Basic concepts

reference．reference，reference，reference．reference，reference，reference．reference，reference1，reference OI reference．reference，reference．

reference，reference：

reference，reference．reference，reference、reference、reference．

reference

reference．

reference

reference GG reference，reference ⋅:G×G→G\cdot:G\times G\rightarrow G，reference，reference (G,⋅)(G,\cdot) reference reference（group）：

  1. reference（associative property）：reference a,b,c∈Ga,b,c\in G，reference a⋅(b⋅c)=(a⋅b)⋅ca\cdot(b\cdot c)=(a\cdot b)\cdot c；
  2. reference：reference e∈Ge\in G，reference a∈Ga\in G，reference a⋅e=e⋅a=aa\cdot e = e\cdot a = a．reference，ee reference GG reference reference（identity element），reference；
  3. reference：reference a∈Ga\in G，reference b∈Gb\in G reference a⋅b=b⋅a=ea\cdot b=b\cdot a=e．reference，bb reference aa reference reference（inverse element）．
reference

reference，reference a,b∈Ga,b\in G，reference a⋅b∈Ga\cdot b\in G．reference．

reference

reference (G,⋅)(G,\cdot)，reference：

  1. reference {gi}i=1k⊆G\{g_i\}_{i=1}^k\subseteq G，reference g1⋅g2⋅⋯⋅gkg_1\cdot g_2\cdot\cdots\cdot g_k reference；
  2. reference ee reference；
  3. reference a∈Ga\in G，reference a−1a^{-1} reference；
  4. reference（cancellation law）：reference a,b,c∈Ga,b,c\in G，reference a⋅c=b⋅ca\cdot c=b\cdot c reference c⋅a=c⋅bc\cdot a=c\cdot b，reference a=ba=b．

reference．reference，reference．reference．

reference
  • reference（symmetric group）：reference MM reference reference，reference MM reference MM reference，reference SMS_M．reference，reference（reference）．reference MM reference，reference nn，reference SnS_n，reference nn reference．
  • reference（symmetry group）：reference，reference．reference．reference reference．
  • reference：reference Z\mathbf Z reference ++ reference (Z,+)(\mathbf Z,+)．reference 00，reference．
  • reference nn reference（multiplicative group of integers modulo nn）：reference nn，reference nn reference reference，reference ((Z/nZ)×,×)((\mathbf Z/n\mathbf Z)^\times,\times)．reference 1ˉ\bar 1，reference nn reference reference（reference），reference reference reference．reference reference nn reference．
  • reference（general linear group）：reference FF reference nn reference GLn(F)GL_n(F)．reference，reference．

reference，reference．

reference
  • reference MM reference（reference），reference．reference．
  • reference，reference 22 reference．
  • reference，reference．
  • reference nn reference．reference (Z/6Z)∖{0‾}(\mathbf Z/6\mathbf Z)\setminus\{\overline 0\} reference，2‾×3‾=0‾\overline 2\times\overline 3=\overline 0 reference，reference（reference，reference）．

reference，reference．reference，reference，reference．

reference

reference GG reference ⋅\cdot，reference，reference (G,⋅)(G,\cdot) reference reference（semigroup）．

reference

reference (G,⋅)(G,\cdot)，reference，reference (G,⋅)(G,\cdot) reference reference（monoid）．

reference

reference，(N+,+)(\mathbf N_+,+) reference，reference (Z,×)(\mathbf Z,\times) reference．

reference，reference，reference．reference，reference Abel reference，reference．

Abel reference

reference (G,⋅)(G,\cdot)，reference ⋅\cdot reference（commutative property），reference a,b∈Ga,b\in G，reference a⋅b=b⋅aa\cdot b=b\cdot a，reference (G,⋅)(G,\cdot) reference Abel reference（Abelian group）reference reference（communicate group）．

Abel reference Abel reference
  • reference (Z,+)(\mathbf Z,+) reference Abel reference．
  • reference n≥3n\ge3 reference，reference SnS_n reference Abel reference．

reference．reference，reference reference reference．

reference

reference．

reference

reference RR reference +:R×R→R+:R\times R\rightarrow R reference ⋅:R×R→R\cdot:R\times R\rightarrow R，reference，reference (R,+,⋅)(R,+,\cdot) reference reference（ring）：

  1. (R,+)(R,+) reference Abel reference，reference 00，reference a∈Ra\in R reference ++ reference −a-a．
  2. (R,⋅)(R,\cdot) reference，reference ⋅\cdot reference．
  3. reference（distributive property）：reference a,b,c∈Ra,b,c\in R，reference a⋅(b+c)=a⋅b+a⋅ca\cdot(b+c)=a\cdot b+a\cdot c reference (a+b)⋅c=a⋅c+b⋅c(a+b)\cdot c=a\cdot c+b\cdot c．

reference，reference ++ reference ⋅\cdot reference，reference，reference reference（zero），reference（reference）reference reference（identity）．reference、reference，reference．

reference

reference，reference；reference，reference reference（rng reference pseudo-ring）．reference．reference2．

reference，reference．reference，reference，reference．

reference

reference (R,+,⋅)(R,+,\cdot)，reference，reference，reference 11，reference (R,+,⋅)(R,+,\cdot) reference reference（ring with identity）．

reference

reference (R,+,⋅)(R,+,\cdot)，reference 00 reference a∈Ra\in R，reference（reference a−1a^{-1}），reference (R,+,⋅)(R,+,\cdot) reference reference（division ring）．

reference

reference (R,+,⋅)(R,+,\cdot)，reference，reference (R,+,⋅)(R,+,\cdot) reference reference（commutative ring）．

reference，reference 00 reference．reference 0=0⋅a=a⋅00 = 0\cdot a = a\cdot 03．reference，reference．reference，reference，reference．reference（reference）．

reference，reference，reference，reference R∖{0}R\setminus\{0\}．reference，reference．

reference

reference (R,+,⋅)(R,+,\cdot)，reference b∈Rb\in R reference b≠0b\ne 0，reference a⋅b=0a\cdot b=0 reference b⋅a=0b\cdot a=0，reference aa reference reference（zero divisor）．

reference（reference）

reference (R,+,⋅)(R,+,\cdot)，reference aa reference，reference b∈Rb\in R，reference a⋅b=b⋅a=1a\cdot b=b\cdot a=1，reference a∈Ra\in R reference reference，reference reference（unit）．

「reference」reference「reference」

reference．reference，reference「reference」reference「reference」．

reference，reference．reference，reference，reference．

reference，reference，reference (R∖{0},⋅)(R\setminus\{0\},\cdot) reference．reference，reference，reference．

reference

reference (R,+,⋅)(R,+,\cdot)，reference，reference，reference，reference（integral domain）．

reference，reference．

reference

reference RR reference a,b,c∈Ra,b,c\in R reference a≠0a\neq 0，reference ab=acab=ac，reference b=cb=c．

reference，reference，reference．reference．

reference（reference）

reference (R,+,⋅)(R,+,\cdot)，reference R×R^\times reference RR reference，reference (R×,⋅)(R^\times,\cdot) reference，reference RR reference reference（multiplicative group），reference reference（unit group）．

reference．

reference

  • reference（zero ring）：reference {0}\{0\} reference ++ reference ×\times reference，reference．reference，reference．

  • reference：reference Z\mathbf Z reference ++ reference ×\times reference (Z,+,×)(\mathbf Z,+,\times)．reference，reference，reference．

  • reference：reference RR，reference reference R[x]R[x]．reference RR reference，reference．

  • reference（quaternion）：reference，reference H={a+bi+cj+dk:a,b,c,d∈R}\mathbf H=\{a+b\mathrm{i}+c\mathrm{j}+d\mathrm{k}:a,b,c,d\in\mathbf R\}，reference，reference，i,j,k\mathrm{i},\mathrm{j},\mathrm{k} reference

    i2=j2=k2=−1, ij=−ji=k, jk=−kj=i, ki=−ik=j.\mathrm{i}^2=\mathrm{j}^2=\mathrm{k}^2=-1,\ \mathrm{i}\mathrm{j}=-\mathrm{j}\mathrm{i}=\mathrm{k},\ \mathrm{j}\mathrm{k}=-\mathrm{k}\mathrm{j}=\mathrm{i},\ \mathrm{k}\mathrm{i}=-\mathrm{i}\mathrm{k}=\mathrm{j}.

    reference，H\mathbf H reference，reference，reference．

  • reference 2Z2\mathbf Z，reference，reference，reference，reference．

  • reference nn reference Z/nZ\mathbf Z/n\mathbf Z reference，reference，reference（reference 1ˉ\bar 1）．reference，reference nn reference．reference，reference nn reference，reference (Z/nZ,+,×)(\mathbf Z/n\mathbf Z, +,\times) reference；reference，reference，reference．reference ((Z/nZ)×,×)((\mathbf Z/n\mathbf Z)^\times,\times) reference nn reference．

  • reference：reference RR reference nn reference Mn(R)M_n(R)．reference，reference，reference．

  • reference AA reference P(A)\mathcal P(A)，reference △\triangle reference ∩\cap reference，reference (P(A),△,∩)(\mathcal P(A),\triangle,\cap) reference．reference，reference，reference，reference．

reference，reference，reference，reference reference reference．

reference

reference．reference，reference．reference．

reference

reference FF reference +:F×F→F+:F\times F\rightarrow F reference ⋅:F×F→F\cdot:F\times F\rightarrow F，reference，reference (F,+,⋅)(F,+,\cdot) reference reference（field）：

  1. (F,+)(F,+) reference Abel reference，reference 00，reference a∈Fa\in F reference ++ reference −a-a．
  2. (F∖{0},⋅)(F\setminus\{0\},\cdot) reference Abel reference，reference 11，reference a∈F∖{0}a\in F\setminus\{0\} reference ⋅\cdot reference a−1a^{-1}．

reference，reference、reference、reference、reference．

reference．

reference

  • reference：reference Q\mathbf Q，reference R\mathbf R reference C\mathbf C reference．

  • reference（finite field）：reference pp reference Z/pZ\mathbf Z/p\mathbf Z reference．reference，reference，reference，reference．

  • reference（fraction field）：reference (R,+,⋅)(R,+,\cdot) reference，reference ab−1ab^{-1} reference QQ．reference，reference R×(R∖{0})R\times(R\setminus\{0\}) reference：(a1,b1)∼(a2,b2)(a_1,b_1)\sim(a_2,b_2) reference a1b2=a2b1a_1b_2=a_2b_1．reference，reference QQ reference R×(R∖{0})/∼R\times(R\setminus\{0\})/\sim，reference，(a,b)(a,b) reference ab−1ab^{-1}．reference

    a1b1−1+a2b2−1=(a1⋅b2+a2⋅b1)(b1⋅b2)−1,(a1b1−1)⋅(a2b2−1)=(a1⋅a2)(b1⋅b2)−1\begin{aligned} a_1b_1^{-1}+a_2b_2^{-1} &= (a_1\cdot b_2+a_2\cdot b_1)(b_1\cdot b_2)^{-1},\\ (a_1b_1^{-1})\cdot(a_2b_2^{-1}) &= (a_1\cdot a_2)(b_1\cdot b_2)^{-1} \end{aligned}

    reference (Q,+,⋅)(Q,+,\cdot) reference，reference RR reference．reference，reference (Q,+,×)(\mathbf Q,+,\times) reference (Z,+,×)(\mathbf Z,+,\times) reference．

  • reference（quadratic field）：reference Q\mathbf Q reference d\sqrt d reference，reference d≠0,1d\neq 0,1 reference．reference reference．

reference，reference．reference，reference．reference，reference，reference Galois reference．reference，reference．reference，reference reference reference．

reference

reference，reference，reference．

???+ note "【reference】"reference DP"& reference（reference）" reference nn reference，reference mm reference．reference．reference．

reference

reference DP reference，reference reference，reference．reference，reference．

reference，reference，reference DP reference．reference [1,n][1,n] reference，reference ii reference（11）reference（00）．reference，[1,i][1,i] reference fi,1f_{i,1} reference fi,0f_{i,0}．reference，reference DP reference

fi,1=wi+fi−1,0,fi,0=max⁡{fi−1,1,fi−1,0}.\begin{aligned} f_{i,1}&=w_{i}+f_{i-1,0},\\ f_{i,0}&=\max\{f_{i-1,1},f_{i-1,0}\}. \end{aligned}

reference (f0,1,f0,0)=(0,0)(f_{0,1},f_{0,0})=(0,0)，reference max⁡{fn,1,fn,0}\max\{f_{n,1},f_{n,0}\}．reference ii reference，reference

(fi,1,fi,0)=g(fi−1,1,fi−1,0;wi).(f_{i,1},f_{i,0})=g(f_{i-1,1},f_{i-1,0};w_i).

reference R2\mathbf R^2 reference R2\mathbf R^2 reference，reference (fi−1,1,fi−1,0)(f_{i-1,1},f_{i-1,0}) reference (fi,1,fi,0)(f_{i,1},f_{i,0})，reference，reference．reference．

reference，reference g(⋅;wi)g(\cdot;w_i) reference，reference R2\mathbf R^2 reference R2\mathbf R^2 reference．reference，reference R∪{−∞}\mathbf R\cup\{-\infty\} reference，reference max⁡\max reference、++ reference，reference R∪{−∞}\mathbf R\cup\{-\infty\} reference，reference，−∞-\infty reference，00 reference．reference，reference．reference4，reference (R∪{−∞},max⁡,+)(\mathbf R\cup\{-\infty\},\max,+) reference reference（tropical semiring）．

reference (R,⊕,⊗)(R,\oplus,\otimes)，reference．reference m×nm\times n reference A=(aij)A=(a_{ij}) reference n×pn\times p reference B=(bjk)B=(b_{jk})，reference ABAB reference (cik)(c_{ik})，reference

cik=⨁j=1n(bij⊗cjk)=max⁡1≤j≤n  (bij+cjk).c_{ik} = \bigoplus_{j=1}^n(b_{ij}\otimes c_{jk}) = \max_{1\le j\le n}\;(b_{ij}+c_{jk}).

reference，reference，reference

(fi,1fi,0)=(−∞wi00)(fi−1,1fi−1,0).\left(\begin{matrix}f_{i,1}\\f_{i,0}\end{matrix}\right) =\left(\begin{matrix}-\infty&w_i\\0&0\end{matrix}\right)\left(\begin{matrix}f_{i-1,1}\\f_{i-1,0}\end{matrix}\right).

reference，reference DP reference．

reference．reference ii，reference S(i)S(i)，reference DP reference

fi,1=wi+∑j∈S(i)fj,0,fi,0=∑j∈S(i)max⁡{fj,1,fj,0}.\begin{aligned} f_{i,1}&=w_i+\sum_{j\in S(i)}f_{j,0},\\ f_{i,0}&=\sum_{j\in S(i)}\max\{f_{j,1},f_{j,0}\}. \end{aligned}

reference，reference．reference hh reference ii reference，reference

fi,1=wi+fh,0+gi,1,fi,0=max⁡{fj,0,fj,1}+gi,0,\begin{aligned} f_{i,1}&=w_i+f_{h,0}+g_{i,1},\\ f_{i,0}&=\max\{f_{j,0},f_{j,1}\}+g_{i,0}, \end{aligned}

reference，

gi,1=∑j∈S(i), j≠hfj,0,gi,0=∑j∈S(i), j≠hmax⁡{fj,1,fj,0}\begin{aligned} g_{i,1}&=\sum_{j\in S(i),\ j\neq h}f_{j,0},\\ g_{i,0}&=\sum_{j\in S(i),\ j\neq h}\max\{f_{j,1},f_{j,0}\} \end{aligned}

reference．reference，reference，reference，reference．reference，reference O(log⁡2n)O(\log^2n) reference，reference O(log⁡n)O(\log n)，reference LCT reference．

reference．reference max⁡\max reference min⁡\min，reference．reference nn reference AA reference nn reference（reference）reference，reference，AkA^k reference (i,j)(i,j) reference ii reference kk reference jj reference；reference，AnA^n reference．reference，reference O(n3)O(n^3) reference Floyd reference．

reference

Footnotes

  1. reference OI Wiki reference． ↩

  2. Ring（mathematics）- Wikipedia ↩

  3. reference 0⋅a+0=0⋅a=(0+0)⋅a=0⋅a+0⋅a0\cdot a+0 = 0\cdot a = (0+0)\cdot a = 0\cdot a + 0\cdot a，reference，reference，reference，reference．reference． ↩

  4. reference（semiring）reference，reference、reference．reference Wikipedia． ↩

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