ΠΎΡ‚ ΠΌΠ½ΠΎΠ³ΠΎΡƒΠ³ΠΎΠ»ΡŒΠ½ΠΈΠΊΠ° $K$ прямой $P_1 P_2$. Π’ΠΎΠ³Π΄Π° ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ + +$$E_n = \cfrac{C_n^2}{F^2} \int\limits_0^{2\pi} \left\{ \int\limits_0^{p(\varphi)} \left(1-\frac{f}{F} \right)^{n-2} ( \int \int |t_1 - t_2| \, dt_1 \, dt_2) \, dp \right\} d\varphi \label{eq:42} \tag{42}$$ + +Π§Π΅Ρ€Π΅Π· $l(p, \phi)$ ΠΎΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Π΄Π»ΠΈΠ½Ρƒ Ρ…ΠΎΡ€Π΄Ρ‹ области $K$, Π»Π΅ΠΆΠ°Ρ‰Π΅ΠΉ Π½Π° прямой с ΠΊΠΎΠΎΡ€Π΄ΠΈΠ½Π°Ρ‚Π°ΠΌΠΈ $p, \varphi$. Π’ΠΎΠ³Π΄Π° + +$$\int \int |t_1 - t_2| \, dt_1 \, dt_2 = \frac{l^3(p, \varphi)}{3} \label{eq:43} \tag{43}$$ + +Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, + +$$E_n = \cfrac{C_n^2}{3F^2} \int\limits_0^{2\pi} \left\{ \int\limits_0^{p(\varphi)} \left(1-\frac{f}{F} \right)^{n-2} l^3(p, \varphi) \, dp \right\} \, d\varphi \label{eq:44} \tag{44}$$ + +ΠŸΡ€ΠΈ расчСтС асимптотичСского значСния $E_n$ ΠΌΡ‹ ΠΌΠΎΠΆΠ΅ΠΌ, ΠΎΡ‡Π΅Π²ΠΈΠ΄Π½ΠΎ, ΠΎΠ³Ρ€Π°Π½ΠΈΡ‡ΠΈΡ‚ΡŒΡΡ ΠΏΠ°Ρ€Π°ΠΌΠΈ Π·Π½Π°Ρ‡Π΅Π½ΠΈΠΉ $(p, \: \varphi)$, для ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… выполняСтся $f/F < \varepsilon$; Π΄Ρ€ΡƒΠ³ΠΈΠ΅ ΠΏΠ°Ρ€Ρ‹ Π·Π½Π°Ρ‡Π΅Π½ΠΈΠΉ Π΄Π°ΡŽΡ‚ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ остаточный Ρ‡Π»Π΅Π½ порядка $n^2(1-\varepsilon)^n = o(1)$. Рассмотрим Ρ‚Π΅ΠΏΠ΅Ρ€ΡŒ фиксированноС Π·Π½Π°Ρ‡Π΅Π½ΠΈΠ΅ $\varphi$. ΠŸΡƒΡΡ‚ΡŒ $P(\varphi)$ - Ρ‚ΠΎΡ‡ΠΊΠ°, Π² ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ ΠΊΠ°ΡΠ°Ρ‚Π΅Π»ΡŒΠ½Π°Ρ ΠΊ $K$ ΠΈΠΌΠ΅Π΅Ρ‚ ΡƒΠ³ΠΎΠ» Π½Π°ΠΊΠ»ΠΎΠ½Π° $\varphi$. Как становится ясно ΠΈΠ· Π½ΠΈΠΆΠ΅ΡΠ»Π΅Π΄ΡƒΡŽΡ‰ΠΈΡ… ΠΎΡ†Π΅Π½ΠΎΠΊ, ΠΌΠΎΠΆΠ½ΠΎ Π·Π°ΠΌΠ΅Π½ΠΈΡ‚ΡŒ $f$ ΠΈ $l$ Π½Π° ΡΠΎΠΎΡ‚Π²Π΅Ρ‚ΡΡ‚Π²ΡƒΡŽΡ‰ΠΈΠ΅ Π²Π΅Π»ΠΈΡ‡ΠΈΠ½Ρ‹ $\bar f$ ΠΈ $\bar l$ окруТности ΠΊΡ€ΠΈΠ²ΠΈΠ·Π½Ρ‹ Π² Ρ‚ΠΎΡ‡ΠΊΠ΅ $P(\varphi)$. ΠŸΡƒΡΡ‚ΡŒ $\alpha$ - Ρ†Π΅Π½Ρ‚Ρ€Π°Π»ΡŒΠ½Ρ‹ΠΉ ΡƒΠ³ΠΎΠ», ΡΠΎΠΎΡ‚Π²Π΅Ρ‚ΡΡ‚Π²ΡƒΡŽΡ‰ΠΈΠΉ Ρ…ΠΎΡ€Π΄Π΅ $l$, ΠΈ $r(\varphi)$ - радиус окруТности ΠΊΡ€ΠΈΠ²ΠΈΠ·Π½Ρ‹ Π² Ρ‚ΠΎΡ‡ΠΊΠ΅ $P(\varphi)$. Π’ΠΎΠ³Π΄Π° + +$$\bar l = 2 \, r(\varphi) \, \sin(\alpha/2) \quad \text{ΠΈ} \quad \bar f = \frac12 (\alpha - \sin\alpha) \, r^2(\varphi) \label{eq:45} \tag{45}$$ + +ΠšΡ€ΠΎΠΌΠ΅ Ρ‚ΠΎΠ³ΠΎ, + +$$|dp| = \frac12 r(\varphi) \sin(\alpha/2) |d\alpha| \label{eq:46} \tag{46}$$ + +НаконСц, + +$$l - \bar l = o(\alpha^2) \quad \text{ΠΈ} \quad f - \bar f = o(\alpha^4) \label{eq:47} \tag{47}$$ + +Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ Ρ€Π΅Π·ΡƒΠ»ΡŒΡ‚Π°Ρ‚ для $E_n$: + +$$E_n = \cfrac{C_n^2}{12F^2} \int\limits_0^{2\pi} r^4 \int\limits_0^{\alpha(\varphi)} \left( 1 - \frac{\alpha^3r^2}{12F^2} + o(\alpha^3) \right)^{n-2} (\alpha^4 + o(\alpha^5)) \, d\alpha \, d\varphi \label{eq:48} \tag{48}$$ + +ΠŸΠΎΠ΄ΡΡ‚Π°Π²ΠΈΠ² + +$$\frac{\alpha^3r^2}{12F} = \frac{X}{n} \label{eq:49} \tag{49}$$ + +ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ + +$$S(\varphi) = \int\limits_0^{\alpha(\varphi)} \left(1 - \frac{\alpha^3r^2}{12F} + o(\alpha^3)\right)^{n-2} (\alpha^4 + o(\alpha^5)) \, d\alpha \label{eq:50} \tag{50}$$ +$$S(\varphi) = \frac13 \left(\frac{12F}{r^2 n}\right)^{5/3} \left(\int\limits_0^\infty \left(1 - \frac{X}{n}\right)^{n-2} x^{2/3} \, dx + o(1)\right) = \\ = \frac13 \, \Gamma \left( \frac{5}{3} \right) \left(\frac{12F}{r^2 n}\right)^{5/3} (1 + o(1)) \notag$$ + +Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ + +$$E_n \sim \Gamma \left(\frac{5}{3}\right) \, \sqrt[3]{\frac{2n}{3F}} \, \int\limits_0^{2\pi} r^{2/3} \, d\varphi \label{eq:51} \tag{51}$$ + +ΠžΠ±ΠΎΠ·Π½Π°Ρ‡ΠΈΠΌ Π·Π° $s$ Π΄Π»ΠΈΠ½Ρƒ Π΄ΡƒΠ³ΠΈ Π³Ρ€Π°Π½ΠΈΡ†Ρ‹ $K$ ΠΈ, считая $d\varphi/ds = 1/r$, ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ ΡΠ»Π΅Π΄ΡƒΡŽΡ‰Π΅Π΅ ΡƒΡ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅: + +**Π£Ρ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 3.** Если $K$ - выпуклая ΠΎΠ±Π»Π°ΡΡ‚ΡŒ, граничная кривая ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ ΠΈΠΌΠ΅Π΅Ρ‚ Π½Π΅ΠΏΡ€Π΅Ρ€Ρ‹Π²Π½ΡƒΡŽ ΠΊΡ€ΠΈΠ²ΠΈΠ·Π½Ρƒ $x = x(s)$, Ρ‚ΠΎ + +$$E_n \sim \Gamma \left(\frac{5}{3}\right) \, \sqrt[3]{\frac23} \, (F^{-1/3} \int\limits_L x^{1/3} \, ds) \sqrt[3]{n} \label{eq:52} \tag{52}$$ + +Π˜Π½Ρ‚Π΅Π³Ρ€Π°Π» $\int\limits_L x^{1/3} \, ds$, входящий Π² $(\ref{eq:52})$, являСтся Π°Ρ„Ρ„ΠΈΠ½Π½ΠΎΠΉ Π΄Π»ΠΈΠ½ΠΎΠΉ ΠΊΡ€ΠΈΠ²ΠΎΠΉ $L$. Π›Π΅Π³ΠΊΠΎ Π²ΠΈΠ΄Π΅Ρ‚ΡŒ, Ρ‡Ρ‚ΠΎ ΠΌΠ½ΠΎΠΆΠΈΡ‚Π΅Π»ΡŒ $\sqrt[3]{n}$ являСтся Π°Ρ„Ρ„ΠΈΠ½Π½ΠΎ-ΠΈΠ½Π²Π°Ρ€ΠΈΠ°Π½Ρ‚Π½Ρ‹ΠΌ. Из Π°Ρ„Ρ„ΠΈΠ½Π½ΠΎΠ³ΠΎ изопСримСтричСского свойства эллипсов (см. [3]) нСпосрСдствСнно слСдуСт, Ρ‡Ρ‚ΠΎ этот ΠΌΠ½ΠΎΠΆΠΈΡ‚Π΅Π»ΡŒ максималСн для эллипсов ΠΈ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ для эллипсов. + +## 4. ΠΠΎΡ€ΠΌΠ°Π»ΡŒΠ½ΠΎΠ΅ распрСдСлСниС случайных Ρ‚ΠΎΡ‡Π΅ΠΊ + +Π’ этом ΠΏΠ°Ρ€Π°Π³Ρ€Π°Ρ„Π΅ ΠΌΡ‹ Π΄ΠΎΠΊΠ°ΠΆΠ΅ΠΌ ΡΠ»Π΅Π΄ΡƒΡŽΡ‰Π΅Π΅ ΡƒΡ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅: + +**Π£Ρ‚Π²Π΅Ρ€ΠΆΠ΄Π΅Π½ΠΈΠ΅ 4.** ΠŸΡƒΡΡ‚ΡŒ Ρ‚ΠΎΡ‡ΠΊΠΈ $P_i \:(i = 1, \:2, \:\ldots, \:n)$ Π²Ρ‹Π±ΠΈΡ€Π°ΡŽΡ‚ΡΡ Π½Π° плоскости нСзависимо Π΄Ρ€ΡƒΠ³ ΠΎΡ‚ Π΄Ρ€ΡƒΠ³Π° Π² соотвСтствии с Π½ΠΎΡ€ΠΌΠ°Π»ΡŒΠ½Ρ‹ΠΌ распрСдСлСниСм. Π’ΠΎΠ³Π΄Π° + +$$E_n \sim 2 \sqrt{2 \pi \log n} \label{eq:53} \tag{53}$$ + +**Π”ΠΎΠΊΠ°Π·Π°Ρ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ:** +Π’Π°ΠΊ ΠΊΠ°ΠΊ $E_n$ ΠΈΠ½Π²Π°Ρ€ΠΈΠ°Π½Ρ‚Π½ΠΎ ΠΎΡ‚Π½ΠΎΡΠΈΡ‚Π΅Π»ΡŒΠ½ΠΎ Π°Ρ„Ρ„ΠΈΠ½Π½Ρ‹Ρ… ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΠΎΠ²Π°Π½ΠΈΠΉ плоскости, ΠΌΠΎΠΆΠ½ΠΎ ΡΡ‡ΠΈΡ‚Π°Ρ‚ΡŒ, Ρ‡Ρ‚ΠΎ функция плотности нашСго Π½ΠΎΡ€ΠΌΠ°Π»ΡŒΠ½ΠΎΠ³ΠΎ распрСдСлСния ΠΈΠΌΠ΅Π΅Ρ‚ Π²ΠΈΠ΄ $\frac{1}{2\pi} \, exp \left(-\frac{x^2 + y^2}{2}\right) \notag$. Как ΠΈ Π² ΠΏΡ€Π΅Π΄Ρ‹Π΄ΡƒΡ‰ΠΈΡ… ΠΏΠ°Ρ€Π°Π³Ρ€Π°Ρ„Π°Ρ…, + +$$E_n = C_n^2 \cdot W_n \notag$$ + +Π³Π΄Π΅ $W_n$ - Π²Π΅Ρ€ΠΎΡΡ‚Π½ΠΎΡΡ‚ΡŒ Ρ‚ΠΎΠ³ΠΎ, Ρ‡Ρ‚ΠΎ всС Ρ‚ΠΎΡ‡ΠΊΠΈ $P_3, \:\ldots, \:P_n$ Π»Π΅ΠΆΠ°Ρ‚ ΠΏΠΎ ΠΎΠ΄Π½Ρƒ сторону ΠΎΡ‚ прямой $P_1 P_2$. +Из Ρ„ΠΎΡ€ΠΌΡƒΠ»Ρ‹ ΠΈΠ½Ρ‚Π΅Π³Ρ€Π°Π»ΡŒΠ½ΠΎΠΉ Π³Π΅ΠΎΠΌΠ΅Ρ‚Ρ€ΠΈΠΈ + +$$dP_1dP_2 = |t_1 - t_2| \, dt_1 \, dt_2 \, dp \, d\varphi \notag$$ + +ΠΈ ΠΈΠ·-Π·Π° ΠΊΡ€ΡƒΠ³ΠΎΠ²ΠΎΠΉ симмСтрии Π½ΠΎΡ€ΠΌΠ°Π»ΡŒΠ½ΠΎΠ³ΠΎ распрСдСлСния ΠΏΠΎΠ»ΡƒΡ‡Π°Π΅ΠΌ + +$$W_n = \frac{1}{2\pi} \iiint (g(p)^{n-2} + (1 - g(p))^{n-2}) \, |t_1 - t_2| \, exp \left(-p^2 - \frac{t_1^2 + t_2^2}{2}\right) \, dt_1 \, dt_2 \, dp \label{eq:54} \tag{54}$$ + +$$\text{Π³Π΄Π΅} \: g(p) = \int\limits_{-\infty}^{+\infty} \int\limits_{P}^{+\infty} \frac{1}{2\pi} \, exp\left(-\frac{x^2+y^2}{2}\right) \, dx \, dy = 1 - \varPhi(p) \label{eq:55} \tag{55}$$ + +$$\varPhi(p) = \frac{1}{\sqrt{2\pi}} \int\limits_{-\infty}^p exp\left(-\frac{u^2}{2}\right) \, du \label{eq:56} \tag{56}$$ + +Π§Π΅Ρ€Π΅Π· Π·Π°ΠΌΠ΅Π½Ρƒ + +$$u = \frac{t_1 - t_2}{\sqrt 2}, \: v = \frac{t_1 + t_2}{\sqrt 2}, \: \frac{\partial(t_1, t_2)}{\partial(u, v)} = 1 \label{eq:57} \tag{57}$$ + +ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ + +$$\frac{1}{2\pi} \int\int |t_1 - t_2| \, exp\left(-\frac{t_1^2 + t_2^2}{2}\right) \, dt_1 \, dt_2 = \frac{2}{\sqrt \pi} \label{eq:58} \tag{58}$$ + +Π‘Π»Π΅Π΄ΠΎΠ²Π°Ρ‚Π΅Π»ΡŒΠ½ΠΎ, + +$$W_n = \frac{2}{\sqrt \pi} \int\limits_0^\infty [\varPhi^{n-2}(p) + (1 - \varPhi(p))^{n-2}] e^{-p^2} dp \label{eq:59} \tag{59}$$ + +ΠΈ, ΠΎΡ‡Π΅Π²ΠΈΠ΄Π½ΠΎ, $1 - \varPhi(p) \leqq 1/2$ для $p \geqq 0$. А Π·Π½Π°Ρ‡ΠΈΡ‚, + +$$W_n \sim \frac{2}{\sqrt \pi} \int\limits_0^\infty \varPhi^{n-2}(p) e^{-p^2} dp \label{eq:60} \tag{60}$$ + +Π’Π΅ΠΏΠ΅Ρ€ΡŒ для $p > 1$ + +$$\varPhi(p) = 1 - \frac{exp \left(-\frac{p^2}{2}\right)}{\sqrt{2\pi}p\left(1 + \frac{\theta p}{p^2}\right)} \label{eq:61} \tag{61}$$ + +Π³Π΄Π΅ $0 < \theta_p < 1$ (см. [4], стр. 136, Π·Π°Π΄Π°Ρ‡Π° 18). + +Π’Ρ‹ΠΏΠΎΠ»Π½ΠΈΠΌ подстановку + +$$p = \sqrt{2\log n - \log(2\log n)} + \frac{u - \log \sqrt{2\pi}}{\sqrt{2 \log n}} \label{eq:62} \tag{62}$$ + +ΠΈ ΠΏΠΎΠ»ΡƒΡ‡ΠΈΠΌ + +$$W_n \sim \frac{4\sqrt{2\pi\log n}}{n^2}$$ + +Π’Π°ΠΊΠΈΠΌ ΠΎΠ±Ρ€Π°Π·ΠΎΠΌ, $E_n \sim 2\sqrt{2\pi \log n}$, Ρ‡Ρ‚ΠΎ ΠΈ Ρ‚Ρ€Π΅Π±ΠΎΠ²Π°Π»ΠΎΡΡŒ Π΄ΠΎΠΊΠ°Π·Π°Ρ‚ΡŒ. + +## Π›ΠΈΡ‚Π΅Ρ€Π°Ρ‚ΡƒΡ€Π° + +1) A. RΓ©nyi, R. Sulanke: Über die konvexe HΓΌlle von n zufΓ€llig gewΓ€hlten Punkten. Z. Wahrscheinlichkeitstheorie 2 (1963), 75--84 +2) W. Blaschke: Integralgeometrie. 3. Aufl. Berlin: VEB Deutscher Verlag der Wissenschaften 1955, 1--130. +3) W. Blaschke, K. Reidemeister: Differentialgeometrie II, Berlin: Springer 1923, 60. +4) A. RΓ©nyi: Wahrscheinlichkeitsrechnung, mit einem Anhang ΓΌber Informationstheorie. Berlin: VEB Deutscher Verlag der Wissenschaften 1962, 1--547. \ No newline at end of file