wpmath0000004_10

Mexican_hat_wavelet.html

  1. ψ ( t ) = 1 2 π σ 3 ( 1 - t 2 σ 2 ) e - t 2 2 σ 2 \psi(t)={1\over{\sqrt{2\pi}\sigma^{3}}}\left(1-{t^{2}\over\sigma^{2}}\right)e^% {-t^{2}\over 2\sigma^{2}}
  2. ψ ( x , y ) = - 1 π σ 4 ( 1 - x 2 + y 2 2 σ 2 ) e - ( x 2 + y 2 ) / 2 σ 2 . \psi(x,y)=-\frac{1}{\pi\sigma^{4}}\left(1-\frac{x^{2}+y^{2}}{2\sigma^{2}}% \right)\mathrm{e}^{-(x^{2}+y^{2})/2\sigma^{2}}.
  3. L 1 L_{1}

Micro_black_hole.html

  1. R = 2 G M / c 2 R=2GM/c^{2}
  2. λ = h / M c \lambda=h/Mc
  3. λ = / M c \lambda\;=\;\hbar/Mc

Microcanonical_ensemble.html

  1. N N
  2. V V
  3. E E
  4. N V E NVE
  5. E E
  6. P P
  7. W W
  8. P = 1 / W , P=1/W,
  9. E E
  10. E E
  11. S = k l o g W S=klogW
  12. W W
  13. ω ω
  14. ω ω
  15. d E = T v d S v - P d V , dE=T_{\rm v}dS_{\rm v}-\langle P\rangle dV,
  16. P ⟨P⟩
  17. T T
  18. T T
  19. T T
  20. f ( H - E ω ) f(\tfrac{H-E}{\omega})
  21. H H
  22. E E
  23. ω ω
  24. f ( x ) = { 1 , if | x | < 1 2 , 0 , otherwise . f(x)=\begin{cases}1,&\mathrm{if}~{}|x|<\tfrac{1}{2},\\ 0,&\mathrm{otherwise.}\end{cases}
  25. f ( x ) = e - π x 2 . f(x)=e^{-\pi x^{2}}.
  26. ρ̂ ρ̂
  27. i i
  28. ρ ^ = 1 W i f ( H i - E ω ) | ψ i ψ i | \hat{\rho}=\frac{1}{W}\sum_{i}f(\tfrac{H_{i}-E}{\omega})|\psi_{i}\rangle% \langle\psi_{i}|
  29. Ĥ Ĥ
  30. W W
  31. ρ̂ ρ̂
  32. W = i f ( H i - E ω ) . W=\sum_{i}f(\tfrac{H_{i}-E}{\omega}).
  33. v ( E ) = H i < E 1. v(E)=\sum_{H_{i}<E}1.
  34. E E
  35. N V E ω NVEω
  36. n n
  37. n n
  38. ρ = 1 h n C 1 W f ( H - E ω ) , \rho=\frac{1}{h^{n}C}\frac{1}{W}f(\tfrac{H-E}{\omega}),
  39. H H
  40. h h
  41. e n e r g y × t i m e energy×time
  42. ρ ρ
  43. C C
  44. W W
  45. ρ ρ
  46. W = 1 h n C f ( H - E ω ) d p 1 d q n W=\int\ldots\int\frac{1}{h^{n}C}f(\tfrac{H-E}{\omega})\,dp_{1}\ldots dq_{n}
  47. v ( E ) = H < E 1 h n C d p 1 d q n . v(E)=\int\ldots\int_{H<E}\frac{1}{h^{n}C}\,dp_{1}\ldots dq_{n}.
  48. ω ω
  49. W W
  50. ω ω
  51. W = ω ( d v / d E ) W=ω(dv/dE)
  52. h = 1 e n e r g y u n i t t × × t i m e u n i t t h=1energyunitt××timeunitt
  53. h h
  54. N N
  55. C = N ! C=N!
  56. N N

Mill_(grinding).html

  1. W K = c k ( ln ( d A ) - ln ( d E ) ) W_{K}=c_{k}(\ln(d_{A})-\ln(d_{E}))\,
  2. W B = c B ( 1 d E - 1 d A ) W_{B}=c_{B}\left(\frac{1}{\sqrt{d}_{E}}-\frac{1}{\sqrt{d}_{A}}\right)\,
  3. W R = c R ( 1 d E - 1 d A ) W_{R}=c_{R}\left(\frac{1}{d_{E}}-\frac{1}{d_{A}}\right)\,
  4. c K = 1.151 c B ( d B U ) - 0.5 c_{K}=1.151c_{B}(d_{BU})^{-0.5}\,
  5. c R = 0.5 c B ( d B L ) 0.5 c_{R}=0.5c_{B}(d_{BL})^{0.5}\,
  6. Z d = d 80 , 1 d 80 , 2 Z_{d}=\frac{d_{80,1}}{d_{80,2}}\,
  7. Z S = S v , 2 S v , 1 = S m , 2 S m , 1 Z_{S}=\frac{S_{v,2}}{S_{v,1}}=\frac{S_{m,2}}{S_{m,1}}\,
  8. Z a = d 1 a Z_{a}=\frac{d_{1}}{a}\,
  9. E = 10 W ( 1 P 80 - 1 F 80 ) E=10W\left(\frac{1}{\sqrt{P_{80}}}-\frac{1}{\sqrt{F_{80}}}\right)\,

Miller_index.html

  1. h 𝐛 𝟏 + k 𝐛 𝟐 + 𝐛 𝟑 h\mathbf{b_{1}}+k\mathbf{b_{2}}+\ell\mathbf{b_{3}}
  2. 𝐛 𝐢 \mathbf{b_{i}}
  3. h 𝐚 𝟏 + k 𝐚 𝟐 + 𝐚 𝟑 h\mathbf{a_{1}}+k\mathbf{a_{2}}+\ell\mathbf{a_{3}}
  4. 3 ¯ \overline{3}
  5. 𝐠 h k = h 𝐛 1 + k 𝐛 2 + 𝐛 3 . \mathbf{g}_{hk\ell}=h\mathbf{b}_{1}+k\mathbf{b}_{2}+\ell\mathbf{b}_{3}.
  6. d = 2 π / | 𝐠 h k | d=2\pi/|\mathbf{g}_{hk\ell}|
  7. h 𝐚 1 + k 𝐚 2 + 𝐚 3 . h\mathbf{a}_{1}+k\mathbf{a}_{2}+\ell\mathbf{a}_{3}.
  8. d h k = a h 2 + k 2 + 2 d_{hk\ell}=\frac{a}{\sqrt{h^{2}+k^{2}+\ell^{2}}}
  9. 2 ¯ \overline{2}
  10. 2 ¯ \overline{2}
  11. 2 ¯ \overline{2}
  12. 1 ¯ \overline{1}
  13. 1 ¯ \overline{1}
  14. 1 ¯ \overline{1}
  15. 1 ¯ \overline{1}
  16. h 𝐛 𝟏 + k 𝐛 𝟐 + 𝐛 𝟑 h\mathbf{b_{1}}+k\mathbf{b_{2}}+\ell\mathbf{b_{3}}
  17. h 𝐛 𝟏 + k 𝐛 𝟐 + 𝐛 𝟑 = 2 3 a 2 ( 2 h + k ) 𝐚 𝟏 + 2 3 a 2 ( h + 2 k ) 𝐚 𝟐 + 1 c 2 ( ) 𝐚 𝟑 . h\mathbf{b_{1}}+k\mathbf{b_{2}}+\ell\mathbf{b_{3}}=\frac{2}{3a^{2}}(2h+k)% \mathbf{a_{1}}+\frac{2}{3a^{2}}(h+2k)\mathbf{a_{2}}+\frac{1}{c^{2}}(\ell)% \mathbf{a_{3}}.
  18. [ 2 h + k , h + 2 k , ( 3 / 2 ) ( a / c ) 2 ] [2h+k,h+2k,\ell(3/2)(a/c)^{2}]
  19. [ h , k , - h - k , ( 3 / 2 ) ( a / c ) 2 ] [h,k,-h-k,\ell(3/2)(a/c)^{2}]

Min-max_theorem.html

  1. A A
  2. n × n n×n
  3. R A ( x ) = ( A x , x ) ( x , x ) R_{A}(x)=\frac{(Ax,x)}{(x,x)}
  4. ( , ) (⋅,⋅)
  5. f ( x ) = ( A x , x ) , x = 1. f(x)=(Ax,x),\;\|x\|=1.
  6. A A
  7. n × n n×n
  8. λ k = min { max { R A ( x ) x U and x 0 } dim ( U ) = k } \lambda_{k}=\min\{\max\{R_{A}(x)\mid x\in U\,\text{ and }x\neq 0\}\mid\dim(U)=k\}
  9. λ k = max { min { R A ( x ) x U and x 0 } dim ( U ) = n - k + 1 } \lambda_{k}=\max\{\min\{R_{A}(x)\mid x\in U\,\text{ and }x\neq 0\}\mid\dim(U)=% n-k+1\}
  10. λ 1 R A ( x ) λ n x 𝐂 n \ { 0 } \lambda_{1}\leq R_{A}(x)\leq\lambda_{n}\quad\forall x\in\mathbf{C}^{n}% \backslash\{0\}
  11. x x
  12. λ n = max { R A ( x ) : x 0 } . \lambda_{n}=\max\{R_{A}(x):x\neq 0\}.
  13. λ 1 = min { R A ( x ) : x 0 } . \lambda_{1}=\min\{R_{A}(x):x\neq 0\}.
  14. A A
  15. v 0 v≠0
  16. v = i = k n α i u i v=\sum_{i=k}^{n}\alpha_{i}u_{i}
  17. R A ( v ) = i = k n λ i α i 2 i = k n α i 2 λ k R_{A}(v)=\frac{\sum_{i=k}^{n}\lambda_{i}\alpha_{i}^{2}}{\sum_{i=k}^{n}\alpha_{% i}^{2}}\geq\lambda_{k}
  18. λ i λ k \lambda_{i}\geq\lambda_{k}
  19. max { R A ( x ) x U } λ k \max\{R_{A}(x)\mid x\in U\}\geq\lambda_{k}
  20. min { max { R A ( x ) x U and x 0 } dim ( U ) = k } λ k \min\{\max\{R_{A}(x)\mid x\in U\,\text{ and }x\neq 0\}\mid\dim(U)=k\}\geq% \lambda_{k}
  21. max { R A ( x ) x V and x 0 } λ k \max\{R_{A}(x)\mid x\in V\,\text{ and }x\neq 0\}\leq\lambda_{k}
  22. λ k \lambda_{k}
  23. min { max { R A ( x ) x U and x 0 } dim ( U ) = k } λ k \min\{\max\{R_{A}(x)\mid x\in U\,\text{ and }x\neq 0\}\mid\dim(U)=k\}\leq% \lambda_{k}
  24. v = i = 1 k α i u i v=\sum_{i=1}^{k}\alpha_{i}u_{i}
  25. R A ( v ) = i = 1 k λ i α i 2 i = 1 k α i 2 λ k R_{A}(v)=\frac{\sum_{i=1}^{k}\lambda_{i}\alpha_{i}^{2}}{\sum_{i=1}^{k}\alpha_{% i}^{2}}\leq\lambda_{k}
  26. min { R A ( x ) x U } λ k \min\{R_{A}(x)\mid x\in U\}\leq\lambda_{k}
  27. max { min { R A ( x ) x U and x 0 } dim ( U ) = n - k + 1 } λ k \max\{\min\{R_{A}(x)\mid x\in U\,\text{ and }x\neq 0\}\mid\dim(U)=n-k+1\}\leq% \lambda_{k}
  28. λ k \lambda_{k}
  29. [ 0 1 0 0 ] . \begin{bmatrix}0&1\\ 0&0\end{bmatrix}.
  30. R N ( x ) R_{N}(x)
  31. 1 2 \frac{1}{2}
  32. σ k = min S : dim ( S ) = k max x S , x = 1 ( M * M x , x ) 1 2 = min S : dim ( S ) = k max x S , x = 1 M x . \sigma_{k}^{\uparrow}=\min_{S:\dim(S)=k}\max_{x\in S,\|x\|=1}(M^{*}Mx,x)^{% \frac{1}{2}}=\min_{S:\dim(S)=k}\max_{x\in S,\|x\|=1}\|Mx\|.
  33. σ k = max S : dim ( S ) = n - k + 1 min x S , x = 1 M x . \sigma_{k}^{\downarrow}=\max_{S:\dim(S)=n-k+1}\min_{x\in S,\|x\|=1}\|Mx\|.
  34. A A
  35. A A
  36. A A
  37. α j β j α n - m + j . \alpha_{j}\leq\beta_{j}\leq\alpha_{n-m+j}.
  38. β j = max x S j , x = 1 ( B x , x ) = max x S j , x = 1 ( P * A P x , x ) min S j max x S j , x = 1 ( A x , x ) = α j . \beta_{j}=\max_{x\in S_{j},\|x\|=1}(Bx,x)=\max_{x\in S_{j},\|x\|=1}(P^{*}APx,x% )\geq\min_{S_{j}}\max_{x\in S_{j},\|x\|=1}(Ax,x)=\alpha_{j}.
  39. β j = min x S m - j + 1 , x = 1 ( B x , x ) = min x S m - j + 1 , x = 1 ( P * A P x , x ) = min x S m - j + 1 , x = 1 ( A x , x ) α n - m + j , \beta_{j}=\min_{x\in S_{m-j+1},\|x\|=1}(Bx,x)=\min_{x\in S_{m-j+1},\|x\|=1}(P^% {*}APx,x)=\min_{x\in S_{m-j+1},\|x\|=1}(Ax,x)\leq\alpha_{n-m+j},
  40. n m = 1 n−m=1
  41. A A
  42. A A
  43. λ k λ 1 , \cdots\leq\lambda_{k}\leq\cdots\leq\lambda_{1},
  44. A A
  45. H H
  46. max S k min x S k , x = 1 ( A x , x ) = λ k , min S k - 1 max x S k - 1 , x = 1 ( A x , x ) = λ k . \begin{aligned}\displaystyle\max_{S_{k}}\min_{x\in S_{k},\|x\|=1}(Ax,x)&% \displaystyle=\lambda_{k}^{\downarrow},\\ \displaystyle\min_{S_{k-1}}\max_{x\in S_{k-1}^{\perp},\|x\|=1}(Ax,x)&% \displaystyle=\lambda_{k}^{\downarrow}.\end{aligned}
  47. min S k - 1 max x S k - 1 , x = 1 ( A x , x ) = λ k . \min_{S_{k-1}}\max_{x\in S_{k-1}^{\perp},\|x\|=1}(Ax,x)=\lambda_{k}.
  48. E 1 E 2 E 3 E_{1}\leq E_{2}\leq E_{3}\leq\cdots
  49. E n = min ψ 1 , , ψ n - 1 max { ψ , A ψ : ψ span ( ψ 1 , , ψ n - 1 ) } E_{n}=\min_{\psi_{1},\ldots,\psi_{n-1}}\max\{\langle\psi,A\psi\rangle:\psi\in% \operatorname{span}(\psi_{1},\ldots,\psi_{n-1})\}
  50. E n := inf σ e s s ( A ) E_{n}:=\inf\sigma_{ess}(A)
  51. E 1 E 2 E 3 E_{1}\leq E_{2}\leq E_{3}\leq\cdots
  52. E n = max ψ 1 , , ψ n - 1 min { ψ , A ψ : ψ ψ 1 , , ψ n - 1 } E_{n}=\max_{\psi_{1},\ldots,\psi_{n-1}}\min\{\langle\psi,A\psi\rangle:\psi% \perp\psi_{1},\ldots,\psi_{n-1}\}
  53. E n := inf σ e s s ( A ) E_{n}:=\inf\sigma_{ess}(A)
  54. ( A - E ) 0 (A-E)\geq 0
  55. E E\in\mathbb{R}
  56. σ ( A ) [ E , ) \sigma(A)\subseteq[E,\infty)
  57. inf σ ( A ) = inf ψ 𝔇 ( A ) , ψ = 1 ψ , A ψ \inf\sigma(A)=\inf_{\psi\in\mathfrak{D}(A),\|\psi\|=1}\langle\psi,A\psi\rangle
  58. sup σ ( A ) = sup ψ 𝔇 ( A ) , ψ = 1 ψ , A ψ \sup\sigma(A)=\sup_{\psi\in\mathfrak{D}(A),\|\psi\|=1}\langle\psi,A\psi\rangle

Minimum_degree_algorithm.html

  1. 𝐀𝐱 = 𝐛 \mathbf{A}\mathbf{x}=\mathbf{b}
  2. n × n n\times n
  3. 𝐏 T 𝐀𝐏 \mathbf{P}^{T}\mathbf{A}\mathbf{P}
  4. ( 𝐏 T 𝐀𝐏 ) ( 𝐏 T 𝐱 ) = 𝐏 T 𝐛 . \left(\mathbf{P}^{T}\mathbf{A}\mathbf{P}\right)\left(\mathbf{P}^{T}\mathbf{x}% \right)=\mathbf{P}^{T}\mathbf{b}.
  5. a i j 0 a_{ij}\neq 0
  6. O ( n 2 m ) O(n^{2}m)
  7. O ( n m ) O(nm)

Mirror_symmetry_(string_theory).html

  1. R R
  2. 1 / R 1/R
  3. α \alpha
  4. β \beta
  5. α \alpha
  6. β \beta
  7. B B
  8. B B
  9. B B
  10. B B
  11. B B
  12. B B
  13. B B
  14. R R
  15. 1 / R 1/R
  16. p p
  17. n n
  18. n n
  19. p p
  20. B B

Mitral_insufficiency.html

  1. V m i t r a l - V a o r t i c V m i t r a l × 100 % \frac{V_{mitral}-V_{aortic}}{V_{mitral}}\times 100\%

Mixed_inhibition.html

  1. K m app > K m K_{m}\text{app}>K_{m}
  2. K m app < K m K_{m}\text{app}<K_{m}
  3. V m a x app < V m a x V_{max}\text{app}<V_{max}
  4. V m a x a p p V_{max}^{app}
  5. K m K_{m}

Mixing_(process_engineering).html

  1. Q = F l * N * D 3 Q=Fl*N*D^{3}
  2. P = P o ρ N 3 D 5 P=Po\rho N^{3}D^{5}
  3. P = K p μ N 2 D 3 P=K_{p}\mu N^{2}D^{3}
  4. P o Po
  5. ρ \rho
  6. N N
  7. D D
  8. K p K_{p}
  9. μ \mu
  10. θ 95 {\theta_{95}}
  11. θ 95 = 5.40 P o 1 3 N ( T D ) 2 {\theta_{95}}=\frac{5.40}{Po^{1\over 3}N}(\frac{T}{D})^{2}
  12. θ 95 = 34596 P o 1 3 N 2 D 2 ( μ ρ ) ( T D ) 2 {\theta_{95}}=\frac{34596}{Po^{1\over 3}N^{2}D^{2}}(\frac{\mu}{\rho})(\frac{T}% {D})^{2}
  13. θ 95 = 896 * 10 3 K p - 1.69 N {\theta_{95}}=\frac{896*10^{3}K_{p}^{-1.69}}{N}
  14. P o 1 3 R e = 6404 Po^{1\over 3}Re=6404
  15. P o 1 3 R e = 186 Po^{1\over 3}Re=186

Mixture_distribution.html

  1. F ( x ) = i = 1 n w i P i ( x ) , F(x)=\sum_{i=1}^{n}\,w_{i}\,P_{i}(x),
  2. f ( x ) = i = 1 n w i p i ( x ) . f(x)=\sum_{i=1}^{n}\,w_{i}\,p_{i}(x).
  3. n = n=\infty\!
  4. f ( x ) = A w ( a ) p ( x ; a ) d a f(x)=\int_{A}\,w(a)\,p(x;a)\,da
  5. f ( x ; a 1 , , a n ) = i = 1 n w i p ( x ; a i ) f(x;a_{1},\ldots,a_{n})=\sum_{i=1}^{n}\,w_{i}\,p(x;a_{i})
  6. f ( x ; a 1 , , a n , b 1 , , b n ) = i = 1 n w i p ( x ; a i , b i ) f(x;a_{1},\ldots,a_{n},b_{1},\ldots,b_{n})=\sum_{i=1}^{n}\,w_{i}\,p(x;a_{i},b_% {i})
  7. E [ H ( X i ) ] \operatorname{E}[H(X_{i})]
  8. E [ H ( X ) ] = - H ( x ) i = 1 n w i p i ( x ) d x = i = 1 n w i - p i ( x ) H ( x ) d x = i = 1 n w i E [ H ( X i ) ] . \begin{aligned}\displaystyle\operatorname{E}[H(X)]&\displaystyle=\int_{-\infty% }^{\infty}H(x)\sum_{i=1}^{n}w_{i}p_{i}(x)\,dx\\ &\displaystyle=\sum_{i=1}^{n}w_{i}\int_{-\infty}^{\infty}p_{i}(x)H(x)\,dx=\sum% _{i=1}^{n}w_{i}\operatorname{E}[H(X_{i})].\end{aligned}
  9. E [ H ( X ) ] = i = 1 n w i E [ H ( X i ) ] , \operatorname{E}[H(X)]=\sum_{i=1}^{n}w_{i}\operatorname{E}[H(X_{i})],
  10. E [ ( X - μ ) j ] = i = 1 n w i E [ ( X i - μ i + μ i - μ ) j ] = i = 1 n k = 0 j ( j k ) ( μ i - μ ) j - k w i E [ ( X i - μ i ) k ] , \begin{aligned}\displaystyle\operatorname{E}[(X-\mu)^{j}]&\displaystyle=\sum_{% i=1}^{n}w_{i}\operatorname{E}[(X_{i}-\mu_{i}+\mu_{i}-\mu)^{j}]\\ &\displaystyle=\sum_{i=1}^{n}\sum_{k=0}^{j}\left(\begin{array}[]{c}j\\ k\end{array}\right)(\mu_{i}-\mu)^{j-k}w_{i}\operatorname{E}[(X_{i}-\mu_{i})^{k% }],\end{aligned}
  11. E [ X ] = μ = i = 1 n w i μ i , \operatorname{E}[X]=\mu=\sum_{i=1}^{n}w_{i}\mu_{i},
  12. E [ ( X - μ ) 2 ] = σ 2 = i = 1 n w i ( ( μ i - μ ) 2 + σ i 2 ) . \operatorname{E}[(X-\mu)^{2}]=\sigma^{2}=\sum_{i=1}^{n}w_{i}((\mu_{i}-\mu)^{2}% +\sigma_{i}^{2}).
  13. x * ( α ) = [ i = 1 n α i Σ i - 1 ] - 1 × [ i = 1 n α i Σ i - 1 μ i ] , x^{*}(\alpha)=\left[\sum_{i=1}^{n}\alpha_{i}\Sigma_{i}^{-1}\right]^{-1}\times% \left[\sum_{i=1}^{n}\alpha_{i}\Sigma_{i}^{-1}\mu_{i}\right],
  14. 𝒮 n = { α n : α i [ 0 , 1 ] , i = 1 n α i = 1 } \mathcal{S}_{n}=\{\alpha\in\mathbb{R}^{n}:\alpha_{i}\in[0,1],\sum_{i=1}^{n}% \alpha_{i}=1\}
  15. d h ( α ) d α = 0 \frac{dh(\alpha)}{d\alpha}=0
  16. d Π ( α ) d α = 0 \frac{d\Pi(\alpha)}{d\alpha}=0
  17. 1 - α ( 1 - α ) d M ( μ 1 , μ 2 , Σ ) 2 1-\alpha(1-\alpha)d_{M}(\mu_{1},\mu_{2},\Sigma)^{2}
  18. | μ 1 - μ 2 | > 2 σ , \left|\mu_{1}-\mu_{2}\right|>2\sigma,
  19. σ , \sigma,

Mixture_model.html

  1. K = number of mixture components N = number of observations θ i = 1 K = parameter of distribution of observation associated with component i ϕ i = 1 K = mixture weight, i.e., prior probability of a particular component i s y m b o l ϕ = K -dimensional vector composed of all the individual ϕ 1 K ; must sum to 1 z i = 1 N = component of observation i x i = 1 N = observation i F ( x | θ ) = probability distribution of an observation, parametrized on θ z i = 1 N Categorical ( s y m b o l ϕ ) x i = 1 N F ( θ z i ) \begin{array}[]{lcl}K&=&\,\text{number of mixture components}\\ N&=&\,\text{number of observations}\\ \theta_{i=1\dots K}&=&\,\text{parameter of distribution of observation % associated with component }i\\ \phi_{i=1\dots K}&=&\,\text{mixture weight, i.e., prior probability of a % particular component }i\\ symbol\phi&=&K\,\text{-dimensional vector composed of all the individual }\phi% _{1\dots K}\,\text{; must sum to 1}\\ z_{i=1\dots N}&=&\,\text{component of observation }i\\ x_{i=1\dots N}&=&\,\text{observation }i\\ F(x|\theta)&=&\,\text{probability distribution of an observation, parametrized% on }\theta\\ z_{i=1\dots N}&\sim&\operatorname{Categorical}(symbol\phi)\\ x_{i=1\dots N}&\sim&F(\theta_{z_{i}})\end{array}
  2. K , N = as above θ i = 1 K , ϕ i = 1 K , s y m b o l ϕ = as above z i = 1 N , x i = 1 N , F ( x | θ ) = as above α = shared hyperparameter for component parameters β = shared hyperparameter for mixture weights H ( θ | α ) = prior probability distribution of component parameters, parametrized on α θ i = 1 K H ( θ | α ) s y m b o l ϕ Symmetric - Dirichlet K ( β ) z i = 1 N Categorical ( s y m b o l ϕ ) x i = 1 N F ( θ z i ) \begin{array}[]{lcl}K,N&=&\,\text{as above}\\ \theta_{i=1\dots K},\phi_{i=1\dots K},symbol\phi&=&\,\text{as above}\\ z_{i=1\dots N},x_{i=1\dots N},F(x|\theta)&=&\,\text{as above}\\ \alpha&=&\,\text{shared hyperparameter for component parameters}\\ \beta&=&\,\text{shared hyperparameter for mixture weights}\\ H(\theta|\alpha)&=&\,\text{prior probability distribution of component % parameters, parametrized on }\alpha\\ \theta_{i=1\dots K}&\sim&H(\theta|\alpha)\\ symbol\phi&\sim&\operatorname{Symmetric-Dirichlet}_{K}(\beta)\\ z_{i=1\dots N}&\sim&\operatorname{Categorical}(symbol\phi)\\ x_{i=1\dots N}&\sim&F(\theta_{z_{i}})\end{array}
  3. K , N = as above ϕ i = 1 K , s y m b o l ϕ = as above z i = 1 N , x i = 1 N = as above θ i = 1 K = { μ i = 1 K , σ i = 1 K 2 } μ i = 1 K = mean of component i σ i = 1 K 2 = variance of component i z i = 1 N Categorical ( s y m b o l ϕ ) x i = 1 N 𝒩 ( μ z i , σ z i 2 ) \begin{array}[]{lcl}K,N&=&\,\text{as above}\\ \phi_{i=1\dots K},symbol\phi&=&\,\text{as above}\\ z_{i=1\dots N},x_{i=1\dots N}&=&\,\text{as above}\\ \theta_{i=1\dots K}&=&\{\mu_{i=1\dots K},\sigma^{2}_{i=1\dots K}\}\\ \mu_{i=1\dots K}&=&\,\text{mean of component }i\\ \sigma^{2}_{i=1\dots K}&=&\,\text{variance of component }i\\ z_{i=1\dots N}&\sim&\operatorname{Categorical}(symbol\phi)\\ x_{i=1\dots N}&\sim&\mathcal{N}(\mu_{z_{i}},\sigma^{2}_{z_{i}})\end{array}
  4. K , N = as above ϕ i = 1 K , s y m b o l ϕ = as above z i = 1 N , x i = 1 N = as above θ i = 1 K = { μ i = 1 K , σ i = 1 K 2 } μ i = 1 K = mean of component i σ i = 1 K 2 = variance of component i μ 0 , λ , ν , σ 0 2 = shared hyperparameters μ i = 1 K 𝒩 ( μ 0 , λ σ i 2 ) σ i = 1 K 2 Inverse - Gamma ( ν , σ 0 2 ) s y m b o l ϕ Symmetric - Dirichlet K ( β ) z i = 1 N Categorical ( s y m b o l ϕ ) x i = 1 N 𝒩 ( μ z i , σ z i 2 ) \begin{array}[]{lcl}K,N&=&\,\text{as above}\\ \phi_{i=1\dots K},symbol\phi&=&\,\text{as above}\\ z_{i=1\dots N},x_{i=1\dots N}&=&\,\text{as above}\\ \theta_{i=1\dots K}&=&\{\mu_{i=1\dots K},\sigma^{2}_{i=1\dots K}\}\\ \mu_{i=1\dots K}&=&\,\text{mean of component }i\\ \sigma^{2}_{i=1\dots K}&=&\,\text{variance of component }i\\ \mu_{0},\lambda,\nu,\sigma_{0}^{2}&=&\,\text{shared hyperparameters}\\ \mu_{i=1\dots K}&\sim&\mathcal{N}(\mu_{0},\lambda\sigma_{i}^{2})\\ \sigma_{i=1\dots K}^{2}&\sim&\operatorname{Inverse-Gamma}(\nu,\sigma_{0}^{2})% \\ symbol\phi&\sim&\operatorname{Symmetric-Dirichlet}_{K}(\beta)\\ z_{i=1\dots N}&\sim&\operatorname{Categorical}(symbol\phi)\\ x_{i=1\dots N}&\sim&\mathcal{N}(\mu_{z_{i}},\sigma^{2}_{z_{i}})\end{array}
  5. s y m b o l x symbol{x}
  6. p ( s y m b o l θ ) = i = 1 K ϕ i 𝒩 ( s y m b o l μ i , Σ i ) p(symbol{\theta})=\sum_{i=1}^{K}\phi_{i}\mathcal{N}(symbol{\mu_{i},\Sigma_{i}})
  7. ϕ i \phi_{i}
  8. s y m b o l μ i symbol{\mu_{i}}
  9. s y m b o l Σ i symbol{\Sigma_{i}}
  10. p ( s y m b o l x | θ ) p(symbol{x|\theta})
  11. s y m b o l x symbol{x}
  12. s y m b o l θ symbol{\theta}
  13. p ( s y m b o l θ | x ) p(symbol{\theta|x})
  14. p ( s y m b o l θ | x ) = i = 1 K ϕ i ~ 𝒩 ( s y m b o l μ i ~ , Σ i ~ ) p(symbol{\theta|x})=\sum_{i=1}^{K}\tilde{\phi_{i}}\mathcal{N}(symbol{\tilde{% \mu_{i}},\tilde{\Sigma_{i}}})
  15. ϕ i ~ , s y m b o l μ i ~ \tilde{\phi_{i}},symbol{\tilde{\mu_{i}}}
  16. s y m b o l Σ i ~ symbol{\tilde{\Sigma_{i}}}
  17. s y m b o l θ symbol{\theta}
  18. s y m b o l Σ i symbol{\Sigma_{i}}
  19. K , N : K,N:
  20. ϕ i = 1 K , s y m b o l ϕ : \phi_{i=1\dots K},symbol\phi:
  21. z i = 1 N , x i = 1 N : z_{i=1\dots N},x_{i=1\dots N}:
  22. V : V:
  23. θ i = 1 K , j = 1 V : \theta_{i=1\dots K,j=1\dots V}:
  24. i i
  25. j j
  26. s y m b o l θ i = 1 K : symbol\theta_{i=1\dots K}:
  27. V , V,
  28. θ i , 1 V ; \theta_{i,1\dots V};
  29. z i = 1 N Categorical ( s y m b o l ϕ ) x i = 1 N Categorical ( s y m b o l θ z i ) \begin{array}[]{lcl}z_{i=1\dots N}&\sim&\operatorname{Categorical}(symbol\phi)% \\ x_{i=1\dots N}&\sim&\,\text{Categorical}(symbol\theta_{z_{i}})\end{array}
  30. K , N : K,N:
  31. ϕ i = 1 K , s y m b o l ϕ : \phi_{i=1\dots K},symbol\phi:
  32. z i = 1 N , x i = 1 N : z_{i=1\dots N},x_{i=1\dots N}:
  33. V : V:
  34. θ i = 1 K , j = 1 V : \theta_{i=1\dots K,j=1\dots V}:
  35. i i
  36. j j
  37. s y m b o l θ i = 1 K : symbol\theta_{i=1\dots K}:
  38. V , V,
  39. θ i , 1 V ; \theta_{i,1\dots V};
  40. α : \alpha:
  41. s y m b o l θ symbol\theta
  42. β : \beta:
  43. s y m b o l ϕ symbol\phi
  44. s y m b o l ϕ Symmetric - Dirichlet K ( β ) s y m b o l θ i = 1 K Symmetric-Dirichlet V ( α ) z i = 1 N Categorical ( s y m b o l ϕ ) x i = 1 N Categorical ( s y m b o l θ z i ) \begin{array}[]{lcl}symbol\phi&\sim&\operatorname{Symmetric-Dirichlet}_{K}(% \beta)\\ symbol\theta_{i=1\dots K}&\sim&\,\text{Symmetric-Dirichlet}_{V}(\alpha)\\ z_{i=1\dots N}&\sim&\operatorname{Categorical}(symbol\phi)\\ x_{i=1\dots N}&\sim&\,\text{Categorical}(symbol\theta_{z_{i}})\end{array}
  45. K = 10 K=10
  46. N 2 N^{2}
  47. p 0 = π ( 1 - θ 1 ) 2 + ( 1 - π ) ( 1 - θ 2 ) 2 p_{0}=\pi(1-\theta_{1})^{2}+(1-\pi)(1-\theta_{2})^{2}
  48. p 1 = 2 π θ 1 ( 1 - θ 1 ) + 2 ( 1 - π ) θ 2 ( 1 - θ 2 ) p_{1}=2\pi\theta_{1}(1-\theta_{1})+2(1-\pi)\theta_{2}(1-\theta_{2})
  49. J = { f ( ; θ ) : θ Ω } J=\{f(\cdot;\theta):\theta\in\Omega\}
  50. K = { p ( ) : p ( ) = i = 1 n a i f i ( ; θ i ) , a i > 0 , i = 1 n a i = 1 , f i ( ; θ i ) J i , n } K=\left\{p(\cdot):p(\cdot)=\sum_{i=1}^{n}a_{i}f_{i}(\cdot;\theta_{i}),a_{i}>0,% \sum_{i=1}^{n}a_{i}=1,f_{i}(\cdot;\theta_{i})\in J\ \forall i,n\right\}
  51. w s ( j + 1 ) = 1 N t = 1 N h s ( j ) ( t ) w_{s}^{(j+1)}=\frac{1}{N}\sum_{t=1}^{N}h_{s}^{(j)}(t)
  52. μ s ( j + 1 ) = t = 1 N h s ( j ) ( t ) x ( t ) t = 1 N h s ( j ) ( t ) \mu_{s}^{(j+1)}=\frac{\sum_{t=1}^{N}h_{s}^{(j)}(t)x^{(t)}}{\sum_{t=1}^{N}h_{s}% ^{(j)}(t)}
  53. Σ s ( j + 1 ) = t = 1 N h s ( j ) ( t ) [ x ( t ) - μ s ( j + 1 ) ] [ x ( t ) - μ s ( j + 1 ) ] t = 1 N h s ( j ) ( t ) \Sigma_{s}^{(j+1)}=\frac{\sum_{t=1}^{N}h_{s}^{(j)}(t)[x^{(t)}-\mu_{s}^{(j+1)}]% [x^{(t)}-\mu_{s}^{(j+1)}]^{\top}}{\sum_{t=1}^{N}h_{s}^{(j)}(t)}
  54. h s ( j ) ( t ) = w s ( j ) p s ( x ( t ) ; μ s ( j ) , Σ s ( j ) ) i = 1 n w i ( j ) p i ( x ( t ) ; μ i ( j ) , Σ i ( j ) ) . h_{s}^{(j)}(t)=\frac{w_{s}^{(j)}p_{s}(x^{(t)};\mu_{s}^{(j)},\Sigma_{s}^{(j)})}% {\sum_{i=1}^{n}w_{i}^{(j)}p_{i}(x^{(t)};\mu_{i}^{(j)},\Sigma_{i}^{(j)})}.
  55. y i , j = a i f Y ( x j ; θ i ) f X ( x j ) . y_{i,j}=\frac{a_{i}f_{Y}(x_{j};\theta_{i})}{f_{X}(x_{j})}.
  56. a i = 1 N j = 1 N y i , j a_{i}=\frac{1}{N}\sum_{j=1}^{N}y_{i,j}
  57. μ i = j y i , j x j j y i , j . \mu_{i}=\frac{\sum_{j}y_{i,j}x_{j}}{\sum_{j}y_{i,j}}.

Mod_n_cryptanalysis.html

  1. X 1 = { 2 X , if X < 2 31 2 X + 1 - 2 32 , if X 2 31 X\lll 1=\left\{\begin{matrix}2X,&\mbox{if }~{}X<2^{31}\\ 2X+1-2^{32},&\mbox{if }~{}X\geq 2^{31}\end{matrix}\right.
  2. 2 32 1 ( mod 3 ) , 2^{32}\equiv 1\;\;(\mathop{{\rm mod}}3),\,
  3. X 1 2 X ( mod 3 ) . X\lll 1\equiv 2X\;\;(\mathop{{\rm mod}}3).

Model_predictive_control.html

  1. t t
  2. [ t , t + T ] [t,t+T]
  3. t + T t+T
  4. J = i = 1 N w x i ( r i - x i ) 2 + i = 1 N w u i Δ u i 2 J=\sum_{i=1}^{N}w_{x_{i}}(r_{i}-x_{i})^{2}+\sum_{i=1}^{N}w_{u_{i}}{\Delta u_{i% }}^{2}
  5. x i x_{i}
  6. r i r_{i}
  7. u i u_{i}
  8. w x i w_{x_{i}}
  9. x i x_{i}
  10. w u i w_{u_{i}}
  11. u i u_{i}

Modern_portfolio_theory.html

  1. E ( R p ) = i w i E ( R i ) \operatorname{E}(R_{p})=\sum_{i}w_{i}\operatorname{E}(R_{i})\quad
  2. R p R_{p}
  3. R i R_{i}
  4. w i w_{i}
  5. i i
  6. σ p 2 = i w i 2 σ i 2 + i j i w i w j σ i σ j ρ i j , \sigma_{p}^{2}=\sum_{i}w_{i}^{2}\sigma_{i}^{2}+\sum_{i}\sum_{j\neq i}w_{i}w_{j% }\sigma_{i}\sigma_{j}\rho_{ij},
  7. ρ i j \rho_{ij}
  8. σ p 2 = i j w i w j σ i σ j ρ i j \sigma_{p}^{2}=\sum_{i}\sum_{j}w_{i}w_{j}\sigma_{i}\sigma_{j}\rho_{ij}
  9. ρ i j = 1 \rho_{ij}=1
  10. σ p = σ p 2 \sigma_{p}=\sqrt{\sigma_{p}^{2}}
  11. E ( R p ) = w A E ( R A ) + w B E ( R B ) = w A E ( R A ) + ( 1 - w A ) E ( R B ) . \operatorname{E}(R_{p})=w_{A}\operatorname{E}(R_{A})+w_{B}\operatorname{E}(R_{% B})=w_{A}\operatorname{E}(R_{A})+(1-w_{A})\operatorname{E}(R_{B}).
  12. σ p 2 = w A 2 σ A 2 + w B 2 σ B 2 + 2 w A w B σ A σ B ρ A B \sigma_{p}^{2}=w_{A}^{2}\sigma_{A}^{2}+w_{B}^{2}\sigma_{B}^{2}+2w_{A}w_{B}% \sigma_{A}\sigma_{B}\rho_{AB}
  13. E ( R p ) = w A E ( R A ) + w B E ( R B ) + w C E ( R C ) \operatorname{E}(R_{p})=w_{A}\operatorname{E}(R_{A})+w_{B}\operatorname{E}(R_{% B})+w_{C}\operatorname{E}(R_{C})
  14. σ p 2 = w A 2 σ A 2 + w B 2 σ B 2 + w C 2 σ C 2 + 2 w A w B σ A σ B ρ A B + 2 w A w C σ A σ C ρ A C + 2 w B w C σ B σ C ρ B C \sigma_{p}^{2}=w_{A}^{2}\sigma_{A}^{2}+w_{B}^{2}\sigma_{B}^{2}+w_{C}^{2}\sigma% _{C}^{2}+2w_{A}w_{B}\sigma_{A}\sigma_{B}\rho_{AB}+2w_{A}w_{C}\sigma_{A}\sigma_% {C}\rho_{AC}+2w_{B}w_{C}\sigma_{B}\sigma_{C}\rho_{BC}
  15. - 1 ρ i j < 1 -1\leq\rho_{ij}<1
  16. q [ 0 , ) q\in[0,\infty)
  17. w T Σ w - q * R T w w^{T}\Sigma w-q*R^{T}w
  18. w w
  19. i w i = 1. \sum_{i}w_{i}=1.
  20. Σ \Sigma
  21. q 0 q\geq 0
  22. \infty
  23. R R
  24. w T Σ w w^{T}\Sigma w
  25. R T w R^{T}w
  26. R T w . R^{T}w.
  27. w T Σ w w^{T}\Sigma w
  28. R T w = μ R^{T}w=\mu
  29. μ \mu
  30. E ( R C ) = R F + σ C E ( R P ) - R F σ P . E(R_{C})=R_{F}+\sigma_{C}\frac{E(R_{P})-R_{F}}{\sigma_{P}}.
  31. E ( R i ) = R f + β i ( E ( R m ) - R f ) \operatorname{E}(R_{i})=R_{f}+\beta_{i}(\operatorname{E}(R_{m})-R_{f})
  32. β \beta
  33. ( E ( R m ) - R f ) (\operatorname{E}(R_{m})-R_{f})
  34. ( w m 2 σ m 2 + [ w a 2 σ a 2 + 2 w m w a ρ a m σ a σ m ] ) (w_{m}^{2}\sigma_{m}^{2}+[w_{a}^{2}\sigma_{a}^{2}+2w_{m}w_{a}\rho_{am}\sigma_{% a}\sigma_{m}])
  35. [ w a 2 σ a 2 + 2 w m w a ρ a m σ a σ m ] [w_{a}^{2}\sigma_{a}^{2}+2w_{m}w_{a}\rho_{am}\sigma_{a}\sigma_{m}]
  36. w a 2 0 w_{a}^{2}\approx 0
  37. [ 2 w m w a ρ a m σ a σ m ] [2w_{m}w_{a}\rho_{am}\sigma_{a}\sigma_{m}]\quad
  38. ( w m E ( R m ) + [ w a E ( R a ) ] ) (w_{m}\operatorname{E}(R_{m})+[w_{a}\operatorname{E}(R_{a})])
  39. [ w a E ( R a ) ] [w_{a}\operatorname{E}(R_{a})]
  40. R f R_{f}
  41. E ( R a ) > R f \operatorname{E}(R_{a})>R_{f}
  42. [ w a ( E ( R a ) - R f ) ] / [ 2 w m w a ρ a m σ a σ m ] = [ w a ( E ( R m ) - R f ) ] / [ 2 w m w a σ m σ m ] [w_{a}(\operatorname{E}(R_{a})-R_{f})]/[2w_{m}w_{a}\rho_{am}\sigma_{a}\sigma_{% m}]=[w_{a}(\operatorname{E}(R_{m})-R_{f})]/[2w_{m}w_{a}\sigma_{m}\sigma_{m}]
  43. [ E ( R a ) ] = R f + [ E ( R m ) - R f ] * [ ρ a m σ a σ m ] / [ σ m σ m ] [\operatorname{E}(R_{a})]=R_{f}+[\operatorname{E}(R_{m})-R_{f}]*[\rho_{am}% \sigma_{a}\sigma_{m}]/[\sigma_{m}\sigma_{m}]
  44. [ E ( R a ) ] = R f + [ E ( R m ) - R f ] * [ σ a m ] / [ σ m m ] [\operatorname{E}(R_{a})]=R_{f}+[\operatorname{E}(R_{m})-R_{f}]*[\sigma_{am}]/% [\sigma_{mm}]
  45. [ σ a m ] / [ σ m m ] [\sigma_{am}]/[\sigma_{mm}]\quad
  46. β \beta
  47. SCL : R i , t - R f = α i + β i ( R M , t - R f ) + ϵ i , t \mathrm{SCL}:R_{i,t}-R_{f}=\alpha_{i}+\beta_{i}\,(R_{M,t}-R_{f})+\epsilon_{i,t% }\frac{}{}
  48. E ( R i ) E(R_{i})

Modular_curve.html

  1. Γ ( N ) = { ( a b c d ) : a d ± 1 mod N and b , c 0 mod N } . \Gamma(N)=\left\{\begin{pmatrix}a&b\\ c&d\\ \end{pmatrix}:\ a\equiv d\equiv\pm 1\mod N\,\text{ and }b,c\equiv 0\mod N% \right\}.
  2. { } { τ 𝐇 Im ( τ ) > r } \{\infty\}\cup\{\tau\in\mathbf{H}\mid\,\text{Im}(\tau)>r\}
  3. { } { τ 𝐇 Im ( τ ) > r } \{\infty\}\cup\{\tau\in\mathbf{H}\mid\,\text{Im}(\tau)>r\}
  4. ( a - m c n ) \begin{pmatrix}a&-m\\ c&n\end{pmatrix}
  5. { ( a b c d ) : c 0 mod N } , \left\{\begin{pmatrix}a&b\\ c&d\end{pmatrix}:\ c\equiv 0\mod N\right\},
  6. { ( a b c d ) : a d 1 mod N , c 0 mod N } . \left\{\begin{pmatrix}a&b\\ c&d\end{pmatrix}:\ a\equiv d\equiv 1\mod N,c\equiv 0\mod N\right\}.
  7. - π χ ( X ( p ) ) = | G | D , -\pi\chi(X(p))=|G|\cdot D,
  8. g = 1 24 ( p + 2 ) ( p - 3 ) ( p - 5 ) . g=\tfrac{1}{24}(p+2)(p-3)(p-5).

Modular_exponentiation.html

  1. c b e d - e ( mod m ) c\equiv b^{e}\equiv d^{-e}\;\;(\mathop{{\rm mod}}m)
  2. b d 1 ( mod m ) b\cdot d\equiv 1\;\;(\mathop{{\rm mod}}m)
  3. c 4 13 ( mod 497 ) c\equiv 4^{13}\;\;(\mathop{{\rm mod}}497)
  4. c ( mod m ) = ( a b ) ( mod m ) c\ (\mbox{mod}~{}\ m)=(a\cdot b)\;\;(\mathop{{\rm mod}}m)
  5. c ( mod m ) = ( a ( mod m ) ( b ( mod m ) ) ) c\ (\mbox{mod}~{}\ m)=(a\ (\mbox{mod}~{}\ m)\cdot(b\ (\mbox{mod}~{}\ m)))
  6. c = ( b c ) mod m c=(b\cdot c)\bmod{m}
  7. c b e ( mod m ) c\equiv b^{e}\;\;(\mathop{{\rm mod}}m)
  8. c b e ( mod m ) c\equiv b^{e^{\prime}}\;\;(\mathop{{\rm mod}}m)
  9. e = i = 0 n - 1 a i 2 i e=\sum_{i=0}^{n-1}a_{i}2^{i}
  10. b e = b ( i = 0 n - 1 a i 2 i ) = i = 0 n - 1 ( b 2 i ) a i b^{e}=b^{\left(\sum_{i=0}^{n-1}a_{i}2^{i}\right)}=\prod_{i=0}^{n-1}\left(b^{2^% {i}}\right)^{a_{i}}
  11. c i = 0 n - 1 ( b 2 i ) a i ( mod m ) c\equiv\prod_{i=0}^{n-1}\left(b^{2^{i}}\right)^{a_{i}}\ (\mbox{mod}~{}\ m)
  12. b b
  13. b 2 i ( mod m ) b^{2^{i}}\ (\mbox{mod}~{}\ m)
  14. i = 0 n - 1 ( b 2 i ) a i ( mod m ) \prod_{i=0}^{n-1}\left(b^{2^{i}}\right)^{a_{i}}\ (\mbox{mod}~{}\ m)
  15. b 2 i ( mod m ) b^{2^{i}}\ (\mbox{mod}~{}\ m)

Modular_representation_theory.html

  1. [ 1 0 0 - 1 ] . \begin{bmatrix}1&0\\ 0&-1\end{bmatrix}.
  2. [ 1 1 0 1 ] . \begin{bmatrix}1&1\\ 0&1\end{bmatrix}.
  3. D C G ( D ) DC_{G}(D)
  4. C G ( D ) C_{G}(D)

Modulatory_space.html

  1. p = 69 + 12 log 2 ( f / 440 ) p=69+12\log_{2}{(f/440)}
  2. 12 \mathbb{Z}_{12}
  3. || ( a , b ) || = a 2 + a b + b 2 . ||(a,b)||=\sqrt{a^{2}+ab+b^{2}}.
  4. || ( a , b , c ) || = a 2 + b 2 + c 2 + a b + b c + c a . ||(a,b,c)||=\sqrt{a^{2}+b^{2}+c^{2}+ab+bc+ca}.

Modulus_of_continuity.html

  1. | f ( x ) - f ( y ) | ω ( | x - y | ) , |f(x)-f(y)|\leq\omega(|x-y|),
  2. d Y ( f ( x ) , f ( x ) ) d X ( x , x ) \frac{d_{Y}(f(x),f(x^{\prime}))}{d_{X}(x,x^{\prime})}
  3. lim t 0 ω ( t ) = ω ( 0 ) = 0. \lim_{t\to 0}\omega(t)=\omega(0)=0.
  4. x X : d Y ( f ( x ) , f ( x ) ) ω ( d X ( x , x ) ) . \forall x^{\prime}\in X:d_{Y}(f(x),f(x^{\prime}))\leq\omega(d_{X}(x,x^{\prime}% )).
  5. x , x X : d Y ( f ( x ) , f ( x ) ) ω ( d X ( x , x ) ) . \forall x,x^{\prime}\in X:d_{Y}(f(x),f(x^{\prime}))\leq\omega(d_{X}(x,x^{% \prime})).
  6. g f : X Z g\circ f:X\to Z
  7. ω 2 ω 1 \omega_{2}\circ\omega_{1}
  8. g ω 1 + f ω 2 \|g\|_{\infty}\omega_{1}+\|f\|_{\infty}\omega_{2}
  9. { f λ } λ Λ \{f_{\lambda}\}_{\lambda\in\Lambda}
  10. inf λ Λ f λ \inf_{\lambda\in\Lambda}f_{\lambda}
  11. sup λ Λ f λ \sup_{\lambda\in\Lambda}f_{\lambda}
  12. ω 1 ( t ) := sup s t ω ( s ) \omega_{1}(t):=\sup_{s\leq t}\omega(s)
  13. ω 2 ( t ) := 1 t t 2 t ω 1 ( s ) d s \omega_{2}(t):=\frac{1}{t}\int_{t}^{2t}\omega_{1}(s)ds
  14. ω f ( t ) := sup { d Y ( f ( x ) , f ( x ) ) : x X , x X , d X ( x , x ) = t } , t 0. \omega_{f}(t):=\sup\{d_{Y}(f(x),f(x^{\prime})):x\in X,x^{\prime}\in X,d_{X}(x,% x^{\prime})=t\},\quad\forall t\geq 0.
  15. ω f ( t ; x ) := sup { d Y ( f ( x ) , f ( x ) ) : x X , d X ( x , x ) = t } , t 0. \omega_{f}(t;x):=\sup\{d_{Y}(f(x),f(x^{\prime})):x^{\prime}\in X,d_{X}(x,x^{% \prime})=t\},\quad\forall t\geq 0.
  16. ω ~ \tilde{\omega}
  17. ω ( t ) ω ~ ( t ) \omega(t)\leq\tilde{\omega}(t)
  18. ω f ( s + t ) \displaystyle\omega_{f}(s+t)
  19. d Y ( f ( x ) , f ( x ) ) / d X ( x , x ) d_{Y}(f(x),f(x^{\prime}))/d_{X}(x,x^{\prime})
  20. f * ( x ) \displaystyle f_{*}(x)
  21. ω ( t ) = inf { a t + b : a > 0 , b > 0 , x X , x X | f ( x ) - f ( x ) | a | x - x | + b } . \omega(t)=\inf\big\{at+b\,:\,a>0,\,b>0,\,\forall x\in X,\,\forall x^{\prime}% \in X\,\,|f(x)-f(x^{\prime})|\leq a|x-x^{\prime}|+b\big\}.
  22. δ ( s ) := inf { f - u , X : u Lip s } + . \delta(s):=\inf\big\{\|f-u\|_{\infty,X}\,:\,u\in\mathrm{Lip}_{s}\big\}\leq+\infty.
  23. 2 δ ( s ) = sup t 0 { ω ( t ) - s t } , 2\delta(s)=\sup_{t\geq 0}\left\{\omega(t)-st\right\},
  24. ω ( t ) = inf s 0 { 2 δ ( s ) + s t } . \omega(t)=\inf_{s\geq 0}\left\{2\delta(s)+st\right\}.
  25. f s := δ ( s ) + inf y X { f ( y ) + s d ( x , y ) } , for s dom ( δ ) : f_{s}:=\delta(s)+\inf_{y\in X}\{f(y)+sd(x,y)\},\quad\mathrm{for}\ s\in\mathrm{% dom}(\delta):
  26. f - f s , X = δ ( s ) ; \|f-f_{s}\|_{\infty,X}=\delta(s);
  27. O ( s - α 1 - α ) , O(s^{-\frac{\alpha}{1-\alpha}}),
  28. O ( e - a s ) . O(e^{-as}).
  29. | P | := max 0 i < n ( t i + 1 - t i ) \scriptstyle|P|:=\max_{0\leq i<n}(t_{i+1}-t_{i})\quad
  30. S * ( f ; P ) - S * ( f ; P ) ( b - a ) ω ( | P | ) . S^{*}(f;P)-S_{*}(f;P)\leq(b-a)\omega(|P|).
  31. τ v + h f - τ v f p = o ( 1 ) , \|\tau_{v+h}f-\tau_{v}f\|_{p}=o(1),
  32. τ h f - f p ω ( h ) . \|\tau_{h}f-f\|_{p}\leq\omega(h).
  33. ω f ( δ ) = ω ( f , δ ) = sup x ; | h | < δ ; | Δ h ( f , x ) | . \omega_{f}(\delta)=\omega(f,\delta)=\sup\limits_{x;|h|<\delta;}\left|\Delta_{h% }(f,x)\right|.
  34. ω n ( f , δ ) = sup x ; | h | < δ ; | Δ h n ( f , x ) | . \omega_{n}(f,\delta)=\sup\limits_{x;|h|<\delta;}\left|\Delta^{n}_{h}(f,x)% \right|.

Mohr–Coulomb_theory.html

  1. τ = σ tan ( ϕ ) + c \tau=\sigma~{}\tan(\phi)+c
  2. τ \tau
  3. σ \sigma
  4. c c
  5. τ \tau
  6. ϕ \phi
  7. c c
  8. ϕ \phi
  9. σ \sigma
  10. - σ -\sigma
  11. ϕ = 0 \phi=0
  12. ϕ = 90 \phi=90^{\circ}
  13. ϕ \phi
  14. σ = σ m - τ m sin ϕ ; τ = τ m cos ϕ \sigma=\sigma_{m}-\tau_{m}\sin\phi~{};~{}~{}\tau=\tau_{m}\cos\phi
  15. τ m = σ 1 - σ 3 2 ; σ m = σ 1 + σ 3 2 \tau_{m}=\cfrac{\sigma_{1}-\sigma_{3}}{2}~{};~{}~{}\sigma_{m}=\cfrac{\sigma_{1% }+\sigma_{3}}{2}
  16. σ 1 \sigma_{1}
  17. σ 3 \sigma_{3}
  18. τ m = σ m sin ϕ + c cos ϕ . \tau_{m}=\sigma_{m}\sin\phi+c\cos\phi~{}.
  19. σ 2 \sigma_{2}
  20. { ± σ 1 - σ 2 2 = [ σ 1 + σ 2 2 ] sin ( ϕ ) + c cos ( ϕ ) ± σ 2 - σ 3 2 = [ σ 2 + σ 3 2 ] sin ( ϕ ) + c cos ( ϕ ) ± σ 3 - σ 1 2 = [ σ 3 + σ 1 2 ] sin ( ϕ ) + c cos ( ϕ ) . \left\{\begin{aligned}\displaystyle\pm\cfrac{\sigma_{1}-\sigma_{2}}{2}&% \displaystyle=\left[\cfrac{\sigma_{1}+\sigma_{2}}{2}\right]\sin(\phi)+c\cos(% \phi)\\ \displaystyle\pm\cfrac{\sigma_{2}-\sigma_{3}}{2}&\displaystyle=\left[\cfrac{% \sigma_{2}+\sigma_{3}}{2}\right]\sin(\phi)+c\cos(\phi)\\ \displaystyle\pm\cfrac{\sigma_{3}-\sigma_{1}}{2}&\displaystyle=\left[\cfrac{% \sigma_{3}+\sigma_{1}}{2}\right]\sin(\phi)+c\cos(\phi).\end{aligned}\right.
  21. τ \tau
  22. σ \sigma
  23. 𝐧 = n 1 𝐞 1 + n 2 𝐞 2 + n 3 𝐞 3 \mathbf{n}=n_{1}~{}\mathbf{e}_{1}+n_{2}~{}\mathbf{e}_{2}+n_{3}~{}\mathbf{e}_{3}
  24. 𝐞 i , i = 1 , 2 , 3 \mathbf{e}_{i},~{}~{}i=1,2,3
  25. σ 1 , σ 2 , σ 3 \sigma_{1},\sigma_{2},\sigma_{3}
  26. 𝐞 1 , 𝐞 2 , 𝐞 3 \mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}
  27. σ , τ \sigma,\tau
  28. σ = n 1 2 σ 1 + n 2 2 σ 2 + n 3 2 σ 3 τ = ( n 1 σ 1 ) 2 + ( n 2 σ 2 ) 2 + ( n 3 σ 3 ) 2 - σ 2 = n 1 2 n 2 2 ( σ 1 - σ 2 ) 2 + n 2 2 n 3 2 ( σ 2 - σ 3 ) 2 + n 3 2 n 1 2 ( σ 3 - σ 1 ) 2 . \begin{aligned}\displaystyle\sigma&\displaystyle=n_{1}^{2}\sigma_{1}+n_{2}^{2}% \sigma_{2}+n_{3}^{2}\sigma_{3}\\ \displaystyle\tau&\displaystyle=\sqrt{(n_{1}\sigma_{1})^{2}+(n_{2}\sigma_{2})^% {2}+(n_{3}\sigma_{3})^{2}-\sigma^{2}}\\ &\displaystyle=\sqrt{n_{1}^{2}n_{2}^{2}(\sigma_{1}-\sigma_{2})^{2}+n_{2}^{2}n_% {3}^{2}(\sigma_{2}-\sigma_{3})^{2}+n_{3}^{2}n_{1}^{2}(\sigma_{3}-\sigma_{1})^{% 2}}.\end{aligned}
  29. τ = σ tan ( ϕ ) + c \tau=\sigma~{}\tan(\phi)+c
  30. 𝐧 = n 1 𝐞 1 + n 2 𝐞 2 + n 3 𝐞 3 \mathbf{n}=n_{1}~{}\mathbf{e}_{1}+n_{2}~{}\mathbf{e}_{2}+n_{3}~{}\mathbf{e}_{3}
  31. 𝐞 i , i = 1 , 2 , 3 \mathbf{e}_{i},~{}~{}i=1,2,3
  32. 𝐭 = n i σ i j 𝐞 j (repeated indices indicate summation) \mathbf{t}=n_{i}~{}\sigma_{ij}~{}\mathbf{e}_{j}~{}~{}~{}\,\text{(repeated % indices indicate summation)}
  33. | 𝐭 | = ( n j σ 1 j ) 2 + ( n k σ 2 k ) 2 + ( n l σ 3 l ) 2 (repeated indices indicate summation) |\mathbf{t}|=\sqrt{(n_{j}~{}\sigma_{1j})^{2}+(n_{k}~{}\sigma_{2k})^{2}+(n_{l}~% {}\sigma_{3l})^{2}}~{}~{}~{}\,\text{(repeated indices indicate summation)}
  34. σ = 𝐭 𝐧 = n i σ i j n j (repeated indices indicate summation) \sigma=\mathbf{t}\cdot\mathbf{n}=n_{i}~{}\sigma_{ij}~{}n_{j}~{}~{}\,\text{(% repeated indices indicate summation)}
  35. τ = | 𝐭 | 2 - σ 2 \tau=\sqrt{|\mathbf{t}|^{2}-\sigma^{2}}
  36. σ = n 1 2 σ 11 + n 2 2 σ 22 + n 3 2 σ 33 + 2 ( n 1 n 2 σ 12 + n 2 n 3 σ 23 + n 3 n 1 σ 31 ) τ = ( n 1 σ 11 + n 2 σ 12 + n 3 σ 31 ) 2 + ( n 1 σ 12 + n 2 σ 22 + n 3 σ 23 ) 2 + ( n 1 σ 31 + n 2 σ 23 + n 3 σ 33 ) 2 - σ 2 \begin{aligned}\displaystyle\sigma&\displaystyle=n_{1}^{2}\sigma_{11}+n_{2}^{2% }\sigma_{22}+n_{3}^{2}\sigma_{33}+2(n_{1}n_{2}\sigma_{12}+n_{2}n_{3}\sigma_{23% }+n_{3}n_{1}\sigma_{31})\\ \displaystyle\tau&\displaystyle=\sqrt{(n_{1}\sigma_{11}+n_{2}\sigma_{12}+n_{3}% \sigma_{31})^{2}+(n_{1}\sigma_{12}+n_{2}\sigma_{22}+n_{3}\sigma_{23})^{2}+(n_{% 1}\sigma_{31}+n_{2}\sigma_{23}+n_{3}\sigma_{33})^{2}-\sigma^{2}}\end{aligned}
  37. σ 1 , σ 2 , σ 3 \sigma_{1},\sigma_{2},\sigma_{3}
  38. 𝐞 1 , 𝐞 2 , 𝐞 3 \mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}
  39. σ , τ \sigma,\tau
  40. σ = n 1 2 σ 1 + n 2 2 σ 2 + n 3 2 σ 3 τ = ( n 1 σ 1 ) 2 + ( n 2 σ 2 ) 2 + ( n 3 σ 3 ) 2 - σ 2 = n 1 2 n 2 2 ( σ 1 - σ 2 ) 2 + n 2 2 n 3 2 ( σ 2 - σ 3 ) 2 + n 3 2 n 1 2 ( σ 3 - σ 1 ) 2 \begin{aligned}\displaystyle\sigma&\displaystyle=n_{1}^{2}\sigma_{1}+n_{2}^{2}% \sigma_{2}+n_{3}^{2}\sigma_{3}\\ \displaystyle\tau&\displaystyle=\sqrt{(n_{1}\sigma_{1})^{2}+(n_{2}\sigma_{2})^% {2}+(n_{3}\sigma_{3})^{2}-\sigma^{2}}\\ &\displaystyle=\sqrt{n_{1}^{2}n_{2}^{2}(\sigma_{1}-\sigma_{2})^{2}+n_{2}^{2}n_% {3}^{2}(\sigma_{2}-\sigma_{3})^{2}+n_{3}^{2}n_{1}^{2}(\sigma_{3}-\sigma_{1})^{% 2}}\end{aligned}
  41. π \pi
  42. c = 2 , ϕ = 20 c=2,\phi=20^{\circ}
  43. σ 1 - σ 2 \sigma_{1}-\sigma_{2}
  44. c = 2 , ϕ = 20 c=2,\phi=20^{\circ}
  45. σ 1 - σ 3 2 = σ 1 + σ 3 2 sin ϕ + c cos ϕ \cfrac{\sigma_{1}-\sigma_{3}}{2}=\cfrac{\sigma_{1}+\sigma_{3}}{2}~{}\sin\phi+c\cos\phi
  46. [ 3 sin ( θ + π 3 ) - sin ϕ cos ( θ + π 3 ) ] ρ - 2 sin ( ϕ ) ξ = 6 c cos ϕ . \left[\sqrt{3}~{}\sin\left(\theta+\cfrac{\pi}{3}\right)-\sin\phi\cos\left(% \theta+\cfrac{\pi}{3}\right)\right]\rho-\sqrt{2}\sin(\phi)\xi=\sqrt{6}c\cos\phi.
  47. p , q , r p,q,r
  48. [ 1 3 cos ϕ sin ( θ + π 3 ) - 1 3 tan ϕ cos ( θ + π 3 ) ] q - p tan ϕ = c \left[\cfrac{1}{\sqrt{3}~{}\cos\phi}~{}\sin\left(\theta+\cfrac{\pi}{3}\right)-% \cfrac{1}{3}\tan\phi~{}\cos\left(\theta+\cfrac{\pi}{3}\right)\right]q-p~{}\tan% \phi=c
  49. θ = 1 3 arccos [ ( r q ) 3 ] . \theta=\cfrac{1}{3}\arccos\left[\left(\cfrac{r}{q}\right)^{3}\right]~{}.
  50. σ 1 - σ 3 2 = σ 1 + σ 3 2 sin ϕ + c cos ϕ \cfrac{\sigma_{1}-\sigma_{3}}{2}=\cfrac{\sigma_{1}+\sigma_{3}}{2}~{}\sin\phi+c\cos\phi
  51. σ 1 ( 1 - sin ϕ ) 2 c cos ϕ - σ 3 ( 1 + sin ϕ ) 2 c cos ϕ = 1 . \sigma_{1}~{}\cfrac{(1-\sin\phi)}{2~{}c~{}\cos\phi}-\sigma_{3}~{}\cfrac{(1+% \sin\phi)}{2~{}c~{}\cos\phi}=1~{}.
  52. σ 1 = 1 3 ξ + 2 3 ρ cos θ ; σ 3 = 1 3 ξ + 2 3 ρ cos ( θ + 2 π 3 ) . \sigma_{1}=\cfrac{1}{\sqrt{3}}~{}\xi+\sqrt{\cfrac{2}{3}}~{}\rho~{}\cos\theta~{% };~{}~{}\sigma_{3}=\cfrac{1}{\sqrt{3}}~{}\xi+\sqrt{\cfrac{2}{3}}~{}\rho~{}\cos% \left(\theta+\cfrac{2\pi}{3}\right)~{}.
  53. - 2 ξ sin ϕ + ρ [ cos θ - cos ( θ + 2 π / 3 ) ] - ρ sin ϕ [ cos θ + cos ( θ + 2 π / 3 ) ] = 6 c cos ϕ -\sqrt{2}~{}\xi~{}\sin\phi+\rho[\cos\theta-\cos(\theta+2\pi/3)]-\rho\sin\phi[% \cos\theta+\cos(\theta+2\pi/3)]=\sqrt{6}~{}c~{}\cos\phi
  54. ξ , ρ , θ \xi,\rho,\theta
  55. p , q p,q
  56. ξ = 3 p ; ρ = 2 3 q \xi=\sqrt{3}~{}p~{};~{}~{}\rho=\sqrt{\cfrac{2}{3}}~{}q
  57. g := ( α c y tan ψ ) 2 + G 2 ( ϕ , θ ) q 2 - p tan ϕ g:=\sqrt{(\alpha c_{\mathrm{y}}\tan\psi)^{2}+G^{2}(\phi,\theta)~{}q^{2}}-p\tan\phi
  58. α \alpha
  59. c y c_{\mathrm{y}}
  60. c c
  61. ψ \psi
  62. p p
  63. G ( ϕ , θ ) G(\phi,\theta)

Molar_refractivity.html

  1. A A
  2. A = 4 π 3 N A α , A=\frac{4\pi}{3}N_{A}\alpha,
  3. N A 6.022 × 10 23 N_{A}\approx 6.022\times 10^{23}
  4. α \alpha
  5. A = R T p n 2 - 1 n 2 + 2 A=\frac{RT}{p}\frac{n^{2}-1}{n^{2}+2}
  6. n 2 1 n^{2}\approx 1
  7. A = R T p n 2 - 1 3 . A=\frac{RT}{p}\frac{n^{2}-1}{3}.
  8. R R
  9. T T
  10. n n
  11. p p
  12. A A
  13. A = M ρ n 2 - 1 n 2 + 2 M ρ n 2 - 1 3 . A=\frac{M}{\rho}\frac{n^{2}-1}{n^{2}+2}\approx\frac{M}{\rho}\frac{n^{2}-1}{3}.

Molecular_geometry.html

  1. β exp ( - Δ E k T ) \beta\equiv\exp\left(-\frac{\Delta E}{kT}\right)
  2. Δ E \Delta E
  3. k k
  4. T T
  5. θ 11 \theta_{11}
  6. θ 22 \theta_{22}
  7. θ 33 \theta_{33}
  8. θ 44 \theta_{44}
  9. 0 = | cos θ 11 cos θ 12 cos θ 13 cos θ 14 cos θ 21 cos θ 22 cos θ 23 cos θ 24 cos θ 31 cos θ 32 cos θ 33 cos θ 34 cos θ 41 cos θ 42 cos θ 43 cos θ 44 | 0=\begin{vmatrix}\cos\theta_{11}&\cos\theta_{12}&\cos\theta_{13}&\cos\theta_{1% 4}\\ \cos\theta_{21}&\cos\theta_{22}&\cos\theta_{23}&\cos\theta_{24}\\ \cos\theta_{31}&\cos\theta_{32}&\cos\theta_{33}&\cos\theta_{34}\\ \cos\theta_{41}&\cos\theta_{42}&\cos\theta_{43}&\cos\theta_{44}\end{vmatrix}

Molecular_modelling.html

  1. E = E bonds + E angle + E dihedral + E non-bonded E=E\text{bonds}+E\text{angle}+E\text{dihedral}+E\text{non-bonded}\,
  2. E non-bonded = E electrostatic + E van der Waals E\text{non-bonded}=E\text{electrostatic}+E\text{van der Waals}\,
  3. 𝐅 = m 𝐚 \mathbf{F}=m\mathbf{a}

Mollweide_projection.html

  1. x = R 2 2 π ( λ - λ 0 ) cos ( θ ) , x=R\frac{2\sqrt{2}}{\pi}\left(\lambda-\lambda_{0}\right)\cos\left(\theta\right),
  2. y = R 2 sin ( θ ) , y=R\sqrt{2}\sin\left(\theta\right),\,
  3. θ \theta\,
  4. 2 θ + sin ( 2 θ ) = π sin ( φ ) ( 1 ) 2\theta+\sin\left(2\theta\right)=\pi\sin\left(\varphi\right)\qquad(1)
  5. θ 0 = φ , \theta_{0}=\varphi,\,
  6. θ n + 1 = θ n - 2 θ n + sin ( 2 θ n ) - π sin ( φ ) 2 + 2 cos ( 2 θ n ) . \theta_{n+1}=\theta_{n}-\frac{2\theta_{n}+\sin\left(2\theta_{n}\right)-\pi\sin% \left(\varphi\right)}{2+2\cos\left(2\theta_{n}\right)}.\,
  7. φ = arcsin [ 2 θ + sin ( 2 θ ) π ] , \varphi=\arcsin\left[\frac{2\theta+\sin\left(2\theta\right)}{\pi}\right],\,
  8. λ = λ 0 + π x 2 R 2 cos θ , \lambda=\lambda_{0}+\frac{\pi x}{2R\sqrt{2}\cos\theta},\,
  9. θ = arcsin ( y R 2 ) . \theta=\arcsin\left(\frac{y}{R\sqrt{2}}\right).\,

Monkey_saddle.html

  1. z = x 3 - 3 x y 2 . z=x^{3}-3xy^{2}.\,
  2. z ( x , y ) = Re ( x + i y ) 3 . z(x,y)=\operatorname{Re}(x+iy)^{3}.

Monomial_basis.html

  1. K x x Kxx
  2. K K
  3. K K
  4. 1 , x , x 2 , x 3 , 1,x,x^{2},x^{3},\ldots
  5. K K
  6. K x x Kxx
  7. d d
  8. 1 , x , x 2 , 1,x,x^{2},\ldots
  9. a 0 + a 1 x + a 2 x 2 + + a d x d , a_{0}+a_{1}x+a_{2}x^{2}+\ldots+a_{d}x^{d},
  10. i = 0 d a i x i . \sum_{i=0}^{d}a_{i}x^{i}.
  11. 1 < x < x 2 < , 1<x<x^{2}<\cdots,
  12. 1 > x > x 2 > . 1>x>x^{2}>\cdots.
  13. x 1 , , x n , x_{1},\ldots,x_{n},
  14. x 1 d 1 x 2 d 2 x n d n , x_{1}^{d_{1}}x_{2}^{d_{2}}\cdots x_{n}^{d_{n}},
  15. d i d_{i}
  16. x i 0 = 1 , x_{i}^{0}=1,
  17. 1 = x 1 0 x 2 0 x n 0 1=x_{1}^{0}x_{2}^{0}\cdots x_{n}^{0}
  18. x 1 , , x n x_{1},\ldots,x_{n}
  19. d d
  20. d = d 1 + + d n d=d_{1}+\cdots+d_{n}
  21. d d
  22. ( d + n - 1 d ) = n ( n + 1 ) ( n + d - 1 ) d ! , {\left({{d+n-1}\atop{d}}\right)}=\frac{n(n+1)\cdots(n+d-1)}{d!},
  23. ( d + n - 1 d ) {\left({{d+n-1}\atop{d}}\right)}
  24. d d
  25. d d
  26. ( d + n d ) = ( d + n n ) = ( d + 1 ) ( d + n ) n ! . {\left({{d+n}\atop{d}}\right)}={\left({{d+n}\atop{n}}\right)}=\frac{(d+1)% \cdots(d+n)}{n!}.
  27. m < n m q < n q m<n\Leftrightarrow mq<nq
  28. 1 m 1\leq m
  29. m , n , q . m,n,q.
  30. Π 4 \Pi_{4}
  31. 1 + x + 3 x 4 1+x+3x^{4}

Monomial_order.html

  1. ( a 1 , , a n ) \R 0 n (a_{1},\ldots,a_{n})\in\R_{\geq 0}^{n}
  2. a i a j \tfrac{a_{i}}{a_{j}}
  3. x y 2 z xy^{2}z
  4. z 2 z^{2}
  5. x 3 x^{3}
  6. x 2 z 2 x^{2}z^{2}
  7. x 3 > x 2 z 2 > x y 2 z > z 2 x^{3}>x^{2}z^{2}>xy^{2}z>z^{2}
  8. x x
  9. x 2 z 2 > x y 2 z > x 3 > z 2 x^{2}z^{2}>xy^{2}z>x^{3}>z^{2}
  10. x x
  11. x y 2 z > x 2 z 2 > x 3 > z 2 xy^{2}z>x^{2}z^{2}>x^{3}>z^{2}
  12. z z
  13. x 2 z 2 > x y 2 z > z 2 > x 3 x^{2}z^{2}>xy^{2}z>z^{2}>x^{3}

Monotonicity_of_entailment.html

  1. \vdash
  2. \vdash

Monstrous_moonshine.html

  1. j ( τ ) = 1 < m t p l > q + 744 + 196884 q + 21493760 q 2 + 864299970 q 3 + 20245856256 q 4 + j(\tau)=\frac{1}{<}mtpl>{{q}}+744+196884{q}+21493760{q}^{2}+864299970{q}^{3}+2% 0245856256{q}^{4}+\cdots
  2. q = e 2 π i τ {q}=e^{2\pi i\tau}
  3. r n r_{n}
  4. 1 = r 1 196884 = r 1 + r 2 21493760 = r 1 + r 2 + r 3 864299970 = 2 r 1 + 2 r 2 + r 3 + r 4 20245856256 = 3 r 1 + 3 r 2 + r 3 + 2 r 4 + r 5 333202640600 = 5 r 1 + 5 r 2 + 2 r 3 + 3 r 4 + 2 r 5 + r 7 \begin{aligned}\displaystyle 1&\displaystyle=r_{1}\\ \displaystyle 196884&\displaystyle=r_{1}+r_{2}\\ \displaystyle 21493760&\displaystyle=r_{1}+r_{2}+r_{3}\\ \displaystyle 864299970&\displaystyle=2r_{1}+2r_{2}+r_{3}+r_{4}\\ \displaystyle 20245856256&\displaystyle=3r_{1}+3r_{2}+r_{3}+2r_{4}+r_{5}\\ \displaystyle 333202640600&\displaystyle=5r_{1}+5r_{2}+2r_{3}+3r_{4}+2r_{5}+r_% {7}\\ \end{aligned}
  5. r 1 - r 3 + r 4 + r 5 - r 6 = 0 r_{1}-r_{3}+r_{4}+r_{5}-r_{6}=0
  6. V V^{\natural}
  7. 𝔪 \mathfrak{m}
  8. 𝔪 \mathfrak{m}
  9. ( a b c d ) S L 2 ( 𝐙 ) \begin{pmatrix}a&b\\ c&d\end{pmatrix}\in SL_{2}(\mathbf{Z})
  10. f ( g , h , a τ + b c τ + d ) f(g,h,\frac{a\tau+b}{c\tau+d})
  11. f ( g a h c , g b h d , τ ) f(g^{a}h^{c},g^{b}h^{d},\tau)
  12. q {q}

Monte_Carlo_integration.html

  1. I = Ω f ( 𝐱 ¯ ) d 𝐱 ¯ I=\int_{\Omega}f(\overline{\mathbf{x}})\,d\overline{\mathbf{x}}
  2. V = Ω d 𝐱 ¯ V=\int_{\Omega}d\overline{\mathbf{x}}
  3. 𝐱 ¯ 1 , , 𝐱 ¯ N Ω , \overline{\mathbf{x}}_{1},\cdots,\overline{\mathbf{x}}_{N}\in\Omega,
  4. I Q N V 1 N i = 1 N f ( 𝐱 ¯ i ) = V f I\approx Q_{N}\equiv V\frac{1}{N}\sum_{i=1}^{N}f(\overline{\mathbf{x}}_{i})=V% \langle f\rangle
  5. lim N Q N = I \lim_{N\to\infty}Q_{N}=I
  6. Var ( f ) σ N 2 = 1 N - 1 i = 1 N ( f ( 𝐱 ¯ i ) - f ) 2 . \mathrm{Var}(f)\equiv\sigma_{N}^{2}=\frac{1}{N-1}\sum_{i=1}^{N}\left(f(% \overline{\mathbf{x}}_{i})-\langle f\rangle\right)^{2}.
  7. Var ( Q N ) = V 2 N 2 i = 1 N Var ( f ) = V 2 Var ( f ) N = V 2 σ N 2 N \mathrm{Var}(Q_{N})=\frac{V^{2}}{N^{2}}\sum_{i=1}^{N}\mathrm{Var}(f)=V^{2}% \frac{\mathrm{Var}(f)}{N}=V^{2}\frac{\sigma_{N}^{2}}{N}
  8. { σ 1 2 , σ 2 2 , σ 3 2 , } \left\{\sigma_{1}^{2},\sigma_{2}^{2},\sigma_{3}^{2},\ldots\right\}
  9. δ Q N Var ( Q N ) = V σ N N , \delta Q_{N}\approx\sqrt{\mathrm{Var}(Q_{N})}=V\frac{\sigma_{N}}{\sqrt{N}},
  10. 1 N \tfrac{1}{\sqrt{N}}
  11. 1 N \tfrac{1}{\sqrt{N}}
  12. H ( x , y ) = { 1 if x 2 + y 2 1 0 else H\left(x,y\right)=\begin{cases}1&\,\text{if }x^{2}+y^{2}\leq 1\\ 0&\,\text{else}\end{cases}
  13. I π = Ω H ( x , y ) d x d y = π . I_{\pi}=\int_{\Omega}H(x,y)dxdy=\pi.
  14. Q N = 4 1 N i = 1 N H ( x i , y i ) Q_{N}=4\frac{1}{N}\sum_{i=1}^{N}H(x_{i},y_{i})
  15. Q N - π π \tfrac{Q_{N}-\pi}{\pi}
  16. 1 N \tfrac{1}{\sqrt{N}}
  17. f ( x ) = 1 1 + sinh ( 2 x ) log ( x ) 2 f(x)=\frac{1}{1+\sinh(2x)\log(x)^{2}}
  18. 0.8 < x < 3 0.8<x<3
  19. f ( x , y ) = { 1 x 2 + y 2 < 1 0 x 2 + y 2 1 f(x,y)=\begin{cases}1&x^{2}+y^{2}<1\\ 0&x^{2}+y^{2}\geq 1\end{cases}
  20. E a ( f ) E_{a}(f)
  21. E b ( f ) E_{b}(f)
  22. σ a 2 ( f ) \sigma_{a}^{2}(f)
  23. σ b 2 ( f ) \sigma_{b}^{2}(f)
  24. E ( f ) = 1 2 ( E a ( f ) + E b ( f ) ) E(f)=\tfrac{1}{2}\left(E_{a}(f)+E_{b}(f)\right)
  25. Var ( f ) = σ a 2 ( f ) 4 N a + σ b 2 ( f ) 4 N b \mathrm{Var}(f)=\frac{\sigma_{a}^{2}(f)}{4N_{a}}+\frac{\sigma_{b}^{2}(f)}{4N_{% b}}
  26. N a N a + N b = σ a σ a + σ b \frac{N_{a}}{N_{a}+N_{b}}=\frac{\sigma_{a}}{\sigma_{a}+\sigma_{b}}
  27. E g ( f ; N ) = E ( f g ; N ) E_{g}(f;N)=E\left(\tfrac{f}{g};N\right)
  28. Var g ( f ; N ) = Var ( f g ; N ) \mathrm{Var}_{g}(f;N)=\mathrm{Var}\left(\tfrac{f}{g};N\right)
  29. g = | f | I ( | f | ) g=\tfrac{|f|}{I(|f|)}
  30. V g ( f ; N ) V_{g}(f;N)
  31. 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})\ldots
  32. 𝐱 ¯ \overline{\mathbf{x}}
  33. p ( 𝐱 ¯ ) p(\overline{\mathbf{x}})
  34. p ( 𝐱 ¯ ) p(\overline{\mathbf{x}})
  35. p ( 𝐱 ¯ ) : 𝐱 ¯ 1 , , 𝐱 ¯ N V , p(\overline{\mathbf{x}}):\qquad\overline{\mathbf{x}}_{1},\cdots,\overline{% \mathbf{x}}_{N}\in V,
  36. Q N 1 N i = 1 N f ( 𝐱 ¯ i ) p ( 𝐱 ¯ i ) Q_{N}\equiv\frac{1}{N}\sum_{i=1}^{N}\frac{f(\overline{\mathbf{x}}_{i})}{p(% \overline{\mathbf{x}}_{i})}
  37. p ( 𝐱 ¯ ) p(\overline{\mathbf{x}})
  38. 𝐱 ¯ \overline{\mathbf{x}}
  39. p ( 𝐱 ¯ ) p(\overline{\mathbf{x}})

Montgomery_modular_multiplication.html

  1. a b mod N . ab\bmod N.
  2. { a + k N : k 𝐙 } , \{a+kN\colon k\in\mathbf{Z}\},
  3. a ¯ \bar{a}
  4. a ¯ b ¯ ( mod N ) . \bar{a}\equiv\bar{b}\;\;(\mathop{{\rm mod}}N).
  5. a ¯ \bar{a}
  6. a b ( mod N ) a\equiv b\;\;(\mathop{{\rm mod}}N)
  7. 7 ¯ \overline{7}
  8. 15 ¯ \overline{15}
  9. 7 ¯ + 15 ¯ 5 ¯ ( mod 17 ) \overline{7}+\overline{15}\equiv\overline{5}\;\;(\mathop{{\rm mod}}17)
  10. a b = q N + r , ab=qN+r,
  11. a b / N \lfloor ab/N\rfloor
  12. 7 ¯ 15 ¯ \overline{7}\cdot\overline{15}
  13. 7 15 = 105 7\cdot 15=105
  14. 105 / 17 = 6 \lfloor 105/17\rfloor=6
  15. 105 - 6 17 = 105 - 102 = 3 105-6\cdot 17=105-102=3
  16. a ¯ \bar{a}
  17. a R ¯ \overline{aR}
  18. a R + b R = ( a + b ) R , aR+bR=(a+b)R,
  19. a R - b R = ( a - b ) R . aR-bR=(a-b)R.
  20. ( a R mod N ) ( b R mod N ) mod N = ( a b R ) R mod N . (aR\bmod N)(bR\bmod N)\bmod N=(abR)R\bmod N.
  21. R R 1 ( mod N ) RR^{\prime}\equiv 1\;\;(\mathop{{\rm mod}}N)
  22. ( a R mod N ) ( b R mod N ) R ( a R ) ( b R ) R - 1 ( a b ) R ( mod N ) . (aR\bmod N)(bR\bmod N)R^{\prime}\equiv(aR)(bR)R^{-1}\equiv(ab)R\;\;(\mathop{{% \rm mod}}N).
  23. R R - N N = 1. RR^{\prime}-NN^{\prime}=1.
  24. N N - 1 ( mod R ) NN^{\prime}\equiv-1\;\;(\mathop{{\rm mod}}R)
  25. S T R - 1 ( mod N ) S\equiv TR^{-1}\;\;(\mathop{{\rm mod}}N)
  26. T + m N T + ( ( ( T mod R ) N ) mod R ) N T + T N N T - T 0 ( mod R ) . T+mN\equiv T+(((T\bmod R)N^{\prime})\bmod R)N\equiv T+TN^{\prime}N\equiv T-T% \equiv 0\;\;(\mathop{{\rm mod}}R).
  27. t ( T + m N ) R - 1 T R - 1 + ( m R - 1 ) N T R - 1 ( mod N ) . t\equiv(T+mN)R^{-1}\equiv TR^{-1}+(mR^{-1})N\equiv TR^{-1}\;\;(\mathop{{\rm mod% }}N).
  28. ( a N ) = ( a R N ) / ( R N ) \textstyle(\frac{a}{N})=(\frac{aR}{N})/(\frac{R}{N})

Morava_K-theory.html

  1. K ( n ) * ( X × Y ) K ( n ) * ( X ) K ( n ) * K ( n ) * ( Y ) . K(n)_{*}(X\times Y)\cong K(n)_{*}(X)\otimes_{K(n)_{*}}K(n)_{*}(Y).

Morley's_trisector_theorem.html

  1. a = b = c = 8 R sin ( A / 3 ) sin ( B / 3 ) sin ( C / 3 ) , a^{{}^{\prime}}=b^{{}^{\prime}}=c^{{}^{\prime}}=8R\sin(A/3)\sin(B/3)\sin(C/3),\,
  2. 3 4 a 2 , \tfrac{\sqrt{3}}{4}a^{\prime 2},
  3. Area = 16 3 R 2 sin 2 ( A / 3 ) sin 2 ( B / 3 ) sin 2 ( C / 3 ) . \,\text{Area}=16\sqrt{3}R^{2}\sin^{2}(A/3)\sin^{2}(B/3)\sin^{2}(C/3).

Morse_potential.html

  1. V ( r ) = D e ( 1 - e - a ( r - r e ) ) 2 V(r)=D_{e}(1-e^{-a(r-r_{e})})^{2}
  2. r r
  3. r e r_{e}
  4. D e D_{e}
  5. a a
  6. a a
  7. E ( 0 ) E(0)
  8. V ( r ) V(r)
  9. r = r e r=r_{e}
  10. a a
  11. a = k e / 2 D e , a=\sqrt{k_{e}/2D_{e}},
  12. k e k_{e}
  13. V ( r ) = D e ( ( 1 - e - a ( r - r e ) ) 2 - 1 ) V(r)=D_{e}((1-e^{-a(r-r_{e})})^{2}-1)
  14. V ( r ) = D e ( e - 2 a ( r - r e ) - 2 e - a ( r - r e ) ) V(r)=D_{e}(e^{-2a(r-r_{e})}-2e^{-a(r-r_{e})})
  15. r r
  16. r r
  17. - D e -D_{e}
  18. r = r e r=r_{e}
  19. Ψ ( v ) \Psi(v)
  20. E ( v ) E(v)
  21. ( - 2 2 m 2 r 2 + V ( r ) ) Ψ ( v ) = E ( v ) Ψ ( v ) , \left(-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial r^{2}}+V(r)\right)\Psi% (v)=E(v)\Psi(v),
  22. x = a r ; x e = a r e ; λ = 2 m D e a ; ε v = 2 m a 2 2 E ( v ) . x=ar\,\text{; }x_{e}=ar_{e}\,\text{; }\lambda=\frac{\sqrt{2mD_{e}}}{a\hbar}\,% \text{; }\varepsilon_{v}=\frac{2m}{a^{2}\hbar^{2}}E(v).
  23. ( - 2 x 2 + V ( x ) ) Ψ n ( x ) = ε n Ψ n ( x ) , \left(-\frac{\partial^{2}}{\partial x^{2}}+V(x)\right)\Psi_{n}(x)=\varepsilon_% {n}\Psi_{n}(x),
  24. V ( x ) = λ 2 ( e - 2 ( x - x e ) - 2 e - ( x - x e ) ) . V(x)=\lambda^{2}\left(e^{-2\left(x-x_{e}\right)}-2e^{-\left(x-x_{e}\right)}% \right).
  25. ε n = 1 - 1 λ 2 ( λ - n - 1 2 ) 2 \varepsilon_{n}=1-\frac{1}{\lambda^{2}}\left(\lambda-n-\frac{1}{2}\right)^{2}
  26. Ψ n ( z ) = N n z λ - n - 1 2 e - 1 2 z L n 2 λ - 2 n - 1 ( z ) , \Psi_{n}(z)=N_{n}z^{\lambda-n-\frac{1}{2}}e^{-\frac{1}{2}z}L_{n}^{2\lambda-2n-% 1}(z),
  27. z = 2 λ e - ( x - x e ) ; N n = [ n ! ( 2 λ - 2 n - 1 ) Γ ( 2 λ - n ) ] 1 2 z=2\lambda e^{-\left(x-x_{e}\right)}\,\text{; }N_{n}=\left[\frac{n!\left(2% \lambda-2n-1\right)}{\Gamma(2\lambda-n)}\right]^{\frac{1}{2}}
  28. L n α ( z ) L_{n}^{\alpha}(z)
  29. L n α ( z ) = z - α e z n ! d n d z n ( z n + α e - z ) = Γ ( α + n + 1 ) / Γ ( α + 1 ) Γ ( n + 1 ) 1 F 1 ( - n , α + 1 , z ) , L_{n}^{\alpha}(z)=\frac{z^{-\alpha}e^{z}}{n!}\frac{d^{n}}{dz^{n}}\left(z^{n+% \alpha}e^{-z}\right)=\frac{\Gamma(\alpha+n+1)/\Gamma(\alpha+1)}{\Gamma(n+1)}\,% _{1}F_{1}(-n,\alpha+1,z),
  30. m > n m>n
  31. N = λ - 1 2 N=\lambda-\frac{1}{2}
  32. Ψ m | x | Ψ n = 2 ( - 1 ) m - n + 1 ( m - n ) ( 2 N - n - m ) ( N - n ) ( N - m ) Γ ( 2 N - m + 1 ) m ! Γ ( 2 N - n + 1 ) n ! . \left\langle\Psi_{m}|x|\Psi_{n}\right\rangle=\frac{2(-1)^{m-n+1}}{(m-n)(2N-n-m% )}\sqrt{\frac{(N-n)(N-m)\Gamma(2N-m+1)m!}{\Gamma(2N-n+1)n!}}.
  33. E ( v ) = h ν 0 ( v + 1 / 2 ) - [ h ν 0 ( v + 1 / 2 ) ] 2 4 D e E(v)=h\nu_{0}(v+1/2)-\frac{\left[h\nu_{0}(v+1/2)\right]^{2}}{4D_{e}}
  34. v v
  35. ν 0 \nu_{0}
  36. m m
  37. ν 0 = a 2 π 2 D e / m . \nu_{0}=\frac{a}{2\pi}\sqrt{2D_{e}/m}.
  38. h ν 0 h\nu_{0}
  39. v v
  40. E ( v + 1 ) - E ( v ) = h ν 0 - ( v + 1 ) ( h ν 0 ) 2 / 2 D e . E(v+1)-E(v)=h\nu_{0}-(v+1)(h\nu_{0})^{2}/2D_{e}.\,
  41. v m v_{m}
  42. E ( v m + 1 ) - E ( v m ) E(v_{m}+1)-E(v_{m})
  43. v m = 2 D e - h ν 0 h ν 0 . v_{m}=\frac{2D_{e}-h\nu_{0}}{h\nu_{0}}.
  44. v m v_{m}
  45. v m v_{m}
  46. E ( v ) E(v)
  47. v m v_{m}
  48. E ( v ) E(v)
  49. E v / h c = ω e ( v + 1 / 2 ) - ω e χ e ( v + 1 / 2 ) 2 E_{v}/hc=\omega_{e}(v+1/2)-\omega_{e}\chi_{e}(v+1/2)^{2}\,
  50. ω e \omega_{e}
  51. ω e χ e \omega_{e}\chi_{e}
  52. ω e \omega_{e}
  53. E = h c ω E=hc\omega
  54. E = ω E=\hbar\omega

Motivic_cohomology.html

  1. D M DM
  2. H p , q ( X ) H^{p,q}(X)
  3. p p
  4. D M DM
  5. D M DM

Motzkin_number.html

  1. M n M_{n}
  2. n = 0 , 1 , n=0,1,\dots
  3. M n + 1 = M n + i = 0 n - 1 M i M n - 1 - i = 2 n + 3 n + 3 M n + 3 n n + 3 M n - 1 . M_{n+1}=M_{n}+\sum_{i=0}^{n-1}M_{i}M_{n-1-i}=\frac{2n+3}{n+3}M_{n}+\frac{3n}{n% +3}M_{n-1}.
  4. M n = k = 0 n / 2 ( n 2 k ) C k . M_{n}=\sum_{k=0}^{\lfloor n/2\rfloor}{\left({{n}\atop{2k}}\right)}C_{k}.

Moufang_loop.html

  1. ( g u ) h = ( g h - 1 ) u (gu)h=(gh^{-1})u
  2. g ( h u ) = ( h g ) u g(hu)=(hg)u
  3. ( g u ) ( h u ) = h - 1 g . (gu)(hu)=h^{-1}g.
  4. u 2 = 1 u^{2}=1
  5. u g = g - 1 u ug=g^{-1}u
  6. L z L x L z ( y ) = L z x z ( y ) L_{z}L_{x}L_{z}(y)=L_{zxz}(y)
  7. R z R y R z ( x ) = R z y z ( x ) R_{z}R_{y}R_{z}(x)=R_{zyz}(x)
  8. L z ( x ) R z ( y ) = B z ( x y ) L_{z}(x)R_{z}(y)=B_{z}(xy)
  9. x , y , z x,y,z
  10. Q Q
  11. B z = L z R z = R z L z B_{z}=L_{z}R_{z}=R_{z}L_{z}
  12. z z
  13. ( L z , R z , B z ) (L_{z},R_{z},B_{z})
  14. Q Q
  15. z z
  16. Q Q
  17. x - 1 ( x y ) = y = ( y x ) x - 1 x^{-1}(xy)=y=(yx)x^{-1}
  18. ( x y ) - 1 = y - 1 x - 1 (xy)^{-1}=y^{-1}x^{-1}
  19. x ( y z ) = e x(yz)=e
  20. ( x y ) z = e (xy)z=e
  21. ( x y ) z = ( x z - 1 ) ( z y z ) (xy)z=(xz^{-1})(zyz)
  22. x ( y z ) = ( x y x ) ( x - 1 z ) . x(yz)=(xyx)(x^{-1}z).
  23. x ( y z ) = ( x y ) z x(yz)=(xy)z
  24. y ( x z ) = ( y x ) z y(xz)=(yx)z
  25. y ( z x ) = ( y z ) x y(zx)=(yz)x
  26. y , z y,z

Moving_average.html

  1. p M , p M - 1 , , p M - ( n - 1 ) p_{M},p_{M-1},\dots,p_{M-(n-1)}
  2. 𝑆𝑀𝐴 = p M + p M - 1 + + p M - ( n - 1 ) n \,\textit{SMA}={p_{M}+p_{M-1}+\cdots+p_{M-(n-1)}\over n}
  3. 𝑆𝑀𝐴 today = 𝑆𝑀𝐴 yesterday + p M - n n - p M n \,\textit{SMA}_{\mathrm{today}}=\,\textit{SMA}_{\mathrm{yesterday}}+{p_{M-n}% \over n}-{p_{M}\over n}
  4. x 1 . , x n x_{1}.\ldots,x_{n}
  5. 𝐶𝑀𝐴 n = x 1 + + x n n . \,\textit{CMA}_{n}={{x_{1}+\cdots+x_{n}}\over n}\,.
  6. x n + 1 x_{n+1}
  7. 𝐶𝑀𝐴 n + 1 = x n + 1 + n 𝐶𝑀𝐴 n n + 1 \,\textit{CMA}_{n+1}={{x_{n+1}+n\cdot\,\textit{CMA}_{n}}\over{n+1}}
  8. x 1 + + x n = n 𝐶𝑀𝐴 n x_{1}+\cdots+x_{n}=n\cdot\,\textit{CMA}_{n}
  9. x n + 1 = ( x 1 + + x n + 1 ) - ( x 1 + + x n ) = ( n + 1 ) 𝐶𝑀𝐴 n + 1 - n 𝐶𝑀𝐴 n x_{n+1}=(x_{1}+\cdots+x_{n+1})-(x_{1}+\cdots+x_{n})=(n+1)\cdot\,\textit{CMA}_{% n+1}-n\cdot\,\textit{CMA}_{n}
  10. 𝐶𝑀𝐴 n + 1 \,\textit{CMA}_{n+1}
  11. 𝐶𝑀𝐴 n + 1 = x n + 1 + n 𝐶𝑀𝐴 n n + 1 = 𝐶𝑀𝐴 n + x n + 1 - 𝐶𝑀𝐴 n n + 1 \,\textit{CMA}_{n+1}={x_{n+1}+n\cdot\,\textit{CMA}_{n}\over{n+1}}={\,\textit{% CMA}_{n}}+{{x_{n+1}-\,\textit{CMA}_{n}}\over{n+1}}
  12. WMA M = n p M + ( n - 1 ) p M - 1 + + 2 p ( M - n + 2 ) + p ( M - n + 1 ) n + ( n - 1 ) + + 2 + 1 \,\text{WMA}_{M}={np_{M}+(n-1)p_{M-1}+\cdots+2p_{(M-n+2)}+p_{(M-n+1)}\over n+(% n-1)+\cdots+2+1}
  13. n ( n + 1 ) 2 . \frac{n(n+1)}{2}.
  14. Total M + 1 = T o t a l M + p M + 1 - p M - n + 1 \,\text{Total}_{M+1}=Total_{M}+p_{M+1}-p_{M-n+1}\,
  15. Numerator M + 1 = Numerator M + n p M + 1 - T o t a l M \,\text{Numerator}_{M+1}=\,\text{Numerator}_{M}+np_{M+1}-Total_{M}\,
  16. WMA M + 1 = Numerator M + 1 n + ( n - 1 ) + + 2 + 1 \,\text{WMA}_{M+1}={\,\text{Numerator}_{M+1}\over n+(n-1)+\cdots+2+1}\,
  17. S 1 = Y 1 S_{1}=Y_{1}
  18. t > 1 , S t = α Y t + ( 1 - α ) S t - 1 t>1,\ \ S_{t}=\alpha\cdot Y_{t}+(1-\alpha)\cdot S_{t-1}
  19. S t = α × ( Y t - 1 + ( 1 - α ) × Y t - 2 + ( 1 - α ) 2 × Y t - 3 + + ( 1 - α ) k × Y t - ( k + 1 ) ) + ( 1 - α ) k + 1 × S t - ( k + 1 ) S_{t}=\alpha\times(Y_{t-1}+(1-\alpha)\times Y_{t-2}+(1-\alpha)^{2}\times Y_{t-% 3}+\cdots+(1-\alpha)^{k}\times Y_{t-(k+1)})+(1-\alpha)^{k+1}\times S_{t-(k+1)}
  20. Y t - i Y_{t-i}
  21. α ( 1 - α ) i - 1 \alpha(1-\alpha)^{i-1}
  22. S t , alternate = α Y t + ( 1 - α ) S t - 1 S_{t,\,\text{ alternate}}=\alpha\cdot Y_{t}+(1-\alpha)\cdot S_{t-1}
  23. EMA today = EMA yesterday + α × ( price today - EMAyesterday ) \,\text{EMA}_{\,\text{today}}=\,\text{EMA}_{\,\text{yesterday}}+\alpha\times(% \,\text{price}_{\,\text{today}}-\,\text{EMA}\text{yesterday})
  24. EMA yesterday \,\text{EMA}_{\,\text{yesterday}}
  25. EMA today = α × ( p 1 + ( 1 - α ) p 2 + ( 1 - α ) 2 p 3 + ( 1 - α ) 3 p 4 + ) \,\text{EMA}_{\,\text{today}}={\alpha\times(p_{1}+(1-\alpha)p_{2}+(1-\alpha)^{% 2}p_{3}+(1-\alpha)^{3}p_{4}+\cdots)}
  26. p 1 p_{1}
  27. price today \,\text{price}_{\,\text{today}}
  28. p 2 p_{2}
  29. price yesterday \,\text{price}_{\,\text{yesterday}}
  30. EMA today = p 1 + ( 1 - α ) p 2 + ( 1 - α ) 2 p 3 + ( 1 - α ) 3 p 4 + 1 + ( 1 - α ) + ( 1 - α ) 2 + ( 1 - α ) 3 + \,\text{EMA}_{\,\text{today}}={p_{1}+(1-\alpha)p_{2}+(1-\alpha)^{2}p_{3}+(1-% \alpha)^{3}p_{4}+\cdots\over 1+(1-\alpha)+(1-\alpha)^{2}+(1-\alpha)^{3}+\cdots}
  31. 1 / α = 1 + ( 1 - α ) + ( 1 - α ) 2 + 1/\alpha=1+(1-\alpha)+(1-\alpha)^{2}+\cdots
  32. α = 2 / ( N + 1 ) \alpha=2/(N+1)
  33. α × ( 1 + ( 1 - α ) + ( 1 - α ) 2 + + ( 1 - α ) N ) α × ( 1 + ( 1 - α ) + ( 1 - α ) 2 + + ( 1 - α ) ) = 1 - ( 1 - 2 N + 1 ) N + 1 {{\alpha\times\left(1+(1-\alpha)+(1-\alpha)^{2}+\cdots+(1-\alpha)^{N}\right)}% \over{\alpha\times\left(1+(1-\alpha)+(1-\alpha)^{2}+\cdots+(1-\alpha)^{\infty}% \right)}}=1-{\left(1-{2\over N+1}\right)}^{N+1}
  34. lim N [ 1 - ( 1 - 2 N + 1 ) N + 1 ] \lim_{N\to\infty}\left[1-{\left(1-{2\over N+1}\right)}^{N+1}\right]
  35. 1 - e - 2 0.8647 1-\,\text{e}^{-2}\approx 0.8647
  36. N N
  37. 1 - ( 1 - α ) N + 1 1-(1-\alpha)^{N+1}
  38. N N
  39. 1 - ( 1 - ( 1 - α ) N + 1 ) = ( 1 - α ) N + 1 1-(1-(1-\alpha)^{N+1})=(1-\alpha)^{N+1}
  40. α \alpha
  41. α = 2 / ( N + 1 ) \alpha=2/(N+1)
  42. N N
  43. α \alpha
  44. α \alpha
  45. lim n ( 1 + a 1 + n ) n = e a \lim_{n\to\infty}\left(1+{a\over{1+n}}\right)^{n}=e^{a}
  46. N N
  47. N N
  48. α × ( ( 1 - α ) k + ( 1 - α ) k + 1 + ( 1 - α ) k + 2 + ) , \alpha\times\left((1-\alpha)^{k}+(1-\alpha)^{k+1}+(1-\alpha)^{k+2}+\cdots% \right),
  49. α × ( 1 - α ) k × ( 1 + ( 1 - α ) + ( 1 - α ) 2 + ) , \alpha\times(1-\alpha)^{k}\times\left(1+(1-\alpha)+(1-\alpha)^{2}+\cdots\right),
  50. weight omitted by stopping after k terms total weight = α × [ ( 1 - α ) k + ( 1 - α ) k + 1 + ( 1 - α ) k + 2 + ] α × [ 1 + ( 1 - α ) + ( 1 - α ) 2 + ] {{\,\text{weight omitted by stopping after k terms}}\over{\,\text{total weight% }}}={{\alpha\times\left[(1-\alpha)^{k}+(1-\alpha)^{k+1}+(1-\alpha)^{k+2}+% \cdots\right]}\over{{\alpha\times\left[1+(1-\alpha)+(1-\alpha)^{2}+\cdots% \right]}}}
  51. = α ( 1 - α ) k × 1 1 - ( 1 - α ) α 1 - ( 1 - α ) ={{\alpha(1-\alpha)^{k}\times{{1}\over{1-(1-\alpha)}}}\over{{{\alpha}\over{1-(% 1-\alpha)}}}}
  52. = ( 1 - α ) k =(1-\alpha)^{k}
  53. k = log ( 0.001 ) log ( 1 - α ) k={\log(0.001)\over\log(1-\alpha)}
  54. log ( 1 - α ) \log\,(1-\alpha)
  55. - 2 N + 1 -2\over N+1
  56. k = 3.45 ( N + 1 ) k=3.45(N+1)\,
  57. MMA today = ( N - 1 ) × MMA yesterday + price N \,\text{MMA}_{\,\text{today}}={(N-1)\times\,\text{MMA}_{\,\text{yesterday}}+\,% \text{price}\over{N}}
  58. α = 1 / N \alpha=1/N
  59. S n = α ( t n - t n - 1 ) × Y n + ( 1 - α ( t n - t n - 1 ) ) × S n - 1 . S_{n}=\alpha(t_{n}-t_{n-1})\times Y_{n}+(1-\alpha(t_{n}-t_{n-1}))\times S_{n-1}.
  60. α \alpha
  61. α ( t n - t n - 1 ) = 1 - exp ( - t n - t n - 1 W × 60 ) \alpha(t_{n}-t_{n-1})=1-\exp\left({-{{t_{n}-t_{n-1}}\over{W\times 60}}}\right)
  62. W W
  63. α \alpha
  64. S n = ( 1 - exp ( - t n - t n - 1 W × 60 ) ) × Y n + exp ( - t n - t n - 1 W × 60 ) × S n - 1 S_{n}=(1-\exp\left({-{{t_{n}-t_{n-1}}\over{W\times 60}}}\right))\times Y_{n}+% \exp\left({-{{t_{n}-t_{n-1}}\over{W\times 60}}}\right)\times S_{n-1}
  65. L n = ( 1 - exp ( - 5 15 × 60 ) ) × Q n + e - 5 15 × 60 × L n - 1 = ( 1 - exp ( - 1 180 ) ) × Q n + e - 1 / 180 × L n - 1 = Q n + e - 1 / 180 × ( L n - 1 - Q n ) L_{n}=(1-\exp\left({-{5\over{15\times 60}}}\right))\times Q_{n}+e^{-{5\over{15% \times 60}}}\times L_{n-1}=(1-\exp\left({-{1\over{180}}}\right))\times Q_{n}+e% ^{-1/180}\times L_{n-1}=Q_{n}+e^{-1/180}\times(L_{n-1}-Q_{n})
  66. 𝑆𝑀𝑀 = Median ( p M , p M - 1 , , p M - n + 1 ) \,\textit{SMM}=\,\text{Median}(p_{M},p_{M-1},\ldots,p_{M-n+1})
  67. α ( 1 - ( 1 - α ) N + 1 1 - ( 1 - α ) ) \alpha\left({1-(1-\alpha)^{N+1}\over 1-(1-\alpha)}\right)
  68. α \alpha
  69. log ( 1 - α ) = - α - α 2 / 2 - \log(1-\alpha)=-\alpha-\alpha^{2}/2-\cdots
  70. - α -\alpha

Mu_Arae.html

  1. Distance in parsecs = 1 parallax in arcseconds \scriptstyle\mathrm{Distance\ in\ parsecs}=\frac{1}{\mathrm{parallax\ in\ % arcseconds}}

Muḥammad_ibn_Mūsā_al-Khwārizmī.html

  1. ( 10 - x ) 2 = 81 x (10-x)^{2}=81x
  2. x 2 - 20 x + 100 = 81 x x^{2}-20x+100=81x
  3. x 2 + 100 = 101 x x^{2}+100=101x
  4. p + q 2 = 50 1 2 \tfrac{p+q}{2}=50\tfrac{1}{2}
  5. p q = 100 pq=100
  6. p - q 2 = ( p + q 2 ) 2 - p q = 2550 1 4 - 100 = 49 1 2 \frac{p-q}{2}=\sqrt{\left(\frac{p+q}{2}\right)^{2}-pq}=\sqrt{2550\tfrac{1}{4}-% 100}=49\tfrac{1}{2}
  7. x = 50 1 2 - 49 1 2 = 1 x=50\tfrac{1}{2}-49\tfrac{1}{2}=1

Muirhead's_inequality.html

  1. a = ( a 1 , , a n ) a=(a_{1},\dots,a_{n})
  2. [ a ] = 1 n ! σ x σ 1 a 1 x σ n a n , [a]={1\over n!}\sum_{\sigma}x_{\sigma_{1}}^{a_{1}}\cdots x_{\sigma_{n}}^{a_{n}},
  3. a 1 + + a n = 1 a_{1}+\cdots+a_{n}=1
  4. [ a ] 1 / ( a 1 + + a n ) [a]^{1/(a_{1}+\cdots+a_{n})}
  5. a 1 a 2 a n a_{1}\geq a_{2}\geq\cdots\geq a_{n}
  6. b 1 b 2 b n . b_{1}\geq b_{2}\geq\cdots\geq b_{n}.
  7. a 1 b 1 a_{1}\leq b_{1}
  8. a 1 + a 2 b 1 + b 2 a_{1}+a_{2}\leq b_{1}+b_{2}
  9. a 1 + a 2 + a 3 b 1 + b 2 + b 3 a_{1}+a_{2}+a_{3}\leq b_{1}+b_{2}+b_{3}
  10. \qquad\vdots\qquad\vdots\qquad\vdots\qquad\vdots
  11. a 1 + + a n - 1 b 1 + + b n - 1 a_{1}+\cdots+a_{n-1}\leq b_{1}+\cdots+b_{n-1}
  12. a 1 + + a n = b 1 + + b n . a_{1}+\cdots+a_{n}=b_{1}+\cdots+b_{n}.
  13. b 1 , , b n b_{1},\ldots,b_{n}
  14. a 1 , , a n a_{1},\ldots,a_{n}
  15. α 1 , , α n \alpha_{1},\ldots,\alpha_{n}
  16. sym x 1 α 1 x n α n \sum\text{sym}x_{1}^{\alpha_{1}}\cdots x_{n}^{\alpha_{n}}
  17. sym x 3 y 2 z 0 = x 3 y 2 z 0 + x 3 z 2 y 0 + y 3 x 2 z 0 + y 3 z 2 x 0 + z 3 x 2 y 0 + z 3 y 2 x 0 \sum\text{sym}x^{3}y^{2}z^{0}=x^{3}y^{2}z^{0}+x^{3}z^{2}y^{0}+y^{3}x^{2}z^{0}+% y^{3}z^{2}x^{0}+z^{3}x^{2}y^{0}+z^{3}y^{2}x^{0}
  18. = x 3 y 2 + x 3 z 2 + y 3 x 2 + y 3 z 2 + z 3 x 2 + z 3 y 2 {}=x^{3}y^{2}+x^{3}z^{2}+y^{3}x^{2}+y^{3}z^{2}+z^{3}x^{2}+z^{3}y^{2}
  19. a G = ( 1 n , , 1 n ) a_{G}=\left(\frac{1}{n},\ldots,\frac{1}{n}\right)
  20. a A = ( 1 , 0 , 0 , , 0 ) a_{A}=(1,0,0,\ldots,0)\,
  21. a A 1 = 1 > a G 1 = 1 n a_{A1}=1>a_{G1}=\frac{1}{n}\,
  22. a A 1 + a A 2 = 1 > a G 1 + a G 2 = 2 n a_{A1}+a_{A2}=1>a_{G1}+a_{G2}=\frac{2}{n}\,
  23. \qquad\vdots\qquad\vdots\qquad\vdots\,
  24. a A 1 + + a A n = a G 1 + + a G n = 1 a_{A1}+\cdots+a_{An}=a_{G1}+\cdots+a_{Gn}=1\,
  25. 1 n ! ( x 1 1 x 2 0 x n 0 + + x 1 0 x n 1 ) ( n - 1 ) ! 1 n ! ( x 1 x n ) 1 n n ! \frac{1}{n!}(x_{1}^{1}\cdot x_{2}^{0}\cdots x_{n}^{0}+\cdots+x_{1}^{0}\cdots x% _{n}^{1})(n-1)!\geq\frac{1}{n!}(x_{1}\cdot\cdots\cdot x_{n})^{\frac{1}{n}}n!
  26. sym x 2 y 0 sym x 1 y 1 . \sum_{\mathrm{sym}}x^{2}y^{0}\geq\sum_{\mathrm{sym}}x^{1}y^{1}.
  27. x 3 + y 3 + z 3 3 x y z x^{3}+y^{3}+z^{3}\geq 3xyz
  28. sym x 3 y 0 z 0 sym x 1 y 1 z 1 \sum_{\mathrm{sym}}x^{3}y^{0}z^{0}\geq\sum_{\mathrm{sym}}x^{1}y^{1}z^{1}
  29. 2 x 3 + 2 y 3 + 2 z 3 6 x y z 2x^{3}+2y^{3}+2z^{3}\geq 6xyz

Multilinear_form.html

  1. f : V n K f:V^{n}\to K
  2. n = 2 n=2
  3. f ( , x , , x , ) = 0. f(\dots,x,\dots,x,\dots)=0.
  4. f ( , x , , y , ) = - f ( , y , , x , ) . f(\dots,x,\dots,y,\dots)=-f(\dots,y,\dots,x,\dots).

Multimodal_distribution.html

  1. R = a + x b + y R=\frac{a+x}{b+y}
  2. Y Y
  3. α \alpha
  4. Z Z
  5. ( 1 - α ) , (1-\alpha),
  6. 0 < α < 1 0<\alpha<1
  7. f ( x ) = p g 1 ( x ) + ( 1 - p ) g 2 ( x ) f(x)=pg_{1}(x)+(1-p)g_{2}(x)\,
  8. μ = p μ 1 + ( 1 - p ) μ 2 \mu=p\mu_{1}+(1-p)\mu_{2}
  9. ν 2 = p [ σ 1 2 + δ 1 2 ] + ( 1 - p ) [ σ 2 2 + δ 2 2 ] \nu_{2}=p[\sigma_{1}^{2}+\delta_{1}^{2}]+(1-p)[\sigma_{2}^{2}+\delta_{2}^{2}]
  10. ν 3 = p [ S 1 σ 1 3 + 3 δ 1 σ 1 2 + δ 1 3 ] + ( 1 - p ) [ S 2 σ 2 3 + 3 δ 2 σ 2 2 + δ 2 3 ] \nu_{3}=p[S_{1}\sigma_{1}^{3}+3\delta_{1}\sigma_{1}^{2}+\delta_{1}^{3}]+(1-p)[% S_{2}\sigma_{2}^{3}+3\delta_{2}\sigma_{2}^{2}+\delta_{2}^{3}]
  11. ν 4 = p [ K 1 σ 1 4 + 4 S 1 δ 1 σ 1 3 + 6 δ 1 2 σ 1 2 + δ 1 4 ] + ( 1 - p ) [ K 2 σ 2 4 + 4 S 2 δ 2 σ 2 3 + 6 δ 2 2 σ 2 2 + δ 2 4 ] \nu_{4}=p[K_{1}\sigma_{1}^{4}+4S_{1}\delta_{1}\sigma_{1}^{3}+6\delta_{1}^{2}% \sigma_{1}^{2}+\delta_{1}^{4}]+(1-p)[K_{2}\sigma_{2}^{4}+4S_{2}\delta_{2}% \sigma_{2}^{3}+6\delta_{2}^{2}\sigma_{2}^{2}+\delta_{2}^{4}]
  12. μ = x f ( x ) d x \mu=\int xf(x)\,dx
  13. δ i = μ i - μ \delta_{i}=\mu_{i}-\mu
  14. ν r = ( x - μ ) r f ( x ) d x \nu_{r}=\int(x-\mu)^{r}f(x)\,dx
  15. d 1 d\leq 1
  16. | log ( 1 - p ) - log ( p ) | 2 log ( d - d 2 - 1 ) + 2 d d 2 - 1 \left|\log(1-p)-\log(p)\right|\geq 2\log(d-\sqrt{d^{2}-1})+2d\sqrt{d^{2}-1}
  17. d = μ 1 - μ 2 2 σ 1 σ 2 d=\frac{\mu_{1}-\mu_{2}}{2\sqrt{\sigma_{1}\sigma_{2}}}
  18. D = 2 1 2 | μ 1 - μ 2 | ( σ 1 2 + σ 2 2 ) D=2^{\frac{1}{2}}\frac{\left|\mu_{1}-\mu_{2}\right|}{\sqrt{(\sigma_{1}^{2}+% \sigma_{2}^{2})}}
  19. A = U ( 1 - S - 1 K - 1 ) A=U(1-\frac{S-1}{K-1})
  20. A o v e r a l l = w i A i A_{overall}=\sum w_{i}A_{i}
  21. S = μ 1 - μ 2 2 ( σ 1 + σ 2 ) S=\frac{\mu_{1}-\mu_{2}}{2(\sigma_{1}+\sigma_{2})}
  22. β = γ 2 + 1 κ \beta=\frac{\gamma^{2}+1}{\kappa}
  23. b = g 2 + 1 k + 3 ( n - 1 ) 2 ( n - 2 ) ( n - 3 ) b=\frac{g^{2}+1}{k+\frac{3(n-1)^{2}}{(n-2)(n-3)}}
  24. A B = A 1 - A a n A 1 A_{B}=\frac{A_{1}-A_{an}}{A_{1}}
  25. R = A r A l R=\frac{A_{r}}{A_{l}}
  26. B = A r A l P i B=\frac{A_{r}}{A_{l}}\sum P_{i}
  27. δ = | μ 1 - μ 2 | σ \delta=\frac{|\mu_{1}-\mu_{2}|}{\sigma}
  28. B I = δ p ( 1 - p ) BI=\delta\sqrt{p(1-p)}
  29. B = 1 N [ ( 1 N cos ( 2 π m γ ) ) 2 + ( 1 N sin ( 2 π m γ ) ) 2 ] B=\frac{1}{N}\left[\left(\sum_{1}^{N}\cos(2\pi m\gamma)\right)^{2}+\left(\sum_% {1}^{N}\sin(2\pi m\gamma)\right)^{2}\right]
  30. B = | μ - μ M | B=|\mu-\mu_{M}|
  31. μ M = i = 1 L m i x i i = 1 L m i \mu_{M}=\frac{\sum_{i=1}^{L}m_{i}x_{i}}{\sum_{i=1}^{L}m_{i}}
  32. B = | ϕ 2 - ϕ 1 | p 2 p 1 B=|\phi_{2}-\phi_{1}|\frac{p_{2}}{p_{1}}
  33. k = n 1 σ 1 2 + n 2 σ 2 2 m σ 2 k=\frac{n_{1}\sigma_{1}^{2}+n_{2}\sigma_{2}^{2}}{m\sigma^{2}}
  34. 𝑀𝑒𝑎𝑛 = ϕ 16 + ϕ 50 + ϕ 84 3 \mathit{Mean}=\frac{\phi_{16}+\phi_{50}+\phi_{84}}{3}
  35. 𝑆𝑡𝑑𝐷𝑒𝑣 = ϕ 84 - ϕ 16 4 + ϕ 95 - ϕ 5 6.6 \mathit{StdDev}=\frac{\phi_{84}-\phi_{16}}{4}+\frac{\phi_{95}-\phi_{5}}{6.6}
  36. 𝑆𝑘𝑒𝑤 = ϕ 84 + ϕ 16 - 2 ϕ 50 2 ( ϕ 84 - ϕ 16 ) + ϕ 95 + ϕ 5 - 2 ϕ 50 2 ( ϕ 95 - ϕ 5 ) \mathit{Skew}=\frac{\phi_{84}+\phi_{16}-2\phi_{50}}{2(\phi_{84}-\phi_{16})}+% \frac{\phi_{95}+\phi_{5}-2\phi_{50}}{2(\phi_{95}-\phi_{5})}
  37. 𝐾𝑢𝑟𝑡 = ϕ 95 - ϕ 5 2.44 ( ϕ 75 - ϕ 25 ) \mathit{Kurt}=\frac{\phi_{95}-\phi_{5}}{2.44(\phi_{75}-\phi_{25})}
  38. b 2 - b 1 1 b_{2}-b_{1}\geq 1

Multinomial_distribution.html

  1. Cov ( X i , X j ) = - n p i p j ( i j ) \textstyle{\mathrm{Cov}}(X_{i},X_{j})=-np_{i}p_{j}~{}~{}(i\neq j)
  2. ( i = 1 k p i e t i ) n \biggl(\sum_{i=1}^{k}p_{i}e^{t_{i}}\biggr)^{n}
  3. ( j = 1 k p j e i t j ) n \left(\sum_{j=1}^{k}p_{j}e^{it_{j}}\right)^{n}
  4. i 2 = - 1 i^{2}=-1
  5. ( i = 1 k p i z i ) n for ( z 1 , , z k ) k \biggl(\sum_{i=1}^{k}p_{i}z_{i}\biggr)^{n}\,\text{ for }(z_{1},\ldots,z_{k})% \in\mathbb{C}^{k}
  6. Dir ( α + β ) \mathrm{Dir}(\alpha+\beta)
  7. i = 1 k p i = 1 \sum_{i=1}^{k}p_{i}=1
  8. 1 K 1\dots K
  9. f ( x 1 , , x k ; n , p 1 , , p k ) \displaystyle f(x_{1},\ldots,x_{k};n,p_{1},\ldots,p_{k})
  10. f ( x 1 , , x k ; p 1 , , p k ) = Γ ( i x i + 1 ) i Γ ( x i + 1 ) i = 1 k p i x i . f(x_{1},\dots,x_{k};p_{1},\ldots,p_{k})=\frac{\Gamma(\sum_{i}x_{i}+1)}{\prod_{% i}\Gamma(x_{i}+1)}\prod_{i=1}^{k}p_{i}^{x_{i}}.
  11. ( p x 1 + ( 1 - p ) x 2 ) n (px_{1}+(1-p)x_{2})^{n}
  12. ( p 1 x 1 + p 2 x 2 + p 3 x 3 + + p k x k ) n (p_{1}x_{1}+p_{2}x_{2}+p_{3}x_{3}+...+p_{k}x_{k})^{n}
  13. E ( X i ) = n p i . \operatorname{E}(X_{i})=np_{i}.\,
  14. var ( X i ) = n p i ( 1 - p i ) . \operatorname{var}(X_{i})=np_{i}(1-p_{i}).\,
  15. cov ( X i , X j ) = - n p i p j \operatorname{cov}(X_{i},X_{j})=-np_{i}p_{j}\,
  16. ρ ( X i , X i ) = 1. \rho(X_{i},X_{i})=1.
  17. ρ ( X i , X j ) = cov ( X i , X j ) var ( X i ) var ( X j ) = - p i p j p i ( 1 - p i ) p j ( 1 - p j ) = - p i p j ( 1 - p i ) ( 1 - p j ) . \rho(X_{i},X_{j})=\frac{\operatorname{cov}(X_{i},X_{j})}{\sqrt{\operatorname{% var}(X_{i})\operatorname{var}(X_{j})}}=\frac{-p_{i}p_{j}}{\sqrt{p_{i}(1-p_{i})% p_{j}(1-p_{j})}}=-\sqrt{\frac{p_{i}p_{j}}{(1-p_{i})(1-p_{j})}}.
  18. { ( n 1 , , n k ) k | n 1 + + n k = n } . \{(n_{1},\dots,n_{k})\in\mathbb{N}^{k}|n_{1}+\cdots+n_{k}=n\}.\,
  19. ( n + k - 1 k - 1 ) . {n+k-1\choose k-1}.
  20. Pr ( A = 1 , B = 2 , C = 3 ) = 6 ! 1 ! 2 ! 3 ! ( 0.2 1 ) ( 0.3 2 ) ( 0.5 3 ) = 0.135 \Pr(A=1,B=2,C=3)=\frac{6!}{1!2!3!}(0.2^{1})(0.3^{2})(0.5^{3})=0.135
  21. p 1 , , p k p_{1},\ldots,p_{k}
  22. j = min { j { 1 , , k } : ( i = 1 j p i ) - X 0 } . j=\min\left\{j^{\prime}\in\{1,\dots,k\}:(\sum_{i=1}^{j^{\prime}}p_{i})-X\geq 0% \right\}.
  23. p 1 , , p k p_{1},\ldots,p_{k}
  24. ( θ 2 , 2 θ ( 1 - θ ) , ( 1 - θ ) 2 ) (\theta^{2},2\theta(1-\theta),(1-\theta)^{2})

Multiplication_operator.html

  1. T ( φ ) ( x ) = f ( x ) φ ( x ) T(\varphi)(x)=f(x)\varphi(x)\quad
  2. T ( φ ) ( x ) = x 2 φ ( x ) T(\varphi)(x)=x^{2}\varphi(x)\quad
  3. ( T - λ ) ( φ ) ( x ) = ( x 2 - λ ) φ ( x ) . (T-\lambda)(\varphi)(x)=(x^{2}-\lambda)\varphi(x).\quad
  4. ( T - λ ) - 1 ( φ ) ( x ) = 1 x 2 - λ φ ( x ) (T-\lambda)^{-1}(\varphi)(x)=\frac{1}{x^{2}-\lambda}\varphi(x)\quad

Multiplicative_group.html

  1. / n \mathbb{Z}/n\mathbb{Z}
  2. + \mathbb{R}^{+}
  3. \mathbb{R}

Multiplicative_group_of_integers_modulo_n.html

  1. / n \mathbb{Z}/n\mathbb{Z}
  2. / ( n ) \mathbb{Z}/(n)
  3. n = ( n ) n\mathbb{Z}=(n)
  4. n \mathbb{Z}_{n}
  5. ( / n ) * , (\mathbb{Z}/n\mathbb{Z})^{*},
  6. ( / n ) × , (\mathbb{Z}/n\mathbb{Z})^{\times},
  7. U ( / n ) , \mathrm{U}(\mathbb{Z}/n\mathbb{Z}),
  8. E ( / n ) \mathrm{E}(\mathbb{Z}/n\mathbb{Z})
  9. ( / n ) × . (\mathbb{Z}/n\mathbb{Z})^{\times}.
  10. C n \mathrm{C}_{n}
  11. ( / 1 ) × C 1 (\mathbb{Z}/1\,\mathbb{Z})^{\times}\cong\mathrm{C}_{1}
  12. ( / n ) × (\mathbb{Z}/n\mathbb{Z})^{\times}
  13. ( / n ) × (\mathbb{Z}/n\mathbb{Z})^{\times}
  14. ( / 2 ) × C 1 (\mathbb{Z}/2\mathbb{Z})^{\times}\cong\mathrm{C}_{1}
  15. ( / 4 ) × C 2 , (\mathbb{Z}/4\mathbb{Z})^{\times}\cong\mathrm{C}_{2},
  16. ( / 8 ) × C 2 × C 2 , (\mathbb{Z}/8\mathbb{Z})^{\times}\cong\mathrm{C}_{2}\times\mathrm{C}_{2},
  17. { ± 1 , ± 7 } C 2 × C 2 , \{\pm 1,\pm 7\}\cong\mathrm{C}_{2}\times\mathrm{C}_{2},
  18. ( / 16 ) × (\mathbb{Z}/16\mathbb{Z})^{\times}
  19. { 1 , 3 , 9 , 11 } \{1,3,9,11\}
  20. { 1 , 5 , 9 , 13 } . \{1,5,9,13\}.
  21. ( / 16 ) × C 2 × C 4 . (\mathbb{Z}/16\mathbb{Z})^{\times}\cong\mathrm{C}_{2}\times\mathrm{C}_{4}.
  22. { ± 1 , 2 k - 1 ± 1 } C 2 × C 2 , \{\pm 1,2^{k-1}\pm 1\}\cong\mathrm{C}_{2}\times\mathrm{C}_{2},
  23. ( / 2 k ) × (\mathbb{Z}/2^{k}\mathbb{Z})^{\times}
  24. ( / 2 k ) × C 2 × C 2 k - 2 . (\mathbb{Z}/2^{k}\mathbb{Z})^{\times}\cong\mathrm{C}_{2}\times\mathrm{C}_{2^{k% -2}}.
  25. ( / p k ) × C p k - 1 ( p - 1 ) C φ ( p k ) . (\mathbb{Z}/p^{k}\mathbb{Z})^{\times}\cong\mathrm{C}_{p^{k-1}(p-1)}\cong% \mathrm{C}_{\varphi(p^{k})}.
  26. n = p 1 k 1 p 2 k 2 p 3 k 3 , \;\;n=p_{1}^{k_{1}}p_{2}^{k_{2}}p_{3}^{k_{3}}\dots,\;
  27. / n \mathbb{Z}/n\mathbb{Z}
  28. / n / p 1 k 1 × / p 2 k 2 × / p 3 k 3 \mathbb{Z}/n\mathbb{Z}\cong\mathbb{Z}/{p_{1}^{k_{1}}}\mathbb{Z}\;\times\;% \mathbb{Z}/{p_{2}^{k_{2}}}\mathbb{Z}\;\times\;\mathbb{Z}/{p_{3}^{k_{3}}}% \mathbb{Z}\dots\;\;
  29. ( / n ) × (\mathbb{Z}/n\mathbb{Z})^{\times}
  30. ( / n ) × ( / p 1 k 1 ) × × ( / p 2 k 2 ) × × ( / p 3 k 3 ) × . (\mathbb{Z}/n\mathbb{Z})^{\times}\cong(\mathbb{Z}/{p_{1}^{k_{1}}}\mathbb{Z})^{% \times}\times(\mathbb{Z}/{p_{2}^{k_{2}}}\mathbb{Z})^{\times}\times(\mathbb{Z}/% {p_{3}^{k_{3}}}\mathbb{Z})^{\times}\dots\;.
  31. | ( / n ) × | = φ ( n ) . |(\mathbb{Z}/n\mathbb{Z})^{\times}|=\varphi(n).
  32. λ ( n ) , \lambda(n),
  33. λ ( n ) \lambda(n)
  34. a λ ( n ) 1 ( mod n ) a^{\lambda(n)}\equiv 1\;\;(\mathop{{\rm mod}}n)
  35. ( / n ) × (\mathbb{Z}/n\mathbb{Z})^{\times}
  36. φ ( n ) \varphi(n)
  37. λ ( n ) \lambda(n)
  38. ( / n ) × , (\mathbb{Z}/n\mathbb{Z})^{\times},
  39. ( / n ) × (\mathbb{Z}/n\mathbb{Z})^{\times}
  40. ( / n ) × (\mathbb{Z}/n\mathbb{Z})^{\times}
  41. ( / n ) × (\mathbb{Z}/n\mathbb{Z})^{\times}
  42. φ ( 20 ) = 8 \varphi(20)=8
  43. ( / 20 ) × (\mathbb{Z}/20\mathbb{Z})^{\times}
  44. λ ( 20 ) = 4 \lambda(20)=4
  45. ( / 20 ) × (\mathbb{Z}/20\mathbb{Z})^{\times}
  46. ( / 20 ) × (\mathbb{Z}/20\mathbb{Z})^{\times}
  47. n n\;
  48. n n\;
  49. n n\;

Multiplier_(Fourier_analysis).html

  1. ( f ) ( n ) = - π π f ( t ) e - i n t d t = - π π i n f ( t ) e - i n t d t = i n ( f ) ( n ) \mathcal{F}(f^{\prime})(n)=\int_{-\pi}^{\pi}f^{\prime}(t)e^{-int}\,dt=\int_{-% \pi}^{\pi}inf(t)e^{-int}\,dt=in\cdot\mathcal{F}(f)(n)
  2. f ^ : G ^ \hat{f}:\hat{G}\to\mathbb{C}
  3. G ^ \hat{G}
  4. m : G ^ m:\hat{G}\to\mathbb{C}
  5. T = T m T=T_{m}
  6. T f ^ ( ξ ) := m ( ξ ) f ^ ( ξ ) . \widehat{Tf}(\xi):=m(\xi)\hat{f}(\xi).
  7. L 2 L^{2}
  8. G = / 2 π G=\mathbb{R}/2\pi\mathbb{Z}
  9. G ^ = \hat{G}=\mathbb{Z}
  10. f ^ ( n ) := 1 2 π 0 2 π f ( t ) e - i n t d t \hat{f}(n):=\frac{1}{2\pi}\int_{0}^{2\pi}f(t)e^{-int}dt
  11. f ( t ) = n = - f ^ ( n ) e i n t . f(t)=\sum_{n=-\infty}^{\infty}\hat{f}(n)e^{int}.
  12. ( m n ) n = - (m_{n})_{n=-\infty}^{\infty}
  13. T = T m T=T_{m}
  14. ( T f ) ( t ) := n = - m n f ^ ( n ) e i n t , (Tf)(t):=\sum_{n=-\infty}^{\infty}m_{n}\widehat{f}(n)e^{int},
  15. ( m n ) n = - (m_{n})_{n=-\infty}^{\infty}
  16. G = n G=\mathbb{R}^{n}
  17. G ^ = n \hat{G}=\mathbb{R}^{n}
  18. f ^ ( ξ ) := n f ( x ) e - 2 π i x ξ d x \hat{f}(\xi):=\int_{\mathbb{R}^{n}}f(x)e^{-2\pi ix\cdot\xi}dx
  19. f ( x ) = n f ^ ( ξ ) e 2 π i x ξ d ξ . f(x)=\int_{\mathbb{R}^{n}}\hat{f}(\xi)e^{2\pi ix\cdot\xi}d\xi.
  20. m : n m:\mathbb{R}^{n}\to\mathbb{C}
  21. T = T m T=T_{m}
  22. T f ( x ) := n m ( ξ ) f ^ ( ξ ) e 2 π i x ξ d ξ , Tf(x):=\int_{\mathbb{R}^{n}}m(\xi)\hat{f}(\xi)e^{2\pi ix\cdot\xi}d\xi,
  23. G = / 2 π G=\mathbb{R}/2\pi\mathbb{Z}
  24. m n m_{n}
  25. T f ( t ) Tf(t)
  26. K ( t ) K(t)
  27. δ ( t ) \delta(t)
  28. c δ ( t ) c\delta(t)
  29. e i n s e^{ins}
  30. δ ( t - s ) \delta(t-s)
  31. δ ( t ) \delta^{\prime}(t)
  32. ( i n ) k (in)^{k}
  33. f ( k ) ( t ) f^{(k)}(t)
  34. δ ( k ) ( t ) \delta^{(k)}(t)
  35. P ( i n ) P(in)
  36. P ( d d t ) f ( t ) P\left(\frac{d}{dt}\right)f(t)
  37. P ( d d t ) δ ( t ) P\left(\frac{d}{dt}\right)\delta(t)
  38. α \alpha
  39. | n | α |n|^{\alpha}
  40. | d d t | α f ( t ) \left|\frac{d}{dt}\right|^{\alpha}f(t)
  41. | d d t | α δ ( t ) \left|\frac{d}{dt}\right|^{\alpha}\delta(t)
  42. 1 n = 0 1_{n=0}
  43. 1 2 π 0 2 π f ( t ) d t \frac{1}{2\pi}\int_{0}^{2\pi}f(t)\,dt
  44. 1 n 0 1_{n\neq 0}
  45. f ( t ) - 1 2 π 0 2 π f ( t ) d t f(t)-\frac{1}{2\pi}\int_{0}^{2\pi}f(t)\,dt
  46. δ ( t ) - 1 \delta(t)-1
  47. 1 i n 1 n 0 \frac{1}{in}1_{n\neq 0}
  48. 1 2 π 0 2 π ( π - s ) f ( t - s ) d s \frac{1}{2\pi}\int_{0}^{2\pi}(\pi-s)f(t-s)ds
  49. 1 2 ( 1 - { t 2 π } ) \frac{1}{2}\left(1-\{\frac{t}{2\pi}\}\right)
  50. 1 n 0 - 1 n < 0 1_{n\geq 0}-1_{n<0}
  51. H f := p . v . 1 π - π π f ( s ) e i ( t - s ) - 1 d s Hf:=p.v.\frac{1}{\pi}\int_{-\pi}^{\pi}\frac{f(s)}{e^{i(t-s)}-1}ds
  52. p . v .2 f ( s ) e i ( t - s ) - 1 d s p.v.2\frac{f(s)}{e^{i(t-s)}-1}ds
  53. D N D_{N}
  54. 1 - N n N 1_{-N\leq n\leq N}
  55. n = - N N f ^ ( n ) e i n t \sum_{n=-N}^{N}\hat{f}(n)e^{int}
  56. sin ( ( N + 1 2 ) t ) / sin ( t / 2 ) \sin((N+\tfrac{1}{2})t)/\sin(t/2)
  57. F N F_{N}
  58. ( 1 - | n | N ) 1 - N n N (1-\frac{|n|}{N})1_{-N\leq n\leq N}
  59. n = - N N ( 1 - | n | N ) f ^ ( n ) e i n t \sum_{n=-N}^{N}(1-\frac{|n|}{N})\hat{f}(n)e^{int}
  60. 1 N ( sin ( N t / 2 ) / sin ( t / 2 ) ) 2 \frac{1}{N}(\sin(Nt/2)/\sin(t/2))^{2}
  61. m n m_{n}
  62. n = - m n f ^ ( n ) e i n t \sum_{n=-\infty}^{\infty}m_{n}\hat{f}(n)e^{int}
  63. T δ ( t ) = n = - m n e i n t T\delta(t)=\sum_{n=-\infty}^{\infty}m_{n}e^{int}
  64. K ^ ( n ) \hat{K}(n)
  65. f * K ( t ) := 1 2 π 0 2 π f ( s ) K ( t - s ) d s f*K(t):=\frac{1}{2\pi}\int_{0}^{2\pi}f(s)K(t-s)ds
  66. K ( t ) K(t)
  67. G = n G=\mathbb{R}^{n}
  68. m ( ξ ) m(\xi)
  69. T f ( x ) Tf(x)
  70. K ( x ) K(x)
  71. δ ( x ) \delta(x)
  72. c δ ( x ) c\delta(x)
  73. e 2 π i y ξ e^{2\pi iy\cdot\xi}
  74. δ ( x - y ) \delta(x-y)
  75. d / d x d/dx
  76. 2 π i ξ 2\pi i\xi
  77. d f d x ( x ) \frac{df}{dx}(x)
  78. δ ( x ) \delta^{\prime}(x)
  79. / x j \partial/\partial x_{j}
  80. 2 π i ξ j 2\pi i\xi_{j}
  81. f x j ( x ) \frac{\partial f}{\partial x_{j}}(x)
  82. δ x j ( x ) \frac{\partial\delta}{\partial x_{j}}(x)
  83. Δ \Delta
  84. - 4 π 2 | ξ | 2 -4\pi^{2}|\xi|^{2}
  85. Δ f ( x ) \Delta f(x)
  86. Δ δ ( x ) \Delta\delta(x)
  87. P ( ) P(\nabla)
  88. P ( i ξ ) P(i\xi)
  89. P ( ) f ( x ) P(\nabla)f(x)
  90. P ( ) δ ( x ) P(\nabla)\delta(x)
  91. α \alpha
  92. ( 2 π | ξ | ) α (2\pi|\xi|)^{\alpha}
  93. ( - Δ ) α / 2 f ( x ) (-\Delta)^{\alpha/2}f(x)
  94. ( - Δ ) α / 2 δ ( x ) (-\Delta)^{\alpha/2}\delta(x)
  95. α \alpha
  96. ( 2 π | ξ | ) - α (2\pi|\xi|)^{-\alpha}
  97. ( - Δ ) - α / 2 f ( x ) (-\Delta)^{-\alpha/2}f(x)
  98. ( - Δ ) - α / 2 δ ( x ) = c n , α | x | α - n (-\Delta)^{-\alpha/2}\delta(x)=c_{n,\alpha}|x|^{\alpha-n}
  99. α \alpha
  100. ( 1 + 4 π 2 | ξ | 2 ) - α / 2 (1+4\pi^{2}|\xi|^{2})^{-\alpha/2}
  101. ( 1 - Δ ) - α / 2 f ( x ) (1-\Delta)^{-\alpha/2}f(x)
  102. 1 ( 4 π ) α / 2 Γ ( α / 2 ) 0 e - π | x | 2 / s e - s / 4 π s ( - n + α ) / 2 d s s \frac{1}{(4\pi)^{\alpha/2}\Gamma(\alpha/2)}\int_{0}^{\infty}e^{-\pi|x|^{2}/s}e% ^{-s/4\pi}s^{(-n+\alpha)/2}\frac{ds}{s}
  103. exp ( t Δ ) \exp(t\Delta)
  104. exp ( - 4 π 2 t | ξ | 2 ) \exp(-4\pi^{2}t|\xi|^{2})
  105. exp ( t Δ ) f ( x ) = 1 ( 4 π t ) n / 2 R n e - | x - y | 2 / 4 t f ( y ) d y \exp(t\Delta)f(x)=\frac{1}{(4\pi t)^{n/2}}\int_{R^{n}}e^{-|x-y|^{2}/4t}f(y)dy
  106. 1 ( 4 π t ) n / 2 e - | x | 2 / 4 t \frac{1}{(4\pi t)^{n/2}}e^{-|x|^{2}/4t}
  107. exp ( i t Δ ) \exp(it\Delta)
  108. exp ( - i 4 π 2 t | ξ | 2 ) \exp(-i4\pi^{2}t|\xi|^{2})
  109. exp ( i t Δ ) f ( x ) = 1 ( 4 π i t ) n / 2 R n e i | x - y | 2 / 4 t f ( y ) d y \exp(it\Delta)f(x)=\frac{1}{(4\pi it)^{n/2}}\int_{R^{n}}e^{i|x-y|^{2}/4t}f(y)dy
  110. 1 ( 4 π i t ) n / 2 e i | x | 2 / 4 t \frac{1}{(4\pi it)^{n/2}}e^{i|x|^{2}/4t}
  111. - i sgn ( ξ ) -i\operatorname{sgn}(\xi)
  112. H f := p . v . 1 π - f ( y ) x - y d y Hf:=p.v.\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{f(y)}{x-y}dy
  113. p . v . 1 π s p.v.\frac{1}{\pi s}
  114. - i ξ j | ξ | -i\frac{\xi_{j}}{|\xi|}
  115. R j f := p . v . c n n f ( y ) ( x j - y j ) | x - y | n d y R_{j}f:=p.v.c_{n}\int_{\mathbb{R}^{n}}\frac{f(y)(x_{j}-y_{j})}{|x-y|^{n}}dy
  116. p . v . c n x j | x | n , c n = Γ ( ( n + 1 ) / 2 ) π ( n + 1 ) / 2 p.v.\frac{c_{n}x_{j}}{|x|^{n}},\quad c_{n}=\frac{\Gamma((n+1)/2)}{\pi^{(n+1)/2}}
  117. S R 0 S^{0}_{R}
  118. 1 - R ξ R 1_{-R\leq\xi\leq R}
  119. - R R f ^ ( ξ ) e 2 π i x ξ d x \int_{-R}^{R}\hat{f}(\xi)e^{2\pi ix\xi}dx
  120. sin ( 2 π R x ) / π x \sin(2\pi Rx)/\pi x
  121. S R 0 S^{0}_{R}
  122. 1 | ξ | R 1_{|\xi|\leq R}
  123. | ξ | R f ^ ( ξ ) e 2 π i x ξ d x \int_{|\xi|\leq R}\hat{f}(\xi)e^{2\pi ix\xi}dx
  124. | x | - n / 2 J n / 2 ( 2 π | x | ) |x|^{-n/2}J_{n/2}(2\pi|x|)
  125. S R δ S^{\delta}_{R}
  126. ( 1 - | ξ | 2 / R 2 ) + δ (1-|\xi|^{2}/R^{2})_{+}^{\delta}
  127. | ξ | R ( 1 - | ξ | 2 R 2 ) δ f ^ ( ξ ) e 2 π i x ξ d ξ \int_{|\xi|\leq R}(1-\frac{|\xi|^{2}}{R^{2}})^{\delta}\hat{f}(\xi)e^{2\pi ix% \cdot\xi}\ d\xi
  128. | ξ | R ( 1 - | ξ | 2 R 2 ) δ e 2 π i x ξ d ξ \int_{|\xi|\leq R}(1-\frac{|\xi|^{2}}{R^{2}})^{\delta}e^{2\pi ix\cdot\xi}\,d\xi
  129. m ( ξ ) m(\xi)
  130. R n m ( ξ ) f ^ ( ξ ) e 2 π i x ξ d ξ \int_{R^{n}}m(\xi)\hat{f}(\xi)e^{2\pi ix\cdot\xi}d\xi
  131. R n m ( ξ ) e 2 π i x ξ d ξ \int_{R^{n}}m(\xi)e^{2\pi ix\cdot\xi}\ d\xi
  132. K ^ ( ξ ) \hat{K}(\xi)
  133. f * K ( x ) := R n f ( y ) K ( x - y ) d y f*K(x):=\int_{R^{n}}f(y)K(x-y)\,dy
  134. K ( x ) K(x)
  135. m T m m\mapsto T_{m}
  136. T m T_{m}
  137. T m T_{m^{\prime}}
  138. m + m m+m^{\prime}
  139. m m mm^{\prime}
  140. T m T_{m}
  141. m ¯ \overline{m}
  142. 1 / p + 1 / q = 1 1/p+1/q=1
  143. m ( ξ ) m(\xi)
  144. m : m:\mathbb{R}\rightarrow\mathbb{R}
  145. ( - 2 j + 1 , - 2 j ) ( 2 j , 2 j + 1 ) (-2^{j+1},-2^{j})\cup(2^{j},2^{j+1})
  146. j j\in\mathbb{Z}
  147. sup j ( - 2 j + 1 - 2 j | m ( ξ ) | d ξ + 2 j 2 j + 1 | m ( ξ ) | d ξ ) < \sup_{j\in\mathbb{Z}}\left(\int_{-2^{j+1}}^{-2^{j}}|m^{\prime}(\xi)|\,d\xi+% \int_{2^{j}}^{2^{j+1}}|m^{\prime}(\xi)|\,d\xi\right)<\infty
  148. | x | k | k m | \scriptstyle|x|^{k}|\nabla^{k}m|
  149. 0 k n / 2 + 1 \scriptstyle 0\leq k\leq n/2+1
  150. [ 2 n , 2 n + 1 - 1 ] [2^{n},2^{n+1}-1]
  151. [ - 2 n + 1 + 1 , - 2 n ] [-2^{n+1}+1,-2^{n}]

Multivariate_analysis_of_variance.html

  1. Σ m o d e l \Sigma_{model}
  2. Σ r e s - 1 \Sigma_{res}^{-1}
  3. A = Σ m o d e l × Σ r e s - 1 A=\Sigma_{model}\times\Sigma_{res}^{-1}
  4. Σ m o d e l = Σ r e s i d u a l \Sigma_{model}=\Sigma_{residual}
  5. A I A\sim I
  6. λ p \lambda_{p}
  7. A A
  8. Λ W i l k s = 1... p ( 1 / ( 1 + λ p ) ) = det ( I + A ) - 1 = det ( Σ r e s ) / det ( Σ r e s + Σ m o d e l ) \Lambda_{Wilks}=\prod_{1...p}(1/(1+\lambda_{p}))=\det(I+A)^{-1}=\det(\Sigma_{% res})/\det(\Sigma_{res}+\Sigma_{model})
  9. Λ P i l l a i = 1... p ( λ p / ( 1 + λ p ) ) = tr ( ( I + A ) - 1 ) \Lambda_{Pillai}=\sum_{1...p}(\lambda_{p}/(1+\lambda_{p}))=\mathrm{tr}((I+A)^{% -1})
  10. Λ L H = 1... p ( λ p ) = tr ( A ) \Lambda_{LH}=\sum_{1...p}(\lambda_{p})=\mathrm{tr}(A)
  11. Λ R o y = m a x p ( λ p ) = A \Lambda_{Roy}=max_{p}(\lambda_{p})=\|A\|_{\infty}

Multivariate_gamma_function.html

  1. Γ p ( a ) = S > 0 exp ( - tr ( S ) ) | S | a - ( p + 1 ) / 2 d S , \Gamma_{p}(a)=\int_{S>0}\exp\left(-{\rm tr}(S)\right)\left|S\right|^{a-(p+1)/2% }dS,
  2. Γ p ( a ) = π p ( p - 1 ) / 4 j = 1 p Γ [ a + ( 1 - j ) / 2 ] . \Gamma_{p}(a)=\pi^{p(p-1)/4}\prod_{j=1}^{p}\Gamma\left[a+(1-j)/2\right].
  3. Γ p ( a ) = π ( p - 1 ) / 2 Γ ( a ) Γ p - 1 ( a - 1 2 ) = π ( p - 1 ) / 2 Γ p - 1 ( a ) Γ [ a + ( 1 - p ) / 2 ] . \Gamma_{p}(a)=\pi^{(p-1)/2}\Gamma(a)\Gamma_{p-1}(a-\tfrac{1}{2})=\pi^{(p-1)/2}% \Gamma_{p-1}(a)\Gamma[a+(1-p)/2].
  4. Γ 1 ( a ) = Γ ( a ) \Gamma_{1}(a)=\Gamma(a)
  5. Γ 2 ( a ) = π 1 / 2 Γ ( a ) Γ ( a - 1 / 2 ) \Gamma_{2}(a)=\pi^{1/2}\Gamma(a)\Gamma(a-1/2)
  6. Γ 3 ( a ) = π 3 / 2 Γ ( a ) Γ ( a - 1 / 2 ) Γ ( a - 1 ) \Gamma_{3}(a)=\pi^{3/2}\Gamma(a)\Gamma(a-1/2)\Gamma(a-1)
  7. ψ p ( a ) = log Γ p ( a ) a = i = 1 p ψ ( a + ( 1 - i ) / 2 ) , \psi_{p}(a)=\frac{\partial\log\Gamma_{p}(a)}{\partial a}=\sum_{i=1}^{p}\psi(a+% (1-i)/2),
  8. ψ p ( n ) ( a ) = n log Γ p ( a ) a n = i = 1 p ψ ( n ) ( a + ( 1 - i ) / 2 ) . \psi_{p}^{(n)}(a)=\frac{\partial^{n}\log\Gamma_{p}(a)}{\partial a^{n}}=\sum_{i% =1}^{p}\psi^{(n)}(a+(1-i)/2).
  9. Γ p ( a ) = π p ( p - 1 ) / 4 j = 1 p Γ ( a + 1 - j 2 ) , \Gamma_{p}(a)=\pi^{p(p-1)/4}\prod_{j=1}^{p}\Gamma(a+\frac{1-j}{2}),
  10. Γ p ( a ) a = π p ( p - 1 ) / 4 i = 1 p Γ ( a + 1 - i 2 ) a j = 1 , j i p Γ ( a + 1 - j 2 ) . \frac{\partial\Gamma_{p}(a)}{\partial a}=\pi^{p(p-1)/4}\sum_{i=1}^{p}\frac{% \partial\Gamma(a+\frac{1-i}{2})}{\partial a}\prod_{j=1,j\neq i}^{p}\Gamma(a+% \frac{1-j}{2}).
  11. Γ ( a + ( 1 - i ) / 2 ) a = ψ ( a + ( i - 1 ) / 2 ) Γ ( a + ( i - 1 ) / 2 ) \frac{\partial\Gamma(a+(1-i)/2)}{\partial a}=\psi(a+(i-1)/2)\Gamma(a+(i-1)/2)
  12. Γ p ( a ) a = π p ( p - 1 ) / 4 j = 1 p Γ ( a + ( 1 - j ) / 2 ) i = 1 p ψ ( a + ( 1 - i ) / 2 ) = Γ p ( a ) i = 1 p ψ ( a + ( 1 - i ) / 2 ) . \frac{\partial\Gamma_{p}(a)}{\partial a}=\pi^{p(p-1)/4}\prod_{j=1}^{p}\Gamma(a% +(1-j)/2)\sum_{i=1}^{p}\psi(a+(1-i)/2)=\Gamma_{p}(a)\sum_{i=1}^{p}\psi(a+(1-i)% /2).

Muzzle_energy.html

  1. E k = 1 2 m v 2 E_{k}=\begin{matrix}\frac{1}{2}\end{matrix}mv^{2}
  2. E k = 1 2 m v 2 × ( 1 ft lbf 7000 gr × 32.1739 ft /s 2 2 ) E_{k}=\begin{matrix}\frac{1}{2}\end{matrix}mv^{2}\times\left(\frac{1\mbox{ ft}% ~{}\cdot\mbox{lbf}~{}}{7000\mbox{ gr}~{}\times 32.1739\mbox{ ft}~{}^{2}\mbox{/% s}~{}^{2}}\right)
  3. E k = 1 2 m v 2 × ( 1 ft lbf 7000 gr × 32.163 ft /s 2 2 ) E_{k}=\begin{matrix}\frac{1}{2}\end{matrix}mv^{2}\times\left(\frac{1\mbox{ ft}% ~{}\cdot\mbox{lbf}~{}}{7000\mbox{ gr}~{}\times 32.163\mbox{ ft}~{}^{2}\mbox{/s% }~{}^{2}}\right)

N-gram.html

  1. n n
  2. n n
  3. x i x_{i}
  4. x i - ( n - 1 ) , , x i - 1 x_{i-(n-1)},\dots,x_{i-1}
  5. P ( x i x i - ( n - 1 ) , , x i - 1 ) P(x_{i}\mid x_{i-(n-1)},\dots,x_{i-1})
  6. 26 3 26^{3}
  7. n n
  8. n n
  9. w w w₁…wₙ
  10. k k
  11. n n
  12. n n
  13. k k

N-vector_model.html

  1. 𝐬 i \mathbf{s}_{i}
  2. H = - J < i , j > 𝐬 i 𝐬 j H=-J{\sum}_{<i,j>}\mathbf{s}_{i}\cdot\mathbf{s}_{j}
  3. < i , j > <i,j>
  4. \cdot
  5. n = 0 n=0
  6. n = 1 n=1
  7. n = 2 n=2
  8. n = 3 n=3
  9. n = 4 n=4
  10. n = 0 n=0
  11. n n

Nambu_mechanics.html

  1. { f 1 , , f N - 1 , { g 1 , , g N } } = { { f 1 , , f N - 1 , g 1 } , g 2 , , g N } + { g 1 , { f 1 , , f N - 1 , g 2 } , , g N } + \{f_{1},\cdots,~{}f_{N-1},~{}\{g_{1},\cdots,~{}g_{N}\}\}=\{\{f_{1},\cdots,~{}f% _{N-1},~{}g_{1}\},~{}g_{2},\cdots,~{}g_{N}\}+\{g_{1},\{f_{1},\cdots,f_{N-1},~{% }g_{2}\},\cdots,g_{N}\}+\dots
  2. + { g 1 , , g N - 1 , { f 1 , , f N - 1 , g N } } , +\{g_{1},\cdots,g_{N-1},\{f_{1},\cdots,f_{N-1},~{}g_{N}\}\},

Nambu–Goto_action.html

  1. S = t i t f L d t . S=\int_{t_{i}}^{t_{f}}L\,dt.
  2. - d s 2 = - ( c d t ) 2 + d x 2 + d y 2 + d z 2 , -ds^{2}=-(c\,dt)^{2}+dx^{2}+dy^{2}+dz^{2},
  3. S = - m c d s . S=-mc\int ds.
  4. x = ( x 0 , x 1 , x 2 , , x d ) . x=(x^{0},x^{1},x^{2},\ldots,x^{d}).
  5. X ( τ , σ ) = ( X 0 ( τ , σ ) , X 1 ( τ , σ ) , X 2 ( τ , σ ) , , X d ( τ , σ ) ) . X(\tau,\sigma)=(X^{0}(\tau,\sigma),X^{1}(\tau,\sigma),X^{2}(\tau,\sigma),% \ldots,X^{d}(\tau,\sigma)).
  6. X μ ( τ , σ ) X^{\mu}(\tau,\sigma)
  7. η μ ν \eta_{\mu\nu}
  8. g a b = η μ ν X μ y a X ν y b g_{ab}=\eta_{\mu\nu}\frac{\partial X^{\mu}}{\partial y^{a}}\frac{\partial X^{% \nu}}{\partial y^{b}}
  9. a , b = 0 , 1 a,b=0,1
  10. y 0 = τ , y 1 = σ y^{0}=\tau,y^{1}=\sigma
  11. 𝒜 \mathcal{A}
  12. d 𝒜 = d 2 Σ - g \mathrm{d}\mathcal{A}=\mathrm{d}^{2}\Sigma\sqrt{-g}
  13. d 2 Σ = d σ d τ \mathrm{d}^{2}\Sigma=\mathrm{d}\sigma\,\mathrm{d}\tau
  14. g = det ( g a b ) g=\mathrm{det}\left(g_{ab}\right)
  15. X ˙ = X τ \dot{X}=\frac{\partial X}{\partial\tau}
  16. X = X σ , X^{\prime}=\frac{\partial X}{\partial\sigma},
  17. g a b g_{ab}
  18. g a b = ( X ˙ 2 X ˙ X X X ˙ X 2 ) g_{ab}=\left(\begin{array}[]{cc}\dot{X}^{2}&\dot{X}\cdot X^{\prime}\\ X^{\prime}\cdot\dot{X}&X^{\prime 2}\end{array}\right)
  19. g = X ˙ 2 X 2 - ( X ˙ X ) 2 g=\dot{X}^{2}X^{\prime 2}-(\dot{X}\cdot X^{\prime})^{2}
  20. 𝒮 \mathcal{S}
  21. = - T 0 c d 𝒜 =-\frac{T_{0}}{c}\int d\mathcal{A}
  22. = - T 0 c d 2 Σ - g =-\frac{T_{0}}{c}\int\mathrm{d}^{2}\Sigma\sqrt{-g}
  23. = - T 0 c d 2 Σ ( X ˙ X ) 2 - ( X ˙ ) 2 ( X ) 2 =-\frac{T_{0}}{c}\int\mathrm{d}^{2}\Sigma\sqrt{(\dot{X}\cdot X^{\prime})^{2}-(% \dot{X})^{2}(X^{\prime})^{2}}
  24. X Y := η μ ν X μ Y ν X\cdot Y:=\eta_{\mu\nu}X^{\mu}Y^{\nu}
  25. \hbar
  26. α \alpha^{\prime}
  27. 𝒮 = - 1 2 π α d 2 Σ ( X ˙ X ) 2 - ( X ˙ ) 2 ( X ) 2 . \mathcal{S}=-\frac{1}{2\pi\alpha^{\prime}}\int\mathrm{d}^{2}\Sigma\sqrt{(\dot{% X}\cdot X^{\prime})^{2}-(\dot{X})^{2}(X^{\prime})^{2}}.
  28. 𝒮 = - 1 2 π α d 2 Σ X ˙ 2 - X 2 , \mathcal{S}=-\frac{1}{2\pi\alpha^{\prime}}\int\mathrm{d}^{2}\Sigma\sqrt{{\dot{% X}}^{2}-{X^{\prime}}^{2}},
  29. 𝒮 = - 1 4 π α d 2 Σ ( X ˙ 2 - X 2 ) . \mathcal{S}=-\frac{1}{4\pi\alpha^{\prime}}\int\mathrm{d}^{2}\Sigma({\dot{X}}^{% 2}-{X^{\prime}}^{2}).

Names_of_large_numbers.html

  1. ( 10 8 ) ( 10 8 ) = 10 8 10 8 , (10^{8})^{(10^{8})}=10^{8\cdot 10^{8}},
  2. ( ( 10 8 ) ( 10 8 ) ) ( 10 8 ) = 10 8 10 16 . \left((10^{8})^{(10^{8})}\right)^{(10^{8})}=10^{8\cdot 10^{16}}.
  3. 10 10 100 \,\!10^{10^{100}}
  4. 10 10 100 10^{10^{100}}

National_savings.html

  1. Y = C + I + G Y=C+I+G
  2. National Savings = Y - C - G = I \,\text{National Savings}=Y-C-G=I
  3. ( Y - T - C ) + ( T - G ) = I (Y-T-C)+(T-G)=I
  4. S = I ( r ) S=I(r)

Natural_number_object.html

  1. 1 q A f A 1\xrightarrow{~{}\quad q\quad~{}}A\xrightarrow{~{}\quad f\quad~{}}A

Neapolitan_chord.html

  1. 4 6 {}^{6}_{4}

Near-infrared_spectroscopy.html

  1. S t O 2 StO_{2}

Needleman–Wunsch_algorithm.html

  1. S ( a , b ) S(a,b)
  2. d d
  3. S ( A , C ) + S ( G , G ) + S ( A , A ) + ( 3 × d ) + S ( G , G ) + S ( T , A ) + S ( T , C ) + S ( A , G ) + S ( C , T ) S(A,C)+S(G,G)+S(A,A)+(3\times d)+S(G,G)+S(T,A)+S(T,C)+S(A,G)+S(C,T)
  4. = - 3 + 7 + 10 - ( 3 × 5 ) + 7 + - 4 + 0 + - 1 + 0 = 1 =-3+7+10-(3\times 5)+7+-4+0+-1+0=1
  5. F i j F_{ij}
  6. O ( n m ) O(nm)
  7. Θ ( min { n , m } ) \Theta(\min\{n,m\})
  8. O ( n m ) O(nm)
  9. F i j F_{ij}
  10. i = 0 , , n i=0,\ldots,n
  11. j = 0 , , m j=0,\ldots,m
  12. F 0 j = d * j F_{0j}=d*j
  13. F i 0 = d * i F_{i0}=d*i
  14. F i j = max ( F i - 1 , j - 1 + S ( A i , B j ) , F i , j - 1 + d , F i - 1 , j + d ) F_{ij}=\max(F_{i-1,j-1}+S(A_{i},B_{j}),\;F_{i,j-1}+d,\;F_{i-1,j}+d)
  15. F n m F_{nm}
  16. A i A_{i}
  17. B j B_{j}
  18. A i A_{i}
  19. B j B_{j}
  20. F i j = max h < i , k < j { F h , j - 1 + S ( A i , B j ) , F i - 1 , k + S ( A i , B j ) } F_{ij}=\max_{h<i,k<j}\{F_{h,j-1}+S(A_{i},B_{j}),F_{i-1,k}+S(A_{i},B_{j})\}

Negative_frequency.html

  1. e i ω t e^{i\omega t}
  2. cos ( ω t ) . \cos(\omega t).
  3. cos ( ω t ) = 1 2 ( e i ω t + e - i ω t ) , \cos(\omega t)=\begin{matrix}\frac{1}{2}\end{matrix}(e^{i\omega t}+e^{-i\omega t% }),
  4. cos ( ω t ) \cos(\omega t)
  5. X ( ω ) = a b x ( t ) e - i ω t d t , X(\omega)=\int_{a}^{b}x(t)\cdot e^{-i\omega t}dt,
  6. e - i ω t e^{-i\omega t}
  7. e i ω t e^{i\omega t}
  8. cos ( ω t ) \cos(\omega t)

Nemeth_Braille.html

  1. \perp

Nephroid.html

  1. x = a ( 3 cos t + cos 3 t ) , y = a ( 3 sin t + sin 3 t ) . x=a(3\cos t+\cos 3t),\quad y=a(3\sin t+\sin 3t).
  2. x = a ( 3 cos t - cos 3 t ) , y = a ( 3 sin t - sin 3 t ) . x=a(3\cos t-\cos 3t),\quad y=a(3\sin t-\sin 3t).
  3. ( x 2 + y 2 - 4 a 2 ) 3 = 108 a 4 x 2 . (x^{2}+y^{2}-4a^{2})^{3}=108a^{4}x^{2}.
  4. L = 24 a , A = 12 π a 2 . L=24a,\quad A=12\pi a^{2}.
  5. ρ = | 3 a cos t | . \rho=|3a\cos t|.
  6. ( a cos t , a sin t ) (a\cos t,a\sin t)
  7. ( a cos 3 t , a sin 3 t ) . (a\cos 3t,a\sin 3t).
  8. ( a 2 - 4 ( x 2 + y 2 ) ) 3 = 108 a 2 x 2 ( x 2 + y 2 ) . (a^{2}-4(x^{2}+y^{2}))^{3}=108a^{2}x^{2}(x^{2}+y^{2}).

Network_analysis_(electrical_circuits).html

  1. V 2 = V 1 V_{2}=V_{1}
  2. I 2 = I 1 I_{2}=I_{1}
  3. V 1 V_{1}
  4. Z eq = Z 1 + Z 2 + + Z n . Z_{\mathrm{eq}}=Z_{1}+Z_{2}+\,\cdots\,+Z_{n}.
  5. 1 Z eq = 1 Z 1 + 1 Z 2 + + 1 Z n . \frac{1}{Z_{\mathrm{eq}}}=\frac{1}{Z_{1}}+\frac{1}{Z_{2}}+\,\cdots\,+\frac{1}{% Z_{n}}.
  6. Z eq = Z 1 Z 2 Z 1 + Z 2 . Z_{\mathrm{eq}}=\frac{Z_{1}Z_{2}}{Z_{1}+Z_{2}}.
  7. R a = R ac R ab R ac + R ab + R bc R_{a}=\frac{R_{\mathrm{ac}}R_{\mathrm{ab}}}{R_{\mathrm{ac}}+R_{\mathrm{ab}}+R_% {\mathrm{bc}}}
  8. R b = R ab R bc R ac + R ab + R bc R_{b}=\frac{R_{\mathrm{ab}}R_{\mathrm{bc}}}{R_{\mathrm{ac}}+R_{\mathrm{ab}}+R_% {\mathrm{bc}}}
  9. R c = R bc R ac R ac + R ab + R bc R_{c}=\frac{R_{\mathrm{bc}}R_{\mathrm{ac}}}{R_{\mathrm{ac}}+R_{\mathrm{ab}}+R_% {\mathrm{bc}}}
  10. R ac = R a R b + R b R c + R c R a R b R_{\mathrm{ac}}=\frac{R_{a}R_{b}+R_{b}R_{c}+R_{c}R_{a}}{R_{b}}
  11. R ab = R a R b + R b R c + R c R a R c R_{\mathrm{ab}}=\frac{R_{a}R_{b}+R_{b}R_{c}+R_{c}R_{a}}{R_{c}}
  12. R bc = R a R b + R b R c + R c R a R a R_{\mathrm{bc}}=\frac{R_{a}R_{b}+R_{b}R_{c}+R_{c}R_{a}}{R_{a}}
  13. N N
  14. R 1 R_{1}
  15. R N R_{N}
  16. ( N 2 ) {N\choose 2}
  17. N N
  18. x x
  19. y y
  20. R xy = R x R y i = 1 N 1 R i R_{\mathrm{xy}}=R_{x}R_{y}\sum_{i=1}^{N}\frac{1}{R_{i}}
  21. N = 3 N=3
  22. R ab = R a R b ( 1 R a + 1 R b + 1 R c ) = R a R b ( R a R b + R a R c + R b R c ) R a R b R c = R a R b + R b R c + R c R a R c R_{\mathrm{ab}}=R_{a}R_{b}(\frac{1}{R}_{a}+\frac{1}{R}_{b}+\frac{1}{R}_{c})=% \frac{R_{a}R_{b}(R_{a}R_{b}+R_{a}R_{c}+R_{b}R_{c})}{R_{a}R_{b}R_{c}}=\frac{R_{% a}R_{b}+R_{b}R_{c}+R_{c}R_{a}}{R_{c}}
  23. N = 2 N=2
  24. R ab = R a R b ( 1 R a + 1 R b ) = R a R b ( R a + R b ) R a R b = R a + R b R_{\mathrm{ab}}=R_{a}R_{b}(\frac{1}{R}_{a}+\frac{1}{R}_{b})=\frac{R_{a}R_{b}(R% _{a}+R_{b})}{R_{a}R_{b}}=R_{a}+R_{b}
  25. N = 1 N=1
  26. ( 1 2 ) = 0 {1\choose 2}=0
  27. V s = R I s V_{\mathrm{s}}=RI_{\mathrm{s}}\,\!
  28. I s = V s R I_{\mathrm{s}}=\frac{V_{\mathrm{s}}}{R}
  29. V i V_{i}
  30. Z i Z_{i}
  31. V i = Z i I = ( Z i Z 1 + Z 2 + + Z n ) V V_{i}=Z_{i}I=\left(\frac{Z_{i}}{Z_{1}+Z_{2}+\cdots+Z_{n}}\right)V
  32. I i I_{i}
  33. Z i Z_{i}
  34. I i = ( ( 1 Z i ) ( 1 Z 1 ) + ( 1 Z 2 ) + + ( 1 Z n ) ) I I_{i}=\left(\frac{\left(\frac{1}{Z_{i}}\right)}{\left(\frac{1}{Z_{1}}\right)+% \left(\frac{1}{Z_{2}}\right)+\,\cdots\,+\left(\frac{1}{Z_{n}}\right)}\right)I
  35. i = 1 , 2 , , n . i=1,2,...,n.
  36. I 1 = ( Z 2 Z 1 + Z 2 ) I I_{1}=\left(\frac{Z_{2}}{Z_{1}+Z_{2}}\right)I
  37. I 2 = ( Z 1 Z 1 + Z 2 ) I I_{2}=\left(\frac{Z_{1}}{Z_{1}+Z_{2}}\right)I
  38. Z ( s ) = R Z(s)=R\,\!
  39. Z ( s ) = s L Z(s)=sL\,\!
  40. Z ( s ) = 1 s C Z(s)=\frac{1}{sC}
  41. Z ( j ω ) = R Z(j\omega)=R\,\!
  42. Z ( j ω ) = j ω L Z(j\omega)=j\omega L\,\!
  43. Z ( j ω ) = 1 j ω C Z(j\omega)=\frac{1}{j\omega C}
  44. Z = R Z=R\,\!
  45. Z = 0 Z=0\,\!
  46. Z = Z=\infty\,\!
  47. A ( j ω ) = V o V i A(j\omega)=\frac{V_{o}}{V_{i}}
  48. A ( ω ) = | V o V i | A(\omega)=\left|{\frac{V_{o}}{V_{i}}}\right|
  49. [ V 1 V 0 ] = [ z ( j ω ) 11 z ( j ω ) 12 z ( j ω ) 21 z ( j ω ) 22 ] [ I 1 I 0 ] \begin{bmatrix}V_{1}\\ V_{0}\end{bmatrix}=\begin{bmatrix}z(j\omega)_{11}&z(j\omega)_{12}\\ z(j\omega)_{21}&z(j\omega)_{22}\end{bmatrix}\begin{bmatrix}I_{1}\\ I_{0}\end{bmatrix}
  50. [ z ( j ω ) ] \left[z(j\omega)\right]
  51. [ z ] \left[z\right]
  52. i = I o ( e v V T - 1 ) i=I_{o}(e^{\frac{v}{V_{T}}}-1)
  53. f ( v , i ) = 0 f(v,i)=0\,
  54. f ( v , φ ) = 0 f(v,\varphi)=0\,
  55. f ( v , q ) = 0 f(v,q)=0\,

Neumann_boundary_condition.html

  1. y ′′ + y = 0 y^{\prime\prime}+y=0~{}
  2. [ a , b ] [a,\,b]
  3. y ( a ) = α and y ( b ) = β y^{\prime}(a)=\alpha\ \,\text{and}\ y^{\prime}(b)=\beta
  4. α \alpha
  5. β \beta
  6. 2 y + y = 0 \nabla^{2}y+y=0
  7. 2 \nabla^{2}
  8. Ω n \Omega\subset\mathbb{R}^{n}
  9. y 𝐧 ( 𝐱 ) = f ( 𝐱 ) 𝐱 Ω . \frac{\partial y}{\partial\mathbf{n}}(\mathbf{x})=f(\mathbf{x})\quad\forall% \mathbf{x}\in\partial\Omega.
  10. 𝐧 \mathbf{n}
  11. Ω \partial\Omega
  12. y 𝐧 ( 𝐱 ) = y ( 𝐱 ) 𝐧 ( 𝐱 ) \frac{\partial y}{\partial\mathbf{n}}(\mathbf{x})=\nabla y(\mathbf{x})\cdot% \mathbf{n}(\mathbf{x})
  13. \nabla

Neutron_transport.html

  1. ( 1 v ( E ) t + 𝛀 ^ + Σ t ( 𝐫 , E , t ) ) ψ ( 𝐫 , E , 𝛀 ^ , t ) = \left(\frac{1}{v(E)}\frac{\partial}{\partial t}+\mathbf{\hat{\Omega}}\cdot% \nabla+\Sigma_{t}(\mathbf{r},E,t)\right)\psi(\mathbf{r},E,\mathbf{\hat{\Omega}% },t)=\quad
  2. χ p ( E ) 4 π 0 d E ν p ( E ) Σ f ( 𝐫 , E , t ) ϕ ( 𝐫 , E , t ) + i = 1 N χ d i ( E ) 4 π λ i C i ( 𝐫 , t ) + \quad\frac{\chi_{p}\left(E\right)}{4\pi}\int_{0}^{\infty}dE^{\prime}\nu_{p}% \left(E^{\prime}\right)\Sigma_{f}\left(\mathbf{r},E^{\prime},t\right)\phi\left% (\mathbf{r},E^{\prime},t\right)+\sum_{i=1}^{N}\frac{\chi_{di}\left(E\right)}{4% \pi}\lambda_{i}C_{i}\left(\mathbf{r},t\right)+\quad
  3. 4 π d Ω 0 d E Σ s ( 𝐫 , E E , 𝛀 ^ 𝛀 ^ , t ) ψ ( 𝐫 , E , 𝛀 ^ , t ) + s ( 𝐫 , E , 𝛀 ^ , t ) \quad\int_{4\pi}d\Omega^{\prime}\int^{\infty}_{0}dE^{\prime}\,\Sigma_{s}(% \mathbf{r},E^{\prime}\rightarrow E,\mathbf{\hat{\Omega}}^{\prime}\rightarrow% \mathbf{\hat{\Omega}},t)\psi(\mathbf{r},E^{\prime},\mathbf{\hat{\Omega}^{% \prime}},t)+s(\mathbf{r},E,\mathbf{\hat{\Omega}},t)
  4. 𝐫 \mathbf{r}
  5. E E
  6. 𝛀 ^ = 𝐯 ( E ) | 𝐯 ( E ) | = 𝐯 ( E ) < m t p l > v ( E ) \mathbf{\hat{\Omega}}=\frac{\mathbf{v}(E)}{|\mathbf{v}(E)|}=\frac{\mathbf{v}(E% )}{<}mtpl>{{v(E)}}
  7. t t
  8. 𝐯 ( E ) \mathbf{v}(E)
  9. ψ ( 𝐫 , E , 𝛀 ^ , t ) d r d E d Ω \psi(\mathbf{r},E,\mathbf{\hat{\Omega}},t)dr\,dE\,d\Omega
  10. d r dr
  11. r r
  12. d E dE
  13. E E
  14. d Ω d\Omega
  15. 𝛀 ^ \mathbf{\hat{\Omega}}
  16. t t
  17. ϕ = 4 π d Ω ψ \phi\ =\ \int_{4\pi}d\Omega\psi
  18. ϕ ( 𝐫 , E , t ) d r d E \phi(\mathbf{r},E,t)dr\,dE
  19. d r dr
  20. r r
  21. d E dE
  22. E E
  23. t t
  24. ν p \nu_{p}
  25. χ p ( E ) \chi_{p}(E)
  26. E E
  27. χ d i ( E ) \chi_{di}(E)
  28. E E
  29. Σ t ( 𝐫 , E , t ) \Sigma_{t}(\mathbf{r},E,t)
  30. Σ f ( 𝐫 , E , t ) \Sigma_{f}(\mathbf{r},E^{\prime},t)
  31. d E dE^{\prime}
  32. E E^{\prime}
  33. Σ s ( 𝐫 , E E , 𝛀 ^ 𝛀 ^ , t ) d E d Ω \Sigma_{s}(\mathbf{r},E^{\prime}\rightarrow E,\mathbf{\hat{\Omega}}^{\prime}% \rightarrow\mathbf{\hat{\Omega}},t)dE^{\prime}d\Omega^{\prime}
  34. E E^{\prime}
  35. d E dE^{\prime}
  36. 𝛀 ^ \mathbf{\hat{\Omega^{\prime}}}
  37. d Ω d\Omega^{\prime}
  38. E E
  39. 𝛀 ^ \mathbf{\hat{\Omega}}
  40. N N
  41. λ i \lambda_{i}
  42. C i ( 𝐫 , t ) C_{i}\left(\mathbf{r},t\right)
  43. 𝐫 \mathbf{r}
  44. t t
  45. s ( 𝐫 , E , 𝛀 ^ , t ) s(\mathbf{r},E,\mathbf{\hat{\Omega}},t)
  46. 𝐫 \mathbf{r}
  47. 𝛀 ^ \mathbf{\hat{\Omega}}
  48. ϕ ( 𝐫 , E ) \phi(\mathbf{r},E)

New_Foundations.html

  1. = =
  2. \in
  3. x n = y n x^{n}=y^{n}\!
  4. x n y n + 1 x^{n}\in y^{n+1}
  5. ϕ ( x n ) \phi(x^{n})\!
  6. { x n ϕ ( x n ) } n + 1 \{x^{n}\mid\phi(x^{n})\}^{n+1}\!
  7. ϕ ( x n ) \phi(x^{n})\!
  8. A n + 1 x n [ x n A n + 1 ϕ ( x n ) ] \exists A^{n+1}\forall x^{n}[x^{n}\in A^{n+1}\leftrightarrow\phi(x^{n})]
  9. A n + 1 A^{n+1}\!
  10. { x n ϕ ( x n ) } n + 1 \{x^{n}\mid\phi(x^{n})\}^{n+1}\!
  11. ϕ \phi
  12. x y x\in y
  13. ϕ \phi
  14. x = y x=y
  15. ϕ \phi
  16. { x ϕ } \{x\mid\phi\}
  17. ϕ \phi
  18. { x x x } \{x\mid x\not\in x\}
  19. x x x\not\in x
  20. x = x x=x
  21. A B A\sim B
  22. ϕ ϕ + \phi\leftrightarrow\phi^{+}
  23. ϕ \phi
  24. ϕ + \phi^{+}
  25. ϕ \phi
  26. N F 3 NF_{3}
  27. N F 3 NF_{3}
  28. N F 3 NF_{3}
  29. N F 3 NF_{3}
  30. N F 4 NF_{4}
  31. x x x\not\in x
  32. { x x x } \{x\mid x\not\in x\}
  33. P ( A ) P(A)
  34. A A
  35. A A
  36. P ( A ) P(A)
  37. A A
  38. P ( V ) P(V)
  39. V V
  40. V V
  41. | A | < | P ( A ) | |A|<|P(A)|
  42. P ( A ) P(A)
  43. A A
  44. | P 1 ( A ) | < | P ( A ) | |P_{1}(A)|<|P(A)|
  45. P 1 ( A ) P_{1}(A)
  46. A A
  47. | P 1 ( V ) | < | P ( V ) | |P_{1}(V)|<|P(V)|
  48. x { x } x\mapsto\{x\}
  49. | P 1 ( V ) | < | P ( V ) | | V | |P_{1}(V)|<|P(V)|<<|V|
  50. | P ( V ) | |P(V)|
  51. | V | |V|
  52. A A
  53. | A | = | P 1 ( A ) | |A|=|P_{1}(A)|
  54. A A
  55. ( x { x } ) A (x\mapsto\{x\})\lceil A
  56. Ω \Omega
  57. α \alpha
  58. α \alpha
  59. Ω \Omega
  60. < Ω <\Omega
  61. Ω \Omega
  62. α \alpha
  63. α \alpha
  64. α \alpha
  65. α \alpha
  66. W α W\in\alpha
  67. T ( α ) T(\alpha)
  68. W ι = { ( { x } , { y } ) x W y } W^{\iota}=\{(\{x\},\{y\})\mid xWy\}
  69. < α <\alpha
  70. T 2 ( α ) T^{2}(\alpha)
  71. T 4 ( α ) T^{4}(\alpha)
  72. < Ω <\Omega
  73. T 2 ( Ω ) T^{2}(\Omega)
  74. T 2 ( Ω ) < Ω T^{2}(\Omega)<\Omega
  75. Ω > T 2 ( Ω ) > T 4 ( Ω ) \Omega>T^{2}(\Omega)>T^{4}(\Omega)\ldots
  76. V α V_{\alpha}
  77. j ( α ) < α j(\alpha)<\alpha
  78. V α V_{\alpha}
  79. x N F U y d e f j ( x ) y y V j ( α ) + 1 . x\in_{NFU}y\equiv_{def}j(x)\in y\wedge y\in V_{j(\alpha)+1}.
  80. ϕ \phi
  81. ϕ \phi
  82. ϕ 1 \phi_{1}
  83. ϕ 1 \phi_{1}
  84. N - i N-i
  85. ( x V α . ψ ( j N - i ( x ) ) ) (\forall x\in V_{\alpha}.\psi(j^{N-i}(x)))
  86. ( x j N - i ( V α ) . ψ ( x ) ) (\forall x\in j^{N-i}(V_{\alpha}).\psi(x))
  87. ϕ 2 \phi_{2}
  88. ϕ \phi
  89. j i - N j^{i-N}
  90. ϕ 3 \phi_{3}
  91. { y V α ϕ 3 } \{y\in V_{\alpha}\mid\phi_{3}\}
  92. V α + 1 V_{\alpha+1}
  93. ϕ \phi
  94. j ( { y V α ϕ 3 } ) j(\{y\in V_{\alpha}\mid\phi_{3}\})
  95. V j ( α ) + 1 V_{j(\alpha)+1}
  96. V α + 1 V_{\alpha+1}
  97. V α V_{\alpha}
  98. V j ( α ) + 1 V_{j(\alpha)+1}
  99. α \alpha
  100. α \alpha
  101. T i T_{i}
  102. P ( T i ) P(T_{i})
  103. P 1 ( T i + 1 ) P_{1}(T_{i+1})
  104. T i T_{i}
  105. T i + 1 T_{i+1}
  106. T 0 T_{0}
  107. T 1 T_{1}
  108. T α T_{\alpha}
  109. ω \beth_{\omega}
  110. T α T_{\alpha}
  111. V α V_{\alpha}
  112. V j ( α ) V_{j(\alpha)}
  113. T 2 ( n ) T^{2}(n)
  114. T ( | A | ) = | P 1 ( A ) | T(|A|)=|P_{1}(A)|
  115. T ( | A | ) = | A | T(|A|)=|A|
  116. n \beth_{n}
  117. ω \beth_{\omega}
  118. n \beth_{\beth_{n}}
  119. ω \beth_{\beth_{\omega}}
  120. N N
  121. ϕ \phi
  122. Σ 2 \Sigma_{2}
  123. ϕ \phi
  124. α \alpha
  125. T n ( Ω ) < α T^{n}(\Omega)<\alpha
  126. Ω \Omega
  127. T n ( Ω ) T^{n}(\Omega)
  128. s n s_{n}
  129. s 0 = Ω s_{0}=\Omega
  130. s i + 1 = T ( s i ) s_{i+1}=T(s_{i})
  131. T n ( Ω ) T^{n}(\Omega)
  132. j - i ( α ) j^{-i}(\alpha)

NewDES.html

  1. E K ( P ) = C , E_{K}(P)=C,
  2. E K ¯ ( P ¯ ) = C ¯ , E_{\overline{K}}(\overline{P})=\overline{C},
  3. x ¯ \overline{x}

Newmark-beta_method.html

  1. u ˙ n + 1 = u ˙ n + Δ t u ¨ γ \dot{u}_{n+1}=\dot{u}_{n}+\Delta t~{}\ddot{u}_{\gamma}\,
  2. u ¨ γ = ( 1 - γ ) u ¨ n + γ u ¨ n + 1 0 γ 1 \ddot{u}_{\gamma}=(1-\gamma)\ddot{u}_{n}+\gamma\ddot{u}_{n+1}~{}~{}~{}~{}0\leq% \gamma\leq 1
  3. u ˙ n + 1 = u ˙ n + ( 1 - γ ) Δ t u ¨ n + γ Δ t u ¨ n + 1 . \dot{u}_{n+1}=\dot{u}_{n}+(1-\gamma)\Delta t~{}\ddot{u}_{n}+\gamma\Delta t~{}% \ddot{u}_{n+1}.
  4. u n + 1 = u n + Δ t u ˙ n + 1 2 Δ t 2 u ¨ β u_{n+1}=u_{n}+\Delta t~{}\dot{u}_{n}+\begin{matrix}\frac{1}{2}\end{matrix}% \Delta t^{2}~{}\ddot{u}_{\beta}
  5. u ¨ β = ( 1 - 2 β ) u ¨ n + 2 β u ¨ n + 1 0 2 β 1 \ddot{u}_{\beta}=(1-2\beta)\ddot{u}_{n}+2\beta\ddot{u}_{n+1}~{}~{}~{}~{}0\leq 2% \beta\leq 1
  6. γ \gamma
  7. u ˙ n + 1 = u ˙ n + Δ t 2 ( u ¨ n + u ¨ n + 1 ) \dot{u}_{n+1}=\dot{u}_{n}+\begin{matrix}\frac{\Delta t}{2}\end{matrix}~{}(% \ddot{u}_{n}+\ddot{u}_{n+1})
  8. u n + 1 = u n + Δ t u ˙ n + 1 - 2 β 2 Δ t 2 u ¨ n + β Δ t 2 u ¨ n + 1 u_{n+1}=u_{n}+{\Delta}t~{}\dot{u}_{n}+\begin{matrix}\frac{1-2\beta}{2}\end{% matrix}\Delta t^{2}\ddot{u}_{n}+\beta{\Delta}t^{2}\ddot{u}_{n+1}

Newton's_cradle.html

  1. F = k x F=k\cdot x
  2. F = k x 1.5 \ F=k\cdot x^{1.5}
  3. F = m a F=m\cdot a

Newton's_rings.html

  1. r N = [ ( N - 1 2 ) λ R ] 1 / 2 , r_{N}=\left[\left(N-{1\over 2}\right)\lambda R\right]^{1/2},
  2. 2 R t = t 2 + x 2 2Rt=t^{2}+x^{2}
  3. t x t\ll x
  4. t 2 x t^{2}\lll x
  5. 2 t = X R 2t={X\over R}
  6. t = X 2 R t={X\over 2R}

Newton_polygon.html

  1. K K
  2. v K v_{K}
  3. f ( x ) = a n x n + + a 1 x + a 0 K [ x ] f(x)=a_{n}x^{n}+\cdots+a_{1}x+a_{0}\in K[x]
  4. a 0 a n 0 a_{0}a_{n}\neq 0
  5. f f
  6. P i = ( i , v K ( a i ) ) , P_{i}=\left(i,v_{K}(a_{i})\right),
  7. a i = 0 a_{i}=0
  8. 3 x 2 y 3 - x y 2 + 2 x 2 y 2 - x 3 y = 0 3x^{2}y^{3}-xy^{2}+2x^{2}y^{2}-x^{3}y=0
  9. μ 1 , μ 2 , , μ r \mu_{1},\mu_{2},\ldots,\mu_{r}
  10. f ( x ) f(x)
  11. λ 1 , λ 2 , , λ r \lambda_{1},\lambda_{2},\ldots,\lambda_{r}
  12. P i P_{i}
  13. P j P_{j}
  14. j - i j-i
  15. 1 κ r 1\leq\kappa\leq r
  16. f ( x ) f(x)
  17. λ κ \lambda_{\kappa}
  18. - μ κ -\mu_{\kappa}

Newtonian_dynamics.html

  1. m 𝐚 = 𝐅 \displaystyle m\,\mathbf{a}=\mathbf{F}
  2. N \displaystyle N
  3. m 1 , , m N \displaystyle m_{1},\,\ldots,\,m_{N}
  4. 𝐫 1 , , 𝐫 N \displaystyle\mathbf{r}_{1},\,\ldots,\,\mathbf{r}_{N}
  5. d 𝐫 i d t = 𝐯 i , d 𝐯 i d t = 𝐅 i ( 𝐫 1 , , 𝐫 N , 𝐯 1 , , 𝐯 N , t ) m i , i = 1 , , N . \frac{d\mathbf{r}_{i}}{dt}=\mathbf{v}_{i},\qquad\frac{d\mathbf{v}_{i}}{dt}=% \frac{\mathbf{F}_{i}(\mathbf{r}_{1},\ldots,\mathbf{r}_{N},\mathbf{v}_{1},% \ldots,\mathbf{v}_{N},t)}{m_{i}},\quad i=1,\ldots,N.
  6. 𝐫 1 , , 𝐫 N \displaystyle\mathbf{r}_{1},\,\ldots,\,\mathbf{r}_{N}
  7. n = 3 N \displaystyle n=3N
  8. 𝐯 1 , , 𝐯 N \displaystyle\mathbf{v}_{1},\,\ldots,\,\mathbf{v}_{N}
  9. n = 3 N \displaystyle n=3N
  10. 𝐫 = 𝐫 1 𝐫 N , 𝐯 = 𝐯 1 𝐯 N . \mathbf{r}=\begin{Vmatrix}\mathbf{r}_{1}\\ \vdots\\ \mathbf{r}_{N}\end{Vmatrix},\qquad\qquad\mathbf{v}=\begin{Vmatrix}\mathbf{v}_{% 1}\\ \vdots\\ \mathbf{v}_{N}\end{Vmatrix}.
  11. d 𝐫 d t = 𝐯 , d 𝐯 d t = 𝐅 ( 𝐫 , 𝐯 , t ) , \frac{d\mathbf{r}}{dt}=\mathbf{v},\qquad\frac{d\mathbf{v}}{dt}=\mathbf{F}(% \mathbf{r},\mathbf{v},t),
  12. m = 1 \displaystyle m=1
  13. 𝐫 \displaystyle\mathbf{r}
  14. ( 𝐫 , 𝐯 ) \displaystyle(\mathbf{r},\mathbf{v})
  15. m = 1 \displaystyle m=1
  16. m 1 , , m N \displaystyle m_{1},\,\ldots,\,m_{N}
  17. m 1 , , m N \displaystyle m_{1},\,\ldots,\,m_{N}
  18. 𝐫 \displaystyle\mathbf{r}
  19. n = 3 N - K \displaystyle n=3\,N-K
  20. n \displaystyle n
  21. M \displaystyle M
  22. M \displaystyle M
  23. T M \displaystyle TM
  24. q 1 , , q n \displaystyle q^{1},\,\ldots,\,q^{n}
  25. M \displaystyle M
  26. 𝐫 \displaystyle\mathbf{r}
  27. q 1 , , q n \displaystyle q^{1},\,\ldots,\,q^{n}
  28. q 1 , , q n \displaystyle q^{1},\,\ldots,\,q^{n}
  29. q ˙ 1 , , q ˙ n \displaystyle\dot{q}^{1},\,\ldots,\,\dot{q}^{n}
  30. T M \displaystyle TM
  31. M \displaystyle M
  32. 3 N \displaystyle 3\,N
  33. M \displaystyle M
  34. ( , ) \displaystyle(\ ,\ )
  35. N \displaystyle N
  36. N \displaystyle N
  37. M \displaystyle M
  38. M \displaystyle M
  39. 𝐅 \displaystyle\mathbf{F}
  40. M \displaystyle M
  41. M \displaystyle M
  42. 𝐍 \displaystyle\mathbf{N}
  43. 𝐅 \displaystyle\mathbf{F}_{\parallel}
  44. F 1 , , F n F^{1},\,\ldots,\,F^{n}
  45. M \displaystyle M
  46. Γ i j s \Gamma^{s}_{ij}
  47. T = T ( q 1 , , q n , w 1 , , w n ) T=T(q^{1},\ldots,q^{n},w^{1},\ldots,w^{n})
  48. Q 1 , , Q n Q_{1},\,\ldots,\,Q_{n}
  49. 𝐅 \mathbf{F}_{\parallel}
  50. F 1 , , F n F^{1},\,\ldots,\,F^{n}
  51. 𝐅 \mathbf{F}_{\parallel}
  52. M \displaystyle M
  53. T \displaystyle T

Newtonian_potential.html

  1. u ( x ) = Γ * f ( x ) = d Γ ( x - y ) f ( y ) d y u(x)=\Gamma*f(x)=\int_{\mathbb{R}^{d}}\Gamma(x-y)f(y)\,dy
  2. Γ ( x ) = { 1 2 π log | x | d = 2 1 d ( 2 - d ) ω d | x | 2 - d d 2. \Gamma(x)=\begin{cases}\frac{1}{2\pi}\log{|x|}&d=2\\ \frac{1}{d(2-d)\omega_{d}}|x|^{2-d}&d\neq 2.\end{cases}
  3. Δ w = f , \Delta w=f,\,
  4. Γ * μ ( x ) = d Γ ( x - y ) d μ ( y ) \Gamma*\mu(x)=\int_{\mathbb{R}^{d}}\Gamma(x-y)\,d\mu(y)
  5. Δ w = μ \Delta w=\mu\,
  6. f * Γ ( x ) = λ Γ ( x ) , λ = d f ( y ) d y . f*\Gamma(x)=\lambda\Gamma(x),\quad\lambda=\int_{\mathbb{R}^{d}}f(y)\,dy.

NEXPTIME.html

  1. NEXPTIME = k NTIME ( 2 n k ) \mbox{NEXPTIME}~{}=\bigcup_{k\in\mathbb{N}}\mbox{NTIME}~{}(2^{n^{k}})
  2. \subseteq
  3. \subseteq
  4. \subseteq
  5. \subsetneq

Neyman–Pearson_lemma.html

  1. Λ ( x ) = L ( θ 0 x ) L ( θ 1 x ) η \Lambda(x)=\frac{L(\theta_{0}\mid x)}{L(\theta_{1}\mid x)}\leq\eta
  2. P ( Λ ( X ) η H 0 ) = α P(\Lambda(X)\leq\eta\mid H_{0})=\alpha
  3. θ 1 Θ 1 \theta_{1}\in\Theta_{1}
  4. Θ 1 \Theta_{1}\,
  5. R N P = { x : L ( θ 0 | x ) L ( θ 1 | x ) η } R_{NP}=\left\{x:\frac{L(\theta_{0}|x)}{L(\theta_{1}|x)}\leq\eta\right\}
  6. η \eta
  7. P ( R N P , θ 0 ) = α P(R_{NP},\theta_{0})=\alpha\,
  8. R A R_{A}
  9. θ \theta
  10. P ( R , θ ) = R L ( θ | x ) d x , P(R,\theta)=\int_{R}L(\theta|x)\,dx,
  11. R A R_{A}
  12. α \alpha
  13. α P ( R A , θ 0 ) \alpha\geq P(R_{A},\theta_{0})
  14. α = P ( R N P , θ 0 ) P ( R A , θ 0 ) . \alpha=P(R_{NP},\theta_{0})\geq P(R_{A},\theta_{0})\,.
  15. P ( R N P , θ ) = P ( R N P R A , θ ) + P ( R N P R A c , θ ) , P(R_{NP},\theta)=P(R_{NP}\cap R_{A},\theta)+P(R_{NP}\cap R_{A}^{c},\theta),
  16. P ( R A , θ ) = P ( R N P R A , θ ) + P ( R N P c R A , θ ) . P(R_{A},\theta)=P(R_{NP}\cap R_{A},\theta)+P(R_{NP}^{c}\cap R_{A},\theta).
  17. θ = θ 0 \theta=\theta_{0}
  18. P ( R N P R A c , θ 0 ) P ( R N P c R A , θ 0 ) . P(R_{NP}\cap R_{A}^{c},\theta_{0})\geq P(R_{NP}^{c}\cap R_{A},\theta_{0}).
  19. P ( R N P , θ 1 ) P(R_{NP},\theta_{1})
  20. P ( R A , θ 1 ) P(R_{A},\theta_{1})
  21. P ( R N P , θ 1 ) P ( R A , θ 1 ) P ( R N P R A c , θ 1 ) P ( R N P c R A , θ 1 ) . P(R_{NP},\theta_{1})\geq P(R_{A},\theta_{1})\iff P(R_{NP}\cap R_{A}^{c},\theta% _{1})\geq P(R_{NP}^{c}\cap R_{A},\theta_{1}).
  22. R N P R_{NP}
  23. P ( R N P R A c , θ 1 ) = R N P R A c L ( θ 1 | x ) d x 1 η R N P R A c L ( θ 0 | x ) d x = 1 η P ( R N P R A c , θ 0 ) P(R_{NP}\cap R_{A}^{c},\theta_{1})=\int_{R_{NP}\cap R_{A}^{c}}L(\theta_{1}|x)% \,dx\geq\frac{1}{\eta}\int_{R_{NP}\cap R_{A}^{c}}L(\theta_{0}|x)\,dx=\frac{1}{% \eta}P(R_{NP}\cap R_{A}^{c},\theta_{0})
  24. 1 η P ( R N P c R A , θ 0 ) = 1 η R N P c R A L ( θ 0 | x ) d x R N P c R A L ( θ 1 | x ) d x = P ( R N P c R A , θ 1 ) . \geq\frac{1}{\eta}P(R_{NP}^{c}\cap R_{A},\theta_{0})=\frac{1}{\eta}\int_{R_{NP% }^{c}\cap R_{A}}L(\theta_{0}|x)\,dx\geq\int_{R_{NP}^{c}\cap R_{A}}L(\theta_{1}% |x)dx=P(R_{NP}^{c}\cap R_{A},\theta_{1}).
  25. X 1 , , X n X_{1},\dots,X_{n}
  26. 𝒩 ( μ , σ 2 ) \mathcal{N}(\mu,\sigma^{2})
  27. μ \mu
  28. H 0 : σ 2 = σ 0 2 H_{0}:\sigma^{2}=\sigma_{0}^{2}
  29. H 1 : σ 2 = σ 1 2 H_{1}:\sigma^{2}=\sigma_{1}^{2}
  30. L ( σ 2 ; 𝐱 ) ( σ 2 ) - n / 2 exp { - i = 1 n ( x i - μ ) 2 2 σ 2 } . L\left(\sigma^{2};\mathbf{x}\right)\propto\left(\sigma^{2}\right)^{-n/2}\exp% \left\{-\frac{\sum_{i=1}^{n}\left(x_{i}-\mu\right)^{2}}{2\sigma^{2}}\right\}.
  31. Λ ( 𝐱 ) = L ( σ 0 2 ; 𝐱 ) L ( σ 1 2 ; 𝐱 ) = ( σ 0 2 σ 1 2 ) - n / 2 exp { - 1 2 ( σ 0 - 2 - σ 1 - 2 ) i = 1 n ( x i - μ ) 2 } . \Lambda(\mathbf{x})=\frac{L\left(\sigma_{0}^{2};\mathbf{x}\right)}{L\left(% \sigma_{1}^{2};\mathbf{x}\right)}=\left(\frac{\sigma_{0}^{2}}{\sigma_{1}^{2}}% \right)^{-n/2}\exp\left\{-\frac{1}{2}(\sigma_{0}^{-2}-\sigma_{1}^{-2})\sum_{i=% 1}^{n}\left(x_{i}-\mu\right)^{2}\right\}.
  32. i = 1 n ( x i - μ ) 2 \sum_{i=1}^{n}\left(x_{i}-\mu\right)^{2}
  33. i = 1 n ( x i - μ ) 2 \sum_{i=1}^{n}\left(x_{i}-\mu\right)^{2}
  34. σ 1 2 > σ 0 2 \sigma_{1}^{2}>\sigma_{0}^{2}
  35. Λ ( 𝐱 ) \Lambda(\mathbf{x})
  36. i = 1 n ( x i - μ ) 2 \sum_{i=1}^{n}\left(x_{i}-\mu\right)^{2}
  37. H 0 H_{0}
  38. i = 1 n ( x i - μ ) 2 \sum_{i=1}^{n}\left(x_{i}-\mu\right)^{2}

NGC_281.html

  1. β 1 \beta 1

Nilpotent_matrix.html

  1. N k = 0 N^{k}=0\,
  2. M = [ 0 1 0 0 ] M=\begin{bmatrix}0&1\\ 0&0\end{bmatrix}
  3. N = [ 0 2 1 6 0 0 1 2 0 0 0 3 0 0 0 0 ] N=\begin{bmatrix}0&2&1&6\\ 0&0&1&2\\ 0&0&0&3\\ 0&0&0&0\end{bmatrix}
  4. N 2 = [ 0 0 2 7 0 0 0 3 0 0 0 0 0 0 0 0 ] ; N 3 = [ 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 ] ; N 4 = [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] . N^{2}=\begin{bmatrix}0&0&2&7\\ 0&0&0&3\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix};\ N^{3}=\begin{bmatrix}0&0&0&6\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix};\ N^{4}=\begin{bmatrix}0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix}.
  5. N = [ 5 - 3 2 15 - 9 6 10 - 6 4 ] N=\begin{bmatrix}5&-3&2\\ 15&-9&6\\ 10&-6&4\end{bmatrix}
  6. S = [ 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 ] . S=\begin{bmatrix}0&1&0&\ldots&0\\ 0&0&1&\ldots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\ldots&1\\ 0&0&0&\ldots&0\end{bmatrix}.
  7. S ( x 1 , x 2 , , x n ) = ( x 2 , , x n , 0 ) . S(x_{1},x_{2},\ldots,x_{n})=(x_{2},\ldots,x_{n},0).
  8. [ S 1 0 0 0 S 2 0 0 0 S r ] \begin{bmatrix}S_{1}&0&\ldots&0\\ 0&S_{2}&\ldots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&S_{r}\end{bmatrix}
  9. [ 0 1 0 0 ] . \begin{bmatrix}0&1\\ 0&0\end{bmatrix}.
  10. { 0 } ker L ker L 2 ker L q - 1 ker L q = n \{0\}\subset\ker L\subset\ker L^{2}\subset\ldots\subset\ker L^{q-1}\subset\ker L% ^{q}=\mathbb{R}^{n}
  11. 0 = n 0 < n 1 < n 2 < < n q - 1 < n q = n , n i = dim ker L i . 0=n_{0}<n_{1}<n_{2}<\ldots<n_{q-1}<n_{q}=n,\qquad n_{i}=\dim\ker L^{i}.
  12. n j + 1 - n j n j - n j - 1 , for all j = 1 , , q - 1. n_{j+1}-n_{j}\leq n_{j}-n_{j-1},\qquad\mbox{for all }~{}j=1,\ldots,q-1.
  13. ( I + N ) - 1 = I - N + N 2 - N 3 + , (I+N)^{-1}=I-N+N^{2}-N^{3}+\cdots,
  14. det ( I + N ) = 1 , \det(I+N)=1,\!\,
  15. det ( I + t A ) = 1 \det(I+tA)=1\!\,
  16. p ( t ) = det ( I + t A ) - 1 p(t)=\det(I+tA)-1
  17. n n
  18. n + 1 n+1
  19. t t
  20. T k ( v ) = 0. T^{k}(v)=0.\!\,

Nilpotent_operator.html

  1. K ( x , y ) = { 1 , if x y 0 , otherwise . K(x,y)=\left\{\begin{matrix}1,&\mbox{if}~{}\;x\geq y\\ 0,&\mbox{otherwise}~{}.\end{matrix}\right.
  2. T f ( x ) = 0 1 K ( x , y ) f ( y ) d y . Tf(x)=\int_{0}^{1}K(x,y)f(y)dy.

Nitric_oxide_synthase.html

  1. \rightleftharpoons

Nitrogen-13.html

  1. K = ( 1 + m / M ) | E | K=(1+m/M)|E|

Nitrogenase.html

  1. Δ H 0 = - 45.2 kJ mol - 1 NH 3 \Delta H^{0}=-45.2\ \mathrm{kJ}\,\mathrm{mol^{-1}}\;\mathrm{NH_{3}}
  2. E A = 230 - 420 kJ mol - 1 E_{A}=230-420\ \mathrm{kJ}\,\mathrm{mol^{-1}}
  3. N 2 + 8 H + + 8 e - + 16 ATP 2 NH 3 + H 2 + 16 ADP + 16 P i \mathrm{N_{2}+8\ H^{+}+8\ e^{-}+16\ ATP\longrightarrow 2\ NH_{3}+H_{2}+16\ ADP% +16\ P_{i}}

NL_(complexity).html

  1. ¬ \neg
  2. ( x 1 ¬ x 3 ) ( ¬ x 2 x 3 ) ( ¬ x 1 ¬ x 2 ) (x_{1}\vee\neg x_{3})\wedge(\neg x_{2}\vee x_{3})\wedge(\neg x_{1}\vee\neg x_{% 2})
  3. 𝐍𝐂 1 𝐋 𝐍𝐋 𝐍𝐂 2 . \mathbf{NC}_{1}\subseteq\mathbf{L}\subseteq\mathbf{NL}\subseteq\mathbf{NC}_{2}.
  4. 𝐍𝐋 𝐒𝐏𝐀𝐂𝐄 ( log 2 n ) equivalently, 𝐍𝐋 𝐋 2 . \mathbf{NL\subseteq SPACE}(\log^{2}n)\ \ \ \ \,\text{equivalently, }\mathbf{NL% \subseteq L}^{2}.

Non-standard_calculus.html

  1. 1 \tfrac{1}{\infty}
  2. f f^{\prime}\,
  3. y = f ( x ) = x 2 y=f(x)=x^{2}
  4. Δ y Δ x = ( x + Δ x ) 2 - x 2 Δ x = 2 x + Δ x 2 x \frac{\Delta y}{\Delta x}=\frac{(x+\Delta x)^{2}-x^{2}}{\Delta x}=2x+\Delta x% \approx 2x
  5. \approx
  6. Δ x \Delta x
  7. Δ x \Delta x
  8. Δ x \Delta x
  9. Δ y / Δ x \Delta y/\Delta x
  10. f ( x ) = st ( f * ( x + ϵ ) - f * ( x ) ϵ ) , f^{\prime}(x)=\mathrm{st}\left(\frac{f^{*}(x+\epsilon)-f^{*}(x)}{\epsilon}% \right),
  11. f * f^{*}
  12. f f
  13. \approx
  14. x x x^{\prime}\approx x
  15. f * ( x ) f * ( x ) f^{*}(x^{\prime})\approx f^{*}(x)
  16. f * ( x ) f * ( y ) f^{*}(x)\approx f^{*}(y)
  17. x 2 x^{2}\,
  18. \forall
  19. \exists
  20. c = st ( x ) = st ( y ) . c=\operatorname{st}(x)=\operatorname{st}(y).\,
  21. f ( x ) f ( c ) f ( y ) , f(x)\approx f(c)\approx f(y),\,
  22. \mathbb{R}
  23. N * N\in\mathbb{R}^{*}
  24. N + 1 N N+\tfrac{1}{N}
  25. f ( N + 1 N ) - f ( N ) = N 2 + 2 + 1 N 2 - N 2 = 2 + 1 N 2 f(N+\tfrac{1}{N})-f(N)=N^{2}+2+\tfrac{1}{N^{2}}-N^{2}=2+\tfrac{1}{N^{2}}
  26. I Q ( x ) := { 1 if x is rational , 0 if x is irrational . I_{Q}(x):=\begin{cases}1&\,\text{ if }x\,\text{ is rational},\\ 0&\,\text{ if }x\,\text{ is irrational}.\end{cases}
  27. lim x a f ( x ) = L \lim_{x\to a}f(x)=L\,
  28. { x n | n } \{x_{n}|n\in\mathbb{N}\}\;
  29. L L\in\mathbb{R}\;
  30. L = lim n x n L=\lim_{n\to\infty}x_{n}
  31. L = lim n x n ϵ > 0 , N , n : n > N | x n - L | < ϵ . L=\lim_{n\to\infty}x_{n}\Longleftrightarrow\forall\epsilon>0\;,\exists N\in% \mathbb{N}\;,\forall n\in\mathbb{N}:n>N\rightarrow|x_{n}-L|<\epsilon.\;
  32. f ( x i 0 ) f ( x i ) f(x_{i_{0}})\geq f(x_{i})
  33. c = st ( x i 0 ) c={\rm st}(x_{i_{0}})
  34. x [ x i , x i + 1 ] x\in[x_{i},x_{i+1}]
  35. f ( x i 0 ) f ( x i ) f(x_{i_{0}})\geq f(x_{i})
  36. st ( f ( x i 0 ) ) st ( f ( x i ) ) {\rm st}(f(x_{i_{0}}))\geq{\rm st}(f(x_{i}))
  37. st ( f ( x i 0 ) ) = f ( st ( x i 0 ) ) = f ( c ) {\rm st}(f(x_{i_{0}}))=f({\rm st}(x_{i_{0}}))=f(c)
  38. c = st ( x i 0 ) c=\mathrm{st}(x_{i_{0}})
  39. k = 0 n - 1 f ( ξ k ) ( x k + 1 - x k ) \sum_{k=0}^{n-1}f(\xi_{k})(x_{k+1}-x_{k})
  40. a = x 0 ξ 0 x 1 x n - 1 ξ n - 1 x n = b . a=x_{0}\leq\xi_{0}\leq x_{1}\leq\ldots x_{n-1}\leq\xi_{n-1}\leq x_{n}=b.
  41. sup k ( x k + 1 - x k ) \sup_{k}(x_{k+1}-x_{k})
  42. S M = st k = 0 n - 1 [ * f ] ( ξ k ) ( x k + 1 - x k ) S_{M}=\operatorname{st}\sum_{k=0}^{n-1}[*f](\xi_{k})(x_{k+1}-x_{k})
  43. Δ h f = st f ( x + h ) - f ( x ) h \Delta_{h}f=\operatorname{st}\frac{f(x+h)-f(x)}{h}
  44. | f ( x ) | M a x b . |f^{\prime}(x)|\leq M\quad a\leq x\leq b.
  45. | f ( b ) - f ( a ) | M ( b - a ) + ϵ . |f(b)-f(a)|\leq M(b-a)+\epsilon.
  46. a = x 0 < x 1 < < x N - 1 < x N = b a=x_{0}<x_{1}<\cdots<x_{N-1}<x_{N}=b
  47. | f ( b ) - f ( a ) | k = 1 N - 1 | f ( x k + 1 ) - f ( x k ) | k = 1 N - 1 { | f ( x k ) | + ϵ k } | x k + 1 - x k | . |f(b)-f(a)|\leq\sum_{k=1}^{N-1}|f(x_{k+1})-f(x_{k})|\leq\sum_{k=1}^{N-1}\left% \{|f^{\prime}(x_{k})|+\epsilon_{k}\right\}|x_{k+1}-x_{k}|.
  48. | f ( b ) - f ( a ) | k = 1 N - 1 ( M + ϵ ) ( x k + 1 - x k ) = M ( b - a ) + ϵ ( b - a ) |f(b)-f(a)|\leq\sum_{k=1}^{N-1}(M+\epsilon)(x_{k+1}-x_{k})=M(b-a)+\epsilon(b-a)

Non-well-founded_set_theory.html

  1. x 2 x 1 x 0 . \in x_{2}\in x_{1}\in x_{0}.

Nonagonal_number.html

  1. n ( 7 n - 5 ) 2 . \frac{n(7n-5)}{2}.
  2. 7 N ( n ) + 3 = T ( 7 n - 3 ) . {7N(n)+3=T(7n-3)}.
  3. x = 56 n + 25 + 5 14 . x=\frac{\sqrt{56n+25}+5}{14}.

Nonassociative_ring.html

  1. a + b = b + a a+b=b+a
  2. ( a + b ) + c = a + ( b + c ) (a+b)+c=a+(b+c)
  3. 0 + a = a + 0 = a 0+a=a+0=a
  4. a + ( - a ) = ( - a ) + a = 0 a+(-a)=(-a)+a=0
  5. ( a + b ) c = a c + b c (a+b)c=ac+bc
  6. a ( b + c ) = a b + a c a(b+c)=ab+ac
  7. 1 x = x 1 = x 1x=x1=x
  8. x x
  9. a L a^{L}
  10. a R a^{R}
  11. a L a^{L}
  12. a R a^{R}

Noncommutative_quantum_field_theory.html

  1. [ x μ , x ν ] = i θ μ ν [x^{\mu},x^{\nu}]=i\theta^{\mu\nu}\,\!

Noncommutative_topology.html

  1. K K ( A , B ) × K K ( B , C ) K K ( A , C ) KK(A,B)\times KK(B,C)\rightarrow KK(A,C)

Noncototient.html

  1. p q - φ ( p q ) = p q - ( p - 1 ) ( q - 1 ) = p + q - 1 = n - 1. pq-\varphi(pq)=pq-(p-1)(q-1)=p+q-1=n-1.\,
  2. 1 = 2 - ϕ ( 2 ) , 3 = 9 - ϕ ( 9 ) 1=2-\phi(2),3=9-\phi(9)
  3. 5 = 25 - ϕ ( 25 ) 5=25-\phi(25)
  4. 2 k 509203 2^{k}\cdot 509203

Noncrossing_partition.html

  1. NC ( S ) \,\text{NC}(S)
  2. NC ( S 1 ) \,\text{NC}(S_{1})
  3. NC ( S 2 ) \,\text{NC}(S_{2})
  4. S 1 , S 2 S_{1},S_{2}
  5. NC ( S ) \,\text{NC}(S)
  6. S S
  7. NC ( n ) \,\text{NC}(n)
  8. ( 𝒜 , ϕ ) (\mathcal{A},\phi)
  9. a 𝒜 a\in\mathcal{A}
  10. ( k n ) n (k_{n})_{n\in\mathbb{N}}
  11. ϕ ( a n ) = π NC ( n ) j k j N j ( π ) \phi(a^{n})=\sum_{\pi\in\,\text{NC}(n)}\prod_{j}k_{j}^{N_{j}(\pi)}
  12. N j ( π ) N_{j}(\pi)
  13. j j
  14. π \pi

Nondeterministic_algorithm.html

  1. a n - 1 1 ( mod n ) a^{n-1}\neq 1\;\;(\mathop{{\rm mod}}n)

Nondeterministic_finite_automaton.html

  1. n - 1 n-1

Nonlinear_autoregressive_exogenous_model.html

  1. y t = F ( y t - 1 , y t - 2 , y t - 3 , , u t , u t - 1 , u t - 2 , u t - 3 , ) + ε t y_{t}=F(y_{t-1},y_{t-2},y_{t-3},\ldots,u_{t},u_{t-1},u_{t-2},u_{t-3},\ldots)+% \varepsilon_{t}

Nonlinear_programming.html

  1. minimize f ( x ) subject to g i ( x ) 0 for each i { 1 , , m } h j ( x ) = 0 for each j { 1 , , p } x X . \begin{aligned}\displaystyle\,\text{minimize }&\displaystyle f(x)\\ \displaystyle\,\text{subject to }&\displaystyle g_{i}(x)\leq 0\,\text{ for % each }i\in\{1,\ldots,m\}\\ &\displaystyle h_{j}(x)=0\,\text{ for each }j\in\{1,\ldots,p\}\\ &\displaystyle x\in X.\end{aligned}

Nonlinear_regression.html

  1. v = V max [ S ] K m + [ S ] v=\frac{V_{\max}\ [\mbox{S}~{}]}{K_{m}+[\mbox{S}~{}]}
  2. f ( x , s y m b o l β ) = β 1 x β 2 + x f(x,symbol\beta)=\frac{\beta_{1}x}{\beta_{2}+x}
  3. β 1 \beta_{1}
  4. V max V_{\max}
  5. β 2 \beta_{2}
  6. K m K_{m}
  7. β \beta
  8. f ( x i , s y m b o l β ) f 0 + j J i j β j f(x_{i},symbol\beta)\approx f^{0}+\sum_{j}J_{ij}\beta_{j}
  9. J i j = f ( x i , s y m b o l β ) β j J_{ij}=\frac{\partial f(x_{i},symbol\beta)}{\partial\beta_{j}}
  10. s y m b o l β ^ ( 𝐉 𝐓 𝐉 ) - 𝟏 𝐉 𝐓 𝐲 . \hat{symbol{\beta}}\approx\mathbf{(J^{T}J)^{-1}J^{T}y}.
  11. y = a e b x U y=ae^{bx}U\,\!
  12. ln ( y ) = ln ( a ) + b x + u , \ln{(y)}=\ln{(a)}+bx+u,\,\!
  13. 1 v = 1 V max + K m V max [ S ] \frac{1}{v}=\frac{1}{V_{\max}}+\frac{K_{m}}{V_{\max}[S]}

Norm_(mathematics).html

  1. | a | |a|
  2. | 𝐯 | |\mathbf{v}|
  3. x = | x | \left\|x\right\|=\left|x\right|
  4. s y m b o l x := x 1 2 + + x n 2 . \left\|symbol{x}\right\|:=\sqrt{x_{1}^{2}+\cdots+x_{n}^{2}}.
  5. s y m b o l z := | z 1 | 2 + + | z n | 2 = z 1 z ¯ 1 + + z n z ¯ n . \left\|symbol{z}\right\|:=\sqrt{\left|z_{1}\right|^{2}+\cdots+\left|z_{n}% \right|^{2}}=\sqrt{z_{1}\bar{z}_{1}+\cdots+z_{n}\bar{z}_{n}}.
  6. s y m b o l x := s y m b o l x * s y m b o l x , \left\|symbol{x}\right\|:=\sqrt{symbol{x}^{*}~{}symbol{x}},
  7. s y m b o l x := s y m b o l x s y m b o l x . \left\|symbol{x}\right\|:=\sqrt{symbol{x}\cdot symbol{x}}.
  8. x 2 + y 2 \sqrt{x^{2}+y^{2}}
  9. s y m b o l x 1 := i = 1 n | x i | . \left\|symbol{x}\right\|_{1}:=\sum_{i=1}^{n}\left|x_{i}\right|.
  10. \ell
  11. \ell
  12. i = 1 n x i \sum_{i=1}^{n}x_{i}
  13. 𝐱 p := ( i = 1 n | x i | p ) 1 / p . \left\|\mathbf{x}\right\|_{p}:=\bigg(\sum_{i=1}^{n}\left|x_{i}\right|^{p}\bigg% )^{1/p}.
  14. \infty
  15. x k 𝐱 2 = x k 𝐱 2 , \frac{\partial}{\partial x_{k}}\left\|\mathbf{x}\right\|_{2}=\frac{x_{k}}{% \left\|\mathbf{x}\right\|_{2}},
  16. 𝐱 𝐱 2 = 𝐱 𝐱 2 . \frac{\partial}{\partial\mathbf{x}}\left\|\mathbf{x}\right\|_{2}=\frac{\mathbf% {x}}{\left\|\mathbf{x}\right\|_{2}}.
  17. 𝐱 := max ( | x 1 | , , | x n | ) . \left\|\mathbf{x}\right\|_{\infty}:=\max\left(\left|x_{1}\right|,\ldots,\left|% x_{n}\right|\right).
  18. ( x n ) n 2 - n x n / ( 1 + x n ) (x_{n})\mapsto\sum_{n}{2^{-n}x_{n}/(1+x_{n})}
  19. x := 2 | x 1 | + 3 | x 2 | 2 + max ( | x 3 | , 2 | x 4 | ) 2 \left\|x\right\|:=2\left|x_{1}\right|+\sqrt{3\left|x_{2}\right|^{2}+\max(\left% |x_{3}\right|,2\left|x_{4}\right|)^{2}}
  20. A x . \left\|Ax\right\|.
  21. x p = ( i | x i | p ) 1 / p resp. f p , X = ( X | f ( x ) | p d x ) 1 / p \left\|x\right\|_{p}=\bigg(\sum_{i\in\mathbb{N}}\left|x_{i}\right|^{p}\bigg)^{% 1/p}\,\text{ resp. }\left\|f\right\|_{p,X}=\bigg(\int_{X}\left|f(x)\right|^{p}% ~{}\mathrm{d}x\bigg)^{1/p}
  22. X X\subset\mathbb{R}
  23. x := x , x . \left\|x\right\|:=\sqrt{\langle x,x\rangle}.
  24. { v n } \{v_{n}\}
  25. v v
  26. v n - v 0 \left\|v_{n}-v\right\|\rightarrow 0
  27. n n\to\infty
  28. C x α x β D x α . C\left\|x\right\|_{\alpha}\leq\left\|x\right\|_{\beta}\leq D\left\|x\right\|_{% \alpha}.
  29. 𝐂 n \mathbf{C}^{n}
  30. x p x r n ( 1 / r - 1 / p ) x p . \left\|x\right\|_{p}\leq\left\|x\right\|_{r}\leq n^{(1/r-1/p)}\left\|x\right\|% _{p}.
  31. x 2 x 1 n x 2 \left\|x\right\|_{2}\leq\left\|x\right\|_{1}\leq\sqrt{n}\left\|x\right\|_{2}
  32. x x 2 n x \left\|x\right\|_{\infty}\leq\left\|x\right\|_{2}\leq\sqrt{n}\left\|x\right\|_% {\infty}
  33. x x 1 n x . \left\|x\right\|_{\infty}\leq\left\|x\right\|_{1}\leq n\left\|x\right\|_{% \infty}.
  34. p ( x ) := i = 0 n p i ( x ) p(x):=\sum_{i=0}^{n}p_{i}(x)
  35. p ( u - 0 ) p(u-0)
  36. p ( v - 0 ) p(v-0)
  37. p ( u - 0 ) p ( u - v ) + p ( v - 0 ) p ( u - v ) p ( u ) - p ( v ) p(u-0)\leq p(u-v)+p(v-0)\Rightarrow p(u-v)\geq p(u)-p(v)
  38. p ( u - 0 ) p ( u + v ) + p ( 0 - v ) p ( u + v ) p ( u ) - p ( v ) p(u-0)\leq p(u+v)+p(0-v)\Rightarrow p(u+v)\geq p(u)-p(v)
  39. p ( v - 0 ) p ( u - v ) + p ( u - 0 ) p ( u - v ) p ( v ) - p ( u ) p(v-0)\leq p(u-v)+p(u-0)\Rightarrow p(u-v)\geq p(v)-p(u)
  40. p ( v - 0 ) p ( u + v ) + p ( 0 - u ) p ( u + v ) p ( v ) - p ( u ) p(v-0)\leq p(u+v)+p(0-u)\Rightarrow p(u+v)\geq p(v)-p(u)
  41. X X
  42. Y Y
  43. u : X Y u:X\to Y
  44. u u
  45. u u
  46. | x 𝖳 y | x p y q 1 p + 1 q = 1. \left|x^{\mathsf{T}}y\right|\leq\left\|x\right\|_{p}\left\|y\right\|_{q}\qquad% \frac{1}{p}+\frac{1}{q}=1.
  47. | x 𝖳 y | x 2 y 2 . \left|x^{\mathsf{T}}y\right|\leq\left\|x\right\|_{2}\left\|y\right\|_{2}.
  48. b 1 b\geq 1
  49. p ( u + v ) b ( p ( u ) + p ( v ) ) p(u+v)\leq b(p(u)+p(v))
  50. u , v V u,v\in V
  51. 0 < k 1 0<k\leq 1
  52. v V v\in V
  53. λ \lambda
  54. p ( λ v ) = | λ | k p ( v ) p(\lambda v)=\left|\lambda\right|^{k}p(v)
  55. 0 < k < log 2 2 b 0<k<\log^{2}_{2}b

Normal_basis.html

  1. { β , β p , β p 2 , , β p m - 1 } \{\beta,\beta^{p},\beta^{p^{2}},\ldots,\beta^{p^{m-1}}\}

Normal_extension.html

  1. ( 2 ) \mathbb{Q}(\sqrt{2})
  2. \mathbb{Q}
  3. ( 2 3 ) \mathbb{Q}(\sqrt[3]{2})
  4. \mathbb{Q}
  5. 2 3 \sqrt[3]{2}
  6. ( 2 3 ) \mathbb{Q}(\sqrt[3]{2})
  7. \mathbb{Q}
  8. 𝔸 \mathbb{A}
  9. \mathbb{Q}
  10. ( 2 3 ) \mathbb{Q}(\sqrt[3]{2})
  11. ( 2 3 ) = { a + b 2 3 + c 4 3 𝔸 | a , b , c } \mathbb{Q}(\sqrt[3]{2})=\{a+b\sqrt[3]{2}+c\sqrt[3]{4}\in\mathbb{A}\,|\,a,b,c% \in\mathbb{Q}\}
  12. σ : ( 2 3 ) 𝔸 a + b 2 3 + c 4 3 a + b ω 2 3 + c ω 2 4 3 \begin{array}[]{rccc}\sigma:&\mathbb{Q}(\sqrt[3]{2})&\longrightarrow&\mathbb{A% }\\ &a+b\sqrt[3]{2}+c\sqrt[3]{4}&\mapsto&a+b\omega\sqrt[3]{2}+c\omega^{2}\sqrt[3]{% 4}\end{array}
  13. ( 2 3 ) \mathbb{Q}(\sqrt[3]{2})
  14. 𝔸 \mathbb{A}
  15. \mathbb{Q}
  16. ( 2 3 ) \mathbb{Q}(\sqrt[3]{2})
  17. ( 2 p , ζ p ) \mathbb{Q}(\sqrt[p]{2},\zeta_{p})
  18. ζ p \zeta_{p}
  19. ( 2 3 , ζ 3 ) \mathbb{Q}(\sqrt[3]{2},\zeta_{3})
  20. ( 2 3 ) \mathbb{Q}(\sqrt[3]{2})

Normal_force.html

  1. F n F_{n}
  2. N = m g N=mg
  3. N = m g cos ( θ ) N=mg\cos(\theta)
  4. 𝐍 = 𝐧 N = 𝐧 ( 𝐓 𝐧 ) = 𝐧 ( 𝐧 τ 𝐧 ) . \mathbf{N}=\mathbf{n}\,N=\mathbf{n}\,(\mathbf{T}\cdot\mathbf{n})=\mathbf{n}\,(% \mathbf{n}\cdot\mathbf{\tau}\cdot\mathbf{n}).
  5. N i = n i N = n i T j n j = n i n k τ j k n j . \ N_{i}=n_{i}N=n_{i}T_{j}n_{j}=n_{i}n_{k}\tau_{jk}n_{j}.
  6. F f r F_{f}r
  7. μ s = tan ( θ ) \mu_{s}=\tan(\theta)
  8. θ \theta
  9. N = m ( g + a ) N=m(g+a)
  10. Σ \Sigma
  11. f ( x , y , z ) = 0 f(x,y,z)=0
  12. m 𝐚 = 𝐅 + 𝐍 m\mathbf{a}=\mathbf{F}+\mathbf{N}
  13. 𝐚 \mathbf{a}

Normal_modal_logic.html

  1. ( A B ) ( A B ) \Box(A\to B)\to(\Box A\to\Box B)
  2. A B , A B A\to B,A\vdash B
  3. A \vdash A
  4. A \vdash\Box A

Normal_order.html

  1. O ^ \hat{O}
  2. O ^ \hat{O}
  3. 𝒩 ( O ^ ) \mathcal{N}(\hat{O})
  4. : O ^ : \mathopen{:}\hat{O}\mathclose{:}
  5. b ^ \hat{b}^{\dagger}
  6. b ^ \hat{b}
  7. [ b ^ , b ^ ] - = 0 \left[\hat{b}^{\dagger},\hat{b}^{\dagger}\right]_{-}=0
  8. [ b ^ , b ^ ] - = 0 \left[\hat{b},\hat{b}\right]_{-}=0
  9. [ b ^ , b ^ ] - = 1 \left[\hat{b},\hat{b}^{\dagger}\right]_{-}=1
  10. [ A , B ] - A B - B A \left[A,B\right]_{-}\equiv AB-BA
  11. b ^ b ^ = b ^ b ^ + 1. \hat{b}\,\hat{b}^{\dagger}=\hat{b}^{\dagger}\,\hat{b}+1.
  12. b ^ b ^ \hat{b}^{\dagger}\hat{b}
  13. : b ^ b ^ := b ^ b ^ . {:\,}\hat{b}^{\dagger}\,\hat{b}{\,:}=\hat{b}^{\dagger}\,\hat{b}.
  14. b ^ b ^ \hat{b}^{\dagger}\,\hat{b}
  15. ( b ^ ) (\hat{b}^{\dagger})
  16. ( b ^ ) (\hat{b})
  17. b ^ b ^ \hat{b}\,\hat{b}^{\dagger}
  18. : b ^ b ^ := b ^ b ^ . {:\,}\hat{b}\,\hat{b}^{\dagger}{\,:}=\hat{b}^{\dagger}\,\hat{b}.
  19. b ^ \hat{b}^{\dagger}
  20. b ^ \hat{b}
  21. b ^ \hat{b}
  22. b ^ \hat{b}^{\dagger}
  23. b ^ b ^ = b ^ b ^ + 1 = : b ^ b ^ : + 1. \hat{b}\,\hat{b}^{\dagger}=\hat{b}^{\dagger}\,\hat{b}+1={:\,}\hat{b}\,\hat{b}^% {\dagger}{\,:}\;+1.
  24. b ^ b ^ - : b ^ b ^ := 1. \hat{b}\,\hat{b}^{\dagger}-{:\,}\hat{b}\,\hat{b}^{\dagger}{\,:}=1.
  25. : b ^ b ^ b ^ b ^ b ^ b ^ b ^ := b ^ b ^ b ^ b ^ b ^ b ^ b ^ = ( b ^ ) 3 b ^ 4 . {:\,}\hat{b}^{\dagger}\,\hat{b}\,\hat{b}\,\hat{b}^{\dagger}\,\hat{b}\,\hat{b}^% {\dagger}\,\hat{b}{\,:}=\hat{b}^{\dagger}\,\hat{b}^{\dagger}\,\hat{b}^{\dagger% }\,\hat{b}\,\hat{b}\,\hat{b}\,\hat{b}=(\hat{b}^{\dagger})^{3}\,\hat{b}^{4}.
  26. : exp ( λ a ^ a ^ ) := n = 0 λ n n ! a ^ n a ^ n {:\,}\exp(\lambda\hat{a}^{\dagger}\hat{a}){\,:}=\sum^{\infty}_{n=0}\frac{% \lambda^{n}}{n!}\hat{a}^{\dagger n}\hat{a}^{n}
  27. : b ^ b ^ := : 1 + b ^ b ^ := : 1 : + : b ^ b ^ := 1 + b ^ b ^ b ^ b ^ = : b ^ b ^ : {:\,}\hat{b}\hat{b}^{\dagger}{\,:}={:\,}1+\hat{b}^{\dagger}\hat{b}{\,:}={:\,}1% {\,:}+{:\,}\hat{b}^{\dagger}\hat{b}{\,:}=1+\hat{b}^{\dagger}\hat{b}\neq\hat{b}% ^{\dagger}\hat{b}={:\,}\hat{b}\hat{b}^{\dagger}{\,:}
  28. N N
  29. 2 N 2N
  30. b ^ i \hat{b}_{i}^{\dagger}
  31. i t h i^{th}
  32. b ^ i \hat{b}_{i}
  33. i t h i^{th}
  34. i = 1 , , N i=1,\ldots,N
  35. [ b ^ i , b ^ j ] - = 0 \left[\hat{b}_{i}^{\dagger},\hat{b}_{j}^{\dagger}\right]_{-}=0
  36. [ b ^ i , b ^ j ] - = 0 \left[\hat{b}_{i},\hat{b}_{j}\right]_{-}=0
  37. [ b ^ i , b ^ j ] - = δ i j \left[\hat{b}_{i},\hat{b}_{j}^{\dagger}\right]_{-}=\delta_{ij}
  38. i , j = 1 , , N i,j=1,\ldots,N
  39. δ i j \delta_{ij}
  40. b ^ i b ^ j = b ^ j b ^ i \hat{b}_{i}^{\dagger}\,\hat{b}_{j}^{\dagger}=\hat{b}_{j}^{\dagger}\,\hat{b}_{i% }^{\dagger}
  41. b ^ i b ^ j = b ^ j b ^ i \hat{b}_{i}\,\hat{b}_{j}=\hat{b}_{j}\,\hat{b}_{i}
  42. b ^ i b ^ j = b ^ j b ^ i + δ i j . \hat{b}_{i}\,\hat{b}_{j}^{\dagger}=\hat{b}_{j}^{\dagger}\,\hat{b}_{i}+\delta_{% ij}.
  43. N = 2 N=2
  44. : b ^ 1 b ^ 2 : = b ^ 1 b ^ 2 :\hat{b}_{1}^{\dagger}\,\hat{b}_{2}:\,=\hat{b}_{1}^{\dagger}\,\hat{b}_{2}
  45. : b ^ 2 b ^ 1 : = b ^ 1 b ^ 2 :\hat{b}_{2}\,\hat{b}_{1}^{\dagger}:\,=\hat{b}_{1}^{\dagger}\,\hat{b}_{2}
  46. N = 3 N=3
  47. : b ^ 1 b ^ 2 b ^ 3 : = b ^ 1 b ^ 2 b ^ 3 :\hat{b}_{1}^{\dagger}\,\hat{b}_{2}\,\hat{b}_{3}:\,=\hat{b}_{1}^{\dagger}\,% \hat{b}_{2}\,\hat{b}_{3}
  48. b ^ 2 b ^ 3 = b ^ 3 b ^ 2 \hat{b}_{2}\,\hat{b}_{3}=\hat{b}_{3}\,\hat{b}_{2}
  49. : b ^ 2 b ^ 1 b ^ 3 : = b ^ 1 b ^ 2 b ^ 3 :\hat{b}_{2}\,\hat{b}_{1}^{\dagger}\,\hat{b}_{3}:\,=\hat{b}_{1}^{\dagger}\,% \hat{b}_{2}\,\hat{b}_{3}
  50. : b ^ 3 b ^ 2 b ^ 1 : = b ^ 1 b ^ 2 b ^ 3 :\hat{b}_{3}\hat{b}_{2}\,\hat{b}_{1}^{\dagger}:\,=\hat{b}_{1}^{\dagger}\,\hat{% b}_{2}\,\hat{b}_{3}
  51. f ^ \hat{f}^{\dagger}
  52. f ^ \hat{f}
  53. [ f ^ , f ^ ] + = 0 \left[\hat{f}^{\dagger},\hat{f}^{\dagger}\right]_{+}=0
  54. [ f ^ , f ^ ] + = 0 \left[\hat{f},\hat{f}\right]_{+}=0
  55. [ f ^ , f ^ ] + = 1 \left[\hat{f},\hat{f}^{\dagger}\right]_{+}=1
  56. [ A , B ] + A B + B A \left[A,B\right]_{+}\equiv AB+BA
  57. f ^ f ^ = 0 \hat{f}^{\dagger}\,\hat{f}^{\dagger}=0
  58. f ^ f ^ = 0 \hat{f}\,\hat{f}=0
  59. f ^ f ^ = 1 - f ^ f ^ . \hat{f}\,\hat{f}^{\dagger}=1-\hat{f}^{\dagger}\,\hat{f}.
  60. : f ^ f ^ : = f ^ f ^ :\hat{f}^{\dagger}\,\hat{f}:\,=\hat{f}^{\dagger}\,\hat{f}
  61. : f ^ f ^ : = - f ^ f ^ :\hat{f}\,\hat{f}^{\dagger}:\,=-\hat{f}^{\dagger}\,\hat{f}
  62. f ^ f ^ = 1 - f ^ f ^ = 1 + : f ^ f ^ : \hat{f}\,\hat{f}^{\dagger}\,=1-\hat{f}^{\dagger}\,\hat{f}=1+:\hat{f}\,\hat{f}^% {\dagger}:
  63. f ^ f ^ - : f ^ f ^ := 1. \hat{f}\,\hat{f}^{\dagger}-:\hat{f}\,\hat{f}^{\dagger}:=1.
  64. : f ^ f ^ f ^ f ^ : = f ^ f ^ f ^ f ^ = 0 :\hat{f}\,\hat{f}^{\dagger}\,\hat{f}\hat{f}^{\dagger}:\,=\hat{f}^{\dagger}\,% \hat{f}^{\dagger}\,\hat{f}\,\hat{f}=0
  65. N N
  66. 2 N 2N
  67. f ^ i \hat{f}_{i}^{\dagger}
  68. i t h i^{th}
  69. f ^ i \hat{f}_{i}
  70. i t h i^{th}
  71. i = 1 , , N i=1,\ldots,N
  72. [ f ^ i , f ^ j ] + = 0 \left[\hat{f}_{i}^{\dagger},\hat{f}_{j}^{\dagger}\right]_{+}=0
  73. [ f ^ i , f ^ j ] + = 0 \left[\hat{f}_{i},\hat{f}_{j}\right]_{+}=0
  74. [ f ^ i , f ^ j ] + = δ i j \left[\hat{f}_{i},\hat{f}_{j}^{\dagger}\right]_{+}=\delta_{ij}
  75. i , j = 1 , , N i,j=1,\ldots,N
  76. δ i j \delta_{ij}
  77. f ^ i f ^ j = - f ^ j f ^ i \hat{f}_{i}^{\dagger}\,\hat{f}_{j}^{\dagger}=-\hat{f}_{j}^{\dagger}\,\hat{f}_{% i}^{\dagger}
  78. f ^ i f ^ j = - f ^ j f ^ i \hat{f}_{i}\,\hat{f}_{j}=-\hat{f}_{j}\,\hat{f}_{i}
  79. f ^ i f ^ j = δ i j - f ^ j f ^ i . \hat{f}_{i}\,\hat{f}_{j}^{\dagger}=\delta_{ij}-\hat{f}_{j}^{\dagger}\,\hat{f}_% {i}.
  80. N = 2 N=2
  81. : f ^ 1 f ^ 2 : = f ^ 1 f ^ 2 :\hat{f}_{1}^{\dagger}\,\hat{f}_{2}:\,=\hat{f}_{1}^{\dagger}\,\hat{f}_{2}
  82. : f ^ 2 f ^ 1 : = - f ^ 1 f ^ 2 :\hat{f}_{2}\,\hat{f}_{1}^{\dagger}:\,=-\hat{f}_{1}^{\dagger}\,\hat{f}_{2}
  83. : f ^ 2 f ^ 1 f ^ 2 : = f ^ 1 f ^ 2 f ^ 2 = - f ^ 2 f ^ 1 f ^ 2 :\hat{f}_{2}\,\hat{f}_{1}^{\dagger}\,\hat{f}^{\dagger}_{2}:\,=\hat{f}_{1}^{% \dagger}\,\hat{f}_{2}^{\dagger}\,\hat{f}_{2}=-\hat{f}_{2}^{\dagger}\,\hat{f}_{% 1}^{\dagger}\,\hat{f}_{2}
  84. N = 3 N=3
  85. : f ^ 1 f ^ 2 f ^ 3 : = f ^ 1 f ^ 2 f ^ 3 = - f ^ 1 f ^ 3 f ^ 2 :\hat{f}_{1}^{\dagger}\,\hat{f}_{2}\,\hat{f}_{3}:\,=\hat{f}_{1}^{\dagger}\,% \hat{f}_{2}\,\hat{f}_{3}=-\hat{f}_{1}^{\dagger}\,\hat{f}_{3}\,\hat{f}_{2}
  86. f ^ 2 f ^ 3 = - f ^ 3 f ^ 2 \hat{f}_{2}\,\hat{f}_{3}=-\hat{f}_{3}\,\hat{f}_{2}
  87. : f ^ 2 f ^ 1 f ^ 3 : = - f ^ 1 f ^ 2 f ^ 3 = f ^ 1 f ^ 3 f ^ 2 :\hat{f}_{2}\,\hat{f}_{1}^{\dagger}\,\hat{f}_{3}:\,=-\hat{f}_{1}^{\dagger}\,% \hat{f}_{2}\,\hat{f}_{3}=\hat{f}_{1}^{\dagger}\,\hat{f}_{3}\,\hat{f}_{2}
  88. : f ^ 3 f ^ 2 f ^ 1 : = f ^ 1 f ^ 3 f ^ 2 = - f ^ 1 f ^ 2 f ^ 3 :\hat{f}_{3}\hat{f}_{2}\,\hat{f}_{1}^{\dagger}:\,=\hat{f}_{1}^{\dagger}\,\hat{% f}_{3}\,\hat{f}_{2}=-\hat{f}_{1}^{\dagger}\,\hat{f}_{2}\,\hat{f}_{3}
  89. | 0 |0\rangle
  90. 0 | a ^ = 0 and a ^ | 0 = 0 \langle 0|\hat{a}^{\dagger}=0\qquad\textrm{and}\qquad\hat{a}|0\rangle=0
  91. a ^ \hat{a}^{\dagger}
  92. a ^ \hat{a}
  93. O ^ \hat{O}
  94. 0 | O ^ | 0 0 \langle 0|\hat{O}|0\rangle\neq 0
  95. 0 | : O ^ : | 0 = 0 \langle 0|:\hat{O}:|0\rangle=0
  96. 0 | H ^ | 0 = 0 \langle 0|\hat{H}|0\rangle=0
  97. : ϕ ( x ) χ ( y ) := ϕ ( x ) χ ( y ) - 0 | ϕ ( x ) χ ( y ) | 0 :\phi(x)\chi(y):=\phi(x)\chi(y)-\langle 0|\phi(x)\chi(y)|0\rangle
  98. | 0 |0\rangle
  99. n n
  100. n n
  101. T [ ϕ ( x 1 ) ϕ ( x n ) ] = : ϕ ( x 1 ) ϕ ( x n ) : + perm 0 | T [ ϕ ( x 1 ) ϕ ( x 2 ) ] | 0 : ϕ ( x 3 ) ϕ ( x n ) : + perm 0 | T [ ϕ ( x 1 ) ϕ ( x 2 ) ] | 0 0 | T [ ϕ ( x 3 ) ϕ ( x 4 ) ] | 0 : ϕ ( x 5 ) ϕ ( x n ) : + perm 0 | T [ ϕ ( x 1 ) ϕ ( x 2 ) ] | 0 0 | T [ ϕ ( x n - 1 ) ϕ ( x n ) ] | 0 \begin{aligned}\displaystyle T\left[\phi(x_{1})\cdots\phi(x_{n})\right]=&% \displaystyle:\phi(x_{1})\cdots\phi(x_{n}):+\sum_{\textrm{perm}}\langle 0|T% \left[\phi(x_{1})\phi(x_{2})\right]|0\rangle:\phi(x_{3})\cdots\phi(x_{n}):\\ &\displaystyle+\sum_{\textrm{perm}}\langle 0|T\left[\phi(x_{1})\phi(x_{2})% \right]|0\rangle\langle 0|T\left[\phi(x_{3})\phi(x_{4})\right]|0\rangle:\phi(x% _{5})\cdots\phi(x_{n}):\\ \displaystyle\vdots\\ &\displaystyle+\sum_{\textrm{perm}}\langle 0|T\left[\phi(x_{1})\phi(x_{2})% \right]|0\rangle\cdots\langle 0|T\left[\phi(x_{n-1})\phi(x_{n})\right]|0% \rangle\end{aligned}
  102. n n
  103. perm 0 | T [ ϕ ( x 1 ) ϕ ( x 2 ) ] | 0 0 | T [ ϕ ( x n - 2 ) ϕ ( x n - 1 ) ] | 0 ϕ ( x n ) . \sum\text{perm}\langle 0|T\left[\phi(x_{1})\phi(x_{2})\right]|0\rangle\cdots% \langle 0|T\left[\phi(x_{n-2})\phi(x_{n-1})\right]|0\rangle\phi(x_{n}).
  104. ϕ i ( x ) = ϕ i + ( x ) + ϕ i - ( x ) \phi_{i}(x)=\phi^{+}_{i}(x)+\phi^{-}_{i}(x)
  105. ϕ + ( x ) \phi^{+}(x)
  106. ϕ - ( x ) \phi^{-}(x)
  107. ϕ + ( x ) \phi^{+}(x)
  108. ϕ - ( x ) \phi^{-}(x)
  109. : ϕ 1 ( x 1 ) ϕ 2 ( x 2 ) ϕ n ( x n ) : = 0 \langle:\phi_{1}(x_{1})\phi_{2}(x_{2})\ldots\phi_{n}(x_{n}):\rangle=0
  110. ϕ i + \phi^{+}_{i}
  111. ϕ j - \phi^{-}_{j}
  112. exp ( - β H ^ ) \exp(-\beta\hat{H})
  113. b ^ b ^ = Tr ( e - β ω b ^ b ^ b ^ b ^ ) Tr ( e - β ω b ^ b ^ ) = 1 e β ω - 1 \langle\hat{b}^{\dagger}\hat{b}\rangle=\frac{\mathrm{Tr}(e^{-\beta\omega\hat{b% }^{\dagger}\hat{b}}\hat{b}^{\dagger}\hat{b})}{\mathrm{Tr}(e^{-\beta\omega\hat{% b}^{\dagger}\hat{b}})}=\frac{1}{e^{\beta\omega}-1}
  114. b ^ b ^ \hat{b}^{\dagger}\hat{b}
  115. ϕ i + \phi^{+}_{i}
  116. ϕ j - \phi^{-}_{j}

Northern_red_snapper.html

  1. W = c L b W=cL^{b}\!\,

NSPACE.html

  1. k NSPACE ( n k ) \bigcup_{k\in\mathbb{N}}\mbox{NSPACE}~{}(n^{k})
  2. k NSPACE ( 2 n k ) \bigcup_{k\in\mathbb{N}}\mbox{NSPACE}~{}(2^{n^{k}})
  3. DSPACE [ s ( n ) ] NSPACE [ s ( n ) ] DSPACE [ ( s ( n ) ) 2 ] . \mbox{DSPACE}~{}[s(n)]\subseteq\mbox{NSPACE}~{}[s(n)]\subseteq\mbox{DSPACE}~{}% [(s(n))^{2}].

NTIME.html

  1. NP = k NTIME ( n k ) \mbox{NP}~{}=\bigcup_{k\in\mathbb{N}}\mbox{NTIME}~{}(n^{k})
  2. NEXP = k NTIME ( 2 n k ) \mbox{NEXP}~{}=\bigcup_{k\in\mathbb{N}}\mbox{NTIME}~{}(2^{n^{k}})
  3. NTIME ( t ( n ) ) DSPACE ( t ( n ) ) \mbox{NTIME}~{}(t(n))\subseteq\mbox{DSPACE}~{}(t(n))
  4. NTIME ( t ( n ) ) ATIME ( t ( n ) ) DSPACE ( t ( n ) ) \mbox{NTIME}~{}(t(n))\subseteq\mbox{ATIME}~{}(t(n))\subseteq\mbox{DSPACE}~{}(t% (n))

Observability.html

  1. n n
  2. 𝒪 = [ C C A C A 2 C A n - 1 ] \mathcal{O}=\begin{bmatrix}C\\ CA\\ CA^{2}\\ \vdots\\ CA^{n-1}\end{bmatrix}
  3. n n
  4. n n
  5. n n
  6. y ( k ) y(k)
  7. v v
  8. rank ( O v ) = rank ( O v + 1 ) \,\text{rank}{(O_{v})}=\,\text{rank}{(O_{v+1})}
  9. 𝒪 v = [ C C A C A 2 C A v - 1 ] . \mathcal{O}_{v}=\begin{bmatrix}C\\ CA\\ CA^{2}\\ \vdots\\ CA^{v-1}\end{bmatrix}.
  10. 𝐱 ˙ ( t ) = A ( t ) 𝐱 ( t ) + B ( t ) 𝐮 ( t ) \dot{\mathbf{x}}(t)=A(t)\mathbf{x}(t)+B(t)\mathbf{u}(t)\,
  11. 𝐲 ( t ) = C ( t ) 𝐱 ( t ) . \mathbf{y}(t)=C(t)\mathbf{x}(t).\,
  12. A , B , and C A,B,\,\text{ and }C
  13. u and y u\,\text{ and }y
  14. t [ t 0 , t 1 ] t\in[t_{0},t_{1}]
  15. x ( t 0 ) x(t_{0})
  16. M ( t 0 , t 1 ) M(t_{0},t_{1})
  17. M ( t 0 , t 1 ) = t 0 t 1 ϕ ( t , t 0 ) T C ( t ) T C ( t ) ϕ ( t , t 0 ) d t M(t_{0},t_{1})=\int_{t_{0}}^{t_{1}}\phi(t,t_{0})^{T}C(t)^{T}C(t)\phi(t,t_{0})dt
  18. ϕ \phi
  19. x ( t 0 ) x(t_{0})
  20. M ( t 0 , t 1 ) M(t_{0},t_{1})
  21. x 1 x_{1}
  22. x 2 x_{2}
  23. x 1 - x 2 x_{1}-x_{2}
  24. M ( t 0 , t 1 ) M(t_{0},t_{1})
  25. M M
  26. M ( t 0 , t 1 ) M(t_{0},t_{1})
  27. M ( t 0 , t 1 ) M(t_{0},t_{1})
  28. t 1 t 0 t_{1}\geq t_{0}
  29. M ( t 0 , t 1 ) M(t_{0},t_{1})
  30. d d t M ( t , t 1 ) = - A ( t ) T M ( t , t 1 ) - M ( t , t 1 ) A ( t ) - C ( t ) T C ( t ) , M ( t 1 , t 1 ) = 0 \frac{d}{dt}M(t,t_{1})=-A(t)^{T}M(t,t_{1})-M(t,t_{1})A(t)-C(t)^{T}C(t),\;M(t_{% 1},t_{1})=0
  31. M ( t 0 , t 1 ) M(t_{0},t_{1})
  32. M ( t 0 , t 1 ) = M ( t 0 , t ) + ϕ ( t , t 0 ) T M ( t , t 1 ) ϕ ( t , t 0 ) M(t_{0},t_{1})=M(t_{0},t)+\phi(t,t_{0})^{T}M(t,t_{1})\phi(t,t_{0})
  33. x ˙ = f ( x ) + j = 1 m g j ( x ) u j \dot{x}=f(x)+\sum_{j=1}^{m}g_{j}(x)u_{j}
  34. y i = h i ( x ) , i p y_{i}=h_{i}(x),i\in p
  35. x n x\in\mathbb{R}^{n}
  36. u m u\in\mathbb{R}^{m}
  37. y p y\in\mathbb{R}^{p}
  38. f , g , h f,g,h
  39. 𝒪 s \mathcal{O}_{s}
  40. x 0 x_{0}
  41. dim ( d 𝒪 s ( x 0 ) ) = n \textrm{dim}(d\mathcal{O}_{s}(x_{0}))=n
  42. d 𝒪 s ( x 0 ) = span ( d h 1 ( x 0 ) , , d h p ( x 0 ) , d L v i L v i - 1 , , L v 1 h j ( x 0 ) ) , j p , k = 1 , 2 , . d\mathcal{O}_{s}(x_{0})=\mathrm{span}(dh_{1}(x_{0}),\ldots,dh_{p}(x_{0}),dL_{v% _{i}}L_{v_{i-1}},\ldots,L_{v_{1}}h_{j}(x_{0})),\ j\in p,k=1,2,\ldots.
  43. n \mathbb{R}^{n}
  44. n \mathbb{R}^{n}

Occurs_check.html

  1. X = f ( X ) X=f(X)
  2. ( x y . p ( x , y ) ) ( y x . p ( x , y ) ) (\forall x\exists y.p(x,y))\rightarrow(\exists y\forall x.p(x,y))
  3. p ( X , f ( X ) ) ¬ p ( g ( Y ) , Y ) p(X,f(X))\land\lnot p(g(Y),Y)
  4. f f
  5. g g
  6. p ( X , f ( X ) ) p(X,f(X))
  7. p ( g ( Y ) , Y ) p(g(Y),Y)
  8. t 1 t_{1}
  9. t 2 t_{2}
  10. O ( size ( t 1 ) + size ( t 2 ) ) O(\,\text{size}(t_{1})+\,\text{size}(t_{2}))
  11. O ( min ( size ( t 1 ) , size ( t 2 ) ) ) O(\,\text{min}(\,\text{size}(t_{1}),\,\text{size}(t_{2})))
  12. O ( 1 ) O(1)
  13. c o n s ( x , y ) = ? c o n s ( 1 , c o n s ( x , c o n s ( 2 , y ) ) ) cons(x,y)\stackrel{?}{=}cons(1,cons(x,cons(2,y)))
  14. p ( x , y ) p(x,y)

Octree.html

  1. 2 3 = 8 2^{3}=8

Old_quantum_theory.html

  1. H ( p , q ) = E p i d q i = n i h \oint\limits_{H(p,q)=E}p_{i}\,dq_{i}=n_{i}h
  2. p i p_{i}
  3. q i q_{i}
  4. n i n_{i}
  5. q i q_{i}
  6. H = p 2 2 m + m ω 2 q 2 2 . H={p^{2}\over 2m}+{m\omega^{2}q^{2}\over 2}.
  7. E = n ω , E=n\hbar\omega,\,
  8. 1 2 ω \frac{1}{2}\hbar\omega
  9. U = n ω n e - β n ω n e - β n ω = ω e - β ω 1 - e - β ω , where β = 1 k T , U={\sum_{n}\hbar\omega ne^{-\beta n\hbar\omega}\over\sum_{n}e^{-\beta n\hbar% \omega}}={\hbar\omega e^{-\beta\hbar\omega}\over 1-e^{-\beta\hbar\omega}},\;\;% \;{\rm where}\;\;\beta=\frac{1}{kT},
  10. β \beta
  11. β \beta
  12. ω \scriptstyle\hbar\omega
  13. exp ( - ω / k T ) \exp(-\hbar\omega/kT)
  14. β \beta
  15. 1 / β = k T 1/\beta=kT
  16. 2 m ( E - V ( q ) ) = p \sqrt{2m(E-V(q))}=p
  17. 2 0 L p d q = n h 2\int_{0}^{L}p\,dq=nh
  18. p = n h 2 L p={nh\over 2L}
  19. E n = p 2 2 m = n 2 h 2 8 m L 2 E_{n}={p^{2}\over 2m}={n^{2}h^{2}\over 8mL^{2}}
  20. 2 0 E F 2 m ( E - F x ) d x = n h 2\int_{0}^{\frac{E}{F}}\sqrt{2m(E-Fx)}\ dx=nh
  21. 4 3 2 m E 3 / 2 F = n h {4\over 3}\sqrt{2m}{E^{3/2}\over F}=nh
  22. E n = ( 3 n h F 4 2 m ) 2 / 3 E_{n}=\left({3nhF\over 4\sqrt{2m}}\right)^{2/3}
  23. 2 - 2 E k 2 E k 2 m ( E - 1 2 k x 2 ) d x = n h 2\int_{-\sqrt{\frac{2E}{k}}}^{\sqrt{\frac{2E}{k}}}\sqrt{2m\left(E-\frac{1}{2}% kx^{2}\right)}\ dx=nh
  24. E = n h 2 π k m = n ω E=n\frac{h}{2\pi}\sqrt{\frac{k}{m}}=n\hbar\omega
  25. ω \omega
  26. L = M R 2 2 θ ˙ 2 L={MR^{2}\over 2}\dot{\theta}^{2}
  27. θ \theta
  28. J = M R 2 θ ˙ \scriptstyle J=MR^{2}\dot{\theta}
  29. θ \theta
  30. 2 π J = n h 2\pi J=nh\,
  31. \scriptstyle\hbar
  32. θ \scriptstyle\theta
  33. ϕ \scriptstyle\phi
  34. θ \scriptstyle\theta
  35. ϕ \scriptstyle\phi
  36. L = M R 2 2 θ ˙ 2 + M R 2 2 ( sin ( θ ) ϕ ˙ ) 2 L={MR^{2}\over 2}\dot{\theta}^{2}+{MR^{2}\over 2}(\sin(\theta)\dot{\phi})^{2}\,
  37. p θ = θ ˙ \scriptstyle p_{\theta}=\dot{\theta}
  38. p ϕ = sin ( θ ) 2 ϕ ˙ \scriptstyle p_{\phi}=\sin(\theta)^{2}\dot{\phi}
  39. ϕ \scriptstyle\phi
  40. p ϕ \scriptstyle p_{\phi}
  41. p ϕ = l ϕ p_{\phi}=l_{\phi}\,
  42. l ϕ \scriptstyle l_{\phi}
  43. ϕ \scriptstyle\phi
  44. 2 π 2\pi
  45. l ϕ = m l_{\phi}=m\hbar\,
  46. H = p 2 2 + l 2 2 r 2 - 1 r . H={p^{2}\over 2}+{l^{2}\over 2r^{2}}-{1\over r}.
  47. 2 2 E - l 2 r 2 + 2 r d r = k h 2\oint\sqrt{2E-{l^{2}\over r^{2}}+{2\over r}}\ dr=kh
  48. E = - 1 2 ( k + l ) 2 E=-{1\over 2(k+l)^{2}}
  49. W = m 0 c 2 ( 1 1 - v 2 c 2 - 1 ) - k Z e 2 r W={m_{\mathrm{0}}c^{2}}\left(\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}-1\right)-k% \frac{Ze^{2}}{r}
  50. u = 1 r u=\frac{1}{r}
  51. 1 1 - v 2 c 2 = 1 + W m 0 c 2 + k Z e 2 m 0 c 2 u \frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}=1+\frac{W}{m_{\mathrm{0}}c^{2}}+k\frac{% Ze^{2}}{m_{\mathrm{0}}c^{2}}u
  52. p r = m r ˙ p_{\mathrm{r}}=m\dot{r}
  53. p φ = m r 2 φ ˙ p_{\mathrm{\varphi}}=mr^{2}\dot{\varphi}
  54. p r p φ = - d u d φ \frac{p_{\mathrm{r}}}{p_{\mathrm{\varphi}}}=-\frac{du}{d\varphi}
  55. d 2 u d φ 2 = - ( 1 - k 2 Z 2 e 4 c 2 p φ 2 ) u + m 0 k Z e 2 p φ 2 ( 1 + W m 0 c 2 ) = - ω 0 2 u + K \frac{d^{2}u}{d\varphi^{2}}=-\left(1-k^{2}\frac{Z^{2}e^{4}}{c^{2}p_{\mathrm{% \varphi}}^{2}}\right)u+\frac{m_{\mathrm{0}}kZe^{2}}{p_{\mathrm{\varphi}}^{2}}% \left(1+\frac{W}{m_{\mathrm{0}}c^{2}}\right)=-\omega_{\mathrm{0}}^{2}u+K
  56. u = 1 r = K + A cos ω 0 φ u=\frac{1}{r}=K+A\cos\omega_{\mathrm{0}}\varphi
  57. φ s = 2 π ( 1 ω 0 - 1 ) 4 π 3 k 2 Z 2 e 4 c 2 n φ 2 h 2 \varphi_{\mathrm{s}}=2\pi\left(\frac{1}{\omega_{\mathrm{0}}}-1\right)\approx 4% \pi^{3}k^{2}\frac{Z^{2}e^{4}}{c^{2}n_{\mathrm{\varphi}}^{2}h^{2}}
  58. p φ d φ = 2 π p φ = n φ h \oint p_{\mathrm{\varphi}}\,d\varphi=2\pi p_{\mathrm{\varphi}}=n_{\mathrm{% \varphi}}h
  59. p r d r = p φ ( 1 r d r d φ ) 2 d φ = n r h \oint p_{\mathrm{r}}\,dr=p_{\mathrm{\varphi}}\oint\left(\frac{1}{r}\frac{dr}{d% \varphi}\right)^{2}\,d\varphi=n_{\mathrm{r}}h
  60. W m 0 c 2 = ( 1 + α 2 Z 2 ( n r + n φ 2 - α 2 Z 2 ) 2 ) - 1 / 2 - 1 \frac{W}{m_{\mathrm{0}}c^{2}}=\left(1+\frac{\alpha^{2}Z^{2}}{(n_{\mathrm{r}}+% \sqrt{n_{\mathrm{\varphi}}^{2}-\alpha^{2}Z^{2}})^{2}}\right)^{-1/2}-1
  61. α \alpha
  62. ω \omega
  63. E = n ω E=n\hbar\omega\,
  64. ω \scriptstyle\hbar\omega
  65. k \scriptstyle\hbar k
  66. k k
  67. p = k p=\hbar k
  68. λ \lambda
  69. p = h λ p={h\over\lambda}
  70. p d x = k d x = 2 π n \int p\,dx=\hbar\int k\,dx=2\pi\hbar n
  71. 2 π 2\pi
  72. n λ = 2 L n\lambda=2L\,
  73. p = n h 2 L p=\frac{nh}{2L}
  74. θ ( J , x ) \theta(J,x)
  75. X n ( t ) = k = - e i k ω t X n ; k X_{n}(t)=\sum_{k=-\infty}^{\infty}e^{ik\omega t}X_{n;k}
  76. ω \omega
  77. 2 π / T n \scriptstyle 2\pi/T_{n}
  78. | X k | 2 |X_{k}|^{2}

Old_Style_and_New_Style_dates.html

  1. 17 33 34 17\tfrac{33}{34}

Olinto_De_Pretto.html

  1. E = m c 2 E=mc^{2}
  2. m v 2 mv^{2}
  3. E = m c 2 E=mc^{2}

On_shell_and_off_shell.html

  1. E 2 - | p | 2 c 2 = m 2 c 4 E^{2}-|\vec{p}\,|^{2}c^{2}=m^{2}c^{4}
  2. p \vec{p}
  3. p μ p μ p 2 = m 2 p^{\mu}p_{\mu}\equiv p^{2}=m^{2}
  4. p μ p μ = - m 2 p^{\mu}p_{\mu}=-m^{2}
  5. q 2 q^{2}
  6. ( ϕ , μ ϕ ) \mathcal{L}(\phi,\partial_{\mu}\phi)
  7. S = d D x ( ϕ , μ ϕ ) S=\int d^{D}x\mathcal{L}(\phi,\partial_{\mu}\phi)
  8. μ ( μ ϕ ) = ϕ \partial_{\mu}\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\phi)}=\frac{% \partial\mathcal{L}}{\partial\phi}
  9. x μ x μ + α μ x^{\mu}\rightarrow x^{\mu}+\alpha^{\mu}
  10. \mathcal{L}
  11. δ = α μ μ \delta\mathcal{L}=\alpha^{\mu}\partial_{\mu}\mathcal{L}
  12. δ \delta\mathcal{L}
  13. δ = ϕ δ ϕ + ( μ ϕ ) δ ( μ ϕ ) \delta\mathcal{L}=\frac{\partial\mathcal{L}}{\partial\phi}\delta\phi+\frac{% \partial\mathcal{L}}{\partial(\partial_{\mu}\phi)}\delta(\partial_{\mu}\phi)
  14. δ \delta\mathcal{L}
  15. δ ( μ ϕ ) = μ ( δ ϕ ) \delta(\partial_{\mu}\phi)=\partial_{\mu}(\delta\phi)
  16. α μ μ = ϕ δ ϕ + ( μ ϕ ) δ ( μ ϕ ) \alpha^{\mu}\partial_{\mu}\mathcal{L}=\frac{\partial\mathcal{L}}{\partial\phi}% \delta\phi+\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\phi)}\delta(% \partial_{\mu}\phi)
  17. \mathcal{L}
  18. α μ μ = ϕ α μ μ ϕ + ( ν ϕ ) α μ μ ν ϕ \alpha^{\mu}\partial_{\mu}\mathcal{L}=\frac{\partial\mathcal{L}}{\partial\phi}% \alpha^{\mu}\partial_{\mu}\phi+\frac{\partial\mathcal{L}}{\partial(\partial_{% \nu}\phi)}\alpha^{\mu}\partial_{\mu}\partial_{\nu}\phi
  19. α μ = ( ϵ , 0 , , 0 ) , ( 0 , ϵ , , 0 ) , \alpha^{\mu}=(\epsilon,0,...,0),(0,\epsilon,...,0),...
  20. α μ \alpha^{\mu}
  21. μ = ϕ μ ϕ + ( ν ϕ ) μ ν ϕ \partial_{\mu}\mathcal{L}=\frac{\partial\mathcal{L}}{\partial\phi}\partial_{% \mu}\phi+\frac{\partial\mathcal{L}}{\partial(\partial_{\nu}\phi)}\partial_{\mu% }\partial_{\nu}\phi
  22. μ = ν ( ν ϕ ) μ ϕ + ( ν ϕ ) μ ν ϕ \partial_{\mu}\mathcal{L}=\partial_{\nu}\frac{\partial\mathcal{L}}{\partial(% \partial_{\nu}\phi)}\partial_{\mu}\phi+\frac{\partial\mathcal{L}}{\partial(% \partial_{\nu}\phi)}\partial_{\mu}\partial_{\nu}\phi
  23. ν ( ( ν ϕ ) μ ϕ - δ μ ν ) = 0 \partial_{\nu}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{\nu}\phi)}% \partial_{\mu}\phi-\delta^{\nu}_{\mu}\mathcal{L}\right)=0
  24. T ν μ T^{\nu}{}_{\mu}
  25. ν T ν = μ 0 \partial_{\nu}T^{\nu}{}_{\mu}=0

One-loop_Feynman_diagram.html

  1. λ 0 \lambda\to 0
  2. Γ [ ϕ ] = S [ ϕ ] + 1 2 Tr [ ln S ( 2 ) [ ϕ ] ] + \Gamma[\phi]=S[\phi]+\frac{1}{2}\mathop{\mathrm{Tr}}{\left[\ln{S^{(2)}[\phi]}% \right]+\dots}

One-parameter_group.html

  1. φ : G \varphi:\mathbb{R}\rightarrow G
  2. \mathbb{R}
  3. G G
  4. φ \varphi
  5. φ ( ) \varphi(\mathbb{R})
  6. G G
  7. \mathbb{R}
  8. φ ( s + t ) = φ ( s ) φ ( t ) \varphi(s+t)=\varphi(s)\varphi(t)
  9. s s
  10. t t
  11. G G
  12. φ ( s ) = e \varphi(s)=e
  13. G G
  14. s 0 s\neq 0
  15. G G
  16. φ ( s ) = e i s \varphi(s)=e^{is}
  17. φ \varphi
  18. 2 π 2\pi
  19. φ ( ) \varphi(\mathbb{R})
  20. G G
  21. \mathbb{R}
  22. φ \varphi
  23. G G
  24. T T
  25. φ \varphi
  26. T T
  27. G G
  28. \mathfrak{R}
  29. 𝔗 \mathfrak{T}
  30. mod 1 \mod 1
  31. \mathbb{R}
  32. ( cosh a + r sinh a ) (\cosh{a}+r\sinh{a})
  33. a a
  34. r 2 = ± 1 r^{2}=\pm{1}

Online_codes.html

  1. F = ln ( ϵ 2 / 4 ) ln ( 1 - ϵ / 2 ) F=\left\lceil\frac{\ln(\epsilon^{2}/4)}{\ln(1-\epsilon/2)}\right\rceil
  2. p 1 = 1 - 1 + 1 / F 1 + ϵ p_{1}=1-\frac{1+1/F}{1+\epsilon}
  3. p i = ( 1 - p 1 ) F ( F - 1 ) i ( i - 1 ) p_{i}=\frac{(1-p_{1})F}{(F-1)i(i-1)}
  4. 2 i F 2\leq i\leq F

Opening_(morphology).html

  1. A B = ( A B ) B , A\circ B=(A\ominus B)\oplus B,\,
  2. \ominus
  3. \oplus
  4. ( A B ) B = A B (A\circ B)\circ B=A\circ B
  5. A C A\subseteq C
  6. A B C B A\circ B\subseteq C\circ B
  7. A B A A\circ B\subseteq A
  8. A B = ( A c B s ) c A\bullet B=(A^{c}\circ B^{s})^{c}
  9. \bullet

OpenMath.html

  1. x = - b ± b 2 - 4 a c 2 a x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}

Operator_theory.html

  1. A = U D U * A=UDU^{*}\;
  2. A * A = ( A * A ) 1 2 ( A * A ) 1 2 , A^{*}A=(A^{*}A)^{\frac{1}{2}}(A^{*}A)^{\frac{1}{2}},
  3. A = U ( A * A ) 1 2 A=U(A^{*}A)^{\frac{1}{2}}
  4. x * * = ( x * ) * = x x^{**}=(x^{*})^{*}=x
  5. ( x + y ) * = x * + y * (x+y)^{*}=x^{*}+y^{*}
  6. ( x y ) * = y * x * (xy)^{*}=y^{*}x^{*}
  7. ( λ x ) * = λ ¯ x * . (\lambda x)^{*}=\overline{\lambda}x^{*}.
  8. x * x = x x * . \|x^{*}x\|=\|x\|\|x^{*}\|.
  9. x x * = x 2 , \|xx^{*}\|=\|x\|^{2},
  10. x 2 = x * x = sup { | λ | : x * x - λ 1 is not invertible } . \|x\|^{2}=\|x^{*}x\|=\sup\{|\lambda|:x^{*}x-\lambda\,1\,\text{ is not % invertible}\}.

Optical_flow.html

  1. t + Δ t t+\Delta t
  2. ( x , y , t ) (x,y,t)
  3. I ( x , y , t ) I(x,y,t)
  4. Δ x \Delta x
  5. Δ y \Delta y
  6. Δ t \Delta t
  7. I ( x , y , t ) = I ( x + Δ x , y + Δ y , t + Δ t ) I(x,y,t)=I(x+\Delta x,y+\Delta y,t+\Delta t)
  8. I ( x , y , t ) I(x,y,t)
  9. I ( x + Δ x , y + Δ y , t + Δ t ) = I ( x , y , t ) + I x Δ x + I y Δ y + I t Δ t + I(x+\Delta x,y+\Delta y,t+\Delta t)=I(x,y,t)+\frac{\partial I}{\partial x}% \Delta x+\frac{\partial I}{\partial y}\Delta y+\frac{\partial I}{\partial t}% \Delta t+
  10. I x Δ x + I y Δ y + I t Δ t = 0 \frac{\partial I}{\partial x}\Delta x+\frac{\partial I}{\partial y}\Delta y+% \frac{\partial I}{\partial t}\Delta t=0
  11. I x Δ x Δ t + I y Δ y Δ t + I t Δ t Δ t = 0 \frac{\partial I}{\partial x}\frac{\Delta x}{\Delta t}+\frac{\partial I}{% \partial y}\frac{\Delta y}{\Delta t}+\frac{\partial I}{\partial t}\frac{\Delta t% }{\Delta t}=0
  12. I x V x + I y V y + I t = 0 \frac{\partial I}{\partial x}V_{x}+\frac{\partial I}{\partial y}V_{y}+\frac{% \partial I}{\partial t}=0
  13. V x , V y V_{x},V_{y}
  14. x x
  15. y y
  16. I ( x , y , t ) I(x,y,t)
  17. I x \tfrac{\partial I}{\partial x}
  18. I y \tfrac{\partial I}{\partial y}
  19. I t \tfrac{\partial I}{\partial t}
  20. ( x , y , t ) (x,y,t)
  21. I x I_{x}
  22. I y I_{y}
  23. I t I_{t}
  24. I x V x + I y V y = - I t I_{x}V_{x}+I_{y}V_{y}=-I_{t}
  25. I T V = - I t \nabla I^{T}\cdot\vec{V}=-I_{t}

Optical_ring_resonators.html

  1. | K | 2 + | t | 2 = 𝟏 |K|^{2}+|t|^{2}=\mathbf{1}
  2. K K
  3. 𝐎𝐏𝐃 = 2 π r n e f f \mathbf{OPD}=2\pi rn_{eff}
  4. n e f f n_{eff}
  5. n e f f n_{eff}
  6. 𝐎𝐏𝐃 = m λ m \mathbf{OPD}=m\lambda_{m}
  7. λ m \lambda_{m}
  8. 𝐐 = m = m ν f δ ν \mathbf{Q}=m\mathcal{F}=m\frac{\nu_{f}}{\delta\nu}
  9. \mathcal{F}
  10. ν f \nu_{f}
  11. δ ν \delta\nu
  12. 2 π n 1 R 1 = m 1 λ 1 \ 2\pi n_{1}R_{1}=m_{1}\lambda_{1}
  13. 2 π n 2 R 2 = m 2 λ 2 \ 2\pi n_{2}R_{2}=m_{2}\lambda_{2}
  14. m 1 m_{1}
  15. m 2 m_{2}
  16. λ 1 = λ 2 \lambda_{1}=\lambda_{2}
  17. n 1 R 1 m 1 = n 2 R 2 m 2 \ \frac{n_{1}R_{1}}{m_{1}}=\frac{n_{2}R_{2}}{m_{2}}
  18. m 1 m_{1}
  19. m 2 m_{2}

Optimization_problem.html

  1. minimize 𝑥 \displaystyle\underset{x}{\operatorname{minimize}}
  2. f ( x ) : n f(x):\mathbb{R}^{n}\to\mathbb{R}
  3. x x
  4. g i ( x ) 0 g_{i}(x)\leq 0
  5. h i ( x ) = 0 h_{i}(x)=0
  6. A A
  7. ( I , f , m , g ) (I,f,m,g)
  8. I I
  9. x I x\in I
  10. f ( x ) f(x)
  11. x x
  12. y y
  13. x x
  14. m ( x , y ) m(x,y)
  15. y y
  16. g g
  17. min \min
  18. max \max
  19. x x
  20. y y
  21. m ( x , y ) = g { m ( x , y ) y f ( x ) } . m(x,y)=g\{m(x,y^{\prime})\mid y^{\prime}\in f(x)\}.
  22. m 0 m_{0}
  23. G G
  24. u u
  25. v v
  26. u u
  27. v v
  28. u u
  29. v v
  30. y f ( x ) \scriptstyle y\in f(x)
  31. x x
  32. { x x I } \scriptstyle\{\,x\,\mid\,x\in I\,\}
  33. { ( x , y ) y f ( x ) } \scriptstyle\{\,(x,y)\,\mid\,y\in f(x)\,\}
  34. 1 / c 1/c

Orbit_equation.html

  1. m 2 m_{2}\,\!
  2. m 1 m_{1}\,\!
  3. m 1 m_{1}\,\!
  4. r = 2 m 2 γ 1 1 + e cos θ r=\frac{\ell^{2}}{m^{2}\gamma}\frac{1}{1+e\cos\theta}
  5. r r
  6. θ \theta
  7. 𝐫 \mathbf{r}
  8. \ell
  9. m r 2 θ ˙ mr^{2}\dot{\theta}
  10. γ \gamma
  11. γ / r 2 \gamma/r^{2}
  12. γ \gamma
  13. G M GM
  14. γ \gamma
  15. e e
  16. e = 1 + 2 E 2 m 3 γ 2 e=\sqrt{1+\frac{2E\ell^{2}}{m^{3}\gamma^{2}}}
  17. E E
  18. r r
  19. θ \theta
  20. e e
  21. e < 1 e<1
  22. e = 1 e=1
  23. e > 1 e>1
  24. r = 2 m 2 γ 1 1 + e r={{\ell^{2}}\over{m^{2}\gamma}}{{1}\over{1+e}}
  25. e < 1 e<1
  26. r = 2 m 2 γ 1 1 - e r={{\ell^{2}}\over{m^{2}\gamma}}{{1}\over{1-e}}
  27. r = 2 m 2 γ 1 1 - e r={{\ell^{2}}\over{m^{2}\gamma}}{{1}\over{1-e}}
  28. r r
  29. v v\,\!
  30. a = R / 2 a=R/2\,\!
  31. R R\,\!
  32. 2 g 2g\,\!
  33. R R\,\!
  34. a a\,\!
  35. g g
  36. v v\,\!
  37. h h
  38. ϵ \epsilon
  39. a a
  40. e e
  41. e = | R a - 1 | e=\left|\frac{R}{a}-1\right|
  42. h h
  43. \ell
  44. h = / m h=\ell/m

Orbit_phasing.html

  1. t = T 1 2 π ( E - e 1 sin E ) t=\frac{T_{1}}{2\pi}(E-e_{1}\sin E)
  2. E = 2 arctan ( 1 - e 1 1 + e 1 tan ϕ 2 ) E=2\arctan(\sqrt{\frac{1-e_{1}}{1+e_{1}}}\tan{\frac{\phi}{2}})
  3. T 2 = T 1 - t T_{2}=T_{1}-t
  4. a 2 = ( μ T 2 2 π ) 2 3 a_{2}=(\frac{\sqrt{\mu}T_{2}}{2\pi})^{\frac{2}{3}}
  5. 2 a 2 = r a + r p 2a_{2}=r_{a}+r_{p}
  6. h 2 = 2 μ r a - r p r a + r p h_{2}=\sqrt{2\mu}\sqrt{\frac{r_{a}-r_{p}}{r_{a}+r_{p}}}
  7. Δ V = v 2 - v 1 = h 2 r - h 1 r \Delta V=v_{2}-v_{1}=\frac{h_{2}}{r}-\frac{h_{1}}{r}

Orbital_eccentricity.html

  1. e = 0 e=0\,\!
  2. 0 < e < 1 0<e<1\,\!
  3. e = 1 e=1\,\!
  4. e > 1 e>1\,\!
  5. e e
  6. e = 1 + 2 E L 2 m red α 2 e=\sqrt{1+\frac{2EL^{2}}{m\text{red}\alpha^{2}}}
  7. L L
  8. m red m\text{red}
  9. α \alpha
  10. F = α r 2 F=\frac{\alpha}{r^{2}}
  11. α \alpha
  12. e = 1 + 2 ϵ h 2 μ 2 e=\sqrt{1+\frac{2\epsilon h^{2}}{\mu^{2}}}
  13. ϵ \epsilon
  14. μ \mu
  15. h h
  16. e e
  17. e e
  18. e e
  19. e = | 𝐞 | e=\left|\mathbf{e}\right|
  20. 𝐞 \mathbf{e}\,\!
  21. r p = a ( 1 - e ) r_{p}=a(1-e)
  22. r a = a ( 1 + e ) r_{a}=a(1+e)
  23. a a
  24. e = r a - r p r a + r p e={{r_{a}-r_{p}}\over{r_{a}+r_{p}}}
  25. = 1 - 2 ( r a / r p ) + 1 =1-\frac{2}{(r_{a}/r_{p})+1}
  26. r a r_{a}\,\!
  27. r p r_{p}\,\!
  28. r p r a = 1 - e 1 + e {{r_{p}}\over{r_{a}}}={{1-e}\over{1+e}}

Orbital_inclination_change.html

  1. v v\,
  2. Δ v i \Delta{v_{i}}\,
  3. Δ i \Delta{i}\,
  4. Δ v i = 2 sin ( Δ i 2 ) 1 - e 2 cos ( w + f ) n a ( 1 + e cos ( f ) ) \Delta{v_{i}}={2\sin(\frac{\Delta{i}}{2})\sqrt{1-e^{2}}\cos(w+f)na\over{(1+e% \cos(f))}}
  5. e e\,
  6. w w\,
  7. f f\,
  8. n n\,
  9. a a\,
  10. e e\,
  11. Δ v i \Delta{v_{i}}\,
  12. Δ i \Delta{i}\,
  13. Δ v i = 2 v sin ( Δ i 2 ) \Delta{v_{i}}={2v\,\sin\left(\frac{\Delta{i}}{2}\right)}
  14. v v\,
  15. Δ v i \Delta{v_{i}}

Orbital_maneuver.html

  1. Δ v = v e ln m 0 m 1 \Delta v=v\text{e}\ln\frac{m_{0}}{m_{1}}
  2. m 0 m_{0}
  3. m 1 m_{1}
  4. v e v\text{e}
  5. v e = I sp g 0 v\text{e}=I\text{sp}\cdot g_{0}
  6. I sp I\text{sp}
  7. g 0 g_{0}
  8. Δ v \Delta v
  9. Δ 𝐯 \Delta\mathbf{v}\,
  10. r b r_{b}
  11. v v\,

Orbital_state_vectors.html

  1. 𝐫 \mathbf{r}
  2. 𝐯 \mathbf{v}
  3. t t\,
  4. 𝐫 \mathbf{r}
  5. 𝐯 \mathbf{v}
  6. 𝐮 ^ t \hat{\mathbf{u}}_{t}
  7. 𝐯 = v 𝐮 ^ t \mathbf{v}=v\hat{\mathbf{u}}_{t}
  8. 𝐯 \mathbf{v}\,
  9. 𝐫 \mathbf{r}\,
  10. 𝐯 = d 𝐫 d t \mathbf{v}={d\mathbf{r}\over{dt}}
  11. 𝐫 \mathbf{r}
  12. 𝐯 \mathbf{v}
  13. 𝐡 = 𝐫 × 𝐯 \mathbf{h}=\mathbf{r}\times\mathbf{v}

Orbiting_body.html

  1. m 2 m_{2}
  2. m 1 m_{1}
  3. m 1 > m 2 m_{1}>m_{2}

Orchestrated_objective_reduction.html

  1. 10 - 35 m 10^{-35}\,\text{m}
  2. τ / E G \tau\approx\hbar/E_{G}
  3. τ \tau
  4. E G E_{G}
  5. \hbar

Order-embedding.html

  1. x y if and only if f ( x ) f ( y ) . x\leq y\,\text{ if and only if }f(x)\leq f(y).
  2. e ( x ) = 2 π arctan x e(x)=\frac{2}{\pi}\arctan x

Order_(ring_theory).html

  1. 𝒪 \mathcal{O}
  2. A A
  3. \mathbb{Q}
  4. 𝒪 \mathcal{O}
  5. \mathbb{Q}
  6. 𝒪 = A \mathbb{Q}\mathcal{O}=A
  7. 𝒪 \mathcal{O}
  8. 𝒪 \mathcal{O}
  9. \mathbb{Q}
  10. 𝒪 \mathcal{O}
  11. 𝒪 \mathcal{O}
  12. a + b i , a+bi,

Order_and_disorder_(physics).html

  1. G ( x , x ) = s ( x ) , s ( x ) . G(x,x^{\prime})=\langle s(x),s(x^{\prime})\rangle.\,
  2. x = x x=x^{\prime}
  3. | x - x | |x-x^{\prime}|
  4. | x - x | |x-x^{\prime}|
  5. | x - x | |x-x^{\prime}|

Orders_of_magnitude_(data).html

  1. log 2 e \log_{2}e
  2. log 2 3 \log_{2}3
  3. × 10 9 \times 10^{9}
  4. × 10 9 \times 10^{9}
  5. × 10 9 \times 10^{9}
  6. × 10 1 0 \times 10^{1}0
  7. × 10 1 1 \times 10^{1}1
  8. × 10 1 1 \times 10^{1}1
  9. × 10 1 2 \times 10^{1}2
  10. × 10 1 2 \times 10^{1}2
  11. × 10 1 2 \times 10^{1}2
  12. × 10 1 2 \times 10^{1}2
  13. × 10 1 4 \times 10^{1}4
  14. × 10 1 6 \times 10^{1}6
  15. × 10 1 7 \times 10^{1}7
  16. × 10 1 7 \times 10^{1}7
  17. × 10 1 8 \times 10^{1}8
  18. × 10 1 8 \times 10^{1}8
  19. × 10 1 8 \times 10^{1}8
  20. × 10 2 0 \times 10^{2}0
  21. × 10 2 1 \times 10^{2}1
  22. × 10 2 1 \times 10^{2}1
  23. × 10 2 3 \times 10^{2}3
  24. × 10 2 3 \times 10^{2}3
  25. × 10 2 4 \times 10^{2}4
  26. × 10 2 5 \times 10^{2}5
  27. × 10 2 5 \times 10^{2}5
  28. × 10 4 5 \times 10^{4}5
  29. × 10 7 7 \times 10^{7}7
  30. 1 300 \scriptstyle\frac{1}{300}
  31. A c 3 / 4 G \scriptstyle Ac^{3}/4G\hbar
  32. A = 16 π G 2 M 2 / c 4 A=16\pi G^{2}M^{2}/c^{4}

Orders_of_magnitude_(energy).html

  1. E p = c 5 G E_{p}=\sqrt{\frac{\hbar c^{5}}{G}}
  2. U = ( 3 / 5 ) G M 2 r U=\frac{(3/5)GM^{2}}{r}

Oregon_Caves_National_Monument_and_Preserve.html

  1. \rightleftarrows

Organic_composition_of_capital.html

  1. c v {c\over v}
  2. c s + v {c\over{s+v}}
  3. c c + v {c\over{c+v}}

Orientifold.html

  1. \mathcal{M}
  2. G 1 G_{1}
  3. G 2 G_{2}
  4. Ω p \Omega_{p}
  5. Ω p : σ 2 π - σ \Omega_{p}:\sigma\to 2\pi-\sigma
  6. / ( G 1 Ω G 2 ) \mathcal{M}/(G_{1}\cup\Omega G_{2})
  7. G 2 G_{2}
  8. G 2 G_{2}
  9. \mathcal{M}
  10. G 1 G_{1}
  11. G 2 G_{2}
  12. σ \sigma
  13. Ω \Omega
  14. σ ( Ω ) = Ω \sigma(\Omega)=\Omega
  15. σ ( Ω ) = - Ω \sigma(\Omega)=-\Omega
  16. σ ( Ω ) = Ω ¯ \sigma(\Omega)=\bar{\Omega}
  17. σ ( Ω ) = Ω \sigma(\Omega)=\Omega
  18. σ ( J ) = J \sigma(J)=J
  19. σ ( J ) = - J \sigma(J)=-J
  20. σ \sigma
  21. ± 1 \pm 1
  22. ω i \omega_{i}
  23. h 1 , 1 h^{1,1}
  24. σ \sigma
  25. A i A_{i}
  26. J = A i ω i J=A_{i}\omega_{i}
  27. σ \sigma
  28. σ \sigma
  29. σ \sigma
  30. h 1 , 1 = h + 1 , 1 + h - 1 , 1 h^{1,1}=h^{1,1}_{+}+h^{1,1}_{-}
  31. h 1 , 1 = h ± 1 , 1 h^{1,1}=h^{1,1}_{\pm}

Orthogonal_basis.html

  1. V V
  2. V V
  3. V V
  4. V V
  5. · , · \langle·,·\rangle
  6. 𝐯 \mathbf{v}
  7. 𝐰 \mathbf{w}
  8. 𝐯 , 𝐰 = 0 \langle\mathbf{v},\mathbf{w}\rangle=0
  9. 𝐞 j , 𝐞 k = { q ( 𝐞 k ) j = k 0 j k , \langle\mathbf{e}_{j},\mathbf{e}_{k}\rangle=\left\{\begin{array}[]{ll}q(% \mathbf{e}_{k})&j=k\\ 0&j\neq k\end{array}\right.\quad,
  10. q q
  11. · , · \langle·,·\rangle
  12. q ( 𝐯 ) = 𝐯 , 𝐯 q(\mathbf{v})=\langle\mathbf{v},\mathbf{v}\rangle
  13. 𝐯 , 𝐰 = k q ( 𝐞 k ) v k w k , \langle\mathbf{v},\mathbf{w}\rangle=\sum\limits_{k}q(\mathbf{e}_{k})v^{k}w^{k}\ ,
  14. v < s u p > k v<sup>k

Orthogonal_complement.html

  1. V V
  2. F F
  3. B B
  4. u u
  5. v v
  6. v v
  7. u u
  8. B ( u , v ) = 0 B(u,v)=0
  9. W W
  10. V V
  11. W W^{\bot}
  12. W = { x V : B ( x , y ) = 0 for all y W } . W^{\bot}=\left\{x\in V:B(x,y)=0\mbox{ for all }~{}y\in W\right\}\,.
  13. B ( u , v ) = 0 B(u,v)=0
  14. B ( v , u ) = 0 B(v,u)=0
  15. u u
  16. v v
  17. V V
  18. B B
  19. V V
  20. V V^{\bot}
  21. V V
  22. W ( W ) W\subset(W^{\bot})^{\bot}
  23. B B
  24. V V
  25. dim ( W ) + dim ( W ) = dim V \dim(W)+\dim(W^{\bot})=\dim V
  26. W W
  27. W W
  28. ( W ) = W ¯ (W^{\bot})^{\bot}=\overline{W}
  29. H H
  30. X X
  31. Y Y
  32. X = X ¯ X^{\bot}=\overline{X}^{\bot}
  33. Y X Y\subset X
  34. X X = { 0 } X\cap X^{\bot}=\{0\}
  35. X ( X ) X\subset(X^{\bot})^{\bot}
  36. X X
  37. H H
  38. X X
  39. H H
  40. W = { x V * : y W , x ( y ) = 0 } . W^{\bot}=\left\{\,x\in V^{*}:\forall y\in W,x(y)=0\,\right\}.\,
  41. i W ¯ = W . i\overline{W}=W^{\bot\,\bot}.