wpmath0000002_0

100-year_flood.html

  1. P e = 1 - [ 1 - ( 1 T ) ] n P_{e}=1-\left[1-\left(\frac{1}{T}\right)\right]^{n}

10_(number).html

  1. 10 × x 10\times x
  2. 10 ÷ x 10\div x
  3. 3. 3 ¯ 3.\overline{3}
  4. 1. 6 ¯ 1.\overline{6}
  5. 1. 428571 ¯ 1.\overline{428571}
  6. 1. 1 ¯ 1.\overline{1}
  7. 0. 90 ¯ 0.\overline{90}
  8. 0.8 3 ¯ 0.8\overline{3}
  9. 0. 769230 ¯ 0.\overline{769230}
  10. 0. 714285 ¯ 0.\overline{714285}
  11. 0. 6 ¯ 0.\overline{6}
  12. x ÷ 10 x\div 10
  13. 10 x 10^{x}\,
  14. x 10 x^{10}\,

11_(number).html

  1. 11 × x 11\times x
  2. 11 ÷ x 11\div x
  3. 3. 6 ¯ 3.\overline{6}
  4. 1.8 3 ¯ 1.8\overline{3}
  5. 1. 571428 ¯ 1.\overline{571428}
  6. 1. 2 ¯ 1.\overline{2}
  7. 0.91 6 ¯ 0.91\overline{6}
  8. 0. 8 ¯ 4615 3 ¯ 0.\overline{8}4615\overline{3}
  9. 0.7 8 ¯ 5714 2 ¯ 0.7\overline{8}5714\overline{2}
  10. 0.7 3 ¯ 0.7\overline{3}
  11. x ÷ 11 x\div 11
  12. 0. 09 ¯ 0.\overline{09}
  13. 0. 18 ¯ 0.\overline{18}
  14. 0. 27 ¯ 0.\overline{27}
  15. 0. 36 ¯ 0.\overline{36}
  16. 0. 45 ¯ 0.\overline{45}
  17. 0. 54 ¯ 0.\overline{54}
  18. 0. 63 ¯ 0.\overline{63}
  19. 0. 72 ¯ 0.\overline{72}
  20. 0. 81 ¯ 0.\overline{81}
  21. 0. 90 ¯ 0.\overline{90}
  22. 1. 09 ¯ 1.\overline{09}
  23. 1. 18 ¯ 1.\overline{18}
  24. 1. 27 ¯ 1.\overline{27}
  25. 1. 36 ¯ 1.\overline{36}
  26. 11 x 11^{x}\,
  27. x 11 x^{11}\,
  28. x 11 x_{11}
  29. A 11 A_{11}
  30. 14 11 14_{11}
  31. 19 11 19_{11}
  32. 23 11 23_{11}
  33. 28 11 28_{11}
  34. 37 11 37_{11}
  35. 46 11 46_{11}
  36. 55 11 55_{11}
  37. 64 11 64_{11}
  38. 73 11 73_{11}
  39. 82 11 82_{11}
  40. 91 11 91_{11}
  41. A 0 11 A0_{11}
  42. A A 11 AA_{11}
  43. 109 11 109_{11}
  44. 118 11 118_{11}
  45. 127 11 127_{11}
  46. 172 11 172_{11}
  47. 208 11 208_{11}
  48. 415 11 415_{11}
  49. 82 A 11 82A_{11}
  50. 7572 11 7572_{11}
  51. 6914 11 6914_{11}
  52. 623351 11 623351_{11}

12_(number).html

  1. 12 × x 12\times x
  2. 12 ÷ x 12\div x
  3. 1. 714285 ¯ \mathrm{1.\overline{714285}}
  4. 1. 3 ¯ \mathrm{1.\overline{3}}
  5. 1. 09 ¯ \mathrm{1.\overline{09}}
  6. 0. 923076 ¯ \mathrm{0.\overline{923076}}
  7. 0. 857142 ¯ \mathrm{0.\overline{857142}}
  8. x ÷ 12 x\div 12
  9. 0.08 3 ¯ \mathrm{0.08\overline{3}}
  10. 0.1 6 ¯ \mathrm{0.1\overline{6}}
  11. 0. 3 ¯ \mathrm{0.\overline{3}}
  12. 0.41 6 ¯ \mathrm{0.41\overline{6}}
  13. 0.58 3 ¯ \mathrm{0.58\overline{3}}
  14. 0. 6 ¯ \mathrm{0.\overline{6}}
  15. 0.8 3 ¯ \mathrm{0.8\overline{3}}
  16. 0.91 6 ¯ \mathrm{0.91\overline{6}}
  17. 1.08 3 ¯ \mathrm{1.08\overline{3}}
  18. 1.1 6 ¯ \mathrm{1.1\overline{6}}
  19. 12 x 12^{x}\,
  20. x 12 x^{12}\,

17_(number).html

  1. 2 2 n + 1 , 2^{2^{n}}+1,

255_(number).html

  1. 255 = 2 8 - 1 = FF = 16 11111111 2 255=2^{8}-1=\mbox{FF}~{}_{16}=11111111_{2}

3-Way.html

  1. 2 22 2^{22}

3_(number).html

  1. 3 × x 3\times x
  2. 3 ÷ x 3\div x
  3. 0. 428571 ¯ 0.\overline{428571}
  4. 0. 3 ¯ 0.\overline{3}
  5. 0. 27 ¯ 0.\overline{27}
  6. 0. 230769 ¯ 0.\overline{230769}
  7. 0.2 142857 ¯ 0.2\overline{142857}
  8. 0.1 7647058823529411 ¯ 0.1\overline{7647058823529411}
  9. 0.1 6 ¯ 0.1\overline{6}
  10. 0.1 57894736842105263 ¯ 0.1\overline{57894736842105263}
  11. x ÷ 3 x\div 3
  12. 0. 3 ¯ 0.\overline{3}
  13. 0. 6 ¯ 0.\overline{6}
  14. 1. 3 ¯ 1.\overline{3}
  15. 1. 6 ¯ 1.\overline{6}
  16. 2. 3 ¯ 2.\overline{3}
  17. 2. 6 ¯ 2.\overline{6}
  18. 3. 3 ¯ 3.\overline{3}
  19. 3. 6 ¯ 3.\overline{6}
  20. 4. 3 ¯ 4.\overline{3}
  21. 4. 6 ¯ 4.\overline{6}
  22. 5. 3 ¯ 5.\overline{3}
  23. 5. 6 ¯ 5.\overline{6}
  24. 6. 3 ¯ 6.\overline{3}
  25. 6. 6 ¯ 6.\overline{6}
  26. 3 x 3^{x}\,
  27. x 3 x^{3}\,

3D_projection.html

  1. a x a_{x}
  2. a y a_{y}
  3. a z a_{z}
  4. b x b_{x}
  5. b y b_{y}
  6. b x = s x a x + c x b_{x}=s_{x}a_{x}+c_{x}
  7. b y = s z a z + c z b_{y}=s_{z}a_{z}+c_{z}
  8. [ b x b y ] = [ s x 0 0 0 0 s z ] [ a x a y a z ] + [ c x c z ] \begin{bmatrix}{b_{x}}\\ {b_{y}}\\ \end{bmatrix}=\begin{bmatrix}{s_{x}}&0&0\\ 0&0&{s_{z}}\\ \end{bmatrix}\begin{bmatrix}{a_{x}}\\ {a_{y}}\\ {a_{z}}\\ \end{bmatrix}+\begin{bmatrix}{c_{x}}\\ {c_{z}}\\ \end{bmatrix}
  9. Z i Z_{i}
  10. Z a v e Z_{ave}
  11. Z a v e Z_{ave}
  12. 𝐚 x , y , z \mathbf{a}_{x,y,z}
  13. 𝐜 x , y , z \mathbf{c}_{x,y,z}
  14. θ x , y , z \mathbf{\theta}_{x,y,z}
  15. 𝐞 x , y , z \mathbf{e}_{x,y,z}
  16. 𝐛 x , y \mathbf{b}_{x,y}
  17. 𝐚 \mathbf{a}
  18. 𝐜 x , y , z = 0 , 0 , 0 , \mathbf{c}_{x,y,z}=\langle 0,0,0\rangle,
  19. θ x , y , z = 0 , 0 , 0 , \mathbf{\theta}_{x,y,z}=\langle 0,0,0\rangle,
  20. 1 , 2 , 0 \langle 1,2,0\rangle
  21. 1 , 2 \langle 1,2\rangle
  22. 𝐛 x , y \mathbf{b}_{x,y}
  23. 𝐝 x , y , z \mathbf{d}_{x,y,z}
  24. θ \mathbf{\theta}
  25. 𝐜 \mathbf{c}
  26. 𝐚 \mathbf{a}
  27. - θ -\mathbf{\theta}
  28. [ 𝐝 x 𝐝 y 𝐝 z ] = [ 1 0 0 0 cos ( - θ x ) - sin ( - θ x ) 0 sin ( - θ x ) cos ( - θ x ) ] [ cos ( - θ y ) 0 sin ( - θ y ) 0 1 0 - sin ( - θ y ) 0 cos ( - θ y ) ] [ cos ( - θ z ) - sin ( - θ z ) 0 sin ( - θ z ) cos ( - θ z ) 0 0 0 1 ] ( [ 𝐚 x 𝐚 y 𝐚 z ] - [ 𝐜 x 𝐜 y 𝐜 z ] ) \begin{bmatrix}\mathbf{d}_{x}\\ \mathbf{d}_{y}\\ \mathbf{d}_{z}\\ \end{bmatrix}=\begin{bmatrix}1&0&0\\ 0&{\cos(\mathbf{-\theta}_{x})}&{-\sin(\mathbf{-\theta}_{x})}\\ 0&{\sin(\mathbf{-\theta}_{x})}&{\cos(\mathbf{-\theta}_{x})}\\ \end{bmatrix}\begin{bmatrix}{\cos(\mathbf{-\theta}_{y})}&0&{\sin(\mathbf{-% \theta}_{y})}\\ 0&1&0\\ {-\sin(\mathbf{-\theta}_{y})}&0&{\cos(\mathbf{-\theta}_{y})}\\ \end{bmatrix}\begin{bmatrix}{\cos(\mathbf{-\theta}_{z})}&{-\sin(\mathbf{-% \theta}_{z})}&0\\ {\sin(\mathbf{-\theta}_{z})}&{\cos(\mathbf{-\theta}_{z})}&0\\ 0&0&1\\ \end{bmatrix}\left({\begin{bmatrix}\mathbf{a}_{x}\\ \mathbf{a}_{y}\\ \mathbf{a}_{z}\\ \end{bmatrix}-\begin{bmatrix}\mathbf{c}_{x}\\ \mathbf{c}_{y}\\ \mathbf{c}_{z}\\ \end{bmatrix}}\right)
  29. θ x , y , z = 0 , 0 , 0 \mathbf{\theta}_{x,y,z}=\langle 0,0,0\rangle
  30. 𝐝 = 𝐚 - 𝐜 . \mathbf{d}=\mathbf{a}-\mathbf{c}.
  31. 𝐝 x = c y ( s z 𝐲 + c z 𝐱 ) - s y 𝐳 𝐝 y = s x ( c y 𝐳 + s y ( s z 𝐲 + c z 𝐱 ) ) + c x ( c z 𝐲 - s z 𝐱 ) 𝐝 z = c x ( c y 𝐳 + s y ( s z 𝐲 + c z 𝐱 ) ) - s x ( c z 𝐲 - s z 𝐱 ) \begin{array}[]{lcl}\mathbf{d}_{x}=c_{y}(s_{z}\mathbf{y}+c_{z}\mathbf{x})-s_{y% }\mathbf{z}\\ \mathbf{d}_{y}=s_{x}(c_{y}\mathbf{z}+s_{y}(s_{z}\mathbf{y}+c_{z}\mathbf{x}))+c% _{x}(c_{z}\mathbf{y}-s_{z}\mathbf{x})\\ \mathbf{d}_{z}=c_{x}(c_{y}\mathbf{z}+s_{y}(s_{z}\mathbf{y}+c_{z}\mathbf{x}))-s% _{x}(c_{z}\mathbf{y}-s_{z}\mathbf{x})\\ \end{array}
  32. 𝐛 x = 𝐞 z 𝐝 z 𝐝 x - 𝐞 x 𝐛 y = 𝐞 z 𝐝 z 𝐝 y - 𝐞 y . \begin{array}[]{lcl}\mathbf{b}_{x}&=&\frac{\mathbf{e}_{z}}{\mathbf{d}_{z}}% \mathbf{d}_{x}-\mathbf{e}_{x}\\ \mathbf{b}_{y}&=&\frac{\mathbf{e}_{z}}{\mathbf{d}_{z}}\mathbf{d}_{y}-\mathbf{e% }_{y}\\ \end{array}.
  33. [ 𝐟 x 𝐟 y 𝐟 z 𝐟 w ] = [ 1 0 - 𝐞 x 𝐞 z 0 0 1 - 𝐞 y 𝐞 z 0 0 0 1 0 0 0 1 / 𝐞 z 0 ] [ 𝐝 x 𝐝 y 𝐝 z 1 ] \begin{bmatrix}\mathbf{f}_{x}\\ \mathbf{f}_{y}\\ \mathbf{f}_{z}\\ \mathbf{f}_{w}\\ \end{bmatrix}=\begin{bmatrix}1&0&-\frac{\mathbf{e}_{x}}{\mathbf{e}_{z}}&0\\ 0&1&-\frac{\mathbf{e}_{y}}{\mathbf{e}_{z}}&0\\ 0&0&1&0\\ 0&0&1/\mathbf{e}_{z}&0\\ \end{bmatrix}\begin{bmatrix}\mathbf{d}_{x}\\ \mathbf{d}_{y}\\ \mathbf{d}_{z}\\ 1\\ \end{bmatrix}
  34. 𝐛 x = 𝐟 x / 𝐟 w 𝐛 y = 𝐟 y / 𝐟 w . \begin{array}[]{lcl}\mathbf{b}_{x}&=&\mathbf{f}_{x}/\mathbf{f}_{w}\\ \mathbf{b}_{y}&=&\mathbf{f}_{y}/\mathbf{f}_{w}\\ \end{array}.
  35. 𝐞 z \mathbf{e}_{z}
  36. α = 2 tan - 1 ( 1 / 𝐞 z ) \alpha=2\cdot\tan^{-1}(1/\mathbf{e}_{z})
  37. 𝐛 x = ( 𝐝 x 𝐬 x ) / ( 𝐝 z 𝐫 x ) 𝐫 z 𝐛 y = ( 𝐝 y 𝐬 y ) / ( 𝐝 z 𝐫 y ) 𝐫 z . \begin{array}[]{lcl}\mathbf{b}_{x}=(\mathbf{d}_{x}\mathbf{s}_{x})/(\mathbf{d}_% {z}\mathbf{r}_{x})\mathbf{r}_{z}\\ \mathbf{b}_{y}=(\mathbf{d}_{y}\mathbf{s}_{y})/(\mathbf{d}_{z}\mathbf{r}_{y})% \mathbf{r}_{z}\\ \end{array}.
  38. 𝐬 x , y \mathbf{s}_{x,y}
  39. 𝐫 x , y \mathbf{r}_{x,y}
  40. 𝐫 z \mathbf{r}_{z}
  41. 𝐝 z \mathbf{d}_{z}
  42. A x , A z A_{x},A_{z}
  43. B x = A x B z A z B_{x}=A_{x}\frac{B_{z}}{A_{z}}
  44. B x B_{x}
  45. A x A_{x}
  46. B z B_{z}
  47. A z A_{z}

42_(number).html

  1. 1 T 0 T | ζ ( 1 2 + i t ) | 6 d t 42 9 ! p { 1 - 1 p } 4 ( 1 + 4 p + 1 p 2 ) log 9 T , {1\over T}\int_{0}^{T}\left|\zeta\left({1\over 2}+it\right)\right|^{6}\,dt\sim% {42\over 9!}\prod_{p}\left\{1-{1\over p}\right\}^{4}\left(1+{4\over p}+{1\over p% ^{2}}\right)\log^{9}T,
  2. σ 2 ( n ) = σ ( σ ( n ) ) = 6 n \sigma^{2}(n)=\sigma(\sigma(n))=6n\,

4_(number).html

  1. ( n - 1 ) ! 0 ( mod n ) (n-1)!\ \equiv\ 0\ ({\rm mod}\ n)
  2. 2 2 = 2 2 = 4 2\uparrow\uparrow 2=2\uparrow\uparrow\uparrow 2=4
  3. n \mathbb{R}^{n}
  4. 4 × x 4\times x
  5. 4 ÷ x 4\div x
  6. 1. 3 ¯ 1.\overline{3}
  7. 0. 6 ¯ 0.\overline{6}
  8. 0. 571428 ¯ 0.\overline{571428}
  9. 0. 4 ¯ 0.\overline{4}
  10. 0. 36 ¯ 0.\overline{36}
  11. 0. 3 ¯ 0.\overline{3}
  12. 0. 307692 ¯ 0.\overline{307692}
  13. 0. 285714 ¯ 0.\overline{285714}
  14. 0.2 6 ¯ 0.2\overline{6}
  15. x ÷ 4 x\div 4
  16. 4 x 4^{x}\,
  17. x 4 x^{4}\,

6_(number).html

  1. 𝒮 \mathcal{S}
  2. 6 = ( 17 21 ) 3 + ( 37 21 ) 3 6=\left(\frac{17}{21}\right)^{3}+\left(\frac{37}{21}\right)^{3}
  3. 6 × x 6\times x
  4. 6 ÷ x 6\div x
  5. 0. 85714 ¯ 2 ¯ 0.\overline{85714}\overline{2}
  6. 0. 6 ¯ 0.\overline{6}
  7. 0. 5 ¯ 4 ¯ 0.\overline{5}\overline{4}
  8. 0. 46153 ¯ 8 ¯ 0.\overline{46153}\overline{8}
  9. 0. 42857 ¯ 1 ¯ 0.\overline{42857}\overline{1}
  10. x ÷ 6 x\div 6
  11. 0.1 6 ¯ 0.1\overline{6}
  12. 0. 3 ¯ 0.\overline{3}
  13. 0. 6 ¯ 0.\overline{6}
  14. 0.8 3 ¯ 0.8\overline{3}
  15. 1.1 6 ¯ 1.1\overline{6}
  16. 1. 3 ¯ 1.\overline{3}
  17. 1. 6 ¯ 1.\overline{6}
  18. 1.8 3 ¯ 1.8\overline{3}
  19. 2.1 6 ¯ 2.1\overline{6}
  20. 2. 3 ¯ 2.\overline{3}
  21. 6 x 6^{x}\,
  22. x 6 x^{6}\,

8_(number).html

  1. 2 3 2^{3}
  2. p 3 p^{3}
  3. 2 x 2^{x}
  4. O ( ) O(\infty)
  5. O ( 1 ) O ( 2 ) O ( k ) O(1)\hookrightarrow O(2)\hookrightarrow\ldots\hookrightarrow O(k)\hookrightarrow\ldots
  6. π k + 8 ( O ( ) ) π k ( O ( ) ) \pi_{k+8}(O(\infty))\cong\pi_{k}(O(\infty))
  7. C l ( p + 8 , q ) Cl(p+8,q)
  8. C l ( p , q ) Cl(p,q)
  9. S p i n ( 8 ) Spin(8)
  10. 8 × x 8\times x
  11. 8 ÷ x 8\div x
  12. 6 ¯ \overline{6}
  13. 3 ¯ \overline{3}
  14. 142857 ¯ \overline{142857}
  15. 8 ¯ \overline{8}
  16. 72 ¯ \overline{72}
  17. 6 ¯ \overline{6}
  18. 615384 ¯ \overline{615384}
  19. 571428 ¯ \overline{571428}
  20. 3 ¯ \overline{3}
  21. x ÷ 8 x\div 8
  22. 8 x 8^{x}\,
  23. x 8 x^{8}\,

9_(number).html

  1. 9 × x 9\times x
  2. 9 ÷ x 9\div x
  3. 285714 ¯ \overline{285714}
  4. 81 ¯ \overline{81}
  5. 692307 ¯ \overline{692307}
  6. 428571 ¯ \overline{428571}
  7. x ÷ 9 x\div 9
  8. 1 ¯ \overline{1}
  9. 2 ¯ \overline{2}
  10. 3 ¯ \overline{3}
  11. 4 ¯ \overline{4}
  12. 5 ¯ \overline{5}
  13. 6 ¯ \overline{6}
  14. 7 ¯ \overline{7}
  15. 8 ¯ \overline{8}
  16. 1 ¯ \overline{1}
  17. 2 ¯ \overline{2}
  18. 3 ¯ \overline{3}
  19. 4 ¯ \overline{4}
  20. 5 ¯ \overline{5}
  21. 6 ¯ \overline{6}
  22. 9 x 9^{x}\,
  23. x 9 x^{9}\,
  24. x 9 x_{9}

A*_search_algorithm.html

  1. f ( n ) = g ( n ) + h ( n ) f(n)=g(n)+h(n)
  2. g ( n ) g(n)
  3. n n
  4. h ( n ) h(n)
  5. n n
  6. h ( x ) d ( x , y ) + h ( y ) h(x)\leq d(x,y)+h(y)

A5::1.html

  1. x 19 + x 18 + x 17 + x 14 + 1 x^{19}+x^{18}+x^{17}+x^{14}+1
  2. x 22 + x 21 + 1 x^{22}+x^{21}+1
  3. x 23 + x 22 + x 21 + x 8 + 1 x^{23}+x^{22}+x^{21}+x^{8}+1
  4. 0 i < 64 0\leq{i}<64
  5. R [ 0 ] = R [ 0 ] K [ i ] . R[0]=R[0]\oplus K[i].

A_New_Kind_of_Science.html

  1. 2 π α . 2\pi\sqrt{\alpha^{\prime}.}

Abc_conjecture.html

  1. = =
  2. c > rad ( a b c ) 1 + ε . c>\operatorname{rad}(abc)^{1+\varepsilon}.
  3. c < K ε rad ( a b c ) 1 + ε c<K_{\varepsilon}\cdot\operatorname{rad}(abc)^{1+\varepsilon}
  4. q ( a , b , c ) = log ( c ) log ( rad ( a b c ) ) . q(a,b,c)=\frac{\log(c)}{\log(\operatorname{rad}(abc))}.
  5. c f = prime p x i ( 1 - ω f ( p ) p 2 + q p ) c_{f}=\prod_{\,\text{prime }p}x_{i}\left(1-\frac{\omega\,\!_{f}(p)}{p^{2+q_{p}% }}\right)
  6. c n < ( rad ( a n b n c n ) ) 2 = ( rad ( a b c ) ) 2 ( a b c ) 2 < ( c 3 ) 2 = c 6 . c^{n}<(\operatorname{rad}(a^{n}b^{n}c^{n}))^{2}=(\operatorname{rad}(abc))^{2}% \leq(abc)^{2}<(c^{3})^{2}=c^{6}.
  7. rad ( x n ) = rad ( x ) , \operatorname{rad}(x^{n})=\operatorname{rad}(x),
  8. rad ( x ) x , \operatorname{rad}(x)\leq x,
  9. a b c < c c c a\cdot b\cdot c<c\cdot c\cdot c
  10. c n < c 6 . c^{n}<c^{6}.
  11. c < exp ( K 1 rad ( a b c ) 15 ) c<\exp{\left(K_{1}\operatorname{rad}(abc)^{15}\right)}
  12. c < exp ( K 2 rad ( a b c ) 2 3 + ε ) c<\exp{\left(K_{2}\operatorname{rad}(abc)^{\frac{2}{3}+\varepsilon}\right)}
  13. c < exp ( K 3 rad ( a b c ) 1 3 + ε ) c<\exp{\left(K_{3}\operatorname{rad}(abc)^{\frac{1}{3}+\varepsilon}\right)}
  14. ( ε - ω rad ( a b c ) ) 1 + ε ({\varepsilon}^{-\omega}\operatorname{rad}(abc))^{1+\varepsilon}
  15. ε > 0 {\varepsilon}>0
  16. ε = ω log ( rad ( a b c ) ) {\varepsilon}=\frac{\omega}{\log(\operatorname{rad}(abc))}
  17. c < κ rad ( a b c ) ( log ( rad ( a b c ) ) ) ω ω ! c<{\kappa}\operatorname{rad}(abc)\frac{(\log(\operatorname{rad}(abc)))^{\omega% }}{\omega!}
  18. κ {\kappa}
  19. κ {\kappa}
  20. 6 5 \tfrac{6}{5}
  21. c < ( rad ( a b c ) ) 1 + 3 4 c<(\operatorname{rad}(abc))^{1+\frac{3}{4}}
  22. K Ω ( a b c ) rad ( a b c ) , K^{\Omega(abc)}\mathrm{rad}(abc),
  23. O ( rad ( a b c ) Θ ( a b c ) ) , O(\mathrm{rad}(abc)\Theta(abc)),
  24. c < k exp ( 4 3 log k log log k ( 1 + log log log k 2 log log k + C 1 log log k ) ) c<k\exp\left(4\sqrt{\frac{3\log k}{\log\log k}}\left(1+\frac{\log\log\log k}{2% \log\log k}+\frac{C_{1}}{\log\log k}\right)\right)
  25. c > k exp ( 4 3 log k log log k ( 1 + log log log k 2 log log k + C 2 log log k ) ) c>k\exp\left(4\sqrt{\frac{3\log k}{\log\log k}}\left(1+\frac{\log\log\log k}{2% \log\log k}+\frac{C_{2}}{\log\log k}\right)\right)

Abel's_theorem.html

  1. G a ( z ) = k = 0 a k z k G_{a}(z)=\sum_{k=0}^{\infty}a_{k}z^{k}\!
  2. k = 0 a k \sum_{k=0}^{\infty}a_{k}\!
  3. lim z 1 - G a ( z ) = k = 0 a k , ( * ) \lim_{z\rightarrow 1^{-}}G_{a}(z)=\sum_{k=0}^{\infty}a_{k},\qquad(*)\!
  4. | 1 - z | M ( 1 - | z | ) |1-z|\leq M(1-|z|)\,
  5. n > 0 ( z 3 n - z 2 × 3 n ) / n \sum_{n>0}(z^{3^{n}}-z^{2\times 3^{n}})/n
  6. G a ( z ) G_{a}(z)
  7. k = 0 a k z k \sum_{k=0}^{\infty}a_{k}z^{k}\!
  8. lim t 1 - G a ( t z ) = k = 0 a k z k \lim_{t\to 1^{-}}G_{a}(tz)=\sum_{k=0}^{\infty}a_{k}z^{k}\!
  9. k = 0 a k = \sum_{k=0}^{\infty}a_{k}=\infty\!
  10. lim z 1 - G a ( z ) \lim_{z\to 1^{-}}G_{a}(z)
  11. 1 1 + z \frac{1}{1+z}
  12. 1 - 1 + 1 - 1 + 1-1+1-1+\cdots
  13. z = 1 z=1
  14. 1 / ( 1 + 1 ) = 1 / 2 1/(1+1)=1/2
  15. z z
  16. R R
  17. a k = ( - 1 ) k / ( k + 1 ) a_{k}=(-1)^{k}/(k+1)
  18. G a ( z ) = ln ( 1 + z ) / z G_{a}(z)=\ln(1+z)/z
  19. 0 < z < 1 0<z<1
  20. [ - z , 0 ] [-z,0]
  21. k = 0 ( - 1 ) k / ( k + 1 ) \sum_{k=0}^{\infty}(-1)^{k}/(k+1)\!
  22. ln ( 2 ) \ln(2)
  23. k = 0 ( - 1 ) k / ( 2 k + 1 ) \sum_{k=0}^{\infty}(-1)^{k}/(2k+1)\!
  24. arctan ( 1 ) = π / 4 \arctan(1)=\pi/4
  25. G a ( z ) G_{a}(z)
  26. a a
  27. a 0 a_{0}\!
  28. k = 0 a k = 0 \sum_{k=0}^{\infty}a_{k}=0\!
  29. s n = k = 0 n a k s_{n}=\sum_{k=0}^{n}a_{k}\!
  30. a k = s k - s k - 1 a_{k}=s_{k}-s_{k-1}\!
  31. G a ( z ) = ( 1 - z ) k = 0 s k z k . G_{a}(z)=(1-z)\sum_{k=0}^{\infty}s_{k}z^{k}.\!
  32. ϵ > 0 \epsilon>0\!
  33. | s k | < ϵ |s_{k}|<\epsilon\!
  34. k n k\geq n\!
  35. | ( 1 - z ) k = n s k z k | ϵ | 1 - z | k = n | z | k = ϵ | 1 - z | | z | n 1 - | z | < ϵ M \left|(1-z)\sum_{k=n}^{\infty}s_{k}z^{k}\right|\leq\epsilon|1-z|\sum_{k=n}^{% \infty}|z|^{k}=\epsilon|1-z|\frac{|z|^{n}}{1-|z|}<\epsilon M\!
  36. | ( 1 - z ) k = 0 n - 1 s k z k | < ϵ , \left|(1-z)\sum_{k=0}^{n-1}s_{k}z^{k}\right|<\epsilon,
  37. | G a ( z ) | < ( M + 1 ) ϵ |G_{a}(z)|<(M+1)\epsilon\!

Abelian_variety.html

  1. End ( A ) \mathrm{End}(A)\otimes\mathbb{Q}

Abel–Ruffini_theorem.html

  1. x = - b ± b 2 - 4 a c 2 a x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}
  2. x 5 - x + 1 = 0 x^{5}-x+1=0
  3. a x 2 + b x + c = 0 , a 0 \textstyle{ax^{2}+bx+c=0,a\neq 0}
  4. x = - b ± b 2 - 4 a c 2 a . x=\frac{-b\pm\sqrt{b^{2}-4ac\ }}{2a}.
  5. x 5 - x + 1 = 0 x^{5}-x+1=0
  6. x 5 - x 4 - x + 1 = 0 x^{5}-x^{4}-x+1=0
  7. ( x - 1 ) ( x - 1 ) ( x + 1 ) ( x + i ) ( x - i ) = 0 (x-1)(x-1)(x+1)(x+i)(x-i)=0
  8. y 1 y_{1}
  9. Q Q
  10. y 2 y_{2}
  11. Q ( y 1 ) Q(y_{1})
  12. y 5 y_{5}
  13. Q ( y 1 , y 2 , y 3 , y 4 ) Q(y_{1},y_{2},y_{3},y_{4})
  14. E = Q ( y 1 , y 2 , y 3 , y 4 , y 5 ) E=Q(y_{1},y_{2},y_{3},y_{4},y_{5})
  15. f ( x ) = ( x - y 1 ) ( x - y 2 ) ( x - y 3 ) ( x - y 4 ) ( x - y 5 ) E [ x ] . f(x)=(x-y_{1})(x-y_{2})(x-y_{3})(x-y_{4})(x-y_{5})\in E[x].
  16. f ( x ) f(x)
  17. y n y_{n}
  18. s 1 = y 1 + y 2 + y 3 + y 4 + y 5 s_{1}=y_{1}+y_{2}+y_{3}+y_{4}+y_{5}
  19. s 2 = y 1 y 2 + y 1 y 3 + y 1 y 4 + y 1 y 5 + y 2 y 3 + y 2 y 4 + y 2 y 5 + y 3 y 4 + y 3 y 5 + y 4 y 5 s_{2}=y_{1}y_{2}+y_{1}y_{3}+y_{1}y_{4}+y_{1}y_{5}+y_{2}y_{3}+y_{2}y_{4}+y_{2}y% _{5}+y_{3}y_{4}+y_{3}y_{5}+y_{4}y_{5}
  20. s 3 = y 1 y 2 y 3 + y 1 y 2 y 4 + y 1 y 2 y 5 + y 1 y 3 y 4 + y 1 y 3 y 5 + y 1 y 4 y 5 + y 2 y 3 y 4 + y 2 y 3 y 5 + y 2 y 4 y 5 + y 3 y 4 y 5 s_{3}=y_{1}y_{2}y_{3}+y_{1}y_{2}y_{4}+y_{1}y_{2}y_{5}+y_{1}y_{3}y_{4}+y_{1}y_{% 3}y_{5}+y_{1}y_{4}y_{5}+y_{2}y_{3}y_{4}+y_{2}y_{3}y_{5}+y_{2}y_{4}y_{5}+y_{3}y% _{4}y_{5}
  21. s 4 = y 1 y 2 y 3 y 4 + y 1 y 2 y 3 y 5 + y 1 y 2 y 4 y 5 + y 1 y 3 y 4 y 5 + y 2 y 3 y 4 y 5 s_{4}=y_{1}y_{2}y_{3}y_{4}+y_{1}y_{2}y_{3}y_{5}+y_{1}y_{2}y_{4}y_{5}+y_{1}y_{3% }y_{4}y_{5}+y_{2}y_{3}y_{4}y_{5}
  22. s 5 = y 1 y 2 y 3 y 4 y 5 . s_{5}=y_{1}y_{2}y_{3}y_{4}y_{5}.
  23. x n x^{n}
  24. f ( x ) f(x)
  25. ( - 1 ) 5 - n s 5 - n (-1)^{5-n}s_{5-n}
  26. F = Q ( s i ) F=Q(s_{i})
  27. s i s_{i}
  28. y i y_{i}
  29. y n y_{n}
  30. Q Q
  31. σ \sigma
  32. S 5 S_{5}
  33. σ \sigma^{\prime}
  34. E E
  35. Q Q
  36. y n y_{n}
  37. ( y - y 3 ) ( y - y 1 ) ( y - y 2 ) ( y - y 5 ) ( y - y 4 ) (y-y_{3})(y-y_{1})(y-y_{2})(y-y_{5})(y-y_{4})
  38. ( y - y 1 ) ( y - y 2 ) ( y - y 3 ) ( y - y 4 ) ( y - y 5 ) (y-y_{1})(y-y_{2})(y-y_{3})(y-y_{4})(y-y_{5})
  39. σ \sigma^{\prime}
  40. f f
  41. G ( E / F ) G(E/F)
  42. S 5 G ( E / F ) S_{5}\subseteq G(E/F)
  43. S 5 S_{5}
  44. G ( E / F ) G(E/F)
  45. S 5 S_{5}
  46. n n
  47. S n S_{n}
  48. S 5 S_{5}
  49. S 5 S_{5}
  50. S 5 A 5 { e } S_{5}\geq A_{5}\geq\{e\}
  51. A 5 A_{5}
  52. A 5 / { e } A_{5}/\{e\}
  53. A 5 A_{5}
  54. S 5 S_{5}
  55. n n
  56. n n
  57. n n
  58. n 5 n\geq 5
  59. x 5 - 1 x^{5}-1
  60. S 5 S_{5}
  61. A 5 A_{5}
  62. x 5 - 1 = 0 x^{5}-1=0

Absolute_continuity.html

  1. π 2 \frac{\pi}{2}
  2. I I
  3. \R \R
  4. f : I \R f\colon I\to\R
  5. I I
  6. ϵ \epsilon
  7. δ \delta
  8. ( x k , y k ) (x_{k},y_{k})
  9. I I
  10. k ( y k - x k ) < δ \sum_{k}\left(y_{k}-x_{k}\right)<\delta
  11. k | f ( y k ) - f ( x k ) | < ϵ . \displaystyle\sum_{k}|f(y_{k})-f(x_{k})|<\epsilon.
  12. I I
  13. AC ( I ) \operatorname{AC}(I)
  14. f ( x ) = f ( a ) + a x f ( t ) d t f(x)=f(a)+\int_{a}^{x}f^{\prime}(t)\,dt
  15. f ( x ) = f ( a ) + a x g ( t ) d t f(x)=f(a)+\int_{a}^{x}g(t)\,dt
  16. L [ a , b ] L\subseteq[a,b]
  17. λ ( L ) = 0 \lambda(L)=0
  18. λ ( f ( L ) ) = 0 \lambda(f(L))=0
  19. λ \lambda
  20. f ( x ) = { 0 , if x = 0 x sin ( 1 / x ) , if x 0 f(x)=\begin{cases}0,&\mbox{if }~{}x=0\\ x\sin(1/x),&\mbox{if }~{}x\neq 0\end{cases}
  21. ϵ \epsilon
  22. δ \delta
  23. k | y k - x k | < δ \sum_{k}\left|y_{k}-x_{k}\right|<\delta
  24. k d ( f ( y k ) , f ( x k ) ) < ϵ . \sum_{k}d\left(f(y_{k}),f(x_{k})\right)<\epsilon.
  25. d ( f ( s ) , f ( t ) ) s t m ( τ ) d τ for all [ s , t ] I d\left(f(s),f(t)\right)\leq\int_{s}^{t}m(\tau)\,\mathrm{d}\tau\mbox{ for all }% ~{}[s,t]\subseteq I
  26. d ( f ( s ) , f ( t ) ) s t m ( τ ) d τ for all [ s , t ] I . d\left(f(s),f(t)\right)\leq\int_{s}^{t}m(\tau)\,\mathrm{d}\tau\mbox{ for all }% ~{}[s,t]\subseteq I.
  27. μ \mu
  28. λ \lambda
  29. λ \lambda
  30. A A
  31. λ ( A ) = 0 \lambda(A)=0
  32. μ ( A ) = 0 \mu(A)=0
  33. μ λ \mu\ll\lambda
  34. n , n = 1 , 2 , 3 , \mathbb{R}^{n},n=1,2,3,\dots
  35. μ ( A ) = A g d λ \mu(A)=\int_{A}g\,\mathrm{d}\lambda
  36. \ll
  37. μ ν ( ν ( A ) = 0 μ ( A ) = 0 ) . \mu\ll\nu\iff\left(\nu(A)=0\ \Rightarrow\ \mu(A)=0\right).
  38. \ll
  39. \ll
  40. μ ( A ) = A f d ν . \mu(A)=\int_{A}f\,\mathrm{d}\nu.
  41. F ( x ) = μ ( ( - , x ] ) F(x)=\mu((-\infty,x])
  42. I I

Absolute_threshold_of_hearing.html

  1. p p
  2. I = p 2 ρ v I=\frac{p^{2}}{\rho v}
  3. v v

Absorption_law.html

  1. \scriptstyle\lor
  2. \scriptstyle\land
  3. \scriptstyle\lor
  4. \scriptstyle\land

Accessibility.html

  1. A c c e s s i b i l i t y i = j O p p o r t u n i t i e s j × f ( C i j ) Accessibility_{i}=\sum_{j}{Opportunities_{j}}\times f\left({C_{ij}}\right)
  2. i i
  3. j j
  4. f ( C i j ) f\left({C_{ij}}\right)

Accretion_disc.html

  1. ( R 2 Ω ) R > 0 , \frac{\partial(R^{2}\Omega)}{\partial R}>0,
  2. Ω \Omega
  3. R R
  4. α \alpha
  5. α \alpha
  6. ν = α c s H \nu=\alpha c_{\rm s}H
  7. c s c_{\rm s}
  8. H H
  9. α \alpha
  10. ν v turb l turb \nu\approx v_{\rm turb}l_{\rm turb}
  11. v turb v_{\rm turb}
  12. l turb l_{\rm turb}
  13. l turb H = c s / Ω l_{\rm turb}\approx H=c_{\rm s}/\Omega
  14. v turb c s v_{\rm turb}\approx c_{\rm s}
  15. Ω = ( G M ) 1 / 2 r - 3 / 2 \Omega=(GM)^{1/2}r^{-3/2}
  16. r r
  17. M M
  18. α \alpha
  19. α \alpha
  20. H = 1.7 × 10 8 α - 1 / 10 M ˙ 16 3 / 20 m 1 - 3 / 8 R 10 9 / 8 f 3 / 5 cm H=1.7\times 10^{8}\alpha^{-1/10}\dot{M}^{3/20}_{16}m_{1}^{-3/8}R^{9/8}_{10}f^{% 3/5}{\rm cm}
  21. T c = 1.4 × 10 4 α - 1 / 5 M ˙ 16 3 / 10 m 1 1 / 4 R 10 - 3 / 4 f 6 / 5 K T_{c}=1.4\times 10^{4}\alpha^{-1/5}\dot{M}^{3/10}_{16}m_{1}^{1/4}R^{-3/4}_{10}% f^{6/5}{\rm K}
  22. ρ = 3.1 × 10 - 8 α - 7 / 10 M ˙ 16 11 / 20 m 1 5 / 8 R 10 - 15 / 8 f 11 / 5 g cm - 3 \rho=3.1\times 10^{-8}\alpha^{-7/10}\dot{M}^{11/20}_{16}m_{1}^{5/8}R^{-15/8}_{% 10}f^{11/5}{\rm g\ cm}^{-3}
  23. T c T_{c}
  24. ρ \rho
  25. M ˙ 16 \dot{M}_{16}
  26. 10 16 g s - 1 10^{16}{\rm g\ s}^{-1}
  27. m 1 m_{1}
  28. M M_{\bigodot}
  29. R 10 R_{10}
  30. 10 10 cm 10^{10}{\rm cm}
  31. f = [ 1 - ( R R ) 1 / 2 ] 1 / 4 f=\left[1-\left(\frac{R_{\star}}{R}\right)^{1/2}\right]^{1/4}
  32. R R_{\star}
  33. β \beta
  34. ν α p gas \nu\propto\alpha p_{\mathrm{gas}}
  35. p tot = p rad + p gas = ρ c s 2 p_{\mathrm{tot}}=p_{\mathrm{rad}}+p_{\mathrm{gas}}=\rho c_{\rm s}^{2}
  36. ν = α c s H = α c s 2 / Ω = α p tot / ( ρ Ω ) \nu=\alpha c_{\rm s}H=\alpha c_{s}^{2}/\Omega=\alpha p_{\mathrm{tot}}/(\rho\Omega)
  37. d Ω 2 d ln R > 0. \frac{d\Omega^{2}}{d\ln R}>0.

Acetaldehyde.html

  1. \overrightarrow{\leftarrow}
  2. × 10 7 \times 10^{−}7

Achromatic_lens.html

  1. ϕ sys \phi_{\,\text{sys}}
  2. n d n_{d}
  3. V V
  4. ϕ 1 + ϕ 2 = ϕ sys ϕ 1 V 1 + ϕ 2 V 2 = 0 , \begin{aligned}\displaystyle\phi_{1}+\phi_{2}&\displaystyle=\phi_{\,\text{sys}% }\\ \displaystyle\frac{\phi_{1}}{V_{1}}+\frac{\phi_{2}}{V_{2}}&\displaystyle=0\ ,% \end{aligned}
  5. ϕ = 1 / f \phi=1/f
  6. f f
  7. ϕ 1 \phi_{1}
  8. ϕ 2 \phi_{2}
  9. ϕ 1 ϕ sys = V 1 V 1 - V 2 and ϕ 2 ϕ sys = - V 2 V 1 - V 2 . \frac{\phi_{1}}{\phi_{\,\text{sys}}}=\frac{V_{1}}{V_{1}-V_{2}}\qquad\,\text{% and}\qquad\frac{\phi_{2}}{\phi_{\,\text{sys}}}=\frac{-V_{2}}{V_{1}-V_{2}}\ .
  10. ϕ 2 = - ϕ 1 V 2 / V 1 \phi_{2}=-\phi_{1}V_{2}/V_{1}

Action_(physics).html

  1. 𝒮 = t 1 t 2 L d t , \mathcal{S}=\int_{t_{1}}^{t_{2}}L\,dt\,,
  2. 𝒮 \mathcal{S}
  3. 𝒮 [ 𝐪 ( t ) ] \mathcal{S}[\mathbf{q}(t)]
  4. 𝒮 [ 𝐪 ( t ) ] = t 1 t 2 L [ 𝐪 ( t ) , 𝐪 ˙ ( t ) , t ] d t \mathcal{S}[\mathbf{q}(t)]=\int_{t_{1}}^{t_{2}}L[\mathbf{q}(t),\dot{\mathbf{q}% }(t),t]\,dt
  5. 𝐪 1 = 𝐪 ( t 1 ) \mathbf{q}_{1}=\mathbf{q}(t_{1})
  6. 𝐪 2 = 𝐪 ( t 2 ) \mathbf{q}_{2}=\mathbf{q}(t_{2})
  7. 𝒮 [ 𝐪 ( t ) ] \mathcal{S}[\mathbf{q}(t)]
  8. 𝒮 0 \mathcal{S}_{0}
  9. 𝒮 0 \mathcal{S}_{0}
  10. 𝒮 0 = 𝐩 d 𝐪 = p i d q i \mathcal{S}_{0}=\int\mathbf{p}\cdot d\mathbf{q}=\int p_{i}\,dq_{i}
  11. 𝒮 0 \mathcal{S}_{0}
  12. 𝒮 \mathcal{S}
  13. S S
  14. S ( q 1 , , q N , t ) = W ( q 1 , , q N ) - E t S(q_{1},\dots,q_{N},t)=W(q_{1},\dots,q_{N})-E\cdot t
  15. d W d t = W q i q ˙ i = p i q ˙ i \frac{dW}{dt}=\frac{\partial W}{\partial q_{i}}\dot{q}_{i}=p_{i}\dot{q}_{i}
  16. W ( q 1 , , q N ) = p i q ˙ i d t = p i d q i W(q_{1},\dots,q_{N})=\int p_{i}\dot{q}_{i}\,dt=\int p_{i}\,dq_{i}
  17. J k = p k d q k J_{k}=\oint p_{k}dq_{k}
  18. 𝒮 = t 1 t 2 L ( x , x ˙ , t ) d t \mathcal{S}=\int_{t_{1}}^{t_{2}}\;L(x,\dot{x},t)\,dt
  19. x 1 = x ( t 1 ) x_{1}=x(t_{1})
  20. x 2 = x ( t 2 ) x_{2}=x(t_{2})
  21. x per ( t ) x_{\mathrm{per}}(t)
  22. x per ( t 1 ) = x 1 x_{\mathrm{per}}(t_{1})=x_{1}
  23. x per ( t 2 ) = x 2 x_{\mathrm{per}}(t_{2})=x_{2}
  24. ε ( t ) \varepsilon(t)
  25. ε ( t ) = x per ( t ) - x true ( t ) \varepsilon(t)=x_{\mathrm{per}}(t)-x_{\mathrm{true}}(t)
  26. ε ( t 1 ) = ε ( t 2 ) = 0 \varepsilon(t_{1})=\varepsilon(t_{2})=0
  27. δ 𝒮 = t 1 t 2 [ L ( x true + ε , x ˙ true + ε ˙ , t ) - L ( x true , x ˙ true , t ) ] d t = t 1 t 2 ( ε L x + ε ˙ L x ˙ ) d t \begin{aligned}\displaystyle\delta\mathcal{S}&\displaystyle=\int_{t_{1}}^{t_{2% }}\;\left[L(x_{\mathrm{true}}+\varepsilon,\dot{x}_{\mathrm{true}}+\dot{% \varepsilon},t)-L(x_{\mathrm{true}},\dot{x}_{\mathrm{true}},t)\right]dt\\ &\displaystyle=\int_{t_{1}}^{t_{2}}\;\left(\varepsilon{\partial L\over\partial x% }+\dot{\varepsilon}{\partial L\over\partial\dot{x}}\right)\,dt\end{aligned}
  28. ε ( t 1 ) = ε ( t 2 ) = 0 \varepsilon(t_{1})=\varepsilon(t_{2})=0
  29. δ 𝒮 = t 1 t 2 ( ε L x - ε d d t L x ˙ ) d t . \delta\mathcal{S}=\int_{t_{1}}^{t_{2}}\;\left(\varepsilon{\partial L\over% \partial x}-\varepsilon{d\over dt}{\partial L\over\partial\dot{x}}\right)\,dt.
  30. 𝒮 \mathcal{S}
  31. δ 𝒮 δ x ( t ) = 0 \frac{\delta\mathcal{S}}{\delta x(t)}=0
  32. L x ˙ \frac{\partial L}{\partial\dot{x}}
  33. L x = 0 \frac{\partial L}{\partial x}=0
  34. L x ˙ \frac{\partial L}{\partial\dot{x}}
  35. L = 1 2 m v 2 = 1 2 m ( x ˙ 2 + y ˙ 2 ) L=\frac{1}{2}mv^{2}=\frac{1}{2}m\left(\dot{x}^{2}+\dot{y}^{2}\right)
  36. L = 1 2 m ( r ˙ 2 + r 2 φ ˙ 2 ) . L=\frac{1}{2}m\left(\dot{r}^{2}+r^{2}\dot{\varphi}^{2}\right).
  37. d d t ( L r ˙ ) - L r = 0 r ¨ - r φ ˙ 2 = 0 d d t ( L φ ˙ ) - L φ = 0 φ ¨ + 2 r r ˙ φ ˙ = 0 \begin{aligned}\displaystyle\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{r% }}\right)-\frac{\partial L}{\partial r}&\displaystyle=0\qquad\Rightarrow\qquad% \ddot{r}-r\dot{\varphi}^{2}&\displaystyle=0\\ \displaystyle\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{\varphi}}\right)% -\frac{\partial L}{\partial\varphi}&\displaystyle=0\qquad\Rightarrow\qquad% \ddot{\varphi}+\frac{2}{r}\dot{r}\dot{\varphi}&\displaystyle=0\end{aligned}
  38. r cos φ = a t + b r sin φ = c t + d \begin{aligned}\displaystyle r\cos\varphi&\displaystyle=at+b\\ \displaystyle r\sin\varphi&\displaystyle=ct+d\end{aligned}
  39. τ \tau
  40. S = - m c 2 C d τ S=-mc^{2}\int_{C}\,d\tau
  41. t 1 t 2 L d t \int_{t1}^{t2}L\,dt
  42. L = - m c 2 1 - v 2 c 2 L=-mc^{2}\sqrt{1-\frac{v^{2}}{c^{2}}}

Action_potential.html

  1. τ V t = λ 2 2 V x 2 - V \tau\frac{\partial V}{\partial t}=\lambda^{2}\frac{\partial^{2}V}{\partial x^{% 2}}-V
  2. τ = r m c m \tau=\ r_{m}c_{m}\,
  3. λ = r m r l \lambda=\sqrt{\frac{r_{m}}{r_{l}}}

Adaptive_filter.html

  1. d k d_{k}
  2. x k x_{k}
  3. d k d_{k}
  4. 𝐖 k = [ w 0 k , w 1 k , , w L k ] T \mathbf{W}_{k}=\left[w_{0k},\,w_{1k},\,...,\,w_{Lk}\right]^{T}
  5. w l k w_{lk}
  6. l l
  7. 𝚫 𝐖 k \mathbf{\Delta W}_{k}
  8. ϵ k \epsilon_{k}
  9. y k y_{k}
  10. d k = g k + u k + v k d_{k}=g_{k}+u_{k}+v_{k}
  11. x k = g k + u k + v k x_{k}=g_{k}^{^{\prime}}+u_{k}^{^{\prime}}+v_{k}^{^{\prime}}
  12. y k = g ^ k + u ^ k + v ^ k y_{k}=\hat{g}_{k}+\hat{u}_{k}+\hat{v}_{k}
  13. ϵ k = d k - y k \epsilon_{k}=d_{k}-y_{k}
  14. g ^ \hat{g}
  15. u ^ \hat{u}
  16. v ^ \hat{v}
  17. y k = l = 0 L w l k x ( k - l ) = g ^ k + u ^ k + v ^ k y_{k}=\sum_{l=0}^{L}w_{lk}\ x_{(k-l)}=\hat{g}_{k}+\hat{u}_{k}+\hat{v}_{k}
  18. w l k w_{lk}
  19. l l
  20. v 0 , v 0 , g 0 v\equiv 0,v^{^{\prime}}\equiv 0,g^{^{\prime}}\equiv 0
  21. d k d_{k}
  22. u k u_{k}
  23. x k \ x_{k}
  24. u k u_{k}
  25. y k = u ^ k y_{k}=\hat{u}_{k}
  26. d k d_{k}
  27. y k y_{k}
  28. ϵ k = d k - y k = g k + u k - u ^ k \epsilon_{k}=d_{k}-y_{k}=g_{k}+u_{k}-\hat{u}_{k}
  29. ϵ k \epsilon_{k}
  30. [ u k - u ^ k ] [u_{k}-\hat{u}_{k}]
  31. u ^ k \hat{u}_{k}
  32. u k u_{k}
  33. u k = u ^ k u_{k}=\hat{u}_{k}
  34. ϵ k = g k \epsilon_{k}=g_{k}
  35. g g
  36. ϵ k = d k - y k = g k - g ^ k + u k - u ^ k \epsilon_{k}=d_{k}-y_{k}=g_{k}-\hat{g}_{k}+u_{k}-\hat{u}_{k}
  37. ρ 𝗈𝗎𝗍 ( z ) = 1 ρ 𝗋𝖾𝖿 ( z ) \rho_{\mathsf{out}}(z)=\frac{1}{\rho_{\mathsf{ref}}(z)}
  38. ρ 𝗈𝗎𝗍 ( z ) \rho_{\mathsf{out}}(z)
  39. ρ 𝗋𝖾𝖿 ( z ) \rho_{\mathsf{ref}}(z)
  40. z z
  41. y k = l = 0 L w l k x l k y_{k}=\sum_{l=0}^{L}w_{lk}\ x_{lk}
  42. w l k w_{lk}
  43. l l
  44. l = 0 L l=0\dots L
  45. w l , k + 1 = w l k + 2 μ ϵ k x k - l w_{l,k+1}=w_{lk}+2\mu\ \epsilon_{k}\ x_{k-l}
  46. w l , k + 1 = w l k + 2 μ ϵ k x l k w_{l,k+1}=w_{lk}+2\mu\ \epsilon_{k}\ x_{lk}
  47. y k y_{k}
  48. 0 < μ < 1 σ 2 0<\mu<\frac{1}{\sigma^{2}}
  49. σ 2 = l = 0 L σ l 2 \sigma^{2}=\sum_{l=0}^{L}\sigma_{l}^{2}
  50. σ l \sigma_{l}
  51. l l
  52. σ 2 = ( L + 1 ) σ 0 2 \sigma^{2}=(L+1)\sigma_{0}^{2}
  53. σ 0 \sigma_{0}
  54. x k x_{k}
  55. w l , k + 1 = w l k + ( 2 μ σ σ 2 ) ϵ k x k - l w_{l,k+1}=w_{lk}+\left(\frac{2\mu_{\sigma}}{\sigma^{2}}\right)\epsilon_{k}\ x_% {k-l}
  56. 0 < μ σ < 1 0<\mu_{\sigma}<1

Additive_white_Gaussian_noise.html

  1. Y i Y_{i}
  2. i i
  3. Y i Y_{i}
  4. X i X_{i}
  5. Z i Z_{i}
  6. Z i Z_{i}
  7. N N
  8. Z i Z_{i}
  9. X i X_{i}
  10. Z i 𝒩 ( 0 , N ) Z_{i}\sim\mathcal{N}(0,N)\,\!
  11. Y i = X i + Z i 𝒩 ( X i , N ) . Y_{i}=X_{i}+Z_{i}\sim\mathcal{N}(X_{i},N).\,\!
  12. X i X_{i}
  13. ( x 1 , x 2 , , x k ) (x_{1},x_{2},\dots,x_{k})
  14. 1 k i = 1 k x i 2 P , \frac{1}{k}\sum_{i=1}^{k}x_{i}^{2}\leq P,
  15. P P
  16. C = max f ( x ) s.t. E ( X 2 ) P I ( X ; Y ) C=\max_{f(x)\,\text{ s.t. }E\left(X^{2}\right)\leq P}I(X;Y)\,\!
  17. f ( x ) f(x)
  18. X X
  19. I ( X ; Y ) I(X;Y)
  20. I ( X ; Y ) = h ( Y ) - h ( Y | X ) = h ( Y ) - h ( X + Z | X ) = h ( Y ) - h ( Z | X ) \begin{aligned}\displaystyle I(X;Y)=h(Y)-h(Y|X)&\displaystyle=h(Y)-h(X+Z|X)&% \displaystyle=h(Y)-h(Z|X)\end{aligned}\,\!
  21. X X
  22. Z Z
  23. I ( X ; Y ) = h ( Y ) - h ( Z ) I(X;Y)=h(Y)-h(Z)\,\!
  24. h ( Z ) = 1 2 log ( 2 π e N ) h(Z)=\frac{1}{2}\log(2\pi eN)\,\!
  25. X X
  26. Z Z
  27. Y Y
  28. E ( Y 2 ) = E ( X + Z ) 2 = E ( X 2 ) + 2 E ( X ) E ( Z ) + E ( Z 2 ) = P + N E(Y^{2})=E(X+Z)^{2}=E(X^{2})+2E(X)E(Z)+E(Z^{2})=P+N\,\!
  29. h ( Y ) 1 2 log ( 2 π e ( P + N ) ) h(Y)\leq\frac{1}{2}\log(2\pi e(P+N))\,\!
  30. I ( X ; Y ) 1 2 log ( 2 π e ( P + N ) ) - 1 2 log ( 2 π e N ) I(X;Y)\leq\frac{1}{2}\log(2\pi e(P+N))-\frac{1}{2}\log(2\pi eN)\,\!
  31. I ( X ; Y ) I(X;Y)
  32. X 𝒩 ( 0 , P ) X\sim\mathcal{N}(0,P)\,\!
  33. C C
  34. C = 1 2 log ( 1 + P N ) C=\frac{1}{2}\log\left(1+\frac{P}{N}\right)\,\!
  35. 1 1
  36. M M
  37. M M
  38. n n
  39. R R
  40. R = log M n R=\frac{\log M}{n}\,\!
  41. n n
  42. C C
  43. n n
  44. N N
  45. N N
  46. n ( N + ϵ ) \sqrt{n(N+\epsilon)}
  47. n n
  48. n ( P + N ) n(P+N)
  49. n ( P + N ) \sqrt{n(P+N)}
  50. n N \sqrt{nN}
  51. r n r^{n}
  52. ( n ( P + N ) ) n 2 ( n N ) n 2 = 2 n 2 log ( 1 + P / N ) \frac{(n(P+N))^{\frac{n}{2}}}{(nN)^{\frac{n}{2}}}=2^{\frac{n}{2}\log(1+P/N)}\,\!
  53. 1 2 log ( 1 + P / N ) \frac{1}{2}\log(1+P/N)
  54. P - ϵ P-\epsilon
  55. X n ( i ) X^{n}(i)
  56. i i
  57. Y n Y^{n}
  58. U U
  59. P P
  60. V V
  61. E j E_{j}
  62. ( X n ( j ) , Y n ) (X^{n}(j),Y^{n})
  63. A ϵ ( n ) A_{\epsilon}^{(n)}
  64. i j i\neq j
  65. U U
  66. V V
  67. E i E_{i}
  68. P ( U ) P(U)
  69. P ( V ) P(V)
  70. n n
  71. P ( U ) P(U)
  72. P ( V ) P(V)
  73. ϵ \epsilon
  74. X n ( i ) X^{n}(i)
  75. X n ( j ) X^{n}(j)
  76. i j i\neq j
  77. X n ( i ) X^{n}(i)
  78. Y n Y^{n}
  79. P ( E j ) = 2 - n ( I ( X ; Y ) - 3 ϵ ) P(E_{j})=2^{-n(I(X;Y)-3\epsilon)}
  80. P e ( n ) P^{(n)}_{e}
  81. P e ( n ) \displaystyle P^{(n)}_{e}
  82. P e ( n ) P^{(n)}_{e}
  83. R < I ( X ; Y ) - 3 ϵ R<I(X;Y)-3\epsilon
  84. C = 1 2 log ( 1 + P N ) C=\frac{1}{2}\log(1+\frac{P}{N})
  85. W W
  86. W ^ \hat{W}
  87. W X ( n ) ( W ) Y ( n ) W ^ W\longrightarrow X^{(n)}(W)\longrightarrow Y^{(n)}\longrightarrow\hat{W}
  88. H ( W | W ^ ) 1 + n R P e ( n ) = n ϵ n H(W|\hat{W})\leq 1+nRP^{(n)}_{e}=n\epsilon_{n}
  89. ϵ n 0 \epsilon_{n}\rightarrow 0
  90. P e ( n ) 0 P^{(n)}_{e}\rightarrow 0
  91. X i X_{i}
  92. n R \displaystyle nR
  93. P i P_{i}
  94. P i = 1 2 n R w x i 2 ( w ) P_{i}=\frac{1}{2^{nR}}\sum_{w}x^{2}_{i}(w)\,\!
  95. w w
  96. X i X_{i}
  97. Z i Z_{i}
  98. Y i Y_{i}
  99. N N
  100. E ( Y i 2 ) = P i + N E(Y_{i}^{2})=P_{i}+N\,\!
  101. Y i Y_{i}
  102. h ( Y i ) 1 2 log 2 π e ( P i + N ) h(Y_{i})\leq\frac{1}{2}\log{2\pi e}(P_{i}+N)\,\!
  103. n R \displaystyle nR
  104. log ( 1 + x ) \log(1+x)
  105. 1 n i = 1 n 1 2 log ( 1 + P i N ) 1 2 log ( 1 + 1 n i = 1 n P i N ) \frac{1}{n}\sum_{i=1}^{n}\frac{1}{2}\log\left(1+\frac{P_{i}}{N}\right)\leq% \frac{1}{2}\log\left(1+\frac{1}{n}\sum_{i=1}^{n}\frac{P_{i}}{N}\right)\,\!
  106. 1 n i = 1 n P i N \frac{1}{n}\sum_{i=1}^{n}\frac{P_{i}}{N}\,\!
  107. 1 2 log ( 1 + 1 n i = 1 n P i N ) 1 2 log ( 1 + P N ) \frac{1}{2}\log\left(1+\frac{1}{n}\sum_{i=1}^{n}\frac{P_{i}}{N}\right)\leq% \frac{1}{2}\log\left(1+\frac{P}{N}\right)\,\!
  108. R 1 2 log ( 1 + P N ) + ϵ n R\leq\frac{1}{2}\log\left(1+\frac{P}{N}\right)+\epsilon_{n}
  109. ϵ n 0 \epsilon_{n}\rightarrow 0
  110. positive zero crossings second = negative zero crossings second \frac{\mathrm{positive\ zero\ crossings}}{\mathrm{second}}=\frac{\mathrm{% negative\ zero\ crossings}}{\mathrm{second}}
  111. = f 0 SNR + 1 + B 2 12 f 0 2 SNR + 1 =f_{0}\sqrt{\frac{\mathrm{SNR}+1+\frac{B^{2}}{12f_{0}^{2}}}{\mathrm{SNR}+1}}

Adele_ring.html

  1. ^ \widehat{\mathbb{Z}}
  2. / n \mathbb{Z}/n\mathbb{Z}
  3. ^ = lim / n . \widehat{\mathbb{Z}}=\underleftarrow{\lim}\,\mathbb{Z}/n\mathbb{Z}.
  4. ^ = p p . \widehat{\mathbb{Z}}=\prod_{p}\mathbb{Z}_{p}.
  5. 𝔸 = × ^ . \mathbb{A}_{\mathbb{Z}}=\mathbb{R}\times\widehat{\mathbb{Z}}.
  6. 𝔸 = 𝔸 \mathbb{A}_{\mathbb{Q}}=\mathbb{Q}\otimes_{\mathbb{Z}}\mathbb{A}_{\mathbb{Z}}
  7. 𝔸 F = F 𝔸 \mathbb{A}_{F}=F\otimes_{\mathbb{Z}}\mathbb{A}_{\mathbb{Z}}
  8. deg ( F ) \deg(F)
  9. 𝔸 = × p p \mathbb{A}_{\mathbb{Q}}=\mathbb{R}\times{\prod_{p}}^{\prime}\mathbb{Q}_{p}

Adjacency_matrix.html

  1. ( 1 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 1 0 0 0 0 0 1 0 0 ) \begin{pmatrix}1&1&0&0&1&0\\ 1&0&1&0&1&0\\ 0&1&0&1&0&0\\ 0&0&1&0&1&1\\ 1&1&0&1&0&0\\ 0&0&0&1&0&0\\ \end{pmatrix}
  2. A A
  3. r r
  4. s s
  5. A = ( 0 r , r B B T 0 s , s ) , A=\begin{pmatrix}0_{r,r}&B\\ B^{T}&0_{s,s}\end{pmatrix},
  6. B B
  7. r × s r\times s
  8. 0
  9. B B
  10. G = ( U , V , E ) G=(U,V,E)
  11. U = u 1 , , u r U={u_{1},...,u_{r}}
  12. V = v 1 , , v s V={v_{1},...,v_{s}}
  13. r × s r\times s
  14. B B
  15. b i , j = 1 b_{i,j}=1
  16. ( u i , v j ) E (u_{i},v_{j})\in E
  17. G G
  18. b i , j b_{i,j}
  19. ( u i , v j ) , (u_{i},v_{j}),
  20. G 1 G_{1}
  21. G 2 G_{2}
  22. A 1 A_{1}
  23. A 2 A_{2}
  24. G 1 G_{1}
  25. G 2 G_{2}
  26. P P
  27. P A 1 P - 1 = A 2 . PA_{1}P^{-1}=A_{2}.
  28. A 1 A_{1}
  29. A 2 A_{2}
  30. ( d ) \left(d\right)
  31. v = ( 1 , , 1 ) v=\left(1,\dots,1\right)
  32. G G
  33. d d
  34. - d -d
  35. n 2 / 8 {n^{2}}/8
  36. n n
  37. 8 e 8e
  38. e e
  39. n 2 n^{2}
  40. d = e / n 2 d=e/n^{2}
  41. 8 e > n 2 / 8 8e>n^{2}/8
  42. d > 1 / 64 d>1/64

Adjoint_representation.html

  1. x x
  2. x g x g - 1 x\mapsto gxg^{-1}
  3. 𝔤 \mathfrak{g}
  4. Ψ : G Aut ( G ) , g Ψ g \Psi:G\to\mathrm{Aut}(G),\,g\mapsto\Psi_{g}
  5. Ψ g ( h ) = g h g - 1 \Psi_{g}(h)=ghg^{-1}\,
  6. 𝔤 \mathfrak{g}
  7. d ( Ψ g ) e = Ad g : 𝔤 𝔤 . d(\Psi_{g})_{e}=\mathrm{Ad}_{g}\colon\mathfrak{g}\to\mathfrak{g}.
  8. 𝔤 \mathfrak{g}
  9. Ad : G Aut ( 𝔤 ) , g Ad g \mathrm{Ad}\colon G\to\mathrm{Aut}(\mathfrak{g}),\,g\mapsto\mathrm{Ad}_{g}
  10. Aut ( 𝔤 ) \mathrm{Aut}(\mathfrak{g})
  11. GL ( 𝔤 ) \mathrm{GL}(\mathfrak{g})
  12. G L n ( ) GL_{n}(\mathbb{C})
  13. exp ( X ) = e X \operatorname{exp}(X)=e^{X}
  14. Ψ g ( exp ( t X ) ) = e t g X g - 1 \Psi_{g}(\operatorname{exp}(tX))=e^{tgXg^{-1}}
  15. 𝔤 \mathfrak{g}
  16. Ad g ( X ) = g X g - 1 \operatorname{Ad}_{g}(X)=gXg^{-1}
  17. Ad : G Aut ( 𝔤 ) \mathrm{Ad}\colon G\to\mathrm{Aut}(\mathfrak{g})
  18. 𝔤 \mathfrak{g}
  19. d ( Ad ) x : T x ( G ) T A d ( x ) ( Aut ( 𝔤 ) ) d(\mathrm{Ad})_{x}:T_{x}(G)\to T_{Ad(x)}(\mathrm{Aut}(\mathfrak{g}))
  20. ad : 𝔤 Der ( 𝔤 ) . \mathrm{ad}\colon\mathfrak{g}\to\mathrm{Der}(\mathfrak{g}).
  21. Der ( 𝔤 ) \mathrm{Der}(\mathfrak{g})
  22. Aut ( 𝔤 ) \mathrm{Aut}(\mathfrak{g})
  23. 𝔤 \mathfrak{g}
  24. ad x ( y ) = [ x , y ] \mathrm{ad}_{x}(y)=[x,y]\,
  25. x , y 𝔤 x,y\in\mathfrak{g}
  26. 𝔤 𝔩 n ( 𝐂 ) \mathfrak{gl}_{n}(\mathbf{C})
  27. Ψ : G Aut ( G ) \Psi\colon G\to\mathrm{Aut}(G)\,
  28. Ψ g : G G \Psi_{g}\colon G\to G\,
  29. Ψ g h = Ψ g Ψ h \Psi_{gh}=\Psi_{g}\Psi_{h}
  30. Ψ g ( a b ) = Ψ g ( a ) Ψ g ( b ) \Psi_{g}(ab)=\Psi_{g}(a)\Psi_{g}(b)
  31. ( Ψ g ) - 1 = Ψ g - 1 (\Psi_{g})^{-1}=\Psi_{g^{-1}}
  32. Ad : G Aut ( 𝔤 ) \mathrm{Ad}\colon G\to\mathrm{Aut}(\mathfrak{g})
  33. Ad g : 𝔤 𝔤 \mathrm{Ad}_{g}\colon\mathfrak{g}\to\mathfrak{g}
  34. Ad g h = Ad g Ad h \mathrm{Ad}_{gh}=\mathrm{Ad}_{g}\mathrm{Ad}_{h}
  35. Ad g \mathrm{Ad}_{g}
  36. ( Ad g ) - 1 = Ad g - 1 (\mathrm{Ad}_{g})^{-1}=\mathrm{Ad}_{g^{-1}}
  37. Ad g [ x , y ] = [ Ad g x , Ad g y ] \mathrm{Ad}_{g}[x,y]=[\mathrm{Ad}_{g}x,\mathrm{Ad}_{g}y]
  38. ad : 𝔤 Der ( 𝔤 ) \mathrm{ad}\colon\mathfrak{g}\to\mathrm{Der}(\mathfrak{g})
  39. ad x : 𝔤 𝔤 \mathrm{ad}_{x}\colon\mathfrak{g}\to\mathfrak{g}
  40. ad \mathrm{ad}
  41. ad [ x , y ] = [ ad x , ad y ] \mathrm{ad}_{[x,y]}=[\mathrm{ad}_{x},\mathrm{ad}_{y}]
  42. ad x \mathrm{ad}_{x}
  43. ad x [ y , z ] = [ ad x y , z ] + [ y , ad x z ] \mathrm{ad}_{x}[y,z]=[\mathrm{ad}_{x}y,z]+[y,\mathrm{ad}_{x}z]
  44. Ad ( G ) G / Z G ( G 0 ) . \mathrm{Ad}(G)\cong G/Z_{G}(G_{0}).
  45. 𝔤 \mathfrak{g}
  46. Int ( 𝔤 ) \operatorname{Int}(\mathfrak{g})
  47. 𝔤 \mathfrak{g}
  48. Lie ( Int ( 𝔤 ) ) = ad ( 𝔤 ) \operatorname{Lie}(\operatorname{Int}(\mathfrak{g}))=\operatorname{ad}(% \mathfrak{g})
  49. 𝔤 \mathfrak{g}
  50. 𝔤 \mathfrak{g}
  51. Int ( 𝔤 ) \operatorname{Int}(\mathfrak{g})
  52. Int ( 𝔤 ) = Ad ( G ) \operatorname{Int}(\mathfrak{g})=\operatorname{Ad}(G)
  53. [ a 11 a 12 a 1 n a 21 a 22 a 2 n a n 1 a n 2 a n n ] [ a 11 t 1 t 2 - 1 a 12 t 1 t n - 1 a 1 n t 2 t 1 - 1 a 21 a 22 t 2 t n - 1 a 2 n t n t 1 - 1 a n 1 t n t 2 - 1 a n 2 a n n ] . \begin{bmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n1}&a_{n2}&\cdots&a_{nn}\\ \end{bmatrix}\mapsto\begin{bmatrix}a_{11}&t_{1}t_{2}^{-1}a_{12}&\cdots&t_{1}t_% {n}^{-1}a_{1n}\\ t_{2}t_{1}^{-1}a_{21}&a_{22}&\cdots&t_{2}t_{n}^{-1}a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ t_{n}t_{1}^{-1}a_{n1}&t_{n}t_{2}^{-1}a_{n2}&\cdots&a_{nn}\\ \end{bmatrix}.
  54. [ a b c d ] \begin{bmatrix}a&b\\ c&d\\ \end{bmatrix}
  55. [ t 1 0 0 t 2 ] = [ t 1 0 0 1 / t 1 ] = [ exp ( θ ) 0 0 exp ( - θ ) ] \begin{bmatrix}t_{1}&0\\ 0&t_{2}\\ \end{bmatrix}=\begin{bmatrix}t_{1}&0\\ 0&1/t_{1}\\ \end{bmatrix}=\begin{bmatrix}\exp(\theta)&0\\ 0&\exp(-\theta)\\ \end{bmatrix}
  56. t 1 t 2 = 1 t_{1}t_{2}=1
  57. [ θ 0 0 - θ ] = θ [ 1 0 0 0 ] - θ [ 0 0 0 1 ] = θ ( e 1 - e 2 ) . \begin{bmatrix}\theta&0\\ 0&-\theta\\ \end{bmatrix}=\theta\begin{bmatrix}1&0\\ 0&0\\ \end{bmatrix}-\theta\begin{bmatrix}0&0\\ 0&1\\ \end{bmatrix}=\theta(e_{1}-e_{2}).
  58. [ t 1 0 0 1 / t 1 ] [ a b c d ] [ 1 / t 1 0 0 t 1 ] = [ a t 1 b t 1 c / t 1 d / t 1 ] [ 1 / t 1 0 0 t 1 ] = [ a b t 1 2 c t 1 - 2 d ] \begin{bmatrix}t_{1}&0\\ 0&1/t_{1}\\ \end{bmatrix}\begin{bmatrix}a&b\\ c&d\\ \end{bmatrix}\begin{bmatrix}1/t_{1}&0\\ 0&t_{1}\\ \end{bmatrix}=\begin{bmatrix}at_{1}&bt_{1}\\ c/t_{1}&d/t_{1}\\ \end{bmatrix}\begin{bmatrix}1/t_{1}&0\\ 0&t_{1}\\ \end{bmatrix}=\begin{bmatrix}a&bt_{1}^{2}\\ ct_{1}^{-2}&d\\ \end{bmatrix}
  59. [ 1 0 0 0 ] [ 0 0 0 1 ] [ 0 1 0 0 ] [ 0 0 1 0 ] \begin{bmatrix}1&0\\ 0&0\\ \end{bmatrix}\begin{bmatrix}0&0\\ 0&1\\ \end{bmatrix}\begin{bmatrix}0&1\\ 0&0\\ \end{bmatrix}\begin{bmatrix}0&0\\ 1&0\\ \end{bmatrix}
  60. 1 , 1 , t 1 2 , t 1 - 2 1,1,t_{1}^{2},t_{1}^{-2}
  61. t 1 2 t_{1}^{2}
  62. ad x ( y ) = d ( Ad x ) e ( y ) = lim ε 0 ( I + ε x ) y ( I + ε x ) - 1 - y ε = lim ε 0 ( I + ε x ) y ( I - ε x + ( ε x ) 2 + O ( ε 3 ) ) - y ε = lim ε 0 ( ( I + ε x ) y I - ( I + ε x ) y ε x + ( I + ε x ) y ( ε x ) 2 + O ( ε 3 ) ) - y ε = lim ε 0 ( I y I + ε x y I - I y ε x - ε x y ε x + I y ( ε x ) 2 + ε x y ( ε x ) 2 + O ( ε 3 ) ) - y ε = lim ε 0 y + x y ε - y x ε - x y x ε 2 + y x 2 ε 2 + x y x 2 ε 2 + O ( ε 3 ) - y ε = lim ε 0 x y - y x - x y x ε + y x 2 ε + x y x 2 ε + O ( ε 2 ) = [ x , y ] \begin{aligned}\displaystyle\mathrm{ad}_{x}(y)&\displaystyle=d(\mathrm{Ad}_{x}% )_{e}(y)\\ &\displaystyle=\lim_{\varepsilon\to 0}\frac{(I+\varepsilon x)y(I+\varepsilon x% )^{-1}-y}{\varepsilon}\\ &\displaystyle=\lim_{\varepsilon\to 0}\frac{(I+\varepsilon x)y(I-\varepsilon x% +(\varepsilon x)^{2}+O(\varepsilon^{3}))-y}{\varepsilon}\\ &\displaystyle=\lim_{\varepsilon\to 0}\frac{((I+\varepsilon x)yI-(I+% \varepsilon x)y\varepsilon x+(I+\varepsilon x)y(\varepsilon x)^{2}+O(% \varepsilon^{3}))-y}{\varepsilon}\\ &\displaystyle=\lim_{\varepsilon\to 0}\frac{(IyI+\varepsilon xyI-Iy\varepsilon x% -\varepsilon xy\varepsilon x+Iy(\varepsilon x)^{2}+\varepsilon xy(\varepsilon x% )^{2}+O(\varepsilon^{3}))-y}{\varepsilon}\\ &\displaystyle=\lim_{\varepsilon\to 0}\frac{y+xy\varepsilon-yx\varepsilon-xyx% \varepsilon^{2}+yx^{2}\varepsilon^{2}+xyx^{2}\varepsilon^{2}+O(\varepsilon^{3}% )-y}{\varepsilon}\\ &\displaystyle=\lim_{\varepsilon\to 0}xy-yx-xyx\varepsilon+yx^{2}\varepsilon+% xyx^{2}\varepsilon+O(\varepsilon^{2})\\ &\displaystyle=[x,y]\end{aligned}

Adjugate_matrix.html

  1. adj ( 𝐀 ) = 𝐂 𝖳 \mathrm{adj}(\mathbf{A})=\mathbf{C}^{\mathsf{T}}
  2. 𝐂 i j = ( - 1 ) i + j 𝐀 i j \mathbf{C}_{ij}=(-1)^{i+j}\mathbf{A}_{ij}\,
  3. adj ( 𝐀 ) i j = 𝐂 j i = ( - 1 ) i + j 𝐀 j i \mathrm{adj}(\mathbf{A})_{ij}=\mathbf{C}_{ji}=(-1)^{i+j}\mathbf{A}_{ji}\,
  4. 𝐀 adj ( 𝐀 ) = det ( 𝐀 ) 𝐈 \mathbf{A}\,\mathrm{adj}(\mathbf{A})=\det(\mathbf{A})\,\mathbf{I}\,
  5. adj ( 𝐀 ) = det ( 𝐀 ) 𝐀 - 1 \mathrm{adj}(\mathbf{A})=\det(\mathbf{A})\mathbf{A}^{-1}\,
  6. 𝐀 - 1 = 1 det ( 𝐀 ) adj ( 𝐀 ) \mathbf{A}^{-1}=\frac{1}{\det(\mathbf{A})}\,\mathrm{adj}(\mathbf{A})\,
  7. 𝐀 = ( < m t p l > a b c d ) \mathbf{A}=\begin{pmatrix}<mtpl>{{a}}&{{b}}\\ {{c}}&{{d}}\end{pmatrix}
  8. adj ( 𝐀 ) = ( < m t p l > d - b - c a ) \operatorname{adj}(\mathbf{A})=\begin{pmatrix}\,\,\,<mtpl>{{d}}&\!\!{{-b}}\\ {{-c}}&{{a}}\end{pmatrix}
  9. 3 × 3 3\times 3
  10. 𝐀 = ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ) = ( 1 2 3 4 5 6 7 8 9 ) \mathbf{A}=\begin{pmatrix}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{pmatrix}=\begin{pmatrix}1&2&3\\ 4&5&6\\ 7&8&9\end{pmatrix}
  11. 𝐂 = ( + | a 22 a 23 a 32 a 33 | - | a 21 a 23 a 31 a 33 | + | a 21 a 22 a 31 a 32 | - | a 12 a 13 a 32 a 33 | + | a 11 a 13 a 31 a 33 | - | a 11 a 12 a 31 a 32 | + | a 12 a 13 a 22 a 23 | - | a 11 a 13 a 21 a 23 | + | a 11 a 12 a 21 a 22 | ) = ( + | 5 6 8 9 | - | 4 6 7 9 | + | 4 5 7 8 | - | 2 3 8 9 | + | 1 3 7 9 | - | 1 2 7 8 | + | 2 3 5 6 | - | 1 3 4 6 | + | 1 2 4 5 | ) \mathbf{C}=\begin{pmatrix}+\left|\begin{matrix}a_{22}&a_{23}\\ a_{32}&a_{33}\end{matrix}\right|&-\left|\begin{matrix}a_{21}&a_{23}\\ a_{31}&a_{33}\end{matrix}\right|&+\left|\begin{matrix}a_{21}&a_{22}\\ a_{31}&a_{32}\end{matrix}\right|\\ &&\\ -\left|\begin{matrix}a_{12}&a_{13}\\ a_{32}&a_{33}\end{matrix}\right|&+\left|\begin{matrix}a_{11}&a_{13}\\ a_{31}&a_{33}\end{matrix}\right|&-\left|\begin{matrix}a_{11}&a_{12}\\ a_{31}&a_{32}\end{matrix}\right|\\ &&\\ +\left|\begin{matrix}a_{12}&a_{13}\\ a_{22}&a_{23}\end{matrix}\right|&-\left|\begin{matrix}a_{11}&a_{13}\\ a_{21}&a_{23}\end{matrix}\right|&+\left|\begin{matrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{matrix}\right|\end{pmatrix}=\begin{pmatrix}+\left|\begin{% matrix}5&6\\ 8&9\end{matrix}\right|&-\left|\begin{matrix}4&6\\ 7&9\end{matrix}\right|&+\left|\begin{matrix}4&5\\ 7&8\end{matrix}\right|\\ &&\\ -\left|\begin{matrix}2&3\\ 8&9\end{matrix}\right|&+\left|\begin{matrix}1&3\\ 7&9\end{matrix}\right|&-\left|\begin{matrix}1&2\\ 7&8\end{matrix}\right|\\ &&\\ +\left|\begin{matrix}2&3\\ 5&6\end{matrix}\right|&-\left|\begin{matrix}1&3\\ 4&6\end{matrix}\right|&+\left|\begin{matrix}1&2\\ 4&5\end{matrix}\right|\end{pmatrix}
  12. adj ( 𝐀 ) = ( + | a 22 a 23 a 32 a 33 | - | a 12 a 13 a 32 a 33 | + | a 12 a 13 a 22 a 23 | - | a 21 a 23 a 31 a 33 | + | a 11 a 13 a 31 a 33 | - | a 11 a 13 a 21 a 23 | + | a 21 a 22 a 31 a 32 | - | a 11 a 12 a 31 a 32 | + | a 11 a 12 a 21 a 22 | ) = ( + | 5 6 8 9 | - | 2 3 8 9 | + | 2 3 5 6 | - | 4 6 7 9 | + | 1 3 7 9 | - | 1 3 4 6 | + | 4 5 7 8 | - | 1 2 7 8 | + | 1 2 4 5 | ) \operatorname{adj}(\mathbf{A})=\begin{pmatrix}+\left|\begin{matrix}a_{22}&a_{2% 3}\\ a_{32}&a_{33}\end{matrix}\right|&-\left|\begin{matrix}a_{12}&a_{13}\\ a_{32}&a_{33}\end{matrix}\right|&+\left|\begin{matrix}a_{12}&a_{13}\\ a_{22}&a_{23}\end{matrix}\right|\\ &&\\ -\left|\begin{matrix}a_{21}&a_{23}\\ a_{31}&a_{33}\end{matrix}\right|&+\left|\begin{matrix}a_{11}&a_{13}\\ a_{31}&a_{33}\end{matrix}\right|&-\left|\begin{matrix}a_{11}&a_{13}\\ a_{21}&a_{23}\end{matrix}\right|\\ &&\\ +\left|\begin{matrix}a_{21}&a_{22}\\ a_{31}&a_{32}\end{matrix}\right|&-\left|\begin{matrix}a_{11}&a_{12}\\ a_{31}&a_{32}\end{matrix}\right|&+\left|\begin{matrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{matrix}\right|\end{pmatrix}=\begin{pmatrix}+\left|\begin{% matrix}5&6\\ 8&9\end{matrix}\right|&-\left|\begin{matrix}2&3\\ 8&9\end{matrix}\right|&+\left|\begin{matrix}2&3\\ 5&6\end{matrix}\right|\\ &&\\ -\left|\begin{matrix}4&6\\ 7&9\end{matrix}\right|&+\left|\begin{matrix}1&3\\ 7&9\end{matrix}\right|&-\left|\begin{matrix}1&3\\ 4&6\end{matrix}\right|\\ &&\\ +\left|\begin{matrix}4&5\\ 7&8\end{matrix}\right|&-\left|\begin{matrix}1&2\\ 7&8\end{matrix}\right|&+\left|\begin{matrix}1&2\\ 4&5\end{matrix}\right|\end{pmatrix}
  13. | a i m a i n a j m a j n | = det ( a i m a i n a j m a j n ) \left|\begin{matrix}a_{im}&a_{in}\\ \,\,a_{jm}&a_{jn}\end{matrix}\right|=\det\left(\begin{matrix}a_{im}&a_{in}\\ \,\,a_{jm}&a_{jn}\end{matrix}\right)
  14. 𝐂 = ( - 3 6 - 3 6 - 12 6 - 3 6 - 3 ) \mathbf{C}=\begin{pmatrix}-3&6&-3\\ 6&-12&6\\ -3&6&-3\end{pmatrix}
  15. adj ( - 3 2 - 5 - 1 0 - 2 3 - 4 1 ) = ( - 8 18 - 4 - 5 12 - 1 4 - 6 2 ) \operatorname{adj}\begin{pmatrix}\!-3&\,2&\!-5\\ \!-1&\,0&\!-2\\ \,3&\!-4&\,1\end{pmatrix}=\begin{pmatrix}\!-8&\,18&\!-4\\ \!-5&\!12&\,-1\\ \,4&\!-6&\,2\end{pmatrix}
  16. ( - 1 ) 2 + 3 det ( - 3 2 3 - 4 ) = - ( ( - 3 ) ( - 4 ) - ( 3 ) ( 2 ) ) = - 6. (-1)^{2+3}\;\operatorname{det}\begin{pmatrix}\!-3&\,2\\ \,3&\!-4\end{pmatrix}=-((-3)(-4)-(3)(2))=-6.
  17. ( - 3 2 3 - 4 ) \begin{pmatrix}\!-3&\,\!2\\ \,\!3&\!-4\end{pmatrix}
  18. adj ( 𝐈 ) = 𝐈 , \mathrm{adj}(\mathbf{I})=\mathbf{I},
  19. adj ( 𝐀𝐁 ) = adj ( 𝐁 ) adj ( 𝐀 ) , \mathrm{adj}(\mathbf{AB})=\mathrm{adj}(\mathbf{B})\,\mathrm{adj}(\mathbf{A}),
  20. adj ( c 𝐀 ) = c n - 1 adj ( 𝐀 ) \mathrm{adj}(c\mathbf{A})=c^{n-1}\mathrm{adj}(\mathbf{A})
  21. adj ( 𝐀 m ) = adj ( 𝐀 ) m . \mathrm{adj}(\mathbf{A}^{m})=\mathrm{adj}(\mathbf{A})^{m}.
  22. adj ( 𝐀 𝖳 ) = adj ( 𝐀 ) 𝖳 . \mathrm{adj}(\mathbf{A}^{\mathsf{T}})=\mathrm{adj}(\mathbf{A})^{\mathsf{T}}.
  23. det ( adj ( 𝐀 ) ) = det ( 𝐀 ) n - 1 , \det\big(\mathrm{adj}(\mathbf{A})\big)=\det(\mathbf{A})^{n-1},
  24. adj ( adj ( 𝐀 ) ) = det ( 𝐀 ) n - 2 𝐀 \mathrm{adj}(\mathrm{adj}(\mathbf{A}))=\det(\mathbf{A})^{n-2}\mathbf{A}
  25. k k
  26. 𝐀 \mathbf{A}
  27. adj k ( 𝐀 ) = det ( 𝐀 ) ( n - 1 ) k - ( - 1 ) k n 𝐀 ( - 1 ) k \mathrm{adj}_{k}(\mathbf{A})=\det(\mathbf{A})^{\frac{(n-1)^{k}-(-1)^{k}}{n}}% \mathbf{A}^{(-1)^{k}}
  28. det ( adj k ( 𝐀 ) ) = det ( 𝐀 ) ( n - 1 ) k \det\big(\mathrm{adj}_{k}(\mathbf{A})\big)=\det(\mathbf{A})^{(n-1)^{k}}
  29. 𝐈 n \mathbf{I}_{n}
  30. R R
  31. R R
  32. 1 = det ( 𝐈 n ) = det ( 𝐀𝐀 - 1 ) = det ( 𝐀 ) det ( 𝐀 - 1 ) , 1=\det(\mathbf{I}_{n})=\det(\mathbf{A}\mathbf{A}^{-1})=\det(\mathbf{A})\det(% \mathbf{A}^{-1})~{},
  33. 𝐀 - 1 = det ( 𝐀 ) - 1 adj ( 𝐀 ) . \mathbf{A}^{-1}=\det(\mathbf{A})^{-1}\,\mathrm{adj}(\mathbf{A}).
  34. R ( t ; 𝐀 ) 𝐈 t 𝐈 - 𝐀 = adj ( t 𝐈 - 𝐀 ) p ( t ) , R(t;\mathbf{A})\equiv\frac{\mathbf{I}}{t\mathbf{\mathbf{I}}-\mathbf{A}}=\frac{% \mathrm{adj}(t\mathbf{I}-\mathbf{A})}{p(t)}~{},
  35. p ( t ) p(t)
  36. q ( t ) ( p ( 0 ) p ( t ) ) / t q(t)≡(p(0) −p(t))/t
  37. adj ( 𝐀 ) = q ( 𝐀 ) = - ( p 1 𝐈 + p 2 𝐀 + p 3 𝐀 2 + + p n 𝐀 n - 1 ) = p 0 𝐀 - 1 , \mathrm{adj}(\mathbf{A})=q(\mathbf{A})=-(p_{1}\mathbf{I}+p_{2}\mathbf{A}+p_{3}% \mathbf{A}^{2}+\cdots+p_{n}\mathbf{A}^{n-1})=p_{0}~{}\mathbf{A}^{-1}~{},
  38. p j p_{j}
  39. p ( t ) = p 0 + p 1 t + p 2 t 2 + + p n t n . p(t)=p_{0}+p_{1}t+p_{2}t^{2}+\cdots+p_{n}t^{n}.
  40. p ( t ) p(t)
  41. d d α det ( A ) = tr ( adj ( A ) d A d α ) . \frac{\mathrm{d}}{\mathrm{d}\alpha}\det(A)=\operatorname{tr}\left(% \operatorname{adj}(A)\frac{\mathrm{d}A}{\mathrm{d}\alpha}\right).
  42. adj ( 𝐀 ) = s = 0 n - 1 𝐀 s k 1 , k 2 , , k n - 1 l = 1 n - 1 ( - 1 ) k l + 1 l k l k l ! tr ( 𝐀 l ) k l , \mathrm{adj}(\mathbf{A})=\sum_{s=0}^{n-1}\mathbf{A}^{s}\sum_{k_{1},k_{2},% \ldots,k_{n-1}}\prod_{l=1}^{n-1}\frac{(-1)^{k_{l}+1}}{l^{k_{l}}k_{l}!}\mathrm{% tr}(\mathbf{A}^{l})^{k_{l}},
  43. s + l = 1 n - 1 l k l = n - 1. s+\sum_{l=1}^{n-1}lk_{l}=n-1.
  44. adj ( 𝐀 ) = 𝐈 2 tr 𝐀 - 𝐀 . \mathrm{adj}(\mathbf{A})=\mathbf{I}_{2}\mathrm{tr}\mathbf{A}-\mathbf{A}.
  45. adj ( 𝐀 ) = 1 2 ( ( tr 𝐀 ) 2 - tr 𝐀 2 ) 𝐈 3 - 𝐀 tr 𝐀 + 𝐀 2 . \mathrm{adj}(\mathbf{A})=\frac{1}{2}\left((\mathrm{tr}\mathbf{A})^{2}-\mathrm{% tr}\mathbf{A}^{2}\right)\mathbf{I}_{3}-\mathbf{A}\mathrm{tr}\mathbf{A}+\mathbf% {A}^{2}.
  46. adj ( 𝐀 ) = 1 6 ( ( tr 𝐀 ) 3 - 3 t r 𝐀 tr 𝐀 2 + 2 t r 𝐀 3 ) 𝐈 4 - 1 2 𝐀 ( ( tr 𝐀 ) 2 - tr 𝐀 2 ) + 𝐀 2 tr 𝐀 - 𝐀 3 . \mathrm{adj}(\mathbf{A})=\frac{1}{6}\left((\mathrm{tr}\mathbf{A})^{3}-3\mathrm% {tr}\mathbf{A}\mathrm{tr}\mathbf{A}^{2}+2\mathrm{tr}\mathbf{A}^{3}\right)% \mathbf{I}_{4}-\frac{1}{2}\mathbf{A}\left((\mathrm{tr}\mathbf{A})^{2}-\mathrm{% tr}\mathbf{A}^{2}\right)+\mathbf{A}^{2}\mathrm{tr}\mathbf{A}-\mathbf{A}^{3}.

Admittance.html

  1. Y 1 Z Y\equiv\frac{1}{Z}\,
  2. Y = G + j B Y=G+jB\,
  3. Y Y
  4. G G
  5. B B
  6. j 2 = - 1 j^{2}=-1
  7. Z = R + j X Z=R+jX\,
  8. Y = Z - 1 = 1 R + j X = ( 1 R 2 + X 2 ) ( R - j X ) Y=Z^{-1}=\frac{1}{R+jX}=\left(\frac{1}{R^{2}+X^{2}}\right)\left(R-jX\right)
  9. Y = G + j B Y=G+jB\,\!
  10. G \displaystyle G
  11. | Y | \displaystyle\left|Y\right|

Adsorption.html

  1. x m = k P 1 n \frac{x}{m}=kP^{\frac{1}{n}}
  2. x {x}
  3. m m
  4. P P
  5. k k
  6. n n
  7. x / m x/m
  8. k k
  9. n n
  10. A g + S A S A_{g}+S\rightleftharpoons AS
  11. θ \theta
  12. K = k k - 1 = θ ( 1 - θ ) P K=\frac{k}{k_{-1}}=\frac{\theta}{(1-\theta)P}
  13. θ = K P 1 + K P \theta=\frac{KP}{1+KP}
  14. P P
  15. θ K P \theta\approx KP
  16. θ 1 \theta\approx 1
  17. θ \theta
  18. θ = v v mon \theta=\frac{v}{v_{\mathrm{mon}}}
  19. 1 v = 1 K v mon 1 P + 1 v mon \frac{1}{v}=\frac{1}{Kv_{\mathrm{mon}}}\frac{1}{P}+\frac{1}{v_{\mathrm{mon}}}
  20. θ E \theta_{E}
  21. θ E = 1 1 + i = 1 n K i P i \theta_{E}=\frac{1}{\displaystyle 1+\sum_{i=1}^{n}K_{i}P_{i}}
  22. θ j \theta_{j}
  23. θ j = K j P j 1 + i = 1 n K i P i \theta_{j}=\frac{K_{j}P_{j}}{\displaystyle 1+\sum_{i=1}^{n}K_{i}P_{i}}
  24. x v ( 1 - x ) = 1 v mon c + x ( c - 1 ) v mon c . \frac{x}{v(1-x)}=\frac{1}{v_{\mathrm{mon}}c}+\frac{x(c-1)}{v_{\mathrm{mon}}c}.
  25. P / P 0 P/P_{0}
  26. k E = S E k ES . S D . k_{\mathrm{E}}=\frac{S_{\mathrm{E}}}{k_{\mathrm{ES}}.S_{\mathrm{D}}}.
  27. d θ ( t ) d t = \R ( 1 - θ ) ( 1 + k E θ ) . \frac{d\theta_{\mathrm{(t)}}}{dt}=\R^{\prime}(1-\theta)(1+k_{\mathrm{E}}\theta).
  28. θ ( t ) = 1 - e - R ( 1 + k E ) t 1 + k E e - R ( 1 + k E ) t . \theta_{\mathrm{(t)}}=\frac{1-e^{-R^{\prime}(1+k_{\mathrm{E}})t}}{1+k_{\mathrm% {E}}e^{-R^{\prime}(1+k_{\mathrm{E}})t}}.
  29. ( ln K 1 T ) θ = - Δ H R . \left(\frac{\partial\ln K}{\partial\frac{1}{T}}\right)_{\theta}=-\frac{\Delta H% }{R}.
  30. Δ H ads = Δ H liq - R T ln c , \Delta H_{\mathrm{ads}}=\Delta H_{\mathrm{liq}}-RT\ln c,

Advection.html

  1. 𝐮 = u x x + u y y + u z z \mathbf{u}\cdot\nabla=u_{x}\frac{\partial}{\partial x}+u_{y}\frac{\partial}{% \partial y}+u_{z}\frac{\partial}{\partial z}
  2. u = 0 \nabla\cdot{u}=0
  3. u ψ = 0 {u}\cdot\nabla\psi=0
  4. ψ / t = 0 , \partial\psi/\partial t=0,
  5. a t + ( u ) a = 0. \frac{\partial{a}}{\partial t}+\left({u}\cdot\nabla\right){a}=0.
  6. ψ t + u x ψ x = 0 \frac{\partial\psi}{\partial t}+u_{x}\frac{\partial\psi}{\partial x}=0
  7. 1 2 u u + 1 2 ( u u ) \frac{1}{2}{u}\cdot\nabla{u}+\frac{1}{2}\nabla({u}{u})
  8. ( u u ) = [ ( u u x ) , ( u u y ) , ( u u z ) ] \nabla({u}{u})=[\nabla({u}u_{x}),\nabla({u}u_{y}),\nabla({u}u_{z})]
  9. 𝐮 𝐮 = ( 𝐮 2 2 ) + ( × 𝐮 ) × 𝐮 \mathbf{u}\cdot\nabla\mathbf{u}=\nabla\left(\frac{\|\mathbf{u}\|^{2}}{2}\right% )+\left(\nabla\times\mathbf{u}\right)\times\mathbf{u}
  10. 1 2 𝐮 𝐮 + 1 2 ( 𝐮𝐮 ) = ( 𝐮 2 2 ) + ( × 𝐮 ) × 𝐮 + 1 2 𝐮 ( 𝐮 ) \frac{1}{2}\mathbf{u}\cdot\nabla\mathbf{u}+\frac{1}{2}\nabla(\mathbf{u}\mathbf% {u})=\nabla\left(\frac{\|\mathbf{u}\|^{2}}{2}\right)+\left(\nabla\times\mathbf% {u}\right)\times\mathbf{u}+\frac{1}{2}\mathbf{u}(\nabla\cdot\mathbf{u})

Adverse_pressure_gradient.html

  1. d P / d x > 0 dP/dx>0
  2. x x

Aeroelasticity.html

  1. G J d 2 θ d y 2 = - M GJ\frac{d^{2}\theta}{dy^{2}}=-M^{\prime}
  2. M = C U 2 ( θ + α 0 ) M^{\prime}=CU^{2}(\theta+\alpha_{0})
  3. d 2 θ d y 2 + λ 2 θ = - λ 2 α 0 \frac{d^{2}\theta}{dy^{2}}+\lambda^{2}\theta=-\lambda^{2}\alpha_{0}
  4. λ 2 = C / ( G J ) \lambda^{2}=C/(GJ)
  5. θ | y = 0 = d θ d y | y = L = 0 \theta|_{y=0}=\frac{d\theta}{dy}\biggl|_{y=L}=0
  6. θ = α 0 [ tan ( λ L ) sin ( λ y ) + cos ( λ y ) - 1 ] \theta=\alpha_{0}[\tan(\lambda L)\sin(\lambda y)+\cos(\lambda y)-1]

Affine_combination.html

  1. i = 1 n α i x i = α 1 x 1 + α 2 x 2 + + α n x n , \sum_{i=1}^{n}{\alpha_{i}\cdot x_{i}}=\alpha_{1}x_{1}+\alpha_{2}x_{2}+\cdots+% \alpha_{n}x_{n},
  2. i = 1 n α i = 1. \sum_{i=1}^{n}{\alpha_{i}}=1.
  3. α i \alpha_{i}
  4. T i = 1 n α i x i = i = 1 n α i T x i T\sum_{i=1}^{n}{\alpha_{i}\cdot x_{i}}=\sum_{i=1}^{n}{\alpha_{i}\cdot Tx_{i}}
  5. T T
  6. T T
  7. T T

Affine_geometry.html

  1. A , B , C A,B,C
  2. A , B , C A^{\prime},B^{\prime},C^{\prime}
  3. ( A B A B and B C B C ) C A C A . (AB^{\prime}\parallel A^{\prime}B\ \and\ BC^{\prime}\parallel B^{\prime}C)% \Rightarrow CA^{\prime}\parallel C^{\prime}A.
  4. V GL ( V ) V\rtimes\mathrm{GL}(V)
  5. 3 4 log e ( 2 ) - 1 2 , \tfrac{3}{4}\log_{e}(2)-\tfrac{1}{2},

Affine_group.html

  1. Aff ( A ) = V GL ( V ) \operatorname{Aff}(A)=V\rtimes\operatorname{GL}(V)
  2. Aff ( n , K ) = K n GL ( n , K ) \operatorname{Aff}(n,K)=K^{n}\rtimes\operatorname{GL}(n,K)
  3. ( A , p ) (A,p)
  4. 1 V V GL ( V ) GL ( V ) 1 1\to V\to V\rtimes\operatorname{GL}(V)\to\operatorname{GL}(V)\to 1
  5. ( M , v ) ( N , w ) = ( M N , v + M w ) . (M,v)\cdot(N,w)=(MN,v+Mw).\,
  6. ( M v 0 1 ) \left(\begin{array}[]{c|c}M&v\\ \hline 0&1\end{array}\right)
  7. GL ( V K ) \operatorname{GL}(V\oplus K)
  8. { ( v , 1 ) | v V } \{(v,1)|v\in V\}
  9. V K V\oplus K
  10. x a x + b y , y - b x + a y , a , b 0 , a 2 + b 2 = 1 , x\mapsto ax+by,\quad y\mapsto-bx+ay,\quad a,b\neq 0,\quad a^{2}+b^{2}=1,
  11. x x + b y , y y , b 0 , x\mapsto x+by,\quad y\mapsto y,\quad b\neq 0,
  12. x a x , y y / a , a 0 , x\mapsto ax,\quad y\mapsto y/a,\quad a\neq 0,
  13. G < G L ( V ) G<GL(V)
  14. Aff ( G ) \operatorname{Aff}(G)
  15. Aff ( G ) := V G \operatorname{Aff}(G):=V\rtimes G
  16. ρ : G GL ( V ) \rho\colon G\to\operatorname{GL}(V)
  17. V ρ G V\rtimes_{\rho}G
  18. 1 V V ρ G G 1. 1\to V\to V\rtimes_{\rho}G\to G\to 1.
  19. 𝔓 \mathfrak{P}
  20. 𝔄 \mathfrak{A}
  21. 𝔓 \mathfrak{P}
  22. 𝔓 \mathfrak{P}
  23. 𝔄 𝔓 \mathfrak{A}\subset\mathfrak{P}
  24. O ( 1 , 3 ) O(1,3)
  25. 𝐑 1 , 3 O ( 1 , 3 ) \mathbf{R}^{1,3}\rtimes\operatorname{O}(1,3)
  26. GL ( V ) < Aut ( V ) \operatorname{GL}(V)<\operatorname{Aut}(V)

Affine_space.html

  1. 𝐩 \mathbf{p}
  2. 𝐚 \mathbf{a}
  3. 𝐛 \mathbf{b}
  4. 𝐩 \mathbf{p}
  5. 𝐚 \mathbf{a}
  6. 𝐩 \mathbf{p}
  7. 𝐛 \mathbf{b}
  8. 𝐚 + 𝐛 \mathbf{a}+\mathbf{b}
  9. s i z e = 120 % 𝐩 + ( 𝐚 𝐩 ) + ( 𝐛 𝐩 ) size=120\%\mathbf{p}+(\mathbf{a}−\mathbf{p})+(\mathbf{b}−\mathbf{p})
  10. 𝐚 \mathbf{a}
  11. 𝐛 \mathbf{b}
  12. s i z e = 120 % λ 𝐚 + ( 1 λ ) 𝐛 size=120\%λ\mathbf{a}+(1−λ)\mathbf{b}
  13. s i z e = 120 % 𝐩 + λ ( 𝐚 𝐩 ) + ( 1 λ ) ( 𝐛 𝐩 ) = λ 𝐚 + ( 1 λ ) 𝐛 size=120\%\mathbf{p}+λ(\mathbf{a}−\mathbf{p})+(1−λ)(\mathbf{b}−\mathbf{p})=λ% \mathbf{a}+(1−λ)\mathbf{b}
  14. λ + ( 1 λ ) = 1 λ+(1−λ)=1
  15. A A
  16. V V
  17. F F
  18. V V
  19. A A
  20. A A
  21. l : V × A A , ( 𝐯 , a ) 𝐯 + a l\colon V\times A\to A,\;(\mathbf{v},a)\mapsto\mathbf{v}+a
  22. a A , 0 + a = a \forall a\in A,\;\mathbf{0}+a=a
  23. 𝐯 , 𝐰 V , a A , 𝐯 + ( 𝐰 + a ) = ( 𝐯 + 𝐰 ) + a \forall\mathbf{v},\mathbf{w}\in V,\forall a\in A,\;\mathbf{v}+(\mathbf{w}+a)=(% \mathbf{v}+\mathbf{w})+a
  24. a A , V A : 𝐯 𝐯 + a \forall a\in A,\;V\to A\colon\mathbf{v}\mapsto\mathbf{v}+a\quad
  25. o o
  26. A A
  27. V V
  28. A A
  29. V V
  30. A A
  31. V V
  32. a - b a\,-\,b\;
  33. V V
  34. ( a - b ) + b = a \left(a\,-\,b\right)\,+\,b\;=\;a
  35. A A
  36. V V
  37. ϕ : A × A V , ( a , b ) b - a a b \operatorname{\phi}:\;A\,\times\,A\;\to\;V,\;\left(a,\,b\right)\,\mapsto\,b\,-% \,a\;\equiv\;\overrightarrow{ab}
  38. p A , 𝐯 V \forall p\,\in\,A,\;\forall\mathbf{v}\,\in\,V
  39. q A q\,\in\,A
  40. q - p = 𝐯 q\,-\,p\;=\;\mathbf{v}
  41. p , q , r A , ( q - p ) + ( r - q ) = r - p \forall p,\,q,\,r\,\in\,A,\;(q\,-\,p)\,+\,(r\,-\,q)\;=\;r\,-\,p
  42. o o
  43. a a
  44. b b
  45. A A
  46. λ λ
  47. A A
  48. λ a + ( 1 - λ ) b \lambda a+(1-\lambda)b
  49. ( λ a + ( 1 - λ ) b ) - o = λ ( a - o ) + ( 1 - λ ) ( b - o ) . (\lambda a+(1-\lambda)b)-o=\lambda(a-o)+(1-\lambda)(b-o).
  50. o o
  51. V V
  52. T T
  53. 𝐛 \mathbf{b}
  54. T 𝐱 = 𝐛 T\mathbf{x}=\mathbf{b}
  55. T 𝐱 = 0 T\mathbf{x}=0
  56. T : V W T:V→W
  57. 𝐲 \mathbf{y}
  58. 𝐱 V \mathbf{x}∈V
  59. T 𝐱 = 𝐲 T\mathbf{x}=\mathbf{y}
  60. T T
  61. K e r T KerT
  62. V V
  63. A = { i N α i 𝐯 i | i N α i = 1 } A=\Bigl\{\sum^{N}_{i}\alpha_{i}\mathbf{v}_{i}\Big|\sum^{N}_{i}\alpha_{i}=1\Bigr\}
  64. { 𝐯 i } i I \scriptstyle\{\mathbf{v}_{i}\}_{i\in I}
  65. V V
  66. W W
  67. V V
  68. W = { i N β i 𝐯 i | i N β i = 0 } . W=\Bigl\{\sum^{N}_{i}\beta_{i}\mathbf{v}_{i}\Big|\sum^{N}_{i}\beta_{i}=0\Bigr\}.
  69. W W
  70. S = 𝐩 + W , S=\mathbf{p}+W,\,
  71. p p
  72. A A
  73. V V / W V→V/W
  74. p p
  75. A A
  76. W W
  77. A A
  78. W W
  79. A A
  80. 3 \scriptstyle{\mathbb{R}^{3}}
  81. i = 1 n a i = 0 \sum_{i=1}^{n}a_{i}=0
  82. A A
  83. V V

Aftershock.html

  1. n ( t ) = K c + t n(t)=\frac{K}{c+t}
  2. n ( t ) = k ( c + t ) p n(t)=\frac{k}{(c+t)^{p}}
  3. N = 10 a - b M \!\,N=10^{a-bM}
  4. N N
  5. M M
  6. M M
  7. a a
  8. b b

Aharonov–Bohm_effect.html

  1. 𝐁 = 0 = × 𝐀 \mathbf{B}=0=\nabla\times\mathbf{A}
  2. φ \varphi
  3. φ = q P 𝐀 d 𝐱 , \varphi=\frac{q}{\hbar}\int_{P}\mathbf{A}\cdot d\mathbf{x},
  4. Δ φ \Delta\varphi
  5. Φ B \Phi_{B}
  6. × 𝐀 = 𝐁 \nabla\times\mathbf{A}=\mathbf{B}
  7. Δ φ = q Φ B . \Delta\varphi=\frac{q\Phi_{B}}{\hbar}.
  8. Δ φ \Delta\varphi
  9. 2 q e q m c 2\frac{q\text{e}q\text{m}}{\hbar c}
  10. exp ( - i E t / ) \exp(-iEt/\hbar)
  11. Δ ϕ = - q V t , \Delta\phi=-\frac{qVt}{\hbar},
  12. / i \hbar/i
  13. i = x i \partial_{i}=\frac{\partial}{\partial x^{i}}
  14. e - i ϕ ( x ) e^{-i\phi(x)}
  15. i = i + i ( i ϕ ) \nabla_{i}=\partial_{i}+i(\partial_{i}\phi)
  16. A = d ϕ A=d\phi
  17. \nabla
  18. i F = iF=\nabla\wedge\nabla
  19. = d + i A \nabla=d+iA\,
  20. γ \gamma
  21. e i γ A e^{i\int_{\gamma}A}
  22. σ \sigma
  23. γ \gamma
  24. σ \sigma
  25. e i σ A = e i σ d A = e i σ F e^{i\int_{\partial\sigma}A}=e^{i\int_{\sigma}dA}=e^{i\int_{\sigma}F}
  26. F = 0 F=0
  27. \nabla
  28. α = \alpha=
  29. ( / e ) (\hbar/e)
  30. F F
  31. H = 1 2 m * H=\frac{1}{2m}\nabla^{*}\nabla
  32. e i α e^{i\alpha}

Airfoil.html

  1. c c
  2. 2 π 2\pi\!
  3. c L = 2 π α \ c_{L}=2\pi\alpha
  4. c L c_{L}\!
  5. α \alpha\!
  6. α \alpha\!
  7. c L = c L 0 + 2 π α \ c_{L}=c_{L_{0}}+2\pi\alpha
  8. c L 0 \ c_{L_{0}}
  9. γ ( s ) \gamma(s)
  10. w ( x ) w(x)
  11. w ( x ) = 1 ( 2 π ) 0 c γ ( x ) ( x - x ) d x w(x)=\frac{1}{(2\pi)}\int_{0}^{c}\frac{\gamma(x^{\prime})}{(x-x^{\prime})}dx^{\prime}
  12. x x
  13. x x^{\prime}
  14. c c
  15. w ( x ) w(x)
  16. V V
  17. α - d y / d x \alpha-dy/dx
  18. V ( α - d y / d x ) = w ( x ) = 1 ( 2 π ) 0 c γ ( x ) ( x - x ) d x V\;(\alpha-dy/dx)=w(x)=\frac{1}{(2\pi)}\int_{0}^{c}\frac{\gamma(x^{\prime})}{(% x-x^{\prime})}dx^{\prime}
  19. γ ( x ) \gamma(x)
  20. x = c ( 1 - cos ( θ ) ) / 2 \ x=c(1-\cos(\theta))/2
  21. A n sin ( n θ ) A_{n}\sin(n\theta)
  22. A 0 ( 1 + cos ( θ ) ) / sin ( θ ) A_{0}(1+\cos(\theta))/\sin(\theta)
  23. γ ( θ ) ( 2 V ) = A 0 ( 1 + cos ( θ ) ) sin ( θ ) + A n sin ( n θ ) ) \frac{\gamma(\theta)}{(2V)}=A_{0}\frac{(1+\cos(\theta))}{\sin(\theta)}+\sum A_% {n}\;\sin(n\theta))
  24. A 0 = α - 1 π 0 π ( d y / d x ) d θ A_{0}=\alpha-\frac{1}{\pi}\int_{0}^{\pi}(dy/dx)\;d\theta
  25. A n = 2 π 0 π cos ( n θ ) ( d y / d x ) d θ A_{n}=\frac{2}{\pi}\int_{0}^{\pi}\cos(n\theta)(dy/dx)\;d\theta
  26. ρ V 0 c γ ( x ) d x \rho V\int_{0}^{c}\gamma(x)\;dx
  27. ρ V 0 c x γ ( x ) d x \rho V\int_{0}^{c}x\;\gamma(x)\;dx
  28. C L = 2 π ( A 0 + A 1 / 2 ) \ C_{L}=2\pi(A_{0}+A_{1}/2)
  29. A 0 , A 1 A_{0},A_{1}
  30. A 2 A_{2}
  31. C M = - 0.5 π ( A 0 + A 1 - A 2 / 2 ) \ C_{M}=-0.5\pi(A_{0}+A_{1}-A_{2}/2)
  32. C M ( 1 / 4 c ) = - π / 4 ( A 1 - A 2 ) \ C_{M}(1/4c)=-\pi/4(A_{1}-A_{2})
  33. Δ x / c = π / 4 ( ( A 1 - A 2 ) / C L ) \ \Delta x/c=\pi/4((A_{1}-A_{2})/C_{L})
  34. ( C M ) ( C L ) = 0 \frac{\partial(C_{M^{\prime}})}{\partial(C_{L})}=0

Akra–Bazzi_method.html

  1. T ( x ) = g ( x ) + i = 1 k a i T ( b i x + h i ( x ) ) for x x 0 . T(x)=g(x)+\sum_{i=1}^{k}a_{i}T(b_{i}x+h_{i}(x))\qquad\,\text{for }x\geq x_{0}.
  2. a i a_{i}
  3. b i b_{i}
  4. a i > 0 a_{i}>0
  5. 0 < b i < 1 0<b_{i}<1
  6. | g ( x ) | O ( x c ) \left|g(x)\right|\in O(x^{c})
  7. | h i ( x ) | O ( x ( log x ) 2 ) \left|h_{i}(x)\right|\in O\left(\frac{x}{(\log x)^{2}}\right)
  8. x 0 x_{0}
  9. i = 1 k a i b i p = 1 \sum_{i=1}^{k}a_{i}b_{i}^{p}=1
  10. T ( x ) Θ ( x p ( 1 + 1 x g ( u ) u p + 1 d u ) ) T(x)\in\Theta\left(x^{p}\left(1+\int_{1}^{x}\frac{g(u)}{u^{p+1}}du\right)\right)
  11. h i ( x ) h_{i}(x)
  12. b i x = b i x + ( b i x - b i x ) \lfloor b_{i}x\rfloor=b_{i}x+(\lfloor b_{i}x\rfloor-b_{i}x)
  13. b i x - b i x \lfloor b_{i}x\rfloor-b_{i}x
  14. h i ( x ) h_{i}(x)
  15. T ( n ) = n + T ( 1 2 n ) T(n)=n+T\left(\frac{1}{2}n\right)
  16. T ( n ) = n + T ( 1 2 n ) T(n)=n+T\left(\left\lfloor\frac{1}{2}n\right\rfloor\right)
  17. T ( n ) T(n)
  18. 0 n 3 0\leq n\leq 3
  19. n 2 + 7 4 T ( 1 2 n ) + T ( 3 4 n ) n^{2}+\frac{7}{4}T\left(\left\lfloor\frac{1}{2}n\right\rfloor\right)+T\left(% \left\lceil\frac{3}{4}n\right\rceil\right)
  20. n > 3 n>3
  21. 7 4 ( 1 2 ) p + ( 3 4 ) p = 1 \frac{7}{4}\left(\frac{1}{2}\right)^{p}+\left(\frac{3}{4}\right)^{p}=1
  22. T ( x ) \displaystyle T(x)
  23. T ( 1 ) = 0 T(1)=0
  24. T ( n ) = T ( 1 2 n ) + T ( 1 2 n ) + n - 1 T(n)=T\left(\left\lfloor\frac{1}{2}n\right\rfloor\right)+T\left(\left\lceil% \frac{1}{2}n\right\rceil\right)+n-1
  25. n > 0 n>0
  26. Θ ( n log n ) \Theta(n\log n)

Aleph_number.html

  1. \aleph
  2. 0 \aleph_{0}
  3. 1 \aleph_{1}
  4. 2 \aleph_{2}
  5. α \aleph_{\alpha}
  6. 0 \aleph_{0}
  7. 0 \aleph_{0}
  8. 0 \aleph_{0}
  9. 0 \aleph_{0}
  10. 0 \aleph_{0}
  11. 1 \aleph_{1}
  12. 1 \aleph_{1}
  13. 0 \aleph_{0}
  14. 1 \aleph_{1}
  15. 0 \aleph_{0}
  16. 1 \aleph_{1}
  17. 1 \aleph_{1}
  18. 0 \aleph_{0}
  19. 2 0 2^{\aleph_{0}}
  20. 2 0 = 1 . 2^{\aleph_{0}}=\aleph_{1}.
  21. ω \aleph_{\omega}
  22. { n : n { 0 , 1 , 2 , } } \left\{\,\aleph_{n}:n\in\left\{\,0,1,2,\dots\,\right\}\,\right\}
  23. 2 0 = n 2^{\aleph_{0}}=\aleph_{n}
  24. 2 0 2^{\aleph_{0}}
  25. 0 \aleph_{0}
  26. 0 \aleph_{0}
  27. α \aleph_{\alpha}
  28. α \alpha
  29. + {}^{+}
  30. 0 = ω \aleph_{0}=\omega
  31. α + 1 = α + \aleph_{\alpha+1}=\aleph_{\alpha}^{+}
  32. λ = β < λ β . \aleph_{\lambda}=\bigcup_{\beta<\lambda}\aleph_{\beta}.
  33. ω α \omega_{\alpha}
  34. α \aleph_{\alpha}
  35. \aleph
  36. α ω α . \alpha\leq\omega_{\alpha}.
  37. ω α \omega_{\alpha}
  38. ω , ω ω , ω ω ω , . \omega,\ \omega_{\omega},\ \omega_{\omega_{\omega}},\ \ldots.
  39. κ = λ \kappa=\aleph_{\lambda}
  40. λ \lambda
  41. λ \aleph_{\lambda}
  42. λ \lambda
  43. κ \kappa
  44. λ \aleph_{\lambda}
  45. κ \kappa
  46. κ \kappa
  47. λ κ \lambda\geq\kappa
  48. λ = κ \lambda=\kappa

Alexandrian_text-type.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}
  5. 𝔓 \mathfrak{P}
  6. 𝔓 \mathfrak{P}
  7. 𝔓 \mathfrak{P}
  8. 𝔓 \mathfrak{P}
  9. 𝔓 \mathfrak{P}
  10. 𝔓 \mathfrak{P}
  11. 𝔓 \mathfrak{P}
  12. 𝔓 \mathfrak{P}
  13. 𝔓 \mathfrak{P}
  14. 𝔓 \mathfrak{P}
  15. 𝔓 \mathfrak{P}
  16. 𝔓 \mathfrak{P}
  17. 𝔓 \mathfrak{P}
  18. 𝔓 \mathfrak{P}
  19. 𝔓 \mathfrak{P}
  20. 𝔓 \mathfrak{P}
  21. 𝔓 \mathfrak{P}
  22. 𝔓 \mathfrak{P}
  23. 𝔓 \mathfrak{P}
  24. 𝔓 \mathfrak{P}
  25. 𝔓 \mathfrak{P}
  26. 𝔓 \mathfrak{P}
  27. 𝔓 \mathfrak{P}
  28. 𝔓 \mathfrak{P}
  29. 𝔓 \mathfrak{P}
  30. 𝔓 \mathfrak{P}
  31. 𝔓 \mathfrak{P}
  32. 𝔓 \mathfrak{P}
  33. 𝔓 \mathfrak{P}
  34. 𝔓 \mathfrak{P}
  35. 𝔓 \mathfrak{P}
  36. 𝔓 \mathfrak{P}
  37. 𝔓 \mathfrak{P}
  38. 𝔓 \mathfrak{P}
  39. 𝔓 \mathfrak{P}
  40. 𝔓 \mathfrak{P}
  41. 𝔓 \mathfrak{P}
  42. 𝔓 \mathfrak{P}
  43. 𝔓 \mathfrak{P}
  44. 𝔓 \mathfrak{P}
  45. 𝔓 \mathfrak{P}
  46. 𝔓 \mathfrak{P}
  47. 𝔓 \mathfrak{P}
  48. 𝔓 \mathfrak{P}
  49. 𝔓 \mathfrak{P}
  50. 𝔓 \mathfrak{P}
  51. 𝔓 \mathfrak{P}
  52. 𝔓 \mathfrak{P}
  53. 𝔓 \mathfrak{P}
  54. 𝔓 \mathfrak{P}
  55. 𝔓 \mathfrak{P}
  56. 𝔓 \mathfrak{P}
  57. 𝔓 \mathfrak{P}
  58. 𝔓 \mathfrak{P}
  59. 𝔓 \mathfrak{P}
  60. 𝔓 \mathfrak{P}
  61. 𝔓 \mathfrak{P}
  62. 𝔓 \mathfrak{P}
  63. 𝔓 \mathfrak{P}
  64. 𝔓 \mathfrak{P}
  65. 𝔓 \mathfrak{P}
  66. 𝔓 \mathfrak{P}
  67. 𝔓 \mathfrak{P}
  68. 𝔓 \mathfrak{P}
  69. 𝔓 \mathfrak{P}
  70. 𝔓 \mathfrak{P}
  71. 𝔓 \mathfrak{P}
  72. 𝔓 \mathfrak{P}
  73. 𝔓 \mathfrak{P}
  74. 𝔓 \mathfrak{P}
  75. 𝔓 \mathfrak{P}
  76. 𝔓 \mathfrak{P}
  77. 𝔓 \mathfrak{P}
  78. 𝔓 \mathfrak{P}
  79. 𝔓 \mathfrak{P}
  80. 𝔓 \mathfrak{P}
  81. 𝔓 \mathfrak{P}
  82. 𝔓 \mathfrak{P}
  83. 𝔓 \mathfrak{P}
  84. 𝔓 \mathfrak{P}

Algebra_over_a_field.html

  1. 𝐇𝐨𝐦 K -alg ( A , B ) . \mathbf{Hom}_{K\,\text{-alg}}(A,B).
  2. V F := V K F V_{F}:=V\otimes_{K}F
  3. A F A_{F}
  4. A × A A A\times A\rightarrow A
  5. η : K Z ( A ) , \eta\colon K\to Z(A),
  6. K × A A K\times A\to A
  7. ( k , a ) η ( k ) a . (k,a)\mapsto\eta(k)a.
  8. f ( k a ) = k f ( a ) f(ka)=kf(a)
  9. k K k\in K
  10. a A a\in A
  11. K η A η B A f B \begin{matrix}&&K&&\\ &\eta_{A}\swarrow&&\eta_{B}\searrow&\\ A&&\begin{matrix}f\\ \longrightarrow\end{matrix}&&B\end{matrix}
  12. 𝐞 i 𝐞 j = k = 1 n c i , j , k 𝐞 k \mathbf{e}_{i}\mathbf{e}_{j}=\sum_{k=1}^{n}c_{i,j,k}\mathbf{e}_{k}
  13. 1 1 = 1 , 1 a = a , a 1 = a . \textstyle 1\cdot 1=1\,,\quad 1\cdot a=a\,,\quad a\cdot 1=a\,.
  14. a a = 1 \textstyle aa=1
  15. a a = 0 \textstyle aa=0
  16. a a = a , b b = b , a b = b a = 0 \textstyle aa=a\,,\quad bb=b\,,\quad ab=ba=0
  17. a a = a , b b = 0 , a b = b a = 0 \textstyle aa=a\,,\quad bb=0\,,\quad ab=ba=0
  18. a a = b , b b = 0 , a b = b a = 0 \textstyle aa=b\,,\quad bb=0\,,\quad ab=ba=0
  19. a a = 1 , b b = 0 , a b = - b a = b \textstyle aa=1\,,\quad bb=0\,,\quad ab=-ba=b
  20. a a = 0 , b b = 0 , a b = b a = 0 \textstyle aa=0\,,\quad bb=0\,,\quad ab=ba=0

Algebraic_curve.html

  1. k k
  2. k k
  3. K K
  4. k k
  5. k k
  6. k k
  7. k k
  8. ( 2 , 3 ) (2,\sqrt{−3})
  9. p h ( x , y , z ) = 0 , {}^{h}p(x,y,z)=0,
  10. p h ( x , y , z ) = z deg ( p ) p ( x z , y z ) {}^{h}p(x,y,z)=z^{\deg(p)}p(\tfrac{x}{z},\tfrac{y}{z})
  11. p h ( x , y , 1 ) = p ( x , y ) {}^{h}p(x,y,1)=p(x,y)
  12. p ( x , y ) = P ( x , y , 1 ) p(x,y)=P(x,y,1)
  13. p h ( x , y , z ) = P ( x , y , z ) , {}^{h}p(x,y,z)=P(x,y,z),
  14. p x p^{\prime}_{x}
  15. p y p^{\prime}_{y}
  16. p ( x , y ) = p z h ( x , y , 1 ) . p^{\prime}_{\infty}(x,y)={{}^{h}p^{\prime}_{z}(x,y,1)}.
  17. x p x ( a , b ) + y p y ( a , b ) + p ( a , b ) = 0. xp^{\prime}_{x}(a,b)+yp^{\prime}_{y}(a,b)+p^{\prime}_{\infty}(a,b)=0.
  18. P ( x , y , z ) = p h ( x , y , z ) P(x,y,z)={}^{h}p(x,y,z)
  19. ( x - a ) p x ( a , b ) + ( y - b ) p y ( a , b ) = 0 (x-a)p^{\prime}_{x}(a,b)+(y-b)p^{\prime}_{y}(a,b)=0
  20. x p x ( a , b ) + y p y ( a , b ) + p ( a , b ) = 0 , xp^{\prime}_{x}(a,b)+yp^{\prime}_{y}(a,b)+p^{\prime}_{\infty}(a,b)=0,
  21. p ( x , y ) = P z ( x , y , 1 ) p^{\prime}_{\infty}(x,y)=P^{\prime}_{z}(x,y,1)
  22. p x ( a , b ) = p y ( a , b ) = 0 , p^{\prime}_{x}(a,b)=p^{\prime}_{y}(a,b)=0,
  23. x P x ( a , b , c ) + y P y ( a , b , c ) + z P z ( a , b , c ) = 0 , xP^{\prime}_{x}(a,b,c)+yP^{\prime}_{y}(a,b,c)+zP^{\prime}_{z}(a,b,c)=0,
  24. P x ( a , b , c ) = P y ( a , b , c ) = P z ( a , b , c ) = 0. P^{\prime}_{x}(a,b,c)=P^{\prime}_{y}(a,b,c)=P^{\prime}_{z}(a,b,c)=0.
  25. p = p d + + p 0 p=p_{d}+\cdots+p_{0}
  26. P = p h = p d + z p d - 1 + + z d p 0 P={{}^{h}p}=p_{d}+zp_{d-1}+\cdots+z^{d}p_{0}
  27. P z ( a , b , 0 ) = p d - 1 ( a , b ) . P^{\prime}_{z}(a,b,0)=p_{d-1}(a,b).
  28. x q x ( a , b ) + y q y ( a , b ) + p d - 1 ( a , b ) = 0. xq^{\prime}_{x}(a,b)+yq^{\prime}_{y}(a,b)+p_{d-1}(a,b)=0.
  29. q x ( a , b ) = q y ( a , b ) = 0 q^{\prime}_{x}(a,b)=q^{\prime}_{y}(a,b)=0
  30. p d - 1 ( a , b ) 0 , p_{d-1}(a,b)\neq 0,
  31. q x ( a , b ) = q y ( a , b ) = p d - 1 ( a , b ) = 0 , q^{\prime}_{x}(a,b)=q^{\prime}_{y}(a,b)=p_{d-1}(a,b)=0,
  32. p x ( x , y ) = p y ( x , y ) = p ( x , y ) = 0. p^{\prime}_{x}(x,y)=p^{\prime}_{y}(x,y)=p(x,y)=0.
  33. p x ( x , y ) = p y ( x , y ) = p ( x , y ) = 0 , p^{\prime}_{x}(x,y)=p^{\prime}_{y}(x,y)=p^{\prime}_{\infty}(x,y)=0,
  34. p ( x , y ) = P z ( x , y , 1 ) . p^{\prime}_{\infty}(x,y)=P^{\prime}_{z}(x,y,1).
  35. P x ( x , y , z ) = P y ( x , y , z ) = P z ( x , y , z ) = 0. P^{\prime}_{x}(x,y,z)=P^{\prime}_{y}(x,y,z)=P^{\prime}_{z}(x,y,z)=0.
  36. P ( x , y , z ) P(x,y,z)
  37. f , g 0 , g 3 , , g n f,g_{0},g_{3},\ldots,g_{n}
  38. f ( x 1 , x 2 ) = 0 g 0 ( x 1 , x 2 ) 0 x 3 = g 3 ( x 1 , x 2 ) g 0 ( x 1 , x 2 ) x n = g n ( x 1 , x 2 ) g 0 ( x 1 , x 2 ) \begin{aligned}&\displaystyle f(x_{1},x_{2})=0\\ &\displaystyle g_{0}(x_{1},x_{2})\neq 0\\ \displaystyle x_{3}&\displaystyle=\frac{g_{3}(x_{1},x_{2})}{g_{0}(x_{1},x_{2})% }\\ &\displaystyle{}\ \vdots\\ \displaystyle x_{n}&\displaystyle=\frac{g_{n}(x_{1},x_{2})}{g_{0}(x_{1},x_{2})% }\end{aligned}
  39. g 0 k h g_{0}^{k}h
  40. f , x 3 g 0 - g 3 , , x n g 0 - g n f,x_{3}g_{0}-g_{3},\ldots,x_{n}g_{0}-g_{n}
  41. f x ( P ) = f y ( P ) = f z ( P ) = 0. \frac{\partial f}{\partial x}(P)=\frac{\partial f}{\partial y}(P)=\frac{% \partial f}{\partial z}(P)=0.
  42. f ( P ) = f x ( P ) = f y ( P ) = 0. f(P)=\frac{\partial f}{\partial x}(P)=\frac{\partial f}{\partial y}(P)=0.
  43. y - x 3 = 0 y-x^{3}=0
  44. 𝒪 P ~ / 𝒪 P \widetilde{\mathcal{O}_{P}}/\mathcal{O}_{P}
  45. 𝒪 P \mathcal{O}_{P}
  46. 𝒪 P ~ \widetilde{\mathcal{O}_{P}}
  47. g r a d f ( grad\frac{f}{(}
  48. g = 1 2 ( d - 1 ) ( d - 2 ) - P δ P , g=\frac{1}{2}(d-1)(d-2)-\sum_{P}\delta_{P},
  49. x = 1 - t 2 1 + t + t 2 . x=\frac{1-t^{2}}{1+t+t^{2}}.
  50. y = t ( x + 1 ) = t ( t + 2 ) 1 + t + t 2 , y=t(x+1)=\frac{t(t+2)}{1+t+t^{2}}\,,
  51. X = U 2 - T 2 , Y = T ( T + 2 U ) , Z = T 2 + T U + U 2 . X=U^{2}-T^{2},\quad Y=T\,(T+2\,U),\quad Z=T^{2}+TU+U^{2}.
  52. X 2 + X Y + Y 2 = Z 2 , X^{2}+X\,Y+Y^{2}=Z^{2},
  53. y 2 z + a 1 x y z + a 3 y z 2 = x 3 + a 2 x 2 z + a 4 x z 2 + a 6 z 3 . y^{2}z+a_{1}xyz+a_{3}yz^{2}=x^{3}+a_{2}x^{2}z+a_{4}xz^{2}+a_{6}z^{3}.

Algebraic_data_type.html

  1. λ α . μ β .1 + α × β \lambda\alpha.\mu\beta.1+\alpha\times\beta
  2. nil α = roll ( inl ) \mathrm{nil}_{\alpha}=\mathrm{roll}\ (\mathrm{inl}\ \langle\rangle)
  3. cons α x l = roll ( inr x , l ) \mathrm{cons}_{\alpha}\ x\ l=\mathrm{roll}\ (\mathrm{inr}\ \langle x,l\rangle)
  4. μ ϕ . λ α .1 + α × ϕ α \mu\phi.\lambda\alpha.1+\alpha\times\phi\ \alpha
  5. μ \mu
  6. λ \lambda
  7. ϕ \phi
  8. β \beta
  9. ϕ \phi
  10. ϕ \phi
  11. α \alpha

Algebraic_group.html

  1. n ! n!
  2. [ n ] q ! [n]_{q}!

Algebraic_independence.html

  1. π \sqrt{\pi}
  2. 2 π + 1 2\pi+1
  3. { π } \{\sqrt{\pi}\}
  4. { 2 π + 1 } \{2\pi+1\}
  5. \mathbb{Q}
  6. { π , 2 π + 1 } \{\sqrt{\pi},2\pi+1\}
  7. P ( x , y ) = 2 x 2 - y + 1 P(x,y)=2x^{2}-y+1
  8. x = π x=\sqrt{\pi}
  9. y = 2 π + 1 y=2\pi+1
  10. π \pi
  11. \mathbb{Q}
  12. π + e \pi+e

Algebraic_integer.html

  1. \mathbb{Q}
  2. K = ( θ ) K=\mathbb{Q}(\theta)
  3. θ \theta\in\mathbb{C}
  4. α K \alpha\in K
  5. f ( x ) [ x ] f(x)\in\mathbb{Z}[x]
  6. f ( α ) = 0 f(\alpha)=0
  7. α K \alpha\in K
  8. α \alpha
  9. \mathbb{Q}
  10. [ x ] \mathbb{Z}[x]
  11. α K \alpha\in K
  12. [ α ] \mathbb{Z}[\alpha]
  13. \mathbb{Z}
  14. α K \alpha\in K
  15. \mathbb{Z}
  16. M M\subset\mathbb{C}
  17. α M M \alpha M\subseteq M
  18. K / K/\mathbb{Q}
  19. d ¯ \overline{d}
  20. d ¯ \overline{d}
  21. d ¯ \overline{d}
  22. d ¯ \overline{d}
  23. d ¯ \overline{d}
  24. F = 𝐐 [ α ] , α = m 3 F=\mathbf{Q}[\alpha],\alpha=\sqrt[3]{m}
  25. m = h k 2 m=hk^{2}
  26. { 1 , α , α 2 ± k 2 α + k 2 3 k m ± 1 mod 9 1 , α , α 2 k else \begin{cases}1,\alpha,\frac{\alpha^{2}\pm k^{2}\alpha+k^{2}}{3k}&m\equiv\pm 1% \mod 9\\ 1,\alpha,\frac{\alpha^{2}}{k}&\mathrm{else}\end{cases}
  27. β = α n \beta=\sqrt[n]{\alpha}

Algebraic_number_theory.html

  1. A = x + y A=x+y
  2. B = x 2 + y 2 . B=x^{2}+y^{2}.
  3. 6 = 2 3 = ( 1 + - 5 ) ( 1 - - 5 ) . 6=2\cdot 3=(1+\sqrt{-5})\cdot(1-\sqrt{-5}).
  4. 2 𝐙 [ i ] = ( ( 1 + i ) 𝐙 [ i ] ) 2 . 2\mathbf{Z}[i]=\left((1+i)\mathbf{Z}[i]\right)^{2}.
  5. p 𝐙 [ i ] is a prime ideal if p 3 ( mod 4 ) p\mathbf{Z}[i]\mbox{ is a prime ideal if }~{}p\equiv 3\,(\operatorname{mod}\,4)
  6. p 𝐙 [ i ] is not a prime ideal if p 1 ( mod 4 ) . p\mathbf{Z}[i]\mbox{ is not a prime ideal if }~{}p\equiv 1\,(\operatorname{mod% }\,4).
  7. 𝔭 \mathfrak{p}
  8. 𝔭 \mathfrak{p}
  9. ( p q ) ( q p ) = ( - 1 ) p - 1 2 q - 1 2 . \left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{\frac{p-1}{2}\frac{q-1}% {2}}.

Algebraic_structure.html

  1. A A
  2. A A
  3. ( , + ) (\mathbb{Z},+)
  4. \mathbb{Z}
  5. + +

Algebraic_variety.html

  1. k k
  2. k k
  3. f f
  4. k k
  5. f f
  6. Z ( S ) = { x 𝐀 n f ( x ) = 0 for all f S } . Z(S)=\left\{x\in\mathbf{A}^{n}\mid f(x)=0\,\text{ for all }f\in S\right\}.
  7. I ( V ) = { f k [ x 1 , , x n ] f ( x ) = 0 for all x V } . I(V)=\left\{f\in k[x_{1},\cdots,x_{n}]\mid f(x)=0\,\text{ for all }x\in V% \right\}.
  8. k k
  9. k k
  10. f f
  11. f f
  12. f f
  13. f f
  14. Z ( S ) = { x 𝐏 n f ( x ) = 0 for all f S } . Z(S)=\{x\in\mathbf{P}^{n}\mid f(x)=0\,\text{ for all }f\in S\}.
  15. k = 𝐂 k=\mathbf{C}
  16. f ( x , y ) f(x,y)
  17. f ( x , y ) = x + y - 1. f(x,y)=x+y-1.
  18. f ( x , y ) f(x,y)
  19. Z ( f ) Z(f)
  20. Z ( f ) = { ( x , 1 - x ) 𝐂 2 } . Z(f)=\{(x,1-x)\in\mathbf{C}^{2}\}.
  21. V = Z ( f ) V=Z(f)
  22. k = 𝐂 k=\mathbf{C}
  23. g ( x , y ) = x 2 + y 2 - 1. g(x,y)=x^{2}+y^{2}-1.
  24. y - x 2 = 0 z - x 3 = 0 \begin{aligned}\displaystyle y-x^{2}&\displaystyle=0\\ \displaystyle z-x^{3}&\displaystyle=0\end{aligned}
  25. y 2 = x 3 - x y^{2}=x^{3}-x
  26. y 2 z = x 3 - x z 2 y^{2}z=x^{3}-xz^{2}
  27. G n ( V ) 𝐏 ( n V ) , b 1 , , b n [ b 1 b n ] G_{n}(V)\hookrightarrow\mathbf{P}(\wedge^{n}V),\,\langle b_{1},\dots,b_{n}% \rangle\mapsto[b_{1}\wedge\dots\wedge b_{n}]
  28. n V \wedge^{n}V
  29. V < s u b > 1 , V 2 V<sub>1,V_{2}

Algor_mortis.html

  1. 98.4 F - rectal temperature in Fahrenheit 1.5 \frac{98.4\,^{\circ}{\rm F}-\,\text{rectal temperature in Fahrenheit}}{1.5}

Algorithmic_efficiency.html

  1. O ( 1 ) O(1)\,
  2. O ( log n ) O(\log n)\,
  3. O ( n ) O(n)\,
  4. O ( n log n ) O(n\log n)\,
  5. O ( n 2 ) O(n^{2})\,
  6. O ( c n ) , c > 1 O(c^{n}),\;c>1
  7. O ( n log n ) O(n\log n)\,

Aliasing.html

  1. f s = 1 f_{s}=1\,
  2. f red = 0.9 f_{\mathrm{red}}=0.9\,
  3. f blue = 0.1 f_{\mathrm{blue}}=0.1\,
  4. f f\,
  5. f s , f_{s},\,
  6. f / f s f/f_{s}
  7. f / f s f/f_{s}
  8. f f
  9. f alias ( N ) = def | f - N f s | , f_{\mathrm{alias}}(N)\ \stackrel{\mathrm{def}}{=}\ |f-Nf_{s}|,\,
  10. f alias ( 0 ) = f \scriptstyle f_{\mathrm{alias}}(0)=f\,
  11. f red \scriptstyle f_{\mathrm{red}}\,
  12. f blue , \scriptstyle f_{\mathrm{blue}},\,
  13. f alias ( N ) \scriptstyle f_{\mathrm{alias}}(N)\,
  14. f alias ( 0 ) \scriptstyle f_{\mathrm{alias}}(0)\,
  15. f s / 2 > | f | , \scriptstyle f_{s}/2\ >\ |f|,\,
  16. f s / 2 \scriptstyle f_{s}/2\,
  17. f s . \scriptstyle f_{s}.\,
  18. f = f blue \scriptstyle f=f_{\mathrm{blue}}
  19. f = f red = 0.9 , \scriptstyle f=f_{\mathrm{red}}=0.9,
  20. f red \scriptstyle f_{\mathrm{red}}\,
  21. f blue \scriptstyle f_{\mathrm{blue}}\,
  22. f s / 2. \scriptstyle f_{s}/2.
  23. f \scriptstyle f
  24. f s / 2 , \scriptstyle f_{s}/2,
  25. f alias ( 1 ) \scriptstyle f_{\mathrm{alias}}(1)
  26. f s \scriptstyle f_{s}
  27. f s / 2. \scriptstyle f_{s}/2.
  28. f \scriptstyle f
  29. f s / 2 \scriptstyle f_{s}/2
  30. f s , \scriptstyle f_{s},
  31. f alias ( 1 ) \scriptstyle f_{\mathrm{alias}}(1)
  32. f s / 2 \scriptstyle f_{s}/2
  33. 0.6 f s \scriptstyle 0.6f_{s}
  34. 0.4 f s , \scriptstyle 0.4f_{s},
  35. 1.4 f s , \scriptstyle 1.4f_{s},
  36. 1.6 f s \scriptstyle 1.6f_{s}
  37. f s / 2 \scriptstyle f_{s}/2
  38. f s \scriptstyle f_{s}
  39. f s . \scriptstyle f_{s}.
  40. f s / 2 \scriptstyle f_{s}/2
  41. f cyan = f alias ( 1 ) = f gold - 1 f s . \scriptstyle f_{\mathrm{cyan}}=f_{\mathrm{alias}}(1)=f_{\mathrm{gold}}-1\cdot f% _{s}.\,
  42. f alias ( N ) = f - N f s . f_{\mathrm{alias}}(N)=f-Nf_{s}.\,
  43. f \scriptstyle f\,
  44. f s / 2 \scriptstyle f_{s}/2\,
  45. f s , \scriptstyle f_{s},\,
  46. f alias ( 1 ) \scriptstyle f_{\mathrm{alias}}(1)
  47. - f s / 2 \scriptstyle-f_{s}/2\,
  48. f s / 2 > f \scriptstyle f_{s}/2\ >\ f

Alignments_of_random_points.html

  1. n ! ( n - k ) ! k ! ( w d ) k - 2 \frac{n!}{(n-k)!k!}\left({\frac{w}{d}}\right)^{k-2}
  2. μ = π 3 w L ( d L ) 3 n ( n - 1 ) ( n - 2 ) \mu=\frac{\pi}{3}\frac{w}{L}\left({\frac{d}{L}}\right)^{3}n\left(n-1\right)% \left(n-2\right)
  3. μ = π 3 w L ( d L ) 3 n ( n - 1 ) ( n - 2 ) ( 1 - 3 π ( d L ) + 3 5 ( 4 π - 1 ) ( d L ) 2 ) \mu=\frac{\pi}{3}\frac{w}{L}\left({\frac{d}{L}}\right)^{3}n\left(n-1\right)% \left(n-2\right)\left(1-\frac{3}{\pi}\left(\frac{d}{L}\right)+\frac{3}{5}\left% (\frac{4}{\pi}-1\right)\left(\frac{d}{L}\right)^{2}\right)
  4. μ = π n ( n - 1 ) ( n - 2 ) ( n - ( k - 1 ) ) k ( k - 2 ) ! ( w L ) k - 2 ( d L ) k \mu=\frac{\pi n\left(n-1\right)\left(n-2\right)\cdots\left(n-\left(k-1\right)% \right)}{k\left(k-2\right)!}\left(\frac{w}{L}\right)^{k-2}\left({\frac{d}{L}}% \right)^{k}

All_horses_are_the_same_color.html

  1. n = 1 n=1
  2. n n
  3. n + 1 n+1
  4. n n
  5. n + 1 n+1
  6. n n
  7. n n
  8. n n
  9. n n
  10. n + 1 n+1
  11. n = 1 n=1
  12. n = 1 n=1
  13. n = 2 n=2
  14. n = 3 n=3

Allele_frequency.html

  1. f ( A A ) f(AA)
  2. f ( A a ) f(Aa)
  3. f ( a a ) f(aa)
  4. p = f ( 𝐀𝐀 ) + 1 2 f ( 𝐀𝐚 ) = frequency of A p=f(\mathbf{AA})+\frac{1}{2}f(\mathbf{Aa})=\mbox{frequency of A}~{}
  5. q = f ( 𝐚𝐚 ) + 1 2 f ( 𝐀𝐚 ) = frequency of a q=f(\mathbf{aa})+\frac{1}{2}f(\mathbf{Aa})=\mbox{frequency of a}~{}
  6. p + q = f ( 𝐀𝐀 ) + f ( 𝐚𝐚 ) + f ( 𝐀𝐚 ) = 1 p+q=f(\mathbf{AA})+f(\mathbf{aa})+f(\mathbf{Aa})=1
  7. q = 1 - p q=1-p
  8. p = 1 - q p=1-q
  9. p = p r o b A = 2 + 1 + 2 + 0 + 1 + 2 + 2 + 1 + 1 + 2 2 * 10 = 0.7 p=prob_{A}=\frac{2+1+2+0+1+2+2+1+1+2}{2*10}=0.7
  10. q = p r o b a = 0 + 1 + 0 + 2 + 1 + 0 + 0 + 1 + 1 + 0 2 * 10 = 0.3 q=prob_{a}=\frac{0+1+0+2+1+0+0+1+1+0}{2*10}=0.3

Almost.html

  1. S = { n 𝐍 | n k } S=\{n\in\mathbf{N}|n\geq k\}

ALOHAnet.html

  1. G k e - G k ! \frac{G^{k}e^{-G}}{k!}
  2. ( 2 G ) k e - 2 G k ! \frac{(2G)^{k}e^{-2G}}{k!}
  3. P r o b p u r e Prob_{pure}
  4. P r o b p u r e = e - 2 G Prob_{pure}=e^{-2G}
  5. S p u r e S_{pure}
  6. S p u r e = G e - 2 G S_{pure}=Ge^{-2G}
  7. P [ X = x ] = G x e - G x ! P[X=x]=\frac{G^{x}e^{-G}}{x!}
  8. P [ X = 1 ] = G 1 e - G 1 ! = G e - G P[X=1]=\frac{G^{1}e^{-G}}{1!}=Ge^{-G}
  9. P [ X = 0 ] = G 0 e - G 0 ! = e - G P[X=0]=\frac{G^{0}e^{-G}}{0!}=e^{-G}
  10. P = P ( 0 ) × P ( 1 ) = G e - G × e - G = G e - 2 G P=P(0)\times P(1)=Ge^{-G}\times e^{-G}=Ge^{-2G}
  11. S p u r e = G e - 2 G S_{pure}=Ge^{-2G}
  12. P [ X = 1 ] = G 1 e - G 1 ! = G e - G P[X=1]=\frac{G^{1}e^{-G}}{1!}=Ge^{-G}
  13. S s l o t t e d = G e - G S_{slotted}=Ge^{-G}
  14. P r o b s l o t t e d = e - G Prob_{slotted}=e^{-G}
  15. P r o b s l o t t e d k = e - G ( 1 - e - G ) k - 1 Prob_{slotted}k=e^{-G}(1-e^{-G})^{k-1}
  16. S s l o t t e d = G e - G S_{slotted}=Ge^{-G}

Alpha_Herculis.html

  1. δ = d R D R \begin{smallmatrix}{\delta}=\frac{d_{R}}{D_{R}}\end{smallmatrix}
  2. d R = δ D R = 0.034 110.0 = 3.740 A U \begin{smallmatrix}d_{R}=\delta\cdot D_{R}={0.034}\cdot 110.0=3.740AU\end{smallmatrix}
  3. R R = ( d R 2 ) = ( 3.740 2 ) = 1.870 A U \begin{smallmatrix}R_{R}={\left({\frac{d_{R}}{2}}\right)}={\left({\frac{3.740}% {2}}\right)}=1.870AU\end{smallmatrix}
  4. d B = ( 1.87 A U ) ( 149 , 597 , 871 k m 696 , 000 k m ) = 280 , 000 , 000 k m = 402 R \begin{smallmatrix}d_{B}={\left(1.87AU\right)}{\left({\frac{149,597,871km}{696% ,000km}}\right)}=280,000,000km=402R_{\odot}\end{smallmatrix}

Alpha_process.html

  1. C 6 12 + He 2 4 O 8 16 + γ + Q \mathrm{{}_{6}^{12}C}+\mathrm{{}_{2}^{4}He}\rightarrow\mathrm{{}_{8}^{16}O}+% \gamma+Q
  2. O 8 16 + He 2 4 Ne 10 20 + γ + Q \mathrm{{}_{8}^{16}O}+\mathrm{{}_{2}^{4}He}\rightarrow\mathrm{{}_{10}^{20}Ne}+% \gamma+Q
  3. Ne 10 20 + He 2 4 Mg 12 24 + γ + Q \mathrm{{}_{10}^{20}Ne}+\mathrm{{}_{2}^{4}He}\rightarrow\mathrm{{}_{12}^{24}Mg% }+\gamma+Q
  4. Mg 12 24 + He 2 4 Si 14 28 + γ + Q \mathrm{{}_{12}^{24}Mg}+\mathrm{{}_{2}^{4}He}\rightarrow\mathrm{{}_{14}^{28}Si% }+\gamma+Q
  5. Si 14 28 + He 2 4 S 16 32 + γ + Q \mathrm{{}_{14}^{28}Si}+\mathrm{{}_{2}^{4}He}\rightarrow\mathrm{{}_{16}^{32}S}% +\gamma+Q
  6. S 16 32 + He 2 4 Ar 18 36 + γ \mathrm{{}_{16}^{32}S}+\mathrm{{}_{2}^{4}He}\rightarrow\mathrm{{}_{18}^{36}Ar}+\gamma
  7. Ar 18 36 + He 2 4 Ca 20 40 + γ \mathrm{{}_{18}^{36}Ar}+\mathrm{{}_{2}^{4}He}\rightarrow\mathrm{{}_{20}^{40}Ca% }+\gamma
  8. Ca 20 40 + He 2 4 Ti 22 44 + γ \mathrm{{}_{20}^{40}Ca}+\mathrm{{}_{2}^{4}He}\rightarrow\mathrm{{}_{22}^{44}Ti% }+\gamma
  9. Ti 22 44 + He 2 4 Cr 24 48 + γ \mathrm{{}_{22}^{44}Ti}+\mathrm{{}_{2}^{4}He}\rightarrow\mathrm{{}_{24}^{48}Cr% }+\gamma
  10. Cr 24 48 + He 2 4 Fe 26 52 + γ \mathrm{{}_{24}^{48}Cr}+\mathrm{{}_{2}^{4}He}\rightarrow\mathrm{{}_{26}^{52}Fe% }+\gamma
  11. Fe 26 52 + He 2 4 Ni 28 56 + γ \mathrm{{}_{26}^{52}Fe}+\mathrm{{}_{2}^{4}He}\rightarrow\mathrm{{}_{28}^{56}Ni% }+\gamma
  12. Ni 28 56 + He 2 4 + γ Zn 30 60 \mathrm{{}_{28}^{56}Ni}+\mathrm{{}_{2}^{4}He}+\gamma\rightarrow\mathrm{{}_{30}% ^{60}Zn}
  13. [ α / F e ] = log 10 ( N α N F e ) S t a r - log 10 ( N α N F e ) S u n [\alpha/Fe]=\log_{10}{\left(\frac{N_{\alpha}}{N_{Fe}}\right)_{Star}}-\log_{10}% {\left(\frac{N_{\alpha}}{N_{Fe}}\right)_{Sun}}
  14. N α N_{\alpha}
  15. N F e N_{Fe}

Alpha–beta_pruning.html

  1. O ( b d / 2 ) = O ( b d ) O(b^{d/2})=O(\sqrt{b^{d}})
  2. O ( b 3 d / 4 ) O(b^{3d/4})

Alternator.html

  1. N = 120 f / P N=120f/P
  2. f f
  3. P P
  4. N N

Altitude_(triangle).html

  1. h c = p q h_{c}=\sqrt{pq}
  2. O H = 2 N H , OH=2NH,
  3. 2 O G = G H . 2OG=GH.
  4. H I < H G , HI<HG,
  5. H G > I G . HG>IG.
  6. O H 2 = R 2 - 8 R 2 cos A cos B cos C = 9 R 2 - ( a 2 + b 2 + c 2 ) , OH^{2}=R^{2}-8R^{2}\cos A\cos B\cos C=9R^{2}-(a^{2}+b^{2}+c^{2}),
  7. H I 2 = 2 r 2 - 4 R 2 cos A cos B cos C . HI^{2}=2r^{2}-4R^{2}\cos A\cos B\cos C.
  8. sec A : sec B : sec C = cos A - sin B sin C : cos B - sin C sin A : cos C - sin A sin B , \sec A:\sec B:\sec C=\cos A-\sin B\sin C:\cos B-\sin C\sin A:\cos C-\sin A\sin B,
  9. ( a 2 + b 2 - c 2 ) ( a 2 - b 2 + c 2 ) : ( a 2 + b 2 - c 2 ) ( - a 2 + b 2 + c 2 ) : ( a 2 - b 2 + c 2 ) ( - a 2 + b 2 + c 2 ) \displaystyle(a^{2}+b^{2}-c^{2})(a^{2}-b^{2}+c^{2}):(a^{2}+b^{2}-c^{2})(-a^{2}% +b^{2}+c^{2}):(a^{2}-b^{2}+c^{2})(-a^{2}+b^{2}+c^{2})
  10. = tan A : tan B : tan C . =\tan A:\tan B:\tan C.
  11. H D A D + H E B E + H F C F = 1. \frac{HD}{AD}+\frac{HE}{BE}+\frac{HF}{CF}=1.
  12. A H A D + B H B E + C H C F = 2. \frac{AH}{AD}+\frac{BH}{BE}+\frac{CH}{CF}=2.
  13. A H H D = B H H E = C H H F . AH\cdot HD=BH\cdot HE=CH\cdot HF.
  14. H D = D P . HD=DP.
  15. a 2 + b 2 + c 2 + A H 2 + B H 2 + C H 2 = 12 R 2 . a^{2}+b^{2}+c^{2}+AH^{2}+BH^{2}+CH^{2}=12R^{2}.
  16. r a + r b + r c + r = A H + B H + C H + 2 R , r_{a}+r_{b}+r_{c}+r=AH+BH+CH+2R,
  17. r a 2 + r b 2 + r c 2 + r 2 = A H 2 + B H 2 + C H 2 + ( 2 R ) 2 . r_{a}^{2}+r_{b}^{2}+r_{c}^{2}+r^{2}=AH^{2}+BH^{2}+CH^{2}+(2R)^{2}.
  18. h a = 2 s ( s - a ) ( s - b ) ( s - c ) a . h_{a}=\frac{2\sqrt{s(s-a)(s-b)(s-c)}}{a}.
  19. 1 r = 1 h a + 1 h b + 1 h c . \displaystyle\frac{1}{r}=\frac{1}{h_{a}}+\frac{1}{h_{b}}+\frac{1}{h_{c}}.
  20. h a = b c 2 R . h_{a}=\frac{bc}{2R}.
  21. p 1 h 1 + p 2 h 2 + p 3 h 3 = 1. \frac{p_{1}}{h_{1}}+\frac{p_{2}}{h_{2}}+\frac{p_{3}}{h_{3}}=1.
  22. h a h_{a}
  23. h b h_{b}
  24. h c h_{c}
  25. H = ( h a - 1 + h b - 1 + h c - 1 ) / 2 H=(h_{a}^{-1}+h_{b}^{-1}+h_{c}^{-1})/2
  26. Area - 1 = 4 H ( H - h a - 1 ) ( H - h b - 1 ) ( H - h c - 1 ) . \mathrm{Area}^{-1}=4\sqrt{H(H-h_{a}^{-1})(H-h_{b}^{-1})(H-h_{c}^{-1})}.
  27. A C 2 + E B 2 = A B 2 + C E 2 . AC^{2}+EB^{2}=AB^{2}+CE^{2}.
  28. 1 h a 2 + 1 h b 2 = 1 h c 2 . \frac{1}{h_{a}^{2}}+\frac{1}{h_{b}^{2}}=\frac{1}{h_{c}^{2}}.
  29. a - 2 + b - 2 = d - 2 a^{-2}+b^{-2}=d^{-2}

AMBER.html

  1. V ( r N ) = bonds k b ( l - l 0 ) 2 + angles k a ( θ - θ 0 ) 2 V(r^{N})=\sum\text{bonds}k_{b}(l-l_{0})^{2}+\sum\text{angles}k_{a}(\theta-% \theta_{0})^{2}
  2. + torsions n 1 2 V n [ 1 + cos ( n ω - γ ) ] +\sum\text{torsions}\sum_{n}\frac{1}{2}V_{n}[1+\cos(n\omega-\gamma)]
  3. + j = 1 N - 1 i = j + 1 N f i j { ϵ i j [ ( r 0 i j r i j ) 12 - 2 ( r 0 i j r i j ) 6 ] + q i q j 4 π ϵ 0 r i j } +\sum_{j=1}^{N-1}\sum_{i=j+1}^{N}f_{ij}\biggl\{\epsilon_{ij}\biggl[\left(\frac% {r_{0ij}}{r_{ij}}\right)^{12}-2\left(\frac{r_{0ij}}{r_{ij}}\right)^{6}\biggr]+% \frac{q_{i}q_{j}}{4\pi\epsilon_{0}r_{ij}}\biggr\}
  4. i i
  5. j j
  6. r 0 i j r_{0ij}
  7. ϵ \epsilon
  8. 2 2
  9. r 0 i j r_{0ij}
  10. σ \sigma
  11. r 0 i j = 2 1 / 6 ( σ ) r_{0ij}=2^{1/6}(\sigma)

Ambisonics.html

  1. W W
  2. X X
  3. Y Y
  4. Z Z
  5. W W
  6. X Y Z XYZ
  7. S S
  8. θ \theta
  9. ϕ \phi
  10. W = S 1 2 W=S\cdot\frac{1}{\sqrt{2}}
  11. X = S cos θ cos ϕ X=S\cdot\cos\theta\cos\phi
  12. Y = S sin θ cos ϕ Y=S\cdot\sin\theta\cos\phi
  13. Z = S sin ϕ Z=S\cdot\sin\phi
  14. W W
  15. X Y Z XYZ
  16. θ \theta
  17. ϕ \phi
  18. p p
  19. 0
  20. [ 0 , 0.5 ] [0,0.5]
  21. 0.5 0.5
  22. [ 0.5 , 1.0 ] [0.5,1.0]
  23. 1.0 1.0
  24. Θ \Theta
  25. 0 p 1 0\leq p\leq 1
  26. M ( Θ , p ) = p 2 W + ( 1 - p ) ( cos Θ X + sin Θ Y ) M(\Theta,p)=p\sqrt{2}W+(1-p)(\cos\Theta X+\sin\Theta Y)
  27. L F = ( 2 W + X + Y ) 8 LF=(2W+X+Y)\sqrt{8}
  28. L B = ( 2 W - X + Y ) 8 LB=(2W-X+Y)\sqrt{8}
  29. R B = ( 2 W - X - Y ) 8 RB=(2W-X-Y)\sqrt{8}
  30. R F = ( 2 W + X - Y ) 8 RF=(2W+X-Y)\sqrt{8}
  31. X X
  32. Y Y
  33. Z Z
  34. \ell
  35. ( + 1 ) 2 (\ell+1)^{2}
  36. 2 + 1 2\ell+1
  37. W W
  38. X Y Z XYZ
  39. W W
  40. X Y Z XYZ
  41. r r
  42. \ell
  43. f f
  44. r c 2 π f r\approx\frac{\ell c}{2\pi f}
  45. c c
  46. r V \vec{r_{V}}
  47. r E \vec{r_{E}}
  48. r V \vec{r_{V}}
  49. r E \vec{r_{E}}
  50. r V = 1 \|\vec{r_{V}}\|=1
  51. r E \vec{r_{E}}
  52. Z Z
  53. W W
  54. X X
  55. Y Y

American_wire_gauge.html

  1. 92 39 \sqrt[39]{92}
  2. d n = 0.005 inch × 92 36 - n 39 = 0.127 mm × 92 36 - n 39 d_{n}=0.005~{}\mathrm{inch}\times 92^{\frac{36-n}{39}}=0.127~{}\mathrm{mm}% \times 92^{\frac{36-n}{39}}
  3. d n = e - 1.12436 - 0.11594 n inch = e 2.1104 - 0.11594 n mm d_{n}=e^{-1.12436-0.11594n}\ \mathrm{inch}=e^{2.1104-0.11594n}\ \mathrm{mm}
  4. n = - 39 log 92 ( d n 0.005 inch ) + 36 = - 39 log 92 ( d n 0.127 mm ) + 36 n=-39\log_{92}\left(\frac{d_{n}}{0.005~{}\mathrm{inch}}\right)+36=-39\log_{92}% \left(\frac{d_{n}}{0.127~{}\mathrm{mm}}\right)+36
  5. A n = π 4 d n 2 = 0.000019635 inch 2 × 92 36 - n 19.5 = 0.012668 mm 2 × 92 36 - n 19.5 A_{n}=\frac{\pi}{4}d_{n}^{2}=0.000019635~{}\mathrm{inch}^{2}\times 92^{\frac{3% 6-n}{19.5}}=0.012668~{}\mathrm{mm}^{2}\times 92^{\frac{36-n}{19.5}}
  6. 92 39 \sqrt[39]{92}

Amortized_analysis.html

  1. O ( n n ) = O ( 1 ) O(\tfrac{n}{n})=O(1)

Ampere-turn.html

  1. 4 π / 10 4\pi/10

Ampère's_circuital_law.html

  1. C 𝐁 d s y m b o l = μ 0 S 𝐉 d 𝐒 = μ 0 I enc \oint_{C}\mathbf{B}\cdot\mathrm{d}symbol{\ell}=\mu_{0}\iint_{S}\mathbf{J}\cdot% \mathrm{d}\mathbf{S}=\mu_{0}I_{\mathrm{enc}}
  2. C 𝐇 d s y m b o l = S 𝐉 f d 𝐒 = I f , enc \oint_{C}\mathbf{H}\cdot\mathrm{d}symbol{\ell}=\iint_{S}\mathbf{J}_{\mathrm{f}% }\cdot\mathrm{d}\mathbf{S}=I_{\mathrm{f,enc}}
  3. C \scriptstyle\oint_{C}
  4. S \scriptstyle\iint_{S}
  5. 𝐁 = μ 0 𝐇 \mathbf{B}=\mu_{0}\mathbf{H}\,\!
  6. C \textstyle\oint_{C}
  7. × 𝐁 = μ 0 𝐉 \mathbf{\nabla}\times\mathbf{B}=\mu_{0}\mathbf{J}
  8. × 𝐇 = 𝐉 f \mathbf{\nabla}\times\mathbf{H}=\mathbf{J}_{\,\text{f}}
  9. 𝐉 = 𝐉 f + 𝐉 M + 𝐉 P \mathbf{J}=\mathbf{J}_{\,\text{f}}+\mathbf{J}_{\,\text{M}}+\mathbf{J}_{\,\text% {P}}
  10. ( × B ) = 0 \nabla\cdot(\nabla\times{B})=0
  11. J = 0. \nabla\cdot{J}=0.
  12. J = - ρ t \nabla\cdot{J}=-\frac{\partial\rho}{\partial t}
  13. J = 0 , {J}={0},
  14. × B = 0 \nabla\times{B}={0}
  15. × B = 1 c 2 E t . \nabla\times{B}=\frac{1}{c^{2}}\frac{\partial{E}}{\partial t}.
  16. 𝐉 D = t 𝐃 ( 𝐫 , t ) , \mathbf{J}_{\,\text{D}}=\frac{\partial}{\partial t}\mathbf{D}(\mathbf{r},\ t)\ ,
  17. 𝐃 = ε 0 𝐄 + 𝐏 = ε 0 ε r 𝐄 , \mathbf{D}=\varepsilon_{0}\mathbf{E}+\mathbf{P}=\varepsilon_{0}\varepsilon_{\,% \text{r}}\mathbf{E}\ ,
  18. 𝐉 D = ε 0 𝐄 t + 𝐏 t . \mathbf{J}_{\mathrm{D}}=\varepsilon_{0}\frac{\partial\mathbf{E}}{\partial t}+% \frac{\partial\mathbf{P}}{\partial t}.
  19. C 𝐇 d s y m b o l = S ( 𝐉 f + t 𝐃 ) d 𝐒 \oint_{C}\mathbf{H}\cdot\mathrm{d}symbol{\ell}=\iint_{S}\left(\mathbf{J}_{% \mathrm{f}}+\frac{\partial}{\partial t}\mathbf{D}\right)\cdot\mathrm{d}\mathbf% {S}
  20. × 𝐇 = 𝐉 f + t 𝐃 . \mathbf{\nabla}\times\mathbf{H}=\mathbf{J}_{\mathrm{f}}+\frac{\partial}{% \partial t}\mathbf{D}\ .
  21. 𝐉 f + 𝐉 D + 𝐉 M = 𝐉 f + 𝐉 P + 𝐉 M + ε 0 𝐄 t = 𝐉 + ε 0 𝐄 t , \mathbf{J}_{\,\text{f}}+\mathbf{J}_{\,\text{D}}+\mathbf{J}_{\,\text{M}}=% \mathbf{J}_{\,\text{f}}+\mathbf{J}_{\,\text{P}}+\mathbf{J}_{\,\text{M}}+% \varepsilon_{0}\frac{\partial\mathbf{E}}{\partial t}=\mathbf{J}+\varepsilon_{0% }\frac{\partial\mathbf{E}}{\partial t}\ ,
  22. × 𝐇 = 𝐉 f + 𝐃 t \nabla\times\mathbf{H}=\mathbf{J}_{\,\text{f}}+\frac{\partial\mathbf{D}}{% \partial t}
  23. × 𝐁 / μ 0 = 𝐉 + ε 0 𝐄 t \mathbf{\nabla}\times\mathbf{B}/\mu_{0}=\mathbf{J}+\varepsilon_{0}\frac{% \partial\mathbf{E}}{\partial t}
  24. 𝐃 = ε 0 𝐄 + 𝐏 \mathbf{D}=\varepsilon_{0}\mathbf{E}+\mathbf{P}
  25. 𝐁 / μ 0 = 𝐇 + 𝐌 \mathbf{B}/\mu_{0}=\mathbf{H}+\mathbf{M}
  26. 𝐉 bound = × 𝐌 + 𝐏 t , \mathbf{J}_{\mathrm{bound}}=\nabla\times\mathbf{M}+\frac{\partial\mathbf{P}}{% \partial t}\ ,
  27. = 𝐉 M + 𝐉 P , =\mathbf{J}_{\mathrm{M}}+\mathbf{J}_{\mathrm{P}}\ ,
  28. 𝐉 M = × 𝐌 , \mathbf{J}_{\mathrm{M}}=\nabla\times\mathbf{M}\ ,
  29. 𝐉 P = 𝐏 t , \mathbf{J}_{\mathrm{P}}=\frac{\partial\mathbf{P}}{\partial t}\ ,
  30. × 𝐁 / μ 0 = × ( 𝐇 + 𝐌 ) \mathbf{\nabla}\times\mathbf{B}/\mu_{0}=\mathbf{\nabla}\times\left(\mathbf{H}+% \mathbf{M}\right)
  31. = × 𝐇 + 𝐉 M =\mathbf{\nabla}\times\mathbf{H}+\mathbf{J}_{\,\text{M}}
  32. = 𝐉 f + 𝐉 P + ε 0 𝐄 t + 𝐉 M =\mathbf{J}_{\,\text{f}}+\mathbf{J}_{\,\text{P}}+\varepsilon_{0}\frac{\partial% \mathbf{E}}{\partial t}+\mathbf{J}_{\,\text{M}}
  33. × 𝐁 / μ 0 = 𝐉 f + 𝐉 bound + ε 0 𝐄 t \mathbf{\nabla}\times\mathbf{B}/\mu_{0}=\mathbf{J}_{\,\text{f}}+\mathbf{J}_{% \mathrm{bound}}+\varepsilon_{0}\frac{\partial\mathbf{E}}{\partial t}
  34. = 𝐉 + ε 0 𝐄 t , =\mathbf{J}+\varepsilon_{0}\frac{\partial\mathbf{E}}{\partial t}\ ,
  35. C 𝐁 d s y m b o l = 1 c S ( 4 π 𝐉 + 𝐄 t ) d 𝐒 \oint_{C}\mathbf{B}\cdot\mathrm{d}symbol{\ell}=\frac{1}{c}\iint_{S}\left(4\pi% \mathbf{J}+\frac{\partial\mathbf{E}}{\partial t}\right)\cdot\mathrm{d}\mathbf{S}
  36. × 𝐁 = 1 c ( 4 π 𝐉 + 𝐄 t ) . \mathbf{\nabla}\times\mathbf{B}=\frac{1}{c}\left(4\pi\mathbf{J}+\frac{\partial% \mathbf{E}}{\partial t}\right).

Analogy.html

  1. 2 \mathbb{R}^{2}
  2. \mathbb{C}
  3. \mathbb{C}
  4. 2 \mathbb{R}^{2}
  5. \mathbb{C}

Analytic_continuation.html

  1. F ( z ) = f ( z ) z U , \displaystyle F(z)=f(z)\qquad\forall z\in U,
  2. f ( z ) = k = 0 α k ( z - z 0 ) k f(z)=\sum_{k=0}^{\infty}\alpha_{k}(z-z_{0})^{k}
  3. D r ( z 0 ) = { z 𝐂 : | z - z 0 | < r } D_{r}(z_{0})=\{z\in\mathbf{C}:|z-z_{0}|<r\}
  4. g = ( z 0 , α 0 , α 1 , α 2 , ) g=(z_{0},\alpha_{0},\alpha_{1},\alpha_{2},\ldots)
  5. 𝒢 \mathcal{G}
  6. 𝒢 \mathcal{G}
  7. U r ( g ) = { h 𝒢 : g h , | g 0 - h 0 | < r } . U_{r}(g)=\{h\in\mathcal{G}:g\geq h,|g_{0}-h_{0}|<r\}.
  8. 𝒢 \mathcal{G}
  9. 𝒢 \mathcal{G}
  10. 𝒢 \mathcal{G}
  11. 𝒢 \mathcal{G}
  12. 𝒢 \mathcal{G}
  13. 𝒢 \mathcal{G}
  14. L ( z ) = k = 1 ( - 1 ) k + 1 k ( z - 1 ) k L(z)=\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}(z-1)^{k}
  15. g = ( 1 , 0 , 1 , - 1 2 , 1 3 , - 1 4 , 1 5 , - 1 6 , ) g=\left(1,0,1,-\frac{1}{2},\frac{1}{3},-\frac{1}{4},\frac{1}{5},-\frac{1}{6},% \cdots\right)
  16. f ( z ) = k = 0 a k z n k f(z)=\sum_{k=0}^{\infty}a_{k}z^{n_{k}}
  17. lim inf k n k + 1 n k > 1 \liminf_{k\to\infty}\frac{n_{k+1}}{n_{k}}>1
  18. f ( z ) = k = 0 α k ( z - z 0 ) k f(z)=\sum_{k=0}^{\infty}\alpha_{k}(z-z_{0})^{k}
  19. f ( z ) = k = 0 ε k α k ( z - z 0 ) k f(z)=\sum_{k=0}^{\infty}\varepsilon_{k}\alpha_{k}(z-z_{0})^{k}

Analytic_number_theory.html

  1. lim x π ( x ) x / ln ( x ) = 1 , \lim_{x\to\infty}\frac{\pi(x)}{x/\ln(x)}=1,
  2. a ( = ) a ln a a(=\infty)\frac{a}{\ln a}
  3. [ i ] \mathbb{Z}[i]
  4. 2 N 1 log t d t . \,\int^{N}_{2}\frac{1}{\log\,t}\,dt.
  5. π ( x ) = ( number of primes x ) , \pi(x)=(\,\text{number of primes }\leq x),
  6. lim x π ( x ) x / log x = 1. \lim_{x\to\infty}\frac{\pi(x)}{x/\log x}=1.
  7. π ( x , a , q ) = ( number of primes x such that p is in the arithmetic progression a + n q , n 𝐙 ) , \pi(x,a,q)=(\text{number of primes }\leq x\,\text{ such that }p\,\text{ is in % the arithmetic progression }a+nq,n\in\mathbf{Z}),
  8. lim x π ( x , a , q ) ϕ ( q ) x / log x = 1. \lim_{x\to\infty}\frac{\pi(x,a,q)\phi(q)}{x/\log x}=1.
  9. n = x 1 k + + x k . n=x_{1}^{k}+\cdots+x_{\ell}^{k}.\,
  10. G ( k ) k ( 3 log k + 11 ) . G(k)\leq k(3\log k+11).\,
  11. x 2 + y 2 r 2 . x^{2}+y^{2}\leq r^{2}.
  12. π r 2 + E ( r ) \,\pi r^{2}+E(r)\,
  13. E ( r ) / r 2 0 \,E(r)/r^{2}\,\to 0\,
  14. r \,r\to\infty\,
  15. E ( r ) = O ( r ) E(r)=O(r)
  16. O ( r δ ) O(r^{\delta})
  17. δ < 1 \delta<1
  18. E ( r ) = O ( r 2 / 3 ) E(r)=O(r^{2/3})
  19. E ( r ) = O ( r 1 / 2 ) E(r)=O(r^{1/2})
  20. ϵ > 0 \epsilon>0
  21. C ( ϵ ) C(\epsilon)
  22. E ( r ) C ( ϵ ) r 1 / 2 + ϵ E(r)\leq C(\epsilon)r^{1/2+\epsilon}
  23. E ( r ) = O ( r 131 / 208 ) E(r)=O(r^{131/208})
  24. f ( s ) = n = 1 a n n - s . f(s)=\sum_{n=1}^{\infty}a_{n}n^{-s}.
  25. a n a_{n}
  26. ( n = 1 a n n - s ) ( n = 1 b n n - s ) = n = 1 ( k = n a k b ) n - s ; \left(\sum_{n=1}^{\infty}a_{n}n^{-s}\right)\left(\sum_{n=1}^{\infty}b_{n}n^{-s% }\right)=\sum_{n=1}^{\infty}\left(\sum_{k\ell=n}a_{k}b_{\ell}\right)n^{-s};
  27. n = 1 1 n s = p 1 1 - p - s for s > 1 ( p is prime number) \sum_{n=1}^{\infty}\frac{1}{n^{s}}=\prod_{p}^{\infty}\frac{1}{1-p^{-s}}\,\text% { for }s>1\,\,\ (p\,\text{ is prime number)}\,
  28. ( s ) = 1 / 2 \,\Re(s)=1/2\,
  29. O ( x 1 / 2 + ε ) O(x^{1/2+\varepsilon})
  30. ( z ) = 1 / 2. \,\Re(z)=1/2.\,

Analytical_Society.html

  1. ϕ ( p , q ) , \phi(p,q),
  2. d ϕ = ϕ p d p + ϕ q d q . d\phi=\frac{\partial\phi}{\partial p}dp+\frac{\partial\phi}{\partial q}dq.

Andrey_Kolmogorov.html

  1. 1 = 1 2 1=1^{2}
  2. 1 + 3 = 2 2 1+3=2^{2}
  3. 1 + 3 + 5 = 3 2 1+3+5=3^{2}
  4. 1 + 3 + 5 + 7 = 4 2 1+3+5+7=4^{2}

Andromeda_Galaxy.html

  1. × 10 1 9 \times 10^{1}9
  2. × 10 1 2 \times 10^{1}2
  3. × 10 1 1 \times 10^{1}1
  4. × 10 9 \times 10^{9}
  5. M \begin{smallmatrix}M_{\odot}\end{smallmatrix}

Anechoic_chamber.html

  1. λ = v / f \lambda=v/f
  2. f f
  3. λ / 4 \lambda/4
  4. λ \lambda

Angle_of_attack.html

  1. α \alpha

Angle_of_view.html

  1. α = 2 arctan d 2 f \alpha=2\arctan\frac{d}{2f}
  2. d d
  3. d = 36 d=36
  4. d = 24 d=24
  5. α d f \alpha\approx\frac{d}{f}
  6. 180 d π f \frac{180d}{\pi f}
  7. f = F ( 1 + m ) f=F\cdot(1+m)
  8. m m
  9. f = 1.5 F f=1.5\cdot F
  10. α = 2 arctan d 2 F ( 1 + m / P ) \alpha=2\arctan\frac{d}{2F\cdot(1+m/P)}
  11. α h = 2 arctan h 2 f = 2 arctan 36 2 × 50 39.6 \alpha_{h}=2\arctan\frac{h}{2f}=2\arctan\frac{36}{2\times 50}\approx 39.6^{\circ}
  12. α v = 2 arctan v 2 f = 2 arctan 24 2 × 50 27.0 \alpha_{v}=2\arctan\frac{v}{2f}=2\arctan\frac{24}{2\times 50}\approx 27.0^{\circ}
  13. α d = 2 arctan d 2 f = 2 arctan 43.3 2 × 50 46.8 \alpha_{d}=2\arctan\frac{d}{2f}=2\arctan\frac{43.3}{2\times 50}\approx 46.8^{\circ}
  14. S 1 S_{1}
  15. d d
  16. S 2 S_{2}
  17. α / 2 \alpha/2
  18. α \alpha
  19. α \alpha
  20. d / 2 d/2
  21. S 2 S_{2}
  22. tan ( α / 2 ) = d / 2 S 2 . \tan(\alpha/2)=\frac{d/2}{S_{2}}.
  23. α = 2 arctan d 2 S 2 \alpha=2\arctan\frac{d}{2S_{2}}
  24. S 2 S_{2}
  25. F F
  26. α = 2 arctan d 2 f \alpha=2\arctan\frac{d}{2f}
  27. f = F f=F
  28. S 2 = S 1 f S 1 - f S_{2}=\frac{S_{1}f}{S_{1}-f}
  29. S 2 S_{2}
  30. F F
  31. 1 F = 1 S 1 + 1 S 2 \frac{1}{F}=\frac{1}{S_{1}}+\frac{1}{S_{2}}
  32. m = S 2 / S 1 m=S_{2}/S_{1}
  33. S 1 S_{1}
  34. S 2 = F ( 1 + m ) S_{2}=F\cdot(1+m)
  35. f = S 2 f=S_{2}
  36. α = 2 arctan d 2 f \alpha=2\arctan\frac{d}{2f}
  37. f = F ( 1 + m ) f=F\cdot(1+m)
  38. α = 2 arctan d 2 F ( 1 + m / P ) \alpha=2\arctan\frac{d}{2F\cdot(1+m/P)}
  39. D D
  40. d d
  41. α = 2 arctan L 2 f c \alpha=2\arctan\frac{L}{2f_{c}}
  42. L L
  43. f c f_{c}
  44. FOV = α D d \mathrm{FOV}=\alpha\frac{D}{d}
  45. FOV = 2 arctan L D 2 f c d \mathrm{FOV}=2\arctan\frac{LD}{2f_{c}d}

Angle_trisection.html

  1. θ \theta
  2. cos θ , \cos\theta,
  3. cos θ = 4 cos 3 ( θ / 3 ) - 3 cos ( θ / 3 ) . \cos\theta=4\cos^{3}(\theta/3)-3\cos(\theta/3).
  4. π / 3 \pi/3
  5. c o s ( 20 ° ) cos(20°)
  6. y = c o s ( 20 ° ) y=cos(20°)
  7. c o s ( 60 ° ) cos(60°)
  8. = cos ( π / 3 ) = 1 / 2 =\cos(\pi/3)=1/2
  9. cos ( π / 3 ) = 1 / 2 = 4 y 3 - 3 y \cos(\pi/3)=1/2=4y^{3}-3y
  10. 4 y 3 - 3 y - 1 / 2 = 0 4y^{3}-3y-1/2=0
  11. 8 y 3 - 6 y - 1 = 0 8y^{3}-6y-1=0
  12. ( 2 y ) 3 - 3 ( 2 y ) - 1 = 0 (2y)^{3}-3(2y)-1=0
  13. x = 2 y x=2y
  14. x 3 - 3 x - 1 = 0 x^{3}-3x-1=0
  15. p ( x ) = x 3 - 3 x - 1 p(x)=x^{3}-3x-1
  16. c o s ( 20 ° ) cos(20°)
  17. p ( x ) p(x)
  18. p ( x ) p(x)
  19. p ( x ) p(x)
  20. c o s ( 20 ° ) cos(20°)
  21. θ \theta
  22. 3 θ 3\theta
  23. θ \theta
  24. 3 π / 7 3\pi/7
  25. 3 π / 7 3\pi/7
  26. 15 π / 7 15\pi/7
  27. π / 7 \pi/7
  28. N N
  29. 2 π / N 2\pi/N
  30. 3 3
  31. N N
  32. N > 2 N>2
  33. N N
  34. θ \theta
  35. q ( t ) = 4 t 3 - 3 t - cos ( θ ) q(t)=4t^{3}-3t-\cos(\theta)
  36. ( cos ( θ ) ) (\cos(\theta))
  37. b = ( 1 / 3 ) a b=(1/3)a
  38. e + c = 180 e+c=180
  39. e + 2 b = 180 e+2b=180
  40. c = 2 b c=2b
  41. d + 2 c = 180 d+2c=180
  42. d = 180 d=180
  43. - 2 c -2c
  44. d = 180 d=180
  45. - 4 b -4b
  46. a + d + b = 180 a+d+b=180
  47. a + ( 180 a+(180
  48. - 4 b ) + b = 180 -4b)+b=180
  49. a - 3 b = 0 a-3b=0
  50. a = 3 b a=3b

Angular_aperture.html

  1. d d
  2. a = 2 arctan ( D / 2 f ) = 2 arctan ( D 2 f ) a=2\arctan\left(\frac{D/2}{f}\right)=2\arctan\left(\frac{D}{2f}\right)
  3. f f
  4. D D
  5. NA = sin a / 2 = sin arctan ( D 2 f ) \mathrm{NA}=\sin a/2=\sin\arctan\left(\frac{D}{2f}\right)
  6. D < f D<f
  7. NA a / 2 \mathrm{NA}\approx a/2

Angular_frequency.html

  1. ω \vec{\omega}
  2. ω = 2 π T = 2 π f , \omega={{2\pi}\over T}={2\pi f},
  3. ω = v / r \omega=v/r
  4. ω = k m , \omega=\sqrt{\frac{k}{m}},
  5. a = - ω 2 x , a=-\omega^{2}x\;,
  6. a = - 4 π 2 f 2 x . a=-4\pi^{2}f^{2}x\;.
  7. ω = 1 L C \omega=\sqrt{1\over LC}

Angular_resolution.html

  1. θ = 1.220 λ D \theta=1.220\frac{\lambda}{D}
  2. J 1 J_{1}
  3. Δ = 1.220 f λ D \Delta\ell=1.220\frac{f\lambda}{D}
  4. Δ 1.220 f λ D = 1.22 λ ( f / # ) \Delta\ell\approx 1.220\frac{f\lambda}{D}=1.22\lambda\cdot(f/\#)
  5. 2.44 λ ( f / # ) 2.44\lambda\cdot(f/\#)
  6. R = λ D R=\frac{\lambda}{D}
  7. R = λ B R=\frac{\lambda}{B}
  8. α \alpha
  9. R = 1.22 λ NA condenser + NA objective R=\frac{1.22\lambda}{\mathrm{NA}\text{condenser}+\mathrm{NA}\text{objective}}
  10. NA = η sin θ \mathrm{NA}=\eta\sin\theta
  11. θ \theta
  12. α \alpha
  13. η \eta
  14. λ \lambda
  15. R = 0.61 λ NA λ 2 N A R=\frac{0.61\lambda}{\mathrm{NA}}\approx\frac{\lambda}{2\mathrm{NA}}
  16. θ \theta
  17. λ \lambda
  18. R = 1.22 × 400 nm 1.45 + 0.95 = 203 nm R=\frac{1.22\times 400\,\mbox{nm}~{}}{1.45\ +\ 0.95}=203\,\mbox{nm}~{}

Anonymous_Internet_banking.html

  1. n = P Q n=PQ
  2. P P
  3. Q Q
  4. n n
  5. e e
  6. d d
  7. R R
  8. R = R e ( mod n ) R^{\prime}=R^{e}\;\;(\mathop{{\rm mod}}n)
  9. R R^{\prime}
  10. R R
  11. R R
  12. R R^{\prime}
  13. d d
  14. R R^{\prime}
  15. w w
  16. n n
  17. n n
  18. R ′′ = w e * R R^{\prime\prime}=w^{e}*R^{\prime}
  19. R ′′′ R^{\prime\prime\prime}
  20. R R
  21. R ′′′ \displaystyle R^{\prime\prime\prime}
  22. R ′′ R^{\prime\prime}
  23. R ′′′ R^{\prime\prime\prime}
  24. R R^{\prime}
  25. R R
  26. R ′′ R^{\prime\prime}
  27. R R^{\prime}
  28. R ′′′ R^{\prime\prime\prime}
  29. R R
  30. R ′′′ R^{\prime\prime\prime}
  31. w w
  32. R R
  33. R R
  34. R = R e ( mod n ) R^{\prime}=R^{e}\;\;(\mathop{{\rm mod}}n)
  35. R R
  36. R R

Antenna_(radio).html

  1. I \scriptstyle I
  2. I iso \scriptstyle I\text{iso}
  3. G dBi = 10 log I I iso G\text{dBi}=10\log{I\over I\text{iso}}\,
  4. I dipole \scriptstyle I\text{dipole}
  5. G dBd = 10 log I I dipole G\text{dBd}=10\log{I\over I\text{dipole}}\,
  6. G dBi = G dBd + 2.15 G\text{dBi}=G\text{dBd}+2.15\,
  7. A e f f = λ 2 4 π G A_{eff}={\lambda^{2}\over 4\pi}\,G
  8. π \scriptstyle{\pi}
  9. | E V | = 2 | E 0 | | cos ( 2 π h λ sin θ ) | \textstyle{\left|E_{V}\right|=2\left|E_{0}\right|\,\left|\cos\left({2\pi h% \over\lambda}\sin\theta\right)\right|}
  10. | E H | = 2 | E 0 | | sin ( 2 π h λ sin θ ) | \textstyle{\left|E_{H}\right|=2\left|E_{0}\right|\,\left|\sin\left({2\pi h% \over\lambda}\sin\theta\right)\right|}
  11. E 0 \scriptstyle{E_{0}}
  12. λ \scriptstyle{\lambda}
  13. h \scriptstyle{h}
  14. θ = 0 \scriptstyle{\theta=0}
  15. θ = 0 \scriptstyle{\theta=0}
  16. Z j i \scriptstyle{Z_{ji}}
  17. j ω M \scriptstyle{j\omega M}
  18. Z 21 \scriptstyle{Z_{21}}
  19. Z j i = v j i i Z_{ji}={v_{j}\over i_{i}}
  20. i i \textstyle{i_{i}}
  21. v j \textstyle{v_{j}}
  22. i 1 \textstyle{i_{1}}
  23. Z i i = v i i i Z_{ii}={v_{i}\over i_{i}}
  24. v 1 = i 1 Z 11 + i 2 Z 12 + + i n Z 1 n v 2 = i 1 Z 21 + i 2 Z 22 + + i n Z 2 n v n = i 1 Z n 1 + i 2 Z n 2 + + i n Z n n \begin{matrix}v_{1}&=&i_{1}Z_{11}&+&i_{2}Z_{12}&+&\cdots&+&i_{n}Z_{1n}\\ v_{2}&=&i_{1}Z_{21}&+&i_{2}Z_{22}&+&\cdots&+&i_{n}Z_{2n}\\ \vdots&&\vdots&&\vdots&&&&\vdots\\ v_{n}&=&i_{1}Z_{n1}&+&i_{2}Z_{n2}&+&\cdots&+&i_{n}Z_{nn}\end{matrix}
  25. v i \scriptstyle{v_{i}}
  26. i i
  27. i i \scriptstyle{i_{i}}
  28. i i
  29. Z i i \scriptstyle{Z_{ii}}
  30. i i
  31. Z i j \scriptstyle{Z_{ij}}
  32. i i
  33. j j
  34. < m t p l > λ 2 \scriptstyle<mtpl>{{\lambda\over 2}}
  35. Z i j = Z j i . \scriptstyle{Z_{ij}\,=\,Z_{ji}}.
  36. i i
  37. i i = 0 i_{i}=0
  38. i i
  39. v i = 0 \textstyle{v_{i}}=0

Anti-de_Sitter_space.html

  1. Λ \Lambda
  2. F = G m 1 m 2 r 2 \textstyle F=G\frac{m_{1}m_{2}}{r^{2}}
  3. p , q + 1 \mathbb{R}^{p,q+1}
  4. d s 2 = i = 1 p d x i 2 - j = 1 q + 1 d t j 2 ds^{2}=\sum_{i=1}^{p}dx_{i}^{2}-\sum_{j=1}^{q+1}dt_{j}^{2}
  5. i = 1 p x i 2 - j = 1 q + 1 t j 2 = - α 2 \sum_{i=1}^{p}x_{i}^{2}-\sum_{j=1}^{q+1}t_{j}^{2}=-\alpha^{2}
  6. α \alpha
  7. t 1 = α sin ( τ ) , t 2 = α cos ( τ ) , t_{1}=\alpha\sin(\tau),t_{2}=\alpha\cos(\tau),
  8. d s 2 = 1 y 2 ( - d t 2 + d y 2 + i d x i 2 ) , ds^{2}=\frac{1}{y^{2}}\left(-dt^{2}+dy^{2}+\sum_{i}dx_{i}^{2}\right),
  9. y > 0 y>0
  10. y 0 y\to 0
  11. d s 2 = - d t 2 + i d x i 2 ds^{2}=-dt^{2}+\sum_{i}dx_{i}^{2}
  12. r 0 r\geqslant 0
  13. d s 2 = - ( k 2 r 2 + 1 ) d t 2 + 1 k 2 r 2 + 1 d r 2 + r 2 d Ω 2 ds^{2}=-\left(k^{2}r^{2}+1\right)dt^{2}+\frac{1}{k^{2}r^{2}+1}dr^{2}+r^{2}d% \Omega^{2}
  14. S 2 = O ( 3 ) O ( 2 ) S^{2}=\frac{\mathrm{O}(3)}{\mathrm{O}(2)}
  15. AdS n = O ( 2 , n - 1 ) O ( 1 , n - 1 ) \mathrm{AdS}_{n}=\frac{\mathrm{O}(2,n-1)}{\mathrm{O}(1,n-1)}
  16. Spin + ( 2 , n - 1 ) Spin + ( 1 , n - 1 ) \frac{\mathrm{Spin}^{+}(2,n-1)}{\mathrm{Spin}^{+}(1,n-1)}
  17. AdS n \mathrm{AdS}_{n}
  18. o ( 1 , n ) o(1,n)
  19. = ( 0 0 0 0 ( 0 v t ) ( 0 v ) B ) \mathcal{H}=\begin{pmatrix}\begin{matrix}0&0\\ 0&0\end{matrix}&\begin{pmatrix}\cdots 0\cdots\\ \leftarrow v^{t}\rightarrow\end{pmatrix}\\ \begin{pmatrix}\vdots&\uparrow\\ 0&v\\ \vdots&\downarrow\end{pmatrix}&B\end{pmatrix}
  20. B B
  21. 𝒢 = O ( 2 , n ) \mathcal{G}=\mathrm{O}(2,n)
  22. 𝒬 = ( 0 a - a 0 ( w t 0 ) ( w 0 ) 0 ) . \mathcal{Q}=\begin{pmatrix}\begin{matrix}0&a\\ -a&0\end{matrix}&\begin{pmatrix}\leftarrow w^{t}\rightarrow\\ \cdots 0\cdots\\ \end{pmatrix}\\ \begin{pmatrix}\uparrow&\vdots\\ w&0\\ \downarrow&\vdots\end{pmatrix}&0\end{pmatrix}.
  23. 𝒢 = 𝒬 \mathcal{G}=\mathcal{H}\oplus\mathcal{Q}
  24. [ , 𝒬 ] 𝒬 [\mathcal{H},\mathcal{Q}]\subseteq\mathcal{Q}
  25. [ 𝒬 , 𝒬 ] [\mathcal{Q},\mathcal{Q}]\subseteq\mathcal{H}
  26. AdS n \mathrm{AdS}_{n}
  27. Λ \Lambda
  28. Λ < 0 \Lambda<0
  29. = 1 16 π G ( n ) ( R - 2 Λ ) \mathcal{L}=\frac{1}{16\pi G_{(n)}}(R-2\Lambda)
  30. G μ ν + Λ g μ ν = 0 G_{\mu\nu}+\Lambda g_{\mu\nu}=0
  31. G μ ν G_{\mu\nu}
  32. g μ ν g_{\mu\nu}
  33. α \alpha
  34. Λ = - ( n - 1 ) ( n - 2 ) 2 α 2 \Lambda=\frac{-(n-1)(n-2)}{2\alpha^{2}}
  35. n + 1 n+1
  36. ( - , - , + , , + ) (-,-,+,\cdots,+)
  37. - X 1 2 - X 2 2 + i = 3 n + 1 X i 2 = - α 2 -X_{1}^{2}-X_{2}^{2}+\sum_{i=3}^{n+1}X_{i}^{2}=-\alpha^{2}
  38. AdS n \mathrm{AdS}_{n}
  39. ( τ , ρ , θ , φ 1 , , φ n - 3 ) (\tau,\rho,\theta,\varphi_{1},\cdots,\varphi_{n-3})
  40. { X 1 = α cosh ρ cos τ X 2 = α cosh ρ sin τ X i = α sinh ρ x ^ i i x ^ i 2 = 1 \begin{cases}X_{1}=\alpha\cosh\rho\cos\tau\\ X_{2}=\alpha\cosh\rho\sin\tau\\ X_{i}=\alpha\sinh\rho\,\hat{x}_{i}\qquad\sum_{i}\hat{x}_{i}^{2}=1\end{cases}
  41. x ^ i \hat{x}_{i}
  42. S n - 2 S^{n-2}
  43. x ^ 1 = sin θ sin φ 1 sin φ n - 3 \hat{x}_{1}=\sin\theta\sin\varphi_{1}\dots\sin\varphi_{n-3}
  44. x ^ 2 = sin θ sin φ 1 cos φ n - 3 \quad\hat{x}_{2}=\sin\theta\sin\varphi_{1}\dots\cos\varphi_{n-3}
  45. AdS n \mathrm{AdS}_{n}
  46. d s 2 = α 2 ( - cosh 2 ρ d τ 2 + d ρ 2 + sinh 2 ρ d Ω n - 2 2 ) \mathrm{d}s^{2}=\alpha^{2}(-\cosh^{2}\rho\,\mathrm{d}\tau^{2}+\,\mathrm{d}\rho% ^{2}+\sinh^{2}\rho\,\mathrm{d}\Omega_{n-2}^{2})
  47. τ [ 0 , 2 π ] \tau\in[0,2\pi]
  48. ρ + \rho\in\mathbb{R}^{+}
  49. τ \tau
  50. τ \tau\in\mathbb{R}
  51. ρ \rho\to\infty
  52. AdS n \mathrm{AdS}_{n}
  53. r α sinh ρ r\equiv\alpha\sinh\rho
  54. t α τ t\equiv\alpha\tau
  55. AdS n \mathrm{AdS}_{n}
  56. d s 2 = - f ( r ) d t 2 + 1 f ( r ) d r 2 + r 2 d Ω n - 2 2 \,\mathrm{d}s^{2}=-f(r)\,\mathrm{d}t^{2}+\frac{1}{f(r)}\,\mathrm{d}r^{2}+r^{2}% \,\mathrm{d}\Omega_{n-2}^{2}
  57. f ( r ) = 1 + r 2 α 2 f(r)=1+\frac{r^{2}}{\alpha^{2}}
  58. { X 1 = α 2 2 r ( 1 + r 2 α 4 ( α 2 + x 2 - t 2 ) ) X 2 = r α t X i = r α x i i { 3 , , n } X n + 1 = α 2 2 r ( 1 - r 2 α 4 ( α 2 - x 2 + t 2 ) ) \begin{cases}X_{1}=\frac{\alpha^{2}}{2r}(1+\frac{r^{2}}{\alpha^{4}}(\alpha^{2}% +\vec{x}^{2}-t^{2}))\\ X_{2}=\frac{r}{\alpha}t\\ X_{i}=\frac{r}{\alpha}x_{i}\qquad i\in\{3,\cdots,n\}\\ X_{n+1}=\frac{\alpha^{2}}{2r}(1-\frac{r^{2}}{\alpha^{4}}(\alpha^{2}-\vec{x}^{2% }+t^{2}))\end{cases}
  59. AdS n \mathrm{AdS}_{n}
  60. d s 2 = - r 2 α 2 d t 2 + α 2 r 2 d r 2 + r 2 α 2 d x 2 \mathrm{d}s^{2}=-\frac{r^{2}}{\alpha^{2}}\,\mathrm{d}t^{2}+\frac{\alpha^{2}}{r% ^{2}}\,\mathrm{d}r^{2}+\frac{r^{2}}{\alpha^{2}}\,\mathrm{d}\vec{x}^{2}
  61. 0 r 0\leq r
  62. r = 0 r=0
  63. r r\to\infty
  64. AdS n \mathrm{AdS}_{n}
  65. AdS n \mathrm{AdS}_{n}
  66. u r α 2 u\equiv\frac{r}{\alpha^{2}}
  67. d s 2 = α 2 ( d u 2 u 2 + u 2 ( d x μ d x μ ) ) \mathrm{d}s^{2}=\alpha^{2}\left(\frac{\,\mathrm{d}u^{2}}{u^{2}}+u^{2}(\,% \mathrm{d}x_{\mu}\,\mathrm{d}x^{\mu})\right)
  68. x μ = ( t , x ) x^{\mu}=(t,\vec{x})
  69. z 1 u z\equiv\frac{1}{u}
  70. d s 2 = α 2 z 2 ( d z 2 + d x μ d x μ ) \,\mathrm{d}s^{2}=\frac{\alpha^{2}}{z^{2}}(\,\mathrm{d}z^{2}+\,\mathrm{d}x_{% \mu}\,\mathrm{d}x^{\mu})
  71. AdS n \mathrm{AdS}_{n}
  72. α \alpha
  73. R μ ν α β = - 1 α 2 ( g μ α g ν β - g μ β g ν α ) R_{\mu\nu\alpha\beta}=\frac{-1}{\alpha^{2}}(g_{\mu\alpha}g_{\nu\beta}-g_{\mu% \beta}g_{\nu\alpha})
  74. R μ ν = - ( n - 1 ) α 2 g μ ν R_{\mu\nu}=\frac{-(n-1)}{\alpha^{2}}g_{\mu\nu}
  75. R = - n ( n - 1 ) α 2 R=\frac{-n(n-1)}{\alpha^{2}}

Anticommutativity.html

  1. n n
  2. * : A n 𝔊 \scriptstyle*:A^{n}\to\mathfrak{G}
  3. 𝔊 \scriptstyle\mathfrak{G}
  4. x 1 * x 2 * * x n = sgn ( σ ) ( x σ ( 1 ) * x σ ( 2 ) * * x σ ( n ) ) \forallsymbol x = ( x 1 , x 2 , , x n ) A n {x_{1}*x_{2}*\dots*x_{n}}=\operatorname{sgn}(\sigma)({x_{\sigma(1)}*x_{\sigma(% 2)}*\dots*x_{\sigma(n)}})\qquad\forallsymbol{x}=(x_{1},x_{2},\dots,x_{n})\in A% ^{n}
  5. σ : ( n ) ( n ) \scriptstyle\sigma:(n)\to(n)
  6. sgn ( σ ) \mathrm{sgn}(\sigma)
  7. 𝔊 \scriptstyle\mathfrak{G}
  8. 𝔊 \scriptstyle\mathfrak{G}
  9. * : A × A 𝔊 \scriptstyle*:A\times A\to\mathfrak{G}
  10. x 1 * x 2 = - ( x 2 * x 1 ) ( x 1 , x 2 ) A × A x_{1}*x_{2}=-(x_{2}*x_{1})\qquad\forall(x_{1},x_{2})\in A\times A
  11. 𝔊 \scriptstyle\mathfrak{G}
  12. 𝔊 \scriptstyle\mathfrak{G}
  13. - 𝔞 = 𝔞 𝔞 = 0 𝔞 𝔊 \mathfrak{-a}=\mathfrak{a}\iff\mathfrak{a}=\mathfrak{0}\qquad\forall\mathfrak{% a}\in\mathfrak{G}
  14. x j = x i x_{j}=x_{i}
  15. i , j i,j
  16. x 1 * x 2 * * x n = 0 x_{1}*x_{2}*\dots*x_{n}=\mathfrak{0}
  17. n = 2 n=2
  18. x 1 * x 1 = x 2 * x 2 = 0 x_{1}*x_{1}=x_{2}*x_{2}=\mathfrak{0}

Antilinear_map.html

  1. f : V W f:V\to W
  2. f ( a x + b y ) = a ¯ f ( x ) + b ¯ f ( y ) f(ax+by)=\bar{a}f(x)+\bar{b}f(y)
  3. a , b a,\,b\,\in\mathbb{C}
  4. x , y V x,\,y\,\in V
  5. a ¯ \bar{a}
  6. b ¯ \bar{b}
  7. a a
  8. b b
  9. f : V W f:V\to W
  10. f ¯ : V W ¯ \bar{f}:V\to\bar{W}
  11. V V
  12. W ¯ \bar{W}

Antiproton.html

  1. E = m c 2 E=mc^{2}

Apparent_wind.html

  1. A = W 2 + V 2 + 2 W V cos α A=\sqrt{W^{2}+V^{2}+2WV\cos{\alpha}}
  2. V V
  3. W W
  4. α \alpha
  5. A A
  6. cos ( α ) = cos ( 180 - α ) = - cos ( α ) \cos(\alpha^{\prime})=\cos(180^{\circ}-\alpha)=-\cos(\alpha)
  7. β \beta
  8. arccos \arccos
  9. β = arccos ( W cos α + V A ) = arccos ( W cos α + V W 2 + V 2 + 2 W V cos α ) \beta=\arccos\left(\frac{W\cos\alpha+V}{A}\right)=\arccos\left(\frac{W\cos% \alpha+V}{\sqrt{W^{2}+V^{2}+2WV\cos{\alpha}}}\right)
  10. W = A 2 + V 2 - 2 A V cos β W=\sqrt{A^{2}+V^{2}-2AV\cos{\beta}}
  11. α = arccos ( A cos β - V W ) = arccos ( A cos β - V A 2 + V 2 - 2 A V cos β ) \alpha=\arccos\left(\frac{A\cos\beta-V}{W}\right)=\arccos\left(\frac{A\cos% \beta-V}{\sqrt{A^{2}+V^{2}-2AV\cos{\beta}}}\right)
  12. α \alpha
  13. α = - arccos ( A cos β - V W ) = - arccos ( A cos β - V A 2 + V 2 - 2 A V cos β ) \alpha=-\arccos\left(\frac{A\cos\beta-V}{W}\right)=-\arccos\left(\frac{A\cos% \beta-V}{\sqrt{A^{2}+V^{2}-2AV\cos{\beta}}}\right)

Apsis.html

  1. v per = ( 1 + e ) μ ( 1 - e ) a v_{\mathrm{per}}=\sqrt{\tfrac{(1+e)\mu}{(1-e)a}}\,
  2. r per = ( 1 - e ) a r_{\mathrm{per}}=(1-e)a\!\,
  3. v ap = ( 1 - e ) μ ( 1 + e ) a v_{\mathrm{ap}}=\sqrt{\tfrac{(1-e)\mu}{(1+e)a}}\,
  4. r ap = ( 1 + e ) a r_{\mathrm{ap}}=(1+e)a\!\,
  5. h = ( 1 - e 2 ) μ a h=\sqrt{(1-e^{2})\mu a}
  6. ϵ = - μ 2 a \epsilon=-\frac{\mu}{2a}
  7. a a\!\,
  8. r per + r ap 2 \frac{r_{\mathrm{per}}+r_{\mathrm{ap}}}{2}
  9. μ \mu\!\,
  10. e e\!\,
  11. e = r ap - r per r ap + r per = 1 - 2 r ap r per + 1 e=\frac{r_{\mathrm{ap}}-r_{\mathrm{per}}}{r_{\mathrm{ap}}+r_{\mathrm{per}}}=1-% \frac{2}{\frac{r_{\mathrm{ap}}}{r_{\mathrm{per}}}+1}
  12. a a
  13. b b
  14. - 2 ϵ = μ / a \sqrt{-2\epsilon}=\sqrt{\mu/a}
  15. a a

Archimedean_property.html

  1. x + + x n terms < y . \underbrace{x+\cdots+x}_{n\,\text{ terms}}<y.\,
  2. | x | |x|
  3. x F x\in F
  4. | x y | = | x | | y | |xy|=|x||y|
  5. | x + y | | x | + | y | |x+y|\leq|x|+|y|
  6. x F x\in F
  7. | x + + x n terms | > 1. |\underbrace{x+\cdots+x}_{n\,\text{ terms}}|>1.\,
  8. n n
  9. x x
  10. n n
  11. | x + y | max ( | x | , | y | ) |x+y|\leq\max(|x|,|y|)
  12. | x | = 1 , |x|=1,
  13. x 0 x\neq 0
  14. | x | = x 2 |x|=\sqrt{x^{2}}
  15. p p
  16. p p
  17. p p
  18. p p
  19. p p
  20. ( < v a r > x , 2 < v a r > x < / v a r > ) (<var>x, 2<var>x</var>)

Arithmetic_progression.html

  1. a 1 a_{1}
  2. a n a_{n}
  3. a n = a 1 + ( n - 1 ) d , \ a_{n}=a_{1}+(n-1)d,
  4. a n = a m + ( n - m ) d . \ a_{n}=a_{m}+(n-m)d.
  5. 2 + 5 + 8 + 11 + 14 2+5+8+11+14
  6. n ( a 1 + a n ) 2 \frac{n(a_{1}+a_{n})}{2}
  7. 2 + 5 + 8 + 11 + 14 = 5 ( 2 + 14 ) 2 = 5 × 16 2 = 40. 2+5+8+11+14=\frac{5(2+14)}{2}=\frac{5\times 16}{2}=40.
  8. a 1 a_{1}
  9. a n a_{n}
  10. ( - 3 2 ) + ( - 1 2 ) + 1 2 = 3 ( - 3 2 + 1 2 ) 2 = - 3 2 . \left(-\frac{3}{2}\right)+\left(-\frac{1}{2}\right)+\frac{1}{2}=\frac{3\left(-% \frac{3}{2}+\frac{1}{2}\right)}{2}=-\frac{3}{2}.
  11. S n = a 1 + ( a 1 + d ) + ( a 1 + 2 d ) + + ( a 1 + ( n - 2 ) d ) + ( a 1 + ( n - 1 ) d ) S_{n}=a_{1}+(a_{1}+d)+(a_{1}+2d)+\cdots+(a_{1}+(n-2)d)+(a_{1}+(n-1)d)
  12. S n = ( a n - ( n - 1 ) d ) + ( a n - ( n - 2 ) d ) + + ( a n - 2 d ) + ( a n - d ) + a n . S_{n}=(a_{n}-(n-1)d)+(a_{n}-(n-2)d)+\cdots+(a_{n}-2d)+(a_{n}-d)+a_{n}.
  13. 2 S n = n ( a 1 + a n ) . \ 2S_{n}=n(a_{1}+a_{n}).
  14. S n = n 2 ( a 1 + a n ) . S_{n}=\frac{n}{2}(a_{1}+a_{n}).
  15. a n = a 1 + ( n - 1 ) d a_{n}=a_{1}+(n-1)d
  16. S n = n 2 [ 2 a 1 + ( n - 1 ) d ] . S_{n}=\frac{n}{2}[2a_{1}+(n-1)d].
  17. S n / n S_{n}/n
  18. n ¯ = a 1 + a n 2 . \overline{n}=\frac{a_{1}+a_{n}}{2}.
  19. a 1 a 2 a n = d a 1 d d ( a 1 d + 1 ) d ( a 1 d + 2 ) d ( a 1 d + n - 1 ) = d n ( a 1 d ) n ¯ = d n Γ ( a 1 / d + n ) Γ ( a 1 / d ) , a_{1}a_{2}\cdots a_{n}=d\frac{a_{1}}{d}d(\frac{a_{1}}{d}+1)d(\frac{a_{1}}{d}+2% )\cdots d(\frac{a_{1}}{d}+n-1)=d^{n}{\left(\frac{a_{1}}{d}\right)}^{\overline{% n}}=d^{n}\frac{\Gamma\left(a_{1}/d+n\right)}{\Gamma\left(a_{1}/d\right)},
  20. x n ¯ x^{\overline{n}}
  21. Γ \Gamma
  22. a 1 / d a_{1}/d
  23. 1 × 2 × × n 1\times 2\times\cdots\times n
  24. n ! n!
  25. m × ( m + 1 ) × ( m + 2 ) × × ( n - 2 ) × ( n - 1 ) × n m\times(m+1)\times(m+2)\times\cdots\times(n-2)\times(n-1)\times n\,\!
  26. m m
  27. n n
  28. n ! ( m - 1 ) ! . \frac{n!}{(m-1)!}.
  29. P 50 = 5 50 Γ ( 3 / 5 + 50 ) Γ ( 3 / 5 ) 3.78438 × 10 98 . P_{50}=5^{50}\cdot\frac{\Gamma\left(3/5+50\right)}{\Gamma\left(3/5\right)}% \approx 3.78438\times 10^{98}.
  30. σ = | d | ( n - 1 ) ( n + 1 ) 12 \sigma=|d|\sqrt{\frac{(n-1)(n+1)}{12}}
  31. n n
  32. d d
  33. a 1 a_{1}
  34. a n a_{n}
  35. d d
  36. n n
  37. S n S_{n}
  38. n ¯ \overline{n}
  39. a n = a 1 + ( n - 1 ) d , \ a_{n}=a_{1}+(n-1)d,
  40. a n = a m + ( n - m ) d . \ a_{n}=a_{m}+(n-m)d.
  41. S n = n 2 [ 2 a 1 + ( n - 1 ) d ] . S_{n}=\frac{n}{2}[2a_{1}+(n-1)d].
  42. S n = n ( a 1 + a n ) 2 S_{n}=\frac{n(a_{1}+a_{n})}{2}
  43. n ¯ \overline{n}
  44. S n / n S_{n}/n
  45. n ¯ = a 1 + a n 2 . \overline{n}=\frac{a_{1}+a_{n}}{2}.

Arithmetical_hierarchy.html

  1. Σ n 0 \Sigma^{0}_{n}
  2. Π n 0 \Pi^{0}_{n}
  3. ϕ \phi
  4. ϕ \phi
  5. Σ 0 0 \Sigma^{0}_{0}
  6. Π 0 0 \Pi^{0}_{0}
  7. Σ n 0 \Sigma^{0}_{n}
  8. Π n 0 \Pi^{0}_{n}
  9. ϕ \phi
  10. n 1 n 2 n k ψ \exists n_{1}\exists n_{2}\cdots\exists n_{k}\psi
  11. ψ \psi
  12. Π n 0 \Pi^{0}_{n}
  13. ϕ \phi
  14. Σ n + 1 0 \Sigma^{0}_{n+1}
  15. ϕ \phi
  16. n 1 n 2 n k ψ \forall n_{1}\forall n_{2}\cdots\forall n_{k}\psi
  17. ψ \psi
  18. Σ n 0 \Sigma^{0}_{n}
  19. ϕ \phi
  20. Π n + 1 0 \Pi^{0}_{n+1}
  21. Σ n 0 \Sigma^{0}_{n}
  22. n - 1 n-1
  23. Π n 0 \Pi^{0}_{n}
  24. Σ n 0 \Sigma^{0}_{n}
  25. Π n 0 \Pi^{0}_{n}
  26. Σ m 0 \Sigma^{0}_{m}
  27. Π m 0 \Pi^{0}_{m}
  28. n X ϕ ( n ¯ ) , n\in X\Leftrightarrow\mathbb{N}\models\phi(\underline{n}),
  29. n ¯ \underline{n}
  30. n n
  31. Σ n 0 \Sigma^{0}_{n}
  32. Π n 0 \Pi^{0}_{n}
  33. Δ n 0 \Delta^{0}_{n}
  34. n n
  35. Σ n 0 \Sigma^{0}_{n}
  36. Σ n 0 \Sigma^{0}_{n}
  37. Π n 0 \Pi^{0}_{n}
  38. Π n 0 \Pi^{0}_{n}
  39. Σ n 0 \Sigma^{0}_{n}
  40. Π n 0 \Pi^{0}_{n}
  41. X X
  42. Δ n 0 \Delta^{0}_{n}
  43. Δ n 0 \Delta^{0}_{n}
  44. Δ n 0 \Delta^{0}_{n}
  45. Δ n 0 \Delta^{0}_{n}
  46. Σ n 0 \Sigma^{0}_{n}
  47. Π n 0 \Pi^{0}_{n}
  48. Σ n 0 \Sigma^{0}_{n}
  49. Δ n 0 \Delta^{0}_{n}
  50. Π n 0 \Pi^{0}_{n}
  51. Σ n 0 , Y \Sigma^{0,Y}_{n}
  52. Δ n 0 , Y \Delta^{0,Y}_{n}
  53. Π n 0 , Y \Pi^{0,Y}_{n}
  54. Σ n 0 , Y \Sigma^{0,Y}_{n}
  55. Σ n 0 \Sigma^{0}_{n}
  56. Σ n 0 , Y \Sigma^{0,Y}_{n}
  57. Σ n 0 \Sigma^{0}_{n}
  58. Σ n 0 , Y \Sigma^{0,Y}_{n}
  59. ϕ ( n ) = m t ( Y ( m ) and m × t = n ) \phi(n)=\exists m\exists t(Y(m)\and m\times t=n)
  60. Σ 1 0 , Y \Sigma^{0,Y}_{1}
  61. Δ 0 0 , Y \Delta^{0,Y}_{0}
  62. Σ n 0 \Sigma^{0}_{n}
  63. Π n 0 \Pi^{0}_{n}
  64. X A Y X\leq_{A}Y
  65. Σ n 0 , Y \Sigma^{0,Y}_{n}
  66. Π n 0 , Y \Pi^{0,Y}_{n}
  67. X A Y X\leq_{A}Y
  68. X A Y X\leq_{A}Y
  69. A \equiv_{A}
  70. X A Y X A Y and Y A X X\equiv_{A}Y\Leftrightarrow X\leq_{A}Y\and Y\leq_{A}X
  71. A \leq_{A}
  72. 2 ω 2^{\omega}
  73. ω ω \omega^{\omega}
  74. 𝒩 \mathcal{N}
  75. Σ n 0 \Sigma^{0}_{n}
  76. Σ n 0 \Sigma^{0}_{n}
  77. Π n 0 \Pi^{0}_{n}
  78. Π n 0 \Pi^{0}_{n}
  79. Σ n 0 \Sigma^{0}_{n}
  80. Π n 0 \Pi^{0}_{n}
  81. Δ n 0 \Delta^{0}_{n}
  82. O 2 ω O\subset 2^{\omega}
  83. O = { X 2 ω | n ( X ( n ) = 1 ) } O=\{X\in 2^{\omega}|\exists n(X(n)=1)\}
  84. O O
  85. Σ 1 0 \Sigma^{0}_{1}
  86. Σ 1 0 \Sigma^{0}_{1}
  87. Π n 0 \Pi^{0}_{n}
  88. X X
  89. { X } \{X\}
  90. Π n 0 \Pi^{0}_{n}
  91. ω \omega
  92. ω \omega
  93. Σ n 1 \Sigma^{1}_{n}
  94. Π n 1 \Pi^{1}_{n}
  95. Δ n 1 \Delta^{1}_{n}
  96. Σ n 0 {\Sigma}^{0}_{n}
  97. Σ n 0 , Y \Sigma^{0,Y}_{n}
  98. Σ 0 0 \Sigma^{0}_{0}
  99. Π 0 0 \Pi^{0}_{0}
  100. Σ n 0 \Sigma^{0}_{n}
  101. Π n 0 \Pi^{0}_{n}
  102. R ( n 1 , , n l , m 1 , , m k ) R(n_{1},\ldots,n_{l},m_{1},\ldots,m_{k})\,
  103. Σ n 0 \Sigma^{0}_{n}
  104. S ( n 1 , , n l ) = m 1 m k R ( n 1 , , n l , m 1 , , m k ) S(n_{1},\ldots,n_{l})=\forall m_{1}\cdots\forall m_{k}R(n_{1},\ldots,n_{l},m_{% 1},\ldots,m_{k})
  105. Π n + 1 0 \Pi^{0}_{n+1}
  106. R ( n 1 , , n l , m 1 , , m k ) R(n_{1},\ldots,n_{l},m_{1},\ldots,m_{k})\,
  107. Π n 0 \Pi^{0}_{n}
  108. S ( n 1 , , n l ) = m 1 m k R ( n 1 , , n l , m 1 , , m k ) S(n_{1},\ldots,n_{l})=\exists m_{1}\cdots\exists m_{k}R(n_{1},\ldots,n_{l},m_{% 1},\ldots,m_{k})
  109. Σ n + 1 0 \Sigma^{0}_{n+1}
  110. n n
  111. Σ n 0 \Sigma^{0}_{n}
  112. Π n 0 \Pi^{0}_{n}
  113. Σ n 0 \Sigma^{0}_{n}
  114. Π n 0 \Pi^{0}_{n}
  115. 0
  116. Σ n 0 \Sigma^{0}_{n}
  117. Π n 0 \Pi^{0}_{n}
  118. Δ n 0 \Delta^{0}_{n}
  119. i + 1 i+1
  120. i i
  121. Σ 1 0 \Sigma^{0}_{1}
  122. n 1 n k ψ ( n 1 , , n k , m ) \exists n_{1}\cdots\exists n_{k}\psi(n_{1},\ldots,n_{k},m)
  123. ψ \psi
  124. Π 2 0 \Pi^{0}_{2}
  125. e e
  126. m m
  127. s s
  128. e e
  129. m m
  130. s s
  131. Σ 1 0 \Sigma^{0}_{1}
  132. Σ 1 0 \Sigma^{0}_{1}
  133. Σ 1 0 \Sigma^{0}_{1}
  134. Π 1 0 \Pi^{0}_{1}
  135. Π 1 0 \Pi^{0}_{1}
  136. Π 2 0 \Pi^{0}_{2}
  137. G δ G_{\delta}
  138. Σ 1 0 \Sigma^{0}_{1}
  139. ϕ ( X , n , m ) \phi(X,n,m)
  140. Σ 0 0 \Sigma^{0}_{0}
  141. n , m n,m
  142. Π 2 0 \Pi^{0}_{2}
  143. { X n m ϕ ( X , n , m ) } \{X\mid\forall n\exists m\phi(X,n,m)\}
  144. Σ 1 0 \Sigma^{0}_{1}
  145. { X m ϕ ( X , n , m ) } \{X\mid\exists m\phi(X,n,m)\}
  146. Π n 0 \Pi^{0}_{n}
  147. Σ n 0 \Sigma^{0}_{n}
  148. Σ n 0 \Sigma^{0}_{n}
  149. Π n 0 \Pi^{0}_{n}
  150. Δ n 0 \Delta^{0}_{n}
  151. Σ n 0 \Sigma^{0}_{n}
  152. Π n 0 \Pi^{0}_{n}
  153. Δ n 0 \Delta^{0}_{n}
  154. Δ n 0 Π n 0 \Delta^{0}_{n}\subsetneq\Pi^{0}_{n}
  155. Δ n 0 Σ n 0 \Delta^{0}_{n}\subsetneq\Sigma^{0}_{n}
  156. n 1 n\geq 1
  157. Π n 0 Π n + 1 0 \Pi^{0}_{n}\subsetneq\Pi^{0}_{n+1}
  158. Σ n 0 Σ n + 1 0 \Sigma^{0}_{n}\subsetneq\Sigma^{0}_{n+1}
  159. n n
  160. Σ n 0 Π n 0 Δ n + 1 0 \Sigma^{0}_{n}\cup\Pi^{0}_{n}\subsetneq\Delta^{0}_{n+1}
  161. n 1 n\geq 1
  162. Δ 1 0 \Delta^{0}_{1}
  163. Σ 1 0 \Sigma^{0}_{1}
  164. Δ n 0 , Y \Delta^{0,Y}_{n}
  165. Σ n + 1 0 , Y \Sigma^{0,Y}_{n+1}
  166. ( n ) \emptyset^{(n)}
  167. Σ n 0 \Sigma^{0}_{n}
  168. ( n ) \mathbb{N}\setminus\emptyset^{(n)}
  169. Π n 0 \Pi^{0}_{n}
  170. ( n - 1 ) \emptyset^{(n-1)}
  171. Δ n 0 \Delta^{0}_{n}
  172. Δ 1 0 \Delta^{0}_{1}

Arrow's_impossibility_theorem.html

  1. A \mathrm{A}
  2. N \mathrm{N}
  3. A \mathrm{A}
  4. L ( A ) \mathrm{L(A)}
  5. F : L ( A ) N L ( A ) F:\mathrm{L(A)}^{N}\to\mathrm{L(A)}
  6. A \mathrm{A}
  7. N \mathrm{N}
  8. ( R 1 , , R N ) (R_{1},\ldots,R_{N})
  9. A \mathrm{A}
  10. R 1 , , R N R_{1},\ldots,R_{N}
  11. F ( R 1 , R 2 , , R N ) F(R_{1},R_{2},\ldots,R_{N})
  12. i { 1 , , N } i\in\{1,\ldots,N\}
  13. ( R 1 , , R N ) (R_{1},\ldots,R_{N})
  14. ( S 1 , , S N ) (S_{1},\ldots,S_{N})
  15. R i R_{i}
  16. S i S_{i}
  17. F ( R 1 , R 2 , , R N ) F(R_{1},R_{2},\ldots,R_{N})
  18. F ( S 1 , S 2 , , S N ) F(S_{1},S_{2},\ldots,S_{N})
  19. \succ
  20. x y x\succ y
  21. y z y\succ z
  22. x z x\succ z
  23. x 1 , , x k x_{1},\ldots,x_{k}
  24. x 1 x 2 , x 2 x 3 , , x k - 1 x k , x k x 1 x_{1}\succ x_{2},\;x_{2}\succ x_{3},\;\ldots,\;x_{k-1}\succ x_{k},\;x_{k}\succ x% _{1}

Arthur_Cayley.html

  1. n n

Articulatory_phonetics.html

  1. P 1 V 1 = P 2 V 2 P_{1}V_{1}=P_{2}V_{2}\,
  2. V 1 ( V 1 + Δ V ) = ( P 1 + Δ P ) P 1 \frac{V_{1}}{(V_{1}+\Delta V)}=\frac{(P_{1}+\Delta P)}{P_{1}}
  3. P 1 P_{1}
  4. V 1 V_{1}
  5. P 2 P_{2}
  6. V 2 V_{2}

Artificial_insemination.html

  1. N = V s × c × r s n r N=\frac{V_{s}\times c\times r_{s}}{n_{r}}
  2. N = V s × f c × c × r s n r N=\frac{V_{s}\times f_{c}\times c\times r_{s}}{n_{r}}

Aryabhata.html

  1. π \pi
  2. π \pi
  3. 1 2 + 2 2 + + n 2 = n ( n + 1 ) ( 2 n + 1 ) 6 1^{2}+2^{2}+\cdots+n^{2}={n(n+1)(2n+1)\over 6}
  4. 1 3 + 2 3 + + n 3 = ( 1 + 2 + + n ) 2 1^{3}+2^{3}+\cdots+n^{3}=(1+2+\cdots+n)^{2}

Ascendant.html

  1. A s c e n d a n t = arctan ( y x ) = arctan ( - cos A sin A cos E + tan L sin E ) Ascendant=\arctan\left(\frac{y}{x}\right)=\arctan\left(\frac{-\cos A}{\sin A% \cos E+\tan L\sin E}\right)

Ashmolean_Museum.html

  1. 𝔓 \mathfrak{P}

Aspect_ratio_(aerodynamics).html

  1. A R = b 2 S AR={b^{2}\over S}
  2. C d C_{d}\;
  3. C d = C d 0 + ( C L ) 2 π e A R C_{d}=C_{d0}+\frac{(C_{L})^{2}}{\pi eAR}
  4. C d C_{d}\;
  5. C d 0 C_{d0}\;
  6. C L C_{L}\;
  7. π \pi\;
  8. e e\;
  9. A R AR
  10. c d c_{d}\;
  11. c d 1 ( chord ) 0.129 . c_{d}\varpropto\frac{1}{(\,\text{chord})^{0.129}}.
  12. 𝐴𝑅 wet = b 2 S w \mathit{AR}_{\mathrm{wet}}={b^{2}\over S_{w}}
  13. b b
  14. S w S_{w}

Assignment_problem.html

  1. a A C ( a , f ( a ) ) \sum_{a\in A}C(a,f(a))
  2. a A C a , f ( a ) \sum_{a\in A}C_{a,f(a)}
  3. i A j T C ( i , j ) x i j \sum_{i\in A}\sum_{j\in T}C(i,j)x_{ij}
  4. j T x i j = 1 for i A , \sum_{j\in T}x_{ij}=1\,\text{ for }i\in A,\,
  5. i A x i j = 1 for j T , \sum_{i\in A}x_{ij}=1\,\text{ for }j\in T,\,
  6. x i j 0 for i , j A , T . x_{ij}\geq 0\,\text{ for }i,j\in A,T.\,
  7. x i j x_{ij}
  8. i i
  9. j j

Associative_array.html

  1. ( k e y , v a l u e ) (key,value)
  2. ( k e y , v a l u e ) (key,value)
  3. ( k e y , v a l u e ) (key,value)

Astronautics.html

  1. m 1 m_{1}
  2. m 0 m_{0}
  3. v e v_{e}
  4. Δ v = v e ln m 0 m 1 \Delta v\ =v_{e}\ln\frac{m_{0}}{m_{1}}

Astronomical_seeing.html

  1. r 0 r_{0}
  2. r 0 r_{0}
  3. r 0 r_{0}
  4. ψ \psi
  5. 𝐤 \mathbf{k}
  6. ψ 0 ( 𝐫 , t ) = A u e i ( ϕ u + 2 π ν t + 𝐤 𝐫 ) \psi_{0}\left(\mathbf{r},t\right)=A_{u}e^{i\left(\phi_{u}+2\pi\nu t+\mathbf{k}% \cdot\mathbf{r}\right)}
  7. ψ 0 \psi_{0}
  8. 𝐫 \mathbf{r}
  9. t t
  10. ϕ u \phi_{u}
  11. ν \nu
  12. ν = c | 𝐤 | / ( 2 π ) \nu=c\left|\mathbf{k}\right|/\left(2\pi\right)
  13. A u A_{u}
  14. A u A_{u}
  15. ψ 0 \psi_{0}
  16. ψ p \psi_{p}
  17. ψ 0 ( 𝐫 ) \psi_{0}\left(\mathbf{r}\right)
  18. ψ p ( 𝐫 ) = ( χ a ( 𝐫 ) e i ϕ a ( 𝐫 ) ) ψ 0 ( 𝐫 ) \psi_{p}\left(\mathbf{r}\right)=\left(\chi_{a}\left(\mathbf{r}\right)e^{i\phi_% {a}\left(\mathbf{r}\right)}\right)\psi_{0}\left(\mathbf{r}\right)
  19. χ a ( 𝐫 ) \chi_{a}\left(\mathbf{r}\right)
  20. ϕ a ( 𝐫 ) \phi_{a}\left(\mathbf{r}\right)
  21. χ a ( 𝐫 ) \chi_{a}\left(\mathbf{r}\right)
  22. ϕ a ( 𝐫 ) \phi_{a}\left(\mathbf{r}\right)
  23. ϕ a ( 𝐫 ) \phi_{a}\left(\mathbf{r}\right)
  24. ϕ a ( 𝐫 ) \phi_{a}\left(\mathbf{r}\right)
  25. χ a ( 𝐫 ) \chi_{a}\left(\mathbf{r}\right)
  26. D ϕ a ( ρ ) = | ϕ a ( 𝐫 ) - ϕ a ( 𝐫 + ρ ) | 2 𝐫 D_{\phi_{a}}\left(\mathbf{\rho}\right)=\left\langle\left|\phi_{a}\left(\mathbf% {r}\right)-\phi_{a}\left(\mathbf{r}+\mathbf{\rho}\right)\right|^{2}\right% \rangle_{\mathbf{r}}
  27. D ϕ a ( ρ ) D_{\phi_{a}}\left({\mathbf{\rho}}\right)
  28. ρ \mathbf{\rho}
  29. < Align g t ; <...&gt;
  30. r 0 r_{0}
  31. D ϕ a ( ρ ) = 6.88 ( | ρ | r 0 ) 5 / 3 D_{\phi_{a}}\left({\mathbf{\rho}}\right)=6.88\left(\frac{\left|\mathbf{\rho}% \right|}{r_{0}}\right)^{5/3}
  32. r 0 r_{0}
  33. r 0 r_{0}
  34. r 0 r_{0}
  35. σ 2 \sigma^{2}
  36. σ 2 = 1.0299 ( d r 0 ) 5 / 3 \sigma^{2}=1.0299\left(\frac{d}{r_{0}}\right)^{5/3}
  37. r 0 r_{0}
  38. r 0 r_{0}
  39. r 0 = ( 16.7 λ - 2 ( cos γ ) - 1 0 d h C N 2 ( h ) ) - 3 / 5 r_{0}=\left(16.7\lambda^{-2}(\cos\gamma)^{-1}\int_{0}^{\infty}dhC_{N}^{2}(h)% \right)^{-3/5}
  40. C N 2 ( h ) C_{N}^{2}(h)
  41. h h
  42. γ \gamma
  43. ϕ a ( 𝐫 ) = Re [ FT [ R ( 𝐤 ) K ( 𝐤 ) ] ] \phi_{a}(\mathbf{r})=\mbox{Re}~{}[\mbox{FT}~{}[R(\mathbf{k})K(\mathbf{k})]]
  44. ϕ a ( 𝐫 ) \phi_{a}(\mathbf{r})
  45. ϕ a ( 𝐫 ) = Re [ FT [ ( R ( 𝐤 ) I ( 𝐤 ) ) K ( 𝐤 ) ] ] \phi_{a}(\mathbf{r})=\mbox{Re}~{}[\mbox{FT}~{}[(R(\mathbf{k})\otimes I(\mathbf% {k}))K(\mathbf{k})]]
  46. \otimes
  47. C n 2 C_{n}^{2}
  48. I ( k ) = δ ( | k | ) I(k)=\delta(|k|)
  49. δ ( ) \delta()
  50. C n 2 C_{n}^{2}
  51. C n 2 C_{n}^{2}
  52. C n 2 C_{n}^{2}
  53. C n 2 C_{n}^{2}
  54. C n 2 C_{n}^{2}

Astronomical_spectroscopy.html

  1. λ max T = b \lambda\text{max}T=b
  2. L = 4 π R 2 σ T 4 L=4\pi R^{2}\sigma T^{4}
  3. λ - λ 0 λ 0 = v 0 c \frac{\lambda-\lambda_{0}}{\lambda_{0}}=\frac{v_{0}}{c}
  4. λ 0 \lambda_{0}
  5. v 0 v_{0}
  6. λ \lambda
  7. v = H 0 d v=H_{0}d
  8. v v
  9. H 0 H_{0}
  10. d d
  11. z z
  12. z = λ obsv - λ emit λ emit z=\frac{\lambda_{\mathrm{obsv}}-\lambda_{\mathrm{emit}}}{\lambda_{\mathrm{emit% }}}
  13. z = f emit - f obsv f obsv z=\frac{f_{\mathrm{emit}}-f_{\mathrm{obsv}}}{f_{\mathrm{obsv}}}
  14. 1 + z = λ obsv λ emit 1+z=\frac{\lambda_{\mathrm{obsv}}}{\lambda_{\mathrm{emit}}}
  15. 1 + z = f emit f obsv 1+z=\frac{f_{\mathrm{emit}}}{f_{\mathrm{obsv}}}
  16. f f
  17. λ \lambda
  18. z = v H u b b l e c z=\frac{v_{Hubble}}{c}
  19. v t o t a l = H 0 d + v p e c v_{total}=H_{0}d+v_{pec}

Asymptotic_equipartition_property.html

  1. - 1 n log p ( X 1 n ) H ( X ) as n -\frac{1}{n}\log p(X_{1}^{n})\to H(X)\quad\,\text{ as }\quad n\to\infty
  2. X 1 n X_{1}^{n}
  3. lim n Pr [ | - 1 n log p ( X 1 , X 2 , , X n ) - H ( X ) | > ϵ ] = 0 ϵ > 0. \lim_{n\to\infty}\Pr\left[\left|-\frac{1}{n}\log p(X_{1},X_{2},\ldots,X_{n})-H% (X)\right|>\epsilon\right]=0\qquad\forall\epsilon>0.
  4. - 1 n log p ( X 1 , X 2 , , X n ) . -\frac{1}{n}\log p(X_{1},X_{2},\ldots,X_{n}).
  5. Pr [ lim n - 1 n log p ( X 1 , X 2 , , X n ) = H ( X ) ] = 1 \Pr\left[\lim_{n\to\infty}-\frac{1}{n}\log p(X_{1},X_{2},\ldots,X_{n})=H(X)% \right]=1
  6. j ( n , x ) := p ( x 0 n - 1 ) . j(n,x):=p\left(x_{0}^{n-1}\right).
  7. c ( i , k , x ) := p ( x i x i - k i - 1 ) . c(i,k,x):=p\left(x_{i}\mid x_{i-k}^{i-1}\right).
  8. c ( i , x ) := p ( x i x - i - 1 ) . c(i,x):=p\left(x_{i}\mid x_{-\infty}^{i-1}\right).
  9. lim n E [ - log j ( n , X ) ] and lim n E [ - log c ( n , n , X ) ] \lim_{n\to\infty}\mathrm{E}[-\log j(n,X)]\quad\,\text{and}\quad\lim_{n\to% \infty}\mathrm{E}[-\log c(n,n,X)]
  10. c ( i , k , X ) \displaystyle c(i,k,X)
  11. a ( n , k , x ) a(n,k,x)
  12. a ( n , k , x ) := p ( X 0 k - 1 ) i = k n - 1 p ( X i X i - k i - 1 ) = j ( k , x ) i = k n - 1 c ( i , k , x ) a(n,k,x):=p\left(X_{0}^{k-1}\right)\prod_{i=k}^{n-1}p\left(X_{i}\mid X_{i-k}^{% i-1}\right)=j(k,x)\prod_{i=k}^{n-1}c(i,k,x)
  13. a ( n , k , X ( Ω ) ) a(n,k,X(\Omega))
  14. - 1 n log a ( n , k , X ) -\frac{1}{n}\log a(n,k,X)
  15. c ( i , k , X ) c(i,k,X)
  16. a ( n , x ) := p ( x 0 n - 1 x - - 1 ) . a(n,x):=p\left(x_{0}^{n-1}\mid x_{-\infty}^{-1}\right).
  17. - 1 n log a ( n , X ) -\frac{1}{n}\log a(n,X)
  18. c ( i , X ) c(i,X)
  19. H k H H^{k}\searrow H
  20. E [ a ( n , k , X ) j ( n , X ) ] = a ( n , k , X ( Ω ) ) \mathrm{E}\left[\frac{a(n,k,X)}{j(n,X)}\right]=a(n,k,X(\Omega))
  21. E [ j ( n , X ) a ( n , X ) ] = 1 \mathrm{E}\left[\frac{j(n,X)}{a(n,X)}\right]=1
  22. X - - 1 X_{-\infty}^{-1}
  23. α : Pr [ a ( n , k , X ) j ( n , X ) α ] a ( n , k , X ( Ω ) ) α \forall\alpha\in\mathbb{R}\ :\ \Pr\left[\frac{a(n,k,X)}{j(n,X)}\geq\alpha% \right]\leq\frac{a(n,k,X(\Omega))}{\alpha}
  24. α : Pr [ j ( n , X ) a ( n , X ) α ] 1 α , \forall\alpha\in\mathbb{R}\ :\ \Pr\left[\frac{j(n,X)}{a(n,X)}\geq\alpha\right]% \leq\frac{1}{\alpha},
  25. α : Pr [ 1 n log j ( n , X ) a ( n , X ) 1 n log α ] 1 α . \forall\alpha\in\mathbb{R}\ :\ \Pr\left[\frac{1}{n}\log\frac{j(n,X)}{a(n,X)}% \geq\frac{1}{n}\log\alpha\right]\leq\frac{1}{\alpha}.
  26. 1 n log a ( n , k , X ) j ( n , X ) and 1 n log j ( n , X ) a ( n , X ) \frac{1}{n}\log\frac{a(n,k,X)}{j(n,X)}\quad\,\text{and}\quad\frac{1}{n}\log% \frac{j(n,X)}{a(n,X)}
  27. - 1 n log j ( n , X ) -\frac{1}{n}\log j(n,X)
  28. Var [ log [ [ p ( X i ) ] ] ] < M \mathrm{Var}[\log[[p(X_{i})]]]<M
  29. lim n Pr [ | - 1 n log p ( X 1 , X 2 , , X n ) - H ¯ n ( X ) | < ϵ ] = 1 ϵ > 0 \lim_{n\to\infty}\Pr\left[\,\left|-\frac{1}{n}\log p(X_{1},X_{2},\ldots,X_{n})% -\overline{H}_{n}(X)\right|<\epsilon\right]=1\qquad\forall\epsilon>0
  30. H ¯ n ( X ) = 1 n H ( X 1 , X 2 , , X n ) \overline{H}_{n}(X)=\frac{1}{n}H(X_{1},X_{2},\ldots,X_{n})
  31. log ( p ( X i ) ) \log(p(X_{i}))
  32. Pr [ | - 1 n log p ( X 1 , X 2 , , X n ) - H ¯ ( X ) | > ϵ ] 1 n 2 ϵ 2 E [ i = 1 n ( log ( p ( X i ) ) 2 ] M n ϵ 2 0 as n \begin{aligned}\displaystyle\Pr\left[\left|-\frac{1}{n}\log p(X_{1},X_{2},% \ldots,X_{n})-\overline{H}(X)\right|>\epsilon\right]&\displaystyle\leq\frac{1}% {n^{2}\epsilon^{2}}\mathrm{E}\left[\sum_{i=1}^{n}\left(\log(p(X_{i})\right)^{2% }\right]\\ &\displaystyle\leq\frac{M}{n\epsilon^{2}}\to 0\,\text{ as }n\to\infty\end{aligned}
  33. E [ | log [ [ p ( X i ) ] ] | r ] \mathrm{E}\left[|\log[[p(Xi)]]|^{r}\right]
  34. \Box{}
  35. X ~ := f X \tilde{X}:=f\circ X
  36. - 1 n log p ( X ~ 0 τ ) H ( X ) -\frac{1}{n}\log p(\tilde{X}_{0}^{\tau})\to H(X)
  37. n H ( X ) / τ nH(X)/\tau
  38. 2 \mathcal{L}_{2}
  39. P N = P × × P P^{N}=P\times\cdots\times P
  40. π : P Q \pi:P\to Q
  41. P P P^{\prime}\subset P
  42. Q Q Q^{\prime}\subset Q
  43. | P - Q | π = | P P | + | Q Q | |P-Q|_{\pi}=|P\smallsetminus P^{\prime}|+|Q\smallsetminus Q^{\prime}|
  44. | P - Q | π |P-Q|_{\pi}
  45. | log P : Q | π = sup p P | log p - log π ( p ) | log min ( | set ( P ) | , | set ( Q ) | ) |\log P:Q|_{\pi}=\frac{\sup_{p\in P^{{}^{\prime}}}|\log p-\log\pi(p)|}{\log% \min\left(|\operatorname{set}(P^{^{\prime}})|,|\operatorname{set}(Q^{^{\prime}% })|\right)}
  46. | set ( P ) | |\operatorname{set}(P)|
  47. dist π ( P , Q ) = | P - Q | π + | log P : Q | π \,\text{dist}_{\pi}(P,Q)=|P-Q|_{\pi}+|\log P:Q|_{\pi}
  48. π N : P N Q N \pi_{N}:P_{N}\to Q_{N}
  49. dist π N ( P N , Q N ) 0 as N \,\text{dist}_{\pi_{N}}(P_{N},Q_{N})\to 0\quad\,\text{ as }\quad N\to\infty
  50. H ( P ) = lim N 1 N | s e t ( H N ) | H(P)=\lim_{N\to\infty}\frac{1}{N}|set(H_{N})|