wpmath0000008_5

Green–Tao_theorem.html

  1. π ( N ) \pi(N)
  2. N N
  3. A A
  4. lim sup N | A [ 1 , N ] | π ( N ) > 0 \limsup_{N\rightarrow\infty}\dfrac{|A\cap[1,N]|}{\pi(N)}>0
  5. k k
  6. A A
  7. k k
  8. ( 𝔖 k + o ( 1 ) ) N 2 ( log N ) k (\mathfrak{S}_{k}+o(1))\frac{N^{2}}{(\log N)^{k}}
  9. p 1 < p 2 < < p k N p_{1}<p_{2}<\cdots<p_{k}\leq N
  10. 𝔖 k \mathfrak{S}_{k}
  11. 𝔖 k := 1 2 ( k - 1 ) ( p k 1 p ( p p - 1 ) k - 1 ) ( p > k ( 1 - k - 1 p ) ( p p - 1 ) k - 1 ) \mathfrak{S}_{k}:=\frac{1}{2(k-1)}\left(\prod_{p\leq k}\frac{1}{p}\left(\frac{% p}{p-1}\right)^{k-1}\right)\left(\prod_{p>k}\left(1-\frac{k-1}{p}\right)\left(% \frac{p}{p-1}\right)^{k-1}\right)

Grimm's_conjecture.html

  1. [ n + 1 , n + k ] [n+1,n+k]
  2. x k ( n + x ) \prod_{x\leq k}(n+x)

Gross_tonnage.html

  1. K = 0.2 + 0.02 × log 10 ( V ) K=0.2+0.02\times\log_{10}(V)\,
  2. G T = K × V GT=K\times V\,
  3. G T = V × ( .02 × log 10 ( V ) + .2 ) GT=V\times(.02\times\log_{10}(V)+.2)
  4. K = 0.2 + 0.02 × log 10 ( V ) K=0.2+0.02\times\log_{10}(V)\,
  5. = 0.2 + 0.02 × log 10 ( 10 , 000 ) =0.2+0.02\times\log_{10}(10,000)\,
  6. = 0.2 + 0.02 × 4 =0.2+0.02\times 4\,
  7. = 0.2 + 0.08 =0.2+0.08\,
  8. = 0.28 =0.28\,
  9. G T = K × V GT=K\times V\,
  10. = 0.28 × 10 , 000 =0.28\times 10,000\,
  11. = 2 , 800 =2,800\,
  12. V = 50 × ln 10 × G T W ( 500 , 000 , 000 , 000 × ln 10 × G T ) V=\frac{50\times\ln 10\times GT}{W(500,000,000,000\times\ln 10\times GT)}

Gross–Pitaevskii_equation.html

  1. Ψ \Psi
  2. N N
  3. ψ \psi
  4. Ψ ( 𝐫 1 , 𝐫 2 , , 𝐫 N ) = ψ ( 𝐫 1 ) ψ ( 𝐫 2 ) ψ ( 𝐫 N ) \Psi(\mathbf{r}_{1},\mathbf{r}_{2},\dots,\mathbf{r}_{N})=\psi(\mathbf{r}_{1})% \psi(\mathbf{r}_{2})\dots\psi(\mathbf{r}_{N})
  5. 𝐫 i \mathbf{r}_{i}
  6. i i
  7. H = i = 1 N ( - 2 2 m 2 𝐫 i 2 + V ( 𝐫 i ) ) + i < j 4 π 2 a s m δ ( 𝐫 i - 𝐫 j ) , H=\sum_{i=1}^{N}\left(-{\hbar^{2}\over 2m}{\partial^{2}\over\partial\mathbf{r}% _{i}^{2}}+V(\mathbf{r}_{i})\right)+\sum_{i<j}{4\pi\hbar^{2}a_{s}\over m}\delta% (\mathbf{r}_{i}-\mathbf{r}_{j}),
  8. m m
  9. V V
  10. a s a_{s}
  11. δ ( 𝐫 ) \delta(\mathbf{r})
  12. ( - 2 2 m 2 𝐫 2 + V ( 𝐫 ) + 4 π 2 a s m | ψ ( 𝐫 ) | 2 ) ψ ( 𝐫 ) = μ ψ ( 𝐫 ) , \left(-\frac{\hbar^{2}}{2m}{\partial^{2}\over\partial\mathbf{r}^{2}}+V(\mathbf% {r})+{4\pi\hbar^{2}a_{s}\over m}|\psi(\mathbf{r})|^{2}\right)\psi(\mathbf{r})=% \mu\psi(\mathbf{r}),
  13. d V | ψ | 2 = N \int dV|\psi|^{2}=N
  14. a s a_{s}
  15. g = 4 π 2 a s m g=\frac{4\pi\hbar^{2}a_{s}}{m}
  16. \hbar
  17. = 2 2 m | Ψ ( 𝐫 ) | 2 + V ( 𝐫 ) | Ψ ( 𝐫 ) | 2 + 1 2 g | Ψ ( 𝐫 ) | 4 , \mathcal{E}=\frac{\hbar^{2}}{2m}|\nabla\Psi(\mathbf{r})|^{2}+V(\mathbf{r})|% \Psi(\mathbf{r})|^{2}+\frac{1}{2}g|\Psi(\mathbf{r})|^{4},
  18. Ψ \Psi
  19. μ Ψ ( 𝐫 ) = ( - 2 2 m 2 + V ( 𝐫 ) + g | Ψ ( 𝐫 ) | 2 ) Ψ ( 𝐫 ) \mu\Psi(\mathbf{r})=\left(-\frac{\hbar^{2}}{2m}\nabla^{2}+V(\mathbf{r})+g|\Psi% (\mathbf{r})|^{2}\right)\Psi(\mathbf{r})
  20. μ \mu
  21. N = | Ψ ( 𝐫 ) | 2 d 3 r . N=\int|\Psi(\mathbf{r})|^{2}\,d^{3}r.
  22. i Ψ ( 𝐫 , t ) t = ( - 2 2 m 2 + V ( 𝐫 ) + g | Ψ ( 𝐫 , t ) | 2 ) Ψ ( 𝐫 , t ) . i\hbar\frac{\partial\Psi(\mathbf{r},t)}{\partial t}=\left(-\frac{\hbar^{2}}{2m% }\nabla^{2}+V(\mathbf{r})+g|\Psi(\mathbf{r},t)|^{2}\right)\Psi(\mathbf{r},t).
  23. V ( 𝐫 ) = 0 V(\mathbf{r})=0
  24. Ψ ( 𝐫 ) = N V e i 𝐤 𝐫 . \Psi(\mathbf{r})=\sqrt{\frac{N}{V}}e^{i\mathbf{k}\cdot\mathbf{r}}.
  25. E ( 𝐤 ) = N [ 2 k 2 2 m + g N 2 V ] . E(\mathbf{k})=N\left[\frac{\hbar^{2}k^{2}}{2m}+g\frac{N}{2V}\right].
  26. g > 0 g>0
  27. ψ ( x ) = ψ 0 tanh ( x 2 ξ ) \psi(x)=\psi_{0}\tanh\left(\frac{x}{\sqrt{2}\xi}\right)
  28. ψ 0 \psi_{0}
  29. \infty
  30. ξ = / 2 m n 0 g \xi=\hbar/\sqrt{2mn_{0}g}
  31. ψ \psi
  32. π \pi
  33. g < 0 g<0
  34. ψ ( x , t ) = ψ ( 0 ) e - i μ t / 1 cosh [ 2 m | μ | / 2 x ] , \psi(x,t)=\psi(0)e^{-i\mu t/\hbar}\frac{1}{\cosh\left[\sqrt{2m|\mu|/\hbar^{2}}% x\right]},
  35. μ = g | ψ ( 0 ) | 2 / 2 \mu=g|\psi(0)|^{2}/2
  36. ψ ( x , t ) = μ - V ( x ) N U 0 \psi(x,t)=\sqrt{\frac{\mu-V(x)}{NU_{0}}}
  37. ψ 0 = n e - i μ t \psi_{0}=\sqrt{n}e^{-i\mu t}
  38. δ ψ \delta\psi
  39. ψ = ψ 0 + δ ψ \psi=\psi_{0}+\delta\psi
  40. δ ψ \delta\psi
  41. - 2 2 m 2 δ ψ + V δ ψ + g ( 2 | ψ 0 | 2 δ ψ + ψ 2 δ ψ * ) = i δ ψ t -\frac{\hbar^{2}}{2m}\nabla^{2}\delta\psi+V\delta\psi+g(2|\psi_{0}|^{2}\delta% \psi+\psi^{2}\delta\psi^{*})=i\hbar\frac{\partial\delta\psi}{\partial t}
  42. - 2 2 m 2 δ ψ * + V δ ψ * + g ( 2 | ψ 0 | 2 δ ψ * + ( ψ * ) 2 δ ψ ) = - i δ ψ * t -\frac{\hbar^{2}}{2m}\nabla^{2}\delta\psi^{*}+V\delta\psi^{*}+g(2|\psi_{0}|^{2% }\delta\psi^{*}+(\psi^{*})^{2}\delta\psi)=-i\hbar\frac{\partial\delta\psi^{*}}% {\partial t}
  43. δ ψ \delta\psi
  44. δ ψ = e - i μ t ( u ( s y m b o l r ) e - i ω t - v * ( s y m b o l r ) e i ω t ) \delta\psi=e^{-i\mu t}(u(symbol{r})e^{-i\omega t}-v^{*}(symbol{r})e^{i\omega t})
  45. u u
  46. v v
  47. e ± i ω t e^{\pm i\omega t}
  48. ( - 2 2 m 2 + V + 2 g n - μ - ω ) u - g n v = 0 (-\frac{\hbar^{2}}{2m}\nabla^{2}+V+2gn-\hbar\mu-\hbar\omega)u-gnv=0
  49. ( - 2 2 m 2 + V + 2 g n - μ + ω ) v - g n u = 0 (-\frac{\hbar^{2}}{2m}\nabla^{2}+V+2gn-\hbar\mu+\hbar\omega)v-gnu=0
  50. V ( s y m b o l r ) = c o n s t . V(symbol{r})=const.
  51. V = μ - g n V=\hbar\mu-gn
  52. u u
  53. v v
  54. s y m b o l q symbol{q}
  55. ω = ϵ s y m b o l q = 2 s y m b o l q 2 2 m ( 2 s y m b o l q 2 2 m + 2 g n ) \hbar\omega=\epsilon_{s}ymbol{q}=\sqrt{\frac{\hbar^{2}symbol{q}^{2}}{2m}\left(% \frac{\hbar^{2}symbol{q}^{2}}{2m}+2gn\right)}
  56. s y m b o l q symbol{q}
  57. s y m b o l q symbol{q}
  58. s y m b o l q symbol{q}
  59. ϵ s y m b o l q = s q \epsilon_{s}ymbol{q}=s\hbar q
  60. s = n g / m s=\sqrt{ng/m}
  61. ϵ s y m b o l q / ( q ) > s \epsilon_{s}ymbol{q}/(\hbar q)>s

Grötzsch_graph.html

  1. - ( x - 1 ) 5 ( x 2 - x - 10 ) ( x 2 + 3 x + 1 ) 2 . -(x-1)^{5}(x^{2}-x-10)(x^{2}+3x+1)^{2}.

Guard_digit.html

  1. 2 1 × 0.100 2 - 2 0 × 0.111 2 2^{1}\times 0.100_{2}-2^{0}\times 0.111_{2}
  2. 2 1 × 0.1000 2 - 2 1 × 0.0111 2 2^{1}\times 0.1000_{2}-2^{1}\times 0.0111_{2}
  3. 2 1 × 0.0001 2 2^{1}\times 0.0001_{2}
  4. 2 - 2 × 0.100 2 2^{-2}\times 0.100_{2}
  5. 2 1 × 0.100 2 - 2 1 × 0.011 2 2^{1}\times 0.100_{2}-2^{1}\times 0.011_{2}
  6. 2 1 × 0.001 2 = 2^{1}\times 0.001_{2}=
  7. 2 - 1 × 0.100 2 2^{-1}\times 0.100_{2}

Guy_Terjanian.html

  1. x 2 p + y 2 p = z 2 p x^{2p}+y^{2p}=z^{2p}
  2. x 2 p + y 2 p = z 2 p x^{2p}+y^{2p}=z^{2p}

GW_approximation.html

  1. = 1 \hbar=1
  2. Σ = i G W - G W G W G + \Sigma=iGW-GWGWG+\cdots
  3. Σ i G W \Sigma\approx iGW
  4. Σ ( 1 , 2 ) = i G ( 1 , 2 ) W ( 1 + , 2 ) - d 3 d 4 G ( 1 , 3 ) G ( 3 , 4 ) G ( 4 , 2 ) W ( 1 , 4 ) W ( 3 , 2 ) + \Sigma(1,2)=iG(1,2)W(1^{+},2)-\int d3\int d4\,G(1,3)G(3,4)G(4,2)W(1,4)W(3,2)+...
  5. Σ ( 1 , 2 ) i G ( 1 , 2 ) W ( 1 + , 2 ) \Sigma(1,2)\approx iG(1,2)W(1^{+},2)
  6. r s r_{s}
  7. 1 / r s 2 1/r_{s}^{2}
  8. 1 / r s 1/r_{s}
  9. q q
  10. ϵ ( q ) = 1 + λ 2 / q 2 \epsilon(q)=1+\lambda^{2}/q^{2}
  11. λ \lambda
  12. r s - 1 / 2 r_{s}^{-1/2}
  13. q q
  14. 1 / r s 1/r_{s}
  15. ϵ 1 + r s \epsilon\sim 1+r_{s}
  16. W ( q ) = V ( q ) / ϵ ( q ) W(q)=V(q)/\epsilon(q)
  17. r s r_{s}
  18. W W
  19. r s / ( 1 + r s ) r_{s}/(1+r_{s})
  20. r s r_{s}

Gyration_tensor.html

  1. S m n = def 1 N i = 1 N r m ( i ) r n ( i ) S_{mn}\ \stackrel{\mathrm{def}}{=}\ \frac{1}{N}\sum_{i=1}^{N}r_{m}^{(i)}r_{n}^% {(i)}
  2. r m ( i ) r_{m}^{(i)}
  3. m th \mathrm{m^{th}}
  4. 𝐫 ( i ) \mathbf{r}^{(i)}
  5. i th \mathrm{i^{th}}
  6. i = 1 N 𝐫 ( i ) = 0 \sum_{i=1}^{N}\mathbf{r}^{(i)}=0
  7. r C M r_{CM}
  8. r C M = 1 N i = 1 N 𝐫 ( i ) r_{CM}=\frac{1}{N}\sum_{i=1}^{N}\mathbf{r}^{(i)}
  9. S m n = def 1 2 N 2 i = 1 N j = 1 N ( r m ( i ) - r m ( j ) ) ( r n ( i ) - r n ( j ) ) S_{mn}\ \stackrel{\mathrm{def}}{=}\ \frac{1}{2N^{2}}\sum_{i=1}^{N}\sum_{j=1}^{% N}(r_{m}^{(i)}-r_{m}^{(j)})(r_{n}^{(i)}-r_{n}^{(j)})
  10. S x y = 1 2 N 2 i = 1 N j = 1 N ( x i - x j ) ( y i - y j ) S_{xy}=\frac{1}{2N^{2}}\sum_{i=1}^{N}\sum_{j=1}^{N}(x_{i}-x_{j})(y_{i}-y_{j})
  11. S m n = def d 𝐫 ρ ( 𝐫 ) r m r n d 𝐫 ρ ( 𝐫 ) S_{mn}\ \stackrel{\mathrm{def}}{=}\ \dfrac{\int d\mathbf{r}\ \rho(\mathbf{r})% \ r_{m}r_{n}}{\int d\mathbf{r}\ \rho(\mathbf{r})}
  12. ρ ( 𝐫 ) \rho(\mathbf{r})
  13. 𝐫 \mathbf{r}
  14. 𝐒 = [ λ x 2 0 0 0 λ y 2 0 0 0 λ z 2 ] \mathbf{S}=\begin{bmatrix}\lambda_{x}^{2}&0&0\\ 0&\lambda_{y}^{2}&0\\ 0&0&\lambda_{z}^{2}\end{bmatrix}
  15. λ x 2 λ y 2 λ z 2 \lambda_{x}^{2}\leq\lambda_{y}^{2}\leq\lambda_{z}^{2}
  16. R g 2 = λ x 2 + λ y 2 + λ z 2 R_{g}^{2}=\lambda_{x}^{2}+\lambda_{y}^{2}+\lambda_{z}^{2}
  17. b b
  18. b = def λ z 2 - 1 2 ( λ x 2 + λ y 2 ) = 3 2 λ z 2 - R g 2 2 b\ \stackrel{\mathrm{def}}{=}\ \lambda_{z}^{2}-\frac{1}{2}\left(\lambda_{x}^{2% }+\lambda_{y}^{2}\right)=\frac{3}{2}\lambda_{z}^{2}-\frac{R_{g}^{2}}{2}
  19. c c
  20. c = def λ y 2 - λ x 2 c\ \stackrel{\mathrm{def}}{=}\ \lambda_{y}^{2}-\lambda_{x}^{2}
  21. κ 2 \kappa^{2}
  22. κ 2 = def b 2 + ( 3 / 4 ) c 2 R g 4 = 3 2 λ x 4 + λ y 4 + λ z 4 ( λ x 2 + λ y 2 + λ z 2 ) 2 - 1 2 \kappa^{2}\ \stackrel{\mathrm{def}}{=}\ \frac{b^{2}+(3/4)c^{2}}{R_{g}^{4}}=% \frac{3}{2}\frac{\lambda_{x}^{4}+\lambda_{y}^{4}+\lambda_{z}^{4}}{(\lambda_{x}% ^{2}+\lambda_{y}^{2}+\lambda_{z}^{2})^{2}}-\frac{1}{2}
  23. κ 2 \kappa^{2}
  24. κ 2 \kappa^{2}

Gysin_sequence.html

  1. π : E M . \pi:E\longrightarrow M.
  2. π * : H * ( M ) H * ( E ) . \pi^{*}:H^{*}(M)\longrightarrow H^{*}(E).\,
  3. π * : H * ( E ) H * ( M ) \pi_{*}:H^{*}(E)\longrightarrow H^{*}(M)
  4. H n ( E ) π * H n - k ( M ) e H n + 1 ( M ) π * H n + 1 ( E ) ...\longrightarrow H^{n}(E)\longrightarrow^{\!\!\!\!\!\!\!\!\!\!\pi_{*}}H^{n-k% }(M)\longrightarrow^{\!\!\!\!\!\!\!\!\!\!e\wedge}H^{n+1}(M)\longrightarrow^{\!% \!\!\!\!\!\!\!\!\!\pi^{*}}H^{n+1}(E)\longrightarrow...
  5. e e\wedge
  6. π * : H * ( E ) H * ( M ) , \pi_{*}\colon H^{*}(E)\longrightarrow H^{*}(M),

H-derivative.html

  1. i : H E i:H\to E
  2. F : E F:E\to\mathbb{R}
  3. D F : E Lin ( E ; ) \mathrm{D}F:E\to\mathrm{Lin}(E;\mathbb{R})
  4. x E x\in E
  5. D F ( x ) \mathrm{D}F(x)
  6. E * E^{*}
  7. E E
  8. H H
  9. D H F \mathrm{D}_{H}F
  10. x E x\in E
  11. D H F ( x ) := D F ( x ) i : H \R \mathrm{D}_{H}F(x):=\mathrm{D}F(x)\circ i:H\to\R
  12. H H
  13. H H
  14. H F : E H \nabla_{H}F:E\to H
  15. H F ( x ) , h H = ( D H F ) ( x ) ( h ) = lim t 0 F ( x + t i ( h ) ) - F ( x ) t \langle\nabla_{H}F(x),h\rangle_{H}=\left(\mathrm{D}_{H}F\right)(x)(h)=\lim_{t% \to 0}\frac{F(x+ti(h))-F(x)}{t}
  16. j : E * H j:E^{*}\to H
  17. i : H E i:H\to E
  18. H F ( x ) := j ( D F ( x ) ) \nabla_{H}F(x):=j\left(\mathrm{D}F(x)\right)

Hack's_law.html

  1. L = C A h L=CA^{h}

Hadamard's_dynamical_system.html

  1. H ( p , q ) = 1 2 m p i p j g i j ( q ) H(p,q)=\frac{1}{2m}p_{i}p_{j}g^{ij}(q)
  2. q i q^{i}
  3. i = 1 , 2 i=1,2
  4. p i p_{i}
  5. p i = m g i j d q j d t p_{i}=mg_{ij}\frac{dq^{j}}{dt}
  6. d s 2 = g i j ( q ) d q i d q j ds^{2}=g_{ij}(q)dq^{i}dq^{j}\,
  7. e λ t e^{\lambda t}
  8. λ = 2 E m R 2 \lambda=\sqrt{\frac{2E}{mR^{2}}}
  9. K = - 1 / R 2 K=-1/R^{2}

Hadamard_code.html

  1. k k
  2. 2 k 2^{k}
  3. 2 k / 2 2^{k}/2
  4. 2 k / 2 2^{k}/2
  5. [ 2 k , k , 2 k / 2 ] 2 [2^{k},k,2^{k}/2]_{2}
  6. 2 k 2^{k}
  7. k k
  8. 2 k / 2 2^{k}/2
  9. [ 2 k - 1 , k , 2 k - 2 ] 2 [2^{k-1},k,2^{k-2}]_{2}
  10. 1 / 2 1/2
  11. ( n , 2 n , n / 2 ) 2 (n,2n,n/2)_{2}
  12. 1 2 - ϵ \frac{1}{2}-\epsilon
  13. x { 0 , 1 } k x\in\{0,1\}^{k}
  14. k k
  15. Had ( x ) \,\text{Had}(x)
  16. Had : { 0 , 1 } k { 0 , 1 } 2 k \,\text{Had}:\{0,1\}^{k}\to\{0,1\}^{2^{k}}
  17. x , y \langle x,y\rangle
  18. x , y { 0 , 1 } k x,y\in\{0,1\}^{k}
  19. x , y = i = 1 k x i y i mod 2 . \langle x,y\rangle=\sum_{i=1}^{k}x_{i}y_{i}\ \bmod\ 2\,.
  20. x x
  21. x x
  22. Had ( x ) = ( x , y ) y { 0 , 1 } k \,\text{Had}(x)=\Big(\langle x,y\rangle\Big)_{y\in\{0,1\}^{k}}
  23. y y
  24. y 1 = 0 y_{1}=0
  25. x 1 x_{1}
  26. x x
  27. y 1 = 1 y_{1}=1
  28. x x
  29. x x
  30. pHad ( x ) = ( x , y ) y { 1 } × { 0 , 1 } k - 1 \,\text{pHad}(x)=\Big(\langle x,y\rangle\Big)_{y\in\{1\}\times\{0,1\}^{k-1}}
  31. G G
  32. Had ( x ) = x G \,\text{Had}(x)=x\cdot G
  33. x { 0 , 1 } k x\in\{0,1\}^{k}
  34. x x
  35. 𝔽 2 \mathbb{F}_{2}
  36. y y
  37. k k
  38. G = ( y 1 y 2 y 2 k ) . G=\begin{pmatrix}\uparrow&\uparrow&&\uparrow\\ y_{1}&y_{2}&\dots&y_{2^{k}}\\ \downarrow&\downarrow&&\downarrow\end{pmatrix}\,.
  39. y i { 0 , 1 } k y_{i}\in\{0,1\}^{k}
  40. i i
  41. k = 3 k=3
  42. G = [ 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 ] . G=\begin{bmatrix}0&0&0&0&1&1&1&1\\ 0&0&1&1&0&0&1&1\\ 0&1&0&1&0&1&0&1\end{bmatrix}.
  43. G G
  44. ( k × 2 k ) (k\times 2^{k})
  45. Had : { 0 , 1 } k { 0 , 1 } 2 k \,\text{Had}:\{0,1\}^{k}\to\{0,1\}^{2^{k}}
  46. G G
  47. k = 3 k=3
  48. G = [ 1 1 1 1 0 0 1 1 0 1 0 1 ] . G^{\prime}=\begin{bmatrix}1&1&1&1\\ 0&0&1&1\\ 0&1&0&1\end{bmatrix}.
  49. pHad : { 0 , 1 } k { 0 , 1 } 2 k - 1 \,\text{pHad}:\{0,1\}^{k}\to\{0,1\}^{2^{k-1}}
  50. pHad ( x ) = x G \,\text{pHad}(x)=x\cdot G^{\prime}
  51. k k
  52. 2 k - 1 2^{k-1}
  53. 2 k - 1 - k 2^{k-1}-k
  54. n = 2 k - 1 n=2^{k-1}
  55. log 2 ( 2 n ) = k \log_{2}(2n)=k
  56. 2 k - 1 2^{k-1}
  57. x { 0 , 1 } k x\in\{0,1\}^{k}
  58. Pr y { 0 , 1 } k [ ( Had ( x ) ) y = 1 ] = Pr y { 0 , 1 } k [ x , y = 1 ] . \Pr_{y\in\{0,1\}^{k}}\big[(\,\text{Had}(x))_{y}=1\big]=\Pr_{y\in\{0,1\}^{k}}% \big[\langle x,y\rangle=1\big]\,.
  59. 1 / 2 1/2
  60. x 1 = 1 x_{1}=1
  61. y 2 , , y k y_{2},\dots,y_{k}
  62. y 1 x 1 = b y_{1}\cdot x_{1}=b
  63. b { 0 , 1 } b\in\{0,1\}
  64. x 2 , , x k x_{2},\dots,x_{k}
  65. y 2 , , y k y_{2},\dots,y_{k}
  66. y 1 = b y_{1}=b
  67. 1 / 2 1/2
  68. 1 / 2 1/2
  69. 1 / 2 1/2
  70. 1 / 2 1/2
  71. 1 / 2 1/2
  72. 1 1
  73. 1 2 k - 1 1^{2^{k-1}}
  74. x = 10 k - 1 x=10^{k-1}
  75. pHad ( 10 k - 1 ) = 1 2 k - 1 \,\text{pHad}(10^{k-1})=1^{2^{k-1}}
  76. x x
  77. 10 k - 1 10^{k-1}
  78. Had ( x ) \,\text{Had}(x)
  79. 1 / 2 1/2
  80. q q
  81. x i x_{i}
  82. q q
  83. C : { 0 , 1 } k { 0 , 1 } n C:\{0,1\}^{k}\rightarrow\{0,1\}^{n}
  84. ( q , δ 0 , ϵ 0 ) (q,\delta\geq 0,\epsilon\geq 0)
  85. D : { 0 , 1 } n { 0 , 1 } k D:\{0,1\}^{n}\rightarrow\{0,1\}^{k}
  86. Δ ( x , y ) \Delta(x,y)
  87. x x
  88. y y
  89. x { 0 , 1 } k , y { 0 , 1 } n \forall x\in\{0,1\}^{k},\forall y\in\{0,1\}^{n}
  90. Δ ( y , C ( x ) ) δ n \Delta(y,C(x))\leq\delta n
  91. P r [ D ( y ) i = x i ] 1 2 + ϵ , i [ k ] Pr[D(y)_{i}=x_{i}]\geq\frac{1}{2}+\epsilon,\forall i\in[k]
  92. ( 2 , δ , 1 2 - 2 δ ) (2,\delta,\frac{1}{2}-2\delta)
  93. 0 δ 1 4 0\leq\delta\leq\frac{1}{4}
  94. c c
  95. C C
  96. c i + c j = c i + j c_{i}+c_{j}=c_{i+j}
  97. c i , c j c_{i},c_{j}
  98. c c
  99. i i
  100. j j
  101. c i + j c_{i+j}
  102. ( i + j ) (i+j)
  103. C ( x ) = c = ( c 0 , , c 2 n - 1 ) C(x)=c=(c_{0},\dots,c_{2^{n}-1})
  104. C C
  105. x x
  106. G = ( g 0 g 1 g 2 n - 1 ) G=\begin{pmatrix}\uparrow&\uparrow&&\uparrow\\ g_{0}&g_{1}&\dots&g_{2^{n}-1}\\ \downarrow&\downarrow&&\downarrow\end{pmatrix}
  107. C C
  108. c i = x g i c_{i}=x\cdot g_{i}
  109. c i + c j = x g i + x g j = x ( g i + g j ) c_{i}+c_{j}=x\cdot g_{i}+x\cdot g_{j}=x\cdot(g_{i}+g_{j})
  110. G G
  111. g i + g j = g i + j g_{i}+g_{j}=g_{i+j}
  112. c i + c j = x g i + j = c i + j c_{i}+c_{j}=x\cdot g_{i+j}=c_{i+j}
  113. y = ( y 0 , , y 2 n - 1 ) y=(y_{0},\dots,y_{2^{n}-1})
  114. i { 1 , , n } i\in\{1,\dots,n\}
  115. j { 0 , , 2 n - 1 } j\in\{0,\dots,2^{n}-1\}
  116. k { 0 , , 2 n - 1 } k\in\{0,\dots,2^{n}-1\}
  117. j + k = e i j+k=e_{i}
  118. j + k j+k
  119. j j
  120. k k
  121. x i y j + y k x_{i}\leftarrow y_{j}+y_{k}
  122. x = ( x 1 , , x n ) x=(x_{1},\dots,x_{n})
  123. x x
  124. y y
  125. y y
  126. c = C ( x ) c=C(x)
  127. δ \delta
  128. x i x_{i}
  129. 1 2 + ( 1 - 2 δ ) \frac{1}{2}+(1-2\delta)
  130. c j + c k = c j + k = x g j + k = x e i = x i c_{j}+c_{k}=c_{j+k}=x\cdot g_{j+k}=x\cdot e_{i}=x_{i}
  131. j j
  132. k k
  133. y j c j y_{j}\not=c_{j}
  134. δ \delta
  135. y k c k y_{k}\not=c_{k}
  136. δ \delta
  137. y j y_{j}
  138. y k y_{k}
  139. c c
  140. 2 δ 2\delta
  141. y j y_{j}
  142. y k y_{k}
  143. c c
  144. x i x_{i}
  145. x i x_{i}
  146. 1 - 2 δ 1-2\delta
  147. ϵ = 1 2 - 2 δ \epsilon=\frac{1}{2}-2\delta
  148. ϵ \epsilon
  149. 0 δ 1 4 0\leq\delta\leq\frac{1}{4}
  150. ( 2 , δ , 1 2 - 2 δ ) (2,\delta,\frac{1}{2}-2\delta)
  151. 0 δ 1 4 0\leq\delta\leq\frac{1}{4}

Half-logistic_distribution.html

  1. 1 - e - k 1 + e - k \frac{1-e^{-k}}{1+e^{-k}}\!
  2. log e ( 4 ) = 1.386 \log_{e}(4)=1.386\ldots
  3. log e ( 3 ) = 1.0986 \log_{e}(3)=1.0986\ldots
  4. π 2 / 3 - ( log e ( 4 ) ) 2 = 1.368 \pi^{2}/3-(\log_{e}(4))^{2}=1.368\ldots
  5. X = | Y | X=|Y|\!
  6. G ( k ) = 1 - e - k 1 + e - k for k 0. G(k)=\frac{1-e^{-k}}{1+e^{-k}}\mbox{ for }~{}k\geq 0.\!
  7. g ( k ) = 2 e - k ( 1 + e - k ) 2 for k 0. g(k)=\frac{2e^{-k}}{(1+e^{-k})^{2}}\mbox{ for }~{}k\geq 0.\!

Half-value_layer.html

  1. H F = 1 s t H V L 2 n d H V L HF=\frac{1^{st}HVL}{2^{nd}HVL}

Hall's_conjecture.html

  1. | y 2 - x 3 | > C | x | . |y^{2}-x^{3}|>C\sqrt{|x|}.
  2. deg ( g ( t ) 2 - f ( t ) 3 ) 1 2 deg f ( t ) + 1. \deg(g(t)^{2}-f(t)^{3})\geq\frac{1}{2}\deg f(t)+1.
  3. | y 2 - x 3 | > c ( ε ) x 1 / 2 - ε . |y^{2}-x^{3}|>c(\varepsilon)x^{1/2-\varepsilon}.

Hall_algebra.html

  1. C p λ i , C_{p^{\lambda_{i}}},
  2. λ = ( λ 1 , λ 2 , ) \lambda=(\lambda_{1},\lambda_{2},\ldots)
  3. n n
  4. g μ , ν λ ( p ) g^{\lambda}_{\mu,\nu}(p)
  5. ν \nu
  6. μ \mu
  7. g μ , ν λ ( q ) [ q ] . g^{\lambda}_{\mu,\nu}(q)\in\mathbb{Z}[q].\,
  8. H H
  9. [ q ] \mathbb{Z}[q]
  10. u λ u_{\lambda}
  11. u μ u ν = λ g μ , ν λ ( q ) u λ . u_{\mu}u_{\nu}=\sum_{\lambda}g^{\lambda}_{\mu,\nu}(q)u_{\lambda}.\,
  12. u 𝟏 n u_{\mathbf{1}^{n}}
  13. u 𝟏 n q - n ( n - 1 ) / 2 e n u_{\mathbf{1}^{n}}\mapsto q^{-n(n-1)/2}e_{n}\,
  14. u λ u_{\lambda}

Hamaker_theory.html

  1. R i j = | R i - R j | R_{ij}=|R_{i}-R_{j}|
  2. V int 1 , 2 , N = 1 2 i = 0 𝒩 j = 0 ( i ) 𝒩 V int i j ( R i j ) V_{\mathrm{int}}^{1,2,...N}=\frac{1}{2}\sum_{i=0}^{\mathcal{N}}\sum_{j=0(\neq i% )}^{\mathcal{N}}V_{\mathrm{int}}^{ij}(R_{ij})
  3. V int i j V_{\mathrm{int}}^{ij}

Hamiltonian_(control_theory).html

  1. u ( t ) u(t)
  2. J ( u ) = Ψ ( x ( T ) ) + 0 T L ( x , u , t ) d t J(u)=\Psi(x(T))+\int^{T}_{0}L(x,u,t)dt
  3. x ( t ) x(t)
  4. x ˙ = f ( x , u , t ) x ( 0 ) = x 0 t [ 0 , T ] \dot{x}=f(x,u,t)\qquad x(0)=x_{0}\quad t\in[0,T]
  5. a u ( t ) b t [ 0 , T ] a\leq u(t)\leq b\quad t\in[0,T]
  6. H ( x , λ , u , t ) = λ T ( t ) f ( x , u , t ) - L ( x , u , t ) H(x,\lambda,u,t)=\lambda^{T}(t)f(x,u,t)-L(x,u,t)\,
  7. λ ( t ) \lambda(t)
  8. x ( t ) x(t)
  9. H ( x , λ , u , t ) = λ T ( t + 1 ) f ( x , u , t ) - L ( x , u , t ) H(x,\lambda,u,t)=\lambda^{T}(t+1)f(x,u,t)-L(x,u,t)\,
  10. λ ( t + 1 ) = - H x d t + λ ( t ) \lambda(t+1)=-\frac{\partial H}{\partial x}dt+\lambda(t)
  11. t t
  12. t + 1. t+1.
  13. x x
  14. λ ( t + 1 ) \lambda(t+1)
  15. = ( p , q , t ) = p , q ˙ - L ( q , q ˙ , t ) \mathcal{H}=\mathcal{H}(p,q,t)=\langle p,\dot{q}\rangle-L(q,\dot{q},t)
  16. q ˙ \dot{q}
  17. p = L q ˙ p=\frac{\partial L}{\partial\dot{q}}
  18. d d t p ( t ) = - q \frac{d}{dt}p(t)=-\frac{\partial}{\partial q}\mathcal{H}
  19. d d t q ( t ) = p \frac{d}{dt}q(t)=~{}~{}\frac{\partial}{\partial p}\mathcal{H}
  20. H ( q , u , p , t ) = p , q ˙ - L ( q , u , t ) H(q,u,p,t)=\langle p,\dot{q}\rangle-L(q,u,t)
  21. d p d t = - H q \frac{dp}{dt}=-\frac{\partial H}{\partial q}
  22. d q d t = H p \frac{dq}{dt}=~{}~{}\frac{\partial H}{\partial p}
  23. H u = 0 \frac{\partial H}{\partial u}=0

Hamiltonian_matrix.html

  1. 2 n 2n
  2. 2 n 2n
  3. A A
  4. J A JA
  5. J J
  6. J = [ 0 I n - I n 0 ] J=\begin{bmatrix}0&I_{n}\\ -I_{n}&0\\ \end{bmatrix}
  7. n n
  8. n n
  9. A A
  10. 2 n 2n
  11. 2 n 2n
  12. A A
  13. A = [ a b c d ] A=\begin{bmatrix}a&b\\ c&d\end{bmatrix}
  14. a a
  15. b b
  16. c c
  17. d d
  18. n n
  19. n n
  20. A A
  21. b b
  22. c c
  23. A A
  24. 1 = A = J S 1=A=JS
  25. S S
  26. s p ( 2 n ) sp(2n)
  27. s p ( 2 n ) sp(2n)
  28. S p ( 2 n ) Sp(2n)
  29. A A
  30. λ λ
  31. λ −λ
  32. A A
  33. A A
  34. V V
  35. Ω Ω
  36. A : V V A:\;V\mapsto V
  37. Ω Ω
  38. x , y Ω ( A ( x ) , y ) x,y\mapsto\Omega(A(x),y)
  39. Ω ( A ( x ) , y ) = - Ω ( x , A ( y ) ) \Omega(A(x),y)=-\Omega(x,A(y))
  40. V V
  41. Ω Ω
  42. i e i e n + i \sum_{i}e_{i}\wedge e_{n+i}
  43. Ω Ω

Hammer_projection.html

  1. x = laea x ( λ 2 , ϕ ) x=\mathrm{laea}_{x}\left(\frac{\lambda}{2},\phi\right)
  2. y = 1 2 laea y ( λ 2 , ϕ ) y=\frac{1}{2}\mathrm{laea}_{y}\left(\frac{\lambda}{2},\phi\right)
  3. laea x \mathrm{laea}_{x}
  4. laea y \mathrm{laea}_{y}
  5. x = 2 2 cos ( ϕ ) sin ( λ 2 ) 1 + cos ( ϕ ) cos ( λ 2 ) x=\frac{2\sqrt{2}\cos(\phi)\sin\left(\frac{\lambda}{2}\right)}{\sqrt{1+\cos(% \phi)\cos\left(\frac{\lambda}{2}\right)}}
  6. y = 2 sin ( ϕ ) 1 + cos ( ϕ ) cos ( λ 2 ) y=\frac{\sqrt{2}\sin(\phi)}{\sqrt{1+\cos(\phi)\cos\left(\frac{\lambda}{2}% \right)}}
  7. z 1 - ( 1 4 x ) 2 - ( 1 2 y ) 2 z\equiv\sqrt{1-\left(\frac{1}{4}x\right)^{2}-\left(\frac{1}{2}y\right)^{2}}
  8. λ = 2 arctan [ z x 2 ( 2 z 2 - 1 ) ] ϕ = arcsin ( z y ) \begin{aligned}\displaystyle\lambda&\displaystyle=2\arctan\left[\frac{zx}{2(2z% ^{2}-1)}\right]\\ \displaystyle\phi&\displaystyle=\arcsin(zy)\end{aligned}
  9. λ \lambda
  10. ϕ \phi

Hammett_acidity_function.html

  1. H 0 = p K B H + + log [ B ] [ B H + ] H_{0}=\mbox{p}~{}K_{BH^{+}}+\log\frac{[B]}{[BH^{+}]}
  2. H 0 = - log ( a H + γ B γ B H + ) H_{0}=-\log\left(a_{H^{+}}\frac{\gamma_{B}}{\gamma_{BH^{+}}}\right)

Hankinson's_equation.html

  1. σ 0 \sigma_{0}
  2. σ 90 \sigma_{90}
  3. α \alpha
  4. σ α = σ 0 σ 90 σ 0 sin 2 α + σ 90 cos 2 α \sigma_{\alpha}=\cfrac{\sigma_{0}~{}\sigma_{90}}{\sigma_{0}~{}\sin^{2}\alpha+% \sigma_{90}~{}\cos^{2}\alpha}
  5. σ α = σ 0 σ 90 σ 0 sin n α + σ 90 cos n α \sigma_{\alpha}=\cfrac{\sigma_{0}~{}\sigma_{90}}{\sigma_{0}~{}\sin^{n}\alpha+% \sigma_{90}~{}\cos^{n}\alpha}
  6. n n
  7. α \alpha
  8. V ( α ) = V 0 V 90 V 0 sin 2 α + V 90 cos 2 α V(\alpha)=\frac{V_{0}V_{90}}{V_{0}\sin^{2}\alpha+V_{90}\cos^{2}\alpha}
  9. V 0 V_{0}
  10. V 90 V_{90}
  11. α \alpha

Hansen_solubility_parameter.html

  1. δ d \ \delta_{d}
  2. δ p \ \delta_{p}
  3. δ h \ \delta_{h}
  4. ( R a ) 2 = 4 ( δ d 2 - δ d 1 ) 2 + ( δ p 2 - δ p 1 ) 2 + ( δ h 2 - δ h 1 ) 2 \ (Ra)^{2}=4(\delta_{d2}-\delta_{d1})^{2}+(\delta_{p2}-\delta_{p1})^{2}+(% \delta_{h2}-\delta_{h1})^{2}
  5. R E D = R a / R 0 \ RED=Ra/R_{0}
  6. δ d \ \delta_{d}

Happiness_economics.html

  1. W i t = α + β x i t + ϵ i t W_{it}=\alpha+\beta{x_{it}}+\epsilon_{it}
  2. W W
  3. i i
  4. t t
  5. x x

Hardy's_inequality.html

  1. a 1 , a 2 , a 3 , a_{1},a_{2},a_{3},\dots
  2. n = 1 ( a 1 + a 2 + + a n n ) p < ( p p - 1 ) p n = 1 a n p . \sum_{n=1}^{\infty}\left(\frac{a_{1}+a_{2}+\cdots+a_{n}}{n}\right)^{p}<\left(% \frac{p}{p-1}\right)^{p}\sum_{n=1}^{\infty}a_{n}^{p}.
  3. 0 ( 1 x 0 x f ( t ) d t ) p d x ( p p - 1 ) p 0 f ( x ) p d x . \int_{0}^{\infty}\left(\frac{1}{x}\int_{0}^{x}f(t)\,dt\right)^{p}\,dx\leq\left% (\frac{p}{p-1}\right)^{p}\int_{0}^{\infty}f(x)^{p}\,dx.

Harish-Chandra_isomorphism.html

  1. x v := χ λ ( x ) v x\cdot v:=\chi_{\lambda}(x)v
  2. χ λ , χ μ \chi_{\lambda},\,\chi_{\mu}
  3. χ λ = χ μ \chi_{\lambda}=\chi_{\mu}

Harmonic_coordinate_condition.html

  1. 0 = ( x α ) ; β ; γ g β γ = ( ( x α ) , β , γ - ( x α ) , σ Γ β γ σ ) g β γ . 0=(x^{\alpha})_{;\beta;\gamma}g^{\beta\gamma}=((x^{\alpha})_{,\beta,\gamma}-(x% ^{\alpha})_{,\sigma}\Gamma^{\sigma}_{\beta\gamma})g^{\beta\gamma}\,.
  2. 0 = ( δ β , γ α - δ σ α Γ β γ σ ) g β γ = ( 0 - Γ β γ α ) g β γ = - Γ β γ α g β γ . 0=(\delta^{\alpha}_{\beta,\gamma}-\delta^{\alpha}_{\sigma}\Gamma^{\sigma}_{% \beta\gamma})g^{\beta\gamma}=(0-\Gamma^{\alpha}_{\beta\gamma})g^{\beta\gamma}=% -\Gamma^{\alpha}_{\beta\gamma}g^{\beta\gamma}\,.
  3. 0 = Γ β γ α g β γ . 0=\Gamma^{\alpha}_{\beta\gamma}g^{\beta\gamma}\,.
  4. 0 = ( g μ ν - g ) ; ρ = ( g μ ν - g ) , ρ + g σ ν Γ σ ρ μ - g + g μ σ Γ σ ρ ν - g - g μ ν Γ σ ρ σ - g . 0=(g^{\mu\nu}\sqrt{-g})_{;\rho}=(g^{\mu\nu}\sqrt{-g})_{,\rho}+g^{\sigma\nu}% \Gamma^{\mu}_{\sigma\rho}\sqrt{-g}+g^{\mu\sigma}\Gamma^{\nu}_{\sigma\rho}\sqrt% {-g}-g^{\mu\nu}\Gamma^{\sigma}_{\sigma\rho}\sqrt{-g}\,.
  5. - g μ ν Γ σ ρ σ - g -g^{\mu\nu}\Gamma^{\sigma}_{\sigma\rho}\sqrt{-g}\!
  6. - g \sqrt{-g}\!
  7. - g ; ρ = 0 \sqrt{-g}_{;\rho}=0\!
  8. g ; ρ μ ν = 0 g^{\mu\nu}_{;\rho}=0\!
  9. - g , ρ = - g Γ σ ρ σ . \sqrt{-g}_{,\rho}=\sqrt{-g}\Gamma^{\sigma}_{\sigma\rho}\,.
  10. 0 = ( g μ ν - g ) , ν + g σ ν Γ σ ν μ - g + g μ σ Γ σ ν ν - g - g μ ν Γ σ ν σ - g 0=(g^{\mu\nu}\sqrt{-g})_{,\nu}+g^{\sigma\nu}\Gamma^{\mu}_{\sigma\nu}\sqrt{-g}+% g^{\mu\sigma}\Gamma^{\nu}_{\sigma\nu}\sqrt{-g}-g^{\mu\nu}\Gamma^{\sigma}_{% \sigma\nu}\sqrt{-g}\,
  11. = ( g μ ν - g ) , ν + 0 + g μ α Γ α β β - g - g μ α Γ β α β - g . =(g^{\mu\nu}\sqrt{-g})_{,\nu}+0+g^{\mu\alpha}\Gamma^{\beta}_{\alpha\beta}\sqrt% {-g}-g^{\mu\alpha}\Gamma^{\beta}_{\beta\alpha}\sqrt{-g}\,.
  12. 0 = ( g μ ν - g ) , ν . 0=(g^{\mu\nu}\sqrt{-g})_{,\nu}\,.
  13. 0 = Γ β γ α g β γ = 1 2 g α δ ( g γ δ , β + g β δ , γ - g β γ , δ ) g β γ . 0=\Gamma^{\alpha}_{\beta\gamma}g^{\beta\gamma}=\tfrac{1}{2}g^{\alpha\delta}(g_% {\gamma\delta,\beta}+g_{\beta\delta,\gamma}-g_{\beta\gamma,\delta})g^{\beta% \gamma}\,.
  14. g α δ g^{\alpha\delta}\,
  15. g α β , γ g β γ = 1 2 g β γ , α g β γ . g_{\alpha\beta,\gamma}\,g^{\beta\gamma}=\tfrac{1}{2}g_{\beta\gamma,\alpha}\,g^% {\beta\gamma}\,.
  16. h α β , γ g β γ = 1 2 h β γ , α g β γ ; h_{\alpha\beta,\gamma}\,g^{\beta\gamma}=\tfrac{1}{2}h_{\beta\gamma,\alpha}\,g^% {\beta\gamma}\,;
  17. g α β , γ η β γ = 1 2 g β γ , α η β γ ; g_{\alpha\beta,\gamma}\,\eta^{\beta\gamma}=\tfrac{1}{2}g_{\beta\gamma,\alpha}% \,\eta^{\beta\gamma}\,;
  18. h α β , γ η β γ = 1 2 h β γ , α η β γ . h_{\alpha\beta,\gamma}\,\eta^{\beta\gamma}=\tfrac{1}{2}h_{\beta\gamma,\alpha}% \,\eta^{\beta\gamma}\,.
  19. 0 = A α ; β ; γ g β γ . 0=A_{\alpha;\beta;\gamma}g^{\beta\gamma}\,.
  20. A α ; β ; γ g β γ = A α ; β , γ g β γ - A σ ; β Γ α γ σ g β γ - A α ; σ Γ β γ σ g β γ . A_{\alpha;\beta;\gamma}g^{\beta\gamma}=A_{\alpha;\beta,\gamma}g^{\beta\gamma}-% A_{\sigma;\beta}\Gamma^{\sigma}_{\alpha\gamma}g^{\beta\gamma}-A_{\alpha;\sigma% }\Gamma^{\sigma}_{\beta\gamma}g^{\beta\gamma}\,.
  21. A α ; β ; γ g β γ = A α ; β , γ g β γ - A σ ; β Γ α γ σ g β γ A_{\alpha;\beta;\gamma}g^{\beta\gamma}=A_{\alpha;\beta,\gamma}g^{\beta\gamma}-% A_{\sigma;\beta}\Gamma^{\sigma}_{\alpha\gamma}g^{\beta\gamma}
  22. = A α , β , γ g β γ - A ρ , γ Γ α β ρ g β γ - A ρ Γ α β , γ ρ g β γ - A σ , β Γ α γ σ g β γ - A ρ Γ σ β ρ Γ α γ σ g β γ . =A_{\alpha,\beta,\gamma}g^{\beta\gamma}-A_{\rho,\gamma}\Gamma^{\rho}_{\alpha% \beta}g^{\beta\gamma}-A_{\rho}\Gamma^{\rho}_{\alpha\beta,\gamma}g^{\beta\gamma% }-A_{\sigma,\beta}\Gamma^{\sigma}_{\alpha\gamma}g^{\beta\gamma}-A_{\rho}\Gamma% ^{\rho}_{\sigma\beta}\Gamma^{\sigma}_{\alpha\gamma}g^{\beta\gamma}\,.

Harmonic_coordinates.html

  1. Δ x i = 0. \Delta x^{i}=0.\,
  2. E [ φ ] = M | d φ | 2 d V . E[\varphi]=\int_{M}|d\varphi|^{2}\,dV.
  3. g i j Γ i j k = 0 g^{ij}\Gamma_{ij}^{k}=0\,
  4. Δ x k = - g i j Γ i j k . \Delta x^{k}=-g^{ij}\Gamma_{ij}^{k}.
  5. Δ u j = 0 \Delta u^{j}=0\,
  6. u j / x i ( p ) \partial u^{j}/\partial x^{i}(p)

Harmonic_drive.html

  1. reduction ratio = flex spline teeth - circular spline teeth flex spline teeth \,\text{reduction ratio}=\frac{\,\text{flex spline teeth}-\,\text{circular % spline teeth}}{\,\text{flex spline teeth}}

Harmonic_superspace.html

  1. S U ( 2 ) R / U ( 1 ) R S 2 1 SU(2)_{R}/U(1)_{R}\approx S^{2}\simeq\mathbb{CP}^{1}
  2. S U ( 2 ) R S 3 SU(2)_{R}\approx S^{3}
  3. U ( 1 ) R S 1 U(1)_{R}\approx S^{1}
  4. u ± i u^{\pm i}
  5. ( u + i ) * = u i - \left(u^{+i}\right)^{*}=u^{-}_{i}
  6. u + i u i - = 1 u^{+i}u^{-}_{i}=1
  7. u ± i e ± i ϕ u ± i u^{\pm i}\to e^{\pm i\phi}u^{\pm i}
  8. Q i α Q^{i\alpha}
  9. θ i α \theta^{i\alpha}
  10. θ ± α = u i ± θ i α \theta^{\pm\alpha}=u^{\pm}_{i}\theta^{i\alpha}
  11. D α ± D^{\pm}_{\alpha}
  12. D α ± f ( u ) = 0 D^{\pm}_{\alpha}f(u)=0
  13. D + + u + i u - i D^{++}\equiv u^{+i}\frac{\partial}{\partial u^{-i}}
  14. D - - u - i u + i D^{--}\equiv u^{-i}\frac{\partial}{\partial u^{+i}}
  15. D α + q = 0 D^{+}_{\alpha}q=0
  16. q + q^{+}
  17. D + + q + = J + + + ( q + , u ) D^{++}q^{+}=J^{+++}(q^{+},\,u)
  18. S U ( 2 ) R SU(2)_{R}
  19. S U ( 2 ) R SU(2)_{R}
  20. S U ( 2 ) R SU(2)_{R}
  21. 𝒩 = ( 1 , 0 ) \mathcal{N}=(1,0)
  22. S U ( 2 ) R SU(2)_{R}

Harnack's_curve_theorem.html

  1. 1 - ( - 1 ) m 2 c ( m - 1 ) ( m - 2 ) 2 + 1. \frac{1-(-1)^{m}}{2}\leq c\leq\frac{(m-1)(m-2)}{2}+1.
  2. y 2 = x 3 - x , y^{2}=x^{3}-x,

Harpoon_reaction.html

  1. - e 2 R x + Δ E 0 = 0 \frac{-e^{2}}{R_{x}}+\Delta E_{0}=0
  2. Δ E 0 = I P - E A \Delta E_{0}=IP-EA

Hash_chain.html

  1. h ( x ) h(x)
  2. x x
  3. h ( h ( h ( h ( x ) ) ) ) h(h(h(h(x))))
  4. h 4 ( x ) h^{4}(x)
  5. h 1000 ( p a s s w o r d ) h^{1000}(password)
  6. h 999 ( p a s s w o r d ) h^{999}(password)
  7. h ( h 999 ( p a s s w o r d ) ) = h 1000 ( p a s s w o r d ) h(h^{999}(password))=h^{1000}(password)
  8. h 999 ( p a s s w o r d ) h^{999}(password)
  9. h 999 ( p a s s w o r d ) h^{999}(password)
  10. h 998 ( p a s s w o r d ) h^{998}(password)

Hata_model_for_open_areas.html

  1. L O = L U - 4.78 ( log 10 f ) 2 + 18.33 log 10 f - 40.94 L_{O}\;=\;L_{U}\;-\;4.78\;(\log_{10}f)^{2}\;+\;18.33\;\log_{10}f\;-\;40.94

Hata_model_for_suburban_areas.html

  1. L S U = L U - 2 ( log 10 f 28 ) 2 - 5.4 L_{SU}\;=\;L_{U}\;-\;2\big(\log_{10}{\frac{f}{28}}\big)^{2}\;-\;5.4

Hata_model_for_urban_areas.html

  1. L U = 69.55 + 26.16 log 10 f - 13.82 log 10 h B - C H + [ 44.9 - 6.55 log 10 h B ] log 10 d L_{U}\;=\;69.55\;+\;26.16\;\log_{10}f\;-\;13.82\;\log_{10}h_{B}\;-\;C_{H}\;+\;% [44.9\;-\;6.55\;\log_{10}h_{B}]\;\log_{10}d
  2. C H = 0.8 + ( 1.1 log 10 f - 0.7 ) h M - 1.56 log 10 f C_{H}\;=\;0.8\;+\;(\;1.1\;\log_{10}f\;-\;0.7\;)\;h_{M}\;-\;1.56\;\log_{10}f
  3. C H = { 8.29 ( log 10 ( 1.54 h M ) ) 2 - 1.1 , if 150 f 200 3.2 ( log 10 ( 11.75 h M ) ) 2 - 4.97 , if 200 < f 1500 C_{H}\;=\begin{cases}\;8.29\;(\;\log_{10}({1.54h_{M}}))^{2}\;-\;1.1\;\mbox{ , % if }~{}150\leq f\leq 200\\ \;3.2\;(\log_{10}({11.75h_{M}}))^{2}\;-\;4.97\;\mbox{ , if }~{}200<f\leq 1500% \end{cases}
  4. L U L_{U}\;
  5. h B h_{B}\;
  6. h M h_{M}\;
  7. f f\;
  8. C H C_{H}\;
  9. d d\;

Haven_(graph_theory).html

  1. S = X ( β ( X ) X ) S=\bigcap_{X}\left(\beta(X)\cup X\right)
  2. X i + 1 β ( X i + 1 ) X_{i+1}\cup\beta(X_{i+1})
  3. κ 1 \kappa\geq\aleph_{1}

Hayashi_track.html

  1. 10 5 10^{5}
  2. 10 5 10^{5}
  3. E = 4 π G 3 / 2 ( μ H / k ) 5 / 2 M 1 / 2 R 3 / 2 P / T 5 / 2 E=4\pi G^{3/2}(\mu H/k)^{5/2}M^{1/2}R^{3/2}P/T^{5/2}
  4. P = K ρ 5 / 3 P=K\rho^{5/3}
  5. j = σ T 4 j^{\star}=\sigma T^{4}
  6. L = 4 π R 2 σ T 4 L=4\pi R^{2}\sigma T^{4}
  7. d ln T d ln P > 0.4 \frac{d\ln T}{d\ln P}>0.4
  8. d ln T d ln P = 0.4 \frac{d\ln T}{d\ln P}=0.4
  9. d T = 0.4 T P d P dT=0.4\frac{T}{P}dP
  10. P = ρ R T μ P=\frac{\rho RT}{\mu}
  11. d ln T d ln P \frac{d\ln T}{d\ln P}
  12. d ln T d ln P = 0.4 \frac{d\ln{T}}{d\ln{P}}=0.4
  13. \nabla
  14. P 1 - γ T γ = C P^{1-\gamma}T^{\gamma}=C
  15. γ \gamma
  16. P = N k T / V P=NkT/V
  17. = ρ k T μ H =\frac{\rho kT}{\mu H}
  18. = ( k ρ C μ H ) γ =(\frac{k\rho C}{\mu H})^{\gamma}
  19. μ \mu
  20. P = K ρ 1 + 1 / n P=K\rho^{1+1/n}
  21. P c = ( k ρ c C μ H ) γ P_{c}=(\frac{k\rho_{c}C}{\mu H})^{\gamma}
  22. C = μ H P c 1 / γ ρ c k C=\frac{\mu HP_{c}^{1/\gamma}}{\rho_{c}k}
  23. P c = W n G M 2 R 4 P_{c}=W_{n}\frac{GM^{2}}{R^{4}}
  24. ρ c = K n ρ a v g \rho_{c}=K_{n}\rho_{avg}
  25. R 3 - n n M n - 1 n = K G N n R^{\frac{3-n}{n}}M^{\frac{n-1}{n}}=\frac{K}{GN_{n}}
  26. W n , K n , N n , W_{n},K_{n},N_{n},
  27. ρ a v g M 4 / 3 π R 3 \rho_{avg}\equiv\frac{M}{4/3\pi R^{3}}
  28. C M 2 - γ R 3 γ - 4 C\sim M^{2-\gamma}R^{3\gamma-4}
  29. P 1 - γ T γ = C P^{1-\gamma}T^{\gamma}=C
  30. P 1 - γ T γ M 2 - γ R 3 γ - 4 P^{1-\gamma}T^{\gamma}\sim M^{2-\gamma}R^{3\gamma-4}
  31. ln P = 2 - γ 1 - γ ln M + 3 γ - 4 1 - γ ln R - γ ln T \ln P=\frac{2-\gamma}{1-\gamma}\ln M+\frac{3\gamma-4}{1-\gamma}\ln R-\gamma\ln T
  32. d τ d r = k ρ \frac{d\tau}{dr}=k\rho
  33. τ = R k ρ d r \tau=\int_{R}^{\infty}k\rho dr
  34. = k R ρ d r =k\int_{R}^{\infty}\rho dr
  35. P 0 = R g ρ d r P_{0}=\int_{R}^{\infty}g\rho dr
  36. = G M R 2 R ρ d r =\frac{GM}{R^{2}}\int_{R}^{\infty}\rho dr
  37. = G M τ k R 2 =\frac{GM\tau}{kR^{2}}
  38. τ = 2 / 3 \tau=2/3
  39. T = T e f f T=T_{eff}
  40. L = 4 π R 2 T e f f 4 L=4\pi R^{2}T_{eff}^{4}
  41. P 0 = G M R 2 2 τ 3 k P_{0}=\frac{GM}{R^{2}}\frac{2\tau}{3k}
  42. k = k 0 P a T b k=k_{0}P^{a}T^{b}
  43. P 0 = c o n s t ( M R 2 T e f f b ) 1 a + 1 P_{0}=const(\frac{M}{R^{2}T_{eff}^{b}})^{\frac{1}{a+1}}
  44. ln P 0 = ln c o n s t + 1 a + 1 ( ln M - 2 ln R - b ln T e f f ) \ln P_{0}=\ln{const}+\frac{1}{a+1}(\ln{M}-2\ln{R}-b\ln{T_{eff}})
  45. L = 4 π R 2 σ T e f f 4 L=4\pi R^{2}\sigma T_{eff}^{4}
  46. ln R = 0.5 ln L - 2 ln T e f f + c o n s t \ln{R}=0.5\ln{L}-2\ln{T_{eff}}+const
  47. T = T e f f T=T_{eff}
  48. P = P 0 P=P_{0}
  49. P 0 P_{0}
  50. γ = 5 / 3 \gamma=5/3
  51. ln T e f f = A ln L + B ln M + c o n s t \ln{T_{eff}}=A\ln{L}+B\ln{M}+const
  52. A = 0.75 a - 0.25 5.5 a + b + 1.5 A=\frac{0.75a-0.25}{5.5a+b+1.5}
  53. B = 0.5 a + 1.5 5.5 a + b + 1.5 B=\frac{0.5a+1.5}{5.5a+b+1.5}
  54. b 3 b\approx 3
  55. A = 0.05 A=0.05
  56. B = 0.2 B=0.2
  57. T e f f = ( 2600 K ) μ 13 / 51 ( M M ) 7 / 51 ( L L ) 1 / 102 T_{eff}=(2600K)\mu^{13/51}(\frac{M}{M_{\odot}})^{7/51}(\frac{L}{L_{\odot}})^{1% /102}
  58. μ \mu
  59. ln T e f f ln M 0.1 \frac{\partial\ln{T_{eff}}}{\partial\ln{M}}\approx 0.1
  60. ln T e f f ln μ - 26 \frac{\partial\ln{T_{eff}}}{\partial\ln{\mu}}\approx-26
  61. ln T e f f \ln{T_{eff}}
  62. 10 6.5 10^{6.5}
  63. 10 6.7 10^{6.7}

Headway.html

  1. T m i n = t r + k V 2 ( 1 a f - 1 a l ) T_{min}=t_{r}+\frac{kV}{2}\left(\frac{1}{a_{f}}-\frac{1}{a_{l}}\right)
  2. T m i n T_{min}
  3. V V
  4. t r t_{r}
  5. a f a_{f}
  6. a l a_{l}
  7. a l a_{l}
  8. k k
  9. T t o t = L V + t r + k V 2 ( 1 a f - 1 a l ) T_{tot}=\frac{L}{V}+t_{r}+\frac{kV}{2}\left(\frac{1}{a_{f}}-\frac{1}{a_{l}}\right)
  10. T t o t T_{tot}
  11. L L
  12. n v e h = 3600 T m i n n_{veh}=\frac{3600}{T_{min}}
  13. n v e h n_{veh}
  14. T m i n T_{min}
  15. n p a s = P 3600 T m i n n_{pas}=P\frac{3600}{T_{min}}
  16. n p a s n_{pas}
  17. P P
  18. T m i n T_{min}
  19. k k
  20. T t o t = 4 28 + 2 + 1.5 × 28 2 ( 1 2.5 ) T_{tot}=\frac{4}{28}+2+\frac{1.5\times 28}{2}\left(\frac{1}{2.5}\right)
  21. n p a s = P × 3600 T t o t n_{pas}={P}\times\frac{3600}{T_{tot}}
  22. T t o t T_{tot}
  23. n p a s n_{pas}
  24. k k
  25. T t o t = 100 11 + 2 + 1.5 × 11 2 ( 1 0.5 ) T_{tot}=\frac{100}{11}+2+\frac{1.5\times 11}{2}\left(\frac{1}{0.5}\right)
  26. n p a s = 1000 × 3600 T t o t n_{pas}={1000}\times\frac{3600}{T_{tot}}
  27. T t o t T_{tot}
  28. n p a s n_{pas}
  29. n p a s = 1000 × 3600 120 n_{pas}={1000}\times\frac{3600}{120}
  30. n p a s n_{pas}
  31. k k
  32. T t o t = 3 8 + 0.01 + 1.1 × 8 2 ( 1 2.5 - 1 2.5 ) T_{tot}=\frac{3}{8}+0.01+\frac{1.1\times 8}{2}\left(\frac{1}{2.5}-\frac{1}{2.5% }\right)
  33. n p a s = 3 × 3600 0.385 n_{pas}={3}\times\frac{3600}{0.385}
  34. T t o t T_{tot}
  35. n p a s n_{pas}
  36. n p a s = 3 × 3600 2 n_{pas}={3}\times\frac{3600}{2}
  37. n p a s n_{pas}

Heap_(mathematics).html

  1. [ x , y , z ] H [x,y,z]\in H
  2. [ [ a , b , c ] , d , e ] = [ a , [ d , c , b ] , e ] = [ a , b , [ c , d , e ] ] a , b , c , d , e H [[a,b,c],d,e]=[a,[d,c,b],e]=[a,b,[c,d,e]]\ \forall\ a,b,c,d,e\in H
  3. [ a , a , x ] = [ x , a , a ] = x a , x H . [a,a,x]=[x,a,a]=x\ \forall\ a,x\in H.
  4. [ x , y , z ] = x y - 1 z [x,y,z]=xy^{-1}z
  5. x * y = [ x , e , y ] x*y=[x,e,y]
  6. x - 1 = [ e , x , e ] x^{-1}=[e,x,e]
  7. [ f , g , h ] = f g - 1 h [f,g,h]=fg^{-1}h
  8. H = { a , b } H=\{a,b\}
  9. [ a , a , a ] = a , [ a , a , b ] = b , [ b , a , a ] = b , [ b , a , b ] = a , [a,a,a]=a,\,[a,a,b]=b,\,[b,a,a]=b,\,[b,a,b]=a,
  10. [ a , b , a ] = b , [ a , b , b ] = a , [ b , b , a ] = a , [ b , b , b ] = b . [a,b,a]=b,\,[a,b,b]=a,\,[b,b,a]=a,\,[b,b,b]=b.
  11. [ x , y , z ] = x y - 1 z . [x,y,z]=xy^{-1}z.
  12. x , y , z x,y,z
  13. [ x , y , z ] = x - y + z [x,y,z]=x-y+z
  14. k k
  15. * *
  16. x * y = x + y - k x*y=x+y-k
  17. x - 1 = 2 k - x x^{-1}=2k-x
  18. [ [ a , b , c ] , d , e ] = [ a , b , [ c , d , e ] ] . [[a,b,c],d,e]=[a,b,[c,d,e]].
  19. [ x , y , z ] = x y T z [x,y,z]=x\cdot y^{\mathrm{T}}\cdot z
  20. [ a , a , a ] = a [a,a,a]=a
  21. [ a , a , [ b , b , x ] ] = [ b , b , [ a , a , x ] ] [a,a,[b,b,x]]=[b,b,[a,a,x]]
  22. [ [ x , a , a ] , b , b ] = [ [ x , b , b ] , a , a ] [[x,a,a],b,b]=[[x,b,b],a,a]
  23. a b [ a , b , a ] = a a\rightarrow b\Leftrightarrow[a,b,a]=a
  24. G G
  25. X X
  26. [ x , y , z ] = ( x / y ) z [x,y,z]=(x/y)\cdot z
  27. X X
  28. x , y X x,y\in X
  29. φ x , y ( z ) = [ x , y , z ] \varphi_{x,y}(z)=[x,y,z]
  30. X X
  31. G G
  32. φ x , y \varphi_{x,y}
  33. G G
  34. X X
  35. G G

Heat_of_formation_group_additivity.html

  1. Δ H f = - 146.0 * n C - C - 124.2 * n C - H - 66.2 * n C = C + 10.2 * n C - C - C + 9.3 * n C - C - H + 6.6 * n H - C - H + f ( C , H ) \ \Delta H_{f}=-146.0*n_{C-C}-124.2*n_{C-H}-66.2*n_{C=C}+10.2*n_{C-C-C}+9.3*n_% {C-C-H}+6.6*n_{H-C-H}+f(C,H)
  2. f ( C , H ) = ( 231.3 * n C + 52.1 * n H ) \ f(C,H)=(231.3*n_{C}+52.1*n_{H})

Heath-Brown–Moroz_constant.html

  1. C = p ( 1 - 1 p ) 7 ( 1 + 7 p + 1 p 2 ) = 0.001317641... C=\prod_{p}\left(1-\frac{1}{p}\right)^{7}\left(1+\frac{7p+1}{p^{2}}\right)=0.0% 01317641...
  2. N ( H ) = C H ( log H ) 6 4 × 6 ! + O ( H ( log H ) 5 ) . N(H)=C\cdot\frac{H(\log H)^{6}}{4\times 6!}+O(H(\log H)^{5}).

Heavy-tailed_distribution.html

  1. lim x e λ x Pr [ X > x ] = for all λ > 0. \lim_{x\to\infty}e^{\lambda x}\Pr[X>x]=\infty\quad\mbox{for all }~{}\lambda>0.\,
  2. F ¯ ( x ) Pr [ X > x ] \overline{F}(x)\equiv\Pr[X>x]\,
  3. lim x e λ x F ¯ ( x ) = for all λ > 0. \lim_{x\to\infty}e^{\lambda x}\overline{F}(x)=\infty\quad\mbox{for all }~{}% \lambda>0.\,
  4. lim x Pr [ X > x + t | X > x ] = 1 , \lim_{x\to\infty}\Pr[X>x+t|X>x]=1,\,
  5. F ¯ ( x + t ) F ¯ ( x ) as x . \overline{F}(x+t)\sim\overline{F}(x)\quad\mbox{as }~{}x\to\infty.\,
  6. X 1 , X 2 X_{1},X_{2}
  7. F F
  8. F F
  9. F * 2 F^{*2}
  10. Pr [ X 1 + X 2 x ] = F * 2 ( x ) = - F ( x - y ) d F ( y ) . \Pr[X_{1}+X_{2}\leq x]=F^{*2}(x)=\int_{-\infty}^{\infty}F(x-y)\,dF(y).
  11. F * n F^{*n}
  12. F ¯ \overline{F}
  13. F ¯ ( x ) = 1 - F ( x ) \overline{F}(x)=1-F(x)
  14. F F
  15. F * 2 ¯ ( x ) 2 F ¯ ( x ) as x . \overline{F^{*2}}(x)\sim 2\overline{F}(x)\quad\mbox{as }~{}x\to\infty.
  16. n 1 n\geq 1
  17. F * n ¯ ( x ) n F ¯ ( x ) as x . \overline{F^{*n}}(x)\sim n\overline{F}(x)\quad\mbox{as }~{}x\to\infty.
  18. n n
  19. X 1 , , X n X_{1},\ldots,X_{n}
  20. F F
  21. Pr [ X 1 + + X n > x ] Pr [ max ( X 1 , , X n ) > x ] as x . \Pr[X_{1}+\cdots+X_{n}>x]\sim\Pr[\max(X_{1},\ldots,X_{n})>x]\quad\,\text{as }x% \to\infty.
  22. F F
  23. F I ( [ 0 , ) ) FI([0,\infty))
  24. I ( [ 0 , ) ) I([0,\infty))
  25. X X
  26. X + = max ( 0 , X ) X^{+}=\max(0,X)
  27. x - a x^{-a}
  28. ( X n , n 1 ) (X_{n},n\geq 1)
  29. F D ( H ( ξ ) ) F\in D(H(\xi))
  30. H H
  31. ξ \xi\in\mathbb{R}
  32. lim n k ( n ) = \lim_{n\to\infty}k(n)=\infty
  33. lim n k ( n ) n = 0 \lim_{n\to\infty}\frac{k(n)}{n}=0
  34. ξ ( k ( n ) , n ) P i c k a n d s = 1 ln 2 ln ( X ( n - k ( n ) + 1 , n ) - X ( n - 2 k ( n ) + 1 , n ) X ( n - 2 k ( n ) + 1 , n ) - X ( n - 4 k ( n ) + 1 , n ) ) \xi^{Pickands}_{(k(n),n)}=\frac{1}{\ln 2}\ln\left(\frac{X_{(n-k(n)+1,n)}-X_{(n% -2k(n)+1,n)}}{X_{(n-2k(n)+1,n)}-X_{(n-4k(n)+1,n)}}\right)
  35. X ( n - k ( n ) + 1 , n ) = max ( X n - k ( n ) + 1 , , X n ) X_{(n-k(n)+1,n)}=\max\left(X_{n-k(n)+1},\ldots,X_{n}\right)
  36. ξ \xi
  37. ( X n , n 1 ) (X_{n},n\geq 1)
  38. F D ( H ( ξ ) ) F\in D(H(\xi))
  39. H H
  40. ξ \xi\in\mathbb{R}
  41. lim n k ( n ) = \lim_{n\to\infty}k(n)=\infty
  42. lim n k ( n ) n = 0 \lim_{n\to\infty}\frac{k(n)}{n}=0
  43. ξ ( k ( n ) , n ) H i l l = 1 k ( n ) i = n - k ( n ) + 1 n ln ( X ( i , n ) ) - ln ( X ( n - k ( n ) + 1 , n ) ) , \xi^{Hill}_{(k(n),n)}=\frac{1}{k(n)}\sum_{i=n-k(n)+1}^{n}\ln(X_{(i,n)})-\ln(X_% {(n-k(n)+1,n)}),
  44. X ( n - k ( n ) + 1 , n ) = min ( X n - k ( n ) + 1 , , X n ) X_{(n-k(n)+1,n)}=\min\left(X_{n-k(n)+1},\ldots,X_{n}\right)
  45. ξ \xi

Hedetniemi's_conjecture.html

  1. κ \kappa
  2. κ \kappa

Hedgehog_space.html

  1. K K
  2. K K
  3. K K
  4. K K
  5. K K
  6. d ( x , y ) = | x - y | d(x,y)=|x-y|
  7. x x
  8. y y
  9. d ( x , y ) = x + y d(x,y)=x+y
  10. x x
  11. y y
  12. K K
  13. K K

Heel_lifts.html

  1. L < [ S B U ] / [ D + C ] L<[SBU]/[D+C]
  2. 0 - 10 y e a r s = 1 p t 0-10years=1pt
  3. 10 - 30 y e a r s = 2 p t s 10-30years=2pts
  4. 30 + y e a r s = 3 p t s 30+years=3pts

Heinz-Jürgen_Kluge.html

  1. ω c = q e B / 2 π m \omega_{c}=qeB/2\pi m
  2. q q
  3. m m
  4. B B

Heinz_mean.html

  1. H x ( A , B ) = A x B 1 - x + A 1 - x B x 2 . H_{x}(A,B)=\frac{A^{x}B^{1-x}+A^{1-x}B^{x}}{2}.

Helly's_selection_theorem.html

  1. sup n ( f n L 1 ( W ) + d f n d t L 1 ( W ) ) < + , \sup_{n\in\mathbb{N}}\left(\left\|f_{n}\right\|_{L^{1}(W)}+\left\|\frac{% \mathrm{d}f_{n}}{\mathrm{d}t}\right\|_{L^{1}(W)}\right)<+\infty,
  2. lim k W | f n k ( x ) - f ( x ) | d x = 0 ; \lim_{k\to\infty}\int_{W}\big|f_{n_{k}}(x)-f(x)\big|\,\mathrm{d}x=0;
  3. d f d t L 1 ( W ) lim inf k d f n k d t L 1 ( W ) . \left\|\frac{\mathrm{d}f}{\mathrm{d}t}\right\|_{L^{1}(W)}\leq\liminf_{k\to% \infty}\left\|\frac{\mathrm{d}f_{n_{k}}}{\mathrm{d}t}\right\|_{L^{1}(W)}.
  4. [ 0 , t ) Δ ( d z n k ) δ ( t ) ; \int_{[0,t)}\Delta(\mathrm{d}z_{n_{k}})\to\delta(t);
  5. z n k ( t ) z ( t ) E ; z_{n_{k}}(t)\rightharpoonup z(t)\in E;
  6. [ s , t ) Δ ( d z ) δ ( t ) - δ ( s ) . \int_{[s,t)}\Delta(\mathrm{d}z)\leq\delta(t)-\delta(s).

Help:Displaying_a_formula.html

  1. e i π + 1 = 0 e^{i\pi}+1=0\,\!
  2. e i π + 1 = 0 e^{i\pi}+1=0\,\!
  3. \definecolor o l i v e R G B 128 , 128 , 0 \pagecolor o l i v e e i π + 1 = 0 \definecolor{olive}{RGB}{128,128,0}\pagecolor{olive}e^{i\pi}+1=0\,\!
  4. π \sqrt{\pi}
  5. π \pi
  6. abc \,\text{abc}
  7. α \alpha
  8. α α
  9. f ( x ) = x 2 f(x)=x^{2}
  10. 2 \sqrt{2}
  11. [ u r a d i c a l , u 2 ] [u^{\prime}radical^{\prime},u^{\prime}2^{\prime}]
  12. 1 - e 2 \sqrt{1-e^{2}}\!
  13. [ u r a d i c a l , u 1 212 , u e , u l e s s t h a n s u p > 2 ] [u^{\prime}radical^{\prime},u^{\prime}1\u{2}212^{\prime},u^{\prime}e^{\prime},% u^{\prime}\\ lessthansup>2^{\prime}]
  14. x x
  15. x x
  16. a ˙ , a ¨ , a ´ , a ` \dot{a},\ddot{a},\acute{a},\grave{a}\!
  17. a ˇ , a ˘ , a ~ , a ¯ \check{a},\breve{a},\tilde{a},\bar{a}\!
  18. a ^ , a ^ , a \hat{a},\widehat{a},\vec{a}\!
  19. exp a b = a b , exp b = e b , 10 m \exp_{a}b=a^{b},\exp b=e^{b},10^{m}\!
  20. ln c , lg d = log e , log 10 f \ln c,\lg d=\log e,\log_{10}f\!
  21. sin a , cos b , tan c , cot d , sec e , csc f \sin a,\cos b,\tan c,\cot d,\sec e,\csc f\!
  22. arcsin h , arccos i , arctan j \arcsin h,\arccos i,\arctan j\!
  23. sinh k , cosh l , tanh m , coth n \sinh k,\cosh l,\tanh m,\coth n\!
  24. sh k , ch l , th m , coth n \operatorname{sh}\,k,\operatorname{ch}\,l,\operatorname{th}\,m,\operatorname{% coth}\,n\!
  25. argsh o , argch p , argth q \operatorname{argsh}\,o,\operatorname{argch}\,p,\operatorname{argth}\,q\!
  26. sgn r , | s | \operatorname{sgn}r,\left|s\right|\!
  27. min ( x , y ) , max ( x , y ) \min(x,y),\max(x,y)\!
  28. min x , max y , inf s , sup t \min x,\max y,\inf s,\sup t\!
  29. lim u , lim inf v , lim sup w \lim u,\liminf v,\limsup w\!
  30. dim p , deg q , det m , ker ϕ \dim p,\deg q,\det m,\ker\phi\!
  31. Pr j , hom l , z , arg z \Pr j,\hom l,\lVert z\rVert,\arg z\!
  32. d t , d t , t , ψ dt,\operatorname{d}\!t,\partial t,\nabla\psi\!
  33. d y / d x , d y / d x , d y d x , d y d x , 2 x 1 x 2 y dy/dx,\operatorname{d}\!y/\operatorname{d}\!x,{dy\over dx},{\operatorname{d}\!% y\over\operatorname{d}\!x},{\partial^{2}\over\partial x_{1}\partial x_{2}}y\!
  34. , , f , f , f ′′ , f ( 3 ) , y ˙ , y ¨ \prime,\backprime,f^{\prime},f^{\prime},f^{\prime\prime},f^{(3)}\!,\dot{y},% \ddot{y}
  35. , , , ϶ , ð , , \infty,\aleph,\complement,\backepsilon,\eth,\Finv,\hbar\!
  36. , ı , ȷ , 𝕜 , , , , , \Im,\imath,\jmath,\Bbbk,\ell,\mho,\wp,\Re,\circledS\!
  37. s k 0 ( mod m ) s_{k}\equiv 0\;\;(\mathop{{\rm mod}}m)\!
  38. a mod b a\,\bmod\,b\!
  39. gcd ( m , n ) , lcm ( m , n ) \gcd(m,n),\operatorname{lcm}(m,n)
  40. , , , \mid,\nmid,\shortmid,\nshortmid\!
  41. , 2 , n , x 3 + y 3 2 3 \surd,\sqrt{2},\sqrt[n]{},\sqrt[3]{x^{3}+y^{3}\over 2}\!
  42. + , - , ± , , +,-,\pm,\mp,\dotplus\!
  43. × , ÷ , , / , \ \times,\div,\divideontimes,/,\backslash\!
  44. , * , , , \cdot,*\ast,\star,\circ,\bullet\!
  45. , , , \boxplus,\boxminus,\boxtimes,\boxdot\!
  46. , , , , \oplus,\ominus,\otimes,\oslash,\odot\!
  47. , , \circleddash,\circledcirc,\circledast\!
  48. , , \bigoplus,\bigotimes,\bigodot\!
  49. { } , Ø , \{\},\O\emptyset,\varnothing\!
  50. , , , ∌ \in,\notin\not\in,\ni,\not\ni\!
  51. , , , \cap,\Cap,\sqcap,\bigcap\!
  52. , , , , , , \cup,\Cup,\sqcup,\bigcup,\bigsqcup,\uplus,\biguplus\!
  53. , , × \setminus,\smallsetminus,\times\!
  54. , , \subset,\Subset,\sqsubset\!
  55. , , \supset,\Supset,\sqsupset\!
  56. , , , , \subseteq,\nsubseteq,\subsetneq,\varsubsetneq,\sqsubseteq\!
  57. , , , , \supseteq,\nsupseteq,\supsetneq,\varsupsetneq,\sqsupseteq\!
  58. , ⫅̸ , , \subseteqq,\nsubseteqq,\subsetneqq,\varsubsetneqq\!
  59. , ⫆̸ , , \supseteqq,\nsupseteqq,\supsetneqq,\varsupsetneqq\!
  60. = , , , , =,\neq,\neq,\equiv,\not\equiv\!
  61. , , = def , := \doteq,\doteqdot,\overset{\underset{\mathrm{def}}{}}{=},:=\!
  62. , , , , , , , , \sim,\nsim,\backsim,\thicksim,\simeq,\backsimeq,\eqsim,\cong,\ncong\!
  63. , , , , , \approx,\thickapprox,\approxeq,\asymp,\propto,\varpropto\!
  64. < , , , ≪̸ , , ⋘̸ , <,\nless,\ll,\not\ll,\lll,\not\lll,\lessdot\!
  65. > , , , ≫̸ , , ⋙̸ , >,\ngtr,\gg,\not\gg,\ggg,\not\ggg,\gtrdot\!
  66. , , , , , ≦̸ , , \leq,\leq,\lneq,\leqq,\nleq,\nleqq,\lneqq,\lvertneqq\!
  67. , , , , , ≧̸ , , \geq,\geq,\gneq,\geqq,\ngeq,\ngeqq,\gneqq,\gvertneqq\!
  68. , , , , , \lessgtr,\lesseqgtr,\lesseqqgtr,\gtrless,\gtreqless,\gtreqqless\!
  69. , ⩽̸ , \leqslant,\nleqslant,\eqslantless\!
  70. , ⩾̸ , \geqslant,\ngeqslant,\eqslantgtr\!
  71. , , , \lesssim,\lnsim,\lessapprox,\lnapprox\!
  72. , , , \gtrsim,\gnsim,\gtrapprox,\gnapprox\,
  73. , , , , \prec,\nprec,\preceq,\npreceq,\precneqq\!
  74. , , , , \succ,\nsucc,\succeq,\nsucceq,\succneqq\!
  75. , \preccurlyeq,\curlyeqprec\,
  76. , \succcurlyeq,\curlyeqsucc\,
  77. , , , \precsim,\precnsim,\precapprox,\precnapprox\,
  78. , , , \succsim,\succnsim,\succapprox,\succnapprox\,
  79. , , , \parallel,\nparallel,\shortparallel,\nshortparallel\!
  80. , , , , 45 \perp,\angle,\sphericalangle,\measuredangle,45^{\circ}\!
  81. , , , , , \Box,\blacksquare,\diamond,\Diamond\lozenge,\blacklozenge,\bigstar\!
  82. , , \bigcirc,\triangle\bigtriangleup,\bigtriangledown\!
  83. , \vartriangle,\triangledown\!
  84. , , , \blacktriangle,\blacktriangledown,\blacktriangleleft,\blacktriangleright\!
  85. , , \forall,\exists,\nexists\!
  86. , , & \therefore,\because,\And\!
  87. , , \lor\vee,\curlyvee,\bigvee\!
  88. and , , \and\land\wedge,\curlywedge,\bigwedge\!
  89. q ¯ , a b c ¯ , q ¯ , a b c ¯ , \bar{q},\bar{abc},\overline{q},\overline{abc},\!
  90. ¬ ¬ , R , , \lnot\neg,\not\operatorname{R},\bot,\top\!
  91. , , , \vdash\dashv,\vDash,\Vdash,\models\!
  92. \Vvdash\nvdash\nVdash\nvDash\nVDash\!
  93. \ulcorner\urcorner\llcorner\lrcorner\,
  94. , \Rrightarrow,\Lleftarrow\!
  95. , , \Rightarrow,\nRightarrow,\Longrightarrow\implies\!
  96. , , \Leftarrow,\nLeftarrow,\Longleftarrow\!
  97. , , \Leftrightarrow,\nLeftrightarrow,\Longleftrightarrow\iff\!
  98. , , \Uparrow,\Downarrow,\Updownarrow\!
  99. , , \rightarrow\to,\nrightarrow,\longrightarrow\!
  100. , , \leftarrow\leftarrow,\nleftarrow,\longleftarrow\!
  101. , , \leftrightarrow,\nleftrightarrow,\longleftrightarrow\!
  102. , , \uparrow,\downarrow,\updownarrow\!
  103. , , , \nearrow,\swarrow,\nwarrow,\searrow\!
  104. , \mapsto,\longmapsto\!
  105. \rightharpoonup\rightharpoondown\leftharpoonup\leftharpoondown\upharpoonleft% \upharpoonright\downharpoonleft\downharpoonright\rightleftharpoons% \leftrightharpoons\,\!
  106. \curvearrowleft\circlearrowleft\Lsh\upuparrows\rightrightarrows% \rightleftarrows\rightarrowtail\looparrowright\,\!
  107. \curvearrowright\circlearrowright\Rsh\downdownarrows\leftleftarrows% \leftrightarrows\leftarrowtail\looparrowleft\,\!
  108. \hookrightarrow\hookleftarrow\multimap\leftrightsquigarrow\rightsquigarrow% \twoheadrightarrow\twoheadleftarrow\!
  109. § % \amalg\P\S\%\dagger\ddagger\ldots\cdots\!
  110. \smile\frown\wr\triangleleft\triangleright\!
  111. , , , , , , , \diamondsuit,\heartsuit,\clubsuit,\spadesuit,\Game,\flat,\natural,\sharp\!
  112. \diagup\diagdown\centerdot\ltimes\rtimes\leftthreetimes\rightthreetimes\!
  113. \eqcirc\circeq\triangleq\bumpeq\Bumpeq\doteqdot\risingdotseq\fallingdotseq\!
  114. \intercal\barwedge\veebar\doublebarwedge\between\pitchfork\!
  115. \vartriangleleft\ntriangleleft\vartriangleright\ntriangleright\!
  116. \trianglelefteq\ntrianglelefteq\trianglerighteq\ntrianglerighteq\!
  117. a 2 a^{2}
  118. a 2 a_{2}
  119. 10 30 a 2 + 2 10^{30}a^{2+2}
  120. a i , j b f a_{i,j}b_{f^{\prime}}
  121. x 2 3 x_{2}^{3}
  122. x 2 3 {x_{2}}^{3}\,\!
  123. 10 10 8 10^{10^{8}}
  124. 3 4 a b 1 2 \sideset{{}_{1}^{2}}{{}_{3}^{4}}{\prod}_{a}^{b}
  125. Ω 3 4 1 2 {}_{1}^{2}\!\Omega_{3}^{4}
  126. ω 𝛼 \overset{\alpha}{\omega}
  127. ω 𝛼 \underset{\alpha}{\omega}
  128. ω 𝛾 𝛼 \overset{\alpha}{\underset{\gamma}{\omega}}
  129. ω α \stackrel{\alpha}{\omega}
  130. x , y ′′ , f , f ′′ x^{\prime},y^{\prime\prime},f^{\prime},f^{\prime\prime}
  131. x , y ′′ x^{\prime},y^{\prime\prime}
  132. x ˙ , x ¨ \dot{x},\ddot{x}
  133. a ^ b ¯ c \hat{a}\ \bar{b}\ \vec{c}
  134. a b c d d e f ^ \overrightarrow{ab}\ \overleftarrow{cd}\ \widehat{def}
  135. g h i ¯ j k l ¯ \overline{ghi}\ \underline{jkl}
  136. A B \overset{\frown}{AB}
  137. A n + μ - 1 B n ± i - 1 𝑇 C A\xleftarrow{n+\mu-1}B\xrightarrow[T]{n\pm i-1}C
  138. 1 + 2 + + 100 5050 \overbrace{1+2+\cdots+100}^{5050}
  139. a + b + + z 26 \underbrace{a+b+\cdots+z}_{26}
  140. k = 1 N k 2 \sum_{k=1}^{N}k^{2}
  141. k = 1 N k 2 \textstyle\sum_{k=1}^{N}k^{2}
  142. k = 1 N k 2 a \frac{\sum_{k=1}^{N}k^{2}}{a}
  143. k = 1 N k 2 a \frac{\displaystyle\sum_{k=1}^{N}k^{2}}{a}
  144. k = 1 N k 2 a \frac{\sum\limits^{{}^{N}}_{k=1}k^{2}}{a}
  145. i = 1 N x i \prod_{i=1}^{N}x_{i}
  146. i = 1 N x i \textstyle\prod_{i=1}^{N}x_{i}
  147. i = 1 N x i \coprod_{i=1}^{N}x_{i}
  148. i = 1 N x i \textstyle\coprod_{i=1}^{N}x_{i}
  149. lim n x n \lim_{n\to\infty}x_{n}
  150. lim n x n \textstyle\lim_{n\to\infty}x_{n}
  151. 1 3 e 3 / x x 2 d x \int\limits_{1}^{3}\frac{e^{3}/x}{x^{2}}\,dx
  152. 1 3 e 3 / x x 2 d x \int_{1}^{3}\frac{e^{3}/x}{x^{2}}\,dx
  153. - N N e x d x \textstyle\int\limits_{-N}^{N}e^{x}\,dx
  154. - N N e x d x \textstyle\int_{-N}^{N}e^{x}\,dx
  155. D d x d y \iint\limits_{D}\,dx\,dy
  156. E d x d y d z \iiint\limits_{E}\,dx\,dy\,dz
  157. F d x d y d z d t \iiiint\limits_{F}\,dx\,dy\,dz\,dt
  158. ( x , y ) C x 3 d x + 4 y 2 d y \int_{(x,y)\in C}x^{3}\,dx+4y^{2}\,dy
  159. ( x , y ) C x 3 d x + 4 y 2 d y \oint_{(x,y)\in C}x^{3}\,dx+4y^{2}\,dy
  160. i = 1 n E i \bigcap_{i=_{1}}^{n}E_{i}
  161. i = 1 n E i \bigcup_{i=_{1}}^{n}E_{i}
  162. i = 0 2 - i \sum_{i=0}^{\infty}2^{-i}
  163. geometric series: i = 0 2 - i = 2 \,\text{geometric series:}\quad\begin{aligned}\displaystyle\sum_{i=0}^{\infty}% 2^{-i}=2\end{aligned}
  164. i = 0 2 - i \sum_{i=0}^{\infty}2^{-i}
  165. geometric series: i = 0 2 - i = 2 \,\text{geometric series:}\quad\sum_{i=0}^{\infty}2^{-i}=2
  166. 2 4 = 0.5 \frac{2}{4}=0.5
  167. 2 4 = 0.5 \tfrac{2}{4}=0.5
  168. 2 4 = 0.5 2 c + 2 d + 2 4 = a \dfrac{2}{4}=0.5\qquad\dfrac{2}{c+\dfrac{2}{d+\dfrac{2}{4}}}=a
  169. 2 c + 2 d + 2 4 = a \cfrac{2}{c+\cfrac{2}{d+\cfrac{2}{4}}}=a
  170. x 1 + \cancel y \cancel y = x 2 \cfrac{x}{1+\cfrac{\cancel{y}}{\cancel{y}}}=\cfrac{x}{2}
  171. ( n k ) {\left({{n}\atop{k}}\right)}
  172. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  173. ( n k ) {\displaystyle\left({{n}\atop{k}}\right)}
  174. x y z v \begin{matrix}x&y\\ z&v\end{matrix}
  175. | x y z v | \begin{vmatrix}x&y\\ z&v\end{vmatrix}
  176. x y z v \begin{Vmatrix}x&y\\ z&v\end{Vmatrix}
  177. [ 0 0 0 0 ] \begin{bmatrix}0&\cdots&0\\ \vdots&\ddots&\vdots\\ 0&\cdots&0\end{bmatrix}
  178. { x y z v } \begin{Bmatrix}x&y\\ z&v\end{Bmatrix}
  179. ( x y z v ) \begin{pmatrix}x&y\\ z&v\end{pmatrix}
  180. ( a b c d ) \bigl(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\bigr)
  181. f ( n ) = { n / 2 , if n is even 3 n + 1 , if n is odd f(n)=\begin{cases}n/2,&\,\text{if }n\,\text{ is even}\\ 3n+1,&\,\text{if }n\,\text{ is odd}\end{cases}
  182. f ( x ) = ( a + b ) 2 = a 2 + 2 a b + b 2 \begin{aligned}\displaystyle f(x)&\displaystyle=(a+b)^{2}\\ &\displaystyle=a^{2}+2ab+b^{2}\\ \end{aligned}
  183. f ( x ) = ( a - b ) 2 = a 2 - 2 a b + b 2 \begin{aligned}\displaystyle f(x)&\displaystyle=(a-b)^{2}\\ &\displaystyle=a^{2}-2ab+b^{2}\\ \end{aligned}
  184. z = a f ( x , y , z ) = x + y + z \begin{array}[]{lcl}z&=&a\\ f(x,y,z)&=&x+y+z\end{array}
  185. z = a f ( x , y , z ) = x + y + z \begin{array}[]{lcr}z&=&a\\ f(x,y,z)&=&x+y+z\end{array}
  186. f ( x ) = n = 0 a n x n = a 0 + a 1 x + a 2 x 2 + f(x)=\sum_{n=0}^{\infty}a_{n}x^{n}=a_{0}+a_{1}x+a_{2}x^{2}+\cdots
  187. { 3 x + 5 y + z 7 x - 2 y + 4 z - 6 x + 3 y + 2 z \begin{cases}3x+5y+z\\ 7x-2y+4z\\ -6x+3y+2z\end{cases}
  188. a b S 0 0 1 0 1 1 1 0 1 1 1 0 \begin{array}[]{|c|c|c|}a&b&S\\ \hline 0&0&1\\ 0&1&1\\ 1&0&1\\ 1&1&0\\ \end{array}
  189. ( 1 2 ) (\frac{1}{2})
  190. ( 1 2 ) \left(\frac{1}{2}\right)
  191. ( a b ) \left(\frac{a}{b}\right)
  192. [ a b ] [ a b ] \left[\frac{a}{b}\right]\quad\left[\frac{a}{b}\right]
  193. { a b } { a b } \left\{\frac{a}{b}\right\}\quad\left\{\frac{a}{b}\right\}
  194. a b \left\langle\frac{a}{b}\right\rangle
  195. | a b | c d \left|\frac{a}{b}\right|\quad\left\|\frac{c}{d}\right\|
  196. a b c d \left\lfloor\frac{a}{b}\right\rfloor\quad\left\lceil\frac{c}{d}\right\rceil
  197. / a b \ \left/\frac{a}{b}\right\backslash
  198. a b a b a b \left\uparrow\frac{a}{b}\right\downarrow\quad\left\Uparrow\frac{a}{b}\right% \Downarrow\quad\left\updownarrow\frac{a}{b}\right\Updownarrow
  199. [ 0 , 1 ) \left[0,1\right)
  200. ψ | \left\langle\psi\right|
  201. A B } X \left.\frac{A}{B}\right\}\to X
  202. ( ( ( ( ( ] ] ] ] ] (\bigl(\Bigl(\biggl(\Biggl(\dots\Biggr]\biggr]\Bigr]\bigr]]
  203. { { { { { \{\bigl\{\Bigl\{\biggl\{\Biggl\{\dots\Biggr\rangle\biggr\rangle\Bigr\rangle% \bigr\rangle\rangle
  204. | | | | | \|\big\|\Big\|\bigg\|\Bigg\|\dots\Bigg|\bigg|\Big|\big||
  205. \lfloor\bigl\lfloor\Bigl\lfloor\biggl\lfloor\Biggl\lfloor\dots\Biggr\rceil% \biggr\rceil\Bigr\rceil\bigr\rceil\rceil
  206. \uparrow\big\uparrow\Big\uparrow\bigg\uparrow\Bigg\uparrow\dots\Bigg\Downarrow% \bigg\Downarrow\Big\Downarrow\big\Downarrow\Downarrow
  207. \updownarrow\big\updownarrow\Big\updownarrow\bigg\updownarrow\Bigg\updownarrow% \dots\Bigg\Updownarrow\bigg\Updownarrow\Big\Updownarrow\big\Updownarrow\Updownarrow
  208. / / / / / \ \ \ \ \ /\big/\Big/\bigg/\Bigg/\dots\Bigg\backslash\bigg\backslash\Big\backslash\big\backslash\backslash
  209. \Alpha B Γ Δ \Epsilon \Zeta \Eta Θ \Alpha B\Gamma\Delta\Epsilon\Zeta\Eta\Theta\!
  210. \Iota K Λ \Mu \Nu Ξ Π \Rho \Iota K\Lambda\Mu\Nu\Xi\Pi\Rho\!
  211. Σ \Tau Υ Φ \Chi Ψ Ω \Sigma\Tau\Upsilon\Phi\Chi\Psi\Omega\!
  212. α β γ δ ϵ ζ η θ \alpha\beta\gamma\delta\epsilon\zeta\eta\theta\!
  213. ι κ λ μ ν ξ π ρ \iota\kappa\lambda\mu\nu\xi\pi\rho\!
  214. σ τ υ ϕ χ ψ ω \sigma\tau\upsilon\phi\chi\psi\omega\!
  215. ε ϝ ϰ ϖ \varepsilon\digamma\varkappa\varpi\!
  216. ϱ ς ϑ φ \varrho\varsigma\vartheta\varphi\!
  217. \aleph\beth\gimel\daleth\!
  218. 𝔸 𝔹 𝔻 𝔼 𝔽 𝔾 𝕀 \mathbb{ABCDEFGHI}\!
  219. 𝕁 𝕂 𝕃 𝕄 𝕆 \mathbb{JKLMNOPQR}\!
  220. 𝕊 𝕋 𝕌 𝕍 𝕎 𝕏 𝕐 \mathbb{STUVWXYZ}\!
  221. 𝐀𝐁𝐂𝐃𝐄𝐅𝐆𝐇𝐈 \mathbf{ABCDEFGHI}\!
  222. 𝐉𝐊𝐋𝐌𝐍𝐎𝐏𝐐𝐑 \mathbf{JKLMNOPQR}\!
  223. 𝐒𝐓𝐔𝐕𝐖𝐗𝐘𝐙 \mathbf{STUVWXYZ}\!
  224. 𝐚𝐛𝐜𝐝𝐞𝐟𝐠𝐡𝐢𝐣𝐤𝐥𝐦 \mathbf{abcdefghijklm}\!
  225. 𝐧𝐨𝐩𝐪𝐫𝐬𝐭𝐮𝐯𝐰𝐱𝐲𝐳 \mathbf{nopqrstuvwxyz}\!
  226. 𝟎𝟏𝟐𝟑𝟒𝟓𝟔𝟕𝟖𝟗 \mathbf{0123456789}\!
  227. s y m b o l \Alpha B Γ Δ \Epsilon \Zeta \Eta Θ symbol{\Alpha B\Gamma\Delta\Epsilon\Zeta\Eta\Theta}\!
  228. s y m b o l \Iota K Λ \Mu \Nu Ξ Π \Rho symbol{\Iota K\Lambda\Mu\Nu\Xi\Pi\Rho}\!
  229. s y m b o l Σ \Tau Υ Φ \Chi Ψ Ω symbol{\Sigma\Tau\Upsilon\Phi\Chi\Psi\Omega}\!
  230. s y m b o l α β γ δ ϵ ζ η θ symbol{\alpha\beta\gamma\delta\epsilon\zeta\eta\theta}\!
  231. s y m b o l ι κ λ μ ν ξ π ρ symbol{\iota\kappa\lambda\mu\nu\xi\pi\rho}\!
  232. s y m b o l σ τ υ ϕ χ ψ ω symbol{\sigma\tau\upsilon\phi\chi\psi\omega}\!
  233. s y m b o l ε ϝ ϰ ϖ symbol{\varepsilon\digamma\varkappa\varpi}\!
  234. s y m b o l ϱ ς ϑ φ symbol{\varrho\varsigma\vartheta\varphi}\!
  235. 0123456789 \mathit{0123456789}\!
  236. \Alpha B Γ Δ \Epsilon \Zeta \Eta Θ \mathit{\Alpha B\Gamma\Delta\Epsilon\Zeta\Eta\Theta}\!
  237. \Iota K Λ \Mu \Nu Ξ Π \Rho \mathit{\Iota K\Lambda\Mu\Nu\Xi\Pi\Rho}\!
  238. Σ \Tau Υ Φ \Chi Ψ Ω \mathit{\Sigma\Tau\Upsilon\Phi\Chi\Psi\Omega}\!
  239. ABCDEFGHI \mathrm{ABCDEFGHI}\!
  240. JKLMNOPQR \mathrm{JKLMNOPQR}\!
  241. STUVWXYZ \mathrm{STUVWXYZ}\!
  242. abcdefghijklm \mathrm{abcdefghijklm}\!
  243. nopqrstuvwxyz \mathrm{nopqrstuvwxyz}\!
  244. 0123456789 \mathrm{0123456789}\!
  245. 𝖠𝖡𝖢𝖣𝖤𝖥𝖦𝖧𝖨 \mathsf{ABCDEFGHI}\!
  246. 𝖩𝖪𝖫𝖬𝖭𝖮𝖯𝖰𝖱 \mathsf{JKLMNOPQR}\!
  247. 𝖲𝖳𝖴𝖵𝖶𝖷𝖸𝖹 \mathsf{STUVWXYZ}\!
  248. 𝖺𝖻𝖼𝖽𝖾𝖿𝗀𝗁𝗂𝗃𝗄𝗅𝗆 \mathsf{abcdefghijklm}\!
  249. 𝗇𝗈𝗉𝗊𝗋𝗌𝗍𝗎𝗏𝗐𝗑𝗒𝗓 \mathsf{nopqrstuvwxyz}\!
  250. 𝟢𝟣𝟤𝟥𝟦𝟧𝟨𝟩𝟪𝟫 \mathsf{0123456789}\!
  251. \Alpha 𝖡 Γ Δ \Epsilon \Zeta \Eta Θ \mathsf{\Alpha B\Gamma\Delta\Epsilon\Zeta\Eta\Theta}\!
  252. \Iota 𝖪 Λ \Mu \Nu Ξ Π \Rho \mathsf{\Iota K\Lambda\Mu\Nu\Xi\Pi\Rho}\!
  253. Σ \Tau Υ Φ \Chi Ψ Ω \mathsf{\Sigma\Tau\Upsilon\Phi\Chi\Psi\Omega}\!
  254. 𝒜 𝒞 𝒟 𝒢 \mathcal{ABCDEFGHI}\!
  255. 𝒥 𝒦 𝒩 𝒪 𝒫 𝒬 \mathcal{JKLMNOPQR}\!
  256. 𝒮 𝒯 𝒰 𝒱 𝒲 𝒳 𝒴 𝒵 \mathcal{STUVWXYZ}\!
  257. 𝔄 𝔅 𝔇 𝔈 𝔉 𝔊 \mathfrak{ABCDEFGHI}\!
  258. 𝔍 𝔎 𝔏 𝔐 𝔑 𝔒 𝔓 𝔔 \mathfrak{JKLMNOPQR}\!
  259. 𝔖 𝔗 𝔘 𝔙 𝔚 𝔛 𝔜 \mathfrak{STUVWXYZ}\!
  260. 𝔞 𝔟 𝔠 𝔡 𝔢 𝔣 𝔤 𝔥 𝔦 𝔧 𝔨 𝔩 𝔪 \mathfrak{abcdefghijklm}\!
  261. 𝔫 𝔬 𝔭 𝔮 𝔯 𝔰 𝔱 𝔲 𝔳 𝔴 𝔵 𝔶 𝔷 \mathfrak{nopqrstuvwxyz}\!
  262. 0123456789 \mathfrak{0123456789}\!
  263. abcdefghijklm {\scriptstyle\,\text{abcdefghijklm}}
  264. x y z xyz
  265. x y z \,\text{x y z}
  266. if n is even \,\text{if}n\,\text{is even}
  267. if n is even \,\text{if }n\,\text{ is even}
  268. if n is even \,\text{if}~{}n\ \,\text{is even}
  269. \color B l u e x 2 + \color O r a n g e 2 x - \color L i m e G r e e n 1 {\color{Blue}{x^{2}}}+{\color{Orange}{2x}}-{\color{LimeGreen}{1}}
  270. x 1 , 2 = \color B l u e - b ± \color R e d b 2 - 4 a c \color G r e e n 2 a x_{1,2}=\frac{{\color{Blue}{-b}}\pm\sqrt{\color{Red}{b^{2}-4ac}}}{\color{Green% }{2a}}
  271. \color B l u e x 2 + \color O r a n g e 2 x - \color L i m e G r e e n 1 {\color{Blue}x^{2}}+{\color{Orange}2x}-{\color{LimeGreen}1}
  272. \color B l u e x 2 \color B l a c k + \color O r a n g e 2 x \color B l a c k - \color L i m e G r e e n 1 \color{Blue}x^{2}\color{Black}+\color{Orange}2x\color{Black}-\color{LimeGreen}1
  273. \color B l u e x 2 + \color O r a n g e 2 x - \color L i m e G r e e n 1 \color{Blue}{x^{2}}+\color{Orange}{2x}-\color{LimeGreen}{1}
  274. \color A p r i c o t Apricot \color{Apricot}{\,\text{Apricot}}
  275. \pagecolor G r a y \color A q u a m a r i n e Aquamarine \pagecolor{Gray}\color{Aquamarine}{\,\text{Aquamarine}}
  276. \color B i t t e r s w e e t Bittersweet \color{Bittersweet}{\,\text{Bittersweet}}
  277. \color B l a c k Black \color{Black}{\,\text{Black}}
  278. \color B l u e Blue \color{Blue}{\,\text{Blue}}
  279. \color B l u e G r e e n BlueGreen \color{BlueGreen}{\,\text{BlueGreen}}
  280. \color B l u e V i o l e t BlueViolet \color{BlueViolet}{\,\text{BlueViolet}}
  281. \color B r i c k R e d BrickRed \color{BrickRed}{\,\text{BrickRed}}
  282. \color B r o w n Brown \color{Brown}{\,\text{Brown}}
  283. \color B u r n t O r a n g e BurntOrange \color{BurntOrange}{\,\text{BurntOrange}}
  284. \color C a d e t B l u e CadetBlue \color{CadetBlue}{\,\text{CadetBlue}}
  285. \color C a r n a t i o n P i n k CarnationPink \color{CarnationPink}{\,\text{CarnationPink}}
  286. \color C e r u l e a n Cerulean \color{Cerulean}{\,\text{Cerulean}}
  287. \color C o r n f l o w e r B l u e CornflowerBlue \color{CornflowerBlue}{\,\text{CornflowerBlue}}
  288. \pagecolor G r a y \color C y a n Cyan \pagecolor{Gray}\color{Cyan}{\,\text{Cyan}}
  289. \color D a n d e l i o n Dandelion \color{Dandelion}{\,\text{Dandelion}}
  290. \color D a r k O r c h i d DarkOrchid \color{DarkOrchid}{\,\text{DarkOrchid}}
  291. \color E m e r a l d Emerald \color{Emerald}{\,\text{Emerald}}
  292. \color F o r e s t G r e e n ForestGreen \color{ForestGreen}{\,\text{ForestGreen}}
  293. \color F u c h s i a Fuchsia \color{Fuchsia}{\,\text{Fuchsia}}
  294. \color G o l d e n r o d Goldenrod \color{Goldenrod}{\,\text{Goldenrod}}
  295. \color G r a y Gray \color{Gray}{\,\text{Gray}}
  296. \color G r e e n Green \color{Green}{\,\text{Green}}
  297. \pagecolor G r a y \color G r e e n Y e l l o w GreenYellow \pagecolor{Gray}\color{GreenYellow}{\,\text{GreenYellow}}
  298. \color J u n g l e G r e e n JungleGreen \color{JungleGreen}{\,\text{JungleGreen}}
  299. \pagecolor G r a y \color L a v e n d e r Lavender \pagecolor{Gray}\color{Lavender}{\,\text{Lavender}}
  300. \color L i m e G r e e n LimeGreen \color{LimeGreen}{\,\text{LimeGreen}}
  301. \color M a g e n t a Magenta \color{Magenta}{\,\text{Magenta}}
  302. \color M a h o g a n y Mahogany \color{Mahogany}{\,\text{Mahogany}}
  303. \color M a r o o n Maroon \color{Maroon}{\,\text{Maroon}}
  304. \color M e l o n Melon \color{Melon}{\,\text{Melon}}
  305. \color M i d n i g h t B l u e MidnightBlue \color{MidnightBlue}{\,\text{MidnightBlue}}
  306. \color M u l b e r r y Mulberry \color{Mulberry}{\,\text{Mulberry}}
  307. \color N a v y B l u e NavyBlue \color{NavyBlue}{\,\text{NavyBlue}}
  308. \color O l i v e G r e e n OliveGreen \color{OliveGreen}{\,\text{OliveGreen}}
  309. \color O r a n g e Orange \color{Orange}{\,\text{Orange}}
  310. \color O r a n g e R e d OrangeRed \color{OrangeRed}{\,\text{OrangeRed}}
  311. \color O r c h i d Orchid \color{Orchid}{\,\text{Orchid}}
  312. \color P e a c h Peach \color{Peach}{\,\text{Peach}}
  313. \color P e r i w i n k l e Periwinkle \color{Periwinkle}{\,\text{Periwinkle}}
  314. \color P i n e G r e e n PineGreen \color{PineGreen}{\,\text{PineGreen}}
  315. \color P l u m Plum \color{Plum}{\,\text{Plum}}
  316. \color P r o c e s s B l u e ProcessBlue \color{ProcessBlue}{\,\text{ProcessBlue}}
  317. \color P u r p l e Purple \color{Purple}{\,\text{Purple}}
  318. \color R a w S i e n n a RawSienna \color{RawSienna}{\,\text{RawSienna}}
  319. \color R e d Red \color{Red}{\,\text{Red}}
  320. \color R e d O r a n g e RedOrange \color{RedOrange}{\,\text{RedOrange}}
  321. \color R e d V i o l e t RedViolet \color{RedViolet}{\,\text{RedViolet}}
  322. \color R h o d a m i n e Rhodamine \color{Rhodamine}{\,\text{Rhodamine}}
  323. \color R o y a l B l u e RoyalBlue \color{RoyalBlue}{\,\text{RoyalBlue}}
  324. \color R o y a l P u r p l e RoyalPurple \color{RoyalPurple}{\,\text{RoyalPurple}}
  325. \color R u b i n e R e d RubineRed \color{RubineRed}{\,\text{RubineRed}}
  326. \color S a l m o n Salmon \color{Salmon}{\,\text{Salmon}}
  327. \color S e a G r e e n SeaGreen \color{SeaGreen}{\,\text{SeaGreen}}
  328. \color S e p i a Sepia \color{Sepia}{\,\text{Sepia}}
  329. \color S k y B l u e SkyBlue \color{SkyBlue}{\,\text{SkyBlue}}
  330. \color S p r i n g G r e e n SpringGreen \color{SpringGreen}{\,\text{SpringGreen}}
  331. \color T a n Tan \color{Tan}{\,\text{Tan}}
  332. \color T e a l B l u e TealBlue \color{TealBlue}{\,\text{TealBlue}}
  333. \pagecolor G r a y \color T h i s t l e Thistle \pagecolor{Gray}\color{Thistle}{\,\text{Thistle}}
  334. \color T u r q u o i s e Turquoise \color{Turquoise}{\,\text{Turquoise}}
  335. \color V i o l e t Violet \color{Violet}{\,\text{Violet}}
  336. \color V i o l e t R e d VioletRed \color{VioletRed}{\,\text{VioletRed}}
  337. \pagecolor G r a y \color W h i t e White \pagecolor{Gray}{\color{White}{\,\text{White}}}
  338. \color W i l d S t r a w b e r r y WildStrawberry \color{WildStrawberry}{\,\text{WildStrawberry}}
  339. \pagecolor G r a y \color Y e l l o w Yellow \pagecolor{Gray}\color{Yellow}{\,\text{Yellow}}
  340. \color Y e l l o w G r e e n YellowGreen \color{YellowGreen}{\,\text{YellowGreen}}
  341. \color Y e l l o w O r a n g e YellowOrange \color{YellowOrange}{\,\text{YellowOrange}}
  342. \pagecolor G r a y x 2 \pagecolor{Gray}x^{2}
  343. \pagecolor G o l d e n r o d y 3 \pagecolor{Goldenrod}y^{3}
  344. x 2 x^{2}
  345. y 3 y^{3}
  346. \definecolor m y o r a n g e r g b 1 , 0.65 , 0.4 \color m y o r a n g e e i π \color B l a c k + 1 = 0 \definecolor{myorange}{rgb}{1,0.65,0.4}\color{myorange}e^{i\pi}\color{Black}+1=0
  347. a b a\qquad b
  348. a b a\quad b
  349. a b a\ b
  350. a b a\mbox{ }~{}b
  351. a b a\;b
  352. a b a\,b
  353. a b ab
  354. a b ab
  355. 𝑎𝑏 \mathit{ab}
  356. a b a\!b
  357. 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 0+1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+\cdots
  358. 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + {0+1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+\cdots}
  359. - N N e x d x \int_{-N}^{N}e^{x}\,dx
  360. S 𝐃 d 𝐀 \iint\limits_{S}\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset\mathbf{D}\cdot\mathrm{d% }\mathbf{A}
  361. V 𝐃 d 𝐀 \int\!\!\!\!\int_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\bigcirc% \,\,\mathbf{D}\cdot\mathrm{d}\mathbf{A}
  362. V 𝐃 d 𝐀 \int\!\!\!\!\!\int\!\!\!\!\!\int_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!% \!\!\!\;\;\;\subset\!\supset\mathbf{D}\cdot\mathrm{d}\mathbf{A}
  363. V 𝐃 d 𝐀 \int\!\!\!\!\!\int\!\!\!\!\!\int_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!% \!\;\;\;\bigcirc\,\,\mathbf{D}\;\cdot\mathrm{d}\mathbf{A}
  364. A B \overset{\frown}{AB}
  365. x \overset{...}{x}
  366. A B C ABC
  367. A B C A\,BC
  368. a x 2 + b x + c = 0 ax^{2}+bx+c=0
  369. x = - b ± b 2 - 4 a c 2 a x={-b\pm\sqrt{b^{2}-4ac}\over 2a}
  370. 2 = ( ( 3 - x ) × 2 3 - x ) 2=\left(\frac{\left(3-x\right)\times 2}{3-x}\right)
  371. S new = S old - ( 5 - T ) 2 2 S_{\,\text{new}}=S_{\,\text{old}}-\frac{\left(5-T\right)^{2}}{2}
  372. a x a s f ( y ) d y d s = a x f ( y ) ( x - y ) d y \int_{a}^{x}\!\!\!\int_{a}^{s}f(y)\,dy\,ds=\int_{a}^{x}f(y)(x-y)\,dy
  373. det ( 𝖠 - λ 𝖨 ) = 0 \det(\mathsf{A}-\lambda\mathsf{I})=0
  374. i = 0 n - 1 i \sum_{i=0}^{n-1}i
  375. m = 1 n = 1 m 2 n 3 m ( m 3 n + n 3 m ) \sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{m^{2}\,n}{3^{m}\left(m\,3^{n}+n\,3% ^{m}\right)}
  376. u ′′ + p ( x ) u + q ( x ) u = f ( x ) , x > a u^{\prime\prime}+p(x)u^{\prime}+q(x)u=f(x),\quad x>a
  377. | z ¯ | = | z | , | ( z ¯ ) n | = | z | n , arg ( z n ) = n arg ( z ) |\bar{z}|=|z|,|(\bar{z})^{n}|=|z|^{n},\arg(z^{n})=n\arg(z)
  378. lim z z 0 f ( z ) = f ( z 0 ) \lim_{z\to z_{0}}f(z)=f(z_{0})
  379. ϕ n ( κ ) = 1 4 π 2 κ 2 0 sin ( κ R ) κ R R [ R 2 D n ( R ) R ] d R \phi_{n}(\kappa)=\frac{1}{4\pi^{2}\kappa^{2}}\int_{0}^{\infty}\frac{\sin(% \kappa R)}{\kappa R}\frac{\partial}{\partial R}\left[R^{2}\frac{\partial D_{n}% (R)}{\partial R}\right]\,dR
  380. ϕ n ( κ ) = 0.033 C n 2 κ - 11 / 3 , 1 L 0 κ 1 l 0 \phi_{n}(\kappa)=0.033C_{n}^{2}\kappa^{-11/3},\quad\frac{1}{L_{0}}\ll\kappa\ll% \frac{1}{l_{0}}
  381. f ( x ) = { 1 - 1 x < 0 1 2 x = 0 1 - x 2 otherwise f(x)=\begin{cases}1&-1\leq x<0\\ \frac{1}{2}&x=0\\ 1-x^{2}&\,\text{otherwise}\end{cases}
  382. F q p ( a 1 , , a p ; c 1 , , c q ; z ) = n = 0 ( a 1 ) n ( a p ) n ( c 1 ) n ( c q ) n z n n ! {}_{p}F_{q}(a_{1},\dots,a_{p};c_{1},\dots,c_{q};z)=\sum_{n=0}^{\infty}\frac{(a% _{1})_{n}\cdots(a_{p})_{n}}{(c_{1})_{n}\cdots(c_{q})_{n}}\frac{z^{n}}{n!}
  383. a b a b \frac{a}{b}\ \tfrac{a}{b}
  384. S = d D sin α S=dD\,\sin\alpha\!
  385. V = 1 6 π h [ 3 ( r 1 2 + r 2 2 ) + h 2 ] V=\frac{1}{6}\pi h\left[3\left(r_{1}^{2}+r_{2}^{2}\right)+h^{2}\right]
  386. u = 1 2 ( x + y ) x = 1 2 ( u + v ) v = 1 2 ( x - y ) y = 1 2 ( u - v ) \begin{aligned}\displaystyle u&\displaystyle=\tfrac{1}{\sqrt{2}}(x+y)&% \displaystyle x&\displaystyle=\tfrac{1}{\sqrt{2}}(u+v)\\ \displaystyle v&\displaystyle=\tfrac{1}{\sqrt{2}}(x-y)&\displaystyle y&% \displaystyle=\tfrac{1}{\sqrt{2}}(u-v)\end{aligned}

Hemicompact_space.html

  1. X X
  2. C ( X , M ) C(X,M)
  3. f : X M f:X\to M
  4. ( M , δ ) (M,\delta)
  5. K 1 , K 2 , K_{1},K_{2},\dots
  6. X X
  7. X X
  8. X X
  9. d n ( f , g ) = sup x K n δ ( f ( x ) , g ( x ) ) d_{n}(f,g)=\sup_{x\in K_{n}}\delta(f(x),g(x))
  10. f , g C ( X , M ) f,g\in C(X,M)
  11. n n\in\mathbb{N}
  12. d ( f , g ) = n = 1 1 2 n d n ( f , g ) 1 + d n ( f , g ) d(f,g)=\sum_{n=1}^{\infty}\frac{1}{2^{n}}\cdot\frac{d_{n}(f,g)}{1+d_{n}(f,g)}
  13. C ( X , M ) C(X,M)

Henyey_track.html

  1. 10 5 10^{5}
  2. 10 5 10^{5}

Herbrand_quotient.html

  1. q ( f , g ) = | ker f : im g | | ker g : im f | q(f,g)=\frac{|\mathrm{ker}f:\mathrm{im}g|}{|\mathrm{ker}g:\mathrm{im}f|}

Herbrandization.html

  1. F F
  2. F F
  3. F F
  4. F F
  5. F F
  6. v v
  7. f v ( x 1 , , x k ) f_{v}(x_{1},\dots,x_{k})
  8. x 1 , , x k x_{1},\dots,x_{k}
  9. v v
  10. F := y x [ R ( y , x ) ¬ z S ( x , z ) ] F:=\forall y\exists x[R(y,x)\wedge\neg\exists zS(x,z)]
  11. y , z y,z
  12. y \forall y
  13. z \exists z
  14. y y
  15. c y c_{y}
  16. y y
  17. z z
  18. f z ( x ) f_{z}(x)
  19. F H = x [ R ( c y , x ) ¬ S ( x , f z ( x ) ) ] . F^{H}=\exists x[R(c_{y},x)\wedge\neg S(x,f_{z}(x))].
  20. F F
  21. F S = y [ R ( y , f x ( y ) ) ¬ z S ( f x ( y ) , z ) ] . F^{S}=\forall y[R(y,f_{x}(y))\wedge\neg\exists zS(f_{x}(y),z)].

Hereditarily_countable_set.html

  1. H 1 H_{\aleph_{1}}
  2. x H 1 x\in H_{\aleph_{1}}
  3. L ω 1 ( x ) H 1 L_{\omega_{1}}(x)\subset H_{\aleph_{1}}
  4. H κ H_{\kappa}\!

Hereditary_ring.html

  1. Ext R i \mathrm{Ext}_{R}^{i}
  2. Tor i R \mathrm{Tor}_{i}^{R}
  3. i > 1 i>1
  4. R x R R\cong xR
  5. r x r r\mapsto xr

Hermetic_detector.html

  1. - π ϕ π -\pi\leq\phi\leq\pi
  2. | η | 0 |\eta|\geq 0
  3. R R
  4. p T p_{T}
  5. p T = q B R p_{T}=qBR
  6. q q
  7. B B

Hermitian_variety.html

  1. θ \theta
  2. 1 \geq 1
  3. e 0 , e 1 , , e n e_{0},e_{1},\ldots,e_{n}
  4. ( X 0 , , X n ) (X_{0},\ldots,X_{n})
  5. i , j = 0 n a i j X i X j θ = 0 \sum_{i,j=0}^{n}a_{ij}X_{i}X_{j}^{\theta}=0
  6. a i j = a j i θ a_{ij}=a_{ji}^{\theta}
  7. a i j = 0 a_{ij}=0
  8. A i j = a i j A_{ij}=a_{ij}
  9. X t A X θ = 0 X^{t}AX^{\theta}=0
  10. X = [ X 0 X 1 X n ] . X=\begin{bmatrix}X_{0}\\ X_{1}\\ \vdots\\ X_{n}\end{bmatrix}.

Heronian_mean.html

  1. H = 1 3 ( A + A B + B ) . H=\frac{1}{3}\left(A+\sqrt{AB}+B\right).
  2. H = 2 3 A + B 2 + 1 3 A B . H=\frac{2}{3}\cdot\frac{A+B}{2}+\frac{1}{3}\cdot\sqrt{AB}.

Hewitt–Savage_zero–one_law.html

  1. ( X n ) n = 1 (X_{n})_{n=1}^{\infty}
  2. 𝕏 \mathbb{X}
  3. \mathcal{E}
  4. ( X n ) n = 1 (X_{n})_{n=1}^{\infty}
  5. ( X n ) n = 1 (X_{n})_{n=1}^{\infty}
  6. A ( A ) { 0 , 1 } A\in\mathcal{E}\implies\mathbb{P}(A)\in\{0,1\}
  7. A A
  8. \mathcal{E}
  9. ( i , j ) (i,j)
  10. i , j i,j\in\mathbb{N}
  11. ( X n ) n = 1 (X_{n})_{n=1}^{\infty}
  12. [ 0 , ) [0,\infty)
  13. n = 1 X n \sum_{n=1}^{\infty}X_{n}
  14. \mathcal{E}
  15. 𝔼 [ X n ] > 0 \mathbb{E}[X_{n}]>0
  16. ( X n = 0 ) < 1 \mathbb{P}(X_{n}=0)<1
  17. ( n = 1 X n = + ) = 1 , \mathbb{P}\left(\sum_{n=1}^{\infty}X_{n}=+\infty\right)=1,
  18. S N = n = 1 N X n , S_{N}=\sum_{n=1}^{N}X_{n},

Hexadecagon.html

  1. A = 4 t 2 cot π 16 = 4 t 2 ( 2 + 1 ) ( 4 - 2 2 + 1 ) A=4t^{2}\cot\frac{\pi}{16}=4t^{2}(\sqrt{2}+1)(\sqrt{4-2\sqrt{2}}+1)
  2. A = r 2 2 1 2 2 2 2 + 2 = 4 r 2 2 - 2 . A=r^{2}\cdot\frac{2}{1}\cdot\frac{2}{\sqrt{2}}\cdot\frac{2}{\sqrt{2+\sqrt{2}}}% =4r^{2}\sqrt{2-\sqrt{2}}.

High-dimensional_model_representation.html

  1. f ( 𝐱 ) = f 0 + i = 1 n f i ( x i ) + i , j = 1 i < j n f i j ( x j , x j ) + + f 12 n ( x 1 , , x n ) . f(\mathbf{x})=f_{0}+\sum_{i=1}^{n}f_{i}(x_{i})+\sum_{i,j=1\atop i<j}^{n}f_{ij}% (x_{j},x_{j})+\cdots+f_{12\ldots n}(x_{1},\ldots,x_{n}).

High-frequency_ventilation.html

  1. Δ p = P I P - P E E P \Delta p=P_{IP}-P_{EEP}
  2. f = H z 60 s e c o n d s f=Hz\cdot 60_{seconds}
  3. O T P = M A P - ( A M P / 3 ) OTP=MAP-(AMP/3)

High-leg_delta.html

  1. V a b = V b c = V c a = 240 V V_{ab}=V_{bc}=V_{ca}=240V
  2. V a n = V c n = V a c 2 = 120 V V_{an}=V_{cn}=\frac{V_{ac}}{2}=120V
  3. V b n = V a b 2 - V a n 2 208 V V_{bn}=\sqrt{{V_{ab}}^{2}-{V_{an}}^{2}}\approx 208V
  4. 0 + 120 0 + 240 120 = 120 3 90 0+120\angle 0^{\circ}+240\angle 120^{\circ}=120\sqrt{3}\angle 90^{\circ}
  5. 0 + 120 sin ( 0 ) + 240 sin ( 120 ) = 120 3 = 207.8 0+120\sin(0^{\circ})+240\sin(120^{\circ})=120\sqrt{3}=207.8

High-speed_craft.html

  1. 3.7 × 0.1667 3.7\times\triangledown^{0.1667}
  2. \triangledown

High_frequency_content_measure.html

  1. HFC = i = 0 N - 1 i | X ( i ) | \mathrm{HFC}=\sum_{i=0}^{N-1}i|X(i)|

Higher_residuosity_problem.html

  1. / n / p × / q \mathbb{Z}/n\mathbb{Z}\simeq\mathbb{Z}/p\mathbb{Z}\times\mathbb{Z}/q\mathbb{Z}
  2. / n \mathbb{Z}/n\mathbb{Z}
  3. ( / n ) * (\mathbb{Z}/n\mathbb{Z})^{*}
  4. ( / n ) * ( / p ) * × ( / q ) * (\mathbb{Z}/n\mathbb{Z})^{*}\simeq(\mathbb{Z}/p\mathbb{Z})^{*}\times(\mathbb{Z% }/q\mathbb{Z})^{*}
  5. ( / p ) * (\mathbb{Z}/p\mathbb{Z})^{*}
  6. ( / q ) * (\mathbb{Z}/q\mathbb{Z})^{*}
  7. ( / p ) * (\mathbb{Z}/p\mathbb{Z})^{*}
  8. ( / q ) * (\mathbb{Z}/q\mathbb{Z})^{*}
  9. ( / n ) * (\mathbb{Z}/n\mathbb{Z})^{*}
  10. ( / q ) * (\mathbb{Z}/q\mathbb{Z})^{*}
  11. ( / n ) * (\mathbb{Z}/n\mathbb{Z})^{*}
  12. ( / n ) * (\mathbb{Z}/n\mathbb{Z})^{*}
  13. ( / n ) * . (\mathbb{Z}/n\mathbb{Z})^{*}.
  14. x ( p - 1 ) / d 1 ( mod p ) x^{(p-1)/d}\equiv 1\;\;(\mathop{{\rm mod}}p)

Higher_spin_alternating_sign_matrix.html

  1. ( 0 0 2 0 0 2 - 1 1 2 - 1 2 - 1 0 1 - 1 2 ) ; ( 0 0 2 0 0 0 1 - 1 2 0 2 - 1 - 1 0 2 0 0 2 0 0 0 2 0 0 0 ) ; ( 0 0 0 2 0 2 0 0 2 - 2 2 0 0 2 0 0 ) ; ( 0 2 0 0 0 0 0 2 2 0 0 0 0 0 2 0 ) . \begin{pmatrix}0&0&2&0\\ 0&2&-1&1\\ 2&-1&2&-1\\ 0&1&-1&2\end{pmatrix};\quad\begin{pmatrix}0&0&2&0&0\\ 0&1&-1&2&0\\ 2&-1&-1&0&2\\ 0&0&2&0&0\\ 0&2&0&0&0\end{pmatrix};\quad\begin{pmatrix}0&0&0&2\\ 0&2&0&0\\ 2&-2&2&0\\ 0&2&0&0\end{pmatrix};\quad\begin{pmatrix}0&2&0&0\\ 0&0&0&2\\ 2&0&0&0\\ 0&0&2&0\end{pmatrix}.

Highest_response_ratio_next.html

  1. P r i o r i t y = w a i t i n g t i m e + e s t i m a t e d r u n t i m e e s t i m a t e d r u n t i m e = 1 + w a i t i n g t i m e e s t i m a t e d r u n t i m e Priority=\frac{waiting\ time+estimated\ run\ time}{estimated\ run\ time}=1+% \frac{waiting\ time}{estimated\ run\ time}

Hilal-i-Jur'at.html

  1. 1 Square of Land = ( 77 acres 3 Squares of Land ) = 25.41 a c r e s \mathrm{1\ Square\ of\ Land}=(\frac{\mathrm{77\ acres}}{\mathrm{3\ Squares\ of% \ Land}})=25.41\ acres
  2. 2 Squares of Land = ( 77 acres 3 Squares of Land ) 2 S q u a r e s o f L a n d = 50.82 a c r e s \mathrm{2\ Squares\ of\ Land}=(\frac{\mathrm{77\ acres}}{\mathrm{3\ Squares\ % of\ Land}})\cdot 2\ Squares\ of\ Land\ =50.82\ acres
  3. 1 Square of Land = ( 77 acres 3 Squares of Land ) 1 = 25.41 a c r e s \mathrm{1\ Square\ of\ Land}=(\frac{\mathrm{77\ acres}}{\mathrm{3\ Squares\ of% \ Land}})\cdot 1=25.41\ acres
  4. 2 Squares of Land = ( 77 acres 3 Squares of Land ) 2 = 50.82 a c r e s \mathrm{2\ Squares\ of\ Land}=(\frac{\mathrm{77\ acres}}{\mathrm{3\ Squares\ % of\ Land}})\cdot 2=50.82\ acres

Hilbert_basis_(linear_programming).html

  1. A = { a 1 , , a n } A=\{a_{1},\ldots,a_{n}\}
  2. C = { λ 1 a 1 + + λ n a n λ 1 , , λ n 0 , λ 1 , , λ n } C=\{\lambda_{1}a_{1}+\ldots+\lambda_{n}a_{n}\mid\lambda_{1},\ldots,\lambda_{n}% \geq 0,\lambda_{1},\ldots,\lambda_{n}\in\mathbb{R}\}
  3. { α 1 a 1 + + α n a n α 1 , , α n 0 , α 1 , , α n } , \{\alpha_{1}a_{1}+\ldots+\alpha_{n}a_{n}\mid\alpha_{1},\ldots,\alpha_{n}\geq 0% ,\alpha_{1},\ldots,\alpha_{n}\in\mathbb{Z}\},

Hilbert_series_and_Hilbert_polynomial.html

  1. S = i 0 S i S=\bigoplus_{i\geq 0}S_{i}
  2. S 0 = K S_{0}=K
  3. H F S : n dim K S n HF_{S}\;:\;n\mapsto\dim_{K}\,S_{n}
  4. H S S ( t ) = n = 0 H F S ( n ) t n . HS_{S}(t)=\sum_{n=0}^{\infty}HF_{S}(n)\,t^{n}.
  5. d 1 , , d h d_{1},\ldots,d_{h}
  6. H S S ( t ) = Q ( t ) i = 1 h ( 1 - t d i ) , HS_{S}(t)=\frac{Q(t)}{\prod_{i=1}^{h}(1-t^{d_{i}})}\,,
  7. H S S ( t ) = P ( t ) ( 1 - t ) δ , HS_{S}(t)=\frac{P(t)}{(1-t)^{\delta}}\,,
  8. H S S ( t ) = P ( t ) ( 1 + δ t + + ( n + δ - 1 δ - 1 ) t n + ) HS_{S}(t)=P(t)\,\left(1+\delta\,t+\cdots+{\left({{n+\delta-1}\atop{\delta-1}}% \right)}\,t^{n}+\cdots\right)
  9. ( n + δ - 1 δ - 1 ) {\left({{n+\delta-1}\atop{\delta-1}}\right)}
  10. ( n + δ - 1 ) ( n + δ - 2 ) n ( δ - 1 ) ! \;\frac{(n+\delta-1)(n+\delta-2)\cdots n}{(\delta-1)!}\;
  11. n > - δ n>-\delta
  12. H P S ( n ) HP_{S}(n)
  13. H F S ( n ) HF_{S}(n)
  14. n deg P - δ + 1 n\geq\deg P-\delta+1
  15. H P S ( n ) = H F S ( n ) HP_{S}(n)=HF_{S}(n)
  16. deg P - δ + 1 \deg P-\delta+1
  17. R n = K [ X 1 , , X n ] R_{n}=K[X_{1},\ldots,X_{n}]
  18. R n / I R_{n}/I
  19. 0 A B C 0 0\;\rightarrow\;A\;\rightarrow\;B\;\rightarrow\;C\;\rightarrow\;0
  20. H S B = H S A + H S C HS_{B}=HS_{A}+HS_{C}
  21. H P B = H P A + H P C . HP_{B}=HP_{A}+HP_{C}.
  22. H S A / ( f ) ( t ) = ( 1 - t d ) H S A ( t ) . HS_{A/(f)}(t)=(1-t^{d})\,HS_{A}(t)\,.
  23. 0 A [ d ] 𝑓 A A / f 0 , 0\;\rightarrow\;A^{[d]}\;\xrightarrow{f}\;A\;\rightarrow\;A/f\rightarrow\;0\,,
  24. A [ d ] A^{[d]}
  25. H S A [ d ] ( t ) = t d H S A ( t ) . HS_{A^{[d]}}(t)=t^{d}\,HS_{A}(t)\,.
  26. R n = K [ x 1 , , x n ] R_{n}=K[x_{1},\ldots,x_{n}]
  27. n n
  28. H S R n ( t ) = 1 ( 1 - t ) n . HS_{R_{n}}(t)=\frac{1}{(1-t)^{n}}\,.
  29. H P R n ( k ) = ( k + n - 1 n - 1 ) = ( k + 1 ) ( k + n - 1 ) ( n - 1 ) ! . HP_{R_{n}}(k)={{k+n-1}\choose{n-1}}=\frac{(k+1)\cdots(k+n-1)}{(n-1)!}\,.
  30. x n x_{n}
  31. H S K ( t ) = 1 . HS_{K}(t)=1\,.
  32. H S A / ( f ) ( t ) = ( 1 - t ) H S A ( t ) HS_{A/(f)}(t)=(1-t)\,HS_{A}(t)
  33. H S A ( t ) = P ( t ) ( 1 - t ) d HS_{A}(t)=\frac{P(t)}{(1-t)^{d}}
  34. V V
  35. I k [ x 0 , x 1 , , x n ] I\subset k[x_{0},x_{1},\ldots,x_{n}]
  36. k k
  37. R = k [ x 1 , , x n ] / I R=k[x_{1},\ldots,x_{n}]/I
  38. V V
  39. R R
  40. d d
  41. V V
  42. V V
  43. d d
  44. h 1 , , h d , h_{1},\ldots,h_{d},
  45. 0 ( R / h 1 , , h k - 1 ) [ 1 ] h k R / h 1 , , h k - 1 R / h 1 , , h k 0 , 0\;\rightarrow\;\left(R/\langle h_{1},\ldots,h_{k-1}\rangle\right)^{[1]}\;% \xrightarrow{h_{k}}\;R/\langle h_{1},\ldots,h_{k-1}\rangle\;\rightarrow\;R/% \langle h_{1},\ldots,h_{k}\rangle\;\rightarrow\;0,
  46. k = 1 , , d . k=1,\ldots,d.
  47. H S R / h 1 , , h d ( t ) = ( 1 - t ) d H S R ( t ) HS_{R/\langle h_{1},\ldots,h_{d}\rangle}(t)=(1-t)^{d}\,HS_{R}(t)
  48. P ( t ) P(t)
  49. R R
  50. x 0 = 1 x_{0}=1
  51. P ( 1 ) P(1)
  52. V V
  53. f f
  54. δ \delta
  55. R R
  56. 0 R [ δ ] 𝑓 R R / f 0 , 0\;\rightarrow\;R^{[\delta]}\;\xrightarrow{f}\;R\;\rightarrow\;R/\langle f% \rangle\;\rightarrow\;0,
  57. H S R / f ( t ) = ( 1 - t δ ) H S R ( t ) . HS_{R/\langle f\rangle}(t)=(1-t^{\delta})HS_{R}(t).
  58. f f
  59. δ \delta
  60. R R
  61. V V
  62. f f
  63. V V
  64. δ \delta
  65. n - 1 n-1
  66. R = K [ x 1 , , x n ] R=K[x_{1},\ldots,x_{n}]

Hilbert_symbol.html

  1. ( a , b ) = { 1 , if z 2 = a x 2 + b y 2 has a non-zero solution ( x , y , z ) K 3 ; - 1 , if not. (a,b)=\begin{cases}1,&\mbox{ if }~{}z^{2}=ax^{2}+by^{2}\mbox{ has a non-zero % solution }~{}(x,y,z)\in K^{3};\\ -1,&\mbox{ if not.}\end{cases}
  2. K 2 M ( K ) K^{M}_{2}(K)
  3. K 2 M ( K ) / 2 K^{M}_{2}(K)/2
  4. i 2 = a i^{2}=a
  5. j 2 = b j^{2}=b
  6. i j = - j i = k ij=-ji=k
  7. a = p α u a=p^{\alpha}u
  8. b = p β v b=p^{\beta}v
  9. ( a , b ) p = ( - 1 ) α β ϵ ( p ) ( u p ) β ( v p ) α (a,b)_{p}=(-1)^{\alpha\beta\epsilon(p)}\left(\frac{u}{p}\right)^{\beta}\left(% \frac{v}{p}\right)^{\alpha}
  10. ϵ ( p ) = ( p - 1 ) / 2 \epsilon(p)=(p-1)/2
  11. a = 2 α u a=2^{\alpha}u
  12. b = 2 β v b=2^{\beta}v
  13. ( a , b ) 2 = ( - 1 ) ϵ ( u ) ϵ ( v ) + α ω ( v ) + β ω ( u ) (a,b)_{2}=(-1)^{\epsilon(u)\epsilon(v)+\alpha\omega(v)+\beta\omega(u)}
  14. ω ( x ) = ( x 2 - 1 ) / 8. \omega(x)=(x^{2}-1)/8.
  15. v ( a , b ) v = 1 \prod_{v}(a,b)_{v}=1
  16. ( , ) : F * / F * 2 × F * / F * 2 B r ( F ) (\cdot,\cdot):F^{*}/F^{*2}\times F^{*}/F^{*2}\rightarrow\mathop{Br}(F)
  17. ( a , b ) b n = ( a , K ( b n ) / K ) b n (a,b)\sqrt[n]{b}=(a,K(\sqrt[n]{b})/K)\sqrt[n]{b}
  18. p ( a , b ) p = 1 \prod_{p}(a,b)_{p}=1
  19. ( a p ) = ( π , a ) p {\left({{a}\atop{p}}\right)}=(\pi,a)_{p}
  20. ( a b ) = ( b a ) p | n , ( a , b ) p {\left({{a}\atop{b}}\right)}={\left({{b}\atop{a}}\right)}\prod_{p|n,\infty}(a,% b)_{p}

HisB.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Histone-arginine_N-methyltransferase.html

  1. \rightleftharpoons

History_of_electromagnetic_theory.html

  1. m = E / c 2 m=E/c^{2}
  2. E = m c 2 E=mc^{2}
  3. S O ( 10 ) SO(10)
  4. 10 16 10^{16}
  5. S O ( 10 ) SO(10)
  6. 𝐑 3 \mathbf{R}^{3}

History_of_information_theory.html

  1. W = K log m W=K\log m\,
  2. H = log S n H=\log S^{n}\,
  3. H = - f i log f i H=-\sum f_{i}\log f_{i}
  4. S = - k B p i ln p i S=-k\text{B}\sum p_{i}\ln p_{i}\,

History_of_Lorentz_transformations.html

  1. x = γ ( x - v t ) , y = y , z = z , t = γ ( t - x v c 2 ) x^{\prime}=\gamma(x-vt),\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t^{\prime}% =\gamma\left(t-x\frac{v}{c^{2}}\right)
  2. γ = 1 1 - v 2 / c 2 \gamma=\frac{1}{\sqrt{1-v^{2}/c^{2}}}
  3. x 2 + y 2 + z 2 - c 2 t 2 x^{2}+y^{2}+z^{2}-c^{2}t^{2}
  4. λ ( δ x 2 + δ y 2 + δ z 2 ) \lambda\left(\delta x^{2}+\delta y^{2}+\delta z^{2}\right)
  5. λ ( δ x 1 2 + + δ x n 2 ) \lambda\left(\delta x_{1}^{2}+\dots+\delta x_{n}^{2}\right)
  6. λ \lambda
  7. x , y , z , R x,y,z,R
  8. x = x , z = 1 + k 2 1 - k 2 z - 2 k R 1 - k 2 , y = y , R = 2 k z 1 - k 2 - 1 + k 2 1 - k 2 R , \begin{aligned}\displaystyle x^{\prime}&\displaystyle=x,&\displaystyle z^{% \prime}&\displaystyle=\frac{1+k^{2}}{1-k^{2}}z-\frac{2kR}{1-k^{2}},\\ \displaystyle y^{\prime}&\displaystyle=y,&\displaystyle R^{\prime}&% \displaystyle=\frac{2kz}{1-k^{2}}-\frac{1+k^{2}}{1-k^{2}}R,\end{aligned}
  9. x 2 + y 2 + z 2 - R 2 = x 2 + y 2 + z 2 - R 2 x^{\prime 2}+y^{\prime 2}+z^{\prime 2}-R^{\prime 2}=x^{2}+y^{2}+z^{2}-R^{2}
  10. R = t R=t
  11. c = 1 c=1
  12. v = 2 k / ( 1 + k 2 ) v=2k/\left(1+k^{2}\right)
  13. 1 - k 2 1 + k 2 = 1 - v 2 = 1 γ , 2 k 1 - k 2 = v γ , \frac{1-k^{2}}{1+k^{2}}=\sqrt{1-v^{2}}=\frac{1}{\gamma},\quad\frac{2k}{1-k^{2}% }=v\gamma,
  14. z z
  15. t t^{\prime}
  16. t - v z t-vz
  17. v z - t vz-t
  18. x = x , y = y , z = γ ( z - v t ) , t = γ ( v z - t ) x^{\prime}=x,\quad y^{\prime}=y,\quad z^{\prime}=\gamma(z-vt),\quad t^{\prime}% =\gamma(vz-t)
  19. x = x - v t , y = y γ , z = z γ , t = t - x v c 2 x^{\prime}=x-vt,\quad y^{\prime}=\frac{y}{\gamma},\quad z^{\prime}=\frac{z}{% \gamma},\quad t^{\prime}=t-x\frac{v}{c^{2}}
  20. γ \gamma
  21. λ \lambda
  22. l = λ l=\sqrt{\lambda}
  23. x = γ l ( x - v t ) , y = l y , z = l z , t = γ l ( t - x v c 2 ) x^{\prime}=\gamma l\left(x-vt\right),\quad y^{\prime}=ly,\quad z^{\prime}=lz,% \quad t^{\prime}=\gamma l\left(t-x\frac{v}{c^{2}}\right)
  24. l = 1 / γ l=1/\gamma
  25. l = 1 l=1
  26. l = 1 l=1
  27. x = γ l ( x - v t ) , y = l y , z = l z , t = γ l ( t - v c 2 x ) x^{\prime}=\gamma l\left(x-vt\right),\ y^{\prime}=ly,\ z^{\prime}=lz,\ t^{% \prime}=\gamma l\left(t-\tfrac{v}{c^{2}}x\right)
  28. E = ( q r r 2 ) ( 1 - v 2 sin 2 θ c 2 ) - 3 / 2 \mathrm{E}=\left(\frac{q\mathrm{r}}{r^{2}}\right)\left(1-\frac{v^{2}\sin^{2}% \theta}{c^{2}}\right)^{-3/2}
  29. x = γ x x^{\prime}=\gamma x
  30. 1 / γ : 1 : 1 1/\gamma:1:1\!
  31. x = γ x * , y = y , z = z , t = t - γ 2 x * v c 2 x^{\prime}=\gamma x^{*},\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t^{\prime}% =t-\gamma^{2}x^{*}\frac{v}{c^{2}}
  32. x = γ x * , y = y , z = z , t = t x^{\prime}=\gamma x^{*},\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t^{\prime}=t
  33. x = x * , y = y , z = z , t = t - x * v c 2 x^{\prime}=x^{*},\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t^{\prime}=t-x^{*% }\frac{v}{c^{2}}
  34. x = x - v t , y = y , z = z , t = t - γ 2 v x * / c 2 x^{\prime}=x-vt,\quad y^{\prime}=y^{\prime},\quad z^{\prime}=z,\quad t^{\prime% }=t-\gamma^{2}vx^{*}/c^{2}
  35. x = γ x * , y = y , z = z , t = t γ - γ x * v c 2 x^{\prime}=\gamma x^{*},\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t^{\prime}% =\frac{t}{\gamma}-\gamma x^{*}\frac{v}{c^{2}}
  36. l l
  37. v v
  38. S S
  39. l γ l\gamma
  40. S 0 S_{0}
  41. S 0 S_{0}
  42. x x^{\ast}
  43. x - v t x-vt
  44. x = γ l x , y = l y , z = l z , t = l γ t - γ l x v c 2 x^{\prime}=\gamma lx^{\ast},\quad y^{\prime}=ly,\quad z^{\prime}=lz,\quad t^{% \prime}=\frac{l}{\gamma}t-\gamma lx^{\ast}\frac{v}{c^{2}}
  45. l = 1 l=1
  46. v = 0 v=0
  47. l = 1 l=1
  48. l l
  49. v / c v/c
  50. δ t o = x * ( c - v ) \delta t_{o}=\frac{x^{*}}{\left(c-v\right)}
  51. δ t b = x * ( c + v ) \delta t_{b}=\frac{x^{*}}{\left(c+v\right)}\cdot
  52. t * = t - ϵ v x * c 2 t^{*}=t-\frac{\epsilon vx^{*}}{c^{2}}\cdot
  53. v 2 c 2 1. \frac{v^{2}}{c^{2}}\ll 1.\,
  54. x = γ l ( x - v t ) , y = l y , z = l z , t = γ l ( t - v x ) x^{\prime}=\gamma l(x-vt),\quad y^{\prime}=ly,\quad z^{\prime}=lz,\quad t^{% \prime}=\gamma l\left(t-vx\right)
  55. l = 1 l=1
  56. x = γ ( x - v t ) , y = y , z = z , t = γ ( t - x v c 2 ) x^{\prime}=\gamma(x-vt),\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t^{\prime}% =\gamma\left(t-x\frac{v}{c^{2}}\right)
  57. x , y , z , i t x,y,z,it
  58. x 1 , x 2 , x 3 , x 4 x_{1},x_{2},x_{3},x_{4}
  59. ψ \psi
  60. z z
  61. x 1 = x 1 , x 2 = x 2 , x 3 = x 3 cos i ψ + x 4 sin i ψ , x 4 = - x 3 sin i ψ + x 4 cos i ψ , x^{\prime}_{1}=x_{1},\quad x^{\prime}_{2}=x_{2},\quad x^{\prime}_{3}=x_{3}\cos i% \psi+x_{4}\sin i\psi,\quad x^{\prime}_{4}=-x_{3}\sin i\psi+x_{4}\cos i\psi,
  62. cos i ψ = 1 / 1 - v 2 \cos i\psi=1/\sqrt{1-v^{2}}
  63. c = 1 c=1
  64. x = p ( x - v t ) , y = y , z = z , t = p ( t - n v x ) x^{\prime}=p(x-vt),\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t^{\prime}=p(t-nvx)
  65. p = 1 / 1 - n v 2 p=1/\sqrt{1-nv^{2}}
  66. n n
  67. x / γ x/\gamma
  68. n = 1 / c 2 n=1/c^{2}
  69. p = γ p=\gamma
  70. n = 0 n=0

History_of_mathematical_notation.html

  1. Δ y \Delta^{y}
  2. K y K^{y}
  3. Δ y Δ \Delta^{y}\Delta
  4. Δ K y \Delta K^{y}
  5. π \pi
  6. e e
  7. d d
  8. S S
  9. Δ \Delta
  10. e e
  11. π \pi
  12. π \pi
  13. i i
  14. f ( x ) f(x)
  15. x x
  16. \aleph
  17. \land
  18. \lor
  19. ¬ \lnot
  20. a ¬ a = 1 a\lor\lnot a=1
  21. \langle
  22. \rangle
  23. a 2 + b 2 = c 2 a^{2}+b^{2}=c^{2}
  24. 2 x 4 + 3 x 3 - 4 x 2 + 5 x - 6 2x^{4}+3x^{3}-4x^{2}+5x-6
  25. \sqrt{~{}}
  26. d = ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 , d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}},\!
  27. {\infty}
  28. 1 . \frac{1}{\infty}.
  29. x ˙ \dot{x}
  30. x ¨ \ddot{x}
  31. d x d t {dx\over dt}
  32. - N N f ( x ) d x \int_{-N}^{N}f(x)\,dx
  33. f ( x ) f(x)
  34. f f
  35. x x
  36. f [ f ( x ) ] f[f(x)]
  37. f 2 ( x ) f^{2}(x)
  38. x x
  39. f f
  40. f - 1 ( x ) f^{-1}(x)
  41. f f
  42. f - 1 f^{-1}
  43. x x
  44. f ( x ) f(x)
  45. f - 1 ( x ) f^{-1}(x)
  46. Δ f ( p ) \Delta f(p)
  47. - 1 \sqrt{-1}
  48. \infty
  49. n = 1 1 n 2 \sum_{n=1}^{\infty}\frac{1}{n^{2}}
  50. n ! = 0 1 ( - ln s ) n d s n!=\int_{0}^{1}(-\ln s)^{n}\,{\rm d}s\,
  51. \nabla
  52. i 2 = j 2 = k 2 = i j k = - 1 i^{2}=j^{2}=k^{2}=ijk=-1
  53. t = w + x i + y j + z k , w , x , y , z t=w+xi+yj+zk,\quad w,x,y,z\in\mathbb{R}
  54. i j = j i = k , i 2 = - 1 , j 2 = + 1. ij=ji=k,\quad i^{2}=-1,\quad j^{2}=+1.
  55. q = w + x i + y j + z k q=w+xi+yj+zk\!
  56. y = i = 1 3 c i x i = c 1 x 1 + c 2 x 2 + c 3 x 3 y=\sum_{i=1}^{3}c_{i}x^{i}=c_{1}x^{1}+c_{2}x^{2}+c_{3}x^{3}
  57. y = c i x i . y=c_{i}x^{i}\,.
  58. = ( \Alpha , Ω , \Zeta , \Iota ) \mathcal{L}=\mathcal{L}\ (\Alpha,\ \Omega,\ \Zeta,\ \Iota)
  59. \Alpha \Alpha
  60. Ω \Omega
  61. \lor
  62. \land
  63. \Zeta \Zeta
  64. \Iota \Iota
  65. ( x ) ( x = ¬ y ) (\exists x)(x=\lnot y)
  66. { 8 , 4 , 11 , 9 , 8 , 11 , 5 , 1 , 13 , 9 } \{8,4,11,9,8,11,5,1,13,9\}
  67. { 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 } \{2,3,5,7,11,13,17,19,23,29\}
  68. 2 8 × 3 4 × 5 11 × 7 9 × 11 8 × 13 11 × 17 5 × 19 1 × 23 13 × 29 9 2^{8}\times 3^{4}\times 5^{11}\times 7^{9}\times 11^{8}\times 13^{11}\times 17% ^{5}\times 19^{1}\times 23^{13}\times 29^{9}
  69. 3.096262735 × 10 78 3.096262735\times 10^{78}
  70. ψ = ψ ( 𝐱 , t ) ψ=ψ(\mathbf{x},t)
  71. 𝐱 \mathbf{x}
  72. t t
  73. m m
  74. p p
  75. c c
  76. ħ = h / 2 π ħ=h/2π
  77. ϕ | ψ \langle\phi|\psi\rangle
  78. \scriptstyle\Box

History_of_string_theory.html

  1. N = 1 N=1

History_of_trigonometry.html

  1. chord θ = 2 sin θ 2 , \mathrm{chord}\ \theta=2\sin\frac{\theta}{2},\,
  2. sin 2 ( x 2 ) = 1 - cos ( x ) 2 . \sin^{2}\left(\frac{x}{2}\right)=\frac{1-\cos(x)}{2}.
  3. sin x 16 x ( π - x ) 5 π 2 - 4 x ( π - x ) , ( 0 x π ) . \sin x\approx\frac{16x(\pi-x)}{5\pi^{2}-4x(\pi-x)},\qquad\left(0\leq x\leq\pi% \right).
  4. 1 - sin 2 ( x ) = cos 2 ( x ) = sin 2 ( π 2 - x ) \ 1-\sin^{2}(x)=\cos^{2}(x)=\sin^{2}\left(\frac{\pi}{2}-x\right)
  5. sin ( a + b ) \sin\left(a+b\right)
  6. sin ( a - b ) \sin\left(a-b\right)
  7. sin ( a + b ) = sin a cos b + cos a sin b \sin\left(a+b\right)=\sin a\cos b+\cos a\sin b
  8. sin ( a - b ) = sin a cos b - cos a sin b \sin\left(a-b\right)=\sin a\cos b-\cos a\sin b
  9. sin ( 2 x ) = 2 sin ( x ) cos ( x ) \ \sin(2x)=2\sin(x)\cos(x)
  10. sin ( α ± β ) = sin 2 α - ( sin α sin β ) 2 ± sin 2 β - ( sin α sin β ) 2 \sin(\alpha\pm\beta)=\sqrt{\sin^{2}\alpha-(\sin\alpha\sin\beta)^{2}}\pm\sqrt{% \sin^{2}\beta-(\sin\alpha\sin\beta)^{2}}
  11. sin ( α ± β ) = sin α cos β ± cos α sin β \sin(\alpha\pm\beta)=\sin\alpha\cos\beta\pm\cos\alpha\sin\beta\,
  12. sin A sin a = sin B sin b = sin C sin c . \frac{\sin A}{\sin a}=\frac{\sin B}{\sin b}=\frac{\sin C}{\sin c}.
  13. cos a cos b = cos ( a + b ) + cos ( a - b ) 2 \cos a\cos b=\frac{\cos(a+b)+\cos(a-b)}{2}
  14. s = c + 2 v 2 d . s=c+\frac{2v^{2}}{d}.

History_of_variational_principles_in_physics.html

  1. M v M\sqrt{v}
  2. M d s v M\,ds\sqrt{v}
  3. M d s v \int Mds\sqrt{v}
  4. d s v \int ds\sqrt{v}
  5. v v
  6. M v M\sqrt{v}
  7. M d s v Mds\sqrt{v}
  8. M d s v \int Mds\sqrt{v}
  9. d s v \int ds\sqrt{v}
  10. M d s v \int Mds\sqrt{v}
  11. v v
  12. p d q \int p\,dq
  13. L = T - V L=T-V
  14. 𝒮 = - 1 4 μ 0 d 4 x F α β F α β - d 4 x j α A α \mathcal{S}=-\int\frac{1}{4\mu_{0}}\,\mathrm{d}^{4}x\,F^{\alpha\beta}F_{\alpha% \beta}-\int\mathrm{d}^{4}x\,j^{\alpha}A_{\alpha}
  15. 𝒮 [ g ] = c 4 16 π G R - g d 4 x \mathcal{S}[g]=\frac{c^{4}}{16\pi G}\int_{\mathcal{M}}R\sqrt{-g}\,\mathrm{d}^{% 4}x
  16. g = det ( g α β ) g=\det(g_{\alpha\beta})
  17. R R
  18. F = m a F=ma
  19. F F
  20. m m
  21. a a
  22. x 1 x_{1}
  23. t 1 t_{1}
  24. x 2 x_{2}
  25. t 2 t_{2}

Hitchin_functional.html

  1. M M
  2. Φ ( Ω ) = M Ω * Ω , \Phi(\Omega)=\int_{M}\Omega\wedge*\Omega,
  3. Ω \Omega
  4. M M
  5. Ω \Omega
  6. Ω \Omega
  7. Φ \Phi
  8. [ Ω ] H 3 ( M , R ) [\Omega]\in H^{3}(M,R)
  9. Ω \Omega
  10. Φ \Phi
  11. Ω * Ω < 0 \Omega\wedge*\Omega<0
  12. Ω \Omega
  13. Ω \Omega
  14. M M
  15. Φ ( Ω ) \Phi(\Omega)
  16. M M
  17. M M
  18. ω \omega
  19. ω = d p 1 d q 1 + + d p m d q m \omega=dp_{1}\wedge dq_{1}+\cdots+dp_{m}\wedge dq_{m}
  20. d ω = 0 d\omega=0
  21. ω \omega
  22. ω Ω p ( M , ) \omega\in\Omega^{p}(M,\mathbb{R})
  23. G L ( n , ) GL(n,\mathbb{R})
  24. ω ω + δ ω \omega\mapsto\omega+\delta\omega
  25. G L ( n , ) GL(n,\mathbb{R})
  26. 3 ( n ) \wedge^{3}(\mathbb{R}^{n})
  27. n 3 n^{3}
  28. G L ( n , ) GL(n,\mathbb{R})
  29. n 2 n^{2}
  30. n = 6 n=6
  31. 3 ( 6 ) = 20 \wedge^{3}(\mathbb{R}^{6})=20
  32. G L ( 6 , ) = 36 GL(6,\mathbb{R})=36
  33. ρ \rho
  34. ρ \rho
  35. G L ( 6 , ) GL(6,\mathbb{R})
  36. S L ( 3 , ) × S L ( 3 , ) SL(3,\mathbb{R})\times SL(3,\mathbb{R})
  37. S L ( 3 , ) S L ( 3 , ) SL(3,\mathbb{C})\cap SL(3,\mathbb{C})
  38. S L ( 3 , ) S L ( 3 , ) SL(3,\mathbb{C})\cap SL(3,\mathbb{C})
  39. ρ \rho
  40. S L ( 3 , ) S L ( 3 , ) SL(3,\mathbb{C})\cap SL(3,\mathbb{C})
  41. ρ = 1 2 ( ζ 1 ζ 2 ζ 3 + ζ 1 ¯ ζ 2 ¯ ζ 3 ¯ ) \rho=\frac{1}{2}(\zeta_{1}\wedge\zeta_{2}\wedge\zeta_{3}+\bar{\zeta_{1}}\wedge% \bar{\zeta_{2}}\wedge\bar{\zeta_{3}})
  42. ζ 1 = e 1 + i e 2 , ζ 2 = e 3 + i e 4 , ζ 3 = e 5 + i e 6 \zeta_{1}=e_{1}+ie_{2},\zeta_{2}=e_{3}+ie_{4},\zeta_{3}=e_{5}+ie_{6}
  43. e i e_{i}
  44. T * M T^{*}M
  45. ζ i \zeta_{i}
  46. M M
  47. ( z 1 , z 2 , z 3 ) (z_{1},z_{2},z_{3})
  48. ζ i = d z i \zeta_{i}=dz_{i}
  49. M M
  50. ρ Ω 3 ( M , ) \rho\in\Omega^{3}(M,\mathbb{R})
  51. ρ = 1 2 ( ζ 1 ζ 2 ζ 3 + ζ 1 ¯ ζ 2 ¯ ζ 3 ¯ ) \rho=\frac{1}{2}(\zeta_{1}\wedge\zeta_{2}\wedge\zeta_{3}+\bar{\zeta_{1}}\wedge% \bar{\zeta_{2}}\wedge\bar{\zeta_{3}})
  52. ρ ~ ( ρ ) = 1 2 ( ζ 1 ζ 2 ζ 3 - ζ 1 ¯ ζ 2 ¯ ζ 3 ¯ ) \tilde{\rho}(\rho)=\frac{1}{2}(\zeta_{1}\wedge\zeta_{2}\wedge\zeta_{3}-\bar{% \zeta_{1}}\wedge\bar{\zeta_{2}}\wedge\bar{\zeta_{3}})
  53. Ω = ρ + i ρ ~ ( ρ ) \Omega=\rho+i\tilde{\rho}(\rho)
  54. ρ \rho
  55. d Ω = 0 d\Omega=0
  56. d ρ = 0 d\rho=0
  57. d ρ ~ ( ρ ) = 0 d\tilde{\rho}(\rho)=0
  58. Ω \Omega
  59. Ω \Omega
  60. κ \kappa
  61. ν \nu
  62. M M
  63. τ \tau
  64. g i j = τ im τ i * ( ν κ τ ) . g_{ij}=\tau\,\text{im}\int\tau i^{*}(\nu\cdot\kappa\tau).
  65. V [ J ] = J J J V[J]=\int J\wedge J\wedge J
  66. G 2 G_{2}
  67. S p i n ( 7 ) Spin(7)
  68. G 2 G_{2}
  69. S p i n ( 7 ) Spin(7)

Hitting_time.html

  1. τ A ( ω ) := inf { t T | X t ( ω ) A } . \tau_{A}(\omega):=\inf\{t\in T|X_{t}(\omega)\in A\}.
  2. τ r \tau_{r}
  3. r > 0 r>0
  4. τ r \tau_{r}
  5. 𝔼 [ τ r ] = r 2 , \mathbb{E}\left[\tau_{r}\right]=r^{2},
  6. Var [ τ r ] = ( 2 / 3 ) r 4 . \mathrm{Var}\left[\tau_{r}\right]=(2/3)r^{4}.
  7. { 0 } \{0\}

Hofmeister_series.html

  1. F - SO 4 2 - > HPO 4 2 - > acetate > Cl - > NO 3 - > Br - > ClO 3 - > I - > ClO 4 - > SCN - \mathrm{F^{-}\approx SO_{4}^{2-}>HPO_{4}^{2-}>acetate>Cl^{-}>NO_{3}^{-}>Br^{-}% >ClO_{3}^{-}>I^{-}>ClO_{4}^{-}>SCN^{-}}
  2. NH 4 + > K + > Na + > Li + > Mg 2 + > Ca 2 + > guanidinium \mathrm{NH_{4}^{+}>K^{+}>Na^{+}>Li^{+}>Mg^{2+}>Ca^{2+}>guanidinium}

Holding_period_return.html

  1. H P R n = I n c o m e + P n + 1 - P n P n HPR_{n}\ =\ \frac{Income+P_{n+1}-P_{n}}{P_{n}}
  2. P n P_{n}
  3. I n c o m e + P n + 1 Income+P_{n+1}
  4. Annualized HPR n = ( I n c o m e + P n + 1 - P n P n + 1 ) 1 t - 1 \mathrm{Annualized\,HPR}_{n}=\left(\frac{Income+P_{n+1}-P_{n}}{P_{n}}+1\right)% ^{\frac{1}{t}}-1
  5. 1 + H P R = ( 1 + H P R 1 ) ( 1 + H P R 2 ) ( 1 + H P R 3 ) ( 1 + H P R 4 ) 1+HPR=\left(1+HPR_{1}\right)\left(1+HPR_{2}\right)\left(1+HPR_{3}\right)\left(% 1+HPR_{4}\right)

Holevo's_theorem.html

  1. ρ = X p X ρ X \rho=\sum_{X}p_{X}\rho_{X}
  2. I ( X : Y ) S ( ρ ) - i p i S ( ρ i ) I(X:Y)\leq S(\rho)-\sum_{i}p_{i}S(\rho_{i})
  3. ρ = i p i ρ i \rho=\sum_{i}p_{i}\rho_{i}
  4. S ( ) S(\cdot)
  5. χ := S ( ρ ) - i p i S ( ρ i ) \chi:=S(\rho)-\sum_{i}p_{i}S(\rho_{i})
  6. P , Q , M P,Q,M
  7. P P
  8. Q Q
  9. M M
  10. P Q M P\otimes Q\otimes M
  11. ρ P Q M := x p x | x x | ρ x | 0 0 | \rho^{PQM}:=\sum_{x}p_{x}|x\rangle\langle x|\otimes\rho_{x}\otimes|0\rangle% \langle 0|
  12. x x
  13. X X
  14. x p x | x x | \sum_{x}p_{x}|x\rangle\langle x|
  15. x p x ρ x \sum_{x}p_{x}\rho_{x}
  16. | 0 |0\rangle
  17. S ( P ; M ) S ( P ; Q ) S(P^{\prime};M^{\prime})\leq S(P;Q)
  18. 2 n - 1 2^{n}-1

Hollomon–Jaffe_parameter.html

  1. H p = ( 273.15 + T ) 1000 ( C + log ( t ) ) H_{p}=\frac{(273.15+T)}{1000}\cdot(C+\log(t))
  2. H p = T ( C + log ( t ) ) H_{p}=T(C+\log(t))\,

Hollow_matrix.html

  1. A n × n = ( a i j ) ; a i j = 0 if i = j , 1 i , j n . \begin{array}[]{rlll}A_{n\times n}&=&(a_{ij});\\ a_{ij}&=&0&\mbox{if}~{}\quad i=j,\quad 1\leq i,j\leq n.\end{array}
  2. ( 0 0 0 0 ) \left(\begin{array}[]{ccccc}0\\ &0\\ &&\ddots\\ &&&0\\ &&&&0\end{array}\right)
  3. ( 0 2 6 1 3 4 2 0 4 8 0 9 4 0 2 933 1 4 4 0 6 7 9 23 8 0 ) \left(\begin{array}[]{ccccc}0&2&6&\frac{1}{3}&4\\ 2&0&4&8&0\\ 9&4&0&2&933\\ 1&4&4&0&6\\ 7&9&23&8&0\end{array}\right)

Holonomic_function.html

  1. 𝕂 \mathbb{K}
  2. 𝕂 = \mathbb{K}=\mathbb{Q}
  3. 𝕂 = \mathbb{K}=\mathbb{C}
  4. f = f ( x ) f=f(x)
  5. a r ( x ) , a r - 1 ( x ) , , a 0 ( x ) 𝕂 [ x ] a_{r}(x),a_{r-1}(x),\ldots,a_{0}(x)\in\mathbb{K}[x]
  6. a r ( x ) f ( r ) ( x ) + a r - 1 ( x ) f ( r - 1 ) ( x ) + + a 1 ( x ) f ( x ) + a 0 ( x ) f ( x ) = 0 a_{r}(x)f^{(r)}(x)+a_{r-1}(x)f^{(r-1)}(x)+\ldots+a_{1}(x)f^{\prime}(x)+a_{0}(x% )f(x)=0
  7. A f = 0 Af=0
  8. A = k = 0 r a k D x k A=\sum_{k=0}^{r}a_{k}D_{x}^{k}
  9. D x D_{x}
  10. f ( x ) f(x)
  11. f ( x ) f^{\prime}(x)
  12. A A
  13. f f
  14. 𝕂 [ x ] [ D x ] \mathbb{K}[x][D_{x}]
  15. f f
  16. c = c 0 , c 1 , c=c_{0},c_{1},\ldots
  17. a r ( n ) , a r - 1 ( n ) , , a 0 ( n ) 𝕂 [ n ] a_{r}(n),a_{r-1}(n),\ldots,a_{0}(n)\in\mathbb{K}[n]
  18. a r ( n ) c n + r + a r - 1 ( n ) c n + r - 1 + + a 0 ( n ) c n = 0 a_{r}(n)c_{n+r}+a_{r-1}(n)c_{n+r-1}+\ldots+a_{0}(n)c_{n}=0
  19. A c = 0 Ac=0
  20. A = k = 0 r a k S n A=\sum_{k=0}^{r}a_{k}S_{n}
  21. S n S_{n}
  22. c 0 , c 1 , c_{0},c_{1},\ldots
  23. c 1 , c 2 , c_{1},c_{2},\ldots
  24. A A
  25. c c
  26. 𝕂 [ n ] [ S n ] \mathbb{K}[n][S_{n}]
  27. c c
  28. f ( x ) f(x)
  29. c n c_{n}
  30. f ( x ) = n = 0 c n x n f(x)=\sum_{n=0}^{\infty}c_{n}x^{n}
  31. c n c_{n}
  32. f ( x ) = n = 0 f n x n f(x)=\sum_{n=0}^{\infty}f_{n}x^{n}
  33. g ( x ) = n = 0 g n x n g(x)=\sum_{n=0}^{\infty}g_{n}x^{n}
  34. h ( x ) = α f ( x ) + β g ( x ) h(x)=\alpha f(x)+\beta g(x)
  35. α \alpha
  36. β \beta
  37. h ( x ) = f ( x ) g ( x ) h(x)=f(x)g(x)
  38. h ( x ) = n = 0 f n g n x n h(x)=\sum_{n=0}^{\infty}f_{n}g_{n}x^{n}
  39. h ( x ) = 0 x f ( t ) d t h(x)=\int_{0}^{x}f(t)dt
  40. h ( x ) = n = 0 ( k = 0 n f k ) x n h(x)=\sum_{n=0}^{\infty}(\sum_{k=0}^{n}f_{k})x^{n}
  41. h ( x ) = f ( a ( x ) ) h(x)=f(a(x))
  42. a ( x ) a(x)
  43. a ( f ( x ) ) a(f(x))
  44. f f
  45. g g
  46. h h
  47. sin ( x ) \sin(x)
  48. cos ( x ) \cos(x)
  49. e x e^{x}
  50. log ( x ) . \log(x).
  51. F q p ( a 1 , , a p , b 1 , , b q , x ) {}_{p}F_{q}(a_{1},\ldots,a_{p},b_{1},\ldots,b_{q},x)
  52. x x
  53. a i a_{i}
  54. b i b_{i}
  55. x x
  56. erf ( x ) \operatorname{erf}(x)
  57. J n ( x ) J_{n}(x)
  58. Y n ( x ) Y_{n}(x)
  59. I n ( x ) I_{n}(x)
  60. K n ( x ) K_{n}(x)
  61. Ai ( x ) \operatorname{Ai}(x)
  62. Bi ( x ) \operatorname{Bi}(x)
  63. P n ( x ) P_{n}(x)
  64. T n ( x ) T_{n}(x)
  65. U n ( x ) U_{n}(x)
  66. F n F_{n}
  67. n ! n!
  68. ( n k ) {n\choose k}
  69. H n = k = 1 n 1 k H_{n}=\sum_{k=1}^{n}\frac{1}{k}
  70. H n , m = k = 1 n 1 k m H_{n,m}=\sum_{k=1}^{n}\frac{1}{k^{m}}
  71. x e x - 1 \frac{x}{e^{x}-1}
  72. log ( n ) \log(n)
  73. n α n^{\alpha}
  74. α \alpha\not\in\mathbb{Z}

Homogeneous_(large_cardinal_property).html

  1. 𝒫 = n ( D ) \mathcal{P}_{=n}(D)
  2. 𝒫 = n ( S ) \mathcal{P}_{=n}(S)

Homogeneous_differential_equation.html

  1. f ( x ) f(x)
  2. n n
  3. λ \lambda
  4. x x
  5. λ x \lambda x
  6. f ( λ x ) = λ n f ( x ) . f(\lambda x)=\lambda^{n}f(x)\,.
  7. f ( x , y ) f(x,y)
  8. n n
  9. x x
  10. y y
  11. λ x \lambda x
  12. λ y \lambda y
  13. f ( λ x , λ y ) = λ n f ( x , y ) . f(\lambda x,\lambda y)=\lambda^{n}f(x,y)\,.
  14. f ( x , y ) = ( 2 x 2 - 3 y 2 + 4 x y ) f(x,y)=(2x^{2}-3y^{2}+4xy)
  15. f ( λ x , λ y ) = [ 2 ( λ x ) 2 - 3 ( λ y ) 2 + 4 ( λ x λ y ) ] = ( 2 λ 2 x 2 - 3 λ 2 y 2 + 4 λ 2 x y ) = λ 2 ( 2 x 2 - 3 y 2 + 4 x y ) = λ 2 f ( x , y ) . f(\lambda x,\lambda y)=[2(\lambda x)^{2}-3(\lambda y)^{2}+4(\lambda x\lambda y% )]=(2\lambda^{2}x^{2}-3\lambda^{2}y^{2}+4\lambda^{2}xy)=\lambda^{2}(2x^{2}-3y^% {2}+4xy)=\lambda^{2}f(x,y).
  16. M ( x , y ) d x + N ( x , y ) d y = 0 M(x,y)\,dx+N(x,y)\,dy=0
  17. λ \lambda
  18. M ( λ x , λ y ) = λ n M ( x , y ) . M(\lambda x,\lambda y)=\lambda^{n}M(x,y)\,.
  19. N ( λ x , λ y ) = λ n N ( x , y ) . N(\lambda x,\lambda y)=\lambda^{n}N(x,y)\,.
  20. M ( λ x , λ y ) N ( λ x , λ y ) = M ( x , y ) N ( x , y ) . \frac{M(\lambda x,\lambda y)}{N(\lambda x,\lambda y)}=\frac{M(x,y)}{N(x,y)}\,.
  21. M ( t x , t y ) N ( t x , t y ) = M ( x , y ) N ( x , y ) \frac{M(tx,ty)}{N(tx,ty)}=\frac{M(x,y)}{N(x,y)}
  22. t = 1 / x t=1/x
  23. f f
  24. y / x y/x
  25. M ( x , y ) N ( x , y ) = M ( t x , t y ) N ( t x , t y ) = M ( 1 , y / x ) N ( 1 , y / x ) = f ( y / x ) . \frac{M(x,y)}{N(x,y)}=\frac{M(tx,ty)}{N(tx,ty)}=\frac{M(1,y/x)}{N(1,y/x)}=f(y/% x)\,.
  26. y = u x y=ux
  27. d ( u x ) d x = x d u d x + u d x d x = x d u d x + u , \frac{d(ux)}{dx}=x\frac{du}{dx}+u\frac{dx}{dx}=x\frac{du}{dx}+u,
  28. x d u d x = f ( u ) - u ; x\frac{du}{dx}=f(u)-u\,;
  29. ( a x + b y + c ) d x + ( e x + f y + g ) d y = 0 , (ax+by+c)dx+(ex+fy+g)dy=0\,,
  30. α \alpha
  31. β \beta
  32. t = x + α ; z = y + β . t=x+\alpha;\,\,\,\,z=y+\beta\,.
  33. ϕ ( x ) \phi(x)
  34. c ϕ ( x ) c\phi(x)
  35. c c
  36. L ( y ) = 0 L(y)=0\,
  37. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  38. f i f_{i}
  39. L = i = 0 n f i ( x ) d i d x i ; L=\sum_{i=0}^{n}f_{i}(x)\frac{d^{i}}{dx^{i}}\,;
  40. f i f_{i}
  41. f i f_{i}
  42. sin ( x ) d 2 y d x 2 + 4 d y d x + y = 0 , \sin(x)\frac{d^{2}y}{dx^{2}}+4\frac{dy}{dx}+y=0\,,
  43. 2 x 2 d 2 y d x 2 + 4 x d y d x + y = cos ( x ) ; 2x^{2}\frac{d^{2}y}{dx^{2}}+4x\frac{dy}{dx}+y=\cos(x)\,;
  44. 2 x 2 d 2 y d x 2 - 3 x d y d x + y = 2 . 2x^{2}\frac{d^{2}y}{dx^{2}}-3x\frac{dy}{dx}+y=2\,.

Homology_manifold.html

  1. H p ( X , X - x , G ) H_{p}(X,X-x,G)

Hopcroft–Karp_algorithm.html

  1. O ( | E | | V | ) O(|E|\sqrt{|V|})
  2. E E
  3. V V
  4. O ( | V | 2.5 ) O(|V|^{2.5})
  5. O ( n ) O(\sqrt{n})
  6. M M
  7. M M
  8. P P
  9. M M
  10. M P M\oplus P
  11. | M | + 1 |M|+1
  12. M M
  13. P P
  14. M M * M\oplus M^{*}
  15. M * M^{*}
  16. P P
  17. M M
  18. M * M^{*}
  19. P P
  20. M M
  21. G ( U V , E ) G(U\cup V,E)
  22. M E M\subseteq E
  23. M M\leftarrow
  24. 𝒫 { P 1 , P 2 , , P k } \mathcal{P}\leftarrow\{P_{1},P_{2},\dots,P_{k}\}
  25. M M ( P 1 P 2 P k ) M\leftarrow M\oplus(P_{1}\cup P_{2}\cup\dots\cup P_{k})
  26. 𝒫 = \mathcal{P}=
  27. U U
  28. V V
  29. G G
  30. U U
  31. V V
  32. M M
  33. U U
  34. U U
  35. U U
  36. V V
  37. k k
  38. V V
  39. V V
  40. k k
  41. F F
  42. v v
  43. F F
  44. k k
  45. F F
  46. U U
  47. F F
  48. M M
  49. | V | \sqrt{|V|}
  50. | V | |V|
  51. | E | |E|
  52. O ( | E | | V | ) O(|E|\sqrt{|V|})
  53. | V | \sqrt{|V|}
  54. | V | \sqrt{|V|}
  55. | V | \sqrt{|V|}
  56. | V | \sqrt{|V|}
  57. M M
  58. | V | \sqrt{|V|}
  59. | V | \sqrt{|V|}
  60. 2 | V | 2\sqrt{|V|}
  61. O ( | E | | V | ) O(|E|\sqrt{|V|})
  62. O ( log | V | ) O(\log|V|)
  63. O ( | E | log | V | ) O(|E|\log|V|)
  64. O ( n 1.5 m log n ) O\left(n^{1.5}\sqrt{\frac{m}{\log n}}\right)
  65. O ( | V | ) O(\sqrt{|V|})

Hopf_manifold.html

  1. ( n \ 0 ) ({\mathbb{C}}^{n}\backslash 0)
  2. Γ \Gamma\cong{\mathbb{Z}}
  3. γ \gamma
  4. Γ \Gamma
  5. γ : n n \gamma:\;{\mathbb{C}}^{n}\mapsto{\mathbb{C}}^{n}
  6. γ N \;\gamma^{N}
  7. n {\mathbb{C}}^{n}
  8. Γ \Gamma
  9. q I d q\cdot Id
  10. q q\in{\mathbb{C}}
  11. 0 < | q | < 1 0<|q|<1
  12. H := ( n \ 0 ) / H:=({\mathbb{C}}^{n}\backslash 0)/{\mathbb{Z}}
  13. S 2 n - 1 × S 1 S^{2n-1}\times S^{1}
  14. n 2 n\geq 2

Hopf_surface.html

  1. H = ( 2 \ 0 ) / Γ , H=\bigg({\mathbb{C}}^{2}\backslash 0\bigg)/\Gamma,
  2. Γ \Gamma
  3. γ \gamma
  4. γ \gamma
  5. γ \gamma
  6. ( x , y ) ( α x + λ y n , β y ) (x,y)\mapsto(\alpha x+\lambda y^{n},\beta y)
  7. α , β \alpha,\beta\in{\mathbb{C}}
  8. 0 < | α | | β | < 1 0<|\alpha|\leq|\beta|<1
  9. λ = 0 \;\lambda=0
  10. α = β n \;\alpha=\beta^{n}

Horizontal_eccentricity.html

  1. arctan ( d 2 - d 1 d 1 ) - arctan ( d 2 - d 1 d 2 ) . \arctan\left(\frac{d_{2}-d_{1}}{d_{1}}\right)-\arctan\left(\frac{d_{2}-d_{1}}{% d_{2}}\right).

Hot_bulb_engine.html

  1. p V T = k \qquad\frac{pV}{T}=k

Householder_operator.html

  1. V V\,
  2. u V u\in V
  3. H u : V V H_{u}:V\to V\,
  4. H u ( x ) = x - 2 x , u u H_{u}(x)=x-2\langle x,u\rangle u\,
  5. , \langle\cdot,\cdot\rangle
  6. V V\,
  7. x x
  8. u u

HRU_(security).html

  1. s s
  2. o o
  3. ( S , O , P ) (S,O,P)
  4. S S
  5. O O
  6. P P
  7. s s
  8. o o
  9. R R
  10. ( s , o ) (s,o)

Hubbard–Stratonovich_transformation.html

  1. exp { - a 2 x 2 } = 1 2 π a - exp [ - y 2 2 a - i x y ] d y , \exp\left\{-\frac{a}{2}x^{2}\right\}=\sqrt{\frac{1}{2\pi a}}\;\int_{-\infty}^{% \infty}\exp\left[-\frac{y^{2}}{2a}-ixy\right]\,dy,
  2. a > 0 a>0

Hubble_volume.html

  1. c / H 0 c/H_{0}
  2. c c
  3. H 0 H_{0}
  4. ( c / H 0 ) 3 (c/H_{0})^{3}
  5. c / H 0 c/H_{0}
  6. c c
  7. 1 / H 0 1/H_{0}
  8. 1 / H 0 1/H_{0}

Hurst_exponent.html

  1. E [ R ( n ) S ( n ) ] = C n H as n , \operatorname{E}\left[\frac{R(n)}{S(n)}\right]=Cn^{H}\,\text{ as }n\to\infty\,,
  2. R ( n ) R(n)
  3. n n
  4. S ( n ) S(n)
  5. E [ x ] \operatorname{E}\left[x\right]\,
  6. n n
  7. C C
  8. n n
  9. X = X 1 , X 2 , , X n X=X_{1},X_{2},\dots,X_{n}\,
  10. m = 1 n i = 1 n X i . m=\frac{1}{n}\sum_{i=1}^{n}X_{i}\,.
  11. Y t = X t - m for t = 1 , 2 , , n . Y_{t}=X_{t}-m\quad\,\text{ for }t=1,2,\dots,n\,.
  12. Z Z
  13. Z t = i = 1 t Y i for t = 1 , 2 , , n . Z_{t}=\sum_{i=1}^{t}Y_{i}\quad\,\text{ for }t=1,2,\dots,n\,.
  14. R R
  15. R ( n ) = max ( Z 1 , Z 2 , , Z n ) - min ( Z 1 , Z 2 , , Z n ) . R(n)=\operatorname{max}\left(Z_{1},Z_{2},\dots,Z_{n}\right)-\operatorname{min}% \left(Z_{1},Z_{2},\dots,Z_{n}\right).
  16. S S
  17. S ( n ) = 1 n i = 1 n ( X i - m ) 2 . S(n)=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(X_{i}-m\right)^{2}}.
  18. R ( n ) / S ( n ) R(n)/S(n)
  19. n . n.
  20. E [ R ( n ) S ( n ) ] = C n H \operatorname{E}\left[\frac{R(n)}{S(n)}\right]=Cn^{H}
  21. E [ R ( n ) S ( n ) ] \operatorname{E}\left[\frac{R(n)}{S(n)}\right]
  22. log n \log n
  23. H H
  24. τ \tau
  25. S q = | g ( t + τ ) - g ( t ) | q t τ q H ( q ) , S_{q}=\langle|g(t+\tau)-g(t)|^{q}\rangle_{t}\sim\tau^{qH(q)},\,
  26. τ \tau
  27. t τ , t\gg\tau,\,
  28. H ( q ) H(q)
  29. H ( q ) H(q)

Hydrodynamic_radius.html

  1. R hyd R_{\rm hyd}
  2. N N
  3. R hyd R_{\rm hyd}
  4. 1 R hyd = def 1 N 2 i j 1 r i j \frac{1}{R_{\rm hyd}}\ \stackrel{\mathrm{def}}{=}\ \frac{1}{N^{2}}\left\langle% \sum_{i\neq j}\frac{1}{r_{ij}}\right\rangle
  5. r i j r_{ij}
  6. i i
  7. j j
  8. \langle\ldots\rangle
  9. R hyd R_{\rm hyd}
  10. R hyd R_{\rm hyd}

Hydrophilic-lipophilic_balance.html

  1. H L B = 20 * M h / M HLB=20*M_{h}/M
  2. M h M_{h}
  3. H L B = 7 + i = 1 m H i - n × 0.475 HLB=7+\sum_{i\mathop{=}1}^{m}H_{i}-n\times 0.475
  4. m m
  5. H i H_{i}
  6. i i
  7. n n

Hydrostatic_stress.html

  1. σ h \sigma_{h}
  2. σ h = i = 1 n ρ i g h i \sigma_{h}=\displaystyle\sum_{i=1}^{n}\rho_{i}gh_{i}
  3. i i
  4. ρ i \rho_{i}
  5. g g
  6. h i h_{i}
  7. σ h , s a n d = ρ w g h w = 1000 kg/m 3 9.8 m/s 2 10 m = 9.8 10 4 k g / m s 2 = 9.8 10 4 N / m 2 \sigma_{h,sand}=\rho_{w}gh_{w}=1000\,\,\text{kg/m}^{3}\cdot 9.8\,\,\text{m/s}^% {2}\cdot 10\,\,\text{m}=9.8\cdot{10^{4}}{kg/ms^{2}}=9.8\cdot 10^{4}{N/m^{2}}
  8. w w
  9. σ h I 3 = [ σ h 0 0 0 σ h 0 0 0 σ h ] \sigma_{h}\cdot I_{3}=\left[\begin{array}[]{ccc}\sigma_{h}&0&0\\ 0&\sigma_{h}&0\\ 0&0&\sigma_{h}\end{array}\right]
  10. I 3 I_{3}

Hydrotrope.html

  1. log S 0 S = K C a \log{\frac{S_{0}}{S}}=K\cdot C_{a}

Hydroxymethylglutaryl-CoA_reductase.html

  1. \rightleftharpoons

Hyperbolic_distribution.html

  1. { ( x - μ ) f ′′ ( x ) ( δ 2 + ( x - μ ) 2 ) + f ( x ) ( - δ 2 - 2 β ( x - μ ) ( δ 2 + ( x - μ ) 2 ) ) + f ( x ) ( α 2 μ 3 - β 2 μ ( δ 2 + μ 2 ) + β δ 2 + x 3 ( β 2 - α 2 ) + 3 μ x 2 ( α - β ) ( α + β ) + 3 μ 2 x ( β 2 - α 2 ) + β 2 δ 2 x ) = 0 , f ( 0 ) = γ e α ( - δ 2 + μ 2 ) - β μ 2 α δ K 1 ( γ δ ) , f ( 0 ) = γ e α ( - δ 2 + μ 2 ) - β μ ( α μ + β δ 2 + μ 2 ) 2 α δ δ 2 + μ 2 K 1 ( γ δ ) } \left\{\begin{array}[]{l}(x-\mu)f^{\prime\prime}(x)\left(\delta^{2}+(x-\mu)^{2% }\right)+f^{\prime}(x)\left(-\delta^{2}-2\beta(x-\mu)\left(\delta^{2}+(x-\mu)^% {2}\right)\right)+f(x)\left(\alpha^{2}\mu^{3}-\beta^{2}\mu\left(\delta^{2}+\mu% ^{2}\right)+\beta\delta^{2}+x^{3}\left(\beta^{2}-\alpha^{2}\right)+3\mu x^{2}(% \alpha-\beta)(\alpha+\beta)+3\mu^{2}x\left(\beta^{2}-\alpha^{2}\right)+\beta^{% 2}\delta^{2}x\right)=0,\\ f(0)=\frac{\gamma e^{\alpha\left(-\sqrt{\delta^{2}+\mu^{2}}\right)-\beta\mu}}{% 2\alpha\delta K_{1}(\gamma\delta)},\\ f^{\prime}(0)=\frac{\gamma e^{\alpha\left(-\sqrt{\delta^{2}+\mu^{2}}\right)-% \beta\mu}\left(\alpha\mu+\beta\sqrt{\delta^{2}+\mu^{2}}\right)}{2\alpha\delta% \sqrt{\delta^{2}+\mu^{2}}K_{1}(\gamma\delta)}\end{array}\right\}

Hypercomplex_manifold.html

  1. I , J , K I,J,K
  2. ( \ 0 ) / \bigg({\mathbb{H}}\backslash 0\bigg)/{\mathbb{Z}}
  3. {\mathbb{Z}}
  4. q q
  5. | q | > 1 |q|>1
  6. S 1 × S 3 , S^{1}\times S^{3},
  7. b 2 p + 1 0 m o d 4 \ b_{2p+1}\equiv 0\ mod\ 4
  8. T 4 , S U ( 2 l + 1 ) , T 1 × S U ( 2 l ) , T l × S O ( 2 l + 1 ) , T^{4},SU(2l+1),T^{1}\times SU(2l),T^{l}\times SO(2l+1),
  9. T 2 l × S O ( 4 l ) , T l × S p ( l ) , T 2 × E 6 , T^{2l}\times SO(4l),T^{l}\times Sp(l),T^{2}\times E_{6},
  10. T 7 × E 7 , T 8 × E 8 , T 4 × F 4 , T 2 × G 2 T^{7}\times E^{7},T^{8}\times E^{8},T^{4}\times F_{4},T^{2}\times G_{2}
  11. T i T^{i}
  12. i i
  13. T 4 T^{4}
  14. I , J , K I,J,K
  15. L L\in{\mathbb{H}}
  16. L 2 = - 1 L^{2}=-1
  17. M × S 2 M\times S^{2}
  18. P 1 = S 2 {\mathbb{C}}P^{1}=S^{2}
  19. ( M , L ) (M,L)
  20. M M
  21. {\mathbb{H}}
  22. P 3 \ P 1 {\mathbb{C}}P^{3}\backslash{\mathbb{C}}P^{1}

Hypercone.html

  1. x 2 + y 2 + z 2 - w 2 = 0. x^{2}+y^{2}+z^{2}-w^{2}=0.
  2. σ ( ϕ , θ , t ) = ( t s cos θ cos ϕ , t s cos θ sin ϕ , t s sin θ , t ) \vec{\sigma}(\phi,\theta,t)=(ts\cos\theta\cos\phi,ts\cos\theta\sin\phi,ts\sin% \theta,t)
  3. σ ( ϕ , θ , t ) = ( v x t + t s cos θ cos ϕ , v y t + t s cos θ sin ϕ , v z t + t s sin θ , t ) \vec{\sigma}(\phi,\theta,t)=(v_{x}t+ts\cos\theta\cos\phi,v_{y}t+ts\cos\theta% \sin\phi,v_{z}t+ts\sin\theta,t)
  4. ( v x , v y , v z ) (v_{x},v_{y},v_{z})
  5. w = 0 w=0
  6. w = r w=r
  7. π r 4 / 3 \pi r^{4}/3
  8. x 2 + y 2 + z 2 - ( c t ) 2 = 0 , x^{2}+y^{2}+z^{2}-(ct)^{2}=0,

Hypercube_graph.html

  1. ( n - 4 ) 2 n - 3 + 1 (n-4)2^{n-3}+1
  2. 2 2 n - n - 1 k = 2 n k < m t p l > ( n k ) 2^{2^{n}-n-1}\prod_{k=2}^{n}k^{<}mtpl>{{n\choose k}}
  3. n 2 n \sqrt{n2^{n}}
  4. i = 0 n ( n n / 2 ) \sum_{i=0}^{n}{\left({{n}\atop{\lfloor n/2\rfloor}}\right)}
  5. ( n k ) {\left({{n}\atop{k}}\right)}

Hypercycle_(hyperbolic_geometry).html

  1. L 1 L_{1}
  2. L 2 L_{2}
  3. R 1 R_{1}
  4. R 2 R_{2}
  5. R 1 R_{1}
  6. R 2 R_{2}
  7. L 1 L_{1}
  8. L 2 L_{2}
  9. C 1 C_{1}
  10. C 2 C_{2}
  11. R 1 R_{1}
  12. C 1 C_{1}
  13. C 2 C_{2}
  14. R 2 R_{2}
  15. R 1 R_{1}
  16. R 2 R_{2}
  17. L 1 L_{1}
  18. L 2 L_{2}
  19. C 1 C_{1}
  20. C 2 C_{2}
  21. L 1 L_{1}
  22. L 2 L_{2}
  23. C 1 C_{1}
  24. C 2 C_{2}
  25. R 1 R_{1}
  26. R 2 R_{2}
  27. R 1 R_{1}
  28. R 2 R_{2}

Hypothesis_Theory.html

  1. { 𝐒𝐡𝐚𝐩𝐞 = s q u a r e , 𝐂𝐨𝐥𝐨𝐫 = b l u e , 𝐒𝐢𝐳𝐞 = s m a l l } { 𝐂𝐥𝐚𝐬𝐬 = g o o d } \{\mathbf{Shape}=square,\mathbf{Color}=blue,\mathbf{Size}=small\}% \Longrightarrow\;\{\mathbf{Class}=good\}

Ideal_triangle.html

  1. d = 4 ln φ 1.925 , d=4\ln\varphi\approx 1.925,
  2. φ = 1 + 5 2 \varphi=\frac{1+\sqrt{5}}{2}

Identity_theorem_for_Riemann_surfaces.html

  1. X X
  2. Y Y
  3. f : X Y f:X\to Y
  4. f | A = g | A f|_{A}=g|_{A}
  5. A X A\subseteq X
  6. f | A : A Y f|_{A}:A\to Y
  7. f f
  8. A A
  9. f = g f=g
  10. X X

Identric_mean.html

  1. I ( x , y ) = 1 e lim ( ξ , η ) ( x , y ) ξ ξ η η ξ - η = lim ( ξ , η ) ( x , y ) exp ( ξ ln ξ - η ln η ξ - η - 1 ) = { x if x = y 1 e x x y y x - y else \begin{aligned}\displaystyle I(x,y)&\displaystyle=\frac{1}{e}\cdot\lim_{(\xi,% \eta)\to(x,y)}\sqrt[\xi-\eta]{\frac{\xi^{\xi}}{\eta^{\eta}}}\\ &\displaystyle=\lim_{(\xi,\eta)\to(x,y)}\exp\left(\frac{\xi\cdot\ln\xi-\eta% \cdot\ln\eta}{\xi-\eta}-1\right)\\ &\displaystyle=\begin{cases}x&\,\text{if }x=y\\ \frac{1}{e}\sqrt[x-y]{\frac{x^{x}}{y^{y}}}&\,\text{else}\end{cases}\end{aligned}
  2. x x ln x x\mapsto x\cdot\ln x

Igusa_zeta-function.html

  1. [ K : p ] < [K:\mathbb{Q}_{p}]<\infty
  2. z K z\in K
  3. ord ( z ) \operatorname{ord}(z)
  4. z = q - ord ( z ) \mid z\mid=q^{-\operatorname{ord}(z)}
  5. a c ( z ) = z π - ord ( z ) ac(z)=z\pi^{-\operatorname{ord}(z)}
  6. ϕ : K n \phi:K^{n}\mapsto\mathbb{C}
  7. χ \chi
  8. K * K*
  9. f ( x 1 , , x n ) K [ x 1 , , x n ] f(x_{1},\ldots,x_{n})\in K[x_{1},\ldots,x_{n}]
  10. Z ϕ ( s , χ ) = K n ϕ ( x 1 , , x n ) χ ( a c ( f ( x 1 , , x n ) ) ) | f ( x 1 , , x n ) | s d x Z_{\phi}(s,\chi)=\int_{K^{n}}\phi(x_{1},\ldots,x_{n})\chi(ac(f(x_{1},\ldots,x_% {n})))|f(x_{1},\ldots,x_{n})|^{s}\,dx
  11. s , Re ( s ) > 0 , s\in\mathbb{C},\operatorname{Re}(s)>0,
  12. R n R^{n}
  13. Z ϕ ( s , χ ) Z_{\phi}(s,\chi)
  14. t = q - s t=q^{-s}
  15. P P
  16. ϕ \phi
  17. R n R^{n}
  18. χ \chi
  19. N i N_{i}
  20. f ( x 1 , , x n ) 0 mod P i f(x_{1},\ldots,x_{n})\equiv 0\mod P^{i}
  21. Z ( t ) = R n | f ( x 1 , , x n ) | s d x Z(t)=\int_{R^{n}}|f(x_{1},\ldots,x_{n})|^{s}\,dx
  22. P ( t ) = i = 0 q - i n N i t i P(t)=\sum_{i=0}^{\infty}q^{-in}N_{i}t^{i}
  23. P ( t ) = 1 - t Z ( t ) 1 - t . P(t)=\frac{1-tZ(t)}{1-t}.

Imaginary_line_(mathematics).html

  1. ( x 1 , x 2 , x 3 ) , x i \isin C . (x_{1},\ x_{2},\ x_{3}),\quad x_{i}\isin C.
  2. a 1 x 1 + a 2 x 2 + a 3 x 3 = 0 a_{1}\ x_{1}+\ a_{2}\ x_{2}\ +a_{3}\ x_{3}\ =\ 0

Imaginary_point.html

  1. ( a 1 , a 2 , , a n ) (a_{1},a_{2},\ldots,a_{n})
  2. ( z a 1 , z a 2 , , z a n ) (za_{1},za_{2},\ldots,za_{n})

Imaginary_Thirteen.html

  1. 7 * 2 = 14 7*2=14
  2. 14 - 13 = 1 14-13=1
  3. 8 * 2 = 16 8*2=16
  4. 16 - 13 = 3 16-13=3

Immanant_of_a_matrix.html

  1. λ = ( λ 1 , λ 2 , ) \lambda=(\lambda_{1},\lambda_{2},\ldots)
  2. n n
  3. χ λ \chi_{\lambda}
  4. S n S_{n}
  5. n × n n\times n
  6. A = ( a i j ) A=(a_{ij})
  7. χ λ \chi_{\lambda}
  8. Imm λ ( A ) = σ S n χ λ ( σ ) a 1 σ ( 1 ) a 2 σ ( 2 ) a n σ ( n ) . {\rm Imm}_{\lambda}(A)=\sum_{\sigma\in S_{n}}\chi_{\lambda}(\sigma)a_{1\sigma(% 1)}a_{2\sigma(2)}\cdots a_{n\sigma(n)}.
  9. χ λ \chi_{\lambda}
  10. sgn \operatorname{sgn}
  11. χ λ \chi_{\lambda}
  12. 3 × 3 3\times 3
  13. S 3 S_{3}
  14. S 3 S_{3}
  15. e e
  16. ( 1 2 ) (1\ 2)
  17. ( 1 2 3 ) (1\ 2\ 3)
  18. χ 1 \chi_{1}
  19. χ 2 \chi_{2}
  20. χ 3 \chi_{3}
  21. χ 1 \chi_{1}
  22. χ 2 \chi_{2}
  23. χ 3 \chi_{3}
  24. ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ) 2 a 11 a 22 a 33 - a 12 a 23 a 31 - a 13 a 21 a 32 \begin{pmatrix}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{pmatrix}\rightsquigarrow 2a_{11}a_{22}a_{33}-a_{12}a_% {23}a_{31}-a_{13}a_{21}a_{32}

Immersion_(mathematics).html

  1. D p f : T p M T f ( p ) N D_{p}f:T_{p}M\to T_{f(p)}N\,
  2. rank D p f = dim M . \operatorname{rank}\,D_{p}f=\dim M.

Impedance_of_free_space.html

  1. Z 0 = E H = μ 0 c 0 = μ 0 ε 0 = 1 ε 0 c 0 Z_{0}=\frac{E}{H}=\mu_{0}c_{0}=\sqrt{\frac{\mu_{0}}{\varepsilon_{0}}}=\frac{1}% {\varepsilon_{0}c_{0}}
  2. μ 0 = \mu_{0}=
  3. ε 0 = \varepsilon_{0}=
  4. c 0 = c_{0}=
  5. Z 0 Z_{0}
  6. Y 0 Y_{0}
  7. Z 0 = μ 0 c 0 = 119.9169832 π Ω Z_{0}=\mu_{0}c_{0}=119.9169832\;\pi\ \Omega
  8. Z 0 376.730 313 461 77 Ω Z_{0}\approx 376.730\ 313\ 461\ 77\ldots\Omega
  9. 120 π 120\pi
  10. Z 0 Z_{0}
  11. R r 80 π 2 ( λ ) 2 R_{r}\approx 80\pi^{2}\left(\frac{\ell}{\lambda}\right)^{2}
  12. R r = 2 π 3 Z 0 ( λ ) 2 R_{r}=\frac{2\pi}{3}Z_{0}\left(\frac{\ell}{\lambda}\right)^{2}

Implicational_propositional_calculus.html

  1. \rightarrow\!
  2. Γ A \Gamma\vdash A
  3. Γ A , \Gamma\vdash A,
  4. σ ( Γ ) σ ( A ) , \sigma(\Gamma)\vdash\sigma(A),
  5. Γ , A B \Gamma,A\vdash B
  6. Γ A B . \Gamma\vdash A\to B.
  7. Γ A \Gamma\vdash A
  8. A C , ( A B ) C , C B C A\to C,(A\to B)\to C,C\to B\vdash C
  9. A C , ( A B ) C ( C B ) C A\to C,(A\to B)\to C\vdash(C\to B)\to C
  10. ( C F ) F ( ( B C ) F ) F (C\to F)\to F\vdash((B\to C)\to F)\to F
  11. B F ( ( B C ) F ) F . B\to F\vdash((B\to C)\to F)\to F.
  12. ( B F ) F , C F , B C \displaystyle(B\to F)\to F,C\to F,B\to C
  13. ( B F ) F , C F ( B C ) F (B\to F)\to F,C\to F\vdash(B\to C)\to F
  14. p 1 e ( p 1 ) , , p k e ( p k ) ( p k + 1 F ) A 1 , p 1 e ( p 1 ) , , p k e ( p k ) ( ( p k + 1 F ) F ) A 1 , \begin{aligned}\displaystyle p_{1}^{e(p_{1})},\dots,p_{k}^{e(p_{k})}&% \displaystyle\vdash(p_{k+1}\to F)\to A^{1},\\ \displaystyle p_{1}^{e(p_{1})},\dots,p_{k}^{e(p_{k})}&\displaystyle\vdash((p_{% k+1}\to F)\to F)\to A^{1},\end{aligned}

Implicit_solvation.html

  1. Δ G solv = i σ i A S A i \Delta G_{\mathrm{solv}}=\sum_{i}\sigma_{i}\ ASA_{i}
  2. A S A i ASA_{i}
  3. σ i \sigma_{i}
  4. [ ϵ ( r ) Ψ ( r ) ] = - 4 π ρ f ( r ) - 4 π i c i z i q λ ( r ) e - z i q Ψ ( r ) k T \vec{\nabla}\cdot\left[\epsilon(\vec{r})\vec{\nabla}\Psi(\vec{r})\right]=-4\pi% \rho^{f}(\vec{r})-4\pi\sum_{i}c_{i}^{\infty}z_{i}q\lambda(\vec{r})e^{\frac{-z_% {i}q\Psi(\vec{r})}{kT}}
  5. [ ϵ ( r ) Ψ ( r ) ] = - ρ f ( r ) - i c i z i q λ ( r ) e - z i q Ψ ( r ) k T \vec{\nabla}\cdot\left[\epsilon(\vec{r})\vec{\nabla}\Psi(\vec{r})\right]=-\rho% ^{f}(\vec{r})-\sum_{i}c_{i}^{\infty}z_{i}q\lambda(\vec{r})e^{\frac{-z_{i}q\Psi% (\vec{r})}{kT}}
  6. ϵ ( r ) \epsilon(\vec{r})
  7. Ψ ( r ) \Psi(\vec{r})
  8. ρ f ( r ) \rho^{f}(\vec{r})
  9. c i c_{i}^{\infty}
  10. z i z_{i}
  11. λ ( r ) \lambda(\vec{r})
  12. G s = 1 8 π ( 1 ϵ 0 - 1 ϵ ) i , j N q i q j f G B G_{s}=\frac{1}{8\pi}\left(\frac{1}{\epsilon_{0}}-\frac{1}{\epsilon}\right)\sum% _{i,j}^{N}\frac{q_{i}q_{j}}{f_{GB}}
  13. f G B = r i j 2 + a i j 2 e - D f_{GB}=\sqrt{r_{ij}^{2}+a_{ij}^{2}e^{-D}}
  14. D = ( r i j 2 a i j ) 2 , a i j = a i a j D=\left(\frac{r_{ij}}{2a_{ij}}\right)^{2},a_{ij}=\sqrt{a_{i}a_{j}}
  15. ϵ 0 \epsilon_{0}
  16. ϵ \epsilon
  17. q i q_{i}
  18. r i j r_{ij}
  19. a i a_{i}
  20. Δ G i s o l v = Δ G i r e f - j V j f i ( r ) d r \Delta G_{i}^{solv}=\Delta G_{i}^{ref}-\sum_{j}\int_{Vj}f_{i}(r)dr

Import.html

  1. N X NX
  2. N X = X - I NX=X-I
  3. I = X - N X I=X-NX
  4. I I
  5. A A
  6. σ \sigma
  7. I = I ( A , σ ) I=I(A,\sigma)

Incremental_capital-output_ratio.html

  1. incremental capital output ratio = Δ K Δ Y = Δ K Y Δ Y Y = I Y Δ Y Y \,\text{incremental capital output ratio}=\frac{\Delta K}{\Delta Y}=\frac{% \frac{\Delta K}{Y}}{\frac{\Delta Y}{Y}}=\frac{\frac{I}{Y}}{\frac{\Delta Y}{Y}}

Indecomposable_distribution.html

  1. X = { 1 with probability p , 0 with probability 1 - p , X=\begin{cases}1&\,\text{with probability }p,\\ 0&\,\text{with probability }1-p,\end{cases}
  2. X = { 2 with probability a , 1 with probability b , 0 with probability c . X=\begin{cases}2&\,\text{with probability }a,\\ 1&\,\text{with probability }b,\\ 0&\,\text{with probability }c.\end{cases}
  3. a + c 1 \sqrt{a}+\sqrt{c}\leq 1
  4. U = { 1 with probability p , 0 with probability 1 - p , and V = { 1 with probability q , 0 with probability 1 - q , \begin{matrix}U=\begin{cases}1&\,\text{with probability }p,\\ 0&\,\text{with probability }1-p,\end{cases}&\mbox{and}&V=\begin{cases}1&\,% \text{with probability }q,\\ 0&\,\text{with probability }1-q,\end{cases}\end{matrix}
  5. a = p q , a=pq,\,
  6. c = ( 1 - p ) ( 1 - q ) , c=(1-p)(1-q),\,
  7. b = 1 - a - c . b=1-a-c.\,
  8. a + c 1. \sqrt{a}+\sqrt{c}\leq 1.
  9. f ( x ) = 1 2 π x 2 e - x 2 / 2 f(x)={1\over\sqrt{2\pi\,}}x^{2}e^{-x^{2}/2}
  10. n = 1 X n 2 n , \sum_{n=1}^{\infty}{X_{n}\over 2^{n}},
  11. Pr ( Y = y ) = ( 1 - p ) n p \Pr(Y=y)=(1-p)^{n}p\,
  12. Y = n = 1 D n 2 n , Y=\sum_{n=1}^{\infty}{D_{n}\over 2^{n}},

Indefinite_inner_product_space.html

  1. ( K , , , J ) (K,\langle\cdot,\,\cdot\rangle,J)
  2. K K
  3. , \langle\cdot,\,\cdot\rangle\,
  4. ( x , y ) = def x , J y , (x,\,y)\ \stackrel{\mathrm{def}}{=}\ \langle x,\,Jy\rangle,
  5. J J
  6. K K
  7. J 3 = J . J^{3}=J.\,
  8. K K
  9. J J
  10. ( x , y ) (x,\,y)
  11. K K
  12. K K
  13. , \langle\cdot,\,\cdot\rangle
  14. K 0 = def { x K : x , x = 0 } K_{0}\ \stackrel{\mathrm{def}}{=}\ \{x\in K:\langle x,\,x\rangle=0\}
  15. K + + = def { x K : x , x > 0 } K_{++}\ \stackrel{\mathrm{def}}{=}\ \{x\in K:\langle x,\,x\rangle>0\}
  16. K - - = def { x K : x , x < 0 } K_{--}\ \stackrel{\mathrm{def}}{=}\ \{x\in K:\langle x,\,x\rangle<0\}
  17. K + 0 = def K + + K 0 K_{+0}\ \stackrel{\mathrm{def}}{=}\ K_{++}\cup K_{0}
  18. K - 0 = def K - - K 0 K_{-0}\ \stackrel{\mathrm{def}}{=}\ K_{--}\cup K_{0}
  19. L K L\subset K
  20. K 0 K_{0}
  21. K + 0 K_{+0}
  22. K - 0 K_{-0}
  23. K + + { 0 } K_{++}\cup\{0\}
  24. K - - { 0 } K_{--}\cup\{0\}
  25. K = K + K - K=K_{+}\oplus K_{-}
  26. K K
  27. P ± P_{\pm}
  28. K ± K_{\pm}
  29. K K_{\mp}
  30. i i
  31. K + K + 0 K_{+}\subset K_{+0}
  32. K - K - 0 K_{-}\subset K_{-0}
  33. K K
  34. K ± K ± ± { 0 } K_{\pm}\subset K_{\pm\pm}\cup\{0\}
  35. K K
  36. K K
  37. J = def P + - P - J\ \stackrel{\mathrm{def}}{=}\ P_{+}-P_{-}
  38. ( , ) (\cdot,\,\cdot)
  39. ( x , y ) = def x , J y = x , P + y - x , P - y (x,\,y)\ \stackrel{\mathrm{def}}{=}\ \langle x,\,Jy\rangle=\langle x,\,P_{+}y% \rangle-\langle x,\,P_{-}y\rangle
  40. K K
  41. K p m K ± 0 K_{p}m\subset K_{\pm 0}
  42. K 0 K_{0}
  43. K 0 K ± K_{0}\cap K_{\pm}
  44. k K 0 K ± k\in K_{0}\cap K_{\pm}
  45. ( k , k ) = def k , J k = ± k , k = 0 (k,\,k)\ \stackrel{\mathrm{def}}{=}\ \langle k,\,Jk\rangle=\pm\langle k,\,k% \rangle=0
  46. k = k + + k - k=k_{+}+k_{-}
  47. k ± K ± k_{\pm}\in K_{\pm}
  48. K 0 K_{0}
  49. k - , k - = - k + , k + \langle k_{-},\,k_{-}\rangle=-\langle k_{+},\,k_{+}\rangle
  50. x , y = 1 4 ( x + y , x + y - x - y , x - y ) \langle x,\,y\rangle=\frac{1}{4}(\langle x+y,\,x+y\rangle-\langle x-y,\,x-y\rangle)
  51. x , y K x,\,y\in K
  52. x - y K 0 x-y\in K_{0}
  53. x + y 2 \frac{x+y}{2}
  54. k 0 ( K 0 K ± ) k_{0}\in(K_{0}\cap K_{\pm})
  55. k ± K ± k_{\pm}\in K_{\pm}
  56. k ± + 2 λ k 0 k_{\pm}+2\lambda k_{0}
  57. k ± k_{\pm}
  58. k ± + λ k 0 K ± k_{\pm}+\lambda k_{0}\in K_{\pm}
  59. K 00 = ( K 0 K + ) ( K 0 K - ) K_{00}=(K_{0}\cap K_{+})\oplus(K_{0}\cap K_{-})
  60. K K
  61. ( , ) (\cdot,\,\cdot)
  62. K ~ = def K / K 00 \tilde{K}\ \stackrel{\mathrm{def}}{=}\ K/K_{00}
  63. K ~ ± = def K ± / ( K 0 K ± ) \tilde{K}_{\pm}\ \stackrel{\mathrm{def}}{=}\ K_{\pm}/(K_{0}\cap K_{\pm})
  64. ( K ~ , ( , ) ) (\tilde{K},\,(\cdot,\,\cdot))
  65. ( , ) (\cdot,\,\cdot)
  66. J J
  67. J J^{\prime}
  68. K K
  69. K ~ \tilde{K}
  70. K ~ \tilde{K}^{\prime}
  71. K ~ ± \tilde{K}_{\pm}
  72. K ~ ± \tilde{K}^{\prime}_{\pm}
  73. k ~ K ~ \tilde{k}\in\tilde{K}
  74. k ~ K ~ \tilde{k}^{\prime}\in\tilde{K}^{\prime}
  75. k K k\in K
  76. K ~ \tilde{K}
  77. L L
  78. L 1 L_{1}
  79. L 2 L_{2}
  80. K K
  81. L [ ] = def { x K : x , y = 0 L^{[\perp]}\ \stackrel{\mathrm{def}}{=}\ \{x\in K:\langle x,\,y\rangle=0
  82. y L } y\in L\}
  83. L L
  84. L 0 = def L L [ ] L^{0}\ \stackrel{\mathrm{def}}{=}\ L\cap L^{[\perp]}
  85. L L
  86. L 0 = { 0 } L^{0}=\{0\}
  87. L L
  88. x , y = 0 \langle x,\,y\rangle=0
  89. x L 1 , y L 2 x\in L_{1},\,\,y\in L_{2}
  90. L 1 [ ] L 2 L_{1}[\perp]L_{2}
  91. L = L 1 + L 2 L=L_{1}+L_{2}
  92. L 1 [ ] L 2 L_{1}[\perp]L_{2}
  93. L = L 1 [ + ] L 2 L=L_{1}[+]L_{2}
  94. L = L 1 [ + ˙ ] L 2 L=L_{1}[\dot{+}]L_{2}
  95. κ := min { dim K + , dim K - } < \kappa:=\min\{\dim K_{+},\dim K_{-}\}<\infty
  96. ( K , , , J ) (K,\langle\cdot,\,\cdot\rangle,J)
  97. Π κ \Pi_{\kappa}
  98. dim K + \dim K_{+}
  99. dim K + \dim K_{+}
  100. , \langle\cdot,\,\cdot\rangle

Independence_of_clones_criterion.html

  1. i { 1 , 2 , 3 } i\in\{1,2,3\}

Independent_goods.html

  1. u ( X 1 , X 2 ) = X 1 a X 2 ( 1 - a ) , u(X_{1},X_{2})=X_{1}^{a}X_{2}^{(1-a)},
  2. X 1 = a m / p 1 , X_{1}=am/p_{1},

Indeterminate_system.html

  1. x + y = 2 x+y=2
  2. x + y = 2 , 2 x + 2 y = 4 x+y=2,\,\,\,\,\,2x+2y=4
  3. x + y = 2 , 2 x + 2 y = 4 , 3 x + 3 y = 6 x+y=2,\,\,\,\,\,2x+2y=4,\,\,\,\,\,3x+3y=6
  4. A x = b Ax=b
  5. x = A + b + [ I - A + A ] w x=A^{+}b+[I-A^{+}A]w
  6. A + A^{+}

Indexed_language.html

  1. { a n b n c n d n | n 1 } \{a^{n}b^{n}c^{n}d^{n}|n\geq 1\}
  2. { a n b m c n d m | m , n 0 } \{a^{n}b^{m}c^{n}d^{m}|m,n\geq 0\}
  3. { a 2 n | n 0 } \{a^{2^{n}}|n\geq 0\}
  4. { w w w | w { a , b } + } \{www|w\in\{a,b\}^{+}\}
  5. { ( a b n ) n | n 0 } \{(ab^{n})^{n}|n\geq 0\}

Inertial_wave.html

  1. u \vec{u}
  2. ν \nu
  3. P P
  4. Ω \Omega
  5. t t
  6. u t + ( u ) u = - 1 ρ P + ν 2 u - 2 Ω × u . \frac{\partial\vec{u}}{\partial t}+(\vec{u}\cdot\vec{\nabla})\vec{u}=-\frac{1}% {\rho}\vec{\nabla}P+\nu\nabla^{2}\vec{u}-2\vec{\Omega}\times\vec{u}.
  7. u \vec{u}
  8. P P
  9. P P
  10. p p
  11. r r
  12. P = p + 1 2 ρ r 2 Ω 2 . P=p+\frac{1}{2}\rho r^{2}\Omega^{2}.
  13. t × u = 2 ( Ω ) u . \frac{\partial}{\partial t}\nabla\times\vec{u}=2(\vec{\Omega}\cdot\vec{\nabla}% )\vec{u}.
  14. k \vec{k}
  15. u k = 0 , \vec{u}\cdot\vec{k}=0,
  16. ω \omega
  17. ω = 2 k ^ Ω = 2 Ω cos θ , \omega=2\hat{k}\cdot\vec{\Omega}=2\Omega\cos{\theta},
  18. θ \theta

Inexact_differential.html

  1. x = d x x=\int dx\,
  2. δ F = F d r \delta F=F\,dr
  3. F = f F=\nabla f
  4. d U = δ Q + δ W \mathrm{d}\,U=\delta\,Q+\delta\,W
  5. d f = 1 d x df=1\,dx
  6. Δ f = x B - x A \Delta f=x_{B}-x_{A}
  7. δ g = sgn ( d x ) d x \delta g=\operatorname{sgn}(dx)dx
  8. Δ g x B - x A \Delta g\neq x_{B}-x_{A}
  9. Δ g = A B d x + B A ( - d x ) = 2 A B d x = 2 ( B - A ) \Delta g=\int_{A}^{B}dx+\int_{B}^{A}(-dx)=2\int_{A}^{B}dx=2(B-A)
  10. d S = δ Q rev T \mathrm{d}\,S=\frac{\delta Q\text{rev}}{T}

Information-based_complexity.html

  1. 0 1 f ( x ) d x . \int_{0}^{1}f(x)\,dx.
  2. f f
  3. [ f ( t 1 ) , , f ( t n ) ] . [f(t_{1}),\dots,f(t_{n})].
  4. f ( x ) . f(x).
  5. ϵ \epsilon
  6. ϵ \epsilon
  7. ϵ - d . \epsilon^{-d}.
  8. d . d.
  9. ϵ \epsilon
  10. ϵ \epsilon
  11. 360 360
  12. 360 360
  13. 30 30
  14. ϵ - d \epsilon^{-d}
  15. ϵ = 10 - 2 , \epsilon=10^{-2},
  16. 10 720 10^{720}
  17. 360 360

Information_algebra.html

  1. ( Φ , D ) (\Phi,D)\,
  2. Φ \Phi\,
  3. D D\,
  4. ( Φ , D ) (\Phi,D)\,
  5. : Φ Φ Φ , ( ϕ , ψ ) ϕ ψ \otimes:\Phi\otimes\Phi\rightarrow\Phi,~{}(\phi,\psi)\mapsto\phi\otimes\psi\,
  6. : Φ D Φ , ( ϕ , x ) ϕ x \Rightarrow:\Phi\otimes D\rightarrow\Phi,~{}(\phi,x)\mapsto\phi^{\Rightarrow x}\,
  7. D D\,
  8. ( Φ , D ) (\Phi,D)\,
  9. D D\,
  10. Φ \Phi\,
  11. ( ϕ x ψ ) x = ϕ x ψ x (\phi^{\Rightarrow x}\otimes\psi)^{\Rightarrow x}=\phi^{\Rightarrow x}\otimes% \psi^{\Rightarrow x}\,
  12. x x\,
  13. x x\,
  14. x x\,
  15. ( ϕ x ) y = ϕ x y (\phi^{\Rightarrow x})^{\Rightarrow y}=\phi^{\Rightarrow x\wedge y}\,
  16. x x\,
  17. y y\,
  18. x y x\wedge y\,
  19. ϕ ϕ x = ϕ \phi\otimes\phi^{\Rightarrow x}=\phi\,
  20. ϕ Φ , x D \forall\phi\in\Phi,~{}\exists x\in D\,
  21. ϕ = ϕ x \phi=\phi^{\Rightarrow x}\,
  22. ( Φ , D ) (\Phi,D)\,
  23. ϕ ψ \phi\leq\psi\,
  24. ϕ ψ = ψ \phi\otimes\psi=\psi\,
  25. ϕ \phi\,
  26. ψ \psi\,
  27. ψ \psi\,
  28. Φ \Phi\,
  29. ϕ ψ = ϕ ψ \phi\otimes\psi=\phi\vee\psi\,
  30. x D x\in D\,
  31. ϕ x ψ \phi\leq_{x}\psi\,
  32. ϕ x ψ x \phi^{\Rightarrow x}\leq\psi^{\Rightarrow x}\,
  33. ϕ \phi\,
  34. ψ \psi\,
  35. x x\,
  36. ( ϕ , x ) (\phi,x)\ \,
  37. ϕ Φ \phi\in\Phi\,
  38. x D x\in D\,
  39. ϕ x = ϕ \phi^{\Rightarrow x}=\phi\,
  40. ( Φ , D ) (\Phi,D)\ \,
  41. d ( ϕ , x ) = x d(\phi,x)=x\ \,
  42. ( ϕ , x ) ( ψ , y ) = ( ϕ ψ , x y ) (\phi,x)\otimes(\psi,y)=(\phi\otimes\psi,x\vee y)~{}~{}~{}~{}\,
  43. ( ϕ , x ) y = ( ϕ y , y ) for y x (\phi,x)^{\downarrow y}=(\phi^{\Rightarrow y},y)\,\text{ for }y\leq x\,
  44. 𝒜 {\mathcal{A}}\,
  45. α 𝒜 \alpha\in{\mathcal{A}}\,
  46. U α U_{\alpha}\,
  47. α \alpha\,
  48. 𝒜 = { 𝚗𝚊𝚖𝚎 , 𝚊𝚐𝚎 , 𝚒𝚗𝚌𝚘𝚖𝚎 } {\mathcal{A}}=\{\texttt{name},\texttt{age},\texttt{income}\}\,
  49. U 𝚗𝚊𝚖𝚎 U_{\texttt{name}}\,
  50. U 𝚊𝚐𝚎 U_{\texttt{age}}\,
  51. U 𝚒𝚗𝚌𝚘𝚖𝚎 U_{\texttt{income}}\,
  52. x 𝒜 x\subseteq{\mathcal{A}}\,
  53. x x\,
  54. f f\,
  55. dom ( f ) = x \hbox{dom}(f)=x\,
  56. f ( α ) U α f(\alpha)\in U_{\alpha}\,
  57. α x \alpha\in x\,
  58. x x\,
  59. E x E_{x}\,
  60. x x\,
  61. f f\,
  62. y x y\subseteq x\,
  63. f [ y ] f[y]\,
  64. y y\,
  65. g g\,
  66. g ( α ) = f ( α ) g(\alpha)=f(\alpha)\,
  67. α y \alpha\in y\,
  68. R R\,
  69. x x\,
  70. x x\,
  71. E x E_{x}\,
  72. x x\,
  73. R R\,
  74. d ( R ) d(R)\,
  75. y d ( R ) y\subseteq d(R)\,
  76. R R\,
  77. y y\,
  78. π y ( R ) := { f [ y ] f R } . \pi_{y}(R):=\{f[y]\mid f\in R\}.\,
  79. R R\,
  80. x x\,
  81. S S\,
  82. y y\,
  83. R S := { f f ( x y ) -tuple , f [ x ] R , f [ y ] S } . R\bowtie S:=\{f\mid f\quad(x\cup y)\hbox{-tuple},\quad f[x]\in R,\;f[y]\in S\}.\,
  84. R R\,
  85. S S\,
  86. R = 𝚗𝚊𝚖𝚎 𝚊𝚐𝚎 𝙰 𝟹𝟺 𝙱 𝟺𝟽 S = 𝚗𝚊𝚖𝚎 𝚒𝚗𝚌𝚘𝚖𝚎 𝙰 20'000 𝙱 32'000 R=\begin{matrix}\texttt{name}&\texttt{age}\\ \texttt{A}&\texttt{34}\\ \texttt{B}&\texttt{47}\\ \end{matrix}\qquad S=\begin{matrix}\texttt{name}&\texttt{income}\\ \texttt{A}&\texttt{20'000}\\ \texttt{B}&\texttt{32'000}\\ \end{matrix}\,
  87. R R\,
  88. S S\,
  89. R S = 𝚗𝚊𝚖𝚎 𝚊𝚐𝚎 𝚒𝚗𝚌𝚘𝚖𝚎 𝙰 𝟹𝟺 20'000 𝙱 𝟺𝟽 32'000 R\bowtie S=\begin{matrix}\texttt{name}&\texttt{age}&\texttt{income}\\ \texttt{A}&\texttt{34}&\texttt{20'000}\\ \texttt{B}&\texttt{47}&\texttt{32'000}\\ \end{matrix}\,
  90. \bowtie\,
  91. π \pi\,
  92. d ( R S ) = d ( R ) d ( S ) d(R\bowtie S)=d(R)\cup d(S)\,
  93. x d ( R ) x\subseteq d(R)\,
  94. d ( π x ( R ) ) = x d(\pi_{x}(R))=x\,
  95. ( R 1 R 2 ) R 3 = R 1 ( R 2 R 3 ) (R_{1}\bowtie R_{2})\bowtie R_{3}=R_{1}\bowtie(R_{2}\bowtie R_{3})\,
  96. R S = S R R\bowtie S=S\bowtie R\,
  97. x y d ( R ) x\subseteq y\subseteq d(R)\,
  98. π x ( π y ( R ) ) = π x ( R ) \pi_{x}(\pi_{y}(R))=\pi_{x}(R)\,
  99. d ( R ) = x d(R)=x\,
  100. d ( S ) = y d(S)=y\,
  101. π x ( R S ) = R π x y ( S ) \pi_{x}(R\bowtie S)=R\bowtie\pi_{x\cap y}(S)\,
  102. x d ( R ) x\subseteq d(R)\,
  103. R π x ( R ) = R R\bowtie\pi_{x}(R)=R\,
  104. x = d ( R ) x=d(R)\,
  105. π x ( R ) = R \pi_{x}(R)=R\,

Information_theory_and_measure_theory.html

  1. H ( X ) H(X)
  2. h ( X ) h(X)
  3. X X
  4. f ( x ) f(x)
  5. h ( X ) = - X f ( x ) log f ( x ) d x . h(X)=-\int_{X}f(x)\log f(x)\,dx.
  6. h ( X ) = - X f ( x ) log f ( x ) d μ ( x ) , h(X)=-\int_{X}f(x)\log f(x)\,d\mu(x),
  7. μ \mu
  8. X X
  9. f f
  10. ν \nu
  11. H ( X ) = - X f ( x ) log f ( x ) d ν ( x ) = - x X f ( x ) log f ( x ) . H(X)=-\int_{X}f(x)\log f(x)\,d\nu(x)=-\sum_{x\in X}f(x)\log f(x).
  12. f f
  13. \mathbb{P}
  14. X X
  15. \mathbb{P}
  16. h ( X ) = - X log d d μ d , h(X)=-\int_{X}\log\frac{\mathrm{d}\mathbb{P}}{\mathrm{d}\mu}\,d\mathbb{P},
  17. \mathbb{Q}
  18. \mathbb{P}
  19. \mathbb{Q}
  20. \mathbb{P}
  21. \mathbb{Q}
  22. , \mathbb{P}<<\mathbb{Q},
  23. d d \frac{\mathrm{d}\mathbb{P}}{\mathrm{d}\mathbb{Q}}
  24. D KL ( ) = supp d d log d d d = supp log d d d , D_{\mathrm{KL}}(\mathbb{P}\|\mathbb{Q})=\int_{\mathrm{supp}\mathbb{P}}\frac{% \mathrm{d}\mathbb{P}}{\mathrm{d}\mathbb{Q}}\log\frac{\mathrm{d}\mathbb{P}}{% \mathrm{d}\mathbb{Q}}\,d\mathbb{Q}=\int_{\mathrm{supp}\mathbb{P}}\log\frac{% \mathrm{d}\mathbb{P}}{\mathrm{d}\mathbb{Q}}\,d\mathbb{P},
  25. . \mathbb{P}.
  26. X ~ \tilde{X}
  27. Y ~ \tilde{Y}
  28. μ ( X ~ Y ~ ) = 0 \mu(\tilde{X}\cap\tilde{Y})=0
  29. X ~ = Y ~ \tilde{X}=\tilde{Y}
  30. μ \mu
  31. H ( X ) = μ ( X ~ ) , H(X)=\mu(\tilde{X}),
  32. H ( Y ) = μ ( Y ~ ) , H(Y)=\mu(\tilde{Y}),
  33. H ( X , Y ) = μ ( X ~ Y ~ ) , H(X,Y)=\mu(\tilde{X}\cup\tilde{Y}),
  34. H ( X | Y ) = μ ( X ~ \ Y ~ ) , H(X|Y)=\mu(\tilde{X}\,\backslash\,\tilde{Y}),
  35. I ( X ; Y ) = μ ( X ~ Y ~ ) ; I(X;Y)=\mu(\tilde{X}\cap\tilde{Y});
  36. μ ( A ) + μ ( B ) = μ ( A B ) + μ ( A B ) \mu(A)+\mu(B)=\mu(A\cup B)+\mu(A\cap B)
  37. μ ( A ) + μ ( B \ A ) = μ ( B ) + μ ( A \ B ) \mu(A)+\mu(B\backslash A)=\mu(B)+\mu(A\backslash B)
  38. μ ( A ) - μ ( B ) = μ ( A \ B ) - μ ( B \ A ) \mu(A)-\mu(B)=\mu(A\backslash B)-\mu(B\backslash A)
  39. H ( X , Y ) = H ( X ) + H ( Y | X ) H(X,Y)=H(X)+H(Y|X)\,\qquad\qquad\qquad\,
  40. μ ( X ~ Y ~ ) = μ ( X ~ ) + μ ( Y ~ \ X ~ ) \mu(\tilde{X}\cup\tilde{Y})=\mu(\tilde{X})+\mu(\tilde{Y}\,\backslash\,\tilde{X})
  41. I ( X ; Y ) = H ( X ) - H ( X | Y ) I(X;Y)=H(X)-H(X|Y)\,
  42. μ ( X ~ Y ~ ) = μ ( X ~ ) - μ ( X ~ \ Y ~ ) \mu(\tilde{X}\cap\tilde{Y})=\mu(\tilde{X})-\mu(\tilde{X}\,\backslash\,\tilde{Y})
  43. H ( X , Y , Z , ) H(X,Y,Z,\cdots)
  44. I ( X ; Y ; Z ; ) I(X;Y;Z;\cdots)
  45. H ( X , Y , Z , ) = μ ( X ~ Y ~ Z ~ ) , H(X,Y,Z,\cdots)=\mu(\tilde{X}\cup\tilde{Y}\cup\tilde{Z}\cup\cdots),
  46. I ( X ; Y ; Z ; ) = μ ( X ~ Y ~ Z ~ ) ; I(X;Y;Z;\cdots)=\mu(\tilde{X}\cap\tilde{Y}\cap\tilde{Z}\cap\cdots);
  47. I ( X ) = H ( X ) I(X)=H(X)
  48. n 2 n\geq 2
  49. I ( X 1 ; ; X n ) = I ( X 1 ; ; X n - 1 ) - I ( X 1 ; ; X n - 1 | X n ) , I(X_{1};\cdots;X_{n})=I(X_{1};\cdots;X_{n-1})-I(X_{1};\cdots;X_{n-1}|X_{n}),
  50. I ( X 1 ; ; X n - 1 | X n ) = 𝔼 X n ( I ( X 1 ; ; X n - 1 ) | X n ) . I(X_{1};\cdots;X_{n-1}|X_{n})=\mathbb{E}_{X_{n}}\big(I(X_{1};\cdots;X_{n-1})|X% _{n}\big).
  51. I ( X 1 ; X 2 ) = H ( X 1 ) - H ( X 1 | X 2 ) . I(X_{1};X_{2})=H(X_{1})-H(X_{1}|X_{2}).
  52. I ( X ; Y ; Z ) = - 1 I(X;Y;Z)=-1
  53. I ( X , Y ; Z ) I(X,Y;Z)
  54. μ ( ( X ~ Y ~ ) Z ~ ) . \mu((\tilde{X}\cup\tilde{Y})\cap\tilde{Z}).
  55. I ( X , Y ; Z | W ) , I(X,Y;Z|W),
  56. H ( X , Z | W , Y ) . H(X,Z|W,Y).

Inhabited_set.html

  1. a A a\in A
  2. ¬ [ z ( z A ) ] . \lnot[\forall z(z\not\in A)].
  3. z ( z A ) . \exists z(z\in A).
  4. z ϕ ( z ) \exists z\phi(z)
  5. ϕ \phi
  6. ϕ \phi

Injective_metric_space.html

  1. B ¯ r ( p ) = { q d ( p , q ) r } {\bar{B}}_{r}(p)=\{q\mid d(p,q)\leq r\}

Inner_measure.html

  1. φ : 2 X [ 0 , ] , \varphi:2^{X}\rightarrow[0,\infty],
  2. φ ( ) = 0 \varphi(\varnothing)=0
  3. φ ( A B ) φ ( A ) + φ ( B ) . \varphi(A\cup B)\geq\varphi(A)+\varphi(B).
  4. A j A j + 1 A_{j}\supseteq A_{j+1}
  5. φ ( A 1 ) < \varphi(A_{1})<\infty
  6. φ ( j = 1 A j ) = lim j φ ( A j ) \varphi\left(\bigcap_{j=1}^{\infty}A_{j}\right)=\lim_{j\to\infty}\varphi(A_{j})
  7. φ ( A ) = \varphi(A)=\infty
  8. c φ ( B ) < c\leq\varphi(B)<\infty
  9. μ * ( T ) = sup { μ ( S ) : S Σ and S T } . \mu_{*}(T)=\sup\{\mu(S):S\in\Sigma\,\text{ and }S\subseteq T\}.
  10. Σ ^ \hat{\Sigma}
  11. Σ Σ ^ \Sigma\subseteq\hat{\Sigma}
  12. μ ^ ( T ) = μ * ( T ) = μ * ( T ) \hat{\mu}(T)=\mu^{*}(T)=\mu_{*}(T)
  13. T Σ ^ T\in\hat{\Sigma}
  14. Σ ^ \hat{\Sigma}

Inner_regular_measure.html

  1. μ ( A ) = sup { μ ( K ) compact K A } . \mu(A)=\sup\{\mu(K)\mid\,\text{compact }K\subseteq A\}.

Integral_representation_theorem_for_classical_Wiener_space.html

  1. C 0 ( [ 0 , T ] ; ) C_{0}([0,T];\mathbb{R})
  2. C 0 C_{0}
  3. γ \gamma
  4. F L 2 ( C 0 ; ) F\in L^{2}(C_{0};\mathbb{R})
  5. α F : [ 0 , T ] × C 0 \alpha^{F}:[0,T]\times C_{0}\to\mathbb{R}
  6. L 2 ( B ) L^{2}(B)
  7. B B
  8. F ( σ ) = C 0 F ( p ) d γ ( p ) + 0 T α F ( σ ) t d σ t F(\sigma)=\int_{C_{0}}F(p)\,\mathrm{d}\gamma(p)+\int_{0}^{T}\alpha^{F}(\sigma)% _{t}\,\mathrm{d}\sigma_{t}
  9. γ \gamma
  10. σ C 0 \sigma\in C_{0}
  11. C 0 F ( p ) d γ ( p ) = 𝔼 [ F ] \int_{C_{0}}F(p)\,\mathrm{d}\gamma(p)=\mathbb{E}[F]
  12. F F
  13. 0 T d σ t \int_{0}^{T}\cdots\,\mathrm{d}\sigma_{t}
  14. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  15. B : [ 0 , T ] × Ω B:[0,T]\times\Omega\to\mathbb{R}
  16. { t | 0 t T } \{\mathcal{F}_{t}|0\leq t\leq T\}
  17. \mathcal{F}
  18. B B
  19. t = σ { B s - 1 ( A ) | A Borel ( ) , 0 s t } . \mathcal{F}_{t}=\sigma\{B_{s}^{-1}(A)|A\in\mathrm{Borel}(\mathbb{R}),0\leq s% \leq t\}.
  20. f L 2 ( Ω ; ) f\in L^{2}(\Omega;\mathbb{R})
  21. T \mathcal{F}_{T}
  22. a f L 2 ( B ) a^{f}\in L^{2}(B)
  23. f = 𝔼 [ f ] + 0 T a t f d B t f=\mathbb{E}[f]+\int_{0}^{T}a_{t}^{f}\,\mathrm{d}B_{t}
  24. \mathbb{P}