wpmath0000004_9

Lipps–Meyer_law.html

  1. F = h 2 / 2 n F=h_{2}/2^{n}
  2. n n
  3. h 2 h_{2}
  4. I = 12 l o g 2 ( h 2 / 2 n ) = 12 l o g 2 ( h 2 ) - 12 n I=12log_{2}(h_{2}/2^{n})=12log_{2}(h_{2})-12n
  5. 12 n 12n

Lissajous_curve.html

  1. x = A sin ( a t + δ ) , y = B sin ( b t ) , x=A\sin(at+\delta),\quad y=B\sin(bt),
  2. δ = N - 1 N π 2 \delta=\frac{N-1}{N}\frac{\pi}{2}
  3. a b \frac{a}{b}
  4. ± 90 \pm 90^{\circ}
  5. \infty

List_coloring.html

  1. f ( v ) = k f(v)=k
  2. n = ( 2 k - 1 k ) n={\left({{2k-1}\atop{k}}\right)}
  3. f : V { j , , k } f:V\to\{j,\dots,k\}
  4. Π 2 p \Pi^{p}_{2}

List_of_important_publications_in_mathematics.html

  1. a x 2 + b y 2 + c x y ax^{2}+by^{2}+cxy
  2. ζ ( 2 k ) ζ(2k)
  3. k k
  4. e e
  5. e \textstyle\sqrt{e}
  6. L 2 L^{2}
  7. ω \omega

List_of_interactive_geometry_software.html

  1. α \alpha

List_of_mathematical_jargon.html

  1. f : A C f\colon A\to C
  2. f = h g f=h\circ g
  3. g : A B g\colon A\to B
  4. h : B C h\colon B\to C
  5. B B
  6. g g
  7. h h
  8. x = y + 1 x=y+1

List_of_mnemonics.html

  1. ( h i g h l o w ) = l o w h i g h - h i l o w l o w 2 \left(\frac{high}{low}\right)^{\prime}=\frac{low\cdot high^{\prime}-hi\cdot low% ^{\prime}}{low^{2}}

List_of_statements_undecidable_in_ZFC.html

  1. 1 \aleph_{1}
  2. 1 \aleph_{1}
  3. 1 \aleph_{1}
  4. ω 2 \omega_{2}
  5. P ( ω ) / P(\omega)/
  6. 1 \aleph_{1}

LM_hash.html

  1. 95 14 2 92 95^{14}\approx 2^{92}
  2. 95 7 2 46 95^{7}\approx 2^{46}
  3. 69 7 2 43 69^{7}\approx 2^{43}

Local_class_field_theory.html

  1. K K
  2. K K
  3. n n
  4. K n M ( K ) \mathrm{K}^{\mathrm{M}}_{n}(K)
  5. n = 1 n=1
  6. n > 1 n>1

Local_hidden_variable_theory.html

  1. P ( a , b ) = d λ ρ ( λ ) p A ( a , λ ) p B ( b , λ ) P(a,b)=\int d\lambda\cdot\rho(\lambda)\cdot p_{A}(a,\lambda)\cdot p_{B}(b,\lambda)
  2. p A ( a , λ ) p_{A}(a,\lambda)
  3. A A
  4. λ \lambda
  5. A A
  6. a a
  7. p B ( b , λ ) p_{B}(b,\lambda)
  8. B B
  9. b b
  10. B B
  11. λ \lambda
  12. λ \lambda
  13. ρ ( λ ) \rho(\lambda)
  14. p A ( a , λ ) p_{A}(a,\lambda)
  15. p B ( a , λ ) p_{B}(a,\lambda)
  16. P ( a , b ) = d λ ρ ( λ ) cos 2 ( a - λ ) cos 2 ( b - λ ) = 1 8 + cos 2 ϕ 4 P(a,b)=\int d\lambda\cdot\rho(\lambda)\cdot\cos^{2}(a-\lambda)\cdot\cos^{2}(b-% \lambda)=\frac{1}{8}+\frac{\cos^{2}\phi}{4}
  17. ϕ = b - a \phi=b-a

Logarithmic_distribution.html

  1. - ln ( 1 - p ) = p + p 2 2 + p 3 3 + . -\ln(1-p)=p+\frac{p^{2}}{2}+\frac{p^{3}}{3}+\cdots.
  2. k = 1 - 1 ln ( 1 - p ) p k k = 1. \sum_{k=1}^{\infty}\frac{-1}{\ln(1-p)}\;\frac{p^{k}}{k}=1.
  3. f ( k ) = - 1 ln ( 1 - p ) p k k f(k)=\frac{-1}{\ln(1-p)}\;\frac{p^{k}}{k}
  4. i = 1 N X i \sum_{i=1}^{N}X_{i}
  5. f ( k + 1 ) = k p k + 1 f ( k ) ; with the initial value f ( 1 ) = - p ln ( 1 - p ) . f(k+1)=\frac{kp}{k+1}f(k);\,\text{ with the initial value }f(1)=\frac{-p}{\ln(% 1-p)}.

Logarithmic_form.html

  1. Ω X p ( log D ) . \Omega^{p}_{X}(\log D).
  2. ω = d f f = ( m z + g ( z ) g ( z ) ) d z \omega=\frac{df}{f}=\left(\frac{m}{z}+\frac{g^{\prime}(z)}{g(z)}\right)dz
  3. f ( z ) = z m g ( z ) f(z)=z^{m}g(z)
  4. Ω X p ( log D ) \Omega^{p}_{X}(\log D)
  5. d Ω X p ( log D ) ( U ) Ω X p + 1 ( log D ) ( U ) d\Omega^{p}_{X}(\log D)(U)\subset\Omega^{p+1}_{X}(\log D)(U)
  6. ( Ω X ( log D ) , d ) (\Omega^{\bullet}_{X}(\log D),d)
  7. j * Ω X - D j_{*}\Omega^{\bullet}_{X-D}
  8. j : X - D X j:X-D\rightarrow X
  9. Ω X - D \Omega^{\bullet}_{X-D}
  10. { D ν } \{D_{\nu}\}
  11. D = D ν D=\sum D_{\nu}
  12. D ν D_{\nu}
  13. z 1 z k = 0 z_{1}\cdots z_{k}=0
  14. Ω X 1 ( log D ) \Omega^{1}_{X}(\log D)
  15. Ω X 1 ( log D ) p = 𝒪 X , p d z 1 z 1 𝒪 X , p d z k z k 𝒪 X , p d z k + 1 𝒪 X , p d z n \Omega_{X}^{1}(\log D)_{p}=\mathcal{O}_{X,p}\frac{dz_{1}}{z_{1}}\oplus\cdots% \oplus\mathcal{O}_{X,p}\frac{dz_{k}}{z_{k}}\oplus\mathcal{O}_{X,p}dz_{k+1}% \oplus\cdots\oplus\mathcal{O}_{X,p}dz_{n}
  16. Ω X k ( log D ) p = j = 1 k Ω X 1 ( log D ) p \Omega_{X}^{k}(\log D)_{p}=\bigwedge^{k}_{j=1}\Omega_{X}^{1}(\log D)_{p}
  17. g ( x , y ) = y 2 - f ( x ) = 0 g(x,y)=y^{2}-f(x)=0
  18. f ( x ) = x ( x - 1 ) ( x - λ ) f(x)=x(x-1)(x-\lambda)
  19. λ 0 , 1 \lambda\neq 0,1
  20. ω = d x d y g ( x , y ) \omega=\frac{dx\wedge dy}{g(x,y)}
  21. Res D ( ω ) = d y g / x | D = - d x g / y | D = - 1 2 d x y | D \,\text{Res}_{D}(\omega)=\frac{dy}{\partial g/\partial x}|_{D}=-\frac{dx}{% \partial g/\partial y}|_{D}=-\frac{1}{2}\frac{dx}{y}|_{D}
  22. d x / y | D dx/y|_{D}
  23. j : X Y j:X\hookrightarrow Y
  24. Ω Y ( log D ) j * Ω X \Omega^{\bullet}_{Y}(\log D)\rightarrow j_{*}\Omega_{X}^{\bullet}
  25. H k ( X ; 𝐂 ) = k ( Y , Ω Y ( log D ) ) H^{k}(X;\mathbf{C})=\mathbb{H}^{k}(Y,\Omega^{\bullet}_{Y}(\log D))
  26. \mathbb{H}^{\bullet}
  27. W Ω Y p ( log D ) W_{\bullet}\Omega^{p}_{Y}(\log D)
  28. W m Ω Y p ( log D ) = { 0 m < 0 Ω Y p ( log D ) m p Ω Y p - m Ω Y m ( log D ) 0 m p W_{m}\Omega^{p}_{Y}(\log D)=\begin{cases}0&m<0\\ \Omega^{p}_{Y}(\log D)&m\geq p\\ \Omega^{p-m}_{Y}\wedge\Omega^{m}_{Y}(\log D)&0\leq m\leq p\end{cases}
  29. F Ω Y p ( log D ) F^{\bullet}\Omega^{p}_{Y}(\log D)
  30. W m H k ( X ; 𝐂 ) = Im ( k ( Y , W m - k Ω Y ( log D ) ) H k ( X ; 𝐂 ) ) W_{m}H^{k}(X;\mathbf{C})=\,\text{Im}(\mathbb{H}^{k}(Y,W_{m-k}\Omega^{\bullet}_% {Y}(\log D))\rightarrow H^{k}(X;\mathbf{C}))
  31. F p H k ( X ; 𝐂 ) = Im ( k ( Y , F p Ω Y ( log D ) ) H k ( X ; 𝐂 ) ) F^{p}H^{k}(X;\mathbf{C})=\,\text{Im}(\mathbb{H}^{k}(Y,F^{p}\Omega^{\bullet}_{Y% }(\log D))\rightarrow H^{k}(X;\mathbf{C}))
  32. W m H k ( X ; 𝐂 ) W_{m}H^{k}(X;\mathbf{C})
  33. W , F W_{\bullet},F^{\bullet}
  34. H k ( X ; 𝐙 ) H^{k}(X;\mathbf{Z})

Logical_framework.html

  1. λ Π \lambda\Pi
  2. λ Π \lambda\Pi
  3. J K J\vdash K
  4. Λ x J . K ( x ) \Lambda x\in J.K(x)
  5. {\mathcal{L}}
  6. Λ x C . J ( x ) K \Lambda x\in C.J(x)\vdash K
  7. λ \lambda
  8. λ Π \lambda\Pi
  9. λ Π \lambda\Pi
  10. λ Π \lambda\Pi

Longitude_of_the_ascending_node.html

  1. 𝐧 = 𝐤 × 𝐡 = ( - h y , h x , 0 ) \mathbf{n}=\mathbf{k}\times\mathbf{h}=(-h_{y},h_{x},0)
  2. Ω = arccos n x | 𝐧 | ( n y 0 ) ; \Omega=\arccos{{n_{x}}\over{\mathbf{\left|n\right|}}}\ \ (n_{y}\geq 0);
  3. Ω = 2 π - arccos n x | 𝐧 | ( n y < 0 ) . \Omega=2\pi-\arccos{{n_{x}}\over{\mathbf{\left|n\right|}}}\ \ (n_{y}<0).

Longitude_of_the_periapsis.html

  1. ϖ \varpi
  2. Ω \Omega
  3. ω \omega
  4. ϖ = Ω + ω \varpi=\Omega+\omega

Loop-erased_random_walk.html

  1. γ \gamma
  2. γ ( 1 ) , , γ ( n ) \gamma(1),\dots,\gamma(n)
  3. γ ( i ) \gamma(i)
  4. γ ( i + 1 ) \gamma(i+1)
  5. γ \gamma
  6. γ \gamma
  7. i j i_{j}
  8. i 1 = 1 i_{1}=1\,
  9. i j + 1 = max { i : γ ( i ) = γ ( i j ) } + 1 i_{j+1}=\max\{i:\gamma(i)=\gamma(i_{j})\}+1\,
  10. γ \gamma
  11. i j i_{j}
  12. γ ( i j ) = γ ( n ) \gamma(i_{j})=\gamma(n)
  13. i J i_{J}
  14. i j i_{j}
  15. γ \gamma
  16. LE ( γ ) \mathrm{LE}(\gamma)
  17. LE ( γ ) ( j ) = γ ( i j ) . \mathrm{LE}(\gamma)(j)=\gamma(i_{j}).\,
  18. γ ( 1 ) , , γ ( n ) \gamma(1),...,\gamma(n)
  19. f ( γ ( i ) ) = 0 f(\gamma(i))=0
  20. i n i\leq n
  21. f ( w ) = 1 f(w)=1
  22. γ ( n + 1 ) \gamma(n+1)
  23. γ ( n ) \gamma(n)
  24. x 1 , , x d x_{1},...,x_{d}
  25. x i x_{i}
  26. f ( x i ) j = 1 d f ( x j ) . \frac{f(x_{i})}{\sum_{j=1}^{d}f(x_{j})}.
  27. ( a 1 , , a d ) (a_{1},...,a_{d})
  28. a i a_{i}
  29. n / log 1 / 3 n n/\log^{1/3}n
  30. G := D ε 2 , G:=D\cap\varepsilon\mathbb{Z}^{2},
  31. S D , x S_{D,x}
  32. ϕ ( S D , x ) = S E , ϕ ( x ) . \phi(S_{D,x})=S_{E,\phi(x)}.\,
  33. r 5 / 4 r^{5/4}
  34. L ( r ) L(r)
  35. c r 1 + ε L ( r ) C r 5 / 3 cr^{1+\varepsilon}\leq L(r)\leq Cr^{5/3}\,
  36. 1 + ε 1+\varepsilon
  37. 1.62400 ± 0.00005 1.62400\pm 0.00005

Loopholes_in_Bell_test_experiments.html

  1. - 2 E ( a , b ) - E ( a , b ) + E ( a , b ) + E ( a , b ) 2 -2\leq E(a,b)-E(a,b^{\prime})+E(a^{\prime},b)+E(a^{\prime},b^{\prime})\leq 2
  2. | E ( A C | coinc. ) + E ( A D | coinc. ) | + | E ( B C | coinc. ) - E ( B D | coinc. ) | 4 η - 2 \big|E(AC^{\prime}|\,\text{coinc.})+E(AD^{\prime}|\,\text{coinc.})\big|+\big|E% (BC^{\prime}|\,\text{coinc.})-E(BD^{\prime}|\,\text{coinc.})\big|\leq\frac{4}{% \eta}-2
  3. η \eta
  4. 2 2 2\sqrt{2}
  5. 2 ( 2 - 1 ) 2(\sqrt{2}-1)

Lorentz_factor.html

  1. γ = 1 1 - v 2 / c 2 = 1 1 - β 2 = d t d τ \gamma=\frac{1}{\sqrt{1-v^{2}/c^{2}}}=\frac{1}{\sqrt{1-\beta^{2}}}=\frac{dt}{d\tau}
  2. α = 1 γ = 1 - v 2 / c 2 , \alpha=\frac{1}{\gamma}=\sqrt{1-v^{2}/c^{2}}\ ,
  3. t = γ ( t - v x c 2 ) t^{\prime}=\gamma\left(t-\frac{vx}{c^{2}}\right)
  4. x = γ ( x - v t ) x^{\prime}=\gamma\left(x-vt\right)
  5. Δ t = γ Δ t . \Delta t^{\prime}=\gamma\Delta t.\,
  6. Δ x = Δ x / γ . \Delta x^{\prime}=\Delta x/\gamma.\,\!
  7. γ \gamma
  8. m = γ m 0 . m=\gamma m_{0}.\,
  9. p = m v = γ m 0 v . \vec{p}=m\vec{v}=\gamma m_{0}\vec{v}.\,
  10. E k = E - E 0 = ( γ - 1 ) m 0 c 2 E_{k}=E-E_{0}=(\gamma-1)m_{0}c^{2}
  11. β = v / c \beta=v/c\,\!
  12. γ \gamma\,\!
  13. 1 / γ 1/\gamma\,\!
  14. γ = 1 + ( p m 0 c ) 2 \gamma=\sqrt{1+\left(\frac{p}{m_{0}c}\right)^{2}}
  15. tanh φ = β \tanh\varphi=\beta\,\!
  16. γ = cosh φ = 1 1 - tanh 2 φ = 1 1 - β 2 \gamma=\cosh\varphi=\frac{1}{\sqrt{1-\tanh^{2}\varphi}}=\frac{1}{\sqrt{1-\beta% ^{2}}}\,\!
  17. γ \displaystyle\gamma
  18. E = γ m c 2 E=\gamma mc^{2}\,
  19. p = m v \vec{p}=m\vec{v}
  20. E = m c 2 + 1 2 m v 2 E=mc^{2}+\tfrac{1}{2}mv^{2}
  21. β = 1 - 1 γ 2 \beta=\sqrt{1-\frac{1}{\gamma^{2}}}
  22. β = 1 - 1 2 γ - 2 - 1 8 γ - 4 - 1 16 γ - 6 - 5 128 γ - 8 + \beta=1-\tfrac{1}{2}\gamma^{-2}-\tfrac{1}{8}\gamma^{-4}-\tfrac{1}{16}\gamma^{-% 6}-\tfrac{5}{128}\gamma^{-8}+\cdots
  23. γ \gamma

Lorentz–Lorenz_equation.html

  1. n 2 - 1 n 2 + 2 = 4 π 3 N α , \frac{n^{2}-1}{n^{2}+2}=\frac{4\pi}{3}N\alpha,
  2. n n
  3. N N
  4. α \alpha
  5. n n
  6. n 1 + 3 A p R T n\approx\sqrt{1+\frac{3Ap}{RT}}
  7. A A
  8. p p
  9. R R
  10. T T

Louis_Bachelier.html

  1. π {\pi}

Louis_Poinsot.html

  1. ω \mathbf{\omega}
  2. ω \mathbf{\omega}

Löb's_theorem.html

  1. if P A ( B e w ( # P ) P ) , then P A P \mathrm{if}\ PA\vdash(Bew(\#P)\rightarrow P)\mathrm{,then}\ PA\vdash P
  2. ϕ \phi
  3. ( P P ) P , \Box(\Box P\rightarrow P)\rightarrow\Box P,
  4. P P
  5. P P . \Box P\rightarrow P.
  6. A A \Box A\rightarrow\Box\Box A
  7. X X
  8. X X
  9. K K
  10. K K
  11. A A
  12. A \Box A
  13. A A
  14. B B
  15. ¬ A \neg A
  16. A B A\rightarrow B
  17. A B A\wedge B
  18. A B A\vee B
  19. A B A\leftrightarrow B
  20. A \vdash A
  21. A A
  22. F ( X ) F(X)
  23. X X
  24. F ( X ) F(X)
  25. Ψ \Psi
  26. Ψ F ( Ψ ) \vdash\Psi\leftrightarrow F(\Box\Psi)
  27. \Box
  28. \Box
  29. A \vdash A
  30. A \vdash\Box A
  31. A A \vdash\Box A\rightarrow\Box\Box A
  32. ( A B ) ( A B ) \vdash\Box(A\rightarrow B)\rightarrow(\Box A\rightarrow\Box B)
  33. P P
  34. P P \vdash\Box P\rightarrow P
  35. P P
  36. X P X\rightarrow P
  37. Ψ \Psi
  38. Ψ ( Ψ P ) \vdash\Psi\leftrightarrow(\Box\Psi\rightarrow P)
  39. Ψ ( Ψ P ) \vdash\Psi\rightarrow(\Box\Psi\rightarrow P)
  40. ( Ψ ( Ψ P ) ) \vdash\Box(\Psi\rightarrow(\Box\Psi\rightarrow P))
  41. Ψ ( Ψ P ) \vdash\Box\Psi\rightarrow\Box(\Box\Psi\rightarrow P)
  42. A = Ψ A=\Box\Psi
  43. B = P B=P
  44. ( Ψ P ) ( Ψ P ) \vdash\Box(\Box\Psi\rightarrow P)\rightarrow(\Box\Box\Psi\rightarrow\Box P)
  45. Ψ ( Ψ P ) \vdash\Box\Psi\rightarrow(\Box\Box\Psi\rightarrow\Box P)
  46. Ψ Ψ \vdash\Box\Psi\rightarrow\Box\Box\Psi
  47. Ψ P \vdash\Box\Psi\rightarrow\Box P
  48. Ψ P \vdash\Box\Psi\rightarrow P
  49. ( Ψ P ) Ψ \vdash(\Box\Psi\rightarrow P)\rightarrow\Psi
  50. Ψ \vdash\Psi
  51. Ψ \vdash\Box\Psi
  52. P \vdash P
  53. p A ( p ) p\leftrightarrow A(p)
  54. ( A A , ) (\Box A\rightarrow\Box\Box A,)

Lubell–Yamamoto–Meshalkin_inequality.html

  1. k = 0 n a k < m t p l > ( n k ) 1. \sum_{k=0}^{n}\frac{a_{k}}{<}mtpl>{{n\choose k}}\leq 1.
  2. S A | S | ! ( n - | S | ) ! = k = 0 n a k k ! ( n - k ) ! . \sum_{S\in A}|S|!(n-|S|)!=\sum_{k=0}^{n}a_{k}k!(n-k)!.
  3. k = 0 n a k k ! ( n - k ) ! n ! . \sum_{k=0}^{n}a_{k}k!(n-k)!\leq n!.

Lucas_pseudoprime.html

  1. D = P 2 - 4 Q D=P^{2}-4Q
  2. ( D n ) \left(\tfrac{D}{n}\right)
  3. δ ( n ) = n - ( D n ) . \delta(n)=n-\left(\tfrac{D}{n}\right).
  4. ( 1 ) U δ ( n ) 0 ( mod n ) . \,\text{ }(1)\,\text{ }U_{\delta(n)}\equiv 0\;\;(\mathop{{\rm mod}}n).
  5. ( D n ) \left(\tfrac{D}{n}\right)
  6. ( D n ) = - 1 , \left(\tfrac{D}{n}\right)=-1,
  7. ( 2 ) U n + 1 0 ( mod n ) . \,\text{ }(2)\,\text{ }U_{n+1}\equiv 0\;\;(\mathop{{\rm mod}}n).
  8. ( 13 19 ) \left(\tfrac{13}{19}\right)
  9. U 20 = 6616217487 0 ( mod 19 ) . U_{20}=6616217487\equiv 0\;\;(\mathop{{\rm mod}}19).
  10. ( 13 119 ) \left(\tfrac{13}{119}\right)
  11. U 120 0 ( mod 119 ) . U_{120}\equiv 0\;\;(\mathop{{\rm mod}}119).
  12. δ ( n ) \delta(n)
  13. d 2 s d\cdot 2^{s}
  14. d d
  15. U d 0 ( mod n ) U_{d}\equiv 0\;\;(\mathop{{\rm mod}}n)
  16. V d 2 r 0 ( mod n ) V_{d\cdot 2^{r}}\equiv 0\;\;(\mathop{{\rm mod}}n)
  17. U n U_{n}
  18. V n V_{n}
  19. U d 0 ( mod n ) and V d ± 2 ( mod n ) U_{d}\equiv 0\;\;(\mathop{{\rm mod}}n)\,\text{ and }V_{d}\equiv\pm 2\;\;(% \mathop{{\rm mod}}n)
  20. V d 2 r 0 ( mod n ) V_{d\cdot 2^{r}}\equiv 0\;\;(\mathop{{\rm mod}}n)
  21. r < s - 1 r<s-1
  22. ( P , Q ) (P,Q)
  23. ( D n ) = - 1 \left(\tfrac{D}{n}\right)=-1
  24. ( D n ) \left(\tfrac{D}{n}\right)
  25. ( D n ) = 0 \left(\tfrac{D}{n}\right)=0
  26. Q = ( 1 - D ) / 4 Q=(1-D)/4
  27. U n + 1 U_{n+1}
  28. V n + 1 V_{n+1}
  29. k k
  30. 2 k 2k
  31. U 2 k = U k V k U_{2k}=U_{k}\cdot V_{k}
  32. V 2 k = V k 2 - 2 Q k V_{2k}=V_{k}^{2}-2Q^{k}
  33. U 2 k + 1 = ( P U 2 k + V 2 k ) / 2 U_{2k+1}=(P\cdot U_{2k}+V_{2k})/2
  34. V 2 k + 1 = ( D U 2 k + P V 2 k ) / 2 V_{2k+1}=(D\cdot U_{2k}+P\cdot V_{2k})/2
  35. D = P 2 - 4 Q D=P^{2}-4Q
  36. ( D n ) = - 1 \left(\tfrac{D}{n}\right)=-1
  37. ( D n ) = - 1 \left(\tfrac{D}{n}\right)=-1
  38. ( 3 ) V n + 1 2 Q ( mod n ) . \,\text{ }(3)\,\text{ }V_{n+1}\equiv 2Q\;\;(\mathop{{\rm mod}}n).
  39. ( D n ) \left(\tfrac{D}{n}\right)
  40. Q n + 1 Q^{n+1}
  41. U n + 1 U_{n+1}
  42. Q Q
  43. Q ( n + 1 ) / 2 Q^{(n+1)/2}
  44. Q ( n - 1 ) / 2 ( Q n ) ( mod n ) Q^{(n-1)/2}\equiv\left(\tfrac{Q}{n}\right)\;\;(\mathop{{\rm mod}}n)
  45. ( Q n ) \left(\tfrac{Q}{n}\right)
  46. ( 4 ) Q ( n + 1 ) / 2 Q Q ( n - 1 ) / 2 Q ( Q n ) ( mod n ) \,\text{ }(4)\,\text{ }Q^{(n+1)/2}\equiv Q\cdot Q^{(n-1)/2}\equiv Q\cdot\left(% \tfrac{Q}{n}\right)\;\;(\mathop{{\rm mod}}n)
  47. ( 5 ) V n P ( mod n ) . \,\text{ }(5)\,\text{ }V_{n}\equiv P\;\;(\mathop{{\rm mod}}n).
  48. U n ( 2 n ) ( mod n ) \,\text{ }U_{n}\equiv\left(\tfrac{2}{n}\right)\;\;(\mathop{{\rm mod}}n)

Lucas_sequence.html

  1. U 0 ( P , Q ) = 0 , U_{0}(P,Q)=0,\,
  2. U 1 ( P , Q ) = 1 , U_{1}(P,Q)=1,\,
  3. U n ( P , Q ) = P U n - 1 ( P , Q ) - Q U n - 2 ( P , Q ) for n > 1 , U_{n}(P,Q)=P\cdot U_{n-1}(P,Q)-Q\cdot U_{n-2}(P,Q)\mbox{ for }~{}n>1,\,
  4. V 0 ( P , Q ) = 2 , V_{0}(P,Q)=2,\,
  5. V 1 ( P , Q ) = P , V_{1}(P,Q)=P,\,
  6. V n ( P , Q ) = P V n - 1 ( P , Q ) - Q V n - 2 ( P , Q ) for n > 1 , V_{n}(P,Q)=P\cdot V_{n-1}(P,Q)-Q\cdot V_{n-2}(P,Q)\mbox{ for }~{}n>1,\,
  7. n > 0 n>0
  8. U n ( P , Q ) = P U n - 1 ( P , Q ) + V n - 1 ( P , Q ) 2 , U_{n}(P,Q)=\frac{P\cdot U_{n-1}(P,Q)+V_{n-1}(P,Q)}{2},\,
  9. V n ( P , Q ) = ( P 2 - 4 Q ) U n - 1 ( P , Q ) + P V n - 1 ( P , Q ) 2 . V_{n}(P,Q)=\frac{(P^{2}-4Q)\cdot U_{n-1}(P,Q)+P\cdot V_{n-1}(P,Q)}{2}.\,
  10. n n\,
  11. U n ( P , Q ) U_{n}(P,Q)\,
  12. V n ( P , Q ) V_{n}(P,Q)\,
  13. 0 0\,
  14. 0 0\,
  15. 2 2\,
  16. 1 1\,
  17. 1 1\,
  18. P P\,
  19. 2 2\,
  20. P P\,
  21. P 2 - 2 Q {P}^{2}-2Q\,
  22. 3 3\,
  23. P 2 - Q {P}^{2}-Q\,
  24. P 3 - 3 P Q {P}^{3}-3PQ\,
  25. 4 4\,
  26. P 3 - 2 P Q {P}^{3}-2PQ\,
  27. P 4 - 4 P 2 Q + 2 Q 2 {P}^{4}-4{P}^{2}Q+2{Q}^{2}\,
  28. 5 5\,
  29. P 4 - 3 P 2 Q + Q 2 {P}^{4}-3{P}^{2}Q+{Q}^{2}\,
  30. P 5 - 5 P 3 Q + 5 P Q 2 {P}^{5}-5{P}^{3}Q+5P{Q}^{2}\,
  31. 6 6\,
  32. P 5 - 4 P 3 Q + 3 P Q 2 {P}^{5}-4{P}^{3}Q+3P{Q}^{2}\,
  33. P 6 - 6 P 4 Q + 9 P 2 Q 2 - 2 Q 3 {P}^{6}-6{P}^{4}Q+9{P}^{2}{Q}^{2}-2{Q}^{3}\,
  34. U n ( P , Q ) U_{n}(P,Q)
  35. V n ( P , Q ) V_{n}(P,Q)
  36. x 2 - P x + Q = 0 x^{2}-Px+Q=0\,
  37. D = P 2 - 4 Q D=P^{2}-4Q
  38. a = P + D 2 and b = P - D 2 . a=\frac{P+\sqrt{D}}{2}\quad\,\text{and}\quad b=\frac{P-\sqrt{D}}{2}.\,
  39. a + b = P , a+b=P\,,
  40. a b = 1 4 ( P 2 - D ) = Q , ab=\frac{1}{4}(P^{2}-D)=Q\,,
  41. a - b = D . a-b=\sqrt{D}\,.
  42. a n a^{n}
  43. b n b^{n}
  44. D 0 D\neq 0
  45. a n = V n + U n D 2 a^{n}=\frac{V_{n}+U_{n}\sqrt{D}}{2}
  46. b n = V n - U n D 2 b^{n}=\frac{V_{n}-U_{n}\sqrt{D}}{2}
  47. U n = a n - b n a - b = a n - b n D U_{n}=\frac{a^{n}-b^{n}}{a-b}=\frac{a^{n}-b^{n}}{\sqrt{D}}
  48. V n = a n + b n V_{n}=a^{n}+b^{n}\,
  49. D = 0 D=0
  50. P = 2 S and Q = S 2 P=2S\,\text{ and }Q=S^{2}
  51. a = b = S a=b=S
  52. U n ( P , Q ) = U n ( 2 S , S 2 ) = n S n - 1 U_{n}(P,Q)=U_{n}(2S,S^{2})=nS^{n-1}\,
  53. V n ( P , Q ) = V n ( 2 S , S 2 ) = 2 S n V_{n}(P,Q)=V_{n}(2S,S^{2})=2S^{n}\,
  54. U n ( P , Q ) U_{n}(P,Q)
  55. V n ( P , Q ) V_{n}(P,Q)
  56. D = P 2 - 4 Q D=P^{2}-4Q
  57. P 2 P_{2}
  58. Q 2 Q_{2}
  59. P 2 = P + 2 P_{2}=P+2
  60. Q 2 = P + Q + 1 Q_{2}=P+Q+1
  61. P 2 2 - 4 Q 2 = ( P + 2 ) 2 - 4 ( P + Q + 1 ) = P 2 - 4 Q = D P_{2}^{2}-4Q_{2}=(P+2)^{2}-4(P+Q+1)=P^{2}-4Q=D
  62. F n = U n ( 1 , - 1 ) F_{n}=U_{n}(1,-1)
  63. L n = V n ( 1 , - 1 ) L_{n}=V_{n}(1,-1)
  64. ( P 2 - 4 Q ) U n = V n + 1 - Q V n - 1 = 2 V n + 1 - P V n (P^{2}-4Q)U_{n}={V_{n+1}-QV_{n-1}}=2V_{n+1}-PV_{n}\,
  65. 5 F n = L n + 1 + L n - 1 = 2 L n + 1 - L n 5F_{n}={L_{n+1}+L_{n-1}}=2L_{n+1}-L_{n}\,
  66. V n = U n + 1 - Q U n - 1 = 2 U n + 1 - P U n V_{n}=U_{n+1}-QU_{n-1}=2U_{n+1}-PU_{n}\,
  67. L n = F n + 1 + F n - 1 = 2 F n + 1 - F n L_{n}=F_{n+1}+F_{n-1}=2F_{n+1}-F_{n}\,
  68. U 2 n = U n V n U_{2n}=U_{n}V_{n}\,
  69. F 2 n = F n L n F_{2n}=F_{n}L_{n}\,
  70. V 2 n = V n 2 - 2 Q n V_{2n}=V_{n}^{2}-2Q^{n}\,
  71. L 2 n = L n 2 - 2 ( - 1 ) n L_{2n}=L_{n}^{2}-2(-1)^{n}\,
  72. U n + m = U n U m + 1 - Q U m U n - 1 = U n V m + U m V n 2 U_{n+m}=U_{n}U_{m+1}-QU_{m}U_{n-1}=\frac{U_{n}V_{m}+U_{m}V_{n}}{2}\,
  73. F n + m = F n F m + 1 + F m F n - 1 = F n L m + F m L n 2 F_{n+m}=F_{n}F_{m+1}+F_{m}F_{n-1}=\frac{F_{n}L_{m}+F_{m}L_{n}}{2}\,
  74. V n + m = V n V m - Q m V n - m V_{n+m}=V_{n}V_{m}-Q^{m}V_{n-m}\,
  75. L n + m = L n L m - ( - 1 ) m L n - m L_{n+m}=L_{n}L_{m}-(-1)^{m}L_{n-m}\,
  76. U k m ( P , Q ) U_{km}(P,Q)
  77. U m ( P , Q ) U_{m}(P,Q)
  78. ( U m ( P , Q ) ) m 1 (U_{m}(P,Q))_{m\geq 1}
  79. U n ( P , Q ) U_{n}(P,Q)
  80. U n ( P , Q ) U_{n}(P,Q)
  81. P P\,
  82. Q Q\,
  83. U n ( P , Q ) U_{n}(P,Q)\,
  84. V n ( P , Q ) V_{n}(P,Q)\,

Luce's_choice_axiom.html

  1. P ( i ) = w i j w j P(i)=\frac{w_{i}}{\sum_{j}{w_{j}}}

Lusternik–Schnirelmann_category.html

  1. X X
  2. k k
  3. { U i } 1 i k \{U_{i}\}_{1\leq i\leq k}
  4. X X
  5. U i X U_{i}\hookrightarrow X
  6. X X

Luttinger_liquid.html

  1. H = k ϵ k c k c k H=\sum_{k}\epsilon_{k}c_{k}^{\dagger}c_{k}
  2. ϵ k ± v F ( k - k F ) \epsilon_{k}\approx\pm v_{F}(k-k_{F})
  3. Λ \Lambda
  4. H = k = k F - Λ k F + Λ v F k ( c k R c k R - c k L c k L ) H=\sum_{k=k_{F}-\Lambda}^{k_{F}+\Lambda}v_{F}k(c_{k}^{R\dagger}c_{k}^{R}-c_{k}% ^{L\dagger}c_{k}^{L})
  5. 2 k F 2k\text{F}

Luyten_726-8.html

  1. R v = + 29 R_{v}=+29

Lyapunov_equation.html

  1. A X A H - X + Q = 0 AXA^{H}-X+Q=0
  2. Q Q
  3. A H A^{H}
  4. A A
  5. A X + X A H + Q = 0 AX+XA^{H}+Q=0
  6. A , P , Q n × n A,P,Q\in\mathbb{R}^{n\times n}
  7. P P
  8. Q Q
  9. P > 0 P>0
  10. P P
  11. Q > 0 Q>0
  12. P > 0 P>0
  13. A T P + P A + Q = 0 A^{T}P+PA+Q=0
  14. x ˙ = A x \dot{x}=Ax
  15. V ( z ) = z T P z V(z)=z^{T}Pz
  16. Q > 0 Q>0
  17. P > 0 P>0
  18. A T P A - P + Q = 0 A^{T}PA-P+Q=0
  19. x ( t + 1 ) = A x ( t ) x(t+1)=Ax(t)
  20. z T P z z^{T}Pz
  21. vec ( A ) \operatorname{vec}(A)
  22. A A
  23. A B A\otimes B
  24. A A
  25. B B
  26. A A
  27. vec ( A B C ) = ( C T A ) vec ( B ) \operatorname{vec}(ABC)=(C^{T}\otimes A)\operatorname{vec}(B)
  28. ( I - A A ) vec ( X ) = vec ( Q ) (I-A\otimes A)\operatorname{vec}(X)=\operatorname{vec}(Q)
  29. I I
  30. vec ( X ) \operatorname{vec}(X)
  31. X X
  32. vec ( X ) \operatorname{vec}(X)
  33. A A
  34. X X
  35. X = k = 0 A k Q ( A H ) k X=\sum_{k=0}^{\infty}A^{k}Q(A^{H})^{k}
  36. ( I n A + A ¯ I n ) vec X = - vec Q , (I_{n}\otimes A+\bar{A}\otimes I_{n})\operatorname{vec}X=-\operatorname{vec}Q,
  37. A ¯ \bar{A}
  38. A A
  39. A A
  40. X X
  41. X = 0 e A τ Q e A H τ d τ X=\int\limits_{0}^{\infty}e^{A\tau}Qe^{A^{H}\tau}d\tau

Mach_principle.html

  1. Ω = def 4 π ρ G T 2 \Omega\ \stackrel{\mathrm{def}}{=}\ 4\pi\rho GT^{2}
  2. ρ \rho
  3. T T

Magdalen_papyrus.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}

Maghemite.html

  1. ( Fe 8 III ) A [ Fe 40 / 3 III 8 / 3 ] B O 32 (\mathrm{Fe}^{\mathrm{III}}_{8})_{A}[\mathrm{Fe}^{\mathrm{III}}_{40/3}\square_% {8/3}]_{B}\mathrm{O}_{32}
  2. \square
  3. A A
  4. B B

Magnetic_circuit.html

  1. \scriptstyle\mathcal{F}
  2. = 𝐇 d 𝐥 \mathcal{F}=\oint\mathbf{H}\cdot\operatorname{d}\mathbf{l}
  3. 1 Gb = 10 4 π At 0.795775 At \begin{aligned}\displaystyle 1\;\,\text{Gb}&\displaystyle=\frac{10}{4\pi}\;\,% \text{At}\\ &\displaystyle\approx 0.795775\;\,\text{At}\end{aligned}
  4. \mathcal{F}
  5. = N I \mathcal{F}=NI
  6. Φ m = S 𝐁 d 𝐒 \Phi_{m}=\int\!\!\!\!\int_{S}\mathbf{B}\cdot\operatorname{d}\mathbf{S}
  7. \scriptstyle\mathcal{E}
  8. = I R \mathcal{E}=IR
  9. = ϕ m \mathcal{F}=\phi\mathcal{R}_{m}
  10. \scriptstyle\mathcal{F}
  11. ϕ \scriptstyle\phi
  12. m \scriptstyle\mathcal{R}_{m}
  13. = Φ \mathcal{R}=\frac{\mathcal{F}}{\Phi}
  14. \scriptstyle\mathcal{R}
  15. 𝒫 = 1 \mathcal{P}=\frac{1}{\mathcal{R}}
  16. = l μ A \mathcal{R}=\frac{l}{\mu A}
  17. μ = μ r μ 0 \scriptstyle\mu\;=\;\mu_{r}\mu_{0}
  18. μ r \scriptstyle\mu_{r}
  19. μ 0 \scriptstyle\mu_{0}
  20. = 𝐇 d 𝐥 \mathcal{F}=\int\mathbf{H}\cdot\operatorname{d}\mathbf{l}
  21. = 𝐄 d 𝐥 \mathcal{E}=\int\mathbf{E}\cdot\operatorname{d}\mathbf{l}
  22. ϕ \phi
  23. = ϕ m \mathcal{F}=\phi\mathcal{R}_{m}
  24. = I R \mathcal{E}=IR
  25. m \mathcal{R}_{m}
  26. 𝒫 = 1 m \mathcal{P}=\frac{1}{\mathcal{R}_{m}}
  27. 𝐁 = μ 𝐇 \mathbf{B}=\mu\mathbf{H}
  28. 𝐉 = σ 𝐄 \mathbf{J}=\sigma\mathbf{E}
  29. T \scriptstyle\mathcal{R}_{T}
  30. 1 , 2 , \scriptstyle\mathcal{R}_{1},\ \mathcal{R}_{2},\ \dots
  31. T = 1 + 2 + \mathcal{R}_{T}=\mathcal{R}_{1}+\mathcal{R}_{2}+\cdots
  32. Φ 1 , Φ 2 , \scriptstyle\Phi_{1},\ \Phi_{2},\ \dots
  33. Φ 1 + Φ 2 + = 0 \Phi_{1}+\Phi_{2}+\cdots=0
  34. m \scriptstyle\mathcal{R}_{m}
  35. F = N I = H d l F=N\,I=\oint\vec{H}\cdot\operatorname{d}\vec{l}

Magnetic_helicity.html

  1. H = 𝐀 𝐁 d 3 𝐫 H=\int{\mathbf{A}}\cdot{\mathbf{B}}\,d^{3}{\mathbf{r}}
  2. 𝐁 {\mathbf{B}}
  3. 𝐁 = × 𝐀 {\mathbf{B}}=\nabla\times{\mathbf{A}}
  4. 𝐀 \mathbf{A}
  5. 𝐁 {\mathbf{B}}
  6. 𝐀 \mathbf{A}

Magnetic_potential.html

  1. 𝐁 = × 𝐀 , 𝐄 = - ϕ - 𝐀 t , \mathbf{B}=\nabla\times\mathbf{A}\,,\quad\mathbf{E}=-\nabla\phi-\frac{\partial% \mathbf{A}}{\partial t}\,,
  2. 𝐁 = ( × 𝐀 ) = 0 \nabla\cdot\mathbf{B}=\nabla\cdot(\nabla\times\mathbf{A})=0
  3. × 𝐄 = × ( - ϕ - 𝐀 t ) = - t ( × 𝐀 ) = - 𝐁 t . \nabla\times\mathbf{E}=\nabla\times\left(-\nabla\phi-\frac{\partial\mathbf{A}}% {\partial t}\right)=-\frac{\partial}{\partial t}(\nabla\times\mathbf{A})=-% \frac{\partial\mathbf{B}}{\partial t}.
  4. 𝐀 + 1 c 2 ϕ t = 0. \nabla\cdot\,\textbf{A}+\frac{1}{c^{2}}\frac{\partial\phi}{\partial t}=0.
  5. 2 ϕ - 1 c 2 2 ϕ t 2 = - ρ / ϵ 0 \nabla^{2}\phi-\frac{1}{c^{2}}\frac{\partial^{2}\phi}{\partial t^{2}}=-\rho/% \epsilon_{0}
  6. 2 𝐀 - 1 c 2 2 𝐀 t 2 = - μ 0 𝐉 \nabla^{2}\,\textbf{A}-\frac{1}{c^{2}}\frac{\partial^{2}\,\textbf{A}}{\partial t% ^{2}}=-\mu_{0}\,\textbf{J}
  7. 𝐀 ( 𝐫 , t ) = μ 0 4 π Ω 𝐉 ( 𝐫 , t ) | 𝐫 - 𝐫 | d 3 𝐫 . \mathbf{A}(\mathbf{r},t)=\frac{\mu_{0}}{4\pi}\int_{\Omega}\frac{\mathbf{J}(% \mathbf{r}^{\prime},t^{\prime})}{|\mathbf{r}-\mathbf{r}^{\prime}|}\,\mathrm{d}% ^{3}\mathbf{r}^{\prime}\,.
  8. ϕ ( 𝐫 , t ) = 1 4 π ϵ 0 Ω ρ ( 𝐫 , t ) | 𝐫 - 𝐫 | d 3 𝐫 \phi(\mathbf{r},t)=\frac{1}{4\pi\epsilon_{0}}\int_{\Omega}\frac{\rho(\mathbf{r% }^{\prime},t^{\prime})}{|\mathbf{r}-\mathbf{r}^{\prime}|}\,\mathrm{d}^{3}% \mathbf{r}^{\prime}
  9. t = t - | 𝐫 - 𝐫 | c t^{\prime}=t-\frac{|\mathbf{r}-\mathbf{r}^{\prime}|}{c}
  10. 𝐀 + 1 c 2 ϕ t = 0 \nabla\cdot\mathbf{A}+\frac{1}{c^{2}}\frac{\partial\phi}{\partial t}=0
  11. A x ( 𝐫 , t ) = μ 0 4 π Ω J x ( 𝐫 , t ) | 𝐫 - 𝐫 | d 3 𝐫 A_{x}(\mathbf{r},t)=\frac{\mu_{0}}{4\pi}\int_{\Omega}\frac{J_{x}(\mathbf{r}^{% \prime},t^{\prime})}{|\mathbf{r}-\mathbf{r}^{\prime}|}\,{\rm d}^{3}\mathbf{r}^% {\prime}
  12. A y ( 𝐫 , t ) = μ 0 4 π Ω J y ( 𝐫 , t ) | 𝐫 - 𝐫 | d 3 𝐫 A_{y}(\mathbf{r},t)=\frac{\mu_{0}}{4\pi}\int_{\Omega}\frac{J_{y}(\mathbf{r}^{% \prime},t^{\prime})}{|\mathbf{r}-\mathbf{r}^{\prime}|}\,{\rm d}^{3}\mathbf{r}^% {\prime}
  13. A z ( 𝐫 , t ) = μ 0 4 π Ω J z ( 𝐫 , t ) | 𝐫 - 𝐫 | d 3 𝐫 A_{z}(\mathbf{r},t)=\frac{\mu_{0}}{4\pi}\int_{\Omega}\frac{J_{z}(\mathbf{r}^{% \prime},t^{\prime})}{|\mathbf{r}-\mathbf{r}^{\prime}|}\,{\rm d}^{3}\mathbf{r}^% {\prime}
  14. × 𝐁 = μ 0 𝐉 \nabla\times\mathbf{B}=\mu_{0}\mathbf{J}
  15. E t 0 × 𝐀 = 𝐁 , \frac{\partial E}{\partial t}\rightarrow 0\,\quad\nabla\times\mathbf{A}=% \mathbf{B}\,,
  16. 1 c 2 ϕ t \frac{1}{c^{2}}\frac{\partial\phi}{\partial t}
  17. μ A μ = 0 \partial^{\mu}A_{\mu}=0\,
  18. A μ = 4 π c J μ \Box A_{\mu}=\frac{4\pi}{c}J_{\mu}
  19. × 𝐇 = 0 , \nabla\times\mathbf{H}=0,
  20. 𝐇 = - ψ . \mathbf{H}=-\nabla\psi.
  21. 𝐁 = μ 0 ( 𝐇 + 𝐌 ) = 0 , \nabla\cdot\mathbf{B}=\mu_{0}\nabla\cdot(\mathbf{H+M})=0,
  22. 2 ψ = - 𝐇 = 𝐌 . \nabla^{2}\psi=-\nabla\cdot\mathbf{H}=\nabla\cdot\mathbf{M}.
  23. ρ m = - 𝐌 \rho_{m}=-\nabla\cdot\mathbf{M}

Magnetic_reconnection.html

  1. × 𝐁 = μ 𝐉 + μ ϵ 𝐄 t . \nabla\times\mathbf{B}=\mu\mathbf{J}+\mu\epsilon\frac{\partial\mathbf{E}}{% \partial t}.
  2. E y = V i n B i n E_{y}=V_{in}B_{in}
  3. E y E_{y}
  4. V i n V_{in}
  5. B i n B_{in}
  6. 𝐉 = × 𝐁 μ 0 \mathbf{J}=\frac{\nabla\times\mathbf{B}}{\mu_{0}}
  7. J y B i n μ 0 δ , J_{y}\sim\frac{B_{in}}{\mu_{0}\delta},
  8. δ \delta
  9. 2 δ \sim 2\delta
  10. 𝐄 = η 𝐉 \mathbf{E}=\eta\mathbf{J}
  11. V i n η μ 0 δ , V_{in}\sim\frac{\eta}{\mu_{0}\delta},
  12. η \eta
  13. V i n L V o u t δ , V_{in}L\sim V_{out}\delta,
  14. L L
  15. V o u t V_{out}
  16. B i n 2 2 μ 0 ρ V o u t 2 2 \frac{B_{in}^{2}}{2\mu_{0}}\sim\frac{\rho V_{out}^{2}}{2}
  17. ρ \rho
  18. V o u t V A B i n μ 0 ρ V_{out}\sim V_{A}\equiv\frac{B_{in}}{\sqrt{\mu_{0}\rho}}
  19. V A V_{A}
  20. V i n V A 1 S 1 / 2 \frac{V_{in}}{V_{A}}\sim\frac{1}{S^{1/2}}
  21. S S
  22. S μ 0 L V A η . S\equiv\frac{\mu_{0}LV_{A}}{\eta}.
  23. V i n V A π 8 ln S . \frac{V_{in}}{V_{A}}\approx\frac{\pi}{8\ln S}.
  24. c / ω p i c/\omega_{pi}
  25. ω p i n i Z 2 e 2 ϵ 0 m i \omega_{pi}\equiv\sqrt{\frac{n_{i}Z^{2}e^{2}}{\epsilon_{0}m_{i}}}
  26. m m
  27. e e
  28. d 𝐯 d t = e m 𝐄 - ν 𝐯 , {d{\mathbf{v}}\over dt}={e\over m}{\mathbf{E}}-\nu{\mathbf{v}},
  29. ν \nu
  30. d 𝐯 / d t = 0 d{\mathbf{v}}/dt=0
  31. 𝐉 = e n 𝐯 {\mathbf{J}}=en{\mathbf{v}}
  32. n n
  33. η = ν c 2 ω p i 2 . \eta=\nu{c^{2}\over\omega_{pi}^{2}}.
  34. η a n o m \eta_{anom}
  35. η a n o m / η \eta_{anom}/\eta
  36. V A 2 ( m c / e B ) V_{A}^{2}(mc/eB)
  37. L L
  38. V = v t u r b m i n [ ( L l ) 1 / 2 , ( l L ) 1 / 2 ] , V=v_{turb}\;min\Big[\Big({L\over l}\Big)^{1/2},\Big({l\over L}\Big)^{1/2}\Big],
  39. v t u r b = V A ( v l / V A ) 2 v_{turb}=V_{A}(v_{l}/V_{A})^{2}
  40. l l
  41. v l v_{l}
  42. V A V_{A}

Magnon.html

  1. M ( T ) = M 0 ( 1 - ( T / T C ) 3 / 2 ) M(T)=M_{0}(1-(T/T_{C})^{3/2})
  2. ω < k B T \hbar\omega<k_{B}T

Mahalanobis_distance.html

  1. x = ( x 1 , x 2 , x 3 , , x N ) T x=(x_{1},x_{2},x_{3},\dots,x_{N})^{T}
  2. μ = ( μ 1 , μ 2 , μ 3 , , μ N ) T \mu=(\mu_{1},\mu_{2},\mu_{3},\dots,\mu_{N})^{T}
  3. D M ( x ) = ( x - μ ) T S - 1 ( x - μ ) . D_{M}(x)=\sqrt{(x-\mu)^{T}S^{-1}(x-\mu)}.\,
  4. x \vec{x}
  5. y \vec{y}
  6. d ( x , y ) = ( x - y ) T S - 1 ( x - y ) . d(\vec{x},\vec{y})=\sqrt{(\vec{x}-\vec{y})^{T}S^{-1}(\vec{x}-\vec{y})}.\,
  7. d ( x , y ) = i = 1 N ( x i - y i ) 2 s i 2 , d(\vec{x},\vec{y})=\sqrt{\sum_{i=1}^{N}{(x_{i}-y_{i})^{2}\over s_{i}^{2}}},
  8. x - μ σ {x-\mu}\over\sigma
  9. d 2 d^{2}
  10. 1 - e - d t h 2 / 2 1-e^{-dth^{2}/2}
  11. d t h 2 = - 2 ln ( 1 - p ) dth^{2}=-2\ln(1-p)
  12. X X
  13. S = 1 S=1
  14. μ = 0 \mu=0
  15. R R
  16. μ 1 \mu_{1}
  17. S 1 S_{1}
  18. X X
  19. R = μ 1 + S 1 X . R=\mu_{1}+\sqrt{S_{1}}X.
  20. X = ( R - μ 1 ) / S 1 X=(R-\mu_{1})/\sqrt{S_{1}}
  21. D = X 2 = ( R - μ 1 ) 2 / S 1 = ( R - μ 1 ) S 1 - 1 ( R - μ 1 ) . D=\sqrt{X^{2}}=\sqrt{(R-\mu_{1})^{2}/S_{1}}=\sqrt{(R-\mu_{1})S_{1}^{-1}(R-\mu_% {1})}.
  22. h h
  23. D 2 = ( N - 1 ) ( h - 1 / N ) . D^{2}=(N-1)(h-1/N).
  24. x 1 = x 2 x_{1}=x_{2}

Mahāvīra_(mathematician).html

  1. 1 + 1 3 + 1 3 4 - 1 3 4 34 1+\tfrac{1}{3}+\tfrac{1}{3\cdot 4}-\tfrac{1}{3\cdot 4\cdot 34}
  2. 1 = 1 1 2 + 1 3 + 1 3 2 + + 1 3 n - 2 + 1 2 3 3 n - 1 1=\frac{1}{1\cdot 2}+\frac{1}{3}+\frac{1}{3^{2}}+\dots+\frac{1}{3^{n-2}}+\frac% {1}{\frac{2}{3}\cdot 3^{n-1}}
  3. 1 = 1 2 3 1 / 2 + 1 3 4 1 / 2 + + 1 ( 2 n - 1 ) 2 n 1 / 2 + 1 2 n 1 / 2 1=\frac{1}{2\cdot 3\cdot 1/2}+\frac{1}{3\cdot 4\cdot 1/2}+\dots+\frac{1}{(2n-1% )\cdot 2n\cdot 1/2}+\frac{1}{2n\cdot 1/2}
  4. 1 / q 1/q
  5. a 1 , a 2 , , a n a_{1},a_{2},\dots,a_{n}
  6. 1 q = a 1 q ( q + a 1 ) + a 2 ( q + a 1 ) ( q + a 1 + a 2 ) + + a n - 1 q + a 1 + + a n - 2 ) ( q + a 1 + + a n - 1 ) + a n a n ( q + a 1 + + a n - 1 ) \frac{1}{q}=\frac{a_{1}}{q(q+a_{1})}+\frac{a_{2}}{(q+a_{1})(q+a_{1}+a_{2})}+% \dots+\frac{a_{n-1}}{q+a_{1}+\dots+a_{n-2})(q+a_{1}+\dots+a_{n-1})}+\frac{a_{n% }}{a_{n}(q+a_{1}+\dots+a_{n-1})}
  7. p / q p/q
  8. q + i p \tfrac{q+i}{p}
  9. p q = 1 r + i r q \frac{p}{q}=\frac{1}{r}+\frac{i}{r\cdot q}
  10. 1 n = 1 p n + 1 p n n - 1 \frac{1}{n}=\frac{1}{p\cdot n}+\frac{1}{\frac{p\cdot n}{n-1}}
  11. p p
  12. p n n - 1 \frac{p\cdot n}{n-1}
  13. p p
  14. n - 1 n-1
  15. 1 a b = 1 a ( a + b ) + 1 b ( a + b ) \frac{1}{a\cdot b}=\frac{1}{a(a+b)}+\frac{1}{b(a+b)}
  16. p / q p/q
  17. a a
  18. b b
  19. p q = a a i + b p q i + b a i + b p q i i \frac{p}{q}=\frac{a}{\frac{ai+b}{p}\cdot\frac{q}{i}}+\frac{b}{\frac{ai+b}{p}% \cdot\frac{q}{i}\cdot{i}}
  20. i i
  21. p p
  22. a i + b ai+b

Mahler's_compactness_theorem.html

  1. GL n ( ) / GL n ( ) \mathrm{GL}_{n}(\mathbb{R})/\mathrm{GL}_{n}(\mathbb{Z})
  2. n \mathbb{R}^{n}
  3. n \mathbb{R}^{n}
  4. n \mathbb{R}^{n}
  5. ϵ > 0 \epsilon>0

Mahler_measure.html

  1. M ( p ) M(p)
  2. p ( z ) p(z)
  3. M ( p ) = | a | | α i | 1 | α i | = | a | i = 1 n max { 1 , | α i | } , M(p)=|a|\prod_{|\alpha_{i}|\geq 1}|\alpha_{i}|=|a|\prod_{i=1}^{n}\max\{1,|% \alpha_{i}|\},
  4. p ( z ) p(z)
  5. \mathbb{C}
  6. p ( z ) = a ( z - α 1 ) ( z - α 2 ) ( z - α n ) . p(z)=a(z-\alpha_{1})(z-\alpha_{2})\cdots(z-\alpha_{n}).
  7. M ( p ) = exp ( 1 2 π 0 2 π ln ( | p ( e i θ ) | ) d θ ) , M(p)=\exp\left(\frac{1}{2\pi}\int_{0}^{2\pi}\ln(|p(e^{i\theta})|)\,d\theta% \right),
  8. | p ( z ) | |p(z)|
  9. z z
  10. | z | = 1. |z|=1.
  11. α \alpha
  12. α \alpha
  13. \mathbb{Q}
  14. α \alpha
  15. α \alpha
  16. M ( p q ) = M ( p ) M ( q ) . M(p\,q)=M(p)\cdot M(q).
  17. M ( p ) = lim τ 0 p τ , M(p)=\lim_{\tau\rightarrow 0}\|p\|_{\tau},
  18. p τ = ( 1 2 π 0 2 π | p ( e i θ ) | τ d θ ) 1 / τ \|p\|_{\tau}=\left(\frac{1}{2\pi}\int_{0}^{2\pi}|p(e^{i\theta})|^{\tau}\,d% \theta\right)^{1/\tau}\,
  19. L τ L_{\tau}
  20. p p
  21. τ < 1 \tau<1
  22. p p
  23. M ( p ) = 1 M(p)=1
  24. p ( z ) = z , p(z)=z,
  25. p p
  26. μ > 1 \mu>1
  27. p p
  28. M ( p ) = 1 M(p)=1
  29. M ( p ) > μ M(p)>\mu
  30. M ( p ) M(p)
  31. p ( x 1 , , x n ) [ x 1 , , x n ] p(x_{1},\ldots,x_{n})\in\mathbb{C}[x_{1},\ldots,x_{n}]
  32. M ( p ) = exp ( 1 ( 2 π ) n 0 2 π 0 2 π 0 2 π log ( | p ( e i θ 1 , e i θ 2 , , e i θ n ) | ) d θ 1 d θ 2 d θ n ) . M(p)=\exp\left(\frac{1}{(2\pi)^{n}}\int_{0}^{2\pi}\int_{0}^{2\pi}\cdots\int_{0% }^{2\pi}\log\Bigl(\bigl|p(e^{i\theta_{1}},e^{i\theta_{2}},\ldots,e^{i\theta_{n% }})\bigr|\Bigr)\,d\theta_{1}\,d\theta_{2}\cdots d\theta_{n}\right).
  33. L L
  34. m ( 1 + x + y ) = 3 3 4 π L ( χ - 3 , 2 ) m(1+x+y)=\frac{3\sqrt{3}}{4\pi}L(\chi_{-3},2)
  35. L ( χ - 3 , s ) L(\chi_{-3},s)
  36. m ( 1 + x + y + z ) = 7 2 π 2 ζ ( 3 ) m(1+x+y+z)=\frac{7}{2\pi^{2}}\zeta(3)
  37. ζ \zeta
  38. m ( P ) = log M ( P ) m(P)=\log{M(P)}
  39. p p
  40. ( S 1 ) n (S^{1})^{n}
  41. M ( p ) M(p)
  42. M ( p ) M(p)
  43. \mathbb{Z}
  44. + N = { r = ( r 1 , , r N ) N : r j 0 for 1 j N } \mathbb{Z}^{N}_{+}=\{r=(r_{1},\dots,r_{N})\in\mathbb{Z}^{N}:r_{j}\geq 0\ \,% \text{for}\ 1\leq j\leq N\}
  45. Q ( z 1 , , z N ) Q(z_{1},\dots,z_{N})
  46. N N
  47. r = ( r 1 , , r N ) + N r=(r_{1},\dots,r_{N})\in\mathbb{Z}^{N}_{+}
  48. Q r ( z ) Q_{r}(z)
  49. Q r ( z ) := Q ( z r 1 , , z r N ) Q_{r}(z):=Q(z^{r_{1}},\dots,z^{r_{N}})
  50. q ( r ) q(r)
  51. q ( r ) := min { H ( s ) : s = ( s 1 , , s N ) N , s ( 0 , , 0 ) and j = 1 N s j r j = 0 } q(r):=\,\text{min}\{H(s):s=(s_{1},\dots,s_{N})\in\mathbb{Z}^{N},s\neq(0,\dots,% 0)\ \,\text{and}\ \sum^{N}_{j=1}s_{j}r_{j}=0\}
  52. H ( s ) = max { | s j | : 1 j N } H(s)=\,\text{max}\{|s_{j}|:1\leq j\leq N\}
  53. Q ( z 1 , , z N ) Q(z_{1},\dots,z_{N})
  54. r i 0 r_{i}\geq 0
  55. lim q ( r ) M ( Q r ) = M ( Q ) \lim_{q(r)\rightarrow\infty}M(Q_{r})=M(Q)
  56. Ψ ( z ) = z 1 b 1 z n b n Φ ( z 1 v 1 z n v n ) , \Psi(z)=z_{1}^{b_{1}}\dots z_{n}^{b_{n}}\Phi(z_{1}^{v_{1}}\dots z_{n}^{v_{n}}),
  57. Φ m ( z ) \Phi_{m}(z)
  58. v i v_{i}
  59. b i = max ( 0 , - v i deg Φ m ) b_{i}=\,\text{max}(0,-v_{i}\,\text{deg}\Phi_{m})
  60. Ψ ( z ) \Psi(z)
  61. z i z_{i}
  62. K n K_{n}
  63. ± z 1 c 1 z n c n \pm z_{1}^{c_{1}}\dots z_{n}^{c_{n}}
  64. F ( z 1 , , z n ) F(z_{1},\dots,z_{n})
  65. M ( F ) = 1 M(F)=1
  66. F F
  67. K n K_{n}
  68. P ( z ) P(z)
  69. L n := { m ( P ( z 1 , , z n ) ) : P [ z ] } , L_{n}:=\{m(P(z_{1},\dots,z_{n})):P\in\mathbb{Z}[z]\}\ ,
  70. L = n = 1 L n {L}_{\infty}=\bigcup^{\infty}_{n=1}L_{n}
  71. L {L}_{\infty}
  72. L 1 L 2 L_{1}\subsetneqq L_{2}
  73. L 1 L 2 L_{1}\subsetneqq L_{2}\subsetneqq\ \cdots

Malliavin_calculus.html

  1. - f ( x ) d λ ( x ) = - f ( x + ε ) d λ ( x ) . \int_{-\infty}^{\infty}f(x)\,d\lambda(x)=\int_{-\infty}^{\infty}f(x+% \varepsilon)\,d\lambda(x).
  2. - f d λ = - ( g h ) d λ = - g h d λ + - g h d λ . \int_{-\infty}^{\infty}f^{\prime}\,d\lambda=\int_{-\infty}^{\infty}(gh)^{% \prime}\,d\lambda=\int_{-\infty}^{\infty}gh^{\prime}\,d\lambda+\int_{-\infty}^% {\infty}g^{\prime}h\,d\lambda.
  3. h s h_{s}
  4. φ ( t ) = 0 t h s d s . \varphi(t)=\int_{0}^{t}h_{s}\,ds.
  5. X X
  6. E ( F ( X + ε φ ) ) = E [ F ( X ) exp ( ε 0 1 h s d X s - 1 2 ε 2 0 1 h s 2 d s ) ] . E(F(X+\varepsilon\varphi))=E\left[F(X)\exp\left(\varepsilon\int_{0}^{1}h_{s}\,% dX_{s}-\frac{1}{2}\varepsilon^{2}\int_{0}^{1}h_{s}^{2}\,ds\right)\right].
  7. E ( D F ( X ) , φ ) = E [ F ( X ) 0 1 h s d X s ] . E(\langle DF(X),\varphi\rangle)=E\Bigl[F(X)\int_{0}^{1}h_{s}\,dX_{s}\Bigr].
  8. F F
  9. φ \varphi
  10. h h
  11. F : C [ 0 , 1 ] \R F:C[0,1]\to\R
  12. E ( F ( X ) 2 ) < E(F(X)^{2})<\infty
  13. φ \varphi
  14. lim ε 0 ( 1 / ε ) ( F ( X + ε φ ) - F ( X ) ) = 0 1 F ( X , d t ) φ ( t ) a . e . X \lim_{\varepsilon\to 0}(1/\varepsilon)(F(X+\varepsilon\varphi)-F(X))=\int_{0}^% {1}F^{\prime}(X,dt)\varphi(t)\ \mathrm{a.e.}\ X
  15. F ( X ) = E ( F ( X ) ) + 0 1 H t d X t , F(X)=E(F(X))+\int_{0}^{1}H_{t}\,dX_{t},
  16. F ( X ) = E ( F ( X ) ) + 0 1 E ( D t F | t ) d X t . F(X)=E(F(X))+\int_{0}^{1}E(D_{t}F|\mathcal{F}_{t})\,dX_{t}.
  17. D t D_{t}
  18. L 2 ( [ 0 , ) × Ω ) L^{2}([0,\infty)\times\Omega)
  19. E ( D F , u ) = E ( F δ ( u ) ) , E(\langle DF,u\rangle)=E(F\delta(u)),
  20. L 2 [ 0 , ) L^{2}[0,\infty)
  21. f , g = 0 f ( s ) g ( s ) d s . \langle f,g\rangle=\int_{0}^{\infty}f(s)g(s)\,ds.
  22. δ ( u ) = 0 u t d W t , \delta(u)=\int_{0}^{\infty}u_{t}\,dW_{t},

Mapping_class_group.html

  1. 1 Aut 0 ( X ) Aut ( X ) MCG ( X ) 1. 1\rightarrow{\rm Aut}_{0}(X)\rightarrow{\rm Aut}(X)\rightarrow{\rm MCG}(X)% \rightarrow 1.
  2. MCG ( 𝐒 2 ) 𝐙 / 2 𝐙 \operatorname{MCG}(\mathbf{S}^{2})\simeq{\mathbf{Z}}/2{\mathbf{Z}}
  3. MCG ( 𝐓 n ) SL ( n , 𝐙 ) . {\rm MCG}(\mathbf{T}^{n})\simeq{\rm SL}(n,{\mathbf{Z}}).
  4. 0 𝐙 2 M C G ( 𝐓 n ) G L ( n , 𝐙 ) 0 0\to\mathbf{Z}_{2}^{\infty}\to MCG(\mathbf{T}^{n})\to GL(n,\mathbf{Z})\to 0
  5. 0 𝐙 2 ( n 2 ) 𝐙 2 M C G ( 𝐓 n ) G L ( n , 𝐙 ) 0 0\to\mathbf{Z}_{2}^{\infty}\oplus{\left({{n}\atop{2}}\right)}\mathbf{Z}_{2}\to MCG% (\mathbf{T}^{n})\to GL(n,\mathbf{Z})\to 0
  6. 0 𝐙 2 ( n 2 ) 𝐙 2 i = 0 n ( n i ) Γ i + 1 M C G ( 𝐓 n ) G L ( n , 𝐙 ) 0 0\to\mathbf{Z}_{2}^{\infty}\oplus{\left({{n}\atop{2}}\right)}\mathbf{Z}_{2}% \oplus\sum_{i=0}^{n}{\left({{n}\atop{i}}\right)}\Gamma_{i+1}\to MCG(\mathbf{T}% ^{n})\to GL(n,\mathbf{Z})\to 0
  7. MCG ( 𝐏 2 ( 𝐑 ) ) = 1. {\rm MCG}(\mathbf{P}^{2}(\mathbf{R}))=1.
  8. MCG ( K ) = 𝐙 2 𝐙 2 . {\rm MCG}(K)=\mathbf{Z}_{2}\oplus\mathbf{Z}_{2}.
  9. MCG ( N 3 ) = GL ( 2 , 𝐙 ) . {\rm MCG(N_{3})}={\rm GL}(2,{\mathbf{Z}}).
  10. 1 Tor ( Σ ) MCG ( Σ ) Sp ( H 1 ( Σ ) ) Sp ( 𝐙 ) 2 g 1 1\to\mbox{Tor}~{}(\Sigma)\to\mbox{MCG}~{}(\Sigma)\to\mbox{Sp}~{}(H^{1}(\Sigma)% )\cong\mbox{Sp}~{}_{2g}(\mathbf{Z})\to 1
  11. 1 Tor ( Σ ) MCG ( Σ ) * Sp ( H 1 ( Σ ) ) ± Sp ( 𝐙 ) 2 g ± 1 1\to\mbox{Tor}~{}(\Sigma)\to\mbox{MCG}~{}^{*}(\Sigma)\to\mbox{Sp}~{}^{\pm}(H^{% 1}(\Sigma))\cong\mbox{Sp}~{}^{\pm}_{2g}(\mathbf{Z})\to 1
  12. Σ g , 1 \Sigma_{g,1}
  13. Σ g + 1 , 1 \Sigma_{g+1,1}
  14. Σ g , 1 \Sigma_{g,1}
  15. Σ 1 , 2 \Sigma_{1,2}

Marcel_Riesz.html

  1. a 0 2 + n = 1 { a n cos ( n x ) + b n sin ( n x ) } . \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left\{a_{n}\cos(nx)+b_{n}\sin(nx)\right\}.\,
  2. n = 1 | a n | + | b n | n 2 < , \sum_{n=1}^{\infty}\frac{|a_{n}|+|b_{n}|}{n^{2}}<\infty,\,
  3. 1 / 4 {1}/{4}
  4. z n d μ ( z ) = 0 , n = 1 , 2 , 3 , \int z^{n}d\mu(z)=0,n=1,2,3\cdots,\,
  5. | x | > R | f ( x ) | p d x < ϵ p \int_{|x|>R}|f(x)|^{p}dx<\epsilon^{p}\,
  6. n | f ( x + y ) - f ( x ) | p d x < ϵ p \int_{\mathbb{R}^{n}}|f(x+y)-f(x)|^{p}dx<\epsilon^{p}\,

Marcinkiewicz_interpolation_theorem.html

  1. λ f ( t ) = ω { x X | f ( x ) | > t } . \lambda_{f}(t)=\omega\left\{x\in X\mid|f(x)|>t\right\}.
  2. L 1 L^{1}
  3. λ f ( t ) C t . \lambda_{f}(t)\leq\frac{C}{t}.
  4. L 1 L^{1}
  5. ( 0 , 1 ) (0,1)
  6. 1 / x 1/x
  7. 1 / ( 1 - x ) 1/(1-x)
  8. L 1 L^{1}
  9. f 1 , w f 1 . \|f\|_{1,w}\leq\|f\|_{1}.
  10. L p L^{p}
  11. | f | p |f|^{p}
  12. L p L^{p}
  13. f p , w = | f | p 1 , w 1 / p . \|f\|_{p,w}=\|\,|f|^{p}\|_{1,w}^{1/p}.
  14. λ f ( t ) C p t p \lambda_{f}(t)\leq\frac{C^{p}}{t^{p}}
  15. L p L^{p}
  16. L p , w L^{p,w}
  17. L q L^{q}
  18. L q , w L^{q,w}
  19. L r L^{r}
  20. L r L^{r}
  21. L r L^{r}
  22. T f p , w N p f p , \|Tf\|_{p,w}\leq N_{p}\|f\|_{p},
  23. T f q , w N q f q , \|Tf\|_{q,w}\leq N_{q}\|f\|_{q},
  24. T f r γ N p δ N q 1 - δ f r \|Tf\|_{r}\leq\gamma N_{p}^{\delta}N_{q}^{1-\delta}\|f\|_{r}
  25. δ = p ( q - r ) r ( q - p ) \delta=\frac{p(q-r)}{r(q-p)}
  26. γ = 2 ( r ( q - p ) ( r - p ) ( q - r ) ) 1 / r . \gamma=2\left(\frac{r(q-p)}{(r-p)(q-r)}\right)^{1/r}.
  27. | T ( f + g ) ( x ) | C ( | T f ( x ) | + | T g ( x ) | ) |T(f+g)(x)|\leq C(|Tf(x)|+|Tg(x)|)
  28. γ = 2 C ( r ( q - p ) ( r - p ) ( q - r ) ) 1 / r . \gamma=2C\left(\frac{r(q-p)}{(r-p)(q-r)}\right)^{1/r}.
  29. T f q , w C f p \|Tf\|_{q,w}\leq C\|f\|_{p}
  30. T f q C f p . \|Tf\|_{q}\leq C\|f\|_{p}.
  31. 1 p = 1 - θ p 0 + θ p 1 , 1 q = 1 - θ q 0 + θ q 1 . \frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}},\quad\frac{1}{q}=\frac% {1-\theta}{q_{0}}+\frac{\theta}{q_{1}}.
  32. L 2 L^{2}
  33. L 2 L^{2}
  34. L 1 L^{1}
  35. L 1 , w L^{1,w}
  36. L p L^{p}
  37. L p L^{p}
  38. L p L^{p}
  39. L 1 L^{1}
  40. L 1 L^{1}
  41. L L^{\infty}
  42. L L^{\infty}
  43. p > 1 p>1

Marginal_likelihood.html

  1. 𝕏 = ( x 1 , , x n ) , \mathbb{X}=(x_{1},\ldots,x_{n}),
  2. x i p ( x i | θ ) x_{i}\sim p(x_{i}|\theta)
  3. θ p ( θ | α ) , \theta\sim p(\theta|\alpha),
  4. p ( 𝕏 | α ) p(\mathbb{X}|\alpha)
  5. p ( 𝕏 | α ) = θ p ( 𝕏 | θ ) p ( θ | α ) d θ p(\mathbb{X}|\alpha)=\int_{\theta}p(\mathbb{X}|\theta)\,p(\theta|\alpha)\ % \operatorname{d}\!\theta
  6. ( ψ ; 𝕏 ) = p ( 𝕏 | ψ ) = λ p ( 𝕏 | ψ , λ ) p ( λ | ψ ) d λ \mathcal{L}(\psi;\mathbb{X})=p(\mathbb{X}|\psi)=\int_{\lambda}p(\mathbb{X}|% \psi,\lambda)\,p(\lambda|\psi)\ \operatorname{d}\!\lambda
  7. p ( x | M ) = p ( x | θ , M ) p ( θ | M ) d θ p(x|M)=\int p(x|\theta,M)\,p(\theta|M)\,\operatorname{d}\!\theta
  8. p ( M 1 | x ) p ( M 2 | x ) = p ( M 1 ) p ( M 2 ) p ( x | M 1 ) p ( x | M 2 ) \frac{p(M_{1}|x)}{p(M_{2}|x)}=\frac{p(M_{1})}{p(M_{2})}\,\frac{p(x|M_{1})}{p(x% |M_{2})}

Marginal_product.html

  1. M P = Δ Y Δ X MP=\frac{\Delta Y}{\Delta X}
  2. Δ X \Delta X
  3. Δ Y \Delta Y
  4. Y Y
  5. Y = F ( K , L ) Y=F(K,L)
  6. M P K = F K MPK=\frac{\partial F}{\partial K}
  7. M P L = F L MPL=\frac{\partial F}{\partial L}

Market_power.html

  1. P M C = P E D 1 + P E D . \frac{P}{MC}=\frac{PED}{1+PED}.
  2. ( P - M C ) P = - 1 P E D . \frac{(P-MC)}{P}=-\frac{1}{PED}.

Markov_blanket.html

  1. A A
  2. A \partial A
  3. A A
  4. M B ( A ) MB(A)
  5. A A
  6. A \partial A
  7. A A
  8. A A
  9. B B
  10. Pr ( A A , B ) = Pr ( A A ) . \Pr(A\mid\partial A,B)=\Pr(A\mid\partial A).\!

Markov_decision_process.html

  1. s s
  2. a a
  3. s s
  4. s s^{\prime}
  5. R a ( s , s ) R_{a}(s,s^{\prime})
  6. s s^{\prime}
  7. P a ( s , s ) P_{a}(s,s^{\prime})
  8. s s^{\prime}
  9. s s
  10. a a
  11. s s
  12. a a
  13. ( S , A , P ( , ) , R ( , ) , γ ) (S,A,P_{\cdot}(\cdot,\cdot),R_{\cdot}(\cdot,\cdot),\gamma)
  14. S S
  15. A A
  16. A s A_{s}
  17. s s
  18. P a ( s , s ) = Pr ( s t + 1 = s s t = s , a t = a ) P_{a}(s,s^{\prime})=\Pr(s_{t+1}=s^{\prime}\mid s_{t}=s,a_{t}=a)
  19. a a
  20. s s
  21. t t
  22. s s^{\prime}
  23. t + 1 t+1
  24. R a ( s , s ) R_{a}(s,s^{\prime})
  25. s s^{\prime}
  26. s s
  27. γ [ 0 , 1 ] \gamma\in[0,1]
  28. S S
  29. A A
  30. π \pi
  31. π ( s ) \pi(s)
  32. s s
  33. π \pi
  34. t = 0 γ t R a t ( s t , s t + 1 ) \sum^{\infty}_{t=0}{\gamma^{t}R_{a_{t}}(s_{t},s_{t+1})}
  35. a t = π ( s t ) a_{t}=\pi(s_{t})
  36. γ \ \gamma
  37. 0 γ < 1 0\leq\ \gamma\ <1
  38. γ = 1 / ( 1 + r ) \gamma=1/(1+r)
  39. γ \gamma
  40. s s
  41. P P
  42. R R
  43. V V
  44. π \pi
  45. π \pi
  46. V ( s ) V(s)
  47. s s
  48. π ( s ) := arg max a { s P a ( s , s ) ( R a ( s , s ) + γ V ( s ) ) } \pi(s):=\arg\max_{a}\left\{\sum_{s^{\prime}}P_{a}(s,s^{\prime})\left(R_{a}(s,s% ^{\prime})+\gamma V(s^{\prime})\right)\right\}
  49. V ( s ) := s P π ( s ) ( s , s ) ( R π ( s ) ( s , s ) + γ V ( s ) ) V(s):=\sum_{s^{\prime}}P_{\pi(s)}(s,s^{\prime})\left(R_{\pi(s)}(s,s^{\prime})+% \gamma V(s^{\prime})\right)
  50. π \pi
  51. π ( s ) \pi(s)
  52. V ( s ) V(s)
  53. π ( s ) \pi(s)
  54. V ( s ) V(s)
  55. V i + 1 ( s ) := max a { s P a ( s , s ) ( R a ( s , s ) + γ V i ( s ) ) } , V_{i+1}(s):=\max_{a}\left\{\sum_{s^{\prime}}P_{a}(s,s^{\prime})\left(R_{a}(s,s% ^{\prime})+\gamma V_{i}(s^{\prime})\right)\right\},
  56. i i
  57. i = 0 i=0
  58. V 0 V_{0}
  59. V i + 1 V_{i+1}
  60. s s
  61. V V
  62. π \pi
  63. V V
  64. π \pi
  65. s s
  66. π ( s ) \pi(s)
  67. a a
  68. Q ( s , a ) = s P a ( s , s ) ( R a ( s , s ) + γ V ( s ) ) . \ Q(s,a)=\sum_{s^{\prime}}P_{a}(s,s^{\prime})(R_{a}(s,s^{\prime})+\gamma V(s^{% \prime})).
  69. ( s , a ) (s,a)
  70. s s^{\prime}
  71. s s
  72. a a
  73. s s^{\prime}
  74. Q Q
  75. ( S , A , P ) (S,A,P)
  76. 𝒜 \mathcal{A}
  77. 𝒜 𝐃𝐢𝐬𝐭 \mathcal{A}\to\mathbf{Dist}
  78. ( 𝒞 , F : 𝒞 𝐃𝐢𝐬𝐭 (\mathcal{C},F:\mathcal{C}\to\mathbf{Dist}
  79. 𝒞 \mathcal{C}
  80. 𝒮 \mathcal{S}
  81. 𝒜 \mathcal{A}
  82. q ( i | j , a ) q(i|j,a)
  83. 𝒮 × 𝒜 𝒮 \mathcal{S}\times\mathcal{A}\rightarrow\triangle\mathcal{S}
  84. R ( i , a ) R(i,a)
  85. 𝒮 × 𝒜 \mathcal{S}\times\mathcal{A}\rightarrow\mathbb{R}
  86. 𝒳 \mathcal{X}
  87. 𝒰 \mathcal{U}
  88. f ( x , u ) f(x,u)
  89. 𝒳 × 𝒰 𝒳 \mathcal{X}\times\mathcal{U}\rightarrow\triangle\mathcal{X}
  90. r ( x , u ) r(x,u)
  91. 𝒳 × 𝒰 \mathcal{X}\times\mathcal{U}\rightarrow\mathbb{R}
  92. r ( x ( t ) , u ( t ) ) d t = d R ( x ( t ) , u ( t ) ) r(x(t),u(t))dt=dR(x(t),u(t))
  93. R ( x , u ) R(x,u)
  94. m a x 𝔼 u [ 0 γ t r ( x ( t ) , u ( t ) ) ) d t | x 0 ] max\quad\mathbb{E}_{u}[\int_{0}^{\infty}\gamma^{t}r(x(t),u(t)))dt|x_{0}]
  95. 0 γ < 1 0\leq\gamma<1
  96. V * V^{*}
  97. g R ( i , a ) + j S q ( j | i , a ) h ( j ) i S a n d a A ( i ) g\geq R(i,a)+\sum_{j\in S}q(j|i,a)h(j)\quad\forall i\in S\,\,and\,\,a\in A(i)
  98. h h
  99. V ¯ * \bar{V}^{*}
  100. V ¯ * \bar{V}^{*}
  101. Minimize g s.t g - j S q ( j | i , a ) h ( j ) R ( i , a ) i S , a A ( i ) \begin{aligned}\displaystyle\,\text{Minimize}&\displaystyle g\\ \displaystyle\,\text{s.t}&\displaystyle g-\sum_{j\in S}q(j|i,a)h(j)\geq R(i,a)% \,\,\forall i\in S,\,a\in A(i)\end{aligned}
  102. Maximize i S a A ( i ) R ( i , a ) y ( i , a ) s.t. i S a A ( i ) q ( j | i , a ) y ( i , a ) = 0 j S , i S a A ( i ) y ( i , a ) = 1 , y ( i , a ) 0 a A ( i ) a n d i S \begin{aligned}\displaystyle\,\text{Maximize}&\displaystyle\sum_{i\in S}\sum_{% a\in A(i)}R(i,a)y(i,a)\\ \displaystyle\,\text{s.t.}&\displaystyle\sum_{i\in S}\sum_{a\in A(i)}q(j|i,a)y% (i,a)=0\quad\forall j\in S,\\ &\displaystyle\sum_{i\in S}\sum_{a\in A(i)}y(i,a)=1,\\ &\displaystyle y(i,a)\geq 0\qquad\forall a\in A(i)\,\,and\,\,\forall i\in S% \end{aligned}
  103. y ( i , a ) y(i,a)
  104. y ( i , a ) y(i,a)
  105. y * ( i , a ) y^{*}(i,a)
  106. i S a A ( i ) R ( i , a ) y * ( i , a ) i S a A ( i ) R ( i , a ) y ( i , a ) \begin{aligned}\displaystyle\sum_{i\in S}\sum_{a\in A(i)}R(i,a)y^{*}(i,a)\geq% \sum_{i\in S}\sum_{a\in A(i)}R(i,a)y(i,a)\end{aligned}
  107. y * ( i , a ) y^{*}(i,a)
  108. V ( x ( 0 ) , 0 ) = max u 0 T r ( x ( t ) , u ( t ) ) d t + D [ x ( T ) ] s . t . d x ( t ) d t = f [ t , x ( t ) , u ( t ) ] \begin{aligned}\displaystyle V(x(0),0)=&\displaystyle\,\text{max}_{u}\int_{0}^% {T}r(x(t),u(t))dt+D[x(T)]\\ \displaystyle s.t.&\displaystyle\frac{dx(t)}{dt}=f[t,x(t),u(t)]\end{aligned}
  109. \cdot
  110. x ( t ) x(t)
  111. u ( t ) u(t)
  112. \cdot
  113. 0 = max u ( r ( t , x , u ) + V ( t , x ) x f ( t , x , u ) ) 0=\,\text{max}_{u}(r(t,x,u)+\frac{\partial V(t,x)}{\partial x}f(t,x,u))
  114. u ( t ) u(t)
  115. V * V^{*}
  116. β \beta
  117. γ \gamma
  118. α \alpha
  119. a a
  120. u u
  121. R R
  122. g g
  123. g g
  124. R R
  125. V V
  126. J J
  127. J J
  128. V V
  129. π \pi
  130. μ \mu
  131. γ \ \gamma
  132. α \alpha
  133. P a ( s , s ) P_{a}(s,s^{\prime})
  134. p s s ( a ) p_{ss^{\prime}}(a)
  135. P r ( s , a , s ) Pr(s,a,s^{\prime})
  136. P r ( s | s , a ) Pr(s^{\prime}|s,a)
  137. p s s ( a ) . p_{s^{\prime}s}(a).

Marshallian_demand_function.html

  1. B ( p , I ) = { x : p , x I } , B(p,I)=\{x:\langle p,x\rangle\leq I\},
  2. p , x \langle p,x\rangle
  3. u : 𝐑 + L 𝐑 . u:\textbf{R}^{L}_{+}\rightarrow\textbf{R}.
  4. x * ( p , I ) = argmax x B ( p , I ) u ( x ) . x^{*}(p,I)=\operatorname{argmax}_{x\in B(p,I)}u(x).
  5. U ( x 1 , x 2 ) = x 1 α x 2 β U(x_{1},x_{2})=x_{1}^{\alpha}x_{2}^{\beta}
  6. x * ( p 1 , p 2 , I ) = ( α I ( α + β ) p 1 , β I ( α + β ) p 2 ) . x^{*}(p_{1},p_{2},I)=\left(\frac{\alpha I}{(\alpha+\beta)p_{1}},\frac{\beta I}% {(\alpha+\beta)p_{2}}\right).
  7. U ( x 1 , x 2 ) = [ x 1 δ δ + x 2 δ δ ] 1 δ U(x_{1},x_{2})=\left[\frac{x_{1}^{\delta}}{\delta}+\frac{x_{2}^{\delta}}{% \delta}\right]^{\frac{1}{\delta}}
  8. x * ( p 1 , p 2 , I ) = ( I p 1 ϵ - 1 p 1 ϵ - 1 + p 2 ϵ - 1 , I p 2 ϵ - 1 p 1 ϵ - 1 + p 2 ϵ - 1 ) , with ϵ = δ δ - 1 . x^{*}(p_{1},p_{2},I)=\left(\frac{Ip_{1}^{\epsilon-1}}{p_{1}^{\epsilon-1}+p_{2}% ^{\epsilon-1}},\frac{Ip_{2}^{\epsilon-1}}{p_{1}^{\epsilon-1}+p_{2}^{\epsilon-1% }}\right),\quad\,\text{with}\quad\epsilon=\frac{\delta}{\delta-1}.

Mashing.html

  1. Lintner = WK + 16 3.5 {}^{\circ}\mbox{Lintner}~{}=\frac{{}^{\circ}\mbox{WK}~{}+16}{3.5}
  2. WK = ( 3.5 × Lintner ) - 16 {}^{\circ}\mbox{WK}~{}=\left(3.5\times{}^{\circ}\mbox{Lintner}~{}\right)-16
  3. 1 / 2 {1}/{2}
  4. 1 / 4 {1}/{4}
  5. 1 / 5 {1}/{5}

Mass_gap.html

  1. ϕ ( x ) \phi(x)
  2. ϕ ( 0 , t ) ϕ ( 0 , 0 ) n A n exp ( - Δ n t ) \langle\phi(0,t)\phi(0,0)\rangle\sim\sum_{n}A_{n}\exp\left(-\Delta_{n}t\right)
  3. Δ 0 > 0 \Delta_{0}>0
  4. lim p 0 Δ ( p ) = constant \lim_{p\rightarrow 0}\Delta(p)=\mathrm{constant}
  5. ϕ + λ ϕ 3 = 0. \Box\phi+\lambda\phi^{3}=0.
  6. ϕ ( x ) = μ ( 2 λ ) 1 4 sn ( p x + θ , - 1 ) \phi(x)=\mu\left(\frac{2}{\lambda}\right)^{\frac{1}{4}}{\rm sn}\left(p\cdot x+% \theta,-1\right)
  7. μ \mu
  8. θ \theta
  9. p 2 = μ 2 λ 2 . p^{2}=\mu^{2}\sqrt{\frac{\lambda}{2}}.
  10. ρ ( μ 2 ) = n = 1 N Z n δ ( μ 2 - m n 2 ) + ρ c ( μ 2 ) \rho(\mu^{2})=\sum_{n=1}^{N}Z_{n}\delta(\mu^{2}-m_{n}^{2})+\rho_{c}(\mu^{2})
  11. ρ c ( μ 2 ) \rho_{c}(\mu^{2})
  12. Δ ( p ) = n = 1 N Z n p 2 - m n 2 + i ϵ + 4 m N 2 d μ 2 ρ c ( μ 2 ) 1 p 2 - μ 2 + i ϵ \Delta(p)=\sum_{n=1}^{N}\frac{Z_{n}}{p^{2}-m^{2}_{n}+i\epsilon}+\int_{4m_{N}^{% 2}}^{\infty}d\mu^{2}\rho_{c}(\mu^{2})\frac{1}{p^{2}-\mu^{2}+i\epsilon}
  13. 4 m N 2 4m_{N}^{2}
  14. 0 d μ 2 ρ ( μ 2 ) = 1 \int_{0}^{\infty}d\mu^{2}\rho(\mu^{2})=1
  15. 1 = n = 1 N Z n + 0 d μ 2 ρ c ( μ 2 ) 1=\sum_{n=1}^{N}Z_{n}+\int_{0}^{\infty}d\mu^{2}\rho_{c}(\mu^{2})
  16. ρ c ( μ 2 ) = 0 \rho_{c}(\mu^{2})=0

Master_equation.html

  1. d P d t = 𝐀 P , \frac{d\vec{P}}{dt}=\mathbf{A}\vec{P},
  2. P \vec{P}
  3. 𝐀 \mathbf{A}
  4. 𝐀 \mathbf{A}
  5. 𝐀 𝐀 ( t ) \mathbf{A}\rightarrow\mathbf{A}(t)
  6. d P d t = 𝐀 ( t ) P . \frac{d\vec{P}}{dt}=\mathbf{A}(t)\vec{P}.
  7. d P d t = 0 t 𝐀 ( t - τ ) P ( τ ) d τ . \frac{d\vec{P}}{dt}=\int^{t}_{0}\mathbf{A}(t-\tau)\vec{P}(\tau)d\tau.
  8. 𝐀 \mathbf{A}
  9. 𝐀 \mathbf{A}
  10. 𝐀 \mathbf{A}
  11. A k P , \sum_{\ell}A_{k\ell}P_{\ell},
  12. P , P_{\ell},
  13. \ell
  14. 𝐀 \mathbf{A}
  15. P k P_{k}
  16. P , P_{\ell},
  17. A k P k , \sum_{\ell}A_{\ell k}P_{k},
  18. 𝐀 \mathbf{A}
  19. d P k d t = ( A k P ) = k ( A k P ) + A k k P k = k ( A k P - A k P k ) . \frac{dP_{k}}{dt}=\sum_{\ell}(A_{k\ell}P_{\ell})=\sum_{\ell\neq k}(A_{k\ell}P_% {\ell})+A_{kk}P_{k}=\sum_{\ell\neq k}(A_{k\ell}P_{\ell}-A_{\ell k}P_{k}).
  20. ( A k ) = d d t ( P k ) = 0 \sum_{\ell}(A_{\ell k})=\frac{d}{dt}\sum_{\ell}(P_{\ell k})=0
  21. P k P_{\ell k}
  22. A k k = - k ( A k ) A k k P k = - k ( A k P k ) A_{kk}=-\sum_{\ell\neq k}(A_{\ell k})\Rightarrow A_{kk}P_{k}=-\sum_{\ell\neq k% }(A_{\ell k}P_{k})
  23. π k \scriptstyle\pi_{k}
  24. π \scriptstyle\pi_{\ell}
  25. A k π = A k π k . A_{k\ell}\pi_{\ell}=A_{\ell k}\pi_{k}.

Matched_filter.html

  1. h h
  2. y [ n ] = k = - h [ n - k ] x [ k ] . \ y[n]=\sum_{k=-\infty}^{\infty}h[n-k]x[k].
  3. h h
  4. x x
  5. s s
  6. v v
  7. x = s + v . \ x=s+v.\,
  8. R v = E { v v H } \ R_{v}=E\{vv^{\mathrm{H}}\}\,
  9. v H v^{\mathrm{H}}
  10. v v
  11. E E
  12. y y
  13. y = k = - h * [ k ] x [ k ] = h H x = h H s + h H v = y s + y v . \ y=\sum_{k=-\infty}^{\infty}h^{*}[k]x[k]=h^{\mathrm{H}}x=h^{\mathrm{H}}s+h^{% \mathrm{H}}v=y_{s}+y_{v}.
  14. SNR = | y s | 2 E { | y v | 2 } . \mathrm{SNR}=\frac{|y_{s}|^{2}}{E\{|y_{v}|^{2}\}}.
  15. SNR = | h H s | 2 E { | h H v | 2 } . \mathrm{SNR}=\frac{|h^{\mathrm{H}}s|^{2}}{E\{|h^{\mathrm{H}}v|^{2}\}}.
  16. h h
  17. E { | h H v | 2 } = E { ( h H v ) ( h H v ) H } = h H E { v v H } h = h H R v h . \ E\{|h^{\mathrm{H}}v|^{2}\}=E\{(h^{\mathrm{H}}v){(h^{\mathrm{H}}v)}^{\mathrm{% H}}\}=h^{\mathrm{H}}E\{vv^{\mathrm{H}}\}h=h^{\mathrm{H}}R_{v}h.\,
  18. SNR \mathrm{SNR}
  19. SNR = | h H s | 2 h H R v h . \mathrm{SNR}=\frac{|h^{\mathrm{H}}s|^{2}}{h^{\mathrm{H}}R_{v}h}.
  20. R v R_{v}
  21. SNR = | ( R v 1 / 2 h ) H ( R v - 1 / 2 s ) | 2 ( R v 1 / 2 h ) H ( R v 1 / 2 h ) , \mathrm{SNR}=\frac{|{(R_{v}^{1/2}h)}^{\mathrm{H}}(R_{v}^{-1/2}s)|^{2}}{{(R_{v}% ^{1/2}h)}^{\mathrm{H}}(R_{v}^{1/2}h)},
  22. | a H b | 2 ( a H a ) ( b H b ) , \ |a^{\mathrm{H}}b|^{2}\leq(a^{\mathrm{H}}a)(b^{\mathrm{H}}b),\,
  23. a a
  24. b b
  25. SNR \mathrm{SNR}
  26. SNR = | ( R v 1 / 2 h ) H ( R v - 1 / 2 s ) | 2 ( R v 1 / 2 h ) H ( R v 1 / 2 h ) [ ( R v 1 / 2 h ) H ( R v 1 / 2 h ) ] [ ( R v - 1 / 2 s ) H ( R v - 1 / 2 s ) ] ( R v 1 / 2 h ) H ( R v 1 / 2 h ) . \mathrm{SNR}=\frac{|{(R_{v}^{1/2}h)}^{\mathrm{H}}(R_{v}^{-1/2}s)|^{2}}{{(R_{v}% ^{1/2}h)}^{\mathrm{H}}(R_{v}^{1/2}h)}\leq\frac{\left[{(R_{v}^{1/2}h)}^{\mathrm% {H}}(R_{v}^{1/2}h)\right]\left[{(R_{v}^{-1/2}s)}^{\mathrm{H}}(R_{v}^{-1/2}s)% \right]}{{(R_{v}^{1/2}h)}^{\mathrm{H}}(R_{v}^{1/2}h)}.
  27. SNR = | ( R v 1 / 2 h ) H ( R v - 1 / 2 s ) | 2 ( R v 1 / 2 h ) H ( R v 1 / 2 h ) s H R v - 1 s . \mathrm{SNR}=\frac{|{(R_{v}^{1/2}h)}^{\mathrm{H}}(R_{v}^{-1/2}s)|^{2}}{{(R_{v}% ^{1/2}h)}^{\mathrm{H}}(R_{v}^{1/2}h)}\leq s^{\mathrm{H}}R_{v}^{-1}s.
  28. R v 1 / 2 h = α R v - 1 / 2 s \ R_{v}^{1/2}h=\alpha R_{v}^{-1/2}s
  29. α \alpha
  30. SNR \mathrm{SNR}
  31. SNR = | ( R v 1 / 2 h ) H ( R v - 1 / 2 s ) | 2 ( R v 1 / 2 h ) H ( R v 1 / 2 h ) = α 2 | ( R v - 1 / 2 s ) H ( R v - 1 / 2 s ) | 2 α 2 ( R v - 1 / 2 s ) H ( R v - 1 / 2 s ) = | s H R v - 1 s | 2 s H R v - 1 s = s H R v - 1 s . \mathrm{SNR}=\frac{|{(R_{v}^{1/2}h)}^{\mathrm{H}}(R_{v}^{-1/2}s)|^{2}}{{(R_{v}% ^{1/2}h)}^{\mathrm{H}}(R_{v}^{1/2}h)}=\frac{\alpha^{2}|{(R_{v}^{-1/2}s)}^{% \mathrm{H}}(R_{v}^{-1/2}s)|^{2}}{\alpha^{2}{(R_{v}^{-1/2}s)}^{\mathrm{H}}(R_{v% }^{-1/2}s)}=\frac{|s^{\mathrm{H}}R_{v}^{-1}s|^{2}}{s^{\mathrm{H}}R_{v}^{-1}s}=% s^{\mathrm{H}}R_{v}^{-1}s.
  32. h = α R v - 1 s . \ h=\alpha R_{v}^{-1}s.
  33. E { | y v | 2 } = 1. \ E\{|y_{v}|^{2}\}=1.\,
  34. α \alpha
  35. E { | y v | 2 } = α 2 s H R v - 1 s = 1 , \ E\{|y_{v}|^{2}\}=\alpha^{2}s^{\mathrm{H}}R_{v}^{-1}s=1,
  36. α = 1 s H R v - 1 s , \ \alpha=\frac{1}{\sqrt{s^{\mathrm{H}}R_{v}^{-1}s}},
  37. h = 1 s H R v - 1 s R v - 1 s . \ h=\frac{1}{\sqrt{s^{\mathrm{H}}R_{v}^{-1}s}}R_{v}^{-1}s.
  38. h h
  39. R v R_{v}
  40. s ( t ) s(t)
  41. v ( t ) v(t)
  42. h ( t ) h(t)
  43. SNR \mathrm{SNR}
  44. x = s + v , \ x=s+v,\,
  45. R v = E { v v H } . \ R_{v}=E\{vv^{\mathrm{H}}\}.\,
  46. SNR = | y s | 2 E { | y v | 2 } . \mathrm{SNR}=\frac{|y_{s}|^{2}}{E\{|y_{v}|^{2}\}}.
  47. | y s | 2 = y s H y s = h H s s H h . \ |y_{s}|^{2}={y_{s}}^{\mathrm{H}}y_{s}=h^{\mathrm{H}}ss^{\mathrm{H}}h.\,
  48. E { | y v | 2 } = E { y v H y v } = E { h H v v H h } = h H R v h . \ E\{|y_{v}|^{2}\}=E\{{y_{v}}^{\mathrm{H}}y_{v}\}=E\{h^{\mathrm{H}}vv^{\mathrm% {H}}h\}=h^{\mathrm{H}}R_{v}h.\,
  49. SNR = h H s s H h h H R v h . \mathrm{SNR}=\frac{h^{\mathrm{H}}ss^{\mathrm{H}}h}{h^{\mathrm{H}}R_{v}h}.
  50. SNR \mathrm{SNR}
  51. h H R v h = 1 \ h^{\mathrm{H}}R_{v}h=1
  52. = h H s s H h + λ ( 1 - h H R v h ) \ \mathcal{L}=h^{\mathrm{H}}ss^{\mathrm{H}}h+\lambda(1-h^{\mathrm{H}}R_{v}h)
  53. h * = s s H h - λ R v h = 0 \ \nabla_{h^{*}}\mathcal{L}=ss^{\mathrm{H}}h-\lambda R_{v}h=0
  54. ( s s H ) h = λ R v h \ (ss^{\mathrm{H}})h=\lambda R_{v}h
  55. h H ( s s H ) h = λ h H R v h . \ h^{\mathrm{H}}(ss^{\mathrm{H}})h=\lambda h^{\mathrm{H}}R_{v}h.
  56. s s H ss^{\mathrm{H}}
  57. λ max = s H R v - 1 s , \ \lambda_{\max}=s^{\mathrm{H}}R_{v}^{-1}s,
  58. h = 1 s H R v - 1 s R v - 1 s . \ h=\frac{1}{\sqrt{s^{\mathrm{H}}R_{v}^{-1}s}}R_{v}^{-1}s.
  59. x k = s k + v k , \ x_{k}=s_{k}+v_{k},\,
  60. v k v_{k}
  61. s k s_{k}
  62. f k f_{k}
  63. s k = μ 0 f k - j 0 \ s_{k}=\mu_{0}\cdot f_{k-j_{0}}
  64. j * j^{*}
  65. μ * \mu^{*}
  66. j 0 j_{0}
  67. μ 0 \mu_{0}
  68. x k x_{k}
  69. h j - k h_{j-k}
  70. j * , μ * = arg min j , μ k ( x k - μ h j - k ) 2 \ j^{*},\mu^{*}=\arg\min_{j,\mu}\sum_{k}\left(x_{k}-\mu\cdot h_{j-k}\right)^{2}
  71. h j - k h_{j-k}
  72. x k x_{k}
  73. j * , μ * = arg min j , μ [ k ( s k + v k ) 2 + μ 2 k h j - k 2 - 2 μ k s k h j - k - 2 μ k v k h j - k ] \ j^{*},\mu^{*}=\arg\min_{j,\mu}\left[\sum_{k}(s_{k}+v_{k})^{2}+\mu^{2}\sum_{k% }h_{j-k}^{2}-2\mu\sum_{k}s_{k}h_{j-k}-2\mu\sum_{k}v_{k}h_{j-k}\right]
  74. j * , μ * = arg max j , μ [ 2 μ k s k h j - k - μ 2 k h j - k 2 ] \ j^{*},\mu^{*}=\arg\max_{j,\mu}\left[2\mu\sum_{k}s_{k}h_{j-k}-\mu^{2}\sum_{k}% h_{j-k}^{2}\right]
  75. μ \mu
  76. μ * \mu^{*}
  77. μ * = k s k h j - k k h j - k 2 \ \mu^{*}=\frac{\sum_{k}s_{k}h_{j-k}}{\sum_{k}h_{j-k}^{2}}
  78. j * j^{*}
  79. j * = arg max j ( k s k h j - k ) 2 k h j - k 2 \ j^{*}=\arg\max_{j}\frac{\left(\sum_{k}s_{k}h_{j-k}\right)^{2}}{\sum_{k}h_{j-% k}^{2}}
  80. ( k s k h j - k ) 2 k h j - k 2 k s k 2 k h j - k 2 k h j - k 2 = k s k 2 = const \ \frac{\left(\sum_{k}s_{k}h_{j-k}\right)^{2}}{\sum_{k}h_{j-k}^{2}}\leq\frac{% \sum_{k}s_{k}^{2}\cdot\sum_{k}h_{j-k}^{2}}{\sum_{k}h_{j-k}^{2}}=\sum_{k}s_{k}^% {2}=\mathrm{const}
  81. h j - k = ν s k = κ f k - j 0 \ h_{j-k}=\nu\cdot s_{k}=\kappa\cdot f_{k-j_{0}}
  82. ν \nu
  83. κ \kappa
  84. j * = j 0 j^{*}=j_{0}
  85. h j - k h_{j-k}
  86. f k - j 0 f_{k-j_{0}}
  87. κ = 1 \kappa=1
  88. h k = f - k \ h_{k}=f_{-k}
  89. k x k h j - k \sum_{k}x_{k}h_{j-k}
  90. x k x_{k}
  91. h k h_{k}
  92. x k x_{k}
  93. f k f_{k}
  94. N N
  95. N N
  96. N N
  97. N N
  98. k th k^{\mathrm{th}}
  99. a k = { 1 , if bit k is 1 , 0 , if bit k is 0 . \ a_{k}=\begin{cases}1,&\mbox{if bit }~{}k\mbox{ is 1}~{},\\ 0,&\mbox{if bit }~{}k\mbox{ is 0}~{}.\end{cases}
  100. M ( t ) M(t)
  101. M ( t ) = k = - a k × Π ( t - k T T ) . \ M(t)=\sum_{k=-\infty}^{\infty}a_{k}\times\Pi\left(\frac{t-kT}{T}\right).
  102. T T
  103. h ( t ) = Π ( t T ) . \ h(t)=\Pi\left(\frac{t}{T}\right).
  104. h ( t ) h(t)
  105. h ( t ) h(t)
  106. h ( t ) h(t)
  107. M filtered ( t ) M_{\mathrm{filtered}}(t)
  108. M filtered ( t ) = M ( t ) * h ( t ) \ M_{\mathrm{filtered}}(t)=M(t)*h(t)
  109. * *

Material_conditional.html

  1. A B A\rightarrow B
  2. A A
  3. A A
  4. B B
  5. B B
  6. p q p\supset q
  7. p q p\Rightarrow q
  8. p q p\rightarrow q
  9. ¬ ( p and ¬ q ) \neg(p\and\neg q)
  10. ¬ p q \neg pq
  11. p q p\rightarrow q
  12. ¬ ( p and ¬ q ) \neg(p\and\neg q)
  13. ¬ p q \neg pq
  14. p q p\rightarrow q
  15. p p
  16. q q
  17. p q p\rightarrow q
  18. ( p q ) ¬ p q (p→q)⇒¬p∨q
  19. \models
  20. A B A\models B
  21. Γ ψ \Gamma\models\psi
  22. ( φ 1 φ n ψ ) \varnothing\models(\varphi_{1}\land\dots\land\varphi_{n}\rightarrow\psi)
  23. φ 1 , , φ n Γ \varphi_{1},\dots,\varphi_{n}\in\Gamma
  24. \rightarrow
  25. \models
  26. Γ ψ \Gamma\models\psi
  27. Δ Γ ψ \Delta\cup\Gamma\models\psi
  28. φ ψ \varphi\rightarrow\psi
  29. ( φ α ) ψ (\varphi\land\alpha)\rightarrow\psi
  30. ( s ( p q ) ) ( ( s p ) ( s q ) ) (s\rightarrow(p\rightarrow q))\rightarrow((s\rightarrow p)\rightarrow(s% \rightarrow q))
  31. ( a b ) ( ( b c ) ( a c ) ) (a\rightarrow b)\rightarrow((b\rightarrow c)\rightarrow(a\rightarrow c))
  32. a a a\rightarrow a
  33. ( a b ) ( b a ) (a\rightarrow b)\vee(b\rightarrow a)
  34. ( a ( b c ) ) ( b ( a c ) ) (a\rightarrow(b\rightarrow c))\equiv(b\rightarrow(a\rightarrow c))
  35. a ( b c ) a\rightarrow(b\rightarrow c)
  36. ( a and b ) c (a\and b)\rightarrow c
  37. a b c a\rightarrow b\rightarrow c
  38. a ( b c ) a\rightarrow(b\rightarrow c)
  39. a b a\rightarrow b
  40. ¬ a b \neg ab

Material_derivative.html

  1. D y D t y t + 𝐮 y , \frac{\mathrm{D}y}{\mathrm{D}t}\equiv\frac{\partial y}{\partial t}+\mathbf{u}% \cdot\nabla y,
  2. y \nabla y
  3. D φ D t φ t + 𝐮 φ , \frac{\mathrm{D}\varphi}{\mathrm{D}t}\equiv\frac{\partial\varphi}{\partial t}+% \mathbf{u}\cdot\nabla\varphi,
  4. D 𝐀 D t 𝐀 t + 𝐮 𝐀 , \frac{\mathrm{D}\mathbf{A}}{\mathrm{D}t}\equiv\frac{\partial\mathbf{A}}{% \partial t}+\mathbf{u}\cdot\nabla\mathbf{A},
  5. φ \nabla\varphi
  6. 𝐀 \nabla\mathbf{A}
  7. 𝐮 φ = u 1 ϕ x 1 + u 2 ϕ x 2 + u 3 ϕ x 3 \mathbf{u}\cdot\nabla\varphi=u_{1}\frac{\partial\phi}{\partial x_{1}}+u_{2}% \frac{\partial\phi}{\partial x_{2}}+u_{3}\frac{\partial\phi}{\partial x_{3}}
  8. d d t φ ( 𝐱 , t ) = φ t + 𝐱 ˙ φ . \frac{\mathrm{d}}{\mathrm{d}t}\varphi(\mathbf{x},t)=\frac{\partial\varphi}{% \partial t}+\dot{\mathbf{x}}\cdot\nabla\varphi.
  9. 𝐱 ˙ d 𝐱 d t \dot{\mathbf{x}}\equiv\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t}
  10. 𝐱 ˙ = 𝟎 \dot{\mathbf{x}}=\mathbf{0}
  11. 𝐱 ˙ = 0 \dot{\mathbf{x}}=0
  12. 𝐱 ˙ = 𝐮 . \dot{\mathbf{x}}=\mathbf{u}.
  13. D φ D t = φ t + 𝐮 φ . \frac{\mathrm{D}\varphi}{\mathrm{D}t}=\frac{\partial\varphi}{\partial t}+% \mathbf{u}\cdot\nabla\varphi.
  14. [ 𝐮 𝐀 ] j = i u i h i A j q i + A i h i h j ( u j h j q i - u i h i q j ) , [\mathbf{u}\cdot\nabla\mathbf{A}]_{j}=\sum_{i}\frac{u_{i}}{h_{i}}\frac{% \partial A_{j}}{\partial q^{i}}+\frac{A_{i}}{h_{i}h_{j}}\left(u_{j}\frac{% \partial h_{j}}{\partial q^{i}}-u_{i}\frac{\partial h_{i}}{\partial q^{j}}% \right),
  15. h i = g i i . h_{i}=\sqrt{g_{ii}}.
  16. 𝐮 𝐀 = ( u x A x x + u y A x y + u z A x z u x A y x + u y A y y + u z A y z u x A z x + u y A z y + u z A z z ) . \mathbf{u}\cdot\nabla\mathbf{A}=\begin{pmatrix}\displaystyle u_{x}\frac{% \partial A_{x}}{\partial x}+u_{y}\frac{\partial A_{x}}{\partial y}+u_{z}\frac{% \partial A_{x}}{\partial z}\\ \displaystyle u_{x}\frac{\partial A_{y}}{\partial x}+u_{y}\frac{\partial A_{y}% }{\partial y}+u_{z}\frac{\partial A_{y}}{\partial z}\\ \displaystyle u_{x}\frac{\partial A_{z}}{\partial x}+u_{y}\frac{\partial A_{z}% }{\partial y}+u_{z}\frac{\partial A_{z}}{\partial z}\end{pmatrix}.

Mathematical_coincidence.html

  1. π 4 / φ = 3.1446 \pi\approx 4/\sqrt{\varphi}=3.1446\dots
  2. 2 10 = 1024 1000 = 10 3 2^{10}=1024\approx 1000=10^{3}
  3. log 10 log 2 3.3219 10 3 \textstyle\frac{\log 10}{\log 2}\approx 3.3219\approx\frac{10}{3}
  4. 2 10 3 / 10 2\approx 10^{3/10}
  5. 5 3 = 125 128 = 2 7 5^{3}=125\approx 128=2^{7}
  6. 2 19 3 12 2^{19}\approx 3^{12}
  7. log 3 log 2 1.5849 19 12 \frac{\log 3}{\log 2}\approx 1.5849\dots\approx\frac{19}{12}
  8. 2 7 / 12 3 / 2 ; 2^{7/12}\approx 3/2;
  9. ( 3 / 2 ) 12 2 7 , {(3/2)}^{12}\approx 2^{7},
  10. 2 12 5 7 = 1.33333319 4 3 \sqrt[12]{2}\sqrt[7]{5}=1.33333319\ldots\approx\frac{4}{3}
  11. 5 8 35 3 = 4.00000559 4 \sqrt[8]{5}\sqrt[3]{35}=4.00000559\ldots\approx 4
  12. 0.6 9 4.9 28 = 0.99999999754 1 \sqrt[9]{0.6}\sqrt[28]{4.9}=0.99999999754\ldots\approx 1
  13. 2 9 3 - 28 5 37 7 - 18 2^{9}3^{-28}5^{37}7^{-18}
  14. ( 5 / 4 ) 3 2 / 1 {(5/4)}^{3}\approx{2/1}
  15. π 2 10 ; \pi^{2}\approx 10;
  16. ζ ( 2 ) = π 2 / 6. \zeta(2)=\pi^{2}/6.
  17. π \pi
  18. 10 , \sqrt{10},
  19. π 2 227 / 23 , \pi^{2}\approx 227/23,
  20. π 3 31 , \pi^{3}\approx 31,
  21. π 3 + 1 5 2 , \sqrt[5]{\pi^{3}+1}\approx 2,
  22. π ( 9 2 + 19 2 22 ) 1 / 4 , \pi\approx\left(9^{2}+\frac{19^{2}}{22}\right)^{1/4},
  23. 22 π 4 2143 ; 22\pi^{4}\approx 2143;
  24. π \pi
  25. 0 cos ( 2 x ) n = 1 cos ( x n ) d x π 8 \int_{0}^{\infty}\cos(2x)\prod_{n=1}^{\infty}\cos\left(\frac{x}{n}\right)dx% \approx\frac{\pi}{8}
  26. π 4 + π 5 e 6 \pi^{4}+\pi^{5}\approx e^{6}
  27. 3 3 e π 4 \sqrt[4]{3^{3}e^{\pi}}
  28. 3 π + e 4 5 {3}^{\frac{\pi+e}{4}}\approx 5\quad
  29. e π - π 19.99909998 e^{\pi}-\pi\approx 19.99909998
  30. ( π + 20 ) i = - 0.9999999992 - i 0.000039 - 1 (\pi+20)^{i}=-0.9999999992\ldots-i\cdot 0.000039\ldots\approx-1
  31. π 3 2 / e 2 3 = 9.9998 10 \pi^{3^{2}}/e^{2^{3}}=9.9998\ldots\approx 10
  32. 163 ( π - e ) 69 {163}\cdot(\pi-e)\approx 69
  33. 163 ln 163 2 5 \frac{163}{\ln 163}\approx 2^{5}
  34. e π 163 ( 2 6 10005 ) 3 + 744 e^{\pi\sqrt{163}}\approx(2^{6}\cdot 10005)^{3}+744
  35. 2.9 10 - 28 % 2.9\cdot 10^{-28}\%
  36. 60 sin ( α ) α 60\sin(\alpha)\approx\alpha
  37. α \alpha
  38. 10 ! = 6 ! 7 ! = 1 ! 3 ! 5 ! 7 ! 10!=6!\cdot 7!=1!\cdot 3!\cdot 5!\cdot 7!
  39. 2 3 = 8 \,2^{3}=8
  40. 3 2 = 9 3^{2}=9\,
  41. 4 2 = 2 4 \,4^{2}=2^{4}
  42. a b = b a , a b a^{b}=b^{a},a\neq b
  43. λ = 1 365 ( 23 2 ) = 253 365 \lambda=\frac{1}{365}{23\choose 2}=\frac{253}{365}
  44. ln ( 2 ) \ln(2)
  45. 2 5 9 2 = 2592 2^{5}\cdot 9^{2}=2592
  46. 1 ! + 4 ! + 5 ! = 145 \,1!+4!+5!=145
  47. 16 64 = 1 64 = 1 4 \frac{16}{64}=\frac{1\!\!\!\not 6}{\not 64}=\frac{1}{4}
  48. 26 65 = 2 65 = 2 5 \frac{26}{65}=\frac{2\!\!\!\not 6}{\not 65}=\frac{2}{5}
  49. 19 95 = 1 95 = 1 5 \frac{19}{95}=\frac{1\!\!\!\not 9}{\not 95}=\frac{1}{5}
  50. 49 98 = 4 98 = 4 8 \frac{49}{98}=\frac{4\!\!\!\not 9}{\not 98}=\frac{4}{8}
  51. ( 4 + 9 + 1 + 3 ) 3 = 4 , 913 \,(4+9+1+3)^{3}=4{,}913
  52. ( 5 + 8 + 3 + 2 ) 3 = 5 , 832 \,(5+8+3+2)^{3}=5{,}832
  53. ( 1 + 9 + 6 + 8 + 3 ) 3 = 19 , 683 \,(1+9+6+8+3)^{3}=19{,}683
  54. 2 7 - 1 = 127 \,2^{7}-1=127
  55. 127 = - 1 + 2 7 \,127=-1+2^{7}
  56. 1 3 + 5 3 + 3 3 = 153 \,1^{3}+5^{3}+3^{3}=153
  57. 3 3 + 7 3 + 0 3 = 370 \,3^{3}+7^{3}+0^{3}=370
  58. 3 3 + 7 3 + 1 3 = 371 \,3^{3}+7^{3}+1^{3}=371
  59. 4 3 + 0 3 + 7 3 = 407 \,4^{3}+0^{3}+7^{3}=407
  60. ( 3 + 4 ) 3 = 343 \,(3+4)^{3}=343
  61. 588 2 + 2353 2 = 5882353 \,588^{2}+2353^{2}=5882353
  62. 1 / 17 = 0.0588235294117647 \,1/17=0.0588235294117647\ldots
  63. 2646798 = 2 1 + 6 2 + 4 3 + 6 4 + 7 5 + 9 6 + 8 7 \,2646798=2^{1}+6^{2}+4^{3}+6^{4}+7^{5}+9^{6}+8^{7}
  64. sin ( 666 ) = cos ( 6 6 6 ) = - φ / 2 \sin(666^{\circ})=\cos(6\cdot 6\cdot 6^{\circ})=-\varphi/2
  65. φ \varphi
  66. ϕ ( 666 ) = 6 6 6 \,\phi(666)=6\cdot 6\cdot 6
  67. ϕ \phi
  68. T 2 π L g T\approx 2\pi\sqrt{\frac{L}{g}}
  69. π 2 3 × 10 15 Hz \frac{\pi^{2}}{3}\times 10^{15}\,\text{Hz}
  70. 3.2898 ¯ 41960364 ( 17 ) × 10 15 Hz = R c \underline{3.2898}41960364(17)\times 10^{15}\,\text{Hz}=R_{\infty}c
  71. 3.2898 ¯ 68133696... = π 2 3 \underline{3.2898}68133696...=\frac{\pi^{2}}{3}
  72. α \alpha
  73. 1 137 \frac{1}{137}
  74. 1 137 \frac{1}{137}
  75. α = 1 137.035999074 \alpha=\frac{1}{137.035999074\dots}
  76. α \alpha
  77. π 2 \pi^{2}

Mathematical_modelling_of_infectious_disease.html

  1. R 0 = 1. \ R_{0}\ =1.
  2. S = A L . S=\frac{A}{L}.
  3. S = 1 R 0 . S=\frac{1}{R_{0}}.
  4. 1 R 0 = A L R 0 = L A . \frac{1}{R_{0}}=\frac{A}{L}\Rightarrow R_{0}=\frac{L}{A}.
  5. R 0 = 1 + L A . R_{0}=1+\frac{L}{A}.
  6. R 0 S = 1. \ R_{0}\cdot S=1.
  7. R 0 ( 1 - q ) = 1 , \ R_{0}\cdot(1-q)=1,
  8. 1 - q = 1 R 0 , 1-q=\frac{1}{R_{0}},
  9. q = 1 - 1 R 0 . q=1-\frac{1}{R_{0}}.
  10. q c = 1 - 1 R 0 q_{c}=1-\frac{1}{R_{0}}
  11. R q = R 0 ( 1 - q ) \ R_{q}=R_{0}(1-q)
  12. R q = L A q , \ R_{q}=\frac{L}{A_{q}},
  13. A q = L R q = L R 0 ( 1 - q ) . \ A_{q}=\frac{L}{R_{q}}=\frac{L}{R_{0}(1-q)}.
  14. A q = L ( L / A ) ( 1 - q ) = A L L ( 1 - q ) = A 1 - q . \ {A_{q}}=\frac{L}{(L/A)(1-q)}=\frac{AL}{L(1-q)}=\frac{A}{1-q}.

Matrix_decoder.html

  1. j = + 90 j=+90^{\circ}
  2. k = - 90 k=-90^{\circ}
  3. j = + 90 j=+90^{\circ}
  4. k = - 90 k=-90^{\circ}
  5. j = + 90 j=+90^{\circ}
  6. k = - 90 k=-90^{\circ}
  7. j = + 90 j=+90^{\circ}
  8. k = - 90 k=-90^{\circ}
  9. j = + 90 j=+90^{\circ}
  10. k = - 90 k=-90^{\circ}
  11. 1 1
  12. 0
  13. 1 2 \sqrt{\frac{1}{2}}
  14. j 1 2 j\sqrt{\frac{1}{2}}
  15. 0
  16. 1 1
  17. 1 2 \sqrt{\frac{1}{2}}
  18. k 1 2 k\sqrt{\frac{1}{2}}
  19. j = + 90 j=+90^{\circ}
  20. k = - 90 k=-90^{\circ}
  21. 1 1
  22. 0
  23. 1 2 \sqrt{\frac{1}{2}}
  24. j 2 3 j\sqrt{\frac{2}{3}}
  25. j 1 3 j\sqrt{\frac{1}{3}}
  26. 0
  27. 1 1
  28. 1 2 \sqrt{\frac{1}{2}}
  29. k 1 3 k\sqrt{\frac{1}{3}}
  30. k 2 3 k\sqrt{\frac{2}{3}}
  31. j = + 90 j=+90^{\circ}
  32. k = - 90 k=-90^{\circ}

Matrix_exponential.html

  1. X X
  2. n × n n×n
  3. X X
  4. e x p ( X ) exp(X)
  5. n × n n×n
  6. e X = k = 0 1 k ! X k . e^{X}=\sum_{k=0}^{\infty}{1\over k!}X^{k}.
  7. X X
  8. X X
  9. X X
  10. X X
  11. X X
  12. Y Y
  13. n × n n×n
  14. a a
  15. b b
  16. n × n n×n
  17. I I
  18. X Y = Y X XY=YX
  19. Y Y
  20. X X
  21. X X
  22. X X
  23. X X
  24. X X
  25. X X
  26. d d t y ( t ) = A y ( t ) , y ( 0 ) = y 0 , \frac{d}{dt}y(t)=Ay(t),\quad y(0)=y_{0},
  27. A A
  28. y ( t ) = e A t y 0 . y(t)=e^{At}y_{0}.\,
  29. d d t y ( t ) = A y ( t ) + z ( t ) , y ( 0 ) = y 0 . \frac{d}{dt}y(t)=Ay(t)+z(t),\quad y(0)=y_{0}.
  30. d d t y ( t ) = A ( t ) y ( t ) , y ( 0 ) = y 0 , \frac{d}{dt}y(t)=A(t)\,y(t),\quad y(0)=y_{0},
  31. A A
  32. x x
  33. y y
  34. X X
  35. Y Y
  36. X Y = Y X XY=YX
  37. e X + Y = e X e Y . e^{X+Y}=e^{X}e^{Y}~{}.
  38. X X
  39. Y Y
  40. A A
  41. H H
  42. tr exp ( A + H ) tr ( exp ( A ) exp ( H ) ) . \operatorname{tr}\exp(A+H)\leq\operatorname{tr}(\exp(A)\exp(H)).
  43. t r ( e x p ( A ) e x p ( B ) e x p ( C ) ) tr(exp(A)exp(B)exp(C))
  44. A A
  45. B B
  46. C C
  47. H H
  48. f ( A ) = tr exp ( H + log A ) f(A)=\operatorname{tr}\,\exp\left(H+\log A\right)
  49. exp : M n ( ) GL ( n , ) \exp\colon M_{n}(\mathbb{C})\to\mathrm{GL}(n,\mathbb{C})
  50. n n
  51. X X
  52. Y Y
  53. e X + Y - e X Y e X e Y , \|e^{X+Y}-e^{X}\|\leq\|Y\|e^{\|X\|}e^{\|Y\|},
  54. t e t X , t t\mapsto e^{tX},\qquad t\in\mathbb{R}
  55. e t X e s X = e ( t + s ) X . e^{tX}e^{sX}=e^{(t+s)X}.\,
  56. d d t e t X = X e t X = e t X X . ( 1 ) \frac{d}{dt}e^{tX}=Xe^{tX}=e^{tX}X.\qquad(1)
  57. t t
  58. X ( t ) X(t)
  59. ( d d t e X ( t ) ) e - X ( t ) = d d t X ( t ) + 1 2 ! [ X ( t ) , d d t X ( t ) ] + 1 3 ! [ X ( t ) , [ X ( t ) , d d t X ( t ) ] ] + \left(\frac{d}{dt}e^{X(t)}\right)e^{-X(t)}=\frac{d}{dt}X(t)+\frac{1}{2!}[X(t),% \frac{d}{dt}X(t)]+\frac{1}{3!}[X(t),[X(t),\frac{d}{dt}X(t)]]+\cdots
  60. exp : M n ( ) GL ( n , ) \exp\colon M_{n}(\mathbb{R})\to\mathrm{GL}(n,\mathbb{R})
  61. A = [ a 1 0 0 0 a 2 0 0 0 a n ] A=\begin{bmatrix}a_{1}&0&\ldots&0\\ 0&a_{2}&\ldots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&a_{n}\end{bmatrix}
  62. e A = [ e a 1 0 0 0 e a 2 0 0 0 e a n ] e^{A}=\begin{bmatrix}e^{a_{1}}&0&\ldots&0\\ 0&e^{a_{2}}&\ldots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&e^{a_{n}}\end{bmatrix}
  63. D D
  64. P P
  65. P P
  66. e P = k = 0 P k k ! = I + ( k = 1 1 k ! ) P = I + ( e - 1 ) P . e^{P}=\sum_{k=0}^{\infty}\frac{P^{k}}{k!}=I+\left(\sum_{k=1}^{\infty}\frac{1}{% k!}\right)P=I+(e-1)P~{}.
  67. a a
  68. b b
  69. R R
  70. G G
  71. θ θ
  72. G = b a 𝖳 - a b 𝖳 a 𝖳 b = 0 G=ba^{\mathsf{T}}-ab^{\mathsf{T}}\qquad a^{\mathsf{T}}b=0
  73. - G 2 = a a 𝖳 + b b 𝖳 = P P 2 = P P G = G P = G , -G^{2}=aa^{\mathsf{T}}+bb^{\mathsf{T}}=P\qquad P^{2}=P\qquad PG=GP=G~{},
  74. R ( θ ) = e G θ = I + G sin ( θ ) + G 2 ( 1 - cos ( θ ) ) = I - P + P cos ( θ ) + G sin ( θ ) . \begin{aligned}\displaystyle R\left(\theta\right)&\displaystyle={{e}^{G\theta}% }=I+G\sin(\theta)+{{G}^{2}}(1-\cos(\theta))\\ &\displaystyle=I-P+P\cos(\theta)+G\sin(\theta)~{}.\\ \end{aligned}
  75. G G
  76. G G
  77. c o s ( θ ) −cos(θ)
  78. s i n ( θ ) sin(θ)
  79. R ( θ ) R(θ)
  80. a = ( 1 0 ) a=\left(\begin{smallmatrix}1\\ 0\end{smallmatrix}\right)
  81. b = ( 0 1 ) b=\left(\begin{smallmatrix}0\\ 1\end{smallmatrix}\right)
  82. G = ( 0 - 1 1 0 ) G=\left(\begin{smallmatrix}0&-1\\ 1&0\end{smallmatrix}\right)
  83. G 2 = ( - 1 0 0 - 1 ) G^{2}=\left(\begin{smallmatrix}-1&0\\ 0&-1\end{smallmatrix}\right)
  84. R ( θ ) = ( cos ( θ ) - sin ( θ ) sin ( θ ) cos ( θ ) ) = I cos ( θ ) + G sin ( θ ) R(\theta)=\left(\begin{matrix}\cos(\theta)&-\sin(\theta)\\ \sin(\theta)&\cos(\theta)\end{matrix}\right)=I\cos(\theta)+G\sin(\theta)
  85. a b ab
  86. 30 ° = π / 6 30°=π/6
  87. a a
  88. b b
  89. a = ( 1 0 0 ) b = 1 5 ( 0 1 2 ) I = ( 1 0 0 0 1 0 0 0 1 ) G = 1 5 ( 0 - 1 - 2 1 0 0 2 0 0 ) P = - G 2 = 1 5 ( 5 0 0 0 1 2 0 2 4 ) P ( 1 2 3 ) = 1 5 ( 5 8 16 ) = a + 8 5 b θ = π 6 R = 1 10 ( 5 3 - 5 - 2 5 5 8 + 3 - 4 + 2 3 2 5 - 4 + 2 3 2 + 4 3 ) \begin{aligned}&\displaystyle a=\left(\begin{matrix}1\\ 0\\ 0\\ \end{matrix}\right)\quad b=\frac{1}{\sqrt{5}}\left(\begin{matrix}0\\ 1\\ 2\\ \end{matrix}\right)\\ &\displaystyle I=\left(\begin{matrix}1&0&0\\ 0&1&0\\ 0&0&1\\ \end{matrix}\right)\quad G=\frac{1}{\sqrt{5}}\left(\begin{matrix}0&-1&-2\\ 1&0&0\\ 2&0&0\\ \end{matrix}\right)\\ &\displaystyle P=-{{G}^{2}}=\frac{1}{5}\left(\begin{matrix}5&0&0\\ 0&1&2\\ 0&2&4\\ \end{matrix}\right)\quad P\left(\begin{matrix}1\\ 2\\ 3\\ \end{matrix}\right)=\frac{1}{5}\left(\begin{matrix}5\\ 8\\ 16\\ \end{matrix}\right)=a+\frac{8}{\sqrt{5}}b\\ &\displaystyle\theta=\frac{\pi}{6}\quad\Rightarrow\quad R=\frac{1}{10}\left(% \begin{matrix}5\sqrt{3}&-\sqrt{5}&-2\sqrt{5}\\ \sqrt{5}&8+\sqrt{3}&-4+2\sqrt{3}\\ 2\sqrt{5}&-4+2\sqrt{3}&2+4\sqrt{3}\\ \end{matrix}\right)\\ \end{aligned}
  90. N = I P N=I−P
  91. P P
  92. G G
  93. R R
  94. R ( π 6 ) = N + P 3 2 + G 1 2 R ( π 6 ) 2 = N + P 1 2 + G 3 2 R ( π 6 ) 3 = N + G R ( π 6 ) 6 = N - P R ( π 6 ) 12 = N + P = I \begin{aligned}&\displaystyle R\left(\frac{\pi}{6}\right)=N+P\frac{\sqrt{3}}{2% }+G\frac{1}{2}\quad\quad R{{\left(\frac{\pi}{6}\right)}^{2}}=N+P\frac{1}{2}+G% \frac{\sqrt{3}}{2}\\ &\displaystyle R{{\left(\frac{\pi}{6}\right)}^{3}}=N+G\quad\quad R{{\left(% \frac{\pi}{6}\right)}^{6}}=N-P\quad\quad R{{\left(\frac{\pi}{6}\right)}^{12}}=% N+P=I\\ \end{aligned}
  95. e N = I + N + 1 2 N 2 + 1 6 N 3 + + 1 ( q - 1 ) ! N q - 1 . e^{N}=I+N+\frac{1}{2}N^{2}+\frac{1}{6}N^{3}+\cdots+\frac{1}{(q-1)!}N^{q-1}~{}.
  96. X = A + N X=A+N\,
  97. e X = e A + N = e A e N . e^{X}=e^{A+N}=e^{A}e^{N}.\,
  98. e X = P e J P - 1 . e^{X}=Pe^{J}P^{-1}.\,
  99. J = J a 1 ( λ 1 ) J a 2 ( λ 2 ) J a n ( λ n ) , J=J_{a_{1}}(\lambda_{1})\oplus J_{a_{2}}(\lambda_{2})\oplus\cdots\oplus J_{a_{% n}}(\lambda_{n}),
  100. e J \displaystyle e^{J}
  101. J a ( λ ) = λ I + N J_{a}(\lambda)=\lambda I+N\,
  102. e λ I + N = e λ e N . e^{\lambda I+N}=e^{\lambda}e^{N}.\,
  103. n n
  104. P P
  105. P ( A ) = 0 P(A)=0
  106. f ( z ) = e t z - Q t ( z ) P ( z ) f(z)=\frac{e^{tz}-Q_{t}(z)}{P(z)}
  107. e t A = Q t ( A ) e^{tA}=Q_{t}(A)
  108. P ( z ) P(z)
  109. z z
  110. A A
  111. a a
  112. P P
  113. P P
  114. f f
  115. a a
  116. a a
  117. P P
  118. P P
  119. P P
  120. A := [ a b c d ] . A:=\begin{bmatrix}a&b\\ c&d\end{bmatrix}.
  121. e t A = s 0 ( t ) I + s 1 ( t ) A e^{tA}=s_{0}(t)\,I+s_{1}(t)\,A
  122. z z
  123. B B
  124. z z
  125. z z
  126. B B
  127. α α
  128. β β
  129. A A
  130. P ( z ) = z 2 - ( a + d ) z + a d - b c = ( z - α ) ( z - β ) . P(z)=z^{2}-(a+d)\ z+ad-bc=(z-\alpha)(z-\beta)~{}.
  131. S t ( z ) = e α t z - β α - β + e β t z - α β - α , S_{t}(z)=e^{\alpha t}\frac{z-\beta}{\alpha-\beta}+e^{\beta t}\frac{z-\alpha}{% \beta-\alpha}~{},
  132. s 0 ( t ) = α e β t - β e α t α - β , s 1 ( t ) = e α t - e β t α - β s_{0}(t)=\frac{\alpha\,e^{\beta t}-\beta\,e^{\alpha t}}{\alpha-\beta},\quad s_% {1}(t)=\frac{e^{\alpha t}-e^{\beta t}}{\alpha-\beta}\quad
  133. α β α≠β
  134. α = β α=β
  135. S t ( z ) = e α t ( 1 + t ( z - α ) ) , S_{t}(z)=e^{\alpha t}(1+t(z-\alpha))~{},
  136. s 0 ( t ) = ( 1 - α t ) e α t , s 1 ( t ) = t e α t . s_{0}(t)=(1-\alpha\,t)\,e^{\alpha t},\quad s_{1}(t)=t\,e^{\alpha t}~{}.
  137. s α + β 2 = tr A 2 , q α - β 2 = ± - det ( A - s I ) , s\equiv\frac{\alpha+\beta}{2}=\frac{\operatorname{tr}A}{2}~{},\qquad\qquad q% \equiv\frac{\alpha-\beta}{2}=\pm\sqrt{-\det\left(A-sI\right)},
  138. s 0 ( t ) = e s t ( cosh ( q t ) - s sinh ( q t ) q ) , s 1 ( t ) = e s t sinh ( q t ) q , s_{0}(t)=e^{st}\left(\cosh(qt)-s\frac{\sinh(qt)}{q}\right),\qquad s_{1}(t)=e^{% st}\frac{\sinh(qt)}{q},
  139. s i n ( q t ) / q sin(qt)/q
  140. t t
  141. t t
  142. q q
  143. A A
  144. A = s I + ( A - s I ) , A=sI+(A-sI)~{},
  145. n n
  146. P P
  147. k k
  148. a a
  149. P P
  150. P P
  151. A A
  152. A A
  153. P P
  154. S t = e a t k = 0 n - 1 t k k ! ( z - a ) k . S_{t}=e^{at}\ \sum_{k=0}^{n-1}\ \frac{t^{k}}{k!}\ (z-a)^{k}~{}.
  155. P = ( z - a ) 2 ( z - b ) P=(z-a)^{2}\,(z-b)
  156. a b a≠b
  157. S t = e a t z - b a - b ( 1 + ( t + 1 b - a ) ( z - a ) ) + e b t ( z - a ) 2 ( b - a ) 2 . S_{t}=e^{at}\ \frac{z-b}{a-b}\ \Bigg(1+\left(t+\frac{1}{b-a}\right)(z-a)\Bigg)% +e^{bt}\ \frac{(z-a)^{2}}{(b-a)^{2}}\quad.
  158. e x p ( t A ) exp(tA)
  159. n n
  160. A A
  161. B B
  162. A A
  163. B B
  164. t t
  165. A A
  166. I I
  167. B B
  168. A = ( 1 1 0 0 0 1 1 0 0 0 1 - 1 / 8 0 0 1 / 2 1 / 2 ) , A=\begin{pmatrix}1&1&0&0\\ 0&1&1&0\\ 0&0&1&-1/8\\ 0&0&1/2&1/2\end{pmatrix}~{},
  169. t t
  170. t t
  171. B i 1 e λ i t , B i 2 t e λ i t , B i 3 t 2 e λ i t B_{i_{1}}e^{\lambda_{i}t},~{}B_{i_{2}}te^{\lambda_{i}t},~{}B_{i_{3}}t^{2}e^{% \lambda_{i}t}
  172. B B
  173. e A t = B 1 1 e λ 1 t + B 1 2 t e λ 1 t + B 2 1 e λ 2 t + B 2 2 t e λ 2 t , e^{At}=B_{1_{1}}e^{\lambda_{1}t}+B_{1_{2}}te^{\lambda_{1}t}+B_{2_{1}}e^{% \lambda_{2}t}+B_{2_{2}}te^{\lambda_{2}t},
  174. e A t = B 1 1 e 3 / 4 t + B 1 2 t e 3 / 4 t + B 2 1 e 1 t + B 2 2 t e 1 t e^{At}=B_{1_{1}}e^{3/4t}+B_{1_{2}}te^{3/4t}+B_{2_{1}}e^{1t}+B_{2_{2}}te^{1t}
  175. B B
  176. A A
  177. t t
  178. t t
  179. A e A t = 3 / 4 B 1 1 e 3 / 4 t + ( 3 / 4 t + 1 ) B 1 2 e 3 / 4 t + 1 B 2 1 e 1 t + ( 1 t + 1 ) B 2 2 e 1 t , Ae^{At}=3/4B_{1_{1}}e^{3/4t}+\left(3/4t+1\right)B_{1_{2}}e^{3/4t}+1B_{2_{1}}e^% {1t}+\left(1t+1\right)B_{2_{2}}e^{1t}~{},
  180. A 2 e A t = ( 3 / 4 ) 2 B 1 1 e 3 / 4 t + ( ( 3 / 4 ) 2 t + ( 3 / 4 + 1 3 / 4 ) ) B 1 2 e 3 / 4 t + B 2 1 e 1 t + ( 1 2 t + ( 1 + 1 1 ) ) B 2 2 e 1 t = ( 3 / 4 ) 2 B 1 1 e 3 / 4 t + ( ( 3 / 4 ) 2 t + 3 / 2 ) B 1 2 e 3 / 4 t + B 2 1 e t + ( t + 2 ) B 2 2 e t , \begin{aligned}\displaystyle A^{2}e^{At}=&\displaystyle(3/4)^{2}B_{1_{1}}e^{3/% 4t}+\left((3/4)^{2}t+(3/4+1\cdot 3/4)\right)B_{1_{2}}e^{3/4t}+B_{2_{1}}e^{1t}% \\ \displaystyle+&\displaystyle\left(1^{2}t+(1+1\cdot 1)\right)B_{2_{2}}e^{1t}\\ \displaystyle=&\displaystyle(3/4)^{2}B_{1_{1}}e^{3/4t}+\left((3/4)^{2}t+3/2% \right)B_{1_{2}}e^{3/4t}+B_{2_{1}}e^{t}+\left(t+2\right)B_{2_{2}}e^{t}~{},\end% {aligned}
  181. A 3 e A t = ( 3 / 4 ) 3 B 1 1 e 3 / 4 t + ( ( 3 / 4 ) 3 t + ( ( 3 / 4 ) 2 + ( 3 / 2 ) 3 / 4 ) ) ) B 1 2 e 3 / 4 t + B 2 1 e 1 t + ( 1 3 t + ( 1 + 2 ) 1 ) B 2 2 e 1 t = ( 3 / 4 ) 3 B 1 1 e 3 / 4 t + ( ( 3 / 4 ) 3 t + 27 / 16 ) ) B 1 2 e 3 / 4 t + B 2 1 e t + ( t + 3 1 ) B 2 2 e t \begin{aligned}\displaystyle A^{3}e^{At}=&\displaystyle(3/4)^{3}B_{1_{1}}e^{3/% 4t}+\left((3/4)^{3}t+((3/4)^{2}+(3/2)\cdot 3/4))\right)B_{1_{2}}e^{3/4t}\\ \displaystyle+&\displaystyle B_{2_{1}}e^{1t}+\left(1^{3}t+(1+2)\cdot 1\right)B% _{2_{2}}e^{1t}\\ \displaystyle=&\displaystyle(3/4)^{3}B_{1_{1}}e^{3/4t}\!+\left((3/4)^{3}t\!+27% /16)\right)B_{1_{2}}e^{3/4t}\!+B_{2_{1}}e^{t}\!+\left(t+3\cdot 1\right)B_{2_{2% }}e^{t}\end{aligned}
  182. n n
  183. t t
  184. B B
  185. I = B 1 1 + B 2 1 A = 3 / 4 B 1 1 + B 1 2 + B 2 1 + B 2 2 A 2 = ( 3 / 4 ) 2 B 1 1 + ( 3 / 2 ) B 1 2 + B 2 1 + 2 B 2 2 A 3 = ( 3 / 4 ) 3 B 1 1 + ( 27 / 16 ) B 1 2 + B 2 1 + 3 B 2 2 \begin{aligned}\displaystyle I=&\displaystyle B_{1_{1}}+B_{2_{1}}\\ \displaystyle A=&\displaystyle 3/4B_{1_{1}}+B_{1_{2}}+B_{2_{1}}+B_{2_{2}}\\ \displaystyle A^{2}=&\displaystyle(3/4)^{2}B_{1_{1}}+(3/2)B_{1_{2}}+B_{2_{1}}+% 2B_{2_{2}}\\ \displaystyle A^{3}=&\displaystyle(3/4)^{3}B_{1_{1}}+(27/16)B_{1_{2}}+B_{2_{1}% }+3B_{2_{2}}\end{aligned}
  186. B 1 1 = 128 A 3 - 366 A 2 + 288 A - 80 I B 1 2 = 16 A 3 - 44 A 2 + 40 A - 12 I B 2 1 = - 128 A 3 + 366 A 2 - 288 A + 80 I B 2 2 = 16 A 3 - 40 A 2 + 33 A - 9 I \begin{aligned}\displaystyle B_{1_{1}}=&\displaystyle 128A^{3}-366A^{2}+288A-8% 0I\\ \displaystyle B_{1_{2}}=&\displaystyle 16A^{3}-44A^{2}+40A-12I\\ \displaystyle B_{2_{1}}=&\displaystyle-128A^{3}+366A^{2}-288A+80I\\ \displaystyle B_{2_{2}}=&\displaystyle 16A^{3}-40A^{2}+33A-9I\end{aligned}
  187. A A
  188. B 1 1 = ( 0 0 48 - 16 0 0 - 8 2 0 0 1 0 0 0 0 1 ) B 1 2 = ( 0 0 4 - 2 0 0 - 1 1 / 2 0 0 1 / 4 - 1 / 8 0 0 1 / 2 - 1 / 4 ) B 2 1 = ( 1 0 - 48 16 0 1 8 - 2 0 0 0 0 0 0 0 0 ) B 2 2 = ( 0 1 8 - 2 0 0 0 0 0 0 0 0 0 0 0 0 ) \begin{aligned}\displaystyle B_{1_{1}}=&\displaystyle\begin{pmatrix}0&0&48&-16% \\ 0&0&-8&2\\ 0&0&1&0\\ 0&0&0&1\end{pmatrix}\\ \displaystyle B_{1_{2}}=&\displaystyle\begin{pmatrix}0&0&4&-2\\ 0&0&-1&1/2\\ 0&0&1/4&-1/8\\ 0&0&1/2&-1/4\end{pmatrix}\\ \displaystyle B_{2_{1}}=&\displaystyle\begin{pmatrix}1&0&-48&16\\ 0&1&8&-2\\ 0&0&0&0\\ 0&0&0&0\end{pmatrix}\\ \displaystyle B_{2_{2}}=&\displaystyle\begin{pmatrix}0&1&8&-2\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{pmatrix}\end{aligned}
  189. e t A = ( e t t e t ( 8 t - 48 ) e t + ( 4 t + 48 ) e 3 t / 4 ( 16 - 2 t ) e t + ( - 2 t - 16 ) e 3 t / 4 0 e t 8 e t + ( - t - 8 ) e 3 t / 4 - 4 e t + ( - t - 4 ) e 3 t / 4 2 0 0 ( t + 4 ) e 3 t / 4 4 - t e 3 t / 4 8 0 0 t e 3 t / 4 2 - ( t - 4 ) e 3 t / 4 4 ) {e}^{tA}\!=\!\begin{pmatrix}{e}^{t}&t{e}^{t}&\left(8t-48\right){e}^{t}\!+\left% (4t+48\right){e}^{3t/4}&\left(16-2\,t\right){e}^{t}\!+\left(-2t-16\right){e}^{% 3t/4}\\ 0&{e}^{t}&8{e}^{t}\!+\left(-t-8\right){e}^{3t/4}&-\frac{4{e}^{t}+\left(-t-4% \right){e}^{3t/4}}{2}\\ 0&0&\frac{\left(t+4\right){e}^{3t/4}}{4}&-\frac{t{e}^{3t/4}}{8}\\ 0&0&\frac{t{e}^{3t/4}}{2}&-\frac{\left(t-4\right){e}^{3t/4}}{4}\end{pmatrix}
  190. B = [ 21 17 6 - 5 - 1 - 6 4 4 16 ] . B=\begin{bmatrix}21&17&6\\ -5&-1&-6\\ 4&4&16\end{bmatrix}.
  191. J = P - 1 B P = [ 4 0 0 0 16 1 0 0 16 ] , J=P^{-1}BP=\begin{bmatrix}4&0&0\\ 0&16&1\\ 0&0&16\end{bmatrix},
  192. P = [ - 1 4 2 5 4 1 4 - 2 - 1 4 0 4 0 ] . P=\begin{bmatrix}-\frac{1}{4}&2&\frac{5}{4}\\ \frac{1}{4}&-2&-\frac{1}{4}\\ 0&4&0\end{bmatrix}.
  193. J = J 1 ( 4 ) J 2 ( 16 ) J=J_{1}(4)\oplus J_{2}(16)\,
  194. exp ( [ 16 1 0 16 ] ) \displaystyle\exp\left(\begin{bmatrix}16&1\\ 0&16\end{bmatrix}\right)
  195. exp ( B ) \displaystyle\exp(B)
  196. 𝐲 = A 𝐲 \mathbf{y}^{\prime}=A\mathbf{y}
  197. 𝐲 ( t ) = ( y 1 ( t ) y n ( t ) ) , \mathbf{y}(t)=\begin{pmatrix}y_{1}(t)\\ \vdots\\ y_{n}(t)\end{pmatrix}~{},
  198. 𝐲 ( t ) = A 𝐲 ( t ) + 𝐛 ( t ) . \mathbf{y}^{\prime}(t)=A\mathbf{y}(t)+\mathbf{b}(t).
  199. e - A t 𝐲 - e - A t A 𝐲 = e - A t 𝐛 e^{-At}\mathbf{y}^{\prime}-e^{-At}A\mathbf{y}=e^{-At}\mathbf{b}
  200. e - A t 𝐲 - A e - A t 𝐲 = e - A t 𝐛 e^{-At}\mathbf{y}^{\prime}-Ae^{-At}\mathbf{y}=e^{-At}\mathbf{b}
  201. d d t ( e - A t 𝐲 ) = e - A t 𝐛 . \frac{d}{dt}(e^{-At}\mathbf{y})=e^{-At}\mathbf{b}~{}.
  202. A B = B A AB=BA
  203. t t
  204. x = 2 x - y + z y = 3 y - 1 z z = 2 x + y + 3 z . \begin{matrix}x^{\prime}&=&2x&-y&+z\\ y^{\prime}&=&&3y&-1z\\ z^{\prime}&=&2x&+y&+3z~{}.\end{matrix}
  205. A = [ 2 - 1 1 0 3 - 1 2 1 3 ] . A=\begin{bmatrix}2&-1&1\\ 0&3&-1\\ 2&1&3\end{bmatrix}~{}.
  206. e t A = 1 2 [ e 2 t ( 1 + e 2 t - 2 t ) - 2 t e 2 t e 2 t ( - 1 + e 2 t ) - e 2 t ( - 1 + e 2 t - 2 t ) 2 ( t + 1 ) e 2 t - e 2 t ( - 1 + e 2 t ) e 2 t ( - 1 + e 2 t + 2 t ) 2 t e 2 t e 2 t ( 1 + e 2 t ) ] , e^{tA}=\frac{1}{2}\begin{bmatrix}e^{2t}(1+e^{2t}-2t)&-2te^{2t}&e^{2t}(-1+e^{2t% })\\ -e^{2t}(-1+e^{2t}-2t)&2(t+1)e^{2t}&-e^{2t}(-1+e^{2t})\\ e^{2t}(-1+e^{2t}+2t)&2te^{2t}&e^{2t}(1+e^{2t})\end{bmatrix}~{},
  207. [ x y z ] = x ( 0 ) 2 [ e 2 t ( 1 + e 2 t - 2 t ) - e 2 t ( - 1 + e 2 t - 2 t ) e 2 t ( - 1 + e 2 t + 2 t ) ] + y ( 0 ) 2 [ - 2 t e 2 t 2 ( t + 1 ) e 2 t 2 t e 2 t ] + z ( 0 ) 2 [ e 2 t ( - 1 + e 2 t ) - e 2 t ( - 1 + e 2 t ) e 2 t ( 1 + e 2 t ) ] , \begin{bmatrix}x\\ y\\ z\end{bmatrix}=\frac{x(0)}{2}\begin{bmatrix}e^{2t}(1+e^{2t}-2t)\\ -e^{2t}(-1+e^{2t}-2t)\\ e^{2t}(-1+e^{2t}+2t)\end{bmatrix}+\frac{y(0)}{2}\begin{bmatrix}-2te^{2t}\\ 2(t+1)e^{2t}\\ 2te^{2t}\end{bmatrix}+\frac{z(0)}{2}\begin{bmatrix}e^{2t}(-1+e^{2t})\\ -e^{2t}(-1+e^{2t})\\ e^{2t}(1+e^{2t})\end{bmatrix}~{},
  208. 2 x = x ( 0 ) ( e 2 t ( 1 + e 2 t - 2 t ) ) + y ( 0 ) ( - 2 t e 2 t ) + z ( 0 ) ( e 2 t ( - 1 + e 2 t ) ) 2 y = x ( 0 ) ( - e 2 t ( - 1 + e 2 t - 2 t ) ) + y ( 0 ) ( 2 ( t + 1 ) e 2 t ) + z ( 0 ) ( - e 2 t ( - 1 + e 2 t ) ) 2 z = x ( 0 ) ( e 2 t ( - 1 + e 2 t + 2 t ) ) + y ( 0 ) ( 2 t e 2 t ) + z ( 0 ) ( e 2 t ( 1 + e 2 t ) ) . \begin{aligned}\displaystyle 2x&\displaystyle=x(0)(e^{2t}(1+e^{2t}-2t))+y(0)(-% 2te^{2t})+z(0)(e^{2t}(-1+e^{2t}))\\ \displaystyle 2y&\displaystyle=x(0)(-e^{2t}(-1+e^{2t}-2t))+y(0)(2(t+1)e^{2t})+% z(0)(-e^{2t}(-1+e^{2t}))\\ \displaystyle 2z&\displaystyle=x(0)(e^{2t}(-1+e^{2t}+2t))+y(0)(2te^{2t})+z(0)(% e^{2t}(1+e^{2t}))~{}.\end{aligned}
  209. x = 2 x - y + z + e 2 t y = 3 y - z z = 2 x + y + 3 z + e 2 t . \begin{matrix}x^{\prime}&=&2x&-&y&+&z&+&e^{2t}\\ y^{\prime}&=&&&3y&-&z&\\ z^{\prime}&=&2x&+&y&+&3z&+&e^{2t}\end{matrix}~{}.
  210. A = [ 2 - 1 1 0 3 - 1 2 1 3 ] , A=\left[\begin{array}[]{rrr}2&-1&1\\ 0&3&-1\\ 2&1&3\end{array}\right]~{},
  211. 𝐛 = e 2 t [ 1 0 1 ] . \mathbf{b}=e^{2t}\begin{bmatrix}1\\ 0\\ 1\end{bmatrix}.
  212. 𝐲 p = e t A 0 t e ( - u ) A [ e 2 u 0 e 2 u ] d u + e t A 𝐜 \mathbf{y}_{p}=e^{tA}\int_{0}^{t}e^{(-u)A}\begin{bmatrix}e^{2u}\\ 0\\ e^{2u}\end{bmatrix}\,du+e^{tA}\mathbf{c}
  213. 𝐲 p = e t A 0 t [ 2 e u - 2 u e 2 u - 2 u e 2 u 0 - 2 e u + 2 ( u + 1 ) e 2 u 2 ( u + 1 ) e 2 u 0 2 u e 2 u 2 u e 2 u 2 e u ] [ e 2 u 0 e 2 u ] d u + e t A 𝐜 \mathbf{y}_{p}=e^{tA}\int_{0}^{t}\begin{bmatrix}2e^{u}-2ue^{2u}&-2ue^{2u}&0\\ \\ -2e^{u}+2(u+1)e^{2u}&2(u+1)e^{2u}&0\\ \\ 2ue^{2u}&2ue^{2u}&2e^{u}\end{bmatrix}\begin{bmatrix}e^{2u}\\ 0\\ e^{2u}\end{bmatrix}\,du+e^{tA}\mathbf{c}
  214. 𝐲 p = e t A 0 t [ e 2 u ( 2 e u - 2 u e 2 u ) e 2 u ( - 2 e u + 2 ( 1 + u ) e 2 u ) 2 e 3 u + 2 u e 4 u ] d u + e t A 𝐜 \mathbf{y}_{p}=e^{tA}\int_{0}^{t}\begin{bmatrix}e^{2u}(2e^{u}-2ue^{2u})\\ \\ e^{2u}(-2e^{u}+2(1+u)e^{2u})\\ \\ 2e^{3u}+2ue^{4u}\end{bmatrix}\,du+e^{tA}\mathbf{c}
  215. 𝐲 p = e t A [ - 1 24 e 3 t ( 3 e t ( 4 t - 1 ) - 16 ) 1 24 e 3 t ( 3 e t ( 4 t + 4 ) - 16 ) 1 24 e 3 t ( 3 e t ( 4 t - 1 ) - 16 ) ] + [ 2 e t - 2 t e 2 t - 2 t e 2 t 0 - 2 e t + 2 ( t + 1 ) e 2 t 2 ( t + 1 ) e 2 t 0 2 t e 2 t 2 t e 2 t 2 e t ] [ c 1 c 2 c 3 ] , \mathbf{y}_{p}=e^{tA}\begin{bmatrix}-{1\over 24}e^{3t}(3e^{t}(4t-1)-16)\\ \\ {1\over 24}e^{3t}(3e^{t}(4t+4)-16)\\ \\ {1\over 24}e^{3t}(3e^{t}(4t-1)-16)\end{bmatrix}+\begin{bmatrix}2e^{t}-2te^{2t}% &-2te^{2t}&0\\ \\ -2e^{t}+2(t+1)e^{2t}&2(t+1)e^{2t}&0\\ \\ 2te^{2t}&2te^{2t}&2e^{t}\end{bmatrix}\begin{bmatrix}c_{1}\\ c_{2}\\ c_{3}\end{bmatrix}~{},
  216. 𝐲 p ( t ) \displaystyle\mathbf{y}_{p}^{\prime}(t)
  217. e t A 𝐳 ( t ) \displaystyle e^{tA}\mathbf{z}^{\prime}(t)
  218. 𝐲 p ( t ) = e t A 0 t e - u A 𝐛 ( u ) d u + e t A 𝐜 = 0 t e ( t - u ) A 𝐛 ( u ) d u + e t A 𝐜 , \begin{aligned}\displaystyle\mathbf{y}_{p}(t)&\displaystyle{}=e^{tA}\int_{0}^{% t}e^{-uA}\mathbf{b}(u)\,du+e^{tA}\mathbf{c}\\ &\displaystyle{}=\int_{0}^{t}e^{(t-u)A}\mathbf{b}(u)\,du+e^{tA}\mathbf{c}\end{% aligned}~{},
  219. 𝐜 \mathbf{c}
  220. Y - A Y = F ( t ) Y^{\prime}-A\ Y=F(t)
  221. A A
  222. n n
  223. n n
  224. F F
  225. I I
  226. t 0 t_{0}
  227. I I
  228. Y 0 Y_{0}
  229. Y ( t ) = e ( t - t 0 ) A Y 0 + t 0 t e ( t - x ) A F ( x ) d x . Y(t)=e^{(t-t_{0})A}\ Y_{0}+\int_{t_{0}}^{t}e^{(t-x)A}\ F(x)\ dx~{}.
  230. P ( d / d t ) y = f ( t ) P(d/dt)\ y=f(t)
  231. y ( k ) ( t 0 ) = y k y^{(k)}(t_{0})=y_{k}
  232. y ( t ) = k = 0 n - 1 y k s k ( t - t 0 ) + t 0 t s n - 1 ( t - x ) f ( x ) d x , y(t)=\sum_{k=0}^{n-1}\ y_{k}\ s_{k}(t-t_{0})+\int_{t_{0}}^{t}s_{n-1}(t-x)\ f(x% )\ dx~{},
  233. P [ X ] P\in\mathbb{C}[X]
  234. n > 0 n>0
  235. f f
  236. I I
  237. t 0 t_{0}
  238. I I
  239. y k y_{k}
  240. X k X^{k}
  241. S t [ X ] S_{t}\in\mathbb{C}[X]
  242. n n
  243. d Y d t - A Y = F ( t ) , Y ( t 0 ) = Y 0 , \frac{dY}{dt}-A\ Y=F(t),\quad Y(t_{0})=Y_{0},
  244. A A
  245. P P
  246. n n
  247. y ′′ - ( α + β ) y + α β y = f ( t ) , y ( t 0 ) = y 0 , y ( t 0 ) = y 1 y^{\prime\prime}-(\alpha+\beta)\ y^{\prime}+\alpha\,\beta\ y=f(t),\quad y(t_{0% })=y_{0},\quad y^{\prime}(t_{0})=y_{1}
  248. y ( t ) = y 0 s 0 ( t - t 0 ) + y 1 s 1 ( t - t 0 ) + t 0 t s 1 ( t - x ) f ( x ) d x , y(t)=y_{0}\ s_{0}(t-t_{0})+y_{1}\ s_{1}(t-t_{0})+\int_{t_{0}}^{t}s_{1}(t-x)\,f% (x)\ dx,
  249. s < s u b > 0 s<sub>0

Matrix_of_ones.html

  1. J 2 = ( 1 1 1 1 ) ; J 3 = ( 1 1 1 1 1 1 1 1 1 ) ; J 2 , 5 = ( 1 1 1 1 1 1 1 1 1 1 ) . J_{2}=\begin{pmatrix}1&1\\ 1&1\end{pmatrix};\quad J_{3}=\begin{pmatrix}1&1&1\\ 1&1&1\\ 1&1&1\end{pmatrix};\quad J_{2,5}=\begin{pmatrix}1&1&1&1&1\\ 1&1&1&1&1\end{pmatrix}.\quad
  2. J k = n k - 1 J , for k = 1 , 2 , . J^{k}=n^{k-1}J,\mbox{ for }~{}k=1,2,\ldots.\,
  3. 1 n J \tfrac{1}{n}J
  4. exp ( J ) = I + e n - 1 n J , \exp(J)=I+\frac{e^{n}-1}{n}J,

Maurer–Cartan_form.html

  1. G G
  2. G G
  3. G G
  4. G G
  5. G G
  6. ω ω
  7. G G
  8. g G g∈G
  9. ω ( v ) = ( L g - 1 ) * v , v T g G . \omega(v)=(L_{g^{-1}})_{*}v,\quad v\in T_{g}G.
  10. G × G ( g , h ) g h G . G\times G\ni(g,h)\mapsto gh\in G.
  11. G G
  12. P P
  13. G G
  14. G / H G/H
  15. e H eH
  16. G G
  17. P P
  18. G G
  19. P P
  20. P P
  21. P P
  22. G G
  23. G G
  24. L : G × G G L:G\times G\to G
  25. g G g∈G
  26. L g : G G where L g ( h ) = g h , L_{g}:G\to G\quad\mbox{where}~{}\quad L_{g}(h)=gh,
  27. ( L g ) * : T h G T g h G . (L_{g})_{*}:T_{h}G\to T_{gh}G.
  28. X X
  29. T G TG
  30. ( L g ) * X = X g G . (L_{g})_{*}X=X\quad\forall\quad g\in G.
  31. ω ω
  32. 𝐠 \mathbf{g}
  33. G G
  34. ω g ( v ) = ( L g - 1 ) * v . \omega_{g}(v)=(L_{g^{-1}})_{*}v.
  35. G G
  36. G L ( n ) GL(n)
  37. ω ω
  38. ω g = g - 1 d g . \omega_{g}=g^{-1}\,dg.
  39. G G
  40. G G
  41. G G
  42. 1 1
  43. G G
  44. ω e = id : T e G 𝔤 , and \omega_{e}=\mathrm{id}:T_{e}G\rightarrow{\mathfrak{g}},\,\text{ and}
  45. g G ω g = Ad ( h ) ( R h * ω e ) , where h = g - 1 , \forall g\in G\quad\omega_{g}=\mathrm{Ad}(h)(R_{h}^{*}\omega_{e}),\,\text{ % where }h=g^{-1},
  46. A d ( h ) Ad(h)
  47. X X
  48. G G
  49. ω ( X ) ω(X)
  50. G G
  51. X X
  52. Y Y
  53. ω ( [ X , Y ] ) = [ ω ( X ) , ω ( Y ) ] \omega([X,Y])=[\omega(X),\omega(Y)]
  54. 𝐠 \mathbf{g}
  55. g g
  56. 𝔤 = T e G { left-invariant vector fields on G } . \mathfrak{g}=T_{e}G\cong\{\hbox{left-invariant vector fields on G}\}.
  57. X X
  58. Y Y
  59. d ω ( X , Y ) = X ( ω ( Y ) ) - Y ( ω ( X ) ) - ω ( [ X , Y ] ) . d\omega(X,Y)=X(\omega(Y))-Y(\omega(X))-\omega([X,Y]).
  60. ω ( Y ) ω(Y)
  61. 𝐠 \mathbf{g}
  62. ω ω
  63. Y Y
  64. X ( ω ( Y ) ) X(ω(Y))
  65. X X
  66. Y ( ω ( X ) ) Y(ω(X))
  67. Y Y
  68. 𝐠 \mathbf{g}
  69. ω ( X ) ω(X)
  70. X X
  71. Y Y
  72. X ( ω ( Y ) ) = Y ( ω ( X ) ) = 0 , X(\omega(Y))=Y(\omega(X))=0,
  73. d ω ( X , Y ) + [ ω ( X ) , ω ( Y ) ] = 0 d\omega(X,Y)+[\omega(X),\omega(Y)]=0
  74. 2 2
  75. X X
  76. Y Y
  77. X X
  78. Y Y
  79. d ω + 1 2 [ ω , ω ] = 0. d\omega+\frac{1}{2}[\omega,\omega]=0.
  80. ω , ω ω ω,ωω
  81. T G TG
  82. [ E i , E j ] = k c i j k E k . [E_{i},E_{j}]=\sum_{k}{c_{ij}}^{k}E_{k}.
  83. d d
  84. d θ i ( E j , E k ) = - θ i ( [ E j , E k ] ) = - r c j k r θ i ( E r ) = - c j k i = - 1 2 ( c j k i - c k j i ) , d\theta^{i}(E_{j},E_{k})=-\theta^{i}([E_{j},E_{k}])=-\sum_{r}{c_{jk}}^{r}% \theta^{i}(E_{r})=-{c_{jk}}^{i}=-\frac{1}{2}({c_{jk}}^{i}-{c_{kj}}^{i}),
  85. d θ i = - 1 2 j k c j k i θ j θ k . d\theta^{i}=-\frac{1}{2}\sum_{jk}{c_{jk}}^{i}\theta^{j}\wedge\theta^{k}.
  86. ω ω
  87. d ω = i E i ( e ) d θ i = - 1 2 i j k c j k i E i ( e ) θ j θ k . d\omega=\sum_{i}E_{i}(e)\otimes d\theta^{i}\,=\,-\frac{1}{2}\sum_{ijk}{c_{jk}}% ^{i}E_{i}(e)\otimes\theta^{j}\wedge\theta^{k}.
  88. d ω ( E j , E k ) = - i c j k i E i ( e ) = - [ E j ( e ) , E k ( e ) ] = - [ ω ( E j ) , ω ( E k ) ] , d\omega(E_{j},E_{k})=-\sum_{i}{c_{jk}}^{i}E_{i}(e)=-[E_{j}(e),E_{k}(e)]=-[% \omega(E_{j}),\omega(E_{k})],
  89. H H
  90. G G
  91. G / H G/H
  92. d i m G d i m H dimG−dimH
  93. G G / H G→G/H
  94. H H
  95. G / H G/H
  96. G G
  97. d ω + ω ω = 0 d\omega+\omega\wedge\omega=0
  98. s : G / H G s:G/H→G
  99. s s
  100. s s
  101. 𝐠 \mathbf{g}
  102. 1 1
  103. d θ + 1 2 [ θ , θ ] = 0. d\theta+\frac{1}{2}[\theta,\theta]=0.
  104. U U
  105. V V
  106. H H
  107. h U V ( x ) = s V s U - 1 ( x ) , x U V . h_{UV}(x)=s_{V}\circ s_{U}^{-1}(x),\quad x\in U\cap V.
  108. h h
  109. θ V = Ad ( h U V - 1 ) θ U + ( h U V ) * ω H \theta_{V}=\operatorname{Ad}(h^{-1}_{UV})\theta_{U}+(h_{UV})^{*}\omega_{H}
  110. ω < s u b > H ω<sub>H
  111. ( L g ) * X (L_{g})_{*}X
  112. T g h G if X T h G T_{gh}G\,\text{ if }X\in T_{h}G

Maximum_modulus_principle.html

  1. | f | |f|
  2. \mathbb{C}
  3. | f ( z 0 ) | | f ( z ) | |f(z_{0})|\geq|f(z)|

Maximum_operating_depth.html

  1. M O D ( f s w ) = 33 feet × [ ( p O 2 F O 2 ) - 1 ] MOD(fsw)=33\mathrm{~{}feet}\times\left[\left({pO_{2}\over FO_{2}}\right)-1\right]
  2. M O D ( m ) = 10 metres × [ ( p O 2 F O 2 ) - 1 ] MOD(m)=10\mathrm{~{}metres}\times\left[\left({pO_{2}\over FO_{2}}\right)-1\right]

Maxwell_relations.html

  1. d U = T d S - P d V \begin{aligned}\displaystyle dU&\displaystyle=&\displaystyle TdS-PdV\\ \end{aligned}\,\!
  2. d z = ( z x ) y d x + ( z y ) x d y dz=\left(\frac{\partial z}{\partial x}\right)_{y}\!dx+\left(\frac{\partial z}{% \partial y}\right)_{x}\!dy
  3. d z = M d x + N d y dz=Mdx+Ndy\,
  4. M = ( z x ) y , N = ( z y ) x M=\left(\frac{\partial z}{\partial x}\right)_{y},\quad N=\left(\frac{\partial z% }{\partial y}\right)_{x}
  5. d U = T d S - P d V dU=TdS-PdV\,
  6. T = ( U S ) V , - P = ( U V ) S T=\left(\frac{\partial U}{\partial S}\right)_{V},\quad-P=\left(\frac{\partial U% }{\partial V}\right)_{S}
  7. y ( z x ) y = x ( z y ) x = 2 z y x = 2 z x y \frac{\partial}{\partial y}\left(\frac{\partial z}{\partial x}\right)_{y}=% \frac{\partial}{\partial x}\left(\frac{\partial z}{\partial y}\right)_{x}=% \frac{\partial^{2}z}{\partial y\partial x}=\frac{\partial^{2}z}{\partial x% \partial y}
  8. V ( U S ) V = S ( U V ) S \frac{\partial}{\partial V}\left(\frac{\partial U}{\partial S}\right)_{V}=% \frac{\partial}{\partial S}\left(\frac{\partial U}{\partial V}\right)_{S}
  9. ( T V ) S = - ( P S ) V \left(\frac{\partial T}{\partial V}\right)_{S}=-\left(\frac{\partial P}{% \partial S}\right)_{V}
  10. d F = - S d T - P d V \begin{aligned}\displaystyle dF&\displaystyle=&\displaystyle-SdT-PdV\\ \end{aligned}\,\!
  11. - S = ( F T ) V , - P = ( F V ) T -S=\left(\frac{\partial F}{\partial T}\right)_{V},\quad-P=\left(\frac{\partial F% }{\partial V}\right)_{T}
  12. V ( F T ) V = T ( F V ) T \frac{\partial}{\partial V}\left(\frac{\partial F}{\partial T}\right)_{V}=% \frac{\partial}{\partial T}\left(\frac{\partial F}{\partial V}\right)_{T}
  13. ( S V ) T = ( P T ) V \left(\frac{\partial S}{\partial V}\right)_{T}=\left(\frac{\partial P}{% \partial T}\right)_{V}
  14. d H = T d S + V d P \begin{aligned}\displaystyle dH&\displaystyle=&\displaystyle TdS+VdP\\ \end{aligned}\,\!
  15. d G = V d P - S d T \begin{aligned}\displaystyle dG&\displaystyle=&\displaystyle VdP-SdT\\ \end{aligned}\,\!
  16. T d S = d U + P d V TdS=dU+PdV
  17. U = U ( x , y ) U=U(x,y)
  18. S = S ( x , y ) S=S(x,y)
  19. V = V ( x , y ) V=V(x,y)
  20. d U = ( U x ) y d x + ( U y ) x d y dU=\left(\frac{\partial U}{\partial x}\right)_{y}\!dx+\left(\frac{\partial U}{% \partial y}\right)_{x}\!dy
  21. d S = ( S x ) y d x + ( S y ) x d y dS=\left(\frac{\partial S}{\partial x}\right)_{y}\!dx+\left(\frac{\partial S}{% \partial y}\right)_{x}\!dy
  22. d V = ( V x ) y d x + ( V y ) x d y dV=\left(\frac{\partial V}{\partial x}\right)_{y}\!dx+\left(\frac{\partial V}{% \partial y}\right)_{x}\!dy
  23. T ( S x ) y d x + T ( S y ) x d y = ( U x ) y d x + ( U y ) x d y + P ( V x ) y d x + P ( V y ) x d y T\left(\frac{\partial S}{\partial x}\right)_{y}\!dx+T\left(\frac{\partial S}{% \partial y}\right)_{x}\!dy=\left(\frac{\partial U}{\partial x}\right)_{y}\!dx+% \left(\frac{\partial U}{\partial y}\right)_{x}\!dy+P\left(\frac{\partial V}{% \partial x}\right)_{y}\!dx+P\left(\frac{\partial V}{\partial y}\right)_{x}\!dy
  24. ( U x ) y d x + ( U y ) x d y = T ( S x ) y d x + T ( S y ) x d y - P ( V x ) y d x - P ( V y ) x d y \left(\frac{\partial U}{\partial x}\right)_{y}\!dx+\left(\frac{\partial U}{% \partial y}\right)_{x}\!dy=T\left(\frac{\partial S}{\partial x}\right)_{y}\!dx% +T\left(\frac{\partial S}{\partial y}\right)_{x}\!dy-P\left(\frac{\partial V}{% \partial x}\right)_{y}\!dx-P\left(\frac{\partial V}{\partial y}\right)_{x}\!dy
  25. ( U x ) y = T ( S x ) y - P ( V x ) y \left(\frac{\partial U}{\partial x}\right)_{y}=T\left(\frac{\partial S}{% \partial x}\right)_{y}-P\left(\frac{\partial V}{\partial x}\right)_{y}
  26. ( U y ) x = T ( S y ) x - P ( V y ) x \left(\frac{\partial U}{\partial y}\right)_{x}=T\left(\frac{\partial S}{% \partial y}\right)_{x}-P\left(\frac{\partial V}{\partial y}\right)_{x}
  27. ( 2 U y x ) = ( T y ) x ( S x ) y + T ( 2 S y x ) - ( P y ) x ( V x ) y - P ( 2 V y x ) \left(\frac{\partial^{2}U}{\partial y\partial x}\right)=\left(\frac{\partial T% }{\partial y}\right)_{x}\left(\frac{\partial S}{\partial x}\right)_{y}+T\left(% \frac{\partial^{2}S}{\partial y\partial x}\right)-\left(\frac{\partial P}{% \partial y}\right)_{x}\left(\frac{\partial V}{\partial x}\right)_{y}-P\left(% \frac{\partial^{2}V}{\partial y\partial x}\right)
  28. ( 2 U x y ) = ( T x ) y ( S y ) x + T ( 2 S x y ) - ( P x ) y ( V y ) x - P ( 2 V x y ) \left(\frac{\partial^{2}U}{\partial x\partial y}\right)=\left(\frac{\partial T% }{\partial x}\right)_{y}\left(\frac{\partial S}{\partial y}\right)_{x}+T\left(% \frac{\partial^{2}S}{\partial x\partial y}\right)-\left(\frac{\partial P}{% \partial x}\right)_{y}\left(\frac{\partial V}{\partial y}\right)_{x}-P\left(% \frac{\partial^{2}V}{\partial x\partial y}\right)
  29. ( 2 U y x ) = ( 2 U x y ) \left(\frac{\partial^{2}U}{\partial y\partial x}\right)=\left(\frac{\partial^{% 2}U}{\partial x\partial y}\right)
  30. ( 2 S y x ) = ( 2 S x y ) : ( 2 V y x ) = ( 2 V x y ) \left(\frac{\partial^{2}S}{\partial y\partial x}\right)=\left(\frac{\partial^{% 2}S}{\partial x\partial y}\right):\left(\frac{\partial^{2}V}{\partial y% \partial x}\right)=\left(\frac{\partial^{2}V}{\partial x\partial y}\right)
  31. ( T y ) x ( S x ) y - ( P y ) x ( V x ) y = ( T x ) y ( S y ) x - ( P x ) y ( V y ) x \left(\frac{\partial T}{\partial y}\right)_{x}\left(\frac{\partial S}{\partial x% }\right)_{y}-\left(\frac{\partial P}{\partial y}\right)_{x}\left(\frac{% \partial V}{\partial x}\right)_{y}=\left(\frac{\partial T}{\partial x}\right)_% {y}\left(\frac{\partial S}{\partial y}\right)_{x}-\left(\frac{\partial P}{% \partial x}\right)_{y}\left(\frac{\partial V}{\partial y}\right)_{x}
  32. ( T V ) S = - ( P S ) V \left(\frac{\partial T}{\partial V}\right)_{S}=-\left(\frac{\partial P}{% \partial S}\right)_{V}
  33. ( S V ) T = ( P T ) V \left(\frac{\partial S}{\partial V}\right)_{T}=\left(\frac{\partial P}{% \partial T}\right)_{V}
  34. ( T P ) S = ( V S ) P \left(\frac{\partial T}{\partial P}\right)_{S}=\left(\frac{\partial V}{% \partial S}\right)_{P}
  35. ( S P ) T = - ( V T ) P \left(\frac{\partial S}{\partial P}\right)_{T}=-\left(\frac{\partial V}{% \partial T}\right)_{P}
  36. ( T P ) V ( S V ) P \left(\frac{\partial T}{\partial P}\right)_{V}\left(\frac{\partial S}{\partial V% }\right)_{P}
  37. - ( T V ) P ( S P ) V -\left(\frac{\partial T}{\partial V}\right)_{P}\left(\frac{\partial S}{% \partial P}\right)_{V}
  38. ( P T ) S ( V S ) T - ( P S ) T ( V T ) S \left(\frac{\partial P}{\partial T}\right)_{S}\left(\frac{\partial V}{\partial S% }\right)_{T}-\left(\frac{\partial P}{\partial S}\right)_{T}\left(\frac{% \partial V}{\partial T}\right)_{S}
  39. ( μ P ) S , N = ( V N ) S , P = 2 H P N \left(\frac{\partial\mu}{\partial P}\right)_{S,N}=\left(\frac{\partial V}{% \partial N}\right)_{S,P}\qquad=\frac{\partial^{2}H}{\partial P\partial N}
  40. ( y x ) z = 1 / ( x y ) z \left(\frac{\partial y}{\partial x}\right)_{z}=1\left/\left(\frac{\partial x}{% \partial y}\right)_{z}\right.

Mayer_f-function.html

  1. V ( 𝐢 , 𝐣 ) V(\mathbf{i},\mathbf{j})
  2. 𝐢 \mathbf{i}
  3. 𝐣 \mathbf{j}
  4. 𝐢 = 𝐫 i \mathbf{i}=\mathbf{r}_{i}
  5. 𝐢 = ( 𝐫 i , Ω i ) \mathbf{i}=(\mathbf{r}_{i},\Omega_{i})
  6. 𝐫 \mathbf{r}
  7. Ω \Omega
  8. f ( 𝐢 , 𝐣 ) = e - β V ( 𝐢 , 𝐣 ) - 1 f(\mathbf{i},\mathbf{j})=e^{-\beta V(\mathbf{i},\mathbf{j})}-1
  9. β = ( k B T ) - 1 \beta=(k_{B}T)^{-1}
  10. k B k_{B}

McDonald's_Monopoly.html

  1. 0 19 $ 50 , 000 = $ 1 , 000 , 000 \sum_{0}^{19}\$50,000=\$1,000,000
  2. $ 50 , 000 ( 1 1 + I ) n \$50,000{\left(\frac{1}{1+I}\right)}^{n}
  3. n = 1 20 $ 50 , 000 ( 1 1 + 0.02 ) n - 1 = 1 - ( 1 1 + 0.02 ) 20 1 - ( 1 1 + 0.02 ) = $ 833 , 923.10 \sum_{n=1}^{20}\$50,000{\left(\frac{1}{1+0.02}\right)}^{n-1}=\frac{1-{\left(% \frac{1}{1+0.02}\right)}^{20}}{1-\left(\frac{1}{1+0.02}\right)}=\$833,923.10

McEliece_cryptosystem.html

  1. t t
  2. G G
  3. S S
  4. P P
  5. n , k , t n,k,t
  6. ( n , k ) (n,k)
  7. C C
  8. t t
  9. k × n k\times n
  10. G G
  11. C C
  12. k × k k\times k
  13. S S
  14. n × n n\times n
  15. P P
  16. k × n k\times n
  17. G ^ = S G P {\hat{G}}=SGP
  18. ( G ^ , t ) ({\hat{G}},t)
  19. ( S , G , P ) (S,G,P)
  20. ( G ^ , t ) ({\hat{G}},t)
  21. m m
  22. k k
  23. c = m G ^ c^{\prime}=m{\hat{G}}
  24. n n
  25. z z
  26. t t
  27. n n
  28. t t
  29. c = c + z c=c^{\prime}+z
  30. c c
  31. P P
  32. P - 1 P^{-1}
  33. c ^ = c P - 1 {\hat{c}}=cP^{-1}
  34. C C
  35. c ^ {\hat{c}}
  36. m ^ {\hat{m}}
  37. m = m ^ S - 1 m={\hat{m}}S^{-1}
  38. c ^ = c P - 1 = m G ^ P - 1 + z P - 1 = m S G + z P - 1 {\hat{c}}=cP^{-1}=m{\hat{G}}P^{-1}+zP^{-1}=mSG+zP^{-1}
  39. P P
  40. z P - 1 zP^{-1}
  41. t t
  42. G G
  43. t t
  44. m S G mSG
  45. t t
  46. c P - 1 cP^{-1}
  47. m ^ = m S {\hat{m}}=mS
  48. S S
  49. m = m ^ S - 1 = m S S - 1 m={\hat{m}}S^{-1}=mSS^{-1}
  50. n = 1024 , k = 524 , t = 50 n=1024,k=524,t=50
  51. n = 2048 , k = 1751 , t = 27 n=2048,k=1751,t=27
  52. n = 1632 , k = 1269 , t = 34 n=1632,k=1269,t=34
  53. ( G ^ , t ) ({\hat{G}},t)
  54. y 𝔽 2 n y\in\mathbb{F}_{2}^{n}
  55. G G
  56. G G
  57. S S
  58. P P
  59. G G
  60. z z
  61. ( n - t k ) / ( n k ) \textstyle{\left({{n-t}\atop{k}}\right)}/{\left({{n}\atop{k}}\right)}

Mean_anomaly.html

  1. 2 π 2\pi
  2. M M
  3. M = n t = G ( M + m ) a 3 t M=n\,t=\sqrt{\frac{G(M_{\star}\!+\!m)}{a^{3}}}\,t
  4. M M_{\star}
  5. 2 π 2\pi
  6. G ( M + m ) a 3 δ t \sqrt{\frac{G(M_{\star}\!+\!m)}{a^{3}}}\,\delta t
  7. δ t \delta t
  8. M = E - e sin E M=E-e\cdot\sin E
  9. M = M 0 + n t M=M_{0}+nt

Mean_arterial_pressure.html

  1. M A P = ( C O S V R ) + C V P MAP=(CO\cdot SVR)+CVP
  2. C O CO
  3. S V R SVR
  4. C V P CVP
  5. M A P MAP
  6. S P SP
  7. D P DP
  8. M A P D P + 1 3 ( S P - D P ) MAP\simeq DP+\frac{1}{3}(SP-DP)
  9. M A P 2 3 ( D P ) + 1 3 ( S P ) MAP\simeq\frac{2}{3}(DP)+\frac{1}{3}(SP)
  10. M A P ( 2 × D P ) + S P 3 MAP\simeq\frac{(2\times DP)+SP}{3}
  11. M A P D P + 1 3 P P MAP\simeq DP+\frac{1}{3}PP
  12. P P PP
  13. S P - D P SP-DP
  14. M A P MAP
  15. M A P MAP
  16. M A P MAP
  17. M A P MAP
  18. M A P MAP

Mean_curvature.html

  1. H H
  2. S S
  3. p p
  4. S S
  5. p p
  6. S S
  7. S S
  8. κ 1 \kappa_{1}
  9. κ 2 \kappa_{2}
  10. S S
  11. p S p\in S
  12. H = 1 2 ( κ 1 + κ 2 ) . H={1\over 2}(\kappa_{1}+\kappa_{2}).
  13. T T
  14. H = 1 n i = 1 n κ i . H=\frac{1}{n}\sum_{i=1}^{n}\kappa_{i}.
  15. H H
  16. \nabla
  17. H n = g i j i j X , H\vec{n}=g^{ij}\nabla_{i}\nabla_{j}X,
  18. X ( x ) X(x)
  19. n \vec{n}
  20. g i j g_{ij}
  21. S S
  22. 2 H = - n ^ 2H=-\nabla\cdot\hat{n}
  23. H = Trace ( ( I I ) ( I - 1 ) ) H=\,\text{Trace}((II)(I^{-1}))
  24. z = S ( x , y ) z=S(x,y)
  25. 2 H = - ( ( z - S ) | ( z - S ) | ) = ( ( S , - 1 ) 1 + | S | 2 ) = ( 1 + ( S x ) 2 ) 2 S y 2 - 2 S x S y 2 S x y + ( 1 + ( S y ) 2 ) 2 S x 2 ( 1 + ( S x ) 2 + ( S y ) 2 ) 3 / 2 . \begin{aligned}\displaystyle 2H&\displaystyle=-\nabla\cdot\left(\frac{\nabla(z% -S)}{|\nabla(z-S)|}\right)\\ &\displaystyle=\nabla\cdot\left(\frac{(\nabla S,-1)}{\sqrt{1+|\nabla S|^{2}}}% \right)\\ &\displaystyle=\frac{\left(1+\left(\frac{\partial S}{\partial x}\right)^{2}% \right)\frac{\partial^{2}S}{\partial y^{2}}-2\frac{\partial S}{\partial x}% \frac{\partial S}{\partial y}\frac{\partial^{2}S}{\partial x\partial y}+\left(% 1+\left(\frac{\partial S}{\partial y}\right)^{2}\right)\frac{\partial^{2}S}{% \partial x^{2}}}{\left(1+\left(\frac{\partial S}{\partial x}\right)^{2}+\left(% \frac{\partial S}{\partial y}\right)^{2}\right)^{3/2}}.\end{aligned}
  26. S = 0 \nabla S=0
  27. S S
  28. z = S ( r ) z=S(r)
  29. 2 H = 2 S r 2 ( 1 + ( S r ) 2 ) 3 / 2 + S r 1 r ( 1 + ( S r ) 2 ) 1 / 2 , 2H=\frac{\frac{\partial^{2}S}{\partial r^{2}}}{\left(1+\left(\frac{\partial S}% {\partial r}\right)^{2}\right)^{3/2}}+{\frac{\partial S}{\partial r}}\frac{1}{% r\left(1+\left(\frac{\partial S}{\partial r}\right)^{2}\right)^{1/2}},
  30. S r 1 r {\frac{\partial S}{\partial r}}\frac{1}{r}
  31. z = S ( r ) = S ( x 2 + y 2 ) z=S(r)=S\left(\scriptstyle\sqrt{x^{2}+y^{2}}\right)
  32. F ( x , y , z ) = 0 F(x,y,z)=0
  33. F = ( F x , F y , F z ) \nabla F=\left(\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},% \frac{\partial F}{\partial z}\right)
  34. Hess ( F ) = ( 2 F x 2 2 F x y 2 F x z 2 F x y 2 F y 2 2 F y z 2 F x z 2 F y z 2 F z 2 ) . \textstyle\mbox{Hess}~{}(F)=\begin{pmatrix}\frac{\partial^{2}F}{\partial x^{2}% }&\frac{\partial^{2}F}{\partial x\partial y}&\frac{\partial^{2}F}{\partial x% \partial z}\\ \frac{\partial^{2}F}{\partial x\partial y}&\frac{\partial^{2}F}{\partial y^{2}% }&\frac{\partial^{2}F}{\partial y\partial z}\\ \frac{\partial^{2}F}{\partial x\partial z}&\frac{\partial^{2}F}{\partial y% \partial z}&\frac{\partial^{2}F}{\partial z^{2}}\end{pmatrix}.
  35. H = F Hess ( F ) F 𝖳 - | F | 2 Trace ( Hess ( F ) ) 2 | F | 3 H=\frac{\nabla F\ \mbox{Hess}~{}(F)\ \nabla F^{\mathsf{T}}-|\nabla F|^{2}\,\,% \text{Trace}(\mbox{Hess}~{}(F))}{2|\nabla F|^{3}}
  36. F | F | \frac{\nabla F}{|\nabla F|}
  37. H = - ( F | F | ) . H=-\nabla\cdot\left(\frac{\nabla F}{|\nabla F|}\right).
  38. H f = ( κ 1 + κ 2 ) H_{f}=(\kappa_{1}+\kappa_{2})\,
  39. H f H_{f}
  40. κ 1 = κ 2 = r - 1 \kappa_{1}=\kappa_{2}=r^{-1}\,

Mean_longitude.html

  1. L L\,
  2. L = M + ϖ = M + Ω + ω L=M+\varpi=M+\Omega+\omega\,
  3. M M\,
  4. ϖ Ω + ω \varpi\equiv\Omega+\omega\,
  5. Ω \Omega\,
  6. ω \omega\,

Mean_width.html

  1. n n
  2. ( n - 1 ) (n-1)
  3. n ^ \hat{n}
  4. S n - 1 S^{n-1}
  5. S n S^{n}
  6. ( n + 1 ) (n+1)
  7. n ^ \hat{n}
  8. n ^ \hat{n}
  9. S n - 1 S^{n-1}
  10. n ^ \hat{n}
  11. n \mathbb{R}^{n}
  12. h B ( n ) = max { n , x | x B } h_{B}(n)=\max\{\langle n,x\rangle|x\in B\}
  13. n n
  14. , \langle,\rangle
  15. n \mathbb{R}^{n}
  16. b ( B ) = 1 S n - 1 S n - 1 h B ( n ^ ) + h B ( - n ^ ) , b(B)=\frac{1}{S_{n-1}}\int_{S^{n-1}}h_{B}(\hat{n})+h_{B}(-\hat{n}),
  17. S n - 1 S_{n-1}
  18. ( n - 1 ) (n-1)
  19. S n - 1 S^{n-1}
  20. δ K H 2 π d S = b ( K ) \int_{\delta K}\frac{H}{2\pi}dS=b(K)
  21. δ K \delta K
  22. K K
  23. d S dS
  24. H H
  25. δ K \delta K

Measuring_network_throughput.html

  1. Throughput RWIN RTT \mathrm{Throughput}\leq\frac{\mathrm{RWIN}}{\mathrm{RTT}}\,\!

Mechanical_energy.html

  1. E mechanical = U + K E_{\mathrm{mechanical}}=U+K\,
  2. U = - x 1 x 2 F d x U=-\int\limits_{x_{1}}^{x_{2}}\vec{F}\cdot d\vec{x}
  3. K = 1 2 m v 2 K={1\over 2}mv^{2}
  4. m m
  5. r r
  6. K K
  7. U U
  8. M M
  9. E mechanical E_{\mathrm{mechanical}}
  10. E mechanical = U + K E_{\mathrm{mechanical}}=U+K
  11. E mechanical = - G M m r + 1 2 m v 2 E_{\mathrm{mechanical}}=-G\frac{Mm}{r}\ +\frac{1}{2}\ mv^{2}
  12. E mechanical = - G M m 2 r E_{\mathrm{mechanical}}=-G\frac{Mm}{2r}
  13. G M m r 2 = m v 2 r G\frac{Mm}{r^{2}}\ =\frac{mv^{2}}{r}

Mechanism_design.html

  1. Θ \Theta
  2. f ( θ ) f(\theta)
  3. M M
  4. g g
  5. ξ ( M , g , θ ) \xi(M,g,\theta)
  6. f ( θ ) f(\theta)
  7. θ \theta
  8. Θ \Theta
  9. θ ^ \hat{\theta}
  10. y ( ) y()
  11. y y
  12. θ ^ \hat{\theta}
  13. y ( θ ^ ) y(\hat{\theta})
  14. y y
  15. y ( θ ) = { x ( θ ) , t ( θ ) } , x X , t T y(\theta)=\{x(\theta),t(\theta)\},\ x\in X,t\in T
  16. x x
  17. t t
  18. f ( θ ) f(\theta)
  19. f ( θ ) : Θ X f(\theta):\Theta\rightarrow X
  20. x x
  21. t t
  22. y ( θ ^ ) : Θ Y y(\hat{\theta}):\Theta\rightarrow Y
  23. θ ^ ( θ ) \hat{\theta}(\theta)
  24. u i ( s i ( θ i ) , s - i ( θ - i ) , θ i ) u_{i}\left(s_{i}(\theta_{i}),s_{-i}(\theta_{-i}),\theta_{i}\right)
  25. s ( θ i ) s(\theta_{i})
  26. s i ( θ i ) arg max s i S i θ - i p ( θ - i θ i ) u i ( s i , s - i ( θ - i ) , θ i ) s_{i}(\theta_{i})\in\arg\max_{s^{\prime}_{i}\in S_{i}}\sum_{\theta_{-i}}\ p(% \theta_{-i}\mid\theta_{i})\ u_{i}\left(s^{\prime}_{i},s_{-i}(\theta_{-i}),% \theta_{i}\right)
  27. y ( θ ^ ) : Θ S ( Θ ) Y y(\hat{\theta}):\Theta\rightarrow S(\Theta)\rightarrow Y
  28. y ( θ ) y(\theta)
  29. θ ^ i ( θ i ) arg max θ i Θ θ - i p ( θ - i θ i ) u i ( y ( θ i , θ - i ) , θ i ) \hat{\theta}_{i}(\theta_{i})\in\arg\max_{\theta^{\prime}_{i}\in\Theta}\sum_{% \theta_{-i}}\ p(\theta_{-i}\mid\theta_{i})\ u_{i}\left(y(\theta^{\prime}_{i},% \theta_{-i}),\theta_{i}\right)
  30. = θ - i p ( θ - i θ i ) u i ( s i ( θ ) , s - i ( θ - i ) , θ i ) =\sum_{\theta_{-i}}\ p(\theta_{-i}\mid\theta_{i})\ u_{i}\left(s_{i}(\theta),s_% {-i}(\theta_{-i}),\theta_{i}\right)
  31. y ( ) y()
  32. y ( ) y()
  33. f ( θ ) f(\theta)
  34. t ( θ ) t(\theta)
  35. x ( θ ) x(\theta)
  36. f ( θ ) = x ( θ ^ ( θ ) ) f(\theta)=x\left(\hat{\theta}(\theta)\right)
  37. t ( θ ) t(\theta)
  38. θ ^ ( θ ) = θ \hat{\theta}(\theta)=\theta
  39. t ( θ ) t(\theta)
  40. x ( θ ) x(\theta)
  41. t ( θ ) t(\theta)
  42. u ( x ( θ ) , t ( θ ) , θ ) u ( x ( θ ^ ) , t ( θ ^ ) , θ ) θ , θ ^ Θ u(x(\theta),t(\theta),\theta)\geq u(x(\hat{\theta}),t(\hat{\theta}),\theta)\ % \forall\theta,\hat{\theta}\in\Theta
  43. t ( θ ) t(\theta)
  44. u ( x , t , θ ) u(x,t,\theta)
  45. x ( θ ) x(\theta)
  46. k k
  47. k k
  48. x ( θ ) x(\theta)
  49. k = 1 n θ ( u / x k | u / t | ) x θ 0 \sum^{n}_{k=1}\frac{\partial}{\partial\theta}\left(\frac{\partial u/\partial x% _{k}}{\left|\partial u/\partial t\right|}\right)\frac{\partial x}{\partial% \theta}\geq 0
  50. x = x ( θ ) x=x(\theta)
  51. t = t ( θ ) t=t(\theta)
  52. θ \theta
  53. θ ( u / x k | u / t | ) = θ M R S x , t \frac{\partial}{\partial\theta}\left(\frac{\partial u/\partial x_{k}}{\left|% \partial u/\partial t\right|}\right)=\frac{\partial}{\partial\theta}MRS_{x,t}
  54. x θ \frac{\partial x}{\partial\theta}
  55. x / θ < 0 \partial x/\partial\theta<0
  56. 1. θ u / x k | u / t | > 0 k 1.\ \frac{\partial}{\partial\theta}\frac{\partial u/\partial x_{k}}{\left|% \partial u/\partial t\right|}>0\ \forall k
  57. 2. K 0 , K 1 such that | u / x k u / t | K 0 + K 1 | t | 2.\ \exists K_{0},K_{1}\,\text{ such that }\left|\frac{\partial u/\partial x_{% k}}{\partial u/\partial t}\right|\leq K_{0}+K_{1}|t|
  58. x ( θ ) x(\theta)
  59. t ( θ ) t(\theta)
  60. x ( θ ) x(\theta)
  61. x ( θ ) x(\theta)
  62. I I
  63. v ( x , t , θ ) v(x,t,\theta)
  64. t t
  65. x I * ( θ ) arg max x X i I v ( x , θ i ) x^{*}_{I}(\theta)\in\arg\max_{x\in X}\sum_{i\in I}v(x,\theta_{i})
  66. t i ( θ ^ ) = j I - i v j ( x I - i * ( θ I - i ) , θ j ) - j I - i v j ( x I * ( θ ^ i , θ I ) , θ j ) t_{i}(\hat{\theta})=\sum_{j\in I-i}v_{j}(x^{*}_{I-i}(\theta_{I-i}),\theta_{j})% -\sum_{j\in I-i}v_{j}(x^{*}_{I}(\hat{\theta}_{i},\theta_{I}),\theta_{j})
  67. for f ( Θ ) , i I such that u i ( x , θ i ) u i ( x , θ i ) x X \,\text{for }f(\Theta)\,\text{, }\exists i\in I\,\text{ such that }u_{i}(x,% \theta_{i})\geq u_{i}(x^{\prime},\theta_{i})\ \forall x^{\prime}\in X
  68. f ( Θ ) = X f(\Theta)=X
  69. θ \theta
  70. u ( x , t , θ ) = V ( x , θ ) - t u(x,t,\theta)=V(x,\theta)-t
  71. P ( θ ) P(\theta)
  72. max x ( θ ) , t ( θ ) 𝔼 θ [ t ( θ ) - c ( x ( θ ) ) ] \max_{x(\theta),t(\theta)}\mathbb{E}_{\theta}\left[t(\theta)-c\left(x(\theta)% \right)\right]
  73. u ( x ( θ ) , t ( θ ) , θ ) u ( x ( θ ) , t ( θ ) , θ ) θ , θ u(x(\theta),t(\theta),\theta)\geq u(x(\theta^{\prime}),t(\theta^{\prime}),% \theta)\ \forall\theta,\theta^{\prime}
  74. u ( x ( θ ) , t ( θ ) , θ ) u ¯ ( θ ) θ u(x(\theta),t(\theta),\theta)\geq\underline{u}(\theta)\ \forall\theta
  75. let U ( θ ) = max θ u ( x ( θ ) , t ( θ ) , θ ) \,\text{let }U(\theta)=\max_{\theta^{\prime}}u\left(x(\theta^{\prime}),t(% \theta^{\prime}),\theta\right)
  76. d U d θ = u θ = V θ \frac{dU}{d\theta}=\frac{\partial u}{\partial\theta}=\frac{\partial V}{% \partial\theta}
  77. U ( θ ) = u ¯ ( θ 0 ) + θ 0 θ V θ ~ d θ ~ U(\theta)=\underline{u}(\theta_{0})+\int^{\theta}_{\theta_{0}}\frac{\partial V% }{\partial\tilde{\theta}}d\tilde{\theta}
  78. θ 0 \theta_{0}
  79. t ( θ ) = V ( x ( θ ) , θ ) - U ( θ ) t(\theta)=V(x(\theta),\theta)-U(\theta)
  80. 𝔼 θ [ V ( x ( θ ) , θ ) - u ¯ ( θ 0 ) - θ 0 θ V θ ~ d θ ~ - c ( x ( θ ) ) ] \mathbb{E}_{\theta}\left[V(x(\theta),\theta)-\underline{u}(\theta_{0})-\int^{% \theta}_{\theta_{0}}\frac{\partial V}{\partial\tilde{\theta}}d\tilde{\theta}-c% \left(x(\theta)\right)\right]
  81. = 𝔼 θ [ V ( x ( θ ) , θ ) - u ¯ ( θ 0 ) - 1 - P ( θ ) p ( θ ) V θ - c ( x ( θ ) ) ] =\mathbb{E}_{\theta}\left[V(x(\theta),\theta)-\underline{u}(\theta_{0})-\frac{% 1-P(\theta)}{p(\theta)}\frac{\partial V}{\partial\theta}-c\left(x(\theta)% \right)\right]
  82. U ( θ ) U(\theta)
  83. x ( θ ) x(\theta)
  84. x ( θ ) x(\theta)
  85. θ 1 \theta_{1}
  86. θ 2 > θ 1 \theta_{2}>\theta_{1}
  87. x x
  88. ϕ 2 ( x ) ϕ 1 ( x ) ( V x ( x , θ ) - 1 - P ( θ ) p ( θ ) 2 V θ x ( x , θ ) - c x ( x ) ) d θ = 0 \int^{\phi_{1}(x)}_{\phi_{2}(x)}\left(\frac{\partial V}{\partial x}(x,\theta)-% \frac{1-P(\theta)}{p(\theta)}\frac{\partial^{2}V}{\partial\theta\,\partial x}(% x,\theta)-\frac{\partial c}{\partial x}(x)\right)d\theta=0
  89. ϕ 1 ( x ) \phi_{1}(x)
  90. θ θ 1 \theta\leq\theta_{1}
  91. ϕ 2 ( x ) \phi_{2}(x)
  92. θ θ 2 \theta\geq\theta_{2}
  93. ϕ 1 \phi_{1}
  94. θ \theta
  95. ϕ 2 \phi_{2}
  96. θ \theta
  97. x ( θ ) x(\theta)
  98. ϕ ( x ) \phi(x)
  99. [ θ ¯ , θ ¯ ] \left[\underline{\theta},\overline{\theta}\right]
  100. x ( θ ) x(\theta)
  101. x ( θ ) x(\theta)
  102. x ( θ ) x(\theta)
  103. x ( θ ) x(\theta)
  104. x θ 0 \frac{\partial x}{\partial\theta}\geq 0
  105. ν ( θ ) \nu(\theta)
  106. H = ( V ( x , θ ) - u ¯ ( θ 0 ) - 1 - P ( θ ) p ( θ ) V θ ( x , θ ) - c ( x ) ) p ( θ ) + ν ( θ ) x θ H=\left(V(x,\theta)-\underline{u}(\theta_{0})-\frac{1-P(\theta)}{p(\theta)}% \frac{\partial V}{\partial\theta}(x,\theta)-c(x)\right)p(\theta)+\nu(\theta)% \frac{\partial x}{\partial\theta}
  107. x x
  108. x / θ \partial x/\partial\theta
  109. ν θ = - H x = - ( V x ( x , θ ) - 1 - P ( θ ) p ( θ ) 2 V θ x ( x , θ ) - c x ( x ) ) p ( θ ) \frac{\partial\nu}{\partial\theta}=-\frac{\partial H}{\partial x}=-\left(\frac% {\partial V}{\partial x}(x,\theta)-\frac{1-P(\theta)}{p(\theta)}\frac{\partial% ^{2}V}{\partial\theta\,\partial x}(x,\theta)-\frac{\partial c}{\partial x}(x)% \right)p(\theta)
  110. θ \theta
  111. ν ( θ ¯ ) = ν ( θ ¯ ) = 0 \nu(\underline{\theta})=\nu(\overline{\theta})=0
  112. θ ¯ θ ¯ ( V x ( x , θ ) - 1 - P ( θ ) p ( θ ) 2 V θ x ( x , θ ) - c x ( x ) ) p ( θ ) d θ = 0 \int^{\overline{\theta}}_{\underline{\theta}}\left(\frac{\partial V}{\partial x% }(x,\theta)-\frac{1-P(\theta)}{p(\theta)}\frac{\partial^{2}V}{\partial\theta\,% \partial x}(x,\theta)-\frac{\partial c}{\partial x}(x)\right)p(\theta)d\theta=0
  113. x x
  114. θ \theta

Median_(geometry).html

  1. A B ¯ \overline{AB}
  2. B C ¯ \overline{BC}
  3. A C ¯ \overline{AC}
  4. A D = D B , A F = F C , B E = E C AD=DB,AF=FC,BE=EC\,
  5. [ A D O ] = [ B D O ] , [ A F O ] = [ C F O ] , [ B E O ] = [ C E O ] , [ADO]=[BDO],[AFO]=[CFO],[BEO]=[CEO],
  6. [ A B E ] = [ A C E ] [ABE]=[ACE]\,
  7. [ A B C ] [ABC]
  8. A B C \triangle ABC
  9. [ A B O ] = [ A B E ] - [ B E O ] [ABO]=[ABE]-[BEO]\,
  10. [ A C O ] = [ A C E ] - [ C E O ] [ACO]=[ACE]-[CEO]\,
  11. [ A B O ] = [ A C O ] [ABO]=[ACO]\,
  12. [ A D O ] = [ D B O ] , [ A D O ] = 1 2 [ A B O ] [ADO]=[DBO],[ADO]=\frac{1}{2}[ABO]
  13. [ A F O ] = [ F C O ] , [ A F O ] = 1 2 [ A C O ] = 1 2 [ A B O ] = [ A D O ] [AFO]=[FCO],[AFO]=\frac{1}{2}[ACO]=\frac{1}{2}[ABO]=[ADO]
  14. [ A F O ] = [ F C O ] = [ D B O ] = [ A D O ] [AFO]=[FCO]=[DBO]=[ADO]\,
  15. [ A F O ] = [ F C O ] = [ D B O ] = [ A D O ] = [ B E O ] = [ C E O ] [AFO]=[FCO]=[DBO]=[ADO]=[BEO]=[CEO]\,
  16. m a = 2 b 2 + 2 c 2 - a 2 4 , m_{a}=\sqrt{\frac{2b^{2}+2c^{2}-a^{2}}{4}},
  17. m b = 2 a 2 + 2 c 2 - b 2 4 , m_{b}=\sqrt{\frac{2a^{2}+2c^{2}-b^{2}}{4}},
  18. m c = 2 a 2 + 2 b 2 - c 2 4 , m_{c}=\sqrt{\frac{2a^{2}+2b^{2}-c^{2}}{4}},
  19. a = 2 3 - m a 2 + 2 m b 2 + 2 m c 2 = 2 ( b 2 + c 2 ) - 4 m a 2 = b 2 2 - c 2 + 2 m b 2 = c 2 2 - b 2 + 2 m c 2 , a=\frac{2}{3}\sqrt{-m_{a}^{2}+2m_{b}^{2}+2m_{c}^{2}}=\sqrt{2(b^{2}+c^{2})-4m_{% a}^{2}}=\sqrt{\frac{b^{2}}{2}-c^{2}+2m_{b}^{2}}=\sqrt{\frac{c^{2}}{2}-b^{2}+2m% _{c}^{2}},
  20. b = 2 3 - m b 2 + 2 m a 2 + 2 m c 2 = 2 ( a 2 + c 2 ) - 4 m b 2 = a 2 2 - c 2 + 2 m a 2 = c 2 2 - a 2 + 2 m c 2 , b=\frac{2}{3}\sqrt{-m_{b}^{2}+2m_{a}^{2}+2m_{c}^{2}}=\sqrt{2(a^{2}+c^{2})-4m_{% b}^{2}}=\sqrt{\frac{a^{2}}{2}-c^{2}+2m_{a}^{2}}=\sqrt{\frac{c^{2}}{2}-a^{2}+2m% _{c}^{2}},
  21. c = 2 3 - m c 2 + 2 m b 2 + 2 m a 2 = 2 ( b 2 + a 2 ) - 4 m c 2 = b 2 2 - a 2 + 2 m b 2 = a 2 2 - b 2 + 2 m a 2 . c=\frac{2}{3}\sqrt{-m_{c}^{2}+2m_{b}^{2}+2m_{a}^{2}}=\sqrt{2(b^{2}+a^{2})-4m_{% c}^{2}}=\sqrt{\frac{b^{2}}{2}-a^{2}+2m_{b}^{2}}=\sqrt{\frac{a^{2}}{2}-b^{2}+2m% _{a}^{2}}.
  22. 3 4 \tfrac{3}{4}
  23. m a , m b , m c m_{a},m_{b},m_{c}
  24. 3 4 ( a 2 + b 2 + c 2 ) = m a 2 + m b 2 + m c 2 . \tfrac{3}{4}(a^{2}+b^{2}+c^{2})=m_{a}^{2}+m_{b}^{2}+m_{c}^{2}.
  25. a 2 + b 2 = 5 c 2 . a^{2}+b^{2}=5c^{2}.
  26. m a 2 + m b 2 = 5 m c 2 . m_{a}^{2}+m_{b}^{2}=5m_{c}^{2}.
  27. m a , m b m_{a},m_{b}
  28. m c m_{c}
  29. T = 4 3 σ ( σ - m a ) ( σ - m b ) ( σ - m c ) . T=\frac{4}{3}\sqrt{\sigma(\sigma-m_{a})(\sigma-m_{b})(\sigma-m_{c})}.

Memorylessness.html

  1. Pr ( X > m + n X > m ) = Pr ( X > n ) . \Pr(X>m+n\mid X>m)=\Pr(X>n).
  2. Pr ( X > 40 X > 30 ) = Pr ( X > 10 ) . \mathrm{}\ \Pr(X>40\mid X>30)=\Pr(X>10).\,
  3. Pr ( X > 40 X > 30 ) = Pr ( X > 40 ) \mathrm{}\ \Pr(X>40\mid X>30)=\Pr(X>40)\,
  4. Pr ( X > 40 X > 30 ) = Pr ( X > 40 , X > 30 ) Pr ( X > 30 ) = Pr ( X > 40 ) Pr ( X > 30 ) \Pr(X>40\mid X>30)=\frac{\Pr(X>40,X>30)}{\Pr(X>30)}=\frac{\Pr(X>40)}{\Pr(X>30)}
  5. Pr ( X > t + s X > t ) = Pr ( X > s ) . \Pr(X>t+s\mid X>t)=\Pr(X>s).\,
  6. G ( t ) = Pr ( X > t ) . G(t)=\Pr(X>t).\,
  7. Pr ( X > t + s | X > t ) = Pr ( X > s ) \Pr(X>t+s|X>t)=\Pr(X>s)\,
  8. Pr ( X > t + s ) Pr ( X > t ) = Pr ( X > s ) . {\Pr(X>t+s)\over\Pr(X>t)}=\Pr(X>s).
  9. G ( t + s ) = G ( t ) G ( s ) G(t+s)=G(t)G(s)\,
  10. G ( 2 ) = G ( 1 ) 2 G(2)=G(1)^{2}
  11. G ( 1 / 2 ) = G ( 1 ) 1 / 2 G(1/2)=G(1)^{1/2}
  12. G ( a ) = G ( 1 ) a G(a)=G(1)^{a}
  13. a a
  14. G ( a ) = G ( 1 ) a = e ln ( G ( 1 ) ) a = e - λ a G(a)=G(1)^{a}=e^{\ln(G(1))a}=e^{-\lambda a}
  15. λ = - ln ( G ( 1 ) ) \lambda=-\ln(G(1))
  16. G ( a ) G(a)
  17. λ > 0 \lambda>0
  18. G G
  19. x y x\leq y
  20. G ( x ) G ( y ) G(x)\geq G(y)

Mertens'_theorems.html

  1. p n p\leq n
  2. p n ln p p - ln n \sum_{p\leq n}\frac{\ln p}{p}-\ln n
  3. n 2 n\geq 2
  4. lim n ( p n 1 p - ln ln n - M ) = 0 , \lim_{n\to\infty}\left(\sum_{p\leq n}\frac{1}{p}-\ln\ln n-M\right)=0,
  5. 4 ln ( n + 1 ) + 2 n ln n \frac{4}{\ln(n+1)}+\frac{2}{n\ln n}
  6. n 2 n\geq 2
  7. lim n ln n p n ( 1 - 1 p ) = e - γ , \lim_{n\to\infty}\ln n\prod_{p\leq n}\left(1-\frac{1}{p}\right)=e^{-\gamma},
  8. p n 1 p - ln ln n - M \sum_{p\leq n}\frac{1}{p}-\ln\ln n-M
  9. ln n p n ( 1 - 1 p ) - e - γ \ln n\prod_{p\leq n}\left(1-\frac{1}{p}\right)-e^{-\gamma}
  10. p x 1 p = log log x + M + O ( 1 / log x ) \sum_{p\leq x}\frac{1}{p}=\log\log x+M+O(1/\log x)
  11. p x 1 p = log log x + M + o ( 1 / log x ) . \sum_{p\leq x}\frac{1}{p}=\log\log x+M+o(1/\log x).
  12. n = 1 a n \sum_{n=1}^{\infty}a_{n}
  13. n = 1 b n \sum_{n=1}^{\infty}b_{n}

Metaballs.html

  1. f ( x , y , z ) f(x,y,z)
  2. i = 0 n metaball ( x , y , z ) i threshold \sum_{i=0}^{n}\mbox{metaball}~{}_{i}(x,y,z)\leq\mbox{threshold}~{}
  3. n n
  4. ( x , y , z ) (x,y,z)
  5. f ( x , y , z ) = 1 / ( ( x - x 0 ) 2 + ( y - y 0 ) 2 + ( z - z 0 ) 2 ) f(x,y,z)=1/((x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2})
  6. ( x 0 , y 0 , z 0 ) (x_{0},y_{0},z_{0})
  7. f ( r ) = ( 1 - r 2 ) 2 f(r)=(1-r^{2})^{2}

Metabolic_theory_of_ecology.html

  1. B = B o M 3 / 4 B=B_{o}M^{3/4}\,
  2. e - E k T e^{-\frac{E}{k\,T}}
  3. B = b o M 3 / 4 e - E k T B=b_{o}M^{3/4}e^{-\frac{E}{k\,T}}
  4. S M R = ( B / M ) = b o M - 1 / 4 e - E k T SMR=(B/M)=b_{o}M^{-1/4}e^{-\frac{E}{k\,T}}

Metallicity.html

  1. X + Y + Z = 1.00 X+Y+Z=1.00
  2. X m H M X\equiv\frac{m_{\mathrm{H}}}{M}
  3. M M
  4. m H m_{\mathrm{H}}
  5. Y m He M Y\equiv\frac{m_{\mathrm{He}}}{M}
  6. Z = i > He m i M = 1 - X - Y . Z=\sum_{i>\mathrm{He}}\frac{m_{i}}{M}=1-X-Y.
  7. Z sun Z_{\mathrm{sun}}
  8. X sun = 0.73 X_{\mathrm{sun}}=0.73
  9. Y sun = 0.25 Y_{\mathrm{sun}}=0.25
  10. Z sun = 0.02 Z_{\mathrm{sun}}=0.02
  11. [ Fe / H ] = log 10 ( N Fe N H ) star - log 10 ( N Fe N H ) sun [\mathrm{Fe}/\mathrm{H}]=\log_{10}{\left(\frac{N_{\mathrm{Fe}}}{N_{\mathrm{H}}% }\right)_{\mathrm{star}}}-\log_{10}{\left(\frac{N_{\mathrm{Fe}}}{N_{\mathrm{H}% }}\right)_{\mathrm{sun}}}
  12. N Fe N_{\mathrm{Fe}}
  13. N H N_{\mathrm{H}}
  14. [ O / Fe ] = log 10 ( N O N Fe ) star - log 10 ( N O N Fe ) sun [\mathrm{O}/\mathrm{Fe}]=\log_{10}{\left(\frac{N_{\mathrm{O}}}{N_{\mathrm{Fe}}% }\right)_{\mathrm{star}}}-\log_{10}{\left(\frac{N_{\mathrm{O}}}{N_{\mathrm{Fe}% }}\right)_{\mathrm{sun}}}
  15. = [ log 10 ( N O N H ) star - log 10 ( N O N H ) sun ] - [ log 10 ( N Fe N H ) star - log 10 ( N Fe N H ) sun ] . =\left[\log_{10}{\left(\frac{N_{\mathrm{O}}}{N_{\mathrm{H}}}\right)_{\mathrm{% star}}}-\log_{10}{\left(\frac{N_{\mathrm{O}}}{N_{\mathrm{H}}}\right)_{\mathrm{% sun}}}\right]-\left[\log_{10}{\left(\frac{N_{\mathrm{Fe}}}{N_{\mathrm{H}}}% \right)_{\mathrm{star}}}-\log_{10}{\left(\frac{N_{\mathrm{Fe}}}{N_{\mathrm{H}}% }\right)_{\mathrm{sun}}}\right].
  16. log 10 ( Z / X Z sun / X sun ) = [ M / H ] \log_{10}\left(\frac{Z/X}{Z_{\mathrm{sun}}/X_{\mathrm{sun}}}\right)=[\mathrm{M% }/\mathrm{H}]
  17. [ M / H ] = log 10 ( N M N H ) star - log 10 ( N M N H ) sun . [\mathrm{M}/\mathrm{H}]=\log_{10}{\left(\frac{N_{\mathrm{M}}}{N_{\mathrm{H}}}% \right)_{\mathrm{star}}}-\log_{10}{\left(\frac{N_{\mathrm{M}}}{N_{\mathrm{H}}}% \right)_{\mathrm{sun}}}.
  18. [ M / H ] = A * [ Fe / H ] [\mathrm{M}/\mathrm{H}]=A*[\mathrm{Fe}/\mathrm{H}]
  19. log 10 ( Z / X Z sun / X sun ) = A * [ Fe / H ] . \log_{10}\left(\frac{Z/X}{Z_{\mathrm{sun}}/X_{\mathrm{sun}}}\right)=A*[\mathrm% {Fe}/\mathrm{H}].

Metamaterial.html

  1. n = ± ϵ r μ r \scriptstyle n=\pm\sqrt{\epsilon_{\mathrm{r}}\mu_{\mathrm{r}}}
  2. k = ω μ ϵ k=\omega\sqrt{\mu\epsilon}
  3. n = ϵ r μ r ± κ n=\sqrt{\epsilon_{r}\mu_{r}}\pm\kappa
  4. [ u r a d i c a l , u 3 b 5 l e s s t h a n s m a l l > r \xb 5 < s m a l l > r < / s m a l l > ] [u^{\prime}radical^{\prime},u^{\prime}\u{0}3b5\\ lessthansmall>r\xb 5<small>r</small>^{\prime}]

Method_of_analytic_tableaux.html

  1. A 1 , , A n A_{1},\ldots,A_{n}
  2. B B
  3. { A 1 , , A n , ¬ B } \{A_{1},\ldots,A_{n},\neg B\}
  4. \top
  5. { ( a ¬ b ) b , ¬ a } \{(a\vee\neg b)\wedge b,\neg a\}
  6. A B A\wedge B
  7. \wedge
  8. A B A\wedge B
  9. A A
  10. B B
  11. ( and ) A B A B (\and)\frac{A\wedge B}{\begin{array}[]{c}A\\ B\end{array}}
  12. { A , B } \{A,B\}
  13. A B A\wedge B
  14. X { A B } X\cup\{A\wedge B\}
  15. X { A , B } X\cup\{A,B\}
  16. A B A\vee B
  17. \vee
  18. A B A\vee B
  19. A A
  20. B B
  21. A B A\vee B
  22. | |
  23. ( ) A B A | B (\vee)\frac{A\vee B}{A|B}
  24. Y { A B } Y\cup\{A\vee B\}
  25. Y { A } Y\cup\{A\}
  26. Y { B } Y\cup\{B\}
  27. ¬ ( A B ) \neg(A\wedge B)
  28. ¬ A ¬ B \neg A\vee\neg B
  29. ¬ ¬ A \neg\neg A
  30. ¬ ¬ ( A B ) \neg\neg(A\wedge B)
  31. ( ¬ 1 ) A ¬ ¬ A (\neg 1)\frac{A}{\neg\neg A}
  32. ( ¬ 2 ) ¬ ¬ A A (\neg 2)\frac{\neg\neg A}{A}
  33. A B A\wedge B
  34. A A
  35. B B
  36. X { A B } X\cup\{A\wedge B\}
  37. X { A , B } X\cup\{A,B\}
  38. ( ) X { A B } X { A , B } (\wedge)\frac{X\cup\{A\wedge B\}}{X\cup\{A,B\}}
  39. X { A B } X\cup\{A\vee B\}
  40. X { A } X\cup\{A\}
  41. X { B } X\cup\{B\}
  42. ( ) X { A B } X { A } | X { B } (\vee)\frac{X\cup\{A\vee B\}}{X\cup\{A\}|X\cup\{B\}}
  43. ( i d ) X { p , ¬ p } c l o s e d (id)\frac{X\cup\{p,\neg p\}}{closed}
  44. \wedge
  45. ¬ \neg
  46. ( ) (\wedge)
  47. ( ) (\vee)
  48. ( i d ) (id)
  49. X X^{\prime}
  50. X X^{\prime}
  51. X X^{\prime}
  52. X X^{\prime}
  53. X X^{\prime}
  54. X X^{\prime}
  55. X X^{\prime}
  56. X = { ¬ A } X=\{\neg A\}
  57. { ¬ A } \{\neg A\}
  58. ¬ A \neg A
  59. { ¬ A } \{\neg A\}
  60. ¬ ( A and B ) = ¬ A ¬ B ¬ ( A B ) = ¬ A and ¬ B ¬ ( ¬ A ) = A ¬ ( A B ) = A and ¬ B A B = ¬ A B A B = ( A and B ) ( ¬ A and ¬ B ) ¬ ( A B ) = ( A and ¬ B ) ( ¬ A and B ) \begin{array}[]{lcl}\neg(A\and B)&=&\neg A\neg B\\ \neg(AB)&=&\neg A\and\neg B\\ \neg(\neg A)&=&A\\ \neg(A\Rightarrow B)&=&A\and\neg B\\ A\Rightarrow B&=&\neg AB\\ A\Leftrightarrow B&=&(A\and B)(\neg A\and\neg B)\\ \neg(A\Leftrightarrow B)&=&(A\and\neg B)(\neg A\and B)\end{array}
  61. x . γ ( x ) \forall x.\gamma(x)
  62. γ ( t ) \gamma(t)
  63. t t
  64. ( ) x . γ ( x ) γ ( t ) (\forall)\frac{\forall x.\gamma(x)}{\gamma(t)}
  65. t t
  66. { ¬ P ( a ) ¬ P ( b ) , x . P ( x ) } \{\neg P(a)\vee\neg P(b),\forall x.P(x)\}
  67. P ( a ) P(a)
  68. P ( b ) P(b)
  69. x . P ( x ) \forall x.P(x)
  70. x . δ ( x ) \exists x.\delta(x)
  71. δ ( c ) \delta(c)
  72. c c
  73. ( ) x . δ ( x ) δ ( c ) (\exists)\frac{\exists x.\delta(x)}{\delta(c)}
  74. c c
  75. c c
  76. x x
  77. x x
  78. y . γ ( y ) \forall y.\gamma(y)
  79. γ ( c ) \gamma(c)
  80. c c
  81. { ¬ P ( f ( c ) ) , x . P ( x ) } \{\neg P(f(c)),\forall x.P(x)\}
  82. x . P ( x ) \forall x.P(x)
  83. P ( c ) , P ( f ( c ) ) , P ( f ( f ( c ) ) ) , P(c),P(f(c)),P(f(f(c))),\ldots
  84. ( ) (\forall)
  85. t t
  86. x . γ ( x ) \forall x.\gamma(x)
  87. γ ( x ) \gamma(x^{\prime})
  88. x x^{\prime}
  89. ( ) x . γ ( x ) γ ( x ) (\forall)\frac{\forall x.\gamma(x)}{\gamma(x^{\prime})}
  90. x x^{\prime}
  91. ( σ ) (\sigma)
  92. σ \sigma
  93. A A
  94. B B
  95. A A
  96. B B
  97. σ \sigma
  98. { ¬ P ( a ) , x . P ( x ) } \{\neg P(a),\forall x.P(x)\}
  99. P ( x 1 ) P(x_{1})
  100. ¬ P ( a ) \neg P(a)
  101. x 1 x_{1}
  102. a a
  103. P ( x 1 ) P(x_{1})
  104. P ( a ) P(a)
  105. x . δ ( x ) \exists x.\delta(x)
  106. ( ) x . δ ( x ) δ ( f ( x 1 , , x n ) ) (\exists)\frac{\exists x.\delta(x)}{\delta(f(x_{1},\ldots,x_{n}))}
  107. f f
  108. x 1 , , x n x_{1},\ldots,x_{n}
  109. δ \delta
  110. x 1 , , x n x_{1},\ldots,x_{n}
  111. δ \delta
  112. δ \delta
  113. F F
  114. x 1 , , x n x_{1},\ldots,x_{n}
  115. x 1 , , x n . F \forall x_{1},\ldots,x_{n}.F
  116. γ ( x ) \gamma(x^{\prime})
  117. γ \gamma
  118. x x^{\prime}
  119. γ ( t ) \gamma(t)
  120. t t
  121. x x
  122. A ( x ) A(x)
  123. B ( x ) B(x)
  124. x x
  125. x . ( A ( x ) B ( x ) ) \forall x.(...A(x)...B(x)...)
  126. ( A ( x ) B ( x ) ) (...A(x)...B(x)...)
  127. x x
  128. ( A ( t ) A ( t ) ) (...A(t)...A(t^{\prime})...)
  129. t t
  130. t t^{\prime}
  131. x x
  132. A ( x ) A(x)
  133. B ( x ) B(x)
  134. P ( x ) P ( c ) P(x)\rightarrow P(c)
  135. D = { 1 , 2 } , P ( 1 ) = , P ( 2 ) = , c = 1 D=\{1,2\},P(1)=\bot,P(2)=\top,c=1
  136. x = 2 x=2
  137. { P ( x ) , ¬ P ( c ) } \{P(x),\neg P(c)\}
  138. P ( x ) P(x)
  139. ¬ P ( c ) \neg P(c)
  140. x x
  141. c c
  142. x . ( P ( x ) P ( c ) ) \forall x.(P(x)\rightarrow P(c))
  143. { P ( f ( x ) ) , R ( c ) , ¬ P ( f ( c ) ) ¬ R ( c ) , x . Q ( x ) } \{P(f(x)),R(c),\neg P(f(c))\vee\neg R(c),\forall x.Q(x)\}
  144. ¬ a ¬ b \neg a\wedge\neg b
  145. a b a\vee b
  146. a a
  147. b b
  148. ( ) (\wedge)
  149. a a
  150. b b
  151. ( ) (\wedge)
  152. a a
  153. b b
  154. a a
  155. b b
  156. x 1 , , x n L 1 L m \forall x_{1},\ldots,x_{n}L_{1}\vee\cdots\vee L_{m}
  157. L i L_{i}
  158. P ( x , y ) Q ( f ( x ) ) P(x,y)\vee Q(f(x))
  159. x , y . P ( x , y ) Q ( f ( x ) ) \forall x,y.P(x,y)\vee Q(f(x))
  160. P ( x , y ) Q ( f ( x ) ) P(x,y)\vee Q(f(x))
  161. x , y . P ( x , y ) Q ( f ( x ) ) \exists x,y.P(x,y)\vee Q(f(x))
  162. ( ) (\forall)
  163. ( ) (\vee)
  164. ( ) (\forall)
  165. ( ) (\vee)
  166. ( C ) L 1 L n L 1 | | L n (C)\frac{L_{1}\vee\cdots\vee L_{n}}{L_{1}^{\prime}|\cdots|L_{n}^{\prime}}
  167. L 1 L n L_{1}^{\prime}\vee\cdots\vee L_{n}^{\prime}
  168. L 1 L n L_{1}\vee\cdots\vee L_{n}
  169. ( C ) (C)
  170. ( σ ) (\sigma)
  171. ( C ) L 1 L n L 1 | | L n (C)\frac{L_{1}\vee\cdots\vee L_{n}}{L_{1}^{\prime}|\cdots|L_{n}^{\prime}}
  172. L 1 L n L_{1}^{\prime}\vee\cdots\vee L_{n}^{\prime}
  173. L 1 L n L_{1}\vee\cdots\vee L_{n}
  174. t r u e true
  175. t r u e - a true-a
  176. { a , ¬ a b , ¬ c d , ¬ b } \{a,\neg a\vee b,\neg c\vee d,\neg b\}
  177. ¬ a b \neg a\vee b
  178. { a , b , ¬ b } \{a,b,\neg b\}
  179. ( C ) (C)
  180. a a
  181. t r u e - a true-a
  182. B B
  183. L L
  184. C C
  185. C C
  186. L L
  187. B - L B-L
  188. L L
  189. B B
  190. B - L B-L
  191. B B
  192. C C
  193. C C
  194. A \Box A
  195. A A
  196. ¬ A \neg\Box A
  197. A A
  198. ¬ A ¬ A \frac{\neg\Box A}{\neg A}
  199. a ¬ a a\wedge\neg\Box a
  200. a a
  201. a a
  202. ( ) (\wedge)
  203. a a
  204. ¬ a \neg a
  205. M M
  206. w w
  207. a a
  208. a a
  209. w w
  210. ¬ a \neg\Box a
  211. ¬ a \neg a
  212. w w^{\prime}
  213. w w
  214. w w
  215. ¬ A ¬ B \neg\Box A\wedge\neg\Box B
  216. ¬ A \neg A
  217. ¬ B \neg B
  218. ¬ A \neg A
  219. ¬ B \neg B
  220. ¬ B \neg B
  221. ¬ A \neg A
  222. ¬ A ¬ B \neg\Box A\wedge\neg\Box B
  223. ¬ A \neg A
  224. ¬ B \neg B
  225. ¬ A \neg\Box A
  226. ¬ B \neg\Box B
  227. A \Box A
  228. A A
  229. A \Box A
  230. A A
  231. A \Box A
  232. ¬ A ¬ A \frac{\neg\Box A}{\neg A}
  233. A \Box A
  234. A A
  235. w : A w:A
  236. A A
  237. w w
  238. w : A B w:A\wedge B
  239. w : A w:A
  240. w : B w:B
  241. w : a w:a
  242. w : ¬ a w:\neg a
  243. w : a w:a
  244. w : ¬ a w^{\prime}:\neg a
  245. ¬ A \neg\Box A
  246. w : ¬ A w : ¬ A \frac{w:\neg\Box A}{w^{\prime}:\neg A}
  247. w w
  248. w w^{\prime}
  249. w R w wRw^{\prime}
  250. w w^{\prime}
  251. w w
  252. ( 1 , 4 , 2 , 3 ) (1,4,2,3)
  253. ( 1 , 4 , 2 ) (1,4,2)
  254. S S
  255. M M
  256. w w
  257. M , w S M,w\models S
  258. ( K ) A 1 ; ; A n ; ¬ B A 1 ; ; A n ; ¬ B (K)\frac{\Box A_{1};\ldots;\Box A_{n};\neg\Box B}{A_{1};\ldots;A_{n};\neg B}
  259. A 1 , , A n A_{1},\ldots,A_{n}
  260. ¬ B \neg B
  261. ¬ B \neg B
  262. M M
  263. w w
  264. A 1 ; ; A n ; ¬ B \Box A_{1};\ldots;\Box A_{n};\neg\Box B
  265. w w
  266. w w^{\prime}
  267. w w
  268. A 1 ; ; A n ; ¬ B A_{1};\ldots;A_{n};\neg B
  269. w w^{\prime}
  270. A 1 ; ; A n ; ¬ B \Box A_{1};\ldots;\Box A_{n};\neg\Box B
  271. M , w M,w
  272. A 1 ; ; A n ; ¬ B A_{1};\ldots;A_{n};\neg B
  273. M , w M,w^{\prime}
  274. ( K ) (K)
  275. A 1 ; ; A n ; ¬ B 1 ; ¬ B 2 \Box A_{1};\ldots;\Box A_{n};\neg\Box B_{1};\neg\Box B_{2}
  276. B 1 B_{1}
  277. B 2 B_{2}
  278. ( K ) (K)
  279. a ; b ; ( b c ) ; ¬ c a;\Box b;\Box(b\rightarrow c);\neg\Box c
  280. ( K ) (K)
  281. a a
  282. ( θ ) A 1 ; ; A n ; B 1 ; ; B m A 1 ; ; A n (\theta)\frac{A_{1};\ldots;A_{n};B_{1};\ldots;B_{m}}{A_{1};\ldots;A_{n}}
  283. ( θ ) (\theta)
  284. ( θ ) (\theta)
  285. ( θ ) (\theta)
  286. ( T ) A 1 ; ; A n ; B A 1 ; ; A n ; B ; B (T)\frac{A_{1};\ldots;A_{n};\Box B}{A_{1};\ldots;A_{n};\Box B;B}
  287. B \Box B
  288. B B
  289. B \Box B
  290. B B
  291. B \Box B
  292. ( a ¬ a ) \Box(a\wedge\neg\Box a)
  293. ( T ) , ( ) , ( θ ) , ( K ) (T),(\wedge),(\theta),(K)
  294. a \Box a
  295. ¬ A \neg\Box A
  296. A A
  297. ¬ A \neg A
  298. M M
  299. A A
  300. w w
  301. B B
  302. B B
  303. A A
  304. G G
  305. M M
  306. G G
  307. B B
  308. { P , ¬ ( P Q ) } \{P,\neg\Box(P\wedge Q)\}
  309. ¬ Q \neg\Box Q
  310. P P
  311. ¬ P , Q \neg P,Q
  312. Q \Box Q
  313. P P
  314. ¬ P \neg P
  315. B B
  316. A A
  317. G G
  318. α \alpha
  319. α 1 α 2 \alpha_{1}\wedge\alpha_{2}
  320. α \alpha
  321. α 1 α 2 \alpha_{1}\wedge\alpha_{2}
  322. ¬ ( α 1 ¯ α 2 ¯ ) \neg(\overline{\alpha_{1}}\vee\overline{\alpha_{2}})
  323. ¬ ( α 1 α 2 ¯ ) \neg(\alpha_{1}\rightarrow\overline{\alpha_{2}})
  324. β \beta
  325. β 1 β 2 \beta_{1}\vee\beta_{2}
  326. β 1 ¯ β 2 \overline{\beta_{1}}\rightarrow\beta_{2}
  327. ¬ ( β 1 ¯ β 2 ¯ ) \neg(\overline{\beta_{1}}\wedge\overline{\beta_{2}})
  328. γ \gamma
  329. x γ 1 ( x ) \forall x\gamma_{1}(x)
  330. ¬ x γ 1 ( x ) ¯ \neg\exists x\overline{\gamma_{1}(x)}
  331. δ \delta
  332. x δ 1 ( x ) \exists x\delta_{1}(x)
  333. ¬ x δ 1 ( x ) ¯ \neg\forall x\overline{\delta_{1}(x)}
  334. π \pi
  335. π 1 \Diamond\pi_{1}
  336. ¬ π 1 ¯ \neg\Box\overline{\pi_{1}}
  337. υ \upsilon
  338. υ 1 \Box\upsilon_{1}
  339. ¬ υ 1 ¯ \neg\Diamond\overline{\upsilon_{1}}
  340. α 1 ¯ \overline{\alpha_{1}}
  341. α 1 \alpha_{1}
  342. ¬ ( a b ) \neg(a\vee b)
  343. α 1 \alpha_{1}
  344. a a
  345. ( α ) α α 1 α 2 (\alpha)\frac{\alpha}{\begin{array}[]{c}\alpha_{1}\\ \alpha_{2}\end{array}}

Method_of_characteristics.html

  1. a ( x , y , z ) z x + b ( x , y , z ) z y = c ( x , y , z ) . a(x,y,z)\frac{\partial z}{\partial x}+b(x,y,z)\frac{\partial z}{\partial y}=c(% x,y,z).
  2. ( z x ( x , y ) , z y ( x , y ) , - 1 ) . \left(\frac{\partial z}{\partial x}(x,y),\frac{\partial z}{\partial y}(x,y),-1% \right).\,
  3. ( a ( x , y , z ) , b ( x , y , z ) , c ( x , y , z ) ) (a(x,y,z),b(x,y,z),c(x,y,z))\,
  4. d x a ( x , y , z ) = d y b ( x , y , z ) = d z c ( x , y , z ) , \frac{dx}{a(x,y,z)}=\frac{dy}{b(x,y,z)}=\frac{dz}{c(x,y,z)},
  5. d x d t = a ( x , y , z ) d y d t = b ( x , y , z ) d z d t = c ( x , y , z ) . \begin{array}[]{rcl}\frac{dx}{dt}&=&a(x,y,z)\\ \frac{dy}{dt}&=&b(x,y,z)\\ \frac{dz}{dt}&=&c(x,y,z).\end{array}
  6. i = 1 n a i ( x 1 , , x n , u ) u x i = c ( x 1 , , x n , u ) . \sum_{i=1}^{n}a_{i}(x_{1},\dots,x_{n},u)\frac{\partial u}{\partial x_{i}}=c(x_% {1},\dots,x_{n},u).
  7. ( x 1 , , x n , u ) = ( x 1 ( s ) , , x n ( s ) , u ( s ) ) (x_{1},\dots,x_{n},u)=(x_{1}(s),\dots,x_{n}(s),u(s))
  8. d x i d s = a i ( x 1 , , x n , u ) \frac{dx_{i}}{ds}=a_{i}(x_{1},\dots,x_{n},u)
  9. d u d s = c ( x 1 , , x n , u ) . \frac{du}{ds}=c(x_{1},\dots,x_{n},u).
  10. p i = u x i . p_{i}=\frac{\partial u}{\partial x_{i}}.
  11. u ( s ) = u ( x 1 ( s ) , , x n ( s ) ) . u(s)=u(x_{1}(s),\dots,x_{n}(s)).
  12. i ( F x i + F u p i ) x ˙ i + i F p i p ˙ i = 0 \sum_{i}(F_{x_{i}}+F_{u}p_{i})\dot{x}_{i}+\sum_{i}F_{p_{i}}\dot{p}_{i}=0
  13. u ˙ - i p i x ˙ i = 0 \dot{u}-\sum_{i}p_{i}\dot{x}_{i}=0
  14. i ( x ˙ i d p i - p ˙ i d x i ) = 0. \sum_{i}(\dot{x}_{i}dp_{i}-\dot{p}_{i}dx_{i})=0.
  15. d u - i p i d x i = 0 du-\sum_{i}p_{i}\,dx_{i}=0
  16. x ˙ i = λ F p i , p ˙ i = - λ ( F x i + F u p i ) , u ˙ = λ i p i F p i \dot{x}_{i}=\lambda F_{p_{i}},\quad\dot{p}_{i}=-\lambda(F_{x_{i}}+F_{u}p_{i}),% \quad\dot{u}=\lambda\sum_{i}p_{i}F_{p_{i}}
  17. x ˙ i F p i = - p ˙ i F x i + F u p i = u ˙ p i F p i . \frac{\dot{x}_{i}}{F_{p_{i}}}=-\frac{\dot{p}_{i}}{F_{x_{i}}+F_{u}p_{i}}=\frac{% \dot{u}}{\sum p_{i}F_{p_{i}}}.
  18. a u x + u t = 0 a\frac{\partial u}{\partial x}+\frac{\partial u}{\partial t}=0\,
  19. a a\,
  20. u u\,
  21. x x\,
  22. t t\,
  23. d d s u ( x ( s ) , t ( s ) ) = F ( u , x ( s ) , t ( s ) ) \frac{d}{ds}u(x(s),t(s))=F(u,x(s),t(s))
  24. ( x ( s ) , t ( s ) ) (x(s),t(s))\,
  25. d d s u ( x ( s ) , t ( s ) ) = u x d x d s + u t d t d s \frac{d}{ds}u(x(s),t(s))=\frac{\partial u}{\partial x}\frac{dx}{ds}+\frac{% \partial u}{\partial t}\frac{dt}{ds}
  26. d x d s = a \frac{dx}{ds}=a
  27. d t d s = 1 \frac{dt}{ds}=1
  28. a u x + u t a\frac{\partial u}{\partial x}+\frac{\partial u}{\partial t}\,
  29. d d s u = a u x + u t = 0. \frac{d}{ds}u=a\frac{\partial u}{\partial x}+\frac{\partial u}{\partial t}=0.
  30. ( x ( s ) , t ( s ) ) (x(s),t(s))\,
  31. u s = F ( u , x ( s ) , t ( s ) ) = 0 u_{s}=F(u,x(s),t(s))=0\,
  32. u ( x s , t s ) = u ( x 0 , 0 ) u(x_{s},t_{s})=u(x_{0},0)\,
  33. ( x s , t s ) (x_{s},t_{s})\,
  34. ( x 0 , 0 ) (x_{0},0)\,
  35. d t d s = 1 \frac{dt}{ds}=1
  36. t ( 0 ) = 0 t(0)=0\,
  37. t = s t=s\,
  38. d x d s = a \frac{dx}{ds}=a
  39. x ( 0 ) = x 0 x(0)=x_{0}\,
  40. x = a s + x 0 = a t + x 0 x=as+x_{0}=at+x_{0}\,
  41. d u d s = 0 \frac{du}{ds}=0
  42. u ( 0 ) = f ( x 0 ) u(0)=f(x_{0})\,
  43. u ( x ( t ) , t ) = f ( x 0 ) = f ( x - a t ) u(x(t),t)=f(x_{0})=f(x-at)\,
  44. a a\,
  45. u u\,
  46. P : C ( X ) C ( X ) P:C^{\infty}(X)\to C^{\infty}(X)
  47. P = | α | k P α ( x ) x α P=\sum_{|\alpha|\leq k}P^{\alpha}(x)\frac{\partial}{\partial x^{\alpha}}
  48. σ P ( x , ξ ) = | α | = k P α ( x ) ξ α \sigma_{P}(x,\xi)=\sum_{|\alpha|=k}P^{\alpha}(x)\xi_{\alpha}
  49. σ P ( x , d F ( x ) ) = 0. \sigma_{P}(x,dF(x))=0.
  50. u u\,

Methods_of_contour_integration.html

  1. z ( a ) = z ( b ) z^{\prime}(a)=z^{\prime}(b)
  2. γ i \gamma_{i}
  3. γ i + 1 \gamma_{i+1}
  4. i , 1 i < n \forall i,1\leq i<n
  5. Γ = γ 1 + γ 2 + + γ n . \Gamma=\gamma_{1}+\gamma_{2}+\cdots+\gamma_{n}.
  6. f ( t ) = u ( t ) + i v ( t ) . f(t)=u(t)+iv(t).
  7. a b f ( t ) d t = a b [ u ( t ) + i v ( t ) ] d t = a b u ( t ) d t + i a b v ( t ) d t . \begin{aligned}\displaystyle\int_{a}^{b}f(t)dt&\displaystyle=\int_{a}^{b}\big[% u(t)+iv(t)\big]\,dt\\ &\displaystyle=\int_{a}^{b}u(t)dt+i\int_{a}^{b}v(t)\,dt.\end{aligned}
  8. γ f ( z ) \int_{\gamma}f(z)\,
  9. γ f ( z ) = a b f ( z ( t ) ) z ( t ) d t . \int_{\gamma}f(z)=\int_{a}^{b}f(z(t))z^{\prime}(t)\,dt.
  10. C 1 z d z . \oint_{C}{1\over z}\,dz.
  11. C 1 z d z = 0 2 π 1 e i t i e i t d t = i 0 2 π 1 d t = [ t ] 0 2 π i = ( 2 π - 0 ) i = 2 π i . \oint_{C}{1\over z}\,dz=\int_{0}^{2\pi}\frac{1}{e^{it}}ie^{it}\,dt=i\int_{0}^{% 2\pi}1\,dt=[t]_{0}^{2\pi}i=(2\pi-0)i=2\pi i.
  12. - 1 ( x 2 + 1 ) 2 d x , \int_{-\infty}^{\infty}{1\over(x^{2}+1)^{2}}dx,
  13. f ( z ) = 1 ( z 2 + 1 ) 2 f(z)={1\over(z^{2}+1)^{2}}
  14. C f ( z ) d z = - a a f ( z ) d z + Arc f ( z ) d z \oint_{C}f(z)\,dz=\int_{-a}^{a}f(z)\,dz+\int\text{Arc}f(z)\,dz
  15. - a a f ( z ) d z = C f ( z ) d z - Arc f ( z ) d z \int_{-a}^{a}f(z)\,dz=\oint_{C}f(z)\,dz-\int\text{Arc}f(z)\,dz
  16. f ( z ) = 1 ( z 2 + 1 ) 2 = 1 ( z + i ) 2 ( z - i ) 2 . f(z)={1\over(z^{2}+1)^{2}}={1\over(z+i)^{2}(z-i)^{2}}.
  17. f ( z ) = 1 ( z + i ) 2 ( z - i ) 2 , f(z)={{1\over(z+i)^{2}}\over(z-i)^{2}},
  18. C f ( z ) d z = C 1 ( z + i ) 2 ( z - i ) 2 d z = 2 π i d d z ( 1 ( z + i ) 2 ) | z = i = 2 π i ( - 2 ( z + i ) 3 ) | z = i = π 2 \oint_{C}f(z)\,dz=\oint_{C}\frac{\frac{1}{(z+i)^{2}}}{(z-i)^{2}}\,dz=2\pi i% \frac{d}{dz}\left({1\over(z+i)^{2}}\right)\Bigg|_{z=i}=2\pi i\left(\frac{-2}{(% z+i)^{3}}\right)\Bigg|_{z=i}=\frac{\pi}{2}
  19. | Arc f ( z ) d z | M L \left|\int\text{Arc}f(z)\,dz\right|\leq ML
  20. | Arc f ( z ) d z | a π ( a 2 - 1 ) 2 0 as a . \left|\int\text{Arc}f(z)\,dz\right|\leq{a\pi\over(a^{2}-1)^{2}}\rightarrow 0\ % \mathrm{as}\ a\rightarrow\infty.
  21. - 1 ( x 2 + 1 ) 2 d x = - f ( z ) d z = lim a + - a a f ( z ) d z = π 2 . \int_{-\infty}^{\infty}{1\over(x^{2}+1)^{2}}\,dx=\int_{-\infty}^{\infty}f(z)\,% dz=\lim_{a\to+\infty}\int_{-a}^{a}f(z)\,dz={\pi\over 2}.\quad\square
  22. f ( z ) = - 1 4 ( z - i ) 2 + - i 4 ( z - i ) + 3 16 + i 8 ( z - i ) + - 5 64 ( z - i ) 2 + f(z)={-1\over 4(z-i)^{2}}+{-i\over 4(z-i)}+{3\over 16}+{i\over 8}(z-i)+{-5% \over 64}(z-i)^{2}+\cdots
  23. C f ( z ) d z = C 1 ( z 2 + 1 ) 2 d z = 2 π i Res z = i f = 2 π i ( - i / 4 ) = π 2 \oint_{C}f(z)\,dz=\oint_{C}{1\over(z^{2}+1)^{2}}\,dz=2\pi i\,\mathrm{Res}_{z=i% }f=2\pi i(-i/4)={\pi\over 2}\quad\square
  24. - e i t x x 2 + 1 d x \int_{-\infty}^{\infty}{e^{itx}\over x^{2}+1}\,dx
  25. C e i t z z 2 + 1 d z . \int_{C}{e^{itz}\over z^{2}+1}\,dz.
  26. lim z i ( z - i ) f ( z ) = lim z i ( z - i ) e i t z z 2 + 1 = lim z i ( z - i ) e i t z ( z - i ) ( z + i ) = lim z i e i t z z + i = e - t 2 i . \lim_{z\to i}(z-i)f(z)=\lim_{z\to i}(z-i){e^{itz}\over z^{2}+1}=\lim_{z\to i}(% z-i){e^{itz}\over(z-i)(z+i)}=\lim_{z\to i}{e^{itz}\over z+i}={e^{-t}\over 2i}.
  27. C f ( z ) d z = ( 2 π i ) Res z = i f ( z ) = 2 π i e - t 2 i = π e - t . \int_{C}f(z)\,dz=(2\pi i)\operatorname{Res}_{z=i}f(z)=2\pi i{e^{-t}\over 2i}=% \pi e^{-t}.
  28. straight + arc = π e - t , \int_{\mbox{straight}~{}}+\int_{\mbox{arc}~{}}=\pi e^{-t},
  29. - a a = π e - t - arc . \int_{-a}^{a}=\pi e^{-t}-\int_{\mbox{arc}~{}}.
  30. arc e i t z z 2 + 1 d z 0 as a . \int_{\mbox{arc}~{}}{e^{itz}\over z^{2}+1}\,dz\rightarrow 0\ \mbox{as}~{}\ a% \rightarrow\infty.
  31. - e i t x x 2 + 1 d x = π e - t . \int_{-\infty}^{\infty}{e^{itx}\over x^{2}+1}\,dx=\pi e^{-t}.
  32. - e i t x x 2 + 1 d x = π e - | t | . \int_{-\infty}^{\infty}{e^{itx}\over x^{2}+1}\,dx=\pi e^{-\left|t\right|}.\quad\square
  33. - π π 1 1 + 3 ( cos t ) 2 d t . \int_{-\pi}^{\pi}{1\over 1+3(\cos{t})^{2}}\,dt.
  34. cos t = 1 2 ( e i t + e - i t ) = 1 2 ( z + 1 z ) \cos t={1\over 2}\left(e^{it}+e^{-it}\right)={1\over 2}\left(z+{1\over z}\right)
  35. d z d t = i z , d t = d z i z . {dz\over dt}=iz,\ dt={dz\over iz}.
  36. C 1 1 + 3 ( 1 2 ( z + 1 z ) ) 2 d z i z \displaystyle\oint_{C}{1\over 1+3({1\over 2}(z+{1\over z}))^{2}}\,{dz\over iz}
  37. - 4 3 i [ C 1 z ( z + 3 i ) ( z - 3 i ) ( z + i 3 ) z - i 3 d z + C 2 z ( z + 3 i ) ( z - 3 i ) ( z - i 3 ) z + i 3 ] \displaystyle-\frac{4}{3}i\left[\oint_{C_{1}}\frac{\frac{z}{(z+\sqrt{3}i)(z-% \sqrt{3}i)\left(z+\frac{i}{\sqrt{3}}\right)}}{z-\frac{i}{\sqrt{3}}}\,dz+\oint_% {C_{2}}\frac{\frac{z}{(z+\sqrt{3}i)(z-\sqrt{3}i)\left(z-\frac{i}{\sqrt{3}}% \right)}}{z+\frac{i}{\sqrt{3}}}\right]
  38. 0 2 π P ( sin ( t ) , sin ( 2 t ) , , cos ( t ) , cos ( 2 t ) , ) Q ( sin ( t ) , sin ( 2 t ) , , cos ( t ) , cos ( 2 t ) , ) d t \int_{0}^{2\pi}\frac{P(\sin(t),\sin(2t),\ldots,\cos(t),\cos(2t),\ldots)}{Q(% \sin(t),\sin(2t),\ldots,\cos(t),\cos(2t),\ldots)}\,dt
  39. z = exp ( i t ) z=\exp(it)
  40. d z = i exp ( i t ) d t dz=i\exp(it)\,dt
  41. 1 i z d z = d t . \frac{1}{iz}\,dz=dt.
  42. sin ( k t ) = exp ( i k t ) - exp ( - i k t ) 2 i = z k - z - k 2 i \sin(kt)=\frac{\exp(ikt)-\exp(-ikt)}{2i}=\frac{z^{k}-z^{-k}}{2i}
  43. cos ( k t ) = exp ( i k t ) + exp ( - i k t ) 2 = z k + z - k 2 \cos(kt)=\frac{\exp(ikt)+\exp(-ikt)}{2}=\frac{z^{k}+z^{-k}}{2}
  44. | z | = 1 f ( z ) 1 i z d z \oint_{|z|=1}f(z)\frac{1}{iz}\,dz
  45. f ( z ) 1 i z f(z)\frac{1}{iz}
  46. I = 0 π 2 1 1 + sin ( t ) 2 d t , I=\int_{0}^{\frac{\pi}{2}}\frac{1}{1+\sin(t)^{2}}\,dt,
  47. I = 1 4 0 2 π 1 1 + sin ( t ) 2 d t . I=\frac{1}{4}\int_{0}^{2\pi}\frac{1}{1+\sin(t)^{2}}\,dt.
  48. 1 4 | z | = 1 4 i z z 4 - 6 z 2 + 1 d z = | z | = 1 i z z 4 - 6 z 2 + 1 d z . \frac{1}{4}\oint_{|z|=1}\frac{4iz}{z^{4}-6z^{2}+1}\,dz=\oint_{|z|=1}\frac{iz}{% z^{4}-6z^{2}+1}\,dz.
  49. I = 2 π i 2 ( - 2 16 i ) = π 2 4 . I=2\pi i\;2\left(-\frac{\sqrt{2}}{16}i\right)=\pi\frac{\sqrt{2}}{4}.
  50. 0 x x 2 + 6 x + 8 d x . \int_{0}^{\infty}{\sqrt{x}\over x^{2}+6x+8}\,dx.
  51. C z z 2 + 6 z + 8 d z = I . \int_{C}{\sqrt{z}\over z^{2}+6z+8}\,dz=I.
  52. C = ε R + Γ + R ε + γ . \int_{C}=\int_{\varepsilon}^{R}+\int_{\Gamma}+\int_{R}^{\varepsilon}+\int_{% \gamma}.
  53. R ε z z 2 + 6 z + 8 d z = R ε e 1 2 Log ( z ) z 2 + 6 z + 8 d z = R ε e 1 2 ( log | z | + i arg z ) z 2 + 6 z + 8 d z = R ε e 1 2 log | z | e 1 / 2 ( 2 π i ) z 2 + 6 z + 8 d z = R ε e 1 2 log | z | e π i z 2 + 6 z + 8 d z = R ε - z z 2 + 6 z + 8 d z = ε R z z 2 + 6 z + 8 d z . \begin{aligned}\displaystyle\int_{R}^{\varepsilon}{\sqrt{z}\over z^{2}+6z+8}\,% dz&\displaystyle=\int_{R}^{\varepsilon}{e^{{1\over 2}\mathrm{Log}(z)}\over z^{% 2}+6z+8}\,dz\\ &\displaystyle=\int_{R}^{\varepsilon}{e^{{1\over 2}(\log{|z|}+i\arg{z})}\over z% ^{2}+6z+8}\,dz\\ &\displaystyle=\int_{R}^{\varepsilon}{e^{{1\over 2}\log{|z|}}e^{1/2(2\pi i)}% \over z^{2}+6z+8}\,dz\\ &\displaystyle=\int_{R}^{\varepsilon}{e^{{1\over 2}\log{|z|}}e^{\pi i}\over z^% {2}+6z+8}\,dz\\ &\displaystyle=\int_{R}^{\varepsilon}{-\sqrt{z}\over z^{2}+6z+8}\,dz\\ &\displaystyle=\int_{\varepsilon}^{R}{\sqrt{z}\over z^{2}+6z+8}\,dz.\end{aligned}
  54. C z z 2 + 6 z + 8 d z = 2 0 x x 2 + 6 x + 8 d x . \int_{C}{\sqrt{z}\over z^{2}+6z+8}\,dz=2\int_{0}^{\infty}{\sqrt{x}\over x^{2}+% 6x+8}\,dx.
  55. π i ( i 2 - i ) = 0 x x 2 + 6 x + 8 d x = π ( 1 - 1 2 ) . \pi i\left({i\over\sqrt{2}}-i\right)=\int_{0}^{\infty}{\sqrt{x}\over x^{2}+6x+% 8}\,dx=\pi\left(1-{1\over\sqrt{2}}\right).\quad\square
  56. 0 log ( x ) ( 1 + x 2 ) 2 d x \int_{0}^{\infty}\frac{\log(x)}{(1+x^{2})^{2}}\,dx
  57. f ( z ) = ( log ( z ) 1 + z 2 ) 2 f(z)=\left(\frac{\log(z)}{1+z^{2}}\right)^{2}
  58. - π < arg ( z ) π -\pi<\arg(z)\leq\pi
  59. ( R + M + N + r ) f ( z ) d z = 2 π i ( Res z = i f ( z ) + Res z = - i f ( z ) ) = 2 π i ( - π 4 + 1 16 i π 2 - π 4 - 1 16 i π 2 ) = - i π 2 . \begin{aligned}\displaystyle\left(\int_{R}+\int_{M}+\int_{N}+\int_{r}\right)f(% z)\,dz&\displaystyle=2\pi i\left(\mathrm{Res}_{z=i}f(z)+\mathrm{Res}_{z=-i}f(z% )\right)\\ &\displaystyle=2\pi i\left(-\frac{\pi}{4}+\frac{1}{16}i\pi^{2}-\frac{\pi}{4}-% \frac{1}{16}i\pi^{2}\right)\\ &\displaystyle=-i\pi^{2}.\end{aligned}
  60. | R f ( z ) d z | 2 π R ( log ( R ) ) 2 + π 2 ( R 2 - 1 ) 2 0. \left|\int_{R}f(z)\,dz\right|\leq 2\pi R\frac{(\log(R))^{2}+\pi^{2}}{(R^{2}-1)% ^{2}}\to 0.
  61. z = - x + i ϵ z=-x+i\epsilon
  62. z = - x - i ϵ z=-x-i\epsilon
  63. 0 log ( x ) ( 1 + x 2 ) 2 d x = - π 4 . \int_{0}^{\infty}\frac{\log(x)}{(1+x^{2})^{2}}\,dx=-\frac{\pi}{4}.
  64. I = 0 3 x 3 4 ( 3 - x ) 1 4 5 - x d x . I=\int_{0}^{3}\frac{x^{\frac{3}{4}}(3-x)^{\frac{1}{4}}}{5-x}\,dx.
  65. f ( z ) = z 3 4 ( 3 - z ) 1 4 . f(z)=z^{\frac{3}{4}}(3-z)^{\frac{1}{4}}.
  66. z 3 4 = exp ( 3 4 log ( z ) ) where - π arg ( z ) < π z^{\frac{3}{4}}=\exp\left(\frac{3}{4}\log(z)\right)\quad\mbox{where}~{}\quad-% \pi\leq\arg(z)<\pi
  67. ( 3 - z ) 1 4 = exp ( 1 4 log ( 3 - z ) ) where 0 arg ( 3 - z ) < 2 π . (3-z)^{\frac{1}{4}}=\exp\left(\frac{1}{4}\log(3-z)\right)\quad\mbox{where}~{}% \quad 0\leq\arg(3-z)<2\pi.
  68. r 3 4 exp ( - 3 π i 4 ) ( 3 + r ) 1 4 exp ( 0 π i 4 ) = r 3 4 ( 3 + r ) 1 4 exp ( - 3 π i 4 ) . r^{\frac{3}{4}}\exp(-\tfrac{3\pi i}{4})(3+r)^{\frac{1}{4}}\exp(\tfrac{0\pi i}{% 4})=r^{\frac{3}{4}}(3+r)^{\frac{1}{4}}\exp(-\tfrac{3\pi i}{4}).
  69. exp ( - 3 π i 4 ) = exp ( 5 π i 4 ) , \exp(-\tfrac{3\pi i}{4})=\exp(\tfrac{5\pi i}{4}),
  70. r 3 4 exp ( 0 π i 4 ) ( 3 - r ) 1 4 exp ( 2 π i 4 ) = i r 3 4 ( 3 - r ) 1 4 r^{\frac{3}{4}}\exp(\tfrac{0\pi i}{4})(3-r)^{\frac{1}{4}}\exp(\tfrac{2\pi i}{4% })=i\,r^{\frac{3}{4}}(3-r)^{\frac{1}{4}}
  71. r 3 4 exp ( 0 π i 4 ) ( 3 - r ) 1 4 exp ( 0 π i 4 ) = r 3 4 ( 3 - r ) 1 4 . r^{\frac{3}{4}}\exp(\tfrac{0\pi i}{4})(3-r)^{\frac{1}{4}}\exp(\tfrac{0\pi i}{4% })=r^{\frac{3}{4}}(3-r)^{\frac{1}{4}}.
  72. f ( z ) 5 - z \frac{f(z)}{5-z}
  73. | C L f ( z ) 5 - z d z | 2 π ρ ρ 3 4 ( 3 + 1 1000 ) 1 4 5 - 1 1000 𝒪 ( ρ 7 4 ) 0. \left|\int_{C_{L}}\frac{f(z)}{5-z}dz\right|\leq 2\pi\rho\frac{\rho^{\frac{3}{4% }}(3+\frac{1}{1000})^{\frac{1}{4}}}{5-\frac{1}{1000}}\in\mathcal{O}\left(\rho^% {\frac{7}{4}}\right)\to 0.
  74. | C R f ( z ) 5 - z d z | 2 π ρ ( 3 + 1 1000 ) 3 4 ρ 1 4 2 - 1 1000 𝒪 ( ρ 5 4 ) 0. \left|\int_{C_{R}}\frac{f(z)}{5-z}dz\right|\leq 2\pi\rho\frac{(3+\frac{1}{1000% })^{\frac{3}{4}}\rho^{\frac{1}{4}}}{2-\frac{1}{1000}}\in\mathcal{O}\left(\rho^% {\frac{5}{4}}\right)\to 0.
  75. ( - i + 1 ) I = - 2 π i ( Res z = 5 f ( z ) 5 - z + Res z = f ( z ) 5 - z ) . (-i+1)I=-2\pi i\left(\mathrm{Res}_{z=5}\frac{f(z)}{5-z}+\mathrm{Res}_{z=\infty% }\frac{f(z)}{5-z}\right).
  76. Res z = 5 f ( z ) 5 - z = - 5 3 4 exp ( log ( - 2 ) 4 ) . \mathrm{Res}_{z=5}\frac{f(z)}{5-z}=-5^{\frac{3}{4}}\exp\left(\tfrac{\log(-2)}{% 4}\right).
  77. - 5 3 4 exp ( log ( 2 ) + π i 4 ) = - exp ( π i 4 ) 5 3 4 2 1 4 . -5^{\frac{3}{4}}\exp\left(\tfrac{\log(2)+\pi i}{4}\right)=-\exp(\tfrac{\pi i}{% 4})5^{\frac{3}{4}}2^{\frac{1}{4}}.
  78. Res z = h ( z ) = Res z = 0 [ - 1 z 2 h ( 1 z ) ] . \mathrm{Res}_{z=\infty}h(z)=\mathrm{Res}_{z=0}\left[-\frac{1}{z^{2}}h\left(% \frac{1}{z}\right)\right].
  79. 1 5 - 1 z = - z ( 1 + 5 z + 5 2 z 2 + 5 3 z 3 + ) \frac{1}{5-\frac{1}{z}}=-z\left(1+5z+5^{2}z^{2}+5^{3}z^{3}+\cdots\right)
  80. ( 1 z 3 ( 3 - 1 z ) ) 1 4 = 1 z ( 3 z - 1 ) 1 4 = 1 z exp ( π i 4 ) ( 1 - 3 z ) 1 4 , \left(\frac{1}{z^{3}}\left(3-\frac{1}{z}\right)\right)^{\frac{1}{4}}=\frac{1}{% z}(3z-1)^{\frac{1}{4}}=\frac{1}{z}\exp(\tfrac{\pi i}{4})(1-3z)^{\frac{1}{4}},
  81. 1 z exp ( π i 4 ) ( 1 - ( 1 4 1 ) 3 z + ( 1 4 2 ) 3 2 z 2 - ( 1 4 3 ) 3 3 z 3 + ) . \frac{1}{z}\exp(\tfrac{\pi i}{4})\left(1-{\frac{1}{4}\choose 1}3z+{\frac{1}{4}% \choose 2}3^{2}z^{2}-{\frac{1}{4}\choose 3}3^{3}z^{3}+\cdots\right).
  82. Res z = f ( z ) 5 - z = exp ( π i 4 ) ( 5 - 3 4 ) = exp ( π i 4 ) 17 4 . \mathrm{Res}_{z=\infty}\frac{f(z)}{5-z}=\exp(\tfrac{\pi i}{4})\left(5-\frac{3}% {4}\right)=\exp(\tfrac{\pi i}{4})\frac{17}{4}.
  83. I = 2 π i exp ( π i 4 ) - 1 + i ( 17 4 - 5 3 4 2 1 4 ) = 2 π 2 - 1 2 ( 17 4 - 5 3 4 2 1 4 ) I=2\pi i\frac{\exp(\tfrac{\pi i}{4})}{-1+i}\left(\frac{17}{4}-5^{\frac{3}{4}}2% ^{\frac{1}{4}}\right)=2\pi 2^{-\frac{1}{2}}\left(\frac{17}{4}-5^{\frac{3}{4}}2% ^{\frac{1}{4}}\right)
  84. I = π 2 2 ( 17 - 5 3 4 2 9 4 ) = π 2 2 ( 17 - 40 3 4 ) . I=\frac{\pi}{2\sqrt{2}}\left(17-5^{\frac{3}{4}}2^{\frac{9}{4}}\right)=\frac{% \pi}{2\sqrt{2}}\left(17-40^{\frac{3}{4}}\right).
  85. ζ ( s ) \zeta(s)
  86. n = 1 1 n s \sum_{n=1}^{\infty}\frac{1}{n^{s}}
  87. ζ ( s ) = - Γ ( 1 - s ) 2 π i ( - t ) s - 1 e t - 1 d t \zeta(s)=-\frac{\Gamma(1-s)}{2\pi i}\int\frac{(-t)^{s-1}}{e^{t}-1}dt
  88. s s