wpmath0000008_8

Moore_space_(topology).html

  1. 2 0 < 2 1 2^{\aleph_{0}}<2^{\aleph_{1}}

Mortgage_calculator.html

  1. c = r P 1 - ( 1 + r ) - N = r P ( 1 + r ) N ( 1 + r ) N - 1 . c=\frac{rP}{1-(1+r)^{-N}}=\frac{rP(1+r)^{N}}{(1+r)^{N}-1}.
  2. P = 200000 P=200000
  3. r = ( 6.5 / 12 ) / 100 r=(6.5/12)/100
  4. N = 30 12 = 360 N=30\cdot 12=360
  5. P P
  6. ( 1 + r ) P - c (1+r)P-c
  7. ( 1 + r ) ( ( 1 + r ) P - c ) - c = ( 1 + r ) 2 P - ( 1 + ( 1 + r ) ) c (1+r)((1+r)P-c)-c=(1+r)^{2}P-(1+(1+r))c
  8. ( 1 + r ) ( ( 1 + r ) ( ( 1 + r ) P - c ) - c ) - c = ( 1 + r ) 3 P - ( 1 + ( 1 + r ) + ( 1 + r ) 2 ) c (1+r)((1+r)((1+r)P-c)-c)-c=(1+r)^{3}P-(1+(1+r)+(1+r)^{2})c
  9. ( 1 + r ) N P - ( 1 + ( 1 + r ) + ( 1 + r ) 2 + + ( 1 + r ) N - 1 ) c (1+r)^{N}P-(1+(1+r)+(1+r)^{2}+\cdots+(1+r)^{N-1})c
  10. p N ( x ) = 1 + x + x 2 + + x N - 1 p_{N}(x)=1+x+x^{2}+\cdots+x^{N-1}
  11. x = 1 + r x=1+r
  12. x p N ( x ) - p N ( x ) = x N - 1 xp_{N}(x)-p_{N}(x)=x^{N}-1
  13. p N ( x ) p_{N}(x)
  14. p N ( x ) = 1 + x + x 2 + + x N - 1 = x N - 1 x - 1 . p_{N}(x)=1+x+x^{2}+\cdots+x^{N-1}=\frac{x^{N}-1}{x-1}.
  15. p N p_{N}
  16. p N ( x ) p_{N}(x)
  17. = ( 1 + r ) N P - p N c \displaystyle{}=(1+r)^{N}P-p_{N}c
  18. c \displaystyle c
  19. I I
  20. c N cN
  21. P P
  22. I = c N - P I=cN-P
  23. c c
  24. N N
  25. P P

Motion_field.html

  1. ( y 1 , y 2 ) (y_{1},y_{2})
  2. ( x 1 , x 2 , x 3 ) (x_{1},x_{2},x_{3})
  3. ( y 1 , y 2 ) (y_{1},y_{2})
  4. m 1 , m 2 m_{1},m_{2}
  5. ( y 1 y 2 ) = ( m 1 ( x 1 , x 2 , x 3 ) m 2 ( x 1 , x 2 , x 3 ) ) \begin{pmatrix}y_{1}\\ y_{2}\end{pmatrix}=\begin{pmatrix}m_{1}(x_{1},x_{2},x_{3})\\ m_{2}(x_{1},x_{2},x_{3})\end{pmatrix}
  6. ( d y 1 d t d y 2 d t ) = ( d m 1 ( x 1 , x 2 , x 3 ) d t d m 2 ( x 1 , x 2 , x 3 ) d t ) = ( d m 1 d x 1 d m 1 d x 2 d m 1 d x 3 d m 2 d x 1 d m 2 d x 2 d m 2 d x 3 ) ( d x 1 d t d x 2 d t d x 3 d t ) \begin{pmatrix}\frac{dy_{1}}{dt}\\ \frac{dy_{2}}{dt}\end{pmatrix}=\begin{pmatrix}\frac{dm_{1}(x_{1},x_{2},x_{3})}% {dt}\\ \frac{dm_{2}(x_{1},x_{2},x_{3})}{dt}\end{pmatrix}=\begin{pmatrix}\frac{dm_{1}}% {dx_{1}}&\frac{dm_{1}}{dx_{2}}&\frac{dm_{1}}{dx_{3}}\\ \frac{dm_{2}}{dx_{1}}&\frac{dm_{2}}{dx_{2}}&\frac{dm_{2}}{dx_{3}}\end{pmatrix}% \,\begin{pmatrix}\frac{dx_{1}}{dt}\\ \frac{dx_{2}}{dt}\\ \frac{dx_{3}}{dt}\end{pmatrix}
  7. 𝐮 = ( d y 1 d t d y 2 d t ) \mathbf{u}=\begin{pmatrix}\frac{dy_{1}}{dt}\\ \frac{dy_{2}}{dt}\end{pmatrix}
  8. ( y 1 , y 2 ) (y_{1},y_{2})
  9. 𝐱 = ( d x 1 d t d x 2 d t d x 3 d t ) \mathbf{x^{\prime}}=\begin{pmatrix}\frac{dx_{1}}{dt}\\ \frac{dx_{2}}{dt}\\ \frac{dx_{3}}{dt}\end{pmatrix}
  10. 𝐮 = 𝐌 𝐱 \mathbf{u}=\mathbf{M}\,\mathbf{x}^{\prime}
  11. 𝐌 \mathbf{M}
  12. 2 × 3 2\times 3
  13. 𝐌 = ( d m 1 d x 1 d m 1 d x 2 d m 1 d x 3 d m 2 d x 1 d m 2 d x 2 d m 2 d x 3 ) \mathbf{M}=\begin{pmatrix}\frac{dm_{1}}{dx_{1}}&\frac{dm_{1}}{dx_{2}}&\frac{dm% _{1}}{dx_{3}}\\ \frac{dm_{2}}{dx_{1}}&\frac{dm_{2}}{dx_{2}}&\frac{dm_{2}}{dx_{3}}\end{pmatrix}
  14. 𝐌 \mathbf{M}
  15. 𝐯 \mathbf{v}
  16. 𝐯 = f Z 𝐕 - V z 𝐏 Z 2 \mathbf{v}=f\frac{Z\mathbf{V}-V_{z}\mathbf{P}}{Z^{2}}
  17. 𝐕 = - 𝐓 - ω × 𝐏 \mathbf{V}=-\mathbf{T}-\mathbf{\omega}\times\mathbf{P}
  18. 𝐏 \mathbf{P}
  19. 𝐕 \mathbf{V}
  20. 𝐓 \mathbf{T}
  21. ω \mathbf{\omega}

Mountain_range_(options).html

  1. S 1 , S 2 , , S n S_{1},S_{2},...,S_{n}
  2. min i = 1... n ( S i T S i 0 ) . \min_{i=1...n}(\frac{S_{i}^{T}}{S_{i}^{0}}).
  3. S 1 , S 2 , , S n S_{1},S_{2},...,S_{n}
  4. R ( 1 ) t = min { S 1 t S 1 0 , S 2 t S 2 0 , , S n t S 1 n } , R_{(1)}^{t}=\min{\{\frac{S_{1}^{t}}{S_{1}^{0}},\frac{S_{2}^{t}}{S_{2}^{0}},...% ,\frac{S_{n}^{t}}{S_{1}^{n}}\}},
  5. R ( n ) t = max { S 1 t S 1 0 , S 2 t S 2 0 , , S n t S n 0 } , R_{(n)}^{t}=\max{\{\frac{S_{1}^{t}}{S_{1}^{0}},\frac{S_{2}^{t}}{S_{2}^{0}},...% ,\frac{S_{n}^{t}}{S_{n}^{0}}\}},
  6. R ( i ) t R_{(i)}^{t}
  7. R ( 1 ) t R ( 2 ) t R ( i ) t R ( n ) t . R_{(1)}^{t}\leq R_{(2)}^{t}\leq\dots\leq R_{(i)}^{t}\leq\dots\leq R_{(n)}^{t}.
  8. n 1 n_{1}
  9. n 2 n_{2}
  10. n 1 + n 2 < n n_{1}+n_{2}<n
  11. K K
  12. j = 1 + n 1 n - n 2 R ( j ) T n - ( n 1 + n 2 ) - K ) + . \sum_{j=1+n_{1}}^{n-n_{2}}{\frac{R_{(j)}^{T}}{n-(n_{1}+n_{2})}-K})^{+}.
  13. N N
  14. T T
  15. m m
  16. m m
  17. t 0 = 0 < t 1 < t 2 < < t m = T t_{0}=0<t_{1}<t_{2}<\dots<t_{m}=T
  18. t i , i = 1 : m t_{i},\ i=1:m
  19. S k i , 1 k i m S_{k_{i}},\ 1\leq k_{i}\leq m
  20. N max ( S k i , t i - S k i , t 0 S k i , t 0 , 0 ) N\max\left(\frac{S_{k_{i},t_{i}}-S_{k_{i},t_{0}}}{S_{k_{i},t_{0}}},\ 0\right)
  21. S k i S_{k_{i}}

Movable_singularity.html

  1. d y d x = 1 2 y \frac{dy}{dx}=\frac{1}{2y}
  2. y = x - c y=\sqrt{x-c}
  3. x = c x=c

Moving_cluster_method.html

  1. distance = ( t a n θ ) v u \mathrm{distance}=\mathrm{(}tan\theta)\frac{\mathrm{v}}{\mathrm{u}}

Moving_equilibrium_theorem.html

  1. x ˙ = f ( x , y ) \dot{x}=f(x,y)
  2. y ˙ = g ( x , y ) \qquad\dot{y}=g(x,y)
  3. x x
  4. y y
  5. x x
  6. y y
  7. y y
  8. x ¯ ( y ) \bar{x}(y)
  9. x x
  10. Y ˙ = g ( x ¯ ( Y ) , Y ) = : G ( Y ) . \qquad\dot{Y}=g(\bar{x}(Y),Y)=:G(Y).
  11. y y
  12. Y Y
  13. Y Y
  14. y y
  15. Y Y
  16. y y
  17. x x
  18. y y
  19. x x
  20. y y

Moving_least_squares.html

  1. f : n f:\mathbb{R}^{n}\to\mathbb{R}
  2. S = { ( x i , f i ) | f ( x i ) = f i } S=\{(x_{i},f_{i})|f(x_{i})=f_{i}\}
  3. x i n x_{i}\in\mathbb{R}^{n}
  4. f i f_{i}
  5. m m
  6. x x
  7. p ~ ( x ) \tilde{p}(x)
  8. p ~ \tilde{p}
  9. i I ( p ( x i ) - f i ) 2 θ ( x - x i ) \sum_{i\in I}(p(x_{i})-f_{i})^{2}\theta(\|x-x_{i}\|)
  10. p p
  11. m m
  12. n \mathbb{R}^{n}
  13. θ ( s ) \theta(s)
  14. s s\to\infty
  15. θ ( s ) = e - s 2 \theta(s)=e^{-s^{2}}

Moving_magnet_and_conductor_problem.html

  1. 𝐅 = q ( 𝐄 + 𝐯 × 𝐁 ) , \mathbf{F}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B}),
  2. q q
  3. 𝐯 \mathbf{v}
  4. 𝐅 = q 𝐯 × 𝐁 . \mathbf{F}=q\mathbf{v}\times\mathbf{B}.
  5. 𝐁 ( 𝐱 , t ) = 𝐁 ( 𝐱 + 𝐯 t ) . \mathbf{B}^{\prime}(\mathbf{x}^{\prime},t)=\mathbf{B}(\mathbf{x}^{\prime}+% \mathbf{v}t).
  6. × 𝐄 = - 𝐁 t . \mathbf{\nabla\times E}^{\prime}=-\frac{\partial\mathbf{B}^{\prime}}{\partial t}.
  7. × 𝐄 = - ( 𝐯 ) 𝐁 = - × ( 𝐁 × 𝐯 ) - 𝐯 ( 𝐁 ) = - × ( 𝐁 × 𝐯 ) \mathbf{\nabla\times E}^{\prime}=-(\mathbf{v}\cdot\nabla)\mathbf{B}=-\nabla% \times(\mathbf{B}\times\mathbf{v})-\mathbf{v}(\nabla\cdot\mathbf{B})=-\nabla% \times(\mathbf{B}\times\mathbf{v})
  8. 𝐄 = - 𝐁 × 𝐯 = 𝐯 × 𝐁 . \mathbf{E}^{\prime}=-\mathbf{B}\times\mathbf{v}=\mathbf{v}\times\mathbf{B}.
  9. 𝐅 = q 𝐄 = q 𝐯 × 𝐁 . \mathbf{F}^{\prime}=q\mathbf{E}^{\prime}=q\mathbf{v}\times\mathbf{B}.
  10. 𝐄 = 𝐄 + 𝐯 × 𝐁 \mathbf{E}^{\prime}=\mathbf{E}+\mathbf{v}\times\mathbf{B}
  11. 𝐁 = 𝐁 - 1 c 2 𝐯 × 𝐄 , \mathbf{B}^{\prime}=\mathbf{B}-\frac{1}{c^{2}}\mathbf{v}\times\mathbf{E},
  12. 𝐄 = γ 𝐯 × 𝐁 \mathbf{E}^{\prime}=\gamma\mathbf{v}\times\mathbf{B}
  13. γ = 1 1 - ( v / c ) 2 \gamma=\frac{1}{\sqrt{1-{(v/c)}^{2}}}
  14. 𝐅 = q 𝐄 = q γ 𝐯 × 𝐁 . \mathbf{F}^{\prime}=q\mathbf{E}^{\prime}=q\gamma\mathbf{v}\times\mathbf{B}.
  15. γ \gamma
  16. 𝐅 = q [ 𝐄 + 𝐯 × 𝐁 ] . \mathbf{F}=q\left[\mathbf{E}+\mathbf{v}\times\mathbf{B}\right].
  17. F y = - q v B , F_{y}=-qvB,
  18. F y = q E = - q γ v B . {F_{y}}^{\prime}=qE^{\prime}=-q\gamma vB.
  19. x = γ ( x - v t ) , x = γ ( x + v t ) , x^{\prime}=\gamma(x-vt),\quad x=\gamma(x^{\prime}+vt^{\prime}),
  20. t = γ ( t - v x c 2 ) , t = γ ( t + v x c 2 ) . t^{\prime}=\gamma(t-\frac{vx}{c^{2}}),\quad t=\gamma(t^{\prime}+\frac{vx^{% \prime}}{c^{2}}).
  21. F y = γ F y . {F_{y}}^{\prime}=\gamma F_{y}.
  22. γ F y = - q γ v B , F y = - q γ v B , \gamma F_{y}=-q\gamma vB,\quad{F_{y}}^{\prime}=-q\gamma vB,
  23. m c d u α d τ = F α β q u β , mc\frac{du^{\alpha}}{d\tau}=F^{\alpha\beta}qu_{\beta},
  24. u β = η β α u α = η β α d x α d τ u_{\beta}=\eta_{\beta\alpha}u^{\alpha}=\eta_{\beta\alpha}\frac{dx^{\alpha}}{d\tau}
  25. τ \tau
  26. η \eta
  27. F α β = ( 0 E x E y E z - E x 0 c B z - c B y - E y - c B z 0 c B x - E z c B y - c B x 0 ) . F^{\alpha\beta}=\left(\begin{matrix}0&{E_{x}}&{E_{y}}&{E_{z}}\\ -{E_{x}}&0&cB_{z}&-cB_{y}\\ -{E_{y}}&-cB_{z}&0&cB_{x}\\ -{E_{z}}&cB_{y}&-cB_{x}&0\end{matrix}\right).
  28. A α = ( ϕ / c , A x , A y , A z ) , A^{\alpha}=\left(\phi/c,A_{x},A_{y},A_{z}\right),
  29. 𝐄 = - ϕ - t 𝐀 , 𝐁 = × 𝐀 , \mathbf{E}=-\nabla\phi-\partial_{t}\mathbf{A},\quad\mathbf{B}=\nabla\times% \mathbf{A},
  30. F α β = A β x α - A α x β , F^{\alpha\beta}=\frac{\partial A^{\beta}}{\partial x_{\alpha}}-\frac{\partial A% ^{\alpha}}{\partial x_{\beta}},
  31. x α = ( - c t , x , y , z ) . x_{\alpha}=\left(-ct,x,y,z\right).
  32. F ´ μ ν = Λ μ α Λ ν β F α β , \acute{F}^{\mu\nu}={\Lambda^{\mu}}_{\alpha}{\Lambda^{\nu}}_{\beta}F^{\alpha% \beta},
  33. Λ μ α {\Lambda^{\mu}}_{\alpha}
  34. 𝐄 = γ 𝐯 c × 𝐁 , \mathbf{E}^{\prime}=\gamma\frac{\mathbf{v}}{c}\times\mathbf{B},

Mössbauer_spectroscopy.html

  1. CS = K ( R e 2 - R g 2 ) ( [ Ψ s 2 ( 0 ) ] b - [ Ψ s 2 ( 0 ) ] a ) \,\text{CS}=K\left(\langle R_{e}^{2}\rangle-\langle R_{g}^{2}\rangle\right)% \left([\Psi_{s}^{2}(0)]_{b}-[\Psi_{s}^{2}(0)]_{a}\right)
  2. V = c B int μ N E γ ( 3 g n e + g n ) V=\frac{c\,B\text{int}\,\mu_{\rm N}}{E_{\gamma}}(3g_{n}^{e}+g_{n})
  3. 1 / 2 {1}/{2}
  4. 3 / 2 {3}/{2}

Multi-commodity_flow_problem.html

  1. G ( V , E ) \,G(V,E)
  2. ( u , v ) E (u,v)\in E
  3. c ( u , v ) \,c(u,v)
  4. k \,k
  5. K 1 , K 2 , , K k K_{1},K_{2},\dots,K_{k}
  6. K i = ( s i , t i , d i ) \,K_{i}=(s_{i},t_{i},d_{i})
  7. s i \,s_{i}
  8. t i \,t_{i}
  9. i \,i
  10. d i \,d_{i}
  11. i \,i
  12. ( u , v ) \,(u,v)
  13. f i ( u , v ) \,f_{i}(u,v)
  14. i = 1 k f i ( u , v ) c ( u , v ) \,\sum_{i=1}^{k}f_{i}(u,v)\leq c(u,v)
  15. w V f i ( u , w ) = 0 when u s i , t i \,\sum_{w\in V}f_{i}(u,w)=0\quad\mathrm{when}\quad u\neq s_{i},t_{i}
  16. v , u , f i ( u , v ) = - f i ( v , u ) \,\forall v,u,\,f_{i}(u,v)=-f_{i}(v,u)
  17. w V f i ( s i , w ) = w V f i ( w , t i ) = d i \,\sum_{w\in V}f_{i}(s_{i},w)=\sum_{w\in V}f_{i}(w,t_{i})=d_{i}
  18. a ( u , v ) f ( u , v ) a(u,v)\cdot f(u,v)
  19. ( u , v ) \,(u,v)
  20. ( u , v ) E ( a ( u , v ) i = 1 k f i ( u , v ) ) \sum_{(u,v)\in E}\left(a(u,v)\sum_{i=1}^{k}f_{i}(u,v)\right)
  21. i = 1 k w V f i ( s i , w ) \sum_{i=1}^{k}\sum_{w\in V}f_{i}(s_{i},w)
  22. min 1 i k w V f i ( s i , w ) d i \min_{1\leq i\leq k}\frac{\sum_{w\in V}f_{i}(s_{i},w)}{d_{i}}

Multi-label_classification.html

  1. k k
  2. 1 N i = 1 N | Y i | \frac{1}{N}\sum_{i=1}^{N}|Y_{i}|
  3. 1 N i = 1 N | Y i | | L | \frac{1}{N}\sum_{i=1}^{N}\frac{|Y_{i}|}{|L|}
  4. L = i = 1 N Y i L=\bigcup_{i=1}^{N}Y_{i}
  5. T T
  6. P P
  7. | T P | | T P | \frac{|T\cap P|}{|T\cup P|}
  8. F 1 F_{1}
  9. | T P | | P | \frac{|T\cap P|}{|P|}
  10. | T P | | T | \frac{|T\cap P|}{|T|}
  11. F 1 F_{1}

Multi-party_fair_exchange_protocol.html

  1. n n
  2. σ \sigma
  3. { 1... n } \{1...n\}
  4. P i P_{i}
  5. K i K_{i}
  6. P σ ( i ) P_{\sigma(i)}
  7. K σ - 1 ( i ) K_{\sigma^{-1}(i)}
  8. P σ - 1 ( i ) P_{\sigma^{-1}(i)}
  9. n n
  10. B i j B_{ij}
  11. i i
  12. j j
  13. P i P_{i}
  14. P j P_{j}

Multi-scale_approaches.html

  1. g ( x , t ) = 1 2 π t exp ( - x 2 / 2 t ) g(x,t)=\frac{1}{\sqrt{2\pi t}}\exp({-x^{2}/2t})
  2. t > 0 t>0
  3. h ( x ) = exp ( - a x ) h(x)=\exp({-ax})
  4. x 0 x\geq 0
  5. a > 0 a>0
  6. h ( x ) = exp ( b x ) h(x)=\exp({bx})
  7. x 0 x\leq 0
  8. b > 0 b>0
  9. T ( n , t ) = I n ( α t ) T(n,t)=I_{n}(\alpha t)
  10. α , t > 0 \alpha,t>0
  11. I n I_{n}
  12. f o u t ( x ) = p f i n ( x ) + q f i n ( x - 1 ) f_{out}(x)=pf_{in}(x)+qf_{in}(x-1)
  13. p , q > 0 p,q>0
  14. f o u t ( x ) = p f i n ( x ) + q f i n ( x + 1 ) f_{out}(x)=pf_{in}(x)+qf_{in}(x+1)
  15. p , q > 0 p,q>0
  16. f o u t ( x ) = f i n ( x ) + α f o u t ( x - 1 ) f_{out}(x)=f_{in}(x)+\alpha f_{out}(x-1)
  17. α > 0 \alpha>0
  18. f o u t ( x ) = f i n ( x ) + β f o u t ( x + 1 ) f_{out}(x)=f_{in}(x)+\beta f_{out}(x+1)
  19. β > 0 \beta>0
  20. p ( n , t ) = e - t t n n ! p(n,t)=e^{-t}\frac{t^{n}}{n!}
  21. n 0 n\geq 0
  22. t 0 t\geq 0
  23. p ( n , t ) = e - t t - n ( - n ) ! p(n,t)=e^{-t}\frac{t^{-n}}{(-n)!}
  24. n 0 n\leq 0
  25. t 0 t\geq 0

Multicomplex_number.html

  1. C n + 1 = { z = x + y i n + 1 : x , y C n } \,\text{C}_{n+1}=\{z=x+yi_{n+1}:x,y\in\,\text{C}_{n}\}
  2. i n i m = i m i n i_{n}i_{m}=i_{m}i_{n}
  3. i n i m + i m i n = 0 i_{n}i_{m}+i_{m}i_{n}=0

Multiple_signal_classification.html

  1. x ( n ) x(n)
  2. p p
  3. M × M M\times M
  4. 𝐑 x \mathbf{R}_{x}
  5. p p
  6. M - p M-p
  7. M = p + 1 M=p+1
  8. P ^ M U ( e j ω ) = 1 i = p + 1 M | 𝐞 H 𝐯 i | 2 , \hat{P}_{MU}(e^{j\omega})=\frac{1}{\sum_{i=p+1}^{M}|\mathbf{e}^{H}\mathbf{v}_{% i}|^{2}},
  9. 𝐯 i \mathbf{v}_{i}
  10. 𝐞 = [ 1 e j ω e j 2 ω e j ( M - 1 ) ω ] T . \mathbf{e}=\begin{bmatrix}1&e^{j\omega}&e^{j2\omega}&\cdots&e^{j(M-1)\omega}% \end{bmatrix}^{T}.
  11. p p
  12. p p

Multivalued_dependency.html

  1. R R
  2. α R \alpha\subseteq R
  3. β R \beta\subseteq R
  4. α β \alpha\twoheadrightarrow\beta
  5. α \alpha
  6. β \beta
  7. R R
  8. r ( R ) r(R)
  9. t 1 t_{1}
  10. t 2 t_{2}
  11. r r
  12. t 1 [ α ] = t 2 [ α ] t_{1}[\alpha]=t_{2}[\alpha]
  13. t 3 t_{3}
  14. t 4 t_{4}
  15. r r
  16. t 1 [ α ] = t 2 [ α ] = t 3 [ α ] = t 4 [ α ] t_{1}[\alpha]=t_{2}[\alpha]=t_{3}[\alpha]=t_{4}[\alpha]
  17. t 3 [ β ] = t 1 [ β ] t_{3}[\beta]=t_{1}[\beta]
  18. t 3 [ R - β ] = t 2 [ R - β ] t_{3}[R-\beta]=t_{2}[R-\beta]
  19. t 4 [ β ] = t 2 [ β ] t_{4}[\beta]=t_{2}[\beta]
  20. t 4 [ R - β ] = t 1 [ R - β ] t_{4}[R-\beta]=t_{1}[R-\beta]
  21. ( x , y , z ) (x,y,z)
  22. α , \alpha,
  23. β , \beta,
  24. R - α - β R-\alpha-\beta
  25. x , x,
  26. y , y,
  27. z , z,
  28. ( a , b , c ) (a,b,c)
  29. ( a , d , e ) (a,d,e)
  30. r r
  31. ( a , b , e ) (a,b,e)
  32. ( a , d , c ) (a,d,c)
  33. r r
  34. \twoheadrightarrow
  35. \twoheadrightarrow
  36. \twoheadrightarrow
  37. \twoheadrightarrow
  38. \twoheadrightarrow
  39. α β \alpha\twoheadrightarrow\beta
  40. α R - β \alpha\twoheadrightarrow R-\beta
  41. α β \alpha\twoheadrightarrow\beta
  42. γ δ \gamma\subseteq\delta
  43. α δ β γ \alpha\delta\twoheadrightarrow\beta\gamma
  44. α β \alpha\twoheadrightarrow\beta
  45. β γ \beta\twoheadrightarrow\gamma
  46. α γ - β \alpha\twoheadrightarrow\gamma-\beta
  47. α β \alpha\rightarrow\beta
  48. α β \alpha\twoheadrightarrow\beta
  49. α β \alpha\twoheadrightarrow\beta
  50. β γ \beta\rightarrow\gamma
  51. α γ - β \alpha\twoheadrightarrow\gamma-\beta
  52. \twoheadrightarrow
  53. \rightarrow
  54. \twoheadrightarrow
  55. \twoheadrightarrow
  56. \twoheadrightarrow
  57. \subseteq
  58. \twoheadrightarrow
  59. \twoheadrightarrow
  60. \twoheadrightarrow
  61. \twoheadrightarrow
  62. \rightarrow
  63. \twoheadrightarrow
  64. \twoheadrightarrow
  65. \exist \exist
  66. \cap
  67. $\empty$
  68. \rightarrow
  69. \subseteq
  70. \rightarrow
  71. R - β R-\beta
  72. R = α β R=\alpha\cup\beta
  73. t 1 t_{1}
  74. t 2 t_{2}
  75. t 3 t_{3}
  76. t 4 t_{4}
  77. t 1 t_{1}
  78. t 2 t_{2}
  79. β α \beta\subseteq\alpha

Multivariate_t-distribution.html

  1. s y m b o l μ symbol\mu
  2. ν > 1 \nu>1
  3. s y m b o l μ symbol\mu
  4. s y m b o l μ symbol\mu
  5. ν ν - 2 s y m b o l Σ \frac{\nu}{\nu-2}symbol\Sigma
  6. ν > 2 \nu>2
  7. p p
  8. 𝐲 \mathbf{y}
  9. u u
  10. 𝒩 ( 𝟎 , s y m b o l Σ ) {\mathcal{N}}({\mathbf{0}},{symbol\Sigma})
  11. χ ν 2 \chi^{2}_{\nu}
  12. 𝚺 \mathbf{\Sigma}\,
  13. 𝐲 ν / u = 𝐱 - s y m b o l μ {\mathbf{y}}\sqrt{\nu/u}={\mathbf{x}}-{symbol\mu}
  14. 𝐱 {\mathbf{x}}
  15. Γ [ ( ν + p ) / 2 ] Γ ( ν / 2 ) ν p / 2 π p / 2 | s y m b o l Σ | 1 / 2 [ 1 + 1 ν ( 𝐱 - s y m b o l μ ) T s y m b o l Σ - 1 ( 𝐱 - s y m b o l μ ) ] ( ν + p ) / 2 \frac{\Gamma\left[(\nu+p)/2\right]}{\Gamma(\nu/2)\nu^{p/2}\pi^{p/2}\left|{% symbol\Sigma}\right|^{1/2}\left[1+\frac{1}{\nu}({\mathbf{x}}-{symbol\mu})^{T}{% symbol\Sigma}^{-1}({\mathbf{x}}-{symbol\mu})\right]^{(\nu+p)/2}}
  16. s y m b o l Σ , s y m b o l μ , ν {symbol\Sigma},{symbol\mu},\nu
  17. ν = 1 \nu=1
  18. p = 1 p=1
  19. t = x - μ t=x-\mu
  20. Σ = 1 \Sigma=1
  21. f ( t ) = Γ [ ( ν + 1 ) / 2 ] ν π Γ [ ν / 2 ] ( 1 + t 2 / ν ) - ( ν + 1 ) / 2 f(t)=\frac{\Gamma[(\nu+1)/2]}{\sqrt{\nu\pi\,}\,\Gamma[\nu/2]}(1+t^{2}/\nu)^{-(% \nu+1)/2}
  22. p p
  23. t i t_{i}
  24. t 2 t^{2}
  25. t i t_{i}
  26. ν \nu
  27. 𝐀 = s y m b o l Σ - 1 \mathbf{A}=symbol\Sigma^{-1}
  28. f ( 𝐭 ) = Γ ( ( ν + p ) / 2 ) | 𝐀 | 1 / 2 ν p π p Γ ( ν / 2 ) ( 1 + i , j = 1 p , p A i j t i t j / ν ) - ( ν + p ) / 2 f(\mathbf{t})=\frac{\Gamma((\nu+p)/2)\left|\mathbf{A}\right|^{1/2}}{\sqrt{\nu^% {p}\pi^{p}\,}\,\Gamma(\nu/2)}(1+\sum_{i,j=1}^{p,p}A_{ij}t_{i}t_{j}/\nu)^{-(\nu% +p)/2}
  29. f ( t 1 , t 2 ) = | 𝐀 | 1 / 2 2 π ( 1 + i , j = 1 2 , 2 A i j t i t j / ν ) - ( ν + 2 ) / 2 f(t_{1},t_{2})=\frac{\left|\mathbf{A}\right|^{1/2}}{2\pi}(1+\sum_{i,j=1}^{2,2}% A_{ij}t_{i}t_{j}/\nu)^{-(\nu+2)/2}
  30. Γ ( ν + 2 2 ) π ν Γ ( ν 2 ) = 1 2 π \frac{\Gamma\left(\frac{\nu+2}{2}\right)}{\pi\ \nu\Gamma\left(\frac{\nu}{2}% \right)}=\frac{1}{2\pi}
  31. 𝐀 \mathbf{A}
  32. f ( t 1 , t 2 ) = 1 2 π ( 1 + ( t 1 2 + t 2 2 ) / ν ) - ( ν + 2 ) / 2 . f(t_{1},t_{2})=\frac{1}{2\pi}(1+(t_{1}^{2}+t_{2}^{2})/\nu)^{-(\nu+2)/2}.
  33. Σ \Sigma

MUSCL_scheme.html

  1. u t + u x = 0 u_{t}+u_{x}=0
  2. u t + u x = 0 u_{t}+u_{x}=0
  3. u t + F x ( u ) = 0. u_{t}+F_{x}\left(u\right)=0.\,
  4. u u
  5. F F
  6. i i
  7. d u i d t + 1 Δ x i [ F ( u i ) - F ( u i + 1 ) ] = 0. \frac{du_{i}}{dt}+\frac{1}{\Delta x_{i}}\left[F\left(u_{i}\right)-F\left(u_{i+% 1}\right)\right]=0.
  8. u ( x ) = u i + ( x - x i ) ( x i + 1 - x i ) ( u i + 1 - u i ) x ( x i , x i + 1 ] . u\left(x\right)=u_{i}+\frac{\left(x-x_{i}\right)}{\left(x_{i+1}-x_{i}\right)}% \left(u_{i+1}-u_{i}\right)\qquad\forall x\in(x_{i},x_{i+1}].
  9. d u i d t + 1 Δ x i [ F ( u i + 1 / 2 ) - F ( u i - 1 / 2 ) ] = 0 , \frac{du_{i}}{dt}+\frac{1}{\Delta x_{i}}\left[F\left(u_{i+1/2}\right)-F\left(u% _{i-1/2}\right)\right]=0,
  10. u t + u x = 0 u_{t}+u_{x}=0
  11. u t + u x = 0 u_{t}+u_{x}=0
  12. u i + 1 / 2 u_{i+1/2}
  13. u i - 1 / 2 u_{i-1/2}
  14. u i + 1 / 2 = 0.5 ( u i + u i + 1 ) , u_{i+1/2}=0.5\left(u_{i}+u_{i+1}\right),
  15. u i - 1 / 2 = 0.5 ( u i - 1 + u i ) . u_{i-1/2}=0.5\left(u_{i-1}+u_{i}\right).
  16. u t + u x = 0 \,u_{t}+u_{x}=0
  17. d u i d t + 1 Δ x i [ F ( u i + 1 / 2 * ) - F ( u i - 1 / 2 * ) ] = 0. \frac{du_{i}}{dt}+\frac{1}{\Delta x_{i}}\left[F\left(u^{*}_{i+1/2}\right)-F% \left(u^{*}_{i-1/2}\right)\right]=0.
  18. d u i d t + 1 Δ x i [ F i + 1 / 2 * - F i - 1 / 2 * ] = 0. \frac{du_{i}}{dt}+\frac{1}{\Delta x_{i}}\left[F^{*}_{i+1/2}-F^{*}_{i-1/2}% \right]=0.
  19. F i ± 1 / 2 * F^{*}_{i\pm 1/2}
  20. u i + 1 / 2 * u^{*}_{i+1/2}
  21. u i - 1 / 2 * u^{*}_{i-1/2}
  22. u i + 1 / 2 * = u i + 1 / 2 * ( u i + 1 / 2 L , u i + 1 / 2 R ) , u i - 1 / 2 * = u i - 1 / 2 * ( u i - 1 / 2 L , u i - 1 / 2 R ) , u^{*}_{i+1/2}=u^{*}_{i+1/2}\left(u^{L}_{i+1/2},u^{R}_{i+1/2}\right),u^{*}_{i-1% /2}=u^{*}_{i-1/2}\left(u^{L}_{i-1/2},u^{R}_{i-1/2}\right),
  23. u i + 1 / 2 L = u i + 0.5 ϕ ( r i ) ( u i - u i - 1 ) , u i + 1 / 2 R = u i + 1 - 0.5 ϕ ( r i ) ( u i + 1 - u i ) , u^{L}_{i+1/2}=u_{i}+0.5\phi\left(r_{i}\right)\left(u_{i}-u_{i-1}\right),u^{R}_% {i+1/2}=u_{i+1}-0.5\phi\left(r_{i}\right)\left(u_{i+1}-u_{i}\right),
  24. u i - 1 / 2 L = u i - 1 + 0.5 ϕ ( r i - 1 ) ( u i - u i - 1 ) , u i - 1 / 2 R = u i - 0.5 ϕ ( r i ) ( u i + 1 - u i ) , u^{L}_{i-1/2}=u_{i-1}+0.5\phi\left(r_{i-1}\right)\left(u_{i}-u_{i-1}\right),u^% {R}_{i-1/2}=u_{i}-0.5\phi\left(r_{i}\right)\left(u_{i+1}-u_{i}\right),
  25. r i = u i - u i - 1 u i + 1 - u i . r_{i}=\frac{u_{i}-u_{i-1}}{u_{i+1}-u_{i}}.
  26. ϕ ( r i ) \phi\left(r_{i}\right)
  27. r 0 r\leq 0
  28. r = 1 r=1
  29. F i + 1 / 2 * F^{*}_{i+1/2}
  30. F i - 1 2 * = 1 2 { [ F ( u i - 1 2 R ) + F ( u i - 1 2 L ) ] - a i - 1 2 [ u i - 1 2 R - u i - 1 2 L ] } . F^{*}_{i-\frac{1}{2}}=\frac{1}{2}\left\{\left[F\left(u^{R}_{i-\frac{1}{2}}% \right)+F\left(u^{L}_{i-\frac{1}{2}}\right)\right]-a_{i-\frac{1}{2}}\left[u^{R% }_{i-\frac{1}{2}}-u^{L}_{i-\frac{1}{2}}\right]\right\}.
  31. F i + 1 2 * = 1 2 { [ F ( u i + 1 2 R ) + F ( u i + 1 2 L ) ] - a i + 1 2 [ u i + 1 2 R - u i + 1 2 L ] } . F^{*}_{i+\frac{1}{2}}=\frac{1}{2}\left\{\left[F\left(u^{R}_{i+\frac{1}{2}}% \right)+F\left(u^{L}_{i+\frac{1}{2}}\right)\right]-a_{i+\frac{1}{2}}\left[u^{R% }_{i+\frac{1}{2}}-u^{L}_{i+\frac{1}{2}}\right]\right\}.
  32. u t + u x = 0 u_{t}+u_{x}=0
  33. u t + u x = 0 u_{t}+u_{x}=0
  34. a i ± 1 2 a_{i\pm\frac{1}{2}}
  35. F ( u ( x , t ) ) F\left(u\left(x,t\right)\right)
  36. i , i ± 1 {i},{i\pm 1}
  37. a i + 1 2 ( t ) = max [ ρ ( F ( u i + 1 / 2 L ( t ) ) u ) , ρ ( F ( u i + 1 / 2 R ( t ) ) u ) , ] a_{i+\frac{1}{2}}\left(t\right)=\max\left[\rho\left(\frac{\partial F\left(u^{L% }_{i+1/2}\left(t\right)\right)}{\partial u}\right),\rho\left(\frac{\partial F% \left(u^{R}_{i+1/2}\left(t\right)\right)}{\partial u}\right),\right]
  38. ρ \rho
  39. F ( u ( t ) ) u . \frac{\partial F\left(u\left(t\right)\right)}{\partial u}.
  40. u t + u x = 0 u_{t}+u_{x}=0
  41. u t + F x ( u ) = Q x ( u , u x ) , u_{t}+F_{x}\left(u\right)=Q_{x}\left(u,u_{x}\right),
  42. d u i d t = - 1 Δ x i [ F i + 1 2 * - F i - 1 2 * ] + 1 Δ x i [ P i + 1 2 - P i - 1 2 ] . \frac{du_{i}}{dt}=-\frac{1}{\Delta x_{i}}\left[F^{*}_{i+\frac{1}{2}}-F^{*}_{i-% \frac{1}{2}}\right]+\frac{1}{\Delta x_{i}}\left[P_{i+\frac{1}{2}}-P_{i-\frac{1% }{2}}\right].
  43. P i + 1 2 = 1 2 [ Q ( u i , u i + 1 - u i Δ x i ) + Q ( u i + 1 , u i + 1 - u i Δ x i ) ] , P_{i+\frac{1}{2}}=\frac{1}{2}\left[Q\left(u_{i},\frac{u_{i+1}-u_{i}}{\Delta x_% {i}}\right)+Q\left(u_{i+1},\frac{u_{i+1}-u_{i}}{\Delta x_{i}}\right)\right],
  44. P i - 1 2 = 1 2 [ Q ( u i - 1 , u i - u i - 1 Δ x i - 1 ) + Q ( u i , u i - u i - 1 Δ x i - 1 ) . ] P_{i-\frac{1}{2}}=\frac{1}{2}\left[Q\left(u_{i-1},\frac{u_{i}-u_{i-1}}{\Delta x% _{i-1}}\right)+Q\left(u_{i},\frac{u_{i}-u_{i-1}}{\Delta x_{i-1}}\right).\right]
  45. u i + 1 2 * u^{*}_{i+\frac{1}{2}}
  46. u i - 1 2 * u^{*}_{i-\frac{1}{2}}
  47. u i + 1 2 * = f ( u i + 1 2 L , u i + 1 2 R ) , u i - 1 2 * = f ( u i - 1 2 L , u i - 1 2 R ) , u^{*}_{i+\frac{1}{2}}=f\left(u^{L}_{i+\frac{1}{2}},u^{R}_{i+\frac{1}{2}}\right% ),u^{*}_{i-\frac{1}{2}}=f\left(u^{L}_{i-\frac{1}{2}},u^{R}_{i-\frac{1}{2}}% \right),
  48. u i + 1 2 L = u i + ϕ ( r i ) 4 [ ( 1 - κ ) δ u i - 1 2 + ( 1 + κ ) δ u i + 1 2 ] , u^{L}_{i+\frac{1}{2}}=u_{i}+\frac{\phi\left(r_{i}\right)}{4}\left[\left(1-% \kappa\right)\delta u_{i-\frac{1}{2}}+\left(1+\kappa\right)\delta u_{i+\frac{1% }{2}}\right],
  49. u i + 1 2 R = u i + 1 - ϕ ( r i + 1 ) 4 [ ( 1 - κ ) δ u i + 3 2 + ( 1 + κ ) δ u i + 1 2 ] , u^{R}_{i+\frac{1}{2}}=u_{i+1}-\frac{\phi\left(r_{i+1}\right)}{4}\left[\left(1-% \kappa\right)\delta u_{i+\frac{3}{2}}+\left(1+\kappa\right)\delta u_{i+\frac{1% }{2}}\right],
  50. u i - 1 2 L = u i - 1 + ϕ ( r i - 1 ) 4 [ ( 1 - κ ) δ u i - 3 2 + ( 1 + κ ) δ u i - 1 2 ] , u^{L}_{i-\frac{1}{2}}=u_{i-1}+\frac{\phi\left(r_{i-1}\right)}{4}\left[\left(1-% \kappa\right)\delta u_{i-\frac{3}{2}}+\left(1+\kappa\right)\delta u_{i-\frac{1% }{2}}\right],
  51. u i - 1 2 R = u i - ϕ ( r i ) 4 [ ( 1 - κ ) δ u i + 1 2 + ( 1 + κ ) δ u i - 1 2 ] . u^{R}_{i-\frac{1}{2}}=u_{i}-\frac{\phi\left(r_{i}\right)}{4}\left[\left(1-% \kappa\right)\delta u_{i+\frac{1}{2}}+\left(1+\kappa\right)\delta u_{i-\frac{1% }{2}}\right].
  52. u t + u x = 0 u_{t}+u_{x}=0
  53. u t + u x = 0 u_{t}+u_{x}=0
  54. κ \kappa
  55. δ u i + 1 2 = ( u i + 1 - u i ) , δ u i - 1 2 = ( u i - u i - 1 ) , \delta u_{i+\frac{1}{2}}=\left(u_{i+1}-u_{i}\right),\delta u_{i-\frac{1}{2}}=% \left(u_{i}-u_{i-1}\right),
  56. δ u i + 3 2 = ( u i + 2 - u i + 1 ) , δ u i - 3 2 = ( u i - 1 - u i - 2 ) , \delta u_{i+\frac{3}{2}}=\left(u_{i+2}-u_{i+1}\right),\delta u_{i-\frac{3}{2}}% =\left(u_{i-1}-u_{i-2}\right),
  57. ϕ ( r ) \phi\left(r\right)
  58. ϕ v a ( r ) = 2 r 1 + r 2 \phi_{va}(r)=\frac{2r}{1+r^{2}}
  59. 𝐔 t + 𝐅 x = 0 , \frac{\partial\mathbf{U}}{\partial t}+\frac{\partial\mathbf{F}}{\partial x}=0,
  60. 𝐔 = ( ρ ρ u E ) 𝐅 = ( ρ u p + ρ u 2 u ( E + p ) ) , \mathbf{U}=\begin{pmatrix}\rho\\ \rho u\\ E\end{pmatrix}\qquad\mathbf{F}=\begin{pmatrix}\rho u\\ p+\rho u^{2}\\ u(E+p)\end{pmatrix},\qquad
  61. U \mbox{U}~{}
  62. F \mbox{F}~{}
  63. ρ \rho
  64. u u
  65. p p
  66. E E
  67. E = ρ e + 1 2 ρ u 2 , E=\rho e+\frac{1}{2}\rho u^{2},
  68. e e
  69. p = ρ ( γ - 1 ) e , p=\rho\left(\gamma-1\right)e,
  70. γ \gamma
  71. [ c p / c v ] \left[c_{p}/c_{v}\right]
  72. ρ i + 1 2 * = ρ i + 1 2 * ( ρ i + 1 2 L , ρ i + 1 2 R ) , ρ i - 1 2 * = ρ i - 1 2 * ( ρ i - 1 2 L , ρ i - 1 2 R ) , \rho^{*}_{i+\frac{1}{2}}=\rho^{*}_{i+\frac{1}{2}}\left(\rho^{L}_{i+\frac{1}{2}% },\rho^{R}_{i+\frac{1}{2}}\right),\rho^{*}_{i-\frac{1}{2}}=\rho^{*}_{i-\frac{1% }{2}}\left(\rho^{L}_{i-\frac{1}{2}},\rho^{R}_{i-\frac{1}{2}}\right),
  73. ρ i + 1 2 L = ρ i + 0.5 ϕ ( r i ) ( ρ i + 1 - ρ i ) , ρ i + 1 2 R = ρ i + 1 - 0.5 ϕ ( r i + 1 ) ( ρ i + 2 - ρ i + 1 ) , \rho^{L}_{i+\frac{1}{2}}=\rho_{i}+0.5\phi\left(r_{i}\right)\left(\rho_{i+1}-% \rho_{i}\right),\rho^{R}_{i+\frac{1}{2}}=\rho_{i+1}-0.5\phi\left(r_{i+1}\right% )\left(\rho_{i+2}-\rho_{i+1}\right),
  74. ρ i - 1 2 L = ρ i - 1 + 0.5 ϕ ( r i - 1 ) ( ρ i - ρ i - 1 ) , ρ i - 1 2 R = ρ i - 0.5 ϕ ( r i ) ( ρ i + 1 - ρ i ) . \rho^{L}_{i-\frac{1}{2}}=\rho_{i-1}+0.5\phi\left(r_{i-1}\right)\left(\rho_{i}-% \rho_{i-1}\right),\rho^{R}_{i-\frac{1}{2}}=\rho_{i}-0.5\phi\left(r_{i}\right)% \left(\rho_{i+1}-\rho_{i}\right).
  75. ρ u \rho u
  76. E E
  77. u u
  78. p p
  79. d 𝐔 i d t = - 1 Δ x i [ 𝐅 i + 1 2 * - 𝐅 i - 1 2 * ] . \frac{d\mathbf{U}_{i}}{dt}=-\frac{1}{\Delta x_{i}}\left[\mathbf{F}^{*}_{i+% \frac{1}{2}}-\mathbf{F}^{*}_{i-\frac{1}{2}}\right].
  80. 𝐅 i ± 1 2 * \mathbf{F}^{*}_{i\pm\frac{1}{2}}
  81. ϕ v a ( r ) = 2 r 1 + r 2 \phi_{va}(r)=\frac{2r}{1+r^{2}}

Musean_hypernumber.html

  1. 1 1
  2. i n i_{n}
  3. i n 2 = - 1 i_{n}^{2}=-1
  4. ε \varepsilon
  5. ε = n 2 + 1 \varepsilon{}_{n}^{2}=+1
  6. i 0 i_{0}
  7. ε 0 := 1 \varepsilon_{0}:=1
  8. ε n \varepsilon_{n}
  9. ε n \varepsilon_{n}
  10. ε n \varepsilon{}_{n}
  11. n = 1 , , 7 n=1,\ldots,7
  12. e ε α n = cosh α + ε ( sinh α ) n e^{\varepsilon{}_{n}\alpha}=\cosh~{}\alpha+\varepsilon{}_{n}(\sinh~{}\alpha)
  13. α \alpha
  14. ε = n α 1 2 [ ( 1 - ε ) n + ( 1 + ε ) n e - π i n α ] \varepsilon{}_{n}^{\alpha}=\frac{1}{2}[(1-\varepsilon{}_{n})+(1+\varepsilon{}_% {n})e^{-\pi i_{n}\alpha}]
  15. ε n \varepsilon{}_{n}
  16. ln ε = n π 2 ( i 0 - i n ) \ln\varepsilon{}_{n}=\frac{\pi}{2}(i_{0}-i_{n})
  17. | z | |z|
  18. | z | = | a + b n i n + c n ε n + d | := ( a 2 + b n 2 - c n 2 - d 2 ) 2 + 4 ( a d - b n c n ) 2 4 |z|=|a+\sum{b_{n}i_{n}}+\sum{c_{n}\varepsilon_{n}}+d|:=\sqrt[4]{(a^{2}+b_{n}^{% 2}-c_{n}^{2}-d^{2})^{2}+4(ad-b_{n}c_{n})^{2}}
  19. i n i_{n}
  20. 1 , ε , 1 ε , 2 i 3 1,\varepsilon{}_{1},\varepsilon{}_{2},i_{3}
  21. 1 , i , ε , i 0 1,i,\varepsilon,i_{0}
  22. - 1 \sqrt{-1}
  23. ε \varepsilon
  24. 1 , i 1 , i 2 , i 3 , ε , 4 ε , 5 ε , 6 ε 7 1,i_{1},i_{2},i_{3},\varepsilon{}_{4},\varepsilon{}_{5},\varepsilon{}_{6},% \varepsilon{}_{7}
  25. { 1 , i 1 , i 2 , i 3 , i 0 , ε , 1 ε , 2 ε } 3 \{1,i_{1},i_{2},i_{3},~{}i_{0},\varepsilon{}_{1},\varepsilon{}_{2},\varepsilon% {}_{3}\}
  26. ε \varepsilon
  27. + 1 +1
  28. w α ~{}w^{\alpha}
  29. α ~{}\alpha
  30. w 0 = w 6 = 1 w^{0}=w^{6}=1
  31. w 1 = w w^{1}=~{}w
  32. w 2 = - 1 + w w^{2}=~{}-1+w
  33. w 3 = - 1 w^{3}=~{}-1
  34. w 4 = - w w^{4}=~{}-w
  35. w 5 = 1 - w . w^{5}=~{}1-w.
  36. || a + b w || = a 2 + a b + b 2 ||a+bw||=\sqrt{a^{2}+ab+b^{2}}
  37. i := - 1 i:=\sqrt{-1}
  38. p 2 = q 2 = 0 p^{2}=q^{2}=0
  39. p 0 = q 0 = p 2 = q 2 = 0 p^{0}=q^{0}=p^{2}=q^{2}=~{}0
  40. p 1 = p p^{1}=~{}p
  41. q 1 = q q^{1}=~{}q
  42. p 3 = q p^{3}=~{}q
  43. q 3 = p q^{3}=~{}p
  44. p α ~{}p^{\alpha}
  45. q α ~{}q^{\alpha}
  46. α ~{}\alpha
  47. a p + b q ap+~{}bq
  48. ( a 2 + b 2 ) 2 = ( a + b ) ( a - b ) 2 . (a^{2}+b^{2})^{2}=~{}(a+b)(a-b)^{2}.
  49. ( q w ) 3 = ( w q ) 3 = ( w 3 ) ( q 3 ) = ( - 1 ) p = - p (qw)^{3}=(wq)^{3}=(w^{3})(q^{3})=(-1)p=~{}-p
  50. p 2 = 0 , p 0 p^{2}=0,p\neq 0
  51. p 0 = 0 p^{0}=~{}0
  52. p - 1 1 / p p^{-1}\neq 1/p
  53. ( 1 / p ) ( 1 / p ) = 1 / p 2 = (1/p)(1/p)=1/p^{2}=\infty
  54. 1 / p ~{}1/p
  55. 1 / ( 1 ± ε ) 1/(1\pm\varepsilon)
  56. m 2 = m m^{2}=~{}m
  57. ( 2 m ) 2 = 0 (\sqrt{2}m)^{2}=~{}0
  58. ( 3 m ) 2 = - 1 (\sqrt{3}m)^{2}=~{}-1
  59. 2 \sqrt{2}
  60. 2 m \sqrt{2}m
  61. s 4 := ( a 2 + b 2 ) 2 + 2 ( a 2 - b 2 ) + 1 s^{4}:=~{}(a^{2}+b^{2})^{2}+2(a^{2}-b^{2})+1
  62. Ω \Omega
  63. Ω n = Ω \Omega^{n}=\Omega
  64. Ω = 0 \Omega^{\infty}=0
  65. Ω - n \Omega^{\infty-n}
  66. a + b Ω a+b\Omega
  67. υ \upsilon
  68. σ \sigma
  69. σ υ \sigma\upsilon
  70. υ \upsilon

Music_and_mathematics.html

  1. / 12 \mathbb{Z}/12\mathbb{Z}
  2. / 12 \mathbb{Z}/12\mathbb{Z}

Myerson–Satterthwaite_theorem.html

  1. v A [ x a , y a ] v_{A}\in[x_{a},y_{a}]
  2. v B [ x b , y b ] v_{B}\in[x_{b},y_{b}]
  3. [ x a , y a ] [x_{a},y_{a}]
  4. [ x b , y b ] [x_{b},y_{b}]
  5. v a v_{a}
  6. v b v_{b}

N-connected.html

  1. π i ( X ) 0 , 1 i n , \pi_{i}(X)\equiv 0~{},\quad 1\leq i\leq n,
  2. π 0 ( X , * ) := [ ( S 0 , * ) , ( X , * ) ] . \pi_{0}(X,*):=[(S^{0},*),(X,*)].
  3. π i ( X ) 0 , 0 i n . \pi_{i}(X)\equiv 0,\quad 0\leq i\leq n.
  4. f : X Y f\colon X\to Y
  5. π i ( f ) : π i ( X ) π i ( Y ) \pi_{i}(f)\colon\pi_{i}(X)\overset{\sim}{\to}\pi_{i}(Y)
  6. i < n i<n
  7. π n ( f ) : π n ( X ) π n ( Y ) \pi_{n}(f)\colon\pi_{n}(X)\twoheadrightarrow\pi_{n}(Y)
  8. π n ( X ) π n ( f ) π n ( Y ) π n - 1 ( F f ) . \pi_{n}(X)\overset{\pi_{n}(f)}{\to}\pi_{n}(Y)\to\pi_{n-1}(Ff).
  9. π n - 1 ( F f ) \pi_{n-1}(Ff)
  10. x 0 X x_{0}\hookrightarrow X
  11. A X A\hookrightarrow X
  12. A X A\hookrightarrow X
  13. π 0 ( X ) , \pi_{0}(X),
  14. π 0 ( A ) π 0 ( X ) , \pi_{0}(A)\to\pi_{0}(X),
  15. π 1 ( X ) . \pi_{1}(X).
  16. π 0 ( A ) π 0 ( X ) \pi_{0}(A)\to\pi_{0}(X)
  17. a , b A a,b\in A
  18. π 1 ( X ) \pi_{1}(X)
  19. π n - 1 ( A ) π n - 1 ( X ) \pi_{n-1}(A)\to\pi_{n-1}(X)
  20. π n - 1 ( A ) \pi_{n-1}(A)
  21. π n ( X ) \pi_{n}(X)
  22. M N , M\to N,
  23. X ( M ) X ( N ) , X(M)\to X(N),

N-jet.html

  1. f ( x ) f(x)
  2. N N
  3. L L
  4. f ( x , y ) f(x,y)
  5. L L

N-vector.html

  1. 𝐧 e = [ cos ( latitude ) cos ( longitude ) cos ( latitude ) sin ( longitude ) sin ( latitude ) ] \mathbf{n}^{e}=\left[\begin{matrix}\cos(\mathrm{latitude})\cos(\mathrm{% longitude})\\ \cos(\mathrm{latitude})\sin(\mathrm{longitude})\\ \sin(\mathrm{latitude})\\ \end{matrix}\right]
  2. n x e n_{x}^{e}
  3. n y e n_{y}^{e}
  4. n z e n_{z}^{e}
  5. latitude = arcsin ( n z e ) = arctan ( n z e n x e 2 + n y e 2 ) \mathrm{latitude}=\arcsin\left(n_{z}^{e}\right)=\arctan\left(\frac{n_{z}^{e}}{% \sqrt{{n_{x}^{e}}^{2}+{n_{y}^{e}}^{2}}}\right)
  6. longitude = arctan ( n y e n x e ) \mathrm{longitude}=\arctan\left(\frac{n_{y}^{e}}{n_{x}^{e}}\right)
  7. arctan ( y / x ) \arctan(y/x)
  8. Δ σ = arccos ( 𝐧 a 𝐧 b ) Δ σ = arcsin ( | 𝐧 a × 𝐧 b | ) Δ σ = arctan ( | 𝐧 a × 𝐧 b | 𝐧 a 𝐧 b ) \begin{aligned}&\displaystyle\Delta\sigma=\arccos\left(\mathbf{n}_{a}\cdot% \mathbf{n}_{b}\right)\\ &\displaystyle\Delta\sigma=\arcsin\left(\left|\mathbf{n}_{a}\times\mathbf{n}_{% b}\right|\right)\\ &\displaystyle\Delta\sigma=\arctan\left(\frac{\left|\mathbf{n}_{a}\times% \mathbf{n}_{b}\right|}{\mathbf{n}_{a}\cdot\mathbf{n}_{b}}\right)\\ \end{aligned}
  9. 𝐧 a \mathbf{n}_{a}
  10. 𝐧 b \mathbf{n}_{b}
  11. Δ σ \Delta\sigma

N4-(beta-N-acetylglucosaminyl)-L-asparaginase.html

  1. \rightleftharpoons

N_(disambiguation).html

  1. \mathbb{N}
  2. F n F_{n}

Naccache–Stern_cryptosystem.html

  1. ( / n ) * (\mathbb{Z}/n\mathbb{Z})^{*}
  2. u = i = 1 k / 2 p i u=\prod_{i=1}^{k/2}p_{i}
  3. v = k / 2 + 1 k p i v=\prod_{k/2+1}^{k}p_{i}
  4. σ = u v = i = 1 k p i \sigma=uv=\prod_{i=1}^{k}p_{i}
  5. / σ \mathbb{Z}/\sigma\mathbb{Z}
  6. x / n x\in\mathbb{Z}/n\mathbb{Z}
  7. E ( m ) = x σ g m mod n E(m)=x^{\sigma}g^{m}\mod n
  8. σ \sigma
  9. c i c ϕ ( n ) / p i mod n c_{i}\equiv c^{\phi(n)/p_{i}}\mod n
  10. c ϕ ( n ) / p i x σ ϕ ( n ) / p i g m ϕ ( n ) / p i mod n g ( m i + y i p i ) ϕ ( n ) / p i mod n g m i ϕ ( n ) / p i mod n \begin{matrix}c^{\phi(n)/p_{i}}&\equiv&x^{\sigma\phi(n)/p_{i}}g^{m\phi(n)/p_{i% }}\mod n\\ &\equiv&g^{(m_{i}+y_{i}p_{i})\phi(n)/p_{i}}\mod n\\ &\equiv&g^{m_{i}\phi(n)/p_{i}}\mod n\end{matrix}
  11. m i m mod p i m_{i}\equiv m\mod p_{i}
  12. c i c_{i}
  13. g j ϕ ( n ) / p i g^{j\phi(n)/p_{i}}

Naccache–Stern_knapsack_cryptosystem.html

  1. x { 0 , 1 } n x\in\{0,1\}^{n}
  2. c i = 0 n v i x i mod p c\equiv\prod_{i=0}^{n}v_{i}^{x_{i}}\mod p
  3. i = 0 n p i < p \prod_{i=0}^{n}p_{i}<p
  4. v i = p i s mod p v_{i}=\sqrt[s]{p_{i}}\mod p
  5. c = i = 0 n v i m i mod p c=\prod_{i=0}^{n}v_{i}^{m_{i}}\mod p
  6. m = i = 0 n 2 i p i - 1 × ( gcd ( p i , c s mod p ) - 1 ) m=\sum_{i=0}^{n}\frac{2^{i}}{p_{i}-1}\times\left(\gcd(p_{i},c^{s}\mod p)-1\right)
  7. gcd ( p i , c s mod p ) - 1 p i - 1 \frac{\gcd(p_{i},c^{s}\mod p)-1}{p_{i}-1}

Nagata's_conjecture_on_curves.html

  1. k k
  2. r > 9 r>9
  3. C C
  4. deg C > 1 r i = 1 r m i . \deg C>\frac{1}{\sqrt{r}}\sum_{i=1}^{r}m_{i}.
  5. r r
  6. r > 9 r>9
  7. r > 9 r>9
  8. r 9 r≤9
  9. 𝐏 < s u p > 2 \mathbf{P}<sup>2

Naimark's_dilation_theorem.html

  1. L ( H ) L(H)
  2. { B i } \{B_{i}\}
  3. E ( i B i ) x , y = i E ( B i ) x , y \langle E(\cup_{i}B_{i})x,y\rangle=\sum_{i}\langle E(B_{i})x,y\rangle
  4. B E ( B ) x , y B\rightarrow\langle E(B)x,y\rangle
  5. | E | = sup B E ( B ) < |E|=\sup_{B}\|E(B)\|<\infty
  6. E ( B 1 B 2 ) = E ( B 1 ) E ( B 2 ) E(B_{1}\cap B_{2})=E(B_{1})E(B_{2})
  7. B 1 , B 2 B_{1},B_{2}
  8. Φ E : C ( X ) L ( H ) \Phi_{E}:C(X)\rightarrow L(H)
  9. Φ E ( f ) h 1 , h 2 = X f ( x ) E ( d x ) h 1 , h 2 \langle\Phi_{E}(f)h_{1},h_{2}\rangle=\int_{X}f(x)\langle E(dx)h_{1},h_{2}\rangle
  10. Φ E ( f ) h , h = X f ( x ) E ( d x ) h , h f | E | . \langle\Phi_{E}(f)h,h\rangle=\int_{X}f(x)\langle E(dx)h,h\rangle\leq\|f\|_{% \infty}\cdot|E|.
  11. Φ E ( f ) \;\Phi_{E}(f)
  12. Φ E \Phi_{E}
  13. Φ E \Phi_{E}
  14. Φ E \Phi_{E}
  15. Φ E \Phi_{E}
  16. h 1 , h 2 H h_{1},h_{2}\in H
  17. Φ E ( f g ) h 1 , h 2 = X f ( x ) g ( x ) E ( d x ) h 1 , h 2 = Φ E ( f ) Φ E ( g ) h 1 , h 2 . \langle\Phi_{E}(fg)h_{1},h_{2}\rangle=\int_{X}f(x)\cdot g(x)\;\langle E(dx)h_{% 1},h_{2}\rangle=\langle\Phi_{E}(f)\Phi_{E}(g)h_{1},h_{2}\rangle.
  18. Φ E \Phi_{E}
  19. Φ E \Phi_{E}
  20. Φ E ( f ¯ ) h 1 , h 2 = Φ E ( f ) * h 1 , h 2 . \langle\Phi_{E}({\bar{f}})h_{1},h_{2}\rangle=\langle\Phi_{E}(f)^{*}h_{1},h_{2}\rangle.
  21. X f ¯ E ( d x ) h 1 , h 2 , \int_{X}{\bar{f}}\;\langle E(dx)h_{1},h_{2}\rangle,
  22. h 1 , Φ E ( f ) h 2 = Φ E ( f ) h 2 , h 1 ¯ = X f ¯ ( x ) E ( d x ) h 2 , h 1 ¯ = X f ¯ ( x ) h 1 , E ( d x ) h 2 \langle h_{1},\Phi_{E}(f)h_{2}\rangle=\overline{\langle\Phi_{E}(f)h_{2},h_{1}% \rangle}=\int_{X}{\bar{f}}(x)\;\overline{\langle E(dx)h_{2},h_{1}\rangle}=\int% _{X}{\bar{f}}(x)\;\langle h_{1},E(dx)h_{2}\rangle
  23. E ( B ) h 1 , h 2 = h 1 , E ( B ) h 2 \langle E(B)h_{1},h_{2}\rangle=\langle h_{1},E(B)h_{2}\rangle
  24. Φ E \Phi_{E}
  25. V : K H V:K\rightarrow H
  26. E ( B ) = V F ( B ) V * . \;E(B)=VF(B)V^{*}.
  27. Φ E \Phi_{E}
  28. Φ E \Phi_{E}
  29. Φ E \Phi_{E}
  30. Φ E \Phi_{E}
  31. π : C ( X ) L ( K ) \pi:C(X)\rightarrow L(K)
  32. V : K H V:K\rightarrow H
  33. Φ E ( f ) = V π ( f ) V * . \;\Phi_{E}(f)=V\pi(f)V^{*}.
  34. X = { 1 , , n } X=\{1,\ldots,n\}
  35. n \mathbb{C}^{n}
  36. E i E_{i}
  37. i E i = I \sum_{i}E_{i}=I
  38. Φ E \Phi_{E}
  39. E i E_{i}
  40. x i m x_{i}\in\mathbb{C}^{m}
  41. n < m n<m
  42. n = m n=m
  43. i = 1 n x i x i * = I \sum_{i=1}^{n}x_{i}x_{i}^{*}=I
  44. { x i } \{x_{i}\}
  45. n > m n>m
  46. { E i } \{E_{i}\}
  47. M = [ x 1 x n ] M=\begin{bmatrix}x_{1}\cdots x_{n}\end{bmatrix}
  48. M M * = I MM^{*}=I
  49. ( n - m ) × n (n-m)\times n
  50. U = [ M N ] U=\begin{bmatrix}M\\ N\end{bmatrix}

Natural_filtration.html

  1. F i X = σ { X j - 1 ( A ) | j I , j i , A Σ } , F_{i}^{X}=\sigma\left\{\left.X_{j}^{-1}(A)\right|j\in I,j\leq i,A\in\Sigma% \right\},

Natural_ventilation.html

  1. q = 1 2 ρ v 2 , q=\tfrac{1}{2}\,\rho\,v^{2},
  2. q q\;
  3. ρ \rho\;
  4. v v\;
  5. Q = U wind C p1 - C p2 1 / ( A 1 2 C 1 2 ) + 1 / ( A 2 2 C 2 2 ) ( 1 ) Q=U_{\textrm{wind}}\sqrt{\frac{C_{\textrm{p1}}-C_{\textrm{p2}}}{1/\left(A_{% \textrm{1}}^{2}C_{\textrm{1}}^{2}\right)+1/\left(A_{\textrm{2}}^{2}C_{\textrm{% 2}}^{2}\right)}}\qquad{}\left(1\right)
  6. U wind U_{\textrm{wind}}
  7. C p1 C_{\textrm{p1}}
  8. C p2 C_{\textrm{p2}}
  9. A 1 A_{\textrm{1}}
  10. A 2 A_{\textrm{2}}
  11. C 1 C_{\textrm{1}}
  12. C 2 C_{\textrm{2}}
  13. Q ¯ = C d l C p z 0 h - 2 Δ P ( z ) ρ d z z r e f 1 / 7 U ¯ \bar{Q}=\frac{C_{d}\;l\;\sqrt{Cp}\;\int\limits_{z_{0}}^{h}\sqrt{-\frac{2\;% \Delta\;P(z)}{\rho}}\,\mathrm{d}z}{z_{ref}^{1/7}}\;\bar{U}
  14. U ¯ \bar{U}
  15. Q S = C d A 2 g H d T I - T O T I Q_{S}=C_{d}\;A\;\sqrt{2\;g\;H_{d}\;\frac{T_{I}-T_{O}}{T_{I}}}

Navier–Stokes_existence_and_smoothness.html

  1. 𝐯 ( s y m b o l x , t ) \mathbf{v}(symbol{x},t)
  2. p ( s y m b o l x , t ) p(symbol{x},t)
  3. 𝐯 t + ( 𝐯 ) 𝐯 = - p + ν Δ 𝐯 + 𝐟 ( s y m b o l x , t ) \frac{\partial\mathbf{v}}{\partial t}+(\mathbf{v}\cdot\nabla)\mathbf{v}=-% \nabla p+\nu\Delta\mathbf{v}+\mathbf{f}(symbol{x},t)
  4. ν > 0 \nu>0
  5. 𝐟 ( s y m b o l x , t ) \mathbf{f}(symbol{x},t)
  6. \nabla
  7. Δ \displaystyle\Delta
  8. \nabla\cdot\nabla
  9. 𝐯 ( s y m b o l x , t ) = ( v 1 ( s y m b o l x , t ) , v 2 ( s y m b o l x , t ) , v 3 ( s y m b o l x , t ) ) , 𝐟 ( s y m b o l x , t ) = ( f 1 ( s y m b o l x , t ) , f 2 ( s y m b o l x , t ) , f 3 ( s y m b o l x , t ) ) \mathbf{v}(symbol{x},t)=\big(\,v_{1}(symbol{x},t),\,v_{2}(symbol{x},t),\,v_{3}% (symbol{x},t)\,\big)\,,\qquad\mathbf{f}(symbol{x},t)=\big(\,f_{1}(symbol{x},t)% ,\,f_{2}(symbol{x},t),\,f_{3}(symbol{x},t)\,\big)
  10. i = 1 , 2 , 3 i=1,2,3
  11. v i t + j = 1 3 v j v i x j = - p x i + ν j = 1 3 2 v i x j 2 + f i ( s y m b o l x , t ) . \frac{\partial v_{i}}{\partial t}+\sum_{j=1}^{3}v_{j}\frac{\partial v_{i}}{% \partial x_{j}}=-\frac{\partial p}{\partial x_{i}}+\nu\sum_{j=1}^{3}\frac{% \partial^{2}v_{i}}{\partial x_{j}^{2}}+f_{i}(symbol{x},t).
  12. 𝐯 ( s y m b o l x , t ) \mathbf{v}(symbol{x},t)
  13. p ( s y m b o l x , t ) p(symbol{x},t)
  14. 𝐯 = 0. \nabla\cdot\mathbf{v}=0.
  15. 3 \mathbb{R}^{3}
  16. 3 \mathbb{R}^{3}
  17. 𝕋 3 = 3 / 3 \mathbb{T}^{3}=\mathbb{R}^{3}/\mathbb{Z}^{3}
  18. 𝐯 0 ( x ) \mathbf{v}_{0}(x)
  19. α \alpha
  20. K > 0 K>0
  21. C = C ( α , K ) > 0 C=C(\alpha,K)>0
  22. | α 𝐯 𝟎 ( x ) | C ( 1 + | x | ) K |\partial^{\alpha}\mathbf{v_{0}}(x)|\leq\frac{C}{(1+|x|)^{K}}\qquad
  23. x 3 . \qquad x\in\mathbb{R}^{3}.
  24. 𝐟 ( x , t ) \mathbf{f}(x,t)
  25. | α 𝐟 ( x ) | C ( 1 + | x | + t ) K |\partial^{\alpha}\mathbf{f}(x)|\leq\frac{C}{(1+|x|+t)^{K}}\qquad
  26. ( x , t ) 3 × [ 0 , ) . \qquad(x,t)\in\mathbb{R}^{3}\times[0,\infty).
  27. | x | |x|\to\infty
  28. 𝐯 ( x , t ) [ C ( 3 × [ 0 , ) ) ] 3 , p ( x , t ) C ( 3 × [ 0 , ) ) \mathbf{v}(x,t)\in\left[C^{\infty}(\mathbb{R}^{3}\times[0,\infty))\right]^{3}% \,,\qquad p(x,t)\in C^{\infty}(\mathbb{R}^{3}\times[0,\infty))
  29. E ( 0 , ) E\in(0,\infty)
  30. 3 | 𝐯 ( x , t ) | 2 d x < E \int_{\mathbb{R}^{3}}|\mathbf{v}(x,t)|^{2}dx<E
  31. t 0 . t\geq 0\,.
  32. 3 \mathbb{R}^{3}
  33. 𝐟 ( x , t ) 0 \mathbf{f}(x,t)\equiv 0
  34. 𝐯 0 ( x ) \mathbf{v}_{0}(x)
  35. 𝐯 ( x , t ) \mathbf{v}(x,t)
  36. p ( x , t ) p(x,t)
  37. 3 \mathbb{R}^{3}
  38. 𝐯 0 ( x ) \mathbf{v}_{0}(x)
  39. 𝐟 ( x , t ) \mathbf{f}(x,t)
  40. 𝐯 ( x , t ) \mathbf{v}(x,t)
  41. p ( x , t ) p(x,t)
  42. e i e_{i}
  43. e 1 = ( 1 , 0 , 0 ) , e 2 = ( 0 , 1 , 0 ) , e 3 = ( 0 , 0 , 1 ) e_{1}=(1,0,0)\,,\qquad e_{2}=(0,1,0)\,,\qquad e_{3}=(0,0,1)
  44. 𝐯 ( x , t ) \mathbf{v}(x,t)
  45. i = 1 , 2 , 3 i=1,2,3
  46. 𝐯 ( x + e i , t ) = 𝐯 ( x , t ) for all ( x , t ) 3 × [ 0 , ) . \mathbf{v}(x+e_{i},t)=\mathbf{v}(x,t)\,\text{ for all }(x,t)\in\mathbb{R}^{3}% \times[0,\infty).
  47. 3 \mathbb{R}^{3}
  48. 3 / 3 \mathbb{R}^{3}/\mathbb{Z}^{3}
  49. 𝕋 3 = { ( θ 1 , θ 2 , θ 3 ) : 0 θ i < 2 π , i = 1 , 2 , 3 } . \mathbb{T}^{3}=\{(\theta_{1},\theta_{2},\theta_{3}):0\leq\theta_{i}<2\pi\,,% \quad i=1,2,3\}.
  50. 𝐯 0 ( x ) \mathbf{v}_{0}(x)
  51. 𝐟 ( x , t ) \mathbf{f}(x,t)
  52. 𝐯 ( x , t ) [ C ( 𝕋 3 × [ 0 , ) ) ] 3 , p ( x , t ) C ( 𝕋 3 × [ 0 , ) ) \mathbf{v}(x,t)\in\left[C^{\infty}(\mathbb{T}^{3}\times[0,\infty))\right]^{3}% \,,\qquad p(x,t)\in C^{\infty}(\mathbb{T}^{3}\times[0,\infty))
  53. E ( 0 , ) E\in(0,\infty)
  54. 𝕋 3 | 𝐯 ( x , t ) | 2 d x < E \int_{\mathbb{T}^{3}}|\mathbf{v}(x,t)|^{2}dx<E
  55. t 0 . t\geq 0\,.
  56. 𝕋 3 \mathbb{T}^{3}
  57. 𝐟 ( x , t ) 0 \mathbf{f}(x,t)\equiv 0
  58. 𝐯 0 ( x ) \mathbf{v}_{0}(x)
  59. 𝐯 ( x , t ) \mathbf{v}(x,t)
  60. p ( x , t ) p(x,t)
  61. 𝕋 3 \mathbb{T}^{3}
  62. 𝐯 0 ( x ) \mathbf{v}_{0}(x)
  63. 𝐟 ( x , t ) \mathbf{f}(x,t)
  64. 𝐯 ( x , t ) \mathbf{v}(x,t)
  65. p ( x , t ) p(x,t)
  66. 𝐯 ( x , t ) \mathbf{v}(x,t)
  67. 𝐯 0 ( x ) \mathbf{v}_{0}(x)
  68. 𝐯 0 ( x ) \mathbf{v}_{0}(x)
  69. 3 × ( 0 , T ) \mathbb{R}^{3}\times(0,T)
  70. 𝐯 ( x , t ) \mathbf{v}(x,t)
  71. p ( x , t ) p(x,t)
  72. p ( 𝐱 , t ) p(\mathbf{x},t)

Nearest_neighbor_search.html

  1. G ( V , E ) G(V,E)
  2. x i S x_{i}\in S
  3. v i V v_{i}\in V
  4. G ( V , E ) G(V,E)
  5. v i V v_{i}\in V
  6. { v j : ( v i , v j ) E } \{v_{j}:(v_{i},v_{j})\in E\}
  7. v i v_{i}
  8. 𝔼 n \mathbb{E}^{n}

Necklace_(combinatorics).html

  1. N k ( n ) = 1 n d n φ ( d ) k n / d N_{k}(n)={1\over n}\sum_{d\mid n}\varphi(d)k^{n/d}
  2. B k ( n ) = { 1 2 N k ( n ) + 1 4 ( k + 1 ) k n / 2 if n is even 1 2 N k ( n ) + 1 2 k ( n + 1 ) / 2 if n is odd B_{k}(n)=\begin{cases}{1\over 2}N_{k}(n)+{1\over 4}(k+1)k^{n/2}&\,\text{if }n% \,\text{ is even}\\ \\ {1\over 2}N_{k}(n)+{1\over 2}k^{(n+1)/2}&\,\text{if }n\,\text{ is odd}\end{cases}
  3. M k ( n ) = 1 n d n μ ( d ) k n / d M_{k}(n)={1\over n}\sum_{d\mid n}\mu(d)k^{n/d}
  4. lim n n = 1 n N k ( n ) = k n n ! 1 ( 1 + X ) ( 1 + X + X 2 ) ( 1 + X + X 2 + + X n - 1 ) \lim_{n\to\infty}\prod_{n=1}^{n}N_{k}(n)=\frac{k^{n}}{n!}1(1+X)(1+X+X^{2})% \cdots(1+X+X^{2}+\cdots+X^{n-1})
  5. X k X^{k}
  6. m = 1 n i = 0 m - 1 X i = 1 ( 1 + X ) ( 1 + X + X 2 ) ( 1 + X + X 2 + + X n - 1 ) \prod_{m=1}^{n}\sum_{i=0}^{m-1}X^{i}=1(1+X)(1+X+X^{2})\cdots(1+X+X^{2}+\cdots+% X^{n-1})

Neighborhood_semantics.html

  1. W , R \langle W,R\rangle
  2. W , N \langle W,N\rangle
  3. N : W 2 2 W N:W\to 2^{2^{W}}
  4. M , w A ( A ) M N ( w ) , M,w\models\square A\Longleftrightarrow(A)^{M}\in N(w),
  5. ( A ) M = { u W M , u A } (A)^{M}=\{u\in W\mid M,u\models A\}
  6. N ( w ) = { ( A ) M : M , w A } . N(w)=\{(A)^{M}:M,w\models\Box A\}.

Neighbourhood_(graph_theory).html

  1. O ( k n ) O(\sqrt{kn})
  2. n 2 - o ( 1 ) n^{2-o(1)}

Nerve_(category_theory).html

  1. A 0 A 1 A 2 A k - 1 A k A_{0}\to A_{1}\to A_{2}\to\cdots\to A_{k-1}\to A_{k}
  2. d i : N ( C ) k N ( C ) k - 1 d_{i}\colon N(C)_{k}\to N(C)_{k-1}
  3. A 0 A i - 1 A i A i + 1 A k A_{0}\to\cdots\to A_{i-1}\to A_{i}\to A_{i+1}\to\cdots\to A_{k}
  4. A 0 A i - 1 A i + 1 A k . A_{0}\to\cdots\to A_{i-1}\to A_{i+1}\to\cdots\to A_{k}.
  5. s i : N ( C ) k N ( C ) k + 1 s_{i}:N(C)_{k}\to N(C)_{k+1}
  6. N ( C ) ( ? ) = Fun ( i ( ? ) , C ) . N(C)(?)=\mathrm{Fun}(i(?),C).\,
  7. X U V Y X\longleftarrow U\longrightarrow V\longleftarrow Y
  8. Π \Pi

Nested_intervals.html

  1. ( - , a n ) ( b n , ) (-\infty,a_{n})\cup(b_{n},\infty)

Nested_loop_join.html

  1. R R
  2. S S
  3. O ( | R | | S | ) O(|R||S|)
  4. | R | |R|
  5. | S | |S|
  6. R R
  7. S S
  8. S S
  9. R R
  10. S S
  11. s s

Nested_sampling_algorithm.html

  1. M 1 M1
  2. M 2 M2
  3. D D
  4. M 1 M1
  5. P ( M 1 | D ) \displaystyle P(M1|D)
  6. M 1 M1
  7. M 2 M2
  8. P ( M 1 ) = P ( M 2 ) = 1 / 2 P(M1)=P(M2)=1/2
  9. P ( M 2 ) / P ( M 1 ) = 1 P(M2)/P(M1)=1
  10. P ( D | M 2 ) / P ( D | M 1 ) P(D|M2)/P(D|M1)
  11. M 1 M1
  12. θ \theta
  13. M 2 M2
  14. θ \theta
  15. M 1 M1
  16. P ( D | M 1 ) = d θ P ( D | θ , M 1 ) P ( θ | M 1 ) P(D|M1)=\int d\theta P(D|\theta,M1)P(\theta|M1)
  17. M 2 M2
  18. P ( θ | D , M 1 ) P(\theta|D,M1)
  19. Z = P ( D | M ) Z=P(D|M)
  20. M M
  21. M 1 M1
  22. M 2 M2
  23. N N
  24. θ 1 , , θ N \theta_{1},...,\theta_{N}
  25. i = 1 i=1
  26. j j
  27. L i := min ( L_{i}:=\min(
  28. ) )
  29. X i := exp ( - i / N ) ; X_{i}:=\exp(-i/N);
  30. w i := X i - 1 - X i w_{i}:=X_{i-1}-X_{i}
  31. Z := Z + L i * w i ; Z:=Z+L_{i}*w_{i};
  32. w i w_{i}
  33. L i L_{i}
  34. Z Z
  35. X i X_{i}
  36. θ i \theta_{i}
  37. w i w_{i}
  38. { θ | P ( D | θ , M ) = P ( D | θ i - 1 , M ) } \{\theta|P(D|\theta,M)=P(D|\theta_{i-1},M)\}
  39. { θ | P ( D | θ , M ) = P ( D | θ i , M ) } \{\theta|P(D|\theta,M)=P(D|\theta_{i},M)\}
  40. Z := Z + L i * w i Z:=Z+L_{i}*w_{i}
  41. i i
  42. L i * w i L_{i}*w_{i}
  43. P ( D | M ) = P ( D | θ , M ) P ( θ | M ) d θ = P ( D | θ , M ) d P ( θ | M ) \begin{array}[]{lcl}P(D|M)&=&\int P(D|\theta,M)P(\theta|M)d\theta\\ &=&\int P(D|\theta,M)dP(\theta|M)\\ \end{array}
  44. f ( θ ) = P ( D | θ , M ) f(\theta)=P(D|\theta,M)
  45. [ f ( θ i - 1 ) , f ( θ i ) ] [f(\theta_{i-1}),f(\theta_{i})]
  46. θ \theta

Neural_coding.html

  1. ξ k \vec{\xi}\in\mathbb{R}^{k}
  2. b 1 , , b n k \vec{b_{1}},\ldots,\vec{b_{n}}\in\mathbb{R}^{k}
  3. s n \vec{s}\in\mathbb{R}^{n}
  4. ξ j = 1 n s j b j \vec{\xi}\approx\sum_{j=1}^{n}s_{j}\vec{b}_{j}

Neutron_magnetic_moment.html

  1. μ N = e 2 m p , \mu_{\mathrm{N}}={{e\hbar}\over{2m_{\mathrm{p}}}},
  2. s y m b o l μ = g μ N s y m b o l I symbol{\mu}=\frac{g\mu_{\mathrm{N}}}{\hbar}symbol{I}
  3. s y m b o l μ = \gammasymbol I symbol{\mu}=\gammasymbol{I}
  4. γ = g μ N = g e 2 m p \gamma=\frac{g\mu_{\mathrm{N}}}{\hbar}=g\frac{e}{2m\text{p}}
  5. μ q = e q 2 m q , \mu_{\mathrm{q}}={{e_{\mathrm{q}}\hbar}\over{2m_{\mathrm{q}}}},
  6. μ N \mu_{\mathrm{N}}
  7. μ N \mu_{\mathrm{N}}

Neutron_reflectometry.html

  1. q z q_{z}
  2. z z
  3. z z
  4. q z = 4 π λ sin ( θ ) q_{z}=\frac{4\pi}{\lambda}\sin(\theta)
  5. λ \lambda
  6. θ \theta
  7. q z q_{z}

Neutron_spin_echo.html

  1. I ( Q , t ) S ( Q ) + cos ( ω t ) S ( Q , ω ) d t I(Q,t)\propto S(Q)+\int\cos(\omega t)\,S(Q,\omega)\,dt
  2. ω Δ v \omega\propto\Delta v
  3. t B × λ 3 t\propto B\times\lambda^{3}

Neville's_algorithm.html

  1. p i , i ( x ) = y i , p_{i,i}(x)=y_{i},\,
  2. 0 i n , 0\leq i\leq n,\,
  3. p i , j ( x ) = ( x j - x ) p i , j - 1 ( x ) + ( x - x i ) p i + 1 , j ( x ) x j - x i , p_{i,j}(x)=\frac{(x_{j}-x)p_{i,j-1}(x)+(x-x_{i})p_{i+1,j}(x)}{x_{j}-x_{i}},\,
  4. 0 i < j n . 0\leq i<j\leq n.\,
  5. p 0 , 0 ( x ) = y 0 p_{0,0}(x)=y_{0}\,
  6. p 0 , 1 ( x ) p_{0,1}(x)\,
  7. p 1 , 1 ( x ) = y 1 p_{1,1}(x)=y_{1}\,
  8. p 0 , 2 ( x ) p_{0,2}(x)\,
  9. p 1 , 2 ( x ) p_{1,2}(x)\,
  10. p 0 , 3 ( x ) p_{0,3}(x)\,
  11. p 2 , 2 ( x ) = y 2 p_{2,2}(x)=y_{2}\,
  12. p 1 , 3 ( x ) p_{1,3}(x)\,
  13. p 0 , 4 ( x ) p_{0,4}(x)\,
  14. p 2 , 3 ( x ) p_{2,3}(x)\,
  15. p 1 , 4 ( x ) p_{1,4}(x)\,
  16. p 3 , 3 ( x ) = y 3 p_{3,3}(x)=y_{3}\,
  17. p 2 , 4 ( x ) p_{2,4}(x)\,
  18. p 3 , 4 ( x ) p_{3,4}(x)\,
  19. p 4 , 4 ( x ) = y 4 p_{4,4}(x)=y_{4}\,
  20. p i , i ( x ) = 0 , p^{\prime}_{i,i}(x)=0,\,
  21. 0 i n , 0\leq i\leq n,\,
  22. p i , j ( x ) = ( x j - x ) p i , j - 1 ( x ) - p i , j - 1 ( x ) + ( x - x i ) p i + 1 , j ( x ) + p i + 1 , j ( x ) x j - x i , p^{\prime}_{i,j}(x)=\frac{(x_{j}-x)p^{\prime}_{i,j-1}(x)-p_{i,j-1}(x)+(x-x_{i}% )p^{\prime}_{i+1,j}(x)+p_{i+1,j}(x)}{x_{j}-x_{i}},\,
  23. 0 i < j n . 0\leq i<j\leq n.\,

Newman–Penrose_formalism.html

  1. Ψ 4 \Psi_{4}
  2. κ , ρ , σ , τ ; λ , μ , ν , π ; ϵ , γ , β , α . \kappa,\rho,\sigma,\tau\,;\lambda,\mu,\nu,\pi\,;\epsilon,\gamma,\beta,\alpha.
  3. Ψ 0 , , Ψ 4 \Psi_{0},\ldots,\Psi_{4}
  4. Φ 00 , Φ 11 , Φ 22 , Λ \Phi_{00},\Phi_{11},\Phi_{22},\Lambda
  5. Φ 01 , Φ 10 , Φ 02 , Φ 20 , Φ 12 , Φ 21 \Phi_{01},\Phi_{10},\Phi_{02},\Phi_{20},\Phi_{12},\Phi_{21}
  6. l μ l^{\mu}
  7. n μ n^{\mu}
  8. l a n a = - 1 l^{a}n_{a}=-1
  9. l a l^{a}
  10. n a n^{a}
  11. m μ = 1 2 ( θ ^ + i ϕ ^ ) μ . m^{\mu}=\frac{1}{\sqrt{2}}\left(\hat{\theta}+i\hat{\phi}\right)^{\mu}\ .
  12. { ( + , - , - , - ) ; l a n a = 1 , m a m ¯ a = - 1 } \{(+,-,-,-);l^{a}n_{a}=1\,,m^{a}\bar{m}_{a}=-1\}
  13. { ( - , + , + , + ) ; l a n a = - 1 , m a m ¯ a = 1 } \{(-,+,+,+);l^{a}n_{a}=-1\,,m^{a}\bar{m}_{a}=1\}
  14. { ( - , + , + , + ) ; l a n a = - 1 , m a m ¯ a = 1 } \{(-,+,+,+);l^{a}n_{a}=-1\,,m^{a}\bar{m}_{a}=1\}
  15. { ( + , - , - , - ) , l a n a = 1 , m a m ¯ a = - 1 } \{(+,-,-,-)\,,l^{a}n_{a}=1\,,m^{a}\bar{m}_{a}=-1\}
  16. { ( - , + , + , + ) , l a n a = - 1 , m a m ¯ a = 1 } \{(-,+,+,+)\,,l^{a}n_{a}=-1\,,m^{a}\bar{m}_{a}=1\}
  17. Ψ i \Psi_{i}
  18. Φ i j \Phi_{ij}
  19. { , n } \{\ell\,,n\}
  20. { m , m ¯ } \{m\,,\bar{m}\}
  21. { , n } \{\ell\,,n\}
  22. 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
  23. l a n a = - 1 = l a n a , m a m ¯ a = 1 = m a m ¯ a , l_{a}n^{a}=-1=l^{a}n_{a}\,,\quad m_{a}\bar{m}^{a}=1=m^{a}\bar{m}_{a}\,,
  24. 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
  25. g a b g_{ab}
  26. 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}\,,\quad g^{ab}% =-l^{a}n^{b}-n^{a}l^{b}+m^{a}\bar{m}^{b}+\bar{m}^{a}m^{b}\,.
  27. D := = l a a , Δ := 𝐧 = n a a , δ := 𝐦 = m a a , δ ¯ := 𝐦 ¯ = m ¯ a a , D:=\nabla_{\ell}=l^{a}\nabla_{a}\,,\;\Delta:=\nabla_{\mathbf{n}}=n^{a}\nabla_{% a}\,,\;\delta:=\nabla_{\mathbf{m}}=m^{a}\nabla_{a}\,,\;\bar{\delta}:=\nabla_{% \mathbf{\bar{m}}}=\bar{m}^{a}\nabla_{a}\,,
  28. { D = l a a , Δ = n a a , δ = m a a , δ ¯ = m ¯ a a } \{D=l^{a}\partial_{a}\,,\Delta=n^{a}\partial_{a}\,,\delta=m^{a}\partial_{a}\,,% \bar{\delta}=\bar{m}^{a}\partial_{a}\}
  29. γ i j k \gamma_{ijk}
  30. κ := - m a D l a = - m a l b b l a , τ := - m a Δ l a = - m a n b b l a , \kappa:=-m^{a}Dl_{a}=-m^{a}l^{b}\nabla_{b}l_{a}\,,\quad\tau:=-m^{a}\Delta l_{a% }=-m^{a}n^{b}\nabla_{b}l_{a}\,,
  31. σ := - m a δ l a = - m a m b b l a , ρ := - m a δ ¯ l a = - m a m ¯ b b l a ; \sigma:=-m^{a}\delta l_{a}=-m^{a}m^{b}\nabla_{b}l_{a}\,,\quad\rho:=-m^{a}\bar{% \delta}l_{a}=-m^{a}\bar{m}^{b}\nabla_{b}l_{a}\,;
  32. π := m ¯ a D n a = m ¯ a l b b n a , ν := m ¯ a Δ n a = m ¯ a n b b n a , \pi:=\bar{m}^{a}Dn_{a}=\bar{m}^{a}l^{b}\nabla_{b}n_{a}\,,\quad\nu:=\bar{m}^{a}% \Delta n_{a}=\bar{m}^{a}n^{b}\nabla_{b}n_{a}\,,
  33. μ := m ¯ a δ n a = m ¯ a m b b n a , λ := m ¯ a δ ¯ n a = m ¯ a m ¯ b b n a ; \mu:=\bar{m}^{a}\delta n_{a}=\bar{m}^{a}m^{b}\nabla_{b}n_{a}\,,\quad\lambda:=% \bar{m}^{a}\bar{\delta}n_{a}=\bar{m}^{a}\bar{m}^{b}\nabla_{b}n_{a}\,;
  34. ε := - 1 2 ( n a D l a - m ¯ a D m a ) = - 1 2 ( n a l b b l a - m ¯ a l b b m a ) , \varepsilon:=-\frac{1}{2}\big(n^{a}Dl_{a}-\bar{m}^{a}Dm_{a}\big)=-\frac{1}{2}% \big(n^{a}l^{b}\nabla_{b}l_{a}-\bar{m}^{a}l^{b}\nabla_{b}m_{a}\big)\,,
  35. γ := - 1 2 ( n a Δ l a - m ¯ a Δ m a ) = - 1 2 ( n a n b b l a - m ¯ a n b b m a ) , \gamma:=-\frac{1}{2}\big(n^{a}\Delta l_{a}-\bar{m}^{a}\Delta m_{a}\big)=-\frac% {1}{2}\big(n^{a}n^{b}\nabla_{b}l_{a}-\bar{m}^{a}n^{b}\nabla_{b}m_{a}\big)\,,
  36. β := - 1 2 ( n a δ l a - m ¯ a δ m a ) = - 1 2 ( n a m b b l a - m ¯ a m b b m a ) , \beta:=-\frac{1}{2}\big(n^{a}\delta l_{a}-\bar{m}^{a}\delta m_{a}\big)=-\frac{% 1}{2}\big(n^{a}m^{b}\nabla_{b}l_{a}-\bar{m}^{a}m^{b}\nabla_{b}m_{a}\big)\,,
  37. α := - 1 2 ( n a δ ¯ l a - m ¯ a δ ¯ m a ) = - 1 2 ( n a m ¯ b b l a - m ¯ a m ¯ b b m a ) . \alpha:=-\frac{1}{2}\big(n^{a}\bar{\delta}l_{a}-\bar{m}^{a}\bar{\delta}m_{a}% \big)=-\frac{1}{2}\big(n^{a}\bar{m}^{b}\nabla_{b}l_{a}-\bar{m}^{a}\bar{m}^{b}% \nabla_{b}m_{a}\big)\,.
  38. D l a = ( ε + ε ¯ ) l a - κ ¯ m a - κ m ¯ a , Dl^{a}=(\varepsilon+\bar{\varepsilon})l^{a}-\bar{\kappa}m^{a}-\kappa\bar{m}^{a% }\,,
  39. Δ l a = ( γ + γ ¯ ) l a - τ ¯ m a - τ m ¯ a , \Delta l^{a}=(\gamma+\bar{\gamma})l^{a}-\bar{\tau}m^{a}-\tau\bar{m}^{a}\,,
  40. δ l a = ( α ¯ + β ) l a - ρ ¯ m a - σ m ¯ a , \delta l^{a}=(\bar{\alpha}+\beta)l^{a}-\bar{\rho}m^{a}-\sigma\bar{m}^{a}\,,
  41. δ ¯ l a = ( α + β ¯ ) l a - σ ¯ m a - ρ m ¯ a ; \bar{\delta}l^{a}=(\alpha+\bar{\beta})l^{a}-\bar{\sigma}m^{a}-\rho\bar{m}^{a}\,;
  42. D n a = π m a + π ¯ m ¯ a - ( ε + ε ¯ ) n a , Dn^{a}=\pi m^{a}+\bar{\pi}\bar{m}^{a}-(\varepsilon+\bar{\varepsilon})n^{a}\,,
  43. Δ n a = ν m a + ν ¯ m ¯ a - ( γ + γ ¯ ) n a , \Delta n^{a}=\nu m^{a}+\bar{\nu}\bar{m}^{a}-(\gamma+\bar{\gamma})n^{a}\,,
  44. δ n a = μ m a + λ ¯ m ¯ a - ( α ¯ + β ) n a , \delta n^{a}=\mu m^{a}+\bar{\lambda}\bar{m}^{a}-(\bar{\alpha}+\beta)n^{a}\,,
  45. δ ¯ n a = λ m a + μ ¯ m ¯ a - ( α + β ¯ ) n a ; \bar{\delta}n^{a}=\lambda m^{a}+\bar{\mu}\bar{m}^{a}-(\alpha+\bar{\beta})n^{a}\,;
  46. D m a = ( ε - ε ¯ ) m a + π ¯ l a - κ n a , Dm^{a}=(\varepsilon-\bar{\varepsilon})m^{a}+\bar{\pi}l^{a}-\kappa n^{a}\,,
  47. Δ m a = ( γ - γ ¯ ) m a + ν ¯ l a - τ n a , \Delta m^{a}=(\gamma-\bar{\gamma})m^{a}+\bar{\nu}l^{a}-\tau n^{a}\,,
  48. δ m a = ( β - α ¯ ) m a + λ ¯ l a - σ n a , \delta m^{a}=(\beta-\bar{\alpha})m^{a}+\bar{\lambda}l^{a}-\sigma n^{a}\,,
  49. δ ¯ m a = ( α - β ¯ ) m a + μ ¯ l a - ρ n a ; \bar{\delta}m^{a}=(\alpha-\bar{\beta})m^{a}+\bar{\mu}l^{a}-\rho n^{a}\,;
  50. D m a = ( ε ¯ - ε ) m a + π l a - κ ¯ n a , Dm^{a}=(\bar{\varepsilon}-\varepsilon)m^{a}+\pi l^{a}-\bar{\kappa}n^{a}\,,
  51. Δ m a = ( γ - γ ¯ ) m a + ν l a - τ ¯ n a , \Delta m^{a}=(\gamma-\bar{\gamma})m^{a}+\nu l^{a}-\bar{\tau}n^{a}\,,
  52. δ m a = ( β - α ¯ ) m a + μ l a - ρ ¯ n a , \delta m^{a}=(\beta-\bar{\alpha})m^{a}+\mu l^{a}-\bar{\rho}n^{a}\,,
  53. δ ¯ m a = ( α - β ¯ ) m a + λ l a - σ ¯ n a . \bar{\delta}m^{a}=(\alpha-\bar{\beta})m^{a}+\lambda l^{a}-\bar{\sigma}n^{a}\,.
  54. Δ D - D Δ = ( γ + γ ¯ ) D + ( ε + ε ¯ ) Δ - ( τ ¯ + π ) δ - ( τ + π ¯ ) δ ¯ , \Delta D-D\Delta=(\gamma+\bar{\gamma})D+(\varepsilon+\bar{\varepsilon})\Delta-% (\bar{\tau}+\pi)\delta-(\tau+\bar{\pi})\bar{\delta}\,,
  55. δ D - D δ = ( α ¯ + β - π ¯ ) D + κ Δ - ( ρ ¯ + ε - ε ¯ ) δ - σ δ ¯ , \delta D-D\delta=(\bar{\alpha}+\beta-\bar{\pi})D+\kappa\Delta-(\bar{\rho}+% \varepsilon-\bar{\varepsilon})\delta-\sigma\bar{\delta}\,,
  56. δ Δ - Δ δ = - ν ¯ D + ( τ - α ¯ - β ) Δ + ( μ - γ + γ ¯ ) δ + λ ¯ δ ¯ , \delta\Delta-\Delta\delta=-\bar{\nu}D+(\tau-\bar{\alpha}-\beta)\Delta+(\mu-% \gamma+\bar{\gamma})\delta+\bar{\lambda}\bar{\delta}\,,
  57. δ ¯ δ - δ δ ¯ = ( μ ¯ - μ ) D + ( ρ ¯ - ρ ) Δ + ( α - β ¯ ) δ - ( α ¯ - β ) δ ¯ , \bar{\delta}\delta-\delta\bar{\delta}=(\bar{\mu}-\mu)D+(\bar{\rho}-\rho)\Delta% +(\alpha-\bar{\beta})\delta-(\bar{\alpha}-\beta)\bar{\delta}\,,
  58. Δ l a - D n a = ( γ + γ ¯ ) l a + ( ε + ε ¯ ) n a - ( τ ¯ + π ) m a - ( τ + π ¯ ) m ¯ a , \Delta l^{a}-Dn^{a}=(\gamma+\bar{\gamma})l^{a}+(\varepsilon+\bar{\varepsilon})% n^{a}-(\bar{\tau}+\pi)m^{a}-(\tau+\bar{\pi})\bar{m}^{a}\,,
  59. δ l a - D m a = ( α ¯ + β - π ¯ ) l a + κ n a - ( ρ ¯ + ε - ε ¯ ) m a - σ m ¯ a , \delta l^{a}-Dm^{a}=(\bar{\alpha}+\beta-\bar{\pi})l^{a}+\kappa n^{a}-(\bar{% \rho}+\varepsilon-\bar{\varepsilon})m^{a}-\sigma\bar{m}^{a}\,,
  60. δ n a - Δ m a = - ν ¯ l a + ( τ - α ¯ - β ) n a + ( μ - γ + γ ¯ ) m a + λ ¯ m ¯ a , \delta n^{a}-\Delta m^{a}=-\bar{\nu}l^{a}+(\tau-\bar{\alpha}-\beta)n^{a}+(\mu-% \gamma+\bar{\gamma})m^{a}+\bar{\lambda}\bar{m}^{a}\,,
  61. δ ¯ m a - δ m ¯ a = ( μ ¯ - μ ) l a + ( ρ ¯ - ρ ) n a + ( α - β ¯ ) m a - ( α ¯ - β ) m ¯ a . \bar{\delta}m^{a}-\delta\bar{m}^{a}=(\bar{\mu}-\mu)l^{a}+(\bar{\rho}-\rho)n^{a% }+(\alpha-\bar{\beta})m^{a}-(\bar{\alpha}-\beta)\bar{m}^{a}\,.
  62. { l a , n a , m a , m ¯ a } \{l^{a},n^{a},m^{a},\bar{m}^{a}\}
  63. Ψ 0 := C a b c d l a m b l c m d , Ψ 1 := C a b c d l a n b l c m d , Ψ 2 := C a b c d l a m b m ¯ c n d , Ψ 3 := C a b c d l a n b m ¯ c n d , Ψ 4 := C a b c d n a m ¯ b n c m ¯ d . \Psi_{0}:=C_{abcd}l^{a}m^{b}l^{c}m^{d}\,,\quad\Psi_{1}:=C_{abcd}l^{a}n^{b}l^{c% }m^{d}\,,\quad\Psi_{2}:=C_{abcd}l^{a}m^{b}\bar{m}^{c}n^{d}\,,\quad\Psi_{3}:=C_% {abcd}l^{a}n^{b}\bar{m}^{c}n^{d}\,,\quad\Psi_{4}:=C_{abcd}n^{a}\bar{m}^{b}n^{c% }\bar{m}^{d}\,.
  64. { Φ 00 \{\Phi_{00}
  65. Φ 11 \Phi_{11}
  66. Φ 22 \Phi_{22}
  67. Λ } \Lambda\}
  68. { Φ 10 , Φ 20 , Φ 21 } \{\Phi_{10},\Phi_{20},\Phi_{21}\}
  69. Φ 00 := 1 2 R a b l a l b , Φ 11 := 1 4 R a b ( l a n b + m a m ¯ b ) , Φ 22 := 1 2 R a b n a n b , Λ := R 24 ; \Phi_{00}:=\frac{1}{2}R_{ab}l^{a}l^{b}\,,\quad\Phi_{11}:=\frac{1}{4}R_{ab}(\,l% ^{a}n^{b}+m^{a}\bar{m}^{b})\,,\quad\Phi_{22}:=\frac{1}{2}R_{ab}n^{a}n^{b}\,,% \quad\Lambda:=\frac{R}{24}\,;
  70. Φ 01 := 1 2 R a b l a m b , Φ 10 := 1 2 R a b l a m ¯ b = Φ 01 ¯ , \Phi_{01}:=\frac{1}{2}R_{ab}l^{a}m^{b}\,,\quad\;\Phi_{10}:=\frac{1}{2}R_{ab}l^% {a}\bar{m}^{b}=\overline{\Phi_{01}}\,,
  71. Φ 02 := 1 2 R a b m a m b , Φ 20 := 1 2 R a b m ¯ a m ¯ b = Φ 02 ¯ , \Phi_{02}:=\frac{1}{2}R_{ab}m^{a}m^{b}\,,\quad\Phi_{20}:=\frac{1}{2}R_{ab}\bar% {m}^{a}\bar{m}^{b}=\overline{\Phi_{02}}\,,
  72. Φ 12 := 1 2 R a b m a n b , Φ 21 := 1 2 R a b m ¯ a n b = Φ 12 ¯ . \Phi_{12}:=\frac{1}{2}R_{ab}m^{a}n^{b}\,,\quad\;\Phi_{21}:=\frac{1}{2}R_{ab}% \bar{m}^{a}n^{b}=\overline{\Phi_{12}}\,.
  73. R a b R_{ab}
  74. Q a b = R a b - 1 4 g a b R \displaystyle Q_{ab}=R_{ab}-\frac{1}{4}g_{ab}R
  75. G a b = R a b - 1 2 g a b R \displaystyle G_{ab}=R_{ab}-\frac{1}{2}g_{ab}R
  76. Φ 11 \Phi_{11}
  77. Φ 11 = 1 2 R a b l a n b = 1 2 R a b m a m ¯ a \Phi_{11}=\frac{1}{2}R_{ab}l^{a}n^{b}=\frac{1}{2}R_{ab}m^{a}\bar{m}^{a}
  78. Λ = 0 \Lambda=0
  79. D ρ - δ ¯ κ = ( ρ 2 + σ σ ¯ ) + ( ε + ε ¯ ) ρ - κ ¯ τ - κ ( 3 α + β ¯ - π ) + Φ 00 , D\rho-\bar{\delta}\kappa=(\rho^{2}+\sigma\bar{\sigma})+(\varepsilon+\bar{% \varepsilon})\rho-\bar{\kappa}\tau-\kappa(3\alpha+\bar{\beta}-\pi)+\Phi_{00}\,,
  80. D σ - δ κ = ( ρ + ρ ¯ ) σ + ( 3 ε - ε ¯ ) σ - ( τ - π ¯ + α ¯ + 3 β ) κ + Ψ 0 , D\sigma-\delta\kappa=(\rho+\bar{\rho})\sigma+(3\varepsilon-\bar{\varepsilon})% \sigma-(\tau-\bar{\pi}+\bar{\alpha}+3\beta)\kappa+\Psi_{0}\,,
  81. D τ - Δ κ = ( τ + π ¯ ) ρ + ( τ ¯ + π ) σ + ( ε - ε ¯ ) τ - ( 3 γ + γ ¯ ) κ + Ψ 1 + Φ 01 , D\tau-\Delta\kappa=(\tau+\bar{\pi})\rho+(\bar{\tau}+\pi)\sigma+(\varepsilon-% \bar{\varepsilon})\tau-(3\gamma+\bar{\gamma})\kappa+\Psi_{1}+\Phi_{01}\,,
  82. D α - δ ¯ ε = ( ρ + ε ¯ - 2 ε ) α + β σ ¯ - β ¯ ε - κ λ - κ ¯ γ + ( ε + ρ ) π + Φ 10 , D\alpha-\bar{\delta}\varepsilon=(\rho+\bar{\varepsilon}-2\varepsilon)\alpha+% \beta\bar{\sigma}-\bar{\beta}\varepsilon-\kappa\lambda-\bar{\kappa}\gamma+(% \varepsilon+\rho)\pi+\Phi_{10}\,,
  83. D β - δ ε = ( α + π ) σ + ( ρ ¯ - ε ¯ ) β - ( μ + γ ) κ - ( α ¯ - π ¯ ) ε + Ψ 1 , D\beta-\delta\varepsilon=(\alpha+\pi)\sigma+(\bar{\rho}-\bar{\varepsilon})% \beta-(\mu+\gamma)\kappa-(\bar{\alpha}-\bar{\pi})\varepsilon+\Psi_{1}\,,
  84. D γ - Δ ε = ( τ + π ¯ ) α + ( τ ¯ + π ) β - ( ε + ε ¯ ) γ - ( γ + γ ¯ ) ε + τ π - ν κ + Ψ 2 + Φ 11 - Λ , D\gamma-\Delta\varepsilon=(\tau+\bar{\pi})\alpha+(\bar{\tau}+\pi)\beta-(% \varepsilon+\bar{\varepsilon})\gamma-(\gamma+\bar{\gamma})\varepsilon+\tau\pi-% \nu\kappa+\Psi_{2}+\Phi_{11}-\Lambda\,,
  85. D λ - δ ¯ π = ( ρ λ + σ ¯ μ ) + π 2 + ( α - β ¯ ) π - ν κ ¯ - ( 3 ε - ε ¯ ) λ + Φ 20 , D\lambda-\bar{\delta}\pi=(\rho\lambda+\bar{\sigma}\mu)+\pi^{2}+(\alpha-\bar{% \beta})\pi-\nu\bar{\kappa}-(3\varepsilon-\bar{\varepsilon})\lambda+\Phi_{20}\,,
  86. D μ - δ π = ( ρ ¯ μ + σ λ ) + π π ¯ - ( ε + ε ¯ ) μ - ( α ¯ - β ) π - ν κ + Ψ 2 + 2 Λ , D\mu-\delta\pi=(\bar{\rho}\mu+\sigma\lambda)+\pi\bar{\pi}-(\varepsilon+\bar{% \varepsilon})\mu-(\bar{\alpha}-\beta)\pi-\nu\kappa+\Psi_{2}+2\Lambda\,,
  87. D ν - Δ π = ( π + τ ¯ ) μ + ( π ¯ + τ ) λ + ( γ - γ ¯ ) π - ( 3 ε + ε ¯ ) ν + Ψ 3 + Φ 21 , D\nu-\Delta\pi=(\pi+\bar{\tau})\mu+(\bar{\pi}+\tau)\lambda+(\gamma-\bar{\gamma% })\pi-(3\varepsilon+\bar{\varepsilon})\nu+\Psi_{3}+\Phi_{21}\,,
  88. Δ λ - δ ¯ ν = - ( μ + μ ¯ ) λ - ( 3 γ - γ ¯ ) λ + ( 3 α + β ¯ + π - τ ¯ ) ν - Ψ 4 , \Delta\lambda-\bar{\delta}\nu=-(\mu+\bar{\mu})\lambda-(3\gamma-\bar{\gamma})% \lambda+(3\alpha+\bar{\beta}+\pi-\bar{\tau})\nu-\Psi_{4}\,,
  89. δ ρ - δ ¯ σ = ρ ( α ¯ + β ) - σ ( 3 α - β ¯ ) + ( ρ - ρ ¯ ) τ + ( μ - μ ¯ ) κ - Ψ 1 + Φ 01 , \delta\rho-\bar{\delta}\sigma=\rho(\bar{\alpha}+\beta)-\sigma(3\alpha-\bar{% \beta})+(\rho-\bar{\rho})\tau+(\mu-\bar{\mu})\kappa-\Psi_{1}+\Phi_{01}\,,
  90. δ α - δ ¯ β = ( μ ρ - λ σ ) + α α ¯ + β β ¯ - 2 α β + γ ( ρ - ρ ¯ ) + ε ( μ - μ ¯ ) - Ψ 2 + Φ 11 + Λ , \delta\alpha-\bar{\delta}\beta=(\mu\rho-\lambda\sigma)+\alpha\bar{\alpha}+% \beta\bar{\beta}-2\alpha\beta+\gamma(\rho-\bar{\rho})+\varepsilon(\mu-\bar{\mu% })-\Psi_{2}+\Phi_{11}+\Lambda\,,
  91. δ λ - δ ¯ μ = ( ρ - ρ ¯ ) ν + ( μ - μ ¯ ) π + ( α + β ¯ ) μ + ( α ¯ - 3 β ) λ - Ψ 3 + Φ 21 , \delta\lambda-\bar{\delta}\mu=(\rho-\bar{\rho})\nu+(\mu-\bar{\mu})\pi+(\alpha+% \bar{\beta})\mu+(\bar{\alpha}-3\beta)\lambda-\Psi_{3}+\Phi_{21}\,,
  92. δ ν - Δ μ = ( μ 2 + λ λ ¯ ) + ( γ + γ ¯ ) μ - ν ¯ π + ( τ - 3 β - α ¯ ) ν + Φ 22 , \delta\nu-\Delta\mu=(\mu^{2}+\lambda\bar{\lambda})+(\gamma+\bar{\gamma})\mu-% \bar{\nu}\pi+(\tau-3\beta-\bar{\alpha})\nu+\Phi_{22}\,,
  93. δ γ - Δ β = ( τ - α ¯ - β ) γ + μ τ - σ ν - ε ν ¯ - ( γ - γ ¯ - μ ) β + α λ ¯ + Φ 12 , \delta\gamma-\Delta\beta=(\tau-\bar{\alpha}-\beta)\gamma+\mu\tau-\sigma\nu-% \varepsilon\bar{\nu}-(\gamma-\bar{\gamma}-\mu)\beta+\alpha\bar{\lambda}+\Phi_{% 12}\,,
  94. δ τ - Δ σ = ( μ σ + λ ¯ ρ ) + ( τ + β - α ¯ ) τ - ( 3 γ - γ ¯ ) σ - κ ν ¯ + Φ 02 , \delta\tau-\Delta\sigma=(\mu\sigma+\bar{\lambda}\rho)+(\tau+\beta-\bar{\alpha}% )\tau-(3\gamma-\bar{\gamma})\sigma-\kappa\bar{\nu}+\Phi_{02}\,,
  95. Δ ρ - δ ¯ τ = - ( ρ μ ¯ + σ λ ) + ( β ¯ - α - τ ¯ ) τ + ( γ + γ ¯ ) ρ + ν κ - Ψ 2 - 2 Λ , \Delta\rho-\bar{\delta}\tau=-(\rho\bar{\mu}+\sigma\lambda)+(\bar{\beta}-\alpha% -\bar{\tau})\tau+(\gamma+\bar{\gamma})\rho+\nu\kappa-\Psi_{2}-2\Lambda\,,
  96. Δ α - δ ¯ γ = ( ρ + ε ) ν - ( τ + β ) λ + ( γ ¯ - μ ¯ ) α + ( β ¯ - τ ¯ ) γ - Ψ 3 . \Delta\alpha-\bar{\delta}\gamma=(\rho+\varepsilon)\nu-(\tau+\beta)\lambda+(% \bar{\gamma}-\bar{\mu})\alpha+(\bar{\beta}-\bar{\tau})\gamma-\Psi_{3}\,.
  97. Ψ i \Psi_{i}
  98. Φ i j \Phi_{ij}
  99. F a b F_{ab}
  100. ϕ 0 := - F a b l a m b , ϕ 1 := - 1 2 F a b ( l a n a - m a m ¯ b ) , ϕ 2 := F a b n a m ¯ b , \phi_{0}:=-F_{ab}l^{a}m^{b}\,,\quad\phi_{1}:=-\frac{1}{2}F_{ab}\big(l^{a}n^{a}% -m^{a}\bar{m}^{b}\big)\,,\quad\phi_{2}:=F_{ab}n^{a}\bar{m}^{b}\,,
  101. d 𝐅 = 0 d\mathbf{F}=0
  102. d 𝐅 = 0 d^{\star}\mathbf{F}=0
  103. 𝐅 = d A \mathbf{F}=dA
  104. D ϕ 1 - δ ¯ ϕ 0 = ( π - 2 α ) ϕ 0 + 2 ρ ϕ 1 - κ ϕ 2 , D\phi_{1}-\bar{\delta}\phi_{0}=(\pi-2\alpha)\phi_{0}+2\rho\phi_{1}-\kappa\phi_% {2}\,,
  105. D ϕ 2 - δ ¯ ϕ 1 = - λ ϕ 0 + 2 π ϕ 1 + ( ρ - 2 ε ) ϕ 2 , D\phi_{2}-\bar{\delta}\phi_{1}=-\lambda\phi_{0}+2\pi\phi_{1}+(\rho-2% \varepsilon)\phi_{2}\,,
  106. Δ ϕ 0 - δ ϕ 1 = ( 2 γ - μ ) ϕ 0 - 2 τ ϕ 1 + σ ϕ 2 , \Delta\phi_{0}-\delta\phi_{1}=(2\gamma-\mu)\phi_{0}-2\tau\phi_{1}+\sigma\phi_{% 2}\,,
  107. Δ ϕ 1 - δ ϕ 2 = ν ϕ 0 - 2 μ ϕ 1 + ( 2 β - τ ) ϕ 2 , \Delta\phi_{1}-\delta\phi_{2}=\nu\phi_{0}-2\mu\phi_{1}+(2\beta-\tau)\phi_{2}\,,
  108. Φ i j \Phi_{ij}
  109. Φ i j = 2 ϕ i ϕ j ¯ , ( i , j { 0 , 1 , 2 } ) . \Phi_{ij}=\,2\,\phi_{i}\,\overline{\phi_{j}}\,,\quad(i,j\in\{0,1,2\})\,.
  110. Φ i j = 2 ϕ i ϕ j ¯ \Phi_{ij}=2\,\phi_{i}\,\overline{\phi_{j}}
  111. Φ i j = Tr ( ϝ i ϝ ¯ j ) \Phi_{ij}=\,\,\text{Tr}\,(\digamma_{i}\,\bar{\digamma}_{j})
  112. ϝ i ( i { 0 , 1 , 2 } ) \digamma_{i}(i\in\{0,1,2\})
  113. Ψ 4 \Psi_{4}
  114. Ψ 4 = - C α β γ δ n α m ¯ β n γ m ¯ δ \Psi_{4}=-C_{\alpha\beta\gamma\delta}n^{\alpha}\bar{m}^{\beta}n^{\gamma}\bar{m% }^{\delta}
  115. ( + , - , - , - ) (+,-,-,-)
  116. R α β = 0 R_{\alpha\beta}=0
  117. C α β γ δ = R α β γ δ C_{\alpha\beta\gamma\delta}=R_{\alpha\beta\gamma\delta}
  118. l μ = 1 2 ( t ^ + r ^ ) , l^{\mu}=\frac{1}{\sqrt{2}}\left(\hat{t}+\hat{r}\right)\ ,
  119. n μ = 1 2 ( t ^ - r ^ ) , n^{\mu}=\frac{1}{\sqrt{2}}\left(\hat{t}-\hat{r}\right)\ ,
  120. m μ = 1 2 ( θ ^ + i ϕ ^ ) . m^{\mu}=\frac{1}{\sqrt{2}}\left(\hat{\theta}+i\hat{\phi}\right)\ .
  121. 1 4 ( h ¨ θ ^ θ ^ - h ¨ ϕ ^ ϕ ^ ) = - R t ^ θ ^ t ^ θ ^ = - R t ^ ϕ ^ r ^ ϕ ^ = - R r ^ θ ^ r ^ θ ^ = R t ^ ϕ ^ t ^ ϕ ^ = R t ^ θ ^ r ^ θ ^ = R r ^ ϕ ^ r ^ ϕ ^ , \frac{1}{4}\left(\ddot{h}_{\hat{\theta}\hat{\theta}}-\ddot{h}_{\hat{\phi}\hat{% \phi}}\right)=-R_{\hat{t}\hat{\theta}\hat{t}\hat{\theta}}=-R_{\hat{t}\hat{\phi% }\hat{r}\hat{\phi}}=-R_{\hat{r}\hat{\theta}\hat{r}\hat{\theta}}=R_{\hat{t}\hat% {\phi}\hat{t}\hat{\phi}}=R_{\hat{t}\hat{\theta}\hat{r}\hat{\theta}}=R_{\hat{r}% \hat{\phi}\hat{r}\hat{\phi}}\ ,
  122. 1 2 h ¨ θ ^ ϕ ^ = - R t ^ θ ^ t ^ ϕ ^ = - R r ^ θ ^ r ^ ϕ ^ = R t ^ θ ^ r ^ ϕ ^ = R r ^ θ ^ t ^ ϕ ^ , \frac{1}{2}\ddot{h}_{\hat{\theta}\hat{\phi}}=-R_{\hat{t}\hat{\theta}\hat{t}% \hat{\phi}}=-R_{\hat{r}\hat{\theta}\hat{r}\hat{\phi}}=R_{\hat{t}\hat{\theta}% \hat{r}\hat{\phi}}=R_{\hat{r}\hat{\theta}\hat{t}\hat{\phi}}\ ,
  123. r ^ \hat{r}
  124. Ψ 4 \Psi_{4}
  125. Ψ 4 = 1 2 ( h ¨ θ ^ θ ^ - h ¨ ϕ ^ ϕ ^ ) + i h ¨ θ ^ ϕ ^ = - h ¨ + + i h ¨ × . \Psi_{4}=\frac{1}{2}\left(\ddot{h}_{\hat{\theta}\hat{\theta}}-\ddot{h}_{\hat{% \phi}\hat{\phi}}\right)+i\ddot{h}_{\hat{\theta}\hat{\phi}}=-\ddot{h}_{+}+i% \ddot{h}_{\times}\ .
  126. h + h_{+}
  127. h × h_{\times}
  128. Ψ 4 \Psi_{4}
  129. Ψ 4 ( t , r , θ , ϕ ) = - 1 r 2 l = 2 m = - l l [ I l m ( l + 2 ) ( t - r ) - i S l m ( l + 2 ) ( t - r ) ] Y l m - 2 ( θ , ϕ ) . \Psi_{4}(t,r,\theta,\phi)=-\frac{1}{r\sqrt{2}}\sum_{l=2}^{\infty}\sum_{m=-l}^{% l}\left[{}^{(l+2)}I^{lm}(t-r)-i\ {}^{(l+2)}S^{lm}(t-r)\right]{}_{-2}Y_{lm}(% \theta,\phi)\ .
  130. G ( l ) ( t ) = ( d d t ) l G ( t ) . {}^{(l)}G(t)=\left(\frac{d}{dt}\right)^{l}G(t)\ .
  131. I l m I^{lm}
  132. S l m S^{lm}
  133. Y l m - 2 {}_{-2}Y_{lm}

Newton–Wigner_localization.html

  1. x x
  2. x x
  3. x x
  4. x x
  5. y y
  6. z z
  7. p p
  8. p p
  9. p p
  10. [ x i , p 0 ] = p i / p 0 . [x_{i}\,,p_{0}]=p_{i}/p_{0}~{}.

Nice_name.html

  1. M M\models
  2. ( , < ) (\mathbb{P},<)
  3. M M
  4. G G\subseteq\mathbb{P}
  5. M M
  6. \mathbb{P}
  7. M M
  8. τ \tau
  9. η \eta
  10. τ \tau
  11. η \eta
  12. \mathbb{P}
  13. dom ( η ) dom ( τ ) \textrm{dom}(\eta)\subseteq\textrm{dom}(\tau)
  14. \mathbb{P}
  15. σ M \sigma\in M
  16. { p | σ , p η } \{p\in\mathbb{P}|\langle\sigma,p\rangle\in\eta\}
  17. σ , p η \langle\sigma,p\rangle\in\eta
  18. q p q\geq p
  19. \mathbb{P}
  20. σ , q τ \langle\sigma,q\rangle\in\tau

Nicola_Cabibbo.html

  1. | d , | s \scriptstyle{|d\rangle,\ |s\rangle}
  2. | d , | s \scriptstyle{|d^{\prime}\rangle,\ |s^{\prime}\rangle}

No-broadcast_theorem.html

  1. ρ 1 , \rho_{1},
  2. ρ A B \rho_{AB}
  3. H A H B H_{A}\otimes H_{B}
  4. T r A ρ A B = ρ 1 Tr_{A}\rho_{AB}=\rho_{1}
  5. T r B ρ A B = ρ 1 Tr_{B}\rho_{AB}=\rho_{1}
  6. ρ 1 . \rho_{1}.

No_Wit,_No_Help_Like_a_Woman's.html

  1. N o { W i t H e l p } L i k e a W o m a n s No\begin{Bmatrix}Wit\\ Help\end{Bmatrix}Like~{}a~{}Woman^{\prime}s

Nodec_space.html

  1. X X
  2. X X

Non-associative_algebra.html

  1. [ , , ] : A × A × A A [\cdot,\cdot,\cdot]:A\times A\times A\to A
  2. [ x , y , z ] = ( x y ) z - x ( y z ) . [x,y,z]=(xy)z-x(yz).\,
  3. A A
  4. [ x , y , x ] = 0 [x,y,x]=0
  5. [ x , y , x 2 ] = 0 [x,y,x^{2}]=0
  6. [ n , A , A ] = [ A , n , A ] = [ A , A , n ] = { 0 } . [n,A,A]=[A,n,A]=[A,A,n]=\{0\}\ .
  7. D ( x y ) = D ( x ) y + x D ( y ) . D(x\cdot y)=D(x)\cdot y+x\cdot D(y)\ .
  8. L ( a ) : x a x ; R ( a ) : x x a . L(a):x\mapsto ax;\ \ R(a):x\mapsto xa\ .
  9. Q ( a ) : x 2 a ( a x ) - ( a a ) x Q(a):x\mapsto 2a\cdot(a\cdot x)-(a\cdot a)\cdot x
  10. Q ( a ) = 2 L 2 ( a ) - L ( a 2 ) . Q(a)=2L^{2}(a)-L(a^{2})\ .

Non-exact_solutions_in_general_relativity.html

  1. g g
  2. γ \gamma
  3. h h
  4. h h
  5. η μ ν \eta_{\mu\nu}
  6. g μ ν = η μ ν + h μ ν + 𝒪 ( h 2 ) g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}+\mathcal{O}(h^{2})
  7. h h

Non-integer_representation.html

  1. x = d n d 2 d 1 d 0 . d - 1 d - 2 d - m x=d_{n}\dots d_{2}d_{1}d_{0}.d_{-1}d_{-2}\dots d_{-m}
  2. x = β n d n + + β 2 d 2 + β d 1 + d 0 + β - 1 d - 1 + β - 2 d - 2 + + β - m d - m . x=\beta^{n}d_{n}+\cdots+\beta^{2}d_{2}+\beta d_{1}+d_{0}+\beta^{-1}d_{-1}+% \beta^{-2}d_{-2}+\cdots+\beta^{-m}d_{-m}.
  3. d k = x / β k d_{k}=\lfloor x/\beta^{k}\rfloor
  4. r k = { x / β k } . r_{k}=\{x/\beta^{k}\}.\,
  5. d j = β r j + 1 , r j = { β r j + 1 } . d_{j}=\lfloor\beta r_{j+1}\rfloor,\quad r_{j}=\{\beta r_{j+1}\}.

Non-interference_(security).html

  1. M M
  2. M L M_{L}
  3. M R M_{R}
  4. M M
  5. = L {=_{L}}
  6. M = L M M\ {=_{L}}\ M^{\prime}
  7. M L = M L M_{L}=M_{L}^{\prime}
  8. ( P , M ) * M (P,M)\rightarrow^{*}M^{\prime}
  9. P P
  10. M M
  11. M M^{\prime}
  12. P P
  13. M 1 , M 2 : M 1 = L M 2 ( P , M 1 ) * M 1 ( P , M 2 ) * M 2 M 1 = L M 2 \begin{array}[]{rrl}\forall M_{1},M_{2}:&M_{1}\ {=_{L}}\ M_{2}&\land\\ &(P,M_{1})\rightarrow^{*}M_{1}^{\prime}&\land\\ &(P,M_{2})\rightarrow^{*}M_{2}^{\prime}&\Rightarrow\\ &M_{1}^{\prime}\ {=_{L}}\ M_{2}^{\prime}\end{array}

Non-logical_symbol.html

  1. {\mathbb{Z}}

Non-standard_model_of_arithmetic.html

  1. \mathbb{N}^{\mathbb{N}}

Nonclassical_light.html

  1. ρ ^ = φ ( α ) | α α | d 2 α , \widehat{\rho}=\int\varphi(\alpha)|{\alpha}\rangle\langle{\alpha}|\rm{d}^{2}\alpha,
  2. | α \scriptstyle|\alpha\rangle
  3. φ ( α ) \scriptstyle\varphi(\alpha)\,
  4. φ ( α ) \scriptstyle\varphi(\alpha)\,
  5. P ( α ) \scriptstyle P(\alpha)\,

Nonoblique_correction.html

  1. Z b b ¯ Zb\bar{b}

NOON_state.html

  1. | ψ NOON = | N a | 0 b + e i N θ | 0 a | N b 2 , |\psi\text{NOON}\rangle=\frac{|N\rangle_{a}|0\rangle_{b}+e^{iN\theta}|{0}% \rangle_{a}|{N}\rangle_{b}}{\sqrt{2}},\,
  2. A = | N , 0 0 , N | + | 0 , N N , 0 | . A=|N,0\rangle\langle 0,N|+|0,N\rangle\langle N,0|.\,
  3. A A
  4. π / N \pi/N
  5. Δ θ = Δ A | d A / d θ | = 1 N . \Delta\theta=\frac{\Delta A}{|d\langle A\rangle/d\theta|}=\frac{1}{N}.

Norm_variety.html

  1. { a 1 , , a n } \{a_{1},\dots,a_{n}\}
  2. p n - 1 - 1. p^{n-1}-1.
  3. s d ( V ) s_{d}(V)

Normal-inverse_Gaussian_distribution.html

  1. K j K_{j}
  2. μ + δ β / γ \mu+\delta\beta/\gamma
  3. δ α 2 / γ 3 \delta\alpha^{2}/\gamma^{3}
  4. 3 β / ( α δ γ ) 3\beta/(\alpha\sqrt{\delta\gamma})
  5. 3 ( 1 + 4 β 2 / α 2 ) / ( δ γ ) 3(1+4\beta^{2}/\alpha^{2})/(\delta\gamma)
  6. e μ z + δ ( γ - α 2 - ( β + z ) 2 ) e^{\mu z+\delta(\gamma-\sqrt{\alpha^{2}-(\beta+z)^{2}})}
  7. e i μ z + δ ( γ - α 2 - ( β + i z ) 2 ) e^{i\mu z+\delta(\gamma-\sqrt{\alpha^{2}-(\beta+iz)^{2}})}
  8. X 1 X_{1}
  9. X 2 X_{2}
  10. α \alpha
  11. β \beta
  12. μ 1 \mu_{1}
  13. δ 1 \delta_{1}
  14. μ 2 , \mu_{2},
  15. δ 2 \delta_{2}
  16. X 1 + X 2 X_{1}+X_{2}
  17. α , \alpha,
  18. β , \beta,
  19. μ 1 + μ 2 \mu_{1}+\mu_{2}
  20. δ 1 + δ 2 . \delta_{1}+\delta_{2}.
  21. N ( μ , σ 2 ) , N(\mu,\sigma^{2}),
  22. β = 0 , δ = σ 2 α , \beta=0,\delta=\sigma^{2}\alpha,
  23. α \alpha\rightarrow\infty
  24. W ( γ ) ( t ) = W ( t ) + γ t W^{(\gamma)}(t)=W(t)+\gamma t
  25. A t = inf { s > 0 : W ( γ ) ( s ) = δ t } . A_{t}=\inf\{s>0:W^{(\gamma)}(s)=\delta t\}.
  26. W ( β ) ( t ) = W ~ ( t ) + β t W^{(\beta)}(t)=\tilde{W}(t)+\beta t
  27. X t = W ( β ) ( A t ) X_{t}=W^{(\beta)}(A_{t})
  28. X ( t ) X(t)

Normal_convergence.html

  1. f n : S f_{n}:S\to\mathbb{C}
  2. n = 0 f n ( x ) \sum_{n=0}^{\infty}f_{n}(x)
  3. n = 0 f n := n = 0 sup S | f n ( x ) | < . \sum_{n=0}^{\infty}\|f_{n}\|:=\sum_{n=0}^{\infty}\sup_{S}|f_{n}(x)|<\infty.
  4. n = 0 | f n ( x ) | \sum_{n=0}^{\infty}|f_{n}(x)|
  5. f n ( x ) = { 1 / n , x = n 0 , x n . f_{n}(x)=\begin{cases}1/n,&x=n\\ 0,&x\neq n.\end{cases}
  6. n = 0 | f n ( x ) | \sum_{n=0}^{\infty}|f_{n}(x)|
  7. n = 0 f n | U \sum_{n=0}^{\infty}f_{n}\mid_{U}
  8. n = 0 f n U < \sum_{n=0}^{\infty}\|f_{n}\|_{U}<\infty
  9. U \|\cdot\|_{U}
  10. n = 0 f n | K \sum_{n=0}^{\infty}f_{n}\mid_{K}
  11. n = 0 f n ( x ) \sum_{n=0}^{\infty}f_{n}(x)
  12. f f
  13. τ : \tau:\mathbb{N}\to\mathbb{N}
  14. n = 0 f τ ( n ) ( x ) \sum_{n=0}^{\infty}f_{\tau(n)}(x)
  15. f f

Normal_coordinates.html

  1. exp p : T p M V M \exp_{p}:T_{p}M\supset V\rightarrow M
  2. E : n T p M E:\mathbb{R}^{n}\rightarrow T_{p}M
  3. φ := E - 1 exp p - 1 : U n \varphi:=E^{-1}\circ\exp_{p}^{-1}:U\rightarrow\mathbb{R}^{n}
  4. γ V \gamma_{V}
  5. γ V \gamma_{V}
  6. γ V ( t ) = ( t V 1 , , t V n ) \gamma_{V}(t)=(tV^{1},...,tV^{n})
  7. δ i j \delta_{ij}
  8. g i j g_{ij}
  9. / r \partial/\partial r
  10. d f , d r = f r \langle df,dr\rangle=\frac{\partial f}{\partial r}
  11. g = [ 1 0 0 0 g ϕ ϕ ( r , ϕ ) 0 ] . g=\begin{bmatrix}1&0&\cdots\ 0\\ 0&&\\ \vdots&&g_{\phi\phi}(r,\phi)\\ 0&&\end{bmatrix}.

Normal_variance-mean_mixture.html

  1. g g
  2. Y Y
  3. Y = α + β V + σ V X , Y=\alpha+\beta V+\sigma\sqrt{V}X,
  4. α \alpha
  5. β \beta
  6. σ > 0 \sigma>0
  7. X X
  8. V V
  9. X X
  10. V V
  11. g g
  12. Y Y
  13. V V
  14. α + β V \alpha+\beta V
  15. σ 2 V \sigma^{2}V
  16. β \beta
  17. σ 2 \sigma^{2}
  18. g g
  19. g g
  20. f ( x ) = 0 1 2 π σ 2 v exp ( - ( x - α - β v ) 2 2 σ 2 v ) g ( v ) d v f(x)=\int_{0}^{\infty}\frac{1}{\sqrt{2\pi\sigma^{2}v}}\exp\left(\frac{-(x-% \alpha-\beta v)^{2}}{2\sigma^{2}v}\right)g(v)\,dv
  21. M ( s ) = exp ( α s ) M g ( β s + 1 2 σ 2 s 2 ) , M(s)=\exp(\alpha s)\,M_{g}\left(\beta s+\frac{1}{2}\sigma^{2}s^{2}\right),
  22. M g M_{g}
  23. g g
  24. M g ( s ) = E ( exp ( s V ) ) = 0 exp ( s v ) g ( v ) d v . M_{g}(s)=E\left(\exp(sV)\right)=\int_{0}^{\infty}\exp(sv)g(v)\,dv.

Notation_for_theoretic_scheduling_problems.html

  1. m m
  2. m m
  3. i i
  4. j j
  5. p i p_{i}
  6. s j s_{j}
  7. m m
  8. p i j p_{ij}
  9. i i
  10. j j
  11. m m
  12. p i = p p_{i}=p
  13. p i j = p p_{ij}=p
  14. p i = 1 p_{i}=1
  15. p i j = 1 p_{ij}=1
  16. r i r_{i}
  17. d i d_{i}
  18. U i \sum U_{i}
  19. s i z e i size_{i}
  20. d i d_{i}
  21. C i C_{i}
  22. i i
  23. L i = C i - d i L_{i}=C_{i}-d_{i}
  24. E i = max { 0 , d i - C i } E_{i}=\max\{0,d_{i}-C_{i}\}
  25. T i = max { 0 , C i - d i } T_{i}=\max\{0,C_{i}-d_{i}\}
  26. U i = 0 U_{i}=0
  27. C i d i C_{i}\leq d_{i}
  28. U i = 1 U_{i}=1
  29. C max , L max , E max , T max , C i , L i , E i , T i C_{\max},L_{\max},E_{\max},T_{\max},\sum C_{i},\sum L_{i},\sum E_{i},\sum T_{i}
  30. w i w_{i}
  31. L max L_{\max}
  32. C i \sum C_{i}
  33. p i j p_{ij}
  34. C max C_{\max}

Novikov_ring.html

  1. Γ \Gamma\subset\mathbb{R}
  2. Nov ( Γ ) \operatorname{Nov}(\Gamma)
  3. Γ \Gamma
  4. [ [ Γ ] ] \mathbb{Z}[\![\Gamma]\!]
  5. n γ i t γ i \sum n_{\gamma_{i}}t^{\gamma_{i}}
  6. γ 1 > γ 2 > \gamma_{1}>\gamma_{2}>\cdots
  7. γ i - \gamma_{i}\to-\infty
  8. Nov ( Γ ) \operatorname{Nov}(\Gamma)
  9. [ Γ ] \mathbb{Z}[\Gamma]
  10. Nov ( Γ ) \operatorname{Nov}(\Gamma)
  11. Nov ( Γ ) [ S - 1 ] \operatorname{Nov}(\Gamma)[S^{-1}]
  12. Nov ( Γ ) \operatorname{Nov}(\Gamma)
  13. Nov ( Γ ) \operatorname{Nov}(\Gamma)
  14. Nov ( Γ ) \operatorname{Nov}(\Gamma)
  15. C * ( f ) C_{*}(f)
  16. C p C_{p}
  17. H * ( C * ( f ) ) H * ( M , 𝐙 ) H^{*}(C_{*}(f))\approx H^{*}(M,\mathbf{Z})
  18. ξ H 1 ( X , ) \xi\in H^{1}(X,\mathbb{R})
  19. H 1 ( X , ) H_{1}(X,\mathbb{R})
  20. ξ : π = π 1 ( X ) \xi:\pi=\pi_{1}(X)\to\mathbb{R}
  21. ϕ ξ : [ π ] Nov = Nov ( ) \phi_{\xi}:\mathbb{Z}[\pi]\to\operatorname{Nov}=\operatorname{Nov}(\mathbb{R})
  22. Nov \operatorname{Nov}
  23. [ π ] \mathbb{Z}[\pi]
  24. [ π ] \mathbb{Z}[\pi]
  25. L ξ L_{\xi}
  26. Nov \operatorname{Nov}
  27. ϕ ξ \phi_{\xi}
  28. H p ( X , L ξ ) H_{p}(X,L_{\xi})
  29. Nov \operatorname{Nov}
  30. b p ( ξ ) b_{p}(\xi)
  31. q p ( ξ ) q_{p}(\xi)
  32. ξ = 0 \xi=0
  33. L ξ L_{\xi}
  34. b p ( 0 ) b_{p}(0)
  35. [ [ Γ ] ] \mathbb{Z}[\![\Gamma]\!]
  36. γ Γ n γ t γ \sum_{\gamma\in\Gamma}n_{\gamma}t^{\gamma}
  37. n γ n_{\gamma}
  38. [ Γ ] \mathbb{Z}[\Gamma]

NOνA.html

  1. ν μ ν e \nu_{\mu}\rightarrow\nu_{e}
  2. ν μ ν e \nu_{\mu}\rightarrow\nu_{e}
  3. ν ¯ μ ν ¯ e . \bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}.

Nuclear_magnetic_resonance_quantum_computer.html

  1. ρ = e - β H Tr ( e - β H ) , \rho=\frac{e^{-\beta H}}{\operatorname{Tr}(e^{-\beta H})},
  2. β = 1 k T \beta=\frac{1}{k\,T}
  3. k k
  4. T T

Nucleon_spin_structure.html

  1. 1 2 = 1 2 Σ q + Σ g + L q + L g . \frac{1}{2}=\frac{1}{2}\Sigma_{q}+\Sigma_{g}+L_{q}+L_{g}.
  2. Σ g \Sigma_{g}

Nuna_3.html

  1. \approx

Nurgaliev's_law.html

  1. d n d t = a n 2 - b n , {dn\over dt}=an^{2}-bn,
  2. n = 0 , b / a n=0,b/a
  3. n = 0 n=0
  4. n = b / a n=b/a

O_(disambiguation).html

  1. 𝒪 \mathcal{O}

Observability_Gramian.html

  1. x ˙ ( t ) = A ( t ) x ( t ) + B ( t ) u ( t ) \dot{x}(t)=A(t)x(t)+B(t)u(t)
  2. y ( t ) = C ( t ) x ( t ) + D ( t ) u ( t ) y(t)=C(t)x(t)+D(t)u(t)\,
  3. W o ( t 0 , t 1 ) = t 0 t 1 Φ T ( s , t 0 ) C T ( s ) C ( s ) Φ ( s , t 0 ) d s W_{o}(t_{0},t_{1})=\int_{t_{0}}^{t_{1}}\Phi^{T}(s,t_{0})C^{T}(s)C(s)\Phi(s,t_{% 0})ds
  4. Φ \Phi
  5. t [ t 0 , t 1 ] t\in[t_{0},t_{1}]
  6. W o ( t 0 , t 1 ) W_{o}(t_{0},t_{1})
  7. x ( t ) x(t)
  8. n n
  9. rank [ C T , A T C T , , ( A T ) n - 1 C T ] = n \,\text{rank}[C^{T},A^{T}C^{T},...,(A^{T})^{n-1}C^{T}]=n

Octacontagon.html

  1. A = 20 t 2 cot π 80 A=20t^{2}\cot\frac{\pi}{80}
  2. r = 1 2 t cot π 80 r=\frac{1}{2}t\cot\frac{\pi}{80}
  3. R = 1 2 t csc π 80 R=\frac{1}{2}t\csc\frac{\pi}{80}
  4. sin π 80 = sin 2.25 = 1 8 ( 1 + 5 ) ( - 2 - 2 + 2 - ( 2 + 2 ) ( 2 - 2 + 2 ) ) \sin\frac{\pi}{80}=\sin 2.25^{\circ}=\frac{1}{8}(1+\sqrt{5})(-\sqrt{2-\sqrt{2+% \sqrt{2}}}-\sqrt{(2+\sqrt{2})(2-\sqrt{2+\sqrt{2}})})
  5. + 1 4 1 2 ( 5 - 5 ) ( ( 2 + 2 ) ( 2 + 2 + 2 ) - 2 + 2 + 2 ) +\frac{1}{4}\sqrt{\frac{1}{2}(5-\sqrt{5})}(\sqrt{(2+\sqrt{2})(2+\sqrt{2+\sqrt{% 2}})}-\sqrt{2+\sqrt{2+\sqrt{2}}})
  6. cos π 80 = cos 2.25 = 1 2 + 1 4 1 2 ( 4 + 2 ( 4 + 2 ( 5 + 5 ) ) ) \cos\frac{\pi}{80}=\cos 2.25^{\circ}=\sqrt{\frac{1}{2}+\frac{1}{4}\sqrt{\frac{% 1}{2}(4+\sqrt{2(4+\sqrt{2(5+\sqrt{5})})})}}

OFFSystem.html

  1. t t
  2. s s
  3. s i s_{i}
  4. t - 1 t-1
  5. o i = s i r 1 r 2 r t - 1 o_{i}=s_{i}\oplus r_{1}\oplus r_{2}\oplus...\oplus r_{t-1}
  6. o i o_{i}
  7. \oplus
  8. s i s_{i}
  9. t t
  10. { o n , r 1 , r 2 r t - 1 } \{o_{n},r_{1},r_{2}...r_{t-1}\}
  11. d i d_{i}
  12. t t
  13. b 1 , b 2 b t b_{1},b_{2}...b_{t}
  14. o i , r 1 , r 2 r t - 1 o_{i},r_{1},r_{2}...r_{t-1}
  15. s i = b 1 b 2 b t s_{i}=b_{1}\oplus b_{2}\oplus...\oplus b_{t}
  16. s i s_{i}
  17. s * ( h i + h o + 1 ) * 2 - s s*(hi+ho+1)*2-s
  18. s s
  19. h i hi
  20. h o ho
  21. o i o_{i}
  22. s * ( t - 1 ) * e 100 s*(t-1)*\frac{e}{100}
  23. s s
  24. t t
  25. e e
  26. e e
  27. t t

Oh-My-God_particle.html

  1. 2 E m c 2 \sqrt{2Emc^{2}}
  2. E E
  3. m c 2 mc^{2}

Ohnesorge_number.html

  1. Oh = μ ρ σ L = We Re viscous forces inertia surface tension \mathrm{Oh}=\frac{\mu}{\sqrt{\rho\sigma L}}=\frac{\sqrt{\mathrm{We}}}{\mathrm{% Re}}\sim\frac{\mbox{viscous forces}~{}}{\sqrt{{\mbox{inertia}~{}}\cdot{\mbox{% surface tension}~{}}}}
  2. Oh = 1 / La \mathrm{Oh}=1/\sqrt{\mathrm{La}}

Okamoto–Uchiyama_cryptosystem.html

  1. ( / n ) * (\mathbb{Z}/n\mathbb{Z})^{*}
  2. ( / n ) * (\mathbb{Z}/n\mathbb{Z})^{*}
  3. n = p 2 q n=p^{2}q
  4. g ( / n ) * g\in(\mathbb{Z}/n\mathbb{Z})^{*}
  5. g p 1 mod p 2 g^{p}\neq 1\mod p^{2}
  6. / p \mathbb{Z}/p\mathbb{Z}
  7. r / n r\in\mathbb{Z}/n\mathbb{Z}
  8. C = g m h r mod n C=g^{m}h^{r}\mod n
  9. L ( x ) = x - 1 p L(x)=\frac{x-1}{p}
  10. m = L ( C p - 1 mod p 2 ) L ( g p - 1 mod p 2 ) mod p m=\frac{L\left(C^{p-1}\mod p^{2}\right)}{L\left(g^{p-1}\mod p^{2}\right)}\mod p
  11. ( \Z / n \Z ) * ( / p 2 ) * × ( / q ) * (\Z/n\Z)^{*}\simeq(\mathbb{Z}/p^{2}\mathbb{Z})^{*}\times(\mathbb{Z}/q\mathbb{Z% })^{*}
  12. ( / p 2 ) * (\mathbb{Z}/p^{2}\mathbb{Z})^{*}
  13. H = { x : x p 1 mod p } H=\{x:x^{p}\equiv 1\mod p\}
  14. ( / p 2 ) * (\mathbb{Z}/p^{2}\mathbb{Z})^{*}
  15. / p \mathbb{Z}/p\mathbb{Z}
  16. ( g m h r ) p - 1 = ( g m g n r ) p - 1 = ( g p - 1 ) m g p ( p - 1 ) r p q = ( g p - 1 ) m mod p 2 . (g^{m}h^{r})^{p-1}=(g^{m}g^{nr})^{p-1}=(g^{p-1})^{m}g^{p(p-1)rpq}=(g^{p-1})^{m% }\mod p^{2}.
  17. L ( ( g p - 1 ) m ) L ( g p - 1 ) = m mod p . \frac{L\left((g^{p-1})^{m}\right)}{L(g^{p-1})}=m\mod p.
  18. ( / n ) * (\mathbb{Z}/n\mathbb{Z})^{*}

Okishio's_theorem.html

  1. x 1 x_{1}
  2. x 2 x_{2}
  3. a 11 a_{11}
  4. a 21 a_{21}
  5. a 12 a_{12}
  6. a 22 a_{22}
  7. w a 21 w\cdot a_{21}
  8. w a 22 w\cdot a_{22}
  9. x 1 x_{1}
  10. x 2 x_{2}
  11. a 11 x 1 a_{11}x_{1}
  12. a 21 w x 1 a_{21}wx_{1}
  13. x 1 x_{1}
  14. a 12 x 2 a_{12}x_{2}
  15. a 22 w x 2 a_{22}wx_{2}
  16. x 2 x_{2}
  17. ( a 11 x 1 p 1 + a 21 w x 1 p 2 ) ( 1 + r ) = x 1 p 1 (a_{11}x_{1}p_{1}+a_{21}wx_{1}p_{2})(1+r)=x_{1}p_{1}
  18. ( a 12 x 2 p 1 + a 22 w x 2 p 2 ) ( 1 + r ) = x 2 p 2 (a_{12}x_{2}p_{1}+a_{22}wx_{2}p_{2})(1+r)=x_{2}p_{2}
  19. p 1 p_{1}
  20. x 1 x_{1}
  21. p 2 p_{2}
  22. x 2 x_{2}
  23. r r
  24. a 11 x 1 p 1 a_{11}x_{1}p_{1}
  25. a 21 w x 1 p 2 a_{21}wx_{1}p_{2}
  26. a 12 x 2 p 1 a_{12}x_{2}p_{1}
  27. a 22 w x 2 p 2 . a_{22}wx_{2}p_{2}.
  28. p 2 = 1 p_{2}=1
  29. x 2 x_{2}
  30. p 2 p_{2}
  31. w = 2 p 2 = 2. w=2p_{2}=2.
  32. x 1 x_{1}
  33. x 2 x_{2}
  34. a 11 = 0.8 a_{11}=0.8
  35. a 21 = 0.1 a_{21}=0.1
  36. a 12 = 0.4 a_{12}=0.4
  37. a 22 = 0.1 a_{22}=0.1
  38. p 1 = 1.78 p_{1}=1.78
  39. r = 0.0961 = 9.61 % r=0.0961=9.61\%
  40. ( 0.8 1 1.78 + 0.1 2 1 1 ) ( 1 + 0.0961 ) = 1 1.78 (0.8\cdot 1\cdot 1.78+0.1\cdot 2\cdot 1\cdot 1)\cdot(1+0.0961)=1\cdot 1.78
  41. ( 0.4 1 1.78 + 0.1 2 1 1 ) ( 1 + 0.0961 ) = 1 1 (0.4\cdot 1\cdot 1.78+0.1\cdot 2\cdot 1\cdot 1)\cdot(1+0.0961)=1\cdot 1
  42. ( a 11 x 1 p 1 + a 21 w x 1 p 2 ) ( 1 + r ) = x 1 p 1 (a_{11}x_{1}p_{1}+a_{21}wx_{1}p_{2})(1+r)=x_{1}p_{1}
  43. = ( 0.8 1 1.78 + 0 , 1 2 1 1 ) ( 1 + 0.0961 ) = 1 1.78 =(0.8\cdot 1\cdot 1.78+0{,}1\cdot 2\cdot 1\cdot 1)\cdot(1+0.0961)=1\cdot 1.78
  44. a 21 = 0.1 a_{21}=0.1
  45. a 21 = 0.05 a_{21}=0.05
  46. a 11 = 0.8 a_{11}=0.8
  47. a 11 = 0.85 a_{11}=0.85
  48. = ( 0.85 1 1.78 + 0.05 2 1 1 ) ( 1 + 0.1036 ) = 1 1.78 =(0.85\cdot 1\cdot 1.78+0.05\cdot 2\cdot 1\cdot 1)\cdot(1+0.1036)=1\cdot 1.78
  49. r = 9 , 61 % r=9{,}61\%
  50. 10 , 36 % 10{,}36\%
  51. p 2 p_{2}
  52. ( 0.85 1 1.77 + 0.05 2 1 1 ) ( 1 + 0.1030 ) = 1 1.77 (0.85\cdot 1\cdot 1.77+0.05\cdot 2\cdot 1\cdot 1)\cdot(1+0.1030)=1\cdot 1.77
  53. ( 0.4 1 1.77 + 0.1 2 1 1 ) ( 1 + 0.1030 ) = 1 1 (0.4\cdot 1\cdot 1.77+0.1\cdot 2\cdot 1\cdot 1)\cdot(1+0.1030)=1\cdot 1
  54. 10.36 % 10.36\%
  55. 10.30 % 10.30\%
  56. 9.61 % 9.61\%
  57. 58.6 % 58.6\%
  58. 41.9 % 41.9\%
  59. x 2 x_{2}
  60. p 2 p_{2}
  61. x 2 x_{2}
  62. x 1 x_{1}
  63. ( a 11 x 1 + K a 12 x 2 ) ( 1 + g ) = x 1 (a_{11}x_{1}+Ka_{12}x_{2})(1+g)=x_{1}
  64. ( a 21 w x 1 + K a 22 w x 2 ) ( 1 + g ) = K x 2 (a_{21}w\cdot x_{1}+Ka_{22}\cdot wx_{2})(1+g)=Kx_{2}
  65. r = 9 , 61 % r=9,61\%
  66. ( 0.8 1 + 0 , 2808 0.4 1 ) ( 1 + 0.0961 ) = 1 (0.8\cdot 1+0{,}2808\cdot 0.4\cdot 1)\cdot(1+0.0961)=1
  67. ( 0.1 2 1 + 0.2808 0.1 2 1 ) ( 1 + 0.0961 ) = 0.2808 1 (0.1\cdot 2\cdot 1+0.2808\cdot 0.1\cdot 2\cdot 1)\cdot(1+0.0961)=0.2808\cdot 1
  68. K = 0.2808 K=0.2808
  69. r = 10.30 % r=10.30\%
  70. ( 0.85 1 + 0.14154 0.4 1 ) ( 1 + 0.1030 ) = 1 (0.85\cdot 1+0.14154\cdot 0.4\cdot 1)\cdot(1+0.1030)=1
  71. ( 0.1 2 1 + 0.14154 0.05 2 1 ) ( 1 + 0.1030 ) = 0.14154 1 (0.1\cdot 2\cdot 1+0.14154\cdot 0.05\cdot 2\cdot 1)(1+0.1030)=0.14154\cdot 1
  72. K = 0.14154 K=0.14154
  73. x 1 x_{1}
  74. x 2 x_{2}
  75. x 1 x_{1}
  76. x 2 x_{2}
  77. x 1 x_{1}
  78. p 1 p_{1}
  79. x 2 x_{2}
  80. p 2 = 1 p_{2}=1
  81. x 1 x_{1}
  82. x 2 x_{2}
  83. p 1 p_{1}
  84. p 2 p_{2}
  85. 58.6 % 58.6\%
  86. 41.9 % 41.9\%
  87. Rate of profit p = s v c v + 1 \,\text{Rate of profit }p={{s\over v}\over{{c\over v}+1}}
  88. 9.61 % 9.61\%
  89. 10.30 % 10.30\%

Okumura_model.html

  1. L = L FSL + A MU - H MG - H BG - K correction L\;=\;L\text{FSL}\;+\;A\text{MU}\;-\;H\text{MG}\;-\;H\text{BG}\;-\;\sum{K\text% {correction}}\;
  2. L 50 % ( d B ) = L F + A m u ( f , d ) - G ( h t e ) - G ( h r e ) - G a r e a L_{50\%}(dB)=LF+A_{mu}(f,d)-G(h_{te})-G(h_{re})-G_{area}

Ole_Barndorff-Nielsen.html

  1. p * p^{*}

Oleg_Firsov.html

  1. E E
  2. U ( r ) U(r)
  3. w = 1 - U E . w=\sqrt{1-\frac{U}{E}}.
  4. θ ( b ) \theta(b)
  5. b b
  6. w = exp ( - 1 π r w θ ( b ) d b b 2 - r 2 w 2 ) . w=\exp\left(-\frac{1}{\pi}\int_{rw}^{\infty}\frac{\theta(b)\,db}{\sqrt{b^{2}-r% ^{2}w^{2}}}\right).

Omnibus_test.html

  1. F = j = 1 k n j ( y ¯ j - y ¯ ) 2 / ( k - 1 ) j = 1 k i = 1 n j ( y i j - y ¯ j ) 2 / ( n - k ) F=\tfrac{{\displaystyle\sum_{j=1}^{k}n_{j}\left(\bar{y}_{j}-\bar{y}\right)^{2}% }/{(k-1)}}{{\displaystyle{\sum_{j=1}^{k}}{\sum_{i=1}^{n_{j}}}\left(y_{ij}-\bar% {y}_{j}\right)^{2}}/{(n-k)}}
  2. y ¯ \bar{y}
  3. y ¯ j \bar{y}_{j}
  4. F = i = 1 n ( y i ^ - y ¯ ) 2 / k j = 1 k i = 1 n j ( y i j - y i ^ ) 2 / ( n - k - 1 ) F=\frac{{\displaystyle\sum_{i=1}^{n}\left(\widehat{y_{i}}-\bar{y}\right)^{2}}/% {k}}{{\displaystyle{\sum_{j=1}^{k}}{\sum_{i=1}^{n_{j}}}\left(y_{ij}-\widehat{y% _{i}}\right)^{2}}/{(n-k-1)}}
  5. P ( y i ) = e β 0 + β 1 x i 1 + + β k x i k 1 + e β 0 + β 1 x i 1 + + β k x i k = 1 1 + e - ( β 0 + β 1 x i 1 + + β k x i k ) P(y_{i})=\frac{e^{\beta_{0}+\beta_{1}x_{i1}+\cdot+\beta_{k}x_{ik}}}{1+e^{\beta% _{0}+\beta_{1}x_{i1}+\cdot+\beta_{k}x_{ik}}}=\frac{1}{1+e^{-(\beta_{0}+\beta_{% 1}x_{i1}+\cdot+\beta_{k}x_{ik})}}
  6. f ( y i ) = l n P ( y i ) 1 - P ( y i ) = β 0 + β 1 x i 1 + + β k x i k f(y_{i})=ln\frac{P(y_{i})}{1-P(y_{i})}=\beta_{0}+\beta_{1}x_{i1}+\cdot+\beta_{% k}x_{ik}
  7. λ ( y i ) = L ( y i | θ 0 ) L ( y i | θ 1 ) \lambda(y_{i})=\frac{L(y_{i}|\theta_{0})}{L(y_{i}|\theta_{1})}
  8. λ ( y i ) > C \lambda(y_{i})>C
  9. λ ( y i ) < C \lambda(y_{i})<C
  10. λ ( y i ) = C \lambda(y_{i})=C
  11. q ( P ( λ ( y i ) = C | H 0 ) + ( P ( λ ( y i ) < C | H 0 ) q\cdot(P(\lambda(y_{i})=C|H_{0})+(P(\lambda(y_{i})<C|H_{0})
  12. D = - 2 l n λ ( y i ) D=-2ln\lambda(y_{i})
  13. D = - 2 l n λ ( y i ) = - 2 l n l i k e l i h o o d u n d e r f i t t e d m o d e l i f n u l l h y p o t h e s i s i s t r u e l i k e l i h o o d u n d e r s a t u r a t e d m o d e l D=-2ln\lambda(y_{i})=-2ln\frac{likelihood\ under\ fitted\ model\ if\ null\ % hypothesis\ is\ true}{likelihood\ under\ saturated\ model\ }

One_woodland_terminal_model.html

  1. A v = A [ 1 - e - d γ A ] A_{v}\;=\;A\;[1\;-\;e^{-}{\frac{d\gamma}{A}}]
  2. γ \gamma\;
  3. γ \gamma

Oort_constants.html

  1. A A
  2. B B
  3. A = 1 2 ( V 0 R 0 - d v d r | R 0 ) \displaystyle A=\frac{1}{2}\left(\frac{V_{0}}{R_{0}}-\frac{dv}{dr}|_{R_{0}}\right)
  4. V 0 V_{0}
  5. R 0 R_{0}
  6. A A
  7. B B
  8. A A
  9. B B
  10. l l
  11. d d
  12. R R
  13. R 0 R_{0}
  14. V V
  15. V 0 V_{0}
  16. V obs, r = V star, r - V sun, r = V cos ( α ) - V 0 sin ( l ) \displaystyle V_{\,\text{obs, r}}=V_{\,\text{star, r}}-V_{\,\text{sun, r}}=V% \cos\left(\alpha\right)-V_{0}\sin\left(l\right)
  17. v = Ω r v=\Omega r
  18. V obs, r = Ω R cos ( α ) - Ω 0 R 0 sin ( l ) \displaystyle V_{\,\text{obs, r}}=\Omega R\cos\left(\alpha\right)-\Omega_{0}R_% {0}\sin\left(l\right)
  19. R cos ( α ) = R 0 sin ( l ) \displaystyle R\cos\left(\alpha\right)=R_{0}\sin\left(l\right)
  20. V obs, r = ( Ω - Ω 0 ) R 0 sin ( l ) V obs, t = ( Ω - Ω 0 ) R 0 cos ( l ) - Ω d \begin{aligned}&\displaystyle V_{\,\text{obs, r}}=\left(\Omega-\Omega_{0}% \right)R_{0}\sin\left(l\right)\\ &\displaystyle V_{\,\text{obs, t}}=\left(\Omega-\Omega_{0}\right)R_{0}\cos% \left(l\right)-\Omega d\\ \end{aligned}
  21. l l
  22. d d
  23. Ω - Ω 0 \Omega-\Omega_{0}
  24. R 0 R_{0}
  25. ( Ω - Ω 0 ) = ( R - R 0 ) d Ω d r | R 0 + \left(\Omega-\Omega_{0}\right)=\left(R-R_{0}\right)\frac{d\Omega}{dr}|_{R_{0}}% +...
  26. R - R 0 R-R_{0}
  27. R R
  28. R 0 R_{0}
  29. R - R 0 = - d cos ( l ) R-R_{0}=-d\cdot\cos\left(l\right)
  30. V obs, r = - R 0 d Ω d r | R 0 d cos ( l ) sin ( l ) \displaystyle V_{\,\text{obs, r}}=-R_{0}\frac{d\Omega}{dr}|_{R_{0}}d\cdot\cos% \left(l\right)\sin\left(l\right)
  31. V obs, r = - R 0 d Ω d r | R 0 d sin ( 2 l ) 2 \displaystyle V_{\,\text{obs, r}}=-R_{0}\frac{d\Omega}{dr}|_{R_{0}}d\frac{\sin% \left(2l\right)}{2}
  32. A A
  33. B B
  34. V obs, r = A d sin ( 2 l ) \displaystyle V_{\,\text{obs, r}}=Ad\sin\left(2l\right)
  35. A = - 1 2 R 0 d Ω d r | R 0 B = - 1 2 R 0 d Ω d r | R 0 - Ω \begin{aligned}&\displaystyle A=-\frac{1}{2}R_{0}\frac{d\Omega}{dr}|_{R_{0}}\\ &\displaystyle B=-\frac{1}{2}R_{0}\frac{d\Omega}{dr}|_{R_{0}}-\Omega\\ \end{aligned}
  36. Ω = v r \displaystyle\Omega=\frac{v}{r}
  37. A = 1 2 ( V 0 R 0 - d v d r | R 0 ) B = - 1 2 ( V 0 R 0 + d v d r | R 0 ) \begin{aligned}&\displaystyle A=\frac{1}{2}\left(\frac{V_{0}}{R_{0}}-\frac{dv}% {dr}|_{R_{0}}\right)\\ &\displaystyle B=-\frac{1}{2}\left(\frac{V_{0}}{R_{0}}+\frac{dv}{dr}|_{R_{0}}% \right)\\ \end{aligned}
  38. A A
  39. B B
  40. A A
  41. B B
  42. V obs, r = A d sin ( 2 l ) \displaystyle V_{\,\text{obs, r}}=A\,d\,\sin\left(2l\right)
  43. A A
  44. B B
  45. A = V obs, r d sin ( 2 l ) \displaystyle A=\frac{V_{\,\text{obs, r}}}{d\,\sin\left(2l\right)}
  46. A A
  47. B B
  48. A A
  49. B B
  50. A A
  51. B B
  52. A A
  53. B B
  54. A A
  55. B B
  56. A A
  57. B B
  58. A A
  59. B B
  60. A A
  61. B B
  62. A A
  63. B B
  64. A A
  65. B B
  66. R R
  67. Ω \Omega
  68. R R
  69. R R
  70. v r v\propto r
  71. d v d r = v r = Ω \begin{aligned}&\displaystyle\frac{dv}{dr}=\frac{v}{r}=\Omega\\ \end{aligned}
  72. A A
  73. B B
  74. A = 1 2 ( Ω 0 R 0 R 0 - Ω | R 0 ) = 0 B = - 1 2 ( Ω 0 R 0 R 0 + Ω | R 0 ) = - Ω 0 \begin{aligned}&\displaystyle A=\frac{1}{2}\left(\frac{\Omega_{0}R_{0}}{R_{0}}% -{\Omega}|_{R_{0}}\right)=0\\ &\displaystyle B=-\frac{1}{2}\left(\frac{\Omega_{0}R_{0}}{R_{0}}+{\Omega}|_{R_% {0}}\right)=-\Omega_{0}\\ \end{aligned}
  75. A = 0 A=0
  76. B = - Ω B=-\Omega
  77. A = 14 A=14
  78. v = G M r v=\sqrt{\frac{GM}{r}}
  79. G G
  80. M M
  81. r r
  82. d v d r = - 1 2 G M R 3 = - 1 2 v r \frac{dv}{dr}=-\frac{1}{2}\sqrt{\frac{GM}{R^{3}}}=-\frac{1}{2}\frac{v}{r}
  83. A = 1 2 ( V 0 R 0 + v 2 r | R 0 ) = 3 V 0 4 R 0 B = - 1 2 ( V 0 R 0 - v 2 r | R 0 ) = - 1 V 0 4 R 0 \begin{aligned}&\displaystyle A=\frac{1}{2}\left(\frac{V_{0}}{R_{0}}+\frac{v}{% 2r}|_{R_{0}}\right)=\frac{3V_{0}}{4R_{0}}\\ &\displaystyle B=-\frac{1}{2}\left(\frac{V_{0}}{R_{0}}-\frac{v}{2r}|_{R_{0}}% \right)=-\frac{1V_{0}}{4R_{0}}\\ \end{aligned}
  84. V 0 = 218 V_{0}=218
  85. R 0 = 8 R_{0}=8
  86. A = 20 A=20
  87. B = - 7 B=-7
  88. A = 14 A=14
  89. B = - 12 B=-12
  90. v v
  91. r r
  92. v v
  93. d v d r = 0 \frac{dv}{dr}=0
  94. A = 1 2 ( V 0 R 0 - 0 | R 0 ) = 1 2 ( V 0 R 0 ) B = - 1 2 ( V 0 R 0 + 0 | R 0 ) = - 1 2 ( V 0 R 0 ) \begin{aligned}&\displaystyle A=\frac{1}{2}\left(\frac{V_{0}}{R_{0}}-0|_{R_{0}% }\right)=\frac{1}{2}\left(\frac{V_{0}}{R_{0}}\right)\\ &\displaystyle B=-\frac{1}{2}\left(\frac{V_{0}}{R_{0}}+0|_{R_{0}}\right)=-% \frac{1}{2}\left(\frac{V_{0}}{R_{0}}\right)\\ \end{aligned}
  95. A = 13.6 A=13.6
  96. B = - 13.6 B=-13.6
  97. V 0 = R 0 ( A - B ) V_{0}=R_{0}(A-B)\,\!
  98. A A
  99. B B
  100. ρ R \rho_{R}
  101. ρ R = B 2 - A 2 2 π G \rho_{R}=\frac{B^{2}-A^{2}}{2\pi G}
  102. κ \kappa
  103. κ 2 = - 4 B Ω \kappa^{2}=-4B\Omega
  104. Ω \Omega

Operational_transconductance_amplifier.html

  1. I out = ( V in + - V in - ) g m I_{\mathrm{out}}=(V_{\mathrm{in+}}-V_{\mathrm{in-}})\cdot g_{\mathrm{m}}
  2. V out = I out R load V_{\mathrm{out}}=I_{\mathrm{out}}\cdot R_{\mathrm{load}}
  3. G voltage = V out ( V in + - V in - ) = R load g m G_{\mathrm{voltage}}={V_{\mathrm{out}}\over(V_{\mathrm{in+}}-V_{\mathrm{in-}})% }=R_{\mathrm{load}}\cdot g_{\mathrm{m}}

Optical_equivalence_theorem.html

  1. g Ω ( a ^ , a ^ ) g_{\Omega}(\hat{a},\hat{a}^{\dagger})
  2. g Ω ( a ^ , a ^ ) = g Ω ( α , α * ) . \langle g_{\Omega}(\hat{a},\hat{a}^{\dagger})\rangle=\langle g_{\Omega}(\alpha% ,\alpha^{*})\rangle.
  3. ρ ^ \hat{\rho}
  4. Ω ¯ {\bar{\Omega}}
  5. f Ω ¯ ( α , α * ) = 1 π ρ Ω ¯ ( α , α * ) | α α | d 2 α . f_{\bar{\Omega}}(\alpha,\alpha^{*})=\frac{1}{\pi}\int\rho_{\bar{\Omega}}(% \alpha,\alpha^{*})|\alpha\rangle\langle\alpha|\,d^{2}\alpha.
  6. tr ( ρ ^ g Ω ( a ^ , a ^ ) ) = f Ω ¯ ( α , α * ) g Ω ( α , α * ) d 2 α . \mathrm{tr}(\hat{\rho}\cdot g_{\Omega}(\hat{a},\hat{a}^{\dagger}))=\int f_{% \bar{\Omega}}(\alpha,\alpha^{*})g_{\Omega}(\alpha,\alpha^{*})\,d^{2}\alpha.
  7. g N ( a ^ , a ^ ) = n , m c n m a ^ n a ^ m . g_{N}(\hat{a}^{\dagger},\hat{a})=\sum_{n,m}c_{nm}\hat{a}^{\dagger n}\hat{a}^{m}.
  8. tr ( ρ ^ g N ( a ^ , a ^ ) ) = P ( α ) g ( α , α * ) d 2 α . \mathrm{tr}(\hat{\rho}\cdot g_{N}(\hat{a},\hat{a}^{\dagger}))=\int P(\alpha)g(% \alpha,\alpha^{*})\,d^{2}\alpha.

Optical_scalars.html

  1. { θ ^ \{\hat{\theta}
  2. σ ^ \hat{\sigma}
  3. ω ^ \hat{\omega}
  4. } \}
  5. { θ ^ , σ ^ , ω ^ } \{\hat{\theta}\,,\hat{\sigma}\,,\hat{\omega}\}
  6. { θ ^ h ^ a b , σ ^ a b , ω ^ a b } \{\hat{\theta}\hat{h}_{ab}\,,\hat{\sigma}_{ab}\,,\hat{\omega}_{ab}\}
  7. { θ ^ , σ ^ , ω ^ } \{\hat{\theta}\,,\hat{\sigma}\,,\hat{\omega}\}
  8. Z a Z^{a}
  9. ( 1 ) h a b = g a b + Z a Z b , h a b = g a b + Z a Z b , h b a = g b a + Z a Z b , (1)\quad h^{ab}=g^{ab}+Z^{a}Z^{b}\;,\quad h_{ab}=g_{ab}+Z_{a}Z_{b}\;,\quad h^{% a}_{\;\;b}=g^{a}_{\;\;b}+Z^{a}Z_{b}\;,
  10. h b a h^{a}_{\;\;b}
  11. h b a h^{a}_{\;\;b}
  12. b Z a \nabla_{b}Z_{a}
  13. B a b B_{ab}
  14. ( 2 ) B a b = h a c h b d d Z c = b Z a + A a Z b , (2)\quad B_{ab}=h^{c}_{\;\;a}\,h^{d}_{\;\;b}\,\nabla_{d}Z_{c}=\nabla_{b}Z_{a}+% A_{a}Z_{b}\;,
  15. A a A_{a}
  16. B a b B_{ab}
  17. B a b Z a = B a b Z b = 0 B_{ab}Z^{a}=B_{ab}Z^{b}=0
  18. ( 3 ) A a = 0 , B a b = b Z a . (3)\quad A_{a}=0\;,\quad\Rightarrow\quad B_{ab}=\nabla_{b}Z_{a}\;.
  19. B a b B_{ab}
  20. θ a b \theta_{ab}
  21. ω a b \omega_{ab}
  22. ( 4 ) θ a b = B ( a b ) , ω a b = B [ a b ] . (4)\quad\theta_{ab}=B_{(ab)}\;,\quad\omega_{ab}=B_{[ab]}\;.
  23. ω a b = B [ a b ] \omega_{ab}=B_{[ab]}
  24. g a b ω a b = 0 g^{ab}\omega_{ab}=0
  25. θ a b \theta_{ab}
  26. g a b θ a b = θ g^{ab}\theta_{ab}=\theta
  27. θ a b \theta_{ab}
  28. ( 5 ) θ a b = 1 3 θ h a b + σ a b . (5)\quad\theta_{ab}=\frac{1}{3}\theta h_{ab}+\sigma_{ab}\;.
  29. ( 6 ) B a b = 1 3 θ h a b + σ a b + ω a b , θ = g a b θ a b = g a b B ( a b ) , σ a b = θ a b - 1 3 θ h a b , ω a b = B [ a b ] . (6)\quad B_{ab}=\frac{1}{3}\theta h_{ab}+\sigma_{ab}+\omega_{ab}\;,\quad\theta% =g^{ab}\theta_{ab}=g^{ab}B_{(ab)}\;,\quad\sigma_{ab}=\theta_{ab}-\frac{1}{3}% \theta h_{ab}\;,\quad\omega_{ab}=B_{[ab]}\;.
  30. k a k^{a}
  31. ( 7 ) B ^ a b := b k a , (7)\quad\hat{B}_{ab}:=\nabla_{b}k_{a}\;,
  32. ( 8 ) B ^ a b = θ ^ a b + ω ^ a b = 1 2 θ ^ h ^ a b + σ ^ a b + ω ^ a b , (8)\quad\hat{B}_{ab}=\hat{\theta}_{ab}+\hat{\omega}_{ab}=\frac{1}{2}\hat{% \theta}\hat{h}_{ab}+\hat{\sigma}_{ab}+\hat{\omega}_{ab}\;,
  33. ( 9 ) θ ^ a b = B ^ ( a b ) , θ ^ = h ^ a b B ^ a b , σ ^ a b = B ^ ( a b ) - 1 2 θ ^ h ^ a b , ω ^ a b = B ^ [ a b ] . (9)\quad\hat{\theta}_{ab}=\hat{B}_{(ab)}\;,\quad\hat{\theta}=\hat{h}^{ab}\hat{% B}_{ab}\;,\quad\hat{\sigma}_{ab}=\hat{B}_{(ab)}-\frac{1}{2}\hat{\theta}\hat{h}% _{ab}\;,\quad\hat{\omega}_{ab}=\hat{B}_{[ab]}\;.
  34. { θ ^ , σ ^ , ω ^ } \{\hat{\theta}\,,\hat{\sigma}\,,\hat{\omega}\}
  35. { θ ^ , σ ^ a b , ω ^ a b } \{\hat{\theta}\,,\hat{\sigma}_{ab}\,,\hat{\omega}_{ab}\}
  36. ; ;
  37. a \nabla_{a}
  38. ( 10 ) θ ^ = 1 2 k a . ; a (10)\quad\hat{\theta}=\frac{1}{2}\,k^{a}{}_{;\,a}\;.
  39. θ ( ) \theta_{(\ell)}
  40. θ ( n ) \theta_{(n)}
  41. ( A .1 ) θ ( ) := h a b a l b , (A.1)\quad\theta_{(\ell)}:=h^{ab}\nabla_{a}l_{b}\;,
  42. ( A .2 ) θ ( n ) := h a b a n b , (A.2)\quad\theta_{(n)}:=h^{ab}\nabla_{a}n_{b}\;,
  43. h a b = g a b + l a n b + n a l b h^{ab}=g^{ab}+l^{a}n^{b}+n^{a}l^{b}
  44. θ ( ) \theta_{(\ell)}
  45. θ ( n ) \theta_{(n)}
  46. ( A .3 ) θ ( ) = g a b a l b - κ ( ) , (A.3)\quad\theta_{(\ell)}=g^{ab}\nabla_{a}l_{b}-\kappa_{(\ell)}\;,
  47. ( A .4 ) θ ( n ) = g a b a n b - κ ( n ) , (A.4)\quad\theta_{(n)}=g^{ab}\nabla_{a}n_{b}-\kappa_{(n)}\;,
  48. κ ( ) \kappa_{(\ell)}
  49. κ ( n ) \kappa_{(n)}
  50. ( A .5 ) l a a l b = κ ( ) l b , (A.5)\quad l^{a}\nabla_{a}l_{b}=\kappa_{(\ell)}l_{b}\;,
  51. ( A .6 ) n a a n b = κ ( n ) n b . (A.6)\quad n^{a}\nabla_{a}n_{b}=\kappa_{(n)}n_{b}\;.
  52. { ( - , + , + , + ) ; l a n a = - 1 , m a m ¯ a = 1 } \{(-,+,+,+);l^{a}n_{a}=-1\,,m^{a}\bar{m}_{a}=1\}
  53. ( A .7 ) θ ( l ) = - ( ρ + ρ ¯ ) = - 2 Re ( ρ ) , θ ( n ) = μ + μ ¯ = 2 Re ( μ ) , (A.7)\quad\theta_{(l)}=-(\rho+\bar{\rho})=-2\,\text{Re}(\rho)\,,\quad\theta_{(% n)}=\mu+\bar{\mu}=2\,\text{Re}(\mu)\,,
  54. θ \theta
  55. θ ( ) \theta_{(\ell)}
  56. θ ( n ) \theta_{(n)}
  57. θ \theta
  58. θ ( ) \theta_{(\ell)}
  59. θ ( n ) \theta_{(n)}
  60. ( 11 ) σ ^ 2 = σ ^ a b σ ¯ ^ a b = 1 2 g c a g d b k ( a ; b ) k c ; d - ( 1 2 k a ) ; a 2 = g c a g d b 1 2 k ( a ; b ) k c ; d - θ ^ 2 . (11)\quad{\hat{\sigma}}^{2}=\hat{\sigma}_{ab}\hat{\bar{\sigma}}^{ab}=\frac{1}{% 2}\,g^{ca}\,g^{db}\,k_{(a\,;\,b)}\,k_{c\,;\,d}-\Big(\frac{1}{2}\,k^{a}{}_{;\,a% }\Big)^{2}=\,g^{ca}\,g^{db}\frac{1}{2}\,k_{(a\,;\,b)}\,k_{c\,;\,d}-{\hat{% \theta}}^{2}\;.
  61. ( 12 ) ω ^ 2 = 1 2 k [ a ; b ] k a ; b = g c a g d b k [ a ; b ] k c ; d . (12)\quad{\hat{\omega}}^{2}=\frac{1}{2}\,k_{[a\,;\,b]}\,k^{a\,;\,b}=g^{ca}\,g^% {db}\,k_{[a\,;\,b]}\,k_{c\,;\,d}\;.
  62. k a = l a k^{a}=l^{a}
  63. k a = n a k^{a}=n^{a}
  64. { θ ^ ( ) , σ ^ ( ) , ω ^ ( ) } \{\hat{\theta}_{(\ell)}\,,\hat{\sigma}_{(\ell)}\,,\hat{\omega}_{(\ell)}\}
  65. { θ ^ ( n ) , σ ^ ( n ) , ω ^ ( n ) } \{\hat{\theta}_{(n)}\,,\hat{\sigma}_{(n)}\,,\hat{\omega}_{(n)}\}
  66. l a l^{a}
  67. n a n^{a}
  68. B a b B_{ab}
  69. Z c Z^{c}
  70. ( 13 ) Z c c B a b = - B b c B a c + R c b a d Z c Z d . (13)\quad Z^{c}\nabla_{c}B_{ab}=-B^{c}_{\;\;b}B_{ac}+R_{cbad}Z^{c}Z^{d}\;.
  71. g a b g^{ab}
  72. ( 14 ) Z c c θ = θ , τ = - 1 3 θ 2 - σ a b σ a b + ω a b ω a b - R a b Z a Z b (14)\quad Z^{c}\nabla_{c}\theta=\theta_{,\,\tau}=-\frac{1}{3}\theta^{2}-\sigma% _{ab}\sigma^{ab}+\omega_{ab}\omega^{ab}-R_{ab}Z^{a}Z^{b}
  73. ( 15 ) Z c c σ a b = - 2 3 θ σ a b - σ a c σ b c - ω a c ω b c + 1 3 h a b ( σ c d σ c d - ω c d ω c d ) + C c b a d Z c Z d + 1 2 R ~ a b . (15)\quad Z^{c}\nabla_{c}\sigma_{ab}=-\frac{2}{3}\theta\sigma_{ab}-\sigma_{ac}% \sigma^{c}_{\;b}-\omega_{ac}\omega^{c}_{\;b}+\frac{1}{3}h_{ab}\,(\sigma_{cd}% \sigma^{cd}-\omega_{cd}\omega^{cd})+C_{cbad}Z^{c}Z^{d}+\frac{1}{2}\tilde{R}_{% ab}\,.
  74. ( 16 ) Z c c ω a b = - 2 3 θ ω a b - 2 σ [ b c ω a ] c . (16)\quad Z^{c}\nabla_{c}\omega_{ab}=-\frac{2}{3}\theta\omega_{ab}-2\sigma^{c}% _{\;[b}\omega_{a]c}\;.
  75. ( 16 ) k c c B ^ a b = - B ^ b c B ^ a c + R c b a d k c k d ^ . (16)\quad k^{c}\nabla_{c}\hat{B}_{ab}=-\hat{B}^{c}_{\;\;b}\hat{B}_{ac}+% \widehat{R_{cbad}k^{c}k^{d}}\;.
  76. ( 17 ) k c c θ ^ = θ ^ , λ = - 1 2 θ ^ 2 - σ ^ a b σ ^ a b + ω ^ a b ω ^ a b - R c d k c k d ^ , (17)\quad k^{c}\nabla_{c}\hat{\theta}=\hat{\theta}_{,\,\lambda}=-\frac{1}{2}% \hat{\theta}^{2}-\hat{\sigma}_{ab}\hat{\sigma}^{ab}+\hat{\omega}_{ab}\hat{% \omega}^{ab}-\widehat{R_{cd}k^{c}k^{d}}\;,
  77. ( 18 ) k c c σ ^ a b = - θ ^ σ ^ a b + C c b a d k c k d ^ , (18)\quad k^{c}\nabla_{c}\hat{\sigma}_{ab}=-\hat{\theta}\hat{\sigma}_{ab}+% \widehat{C_{cbad}k^{c}k^{d}}\;,
  78. ( 19 ) k c c ω ^ a b = - θ ^ ω ^ a b . (19)\quad k^{c}\nabla_{c}\hat{\omega}_{ab}=-\hat{\theta}\hat{\omega}_{ab}\;.
  79. ( 20 ) k c c θ = θ ^ , λ = - 1 2 θ ^ 2 - σ ^ a b σ ^ a b - R c d k c k d ^ + κ ( ) θ ^ , (20)\quad k^{c}\nabla_{c}\theta=\hat{\theta}_{,\,\lambda}=-\frac{1}{2}\hat{% \theta}^{2}-\hat{\sigma}_{ab}\hat{\sigma}^{ab}-\widehat{R_{cd}k^{c}k^{d}}+% \kappa_{(\ell)}\hat{\theta}\;,
  80. ( 21 ) k c c σ ^ a b = - θ ^ σ ^ a b + C c b a d k c k d ^ + κ ( ) σ ^ a b , (21)\quad k^{c}\nabla_{c}\hat{\sigma}_{ab}=-\hat{\theta}\hat{\sigma}_{ab}+% \widehat{C_{cbad}k^{c}k^{d}}+\kappa_{(\ell)}\hat{\sigma}_{ab}\;,
  81. ( 22 ) k c c ω ^ a b = 0 . (22)\quad k^{c}\nabla_{c}\hat{\omega}_{ab}=0\;.
  82. ( 23 ) θ ( ) = - 1 2 θ ( ) 2 + κ ~ ( ) θ ( ) - σ a b σ a b + ω ~ a b ω ~ a b - R a b l a l b , (23)\quad\mathcal{L}_{\ell}\theta_{(\ell)}=-\frac{1}{2}\theta_{(\ell)}^{2}+% \tilde{\kappa}_{(\ell)}\theta_{(\ell)}-\sigma_{ab}\sigma^{ab}+\tilde{\omega}_{% ab}\tilde{\omega}^{ab}-R_{ab}l^{a}l^{b}\,,
  83. κ ~ ( ) \tilde{\kappa}_{(\ell)}
  84. κ ~ ( ) l b := l a a l b \tilde{\kappa}_{(\ell)}l^{b}:=l^{a}\nabla_{a}l^{b}
  85. ( 24 ) θ ( ) = - ( ρ + ρ ¯ ) = - 2 Re ( ρ ) , θ ( n ) = μ + μ ¯ = 2 Re ( μ ) , (24)\quad\theta_{(\ell)}=-(\rho+\bar{\rho})=-2\,\text{Re}(\rho)\,,\quad\theta_% {(n)}=\mu+\bar{\mu}=2\,\text{Re}(\mu)\,,
  86. ( 25 ) σ a b = - σ m ¯ a m ¯ b - σ ¯ m a m b , (25)\quad\sigma_{ab}=-\sigma\bar{m}_{a}\bar{m}_{b}-\bar{\sigma}m_{a}m_{b}\,,
  87. ( 26 ) ω ~ a b = 1 2 ( ρ - ρ ¯ ) ( m a m ¯ b - m ¯ a m b ) = Im ( ρ ) ( m a m ¯ b - m ¯ a m b ) , (26)\quad\tilde{\omega}_{ab}=\frac{1}{2}\,\Big(\rho-\bar{\rho}\Big)\,\Big(m_{a% }\bar{m}_{b}-\bar{m}_{a}m_{b}\Big)=\,\text{Im}(\rho)\cdot\Big(m_{a}\bar{m}_{b}% -\bar{m}_{a}m_{b}\Big)\,,
  88. h ^ a b = h ^ b a = m b m ¯ a + m ¯ b m a \hat{h}^{ab}=\hat{h}^{ba}=m^{b}\bar{m}^{a}+\bar{m}^{b}m^{a}
  89. ( 27 ) θ ( ) = h ^ b a a l b = m b m ¯ a a l b + m ¯ b m a a l b = m b δ ¯ l b + m ¯ b δ l b = - ( ρ + ρ ¯ ) , (27)\quad\theta_{(\ell)}=\hat{h}^{ba}\nabla_{a}l_{b}=m^{b}\bar{m}^{a}\nabla_{a% }l_{b}+\bar{m}^{b}m^{a}\nabla_{a}l_{b}=m^{b}\bar{\delta}l_{b}+\bar{m}^{b}% \delta l_{b}=-(\rho+\bar{\rho})\,,
  90. ( 28 ) θ ( n ) = h ^ b a a n b = m ¯ b m a a n b + m b m ¯ a a n b = m ¯ b δ n b + m b δ ¯ n b = μ + μ ¯ . (28)\quad\theta_{(n)}=\hat{h}^{ba}\nabla_{a}n_{b}=\bar{m}^{b}m^{a}\nabla_{a}n_% {b}+m^{b}\bar{m}^{a}\nabla_{a}n_{b}=\bar{m}^{b}\delta n_{b}+m^{b}\bar{\delta}n% _{b}=\mu+\bar{\mu}\,.

Optimal_stopping.html

  1. X 1 , X 2 , X_{1},X_{2},\ldots
  2. ( y i ) i 1 (y_{i})_{i\geq 1}
  3. y i = y i ( x 1 , , x i ) y_{i}=y_{i}(x_{1},\ldots,x_{i})
  4. i i
  5. i i
  6. y i y_{i}
  7. G = ( G t ) t 0 G=(G_{t})_{t\geq 0}
  8. ( Ω , , ( t ) t 0 , ) (\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\mathbb{P})
  9. G G
  10. τ * \tau^{*}
  11. V t T = 𝔼 G τ * = sup t τ T 𝔼 G τ V_{t}^{T}=\mathbb{E}G_{\tau^{*}}=\sup_{t\leq\tau\leq T}\mathbb{E}G_{\tau}
  12. V t T V_{t}^{T}
  13. T T
  14. \infty
  15. X = ( X t ) t 0 X=(X_{t})_{t\geq 0}
  16. ( Ω , , ( t ) t 0 , x ) (\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\mathbb{P}_{x})
  17. x \mathbb{P}_{x}
  18. x x
  19. M , L M,L
  20. K K
  21. V ( x ) = sup 0 τ T 𝔼 x ( M ( X τ ) + 0 τ L ( X t ) d t + sup 0 t τ K ( X t ) ) . V(x)=\sup_{0\leq\tau\leq T}\mathbb{E}_{x}\left(M(X_{\tau})+\int_{0}^{\tau}L(X_% {t})dt+\sup_{0\leq t\leq\tau}K(X_{t})\right).
  22. T T
  23. Y t Y_{t}
  24. k \mathbb{R}^{k}
  25. d Y t = b ( Y t ) d t + σ ( Y t ) d B t + k γ ( Y t - , z ) N ¯ ( d t , d z ) , Y 0 = y dY_{t}=b(Y_{t})dt+\sigma(Y_{t})dB_{t}+\int_{\mathbb{R}^{k}}\gamma(Y_{t-},z)% \bar{N}(dt,dz),\quad Y_{0}=y
  26. B B
  27. m m
  28. N ¯ \bar{N}
  29. l l
  30. b : k k b:\mathbb{R}^{k}\to\mathbb{R}^{k}
  31. σ : k k × m \sigma:\mathbb{R}^{k}\to\mathbb{R}^{k\times m}
  32. γ : k × k k × l \gamma:\mathbb{R}^{k}\times\mathbb{R}^{k}\to\mathbb{R}^{k\times l}
  33. ( Y t ) (Y_{t})
  34. 𝒮 k \mathcal{S}\subset\mathbb{R}^{k}
  35. τ 𝒮 = inf { t > 0 : Y t 𝒮 } \tau_{\mathcal{S}}=\inf\{t>0:Y_{t}\notin\mathcal{S}\}
  36. V ( y ) = sup τ τ 𝒮 J τ ( y ) = sup τ τ 𝒮 𝔼 y [ M ( Y τ ) + 0 τ L ( Y t ) d t ] . V(y)=\sup_{\tau\leq\tau_{\mathcal{S}}}J^{\tau}(y)=\sup_{\tau\leq\tau_{\mathcal% {S}}}\mathbb{E}_{y}\left[M(Y_{\tau})+\int_{0}^{\tau}L(Y_{t})dt\right].
  37. ϕ : 𝒮 ¯ \phi:\bar{\mathcal{S}}\to\mathbb{R}
  38. ϕ C ( 𝒮 ¯ ) C 1 ( 𝒮 ) C 2 ( 𝒮 D ) \phi\in C(\bar{\mathcal{S}})\cap C^{1}(\mathcal{S})\cap C^{2}(\mathcal{S}% \setminus\partial D)
  39. D = { y 𝒮 : ϕ ( y ) > M ( y ) } D=\{y\in\mathcal{S}:\phi(y)>M(y)\}
  40. ϕ M \phi\geq M
  41. 𝒮 \mathcal{S}
  42. 𝒜 ϕ + L 0 \mathcal{A}\phi+L\leq 0
  43. 𝒮 D \mathcal{S}\setminus\partial D
  44. 𝒜 \mathcal{A}
  45. ( Y t ) (Y_{t})
  46. ϕ ( y ) V ( y ) \phi(y)\geq V(y)
  47. y 𝒮 ¯ y\in\bar{\mathcal{S}}
  48. 𝒜 ϕ + L = 0 \mathcal{A}\phi+L=0
  49. D D
  50. ϕ ( y ) = V ( y ) \phi(y)=V(y)
  51. y 𝒮 ¯ y\in\bar{\mathcal{S}}
  52. τ * = inf { t > 0 : Y t D } \tau^{*}=\inf\{t>0:Y_{t}\notin D\}
  53. max { 𝒜 ϕ + L , M - ϕ } = 0 \max\left\{\mathcal{A}\phi+L,M-\phi\right\}=0
  54. 𝒮 D . \mathcal{S}\setminus\partial D.
  55. 𝔼 ( y i ) \mathbb{E}(y_{i})
  56. Bern ( 1 2 ) , \,\text{Bern}\left(\frac{1}{2}\right),
  57. y i = 1 i k = 1 i X k y_{i}=\frac{1}{i}\sum_{k=1}^{i}X_{k}
  58. ( X i ) i 1 (X_{i})_{i\geq 1}
  59. ( y i ) i 1 (y_{i})_{i\geq 1}
  60. 𝔼 ( y i ) \mathbb{E}(y_{i})
  61. X n X_{n}
  62. k k
  63. n n
  64. y n y_{n}
  65. y n = ( X n - n k ) y_{n}=(X_{n}-nk)
  66. X i X_{i}
  67. ( X i ) (X_{i})
  68. R 1 , , R n R_{1},\ldots,R_{n}
  69. y i y_{i}
  70. ( R i ) (R_{i})
  71. ( y i ) (y_{i})
  72. r r
  73. δ \delta
  74. σ \sigma
  75. S S
  76. S t = S 0 exp { ( r - δ - σ 2 2 ) t + σ B t } S_{t}=S_{0}\exp\left\{\left(r-\delta-\frac{\sigma^{2}}{2}\right)t+\sigma B_{t}\right\}
  77. V ( x ) = sup τ 𝔼 x [ e - r τ g ( S τ ) ] V(x)=\sup_{\tau}\mathbb{E}_{x}\left[e^{-r\tau}g(S_{\tau})\right]
  78. g ( x ) = ( x - K ) + g(x)=(x-K)^{+}
  79. g ( x ) = ( K - x ) + g(x)=(K-x)^{+}
  80. max { 1 2 σ 2 x 2 V ′′ ( x ) + ( r - δ ) x V ( x ) - r V ( x ) , g ( x ) - V ( x ) } = 0 \max\left\{\frac{1}{2}\sigma^{2}x^{2}V^{\prime\prime}(x)+(r-\delta)xV^{\prime}% (x)-rV(x),g(x)-V(x)\right\}=0
  81. x ( 0 , ) { b } x\in(0,\infty)\setminus\{b\}
  82. b b
  83. V ( x ) = { ( b - K ) ( x / b ) γ x ( 0 , b ) x - K x [ b , ) V(x)=\begin{cases}(b-K)(x/b)^{\gamma}&x\in(0,b)\\ x-K&x\in[b,\infty)\end{cases}
  84. γ = ( ν 2 + 2 r - ν ) / σ \gamma=(\sqrt{\nu^{2}+2r}-\nu)/\sigma
  85. ν = ( r - δ ) / σ - σ / 2 , b = γ K / ( γ - 1 ) . \nu=(r-\delta)/\sigma-\sigma/2,\quad b=\gamma K/(\gamma-1).
  86. V ( x ) = { K - x x ( 0 , c ] ( K - c ) ( x / c ) γ ~ x ( c , ) V(x)=\begin{cases}K-x&x\in(0,c]\\ (K-c)(x/c)^{\tilde{\gamma}}&x\in(c,\infty)\end{cases}
  87. γ ~ = - ( ν 2 + 2 r + ν ) / σ \tilde{\gamma}=-(\sqrt{\nu^{2}+2r}+\nu)/\sigma
  88. ν = ( r - δ ) / σ - σ / 2 , c = γ ~ K / ( γ ~ - 1 ) . \nu=(r-\delta)/\sigma-\sigma/2,\quad c=\tilde{\gamma}K/(\tilde{\gamma}-1).

Orbit_portrait.html

  1. f c : z z 2 + c . f_{c}:z\to z^{2}+c.\,
  2. f c : \C \C f_{c}:\mathbb{\C}\to\mathbb{\C}\,
  3. 𝒪 = { z 1 , z n } {\mathcal{O}}=\{z_{1},\ldots z_{n}\}
  4. f f\,
  5. f ( z j ) = z j + 1 f(z_{j})=z_{j+1}\,
  6. n n
  7. A j A_{j}
  8. z j z_{j}\,
  9. 𝒫 = 𝒫 ( 𝒪 ) = { A 1 , A n } {\mathcal{P}}={\mathcal{P}}({\mathcal{O}})=\{A_{1},\ldots A_{n}\}
  10. 𝒪 {\mathcal{O}}
  11. A j A_{j}\,
  12. 𝒫 = { ( 74 511 , 81 511 , 137 511 ) , ( 148 511 , 162 511 , 274 511 ) , ( 296 511 , 324 511 , 37 511 ) } {\mathcal{P}}=\left\{\left(\frac{74}{511},\frac{81}{511},\frac{137}{511}\right% ),\left(\frac{148}{511},\frac{162}{511},\frac{274}{511}\right),\left(\frac{296% }{511},\frac{324}{511},\frac{37}{511}\right)\right\}
  13. 𝒫 = { ( 22 63 , 25 63 , 37 63 ) , ( 11 63 , 44 63 , 50 63 ) } {\mathcal{P}}=\left\{\left(\frac{22}{63},\frac{25}{63},\frac{37}{63}\right),% \left(\frac{11}{63},\frac{44}{63},\frac{50}{63}\right)\right\}
  14. 𝒫 {\mathcal{P}}
  15. A j A_{j}
  16. / {\mathbb{R}}/{\mathbb{Z}}
  17. A j A_{j}
  18. A j + 1 A_{j+1}
  19. A 1 , , A n A_{1},\ldots,A_{n}
  20. n n
  21. r n rn
  22. r r
  23. A j A_{j}
  24. / {\mathbb{R}}/{\mathbb{Z}}
  25. { A 1 , , A n } \{A_{1},\ldots,A_{n}\}
  26. A j A_{j}
  27. < m t p l > 0 <mtpl>{{0}}
  28. { A 1 , , A n } \{A^{\prime}_{1},\ldots,A^{\prime}_{n}\}
  29. A j A j A_{j}\subsetneq A^{\prime}_{j}
  30. f 0 ( z ) = z 2 f_{0}(z)=z^{2}
  31. { A 1 , , A n } \{A_{1},\ldots,A_{n}\}
  32. A j A_{j}
  33. / \mathbb{R}/\mathbb{Z}
  34. A j A_{j}
  35. z j z_{j}
  36. z j z_{j}
  37. 1 2 \frac{1}{2}
  38. z j z_{j}
  39. z j + 1 z_{j+1}
  40. z j + 1 z_{j+1}
  41. z j + 1 z_{j+1}
  42. c c
  43. f c f_{c}
  44. c c
  45. c c
  46. θ c \theta_{c}
  47. θ c \theta_{c}
  48. c c
  49. θ c 2 \frac{\theta_{c}}{2}
  50. θ c + 1 2 \frac{\theta_{c}+1}{2}
  51. A j A_{j}
  52. 𝒫 {\mathcal{I}}_{\mathcal{P}}
  53. z j z_{j}
  54. z j z_{j}
  55. v v
  56. z j z_{j}
  57. 0
  58. c c
  59. f c f_{c}
  60. t - t_{-}
  61. t + t_{+}
  62. 𝒫 \mathcal{P}
  63. [ t - , t + ] [t_{-},t_{+}]
  64. 𝒫 \mathcal{P}
  65. r 𝒫 r_{\mathcal{P}}
  66. 𝒫 \mathcal{P}
  67. 0
  68. 𝒫 \mathcal{P}
  69. 𝒲 𝒫 {\mathcal{W}}_{\mathcal{P}}
  70. f c ( z ) = z 2 + c f_{c}(z)=z^{2}+c
  71. 𝒫 {\mathcal{P}}
  72. c 𝒲 𝒫 c\in{\mathcal{W}}_{\mathcal{P}}
  73. 𝒫 {\mathcal{P}}
  74. c = r 𝒫 c=r_{\mathcal{P}}
  75. v v
  76. 𝒫 \mathcal{P}
  77. r r
  78. r = 1 r=1
  79. v = 2 v=2
  80. f n f^{n}
  81. A j A_{j}
  82. r 𝒫 r_{\mathcal{P}}
  83. r = v 2 r=v\geq 2
  84. r 𝒫 r_{\mathcal{P}}

Ordered_Bell_number.html

  1. A ( B f ( x , y ) d y ) d x = B ( A f ( x , y ) d x ) d y = A × B f ( x , y ) d ( x , y ) , \int_{A}\left(\int_{B}f(x,y)\,\,\text{d}y\right)\,\,\text{d}x=\int_{B}\left(% \int_{A}f(x,y)\,\,\text{d}x\right)\,\,\text{d}y=\int_{A\times B}f(x,y)\,\,% \text{d}(x,y),
  2. a ( n ) = k = 0 n k ! { n k } = k = 0 n j = 0 k ( - 1 ) k - j ( k j ) j n = 1 2 m = 0 m n 2 m . a(n)=\sum_{k=0}^{n}k!\left\{\begin{matrix}n\\ k\end{matrix}\right\}=\sum_{k=0}^{n}\sum_{j=0}^{k}(-1)^{k-j}{\left({{k}\atop{j% }}\right)}j^{n}=\frac{1}{2}\sum_{m=0}^{\infty}\frac{m^{n}}{2^{m}}.
  3. a ( n ) = k = 0 n - 1 2 k n k = A n ( 2 ) , a(n)=\sum_{k=0}^{n-1}2^{k}\left\langle\begin{matrix}n\\ k\end{matrix}\right\rangle=A_{n}(2),
  4. n = 0 a ( n ) x n n ! = 1 2 - e x . \sum_{n=0}^{\infty}a(n)\frac{x^{n}}{n!}=\frac{1}{2-e^{x}}.
  5. a ( n ) = n ! 2 k = - ( log 2 + 2 π i k ) - ( n + 1 ) , n 1 , a(n)=\frac{n!}{2}\sum_{k=-\infty}^{\infty}(\log 2+2\pi ik)^{-(n+1)},\qquad n% \geq 1,
  6. a ( n ) n ! 2 ( log 2 ) n + 1 . a(n)\approx\frac{n!}{2(\log 2)^{n+1}}.
  7. lim n n a ( n - 1 ) a ( n ) = log 2. \lim_{n\to\infty}\frac{n\,a(n-1)}{a(n)}=\log 2.
  8. a ( n ) = i = 1 n ( n i ) a ( n - i ) . a(n)=\sum_{i=1}^{n}{\left({{n}\atop{i}}\right)}a(n-i).
  9. a ( n + 4 ) a ( n ) ( mod 10 ) , a(n+4)\equiv a(n)\;\;(\mathop{{\rm mod}}10),
  10. a ( n + 20 ) a ( n ) ( mod 100 ) , a(n+20)\equiv a(n)\;\;(\mathop{{\rm mod}}100),
  11. a ( n + 100 ) a ( n ) ( mod 1000 ) , a(n+100)\equiv a(n)\;\;(\mathop{{\rm mod}}1000),
  12. a ( n + 500 ) a ( n ) ( mod 10000 ) . a(n+500)\equiv a(n)\;\;(\mathop{{\rm mod}}10000).

Ordered_vector_space.html

  1. V V

Ordinal_notation.html

  1. ω ω ω . \omega^{\omega^{\omega}}.
  2. θ ϵ Ω v + 1 0 = ψ 0 ( ϵ Ω v + 1 ) \theta\epsilon_{\Omega_{v}+1}0=\psi_{0}(\epsilon_{\Omega_{v}+1})
  3. 𝒪 \mathcal{O}
  4. 𝒪 \mathcal{O}
  5. 𝒪 \mathcal{O}

Ornstein_isomorphism_theorem.html

  1. T t T_{t}
  2. T 1 T_{1}
  3. T t T_{t}
  4. S t S_{t}
  5. S t = T c t S_{t}=T_{ct}
  6. T \sqrt{T}
  7. H = - i = 1 N p i log p i . H=-\sum_{i=1}^{N}p_{i}\log p_{i}.

Orthogonality_(term_rewriting).html

  1. ρ 1 : f ( x , y ) g ( y ) \rho_{1}\ :\ f(x,y)\rightarrow g(y)
  2. ρ 2 : h ( y ) f ( g ( y ) , y ) \rho_{2}\ :\ h(y)\rightarrow f(g(y),y)

Outline_of_epistemology.html

  1. a a
  2. b b
  3. a a
  4. b b

Overall_pressure_ratio.html

  1. C R = V 1 V 2 CR=\frac{V_{1}}{V_{2}}
  2. P R = P 2 P 1 PR=\frac{P_{2}}{P_{1}}
  3. P 1 V 1 T 1 = P 2 V 2 T 2 V 1 V 2 = T 1 T 2 P 2 P 1 C R = T 1 T 2 P R \frac{P_{1}V_{1}}{T_{1}}=\frac{P_{2}V_{2}}{T_{2}}\Rightarrow\frac{V_{1}}{V_{2}% }=\frac{T_{1}}{T_{2}}\frac{P_{2}}{P_{1}}\Leftrightarrow CR=\frac{T_{1}}{T_{2}}PR

Overlap_(term_rewriting).html

  1. ρ 1 : f ( g ( x ) , y ) y \rho_{1}\ :\ f(g(x),y)\rightarrow y
  2. ρ 2 : g ( x ) f ( x , x ) \rho_{2}\ :\ g(x)\rightarrow f(x,x)
  3. ρ 1 : g ( g ( x ) ) x \rho_{1}\ :\ g(g(x))\rightarrow x

Ovoid_(polar_space).html

  1. r - 1 r-1
  2. W 2 n - 1 ( q ) W_{2n-1}(q)
  3. q n + 1 q^{n}+1
  4. n = 2 n=2
  5. P G ( 3 , q ) PG(3,q)
  6. H ( 2 n , q 2 ) ( n 2 ) H(2n,q^{2})(n\geq 2)
  7. H ( 2 n + 1 , q 2 ) ( n 1 ) H(2n+1,q^{2})(n\geq 1)
  8. q 2 n + 1 + 1 q^{2n+1}+1
  9. Q + ( 2 n - 1 , q ) ( n 2 ) Q^{+}(2n-1,q)(n\geq 2)
  10. q n - 1 + 1 q^{n-1}+1
  11. Q ( 2 n , q ) ( n 2 ) Q(2n,q)(n\geq 2)
  12. q n + 1 q^{n}+1
  13. n = 2 n=2
  14. Q ( 2 n , q ) Q(2n,q)
  15. W 2 n - 1 ( q ) W_{2n-1}(q)
  16. n 3 n\geq 3
  17. Q - ( 2 n + 1 , q ) ( n 2 ) Q^{-}(2n+1,q)(n\geq 2)
  18. q n + 1 q^{n}+1

P-matrix.html

  1. P P
  2. P 0 P_{0}
  3. P P
  4. \geq
  5. P P
  6. P P
  7. P 0 P_{0}
  8. { u 1 , , u n } \{u_{1},...,u_{n}\}
  9. n n
  10. P P
  11. | a r g ( u i ) | < π - π n , i = 1 , , n |arg(u_{i})|<\pi-\frac{\pi}{n},i=1,...,n
  12. { u 1 , , u n } \{u_{1},...,u_{n}\}
  13. u i 0 u_{i}\neq 0
  14. i = 1 , , n i=1,...,n
  15. n n
  16. P 0 P_{0}
  17. | a r g ( u i ) | π - π n , i = 1 , , n |arg(u_{i})|\leq\pi-\frac{\pi}{n},i=1,...,n
  18. P P
  19. P P
  20. M M
  21. P P
  22. P P
  23. n \mathbb{R}^{n}
  24. P ( - ) P^{(-)}
  25. N - P N-P
  26. A A
  27. P ( - ) P^{(-)}
  28. ( - A ) (-A)
  29. P P
  30. P 0 P_{0}
  31. σ ( A ) = - σ ( - A ) \sigma(A)=-\sigma(-A)
  32. P P
  33. P 0 P_{0}
  34. M M
  35. P P

P-rep.html

  1. p rep = [ 1 + ( p 1 - p ) 2 3 ] - 1 . p\text{rep}=\left[1+\left(\frac{p}{1-p}\right)^{\frac{2}{3}}\right]^{-1}.

Pair_distribution_function.html

  1. g a b ( r ) g_{ab}(\vec{r})
  2. r \vec{r}
  3. r \vec{r}
  4. p ( r ) = 1 / V p(\vec{r})=1/V
  5. V V
  6. g ( r , r ) g(\vec{r},\vec{r^{\prime}})
  7. N N
  8. g ( r , r ) = p ( r , r ) V 2 N - 1 N g(\vec{r},\vec{r}^{\prime})=p(\vec{r},\vec{r}^{\prime})V^{2}\frac{N-1}{N}
  9. g ( r , r ) p ( r , r ) V 2 g(\vec{r},\vec{r}^{\prime})\approx p(\vec{r},\vec{r}^{\prime})V^{2}
  10. g ( r ) = 1 g(\vec{r})=1
  11. r \vec{r}
  12. g ( r ) = { 0 , r < b , 1 , r b , g(r)=\begin{cases}0,&r<b,\\ 1,&r\geq{}b\end{cases},
  13. b b
  14. r = n b r=nb
  15. n n
  16. g ( r ) = i δ ( r - i b ) g(r)=\sum\limits_{i}\delta(r-ib)
  17. lim r g ( r ) = 1 \lim\limits_{r\to\infty}g(r)=1
  18. f f
  19. g a b ( r ) = 1 N a N b i = 1 N a j = 1 N b δ ( | 𝐫 i j | - r ) g_{ab}(r)=\frac{1}{N_{a}N_{b}}\sum\limits_{i=1}^{N_{a}}\sum\limits_{j=1}^{N_{b% }}\langle\delta(|\mathbf{r}_{ij}|-r)\rangle

Paley_construction.html

  1. Q = [ 0 - 1 - 1 1 - 1 1 1 1 0 - 1 - 1 1 - 1 1 1 1 0 - 1 - 1 1 - 1 - 1 1 1 0 - 1 - 1 1 1 - 1 1 1 0 - 1 - 1 - 1 1 - 1 1 1 0 - 1 - 1 - 1 1 - 1 1 1 0 ] . Q=\begin{bmatrix}0&-1&-1&1&-1&1&1\\ 1&0&-1&-1&1&-1&1\\ 1&1&0&-1&-1&1&-1\\ -1&1&1&0&-1&-1&1\\ 1&-1&1&1&0&-1&-1\\ -1&1&-1&1&1&0&-1\\ -1&-1&1&-1&1&1&0\end{bmatrix}.
  2. H = I + [ 0 j T - j Q ] H=I+\begin{bmatrix}0&j^{T}\\ -j&Q\end{bmatrix}
  3. [ 0 j T j Q ] \begin{bmatrix}0&j^{T}\\ j&Q\end{bmatrix}
  4. [ 1 - 1 - 1 - 1 ] \begin{bmatrix}1&-1\\ -1&-1\end{bmatrix}
  5. ± [ 1 1 1 - 1 ] \pm\begin{bmatrix}1&1\\ 1&-1\end{bmatrix}
  6. Q = [ 0 1 1 - 1 - 1 1 - 1 1 - 1 1 0 1 1 - 1 - 1 - 1 - 1 1 1 1 0 - 1 1 - 1 1 - 1 - 1 - 1 1 - 1 0 1 1 - 1 - 1 1 - 1 - 1 1 1 0 1 1 - 1 - 1 1 - 1 - 1 1 1 0 - 1 1 - 1 - 1 - 1 1 - 1 1 - 1 0 1 1 1 - 1 - 1 - 1 - 1 1 1 0 1 - 1 1 - 1 1 - 1 - 1 1 1 0 ] . Q=\begin{bmatrix}0&1&1&-1&-1&1&-1&1&-1\\ 1&0&1&1&-1&-1&-1&-1&1\\ 1&1&0&-1&1&-1&1&-1&-1\\ -1&1&-1&0&1&1&-1&-1&1\\ -1&-1&1&1&0&1&1&-1&-1\\ 1&-1&-1&1&1&0&-1&1&-1\\ -1&-1&1&-1&1&-1&0&1&1\\ 1&-1&-1&-1&-1&1&1&0&1\\ -1&1&-1&1&-1&-1&1&1&0\end{bmatrix}.
  7. H 2 = [ 1 1 1 - 1 ] , H_{2}=\begin{bmatrix}1&1\\ 1&-1\end{bmatrix},
  8. m 0 mod 4 \scriptstyle m\,\equiv\,0\mod 4

Paley_graph.html

  1. E = { ( a , b ) 𝐅 q × 𝐅 q : a - b ( 𝐅 q × ) 2 } E=\left\{(a,b)\in\mathbf{F}_{q}\times\mathbf{F}_{q}\ :\ a-b\in(\mathbf{F}_{q}^% {\times})^{2}\right\}
  2. s r g ( q , 1 2 ( q - 1 ) , 1 4 ( q - 5 ) , 1 4 ( q - 1 ) ) . srg\left(q,\tfrac{1}{2}(q-1),\tfrac{1}{4}(q-5),\tfrac{1}{4}(q-1)\right).
  3. 1 2 ( q - 1 ) \tfrac{1}{2}(q-1)
  4. 1 2 ( - 1 ± q ) \tfrac{1}{2}(-1\pm\sqrt{q})
  5. 1 2 ( q - 1 ) \tfrac{1}{2}(q-1)
  6. q - q 4 i ( G ) ( q + q ) ( q - q 2 ) \frac{q-\sqrt{q}}{4}\leq i(G)\leq\sqrt{\left(q+\sqrt{q}\right)\left(\frac{q-% \sqrt{q}}{2}\right)}
  7. A = { ( a , b ) 𝐅 q × 𝐅 q : b - a ( 𝐅 q × ) 2 } . A=\left\{(a,b)\in\mathbf{F}_{q}\times\mathbf{F}_{q}\ :\ b-a\in(\mathbf{F}_{q}^% {\times})^{2}\right\}.
  8. 1 24 ( q 2 - 13 q + 24 ) \tfrac{1}{24}(q^{2}-13q+24)
  9. ( q 2 - 13 q + 24 ) ( 1 24 + o ( 1 ) ) , (q^{2}-13q+24)\left(\tfrac{1}{24}+o(1)\right),

Paley–Wiener_integral.html

  1. f L 2 ( E , γ ; ) = j ( f ) H \|f\|_{L^{2}(E,\gamma;\mathbb{R})}=\|j(f)\|_{H}

Palindromic_polynomial.html

  1. P ( x ) = i = 0 n a i x i P(x)=\sum_{i=0}^{n}a_{i}x^{i}
  2. ( x + 1 ) 2 = x 2 + 2 x + 1 (x+1)^{2}=x^{2}+2x+1
  3. ( x + 1 ) 3 = x 3 + 3 x 2 + 3 x + 1. (x+1)^{3}=x^{3}+3x^{2}+3x+1.
  4. x 2 + 3 x + 1 x^{2}+3x+1
  5. x 2 - 1 x^{2}-1
  6. x 4 + x 2 + 1 = 0 x^{4}+x^{2}+1=0
  7. X 2 = x 2 + 2 + 1 / x 2 X^{2}=x^{2}+2+1/x^{2}
  8. X 2 - 1 = 0 X^{2}-1=0
  9. ( X - 1 ) ( X + 1 ) = 0 (X-1)(X+1)=0
  10. x + 1 / x = - 1 x+1/x=-1
  11. x 2 + x + 1 = 0 x^{2}+x+1=0
  12. x + 1 / x = 1 x+1/x=1
  13. x 2 - x + 1 = 0 x^{2}-x+1=0
  14. 4 x 2 + 4 x + 1 4x^{2}+4x+1
  15. y 2 + 2 y + 1 y^{2}+2y+1
  16. ( y + 1 ) 2 (y+1)^{2}
  17. ( 2 x + 1 ) 2 (2x+1)^{2}

Pappus_graph.html

  1. ( x - 1 ) x ( x 16 - 26 x 15 + 325 x 14 - 2600 x 13 + 14950 x 12 - 65762 x 11 + (x-1)x(x^{16}-26x^{15}+325x^{14}-2600x^{13}+14950x^{12}-65762x^{11}+
  2. 229852 x 10 - 653966 x 9 + 1537363 x 8 - 3008720 x 7 + 4904386 x 6 - 229852x^{10}-653966x^{9}+1537363x^{8}-3008720x^{7}+4904386x^{6}-
  3. 6609926 x 5 + 7238770 x 4 - 6236975 x 3 + 3989074 x 2 - 1690406 x + 356509 ) 6609926x^{5}+7238770x^{4}-6236975x^{3}+3989074x^{2}-1690406x+356509)
  4. ( x - 3 ) x 4 ( x + 3 ) ( x 2 - 3 ) 6 (x-3)x^{4}(x+3)(x^{2}-3)^{6}

Papyrus_66.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}
  5. 𝔓 \mathfrak{P}

Parabolic_partial_differential_equation.html

  1. A u x x + 2 B u x y + C u y y + D u x + E u y + F = 0 Au_{xx}+2Bu_{xy}+Cu_{yy}+Du_{x}+Eu_{y}+F=0\,
  2. B 2 - A C = 0. B^{2}-AC=0.
  3. u t = k u x x , u_{t}=ku_{xx},
  4. u ( t , x ) u(t,x)
  5. t t
  6. x x
  7. k k
  8. u t u_{t}
  9. t t
  10. u x x u_{xx}
  11. x x
  12. u x x u_{xx}
  13. u t = - L u , u_{t}=-Lu,
  14. L L
  15. L L
  16. L L
  17. ( a ( x ) u ( x ) ) + b ( x ) T u ( x ) + c u ( x ) = f ( x ) \nabla\cdot(a(x)\nabla u(x))+b(x)^{T}\nabla u(x)+cu(x)=f(x)
  18. a ( x ) a(x)
  19. u t = - L ( u ) u_{t}=-L(u)
  20. u t = L u , u_{t}=Lu,
  21. L L
  22. { u t = Δ u on Ω × ( 0 , T ) , u = 0 on Ω × ( 0 , T ) , u = f on Ω × { T } . \begin{cases}u_{t}=\Delta u&\textrm{on}\ \ \Omega\times(0,T),\\ u=0&\textrm{on}\ \ \partial\Omega\times(0,T),\\ u=f&\textrm{on}\ \ \Omega\times\left\{T\right\}.\end{cases}
  23. { u t = - Δ u on Ω × ( 0 , T ) , u = 0 on Ω × ( 0 , T ) , u = f on Ω × { 0 } . \begin{cases}u_{t}=-\Delta u&\textrm{on}\ \ \Omega\times(0,T),\\ u=0&\textrm{on}\ \ \partial\Omega\times(0,T),\\ u=f&\textrm{on}\ \ \Omega\times\left\{0\right\}.\end{cases}

Paradoxical_set.html

  1. G G
  2. G G
  3. G G

Parallax_barrier.html

  1. n sin x = sin y n\sin x=\sin y
  2. sin y e 2 r \sin y\approx\frac{e}{2r}
  3. sin x p 2 d . \sin x\approx\frac{p}{2d}\,.
  4. d = r n p e . d=\frac{rnp}{e}\,.

Parasitic_number.html

  1. x = 0.179487179487179487 = 0. 179487 ¯ has 4 x = 0. 717948 ¯ = 7. 179487 ¯ 10 . x=0.179487179487179487\ldots=0.\overline{179487}\mbox{ has }~{}4x=0.\overline{% 717948}=\frac{7.\overline{179487}}{10}.
  2. 4 x = 7 + x 10 so x = 7 39 . 4x=\frac{7+x}{10}\mbox{ so }~{}x=\frac{7}{39}.
  3. k 10 n - 1 ( 10 m - 1 ) \frac{k}{10n-1}(10^{m}-1)
  4. 1 19 = 0. 052631578947368421 ¯ . \frac{1}{19}=0.\overline{052631578947368421}.
  5. 2 19 = 0. 105263157894736842 ¯ . \frac{2}{19}=0.\overline{105263157894736842}.

Parity_game.html

  1. V = V 0 V 1 V=V_{0}\cup V_{1}
  2. V 0 V_{0}
  3. G = ( V , V 0 , V 1 , E , Ω ) G=(V,V_{0},V_{1},E,\Omega)
  4. V 0 V_{0}
  5. V 1 V_{1}
  6. V = V 0 V 1 V=V_{0}\cup V_{1}
  7. E V × V E\subseteq V\times V
  8. Ω : V \Omega:V\rightarrow\mathbb{N}
  9. U V U\subseteq V
  10. i = 0 , 1 i=0,1
  11. i i
  12. U U
  13. A t t r i ( U ) Attr_{i}(U)
  14. U U
  15. i i
  16. U U
  17. A t t r i ( U ) Attr_{i}(U)
  18. A t t r i ( U ) 0 := U Attr_{i}(U)^{0}:=U
  19. A t t r i ( U ) j + 1 := A t t r i ( U ) j { v V i ( v , w ) E : w A t t r i ( U ) j } { v V 1 - i ( v , w ) E : w A t t r i ( U ) j } Attr_{i}(U)^{j+1}:=Attr_{i}(U)^{j}\cup\{v\in V_{i}\mid\exists(v,w)\in E:w\in Attr% _{i}(U)^{j}\}\cup\{v\in V_{1-i}\mid\forall(v,w)\in E:w\in Attr_{i}(U)^{j}\}
  20. A t t r i ( U ) := j = 0 A t t r i ( U ) j Attr_{i}(U):=\bigcup_{j=0}^{\infty}Attr_{i}(U)^{j}
  21. U U
  22. U U
  23. U U
  24. p p
  25. i = p mod 2 i=p\mod 2
  26. U = { v Ω ( v ) = p } U=\{v\mid\Omega(v)=p\}
  27. p p
  28. A = A t t r i ( U ) A=Attr_{i}(U)
  29. i i
  30. i i
  31. A A
  32. i i
  33. G = G A G^{\prime}=G\setminus A
  34. A A
  35. G G^{\prime}
  36. W i , W 1 - i W^{\prime}_{i},W^{\prime}_{1-i}
  37. W 1 - i W^{\prime}_{1-i}
  38. W 1 - i W_{1-i}
  39. G G
  40. 1 - i 1-i
  41. W i W_{i}
  42. A A
  43. i i
  44. W 1 - i W^{\prime}_{1-i}
  45. 1 - i 1-i
  46. W 1 - i W^{\prime}_{1-i}
  47. i i
  48. W 1 - i W^{\prime}_{1-i}
  49. A A
  50. A A
  51. 1 - i 1-i
  52. B = A t t r 1 - i ( W 1 - i ) B=Attr_{1-i}(W^{\prime}_{1-i})
  53. W 1 - i W^{\prime}_{1-i}
  54. G G
  55. G ′′ = G B G^{\prime\prime}=G\setminus B
  56. W i ′′ , W 1 - i ′′ W^{\prime\prime}_{i},W^{\prime\prime}_{1-i}
  57. W i = W i ′′ W_{i}=W^{\prime\prime}_{i}
  58. W 1 - i = W 1 - i ′′ B W_{1-i}=W^{\prime\prime}_{1-i}\cup B

Parry–Daniels_map.html

  1. Σ := { x = ( x 0 , x 1 , , x n ) n + 1 | 0 x i 1 for each i and x 0 + x 1 + + x n = 1 } . \Sigma:=\{x=(x_{0},x_{1},\dots,x_{n})\in\mathbb{R}^{n+1}|0\leq x_{i}\leq 1% \mbox{ for each }~{}i\mbox{ and }~{}x_{0}+x_{1}+\dots+x_{n}=1\}.
  2. x π ( 0 ) x π ( 1 ) x π ( n ) . x_{\pi(0)}\leq x_{\pi(1)}\leq\dots\leq x_{\pi(n)}.
  3. T π : Σ Σ T_{\pi}:\Sigma\to\Sigma
  4. T π ( x 0 , x 1 , , x n ) := ( x π ( 0 ) x π ( n ) , x π ( 1 ) - x π ( 0 ) x π ( n ) , , x π ( n ) - x π ( n - 1 ) x π ( n ) ) . T_{\pi}(x_{0},x_{1},\dots,x_{n}):=\left(\frac{x_{\pi(0)}}{x_{\pi(n)}},\frac{x_% {\pi(1)}-x_{\pi(0)}}{x_{\pi(n)}},\dots,\frac{x_{\pi(n)}-x_{\pi(n-1)}}{x_{\pi(n% )}}\right).

Parry–Sullivan_invariant.html

  1. PS ( A ) = det ( I - A ) , \mathrm{PS}(A)=\det(I-A),\,

Partial_fractions_in_complex_analysis.html

  1. lim k d ( Γ k ) = \lim_{k\rightarrow\infty}d(\Gamma_{k})=\infty
  2. lim k Γ k | f ( z ) z p + 1 | | d z | < \lim_{k\rightarrow\infty}\oint_{\Gamma_{k}}\left|\frac{f(z)}{z^{p+1}}\right||% dz|<\infty
  3. f ( z ) = k = 0 PP ( f ( z ) ; z = λ k ) , f(z)=\sum_{k=0}^{\infty}\operatorname{PP}(f(z);z=\lambda_{k}),
  4. f ( z ) = k = 0 ( PP ( f ( z ) ; z = λ k ) + c 0 , k + c 1 , k z + + c p , k z p ) , f(z)=\sum_{k=0}^{\infty}(\operatorname{PP}(f(z);z=\lambda_{k})+c_{0,k}+c_{1,k}% z+\cdots+c_{p,k}z^{p}),
  5. c j , k = Res z = λ k f ( z ) z j + 1 c_{j,k}=\operatorname{Res}_{z=\lambda_{k}}\frac{f(z)}{z^{j+1}}
  6. f ( z ) = a - m z m + a - m + 1 z m - 1 + + a 0 + a 1 z + f(z)=\frac{a_{-m}}{z^{m}}+\frac{a_{-m+1}}{z^{m-1}}+\cdots+a_{0}+a_{1}z+\cdots
  7. c j , k = Res z = 0 ( a - m z m + j + 1 + a - m + 1 z m + j + + a j z + ) = a j , c_{j,k}=\operatorname{Res}_{z=0}\left(\frac{a_{-m}}{z^{m+j+1}}+\frac{a_{-m+1}}% {z^{m+j}}+\cdots+\frac{a_{j}}{z}+\cdots\right)=a_{j},
  8. j = 0 p c j , k z j = a 0 + a 1 z + + a p z p \sum_{j=0}^{p}c_{j,k}z^{j}=a_{0}+a_{1}z+\cdots+a_{p}z^{p}
  9. c j , k = 1 λ k j + 1 Res z = λ k f ( z ) c_{j,k}=\frac{1}{\lambda_{k}^{j+1}}\operatorname{Res}_{z=\lambda_{k}}f(z)
  10. j = 0 p c j , k z j = [ Res z = λ k f ( z ) ] j = 0 p 1 λ k j + 1 z j \sum_{j=0}^{p}c_{j,k}z^{j}=[\operatorname{Res}_{z=\lambda_{k}}f(z)]\sum_{j=0}^% {p}\frac{1}{\lambda_{k}^{j+1}}z^{j}
  11. z = t ± π k i , t [ - π k , π k ] , z=t\pm\pi ki,\ \ t\in[-\pi k,\pi k],
  12. | tan ( z ) | 2 = | sin ( t ) cosh ( π k ) ± i cos ( t ) sinh ( π k ) cos ( t ) cosh ( π k ) ± i sin ( t ) sinh ( π k ) | 2 |\tan(z)|^{2}=\left|\frac{\sin(t)\cosh(\pi k)\pm i\cos(t)\sinh(\pi k)}{\cos(t)% \cosh(\pi k)\pm i\sin(t)\sinh(\pi k)}\right|^{2}
  13. | tan ( z ) | 2 = sin 2 ( t ) cosh 2 ( π k ) + cos 2 ( t ) sinh 2 ( π k ) cos 2 ( t ) cosh 2 ( π k ) + sin 2 ( t ) sinh 2 ( π k ) |\tan(z)|^{2}=\frac{\sin^{2}(t)\cosh^{2}(\pi k)+\cos^{2}(t)\sinh^{2}(\pi k)}{% \cos^{2}(t)\cosh^{2}(\pi k)+\sin^{2}(t)\sinh^{2}(\pi k)}
  14. Γ k | tan ( z ) z | d z length ( Γ k ) max z Γ k | tan ( z ) z | < 8 k π coth ( π ) k π = 8 coth ( π ) < . \oint_{\Gamma_{k}}\left|\frac{\tan(z)}{z}\right|dz\leq\operatorname{length}(% \Gamma_{k})\max_{z\in\Gamma_{k}}\left|\frac{\tan(z)}{z}\right|<8k\pi\frac{% \coth(\pi)}{k\pi}=8\coth(\pi)<\infty.
  15. tan ( z ) = k = 0 ( PP ( tan ( z ) ; z = λ k ) + Res z = λ k tan ( z ) z ) . \tan(z)=\sum_{k=0}^{\infty}(\operatorname{PP}(\tan(z);z=\lambda_{k})+% \operatorname{Res}_{z=\lambda_{k}}\frac{\tan(z)}{z}).
  16. PP ( tan ( z ) ; z = ( n + 1 2 ) π ) = - 1 z - ( n + 1 2 ) π \operatorname{PP}(\tan(z);z=(n+\frac{1}{2})\pi)=\frac{-1}{z-(n+\frac{1}{2})\pi}
  17. Res z = ( n + 1 2 ) π tan ( z ) z = - 1 ( n + 1 2 ) π \operatorname{Res}_{z=(n+\frac{1}{2})\pi}\frac{\tan(z)}{z}=\frac{-1}{(n+\frac{% 1}{2})\pi}
  18. tan ( z ) = k = 0 [ ( - 1 z - ( k + 1 2 ) π - 1 ( k + 1 2 ) π ) + ( - 1 z + ( k + 1 2 ) π + 1 ( k + 1 2 ) π ) ] \tan(z)=\sum_{k=0}^{\infty}\left[\left(\frac{-1}{z-(k+\frac{1}{2})\pi}-\frac{1% }{(k+\frac{1}{2})\pi}\right)+\left(\frac{-1}{z+(k+\frac{1}{2})\pi}+\frac{1}{(k% +\frac{1}{2})\pi}\right)\right]
  19. tan ( z ) = k = 0 - 2 z z 2 - ( k + 1 2 ) 2 π 2 \tan(z)=\sum_{k=0}^{\infty}\frac{-2z}{z^{2}-(k+\frac{1}{2})^{2}\pi^{2}}
  20. tan ( z ) = - k = 0 ( 1 z - ( k + 1 2 ) π + 1 z + ( k + 1 2 ) π ) \tan(z)=-\sum_{k=0}^{\infty}\left(\frac{1}{z-(k+\frac{1}{2})\pi}+\frac{1}{z+(k% +\frac{1}{2})\pi}\right)
  21. 0 z tan ( w ) d w = log sec z \int_{0}^{z}\tan(w)dw=\log\sec z
  22. 0 z 1 w ± ( k + 1 2 ) π d w = log ( 1 ± z ( k + 1 2 ) π ) \int_{0}^{z}\frac{1}{w\pm(k+\frac{1}{2})\pi}dw=\log\left(1\pm\frac{z}{(k+\frac% {1}{2})\pi}\right)
  23. log sec z = - k = 0 ( log ( 1 - z ( k + 1 2 ) π ) + log ( 1 + z ( k + 1 2 ) π ) ) \log\sec z=-\sum_{k=0}^{\infty}\left(\log\left(1-\frac{z}{(k+\frac{1}{2})\pi}% \right)+\log\left(1+\frac{z}{(k+\frac{1}{2})\pi}\right)\right)
  24. log cos z = k = 0 log ( 1 - z 2 ( k + 1 2 ) 2 π 2 ) , \log\cos z=\sum_{k=0}^{\infty}\log\left(1-\frac{z^{2}}{(k+\frac{1}{2})^{2}\pi^% {2}}\right),
  25. cos z = k = 0 ( 1 - z 2 ( k + 1 2 ) 2 π 2 ) . \cos z=\prod_{k=0}^{\infty}\left(1-\frac{z^{2}}{(k+\frac{1}{2})^{2}\pi^{2}}% \right).
  26. tan ( z ) = k = 0 - 2 z z 2 - ( k + 1 2 ) 2 π 2 = k = 0 - 8 z 4 z 2 - ( 2 k + 1 ) 2 π 2 . \tan(z)=\sum_{k=0}^{\infty}\frac{-2z}{z^{2}-(k+\frac{1}{2})^{2}\pi^{2}}=\sum_{% k=0}^{\infty}\frac{-8z}{4z^{2}-(2k+1)^{2}\pi^{2}}.
  27. - 8 z 4 z 2 - ( 2 k + 1 ) 2 π 2 = 8 z ( 2 k + 1 ) 2 π 2 1 1 - ( 2 z ( 2 k + 1 ) π ) 2 = 8 ( 2 k + 1 ) 2 π 2 n = 0 2 2 n ( 2 k + 1 ) 2 n π 2 n z 2 n + 1 . \frac{-8z}{4z^{2}-(2k+1)^{2}\pi^{2}}=\frac{8z}{(2k+1)^{2}\pi^{2}}\frac{1}{1-(% \frac{2z}{(2k+1)\pi})^{2}}=\frac{8}{(2k+1)^{2}\pi^{2}}\sum_{n=0}^{\infty}\frac% {2^{2n}}{(2k+1)^{2n}\pi^{2n}}z^{2n+1}.
  28. tan ( z ) = 2 k = 0 n = 0 2 2 n + 2 ( 2 k + 1 ) 2 n + 2 π 2 n + 2 z 2 n + 1 , \tan(z)=2\sum_{k=0}^{\infty}\sum_{n=0}^{\infty}\frac{2^{2n+2}}{(2k+1)^{2n+2}% \pi^{2n+2}}z^{2n+1},
  29. a 2 n + 1 = T 2 n + 1 ( 2 n + 1 ) ! = 2 2 n + 3 π 2 n + 2 k = 0 1 ( 2 k + 1 ) 2 n + 2 a_{2n+1}=\frac{T_{2n+1}}{(2n+1)!}=\frac{2^{2n+3}}{\pi^{2n+2}}\sum_{k=0}^{% \infty}\frac{1}{(2k+1)^{2n+2}}
  30. a 2 n = T 2 n ( 2 n ) ! = 0 , a_{2n}=\frac{T_{2n}}{(2n)!}=0,
  31. tan ( z ) = z + 1 3 z 3 + 2 15 z 5 + \tan(z)=z+\frac{1}{3}z^{3}+\frac{2}{15}z^{5}+\cdots
  32. k = 0 1 ( 2 k + 1 ) 2 = π 2 2 3 = π 2 8 \sum_{k=0}^{\infty}\frac{1}{(2k+1)^{2}}=\frac{\pi^{2}}{2^{3}}=\frac{\pi^{2}}{8}
  33. k = 0 1 ( 2 k + 1 ) 4 = 1 3 π 4 2 5 = π 4 96 . \sum_{k=0}^{\infty}\frac{1}{(2k+1)^{4}}=\frac{1}{3}\frac{\pi^{4}}{2^{5}}=\frac% {\pi^{4}}{96}.

Partial_geometry.html

  1. C = ( P , L , I ) C=(P,L,I)
  2. P P
  3. L L
  4. I P × L I\subseteq P\times L
  5. p p
  6. l l
  7. ( p , l ) I (p,l)\in I
  8. s , t , α 1 s,t,\alpha\geq 1
  9. p p
  10. q q
  11. s + 1 s+1
  12. t + 1 t+1
  13. p p
  14. l l
  15. α \alpha
  16. ( q , m ) I (q,m)\in I
  17. p p
  18. m m
  19. q q
  20. l l
  21. p g ( s , t , α ) pg(s,t,\alpha)
  22. ( s + 1 ) ( s t + α ) α \frac{(s+1)(st+\alpha)}{\alpha}
  23. ( t + 1 ) ( s t + α ) α \frac{(t+1)(st+\alpha)}{\alpha}
  24. p g ( s , t , α ) pg(s,t,\alpha)
  25. s r g ( ( s + 1 ) ( s t + α ) α , s ( t + 1 ) , s - 1 + t ( α - 1 ) , α ( t + 1 ) ) srg((s+1)\frac{(st+\alpha)}{\alpha},s(t+1),s-1+t(\alpha-1),\alpha(t+1))
  26. p g ( s , t , α ) pg(s,t,\alpha)
  27. p g ( t , s , α ) pg(t,s,\alpha)
  28. p g ( s , t , α ) pg(s,t,\alpha)
  29. α = 1 \alpha=1
  30. p g ( s , t , α ) pg(s,t,\alpha)
  31. α = s + 1 \alpha=s+1

Partial_specific_volume.html

  1. v i ¯ , \bar{v_{i}},
  2. V = i = 1 n m i v i ¯ , V=\sum_{i=1}^{n}m_{i}\bar{v_{i}},
  3. v i ¯ \bar{v_{i}}
  4. i i
  5. v i ¯ = ( V m i ) T , P , n j i . \bar{v_{i}}=\left(\frac{\partial V}{\partial m_{i}}\right)_{T,P,n_{j\neq i}}.
  6. v = i w i v i ¯ = 1 ρ v=\sum_{i}w_{i}\cdot\bar{v_{i}}=\frac{1}{\rho}
  7. i ρ i v i ¯ = 1 \sum_{i}\rho_{i}\cdot\bar{v_{i}}=1

Particle_decay.html

  1. c = = 1. c=\hbar=1.\,
  2. P ( t ) = e - t / ( γ τ ) P(t)=e^{-t/(\gamma\tau)}\,
  3. τ \tau
  4. γ = 1 1 - v 2 / c 2 \gamma=\frac{1}{\sqrt{1-v^{2}/c^{2}}}
  5. e - / e + e^{-}\,/\,e^{+}
  6. > 4.6 × 10 26 years >4.6\times 10^{26}\ \mathrm{years}\,
  7. μ - / μ + \mu^{-}\,/\,\mu^{+}
  8. 2.2 × 10 - 6 seconds 2.2\times 10^{-6}\ \mathrm{seconds}\,
  9. τ - / τ + \tau^{-}\,/\,\tau^{+}
  10. 2.9 × 10 - 13 seconds 2.9\times 10^{-13}\ \mathrm{seconds}\,
  11. π 0 \pi^{0}\,
  12. 8.4 × 10 - 17 seconds 8.4\times 10^{-17}\ \mathrm{seconds}\,
  13. π + / π - \pi^{+}\,/\,\pi^{-}
  14. 2.6 × 10 - 8 seconds 2.6\times 10^{-8}\ \mathrm{seconds}\,
  15. p + / p - p^{+}\,/\,p^{-}
  16. > 10 29 years >10^{29}\ \mathrm{years}\,
  17. n / n ¯ n\,/\,\bar{n}
  18. 885.7 seconds 885.7\ \mathrm{seconds}\,
  19. W + / W - W^{+}\,/\,W^{-}
  20. 10 - 25 seconds 10^{-25}\ \mathrm{seconds}\,
  21. Z 0 Z^{0}\,
  22. 10 - 25 seconds 10^{-25}\ \mathrm{seconds}\,
  23. Γ \Gamma
  24. p i p_{i}
  25. d Γ n = S | | 2 2 M d Φ n ( P ; p 1 , p 2 , , p n ) d\Gamma_{n}=\frac{S\left|\mathcal{M}\right|^{2}}{2M}d\Phi_{n}(P;p_{1},p_{2},% \dots,p_{n})\,
  26. \mathcal{M}\,
  27. d Φ n d\Phi_{n}\,
  28. p i p_{i}\,
  29. S = j = 1 m 1 k j ! S=\prod_{j=1}^{m}\frac{1}{k_{j}!}\,
  30. k j k_{j}\,
  31. j = 1 m k j = n \sum_{j=1}^{m}k_{j}=n\,
  32. d Φ n ( P ; p 1 , p 2 , , p n ) = ( 2 π ) 4 δ 4 ( P - i = 1 n p i ) i = 1 n d 3 p i 2 ( 2 π ) 3 E i d\Phi_{n}(P;p_{1},p_{2},\dots,p_{n})=(2\pi)^{4}\delta^{4}\left(P-\sum_{i=1}^{n% }p_{i}\right)\prod_{i=1}^{n}\frac{d^{3}\vec{p}_{i}}{2(2\pi)^{3}E_{i}}
  33. δ 4 \delta^{4}\,
  34. p i \vec{p}_{i}\,
  35. E i E_{i}\,
  36. | p 1 | = | p 2 | = [ ( M 2 - ( m 1 + m 2 ) 2 ) ( M 2 - ( m 1 - m 2 ) 2 ) ] 1 / 2 2 M , |\vec{p}_{1}|=|\vec{p_{2}}|=\frac{[(M^{2}-(m_{1}+m_{2})^{2})(M^{2}-(m_{1}-m_{2% })^{2})]^{1/2}}{2M},\,
  37. ( M , 0 ) = ( E 1 , p 1 ) + ( E 2 , p 2 ) . (M,\vec{0})=(E_{1},\vec{p}_{1})+(E_{2},\vec{p}_{2}).\,
  38. d 3 p = | p | 2 d | p | d ϕ d ( cos θ ) . d^{3}\vec{p}=|\vec{p}\,|^{2}\,d|\vec{p}\,|\,d\phi\,d\left(\cos\theta\right).\,
  39. d 3 p 2 d^{3}\vec{p}_{2}
  40. d | p 1 | d|\vec{p}_{1}|\,
  41. d Γ = | | 2 32 π 2 | p 1 | M 2 d ϕ 1 d ( cos θ 1 ) . d\Gamma=\frac{\left|\mathcal{M}\right|^{2}}{32\pi^{2}}\frac{|\vec{p}_{1}|}{M^{% 2}}\,d\phi_{1}\,d\left(\cos\theta_{1}\right).\,
  42. tan θ = sin θ γ ( β / β + cos θ ) \tan{\theta^{\prime}}=\frac{\sin{\theta}}{\gamma\left(\beta/\beta^{\prime}+% \cos{\theta}\right)}
  43. d Φ 3 = 1 ( 2 π ) 5 δ 4 ( P - p 1 - p 2 - p 3 ) d 3 p 1 2 E 1 d 3 p 2 2 E 2 d 3 p 3 2 E 3 d\Phi_{3}=\frac{1}{(2\pi)^{5}}\delta^{4}(P-p_{1}-p_{2}-p_{3})\frac{d^{3}\vec{p% }_{1}}{2E_{1}}\frac{d^{3}\vec{p}_{2}}{2E_{2}}\frac{d^{3}\vec{p}_{3}}{2E_{3}}\,
  44. M + i Γ \scriptstyle M+i\Gamma
  45. 1 / Γ \scriptstyle 1/\Gamma
  46. Γ > M \scriptstyle\Gamma>M

Particular_point_topology.html

  1. A X A\subset X
  2. x p x\neq p
  3. \emptyset
  4. X X
  5. X X
  6. X X
  7. \emptyset
  8. f ( t ) = { x t = 0 p t ( 0 , 1 ) y t = 1 f(t)=\begin{cases}x&t=0\\ p&t\in(0,1)\\ y&t=1\end{cases}
  9. { x , p } \{x,p\}
  10. x X { p , x } \bigcup_{x\in X}\{p,x\}
  11. p , q X p,q\in X
  12. p q p\neq q
  13. t p = { S X | p S } t_{p}=\{S\subset X\,|\,p\in S\}
  14. t q = { S X | q S } t_{q}=\{S\subset X\,|\,q\in S\}

Partition_topology.html

  1. X = X=\mathbb{N}
  2. P = { { 2 k - 1 , 2 k } , k } . P={\left\{\{2k-1,2k\},k\in\mathbb{N}\right\}}.
  3. X = n ( n - 1 , n ) X=\begin{matrix}\bigcup_{n\in\mathbb{N}}(n-1,n)\subset\mathbb{R}\end{matrix}
  4. P = { ( 0 , 1 ) , ( 1 , 2 ) , ( 2 , 3 ) , } P={\left\{(0,1),(1,2),(2,3),\dots\right\}}
  5. P = { X } P=\{X\}
  6. d ( x , y ) = { 0 if x and y are in the same partition 1 otherwise , d(x,y)=\begin{cases}0&\,\text{if }x\,\text{ and }y\,\text{ are in the same % partition}\\ 1&\,\text{otherwise},\end{cases}

Pascal_matrix.html

  1. U 5 = ( 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 ) ; U_{5}=\begin{pmatrix}1&1&1&1&1\\ 0&1&2&3&4\\ 0&0&1&3&6\\ 0&0&0&1&4\\ 0&0&0&0&1\end{pmatrix};\,\,\,
  2. L 5 = ( 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 ) ; L_{5}=\begin{pmatrix}1&0&0&0&0\\ 1&1&0&0&0\\ 1&2&1&0&0\\ 1&3&3&1&0\\ 1&4&6&4&1\end{pmatrix};\,\,\,
  3. S 5 = ( 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70 ) . S_{5}=\begin{pmatrix}1&1&1&1&1\\ 1&2&3&4&5\\ 1&3&6&10&15\\ 1&4&10&20&35\\ 1&5&15&35&70\end{pmatrix}.
  4. S i j = ( n r ) = n ! r ! ( n - r ) ! , where n = i + j , r = i . S_{ij}={n\choose r}=\frac{n!}{r!(n-r)!},\,\text{ where }n=i+j,\quad r=i.
  5. S i j = 𝐂 i i + j = ( i + j ) ! ( i ) ! ( j ) ! . S_{ij}={}_{i+j}\mathbf{C}_{i}=\frac{(i+j)!}{(i)!(j)!}.
  6. tr ( S n ) = i = 1 n [ 2 ( i - 1 ) ] ! [ ( i - 1 ) ! ] 2 = k = 0 n - 1 ( 2 k ) ! ( k ! ) 2 \,\text{tr}(S_{n})=\sum^{n}_{i=1}\frac{[2(i-1)]!}{[(i-1)!]^{2}}=\sum^{n-1}_{k=% 0}\frac{(2k)!}{(k!)^{2}}
  7. L 7 = exp ( [ . . . . . . . 1 . . . . . . . 2 . . . . . . . 3 . . . . . . . 4 . . . . . . . 5 . . . . . . . 6 . ] ) = [ 1 . . . . . . 1 1 . . . . . 1 2 1 . . . . 1 3 3 1 . . . 1 4 6 4 1 . . 1 5 10 10 5 1 . 1 6 15 20 15 6 1 ] ; U 7 = exp ( [ . 1 . . . . . . . 2 . . . . . . . 3 . . . . . . . 4 . . . . . . . 5 . . . . . . . 6 . . . . . . . ] ) = [ 1 1 1 1 1 1 1 . 1 2 3 4 5 6 . . 1 3 6 10 15 . . . 1 4 10 20 . . . . 1 5 15 . . . . . 1 6 . . . . . . 1 ] ; S 7 = exp ( [ . . . . . . . 1 . . . . . . . 2 . . . . . . . 3 . . . . . . . 4 . . . . . . . 5 . . . . . . . 6 . ] ) exp ( [ . 1 . . . . . . . 2 . . . . . . . 3 . . . . . . . 4 . . . . . . . 5 . . . . . . . 6 . . . . . . . ] ) = [ 1 1 1 1 1 1 1 1 2 3 4 5 6 7 1 3 6 10 15 21 28 1 4 10 20 35 56 84 1 5 15 35 70 126 210 1 6 21 56 126 252 462 1 7 28 84 210 462 924 ] . \begin{array}[]{lll}&L_{7}=\exp\left(\left[\begin{smallmatrix}.&.&.&.&.&.&.\\ 1&.&.&.&.&.&.\\ .&2&.&.&.&.&.\\ .&.&3&.&.&.&.\\ .&.&.&4&.&.&.\\ .&.&.&.&5&.&.\\ .&.&.&.&.&6&.\end{smallmatrix}\right]\right)=\left[\begin{smallmatrix}1&.&.&.&% .&.&.\\ 1&1&.&.&.&.&.\\ 1&2&1&.&.&.&.\\ 1&3&3&1&.&.&.\\ 1&4&6&4&1&.&.\\ 1&5&10&10&5&1&.\\ 1&6&15&20&15&6&1\end{smallmatrix}\right];\\ \\ &U_{7}=\exp\left(\left[\begin{smallmatrix}.&1&.&.&.&.&.\\ .&.&2&.&.&.&.\\ .&.&.&3&.&.&.\\ .&.&.&.&4&.&.\\ .&.&.&.&.&5&.\\ .&.&.&.&.&.&6\\ .&.&.&.&.&.&.\end{smallmatrix}\right]\right)=\left[\begin{smallmatrix}1&1&1&1&% 1&1&1\\ .&1&2&3&4&5&6\\ .&.&1&3&6&10&15\\ .&.&.&1&4&10&20\\ .&.&.&.&1&5&15\\ .&.&.&.&.&1&6\\ .&.&.&.&.&.&1\end{smallmatrix}\right];\\ \\ \therefore&S_{7}=\exp\left(\left[\begin{smallmatrix}.&.&.&.&.&.&.\\ 1&.&.&.&.&.&.\\ .&2&.&.&.&.&.\\ .&.&3&.&.&.&.\\ .&.&.&4&.&.&.\\ .&.&.&.&5&.&.\\ .&.&.&.&.&6&.\end{smallmatrix}\right]\right)\exp\left(\left[\begin{smallmatrix% }.&1&.&.&.&.&.\\ .&.&2&.&.&.&.\\ .&.&.&3&.&.&.\\ .&.&.&.&4&.&.\\ .&.&.&.&.&5&.\\ .&.&.&.&.&.&6\\ .&.&.&.&.&.&.\end{smallmatrix}\right]\right)=\left[\begin{smallmatrix}1&1&1&1&% 1&1&1\\ 1&2&3&4&5&6&7\\ 1&3&6&10&15&21&28\\ 1&4&10&20&35&56&84\\ 1&5&15&35&70&126&210\\ 1&6&21&56&126&252&462\\ 1&7&28&84&210&462&924\end{smallmatrix}\right].\end{array}
  8. L A G 7 = exp ( [ . . . . . . . 1 . . . . . . . 4 . . . . . . . 9 . . . . . . . 16 . . . . . . . 25 . . . . . . . 36 . ] ) = [ 1 . . . . . . 1 1 . . . . . 2 4 1 . . . . 6 18 9 1 . . . 24 96 72 16 1 . . 120 600 600 200 25 1 . 720 4320 5400 2400 450 36 1 ] ; \begin{array}[]{lll}&LAG_{7}=\exp\left(\left[\begin{smallmatrix}.&.&.&.&.&.&.% \\ 1&.&.&.&.&.&.\\ .&4&.&.&.&.&.\\ .&.&9&.&.&.&.\\ .&.&.&16&.&.&.\\ .&.&.&.&25&.&.\\ .&.&.&.&.&36&.\end{smallmatrix}\right]\right)=\left[\begin{smallmatrix}1&.&.&.% &.&.&.\\ 1&1&.&.&.&.&.\\ 2&4&1&.&.&.&.\\ 6&18&9&1&.&.&.\\ 24&96&72&16&1&.&.\\ 120&600&600&200&25&1&.\\ 720&4320&5400&2400&450&36&1\end{smallmatrix}\right];\end{array}
  9. L A H 7 = exp ( [ . . . . . . . 2 . . . . . . . 6 . . . . . . . 12 . . . . . . . 20 . . . . . . . 30 . . . . . . . 42 . ] ) = [ 1 . . . . . . . 2 1 . . . . . . 6 6 1 . . . . . 24 36 12 1 . . . . 120 240 120 20 1 . . . 720 1800 1200 300 30 1 . . 5040 15120 12600 4200 630 42 1 . 40320 141120 141120 58800 11760 1176 56 1 ] ; \begin{array}[]{lll}&LAH_{7}=\exp\left(\left[\begin{smallmatrix}.&.&.&.&.&.&.% \\ 2&.&.&.&.&.&.\\ .&6&.&.&.&.&.\\ .&.&12&.&.&.&.\\ .&.&.&20&.&.&.\\ .&.&.&.&30&.&.\\ .&.&.&.&.&42&.\end{smallmatrix}\right]\right)=\left[\begin{smallmatrix}1&.&.&.% &.&.&.&.\\ 2&1&.&.&.&.&.&.\\ 6&6&1&.&.&.&.&.\\ 24&36&12&1&.&.&.&.\\ 120&240&120&20&1&.&.&.\\ 720&1800&1200&300&30&1&.&.\\ 5040&15120&12600&4200&630&42&1&.\\ 40320&141120&141120&58800&11760&1176&56&1\end{smallmatrix}\right];\end{array}
  10. G S 7 = exp ( [ . . . . . . . . . . . . . . 1 . . . . . . . 3 . . . . . . . 6 . . . . . . . 10 . . . . . . . 15 . . ] ) = [ 1 . . . . . . . 1 . . . . . 1 . 1 . . . . . 3 . 1 . . . 3 . 6 . 1 . . . 15 . 10 . 1 . 15 . 45 . 15 . 1 ] ; \begin{array}[]{lll}&GS_{7}=\exp\left(\left[\begin{smallmatrix}.&.&.&.&.&.&.\\ .&.&.&.&.&.&.\\ 1&.&.&.&.&.&.\\ .&3&.&.&.&.&.\\ .&.&6&.&.&.&.\\ .&.&.&10&.&.&.\\ .&.&.&.&15&.&.\end{smallmatrix}\right]\right)=\left[\begin{smallmatrix}1&.&.&.% &.&.&.\\ .&1&.&.&.&.&.\\ 1&.&1&.&.&.&.\\ .&3&.&1&.&.&.\\ 3&.&6&.&1&.&.\\ .&15&.&10&.&1&.\\ 15&.&45&.&15&.&1\end{smallmatrix}\right];\end{array}
  11. exp ( [ . . . . . . . . . . . . - 5 . . . . . . . . . . . . - 4 . . . . . . . . . . . . - 3 . . . . . . . . . . . . - 2 . . . . . . . . . . . . - 1 . . . . . . . . . . . . 0 . . . . . . . . . . . . 1 . . . . . . . . . . . . 2 . . . . . . . . . . . . 3 . . . . . . . . . . . . 4 . . . . . . . . . . . . 5 . ] ) = [ 1 . . . . . . . . . . . - 5 1 . . . . . . . . . . 10 - 4 1 . . . . . . . . . - 10 6 - 3 1 . . . . . . . . 5 - 4 3 - 2 1 . . . . . . . - 1 1 - 1 1 - 1 1 . . . . . . . . . . . 0 1 . . . . . . . . . . . 1 1 . . . . . . . . . . 1 2 1 . . . . . . . . . 1 3 3 1 . . . . . . . . 1 4 6 4 1 . . . . . . . 1 5 10 10 5 1 ] . \begin{array}[]{lll}&\exp\left(\left[\begin{smallmatrix}.&.&.&.&.&.&.&.&.&.&.&% .\\ -5&.&.&.&.&.&.&.&.&.&.&.\\ .&-4&.&.&.&.&.&.&.&.&.&.\\ .&.&-3&.&.&.&.&.&.&.&.&.\\ .&.&.&-2&.&.&.&.&.&.&.&.\\ .&.&.&.&-1&.&.&.&.&.&.&.\\ .&.&.&.&.&0&.&.&.&.&.&.\\ .&.&.&.&.&.&1&.&.&.&.&.\\ .&.&.&.&.&.&.&2&.&.&.&.\\ .&.&.&.&.&.&.&.&3&.&.&.\\ .&.&.&.&.&.&.&.&.&4&.&.\\ .&.&.&.&.&.&.&.&.&.&5&.\end{smallmatrix}\right]\right)=\left[\begin{% smallmatrix}1&.&.&.&.&.&.&.&.&.&.&.\\ -5&1&.&.&.&.&.&.&.&.&.&.\\ 10&-4&1&.&.&.&.&.&.&.&.&.\\ -10&6&-3&1&.&.&.&.&.&.&.&.\\ 5&-4&3&-2&1&.&.&.&.&.&.&.\\ -1&1&-1&1&-1&1&.&.&.&.&.&.\\ .&.&.&.&.&0&1&.&.&.&.&.\\ .&.&.&.&.&.&1&1&.&.&.&.\\ .&.&.&.&.&.&1&2&1&.&.&.\\ .&.&.&.&.&.&1&3&3&1&.&.\\ .&.&.&.&.&.&1&4&6&4&1&.\\ .&.&.&.&.&.&1&5&10&10&5&1\end{smallmatrix}\right].\end{array}

Pastry_(DHT).html

  1. 2 b - 1 2^{b}-1
  2. ( log 2 N ) / b (\log_{2}{N})/b
  3. b 4 , L 2 b b\approx 4,L\approx 2^{b}
  4. M 2 b M\approx 2^{b}

Peirce_quincuncial_projection.html

  1. tan ( p 2 ) e i θ = cn ( z , 1 2 ) , where w = p e i θ and z = x + i y . \tan\left(\frac{p}{2}\right)e^{i\theta}=\mathrm{cn}\left(z,\frac{1}{2}\right),% \,\text{ where }w=pe^{i\theta}\,\text{ and }z=x+iy.

Penalty_method.html

  1. min f ( x ) \min f(x)
  2. c i ( x ) 0 i I . c_{i}(x)\geq 0~{}\forall i\in I.
  3. min Φ k ( x ) = f ( x ) + σ k i I g ( c i ( x ) ) \min\Phi_{k}(x)=f(x)+\sigma_{k}~{}\sum_{i\in I}~{}g(c_{i}(x))
  4. g ( c i ( x ) ) = min ( 0 , c i ( x ) ) 2 . g(c_{i}(x))=\min(0,~{}c_{i}(x))^{2}.
  5. g ( c i ( x ) ) g(c_{i}(x))
  6. σ k \sigma_{k}
  7. σ k \sigma_{k}

Pendulum_(mathematics).html

  1. g g
  2. \ell
  3. θ \theta
  4. τ = 𝐥 × 𝐅 𝐠 , \mathbf{\tau}=\mathbf{l\times F_{g}},
  5. 𝐥 \mathbf{l}
  6. 𝐅 𝐠 \mathbf{F_{g}}
  7. | τ | = - m g l sin θ , \mathbf{|\tau|}=-mgl\sin\theta,
  8. m m
  9. g g
  10. l l
  11. θ \theta
  12. 𝐋 = 𝐫 × 𝐩 = m 𝐫 × ( ω × 𝐫 ) \mathbf{L}=\mathbf{r\times p}=m\mathbf{r\times(\omega\times r)}
  13. | 𝐋 | = m r 2 ω = m l 2 d θ d t \mathbf{|L|}=mr^{2}\omega=ml^{2}{d\theta\over dt}
  14. d d t | 𝐋 | = m l 2 d 2 θ d t 2 {d\over dt}\mathbf{|L|}=ml^{2}{d^{2}\theta\over dt^{2}}
  15. τ = d 𝐋 d t {\mathbf{\tau}={d\mathbf{L}\over dt}}
  16. - m g l sin θ = m l 2 d 2 θ d t 2 -mgl\sin\theta=ml^{2}{d^{2}\theta\over dt^{2}}
  17. d 2 θ d t 2 + g l sin θ = 0 , {d^{2}\theta\over dt^{2}}+{g\over l}\sin\theta=0,
  18. θ 1 \theta\ll 1\,
  19. sin θ θ \sin\theta\approx\theta\,
  20. d 2 θ d t 2 + g θ = 0. {d^{2}\theta\over dt^{2}}+{g\over\ell}\theta=0.
  21. θ ( t ) = θ 0 cos ( g t ) θ 0 1. \theta(t)=\theta_{0}\cos\left(\sqrt{g\over\ell\,}\,t\right)\quad\quad\quad% \quad\theta_{0}\ll 1.
  22. T 0 = 2 π g θ 0 1 T_{0}=2\pi\sqrt{\frac{\ell}{g}}\quad\quad\quad\quad\quad\theta_{0}\ll 1
  23. T 0 = 2 π g T_{0}=2\pi\sqrt{\frac{\ell}{g}}
  24. = g π 2 × T 0 2 4 . \ell={\frac{g}{\pi^{2}}}\times{\frac{T_{0}^{2}}{4}}.
  25. g 9.81 g\approx 9.81
  26. g / π 2 1 g/\pi^{2}\approx{1}
  27. T 0 2 4 , \ell\approx{\frac{T_{0}^{2}}{4}},
  28. T 0 2 T_{0}\approx 2\sqrt{\ell}
  29. d t d θ = 2 g 1 cos θ - cos θ 0 {dt\over d\theta}=\sqrt{\ell\over 2g}{1\over\sqrt{\cos\theta-\cos\theta_{0}}}
  30. T = t ( θ 0 0 - θ 0 0 θ 0 ) , T=t(\theta_{0}\rightarrow 0\rightarrow-\theta_{0}\rightarrow 0\rightarrow% \theta_{0}),
  31. T = 2 t ( θ 0 0 - θ 0 ) , T=2t\left(\theta_{0}\rightarrow 0\rightarrow-\theta_{0}\right),
  32. T = 4 t ( θ 0 0 ) , T=4t\left(\theta_{0}\rightarrow 0\right),
  33. T = 4 2 g 0 θ 0 1 cos θ - cos θ 0 d θ . T=4\sqrt{\ell\over 2g}\int^{\theta_{0}}_{0}{1\over\sqrt{\cos\theta-\cos\theta_% {0}}}\,d\theta.
  34. θ 0 \theta_{0}
  35. lim θ 0 π T = \lim_{\theta_{0}\rightarrow\pi}T=\infty
  36. T = 4 g F ( θ 0 , csc θ 0 2 ) csc θ 0 2 T=4\sqrt{\ell\over g}F\left({\theta_{0}},\csc{\theta_{0}\over 2}\right)\csc{% \theta_{0}\over 2}
  37. F F
  38. F ( φ , k ) = 0 φ 1 1 - k 2 sin 2 u d u . F(\varphi,k)=\int_{0}^{\varphi}{1\over\sqrt{1-k^{2}\sin^{2}{u}}}\,du\,.
  39. sin u = sin θ 2 sin θ 0 2 \sin{u}=\frac{\sin{\theta\over 2}}{\sin{\theta_{0}\over 2}}
  40. θ \theta
  41. u u
  42. K K
  43. K ( k ) = F ( π 2 , k ) = 0 π / 2 1 1 - k 2 sin 2 u d u . K(k)=F\left({\pi\over 2},k\right)=\int_{0}^{\pi/2}{1\over\sqrt{1-k^{2}\sin^{2}% {u}}}\,du\,.
  44. 4 1 m g K ( sin 10 2 ) 2.0102 s 4\sqrt{1\ \mathrm{m}\over g}K\left({\sin{10^{\circ}\over 2}}\right)\approx 2.0% 102\ \mathrm{s}
  45. 2 π 1 m g 2.0064 s 2\pi\sqrt{1\ \mathrm{m}\over g}\approx 2.0064\ \mathrm{s}
  46. K ( k ) = π 2 { 1 + ( 1 2 ) 2 k 2 + ( 1 3 2 4 ) 2 k 4 + + [ ( 2 n - 1 ) ! ! ( 2 n ) ! ! ] 2 k 2 n + } , K(k)=\frac{\pi}{2}\left\{1+\left(\frac{1}{2}\right)^{2}k^{2}+\left(\frac{1% \cdot 3}{2\cdot 4}\right)^{2}k^{4}+\cdots+\left[\frac{\left(2n-1\right)!!}{% \left(2n\right)!!}\right]^{2}k^{2n}+\cdots\right\},
  47. n ! ! n!!
  48. T \displaystyle T
  49. sin θ 0 2 = 1 2 θ 0 - 1 48 θ 0 3 + 1 3840 θ 0 5 - 1 645120 θ 0 7 + . \sin{\theta_{0}\over 2}=\frac{1}{2}\theta_{0}-\frac{1}{48}\theta_{0}^{3}+\frac% {1}{3840}\theta_{0}^{5}-\frac{1}{645120}\theta_{0}^{7}+\cdots.
  50. T = 2 π g ( 1 + 1 16 θ 0 2 + 11 3072 θ 0 4 + 173 737280 θ 0 6 + 22931 1321205760 θ 0 8 + 1319183 951268147200 θ 0 10 + 233526463 2009078326886400 θ 0 12 + ) . \begin{aligned}\displaystyle T&\displaystyle=2\pi\sqrt{\ell\over g}\left(1+% \frac{1}{16}\theta_{0}^{2}+\frac{11}{3072}\theta_{0}^{4}+\frac{173}{737280}% \theta_{0}^{6}+\frac{22931}{1321205760}\theta_{0}^{8}+\frac{1319183}{951268147% 200}\theta_{0}^{10}+\frac{233526463}{2009078326886400}\theta_{0}^{12}+...% \right)\end{aligned}.
  51. K ( k ) = π / 2 M ( 1 - k , 1 + k ) , K(k)=\frac{\pi/2}{M(1-k,1+k)},
  52. M ( x , y ) M(x,y)
  53. x x
  54. y y
  55. T = 2 π M ( 1 , cos ( θ 0 / 2 ) ) g . T=\frac{2\pi}{M(1,\cos(\theta_{0}/2))}\sqrt{\frac{\ell}{g}}.
  56. τ = I α \tau=I\alpha\,
  57. α \alpha
  58. τ \tau
  59. τ = - m g L sin θ \tau=-mgL\sin\theta\,
  60. sin θ θ \scriptstyle\sin\theta\approx\theta\,
  61. α - m g L θ I \alpha\approx-\frac{mgL\theta}{I}
  62. T = 2 π I m g L T=2\pi\sqrt{\frac{I}{mgL}}
  63. f = 1 T = 1 2 π m g L I f=\frac{1}{T}=\frac{1}{2\pi}\sqrt{\frac{mgL}{I}}

Penetration_depth.html

  1. I ( z ) = I 0 e - α z I(z)=I_{0}\,e^{-\alpha z}
  2. δ p \delta_{p}
  3. δ p = 1 α \delta_{p}=\frac{1}{\alpha}
  4. δ p \delta_{p}
  5. 1 / e 2 1/e^{2}
  6. δ e = 1 α / 2 = 2 α = 2 δ p \delta_{e}=\frac{1}{\alpha/2}=\frac{2}{\alpha}=2\delta_{p}
  7. δ e \delta_{e}
  8. α / 2 \alpha/2
  9. α \alpha
  10. α \alpha
  11. δ \delta
  12. α \alpha
  13. α / 2 = 1 δ e = 1 2 δ p = ω c Im ( n ~ ( ω ) ) \alpha/2=\frac{1}{\delta_{e}}=\frac{1}{2\delta_{p}}=\frac{\omega}{c}\;\mathrm{% Im}(\tilde{n}(\omega))
  14. n ~ \tilde{n}
  15. ω \omega
  16. n ~ ( ω ) \tilde{n}(\omega)
  17. δ e \delta_{e}
  18. n ~ = 1 + .01 j \tilde{n}=1+.01j
  19. δ e 16 λ \delta_{e}\approx 16\lambda
  20. λ \lambda
  21. n ~ = 0 + .01 j \tilde{n}=0+.01j

Pennate_muscle.html

  1. PCSA = muscle volume fiber length = muscle mass ρ fiber length , \,\text{PCSA}={\,\text{muscle volume}\over\,\text{fiber length}}={\,\text{% muscle mass}\over{\rho\cdot\,\text{fiber length}}},
  2. ρ = muscle mass muscle volume . \rho={\,\text{muscle mass}\over\,\text{muscle volume}}.
  3. Total force = PCSA Specific tension \,\text{Total force}=\,\text{PCSA}\cdot\,\text{Specific tension}
  4. Muscle force = Total force cos Φ \,\text{Muscle force}=\,\text{Total force}\cdot\cos\Phi

Penning_ionization.html

  1. G * + M M + + e - + G G^{*}+M\to M^{+\bullet}+e^{-}+G
  2. G * + M M G + + e - G^{*}+M\to MG^{+\bullet}+e^{-}
  3. G * + S G + S + e - G^{*}+S\to G+S+e^{-}
  4. S + S^{+}
  5. * {}^{*}
  6. * {}^{*}
  7. E = E m + I E E=E\text{m}+IE
  8. m {}_{m}
  9. * {}^{*}

Per_Enflo.html

  1. m 2 m\geq 2
  2. C m C_{m}
  3. D D
  4. 2 m 2^{m}
  5. D < m D<\sqrt{m}
  6. { 0 , 1 } m \{0,1\}^{m}
  7. m m
  8. ϵ > 0 \epsilon>0
  9. δ > 0 \delta>0
  10. x 1 \|x\|\leq 1
  11. y 1 , \|y\|\leq 1,
  12. x + y > 2 - δ \|x+y\|>2-\delta
  13. x - y < ϵ . \|x-y\|<\epsilon.
  14. v = n 𝒩 α n b n v=\sum_{n\in\mathcal{N}}\alpha_{n}b_{n}\,
  15. m m
  16. n n
  17. C ( m , n ) > 0 C(m,n)>0
  18. P P
  19. Q Q
  20. m m
  21. n n
  22. k k
  23. | P Q | C ( m , n ) | P | | Q | , |PQ|\geq C(m,n)|P|\,|Q|,
  24. | P | |P|
  25. P P
  26. C ( m , n ) C(m,n)
  27. k k
  28. α , β N \alpha,\beta\in\mathbb{N}^{N}
  29. X α | X β = 0 \langle X^{\alpha}|X^{\beta}\rangle=0
  30. α β \alpha\neq\beta
  31. α N \alpha\in\mathbb{N}^{N}
  32. || X α || 2 = | α | ! α ! , ||X^{\alpha}||^{2}=\frac{|\alpha|!}{\alpha!},
  33. α = ( α 1 , , α N ) N \alpha=(\alpha_{1},\dots,\alpha_{N})\in\mathbb{N}^{N}
  34. | α | = Σ i = 1 N α i |\alpha|=\Sigma_{i=1}^{N}\alpha_{i}
  35. α ! = Π i = 1 N ( α i ! ) \alpha!=\Pi_{i=1}^{N}(\alpha_{i}!)
  36. X α = Π i = 1 N X i α i . X^{\alpha}=\Pi_{i=1}^{N}X_{i}^{\alpha_{i}}.
  37. P , Q P,Q
  38. d ( P ) d^{\circ}(P)
  39. d ( Q ) d^{\circ}(Q)
  40. N N
  41. d ( P ) ! d ( Q ) ! ( d ( P ) + d ( Q ) ) ! || P || 2 || Q || 2 || P Q || 2 || P || 2 || Q || 2 . \frac{d^{\circ}(P)!d^{\circ}(Q)!}{(d^{\circ}(P)+d^{\circ}(Q))!}||P||^{2}\,||Q|% |^{2}\leq||P\cdot Q||^{2}\leq||P||^{2}\,||Q||^{2}.
  42. L p L^{p}

Perceptual_paradox.html

  1. S U N w h i t e SUN_{white}
  2. R G B w h i t e RGB_{white}
  3. C o n t r a s t l o g I a l o g I b Contrast\propto\frac{logI_{a}}{logI_{b}}
  4. I a I_{a}\,
  5. l o g I b logI_{b}\,
  6. R G B w h i t e RGB_{white}
  7. R G B w h i t e RGB_{white}
  8. R G B w h i t e RGB_{white}
  9. R G B w h i t e RGB_{white}

Perceptual_transparency.html

  1. P P
  2. P = t * A + ( 1 - t ) * R P=t*A+(1-t)*R

Perfect_set.html

  1. \mathbb{R}
  2. { X i } i I \{X_{i}\}_{i\in I}
  3. i X i \coprod_{i}X_{i}
  4. X i X_{i}
  5. { X i } i I \{X_{i}\}_{i\in I}
  6. X i X_{i}
  7. I = I=\emptyset
  8. i I i\in I
  9. X i X_{i}
  10. ω \omega
  11. X = { , } X=\{\circ,\bullet\}
  12. { , { } , X } \{\emptyset,\{\circ\},X\}
  13. Aut ( X ) \operatorname{Aut}(X)
  14. 2 0 2^{\aleph_{0}}
  15. 2 0 2^{\aleph_{0}}
  16. 2 0 2^{\aleph_{0}}

Performance_engineering.html

  1. 1 α + 1 - α P \frac{1}{\alpha+\frac{1-\alpha}{P}}

Perifocal_coordinate_system.html

  1. p ^ {\hat{p}}
  2. q ^ {\hat{q}}
  3. p ^ {\hat{p}}
  4. q ^ {\hat{q}}
  5. w ^ {\hat{w}}
  6. w ^ = r × v {\hat{w}}={r}\times{v}
  7. w ^ {\hat{w}}
  8. 𝐰 ^ = 𝐡 𝐡 \mathbf{\hat{w}}=\frac{\mathbf{h}}{\|\mathbf{h}\|}
  9. r = r cos θ 𝐩 ^ + r sin θ 𝐪 ^ {r}=\|r\|\cos\theta\mathbf{\hat{p}}+\|r\|\sin\theta\mathbf{\hat{q}}
  10. v = r ˙ = ( r ˙ cos θ - r θ ˙ sin θ ) p ^ + ( r ˙ sin θ + r θ ˙ cos θ ) q ^ {v}={\dot{r}}=(\dot{r}\cos\theta-r\dot{\theta}\sin\theta){\hat{p}}+(\dot{r}% \sin\theta+r\dot{\theta}\cos\theta){\hat{q}}
  11. r ˙ = μ h e sin θ \dot{r}=\frac{\mu}{h}e\sin\theta
  12. μ \mu
  13. r ˙ \dot{r}
  14. r θ ˙ r\dot{\theta}
  15. r ˙ \dot{r}
  16. r θ ˙ r\dot{\theta}
  17. v = μ h [ - sin θ p ^ + ( e + cos θ ) q ^ ] {v}=\frac{\mu}{h}[-\sin\theta{\hat{p}}+(e+\cos\theta){\hat{q}}]
  18. p i = cos Ω cos ω - sin Ω cos i sin ω p j = sin Ω cos ω + cos Ω cos i sin ω p k = sin i sin ω q i = - cos Ω sin ω - sin Ω cos i cos ω q j = - sin Ω sin ω + cos Ω cos i cos ω q k = sin i cos ω w i = sin i sin Ω w j = - sin i cos Ω w k = cos i \begin{aligned}\displaystyle p_{i}&\displaystyle=\cos\Omega\cos\omega-\sin% \Omega\cos i\sin\omega\\ \displaystyle p_{j}&\displaystyle=\sin\Omega\cos\omega+\cos\Omega\cos i\sin% \omega\\ \displaystyle p_{k}&\displaystyle=\sin i\sin\omega\\ \displaystyle q_{i}&\displaystyle=-\cos\Omega\sin\omega-\sin\Omega\cos i\cos% \omega\\ \displaystyle q_{j}&\displaystyle=-\sin\Omega\sin\omega+\cos\Omega\cos i\cos% \omega\\ \displaystyle q_{k}&\displaystyle=\sin i\cos\omega\\ \displaystyle w_{i}&\displaystyle=\sin i\sin\Omega\\ \displaystyle w_{j}&\displaystyle=-\sin i\cos\Omega\\ \displaystyle w_{k}&\displaystyle=\cos i\end{aligned}
  19. p ^ = p i I ^ + p j J ^ + p k K ^ q ^ = q i I ^ + q j J ^ + q k K ^ w ^ = w i I ^ + w j J ^ + w k K ^ \begin{aligned}\displaystyle{\hat{p}}&\displaystyle=p_{i}{\hat{I}}+p_{j}{\hat{% J}}+p_{k}{\hat{K}}\\ \displaystyle{\hat{q}}&\displaystyle=q_{i}{\hat{I}}+q_{j}{\hat{J}}+q_{k}{\hat{% K}}\\ \displaystyle{\hat{w}}&\displaystyle=w_{i}{\hat{I}}+w_{j}{\hat{J}}+w_{k}{\hat{% K}}\end{aligned}
  20. I ^ {\hat{I}}
  21. J ^ {\hat{J}}
  22. K ^ {\hat{K}}
  23. p ^ {\hat{p}}

Period_mapping.html

  1. X b f - 1 ( U ) X 0 × U X 0 X_{b}\hookrightarrow f^{-1}(U)\cong X_{0}\times U\twoheadrightarrow X_{0}
  2. H k ( X b , 𝐙 ) H k ( X b × U , 𝐙 ) H k ( X 0 × U , 𝐙 ) H k ( X 0 , 𝐙 ) , H^{k}(X_{b},\mathbf{Z})\cong H^{k}(X_{b}\times U,\mathbf{Z})\cong H^{k}(X_{0}% \times U,\mathbf{Z})\cong H^{k}(X_{0},\mathbf{Z}),
  3. 𝒫 : U F = F b 1 , k , , b k , k ( H k ( X 0 , 𝐂 ) ) , \mathcal{P}:U\rightarrow F=F_{b_{1,k},\ldots,b_{k,k}}(H^{k}(X_{0},\mathbf{C})),
  4. b ( F p H k ( X b , 𝐂 ) ) p . b\mapsto(F^{p}H^{k}(X_{b},\mathbf{C}))_{p}.
  5. H k ( X b , 𝐂 ) = F p H k ( X b , 𝐂 ) F k - p + 1 H k ( X b , 𝐂 ) ¯ . H^{k}(X_{b},\mathbf{C})=F^{p}H^{k}(X_{b},\mathbf{C})\oplus\overline{F^{k-p+1}H% ^{k}(X_{b},\mathbf{C})}.
  6. 𝒟 \mathcal{D}
  7. 𝒟 \mathcal{D}
  8. Q ( ξ , η ) = ω b n - k ξ η . Q(\xi,\eta)=\int\omega_{b}^{n-k}\wedge\xi\wedge\eta.
  9. ( - 1 ) k ( k - 1 ) / 2 i p - q Q \textstyle(-1)^{k(k-1)/2}i^{p-q}Q
  10. 𝒟 \mathcal{D}
  11. 𝒫 \mathcal{P}
  12. Ω = ( δ i ω j ) 1 i r , 1 j s . \Omega=\Big(\int_{\delta_{i}}\omega_{j}\Big)_{1\leq i\leq r,1\leq j\leq s}.
  13. y 2 = x ( x - 1 ) ( x - λ ) y^{2}=x(x-1)(x-\lambda)
  14. y = x ( x - 1 ) ( x - λ ) y=\sqrt{x(x-1)(x-\lambda)}
  15. ( γ ω δ ω ) . \begin{pmatrix}\int_{\gamma}\omega\\ \int_{\delta}\omega\end{pmatrix}.
  16. - 1 X 0 ω ω ¯ = - 1 X 0 | f | 2 d z d z ¯ > 0. \sqrt{-1}\int_{X_{0}}\omega\wedge\bar{\omega}=\sqrt{-1}\int_{X_{0}}|f|^{2}\,dz% \wedge d\bar{z}>0.
  17. ω = A γ * + B δ * . \omega=A\gamma^{*}+B\delta^{*}.
  18. - 1 X 0 A B ¯ γ * δ ¯ * + A ¯ B γ ¯ * δ * = X 0 Im ( 2 A ¯ B γ ¯ * δ * ) > 0 \sqrt{-1}\int_{X_{0}}A\bar{B}\gamma^{*}\wedge\bar{\delta}^{*}+\bar{A}B\bar{% \gamma}^{*}\wedge\delta^{*}=\int_{X_{0}}\operatorname{Im}\,(2\bar{A}B\bar{% \gamma}^{*}\wedge\delta^{*})>0
  19. Im 2 A ¯ B \operatorname{Im}\,2\bar{A}B

Peristimulus_time_histogram.html

  1. Δ \Delta
  2. k i n Δ \frac{k_{i}}{n\Delta}
  3. i Δ i\ \Delta

Perl_6_rules.html

  1. { a n b n c n : n 1 } \{a^{n}b^{n}c^{n}:n\geq 1\}

Perrin_friction_factors.html

  1. ξ \xi
  2. ξ = def | p 2 - 1 | p \xi\ \stackrel{\mathrm{def}}{=}\ \frac{\sqrt{\left|p^{2}-1\right|}}{p}
  3. S = def 2 atan ξ ξ S\ \stackrel{\mathrm{def}}{=}\ 2\frac{\mathrm{atan}\ \xi}{\xi}
  4. S = 2 S=2
  5. p 1 p\rightarrow 1
  6. V V
  7. f t o t = f s p h e r e f P f_{tot}=f_{sphere}\ f_{P}
  8. f s p h e r e f_{sphere}
  9. f s p h e r e = 6 π η R e f f = 6 π η ( 3 V 4 π ) ( 1 / 3 ) f_{sphere}=6\pi\eta R_{eff}=6\pi\eta\left(\frac{3V}{4\pi}\right)^{(1/3)}
  10. f P f_{P}
  11. f P = def 2 p 2 / 3 S f_{P}\ \stackrel{\mathrm{def}}{=}\ \frac{2p^{2/3}}{S}
  12. D = k B T f t o t D=\frac{k_{B}T}{f_{tot}}
  13. f t o t f_{tot}
  14. F a x F_{ax}
  15. F e q F_{eq}
  16. F a x = def ( 4 3 ) ξ 2 2 - ( S / p 2 ) F_{ax}\ \stackrel{\mathrm{def}}{=}\ \left(\frac{4}{3}\right)\frac{\xi^{2}}{2-(% S/p^{2})}
  17. F e q = def ( 4 3 ) ( 1 / p ) 2 - p 2 2 - S [ 2 - ( 1 / p ) 2 ] F_{eq}\ \stackrel{\mathrm{def}}{=}\ \left(\frac{4}{3}\right)\frac{(1/p)^{2}-p^% {2}}{2-S\left[2-(1/p)^{2}\right]}
  18. F a x = F e q = 1 F_{ax}=F_{eq}=1
  19. p 1 p\rightarrow 1
  20. p 1 p\approx 1
  21. p 1 p\rightarrow 1
  22. 1 F a x = 1.0 + ( 4 5 ) ( ξ 2 1 + ξ 2 ) + ( 4 6 5 7 ) ( ξ 2 1 + ξ 2 ) 2 + ( 4 6 8 5 7 9 ) ( ξ 2 1 + ξ 2 ) 3 + \frac{1}{F_{ax}}=1.0+\left(\frac{4}{5}\right)\left(\frac{\xi^{2}}{1+\xi^{2}}% \right)+\left(\frac{4\cdot 6}{5\cdot 7}\right)\left(\frac{\xi^{2}}{1+\xi^{2}}% \right)^{2}+\left(\frac{4\cdot 6\cdot 8}{5\cdot 7\cdot 9}\right)\left(\frac{% \xi^{2}}{1+\xi^{2}}\right)^{3}+\ldots
  23. τ a x = ( 1 k B T ) F e q 2 \tau_{ax}=\left(\frac{1}{k_{B}T}\right)\frac{F_{eq}}{2}
  24. τ e q = ( 1 k B T ) F a x F e q F a x + F e q \tau_{eq}=\left(\frac{1}{k_{B}T}\right)\frac{F_{ax}F_{eq}}{F_{ax}+F_{eq}}
  25. ρ \rho
  26. τ a x \tau_{ax}
  27. F e q F_{eq}
  28. τ e q \tau_{eq}

Peskin–Takeuchi_parameter.html

  1. Π γ γ ( q 2 ) = q 2 Π γ γ ( 0 ) + \Pi_{\gamma\gamma}(q^{2})=q^{2}\Pi_{\gamma\gamma}^{\prime}(0)+...
  2. Π Z γ ( q 2 ) = q 2 Π Z γ ( 0 ) + \Pi_{Z\gamma}(q^{2})=q^{2}\Pi_{Z\gamma}^{\prime}(0)+...
  3. Π Z Z ( q 2 ) = Π Z Z ( 0 ) + q 2 Π Z Z ( 0 ) + \Pi_{ZZ}(q^{2})=\Pi_{ZZ}(0)+q^{2}\Pi_{ZZ}^{\prime}(0)+...
  4. Π W W ( q 2 ) = Π W W ( 0 ) + q 2 Π W W ( 0 ) + \Pi_{WW}(q^{2})=\Pi_{WW}(0)+q^{2}\Pi_{WW}^{\prime}(0)+...
  5. Π \Pi^{\prime}
  6. Π γ γ \Pi_{\gamma\gamma}
  7. Π Z γ \Pi_{Z\gamma}
  8. α \alpha
  9. α S = 4 s w 2 c w 2 [ Π Z Z ( 0 ) - c w 2 - s w 2 s w c w Π Z γ ( 0 ) - Π γ γ ( 0 ) ] \alpha S=4s_{w}^{2}c_{w}^{2}\left[\Pi_{ZZ}^{\prime}(0)-\frac{c_{w}^{2}-s_{w}^{% 2}}{s_{w}c_{w}}\Pi_{Z\gamma}^{\prime}(0)-\Pi_{\gamma\gamma}^{\prime}(0)\right]
  10. α T = Π W W ( 0 ) M W 2 - Π Z Z ( 0 ) M Z 2 \alpha T=\frac{\Pi_{WW}(0)}{M_{W}^{2}}-\frac{\Pi_{ZZ}(0)}{M_{Z}^{2}}
  11. α U = 4 s w 2 [ Π W W ( 0 ) - c w 2 Π Z Z ( 0 ) - 2 s w c w Π Z γ ( 0 ) - s w 2 Π γ γ ( 0 ) ] \alpha U=4s_{w}^{2}\left[\Pi_{WW}^{\prime}(0)-c_{w}^{2}\Pi_{ZZ}^{\prime}(0)-2s% _{w}c_{w}\Pi_{Z\gamma}^{\prime}(0)-s_{w}^{2}\Pi_{\gamma\gamma}^{\prime}(0)\right]
  12. | H D μ H | 2 / Λ 2 \left|H^{\dagger}D_{\mu}H\right|^{2}/\Lambda^{2}
  13. H W μ ν B μ ν H / Λ 2 H^{\dagger}W^{\mu\nu}B_{\mu\nu}H/\Lambda^{2}
  14. H B μ ν B μ ν H / Λ 2 H^{\dagger}B^{\mu\nu}B_{\mu\nu}H/\Lambda^{2}
  15. H W μ ν W μ ν H / Λ 2 H^{\dagger}W^{\mu\nu}W_{\mu\nu}H/\Lambda^{2}
  16. ( H W μ ν H ) ( H W μ ν H ) / Λ 4 \left(H^{\dagger}W^{\mu\nu}H\right)\left(H^{\dagger}W_{\mu\nu}H\right)/\Lambda% ^{4}

Petrophysics.html

  1. ϕ \phi
  2. S w S_{w}

Pfister_form.html

  1. a 1 , a 2 , , a n 1 , a 1 1 , a 2 1 , a n , \langle\!\langle a_{1},a_{2},...,a_{n}\rangle\!\rangle\cong\langle 1,a_{1}% \rangle\otimes\langle 1,a_{2}\rangle\otimes...\otimes\langle 1,a_{n}\rangle,
  2. q ( a ) q q\oplus(a)q
  3. a 1 , a x 2 + a y 2 \langle\!\langle a\rangle\!\rangle\cong\langle 1,a\rangle\cong x^{2}+ay^{2}
  4. a , b 1 , a , b , a b x 2 + a y 2 + b z 2 + a b w 2 . \langle\!\langle a,b\rangle\!\rangle\cong\langle 1,a,b,ab\rangle\cong x^{2}+ay% ^{2}+bz^{2}+abw^{2}.
  5. { a 1 , , a n } a 1 , a 2 , , a n , \{a_{1},\ldots,a_{n}\}\mapsto\langle\!\langle a_{1},a_{2},...,a_{n}\rangle\!\rangle,

Phase-comparison_monopulse.html

  1. θ = sin - 1 ( λ Δ Φ 2 π d ) \theta=\sin^{-1}\left(\frac{\lambda\ \Delta\Phi}{2\pi d}\right)

Phase_factor.html

  1. e i θ A e i ( 𝐤 𝐫 - ω t ) = A e i ( 𝐤 𝐫 - ω t + θ ) \,\text{e}^{i\theta}A\,\text{e}^{i\left({\mathbf{k}\cdot\mathbf{r}-\omega t}% \right)}=A\,\text{e}^{i\left({\mathbf{k}\cdot\mathbf{r}-\omega t+\theta}\right)}
  2. | ψ |\psi\rangle
  3. ϕ | \langle\phi|
  4. ϕ | A | ϕ \langle\phi|A|\phi\rangle
  5. ϕ | e - i θ A e i θ | ϕ \langle\phi|e^{-i\theta}Ae^{i\theta}|\phi\rangle

Phi_value_analysis.html

  1. Φ = ( Δ G W T S - D - Δ G M T S - D ) ( Δ G W N - D - Δ G M N - D ) = Δ Δ G T S - D Δ Δ G N - D \Phi=\frac{(\Delta G^{TS-D}_{W}-\Delta G^{TS-D}_{M})}{(\Delta G^{N-D}_{W}-% \Delta G^{N-D}_{M})}=\frac{\Delta\Delta G^{TS-D}}{\Delta\Delta G^{N-D}}
  2. Δ G W T S - D \Delta G^{TS-D}_{W}
  3. Δ G M T S - D \Delta G^{TS-D}_{M}
  4. Δ G N - D \Delta G^{N-D}
  5. ϕ \phi
  6. ψ \psi
  7. ϕ \phi
  8. ϕ \phi
  9. ϕ \phi
  10. ψ \psi
  11. ψ \psi
  12. Φ \Phi
  13. ψ \psi

Philosophy_of_language.html

  1. \land
  2. \lor
  3. \rightarrow

Phosphoenolpyruvate_carboxykinase_(ATP).html

  1. \rightleftharpoons

Phosphoenolpyruvate_carboxykinase_(diphosphate).html

  1. \rightleftharpoons
  2. \rightleftharpoons

Photoelastic_modulator.html

  1. ψ = 1 2 ( 1 e i A sin 2 f t ) \psi=\frac{1}{\sqrt{2}}\left(\begin{matrix}1\\ e^{iA\sin 2ft}\end{matrix}\right)

Photon_gas.html

  1. u ( ν , T ) = 8 π h ν 3 c 3 1 e h ν / k T - 1 u(\nu,T)=\frac{8\pi h\nu^{3}}{c^{3}}~{}\frac{1}{e^{h\nu/kT}-1}
  2. U = ( 8 π 5 k 4 15 c 3 h 3 ) V T 4 U=\left(8\frac{\pi^{5}k^{4}}{15c^{3}h^{3}}\right)VT^{4}
  3. N = ( 16 π k 3 ζ ( 3 ) c 3 h 3 ) V T 3 N=\left(\frac{16\pi k^{3}\zeta(3)}{c^{3}h^{3}}\right)\,VT^{3}
  4. ζ ( n ) \zeta(n)
  5. U = 3 P V U=3PV
  6. P V = ζ ( 4 ) ζ ( 3 ) N k T .9 N k T PV=\frac{\zeta(4)}{\zeta(3)}NkT\approx.9NkT
  7. U = ( π 2 k 4 15 c 3 3 ) V T 4 U=\left(\frac{\pi^{2}k^{4}}{15c^{3}\hbar^{3}}\right)\,VT^{4}
  8. N = ( 16 π k 3 ζ ( 3 ) c 3 h 3 ) V T 3 N=\left(\frac{16\pi k^{3}\zeta(3)}{c^{3}h^{3}}\right)\,VT^{3}
  9. μ = 0 \mu=0\,
  10. P = 1 3 U V P=\frac{1}{3}\,\frac{U}{V}
  11. S = 4 U 3 T S=\frac{4U}{3T}
  12. H = 4 3 U H=\frac{4}{3}\,U
  13. A = - 1 3 U A=-\frac{1}{3}\,U
  14. G = 0 G=0\,
  15. W = - 0 x 0 P ( A d x ) W=-\int_{0}^{x_{0}}P(Adx)
  16. b = 8 π 5 k 4 15 c 3 h 3 b=\frac{8\pi^{5}k^{4}}{15c^{3}h^{3}}
  17. P ( x ) = b T 4 3 P(x)=\frac{bT^{4}}{3}\,
  18. W = - b T 4 A x 0 3 = b T 4 V 0 3 W=-\frac{bT^{4}Ax_{0}}{3}=\frac{bT^{4}V_{0}}{3}
  19. Q = U - W = H 0 Q=U-W=H_{0}\,