wpmath0000003_17

Vandermonde_matrix.html

  1. V = [ 1 α 1 α 1 2 α 1 n - 1 1 α 2 α 2 2 α 2 n - 1 1 α 3 α 3 2 α 3 n - 1 1 α m α m 2 α m n - 1 ] V=\begin{bmatrix}1&\alpha_{1}&\alpha_{1}^{2}&\dots&\alpha_{1}^{n-1}\\ 1&\alpha_{2}&\alpha_{2}^{2}&\dots&\alpha_{2}^{n-1}\\ 1&\alpha_{3}&\alpha_{3}^{2}&\dots&\alpha_{3}^{n-1}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 1&\alpha_{m}&\alpha_{m}^{2}&\dots&\alpha_{m}^{n-1}\end{bmatrix}
  2. V i , j = α i j - 1 V_{i,j}=\alpha_{i}^{j-1}\,
  3. det ( V ) = 1 i < j n ( α j - α i ) . \det(V)=\prod_{1\leq i<j\leq n}(\alpha_{j}-\alpha_{i}).
  4. α i \alpha_{i}
  5. α i \alpha_{i}
  6. V i + k , j = { 0 , if j k ; ( j - 1 ) ! ( j - k - 1 ) ! α i j - k - 1 , if j > k . V_{i+k,j}=\begin{cases}0,&\,\text{if }j\leq k;\\ \frac{(j-1)!}{(j-k-1)!}\alpha_{i}^{j-k-1},&\,\text{if }j>k.\end{cases}
  7. α i \alpha_{i}
  8. α j \alpha_{j}
  9. α i \alpha_{i}
  10. α j \alpha_{j}
  11. det ( V ) = σ S n sgn ( σ ) i = 1 n α i σ ( i ) - 1 , \det(V)=\sum_{\sigma\in S_{n}}\operatorname{sgn}(\sigma)\prod_{i=1}^{n}\alpha_% {i}^{\sigma(i)-1},
  12. { 1 , , n } \{1,\ldots,n\}
  13. sgn ( σ ) \operatorname{sgn}(\sigma)
  14. σ S n sgn ( σ ) i = 1 n α i σ ( i ) - 1 = 1 i < j n ( α j - α i ) . \sum_{\sigma\in S_{n}}\operatorname{sgn}(\sigma)\prod_{i=1}^{n}\alpha_{i}^{% \sigma(i)-1}=\prod_{1\leq i<j\leq n}(\alpha_{j}-\alpha_{i}).
  15. a 0 + a 1 x + a 2 x 2 + + a n - 1 x n - 1 a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{n-1}x^{n-1}
  16. α i . \alpha_{i}.
  17. α i \alpha_{i}
  18. P ( x ) = j = 0 n - 1 u j x j P(x)=\sum_{j=0}^{n-1}u_{j}x^{j}
  19. P ( α i ) = y i for i = 1 , , m . P(\alpha_{i})=y_{i}\quad\,\text{for }i=1,\ldots,m.\,
  20. α k \alpha_{k}

Vanish_at_infinity.html

  1. f ( x ) 0 f(x)\to 0
  2. x . \|x\|\to\infty.
  3. f ( x ) = 1 x 2 + 1 f(x)=\frac{1}{x^{2}+1}
  4. f f
  5. ϵ \epsilon
  6. K K
  7. f ( x ) < ϵ \|f(x)\|<\epsilon
  8. x x
  9. K K
  10. ϵ \epsilon
  11. { x X : f ( x ) ϵ } \left\{x\in X:\|f(x)\|\geq\epsilon\right\}
  12. Ω \Omega
  13. f : Ω 𝕂 f:\Omega\rightarrow\mathbb{K}
  14. 𝕂 \mathbb{K}
  15. \mathbb{R}
  16. \mathbb{C}
  17. 𝕂 \mathbb{K}
  18. C 0 ( Ω ) C_{0}(\Omega)

Variable-frequency_oscillator.html

  1. f I N f_{IN}
  2. f L O f_{LO}
  3. f I N + f L O f_{IN}+f_{LO}
  4. f I N - f L O f_{IN}-f_{LO}
  5. f I N f_{IN}
  6. f L O f_{LO}
  7. f I F = 0 f_{IF}=0

Vector_operator.html

  1. grad \operatorname{grad}\equiv\nabla
  2. div \operatorname{div}\ \equiv\nabla\cdot
  3. curl × \operatorname{curl}\equiv\nabla\times
  4. 2 div grad \nabla^{2}\equiv\operatorname{div}\ \operatorname{grad}\equiv\nabla\cdot\nabla
  5. f \nabla f
  6. f f\nabla

Vector_potential.html

  1. 𝐯 = × 𝐀 . \mathbf{v}=\nabla\times\mathbf{A}.
  2. ( × 𝐀 ) = 0 \nabla\cdot(\nabla\times\mathbf{A})=0
  3. 𝐯 = ( × 𝐀 ) = 0 , \nabla\cdot\mathbf{v}=\nabla\cdot(\nabla\times\mathbf{A})=0,
  4. 𝐯 : 3 3 \mathbf{v}:\mathbb{R}^{3}\to\mathbb{R}^{3}
  5. 𝐀 ( 𝐱 ) = 1 4 π × 3 𝐯 ( 𝐲 ) 𝐱 - 𝐲 d 3 𝐲 . \mathbf{A}(\mathbf{x})=\frac{1}{4\pi}\nabla\times\int_{\mathbb{R}^{3}}\frac{% \mathbf{v}(\mathbf{y})}{\left\|\mathbf{x}-\mathbf{y}\right\|}\,d^{3}\mathbf{y}.
  6. × 𝐀 = 𝐯 . \nabla\times\mathbf{A}=\mathbf{v}.
  7. 𝐀 + m \mathbf{A}+\nabla m

VEGAS_algorithm.html

  1. | f | |f|
  2. f f
  3. g g
  4. E g ( f ; N ) E_{g}(f;N)
  5. E g ( f ; N ) = 1 N i N f ( x i ) / g ( x i ) E_{g}(f;N)={1\over N}\sum_{i}^{N}{f(x_{i})}/g(x_{i})
  6. 𝑉𝑎𝑟 g ( f ; N ) = 𝑉𝑎𝑟 ( f / g ; N ) \mathit{Var}_{g}(f;N)=\mathit{Var}(f/g;N)
  7. 𝑉𝑎𝑟 ( f ; N ) \mathit{Var}(f;N)
  8. 𝑉𝑎𝑟 ( f ; N ) = E ( f 2 ; N ) - ( E ( f ; N ) ) 2 \mathit{Var}(f;N)=E(f^{2};N)-(E(f;N))^{2}
  9. g = | f | / I ( | f | ) g=|f|/I(|f|)
  10. 𝑉𝑎𝑟 g ( f ; N ) \mathit{Var}_{g}(f;N)
  11. K d K^{d}
  12. g ( x 1 , x 2 , ) = g 1 ( x 1 ) g 2 ( x 2 ) g(x_{1},x_{2},\ldots)=g_{1}(x_{1})g_{2}(x_{2})\cdots

Venturi_effect.html

  1. p 1 - p 2 = ρ 2 ( v 2 2 - v 1 2 ) p_{1}-p_{2}=\frac{\rho}{2}\left(v_{2}^{2}-v_{1}^{2}\right)
  2. ρ \scriptstyle\rho\,
  3. v 1 \scriptstyle v_{1}
  4. v 2 \scriptstyle v_{2}
  5. Q \scriptstyle Q
  6. Q = v 1 A 1 = v 2 A 2 p 1 - p 2 = ρ 2 ( v 2 2 - v 1 2 ) \begin{aligned}\displaystyle Q&\displaystyle=v_{1}A_{1}=v_{2}A_{2}\\ \displaystyle p_{1}-p_{2}&\displaystyle=\frac{\rho}{2}(v_{2}^{2}-v_{1}^{2})% \end{aligned}
  7. Q = A 1 2 ρ ( p 1 - p 2 ) ( A 1 A 2 ) 2 - 1 = A 2 2 ρ ( p 1 - p 2 ) 1 - ( A 2 A 1 ) 2 Q=A_{1}\sqrt{\frac{2}{\rho}\cdot\frac{\left(p_{1}-p_{2}\right)}{\left(\frac{A_% {1}}{A_{2}}\right)^{2}-1}}=A_{2}\sqrt{\frac{2}{\rho}\cdot\frac{\left(p_{1}-p_{% 2}\right)}{1-\left(\frac{A_{2}}{A_{1}}\right)^{2}}}

Verbal_arithmetic.html

  1. S E N D + M O R E = M O N E Y \begin{matrix}&&\,\text{S}&\,\text{E}&\,\text{N}&\,\text{D}\\ +&&\,\text{M}&\,\text{O}&\,\text{R}&\,\text{E}\\ \hline=&\,\text{M}&\,\text{O}&\,\text{N}&\,\text{E}&\,\text{Y}\\ \end{matrix}
  2. S E N D + M O R E = M O N E Y \begin{matrix}&&\,\text{S}&\,\text{E}&\,\text{N}&\,\text{D}\\ +&&\,\text{M}&\,\text{O}&\,\text{R}&\,\text{E}\\ \hline=&\,\text{M}&\,\text{O}&\,\text{N}&\,\text{E}&\,\text{Y}\\ \end{matrix}

Versine.html

  1. vercosin ( θ ) \operatorname{vercosin}(\theta)
  2. coversin ( θ ) \operatorname{coversin}(\theta)
  3. cvs ( θ ) \operatorname{cvs}(\theta)
  4. covercosin ( θ ) \operatorname{covercosin}(\theta)
  5. haversin ( θ ) \operatorname{haversin}(\theta)
  6. sem ( θ ) \operatorname{sem}(\theta)
  7. havercosin ( θ ) \operatorname{havercosin}(\theta)
  8. hacoversin ( θ ) \operatorname{hacoversin}(\theta)
  9. hacovercosin ( θ ) \operatorname{hacovercosin}(\theta)
  10. exsec ( θ ) \operatorname{exsec}(\theta)
  11. excosec ( θ ) \operatorname{excosec}(\theta)
  12. versin ( θ ) := 2 sin 2 ( θ 2 ) = 1 - cos ( θ ) \textrm{versin}(\theta):=2\sin^{2}\!\left(\frac{\theta}{2}\right)=1-\cos(% \theta)\,
  13. vercosin ( θ ) := 2 cos 2 ( θ 2 ) = 1 + cos ( θ ) \textrm{vercosin}(\theta):=2\cos^{2}\!\left(\frac{\theta}{2}\right)=1+\cos(% \theta)\,
  14. coversin ( θ ) := versin ( π 2 - θ ) = 1 - sin ( θ ) \textrm{coversin}(\theta):=\textrm{versin}\!\left(\frac{\pi}{2}-\theta\right)=% 1-\sin(\theta)\,
  15. covercosin ( θ ) := vercosin ( π 2 - θ ) = 1 + sin ( θ ) \textrm{covercosin}(\theta):=\textrm{vercosin}\!\left(\frac{\pi}{2}-\theta% \right)=1+\sin(\theta)\,
  16. haversin ( θ ) := versin ( θ ) 2 = 1 - cos ( θ ) 2 \textrm{haversin}(\theta):=\frac{\textrm{versin}(\theta)}{2}=\frac{1-\cos(% \theta)}{2}\,
  17. havercosin ( θ ) := vercosin ( θ ) 2 = 1 + cos ( θ ) 2 \textrm{havercosin}(\theta):=\frac{\textrm{vercosin}(\theta)}{2}=\frac{1+\cos(% \theta)}{2}\,
  18. hacoversin ( θ ) := coversin ( θ ) 2 = 1 - sin ( θ ) 2 \textrm{hacoversin}(\theta):=\frac{\textrm{coversin}(\theta)}{2}=\frac{1-\sin(% \theta)}{2}\,
  19. hacovercosin ( θ ) := covercosin ( θ ) 2 = 1 + sin ( θ ) 2 \textrm{hacovercosin}(\theta):=\frac{\textrm{covercosin}(\theta)}{2}=\frac{1+% \sin(\theta)}{2}\,
  20. d d x versin ( x ) = sin x \frac{\mathrm{d}}{\mathrm{d}x}\mathrm{versin}(x)=\sin{x}
  21. versin ( x ) d x = x - sin x + C \int\mathrm{versin}(x)\,\mathrm{d}x=x-\sin{x}+C
  22. d d x vercosin ( x ) = - sin x \frac{\mathrm{d}}{\mathrm{d}x}\mathrm{vercosin}(x)=-\sin{x}
  23. vercosin ( x ) d x = x + sin x + C \int\mathrm{vercosin}(x)\,\mathrm{d}x=x+\sin{x}+C
  24. d d x coversin ( x ) = - cos x \frac{\mathrm{d}}{\mathrm{d}x}\mathrm{coversin}(x)=-\cos{x}
  25. coversin ( x ) d x = x + cos x + C \int\mathrm{coversin}(x)\,\mathrm{d}x=x+\cos{x}+C
  26. d d x covercosin ( x ) = cos x \frac{\mathrm{d}}{\mathrm{d}x}\mathrm{covercosin}(x)=\cos{x}
  27. covercosin ( x ) d x = x - cos x + C \int\mathrm{covercosin}(x)\,\mathrm{d}x=x-\cos{x}+C
  28. d d x haversin ( x ) = sin x 2 \frac{\mathrm{d}}{\mathrm{d}x}\mathrm{haversin}(x)=\frac{\sin{x}}{2}
  29. haversin ( x ) d x = x - sin x 2 + C \int\mathrm{haversin}(x)\,\mathrm{d}x=\frac{x-\sin{x}}{2}+C
  30. d d x havercosin ( x ) = - sin x 2 \frac{\mathrm{d}}{\mathrm{d}x}\mathrm{havercosin}(x)=\frac{-\sin{x}}{2}
  31. havercosin ( x ) d x = x + sin x 2 + C \int\mathrm{havercosin}(x)\,\mathrm{d}x=\frac{x+\sin{x}}{2}+C
  32. d d x hacoversin ( x ) = - cos x 2 \frac{\mathrm{d}}{\mathrm{d}x}\mathrm{hacoversin}(x)=\frac{-\cos{x}}{2}
  33. hacoversin ( x ) d x = x + cos x 2 + C \int\mathrm{hacoversin}(x)\,\mathrm{d}x=\frac{x+\cos{x}}{2}+C
  34. d d x hacovercosin ( x ) = cos x 2 \frac{\mathrm{d}}{\mathrm{d}x}\mathrm{hacovercosin}(x)=\frac{\cos{x}}{2}
  35. hacovercosin ( x ) d x = x - cos x 2 + C \int\mathrm{hacovercosin}(x)\,\mathrm{d}x=\frac{x-\cos{x}}{2}+C
  36. s L + v 2 r s\approx L+\frac{v^{2}}{r}
  37. v s 3 2 L 1 2 8 r v\approx\frac{s^{\frac{3}{2}}L^{\frac{1}{2}}}{8r}

Vertex-transitive_graph.html

  1. f : V ( G ) V ( G ) f:V(G)\rightarrow V(G)
  2. f ( v 1 ) = v 2 . f(v_{1})=v_{2}.

Vertex_cover.html

  1. G = ( V , E ) G=(V,E)
  2. τ \tau
  3. K m , n K_{m,n}
  4. τ ( K m , n ) = min ( m , n ) \tau(K_{m,n})=\min(m,n)
  5. G G
  6. k k
  7. G G
  8. k k
  9. G G
  10. k k
  11. G G
  12. k k
  13. c ( v ) 0 c(v)\geq 0
  14. v V c ( v ) x v \sum_{v\in V}c(v)x_{v}
  15. x u + x v 1 x_{u}+x_{v}\geq 1
  16. { u , v } E \{u,v\}\in E
  17. x v { 0 , 1 } x_{v}\in\{0,1\}
  18. v V v\in V
  19. 2 2
  20. 2 2
  21. x v x_{v}
  22. 1.2738 k n O ( 1 ) 1.2738^{k}\cdot n^{O(1)}
  23. 2 O ( k ) n O ( 1 ) 2^{O(\sqrt{k})}n^{O(1)}
  24. 2 o ( k ) n O ( 1 ) 2^{o(\sqrt{k})}n^{O(1)}
  25. 2 - Θ ( 1 / log | V | ) 2-\Theta\left(1/\sqrt{\log|V|}\right)
  26. 2 / ( 1 + δ ) 2/(1+\delta)
  27. δ \delta

Very_Long_Baseline_Array.html

  1. θ H P B W \theta_{HPBW}

Vesica_piscis.html

  1. 1351 780 > 3 > 265 153 . \tfrac{1351}{780}>\sqrt{3}>\tfrac{265}{153}\,.

Vestibular_system.html

  1. θ \theta
  2. q ˙ \dot{q}
  3. θ ( s ) = α s ( T 1 s + 1 ) ( T 2 s + 1 ) q ˙ ( s ) \theta(s)=\frac{\alpha s}{(T_{1}s+1)(T_{2}s+1)}\dot{q}(s)

VHF_omnidirectional_range.html

  1. i ( t ) i(t)
  2. a ( t ) a(t)
  3. c ( t ) c(t)
  4. g ( A , t ) g(A,t)
  5. e ( A , t ) = cos ( 2 π F c t ) ( 1 + c ( t ) + g ( A , t ) ) c ( t ) = M i cos ( 2 π F i t ) i ( t ) + M a a ( t ) + M d cos ( 2 π 0 t ( F s + F d cos ( 2 π F n t ) ) d t ) g ( A , t ) = M n cos ( 2 π F n t - A ) \begin{array}[]{rcl}e(A,t)&=&\cos(2\pi F_{c}t)(1+c(t)+g(A,t))\\ c(t)&=&M_{i}\cos(2\pi F_{i}t)~{}i(t)\\ &+&M_{a}~{}a(t)\\ &+&M_{d}\cos(2\pi\int_{0}^{t}(F_{s}+F_{d}\cos(2\pi F_{n}t))dt)\\ g(A,t)&=&M_{n}\cos(2\pi F_{n}t-A)\\ \end{array}
  6. i ( t ) i(t)
  7. a ( t ) a(t)
  8. c ( t ) c(t)
  9. g ( A , t ) g(A,t)
  10. t = t + ( A , t ) - ( R / C ) sin ( 2 π F n t + ( A , t ) + A ) t = t - ( A , t ) + ( R / C ) sin ( 2 π F n t - ( A , t ) + A ) e ( A , t ) = cos ( 2 π F c t ) ( 1 + c ( t ) ) + g ( A , t ) c ( t ) = M i cos ( 2 π F i t ) i ( t ) + M a a ( t ) + M n cos ( 2 π F n t ) g ( A , t ) = ( M d / 2 ) cos ( 2 π ( F c + F s ) t + ( A , t ) ) + ( M d / 2 ) cos ( 2 π ( F c - F s ) t - ( A , t ) ) \begin{array}[]{rcl}t&=&t_{+}(A,t)-(R/C)\sin(2\pi F_{n}t_{+}(A,t)+A)\\ t&=&t_{-}(A,t)+(R/C)\sin(2\pi F_{n}t_{-}(A,t)+A)\\ e(A,t)&=&\cos(2\pi F_{c}t)(1+c(t))\\ &+&g(A,t)\\ c(t)&=&M_{i}\cos(2\pi F_{i}t)~{}i(t)\\ &+&M_{a}~{}a(t)\\ &+&M_{n}\cos(2\pi F_{n}t)\\ g(A,t)&=&(M_{d}/2)\cos(2\pi(F_{c}+F_{s})t_{+}(A,t))\\ &+&(M_{d}/2)\cos(2\pi(F_{c}-F_{s})t_{-}(A,t))\\ \end{array}
  11. R = F < s u b > d C / ( 2 π F n F c ) R=F<sub>dC/(2πF_{n}F_{c})

Virasoro_algebra.html

  1. L n L_{n}
  2. n n∈ℤ
  3. c c
  4. L n + L - n L_{n}+L_{-n}
  5. i ( L n - L - n ) ~{}i(L_{n}-L_{-n})
  6. c c
  7. c c
  8. [ c , L n ] = 0 [c,L_{n}]=0
  9. v v
  10. L i L_{i}
  11. i 1 i\geq 1
  12. L 0 L_{0}
  13. c c
  14. h h
  15. c c
  16. L 0 L_{0}
  17. c c
  18. v v
  19. c c
  20. c c
  21. h h
  22. c c
  23. L n L_{n}
  24. L - n L_{-n}
  25. c = 1 - 6 m ( m + 1 ) = 0 , 1 / 2 , 7 / 10 , 4 / 5 , 6 / 7 , 25 / 28 , c=1-{6\over m(m+1)}=0,\quad 1/2,\quad 7/10,\quad 4/5,\quad 6/7,\quad 25/28,\ldots
  26. h = h r , s ( c ) = ( ( m + 1 ) r - m s ) 2 - 1 4 m ( m + 1 ) h=h_{r,s}(c)={((m+1)r-ms)^{2}-1\over 4m(m+1)}
  27. A N 1 r , s N ( h - h r , s ( c ) ) p ( N - r s ) , A_{N}\prod_{1\leq r,s\leq N}(h-h_{r,s}(c))^{p(N-rs)}~{},
  28. L L
  29. L L

Virtual_ground.html

  1. V o u t V i n = - R f R i n \frac{V_{out}}{V_{in}}=-\frac{R_{f}}{R_{in}}

Visual_cryptography.html

  1. 𝐂 𝟎 = [ 1 0 0 1 0 0 1 0 0 ] . \mathbf{C_{0}=}\begin{bmatrix}1&0&...&0\\ 1&0&...&0\\ ...\\ 1&0&...&0\end{bmatrix}.
  2. 𝐂 𝟏 = [ 1 0 0 0 1 0 0 0 1 ] . \mathbf{C_{1}=}\begin{bmatrix}1&0&...&0\\ 0&1&...&0\\ ...\\ 0&0&...&1\end{bmatrix}.

Vitali_set.html

  1. V V
  2. v V v\in V
  3. u , v V , u v u,v\in V,u\neq v
  4. V k = V + q k = { v + q k : v V } V_{k}=V+q_{k}=\{v+q_{k}:v\in V\}
  5. [ 0 , 1 ] k V k [ - 1 , 2 ] [0,1]\subseteq\bigcup_{k}V_{k}\subseteq[-1,2]
  6. 1 k = 1 λ ( V k ) 3. 1\leq\sum_{k=1}^{\infty}\lambda(V_{k})\leq 3.
  7. λ ( V k ) = λ ( V ) \lambda(V_{k})=\lambda(V)
  8. 1 k = 1 λ ( V ) 3. 1\leq\sum_{k=1}^{\infty}\lambda(V)\leq 3.

VList.html

  1. i = 1 l o g 2 n i - 1 2 i < i = 1 i - 1 2 i = 1. \sum_{i=1}^{\lceil log_{2}n\rceil}\frac{i-1}{2^{i}}<\sum_{i=1}^{\infty}\frac{i% -1}{2^{i}}=1.

Volkswagen_Lupo.html

  1. c w c_{\mathrm{w}}\,

Voltage-controlled_oscillator.html

  1. f ( t ) = f 0 + K 0 v in ( t ) θ ( t ) = - t f ( τ ) d τ \begin{aligned}\displaystyle f(t)&\displaystyle=f_{0}+K_{0}\cdot\ v_{\,\text{% in}}(t)\\ \displaystyle\theta(t)&\displaystyle=\int_{-\infty}^{t}f(\tau)\,d\tau\\ \end{aligned}
  2. f ( t ) f(t)
  3. t t
  4. f 0 f_{0}
  5. K 0 K_{0}
  6. f ( t ) f(t)
  7. θ ( t ) \theta(t)
  8. v in ( t ) v_{\,\text{in}}(t)
  9. F ( s ) = K 0 V in ( s ) Θ ( s ) = F ( s ) s \begin{aligned}\displaystyle F(s)&\displaystyle=K_{0}\cdot\ V_{\,\text{in}}(s)% \\ \displaystyle\Theta(s)&\displaystyle={F(s)\over s}\\ \end{aligned}
  10. L ( f m ) = 10 log [ 1 2 ( ( f 0 2 Q l f m ) 2 + 1 ) ( f c f m + 1 ) ( F k T P s ) ] L(f_{m})=10\log\bigg[\frac{1}{2}\bigg(\bigg(\frac{f_{0}}{2Q_{l}f_{m}}\bigg)^{2% }+1\bigg)\bigg(\frac{f_{c}}{f_{m}}+1\bigg)\bigg(\frac{FkT}{P_{s}}\bigg)\bigg]

Voltage_divider.html

  1. V out = Z 2 Z 1 + Z 2 V in V_{\mathrm{out}}=\frac{Z_{2}}{Z_{1}+Z_{2}}\cdot V_{\mathrm{in}}
  2. V in = I ( Z 1 + Z 2 ) V_{\mathrm{in}}=I\cdot(Z_{1}+Z_{2})
  3. V out = I Z 2 V_{\mathrm{out}}=I\cdot Z_{2}
  4. I = V in Z 1 + Z 2 I=\frac{V_{\mathrm{in}}}{Z_{1}+Z_{2}}
  5. V out = V in Z 2 Z 1 + Z 2 V_{\mathrm{out}}=V_{\mathrm{in}}\cdot\frac{Z_{2}}{Z_{1}+Z_{2}}
  6. H = V out V in = Z 2 Z 1 + Z 2 H=\frac{V_{\mathrm{out}}}{V_{\mathrm{in}}}=\frac{Z_{2}}{Z_{1}+Z_{2}}
  7. V out = R 2 R 1 + R 2 V in V_{\mathrm{out}}=\frac{R_{2}}{R_{1}+R_{2}}\cdot V_{\mathrm{in}}
  8. V out = 1 2 V in V_{\mathrm{out}}=\frac{1}{2}\cdot V_{\mathrm{in}}
  9. V out V in = R 2 R 1 + R 2 = 6 9 = 2 3 \frac{V_{\mathrm{out}}}{V_{\mathrm{in}}}=\frac{R_{2}}{R_{1}+R_{2}}=\frac{6}{9}% =\frac{2}{3}
  10. R 1 = R 2 V in V out - R 2 = R 2 ( V in V out - 1 ) R_{1}=\frac{R_{2}\cdot V_{\mathrm{in}}}{V_{\mathrm{out}}}-R_{2}=R_{2}\cdot({% \frac{V_{\mathrm{in}}}{V_{\mathrm{out}}}-1})
  11. R 2 = R 1 1 ( V in V out - 1 ) R_{2}=R_{1}\cdot\frac{1}{({\frac{V_{\mathrm{in}}}{V_{\mathrm{out}}}-1})}
  12. Z 2 = - j X C = 1 j ω C , Z_{2}=-\mathrm{j}X_{\mathrm{C}}=\frac{1}{\mathrm{j}\omega C}\ ,
  13. V out V in = Z 2 Z 1 + Z 2 = 1 j ω C 1 j ω C + R = 1 1 + j ω R C . \frac{V_{\mathrm{out}}}{V_{\mathrm{in}}}=\frac{Z_{\mathrm{2}}}{Z_{\mathrm{1}}+% Z_{\mathrm{2}}}=\frac{\frac{1}{\mathrm{j}\omega C}}{\frac{1}{\mathrm{j}\omega C% }+R}=\frac{1}{1+\mathrm{j}\omega RC}\ .
  14. | V out V in | = 1 1 + ( ω R C ) 2 . \left|\frac{V_{\mathrm{out}}}{V_{\mathrm{in}}}\right|=\frac{1}{\sqrt{1+(\omega RC% )^{2}}}\ .
  15. V out = L 2 L 1 + L 2 V in V_{\mathrm{out}}=\frac{L_{2}}{L_{1}+L_{2}}\cdot V_{\mathrm{in}}
  16. V out = C 1 C 1 + C 2 V in V_{\mathrm{out}}=\frac{C_{1}}{C_{1}+C_{2}}\cdot V_{\mathrm{in}}

Voltage_regulator.html

  1. R v = V R m i n I D m i n + I L m a x / ( h F E + 1 ) R_{v}=\frac{V_{Rmin}}{I_{Dmin}+I_{Lmax}/(h_{FE}+1)}

Volume_percent.html

  1. volume percent = volume of solute volume of solution * 100 \textrm{volume\ percent}=\frac{\textrm{volume\ of\ solute}}{\textrm{volume\ of% \ solution}}*100

Volumetric_flow_rate.html

  1. Q = V ˙ = lim Δ t 0 Δ V Δ t = d V d t Q=\dot{V}=\lim\limits_{\Delta t\rightarrow 0}\frac{\Delta V}{\Delta t}=\frac{{% \rm d}V}{{\rm d}t}
  2. Q = v A Q={v}\cdot{A}
  3. v {v}
  4. A {A}
  5. Q = A v d A Q=\iint_{A}{v}\cdot{\rm d}{A}
  6. n ^ {\hat{n}}
  7. A = A n ^ {A}=A{\hat{n}}
  8. Q = v A cos θ Q=vA\cos\theta
  9. n ^ {\hat{n}}
  10. cos θ \cos\theta
  11. π / 2 {π}/{2}
  12. Q = v A cos ( π 2 ) = 0 Q=vA\cos\left(\frac{\pi}{2}\right)=0
  13. Q Q
  14. Q = m ˙ ρ Q=\frac{\dot{m}}{\rho}
  15. m ˙ \dot{m}
  16. ρ \rho
  17. L d θ = T 2 π ( - ( cos θ 1 ) R - r θ 1 ) - T 2 π ( - ( cos θ 2 ) R - r θ 2 ) ) \int\!L\,\mathrm{d}\theta=\frac{T}{2\pi}(-(\cos{\theta_{1}})\cdot R-r\cdot% \theta_{1})-\frac{T}{2\pi}(-(\cos{\theta_{2}})\cdot R-r\cdot\theta_{2}))
  18. T T
  19. R R
  20. r r
  21. R - r R-r
  22. θ 1 \theta_{1}
  23. θ 2 \theta_{2}

Von_Neumann_bicommutant_theorem.html

  1. 𝐌 \mathbf{M}
  2. H H
  3. 𝐌 \mathbf{M}
  4. 𝐌 \mathbf{M}′′
  5. 𝐌 \mathbf{M}
  6. 𝐌 \mathbf{M}
  7. 𝐌 \mathbf{M}
  8. H H
  9. L ( H ) L(H)
  10. H H
  11. 𝐌 \mathbf{M}
  12. L ( H ) L(H)
  13. 𝐌 \mathbf{M}
  14. H H
  15. W W
  16. S S
  17. x x
  18. y y
  19. H H
  20. O O
  21. x O x x→Ox
  22. T T x , O * y - T O x , y = O T x , y - T O x , y . T\to\langle Tx,O^{*}y\rangle-\langle TOx,y\rangle=\langle OTx,y\rangle-\langle TOx% ,y\rangle.
  23. L ( H ) L(H)
  24. T T
  25. H H
  26. T T
  27. 𝐌 \mathbf{M}′′
  28. 𝐌 \mathbf{M}
  29. x x
  30. x x
  31. 𝐌 \mathbf{M}
  32. x x
  33. X 𝐌 X∈\mathbf{M}′′
  34. U U
  35. X X
  36. X h \|Xh\|
  37. H H
  38. L ( H ) L(H)
  39. U U
  40. O h \|Oh\|
  41. d > 0 d>0
  42. X h \|Xh\|
  43. c l ( 𝐌 h ) cl(\mathbf{M}h)
  44. H H
  45. P P
  46. P P
  47. L ( H ) L(H)
  48. P 𝐌 P∈\mathbf{M}′
  49. x H x∈H
  50. P x c l ( 𝐌 h ) Px∈cl(\mathbf{M}h)
  51. 𝐌 \mathbf{M}
  52. n n
  53. T 𝐌 T∈\mathbf{M}
  54. 𝐌 h \mathbf{M}h
  55. c l ( 𝐌 h ) cl(\mathbf{M}h)
  56. T T
  57. L ( H ) L(H)
  58. T P x TPx
  59. T P x c l ( 𝐌 h ) TPx∈cl(\mathbf{M}h)
  60. T T
  61. 𝐌 \mathbf{M}
  62. 𝐌 \mathbf{M}
  63. T T
  64. 𝐌 \mathbf{M}
  65. x , y H x,y∈H
  66. x , T P y = x , P T P y = P x , T P y = T * P x , P y = P T * P x , y = T * P x , y = P x , T y = x , P T y \langle x,TPy\rangle=\langle x,PTPy\rangle=\langle Px,TPy\rangle=\langle T^{*}% Px,Py\rangle=\langle PT^{*}Px,y\rangle=\langle T^{*}Px,y\rangle=\langle Px,Ty% \rangle=\langle x,PTy\rangle
  67. 𝐌 \mathbf{M}′
  68. 𝐌 \mathbf{M}
  69. h 𝐌 h h∈\mathbf{M}h
  70. X h = X P h = P X h c l ( 𝐌 h ) Xh=XPh=PXh∈cl(\mathbf{M}h)
  71. ε > 0 ε>0
  72. 𝐌 \mathbf{M}
  73. ε = d ε=d
  74. 𝐌 \mathbf{M}
  75. | X h - T 0 h | < d |\|Xh\|-\|T_{0}h\||<d
  76. T < s u b > 0 T<sub>0

Von_Neumann_cardinal_assignment.html

  1. | U | = card ( U ) = inf { α O N | α = c U } , |U|=\mathrm{card}(U)=\inf\{\alpha\in ON\ |\ \alpha=_{c}U\},
  2. ω α \omega_{\alpha}
  3. φ α ( ω β ) = ω β \varphi_{\alpha}(\omega_{\beta})=\omega_{\beta}\,

Von_Neumann_universe.html

  1. V 0 := { } . V_{0}:=\{\}.
  2. V β + 1 := 𝒫 ( V β ) . V_{\beta+1}:=\mathcal{P}(V_{\beta}).
  3. V λ := β < λ V β . V_{\lambda}:=\bigcup_{\beta<\lambda}V_{\beta}.
  4. V := α V α . V:=\bigcup_{\alpha}V_{\alpha}.
  5. V α := β < α 𝒫 ( V β ) V_{\alpha}:=\bigcup_{\beta<\alpha}\mathcal{P}(V_{\beta})
  6. 𝒫 ( X ) \mathcal{P}(X)\!
  7. X X
  8. S V α . S\subseteq V_{\alpha}\,.

Von_Neumann–Bernays–Gödel_set_theory.html

  1. a s a\in s
  2. a s a\in s
  3. Rp ( A , a ) := x ( x A x a ) . \mathrm{Rp}(A,a):=\forall x(x\in A\leftrightarrow x\in a).
  4. x y x\in y
  5. x Y x\in Y
  6. x = y x=y
  7. X = Y X=Y
  8. a = A a=A
  9. x ( x a x A ) \forall x(x\in a\leftrightarrow x\in A)
  10. a b [ x ( x a x b ) a = b ] . \forall a\forall b[\forall x(x\in a\leftrightarrow x\in b)\rightarrow a=b]\,.
  11. x y z w [ w z ( w = x w = y ) ] . \forall x\forall y\exists z\forall w[w\in z\leftrightarrow(w=xw=y)].
  12. { x , y } \{x,y\}
  13. y { y } y\cup\{y\}
  14. x ( x A x B ) A = B . \forall x(x\in A\leftrightarrow x\in B)\rightarrow A=B.
  15. ϕ \phi
  16. x ( x A ϕ ( x ) ) . \forall x(x\in A\leftrightarrow\phi(x)).
  17. A = { x ϕ } A=\{x\mid\phi\}
  18. B = { x ψ } . B=\{x\mid\psi\}.
  19. { x ¬ ϕ } = V - A \{x\mid\neg\phi\}=V-A
  20. { x ϕ ψ } = A B \{x\mid\phi\wedge\psi\}=A\cap B
  21. V - A = { x x A } V-A=\{x\mid x\not\in A\}
  22. A B = { x x A x B } A\cap B=\{x\mid x\in A\wedge x\in B\}
  23. ( a , b ) (a,b)
  24. { { a } , { a , b } } , \{\{a\},\{a,b\}\},
  25. A × B = { ( a , b ) a A b B } A\times B=\{(a,b)\mid a\in A\wedge b\in B\}
  26. V × A V\times A
  27. 𝐶𝑜𝑛𝑣 1 ( R ) = { ( b , a ) ( a , b ) R } \mathit{Conv}1(R)=\{(b,a)\mid(a,b)\in R\}
  28. 𝐶𝑜𝑛𝑣 2 ( R ) = { ( b , ( a , c ) ) ( a , ( b , c ) ) R } \mathit{Conv}2(R)=\{(b,(a,c))\mid(a,(b,c))\in R\}
  29. 𝐴𝑠𝑠𝑜𝑐 1 ( R ) = { ( ( a , b ) , c ) ( a , ( b , c ) ) R } \mathit{Assoc}1(R)=\{((a,b),c)\mid(a,(b,c))\in R\}
  30. 𝐴𝑠𝑠𝑜𝑐 2 ( R ) = { ( d , ( a , ( b , c ) ) ) ( d , ( ( a , b ) , c ) ) R } \mathit{Assoc}2(R)=\{(d,(a,(b,c)))\mid(d,((a,b),c))\in R\}
  31. ( x 1 , x 2 , , x n ) (x_{1},x_{2},\ldots,x_{n})
  32. ( x 1 , ( x 2 , , x n ) ) (x_{1},(x_{2},\ldots,x_{n}))
  33. { ( x , y ) ϕ ( x , y ) } \{(x,y)\mid\phi(x,y)\}
  34. { y x [ ϕ ( x , y ) ] } , \{y\mid\exists x[\phi(x,y)]\},
  35. 𝑅𝑛𝑔 ( R ) = { y x [ ( x , y ) R ] } \mathit{Rng}(R)=\{y\mid\exists x[(x,y)\in R]\}
  36. [ ] = { ( x , y ) x y } [\in]=\{(x,y)\mid x\in y\}
  37. [ = ] = { ( x , y ) x = y } [=]=\{(x,y)\mid\,x=y\}
  38. X Y u v w [ ( u , v , w ) Y ( v , w , u ) X ] , \forall X\exists Y\forall uvw[(u,v,w)\in Y\leftrightarrow(v,w,u)\in X],
  39. X Y u v w [ ( u , v , w ) Y ( u , w , v ) X ] . \forall X\exists Y\forall uvw[(u,v,w)\in Y\leftrightarrow(u,w,v)\in X].

Voter_turnout.html

  1. P B + D > C , PB+D>C,

Vysochanskij–Petunin_inequality.html

  1. P ( | X - μ | λ σ ) 4 9 λ 2 . P(\left|X-\mu\right|\geq\lambda\sigma)\leq\frac{4}{9\lambda^{2}}.

Walks_plus_hits_per_inning_pitched.html

  1. 2 / 3 {2}/{3}

Wallpaper_group.html

  1. 2 \sqrt{2}

Wall–Sun–Sun_prime.html

  1. F p - ( < m t p l > p 5 ) F_{p-\left(\frac{<}{m}tpl>{{p}}{{5}}\right)}
  2. ( < m t p l > p 5 ) \textstyle\left(\frac{<}{m}tpl>{{p}}{{5}}\right)
  3. ( p 5 ) = { 1 if p ± 1 ( mod 5 ) - 1 if p ± 2 ( mod 5 ) \left(\frac{p}{5}\right)=\begin{cases}1&\,\text{if }p\equiv\pm 1\;\;(\mathop{{% \rm mod}}5)\\ -1&\,\text{if }p\equiv\pm 2\;\;(\mathop{{\rm mod}}5)\end{cases}
  4. F k ( p - ( < m t p l > k 2 + 4 p ) ) F_{k}({p-\left(\frac{<}{m}tpl>{{k^{2}+4}}{{p}}\right)})
  5. ( < m t p l > k 2 + 4 p ) \left(\frac{<}{m}tpl>{{k^{2}+4}}{{p}}\right)
  6. × 10 1 4 \times 10^{1}4
  7. × 10 1 4 \times 10^{1}4
  8. × 10 1 6 \times 10^{1}6

Walsh_function.html

  1. L 2 [ 0 , 1 ] L^{2}[0,1]
  2. \mathbb{R}
  3. W k : [ 0 , 1 ] { - 1 , 1 } W_{k}:[0,1]\rightarrow\{-1,1\}
  4. k 0 k\in\mathbb{N}_{0}
  5. k 0 k\in\mathbb{N}_{0}
  6. x [ 0 , 1 ] x\in[0,1]
  7. k = j = 0 k j 2 j , k j { 0 , 1 } k=\sum_{j=0}^{\infty}k_{j}2^{j},k_{j}\in\{0,1\}
  8. x = j = 1 x j 2 - j , x j { 0 , 1 } x=\sum_{j=1}^{\infty}x_{j}2^{-j},x_{j}\in\{0,1\}
  9. W k ( x ) = ( - 1 ) j = 0 k j x j + 1 W_{k}(x)=(-1)^{\sum_{j=0}^{\infty}k_{j}x_{j+1}}
  10. W 0 ( x ) = 1 W_{0}(x)=1
  11. W 2 m W_{2^{m}}
  12. W k ( x ) = j = 0 r j ( x ) k j W_{k}(x)=\prod_{j=0}^{\infty}r_{j}(x)^{k_{j}}
  13. { W k } , k 0 \{W_{k}\},k\in\mathbb{N}_{0}
  14. n = 0 / 2 \coprod_{n=0}^{\infty}\mathbb{Z}/2\mathbb{Z}
  15. n = 0 / 2 \prod_{n=0}^{\infty}\mathbb{Z}/2\mathbb{Z}
  16. W 0 W_{0}
  17. L 2 [ 0 , 1 ] L^{2}[0,1]
  18. 0 1 W k ( x ) W l ( x ) d x = δ k l \int_{0}^{1}W_{k}(x)W_{l}(x)dx=\delta_{kl}
  19. f L 2 [ 0 , 1 ] f\in L^{2}[0,1]
  20. f k = 0 1 f ( x ) W k ( x ) d x f_{k}=\int_{0}^{1}f(x)W_{k}(x)dx
  21. 0 1 ( f ( x ) - k = 0 N f k W k ( x ) ) 2 d x N 0 \int_{0}^{1}(f(x)-\sum_{k=0}^{N}f_{k}W_{k}(x))^{2}dx\xrightarrow[N\rightarrow% \infty]{}0
  22. f L 2 [ 0 , 1 ] f\in L^{2}[0,1]
  23. k = 0 f k W k ( x ) \sum_{k=0}^{\infty}f_{k}W_{k}(x)
  24. f ( x ) f(x)
  25. x [ 0 , 1 ] x\in[0,1]
  26. L p [ 0 , 1 ] L^{p}[0,1]
  27. 1 < p < 1<p<\infty
  28. L 1 [ 0 , 1 ] L^{1}[0,1]
  29. 𝔻 = n = 1 / 2 \mathbb{D}=\prod_{n=1}^{\infty}\mathbb{Z}/2\mathbb{Z}
  30. 𝔻 ^ = n = 1 / 2 \hat{\mathbb{D}}=\coprod_{n=1}^{\infty}\mathbb{Z}/2\mathbb{Z}
  31. 𝔻 ^ \hat{\mathbb{D}}
  32. 𝔻 \mathbb{D}
  33. ( X , | | | | ) (X,||\cdot||)
  34. { R t } t 𝔻 A u t ( X ) \{R_{t}\}_{t\in\mathbb{D}}\subset Aut(X)
  35. 𝔻 \mathbb{D}
  36. γ 𝔻 ^ \gamma\in\hat{\mathbb{D}}
  37. X γ = { x X : R t x = γ ( t ) x } X_{\gamma}=\{x\in X:R_{t}x=\gamma(t)x\}
  38. X = Span ¯ ( X γ , γ 𝔻 ^ ) X=\overline{\operatorname{Span}}(X_{\gamma},\gamma\in\hat{\mathbb{D}})
  39. w γ X γ w_{\gamma}\in X_{\gamma}
  40. || w γ || = 1 ||w_{\gamma}||=1
  41. { w γ } γ 𝔻 ^ \{w_{\gamma}\}_{\gamma\in\hat{\mathbb{D}}}
  42. { w k } k 0 \{w_{k}\}_{k\in{\mathbb{N}}_{0}}
  43. { R t } t 𝔻 \{R_{t}\}_{t\in\mathbb{D}}
  44. R t : x = j = 1 x j 2 - j j = 1 ( x j t j ) 2 - j R_{t}:x=\sum_{j=1}^{\infty}x_{j}2^{-j}\mapsto\sum_{j=1}^{\infty}(x_{j}\oplus t% _{j})2^{-j}
  45. \oplus
  46. \mathcal{R}
  47. 1 2 \frac{1}{2}
  48. { x , y , z } \{x,y,z\}

Water_hammer.html

  1. δ P δ t = ρ a δ v δ t \frac{\delta P}{\delta t}=\rho a\frac{\delta v}{\delta t}
  2. Δ P = ρ a 0 Δ v \Delta P=\rho a_{0}\Delta v
  3. effective bulk modulus density \sqrt{\frac{\,\text{effective bulk modulus}}{\,\text{density}}}
  4. a 0 = K / ρ ( 1 + V / a ) [ 1 + ( K / E ) ( D / t ) c ] a_{0}=\sqrt{\frac{K/\rho}{(1+V/a)[1+(K/E)(D/t)c]}}
  5. F = m a = P A = ρ L A d v d t . F=ma=PA=\rho LA{dv\over dt}.
  6. P = ρ v L / t . P=\rho vL/t.
  7. P = 0.07 V L / t + P 1 P=0.07VL/t+P_{1}
  8. δ P = Z h Q \delta P=Z_{h}\,Q
  9. Z h = ρ B 𝑒𝑓𝑓 A Z_{h}=\frac{\sqrt{\rho\,B_{\mathit{eff}}}}{A}
  10. B e q = e E D B_{eq}=\frac{e\,E}{D}
  11. B g = γ P α B_{g}=\frac{\gamma\,P}{\alpha}
  12. 1 B 𝑒𝑓𝑓 = 1 B l + 1 B e q + 1 B g \frac{1}{B_{\mathit{eff}}}=\frac{1}{B_{l}}+\frac{1}{B_{eq}}+\frac{1}{B_{g}}
  13. V x + 1 B m P t = 0 \frac{\partial V}{\partial x}+\frac{1}{B_{m}}\frac{\partial P}{\partial t}=0\,
  14. V t + 1 ρ P x + f 2 D V | V | = 0 \frac{\partial V}{\partial t}+\frac{1}{\rho}\frac{\partial P}{\partial x}+% \frac{f}{2D}V|V|=0\,
  15. ρ \rho
  16. B m B_{m}

Wave_packet.html

  1. E = h ν . E=h\nu.
  2. h h
  3. ν ν
  4. 2 u t 2 = c 2 2 u , {\partial^{2}u\over\partial t^{2}}=c^{2}{\nabla^{2}u},
  5. c c
  6. e x p ( i ω t ) exp(−iωt)
  7. u ( x , t ) = e i ( k x - ω t ) , u({x},t)=e^{i{({k\cdot x}}-\omega t)},
  8. ω 2 = | k | 2 c 2 \omega^{2}=|{k}|^{2}c^{2}
  9. | k | 2 = k x 2 + k y 2 + k z 2 . |{k}|^{2}=k_{x}^{2}+k_{y}^{2}+k_{z}^{2}.
  10. ω ω
  11. 𝐤 \mathbf{k}
  12. u ( x , t ) = A e i ( k x - ω t ) + B e - i ( k x + ω t ) , u(x,t)=Ae^{i(kx-\omega t)}+Be^{-i(kx+\omega t)},
  13. ω = k c ω=kc
  14. x c t x−ct
  15. x + c t x+ct
  16. u ( x , t ) = 1 2 π - A ( k ) e i ( k x - ω ( k ) t ) d k . u(x,t)=\frac{1}{\sqrt{2\pi}}\int^{\,\infty}_{-\infty}A(k)~{}e^{i(kx-\omega(k)t% )}dk.
  17. ω ( k ) = k c ω(k)=kc
  18. u ( x , t ) = F ( x c t ) u(x,t)=F(x−ct)
  19. ω ( k ) = k c ω(k)=−kc
  20. u ( x , t ) = F ( x + c t ) u(x,t)=F(x+ct)
  21. 1 / [ u r a d i c a l , u 2 ] {1}/{[u^{\prime}radical^{\prime},u^{\prime}2^{\prime}]}
  22. A ( k ) A(k)
  23. u ( x , t ) u(x,t)
  24. t = 0 t=0
  25. A ( k ) = 1 2 π - u ( x , 0 ) e - i k x d x . A(k)=\frac{1}{\sqrt{2\pi}}\int^{\,\infty}_{-\infty}u(x,0)~{}e^{-ikx}dx.
  26. u ( x , 0 ) = e - x 2 + i k 0 x , u(x,0)=e^{-x^{2}+ik_{0}x},
  27. A ( k ) = 1 2 e - ( k - k 0 ) 2 4 , A(k)=\frac{1}{\sqrt{2}}e^{-\frac{(k-k_{0})^{2}}{4}},
  28. u ( x , t ) = e - ( x - c t ) 2 + i k 0 ( x - c t ) = e - ( x - c t ) 2 [ cos ( 2 π x - c t λ ) + i sin ( 2 π x - c t λ ) ] . u(x,t)=e^{-(x-ct)^{2}+ik_{0}(x-ct)}=e^{-(x-ct)^{2}}\left[\cos\left(2\pi\frac{x% -ct}{\lambda}\right)+i\sin\left(2\pi\frac{x-ct}{\lambda}\right)\right].
  29. i u t = - 1 2 2 u , i{\partial u\over\partial t}=-\frac{1}{2}{\nabla^{2}u},
  30. ω = 1 2 | k | 2 . \omega=\frac{1}{2}|{k}|^{2}.
  31. u ( x , t ) = 1 1 + 2 i t e - 1 4 k 0 2 e - 1 1 + 2 i t ( x - i k 0 2 ) 2 = 1 1 + 2 i t e - 1 1 + 4 t 2 ( x - k 0 t ) 2 e i 1 1 + 4 t 2 ( ( k 0 + 2 t x ) x - 1 2 t k 0 2 ) . u(x,t)=\frac{1}{\sqrt{1+2it}}e^{-\frac{1}{4}k_{0}^{2}}~{}e^{-\frac{1}{1+2it}% \left(x-\frac{ik_{0}}{2}\right)^{2}}=\frac{1}{\sqrt{1+2it}}e^{-\frac{1}{1+4t^{% 2}}(x-k_{0}t)^{2}}~{}e^{i\frac{1}{1+4t^{2}}\left((k_{0}+2tx)x-\frac{1}{2}tk_{0% }^{2}\right)}~{}.
  32. | u ( x , t ) | 2 = 1 1 + 4 t 2 e - 2 ( x - k 0 t ) 2 1 + 4 t 2 . |u(x,t)|^{2}=\frac{1}{\sqrt{1+4t^{2}}}~{}e^{-\frac{2(x-k_{0}t)^{2}}{1+4t^{2}}}% ~{}.
  33. [ u r a d i c a l , u 1 + 4 , u t , u \xb 2 ] 2 t [u^{\prime}radical^{\prime},u^{\prime}1+4^{\prime},u^{\prime}t^{\prime},u^{% \prime}\xb 2^{\prime}]→2t
  34. t t
  35. ψ ( r , 0 ) = e - r r / 2 a , \psi({r},0)=e^{-{r}\cdot{r}/2a},
  36. a a
  37. t t
  38. ψ ( k , 0 ) = ( 2 π a ) 3 / 2 e - a k k / 2 . \psi({k},0)=(2\pi a)^{3/2}e^{-a{k}\cdot{k}/2}.
  39. Ψ ( k , t ) = ( 2 π a ) 3 / 2 e - a k k / 2 e - i E t / = ( 2 π a ) 3 / 2 e - a k k / 2 - i ( 2 k k / 2 m ) t / = ( 2 π a ) 3 / 2 e - ( a + i t / m ) k k / 2 . \begin{aligned}\displaystyle\Psi({k},t)&\displaystyle=(2\pi a)^{3/2}e^{-a{k}% \cdot{k}/2}e^{-iEt/\hbar}\\ &\displaystyle=(2\pi a)^{3/2}e^{-a{k}\cdot{k}/2-i(\hbar^{2}{k}\cdot{k}/2m)t/% \hbar}\\ &\displaystyle=(2\pi a)^{3/2}e^{-(a+i\hbar t/m){k}\cdot{k}/2}.\end{aligned}
  40. a a
  41. Ψ ( r , t ) = ( a a + i t / m ) 3 / 2 e - r r 2 ( a + i t / m ) . \Psi({r},t)=\left({a\over a+i\hbar t/m}\right)^{3/2}e^{-{{r}\cdot{r}\over 2(a+% i\hbar t/m)}}.
  42. Ψ Ψ
  43. Ψ Ψ
  44. η ( x ) η(x)
  45. η | ψ = η ( r ) ψ ( r ) d 3 r , \langle\eta|\psi\rangle=\int\eta({r})\psi({r})d^{3}{r},
  46. η η
  47. η η
  48. P ( r ) = | Ψ | 2 = Ψ * Ψ = ( a a 2 + ( t / m ) 2 ) 3 e - r r a a 2 + ( t / m ) 2 , P(r)=|\Psi|^{2}=\Psi^{*}\Psi=\left({a\over\sqrt{a^{2}+(\hbar t/m)^{2}}}\right)% ^{3}~{}e^{-{{r}\cdot{r}a\over a^{2}+(\hbar t/m)^{2}}},
  49. a a
  50. P ( r ) P(r)
  51. t = 0 t=0
  52. r r
  53. t = 0 t=0
  54. a 2 + ( t / m ) 2 a . \sqrt{a^{2}+(\hbar t/m)^{2}\over a}.
  55. ħ t / ( m a ) ħt/(m√a)
  56. 1 1
  57. Δ x = [ u r a d i c a l , u a , u / 2 ] Δx=[u^{\prime}radical^{\prime},u^{\prime}a^{\prime},u^{\prime}/2^{\prime}]
  58. ħ / [ u r a d i c a l , u 2 , u a ] ħ/[u^{\prime}radical^{\prime},u^{\prime}2^{\prime},u^{\prime}a^{\prime}]
  59. ħ / m [ u r a d i c a l , u 2 , u a ] ħ/m[u^{\prime}radical^{\prime},u^{\prime}2^{\prime},u^{\prime}a^{\prime}]
  60. ħ t / m [ u r a d i c a l , u 2 , u a ] ħt/m[u^{\prime}radical^{\prime},u^{\prime}2^{\prime},u^{\prime}a^{\prime}]
  61. Δ x Δ p = ħ / 2 ΔxΔp=ħ/2
  62. ħ t / m a ħt/ma
  63. ψ = A i ( B ( x B ³ t ² ) ) e x p ( i B ³ t ( x 2 B ³ t ² / 3 ) ) ψ=Ai(B(x−B³t²))exp(iB³t(x−2B³t²/3))
  64. ħ ħ
  65. x ⟨x⟩
  66. p = 0 ⟨p⟩=0
  67. B ( x B ³ t ² ) + ( p / B t B ² ) ² = 0 B(x−B³t²)+(p/B−tB²)²=0
  68. W ( x , p ; t ) = W ( x - B 3 t 2 , p - B 3 t ; 0 ) = 1 2 1 / 3 π B Ai ( 2 2 / 3 ( B x + p 2 B 2 - 2 B p t ) ) . W(x,p;t)=W(x-B^{3}t^{2},p-B^{3}t;0)={1\over 2^{1/3}\pi B}~{}\mathrm{Ai}\left(2% ^{2/3}\left(Bx+{p^{2}\over B^{2}}-2Bpt\right)\right).
  69. x x
  70. K K
  71. K K
  72. a a
  73. ε ε
  74. ψ 0 ( x ) = 1 2 π ϵ e - x 2 2 ϵ \psi_{0}(x)={1\over\sqrt{2\pi\epsilon}}e^{-{x^{2}\over 2\epsilon}}\,
  75. δ ( x ) δ(x)
  76. K t ( x ) = 1 2 π ( i t + ϵ ) e - x 2 2 i t + ϵ K_{t}(x)={1\over\sqrt{2\pi(it+\epsilon)}}e^{-x^{2}\over 2it+\epsilon}\,
  77. ε ε
  78. K K
  79. K K
  80. K K
  81. K t ( x , y ) = K t ( x - y ) = 1 2 π i t e - i ( x - y ) 2 2 t . K_{t}(x,y)=K_{t}(x-y)={1\over\sqrt{2\pi it}}e^{-i(x-y)^{2}\over 2t}\,.
  82. lim t 0 K t ( x - y ) = δ ( x - y ) , \lim_{t\rightarrow 0}K_{t}(x-y)=\delta(x-y)~{},
  83. K K
  84. x K t ( x ) = 1 , \int_{x}K_{t}(x)=1\,,
  85. y y
  86. ψ 0 ( x ) = δ ( x - y ) , \psi_{0}(x)=\delta(x-y)\,,
  87. ψ t ( x ) = 1 2 π i t e - i ( x - y ) 2 / 2 t . \psi_{t}(x)={1\over\sqrt{2\pi it}}e^{-i(x-y)^{2}/2t}\,.
  88. ψ 0 ( x ) = y ψ 0 ( y ) δ ( x - y ) , \psi_{0}(x)=\int_{y}\psi_{0}(y)\delta(x-y)\,,
  89. ψ ψ
  90. K K
  91. x x
  92. t t
  93. y y
  94. y y
  95. x x
  96. K K
  97. ψ ψ
  98. ψ t = K * ψ 0 . \psi_{t}=K*\psi_{0}\,.
  99. x x
  100. y y
  101. t t
  102. t t
  103. y K ( x - y ; t ) K ( y - z ; t ) = K ( x - z ; t + t ) , \int_{y}K(x-y;t)K(y-z;t^{\prime})=K(x-z;t+t^{\prime})~{},
  104. x x
  105. z z
  106. t t
  107. t t
  108. x x
  109. y y
  110. t t
  111. y y
  112. z z
  113. t t
  114. t ρ = 1 2 2 x 2 ρ , {\partial\over\partial t}\rho={1\over 2}{\partial^{2}\over\partial x^{2}}\rho~% {},
  115. ρ t ( x ) = 1 2 π t e - x 2 2 t , \rho_{t}(x)={1\over\sqrt{2\pi t}}e^{-x^{2}\over 2t}~{},
  116. lim t 0 ρ t ( x ) = δ ( x ) \lim_{t\rightarrow 0}\rho_{t}(x)=\delta(x)\,
  117. lim t 0 x f ( x ) ρ t ( x ) = f ( 0 ) \lim_{t\rightarrow 0}\int_{x}f(x)\rho_{t}(x)=f(0)\,
  118. f f
  119. K t + t = K t * K t , K_{t+t^{\prime}}=K_{t}*K_{t^{\prime}}\,,
  120. H H
  121. K t ( x ) = e - t H , K_{t}(x)=e^{-tH}\,,
  122. H = - 2 2 . H=-{\nabla^{2}\over 2}\,.
  123. x x
  124. x x
  125. K K
  126. K t ( x , x ) = K t ( x - x ) . K_{t}(x,x^{\prime})=K_{t}(x-x^{\prime})\,.
  127. C ( x , x ′′ ) = x A ( x , x ) B ( x , x ′′ ) , C(x,x^{\prime\prime})=\int_{x^{\prime}}A(x,x^{\prime})B(x^{\prime},x^{\prime% \prime})\,,
  128. C ( Δ ) = C ( x - x ′′ ) = x A ( x - x ) B ( x - x ′′ ) = y A ( Δ - y ) B ( y ) . C(\Delta)=C(x-x^{\prime\prime})=\int_{x^{\prime}}A(x-x^{\prime})B(x^{\prime}-x% ^{\prime\prime})=\int_{y}A(\Delta-y)B(y)\,.
  129. K z ( x ) = e - z H . K_{z}(x)=e^{-zH}\,.
  130. z z
  131. x x
  132. K K
  133. K K
  134. z z
  135. K t Schr = K i t + ϵ = e - ( i t + ϵ ) H , K_{t}^{\rm Schr}=K_{it+\epsilon}=e^{-(it+\epsilon)H}\,,
  136. K z * K z = K z + z K_{z}*K_{z^{\prime}}=K_{z+z^{\prime}}\,
  137. ψ 0 ( x ) = K a ( x ) = K a * δ ( x ) \psi_{0}(x)=K_{a}(x)=K_{a}*\delta(x)\,
  138. ψ t = K i t * K a = K a + i t . \psi_{t}=K_{it}*K_{a}=K_{a+it}\,.
  139. ψ t ( x ) = 1 2 π ( a + i t ) e - x 2 2 ( a + i t ) . \psi_{t}(x)={1\over\sqrt{2\pi(a+it)}}e^{-{x^{2}\over 2(a+it)}}\,.

Weakly_harmonic_function.html

  1. f f
  2. D D
  3. D f Δ g = 0 \int_{D}f\,\Delta g=0
  4. g g
  5. D D

Weierstrass's_elliptic_functions.html

  1. \wp
  2. | f ( z ) | = | f ( x + i y ) | = 1 . \left|f(z)\right|=\left|f(x+iy)\right|=1\;.
  3. τ \tau
  4. τ \tau
  5. τ \tau
  6. ( z ; ω 1 , ω 2 ) = 1 z 2 + n 2 + m 2 0 { 1 ( z + m ω 1 + n ω 2 ) 2 - 1 ( m ω 1 + n ω 2 ) 2 } . \wp(z;\omega_{1},\omega_{2})=\frac{1}{z^{2}}+\sum_{n^{2}+m^{2}\neq 0}\left\{% \frac{1}{(z+m\omega_{1}+n\omega_{2})^{2}}-\frac{1}{\left(m\omega_{1}+n\omega_{% 2}\right)^{2}}\right\}.
  7. Λ = { m ω 1 + n ω 2 : m , n } \Lambda=\{m\omega_{1}+n\omega_{2}:m,n\in\mathbb{Z}\}
  8. ( z ; Λ ) = ( z ; ω 1 , ω 2 ) \wp(z;\Lambda)=\wp(z;\omega_{1},\omega_{2})
  9. τ \tau
  10. ( z ; τ ) = ( z ; 1 , τ ) = 1 z 2 + n 2 + m 2 0 { 1 ( z + m + n τ ) 2 - 1 ( m + n τ ) 2 } . \wp(z;\tau)=\wp(z;1,\tau)=\frac{1}{z^{2}}+\sum_{n^{2}+m^{2}\neq 0}\left\{{1% \over(z+m+n\tau)^{2}}-{1\over(m+n\tau)^{2}}\right\}.
  11. ( z ; ω 1 , ω 2 ) = ( z ω 1 ; ω 2 ω 1 ) ω 1 2 . \wp(z;\omega_{1},\omega_{2})=\frac{\wp(\frac{z}{\omega_{1}};\frac{\omega_{2}}{% \omega_{1}})}{\omega_{1}^{2}}.
  12. ( z ; τ ) = π 2 ϑ 2 ( 0 ; τ ) ϑ 10 2 ( 0 ; τ ) ϑ 01 2 ( z ; τ ) ϑ 11 2 ( z ; τ ) - π 2 3 [ ϑ 4 ( 0 ; τ ) + ϑ 10 4 ( 0 ; τ ) ] \wp(z;\tau)=\pi^{2}\vartheta^{2}(0;\tau)\vartheta_{10}^{2}(0;\tau){\vartheta_{% 01}^{2}(z;\tau)\over\vartheta_{11}^{2}(z;\tau)}-{\pi^{2}\over{3}}\left[% \vartheta^{4}(0;\tau)+\vartheta_{10}^{4}(0;\tau)\right]
  13. ( z ) \wp(z)
  14. \wp
  15. ( z ; ω 1 , ω 2 ) = z - 2 + 1 20 g 2 z 2 + 1 28 g 3 z 4 + O ( z 6 ) \wp(z;\omega_{1},\omega_{2})=z^{-2}+\frac{1}{20}g_{2}z^{2}+\frac{1}{28}g_{3}z^% {4}+O(z^{6})
  16. g 2 = 60 ( m , n ) ( 0 , 0 ) ( m ω 1 + n ω 2 ) - 4 g_{2}=60\sum_{(m,n)\neq(0,0)}(m\omega_{1}+n\omega_{2})^{-4}
  17. g 3 = 140 ( m , n ) ( 0 , 0 ) ( m ω 1 + n ω 2 ) - 6 . g_{3}=140\sum_{(m,n)\neq(0,0)}(m\omega_{1}+n\omega_{2})^{-6}.
  18. τ \tau
  19. τ \tau
  20. τ \tau
  21. τ \tau
  22. g 2 ( λ ω 1 , λ ω 2 ) = λ - 4 g 2 ( ω 1 , ω 2 ) g_{2}(\lambda\omega_{1},\lambda\omega_{2})=\lambda^{-4}g_{2}(\omega_{1},\omega% _{2})
  23. g 3 ( λ ω 1 , λ ω 2 ) = λ - 6 g 3 ( ω 1 , ω 2 ) . g_{3}(\lambda\omega_{1},\lambda\omega_{2})=\lambda^{-6}g_{3}(\omega_{1},\omega% _{2}).
  24. g 2 g_{2}
  25. g 3 g_{3}
  26. τ = ω 2 / ω 1 \tau=\omega_{2}/\omega_{1}
  27. τ \tau
  28. g 2 ( τ ) = g 2 ( 1 , ω 2 / ω 1 ) g_{2}(\tau)=g_{2}(1,\omega_{2}/\omega_{1})
  29. g 3 ( τ ) = g 3 ( 1 , ω 2 / ω 1 ) g_{3}(\tau)=g_{3}(1,\omega_{2}/\omega_{1})
  30. g 2 g_{2}
  31. g 3 g_{3}
  32. q = exp ( i π τ ) q=\exp(i\pi\tau)
  33. g 2 ( τ ) = 4 π 4 3 [ 1 + 240 k = 1 σ 3 ( k ) q 2 k ] g_{2}(\tau)=\frac{4\pi^{4}}{3}\left[1+240\sum_{k=1}^{\infty}\sigma_{3}(k)q^{2k% }\right]
  34. g 3 ( τ ) = 8 π 6 27 [ 1 - 504 k = 1 σ 5 ( k ) q 2 k ] g_{3}(\tau)=\frac{8\pi^{6}}{27}\left[1-504\sum_{k=1}^{\infty}\sigma_{5}(k)q^{2% k}\right]
  35. σ a ( k ) \sigma_{a}(k)
  36. ω 1 , ω 2 \omega_{1},\omega_{2}
  37. g 2 ( ω 1 , ω 2 ) = π 4 12 ω 1 4 ( a 8 - a 4 b 4 + b 8 ) = π 4 6 ω 1 4 ( a 8 + b 8 + c 8 ) g_{2}(\omega_{1},\omega_{2})=\frac{\pi^{4}}{12\omega_{1}^{4}}(a^{8}-a^{4}b^{4}% +b^{8})=\frac{\pi^{4}}{6\omega_{1}^{4}}(a^{8}+b^{8}+c^{8})
  38. g 3 ( ω 1 , ω 2 ) = π 6 ( 6 ω 1 ) 6 ( a 12 - 3 2 a 8 b 4 - 3 2 a 4 b 8 + b 12 ) g_{3}(\omega_{1},\omega_{2})=\frac{\pi^{6}}{(\sqrt{6}\,\omega_{1})^{6}}(a^{12}% -\frac{3}{2}a^{8}b^{4}-\frac{3}{2}a^{4}b^{8}+b^{12})
  39. a = θ 2 ( 0 ; q ) = ϑ 10 ( 0 ; τ ) a=\theta_{2}(0;q)=\vartheta_{10}(0;\tau)
  40. b = θ 3 ( 0 ; q ) = ϑ 00 ( 0 ; τ ) b=\theta_{3}(0;q)=\vartheta_{00}(0;\tau)
  41. c = θ 4 ( 0 ; q ) = ϑ 01 ( 0 ; τ ) c=\theta_{4}(0;q)=\vartheta_{01}(0;\tau)
  42. τ = ω 2 / ω 1 \tau=\omega_{2}/\omega_{1}
  43. q = e π i τ q=e^{\pi i\tau}
  44. θ m \theta_{m}
  45. ϑ n \vartheta_{n}
  46. [ ( z ) ] 2 = 4 [ ( z ) ] 3 - g 2 ( z ) - g 3 , [\wp^{\prime}(z)]^{2}=4[\wp(z)]^{3}-g_{2}\wp(z)-g_{3},\,
  47. ω 1 \omega_{1}
  48. ω 2 \omega_{2}
  49. [ ( z ) ] 2 | z = 0 4 z 6 - 24 z 2 1 ( m ω 1 + n ω 2 ) 4 - 80 1 ( m ω 1 + n ω 2 ) 6 [\wp^{\prime}(z)]^{2}|_{z=0}\sim\frac{4}{z^{6}}-\frac{24}{z^{2}}\sum\frac{1}{(% m\omega_{1}+n\omega_{2})^{4}}-80\sum\frac{1}{(m\omega_{1}+n\omega_{2})^{6}}
  50. [ ( z ) ] 3 | z = 0 1 z 6 + 9 z 2 1 ( m ω 1 + n ω 2 ) 4 + 15 1 ( m ω 1 + n ω 2 ) 6 . [\wp(z)]^{3}|_{z=0}\sim\frac{1}{z^{6}}+\frac{9}{z^{2}}\sum\frac{1}{(m\omega_{1% }+n\omega_{2})^{4}}+15\sum\frac{1}{(m\omega_{1}+n\omega_{2})^{6}}.
  51. u = y d s 4 s 3 - g 2 s - g 3 . u=\int_{y}^{\infty}\frac{ds}{\sqrt{4s^{3}-g_{2}s-g_{3}}}.
  52. y = ( u ) . y=\wp(u).
  53. Δ = g 2 3 - 27 g 3 2 . \Delta=g_{2}^{3}-27g_{3}^{2}.\,
  54. Δ = ( 2 π ) 12 η 24 \Delta=(2\pi)^{12}\eta^{24}
  55. η \eta
  56. Δ ( a τ + b c τ + d ) = ( c τ + d ) 12 Δ ( τ ) \Delta\left(\frac{a\tau+b}{c\tau+d}\right)=\left(c\tau+d\right)^{12}\Delta(\tau)
  57. τ \tau
  58. Δ \Delta
  59. e 1 + e 2 + e 3 = 0. e_{1}+e_{2}+e_{3}=0.\,
  60. g 2 = - 4 ( e 1 e 2 + e 1 e 3 + e 2 e 3 ) = 2 ( e 1 2 + e 2 2 + e 3 2 ) g_{2}=-4\left(e_{1}e_{2}+e_{1}e_{3}+e_{2}e_{3}\right)=2\left(e_{1}^{2}+e_{2}^{% 2}+e_{3}^{2}\right)
  61. g 3 = 4 e 1 e 2 e 3 g_{3}=4e_{1}e_{2}e_{3}
  62. 4 X 3 - g 2 X - g 3 4X^{3}-g_{2}X-g_{3}
  63. τ \tau
  64. a = θ 2 ( 0 ; e π i τ ) = ϑ 10 ( 0 ; τ ) a=\theta_{2}(0;e^{\pi i\tau})=\vartheta_{10}(0;\tau)
  65. b = θ 3 ( 0 ; e π i τ ) = ϑ 00 ( 0 ; τ ) b=\theta_{3}(0;e^{\pi i\tau})=\vartheta_{00}(0;\tau)
  66. c = θ 4 ( 0 ; e π i τ ) = ϑ 01 ( 0 ; τ ) c=\theta_{4}(0;e^{\pi i\tau})=\vartheta_{01}(0;\tau)
  67. e 1 ( τ ) = 1 3 π 2 ( b 4 + c 4 ) e_{1}(\tau)=\tfrac{1}{3}\pi^{2}(b^{4}+c^{4})
  68. e 2 ( τ ) = 1 3 π 2 ( - a 4 - b 4 ) e_{2}(\tau)=\tfrac{1}{3}\pi^{2}(-a^{4}-b^{4})
  69. e 3 ( τ ) = 1 3 π 2 ( a 4 - c 4 ) e_{3}(\tau)=\tfrac{1}{3}\pi^{2}(a^{4}-c^{4})
  70. g 2 = 2 ( e 1 2 + e 2 2 + e 3 2 ) g_{2}=2\left(e_{1}^{2}+e_{2}^{2}+e_{3}^{2}\right)
  71. g 3 = 4 e 1 e 2 e 3 g_{3}=4e_{1}e_{2}e_{3}
  72. g 2 ( τ ) = 2 3 π 4 ( a 8 + b 8 + c 8 ) g_{2}(\tau)=\tfrac{2}{3}\pi^{4}(a^{8}+b^{8}+c^{8})
  73. g 3 ( τ ) = 4 27 π 6 ( a 8 + b 8 + c 8 ) 3 - 54 ( a b c ) 8 2 g_{3}(\tau)=\tfrac{4}{27}\pi^{6}\sqrt{\frac{(a^{8}+b^{8}+c^{8})^{3}-54(abc)^{8% }}{2}}
  74. Δ = g 2 3 - 27 g 3 2 = ( 2 π ) 12 ( 1 2 a b c ) 8 \Delta=g_{2}^{3}-27g_{3}^{2}=(2\pi)^{12}\left(\tfrac{1}{2}abc\right)^{8}
  75. Δ > 0 \Delta>0
  76. e 1 > e 2 > e 3 e_{1}>e_{2}>e_{3}
  77. Δ < 0 \Delta<0
  78. e 1 = - α + β i e_{1}=-\alpha+\beta i
  79. α 0 \alpha\geq 0
  80. β > 0 \beta>0
  81. e 3 = e 1 ¯ e_{3}=\overline{e_{1}}
  82. e 2 e_{2}
  83. ( ω 1 / 2 ) = e 1 ( ω 2 / 2 ) = e 2 ( ω 3 / 2 ) = e 3 \wp(\omega_{1}/2)=e_{1}\qquad\wp(\omega_{2}/2)=e_{2}\qquad\wp(\omega_{3}/2)=e_% {3}
  84. ω 3 = - ( ω 1 + ω 2 ) \omega_{3}=-(\omega_{1}+\omega_{2})
  85. ( ω i / 2 ) 2 = ( ω i / 2 ) = 0 \wp^{\prime}(\omega_{i}/2)^{2}=\wp^{\prime}(\omega_{i}/2)=0
  86. i = 1 , 2 , 3 i=1,2,3
  87. ( ) \wp()
  88. ω 1 / 2 = e 1 d z 4 z 3 - g 2 z - g 3 \omega_{1}/2=\int_{e_{1}}^{\infty}\frac{dz}{\sqrt{4z^{3}-g_{2}z-g_{3}}}
  89. ω 3 / 2 = i - e 3 d z 4 z 3 - g 2 z - g 3 . \omega_{3}/2=i\int_{-e_{3}}^{\infty}\frac{dz}{\sqrt{4z^{3}-g_{2}z-g_{3}}}.
  90. det [ ( z ) ( z ) 1 ( y ) ( y ) 1 ( z + y ) - ( z + y ) 1 ] = 0 \det\begin{bmatrix}\wp(z)&\wp^{\prime}(z)&1\\ \wp(y)&\wp^{\prime}(y)&1\\ \wp(z+y)&-\wp^{\prime}(z+y)&1\end{bmatrix}=0
  91. det [ ( u ) ( u ) 1 ( v ) ( v ) 1 ( w ) ( w ) 1 ] = 0 \det\begin{bmatrix}\wp(u)&\wp^{\prime}(u)&1\\ \wp(v)&\wp^{\prime}(v)&1\\ \wp(w)&\wp^{\prime}(w)&1\end{bmatrix}=0
  92. ( z + y ) = 1 4 { ( z ) - ( y ) ( z ) - ( y ) } 2 - ( z ) - ( y ) \wp(z+y)=\frac{1}{4}\left\{\frac{\wp^{\prime}(z)-\wp^{\prime}(y)}{\wp(z)-\wp(y% )}\right\}^{2}-\wp(z)-\wp(y)
  93. ( 2 z ) = 1 4 { ′′ ( z ) ( z ) } 2 - 2 ( z ) , \wp(2z)=\frac{1}{4}\left\{\frac{\wp^{\prime\prime}(z)}{\wp^{\prime}(z)}\right% \}^{2}-2\wp(z),
  94. ω 1 = 1 \omega_{1}=1
  95. τ \tau
  96. ω 2 \omega_{2}
  97. τ \tau
  98. τ \tau
  99. ( z ; τ ) = 1 z 2 + ( m , n ) ( 0 , 0 ) 1 ( z + m + n τ ) 2 - 1 ( m + n τ ) 2 . \wp(z;\tau)=\frac{1}{z^{2}}+\sum_{(m,n)\neq(0,0)}{1\over(z+m+n\tau)^{2}}-{1% \over(m+n\tau)^{2}}.
  100. τ \tau
  101. τ \tau
  102. τ \tau
  103. τ \tau
  104. ( z + 1 ) = ( z + τ ) = ( z ) . \wp(z+1)=\wp(z+\tau)=\wp(z).
  105. ( c z ; c τ ) = ( z ; τ ) / c 2 \wp(cz;c\tau)=\wp(z;\tau)/c^{2}
  106. 2 = 4 3 - g 2 - g 3 \wp^{\prime 2}=4\wp^{3}-g_{2}\wp-g_{3}
  107. g 2 g_{2}
  108. g 3 g_{3}
  109. τ \tau
  110. Y 2 = 4 X 3 - g 2 X - g 3 Y^{2}=4X^{3}-g_{2}X-g_{3}
  111. ( , ) (\wp,\wp^{\prime})
  112. ( , ) , \mathbb{C}(\wp,\wp^{\prime}),
  113. ( z ; τ ) = π 2 ϑ 2 ( 0 ; τ ) ϑ 10 2 ( 0 ; τ ) ϑ 01 2 ( z ; τ ) ϑ 11 2 ( z ; τ ) + e 2 ( τ ) . \wp(z;\tau)=\pi^{2}\vartheta^{2}(0;\tau)\vartheta_{10}^{2}(0;\tau){\vartheta_{% 01}^{2}(z;\tau)\over\vartheta_{11}^{2}(z;\tau)}+e_{2}(\tau).
  114. ( z ) = e 3 + e 1 - e 3 sn 2 w = e 2 + ( e 1 - e 3 ) dn 2 w sn 2 w = e 1 + ( e 1 - e 3 ) cn 2 w sn 2 w \wp(z)=e_{3}+\frac{e_{1}-e_{3}}{\operatorname{sn}^{2}w}=e_{2}+(e_{1}-e_{3})% \frac{\operatorname{dn}^{2}w}{\operatorname{sn}^{2}w}=e_{1}+(e_{1}-e_{3})\frac% {\operatorname{cn}^{2}w}{\operatorname{sn}^{2}w}
  115. k e 2 - e 3 e 1 - e 3 k\equiv\sqrt{\frac{e_{2}-e_{3}}{e_{1}-e_{3}}}
  116. w z e 1 - e 3 . w\equiv z\sqrt{e_{1}-e_{3}}.

Weierstrass_function.html

  1. f ( x ) = n = 0 a n cos ( b n π x ) , f(x)=\sum_{n=0}^{\infty}a^{n}\cos(b^{n}\pi x),
  2. 0 < a < 1 0<a<1
  3. b b
  4. a b > 1 + 3 2 π . ab>1+\frac{3}{2}\pi.
  5. b b
  6. b = 7 b=7
  7. x x\in{\mathbb{R}}
  8. lim inf n f ( x n ) - f ( x ) x n - x > lim sup n f ( x n ) - f ( x ) x n - x . \lim\inf_{n}\frac{f(x_{n})-f(x)}{x_{n}-x}>\lim\sup_{n}\frac{f(x^{\prime}_{n})-% f(x)}{x^{\prime}_{n}-x}.
  9. W α ( x ) = n = 0 b - n α cos ( b n x ) W_{\alpha}(x)=\sum_{n=0}^{\infty}b^{-n\alpha}\cos(b^{n}x)
  10. | W α ( x ) - W α ( y ) | C | x - y | α |W_{\alpha}(x)-W_{\alpha}(y)|\leq C|x-y|^{\alpha}

Weierstrass_M-test.html

  1. n 1 , x A : | f n ( x ) | M n , \forall n\geq 1,\forall x\in A:\ |f_{n}(x)|\leq M_{n},
  2. n = 1 M n < \sum_{n=1}^{\infty}M_{n}<\infty
  3. n = 1 f n ( x ) \sum_{n=1}^{\infty}f_{n}(x)
  4. | f n | M n |f_{n}|\leq M_{n}
  5. f n M n \|f_{n}\|\leq M_{n}
  6. \|\cdot\|
  7. n = 1 M n \sum_{n=1}^{\infty}M_{n}
  8. | f n ( x ) | M n |f_{n}(x)|\leq M_{n}
  9. n = 1 f n ( x ) \sum_{n=1}^{\infty}f_{n}(x)
  10. k > K : | M - n = 1 k M n | < ε . \forall k>K:\ \left|M-\sum_{n=1}^{k}M_{n}\right|<\varepsilon.
  11. n = 1 f n ( x ) \sum_{n=1}^{\infty}f_{n}(x)
  12. k > K , x A : | f ( x ) - n = 1 k f n ( x ) | < ε . \forall k>K,\forall x\in A:\ \left|f(x)-\sum_{n=1}^{k}f_{n}(x)\right|<\varepsilon.
  13. x A : | f ( x ) - n = 1 k f n ( x ) | = | n = k + 1 f n ( x ) | n = k + 1 | f n ( x ) | n = k + 1 M n = | M - n = 1 k M n | < ε . \forall x\in A:\ \left|f(x)-\sum_{n=1}^{k}f_{n}(x)\right|=\left|\sum_{n=k+1}^{% \infty}f_{n}(x)\right|\leq\sum_{n=k+1}^{\infty}\left|f_{n}(x)\right|\leq\sum_{% n=k+1}^{\infty}M_{n}=\left|M-\sum_{n=1}^{k}M_{n}\right|<\varepsilon.
  14. k = 1 f k ( x ) \sum_{k=1}^{\infty}f_{k}(x)
  15. S n ( x ) = k = 1 n f k ( x ) S_{n}(x)=\sum_{k=1}^{n}f_{k}(x)
  16. n = 1 M n \sum_{n=1}^{\infty}M_{n}
  17. M n 0 M_{n}\geq 0
  18. n n
  19. ε > 0 \varepsilon>0
  20. N ( ε ) N(\varepsilon)
  21. n > m > N ( ε ) n>m>N(\varepsilon)
  22. k = m + 1 n M k < ε . \sum_{k=m+1}^{n}M_{k}<\varepsilon.
  23. N ( ε ) N(\varepsilon)
  24. x A x\in A
  25. n > m > N ( ε ) n>m>N(\varepsilon)
  26. | S n ( x ) - S m ( x ) | | k = m + 1 n f k ( x ) | ( 1 ) k = m + 1 n | f k ( x ) | k = m + 1 n M k < ε . \left|S_{n}(x)-S_{m}(x)\right|\leq\left|\sum_{k=m+1}^{n}f_{k}(x)\right|% \overset{(1)}{\leq}\sum_{k=m+1}^{n}|f_{k}(x)|\leq\sum_{k=m+1}^{n}M_{k}<\varepsilon.

Weight_Watchers.html

  1. PointsPlus = max { round ( ( 16 protein ) + ( 19 carbohydrates ) + ( 45 fat ) - ( 14 fiber ) 175 ) , 0 } \,\text{PointsPlus}=\max\left\{\mathrm{round}\left(\frac{(16\cdot\,\text{% protein})+(19\cdot\,\text{carbohydrates})+(45\cdot\,\text{fat})-(14\cdot\,% \text{fiber})}{175}\right),0\right\}
  2. PointsPlus = max { round ( protein 10.9375 + carbohydrates 9.2105 + fat 3.8889 - fiber 12.5 ) , 0 } \,\text{PointsPlus}=\max\left\{\mathrm{round}\left(\frac{\,\text{protein}}{10.% 9375}+\frac{\,\text{carbohydrates}}{9.2105}+\frac{\,\text{fat}}{3.8889}-\frac{% \,\text{fiber}}{12.5}\right),0\right\}
  3. PointsPlus = max { round ( ( 16 protein ) + ( 19 ( carbohydrates - sugar alcohol ) ) + ( 45 fat ) - ( 14 fiber ) + ( 58.05 alcohol ) + ( 11.4 sugar alcohol ) 175 ) , 0 } \,\text{PointsPlus}=\max\left\{\mathrm{round}\left(\frac{(16\cdot\,\text{% protein})+(19\cdot(\,\text{carbohydrates}-\,\text{sugar alcohol}))+(45\cdot\,% \text{fat})-(14\cdot\,\text{fiber})+(58.05\cdot\,\text{alcohol})+(11.4\cdot\,% \text{sugar alcohol})}{175}\right),0\right\}
  4. PointsPlus = max { round ( protein 10.9375 + carbohydrates 9.2105 + fat 3.8889 - fiber 12.5 + alcohol 3.0147 - sugar alcohol 23.0263 ) , 0 } \,\text{PointsPlus}=\max\left\{\mathrm{round}\left(\frac{\,\text{protein}}{10.% 9375}+\frac{\,\text{carbohydrates}}{9.2105}+\frac{\,\text{fat}}{3.8889}-\frac{% \,\text{fiber}}{12.5}+\frac{\,\text{alcohol}}{3.0147}-\frac{\,\text{sugar % alcohol}}{23.0263}\right),0\right\}
  5. T E E m = 864 - ( 9.72 a g e ) + 1.12 ( 14.2 w e i g h t + 503 h e i g h t ) TEE_{m}=864-(9.72\cdot age)+1.12\cdot(14.2\cdot weight+503\cdot height)
  6. T E E f = 387 - ( 7.31 a g e ) + 1.14 ( 10.9 w e i g h t + 660.7 h e i g h t ) TEE_{f}=387-(7.31\cdot age)+1.14\cdot(10.9\cdot weight+660.7\cdot height)
  7. T E E m TEE_{m}
  8. T E E f TEE_{f}
  9. T E E TEE
  10. A T E E = 0.9 T E E + 200 ATEE=0.9\cdot TEE+200
  11. T a r g e t = round ( min ( max ( A T E E - 1000 , 1000 ) , 2500 ) 35 ) Target=\mathrm{round}\left(\frac{\min\left(\max\left(ATEE-1000,1000\right),250% 0\right)}{35}\right)
  12. T a r g e t m o d = min ( max ( round ( max ( A T E E - 1000 , 1000 ) 35 ) - 7 - 4 , 26 ) , 71 ) Target_{mod}=\min\left(\max\left(\mathrm{round}\left(\frac{\max\left(ATEE-1000% ,1000\right)}{35}\right)-7-4,26\right),71\right)
  13. P r o P o i n t s = max { round ( ( 16 p r o t e i n ) + ( 19 c a r b o h y d r a t e s ) + ( 45 f a t ) + ( 5 f i b e r ) 175 ) , 0 } ProPoints=\max\left\{\mathrm{round}\left(\frac{(16\cdot protein)+(19\cdot carbohydrates% )+(45\cdot fat)+(5\cdot fiber)}{175}\right),0\right\}
  14. P r o P o i n t s = max { round ( p r o t e i n 10.9375 + c a r b o h y d r a t e s 9.2105 + f a t 3.8889 + f i b e r 35 ) , 0 } ProPoints=\max\left\{\mathrm{round}\left(\frac{protein}{10.9375}+\frac{% carbohydrates}{9.2105}+\frac{fat}{3.8889}+\frac{fiber}{35}\right),0\right\}
  15. p ( c , f , r ) = round ( c 50 + f 12 - min { r , 4 } 5 ) p(c,f,r)=\mathrm{round}\left(\frac{c}{50}+\frac{f}{12}-\frac{\min\{r,4\}}{5}\right)
  16. p p
  17. c c
  18. f f
  19. r r
  20. p = e k 1 + f k 2 p=\frac{e}{k_{1}}+\frac{f}{k_{2}}
  21. p p
  22. e e
  23. f f
  24. k 1 k_{1}
  25. k 2 k_{2}
  26. p = e 290 + f 4 p=\frac{e}{290}+\frac{f}{4}
  27. p = e 70 + f 4 p=\frac{e}{70}+\frac{f}{4}
  28. p p

Weil_restriction.html

  1. Res L / k X \mathrm{Res}_{L/k}X
  2. Res L / k X ( S ) = X ( S × k L ) \mathrm{Res}_{L/k}X(S)=X(S\times_{k}L)
  3. Res L / k X \mathrm{Res}_{L/k}X
  4. \to
  5. T S T\to S
  6. Res L / k \mathrm{Res}_{L/k}
  7. R e s L / k 𝔸 1 Res_{L/k}\mathbb{A}^{1}
  8. 𝔸 s \mathbb{A}^{s}
  9. X = Spec L [ x 1 , , x n ] / ( f 1 , , f m ) X=\,\text{Spec}L[x_{1},\dots,x_{n}]/(f_{1},\dots,f_{m})
  10. Res L / k X \mathrm{Res}_{L/k}X
  11. k [ y i , j ] / ( g l , r ) k[y_{i,j}]/(g_{l,r})
  12. 1 i n , 1 j s 1\leq i\leq n,1\leq j\leq s
  13. y i , j y_{i,j}
  14. e 1 , , e s e_{1},\dots,e_{s}
  15. x i = y i , 1 e 1 + + y i , s e s x_{i}=y_{i,1}e_{1}+\dots+y_{i,s}e_{s}
  16. f t = g t , 1 e 1 + + g t , s e s f_{t}=g_{t,1}e_{1}+\dots+g_{t,s}e_{s}
  17. 𝕊 := Res / 𝔾 m \mathbb{S}:=\mathrm{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}_{m}
  18. × \mathbb{C}^{\times}

Weird_number.html

  1. × 10 9 \times 10^{9}
  2. × 10 1 7 \times 10^{1}7
  3. R = 2 k Q - ( Q + 1 ) ( Q + 1 ) - 2 k R=\frac{2^{k}Q-(Q+1)}{(Q+1)-2^{k}}
  4. n = 2 k - 1 Q R n=2^{k-1}QR
  5. n = 2 56 ( 2 61 - 1 ) 153722867280912929 2 10 52 n=2^{56}\cdot(2^{61}-1)\cdot 153722867280912929\ \approx\ 2\cdot 10^{52}

Well-formed_formula.html

  1. ( A ( B C ) ) (A\land(B\lor C))
  2. ¬ \lnot
  3. ¬ \neg
  4. \wedge
  5. \vee
  6. \rightarrow
  7. \leftrightarrow
  8. \rightarrow
  9. \wedge
  10. \rightarrow
  11. \vee
  12. ¬ \neg
  13. \wedge
  14. ¬ \neg
  15. \rightarrow
  16. \rightarrow
  17. ¬ \neg
  18. \rightarrow
  19. \wedge
  20. \vee
  21. \rightarrow
  22. \wedge
  23. \rightarrow
  24. \vee
  25. ¬ \neg
  26. \wedge
  27. ¬ \neg
  28. \rightarrow
  29. \wedge
  30. \rightarrow
  31. \vee
  32. ¬ \neg
  33. \wedge
  34. ¬ \neg
  35. ¬ \neg
  36. \wedge
  37. \vee
  38. \rightarrow
  39. \rightarrow
  40. \wedge
  41. \rightarrow
  42. \vee
  43. ¬ \neg
  44. \wedge
  45. ¬ \neg
  46. 𝒬 𝒮 \mathcal{QS}
  47. ¬ ϕ \neg\phi
  48. ϕ \ \phi
  49. ( ϕ ψ ) (\phi\land\psi)
  50. ( ϕ ψ ) (\phi\lor\psi)
  51. ϕ \ \phi
  52. ψ \ \psi
  53. x ϕ \exists x\,\phi
  54. x \ x
  55. ϕ \ \phi
  56. x ϕ \forall x\,\phi
  57. x \ x
  58. ϕ \ \phi
  59. x ϕ \forall x\,\phi
  60. ¬ x ¬ ϕ \neg\exists x\,\neg\phi
  61. x \exists x
  62. x \forall x
  63. x \ x
  64. 𝒬 \mathcal{Q}
  65. 𝒬 \mathcal{Q}
  66. 𝒬 \mathcal{Q}
  67. 𝒬 \mathcal{Q}

Well-founded_relation.html

  1. S X ( S m S s S ( s , m ) R ) \forall S\subseteq X\ (S\neq\varnothing\to\exists m\in S\;\;\forall s\in S\;\,% (s,m)\notin R)
  2. x X [ ( y X ( y R x P ( y ) ) ) P ( x ) ] x X P ( x ) . \forall x\in X\,[(\forall y\in X\,(y\,R\,x\to P(y)))\to P(x)]\to\forall x\in X% \,P(x).
  3. G ( x ) = F ( x , G | { y : y R x } ) G(x)=F(x,G|_{\{y:y\,R\,x\}})
  4. \in

Weyl_algebra.html

  1. f n ( X ) X n + f n - 1 ( X ) X n - 1 + + f 1 ( X ) X + f 0 ( X ) . f_{n}(X)\partial_{X}^{n}+f_{n-1}(X)\partial_{X}^{n-1}+\cdots+f_{1}(X)\partial_% {X}+f_{0}(X).
  2. Y X - X Y - 1. YX-XY-1.
  3. X i \partial_{X_{i}}
  4. W ( V ) := T ( V ) / ( ( v u - u v - ω ( v , u ) , for v , u V ) ) , W(V):=T(V)/(\!(v\otimes u-u\otimes v-\omega(v,u),\,\text{ for }v,u\in V)\!),
  5. ( ( ) ) (\!()\!)
  6. i i\hbar
  7. a 1 a n 1 n ! σ S n a σ ( 1 ) a σ ( n ) . a_{1}\cdots a_{n}\mapsto\frac{1}{n!}\sum_{\sigma\in S_{n}}a_{\sigma(1)}\otimes% \cdots\otimes a_{\sigma(n)}.
  8. i i\hbar
  9. i X i i\hbar\partial_{X_{i}}
  10. t r ( [ σ ( X ) , σ ( Y ) ] ) = t r ( 1 ) tr([\sigma(X),\sigma(Y)])=tr(1)
  11. dim ( char ( M ) ) n \dim(\operatorname{char}(M))\geq n

Wheel_theory.html

  1. 0 / 0 0/0
  2. \infty
  3. z / 0 = z/0=\infty
  4. z 0 z\neq 0
  5. 0 / 0 0/0
  6. / x /x
  7. x - 1 x^{-1}
  8. a / b a/b
  9. a / b = / b a a\cdot/b=/b\cdot a
  10. 0 x 0 0x\neq 0
  11. x - x 0 x-x\neq 0
  12. x / x 1 x/x\neq 1
  13. / x /x
  14. x x
  15. + +
  16. \cdot
  17. / /
  18. / ( x y ) = / x / y /(xy)=/x/y
  19. / / x = x //x=x
  20. x z + y z = ( x + y ) z + 0 z xz+yz=(x+y)z+0z
  21. ( x + y z ) / y = x / y + z + 0 y (x+yz)/y=x/y+z+0y
  22. 0 0 = 0 0\cdot 0=0
  23. ( x + 0 y ) z = x z + 0 y (x+0y)z=xz+0y
  24. / ( x + 0 y ) = / x + 0 y /(x+0y)=/x+0y
  25. 0 / 0 + x = 0 / 0 0/0+x=0/0
  26. a a
  27. 1 + a = 0 1+a=0
  28. - x = a x -x=ax
  29. x - y = x + ( - y ) x-y=x+(-y)
  30. 0 x + 0 y = 0 x y 0x+0y=0xy
  31. x - x = 0 x 2 x-x=0x^{2}
  32. x / x = 1 + 0 x / x x/x=1+0x/x
  33. 0 x = 0 0x=0
  34. 0 / x = 0 0/x=0
  35. x - x = 0 x-x=0
  36. x / x = 1 x/x=1
  37. { x 0 x = 0 } \{x\mid 0x=0\}
  38. x x
  39. x - 1 = / x x^{-1}=/x
  40. x - 1 x^{-1}
  41. / x /x
  42. x = 0 x=0

Where's_Willy?.html

  1. 100 × [ ln ( bills entered ) + ln ( hits + 1 ) ] × [ 1 - ( days of inactivity / 100 ) ] 100\times\left[\sqrt{\ln({\rm bills\ entered})}+\ln({\rm hits}+1)\right]\times% [1-({\rm days\ of\ inactivity}/100)]

Whirlpool_(cryptography).html

  1. S S
  2. S = A K M R S C S B ( S ) S=AK\circ MR\circ SC\circ SB(S)
  3. 𝔽 2 8 \mathbb{F}_{2^{8}}
  4. 1 - 10 - 154 1-10^{-154}

Whiteness.html

  1. W 2 = Y 2 + 800 ( x n , 2 - x 2 ) + 1700 ( y n , 2 - y 2 ) W_{2}=Y_{2}+800(x_{n,2}-x_{2})+1700(y_{n,2}-y_{2})
  2. W 10 = Y 10 + 800 ( x n , 10 - x 10 ) + 1700 ( y n , 10 - y 10 ) W_{10}=Y_{10}+800(x_{n,10}-x_{10})+1700(y_{n,10}-y_{10})
  3. T w , 2 = 1000 ( x n , 2 - x 2 ) - 650 ( y n , 2 - y 2 ) T_{w,2}=1000(x_{n,2}-x_{2})-650(y_{n,2}-y_{2})
  4. T w , 10 = 900 ( x n , 10 - x 10 ) - 650 ( y n , 10 - y 10 ) T_{w,10}=900(x_{n,10}-x_{10})-650(y_{n,10}-y_{10})

Whitney_embedding_theorem.html

  1. m m
  2. 2 m 2m
  3. m > 0 m>0
  4. m m
  5. m m
  6. ( 2 m 1 ) (2m−1)
  7. m m
  8. n n
  9. m m
  10. m > 2 n m>2n
  11. m > 2 n 1 m>2n−1
  12. M M
  13. M M
  14. f f
  15. M M
  16. M M
  17. 2 m > 4 2m>4
  18. M M
  19. M M
  20. m = 1 m=1
  21. { α 1 : 𝐑 1 𝐑 2 α 1 ( t 1 ) = ( 1 1 + t 1 2 , t 1 - 2 t 1 1 + t 1 2 ) \begin{cases}\alpha_{1}:\mathbf{R}^{1}\to\mathbf{R}^{2}\\ \alpha_{1}(t_{1})=\left(\frac{1}{1+t_{1}^{2}},t_{1}-\frac{2t_{1}}{1+t_{1}^{2}}% \right)\end{cases}
  22. α 1 ( t 1 ) = ( 1 1 + t 1 2 , t 1 - 2 t 1 1 + t 1 2 , 0 ) \alpha_{1}(t_{1})=\left(\frac{1}{1+t_{1}^{2}},t_{1}-\frac{2t_{1}}{1+t_{1}^{2}}% ,0\right)
  23. β 1 ( t 1 , a ) = ( 1 ( 1 + t 1 2 ) ( 1 + a 2 ) , t 1 - 2 t 1 ( 1 + t 1 2 ) ( 1 + a 2 ) , t 1 a ( 1 + t 1 2 ) ( 1 + a 2 ) ) . \beta_{1}(t_{1},a)=\left(\frac{1}{(1+t_{1}^{2})(1+a^{2})},t_{1}-\frac{2t_{1}}{% (1+t_{1}^{2})(1+a^{2})},\frac{t_{1}a}{(1+t_{1}^{2})(1+a^{2})}\right).
  24. a 0 a≠0
  25. α 2 ( t 1 , t 2 ) = ( β 1 ( t 1 , t 2 ) , t 2 ) = ( 1 ( 1 + t 1 2 ) ( 1 + t 2 2 ) , t 1 - 2 t 1 ( 1 + t 1 2 ) ( 1 + t 2 2 ) , t 1 t 2 ( 1 + t 1 2 ) ( 1 + t 2 2 ) , t 2 ) . \alpha_{2}(t_{1},t_{2})=\left(\beta_{1}(t_{1},t_{2}),t_{2}\right)=\left(\frac{% 1}{(1+t_{1}^{2})(1+t_{2}^{2})},t_{1}-\frac{2t_{1}}{(1+t_{1}^{2})(1+t_{2}^{2})}% ,\frac{t_{1}t_{2}}{(1+t_{1}^{2})(1+t_{2}^{2})},t_{2}\right).
  26. α m ( t 1 , t 2 , , t m ) = ( 1 u , t 1 - 2 t 1 u , t 1 t 2 u , t 2 , t 1 t 3 u , t 3 , , t 1 t m u , t m ) , \alpha_{m}(t_{1},t_{2},\cdots,t_{m})=\left(\frac{1}{u},t_{1}-\frac{2t_{1}}{u},% \frac{t_{1}t_{2}}{u},t_{2},\frac{t_{1}t_{3}}{u},t_{3},\cdots,\frac{t_{1}t_{m}}% {u},t_{m}\right),
  27. u = ( 1 + t 1 2 ) ( 1 + t 2 2 ) ( 1 + t m 2 ) . u=(1+t_{1}^{2})(1+t_{2}^{2})\cdots(1+t_{m}^{2}).
  28. α < s u b > m α<sub>m

Wide_Mouth_Frog_protocol.html

  1. T A T_{A}
  2. T s T_{s}
  3. K A S K_{AS}
  4. K A B K_{AB}
  5. K B S K_{BS}
  6. A S : A , { T A , B , K A B } K A S A\rightarrow S:A,\{T_{A},B,K_{AB}\}_{K_{AS}}
  7. S B : { T S , A , K A B } K B S S\rightarrow B:\{T_{S},A,K_{AB}\}_{K_{BS}}
  8. K A B K_{AB}

Wieferich_prime.html

  1. 2 p - 1 - 1 p \tfrac{2^{p-1}-1}{p}
  2. 2 10 - 1 11 \tfrac{2^{10}-1}{11}
  3. 2 1092 - 1 1093 \tfrac{2^{1092}-1}{1093}
  4. 1 + 1 3 + + 1 p - 2 0 ( mod p 2 ) 1+\tfrac{1}{3}+\dots+\tfrac{1}{p-2}\equiv 0\;\;(\mathop{{\rm mod}}p^{2})
  5. 2 p - 1 - 1 p mod p \tfrac{2^{p-1}-1}{p}\,\bmod\,p
  6. 2 t - 1 p mod p \tfrac{2^{t}-1}{p}\,\bmod\,p
  7. × 10 9 \times 10^{9}
  8. × 10 1 5 \times 10^{1}5
  9. × 10 1 5 \times 10^{1}5
  10. × 10 1 5 \times 10^{1}5
  11. × 10 1 7 \times 10^{1}7
  12. γ ( n ) = p n p \gamma(n)=\prod_{p\mid n}p
  13. γ ( n ) \gamma(n)
  14. Φ n ( 2 ) \Phi_{n}(2)
  15. Φ \Phi
  16. Φ 364 ( 2 ) \Phi_{364}(2)
  17. Φ 1755 ( 2 ) \Phi_{1755}(2)
  18. Φ n ( 2 ) \Phi_{n}(2)
  19. Φ n ( 2 ) \Phi_{n}(2)
  20. Φ 364 ( 2 ) \Phi_{364}(2)
  21. Φ 1755 ( 2 ) \Phi_{1755}(2)
  22. Φ n ( 2 ) \Phi_{n}(2)
  23. Φ n ( 2 ) \Phi_{n}(2)
  24. Φ n ( 2 ) \Phi_{n}(2)
  25. Φ n ( 2 ) \Phi_{n}(2)
  26. × 10 9 \times 10^{9}
  27. 2 n - 1 - 1 n 0 ( mod p ) \tfrac{2^{n-1}-1}{n}\not\equiv 0\;\;(\mathop{{\rm mod}}p)
  28. 2 p - 1 - 1 p 0 ( mod p ) \tfrac{2^{p-1}-1}{p}\not\equiv 0\;\;(\mathop{{\rm mod}}p)
  29. χ D 0 ( p ) = 1 \chi_{D_{0}}\big(p\big)=1
  30. λ p ( ( D 0 ) ) = 1 \lambda\,\!_{p}\big(\mathbb{Q}\big(\sqrt{D_{0}}\big)\big)=1
  31. D 0 < 0 D_{0}<0
  32. ( 1 - p 2 ) \mathbb{Q}\big(\sqrt{1-p^{2}}\big)
  33. D 0 < 0 D_{0}<0
  34. ( 1 - p ) \mathbb{Q}\big(\sqrt{1-p}\big)
  35. D 0 < 0 D_{0}<0
  36. ( 4 - p ) \mathbb{Q}\big(\sqrt{4-p}\big)
  37. χ D 0 ( p ) = 1 \chi_{D_{0}}\big(p\big)=1
  38. λ p ( ( D 0 ) ) = 1 \lambda\,\!_{p}\big(\mathbb{Q}\big(\sqrt{D_{0}}\big)\big)=1
  39. k - 1 2 \tfrac{k-1}{2}
  40. ( ζ p + ζ p - 1 ) \mathbb{Q}\big(\zeta\,\!_{p}+\zeta\,\!_{p}^{-1}\big)
  41. × 10 9 \times 10^{9}
  42. × 10 1 5 \times 10^{1}5
  43. | ω ( p ) p | \left|\tfrac{\omega(p)}{p}\right|
  44. ω ( p ) = 2 p - 1 - 1 p mod p \omega(p)=\tfrac{2^{p-1}-1}{p}\,\bmod\,p
  45. | ω ( p ) p | 10 - 14 \left|\tfrac{\omega(p)}{p}\right|\leq 10^{-14}
  46. ω ( p ) \omega(p)
  47. | ω ( p ) p | × 10 14 \left|\tfrac{\omega(p)}{p}\right|\times 10^{14}
  48. 2 ( p - 1 ) / 2 ± 1 + A p ( mod p 2 ) 2^{(p-1)/2}\equiv\pm 1+Ap\;\;(\mathop{{\rm mod}}p^{2})
  49. 2 p - 1 1 ± 2 A p ( mod p 2 ) 2^{p-1}\equiv 1\pm 2Ap\;\;(\mathop{{\rm mod}}p^{2})
  50. A A
  51. | A | |A|
  52. | ± 2 A | = 2 | A | |\pm 2A|=2|A|
  53. ω ( p ) \omega(p)
  54. A A
  55. p = 3167939147662997 p=3167939147662997
  56. 2 ( p - 1 ) / 2 - 1 - 1583969573831490 p ( mod p 2 ) 2^{(p-1)/2}\equiv-1-1583969573831490p\;\;(\mathop{{\rm mod}}p^{2})
  57. 2 p - 1 1 - 17 p ( mod p 2 ) 2^{p-1}\equiv 1-17p\;\;(\mathop{{\rm mod}}p^{2})
  58. a p - 1 - ( a - 1 ) p - 1 a^{p-1}-(a-1)^{p-1}
  59. a p - 1 - ( a - 1 ) p - 1 a^{p-1}-(a-1)^{p-1}
  60. 2 m - 1 q \tfrac{2^{m}-1}{q}

William_Huggins.html

  1. λ \lambda
  2. λ \lambda

Wilson_loop.html

  1. A μ A_{\mu}
  2. W C := Tr ( 𝒫 exp i C A μ d x μ ) . W_{C}:=\mathrm{Tr}\,(\,\mathcal{P}\exp i\oint_{C}A_{\mu}dx^{\mu}\,)\,.
  3. C C
  4. 𝒫 \mathcal{P}
  5. 𝒫 e i C A μ d x μ g ( x ) 𝒫 e i C A μ d x μ g - 1 ( x ) \mathcal{P}e^{i\oint_{C}A_{\mu}dx^{\mu}}\to g(x)\mathcal{P}e^{i\oint_{C}A_{\mu% }dx^{\mu}}g^{-1}(x)\,
  6. x x\,
  7. g ± 1 ( x ) exp { ± i α j ( x ) σ j 2 } g^{\pm 1}(x)\equiv\exp\{\pm i\alpha^{j}(x)\frac{\sigma^{j}}{2}\}
  8. α j ( x ) \alpha^{j}(x)
  9. x x\,
  10. σ j \sigma^{j}
  11. W C W_{C}
  12. A μ d x μ A_{\mu}\,dx^{\mu}

Wilson_prime.html

  1. × 10 1 3 \times 10^{1}3
  2. × 10 1 1 \times 10^{1}1
  3. ( m - 1 ) ! + 1 m \tfrac{(m-1)!+1}{m}
  4. × 10 8 \times 10^{8}

Wind_gradient.html

  1. v z = v g ( z z g ) 1 α , 0 < z < z g \ v_{z}=v_{g}\cdot\left(\frac{z}{z_{g}}\right)^{\frac{1}{\alpha}},0<z<z_{g}
  2. v z \ v_{z}
  3. z \ z
  4. v g \ v_{g}
  5. z g \ z_{g}
  6. α \ \alpha
  7. v w ( h ) = v 10 ( h h 10 ) a \ v_{w}(h)=v_{10}\cdot\left(\frac{h}{h_{10}}\right)^{a}
  8. v w ( h ) \ v_{w}(h)
  9. h h
  10. v 10 \ v_{10}
  11. h 10 h_{10}
  12. a \ a
  13. U ( h ) = U ( 0 ) h ζ \ U(h)=U(0)h^{\zeta}
  14. d U d H = ζ U ( h ) h \ \frac{dU}{dH}=\zeta\frac{U(h)}{h}
  15. U ( h ) \ U(h)
  16. h \ h
  17. U ( 0 ) \ U(0)
  18. ζ \ \zeta
  19. d U d H \ \frac{dU}{dH}
  20. h h

Wing_loading.html

  1. L A = 1 2 v 2 ρ C L \textstyle\frac{L}{A}=\tfrac{1}{2}v^{2}\rho C_{L}
  2. v 2 = 2 g W S ρ C L \textstyle v^{2}=\frac{2gW_{S}}{\rho C_{L}}
  3. W S \scriptstyle\sqrt{W_{S}}
  4. 150 1.4 = 177 \scriptstyle 150\sqrt{1.4}=177
  5. ρ \rho
  6. M a c = 1 2 v c 2 ρ C L A - M g \textstyle Ma_{c}=\tfrac{1}{2}v_{c}^{2}\rho C_{L}A-Mg
  7. a c = 1 2 W S v c 2 ρ C L - g , \textstyle a_{c}=\frac{1}{2W_{S}}v_{c}^{2}\rho C_{L}-g,
  8. v 2 R \scriptstyle\frac{v^{2}}{R}
  9. L s i n θ \scriptstyle Lsin\theta
  10. θ \theta
  11. M v 2 R = L sin θ = 1 2 v 2 ρ C L A sin θ . \textstyle\frac{Mv^{2}}{R}=L\sin\theta=\frac{1}{2}v^{2}\rho C_{L}A\sin\theta.
  12. R = 2 W s ρ C L sin θ . \textstyle R=\frac{2W_{s}}{\rho C_{L}\sin\theta}.
  13. a = G A M = G W S \textstyle a=\frac{GA}{M}=\frac{G}{W_{S}}

Wireless_power.html

  1. k = M / L 1 L 2 \scriptstyle k\;=\;M/\sqrt{L_{1}L_{2}}

Wishart_distribution.html

  1. t r tr
  2. n 𝐕 n\mathbf{V}
  3. ( n p 1 ) 𝐕 (n−p−1)\mathbf{V}
  4. n p + 1 n≥p+1
  5. Var ( 𝐗 i j ) = n ( v i j 2 + v i i v j j ) \operatorname{Var}(\mathbf{X}_{ij})=n\left(v_{ij}^{2}+v_{ii}v_{jj}\right)
  6. Θ | 𝐈 - 2 i 𝚯 𝐕 | - n 2 \Theta\mapsto\left|{\mathbf{I}}-2i\,{\mathbf{\Theta}}{\mathbf{V}}\right|^{-% \frac{n}{2}}
  7. X X
  8. n × p n×p
  9. p p
  10. X ( i ) = ( x i 1 , , x i p ) T N p ( 0 , V ) . X_{(i)}{=}(x_{i}^{1},\dots,x_{i}^{p})^{T}\sim N_{p}(0,V).
  11. p × p p×p
  12. S S
  13. S W p ( V , n ) . S\sim W_{p}(V,n).
  14. n n
  15. W ( V , p , n ) W(V,p,n)
  16. n p n≥p
  17. S S
  18. 1 1
  19. V V
  20. p = V = 1 p=V=1
  21. n n
  22. 𝐗 \mathbf{X}
  23. p × p p×p
  24. 𝐕 \mathbf{V}
  25. p × p p×p
  26. n p n≥p
  27. 𝐗 \mathbf{X}
  28. n n
  29. 1 2 n p 2 | 𝐕 | n 2 Γ p ( n 2 ) | 𝐗 | n - p - 1 2 e - 1 2 tr ( 𝐕 - 1 𝐗 ) \frac{1}{2^{\frac{np}{2}}\left|{\mathbf{V}}\right|^{\frac{n}{2}}\Gamma_{p}(% \frac{n}{2})}{\left|\mathbf{X}\right|}^{\frac{n-p-1}{2}}e^{-\frac{1}{2}{\rm tr% }({\mathbf{V}}^{-1}\mathbf{X})}
  30. | 𝐗 | \left|{\mathbf{X}}\right|
  31. Γ p ( n 2 ) = π p ( p - 1 ) 4 Π j = 1 p Γ ( n 2 + 1 - j 2 ) . \Gamma_{p}\left(\tfrac{n}{2}\right)=\pi^{\frac{p(p-1)}{4}}\Pi_{j=1}^{p}\Gamma% \left(\tfrac{n}{2}+\tfrac{1-j}{2}\right).
  32. n > p 1 n>p−1
  33. n p 1 n≤p−1
  34. p × p p×p
  35. 𝚺 \mathbf{Σ}
  36. n = p n=p
  37. n 𝐕 n\mathbf{V}
  38. E [ ln | 𝐗 | ] = ψ p ( n / 2 ) + p ln ( 2 ) + ln | 𝐕 | \operatorname{E}[\ln|\mathbf{X}|]=\psi_{p}(n/2)+p\ln(2)+\ln|\mathbf{V}|
  39. ψ p \psi_{p}
  40. H [ 𝐗 ] = - ln ( B ( 𝐕 , n ) ) - n - p - 1 2 E [ ln | 𝐗 | ] + n p 2 \operatorname{H}[\mathbf{X}]=-\ln\left(B(\mathbf{V},n)\right)-\frac{n-p-1}{2}% \operatorname{E}[\ln|\mathbf{X}|]+\frac{np}{2}
  41. B ( 𝐕 , n ) B(\mathbf{V},n)
  42. B ( 𝐕 , n ) = 1 | 𝐕 | n 2 2 n p 2 Γ p ( n 2 ) B(\mathbf{V},n)=\frac{1}{\left|\mathbf{V}\right|^{\frac{n}{2}}2^{\frac{np}{2}}% \Gamma_{p}(\frac{n}{2})}
  43. H [ 𝐗 ] \displaystyle\operatorname{H}[\mathbf{X}]
  44. p 0 p_{0}
  45. n 0 , V 0 n_{0},V_{0}
  46. p 1 p_{1}
  47. n 1 , V 1 n_{1},V_{1}
  48. H ( p 0 , p 1 ) = E p 0 [ - log p 1 ] = E p 0 [ - log | 𝐗 | n 1 - p - 1 2 e - tr ( 𝐕 1 - 1 𝐗 ) 2 2 n 1 p 2 | 𝐕 1 | n 1 2 Γ p ( n 1 2 ) ] = n 1 p 2 log 2 + n 1 2 log | 𝐕 1 | + log Γ p ( n 1 2 ) - n 1 - p - 1 2 E p 0 [ log | 𝐗 | ] + 1 2 E p 0 [ tr ( 𝐕 1 - 1 𝐗 ) ] = n 1 p 2 log 2 + n 1 2 log | 𝐕 1 | + log Γ p ( n 1 2 ) - n 1 - p - 1 2 ( ψ p ( n 0 2 ) + p log 2 + log | 𝐕 0 | ) + 1 2 tr ( 𝐕 1 - 1 n 0 𝐕 0 ) = - n 1 2 log | 𝐕 1 - 1 𝐕 0 | + p + 1 2 log | 𝐕 0 | + n 0 2 tr ( 𝐕 1 - 1 𝐕 0 ) + log Γ p ( n 1 2 ) - n 1 - p - 1 2 ψ p ( n 0 2 ) + p ( p + 1 ) 2 log 2 \begin{aligned}\displaystyle H(p_{0},p_{1})&\displaystyle=\operatorname{E}_{p_% {0}}[-\log p_{1}]\\ &\displaystyle=\operatorname{E}_{p_{0}}\left[-\log\frac{|\mathbf{X}|^{\frac{n_% {1}-p-1}{2}}e^{-\frac{\mathrm{tr}(\mathbf{V}_{1}^{-1}\mathbf{X})}{2}}}{2^{% \frac{n_{1}p}{2}}|\mathbf{V}_{1}|^{\frac{n_{1}}{2}}\Gamma_{p}(\tfrac{n_{1}}{2}% )}\right]\\ &\displaystyle=\tfrac{n_{1}p}{2}\log 2+\tfrac{n_{1}}{2}\log|\mathbf{V}_{1}|+% \log\Gamma_{p}(\tfrac{n_{1}}{2})-\tfrac{n_{1}-p-1}{2}\operatorname{E}_{p_{0}}[% \log|\mathbf{X}|]+\tfrac{1}{2}\operatorname{E}_{p_{0}}[\mathrm{tr}(\mathbf{V}_% {1}^{-1}\mathbf{X})]\\ &\displaystyle=\tfrac{n_{1}p}{2}\log 2+\tfrac{n_{1}}{2}\log|\mathbf{V}_{1}|+% \log\Gamma_{p}(\tfrac{n_{1}}{2})-\tfrac{n_{1}-p-1}{2}\left(\psi_{p}(\tfrac{n_{% 0}}{2})+p\log 2+\log|\mathbf{V}_{0}|\right)+\tfrac{1}{2}\mathrm{tr}(\mathbf{V}% _{1}^{-1}n_{0}\mathbf{V}_{0})\\ &\displaystyle=-\tfrac{n_{1}}{2}\log|\mathbf{V}_{1}^{-1}\mathbf{V}_{0}|+\tfrac% {p+1}{2}\log|\mathbf{V}_{0}|+\tfrac{n_{0}}{2}\mathrm{tr}(\mathbf{V}_{1}^{-1}% \mathbf{V}_{0})+\log\Gamma_{p}(\tfrac{n_{1}}{2})-\tfrac{n_{1}-p-1}{2}\psi_{p}(% \tfrac{n_{0}}{2})+\tfrac{p(p+1)}{2}\log 2\\ \end{aligned}
  49. p 0 = p 1 p_{0}=p_{1}
  50. p 1 p_{1}
  51. p 0 p_{0}
  52. D K L ( p 0 p 1 ) = H ( p 0 , p 1 ) - H ( p 0 ) = - n 1 2 log | 𝐕 1 - 1 𝐕 0 | + n 0 2 ( tr ( 𝐕 1 - 1 𝐕 0 ) - p ) + log Γ p ( n 1 2 ) Γ p ( n 0 2 ) + n 0 - n 1 2 ψ p ( n 0 2 ) D_{KL}(p_{0}\|p_{1})=H(p_{0},p_{1})-H(p_{0})=-\tfrac{n_{1}}{2}\log|\mathbf{V}_% {1}^{-1}\mathbf{V}_{0}|+\tfrac{n_{0}}{2}(\mathrm{tr}(\mathbf{V}_{1}^{-1}% \mathbf{V}_{0})-p)+\log\frac{\Gamma_{p}(\tfrac{n_{1}}{2})}{\Gamma_{p}(\tfrac{n% _{0}}{2})}+\tfrac{n_{0}-n_{1}}{2}\psi_{p}(\tfrac{n_{0}}{2})
  53. Θ | 𝐈 - 2 i 𝚯 𝐕 | - n 2 . \Theta\mapsto\left|{\mathbf{I}}-2i\,{\mathbf{\Theta}}{\mathbf{V}}\right|^{-% \frac{n}{2}}.
  54. Θ E [ exp ( i tr ( 𝐗 𝚯 ) ) ] = | 𝐈 - 2 i 𝚯 𝐕 | - n 2 \Theta\mapsto\operatorname{E}\left[\mathrm{exp}\left(i\mathrm{tr}(\mathbf{X}{% \mathbf{\Theta}})\right)\right]=\left|{\mathbf{I}}-2i{\mathbf{\Theta}}{\mathbf% {V}}\right|^{-\frac{n}{2}}
  55. E E⋅⋅
  56. Θ Θ
  57. 𝐈 \mathbf{I}
  58. 𝐕 \mathbf{V}
  59. 𝐈 \mathbf{I}
  60. i i
  61. p × p p×p
  62. 𝐗 \mathbf{X}
  63. m m
  64. 𝐕 \mathbf{V}
  65. 𝐗 𝒲 p ( 𝐕 , m ) \mathbf{X}\sim\mathcal{W}_{p}({\mathbf{V}},m)
  66. 𝐂 \mathbf{C}
  67. q × p q×p
  68. q q
  69. 𝐂𝐗𝐂 T 𝒲 q ( 𝐂𝐕𝐂 T , m ) . \mathbf{C}\mathbf{X}{\mathbf{C}}^{T}\sim\mathcal{W}_{q}\left({\mathbf{C}}{% \mathbf{V}}{\mathbf{C}}^{T},m\right).
  70. 𝐳 \mathbf{z}
  71. p × 1 p×1
  72. 𝐳 T 𝐗𝐳 σ z 2 χ m 2 . {\mathbf{z}}^{T}\mathbf{X}{\mathbf{z}}\sim\sigma_{z}^{2}\chi_{m}^{2}.
  73. χ m 2 \chi_{m}^{2}
  74. σ z 2 = 𝐳 T 𝐕𝐳 \sigma_{z}^{2}={\mathbf{z}}^{T}{\mathbf{V}}{\mathbf{z}}
  75. σ z 2 \sigma_{z}^{2}
  76. 𝐕 \mathbf{V}
  77. j j
  78. w j j σ j j χ m 2 w_{jj}\sim\sigma_{jj}\chi^{2}_{m}
  79. 𝐗 \mathbf{X}
  80. p p
  81. 𝐕 \mathbf{V}
  82. n n
  83. 𝐗 = 𝐋 𝐀 𝐀 T 𝐋 T , \mathbf{X}={\textbf{L}}{\textbf{A}}{\textbf{A}}^{T}{\textbf{L}}^{T},
  84. 𝐋 \mathbf{L}
  85. 𝐕 \mathbf{V}
  86. 𝐀 = ( c 1 0 0 0 n 21 c 2 0 0 n 31 n 32 c 3 0 n p 1 n p 2 n p 3 c p ) \mathbf{A}=\begin{pmatrix}c_{1}&0&0&\cdots&0\\ n_{21}&c_{2}&0&\cdots&0\\ n_{31}&n_{32}&c_{3}&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ n_{p1}&n_{p2}&n_{p3}&\cdots&c_{p}\end{pmatrix}
  87. c i 2 χ n - i + 1 2 c_{i}^{2}\sim\chi^{2}_{n-i+1}
  88. 𝐕 \mathbf{V}
  89. 2 × 2 2×2
  90. 𝐕 = ( σ 1 2 ρ σ 1 σ 2 ρ σ 1 σ 2 σ 2 2 ) , 𝐋 = ( σ 1 0 ρ σ 2 1 - ρ 2 σ 2 ) \mathbf{V}=\begin{pmatrix}\sigma_{1}^{2}&\rho\sigma_{1}\sigma_{2}\\ \rho\sigma_{1}\sigma_{2}&\sigma_{2}^{2}\end{pmatrix},\qquad\mathbf{L}=\begin{% pmatrix}\sigma_{1}&0\\ \rho\sigma_{2}&\sqrt{1-\rho^{2}}\sigma_{2}\end{pmatrix}
  91. 2 × 2 2×2
  92. 𝐗 = ( σ 1 2 c 1 2 σ 1 σ 2 ( ρ c 1 2 + 1 - ρ 2 c 1 n 21 ) σ 1 σ 2 ( ρ c 1 2 + 1 - ρ 2 c 1 n 21 ) σ 2 2 ( ( 1 - ρ 2 ) c 2 2 + ( 1 - ρ 2 n 21 + ρ c 1 ) 2 ) ) \mathbf{X}=\begin{pmatrix}\sigma_{1}^{2}c_{1}^{2}&\sigma_{1}\sigma_{2}\left(% \rho c_{1}^{2}+\sqrt{1-\rho^{2}}c_{1}n_{21}\right)\\ \sigma_{1}\sigma_{2}\left(\rho c_{1}^{2}+\sqrt{1-\rho^{2}}c_{1}n_{21}\right)&% \sigma_{2}^{2}\left(\left(1-\rho^{2}\right)c_{2}^{2}+\left(\sqrt{1-\rho^{2}}n_% {21}+\rho c_{1}\right)^{2}\right)\end{pmatrix}
  93. n n
  94. f ( x 12 ) = | x 12 | n - 1 2 Γ ( n 2 ) 2 n - 1 π ( 1 - ρ 2 ) ( σ 1 σ 2 ) n + 1 K n - 1 2 ( | x 12 | σ 1 σ 2 ( 1 - ρ 2 ) ) exp ( ρ x 12 σ 1 σ 2 ( 1 - ρ 2 ) ) f(x_{12})=\frac{\left|x_{12}\right|^{\frac{n-1}{2}}}{\Gamma\left(\frac{n}{2}% \right)\sqrt{2^{n-1}\pi\left(1-\rho^{2}\right)\left(\sigma_{1}\sigma_{2}\right% )^{n+1}}}\cdot K_{\frac{n-1}{2}}\left(\frac{\left|x_{12}\right|}{\sigma_{1}% \sigma_{2}\left(1-\rho^{2}\right)}\right)\exp{\left(\frac{\rho x_{12}}{\sigma_% {1}\sigma_{2}(1-\rho^{2})}\right)}
  95. 𝐧 \mathbf{n}
  96. Λ p := { 0 , , p - 1 } ( p - 1 , ) . \Lambda_{p}:=\{0,\cdots,p-1\}\cup\left(p-1,\infty\right).
  97. Λ p * := { 0 , , p - 1 } , \Lambda_{p}^{*}:=\{0,\cdots,p-1\},
  98. W p - 1 W_{p}^{-1}
  99. 𝐂 W p - 1 ( 𝐕 - 1 , n ) \mathbf{C}\sim W_{p}^{-1}(\mathbf{V}^{-1},n)
  100. | 𝐂 | < s u p > p + 1 |\mathbf{C}|<sup>p+1

Witch_of_Agnesi.html

  1. y = 8 a 3 x 2 + 4 a 2 . \!y=\frac{8a^{3}}{x^{2}+4a^{2}}.
  2. y = 1 x 2 + 1 . \!y=\frac{1}{x^{2}+1}.
  3. θ \theta\,
  4. x = 2 a tan θ , y = 2 a cos 2 θ . \!x=2a\tan\theta,\ y=2a\cos^{2}\theta.\,
  5. θ \theta\,
  6. x = 2 a cot θ , y = 2 a sin 2 θ . \!x=2a\cot\theta,\ y=2a\sin^{2}\theta.\,
  7. 4 π a 2 4\pi a^{2}
  8. ( 0 , a ) \!\left(0,a\right)
  9. 4 π 2 a 3 4\pi^{2}a^{3}

Wobble_base_pair.html

  1. 4 3 4^{3}

Woodall_number.html

  1. W n = n 2 n - 1 W_{n}=n\cdot 2^{n}-1
  2. ( 2 p ) \left(\frac{2}{p}\right)
  3. ( 2 p ) \left(\frac{2}{p}\right)

Woodbury_matrix_identity.html

  1. ( A + U C V ) - 1 = A - 1 - A - 1 U ( C - 1 + V A - 1 U ) - 1 V A - 1 , \left(A+UCV\right)^{-1}=A^{-1}-A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1},
  2. I + V A - 1 U I+VA^{-1}U
  3. ( A + U C V ) (A+UCV)
  4. ( A + U C V ) [ A - 1 - A - 1 U ( C - 1 + V A - 1 U ) - 1 V A - 1 ] \displaystyle\left(A+UCV\right)\left[A^{-1}-A^{-1}U\left(C^{-1}+VA^{-1}U\right% )^{-1}VA^{-1}\right]
  5. [ A U V - C - 1 ] [ X Y ] = [ I 0 ] . \begin{bmatrix}A&U\\ V&-C^{-1}\end{bmatrix}\begin{bmatrix}X\\ Y\end{bmatrix}=\begin{bmatrix}I\\ 0\end{bmatrix}.
  6. A X + U Y = I AX+UY=I
  7. V X - C - 1 Y = 0 VX-C^{-1}Y=0
  8. ( A + U C V ) X = I (A+UCV)X=I
  9. X = A - 1 ( I - U Y ) X=A^{-1}(I-UY)
  10. V A - 1 ( I - U Y ) = C - 1 Y VA^{-1}(I-UY)=C^{-1}Y
  11. V A - 1 = ( C - 1 + V A - 1 U ) Y VA^{-1}=(C^{-1}+VA^{-1}U)Y
  12. ( C - 1 + V A - 1 U ) - 1 V A - 1 = Y (C^{-1}+VA^{-1}U)^{-1}VA^{-1}=Y
  13. A X + U Y = I AX+UY=I
  14. A X + U ( C - 1 + V A - 1 U ) - 1 V A - 1 = I AX+U(C^{-1}+VA^{-1}U)^{-1}VA^{-1}=I
  15. ( A + U C V ) - 1 = X = A - 1 - A - 1 U ( C - 1 + V A - 1 U ) - 1 V A - 1 . (A+UCV)^{-1}=X=A^{-1}-A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}.
  16. [ A U V C ] \begin{bmatrix}A&U\\ V&C\end{bmatrix}
  17. [ I 0 - V A - 1 I ] [ A U V C ] = [ A U 0 C - V A - 1 U ] \begin{bmatrix}I&0\\ -VA^{-1}&I\end{bmatrix}\begin{bmatrix}A&U\\ V&C\end{bmatrix}=\begin{bmatrix}A&U\\ 0&C-VA^{-1}U\end{bmatrix}
  18. [ A U V C ] [ I - A - 1 U 0 I ] = [ A 0 V C - V A - 1 U ] \begin{bmatrix}A&U\\ V&C\end{bmatrix}\begin{bmatrix}I&-A^{-1}U\\ 0&I\end{bmatrix}=\begin{bmatrix}A&0\\ V&C-VA^{-1}U\end{bmatrix}
  19. [ I 0 - V A - 1 I ] [ A U V C ] [ I - A - 1 U 0 I ] = [ A 0 0 C - V A - 1 U ] \begin{bmatrix}I&0\\ -VA^{-1}&I\end{bmatrix}\begin{bmatrix}A&U\\ V&C\end{bmatrix}\begin{bmatrix}I&-A^{-1}U\\ 0&I\end{bmatrix}=\begin{bmatrix}A&0\\ 0&C-VA^{-1}U\end{bmatrix}
  20. [ A U V C ] = [ I 0 V A - 1 I ] [ A 0 0 C - V A - 1 U ] [ I A - 1 U 0 I ] \begin{bmatrix}A&U\\ V&C\end{bmatrix}=\begin{bmatrix}I&0\\ VA^{-1}&I\end{bmatrix}\begin{bmatrix}A&0\\ 0&C-VA^{-1}U\end{bmatrix}\begin{bmatrix}I&A^{-1}U\\ 0&I\end{bmatrix}
  21. [ A U V C ] - 1 \displaystyle\begin{bmatrix}A&U\\ V&C\end{bmatrix}^{-1}
  22. [ A U V C ] = [ I U C - 1 0 I ] [ A - U C - 1 V 0 0 C ] [ I 0 C - 1 V I ] \begin{bmatrix}A&U\\ V&C\end{bmatrix}=\begin{bmatrix}I&UC^{-1}\\ 0&I\end{bmatrix}\begin{bmatrix}A-UC^{-1}V&0\\ 0&C\end{bmatrix}\begin{bmatrix}I&0\\ C^{-1}V&I\end{bmatrix}
  23. [ A U V C ] - 1 \displaystyle\begin{bmatrix}A&U\\ V&C\end{bmatrix}^{-1}
  24. ( A - U C - 1 V ) - 1 = A - 1 + A - 1 U ( C - V A - 1 U ) - 1 V A - 1 . \left(A-UC^{-1}V\right)^{-1}=A^{-1}+A^{-1}U(C-VA^{-1}U)^{-1}VA^{-1}.

Work_content.html

  1. U U
  2. T T
  3. S S
  4. A = U - T S A=U-TS
  5. δ W = d A \delta W=dA
  6. δ W = d A = d U - T d S \delta W=dA=dU-T\,dS
  7. H = U + P V H=U+PV
  8. P P
  9. V V
  10. G = H - T S = U + P V - T S G=H-TS=U+PV-TS

World3.html

  1. y = x * e r t y=x*e^{rt}

WWV_(radio_station).html

  1. 2 λ 4 \tfrac{\sqrt{2}\lambda}{4}

X86_instruction_listings.html

  1. ( 2 x ) - 1 (2^{x})-1
  2. 2 x 2^{x}
  3. x x

XY.html

  1. x y xy

YCbCr.html

  1. Y \displaystyle Y^{\prime}
  2. K B \displaystyle K_{B}
  3. Y \displaystyle Y^{\prime}
  4. Y \displaystyle Y^{\prime}
  5. ( Y , C B , C R ) = ( 16 , 128 , 128 ) + ( 219 Y , 224 P B , 224 P R ) \begin{aligned}\displaystyle(Y^{\prime},C_{B},C_{R})&\displaystyle=&% \displaystyle(16,128,128)+(219\cdot Y,224\cdot P_{B},224\cdot P_{R})\\ \end{aligned}
  6. Y \displaystyle Y^{\prime}
  7. 256 255 \frac{256}{255}
  8. 255 219 \frac{255}{219}
  9. R D \displaystyle R^{\prime}_{D}
  10. R D \displaystyle R^{\prime}_{D}
  11. K B \displaystyle K_{B}
  12. K B \displaystyle K_{B}
  13. Y = 0 + ( 0.299 R D ) + ( 0.587 G D ) + ( 0.114 B D ) C B = 128 - ( 0.168736 R D ) - ( 0.331264 G D ) + ( 0.5 B D ) C R = 128 + ( 0.5 R D ) - ( 0.418688 G D ) - ( 0.081312 B D ) \begin{aligned}\displaystyle Y^{\prime}&\displaystyle=&\displaystyle 0&% \displaystyle+(0.299&\displaystyle\cdot R^{\prime}_{D})&\displaystyle+(0.587&% \displaystyle\cdot G^{\prime}_{D})&\displaystyle+(0.114&\displaystyle\cdot B^{% \prime}_{D})\\ \displaystyle C_{B}&\displaystyle=&\displaystyle 128&\displaystyle-(0.168736&% \displaystyle\cdot R^{\prime}_{D})&\displaystyle-(0.331264&\displaystyle\cdot G% ^{\prime}_{D})&\displaystyle+(0.5&\displaystyle\cdot B^{\prime}_{D})\\ \displaystyle C_{R}&\displaystyle=&\displaystyle 128&\displaystyle+(0.5&% \displaystyle\cdot R^{\prime}_{D})&\displaystyle-(0.418688&\displaystyle\cdot G% ^{\prime}_{D})&\displaystyle-(0.081312&\displaystyle\cdot B^{\prime}_{D})\end{aligned}
  14. R \displaystyle R

Yield_curve.html

  1. ( 1 + i l t ) n = ( 1 + i s t year 1 ) ( 1 + i s t year 2 ) ( 1 + i s t year n ) , (1+i_{lt})^{n}=(1+i_{st}^{\,\text{year }1})(1+i_{st}^{\,\text{year }2})\cdots(% 1+i_{st}^{\,\text{year }n}),
  2. ( 1 + i l t ) n = r p n + ( ( 1 + i s t year1 ) ( 1 + i s t year2 ) ( 1 + i s t year n ) ) (1+i_{lt})^{n}=rp_{n}+((1+i_{st}^{\mathrm{year}1})(1+i_{st}^{\mathrm{year}2})% \cdots(1+i_{st}^{\mathrm{year}n}))
  3. r p n rp_{n}
  4. n {n}
  5. Y ( t ) = P ( t ) - 1 / t - 1. Y(t)=P(t)^{-1/t}-1.
  6. A P = F + ε AP=F+\varepsilon\,
  7. ε \varepsilon
  8. ϵ \epsilon

Yield_to_maturity.html

  1. Yield to maturity(YTM) = Face value Present value Time period - 1 \,\text{Yield to maturity(YTM)}=\sqrt[\,\text{Time period}]{\dfrac{\,\text{% Face value}}{\,\text{Present value}}}-1

YIQ.html

  1. R , G , B , Y [ 0 , 1 ] , I [ - 0.5957 , 0.5957 ] , Q [ - 0.5226 , 0.5226 ] R,G,B,Y\in\left[0,1\right],\quad I\in\left[-0.5957,0.5957\right],\quad Q\in% \left[-0.5226,0.5226\right]
  2. [ Y I Q ] = [ 0.299 0.587 0.114 0.596 - 0.274 - 0.322 0.211 - 0.523 0.312 ] [ R G B ] \begin{bmatrix}Y\\ I\\ Q\end{bmatrix}=\begin{bmatrix}0.299&0.587&0.114\\ 0.596&-0.274&-0.322\\ 0.211&-0.523&0.312\end{bmatrix}\begin{bmatrix}R\\ G\\ B\end{bmatrix}
  3. [ R G B ] = [ 1 0.956 0.621 1 - 0.272 - 0.647 1 - 1.106 1.703 ] [ Y I Q ] \begin{bmatrix}R\\ G\\ B\end{bmatrix}=\begin{bmatrix}1&0.956&0.621\\ 1&-0.272&-0.647\\ 1&-1.106&1.703\end{bmatrix}\begin{bmatrix}Y\\ I\\ Q\end{bmatrix}
  4. [ R G B ] = [ 1 1 1 ] [ Y I Q ] = [ 1 0 0 ] \begin{bmatrix}R\\ G\\ B\end{bmatrix}=\begin{bmatrix}1\\ 1\\ 1\end{bmatrix}\implies\begin{bmatrix}Y\\ I\\ Q\end{bmatrix}=\begin{bmatrix}1\\ 0\\ 0\end{bmatrix}
  5. { E Q = 0.41 ( E B - E Y ) + 0.48 ( E R - E Y ) E I = - 0.27 ( E B - E Y ) + 0.74 ( E R - E Y ) E Y = 0.30 E R + 0.59 E G + 0.11 E B \left\{\begin{array}[]{ccl}E_{Q}^{\prime}&=&0.41(E_{B}^{\prime}-E_{Y}^{\prime}% )+0.48(E_{R}^{\prime}-E_{Y}^{\prime})\\ E_{I}^{\prime}&=&-0.27(E_{B}^{\prime}-E_{Y}^{\prime})+0.74(E_{R}^{\prime}-E_{Y% }^{\prime})\\ E_{Y}^{\prime}&=&0.30E_{R}^{\prime}+0.59E_{G}^{\prime}+0.11E_{B}^{\prime}\end{% array}\right.
  6. [ E Y E I E Q ] = [ 0.30 0.59 0.11 0.599 - 0.2773 - 0.3217 0.213 - 0.5251 0.3121 ] [ E R E G E B ] \begin{bmatrix}E_{Y}^{\prime}\\ E_{I}^{\prime}\\ E_{Q}^{\prime}\end{bmatrix}=\begin{bmatrix}0.30&0.59&0.11\\ 0.599&-0.2773&-0.3217\\ 0.213&-0.5251&0.3121\end{bmatrix}\begin{bmatrix}E_{R}^{\prime}\\ E_{G}^{\prime}\\ E_{B}^{\prime}\end{bmatrix}
  7. E Y E_{Y}^{\prime}
  8. E R E_{R}^{\prime}
  9. E G E_{G}^{\prime}
  10. E B E_{B}^{\prime}
  11. E I E_{I}^{\prime}
  12. E Q E_{Q}^{\prime}

Yukawa_potential.html

  1. V Yukawa ( r ) = - g 2 e - k m r r , V\text{Yukawa}(r)=-g^{2}\frac{e^{-kmr}}{r},
  2. m m
  3. m = 0 e - m r = e 0 = 1. m=0\Rightarrow e^{-mr}=e^{0}=1.
  4. V Yukawa ( r ) = - g 2 e - m r r V_{\,\text{Yukawa}}(r)=-g^{2}\;\frac{e^{-mr}}{r}
  5. V Coulomb ( r ) = - g 2 1 r . V_{\,\text{Coulomb}}(r)=-g^{2}\;\frac{1}{r}.
  6. V ( 𝐫 ) = - g 2 ( 2 π ) 3 e i 𝐤 𝐫 4 π k 2 + m 2 d 3 k V(\mathbf{r})=\frac{-g^{2}}{(2\pi)^{3}}\int e^{i\mathbf{k\cdot r}}\frac{4\pi}{% k^{2}+m^{2}}\;d^{3}k
  7. 4 π / ( k 2 + m 2 ) 4\pi/(k^{2}+m^{2})
  8. ψ ( x ) \psi(x)
  9. ϕ ( x ) \phi(x)
  10. int ( x ) = g ψ ¯ ( x ) ϕ ( x ) ψ ( x ) . \mathcal{L}_{\mathrm{int}}(x)=g\overline{\psi}(x)\phi(x)\psi(x).
  11. p 1 p_{1}
  12. p 2 p_{2}
  13. g 2 g^{2}
  14. - 4 π / ( k 2 + m 2 ) -4\pi/(k^{2}+m^{2})
  15. V ( 𝐤 ) = - g 2 4 π k 2 + m 2 . V(\mathbf{k})=-g^{2}\frac{4\pi}{k^{2}+m^{2}}.

Z-test.html

  1. SE = σ n = 12 55 = 12 7.42 = 1.62 \mathrm{SE}=\frac{\sigma}{\sqrt{n}}=\frac{12}{\sqrt{55}}=\frac{12}{7.42}=1.62\,\!
  2. σ {\sigma}
  3. z = M - μ SE = 96 - 100 1.62 = - 2.47 z=\frac{M-\mu}{\mathrm{SE}}=\frac{96-100}{1.62}=-2.47\,\!
  4. θ ^ \hat{\theta}
  5. ( θ ^ - θ 0 ) / SE ( θ ^ ) (\hat{\theta}-\theta_{0})/{\rm SE}(\hat{\theta})

Zariski_tangent_space.html

  1. 𝔪 \mathfrak{m}
  2. 𝔪 / 𝔪 2 \mathfrak{m}/\mathfrak{m}^{2}
  3. 𝔪 \mathfrak{m}
  4. 𝔪 \mathfrak{m}
  5. 𝔪 \mathfrak{m}
  6. T P ( X ) T_{P}(X)
  7. T P * ( X ) T_{P}^{*}(X)
  8. 𝒪 X , P \mathcal{O}_{X,P}
  9. f : R R / I f:R\rightarrow R/I
  10. g : 𝒪 X , f - 1 ( P ) 𝒪 Y , P g:\mathcal{O}_{X,f^{-1}(P)}\rightarrow\mathcal{O}_{Y,P}
  11. T P ( Y ) T_{P}(Y)
  12. T f - 1 P ( X ) T_{f^{-1}P}(X)
  13. 𝔪 P / 𝔪 P 2 \mathfrak{m}_{P}/\mathfrak{m}_{P}^{2}
  14. ( 𝔪 f - 1 P / I ) / ( ( 𝔪 f - 1 P 2 + I ) / I ) \cong(\mathfrak{m}_{f^{-1}P}/I)/((\mathfrak{m}_{f^{-1}P}^{2}+I)/I)
  15. 𝔪 f - 1 P / ( 𝔪 f - 1 P 2 + I ) \cong\mathfrak{m}_{f^{-1}P}/(\mathfrak{m}_{f^{-1}P}^{2}+I)
  16. ( 𝔪 f - 1 P / 𝔪 f - 1 P 2 ) / Ker ( k ) . \cong(\mathfrak{m}_{f^{-1}P}/\mathfrak{m}_{f^{-1}P}^{2})/\mathrm{Ker}(k).
  17. k * : T P ( Y ) T f - 1 P ( X ) k^{*}:T_{P}(Y)\rightarrow T_{f^{-1}P}(X)

Zentrum.html

  1. Z ( G ) Z(G)

Zenzizenzizenzic.html

  1. x 8 = ( ( x 2 ) 2 ) 2 . x^{8}=\left(\left(x^{2}\right)^{2}\right)^{2}.

Zermelo_set_theory.html

  1. \equiv
  2. ω \omega
  3. V ω V_{\omega}
  4. M M
  5. M 0 M_{0}
  6. M M
  7. M 0 M_{0}
  8. M M
  9. x x x\notin x
  10. M 0 M_{0}
  11. M M
  12. M 0 M_{0}
  13. M 0 M_{0}
  14. M 0 M_{0}
  15. M 0 M_{0}
  16. M 0 M_{0}
  17. M 0 M_{0}
  18. M 0 M_{0}
  19. M 0 M_{0}
  20. M 0 M_{0}
  21. x x x\notin x
  22. M 0 M_{0}
  23. M 0 M_{0}
  24. M M

Zero-dimensional_space.html

  1. 2 I 2^{I}
  2. 2 = 0 , 1 2={0,1}
  3. I I
  4. 2 I 2^{I}

Zero-knowledge_proof.html

  1. P P
  2. V V
  3. S S
  4. ( P , V ) (P,V)
  5. L L
  6. V ^ \hat{V}
  7. S S
  8. x L , z { 0 , 1 } * , View V ^ [ P ( x ) V ^ ( x , z ) ] = S ( x , z ) \forall x\in L,z\in\{0,1\}^{*},\,\text{View}_{\hat{V}}[P(x)\leftrightarrow\hat% {V}(x,z)]=S(x,z)
  9. P P
  10. ( P , V ) (P,V)
  11. V ^ \hat{V}
  12. S S
  13. P P
  14. V ^ \hat{V}
  15. z z
  16. V ^ \hat{V}
  17. z z
  18. P P
  19. S S
  20. V ^ \hat{V}
  21. P P
  22. V ^ \hat{V}
  23. y y
  24. p p
  25. g g
  26. x x
  27. g x mod p = y g^{x}~{}\bmod~{}p=y
  28. x x
  29. x x
  30. y = g x mod p y=g^{x}~{}\bmod~{}p
  31. y y
  32. x x
  33. r r
  34. C = g r mod p C=g^{r}~{}\bmod~{}p
  35. C C
  36. r r
  37. ( x + r ) mod ( p - 1 ) (x+r)~{}\bmod~{}(p-1)
  38. r r
  39. g r mod p g^{r}~{}\bmod~{}p
  40. C C
  41. ( x + r ) mod ( p - 1 ) (x+r)~{}\bmod~{}(p-1)
  42. C C
  43. g ( x + r ) mod ( p - 1 ) mod p g^{(x+r)~{}\bmod~{}(p-1)}~{}\bmod~{}p
  44. C y mod p C\cdot y~{}\bmod~{}p
  45. x x
  46. x x
  47. r r
  48. r r
  49. C = g r mod p C=g^{r}~{}\bmod~{}p
  50. C C
  51. ( x + r ) mod ( p - 1 ) (x+r)~{}\bmod~{}(p-1)
  52. r r^{\prime}
  53. C = g r ( g x ) - 1 mod p C^{\prime}=g^{r^{\prime}}\cdot{(g^{x})}^{-1}~{}\bmod~{}p
  54. C C^{\prime}
  55. C C
  56. ( x + r ) mod ( p - 1 ) (x+r)~{}\bmod~{}(p-1)
  57. r r^{\prime}
  58. g r mod p g^{r^{\prime}}~{}\bmod~{}p
  59. C y C^{\prime}\cdot y
  60. y y
  61. r r
  62. C = g r mod p C=g^{r}~{}\bmod~{}p
  63. ( x + r ) mod ( p - 1 ) (x+r)~{}\bmod~{}(p-1)
  64. x x
  65. r r^{\prime}
  66. ( x + r ) mod ( p - 1 ) (x+r)~{}\bmod~{}(p-1)
  67. x 2 x^{2}

Zero_(complex_analysis).html

  1. f ( z ) = ( z - a ) g ( z ) f(z)=(z-a)g(z)\,
  2. f ( z ) = ( z - a ) n g ( z ) and g ( a ) 0. f(z)=(z-a)^{n}g(z)\ \mbox{and}~{}\ g(a)\neq 0.\,

Zipf–Mandelbrot_law.html

  1. H k , q , s H N , q , s \frac{H_{k,q,s}}{H_{N,q,s}}
  2. H N , q , s - 1 H N , q , s - q \frac{H_{N,q,s-1}}{H_{N,q,s}}-q
  3. 1 1\,
  4. s H N , q , s k = 1 N ln ( k + q ) ( k + q ) s + ln ( H N , q , s ) \frac{s}{H_{N,q,s}}\sum_{k=1}^{N}\frac{\ln(k+q)}{(k+q)^{s}}+\ln(H_{N,q,s})
  5. f ( k ; N , q , s ) = 1 / ( k + q ) s H N , q , s f(k;N,q,s)=\frac{1/(k+q)^{s}}{H_{N,q,s}}
  6. H N , q , s H_{N,q,s}
  7. H N , q , s = i = 1 N 1 ( i + q ) s H_{N,q,s}=\sum_{i=1}^{N}\frac{1}{(i+q)^{s}}
  8. k k
  9. q q
  10. s s
  11. N N
  12. ζ ( s , q ) \zeta(s,q)
  13. N N
  14. q = 0 q=0
  15. N N
  16. q = 0 q=0

Zymology.html

  1. C 6 H 12 O 6 2 C O 2 + 2 C 2 H 5 OH \mathrm{C_{6}H_{12}O_{6}\rightarrow 2CO_{2}+2C_{2}H_{5}OH}

Π-calculus.html

  1. P Q P\mid Q
  2. P P
  3. Q Q
  4. c ( x ) . P c\left(x\right).P
  5. c c
  6. c ¯ y . P \overline{c}\langle y\rangle.P
  7. y y
  8. c c
  9. ! P !\,P
  10. ( ν x ) P \left(\nu x\right)P
  11. x x
  12. 0
  13. x x
  14. ( ν x ) ( x ¯ z . 0 | x ( y ) . y ¯ x . x ( y ) . 0 ) | z ( v ) . v ¯ v .0 \begin{aligned}&\displaystyle\begin{aligned}\displaystyle(\nu x)&\displaystyle% (\;\overline{x}\langle z\rangle.\;0\\ &\displaystyle|\;x(y).\;\overline{y}\langle x\rangle.\;x(y).\;0\;)\end{aligned% }\\ \displaystyle|&\displaystyle z(v).\;\overline{v}\langle v\rangle.0\end{aligned}
  15. x x
  16. y y
  17. z z
  18. ( ν x ) ( 0 | z ¯ x . x ( y ) . 0 ) | z ( v ) . v ¯ v . 0 \begin{aligned}&\displaystyle\begin{aligned}\displaystyle(\nu x)\;(&% \displaystyle 0\\ \displaystyle|&\displaystyle\overline{z}\langle x\rangle.\;x(y).\;0\;)\end{% aligned}\\ \displaystyle|&\displaystyle z(v).\;\overline{v}\langle v\rangle.\;0\end{aligned}
  19. y y
  20. z z
  21. v v
  22. x x
  23. ( ν x ) ( 0 | x ( y ) . 0 | x ¯ x . 0 ) \begin{aligned}\displaystyle(\nu x)(&\displaystyle 0\\ \displaystyle|&\displaystyle x(y).\;0\\ \displaystyle|&\displaystyle\overline{x}\langle x\rangle.\;0\;)\end{aligned}
  24. x x
  25. x x
  26. x x
  27. x x
  28. ( ν x ) ( 0 | 0 | 0 ) \begin{aligned}\displaystyle(\nu x)(&\displaystyle 0\\ \displaystyle|\;0\\ \displaystyle|\;0\;)\end{aligned}
  29. P , Q , R : := x ( y ) . P Receive on channel x , bind the result to y , then run P | x ¯ y . P Send the value y over channel x , then run P | P | Q Run P and Q simultaneously | ( ν x ) P Create a new channel x and run P | ! P Repeatedly spawn copies of P | 0 Terminate the process \begin{aligned}\displaystyle P,Q,R::=&\displaystyle x(y).P&\displaystyle\,% \text{Receive on channel }x\,\text{, bind the result to }y\,\text{, then run }% P\\ \displaystyle|&\displaystyle\overline{x}\langle y\rangle.P&\displaystyle\,% \text{Send the value }y\,\text{ over channel }x\,\text{, then run }P\\ \displaystyle|&\displaystyle P|Q&\displaystyle\,\text{Run }P\,\text{ and }Q\,% \text{ simultaneously}\\ \displaystyle|&\displaystyle(\nu x)P&\displaystyle\,\text{Create a new channel% }x\,\text{ and run }P\\ \displaystyle|&\displaystyle!P&\displaystyle\,\text{Repeatedly spawn copies of% }P\\ \displaystyle|&\displaystyle 0&\displaystyle\,\text{Terminate the process}\end% {aligned}
  30. 0
  31. a ¯ x . P \overline{a}\langle x\rangle.P
  32. a ( x ) . P a(x).P
  33. P | Q P|Q
  34. ( ν x ) . P (\nu x).P
  35. ! P !P
  36. P Q P\equiv Q
  37. Q Q
  38. P P
  39. P P
  40. P | Q Q | P P|Q\equiv Q|P
  41. ( P | Q ) | R P | ( Q | R ) (P|Q)|R\equiv P|(Q|R)
  42. P | 0 P P|0\equiv P
  43. ( ν x ) ( ν y ) P ( ν y ) ( ν x ) P (\nu x)(\nu y)P\equiv(\nu y)(\nu x)P
  44. ( ν x ) 0 0 (\nu x)0\equiv 0
  45. ! P P | ! P !P\equiv P|!P
  46. ( ν x ) ( P | Q ) ( ν x ) P | Q (\nu x)(P|Q)\equiv(\nu x)P|Q
  47. x x
  48. Q Q
  49. x x
  50. x x
  51. x x
  52. Q Q
  53. P P P\rightarrow P^{\prime}
  54. P P
  55. P P^{\prime}
  56. \rightarrow
  57. x ¯ z . P | x ( y ) . Q P | Q [ z / y ] \overline{x}\langle z\rangle.P|x(y).Q\rightarrow P|Q[z/y]
  58. Q [ z / y ] Q[z/y]
  59. Q Q
  60. z z
  61. y y
  62. y y
  63. z z
  64. P Q P\rightarrow Q
  65. P | R Q | R P|R\rightarrow Q|R
  66. P Q P\rightarrow Q
  67. ( ν x ) P ( ν x ) Q (\nu x)P\rightarrow(\nu x)Q
  68. P P P\equiv P^{\prime}
  69. P Q P^{\prime}\rightarrow Q^{\prime}
  70. Q Q Q^{\prime}\equiv Q
  71. P Q P\rightarrow Q
  72. ( ν x ) ( x ¯ z .0 | x ( y ) . y ¯ x . x ( y ) .0 ) | z ( v ) . v ¯ v .0 (\nu x)(\overline{x}\langle z\rangle.0|x(y).\overline{y}\langle x\rangle.x(y).% 0)|z(v).\overline{v}\langle v\rangle.0
  73. ( ν x ) ( x ¯ z .0 | x ( y ) . y ¯ x . x ( y ) .0 ) | z ( v ) . v ¯ v .0 ( ν x ) ( 0 | z ¯ x . x ( z ) .0 ) | z ( v ) . v ¯ v .0 (\nu x)(\overline{x}\langle z\rangle.0|x(y).\overline{y}\langle x\rangle.x(y).% 0)|z(v).\overline{v}\langle v\rangle.0\rightarrow(\nu x)(0|\overline{z}\langle x% \rangle.x(z).0)|z(v).\overline{v}\langle v\rangle.0
  74. y y
  75. z z
  76. ( ν x ) ( 0 | z ¯ x . x ( z ) .0 ) | z ( v ) . v ¯ v .0 ( ν x ) ( 0 | x ( z ) .0 | x ¯ x .0 ) (\nu x)(0|\overline{z}\langle x\rangle.x(z).0)|z(v).\overline{v}\langle v% \rangle.0\rightarrow(\nu x)(0|x(z).0|\overline{x}\langle x\rangle.0)
  77. x x
  78. x x
  79. P 𝛼 P P\,\xrightarrow{\alpha}\,P^{\prime}
  80. P P
  81. α \alpha
  82. P P^{\prime}
  83. α \alpha
  84. a ( x ) a(x)
  85. a ¯ x \overline{a}\langle x\rangle
  86. τ \tau
  87. P P P\rightarrow P^{\prime}
  88. P 𝜏 P P\,\xrightarrow{\tau}\,P^{\prime}
  89. τ \tau
  90. P + Q P+Q
  91. [ x = y ] P [x=y]P
  92. P P
  93. x x
  94. y y
  95. x ¯ y \overline{x}\langle y\rangle
  96. x ¯ z 1 , z n . P \overline{x}\langle z_{1},...z_{n}\rangle.P
  97. x ( z 1 , z n ) x(z_{1},...z_{n})
  98. x ¯ y 1 , , y n . P \overline{x}\langle y_{1},\cdots,y_{n}\rangle.P
  99. ( ν w ) x ¯ w . w ¯ y 1 . . w ¯ y n . [ P ] (\nu w)\overline{x}\langle w\rangle.\overline{w}\langle y_{1}\rangle.\cdots.% \overline{w}\langle y_{n}\rangle.[P]
  100. x ( y 1 , , y n ) . P x(y_{1},\cdots,y_{n}).P
  101. x ( w ) . w ( y 1 ) . . w ( y n ) . [ P ] x(w).w(y_{1}).\cdots.w(y_{n}).[P]
  102. [ P ] [P]
  103. P P
  104. ! P !P
  105. ! x ( y ) . P !x(y).P
  106. ! x ( y ) . P x ( y ) . P | ! x ( y ) . P !x(y).P\equiv x(y).P|!x(y).P
  107. ! x ( y ) . P !x(y).P
  108. x x
  109. P [ a / y ] P[a/y]
  110. x ¯ R . P | x ( Y ) . Q P | Q [ R / Y ] \overline{x}\langle R\rangle.P|x(Y).Q\rightarrow P|Q[R/Y]
  111. Y Y
  112. x x
  113. M M
  114. M M
  115. p p
  116. q q
  117. R R
  118. R R
  119. ( p , q ) R (p,q)\in R
  120. p a ( x ) p p\,\xrightarrow{a(x)}\,p^{\prime}
  121. y y
  122. q q^{\prime}
  123. q a ( x ) q q\,\xrightarrow{a(x)}\,q^{\prime}
  124. ( p [ y / x ] , q [ y / x ] ) R (p^{\prime}[y/x],q^{\prime}[y/x])\in R
  125. α \alpha
  126. q q^{\prime}
  127. q 𝛼 q q\xrightarrow{\alpha}q^{\prime}
  128. ( p , q ) R (p^{\prime},q^{\prime})\in R
  129. p p
  130. q q
  131. p p
  132. q q
  133. p e q p\sim_{e}q
  134. ( p , q ) R (p,q)\in R
  135. R R
  136. R R
  137. ( p , q ) R (p,q)\in R
  138. p a ( x ) p p\xrightarrow{a(x)}p^{\prime}
  139. q q^{\prime}
  140. q a ( x ) q q\xrightarrow{a(x)}q^{\prime}
  141. ( p [ y / x ] , q [ y / x ] ) R (p^{\prime}[y/x],q^{\prime}[y/x])\in R
  142. α \alpha
  143. q q^{\prime}
  144. q 𝛼 q q\xrightarrow{\alpha}q^{\prime}
  145. ( p , q ) R (p^{\prime},q^{\prime})\in R
  146. p p
  147. q q
  148. p p
  149. q q
  150. p l q p\sim_{l}q
  151. ( p , q ) R (p,q)\in R
  152. R R
  153. e \sim_{e}
  154. l \sim_{l}
  155. p p
  156. q q
  157. p e q p\sim_{e}q
  158. a ( x ) . p ≁ e a ( x ) . q a(x).p\not\sim_{e}a(x).q
  159. e \sim_{e}
  160. l \sim_{l}
  161. R R
  162. ( p , q ) R (p,q)\in R
  163. σ \sigma
  164. α \alpha
  165. p σ 𝛼 p p\sigma\xrightarrow{\alpha}p^{\prime}
  166. q q^{\prime}
  167. q σ 𝛼 q q\sigma\xrightarrow{\alpha}q^{\prime}
  168. ( p , q ) R (p^{\prime},q^{\prime})\in R
  169. p p
  170. q q
  171. p o q p\sim_{o}q
  172. ( p , q ) R (p,q)\in R
  173. R R
  174. o l e \sim_{o}\subsetneq\sim_{l}\subsetneq\sim_{e}
  175. p a p\Downarrow a
  176. p p
  177. a a
  178. R R
  179. ( p , q ) R (p,q)\in R
  180. p a p\Downarrow a
  181. q a q\Downarrow a
  182. a a
  183. p p p\rightarrow p^{\prime}
  184. q q q\rightarrow q^{\prime}
  185. ( p , q ) R (p^{\prime},q^{\prime})\in R
  186. p p
  187. q q
  188. R R
  189. ( p , q ) R (p,q)\in R
  190. P b Q P\sim_{b}Q\,\!
  191. C [ ] C[]
  192. C [ P ] C[P]
  193. C [ Q ] C[Q]

−1.html

  1. e i π = - 1. e^{i\pi}=-1.\!
  2. x + ( - 1 ) x = 1 x + ( - 1 ) x = ( 1 + ( - 1 ) ) x = 0 x = 0 x+(-1)\cdot x=1\cdot x+(-1)\cdot x=(1+(-1))\cdot x=0\cdot x=0
  3. 0 x = ( 0 + 0 ) x = 0 x + 0 x 0\cdot x=(0+0)\cdot x=0\cdot x+0\cdot x\,
  4. x + ( - 1 ) x = 0 x+(-1)\cdot x=0\,
  5. 0 = - 1 0 = - 1 [ 1 + ( - 1 ) ] 0=-1\cdot 0=-1\cdot[1+(-1)]
  6. 0 = - 1 [ 1 + ( - 1 ) ] = - 1 1 + ( - 1 ) ( - 1 ) = - 1 + ( - 1 ) ( - 1 ) 0=-1\cdot[1+(-1)]=-1\cdot 1+(-1)\cdot(-1)=-1+(-1)\cdot(-1)
  7. ( - 1 ) ( - 1 ) = 1 (-1)\cdot(-1)=1

Positional_notation.html

  1. 2506 = 2 × 10 3 + 5 × 10 2 + 0 × 10 1 + 6 × 10 0 2506=2\times 10^{3}+5\times 10^{2}+0\times 10^{1}+6\times 10^{0}
  2. 171 B = 1 × 16 3 + 7 × 16 2 + 1 × 16 1 + B × 16 0 171\mathrm{B}=1\times 16^{3}+7\times 16^{2}+1\times 16^{1}+\mathrm{B}\times 16% ^{0}
  3. a 3 a 2 a 1 a 0 = a 3 × b 3 + a 2 × b 2 + a 1 × b 1 + a 0 × b 0 a_{3}a_{2}a_{1}a_{0}=a_{3}\times b^{3}+a_{2}\times b^{2}+a_{1}\times b^{1}+a_{% 0}\times b^{0}
  4. a 3 a 2 a 1 a 0 a_{3}a_{2}a_{1}a_{0}
  5. 4 × b 2 + 6 × b 1 + 5 × b 0 4\times b^{2}+6\times b^{1}+5\times b^{0}
  6. 4 × 10 2 + 6 × 10 1 + 5 × 10 0 = 4 × 100 + 6 × 10 + 5 × 1 = 465 4\times 10^{2}+6\times 10^{1}+5\times 10^{0}=4\times 100+6\times 10+5\times 1=% 465
  7. 4 × 7 2 + 6 × 7 1 + 5 × 7 0 = 4 × 49 + 6 × 7 + 5 × 1 = 243 4\times 7^{2}+6\times 7^{1}+5\times 7^{0}=4\times 49+6\times 7+5\times 1=243
  8. 2 × 10 0 + 3 × 10 - 1 + 5 × 10 - 2 2\times 10^{0}+3\times 10^{-1}+5\times 10^{-2}
  9. i = 0 n ( a i × b i ) \sum_{i=0}^{n}\left(a_{i}\times b^{i}\right)
  10. 241 5 = 2 × 5 2 + 4 × 5 1 + 1 × 5 0 = 50 + 20 + 1 = 71 10 241_{5}=2\times 5^{2}+4\times 5^{1}+1\times 5^{0}=50+20+1=71_{10}
  11. 123 10 = 123 / 7 = 17 with a remainder of ( 4 ) \displaystyle 123_{10}=123/7=17\,\text{ with a remainder of }(4)
  12. 456 10 = 456 / 8 = 57 with a remainder of ( 0 ) \displaystyle 456_{10}=456/8=57\,\text{ with a remainder of }(0)
  13. 1 × 3 0 + 1\times 3^{0\,\,\,}+{}
  14. 1 × 3 - 1 + 2 × 3 - 2 + 1\times 3^{-1\,\,}+2\times 3^{-2\,\,\,}+{}
  15. 1 × 3 - 3 + 1 × 3 - 4 + 2 × 3 - 5 + 1\times 3^{-3\,\,}+1\times 3^{-4\,\,\,}+2\times 3^{-5\,\,\,}+{}
  16. 1 × 3 - 6 + 1 × 3 - 7 + 1 × 3 - 8 + 2 × 3 - 9 + 1\times 3^{-6\,\,}+1\times 3^{-7\,\,\,}+1\times 3^{-8\,\,\,}+2\times 3^{-9\,\,% \,}+{}
  17. 1 × 3 - 10 + 1 × 3 - 11 + 1 × 3 - 12 + 1 × 3 - 13 + 2 × 3 - 14 + 1\times 3^{-10}+1\times 3^{-11}+1\times 3^{-12}+1\times 3^{-13}+2\times 3^{-14% }+\cdots
  18. 2.42 314 ¯ 5 = 2.42314314314314314 5 2.42\overline{314}_{5}=2.42314314314314314\dots_{5}
  19. 0.1 3 0.1_{3}\,
  20. 0. 3 ¯ 10 = 0.3333333 10 0.\overline{3}_{10}=0.3333333\dots_{10}
  21. 0. 3 ¯ = 0.3333333 0.\overline{3}=0.3333333\dots
  22. 0. 01 ¯ 2 = 0.010101 2 0.\overline{01}_{2}=0.010101\dots_{2}
  23. 0.2 6 0.2_{6}\,
  24. 3.46 7 = 3.460 7 = 3.460000 7 = 3.46 0 ¯ 7 3.46_{7}=3.460_{7}=3.460000_{7}=3.46\overline{0}_{7}
  25. 3.46 7 = 3.45 6 ¯ 7 3.46_{7}=3.45\overline{6}_{7}
  26. 1 10 = 0. 9 ¯ 10 1_{10}=0.\overline{9}_{10}
  27. 220 5 = 214. 4 ¯ 5 220_{5}=214.\overline{4}_{5}
  28. 1 ¯ \overline{1}
  29. 1 ¯ \overline{1}
  30. ¯ 1.31 30 \overline{$}\par \par \@@section{subsection}{S1.SS31}{1.31}{1.31}{{\@tag[][]{1% .31}30}}{{\@tag[][]{1.31}30}}\par