wpmath0000015_1

Büchi_arithmetic.html

  1. V k ( x ) V_{k}(x)
  2. V k V_{k}
  3. X n X\subseteq\mathbb{N}^{n}
  4. n = 1 n=1
  5. n > 1 n>1
  6. k n = l m k^{n}=l^{m}
  7. V l V_{l}
  8. FO ( V k , + ) \,\text{FO}(V_{k},+)
  9. V k V_{k}
  10. V l V_{l}
  11. FO ( V k , V l , + ) \,\text{FO}(V_{k},V_{l},+)

C::2013_A1.html

  1. ( 100 5 ) 11 - 0.9 10964 (\sqrt[5]{100})^{11-0.9}\approx 10964

Caffeine_dehydrogenase.html

  1. \rightleftharpoons

Calabi_triangle.html

  1. x = 1 3 + ( - 23 + 3 i 237 ) 1 / 3 3 2 2 / 3 + 11 3 ( 2 ( - 23 + 3 i 237 ) ) 1 / 3 x={1\over 3}+{(-23+3i\sqrt{237})^{1/3}\over 3\cdot 2^{2/3}}+{11\over 3\cdot(2% \cdot(-23+3i\sqrt{237}))^{1/3}}
  2. 2 x 3 - 2 x 2 - 3 x + 2 = 0 2x^{3}-2x^{2}-3x+2=0

Cambridge_capital_controversy.html

  1. Q = A . f ( K , L ) Q=A.f(K,L)
  2. Y i = A i . K i a . L i 1 - a Y_{i}=A_{i}.K_{i}^{a}.L_{i}^{1-a}
  3. C o s t = ( 1 + i ) w . L - 1 + ( 1 + i ) 2 w . L - 2 + ( 1 + i ) 3 w . L - 3 + + ( 1 + i ) n w . L - n Cost=(1+i)w.L_{-1}+(1+i)^{2}w.L_{-2}+(1+i)^{3}w.L_{-3}+...+(1+i)^{n}w.L_{-n}

Cannabidiolic_acid_synthase.html

  1. \rightleftharpoons

Canonical_cover.html

  1. F c F_{c}
  2. F c F_{c}
  3. F c F_{c}
  4. F c F_{c}
  5. F c F_{c}
  6. F c F_{c}
  7. a b a\to b
  8. c d c\to d
  9. F c F_{c}
  10. a = c a=c
  11. F c = F F_{c}=F
  12. F c F_{c}
  13. a b a\to b
  14. a d a\to d
  15. a b d a\to bd
  16. F c F_{c}
  17. F c F_{c}
  18. F c F_{c}

Cantelli's_inequality.html

  1. Pr ( X - μ λ ) { σ 2 σ 2 + λ 2 if λ > 0 , 1 - σ 2 σ 2 + λ 2 if λ < 0. \Pr(X-\mu\geq\lambda)\quad\begin{cases}\leq\frac{\sigma^{2}}{\sigma^{2}+% \lambda^{2}}&\,\text{if }\lambda>0,\\ \geq 1-\frac{\sigma^{2}}{\sigma^{2}+\lambda^{2}}&\,\text{if }\lambda<0.\end{cases}
  2. X X
  3. Pr \Pr
  4. μ \mu
  5. X X
  6. σ 2 \sigma^{2}
  7. X X
  8. Pr ( | X - μ | λ ) 2 σ 2 σ 2 + λ 2 . \Pr(|X-\mu|\geq\lambda)\leq\frac{2\sigma^{2}}{\sigma^{2}+\lambda^{2}}.

Cantilever_Magnetometry.html

  1. ω = g / l g = ω 2 l \omega=\sqrt{g/l}\Rightarrow g=\omega^{2}l
  2. L e L_{e}
  3. m m
  4. V V
  5. k k
  6. μ \mu
  7. θ \theta
  8. β \beta
  9. d d t β ˙ = β 0 = - μ B sin ( θ - β ) + 2 K u V sin β cos β β \frac{d}{dt}\frac{\partial\mathcal{L}}{\partial\dot{\beta}}=\frac{\partial% \mathcal{L}}{\partial\beta}\Rightarrow 0=-\mu B\sin{(\theta-\beta)}+2K_{u}V% \underbrace{\sin\beta\cos\beta}_{\approx\beta}\Rightarrow
  10. H k 2 K u V μ H_{k}\equiv\frac{2K_{u}V}{\mu}
  11. θ \theta
  12. ( θ - β ) = θ H k B + H k (\theta-\beta)=\frac{\theta H_{k}}{B+H_{k}}
  13. d d t θ ˙ = θ \frac{d}{dt}\frac{\partial\mathcal{L}}{\partial\dot{\theta}}=\frac{\partial% \mathcal{L}}{\partial\theta}\Rightarrow
  14. m L e 2 θ ¨ = - k L e 2 θ - μ B sin ( θ H k B + H k ) ( H k B + H k ) - 2 K u V sin ( θ B B + H k ) cos ( θ B B + H k ) ( B B + H k ) mL_{e}^{2}\ddot{\theta}=-kL_{e}^{2}\theta-\mu B\sin{\left(\frac{\theta H_{k}}{% B+H_{k}}\right)}\left(\frac{H_{k}}{B+H_{k}}\right)-2K_{u}V\sin{\left(\frac{% \theta B}{B+H_{k}}\right)}\cos{\left(\frac{\theta B}{B+H_{k}}\right)}\left(% \frac{B}{B+H_{k}}\right)\Rightarrow
  15. - k L e 2 θ - μ B θ ( H k B + H k ) 2 - μ H k θ ( B B + H k ) 2 \approx-kL_{e}^{2}\theta-\mu B\theta\left(\frac{H_{k}}{B+H_{k}}\right)^{2}-\mu H% _{k}\theta\left(\frac{B}{B+H_{k}}\right)^{2}\Rightarrow
  16. θ ¨ + θ ( k L e 2 + μ B H k ( B + H k ) m L e 2 ) = θ ¨ + θ ( ω o 2 + μ B H k m L e 2 ( B + H k ) ) = 0 , \ddot{\theta}+\theta\left(\frac{kL_{e}^{2}+\frac{\mu BH_{k}}{(B+H_{k})}}{mL_{e% }^{2}}\right)=\ddot{\theta}+\theta\left(\omega_{o}^{2}+\frac{\mu BH_{k}}{mL_{e% }^{2}(B+H_{k})}\right)=0,
  17. ω 2 ( ω o 2 + μ B H k m L e 2 ( B + H k ) ) = ω o 2 ( 1 + μ B H k k L e 2 ( B + H k ) ) \omega^{2}\equiv\left(\omega_{o}^{2}+\frac{\mu BH_{k}}{mL_{e}^{2}(B+H_{k})}% \right)=\omega_{o}^{2}\left(1+\frac{\mu BH_{k}}{kL_{e}^{2}(B+H_{k})}\right)
  18. θ ( t ) = c 1 cos ω t + c 2 sin ω t \theta(t)=c_{1}\cos\omega t+c_{2}\sin\omega t
  19. c 1 c_{1}
  20. c 2 c_{2}
  21. ω \omega
  22. ω = ω o ( 1 + μ B H k k L e 2 ( B + H k ) ) 1 / 2 ω o ( 1 + μ B H k 2 k L e 2 ( B + H k ) + ) \omega=\omega_{o}\left(1+\frac{\mu BH_{k}}{kL_{e}^{2}(B+H_{k})}\right)^{1/2}% \approx\omega_{o}\left(1+\frac{\mu BH_{k}}{2kL_{e}^{2}(B+H_{k})}+...\right)\Rightarrow

Capacitive_power_supply.html

  1. 2 π 2\pi

Capsule_(geometry).html

  1. V = π r 2 ( 4 3 r + a ) V=\pi r^{2}\left(\frac{4}{3}r+a\right)
  2. r r
  3. a a
  4. S A = 2 π r ( 2 r + a ) . SA=2\pi r(2r+a).

Carbazole_1,9a-dioxygenase.html

  1. \rightleftharpoons

Carboxynorspermidine_decarboxylase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Carboxynorspermidine_synthase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Carhart_four-factor_model.html

  1. E X R t = α J + β m k t 𝐸𝑋𝑀𝐾𝑇 t + ϵ t EXR_{t}=\alpha^{J}+\beta_{mkt}\mathit{EXMKT}_{t}+\epsilon_{t}
  2. E X R t = α F F + β m k t 𝐸𝑋𝑀𝐾𝑇 t + β H M L 𝐻𝑀𝐿 t + β S M B 𝑆𝑀𝐵 t + ϵ t EXR_{t}=\alpha^{FF}+\beta_{mkt}\mathit{EXMKT}_{t}+\beta_{HML}\mathit{HML}_{t}+% \beta_{SMB}\mathit{SMB}_{t}+\epsilon_{t}
  3. E X R t = α c + β m k t 𝐸𝑋𝑀𝐾𝑇 t + β H M L 𝐻𝑀𝐿 t + β S M B 𝑆𝑀𝐵 t + β U M D 𝑈𝑀𝐷 t + ϵ t EXR_{t}=\alpha^{c}+\beta_{mkt}\mathit{EXMKT}_{t}+\beta_{HML}\mathit{HML}_{t}+% \beta_{SMB}\mathit{SMB}_{t}+\beta_{UMD}\mathit{UMD}_{t}+\epsilon_{t}

Carlactone_synthase.html

  1. \rightleftharpoons

Carlitz_exponential.html

  1. [ i ] := T q i - T , [i]:=T^{q^{i}}-T,\,
  2. D i := 1 j i [ j ] q i - j D_{i}:=\prod_{1\leq j\leq i}[j]^{q^{i-j}}
  3. e C ( x ) := j = 0 x q j D i . e_{C}(x):=\sum_{j=0}^{\infty}\frac{x^{q^{j}}}{D_{i}}.
  4. e C ( T x ) = T e C ( x ) + ( e C ( x ) ) q = ( T + τ ) e C ( x ) , e_{C}(Tx)=Te_{C}(x)+\left(e_{C}(x)\right)^{q}=(T+\tau)e_{C}(x),\,
  5. τ \tau
  6. q q
  7. F q ( T ) { τ } F_{q}(T)\{\tau\}

Carotene_epsilon-monooxygenase.html

  1. \rightleftharpoons

Carotenoid-9',10'-cleaving_dioxygenase.html

  1. \rightleftharpoons

Carotenoid_isomerooxygenase.html

  1. \rightleftharpoons

Carvone_reductase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Cascaded_Arc_Plasma_Source.html

  1. 10 21 - 10 24 10^{21}-10^{24}
  2. - 3 {}^{-3}
  3. 10 19 - 10 22 10^{19}-10^{22}
  4. - 3 {}^{-3}
  5. 10 20 10^{20}

Cascades_in_Financial_Networks.html

  1. p k p_{k}
  2. D i k D_{ik}
  3. C i j 0 C_{ij}\geq\ 0
  4. C i i = 0 C_{ii}=0
  5. F i i = 1 - j C j i F_{ii}=1-\sum_{j}C_{ji}
  6. F i i F_{ii}
  7. V i = k D i k p k + j C i j V j V_{i}=\sum_{k}D_{ik}p_{k}+\sum_{j}C_{ij}V_{j}
  8. V = D p + C V V=Dp+CV
  9. V = ( I - C ) - 1 D p V=(I-C)^{-1}Dp
  10. v i = k D i k p k + j C i j - j C j i V i v_{i}=\sum_{k}D_{ik}p_{k}+\sum_{j}C_{ij}-\sum_{j}C_{ji}V_{i}
  11. v = F V = F ( I - C ) - 1 D p = A D p v=FV=F(I-C)^{-1}Dp=ADp
  12. A i j A_{ij}
  13. t i t_{i}
  14. k i k_{i}
  15. I I
  16. V i = k D i k p k + j C i j V j - k i I i V_{i}=\sum_{k}D_{ik}p_{k}+\sum_{j}C_{ij}V_{j}-k_{i}I_{i}
  17. V = ( I - C ) - 1 ( D p - b ( v ) ) V=(I-C)^{-1}(Dp-b(v))
  18. b ( v ) b(v)
  19. b i = k i I i b_{i}=k_{i}I_{i}
  20. v = F ( I - C ) - 1 ( D p - b ( v ) ) = A ( D p - b ( v ) ) v=F(I-C)^{-1}(Dp-b(v))=A(Dp-b(v))
  21. A i j A_{ij}
  22. j j

Cat_state.html

  1. | 00 0 > + | 11 1 Align g t ; |00⋯0>+|11⋯1&gt;
  2. | cat e | α + | - α |\mathrm{cat}_{e}\rangle\propto|\alpha\rangle+|{-}\alpha\rangle
  3. | α = e - | α | 2 2 n = 0 α n n ! | n |\alpha\rangle=e^{-{|\alpha|^{2}\over 2}}\sum_{n=0}^{\infty}{\alpha^{n}\over% \sqrt{n!}}|n\rangle
  4. | - α = e - | - α | 2 2 n = 0 ( - α ) n n ! | n |{-}\alpha\rangle=e^{-{|{-}\alpha|^{2}\over 2}}\sum_{n=0}^{\infty}{({-}\alpha)% ^{n}\over\sqrt{n!}}|n\rangle
  5. | cat e 2 e - | α | 2 2 ( α 0 0 ! | 0 + α 2 2 ! | 2 + α 4 4 ! | 4 + ) |\mathrm{cat}_{e}\rangle\propto 2e^{-{|\alpha|^{2}\over 2}}\left({\alpha^{0}% \over\sqrt{0!}}|0\rangle+{\alpha^{2}\over\sqrt{2!}}|2\rangle+{\alpha^{4}\over% \sqrt{4!}}|4\rangle+\dots\right)
  6. | cat o | α - | - α |\mathrm{cat}_{o}\rangle\propto|\alpha\rangle-|{-}\alpha\rangle
  7. | cat o 2 e - | α | 2 2 ( α 1 1 ! | 1 + α 3 3 ! | 3 + α 5 5 ! | 5 + ) |\mathrm{cat}_{o}\rangle\propto 2e^{-{|\alpha|^{2}\over 2}}\left({\alpha^{1}% \over\sqrt{1!}}|1\rangle+{\alpha^{3}\over\sqrt{3!}}|3\rangle+{\alpha^{5}\over% \sqrt{5!}}|5\rangle+\dots\right)
  8. | c = 1 2 ( 1 + e - 2 | α | 2 ) ( | α + | - α ) |\mathrm{c}\rangle=\frac{1}{\sqrt{2(1+e^{-2|\alpha|^{2}})}}(|\alpha\rangle+|{-% }\alpha\rangle)
  9. | c = 1 2 ( 1 - e - 2 | α | 2 ) ( | α - | - α ) |\mathrm{c}\rangle=\frac{1}{\sqrt{2(1-e^{-2|\alpha|^{2}})}}(|\alpha\rangle-|{-% }\alpha\rangle)

Catalan's_minimal_surface.html

  1. x ( u , v ) = u - cosh ( v ) sin ( u ) y ( u , v ) = 1 - cos ( u ) cosh ( v ) z ( u , v ) = 4 sin ( u / 2 ) sinh ( v / 2 ) \begin{aligned}\displaystyle x(u,v)&\displaystyle=u-\cosh(v)\sin(u)\\ \displaystyle y(u,v)&\displaystyle=1-\cos(u)\cosh(v)\\ \displaystyle z(u,v)&\displaystyle=4\sin(u/2)\sinh(v/2)\end{aligned}

Catalan's_triangle.html

  1. C ( n , k ) C(n,k)
  2. C ( n , n ) C(n,n)
  3. C ( n , k ) C(n,k)
  4. C ( n , k ) = ( n + k ) ! ( n - k + 1 ) k ! ( n + 1 ) ! , C(n,k)=\frac{(n+k)!(n-k+1)}{k!(n+1)!},
  5. ( n = 0 , 1 , 2 , ; k = 0 , 1 , 2 , ) (n=0,1,2,\cdots;k=0,1,2,\cdots)
  6. C m ( n , k ) C_{m}(n,k)
  7. C 1 ( n , k ) = C ( n , k ) C_{1}(n,k)=C(n,k)
  8. C m ( n , k ) C_{m}(n,k)
  9. C m ( n , k ) = { ( n + k k ) 0 k < m ( n + k k ) - ( n + k k - m ) m k n + m - 1 0 k > n + m - 1 C_{m}(n,k)=\begin{cases}\left(\begin{array}[]{c}n+k\\ k\end{array}\right)&\,\,\,0\leq k<m\\ \\ \left(\begin{array}[]{c}n+k\\ k\end{array}\right)-\left(\begin{array}[]{c}n+k\\ k-m\end{array}\right)&\,\,\,m\leq k\leq n+m-1\\ \\ 0&\,\,\,k>n+m-1\end{cases}
  10. ( n = 0 , 1 , 2 , ; k = 0 , 1 , 2 , ; m = 1 , 2 , 3 , ) (n=0,1,2,\cdots;k=0,1,2,\cdots;m=1,2,3,\cdots)

Catalan_pseudoprime.html

  1. ( - 1 ) n - 1 2 C n - 1 2 2 ( mod n ) , (-1)^{\frac{n-1}{2}}\cdot C_{\frac{n-1}{2}}\equiv 2\;\;(\mathop{{\rm mod}}n),

Catalase-peroxidase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Cauchy_process.html

  1. 0
  2. t 2 / 2 t^{2}/2
  3. C C
  4. L L
  5. C ( t ; 0 , 1 ) := W ( L ( t ; 0 , t 2 / 2 ) ) . C(t;0,1)\;:=\;W(L(t;0,t^{2}/2)).
  6. ( 0 , 0 , W ) (0,0,W)
  7. W ( d x ) = d x / ( π x 2 ) W(dx)=dx/(\pi x^{2})
  8. E [ e i θ X t ] = e - t | θ | . \operatorname{E}\Big[e^{i\theta X_{t}}\Big]=e^{-t|\theta|}.
  9. f ( x ; t ) = 1 π [ t x 2 + t 2 ] . f(x;t)={1\over\pi}\left[{t\over x^{2}+t^{2}}\right].
  10. β \beta
  11. β \beta
  12. | β | = 1 |\beta|=1
  13. ( 0 , 0 , W ) (0,0,W)
  14. W ( d x ) = { A x - 2 d x if x > 0 B x - 2 d x if x < 0 W(dx)=\begin{cases}Ax^{-2}\,dx&\,\text{if }x>0\\ Bx^{-2}\,dx&\,\text{if }x<0\end{cases}
  15. A B A\neq B
  16. A > 0 A>0
  17. B > 0 B>0
  18. β \beta
  19. A A
  20. B B
  21. E [ e i θ X t ] = e - t ( | θ | + i β θ ln | θ | / ( 2 π ) ) . \operatorname{E}\Big[e^{i\theta X_{t}}\Big]=e^{-t(|\theta|+i\beta\theta\ln|% \theta|/(2\pi))}.

Causal_inference.html

  1. Y = F ( X ) + E Y=F(X)+E
  2. Y = p X + q E Y=pX+qE
  3. Y = G ( F ( X ) + E ) Y=G(F(X)+E)
  4. Y = F ( X ) + E . G ( X ) Y=F(X)+E.G(X)
  5. Y = F ( X , E ) Y=F(X,E)

CC_(complexity).html

  1. y y
  2. z z
  3. y + z y+z
  4. y y
  5. y / 2 \lceil y/2\rceil
  6. y / 2 \lfloor y/2\rfloor

CDF-based_nonparametric_confidence_interval.html

  1. [ a , b ] [a,b]
  2. F F
  3. x 1 , , x n F x_{1},\ldots,x_{n}\sim F
  4. F ^ n ( t ) = 1 n i = 1 n 1 { x i t } , \hat{F}_{n}(t)=\frac{1}{n}\sum_{i=1}^{n}1\{x_{i}\leq t\},
  5. 1 { A } 1\{A\}
  6. F F
  7. P ( sup x | F ( x ) - F n ( x ) | > ε ) 2 e - 2 n ε 2 . P(\sup_{x}|F(x)-F_{n}(x)|>\varepsilon)\leq 2e^{-2n\varepsilon^{2}}.
  8. X ( j ) X_{(j)}
  9. j th j\text{th}
  10. [ 0 , 1 ] . [0,1].
  11. F F
  12. F F
  13. L ( x ) L(x)
  14. U ( x ) U(x)
  15. E ( X ) = 0 1 ( 1 - F ( x ) ) d x , E(X)=\int_{0}^{1}(1-F(x))\,dx,
  16. [ 0 1 ( 1 - U ( x ) ) d x , 0 1 ( 1 - L ( x ) ) d x ] . \left[\int_{0}^{1}(1-U(x))\,dx,\int_{0}^{1}(1-L(x))\,dx\right].
  17. F F
  18. [ 0 , 1 ] [0,1]
  19. F F
  20. E [ F ] E[F^{\prime}]

CDP-abequose_synthase.html

  1. \rightleftharpoons

CDP-archaeol_synthase.html

  1. \rightleftharpoons

CDP-L-myo-inositol_myo-inositolphosphotransferase.html

  1. \rightleftharpoons

CDP-paratose_synthase.html

  1. \rightleftharpoons

Cell-probe_model.html

  1. S S
  2. c c
  3. w w
  4. s s
  5. s S s\in S
  6. 1 - ε 1-\varepsilon
  7. t t
  8. ε \varepsilon
  9. t t
  10. c c
  11. w w
  12. ( k , v ) (k,v)
  13. A A
  14. k k
  15. v v
  16. ( k ) (k)
  17. A A
  18. 0
  19. k k
  20. O ( 1 ) O(1)
  21. O ( n ) O(n)
  22. O ( log n ) O(\log n)
  23. O ( log n ) O(\log n)
  24. Ω ( log n ) \Omega\left(\log n\right)
  25. d O ( 1 ) d^{O(1)}
  26. Ω ( log log d log log log d ) \Omega\left(\frac{\log\log d}{\log\log\log d}\right)

Central_cylindrical_projection.html

  1. x = R ( λ - λ 0 ) , y = R ( tan ϕ ) \begin{aligned}\displaystyle x&\displaystyle=R(\lambda-\lambda_{0}),\qquad y&% \displaystyle=R(\tan{\phi})\end{aligned}

Central_place_foraging.html

  1. z d 1 z_{d1}
  2. z d 1 z_{d1}
  3. z d 1 z_{d1}
  4. z = z=
  5. x 0 = x_{0}=
  6. x 1 = x_{1}=
  7. y 0 = y_{0}=
  8. y 1 = y_{1}=
  9. z = y 0 x 1 - y 1 x 0 y 1 - y 0 z=\frac{y_{0}x_{1}-y_{1}x_{0}}{y_{1}-y_{0}}
  10. ( y 1 , x 1 ) (y_{1},x_{1})
  11. ( y 0 , x 0 ) (y_{0},x_{0})
  12. z z
  13. x 1 > x 0 x_{1}>x_{0}
  14. y 1 > y 0 y_{1}>y_{0}
  15. y 0 < y 1 y_{0}<y_{1}
  16. y 0 x 0 y 1 x 1 \frac{y_{0}}{x_{0}}\geq\frac{y_{1}}{x_{1}}
  17. y j = i s j A i B i i s j B i y_{j}=\frac{\sum_{i\in s_{j}}A_{i}B_{i}}{\sum_{i\in s_{j}}B_{i}}
  18. A j = A_{j}=
  19. B j = B_{j}=
  20. y j = y_{j}=
  21. x j = ( L P i s j B i ) ( M + i s j D i ) x_{j}=\left(\frac{L}{P\sum_{i\in s_{j}}B_{i}}\right)\left(M+\sum_{i\notin s_{j% }}D_{i}\right)
  22. D j = D_{j}=
  23. L = L=
  24. P = P=
  25. M = M=
  26. x j = x_{j}=
  27. z j z_{j}
  28. z j z_{j}
  29. x 0 x_{0}
  30. x 1 x_{1}
  31. x 1 x_{1}
  32. x 1 x_{1}
  33. z z
  34. y 1 y_{1}
  35. y 0 y_{0}
  36. x 0 x_{0}
  37. z z

Cerium(III)_hydroxide.html

  1. 𝟤 𝖢 𝖾 + 𝟨 𝖧 𝟤 𝖮 𝟫𝟢 𝗈 𝖢 2 𝖢𝖾 ( 𝖮𝖧 ) 𝟥 + 𝟥 𝖧 𝟤 \mathsf{2Ce+6H_{2}O\ \xrightarrow{90^{o}C}\ 2Ce(OH)_{3}\downarrow+3H_{2}\uparrow}

CFD_in_buildings.html

  1. ( ρ ϕ ) t + d i v ( ρ u ϕ ) = d i v ( k g r a d ϕ ) + S ϕ {\frac{\partial{(\rho\phi)}}{\partial t}}+{div\,(\rho u\phi)}={div\,(k\,grad\,% \phi)}+{S_{\phi}}
  2. S ϕ S_{\phi}
  3. Q i Q_{i}
  4. d i v ( ρ u T ) = d i v ( k g r a d T ) + S T {div\,(\rho uT)}={div\,(k\,grad\,T)}+{S_{T}}\,
  5. d i v ( ρ u T ) = d i v ( k g r a d T ) {div\,(\rho uT)}={div\,(k\,grad\,T)}\,
  6. d i v ( k g r a d T ) = 0 {div\,(k\,grad\,T)}=0\,
  7. ( ρ T ) t + d i v ( ρ u T ) = d i v ( k g r a d T ) + S T {\frac{\partial{(\rho T)}}{\partial t}}+{div\,(\rho uT)}={div\,(k\,grad\,T)}+{% S_{T}}\,
  8. ( ρ T ) t + d i v ( ρ u T ) = d i v ( k g r a d T ) {\frac{\partial{(\rho T)}}{\partial t}}+{div\,(\rho uT)}={div\,(k\,grad\,T)}
  9. ( ρ T ) t = d i v ( k g r a d T ) {\frac{\partial{(\rho T)}}{\partial t}}={div\,(k\,grad\,T)}\,
  10. Δ x \Delta x
  11. ( T m - 1 i - 2 T m n + T m i ) Δ x 2 + e k = 0 \frac{(T_{m-1}^{i}-2T_{m}^{n}+T_{m}^{i})}{\Delta{x}^{2}}+\frac{e}{k}=0
  12. q 0 A + k A ( T 1 - T 0 ) Δ x + e 0 2 A Δ x = 0 q_{0}A+kA\frac{(T_{1}-T_{0})}{\Delta{x}}+\frac{e_{0}}{2}A\Delta{x}=0
  13. k A ( T 1 - T 0 ) Δ x + e 0 2 A Δ x = 0 kA\frac{(T_{1}-T_{0})}{\Delta{x}}+\frac{e_{0}}{2}A\Delta{x}=0
  14. h A ( T - T 0 ) + k A ( T 1 - T 0 ) Δ x + e 0 2 A Δ x = 0 hA{(T_{\infty}-T_{0})}+kA\frac{(T_{1}-T_{0})}{\Delta{x}}+\frac{e_{0}}{2}A% \Delta{x}=0
  15. ϵ σ A ( T s u r 4 - T 0 4 ) + k A ( T 1 - T 0 ) Δ x + e 0 2 A Δ x = 0 \epsilon\sigma A{(T_{sur}^{4}-T_{0}^{4})}+kA\frac{(T_{1}-T_{0})}{\Delta{x}}+% \frac{e_{0}}{2}A\Delta{x}=0
  16. h A ( T - T 0 ) + ϵ σ A ( T s u r 4 - T 0 4 ) + k A ( T 1 - T 0 ) Δ x + e 0 2 A Δ x = 0 hA{(T_{\infty}-T_{0})}+\epsilon\sigma A{(T_{sur}^{4}-T_{0}^{4})}+kA\frac{(T_{1% }-T_{0})}{\Delta{x}}+\frac{e_{0}}{2}A\Delta{x}=0
  17. h A c o m b i n e d ( T - T 0 ) + k A ( T 1 - T 0 ) Δ x + e 0 2 A Δ x = 0 hA_{combined}{(T_{\infty}-T_{0})}+kA\frac{(T_{1}-T_{0})}{\Delta{x}}+\frac{e_{0% }}{2}A\Delta{x}=0\,
  18. q 0 A + h A ( T - T 0 ) + ϵ σ A ( T s u r 4 - T 0 4 ) + k A ( T 1 - T 0 ) Δ x + e 0 2 A Δ x = 0 q_{0}A+hA{(T_{\infty}-T_{0})}+\epsilon\sigma A{(T_{sur}^{4}-T_{0}^{4})}+kA% \frac{(T_{1}-T_{0})}{\Delta{x}}+\frac{e_{0}}{2}A\Delta{x}=0
  19. k A A ( T m - 1 - T m ) Δ x + k B A ( T m + 1 - T m ) Δ x + e A , m 2 A Δ x + e B , m 2 A Δ x = 0 k_{A}A\frac{(T_{m-1}-T_{m})}{\Delta{x}}+k_{B}A\frac{(T_{m+1}-T_{m})}{\Delta{x}% }+\frac{e_{A,m}}{2}A\Delta{x}+\frac{e_{B,m}}{2}A\Delta{x}=0
  20. ( W / m 2 ) (W/m^{2})
  21. h c o m b i n e d h_{combined}
  22. T s u r T_{s}ur
  23. T ( ) T_{(}\infty)
  24. T 0 T_{0}
  25. T ( ) T_{(}\infty)
  26. T r T_{r}
  27. T 0 T_{0}
  28. T l T_{l}
  29. k A ( T m - 1 i - T m i ) Δ x + k A ( T m + 1 i - T m i ) Δ x + e m A Δ x = ( ρ c p Δ x A ) ( T m i + 1 - T m i ) Δ x kA\frac{(T_{m-1}^{i}-T_{m}^{i})}{\Delta{x}}+kA\frac{(T_{m+1}^{i}-T_{m}^{i})}{% \Delta{x}}+{e_{m}}A\Delta{x}=(\rho c_{p}\Delta xA)\frac{(T_{m}^{i+1}-T_{m}^{i}% )}{\Delta x}
  30. ( T m i + 1 ) (T_{m}^{i+1})
  31. T m i + 1 = τ ( T m + 1 i - T m i ) + ( 1 - 2 τ ) T m i + τ ( e m Δ x 2 ) k {T_{m}^{i+1}}=\tau{(T_{m+1}^{i}-T_{m}^{i})}+{(1-2\tau)}T_{m}^{i}+\tau\frac{(e_% {m}\Delta{x}^{2})}{k}
  32. τ = ( α Δ t ) Δ x 2 \tau=\frac{(\alpha\Delta t)}{\Delta x^{2}}\,
  33. α = k ρ c p \alpha=\frac{k}{\rho c_{p}}\,
  34. τ \tau
  35. α \alpha
  36. c p c_{p}
  37. Δ t \Delta t
  38. Δ x \Delta x
  39. q s o l a r q_{solar}
  40. ( W / m 2 ) (W/m^{2})
  41. h A ( T i - T 0 i ) + κ A q s o l = ( ρ c p Δ x A ) ( T 1 i - T 0 i ) Δ x hA{(T_{\infty}^{i}-T_{0}^{i})}+\kappa Aq_{sol}=(\rho c_{p}\Delta xA)\frac{(T_{% 1}^{i}-T_{0}^{i})}{\Delta x}

Chalcone_4'-O-glucosyltransferase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Champernowne_distribution.html

  1. f ( y ; α , λ , y 0 ) = n cosh [ α ( y - y 0 ) ] + λ , - < y < , f(y;\alpha,\lambda,y_{0})=\frac{n}{\cosh[\alpha(y-y_{0})]+\lambda},\qquad-% \infty<y<\infty,
  2. α , λ , y 0 \alpha,\lambda,y_{0}
  3. f ( y ) = n 1 / 2 e α ( y - y 0 ) + λ + 1 / 2 e - α ( y - y 0 ) , f(y)=\frac{n}{1/2e^{\alpha(y-y_{0})}+\lambda+1/2e^{-\alpha(y-y_{0})}},
  4. cosh y = ( e y + e - y ) / 2. \cosh y=(e^{y}+e^{-y})/2.
  5. λ = 1 \lambda=1
  6. y 0 = 0 , α = 1 , λ = 1 y_{0}=0,\alpha=1,\lambda=1
  7. f ( y ) = 1 e y + 2 + e - y = e y ( 1 + e y ) 2 , f(y)=\frac{1}{e^{y}+2+e^{-y}}=\frac{e^{y}}{(1+e^{y})^{2}},
  8. f ( x ) = n x [ 1 / 2 ( x / x 0 ) - α + λ + a / 2 ( x / x 0 ) α ] , x > 0 , f(x)=\frac{n}{x[1/2(x/x_{0})^{-\alpha}+\lambda+a/2(x/x_{0})^{\alpha}]},\qquad x% >0,
  9. f ( x ) = α x α - 1 x 0 α [ 1 + ( x / x 0 ) α ] 2 , x > 0. f(x)=\frac{\alpha x^{\alpha-1}}{x_{0}^{\alpha}[1+(x/x_{0})^{\alpha}]^{2}},% \qquad x>0.

Chanoclavine-I_dehydrogenase.html

  1. \rightleftharpoons

Chapman–Kolmogorov_equation.html

  1. p i 1 , , i n ( f 1 , , f n ) p_{i_{1},\ldots,i_{n}}(f_{1},\ldots,f_{n})
  2. p i 1 , , i n - 1 ( f 1 , , f n - 1 ) = - p i 1 , , i n ( f 1 , , f n ) d f n p_{i_{1},\ldots,i_{n-1}}(f_{1},\ldots,f_{n-1})=\int_{-\infty}^{\infty}p_{i_{1}% ,\ldots,i_{n}}(f_{1},\ldots,f_{n})\,df_{n}
  3. p i 1 , , i n ( f 1 , , f n ) = p i 1 ( f 1 ) p i 2 ; i 1 ( f 2 f 1 ) p i n ; i n - 1 ( f n f n - 1 ) , p_{i_{1},\ldots,i_{n}}(f_{1},\ldots,f_{n})=p_{i_{1}}(f_{1})p_{i_{2};i_{1}}(f_{% 2}\mid f_{1})\cdots p_{i_{n};i_{n-1}}(f_{n}\mid f_{n-1}),
  4. p i ; j ( f i f j ) p_{i;j}(f_{i}\mid f_{j})
  5. i > j i>j
  6. p i 3 ; i 1 ( f 3 f 1 ) = - p i 3 ; i 2 ( f 3 f 2 ) p i 2 ; i 1 ( f 2 f 1 ) d f 2 . p_{i_{3};i_{1}}(f_{3}\mid f_{1})=\int_{-\infty}^{\infty}p_{i_{3};i_{2}}(f_{3}% \mid f_{2})p_{i_{2};i_{1}}(f_{2}\mid f_{1})\,df_{2}.
  7. P ( t + s ) = P ( t ) P ( s ) P(t+s)=P(t)P(s)\,
  8. P ( t ) = P t . P(t)=P^{t}.\,

Chebotaryov_theorem_on_roots_of_unity.html

  1. Ω \Omega
  2. a i j = ω i j , 1 i , j n a_{ij}=\omega^{ij},1\leq i,j\leq n
  3. ω = e 2 i π / n , n \omega=e^{2i\pi/n},n\in\mathbb{N}
  4. n n
  5. Ω \Omega

Chemical_milling.html

  1. E = s / t E=s/t

Chemical_plant_cost_indexes.html

  1. Cost at A = Cost at B index at A index at B \,\text{Cost at A}=\,\text{Cost at B}\cdot\frac{\,\text{index at A }}{\,\text{% index at B }}

Chemical_reaction_network_theory.html

  1. 2 H 2 + O 2 2 H 2 O C + O 2 C O 2 \begin{array}[]{rcl}2H_{2}+O_{2}&\rightarrow&2H_{2}O\\ C+O_{2}&\rightarrow&CO_{2}\end{array}
  2. H 2 H_{2}
  3. O 2 O_{2}
  4. C C
  5. H 2 O H_{2}O
  6. C O 2 CO_{2}
  7. a a
  8. H 2 H_{2}
  9. b b
  10. O 2 O_{2}
  11. c c
  12. H 2 O H_{2}O
  13. a ( t ) , b ( t ) a(t),b(t)
  14. x ( t ) = ( a ( t ) b ( t ) c ( t ) ) x(t)=\left(\begin{array}[]{c}a(t)\\ b(t)\\ c(t)\\ \vdots\end{array}\right)
  15. x ˙ d x d t = ( d a d t d b d t d c d t ) . \dot{x}\equiv\frac{dx}{dt}=\left(\begin{array}[]{c}\frac{da}{dt}\\ \frac{db}{dt}\\ \frac{dc}{dt}\\ \vdots\end{array}\right).
  16. x ˙ = f ( x ) \dot{x}=f(x)
  17. x ˙ = Γ V ( x ) \dot{x}=\Gamma V(x)
  18. Γ \Gamma
  19. Γ \Gamma
  20. V ( x ) V(x)
  21. V ( x ) V(x)
  22. A 2 + 2 Z 2 A Z A_{2}+2Z\rightleftharpoons 2AZ
  23. B + Z B Z B+Z\rightleftharpoons BZ
  24. A Z + B Z A B + 2 Z AZ+BZ\to AB+2Z
  25. A 2 , B A_{2},B
  26. O 2 O_{2}
  27. C O 2 CO_{2}
  28. B + Z ( B Z ) B+Z\rightleftharpoons(BZ)

Chemistry_of_photolithography.html

  1. γ SG = γ SL - γ LG cos θ \gamma_{\mathrm{SG}}=\gamma_{\mathrm{SL}}-\gamma_{\mathrm{LG}}\cos\theta
  2. γ SG \gamma_{\mathrm{SG}}
  3. ( J / m 2 ) {(J/m^{2})}
  4. γ SL \gamma_{\mathrm{SL}}
  5. ( J / m 2 ) {(J/m^{2})}
  6. γ LG \gamma_{\mathrm{LG}}
  7. ( J / m 2 ) {(J/m^{2})}
  8. d = ( 4 ρ ω 2 3 μ t ) - 1 / 2 d=({{4\rho\omega^{2}\over\ 3\mu}t})^{-1/2}
  9. ( k g / m 3 ) {(kg/m^{3})}
  10. ( P a s ) {(Pa\cdot s)}
  11. ( r a d / s ) {(rad/s)}
  12. ( s ) {(s)}

Chen–Gackstatter_surface.html

  1. M i j M_{ij}
  2. - 4 π ( i + 1 ) j -4\pi(i+1)j
  3. M 11 M_{11}
  4. - 8 π -8\pi

Chirp_mass.html

  1. m 1 m_{1}
  2. m 2 m_{2}
  3. = ( m 1 m 2 ) 3 / 5 ( m 1 + m 2 ) 1 / 5 \mathcal{M}=\frac{(m_{1}m_{2})^{3/5}}{(m_{1}+m_{2})^{1/5}}

Chitin_disaccharide_deacetylase.html

  1. \rightleftharpoons

Chlorophyll(ide)_b_reductase.html

  1. \rightleftharpoons

Chlorophyllide-a_oxygenase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Cholest-4-en-3-one_26-monooxygenase.html

  1. \rightleftharpoons

Cholesterol-5,6-oxide_hydrolase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Chung–Fuchs_theorem.html

  1. X n X_{n}
  2. X n = Z 1 + + Z n X_{n}=Z_{1}+...+Z_{n}
  3. Z 1 , Z 2 , , Z n Z_{1},Z_{2},...,Z_{n}
  4. m = 1 m=1
  5. E ( | Z i | ) < E(|Z_{i}|)<\infty
  6. E ( Z i ) = 0 E(Z_{i})=0
  7. m = 2 m=2
  8. E ( | Z i 2 | ) < E(|Z^{2}_{i}|)<\infty
  9. E ( Z i ) = 0 E(Z_{i})=0
  10. ϵ > 0 \forall\epsilon>0
  11. P ( n 0 0 , \exist n n 0 , | X n | < ϵ ) = 1 P(\forall n_{0}\geq 0,\exist n\geq n_{0},|X_{n}|<\epsilon)=1
  12. m 3 m\geq 3
  13. A > 0 \forall A>0
  14. P ( \exist n 0 0 , n n 0 , | X n | A ) = 1 P(\exist n_{0}\geq 0,\forall n\geq n_{0},|X_{n}|\geq A)=1

Classical_capacity.html

  1. 𝒩 \mathcal{N}
  2. χ ( 𝒩 ) = max ρ X A I ( X ; B ) 𝒩 ( ρ ) \chi(\mathcal{N})=\max_{\rho^{XA}}I(X;B)_{\mathcal{N}(\rho)}
  3. ρ X A \rho^{XA}
  4. ρ X A = x p X ( x ) | x x | X ρ x A , \rho^{XA}=\sum_{x}p_{X}(x)|x\rangle\langle x|^{X}\otimes\rho_{x}^{A},
  5. p X ( x ) p_{X}(x)
  6. ρ x A \rho_{x}^{A}
  7. 𝒩 \mathcal{N}
  8. I ( X ; B ) I(X;B)
  9. ρ \rho
  10. σ \sigma
  11. ω \omega
  12. x ρ x . x\mapsto\rho_{x}.
  13. x x
  14. ρ x \rho_{x}
  15. ρ x \rho_{x}
  16. Tr { ρ x ρ x } = 0 \mathrm{Tr}\,\left\{\rho_{x}\rho_{x^{\prime}}\right\}=0
  17. x x x\neq x^{\prime}
  18. ρ x \rho_{x}
  19. ρ x \rho_{x}
  20. { Λ m } m \left\{\Lambda_{m}\right\}_{m}
  21. Λ m 0 m \Lambda_{m}\geq 0\ \ \ \ \forall m
  22. m Λ m = I . \sum_{m}\Lambda_{m}=I.
  23. ρ \rho
  24. { Λ m } \left\{\Lambda_{m}\right\}
  25. p ( m ) p\left(m\right)
  26. m m
  27. p ( m ) = Tr { Λ m ρ } , p\left(m\right)=\,\text{Tr}\left\{\Lambda_{m}\rho\right\},
  28. ρ m = 1 p ( m ) Λ m ρ Λ m , \rho_{m}^{\prime}=\frac{1}{p\left(m\right)}\sqrt{\Lambda_{m}}\rho\sqrt{\Lambda% _{m}},
  29. m m
  30. { p X ( x ) , ρ x } \left\{p_{X}\left(x\right),\rho_{x}\right\}
  31. ρ x p X ( x ) ρ x \rho\equiv\sum_{x}p_{X}\left(x\right)\rho_{x}
  32. Λ \Lambda
  33. I Λ 0 I\geq\Lambda\geq 0
  34. ρ \rho
  35. Tr { Λ ρ } 1 - ϵ . \,\text{Tr}\left\{\Lambda\rho\right\}\geq 1-\epsilon.
  36. Λ ρ x Λ \sqrt{\Lambda}\rho_{x}\sqrt{\Lambda}
  37. ρ x \rho_{x}
  38. 𝔼 X { Λ ρ X Λ - ρ X 1 } 2 ϵ . \mathbb{E}_{X}\left\{\left\|\sqrt{\Lambda}\rho_{X}\sqrt{\Lambda}-\rho_{X}% \right\|_{1}\right\}\leq 2\sqrt{\epsilon}.
  39. A 1 \left\|A\right\|_{1}
  40. A A
  41. A 1 \left\|A\right\|_{1}\equiv
  42. { A A } \left\{\sqrt{A^{\dagger}A}\right\}
  43. ρ \rho
  44. σ \sigma
  45. Λ \Lambda
  46. 0 ρ , σ , Λ I 0\leq\rho,\sigma,\Lambda\leq I
  47. Tr { Λ ρ } Tr { Λ σ } + ρ - σ 1 . \,\text{Tr}\left\{\Lambda\rho\right\}\leq\,\text{Tr}\left\{\Lambda\sigma\right% \}+\left\|\rho-\sigma\right\|_{1}.
  48. Λ \Lambda
  49. ρ \rho
  50. Λ \Lambda
  51. σ \sigma
  52. ρ \rho
  53. σ \sigma
  54. σ \sigma
  55. 0 σ 0\leq\sigma
  56. T r { σ } 1 Tr\left\{\sigma\right\}\leq 1
  57. Π 1 \Pi_{1}
  58. Π N \Pi_{N}
  59. Tr { σ } - Tr { Π N Π 1 σ Π 1 Π N } 2 i = 1 N Tr { ( I - Π i ) σ } , \,\text{Tr}\left\{\sigma\right\}-\,\text{Tr}\left\{\Pi_{N}\cdots\Pi_{1}\ % \sigma\ \Pi_{1}\cdots\Pi_{N}\right\}\leq 2\sqrt{\sum_{i=1}^{N}\,\text{Tr}\left% \{\left(I-\Pi_{i}\right)\sigma\right\}},
  60. Pr { ( A 1 A N ) c } = Pr { A 1 c A N c } i = 1 N Pr { A i c } , \Pr\left\{\left(A_{1}\cap\cdots\cap A_{N}\right)^{c}\right\}=\Pr\left\{A_{1}^{% c}\cup\cdots\cup A_{N}^{c}\right\}\leq\sum_{i=1}^{N}\Pr\left\{A_{i}^{c}\right\},
  61. A 1 A_{1}
  62. A N A_{N}
  63. Tr { ( I - Π 1 Π N Π 1 ) ρ } i = 1 N Tr { ( I - Π i ) ρ } , \,\text{Tr}\left\{\left(I-\Pi_{1}\cdots\Pi_{N}\cdots\Pi_{1}\right)\rho\right\}% \leq\sum_{i=1}^{N}\,\text{Tr}\left\{\left(I-\Pi_{i}\right)\rho\right\},
  64. Π 1 Π N \Pi_{1}\cdots\Pi_{N}
  65. Π 1 \Pi_{1}
  66. Π N \Pi_{N}
  67. Π 1 = | + + | \Pi_{1}=\left|+\right\rangle\left\langle+\right|
  68. Π 2 = | 0 0 | \Pi_{2}=\left|0\right\rangle\left\langle 0\right|
  69. ρ = | 0 0 | \rho=\left|0\right\rangle\left\langle 0\right|
  70. x ρ x x\rightarrow\rho_{x}
  71. p X ( x ) p_{X}\left(x\right)
  72. M M
  73. x n x^{n}
  74. p X n ( x n ) p_{X^{n}}\left(x^{n}\right)
  75. { x n ( m ) } m [ M ] \left\{x^{n}\left(m\right)\right\}_{m\in\left[M\right]}
  76. ρ x n ( m ) = ρ x 1 ( m ) ρ x n ( m ) . \rho_{x^{n}\left(m\right)}=\rho_{x_{1}\left(m\right)}\otimes\cdots\otimes\rho_% {x_{n}\left(m\right)}.
  77. { ρ x n ( m ) } \left\{\rho_{x^{n}\left(m\right)}\right\}
  78. 𝔼 X n { ρ X n } = x n p X n ( x n ) ρ x n = ρ n , \mathbb{E}_{X^{n}}\left\{\rho_{X^{n}}\right\}=\sum_{x^{n}}p_{X^{n}}\left(x^{n}% \right)\rho_{x^{n}}=\rho^{\otimes n},
  79. ρ = x p X ( x ) ρ x \rho=\sum_{x}p_{X}\left(x\right)\rho_{x}
  80. { Π ρ , δ n , I - Π ρ , δ n } \left\{\Pi_{\rho,\delta}^{n},I-\Pi_{\rho,\delta}^{n}\right\}
  81. m th m^{\,\text{th}}
  82. m th m^{\,\text{th}}
  83. { Π ρ x n ( m ) , δ , I - Π ρ x n ( m ) , δ } \left\{\Pi_{\rho_{x^{n}\left(m\right)},\delta},I-\Pi_{\rho_{x^{n}\left(m\right% )},\delta}\right\}
  84. 𝔼 X n { Tr { Π ρ , δ ρ X n } } = Tr { Π ρ , δ 𝔼 X n { ρ X n } } \mathbb{E}_{X^{n}}\left\{\,\text{Tr}\left\{\Pi_{\rho,\delta}\ \rho_{X^{n}}% \right\}\right\}=\,\text{Tr}\left\{\Pi_{\rho,\delta}\ \mathbb{E}_{X^{n}}\left% \{\rho_{X^{n}}\right\}\right\}
  85. = Tr { Π ρ , δ ρ n } =\,\text{Tr}\left\{\Pi_{\rho,\delta}\ \rho^{\otimes n}\right\}
  86. 1 - ϵ , \geq 1-\epsilon,
  87. Π ρ x n ( m ) , δ \Pi_{\rho_{x^{n}\left(m\right)},\delta}
  88. ρ x n ( m ) \rho_{x^{n}\left(m\right)}
  89. 𝔼 X n { Tr { Π ρ X n , δ ρ X n } } 1 - ϵ . \mathbb{E}_{X^{n}}\left\{\,\text{Tr}\left\{\Pi_{\rho_{X^{n}},\delta}\ \rho_{X^% {n}}\right\}\right\}\geq 1-\epsilon.
  90. m th m^{\,\text{th}}
  91. Tr { Π ρ X n ( m ) , δ Π ^ ρ X n ( m - 1 ) , δ Π ^ ρ X n ( 1 ) , δ Π ρ , δ n ρ x n ( m ) Π ρ , δ n Π ^ ρ X n ( 1 ) , δ Π ^ ρ X n ( m - 1 ) , δ Π ρ X n ( m ) , δ } , \,\text{Tr}\left\{\Pi_{\rho_{X^{n}\left(m\right)},\delta}\hat{\Pi}_{\rho_{X^{n% }\left(m-1\right)},\delta}\cdots\hat{\Pi}_{\rho_{X^{n}\left(1\right)},\delta}% \ \Pi_{\rho,\delta}^{n}\ \rho_{x^{n}\left(m\right)}\ \Pi_{\rho,\delta}^{n}\ % \hat{\Pi}_{\rho_{X^{n}\left(1\right)},\delta}\cdots\hat{\Pi}_{\rho_{X^{n}\left% (m-1\right)},\delta}\Pi_{\rho_{X^{n}\left(m\right)},\delta}\right\},
  92. Π ^ I - Π \hat{\Pi}\equiv I-\Pi
  93. m th m^{\,\text{th}}
  94. 1 - Tr { Π ρ X n ( m ) , δ Π ^ ρ X n ( m - 1 ) , δ Π ^ ρ X n ( 1 ) , δ Π ρ , δ n ρ x n ( m ) Π ρ , δ n Π ^ ρ X n ( 1 ) , δ Π ^ ρ X n ( m - 1 ) , δ Π ρ X n ( m ) , δ } , 1-\,\text{Tr}\left\{\Pi_{\rho_{X^{n}\left(m\right)},\delta}\hat{\Pi}_{\rho_{X^% {n}\left(m-1\right)},\delta}\cdots\hat{\Pi}_{\rho_{X^{n}\left(1\right)},\delta% }\ \Pi_{\rho,\delta}^{n}\ \rho_{x^{n}\left(m\right)}\ \Pi_{\rho,\delta}^{n}\ % \hat{\Pi}_{\rho_{X^{n}\left(1\right)},\delta}\cdots\hat{\Pi}_{\rho_{X^{n}\left% (m-1\right)},\delta}\Pi_{\rho_{X^{n}\left(m\right)},\delta}\right\},
  95. 1 - 1 M m Tr { Π ρ X n ( m ) , δ Π ^ ρ X n ( m - 1 ) , δ Π ^ ρ X n ( 1 ) , δ Π ρ , δ n ρ x n ( m ) Π ρ , δ n Π ^ ρ X n ( 1 ) , δ Π ^ ρ X n ( m - 1 ) , δ Π ρ X n ( m ) , δ } . 1-\frac{1}{M}\sum_{m}\,\text{Tr}\left\{\Pi_{\rho_{X^{n}\left(m\right)},\delta}% \hat{\Pi}_{\rho_{X^{n}\left(m-1\right)},\delta}\cdots\hat{\Pi}_{\rho_{X^{n}% \left(1\right)},\delta}\ \Pi_{\rho,\delta}^{n}\ \rho_{x^{n}\left(m\right)}\ % \Pi_{\rho,\delta}^{n}\ \hat{\Pi}_{\rho_{X^{n}\left(1\right)},\delta}\cdots\hat% {\Pi}_{\rho_{X^{n}\left(m-1\right)},\delta}\Pi_{\rho_{X^{n}\left(m\right)},% \delta}\right\}.
  96. 1 - 𝔼 X n { 1 M m Tr { Π ρ X n ( m ) , δ Π ^ ρ X n ( m - 1 ) , δ Π ^ ρ X n ( 1 ) , δ Π ρ , δ n ρ X n ( m ) Π ρ , δ n Π ^ ρ X n ( 1 ) , δ Π ^ ρ X n ( m - 1 ) , δ Π ρ X n ( m ) , δ } } . 1-\mathbb{E}_{X^{n}}\left\{\frac{1}{M}\sum_{m}\,\text{Tr}\left\{\Pi_{\rho_{X^{% n}\left(m\right)},\delta}\hat{\Pi}_{\rho_{X^{n}\left(m-1\right)},\delta}\cdots% \hat{\Pi}_{\rho_{X^{n}\left(1\right)},\delta}\ \Pi_{\rho,\delta}^{n}\ \rho_{X^% {n}\left(m\right)}\ \Pi_{\rho,\delta}^{n}\ \hat{\Pi}_{\rho_{X^{n}\left(1\right% )},\delta}\cdots\hat{\Pi}_{\rho_{X^{n}\left(m-1\right)},\delta}\Pi_{\rho_{X^{n% }\left(m\right)},\delta}\right\}\right\}.
  97. 1 = 𝔼 X n { 1 M m Tr { ρ X n ( m ) } } 1=\mathbb{E}_{X^{n}}\left\{\frac{1}{M}\sum_{m}\,\text{Tr}\left\{\rho_{X^{n}% \left(m\right)}\right\}\right\}
  98. = 𝔼 X n { 1 M m Tr { Π ρ , δ n ρ X n ( m ) } + Tr { Π ^ ρ , δ n ρ X n ( m ) } } =\mathbb{E}_{X^{n}}\left\{\frac{1}{M}\sum_{m}\,\text{Tr}\left\{\Pi_{\rho,% \delta}^{n}\rho_{X^{n}\left(m\right)}\right\}+\,\text{Tr}\left\{\hat{\Pi}_{% \rho,\delta}^{n}\rho_{X^{n}\left(m\right)}\right\}\right\}
  99. = 𝔼 X n { 1 M m Tr { Π ρ , δ n ρ X n ( m ) Π ρ , δ n } } + 1 M m Tr { Π ^ ρ , δ n 𝔼 X n { ρ X n ( m ) } } =\mathbb{E}_{X^{n}}\left\{\frac{1}{M}\sum_{m}\,\text{Tr}\left\{\Pi_{\rho,% \delta}^{n}\rho_{X^{n}\left(m\right)}\Pi_{\rho,\delta}^{n}\right\}\right\}+% \frac{1}{M}\sum_{m}\,\text{Tr}\left\{\hat{\Pi}_{\rho,\delta}^{n}\mathbb{E}_{X^% {n}}\left\{\rho_{X^{n}\left(m\right)}\right\}\right\}
  100. = 𝔼 X n { 1 M m Tr { Π ρ , δ n ρ X n ( m ) Π ρ , δ n } } + Tr { Π ^ ρ , δ n ρ n } =\mathbb{E}_{X^{n}}\left\{\frac{1}{M}\sum_{m}\,\text{Tr}\left\{\Pi_{\rho,% \delta}^{n}\rho_{X^{n}\left(m\right)}\Pi_{\rho,\delta}^{n}\right\}\right\}+\,% \text{Tr}\left\{\hat{\Pi}_{\rho,\delta}^{n}\rho^{\otimes n}\right\}
  101. 𝔼 X n { 1 M m Tr { Π ρ , δ n ρ X n ( m ) Π ρ , δ n } } + ϵ \leq\mathbb{E}_{X^{n}}\left\{\frac{1}{M}\sum_{m}\,\text{Tr}\left\{\Pi_{\rho,% \delta}^{n}\rho_{X^{n}\left(m\right)}\Pi_{\rho,\delta}^{n}\right\}\right\}+\epsilon
  102. ϵ \epsilon
  103. 𝔼 X n { 1 M m Tr { Π ρ , δ n ρ X n ( m ) Π ρ , δ n } } \mathbb{E}_{X^{n}}\left\{\frac{1}{M}\sum_{m}\,\text{Tr}\left\{\Pi_{\rho,\delta% }^{n}\rho_{X^{n}\left(m\right)}\Pi_{\rho,\delta}^{n}\right\}\right\}
  104. - 𝔼 X n { 1 M m Tr { Π ρ X n ( m ) , δ Π ^ ρ X n ( m - 1 ) , δ Π ^ ρ X n ( 1 ) , δ Π ρ , δ n ρ X n ( m ) Π ρ , δ n Π ^ ρ X n ( 1 ) , δ Π ^ ρ X n ( m - 1 ) , δ Π ρ X n ( m ) , δ } } . -\mathbb{E}_{X^{n}}\left\{\frac{1}{M}\sum_{m}\,\text{Tr}\left\{\Pi_{\rho_{X^{n% }\left(m\right)},\delta}\hat{\Pi}_{\rho_{X^{n}\left(m-1\right)},\delta}\cdots% \hat{\Pi}_{\rho_{X^{n}\left(1\right)},\delta}\ \Pi_{\rho,\delta}^{n}\ \rho_{X^% {n}\left(m\right)}\ \Pi_{\rho,\delta}^{n}\ \hat{\Pi}_{\rho_{X^{n}\left(1\right% )},\delta}\cdots\hat{\Pi}_{\rho_{X^{n}\left(m-1\right)},\delta}\Pi_{\rho_{X^{n% }\left(m\right)},\delta}\right\}\right\}.
  105. σ = Π ρ , δ n ρ X n ( m ) Π ρ , δ n \sigma=\Pi_{\rho,\delta}^{n}\rho_{X^{n}\left(m\right)}\Pi_{\rho,\delta}^{n}
  106. Π ρ X n ( m ) , δ \Pi_{\rho_{X^{n}\left(m\right)},\delta}
  107. Π ^ ρ X n ( m - 1 ) , δ \hat{\Pi}_{\rho_{X^{n}\left(m-1\right)},\delta}
  108. Π ^ ρ X n ( 1 ) , δ \hat{\Pi}_{\rho_{X^{n}\left(1\right)},\delta}
  109. 𝔼 X n { 1 M m 2 [ Tr { ( I - Π ρ X n ( m ) , δ ) Π ρ , δ n ρ X n ( m ) Π ρ , δ n } + i = 1 m - 1 Tr { Π ρ X n ( i ) , δ Π ρ , δ n ρ X n ( m ) Π ρ , δ n } ] 1 / 2 } . \mathbb{E}_{X^{n}}\left\{\frac{1}{M}\sum_{m}2\left[\,\text{Tr}\left\{\left(I-% \Pi_{\rho_{X^{n}\left(m\right)},\delta}\right)\Pi_{\rho,\delta}^{n}\rho_{X^{n}% \left(m\right)}\Pi_{\rho,\delta}^{n}\right\}+\sum_{i=1}^{m-1}\,\text{Tr}\left% \{\Pi_{\rho_{X^{n}\left(i\right)},\delta}\Pi_{\rho,\delta}^{n}\rho_{X^{n}\left% (m\right)}\Pi_{\rho,\delta}^{n}\right\}\right]^{1/2}\right\}.
  110. 2 [ 𝔼 X n { 1 M m Tr { ( I - Π ρ X n ( m ) , δ ) Π ρ , δ n ρ X n ( m ) Π ρ , δ n } + i = 1 m - 1 Tr { Π ρ X n ( i ) , δ Π ρ , δ n ρ X n ( m ) Π ρ , δ n } } ] 1 / 2 2\left[\mathbb{E}_{X^{n}}\left\{\frac{1}{M}\sum_{m}\,\text{Tr}\left\{\left(I-% \Pi_{\rho_{X^{n}\left(m\right)},\delta}\right)\Pi_{\rho,\delta}^{n}\rho_{X^{n}% \left(m\right)}\Pi_{\rho,\delta}^{n}\right\}+\sum_{i=1}^{m-1}\,\text{Tr}\left% \{\Pi_{\rho_{X^{n}\left(i\right)},\delta}\Pi_{\rho,\delta}^{n}\rho_{X^{n}\left% (m\right)}\Pi_{\rho,\delta}^{n}\right\}\right\}\right]^{1/2}
  111. 2 [ 𝔼 X n { 1 M m Tr { ( I - Π ρ X n ( m ) , δ ) Π ρ , δ n ρ X n ( m ) Π ρ , δ n } + i m Tr { Π ρ X n ( i ) , δ Π ρ , δ n ρ X n ( m ) Π ρ , δ n } } ] 1 / 2 , \leq 2\left[\mathbb{E}_{X^{n}}\left\{\frac{1}{M}\sum_{m}\,\text{Tr}\left\{% \left(I-\Pi_{\rho_{X^{n}\left(m\right)},\delta}\right)\Pi_{\rho,\delta}^{n}% \rho_{X^{n}\left(m\right)}\Pi_{\rho,\delta}^{n}\right\}+\sum_{i\neq m}\,\text{% Tr}\left\{\Pi_{\rho_{X^{n}\left(i\right)},\delta}\Pi_{\rho,\delta}^{n}\rho_{X^% {n}\left(m\right)}\Pi_{\rho,\delta}^{n}\right\}\right\}\right]^{1/2},
  112. m th m^{\,\text{th}}
  113. 𝔼 X n { 1 M m Tr { ( I - Π ρ X n ( m ) , δ ) Π ρ , δ n ρ X n ( m ) Π ρ , δ n } } \mathbb{E}_{X^{n}}\left\{\frac{1}{M}\sum_{m}\,\text{Tr}\left\{\left(I-\Pi_{% \rho_{X^{n}\left(m\right)},\delta}\right)\Pi_{\rho,\delta}^{n}\rho_{X^{n}\left% (m\right)}\Pi_{\rho,\delta}^{n}\right\}\right\}
  114. 𝔼 X n { 1 M m Tr { ( I - Π ρ X n ( m ) , δ ) ρ X n ( m ) } + ρ X n ( m ) - Π ρ , δ n ρ X n ( m ) Π ρ , δ n 1 } \leq\mathbb{E}_{X^{n}}\left\{\frac{1}{M}\sum_{m}\,\text{Tr}\left\{\left(I-\Pi_% {\rho_{X^{n}\left(m\right)},\delta}\right)\rho_{X^{n}\left(m\right)}\right\}+% \left\|\rho_{X^{n}\left(m\right)}-\Pi_{\rho,\delta}^{n}\rho_{X^{n}\left(m% \right)}\Pi_{\rho,\delta}^{n}\right\|_{1}\right\}
  115. ϵ + 2 ϵ . \leq\epsilon+2\sqrt{\epsilon}.
  116. i m 𝔼 X n { Tr { Π ρ X n ( i ) , δ Π ρ , δ n ρ X n ( m ) Π ρ , δ n } } \sum_{i\neq m}\mathbb{E}_{X^{n}}\left\{\,\text{Tr}\left\{\Pi_{\rho_{X^{n}\left% (i\right)},\delta}\ \Pi_{\rho,\delta}^{n}\ \rho_{X^{n}\left(m\right)}\ \Pi_{% \rho,\delta}^{n}\right\}\right\}
  117. = i m Tr { 𝔼 X n { Π ρ X n ( i ) , δ } Π ρ , δ n 𝔼 X n { ρ X n ( m ) } Π ρ , δ n } =\sum_{i\neq m}\,\text{Tr}\left\{\mathbb{E}_{X^{n}}\left\{\Pi_{\rho_{X^{n}% \left(i\right)},\delta}\right\}\ \Pi_{\rho,\delta}^{n}\ \mathbb{E}_{X^{n}}% \left\{\rho_{X^{n}\left(m\right)}\right\}\ \Pi_{\rho,\delta}^{n}\right\}
  118. = i m Tr { 𝔼 X n { Π ρ X n ( i ) , δ } Π ρ , δ n ρ n Π ρ , δ n } =\sum_{i\neq m}\,\text{Tr}\left\{\mathbb{E}_{X^{n}}\left\{\Pi_{\rho_{X^{n}% \left(i\right)},\delta}\right\}\ \Pi_{\rho,\delta}^{n}\ \rho^{\otimes n}\ \Pi_% {\rho,\delta}^{n}\right\}
  119. i m 2 - n [ H ( B ) - δ ] Tr { 𝔼 X n { Π ρ X n ( i ) , δ } Π ρ , δ n } \leq\sum_{i\neq m}2^{-n\left[H\left(B\right)-\delta\right]}\ \,\text{Tr}\left% \{\mathbb{E}_{X^{n}}\left\{\Pi_{\rho_{X^{n}\left(i\right)},\delta}\right\}\ % \Pi_{\rho,\delta}^{n}\right\}
  120. X n ( m ) X^{n}\left(m\right)
  121. X n ( i ) X^{n}\left(i\right)
  122. i m 2 - n [ H ( B ) - δ ] 𝔼 X n { Tr { Π ρ X n ( i ) , δ } } \leq\sum_{i\neq m}2^{-n\left[H\left(B\right)-\delta\right]}\ \mathbb{E}_{X^{n}% }\left\{\,\text{Tr}\left\{\Pi_{\rho_{X^{n}\left(i\right)},\delta}\right\}\right\}
  123. i m 2 - n [ H ( B ) - δ ] 2 n [ H ( B | X ) + δ ] \leq\sum_{i\neq m}2^{-n\left[H\left(B\right)-\delta\right]}\ 2^{n\left[H\left(% B|X\right)+\delta\right]}
  124. = i m 2 - n [ I ( X ; B ) - 2 δ ] =\sum_{i\neq m}2^{-n\left[I\left(X;B\right)-2\delta\right]}
  125. M 2 - n [ I ( X ; B ) - 2 δ ] . \leq M\ 2^{-n\left[I\left(X;B\right)-2\delta\right]}.
  126. Π ρ , δ n I \Pi_{\rho,\delta}^{n}\leq I
  127. 1 - 𝔼 X n { 1 M m Tr { Π ρ X n ( m ) , δ Π ^ ρ X n ( m - 1 ) , δ Π ^ ρ X n ( 1 ) , δ Π ρ , δ n ρ X n ( m ) Π ρ , δ n Π ^ ρ X n ( 1 ) , δ Π ^ ρ X n ( m - 1 ) , δ Π ρ X n ( m ) , δ } } 1-\mathbb{E}_{X^{n}}\left\{\frac{1}{M}\sum_{m}\,\text{Tr}\left\{\Pi_{\rho_{X^{% n}\left(m\right)},\delta}\hat{\Pi}_{\rho_{X^{n}\left(m-1\right)},\delta}\cdots% \hat{\Pi}_{\rho_{X^{n}\left(1\right)},\delta}\ \Pi_{\rho,\delta}^{n}\ \rho_{X^% {n}\left(m\right)}\ \Pi_{\rho,\delta}^{n}\ \hat{\Pi}_{\rho_{X^{n}\left(1\right% )},\delta}\cdots\hat{\Pi}_{\rho_{X^{n}\left(m-1\right)},\delta}\Pi_{\rho_{X^{n% }\left(m\right)},\delta}\right\}\right\}
  128. ϵ + 2 [ ( ϵ + 2 ϵ ) + M 2 - n [ I ( X ; B ) - 2 δ ] ] 1 / 2 . \leq\epsilon+2\left[\left(\epsilon+2\sqrt{\epsilon}\right)+M\ 2^{-n\left[I% \left(X;B\right)-2\delta\right]}\right]^{1/2}.
  129. M = 2 n [ I ( X ; B ) - 3 δ ] M=2^{n\left[I\left(X;B\right)-3\delta\right]}

Clavaminate_synthase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons

Cluster-expansion_approach.html

  1. B ^ 𝐪 \hat{B}^{\dagger}_{\mathbf{q}}
  2. B ^ 𝐪 \hat{B}_{\mathbf{q}}
  3. 𝐪 \hbar\mathbf{q}
  4. B B
  5. a ^ λ , 𝐤 \hat{a}^{\dagger}_{\lambda,\mathbf{k}}
  6. a ^ λ , 𝐤 \hat{a}_{\lambda,\mathbf{k}}
  7. 𝐤 \hbar\mathbf{k}
  8. λ \lambda
  9. N ^ B ^ 1 B ^ K a ^ 1 a ^ N a ^ a ^ N a ^ a ^ 1 B ^ J B ^ 1 \langle\hat{N}\rangle\equiv\langle\hat{B}^{\dagger}_{1}\cdots\hat{B}^{\dagger}% _{K}\ \hat{a}^{\dagger}_{1}\cdots\hat{a}^{\dagger}_{N_{\hat{a}}}\hat{a}_{N_{% \hat{a}}}\cdots\hat{a}_{1}\ \hat{B}_{J}\cdots\hat{B}_{1}\rangle
  10. N = N B ^ + N a ^ N=N_{\hat{B}}+N_{\hat{a}}
  11. N B ^ = J + K N_{\hat{B}}=J+K
  12. N N
  13. N N
  14. ( N + 1 ) (N+1)
  15. i t N ^ = T [ N ^ ] + Hi [ N ^ + 1 ] \mathrm{i}\hbar\frac{\partial}{\partial t}\langle\hat{N}\rangle=\mathrm{T}% \left[\langle\hat{N}\rangle\right]+\mathrm{Hi}\left[\langle\hat{N}+1\rangle\right]
  16. T T
  17. Hi [ N ^ + 1 ] \mathrm{Hi}[\langle\hat{N}+1\rangle]
  18. 1 \langle 1\rangle
  19. 2 \langle 2\rangle
  20. 2 S = 1 1 \langle 2\rangle_{\mathrm{S}}=\langle 1\rangle\langle 1\rangle
  21. 1 \langle 1\rangle
  22. N S \langle N\rangle_{\mathrm{S}}
  23. N N
  24. B ^ B ^ \hat{B}\rightarrow\langle\hat{B}\rangle
  25. 2 \langle 2\rangle
  26. 2 \langle 2\rangle
  27. 2 S \langle 2\rangle_{\mathrm{S}}
  28. 2 = 2 S + Δ 2 \langle 2\rangle=\langle 2\rangle_{\mathrm{S}}+\Delta\langle 2\rangle
  29. Δ \Delta
  30. Δ 2 = 2 - 2 S \Delta\langle 2\rangle=\langle 2\rangle-\langle 2\rangle_{\mathrm{S}}
  31. 3 = 3 S + 1 Δ 2 + Δ 3 , N = N S + N - 2 S Δ 2 + N - 4 S Δ 2 Δ 2 + + N - 3 S Δ 3 + N - 5 S Δ 3 Δ 2 + + Δ N , \begin{aligned}\displaystyle\langle 3\rangle&\displaystyle=\langle 3\rangle_{% \mathrm{S}}+\langle 1\rangle\ \Delta\langle 2\rangle+\Delta\langle 3\rangle\,,% \\ \displaystyle\langle N\rangle&\displaystyle=\langle N\rangle_{\mathrm{S}}\\ &\displaystyle\quad+\langle N-2\rangle_{\mathrm{S}}\ \Delta\langle 2\rangle\\ &\displaystyle\quad+\langle N-4\rangle_{\mathrm{S}}\ \Delta\langle 2\rangle\ % \Delta\langle 2\rangle+\dots\\ &\displaystyle\quad+\langle N-3\rangle_{\mathrm{S}}\ \Delta\langle 3\rangle\\ &\displaystyle\quad+\langle N-5\rangle_{\mathrm{S}}\ \Delta\langle 3\rangle\ % \Delta\langle 2\rangle+\dots\\ &\displaystyle\quad+\Delta\langle N\rangle\,,\end{aligned}
  32. Δ N \Delta\langle N\rangle
  33. Δ 2 \Delta\langle 2\rangle
  34. Δ 3 \Delta\langle 3\rangle
  35. i t Δ N ^ = T [ Δ N ^ ] + NL [ 1 ^ , Δ 2 ^ , , Δ N ^ ] + Hi [ Δ N ^ + 1 ] , \mathrm{i}\hbar\frac{\partial}{\partial t}\Delta\langle\hat{N}\rangle=\mathrm{% T}\left[\Delta\langle\hat{N}\rangle\right]+\mathrm{NL}\left[\langle\hat{1}% \rangle,\Delta\langle\hat{2}\rangle,\cdots,\Delta\langle\hat{N}\rangle\right]+% \mathrm{Hi}\left[\Delta\langle\hat{N}+1\rangle\right]\,,
  36. NL [ ] \mathrm{NL}\left[\cdots\right]
  37. Hi [ Δ C ^ + 1 ] \mathrm{Hi}\left[\Delta\langle\hat{C}+1\rangle\right]
  38. C C
  39. C C
  40. C C
  41. B ^ \hat{B}
  42. [ B ^ ] J B ^ K \langle[\hat{B}^{\dagger}]^{J}\hat{B}^{K}\rangle
  43. [ B ^ ] J B ^ K \langle[\hat{B}^{\dagger}]^{J}\hat{B}^{K}\rangle

CMP-N,N'-diacetyllegionaminic_acid_synthase.html

  1. \rightleftharpoons

Co-citation_Proximity_Analysis.html

  1. 1 2 n \frac{1}{2^{n}}

Cobalt-precorrin-5B_(C1)-methyltransferase.html

  1. \rightleftharpoons

Cobalt-precorrin-7_(C15)-methyltransferase_(decarboxylating).html

  1. \rightleftharpoons

Cobalt-precorrin_5A_hydrolase.html

  1. \rightleftharpoons

Cocaine_esterase.html

  1. \rightleftharpoons

Codazzi_tensor.html

  1. ( M , g ) (M,g)
  2. n 3 n\geq 3
  3. T T
  4. \nabla
  5. T T
  6. ( X T ) g ( Y , Z ) = ( Y T ) g ( X , Z ) (\nabla_{X}T)g(Y,Z)=(\nabla_{Y}T)g(X,Z)

Codeine_3-O-demethylase.html

  1. \rightleftharpoons

CoDel.html

  1. 100 100
  2. 100 2 {100\over\sqrt{2}}
  3. 100 3 {100\over\sqrt{3}}
  4. 100 4 {100\over\sqrt{4}}
  5. 100 5 {100\over\sqrt{5}}

Cohen_ring.html

  1. ( 0 , p ) (0,p)

Coherent_effects_in_semiconductor_optics.html

  1. 𝐏 {\mathbf{P}}
  2. 𝐄 {\mathbf{E}}
  3. ( - 1 c 2 2 t 2 ) 𝐄 ( 𝐫 , t ) = μ 0 2 t 2 𝐏 ( 𝐫 , t ) (\nabla\cdot\nabla-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}){\mathbf% {E}}({\mathbf{r}},t)=\mu_{0}\frac{\partial^{2}}{\partial t^{2}}{\mathbf{P}}({% \mathbf{r}},t)
  4. 𝐏 {\mathbf{P}}
  5. 2 t 2 𝐏 \frac{\partial^{2}}{\partial t^{2}}{\mathbf{P}}
  6. 𝐄 {\mathbf{E}}
  7. λ \lambda
  8. 𝐄 2 t 2 𝐏 {\mathbf{E}}\propto\frac{\partial^{2}}{\partial t^{2}}{\mathbf{P}}
  9. 𝐄 ( t ) {\mathbf{E}}(t)
  10. 𝐏 ( t ) {\mathbf{P}}(t)
  11. v v
  12. c c
  13. 𝐏 {\mathbf{P}}
  14. p c v p_{cv}
  15. 𝐏 = 1 V c , v ( 𝐝 c v p c v + c . c . ) {\mathbf{P}}=\frac{1}{V}\sum_{c,v}({\mathbf{d}}_{cv}p_{cv}+\mathrm{c.c.})
  16. 𝐝 c v {\mathbf{d}}_{cv}
  17. v v
  18. c c
  19. c . c . \mathrm{c.c.}
  20. V V
  21. ϵ c \epsilon_{c}
  22. ϵ v \epsilon_{v}
  23. e - i ϵ c t / \mathrm{e}^{-\mathrm{i}\epsilon_{c}\,t/\hbar}
  24. e - i ϵ v t / \mathrm{e}^{-\mathrm{i}\epsilon_{v}\,t/\hbar}
  25. p c v p_{cv}
  26. e - i ( ϵ c - ϵ v ) t / \mathrm{e}^{-\mathrm{i}(\epsilon_{c}-\epsilon_{v})t/\hbar}
  27. t = 0 t=0
  28. p c v ( t = 0 ) = p c v , 0 p_{cv}(t=0)=p_{cv,0}
  29. 𝐏 ( t ) = c , v ( 𝐝 c v p c v , 0 e - i ( ϵ c - ϵ v ) t / + c . c . ) {\mathbf{P}}(t)=\sum_{c,v}({\mathbf{d}}_{cv}p_{cv,0}\,\mathrm{e}^{-\mathrm{i}(% \epsilon_{c}-\epsilon_{v})t/\hbar}+\mathrm{c.c.})
  30. 𝐏 ( t ) {\mathbf{P}}(t)
  31. 𝐏 ( t ) {\mathbf{P}}(t)
  32. 𝐏 ( t ) {\mathbf{P}}(t)
  33. i t p c v = Δ ϵ p c v + 𝐄 𝐝 I \mathrm{i}\hbar\frac{\partial}{\partial t}p_{cv}=\Delta\epsilon\,p_{cv}+{% \mathbf{E}}\cdot{\mathbf{d}}I
  34. i t I = 2 𝐄 𝐝 ( p c v - p c v ) . \mathrm{i}\hbar\frac{\partial}{\partial t}I=2{\mathbf{E}}\cdot{\mathbf{d}}(p_{% cv}-p_{cv}^{\star}).
  35. Δ ϵ = ( ϵ c - ϵ v ) \Delta\epsilon=(\epsilon_{c}-\epsilon_{v})
  36. I I
  37. 𝐄 {\mathbf{E}}
  38. p p
  39. 𝐄 𝐝 {\mathbf{E}}\cdot{\mathbf{d}}
  40. I I
  41. 𝐄 = 𝟎 {\mathbf{E}}=\mathbf{0}
  42. p p
  43. p c v ( t ) e - i Δ ϵ t / p_{cv}(t)\propto\mathrm{e}^{-\mathrm{i}\Delta\epsilon\,t/\hbar}
  44. i t p 𝐤 = Δ ε 𝐤 p 𝐤 + Ω 𝐤 ( n 𝐤 c - n 𝐤 v ) + i t p 𝐤 | corr , \mathrm{i}\hbar\frac{\partial}{\partial t}p_{\mathbf{k}}=\Delta\varepsilon_{% \mathbf{k}}\,p_{\mathbf{k}}+\Omega_{\mathbf{k}}\,(n^{c}_{\mathbf{k}}-n^{v}_{% \mathbf{k}})+\mathrm{i}\hbar\frac{\partial}{\partial t}p_{\mathbf{k}}|_{\,% \text{corr}}\,,
  45. i t n 𝐤 c = ( Ω 𝐤 p 𝐤 - Ω 𝐤 p 𝐤 ) + i t n 𝐤 c | corr , \mathrm{i}\hbar\frac{\partial}{\partial t}n^{c}_{\mathbf{k}}=(\Omega_{\mathbf{% k}}^{\star}\,p_{\mathbf{k}}-\Omega_{\mathbf{k}}\,p_{\mathbf{k}}^{\star})+% \mathrm{i}\hbar\frac{\partial}{\partial t}n^{c}_{\mathbf{k}}|_{\,\text{corr}}\,,
  46. i t n 𝐤 v = - ( Ω 𝐤 p 𝐤 - Ω 𝐤 p 𝐤 ) + i t n 𝐤 v | corr . \mathrm{i}\hbar\frac{\partial}{\partial t}n^{v}_{\mathbf{k}}=-(\Omega_{\mathbf% {k}}^{\star}\,p_{\mathbf{k}}-\Omega_{\mathbf{k}}\,p_{\mathbf{k}}^{\star})+% \mathrm{i}\hbar\frac{\partial}{\partial t}n^{v}_{\mathbf{k}}|_{\,\text{corr}}\,.
  47. p 𝐤 p_{\mathbf{k}}
  48. n 𝐤 c n^{c}_{\mathbf{k}}
  49. n 𝐤 v n^{v}_{\mathbf{k}}
  50. c c
  51. v v
  52. 𝐤 \hbar{\mathbf{k}}
  53. Δ ε 𝐤 \Delta\varepsilon_{\mathbf{k}}
  54. Ω 𝐤 \Omega_{\mathbf{k}}
  55. p 𝐤 p_{\mathbf{k}^{\prime}}
  56. n 𝐤 c n^{c}_{\mathbf{k}^{\prime}}
  57. n 𝐤 v n^{v}_{\mathbf{k}^{\prime}}
  58. 𝐤 \hbar{\mathbf{k}^{\prime}}
  59. 𝐤 \hbar{\mathbf{k}}
  60. | corr |\text{corr}
  61. t = 0 t=0
  62. 𝐏 ( t ) {\mathbf{P}}(t)
  63. l e - i Δ ω l t \sum_{l}\mathrm{e}^{-\mathrm{i}\Delta\omega_{l}t}
  64. l l
  65. | 𝐏 ( t ) | 2 |{\mathbf{P}}(t)|^{2}
  66. | 𝐄 ( t ) | 2 |{\mathbf{E}}(t)|^{2}
  67. 2 π / ( Δ ω l - Δ ω j ) 2\pi/(\Delta\omega_{l}-\Delta\omega_{j})
  68. [ 1 + cos ( ( Δ ω 1 - Δ ω 2 ) t ) ] [1+\cos((\Delta\omega_{1}-\Delta\omega_{2})t)]
  69. t = 0 t=0
  70. t = τ > 0 t=\tau>0
  71. p p p\rightarrow p^{\star}
  72. t = 2 τ t=2\tau
  73. t = 2 τ t=2\tau
  74. E p E_{p}
  75. E t E_{t}
  76. Δ α ( ω ) \Delta\alpha(\omega)
  77. α pump on ( ω ) \alpha_{\,\text{pump on}}(\omega)
  78. α pump off ( ω ) \alpha_{\,\text{pump off}}(\omega)
  79. Δ α \Delta\alpha
  80. Δ α \Delta\alpha
  81. N N
  82. N N
  83. N N
  84. λ / 2 \lambda/2
  85. λ \lambda
  86. N N
  87. N 2 N^{2}
  88. γ rad \gamma_{\mathrm{rad}}
  89. N N
  90. γ rad = N γ rad , 0 \gamma_{\mathrm{rad}}=N\gamma_{\mathrm{rad},0}
  91. γ rad , 0 \gamma_{\mathrm{rad},0}
  92. N N
  93. e - N γ rad , 0 t \mathrm{e}^{-N\gamma_{\mathrm{rad},0}\,t}
  94. N N
  95. N 2 N^{2}
  96. 1 N \frac{1}{N}

Cohesive_zone_model.html

  1. σ y \sigma_{y}

Colneleate_synthase.html

  1. \rightleftharpoons

Combined_Linear_Congruential_Generator.html

  1. X i ( j = 1 k ( - 1 ) j - 1 Y i , j ) ( mod ( m 1 - 1 ) ) X_{i}\equiv\left(\sum_{j=1}^{k}(-1)^{j-1}Y_{i,j}\right)\;\;(\mathop{{\rm mod}}% (m_{1}-1))
  2. m 1 m_{1}
  3. Y i , j Y_{i,j}
  4. X i X_{i}
  5. R i { X i / m 1 for X i > 0 ( m 1 - 1 ) / m 1 for X i = 0 R_{i}\equiv\begin{cases}X_{i}/m_{1}&\,\text{for }X_{i}>0\\ (m_{1}-1)/m_{1}&\,\text{for }X_{i}=0\end{cases}
  6. R i R_{i}
  7. Z i = ( j = 1 k W i , j ) ( mod ( m 1 - 1 ) ) Z_{i}=\left(\sum_{j=1}^{k}W_{i,j}\right)\;\;(\mathop{{\rm mod}}(m_{1}-1))
  8. P = ( ( m 1 - 1 ) ( m 2 - 1 ) ( m k - 1 ) ) / ( 2 k - 1 ) P=((m_{1}-1)(m_{2}-1)\cdots(m_{k}-1))/(2^{k-1})
  9. k = 2 k=2
  10. a 1 = 40 , 014 a_{1}=40,014
  11. m 1 = 2 , 147 , 483 , 563 m_{1}=2,147,483,563
  12. a 2 = 40 , 692 a_{2}=40,692
  13. m 2 = 2 , 147 , 483 , 399 m_{2}=2,147,483,399
  14. c 1 = c 2 = 0 c_{1}=c_{2}=0
  15. Y 0 , 1 Y_{0,1}
  16. Y 0 , 2 Y_{0,2}
  17. i = 0 i=0
  18. Y i + 1 , 1 = 40 , 014 × Y i , 1 ( mod 2 , 147 , 483 , 563 ) Y_{i+1,1}=40,014\times Y_{i,1}\;\;(\mathop{{\rm mod}}2,147,483,563)
  19. Y i + 1 , 2 = 40 , 692 × Y i , 2 ( mod 2 , 147 , 483 , 399 ) Y_{i+1,2}=40,692\times Y_{i,2}\;\;(\mathop{{\rm mod}}2,147,483,399)
  20. X i + 1 = ( Y i + 1 , 1 - Y i + 1 , 2 ) ( mod 2 , 147 , 483 , 562 ) X_{i+1}=(Y_{i+1,1}-Y_{i+1,2})\;\;(\mathop{{\rm mod}}2,147,483,562)
  21. R i + 1 = { X i + 1 / 2 , 147 , 483 , 563 for X i + 1 > 0 ( X i + 1 / 2 , 147 , 483 , 563 ) + 1 for X i + 1 < 0 2 , 147 , 483 , 562 / 2 , 147 , 483 , 563 for X i + 1 = 0 R_{i+1}=\begin{cases}X_{i+1}/2,147,483,563&\,\text{for }X_{i+1}>0\\ (X_{i+1}/2,147,483,563)+1&\,\text{for }X_{i+1}<0\\ 2,147,483,562/2,147,483,563&\,\text{for }X_{i+1}=0\end{cases}
  22. ( m - 1 ) (m-1)
  23. ( m 1 - 1 ) ( m 2 - 1 ) / 2 = 2.3 × 10 18 (m_{1}-1)(m_{2}-1)/2=2.3\times 10^{18}

Common_Consolidated_Corporate_Tax_Base.html

  1. T i = t i π [ α i K K i K + α i L L i L + α i S S i S ] . T_{i}=t_{i}\cdot\pi\left[\alpha_{i}^{K}\cdot\frac{K_{i}}{K}+\alpha_{i}^{L}% \cdot\frac{L_{i}}{L}+\alpha_{i}^{S}\cdot\frac{S_{i}}{S}\right].
  2. G e r m a n y : 30 % \euro 1 , 000 , 000 [ 1 3 \euro 150 , 000 , 000 \euro 200 , 000 , 000 + 1 3 \euro 3 , 000 , 000 \euro 8 , 000 , 000 + 1 3 \euro 135 , 000 , 000 \euro 200 , 000 , 000 ] = \euro 180 , 000. Germany:30\%\cdot\euro 1,000,000\cdot\left[\frac{1}{3}\cdot\frac{\euro 150,000% ,000}{\euro 200,000,000}+\frac{1}{3}\cdot\frac{\euro 3,000,000}{\euro 8,000,00% 0}+\frac{1}{3}\cdot\frac{\euro 135,000,000}{\euro 200,000,000}\right]=\euro 18% 0,000.
  3. S l o v a k R e p u b l i c : 19 % \euro 1 , 000 , 000 [ 1 3 \euro 50 , 000 , 000 \euro 200 , 000 , 000 + 1 3 \euro 5 , 000 , 000 \euro 8 , 000 , 000 + 1 3 \euro 65 , 000 , 000 \euro 200 , 000 , 000 ] = \euro 76 , 000. SlovakRepublic:19\%\cdot\euro 1,000,000\cdot\left[\frac{1}{3}\cdot\frac{\euro 5% 0,000,000}{\euro 200,000,000}+\frac{1}{3}\cdot\frac{\euro 5,000,000}{\euro 8,0% 00,000}+\frac{1}{3}\cdot\frac{\euro 65,000,000}{\euro 200,000,000}\right]=% \euro 76,000.
  4. t i π K i K t_{i}\cdot\pi\cdot\frac{K_{i}}{K}
  5. G e r m a n y : 30 % \euro 1 , 000 , 000 [ 1 3 \euro 50 , 000 , 000 \euro 200 , 000 , 000 + 1 3 \euro 3 , 000 , 000 \euro 8 , 000 , 000 + 1 3 \euro 135 , 000 , 000 \euro 200 , 000 , 000 ] = \euro 130 , 000. Germany:30\%\cdot\euro 1,000,000\cdot\left[\frac{1}{3}\cdot\frac{\euro 50,000,% 000}{\euro 200,000,000}+\frac{1}{3}\cdot\frac{\euro 3,000,000}{\euro 8,000,000% }+\frac{1}{3}\cdot\frac{\euro 135,000,000}{\euro 200,000,000}\right]=\euro 130% ,000.
  6. S l o v a k R e p u b l i c : 19 % \euro 1 , 000 , 000 [ 1 3 \euro 150 , 000 , 000 \euro 200 , 000 , 000 + 1 3 \euro 5 , 000 , 000 \euro 8 , 000 , 000 + 1 3 \euro 65 , 000 , 000 \euro 200 , 000 , 000 ] = \euro 107 , 667. SlovakRepublic:19\%\cdot\euro 1,000,000\cdot\left[\frac{1}{3}\cdot\frac{\euro 1% 50,000,000}{\euro 200,000,000}+\frac{1}{3}\cdot\frac{\euro 5,000,000}{\euro 8,% 000,000}+\frac{1}{3}\cdot\frac{\euro 65,000,000}{\euro 200,000,000}\right]=% \euro 107,667.

Common_spatial_pattern.html

  1. 𝐗 1 \mathbf{X}_{1}
  2. ( n , t 1 ) (n,t_{1})
  3. 𝐗 2 \mathbf{X}_{2}
  4. ( n , t 2 ) (n,t_{2})
  5. n n
  6. t 1 t_{1}
  7. t 2 t_{2}
  8. 𝐰 T \mathbf{w}\text{T}
  9. 𝐰 = arg max 𝐰 𝐰𝐗 1 2 𝐰𝐗 2 2 \mathbf{w}={\arg\max}_{\mathbf{w}}\frac{\left\|\mathbf{wX}_{1}\right\|^{2}}{% \left\|\mathbf{wX}_{2}\right\|^{2}}
  10. 𝐑 1 = 𝐗 1 𝐗 1 T t 1 \mathbf{R}_{1}=\frac{\mathbf{X}_{1}\mathbf{X}_{1}\text{T}}{t_{1}}
  11. 𝐑 2 = 𝐗 2 𝐗 2 T t 2 \mathbf{R}_{2}=\frac{\mathbf{X}_{2}\mathbf{X}_{2}\text{T}}{t_{2}}
  12. 𝐏 = [ 𝐩 1 𝐩 n ] \mathbf{P}=\begin{bmatrix}\mathbf{p}_{1}&\cdots&\mathbf{p}_{n}\end{bmatrix}
  13. 𝐃 \mathbf{D}
  14. { λ 1 , , λ n } \{\lambda_{1},\cdots,\lambda_{n}\}
  15. 𝐏 - 1 𝐑 1 𝐏 = 𝐃 \mathbf{P}^{-1}\mathbf{R}_{1}\mathbf{P}=\mathbf{D}
  16. 𝐏 - 1 𝐑 2 𝐏 = 𝐈 n \mathbf{P}^{-1}\mathbf{R}_{2}\mathbf{P}=\mathbf{I}_{n}
  17. 𝐈 n \mathbf{I}_{n}
  18. 𝐑 2 - 1 𝐑 1 \mathbf{R}_{2}^{-1}\mathbf{R}_{1}
  19. 𝐑 2 - 1 𝐑 1 = 𝐏𝐃𝐏 - 1 \mathbf{R}_{2}^{-1}\mathbf{R}_{1}=\mathbf{PDP}^{-1}
  20. 𝐰 T \mathbf{w}\text{T}
  21. 𝐏 \mathbf{P}
  22. 𝐰 = 𝐩 1 T \mathbf{w}=\mathbf{p}_{1}\text{T}
  23. 𝐏 \mathbf{P}
  24. λ i = 𝐩 i T 𝐗 1 2 𝐩 i T 𝐗 2 2 \mathbf{\lambda}_{i}=\frac{\left\|\mathbf{p}_{i}\text{T}\mathbf{X}_{1}\right\|% ^{2}}{\left\|\mathbf{p}_{i}\text{T}\mathbf{X}_{2}\right\|^{2}}
  25. E i E_{i}
  26. i i
  27. [ 𝐩 1 𝐩 i ] \begin{bmatrix}\mathbf{p}_{1}&\cdots&\mathbf{p}_{i}\end{bmatrix}
  28. E i = arg max E ( min p E 𝐩 T 𝐗 1 2 𝐩 T 𝐗 2 2 ) E_{i}={\arg\max}_{E}\begin{pmatrix}\min_{p\in E}\frac{\left\|\mathbf{p\text{T}% X}_{1}\right\|^{2}}{\left\|\mathbf{p\text{T}X}_{2}\right\|^{2}}\end{pmatrix}
  29. F j F_{j}
  30. j j
  31. [ 𝐩 n - j + 1 𝐩 n ] \begin{bmatrix}\mathbf{p}_{n-j+1}&\cdots&\mathbf{p}_{n}\end{bmatrix}
  32. F j = arg min F ( max p F 𝐩 T 𝐗 1 2 𝐩 T 𝐗 2 2 ) F_{j}={\arg\min}_{F}\begin{pmatrix}\max_{p\in F}\frac{\left\|\mathbf{p\text{T}% X}_{1}\right\|^{2}}{\left\|\mathbf{p\text{T}X}_{2}\right\|^{2}}\end{pmatrix}
  33. 𝐏 \mathbf{P}
  34. 𝐗 1 \mathbf{X}_{1}
  35. 𝐗 1 \mathbf{X}_{1}
  36. 𝐑 2 \mathbf{R}_{2}

Complex_algebraic_variety.html

  1. 𝐏 n \mathbb{C}\mathbf{P}^{n}

Complexification_(Lie_group).html

  1. g = u e x p i X g=u•expiX
  2. u u
  3. X X
  4. G G
  5. f : G H f:G→H
  6. H H
  7. f = F φ f=F∘φ
  8. G G
  9. 𝖌 𝖌
  10. 𝐆 \mathbf{G}
  11. 𝖌 𝐂 𝖌⊗\mathbf{C}
  12. π : 𝐆 G π:\mathbf{G}→G
  13. f : G H f:G→H
  14. f π = E Φ f∘π=E∘Φ
  15. K K
  16. E E
  17. f f
  18. K K
  19. G G
  20. G G
  21. { 1 } G o G Γ { 1 } \{1\}\rightarrow G^{o}\rightarrow G\rightarrow\Gamma\rightarrow\{1\}
  22. { 1 } ( G o ) 𝐂 G 𝐂 Γ { 1 } \{1\}\rightarrow(G^{o})_{\mathbf{C}}\rightarrow G_{\mathbf{C}}\rightarrow% \Gamma\rightarrow\{1\}
  23. G G
  24. 𝐓 \mathbf{T}
  25. S L ( 2 , 𝐑 ) SL(2,\mathbf{R})
  26. G G
  27. A A
  28. C ( G ) C(G)
  29. G G
  30. Δ f ( g , h ) = f ( g h ) . \displaystyle{\Delta f(g,h)=f(gh).}
  31. A A
  32. A A
  33. 𝐂 \mathbf{C}
  34. f f ( g ) f↦f(g)
  35. g g
  36. G G
  37. G G
  38. A A
  39. 𝐂 \mathbf{C}
  40. G G
  41. A A
  42. σ σ
  43. G G
  44. σ σ
  45. G G
  46. U ( V ) U(V)
  47. V V
  48. X X
  49. e x p t X exptX
  50. G G
  51. t t
  52. W = V 𝐂 W=V⊕\mathbf{C}
  53. G G
  54. G G
  55. u u
  56. G G
  57. g g
  58. G L ( V ) GL(V)
  59. g g
  60. 𝐂 \mathbf{C}
  61. G L ( V ) GL(V)
  62. G G
  63. U ( V ) U(V)
  64. G G
  65. G G
  66. g g
  67. u u
  68. p p
  69. g = u p g=u⋅p
  70. u u
  71. G G
  72. p p
  73. p = e x p T p=expT
  74. T T
  75. h = e x p z T h=expzT
  76. z z
  77. 𝐂 \mathbf{C}
  78. z z
  79. T T
  80. i X iX
  81. X X
  82. G G
  83. G G
  84. G G
  85. G G
  86. p p
  87. G 𝐂 = G P = G exp i 𝔤 , \displaystyle{G_{\mathbf{C}}=G\cdot P=G\cdot\exp i\mathfrak{g},}
  88. 𝖌 𝖌
  89. G G
  90. P P
  91. G G
  92. G G
  93. P P
  94. G L ( V ) GL(V)
  95. g g
  96. G L ( V ) GL(V)
  97. g = X D Y \displaystyle{g=XDY}
  98. X X
  99. Y Y
  100. D D
  101. g g
  102. X , Y X,Y
  103. D D
  104. X X
  105. A A
  106. B B
  107. G G
  108. U ( V ) U(V)
  109. 𝔤 𝐂 = 𝔫 - 𝔱 𝐂 𝔫 + , \displaystyle{\mathfrak{g}_{\mathbf{C}}=\mathfrak{n}_{-}\oplus\mathfrak{t}_{% \mathbf{C}}\oplus\mathfrak{n}_{+},}
  110. 𝖙 𝖙
  111. T T
  112. G G
  113. V V
  114. T , 𝖙 T,𝖙
  115. V V
  116. T T
  117. 𝖆 𝖓 𝖆⊕𝖓
  118. 𝖆 𝖆
  119. 𝖓 𝖓
  120. W W
  121. 𝖆 𝖆
  122. 𝖓 𝖓
  123. w w
  124. 𝖆 𝖆
  125. 𝖓 𝖓
  126. 𝖓 𝖓
  127. d i m 𝖓 dim𝖓
  128. 𝖓 𝖓
  129. 𝖓 / 𝖓 𝖓/𝖓
  130. 𝖆 𝖆
  131. V V
  132. N - T 𝐂 N + \displaystyle{N_{-}T_{\mathbf{C}}N_{+}}
  133. 𝐓 \mathbf{T}
  134. U ( V ) U(V)
  135. N ± = 𝐍 ± G 𝐂 , T 𝐂 = 𝐓 𝐂 G 𝐂 . \displaystyle{N_{\pm}=\mathbf{N}_{\pm}\cap G_{\mathbf{C}},\,\,\,T_{\mathbf{C}}% =\mathbf{T}_{\mathbf{C}}\cap G_{\mathbf{C}}.}
  136. G L ( V ) GL(V)
  137. T T
  138. B B
  139. G 𝐂 = σ W B σ B , \displaystyle{G_{\mathbf{C}}=\bigcup_{\sigma\in W}B\sigma B,}
  140. B B
  141. B σ B BσB
  142. σ σ
  143. W W
  144. B B
  145. S L ( n , 𝐂 ) SL(n,\mathbf{C})
  146. S O ( n , 𝐂 ) SO(n,\mathbf{C})
  147. S p ( n , 𝐂 ) Sp(n,\mathbf{C})
  148. S L ( n , 𝐂 ) SL(n,\mathbf{C})
  149. B B
  150. g g
  151. S L ( n , 𝐂 ) SL(n,\mathbf{C})
  152. b b
  153. B B
  154. b g bg
  155. w w
  156. w b g wbg
  157. B B
  158. g g
  159. N + = N σ M σ . \displaystyle{N_{+}=N_{\sigma}\cdot M_{\sigma}.}
  160. k 1 k−1
  161. k k
  162. ( i , j ) (i,j)
  163. k k
  164. σ σ
  165. S L ( n , 𝐂 ) SL(n,\mathbf{C})
  166. S p ( n , 𝐂 ) Sp(n,\mathbf{C})
  167. J J
  168. n × n n×n
  169. 1 1
  170. 0
  171. A = ( 0 J - J 0 ) . \displaystyle{A=\begin{pmatrix}0&J\\ -J&0\end{pmatrix}.}
  172. S p ( n , 𝐂 ) Sp(n,\mathbf{C})
  173. B B
  174. n , n 1 , , 1 , 1 , , n n,n−1,...,1,−1,...,−n
  175. S p ( n , 𝐂 ) Sp(n,\mathbf{C})
  176. σ σ
  177. σ ( j ) = j σ(j)=−j
  178. θ θ
  179. S p ( n , 𝐂 ) Sp(n,\mathbf{C})
  180. θ θ
  181. g = n σ b = θ ( n ) θ ( σ ) θ ( b ) g=nσb=θ(n)θ(σ)θ(b)
  182. S p ( n , 𝐂 ) Sp(n,\mathbf{C})
  183. S O ( n , 𝐂 ) SO(n,\mathbf{C})
  184. S L ( n , 𝐂 ) SL(n,\mathbf{C})
  185. B = J B=J
  186. G 𝐂 = G A N \displaystyle{G_{\mathbf{C}}=G\cdot A\cdot N}
  187. A N A⋅N
  188. G G
  189. N N
  190. A A
  191. U ( V ) U(V)
  192. G L ( V ) GL(V)
  193. V V
  194. g g
  195. G L ( V ) GL(V)
  196. f i = a i g e i + j < i n j i g e j . \displaystyle{f_{i}=a_{i}ge_{i}+\sum_{j<i}n_{ji}ge_{j}.}
  197. k k
  198. 𝐀𝐍 \mathbf{AN}
  199. 𝐀 \mathbf{A}
  200. 𝐍 \mathbf{N}
  201. A = exp i 𝔱 = 𝐀 G 𝐂 , N = exp 𝔫 + = 𝐍 G 𝐂 . \displaystyle{A=\exp i\mathfrak{t}=\mathbf{A}\cap G_{\mathbf{C}},\,\,\,N=\exp% \mathfrak{n}_{+}=\mathbf{N}\cap G_{\mathbf{C}}.}
  202. G L ( V ) GL(V)
  203. G L ( V ) GL(V)
  204. G × A × N G×A×N
  205. g g
  206. h = g * g h=g*g
  207. h = X D Y h=XDY
  208. X X
  209. D D
  210. Y Y
  211. h h
  212. Y = X * Y=X*
  213. D D
  214. A A
  215. D = e x p i T D=expiT
  216. T T
  217. 𝖙 𝖙
  218. a = e x p i T / 2 a=expiT/2
  219. A A
  220. n = Y n=Y
  221. k k
  222. G G
  223. g = k a n g=kan
  224. G - G-
  225. G G
  226. G / T G/T
  227. G G
  228. T T
  229. G / T G/T
  230. G G
  231. T T
  232. S T S⊆T
  233. x x
  234. X X
  235. 𝖙 𝖙
  236. y y
  237. S S
  238. v v
  239. λ λ
  240. 𝐂 v \mathbf{C}v
  241. G G
  242. H H
  243. v v
  244. T T
  245. H H
  246. T T
  247. V V
  248. P P
  249. v v
  250. α α
  251. P P
  252. B B
  253. v v
  254. α α
  255. ( λ , α ) = 0 (λ,α)=0
  256. 𝐩 \mathbf{p}
  257. P P
  258. v v
  259. 𝔭 = 𝔟 ( α , λ ) = 0 𝔤 - α . \displaystyle{\mathfrak{p}=\mathfrak{b}\oplus\bigoplus_{(\alpha,\lambda)=0}% \mathfrak{g}_{-\alpha}.}
  260. H = P G H=P∩G
  261. 𝐩 𝖌 \mathbf{p}∩𝖌
  262. A N AN
  263. 𝐂 v \mathbf{C}v
  264. G G
  265. v v
  266. G / H = G 𝐂 / P . \displaystyle{G/H=G_{\mathbf{C}}/P.}
  267. G / T = G 𝐂 / B . \displaystyle{G/T=G_{\mathbf{C}}/B.}
  268. T T
  269. H H
  270. λ λ
  271. G G
  272. P P
  273. G / H G/H
  274. G / Z G/Z
  275. Z Z
  276. G G
  277. P P
  278. P P
  279. P P
  280. B B
  281. P = σ W λ B σ B , \displaystyle{P=\bigcup_{\sigma\in W_{\lambda}}B\sigma B,}
  282. λ λ
  283. W W
  284. λ λ
  285. 𝔱 \mathfrak{t}
  286. 𝔱 𝔰 \mathfrak{t}\ominus\mathfrak{s}
  287. 𝔱 \mathfrak{t}
  288. 𝔤 α \mathfrak{g}_{\alpha}
  289. 𝔱 \mathfrak{t}
  290. 𝔤 𝐂 \mathfrak{g}_{\mathbf{C}}
  291. 𝔤 𝐂 \mathfrak{g}_{\mathbf{C}}
  292. 𝔤 𝐂 \mathfrak{g}_{\mathbf{C}}
  293. c 0 = φ c 1 φ - 1 \displaystyle{c_{0}=\varphi\circ c_{1}\circ\varphi^{-1}}
  294. c 0 ( X + i Y ) = σ ( X ) - i σ ( Y ) \displaystyle{c_{0}(X+iY)=\sigma(X)-i\sigma(Y)}
  295. 𝔤 \mathfrak{g}
  296. ( X , Y ) = - B ( X , c ( Y ) ) , \displaystyle{(X,Y)=-B(X,c(Y)),}
  297. 𝔤 𝐂 \mathfrak{g}_{\mathbf{C}}
  298. φ = ( ψ 2 ) 1 / 4 \displaystyle{\varphi=(\psi^{2})^{1/4}}
  299. c 0 c = φ c 1 φ - 1 c = φ c c 1 φ = ( ψ 2 ) 1 / 2 ψ - 1 = φ - 1 c c 1 φ - 1 = c φ c 1 φ - 1 = c c 0 . \displaystyle{c_{0}c=\varphi c_{1}\varphi^{-1}c=\varphi cc_{1}\varphi=(\psi^{2% })^{1/2}\psi^{-1}=\varphi^{-1}cc_{1}\varphi^{-1}=c\varphi c_{1}\varphi^{-1}=cc% _{0}.}
  300. G 𝐂 = G exp i 𝔤 = G P = P G . \displaystyle{G_{\mathbf{C}}=G\cdot\exp i\mathfrak{g}=G\cdot P=P\cdot G.}
  301. 𝔤 = 𝔨 𝔭 \displaystyle{\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}}
  302. 𝔨 \mathfrak{k}
  303. 𝔤 = 𝔨 𝔭 \displaystyle{\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}}
  304. G 0 = K exp i 𝔭 = K P 0 = P 0 K \displaystyle{G_{0}=K\cdot\exp i\mathfrak{p}=K\cdot P_{0}=P_{0}\cdot K}
  305. 𝔞 \mathfrak{a}
  306. 𝔭 \mathfrak{p}
  307. 𝔞 \mathfrak{a}
  308. 𝔞 \mathfrak{a}
  309. 𝔪 \mathfrak{m}
  310. 𝔞 \mathfrak{a}
  311. 𝔞 \mathfrak{a}
  312. 𝔪 \mathfrak{m}
  313. 𝔪 \mathfrak{m}
  314. 𝔨 \mathfrak{k}
  315. 𝔞 \mathfrak{a}
  316. 𝔪 \mathfrak{m}
  317. 𝔞 \mathfrak{a}
  318. 𝔞 \mathfrak{a}
  319. 𝔞 \mathfrak{a}
  320. 𝔞 0 = i 𝔞 \mathfrak{a}_{0}=i\mathfrak{a}
  321. 𝔭 0 \mathfrak{p}_{0}
  322. 𝔞 0 \mathfrak{a}_{0}
  323. G 0 = K A 0 K . \displaystyle{G_{0}=KA_{0}K.}

Compound_matrix.html

  1. C k ( A ) C_{k}(A)
  2. m × n m\times n
  3. ( m k ) × ( n k ) {\left({{m}\atop{k}}\right)}\times{\left({{n}\atop{k}}\right)}
  4. k × k k\times k
  5. k × k k\times k
  6. C 1 ( A ) \displaystyle C_{1}(A)

Compounding_of_steam_turbines.html

  1. V b , o p t i m u m = V a 1 cos θ 1 2 n V_{b,optimum}=\frac{V_{a1}\cos\theta_{1}}{2n}
  2. V 1 2 2 + h 1 = V 2 2 2 + h 2 \frac{V_{1}^{2}}{2}+{h_{1}}=\frac{V_{2}^{2}}{2}+{h_{2}}

Compressed_sensing_in_speech_signals.html

  1. x R N {x\in R^{N}}
  2. M {\it M}
  3. N {\it N}
  4. M N {M\ll N}
  5. x {x}
  6. Ψ {\Psi}
  7. x = Ψ s y m b o l α {x}={\Psi symbol\alpha}
  8. x R N {x\in R^{\it N}}
  9. Ψ R N × N {\Psi\in R^{\it{N\times N}}}
  10. s y m b o l α R N {symbol\alpha\in R^{\it N}}
  11. Ψ {\Psi}
  12. s y m b o l α {symbol\alpha}
  13. K {\it{K}}
  14. K N {\it{K\ll N}}
  15. x {x}
  16. N × 1 {\it{N\times 1}}
  17. s y m b o l α {symbol\alpha}
  18. Φ {\Phi}
  19. y = Φ x {y=\Phi x}
  20. y R M {y\in R^{\it M}}
  21. Φ R M × N {\Phi\in R^{\it M\times N}}
  22. M N {\it{M\ll N}}
  23. l 1 {l_{1}}
  24. s y m b o l 𝐬𝐲𝐦𝐛𝐨𝐥 α ^ = minimize 𝐬𝐲𝐦𝐛𝐨𝐥 α 1 s.t. 𝐲 = 𝚽 𝐱 = 𝚽 𝚿 𝐬𝐲𝐦𝐛𝐨𝐥 α = 𝐀𝐬𝐲𝐦𝐛𝐨𝐥 α , where 𝐀 = 𝚽 𝚿 {symbol{\hat{\mathbf{symbol\alpha}}}=\mbox{minimize}~{}\;\|\mathbf{symbol% \alpha}\|_{1}\;\;\;\;\mbox{s.t.}~{}\;\;\;\;\mathbf{y}=\mathbf{\Phi x}=\mathbf{% \Phi\Psi}\mathbf{symbol\alpha}=\mathbf{Asymbol\alpha},\;\mbox{where}~{}\;\;% \mathbf{A}=\mathbf{\Phi\Psi}}
  25. Φ {\Phi}
  26. Ψ {\Psi}
  27. s y m b o l α ^ {\hat{symbol\alpha}}
  28. y {y}
  29. l 1 {l_{1}}
  30. s y m b o l α ^ {\hat{symbol\alpha}}
  31. Ψ {\Psi}
  32. x ^ {\hat{x}}
  33. Ψ {\Psi}
  34. s y m b o l α ^ {\hat{symbol\alpha}}
  35. y ) {y})

Compressor_characteristic.html

  1. F ( D , N , m ˙ , p 01 , p 02 , R T 01 , R T 02 ) = 0 F(D,N,\dot{m},p_{01},p_{02},RT_{01},RT_{02})=0
  2. D D
  3. N N
  4. m ˙ \dot{m}
  5. p 01 p_{01}
  6. p 02 p_{02}
  7. T 01 T_{01}
  8. T 02 T_{02}
  9. R R
  10. p 02 p 01 \frac{p_{02}}{p_{01}}
  11. T 01 T 02 \frac{T_{01}}{T_{02}}
  12. m ˙ R T 01 D 2 p 01 \frac{{\dot{m}}{\sqrt{RT_{01}}}}{{D^{2}}{p_{01}}}
  13. N D R T 01 \frac{{N}{D}}{\sqrt{RT_{01}}}
  14. F ( p 02 p 01 , T 01 T 02 , m ˙ T 01 p 01 , N T 01 ) = 0 F\left(\frac{p_{02}}{p_{01}}\ ,\frac{T_{01}}{T_{02}}\ ,\frac{{\dot{m}}{\sqrt{T% _{01}}}}{p_{01}}\ ,\frac{N}{\sqrt{T_{01}}}\ \right)=0
  15. m ˙ T 01 p 01 \frac{{\dot{m}}{\sqrt{T_{01}}}}{p_{01}}
  16. N T 01 \frac{N}{\sqrt{T_{01}}}
  17. p 02 p 01 \frac{p_{02}}{p_{01}}
  18. m ˙ T 01 p 01 \frac{{\dot{m}}{\sqrt{T_{01}}}}{p_{01}}
  19. N T 01 \frac{N}{\sqrt{T_{01}}}
  20. Δ p = f ( Q ) {\Delta p=f(Q)}
  21. Δ p = f ( m ˙ ) {\Delta p=f(\dot{m})}
  22. ~{}
  23. Ψ = g h u 2 {\Psi^{\prime}}=\frac{{g}{h}}{u^{2}}
  24. ~{}
  25. Ψ {\Psi}
  26. p 02 - p 01 ρ u 2 \frac{p_{02}-p_{01}}{{\rho}{u^{2}}}
  27. Ψ α Ψ {\Psi}~{}{\alpha}~{}{\Psi^{\prime}}
  28. u u
  29. h h
  30. ϕ α Q N D 3 {\phi}~{}{\alpha}~{}\frac{Q}{{N}{D^{3}}}
  31. Ψ = f ( ϕ ) {\Psi}=f({\phi})
  32. ( t a n β 2 + t a n α 1 ) (tan\beta_{2}+tan\alpha_{1})
  33. α 1 = α 3 \alpha_{1}=\alpha_{3}
  34. A = t a n β 2 + t a n α 3 A=tan\beta_{2}+tan\alpha_{3}
  35. U U
  36. V 1 V_{1}
  37. V 2 V_{2}
  38. V r 1 V_{r1}
  39. V r 2 V_{r2}
  40. V 3 V_{3}
  41. α 1 , α 2 , α 3 , β 1 \alpha_{1},\alpha_{2},\alpha_{3},\beta_{1}
  42. β 2 \beta_{2}
  43. Ψ = 1 - A ϕ {\Psi}^{\ast}=1-A{\phi}^{\ast}
  44. Ψ = 1 - A ϕ {\Psi}=1-A{\phi}^{\ast}
  45. Ψ = 1 - ( 1 - A Ψ ) {\Psi}=1-(1-A{\Psi}^{\ast})
  46. ϕ ϕ \frac{\phi}{{\phi}^{\ast}}
  47. A A
  48. A A
  49. Ψ {\Psi}
  50. ϕ {\phi}
  51. A A
  52. ϕ {\phi}
  53. Ψ {\Psi}
  54. p A , m ˙ A p_{A},{\dot{m}}_{A}
  55. N 4 N_{4}
  56. m ˙ B {\dot{m}}_{B}
  57. p B p_{B}
  58. m ˙ C {\dot{m}}_{C}
  59. m ˙ S {\dot{m}}_{S}
  60. m ˙ B {\dot{m}}_{B}
  61. m ˙ D {\dot{m}}_{D}
  62. p E p_{E}
  63. m ˙ O - m ˙ S m ˙ O \frac{{\dot{m}_{O}}-{\dot{m}_{S}}}{\dot{m}_{O}}
  64. m ˙ O {\dot{m}_{O}}
  65. m ˙ S {\dot{m}_{S}}
  66. N 4 N_{4}
  67. N 0 N_{0}

Computable_topology.html

  1. T o T_{o}
  2. $\empty$
  3. \forall
  4. \exists
  5. \exists
  6. \perp
  7. \perp
  8. \forall
  9. \perp
  10. \cdot
  11. \subseteq
  12. \subseteq
  13. \subseteq
  14. \cap
  15. \neq
  16. $\empty$
  17. \cdot
  18. $\empty$
  19. \cdot
  20. U i U_{i}
  21. \cup
  22. U i U_{i}
  23. U i U_{i}
  24. U i U_{i}
  25. U i U_{i}
  26. U i U_{i}
  27. U i U_{i}
  28. \cap
  29. U i \subseteq U_{i}\subseteq
  30. U i U_{i}
  31. \cdot
  32. \cap
  33. = =
  34. \cap
  35. f : D D f:D\rightarrow D^{{}^{\prime}}
  36. \subseteq
  37. D D^{{}^{\prime}}
  38. f i i [ D D ] {f_{i}}_{i}\subseteq[D\rightarrow D^{{}^{\prime}}]
  39. f ( x ) = i f i ( x ) f(x)=\cup_{i}f_{i}(x)
  40. [ D D ] \subseteq[D\rightarrow D^{{}^{\prime}}]
  41. [ D D ] [D\rightarrow D^{{}^{\prime}}]
  42. [ D D ] × D D [D\rightarrow D^{{}^{\prime}}]\times D\rightarrow D^{{}^{\prime}}
  43. [ D D ] × D D [D\rightarrow D^{{}^{\prime}}]\times D\rightarrow D^{{}^{\prime}}
  44. [ D D ] \subseteq[D\rightarrow D^{{}^{\prime}}]
  45. [ D D ] [D\rightarrow D^{{}^{\prime}}]
  46. D D D\rightarrow D^{{}^{\prime}}
  47. f : D × D D ′′ f:D\times D^{{}^{\prime}}\rightarrow D^{{}^{\prime\prime}}
  48. g = λ x . f ( x , x 0 ) g=\lambda x.f(x,x_{0})
  49. d = λ x . f ( x 0 , x ) d=\lambda x^{{}^{\prime}}.f(x_{0},x^{{}^{\prime}})
  50. D D^{{}^{\prime}}
  51. \subseteq
  52. g ( sup ( X ) ) = f ( sup ( X ) , x 0 ) ) g(\sup(X))=f(\sup(X),x_{0}^{{}^{\prime}}))
  53. x 0 x_{0}^{{}^{\prime}}
  54. x 0 x_{0}^{{}^{\prime}}
  55. x 0 x_{0}^{{}^{\prime}}
  56. f [ D × D D ′′ ] f\in[D\times D^{{}^{\prime}}\rightarrow D^{{}^{\prime\prime}}]
  57. f ˇ \check{f}
  58. D D^{{}^{\prime}}
  59. f ˇ \check{f}
  60. f ˇ \check{f}
  61. [ D [ D D ′′ ] [D\rightarrow[D^{{}^{\prime}}\rightarrow D^{{}^{\prime\prime}}]
  62. f . f ˇ : [ D × D D ′′ ] [ D [ D D ′′ ] f.\check{f}:[D\times D^{{}^{\prime}}\rightarrow D^{{}^{\prime\prime}}]% \rightarrow[D\rightarrow[D^{{}^{\prime}}\rightarrow D^{{}^{\prime\prime}}]
  63. \subseteq
  64. f ˇ \check{f}
  65. sup x \isin X \sup_{x\isin X}
  66. sup x \isin X \sup_{x\isin X}
  67. f ˇ \check{f}
  68. f . f ˇ f.\check{f}
  69. [ D × D D ′′ ] \subseteq[D\times D^{{}^{\prime}}\rightarrow D^{{}^{\prime\prime}}]
  70. sup y \isin F \sup_{y\isin F}
  71. sup y \isin F \sup_{y\isin F}
  72. \mathbb{R}
  73. ψ \psi
  74. Σ \Sigma
  75. \perp\cup
  76. \cdots
  77. , x 1 x n \mathbb{N},x_{1}...x_{n}
  78. \perp
  79. x \vec{x}
  80. M 1 ) M_{1})
  81. M m M_{m}
  82. Σ \Sigma
  83. Σ \Sigma
  84. B T ( M ) ( ) = , BT(M)(\langle\ \rangle)=\perp,
  85. k * α \langle k\rangle*\alpha
  86. k , α \forall k,\alpha
  87. x n . y M 0 M m - 1 \cdots x_{n}.yM_{0}\cdots M_{m-1}
  88. x n . y \cdots x_{n}.y
  89. k * α \langle k\rangle*\alpha
  90. α \alpha
  91. α \forall\alpha
  92. α \forall\alpha
  93. \geq
  94. \subseteq
  95. \subseteq
  96. Γ \Gamma
  97. Γ \Gamma\rightarrow
  98. Γ \Gamma
  99. B T - 1 BT^{-1}
  100. \cdot
  101. A = ( X , ) A=(X,\cdot)
  102. y ^ \hat{y}
  103. \equiv
  104. y ^ \hat{y}
  105. \forall
  106. \rightarrow
  107. \cdot
  108. \cdot
  109. x 1 , , x n x_{1},...,x_{n}
  110. x 1 , , x n {x_{1},...,x_{n}}
  111. f x 1 x n \exists f\forall x_{1}\cdot\cdot\cdot x_{n}
  112. f x 1 x n = A fx_{1}\cdot\cdot\cdot x_{n}=A
  113. \cdot
  114. \cdot
  115. \cdot
  116. \cdot
  117. A ( x , y A(x,\vec{y}
  118. y \vec{y}
  119. y \vec{y}
  120. y \forall\vec{y}
  121. b \exists b
  122. y \vec{y}
  123. y \vec{y}
  124. x \forall x
  125. y \vec{y}
  126. y \vec{y}
  127. y \vec{y}
  128. y ) \vec{y})\equiv
  129. \cdot
  130. \equiv
  131. ( β ) (\beta)
  132. α 1 \alpha_{1}
  133. \equiv
  134. α 2 \alpha_{2}
  135. \equiv
  136. Γ ( W ) \Gamma(W)
  137. c a \isin Γ ( W ) \Rightarrow c_{a}\isin\Gamma(W)
  138. Γ ( W ) \Gamma(W)
  139. v 0 , v 1 , v_{0},v_{1},...
  140. Γ ( W ) \Gamma(W)\Rightarrow
  141. Γ ( W ) \Gamma(W)
  142. Γ ( W ) \Gamma(W)\Rightarrow
  143. Γ ( W ) \Gamma(W)
  144. Γ ( W ) \Gamma(W)
  145. Γ ( W ) \Tau ( W ) \Gamma(W)\rightarrow\Tau(W)
  146. v i * = w i , c a * = c a v_{i}^{*}=w_{i},c_{a}^{*}=c_{a}
  147. Γ ( W ) \Gamma(W)
  148. c a c_{a}
  149. c a c_{a}
  150. Γ \Gamma
  151. \vdash
  152. \Rightarrow
  153. \vDash

Computationally_bounded_adversary.html

  1. E ( ) E()
  2. D ( ) D()
  3. π \pi
  4. R R
  5. X = E ( M ) X=E(M)
  6. Y = π ( X ) R Y=\pi(X)\oplus R
  7. Y Y
  8. Y Y^{\prime}
  9. Z = π - 1 ( Y R ) Z=\pi^{-1}(Y^{\prime}\oplus R)
  10. M = D ( Z ) M=D(Z)
  11. N N
  12. Z = π - 1 ( Y R ) = π - 1 ( Y N R ) , Z=\pi^{-1}(Y^{\prime}\oplus R)=\pi^{-1}(Y\oplus N\oplus R),
  13. Y = Y N Y^{\prime}=Y\oplus N
  14. = π - 1 ( Y R ) N =\pi^{-1}(Y\oplus R)\oplus N^{\prime}
  15. N = π - 1 ( N ) , N^{\prime}=\pi^{-1}(N),
  16. = X N , =X\oplus N^{\prime},
  17. Y Y
  18. π \pi
  19. N N^{\prime}
  20. M M
  21. 1 - R 1-R

Computer_Arimaa.html

  1. ( 16 8 ) ( 8 2 ) ( 6 2 ) ( 4 2 ) ( 2 1 ) ( 1 1 ) = 64 , 864 , 800 {\displaystyle\left({{16}\atop{8}}\right)}\cdot{\displaystyle\left({{8}\atop{2% }}\right)}\cdot{\displaystyle\left({{6}\atop{2}}\right)}\cdot{\displaystyle% \left({{4}\atop{2}}\right)}\cdot{\displaystyle\left({{2}\atop{1}}\right)}\cdot% {\displaystyle\left({{1}\atop{1}}\right)}=64,864,800
  2. 64 , 864 , 800 2 4.2 10 15 64,864,800^{2}\approx 4.2\cdot 10^{15}
  3. 35 8 17000 3 35^{8}\approxeq 17000^{3}

Conceptual_Spaces.html

  1. x x
  2. y y
  3. z z
  4. x x
  5. y y
  6. z z

Concrete_fracture_analysis.html

  1. f t f_{t}
  2. w w
  3. f t f_{t}
  4. f t f_{t}
  5. f t f_{t}
  6. σ \sigma
  7. τ \tau
  8. d d
  9. λ \lambda
  10. δ \delta
  11. σ \sigma
  12. τ \tau
  13. d d
  14. λ \lambda
  15. δ \delta
  16. l l
  17. l = E G σ 2 l=\frac{EG}{\sigma^{2}}
  18. l l
  19. E E
  20. G G
  21. σ \sigma

Conditional_dependence.html

  1. P ( A B ) = P ( A ) or P ( B A ) = P ( B ) P(A\mid B)=P(A)\,\text{ or }P(B\mid A)=P(B)
  2. P ( B C ) > P ( B C , A ) P(B\mid C)>P(B\mid C,A)

Conflict-Driven_Clause_Learning.html

  1. ( A + C ) ( B + C ) (A^{\prime}+C^{\prime})(B+C)
  2. A A^{\prime}
  3. C C^{\prime}
  4. B B
  5. C C
  6. A + C A^{\prime}+C^{\prime}
  7. B + C B+C
  8. A = F a l s e , B = F a l s e , C = T r u e A=False,B=False,C=True
  9. A A^{\prime}
  10. C C
  11. 2 3 = 8 2^{3}=8
  12. A = F a l s e A=False
  13. B = F a l s e B=False
  14. C = T r u e C=True
  15. ( A B C ) (ABC)
  16. ( A B C ) (ABC)
  17. ( A B ¬ C ) (AB\neg C)
  18. ( A B ) (AB)
  19. ¬ C \neg C
  20. C C

Conformal_loop_ensemble.html

  1. φ : D D \varphi:D\to D^{\prime}
  2. φ \varphi

Conformastatic_spacetimes.html

  1. ( 1 ) d s 2 = - e 2 Ψ ( ρ , ϕ , z ) d t 2 + e - 2 Ψ ( ρ , ϕ , z ) ( d ρ 2 + d z 2 + ρ 2 d ϕ 2 ) , (1)\qquad ds^{2}=-e^{2\Psi(\rho,\phi,z)}dt^{2}+e^{-2\Psi(\rho,\phi,z)}\Big(d% \rho^{2}+dz^{2}+\rho^{2}d\phi^{2}\Big)\;,
  2. ( 2 ) R a b - 1 2 R g a b = 8 π T a b . (2)\qquad R_{ab}-\frac{1}{2}Rg_{ab}=8\pi T_{ab}\;.
  3. Ψ ( ρ , ϕ , z ) \Psi(\rho,\phi,z)
  4. Ψ ( ρ , ϕ , z ) \Psi(\rho,\phi,z)
  5. A a A_{a}
  6. ( 3 ) A a = Φ ( ρ , z , ϕ ) [ d t ] a , (3)\qquad A_{a}=\Phi(\rho,z,\phi)[dt]_{a}\;,
  7. F a b F_{ab}
  8. ( 4 ) F a b = A b ; a - A a ; b , (4)\qquad F_{ab}=A_{b\,;a}-A_{a\,;b}\;,
  9. ( 5 ) T a b ( E M ) = 1 4 π ( F a c F b c - 1 4 g a b F c d F c d ) . (5)\qquad T_{ab}^{(EM)}=\frac{1}{4\pi}\Big(F_{ac}F_{b}^{\;\;c}-\frac{1}{4}g_{% ab}F_{cd}F^{cd}\Big)\;.
  10. Ψ ( ρ , ϕ , z ) \Psi(\rho,\phi,z)
  11. ( 6 ) 2 Ψ = e - 2 Ψ Φ Φ (6)\qquad\nabla^{2}\Psi\,=\,e^{-2\Psi}\,\nabla\Phi\,\nabla\Phi
  12. ( 7 ) Ψ i Ψ j = e - 2 Ψ Φ i Φ j (7)\qquad\Psi_{i}\Psi_{j}=e^{-2\Psi}\Phi_{i}\Phi_{j}
  13. 2 = ρ ρ + 1 ρ ρ + 1 ρ 2 ϕ ϕ + z z \nabla^{2}=\partial_{\rho\rho}+\frac{1}{\rho}\,\partial_{\rho}+\frac{1}{\rho^{% 2}}\partial_{\phi\phi}+\partial_{zz}
  14. = ρ e ^ ρ + 1 ρ ϕ e ^ ϕ + z e ^ z \nabla=\partial_{\rho}\,\hat{e}_{\rho}+\frac{1}{\rho}\partial_{\phi}\,\hat{e}_% {\phi}+\partial_{z}\,\hat{e}_{z}
  15. i , j i\,,j
  16. [ ρ , z , ϕ ] [\rho,z,\phi]
  17. ( 8 ) Ψ E R N = ln L L + M , L = ρ 2 + z 2 , (8)\qquad\Psi_{ERN}\,=\,\ln\frac{L}{L+M}\;,\quad L=\sqrt{\rho^{2}+z^{2}}\;,
  18. ( 9 ) d s 2 = - L 2 ( L + M ) 2 d t 2 + ( L + M ) 2 L 2 ( d ρ 2 + d z 2 + ρ 2 d φ 2 ) . (9)\qquad ds^{2}=-\frac{L^{2}}{(L+M)^{2}}dt^{2}+\frac{(L+M)^{2}}{L^{2}}\,\big(% d\rho^{2}+dz^{2}+\rho^{2}d\varphi^{2}\big)\;.
  19. ( 10 ) L = r - M , z = ( r - M ) cos θ , ρ = ( r - M ) sin θ , (10)\;\;\quad L=r-M\;,\quad z=(r-M)\cos\theta\;,\quad\rho=(r-M)\sin\theta\;,
  20. ( 11 ) d s 2 = - ( 1 - M r ) 2 d t 2 + ( 1 - M r ) 2 d r 2 + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) . (11)\;\;\quad ds^{2}=-\Big(1-\frac{M}{r}\Big)^{2}dt^{2}+\Big(1-\frac{M}{r}\Big% )^{2}dr^{2}+r^{2}\Big(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}\Big)\;.
  21. ( 12 ) d s 2 = - e 2 ψ ( ρ , z ) d t 2 + e 2 γ ( ρ , z ) - 2 ψ ( ρ , z ) ( d ρ 2 + d z 2 ) + e - 2 ψ ( ρ , z ) ρ 2 d ϕ 2 . (12)\;\;\quad ds^{2}=-e^{2\psi(\rho,z)}dt^{2}+e^{2\gamma(\rho,z)-2\psi(\rho,z)% }(d\rho^{2}+dz^{2})+e^{-2\psi(\rho,z)}\rho^{2}d\phi^{2}\,.
  22. γ ( ρ , z ) \gamma(\rho,z)
  23. ψ ( ρ , z ) \psi(\rho,z)
  24. ( 13 ) γ ( ρ , z ) 0 , ψ ( ρ , z ) Ψ ( ρ , ϕ , z ) . (13)\;\;\quad\gamma(\rho,z)\equiv 0\;,\quad\psi(\rho,z)\mapsto\Psi(\rho,\phi,z% )\,.
  25. γ ( ρ , z ) \gamma(\rho,z)
  26. ( 14. a ) 2 ψ = ( ψ ) 2 (14.a)\quad\nabla^{2}\psi=\,(\nabla\psi)^{2}
  27. ( 14. b ) 2 ψ = e - 2 ψ ( Φ ) 2 (14.b)\quad\nabla^{2}\psi=\,e^{-2\psi}(\nabla\Phi)^{2}
  28. ( 14. c ) ψ , ρ 2 - ψ , z 2 = e - 2 ψ ( Φ , ρ 2 - Φ , z 2 ) (14.c)\quad\psi^{2}_{,\,\rho}-\psi^{2}_{,\,z}=e^{-2\psi}\big(\Phi^{2}_{,\,\rho% }-\Phi^{2}_{,\,z}\big)
  29. ( 14. d ) 2 ψ , ρ ψ , z = 2 e - 2 ψ Φ , ρ Φ , z (14.d)\quad 2\psi_{,\,\rho}\psi_{,\,z}=2e^{-2\psi}\Phi_{,\,\rho}\Phi_{,\,z}
  30. ( 14. e ) 2 Φ = 2 ψ Φ , (14.e)\quad\nabla^{2}\Phi=\,2\nabla\psi\nabla\Phi\,,
  31. 2 = ρ ρ + 1 ρ ρ + z z \nabla^{2}=\partial_{\rho\rho}+\frac{1}{\rho}\,\partial_{\rho}+\partial_{zz}
  32. = ρ e ^ ρ + z e ^ z \nabla=\partial_{\rho}\,\hat{e}_{\rho}+\partial_{z}\,\hat{e}_{z}

Congruence_coefficient.html

  1. r c = X Y X 2 Y 2 r_{c}=\frac{\sum{XY}}{\sqrt{}}{\sum{X^{2}}\sum{Y^{2}}}

Connection_(affine_bundle).html

  1. Y X , Y\to X,
  2. Y ¯ X \overline{Y}\to X
  3. Γ \Gamma
  4. Y X Y\to X
  5. Γ : Y J 1 Y \Gamma:Y\to J^{1}Y
  6. J 1 Y Y J^{1}Y\to Y
  7. Y Y
  8. X X
  9. T X TX
  10. X X
  11. ( x λ , y i ) (x^{\lambda},y^{i})
  12. Y Y
  13. Γ \Gamma
  14. Y X Y\to X
  15. Γ = d x λ ( λ + Γ λ i i ) , Γ λ i = Γ λ ( x ν ) i j y j + σ λ i ( x ν ) . \Gamma=dx^{\lambda}\otimes(\partial_{\lambda}+\Gamma_{\lambda}^{i}\partial_{i}% ),\qquad\Gamma_{\lambda}^{i}=\Gamma_{\lambda}{}^{i}{}_{j}(x^{\nu})y^{j}+\sigma% _{\lambda}^{i}(x^{\nu}).
  16. G A ( m , ) GA(m,\mathbb{R})
  17. V V
  18. m m
  19. Γ : Y J 1 Y \Gamma:Y\to J^{1}Y
  20. Γ ¯ : Y ¯ J 1 Y ¯ \overline{\Gamma}:\overline{Y}\to J^{1}\overline{Y}
  21. Γ \Gamma
  22. Y ¯ X \overline{Y}\to X
  23. ( x λ , y ¯ i ) (x^{\lambda},\overline{y}^{i})
  24. Y ¯ \overline{Y}
  25. Γ ¯ = d x λ ( λ + Γ λ ( x ν ) i j y ¯ j ¯ i ) . \overline{\Gamma}=dx^{\lambda}\otimes(\partial_{\lambda}+\Gamma_{\lambda}{}^{i% }{}_{j}(x^{\nu})\overline{y}^{j}\overline{\partial}_{i}).
  26. Y X Y\to X
  27. Γ \Gamma
  28. Γ ¯ \overline{\Gamma}
  29. Y X Y\to X
  30. σ = σ λ i ( x ν ) d x λ i \sigma=\sigma_{\lambda}^{i}(x^{\nu})dx^{\lambda}\otimes\partial_{i}
  31. Y X Y\to X
  32. Y X Y\to X
  33. V Y = Y × Y VY=Y\times Y
  34. σ = σ λ i ( x ν ) d x λ e i \sigma=\sigma_{\lambda}^{i}(x^{\nu})dx^{\lambda}\otimes e_{i}
  35. e i e_{i}
  36. Y Y
  37. Γ \Gamma
  38. Y X Y\to X
  39. R R
  40. R ¯ \overline{R}
  41. Γ \Gamma
  42. Γ ¯ \overline{\Gamma}
  43. R = R ¯ + T R=\overline{R}+T
  44. T = 1 2 T λ μ i d x λ d x μ i , T λ μ i = λ σ μ i - μ σ λ i + σ λ h Γ μ - i h σ μ h Γ λ , i h T=\frac{1}{2}T_{\lambda\mu}^{i}dx^{\lambda}\wedge dx^{\mu}\otimes\partial_{i},% \qquad T_{\lambda\mu}^{i}=\partial_{\lambda}\sigma_{\mu}^{i}-\partial_{\mu}% \sigma_{\lambda}^{i}+\sigma_{\lambda}^{h}\Gamma_{\mu}{}^{i}{}_{h}-\sigma_{\mu}% ^{h}\Gamma_{\lambda}{}^{i}{}_{h},
  45. Γ \Gamma
  46. σ \sigma
  47. T X TX
  48. X X
  49. ( x μ , x ˙ μ ) (x^{\mu},\dot{x}^{\mu})
  50. θ = d x μ ˙ μ \theta=dx^{\mu}\otimes\dot{\partial}_{\mu}
  51. T X TX
  52. θ X = d x μ μ \theta_{X}=dx^{\mu}\otimes\partial_{\mu}
  53. X X
  54. V T X = T X × T X VTX=TX\times TX
  55. Γ \Gamma
  56. T X TX
  57. A = Γ + θ , A λ μ = Γ λ x ˙ ν μ ν + δ λ μ , A=\Gamma+\theta,\qquad A_{\lambda}^{\mu}=\Gamma_{\lambda}{}^{\mu}{}_{\nu}\dot{% x}^{\nu}+\delta^{\mu}_{\lambda},
  58. T X TX
  59. A A
  60. θ \theta
  61. Γ \Gamma
  62. R + T R+T
  63. Γ \Gamma

Connection_(composite_bundle).html

  1. Y Σ X Y\to\Sigma\to X
  2. X = X=\mathbb{R}
  3. Y X Y\to X
  4. Y Σ Y\to\Sigma
  5. Σ X \Sigma\to X
  6. π : Y Σ X ( 1 ) \pi:Y\to\Sigma\to X\qquad\qquad(1)
  7. π Y Σ : Y Σ , π Σ X : Σ X . \pi_{Y\Sigma}:Y\to\Sigma,\qquad\pi_{\Sigma X}:\Sigma\to X.
  8. ( x λ , σ m , y i ) (x^{\lambda},\sigma^{m},y^{i})
  9. ( x λ , σ m ) (x^{\lambda},\sigma^{m})
  10. Σ X \Sigma\to X
  11. σ m \sigma^{m}
  12. y i y^{i}
  13. h h
  14. Σ X \Sigma\to X
  15. Y h = h * Y Y^{h}=h^{*}Y
  16. X X
  17. Y X Y\to X
  18. P X P\to X
  19. G G
  20. H H
  21. P P / H X P\to P/H\to X
  22. P P / H P\to P/H
  23. H H
  24. P / H X P/H\to X
  25. P X P\to X
  26. h h
  27. P / H X P/H\to X
  28. h * P h^{*}P
  29. P P
  30. H H
  31. P / H X P/H\to X
  32. Y Σ X Y\to\Sigma\to X
  33. J 1 Σ J^{1}\Sigma
  34. J Σ 1 Y J^{1}_{\Sigma}Y
  35. J 1 Y J^{1}Y
  36. Σ X \Sigma\to X
  37. Y Σ Y\to\Sigma
  38. Y X Y\to X
  39. ( x λ , σ m , σ λ m ) (x^{\lambda},\sigma^{m},\sigma^{m}_{\lambda})
  40. ( x λ , σ m , y i , y ^ λ i , y m i ) , (x^{\lambda},\sigma^{m},y^{i},\widehat{y}^{i}_{\lambda},y^{i}_{m}),
  41. ( x λ , σ m , y i , σ λ m , y λ i ) . (x^{\lambda},\sigma^{m},y^{i},\sigma^{m}_{\lambda},y^{i}_{\lambda}).
  42. J 1 Σ × Σ J Σ 1 Y Y J 1 Y , y λ i = y m i σ λ m + y ^ λ i J^{1}\Sigma\times_{\Sigma}J^{1}_{\Sigma}Y\to_{Y}J^{1}Y,\qquad y^{i}_{\lambda}=% y^{i}_{m}\sigma^{m}_{\lambda}+\widehat{y}^{i}_{\lambda}
  43. Y X Y\to X
  44. Y Σ Y\to\Sigma
  45. Σ X \Sigma\to X
  46. γ = d x λ ( λ + γ λ m m + γ λ i i ) , \gamma=dx^{\lambda}\otimes(\partial_{\lambda}+\gamma_{\lambda}^{m}\partial_{m}% +\gamma_{\lambda}^{i}\partial_{i}),
  47. A Σ = d x λ ( λ + A λ i i ) + d σ m ( m + A m i i ) , A_{\Sigma}=dx^{\lambda}\otimes(\partial_{\lambda}+A_{\lambda}^{i}\partial_{i})% +d\sigma^{m}\otimes(\partial_{m}+A_{m}^{i}\partial_{i}),
  48. Γ = d x λ ( λ + Γ λ m m ) . \Gamma=dx^{\lambda}\otimes(\partial_{\lambda}+\Gamma_{\lambda}^{m}\partial_{m}).
  49. A Σ A_{\Sigma}
  50. Y Σ Y\to\Sigma
  51. Γ \Gamma
  52. Σ X \Sigma\to X
  53. γ = d x λ ( λ + Γ λ m m + ( A λ i + A m i Γ λ m ) i ) \gamma=dx^{\lambda}\otimes(\partial_{\lambda}+\Gamma_{\lambda}^{m}\partial_{m}% +(A_{\lambda}^{i}+A_{m}^{i}\Gamma_{\lambda}^{m})\partial_{i})
  54. Y X Y\to X
  55. γ τ \gamma\tau
  56. Y Y
  57. τ \tau
  58. X X
  59. γ \gamma
  60. A Σ ( Γ τ ) A_{\Sigma}(\Gamma\tau)
  61. τ \tau
  62. Σ \Sigma
  63. Γ \Gamma
  64. Y Y
  65. A Σ A_{\Sigma}
  66. Y Y
  67. Y Y
  68. 0 V Σ Y V Y Y × Σ V Σ 0 , ( 2 ) 0\to V_{\Sigma}Y\to VY\to Y\times_{\Sigma}V\Sigma\to 0,\qquad\qquad(2)
  69. V Σ Y V_{\Sigma}Y
  70. V Σ * Y V_{\Sigma}^{*}Y
  71. Y Σ Y\to\Sigma
  72. A Σ A_{\Sigma}
  73. Y Σ Y\to\Sigma
  74. A Σ : T Y V Y y ˙ i i + σ ˙ m m ( y ˙ i - A m i σ ˙ m ) i A_{\Sigma}:TY\supset VY\ni\dot{y}^{i}\partial_{i}+\dot{\sigma}^{m}\partial_{m}% \to(\dot{y}^{i}-A^{i}_{m}\dot{\sigma}^{m})\partial_{i}
  75. D ~ : J 1 Y T * X Y V Σ Y , D ~ = d x λ ( y λ i - A λ i - A m i σ λ m ) i , \widetilde{D}:J^{1}Y\to T^{*}X\otimes_{Y}V_{\Sigma}Y,\qquad\widetilde{D}=dx^{% \lambda}\otimes(y^{i}_{\lambda}-A^{i}_{\lambda}-A^{i}_{m}\sigma^{m}_{\lambda})% \partial_{i},
  76. Y X Y\to X
  77. h h
  78. Σ X \Sigma\to X
  79. h * Y Y h^{*}Y\subset Y
  80. X X
  81. A Σ A_{\Sigma}
  82. A h = d x λ [ λ + ( ( A m i h ) λ h m + ( A h ) λ i ) i ] A_{h}=dx^{\lambda}\otimes[\partial_{\lambda}+((A^{i}_{m}\circ h)\partial_{% \lambda}h^{m}+(A\circ h)^{i}_{\lambda})\partial_{i}]
  83. h * Y h^{*}Y
  84. D ~ \widetilde{D}
  85. J 1 h * Y J 1 Y J^{1}h^{*}Y\subset J^{1}Y
  86. D A h D^{A_{h}}
  87. h * Y h^{*}Y
  88. A h A_{h}

Connection_(fibred_manifold).html

  1. Y X Y\to X
  2. π : Y X \pi:Y\to X
  3. Y Y
  4. Γ : Y J 1 Y \Gamma:Y\to J^{1}Y
  5. J 1 Y J^{1}Y
  6. Y Y
  7. π : Y X \pi:Y\to X
  8. Y Y
  9. 0 V Y T Y Y × X T X 0 , ( 1 ) 0\to VY\to TY\to Y\times_{X}TX\to 0,\qquad\qquad(1)
  10. T Y TY
  11. T X TX
  12. Y Y
  13. V Y VY
  14. Y Y
  15. Y × X T X Y\times_{X}TX
  16. T X TX
  17. Y Y
  18. Y X Y\to X
  19. Γ : Y × X T X T Y ( 2 ) \Gamma:Y\times_{X}TX\to TY\qquad\qquad(2)
  20. Y Y
  21. Γ \Gamma
  22. H Y = Γ ( Y × X T X ) T Y HY=\Gamma(Y\times_{X}TX)\subset TY
  23. T Y TY
  24. T Y = V Y H Y TY=VY\oplus HY
  25. Γ \Gamma
  26. Y X Y\to X
  27. Γ τ \Gamma\circ\tau
  28. τ \tau
  29. X X
  30. Y Y
  31. X X
  32. Y Y
  33. [ , ] t x ( t ) X \mathbb{R}\supset[,]\ni t\to x(t)\in X
  34. t y ( t ) Y \mathbb{R}\ni t\to y(t)\in Y
  35. X X
  36. Y Y
  37. t y ( t ) t\to y(t)
  38. x ( t ) x(t)
  39. π ( y ( t ) ) = x ( t ) \pi(y(t))=x(t)
  40. y ˙ ( t ) H Y \dot{y}(t)\in HY
  41. t t\in\mathbb{R}
  42. Γ \Gamma
  43. x ( [ 0 , 1 ] ) x([0,1])
  44. X X
  45. y π - 1 ( x ( [ 0 , 1 ] ) ) y\in\pi^{-1}(x([0,1]))
  46. Y X Y\to X
  47. ( x μ , y i ) (x^{\mu},y^{i})
  48. Γ \Gamma
  49. Y X Y\to X
  50. Γ = d x λ ( λ + Γ λ i ( x ν , y j ) i ) ( 3 ) \Gamma=dx^{\lambda}\otimes(\partial_{\lambda}+\Gamma_{\lambda}^{i}(x^{\nu},y^{% j})\partial_{i})\qquad\qquad(3)
  51. Y Y
  52. θ X = d x μ μ \theta_{X}=dx^{\mu}\otimes\partial_{\mu}
  53. X X
  54. Γ : λ λ Γ = λ + Γ i λ i . \Gamma:\partial_{\lambda}\to\partial_{\lambda}\rfloor\Gamma=\partial_{\lambda}% +\Gamma^{i}_{\lambda}\partial_{i}.
  55. Γ \Gamma
  56. τ = τ μ μ \tau=\tau^{\mu}\partial_{\mu}
  57. X X
  58. Γ τ = τ Γ = τ λ ( λ + Γ λ i i ) H Y \Gamma\tau=\tau\rfloor\Gamma=\tau^{\lambda}(\partial_{\lambda}+\Gamma^{i}_{% \lambda}\partial_{i})\subset HY
  59. Y Y
  60. 0 Y × X T * X T * Y V * Y 0 , 0\to Y\times_{X}T^{*}X\to T^{*}Y\to V^{*}Y\to 0,
  61. T * Y T^{*}Y
  62. T * X T^{*}X
  63. Y Y
  64. V * Y Y V^{*}Y\to Y
  65. V Y Y VY\to Y
  66. Γ = ( d y i - Γ λ i d x λ ) i , \Gamma=(dy^{i}-\Gamma^{i}_{\lambda}dx^{\lambda})\otimes\partial_{i},
  67. Y X Y\to X
  68. f : X X f:X^{\prime}\to X
  69. f * Y X f^{*}Y\to X^{\prime}
  70. Y Y
  71. f f
  72. Γ \Gamma
  73. Y X Y\to X
  74. f * Γ = ( d y i - ( Γ f ~ ) λ i f λ x μ d x μ ) i f^{*}\Gamma=(dy^{i}-(\Gamma\circ\widetilde{f})^{i}_{\lambda}\frac{\partial f^{% \lambda}}{\partial x^{\prime\mu}}dx^{\prime\mu})\otimes\partial_{i}
  75. f * Y X f^{*}Y\to X^{\prime}
  76. J 1 Y J^{1}Y
  77. Y X Y\to X
  78. ( x μ , y i , y μ i ) (x^{\mu},y^{i},y^{i}_{\mu})
  79. J 1 Y Y ( Y × X T * X ) Y T Y , ( y μ i ) d x μ ( μ + y μ i i ) , J^{1}Y\to_{Y}(Y\times_{X}T^{*}X)\otimes_{Y}TY,\qquad(y^{i}_{\mu})\to dx^{\mu}% \otimes(\partial_{\mu}+y^{i}_{\mu}\partial_{i}),
  80. Γ \Gamma
  81. Y X Y\to X
  82. Γ : Y J 1 Y , y λ i Γ = Γ λ i , \Gamma:Y\to J^{1}Y,\qquad y_{\lambda}^{i}\circ\Gamma=\Gamma_{\lambda}^{i},
  83. J 1 Y Y J^{1}Y\to Y
  84. ( Y × X T * X ) Y V Y Y . ( 4 ) (Y\times_{X}T^{*}X)\otimes_{Y}VY\to Y.\qquad\qquad(4)
  85. Y X Y\to X
  86. σ = σ μ i d x μ i ( 5 ) \sigma=\sigma^{i}_{\mu}dx^{\mu}\otimes\partial_{i}\qquad\qquad(5)
  87. Y X Y\to X
  88. Γ λ i = x μ x λ ( μ y i + Γ μ j j y i ) . {\Gamma^{\prime}}^{i}_{\lambda}=\frac{\partial x^{\mu}}{\partial{x^{\prime}}^{% \lambda}}(\partial_{\mu}{y^{\prime}}^{i}+\Gamma^{j}_{\mu}\partial_{j}{y^{% \prime}}^{i}).
  89. Γ \Gamma
  90. Y X Y\to X
  91. D Γ : J 1 Y Y T * X Y V Y , D Γ = ( y λ i - Γ λ i ) d x λ i , D_{\Gamma}:J^{1}Y\to_{Y}T^{*}X\otimes_{Y}VY,\qquad D_{\Gamma}=(y^{i}_{\lambda}% -\Gamma^{i}_{\lambda})dx^{\lambda}\otimes\partial_{i},
  92. Y Y
  93. Γ \Gamma
  94. s : X Y s:X\to Y
  95. Γ s = ( λ s i - Γ λ i s ) d x λ i , \nabla^{\Gamma}s=(\partial_{\lambda}s^{i}-\Gamma_{\lambda}^{i}\circ s)dx^{% \lambda}\otimes\partial_{i},
  96. τ Γ s = τ Γ s \nabla_{\tau}^{\Gamma}s=\tau\rfloor\nabla^{\Gamma}s
  97. τ \tau
  98. X X
  99. Γ \Gamma
  100. Y X Y\to X
  101. R = 1 2 d Γ Γ = 1 2 [ Γ , Γ ] F N = 1 2 R λ μ i d x λ d x μ i , R=\frac{1}{2}d_{\Gamma}\Gamma=\frac{1}{2}[\Gamma,\Gamma]_{FN}=\frac{1}{2}R_{% \lambda\mu}^{i}\,dx^{\lambda}\wedge dx^{\mu}\otimes\partial_{i},
  102. R λ μ i = λ Γ μ i - μ Γ λ i + Γ λ j j Γ μ i - Γ μ j j Γ λ i . R_{\lambda\mu}^{i}=\partial_{\lambda}\Gamma_{\mu}^{i}-\partial_{\mu}\Gamma_{% \lambda}^{i}+\Gamma_{\lambda}^{j}\partial_{j}\Gamma_{\mu}^{i}-\Gamma_{\mu}^{j}% \partial_{j}\Gamma_{\lambda}^{i}.
  103. Y Y
  104. Γ \Gamma
  105. σ \sigma
  106. Γ \Gamma
  107. σ \sigma
  108. T = d Γ σ = ( λ σ μ i + Γ λ j j σ μ i - j Γ λ i σ μ j ) d x λ d x μ i . T=d_{\Gamma}\sigma=(\partial_{\lambda}\sigma_{\mu}^{i}+\Gamma_{\lambda}^{j}% \partial_{j}\sigma_{\mu}^{i}-\partial_{j}\Gamma_{\lambda}^{i}\sigma_{\mu}^{j})% \,dx^{\lambda}\wedge dx^{\mu}\otimes\partial_{i}.
  109. π : P M \pi\colon P\to M
  110. G G
  111. P P
  112. P P
  113. P P
  114. J 1 P P J^{1}P\to P
  115. G G
  116. P P
  117. C = J 1 P / G M C=J^{1}P/G\to M
  118. V P / G M VP/G\to M
  119. 𝔤 \mathfrak{g}
  120. G G
  121. G G
  122. C C
  123. T P / G TP/G
  124. { e m } \{{\mathrm{e}}_{m}\}
  125. G G
  126. C C
  127. ( x μ , a μ m ) (x^{\mu},a^{m}_{\mu})
  128. A = d x λ ( λ + a λ m e m ) , A=dx^{\lambda}\otimes(\partial_{\lambda}+a^{m}_{\lambda}{\mathrm{e}}_{m}),
  129. a λ m d x λ e m a^{m}_{\lambda}\,dx^{\lambda}\otimes{\mathrm{e}}_{m}
  130. M M
  131. J 1 C J^{1}C
  132. C C
  133. a λ μ r = 1 2 ( F λ μ r + S λ μ r ) = 1 2 ( a λ μ r + a μ λ r - c p q r a λ p a μ q ) + 1 2 ( a λ μ r - a μ λ r + c p q r a λ p a μ q ) , a_{\lambda\mu}^{r}=\frac{1}{2}(F_{\lambda\mu}^{r}+S_{\lambda\mu}^{r})=\frac{1}% {2}(a_{\lambda\mu}^{r}+a_{\mu\lambda}^{r}-c_{pq}^{r}a_{\lambda}^{p}a_{\mu}^{q}% )+\frac{1}{2}(a_{\lambda\mu}^{r}-a_{\mu\lambda}^{r}+c_{pq}^{r}a_{\lambda}^{p}a% _{\mu}^{q}),
  134. F = 1 2 F λ μ m d x λ d x μ e m F=\frac{1}{2}F_{\lambda\mu}^{m}\,dx^{\lambda}\wedge dx^{\mu}\otimes{\mathrm{e}% }_{m}

Connective_constant.html

  1. 2 + 2 \sqrt{2+\sqrt{2}}
  2. c n c_{n}
  3. c n + m c n c m c_{n+m}\leq c_{n}c_{m}
  4. μ = lim n c n 1 / n \mu=\lim_{n\rightarrow\infty}c_{n}^{1/n}
  5. μ \mu
  6. c n c_{n}
  7. μ \mu
  8. μ \mu
  9. c n μ n n γ - 1 c_{n}\approx\mu^{n}n^{\gamma-1}
  10. μ \mu
  11. γ \gamma
  12. γ = 43 / 32 \gamma=43/32
  13. 2 + 2 \sqrt{2+\sqrt{2}}
  14. ( 3.12 2 ) (3.12^{2})
  15. ( 4.8 2 ) (4.8^{2})
  16. ( 3.12 2 ) (3.12^{2})
  17. x 12 - 4 x 8 - 8 x 7 - 4 x 6 + 2 x 4 + 8 x 3 + 12 x 2 + 8 x + 2 x^{12}-4x^{8}-8x^{7}-4x^{6}+2x^{4}+8x^{3}+12x^{2}+8x+2
  18. μ = 2 + 2 \mu=\sqrt{2+\sqrt{2}}
  19. γ \gamma
  20. a a
  21. b b
  22. ( γ ) \ell(\gamma)
  23. W γ ( a , b ) W_{\gamma}(a,b)
  24. γ \gamma
  25. a a
  26. b b
  27. Z ( x ) = γ : a H x ( γ ) = n = 0 c n x n Z(x)=\sum_{\gamma:a\to H}x^{\ell(\gamma)}=\sum_{n=0}^{\infty}c_{n}x^{n}
  28. x < x c x<x_{c}
  29. x > x c x>x_{c}
  30. x c = 1 / 2 + 2 x_{c}=1/\sqrt{2+\sqrt{2}}
  31. μ = 2 + 2 \mu=\sqrt{2+\sqrt{2}}
  32. Ω \Omega
  33. a a
  34. x x
  35. σ \sigma
  36. F ( z ) = γ Ω : a z e - i σ W γ ( a , z ) x ( γ ) . F(z)=\sum_{\gamma\subset\Omega:a\to z}e^{-i\sigma W_{\gamma}(a,z)}x^{\ell(% \gamma)}.
  37. x = x c = 1 / 2 + 2 x=x_{c}=1/\sqrt{2+\sqrt{2}}
  38. σ = 5 / 8 \sigma=5/8
  39. v v
  40. Ω \Omega
  41. ( p - v ) F ( p ) + ( q - v ) F ( q ) + ( r - v ) F ( r ) = 0 , (p-v)F(p)+(q-v)F(q)+(r-v)F(r)=0,
  42. p , q , r p,q,r
  43. v v
  44. S T , L S_{T,L}
  45. ± π / 3 \pm\pi/3
  46. S T , L S_{T,L}
  47. V ( S T , L ) = { z V ( ) : 0 R e ( z ) 3 T + 1 2 , | 3 I m ( z ) - R e ( z ) | 3 L } . V(S_{T,L})=\{z\in V(\mathbb{H}):0\leq Re(z)\leq\frac{3T+1}{2},\;|\sqrt{3}Im(z)% -Re(z)|\leq 3L\}.
  48. a a
  49. α \alpha
  50. β \beta
  51. ϵ \epsilon
  52. ϵ ¯ \bar{\epsilon}
  53. A T , L x := γ S T , L : a α { a } x ( γ ) , B T , L x := γ S T , L : a β x ( γ ) , E T , L x := γ S T , L : a ϵ ϵ ¯ x ( γ ) . A_{T,L}^{x}:=\sum_{\gamma\in S_{T,L}:a\to\alpha\setminus\{a\}}x^{\ell(\gamma)}% ,\quad B_{T,L}^{x}:=\sum_{\gamma\in S_{T,L}:a\to\beta}x^{\ell(\gamma)},\quad E% _{T,L}^{x}:=\sum_{\gamma\in S_{T,L}:a\to\epsilon\cup\bar{\epsilon}}x^{\ell(% \gamma)}.
  54. ( p - v ) F ( p ) + ( q - v ) F ( q ) + ( r - v ) F ( r ) = 0 (p-v)F(p)+(q-v)F(q)+(r-v)F(r)=0
  55. V ( S T , L ) V(S_{T,L})
  56. 1 = cos ( 3 π / 8 ) A T , L x c + B T , L x c + cos ( π / 4 ) E T , L x c 1=\cos(3\pi/8)A_{T,L}^{x_{c}}+B_{T,L}^{x_{c}}+\cos(\pi/4)E_{T,L}^{x_{c}}
  57. L L\to\infty
  58. S T S_{T}
  59. A T x := γ S T : a α { a } x ( γ ) , B T x := γ S T : a β x ( γ ) , E T x := γ S T : a ϵ ϵ ¯ x ( γ ) . A_{T}^{x}:=\sum_{\gamma\in S_{T}:a\to\alpha\setminus\{a\}}x^{\ell(\gamma)},% \quad B_{T}^{x}:=\sum_{\gamma\in S_{T}:a\to\beta}x^{\ell(\gamma)},\quad E_{T}^% {x}:=\sum_{\gamma\in S_{T}:a\to\epsilon\cup\bar{\epsilon}}x^{\ell(\gamma)}.
  60. E T , L x c = 0 E_{T,L}^{x_{c}}=0
  61. 1 = cos ( 3 π / 8 ) A T , L x c + B T , L x c 1=\cos(3\pi/8)A_{T,L}^{x_{c}}+B_{T,L}^{x_{c}}
  62. A T + 1 x c - A T x c x c ( B T + 1 x c ) 2 A_{T+1}^{x_{c}}-A_{T}^{x_{c}}\leq x_{c}(B_{T+1}^{x_{c}})^{2}
  63. B T x c B_{T}^{x_{c}}
  64. Z ( x c ) T > 0 B T x c = Z(x_{c})\geq\sum_{T>0}B_{T}^{x_{c}}=\infty
  65. μ 1 / 2 + 2 \mu\geq 1/\sqrt{2+\sqrt{2}}
  66. T - I < < T - 1 T_{-I}<\cdots<T_{-1}
  67. T 0 > > T j T_{0}>\cdots>T_{j}
  68. B T x ( x / x c ) T B T x c ( x / x c ) T B_{T}^{x}\leq(x/x_{c})^{T}B_{T}^{x_{c}}\leq(x/x_{c})^{T}
  69. T > 0 ( 1 + B T x ) < \prod_{T>0}(1+B_{T}^{x})<\infty
  70. Z ( x ) T - I < < T - 1 , T 0 > > T j 2 ( k = - I j B T k x ) = 2 ( T > 0 ( 1 + B T x ) ) 2 < . Z(x)\leq\sum_{T_{-I}<\cdots<T_{-1},\;T_{0}>\cdots>T_{j}}2\left(\prod_{k=-I}^{j% }B_{T_{k}}^{x}\right)=2\left(\prod_{T>0}(1+B_{T}^{x})\right)^{2}<\infty.
  71. μ = 2 + 2 \mu=\sqrt{2+\sqrt{2}}
  72. | γ ( n ) | 2 \langle|\gamma(n)|^{2}\rangle
  73. | γ ( n ) | 2 = 1 c n n step SAW | γ ( n ) | 2 = n 2 ν + o ( 1 ) \langle|\gamma(n)|^{2}\rangle=\frac{1}{c_{n}}\sum_{n\;\mathrm{step\;SAW}}|% \gamma(n)|^{2}=n^{2\nu+o(1)}
  74. ν = 3 / 4 \nu=3/4
  75. ν \nu
  76. 11 / 32 11/32
  77. κ = 8 / 3 \kappa=8/3

Consistency_(suspension).html

  1. C = x / ( x + y ) C=x/(x+y)

Constant-mean-curvature_surface.html

  1. S S
  2. \R 3 \R^{3}
  3. \R 3 \R^{3}
  4. H 0 H\neq 0
  5. \R n \R^{n}
  6. n - 1 n-1
  7. \R 4 \R^{4}
  8. \R 3 \R^{3}
  9. \R 3 \R^{3}
  10. n ( 2 π - n ) n(2\pi-n)
  11. k 3 k\geq 3
  12. i = 1 k n i ( k - 1 ) π \sum_{i=1}^{k}n_{i}\leq(k-1)\pi
  13. i = 1 k n i k π \sum_{i=1}^{k}n_{i}\leq k\pi
  14. S S
  15. \R 3 \R^{3}
  16. V V
  17. \C \C
  18. H H
  19. ϕ : V \C \phi:V\rightarrow\C
  20. ϕ z ¯ 0 \phi_{\bar{z}}\neq 0
  21. X : V R X:V\rightarrow R
  22. X ( z ) = z 0 z X z ( z ) d z X(z)=\Re\int_{z_{0}}^{z}X_{z}(z)\,dz
  23. X z ( z ) = - 1 H ( 1 + ϕ ( z ) ϕ ¯ ( z ) ) 2 { ( 1 - ϕ ( z ) 2 , i ( 1 + ϕ ( z ) 2 ) , 2 ϕ ( z ) ) ϕ ¯ z ¯ ( z ) } X_{z}(z)=\frac{-1}{H(1+\phi(z)\bar{\phi}(z))^{2}}\left\{(1-\phi(z)^{2},i(1+% \phi(z)^{2}),2\phi(z))\frac{\bar{\partial\phi}}{\partial\bar{z}}(z)\right\}
  24. z V z\in V
  25. ϕ \phi
  26. H H
  27. ϕ ( z ) = - 1 / z ¯ \phi(z)=-1/\bar{z}
  28. H = 1 H=1
  29. ϕ ( z ) = - e i z \phi(z)=-e^{iz}
  30. H = 1 / 2 H=1/2
  31. \R 3 \R^{3}
  32. 𝕊 3 \mathbb{S}^{3}
  33. 𝕊 3 \mathbb{S}^{3}

Construction_of_a_complex_null_tetrad.html

  1. { l a , n a , m a , m ¯ a } \{l^{a},n^{a},m^{a},\bar{m}^{a}\}
  2. { l a , n a } \{l^{a},n^{a}\}
  3. { m a , m ¯ a } \{m^{a},\bar{m}^{a}\}
  4. ( - , + , + , + ) : (-,+,+,+):
  5. l a l a = n a n a = m a m a = m ¯ a m ¯ a = 0 ; l_{a}l^{a}=n_{a}n^{a}=m_{a}m^{a}=\bar{m}_{a}\bar{m}^{a}=0\,;
  6. l a m a = l a m ¯ a = n a m a = n a m ¯ a = 0 ; l_{a}m^{a}=l_{a}\bar{m}^{a}=n_{a}m^{a}=n_{a}\bar{m}^{a}=0\,;
  7. l a n a = l a n a = - 1 , m a m ¯ a = m a m ¯ a = 1 ; l_{a}n^{a}=l^{a}n_{a}=-1\,,\;\;m_{a}\bar{m}^{a}=m^{a}\bar{m}_{a}=1\,;
  8. g a b = - l a n b - n a l b + m a m ¯ b + m ¯ a m b , g a b = - l a n b - n a l b + m a m ¯ b + m ¯ a m b . g_{ab}=-l_{a}n_{b}-n_{a}l_{b}+m_{a}\bar{m}_{b}+\bar{m}_{a}m_{b}\,,\;\;g^{ab}=-% l^{a}n^{b}-n^{a}l^{b}+m^{a}\bar{m}^{b}+\bar{m}^{a}m^{b}\,.
  9. { l a , n a , m a , m ¯ a } \{l^{a},n^{a},m^{a},\bar{m}^{a}\}
  10. Ψ i \Psi_{i}
  11. Φ i j \Phi_{ij}
  12. ϕ i \phi_{i}
  13. l a l^{a}
  14. n a n^{a}
  15. m a m^{a}
  16. m ¯ a \bar{m}^{a}
  17. { ( - , + , + , + ) ; l a n a = - 1 , m a m ¯ a = 1 } \{(-,+,+,+);l^{a}n_{a}=-1\,,m^{a}\bar{m}_{a}=1\}
  18. { ( + , - , - , - ) ; l a n a = 1 , m a m ¯ a = - 1 } \{(+,-,-,-);l^{a}n_{a}=1\,,m^{a}\bar{m}_{a}=-1\}
  19. g a b g_{ab}
  20. { ω 0 , ω 1 , ω 2 , ω 3 } \{\omega_{0}\,,\omega_{1}\,,\omega_{2}\,,\omega_{3}\}
  21. g a b = - ω 0 ω 0 + ω 1 ω 1 + ω 2 ω 2 + ω 3 ω 3 , g_{ab}=-\omega_{0}\omega_{0}+\omega_{1}\omega_{1}+\omega_{2}\omega_{2}+\omega_% {3}\omega_{3}\,,
  22. { l a , n a , m a , m ¯ a } \{l_{a}\,,n_{a}\,,m_{a}\,,\bar{m}_{a}\}
  23. l a d x a = ω 0 + ω 1 2 , n a d x a = ω 0 - ω 1 2 , l_{a}dx^{a}=\frac{\omega_{0}+\omega_{1}}{\sqrt{2}}\,,\quad n_{a}dx^{a}=\frac{% \omega_{0}-\omega_{1}}{\sqrt{2}}\,,
  24. m a d x a = ω 2 + i ω 3 2 , m ¯ a d x a = ω 2 - i ω 3 2 , m_{a}dx^{a}=\frac{\omega_{2}+i\omega_{3}}{\sqrt{2}}\,,\quad\bar{m}_{a}dx^{a}=% \frac{\omega_{2}-i\omega_{3}}{\sqrt{2}}\,,
  25. { l a , n a , m a , m ¯ a } \{l^{a}\,,n^{a}\,,m^{a}\,,\bar{m}^{a}\}
  26. { l a , n a , m a , m ¯ a } \{l_{a}\,,n_{a}\,,m_{a}\,,\bar{m}_{a}\}
  27. g a b g^{ab}
  28. g a b = - g t t d t 2 + g r r d r 2 + g θ θ d θ 2 + g ϕ ϕ d ϕ 2 , g_{ab}=-g_{tt}dt^{2}+g_{rr}dr^{2}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d% \phi^{2}\,,
  29. ω t = g t t d t , ω r = g r r d r , ω θ = g θ θ d θ , ω ϕ = g ϕ ϕ d ϕ , \omega_{t}=\sqrt{g_{tt}}dt\,,\;\;\omega_{r}=\sqrt{g_{rr}}dr\,,\;\;\omega_{% \theta}=\sqrt{g_{\theta\theta}}d\theta\,,\;\;\omega_{\phi}=\sqrt{g_{\phi\phi}}% d\phi\,,
  30. l a d x a = 1 2 ( g t t d t + g r r d r ) , l_{a}dx^{a}=\frac{1}{\sqrt{2}}(\sqrt{g_{tt}}dt+\sqrt{g_{rr}}dr)\,,
  31. n a d x a = 1 2 ( g t t d t - g r r d r ) , n_{a}dx^{a}=\frac{1}{\sqrt{2}}(\sqrt{g_{tt}}dt-\sqrt{g_{rr}}dr)\,,
  32. m a d x a = 1 2 ( g θ θ d θ + i g ϕ ϕ d ϕ ) , m_{a}dx^{a}=\frac{1}{\sqrt{2}}(\sqrt{g_{\theta\theta}}d\theta+i\sqrt{g_{\phi% \phi}}d\phi)\,,
  33. m ¯ a d x a = 1 2 ( g θ θ d θ - i g ϕ ϕ d ϕ ) . \bar{m}_{a}dx^{a}=\frac{1}{\sqrt{2}}(\sqrt{g_{\theta\theta}}d\theta-i\sqrt{g_{% \phi\phi}}d\phi)\,.
  34. { l a , n a } \{l^{a}\,,n^{a}\}
  35. { l a , n a } \{l^{a}\,,n^{a}\}
  36. { t , r , θ , ϕ } \{t,r,\theta,\phi\}
  37. { u , r , θ , ϕ } \{u,r,\theta,\phi\}
  38. { v , r , θ , ϕ } \{v,r,\theta,\phi\}
  39. u u
  40. v v
  41. d s 2 = - F d v 2 + 2 d v d r + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) , with F := ( 1 - M r ) 2 , ds^{2}=-Fdv^{2}+2dvdr+r^{2}(d\theta^{2}+\sin^{2}\!\theta\,d\phi^{2})\,,\;\;\,% \text{with }F\,:=\,\Big(1-\frac{M}{r}\Big)^{2}\,,
  42. L = - F v ˙ 2 + 2 v ˙ r ˙ , L=-F\dot{v}^{2}+2\dot{v}\dot{r}\,,
  43. v ˙ = 0 \dot{v}=0
  44. r ˙ = F 2 v ˙ \dot{r}=\frac{F}{2}\dot{v}
  45. l a = ( 1 , F 2 , 0 , 0 ) , n a = ( 0 , - 1 , 0 , 0 ) , m a = 1 2 r ( 0 , 0 , 1 , i csc θ ) , l^{a}=(1,\frac{F}{2},0,0)\,,\quad n^{a}=(0,-1,0,0)\,,\quad m^{a}=\frac{1}{% \sqrt{2}\,r}(0,0,1,i\,\csc\theta)\,,
  46. l a = ( - F 2 , 1 , 0 , 0 ) , n a = ( - 1 , 0 , 0 , 0 ) , m a = r 2 ( 0 , 0 , 1 , sin θ ) . l_{a}=(-\frac{F}{2},1,0,0)\,,\quad n_{a}=(-1,0,0,0)\,,\quad m_{a}=\frac{r}{% \sqrt{2}}(0,0,1,\sin\theta)\,.
  47. l a n a = n a l a = - 1 l^{a}n_{a}=n^{a}l_{a}=-1
  48. g a b + l a n b + n a l b g_{ab}+l_{a}n_{b}+n_{a}l_{b}
  49. h A B h_{AB}
  50. d v dv
  51. d r dr
  52. κ = σ = τ = 0 , ν = λ = π = 0 , γ = 0 \kappa=\sigma=\tau=0\,,\quad\nu=\lambda=\pi=0\,,\quad\gamma=0
  53. ρ = - r + 2 M 2 r 2 , μ = - 1 r , α = - β = - 2 cot θ 4 r , ε = M 2 r 2 ; \rho=\frac{-r+2M}{2r^{2}}\,,\quad\mu=-\frac{1}{r}\,,\quad\alpha=-\beta=\frac{-% \sqrt{2}\cot\theta}{4r}\,,\quad\varepsilon=\frac{M}{2r^{2}}\,;
  54. Ψ 0 = Ψ 1 = Ψ 3 = Ψ 4 = 0 , Ψ 2 = - M r 3 , \Psi_{0}=\Psi_{1}=\Psi_{3}=\Psi_{4}=0\,,\quad\Psi_{2}=-\frac{M}{r^{3}}\,,
  55. Φ 00 = Φ 10 = Φ 20 = Φ 11 = Φ 12 = Φ 22 = Λ = 0 . \Phi_{00}=\Phi_{10}=\Phi_{20}=\Phi_{11}=\Phi_{12}=\Phi_{22}=\Lambda=0\,.
  56. d s 2 = - G d v 2 + 2 d v d r + r 2 d θ 2 + r 2 sin 2 θ d ϕ 2 , with G := ( 1 - M r ) 2 , ds^{2}=-Gdv^{2}+2dvdr+r^{2}d\theta^{2}+r^{2}\sin^{2}\!\theta\,d\phi^{2}\,,\;\;% \,\text{with }G\,:=\,\Big(1-\frac{M}{r}\Big)^{2}\,,
  57. 2 L = - G v ˙ 2 + 2 v ˙ r ˙ + r 2 ( θ ˙ 2 + r 2 sin 2 θ ϕ ˙ 2 . 2L=-G\dot{v}^{2}+2\dot{v}\dot{r}+r^{2}({\dot{\theta}}^{2}+r^{2}\sin^{2}\!% \theta\,\dot{\phi}^{2}\,.
  58. { L = 0 , θ ˙ = 0 , ϕ ˙ = 0 } \{L=0\,,\dot{\theta}=0\,,\dot{\phi}=0\}
  59. v ˙ = 0 \dot{v}=0
  60. r ˙ = 2 F v ˙ \dot{r}=2F\dot{v}
  61. l a a = ( 1 , F 2 , 0 , 0 ) , n a a = ( 0 , - 1 , 0 , 0 ) , l^{a}\partial_{a}\,=\,\Big(1\,,\frac{F}{2}\,,0\,,0\Big)\,,\quad n^{a}\partial_% {a}\,=\,\Big(0\,,-1\,,0\,,0\Big)\,,
  62. l a d x a = ( - F 2 , 1 , 0 , 0 ) , n a d x a = ( - 1 , 0 , 0 , 0 ) , l_{a}dx^{a}\,=\,\Big(-\frac{F}{2}\,,1\,,0,0\Big)\,,\quad n_{a}dx^{a}\,=\,\Big(% -1\,,0\,,0\,,0\Big)\,,
  63. m a a = 1 2 ( 0 , 0 , 1 r , i r sin θ ) , m a d x a = 1 2 ( 0 , 0 , r , i sin θ ) . m^{a}\partial_{a}\,=\,\frac{1}{\sqrt{2}}\,\Big(0\,,0\,,\frac{1}{r}\,,\frac{i}{% r\sin\theta}\Big)\,,\quad m_{a}dx^{a}\,=\,\frac{1}{\sqrt{2}}\,\Big(0\,,0\,,r\,% ,i\sin\theta\Big)\,.
  64. κ = σ = τ = 0 , ν = λ = π = 0 , γ = 0 \kappa=\sigma=\tau=0\,,\quad\nu=\lambda=\pi=0\,,\quad\gamma=0
  65. ρ = ( r - M ) 2 2 r 3 , μ = - 1 r , α = - β = - 2 cot θ 4 r , ε = M ( r - M ) 2 r 3 ; \rho=\frac{(r-M)^{2}}{2r^{3}}\,,\quad\mu=-\frac{1}{r}\,,\quad\alpha=-\beta=% \frac{-\sqrt{2}\cot\theta}{4r}\,,\quad\varepsilon=\frac{M(r-M)}{2r^{3}}\,;
  66. Ψ 0 = Ψ 1 = Ψ 3 = Ψ 4 = 0 , Ψ 2 = - ( M r - M ) r 4 , \Psi_{0}=\Psi_{1}=\Psi_{3}=\Psi_{4}=0\,,\quad\Psi_{2}=-\frac{(Mr-M)}{r^{4}}\,,
  67. Φ 00 = Φ 10 = Φ 20 = Φ 12 = Φ 22 = Λ = 0 , Φ 11 = - M 2 2 r 4 . \Phi_{00}=\Phi_{10}=\Phi_{20}=\Phi_{12}=\Phi_{22}=\Lambda=0\,,\quad\Phi_{11}=-% \frac{M^{2}}{2r^{4}}\,.
  68. l a a = r := D , l^{a}\partial_{a}=\partial_{r}:=D\,,
  69. n a a = u + U r + X ς + X ¯ ς ¯ := Δ , n^{a}\partial_{a}=\partial_{u}+U\partial_{r}+X\partial_{\varsigma}+\bar{X}% \partial_{\bar{\varsigma}}:=\Delta\,,
  70. m a a = ω r + ξ 3 ς + ξ 4 ς ¯ := δ , m^{a}\partial_{a}=\omega\partial_{r}+\xi^{3}\partial_{\varsigma}+\xi^{4}% \partial_{\bar{\varsigma}}:=\delta\,,
  71. m ¯ a a = ω ¯ r + ξ ¯ 3 ς ¯ + ξ ¯ 4 ς := δ ¯ , \bar{m}^{a}\partial_{a}=\bar{\omega}\partial_{r}+\bar{\xi}^{3}\partial_{\bar{% \varsigma}}+\bar{\xi}^{4}\partial_{\varsigma}:=\bar{\delta}\,,
  72. { U , X , ω , ξ 3 , ξ 4 } \{U,X,\omega,\xi^{3},\xi^{4}\}
  73. u u
  74. l a = d u l_{a}=du
  75. r r
  76. l a l^{a}
  77. ( D r = l a a r = 1 ) (Dr=l^{a}\partial_{a}r=1)
  78. n a n^{a}
  79. Δ u = n a a u = 1 \Delta u=n^{a}\partial_{a}u=1
  80. { u , r , ς , ς ¯ } \{u,r,\varsigma,\bar{\varsigma}\}
  81. { u , r } \{u,r\}
  82. { ς := e i ϕ cot θ 2 , ς ¯ = e - i ϕ cot θ 2 } \{\varsigma:=e^{i\phi}\cot\frac{\theta}{2},\bar{\varsigma}=e^{-i\phi}\cot\frac% {\theta}{2}\}
  83. { θ , ϕ } \{\theta,\phi\}
  84. Δ ^ u = S u 2 \hat{\Delta}_{u}=S^{2}_{u}
  85. κ = π = ε = 0 , ρ = ρ ¯ , τ = α ¯ + β . \kappa=\pi=\varepsilon=0\,,\quad\rho=\bar{\rho}\,,\quad\tau=\bar{\alpha}+\beta\,.
  86. n a n_{a}
  87. n a = - d v , n_{a}\,=-dv\,,
  88. v v
  89. l a a l^{a}\partial_{a}
  90. D v = 1 , Δ v = δ v = δ ¯ v = 0 . Dv=1\,,\quad\Delta v=\delta v=\bar{\delta}v=0\,.
  91. r r
  92. n a n^{a}
  93. n a a r = - 1 n a a = - r . n^{a}\partial_{a}r\,=\,-1\;\Leftrightarrow\;n^{a}\partial_{a}\,=\,-\partial_{r% }\,.
  94. n a n^{a}
  95. { l a , m a , m ¯ a } \{l^{a},m^{a},\bar{m}^{a}\}
  96. l a l^{a}
  97. { l a , n a , m a , m ¯ a } \{l_{a},n_{a},m_{a},\bar{m}_{a}\}
  98. n a a n^{a}\partial_{a}
  99. { m a , m ¯ a } \{m^{a},\bar{m}^{a}\}
  100. { y , z } \{y,z\}
  101. l a a = v + U r + X 3 y + X 4 z := D , l^{a}\partial_{a}=\partial_{v}+U\partial_{r}+X^{3}\partial_{y}+X^{4}\partial_{% z}\,:=\,D\,,
  102. n a a = - r := Δ , n^{a}\partial_{a}=-\partial_{r}\,:=\,\Delta\,,
  103. m a a = Ω r + ξ 3 y + ξ 4 z := δ , m^{a}\partial_{a}=\Omega\partial_{r}+\xi^{3}\partial_{y}+\xi^{4}\partial_{z}\,% :=\,\delta\,,
  104. m ¯ a a = Ω ¯ r + ξ ¯ 3 y + ξ ¯ 4 z := δ ¯ . \bar{m}^{a}\partial_{a}=\bar{\Omega}\partial_{r}+\bar{\xi}^{3}\partial_{y}+% \bar{\xi}^{4}\partial_{z}\,:=\,\bar{\delta}\,.
  105. ν = τ = γ = 0 , μ = μ ¯ , π = α + β ¯ , \nu=\tau=\gamma=0\,,\quad\mu=\bar{\mu}\,,\quad\pi=\alpha+\bar{\beta}\,,
  106. Q = i = 0 Q ( i ) r i = Q ( 0 ) + Q ( 1 ) r + + Q ( n ) r n + Q=\sum_{i=0}Q^{(i)}r^{i}=Q^{(0)}+Q^{(1)}r+\cdots+Q^{(n)}r^{n}+\ldots
  107. Q ( 0 ) Q^{(0)}

Continuous_foam_separation.html

  1. γ ° γ°
  2. Δ P = 2 γ R \Delta P=\frac{2\gamma^{\circ}}{R}
  3. Δ P = γ ( 1 R 1 + 1 R 2 ) \Delta P=\gamma^{\circ}(\frac{1}{R_{1}}+\frac{1}{R_{2}})
  4. d γ
  5. d γ = Γ 1 d μ 1 + Γ 2 d μ 2 \mathrm{d}\gamma=\Gamma_{1}\mathrm{d}\mu_{1}+\Gamma_{2}\mathrm{d}\mu_{2}
  6. a a
  7. R R
  8. T T
  9. Γ 2 = - a R T ( γ a ) T \Gamma_{2}=-\frac{{a}\,}{RT}\,\left(\frac{\partial\gamma}{\partial a}\right)_{% T}\,
  10. A s = 1 N A Γ 2 A_{s}=\frac{{1}\,}{N_{A}\Gamma_{2}}
  11. Enrichment ratio = ( Protein concentration in the foam Protein concentration in the initial feed ) \text{ Enrichment ratio}=\left(\frac{\text{Protein concentration in the foam}}% {\text{Protein concentration in the initial feed}}\right)
  12. Separation ratio = ( Protein concentration in the foam Protein concentration in the outlet stream ) \text{ Separation ratio}=\left(\frac{\,\text{Protein concentration in the foam% }}{\,\text{Protein concentration in the outlet stream}}\right)
  13. Recovery = ( Mass of protein in the foam Mass of protein in the initial feed ) × 100 % \text{ Recovery}=\left(\frac{\,\text{Mass of protein in the foam}}{\,\text{% Mass of protein in the initial feed}}\right)\times 100\%

Continuous_gusts.html

  1. Φ u g ( Ω ) = σ u 2 2 L u π 1 1 + ( L u Ω ) 2 \Phi_{u_{g}}(\Omega)=\sigma_{u}^{2}\frac{2L_{u}}{\pi}\frac{1}{1+(L_{u}\Omega)^% {2}}
  2. Φ u g ( Ω ) = σ u 2 2 L u π 1 ( 1 + ( 1.339 L u Ω ) 2 ) 5 6 \Phi_{u_{g}}(\Omega)=\sigma_{u}^{2}\frac{2L_{u}}{\pi}\frac{1}{\left(1+(1.339L_% {u}\Omega)^{2}\right)^{\frac{5}{6}}}
  3. 2 L w = h 2L_{w}=h
  4. L u = 2 L v = h ( 0.177 + 0.000823 h ) 1.2 L_{u}=2L_{v}=\frac{h}{(0.177+0.000823h)^{1.2}}
  5. W 20 W_{20}
  6. σ w = 0.1 W 20 \sigma_{w}=0.1W_{20}
  7. σ u σ w = σ v σ w = 1 ( 0.177 + 0.000823 h ) 0.4 \frac{\sigma_{u}}{\sigma_{w}}=\frac{\sigma_{v}}{\sigma_{w}}=\frac{1}{(0.177+0.% 000823h)^{0.4}}
  8. σ u = σ v = σ w = 0.1 W 20 \sigma_{u}=\sigma_{v}=\sigma_{w}=0.1W_{20}
  9. L u = 2 L v = 2 L w = 1750 ft L_{u}=2L_{v}=2L_{w}=1750\,\text{ft}
  10. L u = 2 L v = 2 L w = 2500 ft L_{u}=2L_{v}=2L_{w}=2500\,\text{ft}
  11. σ u = σ v = σ w \sigma_{u}=\sigma_{v}=\sigma_{w}

Contou-Carrère_symbol.html

  1. a = a 0 t w ( a ) i 0 ( 1 - a i t i ) a=a_{0}t^{w(a)}\prod_{i\neq 0}(1-a_{i}t^{i})
  2. a , b = ( - 1 ) w ( a ) w ( b ) a 0 w ( b ) i , j > 0 ( 1 - a i j / ( i , j ) b - j i / ( i , j ) ) ( i , j ) b 0 w ( a ) i , j > 0 ( 1 - b i j / ( i , j ) a - j i / ( i , j ) ) ( i , j ) \langle a,b\rangle=(-1)^{w(a)w(b)}\frac{a_{0}^{w(b)}\prod_{i,j>0}(1-a_{i}^{j/(% i,j)}b_{-j}^{i/(i,j)})^{(i,j)}}{b_{0}^{w(a)}\prod_{i,j>0}(1-b_{i}^{j/(i,j)}a_{% -j}^{i/(i,j)})^{(i,j)}}

Convergent_cross_mapping.html

  1. X X
  2. M X M_{X}
  3. M M
  4. M X M_{X}
  5. M Y M_{Y}
  6. M X M_{X}
  7. M Y M_{Y}
  8. Y Y
  9. M X M_{X}
  10. X X
  11. Y Y
  12. Y Y
  13. X X
  14. X X
  15. M Y M_{Y}
  16. Y Y
  17. M X M_{X}

Convergent_matrix.html

  1. lim k ( T k ) i j = 0 , ( 1 ) \lim_{k\to\infty}(T^{k})_{ij}=0,\quad(1)
  2. 𝐓 = ( 1 4 1 2 0 1 4 ) . \begin{aligned}&\displaystyle\mathbf{T}=\begin{pmatrix}\frac{1}{4}&\frac{1}{2}% \\ 0&\frac{1}{4}\end{pmatrix}.\end{aligned}
  3. 𝐓 2 = ( 1 16 1 4 0 1 16 ) , 𝐓 3 = ( 1 64 3 32 0 1 64 ) , 𝐓 4 = ( 1 256 1 32 0 1 256 ) , 𝐓 5 = ( 1 1024 5 512 0 1 1024 ) , \begin{aligned}&\displaystyle\mathbf{T}^{2}=\begin{pmatrix}\frac{1}{16}&\frac{% 1}{4}\\ 0&\frac{1}{16}\end{pmatrix},\quad\mathbf{T}^{3}=\begin{pmatrix}\frac{1}{64}&% \frac{3}{32}\\ 0&\frac{1}{64}\end{pmatrix},\quad\mathbf{T}^{4}=\begin{pmatrix}\frac{1}{256}&% \frac{1}{32}\\ 0&\frac{1}{256}\end{pmatrix},\quad\mathbf{T}^{5}=\begin{pmatrix}\frac{1}{1024}% &\frac{5}{512}\\ 0&\frac{1}{1024}\end{pmatrix},\end{aligned}
  4. 𝐓 6 = ( 1 4096 3 1024 0 1 4096 ) , \begin{aligned}\displaystyle\mathbf{T}^{6}=\begin{pmatrix}\frac{1}{4096}&\frac% {3}{1024}\\ 0&\frac{1}{4096}\end{pmatrix},\end{aligned}
  5. 𝐓 k = ( ( 1 4 ) k k 2 2 k - 1 0 ( 1 4 ) k ) . \begin{aligned}\displaystyle\mathbf{T}^{k}=\begin{pmatrix}(\frac{1}{4})^{k}&% \frac{k}{2^{2k-1}}\\ 0&(\frac{1}{4})^{k}\end{pmatrix}.\end{aligned}
  6. lim k ( 1 4 ) k = 0 \lim_{k\to\infty}\left(\frac{1}{4}\right)^{k}=0
  7. lim k k 2 2 k - 1 = 0 , \lim_{k\to\infty}\frac{k}{2^{2k-1}}=0,
  8. 1 4 \frac{1}{4}
  9. 1 4 \frac{1}{4}
  10. lim k T k = 0 , \lim_{k\to\infty}\|T^{k}\|=0,
  11. lim k T k = 0 , \lim_{k\to\infty}\|T^{k}\|=0,
  12. ρ ( T ) < 1 \rho(T)<1
  13. lim k T k x = 0 , \lim_{k\to\infty}T^{k}x=0,
  14. A x = b ( 2 ) {Ax}={b}\quad(2)
  15. x = T x + c ( 3 ) {x}={Tx}+{c}\quad(3)
  16. x ( k + 1 ) = T x ( k ) + c ( 4 ) {x}^{(k+1)}={Tx}^{(k)}+{c}\quad(4)
  17. n \mathbb{R}^{n}
  18. { x ( k ) } k = 0 \{{x}^{\left(k\right)}\}_{k=0}^{\infty}
  19. A = B - C ( 5 ) {A}={B}-{C}\quad(5)
  20. lim k T k ( 6 ) \lim_{k\to\infty}T^{k}\quad(6)
  21. n \mathbb{R}^{n}

Conversion_marketing.html

  1. Conversion rate = Number of Goal Achievements Visitors \mathrm{Conversion\ rate}=\frac{\mathrm{Number\ of\ Goal\ Achievements}}{% \mathrm{Visitors}}

Conway_group_Co1.html

  1. × 10 1 8 \times 10^{1}8
  2. 𝟏 / 2 ( 1 - 1 - 1 - 1 - 1 1 - 1 - 1 - 1 - 1 1 - 1 - 1 - 1 - 1 1 ) {\mathbf{1}/2}\left(\begin{matrix}1&-1&-1&-1\\ -1&1&-1&-1\\ -1&-1&1&-1\\ -1&-1&-1&1\end{matrix}\right)

Conway_group_Co3.html

  1. × 10 1 1 \times 10^{1}1
  2. T 4 A ( τ ) T_{4A}(\tau)
  3. j 4 A ( τ ) = T 4 A ( τ ) + 24 = ( η 2 ( 2 τ ) η ( τ ) η ( 4 τ ) ) 24 = ( ( η ( τ ) η ( 4 τ ) ) 4 + 4 2 ( η ( 4 τ ) η ( τ ) ) 4 ) 2 = 1 q + 24 + 276 q + 2048 q 2 + 11202 q 3 + 49152 q 4 + \begin{aligned}\displaystyle j_{4A}(\tau)&\displaystyle=T_{4A}(\tau)+24\\ &\displaystyle=\Big(\tfrac{\eta^{2}(2\tau)}{\eta(\tau)\,\eta(4\tau)}\Big)^{24}% \\ &\displaystyle=\Big(\big(\tfrac{\eta(\tau)}{\eta(4\tau)}\big)^{4}+4^{2}\big(% \tfrac{\eta(4\tau)}{\eta(\tau)}\big)^{4}\Big)^{2}\\ &\displaystyle=\frac{1}{q}+24+276q+2048q^{2}+11202q^{3}+49152q^{4}+\dots\end{aligned}

Coons_patch.html

  1. L c ( s , t ) = ( 1 - t ) c 0 ( s ) + t c 1 ( s ) L_{c}(s,t)=(1-t)c_{0}(s)+tc_{1}(s)\,
  2. L d ( s , t ) = ( 1 - s ) d 0 ( t ) + s d 1 ( t ) L_{d}(s,t)=(1-s)d_{0}(t)+sd_{1}(t)\,
  3. B ( s , t ) = c 0 ( 0 ) ( 1 - s ) ( 1 - t ) + c 0 ( 1 ) s ( 1 - t ) + c 1 ( 0 ) ( 1 - s ) t + c 1 ( 1 ) s t . B(s,t)=c_{0}(0)(1-s)(1-t)+c_{0}(1)s(1-t)+c_{1}(0)(1-s)t+c_{1}(1)st.\,
  4. L c ( s , t ) + L d ( s , t ) - B ( s , t ) . L_{c}(s,t)+L_{d}(s,t)-B(s,t).\,

Cornish–Fisher_expansion.html

  1. y p μ + σ w y_{p}\approx\mu+\sigma w
  2. w = x + [ γ 1 h 1 ( x ) ] + [ γ 2 h 2 ( x ) + γ 1 2 h 11 ( x ) ] + [ γ 3 h 3 ( x ) + γ 1 γ 2 h 12 ( x ) + γ 1 3 h 111 ( x ) ] + \begin{aligned}\displaystyle w&\displaystyle=&\displaystyle x&\displaystyle+% \left[\gamma_{1}h_{1}(x)\right]\\ &&&\displaystyle+\left[\gamma_{2}h_{2}(x)+\gamma_{1}^{2}h_{11}(x)\right]\\ &&&\displaystyle+\left[\gamma_{3}h_{3}(x)+\gamma_{1}\gamma_{2}h_{12}(x)+\gamma% _{1}^{3}h_{111}(x)\right]\\ &&&\displaystyle+\cdots\\ \end{aligned}
  3. x = Φ - 1 ( p ) γ r - 2 = κ r κ 2 r / 2 ; r { 3 , 4 , } h 1 ( x ) = He 2 ( x ) 6 h 2 ( x ) = He 3 ( x ) 24 h 11 ( x ) = - [ 2 H e 3 ( x ) + He 1 ( x ) ] 36 h 3 ( x ) = He 4 ( x ) 120 h 12 ( x ) = - [ He 4 ( x ) + He 2 ( x ) ] 24 h 111 ( x ) = [ 12 H e 4 ( x ) + 19 H e 2 ( x ) ] 324 \begin{aligned}\displaystyle x&\displaystyle=\Phi^{-1}(p)\\ \displaystyle\gamma_{r-2}&\displaystyle=\frac{\kappa_{r}}{\kappa_{2}^{r/2}};\;% r\in\{3,4,\ldots\}\\ \displaystyle h_{1}(x)&\displaystyle=\frac{\mathrm{He}_{2}(x)}{6}\\ \displaystyle h_{2}(x)&\displaystyle=\frac{\mathrm{He}_{3}(x)}{24}\\ \displaystyle h_{11}(x)&\displaystyle=-\frac{\left[2\mathrm{He}_{3}(x)+\mathrm% {He}_{1}(x)\right]}{36}\\ \displaystyle h_{3}(x)&\displaystyle=\frac{\mathrm{He}_{4}(x)}{120}\\ \displaystyle h_{12}(x)&\displaystyle=-\frac{\left[\mathrm{He}_{4}(x)+\mathrm{% He}_{2}(x)\right]}{24}\\ \displaystyle h_{111}(x)&\displaystyle=\frac{\left[12\mathrm{He}_{4}(x)+19% \mathrm{He}_{2}(x)\right]}{324}\end{aligned}
  4. 1.644854 \displaystyle 1.644854

Cortisone_b-reductase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Costunolide_synthase.html

  1. \rightleftharpoons

Courcelle's_theorem.html

  1. card q , r ( S ) \operatorname{card}_{q,r}(S)

Covector_mapping_principle.html

  1. B B
  2. B λ B^{\lambda}
  3. B λ B^{\lambda}
  4. B λ N B^{\lambda N}
  5. N N
  6. N , Problem B λ N Problem B λ N\to\infty,\quad\,\text{Problem }B^{\lambda N}\to\,\text{Problem }B^{\lambda}
  7. B λ N B^{\lambda N}
  8. B λ B^{\lambda}
  9. B B
  10. B N B^{N}
  11. B N λ B^{N\lambda}
  12. B N λ B^{N\lambda}
  13. B λ N B^{\lambda N}

Crack_spacing_of_reinforced_concrete.html

  1. w m = ϵ c f s m {w}_{m}=\epsilon_{cf}s_{m}
  2. s m = 2 ( c + s 10 ) + k 1 k 2 d b ρ e f s_{m}=2(c+\frac{s}{10})+k_{1}k_{2}\frac{d_{b}}{\rho_{ef}}
  3. c c
  4. s s
  5. 15 d b 15d_{b}
  6. d b d_{b}
  7. ρ e f \rho_{ef}
  8. A s / A c e f A_{s}/A_{cef}
  9. A s A_{s}
  10. A c e f A_{cef}
  11. k 1 k_{1}
  12. k 1 k_{1}
  13. k 1 k_{1}
  14. k 2 k_{2}
  15. k 2 = 0.25 ( ϵ 1 + ϵ 2 ) / 2 ϵ 1 k_{2}=0.25(\epsilon_{1}+\epsilon_{2})/2\epsilon_{1}
  16. s m θ = 1 / ( s i n θ s m x + c o s θ s m v ) {s}_{m\theta}=1/(\frac{sin\theta}{s_{mx}}+\frac{cos\theta}{s_{mv}})
  17. s m x s_{mx}
  18. s m v s_{mv}
  19. s m x = 2 ( c x + s x 10 ) + 0.25 k 1 d b x ρ x s_{mx}=2(c_{x}+\frac{s_{x}}{10})+0.25k_{1}\frac{d_{bx}}{\rho_{x}}
  20. s m v = 2 ( c v + s 10 ) + 0.25 k 1 d b v ρ v s_{mv}=2(c_{v}+\frac{s}{10})+0.25k_{1}\frac{d_{bv}}{\rho_{v}}

Crack_tip_opening_displacement.html

  1. K 2 σ y E \frac{K^{2}}{\sigma_{y}E}

Crank_of_a_partition.html

  1. c ( λ ) = { l ( λ ) if ω ( λ ) = 0 μ ( λ ) - ω ( λ ) if ω ( λ ) > 0. c(\lambda)=\begin{cases}l(\lambda)&\,\text{ if }\omega(\lambda)=0\\ \mu(\lambda)-\omega(\lambda)&\,\text{ if }\omega(\lambda)>0.\end{cases}
  2. n = 0 m = - M ( m , n ) z m q n = n = 1 ( 1 - q n ) ( 1 - z q n ) ( 1 - z - 1 q n ) \sum_{n=0}^{\infty}\sum_{m=-\infty}^{\infty}M(m,n)z^{m}q^{n}=\prod_{n=1}^{% \infty}\frac{(1-q^{n})}{(1-zq^{n})(1-z^{-1}q^{n})}

Credal_network.html

  1. ( X 1 , , X n ) (X_{1},\ldots,X_{n})
  2. G G
  3. K ( X i π i ) K(X_{i}\mid\pi_{i})
  4. i = 1 , , n i=1,\ldots,n
  5. X i X_{i}
  6. K ( X 1 , , X n ) = CH { P ( X 1 , , X n ) : P ( x 1 , , x n ) = i = 1 n P ( x i π i ) , P ( X i π i ) K ( X i π i ) } K(X_{1},\ldots,X_{n})=\mathrm{CH}\{P(X_{1},\ldots,X_{n}):P(x_{1},\ldots,x_{n})% =\prod_{i=1}^{n}P(x_{i}\mid\pi_{i}),P(X_{i}\mid\pi_{i})\in K(X_{i}\mid\pi_{i})\}
  7. CH \mathrm{CH}
  8. X q X_{q}
  9. X E X_{E}
  10. P ¯ ( x q x E ) = min P ( X 1 , , X n ) K ( X 1 , , X n ) X q , X E P ( X 1 , , X n ) X q P ( X 1 , , X n ) . \underline{P}(x_{q}\mid x_{E})=\min_{P(X_{1},\ldots,X_{n})\in K(X_{1},\ldots,X% _{n})}\frac{\sum_{X_{q},X_{E}}P(X_{1},\ldots,X_{n})}{\sum_{X_{q}}P(X_{1},% \ldots,X_{n})}.

Creep_and_shrinkage_of_concrete.html

  1. ϵ \epsilon
  2. σ = 1 \sigma=1
  3. t t^{\prime}
  4. t t^{\prime}
  5. t - t t-t^{\prime}
  6. J J
  7. t - t t-t^{\prime}
  8. t t
  9. t t^{\prime}
  10. σ ( t ) \sigma(t)
  11. d σ ( t ) \mbox{d}~{}\sigma(t^{\prime})
  12. t t^{\prime}
  13. d ϵ ( t ) = J ( t , t ) d σ ( t ) \mbox{d}~{}\epsilon(t)=J(t,t^{\prime})\mbox{d}~{}\sigma(t^{\prime})
  14. ϵ ( t ) = t 1 t J ( t , t ) d σ ( t ) + ϵ 0 ( t ) \epsilon(t)=\int_{t_{1}}^{t}J(t,t^{\prime})\mbox{d}~{}\sigma(t^{\prime})+% \epsilon^{0}(t)
  15. ϵ 0 \epsilon^{0}
  16. ϵ s h \epsilon_{sh}
  17. σ ( t ) \sigma(t)
  18. d σ ( t ) = [ d σ ( t ) / d t ] d t \mbox{d}~{}\sigma(t^{\prime})=[\mbox{d}~{}\sigma(t^{\prime})/\mbox{d}~{}t^{% \prime}]\mbox{d}~{}t^{\prime}
  19. ϵ ( t ) \epsilon(t)
  20. σ ( t ) \sigma(t)
  21. J ( t , t ) J(t,t^{\prime})
  22. σ ( t ) \sigma(t)
  23. ϵ = 1 \epsilon=1
  24. t ^ \hat{t}
  25. ϵ 0 = 0 \epsilon^{0}=0
  26. R ( t , t ^ ) R(t,\hat{t})
  27. ν 0.18 \nu\approx 0.18
  28. J ( t , t ) J(t,t^{\prime})
  29. ϵ 0 ( t ) \epsilon^{0}(t)
  30. ϵ 0 ( t ) \epsilon^{0}(t)
  31. f c f_{c}^{\prime}
  32. E E
  33. t t^{\prime}
  34. J ( t , t ) J(t,t^{\prime})
  35. t - t t-t^{\prime}
  36. E ( t ) = 1 / J ( t + δ , t ) E(t^{\prime})=1/J(t^{\prime}+\delta,t^{\prime})
  37. δ \delta
  38. δ 0.01 \delta\approx 0.01
  39. t = 28 t^{\prime}=28
  40. E E
  41. E E
  42. t t^{\prime}
  43. E = 57 , 000 psi E=57,000\mbox{psi}~{}
  44. f c / psi \sqrt{f^{\prime}_{c}/\mbox{psi}~{}}
  45. ( 1 psi = 6895 Pa , f c = uniaxial compressive strength of concrete ) (1\mbox{psi}~{}=6895\mbox{Pa}~{},f_{c}^{\prime}=\mbox{uniaxial compressive % strength of concrete}~{})
  46. q 1 = J ( t , t ) = lim δ 0 J ( t + δ , t ) q_{1}=J(t^{\prime},t^{\prime})=\lim_{\delta\to 0}J(t^{\prime}+\delta,t^{\prime})
  47. q 1 q_{1}
  48. J ( t , t ) J(t,t^{\prime})
  49. J ˙ ( t , t ) = v - 1 ( t ) C ˙ g ( θ ) + 1 / η f , v - 1 ( t ) = q 2 ( λ 0 / t ) m + q 3 \dot{J}(t,t^{\prime})=v^{-1}(t)\ \dot{C}_{g}(\theta)+1/\eta_{f},~{}~{}~{}v^{-1% }(t)=q_{2}\left(\lambda_{0}/t\right)^{m}+q_{3}
  50. C ˙ g ( θ ) = n θ n - 1 λ 0 n + θ n , θ = t - t , 1 / η f = q 4 / t \dot{C}_{g}(\theta)=\frac{n\theta^{n-1}}{\lambda_{0}^{n}+\theta^{n}},~{}~{}~{}% \theta=t-t^{\prime},~{}~{}~{}1/\eta_{f}=q_{4}/t
  51. x ˙ = x / t \dot{x}=\partial x/\partial t
  52. η f \eta_{f}
  53. θ \theta
  54. λ 0 \lambda_{0}
  55. m = 0.5 m=0.5
  56. n = 0.1 n=0.1
  57. v ( t ) MPa - 1 v(t)\rm MPa^{-1}
  58. q 2 , q 3 , q 4 q_{2},q_{3},q_{4}
  59. MPa - 1 \rm MPa^{-1}
  60. C g ( θ ) C_{g}(\theta)
  61. C g ( θ ) = ln [ 1 + ( θ / λ 0 ) n ] C_{g}(\theta)=\mbox{ln}~{}[1+(\theta/\lambda_{0})^{n}]
  62. J ˙ ( t , t ) \dot{J}(t,t^{\prime})
  63. J ( t , t ) J(t,t^{\prime})
  64. J ( t , t ) J(t,t^{\prime})
  65. J ( t , t ) J(t,t^{\prime})
  66. J ˙ ( t , t ) \dot{J}(t,t^{\prime})
  67. θ \theta
  68. J ˙ ( t , t ) \dot{J}(t,t^{\prime})
  69. d v - 1 ( t ) / d t {\rm\mbox{d}~{}v^{-1}(t)/\mbox{d}~{}t}
  70. 2 J ( t , t ) / t t > 0 \partial^{2}J(t,t^{\prime})/\partial t\partial t^{\prime}>0
  71. w w
  72. S S
  73. c 0 , c 1 c_{0},c_{1}
  74. S S
  75. w w
  76. h h
  77. v ( t ) v(t)
  78. t t^{\prime}
  79. t t
  80. t e = β h β T d t t_{e}=\int\beta_{h}\beta_{T}\mbox{d}~{}t
  81. β h \beta_{h}
  82. h h
  83. h < h<
  84. β h e - Q h T / R \beta_{h}\propto\mbox{e}~{}^{-Q_{h}T/R}
  85. ( Q h / R 2700 K ) (Q_{h}/R\approx\mbox{2700 K}~{})
  86. θ = t - t \theta=t-t^{\prime}
  87. t r - t r t_{r}-t^{\prime}_{r}
  88. t r = ψ h ψ T d t t_{r}=\int\psi_{h}\psi_{T}\mbox{d}~{}t
  89. h h
  90. T T
  91. ψ h \psi_{h}
  92. h h
  93. h = 1 h=1
  94. h = 0 h=0
  95. ψ T e - Q v T / R \psi_{T}\propto\mbox{e}~{}^{-Q_{v}T/R}
  96. Q v / R Q_{v}/R\approx
  97. h ( 𝐱 , t ) h(\mathbf{x},t)
  98. 𝐱 \mathbf{x}
  99. h ˙ = \rm\dot{h}=
  100. C ( h ) C(h)
  101. h ] + h ˙ s ( t e ) h]+\dot{h}_{s}(t_{e})
  102. h s ( t e ) h_{s}(t_{e})
  103. C ( h ) C(h)
  104. h h
  105. ϵ ˙ s h = k s h h ˙ \dot{\epsilon}_{sh}=k_{sh}\dot{h}
  106. k s h k_{sh}
  107. ϵ ˙ s h \dot{\epsilon}_{sh}
  108. C g ( θ ) C_{g}(\theta)
  109. t e t_{e}
  110. J ¯ ( t , t , t 0 ) \bar{J}(t,t^{\prime},t_{0})
  111. ϵ ¯ s h ( t , t 0 ) \bar{\epsilon}_{sh}(t,t_{0})
  112. t 0 t_{0}
  113. h e h_{e}
  114. ϵ ¯ s h ( t , t 0 ) = - ϵ s h k h S ( t ) , k h = 1 - h e 3 \bar{\epsilon}_{sh}(t,t_{0})=-\epsilon_{sh\infty}\ k_{h}\ S(t),~{}~{}~{}k_{h}=% 1-{h_{e}}^{3}~{}~{}~{}~{}~{}
  115. S ( t ) = tanh t - t 0 τ s h , τ s h = k t ( k s D ) 2 S(t)=\mbox{tanh}~{}\sqrt{\frac{t-t_{0}}{\tau_{sh}}},~{}~{}~{}\tau_{sh}=k_{t}(k% _{s}D)^{2}~{}~{}~{}~{}~{}
  116. D = 2 v / s D=2v/s
  117. v / s v/s
  118. k t k_{t}
  119. k s k_{s}
  120. ϵ s h ϵ s E ( 607 ) / ( E ( t 0 + τ s h ) \epsilon_{sh\infty}\approx\epsilon_{s\infty}E(607)/(E(t_{0}+\tau_{sh})
  121. ϵ s \epsilon_{s\infty}
  122. E ( t ) E ( 28 ) 4 + 0.85 t E(t)\approx E(28)\sqrt{4+0.85t}
  123. 1 / η f 1/\eta_{f}
  124. 1 η ¯ f = q 4 t + q 5 t F ( t ) - F ( t 0 ) \frac{1}{\bar{\eta}_{f}}=\frac{q_{4}}{t}+q_{5}\frac{\partial}{\partial t}\sqrt% {F(t)-F(t^{\prime}_{0})}
  125. F ( t ) = exp { - 8 [ 1 - ( 1 - h e ) S ( t ) ] } F(t)=\exp\{-8[1-(1-h_{e})S(t)]\}
  126. t 0 = max ( t , t 0 ) t^{\prime}_{0}=\max(t^{\prime},t_{0})
  127. τ s h \tau_{sh}
  128. ϵ ¯ s h t - t 0 \bar{\epsilon}_{sh}\propto\sqrt{t-t_{0}}
  129. ϵ s h \epsilon_{sh\infty}
  130. J ( t , t ) J(t,t^{\prime})
  131. ϵ s h ( t ) \epsilon_{sh}(t)
  132. φ ( t , t ) = E ( t ) J ( t , t ) - 1 \varphi(t,t^{\prime})=E(t^{\prime})J(t,t^{\prime})-1
  133. ϵ creep / ϵ initial \epsilon_{\mbox{creep}~{}}/\epsilon_{\mbox{initial}~{}}
  134. t 1 t_{1}
  135. t t
  136. E E
  137. E ′′ ( t , t 1 ) = [ E ( t 1 ) - R ( t , t 1 ) ] / φ ( t , t 1 ) E^{\prime\prime}(t,t_{1})=[E(t_{1})-R(t,t_{1})]/\varphi(t,t_{1})
  138. J ( t , t ) J(t,t^{\prime})
  139. E k ( t ) E_{k}(t)
  140. k = 1 , 2 , n E k=1,2,...n_{E}

Cremona–Richmond_configuration.html

  1. 15 = ( 6 2 ) 15={\left({{6}\atop{2}}\right)}

Crocetin_glucosyltransferase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons

Cross_Gramian.html

  1. x ˙ = A x + B u \dot{x}=Ax+Bu\,
  2. y = C x y=Cx\,
  3. W X := 0 e A t B C e A t d t W_{X}:=\int_{0}^{\infty}e^{At}BCe^{At}dt\,
  4. A W X + W X A = - B C AW_{X}+W_{X}A=-BC\,
  5. ( A , B , C ) (A,B,C)
  6. W X W_{X}
  7. W X W_{X}
  8. t > 0 t>0
  9. ( A , B , C ) (A,B,C)
  10. J J
  11. A J = J A T AJ=JA^{T}\,
  12. B = J C T B=JC^{T}\,
  13. | λ ( W X ) | = λ ( W C W O ) . |\lambda(W_{X})|=\sqrt{\lambda(W_{C}W_{O})}.\,
  14. W C O W_{CO}

Crotonyl-CoA_carboxylase::reductase.html

  1. \rightleftharpoons

Crotonyl-CoA_reductase.html

  1. \rightleftharpoons

CTP-dependent_riboflavin_kinase.html

  1. \rightleftharpoons

Cyanidin-3-O-glucoside_2-O-glucuronosyltransferase.html

  1. \rightleftharpoons

Cyanophycinase.html

  1. \rightleftharpoons

Cyclic-guanylate-specific_phosphodiesterase.html

  1. \rightleftharpoons

Cyclic_alcohol_dehydrogenase_(quinone).html

  1. \rightleftharpoons

Cyclic_pyranopterin_monophosphate_synthase.html

  1. \rightleftharpoons

Cyclohexane-1,2-dione_hydrolase.html

  1. \rightleftharpoons

Cyclohexanone_monooxygenase.html

  1. \rightleftharpoons

Cyclotruncated_6-simplex_honeycomb.html

  1. A ~ 6 {\tilde{A}}_{6}

Cysteate_synthase.html

  1. \rightleftharpoons

D-Ala-D-Ala_dipeptidase.html

  1. \rightleftharpoons

D-amino_acid_dehydrogenase_(quinone).html

  1. \rightleftharpoons

D-arabinitol_dehydrogenase_(NADP+).html

  1. \rightleftharpoons
  2. \rightleftharpoons

D-arabitol-phosphate_dehydrogenase.html

  1. \rightleftharpoons

D-glycero-alpha-D-manno-heptose-7-phosphate_kinase.html

  1. \rightleftharpoons

D-glycero-alpha-D-manno-heptose_1,7-bisphosphate_7-phosphatase.html

  1. \rightleftharpoons

D-glycero-alpha-D-manno-heptose_1-phosphate_guanylyltransferase.html

  1. \rightleftharpoons

D-glycero-beta-D-manno-heptose-7-phosphate_kinase.html

  1. \rightleftharpoons

D-glycero-beta-D-manno-heptose_1,7-bisphosphate_7-phosphatase.html

  1. \rightleftharpoons

D-glycero-beta-D-manno-heptose_1-phosphate_adenylyltransferase.html

  1. \rightleftharpoons

D-inositol-3-phosphate_glycosyltransferase.html

  1. \rightleftharpoons

D-Man-alpha-(1-3)-D-Glc-beta-(1-4)-D-Glc-alpha-1-diphosphoundecaprenol_2-beta-glucuronyltransferase.html

  1. \rightleftharpoons

D-proline_dehydrogenase.html

  1. \rightleftharpoons

D-threonine_aldolase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

D-xylose_reductase.html

  1. \rightleftharpoons

D::M::1_queue.html

  1. π i = { 0 when i = 0 ( 1 - δ ) δ i - 1 when i > 0 \pi_{i}=\begin{cases}0&\,\text{ when }i=0\\ (1-\delta)\delta^{i-1}&\,\text{ when }i>0\end{cases}

Darken's_equations.html

  1. ν = ( D 1 - D 2 ) N 1 x = ( D 2 - D 1 ) N 2 x \nu=(D_{1}-D_{2})\frac{\partial N_{1}}{\partial x}=(D_{2}-D_{1})\frac{\partial N% _{2}}{\partial x}
  2. ν \textstyle\nu
  3. D ~ = ( N 1 D 2 + N 2 D 1 ) l n a 1 l n N 1 \tilde{D}=(N_{1}D_{2}+N_{2}D_{1})\frac{\partial lna_{1}}{\partial lnN_{1}}
  4. D ~ \textstyle\tilde{D}
  5. ν \textstyle\nu
  6. - D 1 C 1 y \textstyle-D_{1}\frac{\partial C_{1}}{\partial y}
  7. C 1 ν \textstyle C_{1}\nu
  8. - D 1 C 1 y -D_{1}\frac{\partial C_{1}}{\partial y}
  9. - D 1 C 1 x -D_{1}\frac{\partial C_{1}}{\partial x}
  10. ν \textstyle\nu
  11. - [ D 1 C 1 x - C 1 ν ] -[D_{1}\frac{\partial C_{1}}{\partial x}-C_{1}\nu]
  12. C 1 t = x [ D 1 C 1 x - C 1 ν ] \frac{\partial C_{1}}{\partial t}=\frac{\partial}{\partial x}[D_{1}\frac{% \partial C_{1}}{\partial x}-C_{1}\nu]
  13. C 2 t = x [ D 2 C 2 x - C 2 ν ] \frac{\partial C_{2}}{\partial t}=\frac{\partial}{\partial x}[D_{2}\frac{% \partial C_{2}}{\partial x}-C_{2}\nu]
  14. C = C 1 + C 2 C=C_{1}+C_{2}
  15. C 1 t \textstyle\frac{\partial C_{1}}{\partial t}
  16. C 2 t \textstyle\frac{\partial C_{2}}{\partial t}
  17. C t = x [ D 1 C 1 x + D 2 C 2 x - C ν ] \frac{\partial C}{\partial t}=\frac{\partial}{\partial x}[D_{1}\frac{\partial C% _{1}}{\partial x}+D_{2}\frac{\partial C_{2}}{\partial x}-C\nu]
  18. 0 = x [ D 1 C 1 x + D 2 C 2 x - C ν ] 0=\frac{\partial}{\partial x}[D_{1}\frac{\partial C_{1}}{\partial x}+D_{2}% \frac{\partial C_{2}}{\partial x}-C\nu]
  19. D 1 C 1 x + D 2 C 2 x - C ν \textstyle D_{1}\frac{\partial C_{1}}{\partial x}+D_{2}\frac{\partial C_{2}}{% \partial x}-C\nu
  20. D 1 C 1 x + D 2 C 2 x - C ν = I \textstyle D_{1}\frac{\partial C_{1}}{\partial x}+D_{2}\frac{\partial C_{2}}{% \partial x}-C\nu=I
  21. I \textstyle I
  22. ν = 1 C [ D 1 C 1 x + D 2 C 2 x ] \nu=\frac{1}{C}[D_{1}\frac{\partial C_{1}}{\partial x}+D_{2}\frac{\partial C_{% 2}}{\partial x}]
  23. C 1 x = - C 2 x \textstyle\frac{\partial C_{1}}{\partial x}=-\frac{\partial C_{2}}{\partial x}
  24. N 1 = C 1 C \textstyle N_{1}=\frac{C_{1}}{C}
  25. N 2 = C 2 C \textstyle N_{2}=\frac{C_{2}}{C}
  26. ν = ( D 1 - D 2 ) N 1 x = ( D 2 - D 1 ) N 2 x \nu=(D_{1}-D_{2})\frac{\partial N_{1}}{\partial x}=(D_{2}-D_{1})\frac{\partial N% _{2}}{\partial x}
  27. ν \textstyle\nu
  28. ν = 1 C [ D 1 C 1 x + D 2 C 2 x ] \nu=\frac{1}{C}[D_{1}\frac{\partial C_{1}}{\partial x}+D_{2}\frac{\partial C_{% 2}}{\partial x}]
  29. ν \textstyle\nu
  30. C t = x [ D 1 C 1 x - C 1 ν ] \textstyle\frac{\partial C}{\partial t}=\frac{\partial}{\partial x}[D_{1}\frac% {\partial C_{1}}{\partial x}-C_{1}\nu]
  31. C 1 t = x [ D 1 C 1 x - C 1 C [ D 1 C 1 x + D 2 C 2 x ] ] \frac{\partial C_{1}}{\partial t}=\frac{\partial}{\partial x}[D_{1}\frac{% \partial C_{1}}{\partial x}-\frac{C_{1}}{C}[D_{1}\frac{\partial C_{1}}{% \partial x}+D_{2}\frac{\partial C_{2}}{\partial x}]]
  32. C 1 x = - C 2 x \textstyle\frac{\partial C_{1}}{\partial x}=-\frac{\partial C_{2}}{\partial x}
  33. C 1 t = x [ C 1 + C 2 C D 1 C 1 x - C 1 C [ D 1 C 1 x - D 2 C 1 x ] ] \frac{\partial C_{1}}{\partial t}=\frac{\partial}{\partial x}[\frac{C_{1}+C_{2% }}{C}D_{1}\frac{\partial C_{1}}{\partial x}-\frac{C_{1}}{C}[D_{1}\frac{% \partial C_{1}}{\partial x}-D_{2}\frac{\partial C_{1}}{\partial x}]]
  34. N 1 = C 1 C \textstyle N_{1}=\frac{C_{1}}{C}
  35. N 2 = C 2 C \textstyle N_{2}=\frac{C_{2}}{C}
  36. N 1 t = x [ ( N 2 D 1 + N 1 D 2 ) N 1 x ] \frac{\partial N_{1}}{\partial t}=\frac{\partial}{\partial x}[(N_{2}D_{1}+N_{1% }D_{2})\frac{\partial N_{1}}{\partial x}]
  37. λ x t 1 / 2 \textstyle\lambda\equiv\frac{x}{t^{1/2}}
  38. N 1 = f ( λ ) \textstyle N_{1}=f(\lambda)
  39. - 1 2 λ d N 1 = d [ ( N 2 D 1 + N 1 D 2 ) d N 1 d λ ] -\frac{1}{2}\lambda dN_{1}=d[(N_{2}D_{1}+N_{1}D_{2})\frac{dN_{1}}{d\lambda}]
  40. D = D 1 N 2 + D 2 N 1 D=D_{1}N_{2}+D_{2}N_{1}
  41. ν = ( D 2 - D 1 ) N 2 x \textstyle\nu=(D_{2}-D_{1})\frac{\partial N_{2}}{\partial x}
  42. D ~ \textstyle\tilde{D}
  43. J = - 1 N A d F 2 d x B 2 C 2 J=-\frac{1}{N_{A}}\frac{dF_{2}}{dx}B_{2}C_{2}
  44. - D 2 d C 2 d x -D_{2}\frac{dC_{2}}{dx}
  45. D 2 d C 2 d x = 1 N A d F 2 d x B 2 C 2 D_{2}\frac{dC_{2}}{dx}=\frac{1}{N_{A}}\frac{dF_{2}}{dx}B_{2}C_{2}
  46. D 2 = d F 2 d C 2 B 2 C 2 N A D_{2}=\frac{dF_{2}}{dC_{2}}\frac{B_{2}C_{2}}{N_{A}}
  47. 1 N A d F 2 d N 2 B 2 N 2 \frac{1}{N_{A}}\frac{dF_{2}}{dN_{2}}B_{2}N_{2}
  48. d F 2 = R T d l n a 2 \textstyle dF_{2}=RTdlna_{2}
  49. D 2 = k T B 2 l n a 2 d l n N 2 D_{2}=kTB_{2}\frac{lna_{2}}{dlnN_{2}}
  50. γ 2 = a 2 / N 2 \textstyle\gamma_{2}=a_{2}/N_{2}
  51. D 2 = k T B 2 ( 1 + N 2 d l n γ 2 d l n N 2 ) D_{2}=kTB_{2}(1+N_{2}\frac{dln\gamma_{2}}{dlnN_{2}})
  52. D 1 = k T B 1 ( 1 + N 1 d l n γ 1 d l n N 1 ) \textstyle D_{1}=kTB_{1}(1+N_{1}\frac{dln\gamma_{1}}{dlnN_{1}})
  53. D ~ = ( N 1 D 2 + N 2 D 1 ) l n a 1 l n N 1 \tilde{D}=(N_{1}D_{2}+N_{2}D_{1})\frac{\partial lna_{1}}{\partial lnN_{1}}
  54. v = ( D 2 - D 1 ) N 2 x \textstyle v=(D_{2}-D_{1})\frac{\partial N_{2}}{\partial x}
  55. J v = ( D 2 - D 1 ) N 2 x \textstyle J_{v}=(D_{2}-D_{1})\frac{\partial N_{2}}{\partial x}

DATP(dGTP)—DNA_purinetransferase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Davenport_constant.html

  1. G = i C p e i G=\oplus_{i}C_{p^{e_{i}}}
  2. D ( G ) = 1 + i ( p e i - 1 ) . D(G)=1+\sum_{i}\left({p^{e_{i}}-1}\right)\ .
  3. G = i C d i G=\oplus_{i}C_{d_{i}}
  4. d 1 | d 2 | | d r d_{1}|d_{2}|\cdots|d_{r}
  5. i ( d i - 1 ) \sum_{i}(d_{i}-1)
  6. D ( G ) M ( G ) = 1 - r + i d i . D(G)\geq M(G)=1-r+\sum_{i}d_{i}\ .

DBAR_problem.html

  1. ¯ f ( z , z ¯ ) = g ( z ) \bar{\partial}f(z,\bar{z})=g(z)
  2. f ( z , z ¯ ) f(z,\bar{z})
  3. g ( z ) g(z)
  4. z = x + i y z=x+iy
  5. R R\subseteq\mathbb{C}
  6. ¯ \bar{\partial}
  7. ¯ = 1 2 ( x + i y ) \bar{\partial}=\frac{1}{2}\left(\frac{\partial}{\partial x}+i\frac{\partial}{% \partial y}\right)
  8. = z = 1 2 ( x - i y ) \partial=\frac{\partial}{\partial z}=\frac{1}{2}\left(\frac{\partial}{\partial x% }-i\frac{\partial}{\partial y}\right)
  9. z z

Decaprenyl-phosphate_phosphoribosyltransferase.html

  1. \rightleftharpoons

Decaprenylphospho-beta-D-erythro-pentofuranosid-2-ulose_2-reductase.html

  1. \rightleftharpoons

Decaprenylphospho-beta-D-ribofuranose_2-oxidase.html

  1. \rightleftharpoons

Degree_of_reaction.html

  1. R = Isentropic    enthalpy    change   in    rotor Isentropic    enthalpy    change   in   stage R=\frac{\textrm{Isentropic \,\, enthalpy \,\, change\,\, in \,\, rotor}}{% \textrm{Isentropic \,\, enthalpy \,\, change \,\,in \,\,stage}}
  2. R = Isentropic    heat    drop   in    rotor Isentropic    heat    drop   in   stage R=\frac{\textrm{Isentropic \,\, heat \,\, drop\,\, in \,\, rotor}}{\textrm{% Isentropic \,\, heat \,\, drop \,\,in \,\,stage}}
  3. R = Static    pressure    rise   in    rotor Total    pressure    rise   in   stage R=\frac{\textrm{Static \,\, pressure \,\, rise\,\, in \,\, rotor}}{\textrm{% Total \,\, pressure \,\, rise \,\,in \,\,stage}}
  4. T d s = d h - ( P ρ ) d p Tds=dh-(\frac{P}{\rho})dp
  5. R = Δ H    Rotor Δ H    Stage R=\frac{\Delta\textrm{H \,\, Rotor}}{\Delta\textrm{H \,\, Stage}}
  6. R = 3 s s 2 s dh 3 s s 1 dh Or 3 s s 2 s dp 3 s s 1 dp \ R=\frac{\int_{3ss}^{2s}\textrm{dh}}{\int_{3ss}^{1}\textrm{dh}}\,\ \textrm{Or% }\,\ \frac{\int_{3ss}^{2s}\textrm{dp }}{\int_{3ss}^{1}\textrm{dp}}
  7. h 2 - h 3 = 1 2 ( V r 3 2 - V r 2 2 ) + 1 2 ( U 2 2 - U 3 2 ) \,h_{2}-h_{3}={1\over{2}}(V_{r3}^{2}-V_{r2}^{2})+{1\over{2}}(U_{2}^{2}-U_{3}^{% 2})
  8. h 01 - h 03 = h 02 - h 03 = ( U 2 V w 2 - U 1 V w 1 ) \,h_{01}-h_{03}=h_{02}-h_{03}=(U_{2}\,V_{w2}-U_{1}\,V_{w1})
  9. R = [ 1 2 ( V r 3 2 - V r 2 2 ) + 1 2 ( U 2 2 - U 3 2 ) ] ( U 2 V w 2 - U 1 V w 1 ) R=\frac{[{1\over{2}}(V_{r3}^{2}-V_{r2}^{2})+{1\over{2}}(U_{2}^{2}-U_{3}^{2})]}% {(U_{2}\,V_{w2}-U_{1}\,V_{w1})}
  10. U 2 = U 1 = U U2=U1=U
  11. R = ( V r 3 2 - V r 2 2 ) 2 U ( V w 3 + V w 2 ) R=\frac{(V_{r3}^{2}-V_{r2}^{2})}{2U(V_{w3}+V_{w2})}
  12. R = ( V f 2 U ) ( tan β 3 - tan β 2 ) R=(\frac{V_{f}}{2U})(\tan{\beta_{3}}-\tan{\beta_{2}})
  13. β 3 \beta_{3}
  14. β 2 \beta_{2}
  15. ( V f 2 U ) (\frac{V_{f}}{2U})
  16. ( tan β 3 - tan β 2 ) (\tan{\beta_{3}}-\tan{\beta_{2}})
  17. tan β m \tan{\beta_{m}}
  18. R = ϕ tan β m . R=\phi\tan{\beta_{m}}.
  19. tan β m \tan{\beta_{m}}
  20. R = 1 2 + V f 2 U ( tan β 3 - tan α 2 ) R=\frac{1}{2}+\frac{V_{f}}{2U}(\tan{\beta_{3}}-\tan{\alpha_{2}})
  21. R = 1 + Δ W 2 U 2 - C y 2 U R=1+\frac{\Delta W}{2U^{2}}-\frac{C_{y2}}{U}
  22. Δ W 2 U 2 \frac{\Delta W}{2U^{2}}
  23. 1 / 2 {1}/{2}

DeGroot_learning.html

  1. n n
  2. p ( 0 ) = ( p 1 ( 0 ) , , p n ( 0 ) ) p(0)=(p_{1}(0),\dots,p_{n}(0))
  3. T T
  4. T i j T_{ij}
  5. i i
  6. j j
  7. i i
  8. j j
  9. T i j > 0 T_{ij}>0
  10. p ( t ) = T p ( t - 1 ) p(t)=Tp(t-1)
  11. t t
  12. p ( t ) = T t p ( 0 ) p(t)=T^{t}p(0)
  13. p ( ) = lim t p ( t ) = lim t T t p ( 0 ) p(\infty)=\lim_{t\to\infty}p(t)=\lim_{t\to\infty}T^{t}p(0)
  14. p ( 0 ) [ 0 , 1 ] n p(0)\in[0,1]^{n}
  15. T T
  16. s s
  17. T T
  18. p [ 0 , 1 ] n p\in[0,1]^{n}
  19. ( lim t T t p ) i = s p \left(\lim_{t\to\infty}T^{t}p\right)_{i}=s\cdot p
  20. i { 1 , , n } i\in\{1,\dots,n\}
  21. \cdot
  22. C { 1 , , n } C\subseteq\{1,\dots,n\}
  23. i C i\in C
  24. T i j > 0 T_{ij}>0
  25. j C j\in C
  26. C C
  27. p i ( ) = p j ( ) p_{i}(\infty)=p_{j}(\infty)
  28. i , j C i,j\in C
  29. C C
  30. p ( ) = s p ( 0 ) p(\infty)=s\cdot p(0)
  31. s s
  32. T T
  33. s s
  34. s i s_{i}
  35. i i
  36. s = s T s=sT
  37. s i = j = 1 n T j i s j s_{i}=\sum_{j=1}^{n}T_{ji}s_{j}
  38. i i
  39. s j s_{j}
  40. i i
  41. T = ( 0 1 / 2 1 / 2 1 0 0 0 1 0 ) T=\begin{pmatrix}0&1/2&1/2\\ 1&0&0\\ 0&1&0\\ \end{pmatrix}
  42. lim t T t p ( 0 ) = ( lim t T t ) p ( 0 ) = ( 2 / 5 2 / 5 1 / 5 2 / 5 2 / 5 1 / 5 2 / 5 2 / 5 1 / 5 ) p ( 0 ) \lim_{t\to\infty}T^{t}p(0)=\left(\lim_{t\to\infty}T^{t}\right)p(0)=\begin{% pmatrix}2/5&2/5&1/5\\ 2/5&2/5&1/5\\ 2/5&2/5&1/5\\ \end{pmatrix}p(0)
  43. s = ( 2 / 5 , 2 / 5 , 1 / 5 ) s=\left(2/5,2/5,1/5\right)
  44. 2 / 5 p 1 ( 0 ) + 2 / 5 p 2 ( 0 ) + 1 / 5 p 3 ( 0 ) 2/5p_{1}(0)+2/5p_{2}(0)+1/5p_{3}(0)
  45. T = ( 0 1 / 2 1 / 2 1 0 0 1 0 0 ) T=\begin{pmatrix}0&1/2&1/2\\ 1&0&0\\ 1&0&0\\ \end{pmatrix}
  46. k 1 k\geq 1
  47. T 2 k - 1 = ( 0 1 / 2 1 / 2 1 0 0 1 0 0 ) T^{2k-1}=\begin{pmatrix}0&1/2&1/2\\ 1&0&0\\ 1&0&0\\ \end{pmatrix}
  48. T 2 k = ( 1 0 0 0 1 / 2 1 / 2 0 1 / 2 1 / 2 ) T^{2k}=\begin{pmatrix}1&0&0\\ 0&1/2&1/2\\ 0&1/2&1/2\\ \end{pmatrix}
  49. lim t T t \lim_{t\to\infty}T^{t}
  50. n n\to\infty
  51. μ [ 0 , 1 ] \mu\in[0,1]
  52. p i ( 0 ) ( n ) p_{i}^{(0)}(n)
  53. μ \mu
  54. n n
  55. T ( n ) T(n)
  56. p i ( ) ( n ) p_{i}^{(\infty)}(n)
  57. ( T ( n ) ) n = 1 \left(T(n)\right)_{n=1}^{\infty}
  58. max i n | p i ( ) - μ | 𝑝 0 \max_{i\leq n}|p_{i}^{(\infty)}-\mu|\xrightarrow{\ p\ }0
  59. 𝑝 \xrightarrow{\ p\ }
  60. lim n max i n s i ( n ) = 0 \lim_{n\to\infty}\max_{i\leq n}s_{i}(n)=0

Deletion_channel.html

  1. p p
  2. 1 - p 1-p
  3. p p
  4. 0 < p < 1 0<p<1
  5. n n
  6. ( X i ) (X_{i})
  7. X n X_{n}
  8. p p
  9. ( Y i ) (Y_{i})
  10. ( X i ) (X_{i})
  11. p p

Delone_set.html

  1. 𝒜 q n \scriptstyle\mathcal{A}_{q}^{n}

Delphinidin_3',5'-O-glucosyltransferase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons

Delta11-fatty-acid_desaturase.html

  1. \rightleftharpoons

Delta12-fatty-acid_desaturase.html

  1. \rightleftharpoons

Delta8-fatty-acid_desaturase.html

  1. \rightleftharpoons

Demethylmenaquinone_methyltransferase.html

  1. \rightleftharpoons

Demethylrebeccamycin-D-glucose_O-methyltransferase.html

  1. \rightleftharpoons

Demethylspheroidene_O-methyltransferase.html

  1. \rightleftharpoons

Deoxyhypusine_synthase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons
  4. \rightleftharpoons
  5. \rightleftharpoons

Derjaguin_approximation.html

  1. F ( h ) = 2 π R eff W ( h ) , F(h)=2\pi R_{\rm eff}W(h),
  2. R eff - 1 = R 1 - 1 + R 2 - 1 . R_{\rm eff}^{-1}=R_{1}^{-1}+R_{2}^{-1}.
  3. F ( h ) = - d U d h , F(h)=-{dU\over dh},
  4. U ( h ) = h F ( h ) d h . U(h)=\int_{h}^{\infty}F(h^{\prime})\,dh^{\prime}.
  5. Π ( h ) = - d W d h , \Pi(h)=-{dW\over dh},
  6. W ( h ) = h Π ( h ) d h . W(h)=\int_{h}^{\infty}\Pi(h^{\prime})\,dh^{\prime}.
  7. R eff = R / 2. R_{\rm eff}=R/2.
  8. R eff = R . R_{\rm eff}=R.
  9. R eff = R 1 R 2 , R_{\rm eff}=\sqrt{R_{1}R_{2}},
  10. F = Π ( x ) d A , F=\int\Pi(x)\,dA,
  11. R 2 = ( R - y ) 2 + r 2 . R^{2}=(R-y)^{2}+r^{2}.
  12. d A = 2 π r d r = 2 π R d y = π R d x . dA=2\pi r\,dr=2\pi R\,dy=\pi R\,dx.
  13. F ( h ) = π R h Π ( x ) d x = π R W ( h ) , F(h)=\pi R\int_{h}^{\infty}\Pi(x)\,dx=\pi RW(h),
  14. 1 R eff 2 = ( 1 R 1 + 1 R 2 ) ( 1 R 1 ′′ + 1 R 2 ′′ ) + ( 1 R 1 - 1 R 1 ′′ ) ( 1 R 2 - 1 R 2 ′′ ) sin 2 φ , \frac{1}{R_{\rm eff}^{2}}=\left(\frac{1}{R^{\prime}_{1}}+\frac{1}{R^{\prime}_{% 2}}\right)\left(\frac{1}{R^{\prime\prime}_{1}}+\frac{1}{R^{\prime\prime}_{2}}% \right)+\left(\frac{1}{R^{\prime}_{1}}-\frac{1}{R^{\prime\prime}_{1}}\right)% \left(\frac{1}{R^{\prime}_{2}}-\frac{1}{R^{\prime\prime}_{2}}\right)\sin^{2}\varphi,
  15. T = π R eff 3 V ( h ) ( 1 R 1 - 1 R 1 ′′ ) ( 1 R 2 - 1 R 2 ′′ ) sin 2 φ , T=\pi R_{\rm eff}^{3}V(h)\left(\frac{1}{R^{\prime}_{1}}-\frac{1}{R^{\prime% \prime}_{1}}\right)\left(\frac{1}{R^{\prime}_{2}}-\frac{1}{R^{\prime\prime}_{2% }}\right)\sin 2\varphi,
  16. V ( h ) = h W ( h ) d h . V(h)=\int_{h}^{\infty}W(h^{\prime})\,dh^{\prime}.

Dermatan_4-sulfotransferase.html

  1. \rightleftharpoons

Dershowitz–Manna_ordering.html

  1. ( S , < S ) (S,<_{S})
  2. ( S ) \mathcal{M}(S)
  3. S S
  4. M , N ( S ) M,N\in\mathcal{M}(S)
  5. M < D M N M<_{DM}N
  6. M < D M N M<_{DM}N
  7. X , Y ( S ) X,Y\in\mathcal{M}(S)
  8. X X\neq\varnothing
  9. X N X\subseteq N
  10. M = ( N - X ) + Y M=(N-X)+Y
  11. X X
  12. Y Y
  13. y Y y\in Y
  14. x X x\in X
  15. y < S x y<_{S}x
  16. M < D M N M<_{DM}N
  17. M N M\neq N
  18. y y
  19. S S
  20. M ( y ) > N ( y ) M(y)>N(y)
  21. x x
  22. S S
  23. y < S x y<_{S}x
  24. M ( x ) < N ( x ) M(x)<N(x)

Desosaminyl_transferase_EryCIII.html

  1. \rightleftharpoons

Determinant_identities.html

  1. det ( I n ) = 1 \det(I_{n})=1
  2. det ( A T ) = det ( A ) . \det(A^{\rm T})=\det(A).
  3. det ( A - 1 ) = 1 det ( A ) = det ( A ) - 1 . \det(A^{-1})=\frac{1}{\det(A)}=\det(A)^{-1}.
  4. det ( A B ) = det ( A ) det ( B ) . \det(AB)=\det(A)\det(B).
  5. det ( c A ) = c n det ( A ) \det(cA)=c^{n}\det(A)
  6. det ( A ) = a 1 , 1 a 2 , 2 a n , n = i = 1 n a i , i . \det(A)=a_{1,1}a_{2,2}\cdots a_{n,n}=\prod_{i=1}^{n}a_{i,i}.
  7. L = [ I p 0 - D - 1 C I q ] . L=\left[\begin{matrix}I_{p}&0\\ -D^{-1}C&I_{q}\end{matrix}\right].
  8. M L = [ A B C D ] [ I p 0 - D - 1 C I q ] = [ A - B D - 1 C B 0 D ] = [ I p B D - 1 0 I q ] [ A - B D - 1 C 0 0 D ] . \begin{aligned}\displaystyle ML&\displaystyle=\left[\begin{matrix}A&B\\ C&D\end{matrix}\right]\left[\begin{matrix}I_{p}&0\\ -D^{-1}C&I_{q}\end{matrix}\right]=\left[\begin{matrix}A-BD^{-1}C&B\\ 0&D\end{matrix}\right]\\ &\displaystyle=\left[\begin{matrix}I_{p}&BD^{-1}\\ 0&I_{q}\end{matrix}\right]\left[\begin{matrix}A-BD^{-1}C&0\\ 0&D\end{matrix}\right].\end{aligned}
  9. [ A B C D ] = [ I p B D - 1 0 I q ] [ A - B D - 1 C 0 0 D ] [ I p 0 D - 1 C I q ] , \begin{aligned}\displaystyle\left[\begin{matrix}A&B\\ C&D\end{matrix}\right]&\displaystyle=\left[\begin{matrix}I_{p}&BD^{-1}\\ 0&I_{q}\end{matrix}\right]\left[\begin{matrix}A-BD^{-1}C&0\\ 0&D\end{matrix}\right]\left[\begin{matrix}I_{p}&0\\ D^{-1}C&I_{q}\end{matrix}\right],\end{aligned}