wpmath0000001_4

Chirp.html

  1. f ( t ) f(t)
  2. f ( t ) = f 0 + k t f(t)=f_{0}+kt
  3. f 0 f_{0}
  4. t = 0 t=0
  5. k k
  6. k = f 1 - f 0 T k=\frac{f_{1}-f_{0}}{T}
  7. f 1 f_{1}
  8. f 0 f_{0}
  9. T T
  10. f 0 f_{0}
  11. f 1 f_{1}
  12. ϕ ( t + Δ t ) ϕ ( t ) + 2 π f ( t ) Δ t \phi(t+\Delta t)\simeq\phi(t)+2\pi f(t)\,\Delta t
  13. ϕ ( t ) = 2 π f ( t ) \phi^{\prime}(t)=2\pi\,f(t)
  14. ϕ ( t ) \displaystyle\phi(t)
  15. ϕ 0 \phi_{0}
  16. t = 0 t=0
  17. x ( t ) = sin [ ϕ 0 + 2 π ( f 0 t + k 2 t 2 ) ] x(t)=\sin\left[\phi_{0}+2\pi\left(f_{0}t+\frac{k}{2}t^{2}\right)\right]
  18. f ( t ) = f 0 + k t f(t)=f_{0}+kt
  19. t 1 t_{1}
  20. t 2 t_{2}
  21. t 2 - t 1 t_{2}-t_{1}
  22. f ( t 2 ) / f ( t 1 ) f(t_{2})/f(t_{1})
  23. f ( t ) = f 0 k t f(t)=f_{0}k^{t}
  24. f 0 f_{0}
  25. t = 0 t=0
  26. k k
  27. ϕ ( t ) \displaystyle\phi(t)
  28. ϕ 0 \phi_{0}
  29. t = 0 t=0
  30. x ( t ) = sin [ ϕ 0 + 2 π f 0 ( k t - 1 ln ( k ) ) ] x(t)=\sin\left[\phi_{0}+2\pi f_{0}\left(\frac{k^{t}-1}{\ln(k)}\right)\right]
  31. f ( t ) = f 0 k t f(t)=f_{0}k^{t}
  32. g = f [ a x + b c x + 1 ] g=f\left[\frac{a\cdot x+b}{c\cdot x+1}\right]
  33. x ( t ) = sin ( ϕ ( t ) ) x(t)=\sin\left(\phi(t)\right)
  34. f ( t ) = 1 2 π d ϕ ( t ) d t f(t)=\frac{1}{2\pi}\frac{d\phi(t)}{dt}
  35. c ( t ) = 1 2 π d 2 ϕ ( t ) d t 2 c(t)=\frac{1}{2\pi}\frac{d^{2}\phi(t)}{dt^{2}}

Chomsky_hierarchy.html

  1. \rightarrow\,
  2. S S
  3. { a , b } \{a,b\}
  4. { S , A , B } \{S,A,B\}
  5. S S
  6. \rightarrow\,
  7. A B S ABS
  8. S S
  9. \rightarrow\,
  10. B A BA
  11. \rightarrow\,
  12. A B AB
  13. B S BS
  14. \rightarrow\,
  15. b b
  16. B b Bb
  17. \rightarrow\,
  18. b b bb
  19. A b Ab
  20. \rightarrow\,
  21. a b ab
  22. A a Aa
  23. \rightarrow\,
  24. a a aa
  25. S S
  26. a n b n a^{n}b^{n}
  27. n n
  28. a a
  29. n n
  30. b b
  31. { a , b } \{a,b\}
  32. { S } \{S\}
  33. S S
  34. S S
  35. \rightarrow\,
  36. a S b aSb
  37. S S
  38. \rightarrow\,
  39. { g e n e r a t e , h a t e , g r e a t , g r e e n , i d e a s , l i n g u i s t s } \{generate,hate,great,green,ideas,linguists\}
  40. { 𝑆𝐸𝑁𝑇𝐸𝑁𝐶𝐸 , 𝑁𝑂𝑈𝑁𝑃𝐻𝑅𝐴𝑆𝐸 , 𝑉𝐸𝑅𝐵𝑃𝐻𝑅𝐴𝑆𝐸 , 𝑁𝑂𝑈𝑁 , 𝑉𝐸𝑅𝐵 , 𝐴𝐷𝐽 } \{\,\textit{SENTENCE},\,\textit{NOUNPHRASE},\,\textit{VERBPHRASE},\,\textit{% NOUN},\,\textit{VERB},\,\textit{ADJ}\}
  41. 𝑆𝐸𝑁𝑇𝐸𝑁𝐶𝐸 \,\textit{SENTENCE}
  42. \rightarrow\,
  43. 𝑁𝑂𝑈𝑁𝑃𝐻𝑅𝐴𝑆𝐸 𝑉𝐸𝑅𝐵𝑃𝐻𝑅𝐴𝑆𝐸 \,\textit{NOUNPHRASE}\;\,\textit{VERBPHRASE}
  44. 𝑁𝑂𝑈𝑁𝑃𝐻𝑅𝐴𝑆𝐸 \,\textit{NOUNPHRASE}
  45. \rightarrow\,
  46. 𝐴𝐷𝐽 𝑁𝑂𝑈𝑁𝑃𝐻𝑅𝐴𝑆𝐸 \,\textit{ADJ}\;\,\textit{NOUNPHRASE}
  47. 𝑁𝑂𝑈𝑁𝑃𝐻𝑅𝐴𝑆𝐸 \,\textit{NOUNPHRASE}
  48. \rightarrow\,
  49. 𝑁𝑂𝑈𝑁 \,\textit{NOUN}
  50. 𝑉𝐸𝑅𝐵𝑃𝐻𝑅𝐴𝑆𝐸 \,\textit{VERBPHRASE}
  51. \rightarrow\,
  52. 𝑉𝐸𝑅𝐵 𝑁𝑂𝑈𝑁𝑃𝐻𝑅𝐴𝑆𝐸 \,\textit{VERB}\;\,\textit{NOUNPHRASE}
  53. 𝑉𝐸𝑅𝐵𝑃𝐻𝑅𝐴𝑆𝐸 \,\textit{VERBPHRASE}
  54. \rightarrow\,
  55. 𝑉𝐸𝑅𝐵 \,\textit{VERB}
  56. 𝑁𝑂𝑈𝑁 \,\textit{NOUN}
  57. \rightarrow\,
  58. 𝑖𝑑𝑒𝑎𝑠 \,\textit{ideas}
  59. 𝑁𝑂𝑈𝑁 \,\textit{NOUN}
  60. \rightarrow\,
  61. 𝑙𝑖𝑛𝑔𝑢𝑖𝑠𝑡𝑠 \,\textit{linguists}
  62. 𝑉𝐸𝑅𝐵 \,\textit{VERB}
  63. \rightarrow\,
  64. 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑒 \,\textit{generate}
  65. 𝑉𝐸𝑅𝐵 \,\textit{VERB}
  66. \rightarrow\,
  67. ℎ𝑎𝑡𝑒 \,\textit{hate}
  68. 𝐴𝐷𝐽 \,\textit{ADJ}
  69. \rightarrow\,
  70. 𝑔𝑟𝑒𝑎𝑡 \,\textit{great}
  71. 𝐴𝐷𝐽 \,\textit{ADJ}
  72. \rightarrow\,
  73. 𝑔𝑟𝑒𝑒𝑛 \,\textit{green}
  74. 𝑆𝐸𝑁𝑇𝐸𝑁𝐶𝐸 \,\textit{SENTENCE}
  75. \rightarrow
  76. \rightarrow
  77. \rightarrow
  78. \rightarrow
  79. \rightarrow
  80. \rightarrow
  81. \rightarrow
  82. \rightarrow
  83. \rightarrow
  84. \rightarrow
  85. \rightarrow
  86. \rightarrow
  87. \rightarrow
  88. α A β α γ β \alpha A\beta\rightarrow\alpha\gamma\beta
  89. A A
  90. α \alpha
  91. β \beta
  92. γ \gamma
  93. α \alpha
  94. β \beta
  95. γ \gamma
  96. S ϵ S\rightarrow\epsilon
  97. S S
  98. A γ A\rightarrow\gamma
  99. A A
  100. γ \gamma
  101. S ϵ S\rightarrow\epsilon
  102. S S
  103. α β \alpha\rightarrow\beta
  104. α A β α γ β \alpha A\beta\rightarrow\alpha\gamma\beta
  105. A γ A\rightarrow\gamma
  106. A a A\rightarrow a
  107. A a B A\rightarrow aB

Chomsky_normal_form.html

  1. A B C A\rightarrow\,BC
  2. A a A\rightarrow\,a
  3. A A
  4. B B
  5. C C
  6. a a
  7. B B
  8. C C
  9. A : := B C \langle A\rangle::=\,\langle B\rangle\mid\langle C\rangle
  10. A : := B C \langle A\rangle::=\,\langle B\rangle\langle C\rangle
  11. A : := a \langle A\rangle::=\,a
  12. A \langle A\rangle
  13. B \langle B\rangle
  14. C \langle C\rangle
  15. a a

Chord_(aeronautics).html

  1. SMC = S b , \mbox{SMC}~{}=\frac{S}{b},
  2. MAC = 2 S \mbox{MAC}~{}=\frac{2}{S}
  3. 0 b 2 c 2 d y , \int_{0}^{\frac{b}{2}}c^{2}dy,
  4. λ \lambda
  5. c ( y ) = 2 S w ( 1 + λ ) b [ 1 - 2 ( 1 - λ ) b y ] , c(y)=\frac{2\,S_{w}}{(1+\lambda)b}\left[1-\frac{2(1-\lambda)}{b}y\right],
  6. λ = C Tip C Root \lambda=\frac{C_{\rm Tip}}{C_{\rm Root}}

Christiaan_Huygens.html

  1. F c = m v 2 r F_{c}=\frac{m\ v^{2}}{r}
  2. T = 2 π l g T=2\pi\sqrt{\frac{l}{g}}
  3. 1 / 27 , 664 1/27,664

Chromatic_aberration.html

  1. f 1 V 1 + f 2 V 2 = 0 f_{1}\cdot V_{1}+f_{2}\cdot V_{2}=0
  2. 1 f = 1 f 1 + 1 f 2 \frac{1}{f}=\frac{1}{f_{1}}+\frac{1}{f_{2}}

Chromium.html

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

Cigarette.html

  1. \rightleftharpoons

Circle.html

  1. 256 / 81 {256}/{81}
  2. π \pi
  3. π \pi
  4. π \pi
  5. C = 2 π r = π d . C=2\pi r=\pi d.\,
  6. π \pi
  7. Area = π r 2 . \mathrm{Area}=\pi r^{2}.\,
  8. Area = π d 2 4 0.7854 d 2 , \mathrm{Area}=\frac{\pi d^{2}}{4}\approx 0{.}7854d^{2},
  9. ( x - a ) 2 + ( y - b ) 2 = r 2 . \left(x-a\right)^{2}+\left(y-b\right)^{2}=r^{2}.
  10. x 2 + y 2 = r 2 . x^{2}+y^{2}=r^{2}.\!
  11. x = a + r cos t , x=a+r\,\cos t,\,
  12. y = b + r sin t y=b+r\,\sin t\,
  13. π \pi
  14. x = a + r 2 t 1 + t 2 . x=a+r\frac{2t}{1+t^{2}}.\,
  15. y = b + r 1 - t 2 1 + t 2 y=b+r\frac{1-t^{2}}{1+t^{2}}\,
  16. x 2 + y 2 - 2 a x z - 2 b y z + c z 2 = 0. x^{2}+y^{2}-2axz-2byz+cz^{2}=0.\,
  17. r 2 - 2 r r 0 cos ( θ - ϕ ) + r 0 2 = a 2 r^{2}-2rr_{0}\cos(\theta-\phi)+r_{0}^{2}=a^{2}\,
  18. ( r , θ ) (r,\theta)
  19. ( r 0 , ϕ ) (r_{0},\phi)
  20. r = 2 a cos ( θ - ϕ ) . r=2a\cos(\theta-\phi).\,
  21. r = r 0 cos ( θ - ϕ ) ± a 2 - r 0 2 sin 2 ( θ - ϕ ) , r=r_{0}\cos(\theta-\phi)\pm\sqrt{a^{2}-r_{0}^{2}\sin^{2}(\theta-\phi)},
  22. | z - c | = r |z-c|=r\,
  23. z = r e i t + c z=re^{it}+c
  24. p z z ¯ + g z + g z ¯ = q pz\overline{z}+gz+\overline{gz}=q
  25. p = 1 , g = - c ¯ , q = r 2 - | c | 2 p=1,\ g=-\overline{c},\ q=r^{2}-|c|^{2}
  26. | z - c | 2 = z z ¯ - c ¯ z - c z ¯ + c c ¯ |z-c|^{2}=z\overline{z}-\overline{c}z-c\overline{z}+c\overline{c}
  27. ( x 1 - a ) x + ( y 1 - b ) y = ( x 1 - a ) x 1 + ( y 1 - b ) y 1 (x_{1}-a)x+(y_{1}-b)y=(x_{1}-a)x_{1}+(y_{1}-b)y_{1}\,
  28. ( x 1 - a ) ( x - a ) + ( y 1 - b ) ( y - b ) = r 2 . (x_{1}-a)(x-a)+(y_{1}-b)(y-b)=r^{2}.\!
  29. d y d x = - x 1 - a y 1 - b . \frac{dy}{dx}=-\frac{x_{1}-a}{y_{1}-b}.
  30. x 1 x + y 1 y = r 2 , x_{1}x+y_{1}y=r^{2},\!
  31. d y d x = - x 1 y 1 . \frac{dy}{dx}=-\frac{x_{1}}{y_{1}}.
  32. π \pi
  33. π \pi
  34. r = y 2 8 x + x 2 . r=\frac{y^{2}}{8x}+\frac{x}{2}.
  35. A P B P = A C B C . \frac{AP}{BP}=\frac{AC}{BC}.
  36. | [ A , B ; C , P ] | = 1. |[A,B;C,P]|=1.
  37. | A P | | B P | = | A C | | B C | \frac{|AP|}{|BP|}=\frac{|AC|}{|BC|}
  38. | x a | n + | y b | n = 1 \left|\frac{x}{a}\right|^{n}\!+\left|\frac{y}{b}\right|^{n}\!=1
  39. π \pi

Circuit_reliability.html

  1. T s = T a + T o T_{s}=T_{a}+T_{o}

Circular_polarization.html

  1. ( E x , E y , E z ) ( cos 2 π λ ( c t - z ) , sin 2 π λ ( c t - z ) , 0 ) . \left(E_{x},\,E_{y},\,E_{z}\right)\propto\left(\cos\frac{2\pi}{\lambda}\left(% ct-z\right),\,\sin\frac{2\pi}{\lambda}\left(ct-z\right),\,0\right).
  2. g e m = 2 ( θ left - θ right θ left + θ right ) g_{em}\ =\ 2\left({\theta_{\mathrm{left}}-\theta_{\mathrm{right}}\over\theta_{% \mathrm{left}}+\theta_{\mathrm{right}}}\right)
  3. θ left \theta_{\mathrm{left}}
  4. θ right \theta_{\mathrm{right}}
  5. 𝐄 ( 𝐫 , t ) = 𝐄 Re { 𝐐 | ψ exp [ i ( k z - ω t ) ] } \mathbf{E}(\mathbf{r},t)=\mid\mathbf{E}\mid\mathrm{Re}\left\{\mathbf{Q}|\psi% \rangle\exp\left[i\left(kz-\omega t\right)\right]\right\}
  6. 𝐁 ( 𝐫 , t ) = 𝐳 ^ × 𝐄 ( 𝐫 , t ) \mathbf{B}(\mathbf{r},t)=\hat{\mathbf{z}}\times\mathbf{E}(\mathbf{r},t)
  7. ω = c k \omega=ck
  8. 𝐐 = [ 𝐱 ^ , 𝐲 ^ ] \mathbf{Q}=\left[\hat{\mathbf{x}},\hat{\mathbf{y}}\right]
  9. 2 × 2 2\times 2
  10. c c
  11. 𝐄 \mid\mathbf{E}\mid
  12. | ψ = def ( ψ x ψ y ) = ( cos θ exp ( i α x ) sin θ exp ( i α y ) ) |\psi\rangle\ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix}\psi_{x}\\ \psi_{y}\end{pmatrix}=\begin{pmatrix}\cos\theta\exp\left(i\alpha_{x}\right)\\ \sin\theta\exp\left(i\alpha_{y}\right)\end{pmatrix}
  13. α y \alpha_{y}
  14. π / 2 \pi/2
  15. α x \alpha_{x}
  16. | ψ = 1 2 ( 1 ± i ) exp ( i α x ) |\psi\rangle={1\over\sqrt{2}}\begin{pmatrix}1\\ \pm i\end{pmatrix}\exp\left(i\alpha_{x}\right)
  17. | R = def 1 2 ( 1 - i ) |R\rangle\ \stackrel{\mathrm{def}}{=}\ {1\over\sqrt{2}}\begin{pmatrix}1\\ -i\end{pmatrix}
  18. | L = def 1 2 ( 1 i ) |L\rangle\ \stackrel{\mathrm{def}}{=}\ {1\over\sqrt{2}}\begin{pmatrix}1\\ i\end{pmatrix}
  19. | ψ = ψ R | R + ψ L | L |\psi\rangle=\psi_{R}|R\rangle+\psi_{L}|L\rangle
  20. ψ R = def ( cos θ + i sin θ exp ( i δ ) 2 ) exp ( i α x ) \psi_{R}\ \stackrel{\mathrm{def}}{=}\ \left({\cos\theta+i\sin\theta\exp\left(i% \delta\right)\over\sqrt{2}}\right)\exp\left(i\alpha_{x}\right)
  21. ψ L = def ( cos θ - i sin θ exp ( i δ ) 2 ) exp ( i α x ) \psi_{L}\ \stackrel{\mathrm{def}}{=}\ \left({\cos\theta-i\sin\theta\exp\left(i% \delta\right)\over\sqrt{2}}\right)\exp\left(i\alpha_{x}\right)
  22. δ = α y - α x . \delta=\alpha_{y}-\alpha_{x}.
  23. q = 6 × 10 - 4 q=6\times 10^{-4}
  24. p 2 p^{2}
  25. q 2 × 10 - 4 q\sim 2\times 10^{-4}
  26. 3 × 10 - 7 3\times 10^{-7}
  27. 10 - 6 10^{-6}
  28. p p
  29. q q
  30. q p 2 q\sim p^{2}
  31. p p
  32. q 10 - 4 q\sim 10^{-4}
  33. q 7 × 10 - 4 q\sim 7\times 10^{-4}
  34. q = 7 × 10 - 3 q=7\times 10^{-3}
  35. q = 2 × 10 - 3 q=2\times 10^{-3}
  36. q 5 × 10 - 3 q\sim 5\times 10^{-3}

Circulator.html

  1. S = ( 0 0 1 1 0 0 0 1 0 ) S=\begin{pmatrix}0&0&1\\ 1&0&0\\ 0&1&0\end{pmatrix}

Circumference.html

  1. π \pi
  2. π \pi
  3. C C
  4. d d
  5. π = C d \pi=\frac{C}{d}
  6. C = π d = 2 π r . {C}=\pi\cdot{d}=2\pi\cdot{r}.\!
  7. π \pi
  8. C / r = 2 π {C}/{r}=2\pi

Cistron.html

  1. x x
  2. y y
  3. x x
  4. y y
  5. x x
  6. y y

Citroën.html

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

City.html

  1. O = s 2 O=s^{2}
  2. I = 4 s I=4s
  3. O = I 2 / 16 O=I^{2}/16
  4. s s
  5. I = 4 O 1 / 2 I=4O^{1/2}
  6. I I

Class_(set_theory).html

  1. A = { x x = x } A=\{x\mid x=x\}
  2. x . ( x A x = x ) \forall x.(x\in A\leftrightarrow x=x)
  3. 𝒜 \mathcal{A}
  4. { x ϕ } \{x\mid\phi\}
  5. 𝒜 \mathcal{A}
  6. 𝒜 \mathcal{A}
  7. λ x . ϕ \lambda x.\phi
  8. ϕ \phi
  9. ϕ \phi
  10. Φ ( x , y ) \Phi(x,y)
  11. Φ \Phi
  12. y = x { x } y=x\cup\{x\}
  13. Φ \Phi
  14. Φ ( x ) = y \Phi(x)=y

Classical_guitar.html

  1. 2 12 \sqrt[12]{2}
  2. 2 12 = 2 ( 1 12 ) \sqrt[12]{2}=2^{(\frac{1}{12})}
  3. 440 H z ( 2 12 ) - 5 440\rm{Hz}\cdot(\sqrt[12]{2})^{-5}\approx
  4. 440 H z ( 2 12 ) - 10 440\rm{Hz}\cdot(\sqrt[12]{2})^{-10}\approx
  5. 440 H z ( 2 12 ) - 14 440\rm{Hz}\cdot(\sqrt[12]{2})^{-14}\approx
  6. 440 H z ( 2 12 ) - 19 440\rm{Hz}\cdot(\sqrt[12]{2})^{-19}\approx
  7. 440 H z ( 2 12 ) - 24 = 440\rm{Hz}\cdot(\sqrt[12]{2})^{-24}=
  8. 440 H z ( 2 12 ) - 29 440\rm{Hz}\cdot(\sqrt[12]{2})^{-29}\approx

Classical_Kuiper_belt_object.html

  1. e < 0.240 e<0.240

Clathrate_hydrate.html

  1. P m 3 ¯ n Pm\overline{3}n
  2. F d 3 ¯ m Fd\overline{3}m
  3. P 6 / m m m P6/mmm
  4. τ \tau

Clifford_algebra.html

  1. v 2 = Q ( v ) 1 for all v V , v^{2}=Q(v)1\ \,\text{ for all }v\in V,
  2. u v + v u = 2 u , v 1 for all u , v V , uv+vu=2\langle u,v\rangle 1\ \,\text{ for all }u,v\in V,
  3. u , v = 1 2 ( Q ( u + v ) - Q ( u ) - Q ( v ) ) \langle u,v\rangle=\frac{1}{2}\left(Q(u+v)-Q(u)-Q(v)\right)
  4. j ( v ) j ( w ) + j ( w ) j ( v ) = 2 v , w 1 A for all v , w V . j(v)j(w)+j(w)j(v)=2\langle v,w\rangle 1_{A}\quad\mbox{ for all }~{}v,w\in V\ .
  5. v v - Q ( v ) 1 v\otimes v-Q(v)1
  6. v V v\in V
  7. { e i 1 e i 2 e i k 1 i 1 < i 2 < < i k n and 0 k n } \{e_{i_{1}}e_{i_{2}}\cdots e_{i_{k}}\mid 1\leq i_{1}<i_{2}<\cdots<i_{k}\leq n% \mbox{ and }~{}0\leq k\leq n\}
  8. dim C ( V , Q ) = k = 0 n ( n k ) = 2 n . \dim C\ell(V,Q)=\sum_{k=0}^{n}\begin{pmatrix}n\\ k\end{pmatrix}=2^{n}.
  9. e i , e j = 0 \langle e_{i},e_{j}\rangle=0
  10. i j i\neq j
  11. e i , e i = Q ( e i ) . \langle e_{i},e_{i}\rangle=Q(e_{i}).\,
  12. - , - \langle-,-\rangle
  13. e i e j = - e j e i e_{i}e_{j}=-e_{j}e_{i}
  14. i j i\neq j
  15. e i 2 = Q ( e i ) e_{i}^{2}=Q(e_{i})\,
  16. e i 1 e i 2 e i k e_{i_{1}}e_{i_{2}}\cdots e_{i_{k}}
  17. Q ( v ) = v 1 2 + + v p 2 - v p + 1 2 - - v p + q 2 Q(v)=v_{1}^{2}+\cdots+v_{p}^{2}-v_{p+1}^{2}-\cdots-v_{p+q}^{2}
  18. Q ( z ) = z 1 2 + z 2 2 + + z n 2 Q(z)=z_{1}^{2}+z_{2}^{2}+\cdots+z_{n}^{2}
  19. 𝐯 𝐰 = v 1 w 1 + v 2 w 2 + v 3 w 3 . \mathbf{v}\cdot\mathbf{w}=v_{1}w_{1}+v_{2}w_{2}+v_{3}w_{3}.
  20. 𝐯𝐰 + 𝐰𝐯 = - 2 ( 𝐯 𝐰 ) . \mathbf{v}\mathbf{w}+\mathbf{w}\mathbf{v}=-2(\mathbf{v}\cdot\mathbf{w}).\!
  21. 𝐞 2 𝐞 3 = - 𝐞 3 𝐞 2 , 𝐞 3 𝐞 1 = - 𝐞 1 𝐞 3 , 𝐞 1 𝐞 2 = - 𝐞 2 𝐞 1 , \mathbf{e}_{2}\mathbf{e}_{3}=-\mathbf{e}_{3}\mathbf{e}_{2},\,\,\,\mathbf{e}_{3% }\mathbf{e}_{1}=-\mathbf{e}_{1}\mathbf{e}_{3},\,\,\,\mathbf{e}_{1}\mathbf{e}_{% 2}=-\mathbf{e}_{2}\mathbf{e}_{1},\!
  22. 𝐞 1 2 = 𝐞 2 2 = 𝐞 3 2 = - 1. \mathbf{e}_{1}^{2}=\mathbf{e}_{2}^{2}=\mathbf{e}_{3}^{2}=-1.\!
  23. A = a 0 + a 1 𝐞 1 + a 2 𝐞 2 + a 3 𝐞 3 + a 4 𝐞 2 𝐞 3 + a 5 𝐞 3 𝐞 1 + a 6 𝐞 1 𝐞 2 + a 7 𝐞 1 𝐞 2 𝐞 3 . A=a_{0}+a_{1}\mathbf{e}_{1}+a_{2}\mathbf{e}_{2}+a_{3}\mathbf{e}_{3}+a_{4}% \mathbf{e}_{2}\mathbf{e}_{3}+a_{5}\mathbf{e}_{3}\mathbf{e}_{1}+a_{6}\mathbf{e}% _{1}\mathbf{e}_{2}+a_{7}\mathbf{e}_{1}\mathbf{e}_{2}\mathbf{e}_{3}.\!
  24. Q = q 0 + q 1 𝐞 2 𝐞 3 + q 2 𝐞 3 𝐞 1 + q 3 𝐞 1 𝐞 2 . Q=q_{0}+q_{1}\mathbf{e}_{2}\mathbf{e}_{3}+q_{2}\mathbf{e}_{3}\mathbf{e}_{1}+q_% {3}\mathbf{e}_{1}\mathbf{e}_{2}.\!
  25. i = 𝐞 2 𝐞 3 , j = 𝐞 3 𝐞 1 , k = 𝐞 1 𝐞 2 , i=\mathbf{e}_{2}\mathbf{e}_{3},j=\mathbf{e}_{3}\mathbf{e}_{1},k=\mathbf{e}_{1}% \mathbf{e}_{2},
  26. i 2 = ( 𝐞 2 𝐞 3 ) 2 = 𝐞 2 𝐞 3 𝐞 2 𝐞 3 = - 𝐞 2 𝐞 2 𝐞 3 𝐞 3 = - 1 , i^{2}=(\mathbf{e}_{2}\mathbf{e}_{3})^{2}=\mathbf{e}_{2}\mathbf{e}_{3}\mathbf{e% }_{2}\mathbf{e}_{3}=-\mathbf{e}_{2}\mathbf{e}_{2}\mathbf{e}_{3}\mathbf{e}_{3}=% -1,\!
  27. i j = 𝐞 2 𝐞 3 𝐞 3 𝐞 1 = - 𝐞 2 𝐞 1 = 𝐞 1 𝐞 2 = k . ij=\mathbf{e}_{2}\mathbf{e}_{3}\mathbf{e}_{3}\mathbf{e}_{1}=-\mathbf{e}_{2}% \mathbf{e}_{1}=\mathbf{e}_{1}\mathbf{e}_{2}=k.\!
  28. i j k = 𝐞 2 𝐞 3 𝐞 3 𝐞 1 𝐞 1 𝐞 2 = - 1. ijk=\mathbf{e}_{2}\mathbf{e}_{3}\mathbf{e}_{3}\mathbf{e}_{1}\mathbf{e}_{1}% \mathbf{e}_{2}=-1.\!
  29. d ( 𝐯 , 𝐰 ) = v 1 w 1 + v 2 w 2 + v 3 w 3 . d(\mathbf{v},\mathbf{w})=v_{1}w_{1}+v_{2}w_{2}+v_{3}w_{3}.
  30. 𝐯𝐰 + 𝐰𝐯 = - 2 d ( 𝐯 , 𝐰 ) . \mathbf{v}\mathbf{w}+\mathbf{w}\mathbf{v}=-2\,d(\mathbf{v},\mathbf{w}).\!
  31. 𝐞 m 𝐞 n = - 𝐞 n 𝐞 m , m n , \mathbf{e}_{m}\mathbf{e}_{n}=-\mathbf{e}_{n}\mathbf{e}_{m},\,\,\,m\neq n,\!
  32. 𝐞 1 2 = 𝐞 2 2 = 𝐞 3 2 = - 1 , 𝐞 4 2 = 0. \mathbf{e}_{1}^{2}=\mathbf{e}_{2}^{2}=\mathbf{e}_{3}^{2}=-1,\,\,\mathbf{e}_{4}% ^{2}=0.\!
  33. H = h 0 + h 1 𝐞 2 𝐞 3 + h 2 𝐞 3 𝐞 1 + h 3 𝐞 1 𝐞 2 + h 4 𝐞 4 𝐞 1 + h 5 𝐞 4 𝐞 2 + h 6 𝐞 4 𝐞 3 + h 7 𝐞 1 𝐞 2 𝐞 3 𝐞 4 . H=h_{0}+h_{1}\mathbf{e}_{2}\mathbf{e}_{3}+h_{2}\mathbf{e}_{3}\mathbf{e}_{1}+h_% {3}\mathbf{e}_{1}\mathbf{e}_{2}+h_{4}\mathbf{e}_{4}\mathbf{e}_{1}+h_{5}\mathbf% {e}_{4}\mathbf{e}_{2}+h_{6}\mathbf{e}_{4}\mathbf{e}_{3}+h_{7}\mathbf{e}_{1}% \mathbf{e}_{2}\mathbf{e}_{3}\mathbf{e}_{4}.\!
  34. i = 𝐞 2 𝐞 3 , j = 𝐞 3 𝐞 1 , k = 𝐞 1 𝐞 2 , ε = 𝐞 1 𝐞 2 𝐞 3 𝐞 4 . i=\mathbf{e}_{2}\mathbf{e}_{3},j=\mathbf{e}_{3}\mathbf{e}_{1},k=\mathbf{e}_{1}% \mathbf{e}_{2},\,\,\varepsilon=\mathbf{e}_{1}\mathbf{e}_{2}\mathbf{e}_{3}% \mathbf{e}_{4}.\!
  35. ε 2 = ( 𝐞 1 𝐞 2 𝐞 3 𝐞 4 ) 2 = 𝐞 1 𝐞 2 𝐞 3 𝐞 4 𝐞 1 𝐞 2 𝐞 3 𝐞 4 = - 𝐞 1 𝐞 2 𝐞 3 ( 𝐞 4 𝐞 4 ) 𝐞 1 𝐞 2 𝐞 3 = 0 , \varepsilon^{2}=(\mathbf{e}_{1}\mathbf{e}_{2}\mathbf{e}_{3}\mathbf{e}_{4})^{2}% =\mathbf{e}_{1}\mathbf{e}_{2}\mathbf{e}_{3}\mathbf{e}_{4}\mathbf{e}_{1}\mathbf% {e}_{2}\mathbf{e}_{3}\mathbf{e}_{4}=-\mathbf{e}_{1}\mathbf{e}_{2}\mathbf{e}_{3% }(\mathbf{e}_{4}\mathbf{e}_{4})\mathbf{e}_{1}\mathbf{e}_{2}\mathbf{e}_{3}=0,\!
  36. ε i = ( 𝐞 1 𝐞 2 𝐞 3 𝐞 4 ) 𝐞 2 𝐞 3 = 𝐞 1 𝐞 2 𝐞 3 𝐞 4 𝐞 2 𝐞 3 = 𝐞 2 𝐞 3 ( 𝐞 1 𝐞 2 𝐞 3 𝐞 4 ) = i ε . \varepsilon i=(\mathbf{e}_{1}\mathbf{e}_{2}\mathbf{e}_{3}\mathbf{e}_{4})% \mathbf{e}_{2}\mathbf{e}_{3}=\mathbf{e}_{1}\mathbf{e}_{2}\mathbf{e}_{3}\mathbf% {e}_{4}\mathbf{e}_{2}\mathbf{e}_{3}=\mathbf{e}_{2}\mathbf{e}_{3}(\mathbf{e}_{1% }\mathbf{e}_{2}\mathbf{e}_{3}\mathbf{e}_{4})=i\varepsilon.\!
  37. a \sqrt{a}
  38. e i 1 e i 2 e i k e i 1 e i 2 e i k . e_{i_{1}}e_{i_{2}}\cdots e_{i_{k}}\mapsto e_{i_{1}}\wedge e_{i_{2}}\wedge% \cdots\wedge e_{i_{k}}.
  39. f k ( v 1 , , v k ) = 1 k ! σ S k sgn ( σ ) v σ ( 1 ) v σ ( k ) f_{k}(v_{1},\cdots,v_{k})=\frac{1}{k!}\sum_{\sigma\in S_{k}}{\rm sgn}(\sigma)% \,v_{\sigma(1)}\cdots v_{\sigma(k)}
  40. G r F C ( V , Q ) = k F k / F k - 1 Gr_{F}C\ell(V,Q)=\bigoplus_{k}F^{k}/F^{k-1}
  41. C ( V , Q ) = C 0 ( V , Q ) C 1 ( V , Q ) C\ell(V,Q)=C\ell^{0}(V,Q)\oplus C\ell^{1}(V,Q)
  42. C i ( V , Q ) C j ( V , Q ) = C i + j ( V , Q ) C\ell^{\,i}(V,Q)C\ell^{\,j}(V,Q)=C\ell^{\,i+j}(V,Q)
  43. C p , q 0 ( 𝐑 ) C p , q - 1 ( 𝐑 ) C\ell_{p,q}^{0}(\mathbf{R})\cong C\ell_{p,q-1}(\mathbf{R})
  44. C p , q 0 ( 𝐑 ) C q , p - 1 ( 𝐑 ) C\ell_{p,q}^{0}(\mathbf{R})\cong C\ell_{q,p-1}(\mathbf{R})
  45. v 1 v 2 v k v k v 2 v 1 . v_{1}\otimes v_{2}\otimes\cdots\otimes v_{k}\mapsto v_{k}\otimes\cdots\otimes v% _{2}\otimes v_{1}.
  46. x ¯ \bar{x}
  47. x ¯ = α ( x t ) = α ( x ) t . \bar{x}=\alpha(x^{\mathrm{t}})=\alpha(x)^{\mathrm{t}}.
  48. α ( x ) = ± x x t = ± x x ¯ = ± x \alpha(x)=\pm x\qquad x^{\mathrm{t}}=\pm x\qquad\bar{x}=\pm x
  49. α ( x ) \alpha(x)\,
  50. x t x^{\mathrm{t}}\,
  51. x ¯ \bar{x}
  52. Q ( x ) = x t x Q(x)=\langle x^{\mathrm{t}}x\rangle
  53. Q ( v 1 v 2 v k ) = Q ( v 1 ) Q ( v 2 ) Q ( v k ) Q(v_{1}v_{2}\cdots v_{k})=Q(v_{1})Q(v_{2})\cdots Q(v_{k})
  54. x , y = x t y . \langle x,y\rangle=\langle x^{\mathrm{t}}y\rangle.
  55. a x , y = x , a t y , \langle ax,y\rangle=\langle x,a^{\mathrm{t}}y\rangle,
  56. x a , y = x , y a t . \langle xa,y\rangle=\langle x,ya^{\mathrm{t}}\rangle.
  57. C p + 2 , q ( 𝐑 ) = M 2 ( 𝐑 ) C q , p ( 𝐑 ) C\ell_{p+2,q}(\mathbf{R})=M_{2}(\mathbf{R})\otimes C\ell_{q,p}(\mathbf{R})
  58. C p + 1 , q + 1 ( 𝐑 ) = M 2 ( 𝐑 ) C p , q ( 𝐑 ) C\ell_{p+1,q+1}(\mathbf{R})=M_{2}(\mathbf{R})\otimes C\ell_{p,q}(\mathbf{R})
  59. C p , q + 2 ( 𝐑 ) = 𝐇 C q , p ( 𝐑 ) . C\ell_{p,q+2}(\mathbf{R})=\mathbf{H}\otimes C\ell_{q,p}(\mathbf{R}).
  60. x v α ( x ) - 1 V . xv\alpha(x)^{-1}\in V.
  61. 1 K * Γ O ( K ) V 1 , 1\rightarrow K^{*}\rightarrow\Gamma\rightarrow\mbox{O}~{}_{V}(K)\rightarrow 1,\,
  62. 1 K * Γ 0 SO ( K ) V 1. 1\rightarrow K^{*}\rightarrow\Gamma^{0}\rightarrow\mbox{SO}~{}_{V}(K)% \rightarrow 1.\,
  63. Q ( x ) = x t x . Q(x)=x^{t}x.\,
  64. 1 { ± 1 } Pin ( K ) V O ( K ) V K * / K * 2 , 1\to\{\pm 1\}\to\mbox{Pin}~{}_{V}(K)\to\mbox{O}~{}_{V}(K)\to K^{*}/K^{*2},\,
  65. 1 { ± 1 } Spin ( K ) V SO ( K ) V K * / K * 2 . 1\to\{\pm 1\}\to\mbox{Spin}~{}_{V}(K)\to\mbox{SO}~{}_{V}(K)\to K^{*}/K^{*2}.\,
  66. 1 μ 2 Pin V O V 1 1\to\mu_{2}\rightarrow\mbox{Pin}~{}_{V}\rightarrow\mbox{O}~{}_{V}\rightarrow 1\,
  67. 1 H 0 ( μ 2 ; K ) H 0 ( Pin ; V K ) H 0 ( O ; V K ) H 1 ( μ 2 ; K ) . 1\to H^{0}(\mu_{2};K)\to H^{0}(\mbox{Pin}~{}_{V};K)\to H^{0}(\mbox{O}~{}_{V};K% )\to H^{1}(\mu_{2};K).\,
  68. 1 { ± 1 } Pin ( K ) V O ( K ) V K * / K * 2 , 1\to\{\pm 1\}\to\mbox{Pin}~{}_{V}(K)\to\mbox{O}~{}_{V}(K)\to K^{*}/K^{*2},\,
  69. Pin p , q = { v 1 v 2 v r | i v i = ± 1 } . {\mbox{Pin}~{}}_{p,q}=\{v_{1}v_{2}\dots v_{r}|\,\,\forall i\,\|v_{i}\|=\pm 1\}.
  70. C p , q 0 = { x C p , q | α ( x ) = x } . C\ell_{p,q}^{0}=\{x\in C\ell_{p,q}|\,\alpha(x)=x\}.
  71. γ i γ j + γ j γ i = 2 η i j \gamma_{i}\gamma_{j}+\gamma_{j}\gamma_{i}=2\eta_{ij}\,
  72. η η
  73. ( 1 , 3 ) (1,3)
  74. ( 3 , 1 ) (3,1)
  75. 2 2
  76. 4 × 4 4×4
  77. 𝐬𝐨 ( 1 , 3 ) \mathbf{so}(1,3)
  78. σ μ ν = - i 4 [ γ μ , γ ν ] , \sigma^{\mu\nu}=-\frac{i}{4}[\gamma^{\mu},\gamma^{\nu}],
  79. [ σ μ ν , σ ρ τ ] = i ( η τ μ σ ρ ν + η ν τ σ μ ρ - η ρ μ σ τ ν - η ν ρ σ μ τ ) . [\sigma^{\mu\nu},\sigma^{\rho\tau}]=i(\eta^{\tau\mu}\sigma^{\rho\nu}+\eta^{\nu% \tau}\sigma^{\mu\rho}-\eta^{\rho\mu}\sigma^{\tau\nu}-\eta^{\nu\rho}\sigma^{\mu% \tau}).
  80. ( 3 , 1 ) (3,1)
  81. v < s u p > 1 = v t / Q ( v ) v<sup>−1=v^{t} /Q(v)
  82. G n G_{n}

Climate_model.html

  1. ( 1 - a ) S π r 2 = 4 π r 2 ϵ σ T 4 (1-a)S\pi r^{2}=4\pi r^{2}\epsilon\sigma T^{4}
  2. a a
  3. σ \sigma
  4. ϵ \epsilon
  5. ( 1 - a ) S = 4 ϵ σ T 4 (1-a)S=4\epsilon\sigma T^{4}
  6. T = ( 1 - a ) S 4 ϵ σ 4 T=\sqrt[4]{\frac{(1-a)S}{4\epsilon\sigma}}

Closed-loop_transfer_function.html

  1. Y ( s ) X ( s ) = G ( s ) 1 + G ( s ) H ( s ) \dfrac{Y(s)}{X(s)}=\dfrac{G(s)}{1+G(s)H(s)}
  2. Y ( s ) = Z ( s ) G ( s ) Y(s)=Z(s)G(s)
  3. Z ( s ) = X ( s ) - Y ( s ) H ( s ) Z(s)=X(s)-Y(s)H(s)
  4. X ( s ) = Z ( s ) + Y ( s ) H ( s ) X(s)=Z(s)+Y(s)H(s)
  5. X ( s ) = Z ( s ) + Z ( s ) G ( s ) H ( s ) X(s)=Z(s)+Z(s)G(s)H(s)
  6. Y ( s ) X ( s ) = Z ( s ) G ( s ) Z ( s ) + Z ( s ) G ( s ) H ( s ) \Rightarrow\dfrac{Y(s)}{X(s)}=\dfrac{Z(s)G(s)}{Z(s)+Z(s)G(s)H(s)}
  7. Y ( s ) X ( s ) = G ( s ) 1 + G ( s ) H ( s ) \dfrac{Y(s)}{X(s)}=\dfrac{G(s)}{1+G(s)H(s)}

Closure_(topology).html

  1. S ¯ \scriptstyle\bar{S}
  2. S - \scriptstyle S^{-}
  3. = cl ( ) \varnothing=\mathrm{cl}(\varnothing)
  4. 2 . \sqrt{2}.
  5. 𝒫 ( X ) \mathcal{P}(X)
  6. ( X , 𝒯 ) (X,\mathcal{T})
  7. S S
  8. C l ( S ) = S Cl(S)=S
  9. X X
  10. X X
  11. A A
  12. X X
  13. S S
  14. S S
  15. A A
  16. A A
  17. S S
  18. X X
  19. C l A ( S ) = A C l X ( S ) Cl_{A}(S)=A\cap Cl_{X}(S)
  20. S S
  21. A A
  22. A A
  23. C l X ( S ) Cl_{X}(S)
  24. A B A\to B
  25. I : T P I:T\to P
  26. A X A\subseteq X
  27. ( A I ) (A\downarrow I)
  28. A C l ( A ) A\to Cl(A)
  29. ( I X A ) (I\downarrow X\setminus A)
  30. i n t ( A ) int(A)

CMOS.html

  1. P = 0.5 C V 2 f P=0.5CV^{2}f
  2. α \alpha
  3. P = α C V 2 f P=\alpha CV^{2}f

Co-NP.html

  1. 𝒳 {\mathcal{X}}
  2. 𝒳 ¯ \overline{\mathcal{X}}
  3. 𝒳 {\mathcal{X}}
  4. 𝒳 {\mathcal{X}}

Coast.html

  1. L ( ϵ ) F ϵ 1 - D L(\epsilon)\sim F\epsilon^{1-D}\,
  2. F ϵ D ϵ \frac{F}{\epsilon^{D}}\cdot\epsilon

Coaxial_cable.html

  1. Z 0 Z_{0}
  2. h h
  3. d d
  4. D D
  5. ϵ \epsilon
  6. ϵ r \epsilon_{r}
  7. ϵ 0 \epsilon_{0}
  8. ϵ = ϵ r ϵ 0 \epsilon=\epsilon_{r}\epsilon_{0}
  9. ϵ e f f \epsilon_{eff}
  10. μ \mu
  11. μ r \mu_{r}
  12. μ 0 \mu_{0}
  13. μ = μ r μ 0 \mu=\mu_{r}\mu_{0}
  14. ( C h ) = 2 π ϵ ln ( D / d ) = 2 π ϵ 0 ϵ r ln ( D / d ) \left(\frac{C}{h}\right)={2\pi\epsilon\over\ln(D/d)}={2\pi\epsilon_{0}\epsilon% _{r}\over\ln(D/d)}
  15. ( L h ) = μ 2 π ln ( D / d ) = μ 0 μ r 2 π ln ( D / d ) \left(\frac{L}{h}\right)={\mu\over 2\pi}\ln(D/d)={\mu_{0}\mu_{r}\over 2\pi}\ln% (D/d)
  16. C C
  17. L L
  18. Z 0 = L / C Z_{0}=\sqrt{L/C}
  19. ϵ \epsilon
  20. Z 0 = 1 2 π μ ϵ ln D d 60 Ω ϵ r ln D d 138 Ω ϵ r log 10 D d Z_{0}=\frac{1}{2\pi}\sqrt{\frac{\mu}{\epsilon}}\ln\frac{D}{d}\approx\frac{60% \Omega}{\sqrt{\epsilon_{r}}}\ln\frac{D}{d}\approx\frac{138\Omega}{\sqrt{% \epsilon_{r}}}\log_{10}\frac{D}{d}
  21. v = 1 ϵ μ = c ϵ r μ r v={1\over\sqrt{\epsilon\mu}}={c\over\sqrt{\epsilon_{r}\mu_{r}}}
  22. f c 1 π ( D + d 2 ) μ ϵ = c π ( D + d 2 ) μ r ϵ r f_{c}\approx{1\over\pi({D+d\over 2})\sqrt{\mu\epsilon}}={c\over\pi({D+d\over 2% })\sqrt{\mu_{r}\epsilon_{r}}}
  23. V p = 1150 S mils d in log 10 ( D d ) V_{p}=1150\,S\text{mils}\,d\text{in}\,\log_{10}\left(\frac{D}{d}\right)
  24. V p = 0.5 S d ln ( D d ) V_{p}=0.5\,S\,d\,\ln\left(\frac{D}{d}\right)
  25. ϵ r \epsilon_{r}
  26. μ r \mu_{r}

Code.html

  1. C = { a 0 , b 01 , c 011 } C=\{\,a\mapsto 0,b\mapsto 01,c\mapsto 011\,\}
  2. { a , b , c } \{a,b,c\}
  3. { 0 , 1 } \{0,1\}
  4. C : S T * C:\,S\to T^{*}
  5. C C
  6. S * S^{*}
  7. T * T^{*}

Code_coverage.html

  1. n n
  2. 2 n 2^{n}

Code_division_multiple_access.html

  1. T b T_{b}
  2. T c T_{c}
  3. 1 / T 1/T
  4. T T
  5. 1 / T b 1/T_{b}
  6. 1 / T c 1/T_{c}
  7. T c T_{c}
  8. T b T_{b}
  9. T b / T c T_{b}/T_{c}
  10. 𝐚 𝐛 = 0 \scriptstyle\mathbf{a}\cdot\mathbf{b}\,=\,0
  11. 𝐚 ( 𝐚 + 𝐛 ) = 𝐚 2 since 𝐚 𝐚 + 𝐚 𝐛 = 𝐚 2 + 0 \mathbf{a}\cdot(\mathbf{a}+\mathbf{b})=\|\mathbf{a}\|^{2}\quad\mathrm{since}% \quad\mathbf{a}\cdot\mathbf{a}+\mathbf{a}\cdot\mathbf{b}=\|\mathbf{a}\|^{2}+0
  12. 𝐚 ( - 𝐚 + 𝐛 ) = - 𝐚 2 since - 𝐚 𝐚 + 𝐚 𝐛 = - 𝐚 2 + 0 \mathbf{a}\cdot(-\mathbf{a}+\mathbf{b})=-\|\mathbf{a}\|^{2}\quad\mathrm{since}% \quad-\mathbf{a}\cdot\mathbf{a}+\mathbf{a}\cdot\mathbf{b}=-\|\mathbf{a}\|^{2}+0
  13. 𝐛 ( 𝐚 + 𝐛 ) = 𝐛 2 since 𝐛 𝐚 + 𝐛 𝐛 = 0 + 𝐛 2 \mathbf{b}\cdot(\mathbf{a}+\mathbf{b})=\|\mathbf{b}\|^{2}\quad\mathrm{since}% \quad\mathbf{b}\cdot\mathbf{a}+\mathbf{b}\cdot\mathbf{b}=0+\|\mathbf{b}\|^{2}
  14. 𝐛 ( 𝐚 - 𝐛 ) = - 𝐛 2 since 𝐛 𝐚 - 𝐛 𝐛 = 0 - 𝐛 2 \mathbf{b}\cdot(\mathbf{a}-\mathbf{b})=-\|\mathbf{b}\|^{2}\quad\mathrm{since}% \quad\mathbf{b}\cdot\mathbf{a}-\mathbf{b}\cdot\mathbf{b}=0-\|\mathbf{b}\|^{2}

Codex_Sinaiticus.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}

Codomain.html

  1. 𝐟 \mathbf{f}
  2. 𝐗 \mathbf{X}
  3. 𝐘 \mathbf{Y}
  4. 𝐘 \mathbf{Y}
  5. 𝐟 \mathbf{f}
  6. 𝐘 \mathbf{Y}
  7. 𝐟 \mathbf{f}
  8. Y Y
  9. Y Y
  10. f : X Y f:X→Y
  11. f f
  12. ( X , Y , F ) (X,Y,F)
  13. F F
  14. X × Y X×Y
  15. X X
  16. F F
  17. F F
  18. f ( x ) f(x)
  19. x x
  20. X X
  21. f f
  22. y y
  23. f ( x ) = y f(x)=y
  24. X X
  25. ( X , Y , F ) (X,Y,F)
  26. f : X Y f:X→Y
  27. f : f\colon\mathbb{R}\rightarrow\mathbb{R}
  28. f : x x 2 f\colon\,x\mapsto x^{2}
  29. f ( x ) = x 2 f(x)\ =\ x^{2}
  30. f f
  31. \textstyle\mathbb{R}
  32. f f
  33. f f
  34. 0 + \textstyle\mathbb{R}^{+}_{0}
  35. [ 0 , ) [0,∞)
  36. g g
  37. g : 0 + g\colon\mathbb{R}\rightarrow\mathbb{R}^{+}_{0}
  38. g : x x 2 . g\colon\,x\mapsto x^{2}.
  39. f f
  40. g g
  41. x x
  42. h h
  43. h : x x . h\colon\,x\mapsto\sqrt{x}.
  44. h h
  45. 0 + \textstyle\mathbb{R}^{+}_{0}
  46. h : 0 + h\colon\mathbb{R}^{+}_{0}\rightarrow\mathbb{R}
  47. h f h\circ f
  48. h g h\circ g
  49. h f h∘f
  50. f f
  51. \textstyle\mathbb{R}
  52. h h
  53. f f
  54. h h
  55. g g
  56. f f
  57. 2 \textstyle\mathbb{R}^{2}
  58. 2 × 2 2×2
  59. 2 \textstyle\mathbb{R}^{2}
  60. 2 \textstyle\mathbb{R}^{2}
  61. 2 2
  62. 1 1
  63. 0
  64. T T
  65. T = ( 1 0 1 0 ) T=\begin{pmatrix}1&0\\ 1&0\end{pmatrix}
  66. ( x , y ) (x,y)
  67. ( x , x ) (x,x)
  68. ( 2 , 3 ) (2,3)
  69. T T
  70. 2 \textstyle\mathbb{R}^{2}
  71. 2 \textstyle\mathbb{R}^{2}
  72. 2 × 2 2×2
  73. T T
  74. T T

Coefficient.html

  1. 7 x 2 - 3 x y + 1.5 + y 7x^{2}-3xy+1.5+y
  2. a x 2 + b x + c ax^{2}+bx+c
  3. a k x k + + a 1 x 1 + a 0 a_{k}x^{k}+\cdots+a_{1}x^{1}+a_{0}
  4. k k
  5. a k , , a 1 , a 0 a_{k},\ldots,a_{1},a_{0}
  6. i i
  7. a i 0 a_{i}\neq 0
  8. a i a_{i}
  9. 4 x 5 + x 3 + 2 x 2 \,4x^{5}+x^{3}+2x^{2}
  10. M = ( 1 2 0 6 0 2 9 4 0 0 0 4 0 0 0 0 ) M=\begin{pmatrix}1&2&0&6\\ 0&2&9&4\\ 0&0&0&4\\ 0&0&0&0\end{pmatrix}
  11. ( x 1 , x 2 , , x n ) (x_{1},x_{2},\ldots,x_{n})
  12. v v
  13. { e 1 , e 2 , , e n } \{e_{1},e_{2},\ldots,e_{n}\}
  14. v = x 1 e 1 + x 2 e 2 + + x n e n . v=x_{1}e_{1}+x_{2}e_{2}+\cdots+x_{n}e_{n}.
  15. 2 H 2 + O 2 2 H 2 O 2H_{2}+O_{2}\rightarrow 2H_{2}O
  16. H 2 H_{2}
  17. H 2 O H_{2}O

Cofinality.html

  1. ω α + 1 \omega_{\alpha+1}
  2. ω \omega
  3. ω 1 \omega_{1}
  4. ω 2 \omega_{2}
  5. ω ω \omega_{\omega}
  6. cf ( κ ) = min { card ( I ) | κ = i I λ i and ( i ) ( λ i < κ ) } \mathrm{cf}(\kappa)=\min\left\{\mathrm{card}(I)\ |\ \kappa=\sum_{i\in I}% \lambda_{i}\ \mathrm{and}\ (\forall i)(\lambda_{i}<\kappa)\right\}
  7. κ = i κ { i } \kappa=\bigcup_{i\in\kappa}\{i\}
  8. ω = n < ω n \aleph_{\omega}=\bigcup_{n<\omega}\aleph_{n}
  9. ω \aleph_{\omega}
  10. 0 \aleph_{0}
  11. ω \aleph_{\omega}
  12. 2 0 ω . 2^{\aleph_{0}}\neq\aleph_{\omega}.
  13. 2 0 = 1 2^{\aleph_{0}}=\aleph_{1}
  14. cf ( δ ) = cf ( δ ) \mathrm{cf}(\aleph_{\delta})=\mathrm{cf}(\delta)

Coherence_length.html

  1. L = c n Δ f , L={c\over n\,\Delta f},
  2. Δ f \Delta f
  3. L L
  4. L = 2 ln ( 2 ) π n λ 2 Δ λ , L={2\ln(2)\over\pi n}{\lambda^{2}\over\Delta\lambda},
  5. λ \lambda
  6. n n
  7. Δ λ \Delta\lambda
  8. Δ λ \Delta\lambda
  9. L L
  10. 1 / e = 37 % 1/e=37\%
  11. V = I max - I min I max + I min , V={I_{\max}-I_{\min}\over I_{\max}+I_{\min}},\,
  12. I I

Coherence_time.html

  1. τ = 1 Δ ν λ 2 c Δ λ \tau=\frac{1}{\Delta\nu}\approx\frac{\lambda^{2}}{c\,\Delta\lambda}

Collatz_conjecture.html

  1. f ( n ) = { n / 2 if n 0 ( mod 2 ) 3 n + 1 if n 1 ( mod 2 ) . f(n)=\begin{cases}n/2&\,\text{if }n\equiv 0\;\;(\mathop{{\rm mod}}2)\\ 3n+1&\,\text{if }n\equiv 1\;\;(\mathop{{\rm mod}}2).\end{cases}
  2. a i = { n for i = 0 f ( a i - 1 ) for i > 0 a_{i}=\begin{cases}n&\,\text{for }i=0\\ f(a_{i-1})&\,\text{for }i>0\end{cases}
  3. a i a_{i}
  4. f f
  5. n n
  6. i i
  7. a i = f i ( n ) a_{i}=f^{i}(n)
  8. 2 n 2^{n}
  9. n n
  10. f ( n ) = ( 3 n + 1 ) / 2 f(n)=(3n+1)/2
  11. f ( n ) = n / 2 f(n)=n/2
  12. f ( a 0 ) = a 1 f(a_{0})=a_{1}
  13. f ( a 1 ) = a 2 f(a_{1})=a_{2}
  14. f ( a q ) = a 0 f(a_{q})=a_{0}
  15. × 10 1 8 \times 10^{1}8
  16. R ( n ) = { { 2 n } if n 0 , 1 , 2 , 3 , 5 { 2 n , ( n - 1 ) / 3 } if n 4 ( mod 6 ) . R(n)=\begin{cases}\{2n\}&\,\text{if }n\equiv 0,1,2,3,5\\ \{2n,(n-1)/3\}&\,\text{if }n\equiv 4\end{cases}\;\;(\mathop{{\rm mod}}6).
  17. R ( n ) = { { 2 n } if n 0 , 1 { 2 n , ( 2 n - 1 ) / 3 } if n 2 ( mod 3 ) . R(n)=\begin{cases}\{2n\}&\,\text{if }n\equiv 0,1\\ \{2n,(2n-1)/3\}&\,\text{if }n\equiv 2\end{cases}\;\;(\mathop{{\rm mod}}3).
  18. f ( n ) = { n / 2 if n 0 ( 3 n + 1 ) / 2 if n 1. ( mod 2 ) f(n)=\begin{cases}n/2&\,\text{if }n\equiv 0\\ (3n+1)/2&\,\text{if }n\equiv 1.\end{cases}\;\;(\mathop{{\rm mod}}2)
  19. f ( n ) = { n / 2 if n 0 ( 3 n + 1 ) / 2 if n 1. ( mod 2 ) f(n)=\begin{cases}n/2&\,\text{if }n\equiv 0\\ (3n+1)/2&\,\text{if }n\equiv 1.\end{cases}\;\;(\mathop{{\rm mod}}2)
  20. 3 m - 1 2 k 0 + + 3 0 2 k m - 1 2 n - 3 m . \frac{3^{m-1}2^{k_{0}}+\cdots+3^{0}2^{k_{m-1}}}{2^{n}-3^{m}}.
  21. 3 3 2 0 + 3 2 2 2 + 3 1 2 3 + 3 0 2 6 2 7 - 3 4 = 151 47 , \frac{3^{3}2^{0}+3^{2}2^{2}+3^{1}2^{3}+3^{0}2^{6}}{2^{7}-3^{4}}=\frac{151}{47},
  22. 3 3 2 1 + 3 2 2 2 + 3 1 2 5 + 3 0 2 6 2 7 - 3 4 = 250 47 \frac{3^{3}2^{1}+3^{2}2^{2}+3^{1}2^{5}+3^{0}2^{6}}{2^{7}-3^{4}}=\frac{250}{47}
  23. 3 3 2 0 + 3 2 2 1 + 3 1 2 4 + 3 0 2 5 2 7 - 3 4 = < m t p l > 125 \frac{3^{3}2^{0}+3^{2}2^{1}+3^{1}2^{4}+3^{0}2^{5}}{2^{7}-3^{4}}=\frac{<}{m}tpl% >{{125}}
  24. 3 3 2 0 + 3 2 2 3 + 3 1 2 4 + 3 0 2 6 2 7 - 3 4 = 211 47 \frac{3^{3}2^{0}+3^{2}2^{3}+3^{1}2^{4}+3^{0}2^{6}}{2^{7}-3^{4}}=\frac{211}{47}
  25. 3 3 2 2 + 3 2 2 3 + 3 1 2 5 + 3 0 2 6 2 7 - 3 4 = 340 47 \frac{3^{3}2^{2}+3^{2}2^{3}+3^{1}2^{5}+3^{0}2^{6}}{2^{7}-3^{4}}=\frac{340}{47}
  26. 3 3 2 1 + 3 2 2 2 + 3 1 2 4 + 3 0 2 5 2 7 - 3 4 = 170 47 \frac{3^{3}2^{1}+3^{2}2^{2}+3^{1}2^{4}+3^{0}2^{5}}{2^{7}-3^{4}}=\frac{170}{47}
  27. 3 3 2 0 + 3 2 2 1 + 3 1 2 3 + 3 0 2 4 2 7 - 3 4 = 85 47 \frac{3^{3}2^{0}+3^{2}2^{1}+3^{1}2^{3}+3^{0}2^{4}}{2^{7}-3^{4}}=\frac{85}{47}
  28. 3 3 2 0 + 3 2 2 1 + 3 1 2 4 + 3 0 2 5 2 8 - 3 4 = 125 175 \frac{3^{3}2^{0}+3^{2}2^{1}+3^{1}2^{4}+3^{0}2^{5}}{2^{8}-3^{4}}=\frac{125}{175}
  29. 3 1 2 0 + 3 0 2 1 2 4 - 3 2 = 5 7 . \frac{3^{1}2^{0}+3^{0}2^{1}}{2^{4}-3^{2}}=\frac{5}{7}.
  30. f ( z ) = 1 2 z cos 2 ( π 2 z ) + ( 3 z + 1 ) sin 2 ( π 2 z ) , f(z)=\frac{1}{2}z\cos^{2}\left(\frac{\pi}{2}z\right)+(3z+1)\sin^{2}\left(\frac% {\pi}{2}z\right),
  31. 1 4 ( 2 + 7 z - ( 2 + 5 z ) cos ( π z ) ) . \frac{1}{4}(2+7z-(2+5z)\cos(\pi z)).
  32. f ( z ) = 1 2 z cos 2 ( π 2 z ) + ( 3 z + 1 ) 2 sin 2 ( π 2 z ) , f(z)=\frac{1}{2}z\cos^{2}\left(\frac{\pi}{2}z\right)+\frac{(3z+1)}{2}\sin^{2}% \left(\frac{\pi}{2}z\right),
  33. 1 4 ( 1 + 4 z - ( 1 + 2 z ) cos ( π z ) ) \frac{1}{4}(1+4z-(1+2z)\cos(\pi z))
  34. I I
  35. I I
  36. I I
  37. g ( n ) = a i n + b i , n i ( mod P ) g(n)=a_{i}n+b_{i},{n\equiv i\;\;(\mathop{{\rm mod}}P)}
  38. a 0 , b 0 , , a P - 1 , b P - 1 a_{0},b_{0},\dots,a_{P-1},b_{P-1}
  39. g ( n ) g(n)
  40. P = 2 P=2
  41. a 0 = 1 / 2 a_{0}=1/2
  42. b 0 = 0 b_{0}=0
  43. a 1 = 3 a_{1}=3
  44. b 1 = 1 b_{1}=1
  45. g k ( n ) g^{k}(n)
  46. g k ( n ) g^{k}(n)
  47. Π 2 0 \Pi^{0}_{2}

Collision.html

  1. m a 𝐮 a + m b 𝐮 b = ( m a + m b ) 𝐯 m_{a}\mathbf{u}_{a}+m_{b}\mathbf{u}_{b}=\left(m_{a}+m_{b}\right)\mathbf{v}\,
  2. 𝐯 = m a 𝐮 a + m b 𝐮 b m a + m b \mathbf{v}=\frac{m_{a}\mathbf{u}_{a}+m_{b}\mathbf{u}_{b}}{m_{a}+m_{b}}
  3. ( i j k l ) \begin{pmatrix}i&j\\ k&l\end{pmatrix}

Colon_(punctuation).html

  1. G G
  2. H H
  3. ƒ : X Y ƒ:X→Y
  4. f f
  5. X X
  6. Y Y
  7. S = { x : 1 < x < 3 } S=\{x\in\mathbb{R}:1<x<3\}
  8. x x
  9. \mathbb{R}
  10. x x
  11. λ x . x : A A \lambda x.x\mathrel{:}A\to A

Color.html

  1. λ \lambda\,\!
  2. ν \nu\,\!
  3. ν b \nu_{b}\,\!
  4. E E\,\!
  5. E E\,\!

Color_temperature.html

  1. [ R G B ] = [ 3.1956 2.4478 - 0.1434 - 2.5455 7.0492 0.9963 0.0000 0.0000 1.0000 ] [ X Y Z ] . \begin{bmatrix}R\\ G\\ B\end{bmatrix}=\begin{bmatrix}3.1956&2.4478&-0.1434\\ -2.5455&7.0492&0.9963\\ 0.0000&0.0000&1.0000\end{bmatrix}\begin{bmatrix}X\\ Y\\ Z\end{bmatrix}.
  2. u = 0.4661 x + 0.1593 y y - 0.15735 x + 0.2424 , v = 0.6581 y y - 0.15735 x + 0.2424 . u=\frac{0.4661x+0.1593y}{y-0.15735x+0.2424},\quad v=\frac{0.6581y}{y-0.15735x+% 0.2424}.
  3. u = 4 x - 2 x + 12 y + 3 , v = 6 y - 2 x + 12 y + 3 . u=\frac{4x}{-2x+12y+3},\quad v=\frac{6y}{-2x+12y+3}.
  4. Δ u v = ± 0.05 \Delta uv=\pm 0.05
  5. Δ u v \Delta uv
  6. ( u T , v T ) \scriptstyle(u_{T},v_{T})
  7. 1 T c = 1 T i + θ 1 θ 1 + θ 2 ( 1 T i + 1 - 1 T i ) , \frac{1}{T_{c}}=\frac{1}{T_{i}}+\frac{\theta_{1}}{\theta_{1}+\theta_{2}}\left(% \frac{1}{T_{i+1}}-\frac{1}{T_{i}}\right),
  8. T i T_{i}
  9. T i + 1 T_{i+1}
  10. T i < T c < T i + 1 T_{i}<T_{c}<T_{i+1}
  11. d i / d i + 1 < 0 d_{i}/d_{i+1}<0
  12. θ 1 / θ 2 sin θ 1 / sin θ 2 \theta_{1}/\theta_{2}\approx\sin\theta_{1}/\sin\theta_{2}
  13. 1 T c = 1 T i + d i d i - d i + 1 ( 1 T i + 1 - 1 T i ) . \frac{1}{T_{c}}=\frac{1}{T_{i}}+\frac{d_{i}}{d_{i}-d_{i+1}}\left(\frac{1}{T_{i% +1}}-\frac{1}{T_{i}}\right).
  14. d i = ( v T - v i ) - m i ( u T - u i ) 1 + m i 2 , d_{i}=\frac{(v_{T}-v_{i})-m_{i}(u_{T}-u_{i})}{\sqrt{1+m_{i}^{2}}},
  15. ( u i , v i ) (u_{i},v_{i})
  16. m i = - 1 / l i m_{i}=-1/l_{i}
  17. ( u i , v i ) (u_{i},v_{i})
  18. Δ u v = 5 × 10 - 2 \scriptstyle\Delta_{uv}=5\times 10^{-2}
  19. Δ u v \scriptstyle\Delta uv
  20. T C T_{C}
  21. B - V B-V
  22. B - V B-V

Colossus_computer.html

  1. ψ \psi
  2. ψ \psi
  3. ψ \psi
  4. ψ \psi
  5. ψ \psi
  6. μ \mu
  7. μ \mu
  8. χ \chi
  9. χ \chi
  10. χ \chi
  11. χ \chi
  12. χ \chi
  13. χ \chi
  14. ψ \psi
  15. ψ \psi^{\prime}

Column.html

  1. f c r π 3 𝐸 I m i n L 2 ( 1 ) f_{cr}\equiv\frac{\pi^{3}\,\textit{E}I_{min}}{{L}^{2}}\qquad(1)
  2. f c r π 2 E T ( K L r ) 2 ( 2 ) f_{cr}\equiv\frac{\pi^{2}E_{T}}{(\frac{KL}{r})^{2}}\qquad(2)
  3. f c r F y - F y 2 4 π 2 E ( K L r 2 ) ( 3 ) f_{cr}\equiv{F_{y}}-\frac{F^{2}_{y}}{4\pi^{2}E}\left(\frac{KL}{r^{2}}\right)% \qquad(3)

Combination.html

  1. ( n k ) = n ( n - 1 ) ( n - k + 1 ) k ( k - 1 ) 1 , {\left({{n}\atop{k}}\right)}=\frac{n(n-1)\ldots(n-k+1)}{k(k-1)\dots 1},
  2. n ! k ! ( n - k ) ! \frac{n!}{k!(n-k)!}
  3. k n k\leq n
  4. k > n k>n
  5. ( S k ) {\left({{S}\atop{k}}\right)}\,
  6. C ( n , k ) C(n,k)
  7. C k n C^{n}_{k}
  8. C k n {}_{n}C_{k}
  9. C k n {}^{n}C_{k}
  10. C n , k C_{n,k}
  11. C n k C_{n}^{k}
  12. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  13. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  14. ( 1 + X ) n = k 0 ( n k ) X k , \textstyle(1+X)^{n}=\sum_{k\geq 0}{\left({{n}\atop{k}}\right)}X^{k},
  15. ( n 0 ) = ( n n ) = 1 {\textstyle\left({{n}\atop{0}}\right)}={\textstyle\left({{n}\atop{n}}\right)}=1
  16. ( n k ) = 0 {\textstyle\left({{n}\atop{k}}\right)}=0
  17. s S ( 1 + X s ) ; \textstyle\prod_{s\in S}(1+X_{s});
  18. ( n k ) = ( n - 1 k - 1 ) + ( n - 1 k ) , for 0 < k < n , {\left({{n}\atop{k}}\right)}={\left({{n-1}\atop{k-1}}\right)}+{\left({{n-1}% \atop{k}}\right)},\,\text{ for }0<k<n,
  19. ( n k ) = n ( n - 1 ) ( n - 2 ) ( n - k + 1 ) k ! . {\left({{n}\atop{k}}\right)}=\frac{n(n-1)(n-2)\cdots(n-k+1)}{k!}.
  20. ( n k ) = ( n n - k ) , for 0 k n . {\left({{n}\atop{k}}\right)}={\left({{n}\atop{n-k}}\right)},\,\text{ for }0% \leq k\leq n.
  21. ( n k ) = n ! k ! ( n - k ) ! , {\left({{n}\atop{k}}\right)}=\frac{n!}{k!(n-k)!},
  22. ( n k ) = ( n k - 1 ) n - k + 1 k , for k > 0 {\left({{n}\atop{k}}\right)}={\left({{n}\atop{k-1}}\right)}\frac{n-k+1}{k},\,% \text{ for }k>0
  23. ( n k ) = ( n - 1 k ) n n - k , for k < n {\left({{n}\atop{k}}\right)}={\left({{n-1}\atop{k}}\right)}\frac{n}{n-k},\,% \text{ for }{k<n}
  24. ( n k ) = ( n - 1 k - 1 ) n k , for n , k > 0 {\left({{n}\atop{k}}\right)}={\left({{n-1}\atop{k-1}}\right)}\frac{n}{k},\,% \text{ for }n,k>0
  25. ( n 0 ) = 1 = ( n n ) {\textstyle\left({{n}\atop{0}}\right)}=1={\textstyle\left({{n}\atop{n}}\right)}
  26. ( 52 5 ) = 52 × 51 × 50 × 49 × 48 5 × 4 × 3 × 2 × 1 = 311 , 875 , 200 120 = 2 , 598 , 960. {52\choose 5}=\frac{52\times 51\times 50\times 49\times 48}{5\times 4\times 3% \times 2\times 1}=\frac{311{,}875{,}200}{120}=2{,}598{,}960.
  27. ( 52 5 ) \displaystyle{52\choose 5}
  28. ( n k ) = ( n - 0 ) 1 × ( n - 1 ) 2 × ( n - 2 ) 3 × × ( n - ( k - 1 ) ) k , {n\choose k}=\frac{(n-0)}{1}\times\frac{(n-1)}{2}\times\frac{(n-2)}{3}\times% \cdots\times\frac{(n-(k-1))}{k},
  29. ( 52 5 ) = 52 1 × 51 2 × 50 3 × 49 4 × 48 5 = 2 , 598 , 960. {52\choose 5}=\frac{52}{1}\times\frac{51}{2}\times\frac{50}{3}\times\frac{49}{% 4}\times\frac{48}{5}=2{,}598{,}960.
  30. 52 ÷ 1 × 51 ÷ 2 × 50 ÷ 3 × 49 ÷ 4 × 48 ÷ 5 52÷1×51÷2×50÷3×49÷4×48÷5
  31. ( 52 5 ) \displaystyle{52\choose 5}
  32. S S
  33. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  34. S S
  35. S S
  36. 2 n 2^{n}
  37. x i x_{i}
  38. x 1 + x 2 + + x n = k . x_{1}+x_{2}+\ldots+x_{n}=k.
  39. ( ( n k ) ) , \left(\!\!\!{\left({{n}\atop{k}}\right)}\!\!\!\right),
  40. ( ( n k ) ) = ( n + k - 1 k ) . \left(\!\!\!{\left({{n}\atop{k}}\right)}\!\!\!\right)={\left({{n+k-1}\atop{k}}% \right)}.
  41. x 1 x_{1}
  42. x 2 x_{2}
  43. x 1 = 3 , x 2 = 2 , x 3 = 0 , x 4 = 5 x_{1}=3,x_{2}=2,x_{3}=0,x_{4}=5
  44. x 1 + x 2 + x 3 + x 4 = 10 x_{1}+x_{2}+x_{3}+x_{4}=10
  45. | | | \bigstar\bigstar\bigstar|\bigstar\bigstar||\bigstar\bigstar\bigstar\bigstar\bigstar
  46. ( 13 10 ) = ( 13 3 ) = 286 , {\left({{13}\atop{10}}\right)}={\left({{13}\atop{3}}\right)}=286,
  47. n 1 , k 0 n\geq 1,k\geq 0
  48. ( ( n k ) ) = ( ( k + 1 n - 1 ) ) . \left(\!\!\!{\left({{n}\atop{k}}\right)}\!\!\!\right)=\left(\!\!\!{\left({{k+1% }\atop{n-1}}\right)}\!\!\!\right).
  49. ( ( 4 3 ) ) = ( 4 + 3 - 1 3 ) = ( 6 3 ) = 6 × 5 × 4 3 × 2 × 1 = 20. \left(\!\!\!{\left({{4}\atop{3}}\right)}\!\!\!\right)={\left({{4+3-1}\atop{3}}% \right)}={\left({{6}\atop{3}}\right)}=\frac{6\times 5\times 4}{3\times 2\times 1% }=20.
  50. [ x 1 , x 2 , x 3 , x 4 ] [x_{1},x_{2},x_{3},x_{4}]
  51. x 1 + x 2 + x 3 + x 4 = 3 x_{1}+x_{2}+x_{3}+x_{4}=3
  52. | < t d > < / t d > \bigstar\bigstar\bigstar|<td></td>
  53. | | | \bigstar\bigstar|\bigstar||
  54. | | | \bigstar\bigstar||\bigstar|
  55. | < t d > < / t d > \bigstar\bigstar|<td>\bigstar</td>
  56. | | | \bigstar|\bigstar\bigstar||
  57. | | | \bigstar|\bigstar|\bigstar|
  58. | | | \bigstar|\bigstar||\bigstar
  59. | | | \bigstar||\bigstar\bigstar|
  60. | | | \bigstar||\bigstar|\bigstar
  61. | < t d > < / t d > \bigstar|<td>\bigstar\bigstar</td>
  62. | | | |\bigstar\bigstar\bigstar||
  63. | | | |\bigstar\bigstar|\bigstar|
  64. | | | |\bigstar\bigstar||\bigstar
  65. | | | |\bigstar|\bigstar\bigstar|
  66. | | | |\bigstar|\bigstar|\bigstar
  67. | | | |\bigstar||\bigstar\bigstar
  68. | | | ||\bigstar\bigstar\bigstar|
  69. | | | ||\bigstar\bigstar|\bigstar
  70. | | | ||\bigstar|\bigstar\bigstar
  71. | < t d > < / t d > |<td>\bigstar\bigstar\bigstar</td>
  72. 0 k n ( n k ) = 2 n \sum_{0\leq{k}\leq{n}}{\left({{n}\atop{k}}\right)}=2^{n}
  73. | { { } ; { 1 } ; { 2 } ; { 3 } ; { 1 , 2 } ; { 1 , 3 } ; { 2 , 3 } ; { 1 , 2 , 3 } } | = 2 3 = 8 |\{\{\};\{1\};\{2\};\{3\};\{1,2\};\{1,3\};\{2,3\};\{1,2,3\}\}|=2^{3}=8
  74. k - # samples chosen n - # samples visited \frac{k-\mathrm{\#\,samples\ chosen}}{n-\mathrm{\#\,samples\ visited}}

Combinatorial_chemistry.html

  1. N R 1 × N R 2 × N R 3 N_{R_{1}}\times N_{R_{2}}\times N_{R_{3}}
  2. N R 1 N_{R_{1}}
  3. N R 2 N_{R_{2}}
  4. N R 3 N_{R_{3}}

Combustion.html

  1. C x H y + z O 2 x C O 2 + y 2 H 2 O C_{x}H_{y}+zO_{2}\to xCO_{2}+\frac{y}{2}H_{2}O
  2. C 3 H 8 + 5 O 2 3 C O 2 + 4 H 2 O C_{3}H_{8}+5O_{2}\to 3CO_{2}+4H_{2}O
  3. fuel + oxygen water + carbon dioxide \,\text{fuel}+\,\text{oxygen}\to\,\text{water}+\,\text{carbon dioxide}
  4. C x H y + z O 2 + 3.71 z N 2 x C O 2 + y 2 H 2 O + 3.71 z N 2 C_{x}H_{y}+zO_{2}+3.71zN_{2}\to xCO_{2}+\frac{y}{2}H_{2}O+3.71zN_{2}
  5. C 3 H 8 + 5 O 2 + 18.55 N 2 3 C O 2 + 4 H 2 O + 18.55 N 2 C_{3}H_{8}+5O_{2}+18.55N_{2}\to 3CO_{2}+4H_{2}O+18.55N_{2}
  6. fuel + oxygen + nitrogen water + carbon dioxide + nitrogen \,\text{fuel}+\,\text{oxygen}+\,\text{nitrogen}\to\,\text{water}+\,\text{% carbon dioxide}+\,\text{nitrogen}
  7. C x H y + z O 2 a C O 2 + b C O + c H 2 O + d H 2 C_{x}H_{y}+zO_{2}\to aCO_{2}+bCO+cH_{2}O+dH_{2}
  8. fuel + oxygen water + hydrogen + carbon dioxide + carbon monoxide \,\text{fuel}+\,\text{oxygen}\to\,\text{water}+\,\text{hydrogen}+\,\text{% carbon dioxide}+\,\text{carbon monoxide}
  9. Carbon : a + b = 3 \mathrm{Carbon:}\ a+b=3
  10. Hydrogen : 2 c + 2 d = 8 \mathrm{Hydrogen:}\ 2c+2d=8
  11. Oxygen : 2 a + b + c = 8 \mathrm{Oxygen:}\ 2a+b+c=8
  12. C O + H 2 O C O 2 + H 2 ; K e q = a × d b × c CO+H_{2}O\to CO_{2}+H_{2};K_{eq}=\frac{a\times d}{b\times c}
  13. λ {\lambda}
  14. λ = 1.0 \lambda=1.0
  15. G ( x ) = 1 T T q ( x , t ) p ( x , t ) d t G(x)=\frac{1}{T}\int_{T}q^{\prime}(x,t)p^{\prime}(x,t)dt

Communication_complexity.html

  1. f f
  2. × \times
  3. \rightarrow
  4. X = Y = { 0 , 1 } n X=Y=\{0,1\}^{n}
  5. Z = { 0 , 1 } Z=\{0,1\}
  6. x x
  7. \in
  8. y y
  9. \in
  10. f ( x , y ) f(x,y)
  11. D ( f ) D(f)
  12. D ( f ) = D(f)=
  13. f f
  14. A A
  15. x x
  16. \in
  17. y y
  18. \in
  19. A x , y = f ( x , y ) A_{\mathrm{x,y}}=f(x,y)
  20. f f
  21. x x
  22. y y
  23. 2 2 n 2^{2n}
  24. \subseteq
  25. × \times
  26. ( x 1 , y 1 ) (x_{1},y_{1})
  27. \in
  28. ( x 2 , y 2 ) (x_{2},y_{2})
  29. \in
  30. ( x 1 , y 2 ) (x_{1},y_{2})
  31. \in
  32. × \times
  33. \subseteq
  34. \subseteq
  35. k k
  36. h h
  37. \in
  38. { 0 , 1 } k \{0,1\}^{k}
  39. T h = { ( x , y ) : T_{\mathrm{h}}=\{(x,y):
  40. ( x , y ) (x,y)
  41. h } h\}
  42. T h T_{\mathrm{h}}
  43. \subseteq
  44. × \times
  45. T h T_{\mathrm{h}}
  46. x x
  47. y y
  48. n n
  49. x x
  50. y y
  51. x x
  52. y y
  53. x x
  54. y y
  55. x x
  56. y y
  57. y y
  58. y y
  59. D ( E Q ) = n D(EQ)=n
  60. D ( E Q ) n - 1 D(EQ)\leq n-1
  61. ( x , x ) (x,x)
  62. ( x , x ) (x^{\prime},x^{\prime})
  63. h h
  64. f ( x , x ) f(x,x^{\prime})
  65. x x x\neq x^{\prime}
  66. ( a , b ) (a,b)
  67. a = b a=b
  68. D ( E Q ) D(EQ)
  69. n n
  70. f f
  71. R R
  72. f f
  73. Pr [ R ( x , y ) = 0 ] > 1 2 , if f ( x , y ) = 0 \Pr[R(x,y)=0]>\frac{1}{2},\textrm{if }\,f(x,y)=0
  74. Pr [ R ( x , y ) = 1 ] > 1 2 , if f ( x , y ) = 1 \Pr[R(x,y)=1]>\frac{1}{2},\textrm{if }\,f(x,y)=1
  75. z { 0 , 1 } n z\in\{0,1\}^{n}
  76. z x z\cdot x
  77. ( ) (\cdot)
  78. z y z\cdot y
  79. x = y x=y
  80. z x = z y z\cdot x=z\cdot y
  81. P r o b z [ A c c e p t ] = 1 Prob_{z}[Accept]=1
  82. z x = z y z\cdot x=z\cdot y
  83. x = c 1 c 2 p p x n x=c_{1}c_{2}\ldots p\ldots p^{\prime}\ldots x_{n}
  84. y = c 1 c 2 q q y n y=c_{1}c_{2}\ldots q\ldots q^{\prime}\ldots y_{n}
  85. z = z 1 z 2 z i z j z n z=z_{1}z_{2}\ldots z_{i}\ldots z_{j}\ldots z_{n}
  86. x x
  87. y y
  88. z i * x i = z i * c i = z i * y i z_{i}*x_{i}=z_{i}*c_{i}=z_{i}*y_{i}
  89. x x
  90. y y
  91. x i x_{i}
  92. y i y_{i}
  93. x x
  94. y y
  95. x = 00 0 x^{\prime}=00\ldots 0
  96. y = 11 1 y^{\prime}=11\ldots 1
  97. z = z 1 z 2 z n z^{\prime}=z_{1}z_{2}\ldots z_{n^{\prime}}
  98. z x = 0 z^{\prime}\cdot x^{\prime}=0
  99. z y = Σ i z i z^{\prime}\cdot y^{\prime}=\Sigma_{i}z^{\prime}_{i}
  100. z z^{\prime}
  101. Σ i z i = 0 \Sigma_{i}z^{\prime}_{i}=0
  102. z i z^{\prime}_{i}
  103. 0
  104. 1 1
  105. 1 / 2 1/2
  106. x x
  107. y y
  108. P r o b z [ A c c e p t ] = 1 / 2 Prob_{z}[Accept]=1/2
  109. E Q ( x , y ) EQ(x,y)
  110. n n
  111. R R
  112. 100 n 100n
  113. r 1 , r 2 , , r 100 n r_{1},r_{2},\dots,r_{100n}
  114. R R
  115. P R P^{\prime}_{R}
  116. r i r_{i}
  117. r i r_{i}
  118. r i r_{i}
  119. p ( x , y ) p(x,y)
  120. p R ( x , y ) p^{\prime}_{R}(x,y)
  121. P P
  122. P R P^{\prime}_{R}
  123. ( x , y ) (x,y)
  124. ( x , y ) (x,y)
  125. Pr R [ | p R ( x , y ) - p ( x , y ) | 0.1 ] 2 exp ( - 2 ( 0.1 ) 2 100 n ) < 2 - 2 n \Pr_{R}[|p^{\prime}_{R}(x,y)-p(x,y)|\geq 0.1]\leq 2\exp(-2(0.1)^{2}\cdot 100n)% <2^{-2n}
  126. ( x , y ) (x,y)
  127. Pr R [ ( x , y ) : | p R ( x , y ) - p ( x , y ) | 0.1 ] ( x , y ) Pr R [ | p R ( x , y ) - p ( x , y ) | 0.1 ] < ( x , y ) 2 - 2 n = 1 \Pr_{R}[\exists(x,y):\,|p^{\prime}_{R}(x,y)-p(x,y)|\geq 0.1]\leq\sum_{(x,y)}% \Pr_{R}[|p^{\prime}_{R}(x,y)-p(x,y)|\geq 0.1]<\sum_{(x,y)}2^{-2n}=1
  128. 2 2 n 2^{2n}
  129. ( x , y ) (x,y)
  130. R 0 R_{0}
  131. ( x , y ) (x,y)
  132. | p R 0 ( x , y ) - p ( x , y ) | < 0.1 |p^{\prime}_{R_{0}}(x,y)-p(x,y)|<0.1
  133. P P
  134. P R 0 P^{\prime}_{R_{0}}
  135. M f = [ f ( x , y ) ] x , y { 0 , 1 } n M_{f}=[f(x,y)]_{x,y\in\{0,1\}^{n}}
  136. f f
  137. D ( f ) D(f)
  138. M f M_{f}
  139. D ( f ) D(f)
  140. M f M_{f}
  141. M f M_{f}
  142. ( M f ) (M_{f})
  143. ( M f ) (M_{f})
  144. log min ( rank ( M f ) : M f 2 n × 2 n , ( M f - M f ) 1 / 3 ) . \log\min(\textrm{rank}(M^{\prime}_{f}):M^{\prime}_{f}\in\mathbb{R}^{2^{n}% \times 2^{n}},(M_{f}-M^{\prime}_{f})_{\infty}\leq 1/3).

Commutative_ring.html

  1. [ 1 1 0 1 ] [ 1 1 1 0 ] \displaystyle\begin{bmatrix}1&1\\ 0&1\\ \end{bmatrix}\cdot\begin{bmatrix}1&1\\ 1&0\\ \end{bmatrix}
  2. r s \frac{r}{s}
  3. 𝒪 \mathcal{O}
  4. 𝒪 \mathcal{O}
  5. 𝔭 0 𝔭 1 𝔭 n \mathfrak{p}_{0}\subsetneq\mathfrak{p}_{1}\subsetneq\ldots\subsetneq\mathfrak{% p}_{n}
  6. 0 = 𝔭 0 p = 𝔭 1 0=\mathfrak{p}_{0}\subsetneq p\mathbb{Z}=\mathfrak{p}_{1}

Commutator.html

  1. x y = x [ x , y ] . x^{y}=x[x,y].\,
  2. [ y , x ] = [ x , y ] - 1 . [y,x]=[x,y]^{-1}.\,
  3. [ x , z y ] = [ x , y ] [ x , z ] y [x,zy]=[x,y]\cdot[x,z]^{y}
  4. [ x z , y ] = [ x , y ] z [ z , y ] . [xz,y]=[x,y]^{z}\cdot[z,y].
  5. [ x , y - 1 ] = [ y , x ] y - 1 [x,y^{-1}]=[y,x]^{y^{-1}}
  6. [ x - 1 , y ] = [ y , x ] x - 1 . [x^{-1},y]=[y,x]^{x^{-1}}.
  7. [ [ x , y - 1 ] , z ] y [ [ y , z - 1 ] , x ] z [ [ z , x - 1 ] , y ] x = 1 [[x,y^{-1}],z]^{y}\cdot[[y,z^{-1}],x]^{z}\cdot[[z,x^{-1}],y]^{x}=1
  8. [ [ x , y ] , z x ] [ [ z , x ] , y z ] [ [ y , z ] , x y ] = 1. [[x,y],z^{x}]\cdot[[z,x],y^{z}]\cdot[[y,z],x^{y}]=1.
  9. a x {}^{x}a
  10. ( x y ) 2 = x 2 y 2 [ y , x ] [ [ y , x ] , y ] . (xy)^{2}=x^{2}y^{2}[y,x][[y,x],y].\,
  11. ( x y ) n = x n y n [ y , x ] ( n 2 ) . (xy)^{n}=x^{n}y^{n}[y,x]^{{\left({{n}\atop{2}}\right)}}.
  12. [ a , b ] = a b - b a . [a,b]=ab-ba.
  13. { a , b } = a b + b a . \{a,b\}=ab+ba.
  14. [ A + B , C ] = [ A , C ] + [ B , C ] [A+B,C]=[A,C]+[B,C]
  15. [ A , A ] = 0 [A,A]=0
  16. [ A , B ] = - [ B , A ] [A,B]=-[B,A]
  17. [ A , [ B , C ] ] + [ B , [ C , A ] ] + [ C , [ A , B ] ] = 0 [A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0
  18. [ A , B C ] = [ A , B ] C + B [ A , C ] [A,BC]=[A,B]C+B[A,C]
  19. [ A , B C D ] = [ A , B ] C D + B [ A , C ] D + B C [ A , D ] [A,BCD]=[A,B]CD+B[A,C]D+BC[A,D]
  20. [ A , B C D E ] = [ A , B ] C D E + B [ A , C ] D E + B C [ A , D ] E + B C D [ A , E ] [A,BCDE]=[A,B]CDE+B[A,C]DE+BC[A,D]E+BCD[A,E]
  21. [ A B , C ] = A [ B , C ] + [ A , C ] B [AB,C]=A[B,C]+[A,C]B
  22. [ A B C , D ] = A B [ C , D ] + A [ B , D ] C + [ A , D ] B C [ABC,D]=AB[C,D]+A[B,D]C+[A,D]BC
  23. [ A B C D , E ] = A B C [ D , E ] + A B [ C , E ] D + A [ B , E ] C D + [ A , E ] B C D [ABCD,E]=ABC[D,E]+AB[C,E]D+A[B,E]CD+[A,E]BCD
  24. [ A B , C D ] = A [ B , C D ] + [ A , C D ] B = A [ B , C ] D + A C [ B , D ] + [ A , C ] D B + C [ A , D ] B [AB,CD]=A[B,CD]+[A,CD]B=A[B,C]D+AC[B,D]+[A,C]DB+C[A,D]B
  25. [ [ [ A , B ] , C ] , D ] + [ [ [ B , C ] , D ] , A ] + [ [ [ C , D ] , A ] , B ] + [ [ [ D , A ] , B ] , C ] = [ [ A , C ] , [ B , D ] ] [[[A,B],C],D]+[[[B,C],D],A]+[[[C,D],A],B]+[[[D,A],B],C]=[[A,C],[B,D]]
  26. ad A : R R \operatorname{ad}_{A}:R\rightarrow R
  27. ad A ( B ) = [ A , B ] \operatorname{ad}_{A}(B)=[A,B]
  28. e A B e - A = B + [ A , B ] + 1 2 ! [ A , [ A , B ] ] + 1 3 ! [ A , [ A , [ A , B ] ] ] + e ad ( A ) B . e^{A}Be^{-A}=B+[A,B]+\frac{1}{2!}[A,[A,B]]+\frac{1}{3!}[A,[A,[A,B]]]+\cdots% \equiv e^{\operatorname{ad}(A)}B.
  29. ln ( e A e B e - A e - B ) = [ A , B ] + 1 2 ! [ ( A + B ) , [ A , B ] ] + 1 3 ! ( [ A , [ B , [ B , A ] ] ] / 2 + [ ( A + B ) , [ ( A + B ) , [ A , B ] ] ] ) + . \ln\left(e^{A}e^{B}e^{-A}e^{-B}\right)=[A,B]+\frac{1}{2!}[(A+B),[A,B]]+\frac{1% }{3!}\left([A,[B,[B,A]]]/2+[(A+B),[(A+B),[A,B]]]\right)+\cdots.
  30. { A , B C } = { A , B } C - B [ A , C ] \{A,BC\}=\{A,B\}C-B[A,C]
  31. [ A B , C ] = A { B , C } - { A , C } B [AB,C]=A\{B,C\}-\{A,C\}B
  32. [ ω , η ] g r := ω η - ( - 1 ) deg ω deg η η ω . \ [\omega,\eta]_{gr}:=\omega\eta-(-1)^{\deg\omega\deg\eta}\eta\omega.
  33. ad ( x ) ( y ) = [ x , y ] . \operatorname{ad}(x)(y)=[x,y].
  34. 𝐚𝐝 ( x ) \mathbf{ad}(x)
  35. 𝐚𝐝 \mathbf{ad}
  36. ad ( x + y ) = ad ( x ) + ad ( y ) {\rm ad}(x+y)={\rm ad}(x)+{\rm ad}(y)
  37. ad ( λ x ) = λ ad ( x ) , {\rm ad}(\lambda x)=\lambda\,\operatorname{ad}(x)~{},
  38. ad ( [ x , y ] ) = [ ad ( x ) , ad ( y ) ] . {\rm ad}([x,y])=[{\rm ad}(x),{\rm ad}(y)]~{}.
  39. ad ( x y ) = ad ( x ) ad ( y ) \operatorname{ad}(xy)=\operatorname{ad}(x)\operatorname{ad}(y)
  40. ad ( x ) ad ( x ) ( y ) = [ x , [ x , y ] ] {\rm ad}(x){\rm ad}(x)(y)=[x,[x,y]\,]
  41. ad ( x ) ad ( a + b ) ( y ) = [ x , [ a + b , y ] ] {\rm ad}(x){\rm ad}(a+b)(y)=[x,[a+b,y]\,]

Commutator_subgroup.html

  1. [ g , h ] = g - 1 h - 1 g h [g,h]=g^{-1}h^{-1}gh
  2. [ g , h ] - 1 = [ h , g ] . [g,h]^{-1}=[h,g].
  3. [ g , h ] s = [ g s , h s ] [g,h]^{s}=[g^{s},h^{s}]
  4. g s = s - 1 g s g^{s}=s^{-1}gs
  5. x x s x\mapsto x^{s}
  6. [ g 1 , h 1 ] [ g n , h n ] [g_{1},h_{1}]\cdots[g_{n},h_{n}]
  7. ( [ g 1 , h 1 ] [ g n , h n ] ) s = [ g 1 s , h 1 s ] [ g n s , h n s ] ([g_{1},h_{1}]\cdots[g_{n},h_{n}])^{s}=[g_{1}^{s},h_{1}^{s}]\cdots[g_{n}^{s},h% _{n}^{s}]
  8. f ( [ g 1 , h 1 ] [ g n , h n ] ) = [ f ( g 1 ) , f ( h 1 ) ] [ f ( g n ) , f ( h n ) ] f([g_{1},h_{1}]\cdots[g_{n},h_{n}])=[f(g_{1}),f(h_{1})]\cdots[f(g_{n}),f(h_{n})]
  9. f ( [ G , G ] ) [ H , H ] f([G,G])\leq[H,H]
  10. G ( 0 ) := G G^{(0)}:=G
  11. G ( n ) := [ G ( n - 1 ) , G ( n - 1 ) ] n 𝐍 G^{(n)}:=[G^{(n-1)},G^{(n-1)}]\quad n\in\mathbf{N}
  12. G ( 2 ) , G ( 3 ) , G^{(2)},G^{(3)},\ldots
  13. G ( 2 ) G ( 1 ) G ( 0 ) = G \cdots\triangleleft G^{(2)}\triangleleft G^{(1)}\triangleleft G^{(0)}=G
  14. G n := [ G n - 1 , G ] G_{n}:=[G_{n-1},G]
  15. G ( n ) := [ G ( n - 1 ) , G ( n - 1 ) ] G^{(n)}:=[G^{(n-1)},G^{(n-1)}]
  16. φ : G G ab \varphi:G\rightarrow G^{\operatorname{ab}}
  17. φ \varphi
  18. f = F φ f=F\circ\varphi
  19. G ab G^{\operatorname{ab}}
  20. G ( n ) = { e } G^{(n)}=\{e\}
  21. G ( n ) { e } G^{(n)}\neq\{e\}
  22. G ( α ) = { e } G^{(\alpha)}=\{e\}
  23. Out ( G ) Aut ( G ab ) \mbox{Out}~{}(G)\to\mbox{Aut}~{}(G^{\mbox{ab}~{}})

Compact_space.html

  1. A = ( - , - 2 A=(-∞,-2
  2. C = ( 2 , 4 ) C=(2,4)
  3. B = 0 , 11 B=0,11
  4. 0 , 11 0,11
  5. 1 / 2 , 4 / 5 , 1 / 3 , 5 / 6 , 1 / 4 , 6 / 7 , 1/2,4/5,1/3,5/6,1/4,6/7,…
  6. ( 0 , 1 ) (0,1)
  7. 0 , 1 , 2 , 3 , 0, 1, 2, 3, …
  8. [ 0 , 1 ] [0,1]
  9. 1 , 1 / 2 , 1 / 3 , 3 / 4 , 1 / 5 , 5 / 6 , 1 / 7 , 7 / 8 , 1, 1/2, 1/3, 3/4, 1/5, 5/6, 1/7, 7/8, …
  10. [ 0 , ) [0,∞)
  11. 0 , 1 , 2 , 3 , 0, 1, 2, 3, …
  12. { U α } α A \{U_{\alpha}\}_{\alpha\in A}
  13. X X
  14. X = α A U α , X=\bigcup_{\alpha\in A}U_{\alpha},
  15. J J
  16. A A
  17. X = i J U i . X=\bigcup_{i\in J}U_{i}.
  18. δ > 0 δ>0
  19. ev p : C ( X ) 𝐑 \operatorname{ev}_{p}:C(X)\to\mathbf{R}
  20. X X X\subset{}^{\ast}X
  21. { U α } α A \{U_{\alpha}\}_{\alpha\in A}
  22. X X
  23. K α A U α , K\subseteq\bigcup_{\alpha\in A}U_{\alpha},
  24. K i J U i . K\subseteq\bigcup_{i\in J}U_{i}.
  25. [ 0 , 1 ] [0,1]
  26. ( 0 , 1 ) (0,1)
  27. ( 1 n , 1 - 1 n ) \left(\frac{1}{n},1-\frac{1}{n}\right)
  28. n = 3 , 4 , n=3, 4, …
  29. [ 0 , 1 ] [0,1]
  30. [ 0 , 1 π - 1 n ] and [ 1 π + 1 n , 1 ] \left[0,\frac{1}{\pi}-\frac{1}{n}\right]\ \,\text{and}\ \left[\frac{1}{\pi}+% \frac{1}{n},1\right]
  31. n = 4 , 5 , n=4, 5, …
  32. ( n 1 , n + 1 ) (n−1,n+1)
  33. n n
  34. n n
  35. n n
  36. { f n } \{f_{n}\}
  37. f K f\in K
  38. { f n ( x ) } \{f_{n}(x)\}
  39. [ 0 , 1 ] [0,1]
  40. [ 0 , 1 ] [0,1]
  41. [ 0 , 1 ] [0,1]
  42. d ( f , g ) = sup x [ 0 , 1 ] | f ( x ) - g ( x ) | . d(f,g)=\sup_{x\in[0,1]}|f(x)-g(x)|.
  43. 2 \ell^{2}

Comparative_advantage.html

  1. 5 6 \frac{5}{6}
  2. 9 8 \frac{9}{8}
  3. 5 6 \frac{5}{6}
  4. 9 8 \frac{9}{8}
  5. L \textstyle L
  6. a L W \textstyle a_{LW}
  7. a L C \textstyle a_{LC}
  8. Q W Q_{W}
  9. Q C Q_{C}
  10. a L W \textstyle a^{\prime}_{LW}
  11. a L C < a L C a_{LC}<a^{\prime}_{LC}
  12. a L C / a L C < a L W / a L W . a_{LC}/a^{\prime}_{LC}<a_{LW}/a^{\prime}_{LW}.
  13. a L C / a L W < a L C / a L W . a_{LC}/a_{LW}<a^{\prime}_{LC}/a^{\prime}_{LW}.
  14. a L C / a L W a_{LC}/a_{LW}
  15. a L C / a L W a^{\prime}_{LC}/a^{\prime}_{LW}
  16. P C P_{C}
  17. P W P_{W}
  18. P C / P W \textstyle P_{C}/P_{W}
  19. R D \textstyle RD
  20. R S \textstyle RS
  21. P C / P W = a L C / a L W < a L C / a L W \textstyle P_{C}/P_{W}=a_{LC}/a_{LW}<a^{\prime}_{LC}/a^{\prime}_{LW}
  22. P W / a L W P^{\prime}_{W}/a^{\prime}_{LW}
  23. P C / a L C P^{\prime}_{C}/a^{\prime}_{LC}
  24. P C / P W < a L C / a L W < a L C / a L W \textstyle P_{C}/P_{W}<a_{LC}/a_{LW}<a^{\prime}_{LC}/a^{\prime}_{LW}
  25. a L C / a L W < P C / P W < a L C / a L W \textstyle a_{LC}/a_{LW}<P_{C}/P_{W}<a^{\prime}_{LC}/a^{\prime}_{LW}
  26. L / a L C L / a L W \textstyle\frac{L/a_{LC}}{L^{\prime}/a^{\prime}_{LW}}
  27. a L C / a L W < a L C / a L W < P C / P W \textstyle a_{LC}/a_{LW}<a^{\prime}_{LC}/a^{\prime}_{LW}<P_{C}/P_{W}
  28. a L C / a L W < a L C / a L W = P C / P W \textstyle a_{LC}/a_{LW}<a^{\prime}_{LC}/a^{\prime}_{LW}=P_{C}/P_{W}
  29. a L C / a L W P C / P W a L C / a L W . a_{LC}/a_{LW}\leq{P_{C}/P_{W}}\leq{a^{\prime}_{LC}/a^{\prime}_{LW}}.
  30. a L C Q C + a L W Q W L , a_{LC}Q_{C}+a_{LW}Q_{W}\leq L,
  31. Q C = L / a L C - ( a L W / a L C ) Q W Q_{C}=L/a_{LC}-(a_{LW}/a_{LC})Q_{W}
  32. a L C Q C + a L C ( P W / P C ) Q W L a_{LC}Q_{C}+a_{LC}(P_{W}/P_{C})Q_{W}\leq L
  33. Q C = L / a L C - ( P W / P C ) Q W L / a L C - ( a L W / a L C ) Q W Q_{C}=L/a_{LC}-(P_{W}/P_{C})Q_{W}\geq L/a_{LC}-(a_{LW}/a_{LC})Q_{W}

Comparator.html

  1. V + V_{+}\,
  2. V - V_{-}\,
  3. V o V_{o}\,
  4. V o = { 1 , if V + > V - 0 , if V + < V - V_{o}=\begin{cases}1,&\mbox{if }~{}V_{+}>V_{-}\\ 0,&\mbox{if }~{}V_{+}<V_{-}\end{cases}
  5. V S - V + , V - V S + V_{S-}\leq V_{+},V_{-}\leq V_{S+}
  6. 0 V + , V - V c c 0\leq V_{+},V_{-}\leq V_{cc}
  7. V o u t = A o ( V 1 - V 2 ) V_{out}=A_{o}(V_{1}-V_{2})

Compass-and-straightedge_construction.html

  1. π \pi
  2. x + y k x+y{\sqrt{k}}
  3. 2 3 \sqrt[3]{2}
  4. cos ( 2 π 17 ) = - 1 16 + 1 16 17 + 1 16 34 - 2 17 + 1 8 17 + 3 17 - 34 - 2 17 - 2 34 + 2 17 \cos{\left(\frac{2\pi}{17}\right)}=-\frac{1}{16}\;+\;\frac{1}{16}\sqrt{17}\;+% \;\frac{1}{16}\sqrt{34-2\sqrt{17}}\;+\;\frac{1}{8}\sqrt{17+3\sqrt{17}-\sqrt{34% -2\sqrt{17}}-2\sqrt{34+2\sqrt{17}}}
  5. Re ( z ) = z + z ¯ 2 \mathrm{Re}(z)=\frac{z+\bar{z}}{2}\;
  6. Im ( z ) = z - z ¯ 2 i \mathrm{Im}(z)=\frac{z-\bar{z}}{2i}\;
  7. | z | = z z ¯ . \left|z\right|=\sqrt{z\bar{z}}.\;
  8. π {\sqrt{\pi}}

Compass.html

  1. a 0 a_{0}
  2. a 1 , b 1 a_{1},b_{1}
  3. a 2 , b 2 a_{2},b_{2}

Complement_(set_theory).html

  1. B A c = B A B\cap A^{c}~{}~{}~{}~{}=~{}~{}~{}~{}B\setminus A
  2. B A = { x B | x A } . B\setminus A=\{x\in B\,|\,x\notin A\}.
  3. \mathbb{R}
  4. \mathbb{Q}
  5. = 𝕀 \mathbb{R}\setminus\mathbb{Q}=\mathbb{I}
  6. A A
  7. U U
  8. A c = U A A^{c}=U\setminus A
  9. U A \complement_{U}A
  10. A \complement A
  11. ( A B ) c = A c B c . \left(A\cup B\right)^{c}=A^{c}\cap B^{c}.
  12. ( A B ) c = A c B c . \left(A\cap B\right)^{c}=A^{c}\cup B^{c}.
  13. A A c = U . A\cup A^{c}=U.
  14. A A c = . A\cap A^{c}=.
  15. = c U . {}^{c}=U.
  16. U c = . U^{c}=.
  17. If A B , then B c A c . \,\text{If }A\subset B\,\text{, then }B^{c}\subset A^{c}.
  18. ( A c ) c = A . \left(A^{c}\right)^{c}=A.

Complete_lattice.html

  1. A \bigwedge A
  2. A \bigvee A
  3. \bigwedge
  4. \bigvee
  5. { y L | y x } \{y\in L~{}|~{}y\leq x\}\,\!
  6. 1 x L 1\neq x\in L
  7. f ( A ) = { f ( a ) a A } f(\bigwedge A)=\bigwedge\{f(a)\mid a\in A\}
  8. f ( A ) = { f ( a ) a A } f(\bigvee A)=\bigvee\{f(a)\mid a\in A\}
  9. \bigvee

Complete_measure.html

  1. S N Σ and μ ( N ) = 0 S Σ . S\subseteq N\in\Sigma\mbox{ and }~{}\mu(N)=0\ \Rightarrow\ S\in\Sigma.
  2. λ 2 ( { 0 } × A ) = λ ( { 0 } ) λ ( A ) = 0 \lambda^{2}(\{0\}\times A)=\lambda(\{0\})\cdot\lambda(A)=0
  3. { 0 } × A { 0 } × , \{0\}\times A\subseteq\{0\}\times\mathbb{R},
  4. μ 0 ( C ) := inf { μ ( D ) | C D Σ } . \mu_{0}(C):=\inf\{\mu(D)|C\subseteq D\in\Sigma\}.
  5. μ 0 ( A B ) = μ ( A ) . \mu_{0}(A\cup B)=\mu(A).

Complete_metric_space.html

  1. 2 \sqrt{2}
  2. x 1 = 1 \scriptstyle x_{1}\;=\;1
  3. x n + 1 = x n 2 + 1 x n \scriptstyle x_{n+1}\;=\;\frac{x_{n}}{2}\,+\,\frac{1}{x_{n}}
  4. 2 \sqrt{2}
  5. ( 0 , 1 ) (0,1)
  6. 1 n \frac{1}{n}
  7. [ 0 , 1 ] [0,1]
  8. [ a , b ] [a,b]
  9. ( a , b ) (a,b)
  10. ( a , b ) (a,b)
  11. ( a , b ) (a,b)
  12. 1 N \frac{1}{N}
  13. d ( f , g ) sup { d [ f ( x ) , g ( x ) ] : x X } d(f,g)\equiv\sup\left\{d[f(x),g(x)]:x\in X\right\}
  14. μ \scriptstyle\mu
  15. { B ¯ ( x α , r α ) } \scriptstyle\left\{\overline{B}(x_{\alpha},\,r_{\alpha})\right\}
  16. α B ¯ ( x α , μ r α ) \bigcap_{\alpha}\overline{B}(x_{\alpha},\mu r_{\alpha})
  17. M ¯ \overline{M}
  18. d ( x , y ) = lim n d ( x n , y n ) d(x,y)=\lim_{n}d\left(x_{n},y_{n}\right)
  19. ( 0 , 1 ) (0,1)

Complex_analysis.html

  1. z = x + i y z=x+iy\,
  2. w = f ( z ) = u ( x , y ) + i v ( x , y ) w=f(z)=u(x,y)+iv(x,y)\,
  3. x , y x,y\in\mathbb{R}\,
  4. u ( x , y ) , v ( x , y ) u(x,y),v(x,y)\,
  5. u = u ( x , y ) u=u(x,y)\,
  6. v = v ( x , y ) , v=v(x,y),\,

Complex_number.html

  1. ( a , b ) (a,b)
  2. i i
  3. a + b i a+bi
  4. a a
  5. b b
  6. i i
  7. a a
  8. b b
  9. a + b i a+bi
  10. ( a , b ) (a,b)
  11. ( x + 1 ) 2 = - 9 (x+1)^{2}=-9\,
  12. i i
  13. 1 + 3 i −1+3i
  14. 1 3 i −1−3i
  15. ( ( - 1 + 3 i ) + 1 ) 2 = ( 3 i ) 2 = ( 3 2 ) ( i 2 ) = 9 ( - 1 ) = - 9 , ((-1+3i)+1)^{2}=(3i)^{2}=(3^{2})(i^{2})=9(-1)=-9,
  16. ( ( - 1 - 3 i ) + 1 ) 2 = ( - 3 i ) 2 = ( - 3 ) 2 ( i 2 ) = 9 ( - 1 ) = - 9. ((-1-3i)+1)^{2}=(-3i)^{2}=(-3)^{2}(i^{2})=9(-1)=-9.
  17. z = x + i y z=x+iy
  18. x x
  19. y y
  20. a + b i a+bi
  21. a a
  22. b b
  23. i i
  24. 3.5 + 2 i −3.5+2i
  25. a a
  26. a + b i a+bi
  27. b b
  28. a + b i a+bi
  29. b b
  30. b i bi
  31. z z
  32. R e ( z ) Re(z)
  33. ( z ) ℜ(z)
  34. z z
  35. I m ( z ) Im(z)
  36. ( z ) ℑ(z)
  37. Re ( - 3.5 + 2 i ) = - 3.5 Im ( - 3.5 + 2 i ) = 2. \begin{aligned}\displaystyle\operatorname{Re}(-3.5+2i)&\displaystyle=-3.5\\ \displaystyle\operatorname{Im}(-3.5+2i)&\displaystyle=2.\end{aligned}
  38. z z
  39. Re ( z ) + Im ( z ) i \operatorname{Re}(z)+\operatorname{Im}(z)\cdot i
  40. z z
  41. a a
  42. a + 0 i a+0i
  43. b i bi
  44. 0 + b i 0+bi
  45. a a
  46. a + 0 i a+0i
  47. b i bi
  48. 0 + b i 0+bi
  49. a b i a−bi
  50. b > 0 b>0
  51. a + ( b ) i a+(−b)i
  52. 3 4 i 3−4i
  53. 3 + ( 4 ) i 3+(−4)i
  54. 𝐂 \mathbf{C}
  55. \mathbb{C}
  56. a + i b a+ib
  57. a + b i a+bi
  58. j j
  59. i i
  60. i i
  61. a + b j a+bj
  62. a + j b a+jb
  63. i i
  64. ( a + b i ) i (a+bi)i
  65. a i + b i ai+bi
  66. - b + a i -b+ai
  67. z 1 = z 2 ( Re ( z 1 ) = Re ( z 2 ) and Im ( z 1 ) = Im ( z 2 ) ) . z_{1}=z_{2}\,\,\leftrightarrow\,\,(\operatorname{Re}(z_{1})=\operatorname{Re}(% z_{2})\,\and\,\operatorname{Im}(z_{1})=\operatorname{Im}(z_{2})).
  68. 0
  69. z = x + y i z=x+yi
  70. x y i x−yi
  71. z ¯ \bar{z}
  72. z * z*
  73. z ¯ = Re ( z ) - Im ( z ) i . \bar{z}=\operatorname{Re}(z)-\operatorname{Im}(z)\cdot i.
  74. z ¯ \bar{z}
  75. z z
  76. z ¯ ¯ = z \bar{\bar{z}}=z
  77. z z
  78. Re ( z ) = 1 2 ( z + z ¯ ) , \operatorname{Re}\,(z)=\tfrac{1}{2}(z+\bar{z}),\,
  79. Im ( z ) = 1 2 i ( z - z ¯ ) . \operatorname{Im}\,(z)=\tfrac{1}{2i}(z-\bar{z}).\,
  80. z + w ¯ = z ¯ + w ¯ , \overline{z+w}=\bar{z}+\bar{w},\,
  81. z - w ¯ = z ¯ - w ¯ , \overline{z-w}=\bar{z}-\bar{w},\,
  82. z w ¯ = z ¯ w ¯ , \overline{zw}=\bar{z}\bar{w},\,
  83. ( z / w ) ¯ = z ¯ / w ¯ . \overline{(z/w)}=\bar{z}/\bar{w}.\,
  84. z = x + y i z=x+yi
  85. 1 z = z ¯ z z ¯ = z ¯ x 2 + y 2 . \frac{1}{z}=\frac{\bar{z}}{z\bar{z}}=\frac{\bar{z}}{x^{2}+y^{2}}.
  86. ( a + b i ) + ( c + d i ) = ( a + c ) + ( b + d ) i . (a+bi)+(c+di)=(a+c)+(b+d)i.
  87. ( a + b i ) - ( c + d i ) = ( a - c ) + ( b - d ) i . (a+bi)-(c+di)=(a-c)+(b-d)i.
  88. ( a + b i ) ( c + d i ) = ( a c - b d ) + ( b c + a d ) i . (a+bi)(c+di)=(ac-bd)+(bc+ad)i.
  89. i 2 = i × i = - 1. i^{2}=i\times i=-1.
  90. i i
  91. d i di
  92. d d
  93. i i
  94. ( a + b i ) ( c + d i ) = a c + b c i + a d i + b i d i (a+bi)(c+di)=ac+bci+adi+bidi
  95. = a c + b i d i + b c i + a d i =ac+bidi+bci+adi
  96. = a c + b d i 2 + ( b c + a d ) i =ac+bdi^{2}+(bc+ad)i
  97. = ( a c - b d ) + ( b c + a d ) i =(ac-bd)+(bc+ad)i
  98. c c
  99. d d
  100. a + b i c + d i = ( a c + b d c 2 + d 2 ) + ( b c - a d c 2 + d 2 ) i . \,\frac{a+bi}{c+di}=\left({ac+bd\over c^{2}+d^{2}}\right)+\left({bc-ad\over c^% {2}+d^{2}}\right)i.
  101. a + b i c + d i = ( a + b i ) ( c - d i ) ( c + d i ) ( c - d i ) = ( a c + b d c 2 + d 2 ) + ( b c - a d c 2 + d 2 ) i . \,\frac{a+bi}{c+di}=\frac{\left(a+bi\right)\cdot\left(c-di\right)}{\left(c+di% \right)\cdot\left(c-di\right)}=\left({ac+bd\over c^{2}+d^{2}}\right)+\left({bc% -ad\over c^{2}+d^{2}}\right)i.
  102. c d i c−di
  103. c + d i c+di
  104. c c
  105. d d
  106. a + b i a+bi
  107. b 0 b≠0
  108. ± ( γ + δ i ) \pm(\gamma+\delta i)
  109. γ = a + a 2 + b 2 2 \gamma=\sqrt{\frac{a+\sqrt{a^{2}+b^{2}}}{2}}
  110. δ = sgn ( b ) - a + a 2 + b 2 2 , \delta=\operatorname{sgn}(b)\sqrt{\frac{-a+\sqrt{a^{2}+b^{2}}}{2}},
  111. ± ( γ + δ i ) \pm(\gamma+\delta i)
  112. a + b i a+bi
  113. a 2 + b 2 \sqrt{a^{2}+b^{2}}
  114. a + b i a+bi
  115. a 2 + b 2 = z z ¯ \sqrt{a^{2}+b^{2}}=\sqrt{z\bar{z}}
  116. z = a + b i z=a+bi
  117. ( 0 , 0 ) (0, 0)
  118. z = x + y i z=x+yi
  119. r = | z | = x 2 + y 2 . \textstyle r=|z|=\sqrt{x^{2}+y^{2}}.\,
  120. z z
  121. y = 0 y=0
  122. r = | x | r=|x|
  123. r r
  124. z z
  125. | z | 2 = z z ¯ = x 2 + y 2 . \textstyle|z|^{2}=z\bar{z}=x^{2}+y^{2}.\,
  126. z ¯ \bar{z}
  127. z z
  128. z z
  129. arg ( z ) \arg(z)
  130. x + y i x+yi
  131. φ = arg ( z ) = { arctan ( y x ) if x > 0 arctan ( y x ) + π if x < 0 and y 0 arctan ( y x ) - π if x < 0 and y < 0 π 2 if x = 0 and y > 0 - π 2 if x = 0 and y < 0 indeterminate if x = 0 and y = 0. \varphi=\arg(z)=\begin{cases}\arctan(\frac{y}{x})&\mbox{if }~{}x>0\\ \arctan(\frac{y}{x})+\pi&\mbox{if }~{}x<0\mbox{ and }~{}y\geq 0\\ \arctan(\frac{y}{x})-\pi&\mbox{if }~{}x<0\mbox{ and }~{}y<0\\ \frac{\pi}{2}&\mbox{if }~{}x=0\mbox{ and }~{}y>0\\ -\frac{\pi}{2}&\mbox{if }~{}x=0\mbox{ and }~{}y<0\\ \mbox{indeterminate }&\mbox{if }~{}x=0\mbox{ and }~{}y=0.\end{cases}
  132. φ φ
  133. 2 π
  134. ( π , π ] (−π,π]
  135. [ 0 , 2 π ) [0,2π)
  136. 2 π
  137. φ φ
  138. φ = atan2 ( imaginary , real ) \varphi=\mbox{atan2}~{}(\mbox{imaginary}~{},\mbox{real}~{})
  139. r r
  140. φ φ
  141. z = r ( cos φ + i sin φ ) . z=r(\cos\varphi+i\sin\varphi).\,
  142. z = r e i φ . z=re^{i\varphi}.\,
  143. z = r cis φ . z=r\operatorname{cis}\varphi.\,
  144. r r
  145. φ φ
  146. z = r \ang φ . z=r\ang\varphi.\,
  147. cos ( a ) cos ( b ) - sin ( a ) sin ( b ) = cos ( a + b ) \cos(a)\cos(b)-\sin(a)\sin(b)=\cos(a+b)
  148. cos ( a ) sin ( b ) + sin ( a ) cos ( b ) = sin ( a + b ) \cos(a)\sin(b)+\sin(a)\cos(b)=\sin(a+b)
  149. z 1 z 2 = r 1 r 2 ( cos ( φ 1 + φ 2 ) + i sin ( φ 1 + φ 2 ) ) . z_{1}z_{2}=r_{1}r_{2}(\cos(\varphi_{1}+\varphi_{2})+i\sin(\varphi_{1}+\varphi_% {2})).\,
  150. i i
  151. ( 2 + i ) ( 3 + i ) = 5 + 5 i . (2+i)(3+i)=5+5i.\,
  152. 5 + 5 i 5+5i
  153. π 4 = arctan 1 2 + arctan 1 3 \frac{\pi}{4}=\arctan\frac{1}{2}+\arctan\frac{1}{3}
  154. z 1 z 2 = r 1 r 2 ( cos ( φ 1 - φ 2 ) + i sin ( φ 1 - φ 2 ) ) . \frac{z_{1}}{z_{2}}=\frac{r_{1}}{r_{2}}\left(\cos(\varphi_{1}-\varphi_{2})+i% \sin(\varphi_{1}-\varphi_{2})\right).
  155. e i x = cos x + i sin x e^{ix}=\cos x+i\sin x
  156. i 0 \displaystyle i^{0}
  157. e i x \displaystyle e^{ix}
  158. z = r ( cos φ + i sin φ ) . z=r(\cos\varphi+i\sin\varphi).\,
  159. ln ( z ) = ln ( r ) + φ i \ln(z)=\ln(r)+\varphi i
  160. ln ( z ) = { ln ( r ) + ( φ + 2 π k ) i | k } \ln(z)=\left\{\ln(r)+(\varphi+2\pi k)i\;|\;k\in\mathbb{Z}\right\}
  161. ln ( a b ) = b ln ( a ) \ln(a^{b})=b\ln(a)
  162. ln ( z n ) = ln ( ( r ( cos φ + i sin φ ) ) n ) \ln(z^{n})=\ln((r(\cos\varphi+i\sin\varphi))^{n})
  163. = n ln ( r ( cos φ + i sin φ ) ) =n\ln(r(\cos\varphi+i\sin\varphi))
  164. = { n ( ln ( r ) + ( φ + k 2 π ) i ) | k } =\{n(\ln(r)+(\varphi+k2\pi)i)|k\in\mathbb{Z}\}
  165. = { n ln ( r ) + n φ i + n k 2 π i | k } . =\{n\ln(r)+n\varphi i+nk2\pi i|k\in\mathbb{Z}\}.
  166. z n = ( r ( cos φ + i sin φ ) ) n = r n ( cos n φ + i sin n φ ) . z^{n}=(r(\cos\varphi+i\sin\varphi))^{n}=r^{n}\,(\cos n\varphi+i\sin n\varphi).
  167. n n
  168. z z
  169. z n = r n ( cos ( φ + 2 k π n ) + i sin ( φ + 2 k π n ) ) \sqrt[n]{z}=\sqrt[n]{r}\left(\cos\left(\frac{\varphi+2k\pi}{n}\right)+i\sin% \left(\frac{\varphi+2k\pi}{n}\right)\right)
  170. k k
  171. 0 k n 1 0≤k≤n−1
  172. r n \sqrt{rn}
  173. n n
  174. r r
  175. n n
  176. r r
  177. c c
  178. n n
  179. n n
  180. z z
  181. z z
  182. f f
  183. f ( z ) f(z)
  184. z n n = z \sqrt[n]{z^{n}}=z
  185. z z
  186. z −z
  187. z 1 + z 2 = z 2 + z 1 , z_{1}+z_{2}=z_{2}+z_{1},
  188. z 1 z 2 = z 2 z 1 . z_{1}z_{2}=z_{2}z_{1}.
  189. a n z n + + a 1 z + a 0 = 0 a_{n}z^{n}+\cdots+a_{1}z+a_{0}=0
  190. 2 \sqrt{2}
  191. a > 0 a>0
  192. x x
  193. x x
  194. 1 + 1 + + 1 0 1+1+⋯+1≠0
  195. P P
  196. P P
  197. x x
  198. y y
  199. P P
  200. x y x−y
  201. y x y−x
  202. P P
  203. S S
  204. P P
  205. S + P = x + P S+P=x+P
  206. x x
  207. x x * x↦x*
  208. x x * x x*
  209. P P
  210. x x
  211. F F
  212. x x
  213. p p
  214. P P
  215. F F
  216. i i
  217. 𝐂 \mathbf{C}
  218. ( a , b ) (a,b)
  219. ( a , b ) + ( c , d ) \displaystyle(a,b)+(c,d)
  220. ( a , b ) (a,b)
  221. a + b i a+bi
  222. 𝐂 \mathbf{C}
  223. ( x + y ) z = x z + y z (x+y)z=xz+yz
  224. x x
  225. y y
  226. z z
  227. 𝐑 \mathbf{R}
  228. p ( X ) p(X)
  229. a n X n + + a 1 X + a 0 a_{n}X^{n}+\cdots+a_{1}X+a_{0}
  230. 𝐑 X X \mathbf{R}XX
  231. 1 −1
  232. X X
  233. X −X
  234. 1 1
  235. X X
  236. ( a , b ) (a,b)
  237. 𝐂 \mathbf{C}
  238. 𝐂 \mathbf{C}
  239. 𝐂 \mathbf{C}
  240. 𝐑 \mathbf{R}
  241. a + b i a+bi
  242. 2 × 2 2 × 2
  243. ( a - b b a ) . \begin{pmatrix}a&-b\\ b&\;\;a\end{pmatrix}.
  244. a a
  245. b b
  246. | z | 2 = | a - b b a | = ( a 2 ) - ( ( - b ) ( b ) ) = a 2 + b 2 . |z|^{2}=\begin{vmatrix}a&-b\\ b&a\end{vmatrix}=(a^{2})-((-b)(b))=a^{2}+b^{2}.
  247. z ¯ \overline{z}
  248. ( 0 - 1 1 0 ) \bigl(\begin{smallmatrix}0&-1\\ 1&0\end{smallmatrix}\bigr)
  249. d ( z 1 , z 2 ) = | z 1 - z 2 | \operatorname{d}(z_{1},z_{2})=|z_{1}-z_{2}|\,
  250. | z 1 + z 2 | | z 1 | + | z 2 | |z_{1}+z_{2}|\leq|z_{1}|+|z_{2}|
  251. e x p ( z ) exp(z)
  252. exp ( z ) := 1 + z + z 2 2 1 + z 3 3 2 1 + = n = 0 z n n ! . \exp(z):=1+z+\frac{z^{2}}{2\cdot 1}+\frac{z^{3}}{3\cdot 2\cdot 1}+\cdots=\sum_% {n=0}^{\infty}\frac{z^{n}}{n!}.\,
  253. exp ( i φ ) = cos ( φ ) + i sin ( φ ) \exp(i\varphi)=\cos(\varphi)+i\sin(\varphi)\,
  254. exp ( i π ) = - 1 \exp(i\pi)=-1\,
  255. z z
  256. exp ( z ) = w \exp(z)=w\,
  257. w 0 w≠0
  258. z z
  259. a a
  260. log ( x + i y ) = ln | w | + i arg ( w ) , \log(x+iy)=\ln|w|+i\arg(w),\,
  261. ( π , π ] (−π,π]
  262. z ω = exp ( ω log z ) . z^{\omega}=\exp(\omega\log z).\,
  263. ω = 1 / n ω=1/n
  264. n n
  265. n n
  266. a b c = ( a b ) c . \,a^{bc}=(a^{b})^{c}.
  267. f ( z ) = a z + b z ¯ f(z)=az+b\overline{z}
  268. a a
  269. b b
  270. b = 0 b=0
  271. b z ¯ b\overline{z}
  272. f f
  273. g g
  274. f f
  275. s i n ( 1 / z ) sin(1/z)
  276. z = 0 z=0
  277. r r
  278. r r
  279. j j
  280. I I
  281. i i
  282. V ( t ) = V 0 e j ω t = V 0 ( cos ω t + j sin ω t ) , V(t)=V_{0}e^{j\omega t}=V_{0}\left(\cos\omega t+j\sin\omega t\right),
  283. v ( t ) = Re ( V ) = Re [ V 0 e j ω t ] = V 0 cos ω t . v(t)=\mathrm{Re}(V)=\mathrm{Re}\left[V_{0}e^{j\omega t}\right]=V_{0}\cos\omega t.
  284. V ( t ) V(t)
  285. v ( t ) v(t)
  286. | z | |z|
  287. z z
  288. a r g ( z ) arg(z)
  289. x ( t ) = R e { X ( t ) } x(t)=Re\{X(t)\}\,
  290. X ( t ) = A e i ω t = a e i ϕ e i ω t = a e i ( ω t + ϕ ) X(t)=Ae^{i\omega t}=ae^{i\phi}e^{i\omega t}=ae^{i(\omega t+\phi)}\,
  291. cos ( ( ω + α ) t ) + cos ( ( ω - α ) t ) \displaystyle\cos((\omega+\alpha)t)+\cos\left((\omega-\alpha)t\right)
  292. ( x - a ) ( x - b ) ( x - c ) = 0 \scriptstyle(x-a)(x-b)(x-c)=0
  293. 𝐐 ¯ \overline{\mathbf{Q}}
  294. x + i y x+iy
  295. x x
  296. y y
  297. ζ ( s ) ζ(s)
  298. 81 - 144 = 3 i 7 \scriptstyle\sqrt{81-144}=3i\sqrt{7}
  299. 144 - 81 = 3 7 \scriptstyle\sqrt{144-81}=3\sqrt{7}
  300. x 3 = p x + q \scriptstyle x^{3}=px+q
  301. 1 3 ( ( - 1 ) 1 / 3 + 1 ( - 1 ) 1 / 3 ) . \frac{1}{\sqrt{3}}\left((\sqrt{-1})^{1/3}+\frac{1}{(\sqrt{-1})^{1/3}}\right).
  302. i −i
  303. 3 2 + 1 2 i {\scriptstyle\frac{\sqrt{3}}{2}}+{\scriptstyle\frac{1}{2}}i
  304. - 3 2 + 1 2 i {\scriptstyle\frac{-\sqrt{3}}{2}}+{\scriptstyle\frac{1}{2}}i
  305. - 1 1 / 3 {\scriptstyle\sqrt{-1}^{1/3}}
  306. - 1 2 = - 1 - 1 = - 1 \scriptstyle\sqrt{-1}^{2}=\sqrt{-1}\sqrt{-1}=-1
  307. a b = a b \scriptstyle\sqrt{a}\sqrt{b}=\sqrt{ab}
  308. a a
  309. b b
  310. a a
  311. b b
  312. 1 a = 1 a \scriptstyle\frac{1}{\sqrt{a}}=\sqrt{\frac{1}{a}}
  313. a a
  314. b b
  315. i i
  316. 1 \sqrt{−1}
  317. ( cos θ + i sin θ ) n = cos n θ + i sin n θ . (\cos\theta+i\sin\theta)^{n}=\cos n\theta+i\sin n\theta.\,
  318. cos θ + i sin θ = e i θ \cos\theta+i\sin\theta=e^{i\theta}\,
  319. cos ϕ + i sin ϕ \scriptstyle\cos\phi+i\sin\phi
  320. r = a 2 + b 2 \scriptstyle r=\sqrt{a^{2}+b^{2}}
  321. cos ϕ + i sin ϕ \cos\phi+i\sin\phi
  322. i i
  323. - 1 \scriptstyle\sqrt{-1}
  324. a + b i a+bi
  325. cos ϕ + i sin ϕ \cos\phi+i\sin\phi
  326. x , y x,y
  327. x · y y · x x·y≠y·x
  328. x , y , z x,y,z
  329. ( x · y ) · z x · ( y · z ) (x·y)·z≠x·(y·z)
  330. ( 1 , i ) (1,i)
  331. , z w z \mathbb{C}\rightarrow\mathbb{C},z\mapsto wz
  332. w w
  333. 2 × 2 2 × 2
  334. ( 1 , i ) (1,i)
  335. ( Re ( w ) - Im ( w ) Im ( w ) Re ( w ) ) \begin{pmatrix}\operatorname{Re}(w)&-\operatorname{Im}(w)\\ \operatorname{Im}(w)&\;\;\operatorname{Re}(w)\end{pmatrix}
  336. J = ( p q r - p ) , p 2 + q r + 1 = 0 J=\begin{pmatrix}p&q\\ r&-p\end{pmatrix},\quad p^{2}+qr+1=0
  337. { z = a I + b J : a , b R } \{z=aI+bJ:a,b\in R\}
  338. 𝐐 p ¯ \overline{\mathbf{Q}_{p}}
  339. 𝐂 p \mathbf{C}_{p}
  340. 𝐐 p ¯ \overline{\mathbf{Q}_{p}}
  341. ( u 3 + v 3 ) 3 = 3 u v 3 ( u 3 + v 3 ) + u + v \scriptstyle\left(\sqrt[3]{u}+\sqrt[3]{v}\right)^{3}=3\sqrt[3]{uv}\left(\sqrt[% 3]{u}+\sqrt[3]{v}\right)+u+v
  342. x = u 3 + v 3 \scriptstyle x=\sqrt[3]{u}+\sqrt[3]{v}
  343. p = 3 u v 3 \scriptstyle p=3\sqrt[3]{uv}
  344. q = u + v \scriptstyle q=u+v
  345. u u
  346. v v
  347. p p
  348. q q
  349. u = q / 2 + ( q / 2 ) 2 - ( p / 3 ) 3 \scriptstyle u=q/2+\sqrt{(q/2)^{2}-(p/3)^{3}}
  350. v = q / 2 - ( q / 2 ) 2 - ( p / 3 ) 3 \scriptstyle v=q/2-\sqrt{(q/2)^{2}-(p/3)^{3}}
  351. x = q / 2 + ( q / 2 ) 2 - ( p / 3 ) 3 3 + q / 2 - ( q / 2 ) 2 - ( p / 3 ) 3 3 \scriptstyle x=\sqrt[3]{q/2+\sqrt{(q/2)^{2}-(p/3)^{3}}}+\sqrt[3]{q/2-\sqrt{(q/% 2)^{2}-(p/3)^{3}}}
  352. ( q / 2 ) 2 - ( p / 3 ) 3 \scriptstyle(q/2)^{2}-(p/3)^{3}

Complexity.html

  1. O ( n 2 2 n ) O(n^{2}2^{n})

Compression_ratio.html

  1. CR = π 4 b 2 s + V c V c \mbox{CR}~{}=\frac{\tfrac{\pi}{4}b^{2}s+V_{c}}{V_{c}}
  2. b b\;
  3. s s\;
  4. V c V_{c}\;
  5. p = p 0 × CR γ p=p_{0}\times\,\text{CR}^{\gamma}
  6. p 0 p_{0}\;
  7. CR \,\text{CR}
  8. γ \gamma\;
  9. p TDC = 1 bar × 10 1.4 = 25.1 bar p\text{TDC}=1\,\text{ bar}\times 10^{1.4}=25.1\,\text{ bar}
  10. CR = V 1 V 2 \,\text{CR}=\frac{V_{1}}{V_{2}}
  11. PR = P 2 P 1 \,\text{PR}=\frac{P_{2}}{P_{1}}
  12. P 1 V 1 γ = P 2 V 2 γ P 2 P 1 = ( V 1 V 2 ) γ P_{1}V_{1}^{\gamma}=P_{2}V_{2}^{\gamma}\Rightarrow\frac{P_{2}}{P_{1}}=\left(% \frac{V_{1}}{V_{2}}\right)^{\gamma}
  13. γ \gamma

Compton_scattering.html

  1. 17 keV \approx 17\,\mathrm{keV}
  2. θ \theta
  3. θ \theta
  4. λ - λ = h m e c ( 1 - cos θ ) , \lambda^{\prime}-\lambda=\frac{h}{m_{e}c}(1-\cos{\theta}),
  5. λ \lambda
  6. λ \lambda^{\prime}
  7. h h
  8. m e m_{e}
  9. c c
  10. θ \theta
  11. γ \gamma
  12. λ \lambda
  13. e e
  14. γ \gamma^{\prime}
  15. λ {\lambda}^{\prime}
  16. θ \theta
  17. e e^{\prime}
  18. E = m c 2 E=mc^{2}
  19. h f hf
  20. m c 2 = h f mc^{2}=hf
  21. h f / c 2 hf/c^{2}
  22. c c
  23. p = h f / c p=hf/c
  24. h f hf
  25. p c pc
  26. E E
  27. E γ + E e = E γ + E e . E_{\gamma}+E_{e}=E_{\gamma^{\prime}}+E_{e^{\prime}}.\!
  28. 𝐩 γ = 𝐩 γ + 𝐩 e , \mathbf{p}_{\gamma}=\mathbf{p}_{\gamma^{\prime}}+\mathbf{p}_{e^{\prime}},
  29. p e {p_{e}}
  30. E γ = h f E_{\gamma}=hf\!
  31. E γ = h f E_{\gamma^{\prime}}=hf^{\prime}\!
  32. m e m_{e}
  33. E e = m e c 2 . E_{e}=m_{e}c^{2}.\!
  34. E e = ( p e c ) 2 + ( m e c 2 ) 2 . E_{e^{\prime}}=\sqrt{(p_{e^{\prime}}c)^{2}+(m_{e}c^{2})^{2}}.
  35. h f + m e c 2 = h f + ( p e c ) 2 + ( m e c 2 ) 2 . hf+m_{e}c^{2}=hf^{\prime}+\sqrt{(p_{e^{\prime}}c)^{2}+(m_{e}c^{2})^{2}}.
  36. p e 2 c 2 = ( h f - h f + m e c 2 ) 2 - m e 2 c 4 . ( 1 ) p_{e^{\prime}}^{\,2}c^{2}=(hf-hf^{\prime}+m_{e}c^{2})^{2}-m_{e}^{2}c^{4}.% \qquad\qquad(1)\!
  37. 1 c ( h f - h f + m e c 2 ) 2 - m e 2 c 4 > h f - h f c . \frac{1}{c}\sqrt{(hf-hf^{\prime}+m_{e}c^{2})^{2}-m_{e}^{2}c^{4}}>\frac{hf-hf^{% \prime}}{c}.
  38. 𝐩 e = 𝐩 γ - 𝐩 γ . \mathbf{p}_{e^{\prime}}=\mathbf{p}_{\gamma}-\mathbf{p}_{\gamma^{\prime}}.
  39. p e 2 = 𝐩 e 𝐩 e = ( 𝐩 γ - 𝐩 γ ) ( 𝐩 γ - 𝐩 γ ) = p γ 2 + p γ 2 - 2 p γ p γ cos θ . \begin{aligned}\displaystyle p_{e^{\prime}}^{\,2}&\displaystyle=\mathbf{p}_{e^% {\prime}}\cdot\mathbf{p}_{e^{\prime}}=(\mathbf{p}_{\gamma}-\mathbf{p}_{\gamma^% {\prime}})\cdot(\mathbf{p}_{\gamma}-\mathbf{p}_{\gamma^{\prime}})\\ &\displaystyle=p_{\gamma}^{\,2}+p_{\gamma^{\prime}}^{\,2}-2p_{\gamma}\,p_{% \gamma^{\prime}}\cos\theta.\end{aligned}
  40. p γ c p_{\gamma}c
  41. h f hf
  42. c 2 c^{2}
  43. p e 2 c 2 = p γ 2 c 2 + p γ 2 c 2 - 2 c 2 p γ p γ cos θ . p_{e^{\prime}}^{\,2}c^{2}=p_{\gamma}^{\,2}c^{2}+p_{\gamma^{\prime}}^{\,2}c^{2}% -2c^{2}p_{\gamma}\,p_{\gamma^{\prime}}\cos\theta.
  44. h f / c hf/c
  45. p e 2 c 2 = ( h f ) 2 + ( h f ) 2 - 2 ( h f ) ( h f ) cos θ . ( 2 ) p_{e^{\prime}}^{\,2}c^{2}=(hf)^{2}+(hf^{\prime})^{2}-2(hf)(hf^{\prime})\cos{% \theta}.\qquad\qquad(2)
  46. ( h f - h f + m e c 2 ) 2 - m e 2 c 4 = ( h f ) 2 + ( h f ) 2 - 2 h 2 f f cos θ , (hf-hf^{\prime}+m_{e}c^{2})^{2}-m_{e}^{\,2}c^{4}=\left(hf\right)^{2}+\left(hf^% {\prime}\right)^{2}-2h^{2}ff^{\prime}\cos{\theta},
  47. 2 h f m e c 2 - 2 h f m e c 2 = 2 h 2 f f ( 1 - cos θ ) . 2hfm_{e}c^{2}-2hf^{\prime}m_{e}c^{2}=2h^{2}ff^{\prime}\left(1-\cos\theta\right% ).\,
  48. c f - c f = h m e c ( 1 - cos θ ) . \frac{c}{f^{\prime}}-\frac{c}{f}=\frac{h}{m_{e}c}\left(1-\cos\theta\right).\,
  49. f λ = f λ = c , ~{}f\lambda=f^{\prime}\lambda^{\prime}=c,
  50. λ - λ = h m e c ( 1 - cos θ ) . \lambda^{\prime}-\lambda=\frac{h}{m_{e}c}(1-\cos{\theta}).\,
  51. J mag ( 𝐩 z ) J_{\,\text{mag}}(\mathbf{p}_{z})
  52. J mag ( 𝐩 z ) = 1 μ - ( n ( 𝐩 ) - n ( 𝐩 ) ) d 𝐩 x d 𝐩 y J_{\,\text{mag}}(\mathbf{p}_{z})=\frac{1}{\mu}\iint_{-\infty}^{\infty}(n_{% \uparrow}(\mathbf{p})-n_{\downarrow}(\mathbf{p}))d\mathbf{p}_{x}d\mathbf{p}_{y}
  53. μ \mu
  54. n ( 𝐩 ) n_{\uparrow}(\mathbf{p})
  55. n ( 𝐩 ) n_{\downarrow}(\mathbf{p})
  56. φ φ
  57. c o t φ = ( 1 + h f / ( m < s u b > e c 2 ) ) t a n ( θ / 2 ) cotφ=(1+hf/(m<sub>ec^{2}))tan(θ/2)

Computable_number.html

  1. f : f:\mathbb{N}\to\mathbb{Z}
  2. f ( n ) - 1 n a f ( n ) + 1 n . {f(n)-1\over n}\leq a\leq{f(n)+1\over n}.
  3. ε \varepsilon
  4. | r - a | ε . |r-a|\leq\varepsilon.
  5. q i q_{i}
  6. a a
  7. | q i - q i + 1 | < 2 - i |q_{i}-q_{i+1}|<2^{-i}\,
  8. D D\;
  9. r r
  10. D ( r ) = true D(r)=\mathrm{true}\;
  11. D ( r ) = false D(r)=\mathrm{false}\;
  12. r D ( r ) = true \exists rD(r)=\mathrm{true}\;
  13. r D ( r ) = false \exists rD(r)=\mathrm{false}\;
  14. ( D ( r ) = true ) ( D ( s ) = false ) r < s (D(r)=\mathrm{true})\wedge(D(s)=\mathrm{false})\Rightarrow r<s\;
  15. D ( r ) = true \exist s > r , D ( s ) = true . D(r)=\mathrm{true}\Rightarrow\exist s>r,D(s)=\mathrm{true}.\;
  16. q > 0 q>0\;
  17. p 3 < 3 q 3 D ( p / q ) = true p^{3}<3q^{3}\Rightarrow D(p/q)=\mathrm{true}\;
  18. p 3 > 3 q 3 D ( p / q ) = false . p^{3}>3q^{3}\Rightarrow D(p/q)=\mathrm{false}.\;
  19. S S
  20. S S
  21. x , x,
  22. S S
  23. x x
  24. S S S^{\prime}\subset S
  25. S S
  26. S S^{\prime}
  27. ϵ \epsilon
  28. ϵ \epsilon
  29. a a
  30. a > 0 a>0
  31. a 0 a\leq 0
  32. ϵ \epsilon
  33. a < b a<b
  34. a > b a>b
  35. ε < | b - a | / 2 \varepsilon<|b-a|/2
  36. a < b a<b
  37. a > b a>b
  38. Ω \Omega
  39. n 1 n\geq 1
  40. n n
  41. ϵ \epsilon
  42. ϵ \epsilon
  43. n > log 10 ( 1 / ϵ ) n>\log_{10}(1/\epsilon)
  44. ϵ \epsilon
  45. ϵ \epsilon
  46. 2 ω 2^{\omega}
  47. [ 0 , 1 ] [0,1]
  48. 2 ω 2^{\omega}
  49. . d 1 d 2 d n 0111 .d_{1}d_{2}\ldots d_{n}0111\ldots
  50. . d 1 d 2 d n 10 .d_{1}d_{2}\ldots d_{n}10
  51. [ 0 , 1 ] [0,1]
  52. 2 ω 2^{\omega}
  53. ϵ \epsilon
  54. ϵ \epsilon
  55. ϵ \epsilon
  56. 2 ω 2^{\omega}
  57. [ 0 , 1 ] [0,1]
  58. 2 ω 2^{\omega}
  59. [ 0 , 1 ] [0,1]
  60. 2 ω 2^{\omega}
  61. Π 1 0 \Pi^{0}_{1}
  62. 2 ω 2^{\omega}
  63. ω ω \omega^{\omega}
  64. \mathbb{R}
  65. ω ω \omega^{\omega}
  66. x x\in\mathbb{R}
  67. ( n ω ) ϕ ( x , n ) \forall(n\in\omega)\phi(x,n)
  68. ϕ ( x , n ) \phi(x,n)
  69. x ω ω x\in\omega^{\omega}
  70. π \pi

Computational_complexity_theory.html

  1. DTIME ( f ( n ) ) DTIME ( f ( n ) \sdot log 2 ( f ( n ) ) ) \operatorname{DTIME}\big(f(n)\big)\subsetneq\operatorname{DTIME}\big(f(n)\sdot% \log^{2}(f(n))\big)
  2. DSPACE ( f ( n ) ) DSPACE ( f ( n ) \sdot log ( f ( n ) ) ) \operatorname{DSPACE}\big(f(n)\big)\subsetneq\operatorname{DSPACE}\big(f(n)% \sdot\log(f(n))\big)

Computer_number_format.html

  1. b b
  2. 2 b = N 2^{b}=N
  3. octal 756 \,\text{octal }756
  4. = ( 7 * 8 2 ) + ( 5 * 8 1 ) + ( 6 * 8 0 ) =(7*8^{2})+(5*8^{1})+(6*8^{0})
  5. = ( 7 * 64 ) + ( 5 * 8 ) + ( 6 * 1 ) =(7*64)+(5*8)+(6*1)
  6. = 448 + 40 + 6 =448+40+6
  7. = decimal 494 =\,\text{decimal }494
  8. hex 3 b 2 \,\text{hex }3b2
  9. = ( 3 * 16 2 ) + ( 11 * 16 1 ) + ( 2 * 16 0 ) =(3*16^{2})+(11*16^{1})+(2*16^{0})
  10. = ( 3 * 256 ) + ( 11 * 16 ) + ( 2 * 1 ) =(3*256)+(11*16)+(2*1)
  11. = 768 + 176 + 2 =768+176+2
  12. = decimal 946 =\,\text{decimal }946
  13. 1 / 2 {1}/{2}
  14. 11 / 4 1{1}/{4}
  15. 73 / 8 7{3}/{8}
  16. 1 5 \tfrac{1}{5}
  17. 1 3 \tfrac{1}{3}
  18. 1 3 \tfrac{1}{3}

Computer_science.html

  1. O ( n 2 ) O(n^{2})
  2. Γ x : Int \Gamma\vdash x:\,\text{Int}

Concentration.html

  1. ρ i \rho_{i}
  2. m i m_{i}
  3. V V
  4. ρ i = m i V . \rho_{i}=\frac{m_{i}}{V}.
  5. c i c_{i}
  6. n i n_{i}
  7. V V
  8. c i = n i V . c_{i}=\frac{n_{i}}{V}.
  9. C i C_{i}
  10. N i N_{i}
  11. V V
  12. C i = N i V . C_{i}=\frac{N_{i}}{V}.
  13. ϕ i \phi_{i}
  14. V i V_{i}
  15. V V
  16. ϕ i = V i V . \phi_{i}=\frac{V_{i}}{V}.
  17. c i c_{i}
  18. f eq f_{\mathrm{eq}}
  19. b i b_{i}
  20. n i n_{i}
  21. m solvent m_{\mathrm{solvent}}
  22. b i = n i m solvent . b_{i}=\frac{n_{i}}{m_{\mathrm{solvent}}}.
  23. x i x_{i}
  24. n i n_{i}
  25. n tot n_{\mathrm{tot}}
  26. x i = n i n tot . x_{i}=\frac{n_{i}}{n_{\mathrm{tot}}}.
  27. r i r_{i}
  28. n i n_{i}
  29. r i = n i n tot - n i . r_{i}=\frac{n_{i}}{n_{\mathrm{tot}}-n_{i}}.
  30. n i n_{i}
  31. n tot n_{\mathrm{tot}}
  32. w i w_{i}
  33. m i m_{i}
  34. m tot m_{\mathrm{tot}}
  35. w i = m i m tot . w_{i}=\frac{m_{i}}{m_{\mathrm{tot}}}.
  36. ζ i \zeta_{i}
  37. m i m_{i}
  38. ζ i = m i m tot - m i . \zeta_{i}=\frac{m_{i}}{m_{\mathrm{tot}}-m_{i}}.
  39. m i m_{i}
  40. m tot m_{\mathrm{tot}}
  41. ρ i \rho_{i}
  42. γ i \gamma_{i}
  43. m i / V m_{i}/V
  44. c i c_{i}
  45. n i / V n_{i}/V
  46. C i C_{i}
  47. N i / V N_{i}/V
  48. ϕ i \phi_{i}
  49. V i / V V_{i}/V
  50. c i / f eq c_{i}/f_{\mathrm{eq}}
  51. b i b_{i}
  52. n i / m solvent n_{i}/m_{\mathrm{solvent}}
  53. x i x_{i}
  54. n i / n tot n_{i}/n_{\mathrm{tot}}
  55. r i r_{i}
  56. n i / ( n tot - n i ) n_{i}/(n_{\mathrm{tot}}-n_{i})
  57. w i w_{i}
  58. m i / m tot m_{i}/m_{\mathrm{tot}}
  59. ζ i \zeta_{i}
  60. m i / ( m tot - m i ) m_{i}/(m_{\mathrm{tot}}-m_{i})

Condition_number.html

  1. f ( x ) = y , f(x)=y,
  2. κ ( A ) = 10 k \kappa(A)=10^{k}
  3. k k
  4. A - 1 e / A - 1 b e / b . \frac{\left\|A^{-1}e\right\|/\left\|A^{-1}b\right\|}{\left\|e\right\|/\left\|b% \right\|}.
  5. ( A - 1 e / e ) ( b / A - 1 b ) . \left(\left\|A^{-1}e\right\|/\left\|e\right\|\right)\cdot\left(\left\|b\right% \|/\left\|A^{-1}b\right\|\right).
  6. κ ( A ) = A - 1 A . \kappa(A)=\left\|A^{-1}\right\|\cdot\left\|A\right\|.
  7. κ ( A ) 1. \kappa(A)\geq 1.\,
  8. \left\|\cdot\right\|
  9. 2 \left\|\cdot\right\|_{2}
  10. κ ( A ) = σ max ( A ) σ min ( A ) , \kappa(A)=\frac{\sigma_{\max}(A)}{\sigma_{\min}(A)},
  11. σ max ( A ) \sigma_{\max}(A)
  12. σ min ( A ) \sigma_{\min}(A)
  13. A A
  14. A A
  15. κ ( A ) = | λ max ( A ) λ min ( A ) | , \kappa(A)=\left|\frac{\lambda_{\max}(A)}{\lambda_{\min}(A)}\right|,
  16. λ max ( A ) \lambda_{\max}(A)
  17. λ min ( A ) \lambda_{\min}(A)
  18. A A
  19. A A
  20. κ ( A ) = 1. \kappa(A)=1.\,
  21. \left\|\cdot\right\|
  22. \left\|\cdot\right\|_{\infty}
  23. A A
  24. i , a i i 0 \forall i,a_{ii}\neq 0\,
  25. κ ( A ) max i ( | a i i | ) min i ( | a i i | ) . \kappa(A)\geq\frac{\max_{i}(|a_{ii}|)}{\min_{i}(|a_{ii}|)}.
  26. x f / f . xf^{\prime}/f.
  27. x f ( x ) f ( x ) . \frac{xf^{\prime}(x)}{f(x)}.
  28. ( log f ) = f / f (\log f)^{\prime}=f^{\prime}/f
  29. ( log x ) = x / x = 1 / x , (\log x)^{\prime}=x^{\prime}/x=1/x,
  30. x f / f . xf^{\prime}/f.
  31. f f^{\prime}
  32. Δ x \Delta x
  33. [ ( x + Δ x ) - x ] / x = ( Δ x ) / x , [(x+\Delta x)-x]/x=(\Delta x)/x,
  34. f ( x ) f(x)
  35. [ f ( x + Δ x ) - f ( x ) ] / f ( x ) . [f(x+\Delta x)-f(x)]/f(x).
  36. [ f ( x + Δ x ) - f ( x ) ] / f ( x ) ( Δ x ) / x = x f ( x ) f ( x + Δ x ) - f ( x ) ( x + Δ x ) - x . \frac{[f(x+\Delta x)-f(x)]/f(x)}{(\Delta x)/x}=\frac{x}{f(x)}\frac{f(x+\Delta x% )-f(x)}{(x+\Delta x)-x}.
  37. e x e^{x}
  38. x x
  39. ln ( x ) \ln(x)
  40. 1 ln ( x ) \frac{1}{\ln(x)}
  41. sin ( x ) \sin(x)
  42. x cot ( x ) x\cot(x)
  43. cos ( x ) \cos(x)
  44. x tan ( x ) x\tan(x)
  45. tan ( x ) \tan(x)
  46. x ( tan ( x ) + cot ( x ) ) x(\tan(x)+\cot(x))
  47. arcsin ( x ) \arcsin(x)
  48. x 1 - x 2 arcsin ( x ) \frac{x}{\sqrt{1-x^{2}}\arcsin(x)}
  49. arccos ( x ) \arccos(x)
  50. x 1 - x 2 arccos ( x ) \frac{x}{\sqrt{1-x^{2}}\arccos(x)}
  51. arctan ( x ) \arctan(x)
  52. x ( 1 + x 2 ) arctan ( x ) \frac{x}{(1+x^{2})\arctan(x)}
  53. lim ε 0 + sup δ x ε [ f ( x + δ x ) - f ( x ) f ( x ) / δ x x ] , \lim_{\varepsilon\to 0^{+}}\sup_{\|\delta x\|\leq\varepsilon}\left[\frac{\left% \|f(x+\delta x)-f(x)\right\|}{\|f(x)\|}/\frac{\|\delta x\|}{\|x\|}\right],
  54. \|\cdots\|
  55. J ( x ) f ( x ) / x , \frac{\|J(x)\|}{\|f(x)\|/\|x\|},
  56. J ( x ) \|J(x)\|

Conformal_map.html

  1. f : U V f:U\rightarrow V\qquad
  2. U , V n U,V\subset\mathbb{C}^{n}
  3. u 0 u_{0}
  4. u 0 u_{0}
  5. \scriptstyle\mathbb{C}
  6. f : U f:U\rightarrow\mathbb{C}
  7. \scriptstyle\mathbb{C}
  8. \scriptstyle\mathbb{C}
  9. g \scriptstyle g
  10. h \scriptstyle h
  11. M M
  12. g = u h \scriptstyle g=uh
  13. u \scriptstyle u
  14. M \scriptstyle M
  15. u u
  16. 2 f = 0 \scriptstyle\nabla^{2}f=0
  17. E ( z ) , \scriptstyle E(z),
  18. z \scriptstyle z
  19. E ( w ) , \scriptstyle E(w),
  20. w \scriptstyle w
  21. E \scriptstyle E
  22. w \scriptstyle w
  23. z , \scriptstyle z,
  24. E ( w ) \scriptstyle E(w)
  25. E ( w ( z ) ) , \scriptstyle E(w(z)),
  26. z , \scriptstyle z,
  27. f : U 𝕍 \scriptstyle f:U\rightarrow\mathbb{V}
  28. ( x , y ) \scriptstyle(x,y)\,
  29. ( u , v ) \scriptstyle(u,v)\,
  30. f \scriptstyle f
  31. u \scriptstyle u
  32. v \scriptstyle v
  33. x \scriptstyle x
  34. y \scriptstyle y
  35. f f

Congruence_(geometry).html

  1. ABC DEF \triangle\mathrm{ABC}\cong\triangle\mathrm{DEF}
  2. 2 \sqrt{2}

Congruence_relation.html

  1. n n
  2. n n
  3. a a
  4. b b
  5. n n
  6. a b ( mod n ) a\equiv b\;\;(\mathop{{\rm mod}}n)
  7. a - b a-b
  8. n n
  9. a a
  10. b b
  11. n n
  12. 37 37
  13. 57 57
  14. 10 10
  15. 37 57 ( mod 10 ) 37\equiv 57\;\;(\mathop{{\rm mod}}10)
  16. 37 - 57 = - 20 37-57=-20
  17. 37 37
  18. 57 57
  19. 7 7
  20. 10 10
  21. n n
  22. n n
  23. a 1 a 2 ( mod n ) a_{1}\equiv a_{2}\;\;(\mathop{{\rm mod}}n)
  24. b 1 b 2 ( mod n ) b_{1}\equiv b_{2}\;\;(\mathop{{\rm mod}}n)
  25. a 1 + b 1 a 2 + b 2 ( mod n ) a_{1}+b_{1}\equiv a_{2}+b_{2}\;\;(\mathop{{\rm mod}}n)
  26. a 1 b 1 a 2 b 2 ( mod n ) a_{1}b_{1}\equiv a_{2}b_{2}\;\;(\mathop{{\rm mod}}n)
  27. n n
  28. n n
  29. G G
  30. R R
  31. R R

Conjunction_introduction.html

  1. P , Q P and Q \frac{P,Q}{\therefore P\and Q}
  2. P P
  3. Q Q
  4. P and Q P\and Q
  5. P , Q P and Q P,Q\vdash P\and Q
  6. \vdash
  7. P and Q P\and Q
  8. P P
  9. Q Q
  10. P P
  11. Q Q

Connected_space.html

  1. Γ x \Gamma_{x}
  2. Γ x \Gamma_{x}^{\prime}
  3. Γ x Γ x \Gamma_{x}\subset\Gamma^{\prime}_{x}
  4. GL ( n , 𝐑 ) \operatorname{GL}(n,\mathbf{R})
  5. GL ( n , 𝐂 ) \operatorname{GL}(n,\mathbf{C})
  6. 0 , 1 \neq 0,1
  7. { X i } \{X_{i}\}
  8. X = i X i X=\cup_{i}{X_{i}}
  9. X X
  10. U V U\cup V
  11. X X
  12. U , V U,V
  13. X X
  14. X i X_{i}
  15. U U
  16. V V
  17. X i U X_{i}\cap U
  18. X i V X_{i}\cap V
  19. X i X_{i}
  20. X i X_{i}
  21. X X
  22. { X i } \{X_{i}\}
  23. X X
  24. X i \cap X_{i}\neq\emptyset
  25. i , j : X i X j \forall i,j:X_{i}\cap X_{j}\neq\emptyset
  26. i : X i X i + 1 \forall i:X_{i}\cap X_{i+1}\neq\emptyset
  27. X / { X i } X/\{X_{i}\}
  28. X X
  29. U V U\cup V
  30. X X
  31. q ( U ) q ( V ) q(U)\cup q(V)
  32. q ( U ) , q ( V ) q(U),q(V)

Conservation_law.html

  1. ρ t = - 𝐣 \frac{\partial\rho}{\partial t}=-\nabla\cdot\mathbf{j}\,
  2. 𝐣 = ρ 𝐮 \mathbf{j}=\rho\mathbf{u}
  3. ρ t = - ( ρ 𝐮 ) . \frac{\partial\rho}{\partial t}=-\nabla\cdot(\rho\mathbf{u})\,.
  4. y t + A ( y ) y x = 0 y_{t}+A(y)y_{x}=0
  5. y t + A ( y ) y x = s y_{t}+A(y)y_{x}=s
  6. y t + a ( y ) y x = 0 y_{t}+a(y)y_{x}=0
  7. a ( y ) = j y ( y ) a(y)=j_{y}(y)
  8. j x = j y ( y ) y x = a ( y ) y x j_{x}=j_{y}(y)y_{x}=a(y)y_{x}
  9. y t + j x ( y ) = 0 y_{t}+j_{x}(y)=0
  10. y t + 𝐚 ( y ) y = 0 y_{t}+\mathbf{a}(y)\cdot\nabla y=0
  11. \cdot
  12. y t + 𝐣 ( y ) = 0 y_{t}+\nabla\cdot\mathbf{j}(y)=0
  13. ρ t + ( ρ 𝐮 ) = 0 \rho_{t}+\nabla\cdot(\rho\mathbf{u})=0
  14. 𝐲 t + 𝐀 ( 𝐲 ) 𝐲 = 𝟎 \mathbf{y}_{t}+\mathbf{A}(\mathbf{y})\cdot\nabla\mathbf{y}=\mathbf{0}
  15. 𝐀 ( 𝐲 ) = 𝐉 𝐲 ( 𝐲 ) \mathbf{A}(\mathbf{y})=\mathbf{J}_{\mathbf{y}}(\mathbf{y})
  16. 𝐲 t + 𝐉 ( 𝐲 ) = 𝟎 \mathbf{y}_{t}+\nabla\cdot\mathbf{J}(\mathbf{y})=\mathbf{0}
  17. u = 0 \displaystyle\nabla\cdot u=0
  18. y = ( 1 u ) ; J = ( u u u + s I ) ; {y}=\begin{pmatrix}1\\ u\end{pmatrix};\qquad{J}=\begin{pmatrix}u\\ u\otimes u+sI\end{pmatrix};\qquad
  19. \otimes
  20. y t + j x ( y ) = 0 y_{t}+j_{x}(y)=0
  21. - y d x + 0 j ( y ) d t = 0 \int_{-\infty}^{\infty}ydx+\int_{0}^{\infty}j(y)dt=0
  22. [ y d N r + j ( y ) d t ] = 0 \oint[yd^{N}r+j(y)dt]=0
  23. 0 - ϕ t y + ϕ x j ( y ) d x d t = - - ϕ ( x , 0 ) y ( x , 0 ) d t \int_{0}^{\infty}\int_{-\infty}^{\infty}\phi_{t}y+\phi_{x}j(y)dxdt=-\int_{-% \infty}^{\infty}\phi(x,0)y(x,0)dt

Conservation_of_energy.html

  1. i m i v i 2 \sum_{i}m_{i}v_{i}^{2}
  2. i m i v i \,\!\sum_{i}m_{i}v_{i}
  3. 1 2 i m i v i 2 \frac{1}{2}\sum_{i}m_{i}v_{i}^{2}
  4. δ Q = d U + δ W \delta Q=\mathrm{d}U+\delta W
  5. d U = δ Q - δ W , \mathrm{d}U=\delta Q-\delta W,
  6. δ Q \delta Q
  7. δ W \delta W
  8. d U \mathrm{d}U
  9. d U \mathrm{d}U
  10. U U
  11. δ Q \delta Q
  12. δ W \delta W
  13. δ W = P d V , \delta W=P\,\mathrm{d}V,
  14. P P
  15. d V dV
  16. δ Q = T d S , \delta Q=T\,\mathrm{d}S,
  17. T T
  18. d S \mathrm{d}S
  19. d U = δ Q - δ W + u d M , \mathrm{d}U=\delta Q-\delta W+u^{\prime}\,dM,\,
  20. d M dM
  21. u u^{\prime}
  22. δ W = - P ( d V - v d M ) \delta W=-P(dV-v\,dM)
  23. E = m c 2 E=mc^{2}

Conservative_force.html

  1. × F = 0 . \nabla\times\vec{F}=\vec{0}.\,
  2. W C F d r = 0. W\equiv\oint_{C}\vec{F}\cdot\mathrm{d}\vec{r}=0.\,
  3. Φ \Phi
  4. F = - Φ . \vec{F}=-\nabla\Phi.\,
  5. S ( × F ) d a = C F d r \int_{S}(\nabla\times\vec{F})\cdot\mathrm{d}\vec{a}=\oint_{C}\vec{F}\cdot% \mathrm{d}\vec{r}
  6. x x
  7. Φ ( x ) = - c F d r . \Phi(x)=-\int_{c}\vec{F}\cdot\mathrm{d}\vec{r}.
  8. F = - Φ . \vec{F}=-\nabla\Phi.

Constructible_number.html

  1. 2 \sqrt{2}
  2. = K 0 K 1 K n \mathbb{Q}=K_{0}\subseteq K_{1}\subseteq\dots\subseteq K_{n}
  3. { x 3 : x is constructible } \left\{\sqrt[3]{x}:x\mbox{ is constructible}~{}\right\}
  4. 2 3 \sqrt[3]{2}
  5. { cos ( arccos x 3 ) : x is constructible } \left\{\cos\left(\frac{\arccos x}{3}\right):x\mbox{ is constructible}~{}\right\}
  6. cos ( arccos ( 1 / 2 ) 3 ) = 1 2 ( 2 cos ( π 9 ) ) \cos\left(\frac{\arccos(1/2)}{3}\right)=\frac{1}{2}\left(2\cos\left(\frac{\pi}% {9}\right)\right)
  7. 2 cos ( π 9 ) 2\cos\left(\frac{\pi}{9}\right)
  8. { π } \left\{\sqrt{\pi}\right\}
  9. π \sqrt{\pi}
  10. { e 2 π i / n : n , n 3 } \left\{e^{2\pi i/n}:n\in\mathbb{N},n\geq 3\right\}
  11. e 2 π i / 7 e^{2\pi i/7}
  12. 2 k 2^{k}

Constructivism_(mathematics).html

  1. x , y , z , : p ¬ p \forall x,y,z,\ldots\in\mathbb{N}:p\vee\neg p
  2. x X P ( x ) \exists_{x\in X}P(x)
  3. P ( a ) P(a)
  4. a X a\in X
  5. n n
  6. n i , j g ( n ) | f ( i ) - f ( j ) | 1 n \forall n\ \forall i,j\geq g(n)\quad|f(i)-f(j)|\leq{1\over n}
  7. f ( n ) = i = 0 n 1 i ! , g ( n ) = n . f(n)=\sum_{i=0}^{n}{1\over i!},\quad g(n)=n.
  8. n : m : i , j m : | f ( i ) - f ( j ) | 1 n \forall n:\exists m:\forall i,j\geq m:|f(i)-f(j)|\leq{1\over n}

Context-free_grammar.html

  1. G = ( V , Σ , R , S ) G=(V\,,\Sigma\,,R\,,S\,)
  2. V V\,
  3. v V v\in V
  4. G G\,
  5. Σ \Sigma\,
  6. V V\,
  7. G G\,
  8. R R\,
  9. V V\,
  10. ( V Σ ) * (V\cup\Sigma)^{*}
  11. R R\,
  12. P P\,
  13. S S\,
  14. V V\,
  15. R R\,
  16. ( α , β ) R (\alpha,\beta)\in R
  17. α V \alpha\in V
  18. β ( V Σ ) * \beta\in(V\cup\Sigma)^{*}
  19. α \alpha
  20. β \beta
  21. α β \alpha\rightarrow\beta
  22. β \beta
  23. α ε \alpha\rightarrow\varepsilon
  24. α β 1 \alpha\rightarrow\beta_{1}
  25. α β 2 \alpha\rightarrow\beta_{2}
  26. α β 1 β 2 \alpha\rightarrow\beta_{1}\mid\beta_{2}
  27. β 1 \beta_{1}
  28. β 2 \beta_{2}
  29. u , v ( V Σ ) * u,v\in(V\cup\Sigma)^{*}
  30. u u\,
  31. v v\,
  32. u v u\Rightarrow v\,
  33. ( α , β ) R \exists(\alpha,\beta)\in R
  34. α V \alpha\in V
  35. u 1 , u 2 ( V Σ ) * u_{1},u_{2}\in(V\cup\Sigma)^{*}
  36. u = u 1 α u 2 u\,=u_{1}\alpha u_{2}
  37. v = u 1 β u 2 v\,=u_{1}\beta u_{2}
  38. v \!v
  39. ( α , β ) \!(\alpha,\beta)
  40. u \!u
  41. u , v ( V Σ ) * , u,v\in(V\cup\Sigma)^{*},
  42. u u
  43. v v
  44. u * v u\stackrel{*}{\Rightarrow}v
  45. u v u\Rightarrow\Rightarrow v\,
  46. k 1 u 1 , , u k ( V Σ ) * \exists k\geq 1\,\exists\,u_{1},\cdots,u_{k}\in(V\cup\Sigma)^{*}
  47. u = u 1 u 2 u k = v u=\,u_{1}\Rightarrow u_{2}\Rightarrow\cdots\Rightarrow u_{k}\,=v
  48. k 2 k\geq 2
  49. u v u\neq v
  50. u + v u\stackrel{+}{\Rightarrow}v
  51. ( * ) (\stackrel{*}{\Rightarrow})
  52. ( + ) (\stackrel{+}{\Rightarrow})
  53. ( ) (\Rightarrow)
  54. G = ( V , Σ , R , S ) G=(V\,,\Sigma\,,R\,,S\,)
  55. L ( G ) = { w Σ * : S * w } L(G)=\{w\in\Sigma^{*}:S\stackrel{*}{\Rightarrow}w\}
  56. L L\,
  57. G G\,
  58. L = L ( G ) L\,=\,L(G)
  59. N V : α , β ( V Σ ) * : S * α N β \forall N\in V:\exists\alpha,\beta\in(V\cup\Sigma)^{*}:S\stackrel{*}{% \Rightarrow}\alpha{N}\beta
  60. N V : w Σ * : N * w \forall N\in V:\exists w\in\Sigma^{*}:N\stackrel{*}{\Rightarrow}w
  61. ¬ N V : ( N , ε ) R \neg\exists N\in V:(N,\varepsilon)\in R
  62. ¬ N V : N + N \neg\exists N\in V:N\stackrel{+}{\Rightarrow}N
  63. G = ( { S } , { a , b } , P , S ) G=(\{S\},\{a,b\},P,S)
  64. L ( G ) = { w w R : w { a , b } * } L(G)=\{ww^{R}:w\in\{a,b\}^{*}\}
  65. a a
  66. b b
  67. a a
  68. { a n b n : n 1 } \{a^{n}b^{n}:n\geq 1\}
  69. { a n b n : n 0 } \{a^{n}b^{n}:n\geq 0\}
  70. { b n a m b 2 n : n 0 , m 0 } \{b^{n}a^{m}b^{2n}:n\geq 0,m\geq 0\}
  71. S ϵ S\rightarrow\epsilon
  72. α 1 , , α N , β 1 , , β N \alpha_{1},\ldots,\alpha_{N},\beta_{1},\ldots,\beta_{N}
  73. { a 1 , , a k } \{a_{1},\ldots,a_{k}\}
  74. α 1 \alpha_{1}
  75. β 1 r e v \beta_{1}^{rev}
  76. α N \alpha_{N}
  77. β N r e v \beta_{N}^{rev}
  78. b b
  79. β i r e v \beta_{i}^{rev}
  80. β i \beta_{i}
  81. b b
  82. a i a_{i}
  83. a 1 a_{1}
  84. a 1 a_{1}
  85. a k a_{k}
  86. a k a_{k}
  87. b b
  88. α 1 , , α N , β 1 , , β N \alpha_{1},\ldots,\alpha_{N},\beta_{1},\ldots,\beta_{N}

Context-free_language.html

  1. S S S | ( S ) | ε S\to SS~{}|~{}(S)~{}|~{}\varepsilon
  2. L = { a n b n : n 1 } L=\{a^{n}b^{n}:n\geq 1\}
  3. a a
  4. b b
  5. L L
  6. S a S b | a b S\to aSb~{}|~{}ab
  7. M = ( { q 0 , q 1 , q f } , { a , b } , { a , z } , δ , q 0 , z , { q f } ) M=(\{q_{0},q_{1},q_{f}\},\{a,b\},\{a,z\},\delta,q_{0},z,\{q_{f}\})
  8. δ \delta
  9. δ ( q 0 , a , z ) = ( q 0 , a z ) \delta(q_{0},a,z)=(q_{0},az)
  10. δ ( q 0 , a , a ) = ( q 0 , a a ) \delta(q_{0},a,a)=(q_{0},aa)
  11. δ ( q 0 , b , a ) = ( q 1 , ε ) \delta(q_{0},b,a)=(q_{1},\varepsilon)
  12. δ ( q 1 , b , a ) = ( q 1 , ε ) \delta(q_{1},b,a)=(q_{1},\varepsilon)
  13. { a n b m c m d n | n , m > 0 } \{a^{n}b^{m}c^{m}d^{n}|n,m>0\}
  14. { a n b n c m d m | n , m > 0 } \{a^{n}b^{n}c^{m}d^{m}|n,m>0\}
  15. { a n b n c n d n | n > 0 } \{a^{n}b^{n}c^{n}d^{n}|n>0\}
  16. { a n b n c n d n | n > 0 } \{a^{n}b^{n}c^{n}d^{n}|n>0\}
  17. L P L\cup P
  18. L P L\cdot P
  19. L * L^{*}
  20. φ ( L ) \varphi(L)
  21. φ \varphi
  22. φ - 1 ( L ) \varphi^{-1}(L)
  23. φ - 1 \varphi^{-1}
  24. { v u : u v L } \{vu:uv\in L\}
  25. L D L\cap D
  26. L D L\setminus D
  27. A = { a n b n c m m , n 0 } A=\{a^{n}b^{n}c^{m}\mid m,n\geq 0\}
  28. B = { a m b n c n m , n 0 } B=\{a^{m}b^{n}c^{n}\mid m,n\geq 0\}
  29. A B = { a n b n c n n 0 } A\cap B=\{a^{n}b^{n}c^{n}\mid n\geq 0\}
  30. A B = A ¯ B ¯ ¯ A\cap B=\overline{\overline{A}\cup\overline{B}}
  31. L ( A ) = L ( B ) L(A)=L(B)
  32. L ( A ) L ( B ) = L(A)\cap L(B)=\emptyset
  33. L ( A ) L ( B ) L(A)\subseteq L(B)
  34. L ( A ) = Σ * L(A)=\Sigma^{*}
  35. L ( A ) = L(A)=\emptyset
  36. L ( A ) L(A)
  37. w w
  38. w L ( G ) w\in L(G)
  39. w w
  40. w L ( G ) w\in L(G)
  41. L L
  42. G G
  43. δ \delta
  44. δ ( state 1 , read , pop ) = ( state 2 , push ) \delta(\mathrm{state}_{1},\mathrm{read},\mathrm{pop})=(\mathrm{state}_{2},% \mathrm{push})

Context-sensitive_language.html

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

Continued_fraction.html

  1. n n
  2. a a
  3. i i
  4. n n
  5. p q \frac{p}{q}
  6. ( p , q ) (p,q)
  7. α α
  8. α α
  9. 415 93 \frac{415}{93}
  10. 415 93 \frac{415}{93}
  11. 43 93 \frac{43}{93}
  12. 93 43 \frac{93}{43}
  13. 1 2 \frac{1}{2}
  14. 93 43 \frac{93}{43}
  15. 7 43 \frac{7}{43}
  16. 7 43 \frac{7}{43}
  17. 43 7 \frac{43}{7}
  18. 43 7 \frac{43}{7}
  19. 1 6 \frac{1}{6}
  20. 93 43 \frac{93}{43}
  21. 1 2 + 1 6 1\frac{2+}{\frac{1}{6}}
  22. 43 7 \frac{43}{7}
  23. 1 7 \frac{1}{7}
  24. 1 7 \frac{1}{7}
  25. 7 1 \frac{7}{1}
  26. 0 1 \frac{0}{1}
  27. 1 2 + 1 6 + ( 1 / 7 ) 1\frac{2+}{\frac{1}{6+(1 / 7)}}
  28. 415 93 \frac{415}{93}
  29. 1 2 + 1 6 + ( 1 / 7 ) 1\frac{2+}{\frac{1}{6+(1 / 7)}}
  30. 415 93 \frac{415}{93}
  31. ( n + 1 ) (n+1)
  32. 19 = 4 ; 2 , 1 , 3 , 1 , 2 , 8 , 2 , 1 , 3 , 1 , 2 , 8 , \sqrt{19}=4;2,1,3,1,2,8,2,1,3,1,2,8,……
  33. e = 2 ; 1 , 2 , 1 , 1 , 4 , 1 , 1 , 6 , 1 , 1 , 8 , e=2;1,2,1,1,4,1,1,6,1,1,8,……
  34. π = 3 ; 7 , 15 , 1 , 292 , 1 , 1 , 1 , 2 , 1 , 3 , 1 , π=3;7,15,1,292,1,1,1,2,1,3,1,……
  35. ϕ = 1 ; 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , ϕ=1;1,1,1,1,1,1,1,1,1,1,1,……
  36. 137 1600 \frac{137}{1600}
  37. 4 27 \frac{4}{27}
  38. a a
  39. a a
  40. a a
  41. n n
  42. a a
  43. n n
  44. a a
  45. a a
  46. a a
  47. n n
  48. a a
  49. n n
  50. a 0 + b 1 a 1 + b 2 a 2 + b 3 a 3 + a_{0}+\cfrac{b_{1}}{a_{1}+\cfrac{b_{2}}{a_{2}+\cfrac{b_{3}}{a_{3}+\ddots}}}
  51. a 0 \ a_{0}
  52. 2 \ 2
  53. a 0 + 1 a 1 \ a_{0}+\cfrac{1}{a_{1}}
  54. 2 + 1 3 \ 2+\cfrac{1}{3}
  55. a 0 + 1 a 1 + 1 a 2 \ a_{0}+\cfrac{1}{a_{1}+\cfrac{1}{a_{2}}}
  56. - 3 + 1 2 + 1 18 \ -3+\cfrac{1}{2+\cfrac{1}{18}}
  57. a 0 + 1 a 1 + 1 a 2 + 1 a 3 \ a_{0}+\cfrac{1}{a_{1}+\cfrac{1}{a_{2}+\cfrac{1}{a_{3}}}}
  58. 1 15 + 1 1 + 1 102 \ \cfrac{1}{15+\cfrac{1}{1+\cfrac{1}{102}}}
  59. r r
  60. i i
  61. r r
  62. r r
  63. f f
  64. r r
  65. i i
  66. r r
  67. r r
  68. i i
  69. a a
  70. a a
  71. a a
  72. a a
  73. f f
  74. r r
  75. r r
  76. r r
  77. r r
  78. 49 200 \frac{49}{200}
  79. f f
  80. r r
  81. 49 200 \frac{49}{200}
  82. i i
  83. f f
  84. 49 200 \frac{49}{200}
  85. = =
  86. 49 200 \frac{49}{200}
  87. f f
  88. 200 49 \frac{200}{49}
  89. = =
  90. 4 49 \frac{4}{49}
  91. r r
  92. 4 49 \frac{4}{49}
  93. i i
  94. f f
  95. 4 49 \frac{4}{49}
  96. = =
  97. 4 49 \frac{4}{49}
  98. f f
  99. 49 4 \frac{49}{4}
  100. = =
  101. 1 4 \frac{1}{4}
  102. r r
  103. 1 4 \frac{1}{4}
  104. i i
  105. f f
  106. 1 4 \frac{1}{4}
  107. = =
  108. 1 4 \frac{1}{4}
  109. f f
  110. 4 1 \frac{4}{1}
  111. = =
  112. r r
  113. i i
  114. f f
  115. = =
  116. 49 200 \frac{49}{200}
  117. 3 49 200 = 3 + 1 4 + 1 12 + 1 4 3\frac{49}{200}=3+1\frac{4+}{1\frac{12+}{\frac{1}{4}}}
  118. x = a 0 + 1 a 1 + 1 a 2 + 1 a 3 x=a_{0}+\cfrac{1}{a_{1}+\cfrac{1}{a_{2}+\cfrac{1}{a_{3}}}}
  119. x = a 0 + K i = 1 3 1 a i x=a_{0}+\underset{i=1}{\overset{3}{\mathrm{K}}}~{}\frac{1}{a_{i}}\;
  120. x = [ a 0 ; a 1 , a 2 , a 3 ] x=[a_{0};a_{1},a_{2},a_{3}]\;
  121. x = a 0 + 1 a 1 + 1 a 2 + 1 a 3 , x=a_{0}+\frac{1\mid}{\mid a_{1}}+\frac{1\mid}{\mid a_{2}}+\frac{1\mid}{\mid a_% {3}},
  122. x = a 0 + 1 a 1 + 1 a 2 + 1 a 3 + . x=a_{0}+{1\over a_{1}+{}}{1\over a_{2}+{}}{1\over a_{3}+{}}.
  123. x = a 0 ; a 1 , a 2 , a 3 . x=\left\langle a_{0};a_{1},a_{2},a_{3}\right\rangle.
  124. [ a 0 ; a 1 , a 2 , a 3 , ] = lim n [ a 0 ; a 1 , a 2 , , a n ] . [a_{0};a_{1},a_{2},a_{3},\,\ldots]=\lim_{n\to\infty}[a_{0};a_{1},a_{2},\,% \ldots,a_{n}].
  125. a a 0 ; a 1 , a 2 , , a n 1 , a n , 1 = a a 0 ; a 1 , a 2 , , a n 1 , a n + 1 aa_{0};a_{1},a_{2},…,a_{n−1},a_{n},1=aa_{0};a_{1},a_{2},…,a_{n−1},a_{n}+1
  126. a a 0 ; 1 = a a 0 + 1 aa_{0};1=aa_{0}+1
  127. 2.25 = 2 + 1 4 = 2 ; 44 = 2 + 1 3 + 1 / 1 = 2 ; 3 , 11 2.25=2+\frac{1}{4}=2;44=2+\frac{1}{3+1/1}=2;3,11
  128. 4.2 = 5 + 4 5 = 5 + 1 1 + 1 / 4 = 5 ; 1 , 44 = 5 + 1 1 + 1 3 + 1 / 1 = 5 ; 1 , 3 , 11 −4.2=−5+\frac{4}{5}=−5+\frac{1}{1+1/4}=−5;1,44=−5+1\frac{1+}{\frac{1}{3+1/1}}=% −5;1,3,11
  129. a a 0 ; a 1 , a 2 , , a n aa_{0};a_{1},a_{2},…,a_{n}
  130. 0 ; a 0 , a 1 , , a n 0;a_{0},a_{1},…,a_{n}
  131. a a
  132. x = 0 + 1 a + 1 / x=0+1\frac{a}{+1/}
  133. 1 x = a + 1 b \frac{1}{x}=a+\frac{1}{b}
  134. x > 1 x>1
  135. x = a + 1 b x=a+\frac{1}{b}
  136. 1 x = 0 + 1 a + 1 / \frac{1}{x}=0+1\frac{a}{+1/}
  137. x x
  138. 2.25 = 9 4 = 2 ; 44 2.25=\frac{9}{4}=2;44
  139. 1 2.25 = 4 9 = 0 ; 2 , 44 \frac{1}{2.25}=\frac{4}{9}=0;2,44
  140. a a 0 ; a 1 , a 2 , aa_{0};a_{1},a_{2},…
  141. a 0 1 , a 1 a , a 2 ( , a 3 ( a\frac{{}_{0}}{1},a\frac{{}_{1}}{a},a\frac{{}_{2}}{(},a\frac{{}_{3}}{(}
  142. h h
  143. 1 {}_{1}
  144. h h
  145. 2 {}_{2}
  146. k k
  147. 1 {}_{1}
  148. k k
  149. 2 {}_{2}
  150. h n = a n h n 1 + h n 2 h_{n}=a_{n}h_{n−1}+h_{n−2}
  151. k n = a n k n 1 + k n 2 k_{n}=a_{n}k_{n−1}+k_{n−2}
  152. h n k = a n h h\frac{{}_{n}}{k}=a\frac{{}_{n}}{h}
  153. n n
  154. a n a_{n}
  155. h n h_{n}
  156. k n k_{n}
  157. 3 \sqrt{3}
  158. n n
  159. a n a_{n}
  160. h n h_{n}
  161. k n k_{n}
  162. x 0 = 1 = 1 1 x_{0}=1=\frac{1}{1}
  163. x 1 = 1 2 ( 1 + 3 1 ) = 2 1 = 2 x_{1}=\frac{1}{2}(1+\frac{3}{1})=\frac{2}{1}=2
  164. x 2 = 1 2 ( 2 + 3 2 ) = 7 4 x_{2}=\frac{1}{2}(2+\frac{3}{2})=\frac{7}{4}
  165. x 3 = 1 2 ( 7 4 + 3 7 4 ) = 97 56 x_{3}=\frac{1}{2}(\frac{7}{4}+\frac{3}{\frac{7}{4}})=\frac{97}{56}
  166. h n = a n h n - 1 + h n - 2 h_{n}=a_{n}h_{n-1}+h_{n-2}\,
  167. h - 1 = 1 h_{-1}=1\,
  168. h - 2 = 0 h_{-2}=0\,
  169. k n = a n k n - 1 + k n - 2 k_{n}=a_{n}k_{n-1}+k_{n-2}\,
  170. k - 1 = 0 k_{-1}=0\,
  171. k - 2 = 1 k_{-2}=1\,
  172. [ a 0 ; a 1 , , a n - 1 , z ] = z h n - 1 + h n - 2 z k n - 1 + k n - 2 . \left[a_{0};a_{1},\,\dots,a_{n-1},z\right]=\frac{zh_{n-1}+h_{n-2}}{zk_{n-1}+k_% {n-2}}.
  173. [ a 0 ; a 1 , , a n ] = h n k n . \left[a_{0};a_{1},\,\dots,a_{n}\right]=\frac{h_{n}}{k_{n}}.
  174. k n h n - 1 - k n - 1 h n = ( - 1 ) n . k_{n}h_{n-1}-k_{n-1}h_{n}=(-1)^{n}.
  175. h n k n - h n - 1 k n - 1 = h n k n - 1 - k n h n - 1 k n k n - 1 = ( - 1 ) n + 1 k n k n - 1 . \frac{h_{n}}{k_{n}}-\frac{h_{n-1}}{k_{n-1}}=\frac{h_{n}k_{n-1}-k_{n}h_{n-1}}{k% _{n}k_{n-1}}=\frac{(-1)^{n+1}}{k_{n}k_{n-1}}.
  176. a 0 + n = 0 ( - 1 ) n k n k n + 1 . a_{0}+\sum_{n=0}^{\infty}\frac{(-1)^{n}}{k_{n}k_{n+1}}.
  177. [ h n h n - 1 k n k n - 1 ] \begin{bmatrix}h_{n}&h_{n-1}\\ k_{n}&k_{n-1}\end{bmatrix}
  178. | x r - x n | > | x s - x n | \left|x_{r}-x_{n}\right|>\left|x_{s}-x_{n}\right|
  179. 1 k n ( k n + 1 + k n ) < | x - h n k n | < 1 k n k n + 1 . \frac{1}{k_{n}(k_{n+1}+k_{n})}<\left|x-\frac{h_{n}}{k_{n}}\right|<\frac{1}{k_{% n}k_{n+1}}.
  180. h n 1 k , h n k h\frac{{}_{n−1}}{k},h\frac{{}_{n}}{k}
  181. h n 1 + h\frac{{}_{n−1}}{+}
  182. a a
  183. n n
  184. n + 1 n+1
  185. x x
  186. a b \frac{a}{b}
  187. c d \frac{c}{d}
  188. a d b c = ± 1 ad−bc=±1
  189. x x
  190. n d \frac{n}{d}
  191. d > 0 d>0
  192. x x
  193. x x
  194. x x
  195. 3 4 \frac{3}{4}
  196. 4 5 \frac{4}{5}
  197. 5 6 \frac{5}{6}
  198. 11 13 \frac{11}{13}
  199. 16 19 \frac{16}{19}
  200. 27 32 \frac{27}{32}
  201. a a
  202. k {}_{k}
  203. a a
  204. k {}_{k}
  205. | x a a 0 ; a 1 , , a k 1 ! | > | x a a 0 ; a 1 , , a k 1 , a k / 2 ! | |x−aa_{0};a_{1},…,a_{k−1}!|>|x−aa_{0};a_{1},…,a_{k−1},a_{k}/2!|
  206. a a k ; a k 1 , , a 1 > a a k ; a k + 1 , aa_{k};a_{k−1},…,a_{1}>aa_{k};a_{k+1},…
  207. x x
  208. n n
  209. d d
  210. x x
  211. | d x n | |dx−n|
  212. m m
  213. c c
  214. c d c≤d
  215. k k→∞
  216. ( x , y ) (x,y)
  217. y y
  218. x x
  219. y y
  220. x = a a 0 ; a 1 , a 2 , , a k 1 , a k , a k + 1 , x=aa_{0};a_{1},a_{2},…,a_{k−1},a_{k},a_{k+1},…
  221. y = a a 0 ; a 1 , a 2 , , a k 1 , b k , b k + 1 , y=aa_{0};a_{1},a_{2},…,a_{k−1},b_{k},b_{k+1},…
  222. x x
  223. y y
  224. ( x , y ) (x,y)
  225. z ( x , y ) = a a 0 ; a 1 , a 2 , , a k 1 , m i n ( a k , b k ) + 1 z(x,y)=aa_{0};a_{1},a_{2},…,a_{k−1},min(a_{k},b_{k})+1
  226. ( x , y ) (x,y)
  227. x x
  228. y y
  229. + +∞
  230. [ 3.14155 , 3.14165 ] [3.14155, 3.14165]
  231. 3.14155 = 3 ; 7 , 15 , 2 , 7 , 1 , 4 , 1 , 11 = 3 ; 7 , 15 , 2 , 7 , 1 , 4 , 22 3.14155=3;7,15,2,7,1,4,1,11=3;7,15,2,7,1,4,22
  232. 3.14165 = 3 ; 7 , 16 , 1 , 3 , 4 , 2 , 3 , 11 = 3 ; 7 , 16 , 1 , 3 , 4 , 2 , 44 3.14165=3;7,16,1,3,4,2,3,11=3;7,16,1,3,4,2,44
  233. 3 ; 7 , 166 = 355 113 = 3.1415929.... 3;7,166=\frac{355}{113}=3.1415929....
  234. 355 113 \frac{355}{113}
  235. z = a a 0 ; a 1 , , a k 1 , a k , 1 = a a 0 ; a 1 , , a k 1 , a k + 1 z=aa_{0};a_{1},…,a_{k−1},a_{k},1=aa_{0};a_{1},…,a_{k−1},a_{k}+1
  236. x = a a 0 ; a 1 , , a k 1 , a k , 2 x=aa_{0};a_{1},…,a_{k−1},a_{k},2
  237. y = a a 0 ; a 1 , , a k 1 , a k + 2 y=aa_{0};a_{1},…,a_{k−1},a_{k}+2
  238. x x
  239. y y
  240. z z
  241. k k
  242. x > y x>y
  243. k k
  244. 355 113 \frac{355}{113}
  245. 355 113 \frac{355}{113}
  246. 355 113 \frac{355}{113}
  247. 3 ; 7 , 15 , 22 3;7,15,22
  248. = =
  249. 688 219 3.1415525 \frac{688}{219}≈3.1415525
  250. 3 ; 7 , 177 3;7,177
  251. = =
  252. 377 120 3.1416667 \frac{377}{120}≈3.1416667
  253. x = a a 0 ; a 1 , x=aa_{0};a_{1},…
  254. y = b b 0 ; b 1 , y=bb_{0};b_{1},…
  255. k k
  256. a k a_{k}
  257. b k b_{k}
  258. k k
  259. x = a a 0 ; a 1 , , a n x=aa_{0};a_{1},…,a_{n}
  260. y = b b 0 ; b 1 , , b n , b n + 1 , y=bb_{0};b_{1},…,b_{n},b_{n+1},…
  261. a i = b i a_{i}=b_{i}
  262. 0 i n 0≤i≤n
  263. n n
  264. n n
  265. π \pi
  266. π \pi
  267. a 0 = π = 3 a_{0}=⌊\pi⌋=3
  268. u 1 = 1 π 3 7.0625 u_{1}=1\frac{\pi}{−3}≈7.0625
  269. a 1 = u 1 = 7 a_{1}=⌊u_{1}⌋=7
  270. u 2 = 1 u 1 15.9665 u_{2}=1\frac{u}{{}_{1}}≈15.9665
  271. a 2 = u 2 = 15 a_{2}=⌊u_{2}⌋=15
  272. u 3 = 1 u 2 1.003 u_{3}=1\frac{u}{{}_{2}}≈1.003
  273. π \pi
  274. π \pi
  275. 355 113 \frac{355}{113}
  276. π \pi
  277. 3 1 \frac{3}{1}
  278. 22 7 \frac{22}{7}
  279. 333 106 \frac{333}{106}
  280. 3 1 \frac{3}{1}
  281. 22 7 \frac{22}{7}
  282. 333 106 \frac{333}{106}
  283. 355 113 \frac{355}{113}
  284. π \pi
  285. π \pi
  286. π \pi
  287. 22 7 \frac{22}{7}
  288. π \pi
  289. 22 7 \frac{22}{7}
  290. π \pi
  291. 1 7 × 106 \frac{1}{7 × 106}
  292. 1 742 \frac{1}{742}
  293. 22 7 \frac{22}{7}
  294. π \pi
  295. 1 790 \frac{1}{790}
  296. 22 7 \frac{22}{7}
  297. 3 1 \frac{3}{1}
  298. 1 7 \frac{1}{7}
  299. 333 106 \frac{333}{106}
  300. 22 7 \frac{22}{7}
  301. 1 742 \frac{1}{742}
  302. 355 113 \frac{355}{113}
  303. 333 106 \frac{333}{106}
  304. 1 11978 \frac{1}{11978}
  305. 3 1 \frac{3}{1}
  306. 1 1 × 7 \frac{1}{1×7}
  307. 1 7 × 106 \frac{1}{7×106}
  308. 1 106 × 113 \frac{1}{106×113}
  309. 22 7 \frac{22}{7}
  310. 333 106 \frac{333}{106}
  311. x = b 0 + a 1 b 1 + a 2 b 2 + a 3 b 3 + a 4 b 4 + x=b_{0}+\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cfrac{a_{3}}{b_{3}+\cfrac{a_{% 4}}{b_{4}+\ddots\,}}}}
  312. π \pi
  313. π = [ 3 ; 7 , 15 , 1 , 292 , 1 , 1 , 1 , 2 , 1 , 3 , 1 , ] \pi=[3;7,15,1,292,1,1,1,2,1,3,1,\ldots]
  314. π = 3 + 1 7 + 1 15 + 1 1 + 1 292 + 1 1 + 1 1 + 1 1 + 1 2 + 1 1 + 1 3 + 1 1 + \pi=3+\cfrac{1}{7+\cfrac{1}{15+\cfrac{1}{1+\cfrac{1}{292+\cfrac{1}{1+\cfrac{1}% {1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{3+\cfrac{1}{1+\ddots}}}}}}}}}}}
  315. π \pi
  316. π = 4 1 + 1 2 2 + 3 2 2 + 5 2 2 + 7 2 2 + 9 2 2 + = 4 1 + 1 2 3 + 2 2 5 + 3 2 7 + 4 2 9 + = 3 + 1 2 6 + 3 2 6 + 5 2 6 + 7 2 6 + 9 2 6 + \pi=\cfrac{4}{1+\cfrac{1^{2}}{2+\cfrac{3^{2}}{2+\cfrac{5^{2}}{2+\cfrac{7^{2}}{% 2+\cfrac{9^{2}}{2+\ddots}}}}}}=\cfrac{4}{1+\cfrac{1^{2}}{3+\cfrac{2^{2}}{5+% \cfrac{3^{2}}{7+\cfrac{4^{2}}{9+\ddots}}}}}=3+\cfrac{1^{2}}{6+\cfrac{3^{2}}{6+% \cfrac{5^{2}}{6+\cfrac{7^{2}}{6+\cfrac{9^{2}}{6+\ddots}}}}}
  317. π = 2 + 2 1 + 1 1 / 2 + 1 1 / 3 + 1 1 / 4 + = 2 + 2 1 + 1 2 1 + 2 3 1 + 3 4 1 + \displaystyle\pi=2+\cfrac{2}{1+\cfrac{1}{1/2+\cfrac{1}{1/3+\cfrac{1}{1/4+% \ddots}}}}=2+\cfrac{2}{1+\cfrac{1\cdot 2}{1+\cfrac{2\cdot 3}{1+\cfrac{3\cdot 4% }{1+\ddots}}}}
  318. π = 2 + 4 3 + 1 3 4 + 3 5 4 + 5 7 4 + \displaystyle\pi=2+\cfrac{4}{3+\cfrac{1\cdot 3}{4+\cfrac{3\cdot 5}{4+\cfrac{5% \cdot 7}{4+\ddots}}}}
  319. π \pi
  320. 2 \sqrt{2}
  321. 14 \sqrt{14}
  322. 42 \sqrt{42}
  323. 2 \sqrt{2}
  324. 14 \sqrt{14}
  325. k k
  326. m n \frac{m}{n}
  327. | k - m n | < 1 n 2 5 . \left|k-{m\over n}\right|<{1\over n^{2}\sqrt{5}}.
  328. k k
  329. m n \frac{m}{n}
  330. k k
  331. 5 3 \frac{5}{3}
  332. 8 5 \frac{8}{5}
  333. 13 8 \frac{13}{8}
  334. 21 13 \frac{21}{13}
  335. 1 n 2 5 {\scriptstyle{1\over n^{2}\sqrt{5}}}
  336. 355 113 \frac{355}{113}
  337. π \pi
  338. a + b a\frac{+}{b}
  339. a a
  340. b b
  341. c c
  342. d d
  343. a d b c = ± 1 ad−bc=±1
  344. π \pi
  345. e e
  346. e = e 1 = [ 2 ; 1 , 2 , 1 , 1 , 4 , 1 , 1 , 6 , 1 , 1 , 8 , 1 , 1 , 10 , 1 , 1 , 12 , 1 , 1 , ] , e=e^{1}=[2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1,1,\dots],
  347. n n
  348. e 1 / n = [ 1 ; n - 1 , 1 , 1 , 3 n - 1 , 1 , 1 , 5 n - 1 , 1 , 1 , 7 n - 1 , 1 , 1 , ] . e^{1/n}=[1;n-1,1,1,3n-1,1,1,5n-1,1,1,7n-1,1,1,\dots]\,\!.
  349. n n
  350. e 2 / n = [ 1 ; n - 1 2 , 6 n , 5 n - 1 2 , 1 , 1 , 7 n - 1 2 , 18 n , 11 n - 1 2 , 1 , 1 , 13 n - 1 2 , 30 n , 17 n - 1 2 , 1 , 1 , ] , e^{2/n}=\left[1;\frac{n-1}{2},6n,\frac{5n-1}{2},1,1,\frac{7n-1}{2},18n,\frac{1% 1n-1}{2},1,1,\frac{13n-1}{2},30n,\frac{17n-1}{2},1,1,\dots\right]\,\!,
  351. n = 1 n=1
  352. e 2 = [ 7 ; 2 , 1 , 1 , 3 , 18 , 5 , 1 , 1 , 6 , 30 , 8 , 1 , 1 , 9 , 42 , 11 , 1 , 1 , 12 , 54 , 14 , 1 , 1 , 3 k , 12 k + 6 , 3 k + 2 , 1 , 1 ] . e^{2}=[7;2,1,1,3,18,5,1,1,6,30,8,1,1,9,42,11,1,1,12,54,14,1,1\dots,3k,12k+6,3k% +2,1,1\dots]\,\!.
  353. tanh ( 1 / n ) = [ 0 ; n , 3 n , 5 n , 7 n , 9 n , 11 n , 13 n , 15 n , 17 n , 19 n , ] \tanh(1/n)=[0;n,3n,5n,7n,9n,11n,13n,15n,17n,19n,\dots]\,\!
  354. n n
  355. n n
  356. tan ( 1 / n ) = [ 0 ; n - 1 , 1 , 3 n - 2 , 1 , 5 n - 2 , 1 , 7 n - 2 , 1 , 9 n - 2 , 1 , ] , \tan(1/n)=[0;n-1,1,3n-2,1,5n-2,1,7n-2,1,9n-2,1,\dots]\,\!,
  357. n = 1 n=1
  358. tan ( 1 ) = [ 1 ; 1 , 1 , 3 , 1 , 5 , 1 , 7 , 1 , 9 , 1 , 11 , 1 , 13 , 1 , 15 , 1 , 17 , 1 , 19 , 1 , ] . \tan(1)=[1;1,1,3,1,5,1,7,1,9,1,11,1,13,1,15,1,17,1,19,1,\dots]\,\!.
  359. p q \frac{p}{q}
  360. S ( p / q ) = I p / q ( 2 / q ) I 1 + p / q ( 2 / q ) , S(p/q)=\frac{I_{p/q}(2/q)}{I_{1+p/q}(2/q)},
  361. p p
  362. q q
  363. S ( p / q ) = [ p + q ; p + 2 q , p + 3 q , p + 4 q , ] , S(p/q)=[p+q;p+2q,p+3q,p+4q,\dots],
  364. S ( 0 ) = S ( 0 / 1 ) = [ 1 ; 2 , 3 , 4 , 5 , 6 , 7 , ] . S(0)=S(0/1)=[1;2,3,4,5,6,7,\dots].
  365. x x
  366. i = 1 , 2 , 3 , i=1, 2, 3, …
  367. K 2.6854520010 K≈2.6854520010…
  368. x x
  369. n n
  370. n n
  371. n n
  372. n n
  373. x = 1 + x - 1 1 + x \sqrt{x}=1+\frac{x-1}{1+\sqrt{x}}
  374. x = 1 + x - 1 2 + x - 1 2 + x - 1 2 + \sqrt{x}=1+\cfrac{x-1}{2+\cfrac{x-1}{2+\cfrac{x-1}{2+{\ddots}}}}
  375. p p
  376. q q
  377. p < s u p > 2 2 q 2 = ± 1 p<sup>2−2q^{2}=±1

Continuous_function.html

  1. y = f ( x ) y=f(x)
  2. α \alpha
  3. f ( x + α ) - f ( x ) f(x+\alpha)-f(x)
  4. f : I 𝐑 . f\colon I\rightarrow\mathbf{R}.
  5. I = ( a , b ) = { x 𝐑 | a < x < b } , I=(a,b)=\{x\in\mathbf{R}\,|\,a<x<b\},
  6. I = [ a , b ] = { x 𝐑 | a x b } . I=[a,b]=\{x\in\mathbf{R}\,|\,a\leq x\leq b\}.
  7. lim x c f ( x ) = f ( c ) . \lim_{x\to c}{f(x)}=f(c).
  8. ( x n ) n (x_{n})_{n\in\mathbb{N}}
  9. ( f ( x n ) ) n \left(f(x_{n})\right)_{n\in\mathbb{N}}
  10. ( x n ) n I : lim n x n = c lim n f ( x n ) = f ( c ) . \forall(x_{n})_{n\in\mathbb{N}}\subset I:\lim_{n\to\infty}x_{n}=c\Rightarrow% \lim_{n\to\infty}f(x_{n})=f(c)\,.
  11. | x - c | < δ | f ( x ) - f ( c ) | < ε . |x-c|<\delta\Rightarrow|f(x)-f(c)|<\varepsilon.\,
  12. ω f ( x 0 ) = 0. \omega_{f}(x_{0})=0.
  13. f , g : I 𝐑 f,g\colon I\rightarrow\mathbf{R}
  14. f g : { x I | g ( x ) 0 } 𝐑 , x f ( x ) g ( x ) \frac{f}{g}\colon\{x\in I|g(x)\neq 0\}\rightarrow\mathbf{R},x\mapsto\frac{f(x)% }{g(x)}
  15. f ( x ) = 2 x - 1 x + 2 f(x)=\frac{2x-1}{x+2}
  16. G ( x ) = { sin ( x ) x if x 0 1 if x = 0 , G(x)=\begin{cases}\frac{\sin(x)}{x}&\,\text{ if }x\neq 0\\ 1&\,\text{ if }x=0,\end{cases}
  17. f : I J ( 𝐑 ) , g : J 𝐑 , f\colon I\rightarrow J(\subset\mathbf{R}),g\colon J\rightarrow\mathbf{R},
  18. g f : I 𝐑 , x g ( f ( x ) ) g\circ f\colon I\rightarrow\mathbf{R},x\mapsto g(f(x))
  19. 1 / 2 {1}/{2}
  20. f ( x ) = { sin ( 1 x 2 ) if x 0 0 if x = 0 f(x)=\begin{cases}\sin\left(\frac{1}{x^{2}}\right)\,\text{ if }x\neq 0\\ 0\,\text{ if }x=0\end{cases}
  21. f ( x ) = { 1 if x = 0 1 q if x = p q (in lowest terms) is a rational number 0 if x is irrational . f(x)=\begin{cases}1\,\text{ if }x=0\\ \frac{1}{q}\,\text{ if }x=\frac{p}{q}\,\text{(in lowest terms) is a rational % number}\\ 0\,\text{ if }x\,\text{ is irrational}.\end{cases}
  22. D ( x ) = { 0 if x is irrational ( ) 1 if x is rational ( ) D(x)=\begin{cases}0\,\text{ if }x\,\text{ is irrational }(\in\mathbb{R}% \setminus\mathbb{Q})\\ 1\,\text{ if }x\,\text{ is rational }(\in\mathbb{Q})\end{cases}
  23. f : ( a , b ) 𝐑 f\colon(a,b)\rightarrow\mathbf{R}
  24. f ( x ) = | x | = { x if x 0 - x if x < 0 f(x)=|x|=\begin{cases}x\,\text{ if }x\geq 0\\ -x\,\text{ if }x<0\end{cases}
  25. ( a , b ) (a,b)
  26. f : Ω 𝐑 f\colon\Omega\rightarrow\mathbf{R}
  27. f : [ a , b ] 𝐑 f\colon[a,b]\rightarrow\mathbf{R}
  28. f 1 , f 2 , : I 𝐑 f_{1},f_{2},\ldots\colon I\rightarrow\mathbf{R}
  29. f ( x ) := lim n f n ( x ) f(x):=\lim_{n\rightarrow\infty}f_{n}(x)
  30. | f ( x ) - f ( c ) | < ε . |f(x)-f(c)|<\varepsilon.\,
  31. f ( x ) f ( c ) - ϵ . f(x)\geq f(c)-\epsilon.
  32. d X : X × X 𝐑 d_{X}\colon X\times X\rightarrow\mathbf{R}
  33. f : X Y f\colon X\rightarrow Y
  34. T : V W T\colon V\rightarrow W
  35. T ( x ) K x \|T(x)\|\leq K\|x\|
  36. d Y ( f ( b ) , f ( c ) ) K ( d X ( b , c ) ) α d_{Y}(f(b),f(c))\leq K\cdot(d_{X}(b,c))^{\alpha}
  37. d Y ( f ( b ) , f ( c ) ) K d X ( b , c ) d_{Y}(f(b),f(c))\leq K\cdot d_{X}(b,c)
  38. f : X Y f\colon X\rightarrow Y
  39. f - 1 ( V ) = { x X | f ( x ) V } f^{-1}(V)=\{x\in X\;|\;f(x)\in V\}
  40. f : X T f\colon X\rightarrow T
  41. ( X , 𝒯 X ) (X,\mathcal{T}_{X})
  42. ( Y , 𝒯 Y ) (Y,\mathcal{T}_{Y})
  43. f : X Y f:X\rightarrow Y
  44. x X x\in X
  45. f ( x ) Y f(x)\in Y
  46. N 𝒩 f ( x ) : f - 1 ( N ) x \forall N\in\mathcal{N}_{f(x)}:f^{-1}(N)\in\mathcal{M}_{x}
  47. N 𝒩 f ( x ) \exist M x : M f - 1 ( N ) \forall N\in\mathcal{N}_{f(x)}\exist M\in\mathcal{M}_{x}:M\subseteq f^{-1}(N)
  48. N 𝒩 f ( x ) \exist M x : f ( M ) N \forall N\in\mathcal{N}_{f(x)}\exist M\in\mathcal{M}_{x}:f(M)\subseteq N
  49. V 𝒯 Y , f ( x ) V \exist U 𝒯 X , x U : U f - 1 ( V ) \forall V\in\mathcal{T}_{Y},f(x)\in V\exist U\in\mathcal{T}_{X},x\in U:U% \subseteq f^{-1}(V)
  50. V 𝒯 Y , f ( x ) V \exist U 𝒯 X , x U : f ( U ) V \forall V\in\mathcal{T}_{Y},f(x)\in V\exist U\in\mathcal{T}_{X},x\in U:f(U)\subseteq V
  51. f : ( X , cl ) ( X , cl ) f\colon(X,\mathrm{cl})\to(X^{\prime},\mathrm{cl}^{\prime})\,
  52. f ( cl ( A ) ) cl ( f ( A ) ) . f(\mathrm{cl}(A))\subseteq\mathrm{cl}^{\prime}(f(A)).
  53. f - 1 ( cl ( A ) ) cl ( f - 1 ( A ) ) . f^{-1}(\mathrm{cl}^{\prime}(A^{\prime}))\supseteq\mathrm{cl}(f^{-1}(A^{\prime}% )).
  54. f : ( X , int ) ( X , int ) f\colon(X,\mathrm{int})\to(X^{\prime},\mathrm{int}^{\prime})\,
  55. f - 1 ( int ( A ) ) int ( f - 1 ( A ) ) f^{-1}(\mathrm{int}^{\prime}(A^{\prime}))\subseteq\mathrm{int}(f^{-1}(A^{% \prime}))
  56. ( X , τ X ) ( Y , τ Y ) (X,\tau_{X})\rightarrow(Y,\tau_{Y})
  57. f : X S , f\colon X\rightarrow S,\,
  58. S X S\rightarrow X
  59. X S . X\rightarrow S.
  60. F : 𝒞 𝒟 F\colon\mathcal{C}\rightarrow\mathcal{D}
  61. lim i I F ( C i ) F ( lim i I C i ) \underleftarrow{\lim}_{i\in I}F(C_{i})\cong F(\underleftarrow{\lim}_{i\in I}C_% {i})
  62. 𝒞 \mathcal{C}