wpmath0000002_4

Cuboid.html

  1. d = a 2 + b 2 + c 2 . d=\sqrt{a^{2}+b^{2}+c^{2}}.

Cumulative_voting.html

  1. X = S N D + 1 + 1 X={SN\over D+1}+1
  2. N = ( X - 1 ) * ( D + 1 ) S N={(X-1)*(D+1)\over S}

Curie.html

  1. 1 Ci \displaystyle\,\text{1 Ci}

Curie_temperature.html

  1. T < s u b > c T<sub>c
  2. χ χ
  3. χ = M H = M μ 0 B = C T \chi=\frac{M}{H}=\frac{M\mu_{0}}{B}=\frac{C}{T}
  4. χ χ
  5. C = μ 0 μ B 2 3 k B N g 2 J ( J + 1 ) C=\frac{\mu_{0}\mu_{B}^{2}}{3k_{B}}Ng^{2}J(J+1)
  6. χ = C T - T c \chi=\frac{C}{T-T_{c}}
  7. T C = C λ μ 0 T_{C}=\frac{C\lambda}{\mu_{0}}
  8. χ 1 ( T - T c ) γ \chi\sim\frac{1}{(T-T_{c})^{\gamma}}
  9. M ( T - T C ) β M\sim(T-T_{C})^{\beta}
  10. T < s u b > 0 T<sub>0
  11. ϵ = ϵ 0 + C T - T c . \epsilon=\epsilon_{0}+\frac{C}{T-T_{c}}.

Curry's_paradox.html

  1. X = def { x ( x x ) Y } . X\ \stackrel{\mathrm{def}}{=}\ \left\{x\mid(x\in x)\to Y\right\}.
  2. ( X X ) ( ( X X ) Y ) (X\in X)\iff((X\in X)\to Y)
  3. ( X X ) ( ( X X ) Y ) (X\in X)\to((X\in X)\to Y)
  4. ( X X ) Y (X\in X)\to Y
  5. ( ( X X ) Y ) ( X X ) ((X\in X)\to Y)\to(X\in X)
  6. X X X\in X
  7. Y Y
  8. { x ( x x ) Y } \left\{x\mid(x\in x)\to Y\right\}
  9. f = λ x . x y f=\lambda x.x\to y
  10. x = x y x=x\to y
  11. f x = y f = λ x . y f\ x=y\iff f=\lambda x.y
  12. f f
  13. f x = y f\ x=y
  14. y = x 2 x = y y=x^{2}\iff x=\sqrt{y}
  15. 4 \sqrt{4}
  16. + 2 +2
  17. f x = y f\ x=y
  18. f f
  19. A , X , X = A \forall A,\exists X,X=A
  20. X \exists X
  21. A \forall A
  22. f , A , f ( A ) = A \exists f,\forall A,f(A)=A
  23. f ( X Y ) = X Y f(X\to Y)=X\to Y

Curry–Howard_correspondence.html

  1. α Γ Γ α Assum \frac{\alpha\in\Gamma}{\Gamma\vdash\alpha}\qquad\qquad\,\text{Assum}
  2. x : α Γ Γ x : α \frac{x:\alpha\in\Gamma}{\Gamma\vdash x:\alpha}
  3. Γ α ( β α ) Ax K \frac{}{\Gamma\vdash\alpha\rightarrow(\beta\rightarrow\alpha)}\qquad\,\text{Ax% }_{K}
  4. Γ K : α ( β α ) \frac{}{\Gamma\vdash K:\alpha\rightarrow(\beta\rightarrow\alpha)}
  5. Γ ( α ( β γ ) ) ( ( α β ) ( α γ ) ) Ax S \frac{}{\Gamma\vdash(\alpha\!\rightarrow\!(\beta\!\rightarrow\!\gamma))\!% \rightarrow\!((\alpha\!\rightarrow\!\beta)\!\rightarrow\!(\alpha\!\rightarrow% \!\gamma))}\;\,\text{Ax}_{S}
  6. Γ S : ( α ( β γ ) ) ( ( α β ) ( α γ ) ) \frac{}{\Gamma\vdash S:(\alpha\!\rightarrow\!(\beta\!\rightarrow\!\gamma))\!% \rightarrow\!((\alpha\!\rightarrow\!\beta)\!\rightarrow\!(\alpha\!\rightarrow% \!\gamma))}
  7. Γ α β Γ α Γ β Modus Ponens \frac{\Gamma\vdash\alpha\rightarrow\beta\qquad\Gamma\vdash\alpha}{\Gamma\vdash% \beta}\quad\,\text{Modus Ponens}
  8. Γ E 1 : α β Γ E 2 : α Γ E 1 E 2 : β \frac{\Gamma\vdash E_{1}:\alpha\rightarrow\beta\qquad\Gamma\vdash E_{2}:\alpha% }{\Gamma\vdash E_{1}\;E_{2}:\beta}
  9. Γ 1 , α , Γ 2 α Ax \frac{}{\Gamma_{1},\alpha,\Gamma_{2}\vdash\alpha}\,\text{Ax}
  10. Γ 1 , x : α , Γ 2 x : α \frac{}{\Gamma_{1},x:\alpha,\Gamma_{2}\vdash x:\alpha}
  11. Γ , α β Γ α β I \frac{\Gamma,\alpha\vdash\beta}{\Gamma\vdash\alpha\rightarrow\beta}\rightarrow I
  12. Γ , x : α t : β Γ λ x . t : α β \frac{\Gamma,x:\alpha\vdash t:\beta}{\Gamma\vdash\lambda x.t:\alpha\rightarrow\beta}
  13. Γ α β Γ α Γ β E \frac{\Gamma\vdash\alpha\rightarrow\beta\qquad\Gamma\vdash\alpha}{\Gamma\vdash% \beta}\rightarrow E
  14. Γ t : α β Γ u : α Γ t u : β \frac{\Gamma\vdash t:\alpha\rightarrow\beta\qquad\Gamma\vdash u:\alpha}{\Gamma% \vdash t\;u:\beta}
  15. \Box
  16. \Diamond
  17. \top
  18. α \alpha
  19. β \beta
  20. α × β \alpha\times\beta
  21. α β \alpha\rightarrow\beta
  22. i d id
  23. \star
  24. eval \operatorname{eval}
  25. π 1 \pi_{1}
  26. π 2 \pi_{2}
  27. t t
  28. λ t \lambda t
  29. t t
  30. u u
  31. ( t , u ) (t,u)
  32. u t u\circ t
  33. f : α β f:\alpha\to\beta
  34. f : - α β f:\!\!-~{}~{}\alpha~{}\vdash~{}\beta
  35. i d : - α α \frac{}{id:\!\!-~{}~{}\alpha~{}\vdash~{}\alpha}
  36. t : - α β u : - β γ u t : - α γ \frac{t:\!\!-~{}~{}\alpha~{}\vdash~{}\beta\qquad u:\!\!-~{}~{}\beta~{}\vdash~{% }\gamma}{u\circ t:\!\!-~{}\alpha~{}\vdash~{}\gamma}
  37. : - α \frac{}{\star:\!\!-~{}~{}\alpha~{}\vdash~{}\top}
  38. t : - α β u : - α γ ( t , u ) : - α β × γ \frac{t:\!\!-~{}~{}\alpha~{}\vdash~{}\beta\qquad u:\!\!-~{}~{}\alpha~{}\vdash~% {}\gamma}{(t,u):\!\!-~{}~{}\alpha~{}\vdash~{}\beta\times\gamma}
  39. π 1 : - α × β α π 2 : - α × β β \frac{}{\pi_{1}:\!\!-~{}~{}\alpha\times\beta~{}\vdash~{}\alpha}\qquad\frac{}{% \pi_{2}:\!\!-~{}~{}\alpha\times\beta~{}\vdash~{}\beta}
  40. t : - α × β γ λ t : - α β γ \frac{t:\!\!-~{}~{}\alpha\times\beta~{}\vdash~{}\gamma}{\lambda t:\!\!-~{}~{}% \alpha~{}\vdash~{}\beta\rightarrow\gamma}
  41. e v a l : - ( α β ) × α β \frac{}{eval:\!\!-~{}~{}(\alpha\rightarrow\beta)\times\alpha~{}\vdash~{}\beta}
  42. i d t = t id\circ t=t
  43. t i d = t t\circ id=t
  44. ( v u ) t = v ( u t ) (v\circ u)\circ t=v\circ(u\circ t)
  45. t = \star\circ t=\star
  46. π 1 ( t , u ) = t , π 2 ( t , u ) = u , ( π 1 t , π 2 t ) = t \pi_{1}\circ(t,u)=t,\pi_{2}\circ(t,u)=u,(\pi_{1}\circ t,\pi_{2}\circ t)=t
  47. e v a l ( λ t π 1 , π 2 ) = t , λ e v a l = i d eval\circ(\lambda t\circ\pi_{1},\pi_{2})=t,\lambda eval=id
  48. t t
  49. t : - α 1 × × α n β t:\!\!-~{}\alpha_{1}\times\ldots\times\alpha_{n}\vdash\beta
  50. α 1 , , α n β \alpha_{1},\ldots,\alpha_{n}\vdash\beta

Curve.html

  1. I I
  2. \mathbb{R}
  3. γ \!\,\gamma
  4. γ : I X \,\!\gamma:I\rightarrow X
  5. X X
  6. γ \!\,\gamma
  7. x x
  8. y y
  9. I I
  10. γ ( x ) = γ ( y ) \,\!\gamma(x)=\gamma(y)
  11. x = y x=y
  12. I I
  13. [ a , b ] \,\![a,b]
  14. γ ( a ) = γ ( b ) \,\!\gamma(a)=\gamma(b)
  15. γ ( x ) = γ ( y ) \gamma(x)=\gamma(y)
  16. x y x\neq y
  17. I I
  18. γ ( x ) \gamma(x)
  19. γ \!\,\gamma
  20. I = [ a , b ] \,\!I=[a,b]
  21. γ ( a ) = γ ( b ) \!\,\gamma(a)=\gamma(b)
  22. S 1 S^{1}
  23. X X
  24. d d
  25. γ : [ a , b ] X \!\,\gamma:[a,b]\rightarrow X
  26. length ( γ ) = sup { i = 1 n d ( γ ( t i ) , γ ( t i - 1 ) ) : n and a = t 0 < t 1 < < t n = b } . \,\text{length}(\gamma)=\sup\left\{\sum_{i=1}^{n}d(\gamma(t_{i}),\gamma(t_{i-1% })):n\in\mathbb{N}\,\text{ and }a=t_{0}<t_{1}<\cdots<t_{n}=b\right\}.
  27. n n
  28. t 0 < t 1 < < t n t_{0}<t_{1}<\cdots<t_{n}
  29. [ a , b ] [a,b]
  30. γ \!\,\gamma
  31. t 1 t_{1}
  32. t 2 t_{2}
  33. [ a , b ] [a,b]
  34. length ( γ | [ t 1 , t 2 ] ) = | t 2 - t 1 | . \,\text{length}(\gamma|_{[t_{1},t_{2}]})=|t_{2}-t_{1}|.
  35. γ \!\,\gamma
  36. γ \!\,\gamma
  37. t 0 t_{0}
  38. speed ( t 0 ) = lim sup t t 0 d ( γ ( t ) , γ ( t 0 ) ) | t - t 0 | \,\text{speed}(t_{0})=\limsup_{t\to t_{0}}{d(\gamma(t),\gamma(t_{0}))\over|t-t% _{0}|}
  39. length ( γ ) = a b speed ( t ) d t . \,\text{length}(\gamma)=\int_{a}^{b}\,\text{speed}(t)\,dt.
  40. X = n X=\mathbb{R}^{n}
  41. γ : [ a , b ] n \gamma:[a,b]\rightarrow\mathbb{R}^{n}
  42. length ( γ ) = a b | γ ( t ) | d t . \,\text{length}(\gamma)=\int_{a}^{b}|\gamma^{\prime}(t)|\,dt.
  43. X X
  44. X X
  45. X X
  46. X X
  47. X X
  48. X X
  49. γ : I X . \!\,\gamma:I\rightarrow X.
  50. X X
  51. C k C^{k}
  52. k k
  53. C k C^{k}
  54. X X
  55. C k C^{k}
  56. k k
  57. X X
  58. γ \gamma
  59. γ \gamma
  60. C k C^{k}
  61. γ 1 : I X \!\,\gamma_{1}:I\rightarrow X
  62. γ 2 : J X \!\,\gamma_{2}:J\rightarrow X
  63. C k C^{k}
  64. p : J I \!\,p:J\rightarrow I
  65. p - 1 : I J \!\,p^{-1}:I\rightarrow J
  66. C k C^{k}
  67. γ 2 ( t ) = γ 1 ( p ( t ) ) \!\,\gamma_{2}(t)=\gamma_{1}(p(t))
  68. t t
  69. γ 2 \gamma_{2}
  70. γ 1 \gamma_{1}
  71. C k C^{k}
  72. X X
  73. C k C^{k}
  74. C k C^{k}

Curve_sketching.html

  1. x 3 + y 3 - 3 a x y = 0 x^{3}+y^{3}-3axy=0\,
  2. x 2 - 3 a y = 0 x^{2}-3ay=0\,
  3. y 2 - 3 a x = 0 y^{2}-3ax=0\,

CW.html

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

CX.html

  1. c d , c_{\mathrm{d}}\,,
  2. c x c_{\mathrm{x}}\,
  3. c w \!\ c_{\mathrm{w}}

Cycle_space.html

  1. \Z 2 \Z_{2}
  2. S S
  3. S S
  4. S S
  5. S S
  6. G G
  7. G G
  8. H 1 ( G , \Z 2 ) H_{1}(G,\Z_{2})
  9. \Z 2 \Z_{2}
  10. H 1 ( G , \Z ) . H_{1}(G,\Z).
  11. G G
  12. G G
  13. H 1 ( G , \Z ) H_{1}(G,\Z)
  14. H 1 ( G , \Z k ) H_{1}(G,\Z_{k})
  15. k k
  16. k k
  17. n n
  18. m m
  19. c c
  20. m - n + c m-n+c
  21. 2 m - n + c 2^{m-n+c}
  22. m - n + c m-n+c
  23. e e
  24. C e C_{e}
  25. e e
  26. e e
  27. C e C_{e}
  28. e e
  29. m - n + c m-n+c
  30. k k
  31. Z k Z_{k}

Cyclic_quadrilateral.html

  1. α + γ = β + δ = π = 180 . \alpha+\gamma=\beta+\delta=\pi=180^{\circ}.
  2. A C B = A D B . \angle ACB=\angle ADB.
  3. e f = a c + b d . \displaystyle ef=ac+bd.
  4. A X X C = B X X D . \displaystyle AX\cdot XC=BX\cdot XD.
  5. tan α 2 tan γ 2 = tan β 2 tan δ 2 = 1. \tan{\frac{\alpha}{2}}\tan{\frac{\gamma}{2}}=\tan{\frac{\beta}{2}}\tan{\frac{% \delta}{2}}=1.
  6. K = ( s - a ) ( s - b ) ( s - c ) ( s - d ) K=\sqrt{(s-a)(s-b)(s-c)(s-d)}\,
  7. s = 1 2 ( a + b + c + d ) s=\tfrac{1}{2}(a+b+c+d)
  8. K = 1 2 ( a b + c d ) sin B K=\tfrac{1}{2}(ab+cd)\sin{B}
  9. K = 1 2 ( a c + b d ) sin θ K=\tfrac{1}{2}(ac+bd)\sin{\theta}
  10. K = 1 4 ( a 2 - b 2 - c 2 + d 2 ) tan A . K=\tfrac{1}{4}(a^{2}-b^{2}-c^{2}+d^{2})\tan{A}.
  11. K = 2 R 2 sin A sin B sin θ \displaystyle K=2R^{2}\sin{A}\sin{B}\sin{\theta}
  12. K 2 R 2 K\leq 2R^{2}
  13. p = ( a c + b d ) ( a d + b c ) a b + c d p=\sqrt{\frac{(ac+bd)(ad+bc)}{ab+cd}}
  14. q = ( a c + b d ) ( a b + c d ) a d + b c q=\sqrt{\frac{(ac+bd)(ab+cd)}{ad+bc}}
  15. p q = a c + b d . pq=ac+bd.
  16. p q = a d + b c a b + c d \frac{p}{q}=\frac{ad+bc}{ab+cd}
  17. p + q 2 a c + b d . p+q\geq 2\sqrt{ac+bd}.
  18. ( p + q ) 2 ( a + c ) 2 + ( b + d ) 2 . (p+q)^{2}\leq(a+c)^{2}+(b+d)^{2}.
  19. M N E F = 1 2 | A C B D - B D A C | \frac{MN}{EF}=\frac{1}{2}\left|\frac{AC}{BD}-\frac{BD}{AC}\right|
  20. A E C E = A B C B A D C D . \frac{AE}{CE}=\frac{AB}{CB}\cdot\frac{AD}{CD}.
  21. cos A = a 2 + d 2 - b 2 - c 2 2 ( a d + b c ) , \cos A=\frac{a^{2}+d^{2}-b^{2}-c^{2}}{2(ad+bc)},
  22. sin A = 2 ( s - a ) ( s - b ) ( s - c ) ( s - d ) ( a d + b c ) , \sin A=\frac{2\sqrt{(s-a)(s-b)(s-c)(s-d)}}{(ad+bc)},
  23. tan A 2 = ( s - a ) ( s - d ) ( s - b ) ( s - c ) . \tan\frac{A}{2}=\sqrt{\frac{(s-a)(s-d)}{(s-b)(s-c)}}.
  24. tan θ 2 = ( s - b ) ( s - d ) ( s - a ) ( s - c ) . \tan\frac{\theta}{2}=\sqrt{\frac{(s-b)(s-d)}{(s-a)(s-c)}}.
  25. ϕ \phi
  26. cos ϕ 2 = ( s - b ) ( s - d ) ( b + d ) 2 ( a b + c d ) ( a d + b c ) \cos{\frac{\phi}{2}}=\sqrt{\frac{(s-b)(s-d)(b+d)^{2}}{(ab+cd)(ad+bc)}}
  27. R = 1 4 ( a b + c d ) ( a c + b d ) ( a d + b c ) ( s - a ) ( s - b ) ( s - c ) ( s - d ) . R=\frac{1}{4}\sqrt{\frac{(ab+cd)(ac+bd)(ad+bc)}{(s-a)(s-b)(s-c)(s-d)}}.
  28. 4 K R = ( a b + c d ) ( a c + b d ) ( a d + b c ) 4KR=\sqrt{(ab+cd)(ac+bd)(ad+bc)}
  29. P G a = 4 3 P G v . PG_{a}=\tfrac{4}{3}PG_{v}.
  30. a = [ t ( u + v ) + ( 1 - u v ) ] [ u + v - t ( 1 - u v ) ] a=[t(u+v)+(1-uv)][u+v-t(1-uv)]
  31. b = ( 1 + u 2 ) ( v - t ) ( 1 + t v ) b=(1+u^{2})(v-t)(1+tv)
  32. c = t ( 1 + u 2 ) ( 1 + v 2 ) c=t(1+u^{2})(1+v^{2})
  33. d = ( 1 + v 2 ) ( u - t ) ( 1 + t u ) d=(1+v^{2})(u-t)(1+tu)
  34. e = u ( 1 + t 2 ) ( 1 + v 2 ) e=u(1+t^{2})(1+v^{2})
  35. f = v ( 1 + t 2 ) ( 1 + u 2 ) f=v(1+t^{2})(1+u^{2})
  36. K = u v [ 2 t ( 1 - u v ) - ( u + v ) ( 1 - t 2 ) ] [ 2 ( u + v ) t + ( 1 - u v ) ( 1 - t 2 ) ] K=uv[2t(1-uv)-(u+v)(1-t^{2})][2(u+v)t+(1-uv)(1-t^{2})]
  37. 4 R = ( 1 + u 2 ) ( 1 + v 2 ) ( 1 + t 2 ) . 4R=(1+u^{2})(1+v^{2})(1+t^{2}).
  38. D 2 = p 1 2 + p 2 2 + q 1 2 + q 2 2 = a 2 + c 2 = b 2 + d 2 D^{2}=p_{1}^{2}+p_{2}^{2}+q_{1}^{2}+q_{2}^{2}=a^{2}+c^{2}=b^{2}+d^{2}
  39. R = 1 2 p 1 2 + p 2 2 + q 1 2 + q 2 2 R=\tfrac{1}{2}\sqrt{p_{1}^{2}+p_{2}^{2}+q_{1}^{2}+q_{2}^{2}}
  40. R = 1 2 a 2 + c 2 = 1 2 b 2 + d 2 . R=\tfrac{1}{2}\sqrt{a^{2}+c^{2}}=\tfrac{1}{2}\sqrt{b^{2}+d^{2}}.
  41. a 2 + b 2 + c 2 + d 2 = 8 R 2 . a^{2}+b^{2}+c^{2}+d^{2}=8R^{2}.
  42. R = p 2 + q 2 + 4 x 2 8 . R=\sqrt{\frac{p^{2}+q^{2}+4x^{2}}{8}}.
  43. K = 1 2 ( a c + b d ) . K=\tfrac{1}{2}(ac+bd).

Cyclotomic_polynomial.html

  1. x n - 1 x^{n}-1
  2. x k - 1 x^{k}-1
  3. e 2 i π k n e^{2i\pi\frac{k}{n}}
  4. Φ n ( x ) = gcd ( k , n ) = 1 1 k n ( x - e 2 i π k n ) \Phi_{n}(x)=\prod_{\stackrel{1\leq k\leq n}{\gcd(k,n)=1}}\left(x-e^{2i\pi\frac% {k}{n}}\right)
  5. e 2 i π / n e^{2i\pi/n}
  6. x n - 1 = 1 k n ( x - e 2 π i k / n ) = d n 1 k n gcd ( k , n ) = d ( x - e 2 π i k / n ) = d n Φ n / d ( x ) = d n Φ d ( x ) x^{n}-1=\prod_{1\leq k\leq n}\left(x-e^{2\pi\cdot\mathrm{i}k/n}\right)=\prod_{% d\mid n}\prod_{1\leq k\leq n\atop\operatorname{gcd}(k,n)=d}\left(x-e^{2\pi% \cdot\mathrm{i}k/n}\right)=\prod_{d\mid n}\Phi_{n/d}(x)=\prod_{d\mid n}\Phi_{d% }(x)
  7. Φ n ( x ) = 1 + x + x 2 + + x n - 1 = i = 0 n - 1 x i . ~{}\Phi_{n}(x)=1+x+x^{2}+\cdots+x^{n-1}=\sum_{i=0}^{n-1}x^{i}.
  8. Φ 2 p ( x ) = 1 - x + x 2 - + x p - 1 = i = 0 p - 1 ( - x ) i . ~{}\Phi_{2p}(x)=1-x+x^{2}-\cdots+x^{p-1}=\sum_{i=0}^{p-1}(-x)^{i}.
  9. Φ 1 ( x ) = x - 1 ~{}\Phi_{1}(x)=x-1
  10. Φ 2 ( x ) = x + 1 ~{}\Phi_{2}(x)=x+1
  11. Φ 3 ( x ) = x 2 + x + 1 ~{}\Phi_{3}(x)=x^{2}+x+1
  12. Φ 4 ( x ) = x 2 + 1 ~{}\Phi_{4}(x)=x^{2}+1
  13. Φ 5 ( x ) = x 4 + x 3 + x 2 + x + 1 ~{}\Phi_{5}(x)=x^{4}+x^{3}+x^{2}+x+1
  14. Φ 6 ( x ) = x 2 - x + 1 ~{}\Phi_{6}(x)=x^{2}-x+1
  15. Φ 7 ( x ) = x 6 + x 5 + x 4 + x 3 + x 2 + x + 1 ~{}\Phi_{7}(x)=x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1
  16. Φ 8 ( x ) = x 4 + 1 ~{}\Phi_{8}(x)=x^{4}+1
  17. Φ 9 ( x ) = x 6 + x 3 + 1 ~{}\Phi_{9}(x)=x^{6}+x^{3}+1
  18. Φ 10 ( x ) = x 4 - x 3 + x 2 - x + 1 ~{}\Phi_{10}(x)=x^{4}-x^{3}+x^{2}-x+1
  19. Φ 11 ( x ) = x 10 + x 9 + x 8 + x 7 + x 6 + x 5 + x 4 + x 3 + x 2 + x + 1 ~{}\Phi_{11}(x)=x^{10}+x^{9}+x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1
  20. Φ 12 ( x ) = x 4 - x 2 + 1 ~{}\Phi_{12}(x)=x^{4}-x^{2}+1
  21. Φ 13 ( x ) = x 12 + x 11 + x 10 + x 9 + x 8 + x 7 + x 6 + x 5 + x 4 + x 3 + x 2 + x + 1 ~{}\Phi_{13}(x)=x^{12}+x^{11}+x^{10}+x^{9}+x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}% +x^{2}+x+1
  22. Φ 14 ( x ) = x 6 - x 5 + x 4 - x 3 + x 2 - x + 1 ~{}\Phi_{14}(x)=x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+1
  23. Φ 15 ( x ) = x 8 - x 7 + x 5 - x 4 + x 3 - x + 1 ~{}\Phi_{15}(x)=x^{8}-x^{7}+x^{5}-x^{4}+x^{3}-x+1
  24. Φ 16 ( x ) = x 8 + 1 ~{}\Phi_{16}(x)=x^{8}+1
  25. Φ 17 ( x ) = x 16 + x 15 + x 14 + x 13 + x 12 + x 11 + x 10 + x 9 + x 8 + x 7 + x 6 + x 5 + x 4 + x 3 + x 2 + x + 1 ~{}\Phi_{17}(x)=x^{16}+x^{15}+x^{14}+x^{13}+x^{12}+x^{11}+x^{10}+x^{9}+x^{8}+x% ^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1
  26. Φ 18 ( x ) = x 6 - x 3 + 1 ~{}\Phi_{18}(x)=x^{6}-x^{3}+1
  27. Φ 19 ( x ) = x 18 + x 17 + x 16 + x 15 + x 14 + x 13 + x 12 + x 11 + x 10 + x 9 + x 8 + x 7 + x 6 + x 5 + x 4 + x 3 + x 2 + x + 1 ~{}\Phi_{19}(x)=x^{18}+x^{17}+x^{16}+x^{15}+x^{14}+x^{13}+x^{12}+x^{11}+x^{10}% +x^{9}+x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1
  28. Φ 20 ( x ) = x 8 - x 6 + x 4 - x 2 + 1 ~{}\Phi_{20}(x)=x^{8}-x^{6}+x^{4}-x^{2}+1
  29. Φ 21 ( x ) = x 12 - x 11 + x 9 - x 8 + x 6 - x 4 + x 3 - x + 1 ~{}\Phi_{21}(x)=x^{12}-x^{11}+x^{9}-x^{8}+x^{6}-x^{4}+x^{3}-x+1
  30. Φ 22 ( x ) = x 10 - x 9 + x 8 - x 7 + x 6 - x 5 + x 4 - x 3 + x 2 - x + 1 ~{}\Phi_{22}(x)=x^{10}-x^{9}+x^{8}-x^{7}+x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+1
  31. Φ 23 ( x ) = x 22 + x 21 + x 20 + x 19 + x 18 + x 17 + x 16 + x 15 + x 14 + x 13 + x 12 + x 11 + x 10 + x 9 + x 8 + x 7 + x 6 + x 5 + x 4 + x 3 + x 2 + x + 1 ~{}\Phi_{23}(x)=x^{22}+x^{21}+x^{20}+x^{19}+x^{18}+x^{17}+x^{16}+x^{15}+x^{14}% +x^{13}+x^{12}+x^{11}+x^{10}+x^{9}+x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1
  32. Φ 24 ( x ) = x 8 - x 4 + 1 ~{}\Phi_{24}(x)=x^{8}-x^{4}+1
  33. Φ 25 ( x ) = x 20 + x 15 + x 10 + x 5 + 1 ~{}\Phi_{25}(x)=x^{20}+x^{15}+x^{10}+x^{5}+1
  34. Φ 26 ( x ) = x 12 - x 11 + x 10 - x 9 + x 8 - x 7 + x 6 - x 5 + x 4 - x 3 + x 2 - x + 1 ~{}\Phi_{26}(x)=x^{12}-x^{11}+x^{10}-x^{9}+x^{8}-x^{7}+x^{6}-x^{5}+x^{4}-x^{3}% +x^{2}-x+1
  35. Φ 27 ( x ) = x 18 + x 9 + 1 ~{}\Phi_{27}(x)=x^{18}+x^{9}+1
  36. Φ 28 ( x ) = x 12 - x 10 + x 8 - x 6 + x 4 - x 2 + 1 ~{}\Phi_{28}(x)=x^{12}-x^{10}+x^{8}-x^{6}+x^{4}-x^{2}+1
  37. Φ 29 ( x ) = x 28 + x 27 + x 26 + x 25 + x 24 + x 23 + x 22 + x 21 + x 20 + x 19 + x 18 + x 17 + x 16 + x 15 + x 14 + x 13 + x 12 + x 11 + x 10 + x 9 + x 8 + x 7 + x 6 + x 5 + x 4 + x 3 + x 2 + x + 1 ~{}\Phi_{29}(x)=x^{28}+x^{27}+x^{26}+x^{25}+x^{24}+x^{23}+x^{22}+x^{21}+x^{20}% +x^{19}+x^{18}+x^{17}+x^{16}+x^{15}+x^{14}+x^{13}+x^{12}+x^{11}+x^{10}+x^{9}+x% ^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1
  38. Φ 30 ( x ) = x 8 + x 7 - x 5 - x 4 - x 3 + x + 1 ~{}\Phi_{30}(x)=x^{8}+x^{7}-x^{5}-x^{4}-x^{3}+x+1
  39. Φ 105 ( x ) = \displaystyle\Phi_{105}(x)=
  40. Φ n \Phi_{n}
  41. φ ( n ) \varphi(n)
  42. φ \varphi
  43. Φ n \Phi_{n}
  44. φ ( n ) \varphi(n)
  45. [ x ] \mathbb{Z}[x]
  46. d n Φ d ( x ) = x n - 1 , \prod_{d\mid n}\Phi_{d}(x)=x^{n}-1,
  47. Φ n ( x ) \Phi_{n}(x)
  48. Φ n ( x ) = d n ( x d - 1 ) μ ( n / d ) , \Phi_{n}(x)=\prod_{d\mid n}(x^{d}-1)^{\mu(n/d)},
  49. μ \mu
  50. Φ n ( x ) \Phi_{n}(x)
  51. x n - 1 x^{n}-1
  52. Φ n ( x ) = x n - 1 d < n d | n Φ d ( x ) \Phi_{n}(x)=\frac{x^{n}-1}{\prod_{\stackrel{d|n}{{}_{d<n}}}\Phi_{d}(x)}
  53. Φ 1 ( x ) = x - 1 \Phi_{1}(x)=x-1
  54. Φ n ( x ) \Phi_{n}(x)
  55. Φ n ( x ) = 1 + x + x 2 + + x n - 1 = i = 0 n - 1 x i . \Phi_{n}(x)=1+x+x^{2}+\cdots+x^{n-1}=\sum_{i=0}^{n-1}x^{i}.
  56. Φ 2 n ( x ) = Φ n ( - x ) . \Phi_{2n}(x)=\Phi_{n}(-x).
  57. Φ n ( x ) = 1 - x + x 2 - + x p - 1 = i = 0 p - 1 ( - x ) i . \Phi_{n}(x)=1-x+x^{2}-\cdots+x^{p-1}=\sum_{i=0}^{p-1}(-x)^{i}.
  58. Φ n ( x ) = Φ p ( x p m - 1 ) = i = 0 p - 1 x i p m - 1 . \Phi_{n}(x)=\Phi_{p}(x^{p^{m-1}})=\sum_{i=0}^{p-1}x^{ip^{m-1}}.
  59. Φ n ( x ) = Φ q r ( x q m - 1 ) . \Phi_{n}(x)=\Phi_{qr}(x^{q^{m-1}}).
  60. Φ n ( x ) \Phi_{n}(x)
  61. Φ n ( x ) = Φ q ( x n / q ) . \Phi_{n}(x)=\Phi_{q}(x^{n/q}).
  62. Φ 2 h ( x ) = x 2 h - 1 + 1 \Phi_{2^{h}}(x)=x^{2^{h-1}}+1
  63. Φ p k ( x ) = i = 0 p - 1 x i p k - 1 \Phi_{p^{k}}(x)=\sum_{i=0}^{p-1}x^{ip^{k-1}}
  64. Φ 2 h p k ( x ) = i = 0 p - 1 ( - 1 ) i x i 2 h - 1 p k - 1 \Phi_{2^{h}p^{k}}(x)=\sum_{i=0}^{p-1}(-1)^{i}x^{i2^{h-1}p^{k-1}}
  65. Φ q ( x ) , \Phi_{q}(x),
  66. Φ n p ( x ) = Φ n ( x p ) / Φ n ( x ) . \Phi_{np}(x)=\Phi_{n}(x^{p})/\Phi_{n}(x)\,.
  67. Φ n \Phi_{n}
  68. Φ 105 ( x ) ; \Phi_{105}(x);
  69. Φ 231 ( x ) \Phi_{231}(x)
  70. Φ 3 × 7 × 11 ( x ) \Phi_{3\times 7\times 11}(x)
  71. Φ 15015 ( x ) \Phi_{15015}(x)
  72. Φ 3 × 5 × 7 × 11 × 13 ( x ) \Phi_{3\times 5\times 7\times 11\times 13}(x)
  73. Φ 255255 ( x ) \Phi_{255255}(x)
  74. Φ 3 × 5 × 7 × 11 × 13 × 17 ( x ) \Phi_{3\times 5\times 7\times 11\times 13\times 17}(x)
  75. 4 Φ n ( z ) = A n 2 ( z ) - ( - 1 ) n - 1 2 n z 2 B n 2 ( z ) 4\Phi_{n}(z)=A_{n}^{2}(z)-(-1)^{\frac{n-1}{2}}nz^{2}B_{n}^{2}(z)
  76. 4 Φ 5 ( z ) = 4 ( z 4 + z 3 + z 2 + z + 1 ) = ( 2 z 2 + z + 2 ) 2 - 5 z 2 \begin{aligned}\displaystyle 4\Phi_{5}(z)&\displaystyle=4(z^{4}+z^{3}+z^{2}+z+% 1)\\ &\displaystyle=(2z^{2}+z+2)^{2}-5z^{2}\end{aligned}
  77. 4 Φ 7 ( z ) = 4 ( z 6 + z 5 + z 4 + z 3 + z 2 + z + 1 ) = ( 2 z 3 + z 2 - z - 2 ) 2 + 7 z 2 ( z + 1 ) 2 \begin{aligned}\displaystyle 4\Phi_{7}(z)&\displaystyle=4(z^{6}+z^{5}+z^{4}+z^% {3}+z^{2}+z+1)\\ &\displaystyle=(2z^{3}+z^{2}-z-2)^{2}+7z^{2}(z+1)^{2}\end{aligned}
  78. 4 Φ 11 ( z ) = 4 ( z 10 + z 9 + z 8 + z 7 + z 6 + z 5 + z 4 + z 3 + z 2 + z + 1 ) = ( 2 z 5 + z 4 - 2 z 3 + 2 z 2 - z - 2 ) 2 + 11 z 2 ( z 3 + 1 ) 2 \begin{aligned}\displaystyle 4\Phi_{11}(z)&\displaystyle=4(z^{10}+z^{9}+z^{8}+% z^{7}+z^{6}+z^{5}+z^{4}+z^{3}+z^{2}+z+1)\\ &\displaystyle=(2z^{5}+z^{4}-2z^{3}+2z^{2}-z-2)^{2}+11z^{2}(z^{3}+1)^{2}\end{aligned}
  79. Φ n ( z ) = U n 2 ( z ) - ( - 1 ) n - 1 2 n z V n 2 ( z ) \Phi_{n}(z)=U_{n}^{2}(z)-(-1)^{\frac{n-1}{2}}nzV_{n}^{2}(z)
  80. Φ n ( ( - 1 ) n - 1 2 z ) = C n 2 ( z ) - n z D n 2 ( z ) . \Phi_{n}((-1)^{\frac{n-1}{2}}z)=C_{n}^{2}(z)-nzD_{n}^{2}(z).
  81. Φ n / 2 ( - z 2 ) = C n 2 ( z ) - n z D n 2 ( z ) \Phi_{n/2}(-z^{2})=C_{n}^{2}(z)-nzD_{n}^{2}(z)
  82. Φ 3 ( - z ) \displaystyle\Phi_{3}(-z)
  83. Φ 5 ( z ) = z 4 + z 3 + z 2 + z + 1 = ( z 2 + 3 z + 1 ) 2 - 5 z ( z + 1 ) 2 \begin{aligned}\displaystyle\Phi_{5}(z)&\displaystyle=z^{4}+z^{3}+z^{2}+z+1\\ &\displaystyle=(z^{2}+3z+1)^{2}-5z(z+1)^{2}\end{aligned}
  84. Φ 3 ( - z 2 ) = z 4 - z 2 + 1 = ( z 2 + 3 z + 1 ) 2 - 6 z ( z + 1 ) 2 \begin{aligned}\displaystyle\Phi_{3}(-z^{2})&\displaystyle=z^{4}-z^{2}+1\\ &\displaystyle=(z^{2}+3z+1)^{2}-6z(z+1)^{2}\end{aligned}
  85. Φ n \Phi_{n}
  86. φ ( n ) \varphi(n)
  87. Φ n \Phi_{n}
  88. Φ n ( b ) \Phi_{n}(b)
  89. Φ n ( b ) \Phi_{n}(b)
  90. Φ p ( b ) \Phi_{p}(b)
  91. b p - 1 b - 1 \frac{b^{p}-1}{b-1}
  92. Φ p ( b ) \Phi_{p}(b)
  93. Φ n \Phi_{n}

Cyclotron_radiation.html

  1. - d E d t = σ t B 2 V 2 c μ o {-dE\over dt}={\sigma_{t}B^{2}V^{2}\over c\mu_{o}}
  2. σ t \sigma_{t}
  3. μ o \mu_{o}

Cylindrical_coordinate_system.html

  1. x = ρ cos φ x=\rho\cos\varphi
  2. y = ρ sin φ y=\rho\sin\varphi
  3. ρ = x 2 + y 2 \rho=\sqrt{x^{2}+y^{2}}
  4. φ = { 0 if x = 0 and y = 0 arcsin ( y ρ ) if x 0 arctan ( y x ) if x 0 - arcsin ( y ρ ) + π if x < 0 \varphi=\begin{cases}0&\mbox{if }~{}x=0\mbox{ and }~{}y=0\\ \arcsin(\frac{y}{\rho})&\mbox{if }~{}x\geq 0\\ \arctan(\frac{y}{x})&\mbox{if }~{}x\geq 0\\ -\arcsin(\frac{y}{\rho})+\pi&\mbox{if }~{}x<0\end{cases}
  5. ρ = r cos θ \rho=r\cos\theta\,
  6. ρ = r sin θ \rho=r\sin\theta\,
  7. φ = φ \varphi=\varphi\,
  8. φ = φ \varphi=\varphi\,
  9. z = r sin θ z=r\sin\theta\,
  10. z = r cos θ z=r\cos\theta\,
  11. r = ρ 2 + z 2 r=\sqrt{\rho^{2}+z^{2}}
  12. r = ρ 2 + z 2 r=\sqrt{\rho^{2}+z^{2}}
  13. θ = arctan ( z / ρ ) {\theta}=\operatorname{arctan}(z/\rho)
  14. θ = arctan ( ρ / z ) {\theta}=\operatorname{arctan}(\rho/z)
  15. φ = φ {\varphi}=\varphi\quad
  16. φ = φ {\varphi}=\varphi\quad
  17. d 𝐫 = d ρ s y m b o l ρ ^ + ρ d φ s y m b o l φ ^ + d z 𝐳 ^ . \mathrm{d}\mathbf{r}=\mathrm{d}\rho\,symbol{\hat{\rho}}+\rho\,\mathrm{d}% \varphi\,symbol{\hat{\varphi}}+\mathrm{d}z\,\mathbf{\hat{z}}.
  18. d V = ρ d ρ d φ d z . \mathrm{d}V=\rho\,\mathrm{d}\rho\,\mathrm{d}\varphi\,\mathrm{d}z.
  19. ρ \rho
  20. d S ρ = ρ d φ d z . \mathrm{d}S_{\rho}=\rho\,\mathrm{d}\varphi\,\mathrm{d}z.
  21. φ \varphi
  22. d S φ = d ρ d z . \mathrm{d}S_{\varphi}=\mathrm{d}\rho\,\mathrm{d}z.
  23. z z
  24. d S z = ρ d ρ d φ . \mathrm{d}S_{z}=\rho\,\mathrm{d}\rho\,\mathrm{d}\varphi.
  25. f = f ρ s y m b o l ρ ^ + 1 ρ f φ s y m b o l φ ^ + f z 𝐳 ^ , \nabla f=\frac{\partial f}{\partial\rho}symbol{\hat{\rho}}+\frac{1}{\rho}\frac% {\partial f}{\partial\varphi}symbol{\hat{\varphi}}+\frac{\partial f}{\partial z% }\mathbf{\hat{z}},
  26. s y m b o l A = 1 ρ ρ ( ρ A ρ ) + 1 ρ A φ φ + A z z \nabla\cdot symbol{A}=\frac{1}{\rho}\frac{\partial}{\partial\rho}(\rho A_{\rho% })+\frac{1}{\rho}\frac{\partial A_{\varphi}}{\partial\varphi}+\frac{\partial A% _{z}}{\partial z}
  27. × s y m b o l A = ( 1 ρ A z φ - A φ z ) s y m b o l ρ ^ + ( A ρ z - A z ρ ) s y m b o l φ ^ + 1 ρ ( ρ ( ρ A φ ) - A ρ φ ) 𝐳 ^ \nabla\times symbol{A}=\left(\frac{1}{\rho}\frac{\partial A_{z}}{\partial% \varphi}-\frac{\partial A_{\varphi}}{\partial z}\right)symbol{\hat{\rho}}+% \left(\frac{\partial A_{\rho}}{\partial z}-\frac{\partial A_{z}}{\partial\rho}% \right)symbol{\hat{\varphi}}+\frac{1}{\rho}\left(\frac{\partial}{\partial\rho}% (\rho A_{\varphi})-\frac{\partial A_{\rho}}{\partial\varphi}\right)\mathbf{% \hat{z}}
  28. 2 f = 1 ρ ρ ( ρ f ρ ) + 1 ρ 2 2 f φ 2 + 2 f z 2 . \nabla^{2}f={1\over\rho}{\partial\over\partial\rho}\left(\rho{\partial f\over% \partial\rho}\right)+{1\over\rho^{2}}{\partial^{2}f\over\partial\varphi^{2}}+{% \partial^{2}f\over\partial z^{2}}.

Czochralski_process.html

  1. k O k_{O}
  2. V 0 V_{0}
  3. I 0 I_{0}
  4. C 0 C_{0}
  5. V L V_{L}
  6. I L I_{L}
  7. C L C_{L}
  8. V S V_{S}
  9. C S C_{S}
  10. d V dV
  11. d I = - k O C L d V dI=-k_{O}C_{L}dV\;
  12. d I = - k O I L V O - V S d V dI=-k_{O}\frac{I_{L}}{V_{O}-V_{S}}dV
  13. I O I L d I I L = - k O 0 V S d V V O - V S \int_{I_{O}}^{I_{L}}\frac{dI}{I_{L}}=-k_{O}\int_{0}^{V_{S}}\frac{dV}{V_{O}-V_{% S}}
  14. ln ( I L I O ) = ln ( 1 - V S V O ) k O \ln\left(\frac{I_{L}}{I_{O}}\right)=\ln\left(1-\frac{V_{S}}{V_{O}}\right)^{k_{% O}}
  15. I L = I O ( 1 - V S V O ) k O I_{L}=I_{O}\left(1-\frac{V_{S}}{V_{O}}\right)^{k_{O}}
  16. C S = - d I L d V S C_{S}=-\frac{dI_{L}}{dV_{S}}
  17. C S = C O k O ( 1 - f ) k o - 1 C_{S}=C_{O}k_{O}(1-f)^{k_{o}-1}
  18. f = V S / V O f=V_{S}/V_{O}\;

D'Alembert_operator.html

  1. \Box
  2. \displaystyle\Box
  3. 2 \nabla^{2}
  4. g μ ν g^{\mu\nu}
  5. g 00 = 1 g_{00}\,=\,1
  6. g 11 = g 22 = g 33 = - 1 g_{11}\,=\,g_{22}\,=\,g_{33}\,=\,-1
  7. g μ ν = 0 g_{\mu\nu}\,=\,0
  8. μ ν \mu\,\neq\,\nu
  9. c = 1 c\,=\,1
  10. g 00 = - 1 , g 11 = g 22 = g 33 = 1 g_{00}\,=\,-1,\;g_{11}\,=\,g_{22}\,=\,g_{33}\,=\,1
  11. \scriptstyle\Box
  12. 2 \scriptstyle\Box^{2}
  13. Δ M \scriptstyle\Delta_{M}
  14. 2 \scriptstyle\partial^{2}
  15. \scriptstyle\Box
  16. \scriptstyle\nabla
  17. c u ( x , t ) u t t - c 2 u x x = 0 , \Box_{c}u\left(x,t\right)\equiv u_{tt}-c^{2}u_{xx}=0,\,
  18. u ( x , t ) \scriptstyle u\left(x,t\right)
  19. A μ = 0 \Box A^{\mu}=0
  20. A μ A^{\mu}
  21. ( + m 2 ) ψ = 0. (\Box+m^{2})\psi=0.\,
  22. G ( x - x ) \scriptstyle G(x-x^{\prime})
  23. G ( x ~ - x ~ ) = δ ( x ~ - x ~ ) \Box G\left(\tilde{x}-\tilde{x}^{\prime}\right)=\delta\left(\tilde{x}-\tilde{x% }^{\prime}\right)
  24. δ ( x ~ - x ~ ) \scriptstyle\delta(\tilde{x}-\tilde{x}^{\prime})
  25. x ~ \scriptstyle\tilde{x}
  26. x ~ \scriptstyle\tilde{x}^{\prime}
  27. G ( r , t ) = 1 4 π r Θ ( t ) δ ( t - r c ) G(\vec{r},t)=\frac{1}{4\pi r}\Theta(t)\delta\left(t-\frac{r}{c}\right)
  28. Θ \scriptstyle\,\Theta

D'Hondt_method.html

  1. q u o t = V s + 1 quot=\frac{V}{s+1}
  2. V 2 s \textstyle\frac{V}{2^{s}}

Danica_McKellar.html

  1. 2 \mathbb{Z}^{2}

Daraf.html

  1. 1 F - 1 = 1 V 1 C = 1 Ω 1 s 1~{}\,\text{F}^{-1}=\frac{1~{}\,\text{V}}{1~{}\,\text{C}}=\frac{1~{}\Omega}{1~% {}\,\text{s}}

Darcy–Weisbach_equation.html

  1. h f = f D L D u 2 2 g h_{f}=f_{D}\cdot\frac{L}{D}\cdot\frac{u^{2}}{2g}
  2. Δ p = ρ g h f \Delta p=\rho\cdot g\cdot h_{f}
  3. Δ p = f D L D ρ u 2 2 \Delta p=f_{D}\cdot\frac{L}{D}\cdot\frac{\rho u^{2}}{2}
  4. Δ p = f D L D ρ u 2 2 = f L D 2 ρ u 2 \Delta p=f_{D}\cdot\frac{L}{D}\cdot\frac{\rho u^{2}}{2}=f\cdot\frac{L}{D}\cdot% {2\rho u^{2}}
  5. Δ p L D 1 2 ρ u 2 . \Delta p\propto\frac{L}{D}\cdot\frac{1}{2}\rho u^{2}.
  6. u 2 = Q 2 A w 2 u^{2}=\frac{Q^{2}}{A_{w}^{2}}
  7. A w 2 = ( π D 2 4 ) 2 = π 2 D 4 16 A_{w}^{2}=\left(\frac{\pi D^{2}}{4}\right)^{2}=\frac{\pi^{2}D^{4}}{16}
  8. h f = 8 f D L Q 2 g π 2 D 5 h_{f}=\frac{8f_{D}LQ^{2}}{g\pi^{2}D^{5}}

Darlington_transistor.html

  1. β Darlington = β 1 β 2 + β 1 + β 2 \beta_{\mathrm{Darlington}}=\beta_{1}\cdot\beta_{2}+\beta_{1}+\beta_{2}
  2. β Darlington β 1 β 2 \beta_{\mathrm{Darlington}}\approx\beta_{1}\cdot\beta_{2}
  3. V B E = V B E 1 + V B E 2 2 V B E 1 V_{BE}=V_{BE1}+V_{BE2}\approx 2V_{BE1}\!
  4. V CE2 = V CE1 + V BE2 > V BE2 V C2 > V B2 \mathrm{V_{CE2}=V_{CE1}+V_{BE2}>V_{BE2}}\Rightarrow\mathrm{V_{C2}>V_{B2}}

Data_type.html

  1. 2 32 - 1 2^{32}-1
  2. - 2 31 -2^{31}
  3. 2 31 - 1 2^{31}-1

Dating_the_Bible.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}
  5. 𝔓 \mathfrak{P}
  6. 𝔓 \mathfrak{P}
  7. 𝔓 \mathfrak{P}
  8. 𝔓 \mathfrak{P}
  9. 𝔓 \mathfrak{P}
  10. 𝔓 \mathfrak{P}
  11. 𝔓 \mathfrak{P}
  12. 𝔓 \mathfrak{P}
  13. 𝔓 \mathfrak{P}
  14. 𝔓 \mathfrak{P}
  15. 𝔓 \mathfrak{P}
  16. 𝔓 \mathfrak{P}
  17. 𝔓 \mathfrak{P}
  18. 𝔓 \mathfrak{P}
  19. 𝔓 \mathfrak{P}
  20. 𝔓 \mathfrak{P}
  21. 𝔓 \mathfrak{P}
  22. 𝔓 \mathfrak{P}
  23. 𝔓 \mathfrak{P}
  24. 𝔓 \mathfrak{P}
  25. 𝔓 \mathfrak{P}
  26. 𝔓 \mathfrak{P}
  27. 𝔓 \mathfrak{P}
  28. 𝔓 \mathfrak{P}

De_Branges's_theorem.html

  1. f ( z ) = z + n 2 a n z n f(z)=z+\sum_{n\geq 2}a_{n}z^{n}
  2. | a n | n for all n 2. \left|a_{n}\right|\leq n\quad\,\text{for all }n\geq 2.\,
  3. f ( z ) = g ( z ) - g ( 0 ) g ( 0 ) . f(z)=\frac{g(z)-g(0)}{g^{\prime}(0)}.\,
  4. f α ( z ) = z ( 1 - α z ) 2 = n = 1 n α n - 1 z n f_{\alpha}(z)=\frac{z}{(1-\alpha z)^{2}}=\sum_{n=1}^{\infty}n\alpha^{n-1}z^{n}
  5. f ( z ) = z + z 2 = ( z + 1 / 2 ) 2 - 1 / 4 f(z)=z+z^{2}=(z+1/2)^{2}-1/4\;
  6. ϕ ( z ) = b 1 z + b 3 z 3 + b 5 z 5 + \phi(z)=b_{1}z+b_{3}z^{3}+b_{5}z^{5}+\cdots
  7. k = 1 n | b 2 k + 1 | 2 n . \sum_{k=1}^{n}|b_{2k+1}|^{2}\leq n.
  8. k = 1 n ( n - k + 1 ) ( k | γ k | 2 - 1 / k ) 0 \sum^{n}_{k=1}(n-k+1)(k|\gamma_{k}|^{2}-1/k)\leq 0
  9. log ( f ( z ) / z ) = 2 n = 1 γ n z n . \log(f(z)/z)=2\sum^{\infty}_{n=1}\gamma_{n}z^{n}.
  10. ρ n = Γ ( 2 ν + n + 1 ) Γ ( n + 1 ) ( σ n - σ n + 1 ) \rho_{n}=\frac{\Gamma(2\nu+n+1)}{\Gamma(n+1)}(\sigma_{n}-\sigma_{n+1})
  11. F ( z ) ν - z ν ν = n = 1 a n z ν + n \frac{F(z)^{\nu}-z^{\nu}}{\nu}=\sum_{n=1}^{\infty}a_{n}z^{\nu+n}
  12. n = 1 ( ν + n ) σ n | a n | 2 \sum_{n=1}^{\infty}(\nu+n)\sigma_{n}|a_{n}|^{2}

De_Finetti's_theorem.html

  1. X 1 , X 2 , X 3 , X_{1},X_{2},X_{3},\dots\!
  2. X i 1 , , X i n and X j 1 , , X j n X_{i_{1}},\dots,X_{i_{n}}\,\text{ and }X_{j_{1}},\dots,X_{j_{n}}\!
  3. X 1 , X 2 , X 3 , X_{1},X_{2},X_{3},\ldots
  4. X 1 , X 2 , X 3 , X_{1},X_{2},X_{3},\ldots
  5. X 1 , X 2 , X_{1},X_{2},\ldots
  6. X 1 , X 2 , X 3 , X_{1},X_{2},X_{3},\dots\!
  7. lim n X 1 + + X n n = { 2 / 3 with probability 1 / 2 , 9 / 10 with probability 1 / 2. \lim_{n\rightarrow\infty}\frac{X_{1}+\cdots+X_{n}}{n}=\begin{cases}2/3&\,\text% {with probability }1/2,\\ 9/10&\,\text{with probability }1/2.\end{cases}

De_Rham_cohomology.html

  1. M M
  2. 0 Ω 0 ( M ) d Ω 1 ( M ) d Ω 2 ( M ) d Ω 3 ( M ) 0\to\Omega^{0}(M)\ \stackrel{d}{\to}\ \Omega^{1}(M)\ \stackrel{d}{\to}\ \Omega% ^{2}(M)\ \stackrel{d}{\to}\ \Omega^{3}(M)\to\cdots
  3. M M
  4. 1 1
  5. 0
  6. 0
  7. 1 1
  8. d θ
  9. θ θ
  10. d θ
  11. 2 π
  12. θ θ
  13. α β α−β
  14. k k
  15. H dR k ( M ) H^{k}_{\mathrm{dR}}(M)
  16. M M
  17. n n
  18. H dR 0 ( M ) 𝐑 n . H^{0}_{\mathrm{dR}}(M)\cong\mathbf{R}^{n}.
  19. M M
  20. M M
  21. n n
  22. n n
  23. n > 0 , m 0 n>0,m≥0
  24. I I
  25. H dR k ( S n × I m ) { 𝐑 if k = 0 , n , 0 if k 0 , n . H_{\mathrm{dR}}^{k}(S^{n}\times I^{m})\simeq\begin{cases}\mathbf{R}&\mbox{if }% ~{}k=0,n,\\ 0&\mbox{if }~{}k\neq 0,n.\end{cases}
  26. n n
  27. n > 0 n>0
  28. H dR k ( T n ) 𝐑 ( n k ) . H_{\mathrm{dR}}^{k}(T^{n})\simeq\mathbf{R}^{n\choose k}.
  29. n , H dR k ( 𝐑 n { 0 } ) { 𝐑 if k = 0 , n - 1 0 if k 0 , n - 1 H dR k ( S n - 1 ) \begin{aligned}\displaystyle\forall n\in\mathbb{N},H_{\mathrm{dR}}^{k}(\mathbf% {R}^{n}\setminus\{\vec{0}\})&\displaystyle\simeq\begin{cases}\mathbf{R}&\mbox{% if }~{}k=0,n-1\\ 0&\mbox{if }~{}k\neq 0,n-1\end{cases}\\ &\displaystyle\simeq H_{\mathrm{dR}}^{k}(S^{n-1})\end{aligned}
  30. M M
  31. 1 1
  32. H dR k ( M ) H dR k ( S 1 ) . H_{\mathrm{dR}}^{k}(M)\simeq H_{\mathrm{dR}}^{k}(S^{1}).
  33. H dR k ( M ) H^{k}_{\mathrm{dR}}(M)
  34. M M
  35. I : H d R p ( M ) H p ( M ; 𝐑 ) , I:H_{dR}^{p}(M)\to H^{p}(M;\mathbf{R}),
  36. [ ω ] H d R p ( M ) [\omega]\in H_{dR}^{p}(M)
  37. I ( ω ) I(ω)
  38. Hom ( H p ( M ; 𝐑 ) , 𝐑 ) H p ( M ; 𝐑 ) \,\text{Hom}(H_{p}(M;\mathbf{R}),\mathbf{R})\simeq H^{p}(M;\mathbf{R})
  39. H p ( M ) [ c ] c ω . H_{p}(M)\ni[c]\longmapsto\int_{c}\omega.
  40. F F
  41. F ( U ) = 𝐑 F(U)=\mathbf{R}
  42. U M U⊂M
  43. U , V U,V
  44. U V U⊂V
  45. 𝐑 \mathbf{R}
  46. 𝐔 \mathbf{U}
  47. M M
  48. 𝐔 \mathbf{U}
  49. 𝐔 \mathbf{U}
  50. M M
  51. m m
  52. k m k≤m
  53. H dR k ( M ) H ˇ k ( M , 𝐑 ) H^{k}_{\mathrm{dR}}(M)\cong\check{H}^{k}(M,\mathbf{R})
  54. k k
  55. 𝐑 \mathbf{R}
  56. k k
  57. M M
  58. M M
  59. 0 𝐑 Ω 0 𝑑 Ω 1 𝑑 Ω 2 𝑑 𝑑 Ω m 0. 0\to\mathbf{R}\to\Omega^{0}\,\xrightarrow{d}\,\Omega^{1}\,\xrightarrow{d}\,% \Omega^{2}\,\xrightarrow{d}\dots\xrightarrow{d}\,\Omega^{m}\to 0.
  60. 0 d Ω k - 1 Ω k 𝑑 d Ω k 0. 0\to d\Omega^{k-1}\,\xrightarrow{\subset}\,\Omega^{k}\,\xrightarrow{d}\,d% \Omega^{k}\to 0.
  61. i > 0 i>0
  62. M M
  63. H dR k ( M ) H^{k}_{\mathrm{dR}}(M)
  64. ω ω
  65. ω = d α + γ \omega=d\alpha+\gamma
  66. α α
  67. γ γ
  68. Δ γ = 0 Δγ=0
  69. 2 2
  70. 1 1
  71. 2 2
  72. n n
  73. k k
  74. n n
  75. k k
  76. H dR k ( T n ) H^{k}_{\,\text{dR}}(T^{n})
  77. k k
  78. n n
  79. n n
  80. k k
  81. M M
  82. Δ Δ
  83. Δ = d δ + δ d \Delta=d\delta+\delta d
  84. d d
  85. δ δ
  86. k k
  87. M M
  88. k k
  89. k k
  90. k k
  91. M M
  92. k k
  93. δ δ
  94. ω ω
  95. δ ω = 0 δω=0
  96. ω = δ α ω=δα
  97. α α
  98. k k
  99. ω = d α + δ β + γ \omega=d\alpha+\delta\beta+\gamma
  100. γ γ
  101. Δ γ = 0 Δγ=0
  102. ( α , β ) = M α * β . (\alpha,\beta)=\int_{M}\alpha\wedge*\beta.

De_Sitter_space.html

  1. Λ \Lambda
  2. d s 2 = - d x 0 2 + i = 1 n d x i 2 . ds^{2}=-dx_{0}^{2}+\sum_{i=1}^{n}dx_{i}^{2}.
  3. - x 0 2 + i = 1 n x i 2 = α 2 -x_{0}^{2}+\sum_{i=1}^{n}x_{i}^{2}=\alpha^{2}
  4. α \alpha
  5. α 2 \alpha^{2}
  6. - α 2 -\alpha^{2}
  7. R ρ σ μ ν = 1 α 2 ( g ρ μ g σ ν - g ρ ν g σ μ ) R_{\rho\sigma\mu\nu}={1\over\alpha^{2}}(g_{\rho\mu}g_{\sigma\nu}-g_{\rho\nu}g_% {\sigma\mu})
  8. R μ ν = n - 1 α 2 g μ ν R_{\mu\nu}=\frac{n-1}{\alpha^{2}}g_{\mu\nu}
  9. Λ = ( n - 1 ) ( n - 2 ) 2 α 2 . \Lambda=\frac{(n-1)(n-2)}{2\alpha^{2}}.
  10. R = n ( n - 1 ) α 2 = 2 n n - 2 Λ . R=\frac{n(n-1)}{\alpha^{2}}=\frac{2n}{n-2}\Lambda.
  11. ( t , r , ) (t,r,\ldots)
  12. x 0 = α 2 - r 2 sinh ( t / α ) x_{0}=\sqrt{\alpha^{2}-r^{2}}\sinh(t/\alpha)
  13. x 1 = α 2 - r 2 cosh ( t / α ) x_{1}=\sqrt{\alpha^{2}-r^{2}}\cosh(t/\alpha)
  14. x i = r z i 2 i n . x_{i}=rz_{i}\qquad\qquad\qquad\qquad\qquad 2\leq i\leq n.
  15. z i z_{i}
  16. d s 2 = - ( 1 - r 2 α 2 ) d t 2 + ( 1 - r 2 α 2 ) - 1 d r 2 + r 2 d Ω n - 2 2 . ds^{2}=-\left(1-\frac{r^{2}}{\alpha^{2}}\right)dt^{2}+\left(1-\frac{r^{2}}{% \alpha^{2}}\right)^{-1}dr^{2}+r^{2}d\Omega_{n-2}^{2}.
  17. r = α r=\alpha
  18. x 0 = α sinh ( t / α ) + r 2 e t / α / 2 α , x_{0}=\alpha\sinh(t/\alpha)+r^{2}e^{t/\alpha}/2\alpha,
  19. x 1 = α cosh ( t / α ) - r 2 e t / α / 2 α , x_{1}=\alpha\cosh(t/\alpha)-r^{2}e^{t/\alpha}/2\alpha,
  20. x i = e t / α y i , 2 i n x_{i}=e^{t/\alpha}y_{i},\qquad 2\leq i\leq n
  21. r 2 = i y i 2 r^{2}=\sum_{i}y_{i}^{2}
  22. ( t , y i ) (t,y_{i})
  23. d s 2 = - d t 2 + e 2 t / α d y 2 ds^{2}=-dt^{2}+e^{2t/\alpha}dy^{2}
  24. d y 2 = i d y i 2 dy^{2}=\sum_{i}dy_{i}^{2}
  25. y i y_{i}
  26. x 0 = α sinh ( t / α ) cosh ξ , x_{0}=\alpha\sinh(t/\alpha)\cosh\xi,
  27. x 1 = α cosh ( t / α ) , x_{1}=\alpha\cosh(t/\alpha),
  28. x i = α z i sinh ( t / α ) sinh ξ , 2 i n x_{i}=\alpha z_{i}\sinh(t/\alpha)\sinh\xi,\qquad 2\leq i\leq n
  29. i z i 2 = 1 \sum_{i}z_{i}^{2}=1
  30. S n - 2 S^{n-2}
  31. i d z i 2 = d Ω n - 2 2 \sum_{i}dz_{i}^{2}=d\Omega_{n-2}^{2}
  32. d s 2 = - d t 2 + α 2 sinh 2 ( t / α ) d H n - 1 2 , ds^{2}=-dt^{2}+\alpha^{2}\sinh^{2}(t/\alpha)dH_{n-1}^{2},
  33. d H n - 1 2 = d ξ 2 + sinh 2 ξ d Ω n - 2 2 dH_{n-1}^{2}=d\xi^{2}+\sinh^{2}\xi d\Omega_{n-2}^{2}
  34. x 0 = α sinh ( t / α ) , x_{0}=\alpha\sinh(t/\alpha),
  35. x i = α cosh ( t / α ) z i , 1 i n x_{i}=\alpha\cosh(t/\alpha)z_{i},\qquad 1\leq i\leq n
  36. z i z_{i}
  37. S n - 1 S^{n-1}
  38. d s 2 = - d t 2 + α 2 cosh 2 ( t / α ) d Ω n - 1 2 . ds^{2}=-dt^{2}+\alpha^{2}\cosh^{2}(t/\alpha)d\Omega_{n-1}^{2}.
  39. tan ( η / 2 ) = tanh ( t / 2 α ) \tan(\eta/2)=\tanh(t/2\alpha)
  40. d s 2 = α 2 cos 2 η ( - d η 2 + d Ω n - 1 2 ) . ds^{2}=\frac{\alpha^{2}}{\cos^{2}\eta}(-d\eta^{2}+d\Omega_{n-1}^{2}).
  41. x 0 = α sin ( χ / α ) sinh ( t / α ) cosh ξ , x_{0}=\alpha\sin(\chi/\alpha)\sinh(t/\alpha)\cosh\xi,
  42. x 1 = α cos ( χ / α ) , x_{1}=\alpha\cos(\chi/\alpha),
  43. x 2 = α sin ( χ / α ) cosh ( t / α ) , x_{2}=\alpha\sin(\chi/\alpha)\cosh(t/\alpha),
  44. x i = α z i sin ( χ / α ) sinh ( t / α ) sinh ξ , 3 i n x_{i}=\alpha z_{i}\sin(\chi/\alpha)\sinh(t/\alpha)\sinh\xi,\qquad 3\leq i\leq n
  45. z i z_{i}
  46. S n - 3 S^{n-3}
  47. d s 2 = d χ 2 + sin 2 ( χ / α ) d s d S , α , n - 1 2 , ds^{2}=d\chi^{2}+\sin^{2}(\chi/\alpha)ds_{dS,\alpha,n-1}^{2},
  48. d s d S , α , n - 1 2 = - d t 2 + α 2 sinh 2 ( t / α ) d H n - 2 2 ds_{dS,\alpha,n-1}^{2}=-dt^{2}+\alpha^{2}\sinh^{2}(t/\alpha)dH_{n-2}^{2}
  49. n - 1 n-1
  50. α \alpha
  51. d H n - 2 2 = d ξ 2 + sinh 2 ξ d Ω n - 3 2 . dH_{n-2}^{2}=d\xi^{2}+\sinh^{2}\xi d\Omega_{n-3}^{2}.
  52. ( t , ξ , θ , ϕ 1 , ϕ 2 , , ϕ n - 3 ) ( i χ , ξ , i t , θ , ϕ 1 , , ϕ n - 4 ) (t,\xi,\theta,\phi_{1},\phi_{2},\cdots,\phi_{n-3})\to(i\chi,\xi,it,\theta,\phi% _{1},\cdots,\phi_{n-4})
  53. x 0 x_{0}
  54. x 2 x_{2}

Dead_reckoning.html

  1. P t = P 0 + V 0 T + 1 2 A 0 T 2 P_{t}=P_{0}+V_{0}T+\frac{1}{2}A_{0}T^{2}
  2. V b = V 0 + ( V ´ 0 - V 0 ) T ^ V_{b}=V_{0}+\left(\acute{V}_{0}-V_{0}\right)\hat{T}
  3. P t = P 0 + V b T t + 1 2 A ´ 0 T t 2 P_{t}=P_{0}+V_{b}T_{t}+\frac{1}{2}\acute{A}_{0}T_{t}^{2}
  4. P ´ t = P ´ 0 + V ´ 0 T t + 1 2 A ´ 0 T t 2 \acute{P}_{t}=\acute{P}_{0}+\acute{V}_{0}T_{t}+\frac{1}{2}\acute{A}_{0}T_{t}^{2}
  5. P o s = P t + ( P ´ t - P t ) T ^ Pos=P_{t}+\left(\acute{P}_{t}-P_{t}\right)\hat{T}

Dead_space_(physiology).html

  1. V d V t = P a C O 2 - P e C O 2 P a C O 2 \frac{V_{\,d}}{V_{\,t}}=\frac{P_{\,a\,CO_{2}}-P_{\,e\,CO_{2}}}{P_{\,a\,CO_{2}}}
  2. V d V_{\,d}
  3. V t V_{\,t}
  4. P a C O 2 P_{\,a\,CO_{2}}
  5. P e C O 2 P_{\,e\,CO_{2}}
  6. V p h y s i o l o g i c a l d e a d s p a c e V t = P a C O 2 - P m i x e d e x p i r e d C O 2 P a C O 2 \frac{V_{\,physiological\,dead\,space}}{V_{t}}=\frac{P_{\,a\,CO_{2}}-P_{\,% mixed\,expired\,CO_{2}}}{P_{\,a\,CO_{2}}}
  7. V a l v e o l a r d e a d s p a c e V t = P a C O 2 - P e n d t i d a l C O 2 P a C O 2 \frac{V_{\,alveolar\,dead\,space}}{V_{t}}=\frac{P_{\,a\,CO_{2}}-P_{\,end\ % tidal\,CO_{2}}}{P_{\,a\,CO_{2}}}
  8. V a l v e o l a r d e a d s p a c e 500 m L = 42 m m H g - 40 m m H g 42 m m H g \frac{V_{\,alveolar\,dead\,space}}{500\ mL}=\frac{42\ mmHg-40\ mmHg}{42\ mmHg}
  9. V a l v e o l a r d e a d s p a c e = 24 m L . V_{\,alveolar\,dead\,space}=24\ mL.

Debits_and_credits.html

  1. A s s e t s = E q u i t y + L i a b i l i t i e s Assets=Equity+Liabilities
  2. A = E + L A=E+L
  3. A s s e t s + E x p e n s e s = E q u i t y / C a p i t a l + L i a b i l i t i e s + I n c o m e Assets+Expenses=Equity/Capital+Liabilities+Income
  4. A + E x = E + L + I A+Ex=E+L+I

Deborah_number.html

  1. De = t c t p , \mathrm{De}=\frac{t_{\mathrm{c}}}{t_{\mathrm{p}}},

Debt.html

  1. d / ( 1 - d ) % . d/(1-d)\%.

Decay_energy.html

  1. Q = ( Kinetic energy ) after - ( Kinetic energy ) before , Q=\left(\,\text{Kinetic energy}\right)_{\,\text{after}}-\left(\,\text{Kinetic % energy}\right)_{\,\text{before}},
  2. Q = ( ( Rest mass ) before × c 2 ) - ( ( Rest mass ) after × c 2 ) . Q=\left(\left(\,\text{Rest mass}\right)_{\,\text{before}}\times c^{2}\right)-% \left(\left(\,\text{Rest mass}\right)_{\,\text{after}}\times c^{2}\right).
  3. W = d m × ( A M ) . W=dm\times\left(\frac{A}{M}\right).
  4. W = E × ( A M ) . W=E\times\left(\frac{A}{M}\right).

Decay_product.html

  1. U-238 Th-234 daughter of U-238 Pa-234m granddaughter of U-238 Pb-206 decay products of U-238 \mbox{U-238}~{}\rightarrow\overbrace{\underbrace{\mbox{Th-234}~{}}_{\mbox{% daughter of U-238}~{}}\rightarrow\underbrace{\mbox{Pa-234m}~{}}_{\mbox{% granddaughter of U-238}~{}}\rightarrow\ldots\rightarrow\mbox{Pb-206}~{}}^{% \begin{array}[]{c}\mbox{decay products of U-238}\end{array}}

Deconvolution.html

  1. f * g = h f*g=h\,
  2. ( f * g ) + ε = h (f*g)+\varepsilon=h\,
  3. s ( t ) = ( e * w ) ( t ) . s(t)=(e*w)(t).\,
  4. S ( ω ) = E ( ω ) W ( ω ) S(\omega)=E(\omega)W(\omega)\,
  5. | S ( ω ) | k | W ( ω ) | . |S(\omega)|\approx k|W(\omega)|.\,
  6. e ( t ) = i = 1 N r i δ ( t - τ i ) e(t)=\sum_{i=1}^{N}r_{i}\delta(t-\tau_{i})

Dedekind_domain.html

  1. m m
  2. x 2 + m y 2 x^{2}+my^{2}
  3. ( x + - m y ) ( x - - m y ) (x+\sqrt{-m}y)(x-\sqrt{-m}y)
  4. ( - m ) \mathbb{Q}(\sqrt{-m})
  5. n n
  6. z n - y n z^{n}-y^{n}
  7. x n + y n = z n x^{n}+y^{n}=z^{n}
  8. [ ζ n ] \mathbb{Z}[\zeta_{n}]
  9. ζ n \zeta_{n}
  10. n n
  11. m m
  12. n n
  13. m = 1 , n = 4 m=1,n=4
  14. m = 2 , 3 , n = 3 m=2,3,n=3
  15. ( D ) \mathbb{Q}(\sqrt{D})
  16. D < 0 D<0
  17. n > 2 n>2
  18. [ ζ n ] \mathbb{Z}[\zeta_{n}]
  19. n = 23 n=23
  20. [ ζ n ] \mathbb{Z}[\zeta_{n}]
  21. n n
  22. [ ζ n ] \mathbb{Z}[\zeta_{n}]
  23. 𝒪 K \mathcal{O}_{K}
  24. K K
  25. p p
  26. ( p ) \mathbb{Q}(\sqrt{p})
  27. K K
  28. 𝒪 K \mathcal{O}_{K}
  29. R R
  30. M M
  31. R R
  32. R M R_{M}
  33. R R
  34. R R
  35. R R
  36. R R
  37. R = 𝒪 K R=\mathcal{O}_{K}
  38. 𝐐 ¯ \overline{\,\textbf{Q}}
  39. 𝐙 ¯ \overline{\,\textbf{Z}}
  40. 𝐙 ¯ \overline{\,\textbf{Z}}
  41. x I R . xI\subset R.
  42. n i n j n , i n I , j n J \sum_{n}i_{n}j_{n},\,i_{n}\in I,\,j_{n}\in J
  43. I * = ( R : I ) = { x K | x I R } . I^{*}=(R:I)=\{x\in K\ |\ xI\subset R\}.
  44. I * I R I^{*}I\subset R
  45. I * I^{*}
  46. x R xR
  47. x R xR
  48. 1 x R \frac{1}{x}R
  49. K × / R × K^{\times}/R^{\times}
  50. x R xR
  51. y R yR
  52. x y - 1 xy^{-1}
  53. \rightarrow
  54. M M
  55. R R
  56. T T
  57. m m
  58. M M
  59. r m = 0 rm=0
  60. r r
  61. R R
  62. T T
  63. R / I R/I
  64. I I
  65. R R
  66. R / I R/I
  67. R / P i R/P^{i}
  68. P i P^{i}
  69. T R / P 1 a 1 R / P r a r R / Q 1 b 1 R / Q s b s T\cong R/P_{1}^{a_{1}}\oplus\cdots\oplus R/P_{r}^{a_{r}}\cong R/Q_{1}^{b_{1}}% \oplus\cdots\oplus R/Q_{s}^{b_{s}}
  70. P P
  71. M M
  72. M = T P M=T\oplus P
  73. P P
  74. R n R^{n}
  75. n n
  76. P P
  77. M M
  78. R R
  79. P P
  80. R R
  81. P P
  82. P I 1 I r P\cong I_{1}\oplus\cdots\oplus I_{r}
  83. I 1 , , I r , J 1 , , J s I_{1},\ldots,I_{r},J_{1},\ldots,J_{s}
  84. I 1 I r J 1 J s I_{1}\oplus\cdots\oplus I_{r}\cong J_{1}\oplus\cdots\oplus J_{s}
  85. r = s r=s
  86. I 1 I r J 1 J s . I_{1}\otimes\cdots\otimes I_{r}\cong J_{1}\otimes\cdots\otimes J_{s}.\,
  87. [ I 1 I r ] = [ J 1 J s ] C l ( R ) . [I_{1}\cdots I_{r}]=[J_{1}\cdots J_{s}]\in Cl(R).
  88. n > 0 n>0
  89. R n - 1 I R^{n-1}\oplus I
  90. I I
  91. [ I ] [I]
  92. I I
  93. K 0 ( R ) C l ( R ) K_{0}(R)\cong\mathbb{Z}\oplus Cl(R)
  94. R R
  95. R R
  96. R R

Deferent_and_epicycle.html

  1. z 0 = a 0 e i k 0 t z_{0}=a_{0}e^{ik_{0}t}
  2. a 0 a_{0}
  3. k 0 k_{0}
  4. i = - 1 i=\sqrt{-1}
  5. t t
  6. a 0 a_{0}
  7. k 0 = 2 π T k_{0}=\frac{2\pi}{T}
  8. T T
  9. z 1 z_{1}
  10. z 2 = z 0 + z 1 = a 0 e i k 0 t + a 1 e i k 1 t z_{2}=z_{0}+z_{1}=a_{0}e^{ik_{0}t}+a_{1}e^{ik_{1}t}
  11. k j k_{j}
  12. N N
  13. z N = j = 0 N a j e i k j t z_{N}=\sum_{j=0}^{N}a_{j}e^{ik_{j}t}
  14. k j k_{j}
  15. a j a_{j}
  16. z = f ( t ) z=f(t)

Deflagration.html

  1. δ \delta\;
  2. τ d \tau_{d}\;
  3. τ d δ 2 / κ \tau_{d}\simeq\delta^{2}/\kappa
  4. κ \kappa\;
  5. τ b \tau_{b}
  6. τ b exp [ Δ U / ( k B T f ) ] \tau_{b}\propto\exp[\Delta U/(k_{B}T_{f})]
  7. Δ U \Delta U\;
  8. T f T_{f}\;
  9. δ \delta\;
  10. τ b = τ d \tau_{b}=\tau_{d}\;
  11. δ κ τ b \delta\simeq\sqrt{\kappa\tau_{b}}
  12. S l S_{l}\;
  13. S l δ / τ b κ / τ b S_{l}\simeq\delta/\tau_{b}\simeq\sqrt{\kappa/\tau_{b}}
  14. S l S_{l}\;

DEFLATE.html

  1. n 2 - 1 \frac{n}{2}-1

Deformation_(engineering).html

  1. σ = E ε \sigma=E\varepsilon
  2. σ \sigma
  3. E E
  4. E E

Degeneracy_(mathematics).html

  1. S { 1 , 2 , , n } S\subseteq\{1,2,\ldots,n\}
  2. R { 𝐱 n : x i = c i ( for i S ) and a i x i b i ( for i S ) } R\triangleq\left\{\mathbf{x}\in\mathbb{R}^{n}:x_{i}=c_{i}\ (\,\text{for }i\in S% )\,\text{ and }a_{i}\leq x_{i}\leq b_{i}\ (\,\text{for }i\notin S)\right\}
  3. 𝐱 [ x 1 , x 2 , , x n ] \mathbf{x}\triangleq[x_{1},x_{2},\ldots,x_{n}]
  4. a i , b i , c i a_{i},b_{i},c_{i}
  5. a i b i a_{i}\leq b_{i}
  6. i i
  7. R R
  8. S S
  9. n n
  10. R R

Degenerate_distribution.html

  1. F ( k ; k 0 ) = { 1 , if k k 0 0 , if k < k 0 F(k;k_{0})=\left\{\begin{matrix}1,&\mbox{if }~{}k\geq k_{0}\\ 0,&\mbox{if }~{}k<k_{0}\end{matrix}\right.
  2. c c\in\mathbb{R}
  3. Pr ( X = c ) = 1 , \Pr(X=c)=1,
  4. X ( ω ) = c , ω Ω . X(\omega)=c,\quad\forall\omega\in\Omega.
  5. F ( x ) = { 1 , x c , 0 , x < c . F(x)=\begin{cases}1,&x\geq c,\\ 0,&x<c.\end{cases}

Del.html

  1. n n
  2. ( x 1 , , x n ) (x_{1},\dots,x_{n})
  3. { e 1 , , e n } \{\vec{e}_{1},\dots,\vec{e}_{n}\}
  4. = ( x 1 , , x n ) = i = 1 n e i x i \nabla=\left({\partial\over\partial x_{1}},\cdots,{\partial\over\partial x_{n}% }\right)=\sum_{i=1}^{n}\vec{e}_{i}{\partial\over\partial x_{i}}
  5. ( x , y , z ) (x,y,z)
  6. { e x , e y , e z } \{\vec{e}_{x},\vec{e}_{y},\vec{e}_{z}\}
  7. = ( x , y , z ) = e x x + e y y + e z z \nabla=\left({\partial\over\partial x},{\partial\over\partial y},{\partial% \over\partial z}\right)=\vec{e}_{x}{\partial\over\partial x}+\vec{e}_{y}{% \partial\over\partial y}+\vec{e}_{z}{\partial\over\partial z}
  8. f f
  9. grad f = f x e x + f y e y + f z e z = f \operatorname{grad}f={\partial f\over\partial x}\vec{e}_{x}+{\partial f\over% \partial y}\vec{e}_{y}+{\partial f\over\partial z}\vec{e}_{z}=\nabla f
  10. f f
  11. h ( x , y ) h(x,y)
  12. ( f g ) = f g + g f \nabla(fg)=f\nabla g+g\nabla f
  13. ( u v ) = ( u ) v + ( v ) u + u × ( × v ) + v × ( × u ) \nabla(\vec{u}\cdot\vec{v})=(\vec{u}\cdot\nabla)\vec{v}+(\vec{v}\cdot\nabla)% \vec{u}+\vec{u}\times(\nabla\times\vec{v})+\vec{v}\times(\nabla\times\vec{u})
  14. v ( x , y , z ) = v x e x + v y e y + v z e z \vec{v}(x,y,z)=v_{x}\vec{e}_{x}+v_{y}\vec{e}_{y}+v_{z}\vec{e}_{z}
  15. div v = v x x + v y y + v z z = v \operatorname{div}\vec{v}={\partial v_{x}\over\partial x}+{\partial v_{y}\over% \partial y}+{\partial v_{z}\over\partial z}=\nabla\cdot\vec{v}
  16. ( f v ) = f ( v ) + v ( f ) \nabla\cdot(f\vec{v})=f(\nabla\cdot\vec{v})+\vec{v}\cdot(\nabla f)
  17. ( u × v ) = v ( × u ) - u ( × v ) \nabla\cdot(\vec{u}\times\vec{v})=\vec{v}\cdot(\nabla\times\vec{u})-\vec{u}% \cdot(\nabla\times\vec{v})
  18. v ( x , y , z ) = v x e x + v y e y + v z e z \vec{v}(x,y,z)=v_{x}\vec{e}_{x}+v_{y}\vec{e}_{y}+v_{z}\vec{e}_{z}
  19. curl v = ( v z y - v y z ) e x + ( v x z - v z x ) e y + ( v y x - v x y ) e z = × v \operatorname{curl}\vec{v}=\left({\partial v_{z}\over\partial y}-{\partial v_{% y}\over\partial z}\right)\vec{e}_{x}+\left({\partial v_{x}\over\partial z}-{% \partial v_{z}\over\partial x}\right)\vec{e}_{y}+\left({\partial v_{y}\over% \partial x}-{\partial v_{x}\over\partial y}\right)\vec{e}_{z}=\nabla\times\vec% {v}
  20. × v = | e x e y e z x y z v x v y v z | \nabla\times\vec{v}=\left|\begin{matrix}\vec{e}_{x}&\vec{e}_{y}&\vec{e}_{z}\\ {\frac{\partial}{\partial x}}&{\frac{\partial}{\partial y}}&{\frac{\partial}{% \partial z}}\\ v_{x}&v_{y}&v_{z}\end{matrix}\right|
  21. × ( f v ) = ( f ) × v + f ( × v ) \nabla\times(f\vec{v})=(\nabla f)\times\vec{v}+f(\nabla\times\vec{v})
  22. × ( u × v ) = u ( v ) - v ( u ) + ( v ) u - ( u ) v \nabla\times(\vec{u}\times\vec{v})=\vec{u}\,(\nabla\cdot\vec{v})-\vec{v}\,(% \nabla\cdot\vec{u})+(\vec{v}\cdot\nabla)\,\vec{u}-(\vec{u}\cdot\nabla)\,\vec{v}
  23. f ( x , y , z ) f(x,y,z)
  24. a ( x , y , z ) = a x e x + a y e y + a z e z \vec{a}(x,y,z)=a_{x}\vec{e}_{x}+a_{y}\vec{e}_{y}+a_{z}\vec{e}_{z}
  25. a grad f = a x f x + a y f y + a z f z = ( a ) f \vec{a}\cdot\operatorname{grad}f=a_{x}{\partial f\over\partial x}+a_{y}{% \partial f\over\partial y}+a_{z}{\partial f\over\partial z}=(\vec{a}\cdot% \nabla)f
  26. f f
  27. a \vec{a}
  28. ( a ) (\vec{a}\cdot\nabla)
  29. Δ = 2 x 2 + 2 y 2 + 2 z 2 = = 2 \Delta={\partial^{2}\over\partial x^{2}}+{\partial^{2}\over\partial y^{2}}+{% \partial^{2}\over\partial z^{2}}=\nabla\cdot\nabla=\nabla^{2}
  30. v \vec{v}
  31. v \nabla\otimes\vec{v}
  32. \otimes
  33. δ r \delta\vec{r}
  34. δ v = ( v ) \sdot δ r \delta\vec{v}=(\nabla\otimes\vec{v})\sdot\delta\vec{r}
  35. ( f g ) = f g + g f \nabla(fg)=f\nabla g+g\nabla f
  36. ( u v ) = u × ( × v ) + v × ( × u ) + ( u ) v + ( v ) u \nabla(\vec{u}\cdot\vec{v})=\vec{u}\times(\nabla\times\vec{v})+\vec{v}\times(% \nabla\times\vec{u})+(\vec{u}\cdot\nabla)\vec{v}+(\vec{v}\cdot\nabla)\vec{u}
  37. ( f v ) = f ( v ) + v ( f ) \nabla\cdot(f\vec{v})=f(\nabla\cdot\vec{v})+\vec{v}\cdot(\nabla f)
  38. ( u × v ) = v ( × u ) - u ( × v ) \nabla\cdot(\vec{u}\times\vec{v})=\vec{v}\cdot(\nabla\times\vec{u})-\vec{u}% \cdot(\nabla\times\vec{v})
  39. × ( f v ) = ( f ) × v + f ( × v ) \nabla\times(f\vec{v})=(\nabla f)\times\vec{v}+f(\nabla\times\vec{v})
  40. × ( u × v ) = u ( v ) - v ( u ) + ( v ) u - ( u ) v \nabla\times(\vec{u}\times\vec{v})=\vec{u}\,(\nabla\cdot\vec{v})-\vec{v}\,(% \nabla\cdot\vec{u})+(\vec{v}\cdot\nabla)\,\vec{u}-(\vec{u}\cdot\nabla)\,\vec{v}
  41. div ( grad f ) = ( f ) \operatorname{div}(\operatorname{grad}f)=\nabla\cdot(\nabla f)
  42. curl ( grad f ) = × ( f ) \operatorname{curl}(\operatorname{grad}f)=\nabla\times(\nabla f)
  43. Δ f = 2 f \Delta f=\nabla^{2}f
  44. grad ( div v ) = ( v ) \operatorname{grad}(\operatorname{div}\vec{v})=\nabla(\nabla\cdot\vec{v})
  45. div ( curl v ) = ( × v ) \operatorname{div}(\operatorname{curl}\vec{v})=\nabla\cdot(\nabla\times\vec{v})
  46. curl ( curl v ) = × ( × v ) \operatorname{curl}(\operatorname{curl}\vec{v})=\nabla\times(\nabla\times\vec{% v})
  47. Δ v = 2 v \Delta\vec{v}=\nabla^{2}\vec{v}
  48. curl ( grad f ) = × ( f ) = 0 \operatorname{curl}(\operatorname{grad}f)=\nabla\times(\nabla f)=0
  49. div ( curl v ) = × v = 0 \operatorname{div}(\operatorname{curl}\vec{v})=\nabla\cdot\nabla\times\vec{v}=0
  50. div ( grad f ) = ( f ) = 2 f = Δ f \operatorname{div}(\operatorname{grad}f)=\nabla\cdot(\nabla f)=\nabla^{2}f=\Delta f
  51. × × v = ( v ) - 2 v \nabla\times\nabla\times\vec{v}=\nabla(\nabla\cdot\vec{v})-\nabla^{2}\vec{v}
  52. ( v ) = ( v ) \nabla(\nabla\cdot\vec{v})=\nabla\cdot(\nabla\otimes\vec{v})
  53. ( u v ) f = ( v u ) f (\vec{u}\cdot\vec{v})f=(\vec{v}\cdot\vec{u})f
  54. ( v ) f ( v ) f (\nabla\cdot\vec{v})f\neq(\vec{v}\cdot\nabla)f
  55. ( v ) f = ( v x x + v y y + v z z ) f = v x x f + v y y f + v z z f (\nabla\cdot\vec{v})f=\left(\frac{\partial v_{x}}{\partial x}+\frac{\partial v% _{y}}{\partial y}+\frac{\partial v_{z}}{\partial z}\right)f=\frac{\partial v_{% x}}{\partial x}f+\frac{\partial v_{y}}{\partial y}f+\frac{\partial v_{z}}{% \partial z}f
  56. ( v ) f = ( v x x + v y y + v z z ) f = v x f x + v y f y + v z f z (\vec{v}\cdot\nabla)f=\left(v_{x}\frac{\partial}{\partial x}+v_{y}\frac{% \partial}{\partial y}+v_{z}\frac{\partial}{\partial z}\right)f=v_{x}\frac{% \partial f}{\partial x}+v_{y}\frac{\partial f}{\partial y}+v_{z}\frac{\partial f% }{\partial z}
  57. ( x ) × ( y ) \displaystyle(\nabla x)\times(\nabla y)
  58. ( u x ) × ( u y ) = x y ( u × u ) = x y 0 = 0 (\vec{u}x)\times(\vec{u}y)=xy(\vec{u}\times\vec{u})=xy\vec{0}=\vec{0}

Delta-v.html

  1. Δ v = t 0 t 1 | T | m d t \Delta{v}=\int_{t_{0}}^{t_{1}}{\frac{|T|}{m}}\,dt
  2. = t 0 t 1 | a | d t =\int_{t_{0}}^{t_{1}}{|a|}\,dt
  3. = | v 1 - v 0 | =|{v}_{1}-{v}_{0}|\;
  4. T = V e x h ρ T=V_{exh}\ \rho\,
  5. V ˙ \dot{V}\,
  6. V ˙ = T m = V e x h ρ m \dot{V}=\frac{T}{m}=V_{exh}\ \frac{\rho}{m}\,
  7. m ˙ = - ρ \dot{m}=-\rho\,
  8. t 0 t_{0}\,
  9. Δ V = - t 0 t 1 V e x h m ˙ m d t \Delta{V}=-\int_{t_{0}}^{t_{1}}{V_{exh}\ \frac{\dot{m}}{m}}\,dt
  10. Δ V = - m 0 m 1 V e x h d m m \Delta{V}=-\int_{m_{0}}^{m_{1}}{V_{exh}\ \frac{dm}{m}}\,
  11. V e x h V_{exh}\,
  12. Δ V = V e x h ln ( m 0 m 1 ) \Delta{V}=V_{exh}\ \ln(\frac{m_{0}}{m_{1}})\,
  13. V e x h V_{exh}\,
  14. Δ V = 2100 ln ( 1 0.8 ) \Delta{V}=2100\ \ln(\frac{1}{0.8})\,
  15. V e x h V_{exh}\,
  16. V e x h = V e x h ( m ) V_{exh}=V_{exh}(m)\,
  17. Δ V \Delta{V}\,
  18. M 1 M 2 M1M2
  19. = e V 1 / V e e V 2 / V e =e^{V1/V_{e}}e^{V2/V_{e}}
  20. = e ( V 1 + V 2 ) / V e =e^{(V1+V2)/V_{e}}
  21. = e V / V e = M =e^{V/V_{e}}=M
  22. ρ \rho\,
  23. V e x h V_{exh}\,
  24. f ( t ) m ( t ) = V e x h ( t ) m ˙ ( t ) m ( t ) \frac{f(t)}{m(t)}=V_{exh}(t)\frac{\dot{m}(t)}{m(t)}\,
  25. f ( t ) f(t)\,
  26. m ( t ) m(t)\,
  27. t t\,

Delta_operator.html

  1. Q : 𝕂 [ x ] 𝕂 [ x ] Q\colon\mathbb{K}[x]\longrightarrow\mathbb{K}[x]
  2. x x
  3. 𝕂 \mathbb{K}
  4. Q Q
  5. g ( x ) = f ( x + a ) g(x)=f(x+a)
  6. ( Q g ) ( x ) = ( Q f ) ( x + a ) . {(Qg)(x)=(Qf)(x+a)}.\,
  7. f f
  8. g g
  9. Q f Qf
  10. Q g Qg
  11. a a
  12. f f
  13. n n
  14. Q f Qf
  15. n - 1 n-1
  16. n = 0 n=0
  17. Q f Qf
  18. x x
  19. x x
  20. ( Δ f ) ( x ) = f ( x + 1 ) - f ( x ) (\Delta f)(x)=f(x+1)-f(x)\,
  21. k = 1 c k D k \sum_{k=1}^{\infty}c_{k}D^{k}
  22. c 1 0 c_{1}\neq 0
  23. Δ = e D - 1 = k = 1 D k k ! . \Delta=e^{D}-1=\sum_{k=1}^{\infty}\frac{D^{k}}{k!}.
  24. ( δ f ) ( x ) = f ( x + Δ t ) - f ( x ) Δ t , {(\delta f)(x)={{f(x+\Delta t)-f(x)}\over{\Delta t}}},
  25. Δ t \Delta t
  26. Q Q
  27. p 0 ( x ) = 1 ; p_{0}(x)=1;
  28. p n ( 0 ) = 0 ; p_{n}(0)=0;
  29. ( Q p n ) ( x ) = n p n - 1 ( x ) , n . (Qp_{n})(x)=np_{n-1}(x),\;\forall n\in\mathbb{N}.

Deming_regression.html

  1. y i \displaystyle y_{i}
  2. δ = σ ε 2 σ η 2 . \delta=\frac{\sigma_{\varepsilon}^{2}}{\sigma_{\eta}^{2}}.
  3. x x
  4. y y
  5. δ \delta
  6. x x
  7. y y
  8. δ = 1 \delta=1
  9. y * = β 0 + β 1 x * , y^{*}=\beta_{0}+\beta_{1}x^{*},
  10. S S R = i = 1 n ( ε i 2 σ ε 2 + η i 2 σ η 2 ) = 1 σ ε 2 i = 1 n ( ( y i - β 0 - β 1 x i * ) 2 + δ ( x i - x i * ) 2 ) min β 0 , β 1 , x 1 * , , x n * S S R SSR=\sum_{i=1}^{n}\bigg(\frac{\varepsilon_{i}^{2}}{\sigma_{\varepsilon}^{2}}+% \frac{\eta_{i}^{2}}{\sigma_{\eta}^{2}}\bigg)=\frac{1}{\sigma_{\varepsilon}^{2}% }\sum_{i=1}^{n}\Big((y_{i}-\beta_{0}-\beta_{1}x^{*}_{i})^{2}+\delta(x_{i}-x^{*% }_{i})^{2}\Big)\ \to\ \min_{\beta_{0},\beta_{1},x_{1}^{*},\ldots,x_{n}^{*}}SSR
  11. x ¯ = 1 n x i , y ¯ = 1 n y i , \displaystyle\overline{x}=\frac{1}{n}\sum x_{i},\quad\overline{y}=\frac{1}{n}% \sum y_{i},
  12. β ^ 1 = s y y - δ s x x + ( s y y - δ s x x ) 2 + 4 δ s x y 2 2 s x y , \displaystyle\hat{\beta}_{1}=\frac{s_{yy}-\delta s_{xx}+\sqrt{(s_{yy}-\delta s% _{xx})^{2}+4\delta s_{xy}^{2}}}{2s_{xy}},
  13. δ = 1 \delta=1
  14. Z \sqrt{Z}

Dempster–Shafer_theory.html

  1. 2 X 2^{X}\,\!
  2. \emptyset
  3. X = { a , b } X=\left\{a,b\right\}\,\!
  4. 2 X = { , { a } , { b } , X } . 2^{X}=\left\{\emptyset,\left\{a\right\},\left\{b\right\},X\right\}.\,
  5. m : 2 X [ 0 , 1 ] m:2^{X}\rightarrow[0,1]\,\!
  6. m ( ) = 0. m(\emptyset)=0.\,\!
  7. A X m ( A ) = 1 \sum_{A\subseteq X}m(A)=1\,\!
  8. bel ( A ) P ( A ) pl ( A ) . \operatorname{bel}(A)\leq P(A)\leq\operatorname{pl}(A).
  9. bel ( A ) = B B A m ( B ) . \operatorname{bel}(A)=\sum_{B\mid B\subseteq A}m(B).\,
  10. pl ( A ) = B B A m ( B ) . \operatorname{pl}(A)=\sum_{B\mid B\cap A\neq\emptyset}m(B).\,
  11. pl ( A ) = 1 - bel ( A ¯ ) . \operatorname{pl}(A)=1-\operatorname{bel}(\overline{A}).\,
  12. m ( A ) = B B A ( - 1 ) | A - B | bel ( B ) m(A)=\sum_{B\mid B\subseteq A}(-1)^{|A-B|}\operatorname{bel}(B)\,
  13. m 1 , 2 ( ) = 0 m_{1,2}(\emptyset)=0\,
  14. m 1 , 2 ( A ) = ( m 1 m 2 ) ( A ) = 1 1 - K B C = A m 1 ( B ) m 2 ( C ) m_{1,2}(A)=(m_{1}\oplus m_{2})(A)=\frac{1}{1-K}\sum_{B\cap C=A\neq\emptyset}m_% {1}(B)m_{2}(C)\,\!
  15. K = B C = m 1 ( B ) m 2 ( C ) . K=\sum_{B\cap C=\emptyset}m_{1}(B)m_{2}(C).\,
  16. m ( brain tumor ) = Bel ( brain tumor ) = 1. m(\,\text{brain tumor})=\operatorname{Bel}(\,\text{brain tumor})=1.\,
  17. m ( brain tumor ) < 1 and Bel ( brain tumor ) < 1 , m(\,\text{brain tumor})<1\,\text{ and }\operatorname{Bel}(\,\text{brain tumor}% )<1,\,
  18. m : 2 X [ 0 , 1 ] m:2^{X}\rightarrow[0,1]\,\!
  19. bel ( ) = 0 \operatorname{bel}(\emptyset)=0
  20. bel ( X ) = 1 \operatorname{bel}(X)=1
  21. If A B = , then bel ( A B ) = bel ( A ) + bel ( B ) . \,\text{If }A\cap B=\emptyset,\,\text{ then}\operatorname{bel}(A\cup B)=% \operatorname{bel}(A)+\operatorname{bel}(B).
  22. bel ( A ) + bel ( A ¯ ) = 1 for all A X . \operatorname{bel}(A)+\operatorname{bel}(\bar{A})=1\,\text{ for all }A% \subseteq X.

Density_functional_theory.html

  1. Ψ ( r 1 , , r N ) \Psi(\vec{r}_{1},\dots,\vec{r}_{N})
  2. H ^ Ψ = [ T ^ + V ^ + U ^ ] Ψ = [ i N ( - 2 2 m i i 2 ) + i N V ( r i ) + i < j N U ( r i , r j ) ] Ψ = E Ψ \hat{H}\Psi=\left[{\hat{T}}+{\hat{V}}+{\hat{U}}\right]\Psi=\left[\sum_{i}^{N}% \left(-\frac{\hbar^{2}}{2m_{i}}\nabla_{i}^{2}\right)+\sum_{i}^{N}V(\vec{r}_{i}% )+\sum_{i<j}^{N}U(\vec{r}_{i},\vec{r}_{j})\right]\Psi=E\Psi
  3. N \ N
  4. H ^ \hat{H}
  5. E \ E
  6. T ^ \hat{T}
  7. V ^ \hat{V}
  8. U ^ \hat{U}
  9. T ^ \hat{T}
  10. U ^ \hat{U}
  11. N \ N
  12. V ^ \hat{V}
  13. U ^ \hat{U}
  14. U ^ \hat{U}
  15. U ^ \hat{U}
  16. n ( r ) , n(\vec{r}),
  17. Ψ \,\!\Psi
  18. n ( r ) = N d 3 r 2 d 3 r N Ψ * ( r , r 2 , , r N ) Ψ ( r , r 2 , , r N ) . n(\vec{r})=N\int{\rm d}^{3}r_{2}\cdots\int{\rm d}^{3}r_{N}\Psi^{*}(\vec{r},% \vec{r}_{2},\dots,\vec{r}_{N})\Psi(\vec{r},\vec{r}_{2},\dots,\vec{r}_{N}).
  19. n 0 ( r ) n_{0}(\vec{r})
  20. Ψ 0 ( r 1 , , r N ) \Psi_{0}(\vec{r}_{1},\dots,\vec{r}_{N})
  21. Ψ \,\!\Psi
  22. n 0 \,\!n_{0}
  23. Ψ 0 = Ψ [ n 0 ] \,\!\Psi_{0}=\Psi[n_{0}]
  24. O ^ \,\hat{O}
  25. n 0 \,\!n_{0}
  26. O [ n 0 ] = Ψ [ n 0 ] | O ^ | Ψ [ n 0 ] . O[n_{0}]=\left\langle\Psi[n_{0}]\left|\hat{O}\right|\Psi[n_{0}]\right\rangle.
  27. n 0 \,\!n_{0}
  28. E 0 = E [ n 0 ] = Ψ [ n 0 ] | T ^ + V ^ + U ^ | Ψ [ n 0 ] E_{0}=E[n_{0}]=\left\langle\Psi[n_{0}]\left|\hat{T}+\hat{V}+\hat{U}\right|\Psi% [n_{0}]\right\rangle
  29. Ψ [ n 0 ] | V ^ | Ψ [ n 0 ] \left\langle\Psi[n_{0}]\left|\hat{V}\right|\Psi[n_{0}]\right\rangle
  30. n 0 \,\!n_{0}
  31. V [ n 0 ] = V ( r ) n 0 ( r ) d 3 r . V[n_{0}]=\int V(\vec{r})n_{0}(\vec{r}){\rm d}^{3}r.
  32. Ψ | V ^ | Ψ \left\langle\Psi\left|\hat{V}\right|\Psi\right\rangle
  33. n \,\!n
  34. V [ n ] = V ( r ) n ( r ) d 3 r . V[n]=\int V(\vec{r})n(\vec{r}){\rm d}^{3}r.
  35. T [ n ] \,\!T[n]
  36. U [ n ] \,\!U[n]
  37. V [ n ] \,\!V[n]
  38. V ^ \hat{V}
  39. E [ n ] = T [ n ] + U [ n ] + V ( r ) n ( r ) d 3 r E[n]=T[n]+U[n]+\int V(\vec{r})n(\vec{r}){\rm d}^{3}r
  40. n ( r ) n(\vec{r})
  41. T [ n ] \,\!T[n]
  42. U [ n ] \,\!U[n]
  43. n 0 \,\!n_{0}
  44. E [ n ] \,\!E[n]
  45. E s [ n ] = Ψ s [ n ] | T ^ + V ^ s | Ψ s [ n ] E_{s}[n]=\left\langle\Psi_{s}[n]\left|\hat{T}+\hat{V}_{s}\right|\Psi_{s}[n]\right\rangle
  46. T ^ \hat{T}
  47. V ^ s \hat{V}_{s}
  48. n s ( r ) = def n ( r ) n_{s}(\vec{r})\ \stackrel{\mathrm{def}}{=}\ n(\vec{r})
  49. [ - 2 2 m 2 + V s ( r ) ] ϕ i ( r ) = ϵ i ϕ i ( r ) \left[-\frac{\hbar^{2}}{2m}\nabla^{2}+V_{s}(\vec{r})\right]\phi_{i}(\vec{r})=% \epsilon_{i}\phi_{i}(\vec{r})
  50. ϕ i \,\!\phi_{i}
  51. n ( r ) n(\vec{r})
  52. n ( r ) = def n s ( r ) = i N | ϕ i ( r ) | 2 . n(\vec{r})\ \stackrel{\mathrm{def}}{=}\ n_{s}(\vec{r})=\sum_{i}^{N}\left|\phi_% {i}(\vec{r})\right|^{2}.
  53. V s ( r ) = V ( r ) + e 2 n s ( r ) | r - r | d 3 r + V XC [ n s ( r ) ] V_{s}(\vec{r})=V(\vec{r})+\int\frac{e^{2}n_{s}(\vec{r}\,^{\prime})}{|\vec{r}-% \vec{r}\,^{\prime}|}{\rm d}^{3}r^{\prime}+V_{\rm XC}[n_{s}(\vec{r})]
  54. V XC \,\!V_{\rm XC}
  55. V XC \,\!V_{\rm XC}
  56. V XC \,\!V_{\rm XC}
  57. n ( r ) n(\vec{r})
  58. ϕ i \,\!\phi_{i}
  59. V s \,\!V_{s}
  60. n ( r ) n(\vec{r})
  61. V s \,\!V_{s}
  62. ϕ i \,\!\phi_{i}
  63. E s [ n ] E_{s}[n]
  64. G G
  65. n n
  66. E [ G ] E[G]
  67. G G
  68. n ( r , r ) n({\vec{r}},{\vec{r}}^{\prime})
  69. V ( r , r ) V({\vec{r}},{\vec{r}}^{\prime})
  70. n ( r , r ) n({\vec{r}},{\vec{r}}^{\prime})
  71. E XC LDA [ n ] = ϵ XC ( n ) n ( r ) d 3 r . E_{\rm XC}^{\rm LDA}[n]=\int\epsilon_{\rm XC}(n)n(\vec{r}){\rm d}^{3}r.
  72. E XC LSDA [ n , n ] = ϵ XC ( n , n ) n ( r ) d 3 r . E_{\rm XC}^{\rm LSDA}[n_{\uparrow},n_{\downarrow}]=\int\epsilon_{\rm XC}(n_{% \uparrow},n_{\downarrow})n(\vec{r}){\rm d}^{3}r.
  73. ϵ XC ( n , n ) \epsilon_{\rm XC}(n_{\uparrow},n_{\downarrow})
  74. E X C GGA [ n , n ] = ϵ X C ( n , n , n , n ) n ( r ) d 3 r . E_{XC}^{\rm GGA}[n_{\uparrow},n_{\downarrow}]=\int\epsilon_{XC}(n_{\uparrow},n% _{\downarrow},\vec{\nabla}n_{\uparrow},\vec{\nabla}n_{\downarrow})n(\vec{r}){% \rm d}^{3}r.
  75. h 3 h^{3}
  76. d 3 r d^{3}r
  77. p f p_{f}
  78. 4 3 π p f 3 ( r ) . \frac{4}{3}\pi p_{f}^{3}(\vec{r}).
  79. n ( r ) = 8 π 3 h 3 p f 3 ( r ) . n(\vec{r})=\frac{8\pi}{3h^{3}}p_{f}^{3}(\vec{r}).
  80. p f p_{f}
  81. t T F [ n ] = p 2 2 m e ( n 1 3 ) 2 2 m e n 2 3 ( r ) t_{TF}[n]=\frac{p^{2}}{2m_{e}}\propto\frac{(n^{\frac{1}{3}})^{2}}{2m_{e}}% \propto n^{\frac{2}{3}}(\vec{r})
  82. T T F [ n ] = C F n ( r ) n 2 3 ( r ) d 3 r = C F n 5 3 ( r ) d 3 r T_{TF}[n]=C_{F}\int n(\vec{r})n^{\frac{2}{3}}(\vec{r})d^{3}r=C_{F}\int n^{% \frac{5}{3}}(\vec{r})d^{3}r
  83. C F = 3 h 2 10 m e ( 3 8 π ) 2 3 . C_{F}=\frac{3h^{2}}{10m_{e}}\left(\frac{3}{8\pi}\right)^{\frac{2}{3}}.
  84. T W [ n ] = 2 8 m | n ( r ) | 2 n ( r ) d 3 r . T_{W}[n]=\frac{\hbar^{2}}{8m}\int\frac{|\nabla n(\vec{r})|^{2}}{n(\vec{r})}d^{% 3}r.
  85. v 1 ( r ) v_{1}(\vec{r})
  86. v 2 ( r ) v_{2}(\vec{r})
  87. n ( r ) n(\vec{r})
  88. v 1 ( r ) - v 2 ( r ) = c o n s t v_{1}(\vec{r})-v_{2}(\vec{r})=const
  89. F [ n ] = T [ n ] + U [ n ] F[n]=T[n]+U[n]
  90. N N
  91. v ( r ) v(\vec{r})
  92. F [ n ] F[n]
  93. E ( v , N ) [ n ] = F [ n ] + v ( r ) n ( r ) d 3 r E_{(v,N)}[n]=F[n]+\int{v(\vec{r})n(\vec{r})d^{3}r}
  94. N N
  95. v ( r ) v(\vec{r})
  96. E ( v , N ) [ n ] E_{(v,N)}[n]
  97. r l . rl_{.}
  98. R l pp ( r ) = R nl AE ( r ) . R_{\rm l}^{\rm pp}(r)=R_{\rm nl}^{\rm AE}(r).
  99. 0 r l d r | R l PP ( r ) | 2 r 2 = 0 r l d r | R nl AE ( r ) | 2 r 2 . \int_{0}^{rl}dr|R_{\rm l}^{\rm PP}(r)|^{2}r^{2}=\int_{0}^{rl}dr|R_{\rm nl}^{% \rm AE}(r)|^{2}r^{2}.
  100. R l ( r ) . R_{\rm l}(r).
  101. l . l_{.}
  102. p p . pp_{.}
  103. A E . AE_{.}
  104. r l . rl_{.}
  105. l . l_{.}

Dentition.html

  1. ( d i 2 - d c 1 - d p 2 ) / ( d i 2 - d c 1 - d p 2 ) × 2 = 20. (di^{2}-dc^{1}-dp^{2})/(di_{2}-dc_{1}-dp_{2})\times 2=20.
  2. ( I 2 - C 1 - P 2 - M 3 ) / ( I 2 - C 1 - P 2 - M 3 ) × 2 = 32. (I^{2}-C^{1}-P^{2}-M^{3})/(I_{2}-C_{1}-P_{2}-M_{3})\times 2=32.

Depreciation.html

  1. Annual Depreciation Expense = Cost of Fixed Asset - Residual Value Useful Life of Asset ( y e a r s ) \mbox{Annual Depreciation Expense}~{}={\mbox{Cost of Fixed Asset}~{}-\mbox{% Residual Value}~{}\over\mbox{Useful Life of Asset}~{}(years)}
  2. depreciation rate = 1 - residual value cost of fixed asset N \mbox{depreciation rate}~{}=1-\sqrt[N]{\mbox{residual value}~{}\over\mbox{cost% of fixed asset}~{}}
  3. Annual Depreciation Expense = Cost of Fixed Asset - Residual value Estimated Total Production × Actual Production \mbox{Annual Depreciation Expense}~{}={\mbox{Cost of Fixed Asset}~{}-\mbox{% Residual value}~{}\over\mbox{Estimated Total Production}~{}}\times\mbox{Actual% Production}~{}

Depth-first_search.html

  1. u u
  2. v v

Derangement.html

  1. n n
  2. n ! n!
  3. ! n !n
  4. ! n n ! \frac{!n}{n!}
  5. ! n = ( n - 1 ) ( ! ( n - 1 ) + ! ( n - 2 ) ) . !n=(n-1)(!(n-1)+!(n-2)).\,
  6. n ! = ( n - 1 ) ( ( n - 1 ) ! + ( n - 2 ) ! ) n!=(n-1)((n-1)!+(n-2)!)\,
  7. ! n = n ! i = 0 n ( - 1 ) i i ! , !n=n!\sum_{i=0}^{n}\frac{(-1)^{i}}{i!},
  8. ! n = [ n ! e ] = n ! e + 1 2 , n 1 !n=\left[\frac{n!}{e}\right]=\left\lfloor\frac{n!}{e}+\frac{1}{2}\right\rfloor% ,\quad n\geq 1
  9. [ x ] \left[x\right]
  10. x \left\lfloor x\right\rfloor
  11. ! n = ( e + e - 1 ) n ! - e n ! , n 2 , !n=\left\lfloor(e+e^{-1})n!\right\rfloor-\lfloor en!\rfloor,\quad n\geq 2,
  12. ! n = n ! - i = 1 n ( n i ) ! ( n - i ) , !n=n!-\sum_{i=1}^{n}{n\choose i}\cdot!(n-i),
  13. ! n = n [ ! ( n - 1 ) ] + ( - 1 ) n !n=n[!(n-1)]+(-1)^{n}
  14. lim n ! n n ! = 1 e 0.3679 . \lim_{n\to\infty}{!n\over n!}={1\over e}\approx 0.3679\dots.
  15. 0 P n 1 ( x ) P n 2 ( x ) P n r ( x ) e - x d x , \int_{0}^{\infty}P_{n_{1}}(x)P_{n_{2}}(x)\cdots P_{n_{r}}(x)e^{-x}\,dx,

Description_logic.html

  1. 𝒜 \mathcal{AL}
  2. \mathcal{FL}
  3. \mathcal{EL}
  4. \mathcal{F}
  5. \mathcal{E}
  6. \top
  7. 𝒰 \mathcal{U}
  8. 𝒞 \mathcal{C}
  9. \mathcal{H}
  10. \mathcal{R}
  11. 𝒪 \mathcal{O}
  12. \mathcal{I}
  13. 𝒩 \mathcal{N}
  14. 𝒬 \mathcal{Q}
  15. \top
  16. ( 𝒟 ) {}^{\mathcal{(D)}}
  17. 𝒮 \mathcal{S}
  18. 𝒜 𝒞 \mathcal{ALC}
  19. - \mathcal{FL^{-}}
  20. \mathcal{FL}
  21. 𝒜 \mathcal{AL}
  22. o \mathcal{FL}_{o}
  23. - \mathcal{FL^{-}}
  24. + + \mathcal{EL^{++}}
  25. 𝒪 \mathcal{ELRO}
  26. 𝒜 𝒞 \mathcal{ALC}
  27. 𝒜 𝒞 \mathcal{ALC}
  28. 𝒜 \mathcal{AL}
  29. 𝒮 𝒬 \mathcal{SHIQ}
  30. 𝒜 𝒞 \mathcal{ALC}
  31. 𝒜 𝒞 𝒪 𝒩 \mathcal{ALCOIN}
  32. 𝒜 𝒞 𝒩 𝒪 \mathcal{ALCNIO}
  33. 𝒜 𝒞 \mathcal{ALC}
  34. 𝒜 𝒰 \mathcal{ALUE}
  35. 𝒮 𝒪 𝒩 ( 𝒟 ) \mathcal{SHOIN}^{\mathcal{(D)}}
  36. \mathcal{EL}
  37. 𝒮 𝒪 𝒬 ( 𝒟 ) \mathcal{SROIQ}^{\mathcal{(D)}}
  38. 𝒮 𝒪 𝒩 ( 𝒟 ) \mathcal{SHOIN}^{\mathcal{(D)}}
  39. 𝒮 ( 𝒟 ) \mathcal{SHIF}^{\mathcal{(D)}}
  40. 𝒮 𝒬 \mathcal{SHIQ}
  41. 𝒮 \mathcal{SH}
  42. 𝒮 𝒪 𝒩 ( 𝒟 ) \mathcal{SHOIN}^{\mathcal{(D)}}
  43. 𝒮 ( 𝒟 ) \mathcal{SHIF}^{\mathcal{(D)}}
  44. 𝒮 𝒪 𝒬 ( 𝒟 ) \mathcal{SROIQ}^{\mathcal{(D)}}
  45. \top
  46. \top
  47. \bot
  48. \bot
  49. \sqcap
  50. C D C\sqcap D
  51. \sqcup
  52. C D C\sqcup D
  53. ¬ \neg
  54. ¬ C \neg C
  55. \forall
  56. R . C \forall R.C
  57. \exists
  58. R . C \exists R.C
  59. \sqsubseteq
  60. C D C\sqsubseteq D
  61. \equiv
  62. C D C\equiv D
  63. = ˙ \dot{=}
  64. C = ˙ D C\dot{=}D
  65. : :
  66. a : C a:C
  67. : :
  68. ( a , b ) : R (a,b):R
  69. 𝒜 𝒞 \mathcal{ALC}
  70. N C N_{C}
  71. N R N_{R}
  72. N O N_{O}
  73. N C N_{C}
  74. N R N_{R}
  75. N O N_{O}
  76. 𝒜 𝒞 \mathcal{ALC}
  77. \top
  78. \bot
  79. A N C A\in N_{C}
  80. C C
  81. D D
  82. R N R R\in N_{R}
  83. C D C\sqcap D
  84. C D C\sqcup D
  85. ¬ C \neg C
  86. R . C \forall R.C
  87. R . C \exists R.C
  88. C D C\sqsubseteq D
  89. C C
  90. D D
  91. C D C\equiv D
  92. C D C\sqsubseteq D
  93. D C D\sqsubseteq C
  94. a : C a:C
  95. a N O a\in N_{O}
  96. ( a , b ) : R (a,b):R
  97. a , b N O a,b\in N_{O}
  98. ( 𝒯 , 𝒜 ) (\mathcal{T},\mathcal{A})
  99. 𝒯 \mathcal{T}
  100. 𝒜 \mathcal{A}
  101. = ( Δ , ) \mathcal{I}=(\Delta^{\mathcal{I}},\cdot^{\mathcal{I}})
  102. ( N C , N R , N O ) (N_{C},N_{R},N_{O})
  103. Δ \Delta^{\mathcal{I}}
  104. \cdot^{\mathcal{I}}
  105. a a
  106. a Δ a^{\mathcal{I}}\in\Delta^{\mathcal{I}}
  107. Δ \Delta^{\mathcal{I}}
  108. Δ × Δ \Delta^{\mathcal{I}}\times\Delta^{\mathcal{I}}
  109. = Δ \top^{\mathcal{I}}=\Delta^{\mathcal{I}}
  110. = \bot^{\mathcal{I}}=\emptyset
  111. ( C D ) = C D (C\sqcup D)^{\mathcal{I}}=C^{\mathcal{I}}\cup D^{\mathcal{I}}
  112. ( C D ) = C D (C\sqcap D)^{\mathcal{I}}=C^{\mathcal{I}}\cap D^{\mathcal{I}}
  113. ( ¬ C ) = Δ C (\neg C)^{\mathcal{I}}=\Delta^{\mathcal{I}}\setminus C^{\mathcal{I}}
  114. ( R . C ) = { x Δ | 𝚏𝚘𝚛 𝚎𝚟𝚎𝚛𝚢 y , ( x , y ) R 𝚒𝚖𝚙𝚕𝚒𝚎𝚜 y C } (\forall R.C)^{\mathcal{I}}=\{x\in\Delta^{\mathcal{I}}|\texttt{for}\;\texttt{% every}\;y,(x,y)\in R^{\mathcal{I}}\;\texttt{implies}\;y\in C^{\mathcal{I}}\}
  115. ( R . C ) = { x Δ | 𝚝𝚑𝚎𝚛𝚎 𝚎𝚡𝚒𝚜𝚝𝚜 y , ( x , y ) R 𝚊𝚗𝚍 y C } (\exists R.C)^{\mathcal{I}}=\{x\in\Delta^{\mathcal{I}}|\texttt{there}\;\texttt% {exists}\;y,(x,y)\in R^{\mathcal{I}}\;\texttt{and}\;y\in C^{\mathcal{I}}\}
  116. \mathcal{I}\models
  117. C D \mathcal{I}\models C\sqsubseteq D
  118. C D C^{\mathcal{I}}\subseteq D^{\mathcal{I}}
  119. 𝒯 \mathcal{I}\models\mathcal{T}
  120. Φ \mathcal{I}\models\Phi
  121. Φ 𝒯 \Phi\in\mathcal{T}
  122. a : C \mathcal{I}\models a:C
  123. a C a^{\mathcal{I}}\in C^{\mathcal{I}}
  124. ( a , b ) : R \mathcal{I}\models(a,b):R
  125. ( a , b ) R (a^{\mathcal{I}},b^{\mathcal{I}})\in R^{\mathcal{I}}
  126. 𝒜 \mathcal{I}\models\mathcal{A}
  127. ϕ \mathcal{I}\models\phi
  128. ϕ 𝒜 \phi\in\mathcal{A}
  129. 𝒦 = ( 𝒯 , 𝒜 ) \mathcal{K}=(\mathcal{T},\mathcal{A})
  130. 𝒦 \mathcal{I}\models\mathcal{K}
  131. 𝒯 \mathcal{I}\models\mathcal{T}
  132. 𝒜 \mathcal{I}\models\mathcal{A}
  133. 𝒜 𝒞 \mathcal{ALC}
  134. 𝒮 \mathcal{SR}
  135. 𝒮 \mathcal{FSR}
  136. 𝒯 𝒮 , o r 𝒮 \mathcal{TSL},or\mathcal{SRI}
  137. 𝒮 , o r 𝒮 \mathcal{FSL},or\mathcal{FSRI}

Development_economics.html

  1. 1 - i = 1 N s i 2 , 1-\sum_{i=1}^{N}s_{i}^{2},
  2. Q = 1 - i = 1 N ( 1 2 - s i 1 2 ) 2 × s i , Q=1-\sum_{i=1}^{N}\left(\frac{\tfrac{1}{2}-s_{i}}{\tfrac{1}{2}}\right)^{2}% \times s_{i},

Device_independent_file_format.html

  1. ( h , v , w , x , y , z ) (h,v,w,x,y,z)

Diagonal_matrix.html

  1. d i , j = 0 if i j i , j { 1 , 2 , , n } d_{i,j}=0\,\text{ if }i\neq j\ \forall i,j\in\{1,2,\ldots,n\}
  2. [ 1 0 0 0 4 0 0 0 - 2 ] \begin{bmatrix}1&0&0\\ 0&4&0\\ 0&0&-2\end{bmatrix}
  3. [ 1 0 0 0 4 0 0 0 - 3 0 0 0 ] \begin{bmatrix}1&0&0\\ 0&4&0\\ 0&0&-3\\ 0&0&0\\ \end{bmatrix}
  4. [ 1 0 0 0 0 0 4 0 0 0 0 0 - 3 0 0 ] \begin{bmatrix}1&0&0&0&0\\ 0&4&0&0&0\\ 0&0&-3&0&0\end{bmatrix}
  5. [ λ 0 0 0 λ 0 0 0 λ ] λ s y m b o l I 3 \begin{bmatrix}\lambda&0&0\\ 0&\lambda&0\\ 0&0&\lambda\end{bmatrix}\equiv\lambda symbol{I}_{3}
  6. K n K^{n}
  7. R End ( M ) , R\to\operatorname{End}(M),
  8. M R n M\cong R^{n}
  9. a i , j a_{i,j}
  10. a i , i a_{i,i}
  11. A e j = a i , j e i A\vec{e}_{j}=\sum a_{i,j}\vec{e}_{i}
  12. a i , i a_{i,i}
  13. λ i \lambda_{i}
  14. A e i = λ i e i A\vec{e}_{i}=\lambda_{i}\vec{e}_{i}

Diagonalizable_matrix.html

  1. [ - 1 3 - 1 - 3 5 - 1 - 3 3 1 ] , \begin{bmatrix}-1&3&-1\\ -3&5&-1\\ -3&3&1\end{bmatrix},
  2. [ 1 0 0 0 2 0 0 0 2 ] \begin{bmatrix}1&0&0\\ 0&2&0\\ 0&0&2\end{bmatrix}
  3. [ 1 1 - 1 1 1 0 1 0 3 ] . \begin{bmatrix}1&1&-1\\ 1&1&0\\ 1&0&3\end{bmatrix}.
  4. ( x n - λ 1 ) ( x n - λ k ) (x^{n}-\lambda_{1})\cdots(x^{n}-\lambda_{k})
  5. λ j 0 \lambda_{j}\neq 0
  6. P - 1 A P = ( λ 1 λ 2 λ n ) , P^{-1}AP=\begin{pmatrix}\lambda_{1}\\ &\lambda_{2}\\ &&\ddots\\ &&&\lambda_{n}\end{pmatrix},
  7. A P = P ( λ 1 λ 2 λ n ) . AP=P\begin{pmatrix}\lambda_{1}\\ &\lambda_{2}\\ &&\ddots\\ &&&\lambda_{n}\end{pmatrix}.
  8. α i \vec{\alpha}_{i}
  9. P = ( α 1 α 2 α n ) , P=\begin{pmatrix}\vec{\alpha}_{1}&\vec{\alpha}_{2}&\cdots&\vec{\alpha}_{n}\end% {pmatrix},
  10. A α i = λ i α i ( i = 1 , 2 , , n ) . A\vec{\alpha}_{i}=\lambda_{i}\vec{\alpha}_{i}\qquad(i=1,2,\cdots,n).
  11. [ 1 0 0 0 ] and [ 1 1 0 0 ] \begin{bmatrix}1&0\\ 0&0\end{bmatrix}\quad\,\text{and}\quad\begin{bmatrix}1&1\\ 0&0\end{bmatrix}
  12. A = A T A=A^{T}
  13. A A T = A T A AA^{T}=A^{T}A
  14. C = [ 0 1 0 0 ] . C=\begin{bmatrix}0&1\\ 0&0\end{bmatrix}.
  15. B = [ 0 1 - 1 0 ] . B=\begin{bmatrix}0&1\\ -1&0\end{bmatrix}.
  16. Q = [ 1 i i 1 ] , Q=\begin{bmatrix}1&\textrm{i}\\ \textrm{i}&1\end{bmatrix},
  17. A = [ 1 2 0 0 3 0 2 - 4 2 ] . A=\begin{bmatrix}1&2&0\\ 0&3&0\\ 2&-4&2\end{bmatrix}.
  18. λ 1 = 3 , λ 2 = 2 , λ 3 = 1. \lambda_{1}=3,\quad\lambda_{2}=2,\quad\lambda_{3}=1.
  19. v 1 = [ - 1 - 1 2 ] , v 2 = [ 0 0 1 ] , v 3 = [ - 1 0 2 ] . v_{1}=\begin{bmatrix}-1\\ -1\\ 2\end{bmatrix},\quad v_{2}=\begin{bmatrix}0\\ 0\\ 1\end{bmatrix},\quad v_{3}=\begin{bmatrix}-1\\ 0\\ 2\end{bmatrix}.
  20. A v k = λ k v k . Av_{k}=\lambda_{k}v_{k}.
  21. P = [ - 1 0 - 1 - 1 0 0 2 1 2 ] . P=\begin{bmatrix}-1&0&-1\\ -1&0&0\\ 2&1&2\end{bmatrix}.
  22. P - 1 A P = [ 0 - 1 0 2 0 1 - 1 1 0 ] [ 1 2 0 0 3 0 2 - 4 2 ] [ - 1 0 - 1 - 1 0 0 2 1 2 ] = [ 3 0 0 0 2 0 0 0 1 ] . P^{-1}AP=\begin{bmatrix}0&-1&0\\ 2&0&1\\ -1&1&0\end{bmatrix}\begin{bmatrix}1&2&0\\ 0&3&0\\ 2&-4&2\end{bmatrix}\begin{bmatrix}-1&0&-1\\ -1&0&0\\ 2&1&2\end{bmatrix}=\begin{bmatrix}3&0&0\\ 0&2&0\\ 0&0&1\end{bmatrix}.
  23. λ k \lambda_{k}
  24. P - 1 A P = D P^{-1}AP=D
  25. A k = ( P D P - 1 ) k = ( P D P - 1 ) ( P D P - 1 ) ( P D P - 1 ) = P D ( P - 1 P ) D ( P - 1 P ) ( P - 1 P ) D P - 1 = P D k P - 1 \begin{aligned}\displaystyle A^{k}&\displaystyle=(PDP^{-1})^{k}=(PDP^{-1})% \cdot(PDP^{-1})\cdots(PDP^{-1})\\ &\displaystyle=PD(P^{-1}P)D(P^{-1}P)\cdots(P^{-1}P)DP^{-1}\\ &\displaystyle=PD^{k}P^{-1}\end{aligned}
  26. M = [ a b - a 0 b ] . M=\begin{bmatrix}a&b-a\\ 0&b\end{bmatrix}.
  27. M 2 = [ a 2 b 2 - a 2 0 b 2 ] , M 3 = [ a 3 b 3 - a 3 0 b 3 ] , M 4 = [ a 4 b 4 - a 4 0 b 4 ] , M^{2}=\begin{bmatrix}a^{2}&b^{2}-a^{2}\\ 0&b^{2}\end{bmatrix},\quad M^{3}=\begin{bmatrix}a^{3}&b^{3}-a^{3}\\ 0&b^{3}\end{bmatrix},\quad M^{4}=\begin{bmatrix}a^{4}&b^{4}-a^{4}\\ 0&b^{4}\end{bmatrix},\quad\ldots
  28. 𝐮 = [ 1 0 ] = 𝐞 1 , 𝐯 = [ 1 1 ] = 𝐞 1 + 𝐞 2 , \mathbf{u}=\begin{bmatrix}1\\ 0\end{bmatrix}=\mathbf{e}_{1},\quad\mathbf{v}=\begin{bmatrix}1\\ 1\end{bmatrix}=\mathbf{e}_{1}+\mathbf{e}_{2},
  29. 𝐞 1 = 𝐮 , 𝐞 2 = 𝐯 - 𝐮 . \mathbf{e}_{1}=\mathbf{u},\qquad\mathbf{e}_{2}=\mathbf{v}-\mathbf{u}.
  30. M 𝐮 = a 𝐮 , M 𝐯 = b 𝐯 . M\mathbf{u}=a\mathbf{u},\qquad M\mathbf{v}=b\mathbf{v}.
  31. M n 𝐮 = a n 𝐮 , M n 𝐯 = b n 𝐯 . M^{n}\mathbf{u}=a^{n}\,\mathbf{u},\qquad M^{n}\mathbf{v}=b^{n}\,\mathbf{v}.
  32. M n 𝐞 1 = M n 𝐮 = a n 𝐞 1 , M^{n}\mathbf{e}_{1}=M^{n}\mathbf{u}=a^{n}\mathbf{e}_{1},
  33. M n 𝐞 2 = M n ( 𝐯 - 𝐮 ) = b n 𝐯 - a n 𝐮 = ( b n - a n ) 𝐞 1 + b n 𝐞 2 . M^{n}\mathbf{e}_{2}=M^{n}(\mathbf{v}-\mathbf{u})=b^{n}\mathbf{v}-a^{n}\mathbf{% u}=(b^{n}-a^{n})\mathbf{e}_{1}+b^{n}\mathbf{e}_{2}.
  34. M n = [ a n b n - a n 0 b n ] , M^{n}=\begin{bmatrix}a^{n}&b^{n}-a^{n}\\ 0&b^{n}\end{bmatrix},

Dicyclic_group.html

  1. 1 C 2 n Dic n C 2 1. 1\to C_{2n}\to\mbox{Dic}~{}_{n}\to C_{2}\to 1.\,
  2. a = e i π / n = cos π n + i sin π n x = j \begin{aligned}\displaystyle a&\displaystyle=e^{i\pi/n}=\cos\frac{\pi}{n}+i% \sin\frac{\pi}{n}\\ \displaystyle x&\displaystyle=j\end{aligned}
  3. Dic = n a , x a 2 n = 1 , x 2 = a n , x - 1 a x = a - 1 . \mbox{Dic}~{}_{n}=\langle a,x\mid a^{2n}=1,\ x^{2}=a^{n},\ x^{-1}ax=a^{-1}% \rangle.\,\!
  4. a k a m x = a k + m x a^{k}a^{m}x=a^{k+m}x
  5. a k x a m = a k - m x a^{k}xa^{m}=a^{k-m}x
  6. a k x a m x = a k - m + n a^{k}xa^{m}x=a^{k-m+n}

Difference_quotient.html

  1. f ( x + h ) - f ( x ) h \frac{f(x+h)-f(x)}{h}
  2. f ( b ) - f ( a ) b - a \frac{f(b)-f(a)}{b-a}
  3. ι \iota
  4. lim Δ P 0 \lim_{\Delta P\rightarrow 0}\,\!
  5. Δ F ( P ) Δ P = F ( P + Δ P ) - F ( P ) Δ P = F ( P + Δ P ) Δ P . \frac{\Delta F(P)}{\Delta P}=\frac{F(P+\Delta P)-F(P)}{\Delta P}=\frac{\nabla F% (P+\Delta P)}{\Delta P}.\,\!
  6. If | Δ P | = ι : Δ F ( P ) Δ P = d F ( P ) d P = F ( P ) = G ( P ) ; \,\text{If }|\Delta P|=\mathit{\iota}:\quad\frac{\Delta F(P)}{\Delta P}=\frac{% dF(P)}{dP}=F^{\prime}(P)=G(P);\,\!
  7. If | Δ P | > ι : Δ F ( P ) Δ P = D F ( P ) D P = F [ P , P + Δ P ] . \,\text{If }|\Delta P|>\mathit{\iota}:\quad\frac{\Delta F(P)}{\Delta P}=\frac{% DF(P)}{DP}=F[P,P+\Delta P].\,\!
  8. Δ F ( P 0 ) Δ P = F ( P n ´ ) - F ( P 0 ) Δ n ´ P = F ( P 1 ) - F ( P 0 ) Δ 1 P = F ( P 1 ) - F ( P 0 ) P 1 - P 0 . \frac{\Delta F(P_{0})}{\Delta P}=\frac{F(P_{\acute{n}})-F(P_{0})}{\Delta_{% \acute{n}}P}=\frac{F(P_{1})-F(P_{0})}{\Delta_{1}P}=\frac{F(P_{1})-F(P_{0})}{P_% {1}-P_{0}}.\,\!
  9. d F ( P ) d P = F ( P 1 ) - F ( P 0 ) d P = F ( P ) = G ( P ) . \frac{dF(P)}{dP}=\frac{F(P_{1})-F(P_{0})}{dP}=F^{\prime}(P)=G(P).\,\!
  10. P ( t n ) \displaystyle P_{(tn)}
  11. P a ~ := L B < P < U B = P 0 < P < P n ´ P_{\tilde{a}}:=LB<P<UB=P_{0}<P<P_{\acute{n}}\,\!
  12. D F ( P 0 ) D P \displaystyle\frac{DF(P_{0})}{DP}
  13. Δ 2 F ( P 0 ) Δ 1 P 2 \displaystyle\frac{\Delta^{2}F(P_{0})}{\Delta_{1}P^{2}}
  14. d 2 F ( P ) d P 2 \displaystyle\frac{d^{2}F(P)}{dP^{2}}
  15. D 2 F ( P 0 ) D P 2 \displaystyle\frac{D^{2}F(P_{0})}{DP^{2}}
  16. Δ 3 F ( P 0 ) Δ 1 P 3 \displaystyle\frac{\Delta^{3}F(P_{0})}{\Delta_{1}P^{3}}
  17. d 3 F ( P ) d P 3 \displaystyle\frac{d^{3}F(P)}{dP^{3}}
  18. D 3 F ( P 0 ) D P 3 \displaystyle\frac{D^{3}F(P_{0})}{DP^{3}}
  19. Δ n ´ F ( P 0 ) \displaystyle\Delta^{\acute{n}}F(P_{0})
  20. Δ n ´ F ( P 0 ) Δ 1 P n ´ \displaystyle\frac{\Delta^{\acute{n}}F(P_{0})}{\Delta_{1}P^{\acute{n}}}
  21. d n ´ F ( P 0 ) d P n ´ \displaystyle\frac{d^{\acute{n}}F(P_{0})}{dP^{\acute{n}}}
  22. D n ´ F ( P 0 ) D P n ´ \displaystyle\frac{D^{\acute{n}}F(P_{0})}{DP^{\acute{n}}}
  23. L B U B G ( p ) d p \displaystyle\int_{LB}^{UB}G(p)\,dp
  24. π \pi\,\!
  25. 2 π 2\pi\,\!
  26. π 2 \begin{matrix}\frac{\pi}{2}\end{matrix}
  27. 0 2 π F ( p ) d p \displaystyle\int_{0}^{2\pi}F^{\prime}(p)\,dp
  28. C L C U B L B U A L A U F ( r , q , p ) d p d q d r \displaystyle{}\qquad\int_{CL}^{CU}\int_{BL}^{BU}\int_{AL}^{AU}F^{\prime}(r,q,% p)\,dp\,dq\,dr
  29. F ( R , Q : A L < P < A U ) = T A = 1 U A = F ( R , Q : P ( t a ) ) U A ; F^{\prime}(R,Q:AL<P<AU)=\sum_{T\!A=1}^{U\!A=\infty}\frac{F^{\prime}(R,Q:P_{(ta% )})}{U\!A};\,\!
  30. F ( R : B L < Q < B U : A L < P < A U ) = T B = 1 U B = ( T A = 1 U A = F ( R : Q ( t b ) : P ( t a ) ) U A ) 1 U B . F^{\prime}(R:BL<Q<BU:AL<P<AU)=\sum_{T\!B=1}^{U\!B=\infty}\left(\sum_{T\!A=1}^{% U\!A=\infty}\frac{F^{\prime}(R:Q_{(tb)}:P_{(ta)})}{U\!A}\right)\frac{1}{U\!B}.\,\!

Differential_(mechanical_device).html

  1. ω s - ω c ω a - ω c = - 1 , \frac{\omega_{s}-\omega_{c}}{\omega_{a}-\omega_{c}}=-1,
  2. ω c = 1 2 ( ω s + ω a ) . \omega_{c}=\frac{1}{2}(\omega_{s}+\omega_{a}).

Differential_form.html

  1. a b f ( x ) d x \int_{a}^{b}f(x)\,dx
  2. S f ( x , y , z ) d x d y + g ( x , y , z ) d x d z + h ( x , y , z ) d y d z . \int_{S}f(x,y,z)\,dx\wedge dy+g(x,y,z)\,dx\wedge dz+h(x,y,z)\,dy\wedge dz.
  3. ( v f ) ( p ) = d d t f ( p + t v ) | t = 0 . (\partial_{v}f)(p)=\frac{d}{dt}f(p+tv)\Big|_{t=0}.
  4. f x j = i = 1 n y i x j f y i \frac{\partial f}{\partial x^{j}}=\sum_{i=1}^{n}\frac{\partial y^{i}}{\partial x% ^{j}}\frac{\partial f}{\partial y^{i}}
  5. ( v + w f ) ( p ) = ( v f ) ( p ) + ( w f ) ( p ) (\partial_{v+w}f)(p)=(\partial_{v}f)(p)+(\partial_{w}f)(p)
  6. ( c v f ) ( p ) = c ( v f ) ( p ) (\partial_{cv}f)(p)=c(\partial_{v}f)(p)
  7. d f = i = 1 n f x i d x i . df=\sum_{i=1}^{n}\frac{\partial f}{\partial x^{i}}\,dx^{i}.
  8. d f p = i = 1 n f x i ( p ) ( d x i ) p . df_{p}=\sum_{i=1}^{n}\frac{\partial f}{\partial x^{i}}(p)(dx^{i})_{p}.
  9. α p = i g i ( p ) ( d h i ) p \alpha_{p}=\sum_{i}g_{i}(p)(dh_{i})_{p}\,\!
  10. α = i = 1 n f i d x i \alpha=\sum_{i=1}^{n}f_{i}\,dx^{i}
  11. 2 f x i x j = 2 f x j x i , \frac{\partial^{2}f}{\partial x^{i}\,\partial x^{j}}=\frac{\partial^{2}f}{% \partial x^{j}\,\partial x^{i}},
  12. f j x i - f i x j = 0. \frac{\partial f_{j}}{\partial x^{i}}-\frac{\partial f_{i}}{\partial x^{j}}=0.
  13. \wedge
  14. i , j = 1 n f j x i d x i d x j = 0 \sum_{i,j=1}^{n}\frac{\partial f_{j}}{\partial x^{i}}dx^{i}\wedge dx^{j}=0
  15. \wedge
  16. d x i d x j = - d x j d x i . dx^{i}\wedge dx^{j}=-dx^{j}\wedge dx^{i}.
  17. d α = j = 1 n d f j d x j = i , j = 1 n f j x i d x i d x j . d\alpha=\sum_{j=1}^{n}df_{j}\wedge dx^{j}=\sum_{i,j=1}^{n}\frac{\partial f_{j}% }{\partial x^{i}}dx^{i}\wedge dx^{j}.
  18. i 1 , i 2 i k = 1 n f i 1 i 2 i k d x i 1 d x i 2 d x i k \sum_{i_{1},i_{2}\ldots i_{k}=1}^{n}f_{i_{1}i_{2}\ldots i_{k}}dx^{i_{1}}\wedge dx% ^{i_{2}}\wedge\cdots\wedge dx^{i_{k}}
  19. i 1 < i 2 < < i k - 1 < i k i_{1}<i_{2}<\cdots<i_{k-1}<i_{k}\,
  20. β p : T p M × × T p M \beta_{p}\colon T_{p}M\times\cdots\times T_{p}M\to\mathbb{R}
  21. ( α β ) p ( v , w ) = α p ( v ) β p ( w ) - α p ( w ) β p ( v ) (\alpha\wedge\beta)_{p}(v,w)=\alpha_{p}(v)\beta_{p}(w)-\alpha_{p}(w)\beta_{p}(v)
  22. α ( β + γ ) = α β + α γ . \alpha\wedge(\beta+\gamma)=\alpha\wedge\beta+\alpha\wedge\gamma.
  23. α β = ( - 1 ) k l β α . \alpha\wedge\beta=(-1)^{kl}\beta\wedge\alpha.\,
  24. * : Ω k ( M ) Ω n - k ( M ) *\colon\Omega^{k}(M)\overset{\sim}{\to}\Omega^{n-k}(M)
  25. δ : Ω k ( M ) Ω k - 1 ( M ) , \delta\colon\Omega^{k}(M)\rightarrow\Omega^{k-1}(M),
  26. - 1 -1
  27. 0 Ω 0 ( M ) d Ω 1 ( M ) d Ω 2 ( M ) d Ω 3 ( M ) Ω n ( M ) 0. 0\to\Omega^{0}(M)\ \stackrel{d}{\to}\ \Omega^{1}(M)\ \stackrel{d}{\to}\ \Omega% ^{2}(M)\ \stackrel{d}{\to}\ \Omega^{3}(M)\to\cdots\ \to\ \Omega^{n}(M)\ \to\ 0.
  28. ( f * ω ) p ( v 1 , , v k ) = ω f ( p ) ( f * v 1 , , f * v k ) . (f^{*}\omega)_{p}(v_{1},\ldots,v_{k})=\omega_{f(p)}(f_{*}v_{1},\ldots,f_{*}v_{% k}).
  29. M f N ω T * N ( D f ) * T * M . M\stackrel{f}{\to}N\stackrel{\omega}{\to}T^{*}N\stackrel{(Df)^{*}}{% \longrightarrow}T^{*}M.
  30. f * ( ω + η ) = f * ω + f * η , f^{*}(\omega+\eta)=f^{*}\omega+f^{*}\eta,
  31. f * ( ω η ) = f * ω f * η , f^{*}(\omega\wedge\eta)=f^{*}\omega\wedge f^{*}\eta,
  32. f * ( d ω ) = d ( f * ω ) . f^{*}(d\omega)=d(f^{*}\omega).
  33. ω = i 1 < < i k ω i 1 i k d y i 1 d y i k , \omega=\sum_{i_{1}<\cdots<i_{k}}\omega_{i_{1}\cdots i_{k}}dy_{i_{1}}\wedge% \cdots\wedge dy_{i_{k}},
  34. ω i 1 i k \omega_{i_{1}\cdots i_{k}}
  35. f * ω = i 1 < < i k ( ω i 1 i k f ) d f i 1 d f i n . f^{*}\omega=\sum_{i_{1}<\cdots<i_{k}}(\omega_{i_{1}\cdots i_{k}}\circ f)df_{i_% {1}}\wedge\cdots\wedge df_{i_{n}}.
  36. f * ω = i 1 < < i k j 1 < < j k ( ω i 1 i k f ) ( f i 1 , , f i k ) ( x j 1 , , x j k ) d x j 1 d x j k . f^{*}\omega=\sum_{i_{1}<\cdots<i_{k}}\sum_{j_{1}<\cdots<j_{k}}(\omega_{i_{1}% \cdots i_{k}}\circ f)\frac{\partial(f_{i_{1}},\ldots,f_{i_{k}})}{\partial(x_{j% _{1}},\ldots,x_{j_{k}})}dx_{j_{1}}\wedge\cdots\wedge dx_{j_{k}}.
  37. ( f i 1 , , f i k ) ( x j 1 , , x j k ) \frac{\partial(f_{i_{1}},\ldots,f_{i_{k}})}{\partial(x_{j_{1}},\ldots,x_{j_{k}% })}
  38. f i m / x j n \partial f_{i_{m}}/\partial x_{j_{n}}
  39. 1 m , n k 1\leq m,n\leq k
  40. 𝐑 k , \mathbf{R}^{k},
  41. d u 1 d u k . du^{1}\cdots du^{k}.
  42. γ ( t ) : [ 0 , 1 ] 𝐑 2 , \gamma(t)\colon[0,1]\to\mathbf{R}^{2},
  43. [ 0 , 1 ] [0,1]
  44. f ( t ) d t f(t)\,dt
  45. ω = i 1 < < i k a i 1 , , i k ( 𝐱 ) d x i 1 d x i k \omega=\sum_{i_{1}<\cdots<i_{k}}a_{i_{1},\dots,i_{k}}({\mathbf{x}})\,dx^{i_{1}% }\wedge\cdots\wedge dx^{i_{k}}
  46. S ( 𝐮 ) = ( x 1 ( 𝐮 ) , , x k ( 𝐮 ) ) S({\mathbf{u}})=(x^{1}({\mathbf{u}}),\dots,x^{k}({\mathbf{u}}))
  47. S ω = D i 1 < < i k a i 1 , , i k ( S ( 𝐮 ) ) ( x i 1 , , x i k ) ( u 1 , , u k ) d u 1 d u k \int_{S}\omega=\int_{D}\sum_{i_{1}<\cdots<i_{k}}a_{i_{1},\dots,i_{k}}(S({% \mathbf{u}}))\frac{\partial(x^{i_{1}},\dots,x^{i_{k}})}{\partial(u^{1},\dots,u% ^{k})}\,du^{1}\ldots du^{k}
  48. ( x i 1 , , x i k ) ( u 1 , , u k ) \frac{\partial(x^{i_{1}},\dots,x^{i_{k}})}{\partial(u^{1},\dots,u^{k})}
  49. k k
  50. p p
  51. p k p\leq k
  52. ( k - p ) (k-p)
  53. f : M N f:M\to N
  54. f f
  55. ω \omega
  56. M d ω = M ω . \int_{M}d\omega=\oint_{\partial M}\omega.\!\,
  57. ω \omega
  58. M ω = N ω , \textstyle{\int_{M}\omega=\int_{N}\omega},
  59. W d ω = W 0 = 0. \textstyle{\int_{W}d\omega=\int_{W}0=0}.
  60. ω = d f \omega=df
  61. 𝐑 n , \mathbf{R}^{n},
  62. ω \omega
  63. f ( b ) - f ( a ) f(b)-f(a)
  64. 0 1 d x = 1 , \textstyle{\int_{0}^{1}dx=1},
  65. 1 0 d x = - 0 1 d x = - 1. \textstyle{\int_{1}^{0}dx=-\int_{0}^{1}dx=-1}.
  66. | d x | |dx|
  67. | J | |J|
  68. x - x x\mapsto-x
  69. d x dx
  70. - d x -dx
  71. | d x | |dx|
  72. | d x | |dx|
  73. [ M ] [M]
  74. k < n k<n
  75. 𝐅 = 1 2 f a b d x a d x b , \,\textbf{F}=\frac{1}{2}f_{ab}\,dx^{a}\wedge dx^{b}\,,
  76. f a b f_{ab}
  77. E \vec{E}
  78. B \vec{B}
  79. f 12 = E z / c , f_{12}=E_{z}/c\,,
  80. f 23 = - B z \,f_{23}=-B_{z}
  81. 𝐅 = d 𝐀 . \,\textbf{F}=d\,\textbf{A}.
  82. 𝐉 = 1 6 j a ϵ a b c d d x b d x c d x d , \,\textbf{J}=\frac{1}{6}j^{a}\,\epsilon_{abcd}\,dx^{b}\wedge dx^{c}\wedge dx^{% d}\,,
  83. j a j^{a}
  84. F a b \,F_{ab}
  85. f a b , \,f_{ab}\,,
  86. J a J^{a}
  87. j a j^{a}
  88. J \vec{J}
  89. 𝐉 , \mathbf{J}\,,
  90. d 𝐅 = 𝟎 d\,{\,\textbf{F}}=\,\textbf{0}
  91. d * 𝐅 = 𝐉 d\,{*\,\textbf{F}}=\,\textbf{J}
  92. * *
  93. * 𝐅 , *\mathbf{F}\,,
  94. 𝐅 = d 𝐀 + 𝐀 𝐀 . \mathbf{F}=d\mathbf{A}+\mathbf{A}\wedge\mathbf{A}.
  95. 𝐀 𝐀 = 0 \mathbf{A}\wedge\mathbf{A}=0

Differential_operator.html

  1. A A
  2. 1 \mathcal{F}_{1}
  3. 2 \mathcal{F}_{2}
  4. f 2 f\in\mathcal{F}_{2}
  5. f f
  6. u 1 u\in\mathcal{F}_{1}
  7. f = A ( u ) . f=A(u)\ .
  8. u u
  9. P ( x , D ) = | α | m a α ( x ) D α , P(x,D)=\sum_{|\alpha|\leq m}a_{\alpha}(x)D^{\alpha}\ ,
  10. α = ( α 1 , α 2 , , α n ) \alpha=(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})
  11. | α | = α 1 + α 2 + + α n |\alpha|=\alpha_{1}+\alpha_{2}+\cdots+\alpha_{n}
  12. a α ( x ) a_{\alpha}(x)
  13. D α = D α 1 D α 2 D α n . D^{\alpha}=D^{\alpha_{1}}D^{\alpha_{2}}\cdots D^{\alpha_{n}}\ .
  14. D j = - i x j D_{j}=-i\frac{\partial}{\partial x_{j}}
  15. D j = x j D_{j}=\frac{\partial}{\partial x_{j}}
  16. d d x , D , D x , {d\over dx},D,\,D_{x},\,
  17. x . \partial_{x}.
  18. d n d x n , {d^{n}\over dx^{n}},
  19. D n , D^{n}\,,
  20. D x n . D^{n}_{x}.\,
  21. [ f ( x ) ] [f(x)]^{\prime}\,\!
  22. f ( x ) . f^{\prime}(x).\,\!
  23. k = 0 n c k D k \sum_{k=0}^{n}c_{k}D^{k}
  24. Δ = 2 = k = 1 n 2 x k 2 . \Delta=\nabla^{2}=\sum_{k=1}^{n}{\partial^{2}\over\partial x_{k}^{2}}.
  25. Θ = z d d z . \Theta=z{d\over dz}.
  26. Θ ( z k ) = k z k , k = 0 , 1 , 2 , \Theta(z^{k})=kz^{k},\quad k=0,1,2,\dots
  27. Θ = k = 1 n x k x k . \Theta=\sum_{k=1}^{n}x_{k}\frac{\partial}{\partial x_{k}}.
  28. f x g = g x f f\overleftarrow{\partial_{x}}g=g\cdot\partial_{x}f
  29. f x g = f x g f\overrightarrow{\partial_{x}}g=f\cdot\partial_{x}g
  30. f x g = f x g - g x f . f\overleftrightarrow{\partial_{x}}g=f\cdot\partial_{x}g-g\cdot\partial_{x}f.
  31. = 𝐱 ^ x + 𝐲 ^ y + 𝐳 ^ z . \nabla=\mathbf{\hat{x}}{\partial\over\partial x}+\mathbf{\hat{y}}{\partial% \over\partial y}+\mathbf{\hat{z}}{\partial\over\partial z}.
  32. T u = k = 0 n a k ( x ) D k u Tu=\sum_{k=0}^{n}a_{k}(x)D^{k}u
  33. T * T^{*}
  34. T u , v = u , T * v \langle Tu,v\rangle=\langle u,T^{*}v\rangle
  35. , \langle\cdot,\cdot\rangle
  36. f , g = a b f ( x ) g ( x ) ¯ d x , \langle f,g\rangle=\int_{a}^{b}f(x)\,\overline{g(x)}\,dx,
  37. x a x\to a
  38. x b x\to b
  39. T * u = k = 0 n ( - 1 ) k D k [ a k ( x ) ¯ u ] . T^{*}u=\sum_{k=0}^{n}(-1)^{k}D^{k}[\overline{a_{k}(x)}u].\,
  40. T * T^{*}
  41. f , P * g L 2 ( Ω ) = P f , g L 2 ( Ω ) \langle f,P^{*}g\rangle_{L^{2}(\Omega)}=\langle Pf,g\rangle_{L^{2}(\Omega)}
  42. L u = - ( p u ) + q u = - ( p u ′′ + p u ) + q u = - p u ′′ - p u + q u = ( - p ) D 2 u + ( - p ) D u + ( q ) u . Lu=-(pu^{\prime})^{\prime}+qu=-(pu^{\prime\prime}+p^{\prime}u^{\prime})+qu=-pu% ^{\prime\prime}-p^{\prime}u^{\prime}+qu=(-p)D^{2}u+(-p^{\prime})Du+(q)u.\;\!
  43. L * u \displaystyle L^{*}u
  44. D ( f + g ) = ( D f ) + ( D g ) D(f+g)=(Df)+(Dg)\,
  45. D ( a f ) = a ( D f ) D(af)=a(Df)\,
  46. ( D 1 D 2 ) ( f ) = D 1 ( D 2 ( f ) ) . (D_{1}\circ D_{2})(f)=D_{1}(D_{2}(f)).\,
  47. D x - x D = 1. Dx-xD=1.\,
  48. R D , X R\langle D,X\rangle
  49. R D , X / I R\langle D,X\rangle/I
  50. X a D b mod I X^{a}D^{b}\mod{I}
  51. R [ X ] R[X]
  52. R D , X / I R\langle D,X\rangle/I
  53. R D 1 , , D n , X 1 , , X n R\langle D_{1},\ldots,D_{n},X_{1},\ldots,X_{n}\rangle
  54. D 1 , , D n , X 1 , , X n D_{1},\ldots,D_{n},X_{1},\ldots,X_{n}
  55. D i X j - X i D j - δ i , j , D i D j - D j D i , X i X j - X j X i D_{i}X_{j}-X_{i}D_{j}-\delta_{i,j},D_{i}D_{j}-D_{j}D_{i},X_{i}X_{j}-X_{j}X_{i}
  56. 1 i , j n 1\leq i,j\leq n
  57. δ \delta
  58. R D 1 , , D n , X 1 , , X n / I R\langle D_{1},\ldots,D_{n},X_{1},\ldots,X_{n}\rangle/I
  59. X 1 a 1 X n a n D 1 b 1 D n b n X_{1}^{a_{1}}\ldots X_{n}^{a_{n}}D_{1}^{b_{1}}\ldots D_{n}^{b_{n}}
  60. i P : J k ( E ) F i_{P}:J^{k}(E)\rightarrow F\,
  61. P = i P j k P=i_{P}\circ j^{k}
  62. f 0 , , f k C ( M ) f_{0},\ldots,f_{k}\in C^{\infty}(M)
  63. [ f k , [ f k - 1 , [ [ f 0 , P ] ] ] = 0. [f_{k},[f_{k-1},[\cdots[f_{0},P]\cdots]]=0.
  64. [ f , P ] : Γ ( E ) Γ ( F ) [f,P]:\Gamma(E)\rightarrow\Gamma(F)
  65. [ f , P ] ( s ) = P ( f s ) - f P ( s ) . [f,P](s)=P(f\cdot s)-f\cdot P(s).\,
  66. z = 1 2 ( x - i y ) , z ¯ = 1 2 ( x + i y ) . \frac{\partial}{\partial z}=\frac{1}{2}\left(\frac{\partial}{\partial x}-i% \frac{\partial}{\partial y}\right)\quad,\quad\frac{\partial}{\partial\bar{z}}=% \frac{1}{2}\left(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}% \right)\ .

Differential_rotation.html

  1. Ω = Ω 0 - Δ Ω sin 2 Ψ \Omega=\Omega_{0}-\Delta\Omega\sin^{2}\Psi
  2. Ω 0 \Omega_{0}
  3. Δ Ω = ( Ω 0 - Ω pole ) \Delta\Omega=(\Omega_{0}-\Omega_{\mathrm{pole}})
  4. Ψ \Psi
  5. 2 π Δ Ω \frac{2\pi}{\Delta\Omega}
  6. α = Δ Ω Ω 0 \alpha=\frac{\Delta\Omega}{\Omega_{0}}
  7. Ω 2 π ( 451.5 - 65.3 cos 2 θ - 66.7 cos 4 θ ) \frac{\Omega}{2\pi}(451.5-65.3\cos^{2}\theta-66.7\cos^{4}\theta)
  8. v c ( R ) = G M ( < R ) R v_{c}(R)=\frac{\sqrt{GM(<R)}}{R}
  9. v c ( R ) , v_{c}(R),
  10. R R
  11. M ( < R ) , M(<R),
  12. R R

Differential_scanning_calorimetry.html

  1. Δ H = K A \Delta H=KA
  2. Δ H \Delta H
  3. K K
  4. A A

Differintegral.html

  1. 𝔻 q f \mathbb{D}^{q}f
  2. 𝔻 t q a f ( t ) \displaystyle{}_{a}\mathbb{D}^{q}_{t}f(t)
  3. \mathcal{F}
  4. F ( ω ) = { f ( t ) } = 1 2 π - f ( t ) e - i ω t d t F(\omega)=\mathcal{F}\{f(t)\}=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(t)% e^{-i\omega t}\,dt
  5. [ d f ( t ) d t ] = i ω [ f ( t ) ] \mathcal{F}\left[\frac{df(t)}{dt}\right]=i\omega\mathcal{F}[f(t)]
  6. d n f ( t ) d t n = - 1 { ( i ω ) n [ f ( t ) ] } \frac{d^{n}f(t)}{dt^{n}}=\mathcal{F}^{-1}\left\{(i\omega)^{n}\mathcal{F}[f(t)]\right\}
  7. 𝔻 q f ( t ) = - 1 { ( i ω ) q [ f ( t ) ] } . \mathbb{D}^{q}f(t)=\mathcal{F}^{-1}\left\{(i\omega)^{q}\mathcal{F}[f(t)]\right\}.
  8. \mathcal{L}
  9. [ d f ( t ) d t ] = s [ f ( t ) ] . \mathcal{L}\left[\frac{df(t)}{dt}\right]=s\mathcal{L}[f(t)].
  10. 𝔻 q f ( t ) = - 1 { s q [ f ( t ) ] } . \mathbb{D}^{q}f(t)=\mathcal{L}^{-1}\left\{s^{q}\mathcal{L}[f(t)]\right\}.
  11. 𝔻 q ( f + g ) = 𝔻 q ( f ) + 𝔻 q ( g ) \mathbb{D}^{q}(f+g)=\mathbb{D}^{q}(f)+\mathbb{D}^{q}(g)
  12. 𝔻 q ( a f ) = a 𝔻 q ( f ) \mathbb{D}^{q}(af)=a\mathbb{D}^{q}(f)
  13. 𝔻 0 f = f \mathbb{D}^{0}f=f\,
  14. 𝔻 t q ( f g ) = j = 0 ( q j ) 𝔻 t j ( f ) 𝔻 t q - j ( g ) \mathbb{D}^{q}_{t}(fg)=\sum_{j=0}^{\infty}{q\choose j}\mathbb{D}^{j}_{t}(f)% \mathbb{D}^{q-j}_{t}(g)
  15. 𝔻 a 𝔻 b f = 𝔻 a + b f \mathbb{D}^{a}\mathbb{D}^{b}f=\mathbb{D}^{a+b}f
  16. 𝔻 q ( t n ) = Γ ( n + 1 ) Γ ( n + 1 - q ) t n - q \mathbb{D}^{q}(t^{n})=\frac{\Gamma(n+1)}{\Gamma(n+1-q)}t^{n-q}
  17. 𝔻 q ( sin ( t ) ) = sin ( t + q π 2 ) \mathbb{D}^{q}(\sin(t))=\sin\left(t+\frac{q\pi}{2}\right)
  18. 𝔻 q ( e a t ) = a q e a t \mathbb{D}^{q}(e^{at})=a^{q}e^{at}

Diffusing_capacity.html

  1. ( D L O 2 ) (D_{L_{O_{2}}})
  2. V O 2 n O 2 {V_{O_{2}}}\propto{n_{O_{2}}}
  3. ( D L O 2 ) (D_{L_{O_{2}}})
  4. ( V ˙ O 2 ) (\dot{V}_{O_{2}})
  5. V ˙ \dot{V}
  6. V ˙ O 2 \dot{V}_{O_{2}}
  7. P A O 2 P_{A_{O_{2}}}
  8. P a O 2 P_{a_{O_{2}}}
  9. P v O 2 P_{v_{O_{2}}}
  10. D L D_{L}
  11. D L D_{L}
  12. D L D_{L}
  13. ( D L C O ) (D_{L_{CO}})
  14. Δ [ C O ] \Delta{[CO]}
  15. Δ t \Delta{t}
  16. V A V_{A}
  17. F A C O O F_{A_{CO_{O}}}
  18. V B V_{B}
  19. D L C O D_{L_{CO}}
  20. D L C O D_{L_{CO}}
  21. D L C O D_{L_{CO}}
  22. ( D L ) (D_{L})
  23. D M D_{M}
  24. θ \theta
  25. V c V_{c}
  26. V c V_{c}
  27. D L C O D_{L_{CO}}
  28. V c V_{c}
  29. D L C O D_{L_{CO}}
  30. D L C O D_{L_{CO}}
  31. V c V_{c}
  32. D L C O D_{L_{CO}}
  33. θ \theta
  34. θ \theta
  35. D L C O D_{L_{CO}}
  36. D L C O D_{L_{CO}}
  37. θ \theta
  38. θ \theta
  39. D L C O D_{L_{CO}}
  40. θ \theta
  41. D L C O D_{L_{CO}}
  42. D M D_{M}
  43. θ * V c \theta*V_{c}
  44. D M D_{M}
  45. θ * V c \theta*V_{c}
  46. D L C O D_{L_{CO}}
  47. D L C O D_{L_{CO}}

Dihedral_group.html

  1. r i r j = r i + j , r i s j = s i + j , s i r j = s i - j , s i s j = r i - j . \mathrm{r}_{i}\,\mathrm{r}_{j}=\mathrm{r}_{i+j},\quad\mathrm{r}_{i}\,\mathrm{s% }_{j}=\mathrm{s}_{i+j},\quad\mathrm{s}_{i}\,\mathrm{r}_{j}=\mathrm{s}_{i-j},% \quad\mathrm{s}_{i}\,\mathrm{s}_{j}=\mathrm{r}_{i-j}.
  2. r 0 = ( 1 0 0 1 ) , r 1 = ( 0 - 1 1 0 ) , r 2 = ( - 1 0 0 - 1 ) , r 3 = ( 0 1 - 1 0 ) , s 0 = ( 1 0 0 - 1 ) , s 1 = ( 0 1 1 0 ) , s 2 = ( - 1 0 0 1 ) , s 3 = ( 0 - 1 - 1 0 ) . \begin{matrix}\mathrm{r}_{0}=\bigl(\begin{smallmatrix}1&0\\ 0&1\end{smallmatrix}\bigr),&\mathrm{r}_{1}=\bigl(\begin{smallmatrix}0&-1\\ 1&0\end{smallmatrix}\bigr),&\mathrm{r}_{2}=\bigl(\begin{smallmatrix}-1&0\\ 0&-1\end{smallmatrix}\bigr),&\mathrm{r}_{3}=\bigl(\begin{smallmatrix}0&1\\ -1&0\end{smallmatrix}\bigr),\\ \mathrm{s}_{0}=\bigl(\begin{smallmatrix}1&0\\ 0&-1\end{smallmatrix}\bigr),&\mathrm{s}_{1}=\bigl(\begin{smallmatrix}0&1\\ 1&0\end{smallmatrix}\bigr),&\mathrm{s}_{2}=\bigl(\begin{smallmatrix}-1&0\\ 0&1\end{smallmatrix}\bigr),&\mathrm{s}_{3}=\bigl(\begin{smallmatrix}0&-1\\ -1&0\end{smallmatrix}\bigr).\end{matrix}
  3. r k \displaystyle\mathrm{r}_{k}
  4. s r s = r - 1 srs=r^{-1}\,
  5. e 2 π i n e^{2\pi i\over n}
  6. r 1 = [ cos 2 π n - sin 2 π n sin 2 π n cos 2 π n ] s 0 = [ 1 0 0 - 1 ] \mathrm{r}_{1}=\begin{bmatrix}\cos{2\pi\over n}&-\sin{2\pi\over n}\\ \sin{2\pi\over n}&\cos{2\pi\over n}\end{bmatrix}\qquad\mathrm{s}_{0}=\begin{% bmatrix}1&0\\ 0&-1\end{bmatrix}
  7. r j = r 1 j \mathrm{r}_{j}=\mathrm{r}_{1}^{j}
  8. s j = r j s 0 \mathrm{s}_{j}=\mathrm{r}_{j}\,\mathrm{s}_{0}
  9. j { 1 , , n - 1 } j\in\{1,\ldots,n-1\}
  10. r j r k = r ( j + k ) mod n \mathrm{r}_{j}\,\mathrm{r}_{k}=\mathrm{r}_{(j+k)\,\text{ mod }n}
  11. r j s k = s ( j + k ) mod n \mathrm{r}_{j}\,\mathrm{s}_{k}=\mathrm{s}_{(j+k)\,\text{ mod }n}
  12. s j r k = s ( j - k ) mod n \mathrm{s}_{j}\,\mathrm{r}_{k}=\mathrm{s}_{(j-k)\,\text{ mod }n}
  13. s j s k = r ( j - k ) mod n . \mathrm{s}_{j}\,\mathrm{s}_{k}=\mathrm{r}_{(j-k)\,\text{ mod }n}.
  14. D n = r , s o r d ( r ) = n , o r d ( s ) = 2 , s r s = r - 1 D_{n}=\langle r,s\mid ord(r)=n,ord(s)=2,srs=r^{-1}\rangle
  15. D n = x , y x n = y 2 = ( x y ) 2 = 1 . D_{n}=\langle x,y\mid x^{n}=y^{2}=(xy)^{2}=1\rangle.
  16. n φ 2 \mathbb{Z}_{n}\rtimes_{\varphi}\mathbb{Z}_{2}
  17. φ ( 0 ) \varphi(0)
  18. φ ( 1 ) \varphi(1)
  19. 2 = 2 1 2=2^{1}
  20. 2 n = 2 ( 2 k + 1 ) 2n=2(2k+1)
  21. = { a x + b ( a , n ) = 1 } =\{ax+b\mid(a,n)=1\}
  22. n ϕ ( n ) , n\phi(n),
  23. ϕ \phi
  24. 1 , , n - 1 1,\dots,n-1
  25. k ( 2 π / n ) k(2\pi/n)
  26. π / n \pi/n
  27. k = ± 1. k=\pm 1.

Dimension_of_an_algebraic_variety.html

  1. R = K [ x 1 , , x n ] . R=K[x_{1},\ldots,x_{n}].
  2. d d
  3. V 0 V 1 V d V_{0}\subset V_{1}\subset\ldots\subset V_{d}
  4. p 0 p 1 p d p_{0}\subset p_{1}\subset\ldots\subset p_{d}
  5. R = K [ x 0 , x 1 , , x n ] R=K[x_{0},x_{1},\ldots,x_{n}]
  6. x 1 e 1 x n e n {x_{1}}^{e_{1}}\cdots{x_{n}}^{e_{n}}
  7. x 1 min ( e 1 , 1 ) x n min ( e n , 1 ) . x_{1}^{\min(e_{1},1)}\cdots x_{n}^{\min(e_{n},1)}.
  8. x 2 + y 2 + z 2 = 0 x^{2}+y^{2}+z^{2}=0

Diophantine_approximation.html

  1. π \pi
  2. α α
  3. α α
  4. p / q p/q
  5. α α
  6. | α - p q | < | α - p q | , \left|\alpha-\frac{p}{q}\right|<\left|\alpha-\frac{p^{\prime}}{q^{\prime}}% \right|,
  7. / 𝐪 \mathbf{/q}
  8. p / q p/q
  9. | q α - p | < | q α - p | . \left|q\alpha-p\right|<\left|q^{\prime}\alpha-p^{\prime}\right|.
  10. [ 2 ; 1 , 2 , 1 , 1 , 4 , 1 , 1 , 6 , 1 , 1 , 8 , 1 , ] . [2;1,2,1,1,4,1,1,6,1,1,8,1,\ldots\;].
  11. 3 , 8 3 , 11 4 , 19 7 , 87 32 , , 3,\tfrac{8}{3},\tfrac{11}{4},\tfrac{19}{7},\tfrac{87}{32},\ldots\,,
  12. 3 , 5 2 , 8 3 , 11 4 , 19 7 , 49 18 , 68 25 , 87 32 , 106 39 , . 3,\tfrac{5}{2},\tfrac{8}{3},\tfrac{11}{4},\tfrac{19}{7},\tfrac{49}{18},\tfrac{% 68}{25},\tfrac{87}{32},\tfrac{106}{39},\ldots\,.
  13. α α
  14. p / q p/q
  15. | α - p q | . \left|\alpha-\frac{p}{q}\right|.
  16. p p
  17. q q
  18. φ φ
  19. q q
  20. α α
  21. p / q p/q
  22. | α - p q | > ϕ ( q ) \left|\alpha-\frac{p}{q}\right|>\phi(q)
  23. φ φ
  24. α α
  25. α α
  26. p / q p/q
  27. | α - p q | < ϕ ( q ) \left|\alpha-\frac{p}{q}\right|<\phi(q)
  28. | x - p q | > c q 2 . \left|{x-\frac{p}{q}}\right|>\frac{c}{q^{2}}\ .
  29. α = a b \alpha=\frac{a}{b}
  30. p i q i = i a i b \tfrac{p_{i}}{q_{i}}=\tfrac{i\,a}{i\,b}
  31. p q α = a b , \tfrac{p}{q}\not=\alpha=\tfrac{a}{b}\,,
  32. | a b - p q | = | a q - b p b q | 1 b q , \left|\frac{a}{b}-\frac{p}{q}\right|=\left|\frac{aq-bp}{bq}\right|\geq\frac{1}% {bq},
  33. | a q - b p | |aq-bp|
  34. | x - p q | > c ( x ) q n \left|x-\frac{p}{q}\right|>\frac{c(x)}{q^{n}}
  35. j = 1 10 - j ! = 0.110001000000000000000001000 , \sum_{j=1}^{\infty}10^{-j!}=0.110001000000000000000001000\ldots\,,
  36. x x
  37. ε ε
  38. c ( x , ε ) c(x,ε)
  39. | x - p q | > c ( x , ε ) q 2 + ε \left|x-\frac{p}{q}\right|>\frac{c(x,\varepsilon)}{q^{2+\varepsilon}}
  40. p p
  41. q q
  42. q > 0 q>0
  43. ε ε
  44. n n
  45. | x i - p i / q | < q - ( 1 + 1 / n + ε ) , i = 1 , , n . |x_{i}-p_{i}/q|<q^{-(1+1/n+\varepsilon)},\quad i=1,\ldots,n.
  46. ε ε
  47. α α
  48. p q \tfrac{p}{q}\;
  49. | α - p q | < 1 q 2 . \left|\alpha-\frac{p}{q}\right|<\frac{1}{q^{2}}\,.
  50. ε ε
  51. α α
  52. p q \tfrac{p}{q}\;
  53. | α - p q | < 1 5 q 2 . \left|\alpha-\frac{p}{q}\right|<\frac{1}{\sqrt{5}q^{2}}\,.
  54. 1 5 q 2 \frac{1}{\sqrt{5}\,q^{2}}
  55. x , y x,y
  56. a , b , c , d a,b,c,d\;
  57. a d - b c = ± 1 ad-bc=\pm 1\;
  58. y = a x + b c x + d . y=\frac{ax+b}{cx+d}\,.
  59. SL 2 ± ( \Z ) \,\text{SL}_{2}^{\pm}(\Z)
  60. x = [ u 0 ; u 1 , u 2 , ] , x=[u_{0};u_{1},u_{2},\ldots]\,,
  61. y = [ v 0 ; v 1 , v 2 , ] , y=[v_{0};v_{1},v_{2},\ldots]\,,
  62. u h + i = v k + i u_{h+i}=v_{k+i}
  63. ϕ = 1 + 5 2 \phi=\tfrac{1+\sqrt{5}}{2}
  64. c > 5 c>\sqrt{5}\;
  65. p / q p/q
  66. | ϕ - p q | < 1 c q 2 \left|\phi-\frac{p}{q}\right|<\frac{1}{c\,q^{2}}
  67. ϕ \phi
  68. α \alpha
  69. ϕ \phi
  70. p q \tfrac{p}{q}\;
  71. | α - p q | < 1 8 q 2 . \left|\alpha-\frac{p}{q}\right|<\frac{1}{\sqrt{8}q^{2}}.
  72. 2 \sqrt{2}
  73. ψ \psi
  74. ψ \psi
  75. | x - p q | < ψ ( q ) | q | . \left|x-\frac{p}{q}\right|<\frac{\psi(q)}{|q|}.
  76. q ψ ( q ) \sum_{q}\psi(q)
  77. ψ \psi
  78. ψ \psi
  79. ψ \psi
  80. ψ c ( q ) = q - c \psi_{c}(q)=q^{-c}
  81. ψ c \psi_{c}
  82. ψ c \psi_{c}
  83. 1 / c 1/c
  84. ψ c \psi_{c}
  85. c > 1 c>1
  86. ψ c \psi_{c}
  87. c > 1 c>1
  88. ψ ϵ ( q ) = ϵ q - 1 \psi_{\epsilon}(q)=\epsilon q^{-1}
  89. ϵ > 0 \epsilon>0
  90. ψ ϵ \psi_{\epsilon}
  91. f 1 , f 2 , f_{1},f_{2},\ldots
  92. f 1 ( x ) , f 2 ( x ) , f_{1}(x),f_{2}(x),\ldots

Diophantine_set.html

  1. x ¯ \overline{x}
  2. y ¯ \overline{y}
  3. x ¯ \overline{x}
  4. y ¯ \overline{y}
  5. x ¯ \overline{x}
  6. y ¯ \overline{y}
  7. n ¯ S ( m ¯ k ) ( P ( n ¯ , m ¯ ) = 0 ) \bar{n}\in S\iff(\exists\bar{m}\in\mathbb{N}^{k})(P(\bar{n},\bar{m})=0)
  8. \mathbb{Q}
  9. x 2 - d ( y + 1 ) 2 = 1 x^{2}-d(y+1)^{2}=1
  10. x , y x,y
  11. d d
  12. a = ( 2 x + 3 ) y a=(2x+3)y
  13. \mathbb{N}
  14. a = ( x + 2 ) ( y + 2 ) a=(x+2)(y+2)
  15. \mathbb{N}
  16. a + x = b a+x=b
  17. ( a , b ) (a\,,\,b)
  18. a b . a\leq b.\,
  19. Σ 1 0 \Sigma^{0}_{1}

Dioptre.html

  1. 1 / 3 {1}/{3}
  2. V = 0.25 m × φ + 1 V=0.25\ \mathrm{m}\times\varphi+1

Dirac_sea.html

  1. E 2 = p 2 c 2 + m 2 c 4 E^{2}=p^{2}c^{2}+m^{2}c^{4}
  2. E 2 = m 2 c 4 E^{2}=m^{2}c^{4}
  3. E = m c 2 E=mc^{2}
  4. x x = x 2 x\cdot x=x^{2}
  5. ( - x ) ( - x ) = x 2 (-x)\cdot(-x)=x^{2}
  6. E = ± m c 2 . E={\pm}mc^{2}.
  7. ψ ( x ) = a ( k ) e i k x + a ( k ) e - i k x \psi(x)=\sum a^{\dagger}(k)e^{ikx}+a(k)e^{-ikx}
  8. a ( k ) \scriptstyle a^{\dagger}(k)
  9. a ( k ) \scriptstyle a(k)
  10. N = a a = 1 - a a N=a^{\dagger}a=1-aa^{\dagger}

Dirac_spinor.html

  1. ψ = ω p e - i p x \psi=\omega_{\vec{p}}\;e^{-ipx}\;
  2. ( i γ μ μ - m ) ψ = 0 , (i\gamma^{\mu}\partial_{\mu}-m)\psi=0\;,
  3. c = = 1 \scriptstyle c\,=\,\hbar\,=\,1
  4. ψ \scriptstyle\psi
  5. ω p \scriptstyle\omega_{\vec{p}}
  6. p \scriptstyle\vec{p}
  7. p x p μ x μ \scriptstyle px\;\equiv\;p_{\mu}x^{\mu}
  8. p μ = { ± m 2 + p 2 , p } \scriptstyle p^{\mu}\;=\;\{\pm\sqrt{m^{2}+\vec{p}^{2}},\,\vec{p}\}
  9. p \scriptstyle\vec{p}
  10. x μ \scriptstyle x^{\mu}
  11. ω p = [ ϕ σ p E p + m ϕ ] , \omega_{\vec{p}}=\begin{bmatrix}\phi\\ \frac{\vec{\sigma}\vec{p}}{E_{\vec{p}}+m}\phi\end{bmatrix}\;,
  12. ϕ \scriptstyle\phi
  13. σ \scriptstyle\vec{\sigma}
  14. E p \scriptstyle E_{\vec{p}}
  15. E p = + m 2 + p 2 \scriptstyle E_{\vec{p}}\;=\;+\sqrt{m^{2}+\vec{p}^{2}}
  16. ( - i α + β m ) ψ = i ψ t \left(-i\vec{\alpha}\cdot\vec{\nabla}+\beta m\right)\psi=i\frac{\partial\psi}{% \partial t}\,
  17. ω \scriptstyle\omega
  18. α = [ 𝟎 σ σ 𝟎 ] β = [ 𝐈 𝟎 𝟎 - 𝐈 ] \vec{\alpha}=\begin{bmatrix}\mathbf{0}&\vec{\sigma}\\ \vec{\sigma}&\mathbf{0}\end{bmatrix}\quad\quad\beta=\begin{bmatrix}\mathbf{I}&% \mathbf{0}\\ \mathbf{0}&-\mathbf{I}\end{bmatrix}\,
  19. ψ = ω e - i p x \psi=\omega e^{-ip\cdot x}
  20. ω = [ ϕ χ ] \omega=\begin{bmatrix}\phi\\ \chi\end{bmatrix}\,
  21. E [ ϕ χ ] = [ m 𝐈 σ p σ p - m 𝐈 ] [ ϕ χ ] E\begin{bmatrix}\phi\\ \chi\end{bmatrix}=\begin{bmatrix}m\mathbf{I}&\vec{\sigma}\vec{p}\\ \vec{\sigma}\vec{p}&-m\mathbf{I}\end{bmatrix}\begin{bmatrix}\phi\\ \chi\end{bmatrix}\,
  22. ( E - m ) ϕ = ( σ p ) χ \left(E-m\right)\phi=\left(\vec{\sigma}\vec{p}\right)\chi\,
  23. ( E + m ) χ = ( σ p ) ϕ \left(E+m\right)\chi=\left(\vec{\sigma}\vec{p}\right)\phi\,
  24. χ \scriptstyle\chi\,
  25. ω = [ ϕ χ ] = [ ϕ σ p E + m ϕ ] \omega=\begin{bmatrix}\phi\\ \chi\end{bmatrix}=\begin{bmatrix}\phi\\ \frac{\vec{\sigma}\vec{p}}{E+m}\phi\end{bmatrix}\,
  26. ϕ \phi\,
  27. ω = [ ϕ χ ] = [ - σ p - E + m χ χ ] \omega=\begin{bmatrix}\phi\\ \chi\end{bmatrix}=\begin{bmatrix}-\frac{\vec{\sigma}\vec{p}}{-E+m}\chi\\ \chi\end{bmatrix}\,
  28. ϕ 1 = [ 1 0 ] ϕ 2 = [ 0 1 ] \phi^{1}=\begin{bmatrix}1\\ 0\end{bmatrix}\quad\quad\phi^{2}=\begin{bmatrix}0\\ 1\end{bmatrix}\,
  29. χ 1 = [ 0 1 ] χ 2 = [ 1 0 ] \chi^{1}=\begin{bmatrix}0\\ 1\end{bmatrix}\quad\quad\chi^{2}=\begin{bmatrix}1\\ 0\end{bmatrix}\,
  30. σ 1 = [ 0 1 1 0 ] σ 2 = [ 0 - i i 0 ] σ 3 = [ 1 0 0 - 1 ] \sigma_{1}=\begin{bmatrix}0&1\\ 1&0\end{bmatrix}\quad\quad\sigma_{2}=\begin{bmatrix}0&-i\\ i&0\end{bmatrix}\quad\quad\sigma_{3}=\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}
  31. σ p = σ 1 p 1 + σ 2 p 2 + σ 3 p 3 = [ p 3 p 1 - i p 2 p 1 + i p 2 - p 3 ] \vec{\sigma}\vec{p}=\sigma_{1}p_{1}+\sigma_{2}p_{2}+\sigma_{3}p_{3}=\begin{% bmatrix}p_{3}&p_{1}-ip_{2}\\ p_{1}+ip_{2}&-p_{3}\end{bmatrix}
  32. ω ω = 2 E \scriptstyle\omega^{\dagger}\omega\;=\;2E\,
  33. u ( p , s ) = E + m [ ϕ ( s ) σ p E + m ϕ ( s ) ] u(\vec{p},s)=\sqrt{E+m}\begin{bmatrix}\phi^{(s)}\\ \frac{\vec{\sigma}\cdot\vec{p}}{E+m}\phi^{(s)}\end{bmatrix}\,
  34. u ( p , 1 ) = E + m [ 1 0 p 3 E + m p 1 + i p 2 E + m ] and u ( p , 2 ) = E + m [ 0 1 p 1 - i p 2 E + m - p 3 E + m ] u(\vec{p},1)=\sqrt{E+m}\begin{bmatrix}1\\ 0\\ \frac{p_{3}}{E+m}\\ \frac{p_{1}+ip_{2}}{E+m}\end{bmatrix}\quad\mathrm{and}\quad u(\vec{p},2)=\sqrt% {E+m}\begin{bmatrix}0\\ 1\\ \frac{p_{1}-ip_{2}}{E+m}\\ \frac{-p_{3}}{E+m}\end{bmatrix}
  35. E \scriptstyle E
  36. E \scriptstyle E
  37. p \scriptstyle\vec{p}
  38. v ( p , s ) = E + m [ σ p E + m χ ( s ) χ ( s ) ] v(\vec{p},s)=\sqrt{E+m}\begin{bmatrix}\frac{\vec{\sigma}\cdot\vec{p}}{E+m}\chi% ^{(s)}\\ \chi^{(s)}\end{bmatrix}\,
  39. χ \scriptstyle\chi
  40. v ( p , 1 ) = E + m [ p 1 - i p 2 E + m - p 3 E + m 0 1 ] and v ( p , 2 ) = E + m [ p 3 E + m p 1 + i p 2 E + m 1 0 ] v(\vec{p},1)=\sqrt{E+m}\begin{bmatrix}\frac{p_{1}-ip_{2}}{E+m}\\ \frac{-p_{3}}{E+m}\\ 0\\ 1\end{bmatrix}\quad\mathrm{and}\quad v(\vec{p},2)=\sqrt{E+m}\begin{bmatrix}% \frac{p_{3}}{E+m}\\ \frac{p_{1}+ip_{2}}{E+m}\\ 1\\ 0\\ \end{bmatrix}
  41. s = 1 , 2 u p ( s ) u ¯ p ( s ) = p / + m \sum_{s=1,2}{u^{(s)}_{p}\bar{u}^{(s)}_{p}}=p\!\!\!/+m\,
  42. s = 1 , 2 v p ( s ) v ¯ p ( s ) = p / - m \sum_{s=1,2}{v^{(s)}_{p}\bar{v}^{(s)}_{p}}=p\!\!\!/-m\,
  43. p / = γ μ p μ p\!\!\!/=\gamma^{\mu}p_{\mu}\,
  44. u ¯ = u γ 0 \bar{u}=u^{\dagger}\gamma^{0}\,
  45. γ μ \scriptstyle\gamma^{\mu}
  46. μ \scriptstyle\mu
  47. γ μ \scriptstyle\gamma^{\mu}
  48. i \scriptstyle i
  49. ( i γ 2 γ 3 , i γ 3 γ 1 , i γ 1 γ 2 ) = - ( γ 1 , γ 2 , γ 3 ) i γ 1 γ 2 γ 3 (i\gamma^{2}\gamma^{3},\;\;i\gamma^{3}\gamma^{1},\;\;i\gamma^{1}\gamma^{2})=-(% \gamma^{1},\;\gamma^{2},\;\gamma^{3})i\gamma^{1}\gamma^{2}\gamma^{3}
  50. σ ( a , b , c ) = i a γ 2 γ 3 + i b γ 3 γ 1 + i c γ 1 γ 2 \sigma_{(a,b,c)}=ia\gamma^{2}\gamma^{3}+ib\gamma^{3}\gamma^{1}+ic\gamma^{1}% \gamma^{2}
  51. P ( a , b , c ) = 1 2 ( 1 + σ ( a , b , c ) ) P_{(a,b,c)}=\tfrac{1}{2}\left(1+\sigma_{(a,b,c)}\right)
  52. Q = - γ 0 \scriptstyle Q\,=\,-\gamma^{0}
  53. Q \scriptstyle Q
  54. Q \scriptstyle Q
  55. σ ( a , b , c ) \scriptstyle\sigma_{(a,b,c)}
  56. Q \scriptstyle Q
  57. P - Q = 1 2 ( 1 - Q ) = 1 2 ( 1 + γ 0 ) P_{-Q}=\tfrac{1}{2}\left(1-Q\right)=\tfrac{1}{2}\left(1+\gamma^{0}\right)
  58. P ( a , b , c ) P - Q P_{(a,b,c)}\;P_{-Q}
  59. P ( 0 , 0 , 1 ) = 1 2 ( 1 + i γ 1 γ 2 ) P_{(0,0,1)}=\tfrac{1}{2}\left(1+i\gamma_{1}\gamma_{2}\right)
  60. P = 1 2 ( 1 + i γ 1 γ 2 ) 1 2 ( 1 + γ 0 ) = 1 4 ( 1 + γ 0 + i γ 1 γ 2 + i γ 0 γ 1 γ 2 ) P=\tfrac{1}{2}\left(1+i\gamma^{1}\gamma^{2}\right)\cdot\tfrac{1}{2}\left(1+% \gamma^{0}\right)=\tfrac{1}{4}\left(1+\gamma^{0}+i\gamma^{1}\gamma^{2}+i\gamma% ^{0}\gamma^{1}\gamma^{2}\right)
  61. γ 0 = [ 0 1 1 0 ] \gamma_{0}=\begin{bmatrix}0&1\\ 1&0\end{bmatrix}
  62. γ k = [ 0 σ k - σ k 0 ] \gamma_{k}=\begin{bmatrix}0&\sigma^{k}\\ -\sigma^{k}&0\end{bmatrix}
  63. σ i \sigma^{i}
  64. P = 1 4 [ 1 + σ 3 1 + σ 3 1 + σ 3 1 + σ 3 ] = 1 2 [ 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 ] P=\frac{1}{4}\begin{bmatrix}1+\sigma^{3}&1+\sigma^{3}\\ 1+\sigma^{3}&1+\sigma^{3}\end{bmatrix}=\frac{1}{2}\begin{bmatrix}1&0&1&0\\ 0&0&0&0\\ 1&0&1&0\\ 0&0&0&0\end{bmatrix}
  65. | e - , + 1 2 = [ 1 0 1 0 ] \left|e^{-},\,+\tfrac{1}{2}\right\rangle=\begin{bmatrix}1\\ 0\\ 1\\ 0\end{bmatrix}
  66. 1 4 [ 1 + c a - i b ± ( 1 + c ) ± ( a - i b ) a + i b 1 - c ± ( a + i b ) ± ( 1 - c ) ± ( 1 + c ) ± ( a - i b ) 1 + c a - i b ± ( a + i b ) ± ( 1 - c ) a + i b 1 - c ] \frac{1}{4}\begin{bmatrix}1+c&a-ib&\pm(1+c)&\pm(a-ib)\\ a+ib&1-c&\pm(a+ib)&\pm(1-c)\\ \pm(1+c)&\pm(a-ib)&1+c&a-ib\\ \pm(a+ib)&\pm(1-c)&a+ib&1-c\end{bmatrix}
  67. a 2 + b 2 + c 2 = 1 \scriptstyle a^{2}+b^{2}+c^{2}\,=\,1

Direct_limit.html

  1. I , \langle I,\leq\rangle
  2. { A i : i I } \{A_{i}:i\in I\}
  3. I I\,
  4. f i j : A i A j f_{ij}:A_{i}\rightarrow A_{j}
  5. i j i\leq j
  6. f i i f_{ii}\,
  7. A i A_{i}\,
  8. f i k = f j k f i j f_{ik}=f_{jk}\circ f_{ij}
  9. i j k i\leq j\leq k
  10. A i , f i j \langle A_{i},f_{ij}\rangle
  11. I I\,
  12. A A\,
  13. A i , f i j \langle A_{i},f_{ij}\rangle
  14. A i A_{i}\,
  15. \sim\,
  16. lim A i = i A i / . \underrightarrow{\lim}A_{i}=\bigsqcup_{i}A_{i}\bigg/\sim.
  17. x i A i x_{i}\in A_{i}
  18. x j A j x_{j}\in A_{j}
  19. x i x j x_{i}\sim\,x_{j}
  20. k I k\in I
  21. f i k ( x i ) = f j k ( x j ) f_{ik}(x_{i})=f_{jk}(x_{j})\,
  22. x i f i k ( x i ) x_{i}\sim\,f_{ik}(x_{i})
  23. ϕ i : A i A \phi_{i}:A_{i}\rightarrow A
  24. A A\,
  25. 𝒞 \mathcal{C}
  26. X i , f i j \langle X_{i},f_{ij}\rangle
  27. 𝒞 \mathcal{C}
  28. X X\,
  29. 𝒞 \mathcal{C}
  30. ϕ i : X i X \phi_{i}:X_{i}\rightarrow X
  31. ϕ i = ϕ j f i j \phi_{i}=\phi_{j}\circ f_{ij}
  32. X , ϕ i \langle X,\phi_{i}\rangle
  33. Y , ψ i \langle Y,\psi_{i}\rangle
  34. u : X Y u:X\rightarrow Y
  35. X = lim X i X=\underrightarrow{\lim}X_{i}
  36. X i , f i j \langle X_{i},f_{ij}\rangle
  37. 𝒞 \mathcal{C}
  38. I , \langle I,\leq\rangle
  39. \mathcal{I}
  40. i j i\rightarrow j
  41. i j i\leq j
  42. 𝒞 \mathcal{I}\rightarrow\mathcal{C}
  43. M i M_{i}
  44. M i \bigcup M_{i}
  45. Hom ( lim X i , Y ) = lim Hom ( X i , Y ) . \mathrm{Hom}(\underrightarrow{\lim}X_{i},Y)=\underleftarrow{\lim}\mathrm{Hom}(% X_{i},Y).

Direct_proof.html

  1. x = 2 a x=2a
  2. y = 2 b y=2b
  3. x + y = 2 a + 2 b = 2 ( a + b ) x+y=2a+2b=2(a+b)
  4. 1 2 a b \frac{1}{2}ab
  5. 4 ( 1 2 a b ) + c 2 4(\frac{1}{2}ab)+c^{2}
  6. ( a + b ) 2 = 4 ( 1 / 2 a b ) + c 2 (a+b)^{2}=4(1/2ab)+c^{2}
  7. a 2 + 2 a b + b 2 = 2 a b + c 2 a^{2}+2ab+b^{2}=2ab+c^{2}
  8. a 2 + b 2 = c 2 a^{2}+b^{2}=c^{2}
  9. n = 2 k + 1 n=2k+1
  10. n 2 \displaystyle n^{2}
  11. ( 2 k 2 + 2 k ) (2k^{2}+2k)
  12. 2 k + 1 \displaystyle 2k+1

Direct_sum_of_groups.html

  1. e e
  2. G G
  3. H H
  4. G G
  5. G G
  6. K G K\leq G
  7. G G
  8. H H
  9. K K
  10. H H
  11. G G
  12. H H
  13. G G
  14. G = i I H i G=\prod_{i\in I}H_{i}
  15. G G
  16. H i 0 × i i 0 H i H_{i_{0}}\times\prod_{i\not=i_{0}}H_{i}
  17. H H
  18. G G
  19. K G K\leq G
  20. G = K + H G=K+H
  21. G G
  22. G G
  23. \mathbb{R}
  24. K K
  25. G / K G/K
  26. e H i e_{H_{i}}

Directed_acyclic_graph.html

  1. a n = k = 1 n ( - 1 ) k - 1 ( n k ) 2 k ( n - k ) a n - k . a_{n}=\sum_{k=1}^{n}(-1)^{k-1}{n\choose k}2^{k(n-k)}a_{n-k}.

Dirichlet's_theorem_on_arithmetic_progressions.html

  1. a , a + d , a + 2 d , a + 3 d , , a,\ a+d,\ a+2d,\ a+3d,\ \dots,
  2. 1 3 + 1 7 + 1 11 + 1 19 + 1 23 + 1 31 + 1 43 + 1 47 + 1 59 + 1 67 + \frac{1}{3}+\frac{1}{7}+\frac{1}{11}+\frac{1}{19}+\frac{1}{23}+\frac{1}{31}+% \frac{1}{43}+\frac{1}{47}+\frac{1}{59}+\frac{1}{67}+\cdots
  3. φ ( d ) . \varphi(d).
  4. 1 φ ( d ) . \frac{1}{\varphi(d)}.
  5. q , 2 q , 3 q , q,2q,3q,\dots

Dirichlet_character.html

  1. / k \mathbb{Z}/k\mathbb{Z}
  2. χ \chi
  3. L ( s , χ ) = n = 1 χ ( n ) n s L(s,\chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}
  4. χ \chi
  5. \mathbb{Z}
  6. \mathbb{C}
  7. χ \chi
  8. χ \chi
  9. χ \chi
  10. χ \chi
  11. χ ( - 1 ) = - 1 \chi(-1)=-1
  12. χ ( - 1 ) = 1 \chi(-1)=1
  13. n ^ = { m | m n mod k } . \hat{n}=\{m|m\equiv n\mod k\}.
  14. n ^ \hat{n}
  15. φ ( k ) \varphi(k)
  16. m n ^ = m ^ n ^ \widehat{mn}=\hat{m}\hat{n}
  17. φ \varphi
  18. 1 ^ \hat{1}
  19. m ^ \hat{m}
  20. n ^ \hat{n}
  21. m ^ n ^ = 1 ^ \hat{m}\hat{n}=\hat{1}
  22. m n 1 mod k mn\equiv 1\mod k
  23. { 1 ^ , 5 ^ } \{\hat{1},\hat{5}\}
  24. χ \chi
  25. χ : ( / k ) * * \chi:(\mathbb{Z}/k\mathbb{Z})^{*}\to\mathbb{C}^{*}
  26. χ \chi
  27. χ 1 \chi_{1}
  28. χ 1 ( n ) = 1 \chi_{1}(n)=1
  29. χ 1 ( n ) = 0 \chi_{1}(n)=0
  30. φ ( 1 ) = 1 \varphi(1)=1
  31. χ 1 ( n ) \chi_{1}(n)
  32. φ ( 2 ) = 1 \varphi(2)=1
  33. χ 1 ( n ) \chi_{1}(n)
  34. φ ( 3 ) = 2 \varphi(3)=2
  35. χ 1 ( n ) \chi_{1}(n)
  36. χ 2 ( n ) \chi_{2}(n)
  37. φ ( 4 ) = 2 \varphi(4)=2
  38. χ 1 ( n ) \chi_{1}(n)
  39. χ 2 ( n ) \chi_{2}(n)
  40. χ 1 ( n ) \chi_{1}(n)
  41. L ( χ 1 , s ) = ( 1 - 2 - s ) ζ ( s ) L(\chi_{1},s)=(1-2^{-s})\zeta(s)\,
  42. ζ ( s ) \zeta(s)
  43. χ 2 ( n ) \chi_{2}(n)
  44. L ( χ 2 , s ) = β ( s ) . L(\chi_{2},s)=\beta(s).\,
  45. φ ( 5 ) = 4 \varphi(5)=4
  46. χ 1 ( n ) \chi_{1}(n)
  47. χ 2 ( n ) \chi_{2}(n)
  48. χ 3 ( n ) \chi_{3}(n)
  49. χ 4 ( n ) \chi_{4}(n)
  50. φ ( 6 ) = 2 \varphi(6)=2
  51. χ 1 ( n ) \chi_{1}(n)
  52. χ 2 ( n ) \chi_{2}(n)
  53. φ ( 7 ) = 6 \varphi(7)=6
  54. ω = exp ( π i / 3 ) . \omega=\exp(\pi i/3).
  55. χ 1 ( n ) \chi_{1}(n)
  56. χ 2 ( n ) \chi_{2}(n)
  57. χ 3 ( n ) \chi_{3}(n)
  58. χ 4 ( n ) \chi_{4}(n)
  59. χ 5 ( n ) \chi_{5}(n)
  60. χ 6 ( n ) \chi_{6}(n)
  61. φ ( 8 ) = 4 \varphi(8)=4
  62. χ 1 ( n ) \chi_{1}(n)
  63. χ 2 ( n ) \chi_{2}(n)
  64. χ 3 ( n ) \chi_{3}(n)
  65. χ 4 ( n ) \chi_{4}(n)
  66. φ ( 9 ) = 6 \varphi(9)=6
  67. ω = exp ( π i / 3 ) . \omega=\exp(\pi i/3).
  68. χ 1 ( n ) \chi_{1}(n)
  69. χ 2 ( n ) \chi_{2}(n)
  70. χ 3 ( n ) \chi_{3}(n)
  71. χ 4 ( n ) \chi_{4}(n)
  72. χ 5 ( n ) \chi_{5}(n)
  73. χ 6 ( n ) \chi_{6}(n)
  74. φ ( 10 ) = 4 \varphi(10)=4
  75. χ 1 ( n ) \chi_{1}(n)
  76. χ 2 ( n ) \chi_{2}(n)
  77. χ 3 ( n ) \chi_{3}(n)
  78. χ 4 ( n ) \chi_{4}(n)
  79. χ ( n ) = ( n p ) , \chi(n)=\left(\frac{n}{p}\right),
  80. ( n p ) \left(\frac{n}{p}\right)
  81. χ ( n ) = ( n m ) , \chi(n)=\left(\frac{n}{m}\right),
  82. ( n m ) \left(\frac{n}{m}\right)
  83. a mod n χ ( a ) = 0 \sum_{a\bmod n}\chi(a)=0
  84. χ χ ( a ) = 0 \sum_{\chi}\chi(a)=0
  85. a 1 ( mod n ) a\equiv 1\;\;(\mathop{{\rm mod}}n)

Dirichlet_convolution.html

  1. ( f * g ) ( n ) = d n f ( d ) g ( n d ) = a b = n f ( a ) g ( b ) \begin{aligned}\displaystyle(f*g)(n)&\displaystyle=\sum_{d\,\mid\,n}f(d)g\left% (\frac{n}{d}\right)\\ &\displaystyle=\sum_{ab\,=\,n}f(a)g(b)\end{aligned}
  2. C \scriptstyle C\subset\mathbb{Z}
  3. | μ | |μ|
  4. g ( n ) = - 1 f ( 1 ) d < n d n f ( n d ) g ( d ) . g(n)=\frac{-1}{f(1)}\sum_{\stackrel{d\,\mid\,n}{d<n}}f\left(\frac{n}{d}\right)% g(d).
  5. n α n^{\alpha}
  6. μ ( n ) n α \mu(n)\,n^{\alpha}
  7. | μ | |μ|
  8. D G ( f ; s ) = n = 1 f ( n ) n s DG(f;s)=\sum_{n=1}^{\infty}\frac{f(n)}{n^{s}}
  9. D G ( f ; s ) D G ( g ; s ) = D G ( f * g ; s ) DG(f;s)DG(g;s)=DG(f*g;s)\,

Discrete_Hartley_transform.html

  1. H k = n = 0 N - 1 x n [ cos ( 2 π N n k ) + sin ( 2 π N n k ) ] k = 0 , , N - 1 H_{k}=\sum_{n=0}^{N-1}x_{n}\left[\cos\left(\frac{2\pi}{N}nk\right)+\sin\left(% \frac{2\pi}{N}nk\right)\right]\quad\quad k=0,\dots,N-1
  2. cos ( z ) + sin ( z ) \cos(z)+\sin(z)\!
  3. = 2 cos ( z - π 4 ) =\sqrt{2}\cos(z-\frac{\pi}{4})
  4. cas ( z ) \mathrm{cas}(z)\!
  5. e - i z = cos ( z ) - i sin ( z ) e^{-iz}=\cos(z)-i\sin(z)\!
  6. Z k = [ X k ( Y k + Y N - k ) + X N - k ( Y k - Y N - k ) ] / 2 Z N - k = [ X N - k ( Y k + Y N - k ) - X k ( Y k - Y N - k ) ] / 2 \begin{matrix}Z_{k}&=&\left[X_{k}\left(Y_{k}+Y_{N-k}\right)+X_{N-k}\left(Y_{k}% -Y_{N-k}\right)\right]/2\\ Z_{N-k}&=&\left[X_{N-k}\left(Y_{k}+Y_{N-k}\right)-X_{k}\left(Y_{k}-Y_{N-k}% \right)\right]/2\end{matrix}

Discrete_logarithm.html

  1. log b : H 𝐙 . \log_{b}\colon H\rightarrow\mathbf{Z}.
  2. log b : H 𝐙 n , \log_{b}\colon H\rightarrow\mathbf{Z}_{n},
  3. log c ( g ) = log c ( b ) log b ( g ) . \log_{c}(g)=\log_{c}(b)\cdot\log_{b}(g).

Discrete_sine_transform.html

  1. f ( x ) f(x)
  2. x x
  3. x x
  4. f ( x ) f(x)
  5. 2 × 2 × 2 × 2 = 16 2\times 2\times 2\times 2=16
  6. X k = n = 0 N - 1 x n sin [ π N + 1 ( n + 1 ) ( k + 1 ) ] k = 0 , , N - 1 X_{k}=\sum_{n=0}^{N-1}x_{n}\sin\left[\frac{\pi}{N+1}(n+1)(k+1)\right]\quad% \quad k=0,\dots,N-1
  7. X k = n = 0 N - 1 x n sin [ π N ( n + 1 2 ) ( k + 1 ) ] k = 0 , , N - 1 X_{k}=\sum_{n=0}^{N-1}x_{n}\sin\left[\frac{\pi}{N}\left(n+\frac{1}{2}\right)(k% +1)\right]\quad\quad k=0,\dots,N-1
  8. X k = ( - 1 ) k 2 x N - 1 + n = 0 N - 2 x n sin [ π N ( n + 1 ) ( k + 1 2 ) ] k = 0 , , N - 1 X_{k}=\frac{(-1)^{k}}{2}x_{N-1}+\sum_{n=0}^{N-2}x_{n}\sin\left[\frac{\pi}{N}(n% +1)\left(k+\frac{1}{2}\right)\right]\quad\quad k=0,\dots,N-1
  9. X k = n = 0 N - 1 x n sin [ π N ( n + 1 2 ) ( k + 1 2 ) ] k = 0 , , N - 1 X_{k}=\sum_{n=0}^{N-1}x_{n}\sin\left[\frac{\pi}{N}\left(n+\frac{1}{2}\right)% \left(k+\frac{1}{2}\right)\right]\quad\quad k=0,\dots,N-1
  10. 2 / N \sqrt{2/N}

Disjoint_union.html

  1. A 0 A_{0}
  2. A 1 A_{1}
  3. A 0 * \displaystyle A^{*}_{0}
  4. A 0 A 1 = A 0 * A 1 * = { ( 1 , 0 ) , ( 2 , 0 ) , ( 3 , 0 ) , ( 1 , 1 ) , ( 2 , 1 ) } A_{0}\sqcup A_{1}=A^{*}_{0}\cup A^{*}_{1}=\{(1,0),(2,0),(3,0),(1,1),(2,1)\}
  5. i I A i = i I { ( x , i ) : x A i } . \bigsqcup_{i\in I}A_{i}=\bigcup_{i\in I}\{(x,i):x\in A_{i}\}.
  6. A i * = { ( x , i ) : x A i } . A_{i}^{*}=\{(x,i):x\in A_{i}\}.
  7. i I A i = A × I . \bigsqcup_{i\in I}A_{i}=A\times I.
  8. i I A i \sum_{i\in I}A_{i}
  9. i I A i \biguplus_{i\in I}A_{i}
  10. i I A i \cdot\!\!\!\!\!\bigcup_{i\in I}A_{i}
  11. A i * A_{i}^{*}
  12. A i A_{i}
  13. * A C A \underset{A\in C}{\,\,\bigcup\nolimits^{*}\!}A
  14. \coprod
  15. \bigsqcup

Disjunctive_normal_form.html

  1. ( A and ¬ B and ¬ C ) ( ¬ D and E and F ) (A\and\neg B\and\neg C)(\neg D\and E\and F)
  2. ( A and B ) C (A\and B)C
  3. A and B A\and B
  4. A A\!
  5. ¬ ( A B ) \neg(AB)
  6. A ( B and ( C D ) ) A(B\and(CD))
  7. ( X 1 Y 1 ) and ( X 2 Y 2 ) and and ( X n Y n ) (X_{1}Y_{1})\and(X_{2}Y_{2})\and\dots\and(X_{n}Y_{n})

Dispersion_(optics).html

  1. v = c n v=\frac{c}{n}
  2. χ \chi
  3. χ e = n 2 - 1 \chi_{e}=n^{2}-1
  4. 1 < n ( λ red ) < n ( λ yellow ) < n ( λ blue ) , 1<n(\lambda_{\rm red})<n(\lambda_{\rm yellow})<n(\lambda_{\rm blue})\ ,
  5. d n d λ < 0. \frac{{\rm d}n}{{\rm d}\lambda}<0.
  6. v g = c ( n - λ d n d λ ) - 1 = v ( 1 - ( λ / n ) d n d λ ) - 1 . v_{g}=c\left(n-\lambda\frac{dn}{d\lambda}\right)^{-1}=v\left(1-(\lambda/n)% \frac{dn}{d\lambda}\right)^{-1}.
  7. D = - λ c d 2 n d λ 2 . D=-\frac{\lambda}{c}\,\frac{d^{2}n}{d\lambda^{2}}.
  8. e i ( β z - ω t ) e^{i(\beta z-\omega t)}
  9. D = - 2 π c λ 2 d 2 β d ω 2 = 2 π c v g 2 λ 2 d v g d ω D=-\frac{2\pi c}{\lambda^{2}}\frac{d^{2}\beta}{d\omega^{2}}=\frac{2\pi c}{v_{g% }^{2}\lambda^{2}}\frac{dv_{g}}{d\omega}
  10. λ = 2 π c / ω \lambda=2\pi c/\omega
  11. v g = d ω / d β v_{g}=d\omega/d\beta
  12. Δ t \Delta t
  13. Δ λ \Delta\lambda
  14. β ( ω ) \beta(\omega)
  15. ν \nu
  16. t = k DM × ( DM ν 2 ) t=k_{\mathrm{DM}}\times\left(\frac{\mathrm{DM}}{\nu^{2}}\right)
  17. k DM k_{\mathrm{DM}}
  18. k DM = e 2 2 π m e c 4.149 GHz 2 pc - 1 cm 3 ms k_{\mathrm{DM}}=\frac{e^{2}}{2\pi m_{\mathrm{e}}c}\simeq 4.149\mathrm{GHz}^{2}% \mathrm{pc}^{-1}\mathrm{cm}^{3}\mathrm{ms}
  19. DM \mathrm{DM}
  20. n e n_{e}
  21. DM = 0 d n e d l \mathrm{DM}=\int_{0}^{d}{n_{e}\;dl}
  22. Δ t \Delta t
  23. ν h i \nu_{hi}
  24. ν l o \nu_{lo}
  25. Δ t = k DM × DM × ( 1 ν lo 2 - 1 ν hi 2 ) \Delta t=k_{\mathrm{DM}}\times\mathrm{DM}\times\left(\frac{1}{\nu_{\mathrm{lo}% }^{2}}-\frac{1}{\nu_{\mathrm{hi}}^{2}}\right)
  26. Δ t \Delta t
  27. DM \mathrm{DM}

Dispersity.html

  1. M w M_{w}
  2. M n M_{n}
  3. P D I = M w / M n \ PDI=M_{w}/M_{n}
  4. M w M_{w}
  5. M n M_{n}
  6. M n M_{n}
  7. M w M_{w}

Dissipative_system.html

  1. x ( t ) x(t)
  2. u ( t ) u(t)
  3. y ( t ) y(t)
  4. V ( x ) V(x)
  5. d V ( x ( t ) ) d t u ( t ) y ( t ) \frac{dV(x(t))}{dt}\leq u(t)\cdot y(t)
  6. u y u\cdot y
  7. \cdot
  8. V ( x ) V(x)
  9. u y u\cdot y

Distributed_hash_table.html

  1. n n
  2. f i l e n a m e filename
  3. d a t a data
  4. f i l e n a m e filename
  5. k k
  6. p u t ( k , d a t a ) put(k,data)
  7. k k
  8. f i l e n a m e filename
  9. k k
  10. k k
  11. g e t ( k ) get(k)
  12. k k
  13. d a t a data
  14. δ ( k 1 , k 2 ) \delta(k_{1},k_{2})
  15. k 1 k_{1}
  16. k 2 k_{2}
  17. i x i_{x}
  18. k m k_{m}
  19. i x i_{x}
  20. δ ( k m , i x ) \delta(k_{m},i_{x})
  21. δ ( k 1 , k 2 ) \delta(k_{1},k_{2})
  22. k 1 k_{1}
  23. k 2 k_{2}
  24. i 1 i_{1}
  25. i 2 i_{2}
  26. i 1 i_{1}
  27. i 2 i_{2}
  28. i 2 i_{2}
  29. i 1 i_{1}
  30. i 2 i_{2}
  31. S x S_{x}
  32. k m k_{m}
  33. h ( S x , k m ) h(S_{x},k_{m})
  34. k k
  35. k k
  36. k k
  37. k k
  38. k k
  39. k k
  40. n n
  41. O ( 1 ) O(1)
  42. O ( n ) O(n)
  43. O ( log n ) O(\log n)
  44. O ( log n / log ( log n ) ) O(\log n/\log(\log n))
  45. O ( log n ) O(\log n)
  46. O ( log n ) O(\log n)
  47. O ( 1 ) O(1)
  48. O ( log n ) O(\log n)
  49. O ( n ) O(\sqrt{n})
  50. O ( 1 ) O(1)
  51. O ( log n ) O(\log n)
  52. k k
  53. k k
  54. I D ID
  55. I D ID
  56. k k
  57. I D ID
  58. ( k , d a t a ) (k,data)
  59. i i
  60. i i
  61. p u t ( k , d a t a ) put(k,data)
  62. k k
  63. p u t ( k , d a t a ) put(k,data)
  64. p u t ( k , d a t a ) put(k,data)
  65. k k

Distributive_lattice.html

  1. \wedge
  2. \vee
  3. \wedge
  4. \vee
  5. \wedge
  6. \vee
  7. \wedge
  8. \vee
  9. \wedge
  10. \vee
  11. \wedge
  12. \wedge
  13. \vee
  14. \vee
  15. \vee
  16. \wedge
  17. \vee
  18. \vee
  19. \vee
  20. \cup

Distributive_property.html

  1. * \scriptstyle*
  2. + \scriptstyle+
  3. * \scriptstyle*
  4. + \scriptstyle+
  5. x * ( y + z ) = ( x * y ) + ( x * z ) x*(y+z)=(x*y)+(x*z)
  6. + \scriptstyle+
  7. ( y + z ) * x = ( y * x ) + ( z * x ) (y+z)*x=(y*x)+(z*x)
  8. + \scriptstyle+
  9. * \scriptstyle*
  10. + +
  11. \cdot
  12. a ( b ± c ) = a b ± a c a\cdot\left(b\pm c\right)=a\cdot b\pm a\cdot c
  13. ( a ± b ) c = a c ± b c (a\pm b)\cdot c=a\cdot c\pm b\cdot c
  14. ( a ± b ) ÷ c = a ÷ c ± b ÷ c (a\pm b)\div c=a\div c\pm b\div c
  15. a ÷ ( b ± c ) a ÷ b ± a ÷ c a\div(b\pm c)\neq a\div b\pm a\div c
  16. \R \R
  17. 6 16 = 6 ( 10 + 6 ) = 6 10 + 6 6 = 60 + 36 = 96 6\cdot 16=6\cdot(10+6)=6\cdot 10+6\cdot 6=60+36=96
  18. 3 a 2 b ( 4 a - 5 b ) = 3 a 2 b 4 a - 3 a 2 b 5 b = 12 a 3 b - 15 a 2 b 2 3a^{2}b\cdot(4a-5b)=3a^{2}b\cdot 4a-3a^{2}b\cdot 5b=12a^{3}b-15a^{2}b^{2}
  19. ( a + b ) ( a - b ) \displaystyle(a+b)\cdot(a-b)
  20. 12 a 3 b 2 - 30 a 4 b c + 18 a 2 b 3 c 2 . 12a^{3}b^{2}-30a^{4}bc+18a^{2}b^{3}c^{2}\,.
  21. 6 a 2 b 6a^{2}b
  22. 12 a 3 b 2 - 30 a 4 b c + 18 a 2 b 3 c 2 = 6 a 2 b ( 2 a b - 5 a 2 c + 3 b 2 c 2 ) . 12a^{3}b^{2}-30a^{4}bc+18a^{2}b^{3}c^{2}=6a^{2}b(2ab-5a^{2}c+3b^{2}c^{2})\,.
  23. ( A + B ) C = A C + B C (A+B)\cdot C=A\cdot C+B\cdot C
  24. l × m l\times m
  25. A , B A,B
  26. m × n m\times n
  27. C C
  28. A ( B + C ) = A B + A C A\cdot(B+C)=A\cdot B+A\cdot C
  29. l × m l\times m
  30. A A
  31. m × n m\times n
  32. B , C B,C
  33. ( P and ( Q R ) ) ( ( P and Q ) ( P and R ) ) (P\and(QR))\Leftrightarrow((P\and Q)(P\and R))
  34. ( P ( Q and R ) ) ( ( P Q ) and ( P R ) ) (P(Q\and R))\Leftrightarrow((PQ)\and(PR))
  35. \Leftrightarrow
  36. ( P and ( Q and R ) ) ( ( P and Q ) and ( P and R ) ) (P\and(Q\and R))\leftrightarrow((P\and Q)\and(P\and R))
  37. ( P and ( Q R ) ) ( ( P and Q ) ( P and R ) ) (P\and(QR))\leftrightarrow((P\and Q)(P\and R))
  38. ( P ( Q and R ) ) ( ( P Q ) and ( P R ) ) (P(Q\and R))\leftrightarrow((PQ)\and(PR))
  39. ( P ( Q R ) ) ( ( P Q ) ( P R ) ) (P(QR))\leftrightarrow((PQ)(PR))
  40. ( P ( Q R ) ) ( ( P Q ) ( P R ) ) (P\to(Q\to R))\leftrightarrow((P\to Q)\to(P\to R))
  41. ( P ( Q R ) ) ( ( P Q ) ( P R ) ) (P\to(Q\leftrightarrow R))\leftrightarrow((P\to Q)\leftrightarrow(P\to R))
  42. ( P ( Q R ) ) ( ( P Q ) ( P R ) ) (P(Q\leftrightarrow R))\leftrightarrow((PQ)\leftrightarrow(PR))
  43. ( ( P and Q ) ( R and S ) ) ( ( ( P R ) and ( P S ) ) and ( ( Q R ) and ( Q S ) ) ) ( ( P Q ) and ( R S ) ) ( ( ( P and R ) ( P and S ) ) ( ( Q and R ) ( Q and S ) ) ) \begin{aligned}\displaystyle((P\and Q)(R\and S))&\displaystyle\leftrightarrow(% ((PR)\and(PS))\and((QR)\and(QS)))\\ \displaystyle((PQ)\and(RS))&\displaystyle\leftrightarrow(((P\and R)(P\and S))(% (Q\and R)(Q\and S)))\end{aligned}

Divergence_of_the_sum_of_the_reciprocals_of_the_primes.html

  1. p prime 1 p = 1 2 + 1 3 + 1 5 + 1 7 + 1 11 + 1 13 + 1 17 + = \sum_{p\,\text{ prime }}\frac{1}{p}=\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+\frac{% 1}{7}+\frac{1}{11}+\frac{1}{13}+\frac{1}{17}+\cdots=\infty
  2. p prime p n 1 p log log ( n + 1 ) - log π 2 6 \sum_{\scriptstyle p\,\text{ prime }\atop\scriptstyle p\leq n}\frac{1}{p}\geq% \log\log(n+1)-\log\frac{\pi^{2}}{6}
  3. n = 1 1 n = 1 + 1 2 + 1 3 + 1 4 + \sum_{n=1}^{\infty}\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots
  4. n = 1 1 n = p 1 1 - p - 1 = p ( 1 + 1 p + 1 p 2 + ) \sum_{n=1}^{\infty}\frac{1}{n}=\prod_{p}\frac{1}{1-p^{-1}}=\prod_{p}\left(1+% \frac{1}{p}+\frac{1}{p^{2}}+\cdots\right)
  5. ln ( n = 1 1 n ) \displaystyle\ln\left(\sum_{n=1}^{\infty}\frac{1}{n}\right)
  6. log ( n = 1 1 n ) = log ( p 1 1 - p - 1 ) = p log ( p p - 1 ) = p log ( 1 + 1 p - 1 ) . \begin{aligned}&\displaystyle{}\quad\log\left(\sum_{n=1}^{\infty}\frac{1}{n}% \right)=\log\left(\prod_{p}\frac{1}{1-p^{-1}}\right)=\sum_{p}\log\left(\frac{p% }{p-1}\right)=\sum_{p}\log\left(1+\frac{1}{p-1}\right).\end{aligned}
  7. e x e^{x}
  8. e x = 1 + x + n = 2 x n n ! e^{x}=1+x+\sum_{n=2}^{\infty}\frac{x^{n}}{n!}
  9. e x 1 + x e^{x}\,\geq\,1\,+\,x
  10. log ( e x ) log ( 1 + x ) \log(e^{x})\,\geq\,\log(1\,+\,x)
  11. x log ( 1 + x ) x\,\geq\,\log(1\,+\,x)
  12. p 1 p - 1 p log ( 1 + 1 p - 1 ) = log ( n = 1 1 n ) . \sum_{p}\frac{1}{p-1}\geq\sum_{p}\log\left(1+\frac{1}{p-1}\right)=\log\left(% \sum_{n=1}^{\infty}\frac{1}{n}\right).
  13. p 1 p - 1 \sum_{p}\frac{1}{p-1}
  14. p i 3 p_{i}\geq 3
  15. p i p_{i}
  16. i i
  17. 1 p i - 1 1 p i - 1 \frac{1}{p_{i-1}}\,\geq\,\frac{1}{p_{i}-1}
  18. p 1 p \sum_{p}\frac{1}{p}
  19. i = 1 1 p i < \sum_{i=1}^{\infty}{1\over p_{i}}<\infty
  20. i = k + 1 1 p i < 1 2 ( 1 ) \sum_{i=k+1}^{\infty}{1\over p_{i}}<{1\over 2}\qquad(1)
  21. | M x | 2 k x ( 2 ) |M_{x}|\leq 2^{k}\sqrt{x}\qquad(2)
  22. { 1 , 2 , , x } M x = i = k + 1 N i , x \{1,2,\ldots,x\}\setminus M_{x}=\bigcup_{i=k+1}^{\infty}N_{i,x}
  23. x - | M x | i = k + 1 | N i , x | < i = k + 1 x p i x-|M_{x}|\leq\sum_{i=k+1}^{\infty}|N_{i,x}|<\sum_{i=k+1}^{\infty}{x\over p_{i}}
  24. x 2 < | M x | ( 3 ) {x\over 2}<|M_{x}|\qquad(3)
  25. x 2 2 k x \tfrac{x}{2}\geq 2^{k}\sqrt{x}
  26. i = 1 n 1 i p n ( 1 + 1 p ) k = 1 n 1 k 2 \sum_{i=1}^{n}{\frac{1}{i}}\leq\prod_{p\leq n}{\left(1+\frac{1}{p}\right)}\sum% _{k=1}^{n}{\frac{1}{k^{2}}}
  27. log ( n + 1 ) = 1 n + 1 d x x = i = 1 n i i + 1 d x x < 1 / i < i = 1 n 1 i \log(n+1)=\int_{1}^{n+1}\frac{dx}{x}=\sum_{i=1}^{n}\underbrace{\int_{i}^{i+1}% \frac{dx}{x}}_{{}\,<\,1/i}<\sum_{i=1}^{n}{\frac{1}{i}}
  28. k = 1 n 1 k 2 < 1 + k = 2 n ( 1 k - 1 2 - 1 k + 1 2 ) = 1 / ( k 2 - 1 / 4 ) > 1 / k 2 = 1 + 2 3 - 1 n + 1 2 < 5 3 \sum_{k=1}^{n}{\frac{1}{k^{2}}}<1+\sum_{k=2}^{n}\underbrace{\left(\frac{1}{k-% \frac{1}{2}}-\frac{1}{k+\frac{1}{2}}\right)}_{=\,1/(k^{2}-1/4)\,>\,1/k^{2}}=1+% \frac{2}{3}-\frac{1}{n+\frac{1}{2}}<\frac{5}{3}
  29. log ( n + 1 ) < i = 1 n 1 i p n ( 1 + 1 p ) k = 1 n 1 k 2 < 5 3 p n exp ( 1 p ) = 5 3 exp ( p n 1 p ) \begin{aligned}&\displaystyle{}\log(n+1)\\ \displaystyle<&\displaystyle\sum_{i=1}^{n}\frac{1}{i}\\ \displaystyle\leq&\displaystyle\prod_{p\leq n}{\left(1+\frac{1}{p}\right)}\sum% _{k=1}^{n}{\frac{1}{k^{2}}}\\ \displaystyle<&\displaystyle\frac{5}{3}\prod_{p\leq n}{\exp\left(\frac{1}{p}% \right)}\\ \displaystyle=&\displaystyle\frac{5}{3}\exp\left(\sum_{p\leq n}{\frac{1}{p}}% \right)\end{aligned}
  30. 5 3 \frac{5}{3}
  31. log log ( n + 1 ) - log 5 3 < p n 1 p \log\log(n+1)-\log\frac{5}{3}<\sum_{p\leq n}{\frac{1}{p}}
  32. k = 1 1 k 2 = π 2 6 \sum_{k=1}^{\infty}{\frac{1}{k^{2}}}=\frac{\pi^{2}}{6}
  33. 5 3 \frac{5}{3}
  34. lim n ( p n 1 p - log log ( n ) ) = M \lim_{n\to\infty}\left(\sum_{p\leq n}\frac{1}{p}-\log\log(n)\right)=M
  35. p n < n log n + n log log n for n 6 p_{n}<n\log n+n\log\log n\quad\mbox{for }~{}n\geq 6
  36. n = 1 1 p n n = 6 1 p n n = 6 1 n log n + n log log n n = 6 1 2 n log n = \begin{aligned}\displaystyle\sum_{n=1}^{\infty}\frac{1}{p_{n}}&\displaystyle% \geq\sum_{n=6}^{\infty}\frac{1}{p_{n}}\\ &\displaystyle\geq\sum_{n=6}^{\infty}\frac{1}{n\log n+n\log\log n}\\ &\displaystyle\geq\sum_{n=6}^{\infty}\frac{1}{2n\log n}\\ &\displaystyle=\infty\end{aligned}
  37. 1 2 , \frac{1}{2},
  38. odd even = O 1 E 1 \frac{\,\text{odd}}{\,\text{even}}=\frac{O_{1}}{E_{1}}
  39. O n E n , \frac{O_{n}}{E_{n}},
  40. O n E n + 1 p n + 1 = O n p n + 1 + E n E n p n + 1 = O n + 1 E n + 1 \frac{O_{n}}{E_{n}}+\frac{1}{p_{n+1}}=\frac{O_{n}p_{n+1}+E_{n}}{E_{n}p_{n+1}}=% \frac{O_{n+1}}{E_{n+1}}
  41. p n + 1 ; p_{n+1};

Divergence_theorem.html

  1. V V
  2. n \mathbb{R}^{n}
  3. n = 3 , V n=3,V
  4. S S
  5. V = S ∂V=S
  6. 𝐅 \mathbf{F}
  7. V V
  8. V V
  9. V V
  10. V ∂V
  11. V V
  12. 𝐧 \mathbf{n}
  13. V ∂V
  14. d 𝐒 d\mathbf{S}
  15. 𝐧 d S \mathbf{n}dS
  16. V ∂V
  17. V V
  18. V ∂V
  19. g g
  20. 𝐅 \mathbf{F}
  21. 𝐅 = f \mathbf{F}=∇f
  22. 𝐅 × 𝐆 \mathbf{F}×\mathbf{G}
  23. f f
  24. 𝐅 \mathbf{F}
  25. S S
  26. S = { x , y , z 3 : x 2 + y 2 + z 2 = 1 } . S=\left\{x,y,z\in\mathbb{R}^{3}\ :\ x^{2}+y^{2}+z^{2}=1\right\}.
  27. 𝐅 \mathbf{F}
  28. 𝐅 = 2 x 𝐢 + y 2 𝐣 + z 2 𝐤 . \mathbf{F}=2x\mathbf{i}+y^{2}\mathbf{j}+z^{2}\mathbf{k}.
  29. W ( 𝐅 ) d V = 2 W ( 1 + y + z ) d V = 2 W d V + 2 W y d V + 2 W z d V . \iiint_{W}(\nabla\cdot\mathbf{F})\,dV=2\iiint_{W}(1+y+z)\,dV=2\iiint_{W}dV+2% \iiint_{W}y\,dV+2\iiint_{W}z\,dV.
  30. W W
  31. W = { x , y , z 3 : x 2 + y 2 + z 2 1 } . W=\left\{x,y,z\in\mathbb{R}^{3}\ :\ x^{2}+y^{2}+z^{2}\leq 1\right\}.
  32. y y
  33. W W
  34. W W
  35. z z
  36. W y d V = W z d V = 0. \iiint_{W}y\,dV=\iiint_{W}z\,dV=0.
  37. W W
  38. 4 π 3 4\frac{π}{3}
  39. R R
  40. R = { x , y 2 : x 2 + y 2 1 } , R=\left\{x,y\in\mathbb{R}^{2}\ :\ x^{2}+y^{2}\leq 1\right\},
  41. 𝐅 ( x , y ) = 2 y 𝐢 + 5 x 𝐣 . \mathbf{F}(x,y)=2y\mathbf{i}+5x\mathbf{j}.
  42. R R
  43. C C
  44. x = cos ( s ) , y = sin ( s ) x=\cos(s),\quad y=\sin(s)
  45. 0 s 2 π 0≤s≤2π
  46. s s
  47. s = 0 s=0
  48. P P
  49. C C
  50. C C
  51. C ( s ) = cos ( s ) 𝐢 + sin ( s ) 𝐣 . C(s)=\cos(s)\mathbf{i}+\sin(s)\mathbf{j}.
  52. P P
  53. C C
  54. P = ( cos ( s ) , sin ( s ) ) 𝐅 = 2 sin ( s ) 𝐢 + 5 cos ( s ) 𝐣 . P=(\cos(s),\,\sin(s))\,\Rightarrow\,\mathbf{F}=2\sin(s)\mathbf{i}+5\cos(s)% \mathbf{j}.
  55. C 𝐅 𝐧 d s = 0 2 π ( 2 sin ( s ) 𝐢 + 5 cos ( s ) 𝐣 ) ( cos ( s ) 𝐢 + sin ( s ) 𝐣 ) d s = 0 2 π ( 2 sin ( s ) cos ( s ) + 5 sin ( s ) cos ( s ) ) d s = 7 0 2 π sin ( s ) cos ( s ) d s = 0. \begin{aligned}\displaystyle\oint_{C}\mathbf{F}\cdot\mathbf{n}\,ds&% \displaystyle=\int_{0}^{2\pi}(2\sin(s)\mathbf{i}+5\cos(s)\mathbf{j})\cdot(\cos% (s)\mathbf{i}+\sin(s)\mathbf{j})\,ds\\ &\displaystyle=\int_{0}^{2\pi}(2\sin(s)\cos(s)+5\sin(s)\cos(s))\,ds\\ &\displaystyle=7\int_{0}^{2\pi}\sin(s)\cos(s)\,ds\\ &\displaystyle=0.\end{aligned}
  56. M = 2 y , M = 0 M=2y,∂\frac{M}{∂}=0
  57. N = 5 x , N = 0 N=5x,∂\frac{N}{∂}=0
  58. R div 𝐅 d A = R ( M x + N y ) d A = 0. \iint_{R}\,\operatorname{div}\mathbf{F}\,dA=\iint_{R}\left(\frac{\partial M}{% \partial x}+\frac{\partial N}{\partial y}\right)\,dA=0.
  59. n n
  60. 𝐅 \mathbf{F}
  61. U U
  62. ( n 1 ) (n−1)
  63. 𝐅 \mathbf{F}
  64. U U
  65. U 𝐅 d V n = U 𝐅 𝐧 d S n - 1 \int_{U}\nabla\cdot\mathbf{F}\,dV_{n}=\oint_{\partial U}\mathbf{F}\cdot\mathbf% {n}\,dS_{n-1}
  66. n = 2 n=2
  67. n = 1 n=1
  68. 𝐅 \mathbf{F}
  69. n n
  70. T T
  71. S \scriptstyle S
  72. T i 1 i 2 i q i n n i q d S . T_{i_{1}i_{2}\cdots i_{q}\cdots i_{n}}n_{i_{q}}\,dS.

Divide_and_conquer_algorithms.html

  1. O ( n log 2 3 ) O(n^{\log_{2}3})
  2. Ω ( n 2 ) \Omega(n^{2})\,\!
  3. log 2 n \log_{2}n
  4. n n

Division_by_zero.html

  1. 10 5 \textstyle\frac{10}{5}
  2. 10 1 \textstyle\frac{10}{1}
  3. 10 0 \textstyle\frac{10}{0}
  4. a b \textstyle\frac{a}{b}
  5. a b \textstyle\frac{a}{b}
  6. 6 3 = 2 \frac{6}{3}=2
  7. ? × 3 = 6 ?\times 3=6
  8. 6 0 = x \frac{6}{0}=\,x
  9. x × 0 = 6. x\times 0=6.
  10. 0 0 = x \frac{0}{0}=\,x
  11. x × 0 = 0. x\times 0=0.
  12. 0 × 1 = 0 0\times 1=0
  13. 0 × 2 = 0 0\times 2=0
  14. 0 × 1 = 0 × 2. 0\times 1=0\times 2.\,
  15. 0 0 × 1 = 0 0 × 2. \textstyle\frac{0}{0}\times 1=\frac{0}{0}\times 2.
  16. 1 = 2. 1=2.\,
  17. lim b 0 + a b = + \lim_{b\to 0^{+}}{a\over b}=+\infty
  18. lim b 0 - a b = - \lim_{b\to 0^{-}}{a\over b}=-\infty
  19. lim b 0 a b \lim_{b\to 0}{a\over b}
  20. lim ( a , b ) ( 0 , 0 ) a b \lim_{(a,b)\to(0,0)}{a\over b}
  21. lim x 0 f ( x ) g ( x ) \lim_{x\to 0}{f(x)\over g(x)}
  22. \infty
  23. lim x 0 1 x = lim x 0 1 lim x 0 x = . \lim\limits_{x\to 0}{\frac{1}{x}=\frac{\lim\limits_{x\to 0}{1}}{\lim\limits_{x% \to 0}{x}}}=\infty.
  24. lim x 0 + 1 x = + and lim x 0 - 1 x = - . \lim\limits_{x\to 0^{+}}\frac{1}{x}=+\infty\,\text{ and }\lim\limits_{x\to 0^{% -}}\frac{1}{x}=-\infty.
  25. lim x 0 1 lim x 0 x \frac{\lim\limits_{x\to 0}1}{\lim\limits_{x\to 0}x}
  26. { } \mathbb{R}\cup\{\infty\}
  27. \infty
  28. - = -\infty=\infty
  29. a / 0 = \scriptstyle a/0=\infty
  30. a / = 0 \scriptstyle a/\infty=0
  31. + π / 2 \scriptstyle+\pi/2
  32. - π / 2 \scriptstyle-\pi/2
  33. + \infty+\infty
  34. { } \mathbb{C}\cup\{\infty\}
  35. \infty
  36. 1 / 0 = 1/0=\infty
  37. 0 / 0 0/0
  38. 0 × 0\times\infty
  39. 1 x \textstyle\frac{1}{x}
  40. 2 2 \textstyle\frac{2}{2}
  41. 2 x = 2 2x=2
  42. 2 2 \textstyle\frac{2}{2}
  43. a b \textstyle\frac{a}{b}
  44. - 1 -1

DNA_microarray.html

  1. i i
  2. i i
  3. i i
  4. i ! / 2 ! ( i - 2 ) ! i!/2!(i-2)!
  5. i - 1 i-1
  6. Γ \Gamma

Dobson_unit.html

  1. P V = n R T PV=nRT
  2. n air = A a v N V n_{\mathrm{air}}=\frac{A_{av}N}{V}
  3. n air = A a v P R T n_{\mathrm{air}}=\frac{A_{av}P}{RT}
  4. n air = A a v P R T = ( 6.02 10 23 molecules mol ) ( 101325 P a ) 8.314 J m o l K 273 K n_{\mathrm{air}}=\frac{A_{av}P}{RT}=\frac{(6.02\ast 10^{23}\frac{\mathrm{% molecules}}{\mathrm{mol}})\cdot(101325\mathrm{Pa})}{8.314\frac{J}{mol\cdot K}% \cdot 273K}
  5. ( 6.02 10 23 molecules mol ) ( 101325 P a ) 8.314 P a m 3 m o l K 273 K = 2.69 10 25 molecules m - 3 \frac{(6.02\ast 10^{23}\frac{\mathrm{molecules}}{\mathrm{mol}})\cdot(101325Pa)% }{8.314\frac{Pa\cdot m^{3}}{mol\cdot K}\cdot 273K}=2.69\ast 10^{25}\mathrm{% molecules}\cdot m^{-3}
  6. 0 m m 0.01 mm ( 2.69 10 25 molecules m - 3 ) d x = 2.69 10 25 molecules m - 3 0.01 m m - 2.69 10 25 molecules m - 3 0 m m \int_{0\mathrm{mm}}^{0.01\mathrm{mm}}(2.69\ast 10^{25}\,\text{molecules}\cdot m% ^{-3})\mathrm{d}x=2.69\ast 10^{25}\,\text{molecules}\cdot m^{-3}\cdot 0.01mm-2% .69\ast 10^{25}\,\text{molecules}\cdot m^{-3}\cdot 0mm
  7. = 2.69 10 25 molecules m - 3 10 - 5 m = 2.69 10 20 molecules m - 2 =2.69\ast 10^{25}\,\text{molecules}\cdot m^{-3}\cdot 10^{-5}m=2.69\ast 10^{20}% \,\text{molecules}\cdot m^{-2}

Dolly_zoom.html

  1. s i s_{i}
  2. s o s_{o}
  3. f f
  4. 1 s i + 1 s o = 1 f {1\over s_{i}}+{1\over s_{o}}={1\over f}
  5. M M
  6. M = s i s o = f ( s o - f ) M={s_{i}\over s_{o}}={f\over(s_{o}-f)}
  7. M a x M_{ax}
  8. s o s_{o}
  9. s i s_{i}
  10. s o s_{o}
  11. M a x = | d d ( s o ) s i s o | = | d d ( s o ) f ( s o - f ) | = | - f ( s o - f ) 2 | = M 2 f M_{ax}=\left|{d\over d(s_{o})}{s_{i}\over s_{o}}\right|=\left|{d\over d(s_{o})% }{f\over(s_{o}-f)}\right|=\left|{-f\over(s_{o}-f)^{2}}\right|={M^{2}\over f}
  12. distance = width 2 tan ( 1 2 fov ) \mathrm{distance}=\frac{\mathrm{width}}{2\tan\left(\frac{1}{2}\mathrm{fov}% \right)}

Domain_relational_calculus.html

  1. { X 1 , X 2 , . , X n p ( X 1 , X 2 , . , X n ) } \{\langle X_{1},X_{2},....,X_{n}\rangle\mid p(\langle X_{1},X_{2},....,X_{n}% \rangle)\}
  2. p ( X 1 , X 2 , . , X n ) p(\langle X_{1},X_{2},....,X_{n}\rangle)
  3. { A , B , C A , B , C Enterprise A = Captain } \left\{\ {\left\langle A,B,C\right\rangle}\mid{\left\langle A,B,C\right\rangle% \in\mathrm{Enterprise}\ \land\ A=\mathrm{{}^{\prime}Captain^{\prime}}}\ \right\}
  4. { B \displaystyle\{{\left\langle B\right\rangle}
  5. { B A , C A , B , C Enterprise D D , Stellar Cartography , C Departments } \left\{\ {\left\langle B\right\rangle}\mid{\exists A,C\ \left\langle A,B,C% \right\rangle\in\mathrm{Enterprise}}\ \land\ {\exists D\ \left\langle D,% \mathrm{{}^{\prime}Stellar\ Cartography^{\prime}},C\right\rangle\in\mathrm{% Departments}}\ \right\}

Dominical_letter.html

  1. [ y + 11 ( y mod 2 ) 2 + 11 ( y + 11 ( y mod 2 ) 2 mod 2 ) ] mod 7 [\frac{y+11(y\bmod 2)}{2}+11(\frac{y+11(y\bmod 2)}{2}\bmod 2)]\bmod 7
  2. DL = ( y mod 4 × 2 + y mod 7 × 4 + c mod 4 × 2 ) mod 7 , \mbox{DL}~{}=(y\bmod 4\times 2+y\bmod 7\times 4+c\bmod 4\times 2)\bmod 7,
  3. DL = ( y mod 4 × 2 + y mod 7 × 4 + c mod 7 + 2 ) mod 7 , \mbox{DL}~{}=(y\bmod 4\times 2+y\bmod 7\times 4+c\bmod 7+2)\bmod 7,
  4. ( 1 + y e a r + [ y e a r / 4 ] + [ ( y e a r - 1600 ) / 400 ] - [ ( y e a r - 1600 ) / 100 ] ) mod 7 (1+year+[year/4]+[(year-1600)/400]-[(year-1600)/100])\mod 7
  5. ( y e a r + [ y e a r / 4 ] + [ y e a r / 400 ] - [ y e a r / 100 ] - 1 ) mod 7 (year+[year/4]+[year/400]-[year/100]-1)\mod 7
  6. ( y + [ y / 4 ] + 5 ( c mod 4 ) - 1 ) mod 7 (y+[y/4]+5(c\mod 4)-1)\mod 7
  7. C = ( DL + DW ) mod 7 \mbox{C}~{}=(\mbox{DL}~{}+\mbox{DW}~{})\bmod 7
  8. DL = ( C - DW ) mod 7 \mbox{DL}~{}=(\mbox{C}~{}-\mbox{DW}~{})\bmod 7
  9. DW = ( C - DL ) mod 7 \mbox{DW}~{}=(\mbox{C}~{}-\mbox{DL}~{})\bmod 7

Doppler_radar.html

  1. f r f_{r}
  2. f t f_{t}
  3. f r = f t ( 1 + v / c 1 - v / c ) f_{r}=f_{t}\left(\frac{1+v/c}{1-v/c}\right)
  4. f d f_{d}
  5. f d = f r - f t = 2 v f t ( c - v ) f_{d}=f_{r}-f_{t}=2v\frac{f_{t}}{(c-v)}
  6. v c v\ll c
  7. ( c - v ) c \left(c-v\right)\rightarrow c
  8. f d 2 v f t c f_{d}\approx 2v\frac{f_{t}}{c}

Dot_product.html

  1. 𝐀 𝐁 = i = 1 n A i B i = A 1 B 1 + A 2 B 2 + + A n B n \mathbf{A}\cdot\mathbf{B}=\sum_{i=1}^{n}A_{i}B_{i}=A_{1}B_{1}+A_{2}B_{2}+% \cdots+A_{n}B_{n}
  2. [ 1 , 3 , - 5 ] [ 4 , - 2 , - 1 ] \displaystyle\ [1,3,-5]\cdot[4,-2,-1]
  3. 𝐀 \left\|\mathbf{A}\right\|
  4. 𝐀 𝐁 = 𝐀 𝐁 cos θ , \mathbf{A}\cdot\mathbf{B}=\left\|\mathbf{A}\right\|\,\left\|\mathbf{B}\right\|% \cos\theta,
  5. 𝐀 𝐁 = 0. \mathbf{A}\cdot\mathbf{B}=0.
  6. 𝐀 𝐁 = 𝐀 𝐁 \mathbf{A}\cdot\mathbf{B}=\left\|\mathbf{A}\right\|\,\left\|\mathbf{B}\right\|
  7. 𝐀 𝐀 = 𝐀 2 , \mathbf{A}\cdot\mathbf{A}=\left\|\mathbf{A}\right\|^{2},
  8. 𝐀 = 𝐀 𝐀 , \left\|\mathbf{A}\right\|=\sqrt{\mathbf{A}\cdot\mathbf{A}},
  9. A B = 𝐀 cos θ , A_{B}=\left\|\mathbf{A}\right\|\cos\theta,
  10. A B = 𝐀 𝐁 ^ , A_{B}=\mathbf{A}\cdot\widehat{\mathbf{B}},
  11. 𝐁 ^ = 𝐁 / 𝐁 \widehat{\mathbf{B}}=\mathbf{B}/\left\|\mathbf{B}\right\|
  12. 𝐀 𝐁 = A B 𝐁 = B A 𝐀 . \mathbf{A}\cdot\mathbf{B}=A_{B}\left\|\mathbf{B}\right\|=B_{A}\left\|\mathbf{A% }\right\|.
  13. ( α 𝐀 ) 𝐁 = α ( 𝐀 𝐁 ) = 𝐀 ( α 𝐁 ) . (\alpha\mathbf{A})\cdot\mathbf{B}=\alpha(\mathbf{A}\cdot\mathbf{B})=\mathbf{A}% \cdot(\alpha\mathbf{B}).
  14. 𝐀 ( 𝐁 + 𝐂 ) = 𝐀 𝐁 + 𝐀 𝐂 . \mathbf{A}\cdot(\mathbf{B}+\mathbf{C})=\mathbf{A}\cdot\mathbf{B}+\mathbf{A}% \cdot\mathbf{C}.
  15. 𝐀 𝐀 \mathbf{A}\cdot\mathbf{A}
  16. 𝐀 = 0. \mathbf{A}=\mathbf{0}.
  17. 𝐀 = [ A 1 , , A n ] = i A i 𝐞 i 𝐁 = [ B 1 , , B n ] = i B i 𝐞 i . \begin{aligned}\displaystyle\mathbf{A}&\displaystyle=[A_{1},\dots,A_{n}]=\sum_% {i}A_{i}\mathbf{e}_{i}\\ \displaystyle\mathbf{B}&\displaystyle=[B_{1},\dots,B_{n}]=\sum_{i}B_{i}\mathbf% {e}_{i}.\end{aligned}
  18. 𝐞 i 𝐞 i = 1 \mathbf{e}_{i}\cdot\mathbf{e}_{i}=1
  19. 𝐞 i 𝐞 j = 0. \mathbf{e}_{i}\cdot\mathbf{e}_{j}=0.
  20. 𝐀 𝐞 i = 𝐀 𝐞 i cos θ = 𝐀 cos θ = A i , \mathbf{A}\cdot\mathbf{e}_{i}=\left\|\mathbf{A}\right\|\,\left\|\mathbf{e}_{i}% \right\|\cos\theta=\left\|\mathbf{A}\right\|\cos\theta=A_{i},
  21. 𝐀 𝐁 = 𝐀 i B i 𝐞 i = i B i ( 𝐀 𝐞 i ) = i B i A i , \mathbf{A}\cdot\mathbf{B}=\mathbf{A}\cdot\sum_{i}B_{i}\mathbf{e}_{i}=\sum_{i}B% _{i}(\mathbf{A}\cdot\mathbf{e}_{i})=\sum_{i}B_{i}A_{i},
  22. 𝐚 𝐛 = 𝐛 𝐚 , \mathbf{a}\cdot\mathbf{b}=\mathbf{b}\cdot\mathbf{a},
  23. 𝐚 𝐛 = 𝐚 𝐛 cos θ = 𝐛 𝐚 cos θ = 𝐛 𝐚 . \mathbf{a}\cdot\mathbf{b}=\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|% \cos\theta=\left\|\mathbf{b}\right\|\left\|\mathbf{a}\right\|\cos\theta=% \mathbf{b}\cdot\mathbf{a}.
  24. 𝐚 ( 𝐛 + 𝐜 ) = 𝐚 𝐛 + 𝐚 𝐜 . \mathbf{a}\cdot(\mathbf{b}+\mathbf{c})=\mathbf{a}\cdot\mathbf{b}+\mathbf{a}% \cdot\mathbf{c}.
  25. 𝐚 ( r 𝐛 + 𝐜 ) = r ( 𝐚 𝐛 ) + ( 𝐚 𝐜 ) . \mathbf{a}\cdot(r\mathbf{b}+\mathbf{c})=r(\mathbf{a}\cdot\mathbf{b})+(\mathbf{% a}\cdot\mathbf{c}).
  26. ( c 1 𝐚 ) ( c 2 𝐛 ) = c 1 c 2 ( 𝐚 𝐛 ) . (c_{1}\mathbf{a})\cdot(c_{2}\mathbf{b})=c_{1}c_{2}(\mathbf{a}\cdot\mathbf{b}).
  27. 𝐜 𝐜 \displaystyle\mathbf{c}\cdot\mathbf{c}
  28. 𝐚 × ( 𝐛 × 𝐜 ) = 𝐛 ( 𝐚 𝐜 ) - 𝐜 ( 𝐚 𝐛 ) , \mathbf{a}\times(\mathbf{b}\times\mathbf{c})=\mathbf{b}(\mathbf{a}\cdot\mathbf% {c})-\mathbf{c}(\mathbf{a}\cdot\mathbf{b}),
  29. 𝐚 𝐛 = a i b i ¯ , \mathbf{a}\cdot\mathbf{b}=\sum{a_{i}\overline{b_{i}}},
  30. 𝐚 𝐛 = 𝐛 𝐚 ¯ . \mathbf{a}\cdot\mathbf{b}=\overline{\mathbf{b}\cdot\mathbf{a}}.
  31. cos θ = Re ( 𝐚 𝐛 ) 𝐚 𝐛 . \cos\theta=\frac{\operatorname{Re}(\mathbf{a}\cdot\mathbf{b})}{\left\|\mathbf{% a}\right\|\,\left\|\mathbf{b}\right\|}.
  32. \R \R
  33. \C \C
  34. 𝐚 , 𝐛 \left\langle\mathbf{a}\,,\mathbf{b}\right\rangle
  35. n n
  36. u u
  37. i i
  38. u u
  39. a x b a≤x≤b
  40. a a , b aa,b
  41. u , v = a b u ( x ) v ( x ) d x \left\langle u,v\right\rangle=\int_{a}^{b}u(x)v(x)dx
  42. ψ ( x ) ψ(x)
  43. χ ( x ) χ(x)
  44. ψ , χ = a b ψ ( x ) χ ( x ) ¯ d x . \left\langle\psi,\chi\right\rangle=\int_{a}^{b}\psi(x)\overline{\chi(x)}dx.
  45. A : B = i j A i j B i j ¯ = tr ( 𝐁 H 𝐀 ) = tr ( 𝐀𝐁 H ) . {A}:{B}=\sum_{i}\sum_{j}A_{ij}\overline{B_{ij}}=\mathrm{tr}(\mathbf{B}^{% \mathrm{H}}\mathbf{A})=\mathrm{tr}(\mathbf{A}\mathbf{B}^{\mathrm{H}}).
  46. A : B = i j A i j B i j = tr ( 𝐁 T 𝐀 ) = tr ( 𝐀𝐁 T ) = tr ( 𝐀 T 𝐁 ) = tr ( 𝐁𝐀 T ) . {A}:{B}=\sum_{i}\sum_{j}A_{ij}B_{ij}=\mathrm{tr}(\mathbf{B}^{\mathrm{T}}% \mathbf{A})=\mathrm{tr}(\mathbf{A}\mathbf{B}^{\mathrm{T}})=\mathrm{tr}(\mathbf% {A}^{\mathrm{T}}\mathbf{B})=\mathrm{tr}(\mathbf{B}\mathbf{A}^{\mathrm{T}}).

Dots_per_inch.html

  1. 4 3 \tfrac{4}{3}
  2. 72 * ( 1 + 1 3 ) = 96 72*(1+\tfrac{1}{3})=96
  3. 1 3 \tfrac{1}{3}
  4. 1 3 \tfrac{1}{3}

Double_counting_(proof_technique).html

  1. ( n k ) . {n\choose k}.
  2. k = 0 n ( n k ) = 2 n , \sum_{k=0}^{n}{n\choose k}=2^{n},
  3. k = d n ( n k ) ( k d ) = 2 n - d ( n d ) \sum_{k=d}^{n}{n\choose k}{k\choose d}=2^{n-d}{n\choose d}
  4. v d ( v ) = 2 e \sum_{v}d(v)=2e\,
  5. k = 2 n n ( k - 1 ) = n n - 1 ( n - 1 ) ! = n n - 2 n ! . \prod_{k=2}^{n}n(k-1)=n^{n-1}(n-1)!=n^{n-2}n!.
  6. T n n ! = n n - 2 n ! \displaystyle T_{n}n!=n^{n-2}n!
  7. T n = n n - 2 . \displaystyle T_{n}=n^{n-2}.

Doubling_the_cube.html

  1. ( x - k ) (x-k)
  2. a r = r s = s 2 a . \frac{a}{r}=\frac{r}{s}=\frac{s}{2a}.
  3. r = a 2 3 r=a\cdot\sqrt[3]{2}

Doubly_special_relativity.html

  1. λ \lambda
  2. η = 1 / λ \eta=1/\lambda
  3. E 2 - p 2 c 2 = m 2 c 4 E^{2}-p^{2}c^{2}=m^{2}c^{4}
  4. η \eta
  5. η \eta
  6. [ M i , M j ] = i ϵ i j k M k [M_{i},M_{j}]=i\epsilon_{ijk}M_{k}
  7. [ N i , N j ] = - i ϵ i j k M k , [N_{i},N_{j}]=-i\epsilon_{ijk}M_{k},
  8. [ M i , N j ] = - i ϵ i j k N k [M_{i},N_{j}]=-i\epsilon_{ijk}N_{k}
  9. [ M i , p j ] = i ϵ i j k p k [M_{i},p_{j}]=i\epsilon_{ijk}p_{k}
  10. [ M i , p 0 ] = 0 [M_{i},p_{0}]=0
  11. [ N i , p j ] = A ϵ i j + B p i p j + C Δ i j k ϵ N k [N_{i},p_{j}]=A\epsilon_{ij}+Bp_{i}p_{j}+C\Delta^{\epsilon}_{ijk}N_{k}
  12. [ N i , p 0 ] = D p i [N_{i},p_{0}]=Dp_{i}
  13. p i , p 0 , η {p_{i},p_{0},\eta}
  14. η A \eta_{A}
  15. - η 2 = η 0 2 - η 1 2 - η 2 2 - η 3 2 - η 4 2 -\eta^{2}=\eta_{0}^{2}-\eta_{1}^{2}-\eta_{2}^{2}-\eta_{3}^{2}-\eta_{4}^{2}
  16. p μ p_{\mu}
  17. p μ = p μ ( η A , η ) p_{\mu}=p_{\mu}(\eta_{A},\eta)
  18. m 2 = η 0 2 ( p μ , η ) - η i 2 ( p μ , η ) m^{2}=\eta_{0}^{2}(p_{\mu},\eta)-\eta_{i}^{2}(p_{\mu},\eta)
  19. η μ = p μ 1 - P 0 / η \eta_{\mu}=\frac{p_{\mu}}{1-P_{0}/\eta}
  20. [ N i , P 0 ] = i P i ( 1 - P 0 η ) [N_{i},P_{0}]=iP_{i}\left(1-\frac{P_{0}}{\eta}\right)
  21. P 0 = η P_{0}=\eta
  22. [ N i , P 0 = η ] = 0 [N_{i},P_{0}=\eta]=0

Drag_coefficient.html

  1. c d c_{\mathrm{d}}\,
  2. c d = 2 F d ρ v 2 A c_{\mathrm{d}}=\dfrac{2F_{\mathrm{d}}}{\rho v^{2}A}\,
  3. F d F_{\mathrm{d}}\,
  4. ρ \rho\,
  5. v v\,
  6. A A\,
  7. A = π r 2 A=\pi r^{2}\,
  8. 4 π r 2 \!\ 4\pi r^{2}
  9. F d = 1 2 ρ v 2 c d A F_{d}\,=\,\tfrac{1}{2}\,\rho\,v^{2}\,c_{d}\,A
  10. R e \scriptstyle Re\,
  11. C d \scriptstyle C_{\mathrm{d}}\,
  12. R e \scriptstyle Re\,
  13. C d \scriptstyle C_{\mathrm{d}}\,
  14. M a \scriptstyle Ma\,
  15. C d \scriptstyle C_{\mathrm{d}}\,
  16. R e \scriptstyle Re\,
  17. M a \scriptstyle Ma\,
  18. M a \scriptstyle Ma\,
  19. R e \scriptstyle Re\,
  20. C d \scriptstyle C_{\mathrm{d}}\,
  21. C d \scriptstyle C_{\mathrm{d}}\,
  22. F d \scriptstyle F_{\mathrm{d}}\,
  23. v \scriptstyle v\,
  24. v 2 \scriptstyle v^{2}\,
  25. C d \scriptstyle C_{\mathrm{d}}\,
  26. C d \scriptstyle C_{\mathrm{d}}\,
  27. C d \scriptstyle C_{\mathrm{d}}\,
  28. C d \scriptstyle C_{\mathrm{d}}\,
  29. c d c_{\mathrm{d}}\,
  30. R e Re
  31. c d c_{\mathrm{d}}\,
  32. R e < 10 6 \!\ Re<10^{6}
  33. R e > 10 6 \!\ Re>10^{6}
  34. R e = 10 6 \!\ Re=10^{6}
  35. R e = 10 5 \!\ Re=10^{5}
  36. R e = 10 6 \!\ Re=\!\ 10^{6}
  37. c d c_{\mathrm{d}}\,
  38. C d C_{d}
  39. c d = 2 F d ρ v 2 A = c p + c f = 1 ρ v 2 A S ( p - p o ) . n ^ . i ^ d A c p + 1 ρ v 2 A S T w . t ^ . i ^ d A c f c_{\mathrm{d}}=\dfrac{2F_{\mathrm{d}}}{\rho v^{2}A}\ =c_{\mathrm{p}}+c_{% \mathrm{f}}=\underbrace{\dfrac{1}{\rho v^{2}A}\ \textstyle\int\limits_{S}(p-p_% {o}).\hat{n}.\hat{i}dA}_{c_{\mathrm{p}}}+\underbrace{\dfrac{1}{\rho v^{2}A}\ % \textstyle\int\limits_{S}T_{w}.\hat{t}.\hat{i}dA}_{c_{\mathrm{f}}}
  40. c p c_{\mathrm{p}}\,
  41. c f c_{\mathrm{f}}\,
  42. t ^ \hat{t}
  43. n ^ \hat{n}
  44. T w T_{\mathrm{w}}\,
  45. p o p_{\mathrm{o}}\,
  46. p p\,
  47. i ^ \hat{i}
  48. d A ^ d\hat{A}