wpmath0000014_0

(2+1)-dimensional_topological_gravity.html

  1. S O ( 2 , 2 ) SO(2,2)
  2. S O ( 3 , 1 ) SO(3,1)

(419624)_2010_SO16.html

  1. < 0.084 <0.084

1093_(number).html

  1. 1093 = 1111111 3 = 3 6 + 3 5 + 3 4 + 3 3 + 3 2 + 3 1 + 3 0 = 3 7 - 1 2 . 1093=1111111_{3}=3^{6}+3^{5}+3^{4}+3^{3}+3^{2}+3^{1}+3^{0}=\frac{3^{7}-1}{2}\,.

2-EPT_probability_density_function.html

  1. \mathbb{R}
  2. [ 0 , ) [0,\infty)
  3. ( - , 0 ) (-\infty,0)
  4. ( - , 0 ) (-\infty,0)
  5. f ( x ) = 𝐜 N e 𝐀 N x 𝐛 N , f(x)=\,\textbf{c}_{N}e^{\,\textbf{A}_{N}x}\,\textbf{b}_{N},
  6. ( 𝐀 N , 𝐀 P ) (\,\textbf{A}_{N},\,\textbf{A}_{P})
  7. ( 𝐛 N , 𝐛 P ) (\,\textbf{b}_{N},\,\textbf{b}_{P})
  8. ( 𝐜 N , 𝐜 P ) (\,\textbf{c}_{N},\,\textbf{c}_{P})
  9. [ 0 , - ) [0,-\infty)
  10. f ( x ) = 𝐜 P e 𝐀 P x 𝐛 P . f(x)=\,\textbf{c}_{P}e^{\,\textbf{A}_{P}x}\,\textbf{b}_{P}.
  11. ( 𝐀 N , 𝐛 N , 𝐜 N , 𝐀 P , 𝐛 P , 𝐜 P ) (\,\textbf{A}_{N},\,\textbf{b}_{N},\,\textbf{c}_{N},\,\textbf{A}_{P},\,\textbf% {b}_{P},\,\textbf{c}_{P})
  12. \mathbb{R}

2007–08_Paris_Saint-Germain_F.C._season.html

  1. \leftarrow
  2. \rightarrow

2008–09_Paris_Saint-Germain_F.C._(Ladies)_season.html

  1. \leftarrow
  2. \rightarrow

2010–11_US_Créteil-Lusitanos_season.html

  1. \leftarrow
  2. \rightarrow

3-Isopropylmalate_dehydrogenase.html

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

3-transposition_group.html

  1. 2 n 2 ( 2 2 - 1 ) ( 2 4 - 1 ) ( 2 2 n - 1 ) 2^{n^{2}}(2^{2}-1)(2^{4}-1)\cdots(2^{2n}-1)
  2. 2 n ( n - 1 ) / 2 ( 2 2 - 1 ) ( 2 3 + 1 ) ( 2 n - ( - 1 ) n ) 2^{n(n-1)/2}(2^{2}-1)(2^{3}+1)\cdots(2^{n}-(-1)^{n})
  3. 2 × 2 n ( n - 1 ) ( 2 2 - 1 ) ( 2 4 - 1 ) ( 2 2 n - 2 - 1 ) ( 2 n 1 ) 2\times 2^{n(n-1)}(2^{2}-1)(2^{4}-1)\cdots(2^{2n-2}-1)(2^{n}\mp 1)
  4. 2 × 3 m 2 ( 3 2 - 1 ) ( 3 4 - 1 ) ( 3 2 m - 1 ) 2\times 3^{m^{2}}(3^{2}-1)(3^{4}-1)\cdots(3^{2m}-1)
  5. 2 × 3 m ( m - 1 ) ( 3 2 - 1 ) ( 3 4 - 1 ) ( 3 2 m - 2 - 1 ) ( 3 m 1 ) 2\times 3^{m(m-1)}(3^{2}-1)(3^{4}-1)\cdots(3^{2m-2}-1)(3^{m}\mp 1)
  6. S 1 = O 1 ± , ± ( 3 ) = P S U 1 ( 2 ) S_{1}=O_{1}^{\pm,\pm}(3)=PSU_{1}(2)
  7. O 2 + ( 2 ) = O 2 - , ± ( 3 ) = O 1 ± , ( 3 ) = S 2 O_{2}^{+}(2)=O_{2}^{-,\pm}(3)=O_{1}^{\pm,\mp}(3)=S_{2}
  8. O 2 + , ± ( 3 ) O_{2}^{+,\pm}(3)
  9. S p 2 ( 2 ) = P S U 2 ( 2 ) = O 2 - ( 2 ) = S 3 Sp_{2}(2)=PSU_{2}(2)=O_{2}^{-}(2)=S_{3}
  10. O 3 ± , ( 3 ) = 2 3 O_{3}^{\pm,\mp}(3)=2^{3}
  11. O 3 ± , ± ( 3 ) = P O 3 ± , ± ( 3 ) = S 4 O_{3}^{\pm,\pm}(3)=PO_{3}^{\pm,\pm}(3)=S_{4}
  12. P S U 3 ( 2 ) = 3 2 .2 1 + 2 PSU_{3}(2)=3^{2}.2^{1+2}
  13. O 4 + ( 2 ) = ( S 3 × S 3 ) .2 O_{4}^{+}(2)=(S_{3}\times S_{3}).2
  14. O 4 + ± ( 3 ) = ( S L 2 ( 3 ) * S L 2 ( 3 ) ) .2 O_{4}^{+\pm}(3)=(SL_{2}(3)*SL_{2}(3)).2
  15. S 5 = O 4 - ( 2 ) S_{5}=O_{4}^{-}(2)
  16. S 6 = S p 4 ( 2 ) = O 4 - ± ( 3 ) = P O 4 - ± ( 3 ) S_{6}=Sp_{4}(2)=O_{4}^{-\pm}(3)=PO_{4}^{-\pm}(3)
  17. O 6 + ( 2 ) = S 8 O_{6}^{+}(2)=S_{8}
  18. W ( E 6 ) = O 6 - ( 2 ) = O 5 ± ± ( 3 ) = P O 5 ± ± ( 3 ) W(E_{6})=O_{6}^{-}(2)=O_{5}^{\pm\pm}(3)=PO_{5}^{\pm\pm}(3)
  19. P O 5 ± ( 3 ) = P S U 4 ( 2 ) PO_{5}^{\pm\mp}(3)=PSU_{4}(2)
  20. W ( E 7 ) = 2 × S p 6 ( 2 ) W(E_{7})=2\times Sp_{6}(2)
  21. W ( E 8 ) = 2. O 8 + ( 2 ) W(E_{8})=2.O_{8}^{+}(2)
  22. O 2 m + + ( 3 ) = O 2 m + - ( 3 ) = 2. P O 2 m + + ( 3 ) = 2. P O 2 m + - ( 3 ) O_{2m}^{++}(3)=O_{2m}^{+-}(3)=2.PO_{2m}^{++}(3)=2.PO_{2m}^{+-}(3)
  23. O 2 m - + ( 3 ) = O 2 m - - ( 3 ) = P O 2 m - + ( 3 ) = P O 2 m - - ( 3 ) O_{2m}^{-+}(3)=O_{2m}^{--}(3)=PO_{2m}^{-+}(3)=PO_{2m}^{--}(3)
  24. O 2 m + 1 + + ( 3 ) = O 2 m + 1 - - ( 3 ) = P O 2 m + 1 + + ( 3 ) = P O 2 m + 1 - - ( 3 ) O_{2m+1}^{++}(3)=O_{2m+1}^{--}(3)=PO_{2m+1}^{++}(3)=PO_{2m+1}^{--}(3)
  25. O 2 m + 1 - + ( 3 ) = O 2 m + 1 + - ( 3 ) = 2. P O 2 m + 1 - + ( 3 ) = 2. P O 2 m + 1 + - ( 3 ) O_{2m+1}^{-+}(3)=O_{2m+1}^{+-}(3)=2.PO_{2m+1}^{-+}(3)=2.PO_{2m+1}^{+-}(3)
  26. W ( A n ) = S n + 1 W(A_{n})=S_{n+1}
  27. W ( D n ) = 2 n - 1 . S n W(D_{n})=2^{n-1}.S_{n}

3D_reconstruction_from_multiple_images.html

  1. { P i } i = 1 N \{P^{i}\}_{i=1\ldots N}
  2. m j i P i w j m_{j}^{i}\simeq P^{i}w_{j}
  3. j t h j^{th}
  4. i t h i^{th}
  5. { m j i } \{m_{j}^{i}\}
  6. { P i } \{P^{i}\}
  7. { w j } \{w_{j}\}
  8. m j i P i w j m_{j}^{i}\simeq P^{i}w_{j}
  9. { P i } \{P^{i}\}
  10. { w j } \{w_{j}\}
  11. { P i T } \{P^{i}T\}
  12. { T - 1 w j } \{T^{-1}w_{j}\}
  13. K = A A K=AA^{\top}
  14. K = [ k 1 k 2 k 3 k 2 k 4 k 5 k 3 k 5 1 ] K=\begin{bmatrix}k_{1}&k_{2}&k_{3}\\ k_{2}&k_{4}&k_{5}\\ k_{3}&k_{5}&1\\ \end{bmatrix}
  15. F F
  16. F = D U V F=DUV^{\top}
  17. F i j {F}_{i}j
  18. A i {A}_{i}
  19. A j {A}_{j}
  20. n n
  21. A i A_{i}
  22. m m
  23. P j , j = 1 , , m . P_{j},j=1,\ldots,m.
  24. a i j a_{ij}
  25. i t h i^{th}
  26. j t h j^{th}
  27. 2 n m 2nm
  28. 11 m + 3 n 11m+3n
  29. a i j P j A i i = 1 , n , j = 1 , m a_{ij}\sim P_{j}A_{i}\qquad i=1,\ldots n,~{}~{}j=1,\ldots m
  30. P j P_{j}
  31. P j H - 1 P_{j}H^{-1}
  32. A i A_{i}
  33. H A i HA_{i}
  34. Π {\Pi}_{\infty}

5-cell_honeycomb.html

  1. A ~ 4 {\tilde{A}}_{4}
  2. A ~ 3 {\tilde{A}}_{3}
  3. C ~ 2 {\tilde{C}}_{2}
  4. A ~ 4 {\tilde{A}}_{4}
  5. 4 * {}^{*}_{4}
  6. A ~ 4 {\tilde{A}}_{4}
  7. A ~ 4 {\tilde{A}}_{4}
  8. A ~ 4 {\tilde{A}}_{4}
  9. A ~ 4 {\tilde{A}}_{4}
  10. A ~ 4 {\tilde{A}}_{4}
  11. A ~ 4 {\tilde{A}}_{4}
  12. 4 * {}^{*}_{4}

5-simplex_honeycomb.html

  1. A ~ 5 {\tilde{A}}_{5}
  2. A ~ 5 {\tilde{A}}_{5}
  3. 5 2 {}^{2}_{5}
  4. 5 3 {}^{3}_{5}
  5. 5 * {}^{*}_{5}
  6. 5 6 {}^{6}_{5}
  7. A ~ 5 {\tilde{A}}_{5}
  8. C ~ 3 {\tilde{C}}_{3}

500_Keys.html

  1. Y = r 3 / 3 Y=r^{3}/3

56-bit_encryption.html

  1. 2 56 2^{56}

6-simplex_honeycomb.html

  1. A ~ 6 {\tilde{A}}_{6}
  2. A ~ 6 {\tilde{A}}_{6}
  3. 6 * {}^{*}_{6}
  4. 6 7 {}^{7}_{6}
  5. A ~ 6 {\tilde{A}}_{6}
  6. C ~ 3 {\tilde{C}}_{3}

A._H._Lightstone.html

  1. a . a 1 a 2 ; a H - 1 a H a H + 1 . a.a_{1}a_{2}\ldots;\ldots a_{H-1}a_{H}a_{H+1}\ldots\,.
  2. a H a_{H}
  3. H H
  4. 10 - H 10^{-H}
  5. 0. 999 9 H 0.\underbrace{999\ldots 9}_{H}\,

Abel's_irreducibility_theorem.html

  1. 2 \sqrt{2}
  2. 2 \sqrt{2}

Above_threshold_ionization.html

  1. E k E_{k}
  2. E s = ( n + s ) ω - W , E_{s}=(n+s)\hbar\omega-W,
  3. n n
  4. W W
  5. E s E_{s}
  6. ω \omega

Absorbing_Markov_chain.html

  1. P = ( Q R 𝟎 I r ) , P=\left(\begin{array}[]{cc}Q&R\\ \mathbf{0}&I_{r}\end{array}\right),
  2. N = k = 0 Q k = ( I t - Q ) - 1 , N=\sum_{k=0}^{\infty}Q^{k}=(I_{t}-Q)^{-1},
  3. N 2 = N ( 2 N dg - I t ) - N sq , N_{2}=N(2N_{\operatorname{dg}}-I_{t})-N_{\operatorname{sq}},
  4. 𝐭 = N 𝟏 , \mathbf{t}=N\mathbf{1},
  5. ( 2 N - I t ) 𝐭 - 𝐭 sq , (2N-I_{t})\mathbf{t}-\mathbf{t}_{\operatorname{sq}},
  6. H = ( N - I t ) N dg - 1 . H=(N-I_{t})N_{\operatorname{dg}}^{-1}.
  7. B = N R . B=NR.
  8. P = [ 1 / 2 1 / 2 0 0 0 1 / 2 1 / 2 0 1 / 2 0 0 1 / 2 0 0 0 1 ] . P=\begin{bmatrix}1/2&1/2&0&0\\ 0&1/2&1/2&0\\ 1/2&0&0&1/2\\ 0&0&0&1\end{bmatrix}.
  9. N = ( I - Q ) - 1 = ( [ 1 0 0 0 1 0 0 0 1 ] - [ 1 / 2 1 / 2 0 0 1 / 2 1 / 2 1 / 2 0 0 ] ) - 1 = [ 1 / 2 - 1 / 2 0 0 1 / 2 - 1 / 2 - 1 / 2 0 1 ] - 1 = [ 4 4 2 2 4 2 2 2 2 ] . N=(I-Q)^{-1}=\left(\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}-\begin{bmatrix}1/2&1/2&0\\ 0&1/2&1/2\\ 1/2&0&0\end{bmatrix}\right)^{-1}=\begin{bmatrix}1/2&-1/2&0\\ 0&1/2&-1/2\\ -1/2&0&1\end{bmatrix}^{-1}=\begin{bmatrix}4&4&2\\ 2&4&2\\ 2&2&2\end{bmatrix}.
  10. 𝐭 = N 𝟏 = [ 4 4 2 2 4 2 2 2 2 ] [ 1 1 1 ] = [ 10 8 6 ] . \mathbf{t}=N\mathbf{1}=\begin{bmatrix}4&4&2\\ 2&4&2\\ 2&2&2\end{bmatrix}\begin{bmatrix}1\\ 1\\ 1\end{bmatrix}=\begin{bmatrix}10\\ 8\\ 6\end{bmatrix}.

Absorption_(logic).html

  1. P P
  2. Q Q
  3. P P
  4. P P
  5. Q Q
  6. Q Q
  7. P P
  8. P Q P ( P and Q ) \frac{P\to Q}{\therefore P\to(P\and Q)}
  9. P Q P\to Q
  10. P ( P and Q ) P\to(P\and Q)
  11. P Q P ( P and Q ) P\to Q\vdash P\to(P\and Q)
  12. \vdash
  13. P ( P and Q ) P\to(P\and Q)
  14. ( P Q ) (P\rightarrow Q)
  15. ( P Q ) ( P ( P and Q ) ) (P\to Q)\leftrightarrow(P\to(P\and Q))
  16. P P
  17. Q Q
  18. P P\,\!
  19. Q Q\,\!
  20. P Q P\rightarrow Q
  21. P P and Q P\rightarrow P\and Q
  22. P Q P\rightarrow Q
  23. ¬ P Q \neg PQ
  24. ¬ P P \neg PP
  25. ( ¬ P P ) and ( ¬ P Q ) (\neg PP)\and(\neg PQ)
  26. ¬ P ( P and Q ) \neg P(P\and Q)
  27. P ( P and Q ) P\rightarrow(P\and Q)

Acceleration_voltage.html

  1. V V_{\parallel}
  2. β c \beta c
  3. V ( β ) = 1 q e s F L ( s , t ) d s = 1 q e s F L ( s , t = s β c ) d s V_{\parallel}(\beta)=\frac{1}{q}\vec{e}_{s}\cdot\int\vec{F}_{L}(s,t)\,\mathrm{% d}s=\frac{1}{q}\vec{e}_{s}\cdot\int\vec{F}_{L}(s,t=\frac{s}{\beta c})\,\mathrm% {d}s
  4. E , B \vec{E},\vec{B}
  5. F L \vec{F}_{L}
  6. exp ( i ω t ) \exp(i\omega t)
  7. V ( β ) = 1 q e s F L ( s ) exp ( i ω β c s ) d s = 1 q e s F L ( s ) exp ( i k β s ) d s V_{\parallel}(\beta)=\frac{1}{q}\vec{e}_{s}\cdot\int\vec{F}_{L}(s)\exp\left(i% \frac{\omega}{\beta c}s\right)\,\mathrm{d}s=\frac{1}{q}\vec{e}_{s}\cdot\int% \vec{F}_{L}(s)\exp\left(ik_{\beta}s\right)\,\mathrm{d}s
  8. k β = ω β c k_{\beta}=\frac{\omega}{\beta c}
  9. V ( β ) = E s ( s ) exp ( i k β s ) d s V_{\parallel}(\beta)=\int E_{s}(s)\exp\left(ik_{\beta}s\right)\,\mathrm{d}s
  10. V V_{\parallel}
  11. s = 0 s=0
  12. ϕ \phi
  13. E s ( s , t ) = E s ( s ) exp ( i ω t + i ϕ ) E_{s}(s,t)=E_{s}(s)\;\exp\left(i\omega t+i\phi\right)
  14. V ( β ) = e i ϕ E s ( s ) exp ( i k β s ) d s V_{\parallel}(\beta)=e^{i\phi}\int E_{s}(s)\exp\left(ik_{\beta}s\right)\,% \mathrm{d}s
  15. | V ( β ) | |V_{\parallel}(\beta)|
  16. ϕ \phi
  17. T ( β ) = | V | V 0 T(\beta)=\frac{|V_{\parallel}|}{V_{0}}
  18. V ( β ) V_{\parallel}(\beta)
  19. V 0 = E ( s ) d s V_{0}=\int E(s)\,\mathrm{d}s
  20. | V | |V_{\parallel}|
  21. V 0 T V_{0}T
  22. x , y x,y
  23. V x , y = 1 q e x , y F L ( s ) exp ( i k β s ) d s V_{x,y}=\frac{1}{q}\vec{e}_{x,y}\cdot\int\vec{F}_{L}(s)\exp\left(ik_{\beta}s% \right)\,\mathrm{d}s
  24. V 2 ( β ) = V x 2 + V y 2 , α = arctan V ~ y V ~ x V_{\perp}^{2}(\beta)=V_{x}^{2}+V_{y}^{2},\quad\alpha=\arctan\frac{\tilde{V}_{y% }}{\tilde{V}_{x}}
  25. α \alpha
  26. [ - π / 2 , + π / 2 ] [-\pi/2,+\pi/2]
  27. α \alpha
  28. V ~ x = | V x | \tilde{V}_{x}=|V_{x}|
  29. V ~ y = V y exp ( - i arg V x ) \tilde{V}_{y}=V_{y}\cdot\exp(-i\arg V_{x})\in\mathbb{R}

Acceptance_angle_(solar_concentrator).html

  1. C max = n 2 sin 2 θ C_{\mathrm{max}}=\frac{n^{2}}{\sin^{2}\theta}
  2. C A P = C sin θ n CAP=\sqrt{C}\sin\theta\leq n
  3. C max = n sin θ C_{\mathrm{max}}=\frac{n}{\sin\theta}

Acceptance_set.html

  1. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  2. L p = L p ( Ω , , ) L^{p}=L^{p}(\Omega,\mathcal{F},\mathbb{P})
  3. L d p = L d p ( Ω , , ) L_{d}^{p}=L_{d}^{p}(\Omega,\mathcal{F},\mathbb{P})
  4. A A
  5. A L + p A\supseteq L^{p}_{+}
  6. A L - - p = A\cap L^{p}_{--}=\emptyset
  7. L - - p = { X L p : ω Ω , X ( ω ) < 0 } L^{p}_{--}=\{X\in L^{p}:\forall\omega\in\Omega,X(\omega)<0\}
  8. A L - p = { 0 } A\cap L^{p}_{-}=\{0\}
  9. A A
  10. A A
  11. d d
  12. A L d p A\subseteq L^{p}_{d}
  13. u K M u 1 A u\in K_{M}\Rightarrow u1\in A
  14. 1 1
  15. \mathbb{P}
  16. u - int K M u 1 A u\in-\mathrm{int}K_{M}\Rightarrow u1\not\in A
  17. A A
  18. M M
  19. A + u 1 A u K M A+u1\subseteq A\;\forall u\in K_{M}
  20. A + L d p ( K ) A A+L^{p}_{d}(K)\subseteq A
  21. A A
  22. K M = K M K_{M}=K\cap M
  23. K K
  24. M M
  25. m m
  26. R A R ( X ) = R ( X ) R_{A_{R}}(X)=R(X)
  27. A R A = A A_{R_{A}}=A
  28. ρ \rho
  29. A ρ = { X L p : ρ ( X ) 0 } A_{\rho}=\{X\in L^{p}:\rho(X)\leq 0\}
  30. R R
  31. A R = { X L d p : 0 R ( X ) } A_{R}=\{X\in L^{p}_{d}:0\in R(X)\}
  32. A A
  33. ρ A ( X ) = inf { u : X + u 1 A } \rho_{A}(X)=\inf\{u\in\mathbb{R}:X+u1\in A\}
  34. A A
  35. R A ( X ) = { u M : X + u 1 A } R_{A}(X)=\{u\in M:X+u1\in A\}
  36. A = { - V T : ( V t ) t = 0 T is the price of a self-financing portfolio at each time } A=\{-V_{T}:(V_{t})_{t=0}^{T}\,\text{ is the price of a self-financing % portfolio at each time}\}
  37. A = { X L p ( ) : E [ u ( X ) ] 0 } = { X L p ( ) : E [ e - θ X ] 1 } A=\{X\in L^{p}(\mathcal{F}):E[u(X)]\geq 0\}=\{X\in L^{p}(\mathcal{F}):E\left[e% ^{-\theta X}\right]\leq 1\}
  38. u ( X ) u(X)

ACE_Encrypt.html

  1. Z Z\,
  2. F 2 [ T ] F_{2}[T]\,
  3. F 2 F_{2}\,
  4. A r e m n Aremn\,
  5. r { 0 , , n - 1 } r\in\left\{0,...,n-1\right\}
  6. A r ( m o d n ) A\equiv r(modn)
  7. n > 0 n>0\,
  8. A Z A\in Z\,
  9. A r e m f Aremf\,
  10. r F 2 [ T ] r\in F_{2}[T]
  11. d e g ( r ) < d e g ( f ) deg(r)<deg(f)\,
  12. A r ( m o d f ) A\equiv r(modf)
  13. A , f F 2 [ T ] , f 0 A,f\in F_{2}[T],f\neq 0\,
  14. A A^{\ast}\,
  15. A n A^{n}\,
  16. x A L ( x ) x\in A^{\ast}L(x)
  17. x x\,
  18. λ A \lambda_{A}\,
  19. x , y A x,y\in A^{\ast}
  20. x | | y x||y\,
  21. x x\,
  22. y y\,
  23. b = def { 0 , 1 } b\stackrel{\mathrm{def}}{=}\left\{0,1\right\}
  24. b , b n 1 , ( b n 1 ) n 2 , b,b^{n_{1}},(b^{n_{1}})^{n_{2}},...
  25. 0 b = def 0 b 0_{b}\stackrel{\mathrm{def}}{=}0\in b
  26. 0 A n = def ( 0 A , , 0 A ) A n 0_{A^{n}}\stackrel{\mathrm{def}}{=}(0_{A},...,0_{A})\in A^{n}
  27. n > 0 n>0\,
  28. B = def b 8 B\stackrel{\mathrm{def}}{=}b^{8}
  29. W = def b 32 W\stackrel{\mathrm{def}}{=}b^{32}
  30. x A x\in A^{\ast}\,
  31. A { b , B , W } A\in\left\{b,B,W\right\}\,
  32. l > 0 l>0\,
  33. p a d l ( x ) = def { x , L ( x ) l x | | 0 A l - L ( x ) , L ( x ) < l pad_{l}(x)\stackrel{\mathrm{def}}{=}\begin{cases}x,&L(x)\geq l\\ x||0_{A^{l-L(x)}},&L(x)<l\end{cases}
  34. I s r c d s t : s r c d s t I_{src}^{dst}:src\rightarrow dst
  35. Z , F 2 [ T ] , b , B , W Z,F_{2}[T],b^{\ast},B^{\ast},W^{\ast}
  36. ( P , q , g 1 , g 2 , c , d , h 1 , h 2 , k 1 , k 2 ) (P,q,g_{1},g_{2},c,d,h_{1},h_{2},k_{1},k_{2})\,
  37. ( w , x , y , z 1 , z 2 ) (w,x,y,z_{1},z_{2})\,
  38. m m\,
  39. 1024 m 16384 1024\leq m\leq 16384
  40. q q\,
  41. P P\,
  42. P 1 ( m o d q ) P\equiv 1(modq)
  43. g 1 , g 2 , c , d , h 1 , h 2 g_{1},g_{2},c,d,h_{1},h_{2}\,
  44. { 1 , , P - 1 } \left\{1,...,P-1\right\}
  45. P P\,
  46. q q\,
  47. w , x , y , z 1 , z 2 w,x,y,z_{1},z_{2}\,
  48. { 0 , , q - 1 } \left\{0,...,q-1\right\}
  49. k 1 , k 2 k_{1},k_{2}\,
  50. B B^{\ast}
  51. L ( k 1 ) = 20 l + 64 L(k_{1})=20l^{\prime}+64
  52. L ( k 2 ) = 32 l / 16 + 40 L(k_{2})=32\left\lceil l/16\right\rceil+40
  53. l = m / 8 l=\left\lceil m/8\right\rceil
  54. l = L b ( ( 2 l / 4 + 4 ) / 16 ) l^{\prime}=L_{b}(\left\lceil(2\left\lceil l/4\right\rceil+4)/16\right\rceil)
  55. m m\,
  56. 1024 m 16384 1024\leq m\leq 16384
  57. q q\,
  58. 2 255 < q < 2 256 2^{255}<q<2^{256}\,
  59. P P\,
  60. 2 m - 1 < P < 2 m 2^{m-1}<P<2^{m}\,
  61. P 1 ( m o d q ) P\equiv 1(modq)
  62. g 1 { 2 , , P - 1 } g_{1}\in\left\{2,...,P-1\right\}
  63. g 1 q 1 ( m o d P ) g_{1}^{q}\equiv 1(modP)
  64. w { 1 , , q - 1 } w\in\left\{1,...,q-1\right\}
  65. x , y , z 1 , z 2 { 0 , , q - 1 } x,y,z_{1},z_{2}\in\left\{0,...,q-1\right\}
  66. { 1 , , P - 1 } \left\{1,...,P-1\right\}
  67. g 2 g 1 w r e m P g_{2}\leftarrow g_{1}^{w}remP
  68. c g 1 x r e m P c\leftarrow g_{1}^{x}remP
  69. d g 1 y r e m P d\leftarrow g_{1}^{y}remP
  70. h 1 g 1 z 1 r e m P h_{1}\leftarrow g_{1}^{z_{1}}remP
  71. h 2 g 1 z 2 r e m P h_{2}\leftarrow g_{1}^{z_{2}}remP
  72. k 1 B 20 l + 64 k_{1}\in B^{20l^{\prime}+64}
  73. k 2 B 2 l / 16 + 40 k_{2}\in B^{2\left\lceil l/16\right\rceil+40}
  74. l = L B ( P ) l=L_{B}(P)\,
  75. l = L B ( ( 2 l / 4 + 4 ) / 16 ) l^{\prime}=L_{B}(\left\lceil(2\left\lceil l/4\right\rceil+4)/16\right\rceil)
  76. ( ( P , q , g 1 , g 2 , c , d , h 1 , h 2 , k 1 , k 2 ) , ( w , x , y , z 1 , z 2 ) ) ((P,q,g_{1},g_{2},c,d,h_{1},h_{2},k_{1},k_{2}),(w,x,y,z_{1},z_{2}))\,
  77. ( s , u 1 , u 2 , v , e ) (s,u_{1},u_{2},v,e)\,
  78. u 1 , u 2 , v u_{1},u_{2},v\,
  79. { 1 , , P - 1 } \left\{1,...,P-1\right\}
  80. P P\,
  81. q q\,
  82. s s\,
  83. W 4 W^{4}\,
  84. e e\,
  85. B B^{\ast}\,
  86. s , u 1 , u 2 , v s,u_{1},u_{2},v\,
  87. e e\,
  88. l l\,
  89. e e\,
  90. l + 16 l / 1024 l+16\left\lceil l/1024\right\rceil
  91. C E n c o d e CEncode\,
  92. C D e c o d e CDecode\,
  93. l > 0 l>0\,
  94. s W 4 s\in W^{4}
  95. 0 u 1 , u 2 , v < 256 l 0\leq u_{1},u_{2},v<256^{l}
  96. e B e\in B^{\ast}
  97. C E n c o d e ( l , s , u 1 , u 2 , v , e ) = def I W B ( s ) || p a d l ( I Z B ( u 1 ) ) || p a d l ( I Z B ( u 2 ) ) || p a d l ( I Z B ( v ) ) || e B CEncode(l,s,u_{1},u_{2},v,e)\stackrel{\mathrm{def}}{=}I_{W^{\ast}}^{B^{\ast}}(% s)||pad_{l}(I_{Z}^{B^{\ast}}(u_{1}))||pad_{l}(I_{Z}^{B^{\ast}}(u_{2}))||pad_{l% }(I_{Z}^{B^{\ast}}(v))||e\in B^{\ast}
  98. l > 0 l>0\,
  99. ψ B \psi\in B^{\ast}
  100. L ( ψ ) 3 l + 16 L(\psi)\geq 3l+16
  101. C D e c o d e ( l , ψ ) = def ( I B W ( [ ψ ] 0 16 ) , I B Z ( [ ψ ] 16 16 + l ) , I B Z ( [ ψ ] 16 + l 16 + 2 l ) , I B Z ( [ ψ ] 16 + 2 l 16 + 3 l ) , [ ψ ] 16 + 3 l L ( ψ ) ) W 4 × Z × Z × Z × B CDecode(l,\psi)\stackrel{\mathrm{def}}{=}(I_{B^{\ast}}^{W^{\ast}}(\Bigl[\psi% \Bigr]_{0}^{16}),I_{B^{\ast}}^{Z}(\Bigl[\psi\Bigr]_{16}^{16+l}),I_{B^{\ast}}^{% Z}(\Bigl[\psi\Bigr]_{16+l}^{16+2l}),I_{B^{\ast}}^{Z}(\Bigl[\psi\Bigr]_{16+2l}^% {16+3l}),\Bigl[\psi\Bigr]_{16+3l}^{L(\psi)})\in W^{4}\times Z\times Z\times Z% \times B^{\ast}
  102. ( P , q , g 1 , g 2 , c , d , h 1 , h 2 , k 1 , k 2 ) (P,q,g_{1},g_{2},c,d,h_{1},h_{2},k_{1},k_{2})\,
  103. M B M\in B^{\ast}\,
  104. ψ \psi
  105. M M\,
  106. r { 0 , , q - 1 } r\in\left\{0,...,q-1\right\}
  107. s W 4 s\in W^{4}\,
  108. u 1 g 1 r r e m P u_{1}\leftarrow g_{1}^{r}remP
  109. u 2 g 2 r r e m P u_{2}\leftarrow g_{2}^{r}remP
  110. α U O W H a s h ( k 1 , L B ( P ) , s , u 1 , u 2 ) Z \alpha\ \leftarrow UOWHash^{\prime}(k_{1},L_{B}(P),s,u_{1},u_{2})\in Z\,
  111. 0 < α < 2 160 0<\alpha\ <2^{160}\,
  112. v c r d α r r e m P v\leftarrow c^{r}d^{\alpha\ r}remP\,
  113. h 1 ~ h 1 r r e m P \tilde{h_{1}}\leftarrow h_{1}^{r}remP
  114. h 2 ~ h 2 r r e m P \tilde{h_{2}}\leftarrow h_{2}^{r}remP
  115. k E S H a s h ( k , L B ( P ) , s , u 1 , u 2 , h 1 ~ , h 2 ~ ) W 8 k\leftarrow ESHash(k,L_{B}(P),s,u_{1},u_{2},\tilde{h_{1}},\tilde{h_{2}})\in W^% {8}\,
  116. e S E n c ( k , s , 1024 , M ) e\leftarrow SEnc(k,s,1024,M)
  117. ψ C E n c o d e ( L B ( P ) , s , u 1 , u 2 , v , e ) \psi\ \leftarrow CEncode(L_{B}(P),s,u_{1},u_{2},v,e)
  118. ψ \psi
  119. M B M\in B^{\ast}\,
  120. M 1 , , M t M_{1},...,M_{t}\,
  121. E i E_{i}\,
  122. e = E 1 || C 1 || || E t || C t e=E_{1}||C_{1}||...||E_{t}||C_{t}\,
  123. L ( e ) = L ( M ) + 16 L ( M ) / m L(e)=L(M)+16\left\lceil L(M)/m\right\rceil
  124. L ( M ) = 0 L(M)=0\,
  125. L ( e ) = 0 L(e)=0\,
  126. ( k , s , M , m ) W 8 × W 4 × Z × B (k,s,M,m)\in W^{8}\times W^{4}\times Z\times B^{\ast}\,
  127. m > 0 m>0\,
  128. e B l e\in B^{l}
  129. l = L ( M ) + 16 L ( N ) / m l=L(M)+16\left\lceil L(N)/m\right\rceil
  130. M = λ B M=\lambda_{B}\,
  131. λ B \lambda_{B}\,
  132. g e n S t a t e I n i t G e n ( k , s ) G e n S t a t e genState\leftarrow InitGen(k,s)\in GenState
  133. k A X U A X U H a s h k_{AXU}AXUHash\,
  134. ( k A X U , g e n S t a t e ) G e n W o r d s ( ( 5 L b ( m / 64 ) + 24 ) , g e n S t a t e ) . (k_{AXU},genState)\leftarrow GenWords((5L_{b}(\left\lceil m/64\right\rceil)+24% ),genState).
  135. e λ B , i 0 e\leftarrow\lambda_{B},i\leftarrow 0
  136. i < L ( M ) i<L(M)\,
  137. r m i n ( L ( M ) - i , m ) r\leftarrow min(L(M)-i,m)
  138. ( m a s k m , g e n S t a t e ) G e n W o r d s ( 4 , g e n S t a t e ) (mask_{m},genState)\leftarrow GenWords(4,genState)
  139. ( m a s k e , g e n S t a t e ) G e n W o r d s ( r , g e n S t a t e ) (mask_{e},genState)\leftarrow GenWords(r,genState)
  140. e n c [ M ] i i + r m a s k e enc\leftarrow\Bigl[M\Bigr]_{i}^{i+r}\oplus mask_{e}
  141. i + r = L ( M ) i+r=L(M)\,
  142. l a s t B l o c k 1 lastBlock\leftarrow 1
  143. l a s t B l o c k 0 lastBlock\leftarrow 0
  144. m a c A X U H a s h ( k A X U , l a s t B l o c k , e n c ) W 4 mac\leftarrow AXUHash(k_{AXU},lastBlock,enc)\in W^{4}
  145. e e || e n c || I W B ( m a c m a s k m ) e\leftarrow e||enc||I_{W^{\ast}}^{B^{\ast}}(mac\oplus mask_{m})
  146. i i + r i\leftarrow i+r
  147. e e\,
  148. ( P , q , g 1 , g 2 , c , d , h 1 , h 2 , k 1 , k 2 ) (P,q,g_{1},g_{2},c,d,h_{1},h_{2},k_{1},k_{2})\,
  149. ( w , x , y , z 1 , z 2 ) (w,x,y,z_{1},z_{2})\,
  150. ψ B \psi\in B^{\ast}
  151. M B R e j e c t M\in B^{\ast}\cup{Reject}
  152. L ( ψ ) < 3 L B ( P ) + 16 L(\psi)<3L_{B}(P)+16\,
  153. R e j e c t Reject\,
  154. ( s , u 1 , u 2 , v , e ) C D e c o d e ( L B ( P ) , ψ ) W 4 × Z × Z × Z × B (s,u_{1},u_{2},v,e)\leftarrow CDecode(L_{B}(P),\psi)\in W^{4}\times Z\times Z% \times Z\times B^{\ast}
  155. 0 u 1 , u 2 , v < 256 l 0\leq u_{1},u_{2},v<256^{l}
  156. l = L B ( P ) l=L_{B}(P)\,
  157. u 1 P u_{1}\geq P
  158. u 2 P u_{2}\geq P
  159. v P v\geq P
  160. R e j e c t Reject\,
  161. u 1 q 1 r e m P u_{1}^{q}\neq 1remP
  162. R e j e c t Reject\,
  163. r e j e c t 0 reject\leftarrow 0\,
  164. u 2 u 1 w r e m P u_{2}\neq u_{1}^{w}remP
  165. r e j e c t 1 reject\leftarrow 1\,
  166. α U O W H a s h ( k 1 , L B ( P ) , s , u 1 , u 2 ) Z \alpha\leftarrow UOWHash^{\prime}(k_{1},L_{B}(P),s,u_{1},u_{2})\in Z
  167. 0 α 2 160 0\leq\alpha\leq 2^{160}
  168. v u 1 x + α y r e m P v\neq u_{1}^{x+{\alpha}y}remP
  169. r e j e c t 1 reject\leftarrow 1\,
  170. r e j e c t = 1 reject=1\,
  171. R e j e c t Reject\,
  172. h 1 ~ u 1 z 1 r e m P \tilde{h_{1}}\leftarrow u_{1}^{z_{1}}remP
  173. h 2 ~ u 1 z 2 r e m P \tilde{h_{2}}\leftarrow u_{1}^{z_{2}}remP
  174. k E S H a s h ( k 2 , L B ( P ) , s , u 1 , h 1 ~ , h 2 ~ ) W 8 k\leftarrow ESHash(k_{2},L_{B}(P),s,u_{1},\tilde{h_{1}},\tilde{h_{2}})\in W^{8}
  175. M S D e c ( k , s , 1024 , e ) M\leftarrow SDec(k,s,1024,e)
  176. S D e c SDec\,
  177. R e j e c t Reject\,
  178. M M\,
  179. S D e c SDec\,
  180. ( k , s , m , e ) W 8 × W 4 × Z × B (k,s,m,e)\in W^{8}\times W^{4}\times Z\times B^{\ast}\,
  181. m > 0 m>0\,
  182. M B R e j e c t M\in B^{\ast}\cup{Reject}
  183. e = λ B e=\lambda_{B}\,
  184. λ B \lambda_{B}\,
  185. g e n S t a t e I n i t G e n ( k , s ) G e n S t a t e genState\leftarrow InitGen(k,s)\in GenState
  186. k A X U A X U H a s h k_{AXU}AXUHash\,
  187. ( k A X U , g e n S t a t e ) G e n W o r d s ( ( 5 L b ( m / 64 ) + 24 ) , g e n S t a t e ) . (k_{AXU},genState^{\prime})\leftarrow GenWords((5L_{b}(\left\lceil m/64\right% \rceil)+24),genState).
  188. M λ B , i 0 M\leftarrow\lambda_{B},i\leftarrow 0
  189. i < L ( e ) i<L(e)\,
  190. r m i n ( L ( e ) - i , m + 16 ) - 16 r\leftarrow min(L(e)-i,m+16)-16
  191. r 0 r\leq 0
  192. R e j e c t Reject\,
  193. ( m a s k m , g e n S t a t e ) G e n W o r d s ( 4 , g e n S t a t e ) (mask_{m},genState)\leftarrow GenWords(4,genState)
  194. ( m a s k e , g e n S t a t e ) G e n W o r d s ( r , g e n S t a t e ) (mask_{e},genState)\leftarrow GenWords(r,genState)
  195. i + r + 16 = L ( M ) i+r+16=L(M)\,
  196. l a s t b l o c k 1 lastblock\leftarrow 1
  197. l a s t b l o c k 0 lastblock\leftarrow 0
  198. m a c A X U H a s h ( k A X U , l a s t B l o c k , [ e ] i i + r ) W 4 mac\leftarrow AXUHash(k_{AXU},lastBlock,\Bigl[e\Bigr]_{i}^{i+r})\in W^{4}
  199. [ e ] r i + r i + r + 16 I W B ( m a c m a s k m ) \Bigl[e\Big]r_{i+r}^{i+r+16}\neq I_{W^{\ast}}^{B^{\ast}}(mac\oplus mask_{m})
  200. R e j e c t Reject\,
  201. M M | | ( [ e ] i i + r ) m a s k e ) M\leftarrow M||(\Bigl[e\Bigr]_{i}^{i+r})\oplus mask_{e})
  202. i i + r + 16 i\leftarrow i+r+16
  203. M M\,
  204. ( N , h , x , e , k , s ) (N,h,x,e^{\prime},k^{\prime},s)\,
  205. ( p , q , a ) (p,q,a)\,
  206. m m\,
  207. 1024 m 16384 1024\leq m\leq 16384
  208. p p\,
  209. m / 2 \left\lfloor m/2\right\rfloor
  210. ( p - 1 ) / 2 (p-1)/2\,
  211. q q\,
  212. m / 2 \left\lfloor m/2\right\rfloor
  213. ( q - 1 ) / 2 (q-1)/2\,
  214. N N\,
  215. N = p q N=pq\,
  216. m m\,
  217. m - 1 m-1\,
  218. h , x h,x\,
  219. { 1 , , N - 1 } \left\{1,...,N-1\right\}
  220. N N\,
  221. e e^{\prime}\,
  222. a a\,
  223. { 0 , , ( p - 1 ) ( q - 1 ) / 4 - 1 } \left\{0,...,(p-1)(q-1)/4-1\right\}
  224. k k^{\prime}\,
  225. B 184 B^{184}\,
  226. s s\,
  227. B 32 B^{32}\,
  228. m m\,
  229. 1024 m 16384 1024\leq m\leq 16384
  230. p , q p,q\,
  231. ( p - 1 ) / 2 (p-1)/2\,
  232. ( q - 1 ) / 2 (q-1)/2\,
  233. 2 m 1 - 1 < p < 2 m 1 2^{m_{1}-1}<p<2^{m_{1}}
  234. 2 m 2 - 1 < q < 2 m 2 2^{m_{2}-1}<q<2^{m_{2}}
  235. p q p\neq q
  236. m 1 = m / 2 m_{1}=\left\lfloor m/2\right\rfloor
  237. m 1 = m / 2 m_{1}=\left\lceil m/2\right\rceil
  238. N p q N\leftarrow pq
  239. e e^{\prime}\,
  240. 2 160 e 2 161 2^{160}\leq e^{\prime}\leq 2^{161}
  241. h { 1 , , N - 1 } h^{\prime}\in\left\{1,...,N-1\right\}
  242. g c d ( h , N ) = 1 gcd(h^{\prime},N)=1
  243. g c d ( h ± 1 , N ) = 1 gcd(h^{\prime}\pm 1,N)=1
  244. h ( h ) - 2 r e m N h\leftarrow(h^{\prime})^{-2}remN
  245. a { 0 , , ( p - 1 ) ( q - 1 ) / 4 - 1 } a\in\left\{0,...,(p-1)(q-1)/4-1\right\}
  246. x h a r e m N x\leftarrow h^{a}remN
  247. k B 184 k^{\prime}\in B^{184}\,
  248. s B 32 s\in B^{32}\,
  249. ( ( N , h , x , e , k , s ) , ( p , q , a ) ) ((N,h,x,e^{\prime},k^{\prime},s),(p,q,a))\,
  250. ( d , w , y , y , k ~ ) (d,w,y,y^{\prime},\tilde{k})
  251. d d\,
  252. B 64 B^{64}\,
  253. w w\,
  254. 2 160 w 2 161 2^{160}\leq w\leq 2^{161}
  255. y , y y,y^{\prime}\,
  256. { 1 , , N - 1 } \left\{1,...,N-1\right\}
  257. k ~ \tilde{k}\,
  258. B B^{\ast}\,
  259. L ( k ~ ) = 64 + 20 L B ( ( L ( M ) + 8 ) / 64 ) L(\tilde{k})=64+20L_{B}(\left\lceil(L(M)+8)/64\right\rceil)
  260. M M\,
  261. S E n c o d e SEncode\,
  262. S D e c o d e SDecode\,
  263. l > 0 l>0\,
  264. d B 64 d\in B^{64}
  265. 0 w 256 21 0\leq w\leq 256^{21}
  266. 0 y , y < 256 l 0\leq y,y^{\prime}<256^{l}
  267. k ~ B \tilde{k}\in B^{\ast}
  268. S E n c o d e ( l , d , w , y , y , k ~ ) = def d || p a d 21 ( I Z B ( w ) ) || p a d l ( I Z B ( y ) ) || p a d l ( I Z B ( y ) ) || k ~ B SEncode(l,d,w,y,y^{\prime},\tilde{k})\stackrel{\mathrm{def}}{=}d||pad_{21}(I_{% Z}^{B^{\ast}}(w))||pad_{l}(I_{Z}^{B^{\ast}}(y))||pad_{l}(I_{Z}^{B^{\ast}}(y^{% \prime}))||\tilde{k}\in B^{\ast}
  269. l > 0 l>0\,
  270. σ B \sigma\in B^{\ast}
  271. L ( σ ) 2 l + 53 L(\sigma)\geq 2l+53
  272. C S e c o d e ( l , σ ) = def ( [ σ ] 0 64 , I B Z ( [ σ ] 64 85 ) , I B Z ( [ σ ] 85 85 + l ) , I B Z ( [ σ ] 85 + l 85 + 2 l ) , [ σ ] 85 + 2 l L ( σ ) ) B 64 × Z × Z × Z × B CSecode(l,\sigma)\stackrel{\mathrm{def}}{=}(\Bigl[\sigma\Bigr]_{0}^{64},I_{B^{% \ast}}^{Z}(\Bigl[\sigma\Bigr]_{64}^{85}),I_{B^{\ast}}^{Z}(\Bigl[\sigma\Bigr]_{% 85}^{85+l}),I_{B^{\ast}}^{Z}(\Bigl[\sigma\Bigr]_{85+l}^{85+2l}),\Bigl[\sigma% \Bigr]_{85+2l}^{L(\sigma)})\in B^{64}\times Z\times Z\times Z\times B^{\ast}
  273. ( N , h , x , e , k , s ) (N,h,x,e^{\prime},k^{\prime},s)\,
  274. ( p , q , a ) (p,q,a)\,
  275. M B M\in B^{\ast}\,
  276. 0 L ( M ) 2 64 0\leq L(M)\leq 2^{64}
  277. σ B \sigma\in B^{\ast}\,
  278. k ~ B 20 m + 64 \tilde{k}\in B^{20m+64}
  279. m = L b ( ( L ( M ) + 8 ) / 64 ) m=L_{b}(\left\lceil(L(M)+8)/64\right\rceil)
  280. m h I W Z ( U O W H a s h ′′ ( k ~ , M ) ) m_{h}\leftarrow I_{W^{\ast}}^{Z}(UOWHash^{\prime\prime}(\tilde{k},M))
  281. y ~ { 1 , , N - 1 } \tilde{y}\in\left\{1,...,N-1\right\}
  282. y y ~ 2 r e m N y^{\prime}\leftarrow\tilde{y}^{2}remN
  283. x ( y ) r h m h r e m N x^{\prime}\leftarrow(y^{\prime})^{r^{\prime}}h^{m_{h}}remN
  284. e e\,
  285. 2 160 e 2 161 2^{160}\leq e\leq 2^{161}
  286. ( w , d ) (w,d)\,
  287. ( e , w , d ) G e n C e r t P r i m e ( s ) (e,w,d)\leftarrow GenCertPrime(s)\,
  288. e e e\neq e^{\prime}\,
  289. r U O W H a s h ′′′ ( k , L B ( N ) , x , k ~ ) Z r\leftarrow UOWHash^{\prime\prime\prime}(k^{\prime},L_{B}(N),x^{\prime},\tilde% {k})\in Z
  290. 0 r < 2 160 0\leq r<2^{160}
  291. y h b r e m N y\leftarrow h^{b}remN
  292. b e - 1 ( a - r ) r e m ( p q ) b\leftarrow e^{-1}(a-r)rem(p^{\prime}q^{\prime})
  293. p = ( p - 1 ) / 2 p^{\prime}=(p-1)/2
  294. q = ( q - 1 ) / 2 q^{\prime}=(q-1)/2
  295. σ S E n c o d e ( L B ( N ) , d , w , y , y , k ~ ) \sigma\leftarrow SEncode(L_{B}(N),d,w,y,y^{\prime},\tilde{k})
  296. σ \sigma\,

Acid_strength.html

  1. HA ( aq ) H ( aq ) + + A ( aq ) - \mathrm{HA_{(aq)}\,\rightleftharpoons\,H^{+}\,_{(aq)}+\,A^{-}\,_{(aq)}}
  2. K a = [ H + ] [ A - ] [ HA ] \mathrm{K_{a}\,=\,\frac{[H^{+}\,][A^{-}\,]}{[HA]}}
  3. F F
  4. K a K_{a}
  5. K a K_{a}
  6. K a K_{a}
  7. K a = [ H + ] [ A - ] [ H A ] = x 2 F - x K_{a}={{[H^{+}][A^{-}]}\over{[HA]}}={{x^{2}}\over{F-x}}
  8. x 2 + K a x - K a F = 0 x^{2}+K_{a}x-K_{a}F=0
  9. p H = - l o g ( x ) pH=-log(x)
  10. K a = x 2 F K_{a}={{x^{2}}\over{F}}
  11. x = K a F = [ H + ] x=\sqrt{K_{a}F}=\left[H^{+}\right]
  12. p H = - l o g K a F pH=-log\sqrt{K_{a}F}
  13. x 2 + 1 × 10 - 5 x - 1 × 10 - 6 = 0 x^{2}+1\times 10^{-5}x-1\times 10^{-6}=0
  14. p H = - l o g 10 - 5 0.1 = 3.00 pH=-log\sqrt{10^{-5}0.1}=3.00
  15. x 2 + 10 - 5 x - 5 × 10 - 9 = 0 x^{2}+10^{-5}x-5\times 10^{-9}=0
  16. p H = - l o g 10 - 5 ( 5 × 10 - 4 ) = 4.15 pH=-log\sqrt{10^{-5}\left(5\times 10^{-4}\right)}=4.15

Activating_function.html

  1. f n f_{n}
  2. f n = 1 / c ( V n - 1 e - V n e R n - 1 / 2 + R n / 2 + V n + 1 e - V n e R n + 1 / 2 + R n / 2 + ) f_{n}=1/c\left(\frac{V^{e}_{n-1}-V^{e}_{n}}{R_{n-1}/2+R_{n}/2}+\frac{V^{e}_{n+% 1}-V^{e}_{n}}{R_{n+1}/2+R_{n}/2}+...\right)
  3. c c
  4. V n e V^{e}_{n}
  5. n n
  6. R n R_{n}
  7. n n
  8. V m V^{m}
  9. d V n m d t = [ - i i o n , n + d Δ x 4 ρ i L ( V n - 1 m - 2 V n m + V n + 1 m Δ x 2 + V n - 1 e - 2 V n e + V n + 1 e Δ x 2 ) ] / c \frac{dV^{m}_{n}}{dt}=\left[-i_{ion,n}+\frac{d\Delta x}{4\rho_{i}L}\cdot\left(% \frac{V^{m}_{n-1}-2V^{m}_{n}+V^{m}_{n+1}}{\Delta x^{2}}+\frac{V^{e}_{n-1}-2V^{% e}_{n}+V^{e}_{n+1}}{\Delta x^{2}}\right)\right]/c
  10. d d
  11. Δ x \Delta x
  12. L L
  13. ρ i \rho_{i}
  14. c c
  15. i i o n i_{ion}
  16. f n = d Δ x 4 ρ i L c V n - 1 e - 2 V n e + V n + 1 e Δ x 2 f_{n}=\frac{d\Delta x}{4\rho_{i}Lc}\frac{V^{e}_{n-1}-2V^{e}_{n}+V^{e}_{n+1}}{% \Delta x^{2}}
  17. L = Δ x L=\Delta x
  18. Δ x 0 \Delta x\to 0
  19. f = d 4 ρ i c δ 2 V e δ x 2 f=\frac{d}{4\rho_{i}c}\cdot\frac{\delta^{2}V^{e}}{\delta x^{2}}
  20. f f
  21. f f

Active_learning_(machine_learning).html

  1. T T
  2. T T
  3. i i
  4. T T
  5. 𝐓 K , i \mathbf{T}_{K,i}
  6. 𝐓 U , i \mathbf{T}_{U,i}
  7. 𝐓 C , i \mathbf{T}_{C,i}
  8. T U , i T_{U,i}
  9. T C , i T_{C,i}
  10. W W
  11. T U , i T_{U,i}
  12. W W
  13. n n
  14. W W
  15. T C , i T_{C,i}
  16. W W
  17. W W

Actuarial_polynomials.html

  1. n a n ( β ) ( x ) n ! t n = exp ( β t + x ( 1 - e t ) ) \displaystyle\sum_{n}\frac{a_{n}^{(\beta)}(x)}{n!}t^{n}=\exp(\beta t+x(1-e^{t}))

Adams–Williamson_equation.html

  1. v p = K + ( 4 / 3 ) μ ρ v s = μ ρ . \begin{aligned}\displaystyle v_{p}&\displaystyle=\sqrt{\frac{K+(4/3)\mu}{\rho}% }\\ \displaystyle v_{s}&\displaystyle=\sqrt{\frac{\mu}{\rho}}.\end{aligned}
  2. Φ = v p 2 - 4 3 v s 2 = K ρ . \Phi=v_{p}^{2}-\frac{4}{3}v_{s}^{2}=\frac{K}{\rho}.
  3. K = - V d P d V , K=-V\frac{dP}{dV},
  4. K = ρ d P d ρ . K=\rho\frac{dP}{d\rho}.
  5. d P d r = - ρ ( r ) g ( r ) , \frac{dP}{dr}=-\rho(r)g(r),
  6. d ρ d r = - ρ ( r ) g ( r ) Φ ( r ) . \frac{d\rho}{dr}=-\frac{\rho(r)g(r)}{\Phi(r)}.
  7. ln ( ρ ρ 0 ) = - r 0 r g ( r ) Φ ( r ) d r , \ln\left(\frac{\rho}{\rho_{0}}\right)=-\int_{r_{0}}^{r}\frac{g(r)}{\Phi(r)}dr,

Adaptive_comparative_judgement.html

  1. l o g o d d s ( A b e a t s B | v a , v b ) = v a - v b logodds(AbeatsB|v_{a},v_{b})=v_{a}-v_{b}

Adaptive_Gabor_representation.html

  1. s ( t ) = m = - n = - C m , n h ( t - m T ) e j n t Ω s(t)=\sum_{m=-\infty}^{\infty}\sum_{n=-\infty}^{\infty}C_{m,n}h(t-mT)e^{jnt\Omega}
  2. h ( t ) = ( α π ) 1 4 e ( - α 2 t 2 ) h(t)=\left(\frac{\alpha}{\pi}\right)^{\frac{1}{4}}e^{\left(-\frac{\alpha}{2}t^% {2}\right)}
  3. C m , n C_{m,n}
  4. γ ( t ) \gamma(t)
  5. C m , n = s ( t ) γ * ( t - m T ) e - j n t Ω d t . C_{m,n}=\int s(t)\gamma^{*}(t-mT)e^{-jnt\Omega}\,dt.
  6. T T
  7. Ω \Omega
  8. T Ω 2 π T\Omega\leqq 2\pi\,
  9. C m , n C_{m,n}
  10. s ( t ) = p B p h p ( t ) s\left(t\right)=\sum_{p}B_{p}h_{p}(t)
  11. B p B_{p}
  12. h p h_{p}
  13. B p = s , h p B_{p}=\left\langle s,h_{p}\right\rangle\,
  14. B p B_{p}
  15. { h p ( t ) } \left\{h_{p}(t)\right\}
  16. s 0 ( t ) = s ( t ) s_{0}\left(t\right)=s\left(t\right)
  17. h 0 ( t ) h_{0}\left(t\right)
  18. s 0 ( t ) s_{0}\left(t\right)
  19. | B p | 2 = max h | s p ( t ) , h p ( t ) | 2 \left|B_{p}\right|^{2}=\max_{h}\left|\left\langle s_{p}(t),h_{p}(t)\right% \rangle\right|^{2}
  20. s 1 ( t ) = s 0 ( t ) - B 0 h 0 ( t ) , s_{1}\left(t\right)=s_{0}\left(t\right)-B_{0}h_{0}\left(t\right),
  21. s p ( t ) s_{p}\left(t\right)
  22. B p = s p ( t ) , h p ( t ) B_{p}=\left\langle s_{p}(t),h_{p}(t)\right\rangle
  23. h p ( t ) h_{p}\left(t\right)
  24. h p ( t ) h_{p}\left(t\right)
  25. B p B_{p}
  26. s p ( t ) 2 = s p + 1 ( t ) 2 + | B p | 2 , \left\|s_{p}(t)\right\|^{2}=\left\|s_{p+1}(t)\right\|^{2}+\left|B_{p}\right|^{% 2},
  27. s ( t ) 2 = p = o | B p | 2 , \left\|s(t)\right\|^{2}=\sum_{p=o}^{\infty}\left|B_{p}\right|^{2},
  28. h p ( t ) = ( α π ) 1 4 e - α 2 ( t - T p ) 2 e j t Ω p , h_{p}(t)=\left(\frac{\alpha}{\pi}\right)^{\frac{1}{4}}e^{-\frac{\alpha}{2}(t-T% _{p})^{2}}e^{jt\Omega_{p}},
  29. ( T p , Ω p ) \left(T_{p},\Omega_{p}\right)
  30. α p - 1 \alpha_{p}^{-1}
  31. ( T p , Ω p ) \left(T_{p},\Omega_{p}\right)
  32. s ( t ) = p B p h p ( t ) = p B p ( α π ) 1 4 e - α 2 ( t - T p ) 2 e j t Ω p s\left(t\right)=\sum_{p}B_{p}h_{p}(t)=\sum_{p}B_{p}\left(\frac{\alpha}{\pi}% \right)^{\frac{1}{4}}e^{-\frac{\alpha}{2}(t-T_{p})^{2}}e^{jt\Omega_{p}}

Adequality.html

  1. p ( x ) p(x)
  2. p ( x ) p(x)
  3. p ( x + e ) p(x+e)
  4. e , e,
  5. e . e.
  6. p ( x ) = b x - x 2 p(x)=bx-x^{2}
  7. b x - x 2 bx-x^{2}
  8. b ( x + e ) - ( x + e ) 2 = b x - x 2 + b e - 2 e x - e 2 b(x+e)-(x+e)^{2}=bx-x^{2}+be-2ex-e^{2}
  9. \backsim
  10. b x - x 2 b x - x 2 + b e - 2 e x - e 2 . bx-x^{2}\backsim bx-x^{2}+be-2ex-e^{2}.
  11. e e
  12. b 2 x + e . b\backsim 2x+e.
  13. e e
  14. x = b / 2 x=b/2
  15. \backsim
  16. \scriptstyle\sim
  17. \scriptstyle\approx
  18. \scriptstyle\backsim
  19. \scriptstyle\approx
  20. f ( A ) f ( A + E ) \scriptstyle f(A){\sim}f(A+E)
  21. \scriptstyle\approx
  22. \scriptstyle\approx
  23. b \scriptstyle b
  24. x \scriptstyle x
  25. b - x \scriptstyle b-x
  26. x ( b - x ) = b x - x 2 \scriptstyle x(b-x)=bx-x^{2}
  27. 2 b \scriptstyle 2b
  28. x + e \scriptstyle x+e
  29. b ( x + e ) - ( x + e ) 2 = b x + b e - x 2 - 2 x e - e 2 b x - x 2 . \scriptstyle b(x+e)-(x+e)^{2}=bx+be-x^{2}-2xe-e^{2}\;\sim\;bx-x^{2}.
  30. b - 2 x - e 0. \scriptstyle b-2\,x-e\;\sim\;0.
  31. x = b 2 \scriptstyle x=\frac{b}{2}
  32. b x - x 2 \scriptstyle bx-x^{2}
  33. \scriptstyle\approx

Adiabatic_accessibility.html

  1. δ Q = d U - p i d V i \delta Q=dU-\sum p_{i}dV_{i}
  2. S S
  3. d S dS
  4. δ Q \delta Q
  5. Y Y
  6. X X
  7. X Y X\prec Y
  8. X X
  9. Y Y
  10. \prec
  11. \prec
  12. λ \lambda
  13. λ X \lambda X
  14. X Y X\prec Y
  15. Y X Y\prec X
  16. X A Y X\overset{\underset{\mathrm{A}}{}}{\sim}Y
  17. X A X X\overset{\underset{\mathrm{A}}{}}{\sim}X
  18. X Y X\prec Y
  19. Y Z Y\prec Z
  20. X Z X\prec Z
  21. X X X\prec X^{\prime}
  22. Y Y Y\prec Y^{\prime}
  23. ( X , Y ) ( X , Y ) (X,Y)\prec(X^{\prime},Y^{\prime})
  24. λ > 0 \lambda>0
  25. X Y X\prec Y
  26. λ X λ Y \lambda X\prec\lambda Y
  27. X A ( ( 1 - λ ) X , λ X ) X\overset{\underset{\mathrm{A}}{}}{\sim}((1-\lambda)X,\lambda X)
  28. 0 < λ < 1 0<\lambda<1
  29. lim ϵ 0 [ ( X , ϵ Z 0 ) ( Y , ϵ Z 1 ) ] \lim_{\epsilon\to 0}[(X,\epsilon Z_{0})\prec(Y,\epsilon Z_{1})]
  30. X Y X\prec Y
  31. S ( X ) S ( Y ) S(X)\leq S(Y)
  32. X Y X\prec Y
  33. S ( X ) = S ( Y ) S(X)=S(Y)
  34. X A Y X\overset{\underset{\mathrm{A}}{}}{\sim}Y
  35. X 0 X_{0}
  36. X 1 X_{1}
  37. X 0 X 1 X_{0}\prec X_{1}
  38. X 0 X X 1 X_{0}\prec X\prec X_{1}
  39. S ( X ) = sup ( λ : ( ( 1 - λ ) X 0 , λ X 1 ) X ) S(X)=\sup(\lambda:((1-\lambda)X_{0},\lambda X_{1})\prec X)

Admissible_trading_strategy.html

  1. d d
  2. x d x\in\mathbb{R}^{d}
  3. x T S ¯ = x T S 1 + r x^{T}\bar{S}=x^{T}\frac{S}{1+r}
  4. S S
  5. r r
  6. S ¯ \bar{S}

Affine_q-Krawtchouk_polynomials.html

  1. K n a f f ( q - x ; p ; N ; q ) = 2 ϕ 1 ( q - n 0 q - x p q q - N ; q , q ) K^{aff}_{n}(q^{-x};p;N;q)=\;_{2}\phi_{1}\left(\begin{matrix}q^{-n}&0&q^{-x}\\ pq&q^{-N}\end{matrix};q,q\right)
  2. n = 0 , 1 , 2 , N n=0,1,2,\cdots N
  3. $\displaystyle$
  4. lim a 1 = K n a f f ( q x - N ; p , N | q ) = p n ( q x ; p , q ) \lim_{a\to 1}=K_{n}^{aff}(q^{x-N};p,N|q)=p_{n}(q^{x};p,q)

Aggregate_modulus.html

  1. H a = E ( 1 - v ) / [ ( 1 + v ) ( 1 - 2 v ) ] Ha=E(1-v)/[(1+v)(1-2v)]

Aharonov–Casher_effect.html

  1. φ \varphi
  2. φ A B = q P 𝐀 d 𝐱 \varphi_{AB}=\frac{q}{\hbar}\int_{P}\mathbf{A}\cdot d\mathbf{x}
  3. φ A C = 1 c 2 P ( 𝐄 × s y m b o l μ ) d 𝐱 \varphi_{AC}=\frac{1}{\hbar c^{2}}\int_{P}(\mathbf{E}\times symbol\mu)\cdot d% \mathbf{x}
  4. q q
  5. s y m b o l μ symbol\mu

Ahlfors_theory.html

  1. ( γ ) = γ ρ ( z ) | d z | , A ( D ) = D ρ 2 ( x + i y ) d x d y , z = x + i y . \ell(\gamma)=\int_{\gamma}\rho(z)\,|dz|,\quad A(D)=\int_{D}\rho^{2}(x+iy)\,dx% \,dy,\quad z=x+iy.
  2. ρ ( z ) | d z | = σ ( f ( z ) ) | f ( z ) | | d z | . \rho(z)|dz|=\sigma(f(z))|f^{\prime}(z)||dz|.\,
  3. S = A ( X ) A ( Y ) . S=\frac{A(X)}{A(Y)}.
  4. S ( D ) = A ( f - 1 ( D ) ) A ( D ) , S ( γ ) = ( f - 1 ( γ ) ) ( γ ) . S(D)=\frac{A(f^{-1}(D))}{A(D)},\quad S(\gamma)=\frac{\ell(f^{-1}(\gamma))}{% \ell(\gamma)}.
  5. | S - S ( D ) | k L , | S - S ( γ ) | k L , |S-S(D)|\leq kL,\quad|S-S(\gamma)|\leq kL,
  6. max { ρ ( X ) , 0 } S ρ ( Y ) - k L , \max\{\rho(X),0\}\geq S\rho(Y)-kL,\,
  7. ( ( D j ) ) A ( D j ) 0 , j . \frac{\ell(\partial(D_{j}))}{A(D_{j})}\to 0,\;j\to\infty.
  8. | f ( 0 ) | 1 + | f ( 0 ) | 2 c . \frac{|f^{\prime}(0)|}{1+|f(0)|^{2}}\geq c.
  9. ( ) 2 = 4 ( - e 1 ) ( - e 2 ) ( - e 3 ) . (\wp^{\prime})^{2}=4(\wp-e_{1})(\wp-e_{2})(\wp-e_{3}).

Air_core_gauge.html

  1. θ \theta
  2. θ = arctan ( x y ) \theta=\arctan\left(\frac{x}{y}\right)
  3. x x
  4. y y
  5. θ \theta

Air_Quality_Health_Index.html

  1. A Q H I = ( 1000 10.4 ) × [ ( e 0.000537 × O 3 - 1 ) + ( e 0.000871 × N O 2 - 1 ) + ( e 0.000487 × P M 2.5 - 1 ) ] AQHI=(\frac{1000}{10.4})\times[(e^{0.000537\times O_{3}}-1)+(e^{0.000871\times NO% _{2}}-1)+(e^{0.000487\times PM_{2.5}}-1)]

AIXI.html

  1. μ \mu
  2. a t a_{t}
  3. o t o_{t}
  4. r t r_{t}
  5. μ ( o t r t | a 1 o 1 r 1 a t - 1 o t - 1 r t - 1 a t ) \mu(o_{t}r_{t}|a_{1}o_{1}r_{1}...a_{t-1}o_{t-1}r_{t-1}a_{t})
  6. r t + + r m r_{t}+\ldots+r_{m}
  7. a 1 o 1 r 1 a t - 1 o t - 1 r t - 1 a_{1}o_{1}r_{1}...a_{t-1}o_{t-1}r_{t-1}
  8. arg max a t o t r t max a m o m r m [ r t + + r m ] q : U ( q , a 1 a m ) = o 1 r 1 o m r m 2 - length ( q ) , \arg\max_{a_{t}}\sum_{o_{t}r_{t}}\ldots\max_{a_{m}}\sum_{o_{m}r_{m}}[r_{t}+% \ldots+r_{m}]\sum_{q:\;U(q,a_{1}\ldots a_{m})=o_{1}r_{1}\ldots o_{m}r_{m}}2^{-% \textrm{length}(q)},
  9. μ \mu
  10. μ \mu

Akhiezer's_theorem.html

  1. f ( z ) f(z)
  2. τ τ
  3. f ( x ) 0 f(x) ≥ 0
  4. x x
  5. F F
  6. τ / 2 τ/2
  7. f ( z ) = F ( z ) F ( z ¯ ) ¯ f(z)=F(z)\overline{F(\overline{z})}
  8. | Im ( 1 / z n ) | < \sum|\operatorname{Im}(1/z_{n})|<\infty
  9. z < s u b > n z<sub>n

AKLT_model.html

  1. H ^ = j S j S j + 1 + 1 3 ( S j S j + 1 ) 2 \hat{H}=\sum_{j}\vec{S}_{j}\cdot\vec{S}_{j+1}+\frac{1}{3}(\vec{S}_{j}\cdot\vec% {S}_{j+1})^{2}
  2. S z S^{z}
  3. | Ψ = { s } Tr [ A s 1 A s 2 A s N ] | s 1 s 2 s N |\Psi\rangle=\sum_{\{s\}}\,\text{Tr}[A^{s_{1}}A^{s_{2}}\ldots A^{s_{N}}]|s_{1}% s_{2}\ldots s_{N}\rangle
  4. s j s_{j}
  5. A + = 2 3 σ + A^{+}=\sqrt{\frac{2}{3}}\ \sigma^{+}
  6. A 0 = - 1 3 σ z A^{0}=\frac{-1}{\sqrt{3}}\ \sigma^{z}
  7. A - = - 2 3 σ - A^{-}=-\sqrt{\frac{2}{3}}\ \sigma^{-}
  8. σ ’s \sigma\,\text{'s}

Al-Salam–Carlitz_polynomials.html

  1. U n ( a ) ( x ; q ) = ( - a ) n q n ( n - 1 ) / 2 ϕ 1 2 ( q - n , x - 1 ; 0 ; q , q x / a ) U_{n}^{(a)}(x;q)=(-a)^{n}q^{n(n-1)/2}{}_{2}\phi_{1}(q^{-n},x^{-1};0;q,qx/a)
  2. V n ( a ) ( x ; q ) = ( - a ) n q - n ( n - 1 ) / 2 ϕ 0 2 ( q - n , x ; ; q , q n / a ) V_{n}^{(a)}(x;q)=(-a)^{n}q^{-n(n-1)/2}{}_{2}\phi_{0}(q^{-n},x;;q,q^{n}/a)

Al-Salam–Chihara_polynomials.html

  1. Q n ( x ; a , b ; q ) = ( a b ; q ) n a n ϕ 2 3 ( q - n , a e i θ , a e - i θ ; a b , 0 ; q , q ) Q_{n}(x;a,b;q)=\frac{(ab;q)_{n}}{a^{n}}{}_{3}\phi_{2}(q^{-n},ae^{i\theta},ae^{% -i\theta};ab,0;q,q)

Alexander's_Star.html

  1. 30 ! × 2 15 120 7.24 × 10 34 \frac{30!\times 2^{15}}{120}\approx 7.24\times 10^{34}

Alexander_Merkurjev.html

  1. p 2 p^{2}

Alexiewicz_norm.html

  1. f := sup { | I f | : I is an interval } . \|f\|:=\sup\left\{\left|\int_{I}f\right|:I\subseteq\mathbb{R}\,\text{ is an % interval}\right\}.
  2. f := sup x | - x f | . \|f\|^{\prime}:=\sup_{x\in\mathbb{R}}\left|\int_{-\infty}^{x}f\right|.
  3. { F : | F is continuous, lim x - F ( x ) = 0 , lim x + F ( x ) } . \left\{F\colon\mathbb{R}\to\mathbb{R}\,\left|\,F\,\text{ is continuous, }\lim_% {x\to-\infty}F(x)=0,\lim_{x\to+\infty}F(x)\in\mathbb{R}\right.\right\}.
  4. F , φ = - F , φ = - - + F φ = f , φ \langle F^{\prime},\varphi\rangle=-\langle F,\varphi^{\prime}\rangle=-\int_{-% \infty}^{+\infty}F\varphi^{\prime}=\langle f,\varphi\rangle
  5. f = sup x | F ( x ) | = F . \|f\|^{\prime}=\sup_{x\in\mathbb{R}}|F(x)|=\|F\|_{\infty}.
  6. ( T x f ) ( y ) := f ( y - x ) , (T_{x}f)(y):=f(y-x),
  7. T x f - f 0 as x 0. \|T_{x}f-f\|\to 0\,\text{ as }x\to 0.

Algebraic_geometry_of_projective_spaces.html

  1. ( 𝒱 ( I ) ) = I . \mathcal{I}(\mathcal{V}(I))=\sqrt{I}.
  2. 𝔸 k n + 1 \mathbb{A}_{k}^{n+1}
  3. Γ ( D ( P ) , 𝒪 ( V ) ) \Gamma(D(P),\mathcal{O}_{\mathbb{P}(V)})
  4. ( k [ V ] P ) 0 (k[V]_{P})_{0}
  5. ( V ) \mathbb{P}(V)
  6. 𝒪 ( i ) \mathcal{O}(i)
  7. 𝒪 ( 1 ) \mathcal{O}(1)
  8. Pic 𝐏 𝐤 n = \mathrm{Pic}\ \mathbf{P}^{n}_{\mathbf{k}}=\mathbb{Z}
  9. k n , \mathbb{P}^{n}_{k},\,
  10. 𝒪 ( m ) , m , \mathcal{O}(m),\ m\in\mathbb{Z},
  11. k n \mathbb{P}^{n}_{k}
  12. \mathbb{Z}
  13. U ( V ) U\subseteq\mathbb{P}(V)
  14. 𝒪 ( k ) \mathcal{O}(k)
  15. Γ ( , 𝒪 ( m ) ) \Gamma(\mathbb{P},\mathcal{O}(m))
  16. ( m + n m ) = ( m + n n ) {\left({{m+n}\atop{m}}\right)}={\left({{m+n}\atop{n}}\right)}
  17. 𝒪 ( - 1 ) \mathcal{O}(-1)
  18. 𝒦 ( k n ) , \mathcal{K}(\mathbb{P}^{n}_{k}),\,
  19. 𝒪 ( - ( n + 1 ) ) \mathcal{O}(-(n+1))
  20. Ind ( n ) = n + 1 \mathrm{Ind}(\mathbb{P}^{n})=n+1
  21. Ind ( X ) = dim X + 1 \mathrm{Ind}(X)=\mathrm{dim}X+1
  22. 𝐤 n \mathbb{P}^{n}_{\mathbf{k}}
  23. PGL n + 1 ( 𝐤 ) \mathrm{PGL}_{n+1}(\mathbf{k})
  24. j : X 𝐏 n j:X\to\mathbf{P}^{n}
  25. j : X 𝐏 n j:X\to\mathbf{P}^{n}
  26. j * 𝒪 ( 1 ) j^{*}\mathcal{O}(1)
  27. j * ( Γ ( 𝐏 n , 𝒪 ( 1 ) ) ) Γ ( X , j * 𝒪 ( 1 ) ) . j^{*}(\Gamma(\mathbf{P}^{n},\mathcal{O}(1)))\subset\Gamma(X,j^{*}\mathcal{O}(1% )).
  28. j j
  29. j * ( Γ ( 𝐏 n , 𝒪 ( 1 ) ) ) j^{*}(\Gamma(\mathbf{P}^{n},\mathcal{O}(1)))
  30. n N \mathbb{P}^{n}\to\mathbb{P}^{N}
  31. N = ( n + d d ) - 1 N={\left({{n+d}\atop{d}}\right)}-1
  32. P ( X 0 , , X n ) X 0 d e g ( P ) P ( 1 , X 1 , , X n ) \frac{P(X_{0},\ldots,X_{n})}{X_{0}^{deg(P)}}\mapsto P(1,X_{1},\ldots,X_{n})

Algebraic_interior.html

  1. X X
  2. A X A\subseteq X
  3. core ( A ) := { x 0 A : x X , t x > 0 , t [ 0 , t x ] , x 0 + t x A } . \operatorname{core}(A):=\left\{x_{0}\in A:\forall x\in X,\exists t_{x}>0,% \forall t\in[0,t_{x}],x_{0}+tx\in A\right\}.
  4. core ( A ) core ( core ( A ) ) \operatorname{core}(A)\neq\operatorname{core}(\operatorname{core}(A))
  5. A A
  6. core ( A ) = core ( core ( A ) ) \operatorname{core}(A)=\operatorname{core}(\operatorname{core}(A))
  7. A A
  8. x 0 core ( A ) , y A , 0 < λ 1 x_{0}\in\operatorname{core}(A),y\in A,0<\lambda\leq 1
  9. λ x 0 + ( 1 - λ ) y core ( A ) \lambda x_{0}+(1-\lambda)y\in\operatorname{core}(A)
  10. A 2 A\subset\mathbb{R}^{2}
  11. A = { x 2 : x 2 x 1 2 or x 2 0 } A=\{x\in\mathbb{R}^{2}:x_{2}\geq x_{1}^{2}\,\text{ or }x_{2}\leq 0\}
  12. 0 core ( A ) 0\in\operatorname{core}(A)
  13. 0 int ( A ) 0\not\in\operatorname{int}(A)
  14. 0 core ( core ( A ) ) 0\not\in\operatorname{core}(\operatorname{core}(A))
  15. A , B X A,B\subset X
  16. A A
  17. 0 core ( A ) 0\in\operatorname{core}(A)
  18. A + core B core ( A + B ) A+\operatorname{core}B\subset\operatorname{core}(A+B)
  19. A + core B = core ( A + B ) A+\operatorname{core}B=\operatorname{core}(A+B)
  20. B = core B B=\operatorname{core}B
  21. X X
  22. int \operatorname{int}
  23. A X A\subset X
  24. int A core A \operatorname{int}A\subseteq\operatorname{core}A
  25. A A
  26. X X
  27. int A = core A \operatorname{int}A=\operatorname{core}A
  28. A A
  29. int A = core A \operatorname{int}A=\operatorname{core}A
  30. A A
  31. X X
  32. int A = core A \operatorname{int}A=\operatorname{core}A

Algebraic_number_field.html

  1. ( a + b i ) ( a a 2 + b 2 - b a 2 + b 2 i ) = ( a + b i ) ( a - b i ) a 2 + b 2 = 1. (a+bi)\left(\frac{a}{a^{2}+b^{2}}-\frac{b}{a^{2}+b^{2}}i\right)=\frac{(a+bi)(a% -bi)}{a^{2}+b^{2}}=1.
  2. d ¯ \overline{d}
  3. 5 ¯ \overline{−5}
  4. 5 ¯ \overline{−5}
  5. 5 ¯ \overline{−5}
  6. 5 ¯ \overline{−5}
  7. 5 ¯ \overline{−5}
  8. ¯ d \overline{−}{d}
  9. ζ F ( s ) := 𝔭 1 1 - N ( 𝔭 ) - s \zeta_{F}(s):=\prod_{\mathfrak{p}}\frac{1}{1-N(\mathfrak{p})^{-s}}
  10. N ( 𝔭 ) N(\mathfrak{p})
  11. O F / 𝔭 O_{F}/\mathfrak{p}
  12. 2 r 1 ( 2 π ) r 2 h Reg w | D | . \frac{2^{r_{1}}\cdot(2\pi)^{r_{2}}\cdot h\cdot\operatorname{Reg}}{w\cdot\sqrt{% |D|}}.
  13. x e i = j = 1 n a i j e j , a i j 𝐐 . xe_{i}=\sum_{j=1}^{n}a_{ij}e_{j},\quad a_{ij}\in\mathbf{Q}.
  14. [ 3 11 61 11 119 653 61 653 3589 ] . \begin{bmatrix}3&11&61\\ 11&119&653\\ 61&653&3589\end{bmatrix}.

Algimantas_Adolfas_Jucys.html

  1. [ S n ] \mathbb{C}[S_{n}]

Algorithmic_complexity_attack.html

  1. O ( n log n ) O(n\log n)
  2. O ( n 2 ) O(n^{2})

Alias_method.html

  1. O ( n log n ) O(n\log n)
  2. O ( n ) O(n)
  3. O ( 1 ) O(1)

Allometric_engineering.html

  1. l o g ( y ) = m l o g ( x ) + l o g ( b ) log(y)=mlog(x)+log(b)

Alpha_centrality.html

  1. A i , j A_{i,j}
  2. x = ( I - α A T ) - 1 e \vec{x}=(I-\alpha A^{T})^{-1}\vec{e}\,
  3. e j e_{j}
  4. j j
  5. α \alpha
  6. x i = 1 λ A i , j T x j x_{i}=\frac{1}{\lambda}A^{T}_{i,j}x_{j}
  7. x i x_{i}
  8. i i
  9. A i , j A_{i,j}
  10. λ \lambda
  11. i i
  12. e i e_{i}
  13. x i = α A i , j T x j + e i x_{i}=\alpha A^{T}_{i,j}x_{j}+e_{i}\,
  14. α \alpha
  15. α = 0 \alpha=0
  16. α \alpha
  17. x x
  18. x = ( I - α A T ) - 1 e x=(I-\alpha A^{T})^{-1}e\,

Alveolar_air_equation.html

  1. P A O 2 = F I O 2 ( P B - P H 2 0 ) - P A C O 2 ( F I O 2 + 1 - F I O 2 R ) P_{A}O_{2}=F_{I}O_{2}(PB-PH_{2}0)-P_{A}CO_{2}(F_{I}O_{2}+\frac{1-F_{I}O_{2}}{R})
  2. P A O 2 = P I O 2 - P A C O 2 ( F I O 2 + 1 - F I O 2 R ) P_{A}O_{2}=P_{I}O_{2}-P_{A}CO_{2}(F_{I}O_{2}+\frac{1-F_{I}O_{2}}{R})
  3. P A O 2 = P I O 2 - V T V T - V D ( P I O 2 - P E O 2 ) P_{A}O_{2}=P_{I}O_{2}-\frac{V_{T}}{V_{T}-V_{D}}(P_{I}O_{2}-P_{E}O_{2})
  4. P A O 2 = P E O 2 - P I O 2 ( V D V T ) 1 - V D V T P_{A}O_{2}=\frac{P_{E}O_{2}-P_{I}O_{2}(\frac{V_{D}}{V_{T}})}{1-\frac{V_{D}}{V_% {T}}}
  5. P A O 2 = P E O 2 - P i O 2 V D V T 1 - V D V T P_{A}O_{2}=\frac{P_{E}O_{2}-P_{i}O_{2}\frac{V_{D}}{V_{T}}}{1-\frac{V_{D}}{V_{T% }}}
  6. R = P E C O 2 ( 1 - F I O 2 ) P i O 2 - P E O 2 - ( P E C O 2 * F i O 2 ) R=\frac{P_{E}CO_{2}(1-F_{I}O_{2})}{P_{i}O_{2}-P_{E}O_{2}-(P_{E}CO_{2}*F_{i}O_{% 2})}
  7. V D V T = P A C O 2 - P E C O 2 P A C O 2 \frac{V_{D}}{V_{T}}=\frac{P_{A}CO_{2}-P_{E}CO_{2}}{P_{A}CO_{2}}

Alvis–Curtis_duality.html

  1. ζ * = J R ( - 1 ) J ζ P J G \zeta^{*}=\sum_{J\subseteq R}(-1)^{J}\zeta^{G}_{P_{J}}

Amitsur–Levitzki_theorem.html

  1. S n ( x 1 , , x n ) = σ S n sgn ( σ ) x σ 1 x σ n S_{n}(x_{1},\ldots,x_{n})=\sum_{\sigma\in S_{n}}\,\text{sgn}(\sigma)x_{\sigma 1% }\cdots x_{\sigma n}
  2. S 2 n ( A 1 , , A 2 n ) = 0 . S_{2n}(A_{1},\ldots,A_{2n})=0\ .

Amplitude_damping_channel.html

  1. ρ A \rho_{A}
  2. ρ A \rho_{A}
  3. σ 0 \sigma_{0}
  4. R ( t ) = U ( t ) ( ρ A σ 0 ) U ( t ) R(t)=U(t)(\rho_{A}\otimes\sigma_{0})U^{\dagger}(t)
  5. ρ B ( t ) = Tr [ U ( t ) ( ρ A σ 0 ) U ( t ) ] ( B ) \rho_{B}(t)=\mbox{Tr}~{}^{(B)}[U(t)(\rho_{A}\otimes\sigma_{0})U^{\dagger}(t)]
  6. ρ A ( ρ A ) ρ B ( t ) = Tr [ U ( t ) ( ρ A σ 0 ) U ( t ) ] ( B ) \rho_{A}\rightarrow\mathcal{M}(\rho_{A})\equiv\rho_{B}(t)=\mbox{Tr}~{}^{(B)}[U% (t)(\rho_{A}\otimes\sigma_{0})U^{\dagger}(t)]
  7. H = - i , j J i j ( σ x i σ x j + σ y i σ y j + γ σ z i σ z j ) - i = 1 N B i σ z i H=-\sum_{\langle i,j\rangle}\hbar J_{ij}\left({\sigma}_{x}^{i}{\sigma}_{x}^{j}% +{\sigma}_{y}^{i}{\sigma}_{y}^{j}+\gamma{\sigma}_{z}^{i}{\sigma}_{z}^{j}\right% )-\sum_{i=1}^{N}\hbar B_{i}\sigma_{z}^{i}
  8. | j |j\rangle
  9. | j | |{j}\rangle\equiv|\downarrow\downarrow\cdots\downarrow\uparrow\downarrow\cdots\downarrow\rangle
  10. | Ψ A α | A + β | ϕ 1 A |\Psi\rangle_{A}\equiv\alpha|\Downarrow\rangle_{A}+\beta|\phi_{1}\rangle_{A}
  11. | |\Downarrow\rangle
  12. | ϕ 1 |\phi_{1}\rangle
  13. ρ B ( t ) = ( | α | 2 + ( 1 - η ) | β | 2 ) | B | + η | β | 2 | ϕ 1 B ϕ 1 | + η α β * | B ϕ 1 | + η α * β | ϕ 1 B | \rho_{B}(t)=(|\alpha|^{2}+(1-\eta)|\beta|^{2})|\Downarrow\rangle_{B}\langle% \Downarrow|+\eta|\beta|^{2}|\phi_{1}^{\prime}\rangle_{B}\langle\phi_{1}^{% \prime}|+\sqrt{\eta}\alpha\beta^{*}|\Downarrow\rangle_{B}\langle\phi_{1}^{% \prime}|+\sqrt{\eta}\alpha^{*}\beta|\phi_{1}^{\prime}\rangle_{B}\langle\Downarrow|
  14. η \eta
  15. | 1 |1\rangle
  16. | 0 |0\rangle
  17. 𝒟 n \mathcal{D}_{n}
  18. A 0 = | 0 0 | + η | 1 1 | A_{0}=|0\rangle\langle 0|+\sqrt{\eta}|1\rangle\langle 1|
  19. A 1 = 1 - η | 0 1 | A_{1}=\sqrt{1-\eta}|0\rangle\langle 1|
  20. η \eta
  21. η \eta^{\prime}
  22. η \eta
  23. η \eta^{\prime}
  24. 𝒟 η \mathcal{D}_{\eta}
  25. 𝒟 η ( ρ ) Tr [ V ( ρ | 0 C 0 | ) V ] C \mathcal{D}_{\eta}(\rho)\equiv\mbox{Tr}~{}_{C}[V\left(\rho\otimes|0\rangle_{C}% \langle 0|\right)V^{\dagger}]
  26. C \mathcal{H}_{C}
  27. A \mathcal{H}_{A}
  28. 𝒟 ~ η \tilde{\mathcal{D}}_{\eta}
  29. η 0.5 \eta\geqslant 0.5
  30. 𝒟 ~ η ( ρ ) = S 𝒟 ( 1 - η ) / η ( 𝒟 η ( ρ ) ) \tilde{\mathcal{D}}_{\eta}(\rho)=S\mathcal{D}_{(1-\eta)/\eta}\left({\mathcal{D% }}_{\eta}(\rho)\right)
  31. S ( 𝒟 η ( ρ ) ) = H 2 ( ( 1 + ( 1 - 2 η p ) 2 + 4 η | γ | 2 ) / 2 ) S(\mathcal{D}_{\eta}(\rho))=H_{2}(\left(1+\sqrt{(1-2\,\eta\,p)^{2}+4\,\eta\,|% \gamma|^{2}}\right)/2)
  32. p [ 0 , 1 ] p\in[0,1]
  33. | 1 |1\rangle
  34. | γ | ( 1 - p ) p |\gamma|\leqslant\sqrt{(1-p)p}
  35. S ( ( 𝒟 η 1 a n c ) ( Φ ) ) = H 2 ( ( 1 + ( 1 - 2 ( 1 - η ) p ) 2 + 4 ( 1 - η ) | γ | 2 ) / 2 ) S((\mathcal{D}_{\eta}\otimes 1_{anc})(\Phi))=H_{2}(\left(1+\sqrt{(1-2\,(1-\eta% )\,p)^{2}+4\,(1-\eta)\,|\gamma|^{2}}\right)/2)
  36. γ = 0 \gamma=0
  37. Q max p [ 0 , 1 ] { H 2 ( η p ) - H 2 ( ( 1 - η ) p ) } Q\equiv\max_{p\in[0,1]}\;\Big\{\;H_{2}(\eta\,p)-H_{2}((1-\eta)\,p)\;\Big\}\;
  38. η < 0.5 \eta<0.5
  39. η \eta
  40. γ = 0 \gamma=0
  41. C E max p [ 0 , 1 ] { H 2 ( p ) + H 2 ( η p ) - H 2 ( ( 1 - η ) p ) } C_{E}\equiv\max_{p\in[0,1]}\;\Big\{\;H_{2}(p)+H_{2}(\eta\,p)-H_{2}((1-\eta)\,p% )\;\Big\}\;
  42. ξ k \xi_{k}
  43. χ H 2 ( 1 + ( 1 - 2 η p ) 2 + 4 η | γ | 2 2 ) - k ξ k H 2 ( 1 + ( 1 - 2 η p k ) 2 + 4 η | γ k | 2 2 ) \chi\equiv H_{2}\left(\frac{1+\sqrt{(1-2\,\eta\,p)^{2}+4\,\eta\,|\gamma|^{2}}}% {2}\right)-\sum_{k}\xi_{k}H_{2}\left(\frac{1+\sqrt{(1-2\,\eta\,p_{k})^{2}+4\,% \eta\,|\gamma_{k}|^{2}}}{2}\right)\;
  44. p k p_{k}
  45. γ k \gamma_{k}
  46. p p
  47. γ \gamma
  48. p k , γ k , ξ k p_{k},\gamma_{k},\xi_{k}
  49. γ \gamma
  50. H 2 ( z ) H_{2}(z)
  51. | 1 / 2 + z | |1/2+z|
  52. H 2 ( 1 + 1 - z 2 / 2 ) H_{2}(1+\sqrt{1-z^{2}}/2)
  53. k ξ k H 2 ( 1 + ( 1 - 2 η p k ) 2 + 4 η | γ k | 2 2 ) H 2 ( 1 + 1 - 4 η ( 1 - η ) ( k ξ k p k ) 2 2 ) \sum_{k}\xi_{k}H_{2}\left(\frac{1+\sqrt{(1-2\,\eta\,p_{k})^{2}+4\,\eta\,|% \gamma_{k}|^{2}}}{2}\right)\geqslant H_{2}\left(\frac{1+\sqrt{1-4\,\eta\,(1-% \eta)(\sum_{k}\xi_{k}p_{k})^{2}}}{2}\right)
  54. C 1 max p [ 0 , 1 ] { H 2 ( η p ) - H 2 ( 1 + 1 - 4 η ( 1 - η ) p 2 2 ) } C_{1}\leqslant\max_{p\in[0,1]}\Big\{H_{2}\left(\eta\,p\right)-H_{2}\left(\frac% {1+\sqrt{1-4\,\eta\,(1-\eta)\,p^{2}}}{2}\right)\Big\}\;
  55. ξ k = 1 / d \xi_{k}=1/d\,\!
  56. p k = p p_{k}=p\,\!
  57. γ k = e 2 π i k / d ( 1 - p ) p \gamma_{k}=e^{2\pi ik/d}\sqrt{(1-p)p}
  58. η \eta
  59. η \eta
  60. η \eta
  61. η \eta
  62. | r - s | - 2 / 3 |r-s|^{-2/3}
  63. η \eta
  64. η \eta

AN-VI.html

  1. F ( t ) = M a ( t ) + B v ( t ) + ( K + i C ) s ( t ) F(t)=Ma(t)+Bv(t)+(K+iC)s(t)\!
  2. s ( t ) = ( 1 c m ) s i n ( w t ) s(t)=(1cm)sin(wt)\!
  3. F ( t ) = K s ( t ) = K ( ( 1 c m ) s i n ( w t ) ) F(t)=Ks(t)=K((1cm)sin(wt))\!
  4. E e = 1 2 K ( 2 c m ) 2 Ee=\frac{1}{2}K(2cm)^{2}\!
  5. R i = E i E e Ri=\frac{Ei}{Ee}\!
  6. ( C ) (C)
  7. ( K ) (K)
  8. ( F ) (F)
  9. F ( t ) = ( K + i C ) x ( t ) = K ( 1 c m ) s i n ( w t ) + C ( 1 c m ) c o s ( w t ) F(t)=(K+iC)x(t)=K(1cm)sin(wt)+C(1cm)cos(wt)\!
  10. n l = C 2 K nl=\frac{C}{2K}\!
  11. ( R l ) (Rl)
  12. R l = ( 3 , 14 ) n l Rl=(3,14)nl\!
  13. F o r c e ( N ) Force(N)\!
  14. D i s p l a c e m e n t ( c m ) Displacement(cm)\!
  15. A c c e l e r a t i o n ( c m / s 2 ) Acceleration(cm/s^{2})\!

AN_codes.html

  1. A A
  2. N N
  3. x x
  4. r r
  5. w ( x ) = min { t | x = i = 1 t a i r n ( i ) } w(x)=\min\{t|x=\sum_{i=1}^{t}a_{i}r^{n(i)}\}
  6. | a i | |{a_{i}}|
  7. n ( i ) 0 n(i)\geq 0
  8. r , n ( i ) r,n(i)\in\mathbb{Z}
  9. x = i = 1 n b i r i x=\sum_{i=1}^{n}b_{i}r^{i}
  10. b i b_{i}
  11. b i = 0 b_{i}=0
  12. t t
  13. a i a_{i}
  14. x = 29 x=29
  15. 11101 11101
  16. 4 4
  17. x = 2 0 + 2 2 + 2 3 + 2 4 x=2^{0}+2^{2}+2^{3}+2^{4}
  18. a i a_{i}
  19. x = 2 5 - 2 1 - 2 0 x=2^{5}-2^{1}-2^{0}
  20. 3 3
  21. d ( x , y ) = w ( x - y ) d(x,y)=w(x-y)
  22. A A
  23. B B
  24. 0
  25. B - 1 B-1
  26. C = { A N | N , 0 N C=\{AN|N\in\mathbb{Z},0\leq N
  27. A A
  28. B B
  29. B B
  30. A A
  31. R R
  32. R R
  33. 0
  34. B - 1 B-1
  35. R / A R/A
  36. A A
  37. R R
  38. R R
  39. d - 1 2 \lfloor\frac{d-1}{2}\rfloor
  40. d d
  41. A = 3 A=3
  42. 15 15
  43. 16 16
  44. R = 45 + 48 = 93 R=45+48=93
  45. 93 / 3 = 31 93/3=31
  46. B B
  47. 31 31
  48. 45 = 101101 101111 45=101101\rightarrow 101111
  49. 48 = 110000 110001 48=110000\rightarrow 110001
  50. R = 101111 + 110001 = 1100000 R=101111+110001=1100000
  51. 93 = 1011101 93=1011101
  52. 5 5
  53. 2 2
  54. 1100000 - 1011101 = 11 1100000-1011101=11
  55. 11 = 2 0 + 2 1 11=2^{0}+2^{1}
  56. 11 = 2 2 - 2 0 11=2^{2}-2^{0}
  57. 2 2
  58. C C
  59. / m \mathbb{Z}/m\mathbb{Z}
  60. m = A B m=AB
  61. / m \mathbb{Z}/m\mathbb{Z}
  62. x ( mod m ) x\;\;(\mathop{{\rm mod}}m)
  63. x ( mod m ) x^{\prime}\;\;(\mathop{{\rm mod}}m)
  64. x - x ± c r j ( mod m ) x-x^{\prime}\equiv\pm c\cdot r^{j}\;\;(\mathop{{\rm mod}}m)
  65. c , j c,j\in\mathbb{Z}
  66. 0
  67. j 0 j\geq 0
  68. 0
  69. w m ( x ) = m i n { w ( y ) | y , y x ( mod m ) } w_{m}(x)=min\{w(y)|y\in\mathbb{Z},y\equiv x\;\;(\mathop{{\rm mod}}m)\}
  70. m m
  71. m = r n - 1 m=r^{n}-1
  72. mod 2 n - 1 \mod 2^{n}-1
  73. m = r n - 1 m=r^{n}-1
  74. m = r n - 1 m=r^{n}-1
  75. C C
  76. [ r n - 1 ] [r^{n}-1]
  77. [ r n - 1 ] = { 0 , 1 , 2 , , r n - 1 } [r^{n}-1]=\{0,1,2,\dots,r^{n}-1\}
  78. [ r n - 1 ] [r^{n}-1]
  79. A A
  80. B B
  81. A B = r n - 1 AB=r^{n}-1
  82. A , B A,B
  83. B B
  84. r r
  85. / B \mathbb{Z}/B\mathbb{Z}
  86. r r
  87. - 1 -1
  88. m = r n - 1 m=r^{n}-1
  89. n n
  90. r n 1 ( mod B ) r^{n}\equiv 1\;\;(\mathop{{\rm mod}}B)
  91. A = ( r n - 1 ) / B A=(r^{n}-1)/B
  92. r = 2 , B = 5 , n = 4 r=2,B=5,n=4
  93. A = ( r n - 1 ) / B = 3 A=(r^{n}-1)/B=3
  94. C = { 3 N | N , 0 N C=\{3N|N\in\mathbb{Z},0\leq N
  95. 2 2
  96. C [ r n - 1 ] C\subset[r^{n}-1]
  97. A A
  98. B = | C | = ( r n - 1 ) / A B=|C|=(r^{n}-1)/A
  99. x C w m ( x ) = n ( r B r + 1 - B r + 1 ) \sum_{x\in C}w_{m}(x)=n(\lfloor\frac{rB}{r+1}\rfloor-\lfloor\frac{B}{r+1}\rfloor)
  100. x C x\in C
  101. x i = 0 n - 1 c i , x r i ( mod r n - 1 ) x\equiv\sum_{i=0}^{n-1}c_{i,x}r^{i}\;\;(\mathop{{\rm mod}}r^{n}-1)
  102. n × B n\times B
  103. c i , x c_{i,x}
  104. 0 i n - 1 0\leq i\leq n-1
  105. x C x\in C
  106. C C
  107. C C
  108. n | { x C | c n - 1 , x 0 } | n|\{x\in C|c_{n-1,x}\neq 0\}|
  109. n n
  110. 0
  111. c n - 1 , x 0 c_{n-1,x}\neq 0
  112. y y\in\mathbb{Z}
  113. y x ( mod r n - 1 ) , m r + 1 y\equiv x\;\;(\mathop{{\rm mod}}r^{n}-1),\frac{m}{r+1}
  114. x = A N ( mod r n - 1 ) x=AN\;\;(\mathop{{\rm mod}}r^{n}-1)
  115. 0 N 0\leq N
  116. B r + 1 \frac{B}{r+1}
  117. r B r + 1 - B r + 1 \lfloor\frac{rB}{r+1}\rfloor-\lfloor\frac{B}{r+1}\rfloor
  118. n n
  119. n ( r B r + 1 - B r + 1 ) n(\lfloor\frac{rB}{r+1}\rfloor-\lfloor\frac{B}{r+1}\rfloor)
  120. n B - 1 ( r B r + 1 - B r + 1 ) \frac{n}{B-1}(\lfloor\frac{rB}{r+1}\rfloor-\lfloor\frac{B}{r+1}\rfloor)
  121. x C , x 0 x\in C,x\neq 0
  122. x = A N ( mod r n - 1 ) x=AN\;\;(\mathop{{\rm mod}}r^{n}-1)
  123. N N
  124. B B
  125. j ( N ± r j ( mod B ) ) \exists j(N\equiv\pm r^{j}\;\;(\mathop{{\rm mod}}B))
  126. w m ( x ) = w m ( ± r j A ) = w m ( A ) w_{m}(x)=w_{m}(\pm r^{j}A)=w_{m}(A)
  127. C C
  128. A A

An_Essay_towards_solving_a_Problem_in_the_Doctrine_of_Chances.html

  1. f ( p ) = ( n + 1 ) ! k ! ( n - k ) ! p k ( 1 - p ) n - k for 0 p 1 f(p)=\frac{(n+1)!}{k!(n-k)!}p^{k}(1-p)^{n-k}\,\text{ for }0\leq p\leq 1
  2. P ( B | A ) = P ( A B ) P ( A ) , if P ( A ) 0 , P(B|A)=\frac{P(A\cap B)}{P(A)},\,\text{ if }P(A)\neq 0,\!
  3. P ( A | B ) = P ( B | A ) P ( A ) P ( B ) , if P ( B ) 0. \Rightarrow P(A|B)=\frac{P(B|A)\,P(A)}{P(B)},\,\text{ if }P(B)\neq 0.
  4. p p
  5. m m
  6. n n
  7. P ( a < p < b m ; n ) = a b ( n + m m ) p m ( 1 - p ) n d p 0 1 ( n + m m ) p m ( 1 - p ) n d p . P(a<p<b\mid m;n)=\frac{\int_{a}^{b}{n+m\choose m}p^{m}(1-p)^{n}\,dp}{\int_{0}^% {1}{n+m\choose m}p^{m}(1-p)^{n}\,dp}.\!
  8. p p
  9. p p
  10. 1 - p 1-p

Analyst's_traveling_salesman_theorem.html

  1. 2 \mathbb{R}^{2}
  2. β E ( Q ) = 1 ( Q ) inf { δ : there is a line L so that for every x E Q , dist ( x , L ) < δ } , \beta_{E}(Q)=\frac{1}{\ell(Q)}\inf\{\delta:\,\text{ there is a line }L\,\text{% so that for every }x\in E\cap Q,\;\,\text{dist}(x,L)<\delta\},
  3. ( Q ) \ell(Q)
  4. 2 β E ( Q ) ( Q ) 2\beta_{E}(Q)\ell(Q)
  5. β E ( Q ) \beta_{E}(Q)
  6. Δ = { [ i 2 k , ( i + 1 ) 2 k ] × [ j 2 k , ( j + 1 ) 2 k ] : i , j , k } , \Delta=\{[i2^{k},(i+1)2^{k}]\times[j2^{k},(j+1)2^{k}]:i,j,k\in\mathbb{Z}\},
  7. \mathbb{Z}
  8. E 2 E\subseteq\mathbb{R}^{2}
  9. β ( E ) = diam E + Q Δ β E ( 3 Q ) 2 ( Q ) \beta(E)=\,\text{diam}E+\sum_{Q\in\Delta}\beta_{E}(3Q)^{2}\ell(Q)
  10. E d E\subseteq\mathbb{R}^{d}
  11. d \mathbb{R}^{d}
  12. A A\subseteq\mathbb{R}

Anelastic_attenuation_factor.html

  1. Q = 2 π ( E δ E ) Q=2{\pi}\left(\frac{E}{{\delta}E}\right)
  2. E δ E \frac{E}{{\delta}E}
  3. A f i n a l A i n i t i a l = R . G . e - π f t Q \frac{A_{final}}{A_{initial}}=R.G.e^{{\frac{-{\pi}ft}{Q}}}
  4. A f i n a l {A_{final}}
  5. A i n i t i a l {A_{initial}}
  6. f f
  7. R R
  8. G G
  9. t t
  10. l n ( A f i n a l A i n i t i a l ) = ( - π t Q ) f + l n ( R G ) ln\left(\frac{A_{final}}{A_{initial}}\right)=\left(\frac{-{\pi}t}{Q}\right)f+% ln(RG)
  11. Y = m X + C Y=mX+C

Angelescu_polynomials.html

  1. ϕ ( t 1 - t ) exp ( - x t 1 - t ) = n = 0 π n ( x ) t n \displaystyle\phi\left(\frac{t}{1-t}\right)\exp\left(-\frac{xt}{1-t}\right)=% \sum_{n=0}^{\infty}\pi_{n}(x)t^{n}

Angular_momentum_of_light.html

  1. 𝐉 = ϵ 0 𝐫 × ( 𝐄 × 𝐁 ) d 3 𝐫 , \mathbf{J}=\epsilon_{0}\int\mathbf{r}\times\left(\mathbf{E}\times\mathbf{B}% \right)d^{3}\mathbf{r},
  2. 𝐄 \mathbf{E}
  3. 𝐁 \mathbf{B}
  4. ϵ 0 \epsilon_{0}
  5. 𝐉 = ϵ 0 ( 𝐄 × 𝐀 ) d 3 𝐫 + ϵ 0 i = x , y , z ( E i ( 𝐫 × ) A i ) d 3 𝐫 , \mathbf{J}=\epsilon_{0}\int\left(\mathbf{E}\times\mathbf{A}\right)d^{3}\mathbf% {r}+\epsilon_{0}\sum_{i=x,y,z}\int\left({E^{i}}\left(\mathbf{r}\times\mathbf{% \nabla}\right)A^{i}\right)d^{3}\mathbf{r},
  6. 𝐀 \mathbf{A}
  7. 𝐀 \mathbf{A}_{\perp}
  8. 𝐄 \mathbf{E}_{\perp}
  9. 𝐉 = ϵ 0 ( 𝐄 × 𝐀 ) d 3 𝐫 + ϵ 0 i = x , y ( E i ( 𝐫 × ) A i ) d 3 𝐫 . \mathbf{J}_{\perp}=\epsilon_{0}\int\left({\mathbf{E}}_{\perp}\times\mathbf{A}_% {\perp}\right)d^{3}\mathbf{r}+\epsilon_{0}\sum_{i=x,y}\int\left({E^{i}}_{\perp% }\left(\mathbf{r}\times\mathbf{\nabla}\right)A^{i}_{\perp}\right)d^{3}\mathbf{% r}.
  10. 𝐉 = ϵ 0 2 i ω ( 𝐄 × 𝐄 ) d 3 𝐫 + ϵ 0 2 i ω i = x , y , z ( E i ( 𝐫 × ) E i ) d 3 𝐫 . \mathbf{J}=\frac{\epsilon_{0}}{2i\omega}\int\left(\mathbf{E}^{\ast}\times% \mathbf{E}\right)d^{3}\mathbf{r}+\frac{\epsilon_{0}}{2i\omega}\sum_{i=x,y,z}% \int\left({E^{i}}^{\ast}\left(\mathbf{r}\times\mathbf{\nabla}\right)E^{i}% \right)d^{3}\mathbf{r}.
  11. 𝐉 z ^ ϵ 0 2 ω ( | E L | 2 - | E R | 2 ) d 3 𝐫 + z ^ ϵ 0 2 i ω i = x , y , z ( E i ϕ E i ) d 3 𝐫 . \mathbf{J}\approx\frac{\hat{z}\epsilon_{0}}{2\omega}\int\left(|{E}_{L}|^{2}-|{% E}_{R}|^{2}\right)d^{3}\mathbf{r}+\frac{\hat{z}\epsilon_{0}}{2i\omega}\int\sum% _{i=x,y,z}\left({E^{i}}^{\ast}\frac{\partial}{\partial\phi}E^{i}\right)d^{3}% \mathbf{r}.
  12. E L E_{L}
  13. E R E_{R}

Angular_resolution_(graph_drawing).html

  1. d d
  2. 2 π / d 2π/d
  3. v v
  4. d d
  5. v v
  6. v v
  7. d d
  8. 2 π
  9. 2 π / d 2π/d
  10. d d
  11. π d - 1 \frac{\pi}{d-1}
  12. G G
  13. G G
  14. χ χ
  15. π / χ ε π/χ − ε
  16. ε > 0 ε>0
  17. G G
  18. d d
  19. d d
  20. O ( log d d 2 ) O\left(\frac{\log d}{d^{2}}\right)
  21. π / 3 ε π/3−ε
  22. ε > 0 ε>0
  23. π / 3 π/3
  24. d d
  25. 1 / d 1/d
  26. d d
  27. 1 / d 1/d
  28. d d
  29. max ( d + 5 , 3 d 2 + 1 ) \max\left(d+5,\frac{3d}{2}+1\right)
  30. 5 d 3 + O ( 1 ) \frac{5d}{3}+O(1)
  31. O ( 1 / d < s u p > 3 ) O(1/d<sup>3)

Annular_fin.html

  1. q ( 2 π r ) ( 2 t ) | r - q ( 2 π r ) ( 2 t ) | r + Δ r - h c ( 2 ) 2 π r Δ r ( T - T e ) = 0 , q(2\pi r)(2t)\Bigr|_{r}-q(2\pi r)(2t)\Bigr|_{r+\Delta r}-h_{c}(2)2\pi r\,% \Delta r\left(T-T_{e}\right)=0,
  2. q = - k T r q=-k\frac{\partial T}{\partial r}
  3. r ( r T r ) - h c r T - T e k t = 0. \frac{\partial}{\partial r}\left(r\,\frac{\partial T}{\partial r}\right)-h_{c}% \,r\,\frac{T-T_{e}}{k\,t}=0.
  4. z = r h c k t z=r\sqrt{\frac{h_{c}}{k\,t}}
  5. θ = T - T e T b - T e \theta=\frac{T-T_{e}}{T_{b}-T_{e}}
  6. z 2 2 θ z 2 + z θ z - z 2 θ = 0. z^{2}\,\frac{\partial^{2}\theta}{\partial z^{2}}+z\,\frac{\partial\theta}{% \partial z}-z^{2}\,\theta=0.
  7. Q = k ( 4 π r 1 t ) ( T b - T e ) β [ C 2 K 1 ( β r 1 ) - C 1 I 1 ( β r 1 ) ] . Q=k\,\left(4\pi r_{1}t\right)\left(T_{b}-T_{e}\right)\beta\left[C_{2}\,K_{1}% \left(\beta r_{1}\right)-C_{1}\,I_{1}\left(\beta r_{1}\right)\right].
  8. η f = 2 r 1 β K 1 ( β r 1 ) I 1 ( β r 2 ) - I 1 ( β r 1 ) K 1 ( β r 2 ) r 2 2 - r 1 2 K 0 ( β r 1 ) I 1 ( β r 2 ) + I 0 ( β r 1 ) K 1 ( β r 2 ) . \eta_{f}=\frac{\displaystyle\frac{2r_{1}}{\beta}\,K_{1}\left(\beta r_{1}\right% )\,I_{1}\left(\beta r_{2}\right)-I_{1}\left(\beta r_{1}\right)\,K_{1}\left(% \beta r_{2}\right)}{r_{2}^{2}-r_{1}^{2}\,K_{0}\left(\beta r_{1}\right)\,I_{1}% \left(\beta r_{2}\right)+I_{0}\left(\beta r_{1}\right)\,K_{1}\left(\beta r_{2}% \right)}.

Anomalous_Diffraction_Theory.html

  1. Q e x t = 2 - 4 p sin p + 4 p 2 ( 1 - cos p ) Q_{ext}=2-\frac{4}{p}\sin{p}+\frac{4}{p^{2}}(1-\cos{p})
  2. Q e x t = Q a b s + Q s c a = Q s c a Q_{ext}=Q_{abs}+Q_{sca}=Q_{sca}
  3. Q a b s = 0 Q_{abs}=0
  4. Q e x t Q_{ext}

Antieigenvalue_theory.html

  1. x x
  2. A A
  3. μ \mu
  4. ϕ ( A ) \phi(A)

Antimatter_tests_of_Lorentz_violation.html

  1. c c
  2. b b
  3. b b
  4. b b
  5. g - 2 g-2
  6. g - 2 g-2

Antimicrobial_surface.html

  1. c o s Θ o b s = R * c o s Θ cos\Theta_{obs}=R*cos\Theta
  2. 2 H 2 O + 1 2 O 2 M e t a l I o n 2 H 2 O 2 H 2 O + ( O ) \,\text{ }2\,\text{ }H_{2}O\,\text{ }+\,\text{ }\frac{1}{2}O_{2}\stackrel{% MetalIon}{\rightarrow}\,\text{ }2\,\text{ }H_{2}O_{2}{\rightarrow}\,\text{ }H_% {2}O\,\text{ }+\,\text{ }(O)

Antisymmetric_exchange.html

  1. 𝐒 i \mathbf{S}_{i}
  2. 𝐒 j \mathbf{S}_{j}
  3. H D M = 𝐃 i j ( 𝐒 i × 𝐒 j ) H_{DM}=\mathbf{D}_{ij}\cdot(\mathbf{S}_{i}\times\mathbf{S}_{j})
  4. 𝐃 i j \mathbf{D}_{ij}
  5. 𝐃 i j \mathbf{D}_{ij}
  6. 𝐃 i j 𝐫 i × 𝐫 j = 𝐫 i j × 𝐱 \mathbf{D}_{ij}\propto\mathbf{r}_{i}\times\mathbf{r}_{j}=\mathbf{r}_{ij}\times% \mathbf{x}
  7. 𝐃 i j \mathbf{D}_{ij}
  8. 𝐃 i j = 0 \mathbf{D}_{ij}=0

Appert_topology.html

  1. N ( n , S ) = # { m S : m n } . \mathrm{N}(n,S)=\#\{m\in S:m\leq n\}.
  2. lim n N ( n , S ) n = 1. \lim_{n\to\infty}\frac{\,\text{N}(n,S)}{n}=1.
  3. N ( n , Z + ) = n , \,\text{N}\!\left(n,{Z}^{+}\right)=n\ ,
  4. lim n N ( n , S ) n = 0. \lim_{n\to\infty}\frac{\mathrm{N}(n,S)}{n}=0.
  5. lim n N ( n , B ) / n = 0 \scriptstyle\lim_{n\to\infty}\mathrm{N}(n,B)/n\,=\,0

Approximate_entropy.html

  1. Step 1 \,\text{Step 1}
  2. u ( 1 ) , u ( 2 ) , , u ( N ) \ u(1),u(2),\ldots,u(N)
  3. N \,\text{N}
  4. Step 2 \,\text{Step 2}
  5. m \ m
  6. r \ r
  7. m \ m
  8. r \ r
  9. Step 3 \,\text{Step 3}
  10. 𝐱 ( 1 ) \mathbf{x}(1)
  11. 𝐱 ( 2 ) , , 𝐱 ( N - m + 1 ) \mathbf{x}(2),\ldots,\mathbf{x}(N-m+1)
  12. 𝐑 m \mathbf{R}^{m}
  13. m \ m
  14. 𝐱 ( i ) = [ u ( i ) , u ( i + 1 ) , , u ( i + m - 1 ) ] \mathbf{x}(i)=[u(i),u(i+1),\ldots,u(i+m-1)]
  15. Step 4 \,\text{Step 4}
  16. 𝐱 ( 1 ) \mathbf{x}(1)
  17. 𝐱 ( 2 ) , , 𝐱 ( N - m + 1 ) \mathbf{x}(2),\ldots,\mathbf{x}(N-m+1)
  18. i \ i
  19. 1 i N - m + 1 1\leq i\leq N-m+1
  20. C i m ( r ) = ( number of x ( j ) such that d [ x ( i ) , x ( j ) ] < r ) / ( N - m + 1 ) C_{i}^{m}(r)=(\,\text{number of }x(j)\text{ such that }d[x(i),x(j)]<r)/(N-m+1)\,
  21. d [ x , x * ] \ d[x,x^{*}]
  22. d [ x , x * ] = max a | u ( a ) - u * ( a ) | d[x,x^{*}]=\max_{a}|u(a)-u^{*}(a)|\,
  23. u ( a ) \ u(a)
  24. $\text {m}$
  25. 𝐱 \mathbf{x}
  26. d \ d
  27. 𝐱 ( i ) \mathbf{x}(i)
  28. 𝐱 ( j ) \mathbf{x}(j)
  29. j j
  30. i = j i=j
  31. Step 5 \,\text{Step 5}
  32. Φ m ( r ) = ( N - m + 1 ) - 1 i = 1 N - m + 1 l o g ( C i m ( r ) ) \Phi^{m}(r)=(N-m+1)^{-1}\sum_{i=1}^{N-m+1}log(C_{i}^{m}(r))
  33. Step 6 \,\text{Step 6}
  34. ( ApEn ) \ (\mathrm{ApEn})
  35. ApEn = Φ m ( r ) - Φ m + 1 ( r ) . \ \mathrm{ApEn}=\Phi^{m}(r)-\Phi^{m+1}(r).
  36. l o g \ log
  37. m \ m
  38. r \ r
  39. m = 2 \ m=2
  40. m = 3 \ m=3
  41. r \ r
  42. d [ x ( i ) , x ( j ) ] < r d[x(i),x(j)]<r
  43. d [ x ( i ) , x ( j ) ] r d[x(i),x(j)]\leq r
  44. N = 51 \ N=51
  45. S N = { 85 , 80 , 89 , 85 , 80 , 89 , } \ S_{N}=\{85,80,89,85,80,89,\ldots\}
  46. m = 2 \ m=2
  47. r = 3 \ r=3
  48. m \ m
  49. r \ r
  50. 𝐱 ( 1 ) = [ u ( 1 ) u ( 2 ) ] = [ 85 80 ] \mathbf{x}(1)=[u(1)\,u(2)]=[85\,80]
  51. 𝐱 ( 2 ) = [ u ( 2 ) u ( 3 ) ] = [ 80 89 ] \mathbf{x}(2)=[u(2)\,u(3)]=[80\,89]
  52. 𝐱 ( 3 ) = [ u ( 3 ) u ( 4 ) ] = [ 89 85 ] \mathbf{x}(3)=[u(3)\,u(4)]=[89\,85]
  53. 𝐱 ( 4 ) = [ u ( 4 ) u ( 5 ) ] = [ 85 80 ] \mathbf{x}(4)=[u(4)\,u(5)]=[85\,80]
  54. d [ 𝐱 ( 1 ) , 𝐱 ( 1 ) ] = max a | u ( a ) - u * ( a ) | = 0 < r = 3 \ d[\mathbf{x}(1),\mathbf{x}(1)]=\max_{a}|u(a)-u^{*}(a)|=0<r=3
  55. | u ( 2 ) - u ( 3 ) | > | u ( 1 ) - u ( 2 ) | \ |u(2)-u(3)|>|u(1)-u(2)|
  56. d [ 𝐱 ( 1 ) , 𝐱 ( 2 ) ] = max a | u ( a ) - u * ( a ) | = | u ( 2 ) - u ( 3 ) | = 9 > r = 3 \ d[\mathbf{x}(1),\mathbf{x}(2)]=\max_{a}|u(a)-u^{*}(a)|=|u(2)-u(3)|=9>r=3
  57. d [ 𝐱 ( 1 ) , 𝐱 ( 3 ) ] = | u ( 2 ) - u ( 4 ) | = 5 > r \ d[\mathbf{x}(1),\mathbf{x}(3)]=|u(2)-u(4)|=5>r
  58. d [ 𝐱 ( 1 ) , 𝐱 ( 4 ) ] = | u ( 1 ) - u ( 4 ) | = | u ( 2 ) - u ( 5 ) | = 0 < r \ d[\mathbf{x}(1),\mathbf{x}(4)]=|u(1)-u(4)|=|u(2)-u(5)|=0<r
  59. 𝐱 ( j ) s \mathbf{x}(j)\,\text{s}
  60. d [ 𝐱 ( 1 ) , 𝐱 ( j ) ] r \ d[\mathbf{x}(1),\mathbf{x}(j)]\leq r
  61. 𝐱 ( 1 ) , 𝐱 ( 4 ) , 𝐱 ( 7 ) , , 𝐱 ( 49 ) \mathbf{x}(1),\mathbf{x}(4),\mathbf{x}(7),\ldots,\mathbf{x}(49)
  62. C 1 2 ( 3 ) = 17 50 \ C_{1}^{2}(3)=\frac{17}{50}
  63. C 2 2 ( 3 ) = 17 50 \ C_{2}^{2}(3)=\frac{17}{50}
  64. C 3 2 ( 3 ) = 16 50 \ C_{3}^{2}(3)=\frac{16}{50}
  65. C 4 2 ( 3 ) = 17 50 \ C_{4}^{2}(3)=\frac{17}{50}\ \ldots
  66. 𝐱 ( i ) \mathbf{x}(i)
  67. 1 i N - m + 1 \ 1\leq i\leq N-m+1
  68. 𝐱 ( j ) s \mathbf{x}(j)\,\text{s}
  69. d [ 𝐱 ( 3 ) , 𝐱 ( j ) ] < r \ d[\mathbf{x}(3),\mathbf{x}(j)]<r
  70. 𝐱 ( 3 ) , 𝐱 ( 6 ) , 𝐱 ( 9 ) , , 𝐱 ( 48 ) \mathbf{x}(3),\mathbf{x}(6),\mathbf{x}(9),\ldots,\mathbf{x}(48)
  71. Φ 2 ( 3 ) = ( 50 ) - 1 i = 1 50 l o g ( C i 2 ( 3 ) ) - 1.0982 \Phi^{2}(3)=(50)^{-1}\sum_{i=1}^{50}log(C_{i}^{2}(3))\approx-1.0982
  72. 𝐱 ( 1 ) = [ u ( 1 ) u ( 2 ) u ( 3 ) ] = [ 85 80 89 ] \mathbf{x}(1)=[u(1)\,u(2)\,u(3)]=[85\,80\,89]
  73. 𝐱 ( 2 ) = [ u ( 2 ) u ( 3 ) u ( 4 ) ] = [ 80 89 85 ] \mathbf{x}(2)=[u(2)\,u(3)\,u(4)]=[80\,89\,85]
  74. 𝐱 ( 3 ) = [ u ( 3 ) u ( 4 ) u ( 5 ) ] = [ 89 85 80 ] \mathbf{x}(3)=[u(3)\,u(4)\,u(5)]=[89\,85\,80]
  75. 𝐱 ( 4 ) = [ u ( 4 ) u ( 5 ) u ( 6 ) ] = [ 85 80 89 ] \mathbf{x}(4)=[u(4)\,u(5)\,u(6)]=[85\,80\,89]
  76. 𝐱 ( i ) , 𝐱 ( j ) , 1 i 49 \mathbf{x}(i),\mathbf{x}(j),1\leq i\leq 49
  77. d [ 𝐱 ( i ) 𝐱 ( i + 3 ) ] = 0 < r \ d[\mathbf{x}(i)\,\mathbf{x}(i+3)]=0<r
  78. C 1 3 ( 3 ) = 17 49 \ C_{1}^{3}(3)=\frac{17}{49}
  79. C 2 3 ( 3 ) = 16 49 \ C_{2}^{3}(3)=\frac{16}{49}
  80. C 3 3 ( 3 ) = 16 49 \ C_{3}^{3}(3)=\frac{16}{49}
  81. C 4 3 ( 3 ) = 17 49 \ C_{4}^{3}(3)=\frac{17}{49}\ \ldots
  82. Φ 3 ( 3 ) = ( 49 ) - 1 i = 1 49 l o g ( C i 3 ( 3 ) ) - 1.0982 \Phi^{3}(3)=(49)^{-1}\sum_{i=1}^{49}log(C_{i}^{3}(3))\approx-1.0982
  83. ApEn = log ( Φ 2 ( 3 ) ) - log ( Φ 3 ( 3 ) ) 0.000010997 \mathrm{ApEn}=\log(\Phi^{2}(3))-\log(\Phi^{3}(3))\approx 0.000010997

AQUAL.html

  1. m μ ( a / a 0 ) a = G M m / r 2 m\mu(a/a_{0})a=GMm/r^{2}
  2. m M m\neq M
  3. μ ( a m / a 0 ) m a m = G m M / r 2 = G M m / r 2 = μ ( a M / a 0 ) M a M \mu(a_{m}/a_{0})ma_{m}=GmM/r^{2}=GMm/r^{2}=\mu(a_{M}/a_{0})Ma_{M}
  4. m a m = M a M ma_{m}=Ma_{M}
  5. μ ( a m / a 0 ) = μ ( a M / a 0 ) \mu(a_{m}/a_{0})=\mu(a_{M}/a_{0})
  6. a m a M a_{m}\neq a_{M}
  7. μ \mu
  8. ρ Φ + ( 8 π G ) - 1 a 0 2 F ( | Φ | 2 / a 0 2 ) \rho\Phi+(8\pi G)^{-1}a_{0}^{2}F(|\nabla\Phi|^{2}/a_{0}^{2})
  9. ( μ ( | Φ | / a 0 ) Φ ) = 4 π G ρ \nabla\cdot(\mu(|\nabla\Phi|/a_{0})\nabla\Phi)=4\pi G\rho
  10. μ ( x ) = d F ( x 2 ) / d x \mu(x)=dF(x^{2})/dx
  11. - Φ = a -\nabla\Phi=a

Argument_of_a_function.html

  1. f ( x ) = log b ( x ) f(x)=\log_{b}(x)
  2. b b
  3. f ( x ) = x 2 f(x)=x^{2}
  4. f ( x , y ) = x 2 + y 2 f(x,y)=x^{2}+y^{2}
  5. x x
  6. y y
  7. ( x , y ) (x,y)

Arithmetic_surface.html

  1. K K
  2. C / K C/K
  3. S S
  4. R R
  5. p : S Spec ( R ) p:S\rightarrow\mathrm{Spec}(R)
  6. S S
  7. R R
  8. Frac ( R ) \mathrm{Frac}(R)
  9. t t
  10. Spec ( R ) \mathrm{Spec}(R)
  11. S × Spec ( R ) Spec ( k t ) S\underset{\mathrm{Spec}(R)}{\times}\mathrm{Spec}(k_{t})
  12. R / t R/t
  13. X X
  14. S S
  15. O K O_{K}
  16. X X
  17. R R
  18. R R
  19. 𝔪 \mathfrak{m}
  20. R / 𝔪 . R/\mathfrak{m}.

Arithmetic_zeta_function.html

  1. ζ X ( s ) = x 1 1 - N ( x ) - s , {\zeta_{X}(s)}=\prod_{x}\frac{1}{1-N(x)^{-s}},
  2. x x
  3. X X
  4. N ( x ) N(x)
  5. X X
  6. q q
  7. ζ X ( s ) = 1 1 - q - s . \zeta_{X}(s)=\frac{1}{1-q^{-s}}.
  8. X X
  9. X X
  10. X X
  11. ζ 𝐀 n ( X ) ( s ) = ζ X ( s - n ) ζ 𝐏 n ( X ) ( s ) = i = 0 n ζ X ( s - i ) \begin{aligned}\displaystyle\zeta_{\mathbf{A}^{n}(X)}(s)&\displaystyle=\zeta_{% X}(s-n)\\ \displaystyle\zeta_{\mathbf{P}^{n}(X)}(s)&\displaystyle=\prod_{i=0}^{n}\zeta_{% X}(s-i)\end{aligned}
  12. X X
  13. U U
  14. V V
  15. ζ X ( s ) = ζ U ( s ) ζ V ( s ) . \zeta_{X}(s)=\zeta_{U}(s)\zeta_{V}(s).
  16. X X
  17. X X
  18. p p
  19. ζ X ( s ) = p ζ X p ( s ) . \zeta_{X}(s)=\prod_{p}\zeta_{X_{p}}(s).
  20. X X
  21. X X
  22. 𝐙 \mathbf{Z}
  23. p p
  24. 𝐙 \mathbf{Z}
  25. s n s s→n−s
  26. n n
  27. X X
  28. n = 1 n=1
  29. n > 1 n>1
  30. 𝐙 \mathbf{Z}
  31. n n
  32. Re ( s ) > n - 1 2 \mathrm{Re}(s)>n-\tfrac{1}{2}
  33. 0 R e ( s ) n 0≤Re(s)≤n
  34. R e ( s ) = 1 / 2 , 3 / 2 , Re(s)=1/2,3/2,...
  35. 0 R e ( s ) n 0≤Re(s)≤n
  36. R e ( s ) = 0 , 1 , 2 , Re(s)=0,1,2,...
  37. n n
  38. 𝐙 \mathbf{Z}
  39. X X
  40. X X
  41. s = n s=n
  42. X X
  43. ord s = n - 1 ζ X ( s ) = r k 𝒪 X × ( X ) - r k Pic ( X ) \mathrm{ord}_{s=n-1}\zeta_{X}(s)=rk\mathcal{O}_{X}^{\times}(X)-rk\mathrm{Pic}(X)
  44. ord s = n - m ζ X ( s ) = - i ( - 1 ) i r k K i ( X ) ( m ) \mathrm{ord}_{s=n-m}\zeta_{X}(s)=-\sum_{i}(-1)^{i}rkK_{i}(X)^{(m)}
  45. K K
  46. X X
  47. n = 1 n=1
  48. n > 1 n>1
  49. n n
  50. L L
  51. L L
  52. L L
  53. L L

Arterial_resistivity_index.html

  1. R I = v s y s t o l e - v d i a s t o l e v s y s t o l e RI=\frac{v_{systole}-v_{diastole}}{v_{systole}}

Artin_conductor.html

  1. f ( χ ) = i 0 g i g 0 ( χ ( 1 ) - χ ( G i ) ) f(\chi)=\sum_{i\geq 0}\frac{g_{i}}{g_{0}}(\chi(1)-\chi(G_{i}))
  2. f ( χ ) - ( χ ( 1 ) - χ ( G 0 ) ) , f(\chi)-(\chi(1)-\chi(G_{0})),
  3. 𝔣 ( χ ) = p p f ( χ , p ) \mathfrak{f}(\chi)=\prod_{p}p^{f(\chi,p)}
  4. a G = χ f ( χ ) χ a_{G}=\sum_{\chi}f(\chi)\chi
  5. s w G = a G - r G + 1 sw_{G}=a_{G}-r_{G}+1\,

ASAI.html

  1. ASAI = N i × 8760 - U i N i N i × 8760 \mbox{ASAI}~{}=\frac{\sum{N_{i}}\times 8760-\sum{U_{i}N_{i}}}{\sum{N_{i}}% \times 8760}
  2. N i N_{i}
  3. U i U_{i}
  4. i i
  5. ASAI = 1 - SAIDI 8760 \mbox{ASAI}~{}=1-\frac{\mbox{SAIDI}~{}}{8760}

Ash_(Alien).html

  1. s s
  2. - s -s
  3. s ¯ \overline{s}
  4. - s ¯ \overline{-s}

Askey_scheme.html

  1. ϕ \phi
  2. ϕ \phi
  3. ϕ \phi
  4. ϕ \phi
  5. ϕ \phi
  6. ϕ \phi

Aspect_ratio_(image).html

  1. 3 ¯ \overline{3}
  2. 7 ¯ \overline{7}
  3. 3 ¯ \overline{3}
  4. 6 ¯ \overline{6}
  5. 6 ¯ \overline{6}
  6. 3 ¯ \overline{3}
  7. 6 ¯ \overline{6}
  8. 6 ¯ \overline{6}
  9. 3 ¯ \overline{3}
  10. 7 ¯ \overline{7}
  11. h = d r 2 + 1 w = d 1 r 2 + 1 A = d 2 r + 1 r h=\frac{d}{\sqrt{r^{2}+1}}\qquad w=\frac{d}{\sqrt{\frac{1}{r^{2}}+1}}\qquad A=% \frac{d^{2}}{r+\frac{1}{r}}
  12. 3 ¯ \overline{3}
  13. 7 ¯ \overline{7}
  14. 3 ¯ \overline{3}
  15. 7 ¯ \overline{7}
  16. 3 ¯ \overline{3}
  17. 7 ¯ \overline{7}
  18. 3 ¯ \overline{3}
  19. 7 ¯ \overline{7}
  20. 3 ¯ \overline{3}
  21. 3 ¯ \overline{3}
  22. 5 ¯ \overline{5}
  23. 6 ¯ \overline{6}
  24. 7 ¯ \overline{7}
  25. 6 ¯ \overline{6}
  26. 3 ¯ \overline{3}
  27. 3 ¯ \overline{3}
  28. 3 ¯ \overline{3}
  29. 7 ¯ \overline{7}
  30. 3 ¯ \overline{3}
  31. 3 ¯ \overline{3}
  32. 3 ¯ \overline{3}
  33. 6 ¯ \overline{6}
  34. 7 ¯ \overline{7}

ASUI.html

  1. ASUI = U i N i N i × 8760 = 1 - ASAI \mbox{ASUI}~{}=\frac{\sum{U_{i}N_{i}}}{\sum{N_{i}}\times 8760}=1-\mbox{ASAI}~{}
  2. N i N_{i}
  3. U i U_{i}
  4. i i
  5. ASUI = SAIDI 8760 \mbox{ASUI}~{}=\frac{\mbox{SAIDI}~{}}{8760}

Atiyah_algebroid.html

  1. 0 P × G 𝔤 T P / G T M 0. 0\to P\times_{G}\mathfrak{g}\to TP/G\to TM\to 0.
  2. 0 V P T P d π π * T M 0 0\to VP\to TP\xrightarrow{d\pi}\pi^{*}TM\to 0

Atkinson–Mingarelli_theorem.html

  1. - d d x [ p ( x ) d y d x ] + q ( x ) y = λ w ( x ) y , -\frac{d}{dx}\left[p(x)\frac{dy}{dx}\right]+q(x)y=\lambda w(x)y,
  2. α 1 y ( a ) + α 2 y ( a ) = 0 ( α 1 2 + α 2 2 > 0 ) , \alpha_{1}y(a)+\alpha_{2}y^{\prime}(a)=0\qquad\qquad\qquad(\alpha_{1}^{2}+% \alpha_{2}^{2}>0),
  3. β 1 y ( b ) + β 2 y ( b ) = 0 ( β 1 2 + β 2 2 > 0 ) , \beta_{1}y(b)+\beta_{2}y^{\prime}(b)=0\qquad\qquad\qquad(\beta_{1}^{2}+\beta_{% 2}^{2}>0),
  4. { α i , β i } \{\alpha_{i},\beta_{i}\}
  5. ( w / p ) + ( x ) = max { w ( x ) / p ( x ) , 0 } (w/p)_{+}(x)=\max\{w(x)/p(x),0\}
  6. ( w / p ) - ( x ) = max { - w ( x ) / p ( x ) , 0 } ) (w/p)_{-}(x)=\max\{-w(x)/p(x),0\})
  7. λ i + {\lambda_{i}}^{+}
  8. 0 < λ 1 + < λ 2 + < λ 3 + < < λ n + < ; 0<{\lambda_{1}}^{+}<{\lambda_{2}}^{+}<{\lambda_{3}}^{+}<\cdots<{\lambda_{n}}^{% +}<\cdots\to\infty;\,
  9. λ i - {\lambda_{i}}^{-}
  10. 0 > λ 1 - > λ 2 - > λ 3 - > > λ n - > - ; 0>{\lambda_{1}}^{-}>{\lambda_{2}}^{-}>{\lambda_{3}}^{-}>\cdots>{\lambda_{n}}^{% -}>\cdots\to-\infty;\,
  11. λ n + n 2 π 2 ( a b ( w / p ) + ( x ) d x ) 2 , n , {\lambda_{n}}^{+}\sim\frac{n^{2}\pi^{2}}{\left(\int_{a}^{b}\sqrt{(w/p)_{+}(x)}% \,dx\right)^{2}},\quad n\to\infty,
  12. λ n - - n 2 π 2 ( a b ( w / p ) - ( x ) d x ) 2 , n . {\lambda_{n}}^{-}\sim\frac{-n^{2}\pi^{2}}{\left(\int_{a}^{b}\sqrt{(w/p)_{-}(x)% }\,dx\right)^{2}},\quad n\to\infty.
  13. 1 / p , q , w 1/p,\,q,\,w

Atmospheric-pressure_laser_ionization.html

  1. M h ν ( 5 eV ) M * h ν ( 5 eV ) M + + e - \mathrm{M\xrightarrow[]{h\nu\ (5\ \,\text{eV})}M^{*}\xrightarrow[]{h\nu\ (5\ % \,\text{eV})}M^{+\cdot}+e^{-}}

Attractor_network.html

  1. x ( t + 1 ) = f ( W x ( t ) ) x(t+1)=f(Wx(t))
  2. x x
  3. W W
  4. d x d t = - λ x + f ( W x ) \frac{dx}{dt}=-\lambda x+f(Wx)
  5. W W
  6. q i ( t ) = π i ( y ( t ) , w i , σ ( t ) ) j π j g ( y ( t ) , w j , σ ( t ) ) q_{i}(t)=\frac{\pi_{i}(y(t),w_{i},\sigma(t))}{\sum_{j}\pi_{j}g(y(t),w_{j},% \sigma(t))}
  7. y ( t + 1 ) = α ( t ) ξ + ( 1 - α ( t ) ) i q i ( t ) w i y(t+1)=\alpha(t)\xi+(1-\alpha(t))\sum_{i}q_{i}(t)w_{i}\,\!
  8. σ y 2 ( t ) = 1 n i q i ( t ) | y ( t ) - w i | 2 \sigma^{2}_{y}(t)=\frac{1}{n}\sum_{i}q_{i}(t)|y(t)-w_{i}|^{2}
  9. π i \pi_{i}
  10. w i w_{i}
  11. ξ \xi
  12. α \alpha
  13. w i ( t + 1 ) = v q i ( t ) y ( t ) + [ 1 - v q i ( t ) ] w i ( t ) w_{i}(t+1)=vq_{i}(t)\cdot y(t)+[1-vq_{i}(t)]\cdot w_{i}(t)\,\!
  14. v v
  15. w i w_{i}

Augmented_Lagrangian_method.html

  1. min f ( 𝐱 ) \min f(\mathbf{x})
  2. c i ( 𝐱 ) = 0 i I . c_{i}(\mathbf{x})=0~{}\forall i\in I.
  3. min Φ k ( 𝐱 ) = f ( 𝐱 ) + μ k i I c i ( 𝐱 ) 2 \min\Phi_{k}(\mathbf{x})=f(\mathbf{x})+\mu_{k}~{}\sum_{i\in I}~{}c_{i}(\mathbf% {x})^{2}
  4. μ k \mu_{k}
  5. min Φ k ( 𝐱 ) = f ( 𝐱 ) + μ k 2 i I c i ( 𝐱 ) 2 - i I λ i c i ( 𝐱 ) \min\Phi_{k}(\mathbf{x})=f(\mathbf{x})+\frac{\mu_{k}}{2}~{}\sum_{i\in I}~{}c_{% i}(\mathbf{x})^{2}-\sum_{i\in I}~{}\lambda_{i}c_{i}(\mathbf{x})
  6. μ k \mu_{k}
  7. λ \lambda
  8. λ i λ i - μ k c i ( x k ) \lambda_{i}\leftarrow\lambda_{i}-\mu_{k}c_{i}({x}_{k})
  9. x k {x}_{k}
  10. x k = argmin Φ k ( 𝐱 ) {x}_{k}=\,\text{argmin}\Phi_{k}(\mathbf{x})
  11. λ \lambda
  12. μ \mu\rightarrow\infty
  13. μ \mu
  14. μ \mu
  15. min x f ( x ) + g ( x ) . \min_{x}f(x)+g(x).
  16. min x , y f ( x ) + g ( y ) , subject to x = y . \min_{x,y}f(x)+g(y),\quad\,\text{subject to}\quad x=y.
  17. ^ ρ , k = f 1 ( x k ) + < f ( x k , ζ k + 1 ) , x > + g ( y ) - z T ( A x + B y - c ) + ρ 2 A x + B y - c 2 + x - x k 2 2 η k + 1 , \hat{\mathcal{L}}_{\rho,k}=f_{1}(x_{k})+<\nabla f(x_{k},\zeta_{k+1}),x>+g(y)-z% ^{T}(Ax+By-c)+\frac{\rho}{2}\|Ax+By-c\|^{2}+\frac{\|x-x_{k}\|^{2}}{2\eta_{k+1}},
  18. η k + 1 \eta_{k+1}

August_Davidov.html

  1. e m - 1 = cos m + sin m - 1 e^{m\sqrt{-1}}=\cos m+\sin m\sqrt{-1}

Auslander–Buchsbaum_formula.html

  1. pd R ( M ) + depth ( M ) = depth ( R ) . \mathrm{pd}_{R}(M)+\mathrm{depth}(M)=\mathrm{depth}(R).

Auslander–Reiten_theory.html

  1. 0 k [ x ] / ( x m ) k [ x ] / ( x m + 1 ) k [ x ] / ( x m - 1 ) k [ x ] / ( x m ) 0 0\rightarrow k[x]/(x^{m})\rightarrow k[x]/(x^{m+1})\oplus k[x]/(x^{m-1})% \rightarrow k[x]/(x^{m})\rightarrow 0

Autocorrelation_matrix.html

  1. R x x ( j ) R_{xx}(j)
  2. 𝐑 x = E [ 𝐱𝐱 H ] = [ R x x ( 0 ) R x x * ( 1 ) R x x * ( 2 ) R x x * ( N - 1 ) R x x ( 1 ) R x x ( 0 ) R x x * ( 1 ) R x x * ( N - 2 ) R x x ( 2 ) R x x ( 1 ) R x x ( 0 ) R x x * ( N - 3 ) R x x ( N - 1 ) R x x ( N - 2 ) R x x ( N - 3 ) R x x ( 0 ) ] \mathbf{R}_{x}=E[\mathbf{xx}^{H}]=\begin{bmatrix}R_{xx}(0)&R^{*}_{xx}(1)&R^{*}% _{xx}(2)&\cdots&R^{*}_{xx}(N-1)\\ R_{xx}(1)&R_{xx}(0)&R^{*}_{xx}(1)&\cdots&R^{*}_{xx}(N-2)\\ R_{xx}(2)&R_{xx}(1)&R_{xx}(0)&\cdots&R^{*}_{xx}(N-3)\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ R_{xx}(N-1)&R_{xx}(N-2)&R_{xx}(N-3)&\cdots&R_{xx}(0)\\ \end{bmatrix}
  3. 𝐱 \mathbf{x}
  4. 𝐂 x = E [ ( 𝐱 - 𝐦 x ) ( 𝐱 - 𝐦 x ) H ] = 𝐑 x - 𝐦 x 𝐦 x H \mathbf{C}_{x}=\operatorname{E}[(\mathbf{x}-\mathbf{m}_{x})(\mathbf{x}-\mathbf% {m}_{x})^{H}]=\mathbf{R}_{x}-\mathbf{m}_{x}\mathbf{m}_{x}^{H}
  5. 𝐦 x \mathbf{m}_{x}
  6. 𝐱 \mathbf{x}

Automatic_basis_function_construction.html

  1. s , p , γ , r {s,p,\gamma,r}
  2. S = < m t p l > 1 , 2 , , s S=<mtpl>{{1,2,\ldots,s}}
  3. r r
  4. γ [ 0 , 1 ) \gamma\in[0,1)
  5. P P
  6. v = r + γ P v . v=r+\gamma Pv.\,
  7. S S
  8. v v
  9. S S
  10. v v
  11. Φ = ϕ 1 , ϕ 2 , , ϕ n \Phi={\phi_{1},\phi_{2},\ldots,\phi_{n}}
  12. v v ^ = i = 1 n θ n ϕ n v\approx\hat{v}=\sum_{i=1}^{n}\theta_{n}\phi_{n}
  13. Φ \Phi
  14. | S | × n |S|\times n
  15. θ \theta
  16. n | s | n\ll|s|
  17. Φ \Phi
  18. L = I - D - 1 2 W D - 1 2 L=I-D^{-\frac{1}{2}}WD^{-\frac{1}{2}}
  19. W W
  20. y i = P i r y_{i}=P^{i}r
  21. i [ 0 , i n f t y ) i\in[0,infty)
  22. y 0 , y 1 , , y m - 1 y_{0},y_{1},\ldots,y_{m-1}
  23. ( I - γ P ) (I-\gamma P)
  24. p ( A ) = 1 α 0 i = 0 m - 1 α i + 1 A i p(A)=\frac{1}{\alpha_{0}}\sum_{i=0}^{m-1}\alpha_{i+1}A^{i}
  25. B A = I BA=I
  26. v = B r = 1 α 0 i = 0 m - 1 α i + 1 ( I - γ P ) i r = i = 0 m - 1 α i + 1 β i y i . v=Br=\frac{1}{\alpha_{0}}\sum_{i=0}^{m-1}\alpha_{i+1}(I-\gamma P)^{i}r=\sum_{i% =0}^{m-1}\alpha_{i+1}\beta_{i}y_{i}.
  27. z 1 , z 2 , , z k z_{1},z_{2},\ldots,z_{k}
  28. z k + 1 := r z_{k+1}:=r
  29. i := 1 : ( l + k ) i:=1:(l+k)
  30. i > k + 1 i>k+1
  31. z i := P z i - 1 z_{i}:=Pz_{i-1}
  32. j := 1 : ( i - 1 ) j:=1:(i-1)
  33. z i := z i - < z j , z i > z j ; z_{i}:=z_{i}-<z_{j},z_{i}>z_{j};
  34. z i 0 \parallel z_{i}\parallel\approx 0
  35. ε = r + γ P v ^ - v ^ = r + γ P Φ θ - Φ θ \varepsilon=r+\gamma P\hat{v}-\hat{v}=r+\gamma P\Phi\theta-\Phi\theta
  36. ϕ 1 = r \phi_{1}=r
  37. i [ 2 , N ] i\in[2,N]
  38. θ i \theta_{i}
  39. Φ i \Phi_{i}
  40. ε = r + γ P Φ i θ i - Φ i θ i \varepsilon=r+\gamma P\Phi_{i}\theta_{i}-\Phi_{i}\theta_{i}
  41. Φ i + 1 = [ Φ i : ε ] \Phi_{i+1}=[\Phi_{i}:\varepsilon]
  42. P - P * P-P^{*}
  43. P * P^{*}
  44. γ \gamma
  45. P * r P^{*}r
  46. i [ 2 , N ] i\in[2,N]
  47. θ i \theta_{i}
  48. Φ i \Phi_{i}
  49. : ϕ i + 1 = r - P * r + P Φ i θ i - Φ i θ i :\phi_{i+1}=r-P^{*}r+P\Phi_{i}\theta_{i}-\Phi_{i}\theta_{i}
  50. Φ i + 1 = [ Φ i : ϕ i + 1 ] \Phi_{i+1}=[\Phi_{i}:\phi_{i+1}]
  51. γ \gamma
  52. γ \gamma

Automatic_Generation_Control.html

  1. Δ p m = Δ p r e f - 1 / R × Δ f \Delta p_{m}=\Delta p_{ref}-1/R\times\Delta f
  2. Δ p m \Delta p_{m}
  3. Δ p r e f \Delta p_{ref}
  4. R = - Δ f / Δ p m = - s l o p e R=-\Delta f/\Delta p_{m}=-slope
  5. Δ f \Delta f

Availability_(system).html

  1. A o = T m T m + T d { A o = O p e r a t i o n a l A v a i l a b i l i t y T m = M i s s i o n D u r a t i o n T d = O b s e r v e d D o w n T i m e A_{o}=\frac{T_{m}}{T_{m}+T_{d}}\begin{cases}A_{o}=Operational\ Availability\\ T_{m}=Mission\ Duration\\ T_{d}=Observed\ Down\ Time\end{cases}
  2. A p = T m T m + T d { A p = P r e d i c t e d A v a i l a b i l i t y T m = M i s s i o n D u r a t i o n T d = M o d e l D o w n T i m e A_{p}=\frac{T_{m}}{T_{m}+T_{d}}\begin{cases}A_{p}=Predicted\ Availability\\ T_{m}=Mission\ Duration\\ T_{d}=Model\ Down\ Time\end{cases}
  3. T d = T m × M T T R + M L D T + M A M D T M T B F { T d = D o w n T i m e T m = M i s s i o n D u r a t i o n M T T R = M e a n T i m e T o R e c o v e r M L D T = M e a n L o g i s t i c s D e l a y T i m e M A M D T = M e a n A c t i v e M a i n t e n a n c e D o w n T i m e M T B F = M e a n T i m e B e t w e e n F a i l u r e T_{d}=T_{m}\times\frac{MTTR+MLDT+MAMDT}{MTBF}\begin{cases}T_{d}=Down\ Time\\ T_{m}=Mission\ Duration\\ MTTR=Mean\ Time\ To\ Recover\\ MLDT=Mean\ Logistics\ Delay\ Time\\ MAMDT=Mean\ Active\ Maintenance\ Down\ Time\\ MTBF=Mean\ Time\ Between\ Failure\end{cases}
  4. λ \lambda
  5. λ = 1 M T B F \lambda=\frac{1}{MTBF}
  6. P F = 1 - A o { P F = P r o b a b i l i t y o f M i s s i o n F a i l u r e A o = O p e r a t i o n a l A v a i l a b i l i t y P_{F}=1-A_{o}\begin{cases}P_{F}=Probability\ of\ Mission\ Failure\\ A_{o}=Operational\ Availability\end{cases}

Avner_Magen.html

  1. ( O ( l o g l o g n ) ) (O(loglogn))

B_Integral.html

  1. B = 2 π λ n 2 I ( z ) d z B=\frac{2\pi}{\lambda}\int\!n_{2}I(z)\,dz\,
  2. I ( z ) I(z)
  3. z z
  4. n 2 n_{2}
  5. n 2 I ( z ) n_{2}I(z)

Babel_function.html

  1. s y m b o l A symbol{A}
  2. μ ( p ) = max | λ | = p { max j λ { i λ | s y m b o l a i s y m b o l T s y m b o l a j | } } \mu(p)=\max_{|\lambda|=p}\{\max_{j\notin\lambda}\{\sum_{i\in\lambda}{|symbol{a% }_{i}^{symbol{T}}symbol{a}_{j}|}\}\}
  3. s y m b o l a k symbol{a}_{k}
  4. s y m b o l A symbol{A}

Backpressure_routing.html

  1. t { 0 , 1 , 2 , } t\in\{0,1,2,\ldots\}
  2. c { 1 , , N } c\in\{1,\dots,N\}
  3. n { 1 , , N } n\in\{1,\ldots,N\}
  4. c { 1 , , N } c\in\{1,\ldots,N\}
  5. Q n ( c ) ( t ) Q_{n}^{(c)}(t)
  6. Q n ( c ) ( t ) Q_{n}^{(c)}(t)
  7. Q c ( c ) ( t ) = 0 Q_{c}^{(c)}(t)=0
  8. c { 1 , , N } c\in\{1,\ldots,N\}
  9. A n ( c ) ( t ) A_{n}^{(c)}(t)
  10. μ a b ( t ) \mu_{ab}(t)
  11. ( μ a b ( t ) ) (\mu_{ab}(t))
  12. Γ S ( t ) \Gamma_{S(t)}
  13. ( μ a b ( t ) ) (\mu_{ab}(t))
  14. Γ S ( t ) \Gamma_{S(t)}
  15. ( μ a b ( t ) ) (\mu_{ab}(t))
  16. c a b o p t ( t ) c_{ab}^{opt}(t)
  17. ( μ a b ( t ) ) (\mu_{ab}(t))
  18. Γ S ( t ) \Gamma_{S(t)}
  19. c a b o p t ( t ) c_{ab}^{opt}(t)
  20. μ a b ( t ) \mu_{ab}(t)
  21. μ a b ( t ) \mu_{ab}(t)
  22. Γ S ( t ) \Gamma_{S(t)}
  23. Γ S ( t ) \Gamma_{S(t)}
  24. c a b o p t ( t ) c_{ab}^{opt}(t)
  25. c { 1 , , N } c\in\{1,\ldots,N\}
  26. Q a ( c ) ( t ) - Q b ( c ) ( t ) Q_{a}^{(c)}(t)-Q_{b}^{(c)}(t)
  27. Q 1 ( red ) ( t ) - Q 2 ( red ) ( t ) = 1 Q_{1}^{(\,\text{red})}(t)-Q_{2}^{(\,\text{red})}(t)=1
  28. Q 1 ( green ) ( t ) - Q 2 ( green ) ( t ) = 2 Q_{1}^{(\,\text{green})}(t)-Q_{2}^{(\,\text{green})}(t)=2
  29. Q 1 ( blue ) ( t ) - Q 2 ( b l u e ) ( t ) = - 1 Q_{1}^{(\,\text{blue})}(t)-Q_{2}^{(blue)}(t)=-1
  30. W a b ( t ) W_{ab}(t)
  31. W a b ( t ) = max [ Q a ( c a b opt ( t ) ) ( t ) - Q b ( c a b o p t ( t ) ) ( t ) , 0 ] W_{ab}(t)=\max\left[Q_{a}^{(c_{ab}^{\mathrm{opt}}(t))}(t)-Q_{b}^{(c_{ab}^{opt}% (t))}(t),0\right]
  32. W a b ( t ) W_{ab}(t)
  33. (Eq. 1) Maximize: a = 1 N b = 1 N μ a b ( t ) W a b ( t ) \,\text{(Eq. 1)}\qquad\,\text{Maximize: }\sum_{a=1}^{N}\sum_{b=1}^{N}\mu_{ab}(% t)W_{ab}(t)
  34. (Eq. 2) Subject to: ( μ a b ( t ) ) Γ S ( t ) \,\text{(Eq. 2)}\qquad\,\text{Subject to: }(\mu_{ab}(t))\in\Gamma_{S(t)}
  35. W a b ( t ) W_{ab}(t)
  36. ( W a b ( t ) ) = [ 0 2 1 1 6 0 1 0 1 2 5 6 0 7 0 0 0 0 1 0 1 0 0 0 1 0 7 5 0 0 0 0 0 0 5 0 ] (W_{ab}(t))=\left[\begin{array}[]{cccccc}0&2&1&1&6&0\\ 1&0&1&2&5&6\\ 0&7&0&0&0&0\\ 1&0&1&0&0&0\\ 1&0&7&5&0&0\\ 0&0&0&0&5&0\end{array}\right]
  37. Γ S ( t ) \Gamma_{S(t)}
  38. Γ S ( t ) = { s y m b o l μ a , s y m b o l μ b , s y m b o l μ c , s y m b o l μ d } \Gamma_{S(t)}=\{symbol{\mu}_{a},symbol{\mu}_{b},symbol{\mu}_{c},symbol{\mu}_{d}\}
  39. μ 15 = 2 \mu_{15}=2
  40. s y m b o l μ a = [ 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] , s y m b o l μ b = [ 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ] symbol{\mu}_{a}=\left[\begin{array}[]{cccccc}0&0&0&0&2&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\end{array}\right],\quad symbol{\mu}_{b}=\left[\begin{array}[]{% cccccc}0&0&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\end{array}\right]
  41. s y m b o l μ c = [ 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ] , s y m b o l μ d = [ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ] symbol{\mu}_{c}=\left[\begin{array}[]{cccccc}0&0&0&0&0&0\\ 1&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\end{array}\right],\quad symbol{\mu}_{d}=\left[\begin{array}[]{% cccccc}0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&1&0&0\\ 0&0&0&0&0&0\end{array}\right]
  42. a b W a b ( t ) μ a b ( t ) = 12 \sum_{ab}W_{ab}(t)\mu_{ab}(t)=12
  43. a b W a b ( t ) μ a b ( t ) = 1 \sum_{ab}W_{ab}(t)\mu_{ab}(t)=1
  44. a b W a b ( t ) μ a b ( t ) = 1 \sum_{ab}W_{ab}(t)\mu_{ab}(t)=1
  45. a b W a b ( t ) μ a b ( t ) = 12 \sum_{ab}W_{ab}(t)\mu_{ab}(t)=12
  46. s y m b o l μ a symbol{\mu}_{a}
  47. s y m b o l μ d symbol{\mu}_{d}
  48. c a b o p t ( t ) c_{ab}^{opt}(t)
  49. ( μ a b ( t ) ) (\mu_{ab}(t))
  50. μ a b ( t ) \mu_{ab}(t)
  51. c a b opt ( t ) c_{ab}^{\mathrm{opt}}(t)
  52. μ a b ( c ) ( t ) \mu_{ab}^{(c)}(t)
  53. μ a b ( c ) ( t ) = { μ a b ( t ) if c = c a b o p t ( t ) and Q a ( c a b o p t ( t ) ) ( t ) - Q b ( c a b o p t ( t ) ) ( t ) 0 0 otherwise \mu_{ab}^{(c)}(t)=\left\{\begin{array}[]{ll}\mu_{ab}(t)&\mbox{ if }~{}c=c_{ab}% ^{opt}(t)\mbox{ and }~{}Q_{a}^{(c_{ab}^{opt}(t))}(t)-Q_{b}^{(c_{ab}^{opt}(t))}% (t)\geq 0\\ 0&\mbox{ otherwise}\end{array}\right.
  54. μ a b ( c ) ( t ) \mu_{ab}^{(c)}(t)
  55. Q n ( c ) ( t ) < b = 1 N μ n b ( c ) ( t ) Q_{n}^{(c)}(t)<\sum_{b=1}^{N}\mu_{nb}^{(c)}(t)
  56. Q n ( c ) ( t ) Q_{n}^{(c)}(t)
  57. ( μ a b ( t ) ) (\mu_{ab}(t))
  58. μ a b ( c ) ( t ) \mu_{ab}^{(c)}(t)
  59. μ n b ( t ) \mu_{nb}(t)
  60. ( μ n b ( c ) ( t ) ) (\mu_{nb}^{(c)}(t))
  61. ( μ a b ( t ) ) (\mu_{ab}(t))
  62. ( μ a b ( c ) ( t ) ) (\mu_{ab}^{(c)}(t))
  63. (Eq. 3) ( μ a b ( t ) ) Γ S ( t ) \,\text{(Eq. 3)}\qquad(\mu_{ab}(t))\in\Gamma_{S(t)}
  64. (Eq. 4) 0 μ a b ( c ) ( t ) a , b , c , t \,\text{(Eq. 4)}\qquad 0\leq\mu_{ab}^{(c)}(t)\qquad\forall a,b,c,\forall t
  65. (Eq. 5) c = 1 N μ a b ( c ) ( t ) μ a b ( t ) ( a , b ) , t \,\text{(Eq. 5)}\qquad\sum_{c=1}^{N}\mu_{ab}^{(c)}(t)\leq\mu_{ab}(t)\qquad% \forall(a,b),\forall t
  66. (Eq. 6) Q n ( c ) ( t + 1 ) max [ Q n ( c ) ( t ) - b = 1 N μ n b ( c ) ( t ) , 0 ] + a = 1 N μ a n ( c ) ( t ) + A n ( c ) ( t ) \,\text{(Eq. 6)}\qquad Q_{n}^{(c)}(t+1)\leq\max\left[Q_{n}^{(c)}(t)-\sum_{b=1}% ^{N}\mu_{nb}^{(c)}(t),0\right]+\sum_{a=1}^{N}\mu_{an}^{(c)}(t)+A_{n}^{(c)}(t)
  67. A n ( c ) ( t ) A_{n}^{(c)}(t)
  68. μ n b ( c ) ( t ) \mu_{nb}^{(c)}(t)
  69. μ n b ( c ) ( t ) \mu_{nb}^{(c)}(t)
  70. a = 1 N μ a n ( c ) ( t ) \sum_{a=1}^{N}\mu_{an}^{(c)}(t)
  71. μ a b ( c ) ( t ) \mu_{ab}^{(c)}(t)
  72. Q c ( c ) ( t ) = 0 Q_{c}^{(c)}(t)=0
  73. c { 1 , , N } c\in\{1,\ldots,N\}
  74. s y m b o l Q ( t ) = ( Q n ( c ) ( t ) ) symbol{Q}(t)=(Q_{n}^{(c)}(t))
  75. L ( t ) = 1 2 n = 1 N c = 1 N Q n ( c ) ( t ) 2 L(t)=\frac{1}{2}\sum_{n=1}^{N}\sum_{c=1}^{N}Q_{n}^{(c)}(t)^{2}
  76. n c n\neq c
  77. Q c ( c ) ( t ) = 0 Q_{c}^{(c)}(t)=0
  78. c { 1 , , N } c\in\{1,\ldots,N\}
  79. Δ ( t ) \Delta(t)
  80. Δ ( t ) = E [ L ( t + 1 ) - L ( t ) | s y m b o l Q ( t ) ] \Delta(t)=E\left[L(t+1)-L(t)|symbol{Q}(t)\right]
  81. q 0 q\geq 0
  82. a 0 a\geq 0
  83. b 0 b\geq 0
  84. ( max [ q - b , 0 ] + a ) 2 q 2 + b 2 + a 2 + 2 q ( a - b ) (\max[q-b,0]+a)^{2}\leq q^{2}+b^{2}+a^{2}+2q(a-b)
  85. ( μ a b ( t ) ) (\mu_{ab}(t))
  86. ( μ a b ( c ) ( t ) ) (\mu_{ab}^{(c)}(t))
  87. (Eq. 7) Δ ( t ) B + n = 1 N c = 1 N Q n ( c ) ( t ) E [ λ n ( c ) ( t ) + a = 1 N μ a n ( c ) ( t ) - b = 1 N μ n b ( c ) ( t ) | s y m b o l Q ( t ) ] \,\text{(Eq. 7)}\qquad\Delta(t)\leq B+\sum_{n=1}^{N}\sum_{c=1}^{N}Q_{n}^{(c)}(% t)E\left[\lambda_{n}^{(c)}(t)+\sum_{a=1}^{N}\mu_{an}^{(c)}(t)-\sum_{b=1}^{N}% \mu_{nb}^{(c)}(t)|symbol{Q}(t)\right]
  88. s y m b o l Q ( t ) symbol{Q}(t)
  89. ( μ a b ( t ) ) (\mu_{ab}(t))
  90. ( μ a b ( c ) ( t ) ) (\mu_{ab}^{(c)}(t))
  91. λ n ( c ) \lambda_{n}^{(c)}
  92. E [ n = 1 N c = 1 N Q n ( c ) ( t ) [ b = 1 N μ n b ( c ) ( t ) - a = 1 N μ a n ( c ) ( t ) ] | s y m b o l Q ( t ) ] E\left[\sum_{n=1}^{N}\sum_{c=1}^{N}Q_{n}^{(c)}(t)\left[\sum_{b=1}^{N}\mu_{nb}^% {(c)}(t)-\sum_{a=1}^{N}\mu_{an}^{(c)}(t)\right]|symbol{Q}(t)\right]
  93. s y m b o l Q ( t ) symbol{Q}(t)
  94. S ( t ) S(t)
  95. ( μ a b ( t ) ) (\mu_{ab}(t))
  96. ( μ a b ( c ) ( t ) ) (\mu_{ab}^{(c)}(t))
  97. n = 1 N c = 1 N Q n ( c ) ( t ) [ b = 1 N μ n b ( c ) ( t ) - a = 1 N μ a n ( c ) ( t ) ] \sum_{n=1}^{N}\sum_{c=1}^{N}Q_{n}^{(c)}(t)\left[\sum_{b=1}^{N}\mu_{nb}^{(c)}(t% )-\sum_{a=1}^{N}\mu_{an}^{(c)}(t)\right]
  98. a = 1 N b = 1 N c = 1 N μ a b ( c ) ( t ) [ Q a ( c ) ( t ) - Q b ( c ) ( t ) ] \sum_{a=1}^{N}\sum_{b=1}^{N}\sum_{c=1}^{N}\mu_{ab}^{(c)}(t)[Q_{a}^{(c)}(t)-Q_{% b}^{(c)}(t)]
  99. Q a ( c ) ( t ) - Q b ( c ) ( t ) Q_{a}^{(c)}(t)-Q_{b}^{(c)}(t)
  100. ( μ a b ( c ) ( t ) ) (\mu_{ab}^{(c)}(t))
  101. μ a b ( c ) ( t ) = 0 \mu_{ab}^{(c)}(t)=0
  102. Q a ( c ) ( t ) - Q b ( c ) ( t ) < 0 Q_{a}^{(c)}(t)-Q_{b}^{(c)}(t)<0
  103. μ a b ( t ) \mu_{ab}(t)
  104. ( a , b ) (a,b)
  105. μ a b ( c ) ( t ) \mu_{ab}^{(c)}(t)
  106. c a b o p t ( t ) { 1 , , N } c_{ab}^{opt}(t)\in\{1,\ldots,N\}
  107. μ a b ( c ) ( t ) = 0 \mu_{ab}^{(c)}(t)=0
  108. c { 1 , , N } c\in\{1,\ldots,N\}
  109. μ a b ( t ) \mu_{ab}(t)
  110. c a b o p t ( t ) c_{ab}^{opt}(t)
  111. c = 1 N μ a b ( c ) ( t ) [ Q a ( c ) ( t ) - Q b ( c ) ( t ) ] = μ a b ( t ) W a b ( t ) \sum_{c=1}^{N}\mu_{ab}^{(c)}(t)[Q_{a}^{(c)}(t)-Q_{b}^{(c)}(t)]=\mu_{ab}(t)W_{% ab}(t)
  112. W a b ( t ) W_{ab}(t)
  113. W a b ( t ) = max [ Q a ( c a b o p t ( t ) ) ( t ) - Q b ( c a b o p t ( t ) ) ( t ) , 0 ] W_{ab}(t)=\max[Q_{a}^{(c_{ab}^{opt}(t))}(t)-Q_{b}^{(c_{ab}^{opt}(t))}(t),0]
  114. ( μ a b ( t ) ) Γ S ( t ) (\mu_{ab}(t))\in\Gamma_{S(t)}
  115. Maximize : a = 1 N b = 1 N μ a b ( t ) W a b ( t ) \mathrm{Maximize:}\sum_{a=1}^{N}\sum_{b=1}^{N}\mu_{ab}(t)W_{ab}(t)
  116. Subjectto : ( μ a b ( t ) ) Γ S ( t ) \mathrm{Subjectto:}(\mu_{ab}(t))\in\Gamma_{S(t)}
  117. ( μ a b ( t ) ) (\mu_{ab}(t))
  118. ( μ a b ( c ) ( t ) ) (\mu_{ab}^{(c)}(t))
  119. s y m b o l Q ( t ) symbol{Q}(t)
  120. ( λ n ( c ) ) (\lambda_{n}^{(c)})
  121. π S = P r [ S ( t ) = S ] \pi_{S}=Pr[S(t)=S]
  122. ( A n ( c ) ( t ) ) (A_{n}^{(c)}(t))
  123. λ n ( c ) = E [ A n ( c ) ( t ) ] \lambda_{n}^{(c)}=E\left[A_{n}^{(c)}(t)\right]
  124. λ c ( c ) = 0 \lambda_{c}^{(c)}=0
  125. c { 1 , , N } c\in\{1,\ldots,N\}
  126. ( λ n ( c ) ) (\lambda_{n}^{(c)})
  127. N × N N\times N
  128. π S = P r [ S ( t ) = S ] \pi_{S}=Pr[S(t)=S]
  129. π S \pi_{S}
  130. ( μ a b ( t ) ) (\mu_{ab}(t))
  131. ( μ a b ( c ) ( t ) ) (\mu_{ab}^{(c)}(t))
  132. Λ \Lambda
  133. ( λ n ( c ) ) (\lambda_{n}^{(c)})
  134. ( λ n ( c ) ) (\lambda_{n}^{(c)})
  135. Λ \Lambda
  136. ( μ a b * ( t ) ) (\mu_{ab}^{*}(t))
  137. ( μ a b * ( c ) ( t ) ) (\mu_{ab}^{*(c)}(t))
  138. n c n\neq c
  139. (Eq. 8) E [ λ n ( c ) + a = 1 N μ a n * ( c ) ( t ) - b = 1 N μ n b * ( c ) ( t ) ] 0 \,\text{(Eq. 8)}\qquad E\left[\lambda_{n}^{(c)}+\sum_{a=1}^{N}\mu_{an}^{*(c)}(% t)-\sum_{b=1}^{N}\mu_{nb}^{*(c)}(t)\right]\leq 0
  140. ( λ n ( c ) ) (\lambda_{n}^{(c)})
  141. Λ \Lambda
  142. ϵ > 0 \epsilon>0
  143. ( λ n ( c ) + ϵ 1 n ( c ) ) Λ (\lambda_{n}^{(c)}+\epsilon 1_{n}^{(c)})\in\Lambda
  144. 1 n ( c ) 1_{n}^{(c)}
  145. n c n\neq c
  146. n c n\neq c
  147. (Eq. 9) E [ λ n ( c ) + a = 1 N μ a n * ( c ) ( t ) - b = 1 N μ n b * ( c ) ( t ) ] - ϵ \,\text{(Eq. 9)}\qquad E\left[\lambda_{n}^{(c)}+\sum_{a=1}^{N}\mu_{an}^{*(c)}(% t)-\sum_{b=1}^{N}\mu_{nb}^{*(c)}(t)\right]\leq-\epsilon
  148. μ a b ( t ) \mu_{ab}(t)
  149. μ m a x \mu_{max}
  150. s y m b o l Q ( t ) symbol{Q}(t)
  151. ( μ a b ( t ) ) (\mu_{ab}(t))
  152. ( μ a b ( c ) ( t ) ) (\mu_{ab}^{(c)}(t))
  153. (Eq. 10) Δ ( t ) B + n = 1 N c = 1 N Q n ( c ) ( t ) E [ λ n ( c ) ( t ) + a = 1 N μ a n * ( c ) ( t ) - b = 1 N μ n b * ( c ) ( t ) | s y m b o l Q ( t ) ] \,\text{(Eq. 10)}\qquad\Delta(t)\leq B+\sum_{n=1}^{N}\sum_{c=1}^{N}Q_{n}^{(c)}% (t)E\left[\lambda_{n}^{(c)}(t)+\sum_{a=1}^{N}\mu_{an}^{*(c)}(t)-\sum_{b=1}^{N}% \mu_{nb}^{*(c)}(t)|symbol{Q}(t)\right]
  154. ( μ a b * ( t ) ) (\mu_{ab}^{*}(t))
  155. ( μ a b * ( c ) ( t ) ) (\mu_{ab}^{*(c)}(t))
  156. ( λ n ( c ) ) Λ (\lambda_{n}^{(c)})\in\Lambda
  157. s y m b o l Q ( t ) symbol{Q}(t)
  158. Δ ( t ) B \Delta(t)\leq B\,
  159. lim t Q n ( c ) ( t ) t = 0 with probability 1 \lim_{t\rightarrow\infty}\frac{Q_{n}^{(c)}(t)}{t}=0\,\text{ with probability 1}
  160. ( λ n ( c ) ) (\lambda_{n}^{(c)})
  161. Λ \Lambda
  162. ϵ > 0 \epsilon>0
  163. Δ ( t ) B - ϵ n = 1 N c = 1 N Q n ( c ) ( t ) \Delta(t)\leq B-\epsilon\sum_{n=1}^{N}\sum_{c=1}^{N}Q_{n}^{(c)}(t)
  164. lim sup t 1 t τ = 0 t - 1 n = 1 N c = 1 N E [ Q n ( c ) ( τ ) ] B ϵ \limsup_{t\rightarrow\infty}\frac{1}{t}\sum_{\tau=0}^{t-1}\sum_{n=1}^{N}\sum_{% c=1}^{N}E\left[Q_{n}^{(c)}(\tau)\right]\leq\frac{B}{\epsilon}
  165. ϵ \epsilon
  166. Λ \Lambda
  167. λ \lambda
  168. μ \mu
  169. 1 / ϵ 1/\epsilon
  170. ϵ = μ - λ \epsilon=\mu-\lambda
  171. ( λ n ( c ) ) Λ (\lambda_{n}^{(c)})\in\Lambda

Backpropagation_through_time.html

  1. 𝐚 0 , 𝐲 0 , 𝐚 1 , 𝐲 1 , 𝐚 2 , 𝐲 2 , , 𝐚 n - 1 , 𝐲 n - 1 \langle\mathbf{a}_{0},\mathbf{y}_{0}\rangle,\langle\mathbf{a}_{1},\mathbf{y}_{% 1}\rangle,\langle\mathbf{a}_{2},\mathbf{y}_{2}\rangle,...,\langle\mathbf{a}_{n% -1},\mathbf{y}_{n-1}\rangle
  2. 𝐱 0 \mathbf{x}_{0}
  3. 𝐲 t \mathbf{y}_{t}
  4. 𝐱 t , 𝐚 t , 𝐚 t + 1 , 𝐚 t + 2 , , 𝐚 t + k - 1 , 𝐲 t + k \langle\mathbf{x}_{t},\mathbf{a}_{t},\mathbf{a}_{t+1},\mathbf{a}_{t+2},...,% \mathbf{a}_{t+k-1},\mathbf{y}_{t+k}\rangle
  5. f 1 , f 2 , , f k f_{1},f_{2},...,f_{k}
  6. 𝐱 t + 1 \mathbf{x}_{t+1}
  7. 𝐱 t + 1 = f ( 𝐱 t , 𝐚 t ) \mathbf{x}_{t+1}=f(\mathbf{x}_{t},\mathbf{a}_{t})

Baker's_technique.html

  1. G G
  2. w w
  3. ϵ \epsilon
  4. r r
  5. k = 1 / ϵ k=1/\epsilon
  6. G G
  7. r r
  8. ( mod k ) \;\;(\mathop{{\rm mod}}k)
  9. { V 0 , V 1 , , V k - 1 } \{V_{0},V_{1},\ldots,V_{k-1}\}
  10. = 0 , , k - 1 \ell=0,\ldots,k-1
  11. G 1 , G 2 , , G^{\ell}_{1},G^{\ell}_{2},\ldots,
  12. G G
  13. V V_{\ell}
  14. i = 1 , 2 , i=1,2,\ldots
  15. S i S_{i}^{\ell}
  16. G i G_{i}^{\ell}
  17. S = i S i S^{\ell}=\cup_{i}S_{i}^{\ell}
  18. S * S^{\ell^{*}}
  19. { S 0 , S 1 , , S k - 1 } \{S^{0},S^{1},\ldots,S^{k-1}\}
  20. S * S^{\ell^{*}}
  21. S S^{\ell}
  22. G i G_{i}^{\ell}
  23. G i G_{i}^{\ell}
  24. k k
  25. k k

Baker's_theorem.html

  1. L = { λ 𝐂 : e λ 𝐐 ¯ × } . L=\left\{\lambda\in\mathbf{C}:\ e^{\lambda}\in{\overline{\mathbf{Q}}}^{\times}% \right\}.
  2. | β 0 + β 1 λ 1 + + β n λ n | > H - C |\beta_{0}+\beta_{1}\lambda_{1}+\cdots+\beta_{n}\lambda_{n}|>H^{-C}
  3. a 1 b 1 a n b n , a_{1}^{b_{1}}\cdots a_{n}^{b_{n}},
  4. Λ = β 0 + β 1 λ 1 + + β n λ n \Lambda=\beta_{0}+\beta_{1}\lambda_{1}+\cdots+\beta_{n}\lambda_{n}
  5. log | Λ | > ( 16 n d ) 200 n Ω ( log Ω - log log A n ) ( log B + log Ω ) \log|\Lambda|>(16nd)^{200n}\Omega(\log\Omega-\log\log A_{n})(\log B+\log\Omega)
  6. Ω = log A 1 log A 2 log A n \Omega=\log A_{1}\log A_{2}\cdots\log A_{n}
  7. log | Λ | > - C h ( α 1 ) h ( α 2 ) h ( α n ) log ( max { | β 1 | , , | β n | } ) , \log|\Lambda|>-Ch(\alpha_{1})h(\alpha_{2})\cdots h(\alpha_{n})\log\left(\max\{% |\beta_{1}|,\ldots,|\beta_{n}|\}\right),
  8. C = 18 ( n + 1 ) ! n n + 1 ( 32 d ) n + 2 log ( 2 n d ) , C=18(n+1)!\cdot n^{n+1}\cdot(32d)^{n+2}\log(2nd),
  9. β 1 log α 1 + + β n - 1 log α n - 1 = log α n \beta_{1}\log\alpha_{1}+\cdots+\beta_{n-1}\log\alpha_{n-1}=\log\alpha_{n}
  10. Φ ( z 1 , , z n - 1 ) = λ 1 = 0 L λ n = 0 L p ( λ 1 , λ n ) α 1 ( λ 1 + λ n β 1 ) z 1 α n - 1 ( λ n - 1 + λ n β n - 1 ) z n - 1 \Phi(z_{1},\ldots,z_{n-1})=\sum_{\lambda_{1}=0}^{L}\cdots\sum_{\lambda_{n}=0}^% {L}p(\lambda_{1}\ldots,\lambda_{n})\alpha_{1}^{(\lambda_{1}+\lambda_{n}\beta_{% 1})z_{1}}\cdots\alpha_{n-1}^{(\lambda_{n-1}+\lambda_{n}\beta_{n-1})z_{n-1}}
  11. λ 1 = 0 L λ n = 0 L p ( λ 1 , λ n ) α 1 λ 1 l α n λ n l = 0 \sum_{\lambda_{1}=0}^{L}\cdots\sum_{\lambda_{n}=0}^{L}p(\lambda_{1}\ldots,% \lambda_{n})\alpha_{1}^{\lambda_{1}l}\cdots\alpha_{n}^{\lambda_{n}l}=0
  12. α 1 λ 1 α n λ n \alpha_{1}^{\lambda_{1}}\cdots\alpha_{n}^{\lambda_{n}}
  13. β 0 + β 1 log α 1 + + β n - 1 log α n - 1 = log α n \beta_{0}+\beta_{1}\log\alpha_{1}+\cdots+\beta_{n-1}\log\alpha_{n-1}=\log% \alpha_{n}
  14. Φ ( z 0 , , z n - 1 ) = λ 0 = 0 L λ n = 0 L p ( λ 0 , λ n ) z 0 λ 0 e λ n β 0 z 0 α 1 ( λ 1 + λ n β 1 ) z 1 α n - 1 ( λ n - 1 + λ n β n - 1 ) z n - 1 \Phi(z_{0},\ldots,z_{n-1})=\sum_{\lambda_{0}=0}^{L}\cdots\sum_{\lambda_{n}=0}^% {L}p(\lambda_{0}\ldots,\lambda_{n})z_{0}^{\lambda_{0}}e^{\lambda_{n}\beta_{0}z% _{0}}\alpha_{1}^{(\lambda_{1}+\lambda_{n}\beta_{1})z_{1}}\cdots\alpha_{n-1}^{(% \lambda_{n-1}+\lambda_{n}\beta_{n-1})z_{n-1}}
  15. | β 1 λ 1 + β 2 λ 2 | |\beta_{1}\lambda_{1}+\beta_{2}\lambda_{2}|

Balanced_matrix.html

  1. [ 1 0 1 1 1 0 0 1 1 ] \begin{bmatrix}1&0&1\\ 1&1&0\\ 0&1&1\\ \end{bmatrix}
  2. B = [ 1 1 1 1 1 1 0 0 1 0 1 0 1 0 0 1 ] B=\begin{bmatrix}1&1&1&1\\ 1&1&0&0\\ 1&0&1&0\\ 1&0&0&1\\ \end{bmatrix}
  3. [ 1 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 ] \begin{bmatrix}1&0&0&0&1\\ 1&1&0&0&0\\ 0&1&1&0&0\\ 0&0&1&1&0\\ 0&0&0&1&1\\ \end{bmatrix}

Ball_tree.html

  1. t t
  2. D N o d e ( t ) = { m a x ( | t - B . p i v o t | - B . r a d i u s , D B . p a r e n t ) , if N o d e R o o t m a x ( | t - B . p i v o t | - B . r a d i u s , 0 ) , if B = R o o t D^{Node}(t)=\begin{cases}max(|t-B.pivot|-B.radius,D^{B.parent}),&\,\text{if }% Node\neq Root\\ max(|t-B.pivot|-B.radius,0),&\,\text{if }B=Root\\ \end{cases}
  3. D B ( t ) D^{B}(t)
  4. m m
  5. 2 m 2m
  6. O ( n l g n ) O(n\,lg\,n)

Ballistic_conduction_in_single-walled_carbon_nanotubes.html

  1. d I / d V dI/dV
  2. E i = h ν i = h 2 e 2 e T i = h 2 e I i E_{i}=h\nu_{i}=\frac{h}{2e}\frac{2e}{T_{i}}=\frac{h}{2e}I_{i}
  3. I i = 2 e h i E i 1 1 + e E - E f k B T I_{i}=\frac{2e}{h}\sum_{i}E_{i}\frac{1}{1+e^{\frac{E-E_{f}}{k_{B}T}}}
  4. I i = 2 e h 1 1 + e E - E f k B T d E I_{i}=\frac{2e}{h}\int\frac{1}{1+e^{\frac{E-E_{f}}{k_{B}T}}}dE
  5. I d = I o { l n [ 1 + e 2 e ϕ s - E g 2 k T ] - l n [ 1 + e 2 e ϕ s - 2 e V d - E g 2 k T ] } I_{d}=I_{o}\left\{ln\left[1+e^{2e\phi_{s}-\frac{E_{g}}{2kT}}\right]-ln\left[1+% e^{2e\phi_{s}-2eV_{d}-\frac{E_{g}}{2kT}}\right]\right\}
  6. I 0 = 4 e h k T = k T e R 0 I_{0}=\frac{4e}{h}kT=\frac{kT}{eR_{0}}
  7. R 0 R_{0}
  8. I d I_{d}
  9. I d - V g I_{d}-V_{g}
  10. V g = V d V_{g}=V_{d}
  11. I - V I-V

Ballistic_galvanometer.html

  1. K = C V / d K=CV/d
  2. Q = k d Q=kd

Banach_bundle_(non-commutative_geometry).html

  1. X X
  2. X X
  3. 𝔅 = ( B , π ) \mathfrak{B}=(B,\pi)
  4. B B
  5. π : B X \pi\colon B\to X
  6. B x := π - 1 ( x ) B_{x}:=\pi^{-1}(x)
  7. b b b\mapsto\|b\|
  8. b B b\in B
  9. + : { ( b 1 , b 2 ) B × B : π ( b 1 ) = π ( b 2 ) } B +\colon\{(b_{1},b_{2})\in B\times B:\pi(b_{1})=\pi(b_{2})\}\to B
  10. λ \lambda\in\mathbb{C}
  11. b λ b b\mapsto\lambda\cdot b
  12. x X x\in X
  13. { b i } \{b_{i}\}
  14. B B
  15. b i 0 \|b_{i}\|\to 0
  16. π ( b i ) x \pi(b_{i})\to x
  17. b i 0 x B b_{i}\to 0_{x}\in B
  18. 0 x 0_{x}
  19. B x B_{x}
  20. b b b\mapsto\|b\|
  21. 𝔅 \mathfrak{B}
  22. B := A × X B:=A\times X
  23. π : B X \pi\colon B\to X
  24. π ( a , x ) := x \pi(a,x):=x
  25. ( B , π ) (B,\pi)

Bangdiwala's_B.html

  1. B = Σ n i i 2 Σ n i . n . i B=\frac{\Sigma n_{ii}^{2}}{\Sigma n_{i.}n_{.i}}

Bare_mass.html

  1. m = m 0 + δ m m=m_{0}+\delta_{m}
  2. m 0 m_{0}
  3. δ m \delta_{m}

Barratt–Priddy_theorem.html

  1. n n
  2. f ( x ) = x f(x)=x
  3. k k
  4. n n
  5. n > k n>k
  6. k k
  7. n n
  8. n n
  9. n 2 k n≥2k
  10. n 2 k n≥2k
  11. H k ( Σ n ) H k ( Map 0 ( S n , S n ) ) H_{k}(\Sigma_{n})\cong H_{k}(\,\text{Map}_{0}(S^{n},S^{n}))
  12. n 2 n≥2
  13. n = 1 n=1
  14. n 3 n≥3
  15. n 2 k n≥2k
  16. n > k n>k
  17. φ φ
  18. φ φ
  19. φ φ
  20. φ φ
  21. n n
  22. n n
  23. B Σ + Ω 0 S B\Sigma_{\infty}^{+}\simeq\Omega_{0}^{\infty}S^{\infty}
  24. 𝐙 × B Σ + Ω S \,\textbf{Z}\times B\Sigma_{\infty}^{+}\simeq\Omega^{\infty}S^{\infty}
  25. π i ( B Σ + ) π i ( Ω S ) lim n π n + i ( S n ) = π i s \pi_{i}(B\Sigma_{\infty}^{+})\cong\pi_{i}(\Omega^{\infty}S^{\infty})\cong\lim_% {n\rightarrow\infty}\pi_{n+i}(S^{n})=\pi_{i}^{s}
  26. K i ( R ) = π i ( B G L ( R ) + ) K_{i}(R)=\pi_{i}(BGL_{\infty}(R)^{+})
  27. K i ( 𝐅 1 ) = π i ( B G L ( 𝐅 1 ) + ) = π i ( B Σ + ) = π i s . K_{i}(\mathbf{F}_{1})=\pi_{i}(BGL_{\infty}(\mathbf{F}_{1})^{+})=\pi_{i}(B% \Sigma_{\infty}^{+})=\pi_{i}^{s}.

Barrier_cone.html

  1. b ( K ) := { X | sup x K , x < + } . b(K):=\left\{\ell\in X^{\ast}\,\left|\,\sup_{x\in K}\langle\ell,x\rangle<+% \infty\right.\right\}.
  2. σ K : sup x K , x , \sigma_{K}\colon\ell\mapsto\sup_{x\in K}\langle\ell,x\rangle,

Barycentric_Julian_Date.html

  1. B J D T T = J D T T + | r + d n ^ | - d c BJD_{TT}=JD_{TT}+\frac{|\vec{r}+d\,\hat{n}|-d}{c}
  2. r \vec{r}
  3. n ^ \hat{n}
  4. d d
  5. c c
  6. B J D T T = J D T T + r n ^ c BJD_{TT}=JD_{TT}+\frac{\vec{r}\cdot\hat{n}}{c}
  7. B J D T T = J D T T + r n ^ c + r r - ( r n ^ ) 2 2 c d BJD_{TT}=JD_{TT}+\frac{\vec{r}\cdot\hat{n}}{c}+\frac{\vec{r}\cdot\vec{r}-(\vec% {r}\cdot\hat{n})^{2}}{2\,c\,d}

Base_(exponentiation).html

  1. log b a = n . \log_{b}a=n.\,
  2. log 10 10000 = 4 \log_{10}10000=4

Basic_affine_jump_diffusion.html

  1. d Z t = κ ( θ - Z t ) d t + σ Z t d B t + d J t , t 0 , Z 0 0 , dZ_{t}=\kappa(\theta-Z_{t})\,dt+\sigma\sqrt{Z_{t}}\,dB_{t}+dJ_{t},\qquad t\geq 0% ,Z_{0}\geq 0,
  2. B B
  3. J J
  4. l l
  5. μ \mu
  6. κ θ 0 \kappa\theta\geq 0
  7. μ 0 \mu\geq 0
  8. m ( q ) = E ( e q 0 t Z s d s ) , q , m\left(q\right)=\operatorname{E}\left(e^{q\int_{0}^{t}Z_{s}\,ds}\right),\qquad q% \in\mathbb{R},
  9. φ ( u ) = E ( e i u 0 t Z s d s ) , u , \varphi\left(u\right)=\operatorname{E}\left(e^{iu\int_{0}^{t}Z_{s}\,ds}\right)% ,\qquad u\in\mathbb{R},
  10. 0 t Z s d s \int_{0}^{t}Z_{s}\,ds

Basil_Hiley.html

  1. x x
  2. p p
  3. ρ ^ S ^ t + 1 2 [ ρ ^ , H ^ ] + = 0 \hat{\rho}\frac{\partial\hat{S}}{\partial t}+\frac{1}{2}[\hat{\rho},\hat{H}]_{% +}=0
  4. O ( Δ t 2 ) O(\Delta t^{2})

Basis_pursuit.html

  1. min x x 1 subject to y = A x . \min_{x}\|x\|_{1}\quad\mbox{subject to}~{}\quad y=Ax.

Basis_pursuit_denoising.html

  1. min x 1 2 y - A x 2 2 + λ x 1 . \min_{x}\frac{1}{2}\|y-Ax\|^{2}_{2}+\lambda\|x\|_{1}.
  2. λ \lambda
  3. x x
  4. N × 1 N\times 1
  5. y y
  6. M × 1 M\times 1
  7. A A
  8. M × N M\times N
  9. M < N M<N
  10. min x x 1 subject to y - A x 2 2 δ \min_{x}\|x\|_{1}\;\textrm{subject}\ \textrm{to}\;\;\|y-Ax\|^{2}_{2}\leq\delta
  11. λ \lambda
  12. δ \delta
  13. y y
  14. A x Ax
  15. L 2 L_{2}
  16. x x
  17. L 1 L_{1}
  18. min x x 1 \min_{x}\|x\|_{1}
  19. min x y - A x 2 2 \min_{x}\|y-Ax\|^{2}_{2}
  20. λ \lambda\rightarrow\infty
  21. δ = 0 \delta=0

Bat_algorithm.html

  1. v i v_{i}
  2. x i x_{i}
  3. A i A_{i}
  4. r r

Bateman_polynomials.html

  1. F n ( d d x ) cosh - 1 ( x ) = cosh - 1 ( x ) P n ( tanh ( x ) ) = F 2 3 ( - n , n + 1 , ( x + 1 ) / 2 ; 1 , 1 ; 1 ) F_{n}\left(\frac{d}{dx}\right)\cosh^{-1}(x)=\cosh^{-1}(x)P_{n}(\tanh(x))={}_{3% }F_{2}(-n,n+1,(x+1)/2;1,1;1)
  2. F n m ( d d x ) cosh - 1 - m ( x ) = cosh - 1 - m ( x ) P n ( tanh ( x ) ) F_{n}^{m}\left(\frac{d}{dx}\right)\cosh^{-1-m}(x)=\cosh^{-1-m}(x)P_{n}(\tanh(x))
  3. Q n ( x ) = ( - 1 ) n 2 n n ! ( 2 n n ) - 1 F n ( 2 x + 1 ) Q_{n}(x)=(-1)^{n}2^{n}n!{\left({{2n}\atop{n}}\right)}^{-1}F_{n}(2x+1)
  4. F 0 ( x ) = 1 F_{0}(x)=1
  5. F 1 ( x ) = - x F_{1}(x)=-x
  6. F 2 ( x ) = 1 4 + 3 4 x 2 F_{2}(x)=\frac{1}{4}+\frac{3}{4}x^{2}
  7. F 3 ( x ) = - 7 12 x - 5 12 x 3 F_{3}(x)=-\frac{7}{12}x-\frac{5}{12}x^{3}
  8. F 4 ( x ) = 9 64 + 65 96 x 2 + 35 192 x 4 F_{4}(x)=\frac{9}{64}+\frac{65}{96}x^{2}+\frac{35}{192}x^{4}
  9. F 5 ( x ) = 407 960 x - 49 96 x 3 - 21 320 x 5 F_{5}(x)=\frac{407}{960}x-\frac{49}{96}x^{3}-\frac{21}{320}x^{5}

Bates_distribution.html

  1. X = 1 n k = 1 n U k . X=\frac{1}{n}\sum_{k=1}^{n}U_{k}.
  2. f X ( x ; n ) = n 2 ( n - 1 ) ! k = 0 n ( - 1 ) k ( n k ) ( n x - k ) n - 1 sgn ( n x - k ) f_{X}(x;n)=\frac{n}{2\left(n-1\right)!}\sum_{k=0}^{n}\left(-1\right)^{k}{n% \choose k}\left(nx-k\right)^{n-1}\operatorname{sgn}(nx-k)
  3. sgn ( n x - k ) = { - 1 n x < k 0 n x = k 1 n x > k . \operatorname{sgn}\left(nx-k\right)=\begin{cases}-1&nx<k\\ 0&nx=k\\ 1&nx>k.\end{cases}
  4. X ( a , b ) = 1 n k = 1 n U k ( a , b ) . X_{(a,b)}=\frac{1}{n}\sum_{k=1}^{n}U_{k}(a,b).
  5. g ( x ; n , a , b ) = f X ( x - a b - a ; n ) for a x b g(x;n,a,b)=f_{X}\left(\frac{x-a}{b-a};n\right)\,\text{ for }a\leq x\leq b\,

Baux_score.html

  1. Baux  score = Percent  body  surface  burned + Patient’s  age \textrm{Baux \ score}=\textrm{Percent \ body \ surface \ burned}+\textrm{% Patient's \ age}

Bayes_classifier.html

  1. ( X , Y ) (X,Y)
  2. d × { 1 , 2 , , K } \mathbb{R}^{d}\times\{1,2,\dots,K\}
  3. Y Y
  4. X X
  5. X Y = r P r X\mid Y=r\sim P_{r}
  6. r = 1 , 2 , , K r=1,2,\dots,K
  7. \sim
  8. P r P_{r}
  9. C : d { 1 , 2 , , K } C:\mathbb{R}^{d}\to\{1,2,\dots,K\}
  10. ( C ) = P { C ( X ) Y } . \mathcal{R}(C)=\operatorname{P}\{C(X)\neq Y\}.
  11. C Bayes ( x ) = argmax r { 1 , 2 , , K } P ( Y = r X = x ) . C\text{Bayes}(x)=\underset{r\in\{1,2,\dots,K\}}{\operatorname{argmax}}% \operatorname{P}(Y=r\mid X=x).
  12. P ( Y = r X = x ) \operatorname{P}(Y=r\mid X=x)
  13. C C
  14. ( C ) - ( C Bayes ) . \mathcal{R}(C)-\mathcal{R}(C\text{Bayes}).

Bayesian_tool_for_methylation_analysis.html

  1. f ( A m ) = p ϕ ( A p A base + r c C c p , ν - 1 ) , f(A\mid m)=\prod_{p}\phi\left(A_{p}\mid A\text{base}+r\sum_{c}C_{cp},\nu^{-1}% \right),
  2. ϕ \phi

BCS:_50_Years.html

  1. T c T_{c}
  2. T c T_{c}

Belevitch's_theorem.html

  1. 𝐒 ( p ) \mathbf{S}(p)
  2. d d
  3. 𝐒 ( p ) = [ s 11 s 12 s 21 s 22 ] \mathbf{S}(p)=\begin{bmatrix}s_{11}&s_{12}\\ s_{21}&s_{22}\end{bmatrix}
  4. i ω i\omega
  5. 𝐒 ( p ) \scriptstyle\mathbf{S}(p)
  6. 𝐒 ( p ) = 1 g ( p ) [ h ( p ) f ( p ) ± f ( - p ) h ( - p ) ] \mathbf{S}(p)=\frac{1}{g(p)}\begin{bmatrix}h(p)&f(p)\\ \pm f(-p)&\mp h(-p)\end{bmatrix}
  7. f ( p ) f(p)
  8. g ( p ) g(p)
  9. h ( p ) h(p)
  10. g ( p ) g(p)
  11. d d
  12. g ( p ) g ( - p ) = f ( p ) f ( - p ) + h ( p ) h ( - p ) g(p)g(-p)=f(p)f(-p)+h(p)h(-p)
  13. p \scriptstyle p\,\in\,\mathbb{C}

Belinfante–Rosenfeld_stress–energy_tensor.html

  1. M μ ν = d 3 x M 0 μ ν M_{\mu\nu}=\int d^{3}x\,{M^{0}}_{\mu\nu}
  2. M μ ν λ = ( x ν T μ λ - x λ T μ ν ) + S μ ν λ . {M^{\mu}}_{\nu\lambda}=(x_{\nu}{T^{\mu}}_{\lambda}-x_{\lambda}{T^{\mu}}_{\nu})% +{S^{\mu}}_{\nu\lambda}.
  3. T μ λ {T^{\mu}}_{\lambda}
  4. S μ ν λ {S^{\mu}}_{\nu\lambda}
  5. μ M μ ν λ = 0 \partial_{\mu}{M^{\mu}}_{\nu\lambda}=0\,
  6. μ S μ ν λ = T λ ν - T ν λ . \partial_{\mu}{S^{\mu}}_{\nu\lambda}=T_{\lambda\nu}-T_{\nu\lambda}.
  7. T B μ ν = T μ ν + 1 2 λ ( S μ ν λ + S ν μ λ - S λ ν μ ) T_{B}^{\mu\nu}=T^{\mu\nu}+\frac{1}{2}\partial_{\lambda}(S^{\mu\nu\lambda}+S^{% \nu\mu\lambda}-S^{\lambda\nu\mu})
  8. S μ ν λ {S^{\mu}}_{\nu\lambda}
  9. M ν λ = ( x ν T B 0 λ - x λ T B 0 ν ) d 3 x , M^{\nu\lambda}=\int(x^{\nu}T^{0\lambda}_{B}-x^{\lambda}T^{0\nu}_{B})\,d^{3}x,
  10. J bound = × M {J}\text{bound}=\nabla\times{M}
  11. M {M}
  12. T B μ ν T_{B}^{\mu\nu}
  13. T B μ ν = T μ ν - i 2 κ [ ( κ Ψ ) ( 𝒥 μ ν ) m Ψ m - ( μ Ψ ) ( 𝒥 κ ν ) m Ψ m - ( ν Ψ ) ( 𝒥 κ μ ) m Ψ m ] T_{B}^{\mu\nu}=T^{\mu\nu}-\frac{i}{2}\partial_{\kappa}\left[\frac{\partial% \mathcal{L}}{\partial(\partial_{\kappa}\Psi^{\ell})}(\mathcal{J}^{\mu\nu})^{% \ell}_{\,\,m}\Psi^{m}-\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\Psi^{% \ell})}(\mathcal{J}^{\kappa\nu})^{\ell}_{\,\,m}\Psi^{m}-\frac{\partial\mathcal% {L}}{\partial(\partial_{\nu}\Psi^{\ell})}(\mathcal{J}^{\kappa\mu})^{\ell}_{\,% \,m}\Psi^{m}\right]
  14. \mathcal{L}
  15. T μ ν = η μ ν - ( μ Ψ ) ν Ψ T^{\mu\nu}=\eta^{\mu\nu}\mathcal{L}-\frac{\partial\mathcal{L}}{\partial(% \partial_{\mu}\Psi^{\ell})}\partial^{\nu}\Psi^{\ell}
  16. 𝒥 μ ν \mathcal{J^{\mu\nu}}
  17. [ 𝒥 μ ν , 𝒥 ρ σ ] = i 𝒥 ρ ν η μ σ - i 𝒥 σ ν η μ ρ - i 𝒥 μ σ η ν ρ + i 𝒥 μ ρ η ν σ [\mathcal{J}^{\mu\nu},\mathcal{J}^{\rho\sigma}]=i\mathcal{J}^{\rho\nu}\eta^{% \mu\sigma}-i\mathcal{J}^{\sigma\nu}\eta^{\mu\rho}-i\mathcal{J}^{\mu\sigma}\eta% ^{\nu\rho}+i\mathcal{J}^{\mu\rho}\eta^{\nu\sigma}

Bell–Evans–Polanyi_principle.html

  1. E a = E 0 + α Δ H , E_{a}=E_{0}+\alpha\Delta H\,,
  2. 0 α 1 0\leq\alpha\leq 1
  3. A B + C A + B C AB+C\rightarrow A+BC
  4. E A B ( r ) = m 1 ( r - r 1 ) E_{AB}(r)=m_{1}(r-r_{1})
  5. r = E a m 1 + r 1 r^{\ddagger}=\frac{E_{a}}{m_{1}}+r_{1}
  6. E B C ( r ) = m 2 ( r - r ) + E a E_{BC}(r)=m_{2}(r-r^{\ddagger})+E_{a}
  7. E a = m 1 m 1 - m 2 [ Δ H - m 2 ( r 2 - r 1 ) ] E_{a}=\frac{m_{1}}{m_{1}-m_{2}}[\Delta H-m_{2}(r_{2}-r_{1})]

Beltrami_equation.html

  1. w z ¯ = μ w z . {\partial w\over\partial\overline{z}}=\mu{\partial w\over\partial z}.
  2. d s 2 = E d x 2 + 2 F d x d y + G d y 2 \displaystyle{ds^{2}=E\,dx^{2}+2F\,dxdy+G\,dy^{2}}
  3. g ( x , y ) = ( E F F G ) \displaystyle{g(x,y)=\begin{pmatrix}E&F\\ F&G\end{pmatrix}}
  4. μ ( x , y ) = E - G + 2 i F E + G + 2 E G - F 2 \displaystyle{\mu(x,y)={E-G+2iF\over E+G+2\sqrt{EG-F^{2}}}}
  5. | μ | 2 = E + G - 2 E G - F 2 E + G + 2 E G - F 2 < 1. \displaystyle{|\mu|^{2}={E+G-2\sqrt{EG-F^{2}}\over E+G+2\sqrt{EG-F^{2}}}<1.}
  6. z = 1 2 ( x - i y ) , z ¯ = 1 2 ( x + i y ) . \displaystyle{\partial_{z}={1\over 2}(\partial_{x}-i\partial_{y}),\,\,\,% \partial_{\overline{z}}={1\over 2}(\partial_{x}+i\partial_{y}).}
  7. d s 2 = d u 2 + d v 2 = ( u x 2 + v x 2 ) d x 2 + 2 ( u x u y + v x v y ) d x d y + ( u y 2 + v y 2 ) d y 2 . \displaystyle{ds^{2}=du^{2}+dv^{2}=(u_{x}^{2}+v_{x}^{2})\,dx^{2}+2(u_{x}u_{y}+% v_{x}v_{y})\,dxdy+(u_{y}^{2}+v_{y}^{2})\,dy^{2}.}
  8. d s 2 = | f z d z + f z ¯ d z ¯ | 2 = | f z | 2 | d z + f z ¯ f z d z ¯ | 2 . \displaystyle{ds^{2}=|f_{z}dz+f_{\overline{z}}d\overline{z}|^{2}=|f_{z}|^{2}|% dz+{f_{\overline{z}}\over f_{z}}d\overline{z}|^{2}.}
  9. μ ( z ) = f z ¯ f z . \displaystyle{\mu(z)={f_{\overline{z}}\over f_{z}}.}
  10. | f z | 2 - | f z ¯ | 2 = u x v y - v x u y > 0 , \displaystyle{|f_{z}|^{2}-|f_{\overline{z}}|^{2}=u_{x}v_{y}-v_{x}u_{y}>0,}
  11. μ ( z ) = h z ¯ h z , \displaystyle{\mu(z)={h_{\overline{z}}\over h_{z}},}
  12. μ ( z ) = a ( z , z ¯ ) , \displaystyle{\mu(z)=a(z,\overline{z}),}
  13. f ( z ) = g ( z , z ¯ ) \displaystyle{f(z)=g(z,\overline{z})}
  14. h w = h z a + a z . \displaystyle{h_{w}=h_{z}a+a_{z}.}
  15. F ( z , w ) z = - a ( z , F ( z , w ) ) , F ( 0 , w ) = w , \displaystyle{{\partial F(z,w)\over\partial z}=-a(z,F(z,w)),\,\,\,F(0,w)=w,}
  16. k ( z , w ) = h ( z , F ( z , w ) ) , \displaystyle{k(z,w)=h(z,F(z,w)),}
  17. z k ( z , w ) = h z ( z , F ) - a h w = a z ( z , w ) . \displaystyle{\partial_{z}k(z,w)=h_{z}(z,F)-ah_{w}=a_{z}(z,w).}
  18. k ( z , w ) = a ( z , w ) \displaystyle{k(z,w)=a(z,w)}
  19. h ( z , w ) = a ( z , G ( z , w ) ) , \displaystyle{h(z,w)=a(z,G(z,w)),}
  20. T f ^ ( z ) = z ¯ z f ^ ( z ) . \displaystyle{\widehat{Tf}(z)={\overline{z}\over z}\widehat{f}(z).}
  21. T ( f z ¯ ) = f z , \displaystyle{T(f_{\overline{z}})=f_{z},}
  22. D = z ¯ D=\partial_{\overline{z}}
  23. E ( z ) = 1 π z , \displaystyle{E(z)={1\over\pi z},}
  24. z ¯ ( E f ) = f . \displaystyle{\partial_{\overline{z}}(E\star f)=f.}
  25. C f = E f , \displaystyle{Cf=E\star f,}
  26. ( C f ) z ¯ = f . \displaystyle{(Cf)_{\overline{z}}=f.}
  27. ( C f ) z = T f . \displaystyle{(Cf)_{z}=Tf.}
  28. f z ¯ = μ f z , \displaystyle{f_{\overline{z}}=\mu f_{z},}
  29. g ( z ) = f ( z ) - z \displaystyle{g(z)=f(z)-z}
  30. h = g z = T ( g z ¯ ) = T ( f z ¯ ) = T ( μ f z ) = T ( μ h ) + T μ . \displaystyle{h=g_{z}=T(g_{\overline{z}})=T(f_{\overline{z}})=T(\mu f_{z})=T(% \mu h)+T\mu.}
  31. A F = T μ F , B F = μ T F \displaystyle{AF=T\mu F,\,\,\,\,BF=\mu TF}
  32. ( I - A ) h = T μ . \displaystyle{(I-A)h=T\mu.}
  33. h = ( I - A ) - 1 T μ , T * h = ( I - B ) - 1 μ \displaystyle{h=(I-A)^{-1}T\mu,\,\,\,T^{*}h=(I-B)^{-1}\mu}
  34. g z ¯ = T * h , \displaystyle{g_{\overline{z}}=T^{*}h,}
  35. f ( z ) = C T * h ( z ) + z . \displaystyle{f(z)=CT^{*}h(z)+z.}
  36. z ¯ f = 0 , \displaystyle{\partial_{\overline{z}}f=0,}
  37. F z ¯ = μ F z + G F , \displaystyle{F_{\overline{z}}=\mu F_{z}+GF,}
  38. U e m = m ¯ m e m ( m 0 ) , U e 0 = e 0 . \displaystyle{Ue_{m}={\overline{m}\over m}e_{m}\,\,(m\neq 0),Ue_{0}=e_{0}.}
  39. U ( P z ¯ ) = P z . \displaystyle{U(P_{\overline{z}})=P_{z}.}
  40. z ( I - U μ ) F = U G . \displaystyle{\partial_{z}(I-U\mu)F=UG.}
  41. J ( f ) = | f z | 2 - | f z ¯ | 2 = | f z | 2 ( 1 - | μ | 2 ) . \displaystyle{J(f)=|f_{z}|^{2}-|f_{\overline{z}}|^{2}=|f_{z}|^{2}(1-|\mu|^{2}).}
  42. k z ¯ = μ k z + μ z . \displaystyle{k_{\overline{z}}=\mu k_{z}+\mu_{z}.}
  43. J ( f ) = ( 1 - | μ | 2 ) | e 2 k | \displaystyle{J(f)=(1-|\mu|^{2})|e^{2k}|}
  44. μ g f - 1 f = f z f z ¯ μ g - μ f 1 - μ f ¯ μ g . \displaystyle{\mu_{g\circ f^{-1}}\circ f={f_{z}\over\overline{f_{z}}}{\mu_{g}-% \mu_{f}\over 1-\overline{\mu_{f}}\mu_{g}}.}
  45. g ( z ) = f ( z - 1 ) - 1 , \displaystyle{g(z)=f(z^{-1})^{-1},}
  46. μ g ( z ) = z 2 z ¯ 2 μ f ( z - 1 ) . \displaystyle{\mu_{g}(z)={z^{2}\over\overline{z}^{2}}\mu_{f}(z^{-1}).}
  47. μ ~ ( w ) = ( w / w ¯ ) μ ( z ) . \displaystyle{\widetilde{\mu}(w)=(w^{\prime}/\overline{w^{\prime}})\cdot\mu(z).}
  48. μ = ψ μ + ( 1 - ψ ) μ = μ 0 + μ . \displaystyle{\mu=\psi\mu+(1-\psi)\mu=\mu_{0}+\mu_{\infty}.}
  49. λ ( z ) = [ μ 0 1 - μ μ ¯ g z g z ¯ ] g - 1 ( z ) . \displaystyle{\lambda(z)=\left[{\mu_{0}\over 1-\mu\overline{\mu_{\infty}}}{g_{% z}\over\overline{g_{z}}}\right]\circ g^{-1}(z).}
  50. μ ( z ) = μ ( z ¯ ) ¯ , \displaystyle{\mu(z)=\overline{\mu(\overline{z})},}
  51. f ( z ) = f ( z ¯ ) ¯ , \displaystyle{f(z)=\overline{f(\overline{z})},}
  52. f ( z ) = f ( z ¯ - 1 ) ¯ - 1 , \displaystyle{f(z)=\overline{f(\overline{z}^{-1})}^{-1},}
  53. μ ( z ) = z 2 z ¯ 2 μ ( z ¯ - 1 ) ¯ - 1 . \displaystyle{\mu(z)={z^{2}\over\overline{z}^{2}}\overline{\mu(\overline{z}^{-% 1})}^{-1}.}
  54. h z ¯ = μ ( z ) h z \displaystyle{h_{\overline{z}}=\mu(z)h_{z}}
  55. μ F h - 1 = 0 , \displaystyle{\mu_{F\circ h^{-1}}=0,}
  56. G ( r , θ ) = ( x ( θ ) - ( 1 - r ) y ( θ ) , y ( θ ) + ( 1 - r ) x ( θ ) ) . \displaystyle{G(r,\theta)=(x(\theta)-(1-r)y^{\prime}(\theta),y(\theta)+(1-r)x^% {\prime}(\theta)).}
  57. d s 2 = ( 1 + t κ ( θ ) ) 2 d θ 2 + d t 2 , \displaystyle{ds^{2}=(1+t\kappa(\theta))^{2}d\theta^{2}+dt^{2},}
  58. κ ( θ ) = y ′′ ( θ ) x ( θ ) - x ′′ ( θ ) y ( θ ) \displaystyle{\kappa(\theta)=y^{\prime\prime}(\theta)x^{\prime}(\theta)-x^{% \prime\prime}(\theta)y^{\prime}(\theta)}
  59. α = 1 2 π 0 2 π κ ( θ ) d θ , h ( θ ) = κ ( θ ) - α . \displaystyle{\alpha={1\over 2\pi}\int_{0}^{2\pi}\kappa(\theta)\,d\theta,\,\,% \,h(\theta)=\kappa(\theta)-\alpha.}
  60. d s 2 = ( 1 + t ψ ( t ) h ( θ ) ) 2 d θ 2 + d t 2 , \displaystyle{ds^{2}=(1+t\psi(t)h(\theta))^{2}d\theta^{2}+dt^{2},}
  61. ψ ( t ) = 1 + a 1 t + a 2 t 2 + \displaystyle{\psi(t)=1+a_{1}t+a_{2}t^{2}+\cdots}
  62. f ( θ , t ) = θ + f 1 ( θ ) t + f 2 ( θ ) t 2 + = θ + g ( θ , t ) \displaystyle{f(\theta,t)=\theta+f_{1}(\theta)t+f_{2}(\theta)t^{2}+\cdots=% \theta+g(\theta,t)}
  63. [ 1 + t ψ ( t ) ( n 0 h ( n ) g n / n ! ) ] [ ( 1 + g ) d θ + ( n 1 n f n t n - 1 ) d t ] . \displaystyle{[1+t\psi(t)(\sum_{n\geq 0}h^{(n)}g^{n}/n!)]\cdot[(1+g^{\prime})d% \theta+(\sum_{n\geq 1}nf_{n}t^{n-1})dt].}
  64. f z ¯ = μ f z , \displaystyle{f_{\overline{z}}=\mu f_{z},}
  65. g ( z ) = f ( z ) - z \displaystyle{g(z)=f(z)-z}
  66. h = g z = T ( g z ¯ ) = T ( f z ¯ ) = T ( μ f z ) = T ( μ h ) + T μ . \displaystyle{h=g_{z}=T(g_{\overline{z}})=T(f_{\overline{z}})=T(\mu f_{z})=T(% \mu h)+T\mu.}
  67. h = ( I - A ) - 1 T μ , T - 1 = ( I - B ) - 1 μ \displaystyle{h=(I-A)^{-1}T\mu,\,\,\,T^{-1}=(I-B)^{-1}\mu}
  68. g z ¯ = T - 1 h , \displaystyle{g_{\overline{z}}=T^{-1}h,}
  69. f ( z ) = C T - 1 h ( z ) + z . \displaystyle{f(z)=CT^{-1}h(z)+z.}
  70. P f ( w ) = C f ( w ) + π - 1 𝐂 f ( z ) z d x d y = - 1 π 𝐂 f ( z ) ( 1 z - w - 1 z ) d x d y = - 1 π 𝐂 f ( z ) w z ( z - w ) d x d y . \displaystyle{Pf(w)=Cf(w)+\pi^{-1}\iint_{\mathbf{C}}{f(z)\over z}\,dxdy=-{1% \over\pi}\iint_{\mathbf{C}}f(z)\left({1\over z-w}-{1\over z}\right)\,dxdy=-{1% \over\pi}\iint_{\mathbf{C}}{f(z)w\over z(z-w)}\,dxdy.}
  71. ( P f ) z ¯ = f , ( P f ) z = T f . \displaystyle{(Pf)_{\overline{z}}=f,\,\,\,(Pf)_{z}=Tf.}
  72. f ( z ) = P T - 1 h ( z ) + z . \displaystyle{f(z)=PT^{-1}h(z)+z.}
  73. | P f ( w 1 ) - P f ( w 2 ) | K p f p | w 1 - w 2 | 1 - 2 / p , \displaystyle{|Pf(w_{1})-Pf(w_{2})|\leq K_{p}\|f\|_{p}|w_{1}-w_{2}|^{1-2/p},}
  74. K p = z - 1 ( z - 1 ) - 1 q / π . \displaystyle{K_{p}=\|z^{-1}(z-1)^{-1}\|_{q}/\pi.}
  75. | f ( w 1 ) - f ( w 2 ) | K p 1 - μ C p μ p | w 1 - w 2 | 1 - 2 / p + | w 1 - w 2 | . \displaystyle{|f(w_{1})-f(w_{2})|\leq{K_{p}\over 1-\|\mu\|_{\infty}C_{p}}\|\mu% \|_{p}|w_{1}-w_{2}|^{1-2/p}+|w_{1}-w_{2}|.}
  76. μ p ( π R 2 ) 1 / p μ . \displaystyle{\|\mu\|_{p}\leq(\pi R^{2})^{1/p}\|\mu\|_{\infty}.}
  77. | f ( z ) | [ 1 + K p π 1 / p μ 1 - μ C p ] R = C R , \displaystyle{|f(z)|\leq\left[1+{K_{p}\pi^{1/p}\|\mu\|_{\infty}\over 1-\|\mu\|% _{\infty}C_{p}}\right]\cdot R=CR,}
  78. μ n = ( 1 - φ n ) μ . \displaystyle{\mu_{n}=(1-\varphi_{n})\cdot\mu.}
  79. u n = z ¯ f n , v n = z f n - 1. \displaystyle{u_{n}=\partial_{\overline{z}}f_{n},\,\,v_{n}=\partial_{z}f_{n}-1.}
  80. u n p , v n p μ n p 1 - μ n . \displaystyle{\|u_{n}\|_{p},\,\,\|v_{n}\|_{p}\leq{\|\mu_{n}\|_{p}\over 1-\|\mu% _{n}\|_{\infty}}.}
  81. u ψ = lim u n ψ = - lim f n z ¯ ψ = - f z ¯ ψ , \displaystyle{\iint u\cdot\psi=\lim\iint u_{n}\cdot\psi=-\lim\iint f_{n}\cdot% \partial_{\overline{z}}\psi=-\iint f\cdot\partial_{\overline{z}}\psi,}
  82. μ v - μ n v n = μ ( v - v n ) + ( μ - μ n ) v n . \displaystyle{\mu v-\mu_{n}v_{n}=\mu(v-v_{n})+(\mu-\mu_{n})v_{n}.}
  83. A ( f n ( U ) ) = U J ( f n ) d x d y = U | z f n | 2 - | z ¯ f n | 2 d x d y U | z f n | 2 d x d y z f n | U | p 2 A ( U ) 1 - 2 / p . \displaystyle{A(f_{n}(U))=\iint_{U}J(f_{n})\,dx\,dy=\iint_{U}|\partial_{z}f_{n% }|^{2}-|\partial_{\overline{z}}f_{n}|^{2}\,dx\,dy\leq\iint_{U}|\partial_{z}f_{% n}|^{2}\,dx\,dy\leq\|\partial_{z}f_{n}|_{U}\|_{p}^{2}\,A(U)^{1-2/p}.}
  84. z ¯ ( f g n ) = 0 \displaystyle{\partial_{\overline{z}}(f\circ g_{n})=0}
  85. μ ( f ( z ) ) = - f z f z ¯ μ ( z ) \displaystyle{\mu^{\prime}(f(z))=-{f_{z}\over\overline{f_{z}}}\cdot\mu(z)}
  86. μ ( w ) = - g w ¯ g w μ ( g ( w ) ) \displaystyle{\mu^{\prime}(w)=-{\overline{g_{w}}\over g_{w}}\cdot\mu(g(w))}
  87. ν ( f ( z ) ) = f z f z ¯ μ * - μ 1 - μ ¯ μ * , \displaystyle{\nu(f(z))={f_{z}\over\overline{f_{z}}}\,{\mu^{*}-\mu\over 1-% \overline{\mu}\mu^{*}},}
  88. ( P ψ ) z ¯ = ψ , ( P ψ ) z = T ψ \displaystyle{(P\psi)_{\overline{z}}=\psi,\,\,\,(P\psi)_{z}=T\psi}
  89. H = f - P ( f z ¯ ) \displaystyle{H=f-P(f_{\overline{z}})}
  90. z ¯ F = 0. \displaystyle{\partial_{\overline{z}}F=0.}
  91. P ( f z ¯ ) = f ( z ) + z \displaystyle{P(f_{\overline{z}})=f(z)+z}
  92. f z = T ( μ f z ) + 1. \displaystyle{f_{z}=T(\mu f_{z})+1.}
  93. f z - g z = T ( μ ( f z - g z ) ) . \displaystyle{f_{z}-g_{z}=T(\mu(f_{z}-g_{z})).}
  94. f z = g z . \displaystyle{f_{z}=g_{z}.}
  95. f z ¯ = g z ¯ . \displaystyle{f_{\overline{z}}=g_{\overline{z}}.}
  96. f z - 1 , f z ¯ L p ( 𝐂 ) . \displaystyle{f_{z}-1,\,\,f_{\overline{z}}\in L^{p}(\mathbf{C}).}
  97. ν ( g ( z ) ) = g z g z ¯ μ - λ 1 - λ ¯ μ . \displaystyle{\nu(g(z))={g_{z}\over\overline{g_{z}}}\,{\mu-\lambda\over 1-% \overline{\lambda}\mu}.}
  98. μ ^ ( z ) = μ ( z ) ( z 𝐇 ) , μ ^ ( z ) = 0 ( z 𝐑 ) , μ ^ ( z ) = μ ( z ¯ ) ¯ ( z ¯ 𝐇 ) . \displaystyle{\widehat{\mu}(z)=\mu(z)\,\,(z\in\mathbf{H}),\,\,\widehat{\mu}(z)% =0\,\,(z\in\mathbf{R}),\,\,\widehat{\mu}(z)=\overline{\mu(\overline{z})}\,\,(% \overline{z}\in\mathbf{H}).}
  99. μ ^ ( g ( z ) ) = g z ¯ g z μ ^ ( z ) , \displaystyle{\widehat{\mu}(g(z))={\overline{g_{z}}\over g_{z}}\,\widehat{\mu}% (z),}
  100. | f ( z 1 ) - f ( z 2 ) | | f ( z 1 ) - f ( z 3 ) | a | z 1 - z 2 | b | z 1 - z 3 | b . \displaystyle{{{|f(z_{1})-f(z_{2})|\over|f(z_{1})-f(z_{3})|}\leq a\cdot{|z_{1}% -z_{2}|^{b}\over|z_{1}-z_{3}|^{b}}}.}
  101. | z 1 - z 2 | = | z 1 - z 3 | , \displaystyle{|z_{1}-z_{2}|=|z_{1}-z_{3}|,}
  102. a - 1 | f ( z 1 ) - f ( z 2 ) | | f ( z 1 ) - f ( z 3 ) | a . \displaystyle{a^{-1}\leq{|f(z_{1})-f(z_{2})|\over|f(z_{1})-f(z_{3})|}\leq a.}
  103. F ( r , θ ) = r exp [ i ψ ( r ) g ( θ ) + i ( 1 - ψ ( r ) ) θ ] , \displaystyle{F(r,\theta)=r\exp[i\psi(r)g(\theta)+i(1-\psi(r))\theta],}
  104. f ( e i θ ) = e i g ( θ ) , \displaystyle{f(e^{i\theta})=e^{ig(\theta)},}
  105. μ < 1 , μ ( z ) = F z ¯ / F z . \displaystyle{\|\mu\|_{\infty}<1,\,\,\,\mu(z)=F_{\overline{z}}/F_{z}.}
  106. f = f 1 - 1 f 2 . \displaystyle{f=f_{1}^{-1}\circ f_{2}.}

Bel–Robinson_tensor.html

  1. T a b c d = C a e c f C b + d e f 1 4 ϵ a e ϵ b h i C h i c f e j k C j f k d T_{abcd}=C_{aecf}C_{b}{}^{e}{}_{d}{}^{f}+\frac{1}{4}\epsilon_{ae}{}^{hi}% \epsilon_{b}{}^{ej}{}_{k}C_{hicf}C_{j}{}^{k}{}_{d}{}^{f}
  2. T a b c d = C a e c f C b - d e f 3 2 g a [ b C j k ] c f C j k f d T_{abcd}=C_{aecf}C_{b}{}^{e}{}_{d}{}^{f}-\frac{3}{2}g_{a[b}C_{jk]cf}C^{jk}{}_{% d}{}^{f}
  3. C a b c d C_{abcd}
  4. T a b c d = T ( a b c d ) T_{abcd}=T_{(abcd)}
  5. T a = a c d 0 T^{a}{}_{acd}=0
  6. a T a b c d = 0 \nabla^{a}T_{abcd}=0

Bender–Dunne_polynomials.html

  1. P 0 ( x ) = 1 P_{0}(x)=1
  2. P 1 ( x ) = x P_{1}(x)=x
  3. n > 1 n>1
  4. P n ( x ) = x P n - 1 ( x ) + 16 ( n - 1 ) ( n - J - 1 ) ( n + 2 s - 2 ) P n - 2 ( x ) P_{n}(x)=xP_{n-1}(x)+16(n-1)(n-J-1)(n+2s-2)P_{n-2}(x)
  5. J J
  6. s s

Bending_of_plates.html

  1. N α β , α = 0 N_{\alpha\beta,\alpha}=0
  2. M α β , α β - q = 0 M_{\alpha\beta,\alpha\beta}-q=0
  3. N 11 x 1 + N 21 x 2 = 0 ; N 12 x 1 + N 22 x 2 = 0 \cfrac{\partial N_{11}}{\partial x_{1}}+\cfrac{\partial N_{21}}{\partial x_{2}% }=0~{};~{}~{}\cfrac{\partial N_{12}}{\partial x_{1}}+\cfrac{\partial N_{22}}{% \partial x_{2}}=0
  4. 2 M 11 x 1 2 + 2 2 M 12 x 1 x 2 + 2 M 22 x 2 2 = q \cfrac{\partial^{2}M_{11}}{\partial x_{1}^{2}}+2\cfrac{\partial^{2}M_{12}}{% \partial x_{1}\partial x_{2}}+\cfrac{\partial^{2}M_{22}}{\partial x_{2}^{2}}=q
  5. q ( x ) q(x)
  6. H = 2 h H=2h
  7. σ i j \sigma_{ij}
  8. N α β := - h h σ α β d x 3 ; M α β := - h h x 3 σ α β d x 3 . N_{\alpha\beta}:=\int_{-h}^{h}\sigma_{\alpha\beta}~{}dx_{3}~{};~{}~{}M_{\alpha% \beta}:=\int_{-h}^{h}x_{3}~{}\sigma_{\alpha\beta}~{}dx_{3}~{}.
  9. N N
  10. M M
  11. E E
  12. ν \nu
  13. 2 2 w = - q D ; D := 2 h 3 E 3 ( 1 - ν 2 ) = H 3 E 12 ( 1 - ν 2 ) \nabla^{2}\nabla^{2}w=-\cfrac{q}{D}~{};~{}~{}D:=\cfrac{2h^{3}E}{3(1-\nu^{2})}=% \cfrac{H^{3}E}{12(1-\nu^{2})}
  14. w ( x 1 , x 2 ) w(x_{1},x_{2})
  15. 4 w x 1 4 + 2 4 w x 1 2 x 2 2 + 4 w x 2 4 = - q D . \cfrac{\partial^{4}w}{\partial x_{1}^{4}}+2\cfrac{\partial^{4}w}{\partial x_{1% }^{2}\partial x_{2}^{2}}+\cfrac{\partial^{4}w}{\partial x_{2}^{4}}=-\cfrac{q}{% D}\,.
  16. 2 2 w = - q D . \nabla^{2}\nabla^{2}w=-\frac{q}{D}\,.
  17. ( r , θ , z ) (r,\theta,z)
  18. 2 w 1 r r ( r w r ) + 1 r 2 2 w θ 2 + 2 w z 2 . \nabla^{2}w\equiv\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial w% }{\partial r}\right)+\frac{1}{r^{2}}\frac{\partial^{2}w}{\partial\theta^{2}}+% \frac{\partial^{2}w}{\partial z^{2}}\,.
  19. w = w ( r ) w=w(r)
  20. 2 w 1 r d d r ( r d w d r ) . \nabla^{2}w\equiv\frac{1}{r}\cfrac{d}{dr}\left(r\cfrac{dw}{dr}\right)\,.
  21. 1 r d d r [ r d d r { 1 r d d r ( r d w d r ) } ] = - q D . \frac{1}{r}\cfrac{d}{dr}\left[r\cfrac{d}{dr}\left\{\frac{1}{r}\cfrac{d}{dr}% \left(r\cfrac{dw}{dr}\right)\right\}\right]=-\frac{q}{D}\,.
  22. q q
  23. D D
  24. w ( r ) = - q r 4 64 D + C 1 ln r + C 2 r 2 2 + C 3 r 2 4 ( 2 ln r - 1 ) + C 4 w(r)=-\frac{qr^{4}}{64D}+C_{1}\ln r+\cfrac{C_{2}r^{2}}{2}+\cfrac{C_{3}r^{2}}{4% }(2\ln r-1)+C_{4}
  25. C i C_{i}
  26. ϕ ( r ) = d w d r = - q r 3 16 D + C 1 r + C 2 r + C 3 r ln r . \phi(r)=\cfrac{dw}{dr}=-\frac{qr^{3}}{16D}+\frac{C_{1}}{r}+C_{2}r+C_{3}r\ln r\,.
  27. r = 0 r=0
  28. C 1 = C 3 = 0 C_{1}=C_{3}=0
  29. w ( a ) = 0 w(a)=0
  30. ϕ ( a ) = 0 \phi(a)=0
  31. a a
  32. w ( r ) = - q 64 D ( a 2 - r 2 ) 2 and ϕ ( r ) = q r 16 D ( a 2 - r 2 ) . w(r)=-\frac{q}{64D}(a^{2}-r^{2})^{2}\quad\,\text{and}\quad\phi(r)=\frac{qr}{16% D}(a^{2}-r^{2})\,.
  33. u r ( r ) = - z ϕ ( r ) and u θ ( r ) = 0 . u_{r}(r)=-z\phi(r)\quad\,\text{and}\quad u_{\theta}(r)=0\,.
  34. ε r r = d u r d r = - q z 16 D ( a 2 - 3 r 2 ) , ε θ θ = u r r = - q z 16 D ( a 2 - r 2 ) , ε r θ = 0 . \varepsilon_{rr}=\cfrac{du_{r}}{dr}=-\frac{qz}{16D}(a^{2}-3r^{2})~{},~{}~{}% \varepsilon_{\theta\theta}=\frac{u_{r}}{r}=-\frac{qz}{16D}(a^{2}-r^{2})~{},~{}% ~{}\varepsilon_{r\theta}=0\,.
  35. σ r r = E 1 - ν 2 [ ε r r + ν ε θ θ ] ; σ θ θ = E 1 - ν 2 [ ε θ θ + ν ε r r ] ; σ r θ = 0 . \sigma_{rr}=\frac{E}{1-\nu^{2}}\left[\varepsilon_{rr}+\nu\varepsilon_{\theta% \theta}\right]~{};~{}~{}\sigma_{\theta\theta}=\frac{E}{1-\nu^{2}}\left[% \varepsilon_{\theta\theta}+\nu\varepsilon_{rr}\right]~{};~{}~{}\sigma_{r\theta% }=0\,.
  36. 2 h 2h
  37. D = 2 E h 3 / [ 3 ( 1 - ν 2 ) ] D=2Eh^{3}/[3(1-\nu^{2})]
  38. σ r r = - 3 q z 32 h 3 [ ( 1 + ν ) a 2 - ( 3 + ν ) r 2 ] σ θ θ = - 3 q z 32 h 3 [ ( 1 + ν ) a 2 - ( 1 + 3 ν ) r 2 ] σ r θ = 0 . \begin{aligned}\displaystyle\sigma_{rr}&\displaystyle=-\frac{3qz}{32h^{3}}% \left[(1+\nu)a^{2}-(3+\nu)r^{2}\right]\\ \displaystyle\sigma_{\theta\theta}&\displaystyle=-\frac{3qz}{32h^{3}}\left[(1+% \nu)a^{2}-(1+3\nu)r^{2}\right]\\ \displaystyle\sigma_{r\theta}&\displaystyle=0\,.\end{aligned}
  39. M r r = - q 16 [ ( 1 + ν ) a 2 - ( 3 + ν ) r 2 ] ; M θ θ = - q 16 [ ( 1 + ν ) a 2 - ( 1 + 3 ν ) r 2 ] ; M r θ = 0 . M_{rr}=-\frac{q}{16}\left[(1+\nu)a^{2}-(3+\nu)r^{2}\right]~{};~{}~{}M_{\theta% \theta}=-\frac{q}{16}\left[(1+\nu)a^{2}-(1+3\nu)r^{2}\right]~{};~{}~{}M_{r% \theta}=0\,.
  40. z = h z=h
  41. r = a r=a
  42. σ r r | z = h , r = a = 3 q a 2 16 h 2 = 3 q a 2 4 H 2 \left.\sigma_{rr}\right|_{z=h,r=a}=\frac{3qa^{2}}{16h^{2}}=\frac{3qa^{2}}{4H^{% 2}}
  43. H := 2 h H:=2h
  44. M r r | r = a = q a 2 8 , M θ θ | r = a = ν q a 2 8 , M r r | r = 0 = M θ θ | r = 0 = - ( 1 + ν ) q a 2 16 . \left.M_{rr}\right|_{r=a}=\frac{qa^{2}}{8}~{},~{}~{}\left.M_{\theta\theta}% \right|_{r=a}=\frac{\nu qa^{2}}{8}~{},~{}~{}\left.M_{rr}\right|_{r=0}=\left.M_% {\theta\theta}\right|_{r=0}=-\frac{(1+\nu)qa^{2}}{16}\,.
  45. q q
  46. q ( x , y ) = q 0 sin π x a sin π y b . q(x,y)=q_{0}\sin\frac{\pi x}{a}\sin\frac{\pi y}{b}\,.
  47. q 0 q_{0}
  48. a a
  49. x x
  50. b b
  51. y y
  52. w ( x , y ) w(x,y)
  53. M x x M_{xx}
  54. x = 0 x=0
  55. x = a x=a
  56. M y y M_{yy}
  57. y = 0 y=0
  58. y = b y=b
  59. w ( x , y ) = q 0 π 4 D ( 1 a 2 + 1 b 2 ) - 2 sin π x a sin π y b . w(x,y)=\frac{q_{0}}{\pi^{4}D}\,\left(\frac{1}{a^{2}}+\frac{1}{b^{2}}\right)^{-% 2}\,\sin\frac{\pi x}{a}\sin\frac{\pi y}{b}\,.
  60. D = E t 3 12 ( 1 - ν 2 ) D=\frac{Et^{3}}{12(1-\nu^{2})}
  61. q ( x , y ) = q 0 sin m π x a sin n π y b q(x,y)=q_{0}\sin\frac{m\pi x}{a}\sin\frac{n\pi y}{b}
  62. m m
  63. n n
  64. (1) w ( x , y ) = q 0 π 4 D ( m 2 a 2 + n 2 b 2 ) - 2 sin m π x a sin n π y b . \,\text{(1)}\qquad w(x,y)=\frac{q_{0}}{\pi^{4}D}\,\left(\frac{m^{2}}{a^{2}}+% \frac{n^{2}}{b^{2}}\right)^{-2}\,\sin\frac{m\pi x}{a}\sin\frac{n\pi y}{b}\,.
  65. q ( x , y ) q(x,y)
  66. q ( x , y ) = m = 1 n = 1 a m n sin m π x a sin n π y b q(x,y)=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}a_{mn}\sin\frac{m\pi x}{a}\sin% \frac{n\pi y}{b}
  67. a m n a_{mn}
  68. 0 a sin k π x a sin π x a d x = { 0 k a / 2 k = \int_{0}^{a}\sin\frac{k\pi x}{a}\sin\frac{\ell\pi x}{a}\,\text{d}x=\begin{% cases}0&k\neq\ell\\ a/2&k=\ell\end{cases}
  69. a m n a_{mn}
  70. y y
  71. 0 b q ( x , y ) sin π y b d y = m = 1 n = 1 a m n sin m π x a 0 b sin n π y b sin π y b d y = b 2 m = 1 a m sin m π x a . \int_{0}^{b}q(x,y)\sin\frac{\ell\pi y}{b}\,\,\text{d}y=\sum_{m=1}^{\infty}\sum% _{n=1}^{\infty}a_{mn}\sin\frac{m\pi x}{a}\int_{0}^{b}\sin\frac{n\pi y}{b}\sin% \frac{\ell\pi y}{b}\,\,\text{d}y=\frac{b}{2}\sum_{m=1}^{\infty}a_{m\ell}\sin% \frac{m\pi x}{a}\,.
  72. x x
  73. 0 b 0 a q ( x , y ) sin k π x a sin π y b d x d y = b 2 m = 1 a m 0 a sin m π x a sin k π x a d x = a b 4 a k . \int_{0}^{b}\int_{0}^{a}q(x,y)\sin\frac{k\pi x}{a}\sin\frac{\ell\pi y}{b}\,\,% \text{d}x\,\text{d}y=\frac{b}{2}\sum_{m=1}^{\infty}a_{m\ell}\int_{0}^{a}\sin% \frac{m\pi x}{a}\sin\frac{k\pi x}{a}\,\,\text{d}x=\frac{ab}{4}a_{k\ell}\,.
  74. a m n = 4 a b 0 b 0 a q ( x , y ) sin m π x a sin n π y b d x d y . a_{mn}=\frac{4}{ab}\int_{0}^{b}\int_{0}^{a}q(x,y)\sin\frac{m\pi x}{a}\sin\frac% {n\pi y}{b}\,\,\text{d}x\,\text{d}y\,.
  75. a m n a_{mn}
  76. (2) w ( x , y ) = m = 1 n = 1 a m n π 4 D ( m 2 a 2 + n 2 b 2 ) - 2 sin m π x a sin n π y b . \,\text{(2)}\qquad w(x,y)=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{a_{mn}}{% \pi^{4}D}\,\left(\frac{m^{2}}{a^{2}}+\frac{n^{2}}{b^{2}}\right)^{-2}\,\sin% \frac{m\pi x}{a}\sin\frac{n\pi y}{b}\,.
  77. q ( x , y ) = q 0 q(x,y)=q_{0}
  78. a m n = 4 q 0 a b 0 a 0 b sin m π x a sin n π y b d x d y . a_{mn}=\frac{4q_{0}}{ab}\int_{0}^{a}\int_{0}^{b}\sin\frac{m\pi x}{a}\sin\frac{% n\pi y}{b}\,\,\text{d}x\,\text{d}y\,.
  79. 0 a sin m π x a d x = a m π ( 1 - cos m π ) and 0 b sin n π y b d y = b n π ( 1 - cos n π ) . \int_{0}^{a}\sin\frac{m\pi x}{a}\,\,\text{d}x=\frac{a}{m\pi}(1-\cos m\pi)\quad% \,\text{and}\quad\int_{0}^{b}\sin\frac{n\pi y}{b}\,\,\text{d}y=\frac{b}{n\pi}(% 1-\cos n\pi)\,.
  80. a m n a_{mn}
  81. a m n = 4 q 0 m n π 2 ( 1 - cos m π ) ( 1 - cos n π ) . a_{mn}=\frac{4q_{0}}{mn\pi^{2}}(1-\cos m\pi)(1-\cos n\pi)\,.
  82. cos m π = cos n π = 1 \cos m\pi=\cos n\pi=1
  83. ( 1 - cos m π ) = ( 1 - cos n π ) = 0 (1-\cos m\pi)=(1-\cos n\pi)=0
  84. m m
  85. n n
  86. a m n a_{mn}
  87. m m
  88. n n
  89. a m n = { 0 m or n even , 16 q 0 m n π 2 m and n odd . a_{mn}=\begin{cases}0&m~{}\,\text{or}~{}n~{}\,\text{even},\\ \cfrac{16q_{0}}{mn\pi^{2}}&m~{}\,\text{and}~{}n~{}\,\text{odd}\,.\end{cases}
  90. w ( x , y ) = m = 1 n = 1 16 q 0 ( 2 m - 1 ) ( 2 n - 1 ) π 6 D [ ( 2 m - 1 ) 2 a 2 + ( 2 n - 1 ) 2 b 2 ] - 2 × sin ( 2 m - 1 ) π x a sin ( 2 n - 1 ) π y b . \begin{aligned}\displaystyle w(x,y)&\displaystyle=\sum_{m=1}^{\infty}\sum_{n=1% }^{\infty}\frac{16q_{0}}{(2m-1)(2n-1)\pi^{6}D}\,\left[\frac{(2m-1)^{2}}{a^{2}}% +\frac{(2n-1)^{2}}{b^{2}}\right]^{-2}\,\times\\ &\displaystyle\qquad\qquad\quad\sin\frac{(2m-1)\pi x}{a}\sin\frac{(2n-1)\pi y}% {b}\,.\end{aligned}
  91. M x x = - D ( 2 w x 2 + ν 2 w y 2 ) = m = 1 n = 1 16 q 0 ( 2 m - 1 ) ( 2 n - 1 ) π 4 [ ( 2 m - 1 ) 2 a 2 + ν ( 2 n - 1 ) 2 b 2 ] × [ ( 2 m - 1 ) 2 a 2 + ( 2 n - 1 ) 2 b 2 ] - 2 sin ( 2 m - 1 ) π x a sin ( 2 n - 1 ) π y b M y y = - D ( 2 w y 2 + ν 2 w x 2 ) = m = 1 n = 1 16 q 0 ( 2 m - 1 ) ( 2 n - 1 ) π 4 [ ( 2 n - 1 ) 2 b 2 + ν ( 2 m - 1 ) 2 a 2 ] × [ ( 2 m - 1 ) 2 a 2 + ( 2 n - 1 ) 2 b 2 ] - 2 sin ( 2 m - 1 ) π x a sin ( 2 n - 1 ) π y b . \begin{aligned}\displaystyle M_{xx}&\displaystyle=-D\left(\frac{\partial^{2}w}% {\partial x^{2}}+\nu\frac{\partial^{2}w}{\partial y^{2}}\right)\\ &\displaystyle=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{16q_{0}}{(2m-1)(2n-% 1)\pi^{4}}\,\left[\frac{(2m-1)^{2}}{a^{2}}+\nu\frac{(2n-1)^{2}}{b^{2}}\right]% \,\times\\ &\displaystyle\qquad\qquad\left[\frac{(2m-1)^{2}}{a^{2}}+\frac{(2n-1)^{2}}{b^{% 2}}\right]^{-2}\sin\frac{(2m-1)\pi x}{a}\sin\frac{(2n-1)\pi y}{b}\\ \displaystyle M_{yy}&\displaystyle=-D\left(\frac{\partial^{2}w}{\partial y^{2}% }+\nu\frac{\partial^{2}w}{\partial x^{2}}\right)\\ &\displaystyle=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{16q_{0}}{(2m-1)(2n-% 1)\pi^{4}}\,\left[\frac{(2n-1)^{2}}{b^{2}}+\nu\frac{(2m-1)^{2}}{a^{2}}\right]% \,\times\\ &\displaystyle\qquad\qquad\left[\frac{(2m-1)^{2}}{a^{2}}+\frac{(2n-1)^{2}}{b^{% 2}}\right]^{-2}\sin\frac{(2m-1)\pi x}{a}\sin\frac{(2n-1)\pi y}{b}\,.\end{aligned}
  92. σ x x = 3 z 2 h 3 M x x = 12 z H 3 M x x and σ y y = 3 z 2 h 3 M y y = 12 z H 3 M y y . \sigma_{xx}=\frac{3z}{2h^{3}}\,M_{xx}=\frac{12z}{H^{3}}\,M_{xx}\quad\,\text{% and}\quad\sigma_{yy}=\frac{3z}{2h^{3}}\,M_{yy}=\frac{12z}{H^{3}}\,M_{yy}\,.
  93. w ( x , y ) = m = 1 Y m ( y ) sin m π x a . w(x,y)=\sum_{m=1}^{\infty}Y_{m}(y)\sin\frac{m\pi x}{a}\,.
  94. x = 0 x=0
  95. x = a x=a
  96. w = 0 w=0
  97. M x x = 0 M_{xx}=0
  98. 2 w / x 2 = 0 \partial^{2}w/\partial x^{2}=0
  99. Y m ( y ) Y_{m}(y)
  100. y = 0 y=0
  101. y = b y=b
  102. 2 2 w = q / D \nabla^{2}\nabla^{2}w=q/D
  103. q = 0 q=0
  104. w ( x , y ) w(x,y)
  105. 2 2 w = 0 \nabla^{2}\nabla^{2}w=0
  106. 4 w x 4 + 2 4 w x 2 y 2 + 4 w y 4 = 0 . \frac{\partial^{4}w}{\partial x^{4}}+2\frac{\partial^{4}w}{\partial x^{2}% \partial y^{2}}+\frac{\partial^{4}w}{\partial y^{4}}=0\,.
  107. w ( x , y ) w(x,y)
  108. m = 1 [ ( m π a ) 4 Y m sin m π x a - 2 ( m π a ) 2 d 2 Y m d y 2 sin m π x a + d 4 Y m d y 4 sin m π x a ] = 0 \sum_{m=1}^{\infty}\left[\left(\frac{m\pi}{a}\right)^{4}Y_{m}\sin\frac{m\pi x}% {a}-2\left(\frac{m\pi}{a}\right)^{2}\cfrac{d^{2}Y_{m}}{dy^{2}}\sin\frac{m\pi x% }{a}+\frac{d^{4}Y_{m}}{dy^{4}}\sin\frac{m\pi x}{a}\right]=0
  109. d 4 Y m d y 4 - 2 m 2 π 2 a 2 d 2 Y m d y 2 + m 4 π 4 a 4 Y m = 0 . \frac{d^{4}Y_{m}}{dy^{4}}-2\frac{m^{2}\pi^{2}}{a^{2}}\cfrac{d^{2}Y_{m}}{dy^{2}% }+\frac{m^{4}\pi^{4}}{a^{4}}Y_{m}=0\,.
  110. Y m = A m cosh m π y a + B m m π y a cosh m π y a + C m sinh m π y a + D m m π y a sinh m π y a Y_{m}=A_{m}\cosh\frac{m\pi y}{a}+B_{m}\frac{m\pi y}{a}\cosh\frac{m\pi y}{a}+C_% {m}\sinh\frac{m\pi y}{a}+D_{m}\frac{m\pi y}{a}\sinh\frac{m\pi y}{a}
  111. A m , B m , C m , D m A_{m},B_{m},C_{m},D_{m}
  112. w ( x , y ) = m = 1 [ ( A m + B m m π y a ) cosh m π y a + ( C m + D m m π y a ) sinh m π y a ] sin m π x a . w(x,y)=\sum_{m=1}^{\infty}\left[\left(A_{m}+B_{m}\frac{m\pi y}{a}\right)\cosh% \frac{m\pi y}{a}+\left(C_{m}+D_{m}\frac{m\pi y}{a}\right)\sinh\frac{m\pi y}{a}% \right]\sin\frac{m\pi x}{a}\,.
  113. x = 0 x=0
  114. x = a x=a
  115. y = ± b / 2 y=\pm b/2
  116. y = 0 y=0
  117. y = b y=b
  118. y = ± b / 2 y=\pm b/2
  119. w = 0 , - D 2 w y 2 | y = b / 2 = f 1 ( x ) , - D 2 w y 2 | y = - b / 2 = f 2 ( x ) w=0\,,-D\frac{\partial^{2}w}{\partial y^{2}}\Bigr|_{y=b/2}=f_{1}(x)\,,-D\frac{% \partial^{2}w}{\partial y^{2}}\Bigr|_{y=-b/2}=f_{2}(x)
  120. f 1 ( x ) , f 2 ( x ) f_{1}(x),f_{2}(x)
  121. M y y | y = - b / 2 = M y y | y = b / 2 M_{yy}\Bigr|_{y=-b/2}=M_{yy}\Bigr|_{y=b/2}
  122. f 1 ( x ) = f 2 ( x ) = m = 1 E m sin m π x a f_{1}(x)=f_{2}(x)=\sum_{m=1}^{\infty}E_{m}\sin\frac{m\pi x}{a}
  123. w ( x , y ) = a 2 2 π 2 D m = 1 E m m 2 cosh α m sin m π x a ( α m tanh α m cosh m π y a - m π y a sinh m π y a ) w(x,y)=\frac{a^{2}}{2\pi^{2}D}\sum_{m=1}^{\infty}\frac{E_{m}}{m^{2}\cosh\alpha% _{m}}\,\sin\frac{m\pi x}{a}\,\left(\alpha_{m}\tanh\alpha_{m}\cosh\frac{m\pi y}% {a}-\frac{m\pi y}{a}\sinh\frac{m\pi y}{a}\right)
  124. α m = m π b 2 a . \alpha_{m}=\frac{m\pi b}{2a}\,.
  125. M y y | y = - b / 2 = - M y y | y = b / 2 M_{yy}\Bigr|_{y=-b/2}=-M_{yy}\Bigr|_{y=b/2}
  126. w ( x , y ) = a 2 2 π 2 D m = 1 E m m 2 sinh α m sin m π x a ( α m coth α m sinh m π y a - m π y a cosh m π y a ) . w(x,y)=\frac{a^{2}}{2\pi^{2}D}\sum_{m=1}^{\infty}\frac{E_{m}}{m^{2}\sinh\alpha% _{m}}\,\sin\frac{m\pi x}{a}\,\left(\alpha_{m}\coth\alpha_{m}\sinh\frac{m\pi y}% {a}-\frac{m\pi y}{a}\cosh\frac{m\pi y}{a}\right)\,.
  127. y = ± b / 2 y=\pm b/2
  128. M y y = f 1 ( x ) = 4 M 0 π m = 1 1 2 m - 1 sin ( 2 m - 1 ) π x a . M_{yy}=f_{1}(x)=\frac{4M_{0}}{\pi}\sum_{m=1}^{\infty}\frac{1}{2m-1}\,\sin\frac% {(2m-1)\pi x}{a}\,.
  129. w ( x , y ) = 2 M 0 a 2 π 3 D m = 1 1 ( 2 m - 1 ) 3 cosh α m sin ( 2 m - 1 ) π x a × [ α m tanh α m cosh ( 2 m - 1 ) π y a - ( 2 m - 1 ) π y a sinh ( 2 m - 1 ) π y a ] \begin{aligned}\displaystyle w(x,y)&\displaystyle=\frac{2M_{0}a^{2}}{\pi^{3}D}% \sum_{m=1}^{\infty}\frac{1}{(2m-1)^{3}\cosh\alpha_{m}}\sin\frac{(2m-1)\pi x}{a% }\times\\ &\displaystyle\qquad\left[\alpha_{m}\,\tanh\alpha_{m}\cosh\frac{(2m-1)\pi y}{a% }-\frac{(2m-1)\pi y}{a}\sinh\frac{(2m-1)\pi y}{a}\right]\end{aligned}
  130. α m = π ( 2 m - 1 ) b 2 a . \alpha_{m}=\frac{\pi(2m-1)b}{2a}\,.
  131. w w
  132. M x x = - D ( 2 w x 2 + ν 2 w y 2 ) = 2 M 0 ( 1 - ν ) π m = 1 1 ( 2 m - 1 ) cosh α m sin ( 2 m - 1 ) π x a [ - ( 2 m - 1 ) π y a sinh ( 2 m - 1 ) π y a + { 2 ν 1 - ν + α m tanh α m } cosh ( 2 m - 1 ) π y a ] M x y = ( 1 - ν ) D 2 w x y = - 2 M 0 ( 1 - ν ) π m = 1 1 ( 2 m - 1 ) cosh α m cos ( 2 m - 1 ) π x a [ ( 2 m - 1 ) π y a cosh ( 2 m - 1 ) π y a + ( 1 - α m tanh α m ) sinh ( 2 m - 1 ) π y a ] Q z x = M x x x - M x y y = 4 M 0 a m = 1 1 cosh α m cos ( 2 m - 1 ) π x a cosh ( 2 m - 1 ) π y a . \begin{aligned}\displaystyle M_{xx}&\displaystyle=-D\left(\frac{\partial^{2}w}% {\partial x^{2}}+\nu\,\frac{\partial^{2}w}{\partial y^{2}}\right)\\ &\displaystyle=\frac{2M_{0}(1-\nu)}{\pi}\sum_{m=1}^{\infty}\frac{1}{(2m-1)% \cosh\alpha_{m}}\,\sin\frac{(2m-1)\pi x}{a}\left[-\frac{(2m-1)\pi y}{a}\sinh% \frac{(2m-1)\pi y}{a}+\right.\\ &\displaystyle\qquad\qquad\qquad\qquad\left.\left\{\frac{2\nu}{1-\nu}+\alpha_{% m}\tanh\alpha_{m}\right\}\cosh\frac{(2m-1)\pi y}{a}\right]\\ \displaystyle M_{xy}&\displaystyle=(1-\nu)D\frac{\partial^{2}w}{\partial x% \partial y}\\ &\displaystyle=-\frac{2M_{0}(1-\nu)}{\pi}\sum_{m=1}^{\infty}\frac{1}{(2m-1)% \cosh\alpha_{m}}\,\cos\frac{(2m-1)\pi x}{a}\left[\frac{(2m-1)\pi y}{a}\cosh% \frac{(2m-1)\pi y}{a}+\right.\\ &\displaystyle\qquad\qquad\qquad\qquad\left.(1-\alpha_{m}\tanh\alpha_{m})\sinh% \frac{(2m-1)\pi y}{a}\right]\\ \displaystyle Q_{zx}&\displaystyle=\frac{\partial M_{xx}}{\partial x}-\frac{% \partial M_{xy}}{\partial y}\\ &\displaystyle=\frac{4M_{0}}{a}\sum_{m=1}^{\infty}\frac{1}{\cosh\alpha_{m}}\,% \cos\frac{(2m-1)\pi x}{a}\cosh\frac{(2m-1)\pi y}{a}\,.\end{aligned}
  133. σ x x = 12 z h 3 M x x and σ z x = 1 κ h Q z x ( 1 - 4 z 2 h 2 ) . \sigma_{xx}=\frac{12z}{h^{3}}\,M_{xx}\quad\,\text{and}\quad\sigma_{zx}=\frac{1% }{\kappa h}\,Q_{zx}\left(1-\frac{4z^{2}}{h^{2}}\right)\,.
  134. a × b × h a\times b\times h
  135. a b a\ll b
  136. h h
  137. x 1 x_{1}
  138. 2 ( - 1 + ν q ) = - q κ G h ( 2 w + D ) = - ( 1 - c 2 1 + ν ) q 2 ( φ 1 x 2 - φ 2 x 1 ) = c 2 ( φ 1 x 2 - φ 2 x 1 ) \begin{aligned}&\displaystyle\nabla^{2}\left(\mathcal{M}-\frac{\mathcal{B}}{1+% \nu}\,q\right)=-q\\ &\displaystyle\kappa Gh\left(\nabla^{2}w+\frac{\mathcal{M}}{D}\right)=-\left(1% -\cfrac{\mathcal{B}c^{2}}{1+\nu}\right)q\\ &\displaystyle\nabla^{2}\left(\frac{\partial\varphi_{1}}{\partial x_{2}}-\frac% {\partial\varphi_{2}}{\partial x_{1}}\right)=c^{2}\left(\frac{\partial\varphi_% {1}}{\partial x_{2}}-\frac{\partial\varphi_{2}}{\partial x_{1}}\right)\end{aligned}
  139. q q
  140. G G
  141. D = E h 3 / [ 12 ( 1 - ν 2 ) ] D=Eh^{3}/[12(1-\nu^{2})]
  142. h h
  143. c 2 = 2 κ G h / [ D ( 1 - ν ) ] c^{2}=2\kappa Gh/[D(1-\nu)]
  144. κ \kappa
  145. E E
  146. ν \nu
  147. = D [ 𝒜 ( φ 1 x 1 + φ 2 x 2 ) - ( 1 - 𝒜 ) 2 w ] + 2 q 1 - ν 2 . \mathcal{M}=D\left[\mathcal{A}\left(\frac{\partial\varphi_{1}}{\partial x_{1}}% +\frac{\partial\varphi_{2}}{\partial x_{2}}\right)-(1-\mathcal{A})\nabla^{2}w% \right]+\frac{2q}{1-\nu^{2}}\mathcal{B}\,.
  148. w w
  149. φ 1 \varphi_{1}
  150. φ 2 \varphi_{2}
  151. x 2 x_{2}
  152. x 1 x_{1}
  153. 𝒜 = 1 \mathcal{A}=1
  154. = 0 \mathcal{B}=0
  155. κ \kappa
  156. 5 / 6 5/6
  157. w = w K + K κ G h ( 1 - c 2 2 ) - Φ + Ψ φ 1 = - w K x 1 - 1 κ G h ( 1 - 1 𝒜 - c 2 2 ) Q 1 K + x 1 ( D κ G h 𝒜 2 Φ + Φ - Ψ ) + 1 c 2 Ω x 2 φ 2 = - w K x 2 - 1 κ G h ( 1 - 1 𝒜 - c 2 2 ) Q 2 K + x 2 ( D κ G h 𝒜 2 Φ + Φ - Ψ ) + 1 c 2 Ω x 1 \begin{aligned}\displaystyle w&\displaystyle=w^{K}+\frac{\mathcal{M}^{K}}{% \kappa Gh}\left(1-\frac{\mathcal{B}c^{2}}{2}\right)-\Phi+\Psi\\ \displaystyle\varphi_{1}&\displaystyle=-\frac{\partial w^{K}}{\partial x_{1}}-% \frac{1}{\kappa Gh}\left(1-\frac{1}{\mathcal{A}}-\frac{\mathcal{B}c^{2}}{2}% \right)Q_{1}^{K}+\frac{\partial}{\partial x_{1}}\left(\frac{D}{\kappa Gh% \mathcal{A}}\nabla^{2}\Phi+\Phi-\Psi\right)+\frac{1}{c^{2}}\frac{\partial% \Omega}{\partial x_{2}}\\ \displaystyle\varphi_{2}&\displaystyle=-\frac{\partial w^{K}}{\partial x_{2}}-% \frac{1}{\kappa Gh}\left(1-\frac{1}{\mathcal{A}}-\frac{\mathcal{B}c^{2}}{2}% \right)Q_{2}^{K}+\frac{\partial}{\partial x_{2}}\left(\frac{D}{\kappa Gh% \mathcal{A}}\nabla^{2}\Phi+\Phi-\Psi\right)+\frac{1}{c^{2}}\frac{\partial% \Omega}{\partial x_{1}}\end{aligned}
  158. w K w^{K}
  159. Φ \Phi
  160. 2 2 Φ = 0 \nabla^{2}\nabla^{2}\Phi=0
  161. Ψ \Psi
  162. 2 Ψ = 0 \nabla^{2}\Psi=0
  163. = K + 1 + ν q + D 2 Φ ; K := - D 2 w K Q 1 K = - D x 1 ( 2 w K ) , Q 2 K = - D x 2 ( 2 w K ) Ω = φ 1 x 2 - φ 2 x 1 , 2 Ω = c 2 Ω . \begin{aligned}\displaystyle\mathcal{M}&\displaystyle=\mathcal{M}^{K}+\frac{% \mathcal{B}}{1+\nu}\,q+D\nabla^{2}\Phi~{};~{}~{}\mathcal{M}^{K}:=-D\nabla^{2}w% ^{K}\\ \displaystyle Q_{1}^{K}&\displaystyle=-D\frac{\partial}{\partial x_{1}}\left(% \nabla^{2}w^{K}\right)~{},~{}~{}Q_{2}^{K}=-D\frac{\partial}{\partial x_{2}}% \left(\nabla^{2}w^{K}\right)\\ \displaystyle\Omega&\displaystyle=\frac{\partial\varphi_{1}}{\partial x_{2}}-% \frac{\partial\varphi_{2}}{\partial x_{1}}~{},~{}~{}\nabla^{2}\Omega=c^{2}% \Omega\,.\end{aligned}
  164. = 1 1 + ν ( M 11 + M 22 ) = D ( φ 1 x 1 + φ 2 x 2 ) = 0 . \mathcal{M}=\frac{1}{1+\nu}(M_{11}+M_{22})=D\left(\frac{\partial\varphi_{1}}{% \partial x_{1}}+\frac{\partial\varphi_{2}}{\partial x_{2}}\right)=0\,.
  165. Φ \Phi
  166. Ψ \Psi
  167. Ω \Omega
  168. w = w K + K κ G h . w=w^{K}+\frac{\mathcal{M}^{K}}{\kappa Gh}\,.
  169. q x ( y ) q_{x}(y)
  170. x = a x=a
  171. b D d 4 w x d x 4 = 0 b 3 D 12 d 4 θ x d x 4 - 2 b D ( 1 - ν ) d 2 θ x d x 2 = 0 \begin{aligned}&\displaystyle bD\frac{\mathrm{d}^{4}w_{x}}{\mathrm{d}x^{4}}=0% \\ &\displaystyle\frac{b^{3}D}{12}\,\frac{\mathrm{d}^{4}\theta_{x}}{\mathrm{d}x^{% 4}}-2bD(1-\nu)\cfrac{d^{2}\theta_{x}}{dx^{2}}=0\end{aligned}
  172. x = a x=a
  173. b D d 3 w x d x 3 + q x 1 = 0 , b 3 D 12 d 3 θ x d x 3 - 2 b D ( 1 - ν ) d θ x d x + q x 2 = 0 b D d 2 w x d x 2 = 0 , b 3 D 12 d 2 θ x d x 2 = 0 . \begin{aligned}&\displaystyle bD\cfrac{d^{3}w_{x}}{dx^{3}}+q_{x1}=0\quad,\quad% \frac{b^{3}D}{12}\cfrac{d^{3}\theta_{x}}{dx^{3}}-2bD(1-\nu)\cfrac{d\theta_{x}}% {dx}+q_{x2}=0\\ &\displaystyle bD\cfrac{d^{2}w_{x}}{dx^{2}}=0\quad,\quad\frac{b^{3}D}{12}% \cfrac{d^{2}\theta_{x}}{dx^{2}}=0\,.\end{aligned}
  174. w x ( x ) = q x 1 6 b D ( 3 a x 2 - x 3 ) θ x ( x ) = q x 2 2 b D ( 1 - ν ) [ x - 1 ν b ( sinh ( ν b a ) cosh [ ν b ( x - a ) ] + tanh [ ν b ( x - a ) ] ) ] \begin{aligned}\displaystyle w_{x}(x)&\displaystyle=\frac{q_{x1}}{6bD}\,(3ax^{% 2}-x^{3})\\ \displaystyle\theta_{x}(x)&\displaystyle=\frac{q_{x2}}{2bD(1-\nu)}\left[x-% \frac{1}{\nu_{b}}\,\left(\frac{\sinh(\nu_{b}a)}{\cosh[\nu_{b}(x-a)]}+\tanh[\nu% _{b}(x-a)]\right)\right]\end{aligned}
  175. ν b = 24 ( 1 - ν ) / b \nu_{b}=\sqrt{24(1-\nu)}/b
  176. w = w x + y θ x w=w_{x}+y\theta_{x}
  177. M x x = - D ( 2 w x 2 + ν 2 w y 2 ) = q x 1 ( x - a b ) - [ 3 y q x 2 b 3 ν b cosh 3 [ ν b ( x - a ) ] ] × [ 6 sinh ( ν b a ) - sinh [ ν b ( 2 x - a ) ] + sinh [ ν b ( 2 x - 3 a ) ] + 8 sinh [ ν b ( x - a ) ] ] M x y = ( 1 - ν ) D 2 w x y = q x 2 2 b [ 1 - 2 + cosh [ ν b ( x - 2 a ) ] - cosh [ ν b x ] 2 cosh 2 [ ν b ( x - a ) ] ] Q z x = M x x x - M x y y = q x 1 b - ( 3 y q x 2 2 b 3 cosh 4 [ ν b ( x - a ) ] ) × [ 32 + cosh [ ν b ( 3 x - 2 a ) ] - cosh [ ν b ( 3 x - 4 a ) ] - 16 cosh [ 2 ν b ( x - a ) ] + 23 cosh [ ν b ( x - 2 a ) ] - 23 cosh ( ν b x ) ] . \begin{aligned}\displaystyle M_{xx}&\displaystyle=-D\left(\frac{\partial^{2}w}% {\partial x^{2}}+\nu\,\frac{\partial^{2}w}{\partial y^{2}}\right)\\ &\displaystyle=q_{x1}\left(\frac{x-a}{b}\right)-\left[\frac{3yq_{x2}}{b^{3}\nu% _{b}\cosh^{3}[\nu_{b}(x-a)]}\right]\times\\ &\displaystyle\quad\left[6\sinh(\nu_{b}a)-\sinh[\nu_{b}(2x-a)]+\sinh[\nu_{b}(2% x-3a)]+8\sinh[\nu_{b}(x-a)]\right]\\ \displaystyle M_{xy}&\displaystyle=(1-\nu)D\frac{\partial^{2}w}{\partial x% \partial y}\\ &\displaystyle=\frac{q_{x2}}{2b}\left[1-\frac{2+\cosh[\nu_{b}(x-2a)]-\cosh[\nu% _{b}x]}{2\cosh^{2}[\nu_{b}(x-a)]}\right]\\ \displaystyle Q_{zx}&\displaystyle=\frac{\partial M_{xx}}{\partial x}-\frac{% \partial M_{xy}}{\partial y}\\ &\displaystyle=\frac{q_{x1}}{b}-\left(\frac{3yq_{x2}}{2b^{3}\cosh^{4}[\nu_{b}(% x-a)]}\right)\times\left[32+\cosh[\nu_{b}(3x-2a)]-\cosh[\nu_{b}(3x-4a)]\right.% \\ &\displaystyle\qquad\left.-16\cosh[2\nu_{b}(x-a)]+23\cosh[\nu_{b}(x-2a)]-23% \cosh(\nu_{b}x)\right]\,.\end{aligned}
  178. σ x x = 12 z h 3 M x x and σ z x = 1 κ h Q z x ( 1 - 4 z 2 h 2 ) . \sigma_{xx}=\frac{12z}{h^{3}}\,M_{xx}\quad\,\text{and}\quad\sigma_{zx}=\frac{1% }{\kappa h}\,Q_{zx}\left(1-\frac{4z^{2}}{h^{2}}\right)\,.
  179. y y
  180. q x 1 = - b / 2 b / 2 q 0 ( 1 2 - y b ) d y = b q 0 2 ; q x 2 = - b / 2 b / 2 y q 0 ( 1 2 - y b ) d y = - b 2 q 0 12 . q_{x1}=\int_{-b/2}^{b/2}q_{0}\left(\frac{1}{2}-\frac{y}{b}\right)\,\,\text{d}y% =\frac{bq_{0}}{2}~{};~{}~{}q_{x2}=\int_{-b/2}^{b/2}yq_{0}\left(\frac{1}{2}-% \frac{y}{b}\right)\,\,\text{d}y=-\frac{b^{2}q_{0}}{12}\,.

Benini_distribution.html

  1. 1 - e - α log x σ - β [ log x σ ] 2 1-e^{-\alpha\log{\frac{x}{\sigma}}-\beta[\log{\frac{x}{\sigma}}]^{2}}
  2. σ + σ 2 β H - 1 ( - 1 + α 2 β ) \sigma+\tfrac{\sigma}{\sqrt{2\beta}}H_{-1}\left(\tfrac{-1+\alpha}{\sqrt{2\beta% }}\right)
  3. H n ( x ) H_{n}(x)
  4. σ ( e - α + α 2 + β log 16 2 β ) \sigma\left(e^{\frac{-\alpha+\sqrt{\alpha^{2}+\beta\log{16}}}{2\beta}}\right)
  5. ( σ 2 + 2 σ 2 2 β H - 1 ( - 2 + α 2 β ) ) - μ 2 \left(\sigma^{2}+\tfrac{2\sigma^{2}}{\sqrt{2\beta}}H_{-1}\left(\tfrac{-2+% \alpha}{\sqrt{2\beta}}\right)\right)-\mu^{2}
  6. Benini ( α , β , σ ) \mathrm{Benini}(\alpha,\beta,\sigma)
  7. F ( x ) = 1 - exp { - α ( log x - log σ ) - β ( log x - log σ ) 2 } = 1 - ( x σ ) - α - β log ( x σ ) F(x)=1-\exp\{-\alpha(\log x-\log\sigma)-\beta(\log x-\log\sigma)^{2}\}=1-\left% (\frac{x}{\sigma}\right)^{-\alpha-\beta\log{\left(\frac{x}{\sigma}\right)}}
  8. x σ x\geq\sigma
  9. F ( x ) = 1 - exp { - β ( log x - log σ ) 2 } = 1 - ( x σ ) - β ( log x - log σ ) . F(x)=1-\exp\{-\beta(\log x-\log\sigma)^{2}\}=1-\left(\frac{x}{\sigma}\right)^{% -\beta(\log x-\log\sigma)}.
  10. f ( x ) = 2 β x exp { - β [ log ( x σ ) ] 2 } log ( x σ ) , x σ > 0. f(x)=\frac{2\beta}{x}\exp\left\{-\beta\left[\log\left(\frac{x}{\sigma}\right)% \right]^{2}\right\}\cdot\log\left(\frac{x}{\sigma}\right),\qquad x\geq\sigma>0.
  11. F - 1 ( u ) = σ exp - 1 β log ( 1 - u ) , 0 < u < 1. F^{-1}(u)=\sigma\exp\sqrt{-\frac{1}{\beta}\log(1-u)},\quad 0<u<1.
  12. X Benini ( α , 0 , σ ) X\sim\mathrm{Benini}(\alpha,0,\sigma)\,
  13. x m = σ x_{\mathrm{m}}=\sigma
  14. X Benini ( 0 , 1 2 σ 2 , 1 ) , X\sim\mathrm{Benini}(0,\tfrac{1}{2\sigma^{2}},1),
  15. X e U X\sim e^{U}
  16. U Rayleigh ( σ ) U\sim\mathrm{Rayleigh}(\sigma)

Benktander_type_II_distribution.html

  1. 𝐖 ( x ) \mathbf{W}(x)
  2. 1 1
  3. - b + 2 a e a b 𝐄 1 - 1 b ( a b ) a 2 b \frac{-b+2ae^{\frac{a}{b}}\mathbf{E}_{1-\frac{1}{b}}\left(\frac{a}{b}\right)}{% a^{2}b}
  4. 𝐄 n ( x ) \mathbf{E}_{n}(x)

Bennett's_inequality.html

  1. i i
  2. σ 2 = 1 n i = 1 n Var ( X i ) . \sigma^{2}=\frac{1}{n}\sum_{i=1}^{n}\operatorname{Var}(X_{i}).
  3. t 0 t≥0
  4. Pr ( i = 1 n X i > t ) exp ( - n σ 2 a 2 h ( a t n σ 2 ) ) , \Pr\left(\sum_{i=1}^{n}X_{i}>t\right)\leq\exp\left(-\frac{n\sigma^{2}}{a^{2}}h% \left(\frac{at}{n\sigma^{2}}\right)\right),
  5. h ( u ) = ( 1 + u ) l o g ( 1 + u ) u h(u)=(1+u)log(1+u)–u

Benson's_algorithm.html

  1. min C P x subject to A x b \min_{C}Px\;\,\text{ subject to }Ax\geq b
  2. P q × n P\in\mathbb{R}^{q\times n}
  3. A m × n A\in\mathbb{R}^{m\times n}
  4. b m b\in\mathbb{R}^{m}
  5. C C
  6. S = { x n : A x b } S=\{x\in\mathbb{R}^{n}:\;Ax\geq b\}
  7. P [ S ] + C P[S]+C
  8. C = + q := { y q : y 1 0 , , y q 0 } C=\mathbb{R}^{q}_{+}:=\{y\in\mathbb{R}^{q}:y_{1}\geq 0,\dots,y_{q}\geq 0\}

Berlekamp–Welch_algorithm.html

  1. 𝒪 ( N 3 ) \mathcal{O}(N^{3})
  2. α i \alpha_{i}
  3. α i 𝔽 \alpha_{i}\in\mathbb{F}
  4. K K
  5. D = N - K + 1 D=N-K+1
  6. y y
  7. ( y 1 , , y n ) 𝔽 n (y_{1},\ldots,y_{n})\in\mathbb{F}_{n}
  8. e < N - K + 1 2 e<{N-K+1\over 2}
  9. P P
  10. 𝔽 \mathbb{F}
  11. P P
  12. k - 1 k-1
  13. i i
  14. P ( α i ) y i e P(\alpha_{i})\neq y_{i}\leq e
  15. P ( X ) P(X)
  16. Δ ( y , ( P ( α i ) ) i = 1 N ) \Delta(y,(P(\alpha_{i}))^{N}_{i=1})
  17. e D 2 e\leq{D\over 2}
  18. N - K + 1 2 {N-K+1\over 2}
  19. P P
  20. i i
  21. E ( X ) E(X)
  22. 𝔽 \mathbb{F}
  23. E ( α i ) = 0 E(\alpha_{i})=0
  24. y i P ( α i ) y_{i}\neq P(\alpha_{i})
  25. E E
  26. E n - k 2 E\leq{n-k\over 2}
  27. E ( X ) = α i S ( X - α i ) E(X)=\prod_{\alpha_{i}\in S}(X-\alpha_{i})
  28. S = { α i | P ( α i ) y i } S=\{\alpha_{i}|P(\alpha_{i})\neq y_{i}\}
  29. 1 i N 1\leq i\leq N
  30. y i E ( α i ) = P ( α i ) E ( α i ) y_{i}E(\alpha_{i})=P(\alpha_{i})E(\alpha_{i})
  31. y i P ( α i ) y_{i}\neq P(\alpha_{i})
  32. 0
  33. E ( α i ) = 0 E(\alpha_{i})=0
  34. E ( X ) E(X)
  35. P ( X ) P(X)
  36. P ( X ) P(X)
  37. Q ( X ) Q(X)
  38. Q ( X ) Q(X)
  39. P ( X ) E ( X ) P(X)E(X)
  40. n n
  41. e + k e+k
  42. Q ( X ) Q(X)
  43. 𝔽 \mathbb{F}
  44. deg ( Q ) n - k 2 + k - 1 \deg(Q)\leq{{n-k\over 2}+k-1}
  45. Q ( α i ) = E ( α i ) y i \forall Q(\alpha_{i})=E(\alpha_{i})y_{i}
  46. Q ( X ) Q(X)
  47. E ( X ) E(X)
  48. P ( X ) P(X)
  49. P ( X ) = Q ( X ) E ( X ) P(X)={Q(X)\over E(X)}
  50. P ( X ) P(X)
  51. Q ( X ) Q(X)
  52. E ( X ) E(X)
  53. E E
  54. N N
  55. E ( X ) E(X)
  56. P ( X ) P(X)
  57. E ( X ) E(X)
  58. P ( X ) P(X)
  59. Q ( X ) Q(X)
  60. E ( X ) E(X)
  61. Q ( X ) Q(X)
  62. y y
  63. y i 0 y_{i}\neq 0
  64. 1 i n 1\leq i\leq n
  65. Q ( i ) = 0 Q(i)=0
  66. E ( X ) E(X)
  67. Q ( X ) Q(X)
  68. N N
  69. e e
  70. e e
  71. ( y i , α i ) i = 1 N (y_{i},\alpha_{i})^{N}_{i=1}
  72. P ( X ) P(X)
  73. d e g ( P ( X ) ) k - 1 deg(P(X))\leq{k-1}
  74. Δ ( y , P ( α i ) i ) e \Delta(y,{P(\alpha_{i})_{i}})\leq e
  75. P ( X ) P(X)
  76. E ( X ) E(X)
  77. X e X^{e}
  78. Q ( X ) Q(X)
  79. deg ( Q ( X ) ) e + K - 1 \deg(Q(X))\leq{e+K-1}
  80. y i E ( α i ) = Q ( α i ) y_{i}E(\alpha_{i})=Q(\alpha_{i})
  81. 1 i n 1\leq i\leq n
  82. E ( X ) E(X)
  83. Q ( X ) Q(X)
  84. P P
  85. ( X ) (X)
  86. Q ( X ) E ( X ) Q(X)\over E(X)
  87. Δ ( ( y , ( P \Delta((y,(P
  88. ( α i ) i ) e ) (\alpha_{i})_{i})\leq e)
  89. P P
  90. ( X ) (X)
  91. E ( X ) E(X)
  92. Q ( X ) Q(X)
  93. P ( X ) P(X)
  94. E ( X ) E(X)
  95. Q ( X ) Q(X)
  96. Q ( X ) E ( X ) = P ( X ) {Q(X)\over E(X)}=P(X)
  97. P ( X ) P(X)
  98. E ( X ) = X e - Δ ( y , P ( α i ) i ) 1 i n | y i P ( α i ) ( X - α i ) E(X)=X^{e-\Delta(y,P(\alpha_{i})_{i})}\prod_{1\leq i\leq n|y_{i}\neq P(\alpha_% {i})}(X-\alpha_{i})
  99. Q ( X ) = P ( X ) E ( X ) Q(X)=P(X)E(X)
  100. d e g ( Q ( X ) ) d e g ( P ( X ) ) + d e g ( E ( X ) ) e + k - 1 deg(Q(X))\leq{deg(P(X))+deg(E(X))}\leq{e+k-1}
  101. E ( X ) E(X)
  102. e e
  103. E ( X ) E(X)
  104. E ( α i ) = 0 E(\alpha_{i})=0
  105. y i P ( α i ) y_{i}\neq P(\alpha_{i})
  106. E ( X ) E(X)
  107. Q ( X ) Q(X)
  108. y i E ( α i ) = Q ( α i ) y_{i}E(\alpha_{i})=Q(\alpha_{i})
  109. E ( α i ) = 0 E(\alpha_{i})=0
  110. Q ( α i ) = P ( α i ) E ( α i ) = y i E ( α i ) = 0 Q(\alpha_{i})=P(\alpha_{i})E(\alpha_{i})=y_{i}E(\alpha_{i})=0
  111. E ( α i ) 0 E(\alpha_{i})\neq 0
  112. P ( α i ) = y i P(\alpha_{i})=y_{i}
  113. P ( α i ) E ( α i ) = y i E ( α i ) P(\alpha_{i})E(\alpha_{i})=y_{i}E(\alpha_{i})
  114. E ( X ) E(X)
  115. Q ( X ) Q(X)
  116. P ( X ) P(X)
  117. Q ( X ) / E ( X ) Q(X)/E(X)
  118. ( E 1 ( X ) , Q 1 ( X ) ) ( E 2 ( X ) , Q 2 ( X ) ) (E_{1}(X),Q_{1}(X))\neq(E_{2}(X),Q_{2}(X))
  119. Q 1 ( X ) E 1 ( X ) = Q 2 ( X ) E 2 ( X ) {Q_{1}(X)\over E_{1}(X)}={Q_{2}(X)\over E_{2}(X)}
  120. Q 1 ( X ) E 1 ( X ) Q_{1}(X)E_{1}(X)
  121. Q 2 ( X ) E 2 ( X ) 2 e + k - 1 Q_{2}(X)E_{2}(X)\leq{2e+k-1}
  122. R ( X ) = Q 1 ( X ) E 2 ( X ) - Q 2 ( X ) E 1 ( X ) R(X)=Q_{1}(X)E_{2}(X)-Q_{2}(X)E_{1}(X)
  123. R ( X ) R(X)
  124. d e g ( R ( X ) ) 2 e + k - 1 deg(R(X))\leq{2e+k-1}
  125. y i E 1 ( α i ) = Q 1 ( α i ) y_{i}E_{1}(\alpha_{i})=Q_{1}(\alpha_{i})
  126. y i E 2 ( α i ) = Q 2 ( α i y_{i}E_{2}(\alpha_{i})=Q_{2}(\alpha_{i}
  127. Q ( X ) Q(X)
  128. R ( α i ) = y i E 1 ( α i ) E 2 ( α i ) - y i E 2 ( α i ) E 1 ( α i ) = 0 R(\alpha_{i})=y_{i}E_{1}(\alpha_{i})E_{2}(\alpha_{i})-y_{i}E_{2}(\alpha_{i})E_% {1}(\alpha_{i})=0
  129. 1 i n 1\leq i\leq n
  130. R ( X ) R(X)
  131. n n
  132. d e g ( R ( X ) ) 2 e + k - 1 deg(R(X))\leq{2e+k-1}
  133. d e g ( R ( X ) ) deg(R(X))
  134. e e
  135. d e g ( R ( X ) ) deg(R(X))
  136. Q 1 ( X ) E 2 ( X ) Q_{1}(X)E_{2}(X)
  137. Q 2 ( X ) E 1 ( X ) Q_{2}(X)E_{1}(X)
  138. E 1 ( X ) 0 E_{1}(X)\neq 0
  139. E 2 ( X ) 0 E_{2}(X)\neq 0
  140. Q 1 ( X ) E 1 ( X ) = Q 2 ( X ) E 2 ( X ) {Q_{1}(X)\over E_{1}(X)}={Q_{2}(X)\over E_{2}(X)}
  141. P ( X ) P(X)
  142. O ( n 3 ) O(n^{3})
  143. Q ( X ) Q(X)
  144. E ( X ) E(X)
  145. e + k e+k
  146. e + 1 e+1
  147. y i E ( α i ) = Q ( α i ) y_{i}E(\alpha_{i})=Q(\alpha_{i})
  148. 1 i n 1\leq i\leq n
  149. n n
  150. 2 e + k + 1 2e+k+1
  151. E ( X ) E(X)
  152. e e
  153. O ( n 3 ) O(n^{3})
  154. O ( n 3 ) O(n^{3})
  155. [ n , k ] q [n,k]_{q}
  156. O ( n 3 ) O(n^{3})
  157. n - k + 1 2 {n-k+1}\over 2
  158. y = 5 - x y=5-x
  159. ( 1 , 4 ) , ( 2 , 3 ) , ( 3 , 4 ) , ( 4 , 1 ) (1,4),(2,3),(3,4),(4,1)
  160. ( 3 , 4 ) (3,4)
  161. Q ( 1 ) = 4 * E ( 1 ) Q ( 2 ) = 3 * E ( 2 ) Q ( 3 ) = 4 * E ( 3 ) Q ( 4 ) = 1 * E ( 4 ) \begin{aligned}\displaystyle Q(1)&\displaystyle=4*E(1)\\ \displaystyle Q(2)&\displaystyle=3*E(2)\\ \displaystyle Q(3)&\displaystyle=4*E(3)\\ \displaystyle Q(4)&\displaystyle=1*E(4)\\ \end{aligned}
  162. Q Q
  163. E E
  164. x = 1 , 2 , 3 , 4 x=1,2,3,4
  165. Q ( x i ) = E ( x i ) = 0 Q(x_{i})=E(x_{i})=0
  166. P ( x i ) = Q ( x i ) E ( x i ) = y i P(x_{i})={Q(x_{i})\over E(x_{i})}=y_{i}
  167. E E
  168. P ( x i ) P(x_{i})
  169. y i y_{i}
  170. E ( x ) = x + e 0 E(x)=x+e_{0}
  171. Q ( x ) = q 0 + q 1 x + q 2 x 2 Q(x)=q_{0}+q_{1}x+q_{2}x^{2}
  172. E ( x ) E(x)
  173. q 0 + q 1 + q 2 - 4 e 0 - 4 = 0 q 0 + 2 q 1 + 4 q 2 - 3 e 0 - 6 = 0 q 0 + 3 q 1 + 9 q 2 - 4 e 0 - 12 = 0 q 0 + 4 q 1 + 16 q 2 - e 0 - 4 = 0 \begin{aligned}\displaystyle q_{0}&\displaystyle+&\displaystyle q_{1}&% \displaystyle+&\displaystyle q_{2}&\displaystyle-&\displaystyle 4e_{0}&% \displaystyle-&\displaystyle 4&\displaystyle=&\displaystyle 0\\ \displaystyle q_{0}&\displaystyle+&\displaystyle 2q_{1}&\displaystyle+&% \displaystyle 4q_{2}&\displaystyle-&\displaystyle 3e_{0}&\displaystyle-&% \displaystyle 6&\displaystyle=&\displaystyle 0\\ \displaystyle q_{0}&\displaystyle+&\displaystyle 3q_{1}&\displaystyle+&% \displaystyle 9q_{2}&\displaystyle-&\displaystyle 4e_{0}&\displaystyle-&% \displaystyle 12&\displaystyle=&\displaystyle 0\\ \displaystyle q_{0}&\displaystyle+&\displaystyle 4q_{1}&\displaystyle+&% \displaystyle 16q_{2}&\displaystyle-&\displaystyle e_{0}&\displaystyle-&% \displaystyle 4&\displaystyle=&\displaystyle 0\end{aligned}
  174. q 0 = - 15 , q 1 = 8 , q 2 = - 1 , e 0 = - 3 q_{0}=-15,q_{1}=8,q_{2}=-1,e_{0}=-3
  175. Q ( x ) = - x 2 + 8 x - 15 , E ( x ) = x - 3 Q(x)=-x^{2}+8x-15,E(x)=x-3
  176. Q ( x ) E ( x ) = P ( x ) = 5 - x {Q(x)\over E(x)}=P(x)=5-x
  177. 5 - x 5-x

Bernoulli_space.html

  1. P X P_{X}
  2. ( X , D ) (X,D)
  3. ( X , D ) (X,D)
  4. X , D \mathcal{B}_{X,D}
  5. X , D = ( 𝒟 , 𝒳 , 𝒫 ) \mathcal{B}_{X,D}=(\mathcal{D},\mathcal{X},\mathcal{P})
  6. 𝒟 \mathcal{D}
  7. 𝒟 = { d } \mathcal{D}=\{d\}
  8. 𝒟 \mathcal{D}
  9. 𝒳 \mathcal{X}
  10. 𝒫 \mathcal{P}