[mathjax]<\/p>\n
…. \u7ed9\u5b9a\u4e00\u4e2a\u5355\u4f4d\u5706\u5468\uff0c\u968f\u673a\u5728\u91cc\u9762\u70b9 $latex n$ \u4e2a\u70b9\u3002\u3002\u786e\u7acb $latex n$ \u6bb5\u5706\u5f27\u3002
\n\u3002\u3002\u6c42\u7b2c $latex k$ \u77ed\u7684\u5706\u5f27\u7684\u957f\u5ea6\u7684\u671f\u671b\u3002
\n<\/p>\n
\u3002\u3002\u3002\u7528\u4ee3\u6570\u8bed\u8a00\u53d9\u8ff0\u5c31\u662f\uff1a
\n\u3002\u3002n \u4e2a\u53d8\u91cf $$x_1, x_2, \\ldots, x_n$$ \uff0c\u6ee1\u8db3 $$0 \\leq x_i$$ \u4e14 $$\\sum_{i=1}^{n} x_i = 1 $$ \u3002\u6c42 $$x_i$$ \u4e2d\u7b2c $$k$$ \u5c0f\u7684\u671f\u671b\u3002<\/p>\n
fotile \u63d0\u51fa\u7684\u4e00\u4e2a\u95ee\u9898\u3002\u3002\u3002\u6211\u524d\u4e00\u9635\u5b50\u4e5f\u4e5f\u8003\u8651\u8fc7\u3002\u3002\u5f53\u65f6\u5f97\u51fa\u7684\u7ed3\u8bba\u662f\u3002\u3002\u628a\u5706\u5468\u63a8\u5e7f\u5230\u9ad8\u7ef4\u7403\u9762\u540e\u3002\u3002\u548c\u4e8c\u7ef4\u60c5\u51b5\u4e0d\u5728\u4e00\u4e2a\u96be\u5ea6\u7b49\u7ea7\u3002\u3002\u3002
\n\u3002\u3002\u3002\u3002\u73b0\u5728\u4e8c\u7ef4\u5df2\u7ecf\u5f97\u5230\u4e86\u6bd4\u8f83\u597d\u7684\u5904\u7406\u3002\u3002\u3002\u8fd9\u91cc\u8bb0\u5f55\u4e00\u4e0b\u3002\u3002\u3002<\/p>\n
\u3002\u3002\u3002\u4f5c\u4e3a “k-th xxx in circle” \u7cfb\u5217\u7b2c\u4e8c\u5f39\u3002\u3002\u3002 $$!P_n (d_1\\geq x) = (1 – nx)^{n-1} $$<\/p>\n \u5fae\u5206\u5f97\u5230\u5bc6\u5ea6\u51fd\u6570\u3002\u3002\u3002<\/p>\n $$!P_n (d_1=x) = (n-1)(1 – nx)^{n-2} $$<\/p>\n \u4e58\u4ee5 $$x$$ \u79ef\u5206\u641e\u51fa\u671f\u671b\u3002\u3002\u6ce8\u610f\u8981\u4f7f\u5f97 $$(1 – nx)$$ \u6709\u610f\u4e49\u3002\u3002\u79ef\u5206\u4e0a\u9650\u53ea\u80fd\u5230 $$\\frac{1}{n}$$<\/p>\n $$!\\begin{align} E_n(d_1) &= \\int_0^{\\frac{1}{n}} xP_n(d_1 = x) \\mathrm{d}x\ &= \\int_0^{\\frac{1}{n}} x(n-1)(1-nx)^{n-2}\\mathrm{d}x \ &= \\frac{1-n}{n} \\int_0^1 nx(1-nx)^{n-2} \\mathrm{d}(1-nx) \ &= \\frac{n-1}{n}\\int_0^1 (1-x)x^{n-2} \\mathrm{d}x \ &= \\frac{n-1}{n} (\\frac{1}{n-1} – \\frac{1}{n}) \ &= \\frac{1}{n^2} \\end{align<\/em>} $$<\/p>\n \u3002\u3002\u3002\u5bf9\u4e8e\u3002\u3002\u3002$$E_n(d_k) $$ … \u679a\u4e3e $$d_1$$ \u7684\u503c\u3002\u3002\u8bbe\u4e3a $$x$$\u3002\u3002\u4e3a\u4e86\u4fdd\u8bc1\u5269\u4e0b $$n-1$$ \u4e2a\u6570\u6bd4 $$x$$ \u5927\u3002\u3002\u6211\u4eec\u540c\u6837\u628a\u8fd9\u4e9b\u503c\u9884\u5b58\u8d77\u6765\u3002\u3002\u5f97\u5230\u5269\u4e0b $$n-1$$ \u4e2a\u53d8\u91cf\u7684\u548c\u662f $$1 – nx$$ \u3002\u3002\u5e76\u4e14\u4ecd\u7136\u4fdd\u6301\u5747\u5300\u5206\u5e03\u3002\u3002\u53ef\u4ee5\u7528 $$E_{n-1}(d_{k-1})$$ \u8868\u8fbe\u3002\u3002\u3002\u4e8e\u662f\u6b64\u65f6\u7b2c $$k$$ \u5c0f\u7684\u671f\u671b\u662f $$x + (1-nx)E_{n-1}(d_{k-1})$$ \u3002\u3002\u518d\u4e58\u4ee5 $$x$$ \u5bf9\u5e94\u7684\u6982\u7387\u3002\u3002\u3002\u4e5f\u5c31\u662f\u3002\u3002\u3002<\/p>\n $$!\\begin{align} E_n(d_k) &= \\int_0^{\\frac{1}{n}} (x + (1-nx)E_{n-1}(d_{k-1})) P_n(d_1=x) \\mathrm{d}x \ &= \\int_0^{\\frac{1}{n}} x P_n(d_1=x) \\mathrm{d}x + E_{n-1}(d_{k-1})\\int_0^{\\frac{1}{n}} (1-nx)P_n(d_1=x) \\mathrm{d}x \ &= \\frac{1}{n^2} + \\frac{n-1}{n} E_{n-1}(d_{k-1}) \\end{align<\/em>} $$<\/p>\n \u3002\u3002\u4ee5\u4e0a\u5f97\u5230\u4e86\u9012\u63a8\u5f0f\u3002\u3002\u3002<\/p>\n \u3002\u3002\u3002\u6700\u540e\u5c06\u9012\u63a8\u5f0f\u5c55\u5f00\u5f97\u5230 close-form \u3002\u3002\u3002<\/p>\n $$! E_n(d_k) = \\frac{1}{n} \\sum_{i=n-k}^{n} \\frac{1}{i} = \\frac{H_n – H_{n-k}}{n} $$<\/p>\n \u3002\u3002\u3002\u8c03\u548c\u7ea7\u6570\u5b9e\u5728\u662f\u592a\u4f18\u7f8e\u4e86\uff01\u3002\u3002\uff09<\/p>\n https:\/\/www.13331.org\/495.html<\/a> [mathjax] Brief description: …. \u7ed9\u5b9a\u4e00\u4e2a\u5355\u4f4d\u5706\u5468\uff0c\u968f\u673a\u5728\u91cc\u9762\u70b9 $latex n$ \u4e2a\u70b9\u3002\u3002\u786e\u7acb $latex n$ \u6bb5\u5706\u5f27\u3002 \u3002\u3002\u6c42\u7b2c $latex k$ \u77ed\u7684\u5706\u5f27\u7684\u957f\u5ea6\u7684\u671f\u671b\u3002<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false}}},"categories":[122],"tags":[],"class_list":["post-808","post","type-post","status-publish","format-standard","hentry","category-122"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p2tdP7-d2","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/posts\/808","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/comments?post=808"}],"version-history":[{"count":1,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/posts\/808\/revisions"}],"predecessor-version":[{"id":809,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/posts\/808\/revisions\/809"}],"wp:attachment":[{"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/media?parent=808"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/categories?post=808"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/tags?post=808"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\u3002\u3002\u548c\u4e4b\u524d<\/a>\u4e00\u6837\u3002\u3002\u8bbe $$d_k$$ \u4e3a\u7b2c $$k$$ \u77ed\u7684\u957f\u5ea6\u3002\u3002\u3002\u5148\u8003\u8651 $$d_1$$ \u7684\u5206\u5e03\u51fd\u6570\u3002\u3002\u3002\u8bbe $$d_1$$ \u81f3\u5c11\u662f $$x$$ \u3002\u90a3\u4e48\u6211\u4eec\u5148\u53d6\u51fa $$nx$$ \u9884\u5b58\u8d77\u6765\u3002\u3002\u3002\u3002\u4e4b\u540e\u5728\u5269\u4e0b\u7684 $$(1 – nx)$$ \u91cc\u3002\u3002\u968f\u673a\u8bbe\u7f6e $$n-1$$ \u4e2a\u70b9\u4ee5\u5f97\u5230 $$n$$ \u6bb5\u5f27\u3002\u3002\u3002<\/p>\nExternal link: <\/h3>\n
\nhttp:\/\/roosephu.github.io\/2013\/09\/02\/kth-min\/<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"