wpmath0000001_16

Octahedron.html

  1. 1 \ell_{1}
  2. r u = a 2 2 0.7071067 a r_{u}=\frac{a}{2}\sqrt{2}\approx 0.7071067\cdot a
  3. r i = a 6 6 0.4082482 a r_{i}=\frac{a}{6}\sqrt{6}\approx 0.4082482\cdot a
  4. r m = a 2 = 0.5 a r_{m}=\frac{a}{2}=0.5\cdot a
  5. | x - a | + | y - b | + | z - c | = r . \left|x-a\right|+\left|y-b\right|+\left|z-c\right|=r.
  6. A = 2 3 a 2 3.46410162 a 2 A=2\sqrt{3}a^{2}\approx 3.46410162a^{2}
  7. V = 1 3 2 a 3 0.471404521 a 3 V=\frac{1}{3}\sqrt{2}a^{3}\approx 0.471404521a^{3}
  8. | x x m | + | y y m | + | z z m | = 1 \left|\frac{x}{x_{m}}\right|+\left|\frac{y}{y_{m}}\right|+\left|\frac{z}{z_{m}% }\right|=1
  9. A = 4 x m y m z m × 1 x m 2 + 1 y m 2 + 1 z m 2 A=4\,x_{m}\,y_{m}\,z_{m}\times\sqrt{\frac{1}{x_{m}^{2}}+\frac{1}{y_{m}^{2}}+% \frac{1}{z_{m}^{2}}}
  10. V = 4 3 x m y m z m V=\frac{4}{3}\,x_{m}\,y_{m}\,z_{m}
  11. I = [ 1 10 m ( y m 2 + z m 2 ) 0 0 0 1 10 m ( x m 2 + z m 2 ) 0 0 0 1 10 m ( x m 2 + y m 2 ) ] I=\begin{bmatrix}\frac{1}{10}m(y_{m}^{2}+z_{m}^{2})&0&0\\ 0&\frac{1}{10}m(x_{m}^{2}+z_{m}^{2})&0\\ 0&0&\frac{1}{10}m(x_{m}^{2}+y_{m}^{2})\end{bmatrix}
  12. x m = y m = z m = a 2 2 x_{m}=y_{m}=z_{m}=a\,\frac{\sqrt{2}}{2}

Octal.html

  1. 𝟕𝟒 10 = 𝟕 × 10 1 + 𝟒 × 10 0 \mathbf{74}_{10}=\mathbf{7}\times 10^{1}+\mathbf{4}\times 10^{0}
  2. 𝟏𝟏𝟐 8 = 𝟏 × 8 2 + 𝟏 × 8 1 + 𝟐 × 8 0 \mathbf{112}_{8}=\mathbf{1}\times 8^{2}+\mathbf{1}\times 8^{1}+\mathbf{2}% \times 8^{0}
  3. k k
  4. k = i = 0 n ( a i × 8 i ) k=\sum_{i=0}^{n}\left(a_{i}\times 8^{i}\right)
  5. a < s u b > i a<sub>i

Octonion.html

  1. 𝕆 \mathbb{O}
  2. { e 0 , e 1 , e 2 , e 3 , e 4 , e 5 , e 6 , e 7 } , \{e_{0},e_{1},e_{2},e_{3},e_{4},e_{5},e_{6},e_{7}\},\,
  3. x = x 0 e 0 + x 1 e 1 + x 2 e 2 + x 3 e 3 + x 4 e 4 + x 5 e 5 + x 6 e 6 + x 7 e 7 , x=x_{0}e_{0}+x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3}+x_{4}e_{4}+x_{5}e_{5}+x_{6}e_{6}% +x_{7}e_{7},\,
  4. × ×
  5. e < s u b > 0 e<sub>0
  6. e i e j = - δ i j e 0 + ε i j k e k , e_{i}e_{j}=-\delta_{ij}e_{0}+\varepsilon_{ijk}e_{k},\,
  7. ε i j k \varepsilon_{ijk}
  8. e i e 0 = e 0 e i = e i ; e 0 e 0 = e 0 , e_{i}e_{0}=e_{0}e_{i}=e_{i};\,\,\,\,e_{0}e_{0}=e_{0},\,
  9. ( a , b ) ( c , d ) = ( a c - d * b , d a + b c * ) \ (a,b)(c,d)=(ac-d^{*}b,da+bc^{*})
  10. z * z^{*}
  11. x = x 0 e 0 + x 1 e 1 + x 2 e 2 + x 3 e 3 + x 4 e 4 + x 5 e 5 + x 6 e 6 + x 7 e 7 x=x_{0}\,e_{0}+x_{1}\,e_{1}+x_{2}\,e_{2}+x_{3}\,e_{3}+x_{4}\,e_{4}+x_{5}\,e_{5% }+x_{6}\,e_{6}+x_{7}\,e_{7}
  12. x * = x 0 e 0 - x 1 e 1 - x 2 e 2 - x 3 e 3 - x 4 e 4 - x 5 e 5 - x 6 e 6 - x 7 e 7 . x^{*}=x_{0}\,e_{0}-x_{1}\,e_{1}-x_{2}\,e_{2}-x_{3}\,e_{3}-x_{4}\,e_{4}-x_{5}\,% e_{5}-x_{6}\,e_{6}-x_{7}\,e_{7}.
  13. x + x * 2 = x 0 e 0 \frac{x+x^{*}}{2}=x_{0}\,e_{0}
  14. x - x * 2 = x 1 e 1 + x 2 e 2 + x 3 e 3 + x 4 e 4 + x 5 e 5 + x 6 e 6 + x 7 e 7 . \frac{x-x^{*}}{2}=x_{1}\,e_{1}+x_{2}\,e_{2}+x_{3}\,e_{3}+x_{4}\,e_{4}+x_{5}\,e% _{5}+x_{6}\,e_{6}+x_{7}\,e_{7}.
  15. x * = - 1 6 ( x + ( e 1 x ) e 1 + ( e 2 x ) e 2 + ( e 3 x ) e 3 + ( e 4 x ) e 4 + ( e 5 x ) e 5 + ( e 6 x ) e 6 + ( e 7 x ) e 7 ) . x^{*}=-\frac{1}{6}(x+(e_{1}x)e_{1}+(e_{2}x)e_{2}+(e_{3}x)e_{3}+(e_{4}x)e_{4}+(% e_{5}x)e_{5}+(e_{6}x)e_{6}+(e_{7}x)e_{7}).
  16. x * x = x 0 2 + x 1 2 + x 2 2 + x 3 2 + x 4 2 + x 5 2 + x 6 2 + x 7 2 . x^{*}x=x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x% _{7}^{2}.
  17. x = x * x . \|x\|=\sqrt{x^{*}x}.
  18. x - 1 = x * x 2 . x^{-1}=\frac{x^{*}}{\|x\|^{2}}.
  19. e i e j = - e j e i e j e i e_{i}e_{j}=-e_{j}e_{i}\neq e_{j}e_{i}\,
  20. i , j i,j
  21. ( e i e j ) e k = - e i ( e j e k ) e i ( e j e k ) (e_{i}e_{j})e_{k}=-e_{i}(e_{j}e_{k})\neq e_{i}(e_{j}e_{k})\,
  22. i , j , k i,j,k
  23. e i e j ± e k e_{i}e_{j}\neq\pm e_{k}
  24. x y = x y \|xy\|=\|x\|\|y\|
  25. [ x , y ] = x y - y x . [x,y]=xy-yx.\,
  26. x × y = 1 2 ( x y - y x ) . x\times y=\frac{1}{2}(xy-yx).
  27. x × y = x y sin θ . \|x\times y\|=\|x\|\|y\|\sin\theta.
  28. A ( x y ) = A ( x ) A ( y ) . A(xy)=A(x)A(y).\,

Oersted.html

  1. μ \mu
  2. B ( G ) = μ r H ( Oe ) B(\mbox{G}~{})=\mu_{r}H(\mbox{Oe}~{})
  3. H ( Oe ) = 4 π 1000 I ( A ) l ( m ) H(\mbox{Oe}~{})=\frac{4\pi}{1000}\frac{I(\mbox{A}~{})}{l(\mbox{m}~{})}

Ohm's_law.html

  1. I = V R , I=\frac{V}{R},
  2. I I
  3. 𝐉 = σ 𝐄 , \mathbf{J}=\sigma\mathbf{E},
  4. x = a b + l , x=\frac{a}{b+l},
  5. I = V R or V = I R or R = V I . I=\frac{V}{R}\quad\,\text{or}\quad V=IR\quad\,\text{or}\quad R=\frac{V}{I}.
  6. A e s t Ae^{st}
  7. Z = s L Z=sL\,
  8. Z = 1 s C . Z=\frac{1}{sC}.
  9. s y m b o l V = s y m b o l I s y m b o l Z symbol{V}=symbol{I}\cdot symbol{Z}
  10. j ω j\omega
  11. A e j ω t Ae^{\mbox{ }~{}j\omega t}
  12. 𝐄 = ρ 𝐉 \mathbf{E}=\rho\mathbf{J}
  13. σ \sigma
  14. Δ V = - 𝐄 d 𝐥 {\Delta V}=-\int{\mathbf{E}\cdot d\mathbf{l}}
  15. d 𝐥 d\mathbf{l}
  16. V = E l or E = V l . V={E}{l}\ \ \,\text{or}\ \ E=\frac{V}{l}.
  17. J = I a . J=\frac{I}{a}.
  18. V l = I a ρ or V = I ρ l a . \frac{V}{l}=\frac{I}{a}\rho\qquad\,\text{or}\qquad V=I\rho\frac{l}{a}.
  19. R = ρ l a {R}=\rho\frac{l}{a}
  20. V = I R . {V}={I}{R}.
  21. 𝐉 = σ ( 𝐄 + 𝐯 × 𝐁 ) \mathbf{J}=\sigma(\mathbf{E}+\mathbf{v}\times\mathbf{B})
  22. v \vec{v}
  23. B \vec{B}
  24. E \vec{E}
  25. J \vec{J}
  26. n e n_{e}
  27. m e n e d v e d t = - n e e E + n e m e ν ( v i - v e ) - e n e v e × B , m_{e}n_{e}{d\vec{v}_{e}\over dt}=-n_{e}e\vec{E}+n_{e}m_{e}\nu(v_{i}-v_{e})-en_% {e}\vec{v}_{e}\times\vec{B},
  28. e e
  29. m e m_{e}
  30. v e \vec{v}_{e}
  31. ν \nu
  32. v i \vec{v}_{i}
  33. σ ( E + v × B ) = J , \sigma(\vec{E}+\vec{v}\times\vec{B})=\vec{J},
  34. σ = n e e 2 ν m e \sigma={n_{e}e^{2}\over\nu m_{e}}
  35. E + v × B = ρ J , \vec{E}+\vec{v}\times\vec{B}=\rho\vec{J},
  36. ρ = σ - 1 \rho=\sigma^{-1}
  37. η \eta
  38. ρ \rho
  39. η = 1 / μ 0 σ \eta=1/\mu_{0}\sigma

Ohmmeter.html

  1. R = V I R=\frac{V}{I}

Olbers'_paradox.html

  1. U V = 8 π 5 ( k T ) 4 15 ( h c ) 3 , {U\over V}=\frac{8\pi^{5}(kT)^{4}}{15(hc)^{3}},
  2. light = r 0 L ( r ) N ( r ) d r \,\text{light}=\int_{r_{0}}^{\infty}L(r)N(r)\,dr
  3. light \,\text{light}
  4. light \,\text{light}

Oligopoly.html

  1. R M = R T Q 1 = M - Q 2 - 2 Q 1 R_{M}=\frac{\partial R_{T}}{\partial Q_{1}}=M-Q_{2}-2Q_{1}
  2. R M = ( M - Q 2 ) - 2 Q 1 R_{M}=(M-Q_{2})-2Q_{1}

Omega.html

  1. ω 0 \omega_{0}

On-base_percentage.html

  1. O B P = H + B B + H B P A B + B B + H B P + S F OBP=\frac{H+BB+HBP}{AB+BB+HBP+SF}

On-base_plus_slugging.html

  1. O P S = O B P + S L G OPS=OBP+SLG\,
  2. S L G = T B A B SLG=\frac{TB}{AB}
  3. O B P = H + B B + H B P A B + B B + S F + H B P OBP=\frac{H+BB+HBP}{AB+BB+SF+HBP}
  4. O P S = A B * ( H + B B + H B P ) + T B * ( A B + B B + S F + H B P ) A B * ( A B + B B + S F + H B P ) OPS=\frac{AB*(H+BB+HBP)+TB*(AB+BB+SF+HBP)}{AB*(AB+BB+SF+HBP)}
  5. O P S + = 100 * ( O B P * l g O B P + S L G * l g S L G - 1 ) OPS+=100*(\frac{OBP}{*lgOBP}+\frac{SLG}{*lgSLG}-1)

Opal.html

  1. α \alpha

Open_set.html

  1. X X
  2. τ \tau
  3. τ \tau
  4. X X
  5. X τ , τ X\in\tau,\,\emptyset\in\tau
  6. X X
  7. \emptyset
  8. τ \tau
  9. { O i } i I τ i I O i τ \{O_{i}\}_{i\in I}\subseteq\tau\Rightarrow\cup_{i\in I}O_{i}\in\tau
  10. τ \tau
  11. τ \tau
  12. { O i } i I τ i = 1 n O i τ \{O_{i}\}_{i\in I}\subseteq\tau\Rightarrow\cap_{i=1}^{n}O_{i}\in\tau
  13. τ \tau
  14. τ \tau
  15. τ \tau
  16. V Y V\cap Y
  17. V Y V\cap Y
  18. ( a , b ) ( c , d ) (a,b)\cup(c,d)
  19. I = ( 0 , 1 ) I=(0,1)
  20. I C = ( - , 0 ] [ 1 , ) I^{C}=(-\infty,0]\cup[1,\infty)
  21. ( a , b ) (a,b)
  22. J = [ 0 , 1 ] J=[0,1]
  23. K = [ 0 , 1 ) K=[0,1)
  24. K C = ( - , 0 ) [ 1 , ) K^{C}=(-\infty,0)\cup[1,\infty)

Operand.html

  1. 3 + 6 = 9 3+6=9\;
  2. ( 3 + 5 ) × 2 (3+5)\times 2\;
  3. 3 + 5 × 2 3+5\times 2
  4. 1 + 2 1+2\;
  5. + 1 2 +\;1\;2
  6. 1 2 + 1\;2\;+
  7. 4 × 2 2 - ( 2 + 2 2 ) 4\times 2^{2}-(2+2^{2})
  8. 4 × 2 2 - ( 2 + 4 ) 4\times 2^{2}-(2+4)
  9. 4 × 2 2 - 6 4\times 2^{2}-6
  10. 4 × 4 - 6 4\times 4-6\;
  11. 16 - 6 16-6\;
  12. 16 - 6 = 10 16-6=10\;

Operational_amplifier.html

  1. V out = A OL ( V + - V - ) V_{\!\,\text{out}}=A\text{OL}\,(V_{\!+}-V_{\!-})
  2. i = V in R g i=\frac{V\text{in}}{R_{g}}
  3. V out = V in + i × R f = V in + ( V in R g × R f ) = V in + V in × R f R g = V in ( 1 + R f R g ) V\text{out}=V\text{in}+i\times R_{f}=V\text{in}+\left(\frac{V\text{in}}{R_{g}}% \times R_{f}\right)=V\text{in}+\frac{V\text{in}\times R_{f}}{R_{g}}=V\text{in}% \left(1+\frac{R_{f}}{R_{g}}\right)
  4. A CL = V out V in = 1 + R f R g A\text{CL}=\frac{V\text{out}}{V\text{in}}=1+\frac{R_{f}}{R_{g}}
  5. i i
  6. C C
  7. d v / d t = i / C dv/dt=i/C
  8. V out = A O L ( V + - V - ) V_{\,\text{out}}=A_{OL}\,(V_{\!+}-V_{\!-})
  9. V - = β V out V_{\!-}\,\,=\beta\cdot V_{\,\text{out}}
  10. β = R 1 R 1 + R 2 \beta=\frac{R_{1}}{R_{1}+R_{2}}
  11. V out = A O L ( V in - β V out ) V_{\,\text{out}}=A_{OL}(V_{\,\text{in}}-\beta\cdot V_{\,\text{out}})
  12. V out V_{\,\text{out}}
  13. V out = V in ( 1 β + 1 / A O L ) V_{\,\text{out}}=V_{\,\text{in}}\left(\frac{1}{\beta+1/A_{OL}}\right)
  14. A O L A_{OL}
  15. V out V in β = V in R 1 R 1 + R 2 = V in ( 1 + R 2 R 1 ) V_{\,\text{out}}\approx\frac{V_{\,\text{in}}}{\beta}=\frac{V_{\,\text{in}}}{% \frac{R_{\,\text{1}}}{R_{\,\text{1}}+R_{\,\text{2}}}}=V_{\,\text{in}}\left(1+% \frac{R_{2}}{R_{1}}\right)
  16. V out = A O L ( V + - V - ) V_{\,\text{out}}=A_{OL}\,(V_{\!+}-V_{\!-})
  17. V - = 1 R f + R in ( R f V in + R in V out ) V_{\!-}\,\,=\frac{1}{R_{\,\text{f}}+R_{\,\text{in}}}\left(R\text{f}V_{\,\text{% in}}+R_{\,\text{in}}V_{\,\text{out}}\right)
  18. V out V_{\,\text{out}}
  19. V out = - V in A O L R f R f + R in + A O L R in V_{\,\text{out}}=-V_{\,\text{in}}\cdot\frac{A_{OL}R_{\,\text{f}}}{R_{\,\text{f% }}+R_{\,\text{in}}+A_{OL}R_{\,\text{in}}}
  20. A O L A_{OL}
  21. V out - V in R f R in V_{\,\text{out}}\approx-V_{\,\text{in}}\frac{R_{\,\text{f}}}{R_{\,\text{in}}}

Operator_(mathematics).html

  1. ( A + B ) 𝐱 := A 𝐱 + B 𝐱 , (A+B)\mathbf{x}:=A\mathbf{x}+B\mathbf{x},
  2. ( α A ) 𝐱 := α A 𝐱 (\alpha A)\mathbf{x}:=\alpha A\mathbf{x}
  3. ( A B ) 𝐱 := A ( B 𝐱 ) (AB)\mathbf{x}:=A(B\mathbf{x})
  4. 𝐑 \mathbf{R}
  5. || A 𝐱 || V C || 𝐱 || U ||A\mathbf{x}||_{V}\leq C||\mathbf{x}||_{U}
  6. || A || = inf { C : || A 𝐱 || V C || 𝐱 || U } ||A||=\inf\{C:||A\mathbf{x}||_{V}\leq C||\mathbf{x}||_{U}\}
  7. || A B || || A || || B || ||AB||\leq||A||\cdot||B||
  8. A ( α 𝐱 + β 𝐲 ) = α A 𝐱 + β A 𝐲 A(\alpha\mathbf{x}+\beta\mathbf{y})=\alpha A\mathbf{x}+\beta A\mathbf{y}
  9. K K
  10. U U
  11. V V
  12. K K
  13. 𝐮 1 , , 𝐮 n \mathbf{u}_{1},\ldots,\mathbf{u}_{n}
  14. U U
  15. 𝐯 1 , , 𝐯 m \mathbf{v}_{1},\ldots,\mathbf{v}_{m}
  16. V V
  17. 𝐱 = x i 𝐮 i \mathbf{x}=x^{i}\mathbf{u}_{i}
  18. U U
  19. A : U V A:U\to V
  20. A 𝐱 = x i A 𝐮 i = x i ( A 𝐮 i ) j 𝐯 j A\mathbf{x}=x^{i}A\mathbf{u}_{i}=x^{i}(A\mathbf{u}_{i})^{j}\mathbf{v}_{j}
  21. a i j := ( A 𝐮 i ) j K a_{i}^{j}:=(A\mathbf{u}_{i})^{j}\in K
  22. A A
  23. a i j a_{i}^{j}
  24. x x
  25. A 𝐱 = 𝐲 A\mathbf{x}=\mathbf{y}
  26. a i j x i = y j a_{i}^{j}x^{i}=y^{j}
  27. U U
  28. V V
  29. d d t \frac{\mathrm{d}}{\mathrm{d}t}
  30. 0 t \int_{0}^{t}
  31. f ( t ) = a 0 2 + n = 1 a n cos ( ω n t ) + b n sin ( ω n t ) f(t)={a_{0}\over 2}+\sum_{n=1}^{\infty}{a_{n}\cos(\omega nt)+b_{n}\sin(\omega nt)}
  32. f ( t ) = 1 2 π - + g ( ω ) e i ω t d ω . f(t)={1\over\sqrt{2\pi}}\int_{-\infty}^{+\infty}{g(\omega)e^{i\omega t}\,d% \omega}.
  33. F ( s ) = { f } ( s ) = 0 e - s t f ( t ) d t . F(s)=\mathcal{L}\{f\}(s)=\int_{0}^{\infty}e^{-st}f(t)\,dt.
  34. \nabla
  35. \nabla\cdot
  36. × \nabla\times

Optical_aberration.html

  1. Z n m ( ρ , ϕ ) = R n m ( ρ ) cos ( m ϕ ) Z^{m}_{n}(\rho,\phi)=R^{m}_{n}(\rho)\,\cos(m\,\phi)\!
  2. Z n - m ( ρ , ϕ ) = R n m ( ρ ) sin ( m ϕ ) , Z^{-m}_{n}(\rho,\phi)=R^{m}_{n}(\rho)\,\sin(m\,\phi),\!
  3. n m n\geq m
  4. ϕ \phi
  5. ρ \rho
  6. R n m R^{m}_{n}
  7. R n m ( ρ ) = k = 0 ( n - m ) / 2 ( - 1 ) k ( n - k ) ! k ! ( ( n + m ) / 2 - k ) ! ( ( n - m ) / 2 - k ) ! ρ n - 2 k if n - m is even R^{m}_{n}(\rho)=\!\sum_{k=0}^{(n-m)/2}\!\!\!\frac{(-1)^{k}\,(n-k)!}{k!\,((n+m)% /2-k)!\,((n-m)/2-k)!}\;\rho^{n-2\,k}\quad\mbox{if }~{}n-m\mbox{ is even}~{}
  8. R n m ( ρ ) = 0 R^{m}_{n}(\rho)=0
  9. n - m n-m
  10. a 0 a_{0}
  11. a 1 × ρ cos ( θ ) a_{1}\times\rho\cos(\theta)
  12. a 2 × ρ sin ( θ ) a_{2}\times\rho\sin(\theta)
  13. a 3 × ( 2 ρ 2 - 1 ) a_{3}\times(2\rho^{2}-1)
  14. a 4 × ρ 2 cos ( 2 θ ) a_{4}\times\rho^{2}\cos(2\theta)
  15. a 5 × ρ 2 sin ( 2 θ ) a_{5}\times\rho^{2}\sin(2\theta)
  16. a 6 × ( 3 ρ 2 - 2 ) ρ cos ( θ ) a_{6}\times(3\rho^{2}-2)\rho\cos(\theta)
  17. a 7 × ( 3 ρ 2 - 2 ) ρ sin ( θ ) a_{7}\times(3\rho^{2}-2)\rho\sin(\theta)
  18. a 8 × ( 6 ρ 4 - 6 ρ 2 + 1 ) a_{8}\times(6\rho^{4}-6\rho^{2}+1)
  19. ρ \rho
  20. 0 ρ 1 0\leq\rho\leq 1
  21. θ \theta
  22. 0 θ 2 π 0\leq\theta\leq 2\pi
  23. a 0 , , a 8 a_{0},\ldots,a_{8}
  24. n n
  25. n + d n n+dn
  26. f f
  27. f + d f f+df
  28. d f f = d n ( n - 1 ) = 1 n \dfrac{df}{f}=\dfrac{dn}{(n-1)}=\dfrac{1}{n}
  29. d n dn
  30. n n
  31. f 1 f_{1}
  32. f 2 f_{2}
  33. n 1 n_{1}
  34. n 2 n_{2}
  35. r 1 r^{\prime}_{1}
  36. r 1 ′′ r^{\prime\prime}_{1}
  37. r 2 r^{\prime}_{2}
  38. r 2 ′′ r^{\prime\prime}_{2}
  39. f f
  40. d f df
  41. d n 1 dn_{1}
  42. d n 2 dn_{2}
  43. f f
  44. n 1 n_{1}
  45. n 2 n_{2}
  46. f = f 1 - f 2 = ( n 1 - 1 ) ( 1 / r 1 - 1 / r 1 ′′ ) + ( n 2 - 1 ) ( 1 / r 2 - 1 / r 2 ′′ ) = ( n 1 - 1 ) k 1 + ( n 2 - 1 ) k 2 f=f_{1}-f_{2}=(n_{1}-1)(1/r^{\prime}_{1}-1/r^{\prime\prime}_{1})+(n2-1)(1/r^{% \prime}_{2}-1/r^{\prime\prime}_{2})=(n_{1}-1)k_{1}+(n_{2}-1)k_{2}
  47. d f = k 1 d n 1 + k 2 d n 2 df=k_{1}dn_{1}+k_{2}dn_{2}
  48. d f = 0 df=0
  49. k 1 / k 2 = - d n 2 / d n 1 k_{1}/k_{2}=-dn_{2}/dn_{1}
  50. f 1 / f 2 = - n 1 / n 2 f_{1}/f_{2}=-n_{1}/n_{2}
  51. f 1 f_{1}
  52. f 2 f_{2}
  53. f f
  54. v v
  55. f f
  56. v v
  57. v v
  58. n n
  59. d f df
  60. d f / f df/f
  61. D D
  62. D = v 1 f 1 + v 2 f 2 D=v_{1}f_{1}+v_{2}f_{2}
  63. v 1 = v 2 v_{1}=v_{2}
  64. D = ( f 1 + f 2 ) / 2 D=(f_{1}+f_{2})/2
  65. d n 2 / d n 1 dn_{2}/dn_{1}
  66. f a = f b = f f_{a}=f_{b}=f
  67. f c < f f_{c}<f
  68. f C = f F f_{C}=f_{F}
  69. f a = f b = f c = f f_{a}=f_{b}=f_{c}=f
  70. ( n c - n b ) ( n a - n b ) (n_{c}-n_{b})(n_{a}-n_{b})

Optical_depth.html

  1. τ = ln ( Φ e i Φ e t ) = - ln T , \tau=\ln\!\left(\frac{\Phi_{\mathrm{e}}^{\mathrm{i}}}{\Phi_{\mathrm{e}}^{% \mathrm{t}}}\right)=-\ln T,
  2. τ = A ln 10 , \tau=A\ln 10,
  3. τ ν = ln ( Φ e , ν i Φ e , ν t ) = - ln T ν , \tau_{\nu}=\ln\!\left(\frac{\Phi_{\mathrm{e},\nu}^{\mathrm{i}}}{\Phi_{\mathrm{% e},\nu}^{\mathrm{t}}}\right)=-\ln T_{\nu},
  4. τ λ = ln ( Φ e , λ i Φ e , λ t ) = - ln T λ , \tau_{\lambda}=\ln\!\left(\frac{\Phi_{\mathrm{e},\lambda}^{\mathrm{i}}}{\Phi_{% \mathrm{e},\lambda}^{\mathrm{t}}}\right)=-\ln T_{\lambda},
  5. τ ν = A ν ln 10 , \tau_{\nu}=A_{\nu}\ln 10,
  6. τ λ = A λ ln 10 , \tau_{\lambda}=A_{\lambda}\ln 10,
  7. Φ e t + Φ e att = Φ e i + Φ e e , \Phi_{\mathrm{e}}^{\mathrm{t}}+\Phi_{\mathrm{e}}^{\mathrm{att}}=\Phi_{\mathrm{% e}}^{\mathrm{i}}+\Phi_{\mathrm{e}}^{\mathrm{e}},
  8. T + A T T = 1 + E , T+ATT=1+E,
  9. T = e - τ , T=e^{-\tau},
  10. A T T = 1 - e - τ + E τ + E τ , if τ 1 and E τ . ATT=1-e^{-\tau}+E\approx\tau+E\approx\tau,\quad\,\text{if}\ \tau\ll 1\ \,\text% {and}\ E\ll\tau.
  11. τ = 0 l α ( z ) d z , \tau=\int_{0}^{l}\alpha(z)\,\mathrm{d}z,
  12. τ = α l . \tau=\alpha l.
  13. τ = 0 l σ N ( z ) d z , \tau=\int_{0}^{l}\sigma N(z)\,\mathrm{d}z,
  14. τ = σ N l . \tau=\sigma Nl.
  15. τ ν = d 2 N ν 2 c ε 0 σ γ , \tau_{\nu}=\frac{d^{2}N\nu}{2\mathrm{c}\hbar\varepsilon_{0}\sigma\gamma},
  16. T = e - τ = e - m τ . T=e^{-\tau}=e^{-m\tau^{\prime}}.

Optical_isolator.html

  1. B B
  2. β \beta
  3. β = ν B d \beta=\nu Bd\,
  4. ν \nu
  5. d d
  6. β \beta

Optical_path_length.html

  1. OPD = d 1 n 1 - d 2 n 2 \mathrm{OPD}=d_{1}n_{1}-d_{2}n_{2}
  2. OPL = n d . \mathrm{OPL}=nd.\,
  3. OPL = C n ( s ) d s , \mathrm{OPL}=\int_{C}n(s)\mathrm{d}s,\quad

Optical_rotation.html

  1. 𝐄 θ 0 = 𝐄 R H C + e i 2 θ 0 𝐄 L H C , \mathbf{E}_{\theta_{0}}=\mathbf{E}_{RHC}+e^{i2\theta_{0}}\mathbf{E}_{LHC},
  2. 𝐄 \mathbf{E}
  3. 2 θ 0 2\theta_{0}
  4. θ 0 \theta_{0}
  5. Δ n = n R H C - n L H C \Delta n=n_{RHC}-n_{LHC}\,
  6. L L
  7. 2 Δ θ = Δ n L 2 π λ 2\Delta\theta=\frac{\Delta nL2\pi}{\lambda}
  8. λ \lambda
  9. θ 0 + Δ θ \theta_{0}+\Delta\theta
  10. L L
  11. Δ n \Delta n

Optics.html

  1. sin θ 1 sin θ 2 = n \frac{\sin{\theta_{1}}}{\sin{\theta_{2}}}=n
  2. n n
  3. n 1 n_{1}
  4. n 2 n_{2}
  5. n 1 sin θ 1 = n 2 sin θ 2 n_{1}\sin\theta_{1}=n_{2}\sin\theta_{2}
  6. θ 1 \theta_{1}
  7. θ 2 \theta_{2}
  8. v 1 sin θ 2 = v 2 sin θ 1 v_{1}\sin\theta_{2}\ =v_{2}\sin\theta_{1}
  9. v 1 v_{1}
  10. v 2 v_{2}
  11. f f
  12. S 1 S_{1}
  13. 1 S 1 + 1 S 2 = 1 f \frac{1}{S_{1}}+\frac{1}{S_{2}}=\frac{1}{f}
  14. S 2 S_{2}
  15. M = - S 2 S 1 = f f - S 1 M=-\frac{S_{2}}{S_{1}}=\frac{f}{f-S_{1}}
  16. m λ = d sin θ m\lambda=d\sin\theta
  17. d d
  18. θ \theta
  19. m m
  20. m = 0 m=0
  21. d d
  22. sin θ = 1.22 λ D \sin\theta=1.22\frac{\lambda}{D}
  23. D = 1 v g 2 d v g d λ D=\frac{1}{v_{g}^{2}}\frac{dv_{g}}{d\lambda}
  24. v g v_{g}
  25. v g = c ( n - λ d n d λ ) - 1 v_{g}=c\left(n-\lambda\frac{dn}{d\lambda}\right)^{-1}
  26. D = - λ c d 2 n d λ 2 . D=-\frac{\lambda}{c}\,\frac{d^{2}n}{d\lambda^{2}}.
  27. I = I 0 cos 2 θ i , I=I_{0}\cos^{2}\theta_{i}\quad,
  28. cos 2 θ \cos^{2}\theta
  29. I I 0 = 1 2 \frac{I}{I_{0}}=\frac{1}{2}\quad
  30. N N
  31. f / # = N = f D f/\#=N=\frac{f}{D}
  32. f f
  33. D D

Oracle_machine.html

  1. A B = L B A L A^{B}=\bigcup_{L\in B}A^{L}

Orbit.html

  1. A = F m 2 = - 1 m 2 G m 1 m 2 r 2 = - μ r 2 A=\frac{F}{m_{2}}=-\frac{1}{m_{2}}\frac{Gm_{1}m_{2}}{r^{2}}=-\frac{\mu}{r^{2}}
  2. μ \mu\,
  3. G m 1 Gm_{1}
  4. A = F / m = - k r . A=F/m=-kr.
  5. x ^ \hat{{x}}
  6. y ^ \hat{{y}}
  7. r x ′′ = A x = - k r x r^{\prime\prime}_{x}=A_{x}=-kr_{x}
  8. x = A c o s ( t ) x=A\ cos(t)
  9. y = B s i n ( t ) y=B\ sin(t)
  10. A = μ / r 2 A=\mu/r^{2}
  11. t t
  12. r r
  13. θ \theta
  14. x ^ \hat{{x}}
  15. y ^ \hat{{y}}
  16. r ^ = cos ( θ ) x ^ + sin ( θ ) y ^ \hat{{r}}=\cos(\theta)\hat{{x}}+\sin(\theta)\hat{{y}}
  17. s y m b o l θ ^ = - sin ( θ ) x ^ + cos ( θ ) y ^ \hat{symbol\theta}=-\sin(\theta)\hat{{x}}+\cos(\theta)\hat{{y}}
  18. O ^ = r cos ( θ ) x ^ + r sin ( θ ) y ^ = r r ^ \hat{{O}}=r\cos(\theta)\hat{{x}}+r\sin(\theta)\hat{{y}}=r\hat{{r}}
  19. r ˙ \dot{r}
  20. θ ˙ \dot{\theta}
  21. t t
  22. t + δ t t+\delta t
  23. δ t \delta t
  24. r ^ \hat{{r}}
  25. t t
  26. t + δ t t+\delta t
  27. r ^ \hat{{r}}
  28. θ \theta
  29. θ + θ ˙ δ t \theta+\dot{\theta}\ \delta t
  30. θ ˙ δ t \dot{\theta}\ \delta t
  31. s y m b o l θ ^ \hat{symbol\theta}
  32. θ ˙ s y m b o l θ ^ \dot{\theta}\hat{symbol\theta}
  33. r ^ = cos ( θ ) x ^ + sin ( θ ) y ^ \hat{{r}}=\cos(\theta)\hat{{x}}+\sin(\theta)\hat{{y}}
  34. δ r ^ δ t = r ˙ = - sin ( θ ) θ ˙ x ^ + cos ( θ ) θ ˙ y ^ = θ ˙ s y m b o l θ ^ \frac{\delta\hat{{r}}}{\delta t}=\dot{r}=-\sin(\theta)\dot{\theta}\hat{{x}}+% \cos(\theta)\dot{\theta}\hat{{y}}=\dot{\theta}\hat{symbol\theta}
  35. s y m b o l θ ^ = - sin ( θ ) x ^ + cos ( θ ) y ^ \hat{symbol\theta}=-\sin(\theta)\hat{{x}}+\cos(\theta)\hat{{y}}
  36. δ s y m b o l θ ^ δ t = s y m b o l θ ˙ = - cos ( θ ) θ ˙ x ^ - sin ( θ ) θ ˙ y ^ = - θ ˙ r ^ \frac{\delta\hat{symbol\theta}}{\delta t}=\dot{symbol\theta}=-\cos(\theta)\dot% {\theta}\hat{{x}}-\sin(\theta)\dot{\theta}\hat{{y}}=-\dot{\theta}\hat{r}
  37. O ^ = r r ^ \hat{{O}}=r\hat{{r}}
  38. O ˙ = δ r δ t r ^ + r δ r ^ δ t = r ˙ r ^ + r [ θ ˙ s y m b o l θ ^ ] \dot{{O}}=\frac{\delta r}{\delta t}\hat{{r}}+r\frac{\delta\hat{{r}}}{\delta t}% =\dot{r}\hat{r}+r[\dot{\theta}\hat{symbol\theta}]
  39. O ¨ = [ r ¨ r ^ + r ˙ θ ˙ s y m b o l θ ^ ] + [ r ˙ θ ˙ s y m b o l θ ^ + r θ ¨ s y m b o l θ ^ - r θ ˙ 2 r ^ ] \ddot{{O}}=[\ddot{r}\hat{r}+\dot{r}\dot{\theta}\hat{symbol\theta}]+[\dot{r}% \dot{\theta}\hat{symbol\theta}+r\ddot{\theta}\hat{symbol\theta}-r\dot{\theta}^% {2}\hat{r}]
  40. = [ r ¨ - r θ ˙ 2 ] r ^ + [ r θ ¨ + 2 r ˙ θ ˙ ] s y m b o l θ ^ =[\ddot{r}-r\dot{\theta}^{2}]\hat{{r}}+[r\ddot{\theta}+2\dot{r}\dot{\theta}]% \hat{symbol\theta}
  41. r ^ \hat{{r}}
  42. s y m b o l θ ^ \hat{symbol\theta}
  43. - μ / r 2 -\mu/r^{2}
  44. r θ ¨ + 2 r ˙ θ ˙ = 1 r d d t ( r 2 θ ˙ ) = 0 r\ddot{\theta}+2\dot{r}\dot{\theta}=\frac{1}{r}\frac{d}{dt}\left(r^{2}\dot{% \theta}\right)=0
  45. r r
  46. r r
  47. θ \theta
  48. u = 1 / r u=1/r
  49. u u
  50. θ \theta
  51. r r
  52. u u
  53. u = 1 r u={1\over r}
  54. θ ˙ = h r 2 = h u 2 \dot{\theta}=\frac{h}{r^{2}}=hu^{2}
  55. δ u δ θ = δ δ t ( 1 r ) δ t δ θ = - r ˙ r 2 θ ˙ = - r ˙ h \displaystyle\frac{\delta u}{\delta\theta}=\frac{\delta}{\delta t}\left(\frac{% 1}{r}\right)\frac{\delta t}{\delta\theta}=-\frac{{\dot{r}}}{r^{2}\dot{\theta}}% =-\frac{{\dot{r}}}{h}
  56. r ¨ - r θ ˙ 2 = - μ r 2 \ddot{r}-r\dot{\theta}^{2}=-\frac{\mu}{r^{2}}
  57. - h 2 u 2 δ 2 u δ θ 2 - 1 u ( h u 2 ) 2 = - μ u 2 -h^{2}u^{2}\frac{\delta^{2}u}{\delta\theta^{2}}-\frac{1}{u}(hu^{2})^{2}=-\mu u% ^{2}
  58. δ 2 u δ θ 2 + u = μ h 2 \frac{\delta^{2}u}{\delta\theta^{2}}+u=\frac{\mu}{h^{2}}
  59. u ( θ ) = μ h 2 - A cos ( θ - θ 0 ) u(\theta)=\frac{\mu}{h^{2}}-A\cos(\theta-\theta_{0})
  60. e h 2 A / μ e\equiv h^{2}A/\mu
  61. a h 2 / ( μ ( 1 - e 2 ) ) a\equiv h^{2}/(\mu(1-e^{2}))
  62. θ 0 0 \theta_{0}\equiv 0
  63. r ( θ ) = a ( 1 - e 2 ) 1 - e cos θ r(\theta)=\frac{a(1-e^{2})}{1-e\cos\theta}
  64. G T 2 σ = 3 π ( a r ) 3 , GT^{2}\sigma=3\pi\left(\frac{a}{r}\right)^{3},

Orbital_inclination.html

  1. i i
  2. 𝐡 \mathbf{h}\,
  3. i = arccos h z | 𝐡 | i=\arccos{h_{\mathrm{z}}\over\left|\mathbf{h}\right|}
  4. h z h_{\mathrm{z}}
  5. 𝐡 \mathbf{h}

Orbital_resonance.html

  1. n n\,\!
  2. n Io - 2 n Eu = 0 n_{\rm Io}-2\cdot n_{\rm Eu}=0
  3. ω ˙ \dot{\omega}
  4. n Io - 2 n Eu + ω ˙ Io = 0 n_{\rm Io}-2\cdot n_{\rm Eu}+\dot{\omega}_{\rm Io}=0
  5. 4 n Te - 2 n Mi - Ω ˙ Te - Ω ˙ Mi = 0 4\cdot n_{\rm Te}-2\cdot n_{\rm Mi}-\dot{\Omega}_{\rm Te}-\dot{\Omega}_{\rm Mi% }=0
  6. Φ L = λ Io - 3 λ Eu + 2 λ Ga = 180 \Phi_{L}=\lambda_{\rm Io}-3\cdot\lambda_{\rm Eu}+2\cdot\lambda_{\rm Ga}=180^{\circ}
  7. λ \lambda
  8. Φ L \Phi_{L}
  9. Φ = 3 λ S - 5 λ N + 2 λ H = 180 \Phi=3\cdot\lambda_{\rm S}-5\cdot\lambda_{\rm N}+2\cdot\lambda_{\rm H}=180^{\circ}
  10. Φ \Phi
  11. Φ L \Phi_{L}
  12. Φ L = λ c - 3 λ d + 2 λ e = 0 \Phi_{L}=\lambda_{\rm c}-3\cdot\lambda_{\rm d}+2\cdot\lambda_{\rm e}=0^{\circ}

Order_of_magnitude.html

  1. 100 5 2.512 \sqrt[5]{100}\approx 2.512
  2. 0 - 1 , 1 - 10 , 10 - 10 10 , 10 10 - 10 10 10 , 10 10 10 - 10 10 10 10 , 0-1,1-10,10-10^{10},10^{10}-10^{10^{10}},10^{10^{10}}-10^{10^{10^{10}}},\dots
  3. ( 10 ) 1 10 1453 (10\uparrow)^{1}10^{1453}
  4. ( 10 ) 2 10 1453 (10\uparrow)^{2}10^{1453}

Ordered_field.html

  1. p ( x ) q ( x ) \frac{p(x)}{q(x)}\,
  2. p ( x ) p(x)
  3. q ( x ) q(x)
  4. q ( x ) 0 q(x)\neq 0\,
  5. p ( x ) = x p(x)=x
  6. p ( x ) q ( x ) > 0 \frac{p(x)}{q(x)}>0\,
  7. p 0 q 0 > 0 \frac{p_{0}}{q_{0}}>0\,
  8. p ( x ) = p 0 x n + p(x)=p_{0}x^{n}+\cdots
  9. q ( x ) = q 0 x m + q(x)=q_{0}x^{m}+\cdots\,
  10. ( ( x ) ) \mathbb{R}((x))
  11. H ( a ) = { P X F : a P } H(a)=\{P\in X_{F}:a\in P\}

Ordered_pair.html

  1. ( a 1 , b 1 ) (a_{1},b_{1})
  2. ( a 2 , b 2 ) (a_{2},b_{2})
  3. ( a 1 , b 1 ) = ( a 2 , b 2 ) if and only if a 1 = a 2 and b 1 = b 2 . (a_{1},b_{1})=(a_{2},b_{2})\quad\,\text{if and only if}\quad a_{1}=a_{2}\,% \text{ and }b_{1}=b_{2}.\!
  4. ( a , b ) \ (a,b)
  5. a , b . \left\langle a,b\right\rangle.
  6. ( a , b ) := { { { a } , } , { { b } } } . \left(a,b\right):=\left\{\left\{\left\{a\right\},\,\emptyset\right\},\,\left\{% \left\{b\right\}\right\}\right\}.
  7. { { a } , } \{\{a\},\emptyset\}
  8. ( a , b ) := { { a , 1 } , { b , 2 } } (a,b):=\left\{\{a,1\},\{b,2\}\right\}
  9. ( a , b ) K := { { a } , { a , b } } . (a,\ b)_{K}\;:=\ \{\{a\},\ \{a,\ b\}\}.
  10. ( x , x ) K = { { x } , { x , x } } = { { x } , { x } } = { { x } } (x,\ x)_{K}=\{\{x\},\{x,\ x\}\}=\{\{x\},\ \{x\}\}=\{\{x\}\}
  11. Y p : x Y . \forall{Y}{\in}{p}:{x}{\in}{Y}.
  12. ( \exist Y p : x Y ) and ( Y 1 , Y 2 p : Y 1 Y 2 ( x Y 1 x Y 2 ) ) . (\exist{Y}{\in}{p}:{x}{\in}{Y})\and(\forall{Y_{1},Y_{2}}{\in}{p}:Y_{1}\neq Y_{% 2}\rightarrow({x}{\notin}{Y_{1}}{x}{\notin}{Y_{2}})).
  13. ( Y 1 , Y 2 p : Y 1 Y 2 ( x Y 1 x Y 2 ) ) (\forall{Y_{1},Y_{2}}{\in}{p}:Y_{1}\neq Y_{2}\rightarrow({x}{\notin}{Y_{1}}{x}% {\notin}{Y_{2}}))
  14. π 1 ( p ) = p \pi_{1}(p)=\bigcup\bigcap p
  15. π 2 ( p ) = { x p p p x p } \pi_{2}(p)=\bigcup\{x\in\bigcup p\mid\bigcup p\not=\bigcap p\rightarrow x% \notin\bigcap p\}
  16. ( a , b ) = ( x , y ) ( a = x ) and ( b = y ) (a,b)=(x,y)\leftrightarrow(a=x)\and(b=y)
  17. ( a , b ) = ( b , a ) (a,b)=(b,a)
  18. b = a b=a
  19. ( a , b ) reverse := { { b } , { a , b } } ; (a,b)_{\,\text{reverse}}:=\{\{b\},\{a,b\}\};
  20. ( a , b ) short := { a , { a , b } } ; (a,b)_{\,\text{short}}:=\{a,\{a,b\}\};
  21. ( a , b ) 01 := { { 0 , a } , { 1 , b } } . (a,b)_{\,\text{01}}:=\{\{0,a\},\{1,b\}\}.
  22. 𝒩 \mathcal{N}
  23. x 𝒩 x\setminus\mathcal{N}
  24. x x
  25. 𝒩 \mathcal{N}
  26. φ ( x ) = ( x 𝒩 ) { n + 1 : n ( x 𝒩 ) } . \varphi(x)=(x\setminus\mathcal{N})\cup\{n+1:n\in(x\cap\mathcal{N})\}.
  27. φ ( x ) \varphi(x)
  28. φ ( x ) { 0 } φ ( y ) . \varphi(x)\not=\{0\}\cup\varphi(y).
  29. ( A , B ) = { φ ( a ) : a A } { φ ( b ) { 0 } : b B } . (A,B)=\{\varphi(a):a\in A\}\cup\{\varphi(b)\cup\{0\}:b\in B\}.
  30. φ \varphi
  31. ( x , y ) = ( { 0 } × s ( x ) ) ( { 1 } × s ( y ) ) (x,y)=(\{0\}\times s(x))\cup(\{1\}\times s(y))
  32. s ( x ) = { } { { t } | t x } s(x)=\{\emptyset\}\cup\{\{t\}|t\in x\}
  33. ( x , y , z ) = ( { 0 } × s ( x ) ) ( { 1 } × s ( y ) ) ( { 2 } × s ( z ) ) (x,y,z)=(\{0\}\times s(x))\cup(\{1\}\times s(y))\cup(\{2\}\times s(z))
  34. s ( x ) s(x)

Original_proof_of_Gödel's_completeness_theorem.html

  1. ψ = x 1 x n ϕ \psi=\exists x_{1}...\exists x_{n}\phi
  2. ¬ ψ = x 1 x n ¬ ϕ \neg\psi=\forall x_{1}...\forall x_{n}\neg\phi
  3. ψ = y ( P ) z ( Φ [ F ( y ) ¬ F ( z ) ] ) \psi=\forall y(P)\exists z(\Phi\wedge[F(y)\vee\neg F(z)])
  4. y z ( F ( y ) ¬ F ( z ) ) \forall y\exists z(F(y)\vee\neg F(z))
  5. ϕ = ψ \phi=\psi
  6. ϕ = ( x 1 x k 1 ) ( x k 1 + 1 x k 2 ) . ( x k n - 2 + 1 x k n - 1 ) ( x k n - 1 + 1 x k n ) ( Φ ) \phi=(\forall x_{1}...\forall x_{k_{1}})(\exists x_{k_{1}+1}...\exists x_{k_{2% }}).......(\forall x_{k_{n-2}+1}...\forall x_{k_{n-1}})(\exists x_{k_{n-1}+1}.% ..\exists x_{k_{n}})(\Phi)
  7. ϕ \phi
  8. ϕ \phi
  9. ϕ \phi
  10. ϕ = ( x ) ( y ) ( u ) ( \exist v ) ( P ) ψ \phi=(\forall x)(\exists y)(\forall u)(\exist v)(P)\psi
  11. ( P ) ψ (P)\psi
  12. ϕ \phi
  13. ( x ) ( x ) ( y ) ( u ) ( \exist v ) ( \exist y ) ( P ) Q ( x , y ) ( Q ( x , y ) ψ ) (\forall x^{\prime})(\forall x)(\forall y)(\forall u)(\exist v)(\exist y^{% \prime})(P)Q^{\prime}(x^{\prime},y^{\prime})\wedge(Q^{\prime}(x,y)\rightarrow\psi)
  14. ϕ = ( x ) ( y ) ( u ) ( \exist v ) ( P ) ψ \phi=(\forall x)(\exists y)(\forall u)(\exist v)(P)\psi
  15. ϕ \phi
  16. ψ \psi
  17. ϕ \phi
  18. ( x ) (\forall x)
  19. x 1 x 2 x n \forall x_{1}\forall x_{2}...\forall x_{n}
  20. x 1 x n x_{1}...x_{n}
  21. Φ = ( x ) ( y ) Q ( x , y ) ( x ) ( y ) ( Q ( x , y ) ( u ) ( \exist v ) ( P ) ψ ) \Phi=(\forall x^{\prime})(\exists y^{\prime})Q(x^{\prime},y^{\prime})\wedge(% \forall x)(\forall y)(Q(x,y)\rightarrow(\forall u)(\exist v)(P)\psi)
  22. Φ ϕ \Phi\rightarrow\phi
  23. ( u ) ( v ) ( P ) (\forall u)(\exists v)(P)
  24. ( Q ( x , y ) ( u ) ( v ) ( P ) ψ ) ( u ) ( v ) ( P ) ( Q ( x , y ) ψ ) (Q(x,y)\rightarrow(\forall u)(\exists v)(P)\psi)\equiv(\forall u)(\exists v)(P% )(Q(x,y)\rightarrow\psi)
  25. Φ = ( x ) ( \exist y ) Q ( x , y ) ( x ) ( y ) ( u ) ( v ) ( P ) ( Q ( x , y ) ψ ) \Phi^{\prime}=(\forall x^{\prime})(\exist y^{\prime})Q(x^{\prime},y^{\prime})% \wedge(\forall x)(\forall y)(\forall u)(\exists v)(P)(Q(x,y)\rightarrow\psi)
  26. ( S ) ρ ( S ) ρ (S)\rho\wedge(S^{\prime})\rho^{\prime}
  27. ( T ) ( ρ ρ ) (T)(\rho\wedge\rho^{\prime})
  28. Ψ = ( x ) ( x ) ( y ) ( u ) ( y ) ( v ) ( P ) Q ( x , y ) ( Q ( x , y ) ψ ) \Psi=(\forall x^{\prime})(\forall x)(\forall y)(\forall u)(\exists y^{\prime})% (\exists v)(P)Q(x^{\prime},y^{\prime})\wedge(Q(x,y)\rightarrow\psi)
  29. Φ Ψ \Phi^{\prime}\equiv\Psi
  30. Ψ \Psi
  31. Ψ \Psi
  32. Ψ Φ Φ Φ ϕ \Psi\equiv\Phi^{\prime}\equiv\Phi\wedge\Phi\rightarrow\phi
  33. ϕ \phi
  34. Ψ \Psi
  35. Φ \Phi
  36. ¬ Φ \neg\Phi
  37. ¬ Φ \neg\Phi
  38. ¬ Φ \neg\Phi
  39. ¬ Φ \neg\Phi
  40. ( u ) ( v ) ( P ) ψ ( x , y | x , y ) (\forall u)(\exists v)(P)\psi(x,y|x^{\prime},y^{\prime})
  41. ( u ) ( v ) ( P ) ψ (\forall u)(\exists v)(P)\psi
  42. ¬ Φ \neg\Phi
  43. ¬ ( ( x ) ( y ) ( u ) ( v ) ( P ) ψ ( x , y | x , y ) ( x ) ( y ) ( ( u ) ( v ) ( P ) ψ ( u ) ( v ) ( P ) ψ ) ) \neg((\forall x^{\prime})(\exists y^{\prime})(\forall u)(\exists v)(P)\psi(x,y% |x^{\prime},y^{\prime})\wedge(\forall x)(\forall y)((\forall u)(\exists v)(P)% \psi\rightarrow(\forall u)(\exists v)(P)\psi))
  44. \wedge
  45. \wedge
  46. ¬ ϕ \neg\phi
  47. ( u ) ( v ) ( P ) ψ ( x , y | x , y ) (\forall u)(\exists v)(P)\psi(x,y|x^{\prime},y^{\prime})
  48. Ψ \Psi
  49. ( x 1 x k ) ( y 1 y m ) ϕ ( x 1 x k , y 1 y m ) . (\forall x_{1}...x_{k})(\exists y_{1}...y_{m})\phi(x_{1}...x_{k},y_{1}...y_{m}).
  50. ( x 1 x k ) < ( y 1 y k ) (x_{1}...x_{k})<(y_{1}...y_{k})
  51. Σ k ( x 1 x k ) < Σ k ( y 1 y k ) \Sigma_{k}(x_{1}...x_{k})<\Sigma_{k}(y_{1}...y_{k})
  52. Σ k ( x 1 x k ) = Σ k ( y 1 y k ) \Sigma_{k}(x_{1}...x_{k})=\Sigma_{k}(y_{1}...y_{k})
  53. ( x 1 x k ) (x_{1}...x_{k})
  54. ( y 1 y k ) (y_{1}...y_{k})
  55. Σ k ( x 1 x k ) \Sigma_{k}(x_{1}...x_{k})
  56. ( a 1 n a k n ) (a^{n}_{1}...a^{n}_{k})
  57. B n B_{n}
  58. ϕ ( z a 1 n z a k n , z ( n - 1 ) m + 2 , z ( n - 1 ) m + 3 z n m + 1 ) \phi(z_{a^{n}_{1}}...z_{a^{n}_{k}},z_{(n-1)m+2},z_{(n-1)m+3}...z_{nm+1})
  59. D n D_{n}
  60. ( z 1 z n m + 1 ) ( B 1 B 2 B n ) . (\exists z_{1}...z_{nm+1})(B_{1}\wedge B_{2}...\wedge B_{n}).
  61. D n \rightarrow D_{n}
  62. D k D k - 1 ( z 1 z ( n - 1 ) m + 1 ) ( z ( n - 1 ) m + 2 z n m + 1 ) B n D k - 1 ( z a 1 n z a k n ) ( y 1 y m ) ϕ ( z a 1 n z a k n , y 1 y m ) D_{k}\Leftarrow D_{k-1}\wedge(\forall z_{1}...z_{(n-1)m+1})(\exists z_{(n-1)m+% 2}...z_{nm+1})B_{n}\Leftarrow D_{k-1}\wedge(\forall z_{a^{n}_{1}}...z_{a^{n}_{% k}})(\exists y_{1}...y_{m})\phi(z_{a^{n}_{1}}...z_{a^{n}_{k}},y_{1}...y_{m})
  63. ( k ) ( a 1 n a k n ) < ( n - 1 ) m + 2 (\forall k)({a^{n}_{1}}...{a^{n}_{k}})<(n-1)m+2
  64. D k - 1 D_{k-1}\wedge
  65. D 1 ( z 1 z m + 1 ) ϕ ( z a 1 1 z a k 1 , z 2 , z 3 z m + 1 ) ( z 1 z m + 1 ) ϕ ( z 1 z 1 , z 2 , z 3 z m + 1 ) D_{1}\equiv(\exists z_{1}...z_{m+1})\phi(z_{a^{1}_{1}}...z_{a^{1}_{k}},z_{2},z% _{3}...z_{m+1})\equiv(\exists z_{1}...z_{m+1})\phi(z_{1}...z_{1},z_{2},z_{3}..% .z_{m+1})
  66. D n D_{n}
  67. D n D_{n}
  68. E h E_{h}
  69. D n D_{n}
  70. B k B_{k}
  71. D n D_{n}
  72. E h E_{h}
  73. D n D_{n}
  74. E h E_{h}
  75. D n D_{n}
  76. D n D_{n}
  77. E 1 E_{1}
  78. E 1 E_{1}
  79. E 1 E_{1}
  80. E 1 E_{1}
  81. E h - 1 E_{h-1}
  82. E h E_{h}
  83. B k B_{k}
  84. D k D_{k}
  85. B k B_{k}
  86. D n D_{n}
  87. E h E_{h}
  88. D k D_{k}
  89. D n D_{n}
  90. D k D_{k}
  91. Ψ \Psi
  92. ( u 1 u i ) (u_{1}...u_{i})
  93. Ψ ( z u 1 z u i ) \Psi(z_{u_{1}}...z_{u_{i}})
  94. D k D_{k}
  95. ( y 1 y m ) ϕ ( a 1 n a k n , y 1 y m ) (\exists y_{1}...y_{m})\phi(a^{n}_{1}...a^{n}_{k},y_{1}...y_{m})
  96. a n a^{n}
  97. ( x ) E q ( x , x ) ( x , y , z ) [ E q ( x , y ) ( E q ( x , z ) E q ( y , z ) ) ] (\forall x)Eq(x,x)\wedge(\forall x,y,z)[Eq(x,y)\rightarrow(Eq(x,z)\rightarrow Eq% (y,z))]
  98. ( x , y , z ) [ E q ( x , y ) ( E q ( z , x ) E q ( z , y ) ) ] \wedge(\forall x,y,z)[Eq(x,y)\rightarrow(Eq(z,x)\rightarrow Eq(z,y))]
  99. ( x 1 x k , y 1 x k ) [ ( E q ( x 1 , y 1 ) E q ( x k , y k ) ) ( A ( x 1 x k ) A ( y 1 y k ) ) ] \wedge(\forall x_{1}...x_{k},y_{1}...x_{k})[(Eq(x_{1},y_{1})\wedge...\wedge Eq% (x_{k},y_{k}))\rightarrow(A(x_{1}...x_{k})\equiv A(y_{1}...y_{k}))]
  100. ( x 1 x m , y 1 x m ) [ ( E q ( x 1 , y 1 ) E q ( x m , y m ) ) ( Z ( x 1 x m ) Z ( y 1 y m ) ) ] \wedge...\wedge(\forall x_{1}...x_{m},y_{1}...x_{m})[(Eq(x_{1},y_{1})\wedge...% \wedge Eq(x_{m},y_{m}))\rightarrow(Z(x_{1}...x_{m})\equiv Z(y_{1}...y_{m}))]
  101. φ . \wedge\varphi^{\prime}.
  102. A Z A...Z
  103. k m k...m
  104. ϕ i \phi^{i}
  105. B k i B^{i}_{k}
  106. D k D_{k}
  107. B 1 1 B k 1 , , B 1 k B k k B^{1}_{1}...B^{1}_{k},...,B^{k}_{1}...B^{k}_{k}

Ornament_(music).html

  1. t r tr~{}~{}~{}
  2. t r tr~{}~{}~{}

Orrery.html

  1. 4 + 0 10 , 4 + 3 10 , 4 + 6 10 , 4 + 12 10 , 4 + 24 10 , 4 + 48 10 , . . \frac{4+0}{10},\frac{4+3}{10},\frac{4+6}{10},\frac{4+12}{10},\frac{4+24}{10},% \frac{4+48}{10},.....

Orthogonal_frequency-division_multiplexing.html

  1. Δ f = k T U \scriptstyle\Delta f\,=\,\frac{k}{T_{U}}
  2. Δ f = 1 1 ms = 1 kHz \scriptstyle\Delta f\,=\,\frac{1}{1\,\mathrm{ms}}\,=\,1\,\mathrm{kHz}
  3. MIPS = computational complexity T symbol × 1.3 × 10 - 6 = 147 456 × 2 896 × 10 - 6 × 1.3 × 10 - 6 = 428 \begin{aligned}\displaystyle\mathrm{MIPS}&\displaystyle=\frac{\mathrm{% computational\ complexity}}{T_{\mathrm{symbol}}}\times 1.3\times 10^{-6}\\ &\displaystyle=\frac{147\;456\times 2}{896\times 10^{-6}}\times 1.3\times 10^{% -6}\\ &\displaystyle=428\end{aligned}
  4. N = 1000 \scriptstyle N\,=\,1000
  5. N log 2 N = 10 , 000 \scriptstyle N\log_{2}N\,=\,10,000
  6. η = 2. R s B O F D M \eta=2.\frac{R_{s}}{B_{OFDM}}
  7. R s R_{s}
  8. B O F D M B_{OFDM}
  9. s [ n ] \scriptstyle s[n]
  10. N \scriptstyle N
  11. f c \scriptstyle f_{c}
  12. s ( t ) \scriptstyle s(t)
  13. r ( t ) \scriptstyle r(t)
  14. 2 f c \scriptstyle 2f_{c}
  15. N \scriptstyle N
  16. s ^ [ n ] \scriptstyle{\hat{s}}[n]
  17. N \scriptstyle N
  18. M \scriptstyle M
  19. M N \scriptstyle M^{N}
  20. ν ( t ) = k = 0 N - 1 X k e j 2 π k t / T , 0 t < T , \ \nu(t)=\sum_{k=0}^{N-1}X_{k}e^{j2\pi kt/T},\quad 0\leq t<T,
  21. { X k } \scriptstyle\{X_{k}\}
  22. N \scriptstyle N
  23. T \scriptstyle T
  24. 1 T \scriptstyle\frac{1}{T}
  25. 1 T 0 T ( e j 2 π k 1 t / T ) * ( e j 2 π k 2 t / T ) d t \displaystyle\frac{1}{T}\int_{0}^{T}\left(e^{j2\pi k_{1}t/T}\right)^{*}\left(e% ^{j2\pi k_{2}t/T}\right)dt
  26. ( ) * \scriptstyle(\cdot)^{*}
  27. δ \scriptstyle\delta\,
  28. T g \scriptstyle T_{\mathrm{g}}
  29. - T g t < 0 \scriptstyle-T_{\mathrm{g}}\,\leq\,t\,<\,0
  30. ( T - T g ) t < T \scriptstyle(T-T_{\mathrm{g}})\,\leq\,t\,<\,T
  31. ν ( t ) = k = 0 N - 1 X k e j 2 π k t / T , - T g t < T \ \nu(t)=\sum_{k=0}^{N-1}X_{k}e^{j2\pi kt/T},\quad-T_{\mathrm{g}}\leq t<T
  32. f c \scriptstyle f_{c}
  33. s ( t ) \displaystyle s(t)
  34. Δ f = 1 T U B N \scriptstyle\Delta f\,=\,\frac{1}{T_{U}}\,\approx\,\frac{B}{N}

Osmium_tetroxide.html

  1. Os + 2 O 2 Δ T OsO 4 \mathrm{Os+2\ O_{2}\ \xrightarrow{\Delta T}\ OsO_{4}}

Osmotic_pressure.html

  1. μ 0 ( l , p ) \mu^{0}(l,p)
  2. x s x_{s}
  3. p p
  4. l l
  5. x s x_{s}
  6. Π \Pi
  7. x s x_{s}
  8. Π \Pi
  9. μ s ( l , x s , p + Π ) = μ s 0 ( l , p ) \mu_{s}(l,x_{s},p+\Pi)=\mu_{s}^{0}(l,p)
  10. μ s ( l , x s , p + Π ) = μ s 0 ( l , p + Π ) + R T ln γ s x s \mu_{s}(l,x_{s},p+\Pi)=\mu_{s}^{0}(l,p+\Pi)+RT\ln\gamma_{s}x_{s}
  11. γ s \gamma_{s}
  12. γ s x s \gamma_{s}x_{s}
  13. a w a_{w}
  14. μ s o ( l , p + Π ) = μ s 0 ( l , p ) + p p + Π V d p \mu_{s}^{o}(l,p+\Pi)=\mu_{s}^{0}(l,p)+\int_{p}^{p+\Pi}\!V\,\mathrm{d}p
  15. V V
  16. - R T ln γ s x s = p p + Π V d p -RT\ln\gamma_{s}x_{s}=\int_{p}^{p+\Pi}\!V\,\mathrm{d}p
  17. Π V \Pi V
  18. Π = - ( R T / V ) ln ( γ s x s ) \Pi=-(RT/V)\ln(\gamma_{s}x_{s})
  19. Π = - ( R T / V ) ln ( x s ) \Pi=-(RT/V)\ln(x_{s})
  20. Π = i M R T \Pi=iMRT

Ostrich.html

  1. Metabolic Rate = 70 M 0.75 \mathrm{Metabolic\ Rate}=70M^{0.75}
  2. M M

Ovality.html

  1. 2 ( a - b ) a + b \frac{2(a-b)}{a+b}
  2. O = A 4 π ( 3 V 4 π ) 2 3 O=\frac{A}{4\pi(\frac{3V}{4\pi})^{\frac{2}{3}}}

Oxidative_phosphorylation.html

  1. × 10 9 \times 10^{−}9
  2. NADH + Q + 5 H matrix + NAD + + QH 2 + 4 H intermembrane + \rm NADH+Q+5\;H^{+}_{matrix}\rightarrow NAD^{+}+QH_{2}+4\;H^{+}_{intermembrane}\!
  3. Succinate + Q Fumarate + QH 2 \rm Succinate+Q\rightarrow Fumarate+QH_{2}\!
  4. ETF red + Q ETF ox + QH 2 \rm ETF_{red}+Q\rightarrow ETF_{ox}+QH_{2}\!
  5. QH 2 + 2 Cyt c ox + 2 H matrix + Q + 2 Cyt c red + 4 H intermembrane + \rm QH_{2}+2\;Cyt\,c_{ox}+2\;H^{+}_{matrix}\rightarrow Q+2\;Cyt\,c_{red}+4\;H^% {+}_{intermembrane}\!
  6. 4 Cyt c red + O 2 + 8 H matrix + 4 Cyt c ox + 2 H 2 O + 4 H intermembrane + \rm 4\;Cyt\,c_{red}+O_{2}+8\;H^{+}_{matrix}\rightarrow 4\;Cyt\,c_{ox}+2\;H_{2}% O+4\;H^{+}_{intermembrane}\!
  7. ADP + P i + 4 H intermembrane + ATP + H 2 O + 4 H matrix + \rm ADP+P_{i}+4\;H^{+}_{intermembrane}\rightleftharpoons ATP+H_{2}O+4\;H^{+}_{% matrix}\!
  8. e - e - O 2 O 2 ¯ O 2 2 - Superoxide Peroxide \begin{matrix}&{\mathrm{e}^{-}}&&{\mathrm{e}^{-}}\\ {\mbox{O}~{}_{2}}&\longrightarrow&\mbox{O}~{}_{2}^{\underline{\bullet}}&% \longrightarrow&\mbox{O}~{}_{2}^{2-}\\ &&\mbox{Superoxide}&&\mbox{Peroxide}\\ &\end{matrix}

P-adic_number.html

  1. p p
  2. p p
  3. p p
  4. p p
  5. p p
  6. p p
  7. p p
  8. p p
  9. p p
  10. p p
  11. p p
  12. p p
  13. p p
  14. 1 3 = 0.333333 . \frac{1}{3}=0.333333\ldots.
  15. r r
  16. p p
  17. q q
  18. q q
  19. p p
  20. r 0 r≠0
  21. e e
  22. r r
  23. | r | 10 = 1 10 e |r|_{10}=\frac{1}{10^{e}}
  24. x x
  25. y y
  26. p p
  27. 9 = - 1 + 10 9=-1+10\,
  28. | 9 - ( - 1 ) | 10 = 1 10 |9-(-1)|_{10}=\frac{1}{10}
  29. 99 = - 1 + 10 2 99=-1+10^{2}\,
  30. | 99 - ( - 1 ) | 10 = 1 100 |99-(-1)|_{10}=\frac{1}{100}
  31. 999 = - 1 + 10 3 999=-1+10^{3}\,
  32. | 999 - ( - 1 ) | 10 = 1 1000 |999-(-1)|_{10}=\frac{1}{1000}
  33. 9999 = - 1 + 10 4 9999=-1+10^{4}\,
  34. | 9999 - ( - 1 ) | 10 = 1 10000 |9999-(-1)|_{10}=\frac{1}{10000}
  35. 9999 = - 1. \dots 9999=-1.\,
  36. p p
  37. i = n a i 10 i \sum_{i=n}^{\infty}a_{i}10^{i}
  38. n n
  39. - 100 = - 1 × 100 = 9999 × 100 = 9900 -100=-1\times 100=\dots 9999\times 100=\dots 9900\,
  40. - 35 = - 100 + 65 = 9900 + 65 = 9965 \Rightarrow-35=-100+65=\dots 9900+65=\dots 9965\,
  41. - ( 3 + 1 2 ) = - 35 10 = 9965 10 = 9996.5 \Rightarrow-\left(3+\dfrac{1}{2}\right)=\dfrac{-35}{10}=\dfrac{\dots 9965}{10}% =\dots 9996.5
  42. 10 6 - 1 7 = 142857 ; 10 12 - 1 7 = 142857142857 ; 10 18 - 1 7 = 142857142857142857 \dfrac{10^{6}-1}{7}=142857;\dfrac{10^{12}-1}{7}=142857142857;\dfrac{10^{18}-1}% {7}=142857142857142857
  43. - 1 7 = 142857142857142857 \Rightarrow-\dfrac{1}{7}=\dots 142857142857142857
  44. - 6 7 = 142857142857142857 × 6 = 857142857142857142 \Rightarrow-\dfrac{6}{7}=\dots 142857142857142857\times 6=\dots 85714285714285% 7142
  45. 1 7 = - 6 7 + 1 = 857142857142857143. \Rightarrow\dfrac{1}{7}=-\dfrac{6}{7}+1=\dots 857142857142857143.
  46. p q p⁄q
  47. q q
  48. q q
  49. n n
  50. q q
  51. p p
  52. p p
  53. p p
  54. p p
  55. p p
  56. i = 0 n a i p i \sum_{i=0}^{n}a_{i}p^{i}
  57. p 1 p− 1
  58. ± i = - n a i p i . \pm\sum_{i=-\infty}^{n}a_{i}p^{i}.
  59. p p
  60. p p
  61. p p
  62. p p
  63. i = k a i p i \sum_{i=k}^{\infty}a_{i}p^{i}
  64. a i a_{i}
  65. p p
  66. p p
  67. p p
  68. p p
  69. p p
  70. p p
  71. p p
  72. p p
  73. 5 {}_{5}
  74. 5 {}_{5}
  75. 5 {}_{5}
  76. 5 {}_{5}
  77. 5 {}_{5}
  78. 5 {}_{5}
  79. 5 {}_{5}
  80. 5 {}_{5}
  81. 5 2 - 1 3 = 44 5 3 = 13 5 ; 5 4 - 1 3 = 4444 5 3 = 1313 5 \dfrac{5^{2}-1}{3}=\dfrac{44_{5}}{3}=13_{5};\,\dfrac{5^{4}-1}{3}=\dfrac{4444_{% 5}}{3}=1313_{5}
  82. - 1 3 = 1313 5 \Rightarrow-\dfrac{1}{3}=\dots 1313_{5}
  83. - 2 3 = 1313 5 × 2 = 3131 5 \Rightarrow-\dfrac{2}{3}=\dots 1313_{5}\times 2=\dots 3131_{5}
  84. 1 3 = - 2 3 + 1 = 3132 5 . \Rightarrow\dfrac{1}{3}=-\dfrac{2}{3}+1=\dots 3132_{5}.
  85. 5 {}_{5}
  86. p p
  87. p p
  88. p p
  89. p p
  90. p p
  91. p p
  92. p p
  93. p p
  94. p p
  95. p p
  96. p p
  97. 1 / 5 {1}/{5}
  98. 1 5 = 121012102 3 . \dfrac{1}{5}=\dots 121012102_{3}.
  99. p p
  100. p p
  101. 1 / 5 {1}/{5}
  102. 1 5 = 2.01210121 3 or 1 15 = 20.1210121 3 . \dfrac{1}{5}=2.01210121\dots_{3}\mbox{ or }~{}\dfrac{1}{15}=20.1210121\dots_{3}.
  103. p p
  104. p 1 p− 1
  105. 1 5 = 1 ¯ 11 11 ¯ 11 11 ¯ 11 1 ¯ 3 . \dfrac{1}{5}=\dots\underline{1}11\underline{11}11\underline{11}11\underline{1}% _{3}.
  106. p p
  107. p p
  108. p p
  109. p p
  110. p = 2 p=2
  111. p p
  112. p p
  113. x x
  114. n n
  115. p p
  116. x x
  117. p p
  118. n n
  119. < t d > x | 2 = 2 < / t d > < t d > x | 3 = 1 / 9 < / t d > < t d > x | 5 = 25 < / t d > < t d > x | 7 = 1 / 7 < / t d > < t d > x | 11 = 11 < / t d > < t d > x | any other prime = 1. T h i s d e f i n i t i o n o f m a t h < t d > ′′ x ′′ | < s u b Align g t ; ′′ p ′′ h a s t h e e f f e c t t h a t h i g h p o w e r s o f < m a t h > p \begin{aligned}\displaystyle<td>x|_{2}&\displaystyle=2\\ \displaystyle</td><td>x|_{3}&\displaystyle=1/9\\ \displaystyle</td><td>x|_{5}&\displaystyle=25\\ \displaystyle</td><td>x|_{7}&\displaystyle=1/7\\ \displaystyle</td><td>x|_{11}&\displaystyle=11\\ \displaystyle</td><td>x|_{\,\text{any other prime}}&\displaystyle=1.\end{% aligned}Thisdefinitionof{{math<td>^{\prime\prime}x^{\prime\prime}|<sub&gt;^{% \prime\prime}p^{\prime\prime}}}hastheeffectthathighpowersof <math>p
  120. p 1 , , p r p_{1},\ldots,p_{r}
  121. a 1 , , a r a_{1},\ldots,a_{r}
  122. | x | = p 1 a 1 p r a r . |x|=p_{1}^{a_{1}}\ldots p_{r}^{a_{r}}.
  123. | x | p i = p i - a i |x|_{p_{i}}=p_{i}^{-a_{i}}
  124. 1 i r 1\leq i\leq r
  125. | x | p = 1 |x|_{p}=1\,
  126. p { p 1 , , p r } . p\notin\{p_{1},\ldots,p_{r}\}.
  127. p p
  128. p p
  129. p p
  130. p p
  131. d p ( x , y ) = | x - y | p d_{p}(x,y)=|x-y|_{p}\,\!
  132. p p
  133. i = k a i p i \sum_{i=k}^{\infty}a_{i}p^{i}
  134. p 1 p− 1
  135. p p
  136. p p
  137. p p
  138. p p
  139. p p
  140. p p
  141. p p
  142. p p
  143. p p
  144. 𝐐 p = Quot ( 𝐙 p ) ( p 𝐍 ) - 1 𝐙 p . \mathbf{Q}_{p}=\operatorname{Quot}\left(\mathbf{Z}_{p}\right)\cong(p^{\mathbf{% N}})^{-1}\mathbf{Z}_{p}.
  145. S = p 𝐍 = { p n : n 𝐍 } S=p^{\mathbf{N}}=\{p^{n}:n\in\mathbf{N}\}
  146. A A
  147. A A
  148. S S
  149. p p
  150. n n
  151. n × n n×n
  152. 𝐙 \mathbf{Z}
  153. 𝐙 \mathbf{Z}
  154. 𝐙 \mathbf{Z}
  155. p p
  156. 𝐙 \mathbf{Z}
  157. d 2 d 1 d 0 \cdots d_{2}d_{1}d_{0}
  158. 0. e 0 e 1 e 2 3 0.e_{0}e_{1}e_{2}\cdots_{3}
  159. 𝐂 \mathbf{C}
  160. e n = 2 d n . e_{n}=2d_{n}.
  161. 𝐐 \mathbf{Q}
  162. 0
  163. 𝐑 \mathbf{R}
  164. 𝐂 \mathbf{C}
  165. p p
  166. 𝐂 \mathbf{C}
  167. 𝐂 \mathbf{C}
  168. 𝐂 \mathbf{C}
  169. 𝐊 \mathbf{K}
  170. n n
  171. n > 2 n>2
  172. n | p 1 n|p−1
  173. n n
  174. n = 1 , 2 , 3 , 4 , 6 n=1,2,3,4,6
  175. 12 12
  176. p p
  177. p > 2 p>2
  178. 1 −1
  179. k k
  180. k k
  181. 𝐐 [ u s u , u b = , u p , u p = \xd 7 ] \mathbf{Q}[u^{\prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}p^{\prime},u^% {\prime}p=\xd 7^{\prime}]
  182. e e
  183. p p
  184. p = 2 p=2
  185. e e
  186. p p
  187. p p
  188. { f : 𝐐 p 𝐐 p f ( x ) = { | x | p - 2 x 0 0 x = 0 \begin{cases}f:\mathbf{Q}_{p}\to\mathbf{Q}_{p}\\ f(x)=\begin{cases}|x|_{p}^{-2}&x\neq 0\\ 0&x=0\end{cases}\end{cases}
  189. 0
  190. 𝐐 \mathbf{Q}
  191. p p
  192. p p
  193. p p
  194. | x | P = c - ord P ( x ) . |x|_{P}=c^{-\operatorname{ord}_{P}(x)}.
  195. p p
  196. p p

P-group.html

  1. 2 n , n 6 2^{n},n\leq 6
  2. p 2 27 n 3 + O ( n 8 / 3 ) p^{\frac{2}{27}n^{3}+O(n^{8/3})}
  3. | G | = n = p k m |G|=n=p^{k}m
  4. p k , p^{k},
  5. O p ( G ) O_{p}(G)
  6. O p ( G ) . O^{p}(G).
  7. G = C p 2 G=C_{p^{2}}
  8. p 2 p^{2}
  9. G = C p × C p . G=C_{p}\times C_{p}.

P_versus_NP_problem.html

  1. 2 2 c n 2^{2^{cn}}
  2. O ( exp ( ( 64 n 9 log ( 2 ) ) 1 3 ( log ( n log ( 2 ) ) ) 2 3 ) ) O\left(\exp\left(\left(\tfrac{64n}{9}\log(2)\right)^{\frac{1}{3}}\left(\log(n% \log(2))\right)^{\frac{2}{3}}\right)\right)
  3. 𝐏 = { L : L = L ( M ) for some deterministic polynomial-time Turing machine M } \mathbf{P}=\{L:L=L(M)\,\text{ for some deterministic polynomial-time Turing % machine }M\}
  4. L ( M ) = { w Σ * : M accepts w } L(M)=\{w\in\Sigma^{*}:M\,\text{ accepts }w\}
  5. k N k\in N
  6. T M ( n ) O ( n k ) T_{M}(n)\in O(n^{k})
  7. T M ( n ) = max { t M ( w ) : w Σ * , | w | = n } T_{M}(n)=\max\{t_{M}(w):w\in\Sigma^{*},|w|=n\}
  8. t M ( w ) = number of steps M takes to halt on input w . t_{M}(w)=\,\text{ number of steps }M\,\text{ takes to halt on input }w.
  9. R Σ * × Σ * R\subset\Sigma^{*}\times\Sigma^{*}
  10. x Σ * x\in\Sigma^{*}
  11. x L y Σ * x\in L\Leftrightarrow\exists y\in\Sigma^{*}
  12. | y | O ( | x | k ) |y|\in O(|x|^{k})
  13. L R = { x # y : ( x , y ) R } L_{R}=\{x\#y:(x,y)\in R\}
  14. Σ { # } \Sigma\cup\{\#\}
  15. COMPOSITE = { x x = p q for integers p , q > 1 } \mathrm{COMPOSITE}=\left\{x\in\mathbb{N}\mid x=pq\,\text{ for integers }p,q>1\right\}
  16. R = { ( x , y ) × 1 < y x and y divides x } . R=\left\{(x,y)\in\mathbb{N}\times\mathbb{N}\mid 1<y\leq\sqrt{x}\,\text{ and }y% \,\text{ divides }x\right\}.
  17. L p L L^{\prime}\leq_{p}L
  18. L p L L^{\prime}\leq_{p}L
  19. ( w L f ( w ) L ) (w\in L^{\prime}\Leftrightarrow f(w)\in L)
  20. O ( N 2 ) O(N^{2})
  21. Ω ( N 4 ) \Omega(N^{4})

PAL.html

  1. Y Y
  2. R G B R^{\prime}G^{\prime}B^{\prime}
  3. Y = 0.299 R + 0.587 G + 0.114 B Y=0.299R^{\prime}+0.587G^{\prime}+0.114B^{\prime}
  4. U U
  5. V V
  6. U = 0.492 ( B - Y ) U=0.492(B^{\prime}-Y)
  7. V = 0.877 ( R - Y ) V=0.877(R^{\prime}-Y)
  8. = Y + U sin ( ω t ) + V cos ( ω t ) + =Y+U\sin(\omega t)+V\cos(\omega t)+
  9. ω = 2 π F S C \omega=2\pi F_{SC}
  10. F S C F_{SC}

Palermo_Technical_Impact_Hazard_Scale.html

  1. P log 10 p i f B T P\equiv\log_{10}\frac{p_{i}}{f_{B}T}
  2. f B = 3 100 E - 4 / 5 y r - 1 f_{B}=\frac{3}{100}E^{-4/5}yr^{-1}\;

Palindrome.html

  1. n + n \sqrt{n}+\lfloor\sqrt{n}\rfloor
  2. n n
  3. d P ( w ) = ( lim sup k n k + 1 n k ) - 1 . d_{P}(w)=\left({\limsup_{k\rightarrow\infty}\frac{n_{k+1}}{n_{k}}}\right)^{-1}\ .

Panspermia.html

  1. P ( t a r g e t ) = A ( t a r g e t ) π ( d y ) 2 = a r ( t a r g e t ) 2 v 2 ( t p ) 2 d 4 P(target)=\frac{A(target)}{\pi(dy)^{2}}=\frac{ar(target)^{2}v^{2}}{(tp)^{2}d^{% 4}}

Parabola.html

  1. y = x 2 . y=x^{2}.
  2. | x + p | = ( x - p ) 2 + y 2 |x+p|=\sqrt{(x-p)^{2}+y^{2}}
  3. y 2 = 4 p x y^{2}=4px
  4. x 2 = 4 p y x^{2}=4py
  5. ( x - h ) 2 = 4 p ( y - k ) (x-h)^{2}=4p(y-k)\,
  6. y = a x 2 + b x + c y=ax^{2}+bx+c\,
  7. A x 2 + B x y + C y 2 + D x + E y + F = 0 Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0\,
  8. B 2 = 4 A C B^{2}=4AC\,
  9. [ A B / 2 D / 2 B / 2 C E / 2 D / 2 E / 2 F ] \begin{bmatrix}A&B/2&D/2\\ B/2&C&E/2\\ D/2&E/2&F\end{bmatrix}
  10. P K ¯ \overline{PK}
  11. B C ¯ \overline{BC}
  12. E D ¯ \overline{ED}
  13. E M ¯ \overline{EM}
  14. D M ¯ \overline{DM}
  15. P M ¯ \overline{PM}
  16. B M = 2 y sin θ . BM=2y\sin{\theta}.
  17. C M = 2 r . CM=2r.
  18. E M D M = B M C M EM\cdot DM=BM\cdot CM
  19. x 2 = 4 r y sin θ x^{2}=4ry\sin{\theta}
  20. y = x 2 4 r sin θ y=\frac{x^{2}}{4r\sin{\theta}}
  21. y = x 2 4 f y=\frac{x^{2}}{4f}
  22. f f
  23. r sin θ r\sin{\theta}
  24. P V ¯ \overline{PV}
  25. h h
  26. k k
  27. ( h , k ) (h,k)
  28. p p
  29. ( x - h ) 2 = 4 p ( y - k ) (x-h)^{2}=4p(y-k)\,
  30. y = ( x - h ) 2 4 p + k y=\frac{(x-h)^{2}}{4p}+k\,
  31. y = a x 2 + b x + c y=ax^{2}+bx+c\,
  32. a = 1 4 p ; b = - h 2 p ; c = h 2 4 p + k ; a=\frac{1}{4p};\ \ b=\frac{-h}{2p};\ \ c=\frac{h^{2}}{4p}+k;\
  33. h = - b 2 a ; k = 4 a c - b 2 4 a h=\frac{-b}{2a};\ \ k=\frac{4ac-b^{2}}{4a}
  34. x ( t ) = 2 p t + h ; y ( t ) = p t 2 + k x(t)=2pt+h;\ \ y(t)=pt^{2}+k\,
  35. ( y - k ) 2 = 4 p ( x - h ) (y-k)^{2}=4p(x-h)\,
  36. x = ( y - k ) 2 4 p + h ; x=\frac{(y-k)^{2}}{4p}+h;\ \,
  37. x = a y 2 + b y + c x=ay^{2}+by+c\,
  38. a = 1 4 p ; b = - k 2 p ; c = k 2 4 p + h ; a=\frac{1}{4p};\ \ b=\frac{-k}{2p};\ \ c=\frac{k^{2}}{4p}+h;\
  39. h = 4 a c - b 2 4 a ; k = - b 2 a h=\frac{4ac-b^{2}}{4a};\ \ k=\frac{-b}{2a}
  40. x ( t ) = p t 2 + h ; y ( t ) = 2 p t + k x(t)=pt^{2}+h;\ \ y(t)=2pt+k\,
  41. ( α x + β y ) 2 + γ x + δ y + ϵ = 0 (\alpha x+\beta y)^{2}+\gamma x+\delta y+\epsilon=0\,
  42. A x 2 + B x y + C y 2 + D x + E y + F = 0 Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0\,
  43. B 2 = 4 A C B^{2}=4AC\,
  44. a x + b y + c = 0 ax+by+c=0\,
  45. ( a x + b y + c ) 2 a 2 + b 2 = ( x - u ) 2 + ( y - v ) 2 \frac{\left(ax+by+c\right)^{2}}{{a}^{2}+{b}^{2}}=\left(x-u\right)^{2}+\left(y-% v\right)^{2}\,
  46. r ( 1 + cos θ ) = l r(1+\cos\theta)=l\,
  47. y = a x 2 + b x + c y=ax^{2}+bx+c
  48. x x
  49. y y
  50. y = a x 2 + b x + c y=ax^{2}+bx+c
  51. d y d x = 2 a x + b \frac{dy}{dx}=2ax+b
  52. x x
  53. x = - b 2 a x=\frac{-b}{2a}
  54. y = a ( - b 2 a ) 2 + b ( - b 2 a ) + c y=a\left(-\frac{b}{2a}\right)^{2}+b\left(-\frac{b}{2a}\right)+c
  55. = a b 2 4 a 2 - b 2 2 a + c = c - b 2 4 a =\frac{ab^{2}}{4a^{2}}-\frac{b^{2}}{2a}+c=c-\frac{b^{2}}{4a}
  56. y = 4 a c - b 2 4 a = - b 2 - 4 a c 4 a = - D 4 a y=\frac{4ac-b^{2}}{4a}=-\frac{b^{2}-4ac}{4a}=-\frac{D}{4a}
  57. D = ( b 2 - 4 a c ) D=(b^{2}-4ac)
  58. ( - b 2 a , - D 4 a ) \left(-\frac{b}{2a},-\frac{D}{4a}\right)
  59. - b 2 a . -\frac{b}{2a}.
  60. y = a x 2 + b x + c , y=ax^{2}+bx+c,
  61. d y d x = 2 a x + b = 1 \frac{dy}{dx}=2ax+b=1
  62. x = 1 - b 2 a \therefore x=\frac{1-b}{2a}
  63. x x
  64. y = a ( 1 - b 2 a ) 2 + b ( 1 - b 2 a ) + c y=a\left(\frac{1-b}{2a}\right)^{2}+b\left(\frac{1-b}{2a}\right)+c
  65. = a ( 1 - 2 b + b 2 4 a 2 ) + ( b - b 2 2 a ) + c =a\left(\frac{1-2b+b^{2}}{4a^{2}}\right)+\left(\frac{b-b^{2}}{2a}\right)+c
  66. = ( 1 - 2 b + b 2 4 a ) + ( 2 b - 2 b 2 4 a ) + c =\left(\frac{1-2b+b^{2}}{4a}\right)+\left(\frac{2b-2b^{2}}{4a}\right)+c
  67. = 1 - b 2 4 a + c = 1 - ( b 2 - 4 a c ) 4 a = 1 - D 4 a =\frac{1-b^{2}}{4a}+c=\frac{1-(b^{2}-4ac)}{4a}=\frac{1-D}{4a}
  68. D = ( b 2 - 4 a c ) D=(b^{2}-4ac)
  69. ( - b 2 a , 1 - D 4 a ) \left(-\frac{b}{2a},\frac{1-D}{4a}\right)
  70. y = a x 2 + b x + c y=ax^{2}+bx+c
  71. ( - b 2 a , - D 4 a ) \left(\frac{-b}{2a},\frac{-D}{4a}\right)
  72. D = b 2 - 4 a c . D=b^{2}-4ac.
  73. x = - b 2 a x=-\frac{b}{2a}
  74. f = ( 1 - D 4 a ) - ( - D 4 a ) f=\left(\frac{1-D}{4a}\right)-\left(\frac{-D}{4a}\right)
  75. = 1 4 a =\frac{1}{4a}
  76. a = 1 4 f . a=\frac{1}{4f}.
  77. 1 - b 2 a + b 2 a \frac{1-b}{2a}+\frac{b}{2a}
  78. = 1 2 a =\frac{1}{2a}
  79. = 2 f =2f
  80. f f
  81. y = - D 4 a - 1 4 a = - 1 + D 4 a y=-\frac{D}{4a}-\frac{1}{4a}=-\frac{1+D}{4a}
  82. y = x 2 . y=x^{2}.
  83. ( x , x 2 ) . (x,x^{2}).
  84. < m t p l > x 2. \frac{<}{m}tpl>{{x}}{{2}}.
  85. x 2 ( x 2 ) , \frac{x^{2}}{\left(\frac{x}{2}\right)},
  86. 2 x . 2x.
  87. 2 x 2x
  88. α \alpha
  89. α \alpha
  90. y = a x 2 , a 0. y=ax^{2},\ a\neq 0.
  91. ( p , a p 2 ) (p,ap^{2})
  92. ( q , a q 2 ) . (q,aq^{2}).
  93. 2 a p 2ap
  94. 2 a q , 2aq,
  95. y = 2 a p x + C , y=2apx+C,
  96. C C
  97. ( p , a p 2 ) , (p,ap^{2}),
  98. C C
  99. - a p 2 , -ap^{2},
  100. y = 2 a p x - a p 2 . y=2apx-ap^{2}.
  101. y = 2 a q x - a q 2 . y=2aqx-aq^{2}.
  102. 2 a p x - a p 2 = 2 a q x - a q 2 . 2apx-ap^{2}=2aqx-aq^{2}.
  103. 2 x ( p - q ) = p 2 - q 2 . 2x(p-q)=p^{2}-q^{2}.
  104. x = p + q 2 . x=\frac{p+q}{2}.
  105. y = 2 a p ( p + q 2 ) - a p 2 . y=2ap\left(\frac{p+q}{2}\right)-ap^{2}.
  106. y = a p q . y=apq.
  107. 2 a p 2ap
  108. 2 a q , 2aq,
  109. ( 2 a p ) ( 2 a q ) = - 1 , (2ap)(2aq)=-1,
  110. p q = - 1 4 a 2 . pq=-\frac{1}{4a^{2}}.
  111. y = - 1 4 a . y=-\frac{1}{4a}.
  112. c , c,
  113. d . d.
  114. f , f,
  115. f = c 2 16 d f=\frac{c^{2}}{16d}
  116. A = 2 3 b h A=\frac{2}{3}bh
  117. 1 2 b h \frac{1}{2}bh
  118. 2 3 \textstyle\frac{2}{3}
  119. f , f,
  120. p p
  121. f f
  122. p p
  123. h = p 2 h=\frac{p}{2}
  124. q = f 2 + h 2 q=\sqrt{f^{2}+h^{2}}
  125. s = h q f + f ln ( h + q f ) s=\frac{hq}{f}+f\ln\left(\frac{h+q}{f}\right)
  126. s s
  127. 2 s . 2s.
  128. p p
  129. p p
  130. h h
  131. s s
  132. s . s.
  133. s 1 - s 2 = h 1 q 1 - h 2 q 2 f + f ln ( h 1 + q 1 h 2 + q 2 ) s_{1}-s_{2}=\frac{h_{1}q_{1}-h_{2}q_{2}}{f}+f\ln\left(\frac{h_{1}+q_{1}}{h_{2}% +q_{2}}\right)
  134. ( x , y ) (x,y)
  135. R R
  136. ( 0 , R ) . (0,R).
  137. x 2 + ( R - y ) 2 = R 2 x^{2}+(R-y)^{2}=R^{2}
  138. x 2 + R 2 - 2 R y + y 2 = R 2 \therefore x^{2}+R^{2}-2Ry+y^{2}=R^{2}
  139. x 2 + y 2 = 2 R y . \therefore x^{2}+y^{2}=2Ry.
  140. ( x , y ) (x,y)
  141. y y
  142. x , x,
  143. y 2 y^{2}
  144. x 2 = 2 R y . x^{2}=2Ry.
  145. x 2 = 4 f y x^{2}=4fy
  146. f . f.
  147. R = 2 f . R=2f.
  148. ( x , x 2 , x 3 , , x n ) ; (x,x^{2},x^{3},\dots,x^{n});
  149. n = 2 , n=2,
  150. n = 3 n=3
  151. x 2 x^{2}
  152. x 2 + y 2 x^{2}+y^{2}
  153. x 2 - y 2 . x^{2}-y^{2}.
  154. y = x p y=x^{p}
  155. x p = k y q x^{p}=ky^{q}
  156. y = x p / q y=x^{p/q}
  157. x p y q = k , x^{p}y^{q}=k,

Paracompact_space.html

  1. X α A U α . X\subseteq\bigcup_{\alpha\in A}U_{\alpha}.
  2. { α A : U α V ( x ) } \left\{\alpha\in A:U_{\alpha}\cap V(x)\neq\varnothing\right\}
  3. X X\,
  4. 𝒪 \mathcal{O}\,
  5. W U W_{U}\,
  6. U 𝒪 U\in\mathcal{O}\,
  7. W U ¯ U \bar{W_{U}}\subseteq U\,
  8. { W U : U 𝒪 } \{W_{U}:U\in\mathcal{O}\}\,
  9. 𝒪 \mathcal{O}\,
  10. f U : X [ 0 , 1 ] f_{U}:X\to[0,1]\,
  11. supp f U U \operatorname{supp}~{}f_{U}\subseteq U\,
  12. f := U 𝒪 f U f:=\sum_{U\in\mathcal{O}}f_{U}\,
  13. X X\,
  14. 𝒪 \mathcal{O}\,
  15. 𝒱 \mathcal{V}\,
  16. 𝒪 \mathcal{O}\,
  17. 𝒪 \mathcal{O}
  18. 𝒪 \mathcal{O}\,
  19. 𝒱 \mathcal{V}\,
  20. W U = { A 𝒱 : A ¯ U } W_{U}=\bigcup\{A\in\mathcal{V}:\bar{A}\subseteq U\}\,
  21. W U ¯ U \bar{W_{U}}\subseteq U\,
  22. x W U ¯ U x\in\bar{W_{U}}\setminus U
  23. C W U C\supset W_{U}
  24. x C x\notin C
  25. x W U ¯ x\notin\bar{W_{U}}
  26. 𝒱 \mathcal{V}
  27. V [ x ] V[x]
  28. x x
  29. U 1 , , U n { A 𝒱 : A ¯ U } U_{1},...,U_{n}\in\{A\in\mathcal{V}:\bar{A}\subseteq U\}
  30. V [ x ] V[x]
  31. U 1 ¯ , , U n ¯ \bar{U_{1}},...,\bar{U_{n}}
  32. V := V [ x ] U i ¯ V:=V[x]\setminus\cup\bar{U_{i}}
  33. V W U = V\cap W_{U}=\varnothing
  34. x V x\in V
  35. i = { 1 , , n } \forall i=\{1,...,n\}
  36. U i ¯ U \bar{U_{i}}\subseteq U
  37. x U x\notin U
  38. C := X V C:=X\setminus V
  39. x x
  40. W U W_{U}
  41. x W U ¯ x\notin\bar{W_{U}}
  42. { W U : U 𝒪 } \{W_{U}:U\in\mathcal{O}\}\,
  43. 𝒪 \mathcal{O}\,
  44. x X x\in X\,
  45. N N\,
  46. x x\,
  47. U U
  48. W U U W_{U}\subseteq U
  49. O O
  50. U 1 , , U k U_{1},...,U_{k}
  51. N N
  52. W U 1 , , W U k W_{U_{1}},...,W_{U_{k}}
  53. N N
  54. U U^{\prime}
  55. N W U N U = N\cap W_{U^{\prime}}\subseteq N\cap U^{\prime}=\varnothing
  56. f U : X [ 0 , 1 ] f_{U}:X\to[0,1]\,
  57. f U W ¯ U = 1 f_{U}\upharpoonright\bar{W}_{U}=1\,
  58. supp f U U \operatorname{supp}~{}f_{U}\subseteq U\,
  59. f = U 𝒪 f U f=\sum_{U\in\mathcal{O}}f_{U}\,
  60. x X x\in X\,
  61. N N\,
  62. x x\,
  63. 𝒪 \mathcal{O}\,
  64. x x\,
  65. 𝒪 \mathcal{O}\,
  66. f U ( x ) = 0 f_{U}(x)=0\,
  67. U U\,
  68. x W U x\in W_{U}\,
  69. U U\,
  70. f U ( x ) = 1 f_{U}(x)=1\,
  71. f ( x ) f(x)\,
  72. 1 \geq 1\,
  73. x , N x,N\,
  74. S = { U 𝒪 : N meets U } S=\{U\in\mathcal{O}:N\,\text{ meets }U\}\,
  75. f N = U S f U N f\upharpoonright N=\sum_{U\in S}f_{U}\upharpoonright N\,
  76. f f\,
  77. f ( x ) f(x)\,
  78. x x\,
  79. 𝒪 * \mathcal{O}*\,
  80. { V open : ( U 𝒪 ) V ¯ U } \{V\,\text{ open }:(\exists{U\in\mathcal{O}})\bar{V}\subseteq U\}\,
  81. f W : X [ 0 , 1 ] f_{W}:X\to[0,1]\,
  82. supp f W W \operatorname{supp}~{}f_{W}\subseteq W\,
  83. U 𝒪 U\in\mathcal{O}\,
  84. W 𝒪 * W\in\mathcal{O}*\,
  85. 1 / f 1/f\,
  86. f W f_{W}\,
  87. f W / f f_{W}/f\,
  88. 1 1\,
  89. x X x\in X\,
  90. N N\,
  91. x x\,
  92. 𝒪 * \mathcal{O}*\,
  93. f W N = 0 f_{W}\upharpoonright N=0\,
  94. W 𝒪 * W\in\mathcal{O}*\,
  95. supp f W W \operatorname{supp}~{}f_{W}\subseteq W\,
  96. 𝐔 * ( x ) := U α x U α . \mathbf{U}^{*}(x):=\bigcup_{U_{\alpha}\ni x}U_{\alpha}.
  97. { α A : x U α } \left\{\alpha\in A:x\in U_{\alpha}\right\}

Parallax.html

  1. d ( pc ) = 1 / p ( arcsec ) . d(\mathrm{pc})=1/p(\mathrm{arcsec}).
  2. distance moon = distance observerbase tan ( angle ) \mathrm{distance}_{\mathrm{moon}}=\frac{\mathrm{distance}_{\mathrm{% observerbase}}}{\tan(\mathrm{angle})}
  3. sin p = 1 AU d , \sin p=\frac{1\,\text{ AU}}{d},
  4. p p
  5. d d
  6. sin x x radians = x 180 π degrees = x 180 3600 π arcseconds , \sin x\approx x\,\text{ radians}=x\cdot\frac{180}{\pi}\,\text{ degrees}=x\cdot 1% 80\cdot\frac{3600}{\pi}\,\text{ arcseconds},
  7. p ′′ 1 AU d 180 3600 π . p^{\prime\prime}\approx\frac{1\,\text{ AU}}{d}\cdot 180\cdot\frac{3600}{\pi}.
  8. d = 1 AU 180 3600 π 206 , 265 AU 3.2616 ly 1 parsec . d=1\,\text{ AU}\cdot 180\cdot\frac{3600}{\pi}\approx 206,265\,\text{ AU}% \approx 3.2616\,\text{ ly}\equiv 1\,\text{ parsec}.
  9. d = 1 / p d=1/p
  10. δ d = δ ( 1 p ) = | p ( 1 p ) | δ p = δ p p 2 \delta d=\delta\left({1\over p}\right)=\left|{\partial\over\partial p}\left({1% \over p}\right)\right|\delta p={\delta p\over p^{2}}

Parallelepiped.html

  1. V = | 𝐚 ( 𝐛 × 𝐜 ) | = | 𝐛 ( 𝐜 × 𝐚 ) | = | 𝐜 ( 𝐚 × 𝐛 ) | V=|\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})|=|\mathbf{b}\cdot(\mathbf{c}% \times\mathbf{a})|=|\mathbf{c}\cdot(\mathbf{a}\times\mathbf{b})|
  2. V = a b c 1 + 2 cos ( α ) cos ( β ) cos ( γ ) - cos 2 ( α ) - cos 2 ( β ) - cos 2 ( γ ) . V=abc\sqrt{1+2\cos(\alpha)\cos(\beta)\cos(\gamma)-\cos^{2}(\alpha)-\cos^{2}(% \beta)-\cos^{2}(\gamma)}.
  3. m \mathbb{R}^{m}
  4. m n m\geq n
  5. V = v 1 v n . V=\|v_{1}\wedge\cdots\wedge v_{n}\|.

Paramagnetism.html

  1. χ v \chi_{v}
  2. χ \chi
  3. s y m b o l M = \chisymbol H = C T s y m b o l H symbol{M}=\chisymbol{H}=\frac{C}{T}symbol{H}
  4. M M
  5. χ \chi
  6. H H
  7. T T
  8. C C
  9. C = N A 3 k B μ eff 2 where μ eff = g J μ B J ( J + 1 ) C=\frac{N_{A}}{3k_{B}}\mu_{\mathrm{eff}}^{2}\,\text{ where }\mu_{\mathrm{eff}}% =g_{J}\mu_{B}\sqrt{J(J+1)}
  10. N A m ¯ = N A M J = - J J μ M J e - E M J / k B T M J = - J J e - E M J / k B T = N A M J = - J J M J g J μ B e M J g J μ B H / k B T M J = - J J e M J g J μ B H / k B T N_{A}\bar{m}=\frac{N_{A}\sum\limits_{M_{J}=-J}^{J}{\mu_{M_{J}}e^{{-E_{M_{J}}}/% {k_{B}T}\;}}}{\sum\limits_{M_{J}=-J}^{J}{e^{{-E_{M_{J}}}/{k_{B}T}\;}}}=\frac{N% _{A}\sum\limits_{M_{J}=-J}^{J}{M_{J}g_{J}\mu_{B}e^{{M_{J}g_{J}\mu_{B}H}/{k_{B}% T}\;}}}{\sum\limits_{M_{J}=-J}^{J}{e^{{M_{J}g_{J}\mu_{B}H}/{k_{B}T}\;}}}
  11. μ M J \mu_{M_{J}}
  12. μ M J = M J g J μ B \mu_{M_{J}}=M_{J}g_{J}\mu_{B}
  13. E M J = - M J g J μ B H E_{M_{J}}=-M_{J}g_{J}\mu_{B}H
  14. M J g J μ B H / k B T 1 {M_{J}g_{J}\mu_{B}H}/{k_{B}T}\;\ll 1
  15. e M J g J μ B H / k B T 1 + M J g J μ B H / k B T e^{{M_{J}g_{J}\mu_{B}H}/{k_{B}T}\;}\simeq 1+{M_{J}g_{J}\mu_{B}H}/{k_{B}T}\;
  16. m ¯ = M J = - J J M J g J μ B e M J g J μ B H / k B T M J = - J J e M J g J μ B H / k B T g J μ B M J = - J J M J ( 1 + M J g J μ B H / k B T ) M J = - J J ( 1 + M J g J μ B H / k B T ) = g J 2 μ B 2 H k B T - J J M J 2 M J = - J J ( 1 ) \bar{m}=\frac{\sum\limits_{M_{J}=-J}^{J}{M_{J}g_{J}\mu_{B}e^{{M_{J}g_{J}\mu_{B% }H}/{k_{B}T}\;}}}{\sum\limits_{M_{J}=-J}^{J}{e^{{M_{J}g_{J}\mu_{B}H}/{k_{B}T}% \;}}}\simeq g_{J}\mu_{B}\frac{\sum\limits_{M_{J}=-J}^{J}{M_{J}\left(1+{M_{J}g_% {J}\mu_{B}H}/{k_{B}T}\;\right)}}{\sum\limits_{M_{J}=-J}^{J}{\left(1+{M_{J}g_{J% }\mu_{B}H}/{k_{B}T}\;\right)}}=\frac{g_{J}^{2}\mu_{B}^{2}H}{k_{B}T}\frac{\sum% \limits_{-J}^{J}{M_{J}^{2}}}{\sum\limits_{M_{J}=-J}^{J}{\left(1\right)}}
  17. m ¯ = g J 2 μ B 2 H 3 k B T J ( J + 1 ) \bar{m}=\frac{g_{J}^{2}\mu_{B}^{2}H}{3k_{B}T}J(J+1)
  18. M = N A m ¯ = N A 3 k B T [ g J 2 J ( J + 1 ) μ B 2 ] H M=N_{\,\text{A}}\bar{m}=\frac{N_{\,\text{A }}}{3k_{B}T}\left[g_{J}^{2}J(J+1)% \mu_{B}^{2}\right]H
  19. χ m = M H = N A 3 k B T μ eff 2 ; and μ eff = g J J ( J + 1 ) μ B \chi_{m}=\frac{\partial M}{\partial H}=\frac{N_{\,\text{A }}}{3k_{B}T}\mu_{% \mathrm{eff}}^{2}\,\text{ ; and }\mu_{\mathrm{eff}}=g_{J}\sqrt{J(J+1)}\mu_{B}
  20. μ eff 2 S ( S + 1 ) μ B = n ( n + 2 ) μ B \mu_{\mathrm{eff}}\simeq 2\sqrt{S(S+1)}\mu_{B}=\sqrt{n(n+2)}\mu_{B}
  21. s y m b o l M = C T - θ s y m b o l H symbol{M}=\frac{C}{T-\theta}symbol{H}

Parameter.html

  1. f ( x ) = a x 2 + b x + c f(x)=ax^{2}+bx+c
  2. log b ( x ) = log ( x ) log ( b ) \log_{b}(x)=\frac{\log(x)}{\log(b)}
  3. log b ( x ) \textstyle\log_{b}^{\prime}(x)
  4. n k ¯ = n ( n - 1 ) ( n - 2 ) ( n - k + 1 ) n^{\underline{k}}=n(n-1)(n-2)\cdots(n-k+1)
  5. ( n , k ) n k ¯ (n,k)\mapsto n^{\underline{k}}
  6. x 2 + y 2 = 1 x^{2}+y^{2}=1
  7. ( x , y ) = ( cos t , sin t ) (x,y)=(\cos\;t,\sin\;t)
  8. F ( t ) = x 0 ( t ) x 1 ( t ) f ( x ; t ) d x . F(t)=\int_{x_{0}(t)}^{x_{1}(t)}f(x;t)\,dx.
  9. X ¯ \overline{X}
  10. f ( k ; λ ) = e - λ λ k k ! . f(k;\lambda)=\frac{e^{-\lambda}\lambda^{k}}{k!}.
  11. f ( k 1 ; λ ) f(k_{1};\lambda)

Pareto_distribution.html

  1. 1 - ( x m x ) α for x x m 1-\left(\frac{x_{\mathrm{m}}}{x}\right)^{\alpha}\,\text{ for }x\geq x_{m}
  2. { for α 1 α x m α - 1 for α > 1 \begin{cases}\infty&\,\text{for }\alpha\leq 1\\ \frac{\alpha\,x_{\mathrm{m}}}{\alpha-1}&\,\text{for }\alpha>1\end{cases}
  3. x m 2 α x_{\mathrm{m}}\sqrt[\alpha]{2}
  4. x m x_{\mathrm{m}}
  5. { for α ( 1 , 2 ] x m 2 α ( α - 1 ) 2 ( α - 2 ) for α > 2 \begin{cases}\infty&\,\text{for }\alpha\in(1,2]\\ \frac{x_{\mathrm{m}}^{2}\alpha}{(\alpha-1)^{2}(\alpha-2)}&\,\text{for }\alpha>% 2\end{cases}
  6. 2 ( 1 + α ) α - 3 α - 2 α for α > 3 \frac{2(1+\alpha)}{\alpha-3}\,\sqrt{\frac{\alpha-2}{\alpha}}\,\text{ for }% \alpha>3
  7. 6 ( α 3 + α 2 - 6 α - 2 ) α ( α - 3 ) ( α - 4 ) for α > 4 \frac{6(\alpha^{3}+\alpha^{2}-6\alpha-2)}{\alpha(\alpha-3)(\alpha-4)}\,\text{ % for }\alpha>4
  8. ln ( x m α ) + 1 α + 1 \ln\left(\frac{x_{\mathrm{m}}}{\alpha}\right)+\frac{1}{\alpha}+1
  9. α ( - x m t ) α Γ ( - α , - x m t ) for t < 0 \alpha(-x_{\mathrm{m}}t)^{\alpha}\Gamma(-\alpha,-x_{\mathrm{m}}t)\,\text{ for % }t<0
  10. α ( - i x m t ) α Γ ( - α , - i x m t ) \alpha(-ix_{\mathrm{m}}t)^{\alpha}\Gamma(-\alpha,-ix_{\mathrm{m}}t)
  11. ( α x m 2 - 1 x m - 1 x m 1 α 2 ) \begin{pmatrix}\frac{\alpha}{x_{m}^{2}}&-\frac{1}{x_{m}}\\ -\frac{1}{x_{m}}&\frac{1}{\alpha^{2}}\end{pmatrix}
  12. F ¯ ( x ) = Pr ( X > x ) = { ( x m x ) α x x m , 1 x < x m . \overline{F}(x)=\Pr(X>x)=\begin{cases}\left(\frac{x_{\mathrm{m}}}{x}\right)^{% \alpha}&x\geq x_{\mathrm{m}},\\ 1&x<x_{\mathrm{m}}.\end{cases}
  13. F X ( x ) = { 1 - ( x m x ) α x x m , 0 x < x m . F_{X}(x)=\begin{cases}1-\left(\frac{x_{\mathrm{m}}}{x}\right)^{\alpha}&x\geq x% _{\mathrm{m}},\\ 0&x<x_{\mathrm{m}}.\end{cases}
  14. f X ( x ) = { α x m α x α + 1 x x m , 0 x < x m . f_{X}(x)=\begin{cases}\frac{\alpha x_{\mathrm{m}}^{\alpha}}{x^{\alpha+1}}&x% \geq x_{\mathrm{m}},\\ 0&x<x_{\mathrm{m}}.\end{cases}
  15. E ( X ) = { α 1 , α x m α - 1 α > 1. E(X)=\begin{cases}\infty&\alpha\leq 1,\\ \frac{\alpha x_{\mathrm{m}}}{\alpha-1}&\alpha>1.\end{cases}
  16. Var ( X ) = { α ( 1 , 2 ] , ( x m α - 1 ) 2 α α - 2 α > 2. \mathrm{Var}(X)=\begin{cases}\infty&\alpha\in(1,2],\\ \left(\frac{x_{\mathrm{m}}}{\alpha-1}\right)^{2}\frac{\alpha}{\alpha-2}&\alpha% >2.\end{cases}
  17. μ n = { α n , α x m n α - n α > n . \mu_{n}^{\prime}=\begin{cases}\infty&\alpha\leq n,\\ \frac{\alpha x_{\mathrm{m}}^{n}}{\alpha-n}&\alpha>n.\end{cases}
  18. M ( t ; α , x m ) = E [ e t X ] = α ( - x m t ) α Γ ( - α , - x m t ) M\left(t;\alpha,x_{\mathrm{m}}\right)=E\left[e^{tX}\right]=\alpha(-x_{\mathrm{% m}}t)^{\alpha}\Gamma(-\alpha,-x_{\mathrm{m}}t)
  19. M ( 0 , α , x m ) = 1. M\left(0,\alpha,x_{\mathrm{m}}\right)=1.
  20. φ ( t ; α , x m ) = α ( - i x m t ) α Γ ( - α , - i x m t ) , \varphi(t;\alpha,x_{\mathrm{m}})=\alpha(-ix_{\mathrm{m}}t)^{\alpha}\Gamma(-% \alpha,-ix_{\mathrm{m}}t),
  21. x 1 x_{1}
  22. x m x\text{m}
  23. α \alpha
  24. x 1 x_{1}
  25. x m x\text{m}
  26. X 1 , X 2 , X 3 , X_{1},X_{2},X_{3},\ldots
  27. [ x m , ) [x\text{m},\infty)
  28. x m > 0 x\text{m}>0
  29. n n
  30. min { X 1 , , X n } \min\{X_{1},\ldots,X_{n}\}
  31. ( X 1 + + X n ) / min { X 1 , , X n } (X_{1}+\cdots+X_{n})/\min\{X_{1},\ldots,X_{n}\}
  32. G = x m exp ( 1 α ) G=x\text{m}\exp\left(\frac{1}{\alpha}\right)
  33. H = x m ( 1 + 1 α ) H=x\text{m}\left(1+\frac{1}{\alpha}\right)
  34. F ¯ ( x ) = 1 - F ( x ) \overline{F}(x)=1-F(x)
  35. [ x σ ] - α \left[\frac{x}{\sigma}\right]^{-\alpha}
  36. x > σ x>\sigma
  37. σ > 0 , α \sigma>0,\alpha
  38. [ 1 + x - μ σ ] - α \left[1+\frac{x-\mu}{\sigma}\right]^{-\alpha}
  39. x > μ x>\mu
  40. μ , σ > 0 , α \mu\in\mathbb{R},\sigma>0,\alpha
  41. [ 1 + x σ ] - α \left[1+\frac{x}{\sigma}\right]^{-\alpha}
  42. x > 0 x>0
  43. σ > 0 , α \sigma>0,\alpha
  44. [ 1 + ( x - μ σ ) 1 / γ ] - 1 \left[1+\left(\frac{x-\mu}{\sigma}\right)^{1/\gamma}\right]^{-1}
  45. x > μ x>\mu
  46. μ , σ , γ > 0 \mu\in\mathbb{R},\sigma,\gamma>0
  47. [ 1 + ( x - μ σ ) 1 / γ ] - α \left[1+\left(\frac{x-\mu}{\sigma}\right)^{1/\gamma}\right]^{-\alpha}
  48. x > μ x>\mu
  49. μ , σ , γ > 0 , α \mu\in\mathbb{R},\sigma,\gamma>0,\alpha
  50. P ( I V ) ( σ , σ , 1 , α ) = P ( I ) ( σ , α ) , P(IV)(\sigma,\sigma,1,\alpha)=P(I)(\sigma,\alpha),
  51. P ( I V ) ( μ , σ , 1 , α ) = P ( I I ) ( μ , σ , α ) , P(IV)(\mu,\sigma,1,\alpha)=P(II)(\mu,\sigma,\alpha),
  52. P ( I V ) ( μ , σ , γ , 1 ) = P ( I I I ) ( μ , σ , γ ) . P(IV)(\mu,\sigma,\gamma,1)=P(III)(\mu,\sigma,\gamma).
  53. E [ X ] E[X]
  54. E [ X δ ] E[X^{\delta}]
  55. σ α α - 1 \frac{\sigma\alpha}{\alpha-1}
  56. α > 1 \alpha>1
  57. σ δ α α - δ \frac{\sigma^{\delta}\alpha}{\alpha-\delta}
  58. δ < α \delta<\alpha
  59. σ α - 1 \frac{\sigma}{\alpha-1}
  60. α > 1 \alpha>1
  61. σ δ Γ ( α - δ ) Γ ( 1 + δ ) Γ ( α ) \frac{\sigma^{\delta}\Gamma(\alpha-\delta)\Gamma(1+\delta)}{\Gamma(\alpha)}
  62. - 1 < δ < α -1<\delta<\alpha
  63. σ Γ ( 1 - γ ) Γ ( 1 + γ ) \sigma\Gamma(1-\gamma)\Gamma(1+\gamma)
  64. - 1 < γ < 1 -1<\gamma<1
  65. σ δ Γ ( 1 - γ δ ) Γ ( 1 + γ δ ) \sigma^{\delta}\Gamma(1-\gamma\delta)\Gamma(1+\gamma\delta)
  66. - γ - 1 < δ < γ - 1 -\gamma^{-1}<\delta<\gamma^{-1}
  67. σ Γ ( α - γ ) Γ ( 1 + γ ) Γ ( α ) \frac{\sigma\Gamma(\alpha-\gamma)\Gamma(1+\gamma)}{\Gamma(\alpha)}
  68. - 1 < γ < α -1<\gamma<\alpha
  69. σ δ Γ ( α - γ δ ) Γ ( 1 + γ δ ) Γ ( α ) \frac{\sigma^{\delta}\Gamma(\alpha-\gamma\delta)\Gamma(1+\gamma\delta)}{\Gamma% (\alpha)}
  70. - γ - 1 < δ < α / γ -\gamma^{-1}<\delta<\alpha/\gamma
  71. f ( y ) = y γ 1 - 1 ( 1 - y ) γ 2 - 1 B ( γ 1 , γ 2 ) , 0 < y < 1 ; γ 1 , γ 2 > 0 , f(y)=\frac{y^{\gamma_{1}-1}(1-y)^{\gamma_{2}-1}}{B(\gamma_{1},\gamma_{2})},% \qquad 0<y<1;\gamma_{1},\gamma_{2}>0,
  72. W = μ + σ ( Y - 1 - 1 ) γ , σ > 0 , γ > 0 , W=\mu+\sigma(Y^{-1}-1)^{\gamma},\qquad\sigma>0,\gamma>0,
  73. U 1 Γ ( δ 1 , 1 ) U_{1}\sim\Gamma(\delta_{1},1)
  74. U 2 Γ ( δ 2 , 1 ) U_{2}\sim\Gamma(\delta_{2},1)
  75. W = μ + σ ( U 1 U 2 ) γ W=\mu+\sigma\left(\frac{U_{1}}{U_{2}}\right)^{\gamma}
  76. F P ( σ , σ , 1 , 1 , α ) = P ( I ) ( σ , α ) FP(\sigma,\sigma,1,1,\alpha)=P(I)(\sigma,\alpha)
  77. F P ( μ , σ , 1 , 1 , α ) = P ( I I ) ( μ , σ , α ) FP(\mu,\sigma,1,1,\alpha)=P(II)(\mu,\sigma,\alpha)
  78. F P ( μ , σ , γ , 1 , 1 ) = P ( I I I ) ( μ , σ , γ ) FP(\mu,\sigma,\gamma,1,1)=P(III)(\mu,\sigma,\gamma)
  79. F P ( μ , σ , γ , 1 , α ) = P ( I V ) ( μ , σ , γ , α ) . FP(\mu,\sigma,\gamma,1,\alpha)=P(IV)(\mu,\sigma,\gamma,\alpha).
  80. Y = log ( X x m ) Y=\log\left(\frac{X}{x_{\mathrm{m}}}\right)
  81. x m e Y x_{\mathrm{m}}e^{Y}
  82. Pr ( Y < y ) = Pr ( log ( X x m ) < y ) = Pr ( X < x m e y ) = 1 - ( x m x m e y ) α = 1 - e - α y . \Pr(Y<y)=\Pr\left(\log\left(\frac{X}{x_{\mathrm{m}}}\right)<y\right)=\Pr(X<x_{% \mathrm{m}}e^{y})=1-\left(\frac{x_{\mathrm{m}}}{x_{\mathrm{m}}e^{y}}\right)^{% \alpha}=1-e^{-\alpha y}.
  83. x m x_{m}
  84. α \alpha
  85. μ = x m \mu=x_{m}
  86. σ = x m / α \sigma=x_{m}/\alpha
  87. ξ = 1 / α \xi=1/\alpha
  88. x m = σ / ξ x_{m}=\sigma/\xi
  89. α = 1 / ξ \alpha=1/\xi
  90. x ( F ) = x m ( 1 - F ) 1 α x(F)=\frac{x_{\mathrm{m}}}{(1-F)^{\frac{1}{\alpha}}}
  91. L ( F ) = 1 - ( 1 - F ) 1 - 1 α , L(F)=1-(1-F)^{1-\frac{1}{\alpha}},
  92. L ( F ) L(F)
  93. 0 α < 1 0\leq\alpha<1
  94. α 1 \alpha\geq 1
  95. G = 1 - 2 ( 0 1 L ( F ) d F ) = 1 2 α - 1 G=1-2\left(\int_{0}^{1}L(F)dF\right)=\frac{1}{2\alpha-1}
  96. L ( α , x m ) = i = 1 n α x m α x i α + 1 = α n x m n α i = 1 n 1 x i α + 1 . L(\alpha,x_{\mathrm{m}})=\prod_{i=1}^{n}\alpha\frac{x_{\mathrm{m}}^{\alpha}}{x% _{i}^{\alpha+1}}=\alpha^{n}x_{\mathrm{m}}^{n\alpha}\prod_{i=1}^{n}\frac{1}{x_{% i}^{\alpha+1}}.
  97. ( α , x m ) = n ln α + n α ln x m - ( α + 1 ) i = 1 n ln x i . \ell(\alpha,x_{\mathrm{m}})=n\ln\alpha+n\alpha\ln x_{\mathrm{m}}-(\alpha+1)% \sum_{i=1}^{n}\ln x_{i}.
  98. ( α , x m ) \ell(\alpha,x_{\mathrm{m}})
  99. x ^ m = min i x i . \widehat{x}_{\mathrm{m}}=\min_{i}{x_{i}}.
  100. α = n α + n ln x m - i = 1 n ln x i = 0. \frac{\partial\ell}{\partial\alpha}=\frac{n}{\alpha}+n\ln x_{\mathrm{m}}-\sum_% {i=1}^{n}\ln x_{i}=0.
  101. α ^ = n i ( ln x i - ln x ^ m ) . \widehat{\alpha}=\frac{n}{\sum_{i}\left(\ln x_{i}-\ln\widehat{x}_{\mathrm{m}}% \right)}.
  102. σ = α ^ n . \sigma=\frac{\widehat{\alpha}}{\sqrt{n}}.
  103. ( x ^ m , α ^ ) (\hat{x}_{\mathrm{m}},\hat{\alpha})
  104. x ^ m \hat{x}_{\mathrm{m}}
  105. α ^ \hat{\alpha}
  106. x ^ m \hat{x}_{\mathrm{m}}
  107. α ^ \hat{\alpha}
  108. log f X ( x ) = log ( α x m α x α + 1 ) = log ( α x m α ) - ( α + 1 ) log x . \log f_{X}(x)=\log\left(\alpha\frac{x_{\mathrm{m}}^{\alpha}}{x^{\alpha+1}}% \right)=\log(\alpha x_{\mathrm{m}}^{\alpha})-(\alpha+1)\log x.
  109. T = x m U 1 α T=\frac{x_{\mathrm{m}}}{U^{\frac{1}{\alpha}}}
  110. 1 - L α x - α 1 - ( L H ) α \frac{1-L^{\alpha}x^{-\alpha}}{1-\left(\frac{L}{H}\right)^{\alpha}}
  111. L α 1 - ( L H ) α ( α α - 1 ) ( 1 L α - 1 - 1 H α - 1 ) , α 1 \frac{L^{\alpha}}{1-\left(\frac{L}{H}\right)^{\alpha}}\cdot\left(\frac{\alpha}% {\alpha-1}\right)\cdot\left(\frac{1}{L^{\alpha-1}}-\frac{1}{H^{\alpha-1}}% \right),\alpha\neq 1
  112. L ( 1 - 1 2 ( 1 - ( L H ) α ) ) - 1 α L\left(1-\frac{1}{2}\left(1-\left(\frac{L}{H}\right)^{\alpha}\right)\right)^{-% \frac{1}{\alpha}}
  113. L α 1 - ( L H ) α ( α α - 2 ) ( 1 L α - 2 - 1 H α - 2 ) , α 2 \frac{L^{\alpha}}{1-\left(\frac{L}{H}\right)^{\alpha}}\cdot\left(\frac{\alpha}% {\alpha-2}\right)\cdot\left(\frac{1}{L^{\alpha-2}}-\frac{1}{H^{\alpha-2}}% \right),\alpha\neq 2
  114. L α 1 - ( L H ) α α * ( L k - α - H k - α ) ( α - k ) , α j \frac{L^{\alpha}}{1-\left(\frac{L}{H}\right)^{\alpha}}\cdot\frac{\alpha*(L^{k-% \alpha}-H^{k-\alpha})}{(\alpha-k)},\alpha\neq j
  115. α L α x - α - 1 1 - ( L H ) α \frac{\alpha L^{\alpha}x^{-\alpha-1}}{1-\left(\frac{L}{H}\right)^{\alpha}}
  116. U = 1 - L α x - α 1 - ( L H ) α U=\frac{1-L^{\alpha}x^{-\alpha}}{1-(\frac{L}{H})^{\alpha}}
  117. x = ( - U H α - U L α - H α H α L α ) - 1 α x=\left(-\frac{UH^{\alpha}-UL^{\alpha}-H^{\alpha}}{H^{\alpha}L^{\alpha}}\right% )^{-\frac{1}{\alpha}}
  118. f ( x ; α , x m ) = { 1 2 α x m α | x | - α - 1 | x | > x m 0 otherwise . f(x;\alpha,x_{\mathrm{m}})=\begin{cases}\tfrac{1}{2}\alpha x_{\mathrm{m}}^{% \alpha}|x|^{-\alpha-1}&|x|>x_{\mathrm{m}}\\ 0&\,\text{otherwise}.\end{cases}

Pareto_efficiency.html

  1. f : n m f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}
  2. n \mathbb{R}^{n}
  3. m \mathbb{R}^{m}
  4. Y = { y m : y = f ( x ) , x X } Y=\{y\in\mathbb{R}^{m}:\;y=f(x),x\in X\;\}
  5. y ′′ m y^{\prime\prime}\in\mathbb{R}^{m}\;
  6. y m y^{\prime}\in\mathbb{R}^{m}\;
  7. y ′′ y y^{\prime\prime}\succ y^{\prime}
  8. P ( Y ) = { y Y : { y ′′ Y : y ′′ y , y ′′ y } = } . P(Y)=\{y^{\prime}\in Y:\;\{y^{\prime\prime}\in Y:\;y^{\prime\prime}\succ y^{% \prime},y^{\prime\prime}\neq y^{\prime}\;\}=\}.
  9. z i = f i ( x i ) z_{i}=f^{i}(x^{i})
  10. x i = ( x 1 i , x 2 i , , x n i ) x^{i}=(x_{1}^{i},x_{2}^{i},\ldots,x_{n}^{i})
  11. i = 1 m x j i = b j \sum_{i=1}^{m}x_{j}^{i}=b_{j}
  12. j = 1 , , n j=1,\ldots,n
  13. L i ( ( x j k ) k , j , ( λ k ) k , ( μ j ) j ) = f i ( x i ) + k = 2 m λ k ( z k - f k ( x k ) ) + j = 1 n μ j ( b j - k = 1 m x j k ) L_{i}((x_{j}^{k})_{k,j},(\lambda_{k})_{k},(\mu_{j})_{j})=f^{i}(x^{i})+\sum_{k=% 2}^{m}\lambda_{k}(z_{k}-f^{k}(x^{k}))+\sum_{j=1}^{n}\mu_{j}\left(b_{j}-\sum_{k% =1}^{m}x_{j}^{k}\right)
  14. ( λ k ) k (\lambda_{k})_{k}
  15. ( μ j ) j (\mu_{j})_{j}
  16. x j k x_{j}^{k}
  17. j = 1 , , n j=1,\ldots,n
  18. k = 1 , , m k=1,\ldots,m
  19. L i x j i = f x j i 1 - μ j = 0 for j = 1 , , n , \frac{\partial L_{i}}{\partial x_{j}^{i}}=f_{x^{i}_{j}}^{1}-\mu_{j}=0\,\text{ % for }j=1,\ldots,n,
  20. L i x j k = - λ k f x j k i - μ j = 0 for k = 2 , , m and j = 1 , , n , \frac{\partial L_{i}}{\partial x_{j}^{k}}=-\lambda_{k}f_{x^{k}_{j}}^{i}-\mu_{j% }=0\,\text{ for }k=2,\ldots,m\,\text{ and }j=1,\ldots,n,
  21. f x j i f_{x^{i}_{j}}
  22. f f
  23. x j i x_{j}^{i}
  24. k i k\neq i
  25. j , s { 1 , , n } j,s\in\{1,\ldots,n\}
  26. f x j i i f x s i i = μ j μ s = f x j k k f x s k k . \frac{f_{x_{j}^{i}}^{i}}{f_{x_{s}^{i}}^{i}}=\frac{\mu_{j}}{\mu_{s}}=\frac{f_{x% _{j}^{k}}^{k}}{f_{x_{s}^{k}}^{k}}.

Pareto_index.html

  1. ( x m x ) α \left(\frac{x_{\mathrm{m}}}{x}\right)^{\alpha}
  2. p + q = 1 p+q=1
  3. p > q p>q
  4. α = log p / q 1 / q = log ( 1 / q ) / log ( p / q ) = log ( q ) / log ( q / p ) . \alpha=\log_{p/q}1/q=\log(1/q)/\log(p/q)=\log(q)/\log(q/p).
  5. q = 1 / n q=1/n
  6. α = log n - 1 ( n ) . \alpha=\log_{n-1}(n).
  7. α = log X / Y ( X + Y ) / Y , \alpha=\log_{X/Y}(X+Y)/Y,
  8. α = log X ( X + 1 ) . \alpha=\log_{X}(X+1).

Pareto_principle.html

  1. H = G = | 2 A - 1 | = | 1 - 2 B | H=G=\left|2A-1\right|=\left|1-2B\right|\,
  2. A : B = ( 1 + H 2 ) : ( 1 - H 2 ) A:B=\left(\tfrac{1+H}{2}\right):\left(\tfrac{1-H}{2}\right)
  3. T T = T L = T s = 2 H arctanh ( H ) . T_{T}=T_{L}=T_{s}=2H\,\operatorname{arctanh}\left(H\right).\,

Parimutuel_betting.html

  1. W T = i = 1 n W i . W_{T}=\sum^{n}_{i=1}W_{i}.

Parity_of_a_permutation.html

  1. σ \sigma
  2. x , y x,y
  3. x < y x<y
  4. σ ( x ) > σ ( y ) \sigma(x)>\sigma(y)
  5. ϵ σ \epsilon_{\sigma}
  6. m m
  7. σ = ( 1 2 3 4 5 3 4 5 2 1 ) = ( 1 3 5 ) ( 2 4 ) = ( 1 5 ) ( 1 3 ) ( 2 4 ) . \sigma=\begin{pmatrix}1&2&3&4&5\\ 3&4&5&2&1\end{pmatrix}=\begin{pmatrix}1&3&5\end{pmatrix}\begin{pmatrix}2&4\end% {pmatrix}=\begin{pmatrix}1&5\end{pmatrix}\begin{pmatrix}1&3\end{pmatrix}\begin% {pmatrix}2&4\end{pmatrix}.
  8. σ = ( 2 3 ) ( 1 2 ) ( 2 4 ) ( 3 5 ) ( 4 5 ) σ=(2 3) (1 2) (2 4) (3 5) (4 5)
  9. n n
  10. n n
  11. n n
  12. n n
  13. n n
  14. n n
  15. n n
  16. n n
  17. n n
  18. n n
  19. n n
  20. ( 1 2 ) σ (1 2) σ
  21. ( 1 2 ) σ (1 2) σ
  22. P ( x 1 , , x n ) = i < j ( x i - x j ) P(x_{1},\ldots,x_{n})=\prod_{i<j}(x_{i}-x_{j})\;
  23. n n
  24. P ( x 1 , x 2 , x 3 ) = ( x 1 - x 2 ) ( x 2 - x 3 ) ( x 1 - x 3 ) P(x_{1},x_{2},x_{3})=(x_{1}-x_{2})(x_{2}-x_{3})(x_{1}-x_{3})\;
  25. n n
  26. sgn ( σ ) = P ( x σ ( 1 ) , , x σ ( n ) ) P ( x 1 , , x n ) \operatorname{sgn}(\sigma)=\frac{P(x_{\sigma(1)},\ldots,x_{\sigma(n)})}{P(x_{1% },\ldots,x_{n})}
  27. P ( x σ ( 1 ) , , x σ ( n ) ) P(x_{\sigma(1)},\dots,x_{\sigma(n)})
  28. P ( x 1 , , x n ) P(x_{1},\dots,x_{n})
  29. sgn ( σ τ ) = P ( x σ ( τ ( 1 ) ) , , x σ ( τ ( n ) ) ) P ( x 1 , , x n ) \operatorname{sgn}(\sigma\tau)=\frac{P(x_{\sigma(\tau(1))},\ldots,x_{\sigma(% \tau(n))})}{P(x_{1},\ldots,x_{n})}
  30. = P ( x σ ( 1 ) , , x σ ( n ) ) P ( x 1 , , x n ) P ( x σ ( τ ( 1 ) ) , , x σ ( τ ( n ) ) ) P ( x σ ( 1 ) , , x σ ( n ) ) =\frac{P(x_{\sigma(1)},\ldots,x_{\sigma(n)})}{P(x_{1},\ldots,x_{n})}\cdot\frac% {P(x_{\sigma(\tau(1))},\ldots,x_{\sigma(\tau(n))})}{P(x_{\sigma(1)},\ldots,x_{% \sigma(n)})}
  31. = sgn ( σ ) sgn ( τ ) =\operatorname{sgn}(\sigma)\cdot\operatorname{sgn}(\tau)
  32. τ 1 , , τ n - 1 \tau_{1},\dots,\tau_{n-1}
  33. τ i 2 = 1 \tau_{i}^{2}=1
  34. τ i τ i + 1 τ i = τ i + 1 τ i τ i + 1 \tau_{i}\tau_{i+1}\tau_{i}=\tau_{i+1}\tau_{i}\tau_{i+1}
  35. τ i τ j = τ j τ i \tau_{i}\tau_{j}=\tau_{j}\tau_{i}
  36. τ i \tau_{i}
  37. n n
  38. σ = ( i 1 i 2 i r + 1 ) ( j 1 j 2 j s + 1 ) ( l 1 l 2 l u + 1 ) \sigma=(i_{1}i_{2}\dots i_{r+1})(j_{1}j_{2}\dots j_{s+1})\dots(l_{1}l_{2}\dots l% _{u+1})
  39. σ \sigma
  40. ( a b c x y z ) (abc\dots xyz)
  41. k + 1 k+1
  42. k k
  43. ( a b c x y z ) = ( a b ) ( b c ) ( x y ) ( y z ) (abc\dots xyz)=(ab)(bc)\dots(xy)(yz)
  44. k k
  45. σ \sigma
  46. r + s + + u r+s+\dots+u
  47. N ( σ ) = r + s + + u N(\sigma)=r+s+\dots+u
  48. σ \sigma
  49. n - number of disjoint cycles in the decomposition of σ n-\,\text{number of disjoint cycles in the decomposition of }\sigma
  50. σ \sigma
  51. ( a b ) (ab)
  52. σ \sigma
  53. a a
  54. b b
  55. σ \sigma
  56. ( a b ) ( a c 1 c 2 c r ) ( b d 1 d 2 d s ) = ( a c 1 c 2 c r b d 1 d 2 d s ) (ab)(ac_{1}c_{2}\dots c_{r})(bd_{1}d_{2}\dots d_{s})=(ac_{1}c_{2}\dots c_{r}bd% _{1}d_{2}\dots d_{s})
  57. a a
  58. b b
  59. σ \sigma
  60. ( a b ) ( a c 1 c 2 c r b d 1 d 2 d s ) = ( a c 1 c 2 c r ) ( b d 1 d 2 d s ) (ab)(ac_{1}c_{2}\dots c_{r}bd_{1}d_{2}\dots d_{s})=(ac_{1}c_{2}\dots c_{r})(bd% _{1}d_{2}\dots d_{s})
  61. N ( ( a b ) σ ) = N ( σ ) ± 1 N((ab)\sigma)=N(\sigma)\pm 1
  62. N ( ( a b ) σ ) N((ab)\sigma)
  63. N ( σ ) N(\sigma)
  64. σ = t 1 t 2 t m \sigma=t_{1}t_{2}\dots t_{m}
  65. σ \sigma
  66. m m
  67. t 1 t_{1}
  68. t 2 t_{2}
  69. t m t_{m}
  70. N N
  71. N ( σ ) N(\sigma)
  72. m m
  73. σ \sigma
  74. N ( σ ) N(\sigma)
  75. m N ( σ ) m\geq N(\sigma)
  76. σ \sigma
  77. m m
  78. m m
  79. N ( σ ) N(\sigma)
  80. σ \sigma
  81. σ \sigma
  82. l ( v ) , l(v),
  83. v ( - 1 ) l ( v ) v\mapsto(-1)^{l(v)}

Parsec.html

  1. 1 / 3600 {1}/{3600}
  2. S D = ES tan 1 ′′ SD=\frac{\mathrm{ES}}{\tan 1^{\prime\prime}}
  3. S D ES 1 ′′ = 1 AU ( 1 60 × 60 × π 180 ) = 648 000 π AU 206 264.81 AU . SD\approx\frac{\mathrm{ES}}{1^{\prime\prime}}=\frac{1\,\mbox{AU}~{}}{(\tfrac{1% }{60\times 60}\times\tfrac{\pi}{180})}=\frac{648\,000}{\pi}\,\mbox{AU}~{}% \approx 206\,264.81\mbox{ AU}~{}.

Partial_derivative.html

  1. f x , f x , x f , x f , or f x . f^{\prime}_{x},\ f_{x},\ \partial_{x}f,\frac{\partial}{\partial x}f,\,\text{ % or }\frac{\partial f}{\partial x}.
  2. f x ( x , y , ) , < m t p l > f x ( x , y , ) . f_{x}(x,y,...),\ \frac{<}{m}tpl>{{\partial f}}{{\partial x}}(x,y,...).
  3. z = f ( x , y ) = x 2 + x y + y 2 . z=f(x,y)=\,\!x^{2}+xy+y^{2}.\,
  4. z x = 2 x + y \frac{\partial z}{\partial x}=2x+y
  5. z x = 3 \frac{\partial z}{\partial x}=3
  6. f ( x , y ) = f y ( x ) = x 2 + x y + y 2 . f(x,y)=f_{y}(x)=\,\!x^{2}+xy+y^{2}.\,
  7. f y ( x ) = x 2 + x y + y 2 . f_{y}(x)=x^{2}+xy+y^{2}.\,
  8. f a ( x ) = x 2 + a x + a 2 . f_{a}(x)=x^{2}+ax+a^{2}.\,
  9. f a ( x ) = 2 x + a . f_{a}^{\prime}(x)=2x+a.\,
  10. f x ( x , y ) = 2 x + y . \frac{\partial f}{\partial x}(x,y)=2x+y.\,
  11. f x i ( a 1 , , a n ) = lim h 0 f ( a 1 , , a i + h , , a n ) - f ( a 1 , , a i , , a n ) h . \frac{\partial f}{\partial x_{i}}(a_{1},\ldots,a_{n})=\lim_{h\to 0}\frac{f(a_{% 1},\ldots,a_{i}+h,\ldots,a_{n})-f(a_{1},\ldots,a_{i},\dots,a_{n})}{h}.
  12. f a 1 , , a i - 1 , a i + 1 , , a n ( x i ) = f ( a 1 , , a i - 1 , x i , a i + 1 , , a n ) f_{a_{1},\ldots,a_{i-1},a_{i+1},\ldots,a_{n}}(x_{i})=f(a_{1},\ldots,a_{i-1},x_% {i},a_{i+1},\ldots,a_{n})
  13. d f a 1 , , a i - 1 , a i + 1 , , a n d x i ( a i ) = f x i ( a 1 , , a n ) . \frac{df_{a_{1},\ldots,a_{i-1},a_{i+1},\ldots,a_{n}}}{dx_{i}}(a_{i})=\frac{% \partial f}{\partial x_{i}}(a_{1},\ldots,a_{n}).
  14. n \mathbb{R}^{n}
  15. 2 \mathbb{R}^{2}
  16. 3 \mathbb{R}^{3}
  17. f ( a ) = ( f x 1 ( a ) , , f x n ( a ) ) . \nabla f(a)=\left(\frac{\partial f}{\partial x_{1}}(a),\ldots,\frac{\partial f% }{\partial x_{n}}(a)\right).
  18. 3 \mathbb{R}^{3}
  19. 𝐢 ^ , 𝐣 ^ , 𝐤 ^ \mathbf{\hat{i}},\mathbf{\hat{j}},\mathbf{\hat{k}}
  20. = [ x ] 𝐢 ^ + [ y ] 𝐣 ^ + [ z ] 𝐤 ^ \nabla=\bigg[{\frac{\partial}{\partial x}}\bigg]\mathbf{\hat{i}}+\bigg[{\frac{% \partial}{\partial y}}\bigg]\mathbf{\hat{j}}+\bigg[{\frac{\partial}{\partial z% }}\bigg]\mathbf{\hat{k}}
  21. n \mathbb{R}^{n}
  22. 𝐞 ^ 𝟏 , 𝐞 ^ 𝟐 , 𝐞 ^ 𝟑 , , 𝐞 ^ 𝐧 \mathbf{\hat{e}_{1}},\mathbf{\hat{e}_{2}},\mathbf{\hat{e}_{3}},\dots,\mathbf{% \hat{e}_{n}}
  23. = j = 1 n [ x j ] 𝐞 ^ 𝐣 = [ x 1 ] 𝐞 ^ 𝟏 + [ x 2 ] 𝐞 ^ 𝟐 + [ x 3 ] 𝐞 ^ 𝟑 + + [ x n ] 𝐞 ^ 𝐧 \nabla=\sum_{j=1}^{n}\bigg[{\frac{\partial}{\partial x_{j}}}\bigg]\mathbf{\hat% {e}_{j}}=\bigg[{\frac{\partial}{\partial x_{1}}}\bigg]\mathbf{\hat{e}_{1}}+% \bigg[{\frac{\partial}{\partial x_{2}}}\bigg]\mathbf{\hat{e}_{2}}+\bigg[{\frac% {\partial}{\partial x_{3}}}\bigg]\mathbf{\hat{e}_{3}}+\dots+\bigg[{\frac{% \partial}{\partial x_{n}}}\bigg]\mathbf{\hat{e}_{n}}
  24. a i f ( 𝐚 ) = lim h 0 f ( a 1 , , a i - 1 , a i + h , a i + 1 , , a n ) - f ( a 1 , , a i , , a n ) h \frac{\partial}{\partial a_{i}}f(\mathbf{a})=\lim_{h\rightarrow 0}{f(a_{1},% \dots,a_{i-1},a_{i}+h,a_{i+1},\dots,a_{n})-f(a_{1},\dots,a_{i},\dots,a_{n})% \over h}
  25. f x \frac{\partial f}{\partial x}
  26. 2 f x i x j = 2 f x j x i . \frac{\partial^{2}f}{\partial x_{i}\,\partial x_{j}}=\frac{\partial^{2}f}{% \partial x_{j}\,\partial x_{i}}.
  27. V ( r , h ) = π r 2 h 3 . V(r,h)=\frac{\pi r^{2}h}{3}.
  28. V r = 2 π r h 3 , \frac{\partial V}{\partial r}=\frac{2\pi rh}{3},
  29. V h = π r 2 3 , \frac{\partial V}{\partial h}=\frac{\pi r^{2}}{3},
  30. d V d r = 2 π r h 3 V r + π r 2 3 V h d h d r \frac{\operatorname{d}V}{\operatorname{d}r}=\overbrace{\frac{2\pi rh}{3}}^{% \frac{\partial V}{\partial r}}+\overbrace{\frac{\pi r^{2}}{3}}^{\frac{\partial V% }{\partial h}}\frac{\operatorname{d}h}{\operatorname{d}r}
  31. d V d h = π r 2 3 V h + 2 π r h 3 V r d r d h \frac{\operatorname{d}V}{\operatorname{d}h}=\overbrace{\frac{\pi r^{2}}{3}}^{% \frac{\partial V}{\partial h}}+\overbrace{\frac{2\pi rh}{3}}^{\frac{\partial V% }{\partial r}}\frac{\operatorname{d}r}{\operatorname{d}h}
  32. k = h r = d h d r . k=\frac{h}{r}=\frac{\operatorname{d}h}{\operatorname{d}r}.
  33. d V d r = 2 π r h 3 + π r 2 3 k \frac{\operatorname{d}V}{\operatorname{d}r}=\frac{2\pi rh}{3}+\frac{\pi r^{2}}% {3}k
  34. d V d r = k π r 2 \frac{\operatorname{d}V}{\operatorname{d}r}=k\pi r^{2}
  35. d V d h = π r 2 \frac{\operatorname{d}V}{\operatorname{d}h}=\pi r^{2}
  36. V = ( V r , V h ) = ( 2 3 π r h , 1 3 π r 2 ) \nabla V=\left(\frac{\partial V}{\partial r},\frac{\partial V}{\partial h}% \right)=\left(\frac{2}{3}\pi rh,\frac{1}{3}\pi r^{2}\right)
  37. f x = f x = x f . \frac{\partial f}{\partial x}=f_{x}=\partial_{x}f.
  38. 2 f x 2 = f x x = x x f . \frac{\partial^{2}f}{\partial x^{2}}=f_{xx}=\partial_{xx}f.
  39. 2 f y x = y ( f x ) = ( f x ) y = f x y = y x f . \frac{\partial^{2}f}{\partial y\,\partial x}=\frac{\partial}{\partial y}\left(% \frac{\partial f}{\partial x}\right)=(f_{x})_{y}=f_{xy}=\partial_{yx}f.
  40. i + j + k f x i y j z k = f ( i , j , k ) . \frac{\partial^{i+j+k}f}{\partial x^{i}\,\partial y^{j}\,\partial z^{k}}=f^{(i% ,j,k)}.
  41. ( f x ) y , z . \left(\frac{\partial f}{\partial x}\right)_{y,z}.
  42. z x = 2 x + y \frac{\partial z}{\partial x}=2x+y
  43. z = z x d x = x 2 + x y + g ( y ) z=\int\frac{\partial z}{\partial x}\,dx=x^{2}+xy+g(y)
  44. x x
  45. x 2 + x y + g ( y ) x^{2}+xy+g(y)
  46. f ( x , y , ) f(x,y,...)
  47. < m t p l > 2 f x 2 f / x x f x x f x x . \frac{<}{m}tpl>{{\partial^{2}f}}{{\partial x^{2}}}\equiv\partial\frac{{% \partial f/\partial x}}{{\partial x}}\equiv\frac{{\partial f_{x}}}{{\partial x% }}\equiv f_{{xx}}.
  48. < m t p l > 2 f x y f / x y f x y f x y . \frac{<}{m}tpl>{{\partial^{2}f}}{{\partial x\partial y}}\equiv\partial\frac{{% \partial f/\partial x}}{{\partial y}}\equiv\frac{{\partial f_{x}}}{{\partial y% }}\equiv f_{{xy}}.
  49. < m t p l > 2 f x y = 2 f y x \frac{<}{m}tpl>{{\partial^{2}f}}{{\partial x\partial y}}=\frac{{\partial^{2}f}% }{{\partial y\partial x}}
  50. f < m t p l > x y = f y x . f_{<}mtpl>{{xy}}=f_{{yx}}.

Partial_differential_equation.html

  1. u ( x 1 , , x n ) u(x_{1},\cdots,x_{n})
  2. f ( x 1 , , x n , u , u x 1 , , u x n , 2 u x 1 x 1 , , 2 u x 1 x n , ) = 0. f\left(x_{1},\ldots,x_{n},u,\frac{\partial u}{\partial x_{1}},\ldots,\frac{% \partial u}{\partial x_{n}},\frac{\partial^{2}u}{\partial x_{1}\partial x_{1}}% ,\ldots,\frac{\partial^{2}u}{\partial x_{1}\partial x_{n}},\ldots\right)=0.
  3. u x ( x , y ) = 0. \frac{\partial u}{\partial x}(x,y)=0.~{}
  4. u ( x , y ) = f ( y ) , u(x,y)=f(y),
  5. d u d x ( x ) = 0 , \frac{\mathrm{d}u}{\mathrm{d}x}(x)=0,
  6. u ( x ) = c , u(x)=c,
  7. 2 u x 2 + 2 u y 2 = 0 , \frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}=0,~{}
  8. u ( x , 0 ) = 0 , u(x,0)=0,
  9. u y ( x , 0 ) = sin ( n x ) n , \frac{\partial u}{\partial y}(x,0)=\frac{\sin(nx)}{n},
  10. u ( x , y ) = sinh ( n y ) sin ( n x ) n 2 . u(x,y)=\frac{\sinh(ny)\sin(nx)}{n^{2}}.
  11. u x = u x u_{x}={\partial u\over\partial x}
  12. u x x = 2 u x 2 u_{xx}={\partial^{2}u\over\partial x^{2}}
  13. u x y = 2 u y x = y ( u x ) . u_{xy}={\partial^{2}u\over\partial y\,\partial x}={\partial\over\partial y}% \left({\partial u\over\partial x}\right).
  14. u ˙ u ¨ \dot{u}\,\ddot{u}\,
  15. u ¨ = c 2 2 u \ddot{u}=c^{2}\nabla^{2}u
  16. u ¨ = c 2 Δ u \ddot{u}=c^{2}\Delta u
  17. u t = α u x x u_{t}=\alpha u_{xx}
  18. u ( t , x ) = 1 2 π - F ( ξ ) e - α ξ 2 t e i ξ x d ξ , u(t,x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}F(\xi)e^{-\alpha\xi^{2}t}e^% {i\xi x}d\xi,\,
  19. F ( ξ ) = 1 2 π - f ( x ) e - i ξ x d x . F(\xi)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x)e^{-i\xi x}\,dx.\,
  20. F ( ξ ) = 1 2 π , F(\xi)=\frac{1}{\sqrt{2\pi}},\,
  21. u ( t , x ) = 1 2 π - e - α ξ 2 t e i ξ x d ξ . u(t,x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-\alpha\xi^{2}t}e^{i\xi x}d\xi.\,
  22. u ( t , x ) = 1 2 π α t exp ( - x 2 4 α t ) . u(t,x)=\frac{1}{2\sqrt{\pi\alpha t}}\exp\left(-\frac{x^{2}}{4\alpha t}\right).\,
  23. u k k = m 2 u x x . u_{kk}=m^{2}u_{xx}.
  24. u ( 0 , x ) = f ( x ) , u(0,x)=f(x),
  25. u k ( 0 , x ) = g ( x ) , u_{k}(0,x)=g(x),
  26. u ( k , x ) = 1 2 [ f ( x - m k ) + f ( x + m k ) ] + 1 2 m x - m k x + m k g ( y ) d y . u(k,x)=\tfrac{1}{2}\left[f(x-mk)+f(x+mk)\right]+\frac{1}{2m}\int_{x-mk}^{x+mk}% g(y)\,dy.
  27. x - m k = constant, x + m k = constant , x-mk=\,\text{constant,}\quad x+mk=\,\text{constant},
  28. H ^ \hat{H}
  29. u ( x , 0 ) = h ( x ) . u(x,0)=h(x).
  30. H ^ X n = λ n X n \hat{H}X_{n}=\lambda_{n}X_{n}
  31. X n ( a ) = X n ( b ) = 0 X_{n}(a)=X_{n}(b)=0
  32. a ˙ n ( t ) - λ n a n ( t ) - m ( X n f ( x , t ) , X m ) a m ( t ) = ( g ( x , t ) , X n ) \dot{a}_{n}(t)-\lambda_{n}a_{n}(t)-\sum_{m}(X_{n}f(x,t),X_{m})a_{m}(t)=(g(x,t)% ,X_{n})
  33. a n ( 0 ) = ( h ( x ) , X n ) ( X n , X n ) a_{n}(0)=\frac{(h(x),X_{n})}{(X_{n},X_{n})}
  34. u ( x , t ) = n a n ( t ) X n ( x ) u(x,t)=\sum_{n}a_{n}(t)X_{n}(x)
  35. ( f , g ) = a b f ( x ) g ( x ) w ( x ) d x . (f,g)=\int_{a}^{b}f(x)g(x)w(x)\,dx.
  36. u t t = c 2 [ u r r + 2 r u r ] . u_{tt}=c^{2}\left[u_{rr}+\frac{2}{r}u_{r}\right].
  37. ( r u ) t t = c 2 [ ( r u ) r r ] , (ru)_{tt}=c^{2}\left[(ru)_{rr}\right],
  38. u ( t , r ) = 1 r [ F ( r - c t ) + G ( r + c t ) ] , u(t,r)=\frac{1}{r}\left[F(r-ct)+G(r+ct)\right],
  39. φ x x + φ y y = 0. \varphi_{xx}+\varphi_{yy}=0.
  40. u x = v y , v x = - u y , u_{x}=v_{y},\quad v_{x}=-u_{y},\,
  41. u x x + u y y = 0 , v x x + v y y = 0. u_{xx}+u_{yy}=0,\quad v_{xx}+v_{yy}=0.\,
  42. φ ( r , θ ) = 1 2 π 0 2 π 1 - r 2 1 + r 2 - 2 r cos ( θ - θ ) u ( θ ) d θ . \varphi(r,\theta)=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{1-r^{2}}{1+r^{2}-2r\cos(% \theta-\theta^{\prime})}u(\theta^{\prime})d\theta^{\prime}.\,
  43. ψ t + ( u ψ ) x + ( v ψ ) y + ( w ψ ) z = 0. \psi_{t}+(u\psi)_{x}+(v\psi)_{y}+(w\psi)_{z}=0.
  44. ψ t + u ψ x + v ψ y + w ψ z = 0. \psi_{t}+u\psi_{x}+v\psi_{y}+w\psi_{z}=0.
  45. i u t + p u x x + q | u | 2 u = i γ u iu_{t}+pu_{xx}+q|u|^{2}u=i\gamma u
  46. u t = u 3 u x x x . u_{t}\,=u^{3}u_{xxx}.
  47. u ( 0 , x ) = f ( x ) , u t ( 0 , x ) = g ( x ) . u(0,x)=f(x),\quad u_{t}(0,x)=g(x).
  48. u t t = c 2 u x x , u_{tt}=c^{2}u_{xx},\,
  49. u ( t , x ) = T ( t ) X ( x ) , u(t,x)=T(t)X(x),\,
  50. T ′′ + k 2 c 2 T = 0 , X ′′ + k 2 X = 0 , T^{\prime\prime}+k^{2}c^{2}T=0,\quad X^{\prime\prime}+k^{2}X=0,\,
  51. k = n π L , k=\frac{n\pi}{L},
  52. X ( 0 ) = 0 , X ( L ) = 0. X(0)=0,\quad X^{\prime}(L)=0.
  53. 1 c 2 u t t = u x x + u y y , \frac{1}{c^{2}}u_{tt}=u_{xx}+u_{yy},
  54. u ( t , x , y ) = T ( t ) v ( x , y ) , u(t,x,y)=T(t)v(x,y),
  55. 1 c 2 T ′′ + k 2 T = 0 , \frac{1}{c^{2}}T^{\prime\prime}+k^{2}T=0,
  56. v x x + v y y + k 2 v = 0. v_{xx}+v_{yy}+k^{2}v=0.
  57. u x y = u y x u_{xy}=u_{yx}
  58. A u x x + 2 B u x y + C u y y + (lower order terms) = 0 , Au_{xx}+2Bu_{xy}+Cu_{yy}+\cdots\mbox{(lower order terms)}~{}=0,
  59. A 2 + B 2 + C 2 > 0 A^{2}+B^{2}+C^{2}>0
  60. A x 2 + 2 B x y + C y 2 + = 0. Ax^{2}+2Bxy+Cy^{2}+\cdots=0.
  61. B 2 - 4 A C B^{2}-4AC
  62. B 2 - A C , B^{2}-AC,
  63. ( 2 B ) 2 - 4 A C = 4 ( B 2 - A C ) , (2B)^{2}-4AC=4(B^{2}-AC),
  64. B 2 - A C < 0 B^{2}-AC<0
  65. B 2 - A C = 0 B^{2}-AC=0
  66. B 2 - A C > 0 B^{2}-AC>0
  67. L u = i = 1 n j = 1 n a i , j 2 u x i x j plus lower-order terms = 0. Lu=\sum_{i=1}^{n}\sum_{j=1}^{n}a_{i,j}\frac{\partial^{2}u}{\partial x_{i}% \partial x_{j}}\quad\,\text{ plus lower-order terms}=0.
  68. L u = ν = 1 n A ν u x ν + B = 0 , Lu=\sum_{\nu=1}^{n}A_{\nu}\frac{\partial u}{\partial x_{\nu}}+B=0,
  69. φ ( x 1 , x 2 , , x n ) = 0 , \varphi(x_{1},x_{2},\ldots,x_{n})=0,\,
  70. Q ( φ x 1 , , φ x n ) = det [ ν = 1 n A ν φ x ν ] = 0. Q\left(\frac{\partial\varphi}{\partial x_{1}},\ldots,\frac{\partial\varphi}{% \partial x_{n}}\right)=\det\left[\sum_{\nu=1}^{n}A_{\nu}\frac{\partial\varphi}% {\partial x_{\nu}}\right]=0.\,
  71. Q ( λ ξ + η ) = 0 , Q(\lambda\xi+\eta)=0,
  72. u x x = x u y y , u_{xx}\,=xu_{yy},
  73. V t + 1 2 σ 2 S 2 2 V S 2 + r S V S - r V = 0 \frac{\partial V}{\partial t}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}V}{% \partial S^{2}}+rS\frac{\partial V}{\partial S}-rV=0
  74. u τ = 2 u x 2 \frac{\partial u}{\partial\tau}=\frac{\partial^{2}u}{\partial x^{2}}
  75. V ( S , t ) = K v ( x , τ ) V(S,t)=Kv(x,\tau)
  76. x = ln ( S K ) x=\ln\left(\tfrac{S}{K}\right)
  77. τ = 1 2 σ 2 ( T - t ) \tau=\tfrac{1}{2}\sigma^{2}(T-t)
  78. v ( x , τ ) = exp ( - α x - β τ ) u ( x , τ ) . v(x,\tau)=\exp(-\alpha x-\beta\tau)u(x,\tau).

Partial_evaluation.html

  1. p r o g : I s t a t i c × I d y n a m i c O prog:I_{static}\times I_{dynamic}\to O
  2. I s t a t i c I_{static}
  3. p r o g , I s t a t i c \langle prog,I_{static}\rangle
  4. p r o g * : I d y n a m i c O prog^{*}:I_{dynamic}\to O
  5. p r o g * prog^{*}
  6. p r o g prog
  7. p r o g * prog^{*}

Partial_function.html

  1. g : g\colon\mathbb{Z}\to\mathbb{Z}
  2. g ( n ) = n . g(n)=\sqrt{n}.
  3. : Hom ( C ) × Hom ( C ) Hom ( C ) \circ:\operatorname{Hom}(C)\times\operatorname{Hom}(C)\to\operatorname{Hom}(C)
  4. Ob ( C ) \operatorname{Ob}(C)
  5. f : X Y f:X\to Y
  6. g : U V g:U\to V
  7. g f g\circ f
  8. Y = U Y=U
  9. f f
  10. g g
  11. f : × f:\mathbb{N}\times\mathbb{N}\to\mathbb{N}
  12. f ( x , y ) = x - y . f(x,y)=x-y.
  13. x y x\geq y
  14. 𝒫 𝒯 X \mathcal{PT}_{X}

Partial_pressure.html

  1. V x V t o t = p x p t o t = n x n t o t \frac{V_{x}}{V_{tot}}=\frac{p_{x}}{p_{tot}}=\frac{n_{x}}{n_{tot}}
  2. p = p N 2 + p H 2 + p NH 3 p=p_{{\mathrm{N}}_{2}}+p_{{\mathrm{H}}_{2}}+p_{{\mathrm{NH}}_{3}}
  3. p p\,
  4. p N 2 p_{{\mathrm{N}}_{2}}
  5. p H 2 p_{{\mathrm{H}}_{2}}
  6. p NH 3 p_{{\mathrm{NH}}_{3}}
  7. x i = p i p = n i n x_{\mathrm{i}}=\frac{p_{\mathrm{i}}}{p}=\frac{n_{\mathrm{i}}}{n}
  8. p i = x i p p_{\mathrm{i}}=x_{\mathrm{i}}\cdot p
  9. x i x_{\mathrm{i}}
  10. p i p_{\mathrm{i}}
  11. n i n_{\mathrm{i}}
  12. n n
  13. p p
  14. V x = V t o t × p x p t o t = V t o t × n x n t o t V_{x}=V_{tot}\times\frac{p_{x}}{p_{tot}}=V_{tot}\times\frac{n_{x}}{n_{tot}}
  15. a A + b B c C + d D a\,A+b\,B\leftrightarrow c\,C+d\,D
  16. K p = p C c p D d p A a p B b K_{p}=\frac{p_{C}^{c}\,p_{D}^{d}}{p_{A}^{a}\,p_{B}^{b}}
  17. K p K_{p}
  18. a a
  19. A A
  20. b b
  21. B B
  22. c c
  23. C C
  24. d d
  25. D D
  26. p C c p_{C}^{c}
  27. C C
  28. c c
  29. p D d p_{D}^{d}
  30. D D
  31. d d
  32. p A a p_{A}^{a}
  33. A A
  34. a a
  35. p B b p_{B}^{b}
  36. B B
  37. b b
  38. k = p x C x k=\frac{p_{x}}{C_{x}}
  39. k k
  40. p x p_{x}
  41. x x
  42. C x C_{x}
  43. x x
  44. k k
  45. k = C x p x k^{\prime}=\frac{C_{x}}{p_{x}}
  46. k k^{\prime}
  47. k k^{\prime}
  48. k k
  49. P i P_{\mathrm{i}}
  50. P P
  51. x i x_{\mathrm{i}}
  52. P N 2 P_{{\mathrm{N}}_{2}}
  53. P O 2 P_{{\mathrm{O}}_{2}}
  54. p O 2 p_{{\mathrm{O}}_{2}}
  55. p CO 2 p_{{\mathrm{CO}}_{2}}
  56. p O 2 p_{{\mathrm{O}}_{2}}
  57. p CO 2 p_{{\mathrm{CO}}_{2}}
  58. p O 2 p_{{\mathrm{O}}_{2}}
  59. p CO 2 p_{{\mathrm{CO}}_{2}}

Partially_ordered_set.html

  1. P × P P\times P
  2. a i a a_{i}\to a
  3. b i b b_{i}\to b
  4. [ a , b ] [a,b]
  5. [ a , b ) [a,b)
  6. ( a , b ] (a,b]

Particle_in_a_box.html

  1. V ( x ) = { 0 , 0 < x < L , , otherwise, , V(x)=\begin{cases}0,&0<x<L,\\ \infty,&\,\text{otherwise,}\end{cases},
  2. L L
  3. x x
  4. ψ ( x , t ) \psi(x,t)
  5. i t ψ ( x , t ) = - 2 2 m 2 x 2 ψ ( x , t ) + V ( x ) ψ ( x , t ) , i\hbar\frac{\partial}{\partial t}\psi(x,t)=-\frac{\hbar^{2}}{2m}\frac{\partial% ^{2}}{\partial x^{2}}\psi(x,t)+V(x)\psi(x,t),
  6. \hbar
  7. m m
  8. i i
  9. t t
  10. ψ ( x , t ) = [ A sin ( k x ) + B cos ( k x ) ] e - i ω t , \psi(x,t)=[A\sin(kx)+B\cos(kx)]\mathrm{e}^{-i\omega t},
  11. A A
  12. B B
  13. k k
  14. ω \omega
  15. E = ω = 2 k 2 2 m , E=\hbar\omega=\frac{\hbar^{2}k^{2}}{2m},
  16. p 2 2 m \frac{p^{2}}{2m}
  17. p = k p=\hbar k
  18. E = p 2 2 m E=\frac{p^{2}}{2m}
  19. E = T + V E=T+V
  20. P ( x , t ) = | ψ ( x , t ) | 2 P(x,t)=|\psi(x,t)|^{2}
  21. ψ n ( x , t ) = { A sin ( k n x ) e - i ω n t , 0 < x < L , 0 , otherwise, \psi_{n}(x,t)=\begin{cases}A\sin(k_{n}x)\mathrm{e}^{-i\omega_{n}t},&0<x<L,\\ 0,&\,\text{otherwise,}\end{cases}
  22. n n
  23. x = 0 x=0
  24. x = L x=L
  25. k n = n π L , where n = { 1 , 2 , 3 , 4 , } , k_{n}=\frac{n\pi}{L},\quad\mathrm{where}\quad n=\{1,2,3,4,\ldots\},
  26. L L
  27. A = 0 A=0
  28. ψ ( x ) = 0 \psi(x)=0
  29. n n
  30. n n
  31. A A
  32. | A | = 2 L . \left|A\right|=\sqrt{\frac{2}{L}}.
  33. 0
  34. L L
  35. E n E_{n}
  36. ψ n ( x , t ) \psi_{n}(x,t)
  37. ψ n ( x , t ) \psi_{n}(x,t)
  38. ψ n ( x , t ) = { 2 L sin ( k n x - n π x 0 L ) e - i ω n t , x 0 < x < x 0 + L , 0 , otherwise, \psi_{n}(x,t)=\begin{cases}\sqrt{\frac{2}{L}}\sin(k_{n}x-\frac{n\pi{x_{0}}}{L}% )\mathrm{e}^{-i\omega_{n}t},&x_{0}<x<x_{0}+L,\\ 0,&\,\text{otherwise,}\end{cases}
  39. x 0 x_{0}
  40. n π x 0 L \frac{n\pi x_{0}}{L}
  41. ψ n ( x , t ) \psi_{n}(x,t)
  42. x 0 = 0 x_{0}=0
  43. k = p / k^{\prime}=p/\hbar
  44. k k^{\prime}
  45. k k
  46. ω n = π h n 2 4 L 2 m \omega_{n}=\frac{\pi hn^{2}}{4L^{2}m}
  47. ϕ n ( p , t ) = 1 2 π - ψ n ( x , t ) e - i k x d x = π L n ( 1 - ( - 1 ) n e - i k L ) e - i ω n t π 2 n 2 - ( k ) 2 L 2 \phi_{n}(p,t)=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}\psi_{n}(x,t)e^% {-ik^{\prime}x}\,dx=\sqrt{\frac{\pi L}{\hbar}}\,\,\frac{n\left(1-(-1)^{n}e^{-% ik^{\prime}L}\right)e^{-i\omega_{n}t}}{\pi^{2}n^{2}-(k^{\prime})^{2}L^{2}}
  48. k n k_{n}
  49. p = ± k n p=\pm\hbar k_{n}
  50. E n = 2 k n 2 2 m E_{n}=\frac{\hbar^{2}k_{n}^{2}}{2m}
  51. E = p 2 2 m E=\frac{p^{2}}{2m}
  52. ψ n \psi_{n}
  53. P ( x ) = | ψ ( x ) | 2 . P(x)=|\psi(x)|^{2}.
  54. P n ( x ) = { 2 L sin 2 ( n π x L ) ; 0 < x < L 0 ; otherwise . P_{n}(x)=\begin{cases}\frac{2}{L}\sin^{2}\left(\frac{n\pi x}{L}\right);&0<x<L% \\ 0;&\,\text{otherwise}.\end{cases}
  55. P ( x ) = 0 P(x)=0
  56. x = - x P n ( x ) d x . \langle x\rangle=\int_{-\infty}^{\infty}xP_{n}(x)\,\mathrm{d}x.
  57. x = L / 2 \langle x\rangle=L/2
  58. c o s ( ω t ) cos(\omega t)
  59. Var ( x ) = - ( x - x ) 2 P n ( x ) d x = L 2 12 ( 1 - 6 n 2 π 2 ) \mathrm{Var}(x)=\int_{-\infty}^{\infty}(x-\langle x\rangle)^{2}P_{n}(x)\,dx=% \frac{L^{2}}{12}\left(1-\frac{6}{n^{2}\pi^{2}}\right)
  60. P ( x ) = | ϕ ( x ) | 2 . P(x)=|\phi(x)|^{2}.
  61. P n ( p ) = 2 π L n 2 ( 1 - ( - 1 ) n cos ( k L ) ) ( k 2 L 2 - π 2 n 2 ) 2 P_{n}(p)=\frac{2\pi L}{\hbar}\,\frac{n^{2}\left(1-(-1)^{n}\cos(kL)\right)}{% \left(k^{2}L^{2}-\pi^{2}n^{2}\right)^{2}}
  62. k = p / k=p/\hbar
  63. Var ( p ) = ( n π L ) 2 \mathrm{Var}(p)=\left(\frac{\hbar n\pi}{L}\right)^{2}
  64. Δ x \Delta x
  65. Δ p \Delta p
  66. Δ x Δ p = 2 n 2 π 2 3 - 2 \Delta x\Delta p=\frac{\hbar}{2}\sqrt{\frac{n^{2}\pi^{2}}{3}-2}
  67. \hbar
  68. / 2 \hbar/2
  69. E n = n 2 2 π 2 2 m L 2 = n 2 h 2 8 m L 2 E_{n}=\frac{n^{2}\hbar^{2}\pi^{2}}{2mL^{2}}=\frac{n^{2}h^{2}}{8mL^{2}}
  70. n 2 n^{2}
  71. E 1 = 2 π 2 2 m L 2 . E_{1}=\frac{\hbar^{2}\pi^{2}}{2mL^{2}}.
  72. Δ x Δ p 2 \Delta x\Delta p\geq\frac{\hbar}{2}
  73. E = p 2 / ( 2 m ) E=p^{2}/(2m)
  74. x x
  75. y y
  76. L x L_{x}
  77. L y L_{y}
  78. ψ n x , n y = 4 L x L y sin ( k n x x ) sin ( k n y y ) \psi_{n_{x},n_{y}}=\sqrt{\frac{4}{L_{x}L_{y}}}\sin\left(k_{n_{x}}x\right)\sin% \left(k_{n_{y}}y\right)
  79. E n x , n y = 2 k n x , n y 2 2 m E_{n_{x},n_{y}}=\frac{\hbar^{2}k_{n_{x},n_{y}}^{2}}{2m}
  80. 𝐤 𝐧 𝐱 , 𝐧 𝐲 = k n x 𝐱 ^ + k n y 𝐲 ^ = n x π L x 𝐱 ^ + n y π L y 𝐲 ^ \mathbf{k_{n_{x},n_{y}}}=k_{n_{x}}\mathbf{\hat{x}}+k_{n_{y}}\mathbf{\hat{y}}=% \frac{n_{x}\pi}{L_{x}}\mathbf{\hat{x}}+\frac{n_{y}\pi}{L_{y}}\mathbf{\hat{y}}
  81. ψ n x , n y , n z = 8 L x L y L z sin ( k n x x ) sin ( k n y y ) sin ( k n z z ) \psi_{n_{x},n_{y},n_{z}}=\sqrt{\frac{8}{L_{x}L_{y}L_{z}}}\sin\left(k_{n_{x}}x% \right)\sin\left(k_{n_{y}}y\right)\sin\left(k_{n_{z}}z\right)
  82. E n x , n y , n z = 2 k n x , n y , n z 2 2 m E_{n_{x},n_{y},n_{z}}=\frac{\hbar^{2}k_{n_{x},n_{y},n_{z}}^{2}}{2m}
  83. 𝐤 𝐧 𝐱 , 𝐧 𝐲 , 𝐧 𝐳 = k n x 𝐱 ^ + k n y 𝐲 ^ + k n z 𝐳 ^ = n x π L x 𝐱 ^ + n y π L y 𝐲 ^ + n z π L z 𝐳 ^ \mathbf{k_{n_{x},n_{y},n_{z}}}=k_{n_{x}}\mathbf{\hat{x}}+k_{n_{y}}\mathbf{\hat% {y}}+k_{n_{z}}\mathbf{\hat{z}}=\frac{n_{x}\pi}{L_{x}}\mathbf{\hat{x}}+\frac{n_% {y}\pi}{L_{y}}\mathbf{\hat{y}}+\frac{n_{z}\pi}{L_{z}}\mathbf{\hat{z}}
  84. ψ = 2 n i L i i sin ( k i i ) \psi=\sqrt{\frac{2^{n}}{\prod_{i}L_{i}}}\prod_{i}\sin(k_{i}i)
  85. L x = L y L_{x}=L_{y}
  86. n x = 2 , n y = 1 n_{x}=2,n_{y}=1
  87. n x = 1 , n y = 2 n_{x}=1,n_{y}=2
  88. V 0 V_{0}
  89. V 0 V_{0}

Partition_of_unity.html

  1. x X x\in X
  2. ρ R ρ ( x ) = 1 \;\sum_{\rho\in R}\rho(x)=1
  3. { ρ σ : ρ R σ S } \{\;\rho\sigma\;:\;\rho\in R\wedge\sigma\in S\;\}
  4. [ 0 , 1 ] [0,1]

Pascal's_triangle.html

  1. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  2. ( 0 0 ) = 1 {\textstyle\left({{0}\atop{0}}\right)}=1
  3. ( x + y ) n = k = 0 n ( n k ) x n - k y k (x+y)^{n}=\sum_{k=0}^{n}{n\choose k}x^{n-k}y^{k}
  4. ( n k ) = ( n - 1 k - 1 ) + ( n - 1 k ) {n\choose k}={n-1\choose k-1}+{n-1\choose k}
  5. ( n r ) = ( n - 1 r ) + ( n - 1 r - 1 ) {\textstyle\left({{n}\atop{r}}\right)}={\textstyle\left({{n-1}\atop{r}}\right)% }+{\textstyle\left({{n-1}\atop{r-1}}\right)}
  6. ( n r ) = n ! r ! ( n - r ) ! {\textstyle\left({{n}\atop{r}}\right)}=\tfrac{n!}{r!(n-r)!}
  7. a i = ( n i ) . a_{i}={n\choose i}.
  8. ( x + 1 ) n = i = 0 n a i x i . (x+1)^{n}=\sum_{i=0}^{n}a_{i}x^{i}.
  9. ( x + 1 ) n + 1 = ( x + 1 ) ( x + 1 ) n = x ( x + 1 ) n + ( x + 1 ) n = i = 0 n a i x i + 1 + i = 0 n a i x i . (x+1)^{n+1}=(x+1)(x+1)^{n}=x(x+1)^{n}+(x+1)^{n}=\sum_{i=0}^{n}a_{i}x^{i+1}+% \sum_{i=0}^{n}a_{i}x^{i}.
  10. i = 0 n a i x i + 1 + i = 0 n a i x i \displaystyle\sum_{i=0}^{n}a_{i}x^{i+1}+\sum_{i=0}^{n}a_{i}x^{i}
  11. ( n 0 ) + ( n 1 ) + + ( n n - 1 ) + ( n n ) = 2 n . {n\choose 0}+{n\choose 1}+\cdots+{n\choose n-1}+{n\choose n}=2^{n}.
  12. 𝐂 ( n , k ) = 𝐂 k n = C k n = ( n k ) = n ! k ! ( n - k ) ! . \mathbf{C}(n,k)=\mathbf{C}_{k}^{n}={{}_{n}C_{k}}={n\choose k}=\frac{n!}{k!(n-k% )!}.
  13. n n
  14. s n = k = 0 n ( n k ) = k = 0 n n ! k ! ( n - k ) ! , n 0. s_{n}=\prod_{k=0}^{n}{\left({{n}\atop{k}}\right)}=\prod_{k=0}^{n}\frac{n!}{k!(% n-k)!}~{},~{}n\geq 0.
  15. s n + 1 s n = ( n + 1 ) ! ( n + 2 ) k = 0 n + 1 k ! - 2 n ! ( n + 1 ) k = 0 n k ! - 2 = ( n + 1 ) n n ! \frac{s_{n+1}}{s_{n}}=\frac{(n+1)!^{(n+2)}\prod_{k=0}^{n+1}{k!^{-2}}}{n!^{(n+1% )}\prod_{k=0}^{n}{k!^{-2}}}=\frac{(n+1)^{n}}{n!}
  16. ( s n + 1 ) ( s n - 1 ) ( s n ) 2 = ( n + 1 n ) n , n 1. \frac{(s_{n+1})(s_{n-1})}{(s_{n})^{2}}=\left(\frac{n+1}{n}\right)^{n},~{}n\geq 1.
  17. 𝑒 = lim n ( 1 + 1 n ) n . \,\textit{e}=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}.
  18. 11 < s u p > n 11<sup>n
  19. n n
  20. 2 n 2n
  21. k = 0 n ( n k ) 2 = ( 2 n n ) . \sum_{k=0}^{n}{n\choose k}^{2}={2n\choose n}.
  22. n n
  23. n n
  24. ( n / 2 + 1 ) (n/2+1)
  25. 6 1 = 5 6−1=5
  26. 4 / 2 + 1 = 3 4/2+1=3
  27. p p
  28. p p
  29. p p
  30. p p\in\mathbb{P}
  31. p p
  32. ( p - k ) ! (p-k)!
  33. k ! k!
  34. p ! p!\,
  35. p p
  36. p p
  37. p p
  38. n n
  39. n n
  40. x x
  41. P 0 ( n ) \displaystyle P_{0}(n)
  42. P d ( n ) = 1 d ! k = 0 d - 1 ( n + k ) = n ( d ) d ! = ( n + d - 1 d ) P_{d}(n)=\frac{1}{d!}\prod_{k=0}^{d-1}(n+k)={n^{(d)}\over d!}={\left({{n+d-1}% \atop{d}}\right)}
  43. n n
  44. ( n 0 ) {\textstyle\left({{n}\atop{0}}\right)}
  45. ( n 1 ) {\textstyle\left({{n}\atop{1}}\right)}
  46. ( n n ) {\textstyle\left({{n}\atop{n}}\right)}
  47. ( n 0 ) = 1 {\textstyle\left({{n}\atop{0}}\right)}=1
  48. ( n k ) = ( n k - 1 ) × n + 1 - k k . {n\choose k}={n\choose k-1}\times\frac{n+1-k}{k}.
  49. 5 1 \tfrac{5}{1}
  50. 4 2 \tfrac{4}{2}
  51. 3 3 \tfrac{3}{3}
  52. 2 4 \tfrac{2}{4}
  53. 1 5 \tfrac{1}{5}
  54. ( 5 0 ) = 1 {\textstyle\left({{5}\atop{0}}\right)}=1
  55. ( 5 1 ) = 1 × 5 1 = 5 {\textstyle\left({{5}\atop{1}}\right)}=1\times\tfrac{5}{1}=5
  56. ( 5 2 ) = 5 × 4 2 = 10 {\textstyle\left({{5}\atop{2}}\right)}=5\times\tfrac{4}{2}=10
  57. ( n 0 ) {\textstyle\left({{n}\atop{0}}\right)}
  58. ( n + 1 1 ) {\textstyle\left({{n+1}\atop{1}}\right)}
  59. ( n + 2 2 ) {\textstyle\left({{n+2}\atop{2}}\right)}
  60. ( n 0 ) = 1 {\textstyle\left({{n}\atop{0}}\right)}=1
  61. ( n + k k ) = ( n + k - 1 k - 1 ) × n + k k . {n+k\choose k}={n+k-1\choose k-1}\times\frac{n+k}{k}.
  62. ( 5 0 ) {\textstyle\left({{5}\atop{0}}\right)}
  63. 6 1 \tfrac{6}{1}
  64. 7 2 \tfrac{7}{2}
  65. 8 3 \tfrac{8}{3}
  66. ( 5 0 ) = 1 {\textstyle\left({{5}\atop{0}}\right)}=1
  67. ( 6 1 ) = 1 × 6 1 = 6 {\textstyle\left({{6}\atop{1}}\right)}=1\times\tfrac{6}{1}=6
  68. ( 7 2 ) = 6 × 7 2 = 21 {\textstyle\left({{7}\atop{2}}\right)}=6\times\tfrac{7}{2}=21
  69. ( 5 5 ) {\textstyle\left({{5}\atop{5}}\right)}
  70. ( 6 5 ) {\textstyle\left({{6}\atop{5}}\right)}
  71. ( 7 5 ) {\textstyle\left({{7}\atop{5}}\right)}
  72. ( n k ) = 2 × ( n - 1 k - 1 ) + ( n - 1 k ) . {n\choose k}=2\times{n-1\choose k-1}+{n-1\choose k}.
  73. 1 + 8 + 24 + 32 + 16 = 81 1+8+24+32+16=81
  74. 3 4 = 81 3^{4}=81
  75. 𝔢 ( Fourier [ sin ( x ) 5 x ] ) \,\mathfrak{Re}\left(\,\text{Fourier}\left[\frac{\sin(x)^{5}}{x}\right]\right)
  76. 𝔢 ( Fourier [ sin ( x ) 1 x ] ) \,\mathfrak{Re}\left(\,\text{Fourier}\left[\frac{\sin(x)^{1}}{x}\right]\right)
  77. + i , - 1 , - i , + 1 , + i , \,+i,-1,-i,+1,+i,\ldots\,
  78. ( n m ) = ( n - 1 m - 1 ) + ( n - 1 m ) {n\choose m}={n-1\choose m-1}+{n-1\choose m}
  79. ( n - 1 m ) = ( n m ) - ( n - 1 m - 1 ) {n-1\choose m}={n\choose m}-{n-1\choose m-1}
  80. ( n m ) = 1 m ! k = 0 m - 1 ( n - k ) = 1 m ! k = 1 m ( n - k + 1 ) {n\choose m}=\frac{1}{m!}\prod_{k=0}^{m-1}(n-k)=\frac{1}{m!}\prod_{k=1}^{m}(n-% k+1)
  81. ( 1 + x ) n = k = 0 ( n k ) x k | x | < 1 (1+x)^{n}=\sum_{k=0}^{\infty}{n\choose k}x^{k}\quad|x|<1
  82. ( 1 + x ) - 2 = 1 - 2 x + 3 x 2 - 4 x 3 + | x | < 1 (1+x)^{-2}=1-2x+3x^{2}-4x^{3}+\cdots\quad|x|<1
  83. exp ( . . . . . 1 . . . . . 2 . . . . . 3 . . . . . 4 . ) = ( 1 . . . . 1 1 . . . 1 2 1 . . 1 3 3 1 . 1 4 6 4 1 ) , \exp\begin{pmatrix}.&.&.&.&.\\ 1&.&.&.&.\\ .&2&.&.&.\\ .&.&3&.&.\\ .&.&.&4&.\end{pmatrix}=\begin{pmatrix}1&.&.&.&.\\ 1&1&.&.&.\\ 1&2&1&.&.\\ 1&3&3&1&.\\ 1&4&6&4&1\end{pmatrix},
  84. exp ( . . . . . . . . . . - 4 . . . . . . . . . . - 3 . . . . . . . . . . - 2 . . . . . . . . . . - 1 . . . . . . . . . . 0 . . . . . . . . . . 1 . . . . . . . . . . 2 . . . . . . . . . . 3 . . . . . . . . . . 4 . ) = ( 1 . . . . . . . . . - 4 1 . . . . . . . . 6 - 3 1 . . . . . . . - 4 3 - 2 1 . . . . . . 1 - 1 1 - 1 1 . . . . . . . . . . 1 . . . . . . . . . 1 1 . . . . . . . . 1 2 1 . . . . . . . 1 3 3 1 . . . . . . 1 4 6 4 1 ) \exp\begin{pmatrix}.&.&.&.&.&.&.&.&.&.\\ -4&.&.&.&.&.&.&.&.&.\\ .&-3&.&.&.&.&.&.&.&.\\ .&.&-2&.&.&.&.&.&.&.\\ .&.&.&-1&.&.&.&.&.&.\\ .&.&.&.&0&.&.&.&.&.\\ .&.&.&.&.&1&.&.&.&.\\ .&.&.&.&.&.&2&.&.&.\\ .&.&.&.&.&.&.&3&.&.\\ .&.&.&.&.&.&.&.&4&.\end{pmatrix}=\begin{pmatrix}1&.&.&.&.&.&.&.&.&.\\ -4&1&.&.&.&.&.&.&.&.\\ 6&-3&1&.&.&.&.&.&.&.\\ -4&3&-2&1&.&.&.&.&.&.\\ 1&-1&1&-1&1&.&.&.&.&.\\ .&.&.&.&.&1&.&.&.&.\\ .&.&.&.&.&1&1&.&.&.\\ .&.&.&.&.&1&2&1&.&.\\ .&.&.&.&.&1&3&3&1&.\\ .&.&.&.&.&1&4&6&4&1\end{pmatrix}
  85. ( n k ) = n ! ( n - k ) ! k ! Γ ( n + 1 ) Γ ( n - k + 1 ) Γ ( k + 1 ) {n\choose k}=\frac{n!}{(n-k)!k!}\equiv\frac{\Gamma(n+1)}{\Gamma(n-k+1)\Gamma(k% +1)}
  86. Γ ( z ) \Gamma(z)
  87. ( n k ) \scriptstyle{n\choose k}

Pascal_(unit).html

  1. 1 Pa = 1 N m 2 = 1 kg m s 2 {\rm 1~{}Pa=1~{}\frac{N}{m^{2}}=1~{}\frac{kg}{m\cdot s^{2}}}

Password_length_parameter.html

  1. P = L × R / S P=L\times R/S
  2. P P
  3. L L
  4. R R
  5. S S

Path_loss.html

  1. L = 10 n log 10 ( d ) + C L=10\ n\ \log_{10}(d)+C
  2. L L
  3. n n
  4. d d
  5. C C
  6. L = 20 log 10 ( 4 π d λ ) L=20\ \log_{10}\left(\frac{4\pi d}{\lambda}\right)
  7. L L
  8. λ \lambda
  9. d d

Paul_Cohen_(mathematician).html

  1. 1 \aleph_{1}
  2. C C
  3. C C
  4. C C
  5. n , ω , a \aleph_{n},\aleph_{\omega},\aleph_{a}
  6. a = ω a=\aleph_{\omega}
  7. C C

Paul_Dirac.html

  1. g r s g_{rs}
  2. p r s p^{rs}
  3. g m 0 g_{m0}
  4. g r 0 g^{r0}
  5. ( - g 00 ) - 1 / 2 (-{g^{00}})^{-1/2}
  6. x 0 x^{0}
  7. H r H_{r}
  8. H L H_{L}

Pauli_exclusion_principle.html

  1. = h / 2 π \hbar=h/2\pi
  2. | x \scriptstyle|x\rangle
  3. | y \scriptstyle|y\rangle
  4. | ψ = x , y A ( x , y ) | x , y , |\psi\rangle=\sum_{x,y}A(x,y)|x,y\rangle,
  5. A ( x , y ) = ψ | x , y = ψ | ( | x | y ) A(x,y)=\langle\psi|x,y\rangle=\langle\psi|(|x\rangle\otimes|y\rangle)
  6. ψ | ( ( | x + | y ) ( | x + | y ) ) . \langle\psi|\Big((|x\rangle+|y\rangle)\otimes(|x\rangle+|y\rangle)\Big).
  7. | x + | y |x\rangle+|y\rangle
  8. ψ | x , x + ψ | x , y + ψ | y , x + ψ | y , y . \langle\psi|x,x\rangle+\langle\psi|x,y\rangle+\langle\psi|y,x\rangle+\langle% \psi|y,y\rangle.
  9. ψ | x , y + ψ | y , x = 0 , \langle\psi|x,y\rangle+\langle\psi|y,x\rangle=0,
  10. A ( x , y ) = - A ( y , x ) . A(x,y)=-A(y,x).

Pauli_matrices.html

  1. 2 × 2 2×2
  2. σ σ
  3. τ τ
  4. σ 1 = σ x = ( 0 1 1 0 ) σ 2 = σ y = ( 0 - i i 0 ) σ 3 = σ z = ( 1 0 0 - 1 ) . \begin{aligned}\displaystyle\sigma_{1}=\sigma_{x}&\displaystyle=\begin{pmatrix% }0&1\\ 1&0\end{pmatrix}\\ \displaystyle\sigma_{2}=\sigma_{y}&\displaystyle=\begin{pmatrix}0&-i\\ i&0\end{pmatrix}\\ \displaystyle\sigma_{3}=\sigma_{z}&\displaystyle=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}\,.\end{aligned}
  5. I I
  6. 2 × 2 2×2
  7. 2 2
  8. k k
  9. i i
  10. 𝐬𝐮 ( 2 ) \mathbf{su}(2)
  11. σ a = ( δ a 3 δ a 1 - i δ a 2 δ a 1 + i δ a 2 - δ a 3 ) \sigma_{a}=\begin{pmatrix}\delta_{a3}&\delta_{a1}-i\delta_{a2}\\ \delta_{a1}+i\delta_{a2}&-\delta_{a3}\end{pmatrix}
  12. i = 1 i=\sqrt{−1}
  13. a = b a=b
  14. a = 1 , 2 , 3 a=1,2,3
  15. σ 1 2 = σ 2 2 = σ 3 2 = - i σ 1 σ 2 σ 3 = ( 1 0 0 1 ) = I \sigma_{1}^{2}=\sigma_{2}^{2}=\sigma_{3}^{2}=-i\sigma_{1}\sigma_{2}\sigma_{3}=% \begin{pmatrix}1&0\\ 0&1\end{pmatrix}=I
  16. I I
  17. det σ i \displaystyle\det\sigma_{i}
  18. ± 1 ±1
  19. 2 × 2 2×2
  20. I I
  21. 2 × 2 2×2
  22. 2 × 2 2×2
  23. + 1 +1
  24. 1 −1
  25. ψ x + = 1 2 \displaystyle\psi_{x+}=\frac{1}{\sqrt{}}{2}
  26. σ = σ 1 x ^ + σ 2 y ^ + σ 3 z ^ \vec{\sigma}=\sigma_{1}\hat{x}+\sigma_{2}\hat{y}+\sigma_{3}\hat{z}\,
  27. a σ = ( a i x ^ i ) ( σ j x ^ j ) = a i σ j x ^ i x ^ j = a i σ j δ i j = a i σ i = ( a 3 a 1 - i a 2 a 1 + i a 2 - a 3 ) \begin{aligned}\displaystyle\vec{a}\cdot\vec{\sigma}&\displaystyle=(a_{i}\hat{% x}_{i})\cdot(\sigma_{j}\hat{x}_{j})\\ &\displaystyle=a_{i}\sigma_{j}\hat{x}_{i}\cdot\hat{x}_{j}\\ &\displaystyle=a_{i}\sigma_{j}\delta_{ij}\\ &\displaystyle=a_{i}\sigma_{i}=\begin{pmatrix}a_{3}&a_{1}-ia_{2}\\ a_{1}+ia_{2}&-a_{3}\end{pmatrix}\end{aligned}
  28. det a σ = - a a = - | a | 2 , \det\vec{a}\cdot\vec{\sigma}=-\vec{a}\cdot\vec{a}=-|\vec{a}|^{2},
  29. 1 2 tr [ ( a σ ) σ ] = a . \frac{1}{2}\mathrm{tr}[(\vec{a}\cdot\vec{\sigma})\vec{\sigma}]=\vec{a}.
  30. [ σ a , σ b ] = 2 i ε a b c σ c , [\sigma_{a},\sigma_{b}]=2i\varepsilon_{abc}\,\sigma_{c}\,,
  31. { σ a , σ b } = 2 δ a b I . \{\sigma_{a},\sigma_{b}\}=2\delta_{ab}\,I.
  32. I I
  33. 2 × 2 2×2
  34. [ σ 1 , σ 2 ] = 2 i σ 3 [ σ 2 , σ 3 ] = 2 i σ 1 [ σ 3 , σ 1 ] = 2 i σ 2 [ σ 1 , σ 1 ] = 0 { σ 1 , σ 1 } = 2 I { σ 1 , σ 2 } = 0 . \begin{aligned}\displaystyle\left[\sigma_{1},\sigma_{2}\right]&\displaystyle=2% i\sigma_{3}\\ \displaystyle\left[\sigma_{2},\sigma_{3}\right]&\displaystyle=2i\sigma_{1}\\ \displaystyle\left[\sigma_{3},\sigma_{1}\right]&\displaystyle=2i\sigma_{2}\\ \displaystyle\left[\sigma_{1},\sigma_{1}\right]&\displaystyle=0\\ \displaystyle\left\{\sigma_{1},\sigma_{1}\right\}&\displaystyle=2I\\ \displaystyle\left\{\sigma_{1},\sigma_{2}\right\}&\displaystyle=0\,.\\ \end{aligned}
  35. [ σ a , σ b ] + { σ a , σ b } = ( σ a σ b - σ b σ a ) + ( σ a σ b + σ b σ a ) 2 i c ε a b c σ c + 2 δ a b I = 2 σ a σ b \begin{aligned}\displaystyle\left[\sigma_{a},\sigma_{b}\right]+\{\sigma_{a},% \sigma_{b}\}&\displaystyle=(\sigma_{a}\sigma_{b}-\sigma_{b}\sigma_{a})+(\sigma% _{a}\sigma_{b}+\sigma_{b}\sigma_{a})\\ \displaystyle 2i\sum_{c}\varepsilon_{abc}\,\sigma_{c}+2\delta_{ab}I&% \displaystyle=2\sigma_{a}\sigma_{b}\end{aligned}
  36. 3 3
  37. a , b , c p , q , r a,b,c→p,q,r
  38. a p b q σ p σ q = a p b q ( i r ε p q r σ r + δ p q I ) a p σ p b q σ q = i r ε p q r a p b q σ r + a p b q δ p q I . \begin{aligned}\displaystyle a_{p}b_{q}\sigma_{p}\sigma_{q}&\displaystyle=a_{p% }b_{q}\left(i\sum_{r}\varepsilon_{pqr}\,\sigma_{r}+\delta_{pq}I\right)\\ \displaystyle a_{p}\sigma_{p}b_{q}\sigma_{q}&\displaystyle=i\sum_{r}% \varepsilon_{pqr}\,a_{p}b_{q}\sigma_{r}+a_{p}b_{q}\delta_{pq}I~{}.\end{aligned}
  39. a = a n ^ , | n ^ | = 1 , \vec{a}=a\hat{n},\quad|\hat{n}|=1,
  40. ( n ^ σ ) 2 n = I (\hat{n}\cdot\vec{\sigma})^{2n}=I\,
  41. n = 1 n=1
  42. ( n ^ σ ) 2 n + 1 = n ^ σ . (\hat{n}\cdot\vec{\sigma})^{2n+1}=\hat{n}\cdot\vec{\sigma}\,.
  43. e i a ( n ^ σ ) = n = 0 i n [ a ( n ^ σ ) ] n n ! = n = 0 ( - 1 ) n ( a n ^ σ ) 2 n ( 2 n ) ! + i n = 0 ( - 1 ) n ( a n ^ σ ) 2 n + 1 ( 2 n + 1 ) ! = I n = 0 ( - 1 ) n a 2 n ( 2 n ) ! + i ( n ^ σ ) n = 0 ( - 1 ) n a 2 n + 1 ( 2 n + 1 ) ! \begin{aligned}\displaystyle e^{ia(\hat{n}\cdot\vec{\sigma})}&\displaystyle=% \sum_{n=0}^{\infty}{\frac{i^{n}\left[a(\hat{n}\cdot\vec{\sigma})\right]^{n}}{n% !}}\\ &\displaystyle=\sum_{n=0}^{\infty}{\frac{(-1)^{n}(a\hat{n}\cdot\vec{\sigma})^{% 2n}}{(2n)!}}+i\sum_{n=0}^{\infty}{\frac{(-1)^{n}(a\hat{n}\cdot\vec{\sigma})^{2% n+1}}{(2n+1)!}}\\ &\displaystyle=I\sum_{n=0}^{\infty}{\frac{(-1)^{n}a^{2n}}{(2n)!}}+i(\hat{n}% \cdot\vec{\sigma})\sum_{n=0}^{\infty}{\frac{(-1)^{n}a^{2n+1}}{(2n+1)!}}\\ \end{aligned}
  44. det [ i a ( n ^ σ ) ] = a 2 \det[ia(\hat{n}\cdot\vec{\sigma})]=a^{2}
  45. 1 1
  46. 2 × 2 2×2
  47. S U ( 2 ) SU(2)
  48. S U ( 2 ) SU(2)
  49. c c
  50. e i a ( n ^ σ ) e i b ( m ^ σ ) = I ( cos a cos b - n ^ m ^ sin a sin b ) + i ( n ^ sin a cos b + m ^ sin b cos a - n ^ × m ^ sin a sin b ) σ = I cos c + i ( k ^ σ ) sin c = e i c ( k ^ σ ) , \begin{aligned}\displaystyle e^{ia(\hat{n}\cdot\vec{\sigma})}e^{ib(\hat{m}% \cdot\vec{\sigma})}&\displaystyle=I(\cos a\cos b-\hat{n}\cdot\hat{m}\sin a\sin b% )+i(\hat{n}\sin a\cos b+\hat{m}\sin b\cos a-\hat{n}\times\hat{m}~{}\sin a\sin b% )\cdot\vec{\sigma}\\ &\displaystyle=I\cos{c}+i(\hat{k}\cdot\vec{\sigma})\sin{c}\\ &\displaystyle=e^{ic\left(\hat{k}\cdot\vec{\sigma}\right)},\end{aligned}
  51. cos c = cos a cos b - n ^ m ^ sin a sin b , \cos c=\cos a\cos b-\hat{n}\cdot\hat{m}\sin a\sin b~{},
  52. c c
  53. k ^ = 1 sin c ( n ^ sin a cos b + m ^ sin b cos a - n ^ × m ^ sin a sin b ) . \hat{k}=\frac{1}{\sin c}\left(\hat{n}\sin a\cos b+\hat{m}\sin b\cos a-\hat{n}% \times\hat{m}\sin a\sin b\right)~{}.
  54. e i c k ^ σ = exp ( i c sin c ( n ^ sin a cos b + m ^ sin b cos a - n ^ × m ^ sin a sin b ) σ ) . e^{ic\hat{k}\cdot\vec{\sigma}}=\exp\left(i\frac{c}{\sin c}(\hat{n}\sin a\cos b% +\hat{m}\sin b\cos a-\hat{n}\times\hat{m}~{}\sin a\sin b)\cdot\vec{\sigma}% \right)~{}.
  55. c = a + b c=a+b
  56. 2 × 2 2×2
  57. 2 × 2 2×2
  58. 2 × 2 2×2
  59. 1 1
  60. 1 1
  61. i i
  62. α α
  63. β β
  64. i i
  65. σ α β σ γ δ i = 1 3 σ α β i σ γ δ i = 2 δ α δ δ β γ - δ α β δ γ δ . \vec{\sigma}_{\alpha\beta}\cdot\vec{\sigma}_{\gamma\delta}\equiv\sum_{i=1}^{3}% \sigma^{i}_{\alpha\beta}\sigma^{i}_{\gamma\delta}=2\delta_{\alpha\delta}\delta% _{\beta\gamma}-\delta_{\alpha\beta}\delta_{\gamma\delta}.\,
  66. M = c I + i a i σ i M=cI+\sum_{i}a_{i}\sigma^{i}
  67. tr σ i σ j = 2 δ i j \mathrm{tr}\,\sigma^{i}\sigma^{j}=2\delta_{ij}
  68. c = 1 2 tr M c=\frac{1}{2}\mathrm{tr}\,M
  69. a i = 1 2 tr σ i M a_{i}=\frac{1}{2}\mathrm{tr}\,\sigma^{i}M
  70. 2 M = I tr M + i σ i tr σ i M 2M=I\mathrm{tr}\,M+\sum_{i}\sigma^{i}\mathrm{tr}\,\sigma^{i}M
  71. 2 M α β = δ α β M γ γ + i σ α β i σ γ δ i M δ γ 2M_{\alpha\beta}=\delta_{\alpha\beta}M_{\gamma\gamma}+\sum_{i}\sigma^{i}_{% \alpha\beta}\sigma^{i}_{\gamma\delta}M_{\delta\gamma}
  72. i = 0 3 σ α β i σ γ δ i = 2 δ α δ δ β γ \sum_{i=0}^{3}\sigma^{i}_{\alpha\beta}\sigma^{i}_{\gamma\delta}=2\delta_{% \alpha\delta}\delta_{\beta\gamma}\,
  73. P i j | σ i σ j = | σ j σ i . P_{ij}|\sigma_{i}\sigma_{j}\rangle=|\sigma_{j}\sigma_{i}\rangle\,.
  74. P i j = 1 2 ( σ i σ j + 1 ) . P_{ij}=\tfrac{1}{2}(\vec{\sigma}_{i}\cdot\vec{\sigma}_{j}+1)\,.
  75. 𝔰 𝔲 2 \mathfrak{su}_{2}
  76. 𝔰 𝔲 ( 2 ) = span { i σ 1 , i σ 2 , i σ 3 } . \mathfrak{su}(2)=\operatorname{span}\{i\sigma_{1},i\sigma_{2},i\sigma_{3}\}.
  77. s u ( 2 ) su(2)
  78. λ = λ=
  79. 1 2 \frac{1}{2}
  80. 𝔰 𝔲 ( 2 ) = span { i σ 1 2 , i σ 2 2 , i σ 3 2 } . \mathfrak{su}(2)=\operatorname{span}\left\{\frac{i\sigma_{1}}{2},\frac{i\sigma% _{2}}{2},\frac{i\sigma_{3}}{2}\right\}.
  81. 𝐬𝐮 ( 2 ) \mathbf{su}(2)
  82. 𝐬𝐨 ( 3 ) \mathbf{so}(3)
  83. 𝐬𝐮 ( 2 ) \mathbf{su}(2)
  84. 𝐬𝐨 ( 3 ) \mathbf{so}(3)
  85. S U ( 2 ) SU(2)
  86. S O ( 3 ) SO(3)
  87. S U ( 2 ) SU(2)
  88. S O ( 3 ) SO(3)
  89. S U ( 2 ) SU(2)
  90. S O ( 3 ) SO(3)
  91. 1 I , i - i σ 1 , j - i σ 2 , k - i σ 3 . 1\mapsto I,\quad i\mapsto-i\sigma_{1},\quad j\mapsto-i\sigma_{2},\quad k% \mapsto-i\sigma_{3}.
  92. 1 I , i i σ 3 , j i σ 2 , k i σ 1 . 1\mapsto I,\quad i\mapsto i\sigma_{3},\quad j\mapsto i\sigma_{2},\quad k% \mapsto i\sigma_{1}.
  93. S U ( 2 ) SU(2)
  94. S U ( 2 ) SU(2)
  95. S U ( 2 ) SU(2)
  96. S O ( 3 ) SO(3)
  97. i σ < s u b > j iσ<sub>j
  98. M < s u b > 2 ( ) 3 M<sub>2(ℂ)⊗ℝ^{3}
  99. a , b , c , 𝐧 , 𝐦 , 𝐤 a,b,c,\mathbf{n,m,k}
  100. 2 × 2 2×2
  101. S U ( 2 ) SU(2)
  102. ( 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 ) . \quad\left(\begin{smallmatrix}1&0&0&0\\ 0&0&1&0\\ 0&1&0&0\\ 0&0&0&1\end{smallmatrix}\right)~{}.

Pea.html

  1. N 2 + 8 H + + 8 e - 2 N H 3 + H 2 N_{2}+8H^{+}+8e^{-}\to 2NH_{3}+H_{2}
  2. N H 3 + H + N H 4 + NH_{3}+H^{+}\to NH_{4}^{+}

Peano_axioms.html

  1. . \mathbb{N}.
  2. a + 0 \displaystyle a+0
  3. a 0 \displaystyle a\cdot 0
  4. a , b , c N a,b,c\in N
  5. y ¯ ( φ ( 0 , y ¯ ) x ( φ ( x , y ¯ ) φ ( S ( x ) , y ¯ ) ) x φ ( x , y ¯ ) ) \forall\bar{y}(\varphi(0,\bar{y})\land\forall x(\varphi(x,\bar{y})\Rightarrow% \varphi(S(x),\bar{y}))\Rightarrow\forall x\varphi(x,\bar{y}))
  6. y ¯ \bar{y}
  7. x , y , z N \forall x,y,z\in N
  8. ( x + y ) + z = x + ( y + z ) (x+y)+z=x+(y+z)
  9. x , y N \forall x,y\in N
  10. x + y = y + x x+y=y+x
  11. x , y , z N \forall x,y,z\in N
  12. ( x y ) z = x ( y z ) (x\cdot y)\cdot z=x\cdot(y\cdot z)
  13. x , y N \forall x,y\in N
  14. x y = y x x\cdot y=y\cdot x
  15. x , y , z N \forall x,y,z\in N
  16. x ( y + z ) = ( x y ) + ( x z ) x\cdot(y+z)=(x\cdot y)+(x\cdot z)
  17. x N \forall x\in N
  18. x + 0 = x and x 0 = 0 x+0=x\and x\cdot 0=0
  19. x N \forall x\in N
  20. x 1 = x x\cdot 1=x
  21. x , y , z N \forall x,y,z\in N
  22. x < y and y < z x < z x<y\and y<z\Rightarrow x<z
  23. x N \forall x\in N
  24. ¬ ( x < x ) \neg(x<x)
  25. x , y N \forall x,y\in N
  26. x < y x = y y < x x<yx=yy<x
  27. x , y , z N \forall x,y,z\in N
  28. x < y x + z < y + z x<y\Rightarrow x+z<y+z
  29. x , y , z N \forall x,y,z\in N
  30. 0 < z and x < y x z < y z 0<z\and x<y\Rightarrow x\cdot z<y\cdot z
  31. x , y N x < y z N x + z = y \forall x,y\in Nx<y\Rightarrow\exists z\in Nx+z=y
  32. 0 < 1 and x N 0<1\and\forall x\in N
  33. x > 0 x 1 x>0\Rightarrow x\geq 1
  34. x N \forall x\in N
  35. x 0 x\geq 0
  36. f ( 0 A ) = 0 B f ( S A ( n ) ) = S B ( f ( n ) ) \begin{aligned}\displaystyle f(0_{A})&\displaystyle=0_{B}\\ \displaystyle f(S_{A}(n))&\displaystyle=S_{B}(f(n))\end{aligned}
  37. 0 \displaystyle 0
  38. u 0 = 0 X , u ( S x ) = S X ( u x ) . \begin{aligned}\displaystyle u0&\displaystyle=0_{X},\\ \displaystyle u(Sx)&\displaystyle=S_{X}(ux).\end{aligned}

Pearl_Index.html

  1. Pearl-Index = Number of Pregnancies 12 Number of Women Number of Months 100 \mbox{Pearl-Index}~{}=\frac{\mbox{Number of Pregnancies}~{}\cdot 12}{\mbox{% Number of Women}~{}\cdot\mbox{Number of Months}~{}}\cdot 100

Pell's_equation.html

  1. x 2 - n y 2 = 1 x^{2}-ny^{2}=1\,
  2. x 2 - 2 y 2 = 1 x^{2}-2y^{2}=1
  3. x 2 - 2 y 2 = - 1 x^{2}-2y^{2}=-1
  4. a 2 x 2 + c = y 2 , a^{2}x^{2}+c=y^{2},
  5. ( x 1 2 - N y 1 2 ) ( x 2 2 - N y 2 2 ) = ( x 1 x 2 + N y 1 y 2 ) 2 - N ( x 1 y 2 + x 2 y 1 ) 2 = ( x 1 x 2 - N y 1 y 2 ) 2 - N ( x 1 y 2 - x 2 y 1 ) 2 (x_{1}^{2}-Ny_{1}^{2})(x_{2}^{2}-Ny_{2}^{2})=(x_{1}x_{2}+Ny_{1}y_{2})^{2}-N(x_% {1}y_{2}+x_{2}y_{1})^{2}=(x_{1}x_{2}-Ny_{1}y_{2})^{2}-N(x_{1}y_{2}-x_{2}y_{1})% ^{2}
  6. ( x 1 , y 1 , k 1 ) (x_{1},y_{1},k_{1})
  7. ( x 2 , y 2 , k 2 ) (x_{2},y_{2},k_{2})
  8. x 2 - N y 2 = k x^{2}-Ny^{2}=k
  9. ( x 1 x 2 + N y 1 y 2 , x 1 y 2 + x 2 y 1 , k 1 k 2 ) (x_{1}x_{2}+Ny_{1}y_{2}\,,\,x_{1}y_{2}+x_{2}y_{1}\,,\,k_{1}k_{2})
  10. ( x 1 x 2 - N y 1 y 2 , x 1 y 2 - x 2 y 1 , k 1 k 2 ) . (x_{1}x_{2}-Ny_{1}y_{2}\,,\,x_{1}y_{2}-x_{2}y_{1}\,,\,k_{1}k_{2}).
  11. x 2 - N y 2 = 1 x^{2}-Ny^{2}=1
  12. k 1 k 2 k_{1}k_{2}
  13. N = 92 N=92
  14. ( 10 , 1 , 8 ) (10,1,8)
  15. 10 2 - 92 ( 1 2 ) = 8 10^{2}-92(1^{2})=8
  16. ( 192 , 20 , 64 ) (192,20,64)
  17. ( 24 , 5 / 2 , 1 ) (24,5/2,1)
  18. ( 1151 , 120 , 1 ) (1151,120,1)
  19. x 2 - N y 2 = k x^{2}-Ny^{2}=k
  20. ( a , b , k ) (a,b,k)
  21. a 2 - N b 2 = k a^{2}-Nb^{2}=k
  22. ( m , 1 , m 2 - N ) (m,1,m^{2}-N)
  23. ( a m + N b , a + b m , k ( m 2 - N ) ) (am+Nb,a+bm,k(m^{2}-N))
  24. ( a m + N b k , a + b m k , m 2 - N k ) . \left(\frac{am+Nb}{k}\,,\,\frac{a+bm}{k}\,,\,\frac{m^{2}-N}{k}\right).
  25. P + Q a , P+Q\sqrt{a},
  26. h i k i \tfrac{h_{i}}{k_{i}}
  27. n \sqrt{n}
  28. x k + y k n = ( x 1 + y 1 n ) k . x_{k}+y_{k}\sqrt{n}=(x_{1}+y_{1}\sqrt{n})^{k}.
  29. x k + 1 = x 1 x k + n y 1 y k , \displaystyle x_{k+1}=x_{1}x_{k}+ny_{1}y_{k},
  30. y k + 1 = x 1 y k + y 1 x k . \displaystyle y_{k+1}=x_{1}y_{k}+y_{1}x_{k}.
  31. x 2 - n y 2 = 1 x^{2}-ny^{2}=1
  32. ( x + y n ) ( x - y n ) = 1. (x+y\sqrt{n})(x-y\sqrt{n})=1.
  33. s s
  34. ( x - s ) k + n ( y - s ) k = 1. (x-s)^{k}+n\cdot(y-s)^{k}=1.
  35. x 1 + y 1 n = i = 1 t ( a i + b i n ) c i x_{1}+y_{1}\sqrt{n}=\prod_{i=1}^{t}(a_{i}+b_{i}\sqrt{n})^{c_{i}}
  36. x 1 + y 1 n = u 2329 , x_{1}+y_{1}\sqrt{n}=u^{2329},
  37. u = x 1 + y 1 4729494 = x 1 + y 1 609 7766 u=x^{\prime}_{1}+y^{\prime}_{1}\sqrt{4729494}\,=x^{\prime}_{1}+y^{\prime}_{1}% \sqrt{609\cdot 7766}\,
  38. x 1 \scriptstyle x^{\prime}_{1}
  39. y 1 \scriptstyle y^{\prime}_{1}
  40. u = ( 300426607914281713365 609 + 84129507677858393258 7766 ) 2 . u=(300426607914281713365\sqrt{609}+84129507677858393258\sqrt{7766})^{2}.
  41. x 2 - 410286423278424 y 2 = 1 x^{2}-410286423278424y^{2}=1
  42. 410286423278424 = 9314 4729494 \sqrt{410286423278424}=9314\sqrt{4729494}
  43. exp O ( log N log log N ) , \exp O(\sqrt{\log N\log\log N}),
  44. x 2 - 7 y 2 = 1 x^{2}-7y^{2}=1
  45. x 2 - 313 y 2 = 1 x^{2}-313y^{2}=1
  46. x 2 - n y 2 = 1 x^{2}-ny^{2}=1
  47. x 2 - n y 2 = 1 x^{2}-ny^{2}=1
  48. x 2 - n y 2 = ( x + y n ) ( x - y n ) x^{2}-ny^{2}=(x+y\sqrt{n})(x-y\sqrt{n})
  49. [ n ] \mathbb{Z}[\sqrt{n}]
  50. ( n ) \mathbb{Q}(\sqrt{n})
  51. ( x , y ) (x,y)
  52. x + y n x+y\sqrt{n}
  53. [ n ] \mathbb{Z}[\sqrt{n}]
  54. [ n ] \mathbb{Z}[\sqrt{n}]
  55. T i 2 - ( x 2 - 1 ) U i - 1 2 = 1. T_{i}^{2}-(x^{2}-1)U_{i-1}^{2}=1.\,
  56. T i + U i - 1 x 2 - 1 = ( x + x 2 - 1 ) i . T_{i}+U_{i-1}\sqrt{x^{2}-1}=(x+\sqrt{x^{2}-1})^{i}.\,
  57. x 2 - n y 2 = 1 x^{2}-ny^{2}=1
  58. n \sqrt{n}
  59. ( p q n q p ) \begin{pmatrix}p&q\\ nq&p\end{pmatrix}
  60. ( p k - 1 p k q k - 1 q k ) \begin{pmatrix}p_{k-1}&p_{k}\\ q_{k-1}&q_{k}\end{pmatrix}
  61. x 2 - n y 2 = - 1 x^{2}-ny^{2}=-1
  62. n \sqrt{n}
  63. ( x 2 - n y 2 ) 2 = ( - 1 ) 2 (x^{2}-ny^{2})^{2}=(-1)^{2}\,
  64. ( x 2 + n y 2 ) 2 - n ( 2 x y ) 2 = 1. (x^{2}+ny^{2})^{2}-n(2xy)^{2}=1.\,
  65. ( 2 x 2 + 1 ) 2 - n ( 2 x y ) 2 = 1 (2x^{2}+1)^{2}-n(2xy)^{2}=1\,
  66. u 2 - d v 2 = ± 2 u^{2}-dv^{2}=\pm 2\,
  67. ( u 2 - d v 2 ) 2 = ( ± 2 ) 2 (u^{2}-dv^{2})^{2}=(\pm 2)^{2}\,
  68. ( u 2 + d v 2 ) 2 - d ( 2 u v ) 2 = 4. (u^{2}+dv^{2})^{2}-d(2uv)^{2}=4.\,
  69. d v 2 = u 2 2 dv^{2}=u^{2}\mp 2
  70. ( 2 u 2 2 ) 2 - d ( 2 u v ) 2 = 4 (2u^{2}\mp 2)^{2}-d(2uv)^{2}=4\,
  71. ( u 2 1 ) 2 - d ( u v ) 2 = 1 (u^{2}\mp 1)^{2}-d(uv)^{2}=1\,
  72. u 2 - d v 2 = ± 4 u^{2}-dv^{2}=\pm 4\,

Pelton_wheel.html

  1. η s \eta_{s}
  2. n s = n P / ρ ( g H ) 5 / 4 n_{s}=n\sqrt{P}/\sqrt{\rho}(gH)^{5/4}
  3. n n
  4. P P
  5. H H
  6. ρ \rho

Pendulum.html

  1. T 2 π L g θ 0 1 ( 1 ) T\approx 2\pi\sqrt{\frac{L}{g}}\qquad\qquad\qquad\theta_{0}\ll 1\qquad(1)\,
  2. T = 2 π L g ( 1 + 1 16 θ 0 2 + 11 3072 θ 0 4 + ) T=2\pi\sqrt{L\over g}\left(1+\frac{1}{16}\theta_{0}^{2}+\frac{11}{3072}\theta_% {0}^{4}+\cdots\right)
  3. θ ( t ) = θ 0 cos ( 2 π T t + φ ) \theta(t)=\theta_{0}\cos\left(\frac{2\pi}{T}\,t+\varphi\right)\,
  4. φ \varphi
  5. L = I m R L=\frac{I}{mR}
  6. T = 2 π I m g R T=2\pi\sqrt{\frac{I}{mgR}}
  7. 2 L / 3 g \sqrt{2}{L}{/3}{g}
  8. 21 / 2 2{1}/{2}
  9. Q = M ω Γ Q=\frac{M\omega}{\Gamma}\,
  10. g L . g\propto L.\,
  11. 21 / 2 2{1}/{2}
  12. 11 / 4 1{1}/{4}
  13. 11 / 2 1{1}/{2}
  14. 1 Δ T = 1 T 1 - 1 T 2 \frac{1}{\Delta T}=\frac{1}{T_{1}}-\frac{1}{T_{2}}\,
  15. 3 / 4 {3}/{4}

Pendulum_clock.html

  1. T = 2 π L g T=2\pi\sqrt{\frac{L}{g}}\,

Pentium_FDIV_bug.html

  1. 4 , 195 , 835 3 , 145 , 727 = 1.333820449136241002 \textstyle\frac{4,195,835}{3,145,727}=1.333820449136241002
  2. 4 , 195 , 835 3 , 145 , 727 = 1.333 \color R e d 739068902037589 \textstyle\frac{4,195,835}{3,145,727}=1.333{\color{Red}{739068902037589}}

Peptide_bond.html

  1. C α \mathrm{C^{\alpha}}
  2. C δ \mathrm{C^{\delta}}
  3. C α - C - N - C α C^{\alpha}-C^{\prime}-N-C^{\alpha}
  4. ω \omega
  5. ω = 0 \omega=0^{\circ}
  6. ω = 180 \omega=180^{\circ}
  7. τ \tau\sim
  8. ω = ± 90 \omega=\pm 90^{\circ}

Percentage.html

  1. x x
  2. 100 + x 100+x
  3. 1 + 0.01 x 1+0.01x
  4. x x
  5. x x
  6. p p
  7. p ( ( 1 + 0.01 x ) ( 1 - 0.01 x ) ) = p ( 1 - ( 0.01 x ) 2 ) p((1+0.01x)(1-0.01x))=p(1-(0.01x)^{2})
  8. x x
  9. x x
  10. x = 10 x=10
  11. x x
  12. x x
  13. p ( ( 1 - 0.01 x ) ( 1 + 0.01 x ) ) = p ( 1 - ( 0.01 x ) 2 ) p((1-0.01x)(1+0.01x))=p(1-(0.01x)^{2})
  14. x x
  15. y y
  16. p p
  17. p ( ( 1 + 0.01 x ) ( 1 + 0.01 y ) ) p((1+0.01x)(1+0.01y))
  18. x = 10 x=10
  19. y = - 5 y=-5
  20. 100 rise run 100\frac{\,\text{rise}}{\,\text{run}}

Perfect_number.html

  1. p ( p + 1 ) / 2 p(p+1)/2
  2. p p
  3. 6 \displaystyle 6
  4. T 2 p - 1 = 1 + ( 2 p - 2 ) × ( 2 p + 1 ) 2 = 1 + 9 × T ( 2 p - 2 ) / 3 T_{2^{p}-1}=1+\frac{(2^{p}-2)\times(2^{p}+1)}{2}=1+9\times T_{(2^{p}-2)/3}
  5. N = q α p 1 2 e 1 p k 2 e k , N=q^{\alpha}p_{1}^{2e_{1}}\cdots p_{k}^{2e_{k}},
  6. σ 1 ( n ) = 2 n \sigma_{1}(n)=2n
  7. 1 / 6 + 1 / 3 + 1 / 2 + 1 / 1 = 2 1/6+1/3+1/2+1/1=2
  8. 1 / 28 + 1 / 14 + 1 / 7 + 1 / 4 + 1 / 2 + 1 / 1 = 2 1/28+1/14+1/7+1/4+1/2+1/1=2
  9. 2 n - 1 ( 2 n + 1 ) 2^{n-1}(2^{n}+1)
  10. 2 n + 1 2^{n}+1
  11. c n c\sqrt{n}
  12. o ( n ) o(\sqrt{n})
  13. 𝒮 \mathcal{S}

Perimeter.html

  1. 2 π r = π d 2\pi r=\pi d
  2. r r
  3. d d
  4. a + b + c a+b+c\,
  5. a a
  6. b b
  7. c c
  8. 4 a 4a
  9. a a
  10. 2 ( l + w ) 2(l+w)
  11. l l
  12. w w
  13. n × a n\times a\,
  14. n n
  15. a a
  16. 2 n b sin ( π n ) 2nb\sin\left(\frac{\pi}{n}\right)
  17. n n
  18. b b
  19. a 1 + a 2 + a 3 + + a n = i = 1 n a i a_{1}+a_{2}+a_{3}+\cdots+a_{n}=\sum_{i=1}^{n}a_{i}
  20. a i a_{i}
  21. i i
  22. 0 L d s \int_{0}^{L}\mathrm{d}s
  23. L L
  24. d s ds
  25. n n
  26. w w
  27. \ell
  28. 2 w + 2 2w+2\ell
  29. 2 n R sin ( 180 n ) . 2nR\sin\left(\frac{180^{\circ}}{n}\right).
  30. π \pi
  31. P = π D . P=\pi\cdot{D}.\!
  32. P = 2 π r . {P}={2}\pi\cdot{r}.\!
  33. π \pi
  34. π \pi
  35. π \pi
  36. π \pi
  37. 2 {}^{2}

Permittivity.html

  1. ε = ε r ε 0 = ( 1 + χ ) ε 0 \varepsilon=\varepsilon_{\,\text{r}}\varepsilon_{0}=(1+\chi)\varepsilon_{0}
  2. 𝐃 = ε 𝐄 \mathbf{D}=\varepsilon\mathbf{E}
  3. ε 0 = def 1 c 0 2 μ 0 = 1 35950207149.4727056 π F m 8.8541878176 × 10 - 12 F/m \varepsilon_{0}\stackrel{\mathrm{def}}{=}\ \frac{1}{c_{0}^{2}\mu_{0}}=\frac{1}% {35950207149.4727056\pi}\ \frac{\,\text{F}}{\,\text{m}}\approx 8.8541878176% \ldots\times 10^{-12}\ \,\text{F/m}
  4. ε = ε r ε 0 = ( 1 + χ ) ε 0 , \varepsilon=\varepsilon_{\,\text{r}}\varepsilon_{0}=(1+\chi)\varepsilon_{0},
  5. 𝐏 = ε 0 χ 𝐄 , \mathbf{P}=\varepsilon_{0}\chi\mathbf{E},
  6. χ = ε r - 1. \chi=\varepsilon_{\,\text{r}}-1.
  7. χ = 0. \chi=0.
  8. 𝐃 = ε 0 𝐄 + 𝐏 = ε 0 ( 1 + χ ) 𝐄 = ε r ε 0 𝐄 . \mathbf{D}=\varepsilon_{0}\mathbf{E}+\mathbf{P}=\varepsilon_{0}(1+\chi)\mathbf% {E}=\varepsilon_{\,\text{r}}\varepsilon_{0}\mathbf{E}.
  9. ε μ = 1 v 2 . \varepsilon\mu=\frac{1}{v^{2}}.
  10. 𝐏 ( t ) = ε 0 - t χ ( t - t ) 𝐄 ( t ) d t . \mathbf{P}(t)=\varepsilon_{0}\int_{-\infty}^{t}\chi(t-t^{\prime})\mathbf{E}(t^% {\prime})\,dt^{\prime}.
  11. ε ε ^ ( ω ) \varepsilon\rightarrow\widehat{\varepsilon}(\omega)
  12. D 0 e - i ω t = ε ^ ( ω ) E 0 e - i ω t , D_{0}e^{-i\omega t}=\widehat{\varepsilon}(\omega)E_{0}e^{-i\omega t},
  13. ε s = lim ω 0 ε ^ ( ω ) . \varepsilon_{\,\text{s}}=\lim_{\omega\rightarrow 0}\widehat{\varepsilon}(% \omega).
  14. ε ^ = D 0 E 0 = | ε | e i δ . \widehat{\varepsilon}=\frac{D_{0}}{E_{0}}=|\varepsilon|e^{i\delta}.
  15. ε ^ ( ω ) = ε ( ω ) + i ε ′′ ( ω ) = D 0 E 0 ( cos δ + i sin δ ) . \widehat{\varepsilon}(\omega)=\varepsilon^{\prime}(\omega)+i\varepsilon^{% \prime\prime}(\omega)=\frac{D_{0}}{E_{0}}\left(\cos\delta+i\sin\delta\right).
  16. ε ^ \widehat{\varepsilon}
  17. ε ( ω ) = 1 + 8 π 2 e 2 m 2 c , v W c , v ( E ) [ φ ( ω - E ) - φ ( ω + E ) ] d x . \varepsilon(\omega)=1+\frac{8\pi^{2}e^{2}}{m^{2}}\sum_{c,v}\int W_{c,v}(E)% \left[\varphi(\hbar\omega-E)-\varphi(\hbar\omega+E)\right]\,dx.
  18. 𝐃 ( ω ) = | ε 1 - i ε 2 0 i ε 2 ε 1 0 0 0 ε z | 𝐄 ( ω ) \begin{aligned}\displaystyle\mathbf{D}(\omega)&\displaystyle=\begin{vmatrix}% \varepsilon_{1}&-i\varepsilon_{2}&0\\ i\varepsilon_{2}&\varepsilon_{1}&0\\ 0&0&\varepsilon_{z}\\ \end{vmatrix}\mathbf{E}(\omega)\\ \end{aligned}
  19. ε 2 \varepsilon_{2}
  20. J tot = J c + J d = σ E - i ω ε E = - i ω ε ^ E J\text{tot}=J_{\,\text{c}}+J_{\,\text{d}}=\sigma E-i\omega\varepsilon^{\prime}% E=-i\omega\widehat{\varepsilon}E
  21. ε ^ \widehat{\varepsilon}
  22. ε ^ = ε + i σ ω \widehat{\varepsilon}=\varepsilon^{\prime}+i\frac{\sigma}{\omega}

Permutation.html

  1. n n
  2. n n
  3. n ! n!
  4. n n
  5. S S
  6. S S
  7. S S
  8. S S
  9. S S
  10. s s
  11. f ( s ) f(s)
  12. S S
  13. k k
  14. k k
  15. k k
  16. σ = ( 1 2 3 4 5 2 5 4 3 1 ) ; \sigma=\begin{pmatrix}1&2&3&4&5\\ 2&5&4&3&1\end{pmatrix};
  17. σ = ( 3 2 5 1 4 4 5 1 2 3 ) . \sigma=\begin{pmatrix}3&2&5&1&4\\ 4&5&1&2&3\end{pmatrix}.
  18. x 1 , x 2 , , x n x_{1},x_{2},\ldots,x_{n}
  19. σ = ( x 1 x 2 x 3 x n σ ( x 1 ) σ ( x 2 ) σ ( x 3 ) σ ( x n ) ) . \sigma=\begin{pmatrix}x_{1}&x_{2}&x_{3}&\cdots&x_{n}\\ \sigma(x_{1})&\sigma(x_{2})&\sigma(x_{3})&\cdots&\sigma(x_{n})\end{pmatrix}.
  20. σ ( x 1 ) σ ( x 2 ) σ ( x 3 ) σ ( x n ) \sigma(x_{1})\;\sigma(x_{2})\;\sigma(x_{3})\;\cdots\;\sigma(x_{n})
  21. k k
  22. n n
  23. P k n P^{n}_{k}
  24. P k n {}_{n}P_{k}
  25. P k n {}^{n}P_{k}
  26. P n , k P_{n,k}
  27. P ( n , k ) P(n,k)
  28. P ( n , k ) = n ( n - 1 ) ( n - 2 ) ( n - k + 1 ) k factors P(n,k)=\underbrace{n\cdot(n-1)\cdot(n-2)\cdots(n-k+1)}_{k\ \mathrm{factors}}
  29. n ! ( n - k ) ! . \frac{n!}{(n-k)!}.
  30. n n
  31. ( n ) k (n)_{k}
  32. k k
  33. n k ¯ n^{\underline{k}}
  34. n n
  35. C ( n , k ) = P ( n , k ) P ( k , k ) = n ! ( n - k ) ! k ! . C(n,k)=\frac{P(n,k)}{P(k,k)}=\frac{n!}{(n-k)!k!}.
  36. ( n k ) {\left({{n}\atop{k}}\right)}
  37. k n . k^{n}.
  38. m 1 m_{1}
  39. m 2 m_{2}
  40. m l m_{l}
  41. ( n m 1 , m 2 , , m l ) = n ! m 1 ! m 2 ! m l ! . {n\choose m_{1},m_{2},\ldots,m_{l}}=\frac{n!}{m_{1}!\,m_{2}!\,\cdots\,m_{l}!}.
  42. 11 ! 1 ! 4 ! 4 ! 2 ! \frac{11!}{1!4!4!2!}
  43. ( 1 2 3 4 5 2 5 4 3 1 ) = ( 1 2 5 ) ( 3 4 ) = ( 3 4 ) ( 1 2 5 ) = ( 3 4 ) ( 5 1 2 ) . \begin{pmatrix}1&2&3&4&5\\ 2&5&4&3&1\end{pmatrix}=\begin{pmatrix}1&2&5\end{pmatrix}\begin{pmatrix}3&4\end% {pmatrix}=\begin{pmatrix}3&4\end{pmatrix}\begin{pmatrix}1&2&5\end{pmatrix}=% \begin{pmatrix}3&4\end{pmatrix}\begin{pmatrix}5&1&2\end{pmatrix}.
  44. c ( n , k ) c(n,k)
  45. P = ( 1 2 3 4 5 2 4 1 3 5 ) and Q = ( 1 2 3 4 5 5 4 3 2 1 ) , P=\begin{pmatrix}1&2&3&4&5\\ 2&4&1&3&5\end{pmatrix}\quad\,\text{ and }\quad Q=\begin{pmatrix}1&2&3&4&5\\ 5&4&3&2&1\end{pmatrix},
  46. Q P = ( 1 2 3 4 5 5 4 3 2 1 ) ( 1 2 3 4 5 2 4 1 3 5 ) = ( 2 4 1 3 5 4 2 5 3 1 ) ( 1 2 3 4 5 2 4 1 3 5 ) = ( 1 2 3 4 5 4 2 5 3 1 ) . QP=\begin{pmatrix}1&2&3&4&5\\ 5&4&3&2&1\end{pmatrix}\begin{pmatrix}1&2&3&4&5\\ 2&4&1&3&5\end{pmatrix}=\begin{pmatrix}2&4&1&3&5\\ 4&2&5&3&1\end{pmatrix}\begin{pmatrix}1&2&3&4&5\\ 2&4&1&3&5\end{pmatrix}=\begin{pmatrix}1&2&3&4&5\\ 4&2&5&3&1\end{pmatrix}.
  47. Q P = ( 15 ) ( 24 ) ( 1243 ) = ( 1435 ) . Q\cdot P=(15)(24)\cdot(1243)=(1435).
  48. ( 1 2 3 n 1 2 3 n ) . \begin{pmatrix}1&2&3&\cdots&n\\ 1&2&3&\cdots&n\end{pmatrix}.
  49. ( 1 2 3 4 5 2 5 4 3 1 ) - 1 = ( 2 5 4 3 1 1 2 3 4 5 ) = ( 1 2 3 4 5 5 1 4 3 2 ) . \begin{pmatrix}1&2&3&4&5\\ 2&5&4&3&1\end{pmatrix}^{-1}=\begin{pmatrix}2&5&4&3&1\\ 1&2&3&4&5\end{pmatrix}=\begin{pmatrix}1&2&3&4&5\\ 5&1&4&3&2\end{pmatrix}.
  50. [ ( 125 ) ( 34 ) ] - 1 = ( 521 ) ( 43 ) = ( 152 ) ( 34 ) . [(125)(34)]^{-1}=(521)(43)=(152)(34).
  51. σ ( x 1 , , x n ) = ( x σ - 1 ( 1 ) , , x σ - 1 ( n ) ) . \sigma\cdot(x_{1},\ldots,x_{n})=(x_{\sigma^{-1}(1)},\ldots,x_{\sigma^{-1}(n)}).
  52. 1 i < n 1\leq i<n
  53. n k \textstyle\left\langle{n\atop k}\right\rangle
  54. n k \textstyle\left\langle{n\atop k}\right\rangle
  55. f f
  56. q = f ( p ) q=f(p)
  57. q = q 1 q 2 q n q=q_{1}q_{2}\cdots q_{n}
  58. q 1 q_{1}
  59. q 1 q_{1}
  60. j j
  61. q j > q 1 q_{j}>q_{1}
  62. q j q_{j}
  63. c ( n , k ) c(n,k)
  64. i < j i<j
  65. σ i > σ j \sigma_{i}>\sigma_{j}
  66. m = 1 n i = 0 m - 1 X i = 1 ( 1 + X ) ( 1 + X + X 2 ) ( 1 + X + X 2 + + X n - 1 ) , \prod_{m=1}^{n}\sum_{i=0}^{m-1}X^{i}=1(1+X)(1+X+X^{2})\cdots(1+X+X^{2}+\cdots+% X^{n-1}),
  67. σ = ( 6 , 3 , 8 , 1 , 4 , 9 , 7 , 2 , 5 ) \sigma=(6,3,8,1,4,9,7,2,5)

Permutation_group.html

  1. σ \sigma
  2. M = { x 1 , x 2 , , x n } M=\{x_{1},x_{2},\ldots,x_{n}\}
  3. σ = ( x 1 x 2 x 3 x n σ ( x 1 ) σ ( x 2 ) σ ( x 3 ) σ ( x n ) ) . \sigma=\begin{pmatrix}x_{1}&x_{2}&x_{3}&\cdots&x_{n}\\ \sigma(x_{1})&\sigma(x_{2})&\sigma(x_{3})&\cdots&\sigma(x_{n})\end{pmatrix}.
  4. σ = ( 1 2 3 4 5 2 5 4 3 1 ) ; \sigma=\begin{pmatrix}1&2&3&4&5\\ 2&5&4&3&1\end{pmatrix};
  5. σ = ( 3 2 5 1 4 4 5 1 2 3 ) . \sigma=\begin{pmatrix}3&2&5&1&4\\ 4&5&1&2&3\end{pmatrix}.
  6. σ = ( 125 ) ( 34 ) . \sigma=(125)(34).
  7. P = ( 1 2 3 4 5 2 4 1 3 5 ) and Q = ( 1 2 3 4 5 5 4 3 2 1 ) , P=\begin{pmatrix}1&2&3&4&5\\ 2&4&1&3&5\end{pmatrix}\quad\,\text{ and }\quad Q=\begin{pmatrix}1&2&3&4&5\\ 5&4&3&2&1\end{pmatrix},
  8. Q P = ( 1 2 3 4 5 5 4 3 2 1 ) ( 1 2 3 4 5 2 4 1 3 5 ) = ( 2 4 1 3 5 4 2 5 3 1 ) ( 1 2 3 4 5 2 4 1 3 5 ) = ( 1 2 3 4 5 4 2 5 3 1 ) . QP=\begin{pmatrix}1&2&3&4&5\\ 5&4&3&2&1\end{pmatrix}\begin{pmatrix}1&2&3&4&5\\ 2&4&1&3&5\end{pmatrix}=\begin{pmatrix}2&4&1&3&5\\ 4&2&5&3&1\end{pmatrix}\begin{pmatrix}1&2&3&4&5\\ 2&4&1&3&5\end{pmatrix}=\begin{pmatrix}1&2&3&4&5\\ 4&2&5&3&1\end{pmatrix}.
  9. Q P = ( 15 ) ( 24 ) ( 1243 ) = ( 1435 ) . Q\cdot P=(15)(24)\cdot(1243)=(1435).
  10. ( 1 2 3 n 1 2 3 n ) . \begin{pmatrix}1&2&3&\cdots&n\\ 1&2&3&\cdots&n\end{pmatrix}.
  11. ( 1 2 3 4 5 2 5 4 3 1 ) - 1 = ( 2 5 4 3 1 1 2 3 4 5 ) = ( 1 2 3 4 5 5 1 4 3 2 ) . \begin{pmatrix}1&2&3&4&5\\ 2&5&4&3&1\end{pmatrix}^{-1}=\begin{pmatrix}2&5&4&3&1\\ 1&2&3&4&5\end{pmatrix}=\begin{pmatrix}1&2&3&4&5\\ 5&1&4&3&2\end{pmatrix}.
  12. ( 125 ) - 1 = ( 521 ) = ( 152 ) . (125)^{-1}=(521)=(152).
  13. [ ( 125 ) ( 34 ) ] - 1 = ( 34 ) - 1 ( 125 ) - 1 = ( 43 ) ( 521 ) = ( 34 ) ( 152 ) . [(125)(34)]^{-1}=(34)^{-1}(125)^{-1}=(43)(521)=(34)(152).
  14. \mapsto
  15. \mapsto
  16. \mapsto

Peroxide.html

  1. 2 O 2 - + 2 H + SOD H 2 O 2 + O 2 \mathrm{2\ O_{2}^{-}\ +\ 2\ H^{+}\ \xrightarrow{SOD}\ \ H_{2}O_{2}+\ O_{2}}
  2. R - CH 2 - CH 2 - CO - SCoA + O 2 FAD R - CH = CH - CO - SCoA + H 2 O 2 \mathrm{R{-}CH_{2}{-}CH_{2}{-}CO{-}SCoA\ +\ O_{2}\ \xrightarrow{FAD}\ \ R{-}CH% {=}CH{-}CO{-}SCoA\ +\ H_{2}O_{2}}
  3. H 2 O 2 + R H 2 R + 2 H 2 O \mathrm{H}_{2}\mathrm{O}_{2}+\mathrm{R^{\prime}H}_{2}\rightarrow\mathrm{R^{% \prime}}+2\mathrm{H}_{2}\mathrm{O}
  4. H 2 O 2 CAT 1 2 O 2 + H 2 O \mathrm{H_{2}O_{2}\ \xrightarrow{CAT}\ \textstyle\frac{1}{2}O_{2}+H_{2}O}
  5. 2 HSO 4 - 2 H + + S 2 O 8 2 - + 2 e - E 0 = 2.123 V \mathrm{2\ HSO_{4}^{-}\ \xrightarrow{\ }\ 2\ H^{+}\ +\ S_{2}O_{8}^{2-}\ +\ 2\ % e^{-}\ \ \ \ E^{0}=2.123V}
  6. 2 SO 4 2 - S 2 O 8 2 - + 2 e - E 0 = 2.01 V \mathrm{2\ SO_{4}^{2-}\ \xrightarrow{\ }\ S_{2}O_{8}^{2-}\ +\ 2\ e^{-}\ \ \ \ % E^{0}=2.01V}
  7. H 2 S 2 O 8 + 2 H 2 O H 2 O 2 + 2 H 2 SO 4 \mathrm{H_{2}S_{2}O_{8}\ +\ 2\ H_{2}O\longrightarrow\ H_{2}O_{2}\ +\ 2\ H_{2}% SO_{4}}
  8. R - O - O - R 2 R - O \mathrm{R{-}O{-}O{-}R\longrightarrow\ 2\ R{-}O\cdot}
  9. 2 BaO + air 500 C 2 BaO 2 700 C 2 BaO + O 2 ( pure ) \mathrm{2\ BaO\ +\ air\ \xrightarrow{500^{\circ}C}\ \ 2\ BaO_{2}\ \xrightarrow% {700^{\circ}C}\ 2\ BaO\ +\ O_{2}\ (pure)}
  10. Na 2 O 2 + 2 H 2 O H 3 O + 2 NaOH + H 2 O 2 \mathrm{Na_{2}O_{2}\ +\ 2\ H_{2}O\ \xrightarrow{H_{3}O^{+}}\ \ 2\ NaOH\ +\ H_{% 2}O_{2}}
  11. 2 Na 2 O 2 + 2 H 2 O 4 NaOH + O 2 \mathrm{2\ Na_{2}O_{2}\ +2\ H_{2}O\longrightarrow\ 4\ NaOH\ +\ O_{2}}
  12. 2 Na 2 O 2 Δ T 2 Na 2 O + O 2 \mathrm{2\ Na_{2}O_{2}\ \xrightarrow{\Delta T}\ \ 2\ Na_{2}O\ +\ O_{2}}
  13. 2 Na 2 O 2 + 2 CO 2 2 Na 2 CO 3 + O 2 \mathrm{2\ Na_{2}O_{2}\ +\ 2\ CO_{2}\longrightarrow\ 2\ Na_{2}CO_{3}\ +\ O_{2}}
  14. R - COOH + H 2 O 2 R - COOOH + H 2 O \mathrm{R{-}COOH\ +\ H_{2}O_{2}\longrightarrow\ R{-}COOOH\ +\ H_{2}O}
  15. R - COCl + H 2 O 2 R - COOOH + HCl \mathrm{R{-}COCl\ +\ H_{2}O_{2}\longrightarrow\ R{-}COOOH\ +\ HCl}
  16. Ar - CHO + O 2 Ar - COOOH \mathrm{Ar{-}CHO\ +\ O_{2}\longrightarrow\ Ar{-}COOOH}
  17. Ar - COOOH + Ar - CHO 2 Ar - COOH \mathrm{Ar{-}COOOH\ +\ Ar{-}CHO\longrightarrow\ 2\ Ar{-}COOH}
  18. R 2 SO 4 + H 2 O 2 R - O - O - R + H 2 SO 4 \mathrm{R_{2}SO_{4}\ +\ H_{2}O_{2}\longrightarrow\ R{-}O{-}O{-}R\ +\ H_{2}SO_{% 4}}

Peroxisome.html

  1. RH 2 + O 2 R + H 2 O 2 \mathrm{RH}_{\mathrm{2}}+\mathrm{O}_{\mathrm{2}}\rightarrow\mathrm{R}+\mathrm{% H}_{2}\mathrm{O}_{2}
  2. H 2 O 2 + R H 2 R + 2 H 2 O \mathrm{H}_{2}\mathrm{O}_{2}+\mathrm{R^{\prime}H}_{2}\rightarrow\mathrm{R^{% \prime}}+2\mathrm{H}_{2}\mathrm{O}
  3. 2 H 2 O 2 2 H 2 O + O 2 2\mathrm{H}_{2}\mathrm{O}_{2}\rightarrow 2\mathrm{H}_{2}\mathrm{O}+\mathrm{O}_% {2}

Petroleum.html

  1. Q v = 12400 , - 2 , 100 d 2 Q_{v}=12400,-2,100d^{2}
  2. Q v Q_{v}
  3. K = 1.62 A P I [ 1 - 0.0003 ( t - 32 ) ] K=\frac{1.62}{API}[1-0.0003(t-32)]
  4. c = 1 d [ 0.388 + 0.00046 t ] c=\frac{1}{d}[0.388+0.00046t]
  5. c = 1 d [ 0.4024 + 0.00081 t ] c=\frac{1}{d}[0.4024+0.00081t]
  6. L = 1 d [ 110.9 - 0.09 t ] L=\frac{1}{d}[110.9-0.09t]
  7. L = 1 d [ 194.4 - 0.162 t ] L=\frac{1}{d}[194.4-0.162t]

Péclet_number.html

  1. Pe = advective transport rate diffusive transport rate \mathrm{Pe}=\dfrac{\mbox{advective transport rate}~{}}{\mbox{diffusive % transport rate}~{}}
  2. Pe L = L u D = Re L Sc \mathrm{Pe}_{L}=\frac{Lu}{D}=\mathrm{Re}_{L}\,\mathrm{Sc}
  3. Pe L = L u α = Re L Pr . \mathrm{Pe}_{L}=\frac{Lu}{\alpha}=\mathrm{Re}_{L}\,\mathrm{Pr}.
  4. α = k ρ c p \alpha=\frac{k}{\rho c_{p}}

PH.html

  1. pH = - log 10 ( a H + ) = log 10 ( 1 a H + ) \mathrm{pH}=-\log_{10}(a_{\textrm{H}^{+}})=\log_{10}\left(\frac{1}{a_{\textrm{% H}^{+}}}\right)
  2. E = E 0 + R T F ln ( a H + ) = E 0 - 2.303 R T F pH E=E^{0}+\frac{RT}{F}\ln(a_{\textrm{H}^{+}})=E^{0}-\frac{2.303RT}{F}\mathrm{pH}
  3. pH(X) = pH(S) + E S - E X z \,\text{pH(X)}=\,\text{pH(S)}+\frac{E\text{S}-E\text{X}}{z}
  4. 1 2.303 R T / F \frac{1}{2.303RT/F}
  5. E = E 0 + f 2.303 R T F log [ H ] + E=E^{0}+f\frac{2.303RT}{F}\log[\mbox{H}~{}^{+}]
  6. [ OH - ] = K W [ H + ] [\mathrm{OH}^{-}]=\frac{K_{W}}{[\mathrm{H}^{+}]}
  7. pOH = pK W - pH \mathrm{pOH}=\mathrm{pK_{W}}-\mathrm{pH}
  8. a H + = exp ( μ H + - μ H + R T ) a_{H^{+}}=\exp\left(\frac{\mu_{H^{+}}-\mu^{\ominus}_{H^{+}}}{RT}\right)
  9. CO 2 + H 2 O HCO 3 - + H + \mathrm{CO_{2}+H_{2}O\rightleftharpoons HCO_{3}^{-}+H^{+}}
  10. 2 H 2 O H 3 O + ( a q ) + O H - ( a q ) 2H_{2}O\rightleftharpoons H_{3}O^{+}(aq)+OH^{-}(aq)
  11. K w = [ H + ] [ O H - ] K_{w}=[H^{+}][OH^{-}]
  12. H A H + + A - HA\rightleftharpoons H^{+}+A^{-}
  13. H A + H + + A HA^{+}\rightleftharpoons H^{+}+A
  14. K a = [ H ] [ A ] [ H A ] K_{a}=\frac{[H][A]}{[HA]}
  15. C A = [ A ] + [ H A ] C_{A}=[A]+[HA]
  16. C H = [ H ] + [ H A ] C_{H}=[H]+[HA]
  17. [ H ] 2 + K a [ H ] - K a C a = 0 [H]^{2}+K_{a}[H]-K_{a}C_{a}=0
  18. K a = 10 - 4.19 = 6.46 × 10 - 5 \mathrm{K_{a}=10^{-4.19}=6.46\times 10^{-5}}
  19. [ H ] 2 + 6.46 × 10 - 5 [ H ] - 6.46 × 10 - 7 = 0 \mathrm{[H]^{2}+6.46\times 10^{-5}[H]-6.46\times 10^{-7}=0}
  20. [ H + ] = 7.74 × 10 - 4 \mathrm{[H^{+}]=7.74\times 10^{-4}}
  21. K w [ H + ] \frac{K_{w}}{[H^{+}]}
  22. C H = [ H ] + [ H A ] - K w [ H ] C_{H}=\frac{[H]+[HA]-K_{w}}{[H]}
  23. [ A p B q H r ] = β pqr [ A ] p [ B ] q [ H ] r \mathrm{[A_{p}B_{q}H_{r}]=\beta_{pqr}[A]^{p}[B]^{q}[H]^{r}}
  24. C A = [ A ] + Σ p β pqr [ A ] p [ B ] q [ H ] r \mathrm{C_{A}=[A]+\Sigma p\beta_{pqr}[A]^{p}[B]^{q}[H]^{r}}
  25. C B = [ B ] + Σ q β pqr [ A ] p [ B ] q [ H ] r \mathrm{C_{B}=[B]+\Sigma q\beta_{pqr}[A]^{p}[B]^{q}[H]^{r}}
  26. C H = [ H ] + Σ r β pqr [ A ] p [ B ] q [ H ] r - K w [ H ] - 1 \mathrm{C_{H}=[H]+\Sigma r\beta_{pqr}[A]^{p}[B]^{q}[H]^{r}-K_{w}[H]^{-1}}

Phase-locked_loop.html

  1. N N
  2. M M
  3. N / M N/M
  4. f 1 ( θ 1 ( t ) ) f_{1}(\theta_{1}(t))
  5. f 2 ( θ 2 ( t ) ) f_{2}(\theta_{2}(t))
  6. θ 1 ( t ) \theta_{1}(t)
  7. θ 2 ( t ) \theta_{2}(t)
  8. f 1 ( θ ) f_{1}(\theta)
  9. f 2 ( θ ) f_{2}(\theta)
  10. ϕ ( t ) \phi(t)
  11. ϕ ( t ) = f 1 ( θ 1 ( t ) ) f 2 ( θ 2 ( t ) ) \phi(t)=f_{1}(\theta_{1}(t))f_{2}(\theta_{2}(t))
  12. g ( t ) g(t)
  13. θ ˙ 2 ( t ) = ω 2 ( t ) = ω f r e e + g v g ( t ) \dot{\theta}_{2}(t)=\omega_{2}(t)=\omega_{free}+g_{v}g(t)\,
  14. g v g_{v}
  15. ω f r e e \omega_{free}
  16. x ˙ = A x + b ϕ ( t ) , g ( t ) = c * x , x ( 0 ) = x 0 , \begin{array}[]{rcl}\dot{x}&=&Ax+b\phi(t),\\ g(t)&=&c^{*}x,\end{array}\quad x(0)=x_{0},
  17. ϕ ( t ) \phi(t)
  18. g ( t ) g(t)
  19. A A
  20. n n
  21. n n
  22. x n , b n , c n , x\in\mathbb{C}^{n},\quad b\in\mathbb{R}^{n},\quad c\in\mathbb{C}^{n},\quad
  23. x 0 n x_{0}\in\mathbb{C}^{n}
  24. x ˙ = A x + b f 1 ( θ 1 ( t ) ) f 2 ( θ 2 ( t ) ) , θ ˙ 2 = ω f r e e + g v ( c * x ) x ( 0 ) = x 0 , θ 2 ( 0 ) = θ 0 . \begin{array}[]{rcl}\dot{x}&=&Ax+bf_{1}(\theta_{1}(t))f_{2}(\theta_{2}(t)),\\ \dot{\theta}_{2}&=&\omega_{free}+g_{v}(c^{*}x)\\ \end{array}\quad x(0)=x_{0},\quad\theta_{2}(0)=\theta_{0}.
  25. θ 0 \theta_{0}
  26. f 1 ( θ 1 ( t ) ) f_{1}(\theta_{1}(t))
  27. f 2 ( θ 2 ( t ) ) f_{2}(\theta_{2}(t))
  28. 2 π 2\pi
  29. f 1 ( θ ) f_{1}(\theta)
  30. f 2 ( θ ) f_{2}(\theta)
  31. φ ( θ ) \varphi(\theta)
  32. G ( t ) G(t)
  33. x ˙ = A x + b φ ( θ 1 ( t ) - θ 2 ( t ) ) , G ( t ) = c * x , x ( 0 ) = x 0 , \begin{array}[]{rcl}\dot{x}&=&Ax+b\varphi(\theta_{1}(t)-\theta_{2}(t)),\\ G(t)&=&c^{*}x,\end{array}\quad x(0)=x_{0},
  34. G ( t ) - g ( t ) G(t)-g(t)
  35. φ ( θ ) \varphi(\theta)
  36. θ Δ ( t ) \theta_{\Delta}(t)
  37. θ Δ = θ 1 ( t ) - θ 2 ( t ) . \theta_{\Delta}=\theta_{1}(t)-\theta_{2}(t).
  38. x ˙ = A x + b φ ( θ Δ ) , θ ˙ Δ = ω Δ - g v ( c * x ) . x ( 0 ) = x 0 , θ Δ ( 0 ) = θ 1 ( 0 ) - θ 2 ( 0 ) . \begin{array}[]{rcl}\dot{x}&=&Ax+b\varphi(\theta_{\Delta}),\\ \dot{\theta}_{\Delta}&=&\omega_{\Delta}-g_{v}(c^{*}x).\\ \end{array}\quad x(0)=x_{0},\quad\theta_{\Delta}(0)=\theta_{1}(0)-\theta_{2}(0).
  39. ω Δ = ω 1 - ω f r e e \omega_{\Delta}=\omega_{1}-\omega_{free}
  40. ω 1 \omega_{1}
  41. ω f r e e \omega_{free}
  42. f 1 ( θ 1 ( t ) ) = A 1 sin ( θ 1 ( t ) ) , f 2 ( θ 2 ( t ) ) = A 2 cos ( θ 2 ( t ) ) f_{1}(\theta_{1}(t))=A_{1}\sin(\theta_{1}(t)),\quad f_{2}(\theta_{2}(t))=A_{2}% \cos(\theta_{2}(t))
  43. x ˙ = - 1 R C x + 1 R C A 1 A 2 sin ( θ 1 ( t ) ) cos ( θ 2 ( t ) ) , θ ˙ 2 = ω f r e e + g v ( c * x ) \begin{aligned}\displaystyle\dot{x}&\displaystyle=-\frac{1}{RC}x+\frac{1}{RC}A% _{1}A_{2}\sin(\theta_{1}(t))\cos(\theta_{2}(t)),\\ \displaystyle\dot{\theta}_{2}&\displaystyle=\omega_{free}+g_{v}(c^{*}x)\end{aligned}
  44. φ ( θ 1 - θ 2 ) = A 1 A 2 2 sin ( θ 1 - θ 2 ) \varphi(\theta_{1}-\theta_{2})=\frac{A_{1}A_{2}}{2}\sin(\theta_{1}-\theta_{2})
  45. x ˙ = - 1 R C x + 1 R C A 1 A 2 2 sin ( θ Δ ) , θ ˙ Δ = ω Δ - g v ( c * x ) . \begin{aligned}\displaystyle\dot{x}&\displaystyle=-\frac{1}{RC}x+\frac{1}{RC}% \frac{A_{1}A_{2}}{2}\sin(\theta_{\Delta}),\\ \displaystyle\dot{\theta}_{\Delta}&\displaystyle=\omega_{\Delta}-g_{v}(c^{*}x)% .\end{aligned}
  46. x = θ ˙ 2 - ω 2 g v c * = ω 1 - θ ˙ Δ - ω 2 g v c * , x ˙ = θ ¨ 2 g v c * , θ 1 = ω 1 t + Ψ , θ Δ = θ 1 - θ 2 , θ ˙ Δ = θ ˙ 1 - θ ˙ 2 = ω 1 - θ ˙ 2 , 1 g v c * θ ¨ Δ - 1 g v c * R C θ ˙ Δ - A 1 A 2 2 R C sin θ Δ = ω 2 - ω 1 g v c * R C . \begin{aligned}\displaystyle x&\displaystyle=\frac{\dot{\theta}_{2}-\omega_{2}% }{g_{v}c^{*}}=\frac{\omega_{1}-\dot{\theta}_{\Delta}-\omega_{2}}{g_{v}c^{*}},% \\ \displaystyle\dot{x}&\displaystyle=\frac{\ddot{\theta}_{2}}{g_{v}c^{*}},\\ \displaystyle\theta_{1}&\displaystyle=\omega_{1}t+\Psi,\\ \displaystyle\theta_{\Delta}&\displaystyle=\theta_{1}-\theta_{2},\\ \displaystyle\dot{\theta}_{\Delta}&\displaystyle=\dot{\theta}_{1}-\dot{\theta}% _{2}=\omega_{1}-\dot{\theta}_{2},\\ &\displaystyle\frac{1}{g_{v}c^{*}}\ddot{\theta}_{\Delta}-\frac{1}{g_{v}c^{*}RC% }\dot{\theta}_{\Delta}-\frac{A_{1}A_{2}}{2RC}\sin\theta_{\Delta}=\frac{\omega_% {2}-\omega_{1}}{g_{v}c^{*}RC}.\end{aligned}
  47. θ o θ i = K p K v F ( s ) s + K p K v F ( s ) \frac{\theta_{o}}{\theta_{i}}=\frac{K_{p}K_{v}F(s)}{s+K_{p}K_{v}F(s)}
  48. θ o \theta_{o}
  49. θ i \theta_{i}
  50. K p K_{p}
  51. K v K_{v}
  52. F ( s ) F(s)
  53. F ( s ) = 1 1 + s R C F(s)=\frac{1}{1+sRC}
  54. θ o θ i = K p K v R C s 2 + s R C + K p K v R C \frac{\theta_{o}}{\theta_{i}}=\frac{\frac{K_{p}K_{v}}{RC}}{s^{2}+\frac{s}{RC}+% \frac{K_{p}K_{v}}{RC}}
  55. s 2 + 2 s ζ ω n + ω n 2 s^{2}+2s\zeta\omega_{n}+\omega_{n}^{2}
  56. ζ \zeta
  57. ω n \omega_{n}
  58. ω n = K p K v R C \omega_{n}=\sqrt{\frac{K_{p}K_{v}}{RC}}
  59. ζ = 1 2 K p K v R C \zeta=\frac{1}{2\sqrt{K_{p}K_{v}RC}}
  60. R C = 1 2 K p K v RC=\frac{1}{2K_{p}K_{v}}
  61. ω c = K p K v 2 \omega_{c}=K_{p}K_{v}\sqrt{2}
  62. F ( s ) = 1 + s C R 2 1 + s C ( R 1 + R 2 ) F(s)=\frac{1+sCR_{2}}{1+sC(R_{1}+R_{2})}
  63. τ 1 = C ( R 1 + R 2 ) \tau_{1}=C(R_{1}+R_{2})
  64. τ 2 = C R 2 \tau_{2}=CR_{2}
  65. ω n = K p K v τ 1 \omega_{n}=\sqrt{\frac{K_{p}K_{v}}{\tau_{1}}}
  66. ζ = 1 2 ω n τ 1 + ω n τ 2 2 \zeta=\frac{1}{2\omega_{n}\tau_{1}}+\frac{\omega_{n}\tau_{2}}{2}
  67. τ 1 = K p K v ω n 2 \tau_{1}=\frac{K_{p}K_{v}}{\omega_{n}^{2}}
  68. τ 2 = 2 ζ ω n - 1 K p K v \tau_{2}=\frac{2\zeta}{\omega_{n}}-\frac{1}{K_{p}K_{v}}

Phase-shift_keying.html

  1. E b E_{b}
  2. E s E_{s}
  3. n E b nE_{b}
  4. T b T_{b}
  5. T s T_{s}
  6. N 0 / 2 N_{0}/2
  7. P b P_{b}
  8. P s P_{s}
  9. Q ( x ) Q(x)
  10. x x
  11. Q ( x ) = 1 2 π x e - t 2 / 2 d t = 1 2 erfc ( x 2 ) , x 0 Q(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}e^{-t^{2}/2}dt=\frac{1}{2}\,% \operatorname{erfc}\left(\frac{x}{\sqrt{2}}\right),\ x\geq{}0
  12. π / 4 \pi/4
  13. s n ( t ) = 2 E b T b cos ( 2 π f c t + π ( 1 - n ) ) , n = 0 , 1. s_{n}(t)=\sqrt{\frac{2E_{b}}{T_{b}}}\cos(2\pi f_{c}t+\pi(1-n)),n=0,1.
  14. s 0 ( t ) = 2 E b T b cos ( 2 π f c t + π ) = - 2 E b T b cos ( 2 π f c t ) s_{0}(t)=\sqrt{\frac{2E_{b}}{T_{b}}}\cos(2\pi f_{c}t+\pi)=-\sqrt{\frac{2E_{b}}% {T_{b}}}\cos(2\pi f_{c}t)
  15. s 1 ( t ) = 2 E b T b cos ( 2 π f c t ) s_{1}(t)=\sqrt{\frac{2E_{b}}{T_{b}}}\cos(2\pi f_{c}t)
  16. ϕ ( t ) = 2 T b cos ( 2 π f c t ) \phi(t)=\sqrt{\frac{2}{T_{b}}}\cos(2\pi f_{c}t)
  17. E b ϕ ( t ) \sqrt{E_{b}}\phi(t)
  18. - E b ϕ ( t ) -\sqrt{E_{b}}\phi(t)
  19. P b = Q ( 2 E b N 0 ) P_{b}=Q\left(\sqrt{\frac{2E_{b}}{N_{0}}}\right)
  20. P e = 1 2 erfc ( E b N 0 ) P_{e}=\frac{1}{2}\operatorname{erfc}\left(\sqrt{\frac{E_{b}}{N_{0}}}\right)
  21. s n ( t ) = 2 E s T s cos ( 2 π f c t + ( 2 n - 1 ) π 4 ) , n = 0 , 1 , 2 , 3. s_{n}(t)=\sqrt{\frac{2E_{s}}{T_{s}}}\cos\left(2\pi f_{c}t+(2n-1)\frac{\pi}{4}% \right),\quad n=0,1,2,3.
  22. ϕ 1 ( t ) = 2 T s cos ( 2 π f c t ) \phi_{1}(t)=\sqrt{\frac{2}{T_{s}}}\cos(2\pi f_{c}t)
  23. ϕ 2 ( t ) = 2 T s sin ( 2 π f c t ) \phi_{2}(t)=\sqrt{\frac{2}{T_{s}}}\sin(2\pi f_{c}t)
  24. ( ± E s / 2 , ± E s / 2 ) . \left(\pm\sqrt{E_{s}/2},\pm\sqrt{E_{s}/2}\right).
  25. P b = Q ( 2 E b N 0 ) . P_{b}=Q\left(\sqrt{\frac{2E_{b}}{N_{0}}}\right).
  26. P s \,\!P_{s}
  27. = 1 - ( 1 - P b ) 2 =1-\left(1-P_{b}\right)^{2}
  28. = 2 Q ( E s N 0 ) - [ Q ( E s N 0 ) ] 2 =2Q\left(\sqrt{\frac{E_{s}}{N_{0}}}\right)-\left[Q\left(\sqrt{\frac{E_{s}}{N_{% 0}}}\right)\right]^{2}
  29. P s 2 Q ( E s N 0 ) P_{s}\approx 2Q\left(\sqrt{\frac{E_{s}}{N_{0}}}\right)
  30. π / 4 \pi/4
  31. π / 4 \pi/4
  32. π / 4 \pi/4
  33. M M
  34. M > 4 M>4
  35. P s = 1 - - π M π M p θ r ( θ r ) d θ r P_{s}=1-\int_{-\frac{\pi}{M}}^{\frac{\pi}{M}}p_{\theta_{r}}\left(\theta_{r}% \right)d\theta_{r}
  36. p θ r ( θ r ) = 1 2 π e - 2 γ s sin 2 θ r 0 V e - ( V - 4 γ s cos θ r ) 2 / 2 d V p_{\theta_{r}}\left(\theta_{r}\right)=\frac{1}{2\pi}e^{-2\gamma_{s}\sin^{2}% \theta_{r}}\int_{0}^{\infty}Ve^{-\left(V-\sqrt{4\gamma_{s}}\cos\theta_{r}% \right)^{2}/2}dV
  37. V = r 1 2 + r 2 2 V=\sqrt{r_{1}^{2}+r_{2}^{2}}
  38. θ r = tan - 1 ( r 2 / r 1 ) \theta_{r}=\tan^{-1}\left(r_{2}/r_{1}\right)
  39. γ s = E s N 0 \gamma_{s}=\frac{E_{s}}{N_{0}}
  40. r 1 N ( E s , N 0 / 2 ) r_{1}\sim{}N\left(\sqrt{E_{s}},N_{0}/2\right)
  41. r 2 N ( 0 , N 0 / 2 ) r_{2}\sim{}N\left(0,N_{0}/2\right)
  42. M M
  43. E b / N 0 E_{b}/N_{0}
  44. P s 2 Q ( 2 γ s sin π M ) P_{s}\approx 2Q\left(\sqrt{2\gamma_{s}}\sin\frac{\pi}{M}\right)
  45. M M
  46. P b 1 k P s P_{b}\approx\frac{1}{k}P_{s}
  47. M M
  48. t = 0 t=0
  49. M M
  50. E b / N 0 E_{b}/N_{0}
  51. k k
  52. r k r_{k}
  53. ϕ k \phi_{k}
  54. n k n_{k}
  55. r k = E s e j ϕ k + n k r_{k}=\sqrt{E_{s}}e^{j\phi_{k}}+n_{k}
  56. k - 1 k-1
  57. k k
  58. r k r_{k}
  59. r k - 1 r_{k-1}
  60. r k r_{k}
  61. r k - 1 r_{k-1}
  62. r k r k - 1 * = E s e j ( θ k - θ k - 1 ) + E s e j θ k n k - 1 * + E s e - j θ k - 1 n k + n k n k - 1 * r_{k}r_{k-1}^{*}=E_{s}e^{j\left(\theta_{k}-\theta_{k-1}\right)}+\sqrt{E_{s}}e^% {j\theta_{k}}n_{k-1}^{*}+\sqrt{E_{s}}e^{-j\theta_{k-1}}n_{k}+n_{k}n_{k-1}^{*}
  63. θ k - θ k - 1 \theta_{k}-\theta_{k-1}
  64. P b = 1 2 e - E b / N 0 , P_{b}=\frac{1}{2}e^{-E_{b}/N_{0}},
  65. E b / N 0 E_{b}/N_{0}
  66. k th k^{\textrm{th}}
  67. b k b_{k}
  68. e k e_{k}
  69. m k ( t ) m_{k}(t)
  70. e k = e k - 1 b k \,e_{k}=e_{k-1}\oplus{}b_{k}
  71. \oplus{}
  72. e k e_{k}
  73. b k b_{k}
  74. e k = e_{k}=
  75. b k = e k e k - 1 \,b_{k}=e_{k}\oplus{}e_{k-1}
  76. b k = 1 b_{k}=1
  77. e k e_{k}
  78. e k - 1 e_{k-1}
  79. b k = 0 b_{k}=0
  80. e k e_{k}
  81. e k - 1 e_{k-1}
  82. b k b_{k}
  83. E b / N 0 E_{b}/N_{0}
  84. E b / N 0 E_{b}/N_{0}