<...",
"image": "https://raineblog.dpdns.org/whk/assets/images/social/science/number/1.png",
"date_modified": "2026-04-05T08:13:00+00:00",
"date_published": "2025-08-19T11:57:00+00:00",
"authors": [],
"tags": null
},
{
"id": "https://raineblog.dpdns.org/whk/science/polynomial/3/",
"url": "https://raineblog.dpdns.org/whk/science/polynomial/3/",
"title": "\u6570\u5217\u8bad\u7ec3",
"content_html": "\u6570\u5217\u8bad\u7ec3 \n\u57fa\u7840\u9898\u76ee \n\u6c42\u4e0b\u5217\u6570\u5217\u7684\u901a\u9879\u516c\u5f0f\u3002
\n\u9898\u76ee\u4e00 \n\u6c42\uff1a$a_n=2a_{n-1}+3(n\\ge2),a_1=1$\u3002
\n\u56e0\u4e3a $q/(p-1)=3$\uff0c\u7b49\u5f0f\u4e24\u8fb9\u540c\u65f6\u52a0\u4e09\uff0c
\n$$\na_n+3=2a_{n-1}+6=2^{n+1}\\\na_n=2^{n+1}-3\n$$
\n\u6ce8\u610f\u5230\u5f53 $n=1,a_1=1$ \u6ee1\u8db3\u8be5\u5f0f\uff0c\u56e0\u6b64\uff0c
\n$$\na_n=2^{n+1}-3\n$$
\n\u9898\u76ee\u4e8c \n\u6c42\uff1a$a_n=a_{n-1}+n(n\\ge2),a_1=1$\u3002
\n\u6ce8\u610f\u5230\uff0c
\n$$\na_n=a_{n-1}+n\\\na_{n-1}=a{n-2}+n-1\\\n\\dots\\\na_2=a_1+2=1+2\n$$
\n\u4e0a\u5f0f\u76f8\u52a0\uff0c\u5f97\uff0c
\n$$\na_n=1+2+3+\\dots+n-1+n={n(n+1)\\over2}\n$$
\n\u6ce8\u610f\u5230\u5f53 $n=1,a_1=1$ \u6ee1\u8db3\u8be5\u5f0f\uff0c\u56e0\u6b64\uff0c
\n$$\na_n={n(n+1)\\over2}\n$$
\n\u9898\u76ee\u4e09 \n\u6c42\uff1a$a_n=2a_{n-1}+n(n\\ge2),a_1=1$\u3002
\n\u7b49\u5f0f\u4e24\u8fb9\u540c\u65f6\u9664\u4ee5 $2^n$\uff0c\u5f97\uff0c
\n$$\n{a_n\\over2^n}={a_...
",
"image": "https://raineblog.dpdns.org/whk/assets/images/social/science/polynomial/3.png",
"date_modified": "2026-04-05T08:40:00+00:00",
"date_published": "2025-08-19T11:57:00+00:00",
"authors": [],
"tags": null
},
{
"id": "https://raineblog.dpdns.org/whk/science/polynomial/4/",
"url": "https://raineblog.dpdns.org/whk/science/polynomial/4/",
"title": "\u591a\u9879\u5f0f\u5165\u95e8",
"content_html": "\u591a\u9879\u5f0f\u5165\u95e8 \n\u57fa\u7840\u65b9\u6cd5 \n\u4e58\u6cd5\u516c\u5f0f \n\u5206\u914d\u7387\uff1a
\n$$\n\\begin{aligned}\n(a+b)(c+d)&=ac+ad+bc+bd\\\n(a+b)(a-b)&=a^2-b^2\n\\end{aligned}\n$$
\n\u548c\u5dee\u65b9\uff1a
\n$$\n\\begin{aligned}\n(a+b)^2&=a^2+2ab+b^2\\\n(a-b)^2&=a^2-2ab+b^2\\\n(a+b)^3&=a^3+3a^2b+3ab^2+b^3\\\n(a-b)^3&=a^3-3a^2b+3ab^2-b^3\\\n(a+b+c)^2&=a^2+b^2+c^2+2ab+2bc+2ac\\\n\\end{aligned}\n$$
\n\u4e8c\u9879\u5f0f\u5b9a\u7406\uff1a
\n$$\n\\begin{aligned}\n(x+y)^n&=\\sum_{k=0}^n{n\\choose k}x^ky^{n-k}\\\n(x+1)^n&=\\sum_{k=0}^n{n\\choose k}x^k\n\\end{aligned}\n$$
\n\u65b9\u548c\u5dee\uff1a
\n$$\n\\begin{aligned}\na^3+b^3&=(a+b)(a^2-ab+b^2)\\\na^3-b^3&=(...
",
"image": "https://raineblog.dpdns.org/whk/assets/images/social/science/polynomial/4.png",
"date_modified": "2026-05-13T14:55:00+00:00",
"date_published": "2025-08-19T11:57:00+00:00",
"authors": [],
"tags": null
},
{
"id": "https://raineblog.dpdns.org/whk/science/function/5/",
"url": "https://raineblog.dpdns.org/whk/science/function/5/",
"title": "\u8fdb\u9636\u5bfc\u6570\u9898\u578b",
"content_html": "\u8fdb\u9636\u5bfc\u6570\u9898\u578b \n\u6052\u6210\u7acb\u95ee\u9898 \n\u51fd\u6570\u6a21\u578b \n\u7b80\u5355\u7684\u6052\u6210\u7acb\u95ee\u9898\uff1a
\n\n\u5bf9\u4e8e\u4efb\u4f55\u5355\u8c03\u51fd\u6570\uff0c\u6700\u503c\u5fc5\u5728\u7aef\u70b9\u5904\u53d6\u5230\uff1a
\n\n\n\u5355\u8c03\u51fd\u6570 $f(x)\\ge0$ \u5728 $[x_1,x_2]$ \u4e0a\u6052\u6210\u7acb\u7b49\u4ef7\u4e8e $\\begin{cases}f(x_1)&\\ge0\\ f(x_2)&\\ge0\\end{cases}$\u3002
\n \n\n\u5355\u8c03\u51fd\u6570 $f(x)\\le0$ \u5728 $[x_1,x_2]$ \u4e0a\u6052\u6210\u7acb\u7b49\u4ef7\u4e8e $\\begin{cases}f(x_1)&\\le0\\ f(x_2)&\\le0\\end{cases}$\u3002
\n \n \n\u5bf9\u4efb\u4f55\u4e00\u4e2a\u51fd\u6570 $f(x)$\uff0c\u82e5\u5b83\u5728\u533a\u95f4\u4e0a\u662f\u5148\u51cf\u540e\u589e\uff0c\u5219\u5176\u6700\u5927\u503c\u5fc5\u5728\u7aef\u70b9\u5904\u53d6\u5f97\uff0c\u540c\u7406\u53ef\u5f97\u82e5\u51fd\u6570\u5728\u533a\u95f4\u4e0a\u5148\u589e\u540e\u51cf\uff0c\u5176\u6700\u5c0f\u503c\u5fc5\u5728\u533a\u95f4\u7aef\u70b9\u5904\u53d6\u5f97\uff1a
\n\n\u82e5 $a > 0$\uff0c\u5219 $f(x) = ax^2 + bx + c \\le 0$ \u5728 $[x_1, x_2]$ \u4e0a\u6052\u6210\u7acb\u7b49\u4ef7\u4e8e $\\begin{cases}f... \n ",
"image": "https://raineblog.dpdns.org/whk/assets/images/social/science/function/5.png",
"date_modified": "2026-05-02T13:04:00+00:00",
"date_published": "2025-08-11T07:26:00+00:00",
"authors": [],
"tags": null
},
{
"id": "https://raineblog.dpdns.org/whk/science/polynomial/5/",
"url": "https://raineblog.dpdns.org/whk/science/polynomial/5/",
"title": "\u7ec4\u5408\u6570\u5b66",
"content_html": "\u7ec4\u5408\u6570\u5b66 \n\u4f20\u8bf4\u65e9\u5728 $\\texttt{114514}$ \u5e74\u524d\uff0c
\n\u5979\u89c9\u5f97\u8fd9\u5f88\u6709\u8da3\uff0c\u4e3a\u4e86\u52a0\u901f\u4eba\u7c7b\u6587\u660e\u7684\u53d1\u5c55\uff0c
\n\u800c\u53ea\u6709\u641e\u660e\u767d\u8fd9\u7c7b\u95ee\u9898\uff0c\u624d\u80fd\u5728\u7ec4\u5408\u6570\u5b66\u4e0a\u7ee7\u7eed\u6df1\u5165\u3002
\n\n\u57fa\u672c\u539f\u7406 \n\u6c42\u8bc1\u4e0d\u7b49\u5f0f\u7684\u65b9\u6cd5\u6709\u51e0\u79cd\uff0c\u6700\u7b80\u5355\u7684\u662f\uff1a
\n\n\n\u8bbe\u51fd\u6570\uff0c\u8bc1\u660e\u51fd\u6570\u6052\u5927\u4e8e\u6216\u5c0f\u4e8e\u96f6\u3002
\n \n\n\u76f4\u63a5\u5bf9\u51fd\u6570\u6c42\u5bfc\uff0c\u5c1d\u8bd5\u8bc1\u660e\u51fd\u6570\u6700\u5c0f\u503c\u5927\u4e8e\u96f6\uff0c\u6216\u6700\u5927\u503c\u5c0f\u4e8e\u96f6\u3002
\n \n\n\u6ce8\u610f\u5230\u591a\u9879\u5f0f\u53ea\u8981\u6c42\u591f\u591a\u6b21\u6570\u591a\u5bfc\u6570\uff0c\u4e00\u5b9a\u4f1a\u53d8\u4e3a\u96f6\uff0c\u56e0\u6b64\u5982\u679c\u4e0d\u597d\u89e3\u51b3\uff0c\u7ee7\u7eed\u6c42\u5bfc\uff0c\u5bf9\u9ad8\u9636\u5bfc\u6570\u5c1d\u8bd5\u5206\u6790\u5176\u662f\u5426\u6052\u6b63\u6216\u6052\u8d1f\u3002
\n \n \n\u6ce8\u610f\uff1a
\n\n\n\u89e3\u4e00\u5143\u4e00\u6b21\u4e0d\u7b49\u5f0f $ax>b$\uff0c\u9700\u8981\u6309\u7167 $a>0,a=0,a<0$ \u5206\u7c7b\u8ba8\u8bba\u3002
\n \n\n\u89e3\u4e00\u5143\u4e00\u6b21\u4e0d\u7b49\u5f0f $ax>b$\uff0c\u5176\u4e2d $x\\in[m,n]$\uff0c\u5148\u6309\u7167 $a>0,a=0,a<0$ \u5206\u7c7b\u8ba8\u8bba\uff0c\u7136\u540e\u6309\u7167 $\\dfrac{b}{a}$ \u662f\u5426\u843d\u5728\u533a\u95f4 $[m,n]$ \u5185\u3002
\n \n \n\u8fb9\u754c\u6761\u4ef6\uff1a
\n\n\u4e8c\u6b21\u9879\u7cfb\u6570\u4e0d\u542b\u53c2\u6570\u4e14\u81ea\u53d8\u91cf\u6ca1\u6709\u9650\u5236\u65f6\uff0c\u4e34\u754c\u6761\u4ef6\u662f $... \n ",
"image": "https://raineblog.dpdns.org/whk/assets/images/social/science/polynomial/5.png",
"date_modified": "2026-05-02T13:04:00+00:00",
"date_published": "2025-08-11T07:26:00+00:00",
"authors": [],
"tags": null
},
{
"id": "https://raineblog.dpdns.org/whk/science/polynomial/1/",
"url": "https://raineblog.dpdns.org/whk/science/polynomial/1/",
"title": "\u6570\u5217\u57fa\u7840",
"content_html": "\u6570\u5217\u57fa\u7840 \n\u6570\u5217\u7684\u5b9a\u4e49 \n\u6570\u5217 \u662f\u7531\u6570\u5b57\u7ec4\u6210\u7684\u6709\u5e8f\u5e8f\u5217\uff0c\u6570\u5217\u4e2d\u7684\u6bcf\u4e00\u4e2a\u6570\u90fd\u53eb\u505a\u8fd9\u4e2a\u6570\u5217\u7684\u9879 \u3002\u9879\u6570 \u6709\u9650\u7684\u6570\u5217\u6210\u4e3a\u6709\u9650\u6570\u5217 \uff0c\u9879\u6570\u65e0\u7a77\u591a\u7684\u6210\u4e3a\u65e0\u7a77\u6570\u5217 \u3002
\n\u6392\u5728\u7b2c\u4e00\u4f4d\u7684\u6570\u79f0\u4e3a\u8fd9\u4e2a\u6570\u5217\u7684\u9996\u9879 \uff0c\u6709\u9650\u6570\u5217\u7684\u6700\u540e\u4e00\u4e2a\u6570\u6210\u4e3a\u8fd9\u4e2a\u6570\u5217\u7684\u672b\u9879 \u3002\u6ce8\u610f\uff1a\u65e0\u7a77\u6570\u5217\u53ea\u6709\u9996\u9879\uff0c\u6ca1\u6709\u672b\u9879\u3002
\n\u5bf9\u4e8e\u6570\u5217\uff0c\u66f4\u4e25\u8c28\u7684\u5b9a\u4e49\uff0c\u8003\u8651\u6700\u4e00\u822c\u7684\u590d\u6570\uff0c\u4e0b\u6587\u518d\u8bf4\u3002
\n\u65e0\u7a77\u6570\u5217\uff1a
\n\n\n\u4e00\u4e2a $(a:\\mathbb N\\to\\mathbb C)$ \u7684\u51fd\u6570\u88ab\u79f0\u4e3a\u65e0\u7a77\u6570\u5217 \u3002
\n \n\n\u53ef\u8bb0\u4e3a ${a_i}{i\\in\\mathbb N}$ \u6216 $(a_i) {i\\in\\mathbb N}$ \u6216 $\\langle a_i\\rangle_{i\\in\\mathbb N}$\u3002
\n \n\n\u4e00\u4e2a\u6570\u5217 $a$ \u7684\u7b2c $i$ \u9879\uff0c\u901a\u5e38\u8bb0\u4e3a $a(i)$\uff0c\u7b80\u8bb0\u4e3a $a_i$\u3002
\n \n \n\u6709\u9650\u6570\u5217\uff1a
\n\n\n\u82e5 $I_n={1,2,\\dots,n}$\uff0c\u5219\u4e00\u4e2a $(a:I_n\\to\\mathbb C)$ \u7684\u51fd\u6570\u88ab\u79f0\u4e3a\u6709\u9650\u6570\u5217 \u3002
\n \n\n\u53ef\u8bb0\u4e3a ${a_i}{i=1}^n$ \u6216 $(a ...
\n \n ",
"image": "https://raineblog.dpdns.org/whk/assets/images/social/science/polynomial/1.png",
"date_modified": "2026-05-16T12:03:00+00:00",
"date_published": "2025-08-04T03:43:00+00:00",
"authors": [],
"tags": null
},
{
"id": "https://raineblog.dpdns.org/whk/science/polynomial/2/",
"url": "https://raineblog.dpdns.org/whk/science/polynomial/2/",
"title": "\u6570\u5217\u63d0\u9ad8",
"content_html": "\u6570\u5217\u63d0\u9ad8 \n\u56de\u987e\u53ca\u8865\u5145 \n\u7ecf\u5178\u5e38\u91cf \n$$\n\\pi\\approx3.14159\n$$
\n$$\ne\\approx2.71828\n$$
\n$$\n\\gamma\\approx0.57721\n$$
\n$$\n\\varphi={1+\\sqrt5\\over2}\\approx1.61803\n$$
\n$$\n\\hat\\varphi={1-\\sqrt5\\over2}\\approx-.61803\n$$
\n\u57fa\u7840\u516c\u5f0f \n\u4e00\u4e9b\u516c\u5f0f\uff0c
\n$$\n\\sum_{i=1}^ni={n(n+1)\\over2}\n$$
\n$$\n\\sum_{i=1}^ni^2={n(n+1)(2n+1)\\over6}={n(n+1/2)(n+1)\\over3}\n$$
\n$$\n\\sum_{i=1}^ni^3=\\left[{n(n+1)\\over2}\\right]^2={n^2(n+1)^2\\over4}\n$$
\n\u53ef\u4ee5\u901a\u8fc7\u6270\u52a8\u6cd5\uff08\u89c1\u4e0b\uff09\u6216\u8005\u5f85\u5b9a\u7cfb\u6570\u5e76\u5f52\u7eb3\u5f97\u51fa\u3002
\n$$\n\\sum_{i=0}^nc^i={c^{n+1}-1\\over c-1},c\\neq1\n$$
\n$$\n\\sum_{i\\ge0}c^i={1\\over 1-c...
",
"image": "https://raineblog.dpdns.org/whk/assets/images/social/science/polynomial/2.png",
"date_modified": "2026-04-05T08:40:00+00:00",
"date_published": "2025-08-04T03:43:00+00:00",
"authors": [],
"tags": null
},
{
"id": "https://raineblog.dpdns.org/whk/science/function/4/",
"url": "https://raineblog.dpdns.org/whk/science/function/4/",
"title": "\u5bfc\u6570\u4e0e\u4e0d\u7b49\u5f0f",
"content_html": "\u5bfc\u6570\u4e0e\u4e0d\u7b49\u5f0f \n
\n
\n\u5bfc\u6570\u65b9\u6cd5 \n\u4e0d\u7b49\u5f0f\u65b9\u6cd5 \n\u6c42\u8bc1\u4e0d\u7b49\u5f0f\u7684\u65b9\u6cd5\u6709\u51e0\u79cd\uff0c\u6700\u7b80\u5355\u7684\u662f\uff1a
\n\n\n\u8bbe\u51fd\u6570\uff0c\u8bc1\u660e\u51fd\u6570\u6052\u5927\u4e8e\u6216\u5c0f\u4e8e\u96f6\u3002
\n \n\n\u76f4\u63a5\u5bf9\u51fd\u6570\u6c42\u5bfc\uff0c\u5c1d\u8bd5\u8bc1\u660e\u51fd\u6570\u6700\u5c0f\u503c\u5927\u4e8e\u96f6\uff0c\u6216\u6700\u5927\u503c\u5c0f\u4e8e\u96f6\u3002
\n \n\n\u6ce8\u610f\u5230\u591a\u9879\u5f0f\u53ea\u8981\u6c42\u591f\u591a\u6b21\u6570\u591a\u5bfc\u6570\uff0c\u4e00\u5b9a\u4f1a\u53d8\u4e3a\u96f6\uff0c\u56e0\u6b64\u5982\u679c\u4e0d\u597d\u89e3\u51b3\uff0c\u7ee7\u7eed\u6c42\u5bfc\uff0c\u5bf9\u9ad8\u9636\u5bfc\u6570\u5c1d\u8bd5\u5206\u6790\u5176\u662f\u5426\u6052\u6b63\u6216\u6052\u8d1f\u3002
\n \n \n\u6ce8\u610f\uff1a
\n\n\n\u89e3\u4e00\u5143\u4e00\u6b21\u4e0d\u7b49\u5f0f $ax>b$\uff0c\u9700\u8981\u6309\u7167 $a>0,a=0,a<0$ \u5206\u7c7b\u8ba8\u8bba\u3002
\n \n\n\u89e3\u4e00\u5143\u4e00\u6b21\u4e0d\u7b49\u5f0f $ax>b$\uff0c\u5176\u4e2d $x\\in[m,n]$\uff0c\u5148\u6309\u7167 $a>0,a=0,a<0$ \u5206\u7c7b\u8ba8\u8bba\uff0c\u7136\u540e\u6309\u7167 $\\dfrac{b}{a}$ \u662f\u5426\u843d\u5728\u533a\u95f4 $[m,n]$ \u5185\u3002
\n \n \n\u8fb9\u754c\u6761\u4ef6\uff1a
\n\n\n\u4e8c\u6b21\u9879\u7cfb\u6570\u4e0d\u542b\u53c2\u6570\u4e14\u81ea\u53d8\u91cf\u6ca1\u6709\u9650\u5236\u65f6\uff0c\u4e34\u754c\u6761\u4ef6\u662f $\\Delta = 0$\u3002
\n \n\n\u4e8c\u6b21\u9879\u7cfb\u6570\u542b\u6709\u53c2\u6570\u4e14\u81ea\u53d8\u91cf\u6ca1\u6709\u9650\u5236\u65f6\uff0c\u4e34\u754c\u6761\u4ef6\u662f\u4e8c\u6b21\u9879\u7cfb\u6570\u4e3a\u96f6\u4e0e $\\Delta = 0$ \u8054\u7acb\u3002
\n \n\n\u4e8c\u6b21\u9879\u7cfb\u6570\u542b\u53c2\u6570\u4e14\u81ea\u53d8\u91cf\u6709\u9650\u5236\u65f6\uff0c\u4e34\u754c\u6761\u4ef6\u662f\u4e8c\u6b21\u9879\u7cfb\u6570\u4e3a\u96f6...
\n \n ",
"image": "https://raineblog.dpdns.org/whk/assets/images/social/science/function/4.png",
"date_modified": "2026-04-05T08:59:00+00:00",
"date_published": "2025-07-29T06:08:00+00:00",
"authors": [],
"tags": null
},
{
"id": "https://raineblog.dpdns.org/whk/science/geometry/4/",
"url": "https://raineblog.dpdns.org/whk/science/geometry/4/",
"title": "\u5706\u9525\u622a\u7ebf\u8be6\u89e3\uff1a\u79bb\u5fc3\u7387\u673a\u5236\u4e0e\u5929\u4f53\u8f68\u9053\u5e94\u7528\u573a\u666f",
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"title": "\u5747\u503c\u4e0d\u7b49\u5f0f",
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\n$$\na>b>0,m>0\\implies\\dfrac{b+m}{a+m}>\\dfrac{b}{a}\n$$
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\n\u4e0d\u7b49\u5f0f\u52a0\u6cd5\uff1a
\n$$\na>b,c>d\\implies a+c>b+d\n$$
\n\u4e0d\u7b49\u5f0f\u51cf\u6cd5\uff1a
\n$$\na>b,cb-d\n$$
\n\u4e0d\u7b49\u5f0f\u8054\u7acb\uff1a
\n$$\n\\begin{cases}\na_1<x+y<a_2\\\nb_1<x-y<b_2\n\\end{cases}\\implies\\begin{cases}\na_1+b_1<2x<a_2+b_2\\\na_1-b_1<2y<a_2-b_2\n\\end{cases}\n$$
\n\u7b49\u5f0f\u7684\u6027\u8d28\u200b\uff1a
\n\n$a=a$\uff08\u81ea\u53cd\u6027\uff09 \n\u200c$a=b\\Rightarrow b=a$\uff08\u5bf9\u79f0\u6027\uff09 \n$a=b,b=c\\Rightarrow a=c$\uff08\u4f20\u9012\u6027\uff09 \n$a=b\\Rightarrow a\\pm c=b\\pm c,ac... \n ",
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\n\n\u78b3\u539f\u5b50\u4f4d\u4e8e\u6b63\u56db\u9762\u4f53\u7684\u4e2d\u5fc3\u3002 \n$4$ \u4e2a\u6c22\u539f\u5b50\u5206\u522b\u4f4d\u4e8e $4$ \u4e2a\u9876\u70b9\u3002 \n$4$ \u4e2a $\\ce{C - H}$ \u952e\u7684\u957f\u5ea6\u548c\u5f3a\u5ea6\u5b8c\u5168\u76f8\u540c\uff0c\u4efb\u610f\u4e24\u4e2a $\\ce{C - H}$ \u952e\u4e4b\u95f4\u7684\u5939\u89d2\u5747\u4e3a $\\pu{109.5^\\circ C}$\u3002 \n \n\u4e0e\u7532\u70f7\u7ed3\u6784\u76f8\u4f3c\u7684\u6709\u673a\u5316\u5408\u7269\u8fd8\u6709\u5f88\u591a\uff0c\u968f\u7740\u5206\u5b50\u4e2d\u78b3\u539f\u5b50\u6570\u7684\u589e\u52a0\uff0c\u8fd8\u6709\u4e59\u70f7\u3001\u4e19\u70f7\u3001\u4e01\u70f7\u7b49\u4e00\u7cfb\u5217\u6709\u673a\u5316\u5408\u7269\u3002\u8fd9\u4e9b\u5316\u5408\u7269\u7684\u78b3\u94fe\u5448\u952f\u9f7f\u72b6\u6216\u6298\u7ebf\u5f62\u3002
\n\u70f7\u70c3\u5206\u5b50\u4e2d\u7684\u78b3\u539f\u5b50\u90fd\u91c7\u53d6 $\\text{sp3}$ \u6742\u5316\uff0c\u5f62\u6210\u56db\u9762\u4f53\u7ed3\u6784\uff1b\u78b3\u539f\u5b50\u4ee5...
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\n\n\u51fd\u6570\u7684\u5b9a\u4e49\u57df\u5c31\u662f\u80fd\u4f7f\u89e3\u6790\u5f0f\u6709\u610f\u4e49\u7684\u6240\u6709\u5b9e\u6570\u52ff\u7684\u96c6\u5408\uff0c\u81ea\u7136\u5b9a\u4e49\u57df\u662f\u5f0f\u5b50\u672c\u8eab\u6240\u8981\u6c42\u7684\u5b9a\u4e49\u57df\u3002 \n\u4e0d\u8981\u8f7b\u6613\u5bf9\u89e3\u6790\u5f0f\u8fdb\u884c\u5316\u7b80\u53d8\u5f62\uff0c\u4ee5\u514d\u5b9a\u4e49\u57df\u53d1\u751f\u53d8\u5316\u3002 \n\u5f53\u4e00\u4e2a\u51fd\u6570\u7531\u4e24\u4e2a\u6216\u4e24\u4e2a\u4ee5\u4e0a\u4ee3\u6570\u5f0f\u7684\u548c\u3001\u5dee\u3001\u79ef\u3001\u5546\u7684\u5f62\u5f0f\u6784\u6210\u65f6\uff0c\u5b9a\u4e49\u57df\u662f\u4f7f\u5f97\u5404\u5f0f\u5b50\u90fd\u6709\u610f\u4e49\u7684\u516c\u5171\u90e8\u5206\u3001\u4e14\uff08\u82e5\u6709\uff09\u5206\u5f0f\u6709\u610f\u4e49\u7684\u96c6\u5408\u3002 \n \n\u590d\u5408\u51fd\u6570\uff1a\u5982\u679c $g$ \u7684\u503c\u57df\u4e3a $f$ \u7684\u5b9a\u4e49\u57df\u7684\u5b50\u96c6\uff0c\u90a3\u4e48\u5b9a\u4e49 $y=(f\\circ g)(x)=f(g(x))$\u3002
\n\u89e3\u6790\u5f0f \n\u5df2\u77e5\u51fd\u6570 $f$ \u7684\u4e00\u4e9b\u5173\u7cfb\u5f0f\uff0c\u6c42 $f(x)$\uff0c\u6700\u5e38\u7528\u7684\u662f\u6362\u5143\u6cd5\u548c\u53d8\u5f62\u6cd5\uff0c\u4f8b\u5982\uff1a
\n$$\nf(x+1)=x^2\n$$
\n\u6362\u5143\u6cd5\uff0c\u8bbe $t=x+1$\uff0c\u5219\uff1a
\n$$\nf...
",
"image": "https://raineblog.dpdns.org/whk/assets/images/social/science/function/1.png",
"date_modified": "2026-03-31T13:44:00+00:00",
"date_published": "2025-05-18T09:51:00+00:00",
"authors": [],
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},
{
"id": "https://raineblog.dpdns.org/whk/science/function/2/",
"url": "https://raineblog.dpdns.org/whk/science/function/2/",
"title": "\u51fd\u6570\u7684\u57fa\u672c\u6027\u8d28",
"content_html": "\u51fd\u6570\u7684\u57fa\u672c\u6027\u8d28 \n\u57fa\u672c\u6027\u8d28 \n\u5947\u5076\u6027 \n\u51fd\u6570\u7684\u5947\u5076\u6027\u5b9a\u4e49\u5728\u5b9a\u4e49\u57df\u5173\u4e8e $0$ \u5bf9\u79f0\u7684\u51fd\u6570\u4e0a\uff0c\u5bf9\u4e8e\u5947\u51fd\u6570\u6709\uff1a
\n$$\nf(x)+f(-x)=0\n$$
\n\u56e0\u6b64\uff0c\u5982\u679c\u5b9a\u4e49\u57df\u4e2d\u6709 $0$\uff0c\u90a3\u4e48\u53ea\u6709 $f(0)=0$ \u7684\u51fd\u6570\u624d\u53ef\u80fd\u4e3a\u5947\u51fd\u6570\u3002
\n\u5076\u51fd\u6570\u4e3a\uff1a
\n$$\nf(x)=f(-x)\n$$
\n\u7279\u522b\u5730\uff0c\u5947\u51fd\u6570\u7684\u7edd\u5bf9\u503c\u4e3a\u5076\u51fd\u6570\u3002
\n\u5947\u5076\u6027\u7684\u4e58\u9664\u6cd5\u5219\uff1a
\n\n\u5947\u51fd\u6570 $\\pm$ \u5947\u51fd\u6570 $=$ \u5947\u51fd\u6570\u3002 \n\u5076\u51fd\u6570 $\\pm$ \u5076\u51fd\u6570 $=$ \u5076\u51fd\u6570\u3002 \n\u5947\u51fd\u6570 $\\pm$ \u5076\u51fd\u6570 $=$ \u975e\u5947\u975e\u5076\u3002 \n\u5947\u51fd\u6570 $\\times$ \u5947\u51fd\u6570 $=$ \u5076\u51fd\u6570\u3002 \n\u5947\u51fd\u6570 $\\times$ \u5076\u51fd\u6570 $=$ \u5947\u51fd\u6570\u3002 \n\u5076\u51fd\u6570 $\\times$ \u5076\u51fd\u6570 $=$ \u5076\u51fd\u6570\u3002 \n \n\u590d\u5408\u51fd\u6570\u7684\u5947\u5076\u6027\uff1a
\n\n\u5947\u51fd\u6570 $\\circ$ \u5947\u51fd\u6570 $=$ \u5947\u51fd\u6570\u3002 \n\u5076\u51fd\u6570 $\\circ$ \u5947\u51fd\u6570 $=$ \u5076\u51fd\u6570\u3002 \n\u5947\u51fd\u6570 $\\circ$ \u5076\u51fd\u6570 $=$ \u5076\u51fd\u6570\u3002 \n\u5076\u51fd\u6570 $\\circ$ \u5076\u51fd\u6570 $=$ \u5076\u51fd\u6570\u3002 \n \n\u7ecf\u5178\u505a\u9898\u5957\u8def\uff1a
\n\n\u82e5 $f(x)$ \u4e3a\u5076\u51fd\u6570\uff0c\u9996\u5148\u5224\u65ad\u662f\u5426\u80fd\u8f6c\u5316\u4e3a\u5947... \n ",
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\n\n\u6240\u8981\u8c03\u67e5\u7684\u5bf9\u8c61\u5168\u4f53\u53eb\u505a\u603b\u4f53 (population)\uff0c\u603b\u4f53\u4e2d\u6bcf\u4e2a\u6210\u5458\u53eb\u505a\u4e2a\u4f53 \u3002 \n\u603b\u4f53\u53c2\u6570\u662f\u63cf\u8ff0\u603b\u4f53\u7279\u6027\u7684\u6307\u6807\uff0c\u7b80\u79f0\u53c2\u6570 \u3002 \n\u5982\u679c\u603b\u4f53\u4e2d\u7684\u4e2a\u4f53\u662f\u6709\u9650\u4e2a\uff0c\u79f0\u4e2a\u4f53\u603b\u6570 $N$ \u4e3a\u603b\u4f53\u5bb9\u91cf\u3002 \n\u603b\u4f53\u5e73\u5747\u6216\u603b\u4f53\u5747\u503c\u662f\u53c2\u6570\uff0c\u5e38\u7528 $\\mu$ \u8868\u793a\u3002 \n\u603b\u4f53\u65b9\u5dee\u662f\u53c2\u6570\uff0c\u5e38\u8bb0\u4e3a $\\sigma^2... \n ",
"image": "https://raineblog.dpdns.org/whk/assets/images/social/science/probability/3.png",
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"date_published": "2025-05-18T09:51:00+00:00",
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"id": "https://raineblog.dpdns.org/whk/science/probability/4/",
"url": "https://raineblog.dpdns.org/whk/science/probability/4/",
"title": "\u968f\u673a\u8fc7\u7a0b\u548c\u9a6c\u5c14\u53ef\u592b\u94fe",
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\n\n\u79bb\u6563\u65f6\u95f4\u968f\u673a\u8fc7\u7a0b \uff1a$T = {0, 1, 2, \\dots}$\uff0c\u901a\u5e38\u8bb0\u4f5c ${X_n}_{n=0}^{\\infty}$\u3002\u4f8b\u5982\uff1a\u6bcf\u5929\u8bb0\u5f55\u4e00\u6b21\u80a1\u7968\u4ef7\u683c\u3001\u6bcf\u8f6e\u6bd4\u8d5b\u540e\u7684\u6bd4\u5206\u72b6\u6001\u8fde\u7eed\u65f6\u95f4\u968f\u673a\u8fc7\u7a0b \uff1a$T = [0, \\infty)$ \u6216 $\\mathbb{R}$\uff0c\u901a\u5e38\u8bb0\u4f5c ${X_t}_{t \\ge 0}$\u3002\u4f8b\u5982\uff1a\u5e03\u6717\u8fd0\u52a8\u3001\u6cca\u677e\u8fc7\u7a0b \n\u968f\u673a\u53d8\u91cf $X_t$ \u6240\u6709\u53ef\u80fd\u53d6\u503c\u7684\u96c6\u5408\u79f0\u4e3a\u72b6\u6001\u7a7a\u95f4 \uff08St...
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"date_modified": "2026-05-25T10:30:00+00:00",
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"image": "https://raineblog.dpdns.org/whk/assets/images/social/science/cell/2.png",
"date_modified": "2026-05-28T13:59:00+00:00",
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"url": "https://raineblog.dpdns.org/whk/science/cell/3/",
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\n\n\u6d41\u52a8\u6027 \uff1a\u78f7\u8102\u5206\u5b50\u548c\u5927\u591a\u6570\u86cb\u767d\u8d28\u5206\u5b50\u90fd\u4e0d\u662f\u9759\u6b62\u7684\uff0c\u53ef\u4ee5\u5728\u819c\u5e73\u9762\u5185\u505a\u6a2a\u5411\u79fb\u52a8\uff0c\u8fd9\u4f7f\u5f97\u819c... ",
"image": "https://raineblog.dpdns.org/whk/assets/images/social/science/cell/3.png",
"date_modified": "2026-05-31T11:56:00+00:00",
"date_published": "2025-04-19T14:19:00+00:00",
"authors": [],
"tags": null
},
{
"id": "https://raineblog.dpdns.org/whk/science/mechanics/1/",
"url": "https://raineblog.dpdns.org/whk/science/mechanics/1/",
"title": "\u9759\u529b\u4e0e\u52a8\u91cf",
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\n\u529b\u6709\u5927\u5c0f\u3001\u65b9\u5411\u3001\u4f5c\u7528\u70b9\u4e09\u4e2a\u8981\u7d20\u3002
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\n\u529b\u662f\u6539\u53d8\u7269\u4f53\u8fd0\u52a8\u72b6\u6001\u7684\u539f\u56e0\u3002
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\n\u53ef\u4ee5\u6cbf\u4f5c\u7528\u7ebf\u65b9\u5411\u6539\u53d8\u4f5c\u7528\u70b9\uff0c\u800c\u4e0d\u6539\u53d8\u8be5\u529b\u5bf9\u521a\u4f53\u7684\u4f5c\u7528\u6548\u679c\u3002
\n\u6240\u4ee5\u4f5c\u7528\u5728\u521a\u4f53\u4e0a\u7684\u529b\uff0c\u4e09\u4e2a\u8981\u7d20\u662f\uff1a\u5927\u5c0f\u3001\u65b9\u5411\u548c\u4f5c\u7528\u7ebf\u3002
\n\u5982\u679c\u7269\u4f53\u5e73\u8861\uff0c\u5373\u7269\u4f53\u9759\u6b62\u6216\u4fdd\u6301\u8fd0\u52a8\u72b6\u6001\u4e0d\u53d8\uff0c\u4f5c\u7528\u4e8e\u7269\u4f53\u7684\u5404\u529b\u76f8\u4e92\u62b5\u6d88\u3002
\n\u53d7\u529b\u5206\u6790\u7684\u987a\u5e8f\uff1a\u5148\u4e3b\u52a8\u529b\uff08\u5916\u529b\u3001\u91cd\u529b\uff09\u3001\u540e\u88ab\u52a8\u529b\uff08\u5f39\u529b\u3001\u6469\u64e6\u529b\uff09\u3002
\n\u6ce8\u610f\uff1a\u6548\u679c\u529b\u7531\u6027\u8d28\u529b\u5145\u5f53\uff0c\u4f8b\u5982\u62c9\u529b\u652f\u6301\u529b\u5411\u5fc3\u529b\u3002
\n\u62c9\u5bc6\u5b9a\u7406 \n\u62c9\u5bc6\u5b9a\u7406\u6307\u51fa\uff0c\u5982\u679c\u4e09\u4e2a\u5171\u70b9\u529b\u7684\u5408\u529b\u4e3a\u96f6\uff0c\u90a3\u4e48\u4efb\u4e00\u529b\u4e0e\u5176\u76f8\u5bf9\u89d2\u7684\u6b63\u5f26\u7684\u6bd4\u503c\u5747\u76f8\u7b49\uff1a
\n$$\n\\dfrac{A}{\\sin\\alpha}=\\dfrac{B}{\\sin\\beta}=\\dfrac{C}{\\sin\\gamma}\n$$
\n\u5176\u4e2d $A$\u3001$B$\u3001$C$ \u4e3a\u4f5c\u7528\u5728\u4e00\u70b9\uff08\u5047\u8bbe\u4e3a $P$\uff09\u529b\u7684\u5927\u5c0f\u3002
\n\u800c $\\alpha$\u3001$\\beta...
",
"image": "https://raineblog.dpdns.org/whk/assets/images/social/science/mechanics/1.png",
"date_modified": "2026-05-16T12:04:00+00:00",
"date_published": "2025-04-13T13:42:00+00:00",
"authors": [],
"tags": null
},
{
"id": "https://raineblog.dpdns.org/whk/science/mechanics/2/",
"url": "https://raineblog.dpdns.org/whk/science/mechanics/2/",
"title": "\u8fd0\u52a8\u4e0e\u80fd\u91cf",
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\n\u725b\u987f\u8fd0\u52a8\u5b9a\u5f8b \n\u725b\u987f\u7b2c\u4e00\u5b9a\u5f8b \n\u725b\u987f\u7b2c\u4e00\u5b9a\u5f8b\u8868\u8ff0\u4e3a\uff1a\u4e0d\u53d7\u5916\u529b\u4f5c\u7528\u7684\u7269\u4f53\u5fc5\u5c06\u4f5c\u5300\u901f\u76f4\u7ebf\u8fd0\u52a8\u6216\u4fdd\u6301\u9759\u6b62\u3002
\n\u8fd9\u6761\u5b9a\u5f8b\u5c31\u5176\u53d9\u8ff0\u7684\u5185\u5bb9\u672c\u8eab\u63d0\u51fa\u4e86\u529b\u548c\u60ef\u6027\u4e24\u4e2a\u91cd\u8981\u6982\u5ff5\uff1a\u725b\u987f\u628a\u6539\u53d8\u7269\u4f53\u8fd0\u52a8\u72b6\u6001\u7684\u56e0\u7d20\u5f52\u4e8e\u7269\u4f53\u53d7\u5230\u4e86\u529b\uff0c\u800c\u8fd0\u52a8\u7269\u4f53\u81ea\u8eab\u53c8\u5177\u6709\u4fdd\u6301\u5176\u5300\u901f\u76f4\u7ebf\u8fd0\u52a8\u6216\u4fdd\u6301\u9759\u6b62\u7684\u5c5e\u6027\uff0c\u5e76\u79f0\u6b64\u5c5e\u6027\u4e3a\u60ef\u6027\u3002
\n\u6839\u636e\u8fd0\u52a8\u7684\u76f8\u5bf9\u6027\uff0c\u5bf9\u4e8e\u4e0d\u540c\u7684\u53c2\u7167\u7cfb\uff0c\u7269\u4f53\u5f80\u5f80\u6709\u4e0d\u540c\u7684\u52a0\u901f\u5ea6\uff0c\u5728\u4e00\u4e2a\u53c2\u7167\u7cfb\u4e2d\u4f5c\u5300\u901f\u8fd0\u52a8\u7684\u7269\u4f53\uff0c\u5728\u53e6\u4e00\u4e2a\u53c2\u7167\u7cfb\u4e2d\u53ef\u80fd\u6709\u52a0\u901f\u5ea6\u3002
\n\u725b\u987f\u7b2c\u4e00\u5b9a\u5f8b\u65ad\u8a00\u5b58\u5728\u7740\u4e00\u4e2a\u53c2\u7167\u7cfb\uff0c\u5728\u6b64\u53c2\u7167\u7cfb\u4e2d\uff0c\u7269\u4f53\u7684\u8fd0\u52a8\u9075\u5faa\u725b\u987f\u7b2c\u4e00\u5b9a\u5f8b\u3002\u8fd9\u4e2a\u53c2\u7167\u7cfb\u79f0\u4e3a\u60ef\u6027\u7cfb\uff0c\u56e0\u6b64\uff0c\u7b2c\u4e00\u5b9a\u5f8b\u53c8\u79f0\u4e3a\u60ef\u6027\u5b9a\u5f8b\u3002
\n\u8fd0\u52a8\u7684\u76f8\u5bf9\u6027\u544a\u8bc9\u6211\u4eec\uff0c\u5bf9\u4e8e\u4efb\u4f55\u4e24\u4e2a\u76f8\u4e92\u4f5c\u5300\u901f\u8fd0\u52a8\u7684\u53c2\u7167\u7cfb\uff0c\u540c\u4e00\u8d28\u70b9\u5177\u6709\u76f8\u540c\u7684\u52a0\u901f\u5ea6\u3002\u56e0\u6b64\uff0c\u53ea\u8981\u5b58\u5728\u4e00\u4e2a\u60ef\u6027\u7cfb\uff0c\u90a3\u4e48\u76f8\u5bf9\u4e8e\u6b64\u60ef\u6027\u7cfb\u4f5c\u5300\u901f\u8fd0\u52a8\u7684\u6240\u6709\u53c2\u7167\u7cfb\u90fd\u662f\u60ef\u6027\u7cfb\u3002
\n\u603b\u7ed3\uff1a
\n\n\u5047\u82e5\u65bd\u52a0\u4e8e\u67d0\u7269\u4f53\u7684\u5916\u529b\u4e3a\u96f6\uff0c\u5219\u8be5... \n ",
"image": "https://raineblog.dpdns.org/whk/assets/images/social/science/mechanics/2.png",
"date_modified": "2026-05-16T12:03:00+00:00",
"date_published": "2025-04-13T13:42:00+00:00",
"authors": [],
"tags": null
},
{
"id": "https://raineblog.dpdns.org/whk/science/mechanics/3/",
"url": "https://raineblog.dpdns.org/whk/science/mechanics/3/",
"title": "\u52a8\u529b\u5b66\u6a21\u578b",
"content_html": "\u52a8\u529b\u5b66\u6a21\u578b \n\u52a8\u529b\u5b66\u6a21\u578b \n\u667e\u8863\u7ef3\u6a21\u578b \n\u7b49\u8170\u4e09\u89d2\u5f62\u3001\u667e\u8863\u6746\u95ee\u9898\uff0c\u7279\u5f81\u4e3a\u52a8\u6ed1\u8f6e\u901a\u8fc7\u521a\u6027\u8f7b\u7ef3\u56fa\u5b9a\uff0c\u6709\u516c\u5f0f\uff1a
\n$$\nF=\\dfrac{G}{2 \\cos \\theta}\n$$
\n\u7279\u5f81\uff1b$F$ \u4ec5\u4e0e $\\theta$ \u6709\u5173\uff0c\u4e0a\u4e0b\u79fb\u52a8\u7ef3\u5b50\u7aef\u70b9\u529b\u4e0d\u53d8\uff0c\u7aef\u70b9\u6c34\u5e73\u9760\u8fd1\u62c9\u529b\u4e0b\u964d\u3001\u8fdc\u79bb\u62c9\u529b\u4e0a\u5347\u3002
\n\u7269\u4f53\u7684\u5e73\u8861\u53ef\u4ee5\u5206\u4e3a\u7a33\u5b9a\u5e73\u8861\u3001\u4e0d\u7a33\u5b9a\u5e73\u8861\u548c\u968f\u9047\u5e73\u8861\u4e09\u79cd\u3002
\n\u5f39\u7c27\u7a81\u53d8 \n\u56e0\u4e3a\u5f39\u7c27\u7684\u5f39\u529b\u65e0\u6cd5\u7a81\u53d8\uff0c\u56e0\u6b64\u6211\u4eec\uff1a
\n\n\n\u53d7\u529b\u5206\u6790\u521d\u72b6\u6001\uff0c\u5f97\u51fa\u5f39\u7c27\u5f39\u529b\u3002
\n \n\n\u628a\u5f39\u7c27\u5f39\u529b\u5f53\u505a\u5916\u529b\uff0c\u91cd\u65b0\u53d7\u529b\u5206\u6790\u3002
\n \n \n\u6cbf\u7ef3\u65b9\u5411\u901f\u5ea6\u3001\u53d7\u529b\u5927\u5c0f\u4e00\u5b9a\u76f8\u7b49\u3002
\n\u659c\u9762\u6a21\u578b \n\u659c\u9762\u6a21\u578b\u300c\u7269\u4f53\u662f\u5426\u4f1a\u4e0b\u6ed1\u300d\uff0c\u8bbe\u659c\u9762\u4e0e\u6c34\u5e73\u9762\u5939\u89d2\u4e3a $\\theta$\uff1a
\n\u53d7\u529b\u5206\u6790\uff0c\u5f97 $G_x=mg \\sin \\theta$\uff0c$f=\\mu mg \\cos \\theta$\u3002
\n\n\n\u82e5\u7269\u4f53\u4e0b\u6ed1\uff1a$G_x>f \\Rightarrow G_x/f>1 \\Rightarrow \\tan \\theta/\\mu>1 \\Rightarrow \\tan \\theta>\\mu$\u3002
\n \n\n\u540c\u7406\uff0c\u82e5\u7269\u4f53\u9759\u6b62\u4e0d\u52a8\uff0c$G_x\\le f \\Rightar...
\n \n ",
"image": "https://raineblog.dpdns.org/whk/assets/images/social/science/mechanics/3.png",
"date_modified": "2026-05-16T12:03:00+00:00",
"date_published": "2025-04-13T13:42:00+00:00",
"authors": [],
"tags": null
},
{
"id": "https://raineblog.dpdns.org/whk/science/structure/3/",
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