wpmath0000004_11

Orthographic_projection_in_cartography.html

  1. x = R cos φ sin ( λ - λ 0 ) y = R [ cos φ 0 sin φ - sin φ 0 cos φ cos ( λ - λ 0 ) ] \begin{aligned}\displaystyle x&\displaystyle=R\,\cos\varphi\sin\left(\lambda-% \lambda_{0}\right)\\ \displaystyle y&\displaystyle=R\big[\cos\varphi_{0}\sin\varphi-\sin\varphi_{0}% \cos\varphi\cos\left(\lambda-\lambda_{0}\right)\big]\end{aligned}
  2. c c
  3. cos c = sin φ 0 sin φ + cos φ 0 cos φ cos ( λ - λ 0 ) \cos c=\sin\varphi_{0}\sin\varphi+\cos\varphi_{0}\cos\varphi\cos\left(\lambda-% \lambda_{0}\right)\,
  4. cos ( c ) \cos(c)
  5. φ \displaystyle\varphi
  6. ρ = x 2 + y 2 c = arcsin ( ρ R ) \begin{aligned}\displaystyle\rho&\displaystyle=\sqrt{x^{2}+y^{2}}\\ \displaystyle c&\displaystyle=\arcsin\left(\frac{\rho}{R}\right)\end{aligned}

Output_impedance.html

  1. D F = Z load Z source DF=\frac{Z_{\mathrm{load}}}{Z_{\mathrm{source}}}
  2. Z source = Z load D F Z_{\mathrm{source}}=\frac{Z_{\mathrm{load}}}{DF}
  3. R B = ( V s I ) - R L R_{B}=\left(\frac{Vs}{I}\right)-R_{L}
  4. R B = ( V S - V I ) R_{B}=\left(\frac{V_{S}-V}{I}\right)
  5. R B R_{B}
  6. V S V_{S}
  7. V V
  8. R L R_{L}
  9. I I

Overspill.html

  1. ϵ + , δ + , | h | δ | f ( x + h ) - f ( x ) | ε \forall\epsilon\in\mathbb{R}^{+},\exists\delta\in\mathbb{R}^{+},|h|\leq\delta% \implies|f(x+h)-f(x)|\leq\varepsilon
  2. h 0 , | f ( x + h ) - f ( x ) | 0 \forall h\cong 0,\ |f(x+h)-f(x)|\cong 0
  3. positive δ 0 , ( | h | δ | f ( x + h ) - f ( x ) | < ε ) . \forall\mbox{ positive }~{}\delta\cong 0,\ (|h|\leq\delta\implies|f(x+h)-f(x)|% <\varepsilon).\,

Oxaloacetic_acid.html

  1. \overrightarrow{\leftarrow}
  2. \overrightarrow{\leftarrow}
  3. \overrightarrow{\leftarrow}
  4. \rightleftharpoons

P_(complexity).html

  1. { C n : n } \{C_{n}:n\in\mathbb{N}\}
  2. n n\in\mathbb{N}
  3. C n C_{n}
  4. C | x | ( x ) = 1 C_{|x|}(x)=1
  5. C | x | ( x ) = 0 C_{|x|}(x)=0
  6. O ( log n ) O(\log n)
  7. 2 O ( log n ) = n O ( 1 ) 2^{O(\log n)}=n^{O(1)}
  8. L AL = P NP PSPACE EXPTIME . \mbox{L}~{}\subseteq\mbox{AL}~{}=\mbox{P}~{}\subseteq\mbox{NP}~{}\subseteq% \mbox{PSPACE}~{}\subseteq\mbox{EXPTIME}~{}.

Page's_trend_test.html

  1. m 1 = m 2 = m 3 = = m n m_{1}=m_{2}=m_{3}=\cdots=m_{n}\,
  2. m 1 > m 2 > m 3 > > m n . m_{1}>m_{2}>m_{3}>\cdots>m_{n}.\,
  3. ( 12 L - 3 k n ( n + 1 ) 2 ) 2 k n 2 ( n 2 - 1 ) ( n + 1 ) {(12L-3kn(n+1)^{2})^{2}\over kn^{2}(n^{2}-1)(n+1)}

Page_replacement_algorithm.html

  1. h k h\leq k
  2. h k h\leq k
  3. h < k h<k
  4. h k h\leq k
  5. k k - h + 1 \dfrac{k}{k-h+1}
  6. k k - h + 1 \dfrac{k}{k-h+1}
  7. h k h\leq k
  8. k k - h + 1 \dfrac{k}{k-h+1}
  9. k k - h + 1 \dfrac{k}{k-h+1}
  10. k k - h + 1 \dfrac{k}{k-h+1}
  11. k k - h + 1 \dfrac{k}{k-h+1}
  12. k k - h + 1 \dfrac{k}{k-h+1}
  13. k k - h + 1 \dfrac{k}{k-h+1}

Pairing.html

  1. e : M × N L e:M\times N\to L
  2. e ( r m , n ) = e ( m , r n ) = r e ( m , n ) e(rm,n)=e(m,rn)=re(m,n)
  3. e ( m 1 + m 2 , n ) = e ( m 1 , n ) + e ( m 2 , n ) e(m_{1}+m_{2},n)=e(m_{1},n)+e(m_{2},n)
  4. e ( m , n 1 + n 2 ) = e ( m , n 1 ) + e ( m , n 2 ) e(m,n_{1}+n_{2})=e(m,n_{1})+e(m,n_{2})
  5. r R r\in R
  6. m , m 1 , m 2 M m,m_{1},m_{2}\in M
  7. n , n 1 , n 2 N n,n_{1},n_{2}\in N
  8. M R N L M\otimes_{R}N\to L
  9. M R N M\otimes_{R}N
  10. Φ : M Hom R ( N , L ) \Phi:M\to\operatorname{Hom}_{R}(N,L)
  11. Φ ( m ) ( n ) := e ( m , n ) \Phi(m)(n):=e(m,n)
  12. Φ \Phi
  13. N = M N=M
  14. e ( m , m ) = 0 e(m,m)=0
  15. e ( m , n ) = 0 e(m,n)=0
  16. m m
  17. n = 0 n=0
  18. k 2 × k 2 k k^{2}\times k^{2}\to k
  19. S 3 S 2 S^{3}\to S^{2}
  20. h : S 2 × S 2 S 2 h:S^{2}\times S^{2}\to S^{2}
  21. G 1 , G 2 \textstyle G_{1},G_{2}
  22. G T \textstyle G_{T}
  23. p \textstyle p
  24. P G 1 , Q G 2 \textstyle P\in G_{1},Q\in G_{2}
  25. G 1 \textstyle G_{1}
  26. G 2 \textstyle G_{2}
  27. e : G 1 × G 2 G T e:G_{1}\times G_{2}\rightarrow G_{T}
  28. a , b p * : e ( P a , Q b ) = e ( P , Q ) a b \textstyle\forall a,b\in\mathbb{Z}_{p}^{*}:\ e\left(P^{a},Q^{b}\right)=e\left(% P,Q\right)^{ab}
  29. e ( P , Q ) 1 \textstyle e\left(P,Q\right)\neq 1
  30. e \textstyle e
  31. G 1 = G 2 = G \textstyle G_{1}=G_{2}=G
  32. G \textstyle G
  33. e e
  34. P , Q G P,Q\in G
  35. e ( P , Q ) = e ( Q , P ) e(P,Q)=e(Q,P)
  36. g G g\in G
  37. p p
  38. q q
  39. P = g p P=g^{p}
  40. Q = g q Q=g^{q}
  41. e ( P , Q ) = e ( g p , g q ) = e ( g , g ) p q = e ( g q , g p ) = e ( Q , P ) e(P,Q)=e(g^{p},g^{q})=e(g,g)^{pq}=e(g^{q},g^{p})=e(Q,P)

Pairing_function.html

  1. π : × . \pi:\mathbb{N}\times\mathbb{N}\to\mathbb{N}.
  2. π : × \pi:\mathbb{N}\times\mathbb{N}\to\mathbb{N}
  3. π ( k 1 , k 2 ) := 1 2 ( k 1 + k 2 ) ( k 1 + k 2 + 1 ) + k 2 . \pi(k_{1},k_{2}):=\frac{1}{2}(k_{1}+k_{2})(k_{1}+k_{2}+1)+k_{2}.
  4. k 1 k_{1}
  5. k 2 k_{2}
  6. k 1 , k 2 . \langle k_{1},k_{2}\rangle\,.
  7. π ( n ) : n \pi^{(n)}:\mathbb{N}^{n}\to\mathbb{N}
  8. π ( n ) ( k 1 , , k n - 1 , k n ) := π ( π ( n - 1 ) ( k 1 , , k n - 1 ) , k n ) . \pi^{(n)}(k_{1},\ldots,k_{n-1},k_{n}):=\pi(\pi^{(n-1)}(k_{1},\ldots,k_{n-1}),k% _{n})\,.
  9. z z\in\mathbb{N}
  10. z = π ( x , y ) z=\pi(x,y)
  11. x , y x,y\in\mathbb{N}
  12. z = π ( x , y ) = ( x + y + 1 ) ( x + y ) 2 + y z=\pi(x,y)=\frac{(x+y+1)(x+y)}{2}+y
  13. π \pi
  14. w = x + y w=x+y\!
  15. t = w ( w + 1 ) 2 = w 2 + w 2 t=\frac{w(w+1)}{2}=\frac{w^{2}+w}{2}
  16. z = t + y z=t+y\!
  17. w 2 + w - 2 t = 0 w^{2}+w-2t=0\!
  18. w = 8 t + 1 - 1 2 w=\frac{\sqrt{8t+1}-1}{2}
  19. t z = t + y < t + ( w + 1 ) = ( w + 1 ) 2 + ( w + 1 ) 2 t\leq z=t+y<t+(w+1)=\frac{(w+1)^{2}+(w+1)}{2}
  20. w 8 z + 1 - 1 2 < w + 1 w\leq\frac{\sqrt{8z+1}-1}{2}<w+1
  21. w = 8 z + 1 - 1 2 w=\left\lfloor\frac{\sqrt{8z+1}-1}{2}\right\rfloor
  22. \left\lfloor\,\right\rfloor
  23. w = 8 z + 1 - 1 2 w=\left\lfloor\frac{\sqrt{8z+1}-1}{2}\right\rfloor
  24. t = w 2 + w 2 t=\frac{w^{2}+w}{2}
  25. y = z - t y=z-t\!
  26. x = w - y x=w-y\!

Pairwise_independence.html

  1. F X , Y ( x , y ) F_{X,Y}(x,y)
  2. F X , Y ( x , y ) = F X ( x ) F Y ( y ) , F_{X,Y}(x,y)=F_{X}(x)F_{Y}(y),
  3. f X , Y ( x , y ) f_{X,Y}(x,y)
  4. f X , Y ( x , y ) = f X ( x ) f Y ( y ) . f_{X,Y}(x,y)=f_{X}(x)f_{Y}(y).
  5. ( X , Y , Z ) = { ( 0 , 0 , 0 ) with probability 1 / 4 , ( 0 , 1 , 1 ) with probability 1 / 4 , ( 1 , 0 , 1 ) with probability 1 / 4 , ( 1 , 1 , 0 ) with probability 1 / 4. (X,Y,Z)=\left\{\begin{matrix}(0,0,0)&\,\text{with probability}\ 1/4,\\ (0,1,1)&\,\text{with probability}\ 1/4,\\ (1,0,1)&\,\text{with probability}\ 1/4,\\ (1,1,0)&\,\text{with probability}\ 1/4.\end{matrix}\right.
  6. f X ( 0 ) = f Y ( 0 ) = f Z ( 0 ) = 1 / 2 , f_{X}(0)=f_{Y}(0)=f_{Z}(0)=1/2,
  7. f X ( 1 ) = f Y ( 1 ) = f Z ( 1 ) = 1 / 2. f_{X}(1)=f_{Y}(1)=f_{Z}(1)=1/2.
  8. f X , Y = f X , Z = f Y , Z , f_{X,Y}=f_{X,Z}=f_{Y,Z},
  9. f X , Y ( 0 , 0 ) = f X , Y ( 0 , 1 ) = f X , Y ( 1 , 0 ) = f X , Y ( 1 , 1 ) = 1 / 4. f_{X,Y}(0,0)=f_{X,Y}(0,1)=f_{X,Y}(1,0)=f_{X,Y}(1,1)=1/4.
  10. f X , Y , Z ( x , y , z ) f X ( x ) f Y ( y ) f Z ( z ) f_{X,Y,Z}(x,y,z)\neq f_{X}(x)f_{Y}(y)f_{Z}(z)
  11. { X , Y , Z } \{X,Y,Z\}

Papyrus_46.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}
  5. 𝔓 \mathfrak{P}
  6. 𝔓 \mathfrak{P}
  7. 𝔓 \mathfrak{P}
  8. 𝔓 \mathfrak{P}
  9. 𝔓 \mathfrak{P}
  10. 𝔓 \mathfrak{P}
  11. 𝔓 \mathfrak{P}
  12. 𝔓 \mathfrak{P}
  13. 𝔓 \mathfrak{P}

PAQ.html

  1. ( n 0 , n 1 ) (n_{0},n_{1})
  2. n 0 n_{0}
  3. n 1 n_{1}
  4. ( n 0 , n 1 ) = ( 12 , 3 ) (n_{0},n_{1})=(12,3)
  5. w i w_{i}

Parabolic_fractal_distribution.html

  1. f ( n ; b , c ) n - b exp ( - c ( log n ) 2 ) , f(n;b,c)\propto n^{-b}\exp(-c(\log n)^{2}),

Parabolic_geometry.html

  1. E ( n ) = O ( n ) n E(n)=O(n)\ltimes\mathbb{R}^{n}

Parabolic_trajectory.html

  1. v v\,
  2. v = 2 μ r v=\sqrt{2\mu\over{r}}
  3. r r\,
  4. μ \mu\,
  5. v v\,
  6. v = 2 v o v=\sqrt{2}\cdot v_{o}
  7. v o v_{o}\,
  8. r = h 2 μ 1 1 + cos ν r={{h^{2}}\over{\mu}}{{1}\over{1+\cos\nu}}
  9. r r\,
  10. h h\,
  11. ν \nu\,
  12. μ \mu\,
  13. ϵ \epsilon\,
  14. ϵ = v 2 2 - μ r = 0 \epsilon={v^{2}\over 2}-{\mu\over{r}}=0
  15. v v\,
  16. r r\,
  17. μ \mu\,
  18. C 3 = 0 C_{3}=0
  19. t - T = 1 2 p 3 μ ( D + 1 3 D 3 ) t-T=\frac{1}{2}\sqrt{\frac{p^{3}}{\mu}}\left(D+\frac{1}{3}D^{3}\right)
  20. t f - t 0 = 1 2 p 3 μ ( D f + 1 3 D f 3 - D 0 - 1 3 D 0 3 ) t_{f}-t_{0}=\frac{1}{2}\sqrt{\frac{p^{3}}{\mu}}\left(D_{f}+\frac{1}{3}D_{f}^{3% }-D_{0}-\frac{1}{3}D_{0}^{3}\right)
  21. t - T = 2 r p 3 μ ( D + 1 3 D 3 ) t-T=\sqrt{\frac{2r_{p}^{3}}{\mu}}\left(D+\frac{1}{3}D^{3}\right)
  22. A = 3 2 μ 2 r p 3 ( t - T ) A=\frac{3}{2}\sqrt{\frac{\mu}{2r_{p}^{3}}}(t-T)
  23. B = A + A 2 + 1 3 B=\sqrt[3]{A+\sqrt{A^{2}+1}}
  24. ν = 2 arctan ( B - 1 / B ) \nu=2\arctan(B-1/B)
  25. r = ( 4.5 μ t 2 ) 1 / 3 r=(4.5\mu t^{2})^{1/3}\!\,
  26. t = 0 t=0\!\,
  27. t = 0 t=0\!\,
  28. t = 0 t=0\!\,

Parallel_(geometry).html

  1. \parallel
  2. A B C D AB\parallel CD
  3. θ \theta
  4. y = m x + b 1 y=mx+b_{1}\,
  5. y = m x + b 2 , y=mx+b_{2}\,,
  6. { y = m x + b 1 y = - x / m \begin{cases}y=mx+b_{1}\\ y=-x/m\end{cases}
  7. { y = m x + b 2 y = - x / m \begin{cases}y=mx+b_{2}\\ y=-x/m\end{cases}
  8. ( x 1 , y 1 ) = ( - b 1 m m 2 + 1 , b 1 m 2 + 1 ) \left(x_{1},y_{1}\right)\ =\left(\frac{-b_{1}m}{m^{2}+1},\frac{b_{1}}{m^{2}+1}% \right)\,
  9. ( x 2 , y 2 ) = ( - b 2 m m 2 + 1 , b 2 m 2 + 1 ) . \left(x_{2},y_{2}\right)\ =\left(\frac{-b_{2}m}{m^{2}+1},\frac{b_{2}}{m^{2}+1}% \right).
  10. d = ( b 1 m - b 2 m m 2 + 1 ) 2 + ( b 2 - b 1 m 2 + 1 ) 2 , d=\sqrt{\left(\frac{b_{1}m-b_{2}m}{m^{2}+1}\right)^{2}+\left(\frac{b_{2}-b_{1}% }{m^{2}+1}\right)^{2}}\,,
  11. d = | b 2 - b 1 | m 2 + 1 . d=\frac{|b_{2}-b_{1}|}{\sqrt{m^{2}+1}}\,.
  12. a x + b y + c 1 = 0 ax+by+c_{1}=0\,
  13. a x + b y + c 2 = 0 , ax+by+c_{2}=0,\,
  14. d = | c 2 - c 1 | a 2 + b 2 . d=\frac{|c_{2}-c_{1}|}{\sqrt{a^{2}+b^{2}}}.

Parallel_curve.html

  1. a a\,
  2. - a -a\,
  3. X [ x , y ] = x + a y x 2 + y 2 X[x,y]=x+\frac{ay^{\prime}}{\sqrt{x^{\prime 2}+y^{\prime 2}}}
  4. Y [ x , y ] = y - a x x 2 + y 2 . Y[x,y]=y-\frac{ax^{\prime}}{\sqrt{x^{\prime 2}+y^{\prime 2}}}.

Parameshvara.html

  1. R = ( a b + c d ) ( a c + b d ) ( a d + b c ) ( a + b + c - d ) ( b + c + d - a ) ( c + d + a - b ) ( d + a + b - c ) . R=\sqrt{\frac{(ab+cd)(ac+bd)(ad+bc)}{(a+b+c-d)(b+c+d-a)(c+d+a-b)(d+a+b-c)}}.

Parastatistics.html

  1. ϕ ( x ) = i = 1 p ϕ ( i ) ( x ) \phi(x)=\sum_{i=1}^{p}\phi^{(i)}(x)
  2. [ ϕ ( i ) ( x ) , ϕ ( i ) ( y ) ] = 0 [\phi^{(i)}(x),\phi^{(i)}(y)]=0
  3. { ϕ ( i ) ( x ) , ϕ ( j ) ( y ) } = 0 \{\phi^{(i)}(x),\phi^{(j)}(y)\}=0
  4. i j i\neq j
  5. ψ ( x ) = i = 1 p ψ ( i ) ( x ) \psi(x)=\sum_{i=1}^{p}\psi^{(i)}(x)
  6. { ψ ( i ) ( x ) , ψ ( i ) ( y ) } = 0 \{\psi^{(i)}(x),\psi^{(i)}(y)\}=0
  7. [ ψ ( i ) ( x ) , ψ ( j ) ( y ) ] = 0 [\psi^{(i)}(x),\psi^{(j)}(y)]=0
  8. i j i\neq j
  9. ϕ ( x 1 ) ϕ ( x n ) | Ω \phi(x_{1})\cdots\phi(x_{n})|\Omega\rangle
  10. ψ ( x 1 ) ψ ( x n ) | Ω \psi(x_{1})\cdots\psi(x_{n})|\Omega\rangle
  11. ϕ ( x π ( 1 ) ) ϕ ( x π ( n ) ) | Ω \phi(x_{\pi(1)})\cdots\phi(x_{\pi(n)})|\Omega\rangle
  12. ψ ( x π ( 1 ) ) ψ ( x π ( n ) ) | Ω \psi(x_{\pi(1)})\cdots\psi(x_{\pi(n)})|\Omega\rangle
  13. ( π ) \mathcal{E}(\pi)
  14. ( π ) [ ϕ ( x 1 ) ϕ ( x n ) | Ω ] = ϕ ( x π - 1 ( 1 ) ) ϕ ( x π - 1 ( n ) ) | Ω \mathcal{E}(\pi)\left[\phi(x_{1})\cdots\phi(x_{n})|\Omega\rangle\right]=\phi(x% _{\pi^{-1}(1)})\cdots\phi(x_{\pi^{-1}(n)})|\Omega\rangle
  15. ( π ) [ ψ ( x 1 ) ψ ( x n ) | Ω ] = ψ ( x π - 1 ( 1 ) ) ψ ( x π - 1 ( n ) ) | Ω \mathcal{E}(\pi)\left[\psi(x_{1})\cdots\psi(x_{n})|\Omega\rangle\right]=\psi(x% _{\pi^{-1}(1)})\cdots\psi(x_{\pi^{-1}(n)})|\Omega\rangle
  16. ( π ) \mathcal{E}(\pi)
  17. \mathcal{E}

Parseval's_theorem.html

  1. A ( x ) = n = - a n e i n x A(x)=\sum_{n=-\infty}^{\infty}a_{n}e^{inx}
  2. B ( x ) = n = - b n e i n x B(x)=\sum_{n=-\infty}^{\infty}b_{n}e^{inx}
  3. n = - a n b n ¯ = 1 2 π - π π A ( x ) B ( x ) ¯ d x , \sum_{n=-\infty}^{\infty}a_{n}\overline{b_{n}}=\frac{1}{2\pi}\int_{-\pi}^{\pi}% A(x)\overline{B(x)}\,dx,
  4. - | x ( t ) | 2 d t = - | X ( f ) | 2 d f \int_{-\infty}^{\infty}|x(t)|^{2}\,dt=\int_{-\infty}^{\infty}|X(f)|^{2}\,df
  5. X ( f ) = { x ( t ) } X(f)=\mathcal{F}\{x(t)\}
  6. n = - | x [ n ] | 2 = 1 2 π - π π | X ( e i ϕ ) | 2 d ϕ \sum_{n=-\infty}^{\infty}|x[n]|^{2}=\frac{1}{2\pi}\int_{-\pi}^{\pi}|X(e^{i\phi% })|^{2}d\phi
  7. n = 0 N - 1 | x [ n ] | 2 = 1 N k = 0 N - 1 | X [ k ] | 2 \sum_{n=0}^{N-1}|x[n]|^{2}=\frac{1}{N}\sum_{k=0}^{N-1}|X[k]|^{2}

Parsing_expression_grammar.html

  1. { a n b n : n 1 } \{a^{n}b^{n}:n\geq 1\}
  2. { a n b n c n : n 1 } \{a^{n}b^{n}c^{n}:n\geq 1\}
  3. O ( | V | 3 ) O(|V|^{3})
  4. O ( | V | * | E | ) O(|V|*|E|)

Partial_least_squares_regression.html

  1. X = T P + E X=TP^{\top}+E
  2. Y = U Q + F Y=UQ^{\top}+F
  3. X X
  4. n × m n\times m
  5. Y Y
  6. n × p n\times p
  7. T T
  8. U U
  9. n × l n\times l
  10. X X
  11. Y Y
  12. P P
  13. Q Q
  14. m × l m\times l
  15. p × l p\times l
  16. E E
  17. F F
  18. X X
  19. Y Y
  20. T T
  21. U U
  22. T , U , P T,U,P
  23. Q Q
  24. X X
  25. Y Y
  26. Y = X B ~ + B ~ 0 Y=X\tilde{B}+\tilde{B}_{0}
  27. Y Y
  28. Y Y
  29. T T
  30. Y Y
  31. T T
  32. X , y , l X,y,l
  33. X ( 0 ) X X^{(0)}\leftarrow X
  34. w ( 0 ) X T y / || X T y || w^{(0)}\leftarrow X^{T}y/||X^{T}y||
  35. w w
  36. t ( 0 ) X w ( 0 ) t^{(0)}\leftarrow Xw^{(0)}
  37. k k
  38. l l
  39. t k t ( k ) T t ( k ) t_{k}\leftarrow{t^{(k)}}^{T}t^{(k)}
  40. t ( k ) t ( k ) / t k t^{(k)}\leftarrow t^{(k)}/t_{k}
  41. p ( k ) X ( k ) T t ( k ) p^{(k)}\leftarrow{X^{(k)}}^{T}t^{(k)}
  42. q k y T t ( k ) q_{k}\leftarrow{y}^{T}t^{(k)}
  43. q k q_{k}
  44. l k l\leftarrow k
  45. k < l k<l
  46. X ( k + 1 ) X ( k ) - t k t ( k ) p ( k ) T X^{(k+1)}\leftarrow X^{(k)}-t_{k}t^{(k)}{p^{(k)}}^{T}
  47. w ( k + 1 ) X ( k + 1 ) T y w^{(k+1)}\leftarrow{X^{(k+1)}}^{T}y
  48. t ( k + 1 ) X ( k + 1 ) w ( k + 1 ) t^{(k+1)}\leftarrow X^{(k+1)}w^{(k+1)}
  49. W W
  50. w ( 0 ) , w ( 1 ) , , w ( l - 1 ) w^{(0)},w^{(1)},...,w^{(l-1)}
  51. P P
  52. q q
  53. B W ( P T W ) - 1 q B\leftarrow W{(P^{T}W)}^{-1}q
  54. B 0 q 0 - P ( 0 ) T B B_{0}\leftarrow q_{0}-{P^{(0)}}^{T}B
  55. B , B 0 B,B_{0}
  56. X X
  57. Y Y
  58. X X
  59. t k t ( k ) p ( k ) T t_{k}t^{(k)}{p^{(k)}}^{T}
  60. y y
  61. y y
  62. l l
  63. X X
  64. B B
  65. B 0 B_{0}

Partial_trace.html

  1. V V
  2. W W
  3. m m
  4. n n
  5. A A
  6. L ( A ) L(A)
  7. A A
  8. W W
  9. Tr W : L ( V W ) L ( V ) \operatorname{Tr}_{W}:\operatorname{L}(V\otimes W)\to\operatorname{L}(V)
  10. T L ( V W ) Tr W ( T ) L ( V ) T\in\operatorname{L}(V\otimes W)\mapsto\operatorname{Tr}_{W}(T)\in% \operatorname{L}(V)
  11. e 1 , , e m e_{1},\ldots,e_{m}
  12. f 1 , , f n f_{1},\ldots,f_{n}
  13. { a k , i j } 1 k , i m , 1 , j n \{a_{k\ell,ij}\}\quad 1\leq k,i\leq m,\quad 1\leq\ell,j\leq n
  14. e k f e_{k}\otimes f_{\ell}
  15. V W V\otimes W
  16. b k , i = j = 1 n a k j , i j . b_{k,i}=\sum_{j=1}^{n}a_{kj,ij}.
  17. Tr W : L ( V W ) L ( V ) \operatorname{Tr}_{W}:\operatorname{L}(V\otimes W)\rightarrow\operatorname{L}(V)
  18. Tr W ( R S ) = R Tr ( S ) R L ( V ) S L ( W ) . \operatorname{Tr}_{W}(R\otimes S)=R\,\operatorname{Tr}(S)\quad\forall R\in% \operatorname{L}(V)\quad\forall S\in\operatorname{L}(W).
  19. v 1 , , v m v_{1},\ldots,v_{m}
  20. V V
  21. w 1 , , w n w_{1},\ldots,w_{n}
  22. W W
  23. E i j : V V E_{ij}\colon V\to V
  24. v i v_{i}
  25. v j v_{j}
  26. F k l : W W F_{kl}\colon W\to W
  27. w k w_{k}
  28. w l w_{l}
  29. v i w k v_{i}\otimes w_{k}
  30. V W V\otimes W
  31. E i j F k l E_{ij}\otimes F_{kl}
  32. L ( V W ) \operatorname{L}(V\otimes W)
  33. Tr W ( I V W ) = dim W I V \operatorname{Tr}_{W}(I_{V\otimes W})=\dim W\ I_{V}
  34. Tr W ( T ( I V S ) ) = Tr W ( ( I V S ) T ) S L ( W ) T L ( V W ) . \operatorname{Tr}_{W}(T(I_{V}\otimes S))=\operatorname{Tr}_{W}((I_{V}\otimes S% )T)\quad\forall S\in\operatorname{L}(W)\quad\forall T\in\operatorname{L}(V% \otimes W).
  35. ( C , , I ) (C,\otimes,I)
  36. Tr X , Y U : Hom C ( X U , Y U ) Hom C ( X , Y ) \operatorname{Tr}^{U}_{X,Y}\colon\operatorname{Hom}_{C}(X\otimes U,Y\otimes U)% \to\operatorname{Hom}_{C}(X,Y)
  37. X + U Y + U X+U\cong Y+U
  38. X Y X\cong Y
  39. { f i } i I \{f_{i}\}_{i\in I}
  40. I ( V f ) V W \bigoplus_{\ell\in I}(V\otimes\mathbb{C}f_{\ell})\rightarrow V\otimes W
  41. T L ( V W ) T\in\operatorname{L}(V\otimes W)
  42. [ T 11 T 12 T 1 j T 21 T 22 T 2 j T k 1 T k 2 T k j ] , \begin{bmatrix}T_{11}&T_{12}&\ldots&T_{1j}&\ldots\\ T_{21}&T_{22}&\ldots&T_{2j}&\ldots\\ \vdots&\vdots&&\vdots\\ T_{k1}&T_{k2}&\ldots&T_{kj}&\ldots\\ \vdots&\vdots&&\vdots\end{bmatrix},
  43. T k L ( V ) T_{k\ell}\in\operatorname{L}(V)
  44. T \sum_{\ell}T_{\ell\ell}
  45. { | } \{|\ell\rangle\}_{\ell}
  46. Tr W ( k , T k | k | ) = j T j j . \operatorname{Tr}_{W}\left(\sum_{k,\ell}T_{k\ell}\,\otimes\,|k\rangle\langle% \ell|\right)=\sum_{j}T_{jj}.
  47. U ( W ) ( I V U * ) T ( I V U ) d μ ( U ) \int_{\operatorname{U}(W)}(I_{V}\otimes U^{*})T(I_{V}\otimes U)\ d\mu(U)
  48. I V S I_{V}\otimes S
  49. R I W R\otimes I_{W}
  50. H A H B H_{A}\otimes H_{B}
  51. H A H B . H_{A}\otimes H_{B}.
  52. ρ A \rho^{A}
  53. ρ A = Tr B ρ . \rho^{A}=\operatorname{Tr}_{B}\rho.
  54. M I M\otimes I
  55. ρ A \rho^{A}
  56. ρ A \rho^{A}
  57. M I M\otimes I
  58. Tr ( M ρ A ) = Tr ( M I ρ ) . \operatorname{Tr}(M\cdot\rho^{A})=\operatorname{Tr}(M\otimes I\cdot\rho).
  59. ρ A \rho^{A}
  60. Tr B : T ( H A H B ) T ( H A ) \operatorname{Tr}_{B}:T(H_{A}\otimes H_{B})\rightarrow T(H_{A})
  61. Tr B * \operatorname{Tr}_{B}^{*}
  62. H A \;H_{A}
  63. H A H B H_{A}\otimes H_{B}
  64. Tr B * ( A ) = A I . \operatorname{Tr}_{B}^{*}(A)=A\otimes I.
  65. Tr B * \operatorname{Tr}_{B}^{*}
  66. Tr B \operatorname{Tr}_{B}
  67. C ( X ) C ( Y ) = C ( X × Y ) . C(X)\otimes C(Y)=C(X\times Y).

Particle_image_velocimetry.html

  1. U x U_{x}
  2. V y V_{y}
  3. U y U_{y}
  4. V x V_{x}
  5. W x W_{x}
  6. W y W_{y}
  7. U z U_{z}
  8. V z V_{z}
  9. W z W_{z}
  10. [ U x U y U z V x V y V z W x W y W z ] \begin{bmatrix}U_{x}&U_{y}&U_{z}\\ V_{x}&V_{y}&V_{z}\\ W_{x}&W_{y}&W_{z}\\ \end{bmatrix}
  11. U z U_{z}
  12. V z V_{z}
  13. W z W_{z}

Partition_coefficient.html

  1. log P oct / wat = log ( [ solute ] octanol un - ionized [ solute ] water un - ionized ) \log\ P_{\rm oct/wat}=\log\Bigg(\frac{\big[\rm{solute}\big]_{\rm octanol}^{\rm un% -ionized}}{\big[\rm{solute}\big]_{\rm water}^{\rm un-ionized}}\Bigg)
  2. log P oct / wat I = log ( [ solute ] octanol I [ solute ] water I ) \log\ P_{\rm oct/wat}^{\rm I}=\log\Bigg(\frac{\big[\rm{solute}\big]_{\rm octanol% }^{\rm I}}{\big[\rm{solute}\big]_{\rm water}^{\rm I}}\Bigg)
  3. log D oct / wat = log ( [ solute ] octanol ionized + [ solute ] octanol un - ionized [ solute ] water ionized + [ solute ] water un - ionized ) \log\ D_{\rm oct/wat}=\log\Bigg(\frac{\big[\rm{solute}\big]_{\rm octanol}^{\rm ionized% }+\big[\rm{solute}\big]_{\rm octanol}^{\rm un-ionized}}{\big[\rm{solute}\big]_% {\rm water}^{\rm ionized}+\big[\rm{solute}\big]_{\rm water}^{\rm un-ionized}}\Bigg)
  4. log D oct / wat = log ( I = 0 M f I P oct / wat I ) \log\ D_{\rm oct/wat}=\log\Bigg(\sum_{I=0}^{M}f^{I}P_{\rm oct/wat}^{I}\Bigg)
  5. f I f^{I}
  6. f 0 f^{0}
  7. log D log P + log ( f 0 ) \log D\cong\log P+\log\left(f^{0}\right)
  8. log D acids log P + log [ 1 ( 1 + 10 p H - p K a ) ] \log D\text{acids}\cong\log P+\log\left[\frac{1}{(1+10^{pH-pK_{a}})}\right]
  9. log D bases log P + log [ 1 ( 1 + 10 p K a - p H ) ] \log D\text{bases}\cong\log P+\log\left[\frac{1}{(1+10^{pK_{a}-pH})}\right]
  10. for acids with ( p H - p K a ) > 1 , log D acids log P + p K a - p H \,\text{for acids with }\big(pH-pK_{a}\big)>1,\log D\text{acids}\cong\log P+pK% _{a}-pH
  11. for bases with ( p K a - p H ) > 1 , log D bases log P - p K a + p H \,\text{for bases with }\big(pK_{a}-pH\big)>1,\log D\text{bases}\cong\log P-pK% _{a}+pH
  12. log D log P \log D\cong\log P
  13. log P OW = log P octanol / water \log\ P_{\rm OW}=\log\ P_{\rm octanol/water}

Pascal's_pyramid.html

  1. ( 4 + 0 + 0 ) ! 4 ! × 0 ! × 0 ! ( 3 + 0 + 1 ) ! 3 ! × 0 ! × 1 ! ( 2 + 0 + 2 ) ! 2 ! × 0 ! × 2 ! ( 1 + 0 + 3 ) ! 1 ! × 0 ! × 3 ! ( 0 + 0 + 4 ) ! 0 ! × 0 ! × 4 ! \textstyle{(4+0+0)!\over 4!\times 0!\times 0!}\ {(3+0+1)!\over 3!\times 0!% \times 1!}\ {(2+0+2)!\over 2!\times 0!\times 2!}\ {(1+0+3)!\over 1!\times 0!% \times 3!}\ {(0+0+4)!\over 0!\times 0!\times 4!}
  2. ( 3 + 1 + 0 ) ! 3 ! × 1 ! × 0 ! ( 2 + 1 + 1 ) ! 2 ! × 1 ! × 1 ! ( 1 + 1 + 2 ) ! 1 ! × 1 ! × 2 ! ( 0 + 1 + 3 ) ! 0 ! × 1 ! × 3 ! \textstyle{(3+1+0)!\over 3!\times 1!\times 0!}\ {(2+1+1)!\over 2!\times 1!% \times 1!}\ {(1+1+2)!\over 1!\times 1!\times 2!}\ {(0+1+3)!\over 0!\times 1!% \times 3!}
  3. ( 2 + 2 + 0 ) ! 2 ! × 2 ! × 0 ! ( 1 + 2 + 1 ) ! 1 ! × 2 ! × 1 ! ( 0 + 2 + 2 ) ! 0 ! × 2 ! × 2 ! \textstyle{(2+2+0)!\over 2!\times 2!\times 0!}\ {(1+2+1)!\over 1!\times 2!% \times 1!}\ {(0+2+2)!\over 0!\times 2!\times 2!}
  4. ( 1 + 3 + 0 ) ! 1 ! × 3 ! × 0 ! ( 0 + 3 + 1 ) ! 0 ! × 3 ! × 1 ! \textstyle{(1+3+0)!\over 1!\times 3!\times 0!}\ {(0+3+1)!\over 0!\times 3!% \times 1!}
  5. ( 0 + 4 + 0 ) ! 0 ! × 4 ! × 0 ! \textstyle{(0+4+0)!\over 0!\times 4!\times 0!}
  6. 4 ! 4 ! × 0 ! × 0 ! 4 ! 3 ! × 0 ! × 1 ! 4 ! 2 ! × 0 ! × 2 ! 4 ! 1 ! × 0 ! × 3 ! 4 ! 0 ! × 0 ! × 4 ! \textstyle{4!\over 4!\times 0!\times 0!}\ {4!\over 3!\times 0!\times 1!}\ {4!% \over 2!\times 0!\times 2!}\ {4!\over 1!\times 0!\times 3!}\ {4!\over 0!\times 0% !\times 4!}
  7. 4 ! 3 ! × 1 ! × 0 ! 4 ! 2 ! × 1 ! × 1 ! 4 ! 1 ! × 1 ! × 2 ! 4 ! 0 ! × 1 ! × 3 ! \textstyle{4!\over 3!\times 1!\times 0!}\ {4!\over 2!\times 1!\times 1!}\ {4!% \over 1!\times 1!\times 2!}\ {4!\over 0!\times 1!\times 3!}
  8. 4 ! 2 ! × 2 ! × 0 ! 4 ! 1 ! × 2 ! × 1 ! 4 ! 0 ! × 2 ! × 2 ! \textstyle{4!\over 2!\times 2!\times 0!}\ {4!\over 1!\times 2!\times 1!}\ {4!% \over 0!\times 2!\times 2!}
  9. 4 ! 1 ! × 3 ! × 0 ! 4 ! 0 ! × 3 ! × 1 ! \textstyle{4!\over 1!\times 3!\times 0!}\ {4!\over 0!\times 3!\times 1!}
  10. 4 ! 0 ! × 4 ! × 0 ! \textstyle{4!\over 0!\times 4!\times 0!}
  11. ( 4 4 , 0 , 0 ) ( 4 3 , 0 , 1 ) ( 4 2 , 0 , 2 ) ( 4 1 , 0 , 3 ) ( 4 0 , 0 , 4 ) \textstyle{4\choose 4,0,0}\ {4\choose 3,0,1}\ {4\choose 2,0,2}\ {4\choose 1,0,% 3}\ {4\choose 0,0,4}
  12. ( 4 3 , 1 , 0 ) ( 4 2 , 1 , 1 ) ( 4 1 , 1 , 2 ) ( 4 0 , 1 , 3 ) \textstyle{4\choose 3,1,0}\ {4\choose 2,1,1}\ {4\choose 1,1,2}\ {4\choose 0,1,3}
  13. ( 4 2 , 2 , 0 ) ( 4 1 , 2 , 1 ) ( 4 0 , 2 , 2 ) \textstyle{4\choose 2,2,0}\ {4\choose 1,2,1}\ {4\choose 0,2,2}
  14. ( 4 1 , 3 , 0 ) ( 4 0 , 3 , 1 ) \textstyle{4\choose 1,3,0}\ {4\choose 0,3,1}
  15. ( 4 0 , 4 , 0 ) \textstyle{4\choose 0,4,0}
  16. x , y , z ( n x , y , z ) \textstyle\sum_{x,y,z}{n\choose x,y,z}
  17. ( b d ( n + 1 ) + b d + 1 ) n , \left(b^{d\left(n+1\right)}+b^{d}+1\right)^{n},
  18. d = 1 + log b ( n k 1 , k 2 , k 3 ) , i = 1 3 k i = n , n 3 k i n 3 , \textstyle d=1+\left\lfloor\log_{b}{n\choose k_{1},k_{2},k_{3}}\right\rfloor,% \ \sum_{i=1}^{3}{k_{i}}=n,\ \left\lfloor\frac{n}{3}\right\rfloor\leq k_{i}\leq% \left\lceil\frac{n}{3}\right\rceil,
  19. ( 10 12 + 10 2 + 1 ) 5 \textstyle\left(10^{12}+10^{2}+1\right)^{5}
  20. ( 10 189 + 10 9 + 1 ) 20 \textstyle\left(10^{189}+10^{9}+1\right)^{20}
  21. ( b d + 2 ) n , \left(b^{d}+2\right)^{n},
  22. ( b 2 d + b d + 1 ) n , \left(b^{2d}+b^{d}+1\right)^{n},

Pascal's_theorem.html

  1. ( A + B ) + C = D + C = Q = A + F = A + ( B + C ) (A+B)+C=D+C=Q=A+F=A+(B+C)
  2. G B ¯ G D ¯ × I D ¯ I F ¯ × H F ¯ H C ¯ × G C ¯ G A ¯ × I A ¯ I E ¯ × H E ¯ H B ¯ = 1 \frac{\overline{GB}}{\overline{GD}}\times\frac{\overline{ID}}{\overline{IF}}% \times\frac{\overline{HF}}{\overline{HC}}\times\frac{\overline{GC}}{\overline{% GA}}\times\frac{\overline{IA}}{\overline{IE}}\times\frac{\overline{HE}}{% \overline{HB}}=1

Path-ordering.html

  1. 𝒫 \mathcal{P}
  2. 𝒫 { O 1 ( σ 1 ) O 2 ( σ 2 ) O N ( σ N ) } O p 1 ( σ p 1 ) O p 2 ( σ p 2 ) O p N ( σ p N ) . \mathcal{P}\left\{O_{1}(\sigma_{1})O_{2}(\sigma_{2})\cdots O_{N}(\sigma_{N})% \right\}\equiv O_{p_{1}}(\sigma_{p_{1}})O_{p_{2}}(\sigma_{p_{2}})\cdots O_{p_{% N}}(\sigma_{p_{N}}).
  3. p : { 1 , 2 , , N } { 1 , 2 , , N } p\mathrel{:}\{1,2,\dots,N\}\to\{1,2,\dots,N\}
  4. σ p 1 σ p 2 σ p N . \sigma_{p_{1}}\leq\sigma_{p_{2}}\leq\cdots\leq\sigma_{p_{N}}.
  5. 𝒫 { O 1 ( 4 ) O 2 ( 2 ) O 3 ( 3 ) O 4 ( 1 ) } = O 4 ( 1 ) O 2 ( 2 ) O 3 ( 3 ) O 1 ( 4 ) . \mathcal{P}\left\{O_{1}(4)O_{2}(2)O_{3}(3)O_{4}(1)\right\}=O_{4}(1)O_{2}(2)O_{% 3}(3)O_{1}(4).
  6. 𝒯 \mathcal{T}
  7. 𝒯 { A ( x ) B ( y ) } := { A ( x ) B ( y ) if x 0 > y 0 , ± B ( y ) A ( x ) if x 0 < y 0 . \mathcal{T}\left\{A(x)B(y)\right\}:=\begin{cases}A(x)B(y)&\,\text{if }x_{0}>y_% {0},\\ \pm B(y)A(x)&\,\text{if }x_{0}<y_{0}.\end{cases}
  8. x 0 x_{0}
  9. y 0 y_{0}
  10. 𝒯 { A ( x ) B ( y ) } := θ ( x 0 - y 0 ) A ( x ) B ( y ) ± θ ( y 0 - x 0 ) B ( y ) A ( x ) , \mathcal{T}\left\{A(x)B(y)\right\}:=\theta(x_{0}-y_{0})A(x)B(y)\pm\theta(y_{0}% -x_{0})B(y)A(x),
  11. θ \theta
  12. ± \pm
  13. 𝒯 { A 1 ( t 1 ) A 2 ( t 2 ) A n ( t n ) } = p θ ( t p 1 > t p 2 > > t p n ) ε ( p ) A p 1 ( t p 1 ) A p 2 ( t p 2 ) A p n ( t p n ) \mathcal{T}\{A_{1}(t_{1})A_{2}(t_{2})\cdots A_{n}(t_{n})\}=\sum_{p}\theta(t_{p% _{1}}>t_{p_{2}}>\cdots>t_{p_{n}})\varepsilon(p)A_{p_{1}}(t_{p_{1}})A_{p_{2}}(t% _{p_{2}})\cdots A_{p_{n}}(t_{p_{n}})
  14. ε ( p ) { 1 for bosonic operators, sign of the permutation for fermionic operators. \varepsilon(p)\equiv\begin{cases}1&\,\text{for bosonic operators,}\\ \,\text{sign of the permutation}&\,\text{for fermionic operators.}\end{cases}
  15. exp h = lim N ( 1 + h N ) N . \exp h=\lim_{N\to\infty}\left(1+\frac{h}{N}\right)^{N}.
  16. S = ( 1 + h + 3 ) ( 1 + h + 2 ) ( 1 + h + 1 ) ( 1 + h 0 ) ( 1 + h - 1 ) ( 1 + h - 2 ) S=\cdots(1+h_{+3})(1+h_{+2})(1+h_{+1})(1+h_{0})(1+h_{-1})(1+h_{-2})\cdots
  17. 1 + h j 1+h_{j}
  18. [ j ε , ( j + 1 ) ε ] [j\varepsilon,(j+1)\varepsilon]
  19. ε 0 \varepsilon\to 0
  20. h j h_{j}
  21. h j = 1 i j ε ( j + 1 ) ε d t d 3 x H ( x , t ) . h_{j}=\frac{1}{i\hbar}\int_{j\varepsilon}^{(j+1)\varepsilon}dt\int d^{3}x\,H(% \vec{x},t).
  22. S = 𝒯 exp ( j = - h j ) = 𝒯 exp ( d t d 3 x H ( x , t ) i ) . S={\mathcal{T}}\exp\left(\sum_{j=-\infty}^{\infty}h_{j}\right)=\mathcal{T}\exp% \left(\int dt\,d^{3}x\,\frac{H(\vec{x},t)}{i\hbar}\right).
  23. 𝒯 \mathcal{T}
  24. 𝒯 \mathcal{T}

Path_(graph_theory).html

  1. k k
  2. v 0 , e 0 , v 1 , e 1 , v 2 , , v k - 1 , e k - 1 , v k v_{0},e_{0},v_{1},e_{1},v_{2},\ldots,v_{k-1},e_{k-1},v_{k}
  3. e i e_{i}
  4. v i v_{i}
  5. v i + 1 v_{i+1}
  6. v 0 v_{0}

Path_(topology).html

  1. f g ( s ) = { f ( 2 s ) 0 s 1 2 g ( 2 s - 1 ) 1 2 s 1. fg(s)=\begin{cases}f(2s)&0\leq s\leq\frac{1}{2}\\ g(2s-1)&\frac{1}{2}\leq s\leq 1.\end{cases}
  2. f g ( s ) = { f ( s ) 0 s | f | g ( s - | f | ) | f | s | f | + | g | fg(s)=\begin{cases}f(s)&0\leq s\leq|f|\\ g(s-|f|)&|f|\leq s\leq|f|+|g|\end{cases}

Patience_sorting.html

  1. 1 , 2 , , n 1,2,\ldots,n
  2. n = 52 n=52
  3. n n
  4. 1 , , n 1,\ldots,n
  5. O ( n log log n ) O(n\cdot\log\log n)
  6. O ( n ) O(n)
  7. O ( n log n ) O(n\log n)
  8. O ( log p ) O(\log p)
  9. p p
  10. O ( n log n ) O(n\log n)
  11. O ( n n ) O(n\sqrt{n})
  12. O ( log n ) O(\log n)
  13. O ( n log n ) O(n\log n)
  14. O ( n log n ) O(n\log n)
  15. O ( n ) O(n)
  16. O ( n log n ) O(n\log n)

Paul_Flory.html

  1. l i l_{i}
  2. θ i \theta_{i}
  3. ϕ i \phi_{i}

Paul_Milgrom.html

  1. x i {{x}_{i}}
  2. ϕ ( x i , x - i ) \phi({{x}_{i}},{{x}_{-i}})
  3. x i {{x}_{i}}
  4. x - i {{x}_{-i}}
  5. v i = E x - i { ϕ ( x i , x - i ) } {{v}_{i}}={{E}_{{{x}_{-i}}}}\{\phi({{x}_{i}},{{x}_{-i}})\}
  6. v 1 {{v}_{1}}
  7. v 2 {{v}_{2}}
  8. f ( v 1 , v 2 ) f({{v}_{1}},{{v}_{2}})
  9. f ( v 1 , v 2 ) f ( v 1 , v 2 ) f ( v 1 , v 2 ) f ( v 1 , v 2 ) f({{v}_{1}}^{\prime},{{v}_{2}}^{\prime})f({{v}_{1}},{{v}_{2}})\geq f({{v}_{1}}% ,{{v}_{2}}^{\prime})f({{v}_{1}}^{\prime},{{v}_{2}})
  10. v v
  11. v < v {v}^{\prime}<v
  12. f ( v 2 | v 1 ) f ( v 2 | v 1 ) f ( v 2 | v 1 ) f ( v 2 | v 1 ) f({{v}_{2}}^{\prime}|{{v}_{1}}^{\prime})f({{v}_{2}}|{{v}_{1}})\geq f({{v}_{2}}% |{{v}_{1}}^{\prime})f({{v}_{2}}^{\prime}|{{v}_{1}})
  13. v v
  14. v < v {v}^{\prime}<v
  15. v 2 {{v}_{2}}^{\prime}
  16. F ( v 2 | v 1 ) f ( v 2 | v 1 ) F ( v 2 | v 1 ) f ( v 2 | v 1 ) \frac{F({{v}_{2}}|{{v}_{1}}^{\prime})}{f({{v}_{2}}|{{v}_{1}}^{\prime})}\geq% \frac{F({{v}_{2}}|{{v}_{1}})}{f({{v}_{2}}|{{v}_{1}})}
  17. v 2 {{v}_{2}}
  18. v 1 < v 1 {{v}_{1}}^{\prime}<{{v}_{1}}
  19. V = { v 1 , , v n } V=\{{{v}_{1}},...,{{v}_{n}}\}
  20. f ( x , y ) f ( x , y ) f ( x , y ) f ( x , y ) f({x}^{\prime},{y}^{\prime})f(x,y)\geq f(x,{y}^{\prime})f({x}^{\prime},y)
  21. ( x , y ) (x,y)
  22. ( x , y ) ({x}^{\prime},{y}^{\prime})
  23. X × Y X\times Y
  24. ( x , y ) < ( x , y ) ({x}^{\prime},{y}^{\prime})<(x,y)
  25. x i {{x}_{i}}
  26. ϕ ( x i , x - i ) \phi({{x}_{i}},{{x}_{-i}})
  27. x i {{x}_{i}}
  28. x - i {{x}_{-i}}
  29. b i = B ( x i ) {{b}_{i}}=B({{x}_{i}})
  30. v i = ϕ ( x i ) {{v}_{i}}=\phi({{x}_{i}})
  31. e ( v ) = 0 v < m t p l > y f ( y | v ) d y F ( v | v ) e(v)=\frac{\int\limits_{0}^{v}<mtpl>{{}}yf(y|v)dy}{F(v|v)}
  32. B ( v 2 ) < B ( z ) B({{v}_{2}})<B(z)
  33. v 2 < z {{v}_{2}}<z
  34. w = F ( z | v ) w=F(z|v)
  35. U ( z ) = w ( v - B ( z ) ) = F ( z | v ) ( v - B ( z ) ) U(z)=w(v-B(z))=F(z|v)(v-B(z))
  36. U ( z ) U ( z ) = w ( z ) w ( z ) - B ( z ) v - B ( z ) = f ( z | v ) F ( z | v ) - B ( z ) v - B ( z ) \frac{{{U}^{\prime}}(z)}{U(z)}=\frac{{w}^{\prime}(z)}{w(z)}-\frac{{B}^{\prime}% (z)}{v-B(z)}=\frac{f(z|v)}{F(z|v)}-\frac{{B}^{\prime}(z)}{v-B(z)}
  37. B ( z ) B(z)
  38. B ( z + Δ z ) B(z+\Delta z)
  39. B ( v ) = f ( v | v ) F ( v | v ) ( v - B ( v ) ) {B}^{\prime}(v)=\frac{f(v|v)}{F(v|v)}(v-B(v))
  40. f ( v 2 | x ) f({{v}_{2}}|x)
  41. U ( z ) U ( z ) = f ( z | x ) F ( z | x ) - B ( z ) v - B ( z ) \frac{{{U}^{\prime}}(z)}{U(z)}=\frac{f(z|x)}{F(z|x)}-\frac{{B}^{\prime}(z)}{v-% B(z)}
  42. B x ( v ) {{B}_{x}}(v)
  43. B x ( v ) = f ( v | x ) F ( v | x ) ( v - B x ( v ) ) {{B}_{x}}^{\prime}(v)=\frac{f(v|x)}{F(v|x)}(v-{{B}_{x}}(v))
  44. B x ( x ) = e ( x ) {{B}_{x}}(x)=e(x)
  45. B ( x ) B x ( x ) B(x)\leq{{B}_{x}}(x)
  46. B ( v ) - B x ( v ) > 0 < m t p l > B ( v ) - B x ( v ) . B(v)-{{B}_{x}}(v)>{{0}_{<}mtpl>{{}}}{{\Rightarrow}}{B}^{\prime}(v)-{{B}_{x}}^{% \prime}(v).
  47. B ( x ) > B x ( x ) B(x)>{{B}_{x}}(x)
  48. ( i i ) < m t p l > B ( v ) - B x ( v ) > 0 , v [ y , x ] {{(ii)}_{<}mtpl>{{}}}B(v)-{{B}_{x}}(v)>0{{,}}\forall v\in[y,x]
  49. B ( v ) - B x ( v ) B(v)-{{B}_{x}}(v)
  50. B x ( x ) = e ( x ) {{B}_{x}}(x)=e(x)

Peccei–Quinn_theory.html

  1. a Tr [ F F ] a\mathrm{Tr}[F\wedge F]
  2. 1 / Λ 1/\Lambda
  3. Λ \Lambda
  4. Λ \Lambda

Pedal_curve.html

  1. cos α x + sin α y = p \cos\alpha x+\sin\alpha y=p
  2. x 2 a 2 + y 2 b 2 = 1 \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1
  3. x 0 x a 2 + y 0 y b 2 = 1 \frac{x_{0}x}{a^{2}}+\frac{y_{0}y}{b^{2}}=1
  4. x 0 a 2 = cos α p , y 0 b 2 = sin α p . \frac{x_{0}}{a^{2}}=\frac{\cos\alpha}{p},\,\frac{y_{0}}{b^{2}}=\frac{\sin% \alpha}{p}.
  5. a 2 cos 2 α + b 2 sin 2 α = p 2 , a^{2}\cos^{2}\alpha+b^{2}\sin^{2}\alpha=p^{2},\,
  6. a 2 cos 2 θ + b 2 sin 2 θ = r 2 , a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta=r^{2},\,
  7. a 2 x 2 + b 2 y 2 = ( x 2 + y 2 ) 2 . a^{2}x^{2}+b^{2}y^{2}=(x^{2}+y^{2})^{2}.\,
  8. r = d r d θ tan ψ . r=\frac{dr}{d\theta}\tan\psi.
  9. p = r sin ψ p=r\sin\psi
  10. α = θ + ψ - π 2 . \alpha=\theta+\psi-\frac{\pi}{2}.
  11. a cos θ = - a sin θ tan ψ a\cos\theta=-a\sin\theta\tan\psi
  12. tan ψ = - cot θ , ψ = π 2 + θ , α = 2 θ . \tan\psi=-\cot\theta,\,\psi=\frac{\pi}{2}+\theta,\alpha=2\theta.
  13. p = r sin ψ = r cos θ = a cos 2 θ = a cos 2 α 2 . p=r\sin\psi\ =r\cos\theta=a\cos^{2}\theta=a\cos^{2}{\alpha\over 2}.
  14. r = a cos 2 θ 2 . r=a\cos^{2}{\theta\over 2}.
  15. p r = q p . \frac{p}{r}=\frac{q}{p}.
  16. f ( r , r 2 p ) = 0 f(r,\frac{r^{2}}{p})=0
  17. v = P - R \vec{v}=P-R
  18. v = v + v \vec{v}=\vec{v}_{\parallel}+\vec{v}_{\perp}
  19. v \vec{v}
  20. v \vec{v}_{\parallel}
  21. t c ( t ) + c ( t ) ( P - c ( t ) ) | c ( t ) | 2 c ( t ) t\mapsto c(t)+{c^{\prime}(t)\cdot(P-c(t))\over|c^{\prime}(t)|^{2}}c^{\prime}(t)
  22. X [ x , y ] = ( x y - y x ) y x 2 + y 2 X[x,y]=\frac{(xy^{\prime}-yx^{\prime})y^{\prime}}{x^{\prime 2}+y^{\prime 2}}
  23. Y [ x , y ] = ( y x - x y ) x x 2 + y 2 . Y[x,y]=\frac{(yx^{\prime}-xy^{\prime})x^{\prime}}{x^{\prime 2}+y^{\prime 2}}.
  24. t P - c ( t ) ( P - c ( t ) ) | c ( t ) | 2 c ( t ) t\mapsto P-{c^{\prime}(t)\cdot(P-c(t))\over|c^{\prime}(t)|^{2}}c^{\prime}(t)
  25. x 2 a 2 ± y 2 b 2 = 1 \frac{x^{2}}{a^{2}}\pm\frac{y^{2}}{b^{2}}=1
  26. a 2 cos 2 θ ± b 2 sin 2 θ = r 2 {a^{2}}\cos^{2}\theta\pm{b^{2}}\sin^{2}\theta=r^{2}
  27. r n = a n cos n θ r^{n}=a^{n}\cos n\theta
  28. r n n + 1 = a n n + 1 cos n n + 1 θ r^{\tfrac{n}{n+1}}=a^{\tfrac{n}{n+1}}\cos\tfrac{n}{n+1}\theta

Pedigree_collapse.html

  1. 2 30 2^{30}

Peierls_bracket.html

  1. [ A , B ] [A,B]
  2. D A ( B ) - D B ( A ) D_{A}(B)-D_{B}(A)

Penman_equation.html

  1. E mass = m R n + ρ a c p ( δ e ) g a λ v ( m + γ ) E_{\mathrm{mass}}=\frac{mR_{n}+\rho_{a}c_{p}\left(\delta e\right)g_{a}}{% \lambda_{v}\left(m+\gamma\right)}
  2. E mass = m R n + γ * 6.43 ( 1 + 0.536 * U 2 ) δ e λ v ( m + γ ) E_{\mathrm{mass}}=\frac{mR_{n}+\gamma*6.43\left(1+0.536*U_{2}\right)\delta e}{% \lambda_{v}\left(m+\gamma\right)}
  3. 0.0016286 * P k P a λ v \frac{0.0016286*P_{kPa}}{\lambda_{v}}
  4. m = Δ = d e s d T a = 5336 T a 2 e ( 21.07 - 5336 T a ) m=\Delta=\frac{de_{s}}{dT_{a}}=\frac{5336}{T_{a}^{2}}e^{\left(21.07-\frac{5336% }{T_{a}}\right)}

Penrose_diagram.html

  1. ( x , t ) (x,t)
  2. ( u , v ) (u,v)
  3. tan ( u ± v ) = x ± t \tan(u\pm v)=x\pm t
  4. π / 2 \pi/2

Pentacontagon.html

  1. A = 25 2 t 2 cot π 50 A=\frac{25}{2}t^{2}\cot\frac{\pi}{50}
  2. r = 1 2 t cot π 50 r=\frac{1}{2}t\cot\frac{\pi}{50}
  3. R = 1 2 t csc π 50 R=\frac{1}{2}t\csc\frac{\pi}{50}

Pentagonal_cupola.html

  1. V = ( 1 6 ( 5 + 4 5 ) ) a 3 2.32405... a 3 V=\left(\frac{1}{6}\left(5+4\sqrt{5}\right)\right)a^{3}\approx 2.32405...a^{3}
  2. A = ( 1 4 ( 20 + 5 3 + 5 ( 145 + 62 5 ) ) ) a 2 = ( 1 4 ( 20 + 10 ( 80 + 31 5 + 15 ( 145 + 62 5 ) ) ) ) a 2 16.5797... a 2 A=\left(\frac{1}{4}\left(20+5\sqrt{3}+\sqrt{5(145+62\sqrt{5})}\right)\right)a^% {2}=\left(\frac{1}{4}\left(20+\sqrt{10\left(80+31\sqrt{5}+\sqrt{15(145+62\sqrt% {5})}\right)}\right)\right)a^{2}\approx 16.5797...a^{2}
  3. C = ( 1 2 11 + 4 5 ) a 2.23295... a C=\left(\frac{1}{2}\sqrt{11+4\sqrt{5}}\right)a\approx 2.23295...a

Pentagonal_gyrobicupola.html

  1. V = 1 3 ( 5 + 4 5 ) a 3 4.64809... a 3 V=\frac{1}{3}(5+4\sqrt{5})a^{3}\approx 4.64809...a^{3}
  2. A = ( 10 + 5 2 ( 10 + 5 + 75 + 30 5 ) ) a 2 17.7711... a 2 A=(10+\sqrt{\frac{5}{2}(10+\sqrt{5}+\sqrt{75+30\sqrt{5}})})a^{2}\approx 17.771% 1...a^{2}

Pentagonal_gyrocupolarotunda.html

  1. V = 5 12 ( 11 + 5 5 ) a 3 9.24181... a 3 V=\frac{5}{12}(11+5\sqrt{5})a^{3}\approx 9.24181...a^{3}
  2. A = ( 5 + 15 4 3 + 7 4 25 + 10 5 ) a 2 23.5385... a 2 A=(5+\frac{15}{4}\sqrt{3}+\frac{7}{4}\sqrt{25+10\sqrt{5}})a^{2}\approx 23.5385% ...a^{2}

Pentagonal_icositetrahedron.html

  1. cos - 1 ( 1 - t 2 ) \cos^{-1}\left(\frac{1-t}{2}\right)\approx
  2. cos - 1 ( 2 - t ) \cos^{-1}(2-t)\approx
  3. t + 1 2 1.42 \frac{t+1}{2}\approx 1.42
  4. 3 22 ( 5 t - 1 ) 4 t - 3 19.29994 \scriptstyle{3}\sqrt{\tfrac{22(5t-1)}{4t-3}}\scriptstyle{\approx 19.29994}
  5. 11 ( t - 4 ) 2 ( 20 t - 37 ) 7.4474 \sqrt{\tfrac{11(t-4)}{2(20t-37)}}\scriptstyle{\approx 7.4474}

Pentagonal_orthobicupola.html

  1. V = 1 3 ( 5 + 4 5 ) a 3 4.64809... a 3 V=\frac{1}{3}(5+4\sqrt{5})a^{3}\approx 4.64809...a^{3}
  2. A = ( 10 + 5 2 ( 10 + 5 + 75 + 30 5 ) ) a 2 17.7711... a 2 A=(10+\sqrt{\frac{5}{2}(10+\sqrt{5}+\sqrt{75+30\sqrt{5})}})a^{2}\approx 17.771% 1...a^{2}

Pentagonal_orthocupolarotunda.html

  1. V = 5 12 ( 11 + 5 5 ) a 3 9.24181... a 3 V=\frac{5}{12}(11+5\sqrt{5})a^{3}\approx 9.24181...a^{3}
  2. A = ( 5 + 1 4 1900 + 490 5 + 210 75 + 30 5 ) a 2 23.5385... a 2 A=(5+\frac{1}{4}\sqrt{1900+490\sqrt{5}+210\sqrt{75+30\sqrt{5}}})a^{2}\approx 2% 3.5385...a^{2}

Pentagonal_pyramid.html

  1. H = 5 - 5 10 a 0.5257 a . H=\sqrt{{\frac{5-\sqrt{5}}{10}}}\,a\approx 0.5257\,a.
  2. A = ( 25 + 10 5 4 + 5 3 4 ) a 2 3.8855 a 2 . A=\left(\frac{\sqrt{25+10\sqrt{5}}}{4}+5\frac{\sqrt{3}}{4}\right)a^{2}\approx 3% .8855\,a^{2}.
  3. V = 5 + 5 24 a 3 0.3015 a 3 . V=\frac{5+\sqrt{5}}{24}\,a^{3}\approx 0.3015\,a^{3}.

Pentagonal_rotunda.html

  1. V = ( 1 12 ( 45 + 17 5 ) ) a 3 6.91776... a 3 V=\left(\frac{1}{12}\left(45+17\sqrt{5}\right)\right)a^{3}\approx 6.91776...a^% {3}
  2. A = ( 1 2 ( 5 3 + 10 ( 65 + 29 5 ) ) ) a 2 = ( 1 2 5 ( 145 + 58 5 + 2 30 ( 65 + 29 5 ) ) ) a 2 22.3472... a 2 A=\left(\frac{1}{2}\left(5\sqrt{3}+\sqrt{10\left(65+29\sqrt{5}\right)}\right)% \right)a^{2}=\left(\frac{1}{2}\sqrt{5\left(145+58\sqrt{5}+2\sqrt{30\left(65+29% \sqrt{5}\right)}\right)}\right)a^{2}\approx 22.3472...a^{2}
  3. R = ( 1 2 ( 1 + 5 ) ) a 1.61803... a R=\left(\frac{1}{2}\left(1+\sqrt{5}\right)\right)a\approx 1.61803...a
  4. H = ( 1 + 2 5 ) a 1.37638... a H=\left(\sqrt{1+\frac{2}{\sqrt{5}}}\right)a\approx 1.37638...a

Pentatope_number.html

  1. ( n + 3 4 ) = n ( n + 1 ) ( n + 2 ) ( n + 3 ) 24 = n 4 ¯ 4 ! . {n+3\choose 4}=\frac{n(n+1)(n+2)(n+3)}{24}={n^{\overline{4}}\over 4!}.
  2. 4 3 4\over 3
  3. n = 1 4 ! n ( n + 1 ) ( n + 2 ) ( n + 3 ) = 4 3 \sum_{n=1}^{\infty}{4!\over{n(n+1)(n+2)(n+3)}}={4\over 3}
  4. 24 n + 1 + 1 2 \frac{\sqrt{24n+1}+1}{2}
  5. 8 * 24 n + 1 + 1 2 + 1 8*\frac{\sqrt{24n+1}+1}{2}+1

Periodic_points_of_complex_quadratic_mappings.html

  1. f c ( z ) = z 2 + c f_{c}(z)=z^{2}+c\,
  2. z z
  3. c c
  4. f \ f
  5. f \ f
  6. f c ( k ) ( z ) \ f^{(k)}_{c}(z)
  7. k \ k
  8. f c f_{c}\,
  9. f c f_{c}\,
  10. f c ( k ) ( z ) = f c ( f c ( k - 1 ) ( z ) ) \ f^{(k)}_{c}(z)=f_{c}(f^{(k-1)}_{c}(z))
  11. p \ p
  12. z \ z
  13. z : f c ( p ) ( z ) = z \ z:f^{(p)}_{c}(z)=z
  14. p \ p
  15. F p ( z , f ) = f c ( p ) ( z ) - z \ F_{p}(z,f)=f^{(p)}_{c}(z)-z
  16. F p ( z , f ) \ F_{p}(z,f)
  17. z : F p ( z , f ) = 0 \ z:F_{p}(z,f)=0
  18. = 2 p \ =2^{p}
  19. m ( f , z 0 ) = λ m(f,z_{0})=\lambda\,
  20. f f\,
  21. z 0 z_{0}\,
  22. m ( f , z 0 ) = λ = { f c ( z 0 ) , if z 0 1 f c ( z 0 ) , if z 0 = m(f,z_{0})=\lambda=\begin{cases}f_{c}^{\prime}(z_{0}),&\mbox{if }~{}z_{0}\neq% \infty\\ \frac{1}{f_{c}^{\prime}(z_{0})},&\mbox{if }~{}z_{0}=\infty\end{cases}
  23. f c ( z 0 ) f_{c}^{\prime}(z_{0})\,
  24. f c \ f_{c}
  25. z z\,
  26. z 0 z_{0}\,
  27. a b s ( λ ) abs(\lambda)\,
  28. a b s ( λ ) < 1 abs(\lambda)<1\,
  29. a b s ( λ ) = 0 abs(\lambda)=0\,
  30. 0 < a b s ( λ ) < 1 0<abs(\lambda)<1\,
  31. a b s ( λ ) = 1 abs(\lambda)=1\,
  32. a b s ( λ ) abs(\lambda)\,
  33. a b s ( λ ) = 1 abs(\lambda)=1\,
  34. a b s ( λ ) > 1 abs(\lambda)>1\,
  35. f f
  36. f c ( z ) = z \ f_{c}(z)=z
  37. z 2 + c = z z^{2}+c=z\,
  38. z 2 - z + c = 0. \ z^{2}-z+c=0.
  39. A x 2 + B x + C = 0 \ Ax^{2}+Bx+C=0
  40. x = - B ± B 2 - 4 A C 2 A x=\frac{-B\pm\sqrt{B^{2}-4AC}}{2A}
  41. A = 1 , B = - 1 , C = c A=1,B=-1,C=c
  42. α 1 = 1 - 1 - 4 c 2 \alpha_{1}=\frac{1-\sqrt{1-4c}}{2}
  43. α 2 = 1 + 1 - 4 c 2 . \alpha_{2}=\frac{1+\sqrt{1-4c}}{2}.
  44. c C [ 1 / 4 , + inf ] c\in C\setminus[1/4,+\inf]
  45. α 1 \alpha_{1}\,
  46. α 2 \alpha_{2}\,
  47. α 1 = 1 2 - m \alpha_{1}=\frac{1}{2}-m
  48. α 2 = 1 2 + m \alpha_{2}=\frac{1}{2}+m
  49. m = 1 - 4 c 2 m=\frac{\sqrt{1-4c}}{2}
  50. α 1 + α 2 = 1 \alpha_{1}+\alpha_{2}=1\,
  51. z = 1 / 2 z=1/2\,
  52. α c = 1 - 1 - 4 c 2 \alpha_{c}=\frac{1-\sqrt{1-4c}}{2}
  53. λ α c = 1 - 1 - 4 c \lambda_{\alpha_{c}}=1-\sqrt{1-4c}\,
  54. β c = 1 + 1 - 4 c 2 \beta_{c}=\frac{1+\sqrt{1-4c}}{2}
  55. λ β c = 1 + 1 - 4 c \lambda_{\beta_{c}}=1+\sqrt{1-4c}\,
  56. α c + β c = - B A = 1 \alpha_{c}+\beta_{c}=-\frac{B}{A}=1
  57. P c ( z ) = d d z P c ( z ) = 2 z P_{c}^{\prime}(z)=\frac{d}{dz}P_{c}(z)=2z
  58. P c ( α c ) + P c ( β c ) = 2 α c + 2 β c = 2 ( α c + β c ) = 2 P_{c}^{\prime}(\alpha_{c})+P_{c}^{\prime}(\beta_{c})=2\alpha_{c}+2\beta_{c}=2(% \alpha_{c}+\beta_{c})=2\,
  59. P c P_{c}\,
  60. β c \beta_{c}\,
  61. c M { 1 4 } c\in M\setminus\left\{\frac{1}{4}\right\}
  62. α c \alpha_{c}\,
  63. c = 0 c=0
  64. α 1 = 0 \alpha_{1}=0
  65. α 2 = 1 \alpha_{2}=1
  66. c c
  67. α 1 = α 2 \alpha_{1}=\alpha_{2}
  68. 1 - 4 c = 0 1-4c=0
  69. c = 1 / 4 c=1/4
  70. α 1 = α 2 = 1 / 2 \alpha_{1}=\alpha_{2}=1/2
  71. c = 1 / 4 c=1/4
  72. \mathbb{C}
  73. ^ \mathbb{\hat{C}}
  74. ^ = { } \mathbb{\hat{C}}=\mathbb{C}\cup\{\infty\}
  75. f c f_{c}\,
  76. f c ( ) = f_{c}(\infty)=\infty\,
  77. f c f_{c}\,
  78. f c ( ) = = f c - 1 ( ) f_{c}(\infty)=\infty=f^{-1}_{c}(\infty)\,
  79. β 1 \beta_{1}
  80. β 2 \beta_{2}
  81. f c ( β 1 ) = β 2 f_{c}(\beta_{1})=\beta_{2}
  82. f c ( β 2 ) = β 1 f_{c}(\beta_{2})=\beta_{1}
  83. f c ( f c ( β n ) ) = β n f_{c}(f_{c}(\beta_{n}))=\beta_{n}
  84. f c ( f c ( z ) ) = ( z 2 + c ) 2 + c = z 4 + 2 z 2 c + c 2 + c . f_{c}(f_{c}(z))=(z^{2}+c)^{2}+c=z^{4}+2z^{2}c+c^{2}+c.\,
  85. z 4 + 2 c z 2 - z + c 2 + c = 0 z^{4}+2cz^{2}-z+c^{2}+c=0
  86. α 1 \alpha_{1}
  87. α 2 \alpha_{2}
  88. f f
  89. ( z - α 1 ) ( z - α 2 ) ( z - β 1 ) ( z - β 2 ) = 0. (z-\alpha_{1})(z-\alpha_{2})(z-\beta_{1})(z-\beta_{2})=0.\,
  90. x 4 - A x 3 + B x 2 - C x + D = 0 x^{4}-Ax^{3}+Bx^{2}-Cx+D=0
  91. D = α 1 α 2 β 1 β 2 D=\alpha_{1}\alpha_{2}\beta_{1}\beta_{2}\,
  92. C = α 1 α 2 β 1 + α 1 α 2 β 2 + α 1 β 1 β 2 + α 2 β 1 β 2 C=\alpha_{1}\alpha_{2}\beta_{1}+\alpha_{1}\alpha_{2}\beta_{2}+\alpha_{1}\beta_% {1}\beta_{2}+\alpha_{2}\beta_{1}\beta_{2}\,
  93. B = α 1 α 2 + α 1 β 1 + α 1 β 2 + α 2 β 1 + α 2 β 2 + β 1 β 2 B=\alpha_{1}\alpha_{2}+\alpha_{1}\beta_{1}+\alpha_{1}\beta_{2}+\alpha_{2}\beta% _{1}+\alpha_{2}\beta_{2}+\beta_{1}\beta_{2}\,
  94. A = α 1 + α 2 + β 1 + β 2 . A=\alpha_{1}+\alpha_{2}+\beta_{1}+\beta_{2}.\,
  95. α 1 + α 2 = 1 - 1 - 4 c 2 + 1 + 1 - 4 c 2 = 1 + 1 2 = 1 \alpha_{1}+\alpha_{2}=\frac{1-\sqrt{1-4c}}{2}+\frac{1+\sqrt{1-4c}}{2}=\frac{1+% 1}{2}=1
  96. α 1 α 2 = ( 1 - 1 - 4 c ) ( 1 + 1 - 4 c ) 4 = 1 2 - ( 1 - 4 c ) 2 4 = 1 - 1 + 4 c 4 = 4 c 4 = c . \alpha_{1}\alpha_{2}=\frac{(1-\sqrt{1-4c})(1+\sqrt{1-4c})}{4}=\frac{1^{2}-(% \sqrt{1-4c})^{2}}{4}=\frac{1-1+4c}{4}=\frac{4c}{4}=c.
  97. D = c β 1 β 2 D=c\beta_{1}\beta_{2}
  98. A = 1 + β 1 + β 2 A=1+\beta_{1}+\beta_{2}
  99. f f
  100. D = c β 1 β 2 = c 2 + c D=c\beta_{1}\beta_{2}=c^{2}+c
  101. A = 1 + β 1 + β 2 = 0. A=1+\beta_{1}+\beta_{2}=0.
  102. β 1 β 2 = c + 1 \beta_{1}\beta_{2}=c+1
  103. β 1 + β 2 = - 1 \beta_{1}+\beta_{2}=-1
  104. A = 1 , B = 1 , C = c + 1 A^{\prime}=1,B=1,C=c+1
  105. β 1 = - 1 - - 3 - 4 c 2 \beta_{1}=\frac{-1-\sqrt{-3-4c}}{2}
  106. β 2 = - 1 + - 3 - 4 c 2 . \beta_{2}=\frac{-1+\sqrt{-3-4c}}{2}.
  107. f c ( β 1 ) = β 2 f_{c}(\beta_{1})=\beta_{2}
  108. f c ( β 2 ) = β 1 f_{c}(\beta_{2})=\beta_{1}
  109. ( z 2 + c ) 2 + c - z = ( z 2 + c - z ) ( z 2 + z + c + 1 ) (z^{2}+c)^{2}+c-z=(z^{2}+c-z)(z^{2}+z+c+1)\,
  110. z 1 , 2 z_{1,2}\,
  111. z 3 , 4 = - 1 2 ± ( - 3 4 - c ) 1 2 . z_{3,4}=-\frac{1}{2}\pm(-\frac{3}{4}-c)^{\frac{1}{2}}.\,
  112. c = 0 c=0
  113. β 1 = - 1 - i 3 2 \beta_{1}=\frac{-1-i\sqrt{3}}{2}
  114. β 2 = - 1 + i 3 2 \beta_{2}=\frac{-1+i\sqrt{3}}{2}
  115. | β 1 | = | β 2 | = 1 |\beta_{1}|=|\beta_{2}|=1
  116. c = - 1 c=-1
  117. β 1 = 0 \beta_{1}=0
  118. β 2 = - 1 \beta_{2}=-1

Periodic_sequence.html

  1. p , f ( p ) , f ( f ( p ) ) , f 3 ( p ) , f 4 ( p ) , p,\,f(p),\,f(f(p)),\,f^{3}(p),\,f^{4}(p),\,\ldots
  2. k = 1 k = 1 cos ( - 2 π n ( k - 1 ) 1 ) / 1 = 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1... \sum_{k=1}^{k=1}\cos(-2\pi\frac{n(k-1)}{1})/1=1,1,1,1,1,1,1,1,1...
  3. k = 1 k = 2 cos ( - 2 π n ( k - 1 ) 2 ) / 2 = 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0... \sum_{k=1}^{k=2}\cos(-2\pi\frac{n(k-1)}{2})/2=0,1,0,1,0,1,0,1,0...
  4. k = 1 k = 3 cos ( - 2 π n ( k - 1 ) 3 ) / 3 = 0 , 0 , 1 , 0 , 0 , 1 , 0 , 0 , 1 , 0 , 0 , 1 , 0 , 0 , 1... \sum_{k=1}^{k=3}\cos(-2\pi\frac{n(k-1)}{3})/3=0,0,1,0,0,1,0,0,1,0,0,1,0,0,1...
  5. ...
  6. k = 1 k = N cos ( - 2 π n ( k - 1 ) N ) / N = 0 , 0 , 0... , 1 sequence with period N \sum_{k=1}^{k=N}\cos(-2\pi\frac{n(k-1)}{N})/N=0,0,0...,1\,\text{ sequence with% period }N
  7. lim n x n - a n = 0. \lim_{n\rightarrow\infty}x_{n}-a_{n}=0.

Permeability_(earth_sciences).html

  1. \approx
  2. \approx
  3. v = κ μ Δ P Δ x v=\frac{\kappa}{\mu}\frac{\Delta P}{\Delta x}
  4. κ = v μ Δ x Δ P \kappa=v\frac{\mu\Delta x}{\Delta P}
  5. v v
  6. κ \kappa
  7. μ \mu
  8. Δ P \Delta P
  9. Δ x \Delta x
  10. κ = K μ ρ g \kappa=K\frac{\mu}{\rho g}
  11. κ \kappa
  12. K K
  13. μ \mu
  14. ρ \rho
  15. g g
  16. κ I = C d 2 {\kappa}_{I}=C\cdot d^{2}
  17. κ I {\kappa}_{I}
  18. C C
  19. d d

Permeability_(electromagnetism).html

  1. 𝐁 = μ 𝐇 , \mathbf{B}=\mu\mathbf{H},
  2. Δ 𝐁 = μ Δ Δ 𝐇 . \Delta\mathbf{B}=\mu_{\Delta}\Delta\mathbf{H}.
  3. 𝐅 = q ( 𝐄 + 𝐯 × 𝐁 ) . \mathbf{F}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B}).
  4. q 𝐯 × 𝐁 = C m s V s m 2 = C ( J / C ) m = J m = N q\mathbf{v}\times\mathbf{B}=C\cdot\dfrac{m}{s}\cdot\dfrac{V\cdot s}{m^{2}}=% \dfrac{C\cdot(J/C)}{m}=\dfrac{J}{m}=N
  5. μ r = μ μ 0 , \mu_{r}=\frac{\mu}{\mu_{0}},
  6. χ m = μ r - 1. \chi_{m}=\mu_{r}-1.
  7. 𝐁 ( ω ) = | μ 1 - i μ 2 0 i μ 2 μ 1 0 0 0 μ z | 𝐇 ( ω ) \begin{aligned}\displaystyle\mathbf{B}(\omega)&\displaystyle=\begin{vmatrix}% \mu_{1}&-i\mu_{2}&0\\ i\mu_{2}&\mu_{1}&0\\ 0&0&\mu_{z}\\ \end{vmatrix}\mathbf{H}(\omega)\\ \end{aligned}
  8. H = H 0 e j ω t B = B 0 e j ( ω t - δ ) H=H_{0}e^{j\omega t}\qquad B=B_{0}e^{j\left(\omega t-\delta\right)}
  9. δ \delta
  10. B B
  11. H H
  12. μ = B H = B 0 e j ( ω t - δ ) H 0 e j ω t = B 0 H 0 e - j δ , \mu=\frac{B}{H}=\frac{B_{0}e^{j\left(\omega t-\delta\right)}}{H_{0}e^{j\omega t% }}=\frac{B_{0}}{H_{0}}e^{-j\delta},
  13. μ = B 0 H 0 cos δ - j B 0 H 0 sin δ = μ - j μ ′′ . \mu=\frac{B_{0}}{H_{0}}\cos\delta-j\frac{B_{0}}{H_{0}}\sin\delta=\mu^{\prime}-% j\mu^{\prime\prime}.
  14. tan δ = μ ′′ μ , \tan\delta=\frac{\mu^{\prime\prime}}{\mu^{\prime}},

Petr_Hořava_(theorist).html

  1. E 8 × E 8 E_{8}\times E_{8}

PH_(complexity).html

  1. PH = k Δ k P \mathrm{PH}=\bigcup_{k\in\mathbb{N}}\Delta_{k}^{\mathrm{P}}

Phakic_intraocular_lens.html

  1. P = 1000 n 1000 n K + R - d - 1000 n 1000 n K - d P={1000n\over{1000n\over K+R}-d}-{1000n\over{1000n\over K}-d}

Pharmacodynamics.html

  1. L + R L R L+R\ \rightleftharpoons\ L\!\cdot\!R
  2. L + R L R L+R\ \leftrightarrow\ L\!\cdot\!R
  3. K d = [ L ] [ R ] [ L R ] K_{d}=\frac{[L][R]}{[L\!\cdot\!R]}
  4. F r a c t i o n B o u n d = [ L R ] [ R ] + [ L R ] = 1 1 + K d [ L ] Fraction\ Bound=\frac{[L\!\cdot\!R]}{[R]+[L\!\cdot\!R]}=\frac{1}{1+\frac{K_{d}% }{[L]}}

Phase-change_memory.html

  1. V t h \,V_{th}
  2. log 2 8 = 3 \log_{2}8=3

Phosphorylase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons
  4. \rightleftharpoons

Phot.html

  1. 1 phot = 1 lumen centimeter 2 = 10 , 000 lumens meter 2 = 10 , 000 lux = 10 kilolux 1\ \mathrm{phot}=1\ \frac{\mathrm{lumen}}{\mathrm{centimeter}^{2}}=10,000\ % \frac{\mathrm{lumens}}{\mathrm{meter}^{2}}=10,000\ \mathrm{lux}=10\ \mathrm{kilolux}

Photoemission_electron_microscopy.html

  1. h f = ϕ + E k m a x hf=\phi+E_{k_{max}}\,
  2. ϕ = h f 0 \phi=hf_{0}
  3. E k m a x = 1 2 m v m 2 E_{k_{max}}=\frac{1}{2}mv_{m}^{2}
  4. r d Δ E e U r\approx\frac{d\,\Delta\,E}{e\,U}

Photoemission_spectroscopy.html

  1. E k = h ν - E B E_{k}=h\nu-E_{B}
  2. h ν h\nu

Picard–Lindelöf_theorem.html

  1. y ( t ) = f ( t , y ( t ) ) , y ( t 0 ) = y 0 , t [ t 0 - ε , t 0 + ε ] . y^{\prime}(t)=f(t,y(t)),\qquad y(t_{0})=y_{0},\quad t\in\left[t_{0}-% \varepsilon,t_{0}+\varepsilon\right].
  2. f f
  3. y y
  4. t t
  5. t t
  6. ε > 0 ε>0
  7. y ( t ) y(t)
  8. [ t 0 - ε , t 0 + ε ] [t_{0}-\varepsilon,t_{0}+\varepsilon]
  9. y ( t ) - y ( t 0 ) = t 0 t f ( s , y ( s ) ) d s . y(t)-y(t_{0})=\int_{t_{0}}^{t}f(s,y(s))\,ds.
  10. φ 0 ( t ) = y 0 \varphi_{0}(t)=y_{0}
  11. φ k + 1 ( t ) = y 0 + t 0 t f ( s , φ k ( s ) ) d s . \varphi_{k+1}(t)=y_{0}+\int_{t_{0}}^{t}f(s,\varphi_{k}(s))\,ds.
  12. | φ ( t ) ψ ( t ) | |φ(t)−ψ(t)|
  13. φ φ
  14. ψ ψ
  15. φ ( t ) = ψ ( t ) φ(t)=ψ(t)
  16. d y d t = a y \frac{dy}{dt}=ay
  17. y ( t ) = 0 y(t)=0
  18. y ( 0 ) = 0 y(0)=0
  19. y ( 0 ) = 0 y(0)=0
  20. y ( t ) = 0 y(t)=0
  21. y ( t ) = { ( a t 3 ) 3 t < 0 0 t 0 y(t)=\begin{cases}\left(\tfrac{at}{3}\right)^{3}&t<0\\ 0&t\geq 0\end{cases}
  22. y = 0 y=0
  23. C a , b = I a ( t 0 ) ¯ × B b ( y 0 ) ¯ C_{a,b}=\overline{I_{a}(t_{0})}\times\overline{B_{b}(y_{0})}
  24. I a ( t 0 ) ¯ \displaystyle\overline{I_{a}(t_{0})}
  25. f f
  26. M = sup C a , b f , M=\sup_{C_{a,b}}\|f\|,
  27. f f
  28. 𝒞 ( I a ( t 0 ) , B b ( y 0 ) ) \mathcal{C}(I_{a}(t_{0}),B_{b}(y_{0}))
  29. φ = sup t I a | φ ( t ) | . \|\varphi\|_{\infty}=\sup_{t\in I_{a}}|\varphi(t)|.
  30. Γ : 𝒞 ( I a ( t 0 ) , B b ( y 0 ) ) 𝒞 ( I a ( t 0 ) , B b ( y 0 ) ) \Gamma:\mathcal{C}(I_{a}(t_{0}),B_{b}(y_{0}))\longrightarrow\mathcal{C}(I_{a}(% t_{0}),B_{b}(y_{0}))
  31. Γ φ ( t ) = y 0 + t 0 t f ( s , φ ( s ) ) d s . \Gamma\varphi(t)=y_{0}+\int_{t_{0}}^{t}f(s,\varphi(s))\,ds.
  32. B b ( y 0 ) B_{b}(y_{0})
  33. Γ φ ( t ) - y 0 \Gamma\varphi(t)-y_{0}
  34. φ 1 b . \|\varphi_{1}\|_{\infty}\leq b.
  35. Γ φ ( t ) - y 0 = t 0 t f ( s , φ ( s ) ) d s t 0 t f ( s , φ ( s ) ) d s M | t - t 0 | M a b \left\|\Gamma\varphi(t)-y_{0}\right\|=\left\|\int_{t_{0}}^{t}f(s,\varphi(s))\,% ds\right\|\leq\int_{t_{0}}^{t}\left\|f(s,\varphi(s))\right\|ds\leq M\left|t-t_% {0}\right|\leq Ma\leq b
  36. φ 1 , φ 2 𝒞 ( I a ( t 0 ) , B b ( y 0 ) ) \varphi_{1},\varphi_{2}\in\mathcal{C}(I_{a}(t_{0}),B_{b}(y_{0}))
  37. Γ φ 1 - Γ φ 2 q φ 1 - φ 2 , \left\|\Gamma\varphi_{1}-\Gamma\varphi_{2}\right\|_{\infty}\leq q\left\|% \varphi_{1}-\varphi_{2}\right\|_{\infty},
  38. ( Γ φ 1 - Γ φ 2 ) ( t ) = t 0 t ( f ( s , φ 1 ( s ) ) - f ( s , φ 2 ( s ) ) ) d s t 0 t f ( s , φ 1 ( s ) ) - f ( s , φ 2 ( s ) ) d s L t 0 t φ 1 ( s ) - φ 2 ( s ) d s f is Lipschitz-continuous L a φ 1 - φ 2 \begin{aligned}\displaystyle\left\|\left(\Gamma\varphi_{1}-\Gamma\varphi_{2}% \right)(t)\right\|&\displaystyle=\left\|\int_{t_{0}}^{t}\left(f(s,\varphi_{1}(% s))-f(s,\varphi_{2}(s))\right)ds\right\|\\ &\displaystyle\leq\int_{t_{0}}^{t}\left\|f\left(s,\varphi_{1}(s)\right)-f\left% (s,\varphi_{2}(s)\right)\right\|ds\\ &\displaystyle\leq L\int_{t_{0}}^{t}\left\|\varphi_{1}(s)-\varphi_{2}(s)\right% \|ds&&\displaystyle f\,\text{ is Lipschitz-continuous}\\ &\displaystyle\leq La\left\|\varphi_{1}-\varphi_{2}\right\|\end{aligned}
  39. φ 𝒞 ( I a ( t 0 ) , B b ( y 0 ) ) \varphi\in\mathcal{C}(I_{a}(t_{0}),B_{b}(y_{0}))
  40. Γ φ = φ Γφ=φ
  41. a < min { b / M , 1 / L } . a<\min\{b/M,1/L\}.
  42. Γ m φ 1 - Γ m φ 2 L m α m m ! φ 1 - φ 2 \left\|\Gamma^{m}\varphi_{1}-\Gamma^{m}\varphi_{2}\right\|\leq\frac{L^{m}% \alpha^{m}}{m!}\left\|\varphi_{1}-\varphi_{2}\right\|
  43. ( m = 1 ) (m=1)
  44. m 1 m−1
  45. Γ m φ 1 - Γ m φ 2 \displaystyle\left\|\Gamma^{m}\varphi_{1}-\Gamma^{m}\varphi_{2}\right\|
  46. L m α m m ! < 1 , \frac{L^{m}\alpha^{m}}{m!}<1,
  47. f f
  48. y y
  49. y ( t ) = 0 , y ( t ) = ± ( 2 3 t ) 3 2 . y(t)=0,\qquad y(t)=\pm\left(\tfrac{2}{3}t\right)^{\frac{3}{2}}.
  50. f f

Piling-up_lemma.html

  1. X 1 X 2 X n = 0 X_{1}\oplus X_{2}\oplus\cdots\oplus X_{n}=0
  2. P ( X 1 = 0 ) = p 1 P(X_{1}=0)=p_{1}
  3. P ( X 2 = 0 ) = p 2 P(X_{2}=0)=p_{2}
  4. P ( X 1 X 2 = 0 ) P(X_{1}\oplus X_{2}=0)
  5. P ( X 1 = X 2 ) P(X_{1}=X_{2})
  6. P ( X 1 = X 2 ) = P ( X 1 = X 2 = 0 ) + P ( X 1 = X 2 = 1 ) = P ( X 1 = 0 , X 2 = 0 ) + P ( X 1 = 1 , X 2 = 1 ) P(X_{1}=X_{2})=P(X_{1}=X_{2}=0)+P(X_{1}=X_{2}=1)=P(X_{1}=0,X_{2}=0)+P(X_{1}=1,% X_{2}=1)
  7. P ( X 1 X 2 = 0 ) P(X_{1}\oplus X_{2}=0)
  8. = P ( X 1 = 0 ) P ( X 2 = 0 ) + P ( X 1 = 1 ) P ( X 2 = 1 ) =P(X_{1}=0)P(X_{2}=0)+P(X_{1}=1)P(X_{2}=1)
  9. = p 1 p 2 + ( 1 - p 1 ) ( 1 - p 2 ) =p_{1}p_{2}+(1-p_{1})(1-p_{2})
  10. = p 1 p 2 + ( 1 - p 1 - p 2 + p 1 p 2 ) =p_{1}p_{2}+(1-p_{1}-p_{2}+p_{1}p_{2})
  11. = 2 p 1 p 2 - p 1 - p 2 + 1 =2p_{1}p_{2}-p_{1}-p_{2}+1
  12. P ( X 1 X 2 = 0 ) P(X_{1}\oplus X_{2}=0)
  13. = 2 ( 1 / 2 + ϵ 1 ) ( 1 / 2 + ϵ 2 ) - ( 1 / 2 + ϵ 1 ) - ( 1 / 2 + ϵ 2 ) + 1 =2(1/2+\epsilon_{1})(1/2+\epsilon_{2})-(1/2+\epsilon_{1})-(1/2+\epsilon_{2})+1
  14. = 1 / 2 + ϵ 1 + ϵ 2 + 2 ϵ 1 ϵ 2 - 1 / 2 - ϵ 1 - 1 / 2 - ϵ 2 + 1 =1/2+\epsilon_{1}+\epsilon_{2}+2\epsilon_{1}\epsilon_{2}-1/2-\epsilon_{1}-1/2-% \epsilon_{2}+1
  15. = 1 / 2 + 2 ϵ 1 ϵ 2 =1/2+2\epsilon_{1}\epsilon_{2}
  16. P ( X 1 X 2 X n = 0 ) = 1 / 2 + 2 n - 1 i = 1 n ϵ i P(X_{1}\oplus X_{2}\oplus\cdots\oplus X_{n}=0)=1/2+2^{n-1}\prod_{i=1}^{n}% \epsilon_{i}
  17. ϵ i = P ( X i = 1 ) - P ( X i = 0 ) , \epsilon_{i}=P(X_{i}=1)-P(X_{i}=0),
  18. ε t o t a l = P ( X 1 X 2 X n = 1 ) - P ( X 1 X 2 X n = 0 ) \varepsilon_{total}=P(X_{1}\oplus X_{2}\oplus\cdots\oplus X_{n}=1)-P(X_{1}% \oplus X_{2}\oplus\cdots\oplus X_{n}=0)
  19. ε t o t a l = ( - 1 ) n + 1 i = 1 n ε i , \varepsilon_{total}=(-1)^{n+1}\prod_{i=1}^{n}\varepsilon_{i},

Pilot_wave.html

  1. L ( t ) = 1 2 m v 2 - ( V + Q ) , L(t)={\frac{1}{2}}mv^{2}-(V+Q),
  2. Q Q
  3. K Q ( X 1 , t 1 ; X 0 , t 0 ) = 1 J ( t ) 1 2 exp [ i t 0 t 1 L ( t ) d t ] . K^{Q}(X_{1},t_{1};X_{0},t_{0})=\frac{1}{J(t)^{\frac{1}{2}}}\exp\left[\frac{i}{% \hbar}\int_{t_{0}}^{t_{1}}L(t)\,dt\right].
  4. Q Q
  5. H ( 𝐪 , S 𝐪 , t ) + S t ( 𝐪 , t ) = 0 H\left(\mathbf{q},{\partial S\over\partial\mathbf{q}},t\right)+{\partial S% \over\partial t}\left(\mathbf{q},t\right)=0
  6. ρ ( x , t ) \rho(x,t)
  7. ρ d 3 x = 1 \int\rho\,d^{3}x=1
  8. t t
  9. ρ / t = - ( ρ v ) ( 1 ) \partial\rho/\partial t=-\nabla\cdot(\rho v)\quad(1)
  10. v ( x , t ) v(x,t)
  11. v ( x , t ) = S ( x , t ) m v(x,t)=\frac{\nabla S(x,t)}{m}
  12. S ( x , t ) S(x,t)
  13. - S t = ( S ) 2 2 m + V ( 2 ) -\frac{\partial S}{\partial t}=\frac{\left(\nabla S\right)^{2}}{2m}+V\quad(2)
  14. ( 1 ) (1)
  15. ( 2 ) (2)
  16. ψ = ρ e i S \psi=\sqrt{\rho}e^{\frac{iS}{\hbar}}
  17. i ψ t = ( - 2 2 m 2 + V - Q ) ψ i\hbar\frac{\partial\psi}{\partial t}=\left(-\frac{\hbar^{2}}{2m}\nabla^{2}+V-% Q\right)\psi\quad
  18. Q = - 2 2 m 2 ρ ρ \quad Q=-\frac{\hbar^{2}}{2m}\frac{\nabla^{2}\sqrt{\rho}}{\sqrt{\rho}}
  19. Q Q
  20. i ψ t = ( - 2 2 m 2 + V ) ψ i\hbar\frac{\partial\psi}{\partial t}=\left(-\frac{\hbar^{2}}{2m}\nabla^{2}+V% \right)\psi\quad
  21. ψ = ρ exp ( i S ) \psi=\sqrt{\rho}\;\exp\left(\frac{i\,S}{\hbar}\right)
  22. ρ \rho
  23. ρ / t + ( ρ v ) = 0 , \partial\rho/\partial t+\nabla\cdot(\rho v)=0\;,
  24. v ( r , t ) = S ( r , t ) m . \vec{v}(\vec{r},t)=\frac{\nabla S(\vec{r},t)}{m}\;.
  25. ρ = | ψ | 2 \rho=|\psi|^{2}
  26. S S
  27. - S t = ( S ) 2 2 m + V + Q , -\frac{\partial S}{\partial t}=\frac{\left(\nabla S\right)^{2}}{2m}+V+Q\;,
  28. Q = - 2 2 m 2 ρ ρ Q=-\frac{\hbar^{2}}{2m}\frac{\nabla^{2}\sqrt{\rho}}{\sqrt{\rho}}
  29. m d d t v = - ( V + Q ) , m\,\frac{d}{dt}\,\vec{v}=-\nabla(V+Q)\;,
  30. d d t = t + v . \frac{d}{dt}=\frac{\partial}{\partial t}+\vec{v}\cdot\nabla\;.
  31. ψ ( r 1 , r 2 , , t ) \psi(\vec{r}_{1},\vec{r}_{2},\cdots,t)
  32. i ψ t = ( - 2 2 i = 1 N i 2 m i + V ( r 1 , r 2 , r N ) ) ψ i\hbar\frac{\partial\psi}{\partial t}=\left(-\frac{\hbar^{2}}{2}\sum_{i=1}^{N}% \frac{\nabla_{i}^{2}}{m_{i}}+V({r}_{1},{r}_{2},\cdots{r}_{N})\right)\psi
  33. ψ = ρ exp ( i S ) \psi=\sqrt{\rho}\;\exp\left(\frac{i\,S}{\hbar}\right)
  34. v j = j S m . \vec{v}_{j}=\frac{\nabla_{j}S}{m}\;.

Pioneer_anomaly.html

  1. a p a_{p}
  2. c c
  3. H 0 H_{0}

Pisano_period.html

  1. π k ( n ) \pi_{k}(n)
  2. π k ( m n ) = lcm ( π k ( m ) , π k ( n ) ) , \pi_{k}(m\cdot n)=\mathrm{lcm}(\pi_{k}(m),\pi_{k}(n)),
  3. π 1 ( 3 ) = 8 \pi_{1}(3)=8
  4. π 1 ( 4 ) = 6 , \pi_{1}(4)=6,
  5. π 1 ( 12 = 3 4 ) = lcm ( π 1 ( 3 ) , π 1 ( 4 ) ) = lcm ( 8 , 6 ) = 24. \pi_{1}(12=3\cdot 4)=\mathrm{lcm}(\pi_{1}(3),\pi_{1}(4))=\mathrm{lcm}(8,6)=24.
  6. q = p n . q=p^{n}.
  7. π k ( p n ) = p n - 1 π k ( p ) \pi_{k}(p^{n})=p^{n-1}\cdot\pi_{k}(p)
  8. F k ( n ) = φ k n - ( k - φ k ) n k 2 + 4 = φ k n - ( - 1 / φ k ) n k 2 + 4 , F_{k}\left(n\right)={{\varphi_{k}^{n}-(k-\varphi_{k})^{n}}\over{\sqrt{k^{2}+4}% }}={{\varphi_{k}^{n}-(-1/\varphi_{k})^{n}}\over{\sqrt{k^{2}+4}}},\,
  9. φ k \varphi_{k}\,
  10. φ k = k + k 2 + 4 2 . \varphi_{k}=\frac{k+\sqrt{k^{2}+4}}{2}.
  11. p > 2 p>2
  12. k 2 + 4 , 1 / 2 , \sqrt{k^{2}+4},1/2,
  13. k / k 2 + 4 k/\sqrt{k^{2}+4}
  14. ϕ ( p ) = p - 1 \phi(p)=p-1
  15. φ k n \varphi_{k}^{n}
  16. ϕ ( p ) , \phi(p),
  17. F 1 ( n ) 3 ( 8 n - 4 n ) ( mod 11 ) . F_{1}\left(n\right)\equiv 3\cdot\left(8^{n}-4^{n}\right)\;\;(\mathop{{\rm mod}% }11).
  18. i = 1 n F i = F n + 2 - 1 \sum_{i=1}^{n}F_{i}=F_{n+2}-1
  19. i = 1 π ( n ) F i = n k \sum_{i=1}^{\pi(n)}F_{i}=nk
  20. i = n n + 5 F i = 4 F n + 4 \sum_{i=n}^{n+5}F_{i}=4F_{n+4}
  21. i = n n + 9 F i = 11 F n + 6 \sum_{i=n}^{n+9}F_{i}=11F_{n+6}
  22. i = n n + 13 F i = 29 F n + 8 \sum_{i=n}^{n+13}F_{i}=29F_{n+8}
  23. i = n n + 17 F i = 76 F n + 10 \sum_{i=n}^{n+17}F_{i}=76F_{n+10}

Pixel_density.html

  1. d p = w p 2 + h p 2 d_{p}=\sqrt{w_{p}^{2}+h_{p}^{2}}
  2. P P I = d p d i PPI=\frac{d_{p}}{d_{i}}
  3. d p d_{p}
  4. w p w_{p}
  5. h p h_{p}
  6. d i d_{i}
  7. w p w_{p}
  8. h p h_{p}
  9. d i d_{i}
  10. w p w_{p}
  11. h p h_{p}
  12. d i d_{i}

Planck_charge.html

  1. q P q\text{P}
  2. q P = 4 π ϵ 0 c = e α = 1.875 5459 × 10 - 18 q\text{P}=\sqrt{4\pi\epsilon_{0}\hbar c}=\frac{e}{\sqrt{\alpha}}=1.875\;5459% \times 10^{-18}
  3. c c
  4. \hbar
  5. ϵ 0 \epsilon_{0}
  6. e e
  7. α \alpha
  8. α - 1 / 2 11.706 \alpha^{-1/2}\approx 11.706
  9. 4 π ϵ 0 = 1 4\pi\epsilon_{0}=1
  10. q P q\text{P}
  11. q P = c . q\text{P}=\sqrt{\hbar c}.
  12. c = 1 c=1
  13. ϵ 0 = μ 0 = 1 \epsilon_{0}=\mu_{0}=1
  14. q P = ϵ 0 c = e 4 π α = 5.291 × 10 - 19 q^{\prime}\text{P}=\sqrt{\epsilon_{0}\hbar c}=\frac{e}{\sqrt{4\pi\alpha}}=5.29% 1\times 10^{-19}
  15. q P q^{\prime}\text{P}
  16. q P q^{\prime}\text{P}
  17. α = e 2 4 π \alpha=\frac{e^{2}}{4\pi}
  18. q P = 1 q\text{P}=1
  19. α = e 2 \alpha=e^{2}

Planck_energy.html

  1. E P = c 5 G 1.956 × 10 9 J 1.22 × 10 28 eV 0.5433 MWh E_{\mathrm{P}}=\sqrt{\frac{\hbar c^{5}}{G}}\approx 1.956\times 10^{9}\ \mathrm% {J}\approx 1.22\times 10^{28}\ \mathrm{eV}\approx 0.5433\ \mathrm{MWh}
  2. E P = t P , E_{\mathrm{P}}={\frac{\hbar}{t_{\mathrm{P}}}},
  3. E P = m P c 2 , E_{\mathrm{P}}={m_{\mathrm{P}}}{c^{2}},
  4. c 5 8 π G 0.390 × 10 9 J 2.43 × 10 18 GeV . \sqrt{\frac{\hbar{}c^{5}}{8\pi G}}\approx 0.390\times 10^{9}\ \mathrm{J}% \approx 2.43\times 10^{18}\ \mathrm{GeV}.

Planck_length.html

  1. P = G c 3 1.616 199 ( 97 ) × 10 - 35 m \ell_{\mathrm{P}}=\sqrt{\frac{\hbar G}{c^{3}}}\approx 1.616\;199(97)\times 10^% {-35}\ \mathrm{m}
  2. c c
  3. G G
  4. ħ ħ
  5. A 4 P 2 \frac{A}{4\ell_{\mathrm{P}}^{2}}
  6. A A
  7. l < s u b > s l<sub>s

Plane_curve.html

  1. a x + b y = c ax+by=c
  2. ( x 0 + α t , y 0 + β t ) (x_{0}+\alpha t,y_{0}+\beta t)
  3. y = m x + c y=mx+c
  4. x 2 + y 2 = r 2 x^{2}+y^{2}=r^{2}
  5. ( r cos t , r sin t ) (r\cos t,r\sin t)
  6. y - x 2 = 0 y-x^{2}=0
  7. ( t , t 2 ) (t,t^{2})
  8. y = x 2 y=x^{2}
  9. x 2 a 2 + y 2 b 2 = 1 \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1
  10. ( a cos t , b sin t ) (a\cos t,b\sin t)
  11. x 2 a 2 - y 2 b 2 = 1 \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1
  12. ( a cosh t , b sinh t ) (a\cosh t,b\sinh t)

Planimeter.html

  1. N ( x , y ) = ( b - y , x ) , \!\,N(x,y)=(b-y,x),
  2. E M N = x N x + ( y - b ) N y = 0 \overrightarrow{EM}\cdot N=xN_{x}+(y-b)N_{y}=0
  3. N = ( b - y ) 2 + x 2 = m \!\,\|N\|=\sqrt{(b-y)^{2}+x^{2}}=m
  4. C ( N x d x + N y d y ) = S ( N y x - N x y ) d x d y \displaystyle\oint_{C}(N_{x}\,dx+N_{y}\,dy)=\iint_{S}\left(\frac{\partial N_{y% }}{\partial x}-\frac{\partial N_{x}}{\partial y}\right)\,dx\,dy
  5. y ( y - b ) = y m 2 - x 2 = 0 , \frac{\partial}{\partial y}(y-b)=\frac{\partial}{\partial y}\sqrt{m^{2}-x^{2}}% =0,
  6. θ 1 2 ( r ( θ ) ) 2 d θ , \scriptstyle\int_{\theta}\tfrac{1}{2}(r(\theta))^{2}\,d\theta,
  7. t 1 2 ( r ( t ) ) 2 d ( θ ( t ) ) = t 1 2 ( r ( t ) ) 2 θ ˙ ( t ) d t . \int_{t}\tfrac{1}{2}(r(t))^{2}\,d(\theta(t))=\int_{t}\tfrac{1}{2}(r(t))^{2}\,% \dot{\theta}(t)\,dt.
  8. t r ( t ) θ ˙ ( t ) d t , \scriptstyle\int_{t}r(t)\,\dot{\theta}(t)\,dt,
  9. r d θ = 2 π r \scriptstyle\int r\,d\theta=2\pi r
  10. r ( t ) θ ˙ ( t ) \scriptstyle r(t)\,\dot{\theta}(t)
  11. 1 2 ( r ( t ) ) 2 θ ˙ ( t ) \scriptstyle\tfrac{1}{2}(r(t))^{2}\dot{\theta}(t)

Plasma_oscillation.html

  1. ω pe = n e e 2 m * ε 0 , [ rad / s ] \omega_{\mathrm{pe}}=\sqrt{\frac{n_{\mathrm{e}}e^{2}}{m^{*}\varepsilon_{0}}},% \left[\mathrm{rad}/\mathrm{s}\right]
  2. ω pe = 4 π n e e 2 m * , \omega_{\mathrm{pe}}=\sqrt{\frac{4\pi n_{\mathrm{e}}e^{2}}{m^{*}}},
  3. n e n_{\mathrm{e}}
  4. ε 0 \varepsilon_{0}
  5. m * m^{*}
  6. ( m * = m e ) (m^{*}=m_{\mathrm{e}})
  7. ω pe \omega_{\mathrm{pe}}
  8. n e n_{\mathrm{e}}
  9. f pe = ω pe / 2 π f_{\mathrm{pe}}=\omega_{\mathrm{pe}}/2\pi
  10. f pe 8980 n e f_{\mathrm{pe}}\approx 8980\sqrt{n_{\mathrm{e}}}
  11. n e n_{\mathrm{e}}
  12. v e , th = k B T e m e v_{\mathrm{e,th}}=\sqrt{\frac{k_{\mathrm{B}}T_{\mathrm{e}}}{m_{\mathrm{e}}}}
  13. ω 2 = ω pe 2 + 3 k B T e m e k 2 = ω pe 2 + 3 k 2 v e , th 2 \omega^{2}=\omega_{\mathrm{pe}}^{2}+\frac{3k_{\mathrm{B}}T_{\mathrm{e}}}{m_{% \mathrm{e}}}k^{2}=\omega_{\mathrm{pe}}^{2}+3k^{2}v_{\mathrm{e,th}}^{2}
  14. 3 v e , th \sqrt{3}\cdot v_{\mathrm{e,th}}
  15. v v ph = def ω k , v\sim v_{\mathrm{ph}}\ \stackrel{\mathrm{def}}{=}\ \frac{\omega}{k},

Plücker_coordinates.html

  1. p i j p_{ij}\,\!
  2. = | x i y i x j y j | = x i y j - x j y i . {}=\begin{vmatrix}x_{i}&y_{i}\\ x_{j}&y_{j}\end{vmatrix}=x_{i}y_{j}-x_{j}y_{i}.\,\!
  3. ( p 01 : p 02 : p 03 : p 23 : p 31 : p 12 ) (p_{01}:p_{02}:p_{03}:p_{23}:p_{31}:p_{12})\,\!
  4. M = [ x 0 y 0 x 1 y 1 x 2 y 2 x 3 y 3 ] M=\begin{bmatrix}x_{0}&y_{0}\\ x_{1}&y_{1}\\ x_{2}&y_{2}\\ x_{3}&y_{3}\end{bmatrix}
  5. M = M Λ . M^{\prime}=M\Lambda.\,\!
  6. [ x i y i x j y j ] = [ x i y i x j y j ] [ λ 00 λ 01 λ 10 λ 11 ] . \begin{bmatrix}x^{\prime}_{i}&y^{\prime}_{i}\\ x^{\prime}_{j}&y^{\prime}_{j}\end{bmatrix}=\begin{bmatrix}x_{i}&y_{i}\\ x_{j}&y_{j}\end{bmatrix}\begin{bmatrix}\lambda_{00}&\lambda_{01}\\ \lambda_{10}&\lambda_{11}\end{bmatrix}.
  7. α : G 1 , 3 \displaystyle\alpha\colon\mathrm{G}_{1,3}
  8. L α = ( p 01 : p 02 : p 03 : p 23 : p 31 : p 12 ) . L^{\alpha}=(p_{01}:p_{02}:p_{03}:p_{23}:p_{31}:p_{12}).\,\!
  9. p i j p^{ij}\,\!
  10. = | a i a j b i b j | = a i b j - a j b i . {}=\begin{vmatrix}a^{i}&a^{j}\\ b^{i}&b^{j}\end{vmatrix}=a^{i}b^{j}-a^{j}b^{i}.\,\!
  11. ( p 01 : p 02 : p 03 : p 23 : p 31 : p 12 ) = ( p 23 : p 31 : p 12 : p 01 : p 02 : p 03 ) (p_{01}:p_{02}:p_{03}:p_{23}:p_{31}:p_{12})=(p^{23}:p^{31}:p^{12}:p^{01}:p^{02% }:p^{03})
  12. λ \lambda
  13. p i j = λ p k l . p_{ij}=\lambda p^{kl}.
  14. M = [ 1 1 x 1 y 1 x 2 y 2 x 3 y 3 ] , M=\begin{bmatrix}1&1\\ x_{1}&y_{1}\\ x_{2}&y_{2}\\ x_{3}&y_{3}\end{bmatrix},
  15. [ x 0 y 0 z 0 x 1 y 1 z 1 x 2 y 2 z 2 x 3 y 3 z 3 ] \begin{bmatrix}x_{0}&y_{0}&z_{0}\\ x_{1}&y_{1}&z_{1}\\ x_{2}&y_{2}&z_{2}\\ x_{3}&y_{3}&z_{3}\end{bmatrix}
  16. 0 0\,\!
  17. = | x 0 y 0 z 0 x 1 y 1 z 1 x 2 y 2 z 2 | {}=\begin{vmatrix}x_{0}&y_{0}&z_{0}\\ x_{1}&y_{1}&z_{1}\\ x_{2}&y_{2}&z_{2}\end{vmatrix}
  18. = | x 1 y 1 x 2 y 2 | z 0 - | x 0 y 0 x 2 y 2 | z 1 + | x 0 y 0 x 1 y 1 | z 2 {}=\begin{vmatrix}x_{1}&y_{1}\\ x_{2}&y_{2}\end{vmatrix}z_{0}-\begin{vmatrix}x_{0}&y_{0}\\ x_{2}&y_{2}\end{vmatrix}z_{1}+\begin{vmatrix}x_{0}&y_{0}\\ x_{1}&y_{1}\end{vmatrix}z_{2}
  19. = p 12 z 0 - p 02 z 1 + p 01 z 2 . {}=p_{12}z_{0}-p_{02}z_{1}+p_{01}z_{2}.\,\!
  20. = p 03 z 0 + p 13 z 1 + p 23 z 2 . {}=p^{03}z_{0}+p^{13}z_{1}+p^{23}z_{2}.\,\!
  21. 0 = + p 12 z 0 - p 02 z 1 + p 01 z 2 0 = - p 31 z 0 - p 03 z 1 + p 01 z 3 0 = + p 23 z 0 - p 03 z 2 + p 02 z 3 0 = + p 23 z 1 + p 31 z 2 + p 12 z 3 \begin{matrix}0&=&{}+p_{12}z_{0}&{}-p_{02}z_{1}&{}+p_{01}z_{2}&\\ 0&=&{}-p_{31}z_{0}&{}-p_{03}z_{1}&&{}+p_{01}z_{3}\\ 0&=&{}+p_{23}z_{0}&&{}-p_{03}z_{2}&{}+p_{02}z_{3}\\ 0&=&&{}+p_{23}z_{1}&{}+p_{31}z_{2}&{}+p_{12}z_{3}\end{matrix}
  22. 0 = i = 0 3 p i j z i , j = 0 , , 3. 0=\sum_{i=0}^{3}p^{ij}z_{i},\qquad j=0,\ldots,3.\,\!
  23. 0 0\,\!
  24. = p 01 p 01 + p 02 p 02 + p 03 p 03 {}=p_{01}p^{01}+p_{02}p^{02}+p_{03}p^{03}\,\!
  25. = p 01 p 23 + p 02 p 31 + p 03 p 12 . {}=p_{01}p_{23}+p_{02}p_{31}+p_{03}p_{12}.\,\!
  26. 0 0\,\!
  27. = | x 0 y 0 x 1 y 1 | | x 2 y 2 x 3 y 3 | + | x 0 y 0 x 2 y 2 | | x 3 y 3 x 1 y 1 | + | x 0 y 0 x 3 y 3 | | x 1 y 1 x 2 y 2 | {}=\begin{vmatrix}x_{0}&y_{0}\\ x_{1}&y_{1}\end{vmatrix}\begin{vmatrix}x_{2}&y_{2}\\ x_{3}&y_{3}\end{vmatrix}+\begin{vmatrix}x_{0}&y_{0}\\ x_{2}&y_{2}\end{vmatrix}\begin{vmatrix}x_{3}&y_{3}\\ x_{1}&y_{1}\end{vmatrix}+\begin{vmatrix}x_{0}&y_{0}\\ x_{3}&y_{3}\end{vmatrix}\begin{vmatrix}x_{1}&y_{1}\\ x_{2}&y_{2}\end{vmatrix}
  28. = ( x 0 y 1 - y 0 x 1 ) | x 2 y 2 x 3 y 3 | - ( x 0 y 2 - y 0 x 2 ) | x 1 y 1 x 3 y 3 | + ( x 0 y 3 - y 0 x 3 ) | x 1 y 1 x 2 y 2 | {}=(x_{0}y_{1}-y_{0}x_{1})\begin{vmatrix}x_{2}&y_{2}\\ x_{3}&y_{3}\end{vmatrix}-(x_{0}y_{2}-y_{0}x_{2})\begin{vmatrix}x_{1}&y_{1}\\ x_{3}&y_{3}\end{vmatrix}+(x_{0}y_{3}-y_{0}x_{3})\begin{vmatrix}x_{1}&y_{1}\\ x_{2}&y_{2}\end{vmatrix}\,\!
  29. = x 0 ( y 1 | x 2 y 2 x 3 y 3 | - y 2 | x 1 y 1 x 3 y 3 | + y 3 | x 1 y 1 x 2 y 2 | ) - y 0 ( x 1 | x 2 y 2 x 3 y 3 | - x 2 | x 1 y 1 x 3 y 3 | + x 3 | x 1 y 1 x 2 y 2 | ) {}=x_{0}\left(y_{1}\begin{vmatrix}x_{2}&y_{2}\\ x_{3}&y_{3}\end{vmatrix}-y_{2}\begin{vmatrix}x_{1}&y_{1}\\ x_{3}&y_{3}\end{vmatrix}+y_{3}\begin{vmatrix}x_{1}&y_{1}\\ x_{2}&y_{2}\end{vmatrix}\right)-y_{0}\left(x_{1}\begin{vmatrix}x_{2}&y_{2}\\ x_{3}&y_{3}\end{vmatrix}-x_{2}\begin{vmatrix}x_{1}&y_{1}\\ x_{3}&y_{3}\end{vmatrix}+x_{3}\begin{vmatrix}x_{1}&y_{1}\\ x_{2}&y_{2}\end{vmatrix}\right)\,\!
  30. = x 0 | x 1 y 1 y 1 x 2 y 2 y 2 x 3 y 3 y 3 | - y 0 | x 1 x 1 y 1 x 2 x 2 y 2 x 3 x 3 y 3 | {}=x_{0}\begin{vmatrix}x_{1}&y_{1}&y_{1}\\ x_{2}&y_{2}&y_{2}\\ x_{3}&y_{3}&y_{3}\end{vmatrix}-y_{0}\begin{vmatrix}x_{1}&x_{1}&y_{1}\\ x_{2}&x_{2}&y_{2}\\ x_{3}&x_{3}&y_{3}\end{vmatrix}\,\!
  31. d = ( p 01 , p 02 , p 03 ) d=\left(p_{01},p_{02},p_{03}\right)
  32. m = ( p 23 , p 31 , p 12 ) m=\left(p_{23},p_{31},p_{12}\right)
  33. M = [ q 01 0 0 q 01 - q 12 q 02 q 31 q 03 ] M=\begin{bmatrix}q_{01}&0\\ 0&q_{01}\\ -q_{12}&q_{02}\\ q_{31}&q_{03}\end{bmatrix}
  34. p 23 = - q 03 q 12 - q 02 q 31 . p_{23}=-q_{03}q_{12}-q_{02}q_{31}.\,\!
  35. p 01 p 23 + p 02 p 31 + p 03 p 12 . p_{01}p_{23}+p_{02}p_{31}+p_{03}p_{12}.\,\!
  36. 0 0\,\!
  37. = p 01 p 23 + p 02 p 31 + p 03 p 12 + p 23 p 01 + p 31 p 02 + p 12 p 03 {}=p_{01}p^{\prime}_{23}+p_{02}p^{\prime}_{31}+p_{03}p^{\prime}_{12}+p_{23}p^{% \prime}_{01}+p_{31}p^{\prime}_{02}+p_{12}p^{\prime}_{03}\,\!
  38. = | x 0 y 0 x 0 y 0 x 1 y 1 x 1 y 1 x 2 y 2 x 2 y 2 x 3 y 3 x 3 y 3 | . {}=\begin{vmatrix}x_{0}&y_{0}&x^{\prime}_{0}&y^{\prime}_{0}\\ x_{1}&y_{1}&x^{\prime}_{1}&y^{\prime}_{1}\\ x_{2}&y_{2}&x^{\prime}_{2}&y^{\prime}_{2}\\ x_{3}&y_{3}&x^{\prime}_{3}&y^{\prime}_{3}\end{vmatrix}.
  39. 0 = a 0 x 0 + a 1 x 1 + a 2 x 2 + a 3 x 3 , 0=a^{0}x_{0}+a^{1}x_{1}+a^{2}x_{2}+a^{3}x_{3},\,\!
  40. x i = j i a j p i j , i = 0 3. x_{i}=\sum_{j\neq i}a^{j}p_{ij},\qquad i=0\ldots 3.\,\!
  41. a i = j i y j p i j , i = 0 3. a^{i}=\sum_{j\neq i}y_{j}p^{ij},\qquad i=0\ldots 3.\,\!

Plücker_embedding.html

  1. ι : 𝐆𝐫 ( r , K n ) 𝐏 ( r K n ) span ( v 1 , , v r ) K ( v 1 v r ) \begin{aligned}\displaystyle\iota\colon\mathbf{Gr}(r,K^{n})&\displaystyle{}% \rightarrow\mathbf{P}(\wedge^{r}K^{n})\\ \displaystyle\operatorname{span}(v_{1},\ldots,v_{r})&\displaystyle{}\mapsto K(% v_{1}\wedge\cdots\wedge v_{r})\end{aligned}
  2. r r
  3. W W
  4. Z Z
  5. V V
  6. k 0 k≥0
  7. ψ ( W ) ψ ( Z ) - i 1 < < i k ( v 1 v i 1 - 1 w 1 v i 1 + 1 v i k - 1 w k v i k + 1 v r ) ( v i 1 v i k w k + 1 w r ) = 0. \psi(W)\cdot\psi(Z)-\sum_{i_{1}<\cdots<i_{k}}(v_{1}\wedge\cdots\wedge v_{i_{1}% -1}\wedge w_{1}\wedge v_{i_{1}+1}\wedge\cdots\wedge v_{i_{k}-1}\wedge w_{k}% \wedge v_{i_{k}+1}\wedge\cdots\wedge v_{r})\cdot(v_{i_{1}}\wedge\cdots\wedge v% _{i_{k}}\wedge w_{k+1}\cdots\wedge w_{r})=0.
  8. d i m ( V ) = 4 dim(V)=4
  9. r = 2 r=2
  10. [ - F o r m u l a E r r o r - ] [-FormulaError-]

Poincaré_map.html

  1. U S U\subset S
  2. P : U S P:U\to S
  3. P : U S P:U\to S
  4. P 0 := i d U P^{0}:=id_{U}
  5. P n + 1 := P P n P^{n+1}:=P\circ P^{n}
  6. P - n - 1 := P - 1 P - n P^{-n-1}:=P^{-1}\circ P^{-n}
  7. P ( n , x ) := P n ( x ) P(n,x):=P^{n}(x)
  8. P : × U U . P:\mathbb{Z}\times U\to U.

Poincaré–Birkhoff–Witt_theorem.html

  1. h ( x 1 , x 2 , , x n ) = h ( x 1 ) h ( x 2 ) h ( x n ) . h(x_{1},x_{2},\ldots,x_{n})=h(x_{1})\cdot h(x_{2})\cdots h(x_{n}).
  2. y 1 k 1 y 2 k 2 y k y_{1}^{k_{1}}y_{2}^{k_{2}}\cdots y_{\ell}^{k_{\ell}}
  3. [ u , v ] = x X c u , v , x x . [u,v]=\sum_{x\in X}c_{u,v,x}\;x.
  4. v 1 v 2 v n v_{1}v_{2}\cdots v_{n}
  5. v i L v_{i}\in L
  6. 1 n ! σ S n v σ ( 1 ) v σ ( 2 ) v σ ( n ) . \frac{1}{n!}\sum_{\sigma\in S_{n}}v_{\sigma(1)}v_{\sigma(2)}\cdots v_{\sigma(n% )}.
  7. v 1 v 2 v n v_{1}v_{2}\cdots v_{n}
  8. n \leq n
  9. Δ ( v ) = v 1 + 1 v \Delta(v)=v\otimes 1+1\otimes v
  10. L = 𝔤 𝔩 n , L=\mathfrak{gl}_{n},

Point_set_triangulation.html

  1. x n + 1 = x 1 2 + + x n 2 x_{n+1}=x_{1}^{2}+\cdots+x_{n}^{2}
  2. P P
  3. 2 n - h - 2 2n-h-2
  4. 3 n - h - 3 3n-h-3
  5. h h
  6. c h ( P ) ch(P)
  7. p p
  8. { p 1 , , p k } \{p_{1},...,p_{k}\}
  9. O ( n log n ) O(n\log n)
  10. O ( n 2 log n ) O(n^{2}\log n)
  11. O ( n 2 ) O(n^{2})
  12. O ( n 2 log n ) O(n^{2}\log n)
  13. O ( n 2 log n ) O(n^{2}\log n)
  14. O ( n 3 ) O(n^{3})
  15. O ( n log n ) O(n\log n)
  16. O ( n 2 ) O(n^{2})
  17. O ( n log n ) O(n\log n)
  18. O ( n 2 log n ) O(n^{2}\log n)
  19. O ( n 3 ) O(n^{3})

Point_spread_function.html

  1. ( x i , y i ) = ( M x o , M y o ) (x_{i},y_{i})=(Mx_{o},My_{o})
  2. O ( x o , y o ) = O ( u , v ) δ ( u - x o , v - y o ) d u d v O(x_{o},y_{o})=\int\!\!\int O(u,v)~{}\delta(u-x_{o},v-y_{o})~{}du\,dv
  3. O ( x o , y o ) O(x_{o},y_{o})
  4. I ( x i , y i ) = O ( u , v ) PSF ( u - x i / M , v - y i / M ) d u d v I(x_{i},y_{i})=\int\!\!\int O(u,v)~{}\mathrm{PSF}(u-x_{i}/M,v-y_{i}/M)\,du\,dv
  5. δ ( x , y ) e j ( k x x + k y y ) d k x d k y \delta(x,y)\propto\int\!\!\int e^{j(k_{x}x+k_{y}y)}\,dk_{x}\,dk_{y}

Poiseuille.html

  1. 1 PI = 1 Pa s = 1 N s / m = 2 10 dyn s / cm = 2 10 P 1\ \mbox{PI}~{}=1\ \mbox{Pa}~{}\cdot\mbox{s}~{}=1\ \mbox{N}~{}\cdot\mbox{s}~{}% /\mbox{m}~{}^{2}=10\ \mbox{dyn}~{}\cdot\mbox{s}~{}/\mbox{cm}~{}^{2}=10\ \mbox{% P}~{}

Poisson_manifold.html

  1. M M
  2. { , } \{\cdot,\cdot\}
  3. C ( M ) {C^{\infty}}(M)
  4. M M
  5. { f g , h } = f { g , h } + g { f , h } \{fg,h\}=f\{g,h\}+g\{f,h\}
  6. M M
  7. X f = df { f , } : C ( M ) C ( M ) X_{f}\stackrel{\,\text{df}}{=}\{f,\cdot\}:{C^{\infty}}(M)\to{C^{\infty}}(M)
  8. f f
  9. f f
  10. M M
  11. C ( M ) {C^{\infty}}(M)
  12. M M
  13. M M
  14. \mathbb{R}
  15. { , } : C ( M ) × C ( M ) C ( M ) \{\cdot,\cdot\}:{C^{\infty}}(M)\times{C^{\infty}}(M)\to{C^{\infty}}(M)
  16. { f , g } = - { g , f } \{f,g\}=-\{g,f\}
  17. { f , { g , h } } + { g , { h , f } } + { h , { f , g } } = 0 \{f,\{g,h\}\}+\{g,\{h,f\}\}+\{h,\{f,g\}\}=0
  18. { f g , h } = f { g , h } + g { f , h } \{fg,h\}=f\{g,h\}+g\{f,h\}
  19. { , } \{\cdot,\cdot\}
  20. C ( M ) {C^{\infty}}(M)
  21. f C ( M ) f\in{C^{\infty}}(M)
  22. { f , } : C ( M ) C ( M ) \{f,\cdot\}:{C^{\infty}}(M)\to{C^{\infty}}(M)
  23. C ( M ) {C^{\infty}}(M)
  24. X f X_{f}
  25. { f , g } \{f,g\}
  26. f f
  27. g g
  28. { f , g } = π ( d f d g ) \{f,g\}=\pi(df\wedge dg)
  29. π Γ ( 2 T M ) \pi\in\Gamma\left(\bigwedge^{2}TM\right)
  30. π \pi
  31. M M
  32. { f , g } = π ( d f d g ) \{f,g\}=\pi(df\wedge dg)
  33. { , } \{\cdot,\cdot\}
  34. { , } \{\cdot,\cdot\}
  35. [ π , π ] = 0 [\pi,\pi]=0
  36. [ , ] : 𝔛 p ( M ) × 𝔛 q ( M ) 𝔛 p + q ( M ) [\cdot,\cdot]:{\mathfrak{X}^{p}}(M)\times{\mathfrak{X}^{q}}(M)\to{\mathfrak{X}% ^{p+q}}(M)
  37. π : T * M T M \pi^{\sharp}:T^{*}M\to TM
  38. π \pi
  39. x M x\in M
  40. π x \pi^{\sharp}_{x}
  41. X f ( x ) {X_{f}}(x)
  42. x x
  43. x M x\in M
  44. π \pi
  45. M M
  46. π \pi
  47. x M x\in M
  48. M reg M M_{\mathrm{reg}}\subseteq M
  49. M reg = M M_{\mathrm{reg}}=M
  50. π ( T * M ) {\pi^{\sharp}}(T^{*}M)
  51. S M S\subseteq M
  52. T x S = π ( T x M ) T_{x}S={\pi^{\sharp}}(T^{\ast}_{x}M)
  53. x S x\in S
  54. π \pi
  55. π \pi
  56. π \pi
  57. S S
  58. ω S Ω 2 ( S ) \omega_{S}\in{\Omega^{2}}(S)
  59. [ ω S ( X f , X g ) ] ( x ) = - { f , g } ( x ) [{\omega_{S}}(X_{f},X_{g})](x)=-\{f,g\}(x)
  60. f , g C ( M ) f,g\in{C^{\infty}}(M)
  61. x S x\in S
  62. π \pi
  63. M reg M_{\mathrm{reg}}
  64. M M
  65. { f , g } = 0 \{f,g\}=0
  66. ( M , ω ) (M,\omega)
  67. π \pi
  68. ω - 1 \omega^{-1}
  69. ω \omega
  70. 𝔤 * \mathfrak{g}^{*}
  71. ( 𝔤 , [ , ] ) (\mathfrak{g},[\cdot,\cdot])
  72. 𝔤 \mathfrak{g}
  73. C ( 𝔤 * ) {C^{\infty}}(\mathfrak{g}^{*})
  74. { X , Y } = df [ X , Y ] \{X,Y\}\stackrel{\,\text{df}}{=}[X,Y]
  75. X , Y 𝔤 X,Y\in\mathfrak{g}
  76. 𝔤 * \mathfrak{g}^{*}
  77. \mathcal{F}
  78. 2 r 2r
  79. M M
  80. ω Ω 2 ( ) \omega\in{\Omega^{2}}(\mathcal{F})
  81. ω r \omega^{r}
  82. M M
  83. π \pi
  84. S S
  85. \mathcal{F}
  86. ω | S \omega|_{S}
  87. ( M , { , } M ) (M,\{\cdot,\cdot\}_{M})
  88. ( M , { , } M ) (M^{\prime},\{\cdot,\cdot\}_{M^{\prime}})
  89. φ : M M \varphi:M\to M^{\prime}
  90. x M x\in M
  91. f , g C ( M ) f,g\in{C^{\infty}}(M^{\prime})
  92. { f , g } M ( φ ( x ) ) = { f φ , g φ } M ( x ) . {\{f,g\}_{M^{\prime}}}(\varphi(x))={\{f\circ\varphi,g\circ\varphi\}_{M}}(x).
  93. π M \pi_{M}
  94. π M \pi_{M^{\prime}}
  95. φ \varphi
  96. 𝔓 𝔬 𝔦 𝔰 𝔰 \mathfrak{Poiss}
  97. ( M 0 × M 1 , π 0 × π 1 ) (M_{0}\times M_{1},\pi_{0}\times\pi_{1})
  98. ( M 0 , π 0 ) (M_{0},\pi_{0})
  99. ( M 1 , π 1 ) (M_{1},\pi_{1})
  100. pr i : M 0 × M 1 M i \mathrm{pr}_{i}:M_{0}\times M_{1}\to M_{i}
  101. i { 0 , 1 } i\in\{0,1\}
  102. 2 4 \mathbb{R}^{2}\to\mathbb{R}^{4}

Poisson_superalgebra.html

  1. [ , ] : A A A [\cdot,\cdot]:A\otimes A\to A
  2. [ x , ] : A A [x,\cdot]:A\to A
  3. [ x , y z ] = [ x , y ] z + ( - 1 ) | x | | y | y [ x , z ] . [x,yz]=[x,y]z+(-1)^{|x||y|}y[x,z].\,

Poisson_supermanifold.html

  1. C ( M ) C^{\infty}(M)

Polarization_density.html

  1. Δ V \Delta V
  2. Δ 𝐩 \Delta\mathbf{p}
  3. 𝐏 = Δ 𝐩 Δ V \mathbf{P}=\frac{\Delta\mathbf{p}}{\Delta V}
  4. Δ 𝐩 \Delta\mathbf{p}
  5. 𝐏 = d 𝐩 d V ( 1 ) \mathbf{P}={\mathrm{d}\mathbf{p}\over\mathrm{d}V}\qquad(1)
  6. Q b Q_{b}
  7. d q b + \mathrm{d}q_{b}^{+}
  8. 𝐝 \mathbf{d}
  9. d q b - \mathrm{d}q_{b}^{-}
  10. d 𝐩 = d q b 𝐝 \mathrm{d}\mathbf{p}=\mathrm{d}q_{b}\mathbf{d}
  11. 𝐏 = d q b d V 𝐝 \mathbf{P}={\mathrm{d}q_{b}\over\mathrm{d}V}\mathbf{d}
  12. d q b \mathrm{d}q_{b}
  13. ρ b d V \rho_{b}\mathrm{d}V
  14. 𝐏 = ρ b 𝐝 ( 2 ) \mathbf{P}=\rho_{b}\mathbf{d}\qquad(2)
  15. ρ b \rho_{b}
  16. Q b Q_{b}
  17. d q b - \mathrm{d}q_{b}^{-}
  18. d q b + \mathrm{d}q_{b}^{+}
  19. d q b - = ρ b - d V 1 = ρ b - d 1 d A \mathrm{d}q_{b}^{-}=\rho_{b}^{-}\ \mathrm{d}V_{1}=\rho_{b}^{-}d_{1}\ \mathrm{d}A
  20. d q b + = ρ b d V 2 = ρ b d 2 d A \mathrm{d}q_{b}^{+}=\rho_{b}\ \mathrm{d}V_{2}=\rho_{b}d_{2}\ \mathrm{d}A
  21. d Q b \mathrm{d}Q_{b}
  22. d V \mathrm{d}V
  23. d Q b \displaystyle\mathrm{d}Q_{b}
  24. ρ b - = - ρ b \begin{aligned}\displaystyle\rho_{b}^{-}&\displaystyle=-\rho_{b}\end{aligned}
  25. d 1 = ( d - a ) cos θ d 2 = a cos θ \begin{aligned}\displaystyle d_{1}&\displaystyle=(d-a)\cos\theta\\ \displaystyle d_{2}&\displaystyle=a\cos\theta\end{aligned}
  26. d Q b = - ρ b ( d - a ) cos θ d A - ρ b a cos θ d A = - ρ b d d A cos θ \begin{aligned}\displaystyle\mathrm{d}Q_{b}&\displaystyle=-\rho_{b}(d-a)\cos% \theta\ \mathrm{d}A-\rho_{b}a\cos\theta\ \mathrm{d}A\\ &\displaystyle=-\rho_{b}d\ \mathrm{d}A\cos\theta\end{aligned}
  27. ρ b d = P \rho_{b}d=P
  28. d Q b \displaystyle\mathrm{d}Q_{b}
  29. - ρ b = 𝐏 -\rho_{b}=\nabla\cdot\mathbf{P}
  30. ρ b \rho_{b}
  31. - Q b = V 𝐏 d V -Q_{b}=\iiint\limits_{V}\nabla\cdot\mathbf{P}\ \mathrm{d}V
  32. Q b Q_{b}
  33. Q b Q_{b}
  34. ρ b \rho_{b}
  35. - V ρ b d V = V 𝐏 d V -\iiint\limits_{V}\rho_{b}\ \mathrm{d}V=\iiint\limits_{V}\nabla\cdot\mathbf{P}% \ \mathrm{d}V
  36. - ρ b = 𝐏 -\rho_{b}=\nabla\cdot\mathbf{P}
  37. 𝐏 = χ ε 0 𝐄 , \mathbf{P}=\chi\varepsilon_{0}\mathbf{E},
  38. ( Q f ) (Q_{f})
  39. ( Q b ) (Q_{b})
  40. - Q b = χ Q total = χ ( Q f + Q b ) Q b = - χ 1 + χ Q f , \begin{aligned}\displaystyle-Q_{b}&\displaystyle=\chi Q_{\mathrm{total}}\\ &\displaystyle=\chi(Q_{f}+Q_{b})\\ \displaystyle Q_{b}&\displaystyle=-\frac{\chi}{1+\chi}Q_{f},\end{aligned}
  41. ρ b = - χ 1 + χ ρ f \rho_{b}=-\frac{\chi}{1+\chi}\rho_{f}
  42. ( ρ f = 0 ) (\rho_{f}=0)
  43. ( ρ b = 0 ) (\rho_{b}=0)
  44. σ b \sigma_{b}
  45. ρ b \rho_{b}
  46. σ b \sigma_{b}
  47. σ b = 𝐏 𝐧 ^ 𝐨𝐮𝐭 \sigma_{b}=\mathbf{P}\cdot\mathbf{\hat{n}_{out}}
  48. 𝐧 ^ 𝐨𝐮𝐭 \mathbf{\hat{n}_{out}}
  49. P i = j ϵ 0 χ i j E j , P_{i}=\sum_{j}\epsilon_{0}\chi_{ij}E_{j},\,\!
  50. P i ϵ 0 = j χ i j ( 1 ) E j + j k χ i j k ( 2 ) E j E k + j k χ i j k ( 3 ) E j E k E + \frac{P_{i}}{\epsilon_{0}}=\sum_{j}\chi^{(1)}_{ij}E_{j}+\sum_{jk}\chi_{ijk}^{(% 2)}E_{j}E_{k}+\sum_{jk\ell}\chi_{ijk\ell}^{(3)}E_{j}E_{k}E_{\ell}+\cdots\!
  51. χ ( 1 ) \chi^{(1)}
  52. χ ( 2 ) \chi^{(2)}
  53. χ ( 3 ) \chi^{(3)}
  54. ρ f \rho_{f}
  55. ρ f = ρ - ρ b \rho_{f}=\rho-\rho_{b}
  56. ρ \rho
  57. 𝐃 = ε 0 𝐄 + 𝐏 . \mathbf{D}=\varepsilon_{0}\mathbf{E}+\mathbf{P}.
  58. 𝐉 p = 𝐏 t \mathbf{J}_{p}=\frac{\partial\mathbf{P}}{\partial t}
  59. 𝐉 = 𝐉 f + × 𝐌 + 𝐏 t \mathbf{J}=\mathbf{J}_{f}+\nabla\times\mathbf{M}+\frac{\partial\mathbf{P}}{% \partial t}
  60. 𝐃 = ε 0 𝐄 + 𝐏 \mathbf{D}=\varepsilon_{0}\mathbf{E}+\mathbf{P}
  61. 𝐃 = ρ f \nabla\cdot\mathbf{D}=\rho_{f}
  62. ρ f \rho_{f}

Polarization_mode_dispersion.html

  1. Δ τ = D PMD L \Delta\tau=D\text{PMD}\sqrt{L}\,

Polaron.html

  1. H = H e + H p h + H e - p h H=H_{e}+H_{ph}+H_{e-ph}
  2. H e = k , s ξ ( k , s ) c k , s c k , s H_{e}=\sum_{k,s}\xi(k,s)c_{k,s}^{\dagger}c_{k,s}
  3. H p h = q , v ω q , v a q , v a q , v H_{ph}=\sum_{q,v}\omega_{q,v}a_{q,v}^{\dagger}a_{q,v}
  4. H e - p h = 1 2 N k , s , q , v γ ( α , q , k , v ) ω q v ( c k , s c k - q , s a q , v + c k - q , s c k , s a q , v ) H_{e-ph}=\frac{1}{\sqrt{2N}}\sum_{k,s,q,v}\gamma(\alpha,q,k,v)\omega_{qv}(c_{k% ,s}^{\dagger}c_{k-q,s}a_{q,v}+c_{k-q,s}^{\dagger}c_{k,s}a^{\dagger}_{q,v})
  5. γ \gamma
  6. Δ E \Delta E
  7. m * m*
  8. α \alpha
  9. Δ E ω - α - 0.015919622 α 2 , \frac{\Delta E}{\hbar\omega}\approx-\alpha-0.015919622\alpha^{2},
  10. ( 1 ) (1)\,
  11. m * m*
  12. m * m 1 + α 6 + 0.0236 α 2 . \frac{m^{*}}{m}\approx 1+\frac{\alpha}{6}+0.0236\alpha^{2}.
  13. ( 2 ) (2)\,
  14. E < - C P L α 2 E<-C_{PL}\alpha^{2}
  15. C P L C_{PL}
  16. Γ ( Ω ) - Im Σ ( Ω ) [ Ω - ω c - Re Σ ( Ω ) ] 2 + [ Im Σ ( Ω ) ] 2 . \Gamma(\Omega)\propto-\frac{\mathrm{Im}\Sigma(\Omega)}{\left[\Omega-\omega_{% \mathrm{c}}-\mathrm{Re}\Sigma(\Omega)\right]^{2}+\left[\mathrm{Im}\Sigma(% \Omega)\right]^{2}}.
  17. ( 3 ) (3)\,
  18. ω c \omega_{c}
  19. ω c \omega_{c}
  20. ω c = 0 \omega_{c}=0
  21. α 5.9 \alpha\geq 5.9
  22. α = 5 \alpha=5
  23. α 3. \alpha\lesssim 3.
  24. 3 < α < 6 , 3<\alpha<6,
  25. α = 6 \alpha=6
  26. Ω = ω c + Re Σ ( Ω ) \Omega=\omega_{\mathrm{c}}+\mathrm{Re}\Sigma(\Omega)
  27. ω c < ω \omega_{c}<\omega
  28. Σ ( Ω ) \Sigma(\Omega)
  29. Δ E ω - π 2 α - 0.06397 α 2 ; \frac{\Delta E}{\hbar\omega}\approx-\frac{\pi}{2}\alpha\ -0.06397\alpha^{2};
  30. ( 4 ) (4)\,
  31. m * m 1 + π 8 α + 0.1272348 α 2 . \frac{m^{*}}{m}\approx 1+\frac{\pi}{8}\alpha\ +0.1272348\alpha^{2}.
  32. ( 5 ) (5)\,
  33. m 2 D * ( α ) m 2 D = m 3 D * ( 3 4 π α ) m 3 D \frac{m^{*}_{2D}(\alpha)}{m_{2D}}=\frac{m^{*}_{3D}(\frac{3}{4}\pi\alpha)}{m_{3% D}}
  34. ( 6 ) (6)\,
  35. m 2 D * m_{\mathrm{2D}}^{*}
  36. m 3 D * m_{\mathrm{3D}}^{*}
  37. m 2 D m_{\mathrm{2D}}
  38. m 3 D m_{\mathrm{3D}}
  39. T c T_{c}
  40. T c T_{c}

Polyakov_action.html

  1. 𝒮 = T 2 d 2 σ - h h a b g μ ν ( X ) a X μ ( σ ) b X ν ( σ ) \mathcal{S}={T\over 2}\int\mathrm{d}^{2}\sigma\sqrt{-h}h^{ab}g_{\mu\nu}(X)% \partial_{a}X^{\mu}(\sigma)\partial_{b}X^{\nu}(\sigma)
  2. T T
  3. g μ ν g_{\mu\nu}
  4. h a b h_{ab}
  5. h a b h^{ab}
  6. h h
  7. h a b h_{ab}
  8. σ \sigma
  9. τ \tau
  10. X α X α + b α X^{\alpha}\rightarrow X^{\alpha}+b^{\alpha}
  11. X α X α + ω β α X β X^{\alpha}\rightarrow X^{\alpha}+\omega^{\alpha}_{\ \beta}X^{\beta}
  12. ω μ ν = - ω ν μ \omega_{\mu\nu}=-\omega_{\nu\mu}
  13. b α b^{\alpha}
  14. 𝒮 \mathcal{S}
  15. X α X^{\alpha}
  16. 𝒮 \mathcal{S}^{\prime}\,
  17. = T 2 d 2 σ - h h a b g μ ν a ( X μ + ω δ μ X δ ) b ( X ν + ω δ ν X δ ) ={T\over 2}\int\mathrm{d}^{2}\sigma\sqrt{-h}h^{ab}g_{\mu\nu}\partial_{a}\left(% X^{\mu}+\omega^{\mu}_{\ \delta}X^{\delta}\right)\partial_{b}\left(X^{\nu}+% \omega^{\nu}_{\ \delta}X^{\delta}\right)\,
  18. = 𝒮 + T 2 d 2 σ - h h a b ( ω μ δ a X μ b X δ + ω ν δ a X δ b X ν ) + O ( ω 2 ) =\mathcal{S}+{T\over 2}\int\mathrm{d}^{2}\sigma\sqrt{-h}h^{ab}\left(\omega_{% \mu\delta}\partial_{a}X^{\mu}\partial_{b}X^{\delta}+\omega_{\nu\delta}\partial% _{a}X^{\delta}\partial_{b}X^{\nu}\right)+O(\omega^{2})\,
  19. = 𝒮 + T 2 d 2 σ - h h a b ( ω μ δ + ω δ μ ) a X μ b X δ + O ( ω 2 ) = 𝒮 + O ( ω 2 ) =\mathcal{S}+{T\over 2}\int\mathrm{d}^{2}\sigma\sqrt{-h}h^{ab}\left(\omega_{% \mu\delta}+\omega_{\delta\mu}\right)\partial_{a}X^{\mu}\partial_{b}X^{\delta}+% O(\omega^{2})=\mathcal{S}+O(\omega^{2})
  20. σ α σ ~ α ( σ , τ ) \sigma^{\alpha}\rightarrow\tilde{\sigma}^{\alpha}\left(\sigma,\tau\right)
  21. h a b h ~ a b = h c d σ ~ a σ c σ ~ b σ d h^{ab}\rightarrow\tilde{h}^{ab}=h^{cd}\frac{\partial\tilde{\sigma}^{a}}{% \partial\sigma^{c}}\frac{\partial\tilde{\sigma}^{b}}{\partial\sigma^{d}}
  22. h ~ a b σ ~ a X μ σ ~ b X ν = h c d σ ~ a σ c σ ~ b σ d σ ~ a X μ σ ~ b X ν = h a b a X μ b X ν \tilde{h}^{ab}\frac{\partial}{\partial\tilde{\sigma}^{a}}X^{\mu}\frac{\partial% }{\partial\tilde{\sigma}^{b}}X^{\nu}=h^{cd}\frac{\partial\tilde{\sigma}^{a}}{% \partial\sigma^{c}}\frac{\partial\tilde{\sigma}^{b}}{\partial\sigma^{d}}\frac{% \partial}{\partial\tilde{\sigma}^{a}}X^{\mu}\frac{\partial}{\partial\tilde{% \sigma}^{b}}X^{\nu}=h^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}
  23. J = det ( σ ~ α σ β ) \mathrm{J}=\mathrm{det}\left(\frac{\partial\tilde{\sigma}^{\alpha}}{\partial% \sigma^{\beta}}\right)
  24. d 2 σ d 2 σ ~ = Jd 2 σ \mathrm{d}^{2}\sigma\rightarrow\mathrm{d}^{2}\tilde{\sigma}=\mathrm{J}\mathrm{% d}^{2}\sigma\,
  25. h = det ( h a b ) h ~ = J - 2 h h=\mathrm{det}\left(h_{ab}\right)\rightarrow\tilde{h}=\mathrm{J}^{-2}h\,
  26. - h ~ d 2 σ ~ = - h d 2 σ \sqrt{-\tilde{h}}\mathrm{d}^{2}\tilde{\sigma}=\sqrt{-h}\mathrm{d}^{2}\sigma
  27. h a b h ~ a b = Λ ( σ ) h a b h_{ab}\rightarrow\tilde{h}_{ab}=\Lambda(\sigma)h_{ab}
  28. h ~ a b = Λ - 1 ( σ ) h a b \tilde{h}^{ab}=\Lambda^{-1}(\sigma)h^{ab}
  29. det ( h ~ a b ) = Λ 2 ( σ ) det ( h a b ) \mathrm{det}(\tilde{h}_{ab})=\Lambda^{2}(\sigma)\mathrm{det}(h_{ab})
  30. 𝒮 \mathcal{S}^{\prime}\,
  31. = T 2 d 2 σ - h ~ h ~ a b g μ ν ( X ) a X μ ( σ ) b X ν ( σ ) ={T\over 2}\int\mathrm{d}^{2}\sigma\sqrt{-\tilde{h}}\tilde{h}^{ab}g_{\mu\nu}(X% )\partial_{a}X^{\mu}(\sigma)\partial_{b}X^{\nu}(\sigma)\,
  32. = T 2 d 2 σ - h ( Λ Λ - 1 ) h a b g μ ν ( X ) a X μ ( σ ) b X ν ( σ ) = 𝒮 ={T\over 2}\int\mathrm{d}^{2}\sigma\sqrt{-h}\left(\Lambda\Lambda^{-1}\right)h^% {ab}g_{\mu\nu}(X)\partial_{a}X^{\mu}(\sigma)\partial_{b}X^{\nu}(\sigma)=% \mathcal{S}
  33. T a b = 2 - h δ S δ h a b T_{ab}=\frac{2}{\sqrt{-h}}\frac{\delta S}{\delta h^{ab}}
  34. h a b = exp ( ϕ ( σ ) ) h ^ a b h_{ab}=\exp\left(\phi(\sigma)\right)\hat{h}_{ab}
  35. ϕ \phi
  36. δ S δ ϕ = δ S δ h a b δ h a b δ ϕ = 1 2 - h T a b h a b = 1 2 - h T a a = 0 T a a = tr ( T a b ) = 0 \frac{\delta S}{\delta\phi}=\frac{\delta S}{\delta h^{ab}}\frac{\delta h^{ab}}% {\delta\phi}=\frac{1}{2}\sqrt{-h}T_{ab}h^{ab}=\frac{1}{2}\sqrt{-h}T_{a}^{\ a}=% 0\rightarrow T_{a}^{\ a}=\mathrm{tr}\left(T_{ab}\right)=0
  37. h a b h^{ab}
  38. δ S δ h a b = T a b = 0 \frac{\delta S}{\delta h^{ab}}=T_{ab}=0
  39. δ - h = - 1 2 - h h a b δ h a b \delta\sqrt{-h}=-\frac{1}{2}\sqrt{-h}h_{ab}\delta h^{ab}
  40. δ S δ h a b = T 2 - h ( G a b - 1 2 h a b h c d G c d ) \frac{\delta S}{\delta h^{ab}}=\frac{T}{2}\sqrt{-h}\left(G_{ab}-\frac{1}{2}h_{% ab}h^{cd}G_{cd}\right)
  41. G a b = g μ ν a X μ b X ν G_{ab}=g_{\mu\nu}\partial_{a}X^{\mu}\partial_{b}X^{\nu}
  42. T a b = T ( G a b - 1 2 h a b h c d G c d ) = 0 T_{ab}=T\left(G_{ab}-\frac{1}{2}h_{ab}h^{cd}G_{cd}\right)=0
  43. G a b = 1 2 h a b h c d G c d G_{ab}=\frac{1}{2}h_{ab}h^{cd}G_{cd}
  44. G = det ( G a b ) = 1 4 h ( h c d G c d ) 2 G=\mathrm{det}\left(G_{ab}\right)=\frac{1}{4}h\left(h^{cd}G_{cd}\right)^{2}
  45. - h \sqrt{-h}
  46. - h = 2 - G h c d G c d \sqrt{-h}=\frac{2\sqrt{-G}}{h^{cd}G_{cd}}
  47. S = T 2 d 2 σ - h h a b G a b = T 2 d 2 σ 2 - G h c d G c d h a b G a b = T d 2 σ - G S={T\over 2}\int\mathrm{d}^{2}\sigma\sqrt{-h}h^{ab}G_{ab}={T\over 2}\int% \mathrm{d}^{2}\sigma\frac{2\sqrt{-G}}{h^{cd}G_{cd}}h^{ab}G_{ab}=T\int\mathrm{d% }^{2}\sigma\sqrt{-G}
  48. - h h a b η a b \sqrt{-h}h^{ab}\rightarrow\eta^{ab}
  49. 𝒮 = T 2 d 2 σ - η η a b g μ ν ( X ) a X μ ( σ ) b X ν ( σ ) = T 2 d 2 σ ( X ˙ 2 - X 2 ) \mathcal{S}={T\over 2}\int\mathrm{d}^{2}\sigma\sqrt{-\eta}\eta^{ab}g_{\mu\nu}(% X)\partial_{a}X^{\mu}(\sigma)\partial_{b}X^{\nu}(\sigma)={T\over 2}\int\mathrm% {d}^{2}\sigma\left(\dot{X}^{2}-X^{\prime 2}\right)
  50. η a b = ( 1 0 0 - 1 ) \eta_{ab}=\left(\begin{array}[]{cc}1&0\\ 0&-1\end{array}\right)
  51. T a b = 0 T_{ab}=0
  52. T 01 = T 10 = X ˙ X = 0 T_{01}=T_{10}=\dot{X}X^{\prime}=0
  53. T 00 = T 11 = 1 2 ( X ˙ 2 + X 2 ) = 0 T_{00}=T_{11}=\frac{1}{2}\left(\dot{X}^{2}+X^{\prime 2}\right)=0
  54. X μ X μ + δ X μ X^{\mu}\rightarrow X^{\mu}+\delta X^{\mu}
  55. δ 𝒮 = T d 2 σ η a b a X μ b δ X μ = \delta\mathcal{S}=T\int\mathrm{d}^{2}\sigma\eta^{ab}\partial_{a}X^{\mu}% \partial_{b}\delta X_{\mu}=
  56. = - T d 2 σ η a b a b X μ δ X μ + ( T d τ X δ X ) σ = π - ( T d τ X δ X ) σ = 0 = 0 =-T\int\mathrm{d}^{2}\sigma\eta^{ab}\partial_{a}\partial_{b}X^{\mu}\delta X_{% \mu}+\left(T\int d\tau X^{\prime}\delta X\right)_{\sigma=\pi}-\left(T\int d% \tau X^{\prime}\delta X\right)_{\sigma=0}=0
  57. X μ = η a b a b X μ = 0 \square X^{\mu}=\eta^{ab}\partial_{a}\partial_{b}X^{\mu}=0
  58. X μ ( τ , σ + π ) = X μ ( τ , σ ) X^{\mu}(\tau,\sigma+\pi)=X^{\mu}(\tau,\sigma)
  59. σ X μ ( τ , 0 ) = 0 , σ X μ ( τ , π ) = 0 \partial_{\sigma}X^{\mu}(\tau,0)=0,\partial_{\sigma}X^{\mu}(\tau,\pi)=0
  60. X μ ( τ , 0 ) = b μ , X μ ( τ , π ) = b μ X^{\mu}(\tau,0)=b^{\mu},X^{\mu}(\tau,\pi)=b^{\prime\mu}
  61. ξ ± = τ ± σ \xi^{\pm}=\tau\pm\sigma
  62. + - X μ = 0 \partial_{+}\partial_{-}X^{\mu}=0
  63. ( + X ) 2 = ( - X ) 2 = 0 (\partial_{+}X)^{2}=(\partial_{-}X)^{2}=0
  64. X μ = X + μ ( ξ + ) + X - μ ( ξ - ) X^{\mu}=X^{\mu}_{+}(\xi^{+})+X^{\mu}_{-}(\xi^{-})

Polygon_triangulation.html

  1. n ( n + 1 ) ( 2 n - 4 ) ( n - 2 ) ! \tfrac{n\cdot(n+1)\cdots(2n-4)}{(n-2)!}
  2. < v a r > P <var>P

Polyhex_(mathematics).html

  1. n n
  2. n n

Polynomial-time_approximation_scheme.html

  1. n p o l y l o g ( n ) n^{polylog(n)}
  2. ϵ > 0 \epsilon>0

Polynomial_basis.html

  1. { 1 , α , , α m - 1 } , \{1,\alpha,\ldots,\alpha^{m-1}\}\,,
  2. { 0 , 1 , α , α 2 , , α p m - 2 } \{0,1,\alpha,\alpha^{2},\ldots,\alpha^{p^{m}-2}\}
  3. ( 2 α 2 + 2 α + 1 ) + ( 2 α + 2 ) = 2 α 2 + 4 α + 3 mod 3 = 2 α 2 + α (2\alpha^{2}+2\alpha+1)+(2\alpha+2)=2\alpha^{2}+4\alpha+3\mod{3}=2\alpha^{2}+\alpha

Polynomial_hierarchy.html

  1. Σ i + 1 P := NP Σ i P \Sigma_{i+1}^{\rm P}:=\mbox{NP}~{}^{\Sigma_{i}^{\rm P}}
  2. Π i + 1 P := coNP Σ i P \Pi_{i+1}^{\rm P}:=\mbox{coNP}~{}^{\Sigma_{i}^{\rm P}}
  3. Σ 1 P = NP , Π 1 P = coNP \Sigma_{1}^{\rm P}={\rm NP},\Pi_{1}^{\rm P}={\rm coNP}
  4. Δ 2 P = P NP \Delta_{2}^{\rm P}={\rm P^{NP}}
  5. L L
  6. p p
  7. p L := { x { 0 , 1 } * | ( w { 0 , 1 } p ( | x | ) ) x , w L } , \exists^{p}L:=\left\{x\in\{0,1\}^{*}\ \left|\ \left(\exists w\in\{0,1\}^{\leq p% (|x|)}\right)\langle x,w\rangle\in L\right.\right\},
  8. x , w { 0 , 1 } * \langle x,w\rangle\in\{0,1\}^{*}
  9. p L \exists^{p}L
  10. | w | p ( | x | ) |w|\leq p(|x|)
  11. p L \exists^{p}L
  12. x p L x\in\exists^{p}L
  13. x , w L \langle x,w\rangle\in L
  14. p L := { x { 0 , 1 } * | ( w { 0 , 1 } p ( | x | ) ) x , w L } \forall^{p}L:=\left\{x\in\{0,1\}^{*}\ \left|\ \left(\forall w\in\{0,1\}^{\leq p% (|x|)}\right)\langle x,w\rangle\in L\right.\right\}
  15. ( p L ) c = p L c \left(\exists^{p}L\right)^{\rm c}=\forall^{p}L^{\rm c}
  16. ( p L ) c = p L c \left(\forall^{p}L\right)^{\rm c}=\exists^{p}L^{\rm c}
  17. 𝒞 \mathcal{C}
  18. P 𝒞 := { p L | p is a polynomial and L 𝒞 } \exists^{\rm P}\mathcal{C}:=\left\{\exists^{p}L\ |\ p\mbox{ is a polynomial % and }~{}L\in\mathcal{C}\right\}
  19. P 𝒞 := { p L | p is a polynomial and L 𝒞 } \forall^{\rm P}\mathcal{C}:=\left\{\forall^{p}L\ |\ p\mbox{ is a polynomial % and }~{}L\in\mathcal{C}\right\}
  20. co P 𝒞 = P co 𝒞 {\rm co}\exists^{\rm P}\mathcal{C}=\forall^{\rm P}{\rm co}\mathcal{C}
  21. co P 𝒞 = P co 𝒞 {\rm co}\forall^{\rm P}\mathcal{C}=\exists^{\rm P}{\rm co}\mathcal{C}
  22. co 𝒞 = { L c | L 𝒞 } {\rm co}\mathcal{C}=\left\{L^{c}|L\in\mathcal{C}\right\}
  23. NP = P P {\rm NP}=\exists^{\rm P}{\rm P}
  24. coNP = P P {\rm coNP}=\forall^{\rm P}{\rm P}
  25. Σ 0 P := Π 0 P := P \Sigma_{0}^{\rm P}:=\Pi_{0}^{\rm P}:={\rm P}
  26. Σ k + 1 P := P Π k P \Sigma_{k+1}^{\rm P}:=\exists^{\rm P}\Pi_{k}^{\rm P}
  27. Π k + 1 P := P Σ k P \Pi_{k+1}^{\rm P}:=\forall^{\rm P}\Sigma_{k}^{\rm P}
  28. NP = Σ 1 P {\rm NP}=\Sigma_{1}^{\rm P}
  29. coNP = Π 1 P {\rm coNP}=\Pi_{1}^{\rm P}
  30. Σ k P \Sigma_{k}^{\rm P}
  31. Π k P \Pi_{k}^{\rm P}
  32. k k
  33. Σ i P Δ i + 1 P Σ i + 1 P \Sigma_{i}^{\rm P}\subseteq\Delta_{i+1}^{\rm P}\subseteq\Sigma_{i+1}^{\rm P}
  34. Π i P Δ i + 1 P Π i + 1 P \Pi_{i}^{\rm P}\subseteq\Delta_{i+1}^{\rm P}\subseteq\Pi_{i+1}^{\rm P}
  35. Σ i P = co Π i P \Sigma_{i}^{\rm P}={\rm co}\Pi_{i}^{\rm P}
  36. Σ k P = Σ k + 1 P \Sigma_{k}^{\rm P}=\Sigma_{k+1}^{\rm P}
  37. Σ k P = Π k P \Sigma_{k}^{\rm P}=\Pi_{k}^{\rm P}
  38. i > k i>k
  39. Σ i P = Σ k P \Sigma_{i}^{\rm P}=\Sigma_{k}^{\rm P}
  40. Σ k P \Sigma_{k}^{\rm P}
  41. m P \leq_{\rm m}^{\rm P}
  42. m P \leq_{\rm m}^{\rm P}
  43. 𝒞 \mathcal{C}
  44. L 𝒞 L\in\mathcal{C}
  45. A m P L A\leq_{\rm m}^{\rm P}L
  46. A 𝒞 A\in\mathcal{C}
  47. K i K_{i}
  48. Σ i P \Sigma_{i}^{\rm P}
  49. Σ i + 1 P = ( Σ i P ) K i \Sigma_{i+1}^{\rm P}=\left(\Sigma_{i}^{\rm P}\right)^{K_{i}}
  50. Π i + 1 P = ( Π i P ) K i c \Pi_{i+1}^{\rm P}=\left(\Pi_{i}^{\rm P}\right)^{K_{i}^{\rm c}}
  51. Σ 2 P = NP SAT \Sigma_{2}^{\rm P}={\rm NP}^{\rm SAT}
  52. 𝒞 \mathcal{C}
  53. 𝒞 \mathcal{C}
  54. Σ 2 \Sigma_{2}

Polynomial_remainder_theorem.html

  1. f ( x ) f(x)
  2. x - a x-a
  3. f ( a ) . f(a).
  4. x - a x-a
  5. f ( x ) f(x)
  6. f ( a ) = 0. f(a)=0.
  7. f ( x ) = x 3 - 12 x 2 - 42 f(x)=x^{3}-12x^{2}-42\,
  8. f ( x ) f(x)\,
  9. x - 3 x-3\,
  10. x 2 - 9 x - 27 x^{2}-9x-27\,
  11. - 123 -123\,
  12. f ( 3 ) = - 123 f(3)=-123\,
  13. f ( x ) = a x 2 + b x + c f(x)=ax^{2}+bx+c
  14. f ( x ) < m t p l > x - r \displaystyle\frac{f(x)}{<}mtpl>{{x-r}}
  15. f ( x ) = a x 2 + b x + c = ( a x + b + a r ) ( x - r ) + a r 2 + b r + c f(x)=ax^{2}+bx+c=(ax+b+ar)(x-r)+{a{r^{2}}+br+c}
  16. R = a r 2 + b r + c R=ar^{2}+br+c
  17. f ( r ) = R f(r)=R
  18. f ( x ) = q ( x ) g ( x ) + r ( x ) and r ( x ) = 0 or deg ( r ) < deg ( g ) . f(x)=q(x)g(x)+r(x)\quad\,\text{and}\quad r(x)=0\;\,\text{or}\;\deg(r)<\deg(g)\,.
  19. g ( x ) = x - a g(x)=x-a
  20. f ( x ) = q ( x ) ( x - a ) + r . f(x)=q(x)(x-a)+r\,.
  21. x = a x=a
  22. f ( a ) = r . f(a)=r\,.
  23. f ( a ) f(a)\,
  24. r r

Pons_asinorum.html

  1. x + y + z = 0 and x = y , x+y+z=0\,\text{ and }\|x\|=\|y\|,\,
  2. x - z = y - z . \|x-z\|=\|y-z\|.\,
  3. x - z 2 = x 2 - 2 x z + z 2 , \|x-z\|^{2}=\|x\|^{2}-2x\cdot z+\|z\|^{2},\,
  4. x z = x z cos θ x\cdot z=\|x\|\|z\|\cos\theta\,

Pontryagin's_minimum_principle.html

  1. 𝒰 \mathcal{U}
  2. H ( x * ( t ) , u * ( t ) , λ * ( t ) , t ) H ( x * ( t ) , u , λ * ( t ) , t ) , u 𝒰 , t [ t 0 , t f ] H(x^{*}(t),u^{*}(t),\lambda^{*}(t),t)\leq H(x^{*}(t),u,\lambda^{*}(t),t),\quad% \forall u\in\mathcal{U},\quad t\in[t_{0},t_{f}]
  3. x * C 1 [ t 0 , t f ] x^{*}\in C^{1}[t_{0},t_{f}]
  4. λ * B V [ t 0 , t f ] \lambda^{*}\in BV[t_{0},t_{f}]
  5. t f t_{f}
  6. ( H t 0 ) \left(\tfrac{\partial H}{\partial t}\equiv 0\right)
  7. H ( x * ( t ) , u * ( t ) , λ * ( t ) ) constant H(x^{*}(t),u^{*}(t),\lambda^{*}(t))\equiv\mathrm{constant}\,
  8. H ( x * ( t ) , u * ( t ) , λ * ( t ) ) 0. H(x^{*}(t),u^{*}(t),\lambda^{*}(t))\equiv 0.\,
  9. Ψ T ( x ( T ) ) = Ψ ( x ) T | x = x ( T ) \Psi_{T}(x(T))=\frac{\partial\Psi(x)}{\partial T}|_{x=x(T)}\,
  10. Ψ x ( x ( T ) ) = [ Ψ ( x ) x 1 | x = x ( T ) Ψ ( x ) x n | x = x ( T ) ] \Psi_{x}(x(T))=\begin{bmatrix}\frac{\partial\Psi(x)}{\partial x_{1}}|_{x=x(T)}% &\cdots&\frac{\partial\Psi(x)}{\partial x_{n}}|_{x=x(T)}\end{bmatrix}
  11. H x ( x * , u * , λ * , t ) = [ H x 1 | x = x * , u = u * , λ = λ * H x n | x = x * , u = u * , λ = λ * ] H_{x}(x^{*},u^{*},\lambda^{*},t)=\begin{bmatrix}\frac{\partial H}{\partial x_{% 1}}|_{x=x^{*},u=u^{*},\lambda=\lambda^{*}}&\cdots&\frac{\partial H}{\partial x% _{n}}|_{x=x^{*},u=u^{*},\lambda=\lambda^{*}}\end{bmatrix}
  12. L x ( x * , u * ) = [ L x 1 | x = x * , u = u * L x n | x = x * , u = u * ] L_{x}(x^{*},u^{*})=\begin{bmatrix}\frac{\partial L}{\partial x_{1}}|_{x=x^{*},% u=u^{*}}&\cdots&\frac{\partial L}{\partial x_{n}}|_{x=x^{*},u=u^{*}}\end{bmatrix}
  13. f x ( x * , u * ) = [ f 1 x 1 | x = x * , u = u * f 1 x n | x = x * , u = u * f n x 1 | x = x * , u = u * f n x n | x = x * , u = u * ] f_{x}(x^{*},u^{*})=\begin{bmatrix}\frac{\partial f_{1}}{\partial x_{1}}|_{x=x^% {*},u=u^{*}}&\cdots&\frac{\partial f_{1}}{\partial x_{n}}|_{x=x^{*},u=u^{*}}\\ \vdots&\ddots&\vdots\\ \frac{\partial f_{n}}{\partial x_{1}}|_{x=x^{*},u=u^{*}}&\ldots&\frac{\partial f% _{n}}{\partial x_{n}}|_{x=x^{*},u=u^{*}}\end{bmatrix}
  14. x x
  15. u u
  16. x ˙ = f ( x , u ) , x ( 0 ) = x 0 , u ( t ) 𝒰 , t [ 0 , T ] \dot{x}=f(x,u),\quad x(0)=x_{0},\quad u(t)\in\mathcal{U},\quad t\in[0,T]
  17. 𝒰 \mathcal{U}
  18. T T
  19. u 𝒰 u\in\mathcal{U}
  20. t [ 0 , T ] t\in[0,T]
  21. J J
  22. J = Ψ ( x ( T ) ) + 0 T L ( x ( t ) , u ( t ) ) d t J=\Psi(x(T))+\int^{T}_{0}L(x(t),u(t))\,dt
  23. L L
  24. λ \lambda
  25. H H
  26. t [ 0 , T ] t\in[0,T]
  27. H ( λ ( t ) , x ( t ) , u ( t ) , t ) = λ ( t ) f ( x ( t ) , u ( t ) ) + L ( x ( t ) , u ( t ) ) H(\lambda(t),x(t),u(t),t)=\lambda^{\prime}(t)f(x(t),u(t))+L(x(t),u(t))\,
  28. λ \lambda^{\prime}
  29. λ \lambda
  30. x * x^{*}
  31. u * u^{*}
  32. λ * \lambda^{*}
  33. H H
  34. ( 1 ) H ( x * ( t ) , u * ( t ) , λ * ( t ) , t ) H ( x * ( t ) , u , λ * ( t ) , t ) (1)\qquad H(x^{*}(t),u^{*}(t),\lambda^{*}(t),t)\leq H(x^{*}(t),u,\lambda^{*}(t% ),t)\,
  35. t [ 0 , T ] t\in[0,T]
  36. u 𝒰 u\in\mathcal{U}
  37. ( 2 ) Ψ T ( x ( T ) ) + H ( T ) = 0 (2)\qquad\Psi_{T}(x(T))+H(T)=0\,
  38. ( 3 ) - λ ˙ ( t ) = H x ( x * ( t ) , u * ( t ) , λ ( t ) , t ) = λ ( t ) f x ( x * ( t ) , u * ( t ) ) + L x ( x * ( t ) , u * ( t ) ) (3)\qquad-\dot{\lambda}^{\prime}(t)=H_{x}(x^{*}(t),u^{*}(t),\lambda(t),t)=% \lambda^{\prime}(t)f_{x}(x^{*}(t),u^{*}(t))+L_{x}(x^{*}(t),u^{*}(t))
  39. x ( T ) x(T)
  40. ( 4 ) λ ( T ) = Ψ x ( x ( T ) ) (4)\qquad\lambda^{\prime}(T)=\Psi_{x}(x(T))\,
  41. x ( T ) x(T)

Population_ecology.html

  1. d N d T = B - D = b N - d N = ( b - d ) N = r N , \frac{dN}{dT}=B-D=bN-dN=(b-d)N=rN,
  2. d N d T = a N ( 1 - N K ) , \frac{dN}{dT}=aN\left(1-\frac{N}{K}\right),

Population_growth.html

  1. P o p u l a t i o n g r o w t h r a t e = P ( t 2 ) - P ( t 1 ) P ( t 1 ) ( t 2 - t 1 ) Population\ growth\ rate=\frac{P(t_{2})-P(t_{1})}{P(t_{1})(t_{2}-t_{1})}

Positive-definite_function.html

  1. f ( 0 ) = 0 f(0)=0
  2. f ( x ) > 0 f(x)>0
  3. x D x\in D
  4. \geq\,
  5. \leq\,
  6. A = ( a i , j ) i , j = 1 n , a i , j = f ( x i - x j ) A=(a_{i,j})_{i,j=1}^{n}~{},\quad a_{i,j}=f(x_{i}-x_{j})
  7. f ( 0 ) 0 , | f ( x ) | f ( 0 ) f(0)\geq 0~{},\quad|f(x)|\leq f(0)
  8. R d R^{d}

Positive_set_theory.html

  1. { x ϕ } \{x\mid\phi\}
  2. ϕ \phi
  3. G P K + GPK^{+}_{\infty}
  4. x = y a ( a x a y ) x=y\Leftrightarrow\forall a\,(a\in x\Leftrightarrow a\in y)
  5. \emptyset
  6. ¬ x x \,\neg\exists x\;x\in\emptyset\,
  7. \perp
  8. ϕ \phi
  9. \vee
  10. \wedge
  11. \exists
  12. \forall
  13. = =
  14. \in
  15. x x
  16. ϕ ( x ) \phi(x)
  17. \forall
  18. \exists
  19. ϕ ( x ) \phi(x)
  20. ϕ ( x ) \phi(x)
  21. { x ϕ ( x ) } \{x\mid\phi(x)\}
  22. ω \omega
  23. ω \omega
  24. ω \omega

Post's_theorem.html

  1. Σ m 0 \Sigma^{0}_{m}
  2. m m
  3. ϕ ( s ) \phi(s)
  4. Σ m 0 \Sigma^{0}_{m}
  5. n 1 n 2 n 3 n 4 Q n m ρ ( n 1 , , n m , x 1 , , x k ) , \exists n_{1}\forall n_{2}\exists n_{3}\forall n_{4}\cdots Qn_{m}\rho(n_{1},% \ldots,n_{m},x_{1},\ldots,x_{k}),
  6. \forall
  7. \exists
  8. ( n 1 1 n 2 1 n j 1 1 ) ( n 1 2 n j 2 2 ) ( n 1 3 ) ( Q 1 n 1 m ) ρ ( n 1 1 , n j m m , x 1 , , x k ) \left(\exists n^{1}_{1}\exists n^{1}_{2}\cdots\exists n^{1}_{j_{1}}\right)% \left(\forall n^{2}_{1}\cdots\forall n^{2}_{j_{2}}\right)\left(\exists n^{3}_{% 1}\cdots\right)\cdots\left(Q_{1}n^{m}_{1}\cdots\right)\rho(n^{1}_{1},\ldots n^% {m}_{j_{m}},x_{1},\ldots,x_{k})
  9. ρ \rho
  10. Σ m 0 \Sigma^{0}_{m}
  11. Σ m 0 \Sigma^{0}_{m}
  12. Σ m 0 \Sigma^{0}_{m}
  13. Σ m 0 \Sigma^{0}_{m}
  14. ϕ ( s ) \phi(s)
  15. ϕ ( n ) \phi(n)
  16. Σ m 0 \Sigma^{0}_{m}
  17. Σ n 0 \Sigma^{0}_{n}
  18. n > m n>m
  19. Σ m + 1 0 \Sigma^{0}_{m+1}
  20. Σ m 0 \Sigma^{0}_{m}
  21. Σ m 0 \Sigma^{0}_{m}
  22. Σ m 0 , B \Sigma^{0,B}_{m}
  23. Σ m 0 \Sigma^{0}_{m}
  24. A T B A\leq_{T}B
  25. A A^{\prime}
  26. A ( n ) A^{(n)}
  27. A ( 0 ) A^{(0)}
  28. A ( n + 1 ) A^{(n+1)}
  29. A ( n ) A^{(n)}
  30. ( n ) \emptyset^{(n)}
  31. Σ n + 1 0 \Sigma^{0}_{n+1}
  32. B B
  33. ( n ) \emptyset^{(n)}
  34. Σ 1 0 , ( n ) \Sigma^{0,\emptyset^{(n)}}_{1}
  35. ( n ) \emptyset^{(n)}
  36. Σ n 0 \Sigma^{0}_{n}
  37. n > 0 n>0
  38. Σ n 0 \Sigma^{0}_{n}
  39. ( n ) \emptyset^{(n)}
  40. Σ n + 1 0 , C \Sigma^{0,C}_{n+1}
  41. Σ 1 0 , C ( n ) \Sigma^{0,C^{(n)}}_{1}
  42. Δ n + 1 \Delta_{n+1}
  43. B T ( n ) B\leq_{T}\emptyset^{(n)}
  44. Δ n + 1 C \Delta^{C}_{n+1}
  45. B T C ( n ) B\leq_{T}C^{(n)}
  46. Σ n 0 \Sigma^{0}_{n}
  47. ( m ) \emptyset^{(m)}

Potts_model.html

  1. θ n = 2 π n q , \theta_{n}=\frac{2\pi n}{q},
  2. H c = J c ( i , j ) cos ( θ s i - θ s j ) H_{c}=J_{c}\sum_{(i,j)}\cos\left(\theta_{s_{i}}-\theta_{s_{j}}\right)
  3. H p = - J p ( i , j ) δ ( s i , s j ) H_{p}=-J_{p}\sum_{(i,j)}\delta(s_{i},s_{j})\,
  4. β H g = - β ( i , j ) J i j δ ( s i , s j ) - i h i s i \beta H_{g}=-\beta\sum_{(i,j)}J_{ij}\delta(s_{i},s_{j})-\sum_{i}h_{i}s_{i}\,
  5. Q 𝐙 = { s = ( , s - 1 , s 0 , s 1 , ) : s k Q k 𝐙 } Q^{\mathbf{Z}}=\{s=(\ldots,s_{-1},s_{0},s_{1},\ldots):s_{k}\in Q\;\forall k\in% \mathbf{Z}\}
  6. τ ( s k ) = s k + 1 \tau(s_{k})=s_{k+1}
  7. C m [ ξ 0 , , ξ k ] = { s Q 𝐙 : s m = ξ 0 , , s m + k = ξ k } C_{m}[\xi_{0},\ldots,\xi_{k}]=\{s\in Q^{\mathbf{Z}}:s_{m}=\xi_{0},\ldots,s_{m+% k}=\xi_{k}\}
  8. V ( s ) = - J δ ( s 0 , s 1 ) V(s)=-J\delta(s_{0},s_{1})
  9. H n ( s ) = k = 0 n V ( τ k s ) H_{n}(s)=\sum_{k=0}^{n}V(\tau^{k}s)
  10. Z n ( V ) = s 0 , , s n Q exp ( - β H n ( C 0 [ s 0 , s 1 , , s n ] ) ) Z_{n}(V)=\sum_{s_{0},\ldots,s_{n}\in Q}\exp(-\beta H_{n}(C_{0}[s_{0},s_{1},% \ldots,s_{n}]))
  11. μ ( C k [ s 0 , s 1 , , s n ] ) = 1 Z n ( V ) exp ( - β H n ( C k [ s 0 , s 1 , , s n ] ) ) \mu(C_{k}[s_{0},s_{1},\ldots,s_{n}])=\frac{1}{Z_{n}(V)}\exp(-\beta H_{n}(C_{k}% [s_{0},s_{1},\ldots,s_{n}]))
  12. A n ( V ) = - k T log Z n ( V ) A_{n}(V)=-kT\log Z_{n}(V)
  13. P ( V ) = lim n 1 n log Z n ( V ) P(V)=\lim_{n\to\infty}\frac{1}{n}\log Z_{n}(V)
  14. Z n ( c ) = e - c β s 0 , , s n Q 1 Z_{n}(c)=e^{-c\beta}\sum_{s_{0},\ldots,s_{n}\in Q}1
  15. Z n ( c ) = e - c β q n + 1 Z_{n}(c)=e^{-c\beta}q^{n+1}
  16. Z n ( c ) = e - c β | Fix τ n | = e - c β Tr A n Z_{n}(c)=e^{-c\beta}|\mbox{Fix}~{}\,\tau^{n}|=e^{-c\beta}\mbox{Tr}~{}A^{n}
  17. Fix τ n = { s Q 𝐙 : τ n s = s } \mbox{Fix}~{}\,\tau^{n}=\{s\in Q^{\mathbf{Z}}:\tau^{n}s=s\}
  18. V ( σ ) = - J p s 0 s 1 V(\sigma)=-J_{p}s_{0}s_{1}\,
  19. M σ σ = exp ( β J p σ σ ) M_{\sigma\sigma^{\prime}}=\exp\left(\beta J_{p}\sigma\sigma^{\prime}\right)
  20. Z n ( V ) = Tr M n Z_{n}(V)=\mbox{Tr}~{}\,M^{n}
  21. P γ ( u ) = γ u 0 + u - f p p = γ # { i : u i u i + 1 } + i = 1 n | u i - f i | p P_{\gamma}(u)=\gamma\|\nabla u\|_{0}+\|u-f\|_{p}^{p}=\gamma\#\{i:u_{i}\neq u_{% i+1}\}+\sum_{i=1}^{n}|u_{i}-f_{i}|^{p}
  22. u 0 \|\nabla u\|_{0}
  23. u - f p p \|u-f\|_{p}^{p}

Poundal.html

  1. 1 pdl = 1 lb m ft s 2 1\,\,\text{pdl}=1\,\tfrac{\,\text{lb}_{m}\cdot\,\text{ft}}{\,\text{s}^{2}}
  2. 150 lb m 8 ft s 2 = 1200 pdl 150\,\,\text{lb}_{m}\cdot 8\,\tfrac{\,\text{ft}}{\,\text{s}^{2}}=1200\,\,\text% {pdl}
  3. 4.66 slug 8 ft s 2 = 37.3 lb F 4.66\,\,\text{slug}\cdot 8\,\tfrac{\,\text{ft}}{\,\text{s}^{2}}=37.3\,\,\text{% lb}_{F}
  4. 150 lb m 0.249 g = 37.3 lb F 150\,\,\text{lb}_{m}\cdot 0.249\,g=37.3\,\,\text{lb}_{F}

Power-flow_study.html

  1. 2 ( N - 1 ) - ( R - 1 ) 2(N-1)-(R-1)
  2. 2 ( N - 1 ) - ( R - 1 ) 2(N-1)-(R-1)
  3. 2 ( N - 1 ) - ( R - 1 ) 2(N-1)-(R-1)
  4. 0 = - P i + k = 1 N | V i | | V k | ( G i k cos θ i k + B i k sin θ i k ) 0=-P_{i}+\sum_{k=1}^{N}|V_{i}||V_{k}|(G_{ik}\cos\theta_{ik}+B_{ik}\sin\theta_{% ik})
  5. P i P_{i}
  6. G i k G_{ik}
  7. B i k B_{ik}
  8. θ i k \theta_{ik}
  9. θ i k = δ i - δ k \theta_{ik}=\delta_{i}-\delta_{k}
  10. 0 = - Q i + k = 1 N | V i | | V k | ( G i k sin θ i k - B i k cos θ i k ) 0=-Q_{i}+\sum_{k=1}^{N}|V_{i}||V_{k}|(G_{ik}\sin\theta_{ik}-B_{ik}\cos\theta_{% ik})
  11. Q i Q_{i}
  12. θ i k \theta_{ik}
  13. [ Δ θ Δ | V | ] = - J - 1 [ Δ P Δ Q ] \begin{bmatrix}\Delta\theta\\ \Delta|V|\end{bmatrix}=-J^{-1}\begin{bmatrix}\Delta P\\ \Delta Q\end{bmatrix}
  14. Δ P \Delta P
  15. Δ Q \Delta Q
  16. Δ P i = - P i + k = 1 N | V i | | V k | ( G i k cos θ i k + B i k sin θ i k ) \Delta P_{i}=-P_{i}+\sum_{k=1}^{N}|V_{i}||V_{k}|(G_{ik}\cos\theta_{ik}+B_{ik}% \sin\theta_{ik})
  17. Δ Q i = - Q i + k = 1 N | V i | | V k | ( G i k sin θ i k - B i k cos θ i k ) \Delta Q_{i}=-Q_{i}+\sum_{k=1}^{N}|V_{i}||V_{k}|(G_{ik}\sin\theta_{ik}-B_{ik}% \cos\theta_{ik})
  18. J J
  19. J = [ Δ P θ Δ P | V | Δ Q θ Δ Q | V | ] J=\begin{bmatrix}\dfrac{\partial\Delta P}{\partial\theta}&\dfrac{\partial% \Delta P}{\partial|V|}\\ \dfrac{\partial\Delta Q}{\partial\theta}&\dfrac{\partial\Delta Q}{\partial|V|}% \end{bmatrix}
  20. θ m + 1 = θ m + Δ θ \theta^{m+1}=\theta^{m}+\Delta\theta\,
  21. | V | m + 1 = | V | m + Δ | V | |V|^{m+1}=|V|^{m}+\Delta|V|\,

Power_function.html

  1. f : x c x r c , r f\colon x\mapsto cx^{r}\;c,r\in\mathbb{R}
  2. c c
  3. r r
  4. x x
  5. x 0 x\geq 0

Power_rating.html

  1. P D , m a x P_{D,max}
  2. T D , m a x T_{D,max}
  3. T A T_{A}
  4. θ D A \theta_{DA}
  5. P D , m a x = T D , m a x - T A θ D A P_{D,max}=\frac{T_{D,max}-T_{A}}{\theta_{DA}}
  6. η \eta
  7. P m a x = P D , m a x 1 - η P_{max}=\frac{P_{D,max}}{1-\eta}

Power_series_solution_of_differential_equations.html

  1. a 2 ( z ) f ′′ ( z ) + a 1 ( z ) f ( z ) + a 0 ( z ) f ( z ) = 0. a_{2}(z)f^{\prime\prime}(z)+a_{1}(z)f^{\prime}(z)+a_{0}(z)f(z)=0.\;\!
  2. f ′′ + a 1 ( z ) a 2 ( z ) f + a 0 ( z ) a 2 ( z ) f = 0. f^{\prime\prime}+{a_{1}(z)\over a_{2}(z)}f^{\prime}+{a_{0}(z)\over a_{2}(z)}f=0.
  3. f = k = 0 A k z k . f=\sum_{k=0}^{\infty}A_{k}z^{k}.
  4. f ′′ - 2 z f + λ f = 0 ; λ = 1 f^{\prime\prime}-2zf^{\prime}+\lambda f=0;\;\lambda=1
  5. f = k = 0 A k z k f=\sum_{k=0}^{\infty}A_{k}z^{k}
  6. f = k = 0 k A k z k - 1 f^{\prime}=\sum_{k=0}^{\infty}kA_{k}z^{k-1}
  7. f ′′ = k = 0 k ( k - 1 ) A k z k - 2 f^{\prime\prime}=\sum_{k=0}^{\infty}k(k-1)A_{k}z^{k-2}
  8. k = 0 k ( k - 1 ) A k z k - 2 - 2 z k = 0 k A k z k - 1 + k = 0 A k z k = 0 \displaystyle{}\quad\sum_{k=0}^{\infty}k(k-1)A_{k}z^{k-2}-2z\sum_{k=0}^{\infty% }kA_{k}z^{k-1}+\sum_{k=0}^{\infty}A_{k}z^{k}=0
  9. = k + 2 = 0 ( k + 2 ) ( ( k + 2 ) - 1 ) A k + 2 z ( k + 2 ) - 2 - k = 0 2 k A k z k + k = 0 A k z k \displaystyle=\sum_{k+2=0}^{\infty}(k+2)((k+2)-1)A_{k+2}z^{(k+2)-2}-\sum_{k=0}% ^{\infty}2kA_{k}z^{k}+\sum_{k=0}^{\infty}A_{k}z^{k}
  10. ( k + 2 ) ( k + 1 ) A k + 2 + ( - 2 k + 1 ) A k = 0 (k+2)(k+1)A_{k+2}+(-2k+1)A_{k}=0\;\!
  11. ( k + 2 ) ( k + 1 ) A k + 2 = - ( - 2 k + 1 ) A k (k+2)(k+1)A_{k+2}=-(-2k+1)A_{k}\;\!
  12. A k + 2 = ( 2 k - 1 ) ( k + 2 ) ( k + 1 ) A k A_{k+2}={(2k-1)\over(k+2)(k+1)}A_{k}\;\!
  13. A 2 = - 1 ( 2 ) ( 1 ) A 0 = - 1 2 A 0 , A 3 = 1 ( 3 ) ( 2 ) A 1 = 1 6 A 1 A_{2}={-1\over(2)(1)}A_{0}={-1\over 2}A_{0},\,A_{3}={1\over(3)(2)}A_{1}={1% \over 6}A_{1}
  14. A 4 \displaystyle A_{4}
  15. f \displaystyle f
  16. f = A 0 ( 1 + - 1 2 x 2 + - 1 8 x 4 + - 7 240 x 6 + ) + A 1 ( x + 1 6 x 3 + 1 24 x 5 + 1 112 x 7 + ) f=A_{0}\left(1+{-1\over 2}x^{2}+{-1\over 8}x^{4}+{-7\over 240}x^{6}+\cdots% \right)+A_{1}\left(x+{1\over 6}x^{3}+{1\over 24}x^{5}+{1\over 112}x^{7}+\cdots\right)
  17. f = k = 0 A k z k k ! f=\sum_{k=0}^{\infty}A_{k}z^{k}\over{k!}
  18. y [ n ] - > A k + n y^{[n]}->A_{k+n}
  19. x m y [ n ] - > ( k ) ( k - 1 ) ( k - m + 1 ) A k + n - m x^{m}y^{[n]}->(k)(k-1)...(k-m+1)A_{k+n-m}
  20. f ′′ - 2 z f + λ f = 0 ; λ = 1 f^{\prime\prime}-2zf^{\prime}+\lambda f=0;\;\lambda=1
  21. A k + 2 - 2 k A k + λ A k = 0 A_{k+2}-2kA_{k}+\lambda A_{k}=0
  22. A k + 2 = ( 2 k - λ ) A k A_{k+2}=(2k-\lambda)A_{k}
  23. f = k = 0 A k z k k ! f={\sum_{k=0}^{\infty}A_{k}z^{k}\over{k!}}
  24. F F ′′ + 2 F 2 + η F = 0 ; F ( 1 ) = 0 , F ( 1 ) = - 1 2 FF^{\prime\prime}+2F^{\prime 2}+\eta F^{\prime}=0\quad;\quad F(1)=0\ ,\ F^{% \prime}(1)=-\frac{1}{2}
  25. η = 1 \eta=1
  26. F ( η ) = i = 0 c i ( η - 1 ) i F(\eta)=\sum_{i=0}^{\infty}c_{i}(\eta-1)^{i}
  27. d n F d η n | η = 1 = n ! c n \frac{d^{n}F}{d\eta^{n}}\Bigg|_{\eta=1}=n!\ c_{n}
  28. F F ′′ + 2 F 2 + ( η - 1 ) F + F = 0 ; F ( 1 ) = 0 , F ( 1 ) = - 1 2 FF^{\prime\prime}+2F^{\prime 2}+(\eta-1)F^{\prime}+F^{\prime}=0\quad;\quad F(1% )=0\ ,\ F^{\prime}(1)=-\frac{1}{2}
  29. η - 1 \eta-1
  30. ( i = 0 a i x i ) ( i = 0 b i x i ) = i = 0 x i j = 0 i a i - j b j \left(\sum_{i=0}^{\infty}a_{i}x^{i}\right)\left(\sum_{i=0}^{\infty}b_{i}x^{i}% \right)=\sum_{i=0}^{\infty}x^{i}\sum_{j=0}^{i}a_{i-j}b_{j}
  31. j = 0 i ( ( j + 1 ) ( j + 2 ) c i - j c j + 2 + 2 ( i - j + 1 ) ( j + 1 ) c i - j + 1 c j + 1 ) + i c i + ( i + 1 ) c i + 1 = 0 \sum_{j=0}^{i}\left((j+1)(j+2)c_{i-j}c_{j+2}+2(i-j+1)(j+1)c_{i-j+1}c_{j+1}% \right)+ic_{i}+(i+1)c_{i+1}=0
  32. c 0 = 0 c_{0}=0
  33. c 1 = - 1 / 2 c_{1}=-1/2
  34. c 2 = - 1 6 ; c 3 = - 1 108 ; c 4 = 7 3240 ; c 5 = - 19 48600 c_{2}=-\frac{1}{6}\quad;\quad c_{3}=-\frac{1}{108}\quad;\quad c_{4}=\frac{7}{3% 240}\quad;\quad c_{5}=-\frac{19}{48600}\ \dots
  35. η = 1 \eta=1
  36. η = 1 \eta=1
  37. η = 1 \eta=1

Poynting's_theorem.html

  1. u = 1 2 ( 𝐄 𝐃 + 𝐁 𝐇 ) u=\frac{1}{2}\left(\mathbf{E}\cdot\mathbf{D}+\mathbf{B}\cdot\mathbf{H}\right)
  2. V \partial V\!
  3. 𝐒 + ϵ 0 𝐄 𝐄 t + 𝐁 μ 0 𝐁 t + 𝐉 𝐄 = 0 , \nabla\cdot\mathbf{S}+\epsilon_{0}\mathbf{E}\cdot\frac{\partial\mathbf{E}}{% \partial t}+\frac{\mathbf{B}}{\mu_{0}}\cdot\frac{\partial\mathbf{B}}{\partial t% }+\mathbf{J}\cdot\mathbf{E}=0,
  4. ϵ 0 𝐄 𝐄 t \epsilon_{0}\mathbf{E}\cdot\frac{\partial\mathbf{E}}{\partial t}
  5. 𝐁 μ 0 𝐁 t \frac{\mathbf{B}}{\mu_{0}}\cdot\frac{\partial\mathbf{B}}{\partial t}
  6. 𝐉 𝐄 \mathbf{J}\cdot\mathbf{E}
  7. - V u t d V = V 𝐒 d V + V 𝐉 𝐄 d V , -\int_{V}\frac{\partial u}{\partial t}dV=\int_{V}\nabla\cdot\mathbf{S}dV+\int_% {V}\mathbf{J}\cdot\mathbf{E}dV,
  8. - u t = 𝐒 + 𝐉 𝐄 , -\frac{\partial u}{\partial t}=\nabla\cdot\mathbf{S}+\mathbf{J}\cdot\mathbf{E},
  9. u t = 1 2 ( 𝐄 𝐃 t + 𝐃 𝐄 t + 𝐇 𝐁 t + 𝐁 𝐇 t ) = 𝐄 𝐃 t + 𝐇 𝐁 t , \frac{\partial u}{\partial t}=\frac{1}{2}\left(\mathbf{E}\cdot\frac{\partial% \mathbf{D}}{\partial t}+\mathbf{D}\cdot\frac{\partial\mathbf{E}}{\partial t}+% \mathbf{H}\cdot\frac{\partial\mathbf{B}}{\partial t}+\mathbf{B}\cdot\frac{% \partial\mathbf{H}}{\partial t}\right)=\mathbf{E}\cdot\frac{\partial\mathbf{D}% }{\partial t}+\mathbf{H}\cdot\frac{\partial\mathbf{B}}{\partial t},
  10. 𝐃 = ϵ 0 𝐄 , 𝐁 = μ 0 𝐇 . \mathbf{D}=\epsilon_{0}\mathbf{E},\quad\mathbf{B}=\mu_{0}\mathbf{H}.
  11. 𝐁 t = - × 𝐄 𝐇 𝐁 t = - 𝐇 × 𝐄 , \frac{\partial\mathbf{B}}{\partial t}=-\nabla\times\mathbf{E}\ \rightarrow\ % \mathbf{H}\cdot\frac{\partial\mathbf{B}}{\partial t}=-\mathbf{H}\cdot\nabla% \times\mathbf{E},
  12. 𝐃 t + 𝐉 = × 𝐇 𝐄 𝐃 t + 𝐄 𝐉 = 𝐄 × 𝐇 . \frac{\partial\mathbf{D}}{\partial t}+\mathbf{J}=\nabla\times\mathbf{H}\ % \rightarrow\ \mathbf{E}\cdot\frac{\partial\mathbf{D}}{\partial t}+\mathbf{E}% \cdot\mathbf{J}=\mathbf{E}\cdot\nabla\times\mathbf{H}.
  13. - 𝐒 \displaystyle-\nabla\cdot\mathbf{S}
  14. 𝐄 × 𝐇 = 𝐇 × 𝐄 - 𝐄 × 𝐇 , \nabla\cdot\mathbf{E}\times\mathbf{H}=\mathbf{H}\cdot\nabla\times\mathbf{E}-% \mathbf{E}\cdot\nabla\times\mathbf{H},
  15. 𝐒 = 𝐄 × 𝐇 , \mathbf{S}=\mathbf{E}\times\mathbf{H},
  16. t u m ( 𝐫 , t ) + 𝐒 m ( 𝐫 , t ) = 𝐉 ( 𝐫 , t ) 𝐄 ( 𝐫 , t ) , \frac{\partial}{\partial t}u_{m}(\mathbf{r},t)+\nabla\cdot\mathbf{S}_{m}(% \mathbf{r},t)=\mathbf{J}(\mathbf{r},t)\cdot\mathbf{E}(\mathbf{r},t),
  17. u m ( 𝐫 , t ) = α m α 2 r ˙ α 2 δ ( 𝐫 - 𝐫 α ( t ) ) , u_{m}(\mathbf{r},t)=\sum_{\alpha}\frac{m_{\alpha}}{2}\dot{r}^{2}_{\alpha}% \delta(\mathbf{r}-\mathbf{r}_{\alpha}(t)),
  18. 𝐒 m ( 𝐫 , t ) = α m α 2 r ˙ α 2 𝐫 ˙ α δ ( 𝐫 - 𝐫 α ( t ) ) . \mathbf{S}_{m}(\mathbf{r},t)=\sum_{\alpha}\frac{m_{\alpha}}{2}\dot{r}^{2}_{% \alpha}\dot{\mathbf{r}}_{\alpha}\delta(\mathbf{r}-\mathbf{r}_{\alpha}(t)).
  19. t ( u e + u m ) + ( 𝐒 e + 𝐒 m ) = 0 , \frac{\partial}{\partial t}\left(u_{e}+u_{m}\right)+\nabla\cdot\left(\mathbf{S% }_{e}+\mathbf{S}_{m}\right)=0,

PP_(complexity).html

  1. L c L^{c}
  2. x L Pr [ A accepts x ] > 1 / 2 x\in L\Rightarrow\mathrm{Pr}[A\,\mathrm{accepts}\,x]>1/2
  3. x L Pr [ A accepts x ] 1 / 2 x\not\in L\Rightarrow\mathrm{Pr}[A\,\mathrm{accepts}\,x]\leq 1/2
  4. A c A^{c}
  5. A c A^{c}
  6. x L c Pr [ A c accepts x ] > 1 / 2 x\in L^{c}\Rightarrow\mathrm{Pr}[A^{c}\,\mathrm{accepts}\,x]>1/2
  7. x L c Pr [ A c accepts x ] < 1 / 2 x\not\in L^{c}\Rightarrow\mathrm{Pr}[A^{c}\,\mathrm{accepts}\,x]<1/2
  8. L c L^{c}
  9. f ( | x | ) f(|x|)
  10. f ( | x | ) f(|x|)
  11. x L Pr [ A accepts x ] 1 / 2 + 1 / 2 f ( | x | ) x\in L\Rightarrow\mathrm{Pr}[A\,\mathrm{accepts}\,x]\geq 1/2+1/2^{f(|x|)}
  12. 1 / 2 f ( | x | ) 1/2^{f(|x|)}
  13. f ( | x | ) + 1 f(|x|)+1
  14. x L Pr [ A accepts x ] 1 / 2 ( 1 - 1 / 2 f ( | x | ) + 1 ) < 1 / 2 x\not\in L\Rightarrow\mathrm{Pr}[A^{\prime}\,\mathrm{accepts}\,x]\leq 1/2\cdot% (1-1/2^{f(|x|)+1})<1/2
  15. x L Pr [ A accepts x ] ( 1 / 2 + 1 / 2 f ( | x | ) ) ( 1 - 1 / 2 f ( | x | ) + 1 ) > 1 / 2 x\in L\Rightarrow\mathrm{Pr}[A^{\prime}\,\mathrm{accepts}\,x]\geq(1/2+1/2^{f(|% x|)})\cdot(1-1/2^{f(|x|)+1})>1/2

Predicate_variable.html

  1. = , , , < , \sub , =,\ \in,\ \leq,\ <,\ \sub,...
  2. = , , , < , \sub , =,\ \in,\ \leq,\ <,\ \sub,
  3. 1 , 2 , 3 , 2 , π , e 1,\ 2,\ 3,\ \sqrt{2},\ \pi,\ e

Prefix_grammar.html

  1. 01 ( 01 ) * 100 * 01(01)^{*}\cup 100^{*}

Preimage_attack.html

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

Pressure-gradient_force.html

  1. ρ \rho
  2. d z dz
  3. d A dA
  4. m = ρ d A d z m=\rho\cdot dA\cdot dz
  5. F = m a F=m\cdot a
  6. d P dP
  7. z z
  8. F = - d P d A = ρ d A d z a F=-dP\cdot dA=\rho\cdot dA\cdot dz\cdot a
  9. a = - 1 ρ d P d z a=\frac{-1}{\rho}\frac{dP}{dz}
  10. P P
  11. a = - 1 ρ P \vec{a}=\frac{-1}{\rho}\vec{\nabla}P

Pressure_coefficient.html

  1. C p C_{p}
  2. C p = p - p 1 2 ρ V 2 C_{p}={p-p_{\infty}\over\frac{1}{2}\rho_{\infty}V_{\infty}^{2}}
  3. p p
  4. p p_{\infty}
  5. ρ \rho_{\infty}
  6. k g / m 3 kg/m^{3}
  7. V V_{\infty}
  8. C p = 1 - ( V V ) 2 C_{p}={1-\bigg(\frac{V}{V_{\infty}}\bigg)^{2}}
  9. C p C_{p}
  10. C p C_{p}
  11. C p C_{p}
  12. ρ v 2 / 2 {\rho v^{2}}/2
  13. C p C_{p}
  14. C p C_{p}
  15. C l C_{l}
  16. C p C_{p}
  17. C l = 1 x T E - x L E x L E x T E ( C p l ( x ) - C p u ( x ) ) d x C_{l}=\frac{1}{x_{TE}-x_{LE}}\int\limits_{x_{LE}}^{x_{TE}}\left(C_{p_{l}}(x)-C% _{p_{u}}(x)\right)dx
  18. C p l C_{p_{l}}
  19. C p u C_{p_{u}}
  20. x L E x_{LE}
  21. x T E x_{TE}
  22. C p C_{p}

Pressure_vessel.html

  1. M = 3 2 P V ρ σ M={3\over 2}PV{\rho\over\sigma}
  2. M M
  3. P P
  4. V V
  5. ρ \rho
  6. σ \sigma
  7. M = 2 π R 2 ( R + W ) P ρ σ M=2\pi R^{2}(R+W)P{\rho\over\sigma}
  8. M = 6 π R 3 P ρ σ M=6\pi R^{3}P{\rho\over\sigma}
  9. M = 3 2 n R T ρ σ M={3\over 2}nRT{\rho\over\sigma}
  10. ρ / σ \rho/\sigma
  11. M / p V M/{pV}
  12. σ θ = σ long = p r 2 t \sigma_{\theta}=\sigma_{\rm long}=\frac{pr}{2t}
  13. σ θ \sigma_{\theta}
  14. σ l o n g \sigma_{long}
  15. σ θ = p r t \sigma_{\theta}=\frac{pr}{t}
  16. σ long = p r 2 t \sigma_{\rm long}=\frac{pr}{2t}
  17. σ θ \sigma_{\theta}
  18. σ l o n g \sigma_{long}
  19. σ θ = σ long = p ( r + 0.2 t ) 2 t E \sigma_{\theta}=\sigma_{\rm long}=\frac{p(r+0.2t)}{2tE}
  20. σ θ = p ( r + 0.6 t ) t E \sigma_{\theta}=\frac{p(r+0.6t)}{tE}
  21. σ long = p ( r - 0.4 t ) 2 t E \sigma_{\rm long}=\frac{p(r-0.4t)}{2tE}

Price_equation.html

  1. n n
  2. n n
  3. n n
  4. n n
  5. n n
  6. i i
  7. n i n_{i}
  8. z i z_{i}
  9. z i z_{i}
  10. w i w_{i}
  11. w i n i w_{i}n_{i}
  12. i i
  13. n i n_{i}^{\prime}
  14. w i = n i / n i w_{i}=n_{i}^{\prime}/n_{i}
  15. z i z_{i}^{\prime}
  16. i i
  17. i i
  18. Δ z i \Delta z_{i}
  19. Δ z i = def z i - z i \Delta{z_{i}}\;\stackrel{\mathrm{def}}{=}\;z_{i}^{\prime}-z_{i}
  20. z z
  21. z z^{\prime}
  22. Δ z \Delta{z}
  23. Δ z = def z - z \Delta{z}\;\stackrel{\mathrm{def}}{=}\;z^{\prime}-z
  24. Δ z i \Delta{z_{i}}
  25. n i n i n_{i}\neq n^{\prime}_{i}
  26. w w
  27. w Δ z = cov ( w i , z i ) + E ( w i Δ z i ) w\,\Delta{z}=\operatorname{cov}(w_{i},z_{i})+\operatorname{E}(w_{i}\,\Delta z_% {i})
  28. E \operatorname{E}
  29. cov \operatorname{cov}
  30. w w
  31. Δ z = 1 w cov ( w i , z i ) + 1 w E ( w i Δ z i ) \Delta{z}=\frac{1}{w}\operatorname{cov}(w_{i},z_{i})+\frac{1}{w}\operatorname{% E}(w_{i}\,\Delta z_{i})
  32. z i = w i z_{i}=w_{i}
  33. n i n_{i}
  34. x i x_{i}
  35. y i y_{i}
  36. x i x_{i}
  37. E ( x i ) = def 1 i n i i x i n i \operatorname{E}(x_{i})\;\stackrel{\mathrm{def}}{=}\;\frac{1}{\sum_{i}n_{i}}% \sum_{i}x_{i}n_{i}
  38. ( 1 ) (1)
  39. x i x_{i}
  40. y i y_{i}
  41. cov ( x i , y i ) = def 1 i n i i n i [ x i - E ( x i ) ] [ y i - E ( y i ) ] = E ( x i y i ) - E ( x i ) E ( y i ) \operatorname{cov}(x_{i},y_{i})\;\stackrel{\mathrm{def}}{=}\;\frac{1}{\sum_{i}% n_{i}}\sum_{i}n_{i}[x_{i}-\operatorname{E}(x_{i})][y_{i}-\operatorname{E}(y_{i% })]=\operatorname{E}(x_{i}y_{i})-\operatorname{E}(x_{i})\operatorname{E}(y_{i})
  42. ( 2 ) (2)
  43. x i = E ( x i ) \langle x_{i}\rangle=\operatorname{E}(x_{i})
  44. i i
  45. z i z_{i}
  46. n i n_{i}
  47. n n
  48. n = i n i n=\sum_{i}n_{i}
  49. z z
  50. z = def E ( z i ) = 1 n i z i n i z\;\stackrel{\mathrm{def}}{=}\;\operatorname{E}(z_{i})=\frac{1}{n}\sum_{i}z_{i% }n_{i}
  51. ( 3 ) (3)
  52. n = i n i n^{\prime}=\sum_{i}n^{\prime}_{i}
  53. w i = n i n i w_{i}=\frac{n_{i}^{\prime}}{n_{i}}
  54. ( 4 ) (4)\,
  55. w = def E ( w i ) = 1 n i w i n i = 1 n i n i n i n i = 1 n i n i = n n w\;\stackrel{\mathrm{def}}{=}\;\operatorname{E}(w_{i})=\frac{1}{n}\sum_{i}w_{i% }n_{i}=\frac{1}{n}\sum_{i}\frac{n_{i}^{\prime}}{n_{i}}n_{i}=\frac{1}{n}\sum_{i% }n_{i}^{\prime}=\frac{n^{\prime}}{n}
  56. ( 5 ) (5)
  57. z = 1 n i z i n i z^{\prime}=\frac{1}{n^{\prime}}\sum_{i}z^{\prime}_{i}n_{i}^{\prime}
  58. ( 6 ) (6)
  59. cov ( w i , z i ) = E ( w i z i ) - w z \operatorname{cov}(w_{i},z_{i})=\operatorname{E}(w_{i}z_{i})-wz
  60. ( 7 ) (7)
  61. Δ z i \Delta z_{i}
  62. Δ z i = z i - z i \Delta z_{i}=z^{\prime}_{i}-z_{i}
  63. E \operatorname{E}
  64. E ( w i Δ z i ) = E ( w i z i ) - E ( w i z i ) \operatorname{E}(w_{i}\,\Delta z_{i})=\operatorname{E}(w_{i}z^{\prime}_{i})-% \operatorname{E}(w_{i}z_{i})
  65. ( 8 ) (8)
  66. cov ( w i , z i ) + E ( w i Δ z i ) = [ E ( w i z i ) - w z ] + [ E ( w i z i ) - E ( w i z i ) ] = E ( w i z i ) - w z \operatorname{cov}(w_{i},z_{i})+\operatorname{E}(w_{i}\,\Delta z_{i})=\left[% \operatorname{E}(w_{i}z_{i})-wz\right]+\left[\operatorname{E}(w_{i}z^{\prime}_% {i})-\operatorname{E}(w_{i}z_{i})\right]=\operatorname{E}(w_{i}z^{\prime}_{i})% -wz
  67. ( 9 ) (9)
  68. E ( w i z i ) = 1 n i w i z i n i \operatorname{E}(w_{i}z^{\prime}_{i})=\frac{1}{n}\sum_{i}w_{i}z^{\prime}_{i}n_% {i}
  69. w i = n i n i w_{i}=\frac{n^{\prime}_{i}}{n_{i}}
  70. E ( w i z i ) = 1 n i n i n i z i n i = 1 n i n i z i = n n 1 n i z i n i \operatorname{E}(w_{i}z^{\prime}_{i})=\frac{1}{n}\sum_{i}\frac{n^{\prime}_{i}}% {n_{i}}z^{\prime}_{i}n_{i}=\frac{1}{n}\sum_{i}n^{\prime}_{i}z^{\prime}_{i}=% \frac{n^{\prime}}{n}~{}\frac{1}{n^{\prime}}\sum_{i}z^{\prime}_{i}n^{\prime}_{i}
  71. ( 10 ) (10)
  72. w = n n w=\frac{n^{\prime}}{n}
  73. z z^{\prime}
  74. cov ( w i , z i ) + E ( w i Δ z i ) = w z - w z = w Δ z \operatorname{cov}(w_{i},z_{i})+\operatorname{E}(w_{i}\,\Delta z_{i})=wz^{% \prime}-wz=w\,\Delta z\,
  75. w Δ z = cov ( w i , z i ) w\,\Delta z=\operatorname{cov}\left(w_{i},z_{i}\right)
  76. Δ z = cov ( v i , z i ) \Delta z=\operatorname{cov}\left(v_{i},z_{i}\right)
  77. Δ z = def z - z = 1 3 \Delta z\;\stackrel{\mathrm{def}}{=}\;z^{\prime}-z=\frac{1}{3}
  78. Δ z = 1 w cov ( w i , z i ) = 1 3 \Delta z=\frac{1}{w}\operatorname{cov}\left(w_{i},z_{i}\right)=\frac{1}{3}
  79. w ( z - z ) = w i z i - w z w(z^{\prime}-z)=\langle w_{i}z_{i}\rangle-wz
  80. w ( z ′′ - z ) = w i z i - w z w^{\prime}(z^{\prime\prime}-z^{\prime})=\langle w^{\prime}_{i}z^{\prime}_{i}% \rangle-w^{\prime}z^{\prime}
  81. w ( w - w ) \displaystyle w(w^{\prime}-w)
  82. b ( 1 2 n 0 2 + 1 2 n 0 n 1 + 1 8 n 1 2 n 0 + n 1 ) b\left(\frac{\frac{1}{2}n_{0}^{2}+\frac{1}{2}n_{0}n_{1}+\frac{1}{8}n_{1}^{2}}{% n_{0}+n_{1}}\right)
  83. b ( 1 2 n 0 n 1 + 1 4 n 1 2 n 0 + n 1 ) b\left(\frac{\frac{1}{2}n_{0}n_{1}+\frac{1}{4}n_{1}^{2}}{n_{0}+n_{1}}\right)
  84. b ( 1 8 n 1 2 n 0 + n 1 ) b\left(\frac{\frac{1}{8}n_{1}^{2}}{n_{0}+n_{1}}\right)
  85. w 0 \displaystyle w_{0}
  86. 0 = cov ( w i w , z i ) = f ( 2 - 2 a - a f ) ( 1 + f ) ( 2 a + 2 f + a f ) 0=\operatorname{cov}\left(\frac{w_{i}}{w},z_{i}\right)=\frac{f(2-2a-af)}{(1+f)% (2a+2f+af)}
  87. f = n 1 n 0 = 2 ( 1 - a ) a f=\frac{n_{1}}{n_{0}}=\frac{2(1-a)}{a}
  88. w i 2 z i \displaystyle\langle w_{i}^{2}z_{i}\rangle
  89. z 1 = 1 z_{1}=1
  90. z 2 = 0 z_{2}=0
  91. z z
  92. a a
  93. b b
  94. w 1 = a ( 1 - z ) w_{1}=a(1-z)
  95. w 2 = b z w_{2}=bz
  96. w = ( a + b ) z ( 1 - z ) w=(a+b)z(1-z)
  97. c o v ( w i , z i ) + w z = E ( w i , z i ) = a z ( 1 - z ) cov(w_{i},z_{i})+wz=E(w_{i},z_{i})=az(1-z)
  98. c o v ( w i , z i ) = a z ( 1 - z ) - ( a + b ) z 2 ( 1 - z ) = z [ 1 - z ] [ a - ( a + b ) z ] cov(w_{i},z_{i})=az(1-z)-(a+b)z^{2}(1-z)=z[1-z][a-(a+b)z]
  99. Δ z = a a + b - z \Delta z=\frac{a}{a+b}-z
  100. z = a a + b z^{\prime}=\frac{a}{a+b}
  101. w i = n i n i = k - a z i + b z w_{i}=\frac{n^{\prime}_{i}}{n_{i}}=k-az_{i}+bz
  102. w Δ z = - a var ( z i ) w\Delta z=-a\operatorname{var}\left(z_{i}\right)
  103. var ( z i ) = def E ( z i 2 ) - E ( z i ) 2 \operatorname{var}(z_{i})\;\stackrel{\mathrm{def}}{=}\;\operatorname{E}(z_{i}^% {2})-\operatorname{E}(z_{i})^{2}
  104. w i j = n i j n i j = k - a z i j + b z i w_{ij}=\frac{n^{\prime}_{ij}}{n_{ij}}=k-az_{ij}+bz_{i}
  105. n i \displaystyle n_{i}
  106. n \displaystyle n
  107. Δ z = cov ( w i w , z i ) + E ( w i Δ z i w ) \Delta z=\operatorname{cov}\left(\frac{w_{i}}{w},z_{i}\right)+\operatorname{E}% \left(w_{i}\,\Delta\frac{z_{i}}{w}\right)
  108. cov ( w i w , z i ) = ( b - a ) var ( z i ) \operatorname{cov}\left(\frac{w_{i}}{w},z_{i}\right)=\left(b-a\right)% \operatorname{var}(z_{i})
  109. 𝐰 = [ α 0 0 0 m α ( n - 3 m ) α m β m β 0 0 β 0 m α m α m β ( n - 3 m ) β ] \mathbf{w}=\begin{bmatrix}\alpha&0&0&0\\ m\alpha&(n-3m)\alpha&m\beta&m\beta\\ 0&0&\beta&0\\ m\alpha&m\alpha&m\beta&(n-3m)\beta\end{bmatrix}
  110. z i = [ 0 , 1 , 0 , 1 ] z_{i}=\left[0,1,0,1\right]
  111. n i = j w j i n j n^{\prime}_{i}=\sum_{j}w_{ji}n_{j}\,
  112. w i = n i n i w_{i}=\frac{n^{\prime}_{i}}{n_{i}}
  113. z \displaystyle z
  114. n i = n i j w i j n^{\prime}_{i}=n_{i}\sum_{j}w_{ij}\,
  115. w i = n i n i = j w i j w_{i}=\frac{n^{\prime}_{i}}{n_{i}}=\sum_{j}w_{ij}
  116. z j = i n i z i w i j i n i w i j z^{\prime}_{j}=\frac{\sum_{i}n_{i}z_{i}w_{ij}}{\sum_{i}n_{i}w_{ij}}
  117. z \displaystyle z

Prime_(symbol).html

  1. 1 / 60 {1}/{60}
  2. ( x , y ) (x,y)
  3. ( x , y ) (x′,y′)
  4. f f^{\prime}\,\!
  5. \prime
  6. f f_{\prime}^{\prime}

Prime_quadruplet.html

  1. B 4 = ( 1 5 + 1 7 + 1 11 + 1 13 ) + ( 1 11 + 1 13 + 1 17 + 1 19 ) + ( 1 101 + 1 103 + 1 107 + 1 109 ) + B_{4}=\left(\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+\frac{1}{13}\right)+\left(% \frac{1}{11}+\frac{1}{13}+\frac{1}{17}+\frac{1}{19}\right)+\left(\frac{1}{101}% +\frac{1}{103}+\frac{1}{107}+\frac{1}{109}\right)+\cdots

Primitive_cell.html

  1. T \vec{T}
  2. a 1 \vec{a}_{1}
  3. a 2 \vec{a}_{2}
  4. a 3 \vec{a}_{3}
  5. u 1 u_{1}
  6. u 2 u_{2}
  7. u 3 u_{3}
  8. T = u 1 a 1 + u 2 a 2 + u 3 a 3 \vec{T}=u_{1}\vec{a}_{1}+u_{2}\vec{a}_{2}+u_{3}\vec{a}_{3}
  9. a 1 \vec{a}_{1}
  10. a 2 \vec{a}_{2}
  11. a 3 \vec{a}_{3}
  12. V p V_{p}
  13. V p = | a 1 ( a 2 × a 3 ) | . V_{p}=|\vec{a}_{1}\cdot(\vec{a}_{2}\times\vec{a}_{3})|.

Primitive_polynomial_(field_theory).html

  1. F ( X ) F(X)
  2. α \alpha
  3. { 0 , 1 , α , α 2 , α 3 , , α p m - 2 } \{0,1,\alpha,\alpha^{2},\alpha^{3},\dots,\alpha^{p^{m}-2}\}
  4. α \alpha
  5. α \alpha
  6. α p m - 1 = 1 \alpha^{p^{m}-1}=1
  7. α i 1 \alpha^{i}\neq 1
  8. G F ( p m ) = { 0 , 1 , α , α 2 , , α p m - 2 } . GF(p^{m})=\{0,1,\alpha,\alpha^{2},\ldots,\alpha^{p^{m}-2}\}.

Primordial_fluctuations.html

  1. δ ( x ) = def ρ ( x ) ρ ¯ - 1 = d k δ k e i k x , \delta(\vec{x})\ \stackrel{\mathrm{def}}{=}\ \frac{\rho(\vec{x})}{\bar{\rho}}-% 1=\int\,\text{d}k\;\delta_{k}\,e^{i\vec{k}\cdot\vec{x}},
  2. ρ \rho
  3. ρ ¯ \bar{\rho}
  4. k k
  5. 𝒫 ( k ) \mathcal{P}(k)
  6. δ k δ k = 2 π 2 k 3 δ ( k - k ) 𝒫 ( k ) . \langle\delta_{k}\delta_{k^{\prime}}\rangle=\frac{2\pi^{2}}{k^{3}}\,\delta(k-k% ^{\prime})\,\mathcal{P}(k).
  7. 𝒫 s ( k ) k n s - 1 . \mathcal{P}_{\mathrm{s}}(k)\propto k^{n_{\mathrm{s}}-1}.
  8. n s n_{\mathrm{s}}
  9. n s = 1 n_{\mathrm{s}}=1

Principal_branch.html

  1. 1 / 2 1/2
  2. x x
  3. y y
  4. x x
  5. x \sqrt{x}
  6. x x
  7. e z = e a cos b + i e a sin b e^{z}=e^{a}\cos b+ie^{a}\sin b
  8. z = a + i b z=a+ib
  9. Re ( log z ) = log a 2 + b 2 \operatorname{Re}(\log z)=\log\sqrt{a^{2}+b^{2}}
  10. Im ( log z ) = atan2 ( b , a ) + 2 π k \operatorname{Im}(\log z)=\operatorname{atan2}(b,a)+2\pi k
  11. k k
  12. a t a n 2 atan2
  13. l o g z logz
  14. e < s u p > l o g z = z e<sup>logz=z

Principal–agent_problem.html

  1. w = a + b ( e + x + g y ) w=a+b(e+x+gy)\,
  2. wage = ( base salary ) + ( incentives ) ( (unobserved) effort + ( (unobserved) effects ) + ( weight Y ) ( observed exogenous effects ) ) \,\text{wage}=(\,\text{base salary})+(\,\text{incentives})\cdot(\,\text{(% unobserved) effort}+(\,\text{(unobserved) effects})+(\,\text{weight }Y)(\,% \text{observed exogenous effects}))

Principle_of_distributivity.html

  1. A ( B C ) ( A B ) ( A C ) A\land(B\lor C)\iff(A\land B)\lor(A\land C)
  2. A ( B C ) ( A B ) ( A C ) A\lor(B\land C)\iff(A\lor B)\land(A\lor C)

Private_information_retrieval.html

  1. n O ( log log k k log k ) n^{O(\frac{\log\log k}{k\log k})}
  2. n n
  3. n ϵ n^{\epsilon}
  4. ϵ \epsilon
  5. n n
  6. O ( log n + k log 2 n ) O(\ell\log n+k\log^{2}n)
  7. \ell
  8. k k
  9. Ω ( n ) \Omega(n)
  10. O ( log n + k log 2 n ) O(\ell\log n+k\log^{2}n)
  11. O ( n / log n ) O(n/\log n)
  12. ν < k - t - 1 \nu<k-t-1
  13. O ( n 1 / ( 2 k - 1 ) ) O(n^{1/(2k-1)})

Private_network.html

  1. 2 22 2^{22}

Probabilistic_encryption.html

  1. Enc ( x ) = ( f ( r ) , x b ( r ) ) {\rm Enc}(x)=(f(r),x\oplus b(r))
  2. Dec ( y , z ) = b ( f - 1 ( y ) ) z {\rm Dec}(y,z)=b(f^{-1}(y))\oplus z
  3. Enc ( x ) = ( f ( r ) , x h ( r ) ) {\rm Enc}(x)=(f(r),x\oplus h(r))
  4. Dec ( y , z ) = h ( f - 1 ( y ) ) z {\rm Dec}(y,z)=h(f^{-1}(y))\oplus z

Problem_of_multiple_generality.html

  1. m . ( Mouse ( m ) c . ( Cat ( c ) Fears ( m , c ) ) ) \forall m.\,(\,\,\text{Mouse}(m)\rightarrow\exists c.\,(\,\text{Cat}(c)\land\,% \text{Fears}(m,c))\,)
  2. c . ( Cat ( c ) m . ( Mouse ( m ) Fears ( m , c ) ) ) \exists c.\,(\,\,\text{Cat}(c)\land\forall m.\,(\,\text{Mouse}(m)\rightarrow\,% \text{Fears}(m,c))\,)

Process_calculus.html

  1. P \mathit{P}
  2. Q \mathit{Q}
  3. P | Q P|Q
  4. P \mathit{P}
  5. Q \mathit{Q}
  6. P \mathit{P}
  7. Q \mathit{Q}
  8. x ( v ) x(v)
  9. x y x\langle y\rangle
  10. x \mathit{x}
  11. x y x\langle y\rangle
  12. y y
  13. x ( v ) x(v)
  14. v v
  15. x \mathit{x}
  16. y \mathit{y}
  17. x ( v ) P x(v)\cdot P
  18. x \mathit{x}
  19. P \mathit{P}
  20. x \mathit{x}
  21. v \mathit{v}
  22. x y P | x ( v ) Q P | Q [ y / v ] x\langle y\rangle\cdot P\;|\;x(v)\cdot Q\longrightarrow P\;|\;Q[^{y}\!/\!_{v}]
  23. x y P x\langle y\rangle\cdot P
  24. y \mathit{y}
  25. x \mathit{x}
  26. x ( v ) Q x(v)\cdot Q
  27. x \mathit{x}
  28. x y P x\langle y\rangle\cdot P
  29. P \mathit{P}
  30. x ( v ) Q x(v)\cdot Q
  31. Q [ y / v ] Q[^{y}\!/\!_{v}]
  32. Q \mathit{Q}
  33. v \mathit{v}
  34. y \mathit{y}
  35. x \mathit{x}
  36. P \mathit{P}
  37. π \pi
  38. x \mathit{x}
  39. P \mathit{P}
  40. ( ν x ) P (\nu\;x)P
  41. P { x } P\setminus\{x\}
  42. ! P !P
  43. P \mathit{P}
  44. ! P = P ! P !P=P\mid!P
  45. 𝑛𝑖𝑙 \mathit{nil}
  46. 0
  47. 𝑆𝑇𝑂𝑃 \mathit{STOP}
  48. δ \delta
  49. π \pi
  50. π \pi
  51. π \pi

Professor's_Cube.html

  1. 8 ! × 3 7 × 12 ! × 2 10 × 24 ! 3 4 ! 12 2.83 × 10 74 \frac{8!\times 3^{7}\times 12!\times 2^{10}\times 24!^{3}}{4!^{12}}\approx 2.8% 3\times 10^{74}

Profit_margin.html

  1. net profit margin = net profit revenue \,\text{net profit margin}={\,\text{net profit}\over\,\text{revenue}}
  2. profit percentage = net profit cost price \,\text{profit percentage}={\,\text{net profit}\over\,\text{cost price}}

Projectile_motion.html

  1. 𝐯 0 \mathbf{v}_{0}
  2. 𝐯 0 = v 0 x 𝐢 + v 0 y 𝐣 \mathbf{v}_{0}=v_{0x}\mathbf{i}+v_{0y}\mathbf{j}
  3. v 0 x v_{0x}
  4. v 0 y v_{0y}
  5. θ \theta
  6. v 0 x = v 0 cos θ v_{0x}=v_{0}\cos\theta
  7. v 0 y = v 0 sin θ v_{0y}=v_{0}\sin\theta
  8. V 0 = R 2 g R sin 2 θ + 2 h cos 2 θ V_{0}=\sqrt{{R^{2}g}\over{R\sin 2\theta+2h\cos^{2}\theta}}
  9. 𝐯 0 cos ( θ ) \mathbf{v}_{0}\cos(\theta)
  10. g g
  11. a x = 0 a_{x}=0
  12. a y = - g a_{y}=-g
  13. x x
  14. y y
  15. t t
  16. v x = v 0 cos ( θ ) v_{x}=v_{0}\cos(\theta)
  17. v y = v 0 sin ( θ ) - g t v_{y}=v_{0}\sin(\theta)-gt
  18. v = v x 2 + v y 2 v=\sqrt{v_{x}^{2}+v_{y}^{2}\ }
  19. t t
  20. x = v 0 t cos ( θ ) x=v_{0}t\cos(\theta)
  21. y = v 0 t sin ( θ ) - 1 2 g t 2 y=v_{0}t\sin(\theta)-\frac{1}{2}gt^{2}
  22. Δ r = x 2 + y 2 \Delta r=\sqrt{x^{2}+y^{2}\ }
  23. x = v 0 t cos ( θ ) x=v_{0}t\cos(\theta)
  24. y = v 0 t sin ( θ ) - 1 2 g t 2 y=v_{0}t\sin(\theta)-\frac{1}{2}gt^{2}
  25. y = tan ( θ ) x - g 2 v 0 2 cos 2 θ x 2 y=\tan(\theta)\cdot x-\frac{g}{2v^{2}_{0}\cos^{2}\theta}\cdot x^{2}
  26. g g
  27. θ \theta
  28. 𝐯 0 \mathbf{v}_{0}
  29. y = a x + b x 2 y=ax+bx^{2}
  30. a a
  31. b b
  32. y = v 0 t sin ( θ ) - 1 2 g t 2 y=v_{0}t\sin(\theta)-\frac{1}{2}gt^{2}
  33. 0 = v 0 t sin ( θ ) - 1 2 g t 2 0=v_{0}t\sin(\theta)-\frac{1}{2}gt^{2}
  34. v 0 t sin ( θ ) = 1 2 g t 2 v_{0}t\sin(\theta)=\frac{1}{2}gt^{2}
  35. v 0 sin ( θ ) = 1 2 g t v_{0}\sin(\theta)=\frac{1}{2}gt
  36. t = 2 v 0 sin ( θ ) g t=\frac{2v_{0}\sin(\theta)}{g}
  37. v y = 0 v_{y}=0
  38. 0 = v 0 sin ( θ ) - g t h 0=v_{0}\sin(\theta)-gt_{h}
  39. t h = v 0 sin ( θ ) g t_{h}=\frac{v_{0}\sin(\theta)}{g}
  40. h = v 0 t h sin ( θ ) - 1 2 g t h 2 h=v_{0}t_{h}\sin(\theta)-\frac{1}{2}gt^{2}_{h}
  41. h = v 0 2 sin 2 ( θ ) 2 g h=\frac{v_{0}^{2}\sin^{2}(\theta)}{2g}
  42. R R
  43. h h
  44. t d 2 \frac{t_{d}}{2}
  45. h = R tan θ 4 h=\frac{R\tan\theta}{4}
  46. 0 = v 0 t d sin ( θ ) - 1 2 g t d 2 0=v_{0}t_{d}\sin(\theta)-\frac{1}{2}gt_{d}^{2}
  47. t d = 2 v 0 sin ( θ ) g t_{d}=\frac{2v_{0}\sin(\theta)}{g}
  48. d = v 0 t d cos ( θ ) d=v_{0}t_{d}\cos(\theta)
  49. d = v 0 2 g sin ( 2 θ ) d=\frac{v_{0}^{2}}{g}\sin(2\theta)
  50. d d
  51. sin 2 θ = 1 \sin 2\theta=1
  52. 2 θ = 90 2\theta=90^{\circ}
  53. θ = 45 \theta=45^{\circ}
  54. v y 2 = ( v 0 sin θ ) 2 - 2 g y v_{y}^{2}=(v_{0}\sin\theta)^{2}-2gy
  55. g g
  56. 9.81 m / s 2 9.81m/s^{2}
  57. 2 sin ( α ) cos ( α ) = sin ( 2 α ) 2\cdot\sin(\alpha)\cdot\cos(\alpha)=\sin(2\alpha)

Projection-valued_measure.html

  1. π ( X ) = id H \pi(X)=\operatorname{id}_{H}\quad
  2. E π ( E ) ξ η E\mapsto\langle\pi(E)\xi\mid\eta\rangle
  3. S π ( ξ , η ) \operatorname{S}_{\pi}(\xi,\eta)
  4. S π ( ξ , ξ ) \operatorname{S}_{\pi}(\xi,\xi)
  5. ξ \xi
  6. E F = , E\cap F=\emptyset,
  7. π ( E ) π ( F ) = π ( E F ) = π ( F ) π ( E ) , \pi(E)\pi(F)=\pi(E\cap F)=\pi(F)\pi(E),
  8. 𝟏 E π ( E ) \mathbf{1}_{E}\mapsto\pi(E)
  9. T π ( f ) ξ η = X f ( x ) d S π ( ξ , η ) ( x ) \langle\operatorname{T}_{\pi}(f)\xi\mid\eta\rangle=\int_{X}f(x)d\operatorname{% S}_{\pi}(\xi,\eta)(x)
  10. S π ( ξ , η ) \operatorname{S}_{\pi}(\xi,\eta)
  11. E π ( E ) ξ η E\mapsto\langle\pi(E)\xi\mid\eta\rangle
  12. π \pi
  13. f T π ( f ) f\mapsto\operatorname{T}_{\pi}(f)
  14. T π ( f ) \operatorname{T}_{\pi}(f)
  15. T π ( f ) = X f ( x ) d π ( x ) = X f d π . \operatorname{T}_{\pi}(f)=\int_{X}f(x)d\pi(x)=\int_{X}fd\pi.
  16. T π ( f ) \operatorname{T}_{\pi}(f)
  17. A : H H A:H\to H
  18. π A \pi_{A}
  19. A = x d π A ( x ) . A=\int_{\mathbb{R}}xd\pi_{A}(x).
  20. g : g:\mathbb{R}\to\mathbb{C}
  21. g ( A ) := g ( x ) d π A ( x ) . g(A):=\int_{\mathbb{R}}g(x)d\pi_{A}(x).
  22. X H x d μ ( x ) . \int_{X}^{\oplus}H_{x}\ d\mu(x).
  23. π ( E ) = U * ρ ( E ) U \pi(E)=U^{*}\rho(E)U\quad
  24. X H x d μ ( x ) . \int_{X}^{\oplus}H_{x}\ d\mu(x).
  25. π = 1 n ω ( π | H n ) \pi=\bigoplus_{1\leq n\leq\omega}(\pi|H_{n})
  26. H n = X n H x d ( μ | X n ) ( x ) H_{n}=\int_{X_{n}}^{\oplus}H_{x}\ d(\mu|X_{n})(x)
  27. X n = { x X : dim H x = n } . X_{n}=\{x\in X:\operatorname{dim}H_{x}=n\}.
  28. P = ϕ , π ( E ) ( ϕ ) = ϕ | π ( E ) | ϕ , P=\langle\phi,\pi(E)(\phi)\rangle=\langle\phi|\pi(E)|\phi\rangle,
  29. A ( ϕ ) = R λ d π ( λ ) ( ϕ ) , A(\phi)=\int_{{R}}\lambda\,d\pi(\lambda)(\phi),
  30. A ( ϕ ) = i λ i π ( λ i ) ( ϕ ) A(\phi)=\sum_{i}\lambda_{i}\pi({\lambda_{i}})(\phi)

Projective_frame.html

  1. λ 0 v 0 + λ 1 v 1 + + λ n v n + λ n + 1 v n + 1 = 0. \lambda_{0}v_{0}+\lambda_{1}v_{1}+\cdots+\lambda_{n}v_{n}+\lambda_{n+1}v_{n+1}% =0.
  2. μ 0 v 0 + μ 1 v 1 + + μ n v n + μ n + 1 v n + 1 . \mu_{0}v_{0}+\mu_{1}v_{1}+\cdots+\mu_{n}v_{n}+\mu_{n+1}v_{n+1}.

Projective_Hilbert_space.html

  1. P ( H ) P(H)
  2. H H
  3. v v
  4. H H
  5. v 0 v\neq 0
  6. \sim
  7. v w v\sim w
  8. v = λ w v=\lambda w
  9. λ \lambda
  10. \sim
  11. ψ \psi
  12. λ ψ \lambda\psi
  13. λ 0 \lambda\neq 0
  14. ψ \psi
  15. ψ | ψ = 1 \langle\psi|\psi\rangle=1
  16. ψ \psi
  17. ψ \psi
  18. λ \lambda
  19. λ \lambda
  20. λ = e i ϕ \lambda=e^{i\phi}
  21. ϕ \phi
  22. λ \lambda
  23. H H
  24. U ( 1 ) U(1)
  25. H H
  26. H H
  27. H = H n H=H_{n}
  28. U ( n ) \mathrm{U}(n)
  29. O ( n ) \mathrm{O}(n)
  30. P ( H n ) = P n - 1 P(H_{n})=\mathbb{C}P^{n-1}
  31. P 1 \mathbb{C}P^{1}

Projectively_extended_real_line.html

  1. { } \mathbb{R}\cup\{\infty\}
  2. \mathbb{R}
  3. ^ . \widehat{\mathbb{R}}.
  4. 0
  5. 1 1
  6. + +∞
  7. −∞
  8. a 0 = \frac{a}{0}=\infty
  9. 1 / 0 = 1/0=∞
  10. 1 / = 0 1/∞=0
  11. 1 / x 1/x
  12. 0 0⋅∞
  13. \infty
  14. + +\infty
  15. - -\infty
  16. ^ \widehat{\mathbb{R}}
  17. a > a>\infty
  18. a < a<\infty
  19. \infty
  20. ^ \widehat{\mathbb{R}}
  21. a + = + a = , a a - = - a = , a a = a = , a , a 0 = a = 0 , a a = , a a 0 = , a , a 0 \begin{aligned}\\ \displaystyle a+\infty=\infty+a&\displaystyle=\infty,&\displaystyle a\in% \mathbb{R}\\ \displaystyle a-\infty=\infty-a&\displaystyle=\infty,&\displaystyle a\in% \mathbb{R}\\ \displaystyle a\cdot\infty=\infty\cdot a&\displaystyle=\infty,&\displaystyle a% \in\mathbb{R},a\neq 0\\ \displaystyle\infty\cdot\infty&\displaystyle=\infty\\ \displaystyle\frac{a}{\infty}&\displaystyle=0,&\displaystyle a\in\mathbb{R}\\ \displaystyle\frac{\infty}{a}&\displaystyle=\infty,&\displaystyle a\in\mathbb{% R}\\ \displaystyle\frac{a}{0}&\displaystyle=\infty,&\displaystyle a\in\mathbb{R},a% \neq 0\end{aligned}
  22. + \displaystyle\infty+\infty
  23. a , b , c ^ a,b,c\in\widehat{\mathbb{R}}
  24. ( a + b ) + c = a + ( b + c ) a + b = b + a ( a b ) c = a ( b c ) a b = b a a = a 0 \begin{aligned}\displaystyle(a+b)+c&\displaystyle=a+(b+c)\\ \displaystyle a+b&\displaystyle=b+a\\ \displaystyle(a\cdot b)\cdot c&\displaystyle=a\cdot(b\cdot c)\\ \displaystyle a\cdot b&\displaystyle=b\cdot a\\ \displaystyle a\cdot\infty&\displaystyle=\frac{a}{0}\\ \end{aligned}
  25. a , b , c ^ a,b,c\in\widehat{\mathbb{R}}
  26. a ( b + c ) = a b + a c a = ( a b ) b = ( a b ) b a = ( a + b ) - b = ( a - b ) + b \begin{aligned}\displaystyle a\cdot(b+c)&\displaystyle=a\cdot b+a\cdot c\\ \displaystyle a&\displaystyle=(\frac{a}{b})\cdot b&\displaystyle=&% \displaystyle\frac{(a\cdot b)}{b}\\ \displaystyle a&\displaystyle=(a+b)-b&\displaystyle=&\displaystyle(a-b)+b\end{aligned}
  27. ^ \widehat{\mathbb{R}}
  28. a , b , a < b a,b\in\mathbb{R},a<b
  29. [ a , a ] \displaystyle\left[a,a\right]
  30. ^ \widehat{\mathbb{R}}
  31. ^ \widehat{\mathbb{R}}
  32. ^ \widehat{\mathbb{R}}
  33. ( b , a ) = { x x , b < x } { } { x x , x < a } (b,a)=\{x\mid x\in\mathbb{R},b<x\}\cup\{\infty\}\cup\{x\mid x\in\mathbb{R},x<a\}
  34. ^ \widehat{\mathbb{R}}
  35. \mathbb{R}
  36. a , b ^ a,b\in\widehat{\mathbb{R}}
  37. x [ a , b ] 1 x [ 1 b , 1 a ] x\in[a,b]\iff\frac{1}{x}\in\left[\frac{1}{b},\frac{1}{a}\right]
  38. ^ \widehat{\mathbb{R}}
  39. x ^ , A ^ x\in\widehat{\mathbb{R}},A\subseteq\widehat{\mathbb{R}}
  40. x B x\in B
  41. y ^ , y > x y\in\widehat{\mathbb{R}},y>x
  42. [ x , y ) [x,y)
  43. y ^ , y < x y\in\widehat{\mathbb{R}},y<x
  44. ( y , x ] (y,x]
  45. B ^ B\subseteq\widehat{\mathbb{R}}
  46. A = B { x } A=B\setminus\{x\}
  47. f : ^ ^ , p ^ , L ^ f:\widehat{\mathbb{R}}\to\widehat{\mathbb{R}},p\in\widehat{\mathbb{R}},L\in% \widehat{\mathbb{R}}
  48. lim x p f ( x ) = L \lim_{x\to p}{f(x)}=L
  49. x B x\in B
  50. f ( x ) A f(x)\in A
  51. lim x p + f ( x ) = L \lim_{x\to p^{+}}{f(x)}=L
  52. ( lim x p - f ( x ) = L ) \left(\lim_{x\to p^{-}}{f(x)}=L\right)
  53. x B x\in B
  54. f ( x ) A f(x)\in A
  55. lim x p f ( x ) = L \lim_{x\to p}{f(x)}=L
  56. lim x p + f ( x ) = L \lim_{x\to p^{+}}{f(x)}=L
  57. lim x p - f ( x ) = L \lim_{x\to p^{-}}{f(x)}=L
  58. \mathbb{R}
  59. p , L p,L\in\mathbb{R}
  60. lim x p f ( x ) = L \lim_{x\to p}{f(x)}=L
  61. lim x p f ( x ) = L \lim_{x\to p}{f(x)}=L
  62. lim x + f ( x ) = L \lim_{x\to\infty^{+}}{f(x)}=L
  63. lim x - f ( x ) = L \lim_{x\to-\infty}{f(x)}=L
  64. lim x - f ( x ) = L \lim_{x\to\infty^{-}}{f(x)}=L
  65. lim x + f ( x ) = L \lim_{x\to+\infty}{f(x)}=L
  66. lim x p f ( x ) = \lim_{x\to p}{f(x)}=\infty
  67. lim x p | f ( x ) | = + \lim_{x\to p}{|f(x)|}=+\infty
  68. lim x + f ( x ) = \lim_{x\to\infty^{+}}{f(x)}=\infty
  69. lim x - | f ( x ) | = + \lim_{x\to-\infty}{|f(x)|}=+\infty
  70. lim x - f ( x ) = \lim_{x\to\infty^{-}}{f(x)}=\infty
  71. lim x + | f ( x ) | = + \lim_{x\to+\infty}{|f(x)|}=+\infty
  72. A ^ A\subseteq\widehat{\mathbb{R}}
  73. y A y\in A
  74. y x y\neq x
  75. f : ^ ^ , A ^ , L ^ , p ^ f:\widehat{\mathbb{R}}\to\widehat{\mathbb{R}},A\subseteq\widehat{\mathbb{R}},L% \in\widehat{\mathbb{R}},p\in\widehat{\mathbb{R}}
  76. x A C x\in A\cap C
  77. f ( x ) B f(x)\in B
  78. A { p } A\cup\{p\}
  79. A { p } A\cup\{p\}
  80. f : ^ ^ , p ^ . f:\widehat{\mathbb{R}}\to\widehat{\mathbb{R}},\quad p\in\widehat{\mathbb{R}}.
  81. lim x p f ( x ) = f ( p ) . \lim_{x\to p}{f(x)}=f(p).
  82. f : ^ ^ , A ^ . f:\widehat{\mathbb{R}}\to\widehat{\mathbb{R}},\quad A\subseteq\widehat{\mathbb% {R}}.
  83. p A p\in A
  84. ^ \widehat{\mathbb{R}}
  85. tan ( π 2 + n π ) = for n , \tan\left(\frac{\pi}{2}+n\pi\right)=\infty\,\text{ for }n\in\mathbb{Z},
  86. \mathbb{R}
  87. \infty
  88. \mathbb{R}
  89. \infty
  90. ^ \widehat{\mathbb{R}}
  91. ^ \widehat{\mathbb{R}}

Pronic_number.html

  1. ( 10 n + 5 ) 2 = 100 n 2 + 100 n + 25 = 100 n ( n + 1 ) + 25 (10n+5)^{2}=100n^{2}+100n+25=100n(n+1)+25\,

Proof_by_contrapositive.html

  1. P Q P\Rightarrow Q
  2. ¬ ( P Q ) ¬ ( ¬ P Q ) P ¬ Q \lnot(P\Rightarrow Q)\equiv\lnot(\lnot P\vee Q)\equiv P\wedge\lnot Q
  3. ¬ Q ¬ P \lnot Q\Rightarrow\lnot P
  4. P ¬ Q P ¬ P P\wedge\lnot Q\Rightarrow P\wedge\lnot P\equiv\bot

Propagator.html

  1. H H
  2. G ( x , t ; x , t ) = 1 i Θ ( t - t ) K ( x , t ; x , t ) G(x,t;x^{\prime},t^{\prime})=\frac{1}{i\hbar}\Theta(t-t^{\prime})K(x,t;x^{% \prime},t^{\prime})
  3. ( i t - H x ) G ( x , t ; x , t ) = δ ( x - x ) δ ( t - t ) , \left(i\hbar\frac{\partial}{\partial t}-H_{x}\right)G(x,t;x^{\prime},t^{\prime% })=\delta(x-x^{\prime})\delta(t-t^{\prime})~{},
  4. x x
  5. δ ( x ) δ(x)
  6. Θ ( x ) Θ(x)
  7. K ( x , t ; x , t ) K(x,t;x′,t′)
  8. G G
  9. K ( x , t ; x , t ) = x | U ^ ( t , t ) | x , K(x,t;x^{\prime},t^{\prime})=\left\langle x|\hat{U}(t,t^{\prime})|x^{\prime}% \right\rangle,
  10. Û ( t , t ) Û(t,t′)
  11. t t
  12. t t′
  13. K ( x , t ; x , t ) = exp [ i t t L ( q ˙ , q , t ) d t ] D [ q ( t ) ] K(x,t;x^{\prime},t^{\prime})=\int\exp\left[\frac{i}{\hbar}\int_{t}^{t^{\prime}% }L(\dot{q},q,t)dt\right]D[q(t)]
  14. q ( t ) = x , q ( t ) = x q(t)=x,q(t′)=x′
  15. L L
  16. ψ ( x , t ) = - ψ ( x , t ) K ( x , t ; x , t ) d x . \psi(x,t)=\int_{-\infty}^{\infty}\psi(x^{\prime},t^{\prime})K(x,t;x^{\prime},t% ^{\prime})dx^{\prime}.
  17. K ( x , t ; x , t t ) ) K(x,t;x,tt))
  18. x x x−x′
  19. t t t−t′
  20. K ( x , t ; x , t ) = K ( x , x ; t - t ) . K(x,t;x^{\prime},t^{\prime})=K(x,x^{\prime};t-t^{\prime}).
  21. exp ( - i t ( 1 2 m 𝗉 2 + 1 2 m ω 2 𝗑 2 ) ) \exp\left(-\frac{it}{\hbar}\left(\frac{1}{2m}~{}\mathsf{p}^{2}+\frac{1}{2}~{}m% \omega^{2}\mathsf{x}^{2}\right)\right)
  22. = exp ( - i m ω 2 𝗑 2 tan ( ω t 2 ) ) exp ( - i 2 m ω 𝗉 2 sin ( ω t ) ) exp ( - i m ω 2 𝗑 2 tan ( ω t 2 ) ) , =\exp\left(-\frac{im\omega}{2\hbar}~{}\mathsf{x}^{2}\tan\left(\frac{\omega t}{% 2}\right)\right)\exp\left(-\frac{i}{2m\omega\hbar}~{}\mathsf{p}^{2}\sin\left(% \omega t\right)\right)\exp\left(-\frac{im\omega}{2\hbar}~{}\mathsf{x}^{2}\tan% \left(\frac{\omega t}{2}\right)\right)~{},
  23. 𝗑 \mathsf{x}
  24. 𝗉 \mathsf{p}
  25. [ 𝗑 , 𝗉 ] = i [\mathsf{x},\mathsf{p}]=i\hbar
  26. N N
  27. K ( x , x ; t ) = q = 1 N K ( x q , x q ; t ) . K(\vec{x},\vec{x}^{\prime};t)=\prod_{q=1}^{N}K(x_{q},x_{q}^{\prime};t)~{}.
  28. G ( x , y ) G(x,y)
  29. ( x + m 2 ) G ( x , y ) = - δ ( x - y ) (\square_{x}+m^{2})G(x,y)=-\delta(x-y)
  30. x , y x,y
  31. x = 2 t 2 - 2 \square_{x}=\tfrac{\partial^{2}}{\partial t^{2}}-\nabla^{2}
  32. x x
  33. δ ( x y ) δ(x−y)
  34. c c
  35. ħ ħ
  36. ( - p 2 + m 2 ) G ( p ) = - 1. \left(-p^{2}+m^{2}\right)G(p)=-1.
  37. x f ( x ) = 1 xf(x)=1
  38. f ( x ) = 1 x ± i ε = 1 x i π δ ( x ) , f(x)=\frac{1}{x\pm i\varepsilon}=\frac{1}{x}\mp i\pi\delta(x),
  39. ε ε
  40. G ( x , y ) = 1 ( 2 π ) 4 d 4 p e - i p ( x - y ) p 2 - m 2 ± i ε G(x,y)=\frac{1}{(2\pi)^{4}}\int d^{4}p\,\frac{e^{-ip(x-y)}}{p^{2}-m^{2}\pm i\varepsilon}
  41. p ( x - y ) := p 0 ( x 0 - y 0 ) - p ( x - y ) p(x-y):=p_{0}(x^{0}-y^{0})-\vec{p}\cdot(\vec{x}-\vec{y})
  42. p 0 p_{0}
  43. p 0 = ± p 2 + m 2 p_{0}=\pm\sqrt{\vec{p}^{2}+m^{2}}
  44. x x
  45. y y
  46. y y
  47. x x
  48. G r e t ( x , y ) = lim ϵ 0 1 ( 2 π ) 4 d 4 p e - i p ( x - y ) ( p 0 + i ϵ ) 2 - p 2 - m 2 = { 1 2 π δ ( τ x y 2 ) - m J 1 ( m τ x y ) 4 π τ x y y x 0 otherwise G_{ret}(x,y)=\lim_{\epsilon\to 0}\frac{1}{(2\pi)^{4}}\int d^{4}p\,\frac{e^{-ip% (x-y)}}{(p_{0}+i\epsilon)^{2}-\vec{p}^{2}-m^{2}}=\begin{cases}\frac{1}{2\pi}% \delta(\tau_{xy}^{2})-\frac{mJ_{1}(m\tau_{xy})}{4\pi\tau_{xy}}&y\prec x\\ 0&\textrm{otherwise}\end{cases}
  49. τ x y := ( x 0 - y 0 ) 2 - ( x - y ) 2 \tau_{xy}:=\sqrt{(x^{0}-y^{0})^{2}-(\vec{x}-\vec{y})^{2}}
  50. x x
  51. y y
  52. J 1 J_{1}
  53. y x y\prec x
  54. y y
  55. x x
  56. y 0 < x 0 y^{0}<x^{0}
  57. τ x y 2 0 . \tau_{xy}^{2}\geq 0~{}.
  58. G r e t ( x , y ) = i 0 | [ Φ ( x ) , Φ ( y ) ] | 0 Θ ( x 0 - y 0 ) G_{ret}(x,y)=i\langle 0|\left[\Phi(x),\Phi(y)\right]|0\rangle\Theta(x^{0}-y^{0})
  59. Θ ( x ) := { 1 x 0 0 x < 0 \Theta(x):=\begin{cases}1&x\geq 0\\ 0&x<0\end{cases}
  60. [ Φ ( x ) , Φ ( y ) ] := Φ ( x ) Φ ( y ) - Φ ( y ) Φ ( x ) \left[\Phi(x),\Phi(y)\right]:=\Phi(x)\Phi(y)-\Phi(y)\Phi(x)
  61. x x
  62. y y
  63. x > y x⁰>y⁰
  64. y y
  65. x x
  66. G a d v ( x , y ) = lim ϵ 0 1 ( 2 π ) 4 d 4 p e - i p ( x - y ) ( p 0 - i ϵ ) 2 - p 2 - m 2 = { - 1 2 π δ ( τ x y 2 ) + m J 1 ( m τ x y ) 4 π τ x y x y 0 otherwise G_{adv}(x,y)=\lim_{\epsilon\to 0}\frac{1}{(2\pi)^{4}}\int d^{4}p\,\frac{e^{-ip% (x-y)}}{(p_{0}-i\epsilon)^{2}-\vec{p}^{2}-m^{2}}=\begin{cases}-\frac{1}{2\pi}% \delta(\tau_{xy}^{2})+\frac{mJ_{1}(m\tau_{xy})}{4\pi\tau_{xy}}&x\prec y\\ 0&\textrm{otherwise}\end{cases}
  67. G a d v ( x , y ) = - i 0 | [ Φ ( x ) , Φ ( y ) ] | 0 Θ ( y 0 - x 0 ) . G_{adv}(x,y)=-i\langle 0|\left[\Phi(x),\Phi(y)\right]|0\rangle\Theta(y^{0}-x^{% 0})~{}.
  68. G F ( x , y ) = lim ϵ 0 1 ( 2 π ) 4 d 4 p e - i p ( x - y ) p 2 - m 2 + i ϵ = { - 1 4 π δ ( s ) + m 8 π s H 1 ( 2 ) ( m s ) s 0 - i m 4 π 2 - s K 1 ( m - s ) s < 0. G_{F}(x,y)=\lim_{\epsilon\to 0}\frac{1}{(2\pi)^{4}}\int d^{4}p\,\frac{e^{-ip(x% -y)}}{p^{2}-m^{2}+i\epsilon}=\begin{cases}-\frac{1}{4\pi}\delta(s)+\frac{m}{8% \pi\sqrt{s}}H_{1}^{(2)}(m\sqrt{s})&s\geq 0\\ -\frac{im}{4\pi^{2}\sqrt{-s}}K_{1}(m\sqrt{-s})&s<0.\end{cases}
  69. s := ( x 0 - y 0 ) 2 - ( x - y ) 2 , s:=(x^{0}-y^{0})^{2}-(\vec{x}-\vec{y})^{2},
  70. x x
  71. y y
  72. G F ( x - y ) = i 0 | T ( Φ ( x ) Φ ( y ) ) | 0 = i 0 | [ Θ ( x 0 - y 0 ) Φ ( x ) Φ ( y ) + Θ ( y 0 - x 0 ) Φ ( y ) Φ ( x ) ] | 0 . G_{F}(x-y)=i\langle 0|T(\Phi(x)\Phi(y))|0\rangle=i\left\langle 0|\left[\Theta(% x^{0}-y^{0})\Phi(x)\Phi(y)+\Theta(y^{0}-x^{0})\Phi(y)\Phi(x)\right]|0\right\rangle.
  73. x x
  74. y y
  75. Θ Θ
  76. ε ε
  77. ε ε
  78. p p
  79. G ~ r e t ( p ) = 1 ( p 0 + i ε ) 2 - p 2 - m 2 \tilde{G}_{ret}(p)=\frac{1}{(p_{0}+i\varepsilon)^{2}-\vec{p}^{2}-m^{2}}
  80. G ~ a d v ( p ) = 1 ( p 0 - i ε ) 2 - p 2 - m 2 \tilde{G}_{adv}(p)=\frac{1}{(p_{0}-i\varepsilon)^{2}-\vec{p}^{2}-m^{2}}
  81. G ~ F ( p ) = 1 p 2 - m 2 + i ε . \tilde{G}_{F}(p)=\frac{1}{p^{2}-m^{2}+i\varepsilon}.
  82. i −i
  83. Φ ( x ) Φ(x)
  84. G F ε ( x , y ) = ε ( x - y ) 2 + i ε 2 G^{\varepsilon}_{F}(x,y)=\frac{\varepsilon}{(x-y)^{2}+i\varepsilon^{2}}
  85. ε \varepsilon
  86. ε 0 \varepsilon\rightarrow 0
  87. G F ε ( x , y ) = 1 ε G^{\varepsilon}_{F}(x,y)=\frac{1}{\varepsilon}
  88. ( x - y ) 2 = 0 (x-y)^{2}=0
  89. lim ε 0 G F ε ( x , y ) = 0 \lim_{\varepsilon\rightarrow 0}G^{\varepsilon}_{F}(x,y)=0
  90. ( x - y ) 2 0 (x-y)^{2}\neq 0
  91. lim ε 0 | G F ε ( 0 , x ) | 2 d x 3 = lim ε 0 ε 2 ( x 2 - t 2 ) 2 + ε 4 d x 3 = 2 π 2 | t | \lim_{\varepsilon\rightarrow 0}\int\left|G^{\varepsilon}_{F}(0,x)\right|^{2}dx% ^{3}=\lim_{\varepsilon\rightarrow 0}\int\frac{\varepsilon^{2}}{({x}^{2}-t^{2})% ^{2}+\varepsilon^{4}}dx^{3}=2\pi^{2}|t|
  92. S ~ F ( p ) = ( γ μ p μ + m ) p 2 - m 2 + i ε \tilde{S}_{F}(p)=\frac{(\gamma^{\mu}p_{\mu}+m)}{p^{2}-m^{2}+i\varepsilon}
  93. γ μ \gamma^{\mu}
  94. S ~ F ( p ) = 1 γ μ p μ - m + i ε = 1 p / - m + i ϵ \tilde{S}_{F}(p)={1\over\gamma^{\mu}p_{\mu}-m+i\varepsilon}={1\over p\!\!\!/-m% +i\epsilon}
  95. S F ( x - y ) = d 4 p ( 2 π ) 4 e - i p ( x - y ) γ μ p μ + m p 2 - m 2 + i ϵ = ( γ μ ( x - y ) μ | x - y | 5 + m | x - y | 3 ) J 1 ( m | x - y | ) . S_{F}(x-y)=\int\frac{d^{4}p}{(2\pi)^{4}}\,e^{-ip\cdot(x-y)}\frac{\gamma^{\mu}p% _{\mu}+m}{p^{2}-m^{2}+i\epsilon}=\left(\frac{\gamma^{\mu}(x-y)_{\mu}}{|x-y|^{5% }}+\frac{m}{|x-y|^{3}}\right)J_{1}(m|x-y|).
  96. S F ( x - y ) = ( i / + m ) G F ( x - y ) S_{F}(x-y)=(i\partial\!\!\!/+m)G_{F}(x-y)
  97. / := γ μ μ \partial\!\!\!/:=\gamma^{\mu}\partial_{\mu}
  98. - i g μ ν p 2 + i ϵ . {-ig^{\mu\nu}\over p^{2}+i\epsilon}.
  99. λ λ
  100. g μ ν - k μ k ν m 2 k 2 - m 2 + i ϵ + k μ k ν m 2 k 2 - m 2 λ + i ϵ . \frac{g_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{m^{2}}}{k^{2}-m^{2}+i\epsilon}+\frac{% \frac{k_{\mu}k_{\nu}}{m^{2}}}{k^{2}-\frac{m^{2}}{\lambda}+i\epsilon}.
  101. λ = 0 λ=0
  102. λ = 1 λ=1
  103. λ = λ=∞
  104. λ λ
  105. g μ ν - k μ k ν m 2 k 2 - m 2 + i ϵ . \frac{g_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{m^{2}}}{k^{2}-m^{2}+i\epsilon}.
  106. g μ ν k 2 - m 2 + i ϵ . \frac{g_{\mu\nu}}{k^{2}-m^{2}+i\epsilon}.
  107. g μ ν - k μ k ν k 2 k 2 - m 2 + i ϵ . \frac{g_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{k^{2}}}{k^{2}-m^{2}+i\epsilon}.
  108. Δ ( x - y ) \Delta(x-y)
  109. 0 | [ Φ ( x ) , Φ ( y ) ] | 0 = i Δ ( x - y ) \langle 0|\left[\Phi(x),\Phi(y)\right]|0\rangle=i\Delta(x-y)
  110. Δ ( x - y ) = G a d v ( x - y ) - G r e t ( x - y ) \,\Delta(x-y)=G_{adv}(x-y)-G_{ret}(x-y)
  111. Δ ( x - y ) = - Δ ( y - x ) \Delta(x-y)=-\Delta(y-x)
  112. ( x - y ) 2 < 0 (x-y)^{2}<0
  113. Δ ( x - y ) \Delta(x-y)
  114. Δ + ( x - y ) = 0 | Φ ( x ) Φ ( y ) | 0 \Delta_{+}(x-y)=\langle 0|\Phi(x)\Phi(y)|0\rangle
  115. Δ - ( x - y ) = 0 | Φ ( y ) Φ ( x ) | 0 \Delta_{-}(x-y)=\langle 0|\Phi(y)\Phi(x)|0\rangle
  116. i Δ = Δ + - Δ - \,i\Delta=\Delta_{+}-\Delta_{-}
  117. ( x + m 2 ) Δ ± ( x - y ) = 0. (\Box_{x}+m^{2})\Delta_{\pm}(x-y)=0.
  118. Δ 1 ( x - y ) \Delta_{1}(x-y)
  119. 0 | { Φ ( x ) , Φ ( y ) } | 0 = Δ 1 ( x - y ) \langle 0|\left\{\Phi(x),\Phi(y)\right\}|0\rangle=\Delta_{1}(x-y)
  120. Δ 1 ( x - y ) = Δ + ( x - y ) + Δ - ( x - y ) . \,\Delta_{1}(x-y)=\Delta_{+}(x-y)+\Delta_{-}(x-y).
  121. Δ 1 ( x - y ) = Δ 1 ( y - x ) . \,\Delta_{1}(x-y)=\Delta_{1}(y-x).
  122. G r e t ( x - y ) = - Δ ( x - y ) Θ ( x 0 - y 0 ) G_{ret}(x-y)=-\Delta(x-y)\Theta(x_{0}-y_{0})
  123. G a d v ( x - y ) = Δ ( x - y ) Θ ( y 0 - x 0 ) G_{adv}(x-y)=\Delta(x-y)\Theta(y_{0}-x_{0})
  124. 2 G F ( x - y ) = - i Δ 1 ( x - y ) + ϵ ( x 0 - y 0 ) Δ ( x - y ) 2G_{F}(x-y)=-i\Delta_{1}(x-y)+\epsilon(x_{0}-y_{0})\Delta(x-y)
  125. ϵ ( x 0 - y 0 ) = 2 Θ ( x 0 - y 0 ) - 1. \,\epsilon(x_{0}-y_{0})=2\Theta(x_{0}-y_{0})-1.

Proper_length.html

  1. L 0 = Δ x L_{0}=\Delta x
  2. L = L 0 / γ L=L_{0}/\gamma
  3. Δ σ = Δ x 2 - c 2 Δ t 2 \Delta\sigma=\sqrt{\Delta x^{2}-c^{2}\Delta t^{2}}
  4. Δ σ = Δ x 2 + Δ y 2 + Δ z 2 - c 2 Δ t 2 \Delta\sigma=\sqrt{\Delta x^{2}+\Delta y^{2}+\Delta z^{2}-c^{2}\Delta t^{2}}
  5. L = c P - g μ ν d x μ d x ν L=c\int_{P}\sqrt{-g_{\mu\nu}dx^{\mu}dx^{\nu}}
  6. c c

Proper_map.html

  1. f : X Y f:X\to Y
  2. f - 1 ( y ) f^{-1}(y)
  3. y Y y\in Y
  4. K K
  5. Y Y
  6. f - 1 ( K ) f^{-1}(K)
  7. { U λ | λ Λ } \{U_{\lambda}|\lambda\ \in\ \Lambda\}
  8. f - 1 ( K ) f^{-1}(K)
  9. k K k\ \in K
  10. f - 1 ( k ) f^{-1}(k)
  11. k K k\ \in K
  12. γ k Λ \gamma_{k}\subset\Lambda
  13. f - 1 ( k ) λ γ k U λ f^{-1}(k)\subset\cup_{\lambda\in\gamma_{k}}U_{\lambda}
  14. X λ γ k U λ X\setminus\cup_{\lambda\in\gamma_{k}}U_{\lambda}
  15. V k = Y f ( X λ γ k U λ ) V_{k}=Y\setminus f(X\setminus\cup_{\lambda\in\gamma_{k}}U_{\lambda})
  16. V k V_{k}
  17. k k
  18. K k K V k K\subset\cup_{k\in K}V_{k}
  19. k 1 , , k s k_{1},\dots,k_{s}
  20. K i = 1 s V k i K\subset\cup_{i=1}^{s}V_{k_{i}}
  21. Γ = i = 1 s γ k i \Gamma=\cup_{i=1}^{s}\gamma_{k_{i}}
  22. Γ \Gamma
  23. f - 1 ( K ) f - 1 ( i = 1 s V k i ) λ Γ U λ f^{-1}(K)\subset f^{-1}(\cup_{i=1}^{s}V_{k_{i}})\subset\cup_{\lambda\in\Gamma}% U_{\lambda}
  24. f - 1 ( K ) f^{-1}(K)

Property_B.html

  1. 2 n - 1 2^{n-1}
  2. 2 - n + 1 2^{-n+1}
  3. 2 n - 1 2 - n + 1 = 1 2^{n-1}2^{-n+1}=1
  4. O ( 2 n n 2 ) O(2^{n}\cdot n^{2})
  5. n / ( n + 4 ) 2 n n/(n+4)\cdot 2^{n}
  6. m ( n ) = θ ( 2 n n ) m(n)=\theta(2^{n}\cdot n)
  7. m ( n ) = Ω ( n 1 / 3 2 n ) m(n)=\Omega(n^{1/3}2^{n})
  8. m ( n ) = Ω ( 2 n n / log n ) m(n)=\Omega(2^{n}\cdot\sqrt{n/\log n})

Proportionality.html

  1. n n
  2. 1 / n 1/n

Proton_exchange_membrane_fuel_cell.html

  1. H 2 2 H + + 2 e - \mathrm{H}_{2}\rightarrow\mathrm{2H}^{+}+\mathrm{2e}^{-}
  2. E o = 0 V E^{o}=0\,V
  3. ( 1 ) \left(1\right)
  4. 1 2 O 2 + 2 H + + 2 e - H 2 O \frac{1}{2}\mathrm{O}_{2}+{2H}^{+}+{2e}^{-}\rightarrow\mathrm{H}_{2}\mathrm{O}
  5. E o = 1.229 V E^{o}=1.229\,V
  6. ( 2 ) \left(2\right)
  7. H 2 + 1 2 O 2 H 2 O \mathrm{H}_{2}+\frac{1}{2}\mathrm{O}_{2}\rightarrow\mathrm{H}_{2}\mathrm{O}
  8. E o = 1.229 V E^{o}=1.229\,V
  9. ( 3 ) \left(3\right)
  10. η = Δ G Δ H = 1 - T Δ S Δ H \eta=\frac{\Delta G}{\Delta H}=1-\frac{T\Delta S}{\Delta H}

Prouhet–Thue–Morse_constant.html

  1. τ \tau
  2. τ = i = 0 t i 2 i + 1 = 0.412454033640 \tau=\sum_{i=0}^{\infty}\frac{t_{i}}{2^{i+1}}=0.412454033640\ldots
  3. t i t_{i}
  4. t i t_{i}
  5. τ ( x ) = i = 0 ( - 1 ) t i x i = 1 1 - x - 2 i = 0 t i x i \tau(x)=\sum_{i=0}^{\infty}(-1)^{t_{i}}\,x^{i}=\frac{1}{1-x}-2\sum_{i=0}^{% \infty}t_{i}\,x^{i}
  6. τ ( x ) = n = 0 ( 1 - x 2 n ) . \tau(x)=\prod_{n=0}^{\infty}(1-x^{2^{n}}).

Proxy_(climate).html

  1. × ( ( [ O 18 ] / [ O 16 ] ) ( [ O 18 ] / [ O 16 ] ) VSMOW - 1 ) \times\left(\frac{([{}^{18}O]/[{}^{16}O])}{([{}^{18}O]/[{}^{16}O])_{\mathrm{% VSMOW}}}-1\right)

Pseudo-differential_operator.html

  1. P ( D ) := α a α D α P(D):=\sum_{\alpha}a_{\alpha}\,D^{\alpha}
  2. u u
  3. P ( ξ ) = α a α ξ α , P(\xi)=\sum_{\alpha}a_{\alpha}\,\xi^{\alpha},
  4. P ( D ) u ( x ) = 1 ( 2 π ) n n n e i ( x - y ) ξ P ( ξ ) u ( y ) d y d ξ \quad P(D)u(x)=\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e% ^{i(x-y)\xi}P(\xi)u(y)\,dy\,d\xi
  5. α = ( α 1 , , α n ) \alpha=(\alpha_{1},\ldots,\alpha_{n})
  6. a α a_{\alpha}
  7. D α = ( - i 1 ) α 1 ( - i n ) α n D^{\alpha}=(-i\partial_{1})^{\alpha_{1}}\cdots(-i\partial_{n})^{\alpha_{n}}
  8. - i -i
  9. u ^ ( ξ ) := e - i y ξ u ( y ) d y \hat{u}(\xi):=\int e^{-iy\xi}u(y)\,dy
  10. u ( x ) = 1 ( 2 π ) n e i x ξ u ^ ( ξ ) d ξ = 1 ( 2 π ) n e i ( x - y ) ξ u ( y ) d y d ξ u(x)=\frac{1}{(2\pi)^{n}}\int e^{ix\xi}\hat{u}(\xi)d\xi=\frac{1}{(2\pi)^{n}}% \iint e^{i(x-y)\xi}u(y)\,dy\,d\xi
  11. P ( D x ) e i ( x - y ) ξ = e i ( x - y ) ξ P ( ξ ) P(D_{x})\,e^{i(x-y)\xi}=e^{i(x-y)\xi}\,P(\xi)
  12. P ( D ) u = f P(D)\,u=f
  13. P ( ξ ) u ^ ( ξ ) = f ^ ( ξ ) . P(\xi)\,\hat{u}(\xi)=\hat{f}(\xi).
  14. u ^ ( ξ ) = 1 P ( ξ ) f ^ ( ξ ) \hat{u}(\xi)=\frac{1}{P(\xi)}\hat{f}(\xi)
  15. u ( x ) = 1 ( 2 π ) n e i x ξ 1 P ( ξ ) f ^ ( ξ ) d ξ . u(x)=\frac{1}{(2\pi)^{n}}\int e^{ix\xi}\frac{1}{P(\xi)}\hat{f}(\xi)\,d\xi.
  16. u ( x ) = 1 ( 2 π ) n e i ( x - y ) ξ 1 P ( ξ ) f ( y ) d y d ξ . u(x)=\frac{1}{(2\pi)^{n}}\iint e^{i(x-y)\xi}\frac{1}{P(\xi)}f(y)\,dy\,d\xi.
  17. P ( x , D ) u ( x ) = 1 ( 2 π ) n n e i x ξ P ( x , ξ ) u ^ ( ξ ) d ξ \quad P(x,D)u(x)=\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}e^{ix\cdot\xi}P(x,% \xi)\hat{u}(\xi)\,d\xi
  18. u ^ ( ξ ) \hat{u}(\xi)
  19. | ξ α x β P ( x , ξ ) | C α , β ( 1 + | ξ | ) m - | α | |\partial_{\xi}^{\alpha}\partial_{x}^{\beta}P(x,\xi)|\leq C_{\alpha,\beta}\,(1% +|\xi|)^{m-|\alpha|}
  20. S 1 , 0 m \scriptstyle{S^{m}_{1,0}}
  21. Ψ 1 , 0 m . \scriptstyle{\Psi^{m}_{1,0}}.
  22. p ( x , D ) p(x,D)\,

Pseudo-Hadamard_transform.html

  1. a = a + b ( mod 2 n ) a^{\prime}=a+b\,\;\;(\mathop{{\rm mod}}2^{n})\,
  2. b = a + 2 b ( mod 2 n ) b^{\prime}=a+2b\,\;\;(\mathop{{\rm mod}}2^{n})\,
  3. b = b - a ( mod 2 n ) b=b^{\prime}-a^{\prime}\,\;\;(\mathop{{\rm mod}}2^{n})
  4. a = 2 a - b ( mod 2 n ) a=2a^{\prime}-b^{\prime}\,\;\;(\mathop{{\rm mod}}2^{n})
  5. H 1 = [ 2 1 1 1 ] H_{1}=\begin{bmatrix}2&1\\ 1&1\end{bmatrix}
  6. H n = [ 2 × H n - 1 H n - 1 H n - 1 H n - 1 ] H_{n}=\begin{bmatrix}2\times H_{n-1}&H_{n-1}\\ H_{n-1}&H_{n-1}\end{bmatrix}
  7. H 2 = [ 4 2 2 1 2 2 1 1 2 1 2 1 1 1 1 1 ] H_{2}=\begin{bmatrix}4&2&2&1\\ 2&2&1&1\\ 2&1&2&1\\ 1&1&1&1\end{bmatrix}

Psychrometrics.html

  1. r = h c k y c s r=\frac{h_{c}}{k_{y}c_{s}}\,
  2. r r
  3. h c h_{c}
  4. k y k_{y}
  5. c s c_{s}
  6. ( 6 2 ) = 15 \left({6\atop 2}\right)=15

Pullback_(category_theory).html

  1. f : X Z f:X→Z
  2. g : Y Z g:Y→Z
  3. X Z Y X→Z←Y
  4. f f
  5. g g
  6. P P
  7. u : Q P u:Q→P
  8. p 2 u = q 2 , p 1 u = q 1 . p_{2}\circ u=q_{2},\qquad p_{1}\circ u=q_{1}.
  9. A A
  10. B B
  11. X Z Y X→Z←Y
  12. u : Q P u:Q→P
  13. f f
  14. g g
  15. Z Z
  16. X X
  17. Y Y
  18. Z Z
  19. f f
  20. g g
  21. Z Z
  22. f f
  23. g g
  24. X X
  25. Y Y
  26. 𝐂𝐑𝐢𝐧𝐠 \mathbf{CRing}
  27. A , B , C O b ( 𝐂𝐑𝐢𝐧𝐠 ) A,B,C∈Ob(\mathbf{CRing})
  28. α : A C H o m ( 𝐂𝐑𝐢𝐧𝐠 ) α:A→C∈Hom(\mathbf{CRing})
  29. β : B C H o m ( 𝐂𝐑𝐢𝐧𝐠 ) β:B→C∈Hom(\mathbf{CRing})
  30. A A
  31. B B
  32. C C
  33. α α
  34. β β
  35. A × B A×B
  36. A × C B = { ( a , b ) A × B | α ( a ) = β ( b ) } A\times_{C}B=\left\{(a,b)\in A\times B\;\big|\;\alpha(a)=\beta(b)\right\}
  37. β : A × C B A , α : A × C B B \beta^{\prime}\colon A\times_{C}B\to A,\qquad\alpha^{\prime}\colon A\times_{C}% B\to B
  38. β ( a , b ) = a \beta^{\prime}(a,b)=a
  39. α ( a , b ) = b \alpha^{\prime}(a,b)=b
  40. ( a , b ) A × C B (a,b)\in A\times_{C}B
  41. α β = β α . \alpha\circ\beta^{\prime}=\beta\circ\alpha^{\prime}.
  42. f f
  43. g g
  44. X × Z Y = { ( x , y ) X × Y | f ( x ) = g ( y ) } , X\times_{Z}Y=\{(x,y)\in X\times Y|f(x)=g(y)\},
  45. 𝐒𝐞𝐭 \mathbf{Set}
  46. X × Z Y x X g - 1 [ { f ( x ) } ] y Y f - 1 [ { g ( y ) } ] X\times_{Z}Y\cong\coprod_{x\in X}g^{-1}[\{f(x)\}]\cong\coprod_{y\in Y}f^{-1}[% \{g(y)\}]
  47. \coprod
  48. f f
  49. g g
  50. π < s u b > 1 π<sub>1

Pulse_repetition_frequency.html

  1. \Tau \Tau
  2. \Tau = 1 PRF \Tau=\frac{1}{\,\text{PRF}}
  3. Pulse Spacing = Propagation Speed PRF \,\text{Pulse Spacing}=\frac{\,\text{Propagation Speed}}{\,\text{PRF}}
  4. Range = c τ 2 \,\text{Range}=\frac{c\tau}{2}
  5. Max Range = c τ PRT 2 = c 2 PRF { τ PRT = 1 PRF \,\text{Max Range}=\frac{c\tau\text{PRT}}{2}=\frac{c}{2\,\,\text{PRF}}\qquad% \begin{cases}\tau\text{PRT}=\frac{1}{\,\text{PRF}}\end{cases}
  6. 300 km range = C 2 × 500 \,\text{300 km range}=\frac{C}{2\times 500}
  7. 75 m/s velocity = 500 × C 2 × 10 9 \,\text{75 m/s velocity}=\frac{500\times C}{2\times 10^{9}}
  8. 450 km = C 0.033 × 2 × 10 , 000 \,\text{450 km}=\frac{C}{0.033\times 2\times 10,000}
  9. 1,500 m/s = 10 , 000 × C 2 × 10 9 \,\text{1,500 m/s}=\frac{10,000\times C}{2\times 10^{9}}
  10. 150 km = 30 × C 2 × 30 , 000 \,\text{150 km}=\frac{30\times C}{2\times 30,000}
  11. 4,500 m/s = 30 , 000 × C 2 × 10 9 \,\text{4,500 m/s}=\frac{30,000\times C}{2\times 10^{9}}

Pupillary_light_reflex.html

  1. M ( D ) = tanh - 1 ( D - 4.9 3 ) M(D)=\tanh^{-1}\left(\frac{D-4.9}{3}\right)
  2. d M d D d D d t + 2.3026 tanh - 1 ( D - 4.9 3 ) = 5.2 - 0.45 ln [ Φ ( t - τ ) 4.8118 × 10 - 10 ] \frac{\mathrm{d}M}{\mathrm{d}D}\frac{\mathrm{d}D}{\mathrm{d}t}+2.3026\;\tanh^{% -1}\left(\frac{D-4.9}{3}\right)=5.2-0.45\;\ln\left[\frac{\Phi(t-\tau)}{4.8118~% {}\times~{}10^{-10}}\right]\;\;
  3. D D
  4. Φ ( t - τ ) \Phi(t-\tau)
  5. t t
  6. Φ = I A \Phi=IA
  7. τ \tau
  8. d M \mathrm{d}M
  9. d D \mathrm{d}D
  10. d t \mathrm{d}t
  11. M M
  12. D D
  13. t t
  14. d t c = T c - T p S \mathrm{d}t_{c}=\frac{T_{c}-T_{p}}{S}
  15. d t d = T c - T p 3 S \mathrm{d}t_{d}=\frac{T_{c}-T_{p}}{3S}
  16. d t c \mathrm{d}t_{c}
  17. d t d \mathrm{d}t_{d}
  18. d t \mathrm{d}t
  19. T c Tc
  20. T p Tp
  21. S S
  22. S S

Pure_submodule.html

  1. j = 1 n a i j x j = y i for i = 1 , , m \sum_{j=1}^{n}a_{ij}x_{j}=y_{i}\qquad\mbox{ for }~{}i=1,\ldots,m
  2. j = 1 n a i j x j = y i for i = 1 , , m \sum_{j=1}^{n}a_{ij}x^{\prime}_{j}=y_{i}\qquad\mbox{ for }~{}i=1,\ldots,m

Pyrgeometer.html

  1. E net = E in - E out \ E_{\mathrm{net}}={\ E_{\mathrm{in}}-\ E_{\mathrm{out}}}
  2. E net E_{\mathrm{net}}
  3. E in E_{\mathrm{in}}
  4. E out E_{\mathrm{out}}
  5. E net = U emf S \ E_{\mathrm{net}}=\frac{\ U_{\mathrm{emf}}}{S}
  6. E net E_{\mathrm{net}}
  7. U emf U_{\mathrm{emf}}
  8. S S
  9. S S
  10. E out = σ T 4 \ E_{\mathrm{out}}={\sigma T^{4}}
  11. E out E_{\mathrm{out}}
  12. σ \sigma
  13. T T
  14. E in = U emf S + σ T 4 \ E_{\mathrm{in}}=\frac{U_{\mathrm{emf}}}{S}+{\sigma T^{4}}