wpmath0000003_12

Passer_rating.html

  1. a = ( COMP ATT - .3 ) × 5 a=\left({\,\text{COMP}\over\,\text{ATT}}-.3\right)\times 5
  2. b = ( YDS ATT - 3 ) × .25 b=\left({\,\text{YDS}\over\,\text{ATT}}-3\right)\times.25
  3. c = ( TD ATT ) × 20 c=\left({\,\text{TD}\over\,\text{ATT}}\right)\times 20
  4. d = 2.375 - ( INT ATT × 25 ) d=2.375-\left({\,\text{INT}\over\,\text{ATT}}\times 25\right)
  5. Passer Rating = ( a + b + c + d 6 ) × 100 \,\text{Passer Rating}=\left({a+b+c+d\over 6}\right)\times 100
  6. Passer Rating NCAA = ( 8.4 × YDS ) + ( 330 × TD ) + ( 100 × COMP ) - ( 200 × INT ) ATT \,\text{Passer Rating}_{\,\text{NCAA}}={(8.4\times\,\text{YDS})+(330\times\,% \text{TD})+(100\times\,\text{COMP})-(200\times\,\text{INT})\over\,\text{ATT}}

Passive_matrix_addressing.html

  1. V i j = V s e l - V o n | o f f V_{ij}=V_{sel}-V_{on|off}
  2. V i j = V u n s e l - V o n | o f f V_{ij}=V_{unsel}-V_{on|off}

Pathological_(mathematics).html

  1. 2 \sqrt{2}
  2. ( - 5 ) \mathbb{Q}(\sqrt{-5})
  3. φ ( t ) = { e - 1 1 - t 2 , - 1 < t < 1 0 , otherwise \varphi(t)=\left\{\begin{array}[]{cc}e^{-\frac{1}{1-t^{2}}},&-1<t<1\\ 0,&\,\text{otherwise}\end{array}\right.

Paul_Richard_Heinrich_Blasius.html

  1. U U
  2. w ( z ) w(z)
  3. C C
  4. F x - i F y = i ρ 2 C ( d w d z ) 2 d z F_{x}-iF_{y}=\frac{i\rho}{2}\oint_{C}\left(\frac{\mathrm{d}w}{\mathrm{d}z}% \right)^{2}\,\mathrm{d}z
  5. ρ \rho
  6. f / 2 = 0.039 R e - 0.26 f/2=0.039Re^{-0.26}\,
  7. f / 2 = 0.023 R e - 0.26 f/2=0.023Re^{-0.26}\,
  8. λ = 0.3164 R e - 0.25 \lambda=0.3164Re^{-0.25}\,

Pál_Turán.html

  1. max ν = m + 1 , , m + n | j = 1 n b j z j ν | , \max_{\nu=m+1,\dots,m+n}\left|\sum_{j=1}^{n}b_{j}z_{j}^{\nu}\right|,

PDF417.html

  1. K = b 1 - b 2 + b 3 - b 4 + 9 ( mod 9 ) K=b_{1}-b_{2}+b_{3}-b_{4}+9\,\,\;\;(\mathop{{\rm mod}}9)
  2. K = E 1 - E 2 + E 5 - E 6 + 9 ( mod 9 ) K=E_{1}-E_{2}+E_{5}-E_{6}+9\,\,\;\;(\mathop{{\rm mod}}9)

Pedoe's_inequality.html

  1. A 2 ( b 2 + c 2 - a 2 ) + B 2 ( a 2 + c 2 - b 2 ) + C 2 ( a 2 + b 2 - c 2 ) 16 F f , A^{2}(b^{2}+c^{2}-a^{2})+B^{2}(a^{2}+c^{2}-b^{2})+C^{2}(a^{2}+b^{2}-c^{2})\geq 1% 6Ff,\,

Penning_trap.html

  1. ω - \omega_{-}
  2. ω + \omega_{+}
  3. ω + / ω - = 8 \omega_{+}/\omega_{-}=8

Pentagonal_number.html

  1. p n = 3 n 2 - n 2 p_{n}=\tfrac{3n^{2}-n}{2}
  2. 3 n ( n - 1 ) + 1 = 1 2 n ( 3 n - 1 ) + 1 2 ( 1 - n ) ( 3 ( 1 - n ) - 1 ) 3n(n-1)+1=\tfrac{1}{2}n(3n-1)+\tfrac{1}{2}(1-n)(3(1-n)-1)
  3. n = 24 x + 1 + 1 6 . n=\frac{\sqrt{24x+1}+1}{6}.
  4. 24 x + 1 24x+1
  5. 24 x + 1 5 mod 6 \sqrt{24x+1}\equiv 5\mod 6

Pentagonal_number_theorem.html

  1. n = 1 ( 1 - x n ) = k = - ( - 1 ) k x k 2 ( 3 k - 1 ) = 1 + k = 1 ( - 1 ) k ( x k ( 3 k + 1 ) / 2 + x k ( 3 k - 1 ) / 2 ) . \prod_{n=1}^{\infty}\left(1-x^{n}\right)=\sum_{k=-\infty}^{\infty}\left(-1% \right)^{k}x^{\frac{k}{2}\left(3k-1\right)}=1+\sum_{k=1}^{\infty}(-1)^{k}\left% (x^{k(3k+1)/2}+x^{k(3k-1)/2}\right).
  2. ( 1 - x ) ( 1 - x 2 ) ( 1 - x 3 ) = 1 - x - x 2 + x 5 + x 7 - x 12 - x 15 + x 22 + x 26 - . (1-x)(1-x^{2})(1-x^{3})\cdots=1-x-x^{2}+x^{5}+x^{7}-x^{12}-x^{15}+x^{22}+x^{26% }-\cdots.
  3. | x | < 1 |x|<1
  4. p ( n ) p(n)
  5. p ( n ) = p ( n - 1 ) + p ( n - 2 ) - p ( n - 5 ) - p ( n - 7 ) + p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+\cdots
  6. p ( n ) = k ( - 1 ) k - 1 p ( n - g k ) p(n)=\sum_{k}(-1)^{k-1}p(n-g_{k})
  7. g k g_{k}
  8. n = m + ( m + 1 ) + ( m + 2 ) + + ( 2 m - 1 ) = m ( 3 m - 1 ) 2 = k ( 3 k - 1 ) 2 n=m+(m+1)+(m+2)+\cdots+(2m-1)=\frac{m(3m-1)}{2}=\frac{k(3k-1)}{2}
  9. n = m + ( m + 1 ) + ( m + 2 ) + + ( 2 m - 2 ) = ( m - 1 ) ( 3 m - 2 ) 2 = k ( 3 k - 1 ) 2 , n=m+(m+1)+(m+2)+\cdots+(2m-2)=\frac{(m-1)(3m-2)}{2}=\frac{k(3k-1)}{2},
  10. g k = k ( 3 k - 1 ) / 2 g_{k}=k(3k-1)/2
  11. n = λ 1 + λ 2 + + λ n=\lambda_{1}+\lambda_{2}+\cdots+\lambda_{\ell}
  12. λ 1 λ 2 λ > 0 \lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{\ell}>0
  13. n 0 p ( n ) x n = k ( 1 - x k ) - 1 \sum\limits_{n\in\mathbb{N}_{0}}p(n)x^{n}=\prod\limits_{k\in\mathbb{N}}(1-x^{k% })^{-1}
  14. ( n = 0 p ( n ) x n ) ( n = 1 ( 1 - x n ) ) = 1 \left(\sum_{n=0}^{\infty}p(n)x^{n}\right)\cdot\left(\prod_{n=1}^{\infty}(1-x^{% n})\right)=1
  15. n = 1 ( 1 - x n ) = n = 0 a n x n \prod_{n=1}^{\infty}(1-x^{n})=\sum_{n=0}^{\infty}a_{n}x^{n}
  16. ( n = 0 p ( n ) x n ) ( n = 0 a n x n ) = 1 \left(\sum_{n=0}^{\infty}p(n)x^{n}\right)\cdot\left(\sum_{n=0}^{\infty}a_{n}x^% {n}\right)=1
  17. i = 0 n p ( n - i ) a i = 0 \sum_{i=0}^{n}p(n{-}i)a_{i}=0
  18. n 1 n\geq 1
  19. a i := { 1 if i = 1 2 ( 3 k 2 ± k ) and k is even - 1 if i = 1 2 ( 3 k 2 ± k ) and k is odd 0 otherwise a_{i}:=\begin{cases}1&\mbox{ if }~{}i=\frac{1}{2}(3k^{2}\pm k)\mbox{ and }~{}k% \mbox{ is even}\\ -1&\mbox{ if }~{}i=\frac{1}{2}(3k^{2}\pm k)\mbox{ and }~{}k\mbox{ is odd }\\ 0&\mbox{ otherwise }\end{cases}
  20. i 1 i\geq 1
  21. i ( - 1 ) i p ( n - g i ) = 0 , \sum_{i}(-1)^{i}p(n{-}g_{i})=0,
  22. g i := 1 2 ( 3 i 2 - i ) g_{i}:=\textstyle\frac{1}{2}(3i^{2}-i)
  23. g i n g_{i}\leq n
  24. i even p ( n - g i ) = i odd p ( n - g i ) , \sum_{i\mathrm{\ even}}p(n{-}g_{i})=\sum_{i\mathrm{\ odd}}p(n{-}g_{i}),
  25. 𝒳 := i even 𝒫 ( n - g i ) \mathcal{X}:=\bigcup_{i\mathrm{\ even}}\mathcal{P}(n-g_{i})
  26. 𝒴 := i odd 𝒫 ( n - g i ) \mathcal{Y}:=\bigcup_{i\mathrm{\ odd}}\mathcal{P}(n-g_{i})
  27. 𝒫 ( n ) \mathcal{P}(n)
  28. n n
  29. 𝒫 ( n - g i ) λ : n - g i = λ 1 + λ 2 + + λ \mathcal{P}(n-g_{i})\ni\lambda:n-g_{i}=\lambda_{1}+\lambda_{2}+\cdots+\lambda_% {\ell}
  30. λ = φ ( λ ) \lambda^{\prime}=\varphi(\lambda)
  31. φ ( λ ) := { λ : n - g i - 1 = ( + 3 i - 2 ) + ( λ 1 - 1 ) + + ( λ - 1 ) if + 3 i > λ 1 λ : n - g i + 1 = ( λ 2 + 1 ) + + ( λ + 1 ) + 1 + + 1 λ 1 - - 3 i if + 3 i λ 1 \varphi(\lambda):=\begin{cases}\lambda^{\prime}:n-g_{i-1}=(\ell+3i-2)+(\lambda% _{1}-1)+\cdots+(\lambda_{\ell}-1)&\mbox{ if }~{}\ell+3i>\lambda_{1}\\ \\ \lambda^{\prime}:n-g_{i+1}=(\lambda_{2}+1)+\cdots+(\lambda_{\ell}+1)+% \underbrace{1+\cdots+1}_{\lambda_{1}-\ell-3i}&\mbox{ if }~{}\ell+3i\leq\lambda% _{1}\end{cases}

Pentagonal_pyramidal_number.html

  1. n 2 ( n + 1 ) 2 \frac{n^{2}(n+1)}{2}
  2. x ( 2 x + 1 ) ( x - 1 ) 4 . \frac{x(2x+1)}{(x-1)^{4}}.

Per-unit_system.html

  1. 3 \scriptstyle\sqrt{3}
  2. P b a s e = 1 p u P_{base}=1\mathrm{pu}
  3. V b a s e = 1 p u V_{base}=1\mathrm{pu}
  4. Q b a s e = 1 p u Q_{base}=1\mathrm{pu}
  5. S b a s e = 1 p u S_{base}=1\mathrm{pu}
  6. S = I V S=IV
  7. P = S cos ( ϕ ) P=S\cos(\phi)
  8. Q = S sin ( ϕ ) Q=S\sin(\phi)
  9. V ¯ = I ¯ Z ¯ \underline{V}=\underline{I}\underline{Z}
  10. Z Z
  11. Z ¯ = R + j X = Z cos ( ϕ ) + j Z sin ( ϕ ) \underline{Z}=R+jX=Z\cos(\phi)+jZ\sin(\phi)
  12. I base = S base V b a s e = 1 p u I_{\mathrm{base}}=\frac{S_{\mathrm{base}}}{V_{base}}=1\mathrm{pu}
  13. Z base = V base I b a s e = V base 2 I base V base = V base 2 S base = 1 p u Z_{\mathrm{base}}=\frac{V_{\mathrm{base}}}{I_{base}}=\frac{V_{\mathrm{base}}^{% 2}}{I_{\mathrm{base}}V_{\mathrm{base}}}=\frac{V_{\mathrm{base}}^{2}}{S_{% \mathrm{base}}}=1\mathrm{pu}
  14. Y base = 1 Z base = 1 p u Y_{\mathrm{base}}=\frac{1}{Z_{\mathrm{base}}}=1\mathrm{pu}
  15. P = S cos ( ϕ ) P=S\cos(\phi)
  16. Q = S sin ( ϕ ) Q=S\sin(\phi)
  17. S S
  18. S base = 3 V base I base S_{\mathrm{base}}=\sqrt{3}V_{\mathrm{base}}I_{\mathrm{base}}
  19. I base = S base V base × 3 = 1 p u I_{\mathrm{base}}=\frac{S_{\mathrm{base}}}{V_{\mathrm{base}}\times\sqrt{3}}=1% \mathrm{pu}
  20. Z base = V base I base × 3 = V base 2 S base = 1 p u Z_{\mathrm{base}}=\frac{V_{\mathrm{base}}}{I_{\mathrm{base}}\times\sqrt{3}}=% \frac{{V_{\mathrm{base}}^{2}}}{S_{\mathrm{base}}}=1\mathrm{pu}
  21. Y base = 1 Z base = 1 p u Y_{\mathrm{base}}=\frac{1}{Z_{\mathrm{base}}}=1\mathrm{pu}
  22. S base = 500 MVA S_{\mathrm{base}}=500\,\mathrm{MVA}
  23. V base V_{\mathrm{base}}
  24. I base = S base V b a s e × 3 = 2.09 kA I_{\mathrm{base}}=\frac{S_{\mathrm{base}}}{V_{base}\times\sqrt{3}}=2.09\,% \mathrm{kA}
  25. Z base = V base I b a s e × 3 = V base 2 S base = 38.1 Ω Z_{\mathrm{base}}=\frac{V_{\mathrm{base}}}{I_{base}\times\sqrt{3}}=\frac{V_{% \mathrm{base}}^{2}}{S_{\mathrm{base}}}=38.1\,\Omega
  26. Y base = 1 Z base = 26.3 mS Y_{\mathrm{base}}=\frac{1}{Z_{\mathrm{base}}}=26.3\,\mathrm{mS}
  27. V pu = V V base = 136 kV 138 kV = 0.9855 pu V_{\mathrm{pu}}=\frac{V}{V_{\mathrm{base}}}=\frac{136\,\mathrm{kV}}{138\,% \mathrm{kV}}=0.9855\,\mathrm{pu}
  28. Base number selection \,\text{Base number selection}
  29. Arbitrarily selecting from ohm’s law the two base numbers: base voltage and base current \,\text{Arbitrarily selecting from ohm's law the two base numbers: base % voltage and base current}
  30. 1 1
  31. We have, Z = E I \,\text{We have, Z}=\frac{E}{I}
  32. 2 2
  33. Base ohms = base volts base amperes \,\text{Base ohms}=\frac{\,\text{base volts}}{\,\text{base amperes}}
  34. 3 3
  35. Per-unit volts = volts base volts \,\text{Per-unit volts}=\frac{\,\text{volts}}{\,\text{base volts}}
  36. 4 4
  37. Per-unit amperes = amperes base amperes \,\text{Per-unit amperes}=\frac{\,\text{amperes}}{\,\text{base amperes}}
  38. 5 5
  39. Per-unit ohms = ohms base ohms \,\text{Per-unit ohms}=\frac{\,\text{ohms}}{\,\text{base ohms}}
  40. Alternatively, choosing base volts and base kva values, we have, \,\text{Alternatively, choosing base volts and base kva values, we have,}
  41. in single-phase systems: \,\text{in single-phase systems:}
  42. 6 6
  43. Base amperes = base kva * 1000 base volts \,\text{Base amperes }=\frac{\,\text{base kva * 1000}}{\,\text{base volts}}
  44. 7 7
  45. Base amperes = base kva base kv L - L \,\text{Base amperes }=\frac{\,\text{base kva}}{\,\text{base kv}_{L-L}}
  46. 8 8
  47. Base ohms = base volts base amperes \,\text{Base ohms }=\frac{\,\text{base volts}}{\,\text{base amperes}}
  48. and in three-phase systems: \,\text{and in three-phase systems:}
  49. 9 9
  50. Base amperes = base kva * 1000 3 * base volts \,\text{Base amperes }=\frac{\,\text{base kva * 1000}}{\sqrt{3}*\,\text{base % volts}}
  51. 10 10
  52. Base amperes = base kva 3 * base kv L - L \,\text{Base amperes }=\frac{\,\text{base kva}}{\sqrt{3}*\,\text{base kv}_{L-L}}
  53. 11 11
  54. Base ohms = base volts 3 * base amperes \,\text{Base ohms }=\frac{\,\text{base volts}}{\sqrt{3}*\,\text{base amperes}}
  55. Working out for convenience per-unit ohms directly, we have \,\text{Working out for convenience per-unit ohms directly, we have}
  56. for single-phase and three-phase systems: \,\text{for single-phase and three-phase systems:}
  57. 12 12
  58. Base ohms = ohms * base kva k v L - L 2 * 1000 \,\text{Base ohms }=\frac{\,\text{ohms * base kva}}{kv_{L-L}^{2}*1000}
  59. Short-Circuit Calculation Formulas \,\text{Short-Circuit Calculation Formulas}
  60. Ohms conversions: \,\text{Ohms conversions:}
  61. 13 13
  62. Per-unit ohms reactance = ohms reactance * kva base k v L - L 2 * 1000 \,\text{Per-unit ohms reactance}=\frac{\,\text{ohms reactance * }\,\text{kva % base}}{kv_{L-L}^{2}*1000}
  63. 14 14
  64. Ohms reactance = % reactance * k v L - L 2 * 10 kva base \,\text{Ohms reactance}=\frac{\%\,\text{ reactance}*kv_{L-L}^{2}*10}{\,\text{% kva base}}
  65. 15 15
  66. Per-unit ohms reactance = per cent ohms reactance 100 \,\text{Per-unit ohms reactance}=\frac{\,\text{per cent ohms reactance}}{100}
  67. Changing ohms from one kva base to another: \,\text{Changing ohms from one kva base to another:}
  68. 16 16
  69. % ohms reactance on kva base 2 = kva base 2 kva base 1 * % ohms reactance on base 1 \%\,\text{ ohms reactance on kva base}_{2}=\frac{\,\text{kva base}_{2}}{\,% \text{kva base}_{1}}*\%\,\text{ ohms reactance on base}_{1}
  70. 17 17
  71. 0/1 ohms reactance on kva base 2 = kva base 2 kva base 1 * 0/1 ohms reactance on base 1 \,\text{0/1 ohms reactance on kva base}_{2}=\frac{\,\text{kva base}_{2}}{\,% \text{kva base}_{1}}\,\text{ * 0/1 ohms reactance on base}_{1}
  72. Changing incoming system reactance: \,\text{Changing incoming system reactance:}
  73. a. If system reactance is given in percent, use Eq. 16 to change from one kva base to another. \,\text{a. If system reactance is given in percent, use Eq. 16 to change from % one kva base to another.}
  74. b. If system reactance is given in short-circuit symmetrical rms kva or current, convert to per-unit as follows: \,\text{b. If system reactance is given in short-circuit symmetrical rms kva % or current, convert to per-unit as follows:}
  75. 18 18
  76. 0/1 reactance = kva base used in reactance in studied calculation system short-circuit kva \,\text{0/1 reactance}=\frac{\,\text{kva base used in reactance in studied % calculation}}{\,\text{system short-circuit kva}}
  77. 19 19
  78. 0/1 reactance = kva base used in reactance in studied calculation system short-circuit current * 3 * system kv L - L \,\text{0/1 reactance}=\frac{\,\text{kva base used in reactance in studied % calculation}}{\,\text{system short-circuit current * }\sqrt{3}\,\text{ * % system kv}_{L-L}}
  79. Calculating approximate motor kva base: \,\text{Calculating approximate motor kva base:}
  80. a. For induction motors and 0.8 power factor synchronous motors \,\text{a. For induction motors and 0.8 power factor synchronous motors}
  81. 20 20
  82. kva base horsepower rating \,\text{kva base}\approx\,\text{ horsepower rating}
  83. b. For unity power factor synchronous motors \,\text{b. For unity power factor synchronous motors}
  84. 21 21
  85. kva base 0.8 * horsepower rating \,\text{kva base}\approx\,\text{ 0.8 * horsepower rating}
  86. Converting ohms from one voltage to another: \,\text{Converting ohms from one voltage to another:}
  87. 22 22
  88. Ohms on basis of voltage 1 = ( voltage 1 voltage 2 ) 2 * ohms on basis of voltage 2 \,\text{Ohms on basis of voltage}_{1}=(\frac{\,\text{voltage}_{1}}{\,\text{% voltage}_{2}})^{2}\,\text{ * ohms on basis of voltage}_{2}
  89. Short-circuit kva and current calculations \,\text{Short-circuit kva and current calculations}
  90. Symmetrical short circuit kva: \,\text{Symmetrical short circuit kva:}
  91. 23 23
  92. = 100 * kva base % X =\frac{\,\text{100 * kva base}}{\%\,\text{ X}}
  93. 24 24
  94. = kva base 0/1 X =\frac{\,\text{kva base}}{\,\text{0/1 X}}
  95. 25 25
  96. = 3 * Voltage L - N 2 ohms reactance * 1000 =3*\frac{\,\text{Voltage}_{L-N}^{2}}{\,\text{ohms reactance}\,\text{ * 1000}}
  97. 26 26
  98. = kv L - L 2 * 1000 ohms reactance =\frac{\,\text{kv}_{L-L}^{2}\,\text{ * 1000}}{\,\text{ohms reactance}}
  99. Symmetrical short circuit current: \,\text{Symmetrical short circuit current:}
  100. 27 27
  101. = 100 * kva base % X * 3 * kv L - L =\frac{\,\text{100 * kva base}}{\%\,\text{ X}*\sqrt{3}*\,\text{kv}_{L-L}}
  102. 28 28
  103. = kva base 0/1 X * 3 * kv L - L =\frac{\,\text{kva base}}{\,\text{0/1 X}*\sqrt{3}*\,\text{kv}_{L-L}}
  104. 29 29
  105. = kv L - L * 1000 3 * ohms reactance =\frac{\,\text{kv}_{L-L}\,\text{ * 1000}}{\sqrt{3}*\,\text{ohms reactance}}
  106. Asymmetrical short-circuit current and kva: \,\text{Asymmetrical short-circuit current and kva:}
  107. 30 30
  108. Asymmetrical short-circuit current = symmetrical current * X/R factor \,\text{Asymmetrical short-circuit current = symmetrical current * X/R factor }
  109. 31 31
  110. Asymmetrical short-circuit kva = symmetrical kva * X/R factor \,\text{Asymmetrical short-circuit kva = symmetrical kva * X/R factor }
  111. P c u , F L \displaystyle P_{cu,FL}
  112. P c u , F L , p u = P c u , F L P b a s e = I R 1 2 R e q 1 V R 1 I R 1 = R e q 1 V R 1 / I R 1 = R e q 1 Z B 1 = R e q 1 , p u \begin{aligned}\displaystyle P_{cu,FL,pu}&\displaystyle=\frac{P_{cu,FL}}{P_{% base}}\\ &\displaystyle=\frac{I_{R1}^{2}R_{eq1}}{V_{R1}I_{R1}}\\ &\displaystyle=\frac{R_{eq1}}{V_{R1}/I_{R1}}\\ &\displaystyle=\frac{R_{eq1}}{Z_{B1}}\\ &\displaystyle=R_{eq1,pu}\\ \end{aligned}

Percentile.html

  1. σ \sigma
  2. σ \sigma
  3. σ \sigma
  4. σ \sigma
  5. σ \sigma
  6. σ \sigma
  7. P P
  8. n = P 100 × N n=\left\lceil\frac{P}{100}\times N\right\rceil
  9. n = 30 100 × 5 = 1.5 = 2. n=\left\lceil\frac{30}{100}\times 5\right\rceil=\lceil 1.5\rceil=2.
  10. n = 40 100 × 5 = 2.0 = 2. n=\left\lceil\frac{40}{100}\times 5\right\rceil=\lceil 2.0\rceil=2.
  11. n = 50 100 × 5 = 2.5 = 3. n=\left\lceil\frac{50}{100}\times 5\right\rceil=\lceil 2.5\rceil=3.
  12. n = 25 100 × 10 = 2.5 = 3. n=\left\lceil\frac{25}{100}\times 10\right\rceil=\lceil 2.5\rceil=3.
  13. n = 50 100 × 10 = 5.0 = 5. n=\left\lceil\frac{50}{100}\times 10\right\rceil=\lceil 5.0\rceil=5.
  14. n = 75 100 × 10 = 7.5 = 8. n=\left\lceil\frac{75}{100}\times 10\right\rceil=\lceil 7.5\rceil=8.
  15. n = 25 100 × 11 = 2.75 = 3. n=\left\lceil\frac{25}{100}\times 11\right\rceil=\lceil 2.75\rceil=3.
  16. n = 50 100 × 11 = 5.50 = 6. n=\left\lceil\frac{50}{100}\times 11\right\rceil=\lceil 5.50\rceil=6.
  17. n = 75 100 × 11 = 8.25 = 9. n=\left\lceil\frac{75}{100}\times 11\right\rceil=\lceil 8.25\rceil=9.
  18. v 1 v 2 v 3 v N v_{1}\leq v_{2}\leq v_{3}\leq\dots\leq v_{N}
  19. p n = 100 N ( n - 1 2 ) . p_{n}=\frac{100}{N}\left(n-\frac{1}{2}\right).
  20. P < p 1 P<p_{1}
  21. v = v 1 v=v_{1}
  22. P > p N P>p_{N}
  23. v = v N v=v_{N}
  24. P = p k P=p_{k}
  25. v = v k v=v_{k}
  26. p k < P < p k + 1 p_{k}<P<p_{k+1}
  27. v = v k + P - p k p k + 1 - p k ( v k + 1 - v k ) = v k + N × P - p k 100 ( v k + 1 - v k ) . v=v_{k}+\frac{P-p_{k}}{p_{k+1}-p_{k}}(v_{k+1}-v_{k})=v_{k}+N\times\frac{P-p_{k% }}{100}(v_{k+1}-v_{k}).
  28. N 2 \frac{N}{2}
  29. p 1 = 100 5 ( 1 - 1 2 ) = 10. p_{1}=\frac{100}{5}\left(1-\frac{1}{2}\right)=10.
  30. p 2 = 100 5 ( 2 - 1 2 ) = 30. p_{2}=\frac{100}{5}\left(2-\frac{1}{2}\right)=30.
  31. p 3 = 100 5 ( 3 - 1 2 ) = 50. p_{3}=\frac{100}{5}\left(3-\frac{1}{2}\right)=50.
  32. p 4 = 100 5 ( 4 - 1 2 ) = 70. p_{4}=\frac{100}{5}\left(4-\frac{1}{2}\right)=70.
  33. p 5 = 100 5 ( 5 - 1 2 ) = 90. p_{5}=\frac{100}{5}\left(5-\frac{1}{2}\right)=90.
  34. v = 20 + 5 × 40 - 30 100 ( 35 - 20 ) = 27.5 v=20+5\times\frac{40-30}{100}(35-20)=27.5
  35. w 1 , w 2 , w 3 , , w N w_{1},w_{2},w_{3},\dots,w_{N}
  36. S n = k = 1 n w k , S_{n}=\sum_{k=1}^{n}w_{k},
  37. n n
  38. p n = 100 S N ( S n - w n 2 ) p_{n}=\frac{100}{S_{N}}\left(S_{n}-\frac{w_{n}}{2}\right)
  39. v = v k + p - p k p k + 1 - p k ( v k + 1 - v k ) . v=v_{k}+\frac{p-p_{k}}{p_{k+1}-p_{k}}(v_{k+1}-v_{k}).
  40. v P v_{P}
  41. v P v_{P}
  42. v 1 v 2 v N v_{1}\leq v_{2}\leq\dots\leq v_{N}
  43. n = P 100 ( N - 1 ) + 1 , 0 P 100 n=\frac{P}{100}(N-1)+1\,\text{, }0\leq P\leq 100
  44. n = k + d n=k+d
  45. v P v_{P}
  46. v P = v k + d ( v k + 1 - v k ) , 1 k N v_{P}=v_{k}+d(v_{k+1}-v_{k})\,\text{, }1\leq k\leq N
  47. n = 40 100 ( 5 - 1 ) + 1 = 2.6 n=\frac{40}{100}(5-1)+1=2.6
  48. v k + d ( v k + 1 - v k ) = v 2 + 0.6 ( v 3 - v 2 ) = 20 + 0.6 ( 35 - 20 ) = 29 v_{k}+d(v_{k+1}-v_{k})=v_{2}+0.6(v_{3}-v_{2})=20+0.6(35-20)=29
  49. n = 75 100 ( 4 - 1 ) + 1 = 3.25 n=\frac{75}{100}(4-1)+1=3.25
  50. v k + d ( v k + 1 - v k ) = v 3 + 0.25 ( v 4 - v 3 ) = 3 + 0.25 ( 4 - 3 ) = 3.25 v_{k}+d(v_{k+1}-v_{k})=v_{3}+0.25(v_{4}-v_{3})=3+0.25(4-3)=3.25
  51. v P v_{P}
  52. v P v_{P}
  53. v 1 v 2 v N v_{1}\leq v_{2}\leq\dots\leq v_{N}
  54. n = P 100 ( N + 1 ) , 0 < P 100 n=\frac{P}{100}(N+1)\,\text{, }0<P\leq 100
  55. n = k + d n=k+d
  56. v P v_{P}
  57. v P = { v 1 , for k = 0 v N , for k N v k + d ( v k + 1 - v k ) , for 0 < k < N v_{P}=\begin{cases}v_{1},&\,\text{for }k=0\\ v_{N},&\,\text{for }k\geq N\\ v_{k}+d(v_{k+1}-v_{k}),&\,\text{for }0<k<N\end{cases}
  58. P 100 N N + 1 P\geq 100\frac{N}{N+1}
  59. n = 40 100 ( 5 + 1 ) = 2.4 n=\frac{40}{100}(5+1)=2.4
  60. v k + d ( v k + 1 - v k ) = v 2 + 0.4 ( v 3 - v 2 ) = 20 + 0.4 ( 35 - 20 ) = 26 v_{k}+d(v_{k+1}-v_{k})=v_{2}+0.4(v_{3}-v_{2})=20+0.4(35-20)=26

Perfect_field.html

  1. A k F A\otimes_{k}F
  2. k ( X ) k(X)
  3. X X
  4. k p - k^{p^{-\infty}}
  5. A k k p - A\otimes_{k}k^{p^{-\infty}}
  6. A A A \cdots\rightarrow A\rightarrow A\rightarrow A\rightarrow\cdots
  7. x i + 1 p = x i x_{i+1}^{p}=x_{i}

Perfect_group.html

  1. ( - 1 0 0 - 1 ) = ( 4 0 0 4 ) \left(\begin{smallmatrix}-1&0\\ 0&-1\end{smallmatrix}\right)=\left(\begin{smallmatrix}4&0\\ 0&4\end{smallmatrix}\right)
  2. [ Z 2 , G , G ] = [ [ Z 2 , G ] , G ] [ Z 1 , G ] = 1 [Z_{2},G,G]=[[Z_{2},G],G]\subseteq[Z_{1},G]=1
  3. [ G , Z 2 , G ] = [ [ G , Z 2 ] , G ] = [ [ Z 2 , G ] , G ] [ Z 1 , G ] = 1. [G,Z_{2},G]=[[G,Z_{2}],G]=[[Z_{2},G],G]\subseteq[Z_{1},G]=1.
  4. H ~ i ( G ; 𝐙 ) = 0. \tilde{H}_{i}(G;\mathbf{Z})=0.

Perfect_magic_cube.html

  1. [ 25 16 80 104 90 115 98 4 1 97 42 111 85 2 75 66 72 27 102 48 67 18 119 106 5 ] \begin{bmatrix}25&16&80&104&90\\ 115&98&4&1&97\\ 42&111&85&2&75\\ 66&72&27&102&48\\ 67&18&119&106&5\\ \end{bmatrix}
  2. [ 91 77 71 6 70 52 64 117 69 13 30 118 21 123 23 26 39 92 44 114 116 17 14 73 95 ] \begin{bmatrix}91&77&71&6&70\\ 52&64&117&69&13\\ 30&118&21&123&23\\ 26&39&92&44&114\\ 116&17&14&73&95\\ \end{bmatrix}
  3. [ 47 61 45 76 86 107 43 38 33 94 89 68 ( 63 ) 58 37 32 93 88 83 19 40 50 81 65 79 ] \begin{bmatrix}47&61&45&76&86\\ 107&43&38&33&94\\ 89&68&(63)&58&37\\ 32&93&88&83&19\\ 40&50&81&65&79\\ \end{bmatrix}
  4. [ 31 53 112 109 10 12 82 34 87 100 103 3 105 8 96 113 57 9 62 74 56 120 55 49 35 ] \begin{bmatrix}31&53&112&109&10\\ 12&82&34&87&100\\ 103&3&105&8&96\\ 113&57&9&62&74\\ 56&120&55&49&35\\ \end{bmatrix}
  5. [ 121 108 7 20 59 29 28 122 125 11 51 15 41 124 84 78 54 99 24 60 36 110 46 22 101 ] \begin{bmatrix}121&108&7&20&59\\ 29&28&122&125&11\\ 51&15&41&124&84\\ 78&54&99&24&60\\ 36&110&46&22&101\\ \end{bmatrix}

Perlin_noise.html

  1. O ( 2 n ) O(2^{n})
  2. f ( x ) = a 0 ( 1 - x ) + a 1 x f(x)=a_{0}(1-x)+a_{1}x
  3. O ( 2 n ) O(2^{n})
  4. n n

Permutable_prime.html

  1. 10 n - 1 9 \tfrac{10^{n}-1}{9}

Perpetual_calendar.html

  1. ( m + 1 ) 26 10 mod 7 , \left\lfloor\frac{(m+1)26}{10}\right\rfloor\mod 7,

Perpetuity.html

  1. P V = A r PV\ =\ {A\over r}

Persistence_of_a_number.html

  1. n n
  2. n n
  3. n n

Perspective_(visual).html

  1. h = a d h={a\over d}

Perspective_distortion_(photography).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}

Pfaffian.html

  1. pf ( A ) 2 = det ( A ) , \operatorname{pf}(A)^{2}=\det(A),
  2. A = [ 0 a - a 0 ] . pf ( A ) = a . A=\begin{bmatrix}0&a\\ -a&0\end{bmatrix}.\qquad\operatorname{pf}(A)=a.
  3. B = [ 0 a b - a 0 c - b - c 0 ] . pf ( B ) = 0. B=\begin{bmatrix}0&a&b\\ -a&0&c\\ -b&-c&0\end{bmatrix}.\qquad\operatorname{pf}(B)=0.
  4. pf [ 0 a b c - a 0 d e - b - d 0 f - c - e - f 0 ] = a f - b e + d c . \operatorname{pf}\begin{bmatrix}0&a&b&c\\ -a&0&d&e\\ -b&-d&0&f\\ -c&-e&-f&0\end{bmatrix}=af-be+dc.
  5. pf [ 0 a 1 0 0 - a 1 0 b 1 0 0 - b 1 0 a 2 0 0 - a 2 b n - 1 - b n - 1 0 a n - a n 0 ] = a 1 a 2 a n . \operatorname{pf}\begin{bmatrix}0&a_{1}&0&0\\ -a_{1}&0&b_{1}&0\\ 0&-b_{1}&0&a_{2}\\ 0&0&-a_{2}&\ddots&\ddots\\ &&&\ddots&&b_{n-1}\\ &&&&-b_{n-1}&0&a_{n}\\ &&&&&-a_{n}&0\end{bmatrix}=a_{1}a_{2}\cdots a_{n}.
  6. b i b_{i}
  7. pf ( A ) = 1 2 n n ! σ S 2 n sgn ( σ ) i = 1 n a σ ( 2 i - 1 ) , σ ( 2 i ) \operatorname{pf}(A)=\frac{1}{2^{n}n!}\sum_{\sigma\in S_{2n}}\operatorname{sgn% }(\sigma)\prod_{i=1}^{n}a_{\sigma(2i-1),\sigma(2i)}
  8. α = { ( i 1 , j 1 ) , ( i 2 , j 2 ) , , ( i n , j n ) } \alpha=\{(i_{1},j_{1}),(i_{2},j_{2}),\cdots,(i_{n},j_{n})\}
  9. i 1 < i 2 < < i n i_{1}<i_{2}<\cdots<i_{n}
  10. π = [ 1 2 3 4 2 n i 1 j 1 i 2 j 2 j n ] \pi=\begin{bmatrix}1&2&3&4&\cdots&2n\\ i_{1}&j_{1}&i_{2}&j_{2}&\cdots&j_{n}\end{bmatrix}
  11. A α = sgn ( π ) a i 1 , j 1 a i 2 , j 2 a i n , j n . A_{\alpha}=\operatorname{sgn}(\pi)a_{i_{1},j_{1}}a_{i_{2},j_{2}}\cdots a_{i_{n% },j_{n}}.
  12. pf ( A ) = α Π A α . \operatorname{pf}(A)=\sum_{\alpha\in\Pi}A_{\alpha}.
  13. det A = det A T = det ( - A ) = ( - 1 ) n det A \det\,A=\det\,A\text{T}=\det\left(-A\right)=(-1)^{n}\det\,A
  14. det A = 0 \det\,A=0
  15. pf ( A ) = j = 1 j i 2 n ( - 1 ) i + j + 1 + θ ( i - j ) a i j pf ( A i ^ j ^ ) , \operatorname{pf}(A)=\sum_{{j=1}\atop{j\neq i}}^{2n}(-1)^{i+j+1+\theta(i-j)}a_% {ij}\operatorname{pf}(A_{\hat{i}\hat{j}}),
  16. θ ( i - j ) \theta(i-j)
  17. A i ^ j ^ A_{\hat{i}\hat{j}}
  18. i = 1 i=1
  19. pf ( A ) = j = 2 2 n ( - 1 ) j a 1 j pf ( A 1 ^ j ^ ) . \operatorname{pf}(A)=\sum_{j=2}^{2n}(-1)^{j}a_{1j}\operatorname{pf}(A_{\hat{1}% \hat{j}}).
  20. ω = i < j a i j e i e j . \omega=\sum_{i<j}a_{ij}\;e^{i}\wedge e^{j}.
  21. 1 n ! ω n = pf ( A ) e 1 e 2 e 2 n , \frac{1}{n!}\omega^{n}=\operatorname{pf}(A)\;e^{1}\wedge e^{2}\wedge\cdots% \wedge e^{2n},
  22. pf ( A T ) = ( - 1 ) n pf ( A ) . \operatorname{pf}(A\text{T})=(-1)^{n}\operatorname{pf}(A).
  23. pf ( λ A ) = λ n pf ( A ) . \operatorname{pf}(\lambda A)=\lambda^{n}\operatorname{pf}(A).
  24. pf ( A ) 2 = det ( A ) . \operatorname{pf}(A)^{2}=\det(A).
  25. pf ( B A B T ) = det ( B ) pf ( A ) . \operatorname{pf}(BAB\text{T})=\det(B)\operatorname{pf}(A).
  26. pf ( A 2 m + 1 ) = ( - 1 ) n m pf ( A ) 2 m + 1 . \operatorname{pf}(A^{2m+1})=(-1)^{nm}\operatorname{pf}(A)^{2m+1}.
  27. A 1 A 2 = [ A 1 0 0 A 2 ] , A_{1}\oplus A_{2}=\begin{bmatrix}A_{1}&0\\ 0&A_{2}\end{bmatrix},
  28. pf ( A 1 A 2 ) = pf ( A 1 ) pf ( A 2 ) . \operatorname{pf}(A_{1}\oplus A_{2})=\operatorname{pf}(A_{1})\operatorname{pf}% (A_{2}).
  29. pf [ 0 M - M T 0 ] = ( - 1 ) n ( n - 1 ) / 2 det M . \operatorname{pf}\begin{bmatrix}0&M\\ -M\text{T}&0\end{bmatrix}=(-1)^{n(n-1)/2}\det M.
  30. 1 pf ( A ) pf ( A ) x i = 1 2 tr ( A - 1 A x i ) , \frac{1}{\operatorname{pf}(A)}\frac{\partial\operatorname{pf}(A)}{\partial x_{% i}}=\frac{1}{2}\operatorname{tr}\left(A^{-1}\frac{\partial A}{\partial x_{i}}% \right),
  31. 1 pf ( A ) 2 pf ( A ) x i x j = 1 2 tr ( A - 1 2 A x i x j ) - 1 2 tr ( A - 1 A x i A - 1 A x j ) + 1 4 tr ( A - 1 A x i ) tr ( A - 1 A x j ) . \frac{1}{\operatorname{pf}(A)}\frac{\partial^{2}\operatorname{pf}(A)}{\partial x% _{i}\partial x_{j}}=\frac{1}{2}\operatorname{tr}\left(A^{-1}\frac{\partial^{2}% A}{\partial x_{i}\partial x_{j}}\right)-\frac{1}{2}\operatorname{tr}\left(A^{-% 1}\frac{\partial A}{\partial x_{i}}A^{-1}\frac{\partial A}{\partial x_{j}}% \right)+\frac{1}{4}\operatorname{tr}\left(A^{-1}\frac{\partial A}{\partial x_{% i}}\right)\operatorname{tr}\left(A^{-1}\frac{\partial A}{\partial x_{j}}\right).
  32. pf ( B A B T ) = det ( B ) pf ( A ) \operatorname{pf}(BAB\text{T})=\det(B)\operatorname{pf}(A)

PFP_(enzyme).html

  1. \rightleftharpoons

Phase_detector.html

  1. α - β sin ( α - β ) = sin α cos β - sin β cos α \alpha-\beta\approx\sin(\alpha-\beta)=\sin\alpha\cos\beta-\sin\beta\cos\alpha
  2. sin α cos β = sin ( α - β ) 2 + sin ( α + β ) 2 α - β 2 + sin ( α + β ) 2 \sin\alpha\cos\beta={\sin(\alpha-\beta)\over 2}+{\sin(\alpha+\beta)\over 2}% \approx{\alpha-\beta\over 2}+{\sin(\alpha+\beta)\over 2}

Phenology.html

  1. NDVI = NIR - red NIR + red \mathrm{NDVI}={\mathrm{NIR}-\mathrm{red}\over\mathrm{NIR}+\mathrm{red}}

Photoevaporation.html

  1. r g r_{g}
  2. r g r_{g}
  3. r g r_{g}
  4. r g r_{g}
  5. r g r_{g}
  6. r g = ( γ - 1 ) 2 γ G M μ k B T 1.4 ( M / M ) ( T / 10 4 K ) AU , r_{g}=\frac{\left(\gamma-1\right)}{2\gamma}\frac{GM\mu}{k_{B}T}\approx 1.4% \frac{\left(M/M_{\odot}\right)}{\left(T/10^{4}\ {\rm K}\right)}\ {\rm AU},\!
  7. γ \gamma
  8. G G
  9. M M
  10. M M_{\odot}
  11. μ \mu
  12. k B k_{B}
  13. T T

Phthalic_anhydride.html

  1. \overrightarrow{\leftarrow}

Pi_function.html

  1. π ( x ) \pi(x)\,\!
  2. Π ( x ) \Pi(x)\,\!

Pick's_theorem.html

  1. A = i + b 2 - 1. A=i+\frac{b}{2}-1.
  2. i P T = ( i P + i T ) + ( c - 2 ) i_{PT}=(i_{P}+i_{T})+(c-2)\,
  3. b P T = ( b P + b T ) - 2 ( c - 2 ) - 2. b_{PT}=(b_{P}+b_{T})-2(c-2)-2.\,
  4. ( i P + i T ) = i P T - ( c - 2 ) (i_{P}+i_{T})=i_{PT}-(c-2)\,
  5. ( b P + b T ) = b P T + 2 ( c - 2 ) + 2. (b_{P}+b_{T})=b_{PT}+2(c-2)+2.\,
  6. A P T \displaystyle A_{PT}

Piecewise.html

  1. | x | = { - x , if x < 0 x , if x 0 |x|=\begin{cases}-x,&\mbox{if }~{}x<0\\ x,&\mbox{if }~{}x\geq 0\end{cases}
  2. x 0 x_{0}

Piecewise_linear_function.html

  1. f ( x ) = { - x - 3 if x - 3 x + 3 if - 3 < x < 0 - 2 x + 3 if 0 x < 3 0.5 x - 4.5 if x 3 f(x)=\begin{cases}-x-3&\,\text{if }x\leq-3\\ x+3&\,\text{if }-3<x<0\\ -2x+3&\,\text{if }0\leq x<3\\ 0.5x-4.5&\,\text{if }x\geq 3\end{cases}
  2. f : n f:\mathbb{R}^{n}\to\mathbb{R}
  3. Π 𝒫 ( 𝒫 ( n + 1 ) ) \Pi\in\mathcal{P}(\mathcal{P}(\mathbb{R}^{n+1}))
  4. f ( x ) = min Σ Π max ( a , b ) Σ a x + b . f(\vec{x})=\min_{\Sigma\in\Pi}\max_{(\vec{a},b)\in\Sigma}\vec{a}\cdot\vec{x}+b.
  5. f f
  6. Σ 𝒫 ( n + 1 ) \Sigma\in\mathcal{P}(\mathbb{R}^{n+1})
  7. f ( x ) = max ( a , b ) Σ a x + b . f(\vec{x})=\max_{(\vec{a},b)\in\Sigma}\vec{a}\cdot\vec{x}+b.

Pierre-Simon_Laplace.html

  1. 1 d = 1 r [ 1 - 2 cos ( θ - θ ) r r + ( r r ) 2 ] - 1 2 . \frac{1}{d}=\frac{1}{r^{\prime}}\left[1-2\cos(\theta^{\prime}-\theta)\frac{r}{% r^{\prime}}+\left(\frac{r}{r^{\prime}}\right)^{2}\right]^{-\tfrac{1}{2}}.
  2. 1 d = 1 r k = 0 P k 0 ( cos ( θ - θ ) ) ( r r ) k . \frac{1}{d}=\frac{1}{r^{\prime}}\sum_{k=0}^{\infty}P^{0}_{k}(\cos(\theta^{% \prime}-\theta))\left(\frac{r}{r^{\prime}}\right)^{k}.
  3. 2 V = 2 V x 2 + 2 V y 2 + 2 V z 2 = 0. \nabla^{2}V={\partial^{2}V\over\partial x^{2}}+{\partial^{2}V\over\partial y^{% 2}}+{\partial^{2}V\over\partial z^{2}}=0.
  4. Pr ( A i | B ) = Pr ( A i ) Pr ( B | A i ) j Pr ( A j ) Pr ( B | A j ) . \Pr(A_{i}|B)=\Pr(A_{i})\frac{\Pr(B|A_{i})}{\sum_{j}\Pr(A_{j})\Pr(B|A_{j})}.
  5. Pr ( next outcome is success ) = s + 1 n + 2 \Pr(\,\text{next outcome is success})=\frac{s+1}{n+2}
  6. Pr ( sun will rise tomorrow ) = d + 1 d + 2 \Pr(\,\text{sun will rise tomorrow})=\frac{d+1}{d+2}
  7. z = X ( x ) e a x d x and z = X ( x ) x a d x . z=\int X(x)e^{ax}\,dx\,\text{ and }z=\int X(x)x^{a}\,dx.

Pierre_François_Verhulst.html

  1. d N d t = r N ( 1 - N K ) \frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)
  2. N ( t ) = K 1 + C K e - r t N(t)=\frac{K}{1+CKe^{-rt}}
  3. 1 N ( t ) = 1 - e - r t K + e - r t N ( 0 ) . \frac{1}{N(t)}=\frac{1-e^{-rt}}{K}+\frac{e^{-rt}}{N(0)}.

Pincherle_derivative.html

  1. T := [ T , x ] = T x - x T = - ad ( x ) T , T^{\prime}:=[T,x]=Tx-xT=-\operatorname{ad}(x)T,\,
  2. T { p ( x ) } = T { x p ( x ) } - x T { p ( x ) } p ( x ) 𝕂 [ x ] . T^{\prime}\{p(x)\}=T\{xp(x)\}-xT\{p(x)\}\qquad\forall p(x)\in\mathbb{K}[x].
  3. S \scriptstyle S
  4. T \scriptstyle T
  5. End ( 𝕂 [ x ] ) \scriptstyle\operatorname{End}\left(\mathbb{K}[x]\right)
  6. ( T + S ) = T + S \scriptstyle{(T+S)^{\prime}=T^{\prime}+S^{\prime}}
  7. ( T S ) = T S + T S \scriptstyle{(TS)^{\prime}=T^{\prime}\!S+TS^{\prime}}
  8. T S = T S \scriptstyle{TS=T\circ S}
  9. [ T , S ] = [ T , S ] + [ T , S ] \scriptstyle{[T,S]^{\prime}=[T^{\prime},S]+[T,S^{\prime}]}
  10. [ T , S ] = T S - S T \scriptstyle{[T,S]=TS-ST}
  11. D = ( d d x ) = Id 𝕂 [ x ] = 1. D^{\prime}=\left({d\over{dx}}\right)^{\prime}=\operatorname{Id}_{\mathbb{K}[x]% }=1.
  12. ( D n ) = ( d n d x n ) = n D n - 1 , (D^{n})^{\prime}=\left({{d^{n}}\over{dx^{n}}}\right)^{\prime}=nD^{n-1},
  13. = a n d n d x n = a n D n \partial=\sum a_{n}{{d^{n}}\over{dx^{n}}}=\sum a_{n}D^{n}
  14. Diff ( 𝕂 [ x ] ) \scriptstyle\operatorname{Diff}(\mathbb{K}[x])
  15. S h ( f ) ( x ) = f ( x + h ) S_{h}(f)(x)=f(x+h)\,
  16. S h = n = 0 h n n ! D n S_{h}=\sum_{n=0}{{h^{n}}\over{n!}}D^{n}
  17. S h = n = 1 h n ( n - 1 ) ! D n - 1 = h S h . S_{h}^{\prime}=\sum_{n=1}{{h^{n}}\over{(n-1)!}}D^{n-1}=h\cdot S_{h}.
  18. 𝕂 \scriptstyle{\mathbb{K}}
  19. [ T , S h ] = 0 \scriptstyle{[T,S_{h}]=0}
  20. [ T , S h ] = 0 \scriptstyle{[T^{\prime},S_{h}]=0}
  21. T \scriptstyle T^{\prime}
  22. h \scriptstyle h
  23. ( δ f ) ( x ) = f ( x + h ) - f ( x ) h (\delta f)(x)={{f(x+h)-f(x)}\over h}
  24. δ = 1 h ( S h - 1 ) , \delta={1\over h}(S_{h}-1),
  25. δ = S h \scriptstyle{\delta^{\prime}=S_{h}}

Pisot–Vijayaraghavan_number.html

  1. α n \|\alpha^{n}\|
  2. ( α ) \mathbb{Q}(\alpha)
  3. λ α n , \|\lambda\alpha^{n}\|,
  4. n = 1 λ α n 2 < . \sum_{n=1}^{\infty}\|\lambda\alpha^{n}\|^{2}<\infty.
  5. ( α ) \mathbb{Q}(\alpha)
  6. λ α n 0 , n . \|\lambda\alpha^{n}\|\to 0,\quad n\to\infty.
  7. ( α ) \mathbb{Q}(\alpha)
  8. S , S , S ′′ , S,S^{\prime},S^{\prime\prime},\ldots
  9. S ( ω ) S^{(\omega)}
  10. α \alpha\,
  11. α \alpha^{\prime}\,
  12. α \alpha\,
  13. α = a + D to α = a - D \alpha=a+\sqrt{D}\,\text{ to }\alpha^{\prime}=a-\sqrt{D}\,
  14. α = a + D 2 to α = a - D 2 . \alpha=\frac{a+\sqrt{D}}{2}\,\text{ to }\alpha^{\prime}=\frac{a-\sqrt{D}}{2}.\,
  15. ( a - 1 ) 2 < D < a 2 (a-1)^{2}<D<a^{2}
  16. a 2 < D < ( a + 1 ) 2 a^{2}<D<(a+1)^{2}
  17. a > 0 a>0
  18. ( a - 2 ) 2 < D < a 2 (a-2)^{2}<D<a^{2}
  19. a 2 < D < ( a + 2 ) 2 a^{2}<D<(a+2)^{2}
  20. 1 + 5 2 \frac{1+\sqrt{5}}{2}
  21. x 2 - x - 1 x^{2}-x-1
  22. 1 + 2 1+\sqrt{2}\,
  23. x 2 - 2 x - 1 x^{2}-2x-1
  24. 3 + 5 2 \frac{3+\sqrt{5}}{2}
  25. x 2 - 3 x + 1 x^{2}-3x+1
  26. 1 + 3 1+\sqrt{3}\,
  27. x 2 - 2 x - 2 x^{2}-2x-2
  28. 3 + 13 2 \frac{3+\sqrt{13}}{2}
  29. x 2 - 3 x - 1 x^{2}-3x-1
  30. 2 + 2 2+\sqrt{2}\,
  31. x 2 - 4 x + 2 x^{2}-4x+2
  32. 3 + 17 2 \frac{3+\sqrt{17}}{2}
  33. x 2 - 3 x - 2 x^{2}-3x-2
  34. 2 + 3 2+\sqrt{3}\,
  35. x 2 - 4 x + 1 x^{2}-4x+1
  36. 3 + 21 2 \frac{3+\sqrt{21}}{2}
  37. x 2 - 3 x - 3 x^{2}-3x-3
  38. 2 + 5 2+\sqrt{5}\,
  39. x 2 - 4 x - 1 x^{2}-4x-1
  40. ( 3 + 10 ) 6 = 27379 + 8658 10 = 54757.9999817 54758 - 1 54758 . (3+\sqrt{10})^{6}=27379+8658\sqrt{10}=54757.9999817\dots\approx 54758-\frac{1}% {54758}.
  41. 27379 27379\,
  42. 8658 10 8658\sqrt{10}\,
  43. 0.0000182 , 0.0000182\dots,\,
  44. 27379 8658 = 3.162277662 \frac{27379}{8658}=3.162277662\dots
  45. 10 = 3.162277660 . \sqrt{10}=3.162277660\dots.
  46. ( 27379 8658 ) 2 = 10 + 1 8658 2 . \left(\frac{27379}{8658}\right)^{2}=10+\frac{1}{8658^{2}}.
  47. x 6 - 2 x 5 + x 4 - x 2 + x - 1 , x^{6}-2x^{5}+x^{4}-x^{2}+x-1,
  48. x n ( x 2 - x - 1 ) + 1 x^{n}(x^{2}-x-1)+1\,
  49. x n ( x 2 - x - 1 ) + ( x 2 - 1 ) . x^{n}(x^{2}-x-1)+(x^{2}-1).
  50. x n ( x 2 - x - 1 ) - 1 x^{n}(x^{2}-x-1)-1
  51. x n ( x 2 - x - 1 ) - ( x 2 - 1 ) x^{n}(x^{2}-x-1)-(x^{2}-1)\,
  52. x ( x 2 - x - 1 ) + ( x 2 - 1 ) x(x^{2}-x-1)+(x^{2}-1)
  53. x 3 - x - 1 x^{3}-x-1
  54. x 2 ( x 2 - x - 1 ) + ( x 2 - 1 ) x^{2}(x^{2}-x-1)+(x^{2}-1)
  55. x 4 - x 3 - 1 x^{4}-x^{3}-1
  56. x 3 ( x 2 - x - 1 ) + ( x 2 - 1 ) x^{3}(x^{2}-x-1)+(x^{2}-1)
  57. x 5 - x 4 - x 3 + x 2 - 1 x^{5}-x^{4}-x^{3}+x^{2}-1
  58. x 3 ( x 2 - x - 1 ) + 1 x^{3}(x^{2}-x-1)+1
  59. x 3 - x 2 - 1 x^{3}-x^{2}-1
  60. x 4 ( x 2 - x - 1 ) + ( x 2 - 1 ) x^{4}(x^{2}-x-1)+(x^{2}-1)
  61. x 6 - x 5 - x 4 + x 2 - 1 x^{6}-x^{5}-x^{4}+x^{2}-1
  62. x 4 ( x 2 - x - 1 ) + 1 x^{4}(x^{2}-x-1)+1
  63. x 5 - x 3 - x 2 - x - 1 x^{5}-x^{3}-x^{2}-x-1
  64. x 5 ( x 2 - x - 1 ) + ( x 2 - 1 ) x^{5}(x^{2}-x-1)+(x^{2}-1)
  65. x 7 - x 6 - x 5 + x 2 - 1 x^{7}-x^{6}-x^{5}+x^{2}-1
  66. x 3 ( x 3 - 2 x 2 + x - 1 ) + ( x - 1 ) ( x 2 + 1 ) x^{3}(x^{3}-2x^{2}+x-1)+(x-1)(x^{2}+1)
  67. x 6 - 2 x 5 + x 4 - x 2 + x - 1 x^{6}-2x^{5}+x^{4}-x^{2}+x-1
  68. x 5 ( x 2 - x - 1 ) + 1 x^{5}(x^{2}-x-1)+1
  69. x 5 - x 4 - x 2 - 1 x^{5}-x^{4}-x^{2}-1
  70. x 6 ( x 2 - x - 1 ) + ( x 2 - 1 ) x^{6}(x^{2}-x-1)+(x^{2}-1)
  71. x 8 - x 7 - x 6 + x 2 - 1 x^{8}-x^{7}-x^{6}+x^{2}-1

Pitch_class.html

  1. f f
  2. p p
  3. p = 69 + 12 log 2 ( f / 440 ) p=69+12\log_{2}{(f/440)}
  4. a / b = 2 n a/b=2^{n}\,
  5. 1 p / q < 2 1\leq p/q<2

Pitch_space.html

  1. p = 49 + 12 log 2 ( f / 440 ) p=49+12\cdot\log_{2}{(f/440)}\,

Pitman_shorthand.html

  1. i e o i 7 o w and e w ie\;\;\;oi\;\;\;^{\mathfrak{7}}\qquad ow\;\;\;_{\and}\qquad ew\;\;\;_{\cap}

Plancherel_theorem.html

  1. - f ( x ) g * ( x ) d x = 1 2 π - F ( k ) G * ( k ) d k , \int_{-\infty}^{\infty}f(x)g^{*}(x)\operatorname{d}\!x=\frac{1}{2\pi}\int_{-% \infty}^{\infty}F(k)G^{*}(k)\operatorname{d}\!k,
  2. - f ( x ) e - i k x d x = F ( k ) \int_{-\infty}^{\infty}f(x)e^{-ikx}\operatorname{d}\!x=F(k)

Plane_at_infinity.html

  1. P 3 P^{3}
  2. 3 \mathbb{R}^{3}
  3. P 2 \mathbb{R}P^{2}
  4. P 3 \mathbb{R}P^{3}
  5. ( X : Y : Z : 0 ) ( a X : a Y : a Z : 0 ) (X:Y:Z:0)\equiv(aX:aY:aZ:0)
  6. a ( 0 : 0 : 0 : 1 ) + b ( X : Y : Z : 1 ) = ( b X : b Y : b Z : a + b ) . a(0:0:0:1)+b(X:Y:Z:1)=(bX:bY:bZ:a+b).
  7. a + b = 0 a+b=0
  8. a = - b a=-b
  9. ( b X : b Y : b Z : 0 ) = ( X : Y : Z : 0 ) (bX:bY:bZ:0)=(X:Y:Z:0)
  10. λ ( 3 : 0 : 1 : 1 ) + μ ( 0 : 0 : 1 : 1 ) \lambda(3:0:1:1)+\mu(0:0:1:1)
  11. = ( 3 λ : 0 : λ + μ : λ + μ ) =(3\lambda:0:\lambda+\mu:\lambda+\mu)
  12. = ( 3 : 0 : 0 : 0 ) =(3:0:0:0)
  13. λ + μ = 0 \lambda+\mu=0

Plasma_diagnostics.html

  1. B ˙ \dot{B}

Plummer_model.html

  1. ρ P ( r ) = ( 3 M 4 π a 3 ) ( 1 + r 2 a 2 ) - 5 2 , \rho_{P}(r)=\bigg(\frac{3M}{4\pi a^{3}}\bigg)\bigg(1+\frac{r^{2}}{a^{2}}\bigg)% ^{-\frac{5}{2}}\,,
  2. Φ P ( r ) = - G M r 2 + a 2 , \Phi_{P}(r)=-\frac{GM}{\sqrt{r^{2}+a^{2}}}\,,
  3. r r
  4. M ( < r ) = 4 π 0 r r 2 ρ P ( r ) d r = M r 3 ( r 2 + a 2 ) 3 / 2 M(<r)=4\pi\int_{0}^{r}r^{2}\rho_{P}(r)dr=M{r^{3}\over\left(r^{2}+a^{2}\right)^% {3/2}}
  5. r c r_{c}
  6. r c = a 2 - 1 0.64 a r_{c}=a\sqrt{\sqrt{2}-1}\approx 0.64a
  7. r h 1.3 a r_{h}\approx 1.3a
  8. r V = 16 3 π a 1.7 a r_{V}=\frac{16}{3\pi}a\approx 1.7a
  9. ρ r - 5 \rho\rightarrow r^{-5}

Plural_quantification.html

  1. F F
  2. G G
  3. x x
  4. y y
  5. x ¯ \bar{x}
  6. y ¯ \bar{y}
  7. F F
  8. x 0 , , x n x_{0},\ldots,x_{n}
  9. F ( x 0 , , x n ) F(x_{0},\ldots,x_{n})
  10. P P
  11. ¬ P \neg P
  12. P P
  13. Q Q
  14. P Q P\land Q
  15. P P
  16. x x
  17. x . P \exists x.P
  18. x x
  19. y ¯ \bar{y}
  20. x y ¯ x\prec\bar{y}
  21. P P
  22. x ¯ \bar{x}
  23. x ¯ . P \exists\bar{x}.P
  24. ( D , V , s , R ) (D,V,s,R)
  25. D D
  26. V V
  27. V F V_{F}
  28. F F
  29. s s
  30. D D
  31. R R
  32. ( D , V , s , R ) F ( x 0 , , x n ) (D,V,s,R)\models F(x_{0},\ldots,x_{n})
  33. ( s x 0 , , s x n ) V F (s_{x_{0}},\ldots,s_{x_{n}})\in V_{F}
  34. ( D , V , s , R ) ¬ P (D,V,s,R)\models\neg P
  35. ( D , V , s , R ) P (D,V,s,R)\nvDash P
  36. ( D , V , s , R ) P Q (D,V,s,R)\models P\land Q
  37. ( D , V , s , R ) P (D,V,s,R)\models P
  38. ( D , V , s , R ) Q (D,V,s,R)\models Q
  39. ( D , V , s , R ) x . P (D,V,s,R)\models\exists x.P
  40. s x s s^{\prime}\approx_{x}s
  41. ( D , V , s , R ) P (D,V,s^{\prime},R)\models P
  42. ( D , V , s , R ) x y ¯ (D,V,s,R)\models x\prec\bar{y}
  43. s x R y ¯ s_{x}R\bar{y}
  44. ( D , V , s , R ) x ¯ . P (D,V,s,R)\models\exists\bar{x}.P
  45. R x ¯ R R^{\prime}\approx_{\bar{x}}R
  46. ( D , V , s , R ) P (D,V,s,R^{\prime})\models P
  47. s x s s\approx_{x}s^{\prime}
  48. y y
  49. x x
  50. s y = s y s_{y}=s^{\prime}_{y}
  51. R x ¯ R R\approx_{\bar{x}}R^{\prime}
  52. y ¯ \bar{y}
  53. x ¯ \bar{x}
  54. d d
  55. d R y ¯ = d R y ¯ dR\bar{y}=dR^{\prime}\bar{y}
  56. R R
  57. x ¯ . y . y x ¯ F ( y ) \exists\bar{x}.\forall y.y\prec\bar{x}\leftrightarrow F(y)
  58. F ( x ¯ ) F(\bar{x})

Plus-minus_sign.html

  1. x = - b ± b 2 - 4 a c 2 a . \displaystyle x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}.
  2. sin ( A ± B ) = sin ( A ) cos ( B ) ± cos ( A ) sin ( B ) . \sin(A\pm B)=\sin(A)\cos(B)\pm\cos(A)\sin(B).\,
  3. sin ( x ) = x - x 3 3 ! + x 5 5 ! - x 7 7 ! + ± 1 ( 2 n + 1 ) ! x 2 n + 1 + . \sin\left(x\right)=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}+\cdots% \pm\frac{1}{(2n+1)!}x^{2n+1}+\cdots.
  4. cos ( A ± B ) = cos ( A ) cos ( B ) sin ( A ) sin ( B ) \cos(A\pm B)=\cos(A)\cos(B)\mp\sin(A)\sin(B)
  5. cos ( A + B ) = cos ( A ) cos ( B ) - sin ( A ) sin ( B ) \cos(A+B)=\cos(A)\cos(B)-\sin(A)\sin(B)\,
  6. cos ( A - B ) = cos ( A ) cos ( B ) + sin ( A ) sin ( B ) \cos(A-B)=\cos(A)\cos(B)+\sin(A)\sin(B)\,
  7. cos ( A + B ) = cos ( A ) cos ( B ) + sin ( A ) sin ( B ) \cos(A+B)=\cos(A)\cos(B)+\sin(A)\sin(B)\,
  8. cos ( A - B ) = cos ( A ) cos ( B ) - sin ( A ) sin ( B ) \cos(A-B)=\cos(A)\cos(B)-\sin(A)\sin(B)\,
  9. x 3 ± 1 = ( x ± 1 ) ( x 2 x + 1 ) x^{3}\pm 1=(x\pm 1)(x^{2}\mp x+1)

Plus_construction.html

  1. X X
  2. X X
  3. R R
  4. G L n ( R ) GL_{n}(R)
  5. n n
  6. n n
  7. R R
  8. G L n ( R ) GL_{n}(R)
  9. G L n + 1 ( R ) GL_{n+1}(R)
  10. 1 1
  11. 0
  12. G L ( R ) GL(R)
  13. B G L ( R ) BGL(R)
  14. E ( R ) E(R)
  15. G L ( R ) = π 1 ( B G L ( R ) ) GL(R)=\pi_{1}(BGL(R))
  16. i > 0 i>0
  17. n n
  18. B G L ( R ) + BGL(R)^{+}
  19. n n
  20. K K
  21. R R
  22. K n ( R ) K_{n}(R)

Pocket_Cube.html

  1. 8 ! × 3 7 24 = 7 ! × 3 6 = 3 , 674 , 160. \frac{8!\times 3^{7}}{24}=7!\times 3^{6}=3,674,160.

Poincaré_duality.html

  1. H k ( M ) H n - k ( M ) . H^{k}(M)\cong H_{n-k}(M).
  2. S T Δ D S \cup_{S\in T}\Delta\cap DS
  3. Δ \Delta
  4. Δ \Delta
  5. Δ D S \Delta\cap DS
  6. Δ \Delta
  7. Δ \Delta
  8. S S
  9. C i M C n - i M C_{i}M\otimes C^{n-i}M\to\mathbb{Z}
  10. C i M C n - i M C_{i}M\to C^{n-i}M
  11. C i C_{i}
  12. C n - i M C_{n-i}M
  13. C n - i M C^{n-i}M
  14. S D S S\longmapsto DS
  15. τ H i M \tau H_{i}M
  16. H i M H_{i}M
  17. f H i M = H i M / τ H i M fH_{i}M=H_{i}M/\tau H_{i}M
  18. f H i M f H n - i M fH_{i}M\otimes fH_{n-i}M\to\mathbb{Z}
  19. τ H i M τ H n - i - 1 M / . \tau H_{i}M\otimes\tau H_{n-i-1}M\to\mathbb{Q}/\mathbb{Z}.
  20. / \mathbb{Q}/\mathbb{Z}
  21. n - 1 , n-1,
  22. n . n.
  23. f H i M Hom ( f H n - i M , ) fH_{i}M\to\mathrm{Hom}_{\mathbb{Z}}(fH_{n-i}M,\mathbb{Z})
  24. τ H i M Hom ( τ H n - i - 1 M , / ) \tau H_{i}M\to\mathrm{Hom}_{\mathbb{Z}}(\tau H_{n-i-1}M,\mathbb{Q}/\mathbb{Z})
  25. H i M H n - i M H_{i}M\simeq H^{n-i}M
  26. f H n - i M Hom ( H n - i M ; ) fH^{n-i}M\equiv\mathrm{Hom}(H_{n-i}M;\mathbb{Z})
  27. τ H n - i M Ext ( H n - i - 1 M ; ) Hom ( τ H n - i - 1 M ; / ) \tau H^{n-i}M\equiv\mathrm{Ext}(H_{n-i-1}M;\mathbb{Z})\equiv\mathrm{Hom}(\tau H% _{n-i-1}M;\mathbb{Q}/\mathbb{Z})
  28. f H i M fH_{i}M
  29. f H n - i M fH_{n-i}M
  30. τ H i M \tau H_{i}M
  31. τ H n - i - 1 M \tau H_{n-i-1}M
  32. n = 2 k , n=2k,
  33. f H k M f H k M fH_{k}M\otimes fH_{k}M\to\mathbb{Z}
  34. n = 2 k + 1 , n=2k+1,
  35. τ H k M τ H k M / . \tau H_{k}M\otimes\tau H_{k}M\to\mathbb{Q}/\mathbb{Z}.
  36. f H k M f H k + 1 M . fH_{k}M\otimes fH_{k+1}M\to\mathbb{Z}.
  37. M M
  38. M × M M\times M
  39. M M
  40. V V
  41. M × M M\times M
  42. H * M H * M H * ( M × M ) H_{*}M\otimes H_{*}M\to H_{*}(M\times M)
  43. H * ( M × M ) H * ( M × M , ( M × M ) V ) H_{*}(M\times M)\to H_{*}\left(M\times M,(M\times M)\setminus V\right)
  44. H * ( M × M , ( M × M ) V ) H * ( ν M , ν M ) H_{*}\left(M\times M,(M\times M)\setminus V\right)\to H_{*}(\nu M,\partial\nu M)
  45. ν M \nu M
  46. M × M M\times M
  47. H * ( ν M , ν M ) H * - n M H_{*}(\nu M,\partial\nu M)\to H_{*-n}M
  48. ν M T M \nu M\equiv TM
  49. H i M H j M H i + j - n M H_{i}M\otimes H_{j}M\to H_{i+j-n}M
  50. s p i n c spin^{c}

Poincaré_half-plane_model.html

  1. { ( x , y ) | y > 0 ; x , y } \{(x,y)|y>0;x,y\in\mathbb{R}\}
  2. { x , y | y > 0 } \{\langle x,y\rangle|y>0\}\,
  3. ( d s ) 2 = ( d x ) 2 + ( d y ) 2 y 2 (ds)^{2}=\frac{(dx)^{2}+(dy)^{2}}{y^{2}}\,
  4. dist ( x 1 , y 1 , x 2 , y 2 ) = arcosh ( 1 + ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 2 y 1 y 2 ) . \operatorname{dist}(\langle x_{1},y_{1}\rangle,\langle x_{2},y_{2}\rangle)=% \operatorname{arcosh}\left(1+\frac{{(x_{2}-x_{1})}^{2}+{(y_{2}-y_{1})}^{2}}{2y% _{1}y_{2}}\right)\,.
  5. dist ( 0 , r , r sin ϕ , r cos ϕ ) = arcosh ( 1 cos ϕ ) = arcosh ( sec ϕ ) . \operatorname{dist}(\langle 0,r\rangle,\langle\ r\sin\phi,r\cos\phi\rangle)=% \operatorname{arcosh}\left(\frac{1}{\cos\phi}\right)=\operatorname{arcosh}% \left(\operatorname{sec}\phi\right)\,.
  6. arcosh x = ln ( x + x 2 - 1 ) ; x 1 . \operatorname{arcosh}{x}=\ln\left(x+\sqrt{x^{2}-1}\right);x\geq 1\,.
  7. y = y=\infty
  8. x , y \langle x,y\rangle\,
  9. R R\,
  10. x , y cosh R \langle x,y\cosh R\rangle\,
  11. y sinh R . y\sinh R\,.
  12. y = y=\infty
  13. z - z ¯ z\rightarrow-\overline{z}
  14. ( a b c d ) z = a z + b c z + d = ( a c | z | 2 + b d + ( a d + b c ) ( z ) ) + i ( z ) | c z + d | 2 . \left(\begin{matrix}a&b\\ c&d\\ \end{matrix}\right)\cdot z=\frac{az+b}{cz+d}={(ac|z|^{2}+bd+(ad+bc)\Re(z))+i% \Im(z)\over|cz+d|^{2}}.
  15. z 1 , z 2 z_{1},z_{2}\in\mathbb{H}
  16. g PSL ( 2 , ) g\in{\rm PSL}(2,\mathbb{R})
  17. g z 1 = z 2 gz_{1}=z_{2}
  18. g z = z gz=z
  19. g PSL ( 2 , ) g\in{\rm PSL}(2,\mathbb{R})
  20. SO ( 2 ) = { ( cos θ sin θ - sin θ cos θ ) : θ 𝐑 } . {\rm SO}(2)=\left\{\left(\begin{matrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\\ \end{matrix}\right)\,:\,\theta\in{\mathbf{R}}\right\}.
  21. γ ( t ) = ( e t / 2 0 0 e - t / 2 ) i = i e t . \gamma(t)=\left(\begin{matrix}e^{t/2}&0\\ 0&e^{-t/2}\\ \end{matrix}\right)\cdot i=ie^{t}.
  22. γ ( t ) = ( a b c d ) ( e t / 2 0 0 e - t / 2 ) i = a i e t + b c i e t + d . \gamma(t)=\left(\begin{matrix}a&b\\ c&d\\ \end{matrix}\right)\left(\begin{matrix}e^{t/2}&0\\ 0&e^{-t/2}\\ \end{matrix}\right)\cdot i=\frac{aie^{t}+b}{cie^{t}+d}.
  23. { x , y , z | z > 0 } \{\langle x,y,z\rangle|z>0\}\,
  24. ( d s ) 2 = ( d x ) 2 + ( d y ) 2 + ( d z ) 2 z 2 (ds)^{2}=\frac{(dx)^{2}+(dy)^{2}+(dz)^{2}}{z^{2}}\,
  25. dist ( x 1 , y 1 , z 1 , x 2 , y 2 , z 2 ) = arcosh ( 1 + ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 + ( z 2 - z 1 ) 2 2 z 1 z 2 ) . \operatorname{dist}(\langle x_{1},y_{1},z_{1}\rangle,\langle x_{2},y_{2},z_{2}% \rangle)=\operatorname{arcosh}\left(1+\frac{{(x_{2}-x_{1})}^{2}+{(y_{2}-y_{1})% }^{2}+{(z_{2}-z_{1})}^{2}}{2z_{1}z_{2}}\right)\,.

Point_(geometry).html

  1. x x
  2. y y
  3. x x
  4. y y
  5. x x
  6. y y
  7. z z
  8. z z
  9. n n
  10. n n
  11. L = { ( a 1 , a 2 , a n ) | a 1 c 1 + a 2 c 2 + a n c n = d } \scriptstyle{L=\{(a_{1},a_{2},...a_{n})|a_{1}c_{1}+a_{2}c_{2}+...a_{n}c_{n}=d\}}
  12. d d
  13. n n
  14. 1 𝟎 = 𝟎 1\cdot\mathbf{0}=\mathbf{0}
  15. 𝒜 \mathcal{A}
  16. \mathcal{B}
  17. 𝒜 \mathcal{A}
  18. { B ( x i , r i ) : i I } \{B(x_{i},r_{i}):i\in I\}
  19. i I r i d < δ \sum_{i\in I}r_{i}^{d}<\delta
  20. dim H ( X ) := inf { d 0 : C H d ( X ) = 0 } . \operatorname{dim}_{\operatorname{H}}(X):=\inf\{d\geq 0:C_{H}^{d}(X)=0\}.
  21. δ δ

Point_and_click.html

  1. T = a + b log 2 ( 1 + D W ) T=a+b\log_{2}\Bigg(1+\frac{D}{W}\Bigg)
  2. T T
  3. a a
  4. b b
  5. D D
  6. W W
  7. W W
  8. ± W 2 \pm\frac{W}{2}

Point_in_polygon.html

  1. 2 π 2\pi
  2. 2 π 2\pi

Point_location.html

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

Pointwise_convergence.html

  1. lim n f n = f pointwise , \lim_{n\rightarrow\infty}f_{n}=f\ \mbox{pointwise}~{},
  2. lim n f n ( x ) = f ( x ) . \lim_{n\rightarrow\infty}f_{n}(x)=f(x).
  3. lim n f n = f uniformly \lim_{n\rightarrow\infty}f_{n}=f\ \mbox{uniformly}~{}
  4. lim n sup { | f n ( x ) - f ( x ) | : x the domain } = 0. \lim_{n\rightarrow\infty}\,\sup\{\,\left|f_{n}(x)-f(x)\right|:x\in\mbox{the % domain}~{}\,\}=0.
  5. lim n x n = 0 pointwise on the interval [ 0 , 1 ) , but not uniformly on the interval [ 0 , 1 ) . \lim_{n\rightarrow\infty}x^{n}=0\ \mbox{pointwise}~{}\ \mbox{on}~{}\ \mbox{the% }~{}\ \mbox{interval}~{}\ [0,1),\ \mbox{but}~{}\ \mbox{not}~{}\ \mbox{% uniformly}~{}\ \mbox{on}~{}\ \mbox{the}~{}\ \mbox{interval}~{}\ [0,1).
  6. f ( x ) = lim n cos ( π x ) 2 n f(x)=\lim_{n\rightarrow\infty}\cos(\pi x)^{2n}

Poise.html

  1. 1 P = 0.100 kg m - 1 s = - 1 1 g cm - 1 s - 1 1\ \mbox{P}~{}=0.100\ \mbox{kg}~{}\cdot\mbox{m}~{}^{-1}\cdot\mbox{s}~{}^{-1}=1% \ \mbox{g}~{}\cdot\mbox{cm}~{}^{-1}\cdot\mbox{s}~{}^{-1}
  2. 1 Pa s = 1 kg m - 1 s = - 1 10 P 1\ \mbox{Pa}~{}\cdot\mbox{s}~{}=1\ \mbox{kg}~{}\cdot\mbox{m}~{}^{-1}\cdot\mbox% {s}~{}^{-1}=10\ \mbox{P}~{}

Poisson_algebra.html

  1. { F , G } = d G ( X F ) = X F ( G ) \{F,G\}=dG(X_{F})=X_{F}(G)\,
  2. X { F , G } = [ X F , X G ] X_{\{F,G\}}=[X_{F},X_{G}]\,
  3. { F , G } = i = 1 n F q i G p i - F p i G q i . \{F,G\}=\sum_{i=1}^{n}\frac{\partial F}{\partial q_{i}}\frac{\partial G}{% \partial p_{i}}-\frac{\partial F}{\partial p_{i}}\frac{\partial G}{\partial q_% {i}}.

Poisson_summation_formula.html

  1. f , f,\,
  2. f f\,
  3. f ^ ( ν ) = { f ( x ) } . \hat{f}(\nu)=\mathcal{F}\{f(x)\}.
  4. g ( x P ) = def f ( x ) , g(xP)\ \stackrel{\,\text{def}}{=}\ f(x),\,
  5. { g ( x P ) } = 1 P g ^ ( ν P ) \mathcal{F}\{g(xP)\}\ =\frac{1}{P}\cdot\hat{g}\left(\frac{\nu}{P}\right)
  6. s ( t + x ) = def g ( x ) , s(t+x)\ \stackrel{\,\text{def}}{=}\ g(x),\,
  7. { s ( t + x ) } = s ^ ( ν ) e i 2 π ν t , \mathcal{F}\{s(t+x)\}\ =\hat{s}(\nu)\cdot e^{i2\pi\nu t},
  8. k = - s ^ ( ν + k / T ) = n = - T s ( n T ) e - i 2 π n T ν { n = - T s ( n T ) δ ( t - n T ) } , \sum_{k=-\infty}^{\infty}\hat{s}(\nu+k/T)=\sum_{n=-\infty}^{\infty}T\cdot s(nT% )\ e^{-i2\pi nT\nu}\equiv\mathcal{F}\left\{\sum_{n=-\infty}^{\infty}T\cdot s(% nT)\ \delta(t-nT)\right\},
  9. f f
  10. n = - δ ( x - n T ) k = - 1 T e - i 2 π k T x 1 T k = - δ ( ν + k / T ) , \sum_{n=-\infty}^{\infty}\delta(x-nT)\equiv\sum_{k=-\infty}^{\infty}\frac{1}{T% }\cdot e^{-i2\pi\frac{k}{T}x}\quad\stackrel{\mathcal{F}}{\Longleftrightarrow}% \quad\frac{1}{T}\cdot\sum_{k=-\infty}^{\infty}\delta(\nu+k/T),
  11. δ \delta
  12. k = - f ^ ( k ) = k = - ( - f ( x ) e - i 2 π k x d x ) = - f ( x ) ( k = - e - i 2 π k x ) n = - δ ( x - n ) d x = n = - ( - f ( x ) δ ( x - n ) d x ) = n = - f ( n ) . \begin{aligned}\displaystyle\sum_{k=-\infty}^{\infty}\hat{f}(k)&\displaystyle=% \sum_{k=-\infty}^{\infty}\left(\int_{-\infty}^{\infty}f(x)\ e^{-i2\pi kx}dx% \right)=\int_{-\infty}^{\infty}f(x)\underbrace{\left(\sum_{k=-\infty}^{\infty}% e^{-i2\pi kx}\right)}_{\sum_{n=-\infty}^{\infty}\delta(x-n)}dx\\ &\displaystyle=\sum_{n=-\infty}^{\infty}\left(\int_{-\infty}^{\infty}f(x)\ % \delta(x-n)\ dx\right)=\sum_{n=-\infty}^{\infty}f(n).\end{aligned}
  13. k = - s ^ ( ν + k / T ) = k = - { s ( t ) e - i 2 π k T t } = { s ( t ) k = - e - i 2 π k T t T n = - δ ( t - n T ) } = { n = - T s ( n T ) δ ( t - n T ) } = n = - T s ( n T ) { δ ( t - n T ) } = n = - T s ( n T ) e - i 2 π n T ν . \begin{aligned}\displaystyle\sum_{k=-\infty}^{\infty}\hat{s}(\nu+k/T)&% \displaystyle=\sum_{k=-\infty}^{\infty}\mathcal{F}\left\{s(t)\cdot e^{-i2\pi% \frac{k}{T}t}\right\}\\ &\displaystyle=\mathcal{F}\bigg\{s(t)\underbrace{\sum_{k=-\infty}^{\infty}e^{-% i2\pi\frac{k}{T}t}}_{T\sum_{n=-\infty}^{\infty}\delta(t-nT)}\bigg\}=\mathcal{F% }\left\{\sum_{n=-\infty}^{\infty}T\cdot s(nT)\cdot\delta(t-nT)\right\}\\ &\displaystyle=\sum_{n=-\infty}^{\infty}T\cdot s(nT)\cdot\mathcal{F}\left\{% \delta(t-nT)\right\}=\sum_{n=-\infty}^{\infty}T\cdot s(nT)\cdot e^{-i2\pi nT% \nu}.\end{aligned}
  14. 1 P s ^ ( k P ) . \scriptstyle\frac{1}{P}\hat{s}\left(\frac{k}{P}\right).
  15. S [ k ] = def 1 P 0 P s P ( t ) e - i 2 π k P t d t = 1 P 0 P ( n = - s ( t + n P ) ) e - i 2 π k P t d t = 1 P n = - 0 P s ( t + n P ) e - i 2 π k P t d t , \begin{aligned}\displaystyle S[k]&\displaystyle\stackrel{\,\text{def}}{=}\ % \frac{1}{P}\int_{0}^{P}s_{P}(t)\cdot e^{-i2\pi\frac{k}{P}t}\,dt\\ &\displaystyle=\ \frac{1}{P}\int_{0}^{P}\left(\sum_{n=-\infty}^{\infty}s(t+nP)% \right)\cdot e^{-i2\pi\frac{k}{P}t}\,dt\\ &\displaystyle=\ \frac{1}{P}\sum_{n=-\infty}^{\infty}\int_{0}^{P}s(t+nP)\cdot e% ^{-i2\pi\frac{k}{P}t}\,dt,\end{aligned}
  16. S [ k ] = 1 P n = - n P n P + P s ( τ ) e - i 2 π k P τ e i 2 π k n 1 d τ = 1 P - s ( τ ) e - i 2 π k P τ d τ = 1 P s ^ ( k P ) \begin{aligned}\displaystyle S[k]=\frac{1}{P}\sum_{n=-\infty}^{\infty}\int_{nP% }^{nP+P}s(\tau)\ e^{-i2\pi\frac{k}{P}\tau}\ \underbrace{e^{i2\pi kn}}_{1}\,d% \tau\ =\ \frac{1}{P}\int_{-\infty}^{\infty}s(\tau)\ e^{-i2\pi\frac{k}{P}\tau}d% \tau=\frac{1}{P}\cdot\hat{s}\left(\frac{k}{P}\right)\end{aligned}
  17. | s ( t ) | + | s ^ ( t ) | C ( 1 + | t | ) - 1 - δ |s(t)|+|\hat{s}(t)|\leq C(1+|t|)^{-1-\delta}
  18. 2 s ( t ) = lim ε 0 s ( t + ε ) + lim ε 0 s ( t - ε ) 2\cdot s(t)=\lim_{\varepsilon\to 0}s(t+\varepsilon)+\lim_{\varepsilon\to 0}s(t% -\varepsilon)
  19. g g\,
  20. g ^ \hat{g}
  21. f f
  22. f f
  23. f o f_{o}
  24. f ^ ( ξ ) = 0 \hat{f}(\xi)=0
  25. | ξ | > f o |\xi|>f_{o}
  26. 2 f o 2f_{o}
  27. f ^ \hat{f}
  28. f f
  29. f f\,
  30. f ^ \hat{f}
  31. f = 0 f=0\,
  32. q = e i π τ q=e^{i\pi\tau}
  33. τ \tau
  34. θ ( τ ) = n q n 2 . \theta(\tau)=\sum_{n}q^{n^{2}}.
  35. θ ( - 1 / τ ) \theta(-1/\tau)
  36. θ ( τ ) \theta(\tau)
  37. f = e - π x 2 f=e^{-\pi x^{2}}
  38. a = 0 a=0
  39. f ^ = e - π ξ 2 \hat{f}=e^{-\pi\xi^{2}}
  40. θ ( - 1 τ ) = τ i θ ( τ ) \theta\left({-1\over\tau}\right)=\sqrt{\tau\over i}\theta(\tau)
  41. 1 / λ = τ / i {1/\lambda}=\sqrt{\tau/i}
  42. θ 8 \theta^{8}
  43. τ - 1 / τ \tau\mapsto{-1/\tau}
  44. ν Λ f ( x + ν ) . \sum_{\nu\in\Lambda}f(x+\nu).
  45. | f ( x ) | + | f ^ ( x ) | C ( 1 + | x | ) - d - δ |f(x)|+|\hat{f}(x)|\leq C(1+|x|)^{-d-\delta}
  46. ν Λ f ( x + ν ) = ν Λ f ^ ( ν ) e 2 π i x ν , \sum_{\nu\in\Lambda}f(x+\nu)=\sum_{\nu\in\Lambda}\hat{f}(\nu)e^{2\pi ix\cdot% \nu},
  47. G G
  48. Γ \Gamma
  49. G / Γ G/\Gamma
  50. G G
  51. G L n GL_{n}
  52. Γ \Gamma
  53. G L n GL_{n}
  54. G G
  55. Γ \Gamma
  56. n n
  57. G G
  58. Γ \Gamma

Poker_probability_(Texas_hold_'em).html

  1. 52 × 51 2 = 1326 \tfrac{52\times 51}{2}=1326
  2. 6 1326 = 1 221 \begin{matrix}\frac{6}{1326}=\frac{1}{221}\end{matrix}
  3. 4 52 = 1 13 . \begin{matrix}\frac{4}{52}=\frac{1}{13}\end{matrix}.
  4. 3 51 = 1 17 . \begin{matrix}\frac{3}{51}=\frac{1}{17}\end{matrix}.
  5. 1 13 × 1 17 = 1 221 . \begin{matrix}\frac{1}{13}\times\frac{1}{17}=\frac{1}{221}\end{matrix}.
  6. 1 6 \begin{matrix}\frac{1}{6}\end{matrix}
  7. ( 52 2 ) = 1 , 326 {52\choose 2}=1,326
  8. ( 4 2 ) = 6 {4\choose 2}=6
  9. 6 1326 0.00452 \begin{matrix}\frac{6}{1326}\approx 0.00452\end{matrix}
  10. 78 1326 = 3 51 0.0588 \begin{matrix}\frac{78}{1326}=\frac{3}{51}\approx 0.0588\end{matrix}
  11. ( 4 1 ) = 4 {4\choose 1}=4
  12. 4 1326 0.00302 \begin{matrix}\frac{4}{1326}\approx 0.00302\end{matrix}
  13. 312 1326 = 12 51 0.2353 \begin{matrix}\frac{312}{1326}=\frac{12}{51}\approx 0.2353\end{matrix}
  14. ( 4 1 ) ( 3 1 ) = 12 {4\choose 1}{3\choose 1}=12
  15. 12 1326 0.00905 \begin{matrix}\frac{12}{1326}\approx 0.00905\end{matrix}
  16. 936 1326 = 36 51 0.7059 \begin{matrix}\frac{936}{1326}=\frac{36}{51}\approx 0.7059\end{matrix}
  17. ( 52 2 ) ( 50 2 ) ÷ 2 = 812 , 175 {52\choose 2}{50\choose 2}\div 2=812,175
  18. ( 48 5 ) = 1 , 712 , 304 {48\choose 5}=1,712,304
  19. ( 50 2 ) ( 48 2 ) = 1 , 381 , 800 {50\choose 2}{48\choose 2}=1,381,800
  20. n n
  21. n ! n!
  22. H H
  23. H = ( 50 2 ) ( 48 2 ) ÷ 2 ! = 690 , 900 H={50\choose 2}{48\choose 2}\div 2!=690,900
  24. H = ( 50 2 ) ( 48 2 ) ( 46 2 ) ÷ 3 ! = 238 , 360 , 500 H={50\choose 2}{48\choose 2}{46\choose 2}\div 3!=238,360,500
  25. n n
  26. H = k = 1 n ( 52 - 2 k 2 ) ÷ n ! , H=\prod_{k=1}^{n}{52-2k\choose 2}\div n!,
  27. H = ( 50 2 n ) × ( 2 n - 1 ) ! ! , H={50\choose 2n}\times(2n-1)!!,
  28. ( 2 n - 1 ) ! ! (2n-1)!!
  29. 2 n 2n
  30. n n
  31. 169 × ( 50 18 ) × 17 ! ! × ( 32 5 ) 2.117 × 10 28 169\times{50\choose 18}\times 17!!\times{32\choose 5}\approx 2.117\times 10^{28}
  32. r r
  33. P P
  34. P = ( 14 - r ) × 4 50 × 3 49 = 84 - 6 r 1225 . \begin{aligned}\displaystyle P&\displaystyle=\frac{(14-r)\times 4}{50}\times% \frac{3}{49}\\ &\displaystyle=\frac{84-6r}{1225}.\\ \end{aligned}
  35. n n
  36. P m a P_{ma}
  37. P = ( 84 - 6 r 1225 ) × n - P m a = n ( 84 - 6 r ) - 1 , 225 P m a 1 , 225 . P=\left(\frac{84-6r}{1225}\right)\times n-P_{ma}=\frac{n(84-6r)-1,225P_{ma}}{1% ,225}.
  38. P m a P_{ma}
  39. P m a = P 2 + 2 P 3 + + ( n - 1 ) P n , P_{ma}=P_{2}+2P_{3}+\cdots+(n-1)P_{n},
  40. P 2 P_{2}
  41. P 3 P_{3}
  42. P 4 < 0.0015 P_{4}<0.0015
  43. P 5 < 0.00009 P_{5}<0.00009
  44. P 2 P_{2}
  45. P 3 P_{3}
  46. P m P_{m}
  47. P m = P 2 + P 3 + + P n . P_{m}=P_{2}+P_{3}+\cdots+P_{n}.
  48. 3 50 × 2 49 0.00245 \begin{matrix}\frac{3}{50}\times\frac{2}{49}\approx 0.00245\end{matrix}
  49. n n
  50. ( 1 - ( 1 - 0.00245 ) n ) (1-(1-0.00245)^{n})
  51. x x
  52. P = ( 3 50 × 2 49 ) + ( 3 50 × ( 13 - x ) × 4 49 × 2 ) = 3 1225 + 12 × ( 13 - x ) 1225 = 159 - 12 x 1225 . \begin{aligned}\displaystyle P&\displaystyle=\left(\frac{3}{50}\times\frac{2}{% 49}\right)+\left(\frac{3}{50}\times\frac{(13-x)\times 4}{49}\times 2\right)\\ &\displaystyle=\frac{3}{1225}+\frac{12\times(13-x)}{1225}\\ &\displaystyle=\frac{159-12x}{1225}.\\ \end{aligned}
  53. 3 50 × ( 13 - x ) × 4 49 \begin{matrix}\frac{3}{50}\times\frac{(13-x)\times 4}{49}\end{matrix}
  54. h 1 h_{1}
  55. h 2 h_{2}
  56. w 1 w_{1}
  57. h 1 h_{1}
  58. h 2 h_{2}
  59. s s
  60. h 1 h_{1}
  61. h 2 h_{2}
  62. w 2 w_{2}
  63. h 2 h_{2}
  64. h 1 h_{1}
  65. w 2 = 1 - ( s + w 1 ) w_{2}=1-(s+w_{1})
  66. h 2 h_{2}
  67. h 1 h_{1}
  68. \prod
  69. ( 50 3 ) = 19 , 600 {50\choose 3}=19,600
  70. ( 50 4 ) = 230 , 300 {50\choose 4}=230,300
  71. ( 50 5 ) = 2 , 118 , 760 {50\choose 5}=2,118,760
  72. x x
  73. ( 14 - x ) × 4 \begin{matrix}(14-x)\times 4\end{matrix}
  74. x x
  75. 50 - ( 14 - x ) × 4 = 4 x - 6 \begin{matrix}50-(14-x)\times 4=4x-6\end{matrix}
  76. P P
  77. P = ( ( 4 x - 6 ) 3 ) ÷ ( 50 3 ) , P={(4x-6)\choose 3}\div{50\choose 3},
  78. P = ( ( 4 x - 6 ) 4 ) ÷ ( 50 4 ) P={(4x-6)\choose 4}\div{50\choose 4}
  79. P = ( ( 4 x - 6 ) 5 ) ÷ ( 50 5 ) , P={(4x-6)\choose 5}\div{50\choose 5},
  80. P = 1 - ( 47 - o u t s 47 × 46 - o u t s 46 ) = 93 o u t s - o u t s 2 2 , 162 . P=1-\left(\frac{47-outs}{47}\times\frac{46-outs}{46}\right)=\frac{93outs-outs^% {2}}{2,162}.
  81. x x
  82. P P
  83. P = x 47 × x - 1 46 = x 2 - x 2 , 162 . P=\frac{x}{47}\times\frac{x-1}{46}=\frac{x^{2}-x}{2,162}.
  84. 10 47 × 9 46 = 90 2162 1 24 0.04163 \begin{matrix}\frac{10}{47}\times\frac{9}{46}=\frac{90}{2162}\approx\frac{1}{2% 4}\approx 0.04163\end{matrix}
  85. x x
  86. y y
  87. P P
  88. P = x 47 × y 46 × 2 = x y 1081 . P=\frac{x}{47}\times\frac{y}{46}\times 2=\frac{xy}{1081}.
  89. 4 × 8 1081 0.0296 \begin{matrix}\frac{4\times 8}{1081}\approx 0.0296\end{matrix}
  90. P P
  91. P = ( 8 47 × 4 46 ) + ( 8 47 × 8 46 ) = 96 2162 0.0444 P=\left(\frac{8}{47}\times\frac{4}{46}\right)+\left(\frac{8}{47}\times\frac{8}% {46}\right)=\frac{96}{2162}\approx 0.0444
  92. P s P_{s}
  93. P f P_{f}
  94. P s f P_{sf}
  95. P P
  96. P = P s + P f - P s f . P=P_{s}+P_{f}-P_{sf}.
  97. P P
  98. P = 0.08326 + 0.00925 + 2 × 6 1081 + ( 0.00278 × 2 ) 0.1092 P=0.08326+0.00925+\frac{2\times 6}{1081}+(0.00278\times 2)\approx 0.1092

Polish_space.html

  1. I N I^{N}
  2. G δ G_{\delta}

Pollard's_p_−_1_algorithm.html

  1. a K ( p - 1 ) 1 ( mod p ) a^{K(p-1)}\equiv 1\;\;(\mathop{{\rm mod}}p)
  2. x w mod n x^{w}\mod n
  3. M = primes q B q log q B M=\prod_{\,\text{primes}~{}q\leq B}q^{\lfloor\log_{q}{B}\rfloor}
  4. M = primes p B 2 q log q B 2 M^{\prime}=\prod_{\,\text{primes}~{}p\leq B_{2}}q^{\lfloor\log_{q}{B_{2}}\rfloor}
  5. Q = primes q ( B 1 , B 2 ] ( H q - 1 ) Q=\prod_{\,\text{primes}~{}q\in(B_{1},B_{2}]}(H^{q}-1)

Pollard's_rho_algorithm.html

  1. gcd ( a , n ) > 1 \gcd(a,n)>1
  2. gcd ( a b , n ) > 1 \gcd(ab,n)>1
  3. gcd ( | x - y | , n ) \gcd(|x-y|,n)
  4. | x - y | |x-y|
  5. gcd ( z , n ) \gcd(z,n)
  6. gcd ( z , n ) = 1 \gcd(z,n)=1
  7. | x < s u b > i y i | |x<sub>i−y_{i}|

Polychord.html

  1. F C \frac{F}{C}
  2. D b G \frac{Db}{G}

Polydivisible_number.html

  1. F ( n ) 9 × 10 n - 1 n ! F(n)\approx\frac{9\times 10^{n-1}}{n!}
  2. 9 ( e 10 - 1 ) 10 19823 \frac{9(e^{10}-1)}{10}\approx 19823

Polygamma_function.html

  1. \C \C
  2. ψ ( m ) ( z ) := d m d z m ψ ( z ) = d m + 1 d z m + 1 ln Γ ( z ) . \psi^{(m)}(z):=\frac{d^{m}}{dz^{m}}\psi(z)=\frac{d^{m+1}}{dz^{m+1}}\ln\Gamma(z).
  3. ψ ( 0 ) ( z ) = ψ ( z ) = Γ ( z ) Γ ( z ) \psi^{(0)}(z)=\psi(z)=\frac{\Gamma^{\prime}(z)}{\Gamma(z)}
  4. \C - 𝒩 0 \C\setminus-\mathcal{N}_{0}
  5. ln Γ ( z ) \ln\Gamma(z)
  6. ψ ( 0 ) ( z ) \psi^{(0)}(z)
  7. ψ ( 1 ) ( z ) \psi^{(1)}(z)
  8. ψ ( 2 ) ( z ) \psi^{(2)}(z)
  9. ψ ( 3 ) ( z ) \psi^{(3)}(z)
  10. ψ ( 4 ) ( z ) \psi^{(4)}(z)
  11. ψ ( m ) ( z ) = ( - 1 ) m + 1 0 t m e - z t 1 - e - t d t = - 0 1 t z - 1 1 - t ln m t d t \begin{aligned}\displaystyle\psi^{(m)}(z)&\displaystyle=(-1)^{m+1}\int_{0}^{% \infty}\frac{t^{m}e^{-zt}}{1-e^{-t}}\ dt\\ &\displaystyle=-\int_{0}^{1}\frac{t^{z-1}}{1-t}\ln^{m}t\ dt\end{aligned}
  12. ψ ( m ) ( z + 1 ) = ψ ( m ) ( z ) + ( - 1 ) m m ! z m + 1 \psi^{(m)}(z+1)=\psi^{(m)}(z)+\frac{(-1)^{m}\,m!}{z^{m+1}}
  13. ψ ( m ) ( n ) ( - 1 ) m + 1 m ! = ζ ( 1 + m ) - k = 1 n - 1 1 k m + 1 = k = n 1 k m + 1 m 1 \frac{\psi^{(m)}(n)}{(-1)^{m+1}\,m!}=\zeta(1+m)-\sum_{k=1}^{n-1}\frac{1}{k^{m+% 1}}=\sum_{k=n}^{\infty}\frac{1}{k^{m+1}}\qquad m\geq 1
  14. ψ ( 0 ) ( n ) = - γ + k = 1 n - 1 1 k \psi^{(0)}(n)=-\gamma\ +\sum_{k=1}^{n-1}\frac{1}{k}
  15. n 𝒩 n\in\mathcal{N}
  16. ln Γ \ln\Gamma
  17. 𝒩 \mathcal{N}
  18. ψ ( m ) ( 1 ) \psi^{(m)}(1)
  19. \R + \R^{+}
  20. \R + \R^{+}
  21. ψ ( 0 ) \psi^{(0)}
  22. ( - 1 ) m ψ ( m ) ( 1 - z ) - ψ ( m ) ( z ) = π d m d z m cot ( π z ) = π m + 1 P m ( cos ( π z ) ) sin m + 1 ( π z ) (-1)^{m}\psi^{(m)}(1-z)-\psi^{(m)}(z)=\pi\frac{d^{m}}{dz^{m}}\cot{(\pi z)}=\pi% ^{m+1}\frac{P_{m}(\cos(\pi z))}{\sin^{m+1}(\pi z)}
  23. P m P_{m}
  24. | m - 1 | |m-1|
  25. ( - 1 ) m 2 m - 1 (-1)^{m}\lceil 2^{m-1}\rceil
  26. P m + 1 ( x ) = - ( ( m + 1 ) x P m ( x ) + ( 1 - x 2 ) P m ( x ) ) P_{m+1}(x)=-\left((m+1)xP_{m}(x)+(1-x^{2})P_{m}^{\prime}(x)\right)
  27. P 0 ( x ) = x P_{0}(x)=x
  28. k m + 1 ψ ( m ) ( k z ) = n = 0 k - 1 ψ ( m ) ( z + n k ) m 1 k^{m+1}\psi^{(m)}(kz)=\sum_{n=0}^{k-1}\psi^{(m)}\left(z+\frac{n}{k}\right)% \qquad m\geq 1
  29. k ψ ( 0 ) ( k z ) = k log ( k ) + n = 0 k - 1 ψ ( 0 ) ( z + n k ) k\psi^{(0)}(kz)=k\log(k)+\sum_{n=0}^{k-1}\psi^{(0)}\left(z+\frac{n}{k}\right)
  30. ψ ( m ) ( z ) = ( - 1 ) m + 1 m ! k = 0 1 ( z + k ) m + 1 \psi^{(m)}(z)=(-1)^{m+1}\;m!\;\sum_{k=0}^{\infty}\frac{1}{(z+k)^{m+1}}
  31. ψ ( m ) ( z ) = ( - 1 ) m + 1 m ! ζ ( m + 1 , z ) . \psi^{(m)}(z)=(-1)^{m+1}\;m!\;\zeta(m+1,z).
  32. 1 Γ ( z ) = z e n = 1 γ z ( 1 + z n ) e - z / n \frac{1}{\Gamma(z)}=z\;\mbox{e}~{}^{\gamma z}\;\prod_{n=1}^{\infty}\left(1+% \frac{z}{n}\right)\;\mbox{e}~{}^{-z/n}
  33. Γ ( z ) = e - γ z z n = 1 ( 1 + z n ) - 1 e z / n \Gamma(z)=\frac{\mbox{e}~{}^{-\gamma z}}{z}\;\prod_{n=1}^{\infty}\left(1+\frac% {z}{n}\right)^{-1}\;\mbox{e}~{}^{z/n}
  34. ln Γ ( z ) = - γ z - ln ( z ) + n = 1 ( z n - ln ( 1 + z n ) ) \ln\Gamma(z)=-\gamma z-\ln(z)+\sum_{n=1}^{\infty}\left(\frac{z}{n}-\ln(1+\frac% {z}{n})\right)
  35. ψ ( n ) ( z ) = d n + 1 d z n + 1 ln Γ ( z ) = - γ δ n 0 - ( - 1 ) n n ! z n + 1 + k = 1 ( 1 k δ n 0 - ( - 1 ) n n ! ( k + z ) n + 1 ) \psi^{(n)}(z)=\frac{d^{n+1}}{dz^{n+1}}\ln\Gamma(z)=-\gamma\delta_{n0}\;-\;% \frac{(-1)^{n}n!}{z^{n+1}}\;+\;\sum_{k=1}^{\infty}\left(\frac{1}{k}\delta_{n0}% \;-\;\frac{(-1)^{n}n!}{(k+z)^{n+1}}\right)
  36. δ n 0 \delta_{n0}
  37. k = 0 ( - 1 ) k ( z + k ) m + 1 \sum_{k=0}^{\infty}\frac{(-1)^{k}}{(z+k)^{m+1}}
  38. k = 0 ( - 1 ) k ( z + k ) m + 1 = 1 ( - 2 ) m + 1 m ! [ ψ ( m ) ( z 2 ) - ψ ( m ) ( z + 1 2 ) ] \sum_{k=0}^{\infty}\frac{(-1)^{k}}{(z+k)^{m+1}}=\frac{1}{(-2)^{m+1}m!}\left[% \psi^{(m)}\left(\frac{z}{2}\right)-\psi^{(m)}\left(\frac{z+1}{2}\right)\right]
  39. ψ ( m ) ( z + 1 ) = k = 0 ( - 1 ) m + k + 1 ( m + k ) ! k ! ζ ( m + k + 1 ) z k m 1 \psi^{(m)}(z+1)=\sum_{k=0}^{\infty}(-1)^{m+k+1}\frac{(m+k)!}{k!}\;\zeta(m+k+1)% \;z^{k}\qquad m\geq 1
  40. ψ ( 0 ) ( z + 1 ) = - γ + k = 1 ( - 1 ) k + 1 ζ ( k + 1 ) z k \psi^{(0)}(z+1)=-\gamma+\sum_{k=1}^{\infty}(-1)^{k+1}\zeta(k+1)\;z^{k}
  41. ψ ( 0 ) ( z ) ln ( z ) - k = 1 B k k z k \psi^{(0)}(z)\sim\ln(z)-\sum_{k=1}^{\infty}\frac{B_{k}}{kz^{k}}
  42. B 1 = 1 / 2 B_{1}=1/2

Polylogarithm.html

  1. Li s ( z ) = k = 1 z k k s = z + z 2 2 s + z 3 3 s + . \operatorname{Li}_{s}(z)=\sum_{k=1}^{\infty}{z^{k}\over k^{s}}=z+{z^{2}\over 2% ^{s}}+{z^{3}\over 3^{s}}+\cdots\,.
  2. Li s + 1 ( z ) = 0 z Li s ( t ) t d t ; \operatorname{Li}_{s+1}(z)=\int_{0}^{z}\frac{\operatorname{Li}_{s}(t)}{t}\,% \mathrm{d}t\,;
  3. Im ( Li s ( z ) ) = - π μ s - 1 Γ ( s ) . \textrm{Im}\left(\operatorname{Li}_{s}(z)\right)=-{{\pi\mu^{s-1}}\over{\Gamma(% s)}}\,.
  4. Im ( Li s ( z + i ϵ ) ) = π μ s - 1 Γ ( s ) . \textrm{Im}\left(\operatorname{Li}_{s}(z+i\epsilon)\right)={{\pi\mu^{s-1}}% \over{\Gamma(s)}}\,.
  5. z Li s ( z ) z = Li s - 1 ( z ) z\,{\partial\operatorname{Li}_{s}(z)\over\partial z}=\operatorname{Li}_{s-1}(z)
  6. Li s ( e μ ) μ = Li s - 1 ( e μ ) . {\partial\operatorname{Li}_{s}(e^{\mu})\over\partial\mu}=\operatorname{Li}_{s-% 1}(e^{\mu})\,.
  7. Li s ( - z ) + Li s ( z ) = 2 1 - s Li s ( z 2 ) . \operatorname{Li}_{s}(-z)+\operatorname{Li}_{s}(z)=2^{1-s}\,\operatorname{Li}_% {s}(z^{2})\,.
  8. m = 0 p - 1 Li s ( z e 2 π i m / p ) = p 1 - s Li s ( z p ) , \sum_{m=0}^{p-1}\operatorname{Li}_{s}(z\,e^{2\pi im/p})=p^{1-s}\,\operatorname% {Li}_{s}(z^{p})\,,
  9. Li 1 ( z ) = - ln ( 1 - z ) \operatorname{Li}_{1}(z)=-\ln(1-z)
  10. Li 0 ( z ) = z 1 - z \operatorname{Li}_{0}(z)={z\over 1-z}
  11. Li - 1 ( z ) = z ( 1 - z ) 2 \operatorname{Li}_{-1}(z)={z\over(1-z)^{2}}
  12. Li - 2 ( z ) = z ( 1 + z ) ( 1 - z ) 3 \operatorname{Li}_{-2}(z)={z\,(1+z)\over(1-z)^{3}}
  13. Li - 3 ( z ) = z ( 1 + 4 z + z 2 ) ( 1 - z ) 4 \operatorname{Li}_{-3}(z)={z\,(1+4z+z^{2})\over(1-z)^{4}}
  14. Li - 4 ( z ) = z ( 1 + z ) ( 1 + 10 z + z 2 ) ( 1 - z ) 5 . \operatorname{Li}_{-4}(z)={z\,(1+z)(1+10z+z^{2})\over(1-z)^{5}}\,.
  15. Li - n ( z ) = ( z z ) n z 1 - z = \operatorname{Li}_{-n}(z)=\left(z\,{\partial\over\partial z}\right)^{n}{z\over% {1-z}}=
  16. = k = 0 n k ! S ( n + 1 , k + 1 ) ( z 1 - z ) k + 1 ( n = 0 , 1 , 2 , ) , =\sum_{k=0}^{n}k!\,S(n\!+\!1,\,k\!+\!1)\left({z\over{1-z}}\right)^{k+1}\qquad(% n=0,1,2,\ldots)\,,
  17. Li - n ( z ) = ( - 1 ) n + 1 k = 0 n k ! S ( n + 1 , k + 1 ) ( - 1 1 - z ) k + 1 ( n = 1 , 2 , 3 , ) , \operatorname{Li}_{-n}(z)=(-1)^{n+1}\sum_{k=0}^{n}k!\,S(n\!+\!1,\,k\!+\!1)% \left({{-1}\over{1-z}}\right)^{k+1}\qquad(n=1,2,3,\ldots)\,,
  18. Li - n ( z ) = 1 ( 1 - z ) n + 1 k = 0 n - 1 n k z n - k ( n = 1 , 2 , 3 , ) , \operatorname{Li}_{-n}(z)={1\over(1-z)^{n+1}}\sum_{k=0}^{n-1}\left\langle{n% \atop k}\right\rangle z^{n-k}\qquad(n=1,2,3,\ldots)\,,
  19. n k \scriptstyle\left\langle{n\atop k}\right\rangle
  20. Li 1 ( 1 2 ) = ln 2 \operatorname{Li}_{1}(\tfrac{1}{2})=\ln 2
  21. Li 2 ( 1 2 ) = 1 12 π 2 - 1 2 ( ln 2 ) 2 \operatorname{Li}_{2}(\tfrac{1}{2})=\tfrac{1}{12}\pi^{2}-\tfrac{1}{2}(\ln 2)^{2}
  22. Li 3 ( 1 2 ) = 1 6 ( ln 2 ) 3 - 1 12 π 2 ln 2 + 7 8 ζ ( 3 ) , \operatorname{Li}_{3}(\tfrac{1}{2})=\tfrac{1}{6}(\ln 2)^{3}-\tfrac{1}{12}\pi^{% 2}\ln 2+\tfrac{7}{8}\,\zeta(3)\,,
  23. Li 4 ( 1 2 ) = 1 360 π 4 - 1 24 ( ln 2 ) 4 + 1 24 π 2 ( ln 2 ) 2 - 1 2 ζ ( 3 ¯ , 1 ¯ ) , \operatorname{Li}_{4}(\tfrac{1}{2})=\tfrac{1}{360}\pi^{4}-\tfrac{1}{24}(\ln 2)% ^{4}+\tfrac{1}{24}\pi^{2}(\ln 2)^{2}-\tfrac{1}{2}\,\zeta(\bar{3},\bar{1})\,,
  24. ζ ( 3 ¯ , 1 ¯ ) = m > n > 0 ( - 1 ) m + n m - 3 n - 1 \scriptstyle\zeta(\bar{3},\bar{1})~{}=\,\sum_{m>n>0}\,(-1)^{m+n}m^{-3}n^{-1}
  25. Li n ( 1 2 ) = - ζ ( 1 ¯ , 1 ¯ , { 1 } n - 2 ) , \operatorname{Li}_{n}(\tfrac{1}{2})=-\zeta(\bar{1},\bar{1},\left\{1\right\}^{n% -2})\,,
  26. Li 5 ( 1 2 ) = - ζ ( 1 ¯ , 1 ¯ , 1 , 1 , 1 ) . \operatorname{Li}_{5}(\tfrac{1}{2})=-\zeta(\bar{1},\bar{1},1,1,1)\,.
  27. Li s ( e 2 π i m / p ) = p - s k = 1 p e 2 π i m k / p ζ ( s , k p ) ( m = 1 , 2 , , p - 1 ) , \operatorname{Li}_{s}(e^{2\pi im/p})=p^{-s}\sum_{k=1}^{p}e^{2\pi imk/p}\,\zeta% (s,\tfrac{k}{p})\qquad(m=1,2,\dots,p-1)\,,
  28. Li s ( 1 ) = ζ ( s ) ( Re ( s ) > 1 ) . \operatorname{Li}_{s}(1)=\zeta(s)\qquad(\textrm{Re}(s)>1)\,.
  29. Li s ( - 1 ) = - η ( s ) , \operatorname{Li}_{s}(-1)=-\eta(s)\,,
  30. Li s ( ± i ) = - 2 - s η ( s ) ± i β ( s ) , \operatorname{Li}_{s}(\pm i)=-2^{-s}\,\eta(s)\pm i\,\beta(s)\,,
  31. F s ( μ ) = - Li s + 1 ( - e μ ) . F_{s}(\mu)=-\operatorname{Li}_{s+1}(-e^{\mu})\,.
  32. Li s ( z ) = Li s ( 0 , z ) . \operatorname{Li}_{s}(z)=\operatorname{Li}_{s}(0,z)\,.
  33. Li s ( z ) = z Φ ( z , s , 1 ) . \operatorname{Li}_{s}(z)=z\,\Phi(z,s,1)\,.
  34. Li s ( z ) = Γ ( 1 - s ) ( 2 π ) 1 - s [ i 1 - s ζ ( 1 - s , 1 2 + ln ( - z ) 2 π i ) + i s - 1 ζ ( 1 - s , 1 2 - ln ( - z ) 2 π i ) ] , \operatorname{Li}_{s}(z)={\Gamma(1\!-\!s)\over(2\pi)^{1-s}}\left[i^{1-s}~{}% \zeta\!\left(1\!-\!s,~{}\frac{1}{2}+{\ln(-z)\over{2\pi i}}\right)+i^{s-1}~{}% \zeta\!\left(1\!-\!s,~{}\frac{1}{2}-{\ln(-z)\over{2\pi i}}\right)\right],
  35. i - s Li s ( e 2 π i x ) + i s Li s ( e - 2 π i x ) = ( 2 π ) s Γ ( s ) ζ ( 1 - s , x ) , i^{-s}\,\operatorname{Li}_{s}(e^{2\pi ix})+i^{s}\,\operatorname{Li}_{s}(e^{-2% \pi ix})={(2\pi)^{s}\over\Gamma(s)}\,\zeta(1\!-\!s,\,x)\,,
  36. Li s ( z ) + ( - 1 ) s Li s ( 1 / z ) = ( 2 π i ) s Γ ( s ) ζ ( 1 - s , 1 2 + ln ( - z ) 2 π i ) , \operatorname{Li}_{s}(z)+(-1)^{s}\,\operatorname{Li}_{s}(1/z)={(2\pi i)^{s}% \over\Gamma(s)}~{}\zeta\!\left(1\!-\!s,~{}\frac{1}{2}+{\ln(-z)\over{2\pi i}}% \right),
  37. Li s ( z ) + ( - 1 ) s Li s ( 1 / z ) = ( 2 π i ) s Γ ( s ) ζ ( 1 - s , 1 2 - ln ( - 1 / z ) 2 π i ) . \operatorname{Li}_{s}(z)+(-1)^{s}\,\operatorname{Li}_{s}(1/z)={(2\pi i)^{s}% \over\Gamma(s)}~{}\zeta\!\left(1\!-\!s,~{}\frac{1}{2}-{\ln(-1/z)\over{2\pi i}}% \right).
  38. Li n ( e 2 π i x ) + ( - 1 ) n Li n ( e - 2 π i x ) = - ( 2 π i ) n n ! B n ( x ) , \operatorname{Li}_{n}(e^{2\pi ix})+(-1)^{n}\,\operatorname{Li}_{n}(e^{-2\pi ix% })=-{(2\pi i)^{n}\over n!}\,B_{n}(x)\,,
  39. Li - n ( z ) + ( - 1 ) n Li - n ( 1 / z ) = 0 ( n = 1 , 2 , 3 , ) . \operatorname{Li}_{-n}(z)+(-1)^{n}\,\operatorname{Li}_{-n}(1/z)=0\qquad(n=1,2,% 3,\ldots)\,.
  40. Li n ( z ) + ( - 1 ) n Li n ( 1 / z ) = - ( 2 π i ) n n ! B n ( 1 2 + ln ( - z ) 2 π i ) ( z ] 0 ; 1 ] ) , \operatorname{Li}_{n}(z)+(-1)^{n}\,\operatorname{Li}_{n}(1/z)=-\frac{(2\pi i)^% {n}}{n!}~{}B_{n}\!\left(\frac{1}{2}+{\ln(-z)\over{2\pi i}}\right)\qquad(z~{}% \not\in~{}]0;1])\,,
  41. Li n ( z ) + ( - 1 ) n Li n ( 1 / z ) = - ( 2 π i ) n n ! B n ( 1 2 - ln ( - 1 / z ) 2 π i ) ( z ] 1 ; [ ) , \operatorname{Li}_{n}(z)+(-1)^{n}\,\operatorname{Li}_{n}(1/z)=-\frac{(2\pi i)^% {n}}{n!}~{}B_{n}\!\left(\frac{1}{2}-{\ln(-1/z)\over{2\pi i}}\right)\qquad(z~{}% \not\in~{}]1;\infty[)\,,
  42. Li s ( e ± i θ ) = C i s ( θ ) ± i S i s ( θ ) . \operatorname{Li}_{s}(e^{\pm i\theta})=Ci_{s}(\theta)\pm i\,Si_{s}(\theta)\,.
  43. T i s ( z ) = 1 2 i [ Li s ( i z ) - Li s ( - i z ) ] . Ti_{s}(z)={1\over 2i}\left[\operatorname{Li}_{s}(iz)-\operatorname{Li}_{s}(-iz% )\right].
  44. T i 0 ( z ) = z 1 + z 2 , T i 1 ( z ) = arctan z , T i 2 ( z ) = 0 z arctan t t d t , Ti_{0}(z)={z\over 1+z^{2}},\quad Ti_{1}(z)=\arctan z,\quad Ti_{2}(z)=\int_{0}^% {z}{\arctan t\over t}\,\mathrm{d}t,
  45. , T i n + 1 ( z ) = 0 z T i n ( t ) t d t , \quad\ldots~{},\quad Ti_{n+1}(z)=\int_{0}^{z}{Ti_{n}(t)\over t}\,\mathrm{d}t\,,
  46. χ s ( z ) = 1 2 [ Li s ( z ) - Li s ( - z ) ] . \chi_{s}(z)=\tfrac{1}{2}\left[\operatorname{Li}_{s}(z)-\operatorname{Li}_{s}(-% z)\right].
  47. Li n ( z ) = z n + 1 F n ( 1 , 1 , , 1 ; 2 , 2 , , 2 ; z ) ( n = 0 , 1 , 2 , ) , \operatorname{Li}_{n}(z)=z\;_{n+1}F_{n}(1,1,\dots,1;\,2,2,\dots,2;\,z)\qquad(n% =0,1,2,\ldots)~{},
  48. Li - n ( z ) = z n F n - 1 ( 2 , 2 , , 2 ; 1 , 1 , , 1 ; z ) ( n = 1 , 2 , 3 , ) . \operatorname{Li}_{-n}(z)=z\;_{n}F_{n-1}(2,2,\dots,2;\,1,1,\dots,1;\,z)\qquad(% n=1,2,3,\ldots)~{}.
  49. Z n ( z ) = 1 ( n - 1 ) ! z t n - 1 e t - 1 d t ( n = 1 , 2 , 3 , ) , Z_{n}(z)={1\over(n\!-\!1)!}\int_{z}^{\infty}{t^{n-1}\over e^{t}-1}\,\mathrm{d}% t\qquad(n=1,2,3,\ldots)\,,
  50. Li n ( e μ ) = k = 0 n - 1 Z n - k ( - μ ) μ k k ! ( n = 1 , 2 , 3 , ) . \operatorname{Li}_{n}(e^{\mu})=\sum_{k=0}^{n-1}Z_{n-k}(-\mu)\,{\mu^{k}\over k!% }\qquad(n=1,2,3,\ldots)\,.
  51. Z n ( z ) = k = 0 n - 1 Li n - k ( e - z ) z k k ! ( n = 1 , 2 , 3 , ) . Z_{n}(z)=\sum_{k=0}^{n-1}\operatorname{Li}_{n-k}(e^{-z})\,{z^{k}\over k!}% \qquad(n=1,2,3,\ldots)\,.
  52. Li s ( z ) = 1 Γ ( s ) 0 t s - 1 e t / z - 1 d t . \operatorname{Li}_{s}(z)={1\over\Gamma(s)}\int_{0}^{\infty}{t^{s-1}\over e^{t}% /z-1}\,\mathrm{d}t\,.
  53. - Li s ( - z ) = 1 Γ ( s ) 0 t s - 1 e t / z + 1 d t . -\operatorname{Li}_{s}(-z)={1\over\Gamma(s)}\int_{0}^{\infty}{t^{s-1}\over e^{% t}/z+1}\,\mathrm{d}t\,.
  54. Li s ( e μ ) = - Γ ( 1 - s ) 2 π i H ( - t ) s - 1 e t - μ - 1 d t \operatorname{Li}_{s}(e^{\mu})=-{{\Gamma(1\!-\!s)}\over{2\pi i}}\oint_{H}{{(-t% )^{s-1}}\over{e^{t-\mu}-1}}\,\mathrm{d}t
  55. R = i 2 π Γ ( 1 - s ) ( - μ ) s - 1 . R={i\over 2\pi}\Gamma(1\!-\!s)\,(-\mu)^{s-1}\,.
  56. Li s ( z ) = 1 2 z + Γ ( 1 - s , - ln z ) ( - ln z ) 1 - s + 2 z 0 sin ( s arctan t - t ln z ) ( 1 + t 2 ) s / 2 ( e 2 π t - 1 ) d t \operatorname{Li}_{s}(z)=\tfrac{1}{2}z+{\Gamma(1\!-\!s,-\ln z)\over(-\ln z)^{1% -s}}+2z\int_{0}^{\infty}\frac{\sin(s\arctan t\,-\,t\ln z)}{(1+t^{2})^{s/2}\,(e% ^{2\pi t}-1)}\,\mathrm{d}t
  57. Li s ( z ) = 1 2 z + z 0 sin [ s arctan t - t ln ( - z ) ] ( 1 + t 2 ) s / 2 sinh ( π t ) d t , \operatorname{Li}_{s}(z)=\tfrac{1}{2}z+z\int_{0}^{\infty}\frac{\sin[s\arctan t% \,-\,t\ln(-z)]}{(1+t^{2})^{s/2}\,\sinh(\pi t)}\,\mathrm{d}t\,,
  58. Li s ( e μ ) = - Γ ( 1 - s ) 2 π i H ( - t ) s - 1 e t - μ - 1 d t , \operatorname{Li}_{s}(e^{\mu})=-{\Gamma(1\!-\!s)\over 2\pi i}\oint_{H}{(-t)^{s% -1}\over e^{t-\mu}-1}\,\mathrm{d}t\,,
  59. Li s ( e μ ) = Γ ( 1 - s ) k = - ( 2 k π i - μ ) s - 1 . \operatorname{Li}_{s}(e^{\mu})=\Gamma(1\!-\!s)\sum_{k=-\infty}^{\infty}(2k\pi i% -\mu)^{s-1}\,.
  60. Li s ( e μ ) = Γ ( 1 - s ) ( 2 π ) 1 - s [ i 1 - s ζ ( 1 - s , μ 2 π i ) + i s - 1 ζ ( 1 - s , 1 - μ 2 π i ) ] ( 0 < Im ( μ ) 2 π ) . \operatorname{Li}_{s}(e^{\mu})={\Gamma(1\!-\!s)\over(2\pi)^{1-s}}\left[i^{1-s}% ~{}\zeta\!\left(1\!-\!s,~{}{\mu\over{2\pi i}}\right)+i^{s-1}~{}\zeta\!\left(1% \!-\!s,~{}1-{\mu\over{2\pi i}}\right)\right]\qquad(0<\textrm{Im}(\mu)\leq 2\pi% )\,.
  61. Li s ( e μ ) = Γ ( 1 - s ) ( - μ ) s - 1 + Γ ( 1 - s ) h = 1 [ ( - 2 h π i - μ ) s - 1 + ( 2 h π i - μ ) s - 1 ] . \operatorname{Li}_{s}(e^{\mu})=\Gamma(1\!-\!s)\,(-\mu)^{s-1}+\Gamma(1\!-\!s)% \sum_{h=1}^{\infty}\left[(-2h\pi i-\mu)^{s-1}+(2h\pi i-\mu)^{s-1}\right].
  62. Li s ( e μ ) = Γ ( 1 - s ) ( - μ ) s - 1 + k = 0 ζ ( s - k ) k ! μ k . \operatorname{Li}_{s}(e^{\mu})=\Gamma(1\!-\!s)\,(-\mu)^{s-1}+\sum_{k=0}^{% \infty}{\zeta(s-k)\over k!}\,\mu^{k}\,.
  63. lim s k + 1 [ ζ ( s - k ) k ! μ k + Γ ( 1 - s ) ( - μ ) s - 1 ] = μ k k ! [ h = 1 k 1 h - ln ( - μ ) ] , \lim_{s\rightarrow k+1}\left[{\zeta(s-k)\over k!}\,\mu^{k}+\Gamma(1\!-\!s)\,(-% \mu)^{s-1}\right]={\mu^{k}\over k!}\left[\,\sum_{h=1}^{k}{1\over h}-\ln(-\mu)% \right],
  64. H n = h = 1 n 1 h , H 0 = 0 . H_{n}=\sum_{h=1}^{n}{1\over h},\qquad H_{0}=0\,.
  65. lim μ 0 Γ ( 1 - s ) ( - μ ) s - 1 = 0 ( Re ( s ) > 1 ) . \lim_{\mu\rightarrow 0}\Gamma(1\!-\!s)\,(-\mu)^{s-1}=0\qquad(\textrm{Re}(s)>1)\,.
  66. 1 = 1 Γ ( s ) 0 e - t t s - 1 d t ( Re ( s ) > 0 ) , 1={1\over\Gamma(s)}\int_{0}^{\infty}e^{-t}\,t^{s-1}\,\mathrm{d}t\qquad(\textrm% {Re}(s)>0)\,,
  67. Li s ( z ) = 1 2 z + z 2 Γ ( s ) 0 e - t t s - 1 coth t - ln z 2 d t ( Re ( s ) > 0 ) . \operatorname{Li}_{s}(z)=\tfrac{1}{2}z+{z\over 2\Gamma(s)}\int_{0}^{\infty}e^{% -t}\,t^{s-1}\coth{t-\ln z\over 2}\,\mathrm{d}t\qquad(\textrm{Re}(s)>0)\,.
  68. coth t - ln z 2 = 2 k = - 1 2 k π i + t - ln z , \coth{t-\ln z\over 2}=2\sum_{k=-\infty}^{\infty}{1\over 2k\pi i+t-\ln z}\,,
  69. Li s ( z ) = 1 2 z + k = - Γ ( 1 - s , 2 k π i - ln z ) ( 2 k π i - ln z ) 1 - s . \operatorname{Li}_{s}(z)=\tfrac{1}{2}z+\sum_{k=-\infty}^{\infty}{\Gamma(1\!-\!% s,\,2k\pi i-\ln z)\over(2k\pi i-\ln z)^{1-s}}\,.
  70. Li - n ( z ) = k = 0 n ( - z 1 - z ) k + 1 j = 0 k ( - 1 ) j + 1 ( k j ) ( j + 1 ) n ( n = 0 , 1 , 2 , ) . \operatorname{Li}_{-n}(z)=\sum_{k=0}^{n}\left({-z\over 1-z}\right)^{k+1}~{}% \sum_{j=0}^{k}(-1)^{j+1}{k\choose j}(j+1)^{n}\qquad(n=0,1,2,\ldots)\,.
  71. Li s ( z ) = k = 0 ( - z 1 - z ) k + 1 j = 0 k ( - 1 ) j + 1 ( k j ) ( j + 1 ) - s , \operatorname{Li}_{s}(z)=\sum_{k=0}^{\infty}\left({-z\over 1-z}\right)^{k+1}~{% }\sum_{j=0}^{k}(-1)^{j+1}{k\choose j}(j+1)^{-s}\,,
  72. k = j ( k j ) ( - z 1 - z ) k + 1 = [ ( - z 1 - z ) - 1 - 1 ] - j - 1 = ( - z ) j + 1 . \sum_{k=j}^{\infty}{k\choose j}\left({-z\over 1-z}\right)^{k+1}=\left[\left({-% z\over 1-z}\right)^{-1}-1\right]^{-j-1}=(-z)^{j+1}\,.
  73. Li s ( z ) = ± i π Γ ( s ) [ ln ( - z ) ± i π ] s - 1 - k = 0 ( - 1 ) k ( 2 π ) 2 k B 2 k ( 2 k ) ! [ ln ( - z ) ± i π ] s - 2 k Γ ( s + 1 - 2 k ) , \operatorname{Li}_{s}(z)={\pm i\pi\over\Gamma(s)}\,[\ln(-z)\pm i\pi]^{s-1}-% \sum_{k=0}^{\infty}(-1)^{k}\,(2\pi)^{2k}\,{B_{2k}\over(2k)!}\,{[\ln(-z)\pm i% \pi]^{s-2k}\over\Gamma(s+1-2k)}~{},
  74. Li s ( z ) = k = 0 ( - 1 ) k ( 1 - 2 1 - 2 k ) ( 2 π ) 2 k B 2 k ( 2 k ) ! [ ln ( - z ) ] s - 2 k Γ ( s + 1 - 2 k ) , \operatorname{Li}_{s}(z)=\sum_{k=0}^{\infty}(-1)^{k}\,(1-2^{1-2k})\,(2\pi)^{2k% }\,{B_{2k}\over(2k)!}\,{[\ln(-z)]^{s-2k}\over\Gamma(s+1-2k)}~{},
  75. lim | z | 0 Li s ( z ) = z \lim_{|z|\rightarrow 0}\operatorname{Li}_{s}(z)=z
  76. lim | μ | 0 Li s ( e μ ) = Γ ( 1 - s ) ( - μ ) s - 1 ( Re ( s ) < 1 ) \lim_{|\mu|\rightarrow 0}\operatorname{Li}_{s}(e^{\mu})=\Gamma(1\!-\!s)\,(-\mu% )^{s-1}\qquad(\mathrm{Re}(s)<1)
  77. lim Re ( μ ) Li s ( - e μ ) = - μ s Γ ( s + 1 ) ( s - 1 , - 2 , - 3 , ) \lim_{\mathrm{Re}(\mu)\rightarrow\infty}\operatorname{Li}_{s}(-e^{\mu})=-{\mu^% {s}\over\Gamma(s+1)}\qquad(s\neq-1,-2,-3,\ldots)
  78. lim Re ( μ ) Li - n ( e μ ) = - ( - 1 ) n e - μ ( n = 1 , 2 , 3 , ) \lim_{\mathrm{Re}(\mu)\rightarrow\infty}\operatorname{Li}_{-n}(e^{\mu})=-(-1)^% {n}\,e^{-\mu}\qquad(n=1,2,3,\ldots)
  79. lim Re ( s ) Li s ( z ) = z \lim_{\mathrm{Re}(s)\rightarrow\infty}\operatorname{Li}_{s}(z)=z
  80. lim Re ( s ) - Li s ( e μ ) = Γ ( 1 - s ) ( - μ ) s - 1 ( - π < Im ( μ ) < π ) \lim_{\mathrm{Re}(s)\rightarrow-\infty}\operatorname{Li}_{s}(e^{\mu})=\Gamma(1% \!-\!s)\,(-\mu)^{s-1}\qquad(-\pi<\mathrm{Im}(\mu)<\pi)
  81. lim Re ( s ) - Li s ( - e μ ) = Γ ( 1 - s ) [ ( - μ - i π ) s - 1 + ( - μ + i π ) s - 1 ] ( Im ( μ ) = 0 ) \lim_{\mathrm{Re}(s)\rightarrow-\infty}\operatorname{Li}_{s}(-e^{\mu})=\Gamma(% 1\!-\!s)\left[(-\mu-i\pi)^{s-1}+(-\mu+i\pi)^{s-1}\right]\qquad(\mathrm{Im}(\mu% )=0)
  82. Li 2 ( z ) = - 0 z ln ( 1 - t ) t d t = - 0 1 ln ( 1 - z t ) t d t . \operatorname{Li}_{2}(z)=-\int_{0}^{z}{\ln(1-t)\over t}\,\mathrm{d}t=-\int_{0}% ^{1}{\ln(1-zt)\over t}\,\mathrm{d}t.
  83. Li 2 ( z ) = π 2 6 - 1 z ln ( t - 1 ) t d t - i π ln z \operatorname{Li}_{2}(z)=\frac{\pi^{2}}{6}-\int_{1}^{z}{\ln(t-1)\over t}\,% \mathrm{d}t-i\pi\ln z
  84. Li 2 ( z ) = π 2 3 - 1 2 ( ln z ) 2 - k = 1 1 k 2 z k - i π ln z ( z 1 ) . \operatorname{Li}_{2}(z)=\frac{\pi^{2}}{3}-\frac{1}{2}(\ln z)^{2}-\sum_{k=1}^{% \infty}{1\over k^{2}z^{k}}-i\pi\ln z\qquad(z\geq 1)\,.
  85. Li 2 ( x 1 - y ) + Li 2 ( y 1 - x ) - Li 2 ( x y ( 1 - x ) ( 1 - y ) ) = Li 2 ( x ) + Li 2 ( y ) + ln ( 1 - x ) ln ( 1 - y ) \operatorname{Li}_{2}\left(\frac{x}{1-y}\right)+\operatorname{Li}_{2}\left(% \frac{y}{1-x}\right)-\operatorname{Li}_{2}\left(\frac{xy}{(1-x)(1-y)}\right)=% \operatorname{Li}_{2}(x)+\operatorname{Li}_{2}(y)+\ln(1-x)\ln(1-y)
  86. ( Re ( x ) 1 2 Re ( y ) 1 2 Im ( x ) > 0 Im ( y ) > 0 Im ( x ) < 0 Im ( y ) < 0 ) . (\mathrm{Re}(x)\leq\tfrac{1}{2}\;\wedge\;\mathrm{Re}(y)\leq\tfrac{1}{2}\;\vee% \;\mathrm{Im}(x)>0\;\wedge\;\mathrm{Im}(y)>0\;\vee\;\mathrm{Im}(x)<0\;\wedge\;% \mathrm{Im}(y)<0\;\vee\;\ldots)\,.
  87. Li 2 ( x ) + Li 2 ( 1 - x ) = 1 6 π 2 - ln ( x ) ln ( 1 - x ) , \operatorname{Li}_{2}\left(x\right)+\operatorname{Li}_{2}\left(1-x\right)=% \frac{1}{6}\pi^{2}-\ln(x)\ln(1-x)\,,
  88. Li 2 ( u ) + Li 2 ( v ) - Li 2 ( u v ) = Li 2 ( u - u v 1 - u v ) + Li 2 ( v - u v 1 - u v ) + ln ( 1 - u 1 - u v ) ln ( 1 - v 1 - u v ) , \operatorname{Li}_{2}(u)+\operatorname{Li}_{2}(v)-\operatorname{Li}_{2}(uv)=% \operatorname{Li}_{2}\left(\frac{u-uv}{1-uv}\right)+\operatorname{Li}_{2}\left% (\frac{v-uv}{1-uv}\right)+\ln\left(\frac{1-u}{1-uv}\right)\ln\left(\frac{1-v}{% 1-uv}\right),
  89. Li 2 ( 1 - z ) + Li 2 ( 1 - 1 z ) = - 1 2 ( ln z ) 2 ( z ] - ; 0 ] ) , \operatorname{Li}_{2}(1-z)+\operatorname{Li}_{2}\left(1-\frac{1}{z}\right)=-% \frac{1}{2}(\ln z)^{2}\qquad(z\not\in~{}]-\infty;0])\,,
  90. Li 2 ( z ) + Li 2 ( 1 / z ) = - 1 6 π 2 - 1 2 [ ln ( - z ) ] 2 ( z [ 0 ; 1 [ ) , \operatorname{Li}_{2}(z)+\operatorname{Li}_{2}(1/z)=-\tfrac{1}{6}\pi^{2}-% \tfrac{1}{2}[\ln(-z)]^{2}\qquad(z\not\in[0;1[)\,,
  91. Li 2 ( z ) + Li 2 ( 1 / z ) = 1 3 π 2 - 1 2 ( ln z ) 2 - i π ln z . \operatorname{Li}_{2}(z)+\operatorname{Li}_{2}(1/z)=\tfrac{1}{3}\pi^{2}-\tfrac% {1}{2}(\ln z)^{2}-i\pi\ln z\,.
  92. x x\,
  93. Li 2 ( x ) \operatorname{Li}_{2}(x)\,
  94. x x\,
  95. Li 2 ( x ) \operatorname{Li}_{2}(x)\,
  96. - 1 -1\,
  97. - 1 12 π 2 -\tfrac{1}{12}\pi^{2}\,
  98. - ϕ -\phi\,
  99. - 1 10 π 2 - ln 2 ϕ -\tfrac{1}{10}\pi^{2}-\ln^{2}\phi\,
  100. 0 0\,
  101. 0 0\,
  102. - 1 / ϕ -1/\phi\,
  103. - 1 15 π 2 + 1 2 ln 2 ϕ -\tfrac{1}{15}\pi^{2}+\tfrac{1}{2}\ln^{2}\phi\,
  104. 1 2 \tfrac{1}{2}\,
  105. 1 12 π 2 - 1 2 ln 2 2 \tfrac{1}{12}\pi^{2}-\tfrac{1}{2}\ln^{2}2\,
  106. 1 / ϕ 2 1/\phi^{2}\,
  107. 1 15 π 2 - ln 2 ϕ \tfrac{1}{15}\pi^{2}-\ln^{2}\phi\,
  108. 1 1\,
  109. 1 6 π 2 \tfrac{1}{6}\pi^{2}\,
  110. 1 / ϕ 1/\phi\,
  111. 1 10 π 2 - ln 2 ϕ \tfrac{1}{10}\pi^{2}-\ln^{2}\phi\,
  112. 2 2\,
  113. 1 4 π 2 - π i ln 2 \tfrac{1}{4}\pi^{2}-\pi i\ln 2\,
  114. ϕ \phi\,
  115. 11 15 π 2 + 1 2 ln 2 ( - 1 / ϕ ) \tfrac{11}{15}\pi^{2}+\tfrac{1}{2}\ln^{2}(-1/\phi)\,
  116. ϕ 2 \phi^{2}\,
  117. - 11 15 π 2 - ln 2 ( - ϕ ) -\tfrac{11}{15}\pi^{2}-\ln^{2}(-\phi)\,
  118. ϕ = 1 2 ( 5 + 1 ) \scriptstyle\phi\,=\,\tfrac{1}{2}(\sqrt{5}+1)
  119. ρ = 1 2 ( 5 - 1 ) \scriptstyle\rho\,=\,\tfrac{1}{2}(\sqrt{5}-1)
  120. Li 2 ( ρ 6 ) = 4 Li 2 ( ρ 3 ) + 3 Li 2 ( ρ 2 ) - 6 Li 2 ( ρ ) + 7 30 π 2 \operatorname{Li}_{2}(\rho^{6})=4\operatorname{Li}_{2}(\rho^{3})+3% \operatorname{Li}_{2}(\rho^{2})-6\operatorname{Li}_{2}(\rho)+\tfrac{7}{30}\pi^% {2}
  121. Li 2 ( ρ ) = 1 10 π 2 - ln 2 ρ \operatorname{Li}_{2}(\rho)=\tfrac{1}{10}\pi^{2}-\ln^{2}\rho
  122. m 0 , m 1 | w = m 0 m 1 m 0 - 1 m 1 - 1 , w m 1 = m 1 w . \langle m_{0},m_{1}\,|\,w=m_{0}m_{1}m^{-1}_{0}m^{-1}_{1},\,wm_{1}=m_{1}w\rangle.

Polylogarithmic_function.html

  1. a k log k ( n ) + + a 1 log ( n ) + a 0 . a_{k}\log^{k}(n)+\cdots+a_{1}\log(n)+a_{0}.\,
  2. P ( x ) = o ( x ε ) P_{\ell}(x)=o(x^{\varepsilon})\,

Polynomial_ring.html

  1. p = p 0 + p 1 X + p 2 X 2 + + p m - 1 X m - 1 + p m X m , p=p_{0}+p_{1}X+p_{2}X^{2}+\cdots+p_{m-1}X^{m-1}+p_{m}X^{m},
  2. X k X l = X k + l , X^{k}\,X^{l}=X^{k+l},
  3. p = p 0 + p 1 X + p 2 X 2 + + p m X m , p=p_{0}+p_{1}X+p_{2}X^{2}+\cdots+p_{m}X^{m},
  4. q = q 0 + q 1 X + q 2 X 2 + + q n X n , q=q_{0}+q_{1}X+q_{2}X^{2}+\cdots+q_{n}X^{n},
  5. p + q = r 0 + r 1 X + r 2 X 2 + + r k X k , p+q=r_{0}+r_{1}X+r_{2}X^{2}+\cdots+r_{k}X^{k},
  6. p q = s 0 + s 1 X + s 2 X 2 + + s l X l , pq=s_{0}+s_{1}X+s_{2}X^{2}+\cdots+s_{l}X^{l},
  7. r i = p i + q i r_{i}=p_{i}+q_{i}
  8. s i = p 0 q i + p 1 q i - 1 + + p i q 0 . s_{i}=p_{0}q_{i}+p_{1}q_{i-1}+\cdots+p_{i}q_{0}.
  9. p 0 + p 1 X + p 2 X 2 + + p m X m p_{0}+p_{1}X+p_{2}X^{2}+\cdots+p_{m}X^{m}
  10. deg ( p q ) = deg ( p ) + deg ( q ) . \operatorname{deg}(pq)=\operatorname{deg}(p)+\operatorname{deg}(q).
  11. p = u q + r , p=uq+r,
  12. φ ( a m X m + a m - 1 X m - 1 + + a 1 X + a 0 ) = a m θ m + a m - 1 θ m - 1 + + a 1 θ + a 0 . \varphi(a_{m}X^{m}+a_{m-1}X^{m-1}+\cdots+a_{1}X+a_{0})=a_{m}\theta^{m}+a_{m-1}% \theta^{m-1}+\cdots+a_{1}\theta+a_{0}.
  13. L K [ X ] / ( p ) . L\simeq K[X]/(p).
  14. [ X ] / ( X 2 + 1 ) . \mathbb{C}\simeq\mathbb{R}[X]/(X^{2}+1).
  15. ϕ : K [ X ] A , ϕ ( X ) = a . \phi:K[X]\to A,\quad\phi(X)=a.
  16. K [ X ] / P k K[X]/\langle P^{k}\rangle
  17. P = X 2 - 1 , P=X^{2}-1,
  18. P ( 3 ) = 3 2 - 1 = 8 , P(3)=3^{2}-1=8,
  19. P ( X 2 + 1 ) = ( X 2 + 1 ) 2 - 1 = X 4 + 2 X 2 P(X^{2}+1)=(X^{2}+1)^{2}-1=X^{4}+2X^{2}
  20. P = P ( X ) , P=P(X),
  21. P P ( a ) P\mapsto P(a)
  22. x P ( x ) . x\mapsto P(x).
  23. X α = i = 1 n X i α i = X 1 α 1 X n α n . X^{\alpha}=\prod_{i=1}^{n}X_{i}^{\alpha_{i}}=X_{1}^{\alpha_{1}}\ldots X_{n}^{% \alpha_{n}}.
  24. p = α p α X α , p=\sum_{\alpha}p_{\alpha}X^{\alpha},
  25. p α = p α 1 , , α n K , p_{\alpha}=p_{\alpha_{1},\ldots,\alpha_{n}}\in{K},
  26. | α | = i = 1 n α i , |\alpha|=\sum_{i=1}^{n}\alpha_{i},
  27. \mathbb{Z}
  28. m = ( X 1 - a 1 , , X n - a n ) , a = ( a 1 , , a n ) K n . m=(X_{1}-a_{1},\ldots,X_{n}-a_{n}),\quad a=(a_{1},\ldots,a_{n})\in{K}^{n}.
  29. f m I , for some m . f^{m}\in I,\,\text{ for some }m\in\mathbb{N}.\,
  30. I V ( I ) , I K [ X 1 , , X n ] , V ( I ) K n . I\mapsto V(I),\quad I\subset K[X_{1},\ldots,X_{n}],\quad V(I)\subset K^{n}.
  31. gl dim R [ X 1 , , X n ] = gl dim R + n . \operatorname{gl}\,\dim R[X_{1},\ldots,X_{n}]=\operatorname{gl}\,\dim R+n.
  32. n N a n X n \sum_{n\in N}a_{n}X^{n}
  33. ( n N a n X n ) + ( n N b n X n ) = n N ( a n + b n ) X n \left(\sum_{n\in N}a_{n}X^{n}\right)+\left(\sum_{n\in N}b_{n}X^{n}\right)=\sum% _{n\in N}(a_{n}+b_{n})X^{n}
  34. ( n N a n X n ) ( n N b n X n ) = n N ( i + j = n a i b j ) X n \left(\sum_{n\in N}a_{n}X^{n}\right)\cdot\left(\sum_{n\in N}b_{n}X^{n}\right)=% \sum_{n\in N}\left(\sum_{i+j=n}a_{i}b_{j}\right)X^{n}
  35. X a = a X + δ ( a ) . X\cdot a=a\cdot X+\delta(a).
  36. Y \tfrac{\partial}{\partial Y}

Polynomially_reflexive_space.html

  1. p ( x ) = M n ( x , , x ) p(x)=M_{n}(x,\dots,x)
  2. p \ell^{p}
  3. n n
  4. p \ell^{p}
  5. \ell^{\infty}
  6. p \ell^{p}

Pontryagin_class.html

  1. p ( E ) = 1 + p 1 ( E ) + p 2 ( E ) + H * ( M , 𝐙 ) , p(E)=1+p_{1}(E)+p_{2}(E)+\cdots\in H^{*}(M,\mathbf{Z}),
  2. 2 p ( E F ) = 2 p ( E ) p ( F ) 2p(E\oplus F)=2p(E)\smile p(F)
  3. 2 p 1 ( E F ) = 2 p 1 ( E ) + 2 p 1 ( F ) , 2p_{1}(E\oplus F)=2p_{1}(E)+2p_{1}(F),
  4. 2 p 2 ( E F ) = 2 p 2 ( E ) + 2 p 1 ( E ) p 1 ( F ) + 2 p 2 ( F ) 2p_{2}(E\oplus F)=2p_{2}(E)+2p_{1}(E)\smile p_{1}(F)+2p_{2}(F)
  5. p k ( E ) = e ( E ) e ( E ) , p_{k}(E)=e(E)\smile e(E),
  6. \smile
  7. p k ( E , 𝐐 ) H 4 k ( M , 𝐐 ) p_{k}(E,\mathbf{Q})\in H^{4k}(M,\mathbf{Q})
  8. p = [ 1 - Tr ( Ω 2 ) 8 π 2 + Tr ( Ω 2 ) 2 - 2 T r ( Ω 4 ) 128 π 4 - Tr ( Ω 2 ) 3 - 6 T r ( Ω 2 ) Tr ( Ω 4 ) + 8 T r ( Ω 6 ) 3072 π 6 + ] H d R * ( M ) , p=\left[1-\frac{{\rm Tr}(\Omega^{2})}{8\pi^{2}}+\frac{{\rm Tr}(\Omega^{2})^{2}% -2{\rm Tr}(\Omega^{4})}{128\pi^{4}}-\frac{{\rm Tr}(\Omega^{2})^{3}-6{\rm Tr}(% \Omega^{2}){\rm Tr}(\Omega^{4})+8{\rm Tr}(\Omega^{6})}{3072\pi^{6}}+\cdots% \right]\in H^{*}_{dR}(M),
  9. P k 1 , k 2 , , k m P_{k_{1},k_{2},\dots,k_{m}}
  10. P k 1 , k 2 , , k m = p k 1 p k 2 p k m ( [ M ] ) P_{k_{1},k_{2},\dots,k_{m}}=p_{k_{1}}\smile p_{k_{2}}\smile\cdots\smile p_{k_{% m}}([M])
  11. A ^ \hat{A}

Pontryagin_duality.html

  1. x { χ χ ( x ) } i.e. x ( χ ) := χ ( x ) . x\mapsto\{\chi\mapsto\chi(x)\}\mbox{ i.e. }~{}x(\chi):=\chi(x).
  2. L μ p ( G ) = { f : G 𝐂 | G | f ( x ) | p d μ ( x ) < } . L^{p}_{\mu}(G)=\left\{f:G\rightarrow\mathbf{C}\;\left|\;\int_{G}|f(x)|^{p}\,d% \mu(x)<\infty\right.\right\}.
  3. f f
  4. f ^ \widehat{f}
  5. f ^ ( χ ) = G f ( x ) χ ( x ) ¯ d μ ( x ) , \widehat{f}(\chi)=\int_{G}f(x)\overline{\chi(x)}\;d\mu(x),
  6. ( f ) ( χ ) ({\mathcal{F}}f)(\chi)
  7. f ^ \widehat{f}
  8. f ( x ) = G ^ f ^ ( χ ) χ ( x ) d ν ( χ ) f(x)=\int_{\widehat{G}}\widehat{f}(\chi)\chi(x)\;d\nu(\chi)
  9. g ˇ ( x ) = G ^ g ( χ ) χ ( x ) d ν ( χ ) , \check{g}(x)=\int_{\widehat{G}}g(\chi)\chi(x)\;d\nu(\chi),
  10. μ ^ \widehat{\mu}
  11. ( 𝐯 , 𝐰 ) e i 𝐯 𝐰 ({\mathbf{v}},{\mathbf{w}})\mapsto e^{i{\mathbf{v}}\cdot{\mathbf{w}}}
  12. μ ^ = 1 / ( 2 π ) n μ \widehat{\mu}=1/(2\pi)^{n}\mu
  13. μ ^ \widehat{\mu}
  14. μ = ( 2 π ) - n / 2 × Lebesgue measure \mu=(2\pi)^{-n/2}\times\mbox{Lebesgue measure}~{}
  15. μ ^ = ( 2 π ) - n / 2 × Lebesgue measure . \widehat{\mu}=(2\pi)^{-n/2}\times\mbox{Lebesgue measure}~{}.
  16. ( 𝐯 , 𝐰 ) e 2 π i 𝐯 𝐰 ({\mathbf{v}},{\mathbf{w}})\mapsto e^{2\pi{i}{\mathbf{v}}\cdot{\mathbf{w}}}
  17. π \pi
  18. π \pi
  19. ( 𝐯 , 𝐰 ) e i 𝐯 𝐰 ({\mathbf{v}},{\mathbf{w}})\mapsto e^{i{\mathbf{v}}\cdot{\mathbf{w}}}
  20. e - x 2 / 2 e^{-x^{2}/2}
  21. ( 𝐯 , 𝐰 ) e 2 π i 𝐯 𝐰 ({\mathbf{v}},{\mathbf{w}})\mapsto e^{2\pi{i}{\mathbf{v}}\cdot{\mathbf{w}}}
  22. e - π x 2 e^{-\pi x^{2}}
  23. [ f g ] ( x ) = G f ( x - y ) g ( y ) d μ ( y ) . [f\star g](x)=\int_{G}f(x-y)g(y)\,d\mu(y).
  24. f e i f . f\star e_{i}\rightarrow f.
  25. ( f g ) ( χ ) = ( f ) ( χ ) ( g ) ( χ ) . \mathcal{F}(f\star g)(\chi)=\mathcal{F}(f)(\chi)\cdot\mathcal{F}(g)(\chi).
  26. f f ^ ( χ ) . f\mapsto\widehat{f}(\chi).
  27. f ^ \widehat{f}
  28. G | f ( x ) | 2 d μ ( x ) = G ^ | f ^ ( χ ) | 2 d ν ( χ ) . \int_{G}|f(x)|^{2}\ d\mu(x)=\int_{\widehat{G}}|\widehat{f}(\chi)|^{2}\ d\nu(% \chi).
  29. : L μ 2 ( G ) L ν 2 ( G ^ ) . \mathcal{F}:L^{2}_{\mu}(G)\rightarrow L^{2}_{\nu}(\widehat{G}).
  30. G | f ( x ) | 2 d μ ( x ) = G ^ | f ^ ( χ ) | 2 d ν ( χ ) \int_{G}|f(x)|^{2}\ d\mu(x)=\int_{\widehat{G}}|\widehat{f}(\chi)|^{2}\ d\nu(\chi)
  31. L ν 2 ( G ^ ) L μ 2 ( G ) L^{2}_{\nu}(\widehat{G})\rightarrow L^{2}_{\mu}(G)
  32. ι : H G ^ \iota:H\rightarrow\widehat{G}
  33. G G ^ ^ H ^ G\sim\widehat{\widehat{G}}{\rightarrow}\widehat{H}

Ponytail.html

  1. l 3 R s s s s - ( L - s ) R s s + R s - π ( R ) = 0 l^{3}R_{ssss}-(L-s)R_{ss}+R_{s}-\pi(R)=0
  2. π ( R ) = 4 l 3 P A ρ R \pi(R)=\frac{4l^{3}P}{A\rho R}
  3. l = ( A λ g ) 1 3 l=(\frac{A}{\lambda g})^{\frac{1}{3}}
  4. A = E π d 4 64 A=\frac{E\pi d^{4}}{64}
  5. ρ = N π R 2 \rho=\frac{N}{\pi R^{2}}
  6. R a = L l Ra=\frac{L}{l}

Population_dynamics.html

  1. d N d t 1 N = r \dfrac{dN}{dt}\dfrac{1}{N}=r
  2. d N d T = r N \dfrac{dN}{dT}=rN
  3. d N d T \dfrac{dN}{dT}
  4. d N d T = r N ( 1 - N K ) \dfrac{dN}{dT}=rN(1-\dfrac{N}{K})
  5. ( 1 - N K ) (1-\dfrac{N}{K})

Positive_element.html

  1. A A
  2. 𝒜 \mathcal{A}
  3. σ ( A ) \sigma(A)
  4. A A
  5. 𝒜 \mathcal{A}
  6. B B
  7. 𝒜 \mathcal{A}
  8. A = B * B A=B^{*}B
  9. T T
  10. H H
  11. T x , x \langle Tx,x\rangle
  12. x x
  13. H H
  14. T x , x \langle Tx,x\rangle
  15. x x
  16. H H
  17. T T
  18. P P
  19. V V
  20. P = S * S P=S^{*}S
  21. S S
  22. V V
  23. S S
  24. P P
  25. V V
  26. P = S * S P=S^{*}S
  27. S S
  28. V V
  29. P = T 2 P=T^{2}
  30. T T
  31. V V
  32. P P
  33. P u , u 0 , u V \langle Pu,u\rangle\geq 0,\forall u\in V
  34. P P
  35. V V
  36. P = S * S P=S^{*}S
  37. S S
  38. V V
  39. P = T 2 P=T^{2}
  40. T T
  41. V V
  42. P P
  43. P u , u > 0 , u 0 \langle Pu,u\rangle>0,\forall u\neq 0
  44. V V
  45. A = [ a b c d ] A=\begin{bmatrix}a&b\\ c&d\end{bmatrix}
  46. A A
  47. a a
  48. d d
  49. det ( A ) = a d - b c \det(A)=ad-bc
  50. X X
  51. Y Y
  52. T : X Y T\colon X\to Y
  53. T T
  54. T x 0 Tx\geq 0
  55. x 0 x\geq 0
  56. X X
  57. T T
  58. T 0 T\geq 0
  59. X X
  60. Y Y
  61. Y Y
  62. T T
  63. - Δ -\Delta
  64. - d 2 d x 2 -\frac{d^{2}}{dx^{2}}
  65. A A
  66. det ( A ) = 0 \det(A)=0
  67. A A
  68. a = 1 a=1
  69. d = 1 d=1
  70. det ( A ) = 0 \det(A)=0
  71. A = [ 1 1 1 1 ] . A=\begin{bmatrix}1&1\\ 1&1\end{bmatrix}.
  72. A B A - B is positive A\geq B\iff A-B\,\text{ is positive}
  73. 𝒜 \mathcal{A}
  74. 𝒜 \mathcal{A}
  75. A 0 A\geq 0
  76. A A
  77. A B C D AB\geq CD
  78. A , B , C , D 𝒜 A,B,C,D\in\mathcal{A}
  79. A C A\geq C
  80. B D B\geq D

Positive_linear_functional.html

  1. f ( v ) 0. f(v)\geq 0.
  2. ψ ( f ) = X f ( x ) d μ ( x ) \psi(f)=\int_{X}f(x)d\mu(x)\quad
  3. | ρ ( b * a ) | 2 ρ ( a * a ) ρ ( b * b ) . |\rho(b^{*}a)|^{2}\leq\rho(a^{*}a)\rho(b^{*}b).

Posterior_probability.html

  1. θ \theta
  2. X X
  3. p ( θ | X ) p(\theta|X)
  4. p ( X | θ ) p(X|\theta)
  5. p ( θ ) p(\theta)
  6. x x
  7. p ( x | θ ) p(x|\theta)
  8. p ( θ | x ) = p ( x | θ ) p ( θ ) p ( x ) . p(\theta|x)=\frac{p(x|\theta)p(\theta)}{p(x)}.
  9. Posterior probability Likelihood × Prior probability \,\text{Posterior probability}\propto\,\text{Likelihood}\times\,\text{Prior probability}
  10. G G
  11. T T
  12. P ( G | T ) P(G|T)
  13. P ( G ) P(G)
  14. P ( B ) P(B)
  15. B B
  16. G G
  17. P ( T | G ) P(T|G)
  18. P ( T | B ) P(T|B)
  19. P ( T ) P(T)
  20. P ( T ) = P ( T | G ) P ( G ) + P ( T | B ) P ( B ) P(T)=P(T|G)P(G)+P(T|B)P(B)
  21. P ( T ) = 0.5 × 0.4 + 1 × 0.6 = 0.8 P(T)=0.5\times 0.4+1\times 0.6=0.8
  22. P ( G | T ) = P ( T | G ) P ( G ) P ( T ) = 0.5 × 0.4 0.8 = 0.25. P(G|T)=\frac{P(T|G)P(G)}{P(T)}=\frac{0.5\times 0.4}{0.8}=0.25.
  23. f X Y = y ( x ) = f X ( x ) L X Y = y ( x ) - f X ( x ) L X Y = y ( x ) d x f_{X\mid Y=y}(x)={f_{X}(x)L_{X\mid Y=y}(x)\over{\int_{-\infty}^{\infty}f_{X}(x% )L_{X\mid Y=y}(x)\,dx}}
  24. X X
  25. Y = y Y=y
  26. f X ( x ) f_{X}(x)
  27. X X
  28. L X Y = y ( x ) = f Y X = x ( y ) L_{X\mid Y=y}(x)=f_{Y\mid X=x}(y)
  29. x x
  30. - f X ( x ) L X Y = y ( x ) d x \int_{-\infty}^{\infty}f_{X}(x)L_{X\mid Y=y}(x)\,dx
  31. f X Y = y ( x ) f_{X\mid Y=y}(x)
  32. X X
  33. Y = y Y=y

Potential_evaporation.html

  1. P E T = 1.6 ( L 12 ) ( N 30 ) ( 10 T a I ) α PET=1.6\left(\frac{L}{12}\right)\left(\frac{N}{30}\right)\left(\frac{10\,T_{a}% }{I}\right)^{\alpha}
  2. P E T PET
  3. T a T_{a}
  4. 0
  5. N N
  6. L L
  7. α = ( 6.75 × 10 - 7 ) I 3 - ( 7.71 × 10 - 5 ) I 2 + ( 1.792 × 10 - 2 ) I + 0.49239 \alpha=(6.75\times 10^{-7})I^{3}-(7.71\times 10^{-5})I^{2}+(1.792\times 10^{-2% })I+0.49239
  8. I = i = 1 12 ( T a i 5 ) 1.514 I=\sum_{i=1}^{12}\left(\frac{T_{ai}}{5}\right)^{1.514}
  9. T a i T_{ai}

Potential_gradient.html

  1. F = ϕ 2 - ϕ 1 x 2 - x 1 = Δ ϕ Δ x F=\frac{\phi_{2}-\phi_{1}}{x_{2}-x_{1}}=\frac{\Delta\phi}{\Delta x}\,\!
  2. ϕ ( x ) ϕ(x)
  3. x x
  4. x x
  5. F = d ϕ d x . F=\frac{{\rm d}\phi}{{\rm d}x}.\,\!
  6. 𝐅 = 𝐞 x ϕ x + 𝐞 y ϕ y + 𝐞 z ϕ z \mathbf{F}=\mathbf{e}_{x}\frac{\partial\phi}{\partial x}+\mathbf{e}_{y}\frac{% \partial\phi}{\partial y}+\mathbf{e}_{z}\frac{\partial\phi}{\partial z}\,\!
  7. x , y , z x,y,z
  8. 𝐅 = ϕ . \mathbf{F}=\nabla\phi.\,\!
  9. 𝐅 \mathbf{F}
  10. 𝐅 \mathbf{F}
  11. ϕ ϕ
  12. × 𝐅 = s y m b o l 0 \nabla\times\mathbf{F}=symbol{0}\,\!
  13. 𝐠 \mathbf{g}
  14. Φ Φ
  15. 𝐠 = - Φ . \mathbf{g}=-\nabla\Phi.\,\!
  16. 𝐄 \mathbf{E}
  17. t t
  18. 𝐁 \mathbf{B}
  19. × 𝐄 = - 𝐁 t = s y m b o l 0 , \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}=symbol{0}\,,
  20. 𝐄 \mathbf{E}
  21. ϕ ϕ
  22. - 𝐄 = V . -\mathbf{E}=\nabla V.\,\!
  23. 𝐄 \mathbf{E}
  24. 𝐁 \mathbf{B}
  25. 𝐄 \mathbf{E}
  26. - 𝐄 = V + 𝐀 t -\mathbf{E}=\nabla V+\frac{\partial\mathbf{A}}{\partial t}\,\!
  27. 𝐀 \mathbf{A}
  28. 𝐯 \mathbf{v}
  29. ω \mathbf{ω}
  30. s y m b o l ω = × 𝐯 = s y m b o l 0. symbol{\omega}=\nabla\times\mathbf{v}=symbol{0}.
  31. 𝐯 = ϕ \mathbf{v}=\nabla\phi
  32. Δ ϕ ( M , M + z ) = Δ ϕ ( M , M + z ) + R T z e N A ln a M + z \Delta\phi_{(M,M^{+z})}=\Delta\phi_{(M,M^{+z})}^{\ominus}+\frac{RT}{zeN_{A}}% \ln a_{M^{+z}}\,\!
  33. 𝐄 \mathbf{E}
  34. 𝐁 \mathbf{B}
  35. 𝐅 = X ρ \nabla\cdot\mathbf{F}=X\rho
  36. ρ ρ
  37. X X
  38. G G
  39. ( ϕ ) = X ρ 2 ϕ = X ρ \nabla\cdot(\nabla\phi)=X\rho\quad\Rightarrow\quad\nabla^{2}\phi=X\rho

Potential_well.html

  1. ψ n x , n y , n z = 8 L x L y L z sin ( n x π x L x ) sin ( n y π y L y ) sin ( n z π z L z ) \psi_{n_{x},n_{y},n_{z}}=\sqrt{\frac{8}{L_{x}L_{y}L_{z}}}\sin\left(\frac{n_{x}% \pi x}{L_{x}}\right)\sin\left(\frac{n_{y}\pi y}{L_{y}}\right)\sin\left(\frac{n% _{z}\pi z}{L_{z}}\right)
  2. E n x , n y , n z = 2 π 2 2 m [ ( n x L x ) 2 + ( n y L y ) 2 + ( n z L z ) 2 ] E_{n_{x},n_{y},n_{z}}=\frac{\hbar^{2}\pi^{2}}{2m}\left[\left(\frac{n_{x}}{L_{x% }}\right)^{2}+\left(\frac{n_{y}}{L_{y}}\right)^{2}+\left(\frac{n_{z}}{L_{z}}% \right)^{2}\right]
  3. Δ p \displaystyle\Delta p

Power-law_fluid.html

  1. τ = K ( u y ) n \tau=K\left(\frac{\partial u}{\partial y}\right)^{n}
  2. μ eff = K ( u y ) n - 1 \mu_{\operatorname{eff}}=K\left(\frac{\partial u}{\partial y}\right)^{n-1}
  3. τ = μ u y \tau=\mu\frac{\partial u}{\partial y}
  4. u ( r ) = n n + 1 ( d p d z 1 2 K ) 1 n [ R n + 1 n - r n + 1 n ] u(r)=\frac{n}{n+1}\left(\frac{dp}{dz}\frac{1}{2K}\right)^{\frac{1}{n}}\left[R^% {\frac{n+1}{n}}-r^{\frac{n+1}{n}}\right]
  5. u ( r ) u(r)
  6. d p / d z dp/dz
  7. R R

Power_of_two.html

  1. n n
  2. n n
  3. n n
  4. H 3 ( 2 , n ) H_{3}(2,n)
  5. 2 n 2\to n
  6. 2 n 1 2\to n\to 1
  7. n n
  8. n n
  9. 640 = 512 + 128 = 128 × 5 640=512+128=128×5
  10. 480 = 32 × 15 480=32×15
  11. n n
  12. n n
  13. p p
  14. p q p q
  15. 16 × 31 16 × 31
  16. q q
  17. p p
  18. p p
  19. q q
  20. q q
  21. q q
  22. q q
  23. q q
  24. p p
  25. p p
  26. 2 2 n 2^{2^{n}}
  27. i = 0 1 2 2 i x i = 1 2 x 0 + 1 4 x 1 + 1 16 x 2 + \sum_{i=0}^{\infty}\frac{1}{2^{2^{i}}x_{i}}=\frac{1}{2x_{0}}+\frac{1}{4x_{1}}+% \frac{1}{16x_{2}}+\cdots
  28. x x
  29. x x
  30. ( x Align ( x 1 ) ) (x&(x−1))
  31. x x
  32. x > 0 x>0
  33. x x
  34. y y
  35. x m o d y = ( x Align ( y 1 ) ) xmody=(x&(y−1))
  36. x > 0 x>0
  37. 2 round [ log 2 ( x ) ] 2^{\mathrm{round}[\log_{2}(x)]}
  38. x x
  39. k k
  40. x x
  41. k = 0 k=0
  42. k 1 k−1

Powerful_number.html

  1. m = p i α i , m=\prod p_{i}^{\alpha_{i}},
  2. m = ( p i β i ) ( p i γ i ) = ( p i β i / 2 ) 2 ( p i γ i / 3 ) 3 m=(\prod p_{i}^{\beta_{i}})(\prod p_{i}^{\gamma_{i}})=(\prod p_{i}^{\beta_{i}/% 2})^{2}(\prod p_{i}^{\gamma_{i}/3})^{3}
  3. m = 21600 = 2 5 × 3 3 × 5 2 , m=21600=2^{5}\times 3^{3}\times 5^{2}\,,
  4. b = 2 × 3 = 6 , b=2\times 3=6\,,
  5. a = m b 3 = 2 2 × 5 2 = 10 , a=\sqrt{\frac{m}{b^{3}}}=\sqrt{2^{2}\times 5^{2}}=10\,,
  6. m = a 2 b 3 = 10 2 × 6 3 . m=a^{2}b^{3}=10^{2}\times 6^{3}\,.
  7. ζ ( 2 s ) ζ ( 3 s ) ζ ( 6 s ) \frac{\zeta(2s)\zeta(3s)}{\zeta(6s)}
  8. p ( 1 + 1 p ( p - 1 ) ) = ζ ( 2 ) ζ ( 3 ) ζ ( 6 ) = 315 2 π 4 ζ ( 3 ) , \prod_{p}\left(1+\frac{1}{p(p-1)}\right)=\frac{\zeta(2)\zeta(3)}{\zeta(6)}=% \frac{315}{2\pi^{4}}\zeta(3),
  9. c x 1 / 2 - 3 x 1 / 3 k ( x ) c x 1 / 2 , c = ζ ( 3 / 2 ) / ζ ( 3 ) = 2.173 cx^{1/2}-3x^{1/3}\leq k(x)\leq cx^{1/2},c=\zeta(3/2)/\zeta(3)=2.173\cdots

Poynting–Robertson_effect.html

  1. F PR = v c 2 W = r 2 L s 4 c 2 G M s R 5 F_{\rm PR}=\frac{v}{c^{2}}W=\frac{r^{2}L_{\rm s}}{4c^{2}}\sqrt{\frac{GM_{\rm s% }}{R^{5}}}
  2. r 3 \propto r^{3}
  3. r r
  4. r 2 \propto r^{2}
  5. 1 R 2 \frac{1}{R^{2}}
  6. 1 R 2.5 \frac{1}{R^{2.5}}
  7. β \beta
  8. β = F r F g = 3 L Q PR 16 π G M c ρ s \beta={F_{\rm r}\over F_{\rm g}}={3LQ_{\rm PR}\over{16\pi GMc\rho s}}
  9. Q PR Q_{\rm PR}
  10. ρ \rho
  11. s s
  12. β 0.5 \beta\geq 0.5
  13. 0.1 < β < 0.5 0.1<\beta<0.5
  14. β 0.1 \beta\approx 0.1
  15. 1 β \propto{1\over\beta}

Pre-money_valuation.html

  1. post-money valuation = new investment total post investment shares outstanding shares issued for new investment \,\text{post-money valuation}=\,\text{new investment}\,\cdot\,\frac{\,\text{% total post investment shares outstanding}}{\,\text{shares issued for new % investment}}
  2. pre-money valuation = post-money valuation - new investment \,\text{pre-money valuation}=\,\text{post-money valuation}-\,\text{new investment}

Prediction_interval.html

  1. X 1 , , X n . X_{1},\dots,X_{n}.
  2. γ = P ( l < X < u ) = P ( l - μ σ < X - μ σ < u - μ σ ) = P ( l - μ σ < Z < u - μ σ ) , \gamma=P(l<X<u)=P\left(\frac{l-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{u-\mu}{% \sigma}\right)=P\left(\frac{l-\mu}{\sigma}<Z<\frac{u-\mu}{\sigma}\right),
  3. Z = X - μ σ Z=\frac{X-\mu}{\sigma}
  4. l - μ σ = - z , u - μ σ = z , \frac{l-\mu}{\sigma}=-z,\frac{u-\mu}{\sigma}=z,
  5. l = μ - z σ , u = μ + z σ , l=\mu-z\sigma,u=\mu+z\sigma,
  6. γ = P ( - z < Z < z ) . \gamma=P(-z<Z<z).
  7. 1 2 ( 1 - γ ) = P ( Z > z ) . \tfrac{1}{2}(1-\gamma)=P(Z>z).
  8. [ μ - z σ , μ + z σ ] . \left[\mu-z\sigma,\ \mu+z\sigma\right].
  9. X ¯ \overline{X}
  10. ( n - 1 ) (n-1)
  11. Φ X ¯ , s 2 - 1 \Phi^{-1}_{\overline{X},s^{2}}
  12. X ¯ = X ¯ n = ( X 1 + + X n ) / n \overline{X}=\overline{X}_{n}=(X_{1}+\cdots+X_{n})/n
  13. s 2 = s n 2 = 1 n - 1 i = 1 n ( X i - X ¯ n ) 2 . s^{2}=s_{n}^{2}={1\over n-1}\sum_{i=1}^{n}(X_{i}-\overline{X}_{n})^{2}.
  14. X ¯ \overline{X}
  15. X 1 , , X n X_{1},\dots,X_{n}
  16. N ( μ , 1 / n ) , N(\mu,1/n),
  17. X n + 1 X_{n+1}
  18. N ( μ , 1 ) . N(\mu,1).
  19. 1 + ( 1 / n ) , 1+(1/n),
  20. X n + 1 - X ¯ 1 + ( 1 / n ) N ( 0 , 1 ) . \frac{X_{n+1}-\overline{X}}{\sqrt{1+(1/n)}}\sim N(0,1).
  21. X n + 1 X_{n+1}
  22. N ( X ¯ , 1 + ( 1 / n ) ) , N(\overline{X},1+(1/n)),
  23. X n + 1 X_{n+1}
  24. X ¯ \overline{X}
  25. 1 + ( 1 / n ) 1+(1/n)
  26. σ 2 \sigma^{2}
  27. s 2 s^{2}
  28. X 1 , , X n X_{1},\dots,X_{n}
  29. χ n - 1 2 \scriptstyle\chi_{n-1}^{2}
  30. ( n - 1 ) s n 2 σ 2 χ n - 1 2 . \frac{(n-1)s_{n}^{2}}{\sigma^{2}}\sim\chi_{n-1}^{2}.
  31. X n + 1 X_{n+1}
  32. N ( 0 , σ 2 ) . N(0,\sigma^{2}).
  33. X n + 1 s T n - 1 . \frac{X_{n+1}}{s}\sim T^{n-1}.
  34. X n + 1 X_{n+1}
  35. s T n - 1 , sT^{n-1},
  36. s s
  37. N ( μ , σ 2 ) N(\mu,\sigma^{2})
  38. X n + 1 - X ¯ n s n 1 + 1 / n T n - 1 . \frac{X_{n+1}-\overline{X}_{n}}{s_{n}\sqrt{1+1/n}}\sim T^{n-1}.
  39. X n + 1 X_{n+1}
  40. X ¯ n + s n 1 + 1 / n T n - 1 . \overline{X}_{n}+s_{n}\sqrt{1+1/n}\cdot T^{n-1}.
  41. X n + 1 X_{n+1}
  42. Pr ( X ¯ n - T a s n 1 + ( 1 / n ) X n + 1 X ¯ n + T a s n 1 + ( 1 / n ) ) = p \Pr\left(\overline{X}_{n}-T_{a}s_{n}\sqrt{1+(1/n)}\leq X_{n+1}\leq\overline{X}% _{n}+T_{a}s_{n}\sqrt{1+(1/n)}\,\right)=p
  43. X ¯ n ± T a s n 1 + ( 1 / n ) \overline{X}_{n}\pm T_{a}{s}_{n}\sqrt{1+(1/n)}
  44. X n + 1 X_{n+1}
  45. X ¯ n \overline{X}_{n}
  46. S n S_{n}
  47. X n + 1 X_{n+1}
  48. y i = α + β x i + ϵ i y_{i}=\alpha+\beta x_{i}+\epsilon_{i}\,
  49. y i y_{i}
  50. x i x_{i}
  51. α \alpha
  52. β \beta
  53. α ^ \hat{\alpha}
  54. β ^ \hat{\beta}
  55. y ^ d = α ^ + β ^ x d , \hat{y}_{d}=\hat{\alpha}+\hat{\beta}x_{d},
  56. y d = α + β x d + ϵ d . y_{d}=\alpha+\beta x_{d}+\epsilon_{d}.\,
  57. y ^ d \hat{y}_{d}
  58. E ( y | x d ) . E(y|x_{d}).
  59. α ^ \hat{\alpha}
  60. β ^ \hat{\beta}

Predominance_diagram.html

  1. 1. CrO 4 2 - + H + HCrO 4 - ; K 1 = [ HCrO 4 - ] [ CrO 4 2 - ] [ H + ] 1.\ \mathrm{CrO_{4}^{2-}+H^{+}\rightleftharpoons HCrO_{4}^{-};K_{1}=\frac{[% HCrO_{4}^{-}]}{[CrO_{4}^{2-}][H^{+}]}}
  2. 2. 2 HCrO 4 - Cr 2 O 7 2 - + H 2 O ; K D = [ Cr 2 O 7 2 - ] [ HCrO 4 - ] 2 2.\ \mathrm{2HCrO_{4}^{-}\rightleftharpoons Cr_{2}O_{7}^{2-}+H_{2}O;K_{D}=% \frac{[Cr_{2}O_{7}^{2-}]}{[HCrO_{4}^{-}]^{2}}}
  3. 3. 2 H + + 2 C r O 4 2 - Cr 2 O 7 2 - + H 2 O ; β 2 = [ Cr 2 O 7 2 - ] [ H + ] 2 [ CrO 4 2 - ] 2 ; β 2 = K 1 2 K D 3.\ \mathrm{2H^{+}+2CrO_{4}^{2-}\rightleftharpoons Cr_{2}O_{7}^{2-}+H_{2}O;% \beta_{2}=\frac{[Cr_{2}O_{7}^{2-}]}{[H^{+}]^{2}[CrO_{4}^{2-}]^{2}};\beta_{2}=K% _{1}^{2}K_{D}}
  4. [ Cr ] = [ CrO 4 2 - ] + [ HCrO 4 - ] + 2 [ Cr 2 O 7 2 - ] ; pCr = - log 10 [ Cr ] \mathrm{[Cr]=[CrO_{4}^{2-}]+[HCrO_{4}^{-}]+2[Cr_{2}O_{7}^{2-}];pCr=-log_{10}[% Cr]}

Prenex_normal_form.html

  1. ϕ ( y ) \phi(y)
  2. ψ ( z ) \psi(z)
  3. ρ ( x ) \rho(x)
  4. x y z ( ϕ ( y ) ( ψ ( z ) ρ ( x ) ) ) \forall x\exists y\forall z(\phi(y)\lor(\psi(z)\rightarrow\rho(x)))
  5. ϕ ( y ) ( ψ ( z ) ρ ( x ) ) \phi(y)\lor(\psi(z)\rightarrow\rho(x))
  6. x ( ( y ϕ ( y ) ) ( ( z ψ ( z ) ) ρ ( x ) ) ) \forall x((\exists y\phi(y))\lor((\exists z\psi(z))\rightarrow\rho(x)))
  7. ( x ϕ ) ψ (\forall x\phi)\land\psi
  8. x ( ϕ ψ ) \forall x(\phi\land\psi)
  9. ( x ϕ ) ψ (\forall x\phi)\lor\psi
  10. x ( ϕ ψ ) \forall x(\phi\lor\psi)
  11. ( x ϕ ) ψ (\exists x\phi)\land\psi
  12. x ( ϕ ψ ) \exists x(\phi\land\psi)
  13. ( x ϕ ) ψ (\exists x\phi)\lor\psi
  14. x ( ϕ ψ ) \exists x(\phi\lor\psi)
  15. ( x ( x 2 = 1 ) ) ( 0 = y ) (\exists x(x^{2}=1))\land(0=y)
  16. x ( x 2 = 1 0 = y ) \exists x(x^{2}=1\land 0=y)
  17. ( x ( x 2 = 1 ) ) ( 0 = x ) (\exists x(x^{2}=1))\land(0=x)
  18. x ( x 2 = 1 0 = x ) \exists x(x^{2}=1\land 0=x)
  19. ¬ x ϕ \lnot\exists x\phi
  20. x ¬ ϕ \forall x\lnot\phi
  21. ¬ x ϕ \lnot\forall x\phi
  22. x ¬ ϕ \exists x\lnot\phi
  23. ϕ ψ \phi\rightarrow\psi
  24. ¬ ϕ ψ \lnot\phi\lor\psi
  25. ( x ϕ ) ψ (\forall x\phi)\rightarrow\psi
  26. x ( ϕ ψ ) \exists x(\phi\rightarrow\psi)
  27. ( x ϕ ) ψ (\exists x\phi)\rightarrow\psi
  28. x ( ϕ ψ ) \forall x(\phi\rightarrow\psi)
  29. ϕ ( x ψ ) \phi\rightarrow(\exists x\psi)
  30. x ( ϕ ψ ) \exists x(\phi\rightarrow\psi)
  31. ϕ ( x ψ ) \phi\rightarrow(\forall x\psi)
  32. x ( ϕ ψ ) \forall x(\phi\rightarrow\psi)
  33. ϕ \phi
  34. ψ \psi
  35. ρ \rho
  36. ( ϕ x ψ ) z ρ (\phi\lor\exists x\psi)\rightarrow\forall z\rho
  37. ( ϕ x ψ ) z ρ (\phi\lor\exists x\psi)\rightarrow\forall z\rho
  38. ( x ( ϕ ψ ) ) z ρ (\exists x(\phi\lor\psi))\rightarrow\forall z\rho
  39. ¬ ( x ( ϕ ψ ) ) z ρ \neg(\exists x(\phi\lor\psi))\lor\forall z\rho
  40. ( x ¬ ( ϕ ψ ) ) z ρ (\forall x\neg(\phi\lor\psi))\lor\forall z\rho
  41. x ( ¬ ( ϕ ψ ) z ρ ) \forall x(\neg(\phi\lor\psi)\lor\forall z\rho)
  42. x ( ( ϕ ψ ) z ρ ) \forall x((\phi\lor\psi)\rightarrow\forall z\rho)
  43. x ( z ( ( ϕ ψ ) ρ ) ) \forall x(\forall z((\phi\lor\psi)\rightarrow\rho))
  44. x z ( ( ϕ ψ ) ρ ) \forall x\forall z((\phi\lor\psi)\rightarrow\rho)
  45. z x ( ( ϕ ψ ) ρ ) \forall z\forall x((\phi\lor\psi)\rightarrow\rho)
  46. z ( ( ϕ x ψ ) ρ ) \forall z((\phi\lor\exists x\psi)\rightarrow\rho)
  47. z ( ( x ( ϕ ψ ) ) ρ ) \forall z((\exists x(\phi\lor\psi))\rightarrow\rho)
  48. z ( x ( ( ϕ ψ ) ρ ) ) \forall z(\forall x((\phi\lor\psi)\rightarrow\rho))
  49. z x ( ( ϕ ψ ) ρ ) \forall z\forall x((\phi\lor\psi)\rightarrow\rho)
  50. ( x ϕ ) y ψ ( 1 ) (\exists x\phi)\rightarrow\exists y\psi\qquad(1)
  51. ϕ ( x ) \phi(x)
  52. y ( x ϕ ψ ) , ( 2 ) \exists y(\exists x\phi\rightarrow\psi),\qquad(2)
  53. x ϕ \exists x\phi
  54. x ( ϕ ψ ) \forall x(\phi\lor\psi)
  55. ( x ϕ ) ψ (\forall x\phi)\lor\psi
  56. x ( ϕ ψ ) \forall x(\phi\lor\psi)
  57. ϕ ( x ψ ) \phi\lor(\forall x\psi)
  58. ( x ϕ ) ψ (\forall x\phi)\rightarrow\psi
  59. x ( ϕ ψ ) \exists x(\phi\rightarrow\psi)
  60. ϕ ( x ψ ) \phi\rightarrow(\exists x\psi)
  61. x ( ϕ ψ ) \exists x(\phi\rightarrow\psi)
  62. ¬ x ϕ \lnot\forall x\phi
  63. x ¬ ϕ \exists x\lnot\phi
  64. ψ \,\psi
  65. ϕ \,\phi

Pressure_altitude.html

  1. ( 1 - ( m i l l i b a r s / 1013.25 ) .190284 ) * 145366.45 (1-(millibars/1013.25)^{.190284})*145366.45

Prettyprint.html

  1. x 2 + 3 x x^{2}+3x

Prime-counting_function.html

  1. π ( x ) \scriptstyle\pi(x)
  2. x / ln ( x ) x/\operatorname{ln}(x)\!
  3. lim x π ( x ) x / ln ( x ) = 1. \lim_{x\rightarrow\infty}\frac{\pi(x)}{x/\operatorname{ln}(x)}=1.\!
  4. lim x π ( x ) / li ( x ) = 1 \lim_{x\rightarrow\infty}\pi(x)/\operatorname{li}(x)=1\!
  5. π ( x ) \pi(x)\!
  6. π ( x ) = li ( x ) + O ( x e - ln x / 15 ) \pi(x)=\operatorname{li}(x)+O\bigl(xe^{-\sqrt{\ln x}/15}\bigr)\!
  7. x x
  8. x x
  9. li ( x ) \operatorname{li}(x)\!
  10. π ( x ) \pi(x)\!
  11. π ( x ) \pi(x)
  12. x x
  13. x x
  14. π ( x ) \pi(x)
  15. x x
  16. p 1 , p 2 , , p n p_{1},p_{2},\ldots,p_{n}
  17. x x
  18. p i p_{i}
  19. x - i x p i + i < j x p i p j - i < j < k x p i p j p k + \lfloor x\rfloor-\sum_{i}\left\lfloor\frac{x}{p_{i}}\right\rfloor+\sum_{i<j}% \left\lfloor\frac{x}{p_{i}p_{j}}\right\rfloor-\sum_{i<j<k}\left\lfloor\frac{x}% {p_{i}p_{j}p_{k}}\right\rfloor+\cdots
  20. \lfloor\cdots\rfloor
  21. π ( x ) - π ( x ) + 1 \pi(x)-\pi\left(\sqrt{x}\right)+1
  22. p 1 , p 2 , , p n p_{1},p_{2},\ldots,p_{n}
  23. x x
  24. π ( x ) \pi(x)
  25. p 1 p_{1}
  26. p 2 , , p n p_{2},\ldots,p_{n}
  27. n n
  28. Φ ( m , n ) \Phi(m,n)
  29. m m
  30. p i p_{i}
  31. Φ ( m , n ) = Φ ( m , n - 1 ) - Φ ( m p n , n - 1 ) \Phi(m,n)=\Phi(m,n-1)-\Phi\left(\frac{m}{p_{n}},n-1\right)
  32. m m
  33. n = π ( m 3 ) n=\pi\left(\sqrt[3]{m}\right)
  34. μ = π ( m ) - n \mu=\pi\left(\sqrt{m}\right)-n
  35. π ( m ) = Φ ( m , n ) + n ( μ + 1 ) + μ 2 - μ 2 - 1 - k = 1 μ π ( m p n + k ) \pi(m)=\Phi(m,n)+n(\mu+1)+\frac{\mu^{2}-\mu}{2}-1-\sum_{k=1}^{\mu}\pi\left(% \frac{m}{p_{n+k}}\right)
  36. π ( x ) \pi(x)
  37. x x
  38. × 10 5 \times 10^{5}
  39. m m
  40. n n
  41. k k
  42. P k ( m , n ) P_{k}(m,n)
  43. p n p_{n}
  44. P 0 ( m , n ) = 1 P_{0}(m,n)=1
  45. Φ ( m , n ) = k = 0 + P k ( m , n ) \Phi(m,n)=\sum_{k=0}^{+\infty}P_{k}(m,n)
  46. y y
  47. m 3 y m \sqrt[3]{m}\leq y\leq\sqrt{m}
  48. n = π ( y ) n=\pi(y)
  49. P 1 ( m , n ) = π ( m ) - n P_{1}(m,n)=\pi(m)-n
  50. P k ( m , n ) = 0 P_{k}(m,n)=0
  51. k k
  52. π ( m ) = Φ ( m , n ) + n - 1 - P 2 ( m , n ) \pi(m)=\Phi(m,n)+n-1-P_{2}(m,n)
  53. P 2 ( m , n ) P_{2}(m,n)
  54. P 2 ( m , n ) = y < p m ( π ( m p ) - π ( p ) + 1 ) P_{2}(m,n)=\sum_{y<p\leq\sqrt{m}}\left(\pi\left(\frac{m}{p}\right)-\pi(p)+1\right)
  55. Φ ( m , n ) \Phi(m,n)
  56. Φ ( m , 0 ) = m \Phi(m,0)=\lfloor m\rfloor
  57. Φ ( m , b ) = Φ ( m , b - 1 ) - Φ ( m p b , b - 1 ) \Phi(m,b)=\Phi(m,b-1)-\Phi\left(\frac{m}{p_{b}},b-1\right)
  58. π ( 10 10 ) \pi\left(10^{10}\right)
  59. Π 0 ( x ) \Pi_{0}(x)
  60. J 0 ( x ) J_{0}(x)
  61. Π 0 ( x ) \Pi_{0}(x)
  62. Π 0 ( x ) = 1 2 ( p n < x 1 n + p n x 1 n ) \Pi_{0}(x)=\frac{1}{2}\bigg(\sum_{p^{n}<x}\frac{1}{n}\ +\sum_{p^{n}\leq x}% \frac{1}{n}\bigg)
  63. Π 0 ( x ) = 2 x Λ ( n ) ln n - 1 2 Λ ( x ) ln x = n = 1 1 n π 0 ( x 1 / n ) \Pi_{0}(x)=\sum_{2}^{x}\frac{\Lambda(n)}{\ln n}-\frac{1}{2}\frac{\Lambda(x)}{% \ln x}=\sum_{n=1}^{\infty}\frac{1}{n}\pi_{0}(x^{1/n})
  64. π 0 ( x ) = lim ε 0 π ( x - ε ) + π ( x + ε ) 2 . \pi_{0}(x)=\lim_{\varepsilon\rightarrow 0}\frac{\pi(x-\varepsilon)+\pi(x+% \varepsilon)}{2}.
  65. π 0 ( x ) = n = 1 μ ( n ) n Π 0 ( x 1 / n ) \pi_{0}(x)=\sum_{n=1}^{\infty}\frac{\mu(n)}{n}\Pi_{0}(x^{1/n})
  66. Λ \Lambda
  67. ln ζ ( s ) = s 0 Π 0 ( x ) x - s - 1 d x \ln\zeta(s)=s\int_{0}^{\infty}\Pi_{0}(x)x^{-s-1}\,dx
  68. θ ( x ) = p x ln p \theta(x)=\sum_{p\leq x}\ln p
  69. ψ ( x ) = p n x ln p = n = 1 θ ( x 1 / n ) = n x Λ ( n ) . \psi(x)=\sum_{p^{n}\leq x}\ln p=\sum_{n=1}^{\infty}\theta(x^{1/n})=\sum_{n\leq x% }\Lambda(n).
  70. n = 1 Π 0 ( n ) x n = a = 2 x a 1 - x - 1 2 a = 2 b = 2 x a b 1 - x + 1 3 a = 2 b = 2 c = 2 x a b c 1 - x - 1 4 a = 2 b = 2 c = 2 d = 2 x a b c d 1 - x + \sum_{n=1}^{\infty}\Pi_{0}(n)x^{n}=\sum_{a=2}^{\infty}\frac{x^{a}}{1-x}-\frac{% 1}{2}\sum_{a=2}^{\infty}\sum_{b=2}^{\infty}\frac{x^{ab}}{1-x}+\frac{1}{3}\sum_% {a=2}^{\infty}\sum_{b=2}^{\infty}\sum_{c=2}^{\infty}\frac{x^{abc}}{1-x}-\frac{% 1}{4}\sum_{a=2}^{\infty}\sum_{b=2}^{\infty}\sum_{c=2}^{\infty}\sum_{d=2}^{% \infty}\frac{x^{abcd}}{1-x}+\cdots
  71. ψ 0 ( x ) = x - ρ x ρ ρ - ln 2 π - 1 2 ln ( 1 - x - 2 ) \psi_{0}(x)=x-\sum_{\rho}\frac{x^{\rho}}{\rho}-\ln 2\pi-\frac{1}{2}\ln(1-x^{-2})
  72. ψ 0 ( x ) = lim ε 0 ψ ( x - ε ) + ψ ( x + ε ) 2 . \psi_{0}(x)=\lim_{\varepsilon\rightarrow 0}\frac{\psi(x-\varepsilon)+\psi(x+% \varepsilon)}{2}.
  73. Π 0 ( x ) \scriptstyle\Pi_{0}(x)
  74. Π 0 ( x ) = li ( x ) - ρ li ( x ρ ) - ln 2 + x d t t ( t 2 - 1 ) ln t . \Pi_{0}(x)=\operatorname{li}(x)-\sum_{\rho}\operatorname{li}(x^{\rho})-\ln 2+% \int_{x}^{\infty}\frac{dt}{t(t^{2}-1)\ln t}.
  75. π 0 ( x ) = R ( x ) - ρ R ( x ρ ) - 1 ln x + 1 π arctan π ln x \pi_{0}(x)=\operatorname{R}(x)-\sum_{\rho}\operatorname{R}(x^{\rho})-\frac{1}{% \ln x}+\frac{1}{\pi}\arctan\frac{\pi}{\ln x}
  76. R ( x ) = n = 1 μ ( n ) n li ( x 1 / n ) = 1 + k = 1 ( ln x ) k k ! k ζ ( k + 1 ) \operatorname{R}(x)=\sum_{n=1}^{\infty}\frac{\mu(n)}{n}\operatorname{li}(x^{1/% n})=1+\sum_{k=1}^{\infty}\frac{(\ln x)^{k}}{k!k\zeta(k+1)}
  77. π 0 ( x ) \scriptstyle\pi_{0}(x)
  78. π 0 ( x ) \scriptstyle\pi_{0}(x)
  79. R ( x ) - 1 ln x + 1 π arctan π ln x \operatorname{R}(x)-\frac{1}{\ln x}+\frac{1}{\pi}\arctan\frac{\pi}{\ln x}
  80. π ( x ) \scriptstyle\pi(x)
  81. x / ln x \scriptstyle\sqrt{x}/\ln x
  82. Δ ( x ) = ( π 0 ( x ) - R ( x ) + 1 ln x - 1 π arctan π ln x ) ln x x . \Delta(x)=\left(\pi_{0}(x)-\operatorname{R}(x)+\frac{1}{\ln x}-\frac{1}{\pi}% \arctan\frac{\pi}{\ln x}\right)\frac{\ln x}{\sqrt{x}}.
  83. x ln x < π ( x ) < 1.25506 x ln x \frac{x}{\ln x}<\pi(x)<1.25506\frac{x}{\ln x}\!
  84. x ln x - 1 < π ( x ) \frac{x}{\ln x-1}<\pi(x)
  85. x 5393 x\geq 5393
  86. π ( x ) < x ln x - 1.1 \pi(x)<\frac{x}{\ln x-1.1}
  87. x 60184 x\geq 60184
  88. n ( ln ( n ln n ) - 1 ) < p n < n ln ( n ln n ) n(\ln(n\ln n)-1)<p_{n}<n{\ln(n\ln n)}\!
  89. p n = n ( ln ( n ln n ) - 1 ) + n ( ln ln n - 2 ) ln n + O ( n ( ln ln n ) 2 ( ln n ) 2 ) . p_{n}=n(\ln(n\ln n)-1)+\frac{n(\ln\ln n-2)}{\ln n}+O\left(\frac{n(\ln\ln n)^{2% }}{(\ln n)^{2}}\right).
  90. π ( x ) \pi(x)
  91. π ( x ) = li ( x ) + O ( x log x ) . \pi(x)=\operatorname{li}(x)+O(\sqrt{x}\log{x}).
  92. | π ( x ) - li ( x ) | < 1 8 π x log x , for all x 2657. |\pi(x)-\operatorname{li}(x)|<\frac{1}{8\pi}\sqrt{x}\,\log{x},\qquad\,\text{% for all }x\geq 2657.

Prime95.html

  1. 2 79 , 300 , 000 - 1 2^{79,300,000}-1
  2. 2 596 , 000 , 000 - 1 2^{596,000,000}-1
  3. 2 2 , 147 , 483 , 647 - 1 2^{2,147,483,647}-1

Prime_factor.html

  1. 360 = 2 × 2 × 2 × 3 × 3 × 5 = 2 3 × 3 2 × 5 , 360=2\times 2\times 2\times 3\times 3\times 5=2^{3}\times 3^{2}\times 5,
  2. 144 = 2 × 2 × 2 × 2 × 3 × 3 = 2 4 × 3 2 . 144=2\times 2\times 2\times 2\times 3\times 3=2^{4}\times 3^{2}.
  3. 144 = 2 × 2 × 2 × 2 × 3 × 3 = ( 2 × 2 × 3 ) × ( 2 × 2 × 3 ) = ( 2 × 2 × 3 ) 2 = ( 12 ) 2 . 144=2\times 2\times 2\times 2\times 3\times 3=(2\times 2\times 3)\times(2% \times 2\times 3)=(2\times 2\times 3)^{2}=(12)^{2}.
  4. ω ( n ) ω(n)
  5. Ω ( n ) Ω(n)
  6. n n
  7. n = i = 1 ω ( n ) p i α i n=\prod_{i=1}^{\omega(n)}p_{i}^{\alpha_{i}}
  8. Ω ( n ) = i = 1 ω ( n ) α i \Omega(n)=\sum_{i=1}^{\omega(n)}\alpha_{i}
  9. 24 = 2 < s u p > 3 × 3 1 24=2<sup>3×3^{1}

Primitive_data_type.html

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

Primitive_element_theorem.html

  1. E F E\supseteq F
  2. α E \alpha\in E
  3. E F E\supseteq F
  4. E = F ( α ) . E=F(\alpha).
  5. E F E\supseteq F
  6. x = f n - 1 α n - 1 + + f 1 α + f 0 , x=f_{n-1}{\alpha}^{n-1}+\cdots+f_{1}{\alpha}+f_{0},
  7. f i F f_{i}\in F
  8. α E \alpha\in E
  9. E F E\supseteq F
  10. α E \alpha\in E
  11. { 1 , α , , α n - 1 } \{1,\alpha,\cdots,{\alpha}^{n-1}\}
  12. ( 2 ) \mathbb{Q}(\sqrt{2})\supseteq\mathbb{Q}
  13. ( x ) \mathbb{Q}(x)\supseteq\mathbb{Q}
  14. 2 \sqrt{2}
  15. ( x ) \mathbb{Q}(x)
  16. \mathbb{Q}
  17. E F E\supseteq F
  18. E = F ( α ) E=F(\alpha)
  19. α E \alpha\in E
  20. E K F E\supseteq K\supseteq F
  21. E F E\supseteq F
  22. E = F ( α ) E=F(\alpha)
  23. α E \alpha\in E
  24. γ = α + c β \gamma=\alpha+c\beta
  25. \mathbb{Q}
  26. x 2 - 2 x^{2}-2
  27. x 2 - 3 , x^{2}-3,
  28. 2 \sqrt{2}
  29. 3 \sqrt{3}
  30. ( 2 , 3 ) \mathbb{Q}(\sqrt{2},\sqrt{3})
  31. \mathbb{Q}
  32. ( γ ) \mathbb{Q}(\gamma)
  33. γ = 2 + 3 \gamma=\sqrt{2}+\sqrt{3}
  34. 2 \sqrt{2}
  35. 3 \sqrt{3}
  36. 2 3 = 6 \sqrt{2}\sqrt{3}=\sqrt{6}
  37. 2 \sqrt{2}
  38. 3 \sqrt{3}
  39. ( γ ) \mathbb{Q}(\gamma)
  40. 2 = γ 3 - 9 γ 2 \sqrt{2}=\scriptstyle\frac{\gamma^{3}-9\gamma}{2}
  41. 2 \sqrt{2}
  42. 3 \sqrt{3}
  43. 2 3 \sqrt{2}\sqrt{3}
  44. 2 \sqrt{2}
  45. 3 \sqrt{3}
  46. 2 3 \sqrt{2}\sqrt{3}

Primitive_ring.html

  1. R = End ( D V ) R=\mathrm{End}(_{D}V)\,
  2. End ( D V ) \mathrm{End}(_{D}V)\,
  3. 𝔽 𝕄 I ( D ) \mathbb{RFM}_{I}(D)\,

Primorial.html

  1. p n # = k = 1 n p k p_{n}\#=\prod_{k=1}^{n}p_{k}
  2. p 5 # = 2 × 3 × 5 × 7 × 11 = 2310. p_{5}\#=2\times 3\times 5\times 7\times 11=2310.
  3. p n # = e ( 1 + o ( 1 ) ) n log n , p_{n}\#=e^{(1+o(1))n\log n},
  4. o ( ) o(\cdot)
  5. n # = i = 1 π ( n ) p i = p π ( n ) # n\#=\prod_{i=1}^{\pi(n)}p_{i}=p_{\pi(n)}\#
  6. π ( n ) \scriptstyle\pi(n)
  7. n # = { 1 if n = 1 n × ( ( n - 1 ) # ) if n > 1 & n is prime ( n - 1 ) # if n > 1 & n is composite . n\#=\begin{cases}1&\,\text{if }n=1\\ n\times((n-1)\#)&\,\text{if }n>1\ \And\ n\,\text{ is prime}\\ (n-1)\#&\,\text{if }n>1\ \And\ n\,\text{ is composite}.\end{cases}
  8. 12 # = 2 × 3 × 5 × 7 × 11 = 2310. 12\#=2\times 3\times 5\times 7\times 11=2310.
  9. π ( 12 ) = 5 \scriptstyle\pi(12)=5
  10. 12 # = p π ( 12 ) # = p 5 # = 2310. 12\#=p_{\pi(12)}\#=p_{5}\#=2310.
  11. θ ( n ) \theta(n)
  12. \thetasym ( n ) \thetasym(n)
  13. ln ( n # ) n . \ln(n\#)\sim n.
  14. ϕ ( n ) / n \phi(n)/n
  15. ϕ \phi
  16. J k ( n ) J_{k}(n)
  17. ζ ( k ) = 2 k 2 k - 1 + r = 2 ( p r - 1 # ) k J k ( p r # ) , k = 2 , 3 , \zeta(k)=\frac{2^{k}}{2^{k}-1}+\sum_{r=2}^{\infty}\frac{(p_{r-1}\#)^{k}}{J_{k}% (p_{r}\#)},\quad k=2,3,\dots

Primorial_prime.html

  1. k ! + 1 k!+1
  2. 2 3 5 p + 1 2\cdot 3\cdot 5\cdot p+1

Principal_homogeneous_space.html

  1. ( x , g ) ( x , x g ) (x,g)\mapsto(x,x\cdot g)
  2. x / y z x/y\cdot z
  3. x / y y = x = y / y x x/y\cdot y=x=y/y\cdot x
  4. v / w ( x / y z ) = ( v / w x ) / y z v/w\cdot(x/y\cdot z)=(v/w\cdot x)/y\cdot z
  5. x / y z = z / y x x/y\cdot z=z/y\cdot x
  6. x \ y x\backslash y
  7. ( x / w y ) \ z = y \ ( w / x z ) (x/w\cdot y)\backslash z=y\backslash(w/x\cdot z)
  8. ( w \ y ) ( x \ z ) = y \ ( w / x z ) = ( x / w y ) \ z (w\backslash y)\cdot(x\backslash z)=y\backslash(w/x\cdot z)=(x/w\cdot y)\backslash z
  9. e = x \ x e=x\backslash x
  10. ( x \ y ) - 1 = y \ x , (x\backslash y)^{-1}=y\backslash x,
  11. x ( y \ z ) = x / y z . x\cdot(y\backslash z)=x/y\cdot z.
  12. V n ( 𝐑 n ) V_{n}(\mathbf{R}^{n})
  13. E × X G E × X E E\times_{X}G\rightarrow E\times_{X}E
  14. ( x , g ) ( x , x g ) (x,g)\mapsto(x,xg)
  15. E G EG
  16. B G BG

Principal_quantum_number.html

  1. E n = E 1 n 2 = - 13.6 eV n 2 , n = 1 , 2 , 3 , E_{n}=\frac{E_{1}}{n^{2}}=\frac{-13.6\,\text{ eV}}{n^{2}},\quad n=1,2,3,\ldots
  2. 𝐋 = n = n h 2 π \mathbf{L}=n\cdot\hbar=n\cdot{h\over 2\pi}
  3. n = n r + + 1 n=n_{r}+\ell+1\,

Principal_value.html

  1. 4 . \sqrt{4}.
  2. e w = z e^{w}=z\,\!
  3. e w = i e^{w}=i\,\!
  4. log z = ln | z | + i ( arg z ) = ln | z | + i ( Arg z + 2 π k ) \log{z}=\ln{|z|}+i\left(\mathrm{arg}\ z\right)=\ln{|z|}+i\left(\mathrm{Arg}\ z% +2\pi k\right)
  5. ( - π , π ] (-\pi,\ \pi]
  6. pv f ( z ) \mathrm{pv}\ f(z)
  7. log z = ln | z | + i ( arg z ) . \log{z}=\ln{|z|}+i\left(\mathrm{arg}\ z\right).
  8. pv log z = Log z = ln | z | + i ( Arg z ) . \mathrm{pv}\ \log{z}=\mathrm{Log}\ z=\ln{|z|}+i\left(\mathrm{Arg}\ z\right).
  9. z = r e ϕ i z=re^{\phi i}\,
  10. pv z = r e i ϕ / 2 \mathrm{pv}\sqrt{z}=\sqrt{r}\,e^{i\phi/2}
  11. - π < ϕ π -\pi<\phi\leq\pi\,

Principle_of_explosion.html

  1. \vdash
  2. \bot
  3. { ϕ , ¬ ϕ } ψ \{\phi,\lnot\phi\}\vdash\psi
  4. P \bot\to P
  5. ϕ \phi\,
  6. ¬ ϕ \lnot\phi
  7. ψ \psi
  8. ψ \psi
  9. Γ \Gamma
  10. Γ \Gamma
  11. ψ \psi
  12. { ϕ , ¬ ϕ } \{\phi,\lnot\phi\}
  13. { ϕ , ¬ ϕ } \{\phi,\lnot\phi\}
  14. ψ \psi
  15. { ϕ , ¬ ϕ } \{\phi,\lnot\phi\}
  16. ψ \psi
  17. ψ \psi
  18. { ϕ , ¬ ϕ } \{\phi,\lnot\phi\}
  19. ϕ ¬ ϕ \phi\wedge\neg\phi\,
  20. ϕ \phi\,
  21. ¬ ϕ \neg\phi\,
  22. ϕ ψ \phi\vee\psi\,
  23. ψ \psi\,
  24. ( ϕ ¬ ϕ ) ψ (\phi\wedge\neg\phi)\to\psi
  25. ϕ \phi
  26. ψ \psi
  27. ϕ ¬ ϕ \phi\wedge\neg\phi\,
  28. ϕ \phi\,
  29. ¬ ϕ \neg\phi\,
  30. ¬ ψ \neg\psi\,
  31. ϕ \phi\,
  32. ¬ ψ ϕ \neg\psi\to\phi
  33. ( ¬ ϕ ¬ ¬ ψ ) (\neg\phi\to\neg\neg\psi)
  34. ¬ ¬ ψ \neg\neg\psi
  35. ψ \psi\,
  36. ( ϕ ¬ ϕ ) ψ (\phi\wedge\neg\phi)\to\psi
  37. ϕ ¬ ϕ \phi\wedge\neg\phi\,
  38. ¬ ψ \neg\psi\,
  39. ϕ \phi\,
  40. ¬ ϕ \neg\phi\,
  41. ¬ ¬ ψ \neg\neg\psi\,
  42. ψ \psi\,
  43. ( ϕ ¬ ϕ ) ψ (\phi\wedge\neg\phi)\to\psi
  44. { ϕ , ¬ ϕ } \{\phi,\lnot\phi\}
  45. \bot
  46. ϕ ¬ ϕ \phi\land\lnot\phi

Principle_of_locality.html

  1. c \scriptstyle c\,
  2. T = D / c \scriptstyle T\;=\;D/c
  3. D \scriptstyle D\,

Prior_probability.html

  1. p = 0 p=0
  2. p = 1 p=1
  3. A 1 , A 2 , , A n A_{1},A_{2},\ldots,A_{n}
  4. P ( A i | B ) = P ( B | A i ) P ( A i ) j P ( B | A j ) P ( A j ) , P(A_{i}|B)=\frac{P(B|A_{i})P(A_{i})}{\sum_{j}P(B|A_{j})P(A_{j})}\,,

Probability_amplitude.html

  1. Q Q
  2. Q Q
  3. | Ψ |Ψ\rangle
  4. X X
  5. ψ ψ
  6. ψ L 2 ( X ) \psi\in L^{2}(X)
  7. X | ψ ( x ) | 2 d μ ( x ) < ; \int\limits_{X}|\psi(x)|^{2}\,\mathrm{d}\mu(x)<\infty;
  8. ψ ψ
  9. X | ψ ( x ) | 2 d μ ( x ) = 1. \int\limits_{X}|\psi(x)|^{2}\,\mathrm{d}\mu(x)=1.
  10. X X
  11. μ μ
  12. μ μ
  13. X X
  14. X X
  15. X X
  16. x X x∈X
  17. X X
  18. x x
  19. | x |x\rangle
  20. x | \langle x|
  21. ψ ( x ) = x | Ψ \psi(x)=\langle x|\Psi\rangle
  22. | Ψ |Ψ\rangle
  23. Q Q
  24. X X
  25. x x
  26. Q Q
  27. X X
  28. Q Q
  29. Q Q
  30. ψ ( x ) \psi(x)
  31. x | \langle x|
  32. Q Q
  33. | ψ ( x ) | 2 |\psi(x)|^{2}
  34. Q Q
  35. | Ψ |Ψ\rangle
  36. X X
  37. x | \langle x|
  38. X X
  39. x X x∈X
  40. x x
  41. x | Q | Ψ = x x Ψ , x 𝐑 \langle x|Q|Ψ\rangle=x⋅\langle xΨ\rangle,x∈\mathbf{R}
  42. x | \langle x|
  43. x Ψ \langle xΨ\rangle
  44. ψ ψ
  45. | Ψ |Ψ\rangle
  46. x x
  47. X X
  48. x x
  49. X X
  50. | Ψ |Ψ\rangle
  51. X X
  52. x X x∈X
  53. ψ ( x ) ψ(x)
  54. | Ψ |Ψ\rangle
  55. x x
  56. X X
  57. | ψ ( x ) | = 1 |ψ(x)|=1
  58. | x |x\rangle
  59. | Ψ |Ψ\rangle
  60. ψ ( x ) = 0 ψ(x)=0
  61. | x |x\rangle
  62. | Ψ |Ψ\rangle
  63. ψ ( x ) ψ(x)
  64. | H |H\rangle
  65. | V |V\rangle
  66. | ψ |ψ\rangle
  67. | ψ = α | H + β | V , |\psi\rangle=\alpha|H\rangle+\beta|V\rangle,\,
  68. | ψ |ψ\rangle
  69. | H |H\rangle
  70. | V |V\rangle
  71. α α
  72. β β
  73. | ψ = 1 3 | H - i 2 3 | V , |\psi\rangle=\sqrt{1\over 3}|H\rangle-i\sqrt{2\over 3}|V\rangle,
  74. | H |H\rangle
  75. | V |V\rangle
  76. | H |H\rangle
  77. | V |V\rangle
  78. 𝐱 \mathbf{x}
  79. t t
  80. 𝐑 n | ψ 0 ( 𝐱 , t 0 ) | 2 d 𝐱 = a 2 < \int_{\mathbf{R}^{n}}|\psi_{0}(\mathbf{x},t_{0})|^{2}\,\mathrm{d\mathbf{x}}=a^% {2}<\infty
  81. V V
  82. 𝐏 ( V ) = V ρ ( 𝐱 ) d 𝐱 = V | ψ ( 𝐱 , t 0 ) | 2 d 𝐱 . \mathbf{P}(V)=\int_{V}\rho(\mathbf{x})\,\mathrm{d\mathbf{x}}=\int_{V}|\psi(% \mathbf{x},t_{0})|^{2}\,\mathrm{d\mathbf{x}}.
  83. ψ ψ
  84. ρ t ( 𝐱 ) = | ψ ( 𝐱 , t ) | 2 = | ψ 0 ( 𝐱 , t ) a | 2 \rho_{t}(\mathbf{x})=\left|\psi(\mathbf{x},t)\right|^{2}=\left|\frac{\psi_{0}(% \mathbf{x},t)}{a}\right|^{2}
  85. t t
  86. ψ ( 𝐱 , 0 ) ψ(\mathbf{x}, 0)
  87. P = | ϕ | 2 P=|\phi|^{2}
  88. ϕ = i ϕ i ; P = | ϕ | 2 = | i ϕ i | 2 \phi=\sum_{i}\phi_{i};P=|\phi|^{2}=|\sum_{i}\phi_{i}|^{2}
  89. ϕ A P B = ϕ A P ϕ P B \phi_{APB}=\phi_{AP}\phi_{PB}
  90. ϕ system ( α , β , γ , δ , ) = ϕ 1 ( α ) ϕ 2 ( β ) ϕ 3 ( γ ) ϕ 4 ( δ ) \phi_{\rm{system}}(\alpha,\beta,\gamma,\delta,\ldots)=\phi_{1}(\alpha)\phi_{2}% (\beta)\phi_{3}(\gamma)\phi_{4}(\delta)\ldots
  91. P = i | ϕ i | 2 P=\sum_{i}|\phi_{i}|^{2}
  92. 𝐏 ( t h r o u g h e i t h e r s l i t ) = 𝐏 ( t h r o u g h f i r s t s l i t ) + 𝐏 ( t h r o u g h s e c o n d s l i t ) \mathbf{P}(througheitherslit)=\mathbf{P}(throughfirstslit)+\mathbf{P}(throughsecondslit)
  93. 𝐏 ( e v e n t ) \mathbf{P}(event)
  94. P = | ψ first + ψ second | 2 = | ψ first | 2 + | ψ second | 2 + 2 | ψ first | | ψ second | cos ( φ 1 - φ 2 ) P=|\psi_{\rm{first}}+\psi_{\rm{second}}|^{2}=|\psi_{\rm{first}}|^{2}+|\psi_{% \rm{second}}|^{2}+2|\psi_{\rm{first}}||\psi_{\rm{second}}|\cos(\varphi_{1}-% \varphi_{2})
  95. φ 1 \varphi_{1}
  96. φ 2 \varphi_{2}
  97. 2 | ψ first | | ψ second | cos ( φ 1 - φ 2 ) 2|\psi_{\rm{first}}||\psi_{\rm{second}}|\cos(\varphi_{1}-\varphi_{2})
  98. 𝐣 \mathbf{j}
  99. 𝐣 = m 1 2 i ( ψ * ψ - ψ ψ * ) = m Im ( ψ * ψ ) , \mathbf{j}={\hbar\over m}{1\over{2i}}\left(\psi^{*}\nabla\psi-\psi\nabla\psi^{% *}\right)={\hbar\over m}\operatorname{Im}\left(\psi^{*}\nabla\psi\right),
  100. 𝐣 + t | ψ | 2 = 0. \nabla\cdot\mathbf{j}+{\partial\over\partial t}|\psi|^{2}=0.
  101. ρ = | ψ | 2 \rho=|\psi|^{2}
  102. 𝐣 \mathbf{j}
  103. L < s u p > 2 ( X 1 ) L<sup>2(X_{1})
  104. X X
  105. x x
  106. L < s u p > 2 ( X ) L<sup>2(X)
  107. X X

Probably_approximately_correct_learning.html

  1. n n
  2. X X
  3. X = { 0 , 1 } n X=\{0,1\}^{n}
  4. X = X=\mathbb{R}
  5. \mathbb{R}
  6. c X c\subset X
  7. X = { 0 , 1 } n X=\{0,1\}^{n}
  8. π / 2 \pi/2
  9. 10 \sqrt{10}
  10. C C
  11. X X
  12. E X ( c , D ) EX(c,D)
  13. x x
  14. D D
  15. c ( x ) c(x)
  16. x c x\in c
  17. A A
  18. E X ( c , D ) EX(c,D)
  19. ϵ \epsilon
  20. δ \delta
  21. 1 - δ 1-\delta
  22. A A
  23. h C h\in C
  24. ϵ \epsilon
  25. X X
  26. D D
  27. c C c\in C
  28. D D
  29. X X
  30. 0 < ϵ < 1 / 2 0<\epsilon<1/2
  31. 0 < δ < 1 / 2 0<\delta<1/2
  32. C C
  33. A A
  34. C C
  35. t t
  36. t t
  37. t t
  38. 1 / ϵ 1/\epsilon
  39. 1 / δ 1/\delta

Product_(chemistry).html

  1. S P S\rightarrow P
  2. S + C P + C S+C\rightarrow P+C
  3. S + E P + E S+E\rightarrow P+E

Product_detector.html

  1. x ( t ) = ( C + m ( t ) ) cos ( ω t ) . \,x(t)=(C+m(t))\cos(\omega t).
  2. y ( t ) = ( C + m ( t ) ) cos ( ω t ) cos ( ω t ) , \,y(t)=(C+m(t))\cos(\omega t)\cos(\omega t),
  3. y ( t ) = ( C + m ( t ) ) ( 1 2 + 1 2 cos ( 2 ω t ) ) . \,y(t)=(C+m(t))\left(\tfrac{1}{2}+\tfrac{1}{2}\cos(2\omega t)\right).
  4. sin 2 θ cos 2 θ = 1 - cos 4 θ 8 \sin^{2}\theta\cos^{2}\theta=\frac{1-\cos 4\theta}{8}
  5. x ( t ) = ( C + m ( t ) ) cos ( ω t ) . \,x(t)=(C+m(t))\cos(\omega t).
  6. y ( t ) = ( C + m ( t ) ) sin 2 ( ω t ) cos 2 ( ω t ) \,y(t)=(C+m(t))\sin^{2}(\omega t)\cos^{2}(\omega t)
  7. = ( C + m ( t ) ) 1 - cos 4 ω t 8 =(C+m(t))\frac{1-\cos 4\omega t}{8}
  8. = ( C + m ( t ) ) 8 - ( C + m ( t ) ) cos 4 ω t 8 . =\frac{(C+m(t))}{8}-\frac{(C+m(t))\cos 4\omega t}{8}.

Production_function.html

  1. Q = f ( X 1 , X 2 , X 3 , , X n ) Q=f(X_{1},X_{2},X_{3},\ldots,X_{n})
  2. Q Q
  3. X 1 , X 2 , X 3 , , X n X_{1},X_{2},X_{3},\ldots,X_{n}
  4. Q Q
  5. f f
  6. R n k R^{n^{k}}
  7. k k
  8. n n
  9. Q = a + b X 1 + c X 2 + d X 3 + Q=a+bX_{1}+cX_{2}+dX_{3}+\cdots
  10. a , b , c , d a,b,c,d
  11. Q = a X 1 b X 2 c . Q=aX_{1}^{b}X_{2}^{c}\cdots.
  12. Q = min ( a X 1 , b X 2 , ) . Q=\min(aX_{1},bX_{2},\ldots).
  13. a , b , c , a,b,c,...
  14. X X
  15. Q = f ( X 1 , X 2 ) Q=f(X_{1},X_{2})
  16. n n
  17. k k
  18. f ( k X 1 , k X 2 ) = k n f ( X 1 , X 2 ) f(kX_{1},kX_{2})=k^{n}f(X_{1},X_{2})
  19. n > 1 n>1
  20. n < 1 n<1
  21. 1 1
  22. b + c + > 1 b+c+\cdots>1
  23. b + c + < 1 b+c+\cdots<1
  24. b + c + = 1 b+c+\cdots=1
  25. F ( h ( X 1 , X 2 ) ) F(h(X_{1},X_{2}))
  26. F ( y ) F(y)
  27. F ( y ) F(y)
  28. d F / d y > 0 \mathrm{d}F/\mathrm{d}y>0
  29. h ( X 1 , X 2 ) h(X_{1},X_{2})

Program_synthesis.html

  1. M M
  2. x x
  3. y y
  4. A = A A=A
  5. A A A\leq A
  6. A B B A A\leq B\lor B\leq A
  7. x M y M ( x = M y = M ) x\leq M\land y\leq M\land(x=M\lor y=M)
  8. ( x M y M x = M ) ( x M y M y = M ) (x\leq M\land y\leq M\land x=M)\lor(x\leq M\land y\leq M\land y=M)
  9. x M y M x = M x\leq M\land y\leq M\land x=M
  10. x M y M y = M x\leq M\land y\leq M\land y=M
  11. x x y x x\leq x\land y\leq x
  12. y x y\leq x
  13. ¬ ( x y ) \lnot(x\leq y)
  14. x y y y x\leq y\land y\leq y
  15. x y x\leq y
  16. true \,\text{true}
  17. X Y M : X M Y M ( X = M Y = M ) \forall X\forall Y\exists M:X\leq M\land Y\leq M\land(X=M\lor Y=M)
  18. M M
  19. M M
  20. ¬ ( \lnot(
  21. ) ( )\land(
  22. ) )
  23. x x y x x\leq x\land y\leq x
  24. x M y M y = M x\leq M\land y\leq M\land y=M
  25. true \,\text{true}
  26. true \,\text{true}
  27. E [ p ] E[p]
  28. F [ p ] F[p]
  29. G [ p ] G[p]
  30. H [ p ] H[p]
  31. E [ true ] F [ false ] E[\,\text{true}]\lor F[\,\text{false}]
  32. ¬ F [ true ] G [ false ] \lnot F[\,\text{true}]\land G[\,\text{false}]
  33. ¬ F [ false ] G [ true ] \lnot F[\,\text{false}]\land G[\,\text{true}]
  34. G [ true ] H [ false ] G[\,\text{true}]\land H[\,\text{false}]

Project_Orion_(nuclear_propulsion).html

  1. I s p = C 0 V e g n I_{sp}=\frac{C_{0}\cdot V_{e}}{g_{n}}

Projection_(linear_algebra).html

  1. P = [ 1 0 0 0 1 0 0 0 0 ] . P=\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&0\end{bmatrix}.
  2. P ( x y z ) = ( x y 0 ) . P\begin{pmatrix}x\\ y\\ z\end{pmatrix}=\begin{pmatrix}x\\ y\\ 0\end{pmatrix}.
  3. P 2 ( x y z ) = P ( x y 0 ) = ( x y 0 ) = P ( x y z ) . P^{2}\begin{pmatrix}x\\ y\\ z\end{pmatrix}=P\begin{pmatrix}x\\ y\\ 0\end{pmatrix}=\begin{pmatrix}x\\ y\\ 0\end{pmatrix}=P\begin{pmatrix}x\\ y\\ z\end{pmatrix}.
  4. P = [ 0 0 α 1 ] . P=\begin{bmatrix}0&0\\ \alpha&1\end{bmatrix}.
  5. P 2 = [ 0 0 α 1 ] [ 0 0 α 1 ] = [ 0 0 α 1 ] = P . P^{2}=\begin{bmatrix}0&0\\ \alpha&1\end{bmatrix}\begin{bmatrix}0&0\\ \alpha&1\end{bmatrix}=\begin{bmatrix}0&0\\ \alpha&1\end{bmatrix}=P.
  6. x U : P x = x \forall x\in U:Px=x
  7. ( λ I - P ) - 1 = 1 λ I + 1 λ ( λ - 1 ) P (\lambda I-P)^{-1}=\frac{1}{\lambda}I+\frac{1}{\lambda(\lambda-1)}P
  8. P x , ( y - P y ) = ( x - P x ) , P y = 0 \langle Px,(y-Py)\rangle=\langle(x-Px),Py\rangle=0
  9. x , P y = P x , P y = P x , y \langle x,Py\rangle=\langle Px,Py\rangle=\langle Px,y\rangle
  10. P x , y - P y = P 2 x , y - P y = P x , P ( I - P ) y = P x , ( P - P 2 ) y = 0 \langle Px,y-Py\rangle=\langle P^{2}x,y-Py\rangle=\langle Px,P(I-P)y\rangle=% \langle Px,(P-P^{2})y\rangle=0\,
  11. , \langle\cdot,\cdot\rangle
  12. x , P y = P x , y = x , P * y \langle x,Py\rangle=\langle Px,y\rangle=\langle x,P^{*}y\rangle
  13. { x - u | u U } \{\|x-u\||u\in U\}
  14. a = x - P x a=x-Px
  15. a - a , v v 2 v 2 = a 2 - a , v 2 v 2 \|a-\frac{\langle a,v\rangle}{\|v\|^{2}}v\|^{2}=\|a\|^{2}-\frac{{\langle a,v% \rangle}^{2}}{\|v\|^{2}}
  16. w = P x + a , v v 2 v w=Px+\frac{\langle a,v\rangle}{\|v\|^{2}}v
  17. x - w < x - P x \|x-w\|<\|x-Px\|
  18. a , v \langle a,v\rangle
  19. a , v \langle a,v\rangle
  20. x - P x , P x = 0 \langle x-Px,Px\rangle=0
  21. x - P x , v \langle x-Px,v\rangle
  22. ( x + y ) - P ( ( x + y ) , v = 0 \langle\left(x+y\right)-P(\left(x+y\right),v\rangle=0
  23. ( x - P x ) + ( y - P y ) , v = 0 \langle\left(x-Px\right)+\left(y-Py\right),v\rangle=0
  24. P x + P y - P ( ( x + y ) , v = 0 \langle Px+Py-P(\left(x+y\right),v\rangle=0
  25. P v 2 = P v , P v = P v , v P v v \|Pv\|^{2}=\langle Pv,Pv\rangle=\langle Pv,v\rangle\leq\|Pv\|\cdot\|v\|
  26. P v v \|Pv\|\leq\|v\|
  27. , \langle\cdot,\cdot\rangle
  28. P u = u u T . P_{u}=uu^{\mathrm{T}}.\,
  29. x x
  30. x = x + x x=x_{\parallel}+x_{\perp}
  31. P u x = u u T x + u u T x = u ( sign ( u T x ) x ) + u 0 = x P_{u}x=uu^{\mathrm{T}}x_{\parallel}+uu^{\mathrm{T}}x_{\perp}=u(\mathrm{sign}(u% ^{\mathrm{T}}x_{\parallel})\|x_{\parallel}\|)+u\cdot\vec{0}=x_{\parallel}
  32. P A = A A T P_{A}=AA^{\mathrm{T}}
  33. P A = i u i , u i . P_{A}=\sum_{i}\langle u_{i},\cdot\rangle u_{i}.
  34. P A = A ( A T A ) - 1 A T . P_{A}=A(A^{\mathrm{T}}A)^{-1}A^{\mathrm{T}}.
  35. P A = A ( A T A ) + A T P_{A}=A(A^{\mathrm{T}}A)^{+}A^{\mathrm{T}}
  36. [ A B ] \,[A\ B]
  37. I \displaystyle I
  38. I = [ A B ] [ ( A T W A ) - 1 A T ( B T W B ) - 1 B T ] W . I=\begin{bmatrix}A&B\end{bmatrix}\begin{bmatrix}(A^{\mathrm{T}}WA)^{-1}A^{% \mathrm{T}}\\ (B^{\mathrm{T}}WB)^{-1}B^{\mathrm{T}}\end{bmatrix}W.
  39. P = A ( B T A ) - 1 B T . P=A(B^{\mathrm{T}}A)^{-1}B^{\mathrm{T}}.
  40. P = I r 0 d - r P=I_{r}\oplus 0_{d-r}
  41. P = [ 1 σ 1 0 0 ] [ 1 σ k 0 0 ] I m 0 s P=\begin{bmatrix}1&\sigma_{1}\\ 0&0\end{bmatrix}\oplus\cdots\oplus\begin{bmatrix}1&\sigma_{k}\\ 0&0\end{bmatrix}\oplus I_{m}\oplus 0_{s}
  42. σ i \sigma_{i}
  43. T : V W , T\colon V\to W,
  44. ( ker T ) W (\ker T)^{\perp}\to W

Projective_line.html

  1. [ x 1 : x 2 ] [x_{1}:x_{2}]
  2. [ x 1 : x 2 ] [ λ x 1 : λ x 2 ] . [x_{1}:x_{2}]\sim[\lambda x_{1}:\lambda x_{2}].
  3. { [ x : 1 ] 𝐏 1 ( K ) x K } . \left\{[x:1]\in\mathbf{P}^{1}(K)\mid x\in K\right\}.
  4. = [ 1 : 0 ] . \infty=[1:0].
  5. 1 0 = , 1 = 0 , \frac{1}{0}=\infty,\qquad\frac{1}{\infty}=0,
  6. x = if x 0 x\cdot\infty=\infty\quad\,\text{if}\quad x\not=0
  7. x + = if x x+\infty=\infty\quad\,\text{if}\quad x\not=\infty
  8. [ x 1 : x 2 ] + [ y 1 : y 2 ] = [ x 1 y 2 + y 1 x 2 : x 2 y 2 ] , [x_{1}:x_{2}]+[y_{1}:y_{2}]=[x_{1}y_{2}+y_{1}x_{2}:x_{2}y_{2}],
  9. [ x 1 : x 2 ] [ y 1 : y 2 ] = [ x 1 y 1 : x 2 y 2 ] , [x_{1}:x_{2}]\cdot[y_{1}:y_{2}]=[x_{1}y_{1}:x_{2}y_{2}],
  10. [ x 1 : x 2 ] - 1 = [ x 2 : x 1 ] . [x_{1}:x_{2}]^{-1}=[x_{2}:x_{1}].
  11. a a : 1 aa:1
  12. a a
  13. F F

Projective_linear_group.html

  1. 𝐅 p n \mathbf{F}_{p^{n}}
  2. 1 + 2 + + n = ( n + 1 2 ) . \textstyle{1+2+\cdots+n={\left({{n+1}\atop{2}}\right)}}.
  3. 1 + 2 + + ( n - 1 ) = ( n 2 ) . \textstyle{1+2+\cdots+(n-1)={\left({{n}\atop{2}}\right)}}.
  4. x - a x - c b - c b - a \frac{x-a}{x-c}\cdot\frac{b-c}{b-a}
  5. ( a b c d ) \left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)
  6. f ( x ) = a x + b c x + d f(x)=\frac{ax+b}{cx+d}
  7. L 2 ( 4 ) A 5 L_{2}(4)\cong A_{5}
  8. L 2 ( 5 ) A 5 L_{2}(5)\cong A_{5}
  9. L 2 ( 9 ) A 6 L_{2}(9)\cong A_{6}
  10. L 4 ( 2 ) A 8 . L_{4}(2)\cong A_{8}.
  11. ( q 2 - 1 ) ( q 2 - q ) / ( q - 1 ) = q 3 - q = ( q - 1 ) q ( q + 1 ) ; (q^{2}-1)(q^{2}-q)/(q-1)=q^{3}-q=(q-1)q(q+1);
  12. L 2 ( 2 ) S 3 S 2 L_{2}(2)\cong S_{3}\twoheadrightarrow S_{2}
  13. L 2 ( 3 ) A 4 A 3 C 3 L_{2}(3)\cong A_{4}\twoheadrightarrow A_{3}\cong C_{3}
  14. L 2 ( 5 ) A 5 . L_{2}(5)\cong A_{5}.
  15. L 2 ( 5 ) L 2 ( 11 ) L_{2}(5)\hookrightarrow L_{2}(11)
  16. x x
  17. 1 / ( 1 - x ) 1/(1-x)
  18. ( x - 1 ) / x (x-1)/x
  19. ( 1 0 0 1 ) \begin{pmatrix}1&0\\ 0&1\end{pmatrix}
  20. ( 0 1 - 1 1 ) \begin{pmatrix}0&1\\ -1&1\end{pmatrix}
  21. ( 1 - 1 1 0 ) \begin{pmatrix}1&-1\\ 1&0\end{pmatrix}
  22. 1 / x 1/x
  23. 1 - x 1-x
  24. x / ( x - 1 ) x/(x-1)
  25. ( 0 1 1 0 ) \begin{pmatrix}0&1\\ 1&0\end{pmatrix}
  26. ( - 1 1 0 1 ) \begin{pmatrix}-1&1\\ 0&1\end{pmatrix}
  27. ( 1 0 1 - 1 ) \begin{pmatrix}1&0\\ 1&-1\end{pmatrix}
  28. ( 0 i i 0 ) \begin{pmatrix}0&i\\ i&0\end{pmatrix}
  29. ( - i i 0 i ) \begin{pmatrix}-i&i\\ 0&i\end{pmatrix}
  30. ( i 0 i - i ) \begin{pmatrix}i&0\\ i&-i\end{pmatrix}
  31. PGL ( 2 , 𝐙 / 2 ) PGL ( 2 , 𝐙 ) \operatorname{PGL}(2,\mathbf{Z}/2)\hookrightarrow\operatorname{PGL}(2,\mathbf{% Z})
  32. PGL ( 2 , 𝐙 ) PGL ( 2 , 𝐙 / 2 ) . \operatorname{PGL}(2,\mathbf{Z})\twoheadrightarrow\operatorname{PGL}(2,\mathbf% {Z}/2).
  33. φ ± = 1 ± 5 2 , \varphi_{\pm}=\frac{1\pm\sqrt{5}}{2},
  34. S 3 Inn ( S 3 ) S 3 . S_{3}\overset{\sim}{\to}\operatorname{Inn}(S_{3})\cong S_{3}.
  35. SL ( 2 n + 1 , 𝐑 ) PSL ( 2 n + 1 , 𝐑 ) \operatorname{SL}(2n+1,\mathbf{R})\overset{\sim}{\to}\operatorname{PSL}(2n+1,% \mathbf{R})
  36. 𝔰 𝔩 ( n ) : \mathfrak{sl}(n)\colon
  37. 𝔰 𝔩 ( n ) \mathfrak{sl}(n)
  38. S L ( 2 , ¯ 𝐑 ) \overline{SL(2,}{\mathbf{R}}{)}
  39. K * GL ( 1 , K ) K^{*}\overset{\sim}{\to}\operatorname{GL}(1,K)
  40. F * x n F * ; F^{*}\overset{x^{n}}{\to}F^{*};

Projective_module.html

  1. 0 A B P 0 0\rightarrow A\rightarrow B\rightarrow P\rightarrow 0\,
  2. { a i P i I } \{a_{i}\in P\mid i\in I\}
  3. { f i Hom ( P , R ) i I } \{f_{i}\in\mathrm{Hom}(P,R)\mid i\in I\}
  4. x = f i ( x ) a i x=\sum f_{i}(x)a_{i}
  5. M M
  6. M M
  7. M 𝔪 M_{\mathfrak{m}}
  8. R 𝔪 R_{\mathfrak{m}}
  9. 𝔪 \mathfrak{m}
  10. M 𝔭 M_{\mathfrak{p}}
  11. R 𝔭 R_{\mathfrak{p}}
  12. 𝔭 \mathfrak{p}
  13. f 1 , , f n R f_{1},\ldots,f_{n}\in R
  14. M [ f i - 1 ] M[f_{i}^{-1}]
  15. R [ f i - 1 ] R[f_{i}^{-1}]
  16. M ~ \widetilde{M}
  17. Spec R \operatorname{Spec}R
  18. k ( 𝔭 ) k(\mathfrak{p})
  19. M R k ( 𝔭 ) M\otimes_{R}k(\mathfrak{p})
  20. 𝔭 \mathfrak{p}
  21. 𝔭 \mathfrak{p}
  22. R 𝔭 R_{\mathfrak{p}}
  23. P 𝔭 P_{\mathfrak{p}}
  24. k ( 𝔭 ) = R 𝔭 / 𝔭 R 𝔭 k(\mathfrak{p})=R_{\mathfrak{p}}/\mathfrak{p}R_{\mathfrak{p}}
  25. R 𝔭 R_{\mathfrak{p}}

Projective_representation.html

  1. G G
  2. V V
  3. GL ( V , F ) PGL ( V , F ) \operatorname{GL}(V,F)\rightarrow\operatorname{PGL}(V,F)
  4. ρ : G P G L ( V ) ρ:G→PGL(V)
  5. G G L ( V ) G→GL(V)
  6. G G
  7. C C
  8. G G
  9. G L ( V ) P G L ( V ) GL(V)→PGL(V)
  10. P G L PGL
  11. G L GL
  12. ρ : G P G L ( V ) ρ:G→PGL(V)
  13. σ : C G L ( V ) σ:C→GL(V)
  14. C C
  15. G G
  16. G G
  17. G G
  18. G G
  19. G G
  20. g g
  21. G G
  22. L ( g ) L(g)
  23. P G L ( V ) PGL(V)
  24. G L ( V ) GL(V)
  25. L ( g h ) = c ( g , h ) L ( g ) L ( h ) L(gh)=c(g,h)L(g)L(h)
  26. c ( g , h ) c(g,h)
  27. c c
  28. c ( h , k ) c ( g , h k ) = c ( g , h ) c ( g h , k ) c(h,k)c(g,hk)=c(g,h)c(gh,k)
  29. g , h , k g,h,k
  30. G G
  31. c c
  32. L L
  33. L ( g ) = f ( g ) L ( g ) L′(g)=f(g)L(g)
  34. c ( g , h ) = f ( g h ) f ( g ) - 1 f ( h ) - 1 c ( g , h ) c^{\prime}(g,h)=f(gh)f(g)^{-1}f(h)^{-1}c(g,h)
  35. c c
  36. L L
  37. H < s u p > 2 ( G , F ) H<sup>2(G,F^{∗})

Projective_variety.html

  1. k [ x 0 , , x n ] / I k[x_{0},\ldots,x_{n}]/I
  2. k [ x 0 , , x n ] k[x_{0},\ldots,x_{n}]
  3. P ( z ) = ( z + n n ) P(z)={\left({{z+n}\atop{n}}\right)}
  4. Pic ( X ) \operatorname{Pic}(X)
  5. H 1 ( X , 𝒪 X * ) H^{1}(X,{\mathcal{O}_{X}}^{*})
  6. deg : Pic ( X ) 𝐙 \operatorname{deg}:\operatorname{Pic}(X)\to\mathbf{Z}
  7. 𝔾 ( k , n ) \mathbb{G}(k,n)
  8. 𝐏 n × 𝐏 m 𝐏 ( n + 1 ) ( m + 1 ) - 1 , ( x i , y j ) x i y j \mathbf{P}^{n}\times\mathbf{P}^{m}\to\mathbf{P}^{(n+1)(m+1)-1},(x_{i},y_{j})% \mapsto x_{i}y_{j}
  9. ( d - 1 n ) {\left({{d-1}\atop{n}}\right)}
  10. 𝐏 n \mathbf{P}^{n}
  11. ( d - 1 ) ( d - 2 ) / 2 (d-1)(d-2)/2
  12. C 𝐏 1 C\to\mathbf{P}^{1}
  13. g / L \mathbb{C}^{g}/L
  14. g = 1 g=1
  15. \wp
  16. / L 𝐏 2 , L ( 0 : 0 : 1 ) , z ( 1 : ( z ) : ( z ) ) \mathbb{C}/L\to\mathbf{P}^{2},L\mapsto(0:0:1),z\mapsto(1:\wp(z):\wp^{\prime}(z))
  17. / L \mathbb{C}/L
  18. g > 1 g>1
  19. g = 1 g=1
  20. k [ x 0 , , x n ] k[x_{0},...,x_{n}]
  21. R = k [ x 0 , , x n ] / P R=k[x_{0},...,x_{n}]/P
  22. 𝒪 x k ( X ) \mathcal{O}_{x}\subset k(X)
  23. 𝒪 X \mathcal{O}_{X}
  24. 𝒪 X ( U ) = x U 𝒪 x . \mathcal{O}_{X}(U)=\bigcap_{x\in U}\mathcal{O}_{x}.
  25. 𝒪 X \mathcal{O}_{X}
  26. 𝒪 x \mathcal{O}_{x}
  27. ( X , 𝒪 X ) (X,\mathcal{O}_{X})
  28. ( U i , 𝒪 X | U i ) , U i = { ( x 0 : x 1 : : x n ) X | x i 0 } (U_{i},\mathcal{O}_{X}|{U_{i}}),\quad U_{i}=\{(x_{0}:x_{1}:\cdots:x_{n})\in X|% x_{i}\neq 0\}
  29. k [ y 1 , , y n ] k ( X ) , y i x i / x 0 k[y_{1},\dots,y_{n}]\to k(X),\quad y_{i}\mapsto x_{i}/x_{0}
  30. ϕ : U 0 X , ( 1 : x 1 : : x n ) ( x 1 , , x n ) \phi:U_{0}\to X^{\prime},\quad(1:x_{1}:...:x_{n})\mapsto(x_{1},\dots,x_{n})
  31. ϕ # : 𝒪 ϕ ( x ) 𝒪 x , s s ϕ \phi^{\#}:\mathcal{O}_{\phi(x)}\overset{\sim}{\to}\mathcal{O}_{x},\,s\mapsto s\circ\phi
  32. ϕ # : k ( X ) k ( X ) \phi^{\#}:k(X^{\prime})\overset{\sim}{\to}k(X)
  33. 𝐏 A n \mathbf{P}^{n}_{A}
  34. U i = Spec A [ x 1 / x i , , x n / x i ] , 0 i n , U_{i}=\operatorname{Spec}A[x_{1}/x_{i},\dots,x_{n}/x_{i}],\quad 0\leq i\leq n,
  35. 𝐏 k n \mathbf{P}^{n}_{k}
  36. 𝐏 A n = Proj A [ x 0 , , x n ] . \mathbf{P}^{n}_{A}=\operatorname{Proj}A[x_{0},\ldots,x_{n}].
  37. k [ x 0 , , x n ] k[x_{0},\ldots,x_{n}]
  38. Proj R 𝐏 k n . \operatorname{Proj}R\to\mathbf{P}^{n}_{k}.
  39. 𝐏 k n \mathbf{P}^{n}_{k}
  40. k [ x 0 , , x n ] k[x_{0},\ldots,x_{n}]
  41. I : ( x 0 , , x n ) = I I:(x_{0},\dots,x_{n})=I
  42. 𝐏 ( V ) = Proj k [ V ] \mathbf{P}(V)=\operatorname{Proj}k[V]
  43. k [ V ] = Sym ( V * ) k[V]=\operatorname{Sym}(V^{*})
  44. V * V^{*}
  45. π : V - 0 𝐏 ( V ) \pi:V-0\to\mathbf{P}(V)
  46. | D | = 𝐏 ( Γ ( X , L ) ) |D|=\mathbf{P}(\Gamma(X,L))
  47. 𝐏 S n = 𝐏 𝐙 n × Spec 𝐙 S . \mathbf{P}^{n}_{S}=\mathbf{P}_{\mathbf{Z}}^{n}\times_{\operatorname{Spec}% \mathbf{Z}}S.
  48. 𝒪 ( 1 ) \mathcal{O}(1)
  49. 𝐏 𝐙 n \mathbf{P}_{\mathbf{Z}}^{n}
  50. 𝒪 ( 1 ) \mathcal{O}(1)
  51. 𝒪 ( 1 ) \mathcal{O}(1)
  52. 𝐏 S n \mathbf{P}^{n}_{S}
  53. 𝒪 ( 1 ) = g * ( 𝒪 ( 1 ) ) \mathcal{O}(1)=g^{*}(\mathcal{O}(1))
  54. g : 𝐏 S n 𝐏 𝐙 n . g:\mathbf{P}^{n}_{S}\to\mathbf{P}^{n}_{\mathbf{Z}}.
  55. X 𝐏 S n X\to\mathbf{P}^{n}_{S}
  56. \mathcal{L}
  57. i : X 𝐏 S n i:X\to\mathbf{P}^{n}_{S}
  58. 𝒪 ( 1 ) \mathcal{O}(1)
  59. . \mathcal{L}.
  60. 𝒪 ( 1 ) \mathcal{O}(1)
  61. Γ ( X , 𝒪 X ) = k \Gamma(X,\mathcal{O}_{X})=k
  62. ϕ : X 𝐏 A n = Proj A [ x 1 , , x n ] \phi:X\to\mathbf{P}^{n}_{A}=\operatorname{Proj}A[x_{1},\dots,x_{n}]
  63. 𝒪 ( 1 ) \mathcal{O}(1)
  64. ϕ * ( x i ) \phi^{*}(x_{i})
  65. s 0 , , s n s_{0},...,s_{n}
  66. ϕ : X 𝐏 A n \phi:X\to\mathbf{P}^{n}_{A}
  67. X i X_{i}
  68. U i U_{i}
  69. s i = 0 s_{i}=0
  70. x i = 0 x_{i}=0
  71. 𝐏 A n \mathbf{P}^{n}_{A}
  72. U i U_{i}
  73. ϕ : X i U i \phi:X_{i}\to U_{i}
  74. x i / x j s i / s j x_{i}/x_{j}\mapsto s_{i}/s_{j}
  75. ϕ * ( 𝒪 ( 1 ) ) \phi^{*}(\mathcal{O}(1))
  76. s i = ϕ * ( x i ) s_{i}=\phi^{*}(x_{i})
  77. ϕ \phi
  78. X i X_{i}
  79. Γ ( U i , 𝒪 𝐏 A n ) Γ ( X i , 𝒪 X i ) \Gamma(U_{i},\mathcal{O}_{\mathbf{P}^{n}_{A}})\to\Gamma(X_{i},\mathcal{O}_{X_{% i}})
  80. X \mathcal{M}_{X}
  81. U U\mapsto
  82. Γ ( U , 𝒪 X ) \Gamma(U,\mathcal{O}_{X})
  83. X * / 𝒪 X * \mathcal{M}_{X}^{*}/\mathcal{O}_{X}^{*}
  84. D ( D ) D\mapsto\mathcal{L}(D)
  85. 𝒪 ( 1 ) \mathcal{O}(1)
  86. \mathcal{F}
  87. i : X 𝐏 A r i:X\to\mathbf{P}^{r}_{A}
  88. H p ( X , ) = H p ( 𝐏 A r , ) , p 0 H^{p}(X,\mathcal{F})=H^{p}(\mathbf{P}^{r}_{A},\mathcal{F}),p\geq 0
  89. \mathcal{F}
  90. ( n ) = 𝒪 ( n ) \mathcal{F}(n)=\mathcal{F}\otimes\mathcal{O}(n)
  91. H p ( X , ) H^{p}(X,\mathcal{F})
  92. n 0 n_{0}
  93. \mathcal{F}
  94. H p ( X , ( n ) ) = 0 H^{p}(X,\mathcal{F}(n))=0
  95. n n 0 n\geq n_{0}
  96. = 𝒪 𝐏 r ( n ) , \mathcal{F}=\mathcal{O}_{\mathbf{P}^{r}}(n),
  97. R p f * R^{p}f_{*}\mathcal{F}
  98. χ ( ) = i = 0 ( - 1 ) i dim H i ( X , ) \chi(\mathcal{F})=\sum_{i=0}^{\infty}(-1)^{i}\operatorname{dim}H^{i}(X,% \mathcal{F})
  99. \mathcal{F}
  100. H i ( X , ( n ) ) H^{i}(X,\mathcal{F}(n))
  101. χ ( ( n ) ) = P ( n ) \chi(\mathcal{F}(n))=P(n)
  102. 𝒪 X \mathcal{O}_{X}
  103. ( - 1 ) r ( χ ( 𝒪 X ) - 1 ) , (-1)^{r}(\chi(\mathcal{O}_{X})-1),
  104. \mathcal{F}
  105. H i ( X , ) H n - i ( X , ω X ) H^{i}(X,\mathcal{F})\simeq H^{n-i}(X,\mathcal{F}^{\vee}\otimes\omega_{X})^{\prime}
  106. \mathcal{F}^{\vee}
  107. \mathcal{F}
  108. H 1 ( X , 𝒪 X ) H^{1}(X,\mathcal{O}_{X})
  109. Cl ( X ) Pic ( X ) , D 𝒪 ( D ) \operatorname{Cl}(X)\to\operatorname{Pic}(X),D\mapsto\mathcal{O}(D)
  110. H 0 ( X , 𝒪 ( D ) ) H^{0}(X,\mathcal{O}(D))
  111. l ( D ) - l ( K - D ) = deg D + 1 - g l(D)-l(K-D)=\operatorname{deg}D+1-g
  112. χ ( 𝒪 ( D ) ) = deg D + 1 - g \chi(\mathcal{O}(D))=\operatorname{deg}D+1-g
  113. \mathcal{L}
  114. H i ( X , ω X ) = 0 H^{i}(X,\mathcal{L}\otimes\omega_{X})=0
  115. H i ( X , - 1 ) = 0 H^{i}(X,\mathcal{L}^{-1})=0
  116. H X P H_{X}^{P}
  117. H X P H_{X}^{P}
  118. T H X P T\to H^{P}_{X}
  119. X × S T X\times_{S}T
  120. X × S H X P X\times_{S}H_{X}^{P}
  121. H X P H X P H_{X}^{P}\to H_{X}^{P}
  122. P ( z ) = ( z + k k ) P(z)={\left({{z+k}\atop{k}}\right)}
  123. H 𝐏 S n P H_{\mathbf{P}^{n}_{S}}^{P}
  124. 𝐏 S n \mathbf{P}^{n}_{S}
  125. H X P H_{X}^{P}
  126. H i ( X , ) H i ( X an , ) H^{i}(X,\mathcal{F})\to H^{i}(X\text{an},\mathcal{F})
  127. \mathcal{F}
  128. f ( x 0 , , x n ) = 0 f(x_{0},...,x_{n})=0
  129. f ( λ x 0 , , λ x n ) = 0 f(\lambda x_{0},...,\lambda x_{n})=0
  130. 𝒪 X ( X f ) \mathcal{O}_{X}(X_{f})
  131. \mathcal{F}

Proof_by_infinite_descent.html

  1. 2 = p q \sqrt{2}=\frac{p}{q}
  2. p p
  3. q q
  4. 2 = p 2 q 2 , 2=\frac{p^{2}}{q^{2}},
  5. 2 q 2 = p 2 , 2q^{2}=p^{2},\,
  6. p = 2 r p=2r
  7. 2 q 2 = ( 2 r ) 2 = 4 r 2 , 2q^{2}=(2r)^{2}=4r^{2},\,
  8. q 2 = 2 r 2 , q^{2}=2r^{2},\,
  9. k = m n = m ( k - q ) n ( k - q ) = m k - m q n k - n q = n k - m q m - n q ( replacing the first m in the numerator with n k and k in the denominator with m / n ) \begin{aligned}\displaystyle\sqrt{k}&\displaystyle=\frac{m}{n}\\ &\displaystyle=\frac{m(\sqrt{k}-q)}{n(\sqrt{k}-q)}\\ &\displaystyle=\frac{m\sqrt{k}-mq}{n\sqrt{k}-nq}\\ &\displaystyle=\frac{nk-mq}{m-nq}\,\text{ }(\,\text{replacing the first }m\,% \text{ in the numerator with }n\sqrt{k}\,\text{ and }\sqrt{k}\,\text{ in the % denominator with }m/n)\end{aligned}
  10. r 2 + s 4 = t 4 r^{2}+s^{4}=t^{4}
  11. q 4 + s 4 = t 4 q^{4}+s^{4}=t^{4}
  12. x = 2 a b , x=2ab,
  13. y = a 2 - b 2 , y=a^{2}-b^{2},
  14. z = a 2 + b 2 z=a^{2}+b^{2}
  15. y z \sqrt{yz}
  16. b 2 b^{2}
  17. a 2 a^{2}
  18. b 2 b^{2}
  19. a 2 a^{2}
  20. a 2 a^{2}
  21. z = a 2 + b 2 z=a^{2}+b^{2}
  22. b b
  23. y \sqrt{y}
  24. a a
  25. a a
  26. z = a 2 + b 2 z=a^{2}+b^{2}
  27. a a
  28. b b
  29. z \sqrt{z}
  30. a a
  31. b b
  32. z \sqrt{z}
  33. r 2 + s 4 = t 4 r^{2}+s^{4}=t^{4}

Proof_that_22::7_exceeds_π.html

  1. π \pi
  2. π \pi
  3. π \pi
  4. 22 7 \displaystyle\frac{22}{7}
  5. π \pi
  6. 0 < 0 1 x 4 ( 1 - x ) 4 1 + x 2 d x = 22 7 - π . 0<\int_{0}^{1}\frac{x^{4}(1-x)^{4}}{1+x^{2}}\,dx=\frac{22}{7}-\pi.
  7. π \pi
  8. 0 \displaystyle 0
  9. x x
  10. x x
  11. 1 1260 = 0 1 x 4 ( 1 - x ) 4 2 d x < 0 1 x 4 ( 1 - x ) 4 1 + x 2 d x < 0 1 x 4 ( 1 - x ) 4 1 d x = 1 630 . \frac{1}{1260}=\int_{0}^{1}\frac{x^{4}(1-x)^{4}}{2}\,dx<\int_{0}^{1}\frac{x^{4% }(1-x)^{4}}{1+x^{2}}\,dx<\int_{0}^{1}\frac{x^{4}(1-x)^{4}}{1}\,dx={1\over 630}.
  12. 22 7 - 1 630 < π < 22 7 - 1 1260 , {22\over 7}-{1\over 630}<\pi<{22\over 7}-{1\over 1260},
  13. π \pi
  14. π \pi
  15. 0 < 0 1 x 8 ( 1 - x ) 8 ( 25 + 816 x 2 ) 3164 ( 1 + x 2 ) d x = 355 113 - π . 0<\int_{0}^{1}\frac{x^{8}(1-x)^{8}(25+816x^{2})}{3164(1+x^{2})}\,dx=\frac{355}% {113}-\pi.
  16. 355 113 = 3.141 592 92 , \frac{355}{113}=3.141\,592\,92\ldots,
  17. π \pi
  18. x x
  19. 0 1 x 8 ( 1 - x ) 8 ( 25 + 816 x 2 ) 6328 d x = 911 5 261 111 856 = 0.000 000 173 , \int_{0}^{1}\frac{x^{8}(1-x)^{8}(25+816x^{2})}{6328}\,dx=\frac{911}{5\,261\,11% 1\,856}=0.000\,000\,173\ldots,
  20. x x
  21. 355 113 - 911 2 630 555 928 < π < 355 113 - 911 5 261 111 856 . \frac{355}{113}-\frac{911}{2\,630\,555\,928}<\pi<\frac{355}{113}-\frac{911}{5% \,261\,111\,856}\,.
  22. π \pi
  23. n 1 n≥1
  24. 1 2 2 n - 1 0 1 x 4 n ( 1 - x ) 4 n d x < 1 2 2 n - 2 0 1 x 4 n ( 1 - x ) 4 n 1 + x 2 d x < 1 2 2 n - 2 0 1 x 4 n ( 1 - x ) 4 n d x , \frac{1}{2^{2n-1}}\int_{0}^{1}x^{4n}(1-x)^{4n}\,dx<\frac{1}{2^{2n-2}}\int_{0}^% {1}\frac{x^{4n}(1-x)^{4n}}{1+x^{2}}\,dx<\frac{1}{2^{2n-2}}\int_{0}^{1}x^{4n}(1% -x)^{4n}\,dx,
  25. 1 2 2 n - 2 0 1 x 4 n ( 1 - x ) 4 n 1 + x 2 d x = j = 0 2 n - 1 ( - 1 ) j 2 2 n - j - 2 ( 8 n - j - 1 ) ( 8 n - j - 2 4 n + j ) + ( - 1 ) n ( π - 4 j = 0 3 n - 1 ( - 1 ) j 2 j + 1 ) \begin{aligned}&\displaystyle\frac{1}{2^{2n-2}}\int_{0}^{1}\frac{x^{4n}(1-x)^{% 4n}}{1+x^{2}}\,dx\\ &\displaystyle\qquad=\sum_{j=0}^{2n-1}\frac{(-1)^{j}}{2^{2n-j-2}(8n-j-1){\left% ({{8n-j-2}\atop{4n+j}}\right)}}+(-1)^{n}\biggl(\pi-4\sum_{j=0}^{3n-1}\frac{(-1% )^{j}}{2j+1}\biggr)\end{aligned}
  26. π \pi
  27. π \pi
  28. 1 2 2 n - 1 0 1 x 4 n ( 1 - x ) 4 n d x = 1 2 2 n - 1 ( 8 n + 1 ) ( 8 n 4 n ) π n 2 10 n - 2 ( 8 n + 1 ) , \begin{aligned}\displaystyle\frac{1}{2^{2n-1}}\int_{0}^{1}x^{4n}(1-x)^{4n}\,dx% &\displaystyle=\frac{1}{2^{2n-1}(8n+1){\left({{8n}\atop{4n}}\right)}}\\ &\displaystyle\sim\frac{\sqrt{\pi n}}{2^{10n-2}(8n+1)},\end{aligned}
  29. n n
  30. π \pi
  31. k 0 k≥0
  32. l 2 l≥2
  33. x k ( 1 - x ) l = ( 1 - 2 x + x 2 ) x k ( 1 - x ) l - 2 = ( 1 + x 2 ) x k ( 1 - x ) l - 2 - 2 x k + 1 ( 1 - x ) l - 2 . \begin{aligned}\displaystyle x^{k}(1-x)^{l}&\displaystyle=(1-2x+x^{2})x^{k}(1-% x)^{l-2}\\ &\displaystyle=(1+x^{2})\,x^{k}(1-x)^{l-2}-2x^{k+1}(1-x)^{l-2}.\end{aligned}
  34. 2 n 2n
  35. x 4 n ( 1 - x ) 4 n = ( 1 + x 2 ) j = 0 2 n - 1 ( - 2 ) j x 4 n + j ( 1 - x ) 4 n - 2 ( j + 1 ) + ( - 2 ) 2 n x 6 n . x^{4n}(1-x)^{4n}=(1+x^{2})\sum_{j=0}^{2n-1}(-2)^{j}x^{4n+j}(1-x)^{4n-2(j+1)}+(% -2)^{2n}x^{6n}.
  36. x 6 n - ( - 1 ) 3 n = j = 1 3 n ( - 1 ) 3 n - j x 2 j - j = 0 3 n - 1 ( - 1 ) 3 n - j x 2 j = j = 0 3 n - 1 ( ( - 1 ) 3 n - ( j + 1 ) x 2 ( j + 1 ) - ( - 1 ) 3 n - j x 2 j ) = - ( 1 + x 2 ) j = 0 3 n - 1 ( - 1 ) 3 n - j x 2 j , \begin{aligned}\displaystyle x^{6n}-(-1)^{3n}&\displaystyle=\sum_{j=1}^{3n}(-1% )^{3n-j}x^{2j}-\sum_{j=0}^{3n-1}(-1)^{3n-j}x^{2j}\\ &\displaystyle=\sum_{j=0}^{3n-1}\bigl((-1)^{3n-(j+1)}x^{2(j+1)}-(-1)^{3n-j}x^{% 2j}\bigr)\\ &\displaystyle=-(1+x^{2})\sum_{j=0}^{3n-1}(-1)^{3n-j}x^{2j},\\ \end{aligned}
  37. 1 j 3 n 1 1≤j≤3n−1
  38. j j + 1 j→j+1
  39. x 4 n ( 1 - x ) 4 n 2 2 n - 2 ( 1 + x 2 ) = j = 0 2 n - 1 ( - 1 ) j 2 2 n - j - 2 x 4 n + j ( 1 - x ) 4 n - 2 j - 2 - 4 j = 0 3 n - 1 ( - 1 ) 3 n - j x 2 j + ( - 1 ) 3 n 4 1 + x 2 . ( * ) \begin{aligned}\displaystyle\frac{x^{4n}(1-x)^{4n}}{2^{2n-2}(1+x^{2})}&% \displaystyle=\sum_{j=0}^{2n-1}\frac{(-1)^{j}}{2^{2n-j-2}}x^{4n+j}(1-x)^{4n-2j% -2}\\ &\displaystyle\qquad{}-4\sum_{j=0}^{3n-1}(-1)^{3n-j}x^{2j}+(-1)^{3n}\frac{4}{1% +x^{2}}.\qquad(*)\end{aligned}
  40. k , l 0 k,l≥0
  41. l l
  42. 0 1 x k ( 1 - x ) l d x = l k + 1 0 1 x k + 1 ( 1 - x ) l - 1 d x = = l k + 1 l - 1 k + 2 1 k + l 0 1 x k + l d x = 1 ( k + l + 1 ) ( k + l k ) . ( * * ) \begin{aligned}\displaystyle\int_{0}^{1}x^{k}(1-x)^{l}\,dx&\displaystyle=\frac% {l}{k+1}\int_{0}^{1}x^{k+1}(1-x)^{l-1}\,dx\\ &\displaystyle=\cdots\\ &\displaystyle=\frac{l}{k+1}\frac{l-1}{k+2}\cdots\frac{1}{k+l}\int_{0}^{1}x^{k% +l}\,dx\\ &\displaystyle=\frac{1}{(k+l+1){\left({{k+l}\atop{k}}\right)}}.\qquad(**)\end{aligned}
  43. k = l = 4 n k=l=4n
  44. 0 1 x 4 n ( 1 - x ) 4 n d x = 1 ( 8 n + 1 ) ( 8 n 4 n ) . \int_{0}^{1}x^{4n}(1-x)^{4n}\,dx=\frac{1}{(8n+1){\left({{8n}\atop{4n}}\right)}}.
  45. a r c t a n ( 1 ) = π / 4 arctan(1)=π/4
  46. π \pi
  47. n = 1 n=1
  48. n = 2 n=2
  49. 1 4 0 1 x 8 ( 1 - x ) 8 1 + x 2 d x = π - 47 171 15 015 \frac{1}{4}\int_{0}^{1}\frac{x^{8}(1-x)^{8}}{1+x^{2}}\,dx=\pi-\frac{47\,171}{1% 5\,015}
  50. 1 8 0 1 x 8 ( 1 - x ) 8 d x = 1 1 750 320 , \frac{1}{8}\int_{0}^{1}x^{8}(1-x)^{8}\,dx=\frac{1}{1\,750\,320},
  51. n = 3 n=3
  52. 1 16 0 1 x 12 ( 1 - x ) 12 1 + x 2 d x = 431 302 721 137 287 920 - π \frac{1}{16}\int_{0}^{1}\frac{x^{12}(1-x)^{12}}{1+x^{2}}\,dx=\frac{431\,302\,7% 21}{137\,287\,920}-\pi
  53. 1 32 0 1 x 12 ( 1 - x ) 12 d x = 1 2 163 324 800 , \frac{1}{32}\int_{0}^{1}x^{12}(1-x)^{12}\,dx=\frac{1}{2\,163\,324\,800},
  54. 1 64 0 1 x 16 ( 1 - x ) 16 1 + x 2 d x = π - 741 269 838 109 235 953 517 800 \frac{1}{64}\int_{0}^{1}\frac{x^{16}(1-x)^{16}}{1+x^{2}}\,dx=\pi-\frac{741\,26% 9\,838\,109}{235\,953\,517\,800}
  55. 1 128 0 1 x 16 ( 1 - x ) 16 d x = 1 2 538 963 567 360 , \frac{1}{128}\int_{0}^{1}x^{16}(1-x)^{16}\,dx=\frac{1}{2\,538\,963\,567\,360},
  56. π \pi
  57. π \pi
  58. π \pi
  59. π \pi

Proof_that_e_is_irrational.html

  1. e = [ 2 ; 1 , 2 , 1 , 1 , 4 , 1 , 1 , 6 , 1 , 1 , 8 , 1 , 1 , , 2 n , 1 , 1 , ] . e=[2;1,2,1,1,4,1,1,6,1,1,8,1,1,\ldots,2n,1,1,\ldots].\,
  2. e = n = 0 1 n ! e=\sum_{n=0}^{\infty}\frac{1}{n!}\cdot
  3. 1 1 + 1 1 < e = 1 1 + 1 1 + 1 1 2 + 1 1 2 3 + < 1 1 + 1 1 + 1 1 2 + 1 1 2 2 + = 3 \frac{1}{1}\ +\frac{1}{1}\ <e=\frac{1}{1}\ +\frac{1}{1}\ +\frac{1}{1\cdot 2}\ % +\frac{1}{1\cdot 2\cdot 3}\ +...<\frac{1}{1}\ +\frac{1}{1}\ +\frac{1}{1\cdot 2% }\ +\frac{1}{1\cdot 2\cdot 2}\ +...=3
  4. x = b ! ( e - n = 0 b 1 n ! ) x=b!\,\biggl(e-\sum_{n=0}^{b}\frac{1}{n!}\biggr)\!
  5. x = b ! ( a b - n = 0 b 1 n ! ) = a ( b - 1 ) ! - n = 0 b b ! n ! . x=b!\,\biggl(\frac{a}{b}-\sum_{n=0}^{b}\frac{1}{n!}\biggr)=a(b-1)!-\sum_{n=0}^% {b}\frac{b!}{n!}\,.
  6. x = n = b + 1 b ! n ! < n = b + 1 1 ( b + 1 ) n - b = k = 1 1 ( b + 1 ) k = 1 b + 1 ( 1 1 - 1 b + 1 ) = 1 b < 1. x=\sum_{n=b+1}^{\infty}\frac{b!}{n!}<\sum_{n=b+1}^{\infty}\frac{1}{(b+1)^{n-b}% }=\sum_{k=1}^{\infty}\frac{1}{(b+1)^{k}}=\frac{1}{b+1}\biggl(\frac{1}{1-\frac{% 1}{b+1}}\biggr)=\frac{1}{b}<1.
  7. ( b + 1 ) x = 1 + 1 b + 2 + 1 ( b + 2 ) ( b + 3 ) + < 1 + 1 b + 1 + 1 ( b + 1 ) ( b + 2 ) + = 1 + x , (b+1)x=1+\frac{1}{b+2}+\frac{1}{(b+2)(b+3)}+\cdots<1+\frac{1}{b+1}+\frac{1}{(b% +1)(b+2)}+\cdots=1+x,
  8. 1 e = e - 1 = n = 0 ( - 1 ) n n ! \frac{1}{e}=e^{-1}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\cdot

Propagation_of_uncertainty.html

  1. Δ x Δx
  2. ( Δ x ) / x (Δx)/x
  3. Δ x Δx
  4. σ σ
  5. x ± Δ x x±Δx
  6. x ± σ x±σ
  7. { f k ( x 1 , x 2 , , x n ) } \{f_{k}(x_{1},x_{2},\dots,x_{n})\}
  8. n n
  9. x 1 , x 2 , , x n x_{1},x_{2},\dots,x_{n}
  10. A k 1 , A k 2 , , A k n , ( k = 1 m ) A_{k1},A_{k2},\dots,A_{kn},(k=1\dots m)
  11. f k = i n A k i x i f_{k}=\sum_{i}^{n}A_{ki}x_{i}
  12. f = Ax \mathrm{f}=\mathrm{Ax}\,
  13. Σ x \mathrm{\Sigma^{x}}\,
  14. Σ x = ( σ 1 2 σ 12 σ 13 σ 12 σ 2 2 σ 23 σ 13 σ 23 σ 3 2 ) = ( Σ 1 x Σ 12 x Σ 13 x Σ 12 x Σ 2 x Σ 23 x Σ 13 x Σ 23 x Σ 3 x ) \mathrm{\Sigma^{x}}=\begin{pmatrix}\sigma^{2}_{1}&\sigma_{12}&\sigma_{13}&% \cdots\\ \sigma_{12}&\sigma^{2}_{2}&\sigma_{23}&\cdots\\ \sigma_{13}&\sigma_{23}&\sigma^{2}_{3}&\cdots\\ \vdots&\vdots&\vdots&\ddots\\ \end{pmatrix}=\begin{pmatrix}\mathit{\Sigma}^{x}_{1}&\mathit{\Sigma}^{x}_{12}&% \mathit{\Sigma}^{x}_{13}&\cdots\\ \mathit{\Sigma}^{x}_{12}&\mathit{\Sigma}^{x}_{2}&\mathit{\Sigma}^{x}_{23}&% \cdots\\ \mathit{\Sigma}^{x}_{13}&\mathit{\Sigma}^{x}_{23}&\mathit{\Sigma}^{x}_{3}&% \cdots\\ \vdots&\vdots&\vdots&\ddots\\ \end{pmatrix}
  15. Σ f \mathrm{\Sigma^{f}}\,
  16. Σ i j f = k n n A i k Σ k x A j : Σ f = A Σ x A \mathit{\Sigma}^{f}_{ij}=\sum_{k}^{n}\sum_{\ell}^{n}A_{ik}\mathit{\Sigma}^{x}_% {k\ell}A_{j\ell}:\mathrm{\Sigma^{f}=A\Sigma^{x}A^{\top}}
  17. Σ i j f = k n A i k Σ k x A j k . \mathit{\Sigma}^{f}_{ij}=\sum_{k}^{n}A_{ik}\mathit{\Sigma}^{x}_{k}A_{jk}.
  18. Σ k x = σ x k 2 \mathit{\Sigma}^{x}_{k}=\sigma^{2}_{x_{k}}
  19. Σ x \mathrm{\Sigma^{x}}
  20. Σ f \mathrm{\Sigma^{f}}
  21. f = i n a i x i : f = ax f=\sum_{i}^{n}a_{i}x_{i}:f=\mathrm{ax}\,
  22. σ f 2 = i n j n a i Σ i j x a j = a Σ x a \sigma^{2}_{f}=\sum_{i}^{n}\sum_{j}^{n}a_{i}\mathit{\Sigma}^{x}_{ij}a_{j}=% \mathrm{a\Sigma^{x}a^{\top}}
  23. σ i j \sigma_{ij}
  24. ρ i j \rho_{ij}\,
  25. σ i j = ρ i j σ i σ j \sigma_{ij}=\rho_{ij}\sigma_{i}\sigma_{j}\,
  26. σ f 2 = i n a i 2 σ i 2 + i n j ( j i ) n a i a j ρ i j σ i σ j . \sigma^{2}_{f}=\sum_{i}^{n}a_{i}^{2}\sigma^{2}_{i}+\sum_{i}^{n}\sum_{j(j\neq i% )}^{n}a_{i}a_{j}\rho_{ij}\sigma_{i}\sigma_{j}.
  27. σ f 2 = i n a i 2 σ i 2 . \sigma^{2}_{f}=\sum_{i}^{n}a_{i}^{2}\sigma^{2}_{i}.
  28. σ f = n a σ . \sigma_{f}=\sqrt{n}a\sigma.
  29. f k f k 0 + i n f k x i x i f_{k}\approx f^{0}_{k}+\sum_{i}^{n}\frac{\partial f_{k}}{\partial{x_{i}}}x_{i}
  30. f k / x i \partial f_{k}/\partial x_{i}
  31. f f 0 + Jx \mathrm{f}\approx\mathrm{f}^{0}+\mathrm{J}\mathrm{x}\,
  32. f k x i \frac{\partial f_{k}}{\partial x_{i}}
  33. f k x j \frac{\partial f_{k}}{\partial x_{j}}
  34. Σ f = J Σ x J \mathrm{\Sigma}^{\mathrm{f}}=\mathrm{J}\mathrm{\Sigma}^{\mathrm{x}}\mathrm{J}^% {\top}
  35. s f = ( f x ) 2 s x 2 + ( f y ) 2 s y 2 + ( f z ) 2 s z 2 + s_{f}=\sqrt{\left(\frac{\partial f}{\partial{x}}\right)^{2}s_{x}^{2}+\left(% \frac{\partial f}{\partial{y}}\right)^{2}s_{y}^{2}+\left(\frac{\partial f}{% \partial{z}}\right)^{2}s_{z}^{2}+...}
  36. s f s_{f}
  37. f f
  38. s x s_{x}
  39. x x
  40. s y s_{y}
  41. y y
  42. f f
  43. f f
  44. s x , s y , s z , s_{x},s_{y},s_{z},...
  45. f f 0 + f a a + f b b f\approx f^{0}+\frac{\partial f}{\partial a}a+\frac{\partial f}{\partial b}b
  46. σ f 2 | f a | 2 σ a 2 + | f b | 2 σ b 2 + 2 f a f b σ a b . \sigma^{2}_{f}\approx\left|\frac{\partial f}{\partial a}\right|^{2}\sigma^{2}_% {a}+\left|\frac{\partial f}{\partial b}\right|^{2}\sigma^{2}_{b}+2\frac{% \partial f}{\partial a}\frac{\partial f}{\partial b}\sigma_{ab}.
  47. f = a b f=ab\!
  48. f a = b , f b = a \frac{\partial f}{\partial a}=b,\frac{\partial f}{\partial b}=a
  49. σ f 2 b 2 σ a 2 + a 2 σ b 2 + 2 a b σ a b \sigma^{2}_{f}\approx b^{2}\sigma^{2}_{a}+a^{2}\sigma_{b}^{2}+2ab\,\sigma_{ab}
  50. ( σ f f ) 2 ( σ a a ) 2 + ( σ b b ) 2 + 2 ( σ a a ) ( σ b b ) ρ a b . \left(\frac{\sigma_{f}}{f}\right)^{2}\approx\left(\frac{\sigma_{a}}{a}\right)^% {2}+\left(\frac{\sigma_{b}}{b}\right)^{2}+2\left(\frac{\sigma_{a}}{a}\right)% \left(\frac{\sigma_{b}}{b}\right)\rho_{ab}.
  51. 1 / B 1/B
  52. B = N ( 0 , 1 ) B=N(0,1)
  53. 1 p - B \frac{1}{p-B}
  54. B = N ( μ , σ ) B=N(\mu,\sigma)
  55. p p
  56. μ \mu
  57. 2 σ F ( p - μ 2 σ ) \frac{\sqrt{2}}{\sigma}F\left(\frac{p-\mu}{\sqrt{2}\sigma}\right)
  58. p - μ p-\mu
  59. Im ( p - μ ) \operatorname{Im}(p-\mu)
  60. p - μ p-\mu
  61. p - μ p-\mu
  62. p 1 p_{1}
  63. p 2 p_{2}
  64. B B
  65. A , B A,B\!
  66. σ A , σ B \sigma_{A},\sigma_{B}\,
  67. σ A B \sigma_{AB}
  68. a , b a,b\,
  69. σ a = σ b = 0 \sigma_{a}=\sigma_{b}=0
  70. f = a A f=aA\,
  71. σ f 2 = a 2 σ A 2 \sigma_{f}^{2}=a^{2}\sigma_{A}^{2}
  72. σ f = a σ A \sigma_{f}=a\sigma_{A}
  73. f = a A + b B f=aA+bB\,
  74. σ f 2 = a 2 σ A 2 + b 2 σ B 2 + 2 a b σ A B \sigma_{f}^{2}=a^{2}\sigma_{A}^{2}+b^{2}\sigma_{B}^{2}+2ab\,\sigma_{AB}
  75. σ f = a 2 σ A 2 + b 2 σ B 2 + 2 a b σ A B \sigma_{f}=\sqrt{a^{2}\sigma_{A}^{2}+b^{2}\sigma_{B}^{2}+2ab\,\sigma_{AB}}
  76. f = a A - b B f=aA-bB\,
  77. σ f 2 = a 2 σ A 2 + b 2 σ B 2 - 2 a b σ A B \sigma_{f}^{2}=a^{2}\sigma_{A}^{2}+b^{2}\sigma_{B}^{2}-2ab\,\sigma_{AB}
  78. σ f = a 2 σ A 2 + b 2 σ B 2 - 2 a b σ A B \sigma_{f}=\sqrt{a^{2}\sigma_{A}^{2}+b^{2}\sigma_{B}^{2}-2ab\,\sigma_{AB}}
  79. f = A B f=AB\,
  80. σ f 2 f 2 [ ( σ A A ) 2 + ( σ B B ) 2 + 2 σ A B A B ] \sigma_{f}^{2}\approx f^{2}\left[\left(\frac{\sigma_{A}}{A}\right)^{2}+\left(% \frac{\sigma_{B}}{B}\right)^{2}+2\frac{\sigma_{AB}}{AB}\right]
  81. σ f | f | ( σ A A ) 2 + ( σ B B ) 2 + 2 σ A B A B \sigma_{f}\approx\left|f\right|\sqrt{\left(\frac{\sigma_{A}}{A}\right)^{2}+% \left(\frac{\sigma_{B}}{B}\right)^{2}+2\frac{\sigma_{AB}}{AB}}
  82. f = A B f=\frac{A}{B}\,
  83. σ f 2 f 2 [ ( σ A A ) 2 + ( σ B B ) 2 - 2 σ A B A B ] \sigma_{f}^{2}\approx f^{2}\left[\left(\frac{\sigma_{A}}{A}\right)^{2}+\left(% \frac{\sigma_{B}}{B}\right)^{2}-2\frac{\sigma_{AB}}{AB}\right]
  84. σ f | f | ( σ A A ) 2 + ( σ B B ) 2 - 2 σ A B A B \sigma_{f}\approx\left|f\right|\sqrt{\left(\frac{\sigma_{A}}{A}\right)^{2}+% \left(\frac{\sigma_{B}}{B}\right)^{2}-2\frac{\sigma_{AB}}{AB}}
  85. f = a A b f=aA^{b}\,
  86. σ f 2 ( a b A b - 1 σ A ) 2 = ( f b σ A A ) 2 \sigma_{f}^{2}\approx\left({a}{b}{A}^{b-1}{\sigma_{A}}\right)^{2}=\left(\frac{% {f}{b}{\sigma_{A}}}{A}\right)^{2}
  87. σ f | a b A b - 1 σ A | = | f b σ A A | \sigma_{f}\approx\left|{a}{b}{A}^{b-1}{\sigma_{A}}\right|=\left|\frac{{f}{b}{% \sigma_{A}}}{A}\right|
  88. f = a ln ( b A ) f=a\ln(bA)\,
  89. σ f 2 ( a σ A A ) 2 \sigma_{f}^{2}\approx\left(a\frac{\sigma_{A}}{A}\right)^{2}
  90. σ f | a σ A A | \sigma_{f}\approx\left|a\frac{\sigma_{A}}{A}\right|
  91. f = a log 10 ( A ) f=a\log_{10}(A)\,
  92. σ f 2 ( a σ A A ln ( 10 ) ) 2 \sigma_{f}^{2}\approx\left(a\frac{\sigma_{A}}{A\ln(10)}\right)^{2}
  93. σ f | a σ A A ln ( 10 ) | \sigma_{f}\approx\left|a\frac{\sigma_{A}}{A\ln(10)}\right|
  94. f = a e b A f=ae^{bA}\,
  95. σ f 2 f 2 ( b σ A ) 2 \sigma_{f}^{2}\approx f^{2}\left(b\sigma_{A}\right)^{2}
  96. σ f | f ( b σ A ) | \sigma_{f}\approx\left|f\left(b\sigma_{A}\right)\right|
  97. f = a b A f=a^{bA}\,
  98. σ f 2 f 2 ( b ln ( a ) σ A ) 2 \sigma_{f}^{2}\approx f^{2}\left(b\ln(a)\sigma_{A}\right)^{2}
  99. σ f | f ( b ln ( a ) σ A ) | \sigma_{f}\approx\left|f\left(b\ln(a)\sigma_{A}\right)\right|
  100. f = A B f=A^{B}\,
  101. σ f 2 f 2 [ ( B A σ A ) 2 + ( ln ( A ) σ B ) 2 + 2 B ln ( A ) A σ A B ] \sigma_{f}^{2}\approx f^{2}\left[\left(\frac{B}{A}\sigma_{A}\right)^{2}+\left(% \ln(A)\sigma_{B}\right)^{2}+2\frac{B\ln(A)}{A}\sigma_{AB}\right]
  102. σ f | f | ( B A σ A ) 2 + ( ln ( A ) σ B ) 2 + 2 B ln ( A ) A σ A B \sigma_{f}\approx\left|f\right|\sqrt{\left(\frac{B}{A}\sigma_{A}\right)^{2}+% \left(\ln(A)\sigma_{B}\right)^{2}+2\frac{B\ln(A)}{A}\sigma_{AB}}
  103. ρ A B = 0 \rho_{AB}=0
  104. σ A B = ρ A B σ A σ B \sigma_{AB}=\rho_{AB}\sigma_{A}\sigma_{B}\,
  105. f = A B C ; ( σ f f ) 2 ( σ A A ) 2 + ( σ B B ) 2 + ( σ C C ) 2 . f=ABC;\left(\frac{\sigma_{f}}{f}\right)^{2}\approx\left(\frac{\sigma_{A}}{A}% \right)^{2}+\left(\frac{\sigma_{B}}{B}\right)^{2}+\left(\frac{\sigma_{C}}{C}% \right)^{2}.
  106. f = A B f=AB
  107. V ( X Y ) = E ( X ) 2 V ( Y ) + E ( Y ) 2 V ( X ) + E ( ( X - E ( X ) ) 2 ( Y - E ( Y ) ) 2 ) V(XY)=E(X)^{2}V(Y)+E(Y)^{2}V(X)+E((X-E(X))^{2}(Y-E(Y))^{2})
  108. σ f 2 = A 2 σ B 2 + B 2 σ A 2 + σ A 2 σ B 2 \sigma_{f}^{2}=A^{2}\sigma_{B}^{2}+B^{2}\sigma_{A}^{2}+\sigma_{A}^{2}\sigma_{B% }^{2}
  109. f ( x ) = arctan ( x ) , f(x)=\arctan(x),
  110. x x
  111. f ( x ) f(x)
  112. x x
  113. d f d x = 1 1 + x 2 . \frac{df}{dx}=\frac{1}{1+x^{2}}.
  114. σ f σ x 1 + x 2 , \sigma_{f}\approx\frac{\sigma_{x}}{1+x^{2}},
  115. I I
  116. V V
  117. R R
  118. R = V / I R=V/I
  119. σ R σ V 2 ( 1 I ) 2 + σ I 2 ( - V I 2 ) 2 . \sigma_{R}\approx\sqrt{\sigma_{V}^{2}\left(\frac{1}{I}\right)^{2}+\sigma_{I}^{% 2}\left(\frac{-V}{I^{2}}\right)^{2}}.

Proper_convex_function.html

  1. f ( x ) < + f(x)<+\infty
  2. f ( x ) > - f(x)>-\infty
  3. - -\infty
  4. f = - g f=-g
  5. f ( x ) x b - β f(x)\geq x\cdot b-\beta
  6. A X A\subset X
  7. B X B\subset X
  8. I A I_{A}
  9. I B I_{B}
  10. A B = A\cap B=\emptyset
  11. I A + I B I_{A}+I_{B}
  12. + +\infty

Proper_morphism.html

  1. X × Y Z Z X\times_{Y}Z\to Z
  2. f × id : 𝔸 1 × 𝔸 1 { x } × 𝔸 1 f\times\textrm{id}:\mathbb{A}^{1}\times\mathbb{A}^{1}\to\{x\}\times\mathbb{A}^% {1}
  3. X Z Y X\to Z\to Y
  4. 𝒪 \mathcal{O}
  5. 𝒪 ( * ) \mathcal{O}(\mathbb{C}^{*})
  6. f : X S f:X\to S
  7. F F
  8. 𝒪 X \mathcal{O}_{X}
  9. i 0 i\geq 0
  10. R i f * F R^{i}f_{*}F
  11. \mathbb{C}
  12. f ( ) : X ( ) Y ( ) f(\mathbb{C}):X(\mathbb{C})\to Y(\mathbb{C})
  13. f ( ) f(\mathbb{C})
  14. x ¯ X ( R ) \overline{x}\in X(R)
  15. x ¯ X ( R ) \overline{x}\in X(R)
  16. f : 𝔛 𝔖 f:\mathfrak{X}\to\mathfrak{S}
  17. 𝔛 \mathfrak{X}
  18. 𝔖 \mathfrak{S}
  19. f 0 : X 0 Y 0 f_{0}:X_{0}\to Y_{0}
  20. X 0 = ( 𝔛 , 𝒪 𝔛 / I ) , S 0 = ( 𝔛 , 𝒪 𝔛 / K ) , I = f * ( K ) 𝒪 𝔛 X_{0}=(\mathfrak{X},\mathcal{O}_{\mathfrak{X}}/I),S_{0}=(\mathfrak{X},\mathcal% {O}_{\mathfrak{X}}/K),I=f^{*}(K)\mathcal{O}_{\mathfrak{X}}
  21. 𝔖 \mathfrak{S}
  22. X n = ( 𝔛 , 𝒪 𝔛 / I n + 1 ) , S n = ( 𝔛 , 𝒪 𝔛 / K n + 1 ) X_{n}=(\mathfrak{X},\mathcal{O}_{\mathfrak{X}}/I^{n+1}),S_{n}=(\mathfrak{X},% \mathcal{O}_{\mathfrak{X}}/K^{n+1})
  23. f n : X n S n f_{n}:X_{n}\to S_{n}
  24. g : Y Z g:Y\to Z
  25. g ^ : Y ^ Z ^ \widehat{g}:\widehat{Y}\to\widehat{Z}
  26. f : 𝔛 𝔖 f:\mathfrak{X}\to\mathfrak{S}
  27. 𝒪 𝔛 \mathcal{O}_{\mathfrak{X}}
  28. R i f * F R^{i}f_{*}F