wpmath0000006_0

100000000_(number).html

  1. 9 2 {}^{2}9

11-Beta_hydroxysteroid_dehydrogenase.html

  1. \rightleftharpoons

12-metre_class.html

  1. L + B + G / 3 + 3 d + S / 3 - F 2 12 metres \frac{L+B+G/3+3d+\sqrt{S}/3-F}{2}\leq 12\mbox{ metres}~{}
  2. L L
  3. B B
  4. G G
  5. d d
  6. S S
  7. F F
  8. L + G / 4 + 2 d + S - F 2.5 12 metres \frac{L+G/4+2d+\sqrt{S}-F}{2.5}\leq 12\mbox{ metres}~{}
  9. L L
  10. G G
  11. d d
  12. S S
  13. F F
  14. L + 2 d + S - F 2.37 12 metres \frac{L+2d+\sqrt{S}-F}{2.37}\leq 12\mbox{ metres}~{}
  15. L L
  16. d d
  17. S S
  18. F F

193_(number).html

  1. 4 p 2 + 1 4p^{2}+1

199_(number).html

  1. [ - F o r m u l a E r r o r - ] [-FormulaError-]

2-category.html

  1. A A
  2. B B
  3. A A
  4. B B
  5. 𝐂 ( A , B ) \mathbf{C}(A,B)
  6. f , g : A B f,g:A\to B
  7. α : f g \alpha:f\Rightarrow g
  8. \circ
  9. 1 \circ_{1}
  10. A A
  11. 𝐂 ( A , A ) \mathbf{C}(A,A)
  12. A A
  13. A A
  14. B B
  15. C C
  16. 0 : 𝐂 ( A , B ) × 𝐂 ( B , C ) 𝐂 ( A , C ) \circ_{0}:\mathbf{C}(A,B)\times\mathbf{C}(B,C)\to\mathbf{C}(A,C)
  17. 0 \circ_{0}
  18. α : f g : A B \alpha:f\Rightarrow g:A\to B
  19. β : f g : B C \beta:f^{\prime}\Rightarrow g^{\prime}:B\to C
  20. β α : f f g g : A C \beta\alpha:f^{\prime}f\Rightarrow g^{\prime}g:A\to C
  21. α , β , γ , δ \alpha,\beta,\gamma,\delta
  22. ( α 0 β ) 1 ( γ 0 δ ) = ( α 1 γ ) 0 ( β 1 δ ) (\alpha\circ_{0}\beta)\circ_{1}(\gamma\circ_{0}\delta)=(\alpha\circ_{1}\gamma)% \circ_{0}(\beta\circ_{1}\delta)
  23. 0 \circ_{0}
  24. 0 \circ_{0}
  25. 1 \circ_{1}
  26. f : A B f\colon A\rightarrow B
  27. A A
  28. B B

216_(number).html

  1. 216 = 3 3 + 4 3 + 5 3 = 6 3 216=3^{3}+4^{3}+5^{3}=6^{3}
  2. ( 2 9 12 36 6 1 3 4 18 ) \begin{pmatrix}2&9&12\\ 36&6&1\\ 3&4&18\end{pmatrix}

251_(number).html

  1. 251 = 2 3 + 3 3 + 6 3 = 1 3 + 5 3 + 5 3 . 251=2^{3}+3^{3}+6^{3}=1^{3}+5^{3}+5^{3}.

257_(number).html

  1. 2 2 n + 1 , 2^{2^{n}}+1,

3D_scanner.html

  1. c c
  2. t t
  3. c t / 2 \textstyle c\!\cdot\!t/2
  4. t t

6b::8b_encoding.html

  1. ( 8 4 ) {\textstyle\left({{8}\atop{4}}\right)}

786_(number).html

  1. C n 2 n {}_{2n}\!C_{n}

Aaron_Gleeman.html

  1. G P A = ( 1.8 × O B P ) + S L G 4 GPA={(1.8\times OBP)+SLG\over 4}

Abel_polynomials.html

  1. p n ( x ) = x ( x - a n ) n - 1 . p_{n}(x)=x(x-an)^{n-1}.\,
  2. a = 1 a=1
  3. p 0 ( x ) = 1 ; p_{0}(x)=1;
  4. p 1 ( x ) = x ; p_{1}(x)=x;
  5. p 2 ( x ) = - 2 x + x 2 ; p_{2}(x)=-2x+x^{2};
  6. p 3 ( x ) = 9 x - 6 x 2 + x 3 ; p_{3}(x)=9x-6x^{2}+x^{3};
  7. p 4 ( x ) = - 64 x + 48 x 2 - 12 x 3 + x 4 ; p_{4}(x)=-64x+48x^{2}-12x^{3}+x^{4};
  8. a = 2 a=2
  9. p 0 ( x ) = 1 ; p_{0}(x)=1;
  10. p 1 ( x ) = x ; p_{1}(x)=x;
  11. p 2 ( x ) = - 4 x + x 2 ; p_{2}(x)=-4x+x^{2};
  12. p 3 ( x ) = 36 x - 12 x 2 + x 3 ; p_{3}(x)=36x-12x^{2}+x^{3};
  13. p 4 ( x ) = - 512 x + 192 x 2 - 24 x 3 + x 4 ; p_{4}(x)=-512x+192x^{2}-24x^{3}+x^{4};
  14. p 5 ( x ) = 10000 x - 4000 x 2 + 600 x 3 - 40 x 4 + x 5 ; p_{5}(x)=10000x-4000x^{2}+600x^{3}-40x^{4}+x^{5};
  15. p 6 ( x ) = - 248832 x + 103680 x 2 - 17280 x 3 + 1440 x 4 - 60 x 5 + x 6 ; p_{6}(x)=-248832x+103680x^{2}-17280x^{3}+1440x^{4}-60x^{5}+x^{6};

Abouabdillah's_theorem.html

  1. E E
  2. f : E E f:E\rightarrow E
  3. f : E E f:E\rightarrow E

Absolute_Galois_group.html

  1. 𝐙 ^ = lim 𝐙 / n 𝐙 . \hat{\mathbf{Z}}=\underleftarrow{\lim}\mathbf{Z}/n\mathbf{Z}.

Abstract_analytic_number_theory.html

  1. a = p 1 α 1 p 2 α 2 p r α r a=p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}\cdots p_{r}^{\alpha_{r}}
  2. | | |\mbox{ }~{}|
  3. | 1 | = 1 |1|=1
  4. | p | > 1 for all p P |p|>1\mbox{ for all }~{}p\in P
  5. | a b | = | a | | b | for all a , b G |ab|=|a||b|\mbox{ for all }~{}a,b\in G
  6. N G ( x ) N_{G}(x)
  7. a G a\in G
  8. | a | x |a|\leq x
  9. x > 0 x>0
  10. A ( x ) = n a ( n ) x n = n ( 1 - x n ) - p ( n ) A(x)=\sum_{n}a(n)x^{n}=\prod_{n}(1-x^{n})^{-p(n)}
  11. A ( x ) = exp ( m 1 P ( x m ) m ) . A(x)=\exp\left({\sum_{m\geq 1}\frac{P(x^{m})}{m}}\right)\ .
  12. | n | = n |n|=n
  13. N G ( x ) = x N_{G}(x)=\lfloor x\rfloor
  14. | A | = card ( A ) |A|=\mbox{ card}~{}(A)
  15. | M | = c dim M |M|=c^{\mbox{dim }~{}M}
  16. | X | = 2 card ( X ) |X|=2^{\mbox{card}~{}(X)}
  17. δ \delta
  18. ν \nu
  19. 0 ν < δ 0\leq\nu<\delta
  20. N G ( x ) = A x δ + O ( x ν ) as x . N_{G}(x)=Ax^{\delta}+O(x^{\nu})\mbox{ as }x\rightarrow\infty.
  21. π G ( x ) x δ δ log x as x \pi_{G}(x)\sim\frac{x^{\delta}}{\delta\log x}\mbox{ as }x\rightarrow\infty
  22. g G χ ( [ g ] ) | g | - s \sum_{g\in G}\chi([g])|g|^{-s}

Abū_Kāmil_Shujāʿ_ibn_Aslam.html

  1. x 2 x^{2}
  2. x 8 x^{8}
  3. x 5 x^{5}
  4. x 2 x 2 x x^{2}\cdot x^{2}\cdot x
  5. x 2 x^{2}
  6. x x
  7. x 4 + 3125 = 125 x 2 x^{4}+3125=125x^{2}

Accounting_equation.html

  1. Assets = Capital + Liabilities \,\text{Assets}=\,\text{Capital}+\,\text{Liabilities}
  2. a = c + l a=c+l

Accumulation::distribution_index.html

  1. C L V = ( c l o s e - l o w ) - ( h i g h - c l o s e ) h i g h - l o w CLV={(close-low)-(high-close)\over high-low}
  2. a c c d i s t = a c c d i s t p r e v + v o l u m e × C L V accdist=accdist_{prev}+volume\times CLV

Acentric_factor.html

  1. ω \omega
  2. ω = - log 10 ( p r sat ) - 1 , at T r = 0.7 \omega=-\log_{10}(p^{\rm{sat}}_{r})-1,{\rm\ at\ }T_{r}=0.7
  3. T r = T T c T_{r}=\frac{T}{T_{c}}
  4. p r sat = p sat p c p^{\rm{sat}}_{r}=\frac{p^{\rm{sat}}}{p_{c}}
  5. p r sat at T r = 0.7 p_{r}^{\rm{sat}}{\rm\ at\ }T_{r}=0.7
  6. ω 0 \omega\to 0
  7. T r = 0.7 T_{r}=0.7
  8. ω \omega
  9. { T r , p r } \{T_{r},p_{r}\}
  10. T r = 0.7 T_{r}=0.7
  11. ω \omega
  12. ω \omega
  13. Z Z
  14. T r T_{r}
  15. p r p_{r}
  16. ω \omega
  17. { T r , p r } = c o n s t . \{T_{r},p_{r}\}=const.
  18. Z Z

Acetone_(data_page).html

  1. P m m H g = 10 7.02447 - 1161.0 224 + T \scriptstyle P_{mmHg}=10^{7.02447-\frac{1161.0}{224+T}}
  2. log 10 P m m H g = 7.02447 - 1161.0 224 + T \scriptstyle\log_{10}P_{mmHg}=7.02447-\frac{1161.0}{224+T}

Acid_catalysis.html

  1. rate = - d [ R 1 ] d t = k [ S H + ] [ R 1 ] [ R 2 ] \,\text{rate}=-\frac{\,\text{d}[R1]}{\,\text{d}t}=k[SH^{+}][R1][R2]
  2. rate = - d [ R 1 ] d t = k 1 [ S H + ] [ R 1 ] [ R 2 ] + k 2 [ A H 1 ] [ R 1 ] [ R 2 ] + k 3 [ A H 2 ] [ R 1 ] [ R 2 ] + \,\text{rate}=-\frac{\,\text{d}[R1]}{\,\text{d}t}=k_{1}[SH^{+}][R1][R2]+k_{2}[% AH^{1}][R1][R2]+k_{3}[AH^{2}][R1][R2]+...

Acnode.html

  1. f ( x , y ) = y 2 + x 2 - x 3 = 0 f(x,y)=y^{2}+x^{2}-x^{3}=0\;
  2. 2 \mathbb{R}^{2}
  3. y 2 = x 2 ( x - 1 ) y^{2}=x^{2}(x-1)
  4. x 2 ( x - 1 ) x^{2}(x-1)
  5. x x
  6. x = 0 x=0
  7. x < 1 x<1
  8. f x \partial f\over\partial x
  9. f y \partial f\over\partial y

Acoustic_metric.html

  1. v \vec{v}
  2. u \vec{u}
  3. ( u - v ( x ) ) 2 = c ( x ) 2 (\vec{u}-\vec{v}(x))^{2}=c(x)^{2}
  4. 𝐠 = g 00 d t d t + 2 g 0 i d x i d t + g i j d x i d x j \mathbf{g}=g_{00}dt\otimes dt+2g_{0i}dx^{i}\otimes dt+g_{ij}dx^{i}\otimes dx^{j}
  5. u \vec{u}
  6. g 00 + 2 g 0 i u i + g i j u i u j = 0 g_{00}+2g_{0i}u^{i}+g_{ij}u^{i}u^{j}=0
  7. g = α 2 ( - ( c 2 - v 2 ) - v - v 𝟏 ) g=\alpha^{2}\begin{pmatrix}-(c^{2}-v^{2})&-\vec{v}\\ -\vec{v}&\mathbf{1}\end{pmatrix}

Acoustic_wave_equation.html

  1. p p
  2. t t
  3. x x
  4. 2 p x 2 - 1 c 2 2 p t 2 = 0 {\partial^{2}p\over\partial x^{2}}-{1\over c^{2}}{\partial^{2}p\over\partial t% ^{2}}=0
  5. p p
  6. c c
  7. c c
  8. p = f ( c t - x ) + g ( c t + x ) p=f(ct-x)+g(ct+x)
  9. f f
  10. g g
  11. f f
  12. g g
  13. c c
  14. f f
  15. g g
  16. p = p 0 sin ( ω t k x ) p=p_{0}\sin(\omega t\mp kx)
  17. ω \omega
  18. k k
  19. P V = n R T PV=nRT
  20. ρ \rho
  21. P = C ρ P=C\rho\,
  22. C = P ρ C=\frac{\partial P}{\partial\rho}
  23. P - P 0 = ( P ρ ) ( ρ - ρ 0 ) P-P_{0}=\left(\frac{\partial P}{\partial\rho}\right)(\rho-\rho_{0})
  24. B = ρ 0 ( P ρ ) a d i a b a t i c B=\rho_{0}\left(\frac{\partial P}{\partial\rho}\right)_{adiabatic}
  25. P - P 0 = B ρ - ρ 0 ρ 0 P-P_{0}=B\frac{\rho-\rho_{0}}{\rho_{0}}
  26. s = ρ - ρ 0 ρ 0 s=\frac{\rho-\rho_{0}}{\rho_{0}}
  27. p = B s p=Bs\,
  28. P - P 0 P-P_{0}
  29. ρ t + x ( ρ u ) = 0 \frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x}(\rho u)=0
  30. t ( ρ 0 + ρ 0 s ) + x ( ρ 0 u + ρ 0 s u ) = 0 \frac{\partial}{\partial t}(\rho_{0}+\rho_{0}s)+\frac{\partial}{\partial x}(% \rho_{0}u+\rho_{0}su)=0
  31. s t + x u = 0 \frac{\partial s}{\partial t}+\frac{\partial}{\partial x}u=0
  32. ρ D u D t + P x = 0 \rho\frac{Du}{Dt}+\frac{\partial P}{\partial x}=0
  33. D / D t D/Dt
  34. ( ρ 0 + ρ 0 s ) ( t + u x ) u + x ( P 0 + p ) = 0 (\rho_{0}+\rho_{0}s)\left(\frac{\partial}{\partial t}+u\frac{\partial}{% \partial x}\right)u+\frac{\partial}{\partial x}(P_{0}+p)=0
  35. ρ 0 u t + p x = 0 \rho_{0}\frac{\partial u}{\partial t}+\frac{\partial p}{\partial x}=0
  36. 2 s t 2 + 2 u x t = 0 \frac{\partial^{2}s}{\partial t^{2}}+\frac{\partial^{2}u}{\partial x\partial t% }=0
  37. ρ 0 2 u x t + 2 p x 2 = 0 \rho_{0}\frac{\partial^{2}u}{\partial x\partial t}+\frac{\partial^{2}p}{% \partial x^{2}}=0
  38. ρ 0 \rho_{0}
  39. - ρ 0 B 2 p t 2 + 2 p x 2 = 0 -\frac{\rho_{0}}{B}\frac{\partial^{2}p}{\partial t^{2}}+\frac{\partial^{2}p}{% \partial x^{2}}=0
  40. 2 p x 2 - 1 c 2 2 p t 2 = 0 {\partial^{2}p\over\partial x^{2}}-{1\over c^{2}}{\partial^{2}p\over\partial t% ^{2}}=0
  41. c = B ρ 0 c=\sqrt{\frac{B}{\rho_{0}}}
  42. 2 p - 1 c 2 2 p t 2 = 0 \nabla^{2}p-{1\over c^{2}}{\partial^{2}p\over\partial t^{2}}=0
  43. 2 \nabla^{2}
  44. p p
  45. c c
  46. 2 𝐮 - 1 c 2 2 𝐮 t 2 = 0 \nabla^{2}\mathbf{u}\;-{1\over c^{2}}{\partial^{2}\mathbf{u}\;\over\partial t^% {2}}=0
  47. 2 Φ - 1 c 2 2 Φ t 2 = 0 \nabla^{2}\Phi-{1\over c^{2}}{\partial^{2}\Phi\over\partial t^{2}}=0
  48. 𝐮 = Φ \mathbf{u}=\nabla\Phi\;
  49. p = - ρ t Φ p=-\rho{\partial\over\partial t}\Phi
  50. e i ω t e^{i\omega t}
  51. ω = 2 π f \omega=2\pi f
  52. p ( r , t , k ) = Real [ p ( r , k ) e i ω t ] p(r,t,k)=\operatorname{Real}\left[p(r,k)e^{i\omega t}\right]
  53. k = ω / c k=\omega/c
  54. p ( r , k ) = A e ± i k r p(r,k)=Ae^{\pm ikr}
  55. p ( r , k ) = A H 0 ( 1 ) ( k r ) + B H 0 ( 2 ) ( k r ) p(r,k)=AH_{0}^{(1)}(kr)+\ BH_{0}^{(2)}(kr)
  56. k r kr\rightarrow\infty
  57. H 0 ( 1 ) ( k r ) 2 π k r e i ( k r - π / 4 ) H_{0}^{(1)}(kr)\simeq\sqrt{\frac{2}{\pi kr}}e^{i(kr-\pi/4)}
  58. H 0 ( 2 ) ( k r ) 2 π k r e - i ( k r - π / 4 ) H_{0}^{(2)}(kr)\simeq\sqrt{\frac{2}{\pi kr}}e^{-i(kr-\pi/4)}
  59. p ( r , k ) = A r e ± i k r p(r,k)=\frac{A}{r}e^{\pm ikr}

Added_mass.html

  1. 𝐅 = ρ c V p 2 ( D 𝐮 D t - d 𝐯 d t ) , \mathbf{F}=\frac{\rho_{\mathrm{c}}V_{\mathrm{p}}}{2}\left(\frac{\mathrm{D}% \mathbf{u}}{\mathrm{D}t}-\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}\right),
  2. 𝐮 \mathbf{u}
  3. 𝐯 \mathbf{v}
  4. ρ c \rho_{\mathrm{c}}
  5. V p V_{\mathrm{p}}
  6. m p d 𝐯 p d t = 𝐅 + ρ c V p 2 ( D 𝐮 D t - d 𝐯 d t ) , m_{\mathrm{p}}\frac{\mathrm{d}\mathbf{v}_{\mathrm{p}}}{\mathrm{d}t}=\sum% \mathbf{F}+\frac{\rho_{\mathrm{c}}V_{\mathrm{p}}}{2}\left(\frac{\mathrm{D}% \mathbf{u}}{\mathrm{D}t}-\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}\right),
  7. 𝐅 \sum\mathbf{F}
  8. ( m p + ρ c V p 2 ) d 𝐯 p d t = 𝐅 + ρ c V p 2 D 𝐮 D t , \left(m_{\mathrm{p}}+\frac{\rho_{\mathrm{c}}V_{\mathrm{p}}}{2}\right)\frac{% \mathrm{d}\mathbf{v}_{\mathrm{p}}}{\mathrm{d}t}=\sum\mathbf{F}+\frac{\rho_{% \mathrm{c}}V_{\mathrm{p}}}{2}\frac{\mathrm{D}\mathbf{u}}{\mathrm{D}t},
  9. F = m a F=m\,a
  10. F = ( m + m added ) a . F=(m+m\text{added})\,a.
  11. r r
  12. 2 3 π r 3 ρ fluid . \tfrac{2}{3}\pi r^{3}\rho\text{fluid}.
  13. 4 3 π r 3 ρ air \tfrac{4}{3}\pi r^{3}\rho\text{air}
  14. 2 3 π r 3 ρ water . \tfrac{2}{3}\pi r^{3}\rho\text{water}.

Adhesion_railway.html

  1. λ = 2 π r d 2 k \lambda={2\pi}\sqrt{\frac{rd}{2k}}
  2. V 2 = W r a d 2 k ( 4 C + m d 2 ) V^{2}=\frac{Wrad^{2}}{k\left(4C+md^{2}\right)}
  3. M M
  4. V V
  5. ω R \omega R
  6. F w F_{w}
  7. creep = ( actual displacement - rolling displacement ) ( rolling displacement ) \mbox{creep}~{}=\frac{(\mbox{actual displacement}~{}-\mbox{rolling % displacement}~{})}{(\mbox{rolling displacement}~{})}\,

Adjunction_formula.html

  1. \mathcal{I}
  2. 0 / 2 i * Ω X Ω Y 0 , 0\to\mathcal{I}/\mathcal{I}^{2}\to i^{*}\Omega_{X}\to\Omega_{Y}\to 0,
  3. ω Y = i * ω X det ( / 2 ) , \omega_{Y}=i^{*}\omega_{X}\otimes\operatorname{det}(\mathcal{I}/\mathcal{I}^{2% })^{\vee},
  4. \vee
  5. 𝒪 ( D ) \mathcal{O}(D)
  6. 𝒪 ( - D ) \mathcal{O}(-D)
  7. / 2 \mathcal{I}/\mathcal{I}^{2}
  8. i * 𝒪 ( - D ) i^{*}\mathcal{O}(-D)
  9. ω D = i * ( ω X 𝒪 ( D ) ) \omega_{D}=i^{*}(\omega_{X}\otimes\mathcal{O}(D))
  10. K D = ( K X + D ) | D K_{D}=(K_{X}+D)|_{D}
  11. ω X 𝒪 ( D ) ω D \omega_{X}\otimes\mathcal{O}(D)\to\omega_{D}
  12. 𝒪 ( D ) \mathcal{O}(D)
  13. η s f s η f | f = 0 , \eta\otimes\frac{s}{f}\mapsto s\frac{\partial\eta}{\partial f}\bigg|_{f=0},
  14. g ( z ) d z 1 d z n f ( z ) ( - 1 ) i - 1 g ( z ) d z 1 d z i ^ d z n f / z i | f = 0 . \frac{g(z)\,dz_{1}\wedge\cdots\wedge dz_{n}}{f(z)}\mapsto(-1)^{i-1}\frac{g(z)% \,dz_{1}\wedge\cdots\wedge\widehat{dz_{i}}\wedge\cdots\wedge dz_{n}}{\partial f% /\partial z_{i}}\bigg|_{f=0}.
  15. ω D i * 𝒪 ( - D ) = i * ω X . \omega_{D}\otimes i^{*}\mathcal{O}(-D)=i^{*}\omega_{X}.
  16. i * 𝒪 ( - D ) i^{*}\mathcal{O}(-D)
  17. i * 𝒪 ( - D ) i^{*}\mathcal{O}(-D)
  18. g = ( d - 1 ) ( d - 2 ) / 2. g=(d-1)(d-2)/2.
  19. ( ( d 1 , d 2 ) , ( e 1 , e 2 ) ) d 1 e 2 + d 2 e 1 ((d_{1},d_{2}),(e_{1},e_{2}))\mapsto d_{1}e_{2}+d_{2}e_{1}
  20. 2 g - 2 = d 1 ( d 2 - 2 ) + d 2 ( d 1 - 2 ) 2g-2=d_{1}(d_{2}-2)+d_{2}(d_{1}-2)
  21. g = d 1 d 2 - d 1 - d 2 + 1 g=d_{1}d_{2}-d_{1}-d_{2}+1
  22. g = d e ( d + e - 4 ) / 2 + 1. g=de(d+e-4)/2+1.

Adjustable-speed_drive.html

  1. n = 120 × f p n={120\times{f}\over{p}}

Adomian_decomposition_method.html

  1. y ( t ) + y 2 ( t ) = - 1 , y^{\prime}(t)+y^{2}(t)=-1,
  2. y ( 0 ) = 0. y(0)=0.
  3. L y = - 1 - y 2 , Ly=-1-y^{2},
  4. L - 1 = 0 t ( ) L^{-1}=\int_{0}^{t}()
  5. y = y 0 + y 1 + y 2 + y 3 + . y=y_{0}+y_{1}+y_{2}+y_{3}+\cdots.
  6. ( y 0 + y 1 + y 2 + y 3 + ) = y ( 0 ) + L - 1 [ - 1 - ( y 0 + y 1 + y 2 + y 3 + ) 2 ] . (y_{0}+y_{1}+y_{2}+y_{3}+\cdots)=y(0)+L^{-1}[-1-(y_{0}+y_{1}+y_{2}+y_{3}+% \cdots)^{2}].
  7. y 0 \displaystyle y_{0}
  8. y \displaystyle y
  9. d 3 u d x 3 + 1 2 u d 2 u d x 2 = 0 \frac{\mathrm{d}^{3}u}{\mathrm{d}x^{3}}+\frac{1}{2}u\frac{\mathrm{d}^{2}u}{% \mathrm{d}x^{2}}=0
  10. u ( 0 ) \displaystyle u(0)
  11. L = d 3 d x 3 L=\frac{\mathrm{d}^{3}}{\mathrm{d}x^{3}}
  12. N u = 1 2 u d 2 d x 2 Nu=\frac{1}{2}u\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}
  13. L u + N u = 0 Lu+Nu=0
  14. u = α + β x + γ x 2 / 2 - L - 1 N u u=\alpha+\beta x+\gamma x^{2}/2-L^{-1}Nu
  15. L - 1 ξ ( x ) = d x d x d x ξ ( x ) L^{-1}\xi(x)=\int dx\int\mathrm{d}x\int\mathrm{d}x\;\;\xi(x)
  16. u \displaystyle u
  17. u 0 \displaystyle u^{0}
  18. A n = 1 n ! d n d λ n f ( u ( λ ) ) | λ = 0 , A_{n}=\frac{1}{n!}\frac{\mathrm{d}^{n}}{\mathrm{d}\lambda^{n}}f(u(\lambda))% \mid_{\lambda=0},
  19. d n d λ n u ( λ ) | λ = 0 = n ! u n \frac{\mathrm{d}^{n}}{\mathrm{d}\lambda^{n}}u(\lambda)\mid_{\lambda=0}=n!u_{n}
  20. u = γ 2 x 2 - γ 2 2 ( x 5 5 ! ) + 11 γ 3 4 ( x 8 8 ! ) - 375 γ 4 8 ( x 11 11 ! ) + u=\frac{\gamma}{2}x^{2}-\frac{\gamma^{2}}{2}\left(\frac{x^{5}}{5!}\right)+% \frac{11\gamma^{3}}{4}\left(\frac{x^{8}}{8!}\right)-\frac{375\gamma^{4}}{8}% \left(\frac{x^{11}}{11!}\right)+\cdots
  21. f ( z ) = n = 0 L + M c n z n = a 0 + a 1 z + + a L z L b 0 + b 1 z + + b M z M f(z)=\sum_{n=0}^{L+M}c_{n}z^{n}=\frac{a_{0}+a_{1}z+\cdots+a_{L}z^{L}}{b_{0}+b_% {1}z+\cdots+b_{M}z^{M}}
  22. \infty
  23. [ c L - M + 1 c L - M + 2 c L c L - M + 2 c L - M + 3 c L + 1 c L c L + 1 c L + M - 1 ] [ b M b M - 1 b 1 ] = - [ c L + 1 c L + 2 c L + M ] \left[\begin{array}[]{cccc}c_{L-M+1}&c_{L-M+2}&\cdots&c_{L}\\ c_{L-M+2}&c_{L-M+3}&\cdots&c_{L+1}\\ \vdots&\vdots&&\vdots\\ c_{L}&c_{L+1}&\cdots&c_{L+M-1}\end{array}\right]\left[\begin{array}[]{c}b_{M}% \\ b_{M-1}\\ \vdots\\ b_{1}\end{array}\right]=-\left[\begin{array}[]{c}c_{L+1}\\ c_{L+2}\\ \vdots\\ c_{L+M}\end{array}\right]
  24. a 0 \displaystyle a_{0}
  25. u ( x ) = γ x - γ 2 2 ( x 4 4 ! ) + 11 γ 3 4 ( x 7 7 ! ) - 375 γ 4 8 ( x 10 10 ! ) u^{\prime}(x)=\gamma x-\frac{\gamma^{2}}{2}\left(\frac{x^{4}}{4!}\right)+\frac% {11\gamma^{3}}{4}\left(\frac{x^{7}}{7!}\right)-\frac{375\gamma^{4}}{8}\left(% \frac{x^{10}}{10!}\right)
  26. u ( x ) = 0.0204 + 0.0379 z - 0.0059 z 2 - 0.00004575 z 3 + 6.357 10 - 6 z 4 - 1.291 10 - 6 z 5 1 - 0.1429 z - 0.0000232 z 2 + 0.0008375 z 3 - 0.0001558 z 4 - 1.2849 10 - 6 z 5 , u^{\prime}(x)=\frac{0.0204+0.0379\,z-0.0059\,z^{2}-0.00004575\,z^{3}+6.357% \cdot 10^{-6}z^{4}-1.291\cdot 10^{-6}z^{5}}{1-0.1429\,z-0.0000232\,z^{2}+0.000% 8375\,z^{3}-0.0001558\,z^{4}-1.2849\cdot 10^{-6}z^{5}},
  27. lim x u ( x ) = 1.004. \lim_{x\to\infty}u^{\prime}(x)=1.004.
  28. 2 u x 2 + 2 u y 2 - b u 2 x = ρ ( x , y ) ( 1 ) \frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}-b% \frac{\partial u^{2}}{\partial x}=\rho(x,y)\qquad(1)
  29. u ( x = 0 ) = f 1 ( y ) and u ( x = x l ) = f 2 ( y ) (1-a) u(x=0)=f_{1}(y)\quad\,\text{and}\quad u(x=x_{l})=f_{2}(y)\qquad\,\text{(1-a)}
  30. u ( y = - y l ) = g 1 ( x ) and u ( y = y l ) = g 2 ( x ) (1-b) u(y=-y_{l})=g_{1}(x)\quad\,\text{and}\quad u(y=y_{l})=g_{2}(x)\qquad\,\text{(1% -b)}
  31. L x u + L y u + N u = ρ ( x , y ) ( 2 ) L_{x}u+L_{y}u+Nu=\rho(x,y)\qquad(2)
  32. u = a ( y ) + b ( y ) x + L x - 1 ρ ( x , y ) - L x - 1 L y u - L x - 1 N u ( 3 ) u=a(y)+b(y)x+L_{x}^{-1}\rho(x,y)-L_{x}^{-1}L_{y}u-L_{x}^{-1}Nu\qquad(3)
  33. u = u 0 + u 1 + u 2 + u 3 + u=u_{0}+u_{1}+u_{2}+u_{3}+\cdots
  34. u 0 \displaystyle u_{0}
  35. φ n ( x = 0 ) \displaystyle\varphi^{n}(x=0)
  36. φ n i = 0 n u i \varphi^{n}\equiv\sum_{i=0}^{n}u_{i}
  37. N u \displaystyle Nu
  38. A n = ν = 1 n C ( ν , n ) f ( ν ) ( u 0 ) A_{n}=\sum_{\nu=1}^{n}C(\nu,n)f^{(\nu)}(u_{0})
  39. n = 0 A n \sum_{n=0}^{\infty}A_{n}
  40. A 0 \displaystyle A_{0}
  41. lim n φ n = u \lim_{n\to\infty}\varphi^{n}=u
  42. u = c ( x ) + d ( x ) y + L y - 1 ρ ( x , y ) - L y - 1 L x u - L y - 1 N u u=c(x)+d(x)y+L_{y}^{-1}\rho(x,y)-L_{y}^{-1}L_{x}u-L_{y}^{-1}Nu
  43. u ( y = - y l ) \displaystyle u(y=-y_{l})
  44. u ( x = 0 ) \displaystyle u(x=0)
  45. g 1 ( x ) = 0.0520833 x - 0.347222 x 3 + 9.25186 × 10 - 17 x 4 + 0.833333 x 5 - 0.555556 x 6 g_{1}(x)=0.0520833\,x-0.347222\,x^{3}+9.25186\times 10^{-17}x^{4}+0.833333\,x^% {5}-0.555556\,x^{6}
  46. f 1 , 2 ( y ) = 0.00413682 - 0.0813801 y 2 + 0.260416 y 4 - 0.277778 y 6 f_{1,2}(y)=0.00413682-0.0813801\,y^{2}+0.260416\,y^{4}-0.277778\,y^{6}
  47. f 1 , 2 ( y ) = 0.00413683 \displaystyle f_{1,2}(y)=0.00413683

AdS_black_hole.html

  1. d s 2 = - ( k 2 r 2 + 1 - C r ) d t 2 + 1 k 2 r 2 + 1 - C r d r 2 + r 2 d Ω 2 ds^{2}=-\left(k^{2}r^{2}+1-\frac{C}{r}\right)dt^{2}+\frac{1}{k^{2}r^{2}+1-% \frac{C}{r}}dr^{2}+r^{2}d\Omega^{2}
  2. d s 2 = - ( k 2 r 2 + 1 - C r d - 2 ) d t 2 + 1 k 2 r 2 + 1 - C r d - 2 d r 2 + r 2 d Ω 2 ds^{2}=-\left(k^{2}r^{2}+1-\frac{C}{r^{d-2}}\right)dt^{2}+\frac{1}{k^{2}r^{2}+% 1-\frac{C}{r^{d-2}}}dr^{2}+r^{2}d\Omega^{2}

Aerial_(album).html

  1. π \pi

Aeroacoustics.html

  1. ρ t + ( ρ 𝐯 ) = D ρ D t + ρ 𝐯 = 0 , \frac{\partial\rho}{\partial t}+\nabla\cdot\left(\rho\mathbf{v}\right)=\frac{D% \rho}{Dt}+\rho\nabla\cdot\mathbf{v}=0,
  2. ρ \rho
  3. 𝐯 \mathbf{v}
  4. D / D t D/Dt
  5. ρ 𝐯 t + ρ ( 𝐯 ) 𝐯 = - p + σ , {\rho}\frac{\partial\mathbf{v}}{\partial t}+{\rho(\mathbf{v}\cdot\nabla)% \mathbf{v}}=-\nabla p+\nabla\cdot\sigma,
  6. p p
  7. σ \sigma
  8. 𝐯 \mathbf{v}
  9. t ( ρ 𝐯 ) + ( ρ 𝐯 𝐯 ) = - p + σ . \frac{\partial}{\partial t}\left(\rho\mathbf{v}\right)+\nabla\cdot(\rho\mathbf% {v}\otimes\mathbf{v})=-\nabla p+\nabla\cdot\sigma.
  10. 𝐯 𝐯 \mathbf{v}\otimes\mathbf{v}
  11. 2 ρ t 2 - 2 p + σ = ( ρ 𝐯 𝐯 ) . \frac{\partial^{2}\rho}{\partial t^{2}}-\nabla^{2}p+\nabla\cdot\nabla\cdot% \sigma=\nabla\cdot\nabla\cdot(\rho\mathbf{v}\otimes\mathbf{v}).
  12. c 0 2 2 ρ c_{0}^{2}\nabla^{2}\rho
  13. c 0 c_{0}
  14. 2 ρ t 2 - c 0 2 2 ρ = [ ( ρ 𝐯 𝐯 ) - σ + p - c 0 2 ρ ] , \frac{\partial^{2}\rho}{\partial t^{2}}-c^{2}_{0}\nabla^{2}\rho=\nabla\cdot% \left[\nabla\cdot(\rho\mathbf{v}\otimes\mathbf{v})-\nabla\cdot\sigma+\nabla p-% c^{2}_{0}\nabla\rho\right],
  15. 2 ρ t 2 - c 0 2 2 ρ = ( ) : [ ρ 𝐯 𝐯 - σ + ( p - c 0 2 ρ ) 𝕀 ] , \frac{\partial^{2}\rho}{\partial t^{2}}-c^{2}_{0}\nabla^{2}\rho=(\nabla\otimes% \nabla):\left[\rho\mathbf{v}\otimes\mathbf{v}-\sigma+(p-c^{2}_{0}\rho)\mathbb{% I}\right],
  16. 𝕀 \mathbb{I}
  17. : :
  18. ρ 𝐯 𝐯 - σ + ( p - c 0 2 ρ ) 𝕀 \rho\mathbf{v}\otimes\mathbf{v}-\sigma+(p-c^{2}_{0}\rho)\mathbb{I}
  19. T T
  20. 2 ρ t 2 - c 0 2 2 ρ = 2 T i j x i x j , ( * ) \frac{\partial^{2}\rho}{\partial t^{2}}-c^{2}_{0}\nabla^{2}\rho=\frac{\partial% ^{2}T_{ij}}{\partial x_{i}\partial x_{j}},\quad(*)
  21. T i j = ρ v i v j - σ i j + ( p - c 0 2 ρ ) δ i j , T_{ij}=\rho v_{i}v_{j}-\sigma_{ij}+(p-c^{2}_{0}\rho)\delta_{ij},
  22. δ i j \delta_{ij}
  23. T i j T_{ij}
  24. ρ v i v j \rho v_{i}v_{j}
  25. σ i j \sigma_{ij}
  26. ( p - c 0 2 ρ ) δ i j (p-c^{2}_{0}\rho)\delta_{ij}
  27. σ = 0 \sigma=0
  28. p p
  29. ρ \rho
  30. p - p 0 = c 0 2 ( ρ - ρ 0 ) p-p_{0}=c_{0}^{2}(\rho-\rho_{0})
  31. ρ 0 \rho_{0}
  32. p 0 p_{0}
  33. ( * ) (*)\,
  34. 1 c 0 2 2 p t 2 - 2 p = 2 T ~ i j x i x j , where T ~ i j = ρ v i v j . \frac{1}{c_{0}^{2}}\frac{\partial^{2}p}{\partial t^{2}}-\nabla^{2}p=\frac{% \partial^{2}\tilde{T}_{ij}}{\partial x_{i}\partial x_{j}},\quad\,\text{where}% \quad\tilde{T}_{ij}=\rho v_{i}v_{j}.
  35. ρ = ρ 0 \rho=\rho_{0}
  36. ρ 0 \rho_{0}
  37. 1 c 0 2 2 p t 2 - 2 p = ρ 0 2 T ^ i j x i x j , where T ^ i j = v i v j . \frac{1}{c_{0}^{2}}\frac{\partial^{2}p}{\partial t^{2}}-\nabla^{2}p=\rho_{0}% \frac{\partial^{2}\hat{T}_{ij}}{\partial x_{i}\partial x_{j}},\quad\,\text{% where}\quad\hat{T}_{ij}=v_{i}v_{j}.
  38. ( * ) (*)\,
  39. T ρ 0 T ^ T\approx\rho_{0}\hat{T}
  40. p - p 0 = c 0 2 ( ρ - ρ 0 ) p-p_{0}=c_{0}^{2}(\rho-\rho_{0})
  41. ρ ρ 0 \rho\ll\rho_{0}
  42. p p 0 p\ll p_{0}
  43. p p
  44. ρ \rho

Affine_arithmetic.html

  1. x = x 0 + x 1 ϵ 1 + x 2 ϵ 2 + x=x_{0}+x_{1}\epsilon_{1}+x_{2}\epsilon_{2}+{}
  2. \cdots
  3. + x n ϵ n {}+x_{n}\epsilon_{n}
  4. x 0 , x 1 , x 2 , x_{0},x_{1},x_{2},
  5. , \dots,
  6. x n x_{n}
  7. ϵ 1 , ϵ 2 , ϵ n \epsilon_{1},\epsilon_{2},\epsilon_{n}
  8. x = 5 + 2 ϵ k x=5+2\epsilon_{k}
  9. x = 10 + 2 ϵ 3 - 5 ϵ 8 x=10+2\epsilon_{3}-5\epsilon_{8}
  10. ϵ j \epsilon_{j}
  11. x x
  12. y y
  13. x = 10 + 2 ϵ 3 - 6 ϵ 8 x=10+2\epsilon_{3}-6\epsilon_{8}
  14. y = 20 + 3 ϵ 4 + 4 ϵ 8 y=20+3\epsilon_{4}+4\epsilon_{8}
  15. x , y x,y
  16. z z
  17. z j z_{j}
  18. \leftarrow
  19. x j + y j x_{j}+y_{j}
  20. z z
  21. α \alpha
  22. α \alpha
  23. z j z_{j}
  24. \leftarrow
  25. α x j \alpha x_{j}
  26. α \alpha
  27. β \beta
  28. γ \gamma
  29. Z Z
  30. \leftarrow
  31. F ( X , Y , F(X,Y,
  32. \dots
  33. ) )
  34. Z Z
  35. \leftarrow
  36. X Y XY
  37. Z Z
  38. \leftarrow
  39. sin ( X ) \sin(X)
  40. ϵ i \epsilon_{i}
  41. x x
  42. y y
  43. z z
  44. \leftarrow
  45. G ( x , y , G(x,y,
  46. \dots
  47. ) + z k ϵ k )+z_{k}\epsilon_{k}
  48. z k z_{k}
  49. | F - G | |F-G|
  50. ϵ k \epsilon_{k}
  51. z z
  52. x , y , x,y,
  53. \dots
  54. , z ,z
  55. z j z_{j}
  56. z j z_{j}
  57. ϵ j \epsilon_{j}
  58. δ j \delta_{j}
  59. z j z_{j}
  60. δ j \delta_{j}
  61. z k z_{k}
  62. ϵ k \epsilon_{k}
  63. α \alpha
  64. z k ϵ k z_{k}\epsilon_{k}
  65. X 1 , X 2 , X_{1},X_{2},
  66. , \dots,
  67. X m X_{m}
  68. A i , j A_{i,j}
  69. ϵ j \epsilon_{j}
  70. X i X_{i}
  71. b i b_{i}
  72. ( X 1 , X 2 , (X_{1},X_{2},
  73. , \dots,
  74. X m ) X_{m})
  75. U n = [ - 1 , + 1 ] n U^{n}=[-1,+1]^{n}
  76. U n U^{n}
  77. R m R^{m}
  78. ϵ \epsilon
  79. \to
  80. A ϵ + b A\epsilon+b
  81. X 1 , X 2 , X_{1},X_{2},
  82. , \dots,
  83. X m X_{m}
  84. ϵ k \epsilon_{k}
  85. ϵ k \epsilon_{k}
  86. ϵ j \epsilon_{j}
  87. x j x_{j}
  88. x j x_{j}

Affine_involution.html

  1. A 2 = I ( 1 ) A^{2}=I\quad\quad\quad\quad(1)
  2. D = ( ± 1 0 0 0 0 ± 1 0 0 0 0 ± 1 0 0 0 0 ± 1 ) D=\begin{pmatrix}\pm 1&0&\cdots&0&0\\ 0&\pm 1&\cdots&0&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\cdots&\pm 1&0\\ 0&0&\cdots&0&\pm 1\end{pmatrix}

Affine_vector_field.html

  1. ( X g a b ) ; c = 0 (\mathcal{L}_{X}g_{ab})_{;c}=0

Affinity_laws.html

  1. Q 1 Q 2 = ( N 1 N 2 ) {Q_{1}\over\ Q_{2}}={\left({N_{1}\over N_{2}}\right)}
  2. H 1 H 2 = ( N 1 N 2 ) 2 {H_{1}\over H_{2}}={\left({N_{1}\over N_{2}}\right)^{2}}
  3. P 1 P 2 = ( N 1 N 2 ) 3 {P_{1}\over P_{2}}={\left({N_{1}\over N_{2}}\right)^{3}}
  4. Q 1 Q 2 = ( D 1 D 2 ) 3 {Q_{1}\over\ Q_{2}}={\left({D_{1}\over D_{2}}\right)^{3}}
  5. H 1 H 2 = ( D 1 D 2 ) 2 {H_{1}\over H_{2}}={\left({D_{1}\over D_{2}}\right)^{2}}
  6. P 1 P 2 = ( D 1 D 2 ) 5 {P_{1}\over P_{2}}={\left({D_{1}\over D_{2}}\right)^{5}}
  7. Q Q
  8. D D
  9. N N
  10. H H
  11. P P
  12. η 1 = η 2 \eta_{1}=\eta_{2}

Aichelburg–Sexl_ultraboost.html

  1. d s 2 = - 8 m δ ( u ) log r d u 2 + 2 d u d v + d r 2 + r 2 d θ 2 , ds^{2}=-8m\,\delta(u)\,\log r\,du^{2}+2\,du\,dv+dr^{2}+r^{2}\,d\theta^{2},
  2. - < u < , 0 < r < , - < v < , - π < θ < π -\infty<u<\infty,\,0<r<\infty,\,-\infty<v<\infty,-\pi<\theta<\pi
  3. d s 2 = - 4 m a log ( r ) π ( 1 + a 2 u 2 ) d u 2 - 2 d u d v + d r 2 + r 2 d θ 2 , ds^{2}=-\frac{4ma\,\log(r)}{\pi\,(1+a^{2}u^{2})}\,du^{2}-2du\,dv+dr^{2}+r^{2}% \,d\theta^{2},
  4. - < u < , 0 < r < , - < v < , - π < θ < π -\infty<u<\infty,\,0<r<\infty,\,-\infty<v<\infty,-\pi<\theta<\pi
  5. u = 0 u=0
  6. a a\rightarrow\infty

Alexander's_trick.html

  1. D n D^{n}
  2. S n - 1 S^{n-1}
  3. f : D n D n f\colon D^{n}\to D^{n}
  4. f ( x ) = x for all x S n - 1 f(x)=x\mbox{ for all }~{}x\in S^{n-1}
  5. J ( x , t ) = { t f ( x / t ) , if 0 x < t , x , if t x 1. J(x,t)=\begin{cases}tf(x/t),&\mbox{if }~{}0\leq\|x\|<t,\\ x,&\mbox{if }~{}t\leq\|x\|\leq 1.\end{cases}
  6. f f
  7. t = 0 t=0
  8. f f
  9. f f
  10. ( x , t ) = ( 0 , 0 ) (x,t)=(0,0)
  11. f , g : D n D n f,g\colon D^{n}\to D^{n}
  12. S n - 1 S^{n-1}
  13. g - 1 f g^{-1}f
  14. S n - 1 S^{n-1}
  15. J J
  16. g - 1 f g^{-1}f
  17. g J gJ
  18. g g
  19. f f
  20. S n - 1 S^{n-1}
  21. D n D^{n}
  22. f : S n - 1 S n - 1 f\colon S^{n-1}\to S^{n-1}
  23. F : D n D n with F ( r x ) = r f ( x ) for all r [ 0 , 1 ] and x S n - 1 F\colon D^{n}\to D^{n}\mbox{ with }~{}F(rx)=rf(x)\mbox{ for all }~{}r\in[0,1]% \mbox{ and }~{}x\in S^{n-1}

Algebra_(disambiguation).html

  1. 2 \mathbb{Z}_{2}

Algebra_of_physical_space.html

  1. x = x 0 + x 1 𝐞 1 + x 2 𝐞 2 + x 3 𝐞 3 , x=x^{0}+x^{1}\mathbf{e}_{1}+x^{2}\mathbf{e}_{2}+x^{3}\mathbf{e}_{3},
  2. x ( x 0 + x 3 x 1 - i x 2 x 1 + i x 2 x 0 - x 3 ) x\rightarrow\begin{pmatrix}x^{0}+x^{3}&&x^{1}-ix^{2}\\ x^{1}+ix^{2}&&x^{0}-x^{3}\end{pmatrix}
  3. L = e 1 2 W L=e^{\frac{1}{2}W}
  4. L L ¯ = L ¯ L = 1 L\bar{L}=\bar{L}L=1
  5. L = B R L=BR
  6. u = d x d τ = d x 0 d τ + d d τ ( x 1 𝐞 1 + x 2 𝐞 2 + x 3 𝐞 3 ) = d x 0 d τ [ 1 + d d x 0 ( x 1 𝐞 1 + x 2 𝐞 2 + x 3 𝐞 3 ) ] . u=\frac{dx}{d\tau}=\frac{dx^{0}}{d\tau}+\frac{d}{d\tau}(x^{1}\mathbf{e}_{1}+x^% {2}\mathbf{e}_{2}+x^{3}\mathbf{e}_{3})=\frac{dx^{0}}{d\tau}\left[1+\frac{d}{dx% ^{0}}(x^{1}\mathbf{e}_{1}+x^{2}\mathbf{e}_{2}+x^{3}\mathbf{e}_{3})\right].
  7. 𝐯 = d d x 0 ( x 1 𝐞 1 + x 2 𝐞 2 + x 3 𝐞 3 ) \mathbf{v}=\frac{d}{dx^{0}}(x^{1}\mathbf{e}_{1}+x^{2}\mathbf{e}_{2}+x^{3}% \mathbf{e}_{3})
  8. γ ( 𝐯 ) = 1 1 - | 𝐯 | 2 c 2 \gamma(\mathbf{v})=\frac{1}{\sqrt{1-\frac{|\mathbf{v}|^{2}}{c^{2}}}}
  9. u = γ ( 𝐯 ) ( 1 + 𝐯 ) u=\gamma(\mathbf{v})(1+\mathbf{v})
  10. u u ¯ = 1 u\bar{u}=1
  11. u u = L u L . u\rightarrow u^{\prime}=LuL^{\dagger}.
  12. p = m u , p=mu,
  13. p ¯ p = m 2 \bar{p}p=m^{2}
  14. F = 𝐄 + i 𝐁 F=\mathbf{E}+i\mathbf{B}
  15. F ( E 3 E 1 - i E 2 E 1 + i E 2 - E 3 ) + i ( B 3 B 1 - i B 2 B 1 + i B 2 - B 3 ) . F\rightarrow\begin{pmatrix}E_{3}&E_{1}-iE_{2}\\ E_{1}+iE_{2}&-E_{3}\end{pmatrix}+i\begin{pmatrix}B_{3}&B_{1}-iB_{2}\\ B_{1}+iB_{2}&-B_{3}\end{pmatrix}\,.
  16. j = ρ + 𝐣 , j=\rho+\mathbf{j}\,,
  17. A = ϕ + 𝐀 , A=\phi+\mathbf{A}\,,
  18. F = A ¯ V . F=\langle\partial\bar{A}\rangle_{V}\,.
  19. A A + χ , A\rightarrow A+\partial\chi\,,
  20. χ \chi
  21. F F = L F L ¯ . F\rightarrow F^{\prime}=LF\bar{L}\,.
  22. ¯ F = 1 ϵ j ¯ , \bar{\partial}F=\frac{1}{\epsilon}\bar{j}\,,
  23. d p d τ = e F u R . \frac{dp}{d\tau}=e\langle Fu\rangle_{R}\,.
  24. L = 1 2 F F S - A j ¯ S , L=\frac{1}{2}\langle FF\rangle_{S}-\langle A\bar{j}\rangle_{S}\,,
  25. i ¯ Ψ 𝐞 3 + e A ¯ Ψ = m Ψ ¯ i\bar{\partial}\Psi\mathbf{e}_{3}+e\bar{A}\Psi=m\bar{\Psi}^{\dagger}
  26. d Λ d τ = e 2 m c F Λ , \frac{d\Lambda}{d\tau}=\frac{e}{2mc}F\Lambda,
  27. u = Λ Λ , u=\Lambda\Lambda^{\dagger},
  28. x ( τ ) x(\tau)
  29. d x d τ = u \frac{dx}{d\tau}=u

Algebraic_space.html

  1. Hom ( Y , X ) Hom ( V , X ) Hom ( S , X ) \mathrm{Hom}(Y,X)\rightarrow\mathrm{Hom}(V,X){{{}\atop\longrightarrow}\atop{% \longrightarrow\atop{}}}\mathrm{Hom}(S,X)
  2. k { x 1 , , x n } k\{x_{1},\ldots,x_{n}\}

Algorithmic_information_theory.html

  1. { 0 , 1 } \{0,1\}

Almost_convergent_sequence.html

  1. ( x n ) (x_{n})
  2. L L
  3. L L
  4. ( x n ) (x_{n})
  5. ( x n ) (x_{n})
  6. lim p x n + + x n + p - 1 p = L \lim\limits_{p\to\infty}\frac{x_{n}+\ldots+x_{n+p-1}}{p}=L
  7. n n
  8. ( ε > 0 ) ( p 0 ) ( p > p 0 ) ( n ) | x n + + x n + p - 1 p - L | < ε . (\forall\varepsilon>0)(\exists p_{0})(\forall p>p_{0})(\forall n)\left|\frac{x% _{n}+\ldots+x_{n+p-1}}{p}-L\right|<\varepsilon.

Ambivalence.html

  1. A m b i v a l e n c e = S + L - | S - L | = S Ambivalence=S+L-|S-L|=S
  2. A m b i v a l e n c e = ( S + L ) / 2 - | S - L | Ambivalence=(S+L)/2-|S-L|
  3. A m b i v a l e n c e = ( P + N ) / 2 - | P - N | Ambivalence=(P+N)/2-|P-N|

Ammonium_hydrosulfide.html

  1. \overrightarrow{\leftarrow}

Ammonium_persulfate.html

  1. \overrightarrow{\leftarrow}

Ancient_Mesopotamian_units_of_measurement.html

  1. 80 / 81 80/81
  2. 24 / 25 24/25
  3. 15 / 16 15/16
  4. 5 / 6 5/6

Anderson_localization.html

  1. i ψ ˙ = H ψ , i\hbar\dot{\psi}=H\psi~{},
  2. ( H ϕ ) ( j ) = E j ϕ ( j ) + k j V ( | k - j | ) ϕ ( k ) , (H\phi)(j)=E_{j}\phi(j)+\sum_{k\neq j}V(|k-j|)\phi(k)~{},
  3. V ( | r | ) = { 1 , | r | = 1 0 , otherwise. V(|r|)=\begin{cases}1,&|r|=1\\ 0,&\,\text{otherwise.}\end{cases}
  4. | ψ | 2 |\psi|^{2}
  5. n d | ψ ( t , n ) | 2 | n | C \sum_{n\in\mathbb{Z}^{d}}|\psi(t,n)|^{2}|n|\leq C
  6. n d | ψ ( t , n ) | 2 | n | D t , \sum_{n\in\mathbb{Z}^{d}}|\psi(t,n)|^{2}|n|\approx D\sqrt{t}~{},

Angle_bisector_theorem.html

  1. | B D | | D C | = | A B | | A C | , {\frac{|BD|}{|DC|}}={\frac{|AB|}{|AC|}},
  2. | B D | | D C | = | A B | sin D A B | A C | sin D A C . {\frac{|BD|}{|DC|}}={\frac{|AB|\sin\angle DAB}{|AC|\sin\angle DAC}}.
  3. | A B | | B D | = sin B D A sin B A D {\frac{|AB|}{|BD|}}={\frac{\sin\angle BDA}{\sin\angle BAD}}
  4. | A C | | D C | = sin A D C sin D A C {\frac{|AC|}{|DC|}}={\frac{\sin\angle ADC}{\sin\angle DAC}}
  5. < m t p l > sin B D A = sin A D C . <mtpl>{{\sin\angle BDA}}={\sin\angle ADC}.
  6. | A B | | B D | = | A C | | D C | , {\frac{|AB|}{|BD|}}={\frac{|AC|}{|DC|}},
  7. | A B | | B D | sin B A D = sin B D A , {\frac{|AB|}{|BD|}\sin\angle\ BAD=\sin\angle BDA},
  8. | A C | | D C | sin D A C = sin A D C . {\frac{|AC|}{|DC|}\sin\angle\ DAC=\sin\angle ADC}.
  9. | A B | | B D | sin B A D = | A C | | D C | sin D A C , {\frac{|AB|}{|BD|}\sin\angle\ BAD=\frac{|AC|}{|DC|}\sin\angle\ DAC},
  10. | B D | | C D | = | B B 1 | | C C 1 | = | A B | sin B A D | A C | sin C A D . {\frac{|BD|}{|CD|}}={\frac{|BB_{1}|}{|CC_{1}|}}=\frac{|AB|\sin\angle BAD}{|AC|% \sin\angle CAD}.
  11. | B D | | A B | = sin B A D and | C D | | A C | = sin D A C , \frac{|BD|}{|AB|}=\sin\angle\ BAD\,\text{ and }\frac{|CD|}{|AC|}=\sin\angle\ DAC,

Angular_diameter_distance.html

  1. x x
  2. θ \theta
  3. d A = x θ d_{A}=\frac{x}{\theta}
  4. z z
  5. χ \chi
  6. d A = r ( χ ) 1 + z d_{A}=\frac{r(\chi)}{1+z}
  7. r ( χ ) r(\chi)
  8. r ( χ ) = { sin ( - Ω k H 0 χ ) / ( H 0 | Ω k | ) Ω k < 0 χ Ω k = 0 sinh ( Ω k H 0 χ ) / ( H 0 | Ω k | ) Ω k > 0 r(\chi)=\begin{cases}\sin\left(\sqrt{-\Omega_{k}}H_{0}\chi\right)/\left(H_{0}% \sqrt{|\Omega_{k}|}\right)&\Omega_{k}<0\\ \chi&\Omega_{k}=0\\ \sinh\left(\sqrt{\Omega_{k}}H_{0}\chi\right)/\left(H_{0}\sqrt{|\Omega_{k}|}% \right)&\Omega_{k}>0\end{cases}
  9. Ω k \Omega_{k}
  10. H 0 H_{0}
  11. d d
  12. tan ( θ ) = x d \tan\left(\theta\right)=\frac{x}{d}
  13. θ \theta
  14. x x
  15. d d
  16. θ \theta
  17. θ x d \theta\approx\frac{x}{d}
  18. θ \theta
  19. x x
  20. d A d_{A}
  21. q 0 q_{0}
  22. q 0 < 0.5 q_{0}<0.5
  23. q 0 > 0.5 q_{0}>0.5
  24. q 0 = 0.5 q_{0}=0.5
  25. d A = c H 0 q 0 2 ( z q 0 + ( q 0 - 1 ) ( 2 q 0 z + 1 - 1 ) ) ( 1 + z ) 2 d_{A}=\cfrac{c}{H_{0}q^{2}_{0}}\cfrac{(zq_{0}+(q_{0}-1)(\sqrt{2q_{0}z+1}-1))}{% (1+z)^{2}}

Anharmonicity.html

  1. 2 ω 2\omega
  2. 3 ω 3\omega
  3. ω \omega
  4. ω \omega
  5. ω 0 \omega_{0}
  6. Δ ω = ω - ω 0 \Delta\omega=\omega-\omega_{0}
  7. A A
  8. Δ ω A 2 \Delta\omega\propto A^{2}
  9. ω α \omega_{\alpha}
  10. ω β \omega_{\beta}
  11. ω α ± ω β \omega_{\alpha}\pm\omega_{\beta}
  12. U ( x ) U(x)
  13. U = U ( x ) U=U(x)
  14. U U
  15. T ( E ) T(E)
  16. E E
  17. x ( U ) = 1 2 π 2 m 0 U T ( E ) d E U - E x(U)=\frac{1}{2\pi\sqrt{2m}}\int_{0}^{U}\frac{T(E)\,dE}{\sqrt{U-E}}

Anionic_addition_polymerization.html

  1. I + - M k i n i t M - M + - M k p r o p M - \textstyle\ \begin{aligned}&\displaystyle\mbox{I}~{}^{-}+\mbox{M}~{}\overset{k% _{init}}{\longrightarrow}\mbox{M}~{}^{-}\\ &\displaystyle\mbox{M}~{}^{-}+\mbox{M}~{}\overset{k_{prop}}{\longrightarrow}% \mbox{M}~{}^{-}\end{aligned}
  2. rate(prop) = k p [ M ] - [ M ] \textstyle\ \mbox{rate(prop)}~{}=k_{p}[\mbox{M}~{}^{-}][\mbox{M}~{}]
  3. rate(prop) = k p [ I ] [ M ] \textstyle\ \mbox{rate(prop)}~{}=k_{p}[\mbox{I}~{}][\mbox{M}~{}]
  4. ν = [ M ] o [ I ] o ρ \nu=\frac{[\mbox{M}~{}]_{o}}{[\mbox{I}~{}]_{o}}\rho
  5. ν = [ M ] o [ I ] o \nu=\frac{[\mbox{M}~{}]_{o}}{[\mbox{I}~{}]_{o}}
  6. M + - H X k t e r m M-H + X - \textstyle\ \mbox{M}~{}^{-}+HX\overset{k_{term}}{\longrightarrow}\mbox{M-H}~{}% +\mbox{X}~{}^{-}
  7. rate(prop) = k i n i t k p r o p [ I ] [ M ] 2 k t e r m [ H-X ] \textstyle\ \mbox{rate(prop)}~{}=\frac{k_{init}k_{prop}[\mbox{I}~{}][\mbox{M}~% {}]^{2}}{k_{term}[\mbox{H-X}~{}]}
  8. ν = r a t e ( p r o p ) r a t e ( t e r m ) = k p r o p [ M ] k t e r m [ H-X ] \textstyle\ \nu=\frac{rate(prop)}{rate(term)}=\frac{k_{prop}[\mbox{M}~{}]}{k_{% term}[\mbox{H-X}~{}]}

Ankle_brachial_pressure_index.html

  1. A B P I L e g = P L e g P A r m ABPI_{Leg}=\frac{P_{Leg}}{P_{Arm}}

Annular_tropical_cyclone.html

  1. θ e \theta_{e}

Annus_Mirabilis_papers.html

  1. h f = Φ + E k hf=\Phi+E_{k}

Anomalous_magnetic_dipole_moment.html

  1. a = g - 2 2 a=\frac{g-2}{2}
  2. a = α 2 π 0.0011614 a=\frac{\alpha}{2\pi}\approx 0.0011614
  3. a = 0.00115965218073 ( 28 ) a=0.00115965218073(28)
  4. α μ SM = α μ QED + α μ EW + α μ Hadron . \alpha_{\mu}^{\mathrm{SM}}=\alpha_{\mu}^{\mathrm{QED}}+\alpha_{\mu}^{\mathrm{% EW}}+\alpha_{\mu}^{\mathrm{Hadron}}.
  5. a = g - 2 2 = 0.00116592091 ( 54 ) ( 33 ) , a=\frac{g-2}{2}=0.00116592091(54)(33),

Anosov_diffeomorphism.html

  1. Ω ( f ) = M \Omega(f)=M
  2. C 1 C^{1}
  3. C 2 C^{2}
  4. C 1 + α C^{1+\alpha}
  5. C 2 C^{2}
  6. f : M M f:M\to M
  7. μ f \mu_{f}
  8. M M
  9. B ( μ f ) B(\mu_{f})
  10. B ( μ f ) = { x M : 1 n k = 0 n - 1 δ f k x μ f } . B(\mu_{f})=\{x\in M:\frac{1}{n}\sum_{k=0}^{n-1}\delta_{f^{k}x}\to\mu_{f}\}.
  11. J = ( 1 / 2 0 0 - 1 / 2 ) X = ( 0 1 0 0 ) Y = ( 0 0 1 0 ) J=\left(\begin{matrix}1/2&0\\ 0&-1/2\\ \end{matrix}\right)\quad\quad X=\left(\begin{matrix}0&1\\ 0&0\\ \end{matrix}\right)\quad\quad Y=\left(\begin{matrix}0&0\\ 1&0\\ \end{matrix}\right)
  12. [ J , X ] = X [ J , Y ] = - Y [ X , Y ] = 2 J [J,X]=X\quad\quad[J,Y]=-Y\quad\quad[X,Y]=2J
  13. g t = exp ( t J ) = ( e t / 2 0 0 e - t / 2 ) h t * = exp ( t X ) = ( 1 t 0 1 ) h t = exp ( t Y ) = ( 1 0 t 1 ) g_{t}=\exp(tJ)=\left(\begin{matrix}e^{t/2}&0\\ 0&e^{-t/2}\\ \end{matrix}\right)\quad\quad h^{*}_{t}=\exp(tX)=\left(\begin{matrix}1&t\\ 0&1\\ \end{matrix}\right)\quad\quad h_{t}=\exp(tY)=\left(\begin{matrix}1&0\\ t&1\\ \end{matrix}\right)
  14. g t g_{t}
  15. g t g_{t}
  16. T Q = E + E 0 E - TQ=E^{+}\oplus E^{0}\oplus E^{-}
  17. g e = q Q g\cdot e=q\in Q
  18. T q Q = E q + E q 0 E q - T_{q}Q=E_{q}^{+}\oplus E_{q}^{0}\oplus E_{q}^{-}
  19. E e + = Y E_{e}^{+}=Y
  20. E e 0 = J E_{e}^{0}=J
  21. E e - = X E_{e}^{-}=X
  22. g = g t g=g_{t}
  23. T q Q T_{q}Q
  24. T e P = s l ( 2 , ) T_{e}P=sl(2,\mathbb{R})
  25. v E q + v\in E^{+}_{q}
  26. g t g_{t}
  27. v E q - v\in E^{-}_{q}
  28. g t g_{t}
  29. E q 0 E^{0}_{q}
  30. g s g t = g t g s = g s + t g_{s}g_{t}=g_{t}g_{s}=g_{s+t}\,
  31. g s h t * = h t exp ( - s ) * g s g_{s}h^{*}_{t}=h^{*}_{t\exp(-s)}g_{s}
  32. g s h t = h t exp ( s ) g s g_{s}h_{t}=h_{t\exp(s)}g_{s}\,
  33. E q + E^{+}_{q}
  34. h t h_{t}
  35. g t g_{t}
  36. g t i = ( exp ( t / 2 ) 0 0 exp ( - t / 2 ) ) i = i exp ( t ) g_{t}\cdot i=\left(\begin{matrix}\exp(t/2)&0\\ 0&\exp(-t/2)\end{matrix}\right)\cdot i=i\exp(t)
  37. ( a b c d ) i exp ( t ) = a i exp ( t ) + b c i exp ( t ) + d \left(\begin{matrix}a&b\\ c&d\end{matrix}\right)\cdot i\exp(t)=\frac{ai\exp(t)+b}{ci\exp(t)+d}
  38. h t * h^{*}_{t}
  39. h t h_{t}

Answer_set_programming.html

  1. p , q , r p,q,r
  2. { p , q , r } \{p,q,r\}
  3. ( p ¬ p ) ( q ¬ q ) ( r ¬ r ) . (p\lor\neg p)\land(q\lor\neg q)\land(r\lor\neg r).
  4. p , q , r p,q,r
  5. ( p ¬ p ) ( q ¬ q ) ( r ¬ r ) (p\lor\neg p)\land(q\lor\neg q)\land(r\lor\neg r)
  6. ( p q r ) ¬ ( p q r ) . \land\,(p\lor q\lor r)\land\neg(p\land q\land r).
  7. p , q , r p,q,r
  8. ¬ ( ( p q ) ( p r ) ( q r ) ) . \neg((p\land q)\lor(p\land r)\lor(q\land r)).
  9. n n
  10. G G
  11. c o l o r color
  12. { 1 , , n } \{1,\dots,n\}
  13. c o l o r ( x ) c o l o r ( y ) color(x)\neq color(y)
  14. x , y x,y
  15. n n
  16. 1 , , n 1,\dots,n
  17. i i
  18. x x
  19. x x
  20. y y
  21. G G
  22. n n
  23. c o l o r ( , ) color(\dots,\dots)
  24. n n
  25. G G
  26. n \geq n
  27. n \geq n
  28. r ( x ) r(x)
  29. x x

Antenna_factor.html

  1. A F = E V AF=\frac{E}{V}
  2. A F dBm - 1 = E dBV / m - V dBV = E dB μ V / m - V dB μ V AF_{\mathrm{dBm}^{-1}}=E_{\mathrm{\mathrm{dBV/m}}}-V_{\mathrm{dBV}}=E_{\mathrm% {\mathrm{dB}\mu\mathrm{V/m}}}-V_{\mathrm{dB}\mu\mathrm{V}}
  3. A F = 377 P D 50 P D A e = 2.75 A e = 9.73 λ G AF=\frac{\sqrt{377P_{D}}}{\sqrt{50P_{D}A_{e}}}=\frac{2.75}{\sqrt{A_{e}}}=\frac% {9.73}{\lambda\sqrt{G}}

Anthropometry_of_the_upper_arm.html

  1. M U A A = M U A C 2 4 π MUAA=\frac{MUAC^{2}}{4\pi}
  2. M U A M C = M U A C - ( π × T S F 10 ) MUAMC=MUAC-\left(\pi\times\frac{TSF}{10}\right)
  3. M U A M A = M U A M C 2 4 π MUAMA=\frac{MUAMC^{2}}{4\pi}
  4. C M U A M A = ( M U A C - ( π × T S F 10 ) ) 2 - 10 4 π CMUAMA=\frac{\left(MUAC-\left(\pi\times\frac{TSF}{10}\right)\right)^{2}-10}{4\pi}
  5. C M U A M A = ( M U A C - ( π × T S F 10 ) ) 2 - 6.5 4 π CMUAMA=\frac{\left(MUAC-\left(\pi\times\frac{TSF}{10}\right)\right)^{2}-6.5}{4\pi}
  6. M U A F A = M U A A - M U A M A MUAFA=MUAA-MUAMA
  7. A F I = 100 × M U A F A M U A A AFI=100\times\frac{MUAFA}{MUAA}

Anti-diagonal_matrix.html

  1. [ 0 0 0 0 1 0 0 0 2 0 0 0 5 0 0 0 7 0 0 0 - 1 0 0 0 0 ] . \begin{bmatrix}0&0&0&0&1\\ 0&0&0&2&0\\ 0&0&5&0&0\\ 0&7&0&0&0\\ -1&0&0&0&0\end{bmatrix}.

Antiisomorphism.html

  1. \rightarrow
  2. 1 2 ; 1\rightarrow 2;
  3. 1 3 ; 1\rightarrow 3;
  4. 2 1. 2\rightarrow 1.
  5. \Rightarrow
  6. b a ; b\Rightarrow a;
  7. c a ; c\Rightarrow a;
  8. a b . a\Rightarrow b.
  9. \Leftarrow
  10. b a ; b\Leftarrow a;
  11. c a ; c\Leftarrow a;
  12. a b . a\Leftarrow b.
  13. ϕ \phi
  14. ϕ ( n ) = { a if n = 1 ; b if n = 2 ; c if n = 3. \phi(n)=\begin{cases}a&\mbox{if }~{}n=1;\\ b&\mbox{if }~{}n=2;\\ c&\mbox{if }~{}n=3.\end{cases}
  15. ϕ \phi
  16. f ( x + R y ) = f ( x ) + S f ( y ) and f ( x R y ) = f ( y ) S f ( x ) for all x , y R f(x+_{R}y)=f(x)+_{S}f(y)\,\text{ and }f(x\cdot_{R}y)=f(y)\cdot_{S}f(x)\,\text{% for all }x,y\in R
  17. x 0 + x 1 𝐢 + x 2 𝐣 + x 3 𝐤 x 0 - x 1 𝐢 - x 2 𝐣 - x 3 𝐤 . x_{0}+x_{1}\mathbf{i}+x_{2}\mathbf{j}+x_{3}\mathbf{k}\quad\mapsto\quad x_{0}-x% _{1}\mathbf{i}-x_{2}\mathbf{j}-x_{3}\mathbf{k}.

Antisymmetry.html

  1. < H , C > <H,C>
  2. < H , C > <H,C>
  3. < S , H Align g t ; <S,H&gt;
  4. < S , H > , < H , C > <S,H>,<H,C>
  5. < S , H > <S,H>
  6. < H , C > <H,C>
  7. < S , H > , < H , C > <S,H>,<H,C>
  8. < S , H , C > <S,H,C>
  9. < S , H > , < H , C > <S,H>,<H,C>
  10. < S , H , C Align g t ; <S,H,C&gt;

Apéry's_constant.html

  1. 1 + 1 4 + 1 1 + 1 18 + 1 1+\frac{1}{4+\cfrac{1}{1+\cfrac{1}{18+\cfrac{1}{\ddots\qquad{}}}}}
  2. ζ ( 3 ) = k = 1 1 k 3 = 1 + 1 2 3 + 1 3 3 + 1 4 3 + 1 5 3 + 1 6 3 + 1 7 3 + 1 8 3 + 1 9 3 + \zeta(3)=\sum_{k=1}^{\infty}\frac{1}{k^{3}}=1+\frac{1}{2^{3}}+\frac{1}{3^{3}}+% \frac{1}{4^{3}}+\frac{1}{5^{3}}+\frac{1}{6^{3}}+\frac{1}{7^{3}}+\frac{1}{8^{3}% }+\frac{1}{9^{3}}+\cdots\,\!
  3. ζ ( 3 ) = π 2 7 [ 1 - 4 k = 1 ζ ( 2 k ) ( 2 k + 1 ) ( 2 k + 2 ) 2 2 k ] \zeta(3)=\frac{\pi^{2}}{7}\left[1-4\sum_{k=1}^{\infty}\frac{\zeta(2k)}{(2k+1)(% 2k+2)2^{2k}}\right]
  4. ζ ( 3 ) = 7 180 π 3 - 2 k = 1 1 k 3 ( e 2 π k - 1 ) \zeta(3)=\frac{7}{180}\pi^{3}-2\sum_{k=1}^{\infty}\frac{1}{k^{3}(e^{2\pi k}-1)}
  5. ζ ( 3 ) = 14 k = 1 1 k 3 sinh ( π k ) - 11 2 k = 1 1 k 3 ( e 2 π k - 1 ) - 7 2 k = 1 1 k 3 ( e 2 π k + 1 ) . \zeta(3)=14\sum_{k=1}^{\infty}\frac{1}{k^{3}\sinh(\pi k)}-\frac{11}{2}\sum_{k=% 1}^{\infty}\frac{1}{k^{3}(e^{2\pi k}-1)}-\frac{7}{2}\sum_{k=1}^{\infty}\frac{1% }{k^{3}(e^{2\pi k}+1)}.
  6. ζ ( 2 n + 1 ) \zeta(2n+1)
  7. ζ ( 3 ) = 8 7 k = 0 1 ( 2 k + 1 ) 3 \zeta(3)=\frac{8}{7}\sum_{k=0}^{\infty}\frac{1}{(2k+1)^{3}}
  8. ζ ( 3 ) = 4 3 k = 0 ( - 1 ) k ( k + 1 ) 3 \zeta(3)=\frac{4}{3}\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(k+1)^{3}}
  9. ζ ( 3 ) = 5 2 k = 1 ( - 1 ) k - 1 k ! 2 k 3 ( 2 k ) ! \zeta(3)=\frac{5}{2}\sum_{k=1}^{\infty}(-1)^{k-1}\frac{k!^{2}}{k^{3}(2k)!}
  10. ζ ( 3 ) = 1 4 k = 1 ( - 1 ) k - 1 56 k 2 - 32 k + 5 ( 2 k - 1 ) 2 ( k - 1 ) ! 3 ( 3 k ) ! \zeta(3)=\frac{1}{4}\sum_{k=1}^{\infty}(-1)^{k-1}\frac{56k^{2}-32k+5}{(2k-1)^{% 2}}\frac{(k-1)!^{3}}{(3k)!}
  11. ζ ( 3 ) = 8 7 - 8 7 k = 1 ( - 1 ) k 2 - 5 + 12 k k ( - 3 + 9 k + 148 k 2 - 432 k 3 - 2688 k 4 + 7168 k 5 ) k ! 3 ( - 1 + 2 k ) ! 6 ( - 1 + 2 k ) 3 ( 3 k ) ! ( 1 + 4 k ) ! 3 \zeta(3)=\frac{8}{7}-\frac{8}{7}\sum_{k=1}^{\infty}\frac{{\left(-1\right)}^{k}% \,2^{-5+12\,k}\,k\,\left(-3+9\,k+148\,k^{2}-432\,k^{3}-2688\,k^{4}+7168\,k^{5}% \right)\,{k!}^{3}\,{\left(-1+2\,k\right)!}^{6}}{{\left(-1+2\,k\right)}^{3}\,% \left(3\,k\right)!\,{\left(1+4\,k\right)!}^{3}}
  12. ζ ( 3 ) = k = 0 ( - 1 ) k 205 k 2 + 250 k + 77 64 k ! 10 ( 2 k + 1 ) ! 5 \zeta(3)=\sum_{k=0}^{\infty}(-1)^{k}\frac{205k^{2}+250k+77}{64}\frac{k!^{10}}{% (2k+1)!^{5}}
  13. ζ ( 3 ) = k = 0 ( - 1 ) k P ( k ) 24 ( ( 2 k + 1 ) ! ( 2 k ) ! k ! ) 3 ( 3 k + 2 ) ! ( 4 k + 3 ) ! 3 \zeta(3)=\sum_{k=0}^{\infty}(-1)^{k}\frac{P(k)}{24}\frac{((2k+1)!(2k)!k!)^{3}}% {(3k+2)!(4k+3)!^{3}}
  14. P ( k ) = 126392 k 5 + 412708 k 4 + 531578 k 3 + 336367 k 2 + 104000 k + 12463. P(k)=126392k^{5}+412708k^{4}+531578k^{3}+336367k^{2}+104000k+12463.\,
  15. ζ ( 3 ) = 0 1 0 1 0 1 1 1 - x y z d x d y d z \zeta(3)=\int\limits_{0}^{1}\int\limits_{0}^{1}\int\limits_{0}^{1}\!\frac{1}{1% -xyz}\,dxdydz
  16. ζ ( 3 ) = 1 2 0 x 2 e x - 1 d x \zeta(3)=\frac{1}{2}\int\limits_{0}^{\infty}\!\frac{x^{2}}{e^{x}-1}\,dx
  17. ζ ( 3 ) = 2 3 0 x 2 e x + 1 d x \zeta(3)=\frac{2}{3}\int\limits_{0}^{\infty}\!\frac{x^{2}}{e^{x}+1}\,dx
  18. ζ ( 3 ) = π 0 cos ( 2 arctan x ) ( x 2 + 1 ) [ cosh 1 2 π x ] 2 d x \zeta(3)=\pi\!\!\int\limits_{0}^{\infty}\!\frac{\cos(2\arctan\,x)}{\left(x^{2}% +1\right)\big[\cosh\frac{1}{2}\pi x\big]^{2}}\,dx
  19. ζ ( 3 ) = - 1 2 0 1 0 1 ln ( x y ) 1 - x y d x d y \zeta(3)=-\frac{1}{2}\int\limits_{0}^{1}\!\!\int\limits_{0}^{1}\frac{\ln(xy)}{% \,1-xy\,}\,dx\,dy
  20. ζ ( 3 ) = 8 π 2 7 0 1 x ( x 4 - 4 x 2 + 1 ) ln ln 1 x ( 1 + x 2 ) 4 d x = 8 π 2 7 1 x ( x 4 - 4 x 2 + 1 ) ln ln x ( 1 + x 2 ) 4 d x \zeta(3)=\,\frac{8\pi^{2}}{7}\!\!\int\limits_{0}^{1}\!\frac{x\left(x^{4}-4x^{2% }+1\right)\ln\ln\frac{1}{x}}{\,(1+x^{2})^{4}\,}\,dx\,=\,\frac{8\pi^{2}}{7}\!\!% \int\limits_{1}^{\infty}\!\frac{x\left(x^{4}-4x^{2}+1\right)\ln\ln{x}}{\,(1+x^% {2})^{4}\,}\,dx
  21. ζ ( 3 ) = - 1 2 Γ ′′′ ( 1 ) + 3 2 Γ ( 1 ) Γ ′′ ( 1 ) - [ Γ ( 1 ) ] 3 = - 1 2 ψ ( 2 ) ( 1 ) \zeta(3)=-\frac{1}{2}\Gamma^{\prime\prime\prime}(1)+\frac{3}{2}\Gamma^{\prime}% (1)\Gamma^{\prime\prime}(1)-[\Gamma^{\prime}(1)]^{3}=-\frac{1}{2}\,\psi^{(2)}(1)

Arbitrage_betting.html

  1. s 1 s_{1}
  2. s 2 s_{2}
  3. o 1 o_{1}
  4. o 2 o_{2}
  5. r 1 r_{1}
  6. r 2 r_{2}
  7. 1.25 - 1 + 3.9 - 1 = 1.056 1.25^{-1}+3.9^{-1}=1.056
  8. 1.43 - 1 + 2.85 - 1 = 1.051 1.43^{-1}+2.85^{-1}=1.051
  9. 1 - ( 1.25 * 3.9 ) / ( 1.25 + 3.9 ) = 5.34 % 1-(1.25*3.9)/(1.25+3.9)=5.34\%
  10. 1.43 - 1 + 3.9 - 1 = 0.956 1.43^{-1}+3.9^{-1}=0.956
  11. $ 100 * 1.43 / 3.9 = 36.67 \$100*1.43/3.9=36.67
  12. r 1 = $ 100 * 1.43 = $ 143 r_{1}=\$100*1.43=\$143
  13. r 2 = $ 36.67 * 3.9 = $ 143 r_{2}=\$36.67*3.9=\$143
  14. o 1 o_{1}
  15. o 2 o_{2}
  16. o 1 - 1 + o 2 - 1 < 1 o_{1}^{-1}+o_{2}^{-1}<1
  17. s 1 s_{1}
  18. s 2 = s 1 * o 1 / o 2 s_{2}=s_{1}*o_{1}/o_{2}
  19. o 1 o_{1}
  20. o 2 o_{2}
  21. o 3 o_{3}
  22. b 1 b_{1}
  23. b 2 b_{2}
  24. b 3 b_{3}
  25. b 1 = B / ( 1 + ( o 1 / o 2 ) + ( o 1 / o 3 ) ) b_{1}=B/(1+(o_{1}/o_{2})+(o_{1}/o_{3}))
  26. b 2 = B / ( 1 + ( o 2 / o 1 ) + ( o 2 / o 3 ) ) b_{2}=B/(1+(o_{2}/o_{1})+(o_{2}/o_{3}))
  27. b 3 = B / ( 1 + ( o 3 / o 1 ) + ( o 3 / o 2 ) ) b_{3}=B/(1+(o_{3}/o_{1})+(o_{3}/o_{2}))

Arboricity.html

  1. m / ( n - 1 ) \lceil m/(n-1)\rceil
  2. max { m S / ( n S - 1 ) } . \max\{\lceil m_{S}/(n_{S}-1)\rceil\}.
  3. n n
  4. 3 n - 6 3n-6
  5. a a
  6. a a
  7. 2 a - 1 2a-1
  8. G G
  9. max { m S / ( n S - 1 ) | S G } . \max\{m_{S}/(n_{S}-1)|S\subseteq G\}.
  10. ( a , b ) (a,b)
  11. a + 1 a+1
  12. b b
  13. a a
  14. ( a , 0 ) (a,0)

Arc_(geometry).html

  1. π \pi
  2. π \pi
  3. r r
  4. θ \theta\,\!
  5. θ r \theta r\,\!
  6. L circumference = θ 2 π . \frac{L}{\mathrm{circumference}}=\frac{\theta}{2\pi}.\,\!
  7. L 2 π r = θ 2 π , \frac{L}{2\pi r}=\frac{\theta}{2\pi},\,\!
  8. α \alpha
  9. θ = α 180 π , \theta=\frac{\alpha}{180}\pi,\,\!
  10. L = α π r 180 . L=\frac{\alpha\pi r}{180}.\,\!
  11. 60 360 = L 24 \frac{60}{360}=\frac{L}{24}
  12. 360 L = 1440 360L=1440
  13. L = 4 L=4
  14. A = 1 2 r 2 θ . A=\frac{1}{2}r^{2}\theta.
  15. A A
  16. θ \theta
  17. A π r 2 = θ 2 π . \frac{A}{\pi r^{2}}=\frac{\theta}{2\pi}.
  18. π \pi
  19. A r 2 = θ 2 . \frac{A}{r^{2}}=\frac{\theta}{2}.
  20. r 2 r^{2}
  21. A = 1 2 r 2 θ . A=\frac{1}{2}r^{2}\theta.
  22. A = α 360 π r 2 . A=\frac{\alpha}{360}\pi r^{2}.
  23. 1 2 r 2 ( θ - sin θ ) . \frac{1}{2}r^{2}(\theta-\sin{\theta}).
  24. A A
  25. r r
  26. H H
  27. W W
  28. W , W,
  29. W 2 . \frac{W}{2}.
  30. 2 r , 2r,
  31. H , H,
  32. ( 2 r - H ) . (2r-H).
  33. H ( 2 r - H ) = ( W 2 ) 2 H(2r-H)=\left(\frac{W}{2}\right)^{2}
  34. 2 r - H = W 2 4 H 2r-H=\frac{W^{2}}{4H}
  35. r = W 2 8 H + H 2 r=\frac{W^{2}}{8H}+\frac{H}{2}

Area_compatibility_factor.html

  1. F = x E x , t c s m x , t s x E x , t c s / x E x , t c m x , t s x E x , t c F=\frac{\sum_{x}{}^{s}E_{x,t}^{c}{}^{s}m_{x,t}}{\sum_{x}{}^{s}E_{x,t}^{c}}% \left/\frac{\sum_{x}E_{x,t}^{c}{}^{s}m_{x,t}}{\sum_{x}E_{x,t}^{c}}\right.
  2. E x , t c s {}^{s}E_{x,t}^{c}
  3. E x , t c E_{x,t}^{c}
  4. m x , t s {}^{s}m_{x,t}

ARGUS_distribution.html

  1. Ψ ( x ) \Psi(x)
  2. c 2 χ ( χ 2 - 2 ) + χ 4 + 4 \frac{c}{\sqrt{2}\chi}\sqrt{(\chi^{2}-2)+\sqrt{\chi^{4}+4}}
  3. c 2 ( 1 - 3 χ 2 + χ ϕ ( χ ) Ψ ( χ ) ) - μ 2 c^{2}\!\left(1-\frac{3}{\chi^{2}}+\frac{\chi\phi(\chi)}{\Psi(\chi)}\right)-\mu% ^{2}
  4. f ( x ; χ , c ) = χ 3 2 π Ψ ( χ ) x c 2 1 - x 2 c 2 exp { - 1 2 χ 2 ( 1 - x 2 c 2 ) } , f(x;\chi,c)=\frac{\chi^{3}}{\sqrt{2\pi}\,\Psi(\chi)}\cdot\frac{x}{c^{2}}\sqrt{% 1-\frac{x^{2}}{c^{2}}}\exp\bigg\{-\frac{1}{2}\chi^{2}\Big(1-\frac{x^{2}}{c^{2}% }\Big)\bigg\},
  5. Ψ ( χ ) = Φ ( χ ) - χ ϕ ( χ ) - 1 2 , \Psi(\chi)=\Phi(\chi)-\chi\phi(\chi)-\tfrac{1}{2},
  6. { c 2 x ( c - x ) ( c + x ) f ( x ) + f ( x ) ( - c 4 - c 2 ( χ 2 - 2 ) x 2 + χ 2 x 4 ) = 0 , f ( 1 ) = - 2 - 2 c 2 χ 3 e χ 2 2 c 2 c 2 ( 2 χ - π e χ 2 2 erf ( χ 2 ) ) } \left\{c^{2}x(c-x)(c+x)f^{\prime}(x)+f(x)\left(-c^{4}-c^{2}\left(\chi^{2}-2% \right)x^{2}+\chi^{2}x^{4}\right)=0,f(1)=-\frac{\sqrt{2-\frac{2}{c^{2}}}\chi^{% 3}e^{\frac{\chi^{2}}{2c^{2}}}}{c^{2}\left(\sqrt{2}\chi-\sqrt{\pi}e^{\frac{\chi% ^{2}}{2}}\,\text{erf}\left(\frac{\chi}{\sqrt{2}}\right)\right)}\right\}
  7. F ( x ) = 1 - Ψ ( χ 1 - x 2 / c 2 ) Ψ ( χ ) . F(x)=1-\frac{\Psi\Big(\chi\sqrt{1-x^{2}/c^{2}}\,\Big)}{\Psi(\chi)}.
  8. 1 - 3 χ 2 + χ ϕ ( χ ) Ψ ( χ ) = 1 n i = 1 n x i 2 c 2 . 1-\frac{3}{\chi^{2}}+\frac{\chi\phi(\chi)}{\Psi(\chi)}=\frac{1}{n}\sum_{i=1}^{% n}\frac{x_{i}^{2}}{c^{2}}.
  9. χ ^ \scriptstyle\hat{\chi}
  10. f ( x ) = 2 - p χ 2 ( p + 1 ) Γ ( p + 1 ) - Γ ( p + 1 , 1 2 χ 2 ) x c 2 ( 1 - x 2 c 2 ) p exp { - 1 2 χ 2 ( 1 - x 2 c 2 ) } , 0 x c , f(x)=\frac{2^{-p}\chi^{2(p+1)}}{\Gamma(p+1)-\Gamma(p+1,\,\tfrac{1}{2}\chi^{2})% }\cdot\frac{x}{c^{2}}\bigg(1-\frac{x^{2}}{c^{2}}\bigg)^{p}\exp\bigg\{-\frac{1}% {2}\chi^{2}\Big(1-\frac{x^{2}}{c^{2}}\Big)\bigg\},\qquad 0\leq x\leq c,
  11. c 2 χ ( χ 2 - 2 p - 1 ) + χ 2 ( χ 2 - 4 p + 2 ) + ( 1 + 2 p ) 2 \frac{c}{\sqrt{2}\chi}\sqrt{(\chi^{2}-2p-1)+\sqrt{\chi^{2}(\chi^{2}-4p+2)+(1+2% p)^{2}}}

Arithmetical_set.html

  1. f : k f:\subseteq\mathbb{N}^{k}\to\mathbb{N}
  2. f f
  3. R ( n 1 , , n k ) R(n_{1},\ldots,n_{k})
  4. ψ ( n 1 , , n k ) \psi(n_{1},\ldots,n_{k})
  5. R ( n 1 , , n k ) ψ ( n 1 , , n k ) R(n_{1},\ldots,n_{k})\Leftrightarrow\psi(n_{1},\ldots,n_{k})
  6. ( n 1 , , n k ) (n_{1},\ldots,n_{k})
  7. θ ( Z ) \theta(Z)
  8. \in
  9. θ ( Z ) \theta(Z)
  10. n [ n Z ϕ ( n ) ] \forall n[n\in Z\Leftrightarrow\phi(n)]

Arrow_(computer_science).html

  1. a r r arr
  2. \to
  3. s s
  4. t t
  5. A A
  6. f i r s t first
  7. u u
  8. \ggg
  9. * * * ***
  10. i d id
  11. f f
  12. g g

Artificial_heart_valve.html

  1. E O A ( cm 2 ) = Q r m s 51.6 Δ p EOA(\mathrm{cm}^{2})=\frac{Q_{rms}}{51.6\sqrt{\Delta p}}
  2. Q r m s Q_{rms}
  3. Δ p \Delta p
  4. A 1 V 1 = A 2 V 2 A_{1}V_{1}=A_{2}V_{2}
  5. P 1 + 1 2 ρ 1 V 1 2 = P 2 + 1 2 ρ 2 V 2 2 P_{1}+\frac{1}{2}\rho_{1}V_{1}^{2}=P_{2}+\frac{1}{2}\rho_{2}V_{2}^{2}
  6. ρ \rho

Artin's_conjecture_on_primitive_roots.html

  1. C Artin = p prime ( 1 - 1 p ( p - 1 ) ) = 0.3739558136 C_{\mathrm{Artin}}=\prod_{p\ \mathrm{prime}}\left(1-\frac{1}{p(p-1)}\right)=0.% 3739558136\ldots

Artin_L-function.html

  1. ρ \rho
  2. G G
  3. V V
  4. G G
  5. L / K L/K
  6. L L
  7. L ( ρ , s ) L(\rho,s)
  8. 𝔭 \mathfrak{p}
  9. K K
  10. 𝔭 \mathfrak{p}
  11. L L
  12. 𝔭 \mathfrak{p}
  13. 𝐅𝐫𝐨𝐛 ( 𝔭 ) \mathbf{Frob}(\mathfrak{p})
  14. G G
  15. ρ ( 𝐅𝐫𝐨𝐛 ( 𝔭 ) ) \rho(\mathbf{Frob}(\mathfrak{p}))
  16. 𝔭 \mathfrak{p}
  17. charpoly ( ρ ( 𝐅𝐫𝐨𝐛 ( 𝔭 ) ) ) - 1 = det [ I - t ρ ( 𝐅𝐫𝐨𝐛 ( 𝔭 ) ) ] - 1 , \operatorname{charpoly}(\rho(\mathbf{Frob}(\mathfrak{p})))^{-1}=\operatorname{% det}\left[I-t\rho(\mathbf{Frob}(\mathfrak{p}))\right]^{-1},
  18. t = N ( 𝔭 ) - s t=N(\mathfrak{p})^{-s}
  19. s s
  20. 𝔭 \mathfrak{p}
  21. L ( ρ , s ) L(\rho,s)
  22. 𝔭 \mathfrak{p}
  23. n 1 n\geq 1

Artin_reciprocity_law.html

  1. 𝔭 \mathfrak{p}
  2. 𝔓 \mathfrak{P}
  3. 𝔭 \mathfrak{p}
  4. 𝔭 \mathfrak{p}
  5. D 𝔭 D_{\mathfrak{p}}
  6. 𝒪 L , 𝔓 / 𝔓 \mathcal{O}_{L,\mathfrak{P}}/\mathfrak{P}
  7. 𝒪 K , 𝔭 / 𝔭 \mathcal{O}_{K,\mathfrak{p}}/\mathfrak{p}
  8. Frob 𝔭 \mathrm{Frob}_{\mathfrak{p}}
  9. ( L / K 𝔭 ) \left(\frac{L/K}{\mathfrak{p}}\right)
  10. I K Δ I_{K}^{\Delta}
  11. ( L / K ) : I K Δ Gal ( L / K ) i = 1 m 𝔭 i n i i = 1 m ( L / K 𝔭 i ) n i . \begin{matrix}\left(\frac{L/K}{\cdot}\right):&I_{K}^{\Delta}&\longrightarrow&% \mathrm{Gal}(L/K)\\ &\displaystyle{\prod_{i=1}^{m}\mathfrak{p}_{i}^{n_{i}}}&\mapsto&\displaystyle{% \prod_{i=1}^{m}\left(\frac{L/K}{\mathfrak{p}_{i}}\right)^{n_{i}}.}\end{matrix}
  12. I K 𝐜 / i ( K 𝐜 , 1 ) N L / K ( I L 𝐜 ) Gal ( L / K ) I_{K}^{\mathbf{c}}/i(K_{\mathbf{c},1})\mathrm{N}_{L/K}(I_{L}^{\mathbf{c}})% \overset{\sim}{\longrightarrow}\mathrm{Gal}(L/K)
  13. I L 𝐜 I_{L}^{\mathbf{c}}
  14. 𝔣 ( L / K ) \scriptstyle\mathfrak{f}(L/K)
  15. d 1 d\neq 1
  16. L = 𝐐 ( d ) \scriptstyle L=\mathbf{Q}(\sqrt{d})
  17. p ( Δ p ) p\mapsto\left(\frac{\Delta}{p}\right)
  18. ( Δ p ) \left(\frac{\Delta}{p}\right)
  19. ( Δ n ) . \left(\frac{\Delta}{n}\right).
  20. ( Δ p ) \left(\frac{\Delta}{p}\right)
  21. σ ( ζ m ) = ζ m a σ . \sigma(\zeta_{m})=\zeta_{m}^{a_{\sigma}}.
  22. ( p ) = ( p ) . \left(\frac{\ell^{\ast}}{p}\right)=\left(\frac{p}{\ell}\right).
  23. F = 𝐐 ( ) \scriptstyle F=\mathbf{Q}(\sqrt{\ell^{\ast}})
  24. L = 𝐐 ( ζ ) \scriptstyle L=\mathbf{Q}(\zeta_{\ell})
  25. ( F / 𝐐 ( n ) ) = ( L / 𝐐 ( n ) ) (mod H ) . \left(\frac{F/\mathbf{Q}}{(n)}\right)=\left(\frac{L/\mathbf{Q}}{(n)}\right)\,% \text{ (mod }H).
  26. ( p ) = 1 \left(\frac{\ell^{\ast}}{p}\right)=1
  27. θ v : K v × / N L v / K v ( L v × ) G ab , \theta_{v}:K_{v}^{\times}/N_{L_{v}/K_{v}}(L_{v}^{\times})\to G^{\,\text{ab}},
  28. θ : C K / N L / K ( C L ) Gal ( L / K ) ab . \theta:C_{K}/{N_{L/K}(C_{L})}\to\,\text{Gal}(L/K)^{\,\text{ab}}.
  29. ( Gal ( K s e p / K ) , lim C L ) (\,\text{Gal}(K^{sep}/K),\underrightarrow{\lim}C_{L})
  30. H ^ 0 ( Gal ( L / K ) , C L ) H ^ - 2 ( Gal ( L / K ) , ) , \hat{H}^{0}(\,\text{Gal}(L/K),C_{L})\simeq\hat{H}^{-2}(\,\text{Gal}(L/K),% \mathbb{Z}),
  31. H ^ i \hat{H}^{i}
  32. L E / K Artin ( σ , s ) = L K Hecke ( χ , s ) L_{E/K}^{\mathrm{Artin}}(\sigma,s)=L_{K}^{\mathrm{Hecke}}(\chi,s)

Artin–Schreier_theory.html

  1. X p - X + α , X^{p}-X+\alpha,\,
  2. α \alpha
  3. α \alpha
  4. { y K | y = x p - x for x K } \{y\in K\,|\,y=x^{p}-x\;\mbox{for }~{}x\in K\}
  5. 1 i p 1\leq i\leq p
  6. K ( β ) K(\beta)

Aspheric_lens.html

  1. z ( r ) = r 2 R ( 1 + 1 - ( 1 + κ ) r 2 R 2 ) + α 1 r 2 + α 2 r 4 + α 3 r 6 + , z(r)=\frac{r^{2}}{R\left(1+\sqrt{1-(1+\kappa)\frac{r^{2}}{R^{2}}}\right)}+% \alpha_{1}r^{2}+\alpha_{2}r^{4}+\alpha_{3}r^{6}+\cdots,
  2. z ( r ) z(r)
  3. r r
  4. α i \alpha_{i}
  5. R R
  6. κ \kappa
  7. α i \alpha_{i}
  8. R R
  9. κ \kappa
  10. r = 0 r=0
  11. κ \kappa
  12. κ \kappa
  13. κ < - 1 \kappa<-1
  14. κ = - 1 \kappa=-1
  15. - 1 < κ < 0 -1<\kappa<0
  16. κ = 0 \kappa=0
  17. κ > 0 \kappa>0

Association_scheme.html

  1. R 0 = { ( x , x ) : x X } R_{0}=\{(x,x):x\in X\}
  2. R * := { ( x , y ) | ( y , x ) R } R^{*}:=\{(x,y)|(y,x)\in R\}
  3. ( x , y ) R k (x,y)\in R_{k}
  4. z X z\in X
  5. ( x , z ) R i (x,z)\in R_{i}
  6. ( z , y ) R j (z,y)\in R_{j}
  7. p i j k p^{k}_{ij}
  8. i i
  9. j j
  10. k k
  11. x x
  12. y y
  13. p i j k = p j i k p_{ij}^{k}=p_{ji}^{k}
  14. i i
  15. j j
  16. k k
  17. R i R_{i}
  18. ( x , y ) R i (x,y)\in R_{i}
  19. i i
  20. z z
  21. x x
  22. y y
  23. p i j k p^{k}_{ij}
  24. v v
  25. X X
  26. x x
  27. y y
  28. i i
  29. x x
  30. y y
  31. i i
  32. k k
  33. i i
  34. j j
  35. p i j k p^{k}_{ij}
  36. i , j , k i,j,k
  37. p i i 0 = v i p^{0}_{ii}=v_{i}
  38. i i
  39. v i v_{i}
  40. R i R_{i}
  41. 0
  42. x x
  43. R 0 R_{0}
  44. A i A_{i}
  45. R i R_{i}
  46. i = 0 , , n i=0,\ldots,n
  47. X X
  48. ( A i ) x , y = { 1 , if ( x , y ) R i , 0 , otherwise. ( 1 ) \left(A_{i}\right)_{x,y}=\left\{\begin{matrix}1,&\mbox{if }~{}\left(x,y\right)% \in R_{i},\\ 0,&\mbox{otherwise.}\end{matrix}\right.\qquad(1)
  49. A i A_{i}
  50. A i A_{i}\,
  51. i = 0 n A i = J \sum_{i=0}^{n}A_{i}=J
  52. A 0 = I A_{0}=I\,
  53. A i A j = k = 0 n p i j k A k = A j A i , i , j = 0 , , n A_{i}A_{j}=\sum_{k=0}^{n}p^{k}_{ij}A_{k}=A_{j}A_{i},i,j=0,\ldots,n
  54. A i A_{i}
  55. v i v_{i}
  56. 1 1
  57. A i J = J A i = v i J . ( 2 ) A_{i}J=JA_{i}=v_{i}J.\qquad(2)
  58. p i j k p_{ij}^{k}
  59. p 00 0 = 1 p_{00}^{0}=1
  60. ( x , y ) R 0 (x,y)\in R_{0}
  61. x = y x=y
  62. z z
  63. ( x , z ) R 0 (x,z)\in R_{0}
  64. z = x z=x
  65. i = 0 k p i i 0 = | X | \sum_{i=0}^{k}p_{ii}^{0}=|X|
  66. R i R_{i}
  67. X X
  68. A i A_{i}
  69. ( X , R i ) \left(X,R_{i}\right)
  70. 𝒜 \mathcal{A}
  71. 𝒜 \mathcal{A}
  72. 𝒜 \mathcal{A}
  73. J 0 , , J n J_{0},\ldots,J_{n}
  74. ( n + 1 ) × ( n + 1 ) \left(n+1\right)\times\left(n+1\right)
  75. 𝒜 \mathcal{A}
  76. ( v k ) {v\choose k}
  77. X = G X=G
  78. g G g\in G
  79. R g = { ( x , y ) | x = g * y } R_{g}=\{(x,y)|x=g*y\}
  80. * *

Asymptotic_distribution.html

  1. Y i = Z i - a i b i Y_{i}=\frac{Z_{i}-a_{i}}{b_{i}}
  2. P ( Z n - a n b n x ) F ( x ) , P\left(\frac{Z_{n}-a_{n}}{b_{n}}\leq x\right)\approx F(x),
  3. P ( Z n z ) F ( z - a n b n ) . P(Z_{n}\leq z)\approx F\left(\frac{z-a_{n}}{b_{n}}\right).
  4. n \sqrt{n}

Atmospheric_escape.html

  1. E 𝑘𝑖𝑛 = 1 2 m v 2 E_{\mathit{kin}}=\frac{1}{2}mv^{2}

Atom-transfer_radical-polymerization.html

  1. [ R-Pn ] = K ATRP [ R-Pn -X ] [ CuI X / L ] [ CuII X 2 / L ] [\,\text{R-P}\text{n}^{\bullet}]=K\text{ATRP}\cdot[\,\text{R-P}\text{n}\,\text% {-X}]\cdot\frac{[\,\text{Cu}\text{I}\,\text{X}/\,\text{L}]}{[\,\text{Cu}\text{% II}\,\text{X}_{2}/\,\text{L}]}

Atom_(measure_theory).html

  1. ( X , Σ ) (X,\Sigma)
  2. μ \mu
  3. A X A\subset X
  4. Σ \Sigma
  5. μ ( A ) > 0 \mu(A)>0
  6. B A B\subset A
  7. μ ( B ) < μ ( A ) \mu(B)<\mu(A)
  8. B B
  9. Σ \Sigma
  10. μ \mu
  11. A A
  12. μ ( A ) > 0 \mu(A)>0
  13. μ ( A ) > μ ( B ) > 0. \mu(A)>\mu(B)>0.\,
  14. μ ( A ) > 0 \mu(A)>0
  15. A = A 1 A 2 A 3 A=A_{1}\supset A_{2}\supset A_{3}\supset\cdots
  16. μ ( A ) = μ ( A 1 ) > μ ( A 2 ) > μ ( A 3 ) > > 0. \mu(A)=\mu(A_{1})>\mu(A_{2})>\mu(A_{3})>\cdots>0.
  17. μ ( A ) > 0 , \mu(A)>0,
  18. μ ( A ) b 0 \mu(A)\geq b\geq 0\,
  19. μ ( B ) = b . \mu(B)=b.\,
  20. ( X , Σ , μ ) (X,\Sigma,\mu)
  21. μ ( X ) = c \mu(X)=c
  22. S : [ 0 , c ] Σ S:[0,c]\to\Sigma
  23. μ : Σ [ 0 , c ] \mu:\Sigma\to[0,\,c]
  24. 0 t t c 0\leq t\leq t^{\prime}\leq c
  25. S ( t ) S ( t ) , S(t)\subset S(t^{\prime}),
  26. μ ( S ( t ) ) = t . \mu\left(S(t)\right)=t.
  27. μ \mu
  28. Γ := { S : D Σ : D [ 0 , c ] , S monotone , t D ( μ ( S ( t ) ) = t ) } , \Gamma:=\{S:D\to\Sigma\;:\;D\subset[0,\,c],\,S\;\mathrm{monotone},\forall t\in D% \;(\mu\left(S(t)\right)=t)\},
  29. graph ( S ) graph ( S ) . \mathrm{graph}(S)\subset\mathrm{graph}(S^{\prime}).
  30. Γ \Gamma
  31. Γ \Gamma
  32. Γ \Gamma
  33. [ 0 , c ] , [0,c],

Atomic_diffusion.html

  1. n n
  2. α \alpha
  3. r = α n . r=\alpha\cdot\sqrt{n}.
  4. T T
  5. t t
  6. r r
  7. T t Tt
  8. r T t . r\sim\sqrt{Tt}.

Atriphtaloid.html

  1. x 4 ( x 2 + y 2 ) - ( a x 2 - b ) 2 = 0 , x^{4}(x^{2}+y^{2})-(ax^{2}-b)^{2}=0,\,

Augmentation_ideal.html

  1. ε \varepsilon
  2. R [ G ] R[G]
  3. r i g i \sum r_{i}g_{i}
  4. r i . \sum r_{i}.
  5. ε ( g ) \varepsilon(g)
  6. ε \varepsilon
  7. ε \varepsilon
  8. g - g g-g^{\prime}
  9. g - 1 , g G g-1,g\in G
  10. ε \varepsilon

Austin_moving-knife_procedures.html

  1. 1 / n 1/n
  2. 1 / n 1/n
  3. n = 2 n=2
  4. n > 2 n>2
  5. 1 / n 1/n
  6. n = 2 n=2
  7. 1 / k 1/k
  8. k 2 k\geq 2
  9. Cut 2 ( 1 / k ) \mathrm{Cut}_{2}(1/k)
  10. k - 1 k-1
  11. k k
  12. 1 / k 1/k
  13. 1 / k 1/k
  14. 1 / k 1/k
  15. 1 / k 1/k
  16. 1 / k 1/k
  17. 1 / k 1/k
  18. Cut 2 \mathrm{Cut}_{2}
  19. k k
  20. 1 / k 1/k
  21. Cut 2 ( 1 / k ) \mathrm{Cut}_{2}(1/k)
  22. 1 / k 1/k
  23. ( k - 1 ) / k (k-1)/k
  24. Cut 2 ( 1 / ( k - 1 ) ) \mathrm{Cut}_{2}(1/(k-1))
  25. 1 / k 1/k
  26. k k
  27. Cut 2 \mathrm{Cut}_{2}
  28. n n
  29. 1 / n 1/n
  30. Cut 2 ( 1 / 2 ) \mathrm{Cut}_{2}(1/2)
  31. Cut 2 ( 1 / 3 ) \mathrm{Cut}_{2}(1/3)
  32. Cut 2 ( 1 / 3 ) \mathrm{Cut}_{2}(1/3)
  33. n > 2 n>2
  34. 1 / n 1/n
  35. n > 2 n>2

Autoepistemic_logic.html

  1. \Box
  2. F F
  3. F \Box F
  4. F F
  5. ¬ F \Box\neg F
  6. ¬ F \neg F
  7. ¬ F \neg\Box F
  8. F F
  9. ¬ F ¬ F \neg\Box F\rightarrow\neg F
  10. F F
  11. F \Box F
  12. T T
  13. F \Box F
  14. T T
  15. T T
  16. \Box
  17. T T
  18. F F
  19. F F
  20. F \Box F
  21. T = x x T=\Box x\rightarrow x
  22. x \Box x
  23. x \Box x
  24. T T
  25. x x \Box x\rightarrow x
  26. ¬ x x \neg\Box x\vee x
  27. ¬ x \neg\Box x
  28. x x
  29. x \Box x
  30. x x
  31. x \Box x
  32. x \Box x
  33. T T
  34. x x
  35. x \Box x
  36. x x
  37. T T
  38. x x
  39. x x
  40. x \Box x
  41. x x

Automated_readability_index.html

  1. 4.71 ( characters words ) + 0.5 ( words sentences ) - 21.43 4.71\left(\frac{\mbox{characters}~{}}{\mbox{words}~{}}\right)+0.5\left(\frac{% \mbox{words}~{}}{\mbox{sentences}~{}}\right)-21.43

Automatic_vectorization.html

  1. c 1 \displaystyle c_{1}

Avalanche_transistor.html

  1. M M
  2. I C - V C E I_{C}-V_{CE}
  3. I C I_{C}
  4. I C = I E - I B I_{C}=I_{E}-I_{B}\,
  5. I C = β I B + ( β + 1 ) I C B O I_{C}=\beta I_{B}+(\beta+1)I_{CBO}\,
  6. I B I_{B}
  7. I C B O I_{CBO}
  8. I E I_{E}
  9. β \beta
  10. I C I_{C}
  11. I E = ( β + 1 ) I B + ( β + 1 ) I C B O I_{E}=(\beta+1)I_{B}+(\beta+1)I_{CBO}\,
  12. α = β ( β + 1 ) - 1 \alpha=\beta{(\beta+1)^{-1}}
  13. α I E = β I B + β I C B O = I C - I C B O I C = α I E + I C B O \alpha I_{E}=\beta I_{B}+\beta I_{CBO}=I_{C}-I_{CBO}\iff I_{C}=\alpha I_{E}+I_% {CBO}
  14. I C I_{C}
  15. I C = M ( α I E + I C B O ) I_{C}=M(\alpha I_{E}+I_{CBO})\,
  16. M M
  17. M = 1 1 - ( V C B B V C B O ) n M={\frac{1}{1-\left(\frac{V_{CB}}{BV_{CBO}}\right)^{n}}}\,
  18. B V C B O BV_{CBO}
  19. n n
  20. V C B V_{CB}
  21. M M
  22. I C I_{C}
  23. I C = M 1 - α M ( I C B O + α I B ) I C = I C B O + α I B 1 - α - ( V C B B V C B O ) n I_{C}=\frac{M}{1-\alpha M}(I_{CBO}+\alpha I_{B})\iff I_{C}=\frac{I_{CBO}+% \alpha I_{B}}{1-\alpha-\left(\frac{V_{CB}}{BV_{CBO}}\right)^{\!n}}
  24. V C B = V C E - V B E V_{CB}=V_{CE}-V_{BE}
  25. V B E = V B E ( I B ) V_{BE}=V_{BE}(I_{B})
  26. V B E V_{BE}
  27. I C = I C B O + α I B 1 - α - ( V C E - V B E ( I B ) B V C B O ) n I C B O + α I B 1 - α - ( V C E B V C B O ) n I_{C}=\frac{I_{CBO}+\alpha I_{B}}{1-\alpha-\left(\frac{V_{CE}-V_{BE}(I_{B})}{% BV_{CBO}}\right)^{\!n}}\cong\frac{I_{CBO}+\alpha I_{B}}{1-\alpha-\left(\frac{V% _{CE}}{BV_{CBO}}\right)^{\!n}}
  28. V C E V B E V_{CE}>>V_{BE}
  29. I C - V C E I_{C}-V_{CE}
  30. I B I_{B}
  31. I C I_{C}
  32. ( V C E B V C B O ) n = 1 - α V C E = B V C E O = ( 1 - α ) n B V C B O = B V C B O β + 1 n \left(\frac{V_{CE}}{BV_{CBO}}\right)^{\!n}=1-\alpha\iff V_{CE}=BV_{CEO}=\sqrt[% n]{(1-\alpha)}BV_{CBO}=\frac{BV_{CBO}}{\sqrt[n]{\beta+1}}
  33. B V C E O BV_{CEO}
  34. V C E V_{CE}
  35. I C I_{C}
  36. V C E V_{CE}
  37. V B B V_{BB}
  38. R E R_{E}
  39. R C RC
  40. I B I_{B}
  41. τ A c e = r A c e C A c e \tau_{Ace}=r_{Ace}C_{Ace}\,
  42. r A c e r_{Ace}
  43. V C E V_{CE}
  44. I C I_{C}
  45. I B I_{B}
  46. r A c e = V C E I C | I B = c o n s t . r_{Ace}=\frac{\partial{V_{CE}}}{\partial{I_{C}}}\Bigg|_{I_{B}=const.}
  47. C A c e C_{Ace}
  48. C A c e = - ( 1 r A c e ω β - C o b ) C_{Ace}=-\left(\frac{1}{r_{Ace}\omega_{\beta}}-C_{ob}\right)
  49. ω β = 2 π f β \omega_{\beta}=2\pi f_{\beta}
  50. C o b C_{ob}
  51. V C C V_{CC}
  52. I C M A X I_{CMAX}
  53. I C - V C E I_{C}-V_{CE}
  54. D S D_{S}
  55. C o b C_{ob}
  56. D S D_{S}
  57. C i b C_{ib}
  58. D S D_{S}
  59. V E V_{E}
  60. V o u t 1 V_{out1}
  61. V o u t 2 V_{out2}
  62. R E R_{E}
  63. R L R_{L}
  64. R C R_{C}
  65. C T C_{T}
  66. T L t f \scriptstyle TL_{t_{f}}
  67. C T C_{T}
  68. R E R_{E}
  69. R L R_{L}
  70. C T C_{T}
  71. t f t_{f}
  72. R E R_{E}
  73. R L R_{L}
  74. t = 2 t f t=2t_{f}\,
  75. V C C V_{CC}
  76. R B R_{B}
  77. V B B V_{BB}
  78. M M
  79. B V C E O BV_{CEO}

Average_propensity_to_consume.html

  1. A P C = C Y APC=\frac{C}{Y}
  2. A P C = C Y - T APC=\frac{C}{Y-T}

Average_true_range.html

  1. high - low \mbox{high}~{}-\mbox{low}~{}
  2. true range = max [ ( high - low ) , abs ( high - close ) prev , abs ( low - close ) prev ] \mbox{true range}~{}={\max[(\mbox{high}~{}-\mbox{low}~{}),\mbox{abs}~{}(\mbox{% high}~{}-\mbox{close}~{}_{\mbox{prev}}~{}),\mbox{abs}~{}(\mbox{low}~{}-\mbox{% close}~{}_{\mbox{prev}}~{})]}\,
  3. A T R t = A T R t - 1 × ( n - 1 ) + T R t n ATR_{t}={{ATR_{t-1}\times(n-1)+TR_{t}}\over n}
  4. A T R = 1 n i = 1 n T R i ATR={1\over n}\sum_{i=1}^{n}TR_{i}

Avoided_crossing.html

  1. H = ( E 1 0 0 E 2 ) H=\begin{pmatrix}E_{1}&0\\ 0&E_{2}\end{pmatrix}\,\!
  2. E 1 \textstyle E_{1}
  3. E 2 \textstyle E_{2}
  4. ( 1 0 ) \textstyle\begin{pmatrix}1\\ 0\end{pmatrix}
  5. ( 0 1 ) \textstyle\begin{pmatrix}0\\ 1\end{pmatrix}
  6. E 1 \textstyle E_{1}
  7. E 2 E_{2}
  8. w ( = | W | ) \textstyle w(=|W|)
  9. Δ E = 0 \textstyle\Delta E=0
  10. H = H + P = ( E 1 0 0 E 2 ) + ( 0 W W * 0 ) = ( E 1 W W * E 2 ) H^{{}^{\prime}}=H+P=\begin{pmatrix}E_{1}&0\\ 0&E_{2}\end{pmatrix}+\begin{pmatrix}0&W\\ W^{*}&0\end{pmatrix}=\begin{pmatrix}E_{1}&W\\ W^{*}&E_{2}\end{pmatrix}\,\!
  11. E + = 1 2 ( E 1 + E 2 ) + 1 2 ( E 1 - E 2 ) 2 + 4 | W | 2 E_{+}=\frac{1}{2}(E_{1}+E_{2})+\frac{1}{2}\sqrt{(E_{1}-E_{2})^{2}+4|W|^{2}}
  12. E - = 1 2 ( E 1 + E 2 ) - 1 2 ( E 1 - E 2 ) 2 + 4 | W | 2 E_{-}=\frac{1}{2}(E_{1}+E_{2})-\frac{1}{2}\sqrt{(E_{1}-E_{2})^{2}+4|W|^{2}}
  13. ( E 1 - E 2 ) \textstyle(E_{1}-E_{2})
  14. E + \textstyle E_{+}
  15. E - \textstyle E_{-}
  16. E 1 = E 2 \textstyle E_{1}=E_{2}
  17. W \textstyle W
  18. ( E 1 - E 2 ) = 0 \textstyle(E_{1}-E_{2})=0
  19. E + = E - \textstyle E_{+}=E_{-}
  20. ( 1 0 ) \textstyle\begin{pmatrix}1\\ 0\end{pmatrix}
  21. ( 0 1 ) \textstyle\begin{pmatrix}0\\ 1\end{pmatrix}
  22. | ψ 1 \textstyle|\psi_{1}\rangle
  23. | ψ 2 \textstyle|\psi_{2}\rangle
  24. H H^{{}^{\prime}}
  25. H i j = ψ i | H | ψ j H^{{}^{\prime}}_{ij}=\langle\psi_{i}|H^{{}^{\prime}}|\psi_{j}\rangle
  26. i , j { 1 , 2 } i,j\in\left\{{1,2}\right\}
  27. H 11 = H 22 = E H^{{}^{\prime}}_{11}=H^{{}^{\prime}}_{22}=E
  28. | ψ + \textstyle|\psi_{+}\rangle
  29. | ψ - \textstyle|\psi_{-}\rangle
  30. H | ψ + = E + | ψ + H^{{}^{\prime}}|\psi_{+}\rangle=E_{+}|\psi_{+}\rangle
  31. H | ψ - = E - | ψ - H^{{}^{\prime}}|\psi_{-}\rangle=E_{-}|\psi_{-}\rangle
  32. | ψ + = 1 2 ( e i ϕ 1 ) = 1 2 ( e i ϕ | ψ 1 + | ψ 2 ) |\psi_{+}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}e^{i\phi}\\ 1\end{pmatrix}=\frac{1}{\sqrt{2}}(e^{i\phi}|\psi_{1}\rangle+|\psi_{2}\rangle)
  33. | ψ - = 1 2 ( - e i ϕ 1 ) = 1 2 ( - e i ϕ | ψ 1 + | ψ 2 ) |\psi_{-}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}-e^{i\phi}\\ 1\end{pmatrix}=\frac{1}{\sqrt{2}}(-e^{i\phi}|\psi_{1}\rangle+|\psi_{2}\rangle)
  34. e i ϕ = W / | W | e^{i\phi}=W/|W|
  35. E - E_{-}
  36. | ψ 1 \textstyle|\psi_{1}\rangle
  37. | ψ 2 \textstyle|\psi_{2}\rangle
  38. ψ 1 | H | ψ 1 = ψ 2 | H | ψ 2 = E \langle\psi_{1}|H|\psi_{1}\rangle=\langle\psi_{2}|H|\psi_{2}\rangle=E
  39. H H
  40. E - < E E_{-}<E
  41. | ψ + |\psi_{+}\rangle
  42. | ψ - |\psi_{-}\rangle
  43. P = ( W 1 W W W 2 ) \textstyle P=\begin{pmatrix}W_{1}&W\\ W&W_{2}\end{pmatrix}
  44. H H
  45. ( E 1 0 0 E 2 ) + ( W 1 W W W 2 ) = ( E 1 + W 1 W W E 2 + W 2 ) \begin{pmatrix}E_{1}&0\\ 0&E_{2}\end{pmatrix}+\begin{pmatrix}W_{1}&W\\ W&W_{2}\end{pmatrix}=\begin{pmatrix}E_{1}+W_{1}&W\\ W&E_{2}+W_{2}\end{pmatrix}
  46. E ± = 1 2 ( E 1 + E 2 + W 1 + W 2 ) ± 1 2 ( E 1 - E 2 + W 1 - W 2 ) 2 + 4 W 2 E_{\pm}=\frac{1}{2}(E_{1}+E_{2}+W_{1}+W_{2})\pm\frac{1}{2}\sqrt{(E_{1}-E_{2}+W% _{1}-W_{2})^{2}+4W^{2}}
  47. ( E 1 - E 2 + W 1 - W 2 ) = 0 (E_{1}-E_{2}+W_{1}-W_{2})=0
  48. W = 0 W=0
  49. P P
  50. k k
  51. α 1 , α 2 , α 3 . . α k {\alpha_{1},\alpha_{2},\alpha_{3}.....\alpha_{k}}
  52. ( E 1 - E 2 + W 1 - W 2 ) = F 1 ( α 1 , α 2 , α 3 . . α k ) = 0 (E_{1}-E_{2}+W_{1}-W_{2})=F_{1}(\alpha_{1},\alpha_{2},\alpha_{3}.....\alpha_{k% })=0
  53. W = F 2 ( α 1 , α 2 , α 3 . . α k ) = 0 W=F_{2}(\alpha_{1},\alpha_{2},\alpha_{3}.....\alpha_{k})=0
  54. α 1 \alpha_{1}
  55. α k - 1 \alpha_{k-1}
  56. α k \alpha_{k}
  57. F 1 ( α k - 1 , α k ) | α 1 , α 2 , , α k - 2 f i x e d = 0 F_{1}(\alpha_{k-1},\alpha_{k})|_{\alpha_{1},\alpha_{2},...,\alpha_{k-2}\,fixed% }=0
  58. F 2 ( α k - 1 , α k ) | α 1 , α 2 , , α k - 2 f i x e d = 0 F_{2}(\alpha_{k-1},\alpha_{k})|_{\alpha_{1},\alpha_{2},...,\alpha_{k-2}\,fixed% }=0
  59. k k
  60. k - 2 k-2
  61. α k \alpha_{k}
  62. E + E_{+}
  63. E - E_{-}
  64. k - 2 k-2
  65. k + 1 k+1
  66. E ± = E ± ( α 1 , α 2 , α 3 . . α k ) E_{\pm}=E_{\pm}(\alpha_{1},\alpha_{2},\alpha_{3}.....\alpha_{k})
  67. k - 2 k-2
  68. k k
  69. k - 2 k-2
  70. ψ 1 | W | ψ 2 ψ 2 | W | ψ 1 \langle\psi_{1}|W|\psi_{2}\rangle\neq\langle\psi_{2}|W|\psi_{1}\rangle
  71. W = 0 W=0
  72. k - 1 k-1
  73. r r
  74. S = 3 N - 6 \textstyle S=3N-6
  75. N 2 \textstyle N\geq 2

Axial-flow_pump.html

  1. V w2 V_{\rm w2}
  2. U = U 2 = U 1 U=U_{\rm 2}=U_{\rm 1}
  3. V w1 = 0 V_{\rm w1}=0
  4. α 1 = 90 deg \alpha_{\rm 1}=90\deg
  5. V w2 = U - V f2 cot β 2 V_{\rm w2}=U-V_{\rm f2}\cot\beta_{\rm 2}
  6. U ( U - V f2 cot β 2 ) g U\frac{(U-V_{\rm f2}\cot\beta_{\rm 2})}{g}
  7. V f1 = V f2 = V f V_{\rm f1}=V_{\rm f2}=V_{\rm f}
  8. U ( U - V f cot β 2 ) g U\frac{(U-V_{\rm f}\cot\beta_{\rm 2})}{g}
  9. r r
  10. U 2 U^{2}
  11. r r
  12. U V f cot β 2 UV_{\rm f}\cot\beta_{\rm 2}
  13. V f V_{\rm f}
  14. cot β 2 \cot\beta_{\rm 2}
  15. r r

Axiality_and_rhombicity.html

  1. A < m t p l > x x A_{<}mtpl>{{xx}}
  2. A < m t p l > y y A_{<}mtpl>{{yy}}
  3. A < m t p l > z z A_{<}mtpl>{{zz}}
  4. A < m t p l > x x A y y A z z . A_{<}mtpl>{{xx}}\leq A_{{yy}}\leq A_{{zz}}.
  5. Δ A = 2 A z z - ( A x x + A y y ) . \Delta A=2A_{zz}-(A_{xx}+A_{yy}).\,
  6. δ A = A x x - A y y . \delta A=A_{xx}-A_{yy}.\,
  7. 6 {\sqrt{6}}
  8. 2 {2}
  9. H ^ = 𝐚 ^ 𝐀 𝐛 ^ = A x x a ^ x b ^ x + A y y a ^ y b ^ y + A z z a ^ z b ^ z \hat{H}=\hat{\vec{\mathbf{a}}}\cdot\mathbf{A}\cdot\hat{\vec{\mathbf{b}}}=A_{xx% }\hat{a}_{x}\hat{b}_{x}+A_{yy}\hat{a}_{y}\hat{b}_{y}+A_{zz}\hat{a}_{z}\hat{b}_% {z}
  10. 𝐚 ^ 𝐀 𝐛 ^ = δ A 2 T ^ 2 , - 2 + δ A 2 T ^ 2 , 2 + Δ A 6 T ^ 2 , - 2 \hat{\vec{\mathbf{a}}}\cdot\mathbf{A}\cdot\hat{\vec{\mathbf{b}}}=\frac{\delta A% }{2}\hat{T}_{2,-2}+\frac{\delta A}{2}\hat{T}_{2,2}+\frac{\Delta A}{\sqrt{6}}% \hat{T}_{2,-2}
  11. R ^ ^ α , β , γ ( T ^ l , m ) = k = - 2 2 T ^ l , k 𝔇 k , m ( l ) ( α , β , γ ) \hat{\hat{R}}_{\alpha,\beta,\gamma}(\hat{T}_{l,m})=\sum_{k=-2}^{2}\hat{T}_{l,k% }\mathfrak{D}_{k,m}^{(l)}(\alpha,\beta,\gamma)
  12. 𝔇 k , m ( l ) ( α , β , γ ) \mathfrak{D}_{k,m}^{(l)}(\alpha,\beta,\gamma)
  13. ( α , β , γ ) (\alpha,\beta,\gamma)
  14. T ^ 2 , 2 = + 1 2 a ^ + b ^ + \hat{T}_{2,2}=+\frac{1}{2}\hat{a}_{+}\hat{b}_{+}
  15. T ^ 2 , 1 = - 1 2 ( a ^ z b ^ + + a ^ + b ^ z ) \hat{T}_{2,1}=-\frac{1}{2}(\hat{a}_{z}\hat{b}_{+}+\hat{a}_{+}\hat{b}_{z})
  16. T ^ 2 , 0 = + 2 3 ( a ^ z b ^ z - 1 4 ( a ^ + b ^ - + a ^ - b ^ + ) ) \hat{T}_{2,0}=+\sqrt{\frac{2}{3}}(\hat{a}_{z}\hat{b}_{z}-\frac{1}{4}(\hat{a}_{% +}\hat{b}_{-}+\hat{a}_{-}\hat{b}_{+}))
  17. T ^ 2 , - 1 = + 1 2 ( a ^ z b ^ - + a ^ - b ^ z ) \hat{T}_{2,-1}=+\frac{1}{2}(\hat{a}_{z}\hat{b}_{-}+\hat{a}_{-}\hat{b}_{z})
  18. T ^ 2 , - 2 = + 1 2 a ^ - b ^ - \hat{T}_{2,-2}=+\frac{1}{2}\hat{a}_{-}\hat{b}_{-}

Azumaya_algebra.html

  1. A R A A\otimes_{R}A^{\circ}
  2. a b a\otimes b
  3. A 1 End ( E 1 ) A 2 End ( E 2 ) , A_{1}\otimes\mathrm{End}(E_{1})\simeq A_{2}\otimes\mathrm{End}(E_{2}),

Back_scattering_alignment.html

  1. S F S A = [ 1 0 0 - 1 ] S B S A S_{FSA}=\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}S_{BSA}

Backward-wave_oscillator.html

  1. n = - + E n e j ( ω t - k n z ) \sum_{n=-\infty}^{+\infty}{E_{n}}e^{j({\omega}t-{k_{n}}z)}
  2. Δ \Delta
  3. Δ \Delta
  4. Δ \Delta
  5. Δ \Delta

Baguenaudier.html

  1. a ( n ) = { 2 n + 1 - 2 3 , when n is even, 2 n + 1 - 1 3 , when n is odd. a(n)=\begin{cases}\frac{2^{n+1}-2}{3},&\,\text{when }n\,\text{ is even,}\\ \frac{2^{n+1}-1}{3},&\,\text{when }n\,\text{ is odd.}\end{cases}

Baire_space_(set_theory).html

  1. 𝒩 \mathcal{N}

Balance_spring.html

  1. κ \kappa\,
  2. I I\,
  3. T T\,
  4. T = 2 π I / κ T=2\pi\sqrt{I/\kappa}\,

Ballistic_coefficient.html

  1. B C P h y s i c s = M C d A = ρ l C d BC_{Physics}=\frac{M}{C_{d}\cdot A}=\frac{\rho\cdot l}{C_{d}}
  2. ρ \rho
  3. B C P r o j e c t i l e = m d 2 i BC_{Projectile}=\frac{m}{d^{2}\cdot i}
  4. i = 2 n 4 n - 1 n i=\frac{2}{n}\cdot\sqrt{\frac{4n-1}{n}}
  5. n = ( 4 l 2 + 1 ) 4 n=\frac{(4\cdot l^{2}+1)}{4}
  6. C d = 8 ρ v 2 π d 2 C_{d}=\frac{8}{\rho\cdot v^{2}\cdot\pi\cdot d^{2}}
  7. ρ \rho
  8. i = C G C p i=\frac{C_{G}}{C_{p}}
  9. B C S m a l l a r m s = S D i BC_{Smallarms}=\frac{SD}{i}
  10. k k
  11. k > i k>i
  12. k k
  13. i i
  14. K v K_{v}
  15. δ w d 2 \tfrac{\delta w}{d^{2}}
  16. δ C \tfrac{\delta}{C}
  17. C C
  18. k k
  19. C C
  20. i i
  21. m d 2 p 0 p \tfrac{m}{d^{2}}\cdot\tfrac{p_{0}}{p}
  22. m d 2 i \tfrac{m}{d^{2}i}
  23. S D i \tfrac{SD}{i}
  24. B C BC
  25. B C = S D i = 1 1 = 1.00 BC=\tfrac{SD}{i}=\tfrac{1}{1}=1.00

Banach_limit.html

  1. ϕ : \phi:\ell^{\infty}\to\mathbb{C}
  2. \ell^{\infty}
  3. x = ( x n ) x=(x_{n})
  4. y = ( y n ) y=(y_{n})
  5. \ell^{\infty}
  6. α \alpha
  7. ϕ ( α x + y ) = α ϕ ( x ) + ϕ ( y ) \phi(\alpha x+y)=\alpha\phi(x)+\phi(y)
  8. x n 0 x_{n}\geq 0
  9. n n\in\mathbb{N}
  10. ϕ ( x ) 0 \phi(x)\geq 0
  11. ϕ ( x ) = ϕ ( S x ) \phi(x)=\phi(Sx)
  12. S S
  13. ( S x ) n = x n + 1 (Sx)_{n}=x_{n+1}
  14. x x
  15. ϕ ( x ) = lim x \phi(x)=\lim x
  16. ϕ \phi
  17. lim x : c \lim x:c\mapsto\mathbb{C}
  18. c c\subset\ell^{\infty}
  19. \mathbb{C}
  20. lim inf n x n ϕ ( x ) lim sup n x n \liminf_{n\to\infty}x_{n}\leq\phi(x)\leq\limsup_{n\to\infty}x_{n}
  21. x = ( 1 , 0 , 1 , 0 , ) x=(1,0,1,0,\ldots)
  22. x + S ( x ) = ( 1 , 1 , 1 , ) x+S(x)=(1,1,1,\ldots)
  23. 2 ϕ ( x ) = ϕ ( x ) + ϕ ( S x ) = ϕ ( x + S x ) = ϕ ( ( 1 , 1 , 1 , ) ) = lim ( ( 1 , 1 , 1 , ) ) = 1 2\phi(x)=\phi(x)+\phi(Sx)=\phi(x+Sx)=\phi((1,1,1,\ldots))=\lim((1,1,1,\ldots))=1
  24. 1 / 2 1/2
  25. x x
  26. ϕ \phi
  27. ϕ ( x ) \phi(x)
  28. x = ( x n ) x=(x_{n})
  29. c c\subset\ell^{\infty}
  30. x x
  31. 1 \ell^{1}
  32. < 1 , > {<}\ell^{1},\ell^{\infty}{>}
  33. \ell^{\infty}
  34. 1 \ell^{1}
  35. 1 \ell^{1}
  36. \ell^{\infty}
  37. \ell^{\infty}
  38. \ell^{\infty}
  39. 1 \ell^{1}
  40. \ell^{\infty}

Banach_manifold.html

  1. φ j φ i - 1 : φ i ( U i U j ) φ j ( U i U j ) \varphi_{j}\circ\varphi_{i}^{-1}:\varphi_{i}(U_{i}\cap U_{j})\to\varphi_{j}(U_% {i}\cap U_{j})
  2. d r ( φ j φ i - 1 ) : φ i ( U i U j ) Lin ( E i r ; E j ) \mathrm{d}^{r}\big(\varphi_{j}\circ\varphi_{i}^{-1}\big):\varphi_{i}(U_{i}\cap U% _{j})\to\mathrm{Lin}\big(E_{i}^{r};E_{j}\big)
  3. φ j φ i - 1 : φ i ( U i U j ) φ j ( U i U j ) \varphi_{j}\circ\varphi_{i}^{-1}:\varphi_{i}(U_{i}\cap U_{j})\to\varphi_{j}(U_% {i}\cap U_{j})
  4. φ i φ - 1 : φ ( U U i ) φ i ( U U i ) \varphi_{i}\circ\varphi^{-1}:\varphi(U\cap U_{i})\to\varphi_{i}(U\cap U_{i})

Bar_product.html

  1. C 1 C 2 = { ( c 1 c 1 + c 2 ) : c 1 C 1 , c 2 C 2 } , C_{1}\mid C_{2}=\{(c_{1}\mid c_{1}+c_{2}):c_{1}\in C_{1},c_{2}\in C_{2}\},
  2. rank ( C 1 C 2 ) = rank ( C 1 ) + rank ( C 2 ) \operatorname{rank}(C_{1}\mid C_{2})=\operatorname{rank}(C_{1})+\operatorname{% rank}(C_{2})\,
  3. { x 1 , , x k } \{x_{1},\ldots,x_{k}\}
  4. C 1 C_{1}
  5. { y 1 , , y l } \{y_{1},\ldots,y_{l}\}
  6. C 2 C_{2}
  7. { ( x i x i ) 1 i k } { ( 0 y j ) 1 j l } \{(x_{i}\mid x_{i})\mid 1\leq i\leq k\}\cup\{(0\mid y_{j})\mid 1\leq j\leq l\}
  8. C 1 C 2 C_{1}\mid C_{2}
  9. w ( C 1 C 2 ) = min { 2 w ( C 1 ) , w ( C 2 ) } . w(C_{1}\mid C_{2})=\min\{2w(C_{1}),w(C_{2})\}.\,
  10. c 1 C 1 c_{1}\in C_{1}
  11. ( c 1 c 1 + 0 ) C 1 C 2 (c_{1}\mid c_{1}+0)\in C_{1}\mid C_{2}
  12. 2 w ( c 1 ) 2w(c_{1})
  13. ( 0 c 2 ) C 1 C 2 (0\mid c_{2})\in C_{1}\mid C_{2}
  14. c 2 C 2 c_{2}\in C_{2}
  15. w ( c 2 ) w(c_{2})
  16. c 1 C 1 , c 2 C 2 c_{1}\in C_{1},c_{2}\in C_{2}
  17. w ( C 1 C 2 ) min { 2 w ( C 1 ) , w ( C 2 ) } w(C_{1}\mid C_{2})\leq\min\{2w(C_{1}),w(C_{2})\}
  18. c 1 C 1 c_{1}\in C_{1}
  19. c 2 C 2 c_{2}\in C_{2}
  20. c 2 0 c_{2}\not=0
  21. w ( c 1 c 1 + c 2 ) \displaystyle w(c_{1}\mid c_{1}+c_{2})
  22. c 2 = 0 c_{2}=0
  23. w ( c 1 c 1 + c 2 ) \displaystyle w(c_{1}\mid c_{1}+c_{2})
  24. w ( C 1 C 2 ) min { 2 w ( C 1 ) , w ( C 2 ) } w(C_{1}\mid C_{2})\geq\min\{2w(C_{1}),w(C_{2})\}

Barnes_G-function.html

  1. G ( 1 + z ) = ( 2 π ) z / 2 exp ( - z + z 2 ( 1 + γ ) 2 ) k = 1 { ( 1 + z k ) k exp ( z 2 2 k - z ) } G(1+z)=(2\pi)^{z/2}\,\text{exp}\left(-\frac{z+z^{2}(1+\gamma)}{2}\right)\,% \prod_{k=1}^{\infty}\left\{\left(1+\frac{z}{k}\right)^{k}\,\text{exp}\left(% \frac{z^{2}}{2k}-z\right)\right\}
  2. γ \,\gamma\,
  3. G ( z + 1 ) = Γ ( z ) G ( z ) G(z+1)=\Gamma(z)\,G(z)
  4. Γ ( z + 1 ) = z Γ ( z ) \Gamma(z+1)=z\,\Gamma(z)
  5. G ( n ) = { 0 if n = - 1 , - 2 , i = 0 n - 2 i ! if n = 0 , 1 , 2 , G(n)=\begin{cases}0&\,\text{if }n=-1,-2,\dots\\ \prod_{i=0}^{n-2}i!&\,\text{if }n=0,1,2,\dots\end{cases}
  6. G ( 0 ) = G ( 1 ) = 1 \,G(0)=G(1)=1\,
  7. G ( n ) = ( Γ ( n ) ) n - 1 K ( n ) G(n)=\frac{(\Gamma(n))^{n-1}}{K(n)}
  8. Γ ( x ) \,\Gamma(x)\,
  9. d 3 d x 3 G ( x ) 0 \,\frac{d^{3}}{dx^{3}}G(x)\geq 0\,
  10. log G ( 1 - z ) = log G ( 1 + z ) - z log 2 π + 0 z π x cot π x d x . \log G(1-z)=\log G(1+z)-z\log 2\pi+\int_{0}^{z}\pi x\cot\pi x\,dx.
  11. 2 π log ( G ( 1 - z ) G ( 1 + z ) ) = 2 π z log ( sin π z π ) + Cl 2 ( 2 π z ) 2\pi\log\left(\frac{G(1-z)}{G(1+z)}\right)=2\pi z\log\left(\frac{\sin\pi z}{% \pi}\right)+\,\text{Cl}_{2}(2\pi z)
  12. L c ( z ) \,Lc(z)\,
  13. ( d / d x ) log ( sin π x ) = π cot π x \,(d/dx)\log(\sin\pi x)=\pi\cot\pi x\,
  14. L c ( z ) = 0 z π x log ( sin π x ) d x = z log ( sin π z ) - 0 z log ( sin π x ) d x = Lc(z)=\int_{0}^{z}\pi x\log(\sin\pi x)\,dx=z\log(\sin\pi z)-\int_{0}^{z}\log(% \sin\pi x)\,dx=
  15. z log ( sin π z ) - 0 z [ log ( 2 sin π x ) - log 2 ] d x = z\log(\sin\pi z)-\int_{0}^{z}\Bigg[\log(2\sin\pi x)-\log 2\Bigg]\,dx=
  16. z log ( 2 sin π z ) - 0 z log ( 2 sin π x ) d x z\log(2\sin\pi z)-\int_{0}^{z}\log(2\sin\pi x)\,dx
  17. y = 2 π x d x = d y / ( 2 π ) \,y=2\pi x\Rightarrow dx=dy/(2\pi)\,
  18. z log ( 2 sin π z ) - 1 2 π 0 2 π z log ( 2 sin π y 2 ) d y z\log(2\sin\pi z)-\frac{1}{2\pi}\int_{0}^{2\pi z}\log\left(2\sin\pi\frac{y}{2}% \right)\,dy
  19. Cl 2 ( θ ) = - 0 θ log | 2 sin x 2 | d x \,\text{Cl}_{2}(\theta)=-\int_{0}^{\theta}\log\Bigg|2\sin\frac{x}{2}\Bigg|\,dx
  20. 0 < θ < 2 π \,0<\theta<2\pi\,
  21. L c ( z ) = z log ( 2 sin π z ) + 1 2 π Cl 2 ( 2 π z ) Lc(z)=z\log(2\sin\pi z)+\frac{1}{2\pi}\,\,\text{Cl}_{2}(2\pi z)
  22. 2 π log ( G ( 1 - z ) G ( 1 + z ) ) = 2 π z log ( sin π z π ) + Cl 2 ( 2 π z ) . 2\pi\log\left(\frac{G(1-z)}{G(1+z)}\right)=2\pi z\log\left(\frac{\sin\pi z}{% \pi}\right)+\,\text{Cl}_{2}(2\pi z)\,.\,\Box
  23. G ( 1 + z ) = Γ ( z ) G ( z ) \,G(1+z)=\Gamma(z)\,G(z)\,
  24. 2 π \,2\pi\,
  25. log ( G ( 1 - z ) G ( z ) ) = z log ( sin π z π ) + log Γ ( z ) + 1 2 π Cl 2 ( 2 π z ) \log\left(\frac{G(1-z)}{G(z)}\right)=z\log\left(\frac{\sin\pi z}{\pi}\right)+% \log\Gamma(z)+\frac{1}{2\pi}\,\text{Cl}_{2}(2\pi z)
  26. log ( G ( 1 2 + z ) G ( 1 2 - z ) ) = \log\left(\frac{G\left(\frac{1}{2}+z\right)}{G\left(\frac{1}{2}-z\right)}% \right)=
  27. log Γ ( 1 2 - z ) + B 1 ( z ) log 2 π - 1 2 log 2 + π 0 z B 1 ( x ) tan π x d x \log\Gamma\left(\frac{1}{2}-z\right)+B_{1}(z)\log 2\pi-\frac{1}{2}\log 2+\pi% \int_{0}^{z}B_{1}(x)\tan\pi x\,dx
  28. log G ( 1 + z ) = z 2 log 2 π - ( z + ( 1 + γ ) z 2 2 ) + k = 2 ( - 1 ) k ζ ( k ) k + 1 z k + 1 \log G(1+z)=\frac{z}{2}\log 2\pi-\left(\frac{z+(1+\gamma)z^{2}}{2}\right)+\sum% _{k=2}^{\infty}(-1)^{k}\frac{\zeta(k)}{k+1}z^{k+1}
  29. 0 < z < 1 \,0<z<1\,
  30. ζ ( x ) \,\zeta(x)\,
  31. ζ ( x ) = k = 1 1 k x \zeta(x)=\sum_{k=1}^{\infty}\frac{1}{k^{x}}
  32. G ( 1 + z ) = exp [ z 2 log 2 π - ( z + ( 1 + γ ) z 2 2 ) + k = 2 ( - 1 ) k ζ ( k ) k + 1 z k + 1 ] = G(1+z)=\exp\left[\frac{z}{2}\log 2\pi-\left(\frac{z+(1+\gamma)z^{2}}{2}\right)% +\sum_{k=2}^{\infty}(-1)^{k}\frac{\zeta(k)}{k+1}z^{k+1}\right]=
  33. ( 2 π ) z / 2 exp ( - z + ( 1 + γ ) z 2 2 ) exp [ k = 2 ( - 1 ) k ζ ( k ) k + 1 z k + 1 ] (2\pi)^{z/2}\,\text{exp}\left(-\frac{z+(1+\gamma)z^{2}}{2}\right)\exp\left[% \sum_{k=2}^{\infty}(-1)^{k}\frac{\zeta(k)}{k+1}z^{k+1}\right]
  34. exp [ k = 2 ( - 1 ) k ζ ( k ) k + 1 z k + 1 ] = k = 1 { ( 1 + z k ) k exp ( z 2 2 k - z ) } \exp\left[\sum_{k=2}^{\infty}(-1)^{k}\frac{\zeta(k)}{k+1}z^{k+1}\right]=\prod_% {k=1}^{\infty}\left\{\left(1+\frac{z}{k}\right)^{k}\,\text{exp}\left(\frac{z^{% 2}}{2k}-z\right)\right\}
  35. G ( n z ) = K ( n ) n n 2 z 2 / 2 - n z ( 2 π ) - n 2 - n 2 z i = 0 n - 1 j = 0 n - 1 G ( z + i + j n ) G(nz)=K(n)n^{n^{2}z^{2}/2-nz}(2\pi)^{-\frac{n^{2}-n}{2}z}\prod_{i=0}^{n-1}% \prod_{j=0}^{n-1}G\left(z+\frac{i+j}{n}\right)
  36. K ( n ) K(n)
  37. K ( n ) = e - ( n 2 - 1 ) ζ ( - 1 ) n 5 12 ( 2 π ) ( n - 1 ) / 2 = ( A e - 1 12 ) n 2 - 1 n 5 12 ( 2 π ) ( n - 1 ) / 2 . K(n)=e^{-(n^{2}-1)\zeta^{\prime}(-1)}\cdot n^{\frac{5}{12}}\cdot(2\pi)^{(n-1)/% 2}\,=\,(Ae^{-\frac{1}{12}})^{n^{2}-1}\cdot n^{\frac{5}{12}}\cdot(2\pi)^{(n-1)/% 2}.
  38. ζ \zeta^{\prime}
  39. A A
  40. log G ( z + 1 ) = \log G(z+1)=
  41. 1 12 - log A + z 2 log 2 π + ( z 2 2 - 1 12 ) log z - 3 z 2 4 + k = 1 N B 2 k + 2 4 k ( k + 1 ) z 2 k + O ( 1 z 2 N + 2 ) . \frac{1}{12}~{}-~{}\log A~{}+~{}\frac{z}{2}\log 2\pi~{}+~{}\left(\frac{z^{2}}{% 2}-\frac{1}{12}\right)\log z~{}-~{}\frac{3z^{2}}{4}~{}+~{}\sum_{k=1}^{N}\frac{% B_{2k+2}}{4k\left(k+1\right)z^{2k}}~{}+~{}O\left(\frac{1}{z^{2N+2}}\right).
  42. B k B_{k}
  43. A A
  44. B 2 k B_{2k}
  45. ( - 1 ) k + 1 B k (-1)^{k+1}B_{k}
  46. z z
  47. | z | |z|
  48. 0 z log Γ ( x ) d x = z ( 1 - z ) 2 + z 2 log 2 π + z log Γ ( z ) - log G ( 1 + z ) \int_{0}^{z}\log\Gamma(x)\,dx=\frac{z(1-z)}{2}+\frac{z}{2}\log 2\pi+z\log% \Gamma(z)-\log G(1+z)
  49. z log Γ ( z ) - log G ( 1 + z ) z\log\Gamma(z)-\log G(1+z)
  50. 1 Γ ( z ) = z e γ z k = 1 { ( 1 + z k ) e - z / k } \frac{1}{\Gamma(z)}=ze^{\gamma z}\prod_{k=1}^{\infty}\left\{\left(1+\frac{z}{k% }\right)e^{-z/k}\right\}
  51. γ \,\gamma\,
  52. z log Γ ( z ) - log G ( 1 + z ) = - z log ( 1 Γ ( z ) ) - log G ( 1 + z ) = z\log\Gamma(z)-\log G(1+z)=-z\log\left(\frac{1}{\Gamma(z)}\right)-\log G(1+z)=
  53. - z [ log z + γ z + k = 1 { log ( 1 + z k ) - z k } ] -z\left[\log z+\gamma z+\sum_{k=1}^{\infty}\Bigg\{\log\left(1+\frac{z}{k}% \right)-\frac{z}{k}\Bigg\}\right]
  54. - [ z 2 log 2 π - z 2 - z 2 2 - z 2 γ 2 + k = 1 { k log ( 1 + z k ) + z 2 2 k - z } ] -\left[\frac{z}{2}\log 2\pi-\frac{z}{2}-\frac{z^{2}}{2}-\frac{z^{2}\gamma}{2}+% \sum_{k=1}^{\infty}\Bigg\{k\log\left(1+\frac{z}{k}\right)+\frac{z^{2}}{2k}-z% \Bigg\}\right]
  55. k = 1 { ( k + z ) log ( 1 + z k ) - z 2 2 k - z } = \sum_{k=1}^{\infty}\Bigg\{(k+z)\log\left(1+\frac{z}{k}\right)-\frac{z^{2}}{2k}% -z\Bigg\}=
  56. - z log z - z 2 log 2 π + z 2 + z 2 2 - z 2 γ 2 - z log Γ ( z ) + log G ( 1 + z ) -z\log z-\frac{z}{2}\log 2\pi+\frac{z}{2}+\frac{z^{2}}{2}-\frac{z^{2}\gamma}{2% }-z\log\Gamma(z)+\log G(1+z)
  57. [ 0 , z ] \,[0,\,z]\,
  58. 0 z log Γ ( x ) d x = - 0 z log ( 1 Γ ( x ) ) d x = \int_{0}^{z}\log\Gamma(x)\,dx=-\int_{0}^{z}\log\left(\frac{1}{\Gamma(x)}\right% )\,dx=
  59. - ( z log z - z ) - z 2 γ 2 - k = 1 { ( k + z ) log ( 1 + z k ) - z 2 2 k - z } -(z\log z-z)-\frac{z^{2}\gamma}{2}-\sum_{k=1}^{\infty}\Bigg\{(k+z)\log\left(1+% \frac{z}{k}\right)-\frac{z^{2}}{2k}-z\Bigg\}
  60. 0 z log Γ ( x ) d x = z ( 1 - z ) 2 + z 2 log 2 π + z log Γ ( z ) - log G ( 1 + z ) . \int_{0}^{z}\log\Gamma(x)\,dx=\frac{z(1-z)}{2}+\frac{z}{2}\log 2\pi+z\log% \Gamma(z)-\log G(1+z)\,.\,\Box
  61. ( 2 , ) (2,\mathbb{Z})

Bartlett's_test.html

  1. n i n_{i}
  2. S i 2 S_{i}^{2}
  3. χ 2 = ( N - k ) ln ( S p 2 ) - i = 1 k ( n i - 1 ) ln ( S i 2 ) 1 + 1 3 ( k - 1 ) ( i = 1 k ( 1 n i - 1 ) - 1 N - k ) \chi^{2}=\frac{(N-k)\ln(S_{p}^{2})-\sum_{i=1}^{k}(n_{i}-1)\ln(S_{i}^{2})}{1+% \frac{1}{3(k-1)}\left(\sum_{i=1}^{k}(\frac{1}{n_{i}-1})-\frac{1}{N-k}\right)}
  4. N = i = 1 k n i N=\sum_{i=1}^{k}n_{i}
  5. S p 2 = 1 N - k i ( n i - 1 ) S i 2 S_{p}^{2}=\frac{1}{N-k}\sum_{i}(n_{i}-1)S_{i}^{2}
  6. χ k - 1 2 \chi^{2}_{k-1}
  7. χ 2 > χ k - 1 , α 2 \chi^{2}>\chi^{2}_{k-1,\alpha}
  8. χ k - 1 , α 2 \chi^{2}_{k-1,\alpha}
  9. χ k - 1 2 \chi^{2}_{k-1}
  10. χ k - 1 2 \chi^{2}_{k-1}

Basic_hypergeometric_series.html

  1. ϕ k j [ a 1 a 2 a j b 1 b 2 b k ; q , z ] = n = 0 ( a 1 , a 2 , , a j ; q ) n ( b 1 , b 2 , , b k , q ; q ) n ( ( - 1 ) n q ( n 2 ) ) 1 + k - j z n \;{}_{j}\phi_{k}\left[\begin{matrix}a_{1}&a_{2}&\ldots&a_{j}\\ b_{1}&b_{2}&\ldots&b_{k}\end{matrix};q,z\right]=\sum_{n=0}^{\infty}\frac{(a_{1% },a_{2},\ldots,a_{j};q)_{n}}{(b_{1},b_{2},\ldots,b_{k},q;q)_{n}}\left((-1)^{n}% q^{n\choose 2}\right)^{1+k-j}z^{n}
  2. ( a 1 , a 2 , , a m ; q ) n = ( a 1 ; q ) n ( a 2 ; q ) n ( a m ; q ) n (a_{1},a_{2},\ldots,a_{m};q)_{n}=(a_{1};q)_{n}(a_{2};q)_{n}\ldots(a_{m};q)_{n}
  3. ( a ; q ) n = k = 0 n - 1 ( 1 - a q k ) = ( 1 - a ) ( 1 - a q ) ( 1 - a q 2 ) ( 1 - a q n - 1 ) . (a;q)_{n}=\prod_{k=0}^{n-1}(1-aq^{k})=(1-a)(1-aq)(1-aq^{2})\cdots(1-aq^{n-1}).
  4. ϕ k k + 1 [ a 1 a 2 a k a k + 1 b 1 b 2 b k ; q , z ] = n = 0 ( a 1 , a 2 , , a k + 1 ; q ) n ( b 1 , b 2 , , b k , q ; q ) n z n . \;{}_{k+1}\phi_{k}\left[\begin{matrix}a_{1}&a_{2}&\ldots&a_{k}&a_{k+1}\\ b_{1}&b_{2}&\ldots&b_{k}\end{matrix};q,z\right]=\sum_{n=0}^{\infty}\frac{(a_{1% },a_{2},\ldots,a_{k+1};q)_{n}}{(b_{1},b_{2},\ldots,b_{k},q;q)_{n}}z^{n}.
  5. ψ k j [ a 1 a 2 a j b 1 b 2 b k ; q , z ] = n = - ( a 1 , a 2 , , a j ; q ) n ( b 1 , b 2 , , b k ; q ) n ( ( - 1 ) n q ( n 2 ) ) k - j z n . \;{}_{j}\psi_{k}\left[\begin{matrix}a_{1}&a_{2}&\ldots&a_{j}\\ b_{1}&b_{2}&\ldots&b_{k}\end{matrix};q,z\right]=\sum_{n=-\infty}^{\infty}\frac% {(a_{1},a_{2},\ldots,a_{j};q)_{n}}{(b_{1},b_{2},\ldots,b_{k};q)_{n}}\left((-1)% ^{n}q^{n\choose 2}\right)^{k-j}z^{n}.
  6. ψ k k [ a 1 a 2 a k b 1 b 2 b k ; q , z ] = n = - ( a 1 , a 2 , , a k ; q ) n ( b 1 , b 2 , , b k ; q ) n z n . \;{}_{k}\psi_{k}\left[\begin{matrix}a_{1}&a_{2}&\ldots&a_{k}\\ b_{1}&b_{2}&\ldots&b_{k}\end{matrix};q,z\right]=\sum_{n=-\infty}^{\infty}\frac% {(a_{1},a_{2},\ldots,a_{k};q)_{n}}{(b_{1},b_{2},\ldots,b_{k};q)_{n}}z^{n}.
  7. z 1 - q 1 / 2 2 ϕ 1 [ q q 1 / 2 q 3 / 2 ; q , z ] = z 1 - q 1 / 2 + z 2 1 - q 3 / 2 + z 3 1 - q 5 / 2 + \frac{z}{1-q^{1/2}}\;_{2}\phi_{1}\left[\begin{matrix}q\;q^{1/2}\\ q^{3/2}\end{matrix}\;;q,z\right]=\frac{z}{1-q^{1/2}}+\frac{z^{2}}{1-q^{3/2}}+% \frac{z^{3}}{1-q^{5/2}}+\ldots
  8. ϕ 1 2 [ q - 1 - q ; q , z ] = 1 + 2 z 1 + q + 2 z 2 1 + q 2 + 2 z 3 1 + q 3 + . \;{}_{2}\phi_{1}\left[\begin{matrix}q\;-1\\ -q\end{matrix}\;;q,z\right]=1+\frac{2z}{1+q}+\frac{2z^{2}}{1+q^{2}}+\frac{2z^{% 3}}{1+q^{3}}+\ldots.
  9. ϕ 0 1 ( a ; q , z ) = ( a z ; q ) ( z ; q ) = n = 0 1 - a q n z 1 - q n z \;{}_{1}\phi_{0}(a;q,z)=\frac{(az;q)_{\infty}}{(z;q)_{\infty}}=\prod_{n=0}^{% \infty}\frac{1-aq^{n}z}{1-q^{n}z}
  10. ϕ 0 1 ( a ; q , z ) = 1 - a z 1 - z 1 ϕ 0 ( a ; q , q z ) . \;{}_{1}\phi_{0}(a;q,z)=\frac{1-az}{1-z}\;_{1}\phi_{0}(a;q,qz).
  11. ψ 1 1 [ a b ; q , z ] = n = - ( a ; q ) n ( b ; q ) n z n = ( b / a , q , q / a z , a z ; q ) ( b , b / a z , q / a , z ; q ) \;{}_{1}\psi_{1}\left[\begin{matrix}a\\ b\end{matrix};q,z\right]=\sum_{n=-\infty}^{\infty}\frac{(a;q)_{n}}{(b;q)_{n}}z% ^{n}=\frac{(b/a,q,q/az,az;q)_{\infty}}{(b,b/az,q/a,z;q)_{\infty}}
  12. ψ 6 6 \;{}_{6}\psi_{6}
  13. n = - q n ( n + 1 ) / 2 z n = ( q ; q ) ( - 1 / z ; q ) ( - z q ; q ) . \sum_{n=-\infty}^{\infty}q^{n(n+1)/2}z^{n}=(q;q)_{\infty}\;(-1/z;q)_{\infty}\;% (-zq;q)_{\infty}.
  14. A ( z ; q ) = def 1 1 + z n = 0 ( z ; q ) n ( - zq ; q ) n z n = n = 0 ( - 1 ) n z 2 n q n 2 . A(z;q)\stackrel{\rm{def}}{=}\frac{1}{1+z}\sum_{n=0}^{\infty}\frac{(z;q)_{n}}{(% -zq;q)_{n}}z^{n}=\sum_{n=0}^{\infty}(-1)^{n}z^{2n}q^{n^{2}}.
  15. ϕ 1 2 ( a , b ; c ; q , z ) = - 1 2 π i ( a , b ; q ) ( q , c ; q ) - i i ( q q s , c q s ; q ) ( a q s , b q s ; q ) π ( - z ) s sin π s d s {}_{2}\phi_{1}(a,b;c;q,z)=\frac{-1}{2\pi i}\frac{(a,b;q)_{\infty}}{(q,c;q)_{% \infty}}\int_{-i\infty}^{i\infty}\frac{(qq^{s},cq^{s};q)_{\infty}}{(aq^{s},bq^% {s};q)_{\infty}}\frac{\pi(-z)^{s}}{\sin\pi s}ds
  16. ( a q s , b q s ; q ) (aq^{s},bq^{s};q)_{\infty}
  17. ψ 1 1 \,{}_{1}\psi_{1}

Batchelor_vortex.html

  1. U U
  2. V V
  3. W W
  4. ( x , r , θ ) (x,r,\theta)
  5. U ( r ) = U + W 0 ( R / R 0 ) 2 e - ( r / R ) 2 , V ( r ) = 0 , W ( r ) = q W 0 1 - e - ( r / R ) 2 ( r / R 0 ) . \begin{array}[]{cl}U(r)&=U_{\infty}+\frac{W_{0}}{(R/R_{0})^{2}}e^{-(r/R)^{2}},% \\ V(r)&=0,\\ W(r)&=qW_{0}\frac{1-e^{-(r/R)^{2}}}{(r/R_{0})}.\end{array}
  6. U U_{\infty}
  7. W 0 W_{0}
  8. R 0 R_{0}
  9. R = R ( t ) = R 0 2 + 4 ν t R=R(t)=\sqrt{R_{0}^{2}+4\nu t}
  10. R 0 R_{0}
  11. ν \nu
  12. q q
  13. r r
  14. R 0 / W 0 R_{0}/W_{0}
  15. { U ( r ) = a + 1 1 + 4 t / R e e - r 2 / ( 1 + 4 t / R e ) , V ( r ) = 0 , W ( r ) = q 1 - e - r 2 / ( 1 + 4 t / R e ) r , \left\{\begin{array}[]{cl}U(r)&=a+\displaystyle{\frac{1}{1+4t/Re}e^{-r^{2}/(1+% 4t/Re)}},\\ V(r)&=0,\\ W(r)&=q\displaystyle{\frac{1-e^{-r^{2}/(1+4t/Re)}}{r}},\end{array}\right.
  16. a = U / W 0 a=U_{\infty}/W_{0}
  17. R e Re
  18. a = 0 a=0
  19. W Θ ( r ) = Γ 2 π r ( 1 - e - r 2 / R c 2 ) W_{\Theta}(r)=\frac{\Gamma}{2\pi r}\left(1-e^{-r^{2}/R_{c}^{2}}\right)
  20. Γ \Gamma

Bathochromic_shift.html

  1. Δ λ = λ observed state2 - λ observed state1 \Delta\lambda=\lambda^{\mathrm{state2}}_{\mathrm{observed}}-\lambda^{\mathrm{% state1}}_{\mathrm{observed}}
  2. λ \lambda
  3. λ observed state2 > λ observed state1 \lambda^{\mathrm{state2}}_{\mathrm{observed}}>\lambda^{\mathrm{state1}}_{% \mathrm{observed}}

Batting_park_factor.html

  1. P F = 100 * ( h o m e R S + h o m e R A h o m e G r o a d R S + r o a d R A r o a d G ) PF=100*({{homeRS+homeRA\over homeG}\over{roadRS+roadRA\over roadG}})

Bauer–Fike_theorem.html

  1. Λ Λ
  2. p p
  3. κ p ( X ) = X p X - 1 p . \kappa_{p}(X)=\|X\|_{p}\left\|X^{-1}\right\|_{p}.
  4. μ μ
  5. A + δ A A+δA
  6. λ σ ( A ) λ∈σ(A)
  7. | λ - μ | κ p ( V ) δ A p |\lambda-\mu|\leq\kappa_{p}(V)\|\delta A\|_{p}
  8. μ σ ( A ) μ∉σ(A)
  9. λ = μ λ=μ
  10. μ μ
  11. A + δ A A+δA
  12. d e t ( A + δ A μ I ) = 0 det(A+δA−μI)=0
  13. 0 = det ( A + δ A - μ I ) = det ( V - 1 ) det ( A + δ A - μ I ) det ( V ) = det ( V - 1 ( A + δ A - μ I ) V ) = det ( V - 1 A V + V - 1 δ A V - V - 1 μ I V ) = det ( Λ + V - 1 δ A V - μ I ) = det ( Λ - μ I ) det ( ( Λ - μ I ) - 1 V - 1 δ A V + I ) \begin{aligned}\displaystyle 0&\displaystyle=\det(A+\delta A-\mu I)\\ &\displaystyle=\det(V^{-1})\det(A+\delta A-\mu I)\det(V)\\ &\displaystyle=\det\left(V^{-1}(A+\delta A-\mu I)V\right)\\ &\displaystyle=\det\left(V^{-1}AV+V^{-1}\delta AV-V^{-1}\mu IV\right)\\ &\displaystyle=\det\left(\Lambda+V^{-1}\delta AV-\mu I\right)\\ &\displaystyle=\det(\Lambda-\mu I)\det\left((\Lambda-\mu I)^{-1}V^{-1}\delta AV% +I\right)\\ \end{aligned}
  14. μ σ ( A ) μ∉σ(A)
  15. d e t ( Λ μ I ) 0 det(Λ−μI)≠0
  16. det ( ( Λ - μ I ) - 1 V - 1 δ A V + I ) = 0. \det\left((\Lambda-\mu I)^{-1}V^{-1}\delta AV+I\right)=0.
  17. 1 −1
  18. ( Λ - μ I ) - 1 V - 1 δ A V . (\Lambda-\mu I)^{-1}V^{-1}\delta AV.
  19. p p
  20. λ λ
  21. A A
  22. 1 = | - 1 | ( Λ - μ I ) - 1 V - 1 δ A V p ( Λ - μ I ) - 1 p V - 1 p V p δ A p = ( Λ - μ I ) - 1 p κ p ( V ) δ A p 1=|-1|\leq\left\|(\Lambda-\mu I)^{-1}V^{-1}\delta AV\right\|_{p}\leq\left\|(% \Lambda-\mu I)^{-1}\right\|_{p}\left\|V^{-1}\right\|_{p}\|V\|_{p}\|\delta A\|_% {p}=\left\|(\Lambda-\mu I)^{-1}\right\|_{p}\ \kappa_{p}(V)\|\delta A\|_{p}
  23. p p
  24. ( Λ - μ I ) - 1 p = max s y m b o l x p 0 ( Λ - μ I ) - 1 s y m b o l x p s y m b o l x p = max λ σ ( A ) 1 | λ - μ | = 1 min λ σ ( A ) | λ - μ | \left\|\left(\Lambda-\mu I\right)^{-1}\right\|_{p}\ =\max_{\|symbol{x}\|_{p}% \neq 0}\frac{\left\|\left(\Lambda-\mu I\right)^{-1}symbol{x}\right\|_{p}}{\|% symbol{x}\|_{p}}=\max_{\lambda\in\sigma(A)}\frac{1}{|\lambda-\mu|}\ =\frac{1}{% \min_{\lambda\in\sigma(A)}|\lambda-\mu|}
  25. min λ σ ( A ) | λ - μ | κ p ( V ) δ A p . \min_{\lambda\in\sigma(A)}|\lambda-\mu|\leq\ \kappa_{p}(V)\|\delta A\|_{p}.
  26. A A
  27. λ σ ( A ) λ∈σ(A)
  28. | λ - λ a | κ p ( V ) s y m b o l r p s y m b o l v a p \left|\lambda-\lambda^{a}\right|\leq\kappa_{p}(V)\frac{\|symbol{r}\|_{p}}{% \left\|symbol{v}^{a}\right\|_{p}}
  29. s y m b o l v a = ( A - λ a I ) - 1 s y m b o l r = V ( D - λ a I ) - 1 V - 1 s y m b o l r symbol{v}^{a}=\left(A-\lambda^{a}I\right)^{-1}symbol{r}=V\left(D-\lambda^{a}I% \right)^{-1}V^{-1}symbol{r}
  30. A A
  31. p p
  32. s y m b o l v a p = V ( D - λ a I ) - 1 V - 1 s y m b o l r p V p ( D - λ a I ) - 1 p V - 1 p s y m b o l r p = κ p ( V ) ( D - λ a I ) - 1 p s y m b o l r p . \left\|symbol{v}^{a}\right\|_{p}=\left\|V\left(D-\lambda^{a}I\right)^{-1}V^{-1% }symbol{r}\right\|_{p}\leq\|V\|_{p}\left\|\left(D-\lambda^{a}I\right)^{-1}% \right\|_{p}\left\|V^{-1}\right\|_{p}\|symbol{r}\|_{p}=\kappa_{p}(V)\left\|% \left(D-\lambda^{a}I\right)^{-1}\right\|_{p}\|symbol{r}\|_{p}.
  33. ( D - λ a I ) - 1 \left(D-\lambda^{a}I\right)^{-1}
  34. p p
  35. ( D - λ a I ) - 1 p = max s y m b o l x p 0 ( D - λ a I ) - 1 s y m b o l x p s y m b o l x p = max λ σ ( A ) 1 | λ - λ a | = 1 min λ σ ( A ) | λ - λ a | \left\|\left(D-\lambda^{a}I\right)^{-1}\right\|_{p}=\max_{\|symbol{x}\|_{p}% \neq 0}\frac{\left\|\left(D-\lambda^{a}I\right)^{-1}symbol{x}\right\|_{p}}{\|% symbol{x}\|_{p}}=\max_{\lambda\in\sigma(A)}\frac{1}{\left|\lambda-\lambda^{a}% \right|}=\frac{1}{\min_{\lambda\in\sigma(A)}\left|\lambda-\lambda^{a}\right|}
  36. min λ σ ( A ) | λ - λ a | κ p ( V ) s y m b o l r p s y m b o l v a p . \min_{\lambda\in\sigma(A)}\left|\lambda-\lambda^{a}\right|\leq\kappa_{p}(V)% \frac{\|symbol{r}\|_{p}}{\left\|symbol{v}^{a}\right\|_{p}}.
  37. A A
  38. μ μ
  39. A + δ A A+δA
  40. λ σ ( A ) λ∈σ(A)
  41. | λ - μ | | λ | κ p ( V ) A - 1 δ A p \frac{|\lambda-\mu|}{|\lambda|}\leq\kappa_{p}(V)\left\|A^{-1}\delta A\right\|_% {p}
  42. A A
  43. | λ |\frac{λ}{−}
  44. λ λ
  45. μ μ
  46. A + δ A A+δA
  47. d e t ( A ) 0 det(A)≠0
  48. - A - 1 ( A + δ A ) s y m b o l v = - μ A - 1 s y m b o l v . -A^{-1}(A+\delta A)symbol{v}=-\mu A^{-1}symbol{v}.
  49. A a = μ A - 1 , ( δ A ) a = - A - 1 δ A A^{a}=\mu A^{-1},\qquad(\delta A)^{a}=-A^{-1}\delta A
  50. ( A a + ( δ A ) a - I ) s y m b o l v = s y m b o l 0 \left(A^{a}+(\delta A)^{a}-I\right)symbol{v}=symbol{0}
  51. 1 1
  52. 𝐯 \mathbf{v}
  53. A A
  54. 1 1
  55. min λ σ ( A ) | μ λ - 1 | = min λ σ ( A ) | λ - μ | | λ | κ p ( V ) A - 1 δ A p \min_{\lambda\in\sigma(A)}\left|\frac{\mu}{\lambda}-1\right|=\min_{\lambda\in% \sigma(A)}\frac{|\lambda-\mu|}{|\lambda|}\leq\kappa_{p}(V)\left\|A^{-1}\delta A% \right\|_{p}
  56. A A
  57. V V
  58. V 2 = V - 1 2 = 1 , \|V\|_{2}=\left\|V^{-1}\right\|_{2}=1,
  59. λ σ ( A ) : | λ - μ | δ A 2 \exists\lambda\in\sigma(A):\quad|\lambda-\mu|\leq\|\delta A\|_{2}
  60. λ σ ( A ) : | λ - λ a | s y m b o l r 2 s y m b o l v a 2 \exists\lambda\in\sigma(A):\quad\left|\lambda-\lambda^{a}\right|\leq\frac{\|% symbol{r}\|_{2}}{\left\|symbol{v}^{a}\right\|_{2}}
  61. A A
  62. A σ ( A ) A↦σ(A)
  63. 𝐂 \mathbf{C}

Baumé_scale.html

  1. s.g. = 145 145 - degrees Baum e ´ \,\text{s.g.}=\frac{145}{145-\mathrm{degrees\ Baum\acute{e}}}
  2. s.g. = 140 130 + degrees Baum e ´ \,\text{s.g.}=\frac{140}{130+\mathrm{degrees\ Baum\acute{e}}}
  3. s.g. = 144 144 - degrees Baum e ´ \,\text{s.g.}=\frac{144}{144-\mathrm{degrees\ Baum\acute{e}}}
  4. s.g. = 144 134 + degrees Baum e ´ \,\text{s.g.}=\frac{144}{134+\mathrm{degrees\ Baum\acute{e}}}

Bayesian_information_criterion.html

  1. BIC = - 2 ln L ^ + k ln ( n ) . \mathrm{BIC}={-2\cdot\ln{\hat{L}}+k\cdot\ln(n)}.
  2. x x
  3. θ \theta
  4. n n
  5. x x
  6. k k
  7. k k
  8. L ^ \hat{L}
  9. M M
  10. L ^ = p ( x | θ ^ , M ) \hat{L}=p(x|\hat{\theta},M)
  11. θ ^ \hat{\theta}
  12. p ( x | θ , M ) p(x|\theta,M)
  13. p ( θ | M ) p(\theta|M)
  14. θ \theta
  15. M M
  16. x x
  17. - 2 ln p ( x | M ) BIC = - 2 ln L ^ + k ( ln ( n ) - ln ( 2 π ) ) . {-2\cdot\ln{p(x|M)}}\approx\mathrm{BIC}={-2\cdot\ln{\hat{L}}+k\cdot(\ln(n)-\ln% (2\pi))}.
  18. n n
  19. n n
  20. k k
  21. BIC = n ln ( σ e 2 ^ ) + k ln ( n ) \mathrm{BIC}=n\cdot\ln(\widehat{\sigma_{e}^{2}})+k\cdot\ln(n)
  22. σ e 2 ^ \widehat{\sigma_{e}^{2}}
  23. σ e 2 ^ = 1 n i = 1 n ( x i - x i ^ ) 2 . \widehat{\sigma_{e}^{2}}=\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\hat{x_{i}})^{2}.
  24. BIC = n ln ( R S S / n ) + k ln ( n ) \mathrm{BIC}=n\cdot\ln(RSS/n)+k\cdot\ln(n)
  25. χ 2 \chi^{2}
  26. BIC = χ 2 + d f ln ( n ) \mathrm{BIC}=\chi^{2}+df\cdot\ln(n)
  27. d f df
  28. σ e 2 \sigma_{e}^{2}

Beal's_conjecture.html

  1. A x + B y = C z , A^{x}+B^{y}=C^{z},
  2. 3 3 + 6 3 = 3 5 3^{3}+6^{3}=3^{5}
  3. 7 3 + 7 4 = 14 3 7^{3}+7^{4}=14^{3}
  4. 2 n + 2 n = 2 n + 1 2^{n}+2^{n}=2^{n+1}
  5. 3 3 n + [ 2 ( 3 n ) ] 3 = 3 3 n + 2 ; 3^{3n}+[2(3^{n})]^{3}=3^{3n+2};
  6. n 1 n\geq 1
  7. ( a n - 1 ) 2 n + ( a n - 1 ) 2 n + 1 = [ a ( a n - 1 ) 2 ] n ; (a^{n}-1)^{2n}+(a^{n}-1)^{2n+1}=[a(a^{n}-1)^{2}]^{n};
  8. a 2 , n 3 a\geq 2,n\geq 3
  9. [ a ( a n + b n ) ] n + [ b ( a n + b n ) ] n = ( a n + b n ) n + 1 ; [a(a^{n}+b^{n})]^{n}+[b(a^{n}+b^{n})]^{n}=(a^{n}+b^{n})^{n+1};
  10. a 1 , b 1 , n 3 a\geq 1,b\geq 1,n\geq 3
  11. A 1 x + B 1 y = C 1 z A_{1}^{x}+B_{1}^{y}=C_{1}^{z}
  12. A n x + B n y = C n z ; A_{n}^{x}+B_{n}^{y}=C_{n}^{z};
  13. n 2 n\geq 2
  14. A n = ( A n - 1 y z + 1 ) ( B n - 1 y z ) ( C n - 1 y z ) A_{n}=(A_{n-1}^{yz+1})(B_{n-1}^{yz})(C_{n-1}^{yz})
  15. B n = ( A n - 1 x z ) ( B n - 1 x z + 1 ) ( C n - 1 x z ) B_{n}=(A_{n-1}^{xz})(B_{n-1}^{xz+1})(C_{n-1}^{xz})
  16. C n = ( A n - 1 x y ) ( B n - 1 x y ) ( C n - 1 x y + 1 ) C_{n}=(A_{n-1}^{xy})(B_{n-1}^{xy})(C_{n-1}^{xy+1})
  17. 271 3 + 2 3 3 5 73 3 = 919 3 \displaystyle 271^{3}+2^{3}3^{5}73^{3}=919^{3}
  18. A n + B n = C n A^{n}+B^{n}=C^{n}
  19. A x + B y = C z A^{x}+B^{y}=C^{z}
  20. 1 x + 1 y + 1 z < 1. \frac{1}{x}+\frac{1}{y}+\frac{1}{z}<1.
  21. 7 3 + 13 2 = 2 9 7^{3}+13^{2}=2^{9}
  22. 1 m + 2 3 = 3 2 1^{m}+2^{3}=3^{2}
  23. 27 4 + 162 3 = 9 7 , 27^{4}+162^{3}=9^{7},
  24. ( - 2 + i ) 3 + ( - 2 - i ) 3 = ( 1 + i ) 4 . (-2+i)^{3}+(-2-i)^{3}=(1+i)^{4}.

Beam_(nautical).html

  1. B e a m = L O A 2 / 3 + 1 Beam=LOA^{2/3}+1

Beam_emittance.html

  1. ϵ = γ x 2 + 2 α x x + β x 2 \epsilon=\gamma x^{2}+2\alpha xx^{\prime}+\beta x^{\prime 2}
  2. β , α , γ \beta,\alpha,\gamma
  3. emittance = 6 π ( width 2 - D 2 ( d p p ) 2 ) B \text{emittance}=\frac{6\pi\left(\,\text{width}^{2}-D^{2}\left(\frac{\mathrm{d% }p}{p}\right)^{2}\right)}{B}
  4. ϵ * = β γ ϵ \epsilon^{*}=\beta\gamma\epsilon
  5. B = η I ϵ x ϵ y B=\frac{{\eta}I}{{\epsilon_{x}}{\epsilon_{y}}}
  6. η = 1 8 π 2 \eta=\frac{1}{8\pi^{2}}

Beat_(acoustics).html

  1. cos ( 2 π f 1 t ) + cos ( 2 π f 2 t ) = 2 cos ( 2 π f 1 + f 2 2 t ) cos ( 2 π f 1 - f 2 2 t ) {\cos(2\pi f_{1}t)+\cos(2\pi f_{2}t)}={2\cos\left(2\pi\frac{f_{1}+f_{2}}{2}t% \right)\cos\left(2\pi\frac{f_{1}-f_{2}}{2}t\right)}
  2. f b e a t = f 1 - f 2 f_{beat}=f_{1}-f_{2}\,
  3. cos ( 2 π f 1 - f 2 2 t ) \cos\left(2\pi\frac{f_{1}-f_{2}}{2}t\right)

Beck's_monadicity_theorem.html

  1. U : C D U:C\to D

Beck's_theorem.html

  1. n 2 / K n^{2}/K
  2. 2 j 2^{j}
  3. 2 j + 1 - 1 2^{j+1}-1
  4. O ( n 2 / 2 3 j + n / 2 j ) O(n^{2}/2^{3j}+n/2^{j})
  5. 2 j 2^{j}
  6. | L | ( 2 j 2 ) ( n 2 ) |L|\cdot{2^{j}\choose 2}\leq{n\choose 2}
  7. O ( n 2 / 2 2 j + n ) O(n^{2}/2^{2j}+n)
  8. 2 j 2^{j}
  9. O ( n 2 / 2 3 j + n / 2 j ) O(n^{2}/2^{3j}+n/2^{j})
  10. Ω ( 2 2 j ) \Omega(2^{2j})
  11. O ( n 2 / 2 j + 2 j n ) O(n^{2}/2^{j}+2^{j}n)
  12. C 2 j n / C C\leq 2^{j}\leq n/C
  13. O ( n 2 / C ) O(n^{2}/C)
  14. n ( n - 1 ) 2 \frac{n(n-1)}{2}
  15. n 2 / 4 n^{2}/4
  16. C 2 j n / C C\leq 2^{j}\leq n/C
  17. n 2 / 4 n^{2}/4
  18. C ( 2 C - 1 ) C(2C-1)
  19. n 2 / 4 C ( 2 C - 1 ) n^{2}/4C(2C-1)
  20. K = 4 C ( 2 C - 1 ) K=4C(2C-1)

Behrens–Fisher_problem.html

  1. T x ¯ 1 - x ¯ 2 s 1 2 / n 1 + s 2 2 / n 2 T\equiv{\bar{x}_{1}-\bar{x}_{2}\over\sqrt{s_{1}^{2}/n_{1}+s_{2}^{2}/n_{2}}}
  2. x ¯ 1 \bar{x}_{1}
  3. x ¯ 2 \bar{x}_{2}
  4. s 1 / n 1 s 1 2 / n 1 + s 2 2 / n 2 . {s_{1}/\sqrt{n_{1}}\over\sqrt{s_{1}^{2}/n_{1}+s_{2}^{2}/n_{2}}}.
  5. d ¯ = x ¯ 1 - x ¯ 2 \bar{d}=\bar{x}_{1}-\bar{x}_{2}\,
  6. s d ¯ 2 = s 1 2 n 1 + s 2 2 n 2 . s_{\bar{d}}^{2}=\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}.
  7. s d ¯ 2 s_{\bar{d}}^{2}
  8. s d ¯ 2 s_{\bar{d}}^{2}
  9. ν ( γ 1 + γ 2 ) 2 γ 1 2 / ( n 1 - 1 ) + γ 2 2 / ( n 2 - 1 ) where γ i = σ i 2 / n i . \nu\approx{(\gamma_{1}+\gamma_{2})^{2}\over\gamma_{1}^{2}/(n_{1}-1)+\gamma_{2}% ^{2}/(n_{2}-1)}\quad\,\text{ where }\gamma_{i}=\sigma_{i}^{2}/n_{i}.\,
  10. ν ^ ( g 1 + g 2 ) 2 g 1 2 / ( n 1 - 1 ) + g 2 2 / ( n 2 - 1 ) where g i = s i 2 / n i . \hat{\nu}\approx{(g_{1}+g_{2})^{2}\over g_{1}^{2}/(n_{1}-1)+g_{2}^{2}/(n_{2}-1% )}\quad\,\text{ where }g_{i}=s_{i}^{2}/n_{i}.
  11. ν ^ \hat{\nu}
  12. ν ^ \hat{\nu}

Bel_decomposition.html

  1. X \vec{X}
  2. E [ X ] a b = R a m b n X m X n E[\vec{X}]_{ab}=R_{ambn}\,X^{m}\,X^{n}
  3. B [ X ] a b = R a m b n X m X n B[\vec{X}]_{ab}={{}^{\star}R}_{ambn}\,X^{m}\,X^{n}
  4. L [ X ] a b = R a m b n X m X n L[\vec{X}]_{ab}={{}^{\star}R^{\star}}_{ambn}\,X^{m}\,X^{n}
  5. K 1 / 4 K_{1}/4
  6. - K 2 / 8 -K_{2}/8
  7. K 3 / 8 K_{3}/8

Bell's_spaceship_paradox.html

  1. t = ( t - v x / c 2 ) / 1 - v 2 / c 2 \scriptstyle t^{\prime}=(t-vx/c^{2})/\sqrt{1-v^{2}/c^{2}}
  2. v x / c 2 vx/c^{2}
  3. v v
  4. U = γ U 0 U=\gamma U_{0}
  5. U U
  6. U 0 U_{0}
  7. γ \gamma
  8. 1 / 1 - v 2 / c 2 1/\sqrt{1-v^{2}/c^{2}}
  9. L L
  10. L L^{\prime}
  11. L = γ L L^{\prime}=\gamma L
  12. x A x_{A}
  13. x B = x A + L x_{B}=x_{A}+L
  14. x A = γ ( x A - v t ) x B = γ ( x A + L - v t ) L = x B - x A = γ L \begin{aligned}\displaystyle x^{\prime}_{A}&\displaystyle=\gamma\left(x_{A}-vt% \right)\\ \displaystyle x^{\prime}_{B}&\displaystyle=\gamma\left(x_{A}+L-vt\right)\\ \displaystyle L^{\prime}&\displaystyle=x^{\prime}_{B}-x^{\prime}_{A}\\ &\displaystyle=\gamma L\end{aligned}
  15. t A = t B t_{A}=t_{B}
  16. Δ t \displaystyle\Delta t^{\prime}
  17. L L
  18. L o l d = L / γ L^{\prime}_{old}=L/\gamma
  19. Δ t \Delta t^{\prime}
  20. L = L o l d + v Δ t = L γ + γ v 2 L c 2 = γ L \begin{aligned}\displaystyle L^{\prime}&\displaystyle=L^{\prime}_{old}+v\Delta t% ^{\prime}=\frac{L}{\gamma}+\frac{\gamma v^{2}L}{c^{2}}\\ &\displaystyle=\gamma L\end{aligned}
  21. L = L / γ L=L^{\prime}/\gamma
  22. L L
  23. γ \gamma
  24. L c o n t r . = L / γ = γ L / γ = L L_{contr.}=L^{\prime}/\gamma=\gamma L/\gamma=L
  25. γ L \gamma L
  26. L L

Belle_experiment.html

  1. B K * l + l - B\to K^{*}l^{+}l^{-}
  2. b s l + l - b\to sl^{+}l^{-}
  3. ϕ 3 \phi_{3}
  4. B D K , D K S π + π - B\to DK,D\to K_{S}\pi^{+}\pi^{-}
  5. | V u b | |V_{ub}|
  6. | V c b | |V_{cb}|
  7. B 0 π + π - B^{0}\to\pi^{+}\pi^{-}
  8. B 0 K - π + B^{0}\to K^{-}\pi^{+}
  9. b d b\to d
  10. B τ ν B\to\tau\nu

Belyi's_theorem.html

  1. { 0 , 1 , } \{0,1,\infty\}

Benedict–Webb–Rubin_equation.html

  1. P = ρ R T + ( B 0 R T - A 0 - C 0 T 2 ) ρ 2 + ( b R T - a ) ρ 3 + α a ρ 6 + c ρ 3 T 2 ( 1 + γ ρ 2 ) exp ( - γ ρ 2 ) P=\rho RT+\left(B_{0}RT-A_{0}-\frac{C_{0}}{T^{2}}\right)\rho^{2}+\left(bRT-a% \right)\rho^{3}+\alpha a\rho^{6}+\frac{c\rho^{3}}{T^{2}}\left(1+\gamma\rho^{2}% \right)\exp\left(-\gamma\rho^{2}\right)
  2. P = ρ R T + ( B 0 R T - A 0 - C 0 T 2 + D 0 T 3 - E 0 T 4 ) ρ 2 + ( b R T - a - d T ) ρ 3 + α ( a + d T ) ρ 6 + c ρ 3 T 2 ( 1 + γ ρ 2 ) exp ( - γ ρ 2 ) P=\rho RT+\left(B_{0}RT-A_{0}-\frac{C_{0}}{T^{2}}+\frac{D_{0}}{T^{3}}-\frac{E_% {0}}{T^{4}}\right)\rho^{2}+\left(bRT-a-\frac{d}{T}\right)\rho^{3}+\alpha\left(% a+\frac{d}{T}\right)\rho^{6}+\frac{c\rho^{3}}{T^{2}}\left(1+\gamma\rho^{2}% \right)\exp\left(-\gamma\rho^{2}\right)
  3. P = n = 1 9 a n ρ n + exp ( - γ ρ 2 ) n = 10 15 a n ρ 2 n - 17 P=\sum_{n=1}^{9}a_{n}\rho^{n}+\exp\left(-\gamma\rho^{2}\right)\sum_{n=10}^{15}% a_{n}\rho^{2n-17}
  4. γ = 1 / ρ c 2 \gamma=1/\rho_{c}^{2}

Berezinian.html

  1. Ber ( X Y ) = Ber ( X ) Ber ( Y ) \operatorname{Ber}(XY)=\operatorname{Ber}(X)\operatorname{Ber}(Y)
  2. Ber ( e X ) = e str ( X ) \operatorname{Ber}(e^{X})=e^{\operatorname{str(X)}}\,
  3. X = [ A 0 0 D ] X=\begin{bmatrix}A&0\\ 0&D\end{bmatrix}
  4. Ber ( X ) = det ( A ) det ( D ) - 1 \operatorname{Ber}(X)=\det(A)\det(D)^{-1}
  5. X = [ A B C D ] X=\begin{bmatrix}A&B\\ C&D\end{bmatrix}
  6. Ber ( X ) = det ( A - B D - 1 C ) det ( D ) - 1 \operatorname{Ber}(X)=\det(A-BD^{-1}C)\det(D)^{-1}
  7. Ber ( X ) = det ( A ) det ( D - C A - 1 B ) - 1 . \operatorname{Ber}(X)=\det(A)\det(D-CA^{-1}B)^{-1}.
  8. D - C A - 1 B D-CA^{-1}B\,
  9. [ A B C D ] . \begin{bmatrix}A&B\\ C&D\end{bmatrix}.
  10. J = [ 0 I - I 0 ] . J=\begin{bmatrix}0&I\\ -I&0\end{bmatrix}.
  11. Ber ( X ) = Ber ( J X ) = det ( C - D B - 1 A ) det ( - B ) - 1 . \operatorname{Ber}(X)=\operatorname{Ber}(JX)=\det(C-DB^{-1}A)\det(-B)^{-1}.
  12. Ber ( X ) - 1 = Ber ( X - 1 ) \operatorname{Ber}(X)^{-1}=\operatorname{Ber}(X^{-1})
  13. Ber ( X s t ) = Ber ( X ) \operatorname{Ber}(X^{st})=\operatorname{Ber}(X)
  14. X s t X^{st}
  15. Ber ( X Y ) = Ber ( X ) Ber ( Y ) \operatorname{Ber}(X\oplus Y)=\operatorname{Ber}(X)\mathrm{Ber}(Y)
  16. E x t A p ( R , A ) Ext_{A}^{p}(R,A)

Bergeron_process.html

  1. p p
  2. p w p_{w}
  3. p w p_{w}
  4. p p
  5. p p
  6. p i p_{i}
  7. u u p = p w - p i p i η N i r i ¯ u_{up}=\frac{p_{w}-p_{i}}{p_{i}}\eta N_{i}\bar{r_{i}}\,
  8. u d n = p i - p w p w χ N w r w ¯ u_{dn}=\frac{p_{i}-p_{w}}{p_{w}}\chi N_{w}\bar{r_{w}}\,
  9. N i N_{i}
  10. N w N_{w}
  11. r i ¯ \bar{r_{i}}
  12. r w ¯ \bar{r_{w}}
  13. N i r i ¯ N_{i}\bar{r_{i}}
  14. u u p u_{up}
  15. N w r w ¯ N_{w}\bar{r_{w}}
  16. m s - 1 ms^{-1}

Bergius_process.html

  1. n C + ( n - x + 1 ) H 2 C n H 2 n - 2 x + 2 n{\rm C}+(n-x+1){\rm H}_{2}\rightarrow{\rm C}_{n}{\rm H}_{2n-2x+2}

Bernoulli_scheme.html

  1. p i p_{i}
  2. i = 1 N p i = 1. \sum_{i=1}^{N}p_{i}=1.
  3. X = { 1 , , N } X=\{1,\ldots,N\}^{\mathbb{Z}}
  4. X = { x = ( , x - 1 , x 0 , x 1 , ) : x k { 1 , , N } k } . X=\{x=(\ldots,x_{-1},x_{0},x_{1},\ldots):x_{k}\in\{1,\ldots,N\}\;\forall k\in% \mathbb{Z}\}.
  5. μ = { p 1 , , p N } \mu=\{p_{1},\ldots,p_{N}\}^{\mathbb{Z}}
  6. 𝒜 \mathcal{A}
  7. ( X , 𝒜 , μ ) (X,\mathcal{A},\mu)
  8. 𝒜 \mathcal{A}
  9. [ x 0 , x 1 , , x n ] [x_{0},x_{1},\ldots,x_{n}]
  10. μ ( [ x 0 , x 1 , , x n ] ) = i = 0 n p x i \mu\left([x_{0},x_{1},\ldots,x_{n}]\right)=\prod_{i=0}^{n}p_{x_{i}}
  11. μ ( [ x 0 , x 1 , , x n ] ) = Pr ( X 0 = x 0 , X 1 = x 1 , , X n = x n ) \mu\left([x_{0},x_{1},\ldots,x_{n}]\right)=\mathrm{Pr}(X_{0}=x_{0},X_{1}=x_{1}% ,\ldots,X_{n}=x_{n})
  12. { X k } \{X_{k}\}
  13. T x k = x k + 1 . Tx_{k}=x_{k+1}.
  14. ( X , 𝒜 , μ , T ) (X,\mathcal{A},\mu,T)
  15. B S ( p ) = B S ( p 1 , , p N ) . BS(p)=BS(p_{1},\ldots,p_{N}).
  16. ( Y , , ν ) (Y,\mathcal{B},\nu)
  17. ( X , 𝒜 , μ ) = ( Y , , ν ) (X,\mathcal{A},\mu)=(Y,\mathcal{B},\nu)^{\mathbb{Z}}
  18. \mathbb{Z}
  19. G G
  20. ( X , 𝒜 , μ ) = ( Y , , ν ) G (X,\mathcal{A},\mu)=(Y,\mathcal{B},\nu)^{G}
  21. g x ( f ) = x ( g - 1 f ) gx(f)=x(g^{-1}f)
  22. f , g G f,g\in G
  23. x Y G x\in Y^{G}
  24. x : G Y x:G\to Y
  25. Y G Y^{G}
  26. [ G Y ] [G\to Y]
  27. μ \mu
  28. μ ( g x ) = μ ( x ) . \mu(gx)=\mu(x).\,
  29. H = - i = 1 N p i log p i . H=-\sum_{i=1}^{N}p_{i}\log p_{i}.
  30. ( Y , , ν ) (Y,\mathcal{B},\nu)
  31. Y Y Y^{\prime}\subset Y
  32. ν ( Y ) = 1 \nu(Y^{\prime})=1
  33. H Y = - y Y ν ( y ) log ν ( y ) . H_{Y^{\prime}}=-\sum_{y^{\prime}\in Y^{\prime}}\nu(y^{\prime})\log\nu(y^{% \prime}).

Bessel_filter.html

  1. H ( s ) = θ n ( 0 ) θ n ( s / ω 0 ) H(s)=\frac{\theta_{n}(0)}{\theta_{n}(s/\omega_{0})}\,
  2. θ n ( s ) \theta_{n}(s)
  3. ω 0 \omega_{0}
  4. 1 / ω 0 1/\omega_{0}
  5. θ n ( 0 ) \theta_{n}(0)
  6. θ n ( 0 ) = lim x 0 θ n ( x ) \theta_{n}(0)=\lim_{x\rightarrow 0}\theta_{n}(x)
  7. n = 1 ; s + 1 n=1;\quad s+1
  8. n = 2 ; s 2 + 3 s + 3 n=2;\quad s^{2}+3s+3
  9. n = 3 ; s 3 + 6 s 2 + 15 s + 15 n=3;\quad s^{3}+6s^{2}+15s+15
  10. n = 4 ; s 4 + 10 s 3 + 45 s 2 + 105 s + 105 n=4;\quad s^{4}+10s^{3}+45s^{2}+105s+105
  11. n = 5 ; s 5 + 15 s 4 + 105 s 3 + 420 s 2 + 945 s + 945 n=5;\quad s^{5}+15s^{4}+105s^{3}+420s^{2}+945s+945
  12. θ n ( s ) = k = 0 n a k s k , \theta_{n}(s)=\sum_{k=0}^{n}a_{k}s^{k},
  13. a k = ( 2 n - k ) ! 2 n - k k ! ( n - k ) ! k = 0 , 1 , , n . a_{k}=\frac{(2n-k)!}{2^{n-k}k!(n-k)!}\quad k=0,1,\ldots,n.
  14. H ( s ) = 15 s 3 + 6 s 2 + 15 s + 15 . H(s)=\frac{15}{s^{3}+6s^{2}+15s+15}.
  15. G ( ω ) = | H ( j ω ) | = 15 ω 6 + 6 ω 4 + 45 ω 2 + 225 . G(\omega)=|H(j\omega)|=\frac{15}{\sqrt{\omega^{6}+6\omega^{4}+45\omega^{2}+225% }}.\,
  16. ϕ ( ω ) = - arg ( H ( j ω ) ) = - arctan ( 15 ω - ω 3 15 - 6 ω 2 ) . \phi(\omega)=-\arg(H(j\omega))=-\arctan\left(\frac{15\omega-\omega^{3}}{15-6% \omega^{2}}\right).\,
  17. D ( ω ) = - d ϕ d ω = 6 ω 4 + 45 ω 2 + 225 ω 6 + 6 ω 4 + 45 ω 2 + 225 . D(\omega)=-\frac{d\phi}{d\omega}=\frac{6\omega^{4}+45\omega^{2}+225}{\omega^{6% }+6\omega^{4}+45\omega^{2}+225}.\,
  18. D ( ω ) = 1 - ω 6 225 + ω 8 1125 + . D(\omega)=1-\frac{\omega^{6}}{225}+\frac{\omega^{8}}{1125}+\cdots.

BET_theory.html

  1. 1 v [ ( p 0 / p ) - 1 ] = c - 1 v m c ( p p 0 ) + 1 v m c , ( 1 ) \frac{1}{v\left[\left({p_{0}}/{p}\right)-1\right]}=\frac{c-1}{v_{\mathrm{m}}c}% \left(\frac{p}{p_{0}}\right)+\frac{1}{v_{m}c},\qquad(1)
  2. p p
  3. p 0 p_{0}
  4. v v
  5. v m v_{\mathrm{m}}
  6. c c
  7. c = exp ( E 1 - E L R T ) , ( 2 ) c=\exp\left(\frac{E_{1}-E_{\mathrm{L}}}{RT}\right),\qquad(2)
  8. E 1 E_{1}
  9. E L E_{\mathrm{L}}
  10. 1 / v [ ( p 0 / p ) - 1 ] {1}/{v[({p_{0}}/{p})-1]}
  11. ϕ = p / p 0 \phi={p}/{p_{0}}
  12. 0.05 < p / p 0 < 0.35 0.05<{p}/{p_{0}}<0.35
  13. A A
  14. I I
  15. v m v_{\mathrm{m}}
  16. c c
  17. v m = 1 A + I ( 3 ) v_{m}=\frac{1}{A+I}\qquad(3)
  18. c = 1 + A I . ( 4 ) c=1+\frac{A}{I}.\qquad(4)
  19. S total S_{\mathrm{total}}
  20. S BET S_{\mathrm{BET}}
  21. S total = ( v m N s ) V , ( 5 ) S_{\mathrm{total}}=\frac{\left(v_{\mathrm{m}}Ns\right)}{V},\qquad(5)
  22. S BET = S total a , ( 6 ) S_{\mathrm{BET}}=\frac{S_{\mathrm{total}}}{a},\qquad(6)
  23. v m v_{\mathrm{m}}
  24. N N
  25. s s
  26. V V
  27. a a
  28. R ads , i - 1 = k i P Θ i - 1 R_{\mathrm{ads},i-1}=k_{i}P\Theta_{i-1}
  29. R des , i = k - i Θ i , R_{\mathrm{des},i}=k_{-i}\Theta_{i},
  30. k i = exp ( - E i / R T ) , k_{i}=\exp(-E_{i}/RT),
  31. A A
  32. I I
  33. v m v_{\mathrm{m}}
  34. c c
  35. A = 24.20 A=24.20
  36. I = 0.33 I=0.33
  37. v m = 1 A + I = 0.0408 , v_{\mathrm{m}}=\frac{1}{A+I}=0.0408,
  38. c = 1 + A I = 73.6. c=1+\frac{A}{I}=73.6.
  39. S BET S_{\mathrm{BET}}
  40. s = 0.114 nm 2 s=0.114\mathrm{nm}^{2}
  41. S BET = 156 m 2 / g S_{\mathrm{BET}}=156\mathrm{m}^{2}/\mathrm{g}
  42. s s

Beta_Cephei.html

  1. L A L = ( R A R ) 2 ( T A T ) 4 \frac{L_{\rm A}}{L_{\odot}}={\left(\frac{R_{\rm A}}{R_{\odot}}\right)}^{2}{% \left(\frac{T_{\rm A}}{T_{\odot}}\right)}^{4}
  2. L A L = ( 9 1 ) 2 ( 26 , 700 5 , 778 ) 4 = 36 , 933 L \frac{L_{\rm A}}{L_{\odot}}={\left({\frac{9}{1}}\right)}^{2}{\left({\frac{26,7% 00}{5,778}}\right)}^{4}=36,933L_{\odot}

Beta_Cephei_variable.html

  1. {}_{\odot}

Beta_prime_distribution.html

  1. I x 1 + x ( α , β ) I_{\frac{x}{1+x}(\alpha,\beta)}
  2. I x ( α , β ) I_{x}(\alpha,\beta)
  3. α β - 1 if β > 1 \frac{\alpha}{\beta-1}\,\text{ if }\beta>1
  4. α - 1 β + 1 if α 1 , 0 otherwise \frac{\alpha-1}{\beta+1}\,\text{ if }\alpha\geq 1\,\text{, 0 otherwise}\!
  5. α ( α + β - 1 ) ( β - 2 ) ( β - 1 ) 2 if β > 2 \frac{\alpha(\alpha+\beta-1)}{(\beta-2)(\beta-1)^{2}}\,\text{ if }\beta>2
  6. 2 ( 2 α + β - 1 ) β - 3 β - 2 α ( α + β - 1 ) if β > 3 \frac{2(2\alpha+\beta-1)}{\beta-3}\sqrt{\frac{\beta-2}{\alpha(\alpha+\beta-1)}% }\,\text{ if }\beta>3
  7. x > 0 x>0
  8. f ( x ) = x α - 1 ( 1 + x ) - α - β B ( α , β ) f(x)=\frac{x^{\alpha-1}(1+x)^{-\alpha-\beta}}{B(\alpha,\beta)}
  9. F ( x ; α , β ) = I x 1 + x ( α , β ) , F(x;\alpha,\beta)=I_{\frac{x}{1+x}}\left(\alpha,\beta\right),
  10. β > 4 \beta>4
  11. γ 2 = 6 α ( α + β - 1 ) ( 5 β - 11 ) + ( β - 1 ) 2 ( β - 2 ) α ( α + β - 1 ) ( β - 3 ) ( β - 4 ) \gamma_{2}=6\frac{\alpha(\alpha+\beta-1)(5\beta-11)+(\beta-1)^{2}(\beta-2)}{% \alpha(\alpha+\beta-1)(\beta-3)(\beta-4)}
  12. β ( α , β ) \beta^{{}^{\prime}}(\alpha,\beta)
  13. X ^ = α - 1 β + 1 \hat{X}=\frac{\alpha-1}{\beta+1}
  14. α β - 1 \frac{\alpha}{\beta-1}
  15. β > 1 \beta>1
  16. β 1 \beta\leq 1
  17. α ( α + β - 1 ) ( β - 2 ) ( β - 1 ) 2 \frac{\alpha(\alpha+\beta-1)}{(\beta-2)(\beta-1)^{2}}
  18. β > 2 \beta>2
  19. - α < k < β -\alpha<k<\beta
  20. E [ X k ] E[X^{k}]
  21. E [ X k ] = B ( α + k , β - k ) B ( α , β ) . E[X^{k}]=\frac{B(\alpha+k,\beta-k)}{B(\alpha,\beta)}.
  22. k k\in\mathbb{N}
  23. k < β k<\beta
  24. E [ X k ] = i = 1 k α + i - 1 β - i . E[X^{k}]=\prod_{i=1}^{k}\frac{\alpha+i-1}{\beta-i}.
  25. x α 2 F 1 ( α , α + β , α + 1 , - x ) α B ( α , β ) \frac{x^{\alpha}\cdot_{2}F_{1}(\alpha,\alpha+\beta,\alpha+1,-x)}{\alpha\cdot B% (\alpha,\beta)}\!
  26. F 1 2 {}_{2}F_{1}
  27. { ( x 2 + x ) f ( x ) + f ( x ) ( - α + β x + x + 1 ) = 0 , f ( 1 ) = 2 - α - β B ( α , β ) } \left\{\left(x^{2}+x\right)f^{\prime}(x)+f(x)(-\alpha+\beta x+x+1)=0,f(1)=% \frac{2^{-\alpha-\beta}}{B(\alpha,\beta)}\right\}
  28. p > 0 p>0
  29. q > 0 q>0
  30. f ( x ; α , β , p , q ) = p ( x q ) α p - 1 ( 1 + ( x q ) p ) - α - β q B ( α , β ) f(x;\alpha,\beta,p,q)=\frac{p{\left({\frac{x}{q}}\right)}^{\alpha p-1}\left({1% +{\left({\frac{x}{q}}\right)}^{p}}\right)^{-\alpha-\beta}}{qB(\alpha,\beta)}
  31. q Γ ( α + 1 p ) Γ ( β - 1 p ) Γ ( α ) Γ ( β ) if β p > 1 \frac{q\Gamma(\alpha+\tfrac{1}{p})\Gamma(\beta-\tfrac{1}{p})}{\Gamma(\alpha)% \Gamma(\beta)}\,\text{ if }\beta p>1
  32. q ( α p - 1 β p + 1 ) 1 p if α p 1 q{\left({\frac{\alpha p-1}{\beta p+1}}\right)}^{\tfrac{1}{p}}\,\text{ if }% \alpha p\geq 1\!
  33. β ( x ; α , β , 1 , q ) = 0 G ( x ; α , p ) G ( p ; β , q ) d p \beta^{\prime}(x;\alpha,\beta,1,q)=\int_{0}^{\infty}G(x;\alpha,p)G(p;\beta,q)% \;dp
  34. X β ( α , β ) X\sim\beta^{{}^{\prime}}(\alpha,\beta)\,
  35. 1 X β ( β , α ) \tfrac{1}{X}\sim\beta^{{}^{\prime}}(\beta,\alpha)
  36. X β ( α , β , p , q ) X\sim\beta^{{}^{\prime}}(\alpha,\beta,p,q)\,
  37. k X β ( α , β , p , k q ) kX\sim\beta^{{}^{\prime}}(\alpha,\beta,p,kq)\,
  38. β ( α , β , 1 , 1 ) = β ( α , β ) \beta^{{}^{\prime}}(\alpha,\beta,1,1)=\beta^{{}^{\prime}}(\alpha,\beta)\,
  39. X F ( α , β ) X\sim F(\alpha,\beta)\,
  40. α β X β ( α 2 , β 2 ) \tfrac{\alpha}{\beta}X\sim\beta^{{}^{\prime}}(\tfrac{\alpha}{2},\tfrac{\beta}{% 2})\,
  41. X Beta ( α , β ) X\sim\textrm{Beta}(\alpha,\beta)\,
  42. X 1 - X β ( α , β ) \frac{X}{1-X}\sim\beta^{{}^{\prime}}(\alpha,\beta)\,
  43. X Γ ( α , 1 ) X\sim\Gamma(\alpha,1)\,
  44. Y Γ ( β , 1 ) Y\sim\Gamma(\beta,1)\,
  45. X Y β ( α , β ) \frac{X}{Y}\sim\beta^{{}^{\prime}}(\alpha,\beta)
  46. β ( p , 1 , a , b ) = Dagum ( p , a , b ) \beta^{{}^{\prime}}(p,1,a,b)=\textrm{Dagum}(p,a,b)\,
  47. β ( 1 , p , a , b ) = SinghMaddala ( p , a , b ) \beta^{{}^{\prime}}(1,p,a,b)=\textrm{SinghMaddala}(p,a,b)\,
  48. β ( 1 , 1 , γ , σ ) = LL ( γ , σ ) \beta^{{}^{\prime}}(1,1,\gamma,\sigma)=\textrm{LL}(\gamma,\sigma)\,

Bézout_matrix.html

  1. f ( z ) = i = 0 n u i z i , g ( z ) = i = 0 n v i z i . f(z)=\sum_{i=0}^{n}u_{i}z^{i},\quad\quad g(z)=\sum_{i=0}^{n}v_{i}z^{i}.
  2. B n ( f , g ) = ( b i j ) i , j = 1 , , n B_{n}(f,g)=\left(b_{ij}\right)_{i,j=1,\dots,n}
  3. f ( x ) g ( y ) - f ( y ) g ( x ) x - y = i , j = 1 n b i j x i - 1 y j - 1 . \frac{f(x)g(y)-f(y)g(x)}{x-y}=\sum_{i,j=1}^{n}b_{ij}\,x^{i-1}\,y^{j-1}.
  4. \C n × n \C^{n\times n}
  5. m i j = min { i , n + 1 - j } m_{ij}=\min\{i,n+1-j\}
  6. b i j = k = 1 m i j u j + k - 1 v i - k - u i - k v j + k - 1 . b_{ij}=\sum_{k=1}^{m_{ij}}u_{j+k-1}v_{i-k}-u_{i-k}v_{j+k-1}.
  7. Bez : \C n × \C n \C : ( x , y ) Bez ( x , y ) = x * B n ( f , g ) y . \operatorname{Bez}:\C^{n}\times\C^{n}\to\C:(x,y)\mapsto\operatorname{Bez}(x,y)% =x^{*}B_{n}(f,g)y.
  8. B 3 ( f , g ) = [ u 1 v 0 - u 0 v 1 u 2 v 0 - u 0 v 2 u 3 v 0 - u 0 v 3 u 2 v 0 - u 0 v 2 u 2 v 1 - u 1 v 2 + u 3 v 0 - u 0 v 3 u 3 v 1 - u 1 v 3 u 3 v 0 - u 0 v 3 u 3 v 1 - u 1 v 3 u 3 v 2 - u 2 v 3 ] . B_{3}(f,g)=\left[\begin{matrix}u_{1}v_{0}-u_{0}v_{1}&u_{2}v_{0}-u_{0}v_{2}&u_{% 3}v_{0}-u_{0}v_{3}\\ u_{2}v_{0}-u_{0}v_{2}&u_{2}v_{1}-u_{1}v_{2}+u_{3}v_{0}-u_{0}v_{3}&u_{3}v_{1}-u% _{1}v_{3}\\ u_{3}v_{0}-u_{0}v_{3}&u_{3}v_{1}-u_{1}v_{3}&u_{3}v_{2}-u_{2}v_{3}\end{matrix}% \right].
  9. f ( x ) = 3 x 3 - x f(x)=3x^{3}-x
  10. g ( x ) = 5 x 2 + 1 g(x)=5x^{2}+1
  11. B 4 ( f , g ) = [ - 1 0 3 0 0 8 0 0 3 0 15 0 0 0 0 0 ] . B_{4}(f,g)=\left[\begin{matrix}-1&0&3&0\\ 0&8&0&0\\ 3&0&15&0\\ 0&0&0&0\end{matrix}\right].
  12. u i u_{i}
  13. v i v_{i}
  14. B n ( f , g ) B_{n}(f,g)
  15. B n ( f , g ) = - B n ( g , f ) B_{n}(f,g)=-B_{n}(g,f)
  16. B n ( f , f ) = 0 B_{n}(f,f)=0
  17. B n ( f , g ) B_{n}(f,g)
  18. B n ( f , g ) B_{n}(f,g)
  19. n × n \mathbb{R}^{n\times n}
  20. B n ( f , g ) B_{n}(f,g)
  21. n = m a x ( d e g ( f ) , d e g ( g ) ) n=max(deg(f),deg(g))
  22. B n ( f , g ) B_{n}(f,g)
  23. n = m a x ( d e g ( f ) , d e g ( g ) ) n=max(deg(f),deg(g))
  24. B n ( p , q ) B_{n}(p,q)
  25. B n ( p , q ) B_{n}(p,q)

Biased_graph.html

  1. C e C\cup e

Bicoherence.html

  1. B ( f 1 , f 2 ) = F ( f 1 ) F ( f 2 ) F * ( f 1 + f 2 ) B(f_{1},f_{2})=F(f_{1})F(f_{2})F^{*}(f_{1}+f_{2})
  2. B B
  3. f 1 f_{1}
  4. f 2 f_{2}
  5. F F
  6. * {}^{*}
  7. F ( f 1 ) F(f_{1})
  8. F ( f 2 ) F(f_{2})
  9. F ( f 1 + f 2 ) F(f_{1}+f_{2})
  10. b ( f 1 , f 2 ) = | n F n ( f 1 ) F n ( f 2 ) F n * ( f 1 + f 2 ) | n | F n ( f 1 ) | 2 | F n ( f 2 ) | 2 | F n * ( f 1 + f 2 ) | 2 b(f_{1},f_{2})=\frac{\left|\sum\limits_{n}F_{n}(f_{1})F_{n}(f_{2})F_{n}^{*}(f_% {1}+f_{2})\right|}{\sqrt{\sum\limits_{n}|F_{n}(f_{1})|^{2}|F_{n}(f_{2})|^{2}|F% _{n}^{*}(f_{1}+f_{2})|^{2}}}
  11. b 2 ( f 1 , f 2 ) = F n ( f 1 ) F n ( f 2 ) F n * ( f 1 + f 2 ) | F n ( f 1 ) F n ( f 2 ) | 2 | F n * ( f 1 + f 2 ) | 2 b^{2}(f_{1},f_{2})=\frac{\langle F_{n}(f_{1})F_{n}(f_{2})F_{n}^{*}(f_{1}+f_{2}% )\rangle}{\langle|F_{n}(f_{1})F_{n}(f_{2})|^{2}\rangle\langle|F_{n}^{*}(f_{1}+% f_{2})|^{2}\rangle}
  12. n n
  13. b ( f 1 , f 2 ) = | n F n ( f 1 ) F n ( f 2 ) F n * ( f 1 + f 2 ) | n | F n ( f 1 ) F n ( f 2 ) F n * ( f 1 + f 2 ) | b(f_{1},f_{2})=\frac{\left|\sum\limits_{n}F_{n}(f_{1})F_{n}(f_{2})F_{n}^{*}(f_% {1}+f_{2})\right|}{\sum\limits_{n}|F_{n}(f_{1})F_{n}(f_{2})F_{n}^{*}(f_{1}+f_{% 2})|}

Bicorn.html

  1. y 2 ( a 2 - x 2 ) = ( x 2 + 2 a y - a 2 ) 2 . y^{2}(a^{2}-x^{2})=(x^{2}+2ay-a^{2})^{2}.
  2. y 4 - x y 3 - 8 x y 2 + 36 x 2 y + 16 x 2 - 27 x 3 = 0 y^{4}-xy^{3}-8xy^{2}+36x^{2}y+16x^{2}-27x^{3}=0
  3. ( x 2 - 2 a z + a 2 z 2 ) 2 = x 2 + a 2 z 2 . (x^{2}-2az+a^{2}z^{2})^{2}=x^{2}+a^{2}z^{2}.\,
  4. x = a sin ( θ ) x=a\sin(\theta)
  5. y = cos 2 ( θ ) ( 2 + cos ( θ ) ) 3 + sin 2 ( θ ) y=\frac{\cos^{2}(\theta)\left(2+\cos(\theta)\right)}{3+\sin^{2}(\theta)}
  6. - π θ π -\pi\leq\theta\leq\pi

Bidirectional_search.html

  1. s s
  2. t t
  3. s s
  4. t t
  5. t t
  6. s s
  7. s s
  8. t t
  9. n n
  10. n n
  11. n n
  12. n n
  13. p p
  14. k 1 ( p , n ) = k 2 ( n , p ) k_{1}(p,n)=k_{2}(n,p)
  15. p p
  16. n n
  17. b b
  18. k ( n , m ) k(n,m)
  19. n n
  20. m m
  21. g ( n ) g(n)
  22. n n
  23. h ( n ) h(n)
  24. n n
  25. s s
  26. t t
  27. g g
  28. d d
  29. d d
  30. d d^{\prime}
  31. d = 3 - d d^{\prime}=3-d
  32. T R E E d TREE_{d}
  33. d = 1 d=1
  34. s s
  35. d = 2 d=2
  36. t t
  37. O P E N d OPEN_{d}
  38. T R E E d TREE_{d}
  39. F R I N G E d FRINGE_{d}
  40. C L O S E D d CLOSED_{d}
  41. T R E E d TREE_{d}
  42. h h
  43. n n
  44. n n
  45. s s
  46. t t
  47. h h
  48. n n
  49. n n
  50. O P E N d OPEN_{d}^{\prime}
  51. h h
  52. h d ( n ) = min i { H ( n , o i ) | o i O P E N d } h_{d}(n)=\min_{i}\left\{H(n,o_{i})|o_{i}\in OPEN_{d^{\prime}}\right\}
  53. H ( n , o ) H(n,o)
  54. n n
  55. o o
  56. n n
  57. f = g + h f=g+h
  58. n n
  59. O P E N OPEN
  60. O P E N OPEN
  61. b > 1 b>1

Bifurcation_theory.html

  1. x ˙ = f ( x , λ ) f : n × n . \dot{x}=f(x,\lambda)\quad f\colon\mathbb{R}^{n}\times\mathbb{R}\rightarrow% \mathbb{R}^{n}.
  2. ( x 0 , λ 0 ) (x_{0},\lambda_{0})
  3. d f x 0 , λ 0 \textrm{d}f_{x_{0},\lambda_{0}}
  4. x n + 1 = f ( x n , λ ) . x_{n+1}=f(x_{n},\lambda)\,.
  5. ( x 0 , λ 0 ) (x_{0},\lambda_{0})
  6. d f x 0 , λ 0 \textrm{d}f_{x_{0},\lambda_{0}}

Biharmonic_equation.html

  1. 4 φ = 0 \nabla^{4}\varphi=0
  2. 2 2 φ = 0 \nabla^{2}\nabla^{2}\varphi=0
  3. Δ 2 φ = 0 \Delta^{2}\varphi=0
  4. 4 \nabla^{4}
  5. 2 \nabla^{2}
  6. Δ \Delta
  7. n n
  8. 4 φ = i = 1 n j = 1 n i i j j φ . \nabla^{4}\varphi=\sum_{i=1}^{n}\sum_{j=1}^{n}\partial_{i}\partial_{i}\partial% _{j}\partial_{j}\varphi.
  9. 4 φ x 4 + 4 φ y 4 + 4 φ z 4 + 2 4 φ x 2 y 2 + 2 4 φ y 2 z 2 + 2 4 φ x 2 z 2 = 0. {\partial^{4}\varphi\over\partial x^{4}}+{\partial^{4}\varphi\over\partial y^{% 4}}+{\partial^{4}\varphi\over\partial z^{4}}+2{\partial^{4}\varphi\over% \partial x^{2}\partial y^{2}}+2{\partial^{4}\varphi\over\partial y^{2}\partial z% ^{2}}+2{\partial^{4}\varphi\over\partial x^{2}\partial z^{2}}=0.
  10. 4 ( 1 r ) = 3 ( 15 - 8 n + n 2 ) r 5 \nabla^{4}\left({1\over r}\right)={3(15-8n+n^{2})\over r^{5}}
  11. r = x 1 2 + x 2 2 + + x n 2 . r=\sqrt{x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}}.
  12. 1 r r ( r r ( 1 r r ( r φ r ) ) ) + 2 r 2 4 φ θ 2 r 2 + 1 r 4 4 φ θ 4 - 2 r 3 3 φ θ 2 r + 4 r 4 2 φ θ 2 = 0 \frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial}{\partial r}\left(% \frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial\varphi}{\partial r% }\right)\right)\right)+\frac{2}{r^{2}}\frac{\partial^{4}\varphi}{\partial% \theta^{2}\partial r^{2}}+\frac{1}{r^{4}}\frac{\partial^{4}\varphi}{\partial% \theta^{4}}-\frac{2}{r^{3}}\frac{\partial^{3}\varphi}{\partial\theta^{2}% \partial r}+\frac{4}{r^{4}}\frac{\partial^{2}\varphi}{\partial\theta^{2}}=0
  13. x v ( x , y ) - y u ( x , y ) + w ( x , y ) xv(x,y)-yu(x,y)+w(x,y)
  14. u ( x , y ) u(x,y)
  15. v ( x , y ) v(x,y)
  16. w ( x , y ) w(x,y)
  17. v ( x , y ) v(x,y)
  18. u ( x , y ) u(x,y)
  19. Im ( z ¯ f ( z ) + g ( z ) ) \operatorname{Im}(\bar{z}f(z)+g(z))
  20. f ( z ) f(z)
  21. g ( z ) g(z)

Bijective_numeration.html

  1. a 0 \displaystyle a_{0}
  2. f ( x ) = x - 1 f(x)=\lceil x\rceil-1
  3. x \lceil x\rceil
  4. log k ( n + 1 ) + log k ( k - 1 ) \lfloor\log_{k}(n+1)+\log_{k}(k-1)\rfloor
  5. log k n + 1 ( n > 0 ) \lfloor\log_{k}n\rfloor+1\ (n>0)
  6. 1 × 26 2 + 2 × 26 1 + 3 × 26 0 1\times 26^{2}+2\times 26^{1}+3\times 26^{0}

Biological_thermodynamics.html

  1. λ λ
  2. ν ν
  3. E = h c λ = h ν , E=\frac{hc}{\lambda}=h\nu,

Birth–death_process.html

  1. { λ i } i = 0 \{\lambda_{i}\}_{i=0\dots\infty}
  2. { μ i } i = 1 \{\mu_{i}\}_{i=1\dots\infty}
  3. μ i = 0 \mu_{i}=0
  4. i 0 i\geq 0
  5. λ i = 0 \lambda_{i}=0
  6. i 0 i\geq 0
  7. λ i = λ \lambda_{i}=\lambda
  8. i 0 i\geq 0
  9. \infty
  10. λ \lambda
  11. 1 / μ 1/\mu
  12. λ i = λ and μ i = μ for all i . \lambda_{i}=\lambda\,\text{ and }\mu_{i}=\mu\,\text{ for all }i.\,
  13. p 0 ( t ) = μ 1 p 1 ( t ) - λ 0 p 0 ( t ) p_{0}^{\prime}(t)=\mu_{1}p_{1}(t)-\lambda_{0}p_{0}(t)\,
  14. p k ( t ) = λ k - 1 p k - 1 ( t ) + μ k + 1 p k + 1 ( t ) - ( λ k + μ k ) p k ( t ) p_{k}^{\prime}(t)=\lambda_{k-1}p_{k-1}(t)+\mu_{k+1}p_{k+1}(t)-(\lambda_{k}+\mu% _{k})p_{k}(t)\,
  15. μ i = i μ for i C \mu_{i}=i\mu\,\text{ for }i\leq C\,
  16. μ i = C μ for i C \mu_{i}=C\mu\,\text{ for }i\geq C\,
  17. λ i = λ for all i . \lambda_{i}=\lambda\,\text{ for all }i.\,
  18. λ i = λ for 0 i < K \lambda_{i}=\lambda\,\text{ for }0\leq i<K\,
  19. λ i = 0 for i K \lambda_{i}=0\,\text{ for }i\geq K\,
  20. μ i = μ for 1 i K . \mu_{i}=\mu\,\text{ for }1\leq i\leq K.\,
  21. λ 0 = 0. \lambda_{0}=0.\,
  22. μ K = 0. \mu_{K}=0.\,
  23. p 0 ( t ) = μ 1 p 1 ( t ) - λ 0 p 0 ( t ) p_{0}^{\prime}(t)=\mu_{1}p_{1}(t)-\lambda_{0}p_{0}(t)
  24. p k ( t ) = λ k - 1 p k - 1 ( t ) + μ k + 1 p k + 1 ( t ) - ( λ k + μ k ) p k ( t ) for k K p_{k}^{\prime}(t)=\lambda_{k-1}p_{k-1}(t)+\mu_{k+1}p_{k+1}(t)-(\lambda_{k}+\mu% _{k})p_{k}(t)\,\text{ for }k\leq K\,
  25. p k ( t ) = 0 for k > K p_{k}^{\prime}(t)=0\,\text{ for }k>K\,
  26. lim t p k ( t ) \lim_{t\to\infty}p_{k}(t)
  27. p k ( t ) p_{k}^{\prime}(t)
  28. λ 0 p 0 ( t ) = μ 1 p 1 ( t ) \lambda_{0}p_{0}(t)=\mu_{1}p_{1}(t)\,
  29. ( λ k + μ k ) p k ( t ) = λ k - 1 p k - 1 ( t ) + μ k + 1 p k + 1 ( t ) (\lambda_{k}+\mu_{k})p_{k}(t)=\lambda_{k-1}p_{k-1}(t)+\mu_{k+1}p_{k+1}(t)\,
  30. λ k = λ \lambda_{k}=\lambda
  31. μ k = μ \mu_{k}=\mu
  32. k k
  33. λ p k ( t ) = μ p k + 1 ( t ) for k 0. \lambda p_{k}(t)=\mu p_{k+1}(t)\,\text{ for }k\geq 0.\,
  34. Δ t \Delta t
  35. λ \lambda
  36. μ \mu
  37. λ Δ t \lambda\Delta t
  38. μ Δ t \mu\Delta t
  39. 1 - ( λ + μ ) Δ t 1-(\lambda+\mu)\Delta t

Bispectrum.html

  1. B ( f 1 , f 2 ) = F * ( f 1 + f 2 ) . F ( f 1 ) . F ( f 2 ) B(f_{1},f_{2})=F^{*}(f_{1}+f_{2}).F(f_{1}).F(f_{2})
  2. F F
  3. F * F^{*}

Bitonic_sorter.html

  1. O ( n log ( n ) 2 ) O(n\log(n)^{2})
  2. O ( log ( n ) 2 ) O(\log(n)^{2})
  3. n n
  4. x 0 x k x n - 1 x_{0}\leq\cdots\leq x_{k}\geq\cdots\geq x_{n-1}
  5. k , 0 k < n k,0\leq k<n

Black_brane.html

  1. d s 2 = ( η a b + r s n - p - 3 r n - p - 3 u a u b ) d σ a d σ b + ( 1 - r s n - p - 3 r n - p - 3 ) - 1 d r 2 + r 2 d Ω n - p - 2 2 {ds}^{2}=\left(\eta_{ab}+\frac{r_{s}^{n-p-3}}{r^{n-p-3}}u_{a}u_{b}\right)d% \sigma^{a}d\sigma^{b}+\left(1-\frac{r_{s}^{n-p-3}}{r^{n-p-3}}\right)^{-1}dr^{2% }+r^{2}d\Omega^{2}_{n-p-2}

Black–Derman–Toy_model.html

  1. u {}_{u}
  2. d {}_{d}
  3. u {}_{u}
  4. d {}_{d}
  5. i {}_{i}
  6. d ln ( r ) = [ θ t + σ t σ t ln ( r ) ] d t + σ t d W t d\ln(r)=[\theta_{t}+\frac{\sigma^{\prime}_{t}}{\sigma_{t}}\ln(r)]dt+\sigma_{t}% \,dW_{t}
  7. r r\,
  8. θ t \theta_{t}\,
  9. σ t \sigma_{t}\,
  10. W t W_{t}\,
  11. d W t dW_{t}\,
  12. σ \sigma\,
  13. d ln ( r ) = θ t d t + σ d W t d\ln(r)=\theta_{t}\,dt+\sigma\,dW_{t}

Bland–Altman_plot.html

  1. S 1 S_{1}
  2. S 2 S_{2}
  3. S ( x , y ) = ( S 1 + S 2 2 , ( S 1 - S 2 ) ) . S(x,y)=\left(\frac{S_{1}+S_{2}}{2},(S_{1}-S_{2})\right).

Blattner's_conjecture.html

  1. w W K ϵ ( ω ) Q ( w ( μ + ρ c ) - λ - ρ n ) \sum_{w\in W_{K}}\epsilon(\omega)Q(w(\mu+\rho_{c})-\lambda-\rho_{n})

Block-matching_algorithm.html

  1. 1 N 2 i = 0 n - 1 j = 0 n - 1 | C i j - R i j | \frac{1}{N^{2}}\sum_{i=0}^{n-1}\sum_{j=0}^{n-1}|C_{ij}-R_{ij}|
  2. 1 N 2 i = 0 n - 1 j = 0 n - 1 ( C i j - R i j ) 2 \frac{1}{N^{2}}\sum_{i=0}^{n-1}\sum_{j=0}^{n-1}(C_{ij}-R_{ij})^{2}
  3. C i j C_{ij}
  4. R i j R_{ij}
  5. PSNR = 10 log 10 ( peak to peak value of original data ) 2 MSE \,\text{PSNR}=10\log_{10}\frac{(\,\text{peak to peak value of original data})^% {2}}{\,\text{MSE}}

Block_cellular_automaton.html

  1. 2 2
  2. 2 × 2 2×2
  3. 2 × 2 × 2 2×2×2
  4. F F
  5. F F
  6. 2 × 2 2×2
  7. 2 × 2 2×2
  8. 2 × 4 2×4

Blood_volume.html

  1. B V = P V 1 - H C BV=\frac{PV}{1-HC}

Bloodstain_pattern_analysis.html

  1. sin α = ( w l ) \sin\alpha=\left(\frac{w}{l}\right)
  2. tan β = tan α sin γ \tan\beta=\frac{\tan\alpha}{\sin\gamma}
  3. 𝐥 \mathbf{l}
  4. 𝐰 \mathbf{w}

Blum's_speedup_theorem.html

  1. ( φ , Φ ) (\varphi,\Phi)
  2. f f
  3. g g
  4. i i
  5. g g
  6. j j
  7. g g
  8. x x
  9. f ( x , Φ j ( x ) ) Φ i ( x ) f(x,\Phi_{j}(x))\leq\Phi_{i}(x)\,
  10. f f

Blum_axioms.html

  1. ( φ , Φ ) (\varphi,\Phi)
  2. φ \varphi
  3. 𝐏 ( 1 ) \mathbf{P}^{(1)}
  4. Φ : 𝐏 ( 1 ) \Phi:\mathbb{N}\to\mathbf{P}^{(1)}
  5. φ i \varphi_{i}
  6. φ \varphi
  7. Φ i \Phi_{i}
  8. Φ ( i ) \Phi(i)
  9. φ i \varphi_{i}
  10. Φ i \Phi_{i}
  11. { ( i , x , t ) 3 | Φ i ( x ) = t } \{(i,x,t)\in\mathbb{N}^{3}|\Phi_{i}(x)=t\}
  12. ( φ , Φ ) (\varphi,\Phi)
  13. Φ \Phi
  14. ( φ , φ ) (\varphi,\varphi)
  15. Φ \Phi
  16. M M
  17. x x
  18. M M
  19. Φ \Phi
  20. Φ ( M , x ) \Phi(M,x)
  21. M ( x ) M(x)
  22. ( M , x , n ) (M,x,n)
  23. Φ ( M , x ) = n \Phi(M,x)=n
  24. Φ ( M , x ) \Phi(M,x)
  25. f f
  26. C ( f ) := { φ i 𝐏 ( 1 ) | x . Φ i ( x ) f ( x ) } C(f):=\{\varphi_{i}\in\mathbf{P}^{(1)}|\forall x.\ \Phi_{i}(x)\leq f(x)\}
  27. C 0 ( f ) := { h C ( f ) | codom ( h ) { 0 , 1 } } C^{0}(f):=\{h\in C(f)|\mathrm{codom}(h)\subseteq\{0,1\}\}
  28. C ( f ) C(f)
  29. f f
  30. C 0 ( f ) C^{0}(f)
  31. f f
  32. C 0 ( f ) C^{0}(f)

Blum_integer.html

  1. ( - 1 n ) = ( - 1 p ) ( - 1 q ) = ( - 1 ) 2 = 1 \left(\frac{-1}{n}\right)=\left(\frac{-1}{p}\right)\left(\frac{-1}{q}\right)=(% -1)^{2}=1

Blum–Goldwasser_cryptosystem.html

  1. N = p q N=pq
  2. p , q p,q
  3. ( p , q ) (p,q)
  4. p p\,
  5. q q\,
  6. p q p\neq q
  7. p q 3 p\equiv q\equiv 3
  8. 4 4
  9. N = p q N=pq
  10. N N
  11. ( p , q ) (p,q)
  12. N N
  13. m m
  14. L L
  15. ( m 0 , , m L - 1 ) (m_{0},\dots,m_{L-1})
  16. r r
  17. 1 < r < N 1<r<N
  18. x 0 = r 2 m o d N x_{0}=r^{2}~{}mod~{}N
  19. L L
  20. b = ( b 0 , , b L - 1 ) {\vec{b}}=(b_{0},\dots,b_{L-1})
  21. i = 0 i=0
  22. L L
  23. b i b_{i}
  24. x i x_{i}
  25. i i
  26. x i = ( x i - 1 ) 2 m o d N x_{i}=(x_{i-1})^{2}~{}mod~{}N
  27. i = 0 i=0
  28. L - 1 L-1
  29. c = m b {\vec{c}}={\vec{m}}\oplus{\vec{b}}
  30. x L x_{L}
  31. ( c 0 , , c L - 1 ) , x L (c_{0},\dots,c_{L-1}),x_{L}
  32. x L x_{L}
  33. x L = x 0 2 L m o d N x_{L}=x_{0}^{2^{L}}~{}mod~{}N
  34. O ( l o g l o g N ) O(loglogN)
  35. x i x_{i}
  36. ( c 0 , , c L - 1 ) , y (c_{0},\dots,c_{L-1}),y
  37. m m
  38. ( p , q ) (p,q)
  39. r p = y ( ( p + 1 ) / 4 ) L m o d p r_{p}=y^{((p+1)/4)^{L}}~{}mod~{}p
  40. r q = y ( ( q + 1 ) / 4 ) L m o d q r_{q}=y^{((q+1)/4)^{L}}~{}mod~{}q
  41. x 0 = ( q ( q - 1 m o d p ) r p + p ( p - 1 m o d q ) r q ) m o d N x_{0}=(q(q^{-1}~{}{mod}~{}p)r_{p}+p(p^{-1}~{}{mod}~{}q)r_{q})~{}{mod}~{}N
  42. x 0 x_{0}
  43. b {\vec{b}}
  44. m = c b {\vec{m}}={\vec{c}}\oplus{\vec{b}}
  45. m = ( m 0 , , m L - 1 ) m=(m_{0},\dots,m_{L-1})
  46. y y
  47. N N
  48. c , y {\vec{c}},y
  49. m m^{\prime}
  50. a , y {\vec{a}},y
  51. m m
  52. a m c {\vec{a}}\oplus m^{\prime}\oplus{\vec{c}}

Bochner's_theorem.html

  1. G ^ \widehat{G}
  2. G ^ \widehat{G}
  3. f ( g ) = G ^ ξ ( g ) d μ ( ξ ) , f(g)=\int_{\widehat{G}}\xi(g)d\mu(\xi),
  4. G ^ \widehat{G}
  5. G ^ \widehat{G}
  6. ( , , f ) (\mathcal{H},\langle\;,\;\rangle_{f})
  7. g U g g\;\mapsto\;U_{g}
  8. ( , , f ) (\mathcal{H},\langle\;,\;\rangle_{f})
  9. U g [ e ] , [ e ] f = f ( g ) \langle U_{g}[e],[e]\rangle_{f}=f(g)
  10. [ e ] , [ e ] f \langle\cdot[e],[e]\rangle_{f}
  11. C 0 ( G ^ ) C_{0}(\widehat{G})
  12. U g [ e ] , [ e ] f = G ^ ξ ( g ) d μ ( ξ ) . \langle U_{g}[e],[e]\rangle_{f}=\int_{\widehat{G}}\xi(g)d\mu(\xi).
  13. G ^ \widehat{G}
  14. f ( g ) = G ^ ξ ( g ) d μ ( ξ ) . f(g)=\int_{\widehat{G}}\xi(g)d\mu(\xi).
  15. C 0 ( G ^ ) C_{0}(\widehat{G})
  16. C b ( G ^ ) C_{b}(\widehat{G})
  17. f ( g ) = U g v , v , f(g)=\langle U_{g}v,v\rangle,
  18. f ( k ) = 𝕋 e - 2 π i k x d μ ( x ) . f(k)=\int_{\mathbb{T}}e^{-2\pi ikx}d\mu(x).
  19. f ( t ) = e - 2 π i ξ t d μ ( ξ ) . f(t)=\int_{\mathbb{R}}e^{-2\pi i\xi t}d\mu(\xi).
  20. { f n } \{f_{n}\}
  21. Cov ( f n , f m ) \mbox{Cov}~{}(f_{n},f_{m})
  22. g ( n - m ) = Cov ( f n , f m ) g(n-m)=\mbox{Cov}~{}(f_{n},f_{m})
  23. g ( n - m ) = f n , f m g(n-m)=\langle f_{n},f_{m}\rangle
  24. g ( k ) = e - 2 π i k x d μ ( x ) g(k)=\int e^{-2\pi ikx}d\mu(x)
  25. { z n f } \{z^{n}f\}
  26. g ( k ) = z k g(k)=z^{k}

Bohm_diffusion.html

  1. D B o h m = 1 16 k B T e B D_{Bohm}=\frac{1}{16}\,\frac{k_{B}T}{eB}
  2. D = δ 2 τ = v 2 τ = δ v , D=\frac{\delta^{2}}{\tau}=v^{2}\tau=\delta\,v,
  3. D = Δ x 2 τ D c 2 δ E 2 B 2 k 2 D c δ E B k D=\left\langle\frac{\Delta x^{2}}{\tau_{D}}\right\rangle\sim\frac{c^{2}\delta E% ^{2}}{B^{2}k_{\perp}^{2}\,D}\sim\frac{c\delta E}{Bk_{\perp}}

Bond_dipole_moment.html

  1. μ = δ d \mu=\delta\,d
  2. δ \delta
  3. × 10 2 9 \times 10^{2}9

Bonnor_beam.html

  1. d s 2 = - 8 π m r 2 d u 2 - 2 d u d v + d r 2 + r 2 d θ 2 , ds^{2}=-8\pi mr^{2}\,du^{2}-2\,du\,dv+dr^{2}+r^{2}\,d\theta^{2},
  2. - < u , v < , 0 < r < r 0 , - π < θ < π -\infty<u,v<\infty,0<r<r_{0},-\pi<\theta<\pi
  3. d s 2 = - 8 π m r 0 2 ( 1 + 2 log ( r / r 0 ) ) d u 2 - 2 d u d v + d r 2 + r 2 d θ 2 ds^{2}=-8\pi mr_{0}^{2}\left(1+2\log(r/r_{0})\right)\,du^{2}-2\,du\,dv+dr^{2}+% r^{2}\,d\theta^{2}
  4. - < u , v < , r 0 < r < , - π < θ < π -\infty<u,v<\infty,r_{0}<r<\infty,-\pi<\theta<\pi

Book_(graph_theory).html

  1. B p B_{p}
  2. p p
  3. K e ( 2 , p ) K_{e}(2,p)
  4. G G
  5. b k ( G ) bk(G)
  6. G G
  7. B p B_{p}
  8. r ( B p , B q ) . r(B_{p},\ B_{q}).
  9. 1 p q 1\leq p\leq q
  10. r ( B p , B q ) = 2 q + 3 r(B_{p},\ B_{q})=2q+3
  11. c = o ( 1 ) c=o(1)
  12. r ( B p , B q ) = 2 q + 3 r(B_{p},\ B_{q})=2q+3
  13. q c p q\geq cp
  14. p q / 6 + o ( q ) p\leq q/6+o(q)
  15. q q
  16. 2 q + 3 2q+3
  17. C C
  18. k = C n k=Cn
  19. n n
  20. m m
  21. B k B_{k}

Boole's_expansion_theorem.html

  1. F = x F x + x F x F=x\cdot F_{x}+x^{\prime}\cdot F_{x}^{\prime}
  2. F F
  3. F x F_{x}
  4. F x F_{x}^{\prime}
  5. F F
  6. x x
  7. 1 1
  8. 0
  9. F x F_{x}
  10. F x F_{x}^{\prime}
  11. F F
  12. x x
  13. r e s t r i c t ( F , x , 0 ) restrict(F,x,0)
  14. r e s t r i c t ( F , x , 1 ) restrict(F,x,1)
  15. f ( X 1 , X 2 , , X n ) = X 1 f ( 1 , X 2 , , X n ) + X 1 f ( 0 , X 2 , , X n ) f(X_{1},X_{2},\dots,X_{n})=X_{1}\cdot f(1,X_{2},\dots,X_{n})+X_{1}^{\prime}% \cdot f(0,X_{2},\dots,X_{n})
  16. f ( X 1 ) = X 1 .1 + X 1 .0 f(X_{1})=X_{1}.1+X_{1}^{\prime}.0

Boolean-valued_model.html

  1. ϕ ( a ) ψ ( b , c ) \phi(a)\land\psi(b,c)
  2. || ϕ ( a ) ψ ( b , c ) || = || ϕ ( a ) || || ψ ( b , c ) || ||\phi(a)\land\psi(b,c)||=||\phi(a)||\ \land\ ||\psi(b,c)||
  3. || x ϕ ( x ) || = a M || ϕ ( a ) || , ||\exists x\phi(x)||=\bigvee_{a\in M}||\phi(a)||,
  4. p ϕ p\Vdash\phi
  5. p ϕ p || ϕ || p\Vdash\phi\iff p\leq||\phi||

Boomerang_attack.html

  1. Δ Δ * \Delta\to\Delta^{*}
  2. * \nabla\to\nabla^{*}
  3. P P
  4. P = P Δ P^{\prime}=P\oplus\Delta
  5. P P
  6. P P^{\prime}
  7. C = E ( P ) C=E(P)
  8. C = E ( P ) C^{\prime}=E(P^{\prime})
  9. D = C D=C\oplus\nabla
  10. D = C D^{\prime}=C^{\prime}\oplus\nabla
  11. D D
  12. D D^{\prime}
  13. Q = E - 1 ( D ) Q=E^{-1}(D)
  14. Q = E - 1 ( D ) Q^{\prime}=E^{-1}(D^{\prime})
  15. Q Q
  16. Q Q^{\prime}
  17. Q Q = Δ Q\oplus Q^{\prime}=\Delta

Borel_summation.html

  1. A ( z ) = k = 0 a k z k A(z)=\sum_{k=0}^{\infty}a_{k}z^{k}
  2. A ( t ) k = 0 a k k ! t k . \mathcal{B}A(t)\equiv\sum_{k=0}^{\infty}\frac{a_{k}}{k!}t^{k}.
  3. A n ( z ) = k = 0 n a k z k . A_{n}(z)=\sum_{k=0}^{n}a_{k}z^{k}.
  4. lim t e - t n = 0 t n n ! A n ( z ) . \lim_{t\rightarrow\infty}e^{-t}\sum_{n=0}^{\infty}\frac{t^{n}}{n!}A_{n}(z).
  5. a k z k = a ( z ) ( s y m b o l w B ) {\textstyle\sum}a_{k}z^{k}=a(z)\,(symbol{wB})
  6. 0 e - t A ( t z ) d t . \int_{0}^{\infty}e^{-t}\mathcal{B}A(tz)\,dt.
  7. a k z k = a ( z ) ( s y m b o l B ) {\textstyle\sum}a_{k}z^{k}=a(z)\,(symbolB)
  8. k = 0 a k z k = A ( z ) < a k z k = A ( z ) ( s y m b o l B , s y m b o l w B ) . \sum_{k=0}^{\infty}a_{k}z^{k}=A(z)<\infty\quad\Rightarrow\quad{\textstyle\sum}% a_{k}z^{k}=A(z)\,\,(symbol{B},\,symbol{wB}).
  9. A ( z ) = k = 0 a k z k = k = 0 a k ( 0 e - t t k d t ) z k k ! = 0 e - t k = 0 a k ( t z ) k k ! d t , A(z)=\sum_{k=0}^{\infty}a_{k}z^{k}=\sum_{k=0}^{\infty}a_{k}\left(\int_{0}^{% \infty}e^{-t}t^{k}dt\right)\frac{z^{k}}{k!}=\int_{0}^{\infty}e^{-t}\sum_{k=0}^% {\infty}a_{k}\frac{(tz)^{k}}{k!}dt,
  10. a k z k = a ( z ) ( s y m b o l w B ) {\textstyle\sum}a_{k}z^{k}=a(z)\,(symbol{wB})
  11. a k z k = a ( z ) ( s y m b o l B ) {\textstyle\sum}a_{k}z^{k}=a(z)\,(symbol{B})
  12. a k z k = a ( z ) ( s y m b o l B ) {\textstyle\sum}a_{k}z^{k}=a(z)\,(symbol{B})
  13. lim t e - t A ( z t ) = 0 , \lim_{t\rightarrow\infty}e^{-t}\mathcal{B}A(zt)=0,
  14. a k z k = a ( z ) ( s y m b o l w B ) {\textstyle\sum}a_{k}z^{k}=a(z)\,(symbol{wB})
  15. | f ( z ) - a 0 - a 1 z - - a n - 1 z n - 1 | |f(z)-a_{0}-a_{1}z-\cdots-a_{n-1}z^{n-1}|
  16. C n + 1 n ! z n C^{n+1}n!z^{n}
  17. A ( z ) = k = 0 z k , A(z)=\sum_{k=0}^{\infty}z^{k},
  18. 0 e - t A ( t z ) d t = 0 e - t e t z d t = 1 1 - z \int_{0}^{\infty}e^{-t}\mathcal{B}A(tz)\,dt=\int_{0}^{\infty}e^{-t}e^{tz}\,dt=% \frac{1}{1-z}
  19. lim t e - t n = 0 1 - z n + 1 1 - z t n n ! = lim t e - t 1 - z ( e t - z e t z ) = 1 1 - z , \lim_{t\rightarrow\infty}e^{-t}\sum_{n=0}^{\infty}\frac{1-z^{n+1}}{1-z}\frac{t% ^{n}}{n!}=\lim_{t\rightarrow\infty}\frac{e^{-t}}{1-z}\big(e^{t}-ze^{tz}\big)=% \frac{1}{1-z},
  20. A ( z ) = k = 0 k ! ( - 1 z ) k , A(z)=\sum_{k=0}^{\infty}k!\left(-1\cdot z\right)^{k},
  21. A ( t ) k = 0 ( - 1 t ) k = 1 1 + t \mathcal{B}A(t)\equiv\sum_{k=0}^{\infty}\left(-1\cdot t\right)^{k}=\frac{1}{1+t}
  22. lim t e - t ( A ) ( z t ) = lim t e - t 1 + z t = 0 , \lim_{t\rightarrow\infty}e^{-t}(\mathcal{B}A)(zt)=\lim_{t\rightarrow\infty}% \frac{e^{-t}}{1+zt}=0,
  23. A ( z ) = k = 0 ( l = 0 ( - 1 ) l ( 2 l + 2 ) k ( 2 l + 1 ) ! ) z k . A(z)=\sum_{k=0}^{\infty}\left(\sum_{l=0}^{\infty}\frac{(-1)^{l}(2l+2)^{k}}{(2l% +1)!}\right)z^{k}.
  24. A ( t ) = l = 0 ( k = 0 ( ( 2 l + 2 ) t ) k k ! ) ( - 1 ) l ( 2 l + 1 ) ! = l = 0 e ( 2 l + 2 ) t ( - 1 ) l ( 2 l + 1 ) ! = e t l = 0 ( e t ) 2 l + 1 ( - 1 ) l ( 2 l + 1 ) ! = e t sin ( e t ) . \begin{aligned}\displaystyle\mathcal{B}A(t)&\displaystyle=\sum_{l=0}^{\infty}% \left(\sum_{k=0}^{\infty}\frac{\big((2l+2)t\big)^{k}}{k!}\right)\frac{(-1)^{l}% }{(2l+1)!}\\ &\displaystyle=\sum_{l=0}^{\infty}e^{(2l+2)t}\frac{(-1)^{l}}{(2l+1)!}\\ &\displaystyle=e^{t}\sum_{l=0}^{\infty}\big(e^{t}\big)^{2l+1}\frac{(-1)^{l}}{(% 2l+1)!}\\ &\displaystyle=e^{t}\sin\left(e^{t}\right).\end{aligned}
  25. 0 e t sin ( e 2 t ) d t = 1 sin ( u 2 ) d u = π 8 - S ( 1 ) < , \int_{0}^{\infty}e^{t}\sin(e^{2t})dt=\int_{1}^{\infty}\sin(u^{2})du=\frac{% \sqrt{\pi}}{8}-S(1)<\infty,
  26. lim t e ( z - 1 ) t sin ( e z t ) = 0 \lim_{t\rightarrow\infty}e^{(z-1)t}\sin\left(e^{zt}\right)=0
  27. a k z k = a ( z ) ( s y m b o l B ) , {\textstyle\sum}a_{k}z^{k}=a(z)\,(symbolB),
  28. Π P = { z : O z L P = } , \Pi_{P}=\{z\in\mathbb{C}\,\colon\,Oz\cap L_{P}=\varnothing\},
  29. Π A = cl ( P S A Π P ) . \Pi_{A}=\,\text{cl}\Big(\bigcap_{P\in S_{A}}\Pi_{P}\Big).
  30. S S\subset\mathbb{C}
  31. Π A \Pi_{A}
  32. S S
  33. P Π A P\in\Pi_{A}
  34. S S
  35. Π A \Pi_{A}
  36. Π A \Pi_{A}
  37. z int ( Π A ) z\in\,\text{int}(\Pi_{A})
  38. z \ Π A z\in\mathbb{C}\backslash\Pi_{A}
  39. z Π A z\in\partial\Pi_{A}
  40. A ( z ) \displaystyle A(z)
  41. Π A \Pi_{A}
  42. A ( z ) = k = 0 z 2 k , A(z)=\sum_{k=0}^{\infty}z^{2^{k}},
  43. | z | < 1 |z|<1
  44. Π A = B ( 0 , 1 ) \Pi_{A}=B(0,1)
  45. a k z 0 k = a ( z 0 ) ( s y m b o l w B ) {\textstyle\sum}a_{k}z_{0}^{k}=a(z_{0})\,(symbol{wB})
  46. a k z 0 k = O ( k - 1 2 ) , k 0 , a_{k}z_{0}^{k}=O\left(k^{-{\textstyle\frac{1}{2}}}\right),\qquad\forall k\geq 0,
  47. k = 0 a k z 0 k = a ( z 0 ) \sum_{k=0}^{\infty}a_{k}z_{0}^{k}=a(z_{0})

Born_approximation.html

  1. | Ψ 𝐩 ( ± ) |{\Psi_{\mathbf{p}}^{(\pm)}}\rangle
  2. | Ψ 𝐩 ( ± ) = | Ψ 𝐩 + G ( E p ± i ϵ ) V | Ψ 𝐩 ( ± ) |{\Psi_{\mathbf{p}}^{(\pm)}}\rangle=|{\Psi_{\mathbf{p}}^{\circ}}\rangle+G^{% \circ}(E_{p}\pm i\epsilon)V|{\Psi_{\mathbf{p}}^{(\pm)}}\rangle
  3. G G^{\circ}
  4. ϵ \epsilon
  5. | Ψ 𝐩 |{\Psi_{\mathbf{p}}^{\circ}}\rangle
  6. | Ψ 𝐩 ( ± ) |{\Psi_{\mathbf{p}}^{(\pm)}}\rangle
  7. | Ψ 𝐩 ( ± ) = | Ψ 𝐩 + G ( E p ± i ϵ ) V | Ψ 𝐩 |{\Psi_{\mathbf{p}}^{(\pm)}}\rangle=|{\Psi_{\mathbf{p}}^{\circ}}\rangle+G^{% \circ}(E_{p}\pm i\epsilon)V|{\Psi_{\mathbf{p}}^{\circ}}\rangle
  8. | Ψ 𝐩 ( ± ) |{\Psi_{\mathbf{p}}^{(\pm)}}\rangle
  9. | Ψ 𝐩 |{\Psi_{\mathbf{p}}^{\circ}}\rangle
  10. | Ψ 𝐩 1 ( ± ) |{\Psi_{\mathbf{p}}^{1}}^{(\pm)}\rangle
  11. V 1 V^{1}
  12. V = V 1 + V 2 V=V^{1}+V^{2}
  13. V V
  14. V 2 V^{2}
  15. V 1 V^{1}
  16. | Ψ 𝐩 1 ( ± ) = | Ψ 𝐩 + G ( E p ± i 0 ) V 1 | Ψ 𝐩 1 ( ± ) |{\Psi_{\mathbf{p}}^{1}}^{(\pm)}\rangle=|{\Psi_{\mathbf{p}}^{\circ}}\rangle+G^% {\circ}(E_{p}\pm i0)V^{1}|{\Psi_{\mathbf{p}}^{1}}^{(\pm)}\rangle
  17. | Ψ 𝐩 ( ± ) = | Ψ 𝐩 1 ( ± ) + G 1 ( E p ± i 0 ) V 2 | Ψ 𝐩 1 ( ± ) |{\Psi_{\mathbf{p}}^{(\pm)}}\rangle=|{\Psi_{\mathbf{p}}^{1}}^{(\pm)}\rangle+G^% {1}(E_{p}\pm i0)V^{2}|{\Psi_{\mathbf{p}}^{1}}^{(\pm)}\rangle

Bosonization.html

  1. ψ , ψ ¯ \psi,\bar{\psi}
  2. ϕ \phi
  3. ψ ¯ - ψ + = : exp ( i ϕ ) : , ψ ¯ - ψ + = : exp ( - i ϕ ) : \bar{\psi}_{-}\psi_{+}=:\exp(i\phi):,\qquad\bar{\psi}_{-}\psi_{+}=:\exp(-i\phi):
  4. ϕ = : ψ ¯ ψ : \partial\phi=:\bar{\psi}\psi:

Boudouard_reaction.html

  1. L o g ( K e q ) = 9141 T + 0.000224 T - 9.595 Log(K_{eq})=\frac{9141}{T}+0.000224T-9.595

Boustrophedon_transform.html

  1. ( a 0 , a 1 , a 2 , ) (a_{0},a_{1},a_{2},\ldots)
  2. ( b 0 , b 1 , b 2 , ) (b_{0},b_{1},b_{2},\ldots)
  3. a k a_{k}
  4. a k a_{k}
  5. b 0 = a 0 b_{0}=a_{0}
  6. b k | k > 0 b_{k}|k>0
  7. a k a_{k}
  8. T k , n T_{k,n}
  9. T k , 0 = a k for k 0 , T_{k,0}=a_{k}\quad\,\text{for }k\geq 0,
  10. T k , n = T k , n - 1 + T k - 1 , k - n for k n > 0. T_{k,n}=T_{k,n-1}+T_{k-1,k-n}\quad\,\text{for }k\geq n>0.
  11. b n = T n , n b_{n}=T_{n,n}
  12. T k , n T_{k,n}
  13. E G ( a n ; x ) = n = 0 a n x n n ! . EG(a_{n};x)=\sum_{n=0}^{\infty}a_{n}\frac{x^{n}}{n!}.
  14. E G ( b n ; x ) = ( sec x + tan x ) E G ( a n ; x ) . EG(b_{n};x)=(\sec x+\tan x)\,EG(a_{n};x).

Boxcars_(slang).html

  1. 1 - ( 35 36 ) 25 0.505532 {1-}\left(\dfrac{35}{36}\right)^{25}\approx 0.505532

Box–Jenkins.html

  1. ± 2 / N \pm 2/\sqrt{N}

Boy_or_Girl_paradox.html

  1. p = 1 / 2 p=1/2
  2. P ( B B | B T ) = P ( B T | B B ) P ( B B ) P ( B T ) P(BB|B_{T})=\frac{P(B_{T}|BB)P(BB)}{P(B_{T})}
  3. 1 - ( 1 - ϵ ) 2 1-(1-\epsilon)^{2}
  4. P ( B T ( G G ) ) P(B_{T}(GG))
  5. P ( B T ( B G ) ) P(B_{T}(BG))
  6. P ( B T ( G B ) ) P(B_{T}(GB))
  7. P ( B T ( B B ) ) P(B_{T}(BB))
  8. ϵ + ϵ - ϵ 2 \epsilon+\epsilon-\epsilon^{2}
  9. P ( B B | B T ) = ( 1 - ( 1 - ϵ ) 2 ) 1 4 1 4 0 + 1 4 ϵ + 1 4 ϵ + 1 4 ( ϵ + ϵ - ϵ 2 ) = ( 1 - ( 1 - ϵ ) 2 ) 4 ϵ - ϵ 2 P(BB|B_{T})=\frac{(1-(1-\epsilon)^{2})\frac{1}{4}}{\frac{1}{4}0+\frac{1}{4}% \epsilon+\frac{1}{4}\epsilon+\frac{1}{4}(\epsilon+\epsilon-\epsilon^{2})}=% \frac{(1-(1-\epsilon)^{2})}{4\epsilon-\epsilon^{2}}

Boyer–Lindquist_coordinates.html

  1. r , θ r,\theta
  2. ϕ \phi
  3. x = r 2 + a 2 sin θ cos ϕ {x}=\sqrt{r^{2}+a^{2}}\sin\theta\cos\phi
  4. y = r 2 + a 2 sin θ sin ϕ {y}=\sqrt{r^{2}+a^{2}}\sin\theta\sin\phi
  5. z = r cos θ {z}=r\cos\theta\quad
  6. M M
  7. J J
  8. Q Q
  9. G = c = 1 G=c=1
  10. d s 2 = - Δ Σ ( d t - a sin 2 θ d ϕ ) 2 + sin 2 θ Σ ( ( r 2 + a 2 ) d ϕ - a d t ) 2 + Σ Δ d r 2 + Σ d θ 2 ds^{2}=-\frac{\Delta}{\Sigma}\left(dt-a\sin^{2}\theta d\phi\right)^{2}+\frac{% \sin^{2}\theta}{\Sigma}\Big((r^{2}+a^{2})d\phi-adt\Big)^{2}+\frac{\Sigma}{% \Delta}dr^{2}+\Sigma d\theta^{2}
  11. Δ = r 2 - 2 M r + a 2 + Q 2 \Delta=r^{2}-2Mr+a^{2}+Q^{2}
  12. Σ = r 2 + a 2 cos 2 θ \Sigma=r^{2}+a^{2}\cos^{2}\theta
  13. a = J / M a=J/M
  14. M M
  15. a a
  16. Q Q

Brauer's_theorem_on_induced_characters.html

  1. λ H G \lambda^{G}_{H}
  2. θ , θ = 1 \langle\theta,\theta\rangle=1
  3. , \langle,\rangle
  4. [ ω ] \mathbb{Z}[\omega]
  5. [ ω ] \mathbb{Z}[\omega]
  6. [ ω ] \mathbb{Z}[\omega]

Breusch–Pagan_test.html

  1. y = β 0 + β 1 x + u , y=\beta_{0}+\beta_{1}x+u,\,
  2. u ^ \hat{u}
  3. u ^ 2 = γ 0 + γ 1 x + v . \hat{u}^{2}=\gamma_{0}+\gamma_{1}x+v.\,
  4. σ i 2 = h ( z i γ ) \sigma_{i}^{2}=h(z_{i}^{\prime}\gamma)
  5. z i = ( 1 , z 2 i , , z p i ) z_{i}=(1,z_{2i},\dots,z_{pi})
  6. ( p - 1 ) (p-1)\,
  7. γ 2 = = γ p = 0. \gamma_{2}=\dots=\gamma_{p}=0.
  8. L M = ( l θ ) ( - E [ 2 l θ θ ] ) - 1 ( l θ ) . LM=\left(\frac{\partial l}{\partial\theta}\right)^{\prime}\left(-E\left[\frac{% \partial^{2}l}{\partial\theta\partial\theta^{\prime}}\right]\right)^{-1}\left(% \frac{\partial l}{\partial\theta}\right).
  9. y = X β + ε . y=X\beta+\varepsilon.
  10. e i 2 = γ 1 + γ 2 z 2 i + + γ p z p i + η i . e_{i}^{2}=\gamma_{1}+\gamma_{2}z_{2i}+\dots+\gamma_{p}z_{pi}+\eta_{i}.
  11. n n\,
  12. L M = n R 2 . LM=nR^{2}\,.
  13. χ p - 1 2 \chi^{2}_{p-1}

Brier_score.html

  1. i 1... N i\in{1...N}
  2. o i o_{i}
  3. B S = 1 N t = 1 N ( f t - o t ) 2 BS=\frac{1}{N}\sum\limits_{t=1}^{N}(f_{t}-o_{t})^{2}\,\!
  4. f t f_{t}
  5. o t o_{t}
  6. B S = 1 N t = 1 N i = 1 R ( f t i - o t i ) 2 BS=\frac{1}{N}\sum\limits_{t=1}^{N}\sum\limits_{i=1}^{R}(f_{ti}-o_{ti})^{2}\,\!
  7. B S = R E L - R E S + U N C BS=REL-RES+UNC
  8. B S = 1 N k = 1 K n k ( 𝐟 𝐤 - 𝐨 ¯ 𝐤 ) 2 - 1 N k = 1 K n k ( 𝐨 ¯ 𝐤 - 𝐨 ¯ ) 2 + 𝐨 ¯ ( 1 - 𝐨 ¯ ) BS=\frac{1}{N}\sum\limits_{k=1}^{K}{n_{k}(\mathbf{f_{k}}-\mathbf{\bar{o}}_{% \mathbf{k}})}^{2}-\frac{1}{N}\sum\limits_{k=1}^{K}{n_{k}(\mathbf{\bar{o}_{k}}-% \bar{\mathbf{o}})}^{2}+\mathbf{\bar{o}}\left({1-\mathbf{\bar{o}}}\right)
  9. N \textstyle N
  10. K \textstyle K
  11. 𝐨 ¯ = t = 1 N < m t p l > o t / N \mathbf{\bar{o}}={\sum_{t=1}^{N}}\mathbf{<}mtpl>{{o_{t}}}/N
  12. n k n_{k}
  13. 𝐨 ¯ 𝐤 \mathbf{\overline{o}}_{\mathbf{k}}
  14. 𝐟 𝐤 \mathbf{f_{k}}
  15. < m t p l > f = ( 0.3 , 0.7 ) \mathbf{<}mtpl>{{f}}=(0.3,0.7)
  16. < m t p l > o = ( 1 , 0 ) \mathbf{<}mtpl>{{o}}=(1,0)
  17. B S = C A L + R E F BS=CAL+REF
  18. B S = 1 N k = 1 K n k ( 𝐟 𝐤 - 𝐨 ¯ 𝐤 ) 2 + 1 N k = 1 K n k ( 𝐨 ¯ 𝐤 ( 1 - 𝐨 ¯ 𝐤 ) ) BS=\frac{1}{N}\sum\limits_{k=1}^{K}{n_{k}(\mathbf{f_{k}}-\mathbf{\bar{o}}_{% \mathbf{k}})}^{2}+\frac{1}{N}\sum\limits_{k=1}^{K}{n_{k}(\mathbf{\bar{o}_{k}}(% 1-\mathbf{\bar{o}_{k}}}))