wpmath0000003_9

Language_of_mathematics.html

  1. . \forall\ \exists\ \nabla\ \wedge\ \infty.
  2. x , y , z x,y,z
  3. sin x + a cos 2 x 0 \sin x+a\cos 2x\geq 0\,
  4. d y d x \tfrac{dy}{dx}

Laplace_number.html

  1. La = Su = σ ρ L μ 2 \mathrm{La}=\mathrm{Su}=\frac{\sigma\rho L}{\mu^{2}}
  2. La = Re 2 We \mathrm{La}=\frac{\mathrm{Re}^{2}}{\mathrm{We}}
  3. La = Oh - 2 \mathrm{La}=\mathrm{Oh}^{-2}

Laplace_transform_applied_to_differential_equations.html

  1. { f } = s { f } - f ( 0 ) \mathcal{L}\{f^{\prime}\}=s\mathcal{L}\{f\}-f(0)
  2. { f ′′ } = s 2 { f } - s f ( 0 ) - f ( 0 ) \mathcal{L}\{f^{\prime\prime}\}=s^{2}\mathcal{L}\{f\}-sf(0)-f^{\prime}(0)
  3. { f ( n ) } = s n { f } - i = 1 n s n - i f ( i - 1 ) ( 0 ) \mathcal{L}\{f^{(n)}\}=s^{n}\mathcal{L}\{f\}-\sum_{i=1}^{n}s^{n-i}f^{(i-1)}(0)
  4. i = 0 n a i f ( i ) ( t ) = ϕ ( t ) \sum_{i=0}^{n}a_{i}f^{(i)}(t)=\phi(t)
  5. f ( i ) ( 0 ) = c i f^{(i)}(0)=c_{i}
  6. i = 0 n a i { f ( i ) ( t ) } = { ϕ ( t ) } \sum_{i=0}^{n}a_{i}\mathcal{L}\{f^{(i)}(t)\}=\mathcal{L}\{\phi(t)\}
  7. { f ( t ) } i = 0 n a i s i - i = 1 n j = 1 i a i s i - j f ( j - 1 ) ( 0 ) = { ϕ ( t ) } \mathcal{L}\{f(t)\}\sum_{i=0}^{n}a_{i}s^{i}-\sum_{i=1}^{n}\sum_{j=1}^{i}a_{i}s% ^{i-j}f^{(j-1)}(0)=\mathcal{L}\{\phi(t)\}
  8. { f ( t ) } \mathcal{L}\{f(t)\}
  9. f ( i ) ( 0 ) f^{(i)}(0)
  10. c i c_{i}
  11. { f ( t ) } = { ϕ ( t ) } + i = 1 n j = 1 i a i s i - j c j - 1 i = 0 n a i s i \mathcal{L}\{f(t)\}=\frac{\mathcal{L}\{\phi(t)\}+\sum_{i=1}^{n}\sum_{j=1}^{i}a% _{i}s^{i-j}c_{j-1}}{\sum_{i=0}^{n}a_{i}s^{i}}
  12. { f ( t ) } . \mathcal{L}\{f(t)\}.
  13. f ( i ) ( 0 ) = c i = 0 i { 0 , 1 , 2 , n } f^{(i)}(0)=c_{i}=0\quad\forall i\in\{0,1,2,...\ n\}
  14. f ( t ) = - 1 { { ϕ ( t ) } i = 0 n a i s i } f(t)=\mathcal{L}^{-1}\left\{{\mathcal{L}\{\phi(t)\}\over\sum_{i=0}^{n}a_{i}s^{% i}}\right\}
  15. f ′′ ( t ) + 4 f ( t ) = sin ( 2 t ) f^{\prime\prime}(t)+4f(t)=\sin(2t)
  16. ϕ ( t ) = sin ( 2 t ) \phi(t)=\sin(2t)
  17. { ϕ ( t ) } = 2 s 2 + 4 \mathcal{L}\{\phi(t)\}=\frac{2}{s^{2}+4}
  18. s 2 { f ( t ) } - s f ( 0 ) - f ( 0 ) + 4 { f ( t ) } = { ϕ ( t ) } s^{2}\mathcal{L}\{f(t)\}-sf(0)-f^{\prime}(0)+4\mathcal{L}\{f(t)\}=\mathcal{L}% \{\phi(t)\}
  19. { f ( t ) } = 2 ( s 2 + 4 ) 2 \mathcal{L}\{f(t)\}=\frac{2}{(s^{2}+4)^{2}}
  20. f ( t ) = 1 8 sin ( 2 t ) - t 4 cos ( 2 t ) f(t)=\frac{1}{8}\sin(2t)-\frac{t}{4}\cos(2t)

Laplace–Stieltjes_transform.html

  1. e - s x d g ( x ) \int\mathrm{e}^{-sx}\,dg(x)
  2. { * g } ( s ) = - e - s x d g ( x ) . \{\mathcal{L}^{*}g\}(s)=\int_{-\infty}^{\infty}\mathrm{e}^{-sx}\,dg(x).
  3. { * g } ( s ) = 0 - e - s x d g ( x ) . \{\mathcal{L}^{*}g\}(s)=\int_{0^{-}}^{\infty}\mathrm{e}^{-sx}\,dg(x).
  4. lim ε 0 + - ε . \lim_{\varepsilon\to 0^{+}}\int_{-\varepsilon}^{\infty}.
  5. * g = ( d g ) . \mathcal{L}^{*}g=\mathcal{L}(dg).
  6. { * ( g * h ) } ( s ) = { * g } ( s ) { * h } ( s ) . \{\mathcal{L}^{*}(g*h)\}(s)=\{\mathcal{L}^{*}g\}(s)\{\mathcal{L}^{*}h\}(s).
  7. { * F } ( s ) = E [ e - s X ] . \{\mathcal{L}^{*}F\}(s)=\mathrm{E}\left[\mathrm{e}^{-sX}\right].
  8. sup i g ( t i ) - g ( t i + 1 ) X < \sup\sum_{i}\|g(t_{i})-g(t_{i+1})\|_{X}<\infty
  9. 0 = t 0 < t 1 < < t n = T . 0=t_{0}<t_{1}<\cdots<t_{n}=T.
  10. 0 T e - s t d g ( t ) \int_{0}^{T}e^{-st}dg(t)
  11. lim | π | 0 i = 0 n - 1 e - s τ i [ g ( t i + 1 ) - g ( t i ) ] \lim_{|\pi|\to 0}\sum_{i=0}^{n-1}e^{-s\tau_{i}}[g(t_{i+1})-g(t_{i})]
  12. lim T 0 T e - s t d g ( t ) \lim_{T\to\infty}\int_{0}^{T}e^{-st}dg(t)
  13. { * g } ( s ) = { g } ( s ) , \{\mathcal{L}^{*}g\}(s)=\{\mathcal{L}g^{\prime}\}(s),
  14. { * g } ( s ) = { * g } ( i s ) , s . \{\mathcal{F}^{*}g\}(s)=\{\mathcal{L}^{*}g\}(\mathrm{i}s),\quad s\in\mathbb{R}.
  15. 𝔼 [ X n ] = ( - 1 ) n d n { * F } ( s ) d s n | s = 0 . \mathbb{E}[X^{n}]=(-1)^{n}\left.\frac{\,\text{d}^{n}\{\mathcal{L}^{*}F\}(s)}{% \,\text{d}s^{n}}\right|_{s=0}.
  16. Y ~ ( s ) = { * F Y } ( s ) = 0 e - s t λ e - λ t d t = λ λ + s \tilde{Y}(s)=\{\mathcal{L}^{*}F_{Y}\}(s)=\int_{0}^{\infty}e^{-st}\lambda e^{-% \lambda t}dt=\frac{\lambda}{\lambda+s}
  17. Z = Y 1 + Y 2 + + Y n Z=Y_{1}+Y_{2}+\ldots+Y_{n}
  18. Z ~ ( s ) = Y ~ 1 ( s ) Y ~ 2 ( s ) Y ~ n ( s ) \tilde{Z}(s)=\tilde{Y}_{1}(s)\cdot\tilde{Y}_{2}(s)\cdot\cdots\cdot\tilde{Y}_{n% }(s)
  19. Z ~ ( s ) = ( λ λ + s ) n . \tilde{Z}(s)=\left(\frac{\lambda}{\lambda+s}\right)^{n}.
  20. U ~ ( s ) = a b e - s t 1 b - a d t = e - s a - e - s b s ( b - a ) . \tilde{U}(s)=\int_{a}^{b}e^{-st}\frac{1}{b-a}\,\text{d}t=\frac{e^{-sa}-e^{-sb}% }{s(b-a)}.

Largest_remainder_method.html

  1. total votes total seats \frac{\mbox{total}~{}\;\mbox{votes}~{}}{\mbox{total}~{}\;\mbox{seats}~{}}
  2. 1 + total votes 1 + total seats 1+\frac{\mbox{total}~{}\;\mbox{votes}~{}}{1+\mbox{total}~{}\;\mbox{seats}~{}}
  3. total votes 1 + total seats \frac{\mbox{total}~{}\;\mbox{votes}~{}}{1+\mbox{total}~{}\;\mbox{seats}~{}}
  4. total votes 2 + total seats \frac{\mbox{total}~{}\;\mbox{votes}~{}}{2+\mbox{total}~{}\;\mbox{seats}~{}}

Las_Vegas_algorithm.html

  1. ZPP = RP co -RP , \textrm{ZPP}=\textrm{RP}\cap\,\,\text{co}\,\textrm{-RP},\,\!

Laser_cutting.html

  1. R z = 12.528 ( S 0.542 ) / ( ( P 0.528 ) ( V 0.322 ) ) Rz=12.528\cdot(S^{0.542})/((P^{0.528})\cdot(V^{0.322}))
  2. S = S=
  3. P = P=
  4. V = V=

Laser_diode_rate_equations.html

  1. d N d t = I e V - N τ n - μ = 1 μ = M G μ P μ \frac{dN}{dt}=\frac{I}{eV}-\frac{N}{\tau_{n}}-\sum_{\mu=1}^{\mu=M}G_{\mu}P_{\mu}
  2. d P μ d t = Γ μ ( G μ - 1 τ p ) P μ + β μ N τ r \frac{dP_{\mu}}{dt}=\Gamma_{\mu}(G_{\mu}-\frac{1}{\tau_{p}})P_{\mu}+\beta_{\mu% }\frac{N}{\tau_{r}}
  3. τ n {\tau_{n}}
  4. Γ \Gamma
  5. τ p {\tau_{p}}
  6. β {\beta}
  7. τ r {\tau_{r}}
  8. τ n {\tau_{n}}
  9. τ p {\tau_{p}}
  10. G μ = α N [ 1 - ( 2 λ ( t ) - λ μ δ λ g ) 2 ] - α N 0 1 + ϵ μ = 1 μ = M P μ G_{\mu}=\frac{\alpha N[1-(2\frac{\lambda(t)-\lambda_{\mu}}{\delta\lambda_{g}})% ^{2}]-\alpha N_{0}}{1+\epsilon\sum_{\mu=1}^{\mu=M}P_{\mu}}
  11. λ ( t ) = λ 0 + k ( N t h - N ( t ) ) N t h \lambda(t)=\lambda_{0}+\frac{k(N_{th}-N(t))}{N_{th}}
  12. N t h = N t r + 1 α τ p Γ N_{th}=N_{tr}+\frac{1}{\alpha\tau_{p}\Gamma}
  13. β μ = β 0 1 + ( 2 ( λ s - λ μ ) / δ λ s ) 2 \beta_{\mu}=\frac{\beta_{0}}{1+(2(\lambda_{s}-\lambda_{\mu})/\delta\lambda_{s}% )^{2}}
  14. λ μ = λ 0 - μ δ λ + ( n - 1 ) δ λ 2 \lambda_{\mu}=\lambda_{0}-\mu\delta\lambda+\frac{(n-1)\delta\lambda}{2}
  15. 1 + ϵ μ = 1 μ = M P μ 1+\epsilon\sum_{\mu=1}^{\mu=M}P_{\mu}
  16. δ λ = k ( I 0 I t h - 1 ) \delta\lambda=k\left(\sqrt{\frac{I_{0}}{I_{th}}}-1\right)

Lasing_threshold.html

  1. R 1 R 2 exp ( 2 g threshold l ) exp ( - 2 α l ) = 1 R_{1}R_{2}\exp(2g\text{threshold}\,l)\exp(-2\alpha l)=1
  2. R 1 R_{1}
  3. R 2 R_{2}
  4. l l
  5. exp ( 2 g threshold l ) \exp(2g\text{threshold}\,l)
  6. exp ( - 2 α l ) \exp(-2\alpha l)
  7. α > 0 \alpha>0
  8. α = α 0 \alpha=\alpha_{0}
  9. g threshold = α 0 - 1 2 l ln ( R 1 R 2 ) g\text{threshold}=\alpha_{0}-\frac{1}{2l}\ln(R_{1}R_{2})
  10. R 1 R 2 < 1 R_{1}R_{2}<1
  11. g threshold g\text{threshold}
  12. l l
  13. l l
  14. α 0 \alpha_{0}
  15. l l
  16. R 1 = 1 R_{1}=1
  17. R OC R\text{OC}
  18. 2 g threshold l = 2 α 0 l - ln R OC 2g\text{threshold}\,l=2\alpha_{0}l-\ln R\text{OC}
  19. P threshold 2 g threshold l P\text{threshold}\propto 2g\text{threshold}\,l
  20. P threshold = K ( L - ln R OC ) P\text{threshold}=K(\,L-\ln R\text{OC}\,)
  21. L L
  22. L = 2 α 0 l L=2\alpha_{0}l
  23. K K
  24. L L
  25. - ln R OC -\ln R\text{OC}
  26. L = 2 α 0 l L=2\alpha_{0}l
  27. g threshold g\text{threshold}
  28. P threshold P\text{threshold}

Lattice_(group).html

  1. n \mathbb{R}^{n}
  2. n \mathbb{R}^{n}
  3. n \mathbb{R}^{n}
  4. n \mathbb{R}^{n}
  5. n \mathbb{Z}^{n}
  6. n \mathbb{R}^{n}
  7. n \mathbb{Z}^{n}
  8. 8 \mathbb{R}^{8}
  9. 24 \mathbb{R}^{24}
  10. 2 \mathbb{R}^{2}
  11. Λ \Lambda
  12. n \mathbb{R}^{n}
  13. Λ = { i = 1 n a i v i | a i } \Lambda=\left\{\left.\sum_{i=1}^{n}a_{i}v_{i}\;\right|\;a_{i}\in\mathbb{Z}\right\}
  14. n \mathbb{R}^{n}
  15. n \mathbb{R}^{n}
  16. T : z z + 1 T:z\mapsto z+1
  17. S : z - 1 / z S:z\mapsto-1/z
  18. n \mathbb{C}^{n}
  19. n \mathbb{C}^{n}
  20. n \mathbb{C}^{n}
  21. n \mathbb{R}^{n}
  22. n \mathbb{C}^{n}
  23. \mathbb{Z}
  24. n \mathbb{R}^{n}
  25. B = { 𝐯 1 , , 𝐯 n } B=\{\mathbf{v}_{1},\ldots,\mathbf{v}_{n}\}
  26. \mathcal{L}
  27. = { i = 1 n a i 𝐯 i | a i R , 𝐯 i B } . \mathcal{L}=\left\{\sum_{i=1}^{n}a_{i}\mathbf{v}_{i}\quad|\quad a_{i}\in R,% \mathbf{v}_{i}\in B\right\}.
  28. G L n ( R ) GL_{n}(R)
  29. T - 1 T^{-1}
  30. R * R^{*}
  31. * = { 𝐯 V | 𝐯 , 𝐱 R , 𝐱 } \mathcal{L}^{*}=\{\mathbf{v}\in V\quad|\quad\langle\mathbf{v},\mathbf{x}% \rangle\in R,\forall\mathbf{x}\in\mathcal{L}\}
  32. * = { 𝐯 V | 𝐯 , 𝐯 i R } . \mathcal{L}^{*}=\{\mathbf{v}\in V\quad|\quad\langle\mathbf{v},\mathbf{v}_{i}% \rangle\in R\}.

Lattice_(order).html

  1. L = a 1 a n \bigvee L=a_{1}\lor\cdots\lor a_{n}
  2. L = a 1 a n \bigwedge L=a_{1}\land\cdots\land a_{n}
  3. L = { a 1 , , a n } L=\{a_{1},\ldots,a_{n}\}
  4. a : x a \forall a\in\varnothing:x\leq a
  5. a : a x \forall a\in\varnothing:a\leq x
  6. = 0 \bigvee\varnothing=0
  7. = 1 \bigwedge\varnothing=1
  8. ( A B ) = ( A ) ( B ) \bigvee\left(A\cup B\right)=\left(\bigvee A\right)\vee\left(\bigvee B\right)
  9. ( A B ) = ( A ) ( B ) \bigwedge\left(A\cup B\right)=\left(\bigwedge A\right)\wedge\left(\bigwedge B\right)
  10. ( A ) = ( A ) ( ) = ( A ) 0 = A \bigvee\left(A\cup\emptyset\right)=\left(\bigvee A\right)\vee\left(\bigvee% \emptyset\right)=\left(\bigvee A\right)\vee 0=\bigvee A
  11. ( A ) = ( A ) ( ) = ( A ) 1 = A \bigwedge\left(A\cup\emptyset\right)=\left(\bigwedge A\right)\wedge\left(% \bigwedge\emptyset\right)=\left(\bigwedge A\right)\wedge 1=\bigwedge A
  12. A = A A\cup\emptyset=A
  13. H L H\subset L
  14. H H
  15. H H
  16. , \lor,\land
  17. \lor
  18. \land
  19. a b = b a a\lor b=b\lor a
  20. a b = b a a\land b=b\land a
  21. a ( b c ) = ( a b ) c a\lor(b\lor c)=(a\lor b)\lor c
  22. a ( b c ) = ( a b ) c a\land(b\land c)=(a\land b)\land c
  23. a ( a b ) = a a\lor(a\land b)=a
  24. a ( a b ) = a a\land(a\lor b)=a
  25. a a = a a\lor a=a
  26. a a = a a\land a=a
  27. \lor
  28. \land
  29. , \lor,\land
  30. , \lor,\land
  31. \lor
  32. \land
  33. a 0 = a a\lor 0=a
  34. a 1 = a a\land 1=a
  35. \lor
  36. \land
  37. \lor
  38. \land
  39. \lor
  40. \land
  41. \land
  42. \lor
  43. \lor
  44. \land
  45. \not=\varnothing
  46. \wedge
  47. \vee
  48. r ( x ) + r ( y ) r ( x y ) + r ( x y ) . r(x)+r(y)\geq r(x\wedge y)+r(x\vee y).
  49. x y x\wedge y
  50. x y x\vee y
  51. \vee
  52. \wedge
  53. x y = 1 x\vee y=1
  54. x y = 0. x\wedge y=0.
  55. \wedge
  56. { x 0 , x 1 , , x n } \{x_{0},x_{1},\ldots,x_{n}\}
  57. x 0 < x 1 < x 2 < < x n x_{0}<x_{1}<x_{2}<\ldots<x_{n}
  58. i I a i \bigvee_{i\in I}a_{i}
  59. x a x\leq a
  60. x b . x\nleq b.
  61. a a = a ( a ( a a ) ) = a a\lor a=a\lor(a\land(a\lor a))=a

Law_of_demand.html

  1. Q x = f ( P x ) , f < 0 , Q_{x}=f(P_{x}),\quad f^{\prime}<0,
  2. Q x Q_{x}
  3. P x P_{x}
  4. f f
  5. f f^{\prime}
  6. P x P_{x}
  7. Q x Q_{x}

Law_of_mass_action.html

  1. \leftrightharpoons
  2. affinity = α [ A ] a [ B ] b \mbox{affinity}~{}=\alpha[A]^{a}[B]^{b}\!
  3. \rightleftharpoons
  4. α \alpha
  5. ξ \xi
  6. α ( p - ξ ) a ( q - ξ ) b = α ( p + ξ ) a ( q + ξ ) b \alpha(p-\xi)^{a}(q-\xi)^{b}=\alpha^{\prime}(p^{\prime}+\xi)^{a^{\prime}}(q^{% \prime}+\xi)^{b^{\prime}}\!
  7. ξ \xi
  8. ( d x d t ) f o r w a r d = k ( p - x ) a ( q - x ) b \left(\frac{dx}{dt}\right)_{forward}=k(p-x)^{a}(q-x)^{b}
  9. ( d x d t ) r e v e r s e = k ( p + x ) a ( q + x ) b \left(\frac{dx}{dt}\right)_{reverse}=k^{\prime}(p^{\prime}+x)^{a^{\prime}}(q^{% \prime}+x)^{b^{\prime}}
  10. ( p - x ) a ( q - x ) b = k k ( p + x ) a ( q + x ) b (p-x)^{a}(q-x)^{b}=\frac{k^{\prime}}{k}(p^{\prime}+x)^{a^{\prime}}(q^{\prime}+% x)^{b^{\prime}}
  11. affinity = α [ A ] [ B ] \mbox{affinity}~{}=\alpha[A][B]\!
  12. k [ A ] eq [ B ] eq = k [ A ] eq [ B ] eq k[A]\text{eq}[B]\text{eq}=k^{\prime}[A^{\prime}]\text{eq}[B^{\prime}]\text{eq}\!
  13. ( p - ξ ) ( q - ξ ) = k k ( p + ξ ) ( q + ξ ) (p-\xi)(q-\xi)=\frac{k^{\prime}}{k}(p^{\prime}+\xi)(q^{\prime}+\xi)
  14. v = ψ ( k ( p - x ) ( q - x ) - k ( p + x ) ( q + x ) ) v=\psi(k(p-x)(q-x)-k^{\prime}(p^{\prime}+x)(q^{\prime}+x))\!
  15. ψ \psi
  16. ξ \xi
  17. affinity = k [ A ] α [ B ] β \mbox{affinity}~{}=k[A]^{\alpha}[B]^{\beta}\dots\!
  18. α A + β B σ S + τ T \alpha A+\beta B\dots\rightleftharpoons\sigma S+\tau T\dots
  19. forward reaction rate = k + [ A ] α [ B ] β \mbox{forward reaction rate}~{}=k_{+}[A]^{\alpha}[B]^{\beta}\dots\,\!
  20. backward reaction rate = k - [ S ] σ [ T ] τ \mbox{backward reaction rate}~{}=k_{-}[S]^{\sigma}[T]^{\tau}\dots\,\!
  21. K = k + k - = [ S ] σ [ T ] τ [ A ] α [ B ] β K=\frac{k_{+}}{k_{-}}=\frac{[S]^{\sigma}[T]^{\tau}\dots}{[A]^{\alpha}[B]^{% \beta}\dots}
  22. K = [ S ] σ [ T ] τ [ A ] α [ B ] β K=\frac{[S]^{\sigma}[T]^{\tau}\dots}{[A]^{\alpha}[B]^{\beta}\dots}\,
  23. r f = k f [ A ] [ B ] r_{f}=k_{f}[A][B]\,
  24. r f r_{f}
  25. r f r_{f}
  26. k B k_{B}
  27. T T
  28. E g E C - E V E_{g}\equiv E_{C}-E_{V}
  29. ( N V ( T ) ) (N_{V}(T))
  30. ( N C ( T ) ) (N_{C}(T))
  31. ( n o ) (n_{o})
  32. ( p o ) (p_{o})
  33. ( n i ) (n_{i})
  34. n o n_{o}
  35. p o p_{o}
  36. ( E F ) (E_{F})
  37. n o p o = ( N C e - E C - E F k B T ) ( N V e - E F - E V k B T ) = N C N V e - E g k B T = n i 2 n_{o}p_{o}=\left(N_{C}e^{-\frac{E_{C}-E_{F}}{k_{B}T}}\right)\left(N_{V}e^{-% \frac{E_{F}-E_{V}}{k_{B}T}}\right)=N_{C}N_{V}e^{-\frac{E_{g}}{k_{B}T}}=n_{i}^{2}

Law_of_tangents.html

  1. a - b a + b = tan [ 1 2 ( α - β ) ] tan [ 1 2 ( α + β ) ] . \frac{a-b}{a+b}=\frac{\tan[\frac{1}{2}(\alpha-\beta)]}{\tan[\frac{1}{2}(\alpha% +\beta)]}.
  2. a sin α = b sin β . \frac{a}{\sin\alpha}=\frac{b}{\sin\beta}.
  3. d = a sin α , d = b sin β d=\frac{a}{\sin\alpha},d=\frac{b}{\sin\beta}
  4. a = d sin α and b = d sin β . a=d\sin\alpha\,\text{ and }b=d\sin\beta.\,
  5. a - b a + b = d sin α - d sin β d sin α + d sin β = sin α - sin β sin α + sin β . \frac{a-b}{a+b}=\frac{d\sin\alpha-d\sin\beta}{d\sin\alpha+d\sin\beta}=\frac{% \sin\alpha-\sin\beta}{\sin\alpha+\sin\beta}.
  6. sin ( α ) ± sin ( β ) = 2 sin ( α ± β 2 ) cos ( α β 2 ) , \sin(\alpha)\pm\sin(\beta)=2\sin\left(\frac{\alpha\pm\beta}{2}\right)\cos\left% (\frac{\alpha\mp\beta}{2}\right),\;
  7. a - b a + b = 2 sin 1 2 ( α - β ) cos 1 2 ( α + β ) 2 sin 1 2 ( α + β ) cos 1 2 ( α - β ) = tan [ 1 2 ( α - β ) ] tan [ 1 2 ( α + β ) ] . \frac{a-b}{a+b}=\frac{2\sin\tfrac{1}{2}\left(\alpha-\beta\right)\cos\tfrac{1}{% 2}\left(\alpha+\beta\right)}{2\sin\tfrac{1}{2}\left(\alpha+\beta\right)\cos% \tfrac{1}{2}\left(\alpha-\beta\right)}=\frac{\tan[\frac{1}{2}(\alpha-\beta)]}{% \tan[\frac{1}{2}(\alpha+\beta)]}.
  8. tan ( α ± β 2 ) = sin α ± sin β cos α + cos β \tan\left(\frac{\alpha\pm\beta}{2}\right)=\frac{\sin\alpha\pm\sin\beta}{\cos% \alpha+\cos\beta}
  9. a , b a,b
  10. γ \gamma
  11. tan [ 1 2 ( α - β ) ] = a - b a + b tan [ 1 2 ( α + β ) ] = a - b a + b cot ( γ 2 ) \tan\left[\frac{1}{2}(\alpha-\beta)\right]=\frac{a-b}{a+b}\tan\left[\frac{1}{2% }(\alpha+\beta)\right]=\frac{a-b}{a+b}\cot\left(\frac{\gamma}{2}\right)
  12. α - β \alpha-\beta
  13. α + β = 180 - γ \alpha+\beta=180^{\circ}-\gamma
  14. α \alpha
  15. β \beta
  16. c c
  17. c = a 2 + b 2 - 2 a b cos γ c=\sqrt{a^{2}+b^{2}-2ab\cos\gamma}
  18. γ \gamma
  19. a a
  20. b b
  21. tan ( A - B 2 ) tan ( A + B 2 ) = tan ( a - b 2 ) tan ( a + b 2 ) . \frac{\tan\left(\dfrac{A-B}{2}\right)}{\tan\left(\dfrac{A+B}{2}\right)}=\frac{% \tan\left(\dfrac{a-b}{2}\right)}{\tan\left(\dfrac{a+b}{2}\right)}.

Law_of_the_iterated_logarithm.html

  1. S n / n S_{n}/n
  2. 1 / n 1/\sqrt{n}
  3. 2 log log n / n \sqrt{2\log\log n/n}
  4. lim sup n S n n log log n = 2 , a.s. , \limsup_{n\to\infty}\frac{S_{n}}{\sqrt{n\log\log n}}=\sqrt{2},\qquad\,\text{a.% s.},
  5. S n n 𝑝 0 , S n n a . s . 0 , as n . \frac{S_{n}}{n}\ \xrightarrow{p}\ 0,\qquad\frac{S_{n}}{n}\ \xrightarrow{a.s.}0% ,\qquad\,\text{as}\ \ n\to\infty.
  6. lim sup n S n n > M \limsup_{n}\frac{S_{n}}{\sqrt{n}}>M
  7. P ( lim sup n S n n > M ) lim sup n P ( S n n > M ) = P ( 𝒩 ( 0 , 1 ) > M ) > 0 P(\limsup_{n}\frac{S_{n}}{\sqrt{n}}>M)\geq\limsup_{n}P(\frac{S_{n}}{\sqrt{n}}>% M)=P(\mathcal{N}(0,1)>M)>0
  8. lim sup n S n n = \limsup_{n}\frac{S_{n}}{\sqrt{n}}=\infty
  9. lim inf n S n n = - \liminf_{n}\frac{S_{n}}{\sqrt{n}}=-\infty
  10. S 2 n 2 n - S n n = 1 2 S 2 n - S n n - ( 1 - 1 2 ) S n n \frac{S_{2n}}{\sqrt{2n}}-\frac{S_{n}}{\sqrt{n}}=\frac{1}{\sqrt{2}}\frac{S_{2n}% -S_{n}}{\sqrt{n}}-(1-\frac{1}{\sqrt{}}2)\frac{S_{n}}{\sqrt{n}}
  11. S n n \frac{S_{n}}{\sqrt{n}}
  12. S 2 n - S n n \frac{S_{2n}-S_{n}}{\sqrt{n}}
  13. 𝒩 ( 0 , 1 ) . \mathcal{N}(0,1).
  14. S n n log log n 𝑝 0 , S n n log log n a . s . 0 , as n . \frac{S_{n}}{\sqrt{n\log\log n}}\ \xrightarrow{p}\ 0,\qquad\frac{S_{n}}{\sqrt{% n\log\log n}}\ \stackrel{a.s.}{\nrightarrow}\ 0,\qquad\,\text{as}\ \ n\to\infty.
  15. S n / n log log n S_{n}/\sqrt{n\log\log n}

LC_circuit.html

  1. ω 0 \omega_{0}
  2. ω 0 = 1 L C \omega_{0}={1\over\sqrt{LC}}
  3. ω 0 \omega_{0}\,
  4. f 0 = ω 0 2 π = 1 2 π L C . f_{0}={\omega_{0}\over 2\pi}={1\over{2\pi\sqrt{LC}}}.
  5. V C + V L = 0. V_{C}+V_{L}=0.\,
  6. i C = i L . i_{C}=i_{L}.\,
  7. V L ( t ) = L d i L d t V_{L}(t)=L\frac{di_{L}}{dt}\,
  8. i C ( t ) = C d V C d t . i_{C}(t)=C\frac{dV_{C}}{dt}.\,
  9. d 2 i ( t ) d t 2 + 1 L C i ( t ) = 0. \frac{d^{2}i(t)}{dt^{2}}+\frac{1}{LC}i(t)=0.\,
  10. ω 0 = 1 L C \omega_{0}={1\over\sqrt{LC}}
  11. d 2 i ( t ) d t 2 + ω 0 2 i ( t ) = 0. \frac{d^{2}i(t)}{dt^{2}}+\omega_{0}^{2}i(t)=0.\,
  12. s 2 + ω 0 2 = 0 s^{2}+\omega_{0}^{2}=0
  13. s = + j ω 0 s=+j\omega_{0}\,
  14. s = - j ω 0 s=-j\omega_{0}\,
  15. i ( t ) = A e + j ω 0 t + B e - j ω 0 t i(t)=Ae^{+j\omega_{0}t}+Be^{-j\omega_{0}t}\,
  16. A = B * A=B^{*}
  17. A = I 0 2 e j ϕ A={I_{0}\over 2}e^{j\phi}
  18. B = I 0 2 e - j ϕ B={I_{0}\over 2}e^{-j\phi}
  19. ϕ \phi
  20. i ( t ) = I 0 cos ( ω 0 t + ϕ ) . i(t)=I_{0}\cos(\omega_{0}t+\phi).\,
  21. V ( t ) = L d i d t = - ω 0 L I 0 sin ( ω 0 t + ϕ ) V(t)=L\frac{di}{dt}=-\omega_{0}LI_{0}\sin(\omega_{0}t+\phi)\,
  22. i ( t = 0 ) = I 0 cos ( ϕ ) . i(t=0)=I_{0}\cos(\phi).\,
  23. V ( t = 0 ) = L d i d t ( t = 0 ) = - ω 0 L I 0 sin ( ϕ ) . V(t=0)=L\frac{di}{dt}(t=0)=-\omega_{0}LI_{0}\sin(\phi).\,
  24. v = v L + v C v=v_{L}+v_{C}\,
  25. i = i L = i C i=i_{L}=i_{C}\,
  26. X L \scriptstyle X_{L}\,
  27. X C \scriptstyle X_{C}\,
  28. f 0 \scriptstyle f_{0}\,
  29. X L = - X C X_{L}=-X_{C}\,
  30. ω L = 1 ω C {\omega{L}}={{1}\over{\omega}{C}}\,
  31. ω \scriptstyle\omega
  32. ω = ω 0 = 1 L C \omega=\omega_{0}={1\over\sqrt{LC}}
  33. f 0 = ω 0 2 π = 1 2 π L C f_{0}={\omega_{0}\over 2\pi}={1\over{2\pi\sqrt{LC}}}
  34. f f 0 \scriptstyle f\to f_{0}
  35. f < f 0 \scriptstyle f<f_{0}
  36. X L ( - X C ) \scriptstyle X_{L}\;\ll\;(-X_{C})\,
  37. f > f 0 \scriptstyle f>f_{0}
  38. X L ( - X C ) \scriptstyle X_{L}\;\gg\;(-X_{C})\,
  39. Z = Z L + Z C Z=Z_{L}+Z_{C}
  40. Z ( ω ) = j ω L + 1 j ω C Z(\omega)=j\omega L+\frac{1}{j{\omega C}}
  41. Z ( ω ) = j ( ω 2 L C - 1 ) ω C Z(\omega)=j\frac{(\omega^{2}LC-1)}{\omega C}
  42. ω 0 = 1 L C \omega_{0}={1\over\sqrt{LC}}
  43. Z ( ω ) = j L ( ω 2 - ω 0 2 ω ) Z(\omega)=jL\bigg({\omega^{2}-\omega_{0}^{2}\over\omega}\bigg)
  44. ω ± ω 0 \omega\to\pm\omega_{0}
  45. v = v L = v C v=v_{L}=v_{C}\,
  46. i = i L + i C i=i_{L}+i_{C}\,
  47. f 0 = ω 0 2 π = 1 2 π L C f_{0}={\omega_{0}\over 2\pi}={1\over{2\pi\sqrt{LC}}}
  48. Z = Z L Z C Z L + Z C Z=\frac{Z_{L}Z_{C}}{Z_{L}+Z_{C}}
  49. Z L \scriptstyle Z_{L}
  50. Z C \scriptstyle Z_{C}
  51. Z ( ω ) = - j ω L ω 2 L C - 1 Z(\omega)=-j\frac{\omega L}{\omega^{2}LC-1}
  52. Z ( ω ) = - j ( 1 C ) ( ω ω 2 - ω 0 2 ) Z(\omega)=-j\bigg({1\over C}\bigg)\bigg(\frac{\omega}{\omega^{2}-\omega_{0}^{2% }}\bigg)
  53. ω 0 = 1 L C \omega_{0}={1\over\sqrt{LC}}
  54. lim ω ± ω 0 Z ( ω ) = \lim_{\omega\to\pm\omega_{0}}Z(\omega)=\infty
  55. ω \scriptstyle\omega

Lebesgue_covering_dimension.html

  1. 𝔼 n \mathbb{E}^{n}
  2. n \leq n
  3. f : A S n f:A\rightarrow S^{n}
  4. f f
  5. g : X S n g:X\rightarrow S^{n}
  6. S n S^{n}
  7. X X
  8. 0 dim X n 0\leq\dim X\leq n
  9. 𝒰 = { U α } α 𝒜 \mathcal{U}=\{U_{\alpha}\}_{\alpha\in\mathcal{A}}
  10. X X
  11. 𝒱 \mathcal{V}
  12. X X
  13. n + 1 n+1
  14. 𝒱 1 , 𝒱 2 , , 𝒱 n + 1 \mathcal{V}_{1},\mathcal{V}_{2},\dots,\mathcal{V}_{n+1}
  15. 𝒱 i = { V i , α } α 𝒜 \mathcal{V}_{i}=\{V_{i,\alpha}\}_{\alpha\in\mathcal{A}}
  16. 𝒱 i \mathcal{V}_{i}
  17. V i , α U α V_{i,\alpha}\subset U_{\alpha}
  18. i i
  19. α \alpha
  20. X X
  21. H i ( X , A ) = 0 H^{i}(X,A)=0
  22. A A
  23. X X
  24. i i
  25. X X

Lebesgue–Stieltjes_integration.html

  1. a b f ( x ) d g ( x ) \int_{a}^{b}f(x)\,dg(x)
  2. f : a a , b 𝐑 f:aa,b→\mathbf{R}
  3. g : a a , b 𝐑 g:aa,b→\mathbf{R}
  4. a a , b aa,b
  5. f f
  6. g g
  7. f f
  8. g g
  9. w ( ( s , t ) = g ( t ) g ( s ) w((s,t)=g(t)−g(s)
  10. g g
  11. w ( s s , t ) ) = g ( t ) g ( s ) w(ss,t))=g(t)−g(s)
  12. a a , b aa,b
  13. w w
  14. I I
  15. μ g ( E ) = inf { i μ g ( I i ) : E i I i } \mu_{g}(E)=\inf\left\{\sum_{i}\mu_{g}(I_{i})\ :\ E\subset\bigcup_{i}I_{i}\right\}
  16. E E
  17. g g
  18. a b f ( x ) d g ( x ) \int_{a}^{b}f(x)\,dg(x)
  19. f f
  20. g g
  21. a b f ( x ) d g ( x ) := - a b f ( x ) d ( - g ) ( x ) , \int_{a}^{b}f(x)\,dg(x):=-\int_{a}^{b}f(x)\,d(-g)(x),
  22. g g
  23. f f
  24. d g ( x ) = d g 1 ( x ) - d g 2 ( x ) dg(x)=dg_{1}(x)-dg_{2}(x)
  25. g g
  26. a a , x aa,x
  27. g g
  28. a b f ( x ) d g ( x ) = a b f ( x ) d g 1 ( x ) - a b f ( x ) d g 2 ( x ) , \int_{a}^{b}f(x)\,dg(x)=\int_{a}^{b}f(x)\,dg_{1}(x)-\int_{a}^{b}f(x)\,dg_{2}(x),
  29. g g
  30. a a , b aa,b
  31. I ( f ) I(f)
  32. I ( f ) = a b f ( x ) d g ( x ) I(f)=\int_{a}^{b}f(x)\,dg(x)
  33. f f
  34. I I
  35. a a , b aa,b
  36. I ¯ ( h ) = sup { I ( f ) : f C [ a , b ] , 0 f h } I ¯ ¯ ( h ) = inf { I ( f ) : f C [ a , b ] , h f } . \begin{aligned}\displaystyle\overline{I}(h)&\displaystyle=\sup\left\{I(f)\ :\ % f\in C[a,b],0\leq f\leq h\right\}\\ \displaystyle\overline{\overline{I}}(h)&\displaystyle=\inf\left\{I(f)\ :\ f\in C% [a,b],h\leq f\right\}.\end{aligned}
  37. I ¯ ( h ) = I ¯ ¯ ( h ) , \overline{I}(h)=\overline{\overline{I}}(h),
  38. h h
  39. μ g ( A ) = I ¯ ¯ ( χ A ) \mu_{g}(A)=\overline{\overline{I}}(\chi_{A})
  40. A A
  41. γ γ
  42. a b ρ ( γ ( t ) ) d ( t ) , \int_{a}^{b}\rho(\gamma(t))\,d\ell(t),
  43. ( t ) \ell(t)
  44. γ γ
  45. a a , t aa,t
  46. ρ ρ
  47. γ γ
  48. ρ ( z ) ρ(z)
  49. z z
  50. ρ ρ
  51. γ γ
  52. γ γ
  53. ρ ρ
  54. f f
  55. a a
  56. f ( a + ) f(a+)
  57. f ( a ) f(a−)
  58. f ( a ) = f ( a - ) + f ( a + ) 2 , f(a)=\frac{f(a-)+f(a+)}{2},
  59. U U
  60. V V
  61. U U
  62. V V
  63. U U
  64. V V
  65. a b U d V + a b V d U = U ( b + ) V ( b + ) - U ( a - ) V ( a - ) , b > a . \int_{a}^{b}U\,dV+\int_{a}^{b}V\,dU=U(b+)V(b+)-U(a-)V(a-),\qquad b>a.
  66. U U
  67. V V
  68. U U
  69. V V
  70. U ( t ) V ( t ) = U ( 0 ) V ( 0 ) + ( 0 , t ] U ( s - ) d V ( s ) + ( 0 , t ] V ( s - ) d U ( s ) + u ( 0 , t ] Δ U u Δ V u , U(t)V(t)=U(0)V(0)+\int_{(0,t]}U(s-)\,dV(s)+\int_{(0,t]}V(s-)\,dU(s)+\sum_{u\in% (0,t]}\Delta U_{u}\Delta V_{u},
  71. Δ U ( t ) Δ V ( t ) = d U U , V , , ΔU(t)ΔV(t)=dUU,V,,
  72. U U
  73. V V
  74. g ( x ) = x g(x)=x
  75. x x
  76. f f
  77. g g
  78. f f
  79. f f
  80. v v
  81. a b f ( x ) d v ( x ) \int_{a}^{b}f(x)\,dv(x)
  82. v v
  83. X X
  84. - f ( x ) d v ( x ) = E [ f ( X ) ] . \int_{-\infty}^{\infty}f(x)\,dv(x)=\mathrm{E}[f(X)].

Leech_lattice.html

  1. π 12 12 ! \tfrac{\pi^{12}}{12!}
  2. a 1 + a 2 + + a 24 4 a 1 4 a 2 4 a 24 ( mod 8 ) a_{1}+a_{2}+\cdots+a_{24}\equiv 4a_{1}\equiv 4a_{2}\equiv\cdots\equiv 4a_{24}% \;\;(\mathop{{\rm mod}}8)
  3. w / w w^{\perp}/w
  4. ( 0 , 1 , 2 , 3 , , 22 , 23 , 24 ; 70 ) (0,1,2,3,\dots,22,23,24;70)
  5. ( 0 , 1 , 2 , 3 , , 22 , 23 , 24 ) (0,1,2,3,\dots,22,23,24)
  6. ( I a H / 2 H / 2 I b ) \begin{pmatrix}Ia&H/2\\ H/2&Ib\end{pmatrix}
  7. D 24 D_{24}
  8. ( - 23 ) \mathbb{Q}(\sqrt{-23})
  9. 2 \sqrt{2}
  10. 2 \sqrt{2}
  11. π 12 12 ! 0.001930 \tfrac{\pi^{12}}{12!}\approx 0.001930
  12. Θ Λ ( τ ) = x Λ e i π τ x 2 Im τ > 0. \Theta_{\Lambda}(\tau)=\sum_{x\in\Lambda}e^{i\pi\tau\|x\|^{2}}\qquad\mathrm{Im% }\,\tau>0.
  13. q = e 2 i π τ q=e^{2i\pi\tau}
  14. m = 0 65520 691 ( σ 11 ( m ) - τ ( m ) ) q 2 m = 1 + 196560 q 4 + 16773120 q 6 + 398034000 q 8 + \sum_{m=0}^{\infty}\frac{65520}{691}\left(\sigma_{11}(m)-\tau(m)\right)q^{2m}=% 1+196560q^{4}+16773120q^{6}+398034000q^{8}+\cdots
  15. τ ( n ) \tau(n)
  16. σ 11 ( n ) \sigma_{11}(n)
  17. 65520 691 ( σ 11 ( m ) - τ ( m ) ) . \frac{65520}{691}\left(\sigma_{11}(m)-\tau(m)\right).

Lefschetz_fixed-point_theorem.html

  1. f : X X f:X\rightarrow X\,
  2. Λ f := k 0 ( - 1 ) k Tr ( f * | H k ( X , ) ) , \Lambda_{f}:=\sum_{k\geq 0}(-1)^{k}\mathrm{Tr}(f_{*}|H_{k}(X,\mathbb{Q})),
  3. Λ f 0 \Lambda_{f}\neq 0\,
  4. x Fix ( f ) i ( f , x ) = Λ f , \sum_{x\in\mathrm{Fix}(f)}i(f,x)=\Lambda_{f},
  5. f \scriptstyle f_{\ast}
  6. Λ id = χ ( X ) . \Lambda_{\mathrm{id}}=\chi(X).
  7. Λ f , g = ( - 1 ) k Tr ( D X g * D Y - 1 f * ) , \Lambda_{f,g}=\sum(-1)^{k}\mathrm{Tr}(D_{X}\circ g^{*}\circ D_{Y}^{-1}\circ f_% {*}),
  8. X X\,
  9. k k
  10. q q
  11. X ¯ \bar{X}
  12. X X\,
  13. k k
  14. F q F_{q}
  15. X ¯ \bar{X}
  16. x 1 , , x n x_{1},\ldots,x_{n}
  17. x 1 q , , x n q x_{1}^{q},\ldots,x_{n}^{q}
  18. F q F_{q}
  19. F q F_{q}
  20. X X
  21. k k
  22. X ( k ) X(k)
  23. # X ( k ) = i ( - 1 ) i tr F q | H c i ( X ¯ , ) . \#X(k)=\sum_{i}(-1)^{i}\mathop{\rm tr}F_{q}|H^{i}_{c}(\bar{X},{\mathbb{Q}}_{% \ell}).
  24. X ¯ \bar{X}
  25. \ell
  26. \ell
  27. q q
  28. X X
  29. Φ q \Phi_{q}
  30. F q F_{q}
  31. # X ( k ) = q dim X i ( - 1 ) i tr Φ q - 1 | H i ( X ¯ , ) . \#X(k)=q^{\dim X}\sum_{i}(-1)^{i}\mathop{\rm tr}\Phi_{q}^{-1}|H^{i}(\bar{X},{% \mathbb{Q}}_{\ell}).

Legendre's_constant.html

  1. π ( x ) \scriptstyle\pi(x)
  2. π ( x ) \scriptstyle\pi(x)
  3. π ( x ) = x ln ( x ) - B ( x ) \pi(x)=\frac{x}{\ln(x)-B(x)}
  4. lim x B ( x ) = 1.08366 \lim_{x\to\infty}B(x)=1.08366
  5. lim n ( ln ( n ) - n π ( n ) ) = B \lim_{n\to\infty}\left(\ln(n)-{n\over\pi(n)}\right)=B
  6. π ( x ) = Li ( x ) + O ( x e - a ln x ) as x \pi(x)={\rm Li}(x)+O\left(x\mathrm{e}^{-a\sqrt{\ln x}}\right)\quad\,\text{as }% x\to\infty
  7. x ln x - 1 < π ( x ) \frac{x}{\ln x-1}<\pi(x)
  8. x 5393 x\geq 5393
  9. π ( x ) < x ln x - 1.1 \pi(x)<\frac{x}{\ln x-1.1}
  10. x 60184 x\geq 60184
  11. π ( x ) = x ln ( x ) - A ( x ) \pi(x)=\frac{x}{\ln(x)-A(x)}
  12. 1 A ( x ) < 1.1 1\leq A(x)<1.1
  13. ζ ( s ) \zeta(s)

Legendre_form.html

  1. k \scriptstyle{k}
  2. x = 1 - k 2 cos ( t ) \scriptstyle{x=\sqrt{1-k^{2}}\cos(t)}
  3. y = sin ( t ) \scriptstyle{y=\sin(t)}
  4. F ( ϕ , k ) = 0 ϕ 1 1 - k 2 sin 2 ( t ) d t , F(\phi,k)=\int_{0}^{\phi}\frac{1}{\sqrt{1-k^{2}\sin^{2}(t)}}dt,
  5. E ( ϕ , k ) = 0 ϕ 1 - k 2 sin 2 ( t ) d t , E(\phi,k)=\int_{0}^{\phi}\sqrt{1-k^{2}\sin^{2}(t)}\,dt,
  6. Π ( ϕ , n , k ) = 0 ϕ 1 ( 1 - n sin 2 ( t ) ) 1 - k 2 sin 2 ( t ) d t . \Pi(\phi,n,k)=\int_{0}^{\phi}\frac{1}{(1-n\sin^{2}(t))\sqrt{1-k^{2}\sin^{2}(t)% }}\,dt.
  7. Π ( ϕ , - n , k ) \scriptstyle{\Pi(\phi,-n,k)}
  8. ϕ \scriptstyle{\phi}
  9. π / 2 \scriptstyle{\pi/2}
  10. y 2 = x ( x - 1 ) ( x - λ ) y^{2}=x(x-1)(x-\lambda)
  11. k \scriptstyle{k}
  12. ϕ \scriptstyle{\phi}
  13. k \scriptstyle{k}

Legendre_transformation.html

  1. f ( x ) f(x)
  2. a a , b aa,b
  3. p x f ( x ) px−f(x)
  4. $\mathbf{ }$
  5. f * ( p ) = p x f ( x x ) ) f*(p)=px−f(xx))
  6. g g
  7. f f
  8. D f = ( D g ) - 1 . Df=\left(Dg\right)^{-1}~{}.
  9. I 𝐑 I⊂\mathbf{R}
  10. f : I 𝐑 f:I→\mathbf{R}
  11. f * : I * 𝐑 f*:I*→\mathbf{R}
  12. f * ( x * ) = sup x I ( x * x - f ( x ) ) , x * I * f^{*}(x^{*})=\sup_{x\in I}(x^{*}x-f(x)),\quad x^{*}\in I^{*}
  13. I * = { x * 𝐑 : sup x I ( x * x - f ( x ) ) < } . I^{*}=\left\{x^{*}\in\mathbf{R}:\sup_{x\in I}(x^{*}x-f(x))<\infty\right\}~{}.
  14. f ( x ) f(x)
  15. f : X 𝐑 f:X→\mathbf{R}
  16. X * = { x * 𝐑 n : sup x X ( x * , x - f ( x ) ) < } X^{*}=\left\{x^{*}\in\mathbf{R}^{n}:\sup_{x\in X}(\langle x^{*},x\rangle-f(x))% <\infty\right\}
  17. f * ( x * ) = sup x X ( x * , x - f ( x ) ) , x * X * , f^{*}(x^{*})=\sup_{x\in X}(\langle x^{*},x\rangle-f(x)),\quad x^{*}\in X^{*}~{},
  18. x * , x \langle x^{*},x\rangle
  19. x * x*
  20. x x
  21. f * f*
  22. f f
  23. p p
  24. x * x*
  25. f f
  26. f * ( p ) = sup x I ( p x - f ( x ) ) f^{*}(p)=\sup_{x\in I}(px-f(x))
  27. y y
  28. f f
  29. p p
  30. f f
  31. ( x , y ) (x,y)
  32. f f
  33. p p
  34. x x
  35. p x f ( x ) px−f(x)
  36. f * ( p ) = p x f ( x ) f*(p)=px−f(x)
  37. x x
  38. p p
  39. f ( x ) = p . f^{\prime}(x)=p~{}.
  40. f f
  41. x = g ( p ) x=g(p)
  42. g ( f ) - 1 ( p ) g\equiv(f^{\prime})^{-1}(p)
  43. g g
  44. f ( g ( p ) ) = p f^{\prime}(g(p))=p
  45. g g
  46. d g ( p ) d p = 1 f ′′ ( g ( p ) ) . \frac{dg(p)}{dp}=\frac{1}{f^{\prime\prime}(g(p))}~{}.
  47. f * ( p ) = p g ( p ) f ( g ( p ) ) f*(p)=pg(p)−f(g(p))
  48. d ( f * ) d p = g ( p ) + ( p - f ( g ( p ) ) ) d g ( p ) d p = g ( p ) , , \begin{aligned}\displaystyle\frac{d(f^{*})}{dp}&\displaystyle=g(p)+\left(p-f^{% \prime}(g(p))\right)\cdot\frac{dg(p)}{dp}\\ &\displaystyle=g(p),\end{aligned}~{},
  49. d 2 ( f * ) d p 2 = d g ( p ) d p = 1 f ′′ ( g ( p ) ) > 0 , \begin{aligned}\displaystyle\frac{d^{2}(f^{*})}{dp^{2}}&\displaystyle=\frac{dg% (p)}{dp}\\ &\displaystyle{}=\frac{1}{f^{\prime\prime}(g(p))}\\ &\displaystyle{}>0,\end{aligned}
  50. f * f*
  51. f * * = f f**=f
  52. g ( p ) g(p)
  53. f * ( p ) f*(p)
  54. f * * ( x ) = ( x p s - f * ( p s ) ) | d d p f * ( p = p s ) = x = g ( p s ) p s - f * ( p s ) = f ( g ( p s ) ) = f ( x ) . \begin{aligned}\displaystyle f^{**}(x)&\displaystyle{}={\left(x\cdot p_{s}-f^{% *}(p_{s})\right)}_{|\frac{d}{dp}f^{*}(p=p_{s})=x}\\ &\displaystyle{}=g(p_{s})\cdot p_{s}-f^{*}(p_{s})\\ &\displaystyle{}=f(g(p_{s}))\\ &\displaystyle{}=f(x)~{}.\end{aligned}
  55. 𝐑 \mathbf{R}
  56. c > 0 c>0
  57. x * x*
  58. x x
  59. x * 2 c x x*–2cx
  60. 2 c −2c
  61. x = x * / 2 c x=x*/2c
  62. I * = 𝐑 I*=\mathbf{R}
  63. f * ( x * ) = x * 2 4 c f^{*}(x^{*})=\frac{{x^{*}}^{2}}{4c}
  64. f * * ( x ) = 1 4 ( 1 / 4 c ) x 2 = c x 2 , f^{**}(x)=\frac{1}{4(1/4c)}x^{2}=cx^{2},
  65. f * * = f f**=f
  66. x I = 2 , 33 x∈I=2,33
  67. x * x*
  68. x * x f ( x ) x*x−f(x)
  69. I I
  70. I * = 𝐑 I*=\mathbf{R}
  71. x = x * / 2 x=x*/2
  72. 2 , 33 2,33
  73. 4 x * 6 4≤x*≤6
  74. x = 2 x=2
  75. x = 3 x=3
  76. f * ( x * ) = { 2 x * - 4 , x * < 4 x * 2 4 , 4 x * 6 , 3 x * - 9 , x * > 6 f^{*}(x^{*})=\begin{cases}2x^{*}-4,&x^{*}<4\\ \frac{{x^{*}}^{2}}{4},&4\leqslant x^{*}\leqslant 6,\\ 3x^{*}-9,&x^{*}>6\end{cases}
  77. f ( x ) = c x f(x)=cx
  78. x x
  79. x * x f ( x ) = ( x * c ) x x*x−f(x)=(x*−c)x
  80. x x
  81. x * c = 0 x*−c=0
  82. f * f*
  83. f * ( c ) = 0 f*(c)=0
  84. x * x f * ( x * ) x*x−f*(x*)
  85. I * * = 𝐑 I**=\mathbf{R}
  86. x x
  87. sup x * { c } ( x x * - f * ( x * ) ) = x c , \sup_{x^{*}\in\{c\}}(xx^{*}-f^{*}(x^{*}))=xc,
  88. f * * ( x ) = c x = f ( x ) f**(x)=cx=f(x)
  89. f ( x ) = x , A x + c f(x)=\langle x,Ax\rangle+c
  90. A A
  91. f f
  92. p , x - f ( x ) = p , x - x , A x - c , \langle p,x\rangle-f(x)=\langle p,x\rangle-\langle x,Ax\rangle-c,
  93. p 2 A x p−2Ax
  94. 2 A −2A
  95. f * ( p ) = 1 4 p , A - 1 p - c f^{*}(p)=\frac{1}{4}\langle p,A^{-1}p\rangle-c
  96. f f
  97. g g
  98. D f = ( D g ) - 1 , Df=\left(Dg\right)^{-1}~{},
  99. f * = g f*=g
  100. g * = f g*=f
  101. f * f*
  102. d f ( p ) d p = d d p ( x p - f ( x ) ) = x + p d x d p - d f d x d x d p = x . {df^{\star}(p)\over dp}={d\over dp}(xp-f(x))=x+p{dx\over dp}-{df\over dx}{dx% \over dp}=x~{}.
  103. p = d f d x ( x ) , p={df\over dx}(x),
  104. x = d f d p ( p ) . x={df^{\star}\over dp}(p).
  105. D f Df
  106. D f * Df*
  107. f ( x ) = e x p x f(x)=expx
  108. f * ( p ) = p l o g p p f*(p)=plogp−p
  109. f ( x ) + f ( p ) = x p , f(x)+f^{\star}(p)=x\,p~{},
  110. f ( x ) + f ( f ( x ) ) = x f ( x ) , f(x)+f^{\star}(f^{\prime}(x))=x\,f^{\prime}(x)~{},
  111. f ( p ) + f ( ( f ) ( p ) ) = p ( f ) ( p ) . f^{\star}(p)+f((f^{\star})^{\prime}(p))=p~{}(f^{\star})^{\prime}(p)~{}.
  112. f ( x ) f(x)
  113. x f ( ( x ) f ( x ) xf((x)−f(x)
  114. f ( ( x ) f((x)
  115. g ( p ) g(p)
  116. p p
  117. f * f*
  118. f ( x ) - f ( p ) = x p . f(x)-f^{\star}(p)=xp.
  119. p d x = d ( p x ) x d p pdx=d(px)−xdp
  120. f f
  121. x x
  122. y y
  123. d f = f x d x + f y d y = p d x + v d y df={\partial f\over\partial x}dx+{\partial f\over\partial y}dy=pdx+vdy
  124. x x
  125. y y
  126. x x
  127. p p
  128. x x
  129. p p
  130. d x dx
  131. d y dy
  132. d p dp
  133. d y dy
  134. d p dp
  135. d y dy
  136. g ( p , y ) = f p x g(p,y)=f−px
  137. d g = d f - p d x - x d p = p d x + v d y - p d x - x d p = - x d p + v d y dg=df-pdx-xdp=pdx+vdy-pdx-xdp=-xdp+vdy
  138. x = - g p x=-\frac{\partial g}{\partial p}
  139. v = g y . v=\frac{\partial g}{\partial y}.
  140. g ( p , y ) g(p,y)
  141. f ( x , y ) f(x,y)
  142. x x
  143. p p
  144. L ( v , q ) = 1 2 v , M v - V ( q ) L(v,q)=\tfrac{1}{2}\langle v,Mv\rangle-V(q)
  145. ( v , q ) (v,q)
  146. M M
  147. x , y = j x j y j . \langle x,y\rangle=\sum_{j}x_{j}y_{j}.
  148. q q
  149. L ( v , q ) L(v,q)
  150. v v
  151. V ( q ) V(q)
  152. L ( v , q ) L(v,q)
  153. v v
  154. H ( p , q ) = 1 2 p , M - 1 p + V ( q ) H(p,q)=\tfrac{1}{2}\langle p,M^{-1}p\rangle+V(q)
  155. ( v , q ) (v,q)
  156. T T\mathcal{M}
  157. \mathcal{M}
  158. q q
  159. L ( v , q ) L(v,q)
  160. H ( p , q ) H(p,q)
  161. ( p , q ) (p,q)
  162. T * T^{*}\mathcal{M}
  163. U = U ( S , V , { N i } ) , U=U\left(S,V,\{N_{i}\}\right),
  164. d U = T d S - P d V + μ i d N i dU=TdS-PdV+\sum\mu_{i}dN_{i}
  165. U U
  166. V V
  167. H = U + P V = H ( S , P , { N i } ) , H=U+PV\,=H\left(S,P,\{N_{i}\}\right),
  168. P P
  169. S S
  170. T T
  171. A A
  172. G G
  173. A = U - T S , A=U-TS~{},
  174. G = H - T S = U + P V - T S . G=H-TS=U+PV-TS~{}.
  175. 𝐱 \mathbf{x}
  176. C ( 𝐱 ) C(\mathbf{x})
  177. Q Q
  178. U ( Q , 𝐱 ) = 1 2 Q V = 1 2 Q 2 C ( 𝐱 ) U(Q,\mathbf{x})=\frac{1}{2}QV=\frac{1}{2}\frac{Q^{2}}{C(\mathbf{x})}~{}
  179. 𝐱 \mathbf{x}
  180. C ( 𝐱 ) C(\mathbf{x})
  181. 𝐅 \mathbf{F}
  182. 𝐅 ( 𝐱 ) = - d U d 𝐱 . \mathbf{F}(\mathbf{x})=-\frac{dU}{d\mathbf{x}}~{}.
  183. 𝐅 ( 𝐱 ) = 1 2 d C d 𝐱 Q 2 C 2 . \mathbf{F}(\mathbf{x})=\frac{1}{2}\frac{dC}{d\mathbf{x}}\frac{Q^{2}}{C^{2}}.
  184. V V
  185. U * = U - Q V = 1 2 Q V - Q V = - 1 2 Q V = - 1 2 V 2 C ( 𝐱 ) . U^{*}=U-QV=\frac{1}{2}QV-QV=-\frac{1}{2}QV=-\tfrac{1}{2}V^{2}C(\mathbf{x}).
  186. 𝐅 ( 𝐱 ) = - d U * d 𝐱 . \mathbf{F}(\mathbf{x})=-\frac{dU^{*}}{d\mathbf{x}}~{}.
  187. Q Q
  188. p p
  189. y y
  190. b b
  191. y = p x + b y=px+b
  192. f f
  193. f ( x 0 ) = p x 0 + b f\left(x_{0}\right)=px_{0}+b
  194. p = f ˙ ( x 0 ) . p=\dot{f}\left(x_{0}\right).
  195. $\mathbf{ }$
  196. y y
  197. b b
  198. p p
  199. b = f ( f ˙ - 1 ( p ) ) - p f ˙ - 1 ( p ) = - f ( p ) . b=f\left(\dot{f}^{-1}\left(p\right)\right)-p\cdot\dot{f}^{-1}\left(p\right)=-f% ^{\star}(p).
  200. f f^{\star}
  201. f f
  202. f f
  203. p p
  204. y = p x - f ( p ) y=px-f^{\star}(p)
  205. F ( x , y , p ) = y + f ( p ) - p x = 0 . F(x,y,p)=y+f^{\star}(p)-px=0~{}.
  206. F ( x , y , p ) p = f ˙ ( p ) - x = 0 . {\partial F(x,y,p)\over\partial p}=\dot{f}^{\star}(p)-x=0~{}.
  207. p p
  208. y = x f ˙ - 1 ( x ) - f ( f ˙ - 1 ( x ) ) . y=x\cdot\dot{f}^{\star-1}(x)-f^{\star}\left(\dot{f}^{\star-1}(x)\right).
  209. y y
  210. f ( x ) f(x)
  211. f * f*
  212. f ( x ) = f ( x ) . f(x)=f^{\star\star}(x)~{}.
  213. U U
  214. ( U , f ) (U,f)
  215. ( V , g ) (V,g)
  216. V V
  217. U U
  218. D f Df
  219. g g
  220. V V
  221. g ( y ) = y , x - f ( x ) , x = ( D f ) - 1 ( y ) g(y)=\left\langle y,x\right\rangle-f(x),\qquad x=\left(Df\right)^{-1}(y)
  222. u , v = k = 1 n u k v k \left\langle u,v\right\rangle=\sum_{k=1}^{n}u_{k}\cdot v_{k}
  223. X X
  224. Y Y
  225. x x
  226. X X
  227. y y
  228. Y Y
  229. Y Y
  230. X X
  231. f f
  232. X X
  233. f ∇f
  234. T * X T*X
  235. X X
  236. Y Y
  237. g g
  238. Y Y
  239. g ∇g
  240. Y Y
  241. $\mathbf{ }$
  242. a > 0 a>0
  243. f ( x ) = a g ( x ) f ( p ) = a g ( p a ) f(x)=a\cdot g(x)\Rightarrow f^{\star}(p)=a\cdot g^{\star}\left(\frac{p}{a}\right)
  244. f ( x ) = g ( a x ) f ( p ) = g ( p a ) . f(x)=g(a\cdot x)\Rightarrow f^{\star}(p)=g^{\star}\left(\frac{p}{a}\right).
  245. r r
  246. s s
  247. 1 / r + 1 / s = 1 1/r+1/s=1
  248. r > 1 r>1
  249. f ( x ) = g ( x ) + b f ( p ) = g ( p ) - b f(x)=g(x)+b\Rightarrow f^{\star}(p)=g^{\star}(p)-b
  250. f ( x ) = g ( x + y ) f ( p ) = g ( p ) - p y f(x)=g(x+y)\Rightarrow f^{\star}(p)=g^{\star}(p)-p\cdot y
  251. f ( x ) = g - 1 ( x ) f ( p ) = - p g ( 1 p ) f(x)=g^{-1}(x)\Rightarrow f^{\star}(p)=-p\cdot g^{\star}\left(\frac{1}{p}\right)
  252. f f
  253. ( A f ) = f A (Af)^{\star}=f^{\star}A^{\star}
  254. A * A*
  255. A A
  256. A x , y = x , A y , \left\langle Ax,y^{\star}\right\rangle=\left\langle x,A^{\star}y^{\star}\right\rangle,
  257. A f Af
  258. f f
  259. A A
  260. ( A f ) ( y ) = inf { f ( x ) : x X , A x = y } . (Af)(y)=\inf\{f(x):x\in X,Ax=y\}.
  261. f f
  262. G G
  263. f ( A x ) = f ( x ) , x , A G f(Ax)=f(x),\;\forall x,\;\forall A\in G
  264. f * f*
  265. G G
  266. f f
  267. g g
  268. ( f inf g ) ( x ) = inf { f ( x - y ) + g ( y ) | y 𝐑 n } . \left(f\star_{\inf}g\right)(x)=\inf\left\{f(x-y)+g(y)\,|\,y\in\mathbf{R}^{n}% \right\}.
  269. ( f 1 inf inf f m ) = f 1 + + f m . \left(f_{1}\star_{\inf}\cdots\star_{\inf}f_{m}\right)^{\star}=f_{1}^{\star}+% \cdots+f_{m}^{\star}.
  270. f f
  271. f * f*
  272. x X x∈X
  273. p X * p∈X*
  274. x , p x,p
  275. p , x f ( x ) + f ( p ) . \left\langle p,x\right\rangle\leq f(x)+f^{\star}(p).

Lehmann–Scheffé_theorem.html

  1. X = X 1 , X 2 , , X n \vec{X}=X_{1},X_{2},\dots,X_{n}
  2. f ( x : θ ) f(x:\theta)
  3. θ Ω \theta\in\Omega
  4. Y = u ( X ) Y=u(\vec{X})
  5. { f Y ( y : θ ) : θ Ω } \{f_{Y}(y:\theta):\theta\in\Omega\}
  6. ϕ : 𝔼 [ ϕ ( Y ) ] = θ \phi:\mathbb{E}[\phi(Y)]=\theta
  7. ϕ ( Y ) \phi(Y)
  8. Z Z
  9. ϕ ( Y ) := 𝔼 [ Z | Y ] \phi(Y):=\mathbb{E}[Z|Y]
  10. Z Z
  11. W W
  12. ψ ( Y ) := 𝔼 [ W | Y ] \psi(Y):=\mathbb{E}[W|Y]
  13. W W
  14. 𝔼 [ ϕ ( Y ) - ψ ( Y ) ] = 0 , θ Ω . \mathbb{E}[\phi(Y)-\psi(Y)]=0,\theta\in\Omega.
  15. { f Y ( y : θ ) : θ Ω } \{f_{Y}(y:\theta):\theta\in\Omega\}
  16. 𝔼 [ ϕ ( Y ) - ψ ( Y ) ] = 0 ϕ ( y ) - ψ ( y ) = 0 , θ Ω \mathbb{E}[\phi(Y)-\psi(Y)]=0\implies\phi(y)-\psi(y)=0,\theta\in\Omega
  17. ϕ \phi
  18. ϕ ( Y ) \phi(Y)

Leibniz's_notation.html

  1. d x dx
  2. d y dy
  3. x x
  4. y y
  5. Δ x Δx
  6. Δ y Δy
  7. x x
  8. y y
  9. y y
  10. x x
  11. y y
  12. f ( x ) f(x)
  13. y y
  14. x x
  15. lim Δ x 0 Δ y Δ x = lim Δ x 0 f ( x + Δ x ) - f ( x ) Δ x , \lim_{\Delta x\rightarrow 0}\frac{\Delta y}{\Delta x}=\lim_{\Delta x% \rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x},
  16. y y
  17. x x
  18. d y d x = f ( x ) , \frac{dy}{dx}=f^{\prime}(x),
  19. f f
  20. x x
  21. Δ x Δx
  22. x x
  23. Δ y Δy
  24. y y
  25. f ( x ) = st ( Δ y Δ x ) f^{\prime}(x)={\rm st}\Bigg(\frac{\Delta y}{\Delta x}\Bigg)
  26. d x = Δ x dx=\Delta x
  27. d y = f ( x ) d x dy=f^{\prime}(x)dx\,
  28. f ( x ) f^{\prime}(x)\,
  29. f ( x ) d x \int f(x)\,dx
  30. lim Δ x 0 i f ( x i ) Δ x , \lim_{\Delta x\rightarrow 0}\sum_{i}f(x_{i})\,\Delta x,
  31. Δ x Δx
  32. f ( x ) d x f(x)dx
  33. \int
  34. y y
  35. f f
  36. x x
  37. y = f ( x ) y=f(x)\,
  38. f ( x ) f(x)
  39. d y d x or d d x y or d ( f ( x ) ) d x \frac{dy}{dx}\,\,\text{ or }\frac{d}{dx}y\,\,\text{ or }\frac{d\bigl(f(x)\bigr% )}{dx}\,
  40. d y d x \frac{dy}{dx}
  41. d y d x = y \frac{dy}{dx}\,=y^{\prime}
  42. f ( x ) f(x)
  43. y y
  44. d ( f ( x ) ) d x = f ( x ) \frac{d\bigl(f(x)\bigr)}{dx}=f^{\prime}(x)\,
  45. f ( x ) f′(x)
  46. x x
  47. d x d t = x ˙ \frac{dx}{dt}=\dot{x}\,
  48. d y d x \frac{dy}{dx}
  49. d x dx
  50. d y dy
  51. lim δ x 0 δ y δ x \lim_{\delta x\to 0}\frac{\delta y}{\delta x}
  52. δ δ
  53. d d
  54. d d x \frac{d}{dx}
  55. y y
  56. x x
  57. D D
  58. d n ( f ( x ) ) d x n or d n y d x n \frac{d^{n}\bigl(f(x)\bigr)}{dx^{n}}\,\text{ or }\frac{d^{n}y}{dx^{n}}
  59. n n
  60. f ( x ) f(x)
  61. y y
  62. d y d x \frac{dy}{dx}
  63. d 2 y d x 2 = d d x ( d y d x ) . \frac{d^{2}y}{dx^{2}}\,=\,\frac{d}{dx}\left(\frac{dy}{dx}\right).
  64. d ( d ( d ( f ( x ) ) d x ) d x ) d x , \frac{d\left(\frac{d\left(\frac{d\left(f(x)\right)}{dx}\right)}{dx}\right)}{dx% }\,,
  65. ( d d x ) 3 ( f ( x ) ) = d 3 ( d x ) 3 ( f ( x ) ) . \left(\frac{d}{dx}\right)^{3}\bigl(f(x)\bigr)=\frac{d^{3}}{\left(dx\right)^{3}% }\bigl(f(x)\bigr)\,.
  66. d 3 d x 3 ( f ( x ) ) or d 3 y d x 3 . \frac{d^{3}}{dx^{3}}\bigl(f(x)\bigr)\ \mbox{or}~{}\ \frac{d^{3}y}{dx^{3}}\,.
  67. d y d x = d y d u 1 d u 1 d u 2 d u 2 d u 3 d u n d x , \frac{dy}{dx}=\frac{dy}{du_{1}}\cdot\frac{du_{1}}{du_{2}}\cdot\frac{du_{2}}{du% _{3}}\cdots\frac{du_{n}}{dx}\,,
  68. y d x = y d x d u d u . \int y\,dx=\int y\frac{dx}{du}\,du.

Length_contraction.html

  1. c c
  2. c c
  3. L = L 0 γ ( v ) = L 0 1 - v 2 / c 2 L=\frac{L_{0}}{\gamma(v)}=L_{0}\sqrt{1-v^{2}/c^{2}}
  4. L L
  5. v v
  6. c c
  7. γ ( v ) γ(v)
  8. γ ( v ) 1 1 - v 2 / c 2 \gamma(v)\equiv\frac{1}{\sqrt{1-v^{2}/c^{2}}}
  9. L 0 L_{0}
  10. L L
  11. T 0 T_{0}
  12. T T
  13. L 0 = T v L_{0}=T\cdot v
  14. L = T 0 v L=T_{0}\cdot v
  15. L L
  16. L 0 L_{0}
  17. T T
  18. T 0 T_{0}
  19. v v
  20. c c
  21. L = L 0 / γ . L=L_{0}/\gamma.
  22. v = 0.8 c v=0{.}8c
  23. L 0 = AB = 30 cm L_{0}=\mathrm{AB}=30\ \mathrm{cm}
  24. L L^{\prime}
  25. L = AC = L 0 / γ = 18 cm . L^{\prime}=\mathrm{AC}=L_{0}/\gamma=18\ \mathrm{cm}.
  26. L 0 = EF = 30 cm L^{\prime}_{0}=\mathrm{EF}=30\ \mathrm{cm}
  27. L = DE = L 0 / γ = 18 cm . L=\mathrm{DE}=L^{\prime}_{0}/\gamma=18\ \mathrm{cm}.
  28. x \displaystyle x^{\prime}
  29. x 1 x_{1}
  30. x 2 x_{2}
  31. L L
  32. t 1 = t 2 t_{1}=t_{2}\,
  33. x 1 = γ ( x 1 - v t 1 ) and x 2 = γ ( x 2 - v t 2 ) . x^{\prime}_{1}=\gamma\left(x_{1}-vt_{1}\right)\quad\mathrm{and}\quad x^{\prime% }_{2}=\gamma\left(x_{2}-vt_{2}\right).
  34. t 1 = t 2 t_{1}=t_{2}\,
  35. L = x 2 - x 1 L=x_{2}-x_{1}\,
  36. L 0 = x 2 - x 1 L_{0}^{{}^{\prime}}=x_{2}^{{}^{\prime}}-x_{1}^{{}^{\prime}}
  37. L 0 = L γ . (1) , L_{0}^{{}^{\prime}}=L\cdot\gamma.\qquad\qquad\,\text{(1)},
  38. L = L 0 / γ . (2) L=L_{0}^{{}^{\prime}}/\gamma.\qquad\qquad\,\text{(2)}
  39. L 0 = L γ . (3) L_{0}=L^{\prime}\cdot\gamma.\qquad\qquad\,\text{(3)}
  40. L = L 0 / γ . (4) L^{\prime}=L_{0}/\gamma.\qquad\qquad\,\text{(4)}
  41. x 1 = γ ( x 1 - v t 1 ) and x 2 = γ ( x 2 - v t 2 ) t 1 = γ ( t 1 - v x 1 / c 2 ) and t 2 = γ ( t 2 - v x 2 / c 2 ) . \begin{aligned}\displaystyle x_{1}^{{}^{\prime}}&\displaystyle=\gamma\left(x_{% 1}-vt_{1}\right)&\displaystyle\quad\mathrm{and}&&\displaystyle x_{2}^{{}^{% \prime}}&\displaystyle=\gamma\left(x_{2}-vt_{2}\right)\\ \displaystyle t_{1}^{{}^{\prime}}&\displaystyle=\gamma\left(t_{1}-vx_{1}/c^{2}% \right)&\displaystyle\quad\mathrm{and}&&\displaystyle t_{2}^{{}^{\prime}}&% \displaystyle=\gamma\left(t_{2}-vx_{2}/c^{2}\right).\end{aligned}
  42. t 1 = t 2 t_{1}=t_{2}
  43. L 0 = x 2 - x 1 L_{0}=x_{2}-x_{1}
  44. Δ x \displaystyle\Delta x^{\prime}
  45. v v
  46. Δ t \Delta t^{\prime}
  47. Δ x \Delta x^{\prime}
  48. L \displaystyle L^{\prime}
  49. L = L 0 / γ L=L^{{}^{\prime}}_{0}/\gamma
  50. T 0 T_{0}
  51. T T
  52. T = T 0 γ T=T_{0}\cdot\gamma
  53. L 0 L_{0}
  54. S S
  55. S S^{\prime}
  56. T = L 0 / v T=L_{0}/v
  57. S S
  58. T 0 = L / v T^{\prime}_{0}=L^{\prime}/v
  59. S S^{\prime}
  60. L 0 = T v L_{0}=Tv
  61. L = T 0 v L^{\prime}=T^{\prime}_{0}v
  62. L L 0 = T 0 v T v = 1 / γ \frac{L^{\prime}}{L_{0}}=\frac{T^{\prime}_{0}v}{Tv}=1/\gamma
  63. S S^{\prime}
  64. L = L 0 / γ L^{\prime}=L_{0}/\gamma
  65. S S
  66. S S^{\prime}
  67. S S
  68. S S^{\prime}
  69. L = L 0 / γ L=L^{\prime}_{0}/\gamma

Level.html

  1. - 1 -1

Level_set.html

  1. L c ( f ) = { ( x 1 , , x n ) f ( x 1 , , x n ) = c } , L_{c}(f)=\left\{(x_{1},\cdots,x_{n})\,\mid\,f(x_{1},\cdots,x_{n})=c\right\}~{},
  2. r 2 = x 2 + y 2 r^{2}=x^{2}+y^{2}
  3. r = 5 r=5
  4. c = 5 2 = 25 c=5^{2}=25
  5. L c - ( f ) = { ( x 1 , , x n ) f ( x 1 , , x n ) c } L_{c}^{-}(f)=\left\{(x_{1},\cdots,x_{n})\,\mid\,f(x_{1},\cdots,x_{n})\leq c\right\}
  6. L c + ( f ) = { ( x 1 , , x n ) f ( x 1 , , x n ) c } L_{c}^{+}(f)=\left\{(x_{1},\cdots,x_{n})\,\mid\,f(x_{1},\cdots,x_{n})\geq c\right\}

Levenshtein_distance.html

  1. a , b a,b
  2. | a | |a|
  3. | b | |b|
  4. lev a , b ( | a | , | b | ) \operatorname{lev}_{a,b}(|a|,|b|)
  5. lev a , b ( i , j ) = { max ( i , j ) if min ( i , j ) = 0 , min { lev a , b ( i - 1 , j ) + 1 lev a , b ( i , j - 1 ) + 1 lev a , b ( i - 1 , j - 1 ) + 1 ( a i b j ) otherwise. \qquad\operatorname{lev}_{a,b}(i,j)=\begin{cases}\max(i,j)&\,\text{ if}\min(i,% j)=0,\\ \min\begin{cases}\operatorname{lev}_{a,b}(i-1,j)+1\\ \operatorname{lev}_{a,b}(i,j-1)+1\\ \operatorname{lev}_{a,b}(i-1,j-1)+1_{(a_{i}\neq b_{j})}\end{cases}&\,\text{ % otherwise.}\end{cases}
  6. 1 ( a i b j ) 1_{(a_{i}\neq b_{j})}
  7. a i = b j a_{i}=b_{j}
  8. a a
  9. b b

Lewis_structure.html

  1. C f = N v - U e - B n 2 C_{f}=N_{v}-U_{e}-\frac{B_{n}}{2}
  2. C f C_{f}
  3. N v N_{v}
  4. U e U_{e}
  5. B n B_{n}
  6. N O 2 - {NO_{2}}^{-}

Lexicographical_order.html

  1. { A 1 , A 2 , , A n } \{A_{1},A_{2},\cdots,A_{n}\}
  2. { < 1 , < 2 , , < n } \{<_{1},<_{2},\cdots,<_{n}\}
  3. < d \ \ <^{d}
  4. A 1 × A 2 × × A n A_{1}\times A_{2}\times\cdots\times A_{n}
  5. ( a 1 , a 2 , , a n ) < d ( b 1 , b 2 , , b n ) ( m > 0 ) ( i < m ) ( a i = b i ) ( a m < m b m ) (a_{1},a_{2},\dots,a_{n})<^{d}(b_{1},b_{2},\dots,b_{n})\iff(\exists\ m>0)\ (% \forall\ i<m)(a_{i}=b_{i})\land(a_{m}<_{m}b_{m})
  6. a m < m b m \ \ a_{m}<_{m}b_{m}
  7. a 1 \ \ a_{1}
  8. a 2 \ \ a_{2}
  9. C = A j × A j + 1 × × A k \ \ C=A_{j}\times A_{j+1}\times\cdots\times A_{k}
  10. < d ( C ) \ \ <^{d}(C)
  11. a < d ( A i ) a ( a < i a ) a<^{d}(A_{i})a^{\prime}\iff(a<_{i}a^{\prime})
  12. ( a , b ) < d ( A i × B ) ( a , b ) a < d ( A i ) a ( a = a b < d ( B ) b ) (a,b)<^{d}(A_{i}\times B)(a^{\prime},b^{\prime})\iff a<^{d}(A_{i})a^{\prime}% \lor(a=a^{\prime}\ \land\ b<^{d}(B)b^{\prime})
  13. B = A i + 1 × A i + 2 × × A n . B=A_{i+1}\times A_{i+2}\times\cdots\times A_{n}.
  14. u < d v u<^{\mathrm{d}}v
  15. u u
  16. v v
  17. u = w a u u=wau^{\prime}
  18. v = w b v v=wbv^{\prime}
  19. w w
  20. u u
  21. v v
  22. a a
  23. b b
  24. a < b a<b
  25. u u^{\prime}
  26. v v^{\prime}
  27. x 1 x 2 3 x 4 x 5 2 x_{1}x_{2}^{3}x_{4}x_{5}^{2}
  28. { A 1 , A 2 , , A n } \{A_{1},A_{2},\ldots,A_{n}\}
  29. { < 1 , < 2 , , < n } \{<_{1},<_{2},\ldots,<_{n}\}
  30. < colex \ \ <\text{colex}
  31. A 1 × A 2 × × A n A_{1}\times A_{2}\times\cdots\times A_{n}
  32. ( a 1 , a 2 , , a n ) < colex ( b 1 , b 2 , , b n ) (a_{1},a_{2},\dots,a_{n})<\text{colex}(b_{1},b_{2},\dots,b_{n})\iff
  33. ( m Align g t ; 0 ) ( i Align g t ; m ) ( a i = b i ) ( a m < m b m = " " > T h e f o l l o w i n g i s a n o r d e r i n g o n t h e 3 - e l e m e n t s u b s e t s o f < m a t h > { 1 , 2 , 3 , 4 , 5 , 6 } (\exists\ m&gt;0)\ (\forall\ i&gt;m)(a_{i}=b_{i})\land(a_{m}<_{m}b_{m}="">% \par Thefollowingisanorderingonthe3-elementsubsetsof<math>\{1,2,3,4,5,6\}
  34. x 2 y z 2 < x y 3 z 2 x^{2}yz^{2}<xy^{3}z^{2}

Lévy_flight.html

  1. Pr ( U > u ) = { 1 : u < 1 , u - D : u 1. \Pr(U>u)=\begin{cases}1&:\ u<1,\\ u^{-D}&:\ u\geq 1.\end{cases}
  2. Pr ( U > u ) = O ( u - k ) , \Pr(U>u)=O(u^{-k}),
  3. ϕ ( x , t ) t = - x f ( x , t ) ϕ ( x , t ) + γ α ϕ ( x , t ) | x | α \frac{\partial\phi(x,t)}{\partial t}=-\frac{\partial}{\partial x}f(x,t)\phi(x,% t)+\gamma\frac{\partial^{\alpha}\phi(x,t)}{\partial|x|^{\alpha}}
  4. F k [ α ϕ ( x , t ) | x | α ] = k α F k [ ϕ ( x , t ) ] F_{k}[\frac{\partial^{\alpha}\phi(x,t)}{\partial|x|^{\alpha}}]=k^{\alpha}F_{k}% [\phi(x,t)]
  5. < | x | θ > t θ α <|x|^{\theta}>\propto t^{\frac{\theta}{\alpha}}

Lie_algebra_representation.html

  1. 𝔤 \mathfrak{g}
  2. ρ : 𝔤 𝔤 𝔩 ( V ) \rho\colon\mathfrak{g}\to\mathfrak{gl}(V)
  3. 𝔤 \mathfrak{g}
  4. 𝔤 \mathfrak{g}
  5. 𝔤 𝔩 ( V ) \mathfrak{gl}(V)
  6. ρ [ x , y ] = [ ρ x , ρ y ] = ρ x ρ y - ρ y ρ x \rho_{[x,y]}=[\rho_{x},\rho_{y}]=\rho_{x}\rho_{y}-\rho_{y}\rho_{x}\,
  7. 𝔤 \mathfrak{g}
  8. 𝔤 \mathfrak{g}
  9. ρ \rho
  10. 𝔤 \mathfrak{g}
  11. 𝔤 × V V \mathfrak{g}\times V\to V
  12. [ x , y ] v = x ( y v ) - y ( x v ) [x,y]\cdot v=x\cdot(y\cdot v)-y\cdot(x\cdot v)
  13. 𝔤 \mathfrak{g}
  14. 𝔤 \mathfrak{g}
  15. ad : 𝔤 𝔤 𝔩 ( 𝔤 ) , x ad x , ad x ( y ) = [ x , y ] . \textrm{ad}:\mathfrak{g}\to\mathfrak{gl}(\mathfrak{g}),\quad x\mapsto% \operatorname{ad}_{x},\quad\operatorname{ad}_{x}(y)=[x,y].
  16. ad \operatorname{ad}
  17. 𝔤 \mathfrak{g}
  18. 𝔥 \mathfrak{h}
  19. d e ϕ : 𝔤 𝔥 d_{e}\phi:\mathfrak{g}\to\mathfrak{h}
  20. ϕ : G GL ( V ) \phi:G\to\mathrm{GL}(V)\,
  21. d ϕ : 𝔤 𝔤 𝔩 ( V ) d\phi:\mathfrak{g}\to\mathfrak{gl}(V)
  22. 𝔤 \mathfrak{g}
  23. c g ( x ) = g x g - 1 c_{g}(x)=gxg^{-1}
  24. c g : G G c_{g}:G\to G
  25. GL ( 𝔤 ) \mathrm{GL}(\mathfrak{g})
  26. Ad ( g ) \operatorname{Ad}(g)
  27. Ad \operatorname{Ad}
  28. 𝔤 \mathfrak{g}
  29. d Ad d\operatorname{Ad}
  30. d e Ad = ad . d_{e}\operatorname{Ad}=\operatorname{ad}.
  31. 𝔤 \mathfrak{g}
  32. 𝔤 \mathfrak{g}
  33. f : V W f:V\to W
  34. 𝔤 \mathfrak{g}
  35. 𝔤 \mathfrak{g}
  36. f ( x v ) = x f ( v ) f(xv)=xf(v)
  37. x 𝔤 , v V x\in\mathfrak{g},v\in V
  38. V , W V,W
  39. 𝔤 \mathfrak{g}
  40. 𝔤 \mathfrak{g}
  41. 𝔤 \mathfrak{g}
  42. x v = 0 xv=0
  43. x 𝔤 x\in\mathfrak{g}
  44. V 𝔤 V^{\mathfrak{g}}
  45. V V 𝔤 V\mapsto V^{\mathfrak{g}}
  46. x [ v 1 v 2 ] = x [ v 1 ] v 2 + v 1 x [ v 2 ] . x[v_{1}\otimes v_{2}]=x[v_{1}]\otimes v_{2}+v_{1}\otimes x[v_{2}].
  47. ( z ω ) [ X ] = z ¯ ω [ X ] (z\omega)[X]=\bar{z}\omega[X]
  48. ρ ¯ \bar{\rho}
  49. ρ ¯ \bar{\rho}
  50. ρ ¯ \bar{\rho}
  51. V , W V,W
  52. 𝔤 \mathfrak{g}
  53. 𝔤 \mathfrak{g}
  54. Hom ( V , W ) \operatorname{Hom}(V,W)
  55. 𝔤 \mathfrak{g}
  56. ( x f ) ( v ) = x f ( v ) - f ( x v ) (x\cdot f)(v)=xf(v)-f(xv)
  57. Hom 𝔤 ( V , W ) = Hom ( V , W ) 𝔤 \operatorname{Hom}_{\mathfrak{g}}(V,W)=\operatorname{Hom}(V,W)^{\mathfrak{g}}
  58. 𝔤 \mathfrak{g}
  59. V * V^{*}
  60. 𝔤 \mathfrak{g}
  61. 𝔤 \mathfrak{g}
  62. 𝔤 \mathfrak{g}
  63. 𝔤 \mathfrak{g}
  64. 𝔤 \mathfrak{g}
  65. 𝔤 \mathfrak{g}
  66. T = n = 0 1 n 𝔤 T=\oplus_{n=0}^{\infty}\otimes_{1}^{n}\mathfrak{g}
  67. \otimes
  68. U ( 𝔤 ) U(\mathfrak{g})
  69. [ x , y ] - x y + y x [x,y]-x\otimes y+y\otimes x
  70. U ( 𝔤 ) U(\mathfrak{g})
  71. [ x , y ] = x y - y x [x,y]=xy-yx
  72. \otimes
  73. 𝔤 U ( 𝔤 ) \mathfrak{g}\to U(\mathfrak{g})
  74. T U ( 𝔤 ) T\to U(\mathfrak{g})
  75. 𝔤 \mathfrak{g}
  76. U ( 𝔤 ) U(\mathfrak{g})
  77. 𝔤 \mathfrak{g}
  78. 𝔤 \mathfrak{g}
  79. U ( 𝔤 ) U(\mathfrak{g})
  80. 𝔤 \mathfrak{g}
  81. 𝔤 \mathfrak{g}
  82. l x ( y ) = x y , x 𝔤 , y U ( 𝔤 ) l_{x}(y)=xy,x\in\mathfrak{g},y\in U(\mathfrak{g})
  83. x l x x\mapsto l_{x}
  84. 𝔤 \mathfrak{g}
  85. U ( 𝔤 ) U(\mathfrak{g})
  86. 𝔤 \mathfrak{g}
  87. 𝔥 𝔤 \mathfrak{h}\subset\mathfrak{g}
  88. U ( 𝔥 ) U(\mathfrak{h})
  89. U ( 𝔤 ) U(\mathfrak{g})
  90. 𝔥 \mathfrak{h}
  91. U ( 𝔤 ) U(\mathfrak{g})
  92. U ( 𝔤 ) U ( 𝔥 ) W U(\mathfrak{g})\otimes_{U(\mathfrak{h})}W
  93. 𝔤 \mathfrak{g}
  94. Ind 𝔥 𝔤 W \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}}W
  95. 𝔤 \mathfrak{g}
  96. 𝔤 \mathfrak{g}
  97. Hom 𝔤 ( Ind 𝔥 𝔤 W , E ) Hom 𝔥 ( W , Res 𝔥 𝔤 E ) \operatorname{Hom}_{\mathfrak{g}}(\operatorname{Ind}_{\mathfrak{h}}^{\mathfrak% {g}}W,E)\simeq\operatorname{Hom}_{\mathfrak{h}}(W,\operatorname{Res}^{% \mathfrak{g}}_{\mathfrak{h}}E)
  98. Ind 𝔥 𝔤 \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}}
  99. 𝔥 \mathfrak{h}
  100. 𝔤 \mathfrak{g}
  101. U ( 𝔤 ) U(\mathfrak{g})
  102. U ( 𝔥 ) U(\mathfrak{h})
  103. Ind 𝔥 𝔤 W \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}}W
  104. 𝔤 \mathfrak{g}
  105. V k F V\otimes_{k}F
  106. F / k F/k
  107. Ind 𝔥 𝔤 Ind 𝔥 𝔤 Ind 𝔥 𝔥 \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}}\simeq\operatorname{Ind}_{% \mathfrak{h^{\prime}}}^{\mathfrak{g}}\circ\operatorname{Ind}_{\mathfrak{h}}^{% \mathfrak{h^{\prime}}}
  108. 𝔥 𝔤 \mathfrak{h^{\prime}}\subset\mathfrak{g}
  109. 𝔥 𝔥 \mathfrak{h}\subset\mathfrak{h}^{\prime}
  110. 𝔥 𝔤 \mathfrak{h}\subset\mathfrak{g}
  111. 𝔫 \mathfrak{n}
  112. 𝔤 \mathfrak{g}
  113. 𝔥 \mathfrak{h}
  114. 𝔤 1 = 𝔤 / 𝔫 \mathfrak{g}_{1}=\mathfrak{g}/\mathfrak{n}
  115. 𝔥 1 = 𝔥 / 𝔫 \mathfrak{h}_{1}=\mathfrak{h}/\mathfrak{n}
  116. Ind 𝔥 𝔤 Res 𝔥 Res 𝔤 Ind 𝔥 1 𝔤 1 \operatorname{Ind}^{\mathfrak{g}}_{\mathfrak{h}}\circ\operatorname{Res}_{% \mathfrak{h}}\simeq\operatorname{Res}_{\mathfrak{g}}\circ\operatorname{Ind}^{% \mathfrak{g_{1}}}_{\mathfrak{h_{1}}}
  117. 𝔤 \mathfrak{g}
  118. 𝔤 \mathfrak{g}
  119. π \pi
  120. 𝔤 \mathfrak{g}
  121. 𝔤 \mathfrak{g}
  122. π \pi
  123. K K

Liebig's_law_of_the_minimum.html

  1. O O
  2. I I
  3. N N
  4. P P
  5. d O d t = O ( m i n ( μ I I k I + I , μ N N k N + N , μ P P k P + P ) - m ) \frac{dO}{dt}=O\left(min\left(\frac{\mu_{I}I}{k_{I}+I},\frac{\mu_{N}N}{k_{N}+N% },\frac{\mu_{P}P}{k_{P}+P}\right)-m\right)

Limit-preserving_function_(order_theory).html

  1. x ( y z ) = ( x y ) ( x z ) x\wedge\left(y\vee z\right)=\left(x\wedge y\right)\vee\left(x\wedge z\right)
  2. x S = { x s s S } x\wedge\bigvee S=\bigvee\left\{x\wedge s\mid s\in S\right\}

Limit_cardinal.html

  1. 0 \aleph_{0}
  2. ω \aleph_{\omega}
  3. λ \aleph_{\lambda}
  4. 0 = 0 , \beth_{0}=\aleph_{0},
  5. α + 1 = 2 α , \beth_{\alpha+1}=2^{\beth_{\alpha}},
  6. λ = { α : α < λ } . \beth_{\lambda}=\bigcup\{\beth_{\alpha}:\alpha<\lambda\}.
  7. ω = { 0 , 1 , 2 , } = n < ω n \beth_{\omega}=\bigcup\{\beth_{0},\beth_{1},\beth_{2},\ldots\}=\bigcup_{n<% \omega}\beth_{n}
  8. α + ω = n < ω α + n \beth_{\alpha+\omega}=\bigcup_{n<\omega}\beth_{\alpha+n}
  9. ω λ , \omega_{\lambda}\,,
  10. λ \aleph_{\lambda}
  11. λ \aleph_{\lambda}
  12. α + = ( α ) + , \aleph_{\alpha^{+}}=(\aleph_{\alpha})^{+}\,,
  13. λ \aleph_{\lambda}
  14. κ = ( α ) + , \kappa=(\aleph_{\alpha})^{+}\,,
  15. κ = α + . \kappa=\aleph_{\alpha^{+}}\,.
  16. λ \aleph_{\lambda}
  17. ω \aleph_{\omega}
  18. ω \aleph_{\omega}
  19. κ + = 2 κ \kappa^{+}=2^{\kappa}\,
  20. 0 \aleph_{0}
  21. 0 \aleph_{0}
  22. κ \kappa
  23. L κ Z F C L_{\kappa}\models ZFC

Limit_cycle.html

  1. x ( t ) = V ( x ( t ) ) x^{\prime}(t)=V(x(t))
  2. V : 2 2 V:\mathbb{R}^{2}\to\mathbb{R}^{2}
  3. x ( t ) x(t)
  4. 2 \mathbb{R}^{2}
  5. t 0 > 0 t_{0}>0
  6. x ( t + t 0 ) = x ( t ) x(t+t_{0})=x(t)
  7. t t\in\mathbb{R}
  8. 2 \mathbb{R}^{2}
  9. p p
  10. V ( p ) = 0 V(p)=0
  11. x = V ( x ) x^{\prime}=V(x)
  12. V V

Limit_ordinal.html

  1. ω 3 , ω 4 , , ω ω , ω ω ω , , ϵ 0 = ω ω ω , \omega^{3},\omega^{4},\ldots,\omega^{\omega},\omega^{\omega^{\omega}},\ldots,% \epsilon_{0}=\omega^{\omega^{\omega^{~{}\cdot^{~{}\cdot^{~{}\cdot}}}}},\ldots
  2. ω 2 , ω 3 , , ω ω , ω ω + 1 , , ω ω ω , \omega_{2},\omega_{3},\ldots,\omega_{\omega},\omega_{\omega+1},\ldots,\omega_{% \omega_{\omega}},\ldots

Limit_price.html

  1. 𝖰 𝖣 = a - b P \mathsf{Q^{D}}=a-bP
  2. c > 0 c>0
  3. g c gc
  4. g > 1 g>1
  5. p M = a + c b 2 b p^{M}=\frac{a+cb}{2b}
  6. p M g c p^{M}\leq gc
  7. p M p^{M}
  8. g a + c b 2 c b g\geq\frac{a+cb}{2cb}
  9. g < a + c b 2 c b g<\frac{a+cb}{2cb}
  10. p M p^{M}
  11. g c gc
  12. p L = g c p^{L}=gc

Lindblad_equation.html

  1. ρ ρ
  2. N N
  3. ρ ρ
  4. ρ ˙ = - i [ H , ρ ] + n , m = 1 N 2 - 1 h n , m ( L n ρ L m - 1 2 ( ρ L m L n + L m L n ρ ) ) \dot{\rho}=-{i\over\hbar}[H,\rho]+\sum_{n,m=1}^{N^{2}-1}h_{n,m}\left(L_{n}\rho L% _{m}^{\dagger}-\frac{1}{2}\left(\rho L_{m}^{\dagger}L_{n}+L_{m}^{\dagger}L_{n}% \rho\right)\right)
  5. H H
  6. L ( C ) ρ = C ρ C - 1 2 ( C C ρ + ρ C C ) . L(C)\rho=C\rho C^{\dagger}-\frac{1}{2}\left(C^{\dagger}C\rho+\rho C^{\dagger}C% \right).
  7. H H
  8. d d t A = - 1 i [ H , A ] + 1 2 k = 1 ( V k [ A , V k ] + [ V k , A ] V k ) , \frac{d}{dt}A=-\frac{1}{i\hbar}[H,A]+\frac{1}{2\hbar}\sum^{\infty}_{k=1}\big(V% ^{\dagger}_{k}[A,V_{k}]+[V^{\dagger}_{k},A]V_{k}\big),
  9. A A
  10. u u
  11. u h u = [ γ 1 0 0 0 γ 2 0 0 0 γ N 2 - 1 ] u^{\dagger}hu=\begin{bmatrix}\gamma_{1}&0&\cdots&0\\ 0&\gamma_{2}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&\gamma_{N^{2}-1}\end{bmatrix}
  12. A i = j = 1 N 2 - 1 u j , i L j A_{i}=\sum_{j=1}^{N^{2}-1}u_{j,i}L_{j}
  13. ρ ˙ = - i [ H , ρ ] + i = 1 N 2 - 1 γ i ( A i ρ A i - 1 2 ρ A i A i - 1 2 A i A i ρ ) . \dot{\rho}=-{i\over\hbar}[H,\rho]+\sum_{i=1}^{N^{2}-1}\gamma_{i}\left(A_{i}% \rho A_{i}^{\dagger}-\frac{1}{2}\rho A_{i}^{\dagger}A_{i}-\frac{1}{2}A_{i}^{% \dagger}A_{i}\rho\right).
  14. γ i A i γ i A i = j = 1 N 2 - 1 v j , i δ i A j , \sqrt{\gamma_{i}}A_{i}\to\sqrt{\gamma_{i}^{\prime}}A_{i}^{\prime}=\sum_{j=1}^{% N^{2}-1}v_{j,i}\sqrt{\delta_{i}}A_{j},
  15. A i A i = A i + a i , A_{i}\to A_{i}^{\prime}=A_{i}+a_{i},
  16. H H = H + 1 2 i j = 1 N 2 - 1 γ j ( a j * A j - a j A J ) . H\to H^{\prime}=H+\frac{1}{2i}\sum_{j=1}^{N^{2}-1}\gamma_{j}\left(a_{j}^{*}A_{% j}-a_{j}A_{J}^{\dagger}\right).
  17. L 1 = a L 2 = a h n , m = { γ 2 ( n ¯ + 1 ) n = m = 1 γ 2 n ¯ n = m = 2 0 else \begin{aligned}\displaystyle L_{1}&\displaystyle=a\\ \displaystyle L_{2}&\displaystyle=a^{\dagger}\\ \displaystyle h_{n,m}&\displaystyle=\begin{cases}\tfrac{\gamma}{2}\left(% \overline{n}+1\right)&n=m=1\\ \tfrac{\gamma}{2}\overline{n}&n=m=2\\ 0&\,\text{else}\end{cases}\end{aligned}
  18. n ¯ \overline{n}
  19. γ γ

Lindemann–Weierstrass_theorem.html

  1. , ℚ,
  2. . ℚ.
  3. α , α,
  4. π π
  5. e e
  6. π π
  7. e e
  8. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  9. α α
  10. e e
  11. α α
  12. π π
  13. π π
  14. π i πi
  15. α α
  16. s i n ( α ) , c o s ( α ) , t a n ( α ) sin(α),cos(α),tan(α)
  17. p p
  18. p p
  19. p p
  20. α < s u b > 1 , , α n α<sub>1, ..., α_{n}
  21. j j
  22. j ( τ ) = J ( q ) , j(τ)=J(q),
  23. 3 n 3n
  24. { J ( q 1 ) , J ( q 1 ) , J ′′ ( q 1 ) , , J ( q n ) , J ( q n ) , J ′′ ( q n ) } \left\{J(q_{1}),J^{\prime}(q_{1}),J^{\prime\prime}(q_{1}),\ldots,J(q_{n}),J^{% \prime}(q_{n}),J^{\prime\prime}(q_{n})\right\}
  25. . ℚ.
  26. a 1 e α 1 + + a n e α n 0. a_{1}e^{\alpha_{1}}+\cdots+a_{n}e^{\alpha_{n}}\neq 0.
  27. c ( 1 ) , , c ( r ) c(1), ...,c(r)
  28. k k
  29. 1 1
  30. r , r,
  31. w i t h v ( k ) , u ( k ) 0 withv(k),u(k) ≠ 0
  32. ( k , i ) ( u , v ) , (k,i) ≠ (u,v),
  33. c ( 1 ) ( e γ ( 1 ) 1 + + e γ ( 1 ) m ( 1 ) ) + + c ( r ) ( e γ ( r ) 1 + + e γ ( r ) m ( r ) ) 0. c(1)\left(e^{\gamma(1)_{1}}+\cdots+e^{\gamma(1)_{m(1)}}\right)+\cdots+c(r)% \left(e^{\gamma(r)_{1}}+\cdots+e^{\gamma(r)_{m(r)}}\right)\neq 0.
  34. n 0 = 0 n_{0}=0
  35. n i = k = 1 i m ( k ) n_{i}=\sum_{k=1}^{i}m(k)
  36. i = 1 , , r i=1,\dots,r
  37. n = n r n=n_{r}
  38. α n i + j = γ ( i + 1 ) j \alpha_{n_{i}+j}=\gamma(i+1)_{j}
  39. 0 i < n r 0\leq i<n_{r}
  40. 1 j m ( i ) 1\leq j\leq m(i)
  41. β n i + j = c ( i + 1 ) \beta_{n_{i}+j}=c(i+1)
  42. k = 1 n β k e α k 0 \sum_{k=1}^{n}\beta_{k}e^{\alpha_{k}}\neq 0
  43. p p
  44. f i ( x ) = l n p ( x - α 1 ) p ( x - α n ) p ( x - α i ) , f_{i}(x)=\frac{l^{np}(x-\alpha_{1})^{p}\cdots(x-\alpha_{n})^{p}}{(x-\alpha_{i}% )},
  45. l l
  46. l α 1 , , l α n l\alpha_{1},\dots,l\alpha_{n}
  47. I i ( s ) = 0 s e s - x f i ( x ) d x . I_{i}(s)=\int^{s}_{0}e^{s-x}f_{i}(x)\,dx.
  48. e e
  49. I i ( s ) = e s j = 0 n p - 1 f i ( j ) ( 0 ) - j = 0 n p - 1 f i ( j ) ( s ) , I_{i}(s)=e^{s}\sum_{j=0}^{np-1}f_{i}^{(j)}(0)-\sum_{j=0}^{np-1}f_{i}^{(j)}(s),
  50. n p - 1 np-1
  51. f i f_{i}
  52. f i ( j ) f_{i}^{(j)}
  53. f i f_{i}
  54. - e s - x j = 0 n p - 1 f i ( j ) ( x ) -e^{s-x}\sum_{j=0}^{np-1}f_{i}^{(j)}(x)
  55. e s - x f i ( x ) e^{s-x}f_{i}(x)
  56. J i = k = 1 n β k I i ( α k ) = k = 1 n ( β k e α k j = 0 n p - 1 f i ( j ) ( 0 ) ) - k = 1 n ( β k j = 0 n p - 1 f i ( j ) ( α k ) ) = ( j = 0 n p - 1 f i ( j ) ( 0 ) ) ( k = 1 n β k e α k ) - k = 1 n ( β k j = 0 n p - 1 f i ( j ) ( α k ) ) J_{i}=\sum_{k=1}^{n}\beta_{k}I_{i}(\alpha_{k})=\sum_{k=1}^{n}\left(\beta_{k}e^% {\alpha_{k}}\sum_{j=0}^{np-1}f_{i}^{(j)}(0)\right)-\sum_{k=1}^{n}\left(\beta_{% k}\sum_{j=0}^{np-1}f_{i}^{(j)}(\alpha_{k})\right)=\left(\sum_{j=0}^{np-1}f_{i}% ^{(j)}(0)\right)\left(\sum_{k=1}^{n}\beta_{k}e^{\alpha_{k}}\right)-\sum_{k=1}^% {n}\left(\beta_{k}\sum_{j=0}^{np-1}f_{i}^{(j)}(\alpha_{k})\right)
  57. k = 1 n β k e α k = 0 \sum_{k=1}^{n}\beta_{k}e^{\alpha_{k}}=0
  58. | J 1 J n | |J_{1}\cdots J_{n}|
  59. J i = - j = 0 n p - 1 k = 1 n β k f i ( j ) ( α k ) J_{i}=-\sum_{j=0}^{np-1}\sum_{k=1}^{n}\beta_{k}f_{i}^{(j)}(\alpha_{k})
  60. f i ( j ) ( α k ) f_{i}^{(j)}(\alpha_{k})
  61. j p j\geq p
  62. j < p j<p
  63. l n p ( p - 1 ) ! k i ( α i - α k ) p l^{np}(p-1)!\prod_{k\neq i}(\alpha_{i}-\alpha_{k})^{p}
  64. δ i = k i ( l α i - l α k ) \delta_{i}=\prod_{k\neq i}(l\alpha_{i}-l\alpha_{k})
  65. d i d_{i}
  66. l p ( p - 1 ) ! d i p l^{p}(p-1)!d_{i}^{p}
  67. d i d_{i}
  68. l ( p - 1 ) ! d i l(p-1)!d_{i}
  69. J i J_{i}
  70. J i = - j = 0 n p - 1 t = 0 r - 1 c ( t + 1 ) ( f i ( j ) ( α n t + 1 ) + + f i ( j ) ( α n t + 1 ) ) . J_{i}=-\sum_{j=0}^{np-1}\sum_{t=0}^{r-1}c(t+1)\left(f_{i}^{(j)}(\alpha_{n_{t}+% 1})+\dots+f_{i}^{(j)}(\alpha_{n_{t+1}})\right).
  71. f i ( x ) f_{i}(x)
  72. ( x - α i ) (x-\alpha_{i})
  73. f i ( x ) = m = 0 n p - 1 g m ( α i ) x m f_{i}(x)=\sum_{m=0}^{np-1}g_{m}(\alpha_{i})x^{m}
  74. g m g_{m}
  75. f i ( j ) ( x ) f_{i}^{(j)}(x)
  76. f i ( j ) ( α n t + 1 ) + + f i ( j ) ( α n t + 1 ) f_{i}^{(j)}(\alpha_{n_{t}+1})+\dots+f_{i}^{(j)}(\alpha_{n_{t+1}})
  77. α i \alpha_{i}
  78. α n t + 1 , , α n t + 1 \alpha_{n_{t}+1},\dots,\alpha_{n_{t+1}}
  79. α n t + 1 , , α n t + 1 \alpha_{n_{t}+1},\dots,\alpha_{n_{t+1}}
  80. J i J_{i}
  81. G ( α i ) G(\alpha_{i})
  82. J 1 J n = G ( α 1 ) G ( α n ) J_{1}\dots J_{n}=G(\alpha_{1})\dots G(\alpha_{n})
  83. J i J_{i}
  84. ( p - 1 ) ! n (p-1)!^{n}
  85. | J 1 J n | ( p - 1 ) ! n |J_{1}\dots J_{n}|\geq(p-1)!^{n}
  86. | I i ( α k ) | | α k | e | α k | F i ( | α k | ) |I_{i}(\alpha_{k})|\leq|\alpha_{k}|e^{|\alpha_{k}|}F_{i}(|\alpha_{k}|)
  87. I i ( s ) I_{i}(s)
  88. | J i | k = 1 n | β k α k | e | α k | F i ( | α k | ) |J_{i}|\leq\sum_{k=1}^{n}|\beta_{k}\alpha_{k}|e^{|\alpha_{k}|}F_{i}(|\alpha_{k% }|)
  89. f i f_{i}
  90. | J 1 J n | C p |J_{1}\dots J_{n}|\leq C^{p}
  91. b ( 1 ) e γ ( 1 ) + + b ( n ) e γ ( n ) 0. b(1)e^{\gamma(1)}+\cdots+b(n)e^{\gamma(n)}\neq 0.
  92. b ( 1 ) e γ ( 1 ) + + b ( n ) e γ ( n ) = 0 , b(1)e^{\gamma(1)}+\cdots+b(n)e^{\gamma(n)}=0,
  93. γ ( k ) \gamma(k)
  94. γ ( 1 ) , , γ ( n ) , γ ( n + 1 ) , , γ ( N ) \gamma(1),\dots,\gamma(n),\gamma(n+1),\dots,\gamma(N)
  95. σ S N ( b ( 1 ) e γ ( σ ( 1 ) ) + + b ( N ) e γ ( σ ( N ) ) ) \prod_{\sigma\in S_{N}}(b(1)e^{\gamma(\sigma(1))}+\dots+b(N)e^{\gamma(\sigma(N% ))})
  96. e h 1 γ ( 1 ) + + h N γ ( N ) e^{h_{1}\gamma(1)+\dots+h_{N}\gamma(N)}
  97. τ S n \tau\in S_{n}
  98. e h 1 γ ( τ ( 1 ) ) + + h N γ ( τ ( N ) ) e^{h_{1}\gamma(\tau(1))+\dots+h_{N}\gamma(\tau(N))}
  99. e h 1 γ ( 1 ) + + h N γ ( N ) e^{h_{1}\gamma(1)+\dots+h_{N}\gamma(N)}
  100. \mathbb{C}
  101. a ( 1 ) e α ( 1 ) + + a ( n ) e α ( n ) = 0. a(1)e^{\alpha(1)}+\cdots+a(n)e^{\alpha(n)}=0.
  102. { σ } ( a ( 1 ) σ ( 1 ) e α ( 1 ) + + a ( n ) σ ( n ) e α ( n ) ) = 0 \prod\nolimits_{\{\sigma\}}\left(a(1)_{\sigma(1)}e^{\alpha(1)}+\cdots+a(n)_{% \sigma(n)}e^{\alpha(n)}\right)=0
  103. b ( 1 ) e β ( 1 ) + b ( 2 ) e β ( 2 ) + + b ( N ) e β ( N ) = 0. b(1)e^{\beta(1)}+b(2)e^{\beta(2)}+\cdots+b(N)e^{\beta(N)}=0.

Line_at_infinity.html

  1. P 2 \mathbb{R}P^{2}

Line_bundle.html

  1. V { 0 } V\setminus\{0\}
  2. 𝒪 ( - 1 ) \mathcal{O}(-1)
  3. 𝒪 ( 1 ) \mathcal{O}(1)

Linear_algebraic_group.html

  1. 𝔾 m = G L 1 \mathbb{G}_{m}=GL_{1}
  2. 𝔾 m \mathbb{G}_{m}
  3. x [ 1 x 0 1 ] x\mapsto\begin{bmatrix}1&x\\ 0&1\end{bmatrix}
  4. 𝔾 a \mathbb{G}_{a}

Linear_combination_of_atomic_orbitals.html

  1. ϕ i = c 1 i χ 1 + c 2 i χ 2 + c 3 i χ 3 + + c n i χ n \ \phi_{i}=c_{1i}\chi_{1}+c_{2i}\chi_{2}+c_{3i}\chi_{3}+\cdots+c_{ni}\chi_{n}
  2. ϕ i = r c r i χ r \ \phi_{i}=\sum_{r}c_{ri}\chi_{r}
  3. ϕ i \ \phi_{i}
  4. χ r \ \chi_{r}
  5. c r i \ c_{ri}

Linear_differential_equation.html

  1. L y = f Ly=f
  2. L y ( t ) = f ( t ) Ly(t)=f(t)
  3. L [ y ] ( t ) = f ( t ) L[y](t)=f(t)
  4. L n ( y ) d n y d t n + A 1 ( t ) d n - 1 y d t n - 1 + + A n - 1 ( t ) d y d t + A n ( t ) y L_{n}(y)\equiv\frac{d^{n}y}{dt^{n}}+A_{1}(t)\frac{d^{n-1}y}{dt^{n-1}}+\cdots+A% _{n-1}(t)\frac{dy}{dt}+A_{n}(t)y
  5. L n ( y ) [ D n + A 1 ( t ) D n - 1 + + A n - 1 ( t ) D + A n ( t ) ] y L_{n}(y)\equiv\left[\,D^{n}+A_{1}(t)D^{n-1}+\cdots+A_{n-1}(t)D+A_{n}(t)\right]y
  6. d N d t = - k N \frac{dN}{dt}=-kN
  7. y ( n ) + A 1 y ( n - 1 ) + + A n y = 0 , y^{(n)}+A_{1}y^{(n-1)}+\cdots+A_{n}y=0\,,
  8. z n e z x + A 1 z n - 1 e z x + + A n e z x = 0. z^{n}e^{zx}+A_{1}z^{n-1}e^{zx}+\cdots+A_{n}e^{zx}=0.
  9. F ( z ) = z n + A 1 z n - 1 + + A n = 0. F(z)=z^{n}+A_{1}z^{n-1}+\cdots+A_{n}=0.\,
  10. y ( k ) ( k = 1 , 2 , , n ) . y^{(k)}\quad\quad(k=1,2,\dots,n).
  11. k { 0 , 1 , , m - 1 } k\in\{0,1,\dots,m-1\}\,
  12. y = x k e z x y=x^{k}e^{zx}
  13. y ′′ - 4 y + 5 y = 0 y^{\prime\prime}-4y^{\prime}+5y=0
  14. z 2 - 4 z + 5 = 0 z^{2}-4z+5=0
  15. { y 1 , y 2 } \{y_{1},y_{2}\}
  16. { e ( 2 + i ) x , e ( 2 - i ) x } \{e^{(2+i)x},e^{(2-i)x}\}
  17. y = c 1 y 1 + c 2 y 2 y=c_{1}y_{1}+c_{2}y_{2}
  18. c 1 , c 2 𝐂 c_{1},c_{2}\in\mathbf{C}
  19. u 1 = Re ( y 1 ) = 1 2 ( y 1 + y 2 ) = e 2 x cos ( x ) , u_{1}=\mbox{Re}~{}(y_{1})=\tfrac{1}{2}(y_{1}+y_{2})=e^{2x}\cos(x),
  20. u 2 = Im ( y 1 ) = 1 2 i ( y 1 - y 2 ) = e 2 x sin ( x ) , u_{2}=\mbox{Im}~{}(y_{1})=\tfrac{1}{2i}(y_{1}-y_{2})=e^{2x}\sin(x),
  21. { u 1 , u 2 } \{u_{1},u_{2}\}
  22. D 2 y = - k 2 y , D^{2}y=-k^{2}y,
  23. ( D 2 + k 2 ) y = 0. (D^{2}+k^{2})y=0.
  24. ( D + i k ) ( D - i k ) y = 0 , (D+ik)(D-ik)y=0,
  25. ( D - i k ) y = 0 (D-ik)y=0
  26. ( D + i k ) y = 0. (D+ik)y=0.
  27. y 0 = A 0 e i k x y_{0}=A_{0}e^{ikx}
  28. y 1 = A 1 e - i k x . y_{1}=A_{1}e^{-ikx}.
  29. y 0 = C 0 e i k x + C 0 e - i k x 2 = C 0 cos ( k x ) y_{0^{\prime}}={C_{0}e^{ikx}+C_{0}e^{-ikx}\over 2}=C_{0}\cos(kx)
  30. y 1 = C 1 e i k x - C 1 e - i k x 2 i = C 1 sin ( k x ) . y_{1^{\prime}}={C_{1}e^{ikx}-C_{1}e^{-ikx}\over 2i}=C_{1}\sin(kx).
  31. y H = C 0 cos ( k x ) + C 1 sin ( k x ) . y_{H}=C_{0}\cos(kx)+C_{1}\sin(kx).
  32. ( D 2 + b m D + ω 0 2 ) y = 0 , \left(D^{2}+\frac{b}{m}D+\omega_{0}^{2}\right)y=0,
  33. λ 2 + b m λ + ω 0 2 = 0. \lambda^{2}+\frac{b}{m}\lambda+\omega_{0}^{2}=0.
  34. λ = 1 2 ( - b m ± b 2 m 2 - 4 ω 0 2 ) . \lambda=\tfrac{1}{2}\left(-\frac{b}{m}\pm\sqrt{\frac{b^{2}}{m^{2}}-4\omega_{0}% ^{2}}\right).
  35. ( D + b 2 m - b 2 4 m 2 - ω 0 2 ) ( D + b 2 m + b 2 4 m 2 - ω 0 2 ) y = 0. \left(D+\frac{b}{2m}-\sqrt{\frac{b^{2}}{4m^{2}}-\omega_{0}^{2}}\right)\left(D+% \frac{b}{2m}+\sqrt{\frac{b^{2}}{4m^{2}}-\omega_{0}^{2}}\right)y=0.
  36. ( D + b 2 m - b 2 4 m 2 - ω 0 2 ) y = 0 \left(D+\frac{b}{2m}-\sqrt{\frac{b^{2}}{4m^{2}}-\omega_{0}^{2}}\right)y=0
  37. ( D + b 2 m + b 2 4 m 2 - ω 0 2 ) y = 0 \left(D+\frac{b}{2m}+\sqrt{\frac{b^{2}}{4m^{2}}-\omega_{0}^{2}}\right)y=0
  38. y 0 = A 0 e - ω x + ω 2 - ω 0 2 x = A 0 e - ω x e ω 2 - ω 0 2 x y_{0}=A_{0}e^{-\omega x+\sqrt{\omega^{2}-\omega_{0}^{2}}x}=A_{0}e^{-\omega x}e% ^{\sqrt{\omega^{2}-\omega_{0}^{2}}x}
  39. y 1 = A 1 e - ω x - ω 2 - ω 0 2 x = A 1 e - ω x e - ω 2 - ω 0 2 x y_{1}=A_{1}e^{-\omega x-\sqrt{\omega^{2}-\omega_{0}^{2}}x}=A_{1}e^{-\omega x}e% ^{-\sqrt{\omega^{2}-\omega_{0}^{2}}x}
  40. y H ( A 0 , A 1 ) ( x ) = ( A 0 sinh ( ω 2 - ω 0 2 x ) + A 1 cosh ( ω 2 - ω 0 2 x ) ) e - ω x . y_{H}(A_{0},A_{1})(x)=\left(A_{0}\sinh\left(\sqrt{\omega^{2}-\omega_{0}^{2}}x% \right)+A_{1}\cosh\left(\sqrt{\omega^{2}-\omega_{0}^{2}}x\right)\right)e^{-% \omega x}.
  41. y H ( A 0 , A 1 ) ( x ) = ( A 0 sin ( ω 0 2 - ω 2 x ) + A 1 cos ( ω 0 2 - ω 2 x ) ) e - ω x . y_{H}(A_{0},A_{1})(x)=\left(A_{0}\sin\left(\sqrt{\omega_{0}^{2}-\omega^{2}}x% \right)+A_{1}\cos\left(\sqrt{\omega_{0}^{2}-\omega^{2}}x\right)\right)e^{-% \omega x}.
  42. d n y ( x ) d x n + A 1 d n - 1 y ( x ) d x n - 1 + + A n y ( x ) = f ( x ) . \frac{d^{n}y(x)}{dx^{n}}+A_{1}\frac{d^{n-1}y(x)}{dx^{n-1}}+\cdots+A_{n}y(x)=f(% x).
  43. P ( v ) = v n + A 1 v n - 1 + + A n . P(v)=v^{n}+A_{1}v^{n-1}+\cdots+A_{n}.
  44. { y 1 ( x ) , y 2 ( x ) , , y n ( x ) } \{y_{1}(x),y_{2}(x),\ldots,y_{n}(x)\}
  45. y p ( x ) = u 1 ( x ) y 1 ( x ) + u 2 ( x ) y 2 ( x ) + + u n ( x ) y n ( x ) . y_{p}(x)=u_{1}(x)y_{1}(x)+u_{2}(x)y_{2}(x)+\cdots+u_{n}(x)y_{n}(x).
  46. f = P ( D ) y p = P ( D ) ( u 1 y 1 ) + P ( D ) ( u 2 y 2 ) + + P ( D ) ( u n y n ) . f=P(D)y_{p}=P(D)(u_{1}y_{1})+P(D)(u_{2}y_{2})+\cdots+P(D)(u_{n}y_{n}).
  47. 0 = u 1 y 1 + u 2 y 2 + + u n y n 0=u^{\prime}_{1}y_{1}+u^{\prime}_{2}y_{2}+\cdots+u^{\prime}_{n}y_{n}
  48. 0 = u 1 y 1 + u 2 y 2 + + u n y n 0=u^{\prime}_{1}y^{\prime}_{1}+u^{\prime}_{2}y^{\prime}_{2}+\cdots+u^{\prime}_% {n}y^{\prime}_{n}
  49. \cdots
  50. 0 = u 1 y 1 ( n - 2 ) + u 2 y 2 ( n - 2 ) + + u n y n ( n - 2 ) 0=u^{\prime}_{1}y^{(n-2)}_{1}+u^{\prime}_{2}y^{(n-2)}_{2}+\cdots+u^{\prime}_{n% }y^{(n-2)}_{n}
  51. f = u 1 P ( D ) y 1 + u 2 P ( D ) y 2 + + u n P ( D ) y n + u 1 y 1 ( n - 1 ) + u 2 y 2 ( n - 1 ) + + u n y n ( n - 1 ) . f=u_{1}P(D)y_{1}+u_{2}P(D)y_{2}+\cdots+u_{n}P(D)y_{n}+u^{\prime}_{1}y^{(n-1)}_% {1}+u^{\prime}_{2}y^{(n-1)}_{2}+\cdots+u^{\prime}_{n}y^{(n-1)}_{n}.
  52. f = u 1 y 1 ( n - 1 ) + u 2 y 2 ( n - 1 ) + + u n y n ( n - 1 ) . f=u^{\prime}_{1}y^{(n-1)}_{1}+u^{\prime}_{2}y^{(n-1)}_{2}+\cdots+u^{\prime}_{n% }y^{(n-1)}_{n}.
  53. u j = ( - 1 ) n + j W ( y 1 , , y j - 1 , y j + 1 , y n ) ( 0 f ) W ( y 1 , y 2 , , y n ) . u^{\prime}_{j}=(-1)^{n+j}\frac{W(y_{1},\ldots,y_{j-1},y_{j+1}\ldots,y_{n})_{0% \choose f}}{W(y_{1},y_{2},\ldots,y_{n})}.
  54. y p + c 1 y 1 + + c n y n y_{p}+c_{1}y_{1}+\cdots+c_{n}y_{n}
  55. y ′′ - 4 y + 5 y = sin ( k x ) y^{\prime\prime}-4y^{\prime}+5y=\sin(kx)
  56. { e ( 2 + i ) x = y 1 ( x ) , e ( 2 - i ) x = y 2 ( x ) } \{e^{(2+i)x}=y_{1}(x),e^{(2-i)x}=y_{2}(x)\}
  57. W = | e ( 2 + i ) x e ( 2 - i ) x ( 2 + i ) e ( 2 + i ) x ( 2 - i ) e ( 2 - i ) x | = e 4 x | 1 1 2 + i 2 - i | = - 2 i e 4 x u 1 = 1 W | 0 e ( 2 - i ) x sin ( k x ) ( 2 - i ) e ( 2 - i ) x | = - i 2 sin ( k x ) e ( - 2 - i ) x u 2 = 1 W | e ( 2 + i ) x 0 ( 2 + i ) e ( 2 + i ) x sin ( k x ) | = i 2 sin ( k x ) e ( - 2 + i ) x . \begin{aligned}\displaystyle W&\displaystyle=\begin{vmatrix}e^{(2+i)x}&e^{(2-i% )x}\\ (2+i)e^{(2+i)x}&(2-i)e^{(2-i)x}\end{vmatrix}=e^{4x}\begin{vmatrix}1&1\\ 2+i&2-i\end{vmatrix}=-2ie^{4x}\\ \displaystyle u^{\prime}_{1}&\displaystyle=\frac{1}{W}\begin{vmatrix}0&e^{(2-i% )x}\\ \sin(kx)&(2-i)e^{(2-i)x}\end{vmatrix}=-\tfrac{i}{2}\sin(kx)e^{(-2-i)x}\\ \displaystyle u^{\prime}_{2}&\displaystyle=\frac{1}{W}\begin{vmatrix}e^{(2+i)x% }&0\\ (2+i)e^{(2+i)x}&\sin(kx)\end{vmatrix}=\tfrac{i}{2}\sin(kx)e^{(-2+i)x}.\end{aligned}
  58. u 1 = - i 2 sin ( k x ) e ( - 2 - i ) x d x = i e ( - 2 - i ) x 2 ( 3 + 4 i + k 2 ) ( ( 2 + i ) sin ( k x ) + k cos ( k x ) ) u_{1}=-\tfrac{i}{2}\int\sin(kx)e^{(-2-i)x}\,dx=\frac{ie^{(-2-i)x}}{2(3+4i+k^{2% })}\left((2+i)\sin(kx)+k\cos(kx)\right)
  59. u 2 = i 2 sin ( k x ) e ( - 2 + i ) x d x = i e ( i - 2 ) x 2 ( 3 - 4 i + k 2 ) ( ( i - 2 ) sin ( k x ) - k cos ( k x ) ) . u_{2}=\tfrac{i}{2}\int\sin(kx)e^{(-2+i)x}\,dx=\frac{ie^{(i-2)x}}{2(3-4i+k^{2})% }\left((i-2)\sin(kx)-k\cos(kx)\right).
  60. y p = u 1 ( x ) y 1 ( x ) + u 2 ( x ) y 2 ( x ) = i 2 ( 3 + 4 i + k 2 ) ( ( 2 + i ) sin ( k x ) + k cos ( k x ) ) + i 2 ( 3 - 4 i + k 2 ) ( ( i - 2 ) sin ( k x ) - k cos ( k x ) ) = ( 5 - k 2 ) sin ( k x ) + 4 k cos ( k x ) ( 3 + k 2 ) 2 + 16 . \begin{aligned}\displaystyle y_{p}&\displaystyle=u_{1}(x)y_{1}(x)+u_{2}(x)y_{2% }(x)=\frac{i}{2(3+4i+k^{2})}\left((2+i)\sin(kx)+k\cos(kx)\right)+\frac{i}{2(3-% 4i+k^{2})}\left((i-2)\sin(kx)-k\cos(kx)\right)\\ &\displaystyle=\frac{(5-k^{2})\sin(kx)+4k\cos(kx)}{(3+k^{2})^{2}+16}.\end{aligned}
  61. c 1 y 1 + c 2 y 2 c_{1}y_{1}+c_{2}y_{2}
  62. p n ( x ) y ( n ) ( x ) + p n - 1 ( x ) y ( n - 1 ) ( x ) + + p 0 ( x ) y ( x ) = r ( x ) . p_{n}(x)y^{(n)}(x)+p_{n-1}(x)y^{(n-1)}(x)+\cdots+p_{0}(x)y(x)=r(x).
  63. x n y ( n ) ( x ) + a n - 1 x n - 1 y ( n - 1 ) ( x ) + + a 0 y ( x ) = 0. x^{n}y^{(n)}(x)+a_{n-1}x^{n-1}y^{(n-1)}(x)+\cdots+a_{0}y(x)=0.
  64. D y ( x ) + f ( x ) y ( x ) = g ( x ) . Dy(x)+f(x)y(x)=g(x).
  65. e f ( x ) d x e^{\int f(x)\,dx}
  66. D y ( x ) e f ( x ) d x + f ( x ) y ( x ) e f ( x ) d x = g ( x ) e f ( x ) d x , Dy(x)e^{\int f(x)\,dx}+f(x)y(x)e^{\int f(x)\,dx}=g(x)e^{\int f(x)\,dx},
  67. D ( y ( x ) e f ( x ) d x ) = g ( x ) e f ( x ) d x D\left(y(x)e^{\int f(x)\,dx}\right)=g(x)e^{\int f(x)\,dx}
  68. y ( x ) = e - f ( x ) d x ( g ( x ) e f ( x ) d x d x + κ ) . y(x)=e^{-\int f(x)\,dx}\left(\int g(x)e^{\int f(x)\,dx}\,dx+\kappa\right).
  69. y ( x ) + f ( x ) y ( x ) = g ( x ) , y^{\prime}(x)+f(x)y(x)=g(x),
  70. y = e - a ( x ) ( g ( x ) e a ( x ) d x + κ ) y=e^{-a(x)}\left(\int g(x)e^{a(x)}\,dx+\kappa\right)
  71. a ( x ) = f ( x ) d x . a(x)=\int{f(x)\,dx}.
  72. y ( x ) = a x [ y ( a ) δ ( t - a ) + g ( t ) ] e - t x f ( u ) d u d t . y(x)=\int_{a}^{x}\!{[y(a)\delta(t-a)+g(t)]e^{-\int_{t}^{x}\!f(u)du}\,dt}\,.
  73. d y d x + b y = 1. \frac{dy}{dx}+by=1.
  74. y ( x ) = e - b x ( e b x b + C ) = 1 b + C e - b x . y(x)=e^{-bx}\left(\frac{e^{bx}}{b}+C\right)=\frac{1}{b}+Ce^{-bx}.
  75. { 𝐲 ( x ) = A ( x ) 𝐲 ( x ) + 𝐛 ( x ) 𝐲 ( x 0 ) = 𝐲 0 \left\{\begin{array}[]{rl}\mathbf{y}^{\prime}(x)&=A(x)\mathbf{y}(x)+\mathbf{b}% (x)\\ \mathbf{y}(x_{0})&=\mathbf{y}_{0}\end{array}\right.
  76. 𝐲 ( x ) \mathbf{y}(x)
  77. A ( x ) A(x)
  78. U ( x ) U(x)
  79. 𝐲 ( x ) = A ( x ) 𝐲 ( x ) \mathbf{y}^{\prime}(x)=A(x)\mathbf{y}(x)
  80. U ( x 0 ) = I U(x_{0})=I
  81. U U
  82. U U
  83. 𝐲 ( x ) = U ( x ) 𝐳 ( x ) \mathbf{y}(x)=U(x)\mathbf{z}(x)
  84. 𝐲 ( x ) = A ( x ) 𝐲 ( x ) + 𝐛 ( x ) \mathbf{y}^{\prime}(x)=A(x)\mathbf{y}(x)+\mathbf{b}(x)
  85. U ( x ) 𝐳 ( x ) = 𝐛 ( x ) . U(x)\mathbf{z}^{\prime}(x)=\mathbf{b}(x).
  86. 𝐲 ( x ) = U ( x ) 𝐲 𝟎 + U ( x ) x 0 x U - 1 ( t ) 𝐛 ( t ) d t \mathbf{y}(x)=U(x)\mathbf{y_{0}}+U(x)\int_{x_{0}}^{x}U^{-1}(t)\mathbf{b}(t)\,dt
  87. A ( x 1 ) A(x_{1})
  88. A ( x 2 ) A(x_{2})
  89. x 1 x_{1}
  90. x 2 x_{2}
  91. U ( x ) = e x 0 x A ( x ) d x U(x)=e^{\int_{x_{0}}^{x}A(x)\,dx}
  92. U - 1 ( x ) = e - x 0 x A ( x ) d x , U^{-1}(x)=e^{-\int_{x_{0}}^{x}A(x)\,dx},

Linear_fractional_transformation.html

  1. z a z + b c z + d , z , a , b , c , d \isin A . z\mapsto\frac{az+b}{cz+d},\quad z,a,b,c,d\isin A.
  2. U ( a z + b , c z + d ) U ( ( c z + d ) - 1 ( a z + b ) , 1 ) . U(az+b,cz+d)\sim U((cz+d)^{-1}(az+b),1).
  3. exp ( y j ) = cosh y + j sinh y , j 2 = + 1 , \exp(yj)=\cosh y+j\sinh y,\quad j^{2}=+1,
  4. exp ( y ϵ ) = 1 + y ϵ , ϵ 2 = 0 , \exp(y\epsilon)=1+y\epsilon,\quad\epsilon^{2}=0,
  5. exp ( y i ) = cos y + i sin y , i 2 = - 1. \exp(yi)=\cos y+i\sin y,\quad i^{2}=-1.
  6. exp ( y b ) exp ( - y b ) , b 2 = 1 , 0 , - 1. \exp(yb)\mapsto\exp(-yb),\quad b^{2}=1,0,-1.

Linear_induction_motor.html

  1. n s = 2 f s / p n_{s}=2f_{s}/p
  2. f s f_{s}
  3. n s n_{s}
  4. v s = 2 t f s v_{s}=2tf_{s}
  5. v s v_{s}
  6. v r = ( 1 - s ) v s v_{r}=(1-s)v_{s}

Linear_least_squares_(mathematics).html

  1. ( x , y ) (x,y)
  2. ( 1 , 6 ) , (1,6),
  3. ( 2 , 5 ) , (2,5),
  4. ( 3 , 7 ) , (3,7),
  5. ( 4 , 10 ) (4,10)
  6. y = β 1 + β 2 x y=\beta_{1}+\beta_{2}x
  7. β 1 \beta_{1}
  8. β 2 \beta_{2}
  9. β 1 + 1 β 2 = 6 β 1 + 2 β 2 = 5 β 1 + 3 β 2 = 7 β 1 + 4 β 2 = 10 \begin{aligned}\displaystyle\beta_{1}+1\beta_{2}&&\displaystyle\;=&&% \displaystyle 6&\\ \displaystyle\beta_{1}+2\beta_{2}&&\displaystyle\;=&&\displaystyle 5&\\ \displaystyle\beta_{1}+3\beta_{2}&&\displaystyle\;=&&\displaystyle 7&\\ \displaystyle\beta_{1}+4\beta_{2}&&\displaystyle\;=&&\displaystyle 10&\\ \end{aligned}
  10. S ( β 1 , β 2 ) = \displaystyle S(\beta_{1},\beta_{2})=
  11. S ( β 1 , β 2 ) S(\beta_{1},\beta_{2})
  12. β 1 \beta_{1}
  13. β 2 \beta_{2}
  14. S β 1 = 0 = 8 β 1 + 20 β 2 - 56 \frac{\partial S}{\partial\beta_{1}}=0=8\beta_{1}+20\beta_{2}-56
  15. S β 2 = 0 = 20 β 1 + 60 β 2 - 154. \frac{\partial S}{\partial\beta_{2}}=0=20\beta_{1}+60\beta_{2}-154.
  16. β 1 = 3.5 \beta_{1}=3.5
  17. β 2 = 1.4 \beta_{2}=1.4
  18. y = 3.5 + 1.4 x y=3.5+1.4x
  19. y y
  20. y y
  21. 1.1 , 1.1,
  22. - 1.3 , -1.3,
  23. - 0.7 , -0.7,
  24. 0.9 0.9
  25. S ( 3.5 , 1.4 ) = 1.1 2 + ( - 1.3 ) 2 + ( - 0.7 ) 2 + 0.9 2 = 4.2. S(3.5,1.4)=1.1^{2}+(-1.3)^{2}+(-0.7)^{2}+0.9^{2}=4.2.
  26. y = β 1 + β 2 x + β 3 x 2 y=\beta_{1}+\beta_{2}x+\beta_{3}x^{2}\,
  27. ( x i , y i ) (x_{i},y_{i})
  28. x , x,
  29. β j \beta_{j}
  30. y = β 1 x 2 y=\beta_{1}x^{2}
  31. β 1 \beta_{1}
  32. 6 \displaystyle 6
  33. S β 1 = 0 = 708 β 1 - 498 \frac{\partial S}{\partial\beta_{1}}=0=708\beta_{1}-498
  34. β 1 = .703 \beta_{1}=.703
  35. y = .703 x 2 y=.703x^{2}
  36. j = 1 n X i j β j = y i , ( i = 1 , 2 , , m ) , \sum_{j=1}^{n}X_{ij}\beta_{j}=y_{i},\ (i=1,2,\dots,m),
  37. 𝐗 s y m b o l β = 𝐲 , \mathbf{X}symbol{\beta}=\mathbf{y},
  38. 𝐗 = [ X 11 X 12 X 1 n X 21 X 22 X 2 n X m 1 X m 2 X m n ] , s y m b o l β = [ β 1 β 2 β n ] , 𝐲 = [ y 1 y 2 y m ] . \mathbf{X}=\begin{bmatrix}X_{11}&X_{12}&\cdots&X_{1n}\\ X_{21}&X_{22}&\cdots&X_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ X_{m1}&X_{m2}&\cdots&X_{mn}\end{bmatrix},\qquad symbol\beta=\begin{bmatrix}% \beta_{1}\\ \beta_{2}\\ \vdots\\ \beta_{n}\end{bmatrix},\qquad\mathbf{y}=\begin{bmatrix}y_{1}\\ y_{2}\\ \vdots\\ y_{m}\end{bmatrix}.
  39. s y m b o l β ^ = arg min s y m b o l β S ( s y m b o l β ) , \hat{symbol{\beta}}=\underset{symbol{\beta}}{\operatorname{arg\,min}}\,S(% symbol{\beta}),
  40. S ( s y m b o l β ) = i = 1 m | y i - j = 1 n X i j β j | 2 = 𝐲 - 𝐗 s y m b o l β 2 . S(symbol{\beta})=\sum_{i=1}^{m}\bigl|y_{i}-\sum_{j=1}^{n}X_{ij}\beta_{j}\bigr|% ^{2}=\bigl\|\mathbf{y}-\mathbf{X}symbol\beta\bigr\|^{2}.
  41. ( 𝐗 T 𝐗 ) s y m b o l β ^ = 𝐗 T 𝐲 . (\mathbf{X}^{\rm T}\mathbf{X})\hat{symbol{\beta}}=\mathbf{X}^{\rm T}\mathbf{y}.
  42. 𝐗 T 𝐗 \mathbf{X}^{\rm T}\mathbf{X}
  43. 𝐗 \mathbf{X}
  44. ( 𝐀 T 𝐀 ) s y m b o l β ^ = 𝐀 T 𝐲 . (\mathbf{A}^{\rm T}\mathbf{A})\hat{symbol{\beta}}=\mathbf{A}^{\rm T}\mathbf{y}.
  45. s y m b o l β ^ \hat{symbol{\beta}}
  46. i i
  47. r i = y i - j = 1 n X i j β j r_{i}=y_{i}-\sum_{j=1}^{n}X_{ij}\beta_{j}
  48. S S
  49. S = i = 1 m r i 2 . S=\sum_{i=1}^{m}r_{i}^{2}.
  50. S β j = 2 i = 1 m r i r i β j ( j = 1 , 2 , , n ) . \frac{\partial S}{\partial\beta_{j}}=2\sum_{i=1}^{m}r_{i}\frac{\partial r_{i}}% {\partial\beta_{j}}\ (j=1,2,\dots,n).
  51. r i β j = - X i j . \frac{\partial r_{i}}{\partial\beta_{j}}=-X_{ij}.
  52. S β j = 2 i = 1 m ( y i - k = 1 n X i k β k ) ( - X i j ) ( j = 1 , 2 , , n ) . \frac{\partial S}{\partial\beta_{j}}=2\sum_{i=1}^{m}\left(y_{i}-\sum_{k=1}^{n}% X_{ik}\beta_{k}\right)(-X_{ij})\ (j=1,2,\dots,n).
  53. β ^ \hat{\beta}
  54. 2 i = 1 m ( y i - k = 1 n X i k β ^ k ) ( - X i j ) = 0 ( j = 1 , 2 , , n ) . 2\sum_{i=1}^{m}\left(y_{i}-\sum_{k=1}^{n}X_{ik}\hat{\beta}_{k}\right)(-X_{ij})% =0\ (j=1,2,\dots,n).
  55. i = 1 m k = 1 n X i j X i k β ^ k = i = 1 m X i j y i ( j = 1 , 2 , , n ) . \sum_{i=1}^{m}\sum_{k=1}^{n}X_{ij}X_{ik}\hat{\beta}_{k}=\sum_{i=1}^{m}X_{ij}y_% {i}\ (j=1,2,\dots,n).
  56. ( 𝐗 T 𝐗 ) s y m b o l β ^ = 𝐗 T 𝐲 (\mathbf{X}^{\mathrm{T}}\mathbf{X})\hat{symbol{\beta}}=\mathbf{X}^{\mathrm{T}}% \mathbf{y}
  57. s y m b o l β ^ \hat{symbol{\beta}}
  58. S ( s y m b o l β ) = 𝐲 - 𝐗 s y m b o l β 2 = ( 𝐲 - 𝐗 s y m b o l β ) T ( 𝐲 - 𝐗 s y m b o l β ) = 𝐲 T 𝐲 - s y m b o l β T 𝐗 T 𝐲 - 𝐲 T 𝐗 s y m b o l β + s y m b o l β T 𝐗 T 𝐗 s y m b o l β . S(symbol{\beta})=\bigl\|\mathbf{y}-\mathbf{X}symbol\beta\bigr\|^{2}=(\mathbf{y% }-\mathbf{X}symbol\beta)^{\rm T}(\mathbf{y}-\mathbf{X}symbol\beta)=\mathbf{y}^% {\rm T}\mathbf{y}-symbol\beta^{\rm T}\mathbf{X}^{\rm T}\mathbf{y}-\mathbf{y}^{% \rm T}\mathbf{X}symbol\beta+symbol\beta^{\rm T}\mathbf{X}^{\rm T}\mathbf{X}% symbol\beta.
  59. ( s y m b o l β T 𝐗 T 𝐲 ) T = 𝐲 T 𝐗 s y m b o l β (symbol\beta^{\rm T}\mathbf{X}^{\rm T}\mathbf{y})^{\rm T}=\mathbf{y}^{\rm T}% \mathbf{X}symbol\beta
  60. 𝐲 \mathbf{y}
  61. s y m b o l β T 𝐗 T 𝐲 = 𝐲 T 𝐗 s y m b o l β symbol\beta^{\rm T}\mathbf{X}^{\rm T}\mathbf{y}=\mathbf{y}^{\rm T}\mathbf{X}symbol\beta
  62. S ( s y m b o l β ) = 𝐲 T 𝐲 - 2 s y m b o l β T 𝐗 T 𝐲 + s y m b o l β T 𝐗 T 𝐗 s y m b o l β . S(symbol{\beta})=\mathbf{y}^{\rm T}\mathbf{y}-2symbol\beta^{\rm T}\mathbf{X}^{% \rm T}\mathbf{y}+symbol\beta^{\rm T}\mathbf{X}^{\rm T}\mathbf{X}symbol\beta.
  63. s y m b o l β symbol\beta
  64. - 𝐗 T 𝐲 + ( 𝐗 T 𝐗 ) s y m b o l β = 0 , -\mathbf{X}^{\rm T}\mathbf{y}+(\mathbf{X}^{\rm T}\mathbf{X}){symbol{\beta}}=0,
  65. 𝐗 \mathbf{X}
  66. 𝐗 T 𝐗 \mathbf{X}^{\rm T}\mathbf{X}
  67. 𝐗 T 𝐗 \mathbf{X}^{\rm T}\mathbf{X}
  68. s y m b o l β symbol\beta
  69. S ( s y m b o l β ) = 𝐲 T 𝐲 - 2 s y m b o l β T 𝐗 T 𝐲 + s y m b o l β T 𝐗 T 𝐗 s y m b o l β S(symbol{\beta})=\mathbf{y}^{\rm T}\mathbf{y}-2symbol\beta^{\rm T}\mathbf{X}^{% \rm T}\mathbf{y}+symbol\beta^{\rm T}\mathbf{X}^{\rm T}\mathbf{X}symbol\beta
  70. s y m b o l β , s y m b o l β - 2 s y m b o l β , ( 𝐗 T 𝐗 ) - 1 𝐗 T 𝐲 + ( 𝐗 T 𝐗 ) - 1 𝐗 T 𝐲 , ( 𝐗 T 𝐗 ) - 1 𝐗 T 𝐲 + C , \langle symbol\beta,symbol\beta\rangle-2\langle symbol\beta,(\mathbf{X}^{\rm T% }\mathbf{X})^{-1}\mathbf{X}^{\rm T}\mathbf{y}\rangle+\langle(\mathbf{X}^{\rm T% }\mathbf{X})^{-1}\mathbf{X}^{\rm T}\mathbf{y},(\mathbf{X}^{\rm T}\mathbf{X})^{% -1}\mathbf{X}^{\rm T}\mathbf{y}\rangle+C,
  71. C C
  72. 𝐲 \mathbf{y}
  73. 𝐗 \mathbf{X}
  74. , \langle\cdot,\cdot\rangle
  75. x , y = x T ( 𝐗 T 𝐗 ) y . \langle x,y\rangle=x^{\rm T}(\mathbf{X}^{\rm T}\mathbf{X})y.
  76. S ( s y m b o l β ) S(symbol{\beta})
  77. s y m b o l β - ( 𝐗 T 𝐗 ) - 1 𝐗 T 𝐲 , s y m b o l β - ( 𝐗 T 𝐗 ) - 1 𝐗 T 𝐲 + C \langle symbol\beta-(\mathbf{X}^{\rm T}\mathbf{X})^{-1}\mathbf{X}^{\rm T}% \mathbf{y},symbol\beta-(\mathbf{X}^{\rm T}\mathbf{X})^{-1}\mathbf{X}^{\rm T}% \mathbf{y}\rangle+C
  78. s y m b o l β - ( 𝐗 T 𝐗 ) - 1 𝐗 T 𝐲 = 0. symbol\beta-(\mathbf{X}^{\rm T}\mathbf{X})^{-1}\mathbf{X}^{\rm T}\mathbf{y}=0.
  79. min 𝐲 - X s y m b o l β 2 \operatorname{\,min}\,\big\|\mathbf{y}-Xsymbol\beta\big\|^{2}
  80. s y m b o l β = S 𝐲 symbol\beta=S\mathbf{y}
  81. X s y m b o l β = X ( S 𝐲 ) = ( X S ) 𝐲 Xsymbol\beta=X(S\mathbf{y})=(XS)\mathbf{y}
  82. 𝐲 \mathbf{y}
  83. s y m b o l β ^ = ( 𝐗 T 𝐗 ) - 1 𝐗 T 𝐲 = 𝐗 + 𝐲 \hat{symbol{\beta}}=(\mathbf{X}^{\rm T}\mathbf{X})^{-1}\mathbf{X}^{\rm T}% \mathbf{y}=\mathbf{X}^{+}\mathbf{y}
  84. R T R s y m b o l β ^ = X T 𝐲 . R^{\rm T}R\hat{symbol{\beta}}=X^{\rm T}\mathbf{y}.
  85. R T 𝐳 = X T 𝐲 , R^{\rm T}\mathbf{z}=X^{\rm T}\mathbf{y},
  86. s y m b o l β ^ \hat{symbol{\beta}}
  87. R s y m b o l β ^ = 𝐳 . R\hat{symbol{\beta}}=\mathbf{z}.
  88. 𝐫 = 𝐲 - X s y m b o l β ^ . \mathbf{r}=\mathbf{y}-X\hat{symbol{\beta}}.
  89. X = Q R X=QR
  90. r i i > 0 r_{ii}>0
  91. Q T 𝐫 = Q T 𝐲 - ( Q T Q ) R s y m b o l β ^ = [ ( Q T 𝐲 ) n - R s y m b o l β ^ ( Q T 𝐲 ) m - n ] = [ 𝐮 𝐯 ] Q^{\rm T}\mathbf{r}=Q^{\rm T}\mathbf{y}-\left(Q^{\rm T}Q\right)R\hat{symbol{% \beta}}=\begin{bmatrix}\left(Q^{\rm T}\mathbf{y}\right)_{n}-R\hat{symbol{\beta% }}\\ \left(Q^{\rm T}\mathbf{y}\right)_{m-n}\end{bmatrix}=\begin{bmatrix}\mathbf{u}% \\ \mathbf{v}\end{bmatrix}
  92. s = 𝐫 2 = 𝐫 T 𝐫 = 𝐫 T Q Q T 𝐫 = 𝐮 T 𝐮 + 𝐯 T 𝐯 s=\|\mathbf{r}\|^{2}=\mathbf{r}^{\rm T}\mathbf{r}=\mathbf{r}^{\rm T}QQ^{\rm T}% \mathbf{r}=\mathbf{u}^{\rm T}\mathbf{u}+\mathbf{v}^{\rm T}\mathbf{v}
  93. R s y m b o l β ^ = ( Q T 𝐲 ) n . R\hat{symbol{\beta}}=\left(Q^{\rm T}\mathbf{y}\right)_{n}.
  94. X = U Σ V T X=U\Sigma V^{\rm T}
  95. Σ \Sigma
  96. Σ \Sigma
  97. 𝐗𝐗 + = U Σ V T V Σ + U T = U P U T , \mathbf{X}\mathbf{X}^{+}=U\Sigma V^{\rm T}V\Sigma^{+}U^{\rm T}=UPU^{\rm T},
  98. Σ \Sigma
  99. ( 𝐗𝐗 + ) * = 𝐗𝐗 + (\mathbf{X}\mathbf{X}^{+})^{*}=\mathbf{X}\mathbf{X}^{+}
  100. U P U T UPU^{\rm T}
  101. S = 𝐗 + S=\mathbf{X}^{+}
  102. β = V Σ + U T 𝐲 \beta=V\Sigma^{+}U^{\rm T}\mathbf{y}
  103. ( 𝐲 - X s y m b o l β ^ ) T X = 0. (\mathbf{y}-X\hat{symbol{\beta}})^{\rm T}X=0.
  104. 𝐲 - X s y m b o l β ^ \mathbf{y}-X\hat{symbol{\beta}}
  105. ( 𝐲 - X s y m b o l β ^ ) X 𝐯 (\mathbf{y}-X\hat{symbol{\beta}})\cdot X\mathbf{v}
  106. 𝐲 - X s y m b o l β ^ \mathbf{y}-Xsymbol{\hat{\beta}}
  107. 𝐲 - X s y m b o l β \mathbf{y}-Xsymbol\beta
  108. s y m b o l γ ^ \hat{symbol{\gamma}}
  109. [ X K ] [X\ K]
  110. 𝐫 ^ 𝐲 - X s y m b o l β ^ = K s y m b o l γ ^ . \hat{\mathbf{r}}\triangleq\mathbf{y}-X\hat{symbol{\beta}}=K\hat{{symbol{\gamma% }}}.
  111. 𝐲 = [ X K ] ( s y m b o l β ^ s y m b o l γ ^ ) , \mathbf{y}=\begin{bmatrix}X&K\end{bmatrix}\begin{pmatrix}\hat{symbol{\beta}}\\ \hat{symbol{\gamma}}\end{pmatrix},
  112. ( s y m b o l β ^ s y m b o l γ ^ ) = [ X K ] - 1 𝐲 = [ ( X T X ) - 1 X T ( K T K ) - 1 K T ] 𝐲 . \begin{pmatrix}\hat{symbol{\beta}}\\ \hat{symbol{\gamma}}\end{pmatrix}=\begin{bmatrix}X&K\end{bmatrix}^{-1}\mathbf{% y}=\begin{bmatrix}(X^{\rm T}X)^{-1}X^{\rm T}\\ (K^{\rm T}K)^{-1}K^{\rm T}\end{bmatrix}\mathbf{y}.
  113. ϵ \epsilon\,
  114. σ \sigma
  115. s y m b o l β ^ \hat{symbol{\beta}}
  116. 𝐬𝐲𝐦𝐛𝐨𝐥 β ^ \mathbf{\hat{symbol{\beta}}}
  117. 𝐲 - X s y m b o l β ^ \|\mathbf{y}-X\hat{symbol{\beta}}\|
  118. 𝐬𝐲𝐦𝐛𝐨𝐥 β ^ \mathbf{\hat{symbol{\beta}}}
  119. s y m b o l β - s y m b o l β ^ \|{symbol{\beta}}-\hat{symbol{\beta}}\|
  120. s y m b o l β {symbol{\beta}}
  121. s y m b o l β ^ \hat{symbol{\beta}}
  122. E { s y m b o l β - s y m b o l β ^ 2 } E\left\{\|{symbol{\beta}}-\hat{symbol{\beta}}\|^{2}\right\}
  123. arg min s y m b o l β i = 1 m w i | y i - j = 1 n X i j β j | 2 = arg min s y m b o l β W 1 / 2 ( 𝐲 - X s y m b o l β ) 2 . \underset{symbol\beta}{\operatorname{arg\,min}}\,\sum_{i=1}^{m}w_{i}\left|y_{i% }-\sum_{j=1}^{n}X_{ij}\beta_{j}\right|^{2}=\underset{symbol\beta}{% \operatorname{arg\,min}}\,\big\|W^{1/2}(\mathbf{y}-Xsymbol\beta)\big\|^{2}.
  124. ( X T W X ) s y m b o l β ^ = X T W 𝐲 . \left(X^{\rm T}WX\right)\hat{symbol{\beta}}=X^{\rm T}W\mathbf{y}.
  125. s y m b o l β ^ = ( X T W X ) - 1 X T W 𝐲 . \hat{symbol{\beta}}=(X^{\rm T}WX)^{-1}X^{\rm T}W\mathbf{y}.\,
  126. M β = ( X T W X ) - 1 X T W M W T X ( X T W T X ) - 1 . M^{\beta}=(X^{\rm T}WX)^{-1}X^{\rm T}WMW^{\rm T}X(X^{\rm T}W^{\rm T}X)^{-1}.
  127. M β = ( X T W X ) - 1 . M^{\beta}=(X^{\rm T}WX)^{-1}.
  128. S m - n \frac{S}{m-n}
  129. M β = S m - n ( X T X ) - 1 . M^{\beta}=\frac{S}{m-n}(X^{\rm T}X)^{-1}.
  130. β i \beta_{i}
  131. M i i β M^{\beta}_{ii}
  132. β i \beta_{i}
  133. β j \beta_{j}
  134. M i j β M^{\beta}_{ij}
  135. ρ i j = M i j β / ( σ i σ j ) \rho_{ij}=M^{\beta}_{ij}/(\sigma_{i}\sigma_{j})
  136. σ \sigma
  137. s e β se_{\beta}
  138. β ^ ± s e β \hat{\beta}\pm se_{\beta}
  139. β ^ ± 2 s e β \hat{\beta}\pm 2se_{\beta}
  140. β ^ ± 2.5 s e β \hat{\beta}\pm 2.5se_{\beta}
  141. 𝐫 ^ = 𝐲 - X s y m b o l β ^ = 𝐲 - H 𝐲 = ( I - H ) 𝐲 \mathbf{\hat{r}}=\mathbf{y}-X\hat{symbol{\beta}}=\mathbf{y}-H\mathbf{y}=(I-H)% \mathbf{y}
  142. H = X ( X T W X ) - 1 X T W H=X\left(X^{\rm T}WX\right)^{-1}X^{\rm T}W
  143. M 𝐫 = ( I - H ) M ( I - H ) T . M^{\mathbf{r}}=\left(I-H\right)M\left(I-H\right)^{\rm T}.
  144. W = M - 1 W=M^{-1}
  145. M 𝐫 = ( I - H ) M . M^{\mathbf{r}}=\left(I-H\right)M.
  146. X T 𝐫 ^ = X T 𝐲 - X T X s y m b o l β ^ = X T 𝐲 - ( X T X ) ( X T X ) - 1 X T 𝐲 = 𝟎 X^{\rm T}\hat{\mathbf{r}}=X^{\rm T}\mathbf{y}-X^{\rm T}X\hat{symbol{\beta}}=X^% {\rm T}\mathbf{y}-(X^{\rm T}X)(X^{\rm T}X)^{-1}X^{\rm T}\mathbf{y}=\mathbf{0}
  147. X i 1 = 1 X_{i1}=1
  148. i m X i 1 r ^ i = i m r ^ i = 0. \sum_{i}^{m}X_{i1}\hat{r}_{i}=\sum_{i}^{m}\hat{r}_{i}=0.
  149. S = 𝐲 T ( I - H ) T ( I - H ) 𝐲 = 𝐲 T ( I - H ) 𝐲 , S=\mathbf{y}^{\rm T}(I-H)^{\rm T}(I-H)\mathbf{y}=\mathbf{y}^{\rm T}(I-H)% \mathbf{y},
  150. ( m - n ) σ 2 (m-n)\sigma^{2}
  151. σ 2 \sigma^{2}
  152. χ 2 \chi^{2}
  153. χ 2 \chi^{2}
  154. χ 0.50 2 \chi^{2}_{0.50}
  155. χ 0.95 2 \chi^{2}_{0.95}
  156. χ 0.99 2 \chi^{2}_{0.99}
  157. 𝐗 s y m b o l β = 𝐲 \mathbf{X}symbol{\beta}=\mathbf{y}
  158. s y m b o l β symbol{\beta}
  159. s y m b o l β symbol{\beta}
  160. 𝐋 s y m b o l β = 𝐝 \mathbf{L}symbol{\beta}=\mathbf{d}
  161. s y m b o l β symbol{\beta}
  162. 𝐋 s y m b o l β - 𝐝 ρ \|\mathbf{L}symbol{\beta}-\mathbf{d}\|\leq\rho
  163. s y m b o l β symbol{\beta}
  164. s y m b o l β s y m b o l 0 symbol{\beta}\geq symbol{0}
  165. s y m b o l β symbol{\beta}
  166. s y m b o l l b s y m b o l β s y m b o l u b symbol{lb}\leq symbol{\beta}\leq symbol{ub}
  167. s y m b o l β symbol{\beta}
  168. s y m b o l β symbol{\beta}
  169. 𝐗 = [ 𝐗 𝟏 𝐗 𝟐 ] \mathbf{X}=[\mathbf{X_{1}}\mathbf{X_{2}}]
  170. β T = [ β 𝟏 T β 𝟐 T ] \mathbf{\beta}^{\rm T}=[\mathbf{\beta_{1}}^{\rm T}\mathbf{\beta_{2}}^{\rm T}]
  171. β 𝟏 \mathbf{\beta_{1}}
  172. s y m b o l β 1 ^ = 𝐗 𝟏 + ( 𝐲 - 𝐗 𝟐 s y m b o l β 2 ) \hat{symbol{\beta_{1}}}=\mathbf{X_{1}}^{+}(\mathbf{y}-\mathbf{X_{2}}symbol{% \beta_{2}})
  173. β 𝟐 \mathbf{\beta_{2}}
  174. 𝐏𝐗 𝟐 s y m b o l β 2 = 𝐏𝐲 \mathbf{P}\mathbf{X_{2}}symbol{\beta_{2}}=\mathbf{P}\mathbf{y}
  175. 𝐏 := 𝐈 - 𝐗 𝟏 𝐗 𝟏 + \mathbf{P}:=\mathbf{I}-\mathbf{X_{1}}\mathbf{X_{1}}^{+}
  176. s y m b o l β 2 ^ \hat{symbol{\beta_{2}}}
  177. s y m b o l β 1 ^ \hat{symbol{\beta_{1}}}
  178. f ( x , s y m b o l β ) = β 1 + β 2 x f(x,symbol\beta)=\beta_{1}+\beta_{2}x
  179. f ( x , s y m b o l β ) = β 1 + β 2 x + β 3 x 2 f(x,symbol\beta)=\beta_{1}+\beta_{2}x+\beta_{3}x^{2}
  180. y 1 , y 2 , , y m , y_{1},y_{2},\dots,y_{m},
  181. x 1 , x 2 , , x m x_{1},x_{2},\dots,x_{m}
  182. x i x_{i}
  183. y = f ( x , s y m b o l β ) , y=f(x,symbol\beta),
  184. s y m b o l β = ( β 1 , β 2 , , β n ) , symbol\beta=(\beta_{1},\beta_{2},\dots,\beta_{n}),
  185. β j \beta_{j}
  186. β j , \beta_{j},
  187. f ( x , s y m b o l β ) = j = 1 n β j ϕ j ( x ) . f(x,symbol\beta)=\sum_{j=1}^{n}\beta_{j}\phi_{j}(x).
  188. ϕ j \phi_{j}
  189. y i = f ( x i , s y m b o l β ) y_{i}=f(x_{i},symbol\beta)
  190. i = 1 , 2 , , m . i=1,2,\dots,m.
  191. r i ( s y m b o l β ) = y i - f ( x i , s y m b o l β ) , ( i = 1 , 2 , , m ) r_{i}(symbol\beta)=y_{i}-f(x_{i},symbol\beta),\ (i=1,2,\dots,m)
  192. S ( s y m b o l β ) = i = 1 m r i 2 ( s y m b o l β ) . S(symbol\beta)=\sum_{i=1}^{m}r_{i}^{2}(symbol\beta).
  193. r i r_{i}
  194. f f
  195. X i j = ϕ j ( x i ) , X_{ij}=\phi_{j}(x_{i}),
  196. S = k j r k W k j r j \textstyle S=\sum_{k}\sum_{j}r_{k}W_{kj}r_{j}\,

Linear_logic.html

  1. < V A R > A <VAR>A
  2. ( < V A R > p ) = < v a r > p < / v a r > (<VAR>p)^{⊥}=<var>p</var>^{⊥}
  3. Γ Γ
  4. Δ Δ
  5. Γ [ u t e e ] Δ Γ[u^{\prime}tee^{\prime}]Δ
  6. Γ Γ
  7. Δ Δ
  8. [ u t e e ] Γ [u^{\prime}tee^{\prime}]Γ
  9. [ u t e e ] Γ [u^{\prime}tee^{\prime}]Γ
  10. ( Γ a p e r m u t a t i o n o f Γ ) ) (ΓapermutationofΓ))
  11. [ u t e e ] < V A R > A , < v a r > A < / v a r > [u^{\prime}tee^{\prime}]<VAR>A,<var>A</var>^{⊥}
  12. [ u t e e ] Γ , < V A R > A [u^{\prime}tee^{\prime}]Γ,<VAR>A
  13. [ u t e e ] Γ , < V A R > A [u^{\prime}tee^{\prime}]Γ,<VAR>A
  14. [ u t e e ] Γ , < V A R > A , < v a r > B < / v a r > [u^{\prime}tee^{\prime}]Γ,<VAR>A,<var>B</var>
  15. [ u t e e ] 1 [u^{\prime}tee^{\prime}]1
  16. [ u t e e ] Γ [u^{\prime}tee^{\prime}]Γ
  17. [ u t e e ] Γ , [u^{\prime}tee^{\prime}]Γ,⊥
  18. [ u t e e ] Γ , < V A R > A [u^{\prime}tee^{\prime}]Γ,<VAR>A
  19. [ u t e e ] Γ , < V A R > A [u^{\prime}tee^{\prime}]Γ,<VAR>A
  20. [ u t e e ] Γ , < V A R > B [u^{\prime}tee^{\prime}]Γ,<VAR>B
  21. [ u t e e ] Γ , [ u U n i c o d e , u 2 a 4 ] [u^{\prime}tee^{\prime}]Γ,[u^{\prime}Unicode^{\prime},u^{\prime}\u{2}2a4^{% \prime}]
  22. 0
  23. Γ , Δ Γ,Δ
  24. Γ Γ
  25. [ u t e e ] Γ [u^{\prime}tee^{\prime}]Γ
  26. [ u t e e ] Γ , ? < V A R > A [u^{\prime}tee^{\prime}]Γ,?<VAR>A
  27. [ u t e e ] Γ , ? < V A R > A , ? < v a r > A < / v a r > [u^{\prime}tee^{\prime}]Γ,?<VAR>A,?<var>A</var>
  28. [ u t e e ] ? Γ , < V A R > A [u^{\prime}tee^{\prime}]?Γ,<VAR>A
  29. [ u t e e ] Γ , < V A R > A [u^{\prime}tee^{\prime}]Γ,<VAR>A
  30. A ( B C ) ( A B ) ( A C ) A\otimes(B\oplus C)\equiv(A\otimes B)\oplus(A\otimes C)
  31. ! ( A & B ) ! A ! B \,!(A\&B)\equiv\,!A\otimes\,!B
  32. A B = ( A B ) & ( B A ) A\equiv B\quad=\quad(A\multimap B)\&(B\multimap A)
  33. ( < V A R > A [ u U n i c o d e , u 297 ] ( < v a r > B < / v a r > [ u U n i c o d e , u 14 b ] < v a r > C < / v a r > ) ) [ u U n i c o d e , u 2 b 8 ] ( ( < v a r > A < / v a r > [ u U n i c o d e , u 297 ] < v a r > B < / v a r > ) [ u U n i c o d e , u 14 b ] < v a r > C < / v a r > ) (<VAR>A[u^{\prime}Unicode^{\prime},u^{\prime}\u{2}297^{\prime}](<var>B</var>[u% ^{\prime}Unicode^{\prime},u^{\prime}\u{2}14b^{\prime}]<var>C</var>))[u^{\prime% }Unicode^{\prime},u^{\prime}\u{2}2b8^{\prime}]((<var>A</var>[u^{\prime}Unicode% ^{\prime},u^{\prime}\u{2}297^{\prime}]<var>B</var>)[u^{\prime}Unicode^{\prime}% ,u^{\prime}\u{2}14b^{\prime}]<var>C</var>)

Linear_separability.html

  1. X 0 X_{0}
  2. X 1 X_{1}
  3. X 0 X_{0}
  4. X 1 X_{1}
  5. w 1 , w 2 , . . , w n , k w_{1},w_{2},..,w_{n},k
  6. x X 0 x\in X_{0}
  7. i = 1 n w i x i > k \sum^{n}_{i=1}w_{i}x_{i}>k
  8. x X 1 x\in X_{1}
  9. i = 1 n w i x i < k \sum^{n}_{i=1}w_{i}x_{i}<k
  10. x i x_{i}
  11. i i
  12. x x
  13. 𝒟 \mathcal{D}
  14. 𝒟 = { ( 𝐱 i , y i ) 𝐱 i p , y i { - 1 , 1 } } i = 1 n \mathcal{D}=\left\{(\mathbf{x}_{i},y_{i})\mid\mathbf{x}_{i}\in\mathbb{R}^{p},% \,y_{i}\in\{-1,1\}\right\}_{i=1}^{n}
  15. 𝐱 i \mathbf{x}_{i}
  16. 𝐱 i \mathbf{x}_{i}
  17. y i = 1 y_{i}=1
  18. y i = - 1 y_{i}=-1
  19. 𝐱 \mathbf{x}
  20. 𝐰 𝐱 - b = 0 , \mathbf{w}\cdot\mathbf{x}-b=0,\,
  21. \cdot
  22. 𝐰 {\mathbf{w}}
  23. b 𝐰 \tfrac{b}{\|\mathbf{w}\|}
  24. 𝐰 {\mathbf{w}}

Linearly_ordered_group.html

  1. | a | := { a , if a 0 , - a , otherwise . |a|:=\begin{cases}a,&\,\text{if }a\geqslant 0,\\ -a,&\,\text{otherwise}.\end{cases}
  2. G ^ \widehat{G}
  3. n n
  4. g G ^ g\in\widehat{G}
  5. g : ( , + ) ( G ^ , ) : lim i q i lim i g q i g^{\cdot}:(\mathbb{R},+)\to(\widehat{G},\cdot):\lim_{i}q_{i}\in\mathbb{Q}% \mapsto\lim_{i}g^{q_{i}}

Lineweaver–Burk_plot.html

  1. V = V max [ S ] K m + [ S ] V=\frac{V_{\max}[S]}{K_{m}+[S]}
  2. 1 V = K m + [ S ] V max [ S ] = K m V max 1 [ S ] + 1 V max {1\over V}={{K_{m}+[S]}\over V_{\max}[S]}={K_{m}\over V_{\max}}{1\over[S]}+{1% \over V_{\max}}

Linking_number.html

  1. \cdots
  2. \cdots
  3. linking number = n 1 + n 2 - n 3 - n 4 2 \,\text{linking number}=\frac{n_{1}+n_{2}-n_{3}-n_{4}}{2}
  4. n 1 + n 3 n_{1}+n_{3}\,\!
  5. n 2 + n 4 n_{2}+n_{4}\,\!
  6. linking number = n 1 - n 4 = n 2 - n 3 . \,\text{linking number}\,=\,n_{1}-n_{4}\,=\,n_{2}-n_{3}.
  7. n 1 - n 4 n_{1}-n_{4}
  8. n 2 - n 3 n_{2}-n_{3}
  9. γ 1 , γ 2 : S 1 3 \gamma_{1},\gamma_{2}\colon S^{1}\rightarrow\mathbb{R}^{3}
  10. Γ \Gamma
  11. Γ ( s , t ) = γ 1 ( s ) - γ 2 ( t ) | γ 1 ( s ) - γ 2 ( t ) | . \Gamma(s,t)=\frac{\gamma_{1}(s)-\gamma_{2}(t)}{|\gamma_{1}(s)-\gamma_{2}(t)|}.
  12. γ 1 \gamma_{1}
  13. γ 2 \gamma_{2}
  14. linking number = 1 4 π γ 1 γ 2 𝐫 1 - 𝐫 2 | 𝐫 1 - 𝐫 2 | 3 ( d 𝐫 1 × d 𝐫 2 ) . \,\text{linking number}\,=\,\frac{1}{4\pi}\oint_{\gamma_{1}}\oint_{\gamma_{2}}% \frac{\mathbf{r}_{1}-\mathbf{r}_{2}}{|\mathbf{r}_{1}-\mathbf{r}_{2}|^{3}}\cdot% (d\mathbf{r}_{1}\times d\mathbf{r}_{2}).
  15. m + n + 1 m+n+1

List_of_commutative_algebra_topics.html

  1. \mathbb{Z}

List_of_moments_of_inertia.html

  1. I I
  2. I = m r 2 I=mr^{2}
  3. I = M m M + m x 2 = μ x 2 I=\frac{Mm}{M\!+\!m}x^{2}=\mu x^{2}
  4. I end = m L 2 3 I_{\mathrm{end}}=\frac{mL^{2}}{3}\,\!
  5. I center = m L 2 12 I_{\mathrm{center}}=\frac{mL^{2}}{12}\,\!
  6. I z = m r 2 I_{z}=mr^{2}\!
  7. I x = I y = m r 2 2 I_{x}=I_{y}=\frac{mr^{2}}{2}\,\!
  8. I x = I y = I z 2 I_{x}=I_{y}=\frac{I_{z}}{2}\,
  9. I z = m r 2 2 I_{z}=\frac{mr^{2}}{2}\,\!
  10. I x = I y = m r 2 4 I_{x}=I_{y}=\frac{mr^{2}}{4}\,\!
  11. I = m r 2 I=mr^{2}\,\!
  12. I z = m r 2 2 I_{z}=\frac{mr^{2}}{2}\,\!
  13. I x = I y = 1 12 m ( 3 r 2 + h 2 ) I_{x}=I_{y}=\frac{1}{12}m\left(3r^{2}+h^{2}\right)
  14. I z = 1 2 m ( r 1 2 + r 2 2 ) = m r 2 2 ( 1 - t + 1 2 t 2 ) I_{z}=\frac{1}{2}m\left(r_{1}^{2}+r_{2}^{2}\right)=mr_{2}^{2}\left(1-t+\frac{1% }{2}{t}^{2}\right)
  15. I x = I y = 1 12 m [ 3 ( r 2 2 + r 1 2 ) + h 2 ] I_{x}=I_{y}=\frac{1}{12}m\left[3\left({r_{2}}^{2}+{r_{1}}^{2}\right)+h^{2}\right]
  16. I z = 1 2 π ρ h ( r 2 4 - r 1 4 ) I_{z}=\frac{1}{2}\pi\rho h\left({r_{2}}^{4}-{r_{1}}^{4}\right)
  17. I x = I y = 1 12 π ρ h ( 3 ( r 2 4 - r 1 4 ) + h 2 ( r 2 2 - r 1 2 ) ) I_{x}=I_{y}=\frac{1}{12}\pi\rho h\left(3({r_{2}}^{4}-{r_{1}}^{4})+h^{2}({r_{2}% }^{2}-{r_{1}}^{2})\right)
  18. I s o l i d = m s 2 20 I_{solid}=\frac{ms^{2}}{20}\,\!
  19. I h o l l o w = m s 2 12 I_{hollow}=\frac{ms^{2}}{12}\,\!
  20. I z = I x = I y = 5 m s 2 9 I_{z}=I_{x}=I_{y}=\frac{5ms^{2}}{9}\,\!
  21. I z = I x = I y = m s 2 10 I_{z}=I_{x}=I_{y}=\frac{ms^{2}}{10}\,\!
  22. I = 2 m r 2 3 I=\frac{2mr^{2}}{3}\,\!
  23. I = 2 m r 2 5 I=\frac{2mr^{2}}{5}\,\!
  24. [ r 2 5 - r 1 5 r 2 3 - r 1 3 ] = 5 3 r 2 2 \left[\frac{{r_{2}}^{5}-{r_{1}}^{5}}{{r_{2}}^{3}-{r_{1}}^{3}}\right]=\frac{5}{% 3}{r_{2}}^{2}
  25. I = 2 m 5 [ r 2 5 - r 1 5 r 2 3 - r 1 3 ] I=\frac{2m}{5}\left[\frac{{r_{2}}^{5}-{r_{1}}^{5}}{{r_{2}}^{3}-{r_{1}}^{3}}% \right]\,\!
  26. I z = 3 10 m r 2 I_{z}=\frac{3}{10}mr^{2}\,\!
  27. I x = I y = 3 5 m ( r 2 4 + h 2 ) I_{x}=I_{y}=\frac{3}{5}m\left(\frac{r^{2}}{4}+h^{2}\right)\,\!
  28. ( a 2 + 3 4 b 2 ) m \left(a^{2}+\frac{3}{4}b^{2}\right)m
  29. 1 8 ( 4 a 2 + 5 b 2 ) m \frac{1}{8}\left(4a^{2}+5b^{2}\right)m
  30. I a = m ( b 2 + c 2 ) 5 I_{a}=\frac{m(b^{2}+c^{2})}{5}\,\!
  31. I b = m ( a 2 + c 2 ) 5 I_{b}=\frac{m(a^{2}+c^{2})}{5}\,\!
  32. I c = m ( a 2 + b 2 ) 5 I_{c}=\frac{m(a^{2}+b^{2})}{5}\,\!
  33. I e = m h 2 3 + m w 2 12 I_{e}=\frac{mh^{2}}{3}+\frac{mw^{2}}{12}\,\!
  34. I c = m ( h 2 + w 2 ) 12 I_{c}=\frac{m(h^{2}+w^{2})}{12}\,\!
  35. s s
  36. I C M = m s 2 6 I_{CM}=\frac{ms^{2}}{6}\,\!
  37. I h = 1 12 m ( w 2 + d 2 ) I_{h}=\frac{1}{12}m\left(w^{2}+d^{2}\right)
  38. I w = 1 12 m ( h 2 + d 2 ) I_{w}=\frac{1}{12}m\left(h^{2}+d^{2}\right)
  39. I d = 1 12 m ( h 2 + w 2 ) I_{d}=\frac{1}{12}m\left(h^{2}+w^{2}\right)
  40. s s
  41. I = m s 2 6 I=\frac{ms^{2}}{6}\,\!
  42. I = m ( W 2 D 2 + L 2 D 2 + L 2 W 2 ) 6 ( L 2 + W 2 + D 2 ) I=\frac{m\left(W^{2}D^{2}+L^{2}D^{2}+L^{2}W^{2}\right)}{6\left(L^{2}+W^{2}+D^{% 2}\right)}
  43. I = m 6 ( 𝐏 𝐏 + 𝐏 𝐐 + 𝐐 𝐐 ) I=\frac{m}{6}(\mathbf{P}\cdot\mathbf{P}+\mathbf{P}\cdot\mathbf{Q}+\mathbf{Q}% \cdot\mathbf{Q})
  44. I = m 6 n = 1 N 𝐏 n + 1 × 𝐏 n ( ( 𝐏 n + 1 𝐏 n + 1 ) + ( 𝐏 n + 1 𝐏 n ) + ( 𝐏 n 𝐏 n ) ) n = 1 N 𝐏 n + 1 × 𝐏 n I=\frac{m}{6}\frac{\sum\limits_{n=1}^{N}\|\mathbf{P}_{n+1}\times\mathbf{P}_{n}% \|((\mathbf{P}_{n+1}\cdot\mathbf{P}_{n+1})+(\mathbf{P}_{n+1}\cdot\mathbf{P}_{n% })+(\mathbf{P}_{n}\cdot\mathbf{P}_{n}))}{\sum\limits_{n=1}^{N}\|\mathbf{P}_{n+% 1}\times\mathbf{P}_{n}\|}
  45. I = m a 2 24 [ 1 + 3 cot 2 ( π n ) ] I=\frac{ma^{2}}{24}[1+3\cot^{2}(\tfrac{\pi}{n})]
  46. ρ ( x , y ) = m 2 π a b e - ( ( x / a ) 2 + ( y / b ) 2 ) / 2 , \rho(x,y)=\tfrac{m}{2\pi ab}\,e^{-((x/a)^{2}+(y/b)^{2})/2}\,,
  47. I = m ( a 2 + b 2 ) I=m(a^{2}+b^{2})\,\!
  48. I = 3 m R 2 2 I=\frac{3mR^{2}}{2}
  49. 𝐧 𝐈 𝐧 n i I i j n j , \mathbf{n}\cdot\mathbf{I}\cdot\mathbf{n}\equiv n_{i}I_{ij}n_{j}\,,
  50. I = [ 2 5 m r 2 0 0 0 2 5 m r 2 0 0 0 2 5 m r 2 ] I=\begin{bmatrix}\frac{2}{5}mr^{2}&0&0\\ 0&\frac{2}{5}mr^{2}&0\\ 0&0&\frac{2}{5}mr^{2}\end{bmatrix}
  51. I = [ 2 3 m r 2 0 0 0 2 3 m r 2 0 0 0 2 3 m r 2 ] I=\begin{bmatrix}\frac{2}{3}mr^{2}&0&0\\ 0&\frac{2}{3}mr^{2}&0\\ 0&0&\frac{2}{3}mr^{2}\end{bmatrix}
  52. I = [ 1 5 m ( b 2 + c 2 ) 0 0 0 1 5 m ( a 2 + c 2 ) 0 0 0 1 5 m ( a 2 + b 2 ) ] I=\begin{bmatrix}\frac{1}{5}m(b^{2}+c^{2})&0&0\\ 0&\frac{1}{5}m(a^{2}+c^{2})&0\\ 0&0&\frac{1}{5}m(a^{2}+b^{2})\end{bmatrix}
  53. I = [ 3 5 m h 2 + 3 20 m r 2 0 0 0 3 5 m h 2 + 3 20 m r 2 0 0 0 3 10 m r 2 ] I=\begin{bmatrix}\frac{3}{5}mh^{2}+\frac{3}{20}mr^{2}&0&0\\ 0&\frac{3}{5}mh^{2}+\frac{3}{20}mr^{2}&0\\ 0&0&\frac{3}{10}mr^{2}\end{bmatrix}
  54. I = [ 1 12 m ( h 2 + d 2 ) 0 0 0 1 12 m ( w 2 + d 2 ) 0 0 0 1 12 m ( w 2 + h 2 ) ] I=\begin{bmatrix}\frac{1}{12}m(h^{2}+d^{2})&0&0\\ 0&\frac{1}{12}m(w^{2}+d^{2})&0\\ 0&0&\frac{1}{12}m(w^{2}+h^{2})\end{bmatrix}
  55. I = [ 1 3 m l 2 0 0 0 0 0 0 0 1 3 m l 2 ] I=\begin{bmatrix}\frac{1}{3}ml^{2}&0&0\\ 0&0&0\\ 0&0&\frac{1}{3}ml^{2}\end{bmatrix}
  56. I = [ 1 12 m l 2 0 0 0 0 0 0 0 1 12 m l 2 ] I=\begin{bmatrix}\frac{1}{12}ml^{2}&0&0\\ 0&0&0\\ 0&0&\frac{1}{12}ml^{2}\end{bmatrix}
  57. I = [ 1 12 m ( 3 r 2 + h 2 ) 0 0 0 1 12 m ( 3 r 2 + h 2 ) 0 0 0 1 2 m r 2 ] I=\begin{bmatrix}\frac{1}{12}m(3r^{2}+h^{2})&0&0\\ 0&\frac{1}{12}m(3r^{2}+h^{2})&0\\ 0&0&\frac{1}{2}mr^{2}\end{bmatrix}
  58. I = [ 1 12 m ( 3 ( r 1 2 + r 2 2 ) + h 2 ) 0 0 0 1 12 m ( 3 ( r 1 2 + r 2 2 ) + h 2 ) 0 0 0 1 2 m ( r 1 2 + r 2 2 ) ] I=\begin{bmatrix}\frac{1}{12}m(3({r_{1}}^{2}+{r_{2}}^{2})+h^{2})&0&0\\ 0&\frac{1}{12}m(3({r_{1}}^{2}+{r_{2}}^{2})+h^{2})&0\\ 0&0&\frac{1}{2}m({r_{1}}^{2}+{r_{2}}^{2})\end{bmatrix}

List_of_numerical_analysis_topics.html

  1. π \pi
  2. f ( x , y ) = ( x 2 + y - 11 ) 2 + ( x + y 2 - 7 ) 2 f(x,y)=(x^{2}+y-11)^{2}+(x+y^{2}-7)^{2}
  3. y ′′ = f ( t , y ) y^{\prime\prime}=f(t,y)

List_of_prime_numbers.html

  1. 18 {}^{18}
  2. 17 {}^{17}
  3. × 10 2 1 \times 10^{2}1
  4. 23 {}^{23}
  5. × 10 2 2 \times 10^{2}2
  6. 24 {}^{24}
  7. n {}^{n}
  8. 2 {}^{2}
  9. 2 {}^{2}
  10. 2 {}^{2}
  11. 2 {}^{2}
  12. 2 {}^{2}
  13. 2 {}^{2}
  14. x 3 - y 3 x - y , \tfrac{x^{3}-y^{3}}{x-y},
  15. x 3 - y 3 x - y , \tfrac{x^{3}-y^{3}}{x-y},
  16. n {}^{n}
  17. 2 p 1 {}^{2^{p}−1}
  18. n {}_{n}
  19. 2 n {}^{2^{n}}
  20. 0 {}_{0}
  21. 1 {}_{1}
  22. n {}_{n}
  23. n 1 {}_{n−1}
  24. n 2 {}_{n−2}
  25. 2 n {}^{2^{n}}
  26. n {}_{n}
  27. n {}_{n}
  28. 2 {}^{2}
  29. n i {}_{n−i}
  30. n + i {}_{n+i}
  31. n {}_{n}
  32. k {}_{k}
  33. k {}_{k}
  34. p {}_{p}
  35. k {}_{k}
  36. p {}_{p}
  37. n {}^{n}
  38. 2 {}^{2}
  39. y {}^{y}
  40. x {}^{x}
  41. b p - 1 - 1 p \frac{b^{p-1}-1}{p}
  42. 0 {}_{0}
  43. 1 {}_{1}
  44. n {}_{n}
  45. n 1 {}_{n−1}
  46. n 2 {}_{n−2}
  47. 2 {}^{2}
  48. 2 {}^{2}
  49. 2 {}^{2}
  50. n {}^{n}
  51. p {}^{p}
  52. 3 n {}^{3^{n}}
  53. 2 {}^{2}
  54. a ( 10 m - 1 ) 9 ± b × 10 m 2 \frac{a\big(10^{m}-1\big)}{9}\pm b\times 10^{\frac{m}{2}}
  55. 0 a ± b < 10 0\leq a\pm b<10
  56. 0 {}_{0}
  57. 1 {}_{1}
  58. n {}_{n}
  59. n 1 {}_{n−1}
  60. n 2 {}_{n−2}
  61. u {}^{u}
  62. v {}^{v}
  63. n {}_{n}
  64. n {}^{n}
  65. 2 {}^{2}
  66. 2 {}^{2}
  67. 4 {}^{4}
  68. 4 {}^{4}
  69. n {}_{n}
  70. n {}_{n}
  71. a {}^{a}
  72. b {}^{b}
  73. n ± 1 n\wr\pm 1
  74. n n\wr
  75. n = ( n - 1 ) n n\wr=(n-1)\wr\wr n\wr\wr
  76. n = { 1 n 0 ( n - 2 ) n [ n odd ] ( 4 / n ) [ n even ] n > 0 n\wr\wr=\begin{cases}1\qquad\qquad\qquad\qquad\qquad\qquad\quad\ n\leqslant 0% \\ (n-2)\wr\wr n^{\big[\,\text{n odd}\big]}(4/n)^{\big[\,\text{n even}\big]}\quad n% >0\end{cases}
  77. n {}^{n}
  78. n {}^{n}
  79. n {}^{n}
  80. 2 {}^{2}
  81. F p - ( < m t p l > p 5 ) F_{p-\left(\frac{<}{m}tpl>{{p}}{{5}}\right)}
  82. ( < m t p l > p 5 ) \left(\frac{<}{m}tpl>{{p}}{{5}}\right)
  83. ( p 5 ) = { 1 if p ± 1 ( mod 5 ) - 1 if p ± 2 ( mod 5 ) . \left(\frac{p}{5}\right)=\begin{cases}1&\textrm{if}\;p\equiv\pm 1\;\;(\mathop{% {\rm mod}}5)\\ -1&\textrm{if}\;p\equiv\pm 2\;\;(\mathop{{\rm mod}}5).\end{cases}
  84. 2 {}^{2}
  85. ( 2 p - 1 p - 1 ) 1 ( mod p 4 ) . {{2p-1}\choose{p-1}}\equiv 1\;\;(\mathop{{\rm mod}}p^{4}).
  86. n {}^{n}
  87. 0 {}_{0}

List_of_real_analysis_topics.html

  1. f ( x ) = n = 0 a n ( x - c ) n = a 0 + a 1 ( x - c ) 1 + a 2 ( x - c ) 2 + a 3 ( x - c ) 3 + f(x)=\sum_{n=0}^{\infty}a_{n}\left(x-c\right)^{n}=a_{0}+a_{1}(x-c)^{1}+a_{2}(x% -c)^{2}+a_{3}(x-c)^{3}+\cdots
  2. f ( a ) + f ( a ) 1 ! ( x - a ) + f ′′ ( a ) 2 ! ( x - a ) 2 + f ( 3 ) ( a ) 3 ! ( x - a ) 3 + . f(a)+\frac{f^{\prime}(a)}{1!}(x-a)+\frac{f^{\prime\prime}(a)}{2!}(x-a)^{2}+% \frac{f^{(3)}(a)}{3!}(x-a)^{3}+\cdots.

List_of_refractive_indices.html

  1. k k

List_of_regular_polytopes_and_compounds.html

  1. 1 p + 1 q > 1 2 \frac{1}{p}+\frac{1}{q}>\frac{1}{2}
  2. 1 p + 1 q = 1 2 \frac{1}{p}+\frac{1}{q}=\frac{1}{2}
  3. 1 p + 1 q < 1 2 \frac{1}{p}+\frac{1}{q}<\frac{1}{2}
  4. { p , q , r } \{p,q,r\}
  5. { p , q } \{p,q\}
  6. { p } \{p\}
  7. { r } \{r\}
  8. { q , r } \{q,r\}
  9. { p , q , r } \{p,q,r\}
  10. { p , q } , { q , r } \{p,q\},\{q,r\}
  11. sin ( π p ) sin ( π r ) - cos ( π q ) \sin\left(\frac{\pi}{p}\right)\sin\left(\frac{\pi}{r}\right)-\cos\left(\frac{% \pi}{q}\right)
  12. > 0 >0
  13. = 0 =0
  14. < 0 <0
  15. χ \chi
  16. χ = V + F - E - C \chi=V+F-E-C
  17. { p , q , r , s } \{p,q,r,s\}
  18. { p , q , r } \{p,q,r\}
  19. { p , q } \{p,q\}
  20. { p } \{p\}
  21. { s } \{s\}
  22. { r , s } \{r,s\}
  23. { q , r , s } \{q,r,s\}
  24. { p , q , r , s } \{p,q,r,s\}
  25. { p , q , r } \{p,q,r\}
  26. { q , r , s } \{q,r,s\}
  27. cos 2 ( π q ) sin 2 ( π p ) + cos 2 ( π r ) sin 2 ( π s ) \frac{\cos^{2}\left(\frac{\pi}{q}\right)}{\sin^{2}\left(\frac{\pi}{p}\right)}+% \frac{\cos^{2}\left(\frac{\pi}{r}\right)}{\sin^{2}\left(\frac{\pi}{s}\right)}
  28. < 1 <1
  29. = 1 =1
  30. > 1 >1
  31. ( n + 1 k + 1 ) {{n+1}\choose{k+1}}
  32. 2 n - k ( n k ) 2^{n-k}{n\choose k}
  33. 2 k + 1 ( n k + 1 ) 2^{k+1}{n\choose{k+1}}

List_of_rules_of_inference.html

  1. φ ψ \varphi\vdash\psi\,\!
  2. φ \varphi\,\!
  3. ψ \psi\,\!
  4. φ ψ \varphi\vdash\psi\,\!
  5. φ ¬ ψ ¯ \underline{\varphi\vdash\lnot\psi}\,\!
  6. ¬ φ \lnot\varphi\,\!
  7. ¬ φ ψ \lnot\varphi\vdash\psi\,\!
  8. ¬ φ ¬ ψ ¯ \underline{\lnot\varphi\vdash\lnot\psi}\,\!
  9. φ \varphi\,\!
  10. φ \varphi\,\!
  11. ¬ φ ¯ \underline{\lnot\varphi}\,\!
  12. ψ \psi\,\!
  13. ¬ ¬ φ ¯ \underline{\lnot\lnot\varphi}\,\!
  14. φ \varphi\,\!
  15. φ ¯ \underline{\varphi\quad\quad}\,\!
  16. ¬ ¬ φ \lnot\lnot\varphi\,\!
  17. φ ψ ¯ \underline{\varphi\vdash\psi}\,\!
  18. φ ψ \varphi\rightarrow\psi\,\!
  19. φ ψ \varphi\rightarrow\psi\,\!
  20. φ ¯ \underline{\varphi\quad\quad\quad}\,\!
  21. ψ \psi\,\!
  22. φ ψ \varphi\rightarrow\psi\,\!
  23. ¬ ψ ¯ \underline{\lnot\psi\quad\quad\quad}\,\!
  24. ¬ φ \lnot\varphi\,\!
  25. φ \varphi\,\!
  26. ψ ¯ \underline{\psi\quad\quad\ \ }\,\!
  27. φ ψ \varphi\land\psi\,\!
  28. φ ψ ¯ \underline{\varphi\land\psi}\,\!
  29. φ \varphi\,\!
  30. φ ψ ¯ \underline{\varphi\land\psi}\,\!
  31. ψ \psi\,\!
  32. φ ¯ \underline{\varphi\quad\quad\ \ }\,\!
  33. φ ψ \varphi\lor\psi\,\!
  34. ψ ¯ \underline{\psi\quad\quad\ \ }\,\!
  35. φ ψ \varphi\lor\psi\,\!
  36. φ χ \varphi\rightarrow\chi\,\!
  37. ψ χ \psi\rightarrow\chi\,\!
  38. φ ψ ¯ \underline{\varphi\lor\psi}\,\!
  39. χ \chi\,\!
  40. φ ψ \varphi\lor\psi\,\!
  41. ¬ φ ¯ \underline{\lnot\varphi\quad\quad}\,\!
  42. ψ \psi\,\!
  43. φ ψ \varphi\lor\psi\,\!
  44. ¬ ψ ¯ \underline{\lnot\psi\quad\quad}\,\!
  45. φ \varphi\,\!
  46. φ χ \varphi\rightarrow\chi\,\!
  47. ψ ξ \psi\rightarrow\xi\,\!
  48. φ ψ ¯ \underline{\varphi\lor\psi}\,\!
  49. χ ξ \chi\lor\xi\,\!
  50. φ ψ \varphi\rightarrow\psi\,\!
  51. ψ φ ¯ \underline{\psi\rightarrow\varphi}\,\!
  52. φ ψ \varphi\leftrightarrow\psi\,\!
  53. φ ψ \varphi\leftrightarrow\psi\,\!
  54. φ ¯ \underline{\varphi\quad\quad}\,\!
  55. ψ \psi\,\!
  56. φ ψ \varphi\leftrightarrow\psi\,\!
  57. ψ ¯ \underline{\psi\quad\quad}\,\!
  58. φ \varphi\,\!
  59. φ ψ \varphi\leftrightarrow\psi\,\!
  60. ¬ φ ¯ \underline{\lnot\varphi\quad\quad}\,\!
  61. ¬ ψ \lnot\psi\,\!
  62. φ ψ \varphi\leftrightarrow\psi\,\!
  63. ¬ ψ ¯ \underline{\lnot\psi\quad\quad}\,\!
  64. ¬ φ \lnot\varphi\,\!
  65. φ ψ \varphi\leftrightarrow\psi\,\!
  66. ψ φ ¯ \underline{\psi\lor\varphi}\,\!
  67. ψ φ \psi\land\varphi\,\!
  68. φ ψ \varphi\leftrightarrow\psi\,\!
  69. ¬ ψ ¬ φ ¯ \underline{\lnot\psi\lor\lnot\varphi}\,\!
  70. ¬ ψ ¬ φ \lnot\psi\land\lnot\varphi\,\!
  71. φ ( β / α ) \varphi(\beta/\alpha)\,\!
  72. φ \varphi\,\!
  73. β \beta\,\!
  74. φ \varphi\,\!
  75. α \alpha\,\!
  76. φ ( β / α ) ¯ \underline{\varphi{(\beta/\alpha)}}\,\!
  77. α φ \forall\alpha\,\varphi\,\!
  78. β \beta
  79. φ \varphi
  80. β \beta
  81. α φ \forall\alpha\,\varphi\!
  82. φ ( β / α ) ¯ \overline{\varphi{(\beta/\alpha)}}\!
  83. α \alpha\,\!
  84. φ \varphi\,\!
  85. β \beta\,\!
  86. φ ( β / α ) ¯ \underline{\varphi(\beta/\alpha)}\,\!
  87. α φ \exists\alpha\,\varphi\,\!
  88. α \alpha\,\!
  89. φ \varphi\,\!
  90. β \beta\,\!
  91. α φ \exists\alpha\,\varphi\,\!
  92. φ ( β / α ) ψ ¯ \underline{\varphi(\beta/\alpha)\vdash\psi}\,\!
  93. ψ \psi\,\!
  94. β \beta
  95. φ \varphi\,\!
  96. β \beta\,\!
  97. ψ \psi\,\!
  98. β \beta
  99. p p q q ¯ \begin{aligned}\displaystyle p\\ \displaystyle p\rightarrow q\\ \displaystyle\therefore\overline{q\quad\quad\quad}\\ \end{aligned}
  100. ( ( p ( p q ) ) q ((p\wedge(p\rightarrow q))\rightarrow q
  101. ¬ q p q ¬ p ¯ \begin{aligned}\displaystyle\neg q\\ \displaystyle p\rightarrow q\\ \displaystyle\therefore\overline{\neg p\quad\quad\quad}\\ \end{aligned}
  102. ( ( ¬ q ( p q ) ) ¬ p ((\neg q\wedge(p\rightarrow q))\rightarrow\neg p
  103. ( p q ) r p ( q r ) ¯ \begin{aligned}\displaystyle(p\vee q)\vee r\\ \displaystyle\therefore\overline{p\vee(q\vee r)}\\ \end{aligned}
  104. ( ( p q ) r ) ( p ( q r ) ) ((p\vee q)\vee r)\rightarrow(p\vee(q\vee r))
  105. p q q p ¯ \begin{aligned}\displaystyle p\wedge q\\ \displaystyle\therefore\overline{q\wedge p}\\ \end{aligned}
  106. ( p q ) ( q p ) (p\wedge q)\rightarrow(q\wedge p)
  107. p q q p p q ¯ \begin{aligned}\displaystyle p\rightarrow q\\ \displaystyle q\rightarrow p\\ \displaystyle\therefore\overline{p\leftrightarrow q}\\ \end{aligned}
  108. ( ( p q ) ( q p ) ) ( p q ) ((p\rightarrow q)\wedge(q\rightarrow p))\rightarrow(\ p\leftrightarrow q)
  109. ( p q ) r p ( q r ) ¯ \begin{aligned}\displaystyle(p\wedge q)\rightarrow r\\ \displaystyle\therefore\overline{p\rightarrow(q\rightarrow r)}\\ \end{aligned}
  110. ( ( p q ) r ) ( p ( q r ) ) ((p\wedge q)\rightarrow r)\rightarrow(p\rightarrow(q\rightarrow r))
  111. p q ¬ q ¬ p ¯ \begin{aligned}\displaystyle p\rightarrow q\\ \displaystyle\therefore\overline{\neg q\rightarrow\neg p}\\ \end{aligned}
  112. ( p q ) ( ¬ q ¬ p ) (p\rightarrow q)\rightarrow(\neg q\rightarrow\neg p)
  113. p q q r p r ¯ \begin{aligned}\displaystyle p\rightarrow q\\ \displaystyle q\rightarrow r\\ \displaystyle\therefore\overline{p\rightarrow r}\\ \end{aligned}
  114. ( ( p q ) ( q r ) ) ( p r ) ((p\rightarrow q)\wedge(q\rightarrow r))\rightarrow(p\rightarrow r)
  115. p q ¬ p q ¯ \begin{aligned}\displaystyle p\rightarrow q\\ \displaystyle\therefore\overline{\neg p\vee q}\\ \end{aligned}
  116. ( p q ) ( ¬ p q ) (p\rightarrow q)\rightarrow(\neg p\vee q)
  117. ( p q ) r ( p r ) ( q r ) ¯ \begin{aligned}\displaystyle(p\vee q)\wedge r\\ \displaystyle\therefore\overline{(p\wedge r)\vee(q\wedge r)}\\ \end{aligned}
  118. ( ( p q ) r ) ( ( p r ) ( q r ) ) ((p\vee q)\wedge r)\rightarrow((p\wedge r)\vee(q\wedge r))
  119. p q p ( p q ) ¯ \begin{aligned}\displaystyle p\rightarrow q\\ \displaystyle\therefore\overline{p\rightarrow(p\wedge q)}\\ \end{aligned}
  120. ( p q ) ( p ( p q ) ) (p\rightarrow q)\rightarrow(p\rightarrow(p\wedge q))
  121. p q ¬ p q ¯ \begin{aligned}\displaystyle p\vee q\\ \displaystyle\neg p\\ \displaystyle\therefore\overline{q\quad\quad\quad}\\ \end{aligned}
  122. ( ( p q ) ¬ p ) q ((p\vee q)\wedge\neg p)\rightarrow q
  123. p p q ¯ \begin{aligned}\displaystyle p\\ \displaystyle\therefore\overline{p\vee q}\\ \end{aligned}
  124. p ( p q ) p\rightarrow(p\vee q)
  125. p q p ¯ \begin{aligned}\displaystyle p\wedge q\\ \displaystyle\therefore\overline{p\quad\quad\quad}\\ \end{aligned}
  126. ( p q ) p (p\wedge q)\rightarrow p
  127. p q p q ¯ \begin{aligned}\displaystyle p\\ \displaystyle q\\ \displaystyle\therefore\overline{p\wedge q}\\ \end{aligned}
  128. ( ( p ) ( q ) ) ( p q ) ((p)\wedge(q))\rightarrow(p\wedge q)
  129. p ¬ ¬ p ¯ \begin{aligned}\displaystyle p\\ \displaystyle\therefore\overline{\neg\neg p}\\ \end{aligned}
  130. p ( ¬ ¬ p ) p\rightarrow(\neg\neg p)
  131. p p p ¯ \begin{aligned}\displaystyle p\vee p\\ \displaystyle\therefore\overline{p\quad\quad\quad}\\ \end{aligned}
  132. ( p p ) p (p\vee p)\rightarrow p
  133. p q ¬ p r q r ¯ \begin{aligned}\displaystyle p\vee q\\ \displaystyle\neg p\vee r\\ \displaystyle\therefore\overline{q\vee r}\\ \end{aligned}
  134. ( ( p q ) ( ¬ p r ) ) ( q r ) ((p\vee q)\wedge(\neg p\vee r))\rightarrow(q\vee r)
  135. \therefore
  136. p p
  137. q q
  138. r r
  139. p q \displaystyle p\rightarrow q
  140. p p
  141. q q
  142. r r
  143. s s
  144. t t
  145. ¬ p q , r p , ¬ r s \neg p\wedge q,r\rightarrow p,\neg r\rightarrow s
  146. s t s\rightarrow t
  147. t t
  148. ¬ p q \neg p\wedge q
  149. ¬ p \neg p
  150. r p r\rightarrow p
  151. ¬ r \neg r
  152. ¬ r s \neg r\rightarrow s
  153. s s
  154. s t s\rightarrow t
  155. t t

List_of_scientific_constants_named_after_people.html

  1. π \pi
  2. k e s = 0 {k}_{e}^{s=0}
  3. γ \gamma
  4. e e
  5. ϕ \phi
  6. G G
  7. h h
  8. h h
  9. ħ ħ

List_of_Solar_System_objects_by_size.html

  1. × 10 2 0 \times 10^{2}0
  2. × 10 1 9 \times 10^{1}9
  3. ± \pm
  4. × 10 1 8 \times 10^{1}8
  5. × 10 1 5 \times 10^{1}5
  6. a g = G m r 2 a_{g}=G\frac{m}{r^{2}}
  7. a c = 4 π 2 r T 2 a_{c}=4\pi^{2}\frac{r}{T^{2}}
  8. g = a g - a c = G m r 2 - 4 π 2 r T 2 g=a_{g}-a_{c}=\frac{Gm}{r^{2}}-\frac{4\pi^{2}r}{T^{2}}

Littlewood_conjecture.html

  1. lim inf n n n α n β = 0 , \liminf_{n\to\infty}\ n\,\|n\alpha\|\,\|n\beta\|=0,
  2. \|\,\|
  3. inf n 1 n || n α || > 0 \inf_{n\geq 1}n\cdot||n\alpha||>0

LIX.html

  1. LIX = A B + C 100 A \,\text{LIX}=\frac{A}{B}+\frac{C\cdot 100}{A}
  2. A A
  3. B B
  4. C C

Local_quantum_field_theory.html

  1. 𝒜 \mathcal{A}
  2. 𝒜 ( M ) \mathcal{A}(M)
  3. 𝒜 ( i U , U V ) \mathcal{A}(i_{U,U\cup V})
  4. 𝒜 ( i V , U V ) \mathcal{A}(i_{V,U\cup V})
  5. U ¯ \bar{U}
  6. 𝒜 ( i U , U ¯ ) \mathcal{A}(i_{U,\bar{U}})
  7. 𝒜 ( M ) \mathcal{A}(M)
  8. 𝒜 ( U ) \mathcal{A}(U)
  9. 𝒜 ( M ) . \mathcal{A}(M).

Local_zeta-function.html

  1. Z ( V , s ) = exp ( m = 1 N m m ( q - s ) m ) Z(V,s)=\exp\left(\sum_{m=1}^{\infty}\frac{N_{m}}{m}(q^{-s})^{m}\right)
  2. u = q - s u=q^{-s}
  3. Z ( V , u ) = exp ( m = 1 N m u m m ) \mathit{Z}(V,u)=\exp\left(\sum_{m=1}^{\infty}N_{m}\frac{u^{m}}{m}\right)
  4. ( 1 ) Z ( V , 0 ) = 1 (1)\ \ \mathit{Z}(V,0)=1\,
  5. ( 2 ) d d u log Z ( V , u ) = m = 1 N m u m - 1 . (2)\ \ \frac{d}{du}\log\mathit{Z}(V,u)=\sum_{m=1}^{\infty}N_{m}u^{m-1}\ .
  6. [ F k : F ] = k [F_{k}:F]=k\,
  7. N k N_{k}\,
  8. G ( t ) = N 1 t + N 2 t 2 / 2 + N 3 t 3 / 3 + G(t)=N_{1}t+N_{2}t^{2}/2+N_{3}t^{3}/3+\cdots\,
  9. Z = exp ( G ( t ) ) Z=\exp(G(t))\,
  10. Z ( t ) / Z ( t ) Z^{\prime}(t)/Z(t)\,
  11. G ( t ) = N 1 + N 2 t 1 + N 3 t 2 + G^{\prime}(t)=N_{1}+N_{2}t^{1}+N_{3}t^{2}+\cdots\,
  12. G ( t ) = - log ( 1 - t ) G(t)=-\log(1-t)
  13. N k = q k + 1 N_{k}=q^{k}+1
  14. G ( t ) = - log ( 1 - t ) - log ( 1 - q t ) G(t)=-\log(1-t)-\log(1-qt)
  15. Z ( t ) = 1 ( 1 - t ) ( 1 - q t ) . Z(t)=\frac{1}{(1-t)(1-qt)}\ .
  16. ζ ( s ) \zeta(s)
  17. ζ ( s ) ζ ( s - 1 ) \zeta(s)\zeta(s-1)
  18. Z ( t ) = P ( t ) ( 1 - t ) ( 1 - q t ) , Z(t)=\frac{P(t)}{(1-t)(1-qt)}\ ,
  19. P ( t ) = i = 1 2 g ( 1 - ω i u ) , P(t)=\prod^{2g}_{i=1}(1-\omega_{i}u)\ ,
  20. | ω i | = q 1 / 2 . |\omega_{i}|=q^{1/2}\ .
  21. Z ( X , t ) = i = 0 2 dim X det ( 1 - t Frob | q H c i ( X ¯ , ) ) ( - 1 ) i + 1 . Z(X,t)=\prod_{i=0}^{2\dim X}\det\big(1-t\mbox{Frob}~{}_{q}|H^{i}_{c}(\overline% {X},{\mathbb{Q}}_{\ell})\big)^{(-1)^{i+1}}.
  22. X X
  23. q q
  24. \ell
  25. X ¯ \overline{X}
  26. X X
  27. t t
  28. Z ( X , t ) Z(X,t)
  29. Z ( X , t ) = ( 1 - t deg ( x ) ) - 1 . Z(X,t)=\prod\ (1-t^{\deg(x)})^{-1}.
  30. V ¯ \overline{V}
  31. N 1 + N 2 t 1 + N 3 t 2 + N_{1}+N_{2}t^{1}+N_{3}t^{2}+\cdots\,

Localization_of_a_category.html

  1. M M m r m . M\to M\quad m\mapsto r\cdot m.
  2. R [ S - 1 ] R[S^{-1}]
  3. R [ S - 1 ] R[S^{-1}]
  4. S = { 1 , r , r 2 , r 3 , } S=\{1,r,r^{2},r^{3},\dots\}
  5. φ : Mod R Mod R [ S - 1 ] M M [ S - 1 ] \varphi:\,\text{Mod}_{R}\to\,\text{Mod}_{R[S^{-1}]}\quad M\mapsto M[S^{-1}]
  6. F : Mod R C F:\,\text{Mod}_{R}\to C
  7. G : Mod R [ S - 1 ] C G:\,\text{Mod}_{R[S^{-1}]}\to C
  8. F = G φ F=G\circ\varphi
  9. X f X Y X\stackrel{f}{\leftarrow}X^{\prime}\rightarrow Y
  10. C C [ W - 1 ] C\to C[W^{-1}]
  11. C [ W - 1 ] C C[W^{-1}]\to C
  12. R [ S - 1 ] R[S^{-1}]
  13. R [ S - 1 ] - M o d R - M o d . R[S^{-1}]-Mod\to R-Mod.
  14. L : C C [ W - 1 ] C L:C\to C[W^{-1}]\to C

Localization_of_a_module.html

  1. m s \frac{m}{s}
  2. m s \frac{m}{s}
  3. m s + n t := t m + s n s t \frac{m}{s}+\frac{n}{t}:=\frac{tm+sn}{st}
  4. a m s := a m s a\cdot\frac{m}{s}:=\frac{am}{s}
  5. p M p p\mapsto M_{p}

Locally_convex_topological_vector_space.html

  1. V V
  2. 𝐊 \mathbf{K}
  3. 𝐂 \mathbf{C}
  4. 𝐑 \mathbf{R}
  5. C C
  6. V V
  7. x , y x,y
  8. C C
  9. 0 t 1 , 0≤t≤1,
  10. t x + ( 1 t ) y tx+(1–t)y
  11. C C
  12. C C
  13. C C
  14. x x
  15. C C
  16. λ x λx
  17. C C
  18. | λ | = 1 |λ|=1
  19. 𝐊 = 𝐑 \mathbf{K}=\mathbf{R}
  20. C C
  21. 𝐊 = 𝐂 \mathbf{K}=\mathbf{C}
  22. x x
  23. C C
  24. C C
  25. x x
  26. x x
  27. x x
  28. C C
  29. 0 λ 1 , 0≤λ≤1,
  30. λ x λx
  31. C C
  32. x x
  33. C C
  34. λ x λx
  35. C C
  36. | λ | 1 |λ|≤1
  37. 𝐊 = 𝐑 \mathbf{K}=\mathbf{R}
  38. x x
  39. C C
  40. C C
  41. x x
  42. x −x
  43. 𝐊 = 𝐂 \mathbf{K}=\mathbf{C}
  44. x x
  45. C C
  46. C C
  47. x x
  48. x x
  49. t C tC
  50. t > 0 t>0
  51. V V
  52. x x
  53. V V
  54. t x tx
  55. C C
  56. t > 0 t>0
  57. C C
  58. V V
  59. 1 ≤1
  60. V V
  61. V V
  62. p : V 𝐑 p:V→\mathbf{R}
  63. p p
  64. p ( x ) 0 p(x)≥0
  65. p p
  66. p ( λ x ) = | λ | p ( x ) p(λx)=|λ|p(x)
  67. λ λ
  68. p ( 0 ) = 0 p(0)=0
  69. p p
  70. p ( x + y ) p ( x ) + p ( y ) p(x+y)≤p(x)+p(y)
  71. p p
  72. p ( x ) = 0 p(x)=0
  73. x = 0 x=0
  74. p p
  75. V V
  76. V V
  77. { p α , y : V 𝐑 x p α ( x - y ) y V , α A \begin{cases}p_{\alpha,y}:V\to\mathbf{R}\\ x\mapsto p_{\alpha}(x-y)&y\in V,\alpha\in A\end{cases}
  78. y y
  79. B B
  80. A A
  81. ε > 0 ε> 0
  82. U B , ε ( y ) = { x V : p α ( x - y ) < ε α B } . U_{B,\varepsilon}(y)=\{x\in V:p_{\alpha}(x-y)<\varepsilon\ \forall\alpha\in B\}.
  83. U B , ε ( y ) = α B ( p α , y ) - 1 ( [ 0 , ε ) ) . U_{B,\varepsilon}(y)=\bigcap_{\alpha\in B}(p_{\alpha,y})^{-1}([0,\varepsilon)).
  84. ε ε
  85. C C
  86. x x
  87. C C
  88. t x tx
  89. C C
  90. 0 t 1 0≤t≤1
  91. C C
  92. μ C ( x ) = inf { λ > 0 : x \isin λ C } . \mu_{C}(x)=\inf\{\lambda>0:x\isin\lambda C\}.
  93. C C
  94. { x : p α 1 ( x ) < ε , , p α n ( x ) < ε } \left\{x:p_{\alpha_{1}}(x)<\varepsilon,\cdots,p_{\alpha_{n}}(x)<\varepsilon\right\}
  95. α α
  96. x x
  97. 0
  98. d ( x , y ) = 0 d(x,y)=0
  99. x = y x=y
  100. d ( x , y ) = n 1 2 n p n ( x - y ) 1 + p n ( x - y ) d(x,y)=\sum^{\infty}_{n}\frac{1}{2^{n}}\frac{p_{n}(x-y)}{1+p_{n}(x-y)}
  101. d ( k x , k y ) | k | d ( x , y ) d(kx,ky)≠|k|d(x,y)
  102. ε > 0 ε>0
  103. κ κ
  104. λ , μ > κ λ,μ>κ
  105. p 1 p≥1
  106. p i ( { x n } n ) = | x i | , i 𝐍 . p_{i}\left(\left\{x_{n}\right\}_{n}\right)=\left|x_{i}\right|,\qquad i\in% \mathbf{N}.
  107. 0
  108. V V
  109. F F
  110. V V
  111. F F
  112. F F
  113. F F
  114. V V
  115. f f
  116. F F
  117. D ( U ) D(U)
  118. C [ u s u , u p = 21 e , u b = 0 ] ( U ) C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime},u^{\prime}b=0^{\prime}](U)
  119. D ( U ) D(U)
  120. K U K⊂U
  121. C [ u s u , u p = 21 e , u b = 0 ] ( K ) C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime},u^{\prime}b=0^{\prime}](K)
  122. f C [ u s u , u p = 21 e , u b = 0 ] ( U ) f∈C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime},u^{\prime}b=0^{\prime}% ](U)
  123. s u p p ( f ) K supp(f)⊂K
  124. C [ u s u , u p = 21 e , u b = 0 ] ( K ) C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime},u^{\prime}b=0^{\prime}](K)
  125. U U
  126. D ( U ) D(U)
  127. D ( U ) D(U)
  128. X X
  129. C ( X ) C(X)
  130. X X
  131. K K
  132. X X
  133. X X
  134. C ( X ) C(X)
  135. f p = 0 1 | f ( x ) | p d x \|f\|_{p}=\int_{0}^{1}|f(x)|^{p}\,dx
  136. μ μ
  137. 0 , 11 0,11
  138. d ( f , g ) = 0 1 | f ( x ) - g ( x ) | 1 + | f ( x ) - g ( x ) | d x . d(f,g)=\int_{0}^{1}\frac{|f(x)-g(x)|}{1+|f(x)-g(x)|}\,dx.
  139. 0
  140. V V
  141. W W
  142. T : V W T:V→W
  143. β β
  144. M > 0 M>0
  145. v v
  146. V V
  147. q β ( T v ) M ( p α 1 ( v ) + + p α n ( v ) ) . q_{\beta}(Tv)\leq M\left(p_{\alpha_{1}}(v)+\cdots+p_{\alpha_{n}}(v)\right).
  148. T T
  149. q β ( T v ) M p α ( v ) . q_{\beta}(Tv)\leq Mp_{\alpha}(v).

Locally_cyclic_group.html

  1. μ p = { exp ( 2 π i m p k ) : m , k } \mu_{p^{\infty}}=\left\{\exp\left(\frac{2\pi im}{p^{k}}\right):m,k\in\mathbb{Z% }\right\}

Locus_(mathematics).html

  1. | P A | = 3 | P B | \Leftrightarrow|PA|=3|PB|
  2. | P A | 2 = 9 | P B | 2 \Leftrightarrow|PA|^{2}=9|PB|^{2}
  3. ( x + 1 ) 2 + ( y - 0 ) 2 = 9 ( x - 0 ) 2 + 9 ( y - 2 ) 2 \Leftrightarrow(x+1)^{2}+(y-0)^{2}=9(x-0)^{2}+9(y-2)^{2}
  4. 8 ( x 2 + y 2 ) - 2 x - 36 y + 35 = 0 \Leftrightarrow 8(x^{2}+y^{2})-2x-36y+35=0
  5. ( x - 1 8 ) 2 + ( y - 9 4 ) 2 = 45 64 \Leftrightarrow\left(x-\frac{1}{8}\right)^{2}+\left(y-\frac{9}{4}\right)^{2}=% \frac{45}{64}
  6. 3 8 5 \frac{3}{8}\sqrt{5}
  7. \Leftrightarrow
  8. y x 2 y 2 x + 3 c = - 1 \Leftrightarrow\frac{y}{x}\cdot\frac{2y}{2x+3c}=-1
  9. 2 y 2 + 2 x 2 + 3 c x = 0 \Leftrightarrow 2y^{2}+2x^{2}+3cx=0
  10. x 2 + y 2 + ( 3 c / 2 ) x = 0 \Leftrightarrow x^{2}+y^{2}+(3c/2)x=0
  11. ( x + 3 c / 4 ) 2 + y 2 = 9 c 2 / 16 \Leftrightarrow(x+3c/4)^{2}+y^{2}=9c^{2}/16
  12. α \alpha

Logarithmic_derivative.html

  1. f f \frac{f^{\prime}}{f}\!
  2. f f^{\prime}
  3. f , f^{\prime},
  4. ( log u v ) = ( log u + log v ) = ( log u ) + ( log v ) . (\log uv)^{\prime}=(\log u+\log v)^{\prime}=(\log u)^{\prime}+(\log v)^{\prime% }.\!
  5. ( u v ) u v = u v + u v u v = u u + v v . \frac{(uv)^{\prime}}{uv}=\frac{u^{\prime}v+uv^{\prime}}{uv}=\frac{u^{\prime}}{% u}+\frac{v^{\prime}}{v}.\!
  6. ( 1 / u ) 1 / u = - u / u 2 1 / u = - u u , \frac{(1/u)^{\prime}}{1/u}=\frac{-u^{\prime}/u^{2}}{1/u}=-\frac{u^{\prime}}{u},\!
  7. ( u / v ) u / v = ( u v - u v ) / v 2 u / v = u u - v v , \frac{(u/v)^{\prime}}{u/v}=\frac{(u^{\prime}v-uv^{\prime})/v^{2}}{u/v}=\frac{u% ^{\prime}}{u}-\frac{v^{\prime}}{v},\!
  8. ( u k ) u k = k u k - 1 u u k = k u u , \frac{(u^{k})^{\prime}}{u^{k}}=\frac{ku^{k-1}u^{\prime}}{u^{k}}=k\frac{u^{% \prime}}{u},\!
  9. f f = u u + v v . \frac{f^{\prime}}{f}=\frac{u^{\prime}}{u}+\frac{v^{\prime}}{v}.
  10. f = f ( u u + v v ) . f^{\prime}=f\left(\frac{u^{\prime}}{u}+\frac{v^{\prime}}{v}\right).
  11. m ( r , h / h ) = S ( r , h ) = o ( T ( r , h ) ) m(r,h^{\prime}/h)=S(r,h)=o(T(r,h))
  12. X d d X X\frac{d}{dX}

Logical_effort.html

  1. d a b s = d τ d_{abs}=d\cdot\tau
  2. d = f + p d=f+p
  3. f = g h f=gh
  4. d = g h + p d=gh+p
  5. D = N F 1 / N + P D=NF^{1/N}+P
  6. F = G H F=GH
  7. b = C o n p a t h + C o f f p a t h C o n p a t h b=\frac{C_{onpath}+C_{offpath}}{C_{onpath}}
  8. F = G H B F=GHB
  9. f = F 1 / N f=F^{1/N}
  10. d = g h + p = ( 1 ) ( 1 ) + 1 = 2 d=gh+p=(1)(1)+1=2
  11. 4 3 \frac{4}{3}
  12. 5 3 \frac{5}{3}
  13. 6 3 \frac{6}{3}
  14. 7 3 \frac{7}{3}
  15. n + 2 3 \frac{n+2}{3}
  16. 5 3 \frac{5}{3}
  17. 7 3 \frac{7}{3}
  18. 9 3 \frac{9}{3}
  19. 11 3 \frac{11}{3}
  20. 2 n + 1 3 \frac{2n+1}{3}
  21. d = g h + p = ( 4 / 3 ) ( 1 ) + 2 = 10 / 3 d=gh+p=(4/3)(1)+2=10/3
  22. d = g h + p = ( 5 / 3 ) ( 2 ) + 1 = 13 / 3 d=gh+p=(5/3)(2)+1=13/3

Logistic_distribution.html

  1. 1 1 + e - x - μ s \frac{1}{1+e^{-\frac{x-\mu}{s}}}\!
  2. μ \mu
  3. μ \mu
  4. μ \mu
  5. s 2 π 2 3 \tfrac{s^{2}\pi^{2}}{3}
  6. 0
  7. 1.2 1.2
  8. ln ( s ) + 2 = ln ( σ ) + 1.404576 , \ln(s)+2=\ln(\sigma)+1.404576,
  9. e μ t B ( 1 - s t , 1 + s t ) e^{\mu t}\operatorname{B}(1-st,1+st)
  10. e i t μ π s t sinh ( π s t ) e^{it\mu}\frac{\pi st}{\sinh(\pi st)}
  11. f ( x ; μ , s ) = e - x - μ s s ( 1 + e - x - μ s ) 2 = 1 4 s sech 2 ( x - μ 2 s ) . f(x;\mu,s)=\frac{e^{-\frac{x-\mu}{s}}}{s\left(1+e^{-\frac{x-\mu}{s}}\right)^{2% }}=\frac{1}{4s}\operatorname{sech}^{2}\!\left(\frac{x-\mu}{2s}\right).
  12. F ( x ; μ , s ) = 1 1 + e - x - μ s = 1 2 + 1 2 tanh ( x - μ 2 s ) . F(x;\mu,s)=\frac{1}{1+e^{-\frac{x-\mu}{s}}}=\frac{1}{2}+\frac{1}{2}\;% \operatorname{tanh}\!\left(\frac{x-\mu}{2s}\right).
  13. Q ( p ; μ , s ) = μ + s ln ( p 1 - p ) . Q(p;\mu,s)=\mu+s\,\ln\left(\frac{p}{1-p}\right).
  14. Q ( p ; s ) = s p ( 1 - p ) . Q^{\prime}(p;s)=\frac{s}{p(1-p)}.
  15. s s
  16. σ \sigma
  17. s = q σ s\,=\,q\,\sigma
  18. q = 3 / π = 0.551328895 + q\,=\,\sqrt{3}/{\pi}\,=\,0.551328895+
  19. μ + β log ( e X - 1 ) Logistic ( μ , β ) . \mu+\beta\log\left(e^{X}-1\right)\sim\mathrm{Logistic}(\mu,\beta).
  20. μ - β log ( X Y ) Logistic ( μ , β ) . \mu-\beta\log\left(\frac{X}{Y}\right)\sim\mathrm{Logistic}(\mu,\beta).
  21. E [ ( X - μ ) n ] = - ( x - μ ) n d F ( x ) = 0 1 ( Q ( p ) - μ ) n d p = s n 0 1 [ ln ( p 1 - p ) ] n d p . \operatorname{E}[(X-\mu)^{n}]=\int_{-\infty}^{\infty}(x-\mu)^{n}dF(x)=\int_{0}% ^{1}\big(Q(p)-\mu\big)^{n}dp=s^{n}\int_{0}^{1}\left[\ln\!\left(\frac{p}{1-p}% \right)\right]^{n}dp.
  22. E [ ( X - μ ) n ] = s n π n ( 2 n - 2 ) | B n | . \operatorname{E}[(X-\mu)^{n}]=s^{n}\pi^{n}(2^{n}-2)\cdot|B_{n}|.

London_dispersion_force.html

  1. 1 R \frac{1}{R}
  2. R R
  3. E A B disp E_{AB}^{\rm disp}
  4. A A
  5. B B
  6. α A \alpha^{A}
  7. α B \alpha^{B}
  8. I A I_{A}
  9. I B I_{B}
  10. R R
  11. E A B disp - 3 2 I A I B I A + I B α A α B R 6 E_{AB}^{\rm disp}\approx-{3\over 2}{I_{A}I_{B}\over I_{A}+I_{B}}{\alpha^{A}% \alpha^{B}\over{R^{6}}}
  12. R R
  13. E A B disp - 3 4 I A I B I A + I B α A α B E_{AB}^{\rm disp}\approx-{3\over 4}{I_{A}I_{B}\over{I_{A}+I_{B}}}\alpha^{A}% \alpha^{\dagger B}
  14. α B \alpha^{\dagger B}

Look-and-say_sequence.html

  1. L n + 1 L n \frac{L_{n+1}}{L_{n}}
  2. lim n L n + 1 L n = λ \lim_{n\to\infty}\frac{L_{n+1}}{L_{n}}=\lambda
  3. x 71 \displaystyle\,\,\,\,\,\,\,x^{71}

Lookback_option.html

  1. L C f l o a t = max ( S T - S m i n , 0 ) = S T - S m i n , and L P f l o a t = max ( S m a x - S T , 0 ) = S m a x - S T , LC_{float}=\max(S_{T}-S_{min},0)=S_{T}-S_{min},~{}~{}\,\text{and}~{}~{}LP_{% float}=\max(S_{max}-S_{T},0)=S_{max}-S_{T},
  2. S m a x S_{max}
  3. S m i n S_{min}
  4. S T S_{T}
  5. T T
  6. L C f i x = max ( S m a x - K , 0 ) , and L P f i x = max ( K - S m i n , 0 ) , LC_{fix}=\max(S_{max}-K,0),~{}~{}\,\text{and}~{}~{}LP_{fix}=\max(K-S_{min},0),
  7. S m a x S_{max}
  8. S m i n S_{min}
  9. K K
  10. r > 0 r>0
  11. σ > 0 \sigma>0
  12. T > 0 T>0
  13. t < T t<T
  14. τ = T - t \tau=T-t
  15. M = max 0 u t S u , m = min 0 u t S u and S t = S . M=\max_{0\leq u\leq t}S_{u},~{}~{}m=\min_{0\leq u\leq t}S_{u}\,\text{ and }S_{% t}=S.
  16. L C t = S Φ ( a 1 ( S , m ) ) - m e - r τ Φ ( a 2 ( S , m ) ) - S σ 2 2 r ( Φ ( - a 1 ( S , m ) ) - e - r τ ( m / S ) 2 r σ 2 Φ ( - a 3 ( S , m ) ) ) , LC_{t}=S\Phi(a_{1}(S,m))-me^{-r\tau}\Phi(a_{2}(S,m))-\frac{S\sigma^{2}}{2r}(% \Phi(-a_{1}(S,m))-e^{-r\tau}(m/S)^{\frac{2r}{\sigma^{2}}}\Phi(-a_{3}(S,m))),
  17. a 1 ( S , H ) = ln ( S / H ) + ( r + 1 2 σ 2 ) τ σ τ a_{1}(S,H)=\frac{\ln(S/H)+(r+\frac{1}{2}\sigma^{2})\tau}{\sigma\sqrt{\tau}}
  18. a 2 ( S , H ) = ln ( S / H ) + ( r - 1 2 σ 2 ) τ σ τ = a 1 ( S , H ) - σ τ a_{2}(S,H)=\frac{\ln(S/H)+(r-\frac{1}{2}\sigma^{2})\tau}{\sigma\sqrt{\tau}}=a_% {1}(S,H)-\sigma\sqrt{\tau}
  19. a 3 ( S , H ) = ln ( S / H ) - ( r - 1 2 σ 2 ) τ σ τ = a 1 ( S , H ) - 2 r τ σ , with H > 0 , S > 0 , a_{3}(S,H)=\frac{\ln(S/H)-(r-\frac{1}{2}\sigma^{2})\tau}{\sigma\sqrt{\tau}}=a_% {1}(S,H)-\frac{2r\sqrt{\tau}}{\sigma},\,\text{ with }H>0,S>0,
  20. Φ \Phi
  21. Φ ( a ) = 1 2 π - a e - x 2 2 d x \Phi(a)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{a}e^{-\frac{x^{2}}{2}}\,dx
  22. L P t = - S Φ ( - a 1 ( S , M ) ) + M e - r τ Φ ( - a 2 ( S , M ) ) + S σ 2 2 r ( Φ ( a 1 ( S , M ) ) - e - r τ ( M / S ) 2 r σ 2 Φ ( a 3 ( S , M ) ) ) . LP_{t}=-S\Phi(-a_{1}(S,M))+Me^{-r\tau}\Phi(-a_{2}(S,M))+\frac{S\sigma^{2}}{2r}% (\Phi(a_{1}(S,M))-e^{-r\tau}(M/S)^{\frac{2r}{\sigma^{2}}}\Phi(a_{3}(S,M))).

Lookup_table.html

  1. sin ( x ) x - x 3 6 + x 5 120 - x 7 5040 \operatorname{sin}(x)\approx x-\frac{x^{3}}{6}+\frac{x^{5}}{120}-\frac{x^{7}}{% 5040}

Lorentz_covariance.html

  1. Δ s 2 = Δ x a Δ x b η a b = c 2 Δ t 2 - Δ x 2 - Δ y 2 - Δ z 2 \Delta s^{2}=\Delta x^{a}\Delta x^{b}\eta_{ab}=c^{2}\Delta t^{2}-\Delta x^{2}-% \Delta y^{2}-\Delta z^{2}
  2. Δ τ = Δ s 2 c 2 , Δ s 2 > 0 \Delta\tau=\sqrt{\frac{\Delta s^{2}}{c^{2}}},\,\Delta s^{2}>0
  3. L = - Δ s 2 , Δ s 2 < 0 L=\sqrt{-\Delta s^{2}},\,\Delta s^{2}<0
  4. m 0 2 c 2 = P a P b η a b = E 2 c 2 - p x 2 - p y 2 - p z 2 m_{0}^{2}c^{2}=P^{a}P^{b}\eta_{ab}=\frac{E^{2}}{c^{2}}-p_{x}^{2}-p_{y}^{2}-p_{% z}^{2}
  5. F a b F a b = 2 ( B 2 - E 2 c 2 ) F_{ab}F^{ab}=\ 2\left(B^{2}-\frac{E^{2}}{c^{2}}\right)
  6. G c d F c d = 1 2 ϵ a b c d F a b F c d = - 4 c ( B E ) G_{cd}F^{cd}=\frac{1}{2}\epsilon_{abcd}F^{ab}F^{cd}=-\frac{4}{c}\left(\vec{B}% \cdot\vec{E}\right)
  7. = η μ ν μ ν = 1 c 2 2 t 2 - 2 x 2 - 2 y 2 - 2 z 2 \Box=\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}=\frac{1}{c^{2}}\frac{\partial^{% 2}}{\partial t^{2}}-\frac{\partial^{2}}{\partial x^{2}}-\frac{\partial^{2}}{% \partial y^{2}}-\frac{\partial^{2}}{\partial z^{2}}
  8. X a = [ c t , x , y , z ] X^{a}=\left[ct,x,y,z\right]
  9. a = [ 1 c t , x , y , z ] \partial_{a}=\left[\frac{1}{c}\frac{\partial}{\partial t},\frac{\partial}{% \partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right]
  10. U a = d X a d τ = γ [ c , d x d t , d y d t , d z d t ] U^{a}=\frac{dX^{a}}{d\tau}=\gamma\left[c,\frac{dx}{dt},\frac{dy}{dt},\frac{dz}% {dt}\right]
  11. P a = m 0 U a = [ E c , p x , p y , p z ] P^{a}=m_{0}U^{a}=\left[\frac{E}{c},p_{x},p_{y},p_{z}\right]
  12. j a = [ c ρ , j x , j y , j z ] j^{a}=\left[c\rho,j_{x},j_{y},j_{z}\right]
  13. δ b a = { 1 if a = b , 0 if a b . \delta^{a}_{b}=\begin{cases}1&\mbox{if }~{}a=b,\\ 0&\mbox{if }~{}a\neq b.\end{cases}
  14. η a b = η a b = { 1 if a = b = 0 , - 1 if a = b = 1 , 2 , 3 , 0 if a b . \eta_{ab}=\eta^{ab}=\begin{cases}1&\mbox{if }~{}a=b=0,\\ -1&\mbox{if }~{}a=b=1,2,3,\\ 0&\mbox{if }~{}a\neq b.\end{cases}
  15. ϵ a b c d = - ϵ a b c d = { + 1 if { a b c d } is an even permutation of { 0123 } , - 1 if { a b c d } is an odd permutation of { 0123 } , 0 otherwise. \epsilon_{abcd}=-\epsilon^{abcd}=\begin{cases}+1&\mbox{if }~{}\{abcd\}\mbox{ % is an even permutation of }~{}\{0123\},\\ -1&\mbox{if }~{}\{abcd\}\mbox{ is an odd permutation of }~{}\{0123\},\\ 0&\mbox{otherwise.}\end{cases}
  16. F a b = [ 0 E x / c E y / c E z / c - E x / c 0 - B z B y - E y / c B z 0 - B x - E z / c - B y B x 0 ] F_{ab}=\begin{bmatrix}0&E_{x}/c&E_{y}/c&E_{z}/c\\ -E_{x}/c&0&-B_{z}&B_{y}\\ -E_{y}/c&B_{z}&0&-B_{x}\\ -E_{z}/c&-B_{y}&B_{x}&0\end{bmatrix}
  17. G c d = 1 2 ϵ a b c d F a b = [ 0 B x B y B z - B x 0 E z / c - E y / c - B y - E z / c 0 E x / c - B z E y / c - E x / c 0 ] G_{cd}=\frac{1}{2}\epsilon_{abcd}F^{ab}=\begin{bmatrix}0&B_{x}&B_{y}&B_{z}\\ -B_{x}&0&E_{z}/c&-E_{y}/c\\ -B_{y}&-E_{z}/c&0&E_{x}/c\\ -B_{z}&E_{y}/c&-E_{x}/c&0\end{bmatrix}

Loss_function.html

  1. X = ( X 1 , , X n ) X=(X_{1},\ldots,X_{n})
  2. X i F θ X_{i}\sim F_{\theta}
  3. F θ F_{\theta}
  4. 𝒳 \scriptstyle\mathcal{X}
  5. 𝔼 θ {\mathbb{E}}_{\theta}
  6. R ( θ , δ ) = 𝔼 θ L ( θ , δ ( X ) ) = X L ( θ , δ ( x ) ) d P θ ( x ) . R(\theta,\delta)=\mathbb{E}_{\theta}L\big(\theta,\delta(X)\big)=\int_{X}L\big(% \theta,\delta(x)\big)\,\operatorname{d}P_{\theta}(x).
  7. ρ ( π * , a ) = Θ L ( θ , a ) d π * ( θ ) \rho(\pi^{*},a)=\int_{\Theta}L(\theta,a)\,\operatorname{d}\pi^{*}(\theta)
  8. θ ^ \hat{\theta}
  9. L ( θ , θ ^ ) = ( θ - θ ^ ) 2 , L(\theta,\hat{\theta})=(\theta-\hat{\theta})^{2},
  10. R ( θ , θ ^ ) = E θ ( θ - θ ^ ) 2 . R(\theta,\hat{\theta})=E_{\theta}(\theta-\hat{\theta})^{2}.
  11. L ( f , f ^ ) = f - f ^ 2 2 , L(f,\hat{f})=\|f-\hat{f}\|_{2}^{2}\,,
  12. R ( f , f ^ ) = E f - f ^ 2 . R(f,\hat{f})=E\|f-\hat{f}\|^{2}.\,
  13. arg min 𝛿 max θ Θ R ( θ , δ ) . \underset{\delta}{\operatorname{arg\,min}}\ \max_{\theta\in\Theta}\ R(\theta,% \delta).
  14. arg min 𝛿 𝔼 θ Θ [ R ( θ , δ ) ] = arg min 𝛿 θ Θ R ( θ , δ ) p ( θ ) d θ . \underset{\delta}{\operatorname{arg\,min}}\ \mathbb{E}_{\theta\in\Theta}[R(% \theta,\delta)]=\underset{\delta}{\operatorname{arg\,min}}\ \int_{\theta\in% \Theta}R(\theta,\delta)\,p(\theta)\,d\theta.
  15. L ( a ) = a 2 L(a)=a^{2}
  16. L ( a ) = | a | L(a)=|a|
  17. a = 0 a=0
  18. a a
  19. i = 1 n L ( a i ) \sum_{i=1}^{n}L(a_{i})
  20. λ ( x ) = C ( t - x ) 2 \lambda(x)=C(t-x)^{2}\;
  21. L ( y ^ , y ) = I ( y ^ y ) , L(\hat{y},y)=I(\hat{y}\neq y),\,
  22. I I

Lotka–Volterra_equation.html

  1. d x d t = α x - β x y \displaystyle\frac{dx}{dt}=\alpha x-\beta xy
  2. d y d t \tfrac{dy}{dt}
  3. d x d t \tfrac{dx}{dt}
  4. t t
  5. α , β , γ , δ α,β,γ,δ
  6. d x d t = α x - β x y . \frac{dx}{dt}=\alpha x-\beta xy.
  7. d y d t = δ x y - γ y . \frac{dy}{dt}=\delta xy-\gamma y.
  8. δ x y \displaystyle\delta xy
  9. γ y \displaystyle\gamma y
  10. α , β , γ , δ α,β,γ,δ
  11. x x
  12. y y
  13. y y
  14. x x
  15. γ γ
  16. t t
  17. α / γ α/γ
  18. d y d x = - y x δ x - γ β y - α , \frac{dy}{dx}=-\frac{y}{x}\frac{\delta x-\gamma}{\beta y-\alpha}~{},
  19. d ln y ( α - β y ) - d ln x ( γ - δ x ) d\ln y~{}~{}(\alpha-\beta y)-d\ln x~{}~{}(\gamma-\delta x)
  20. V = - δ x + γ ln ( x ) - β y + α ln ( y ) . V=-\delta\,x+\gamma\,\ln(x)-\beta\,y+\alpha\,\ln(y)~{}.
  21. x ( α - β y ) = 0 x(\alpha-\beta y)=0\,
  22. - y ( γ - δ x ) = 0 -y(\gamma-\delta x)=0\,
  23. { y = 0 , x = 0 } \left\{y=0,x=0\right\}\,
  24. { y = α β , x = γ δ } , \left\{y=\frac{\alpha}{\beta},x=\frac{\gamma}{\delta}\right\},\,
  25. J ( x , y ) = [ α - β y - β x δ y δ x - γ ] . J(x,y)=\begin{bmatrix}\alpha-\beta y&-\beta x\\ \delta y&\delta x-\gamma\end{bmatrix}.
  26. J ( 0 , 0 ) = [ α 0 0 - γ ] . J(0,0)=\begin{bmatrix}\alpha&0\\ 0&-\gamma\end{bmatrix}.
  27. λ 1 = α , λ 2 = - γ . \lambda_{1}=\alpha,\quad\lambda_{2}=-\gamma.\,
  28. J ( γ δ , α β ) = [ 0 - β γ δ α δ β 0 ] . J\left(\frac{\gamma}{\delta},\frac{\alpha}{\beta}\right)=\begin{bmatrix}0&-% \frac{\beta\gamma}{\delta}\\ \frac{\alpha\delta}{\beta}&0\end{bmatrix}~{}.
  29. λ 1 = i α γ , λ 2 = - i α γ . \lambda_{1}=i\sqrt{\alpha\gamma},\quad\lambda_{2}=-i\sqrt{\alpha\gamma}~{}.
  30. K = y α e - β y x γ e - δ x , K=y^{\alpha}e^{-\beta y}\,x^{\gamma}e^{-\delta x}~{},
  31. y α e - β y x γ e - δ x = y α x γ e δ x + β y max x , y > 0 . y^{\alpha}e^{-\beta y}\,x^{\gamma}e^{-\delta x}=\frac{y^{\alpha}x^{\gamma}}{e^% {\delta x+\beta y}}\longrightarrow\max\limits_{x,y>0}~{}.
  32. ( γ δ , α β ) \left(\frac{\gamma}{\delta},\frac{\alpha}{\beta}\right)
  33. K * = ( α β e ) α ( γ δ e ) γ , K^{*}=\left(\frac{\alpha}{\beta e}\right)^{\alpha}\left(\frac{\gamma}{\delta e% }\right)^{\gamma}~{},

Low-density_parity-check_code.html

  1. 𝐇 = ( 1 1 1 1 0 0 0 0 1 1 0 1 1 0 0 1 1 0 ) . \mathbf{H}=\begin{pmatrix}1&1&1&1&0&0\\ 0&0&1&1&0&1\\ 1&0&0&1&1&0\\ \end{pmatrix}.
  2. [ - P T | I n - k ] \begin{bmatrix}-P^{T}|I_{n-k}\end{bmatrix}
  3. 𝐇 = ( 1 1 1 1 0 0 0 0 1 1 0 1 1 0 0 1 1 0 ) 1 ( 1 1 1 1 0 0 0 0 1 1 0 1 0 1 1 0 1 0 ) 2 ( 1 1 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1 ) 3 ( 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 ) 4 . \mathbf{H}=\begin{pmatrix}1&1&1&1&0&0\\ 0&0&1&1&0&1\\ 1&0&0&1&1&0\\ \end{pmatrix}_{1}\sim\begin{pmatrix}1&1&1&1&0&0\\ 0&0&1&1&0&1\\ 0&1&1&0&1&0\\ \end{pmatrix}_{2}\sim\begin{pmatrix}1&1&1&1&0&0\\ 0&1&1&0&1&0\\ 0&0&1&1&0&1\\ \end{pmatrix}_{3}\sim\begin{pmatrix}1&1&1&1&0&0\\ 0&1&1&0&1&0\\ 1&1&0&0&0&1\\ \end{pmatrix}_{4}.
  4. [ I k | P ] \begin{bmatrix}I_{k}|P\end{bmatrix}
  5. P = - P P=-P
  6. 𝐆 = ( 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 0 ) . \mathbf{G}=\begin{pmatrix}1&0&0&1&0&1\\ 0&1&0&1&1&1\\ 0&0&1&1&1&0\\ \end{pmatrix}.
  7. ( 1 0 1 ) ( 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 0 ) = ( 1 0 1 0 1 1 ) . \begin{pmatrix}1&0&1\\ \end{pmatrix}\cdot\begin{pmatrix}1&0&0&1&0&1\\ 0&1&0&1&1&1\\ 0&0&1&1&1&0\\ \end{pmatrix}=\begin{pmatrix}1&0&1&0&1&1\\ \end{pmatrix}.
  8. G H T = 0 GH^{T}=0
  9. 𝐳 = 𝐇𝐫 = ( 1 1 1 1 0 0 0 0 1 1 0 1 1 0 0 1 1 0 ) ( 1 0 1 0 1 1 ) = ( 0 0 0 ) . \mathbf{z}=\mathbf{Hr}=\begin{pmatrix}1&1&1&1&0&0\\ 0&0&1&1&0&1\\ 1&0&0&1&1&0\\ \end{pmatrix}\begin{pmatrix}1\\ 0\\ 1\\ 0\\ 1\\ 1\\ \end{pmatrix}=\begin{pmatrix}0\\ 0\\ 0\\ \end{pmatrix}.

Low-dimensional_topology.html

  1. g 1 g\geq 1
  2. k 1 k\geq 1
  3. X K = M - interior ( N ) . X_{K}=M-\mbox{interior}~{}(N).

Low-discrepancy_sequence.html

  1. 0 1 f ( u ) d u 1 N i = 1 N f ( x i ) . \int_{0}^{1}f(u)\,du\approx\frac{1}{N}\,\sum_{i=1}^{N}f(x_{i}).
  2. D N ( P ) = sup B J | A ( B ; P ) N - λ s ( B ) | D_{N}(P)=\sup_{B\in J}\left|\frac{A(B;P)}{N}-\lambda_{s}(B)\right|
  3. i = 1 s [ a i , b i ) = { 𝐱 𝐑 s : a i x i < b i } \prod_{i=1}^{s}[a_{i},b_{i})=\{\mathbf{x}\in\mathbf{R}^{s}:a_{i}\leq x_{i}<b_{% i}\}\,
  4. 0 a i < b i 1 0\leq a_{i}<b_{i}\leq 1
  5. i = 1 s [ 0 , u i ) \prod_{i=1}^{s}[0,u_{i})
  6. D N * D N 2 s D N * . D^{*}_{N}\leq D_{N}\leq 2^{s}D^{*}_{N}.\,
  7. | 1 N i = 1 N f ( x i ) - I ¯ s f ( u ) d u | V ( f ) D N * ( x 1 , , x N ) . \left|\frac{1}{N}\sum_{i=1}^{N}f(x_{i})-\int_{\bar{I}^{s}}f(u)\,du\right|\leq V% (f)\,D_{N}^{*}(x_{1},\ldots,x_{N}).
  8. ϵ > 0 \epsilon>0
  9. | 1 N i = 1 N f ( x i ) - I ¯ s f ( u ) d u | > D N * ( x 1 , , x N ) - ϵ . \left|\frac{1}{N}\sum_{i=1}^{N}f(x_{i})-\int_{\bar{I}^{s}}f(u)\,du\right|>D_{N% }^{*}(x_{1},\ldots,x_{N})-\epsilon.
  10. D = { 1 , 2 , , d } D=\{1,2,\ldots,d\}
  11. u D \emptyset\neq u\subseteq D
  12. d x u := j u d x j dx_{u}:=\prod_{j\in u}dx_{j}
  13. ( x u , 1 ) (x_{u},1)
  14. 1 1
  15. 1 N i = 1 N f ( x i ) - I ¯ s f ( u ) d u = u D ( - 1 ) | u | [ 0 , 1 ] | u | disc ( x u , 1 ) | u | x u f ( x u , 1 ) d x u . \frac{1}{N}\sum_{i=1}^{N}f(x_{i})-\int_{\bar{I}^{s}}f(u)\,du=\sum_{\emptyset% \neq u\subseteq D}(-1)^{|u|}\int_{[0,1]^{|u|}}{\rm disc}(x_{u},1)\frac{% \partial^{|u|}}{\partial x_{u}}f(x_{u},1)dx_{u}.
  16. L 2 L^{2}
  17. L 2 L^{2}
  18. | 1 N i = 1 N f ( x i ) - I ¯ s f ( u ) d u | f d disc d ( { t i } ) , \left|\frac{1}{N}\sum_{i=1}^{N}f(x_{i})-\int_{\bar{I}^{s}}f(u)\,du\right|\leq% \|f\|_{d}\,{\rm disc}_{d}(\{t_{i}\}),
  19. disc d ( { t i } ) = ( u D [ 0 , 1 ] | u | disc ( x u , 1 ) 2 d x u ) 1 / 2 {\rm disc}_{d}(\{t_{i}\})=\left(\sum_{\emptyset\neq u\subseteq D}\int_{[0,1]^{% |u|}}{\rm disc}(x_{u},1)^{2}dx_{u}\right)^{1/2}
  20. f d = ( u D [ 0 , 1 ] | u | | | u | x u f ( x u , 1 ) | 2 d x u ) 1 / 2 . \|f\|_{d}=\left(\sum_{u\subseteq D}\int_{[0,1]^{|u|}}\left|\frac{\partial^{|u|% }}{\partial x_{u}}f(x_{u},1)\right|^{2}dx_{u}\right)^{1/2}.
  21. D N * ( x 1 , , x N ) ( 3 2 ) s ( 2 H + 1 + 0 < h H 1 r ( h ) | 1 N n = 1 N e 2 π i h , x n | ) D_{N}^{*}(x_{1},\ldots,x_{N})\leq\left(\frac{3}{2}\right)^{s}\left(\frac{2}{H+% 1}+\sum_{0<\|h\|_{\infty}\leq H}\frac{1}{r(h)}\left|\frac{1}{N}\sum_{n=1}^{N}e% ^{2\pi i\langle h,x_{n}\rangle}\right|\right)
  22. r ( h ) = i = 1 s max { 1 , | h i | } for h = ( h 1 , , h s ) \Z s . r(h)=\prod_{i=1}^{s}\max\{1,|h_{i}|\}\quad\mbox{for}~{}\quad h=(h_{1},\ldots,h% _{s})\in\Z^{s}.
  23. D N * ( x 1 , , x N ) c s ( ln N ) s - 1 N D_{N}^{*}(x_{1},\ldots,x_{N})\geq c_{s}\frac{(\ln N)^{s-1}}{N}
  24. D N * ( x 1 , , x N ) c s ( ln N ) s N D_{N}^{*}(x_{1},\ldots,x_{N})\geq c^{\prime}_{s}\frac{(\ln N)^{s}}{N}
  25. D N * ( x 1 , , x N ) 1 2 N D_{N}^{*}(x_{1},\ldots,x_{N})\geq\frac{1}{2N}
  26. D N * ( x 1 , , x N ) C log N N D_{N}^{*}(x_{1},\ldots,x_{N})\geq C\frac{\log N}{N}
  27. C = max a 3 1 16 a - 2 a log a = 0.023335 C=\max_{a\geq 3}\frac{1}{16}\frac{a-2}{a\log a}=0.023335\dots
  28. D N * ( x 1 , , x N ) 1 2 4 s 1 ( ( s - 1 ) log 2 ) s - 1 2 log s - 1 2 N N D_{N}^{*}(x_{1},\ldots,x_{N})\geq\frac{1}{2^{4s}}\frac{1}{((s-1)\log 2)^{\frac% {s-1}{2}}}\frac{\log^{\frac{s-1}{2}}N}{N}
  29. D N * ( x 1 , , x N ) C ( ln N ) s N . D_{N}^{*}(x_{1},\ldots,x_{N})\leq C\frac{(\ln N)^{s}}{N}.
  30. r i r_{i}
  31. [ 0 , 0.5 ) [0,0.5)
  32. s i s_{i}
  33. [ 0 , 1 ) [0,1)
  34. s i = r i s_{i}=r_{i}
  35. i i
  36. s i = 0.5 + r i s_{i}=0.5+r_{i}
  37. i i
  38. s i = s i - 1 + 0.5 + r i ( mod 1 ) . s_{i}=s_{i-1}+0.5+r_{i}\;\;(\mathop{{\rm mod}}1).\,
  39. α \alpha
  40. s n = { s 0 + n α } s_{n}=\{s_{0}+n\alpha\}
  41. s n + 1 = ( s n + α ) mod 1 s_{n+1}=(s_{n}+\alpha)\bmod\,1
  42. α \alpha
  43. α \alpha
  44. μ \mu
  45. ϵ > 0 \epsilon>0
  46. D N ( ( s n ) ) = O ϵ ( N - 1 / ( μ - 1 ) + ϵ ) . D_{N}((s_{n}))=O_{\epsilon}(N^{-1/(\mu-1)+\epsilon}).
  47. N - 1 + ϵ N^{-1+\epsilon}
  48. c c
  49. c = 5 - 1 2 0.618034. c=\frac{\sqrt{5}-1}{2}\approx 0.618034.
  50. c = 2 - 1 0.414214. c=\sqrt{2}-1\approx 0.414214.\,
  51. c = 2 , 3 , 5 , 7 , 11 , c=\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{7},\sqrt{11},\ldots\,
  52. r i = mod ( a r i - 1 + c , m ) r_{i}=\bmod(ar_{i-1}+c,m)
  53. w w
  54. w w
  55. V i , i = 1 , 2 , , w V_{i},i=1,2,\dots,w
  56. i i
  57. G ( i ) G(i)
  58. s i s_{i}
  59. i i
  60. V i V_{i}
  61. n = k = 0 L - 1 d k ( n ) b k n=\sum_{k=0}^{L-1}d_{k}(n)b^{k}
  62. D N * ( g b ( 1 ) , , g b ( N ) ) C log N N , D^{*}_{N}(g_{b}(1),\dots,g_{b}(N))\leq C\frac{\log N}{N},
  63. x ( n ) = ( g b 1 ( n ) , , g b s ( n ) ) . x(n)=(g_{b_{1}}(n),\dots,g_{b_{s}}(n)).
  64. D N * ( x ( 1 ) , , x ( N ) ) C ( log N ) s N . D^{*}_{N}(x(1),\dots,x(N))\leq C^{\prime}\frac{(\log N)^{s}}{N}.
  65. x ( n ) = ( g b 1 ( n ) , , g b s - 1 ( n ) , n N ) x(n)=(g_{b_{1}}(n),\dots,g_{b_{s-1}}(n),\frac{n}{N})
  66. D N * ( x ( 1 ) , , x ( N ) ) C ( log N ) s - 1 N D^{*}_{N}(x(1),\dots,x(N))\leq C\frac{(\log N)^{s-1}}{N}