wpmath0000012_11

Rational_number.html

  1. \mathbb{Q}
  2. 2 \sqrt{2}
  3. a b = c d \frac{a}{b}=\frac{c}{d}
  4. a d = b c . ad=bc.
  5. a b < c d \frac{a}{b}<\frac{c}{d}
  6. a d < b c . ad<bc.
  7. - a - b = a b \frac{-a}{-b}=\frac{a}{b}
  8. a - b = - a b . \frac{a}{-b}=\frac{-a}{b}.
  9. a b + c d = a d + b c b d . \frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}.
  10. a b - c d = a d - b c b d . \frac{a}{b}-\frac{c}{d}=\frac{ad-bc}{bd}.
  11. a b c d = a c b d . \frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}.
  12. a b ÷ c d = a d b c . \frac{a}{b}\div\frac{c}{d}=\frac{ad}{bc}.
  13. a d b c = a b × d c . \frac{ad}{bc}=\frac{a}{b}\times\frac{d}{c}.
  14. - ( a b ) = - a b = a - b and ( a b ) - 1 = b a if a 0. -\left(\frac{a}{b}\right)=\frac{-a}{b}=\frac{a}{-b}\quad\mbox{and}~{}\quad% \left(\frac{a}{b}\right)^{-1}=\frac{b}{a}\mbox{ if }~{}a\neq 0.
  15. ( a b ) n = a n b n \left(\frac{a}{b}\right)^{n}=\frac{a^{n}}{b^{n}}
  16. ( a b ) - n = b n a n . \left(\frac{a}{b}\right)^{-n}=\frac{b^{n}}{a^{n}}.
  17. a 0 + 1 a 1 + 1 a 2 + 1 + 1 a n , a_{0}+\cfrac{1}{a_{1}+\cfrac{1}{a_{2}+\cfrac{1}{\ddots+\cfrac{1}{a_{n}}}}},
  18. ( m 1 , n 1 ) + ( m 2 , n 2 ) ( m 1 n 2 + n 1 m 2 , n 1 n 2 ) \left(m_{1},n_{1}\right)+\left(m_{2},n_{2}\right)\equiv\left(m_{1}n_{2}+n_{1}m% _{2},n_{1}n_{2}\right)
  19. ( m 1 , n 1 ) × ( m 2 , n 2 ) ( m 1 m 2 , n 1 n 2 ) \left(m_{1},n_{1}\right)\times\left(m_{2},n_{2}\right)\equiv\left(m_{1}m_{2},n% _{1}n_{2}\right)
  20. ( m 1 , n 1 ) ( m 2 , n 2 ) ( m 1 n 2 , n 1 m 2 ) . \frac{\left(m_{1},n_{1}\right)}{\left(m_{2},n_{2}\right)}\equiv\left(m_{1}n_{2% },n_{1}m_{2}\right).
  21. = [ ( - 2 m , - 2 n ) ] = [ ( - m , - n ) ] = [ ( m , n ) ] = [ ( 2 m , 2 n ) ] = . \cdots=[(-2m,-2n)]=[(-m,-n)]=[(m,n)]=[(2m,2n)]=\cdots.
  22. ( n 1 n 2 > 0 and m 1 n 2 n 1 m 2 ) ( n 1 n 2 < 0 and m 1 n 2 n 1 m 2 ) . (n_{1}n_{2}>0\ \and\ m_{1}n_{2}\leq n_{1}m_{2})\ \ (n_{1}n_{2}<0\ \and\ m_{1}n% _{2}\geq n_{1}m_{2}).
  23. a b < c d \frac{a}{b}<\frac{c}{d}
  24. b , d b,d
  25. a b < a d + b c 2 b d < c d . \frac{a}{b}<\frac{ad+bc}{2bd}<\frac{c}{d}.

Rationalisation_(mathematics).html

  1. 10 a \frac{10}{\sqrt{a}}
  2. a \sqrt{a}
  3. 10 a = 10 a a a = 10 a a 2 \frac{10}{\sqrt{a}}=\frac{10}{\sqrt{a}}\cdot\frac{\sqrt{a}}{\sqrt{a}}=\frac{{1% 0\sqrt{a}}}{\sqrt{a}^{2}}
  4. 10 a a 2 = 10 a a \frac{{10\sqrt{a}}}{\sqrt{a}^{2}}=\frac{10\sqrt{a}}{a}
  5. 10 a < m t p l > a \frac{{10\sqrt{a}}}{<}mtpl>{{a}}
  6. 10 b 3 \frac{10}{\sqrt[3]{b}}
  7. b 3 2 \sqrt[3]{b}^{2}
  8. 10 b 3 = 10 b 3 b 3 2 b 3 2 = 10 b 3 2 b 3 3 \frac{10}{\sqrt[3]{b}}=\frac{10}{\sqrt[3]{b}}\cdot\frac{\sqrt[3]{b}^{2}}{\sqrt% [3]{b}^{2}}=\frac{{10\sqrt[3]{b}^{2}}}{\sqrt[3]{b}^{3}}
  9. 10 b 3 2 b 3 3 = 10 b 3 2 b \frac{{10\sqrt[3]{b}^{2}}}{\sqrt[3]{b}^{3}}=\frac{10\sqrt[3]{b}^{2}}{b}
  10. 10 b 3 2 < m t p l > b \frac{{10\sqrt[3]{b}^{2}}}{<}mtpl>{{b}}
  11. 2 + 3 \sqrt{2}+\sqrt{3}\,
  12. 2 - 3 \sqrt{2}-\sqrt{3}\,
  13. 2 - 3 2 - 3 = 1. \frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}-\sqrt{3}}=1.
  14. x + y x+\sqrt{y}\,
  15. x - y x-\sqrt{y}
  16. 3 3 + 5 \frac{3}{\sqrt{3}+\sqrt{5}}
  17. 3 - 5 {\sqrt{3}-\sqrt{5}}
  18. 3 3 + 5 3 - 5 3 - 5 = 3 ( 3 - 5 ) 3 2 - 5 2 \frac{3}{\sqrt{3}+\sqrt{5}}\cdot\frac{\sqrt{3}-\sqrt{5}}{\sqrt{3}-\sqrt{5}}=% \frac{3(\sqrt{3}-\sqrt{5})}{\sqrt{3}^{2}-\sqrt{5}^{2}}
  19. 3 ( 3 - 5 ) 3 2 - 5 2 = 3 ( 3 - 5 ) 3 - 5 = 3 ( 3 - 5 ) - 2 \frac{{3(\sqrt{3}-\sqrt{5})}}{\sqrt{3}^{2}-\sqrt{5}^{2}}=\frac{3(\sqrt{3}-% \sqrt{5})}{3-5}=\frac{3(\sqrt{3}-\sqrt{5})}{-2}

Rauch_comparison_theorem.html

  1. M M
  2. M ~ \widetilde{M}
  3. γ : [ 0 , T ] M \gamma:[0,T]\to M
  4. γ ~ : [ 0 , T ] M ~ \widetilde{\gamma}:[0,T]\to\widetilde{M}
  5. γ ~ ( 0 ) \widetilde{\gamma}(0)
  6. γ ~ \widetilde{\gamma}
  7. J J
  8. J ~ \widetilde{J}
  9. γ \gamma
  10. γ ~ \widetilde{\gamma}
  11. J ( 0 ) = J ~ ( 0 ) = 0 J(0)=\widetilde{J}(0)=0
  12. | D t J ( 0 ) | = | D ~ t J ~ ( 0 ) | |D_{t}J(0)|=|\widetilde{D}_{t}\widetilde{J}(0)|
  13. M M
  14. M ~ \widetilde{M}
  15. K ( Π ) K ~ ( Π ~ ) K(\Pi)\leq\widetilde{K}(\widetilde{\Pi})
  16. Π T γ ( t ) M \Pi\subset T_{\gamma(t)}M
  17. γ ˙ ( t ) \dot{\gamma}(t)
  18. Π ~ T γ ~ ( t ) M ~ \widetilde{\Pi}\subset T_{\tilde{\gamma}(t)}\widetilde{M}
  19. γ ~ ˙ ( t ) \dot{\widetilde{\gamma}}(t)
  20. | J ( t ) | | J ~ ( t ) | |J(t)|\geq|\widetilde{J}(t)|
  21. t [ 0 , T ] t\in[0,T]

Rayleigh's_method_of_dimensional_analysis.html

  1. X = C X 1 a X 2 b X 3 c X n m X=CX_{1}^{a}X_{2}^{b}X_{3}^{c}\cdots X_{n}^{m}\,

Rayleigh_wave.html

  1. 1 / r {1}/{\sqrt{r}}
  2. r r

Real_number.html

  1. 2 \sqrt{2}
  2. π \pi
  3. 𝔠 \mathfrak{c}
  4. 0 \aleph_{0}
  5. 0 \aleph_{0}
  6. 𝔠 \mathfrak{c}
  7. π \pi
  8. π \pi
  9. π \pi
  10. ( ; + ; ; < ) (\mathbb{R};+;\cdot;<)
  11. π \pi
  12. e x = n = 0 x n n ! \mathrm{e}^{x}=\sum_{n=0}^{\infty}\frac{x^{n}}{n!}
  13. n = N M x n n ! \sum_{n=N}^{M}\frac{x^{n}}{n!}
  14. 1 \aleph_{1}
  15. 0 \aleph_{0}
  16. 2 \textstyle\sqrt{2}
  17. arcsin ( 2 23 ) \textstyle\arcsin\left({{2}\over{23}}\right)
  18. 0 1 x x d x \textstyle\int_{0}^{1}{x^{x}}\;dx

Redescending_M-estimator.html

  1. Ψ 2 d F ( Ψ d F ) 2 \frac{\int\Psi^{2}\,dF\,\!}{(\int\Psi^{\prime}\,dF\,\!)^{2}}
  2. Ψ ( x ) = { x , 0 | x | a (central segment) a sign ( x ) , a | x | b (high and low flat segments) a ( r - | x | ) r - b sign ( x ) , b | x | r (end slopes) 0 , r | x | (left and right tails) \Psi(x)=\begin{cases}x,&0\leq|x|\leq a\,\text{ (central segment)}\\ a\,\operatorname{sign}(x),&a\leq|x|\leq b\,\text{ (high and low flat segments)% }\\ \frac{a(r-|x|)}{r-b}\,\operatorname{sign}(x),&b\leq|x|\leq r\,\text{ (end % slopes)}\\ 0,&r\leq|x|\qquad\,\,\text{(left and right tails)}\end{cases}
  3. Ψ ( x ) = x ( 1 - ( x / k ) 2 ) 2 ; | x | k \Psi(x)=x(1-(x/k)^{2})^{2};\ |x|\leq k
  4. Ψ ( x ) = sin ( x ) ; - π x π \Psi(x)=\sin{(x)};\ -\pi\leq x\leq\pi

Redlich–Kwong_equation_of_state.html

  1. P = R T V m - b - a T V m ( V m + b ) , P=\frac{R\,T}{V_{m}-b}-\frac{a}{\sqrt{T}\;V_{m}\,(V_{m}+b)},
  2. a = 0.4275 R 2 T c 5 / 2 P c , b = 0.08664 R T c P c , a=\frac{0.4275\,R^{2}\,T_{c}^{5/2}}{P_{c}},\qquad b=\frac{0.08664\,R\,T_{c}}{P% _{c}},
  3. p p c < T 2 T c . \frac{p}{p_{c}}<\frac{T}{2T_{c}}.
  4. Z = P V m R T = 1 1 - h - A 2 B h 1 + h Z=\frac{P\,V_{m}}{R\,T}=\frac{1}{1-h}\ -\frac{A^{2}}{B}\frac{h}{1+h}
  5. A 2 = a R 2 T 2.5 = 0.4278 T c 2.5 P c T 2.5 A^{2}=\frac{a}{R^{2}\,T^{2.5}}=\frac{0.4278\,T_{c}^{2.5}}{P_{c}\,T^{2.5}}
  6. B = b R T = 0.0867 T c P c T B=\frac{b}{R\,T}=\frac{0.0867\,T_{c}}{P_{c}\,T}
  7. h = B P Z = b V m h=\frac{B\,P}{Z}=\frac{b}{V_{m}}
  8. Z c = 1 3 Z_{c}={1\over 3}
  9. ln ϕ = 0 P Z - 1 p d P = Z - 1 - ln ( Z - B P ) - A 2 B ln ( 1 + B P Z ) \ln\phi=\int_{0}^{P}{\frac{Z-1}{p}dP}=Z-1-\ln{(Z-B\,P)}-\frac{A^{2}}{B}\,\ln{(% 1+\frac{B\,P}{Z})}
  10. b = i x i b i b=\sum_{i}{x_{i}\,b_{i}}
  11. B = i x i B i B=\sum_{i}{x_{i}\,B_{i}}
  12. a = i j x i x j a i j a=\sum_{i}{\sum_{j}{x_{i}\,x_{j}\,a_{i\,j}}}
  13. a i j = ( a i a j ) 1 / 2 a_{i\,j}=({a_{i}\,a_{j}})^{1/2}
  14. A = i x i A i A=\sum_{i}{x_{i}\,A_{i}}
  15. p = R T V m - b - a V m 2 p=\frac{RT}{V_{\mathrm{m}}-b}-\frac{a}{V_{\mathrm{m}}^{2}}
  16. b = 0.26 V c b=0.26\ V_{c}
  17. a = α + γ T - 1.5 a=\alpha+\gamma\,T^{-1.5}
  18. P = R T V m - b - a α V m ( V m + b ) P=\frac{R\,T}{V_{m}-b}-\frac{a\,\alpha}{V_{m}\left(V_{m}+b\right)}
  19. α = ( 1 + ( 0.480 + 1.574 ω - 0.176 ω 2 ) ( 1 - T r 0.5 ) ) 2 \alpha=\left(1+\left(0.480+1.574\,\omega-0.176\,\omega^{2}\right)\left(1-T_{r}% ^{\,0.5}\right)\right)^{2}
  20. p = R T V m - b - a α V m ( V m + b ) + b ( V m - b ) p=\frac{R\,T}{V_{m}-b}-\frac{a\,\alpha}{V_{m}\,(V_{m}+b)+b\,(V_{m}-b)}
  21. a = 0.457235 R 2 T c 2 p c a=\frac{0.457235\,R^{2}\,T_{c}^{2}}{p_{c}}
  22. b = 0.077796 R T c p c b=\frac{0.077796\,R\,T_{c}}{p_{c}}
  23. α = ( 1 + ( 0.37464 + 1.54226 ω - 0.26992 ω 2 ) ( 1 - T r 0.5 ) ) 2 \alpha=\left(1+\left(0.37464+1.54226\,\omega-0.26992\,\omega^{2}\right)\left(1% -T_{r}^{\,0.5}\right)\right)^{2}
  24. P h s = R T V m - b = R T V m 1 1 - b V m P_{hs}=\frac{R\,T}{V_{m}-b}=\frac{R\,T}{V_{m}}\,\frac{1}{1-\frac{b}{V_{m}}}
  25. P h s = R T V m 1 - η 3 ( 1 - η ) 4 P_{hs}=\frac{R\,T}{V_{m}}\,\frac{1-\eta^{3}}{(1-\eta)^{4}}
  26. η = b 4 V m \eta=\frac{b}{4\,V_{m}}
  27. P h s = R T V m 1 + η + η 2 - η 3 ( 1 - η ) 3 P_{hs}=\frac{R\,T}{V_{m}}\,\frac{1+\eta+\eta^{2}-\eta^{3}}{(1-\eta)^{3}}
  28. P = R T V m - b - a V m 2 + ( 1 + 3 ω ) b V m - 3 ω b 2 P=\frac{R\,T}{V_{m}-b}-\frac{a}{V_{m}^{2}+(1+3\,\omega)\,b\,V_{m}-3\,\omega\,b% ^{2}}

Reduced_cost.html

  1. 𝐜 T 𝐱 \mathbf{c}^{T}\mathbf{x}
  2. 𝐀𝐱 𝐛 , 𝐱 0 \mathbf{Ax}\leq\mathbf{b},\mathbf{x}\geq 0
  3. 𝐜 - 𝐀 T 𝐲 \mathbf{c}-\mathbf{A}^{T}\mathbf{y}
  4. 𝐲 \mathbf{y}

Reduced_derivative.html

  1. f + ( t ) = lim s t f ( s ) ; f_{+}(t)=\lim_{s\downarrow t}f(s);
  2. f - ( t ) = lim s t f ( s ) . f_{-}(t)=\lim_{s\uparrow t}f(s).
  3. Var ( z , [ a , b ] ) = sup { i = 1 k z ( t i ) - z ( t i - 1 ) | a = t 0 < t 1 < < t k = b , k } . \mathrm{Var}(z,[a,b])=\sup\left\{\left.\sum_{i=1}^{k}\|z(t_{i})-z(t_{i-1})\|% \right|a=t_{0}<t_{1}<\cdots<t_{k}=b,k\in\mathbb{N}\right\}.
  4. τ ^ : [ 0 , T ] [ 0 , + ) ; \hat{\tau}\colon[0,T]\to[0,+\infty);
  5. τ ^ ( t ) = t + [ 0 , t ] d z = t + Var ( z , [ 0 , t ] ) . \hat{\tau}(t)=t+\int_{[0,t]}\|\mathrm{d}z\|=t+\mathrm{Var}(z,[0,t]).
  6. t ^ : [ 0 , T ^ ] [ 0 , T ] ; \hat{t}\colon[0,\hat{T}]\to[0,T];
  7. t ^ ( τ ) = max { t [ 0 , T ] | τ ^ ( t ) τ } . \hat{t}(\tau)=\max\{t\in[0,T]|\hat{\tau}(t)\leq\tau\}.
  8. z ^ C 0 ( [ 0 , T ^ ] ; X ) ; \hat{z}\in C^{0}([0,\hat{T}];X);
  9. z ^ ( τ ) = ( 1 - θ ) z - ( t ) + θ z + ( t ) \hat{z}(\tau)=(1-\theta)z_{-}(t)+\theta z_{+}(t)
  10. τ = ( 1 - θ ) τ ^ - ( t ) + θ τ ^ + ( t ) . \tau=(1-\theta)\hat{\tau}_{-}(t)+\theta\hat{\tau}_{+}(t).
  11. z ^ W 1 , ( [ 0 , T ^ ] ; X ) ; \hat{z}\in W^{1,\infty}([0,\hat{T}];X);
  12. d z ^ d τ L ( [ 0 , T ^ ] ; X ) 1. \left\|\frac{\mathrm{d}\hat{z}}{\mathrm{d}\tau}\right\|_{L^{\infty}([0,\hat{T}% ];X)}\leq 1.
  13. rd ( z ) : [ 0 , T ] { x X | x 1 } ; \mathrm{rd}(z)\colon[0,T]\to\{x\in X|\|x\|\leq 1\};
  14. rd ( z ) ( t ) = d z ^ d τ ( τ ^ - ( t ) + τ ^ + ( t ) 2 ) . \mathrm{rd}(z)(t)=\frac{\mathrm{d}\hat{z}}{\mathrm{d}\tau}\left(\frac{\hat{% \tau}_{-}(t)+\hat{\tau}_{+}(t)}{2}\right).
  15. μ z ( [ t 1 , t 2 ) ) = λ ( [ τ ^ ( t 1 ) , τ ^ ( t 2 ) ) = τ ^ ( t 2 ) - τ ^ ( t 1 ) = t 2 - t 1 + [ t 1 , t 2 ] d z . \mu_{z}([t_{1},t_{2}))=\lambda([\hat{\tau}(t_{1}),\hat{\tau}(t_{2}))=\hat{\tau% }(t_{2})-\hat{\tau}(t_{1})=t_{2}-t_{1}+\int_{[t_{1},t_{2}]}\|\mathrm{d}z\|.
  16. μ z ( { t } ) = z + ( t ) - z - ( t ) and rd ( z ) ( t ) = z + ( t ) - z - ( t ) z + ( t ) - z - ( t ) . \mu_{z}(\{t\})=\|z_{+}(t)-z_{-}(t)\|\mbox{ and }~{}\mathrm{rd}(z)(t)=\frac{z_{% +}(t)-z_{-}(t)}{\|z_{+}(t)-z_{-}(t)\|}.
  17. μ z ( ( t 1 , t 2 ) ) = t 1 t 2 1 + z ˙ ( t ) d t \mu_{z}((t_{1},t_{2}))=\int_{t_{1}}^{t_{2}}1+\|\dot{z}(t)\|\,\mathrm{d}t
  18. rd ( z ) ( t ) = z ˙ ( t ) 1 + z ˙ ( t ) \mathrm{rd}(z)(t)=\frac{\dot{z}(t)}{1+\|\dot{z}(t)\|}
  19. [ s , t ) rd ( z ) ( r ) d μ z ( r ) = [ s , t ) d z = z ( t ) - z ( s ) . \int_{[s,t)}\mathrm{rd}(z)(r)\,\mathrm{d}\mu_{z}(r)=\int_{[s,t)}\mathrm{d}z=z(% t)-z(s).

Redundant_binary_representation.html

  1. k = 0 n - 1 d k 2 k \sum_{k=0}^{n-1}d_{k}2^{k}
  2. d k \isin { - 1 , 0 , 1 } d_{k}\isin\{-1,0,1\}

Reeve_tetrahedron.html

  1. 3 \mathbb{R}^{3}
  2. ( 0 , 0 , 0 ) (0,0,0)
  3. ( 1 , 0 , 0 ) (1,0,0)
  4. ( 0 , 1 , 0 ) (0,1,0)
  5. ( 1 , 1 , r ) (1,1,r)
  6. r r
  7. 3 \mathbb{Z}^{3}
  8. 3 \mathbb{R}^{3}
  9. r r
  10. 𝒯 r \mathcal{T}_{r}
  11. r r
  12. L ( 𝒯 r , t ) = r 6 t 3 + t 2 + ( 2 - r 6 ) t + 1. L(\mathcal{T}_{r},t)=\frac{r}{6}t^{3}+t^{2}+\left(2-\frac{r}{6}\right)t+1.
  13. r 13 r\geq 13
  14. 𝒯 r \mathcal{T}_{r}

Reflectron.html

  1. t ( U ) = L 2 U m q + 2 L m 2 U U m m q t(U)=\frac{L}{\sqrt{2U}}\sqrt{\frac{m}{q}}\ +\frac{2L_{m}\sqrt{2U}}{U_{m}}% \sqrt{\frac{m}{q}}
  2. d t d U = 0 \frac{dt}{dU}=0
  3. L m = L 4 L_{m}=\frac{L}{4}
  4. E m = 4 U L E_{m}=\frac{4U}{L}
  5. d t t = k d 2 t d U 2 \frac{dt}{t}=k\frac{d^{2}t}{dU^{2}}

Regiomontanus'_angle_maximization_problem.html

  1. tan ( β - α ) = tan β - tan α 1 + tan β tan α = b x - a x 1 + b x a x = ( b - a ) x x 2 + a b . \tan(\beta-\alpha)=\frac{\tan\beta-\tan\alpha}{1+\tan\beta\tan\alpha}=\frac{% \frac{b}{x}-\frac{a}{x}}{1+\frac{b}{x}\cdot\frac{a}{x}}=(b-a)\frac{x}{x^{2}+ab}.
  2. d d x ( x x 2 + a b ) = a b - x 2 ( x 2 + a b ) 2 { > 0 if 0 x < a b , = 0 if x = a b , < 0 if x > a b . {d\over dx}\left(\frac{x}{x^{2}+ab}\right)=\frac{ab-x^{2}}{(x^{2}+ab)^{2}}% \qquad\begin{cases}{}>0&\,\text{if }0\leq x<\sqrt{ab\,{}},\\ {}=0&\,\text{if }x=\sqrt{ab\,{}},\\ {}<0&\,\text{if }x>\sqrt{ab\,{}}.\end{cases}
  3. x x 2 + a b . \frac{x}{x^{2}+ab}.
  4. x 2 + a b x = x + a b x . \frac{x^{2}+ab}{x}=x+\frac{ab}{x}.
  5. ( x - a b x ) 2 + 2 a b . \left(\sqrt{x}-\sqrt{\frac{ab}{x}}\,\right)^{2}+2\sqrt{ab\,{}}.
  6. x + a b x \displaystyle x+\frac{ab}{x}

Region_growing.html

  1. ( a ) i = 1 n R i = R . (a)\,\text{ }\bigcup\nolimits_{i=1}^{n}{R_{i}=R.}
  2. ( b ) R i is a connected region , i = 1 , 2 , , n (b)\,\text{ }R_{i}\,\text{ is a connected region},\,\text{ i}=\,\text{1},\,% \text{ 2},\,\text{ }...,\,\text{n}
  3. ( c ) R i R j = for all i = 1 , 2 , , n . (c)\,\text{ }R_{i}\bigcap R_{j}=\varnothing\,\text{ for all }i=1,2,...,n.
  4. ( d ) P ( R i ) = T R U E for i = 1 , 2 , , n . (d)\,\text{ }P(R_{i})=TRUE\,\text{ for }i=1,2,...,n.
  5. ( e ) P ( R i R j ) = F A L S E for any adjacent region R i and R j . (e)\,\text{ }P(R_{i}\bigcup R_{j})=FALSE\,\text{ for any adjacent region }R_{i% }\,\text{ and }R_{j}.
  6. P ( R i ) P(R_{i})
  7. R i R_{i}
  8. \varnothing
  9. P ( R i ) = TRUE P(R_{i})=\,\text{TRUE}
  10. R i R_{i}
  11. R i R_{i}
  12. R j R_{j}
  13. P P

Regression_discontinuity_design.html

  1. Y = α + τ D + β 1 ( X - c ) + β 2 D ( X - c ) + ε Y=\alpha+\tau D+\beta_{1}(X-c)+\beta_{2}D(X-c)+\varepsilon
  2. c - h X c + h c-h\leq X\leq c+h
  3. c c
  4. D D
  5. X c X\geq c
  6. h h

Regression_validation.html

  1. e i = y i - f ( x i ; β ^ ) , e_{i}=y_{i}-f(x_{i};\hat{\beta}),

Regret_(decision_theory).html

  1. x x
  2. x x
  3. x x
  4. y = H x + w y=Hx+w
  5. H H
  6. n × m n\times m
  7. m m
  8. w w
  9. C w C_{w}
  10. x ^ = G y \hat{x}=Gy
  11. x x
  12. y y
  13. G G
  14. m × n m\times n
  15. M S E = E ( || x ^ - x || 2 ) = T r ( G C w G * ) + x * ( I - G H ) * ( I - G H ) x . MSE=E\left(||\hat{x}-x||^{2}\right)=Tr(GC_{w}G^{*})+x^{*}(I-GH)^{*}(I-GH)x.
  16. x x
  17. x x
  18. G G
  19. x x
  20. x ^ o = G ( x ) y . \hat{x}^{o}=G(x)y.
  21. x ^ o \hat{x}^{o}
  22. M S E o = E ( || x ^ o - x || 2 ) = T r ( G ( x ) C w G ( x ) * ) + x * ( I - G ( x ) H ) * ( I - G ( x ) H ) x . MSE^{o}=E\left(||\hat{x}^{o}-x||^{2}\right)=Tr(G(x)C_{w}G(x)^{*})+x^{*}(I-G(x)% H)^{*}(I-G(x)H)x.
  23. G ( x ) G(x)
  24. G G
  25. G ( x ) = x x * H * ( C w + H x x * H * ) - 1 G(x)=xx^{*}H^{*}(C_{w}+Hxx^{*}H^{*})^{-1}
  26. G ( x ) = 1 1 + x * H * C w - 1 H x x x * H * C w - 1 . G(x)=\frac{1}{1+x^{*}H^{*}C_{w}^{-1}Hx}xx^{*}H^{*}C_{w}^{-1}.
  27. G ( x ) G(x)
  28. M S E o MSE^{o}
  29. M S E o = x * x 1 + x * H * C w - 1 H x . MSE^{o}=\frac{x^{*}x}{1+x^{*}H^{*}C_{w}^{-1}Hx}.
  30. x x
  31. R ( x , G ) = M S E - M S E o = T r ( G C w G * ) + x * ( I - G H ) * ( I - G H ) x - x * x 1 + x * H * C w - 1 H x . R(x,G)=MSE-MSE^{o}=Tr(GC_{w}G^{*})+x^{*}(I-GH)^{*}(I-GH)x-\frac{x^{*}x}{1+x^{*% }H^{*}C_{w}^{-1}Hx}.
  32. x x
  33. x x

Regular_4-polytope.html

  1. { p , q , r } \{p,q,r\}
  2. { p , q } , { q , r } \{p,q\},\{q,r\}
  3. sin ( π p ) sin ( π r ) < cos ( π q ) . \sin\left(\frac{\pi}{p}\right)\sin\left(\frac{\pi}{r}\right)<\cos\left(\frac{% \pi}{q}\right).
  4. N 0 - N 1 + N 2 - N 3 = 0 N_{0}-N_{1}+N_{2}-N_{3}=0\,

Regular_chain.html

  1. T = { x 2 2 - x 1 2 , x 2 ( x 3 - x 1 ) } T=\{x_{2}^{2}-x_{1}^{2},x_{2}(x_{3}-x_{1})\}
  2. R = k [ x 1 , , x n ] R=k[x_{1},\ldots,x_{n}]
  3. R R
  4. f = a e u e + + a 0 f=a_{e}u^{e}+\cdots+a_{0}
  5. a e a_{e}
  6. a e a_{e}
  7. R R
  8. h i h_{i}
  9. resultant ( h , T ) = resultant ( ( resultant ( h , t s ) , , t i ) ) 0 \mathrm{resultant}(h,T)=\mathrm{resultant}(\cdots(\mathrm{resultant}(h,t_{s}),% \ldots,t_{i})\cdots)\neq 0
  10. W ( T ) = V ( T ) V ( h ) W(T)=V(T)\setminus V(h)
  11. sat ( T ) = ( T ) : h \mathrm{sat}(T)=(T):h^{\infty}
  12. W ( T ) ¯ = V ( sat ( T ) ) \overline{W(T)}=V(\mathrm{sat}(T))
  13. ( F ) = i = 1 e sat ( T i ) \sqrt{(F)}=\cap_{i=1}^{e}\sqrt{\mathrm{sat}(T_{i})}
  14. V ( F ) = i = 1 e W ( T i ) V(F)=\cup_{i=1}^{e}W(T_{i})
  15. sat ( T k [ x 1 , , x i ] ) = sat ( T ) k [ x 1 , , x i ] \mathrm{sat}(T\cap k[x_{1},\ldots,x_{i}])=\mathrm{sat}(T)\cap k[x_{1},\ldots,x% _{i}]
  16. p sat ( T ) prem ( p , T ) = 0 p\in\mathrm{sat}(T)\iff\mathrm{prem}(p,T)=0
  17. prem ( p , T ) 0 \mathrm{prem}(p,T)\neq 0
  18. resultant ( p , T ) = 0 \mathrm{resultant}(p,T)=0

Regular_conditional_probability.html

  1. P ( A | B ) = P ( A B ) P ( B ) . P(A|B)=\frac{P(A\cap B)}{P(B)}.
  2. [ 0 , 1 ] , [0,1],
  3. X = 2 / 3. X=2/3.
  4. P ( B ) = 0 , P(B)=0,
  5. P ( A | X = 2 / 3 ) . P(A|X=2/3).
  6. ( Ω , , P ) (\Omega,\mathcal{F},P)
  7. T : Ω E T:\Omega\rightarrow E
  8. Ω \Omega
  9. ( E , ) . (E,\mathcal{E}).
  10. ν : E × [ 0 , 1 ] , \nu:E\times\mathcal{F}\rightarrow[0,1],
  11. ν ( x , A ) \nu(x,A)
  12. \mathcal{F}
  13. x E x\in E
  14. A , A\in\mathcal{F},
  15. A A\in\mathcal{F}
  16. B B\in\mathcal{E}
  17. P ( A T - 1 ( B ) ) = B ν ( x , A ) P ( T - 1 ( d x ) ) . P\big(A\cap T^{-1}(B)\big)=\int_{B}\nu(x,A)\,P\big(T^{-1}(dx)\big).
  18. P ( A | T = x ) = ν ( x , A ) , P(A|T=x)=\nu(x,A),
  19. x supp T , x\in\mathrm{supp}\,T,
  20. T * P = P ( T - 1 ( ) ) . T_{*}P=P\big(T^{-1}(\cdot)\big).
  21. ν \nu
  22. ( Ω , ) (\Omega,\mathcal{F})
  23. P P
  24. ( Ω , ) , (\Omega,\mathcal{F}),
  25. ( Ω , , P ) (\Omega,\mathcal{F},P)
  26. Ω \Omega
  27. P ( A | T = t ) = lim U { T = t } P ( A U ) P ( U ) , P(A|T=t)=\lim_{U\supset\{T=t\}}\frac{P(A\cap U)}{P(U)},
  28. ϵ > 0 , \epsilon>0,
  29. t V U , t\in V\subset U,
  30. | P ( A V ) P ( V ) - L | < ϵ , \left|\frac{P(A\cap V)}{P(V)}-L\right|<\epsilon,
  31. L = P ( A | T = t ) L=P(A|T=t)
  32. P ( A | X = x 0 ) = ν ( x 0 , A ) = lim ϵ 0 + P ( A { x 0 - ϵ < X < x 0 + ϵ } ) P ( { x 0 - ϵ < X < x 0 + ϵ } ) , P(A|X=x_{0})=\nu(x_{0},A)=\lim_{\epsilon\rightarrow 0+}\frac{P(A\cap\{x_{0}-% \epsilon<X<x_{0}+\epsilon\})}{P(\{x_{0}-\epsilon<X<x_{0}+\epsilon\})},
  33. x 0 = 2 / 3 x_{0}=2/3
  34. supp X . \mathrm{supp}\,X.
  35. x 0 x_{0}
  36. x 0 x_{0}
  37. ϵ > 0 \epsilon>0
  38. P ( { x 0 - ϵ < X < x 0 + ϵ } ) = 0. P(\{x_{0}-\epsilon<X<x_{0}+\epsilon\})=0.
  39. [ 0 , 1 ] , [0,1],
  40. X = 3 / 2 X=3/2

Regular_paperfolding_sequence.html

  1. t n = { 1 if m = 1 mod 4 0 if m = 3 mod 4 t_{n}=\begin{cases}1&\,\text{if }m=1\mod 4\\ 0&\,\text{if }m=3\mod 4\end{cases}
  2. t n = { 1 if n = 2 k 1 - t 2 k - n if 2 k - 1 < n < 2 k t_{n}=\begin{cases}1&\,\text{if }n=2^{k}\\ 1-t_{2^{k}-n}&\,\text{if }2^{k-1}<n<2^{k}\end{cases}
  3. G ( t n ; x ) = n = 0 t n x n . G(t_{n};x)=\sum_{n=0}^{\infty}t_{n}x^{n}\,.
  4. G ( t n ; x ) = G ( t n ; x 2 ) + n = 0 x 4 n + 1 = G ( t n ; x 2 ) + x 1 - x 4 . G(t_{n};x)=G(t_{n};x^{2})+\sum_{n=0}^{\infty}x^{4n+1}=G(t_{n};x^{2})+\frac{x}{% 1-x^{4}}\,.
  5. x = 0.5 x=0.5
  6. 0
  7. 1 1
  8. G ( t n ; 1 2 ) = n = 1 t n 2 n G(t_{n};\frac{1}{2})=\sum_{n=1}^{\infty}\frac{t_{n}}{2^{n}}
  9. k = 0 8 2 k 2 2 k + 2 - 1 = 0.85073618820186... \sum_{k=0}^{\infty}\frac{8^{2^{k}}}{2^{2^{k+2}}-1}=0.85073618820186...
  10. F a : w w a w F_{a}:w\mapsto waw^{\ddagger}
  11. w n = F f 1 ( F f 2 ( F f n ( ε ) ) ) . w_{n}=F_{f_{1}}(F_{f_{2}}(\cdots F_{f_{n}}(\varepsilon)\cdots))\ .
  12. t n = { f j if m = 1 mod 4 1 - f j if m = 3 mod 4 t_{n}=\begin{cases}f_{j}&\,\text{if }m=1\mod 4\\ 1-f_{j}&\,\text{if }m=3\mod 4\end{cases}

Relation_between_Schrödinger's_equation_and_the_path_integral_formulation_of_quantum_mechanics.html

  1. i d d t | ψ = H ^ | ψ i\hbar\frac{d}{dt}|\psi\rangle=\hat{H}|\psi\rangle
  2. H ^ \hat{H}
  3. H ^ = p ^ 2 2 m + V ( q ^ ) \hat{H}=\frac{\hat{p}^{2}}{2m}+V(\hat{q})
  4. V ( q ^ ) V(\hat{q})
  5. q q
  6. | ψ ( t ) = exp ( - i H ^ t ) | q 0 exp ( - i H ^ t ) | 0 |\psi(t)\rangle=\exp\left(-\frac{i}{\hbar}\hat{H}t\right)|q_{0}\rangle\equiv% \exp\left(-\frac{i}{\hbar}\hat{H}t\right)|0\rangle
  7. | q 0 |q_{0}\rangle
  8. | 0 |0\rangle
  9. | F |F\rangle
  10. T T
  11. F | ψ ( t ) = F | exp ( - i H ^ T ) | 0 . \langle F|\psi(t)\rangle=\left\langle F\bigg|\exp\left(-\frac{i}{\hbar}\hat{H}% T\right)\bigg|0\right\rangle.
  12. exp ( i S ) \exp\left(\frac{i}{\hbar}S\right)
  13. V ( q ) V(q)
  14. p p
  15. p p
  16. m m
  17. δ t δt
  18. p = m ( x b - x a δ t ) p=m\left(\frac{x_{b}-x_{a}}{\delta t}\right)
  19. 0 , T 0,T
  20. N N
  21. δ t = T N . \delta t=\frac{T}{N}.
  22. F | exp ( - i H ^ T ) | 0 = F | exp ( - i H ^ δ t ) exp ( - i H ^ δ t ) exp ( - i H ^ δ t ) | 0 . \left\langle F\bigg|\exp\left(-\frac{i}{\hbar}\hat{H}T\right)\bigg|0\right% \rangle=\left\langle F\bigg|\exp\left(-\frac{i}{\hbar}\hat{H}\delta t\right)% \exp\left(-\frac{i}{\hbar}\hat{H}\delta t\right)\cdots\exp\left(-\frac{i}{% \hbar}\hat{H}\delta t\right)\bigg|0\right\rangle.
  23. I = d q | q q | I=\int dq|q\rangle\langle q|
  24. N 1 N−1
  25. F | exp ( - i H ^ T ) | 0 = ( j = 1 N - 1 d q j ) F | exp ( - i H ^ δ t ) | q N - 1 q N - 1 | exp ( - i H ^ δ t ) | q N - 2 q 1 | exp ( - i H ^ δ t ) | 0 . \left\langle F\bigg|\exp\left(-\frac{i}{\hbar}\hat{H}T\right)\bigg|0\right% \rangle=\left(\prod_{j=1}^{N-1}\int dq_{j}\right)\left\langle F\bigg|\exp\left% (-\frac{i}{\hbar}\hat{H}\delta t\right)\bigg|q_{N-1}\right\rangle\left\langle q% _{N-1}\bigg|\exp\left(-\frac{i}{\hbar}\hat{H}\delta t\right)\bigg|q_{N-2}% \right\rangle\cdots\left\langle q_{1}\bigg|\exp\left(-\frac{i}{\hbar}\hat{H}% \delta t\right)\bigg|0\right\rangle.
  26. q j + 1 | exp ( - i H ^ δ t ) | q j = q j + 1 | exp ( - i p ^ 2 2 m δ t ) exp ( - i V ( q j ) δ t ) | q j . \left\langle q_{j+1}\bigg|\exp\left(-\frac{i}{\hbar}\hat{H}\delta t\right)% \bigg|q_{j}\right\rangle=\left\langle q_{j+1}\Bigg|\exp\left({-{i\over\hbar}{{% \hat{p}}^{2}\over 2m}\delta t}\right)\exp\left({-{i\over\hbar}V\left(q_{j}% \right)\delta t}\right)\Bigg|q_{j}\right\rangle.
  27. I = d p 2 π | p p | I=\int{dp\over 2\pi}|p\rangle\langle p|
  28. q j + 1 | exp ( - i H ^ δ t ) | q j = exp ( - i V ( q j ) δ t ) d p 2 π q j + 1 | exp ( - i p 2 2 m δ t ) | p p | q j = exp ( - i V ( q j ) δ t ) d p 2 π exp ( - i p 2 2 m δ t ) q j + 1 | p p | q j = exp ( - i V ( q j ) δ t ) d p 2 π exp ( - i p 2 2 m δ t - i p ( q j + 1 - q j ) ) \begin{aligned}\displaystyle\left\langle q_{j+1}\bigg|\exp\left(-\frac{i}{% \hbar}\hat{H}\delta t\right)\bigg|q_{j}\right\rangle&\displaystyle=\exp\left(-% \frac{i}{\hbar}V\left(q_{j}\right)\delta t\right)\int\frac{dp}{2\pi}\left% \langle q_{j+1}\bigg|\exp\left(-\frac{i}{\hbar}\frac{p^{2}}{2m}\delta t\right)% \bigg|p\right\rangle\langle p|q_{j}\rangle\\ &\displaystyle=\exp\left(-\frac{i}{\hbar}V\left(q_{j}\right)\delta t\right)% \int\frac{dp}{2\pi}\exp\left(-\frac{i}{\hbar}\frac{p^{2}}{2m}\delta t\right)% \left\langle q_{j+1}|p\right\rangle\left\langle p|q_{j}\right\rangle\\ &\displaystyle=\exp\left(-\frac{i}{\hbar}V\left(q_{j}\right)\delta t\right)% \int\frac{dp}{2\pi}\exp\left(-\frac{i}{\hbar}\frac{p^{2}}{2m}\delta t-\frac{i}% {\hbar}p\left(q_{j+1}-q_{j}\right)\right)\end{aligned}
  29. p | q j = exp ( i p q j ) \langle p|q_{j}\rangle=\frac{\exp\left(\frac{i}{\hbar}pq_{j}\right)}{\sqrt{% \hbar}}
  30. q j + 1 | exp ( - i H ^ δ t ) | q j = ( - i m 2 π δ t ) 1 2 exp [ i δ t ( 1 2 m ( q j + 1 - q j δ t ) 2 - V ( q j ) ) ] \left\langle q_{j+1}\bigg|\exp\left(-\frac{i}{\hbar}\hat{H}\delta t\right)% \bigg|q_{j}\right\rangle=\left({-im\over 2\pi\delta t\hbar}\right)^{1\over 2}% \exp\left[{i\over\hbar}\delta t\left({1\over 2}m\left({q_{j+1}-q_{j}\over% \delta t}\right)^{2}-V\left(q_{j}\right)\right)\right]
  31. F | exp ( - i H ^ T ) | 0 = ( - i m 2 π δ t ) N 2 ( j = 1 N - 1 d q j ) exp [ i j = 0 N - 1 δ t ( 1 2 m ( q j + 1 - q j δ t ) 2 - V ( q j ) ) ] . \left\langle F\bigg|\exp\left(-\frac{i}{\hbar}\hat{H}T\right)\bigg|0\right% \rangle=\left({-im\over 2\pi\delta t\hbar}\right)^{N\over 2}\left(\prod_{j=1}^% {N-1}\int dq_{j}\right)\exp\left[{i\over\hbar}\sum_{j=0}^{N-1}\delta t\left({1% \over 2}m\left({q_{j+1}-q_{j}\over\delta t}\right)^{2}-V\left(q_{j}\right)% \right)\right].
  32. N N
  33. F | exp ( - i H ^ T ) | 0 = D q ( t ) exp [ i S ] \left\langle F\bigg|\exp\left({-{i\over\hbar}\hat{H}T}\right)\bigg|0\right% \rangle=\int Dq(t)\exp\left[{i\over\hbar}S\right]
  34. S = 0 T d t L ( q ( t ) , q ˙ ( t ) ) S=\int_{0}^{T}dtL\left(q(t),\dot{q}(t)\right)
  35. L ( q , q ˙ ) = 1 2 m q ˙ 2 - V ( q ) L\left(q,\dot{q}\right)={1\over 2}m{\dot{q}}^{2}-V(q)
  36. D q ( t ) = lim N ( - i m 2 π δ t ) N 2 ( j = 1 N - 1 d q j ) \int Dq(t)=\lim_{N\to\infty}\left(\frac{-im}{2\pi\delta t\hbar}\right)^{\frac{% N}{2}}\left(\prod_{j=1}^{N-1}\int dq_{j}\right)

Relational_approach_to_quantum_physics.html

  1. E ( 𝐫 , t ) = E ( + ) ( 𝐫 , t ) + E ( - ) ( 𝐫 , t ) E(\mathbf{r},t)=E^{(+)}(\mathbf{r},t)+E^{(-)}(\mathbf{r},t)
  2. E ( - ) ( 𝐫 , t ) = E ( + ) ( 𝐫 , t ) E^{(-)}(\mathbf{r},t)=E^{(+)}(\mathbf{r},t)^{\dagger}
  3. E ( + ) ( 𝐫 , t ) E^{(+)}(\mathbf{r},t)
  4. E ( + ) ( 𝐫 , t ) = i i [ ω i 2 ] 1 / 2 a ^ i ε i e i ( 𝐤 i 𝐫 - ω i t ) E^{(+)}(\mathbf{r},t)=i\sum_{i}[\frac{\hbar\omega_{i}}{2}]^{1/2}\hat{a}_{i}% \mathbf{\varepsilon}_{i}e^{i(\mathbf{k}_{i}\cdot\mathbf{r}-\omega_{i}t)}
  5. ε i \mathbf{\varepsilon}_{i}
  6. a ^ i \hat{a}_{i}
  7. 𝐫 \mathbf{r}
  8. t {t}
  9. t + d t {\it t}+d{\it t}
  10. W I ( 𝐫 , t ) d t W_{I}(\mathbf{r},t)d{\it t}
  11. W I ( 𝐫 , t ) = ψ | E ( - ) ( 𝐫 , t ) E ( + ) ( 𝐫 , t ) | ψ {W_{I}(\mathbf{r},t)}=\langle\psi|{E^{(-)}(\mathbf{r},t)}\cdot{E^{(+)}(\mathbf% {r},t)}|\psi\rangle
  12. | ψ |\psi\rangle
  13. | ψ = k c k a ^ k | 0 {|\psi\rangle}=\sum_{k}c_{k}{\hat{a}_{k}}^{\dagger}|0\rangle
  14. E ( + ) ( x , t ) = c 2 k k a ^ k e i ( k x - ω t ) {E^{(+)}({x,t)}={\sqrt{\frac{\hbar c}{2}}}\sum_{k}\sqrt{k}{\hat{a}_{k}}e^{i(kx% -\omega t)}}
  15. x {x}
  16. W I ( x , t ) = c 2 k k c k e i ( k x - ω t ) 2 {W_{I}(x,t)}=\frac{\hbar c}{2}\mid\sum_{k}\sqrt{k}c_{k}e^{i(kx-\omega t)}\mid^% {2}
  17. E ( + ) ( 𝐱 , 0 ) E^{(+)}(\mathbf{x},0)
  18. Δ k \Delta k
  19. Δ x \Delta x
  20. Δ k Δ x 1 \Delta k\ \Delta x\geq 1
  21. Δ x \Delta x
  22. Δ k \Delta k
  23. Δ x \Delta x
  24. Δ k \Delta k
  25. Δ x \Delta x
  26. Δ x \Delta x
  27. E ( 𝐫 , t ) E(\mathbf{r},t)
  28. W I ( 𝐫 , t ) {W_{I}(\mathbf{r},t)}
  29. E ( 𝐫 , t ) E(\mathbf{r},t)
  30. a ^ i {\hat{a}}_{i}
  31. a ^ i {\hat{a}}_{i}^{\dagger}
  32. e i ( 𝐤 i 𝐫 - ω i t ) e^{i(\mathbf{k}_{i}\cdot\mathbf{r}-\omega_{i}t)}
  33. e - i ( 𝐤 i 𝐫 - ω i t ) e^{-i(\mathbf{k}_{i}\cdot\mathbf{r}-\omega_{i}t)}
  34. E ( 𝐫 , t ) E(\mathbf{r},t)
  35. a ^ i {\hat{a}}_{i}
  36. a ^ i {\hat{a}}_{i}^{\dagger}
  37. e i ( 𝐤 i 𝐫 - ω i t ) e^{i(\mathbf{k}_{i}\cdot\mathbf{r}-\omega_{i}t)}
  38. a ^ i {\hat{a}}_{i}
  39. a ^ i {\hat{a}}_{i}^{\dagger}
  40. e i ( 𝐤 i 𝐫 - ω i t ) e^{i(\mathbf{k}_{i}\cdot\mathbf{r}-\omega_{i}t)}
  41. e - i ( 𝐤 i 𝐫 - ω i t ) e^{-i(\mathbf{k}_{i}\cdot\mathbf{r}-\omega_{i}t)}
  42. a ^ i {\hat{a}}_{i}
  43. a ^ i {\hat{a}}_{i}^{\dagger}
  44. a ^ i {\hat{a}}_{i}
  45. a ^ i {\hat{a}}_{i}^{\dagger}
  46. e i ( 𝐤 i 𝐫 - ω i t ) e^{i(\mathbf{k}_{i}\cdot\mathbf{r}-\omega_{i}t)}
  47. e - i ( 𝐤 i 𝐫 - ω i t ) e^{-i(\mathbf{k}_{i}\cdot\mathbf{r}-\omega_{i}t)}
  48. p = h / λ p=h/\lambda

Relations_between_heat_capacities.html

  1. C V C_{V}
  2. C P C_{P}
  3. C P - C V = V T α 2 β T C_{P}-C_{V}=VT\frac{\alpha^{2}}{\beta_{T}}\,
  4. C P C V = β T β S \frac{C_{P}}{C_{V}}=\frac{\beta_{T}}{\beta_{S}}\,
  5. α \alpha
  6. α = 1 V ( V T ) P \alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P}\,
  7. β T \beta_{T}
  8. β T = - 1 V ( V P ) T \beta_{T}=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}\,
  9. β S \beta_{S}
  10. β S = - 1 V ( V P ) S \beta_{S}=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{S}\,
  11. c p - c v = α 2 T ρ β T c_{p}-c_{v}=\frac{\alpha^{2}T}{\rho\beta_{T}}
  12. c p c v = β T β S \frac{c_{p}}{c_{v}}=\frac{\beta_{T}}{\beta_{S}}\,
  13. δ Q \delta Q
  14. d S = δ Q T dS=\frac{\delta Q}{T}\,
  15. δ Q = C d T \delta Q=CdT\,
  16. T d S = C d T TdS=CdT\,
  17. d S = ( S T ) V d T + ( S V ) T d V dS=\left(\frac{\partial S}{\partial T}\right)_{V}dT+\left(\frac{\partial S}{% \partial V}\right)_{T}dV
  18. C V = T ( S T ) V C_{V}=T\left(\frac{\partial S}{\partial T}\right)_{V}\,
  19. C P = T ( S T ) P C_{P}=T\left(\frac{\partial S}{\partial T}\right)_{P}\,
  20. C P - C V C_{P}-C_{V}
  21. d V = ( V T ) P d T + ( V P ) T d P dV=\left(\frac{\partial V}{\partial T}\right)_{P}dT+\left(\frac{\partial V}{% \partial P}\right)_{T}dP\,
  22. d S = [ ( S T ) V + ( S V ) T ( V T ) P ] d T + ( S V ) T ( V P ) T d P dS=\left[\left(\frac{\partial S}{\partial T}\right)_{V}+\left(\frac{\partial S% }{\partial V}\right)_{T}\left(\frac{\partial V}{\partial T}\right)_{P}\right]% dT+\left(\frac{\partial S}{\partial V}\right)_{T}\left(\frac{\partial V}{% \partial P}\right)_{T}dP
  23. ( S T ) P = ( S T ) V + ( S V ) T ( V T ) P \left(\frac{\partial S}{\partial T}\right)_{P}=\left(\frac{\partial S}{% \partial T}\right)_{V}+\left(\frac{\partial S}{\partial V}\right)_{T}\left(% \frac{\partial V}{\partial T}\right)_{P}\,
  24. C P - C V = T ( S V ) T ( V T ) P = V T α ( S V ) T C_{P}-C_{V}=T\left(\frac{\partial S}{\partial V}\right)_{T}\left(\frac{% \partial V}{\partial T}\right)_{P}=VT\alpha\left(\frac{\partial S}{\partial V}% \right)_{T}\,
  25. ( S V ) T \left(\frac{\partial S}{\partial V}\right)_{T}
  26. d E = T d S - P d V dE=TdS-PdV\,
  27. F = E - T S F=E-TS
  28. d F = - S d T - P d V dF=-SdT-PdV\,
  29. - S = ( F T ) V -S=\left(\frac{\partial F}{\partial T}\right)_{V}\,
  30. - P = ( F V ) T -P=\left(\frac{\partial F}{\partial V}\right)_{T}\,
  31. ( S V ) T = ( P T ) V \left(\frac{\partial S}{\partial V}\right)_{T}=\left(\frac{\partial P}{% \partial T}\right)_{V}\,
  32. C P - C V = V T α ( P T ) V C_{P}-C_{V}=VT\alpha\left(\frac{\partial P}{\partial T}\right)_{V}\,
  33. d V = ( V P ) T d P + ( V T ) P d T dV=\left(\frac{\partial V}{\partial P}\right)_{T}dP+\left(\frac{\partial V}{% \partial T}\right)_{P}dT\,
  34. ( P T ) V \left(\frac{\partial P}{\partial T}\right)_{V}
  35. ( P T ) V = - ( V T ) P ( V P ) T = α β T \left(\frac{\partial P}{\partial T}\right)_{V}=-\frac{\left(\frac{\partial V}{% \partial T}\right)_{P}}{\left(\frac{\partial V}{\partial P}\right)_{T}}=\frac{% \alpha}{\beta_{T}}\,
  36. C P - C V = V T α 2 β T C_{P}-C_{V}=VT\frac{\alpha^{2}}{\beta_{T}}\,
  37. C P C V = ( S T ) P ( S T ) V \frac{C_{P}}{C_{V}}=\frac{\left(\frac{\partial S}{\partial T}\right)_{P}}{% \left(\frac{\partial S}{\partial T}\right)_{V}}\,
  38. d P = ( P S ) T d S + ( P T ) S d T dP=\left(\frac{\partial P}{\partial S}\right)_{T}dS+\left(\frac{\partial P}{% \partial T}\right)_{S}dT\,
  39. d P = 0 dP=0
  40. d S d T \frac{dS}{dT}
  41. ( S T ) P \left(\frac{\partial S}{\partial T}\right)_{P}
  42. ( S T ) P = - ( P T ) S ( P S ) T \left(\frac{\partial S}{\partial T}\right)_{P}=-\frac{\left(\frac{\partial P}{% \partial T}\right)_{S}}{\left(\frac{\partial P}{\partial S}\right)_{T}}\,
  43. ( S T ) V \left(\frac{\partial S}{\partial T}\right)_{V}
  44. d S d T \frac{dS}{dT}
  45. C P C V = ( P T ) S ( P S ) T ( V S ) T ( V T ) S \frac{C_{P}}{C_{V}}=\frac{\left(\frac{\partial P}{\partial T}\right)_{S}}{% \left(\frac{\partial P}{\partial S}\right)_{T}}\frac{\left(\frac{\partial V}{% \partial S}\right)_{T}}{\left(\frac{\partial V}{\partial T}\right)_{S}}\,
  46. ( P T ) S ( V T ) S = ( P T ) S ( T V ) S = ( P V ) S \frac{\left(\frac{\partial P}{\partial T}\right)_{S}}{\left(\frac{\partial V}{% \partial T}\right)_{S}}=\left(\frac{\partial P}{\partial T}\right)_{S}\left(% \frac{\partial T}{\partial V}\right)_{S}=\left(\frac{\partial P}{\partial V}% \right)_{S}\,
  47. ( V S ) T ( P S ) T = ( V S ) T ( S P ) T = ( V P ) T \frac{\left(\frac{\partial V}{\partial S}\right)_{T}}{\left(\frac{\partial P}{% \partial S}\right)_{T}}=\left(\frac{\partial V}{\partial S}\right)_{T}\left(% \frac{\partial S}{\partial P}\right)_{T}=\left(\frac{\partial V}{\partial P}% \right)_{T}\,
  48. C P C V = ( P V ) S ( V P ) T = β T β S \frac{C_{P}}{C_{V}}=\left(\frac{\partial P}{\partial V}\right)_{S}\left(\frac{% \partial V}{\partial P}\right)_{T}=\frac{\beta_{T}}{\beta_{S}}\,
  49. C P - C V C_{P}-C_{V}\,
  50. P V = n R T PV=nRT\,
  51. V = n R T / P V=nRT/P\,
  52. n R = P V / T \,nR=PV/T
  53. ( V T ) P = n R P = ( V P T ) ( 1 P ) = V T \left(\frac{\partial V}{\partial T}\right)_{P}\ =\frac{nR}{P}\ =\left(\frac{VP% }{T}\right)\left(\frac{1}{P}\right)=\frac{V}{T}
  54. ( V P ) T = - n R T P 2 = - P V P 2 = - V P \left(\frac{\partial V}{\partial P}\right)_{T}\ =-\frac{nRT}{P^{2}}\ =-\frac{% PV}{P^{2}}\ =-\frac{V}{P}
  55. α \alpha
  56. α = 1 V ( V T ) P = 1 V ( V T ) \alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P}\ =\frac{1}{V}% \left(\frac{V}{T}\right)
  57. α = 1 / T \alpha=1/T\,
  58. β T \beta_{T}
  59. β T = - 1 V ( V P ) T = - 1 V ( - V P ) \beta_{T}=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}\ =-\frac{% 1}{V}\left(-\frac{V}{P}\right)
  60. β T = 1 / P \beta_{T}=1/P\,
  61. C P - C V C_{P}-C_{V}\,
  62. C P - C V = V T α 2 β T = V T ( 1 / T ) 2 1 / P = V P T C_{P}-C_{V}=VT\frac{\alpha^{2}}{\beta_{T}}\ =VT\frac{(1/T)^{2}}{1/P}=\frac{VP}% {T}
  63. C P - C V = n R C_{P}-C_{V}=nR\,
  64. C P , m - C V , m = C P - C V n = n R n = R C_{P,m}-C_{V,m}=\frac{C_{P}-C_{V}}{n}=\frac{nR}{n}=R
  65. c p - c v c_{p}-c_{v}\,

Relationships_among_probability_distributions.html

  1. X ( U / ν ) \frac{X}{\sqrt{(U/\nu)}}

Relativistic_heat_conduction.html

  1. θ t = α 2 θ , \frac{\partial\theta}{\partial t}~{}=~{}\alpha~{}\nabla^{2}\theta,
  2. 2 \scriptstyle\nabla^{2}
  3. 2 = 2 x 2 + 2 y 2 + 2 z 2 . \nabla^{2}~{}=~{}\frac{\partial^{2}}{\partial x^{2}}~{}+~{}\frac{\partial^{2}}% {\partial y^{2}}~{}+~{}\frac{\partial^{2}}{\partial z^{2}}.
  4. 𝐪 = - k θ , \mathbf{q}~{}=~{}-k~{}\nabla\theta,
  5. ρ c θ t + 𝐪 = 0 , \rho~{}c~{}\frac{\partial\theta}{\partial t}~{}+~{}\nabla\cdot\mathbf{q}~{}=~{% }0,
  6. = x 𝐢 + y 𝐣 + z 𝐤 . \nabla~{}=~{}\frac{\partial}{\partial x}~{}\mathbf{i}~{}+~{}\frac{\partial}{% \partial y}~{}\mathbf{j}~{}+~{}\frac{\partial}{\partial z}~{}\mathbf{k}.
  7. ( 𝐪 θ ) + ρ s t = σ , \nabla\cdot\left(\frac{\mathbf{q}}{\theta}\right)~{}+~{}\rho~{}\frac{\partial s% }{\partial t}~{}=~{}\sigma,
  8. σ = - 1 θ 2 𝐪 θ , \sigma~{}=~{}\frac{-1}{\theta^{2}}~{}\mathbf{q}\cdot\nabla\theta,
  9. σ = k θ 2 [ ( θ x ) 2 + ( θ y ) 2 + ( θ z ) 2 ] , \sigma~{}=~{}\frac{k}{\theta^{2}}~{}\left[{\left(\frac{\partial\theta}{% \partial x}\right)}^{2}~{}+~{}{\left(\frac{\partial\theta}{\partial y}\right)}% ^{2}~{}+~{}{\left(\frac{\partial\theta}{\partial z}\right)}^{2}\right],
  10. 1 C 2 2 θ t 2 + 1 α θ t = 2 θ , \frac{1}{C^{2}}~{}\frac{\partial^{2}\theta}{\partial t^{2}}~{}+~{}\frac{1}{% \alpha}~{}\frac{\partial\theta}{\partial t}~{}=~{}\nabla^{2}\theta,
  11. τ 0 𝐪 t + 𝐪 = - k θ , \tau_{{}_{0}}~{}\frac{\partial\mathbf{q}}{\partial t}~{}+~{}\mathbf{q}~{}=~{}-% k~{}\nabla\theta,
  12. τ 0 \scriptstyle\tau_{{}_{0}}
  13. C 2 = α / τ 0 . \scriptstyle C^{2}~{}=~{}\alpha/\tau_{{}_{0}}.
  14. τ 0 \scriptstyle\tau_{{}_{0}}
  15. τ 0 \scriptstyle\tau_{{}_{0}}
  16. d s 2 = d x 2 + d y 2 + d z 2 , ds^{2}~{}=~{}dx^{2}~{}+~{}dy^{2}~{}+~{}dz^{2},
  17. d s 2 = d τ 2 + d x 2 + d y 2 + d z 2 , ds^{2}~{}=~{}d\tau^{2}~{}+~{}dx^{2}~{}+~{}dy^{2}~{}+~{}dz^{2},
  18. τ = i C t , \tau~{}=~{}i~{}C~{}t,
  19. i = - 1 \scriptstyle i~{}=~{}\sqrt{-1}
  20. d s 2 = d x 2 + d y 2 + d z 2 - C 2 d t 2 . ds^{2}~{}=~{}dx^{2}~{}+~{}dy^{2}~{}+~{}dz^{2}~{}-~{}C^{2}~{}dt^{2}.
  21. \scriptstyle\square
  22. = τ 𝐨 + x 𝐢 + y 𝐣 + z 𝐤 = τ 𝐨 + = - i C t 𝐨 + . \square~{}=~{}\frac{\partial}{\partial\tau}~{}\mathbf{o}~{}+~{}\frac{\partial}% {\partial x}~{}\mathbf{i}~{}+~{}\frac{\partial}{\partial y}~{}\mathbf{j}~{}+~{% }\frac{\partial}{\partial z}~{}\mathbf{k}~{}=~{}\frac{\partial}{\partial\tau}~% {}\mathbf{o}~{}+~{}\nabla~{}=~{}\frac{-i}{C}~{}\frac{\partial}{\partial t}~{}% \mathbf{o}~{}+~{}\nabla.
  23. 2 \scriptstyle\nabla^{2}
  24. 2 , \scriptstyle\square^{2},
  25. 2 = 2 τ 2 + 2 x 2 + 2 y 2 + 2 z 2 = 2 τ 2 + 2 = - 1 C 2 2 t 2 + 2 . \square^{2}~{}=~{}\frac{\partial^{2}}{\partial\tau^{2}}~{}+\frac{\partial^{2}}% {\partial x^{2}}~{}+~{}\frac{\partial^{2}}{\partial y^{2}}~{}+~{}\frac{% \partial^{2}}{\partial z^{2}}~{}=~{}\frac{\partial^{2}}{\partial\tau^{2}}~{}+~% {}\nabla^{2}~{}=~{}\frac{-1}{C^{2}}~{}\frac{\partial^{2}}{\partial t^{2}}~{}+~% {}\nabla^{2}.
  26. θ t = α 2 θ = - α C 2 2 θ t 2 + α 2 θ , \frac{\partial\theta}{\partial t}~{}=~{}\alpha~{}\square^{2}\theta~{}=~{}\frac% {-\alpha}{C^{2}}~{}\frac{\partial^{2}\theta}{\partial t^{2}}~{}+~{}\alpha~{}% \nabla^{2}\theta,
  27. 𝐪 = - k θ = - k θ + i k C θ t 𝐨 . \mathbf{q}~{}=~{}-k~{}\square\theta~{}=~{}-k~{}\nabla\theta~{}+~{}\frac{i~{}k}% {C}~{}\frac{\partial\theta}{\partial t}~{}\mathbf{o}.
  28. ρ c θ t + 𝐪 = 0 = ρ c θ t + 𝐪 + - i C 𝐪 t 𝐨 , \rho~{}c~{}\frac{\partial\theta}{\partial t}~{}+~{}\square\cdot\mathbf{q}~{}=~% {}0~{}=~{}\rho~{}c~{}\frac{\partial\theta}{\partial t}~{}+~{}\nabla\cdot% \mathbf{q}~{}+~{}\frac{-i}{C}~{}\frac{\partial\mathbf{q}}{\partial t}\cdot% \mathbf{o},
  29. ( 𝐪 θ ) + ρ s t = σ , \square\cdot\left(\frac{\mathbf{q}}{\theta}\right)~{}+~{}\rho~{}\frac{\partial s% }{\partial t}~{}=~{}\sigma,
  30. ( d t d x ) 2 + ( d t d y ) 2 + ( d t d z ) 2 1 C 2 , {\left(\frac{dt}{dx}\right)}^{2}~{}+~{}{\left(\frac{dt}{dy}\right)}^{2}~{}+~{}% {\left(\frac{dt}{dz}\right)}^{2}~{}\geqslant~{}\frac{1}{C^{2}},
  31. \scriptstyle\square
  32. 2 \scriptstyle\square^{2}

Relativistic_quantum_mechanics.html

  1. i t ψ = H ^ ψ i\hbar\frac{\partial}{\partial t}\psi=\widehat{H}\psi
  2. Ĥ Ĥ
  3. ψ ( 𝐫 , t ) ψ(\mathbf{r},t)
  4. 𝐫 \mathbf{r}
  5. t t
  6. s s
  7. 2 s 2s
  8. s s
  9. 2 s + 1 2s+1
  10. σ = s , s 1 , , s + 1 , s σ=s,s−1,...,−s+1,−s
  11. ψ ( 𝐫 , t , σ ) ψ(\mathbf{r},t,σ)
  12. m m
  13. E E
  14. 𝐩 \mathbf{p}
  15. p = 𝐩 · 𝐩 p=\sqrt{\mathbf{p}}{·}{\mathbf{p}}
  16. E 2 = c 2 𝐩 𝐩 + ( m c 2 ) 2 . E^{2}=c^{2}\mathbf{p}\cdot\mathbf{p}+(mc^{2})^{2}\,.
  17. E ^ = i t , 𝐩 ^ = - i , \widehat{E}=i\hbar\frac{\partial}{\partial t}\,,\quad\widehat{\mathbf{p}}=-i% \hbar\nabla\,,
  18. ψ ψ
  19. ψ ψ
  20. A ( t ) A(t)
  21. d d t A = 1 i [ A , H ^ ] + t A , \frac{d}{dt}A=\frac{1}{i\hbar}[A,\widehat{H}]+\frac{\partial}{\partial t}A\,,
  22. 𝐗 = ( c t , 𝐫 ) \mathbf{X}=(ct,\mathbf{r})
  23. 𝐏 = ( E / c , 𝐩 ) \mathbf{P}=(E/c,\mathbf{p})
  24. ( 𝐫 , t ) Λ ( 𝐫 , t ) (\mathbf{r},t)→Λ(\mathbf{r},t)
  25. D D
  26. ψ σ ( 𝐫 , t ) D ( Λ ) ψ σ ( Λ - 1 ( 𝐫 , t ) ) \psi_{\sigma}(\mathbf{r},t)\rightarrow D(\Lambda)\psi_{\sigma}(\Lambda^{-1}(% \mathbf{r},t))
  27. D ( Λ ) D(Λ)
  28. ( 2 s + 1 ) × ( 2 s + 1 ) (2s+1)×(2s+1)
  29. ψ ψ
  30. ( 2 s + 1 ) (2s+1)
  31. σ σ
  32. s s
  33. σ σ
  34. σ σ
  35. 𝐩 · 𝐩 / 2 m \mathbf{p}·\mathbf{p}/2m
  36. V ( 𝐫 , t ) V(\mathbf{r},t)
  37. H ^ = 𝐩 ^ 𝐩 ^ 2 m + V ( 𝐫 , t ) \widehat{H}=\frac{\widehat{\mathbf{p}}\cdot\widehat{\mathbf{p}}}{2m}+V(\mathbf% {r},t)
  38. H ^ = E ^ = c 2 𝐩 ^ 𝐩 ^ + ( m c 2 ) 2 i t ψ = c 2 𝐩 ^ 𝐩 ^ + ( m c 2 ) 2 ψ \widehat{H}=\widehat{E}=\sqrt{c^{2}\widehat{\mathbf{p}}\cdot\widehat{\mathbf{p% }}+(mc^{2})^{2}}\quad\Rightarrow\quad i\hbar\frac{\partial}{\partial t}\psi=% \sqrt{c^{2}\widehat{\mathbf{p}}\cdot\widehat{\mathbf{p}}+(mc^{2})^{2}}\,\psi
  39. ψ ψ
  40. ψ ( 𝐫 , t ) 0 ψ(\mathbf{r},t)≠0
  41. | 𝐫 | > c t |\mathbf{r}|>ct
  42. ψ ( | 𝐫 | > c t , t ) = 0 ψ(|\mathbf{r}|>ct,t)=0
  43. s y m b o l μ ^ S = - g μ B 𝐒 ^ , | s y m b o l μ S | = - g μ B σ , \widehat{symbol{\mu}}_{S}=-\frac{g\mu_{B}}{\hbar}\widehat{\mathbf{S}}\,,\quad% \left|symbol{\mu}_{S}\right|=-g\mu_{B}\sigma\,,
  44. g g
  45. 𝐒 \mathbf{S}
  46. 𝐁 \mathbf{B}
  47. H ^ B = - 𝐁 s y m b o l μ ^ S \widehat{H}_{B}=-\mathbf{B}\cdot\widehat{symbol{\mu}}_{S}
  48. σ σ
  49. H ^ = H ^ ( 𝐫 , t , 𝐩 ^ , 𝐒 ^ ) \widehat{H}=\widehat{H}(\mathbf{r},t,\widehat{\mathbf{p}},\widehat{\mathbf{S}})
  50. E ^ 2 ψ = c 2 𝐩 ^ 𝐩 ^ ψ + ( m c 2 ) 2 ψ , \widehat{E}^{2}\psi=c^{2}\widehat{\mathbf{p}}\cdot\widehat{\mathbf{p}}\psi+(mc% ^{2})^{2}\psi\,,
  51. ( E ^ - c s y m b o l α 𝐩 ^ - β m c 2 ) ( E ^ + c s y m b o l α 𝐩 ^ + β m c 2 ) ψ = 0 , \left(\widehat{E}-csymbol{\alpha}\cdot\widehat{\mathbf{p}}-\beta mc^{2}\right)% \left(\widehat{E}+csymbol{\alpha}\cdot\widehat{\mathbf{p}}+\beta mc^{2}\right)% \psi=0\,,
  52. β β
  53. i j i≠j
  54. α i β = - β α i , α i α j = - α j α i , \alpha_{i}\beta=-\beta\alpha_{i},\quad\alpha_{i}\alpha_{j}=-\alpha_{j}\alpha_{% i}\,,
  55. α i 2 = β 2 = I , \alpha_{i}^{2}=\beta^{2}=I\,,
  56. ( E ^ - c s y m b o l α 𝐩 ^ - β m c 2 ) ψ = 0 H ^ = c s y m b o l α 𝐩 ^ + β m c 2 \left(\widehat{E}-csymbol{\alpha}\cdot\widehat{\mathbf{p}}-\beta mc^{2}\right)% \psi=0\quad\Leftrightarrow\quad\widehat{H}=csymbol{\alpha}\cdot\widehat{% \mathbf{p}}+\beta mc^{2}
  57. α \mathbf{α}
  58. β β
  59. ψ ψ
  60. ψ ψ
  61. ψ ( 𝐫 , t ) ψ(\mathbf{r},t)
  62. ρ ρ
  63. 𝐣 \mathbf{j}
  64. ρ = ψ ψ , 𝐣 = ψ γ 0 s y m b o l γ ψ J μ = ψ γ 0 γ μ ψ \rho=\psi^{\dagger}\psi,\quad\mathbf{j}=\psi^{\dagger}\gamma^{0}symbol{\gamma}% \psi\quad\rightleftharpoons\quad J^{\mu}=\psi^{\dagger}\gamma^{0}\gamma^{\mu}\psi
  65. ρ = i 2 m c 2 ( ψ * ψ t - ψ ψ * t ) , 𝐣 = - i 2 m ( ψ * ψ - ψ ψ * ) J μ = i 2 m ( ψ * μ ψ - ψ μ ψ * ) \rho=\frac{i\hbar}{2mc^{2}}\left(\psi^{*}\frac{\partial\psi}{\partial t}-\psi% \frac{\partial\psi^{*}}{\partial t}\right)\,,\quad\mathbf{j}=-\frac{i\hbar}{2m% }\left(\psi^{*}\nabla\psi-\psi\nabla\psi^{*}\right)\quad\rightleftharpoons% \quad J^{\mu}=\frac{i\hbar}{2m}(\psi^{*}\partial^{\mu}\psi-\psi\partial^{\mu}% \psi^{*})
  66. ψ ψ
  67. ψ / t ∂ψ/∂t
  68. ψ ψ
  69. ρ t + 𝐉 = 0 μ J μ = 0 , \frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf{J}=0\quad\rightleftharpoons% \quad\partial_{\mu}J^{\mu}=0\,,
  70. q q
  71. 𝐀 ( 𝐫 , t ) \mathbf{A}(\mathbf{r},t)
  72. 𝐁 = × 𝐀 \mathbf{B}=∇×\mathbf{A}
  73. ϕ ( 𝐫 , t ) ϕ(\mathbf{r},t)
  74. E ^ E ^ - q ϕ , 𝐩 ^ 𝐩 ^ - q 𝐀 P ^ μ P ^ μ - q A μ \widehat{E}\rightarrow\widehat{E}-q\phi\,,\quad\widehat{\mathbf{p}}\rightarrow% \widehat{\mathbf{p}}-q\mathbf{A}\quad\rightleftharpoons\quad\widehat{P}_{\mu}% \rightarrow\widehat{P}_{\mu}-qA_{\mu}
  75. E - e ϕ m c 2 , 𝐩 m 𝐯 , E-e\phi\approx mc^{2}\,,\quad\mathbf{p}\approx m\mathbf{v}\,,
  76. ( E ^ - q ϕ ) 2 ψ = c 2 ( 𝐩 ^ - q 𝐀 ) 2 ψ + ( m c 2 ) 2 ψ [ ( P ^ μ - q A μ ) ( P ^ μ - q A μ ) - ( m c 2 ) 2 ] ψ = 0. {(\widehat{E}-q\phi)}^{2}\psi=c^{2}{(\widehat{\mathbf{p}}-q\mathbf{A})}^{2}% \psi+(mc^{2})^{2}\psi\quad\rightleftharpoons\quad\left[{(\widehat{P}_{\mu}-qA_% {\mu})}{(\widehat{P}^{\mu}-qA^{\mu})}-{(mc^{2})}^{2}\right]\psi=0.
  77. ( 0 , 0 ) (0,0)
  78. ( 0 , 0 ) (0,0)
  79. ( 0 , 0 ) (0,0)
  80. ( i t - q ϕ ) ψ = 1 2 m ( 𝐩 ^ - q 𝐀 ) 2 ψ H ^ = 1 2 m ( 𝐩 ^ - q 𝐀 ) 2 + q ϕ . \left(i\hbar\frac{\partial}{\partial t}-q\phi\right)\psi=\frac{1}{2m}{(% \widehat{\mathbf{p}}-q\mathbf{A})}^{2}\psi\quad\Leftrightarrow\quad\widehat{H}% =\frac{1}{2m}{(\widehat{\mathbf{p}}-q\mathbf{A})}^{2}+q\phi.
  81. ( i t - q ϕ ) ψ = [ 1 2 m ( s y m b o l σ ( 𝐩 - q 𝐀 ) ) 2 ] ψ H ^ = 1 2 m ( s y m b o l σ ( 𝐩 - q 𝐀 ) ) 2 + q ϕ \left(i\hbar\frac{\partial}{\partial t}-q\phi\right)\psi=\left[\frac{1}{2m}{(% symbol{\sigma}\cdot(\mathbf{p}-q\mathbf{A}))}^{2}\right]\psi\quad% \Leftrightarrow\quad\widehat{H}=\frac{1}{2m}{(symbol{\sigma}\cdot(\mathbf{p}-q% \mathbf{A}))}^{2}+q\phi
  82. ψ ψ
  83. ψ = ( ψ ψ ) \psi=\begin{pmatrix}\psi_{\uparrow}\\ \psi_{\downarrow}\end{pmatrix}
  84. σ = + 1 / 2 σ=+1/2
  85. σ = 1 / 2 σ=−1/2
  86. ( i t - q ϕ ) ψ = γ 0 [ c s y m b o l γ ( 𝐩 ^ - q 𝐀 ) - m c 2 ] ψ [ γ μ ( P ^ μ - q A μ ) - m c 2 ] ψ = 0 \left(i\hbar\frac{\partial}{\partial t}-q\phi\right)\psi=\gamma^{0}\left[% csymbol{\gamma}\cdot{(\widehat{\mathbf{p}}-q\mathbf{A})}-mc^{2}\right]\psi% \quad\rightleftharpoons\quad\left[\gamma^{\mu}(\widehat{P}_{\mu}-qA_{\mu})-mc^% {2}\right]\psi=0
  87. ψ ψ
  88. ψ = ( u v ) = ( u 1 u 2 v 1 v 2 ) \psi=\begin{pmatrix}u\\ v\end{pmatrix}=\begin{pmatrix}u^{1}\\ u^{2}\\ v^{1}\\ v^{2}\end{pmatrix}
  89. ψ = ( ψ + ψ - ) = ( ψ + ψ + ψ - ψ - ) \psi=\begin{pmatrix}\psi_{+}\\ \psi_{-}\end{pmatrix}=\begin{pmatrix}\psi_{+\uparrow}\\ \psi_{+\downarrow}\\ \psi_{-\uparrow}\\ \psi_{-\downarrow}\end{pmatrix}
  90. ( E , 𝐩 ) (E,\mathbf{p})
  91. q q
  92. σ = ± 1 / 2 σ=±1/2
  93. ( E , 𝐩 ) −(E,\mathbf{p})
  94. q −q
  95. 𝐀 = 𝟎 \mathbf{A}=\mathbf{0}
  96. ϕ ϕ
  97. ( E ^ c + s y m b o l σ 𝐩 ^ ) ψ + = 0 , ( E ^ c - s y m b o l σ 𝐩 ^ ) ψ - = 0 σ μ P ^ μ ψ + = 0 , σ μ P ^ μ ψ - = 0 , \left(\frac{\widehat{E}}{c}+symbol{\sigma}\cdot\widehat{\mathbf{p}}\right)\psi% _{+}=0\,,\quad\left(\frac{\widehat{E}}{c}-symbol{\sigma}\cdot\widehat{\mathbf{% p}}\right)\psi_{-}=0\quad\rightleftharpoons\quad\sigma^{\mu}\widehat{P}_{\mu}% \psi_{+}=0\,,\quad\sigma_{\mu}\widehat{P}^{\mu}\psi_{-}=0\,,
  98. [ σ a , σ b ] = 2 i ε a b c σ c , [ σ a , σ b ] + = 2 δ a b σ 0 \left[\sigma_{a},\sigma_{b}\right]=2i\varepsilon_{abc}\sigma_{c}\,,\quad\left[% \sigma_{a},\sigma_{b}\right]_{+}=2\delta_{ab}\sigma_{0}
  99. [ γ α , γ β ] + = γ α γ β + γ β γ α = 2 η α β , \left[\gamma^{\alpha},\gamma^{\beta}\right]_{+}=\gamma^{\alpha}\gamma^{\beta}+% \gamma^{\beta}\gamma^{\alpha}=2\eta^{\alpha\beta}\,,
  100. h ^ = 𝐒 ^ 𝐩 ^ | 𝐩 | = 𝐒 ^ c 𝐩 ^ E 2 - ( m 0 c 2 ) 2 \widehat{h}=\widehat{\mathbf{S}}\cdot\frac{\widehat{\mathbf{p}}}{|\mathbf{p}|}% =\widehat{\mathbf{S}}\cdot\frac{c\widehat{\mathbf{p}}}{\sqrt{E^{2}-(m_{0}c^{2}% )^{2}}}
  101. σ · c 𝐩 \mathbf{σ}·c\mathbf{p}
  102. E 2 - ( m 0 c 2 ) 2 \sqrt{E^{2}-(m_{0}c^{2})^{2}}
  103. h ^ = 𝐒 ^ c 𝐩 ^ E \widehat{h}=\widehat{\mathbf{S}}\cdot\frac{c\widehat{\mathbf{p}}}{E}
  104. ψ ( 𝐫 , t ) = [ ψ σ = s ( 𝐫 , t ) ψ σ = s - 1 ( 𝐫 , t ) ψ σ = - s + 1 ( 𝐫 , t ) ψ σ = - s ( 𝐫 , t ) ] ψ ( 𝐫 , t ) = [ ψ σ = s ( 𝐫 , t ) ψ σ = s - 1 ( 𝐫 , t ) ψ σ = - s + 1 ( 𝐫 , t ) ψ σ = - s ( 𝐫 , t ) ] \psi(\mathbf{r},t)=\begin{bmatrix}\psi_{\sigma=s}(\mathbf{r},t)\\ \psi_{\sigma=s-1}(\mathbf{r},t)\\ \vdots\\ \psi_{\sigma=-s+1}(\mathbf{r},t)\\ \psi_{\sigma=-s}(\mathbf{r},t)\end{bmatrix}\quad\rightleftharpoons\quad{\psi(% \mathbf{r},t)}^{\dagger}=\begin{bmatrix}{\psi_{\sigma=s}(\mathbf{r},t)}^{\star% }&{\psi_{\sigma=s-1}(\mathbf{r},t)}^{\star}&\cdots&{\psi_{\sigma=-s+1}(\mathbf% {r},t)}^{\star}&{\psi_{\sigma=-s}(\mathbf{r},t)}^{\star}\end{bmatrix}
  105. s s
  106. 2 s + 1 2s+1
  107. 2 s + 1 2s+1
  108. 2 s + 1 2s+1
  109. σ σ
  110. 2 ( 2 s + 1 ) 2(2s+1)
  111. ψ ( 𝐫 , t ) = [ ψ + , σ = s ( 𝐫 , t ) ψ + , σ = s - 1 ( 𝐫 , t ) ψ + , σ = - s + 1 ( 𝐫 , t ) ψ + , σ = - s ( 𝐫 , t ) ψ - , σ = s ( 𝐫 , t ) ψ - , σ = s - 1 ( 𝐫 , t ) ψ - , σ = - s + 1 ( 𝐫 , t ) ψ - , σ = - s ( 𝐫 , t ) ] ψ ( 𝐫 , t ) [ ψ + , σ = s ( 𝐫 , t ) ψ + , σ = s - 1 ( 𝐫 , t ) ψ - , σ = - s ( 𝐫 , t ) ] \psi(\mathbf{r},t)=\begin{bmatrix}\psi_{+,\,\sigma=s}(\mathbf{r},t)\\ \psi_{+,\,\sigma=s-1}(\mathbf{r},t)\\ \vdots\\ \psi_{+,\,\sigma=-s+1}(\mathbf{r},t)\\ \psi_{+,\,\sigma=-s}(\mathbf{r},t)\\ \psi_{-,\,\sigma=s}(\mathbf{r},t)\\ \psi_{-,\,\sigma=s-1}(\mathbf{r},t)\\ \vdots\\ \psi_{-,\,\sigma=-s+1}(\mathbf{r},t)\\ \psi_{-,\,\sigma=-s}(\mathbf{r},t)\end{bmatrix}\quad\rightleftharpoons\quad{% \psi(\mathbf{r},t)}^{\dagger}\begin{bmatrix}{\psi_{+,\,\sigma=s}(\mathbf{r},t)% }^{\star}&{\psi_{+,\,\sigma=s-1}(\mathbf{r},t)}^{\star}&\cdots&{\psi_{-,\,% \sigma=-s}(\mathbf{r},t)}^{\star}\end{bmatrix}
  112. ψ ( 𝐫 , t ) = ( ψ + ( 𝐫 , t ) ψ - ( 𝐫 , t ) ) \psi(\mathbf{r},t)=\begin{pmatrix}\psi_{+}(\mathbf{r},t)\\ \psi_{-}(\mathbf{r},t)\end{pmatrix}
  113. 𝐩 = m 𝐯 \mathbf{p}=m\mathbf{v}
  114. 𝐯 ^ = 1 m 𝐩 ^ \widehat{\mathbf{v}}=\frac{1}{m}\widehat{\mathbf{p}}
  115. 𝐯 ^ = i [ H ^ , 𝐫 ^ ] \widehat{\mathbf{v}}=\frac{i}{\hbar}\left[\widehat{H},\widehat{\mathbf{r}}\right]
  116. ψ ψ
  117. μ ( ( μ ψ ) ) - ψ = 0 \partial_{\mu}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\psi)}% \right)-\frac{\partial\mathcal{L}}{\partial\psi}=0\,
  118. = ψ ¯ ( γ μ P μ - m c ) ψ \mathcal{L}=\overline{\psi}(\gamma^{\mu}P_{\mu}-mc)\psi
  119. = - 2 m η μ ν μ ψ * ν ψ - m c 2 ψ * ψ . \mathcal{L}=-\frac{\hbar^{2}}{m}\eta^{\mu\nu}\partial_{\mu}\psi^{*}\partial_{% \nu}\psi-mc^{2}\psi^{*}\psi\,.
  120. ψ ψ
  121. 𝐋 = 𝐫 × 𝐩 \mathbf{L}=\mathbf{r}×\mathbf{p}
  122. M α β = X α P β - X β P α = 2 X [ α P β ] 𝐌 = 𝐗 𝐏 , M^{\alpha\beta}=X^{\alpha}P^{\beta}-X^{\beta}P^{\alpha}=2X^{[\alpha}P^{\beta]}% \quad\rightleftharpoons\quad\mathbf{M}=\mathbf{X}\wedge\mathbf{P}\,,
  123. m m
  124. J α β = 2 X [ α P β ] + 1 m 2 ε α β γ δ W γ p δ 𝐉 = 𝐗 𝐏 + 1 m 2 ( 𝐖 𝐏 ) J^{\alpha\beta}=2X^{[\alpha}P^{\beta]}+\frac{1}{m^{2}}\varepsilon^{\alpha\beta% \gamma\delta}W_{\gamma}p_{\delta}\quad\rightleftharpoons\quad\mathbf{J}=% \mathbf{X}\wedge\mathbf{P}+\frac{1}{m^{2}}\star(\mathbf{W}\wedge\mathbf{P})
  125. W α = 1 2 ε α β γ δ M β γ p δ 𝐖 = ( 𝐌 𝐏 ) W_{\alpha}=\frac{1}{2}\varepsilon_{\alpha\beta\gamma\delta}M^{\beta\gamma}p^{% \delta}\quad\rightleftharpoons\quad\mathbf{W}=\star(\mathbf{M}\wedge\mathbf{P})
  126. 𝐯 \mathbf{v}
  127. 𝐄 \mathbf{E}
  128. 𝐁 \mathbf{B}
  129. 𝐁 \mathbf{B′}
  130. 𝐁 = 𝐄 × 𝐯 c 2 1 - ( v / c ) 2 . \mathbf{B}^{\prime}=\frac{\mathbf{E}\times\mathbf{v}}{c^{2}\sqrt{1-\left(v/c% \right)^{2}}}\,.
  131. H ^ = - 𝐁 s y m b o l μ ^ S = - ( 𝐁 + 𝐄 × 𝐯 c 2 ) s y m b o l μ ^ S , \widehat{H}=-\mathbf{B}^{\prime}\cdot\widehat{symbol{\mu}}_{S}=-\left(\mathbf{% B}+\frac{\mathbf{E}\times\mathbf{v}}{c^{2}}\right)\cdot\widehat{symbol{\mu}}_{% S}\,,
  132. H ^ = - 𝐁 s y m b o l μ ^ S = - ( 𝐁 + 𝐄 × 𝐯 2 c 2 ) s y m b o l μ ^ S . \widehat{H}=-\mathbf{B}^{\prime}\cdot\widehat{symbol{\mu}}_{S}=-\left(\mathbf{% B}+\frac{\mathbf{E}\times\mathbf{v}}{2c^{2}}\right)\cdot\widehat{symbol{\mu}}_% {S}\,.
  133. m < s u b > s m<sub>s
  134. ψ = ( u 1 u 2 ) \psi=\begin{pmatrix}u^{1}\\ u^{2}\end{pmatrix}
  135. ψ = ( χ η ) \psi=\begin{pmatrix}\chi\\ \eta\end{pmatrix}

Reo_Coordination_Language.html

  1. R = ( N , B , C , t ) R=(N,B,C,t)
  2. N N
  3. B N B\subseteq N
  4. C 2 N × 2 N C\subseteq 2^{N}\times 2^{N}
  5. t : C T t:C\rightarrow T
  6. | I O | = 2 |I\cup O|=2
  7. ( I , O ) C (I,O)\in C
  8. c = ( I , O ) C c=(I,O)\in C
  9. I I
  10. c c
  11. O O
  12. c c
  13. n \langle n\rangle
  14. n n
  15. c \langle c\rangle
  16. c c

Representation_ring.html

  1. Ψ k χ ( g ) = χ ( g k ) . \Psi^{k}\chi(g)=\chi(g^{k})\ .
  2. Ψ k ( ρ ) = N k ( Λ 1 ρ , Λ 2 ρ , , Λ d ρ ) \Psi^{k}(\rho)=N_{k}(\Lambda^{1}\rho,\Lambda^{2}\rho,\ldots,\Lambda^{d}\rho)

Representation_theory.html

  1. Φ : G × V V or Φ : A × V V \Phi\colon G\times V\to V\quad\,\text{or}\quad\Phi\colon A\times V\to V
  2. φ ( g ) : V V v Φ ( g , v ) \begin{aligned}\displaystyle\varphi(g)\colon V&\displaystyle\to V\\ \displaystyle v&\displaystyle\mapsto\Phi(g,v)\end{aligned}
  3. ( 1 ) e v = v (1)\quad e\cdot v=v
  4. ( 2 ) g 1 ( g 2 v ) = ( g 1 g 2 ) v (2)\quad g_{1}\cdot(g_{2}\cdot v)=(g_{1}g_{2})\cdot v
  5. ( 2 ) x 1 ( x 2 v ) - x 2 ( x 1 v ) = [ x 1 , x 2 ] v (2^{\prime})\quad x_{1}\cdot(x_{2}\cdot v)-x_{2}\cdot(x_{1}\cdot v)=[x_{1},x_{% 2}]\cdot v
  6. φ ( g 1 g 2 ) = φ ( g 1 ) φ ( g 2 ) for all g 1 , g 2 G \varphi(g_{1}g_{2})=\varphi(g_{1})\circ\varphi(g_{2})\quad\,\text{for all }g_{% 1},g_{2}\in G\,\!
  7. α ( g v ) = g α ( v ) \alpha(g\cdot v)=g\cdot\alpha(v)
  8. α ϕ ( g ) = ψ ( g ) α \alpha\circ\phi(g)=\psi(g)\circ\alpha
  9. g ( v , w ) = ( g v , g w ) . g\cdot(v,w)=(g\cdot v,g\cdot w).
  10. π G ( x ) = 1 | G | g G g π ( g - 1 x ) . \pi_{G}(x)=\frac{1}{|G|}\sum_{g\in G}g\cdot\pi(g^{-1}\cdot x).
  11. χ φ ( g ) = Tr ( φ ( g ) ) \chi_{\varphi}(g)=\mathrm{Tr}(\varphi(g))\,
  12. Tr \mathrm{Tr}
  13. , \langle\cdot,\cdot\rangle
  14. g v , g w = v , w \langle g\cdot v,g\cdot w\rangle=\langle v,w\rangle
  15. ρ ( 1 ) [ x ] = x \rho(1)[x]=x
  16. ρ ( g 1 g 2 ) [ x ] = ρ ( g 1 ) [ ρ ( g 2 ) [ x ] ] . \rho(g_{1}g_{2})[x]=\rho(g_{1})[\rho(g_{2})[x]].

Residual_property_(mathematics).html

  1. h ( g ) e h(g)\neq e
  2. ϕ : G H \phi\colon G\to H

Residuated_mapping.html

  1. x x x\mapsto\lceil x\rceil
  2. x x x\mapsto\lfloor x\rfloor

Resolution_(algebra).html

  1. d n + 1 E n d n d 3 E 2 d 2 E 1 d 1 E 0 ϵ M 0. \cdots\overset{d_{n+1}}{\longrightarrow}E_{n}\overset{d_{n}}{\longrightarrow}% \cdots\overset{d_{3}}{\longrightarrow}E_{2}\overset{d_{2}}{\longrightarrow}E_{% 1}\overset{d_{1}}{\longrightarrow}E_{0}\overset{\epsilon}{\longrightarrow}M% \longrightarrow 0.
  2. E ϵ M 0. E_{\bullet}\overset{\epsilon}{\longrightarrow}M\longrightarrow 0.
  3. 0 M ϵ C 0 d 0 C 1 d 1 C 2 d 2 d n - 1 C n d n , 0\longrightarrow M\overset{\epsilon}{\longrightarrow}C^{0}\overset{d^{0}}{% \longrightarrow}C^{1}\overset{d^{1}}{\longrightarrow}C^{2}\overset{d^{2}}{% \longrightarrow}\cdots\overset{d^{n-1}}{\longrightarrow}C^{n}\overset{d^{n}}{% \longrightarrow}\cdots,
  4. 0 M ϵ C . 0\longrightarrow M\overset{\epsilon}{\longrightarrow}C^{\bullet}.
  5. 0 M I * , 0 M I * , 0\rightarrow M\rightarrow I_{*},\ \ 0\rightarrow M^{\prime}\rightarrow I^{% \prime}_{*},
  6. I * I_{*}
  7. I * I^{\prime}_{*}
  8. 0 M E 0 E 1 E 2 0\rightarrow M\rightarrow E_{0}\rightarrow E_{1}\rightarrow E_{2}\rightarrow\dots
  9. R M \otimes_{R}M
  10. E * E_{*}
  11. R i F ( M ) = H i F ( E * ) , R_{i}F(M)=H_{i}F(E_{*}),
  12. F ( E * ) . F(E_{*}).
  13. 𝒞 * ( M ) \mathcal{C}^{*}(M)
  14. 0 R 𝒞 0 ( M ) d 𝒞 1 ( M ) d 𝒞 d i m M ( M ) 0. 0\rightarrow R\subset\mathcal{C}^{0}(M)\stackrel{d}{\rightarrow}\mathcal{C}^{1% }(M)\stackrel{d}{\rightarrow}\dots\mathcal{C}^{dimM}(M)\rightarrow 0.
  15. 𝒞 * ( M ) \mathcal{C}^{*}(M)
  16. Γ : ( M ) \Gamma:\mathcal{F}\mapsto\mathcal{F}(M)
  17. H i ( M , 𝐑 ) = H i ( 𝒞 * ( M ) ) . \mathrm{H}^{i}(M,\mathbf{R})=\mathrm{H}^{i}(\mathcal{C}^{*}(M)).

Resolution_(mass_spectrometry).html

  1. r = d ( w 1 - w 2 ) / 2 r=\cfrac{d}{(w_{1}-w_{2})/2}
  2. R = t w R=\cfrac{t}{w}
  3. R = M Δ M R=\cfrac{M}{\Delta M}
  4. Δ M \Delta M
  5. M M
  6. R = M Δ M R=\cfrac{M}{\Delta M}
  7. Δ M \Delta M

Response_factor.html

  1. f i f_{i}
  2. f i = A i A s t f s t f_{i}=\frac{A_{i}}{A_{st}}f_{st}
  3. A r e a o c t a n e A r e a n o n a n e = k o c t a n e × M o c t a n e × V o c t a n e k n o n a n e × M n o n a n e × V n o n a n e {{Area_{octane}}\over{Area_{nonane}}}={{k_{octane}\times M_{octane}\times V_{% octane}}\over{k_{nonane}\times M_{nonane}\times V_{nonane}}}
  4. F = k o c t a n e k n o n a n e = A r e a o c t a n e / M o c t a n e A r e a n o n a n e / M n o n a n e F={k_{octane}\over k_{nonane}}={{Area_{octane}/M_{octane}}\over{Area_{nonane}/% M_{nonane}}}
  5. ( A r e a o c t a n e / M o c t a n e A r e a n o n a n e / M n o n a n e ) 1 = F = ( A r e a o c t a n e / M o c t a n e A r e a n o n a n e / M n o n a n e ) 2 {\left({{Area_{octane}/M_{octane}}\over{Area_{nonane}/M_{nonane}}}\right)}_{1}% =F={\left({{Area_{octane}/M_{octane}}\over{Area_{nonane}/M_{nonane}}}\right)}_% {2}

Restricted_Lie_algebra.html

  1. X X [ p ] X\mapsto X^{[p]}
  2. ad ( X [ p ] ) = ad ( X ) p \mathrm{ad}(X^{[p]})=\mathrm{ad}(X)^{p}
  3. X L X\in L
  4. ( t X ) [ p ] = t p X [ p ] (tX)^{[p]}=t^{p}X^{[p]}
  5. t k , X L t\in k,X\in L
  6. ( X + Y ) [ p ] = X [ p ] + Y [ p ] + i = 1 p - 1 s i ( X , Y ) i (X+Y)^{[p]}=X^{[p]}+Y^{[p]}+\sum_{i=1}^{p-1}\frac{s_{i}(X,Y)}{i}
  7. X , Y L X,Y\in L
  8. s i ( X , Y ) s_{i}(X,Y)
  9. t i - 1 t^{i-1}
  10. ad ( t X + Y ) p - 1 ( X ) \mathrm{ad}(tX+Y)^{p-1}(X)
  11. [ X , Y ] := X Y - Y X [X,Y]:=XY-YX
  12. X [ p ] := X p X^{[p]}:=X^{p}
  13. Lie ( A ) \mathrm{Lie}(A)
  14. Lie ( G ) \mathrm{Lie}(G)
  15. Lie ( G ) \mathrm{Lie}(G)
  16. Lie ( G ) \mathrm{Lie}(G)
  17. x x p x\mapsto x^{p}
  18. Lie ( G ) \mathrm{Lie}(G)
  19. A Lie ( A ) A\mapsto\mathrm{Lie}(A)
  20. L U [ p ] ( L ) L\mapsto U^{[p]}(L)
  21. U ( L ) U(L)
  22. x p - x [ p ] x^{p}-x^{[p]}
  23. U [ p ] ( L ) = U ( L ) / I U^{[p]}(L)=U(L)/I

Richard_Shore.html

  1. 𝒟 \mathcal{D}
  2. a a
  3. b b
  4. 𝒟 a \mathcal{D}_{a}
  5. 𝒟 b \mathcal{D}_{b}
  6. a a
  7. b b
  8. 𝒟 \mathcal{D}

Richter_magnitude_scale.html

  1. M L M\text{L}
  2. M L M\text{L}
  3. M L M\text{L}
  4. M s M\text{s}
  5. M b M\text{b}
  6. M L M\text{L}
  7. M s M\text{s}
  8. M b M\text{b}
  9. M w M\text{w}
  10. M w M_{w}
  11. M L M\text{L}
  12. M s M\text{s}
  13. m b m\text{b}
  14. M w M\text{w}
  15. M L M\text{L}
  16. M s M\text{s}
  17. M w M\text{w}
  18. m b m\text{b}
  19. M L M\text{L}
  20. M s M\text{s}
  21. M o M_{o}
  22. M w M\text{w}
  23. M s M\text{s}
  24. M o M_{o}
  25. M w M\text{w}
  26. M L M_{L}
  27. M s M\text{s}
  28. 3 / 2 {3}/{2}
  29. = ( 10 1.0 ) ( 3 / 2 ) =({10^{1.0}})^{(3/2)}
  30. = ( 10 2.0 ) ( 3 / 2 ) =({10^{2.0}})^{(3/2)}
  31. m b m\text{b}
  32. M L = log 10 A - log 10 A 0 ( δ ) = log 10 [ A / A 0 ( δ ) ] , M_{\mathrm{L}}=\log_{10}A-\log_{10}A_{\mathrm{0}}(\delta)=\log_{10}[A/A_{% \mathrm{0}}(\delta)],
  33. δ \delta
  34. M L M\text{L}
  35. M w M\text{w}
  36. M w M\text{w}
  37. M w M\text{w}
  38. M w M\text{w}
  39. M s M\text{s}
  40. M w M\text{w}
  41. M w M\text{w}
  42. M w M\text{w}
  43. M w M\text{w}
  44. M w M\text{w}
  45. M w M\text{w}
  46. M w M\text{w}
  47. M w M\text{w}
  48. M w M\text{w}
  49. M w M\text{w}
  50. M w M\text{w}
  51. M w M\text{w}
  52. M w M\text{w}
  53. M w M\text{w}
  54. M w M\text{w}
  55. M w M\text{w}
  56. M w M\text{w}
  57. M w M\text{w}
  58. M w M\text{w}
  59. M w M\text{w}
  60. M w M\text{w}
  61. M w M\text{w}
  62. M w M\text{w}
  63. M w M\text{w}
  64. M w M\text{w}
  65. M w M\text{w}
  66. M s M\text{s}
  67. M s M\text{s}
  68. M w M\text{w}
  69. M w M\text{w}
  70. M w M\text{w}
  71. M s M\text{s}
  72. M s M\text{s}
  73. M w M\text{w}
  74. M s M\text{s}
  75. M w M\text{w}
  76. M w M\text{w}
  77. M w M\text{w}
  78. M w M\text{w}
  79. M w M\text{w}
  80. M w M\text{w}
  81. M w M\text{w}
  82. M w M\text{w}
  83. M w M\text{w}
  84. M w M\text{w}
  85. M w M\text{w}
  86. M s M\text{s}
  87. M L M\text{L}
  88. M w M\text{w}
  89. M L M_{\mathrm{L}}
  90. M L = log 10 A - 2.48 + 2.76 log 10 Δ M_{\mathrm{L}}=\log_{10}A-2.48+2.76\log_{10}\Delta
  91. Δ \Delta
  92. M L = log 10 A + 1.6 log 10 D - 0.15 M_{\mathrm{L}}=\log_{10}A+1.6\log_{10}D-0.15
  93. M L = log 10 A + 3.0 log 10 D - 3.38 M_{\mathrm{L}}=\log_{10}A+3.0\log_{10}D-3.38
  94. M L = 2.92 + 2.25 log 10 ( τ ) - 0.001 Δ M_{\mathrm{L}}=2.92+2.25\log_{10}(\tau)-0.001\Delta^{\circ}
  95. M L M_{\mathrm{L}}
  96. τ \tau
  97. Δ \Delta
  98. M L = - 2.53 + 2.85 log 10 ( F - P ) + 0.0014 Δ M_{\mathrm{L}}=-2.53+2.85\log_{10}(F-P)+0.0014\Delta^{\circ}
  99. M L M_{\mathrm{L}}
  100. F - P F-P
  101. Δ \Delta
  102. M L = log 10 A + 1.73 log 10 Δ - 0.83 M_{\mathrm{L}}=\log_{10}A+1.73\log_{10}\Delta-0.83
  103. M L M_{\mathrm{L}}
  104. A A
  105. Δ \Delta

Ridders'_method.html

  1. x 4 = x 3 + ( x 3 - x 1 ) sign [ f ( x 1 ) - f ( x 2 ) ] f ( x 3 ) f ( x 3 ) 2 - f ( x 1 ) f ( x 2 ) . x_{4}=x_{3}+(x_{3}-x_{1})\frac{\operatorname{sign}[f(x_{1})-f(x_{2})]f(x_{3})}% {\sqrt{f(x_{3})^{2}-f(x_{1})f(x_{2})}}.

Riemann_hypothesis.html

  1. 1 2 \frac{1}{2}
  2. ζ ( s ) = n = 1 1 n s = 1 1 s + 1 2 s + 1 3 s + . \zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}=\frac{1}{1^{s}}+\frac{1}{2^{s}}+% \frac{1}{3^{s}}+\cdots.
  3. ζ ( s ) = p prime 1 1 - p - s = 1 1 - 2 - s 1 1 - 3 - s 1 1 - 5 - s 1 1 - 7 - s 1 1 - p - s \zeta(s)=\prod_{p\,\text{ prime}}\frac{1}{1-p^{-s}}=\frac{1}{1-2^{-s}}\cdot% \frac{1}{1-3^{-s}}\cdot\frac{1}{1-5^{-s}}\cdot\frac{1}{1-7^{-s}}\cdots\frac{1}% {1-p^{-s}}\cdots
  4. ( 1 - 2 2 s ) ζ ( s ) = n = 1 ( - 1 ) n + 1 n s = 1 1 s - 1 2 s + 1 3 s - . \left(1-\frac{2}{2^{s}}\right)\zeta(s)=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^% {s}}=\frac{1}{1^{s}}-\frac{1}{2^{s}}+\frac{1}{3^{s}}-\cdots.
  5. s = 1 + 2 π i n / ln ( 2 ) s=1+2\pi in/\ln(2)
  6. 1 - 2 / 2 s 1-2/2^{s}
  7. π ( x ) = n = 1 μ ( n ) n Π ( x 1 n ) = Π ( x ) - 1 2 Π ( x 1 2 ) - 1 3 Π ( x 1 3 ) - 1 5 Π ( x 1 5 ) + 1 6 Π ( x 1 6 ) - , \begin{aligned}\displaystyle\pi(x)&\displaystyle=\sum_{n=1}^{\infty}\frac{\mu(% n)}{n}\Pi(x^{\frac{1}{n}})\\ &\displaystyle=\Pi(x)-\frac{1}{2}\Pi(x^{\frac{1}{2}})-\frac{1}{3}\Pi(x^{\frac{% 1}{3}})-\frac{1}{5}\Pi(x^{\frac{1}{5}})\\ &\displaystyle\ \ \ \ +\frac{1}{6}\Pi(x^{\frac{1}{6}})-\cdots,\end{aligned}
  8. Π 0 ( x ) = Li ( x ) - ρ Li ( x ρ ) - log ( 2 ) + x d t t ( t 2 - 1 ) log ( t ) \begin{aligned}\displaystyle\Pi_{0}(x)&\displaystyle=\operatorname{Li}(x)-\sum% _{\rho}\operatorname{Li}(x^{\rho})-\log(2)\\ &\displaystyle\ \ \ \ +\int_{x}^{\infty}\frac{dt}{t(t^{2}-1)\log(t)}\end{aligned}
  9. Π 0 ( x ) = lim ε 0 Π ( x - ε ) + Π ( x + ε ) 2 . \Pi_{0}(x)=\lim_{\varepsilon\to 0}\frac{\Pi(x-\varepsilon)+\Pi(x+\varepsilon)}% {2}.
  10. Li ( x ) = 0 x d t log ( t ) . \operatorname{Li}(x)=\int_{0}^{x}\frac{dt}{\log(t)}.
  11. | π ( x ) - Li ( x ) | < 1 8 π x log ( x ) , for all x 2657. |\pi(x)-\operatorname{Li}(x)|<\frac{1}{8\pi}\sqrt{x}\log(x),\qquad\,\text{for % all }x\geq 2657.
  12. | ψ ( x ) - x | < 1 8 π x log 2 ( x ) , for all x 73.2 , |\psi(x)-x|<\frac{1}{8\pi}\sqrt{x}\log^{2}(x),\qquad\,\text{for all }x\geq 73.2,
  13. 1 ζ ( s ) = n = 1 μ ( n ) n s \frac{1}{\zeta(s)}=\sum_{n=1}^{\infty}\frac{\mu(n)}{n^{s}}
  14. M ( x ) = n x μ ( n ) M(x)=\sum_{n\leq x}\mu(n)
  15. M ( x ) = O ( x 1 2 + ε ) M(x)=O(x^{\frac{1}{2}+\varepsilon})
  16. | M ( x ) | x . |M(x)|\leq\sqrt{x}.
  17. σ ( n ) = d n d \sigma(n)=\sum_{d\mid n}d
  18. σ ( n ) < e γ n log log n \sigma(n)<e^{\gamma}n\log\log n
  19. i = 1 m | F n ( i ) - i m | = O ( n 1 2 + ϵ ) \sum_{i=1}^{m}|F_{n}(i)-\tfrac{i}{m}|=O(n^{\frac{1}{2}+\epsilon})
  20. m = i = 1 n ϕ ( i ) m=\sum_{i=1}^{n}\phi(i)
  21. log g ( n ) < Li - 1 ( n ) \log g(n)<\sqrt{\operatorname{Li}^{-1}(n)}
  22. ζ ( 1 2 + i t ) = O ( t ε ) , \zeta\left(\frac{1}{2}+it\right)=O(t^{\varepsilon}),
  23. e γ lim sup t + | ζ ( 1 + i t ) | log log t 2 e γ e^{\gamma}\leq\limsup_{t\rightarrow+\infty}\frac{|\zeta(1+it)|}{\log\log t}% \leq 2e^{\gamma}
  24. 6 π 2 e γ lim sup t + 1 / | ζ ( 1 + i t ) | log log t 12 π 2 e γ \frac{6}{\pi^{2}}e^{\gamma}\leq\limsup_{t\rightarrow+\infty}\frac{1/|\zeta(1+% it)|}{\log\log t}\leq\frac{12}{\pi^{2}}e^{\gamma}
  25. - k = 1 ( - x ) k ( k - 1 ) ! ζ ( 2 k ) = O ( x 1 4 + ϵ ) -\sum_{k=1}^{\infty}\frac{(-x)^{k}}{(k-1)!\zeta(2k)}=O\left(x^{\frac{1}{4}+% \epsilon}\right)
  26. f ( x ) = ν = 1 n c ν ρ ( θ ν x ) f(x)=\sum_{\nu=1}^{n}c_{\nu}\rho\left(\frac{\theta_{\nu}}{x}\right)
  27. ν = 1 n c ν θ ν = 0 \sum_{\nu=1}^{n}c_{\nu}\theta_{\nu}=0
  28. 0 z - σ - 1 ϕ ( z ) d z e x / z + 1 = 0 \int_{0}^{\infty}\frac{z^{-\sigma-1}\phi(z)\,dz}{{e^{x/z}}+1}=0
  29. ϕ \phi
  30. 1 / 2 < σ < 1 1/2<\sigma<1
  31. ζ ( s ) \zeta^{\prime}(s)
  32. 0 < ( s ) < 1 2 . 0<\Re(s)<\frac{1}{2}.
  33. lim x 1 - p > 2 ( - 1 ) ( p + 1 ) / 2 x p = + , \lim_{x\to 1^{-}}\sum_{p>2}(-1)^{(p+1)/2}x^{p}=+\infty,
  34. π ( x ) > Li ( x ) + 1 3 x log x log log log x , \pi(x)>\operatorname{Li}(x)+\frac{1}{3}\frac{\sqrt{x}}{\log x}\log\log\log x,
  35. π ( x ) < Li ( x ) - 1 3 x log x log log log x . \pi(x)<\operatorname{Li}(x)-\frac{1}{3}\frac{\sqrt{x}}{\log x}\log\log\log x.
  36. φ ( n ) < e - γ n log log n \varphi(n)<e^{-\gamma}\frac{n}{\log\log n}
  37. q \sqrt{q}
  38. ( s ) ( 0 , n ) \Re(s)\in(0,n)
  39. ( s ) = 1 / 2 , 3 / 2 , , n - 1 / 2 \Re(s)=1/2,3/2,\dots,n-1/2
  40. ( s ) = 1 , 2 , , n - 1 \Re(s)=1,2,\dots,n-1
  41. H ^ \hat{H}
  42. ζ ( 1 / 2 + i H ^ ) = 0 \zeta(1/2+i\hat{H})=0
  43. 1 / 2 + i H ^ 1/2+i\hat{H}
  44. [ x , p ] = 1 / 2 [x,p]=1/2
  45. N ( s ) = 1 π Arg ξ ( 1 / 2 + i s ) N(s)=\frac{1}{\pi}\operatorname{Arg}\xi(1/2+i\sqrt{s})
  46. V - 1 ( x ) = 4 π d 1 / 2 N ( x ) d x 1 / 2 V^{-1}(x)=\sqrt{4\pi}\frac{d^{1/2}N(x)}{dx^{1/2}}
  47. det ( H + 1 / 4 + s ( s - 1 ) ) \det(H+1/4+s(s-1))
  48. ξ ( s ) ξ ( 0 ) = det ( H + s ( s - 1 ) + 1 / 4 ) det ( H + 1 / 4 ) . \frac{\xi(s)}{\xi(0)}=\frac{\det(H+s(s-1)+1/4)}{\det(H+1/4)}.
  49. n = 1 N n - s \sum_{n=1}^{N}n^{-s}
  50. T ( x ) = n x λ ( n ) n 0 for x > 0 , T(x)=\sum_{n\leq x}\frac{\lambda(n)}{n}\geq 0\,\text{ for }x>0,
  51. N ( T ) = 1 π Arg ( ξ ( s ) ) = 1 π Arg ( Γ ( s 2 ) π - s 2 ζ ( s ) s ( s - 1 ) / 2 ) N(T)=\frac{1}{\pi}\mathop{\mathrm{Arg}}(\xi(s))=\frac{1}{\pi}\mathop{\mathrm{% Arg}}(\Gamma(\tfrac{s}{2})\pi^{-\frac{s}{2}}\zeta(s)s(s-1)/2)
  52. 1 π Arg ( Γ ( s 2 ) π - s / 2 s ( s - 1 ) / 2 ) = T 2 π log T 2 π - T 2 π + 7 / 8 + O ( 1 / T ) \frac{1}{\pi}\mathop{\mathrm{Arg}}(\Gamma(\tfrac{s}{2})\pi^{-s/2}s(s-1)/2)=% \frac{T}{2\pi}\log\frac{T}{2\pi}-\frac{T}{2\pi}+7/8+O(1/T)
  53. S ( T ) = 1 π Arg ( ζ ( 1 / 2 + i T ) ) = O ( log ( T ) ) . S(T)=\frac{1}{\pi}\mathop{\mathrm{Arg}}(\zeta(1/2+iT))=O(\log(T)).
  54. H T 27 82 + ε H\geq T^{\frac{27}{82}+\varepsilon}
  55. H ( ln T ) 1 3 e - c ln ln T H(\ln T)^{\frac{1}{3}}e^{-c\sqrt{\ln\ln T}}
  56. 0 T | S ( t ) | 2 k d t = ( 2 k ) ! k ! ( 2 π ) 2 k T ( log log T ) k + O ( T ( log log T ) k - 1 / 2 ) . \int_{0}^{T}|S(t)|^{2k}dt=\frac{(2k)!}{k!(2\pi)^{2k}}T(\log\log T)^{k}+O(T(% \log\log T)^{k-1/2}).
  57. | ζ ( σ ) 3 ζ ( σ + i t ) 4 ζ ( σ + 2 i t ) | = exp p n p - n σ 3 + 4 cos ( t log p n ) + cos ( 2 t log p n ) n |\zeta(\sigma)^{3}\zeta(\sigma+it)^{4}\zeta(\sigma+2it)|=\exp\sum_{p^{n}}p^{-n% \sigma}\frac{3+4\cos(t\log p^{n})+\cos(2t\log p^{n})}{n}
  58. 3 + 4 cos ( θ ) + cos ( 2 θ ) = 2 ( 1 + cos ( θ ) ) 2 0. 3+4\cos(\theta)+\cos(2\theta)=2(1+\cos(\theta))^{2}\geq 0.
  59. σ 1 - 1 57.54 ( log | t | ) 2 / 3 ( log log | t | ) 1 / 3 . \sigma\geq 1-\frac{1}{57.54(\log{|t|})^{2/3}(\log{\log{|t|}})^{1/3}}.
  60. ζ ( 1 2 + i t ) \zeta\left(\tfrac{1}{2}+it\right)
  61. N 0 ( T ) N_{0}(T)
  62. ζ ( 1 2 + i t ) , \zeta\left(\tfrac{1}{2}+it\right),
  63. ζ ( 1 2 + i t ) \zeta\left(\tfrac{1}{2}+it\right)
  64. ζ ( 1 2 + i t ) \zeta\left(\tfrac{1}{2}+it\right)
  65. H = T a + ε H=T^{a+\varepsilon}
  66. T 0 = T 0 ( ε ) > 0 T_{0}=T_{0}(\varepsilon)>0
  67. T T 0 T\geq T_{0}
  68. H = T 0.25 + ε H=T^{0.25+\varepsilon}
  69. ( T , T + H ] (T,T+H]
  70. ζ ( 1 2 + i t ) \zeta\bigl(\tfrac{1}{2}+it\bigr)
  71. T 0 = T 0 ( ε ) > 0 T_{0}=T_{0}(\varepsilon)>0
  72. T T 0 T\geq T_{0}
  73. H = T 0.5 + ε H=T^{0.5+\varepsilon}
  74. N 0 ( T + H ) - N 0 ( T ) c H N_{0}(T+H)-N_{0}(T)\geq cH
  75. T 0 = T 0 ( ε ) > 0 T_{0}=T_{0}(\varepsilon)>0
  76. T T 0 T\geq T_{0}
  77. H = T 0.5 + ε H=T^{0.5+\varepsilon}
  78. N ( T + H ) - N ( T ) c H log T N(T+H)-N(T)\geq cH\log T
  79. H = T 0.5 H=T^{0.5}
  80. a = 27 82 = 1 3 - 1 246 a=\tfrac{27}{82}=\tfrac{1}{3}-\tfrac{1}{246}
  81. ζ ( 1 2 + i t ) \zeta\left(\tfrac{1}{2}+it\right)
  82. H = T ε H=T^{\varepsilon}
  83. ε 1 \varepsilon_{1}
  84. 0 < ε , ε 1 < 1 0<\varepsilon,\varepsilon_{1}<1
  85. H exp { ( ln T ) ε } H\geq\exp{\{(\ln T)^{\varepsilon}\}}
  86. H ( ln T ) 1 - ε 1 H(\ln T)^{1-\varepsilon_{1}}
  87. ζ ( 1 2 + i t ) \zeta\left(\tfrac{1}{2}+it\right)
  88. π - s 2 Γ ( s 2 ) ζ ( s ) \pi^{-\frac{s}{2}}\Gamma(\tfrac{s}{2})\zeta(s)
  89. ζ ( 1 2 + i t ) = Z ( t ) e - i θ ( t ) \zeta(\tfrac{1}{2}+it)=Z(t)e^{-i\theta(t)}
  90. × 10 2 0 \times 10^{2}0
  91. M ( x ) = n x μ ( n ) M(x)=\sum_{n\leq x}\mu(n)
  92. M ( x ) = O ( x 1 / 2 + ε ) M(x)=O(x^{1/2+\varepsilon})

Riemannian_connection_on_a_surface.html

  1. \mapsto
  2. X ( f g ) = ( X f ) g + f ( X g ) . X(fg)=(Xf)g+f(Xg).
  3. 𝒳 \mathcal{X}
  4. [ X , Y ] f = X ( Y f ) - Y ( X f ) [X,Y]f=X(Yf)-Y(Xf)
  5. 𝒳 \mathcal{X}
  6. \otimes
  7. \otimes
  8. 𝒳 \mathcal{X}
  9. 𝒳 \mathcal{X}
  10. X Y = P ( ( X I ) Y ) \nabla_{X}Y=P((X\otimes I)Y)
  11. X \nabla_{X}
  12. 𝒳 \mathcal{X}
  13. X \nabla_{X}
  14. X ( f Y ) = ( X f ) Y + f X Y \nabla_{X}(fY)=(Xf)Y+f\nabla_{X}Y
  15. X ( Y , Z ) = ( X Y , Z ) + ( Y , X Z ) X(Y,Z)=(\nabla_{X}Y,Z)+(Y,\nabla_{X}Z)
  16. X Y - Y X = [ X , Y ] \nabla_{X}Y-\nabla_{Y}X=[X,Y]
  17. \nabla
  18. T ( X , Y ) = X Y - Y X - [ X , Y ] T(X,Y)=\nabla_{X}Y-\nabla_{Y}X-[X,Y]
  19. 2 ( X Y , Z ) = X ( Y , Z ) + Y ( X , Z ) - Z ( X , Y ) + ( [ X , Y ] , Z ) + ( [ Z , X ] , Y ) + ( X , [ Z , Y ] ) , 2(\nabla_{X}Y,Z)=X\cdot(Y,Z)+Y\cdot(X,Z)-Z\cdot(X,Y)+([X,Y],Z)+([Z,X],Y)+(X,[Z% ,Y]),
  20. X Y \nabla_{X}Y
  21. X Y \nabla_{X}Y
  22. x \partial_{x}
  23. y \partial_{y}
  24. \nabla
  25. 𝐞 i 𝐞 j = k Γ i j k 𝐞 k . \nabla_{{\mathbf{e}}_{i}}{\mathbf{e}}_{j}=\sum_{k}\Gamma^{k}_{ij}{\mathbf{e}}_% {k}.
  26. c ˙ c ˙ = 0. \nabla_{\dot{c}}\dot{c}=0.
  27. c ˙ ( t ) \dot{c}(t)
  28. θ ˙ ( t ) = - k g ( t ) \dot{\theta}(t)=-k_{g}(t)
  29. c ˙ v = 0 \nabla_{\dot{c}}v=0
  30. P ( c ( t ) ) v ˙ ( t ) = 0 P(c(t))\dot{v}(t)=0
  31. X Y \nabla_{X}Y
  32. det ( 𝐯 , 𝐰 ) = ( 𝐯 × 𝐰 ) 𝐧 , \mathrm{det}({\mathbf{v}},{\mathbf{w}})=({\mathbf{v}}\times{\mathbf{w}})\cdot{% \mathbf{n}},
  33. \mapsto
  34. \subset
  35. C ( E , V ) K , C^{\infty}(E,V)^{K},
  36. ξ ( x g ) = σ ( g - 1 ) ξ ( x ) \xi(x\cdot g)=\sigma(g^{-1})\xi(x)
  37. X \nabla_{X}
  38. X ξ = ( X * I ) ξ . \nabla_{X}\xi=(X^{*}\otimes I)\xi.
  39. \mapsto
  40. Λ 1 ( E ) K \Lambda^{1}(E)^{K}
  41. ω ( X * ) = 0 \omega(X^{*})=0
  42. ω ( A * ) = 1 , \omega(A^{*})=1,
  43. 𝔨 \mathfrak{k}
  44. \rightarrow
  45. θ i ( Y ) = ( d π ( Y ) , e i ) \theta_{i}(Y)=(d\pi(Y),e_{i})
  46. d θ 1 = ω θ 2 , d θ 2 = - ω θ 1 d\theta_{1}=\omega\wedge\theta_{2},\,\,d\theta_{2}=-\omega\wedge\theta_{1}
  47. d ω = - ( K π ) θ 1 θ 2 d\omega=-(K\circ\pi)\theta_{1}\wedge\theta_{2}
  48. s \sqrt{s}
  49. α s ( x ) \alpha_{\sqrt{s}}(x)
  50. s \sqrt{s}
  51. β s α s ( x ) \beta_{\sqrt{s}}\alpha_{\sqrt{s}}(x)
  52. - s -\sqrt{s}
  53. α - s β s α s ( x ) \alpha_{-\sqrt{s}}\beta_{\sqrt{s}}\alpha_{\sqrt{s}}(x)
  54. - s -\sqrt{s}
  55. β - s α - s β s α s ( x ) \beta_{-\sqrt{s}}\alpha_{-\sqrt{s}}\beta_{\sqrt{s}}\alpha_{\sqrt{s}}(x)
  56. \rightarrow
  57. ω ( U i ) = 0 , θ i ( V ) = 0 , ω ( V ) = 1 , θ i ( U j ) = δ i j . \omega(U_{i})=0,\,\theta_{i}(V)=0,\,\omega(V)=1,\,\theta_{i}(U_{j})=\delta_{ij}.
  58. [ V , U 1 ] = U 2 , [ V , U 2 ] = - U 1 , [ U 1 , U 2 ] = ( K π ) V [V,U_{1}]=U_{2},\,\,\,\,[V,U_{2}]=-U_{1},\,\,\,\,[U_{1},U_{2}]=(K\circ\pi)V
  59. s s = s \sqrt{s}\cdot\sqrt{s}=s
  60. \rightarrow
  61. θ = f ( Δ ) K \theta=\int_{f(\Delta)}K
  62. X \nabla_{X}
  63. R ( X , Y ) = X Y - Y X - [ X , Y ] R(X,Y)=\nabla_{X}\nabla_{Y}-\nabla_{Y}\nabla_{X}-\nabla_{[X,Y]}
  64. X \nabla_{X}
  65. 𝔨 \mathfrak{k}
  66. S 2 = { a E 3 : a = 1 } . S^{2}=\{a\in E^{3}\colon\|a\|=1\}.
  67. T = { ( a , v ) : a = 1 , a v = 0 } , T=\{(a,v)\colon\|a\|=1,\,a\cdot v=0\},
  68. U = { ( a , v ) : a = 1 , v = 1 , a v = 0 } , U=\{(a,v)\colon\|a\|=1,\,\|v\|=1,\,a\cdot v=0\},
  69. E = { ( a , e 1 , e 2 ) : ( e 1 × e 2 ) a = 1 , a = 1 , e i = 1 , a e i = 0 , e 1 e 2 = 0 } . E=\{(a,e_{1},e_{2})\colon(e_{1}\times e_{2})\cdot a=1,\,\|a\|=1,\,\|e_{i}\|=1,% \,a\cdot e_{i}=0,\,e_{1}\cdot e_{2}=0\}.
  70. Q ( a ) v = ( v a ) a Q(a)v=(v\cdot a)a
  71. P ( a ) = I - Q ( a ) P(a)=I-Q(a)
  72. ( e 1 , e 2 ) ( cos θ e 1 - sin θ e 2 , sin θ e 1 + cos θ e 2 ) . (e_{1},e_{2})\mapsto(\cos\theta\,e_{1}-\sin\theta\,e_{2},\sin\theta\,e_{1}+% \cos\theta\,e_{2}).
  73. 𝔤 \mathfrak{g}
  74. 𝔤 \mathfrak{g}
  75. 𝔤 \mathfrak{g}
  76. 𝔤 \mathfrak{g}
  77. λ ( X ) f ( g ) = d d t f ( e - X t g ) | t = 0 , ρ ( X ) f ( g ) = d d t f ( g e X t ) | t = 0 . \lambda(X)f(g)={d\over dt}f(e^{-Xt}g)|_{t=0},\,\,\rho(X)f(g)={d\over dt}f(ge^{% Xt})|_{t=0}.
  78. λ ( X ) f ( g ) = - ρ ( g - 1 X g ) f ( g ) . \lambda(X)f(g)=-\rho(g^{-1}Xg)f(g).
  79. 𝔤 \mathfrak{g}
  80. A = ( 0 1 0 - 1 0 0 0 0 0 ) , B = ( 0 0 0 0 0 1 0 - 1 0 ) , C = ( 0 0 1 0 0 0 - 1 0 0 ) . A=\begin{pmatrix}0&1&0\\ -1&0&0\\ 0&0&0\end{pmatrix},\,\,B=\begin{pmatrix}0&0&0\\ 0&0&1\\ 0&-1&0\end{pmatrix},\,\,C=\begin{pmatrix}0&0&1\\ 0&0&0\\ -1&0&0\end{pmatrix}.
  81. [ A , B ] = C , [ B , C ] = A , [ C , A ] = B . [A,B]=C,\,\,[B,C]=A,\,\,[C,A]=B.
  82. d α = β γ , d β = γ α , d γ = α β . d\alpha=\beta\wedge\gamma,\,\,d\beta=\gamma\wedge\alpha,\,\,d\gamma=\alpha% \wedge\beta.
  83. ω G = g - 1 d g , \omega_{G}=g^{-1}dg,
  84. d ω G = - ( g - 1 d g g - 1 ) d g = - ω G ω G . d\omega_{G}=-(g^{-1}dg\,g^{-1})dg=-\omega_{G}\wedge\omega_{G}.
  85. 𝔤 \mathfrak{g}
  86. ( X , Y ) = Tr X Y T (X,Y)=\mathrm{Tr}\,XY^{T}
  87. 𝔨 \mathfrak{k}
  88. 𝔤 \mathfrak{g}
  89. Π ( X ) * f ( g ) = - ρ ( π ( g - 1 X g ) ) f ( g ) \Pi(X)^{*}f(g)=-\rho(\pi(g^{-1}Xg))f(g)
  90. Q = g Q 0 g - 1 , Q 0 g - 1 g ˙ Q 0 = 0 Q=gQ_{0}g^{-1},\,\,Q_{0}g^{-1}\dot{g}Q_{0}=0
  91. g ˙ = A g \dot{g}=Ag\,
  92. d d t ( g T g ) = g ˙ T g + g T g ˙ = g T ( A T + A ) g = 0. {d\over dt}(g^{T}g)=\dot{g}^{T}g+g^{T}\dot{g}=g^{T}(A^{T}+A)g=0.
  93. F ( t ) = g ( t ) F ( 0 ) g ( t ) - 1 F(t)=g(t)F(0)g(t)^{-1}\,
  94. d d t ( g - 1 F g ) = - g - 1 g ˙ g - 1 F g + g - 1 F ˙ g + g - 1 F g ˙ = g - 1 ( - g ˙ g - 1 F + F ˙ + F g ˙ g - 1 ) g = 0. {d\over dt}(g^{-1}Fg)=-g^{-1}\dot{g}g^{-1}Fg+g^{-1}\dot{F}g+g^{-1}F\dot{g}=g^{% -1}(-\dot{g}g^{-1}F+\dot{F}+F\dot{g}g^{-1})g=0.
  95. \rightarrow
  96. \rightarrow
  97. v ˙ ( t ) \dot{v}(t)
  98. Q = g Q 0 g - 1 , Q 0 g - 1 g ˙ Q 0 = 0. Q=gQ_{0}g^{-1},\,\,\,Q_{0}g^{-1}\dot{g}Q_{0}=0.
  99. P ( c ( t ) ) v ˙ ( t ) = 0. P(c(t))\dot{v}(t)=0.
  100. v ˙ ( t ) \dot{v}(t)
  101. c ˙ v = 0 \nabla_{\dot{c}}v=0
  102. ω = d e 1 e 2 \omega=de_{1}\cdot e_{2}
  103. ψ θ 1 + χ θ 2 = 0 \psi\wedge\theta_{1}+\chi\wedge\theta_{2}=0
  104. d ω = ψ χ d\omega=\psi\wedge\chi
  105. d ψ = χ ω , d χ = ω ψ d\psi=\chi\wedge\omega,\,\,d\chi=\omega\wedge\psi
  106. θ t = a ( t ) \theta_{t}=a(t)
  107. 𝔥 \mathfrak{h}

Riesz_transform.html

  1. R j f ( x ) = c d lim ϵ 0 𝐑 d \ B ϵ ( x ) ( t j - x j ) f ( t ) | x - t | d + 1 d t R_{j}f(x)=c_{d}\lim_{\epsilon\to 0}\int_{\mathbf{R}^{d}\backslash B_{\epsilon}% (x)}\frac{(t_{j}-x_{j})f(t)}{|x-t|^{d+1}}\,dt
  2. c d = 1 π ω d - 1 = Γ [ ( d + 1 ) / 2 ] π ( d + 1 ) / 2 . c_{d}=\frac{1}{\pi\omega_{d-1}}=\frac{\Gamma[(d+1)/2]}{\pi^{(d+1)/2}}.
  3. K ( x ) = 1 π ω d - 1 p . v . x j | x | d + 1 . K(x)=\frac{1}{\pi\omega_{d-1}}\,p.v.\frac{x_{j}}{|x|^{d+1}}.
  4. ( R j f ) ( x ) = i x j | x | ( f ) ( x ) \mathcal{F}(R_{j}f)(x)=i\frac{x_{j}}{|x|}(\mathcal{F}f)(x)
  5. σ s * f = f σ s . \sigma_{s}^{*}f=f\circ\sigma_{s}.
  6. σ s * ( R j f ) = R j ( σ x * f ) . \sigma_{s}^{*}(R_{j}f)=R_{j}(\sigma_{x}^{*}f).
  7. τ a * ( R j f ) = R j ( τ a * f ) . \tau_{a}^{*}(R_{j}f)=R_{j}(\tau_{a}^{*}f).
  8. ρ * R j [ ( ρ - 1 ) * f ] = k = 1 d ρ j k R k f . \rho^{*}R_{j}[(\rho^{-1})^{*}f]=\sum_{k=1}^{d}\rho_{jk}R_{k}f.
  9. ( - Δ ) 1 2 u = f , {(-\Delta)^{\frac{1}{2}}u=f},
  10. R f = ( - Δ ) - 1 2 f {Rf=\nabla(-\Delta)^{-\frac{1}{2}}f}
  11. R i R j Δ u = - 2 u x i x j , R_{i}R_{j}\Delta u=-\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}},
  12. R i R j ( Δ u ) = - 2 u x i x j . R_{i}R_{j}(\Delta u)=-\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}.
  13. 2 u x i x j = - R i R j Δ u + P i j ( x ) \frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}=-R_{i}R_{j}\Delta u+P_{ij}(x)

Rising_sun_lemma.html

  1. E = k ( a k , b k ) E=\bigcup_{k}(a_{k},b_{k})

Roe_solver.html

  1. F i + 1 2 F_{i+\frac{1}{2}}
  2. U i U_{i}
  3. U i + 1 U_{i+1}
  4. s y m b o l U t + s y m b o l F ( s y m b o l U ) x = 0. \frac{\partial symbol{U}}{\partial t}+\frac{\partial symbol{F}(symbol{U})}{% \partial x}=0.
  5. s y m b o l U t + A ( s y m b o l U ) s y m b o l U x = 0 , \frac{\partial symbol{U}}{\partial t}+A(symbol{U})\frac{\partial symbol{U}}{% \partial x}=0,
  6. A A
  7. s y m b o l F ( s y m b o l U ) symbol{F}(symbol{U})
  8. A ~ ( s y m b o l U i , s y m b o l U i + 1 ) \tilde{A}(symbol{U}_{i},symbol{U}_{i+1})
  9. s y m b o l U i , s y m b o l U i + 1 s y m b o l U symbol{U}_{i},symbol{U}_{i+1}\rightarrow symbol{U}
  10. A ~ ( s y m b o l U i , s y m b o l U i + 1 ) = A ( s y m b o l U ) \tilde{A}(symbol{U}_{i},symbol{U}_{i+1})=A(symbol{U})
  11. s y m b o l F i + 1 - s y m b o l F i = A ~ ( s y m b o l U i + 1 - s y m b o l U i ) symbol{F}_{i+1}-symbol{F}_{i}=\tilde{A}(symbol{U}_{i+1}-symbol{U}_{i})

Rogers–Ramanujan_continued_fraction.html

  1. q 1 / 5 A 400 ( q ) / B 400 ( q ) q^{1/5}A_{400}(q)/B_{400}(q)
  2. G ( q ) = n = 0 q n 2 ( 1 - q ) ( 1 - q 2 ) ( 1 - q k ) = n = 0 q n 2 ( q ; q ) n = 1 ( q ; q 5 ) ( q 4 ; q 5 ) = n = 1 1 ( 1 - q 5 n - 1 ) ( 1 - q 5 n - 4 ) = q j 60 2 F 1 ( - 1 60 , 19 60 ; 4 5 ; 1728 j ) = q ( j - 1728 ) 60 2 F 1 ( - 1 60 , 29 60 ; 4 5 ; - 1728 j - 1728 ) = 1 + q + q 2 + q 3 + 2 q 4 + 2 q 5 + 3 q 6 + \begin{aligned}\displaystyle G(q)&\displaystyle=\sum_{n=0}^{\infty}\frac{q^{n^% {2}}}{(1-q)(1-q^{2})\dots(1-q^{k})}=\sum_{n=0}^{\infty}\frac{q^{n^{2}}}{(q;q)_% {n}}=\frac{1}{(q;q^{5})_{\infty}(q^{4};q^{5})_{\infty}}\\ &\displaystyle=\prod_{n=1}^{\infty}\frac{1}{(1-q^{5n-1})(1-q^{5n-4})}\\ &\displaystyle=\sqrt[60]{qj}\,_{2}F_{1}\left(-\tfrac{1}{60},\tfrac{19}{60};% \tfrac{4}{5};\tfrac{1728}{j}\right)\\ &\displaystyle=\sqrt[60]{q\left(j-1728\right)}\,_{2}F_{1}\left(-\tfrac{1}{60},% \tfrac{29}{60};\tfrac{4}{5};-\tfrac{1728}{j-1728}\right)\\ &\displaystyle=1+q+q^{2}+q^{3}+2q^{4}+2q^{5}+3q^{6}+\cdots\end{aligned}
  3. H ( q ) = n = 0 q n 2 + n ( 1 - q ) ( 1 - q 2 ) ( 1 - q k ) = n = 0 q n 2 + n ( q ; q ) n = 1 ( q 2 ; q 5 ) ( q 3 ; q 5 ) = n = 1 1 ( 1 - q 5 n - 2 ) ( 1 - q 5 n - 3 ) = 1 q 11 j 11 60 2 F 1 ( 11 60 , 31 60 ; 6 5 ; 1728 j ) = 1 q 11 ( j - 1728 ) 11 60 2 F 1 ( 11 60 , 41 60 ; 6 5 ; - 1728 j - 1728 ) = 1 + q 2 + q 3 + q 4 + q 5 + 2 q 6 + 2 q 7 + \begin{aligned}\displaystyle H(q)&\displaystyle=\sum_{n=0}^{\infty}\frac{q^{n^% {2}+n}}{(1-q)(1-q^{2})\dots(1-q^{k})}=\sum_{n=0}^{\infty}\frac{q^{n^{2}+n}}{(q% ;q)_{n}}=\frac{1}{(q^{2};q^{5})_{\infty}(q^{3};q^{5})_{\infty}}\\ &\displaystyle=\prod_{n=1}^{\infty}\frac{1}{(1-q^{5n-2})(1-q^{5n-3})}\\ &\displaystyle=\frac{1}{\sqrt[60]{q^{11}j^{11}}}\,_{2}F_{1}\left(\tfrac{11}{60% },\tfrac{31}{60};\tfrac{6}{5};\tfrac{1728}{j}\right)\\ &\displaystyle=\frac{1}{\sqrt[60]{q^{11}\left(j-1728\right)^{11}}}\,_{2}F_{1}% \left(\tfrac{11}{60},\tfrac{41}{60};\tfrac{6}{5};-\tfrac{1728}{j-1728}\right)% \\ &\displaystyle=1+q^{2}+q^{3}+q^{4}+q^{5}+2q^{6}+2q^{7}+\cdots\end{aligned}
  4. ( a ; q ) (a;q)_{\infty}
  5. R ( q ) = q 11 60 H ( q ) q - 1 60 G ( q ) = q 1 5 n = 1 ( 1 - q 5 n - 1 ) ( 1 - q 5 n - 4 ) ( 1 - q 5 n - 2 ) ( 1 - q 5 n - 3 ) = q 1 / 5 1 + q 1 + q 2 1 + q 3 1 + \begin{aligned}\displaystyle R(q)&\displaystyle=\frac{q^{\frac{11}{60}}H(q)}{q% ^{-\frac{1}{60}}G(q)}=q^{\frac{1}{5}}\prod_{n=1}^{\infty}\frac{(1-q^{5n-1})(1-% q^{5n-4})}{(1-q^{5n-2})(1-q^{5n-3})}\\ &\displaystyle=\cfrac{q^{1/5}}{1+\cfrac{q}{1+\cfrac{q^{2}}{1+\cfrac{q^{3}}{1+% \ddots}}}}\end{aligned}
  6. q = e 2 π i τ q=e^{2\pi{\rm{i}}\tau}
  7. q - 1 60 G ( q ) q^{-\frac{1}{60}}G(q)
  8. q 11 60 H ( q ) q^{\frac{11}{60}}H(q)
  9. R ( q ) R(q)
  10. τ \tau
  11. τ \tau
  12. R ( e - 2 π ) = e - 2 π 5 1 + e - 2 π 1 + e - 4 π 1 + = 5 + 5 2 - ϕ R\big(e^{-2\pi}\big)=\cfrac{e^{-\frac{2\pi}{5}}}{1+\cfrac{e^{-2\pi}}{1+\cfrac{% e^{-4\pi}}{1+\ddots}}}={\sqrt{5+\sqrt{5}\over 2}-\phi}
  13. R ( e - 2 5 π ) = e - 2 π 5 1 + e - 2 π 5 1 + e - 4 π 5 1 + = 5 1 + ( 5 3 / 4 ( ϕ - 1 ) 5 / 2 - 1 ) 1 / 5 - ϕ R\big(e^{-2\sqrt{5}\pi}\big)=\cfrac{e^{-\frac{2\pi}{\sqrt{5}}}}{1+\cfrac{e^{-2% \pi\sqrt{5}}}{1+\cfrac{e^{-4\pi\sqrt{5}}}{1+\ddots}}}=\frac{\sqrt{5}}{1+\big(5% ^{3/4}(\phi-1)^{5/2}-1\big)^{1/5}}-{\phi}
  14. ϕ = 1 + 5 2 \phi=\frac{1+\sqrt{5}}{2}
  15. 1 R ( q ) - R ( q ) = η ( τ 5 ) η ( 5 τ ) + 1 \frac{1}{R(q)}-R(q)=\frac{\eta(\frac{\tau}{5})}{\eta(5\tau)}+1
  16. 1 R 5 ( q ) - R 5 ( q ) = [ η ( τ ) η ( 5 τ ) ] 6 + 11 \frac{1}{R^{5}(q)}-R^{5}(q)=\left[\frac{\eta(\tau)}{\eta(5\tau)}\right]^{6}+11
  17. j ( τ ) = ( x 2 + 10 x + 5 ) 3 x j(\tau)=\frac{(x^{2}+10x+5)^{3}}{x}
  18. x = [ 5 η ( 5 τ ) η ( τ ) ] 6 x=\left[\frac{\sqrt{5}\,\eta(5\tau)}{\eta(\tau)}\right]^{6}
  19. r = R ( q ) r=R(q)
  20. j ( τ ) = - ( r 20 - 228 r 15 + 494 r 10 + 228 r 5 + 1 ) 3 r 5 ( r 10 + 11 r 5 - 1 ) 5 j(\tau)=-\frac{(r^{20}-228r^{15}+494r^{10}+228r^{5}+1)^{3}}{r^{5}(r^{10}+11r^{% 5}-1)^{5}}
  21. j ( τ ) - 1728 = - ( r 30 + 522 r 25 - 10005 r 20 - 10005 r 10 - 522 r 5 + 1 ) 2 r 5 ( r 10 + 11 r 5 - 1 ) 5 j(\tau)-1728=-\frac{(r^{30}+522r^{25}-10005r^{20}-10005r^{10}-522r^{5}+1)^{2}}% {r^{5}(r^{10}+11r^{5}-1)^{5}}
  22. R ( q ) R(q)
  23. R ( q 5 ) R(q^{5})
  24. j ( 5 τ ) = - ( r 20 + 12 r 15 + 14 r 10 - 12 r 5 + 1 ) 3 r 25 ( r 10 + 11 r 5 - 1 ) j(5\tau)=-\frac{(r^{20}+12r^{15}+14r^{10}-12r^{5}+1)^{3}}{r^{25}(r^{10}+11r^{5% }-1)}
  25. z = r 5 - 1 r 5 z=r^{5}-\frac{1}{r^{5}}
  26. j ( 5 τ ) = - ( z 2 + 12 z + 16 ) 3 z + 11 j(5\tau)=-\frac{\left(z^{2}+12z+16\right)^{3}}{z+11}
  27. z = - [ 5 η ( 25 τ ) η ( 5 τ ) ] 6 - 11 , z 0 = - [ η ( τ ) η ( 5 τ ) ] 6 - 11 , z 1 = [ η ( 5 τ + 2 5 ) η ( 5 τ ) ] 6 - 11 , z 2 = - [ η ( 5 τ + 4 5 ) η ( 5 τ ) ] 6 - 11 , z 3 = [ η ( 5 τ + 6 5 ) η ( 5 τ ) ] 6 - 11 , z 4 = - [ η ( 5 τ + 8 5 ) η ( 5 τ ) ] 6 - 11 z_{\infty}=-\left[\frac{\sqrt{5}\,\eta(25\tau)}{\eta(5\tau)}\right]^{6}-11,z_{% 0}=-\left[\frac{\eta(\tau)}{\eta(5\tau)}\right]^{6}-11,z_{1}=\left[\frac{\eta(% \frac{5\tau+2}{5})}{\eta(5\tau)}\right]^{6}-11,z_{2}=-\left[\frac{\eta(\frac{5% \tau+4}{5})}{\eta(5\tau)}\right]^{6}-11,z_{3}=\left[\frac{\eta(\frac{5\tau+6}{% 5})}{\eta(5\tau)}\right]^{6}-11,z_{4}=-\left[\frac{\eta(\frac{5\tau+8}{5})}{% \eta(5\tau)}\right]^{6}-11
  28. y 2 + ( 1 + r 5 ) x y + r 5 y = x 3 + r 5 x 2 y^{2}+(1+r^{5})xy+r^{5}y=x^{3}+r^{5}x^{2}
  29. X 1 ( 5 ) X_{1}(5)
  30. r ( τ ) = R ( q ) r(\tau)=R(q)
  31. j ( - 1 τ ) = j ( τ ) j(-\tfrac{1}{\tau})=j(\tau)
  32. η ( - 1 τ ) = - i τ η ( τ ) \eta(-\tfrac{1}{\tau})=\sqrt{-i\tau}\,\eta(\tau)
  33. ϕ \phi
  34. r ( - 1 τ ) = 1 - ϕ r ( τ ) ϕ + r ( τ ) r(-\tfrac{1}{\tau})=\frac{1-\phi\,r(\tau)}{\phi+r(\tau)}
  35. r ( 7 + i 10 ) = i r(\tfrac{7+i}{10})=i
  36. R ( q ) R(q)
  37. R ( q n ) R(q^{n})
  38. n = 2 n=2
  39. u = R ( q ) u=R(q)
  40. v = R ( q 2 ) v=R(q^{2})
  41. v - u 2 = ( v + u 2 ) u v 2 . v-u^{2}=(v+u^{2})uv^{2}.
  42. n = 3 n=3
  43. u = R ( q ) u=R(q)
  44. v = R ( q 3 ) v=R(q^{3})
  45. ( v - u 3 ) ( 1 + u v 3 ) = 3 u 2 v 2 . (v-u^{3})(1+uv^{3})=3u^{2}v^{2}.
  46. n = 5 n=5
  47. u = R ( q ) u=R(q)
  48. v = R ( q 5 ) v=R(q^{5})
  49. ( v 4 - 3 v 3 + 4 v 2 - 2 v + 1 ) v = ( v 4 + 2 v 3 + 4 v 2 + 3 v + 1 ) u 5 . (v^{4}-3v^{3}+4v^{2}-2v+1)v=(v^{4}+2v^{3}+4v^{2}+3v+1)u^{5}.
  50. n = 11 n=11
  51. u = R ( q ) u=R(q)
  52. v = R ( q 11 ) v=R(q^{11})
  53. u v ( u 10 + 11 u 5 - 1 ) ( v 10 + 11 v 5 - 1 ) = ( u - v ) 12 . uv(u^{10}+11u^{5}-1)(v^{10}+11v^{5}-1)=(u-v)^{12}.
  54. n = 5 n=5
  55. v 10 + 11 v 5 - 1 = ( v 2 + v - 1 ) ( v 4 - 3 v 3 + 4 v 2 - 2 v + 1 ) ( v 4 + 2 v 3 + 4 v 2 + 3 v + 1 ) . v^{10}+11v^{5}-1=(v^{2}+v-1)(v^{4}-3v^{3}+4v^{2}-2v+1)(v^{4}+2v^{3}+4v^{2}+3v+% 1).
  56. u = R ( q a ) u=R(q^{a})
  57. v = R ( q b ) v=R(q^{b})
  58. ϕ \phi
  59. a b = 4 π 2 ab=4\pi^{2}
  60. ( u + ϕ ) ( v + ϕ ) = 5 ϕ . (u+\phi)(v+\phi)=\sqrt{5}\,\phi.
  61. 5 a b = 4 π 2 5ab=4\pi^{2}
  62. ( u 5 + ϕ 5 ) ( v 5 + ϕ 5 ) = 5 5 ϕ 5 . (u^{5}+\phi^{5})(v^{5}+\phi^{5})=5\sqrt{5}\,\phi^{5}.
  63. R 3 ( q ) = n = 0 q 2 n 1 - q 5 n + 2 - n = 0 q 3 n + 1 1 - q 5 n + 3 n = 0 q n 1 - q 5 n + 1 - n = 0 q 4 n + 3 1 - q 5 n + 4 R^{3}(q)=\frac{\sum_{n=0}^{\infty}\frac{q^{2n}}{1-q^{5n+2}}-\sum_{n=0}^{\infty% }\frac{q^{3n+1}}{1-q^{5n+3}}}{\sum_{n=0}^{\infty}\frac{q^{n}}{1-q^{5n+1}}-\sum% _{n=0}^{\infty}\frac{q^{4n+3}}{1-q^{5n+4}}}
  64. w = R ( q ) R 2 ( q 2 ) w=R(q)R^{2}(q^{2})
  65. R 5 ( q ) = w ( 1 - w 1 + w ) 2 , R 5 ( q 2 ) = w 2 ( 1 + w 1 - w ) R^{5}(q)=w\left(\frac{1-w}{1+w}\right)^{2},\;\;R^{5}(q^{2})=w^{2}\left(\frac{1% +w}{1-w}\right)

Roll's_critique.html

  1. E ( R i ) = R f + β i m [ E ( R m ) - R f ] . E(R_{i})=R_{f}+\beta_{im}[E(R_{m})-R_{f}].\,
  2. E ( R i ) E(R_{i})
  3. β i m \beta_{im}
  4. R m R_{m}
  5. R p R_{p}
  6. E ( R i ) = R f + β i p [ E ( R p ) - R f ] E(R_{i})=R_{f}+\beta_{ip}[E(R_{p})-R_{f}]\,
  7. E ( R i ) = α + β 1 F 1 + + β N F N . E(R_{i})=\alpha+\beta_{1}F_{1}+...+\beta_{N}F_{N}.\,
  8. F 1 , , F N F_{1},...,F_{N}\,

Roman_numeral_analysis.html

  1. 1 ^ \hat{1}

Root-raised-cosine_filter.html

  1. H r r c ( f ) H_{rrc}(f)
  2. H r c ( f ) H_{rc}(f)
  3. H r c ( f ) = H r r c ( f ) H r r c ( f ) H_{rc}(f)=H_{rrc}(f)\cdot H_{rrc}(f)
  4. | H r r c ( f ) | = | H r c ( f ) | |H_{rrc}(f)|=\sqrt{|H_{rc}(f)|}
  5. h ( t ) = { 1 T s ( 1 - β + 4 β π ) , t = 0 β 2 T s [ ( 1 + 2 π ) sin ( π 4 β ) + ( 1 - 2 π ) cos ( π 4 β ) ] , t = ± T s 4 β 1 T s sin [ π t T s ( 1 - β ) ] + 4 β t T s cos [ π t T s ( 1 + β ) ] π t T s [ 1 - ( 4 β t T s ) 2 ] , otherwise h(t)=\begin{cases}\dfrac{1}{\sqrt{T_{s}}}\left(1-\beta+4\dfrac{\beta}{\pi}% \right),&t=0\\ \dfrac{\beta}{\sqrt{2T_{s}}}\left[\left(1+\dfrac{2}{\pi}\right)\sin\left(% \dfrac{\pi}{4\beta}\right)+\left(1-\dfrac{2}{\pi}\right)\cos\left(\dfrac{\pi}{% 4\beta}\right)\right],&t=\pm\dfrac{T_{s}}{4\beta}\\ \dfrac{1}{\sqrt{T_{s}}}\dfrac{\sin\left[\pi\dfrac{t}{T_{s}}\left(1-\beta\right% )\right]+4\beta\dfrac{t}{T_{s}}\cos\left[\pi\dfrac{t}{T_{s}}\left(1+\beta% \right)\right]}{\pi\dfrac{t}{T_{s}}\left[1-\left(4\beta\dfrac{t}{T_{s}}\right)% ^{2}\right]},&\mbox{otherwise}\end{cases}

Rossby-gravity_waves.html

  1. f y = β \frac{\partial f}{\partial y}=\beta
  2. ϕ t + c 2 ( v y + u x ) = 0 \frac{\partial\phi}{\partial t}+c^{2}\left(\frac{\partial v}{\partial y}+\frac% {\partial u}{\partial x}\right)=0
  3. u t - v β y = - ϕ x \frac{\partial u}{\partial t}-v\beta y=-\frac{\partial\phi}{\partial x}
  4. v t + u β y = - ϕ y \frac{\partial v}{\partial t}+u\beta y=-\frac{\partial\phi}{\partial y}
  5. { u , v , ϕ } = { u ^ ( y ) , v ^ ( y ) , ϕ ^ ( y ) } e i ( k x - ω t ) \begin{Bmatrix}u,v,\phi\end{Bmatrix}=\begin{Bmatrix}\hat{u}(y),\hat{v}(y),\hat% {\phi}(y)\end{Bmatrix}e^{i(kx-\omega t)}

Rosser's_trick.html

  1. Proof T R ( x , y ) Proof T ( x , y ) ¬ z x [ Proof T ( z , neg ( y ) ) ] , \mathrm{Proof}^{R}_{T}(x,y)\equiv\mathrm{Proof}_{T}(x,y)\land\lnot\exists z% \leq x[\mathrm{Proof}_{T}(z,\mathrm{neg}(y))],
  2. ¬ Proof T R ( x , y ) Proof T ( x , y ) z x [ Proof T ( z , neg ( y ) ) ] . \lnot\mathrm{Proof}^{R}_{T}(x,y)\equiv\mathrm{Proof}_{T}(x,y)\to\exists z\leq x% [\mathrm{Proof}_{T}(z,\mathrm{neg}(y))].
  3. Pvbl T R ( y ) x Proof T R ( x , y ) . \mathrm{Pvbl}^{R}_{T}(y)\equiv\exists x\mathrm{Proof}^{R}_{T}(x,y).
  4. ¬ x < e Proof T ( x , # ρ ) . \lnot\exists x<e\,\mathrm{Proof}_{T}(x,\#\rho).
  5. Proof T ( x , # ρ ) e x . \mathrm{Proof}_{T}(x,\#\rho)\to e\leq x.
  6. x [ Proof T ( x , # ρ ) z x Proof T ( z , neg ( # ρ ) ) ] . \forall x[\mathrm{Proof}_{T}(x,\#\rho)\to\exists z\leq x\mathrm{Proof}_{T}(z,% \mathrm{neg}(\#\rho))].

Rotating_calipers.html

  1. O ( n ) O(n)

Rotating_spheres.html

  1. 𝐅 centripetal = - m 𝛀 × ( 𝛀 × 𝐱 𝐁 ) \mathbf{F}_{\mathrm{centripetal}}=-m\mathbf{\Omega\ \times}\left(\mathbf{% \Omega\times x_{B}}\right)
  2. = - m ω 2 R 𝐮 R , =-m\omega^{2}R\ \mathbf{u}_{R}\ ,
  3. 𝐅 fict = - 2 m s y m b o l Ω × 𝐯 B \mathbf{F}_{\mathrm{fict}}=-2msymbol\Omega\times\mathbf{v}_{B}
  4. = - 2 m ω ( ω R ) 𝐮 R , =-2m\omega\left(\omega R\right)\ \mathbf{u}_{R},
  5. 𝐓 = - m ω I 2 R 𝐮 R . \mathbf{T}=-m\omega_{I}^{2}R\mathbf{u}_{R}\ .
  6. 𝐅 Centripetal = 𝐓 + 𝐅 Fict , \mathbf{F}_{\mathrm{Centripetal}}=\mathbf{T}+\mathbf{F}_{\mathrm{Fict}}\ ,
  7. 𝐅 Fict = - m ( ω S 2 R - ω I 2 R ) 𝐮 R . \mathbf{F}_{\mathrm{Fict}}=-m\left(\omega_{S}^{2}R-\omega_{I}^{2}R\right)% \mathbf{u}_{R}\ .
  8. 𝐅 Fict = - 2 m s y m b o l Ω × 𝐯 B - m s y m b o l Ω × ( s y m b o l Ω × 𝐱 B ) \mathbf{F}_{\mathrm{Fict}}=-2msymbol\Omega\times\mathbf{v}_{B}-msymbol\Omega% \times(symbol\Omega\times\mathbf{x}_{B})
  9. - m d s y m b o l Ω d t × 𝐱 B . \ -m\frac{dsymbol\Omega}{dt}\times\mathbf{x}_{B}\ .
  10. 𝐱 B = R 𝐮 R , \mathbf{x}_{B}=R\mathbf{u}_{R}\ ,
  11. 𝐯 B = ω S R 𝐮 θ , \mathbf{v}_{B}=\omega_{S}R\mathbf{u}_{\theta}\ ,
  12. 𝐅 Cfgl = - m s y m b o l Ω × ( s y m b o l Ω × 𝐱 B ) = m ω R 2 R 𝐮 R , \mathbf{F}_{\mathrm{Cfgl}}=-msymbol\Omega\times(symbol\Omega\times\mathbf{x}_{% B})=m\omega_{R}^{2}R\mathbf{u}_{R}\ ,
  13. 𝐅 Cor = - 2 m s y m b o l Ω × 𝐯 B = 2 m ω S ω R R 𝐮 R \mathbf{F}_{\mathrm{Cor}}=-2msymbol\Omega\times\mathbf{v}_{B}=2m\omega_{S}% \omega_{R}R\mathbf{u}_{R}
  14. 𝐅 Fict = 𝐅 Cfgl + 𝐅 Cor \mathbf{F}_{\mathrm{Fict}}=\mathbf{F}_{\mathrm{Cfgl}}+\mathbf{F}_{\mathrm{Cor}}
  15. = ( m ω R 2 R + 2 m ω S ω R R ) 𝐮 R = m ω R ( ω R + 2 ω S ) R 𝐮 R =\left(m\omega_{R}^{2}R+2m\omega_{S}\omega_{R}R\right)\mathbf{u}_{R}=m\omega_{% R}\left(\omega_{R}+2\omega_{S}\right)R\mathbf{u}_{R}
  16. = m ( ω I - ω S ) ( ω I + ω S ) R 𝐮 R = - m ( ω S 2 - ω I 2 ) R 𝐮 R . =m(\omega_{I}-\omega_{S})(\omega_{I}+\omega_{S})\ R\mathbf{u}_{R}=-m\left(% \omega_{S}^{2}-\omega_{I}^{2}\right)\ R\mathbf{u}_{R}.
  17. r ¨ - ω S 2 r = 2 ω S ω R r + ω R 2 r \ddot{r}-\omega_{S}^{2}r=2\omega_{S}\omega_{R}r+\omega_{R}^{2}r
  18. ω S = ϕ ˙ \omega_{S}=\dot{\phi}^{\prime}
  19. ω R = Ω \omega_{R}=\Omega

Rotational_Brownian_motion_(astronomy).html

  1. M 1 M_{1}
  2. M 2 M_{2}
  3. M 12 = M 1 + M 2 M_{12}=M_{1}+M_{2}
  4. p p
  5. V V
  6. r p r_{p}
  7. p 2 = r p 2 ( 1 + 2 G M 12 / V 2 r p ) 2 G M 12 r p / V 2 ; p^{2}=r_{p}^{2}\left(1+2GM_{12}/V^{2}r_{p}\right)\approx 2GM_{12}r_{p}/V^{2};
  8. r p < a r_{p}<a
  9. n π p 2 σ = 2 π G M 12 n a / σ n\pi p^{2}\sigma=2\pi GM_{12}na/\sigma
  10. n n
  11. σ \sigma
  12. a a
  13. Δ V V bin = G M 12 / a \Delta V\approx V_{\rm bin}=\sqrt{GM_{12}/a}
  14. V bin V_{\rm bin}
  15. l l
  16. Δ ξ 2 = Δ l bin 2 / l bin 2 ( m M 12 ) 2 Δ l 2 / G M 12 a m M 12 G ρ a σ \langle\Delta\xi^{2}\rangle=\langle\Delta l_{\rm bin}^{2}\rangle/l_{\rm bin}^{% 2}\approx\left({m\over M_{12}}\right)^{2}\langle\Delta l^{2}\rangle/GM_{12}a% \approx{m\over M_{12}}{G\rho a\over\sigma}
  17. F t = 1 sin θ θ ( sin θ Δ ξ 2 4 F θ ) . {\partial F\over\partial t}={1\over\sin\theta}{\partial\over\partial\theta}% \left(\sin\theta{\langle\Delta\xi^{2}\rangle\over 4}{\partial F\over\partial% \theta}\right).
  18. F τ = 1 2 μ [ ( 1 - μ 2 ) F μ ] {\partial F\over\partial\tau}={1\over 2}{\partial\over\partial\mu}\left[(1-\mu% ^{2}){\partial F\over\partial\mu}\right]
  19. t rel M 12 m σ G ρ a . t_{\rm rel}\approx{M_{12}\over m}{\sigma\over G\rho a}.
  20. μ ¯ = μ ¯ 0 e - τ . \overline{\mu}=\overline{\mu}_{0}e^{-\tau}.
  21. δ θ 20 m / M 12 . \delta\theta\approx\sqrt{20m/M_{12}}.

Rota–Baxter_algebra.html

  1. R ( x ) R ( y ) + θ R ( x y ) = R ( R ( x ) y + x R ( y ) ) . R(x)R(y)+\theta R(xy)=R(R(x)y+xR(y)).\,
  2. C ( R ) C(R)
  3. f ( x ) C ( R ) f(x)\in C(R)
  4. I ( f ) ( x ) = 0 x f ( t ) d t . I(f)(x)=\int_{0}^{x}f(t)dt\;.
  5. F ( x ) G ( x ) = 0 x f ( t ) G ( t ) d t + 0 x F ( t ) g ( t ) d t . F(x)G(x)=\int_{0}^{x}f(t)G(t)dt+\int_{0}^{x}F(t)g(t)dt\;.
  6. I ( f ) ( x ) I ( g ) ( x ) = I ( f I ( g ) ( t ) ) ( x ) + I ( I ( f ) ( t ) g ) ( x ) , I(f)(x)I(g)(x)=I(fI(g)(t))(x)+I(I(f)(t)g)(x)\;,

Rouse_number.html

  1. w s w_{s}
  2. κ \kappa
  3. u * u_{*}
  4. P = w s κ u * \mathrm{P}=\frac{w_{s}}{\kappa u_{*}}
  5. P = w s β κ u * \mathrm{P}=\frac{w_{s}}{\beta\kappa u_{*}}

Routing_in_delay-tolerant_networking.html

  1. n - 1 n-1
  2. n n
  3. n - 1 n-1
  4. 1 | n | - 1 \frac{1}{|n|-1}
  5. j j
  6. j th j\text{th}
  7. U i U_{i}
  8. i i
  9. U i U_{i}
  10. i i
  11. U i = - D ( i ) U_{i}=-D(i)
  12. L L
  13. L L
  14. L L
  15. L L
  16. L - 1 L-1
  17. L L
  18. floor ( L / 2 ) \,\text{floor}(L/2)

Rudin–Shapiro_sequence.html

  1. a n = ε i ε i + 1 a_{n}=\textstyle\sum\varepsilon_{i}\varepsilon_{i+1}
  2. b n = ( - 1 ) a n b_{n}=(-1)^{a_{n}}
  3. { b 2 n = b n b 2 n + 1 = ( - 1 ) n b n \begin{cases}b_{2n}&=b_{n}\\ b_{2n+1}&=(-1)^{n}b_{n}\end{cases}
  4. a n = { a ( m - 1 ) / 4 if m = 1 mod 4 a ( m - 1 ) / 2 + 1 if m = 3 mod 4 a_{n}=\begin{cases}a_{(m-1)/4}&\,\text{if }m=1\mod 4\\ a_{(m-1)/2}+1&\,\text{if }m=3\mod 4\end{cases}
  5. b n = { b ( m - 1 ) / 4 if m = 1 mod 4 - b ( m - 1 ) / 2 if m = 3 mod 4 b_{n}=\begin{cases}b_{(m-1)/4}&\,\text{if }m=1\mod 4\\ -b_{(m-1)/2}&\,\text{if }m=3\mod 4\end{cases}
  6. s n = k = 0 n b k , s_{n}=\sum_{k=0}^{n}b_{k}\,,
  7. 3 n 5 < s n < 6 n for n 1 . \sqrt{\frac{3n}{5}}<s_{n}<\sqrt{6n}\,\text{ for }n\geq 1\,.

Rule_complex.html

  1. r := ( X , Y , α ) r:=(X,Y,\alpha)
  2. α \alpha
  3. C 1 , C 2 C_{1},C_{2}
  4. C 1 C 2 C_{1}\cup C_{2}
  5. P ( C 1 ) P(C_{1})
  6. C 1 C 2 C_{1}\subseteq C_{2}
  7. C 2 C_{2}
  8. C 1 C_{1}
  9. C 1 C_{1}
  10. C 2 C_{2}
  11. C 1 C 2 , C 1 - C 2 C_{1}\cap C_{2},C_{1}-C_{2}
  12. B B
  13. A A
  14. B = A B=A
  15. B B
  16. A A
  17. A A

Rule_of_three_(statistics).html

  1. p ^ = 0 \hat{p}=0
  2. p ^ = 1 \hat{p}=1

Russo–Dye_theorem.html

  1. sup U U ( A ) || f ( U ) || . \sup_{U\in U(A)}||f(U)||.

Ryd.html

  1. = 2 m e = e 2 / 2 = 1 \scriptstyle\hbar\;=\;2m_{e}\;=\;e^{2}/2\;=\;1
  2. 4 π ε 0 = 1 \scriptstyle 4\pi\varepsilon_{0}\;=\;1

Salinon.html

  1. A = 1 4 π ( r 1 + r 2 ) 2 . A=\frac{1}{4}\pi\left(r_{1}+r_{2}\right)^{2}.
  2. A B = 1 2 π ( 2 x + y ) 2 AB=\frac{1}{2}\pi\left(2x+y\right)^{2}
  3. D E = 1 2 π y 2 DE=\frac{1}{2}\pi y^{2}
  4. A D = E B = 1 2 π x 2 AD=EB=\frac{1}{2}\pi x^{2}
  5. 1 2 π ( ( 2 x + y ) 2 - 2 x 2 + y 2 ) = 1 2 π ( 2 x 2 + 4 x y + 2 y 2 ) = π ( x 2 + 2 x y + y 2 ) = π ( x + y ) 2 = π ( r 1 + r 2 ) 2 \frac{1}{2}\pi\left(\left(2x+y\right)^{2}-2x^{2}+y^{2}\right)=\frac{1}{2}\pi% \left(2x^{2}+4xy+2y^{2}\right)=\pi\left(x^{2}+2xy+y^{2}\right)=\pi\left(x+y% \right)^{2}=\pi\left(r_{1}+r_{2}\right)^{2}

Sandwich_theory.html

  1. ε x x ( x , z ) = - z d 2 w d x 2 \varepsilon_{xx}(x,z)=-z~{}\cfrac{\mathrm{d}^{2}w}{\mathrm{d}x^{2}}
  2. σ x x ( x , z ) = - z E ( z ) d 2 w d x 2 \sigma_{xx}(x,z)=-z~{}E(z)~{}\cfrac{\mathrm{d}^{2}w}{\mathrm{d}x^{2}}
  3. E ( z ) E(z)
  4. M x ( x ) = z σ x x d z d y = - ( z 2 E ( z ) d z d y ) d 2 w d x 2 = : - D d 2 w d x 2 M_{x}(x)=\int\int z~{}\sigma_{xx}~{}\mathrm{d}z\,\mathrm{d}y=-\left(\int\int z% ^{2}E(z)~{}\mathrm{d}z\,\mathrm{d}y\right)~{}\cfrac{\mathrm{d}^{2}w}{\mathrm{d% }x^{2}}=:-D~{}\cfrac{\mathrm{d}^{2}w}{\mathrm{d}x^{2}}
  5. D D
  6. Q x Q_{x}
  7. Q x = d M x d x . Q_{x}=\frac{\mathrm{d}M_{x}}{\mathrm{d}x}~{}.
  8. 2 h 2h
  9. E c E^{c}
  10. f f
  11. E f E^{f}
  12. σ x x f = z E f M x D ; σ x x c = z E c M x D τ x z f = Q x E f 2 D [ ( h + f ) 2 - z 2 ] ; τ x z c = Q x 2 D [ E c ( h 2 - z 2 ) + E f f ( f + 2 h ) ] \begin{aligned}\displaystyle\sigma_{xx}^{\mathrm{f}}&\displaystyle=\cfrac{zE^{% \mathrm{f}}M_{x}}{D}~{};&\displaystyle\sigma_{xx}^{\mathrm{c}}&\displaystyle=% \cfrac{zE^{\mathrm{c}}M_{x}}{D}\\ \displaystyle\tau_{xz}^{\mathrm{f}}&\displaystyle=\cfrac{Q_{x}E^{\mathrm{f}}}{% 2D}\left[(h+f)^{2}-z^{2}\right]~{};&\displaystyle\tau_{xz}^{\mathrm{c}}&% \displaystyle=\cfrac{Q_{x}}{2D}\left[E^{\mathrm{c}}\left(h^{2}-z^{2}\right)+E^% {\mathrm{f}}f(f+2h)\right]\end{aligned}
  13. d 2 w d x 2 = - M x ( x ) D \cfrac{d^{2}w}{dx^{2}}=-\cfrac{M_{x}(x)}{D}
  14. σ x x ( x , z ) = z E ( z ) M x ( x ) D \sigma_{xx}(x,z)=\cfrac{z~{}E(z)~{}M_{x}(x)}{D}
  15. σ x x x + τ x z z = 0 \frac{\partial\sigma_{xx}}{\partial x}+\frac{\partial\tau_{xz}}{\partial z}=0
  16. τ x z \tau_{xz}
  17. τ x z ( x , z ) = σ x x x d z + C ( x ) = z E ( z ) D d M x d x d z + C ( x ) \tau_{xz}(x,z)=\int\frac{\partial\sigma_{xx}}{\partial x}~{}\mathrm{d}z+C(x)=% \int\cfrac{z~{}E(z)}{D}~{}\frac{\mathrm{d}M_{x}}{\mathrm{d}x}~{}\mathrm{d}z+C(x)
  18. C ( x ) C(x)
  19. τ x z ( x , z ) = Q x D z E ( z ) d z + C ( x ) \tau_{xz}(x,z)=\cfrac{Q_{x}}{D}\int z~{}E(z)~{}\mathrm{d}z+C(x)
  20. τ x z face ( x , z ) = Q x E f D z h + f z d z + C ( x ) = Q x E f 2 D [ ( h + f ) 2 - z 2 ] + C ( x ) \tau^{\mathrm{face}}_{xz}(x,z)=\cfrac{Q_{x}E^{f}}{D}\int_{z}^{h+f}z~{}\mathrm{% d}z+C(x)=\cfrac{Q_{x}E^{f}}{2D}\left[(h+f)^{2}-z^{2}\right]+C(x)
  21. z = h + f z=h+f
  22. τ x z ( x , h + f ) = 0 \tau_{xz}(x,h+f)=0
  23. C ( x ) = 0 C(x)=0
  24. z = h z=h
  25. τ x z ( x , h ) = Q x E f f ( f + 2 h ) 2 D \tau_{xz}(x,h)=\cfrac{Q_{x}E^{f}f(f+2h)}{2D}
  26. τ x z core ( x , z ) = Q x E c D z h z d z + C ( x ) = Q x E c 2 D ( h 2 - z 2 ) + C ( x ) \tau^{\mathrm{core}}_{xz}(x,z)=\cfrac{Q_{x}E^{c}}{D}\int_{z}^{h}z~{}\mathrm{d}% z+C(x)=\cfrac{Q_{x}E^{c}}{2D}\left(h^{2}-z^{2}\right)+C(x)
  27. C ( x ) C(x)
  28. C ( x ) = Q x E f f ( f + 2 h ) 2 D C(x)=\cfrac{Q_{x}E^{f}f(f+2h)}{2D}
  29. τ x z core ( x , z ) = Q x 2 D [ E c ( h 2 - z 2 ) + E f f ( f + 2 h ) ] \tau^{\mathrm{core}}_{xz}(x,z)=\cfrac{Q_{x}}{2D}\left[E^{c}\left(h^{2}-z^{2}% \right)+E^{f}f(f+2h)\right]
  30. D D
  31. D = E f w - h - f - h z 2 d z d y + E c w - h h z 2 d z d y + E f w h h + f z 2 d z d y = 2 3 E f f 3 + 2 3 E c h 3 + 2 E f f h ( f + h ) . \begin{aligned}\displaystyle D&\displaystyle=E^{f}\int_{w}\int_{-h-f}^{-h}z^{2% }~{}\mathrm{d}z\,\mathrm{d}y+E^{c}\int_{w}\int_{-h}^{h}z^{2}~{}\mathrm{d}z\,% \mathrm{d}y+E^{f}\int_{w}\int_{h}^{h+f}z^{2}~{}\mathrm{d}z\,\mathrm{d}y\\ &\displaystyle=\frac{2}{3}E^{f}f^{3}+\frac{2}{3}E^{c}h^{3}+2E^{f}fh(f+h)~{}.% \end{aligned}
  32. E f E c E^{f}\gg E^{c}
  33. D D
  34. D 2 3 E f f 3 + 2 E f f h ( f + h ) = 2 f E f ( 1 3 f 2 + h ( f + h ) ) D\approx\frac{2}{3}E^{f}f^{3}+2E^{f}fh(f+h)=2fE^{f}\left(\frac{1}{3}f^{2}+h(f+% h)\right)
  35. σ x x f z M x 2 3 f 3 + 2 f h ( f + h ) ; σ x x c 0 τ x z f Q x 4 3 f 3 + 4 f h ( f + h ) [ ( h + f ) 2 - z 2 ] ; τ x z c Q x ( f + 2 h ) 2 3 f 2 + h ( f + h ) \begin{aligned}\displaystyle\sigma_{xx}^{\mathrm{f}}&\displaystyle\approx% \cfrac{zM_{x}}{\frac{2}{3}f^{3}+2fh(f+h)}~{};&\displaystyle\sigma_{xx}^{% \mathrm{c}}&\displaystyle\approx 0\\ \displaystyle\tau_{xz}^{\mathrm{f}}&\displaystyle\approx\cfrac{Q_{x}}{\frac{4}% {3}f^{3}+4fh(f+h)}\left[(h+f)^{2}-z^{2}\right]~{};&\displaystyle\tau_{xz}^{% \mathrm{c}}&\displaystyle\approx\cfrac{Q_{x}(f+2h)}{\frac{2}{3}f^{2}+h(f+h)}% \end{aligned}
  36. f 2 h f\ll 2h
  37. D 2 E f f h ( f + h ) D\approx 2E^{f}fh(f+h)
  38. σ x x f z M x 2 f h ( f + h ) ; σ x x c 0 τ x z f Q x 4 f h ( f + h ) [ ( h + f ) 2 - z 2 ] ; τ x z c Q x ( f + 2 h ) 4 h ( f + h ) Q x 2 h \begin{aligned}\displaystyle\sigma_{xx}^{\mathrm{f}}&\displaystyle\approx% \cfrac{zM_{x}}{2fh(f+h)}~{};&\displaystyle\sigma_{xx}^{\mathrm{c}}&% \displaystyle\approx 0\\ \displaystyle\tau_{xz}^{\mathrm{f}}&\displaystyle\approx\cfrac{Q_{x}}{4fh(f+h)% }\left[(h+f)^{2}-z^{2}\right]~{};&\displaystyle\tau_{xz}^{\mathrm{c}}&% \displaystyle\approx\cfrac{Q_{x}(f+2h)}{4h(f+h)}\approx\cfrac{Q_{x}}{2h}\end{aligned}
  39. σ x x f ± M x 2 f h ; σ x x c 0 τ x z f 0 ; τ x z c Q x 2 h \begin{aligned}\displaystyle\sigma_{xx}^{\mathrm{f}}&\displaystyle\approx\pm% \cfrac{M_{x}}{2fh}~{};&\displaystyle\sigma_{xx}^{\mathrm{c}}&\displaystyle% \approx 0\\ \displaystyle\tau_{xz}^{\mathrm{f}}&\displaystyle\approx 0~{};&\displaystyle% \tau_{xz}^{\mathrm{c}}&\displaystyle\approx\cfrac{Q_{x}}{2h}\end{aligned}
  40. [ σ x x σ z z σ z x ] = [ C 11 C 13 0 C 13 C 33 0 0 0 C 55 ] [ ε x x ε z z ε z x ] \begin{bmatrix}\sigma_{xx}\\ \sigma_{zz}\\ \sigma_{zx}\end{bmatrix}=\begin{bmatrix}C_{11}&C_{13}&0\\ C_{13}&C_{33}&0\\ 0&0&C_{55}\end{bmatrix}\begin{bmatrix}\varepsilon_{xx}\\ \varepsilon_{zz}\\ \varepsilon_{zx}\end{bmatrix}
  41. σ x x face = C 11 face ε x x face ; σ z x core = C 55 core ε z x core ; σ z z face = σ x z face = 0 ; σ z z core = σ x x core = 0 \sigma_{xx}^{\mathrm{face}}=C_{11}^{\mathrm{face}}~{}\varepsilon_{xx}^{\mathrm% {face}}~{};~{}~{}\sigma_{zx}^{\mathrm{core}}=C_{55}^{\mathrm{core}}~{}% \varepsilon_{zx}^{\mathrm{core}}~{};~{}~{}\sigma_{zz}^{\mathrm{face}}=\sigma_{% xz}^{\mathrm{face}}=0~{};~{}~{}\sigma_{zz}^{\mathrm{core}}=\sigma_{xx}^{% \mathrm{core}}=0
  42. ε z z face = ε x z face = 0 ; ε z z core = ε x x core = 0 \varepsilon_{zz}^{\mathrm{face}}=\varepsilon_{xz}^{\mathrm{face}}=0~{};~{}~{}% \varepsilon_{zz}^{\mathrm{core}}=\varepsilon_{xx}^{\mathrm{core}}=0
  43. σ x x x + σ z x z = 0 ; σ z x x + σ z z z = 0 \cfrac{\partial\sigma_{xx}}{\partial x}+\cfrac{\partial\sigma_{zx}}{\partial z% }=0~{};~{}~{}\cfrac{\partial\sigma_{zx}}{\partial x}+\cfrac{\partial\sigma_{zz% }}{\partial z}=0
  44. σ x x face σ x x face ( z ) ; σ z x core = constant \sigma_{xx}^{\mathrm{face}}\equiv\sigma_{xx}^{\mathrm{face}}(z)~{};~{}~{}% \sigma_{zx}^{\mathrm{core}}=\mathrm{constant}
  45. ε x x face ε x x face ( z ) ; ε z x core = constant \varepsilon_{xx}^{\mathrm{face}}\equiv\varepsilon_{xx}^{\mathrm{face}}(z)~{};~% {}~{}\varepsilon_{zx}^{\mathrm{core}}=\mathrm{constant}
  46. M M
  47. Q Q
  48. w w
  49. w b w_{b}
  50. w s w_{s}
  51. w ( x ) = w b ( x ) + w s ( x ) w(x)=w_{b}(x)+w_{s}(x)
  52. γ \gamma
  53. γ z x core = 2 h + f 2 h γ z x beam \gamma_{zx}^{\mathrm{core}}=\tfrac{2h+f}{2h}~{}\gamma_{zx}^{\mathrm{beam}}
  54. tan γ = γ \tan\gamma=\gamma
  55. γ z x beam = d w s d x \gamma_{zx}^{\mathrm{beam}}=\cfrac{\mathrm{d}w_{s}}{\mathrm{d}x}
  56. x x
  57. u b face ( x , z ) = - z d w b d x u_{b}^{\mathrm{face}}(x,z)=-z~{}\cfrac{\mathrm{d}w_{b}}{\mathrm{d}x}
  58. u s topface ( x , z ) = - ( z - h - f 2 ) d w s d x u_{s}^{\mathrm{topface}}(x,z)=-\left(z-h-\tfrac{f}{2}\right)~{}\cfrac{\mathrm{% d}w_{s}}{\mathrm{d}x}
  59. u s botface ( x , z ) = - ( z + h + f 2 ) d w s d x u_{s}^{\mathrm{botface}}(x,z)=-\left(z+h+\tfrac{f}{2}\right)~{}\cfrac{\mathrm{% d}w_{s}}{\mathrm{d}x}
  60. ε x x = u b x + u s x \varepsilon_{xx}=\cfrac{\partial u_{b}}{\partial x}+\cfrac{\partial u_{s}}{% \partial x}
  61. ε x x topface = - z d 2 w b d x 2 - ( z - h - f 2 ) d 2 w s d x 2 ; ε x x botface = - z d 2 w b d x 2 - ( z + h + f 2 ) d 2 w s d x 2 \varepsilon_{xx}^{\mathrm{topface}}=-z~{}\cfrac{\mathrm{d}^{2}w_{b}}{\mathrm{d% }x^{2}}-\left(z-h-\tfrac{f}{2}\right)~{}\cfrac{\mathrm{d}^{2}w_{s}}{\mathrm{d}% x^{2}}~{};~{}~{}\varepsilon_{xx}^{\mathrm{botface}}=-z~{}\cfrac{\mathrm{d}^{2}% w_{b}}{\mathrm{d}x^{2}}-\left(z+h+\tfrac{f}{2}\right)~{}\cfrac{\mathrm{d}^{2}w% _{s}}{\mathrm{d}x^{2}}
  62. σ z x core = C 55 core ε z x core = C 55 core 2 γ z x core = 2 h + f 4 h C 55 core γ z x beam \sigma_{zx}^{\mathrm{core}}=C^{\mathrm{core}}_{55}~{}\varepsilon_{zx}^{\mathrm% {core}}=\cfrac{C_{55}^{\mathrm{core}}}{2}~{}\gamma_{zx}^{\mathrm{core}}=\tfrac% {2h+f}{4h}~{}C_{55}^{\mathrm{core}}~{}\gamma_{zx}^{\mathrm{beam}}
  63. σ z x core = 2 h + f 4 h C 55 core d w s d x \sigma_{zx}^{\mathrm{core}}=\tfrac{2h+f}{4h}~{}C_{55}^{\mathrm{core}}~{}\cfrac% {\mathrm{d}w_{s}}{\mathrm{d}x}
  64. σ x x face = C 11 face ε x x face \sigma_{xx}^{\mathrm{face}}=C_{11}^{\mathrm{face}}~{}\varepsilon_{xx}^{\mathrm% {face}}
  65. σ x x topface = - z C 11 face d 2 w b d x 2 - ( z - h - f 2 ) C 11 face d 2 w s d x 2 = - z C 11 face d 2 w d x 2 + ( 2 h + f 2 ) C 11 face d 2 w s d x 2 σ x x botface = - z C 11 face d 2 w b d x 2 - ( z + h + f 2 ) C 11 face d 2 w s d x 2 = - z C 11 face d 2 w d x 2 - ( 2 h + f 2 ) C 11 face d 2 w s d x 2 \begin{aligned}\displaystyle\sigma_{xx}^{\mathrm{topface}}&\displaystyle=-z~{}% C_{11}^{\mathrm{face}}~{}\cfrac{\mathrm{d}^{2}w_{b}}{\mathrm{d}x^{2}}-\left(z-% h-\tfrac{f}{2}\right)~{}C_{11}^{\mathrm{face}}~{}\cfrac{\mathrm{d}^{2}w_{s}}{% \mathrm{d}x^{2}}&\displaystyle=&\displaystyle-z~{}C_{11}^{\mathrm{face}}~{}% \cfrac{\mathrm{d}^{2}w}{\mathrm{d}x^{2}}+\left(\tfrac{2h+f}{2}\right)~{}C_{11}% ^{\mathrm{face}}~{}\cfrac{\mathrm{d}^{2}w_{s}}{\mathrm{d}x^{2}}\\ \displaystyle\sigma_{xx}^{\mathrm{botface}}&\displaystyle=-z~{}C_{11}^{\mathrm% {face}}~{}\cfrac{\mathrm{d}^{2}w_{b}}{\mathrm{d}x^{2}}-\left(z+h+\tfrac{f}{2}% \right)~{}C_{11}^{\mathrm{face}}~{}\cfrac{\mathrm{d}^{2}w_{s}}{\mathrm{d}x^{2}% }&\displaystyle=&\displaystyle-z~{}C_{11}^{\mathrm{face}}~{}\cfrac{\mathrm{d}^% {2}w}{\mathrm{d}x^{2}}-\left(\tfrac{2h+f}{2}\right)~{}C_{11}^{\mathrm{face}}~{% }\cfrac{\mathrm{d}^{2}w_{s}}{\mathrm{d}x^{2}}\end{aligned}
  66. N x x face := - f / 2 f / 2 σ x x face d z f N^{\mathrm{face}}_{xx}:=\int_{-f/2}^{f/2}\sigma^{\mathrm{face}}_{xx}~{}\mathrm% {d}z_{f}
  67. M x x face := - f / 2 f / 2 z f σ x x face d z f M^{\mathrm{face}}_{xx}:=\int_{-f/2}^{f/2}z_{f}~{}\sigma^{\mathrm{face}}_{xx}~{% }\mathrm{d}z_{f}
  68. z f topface := z - h - f 2 ; z f botface := z + h + f 2 z_{f}^{\mathrm{topface}}:=z-h-\tfrac{f}{2}~{};~{}~{}z_{f}^{\mathrm{botface}}:=% z+h+\tfrac{f}{2}
  69. N x x topface = - f ( h + f 2 ) C 11 face d 2 w b d x 2 = - N x x botface M x x topface = - f 3 C 11 face 12 ( d 2 w b d x 2 + d 2 w s d x 2 ) = - f 3 C 11 face 12 d 2 w d x 2 = M x x botface \begin{aligned}\displaystyle N^{\mathrm{topface}}_{xx}&\displaystyle=-f\left(h% +\tfrac{f}{2}\right)~{}C_{11}^{\mathrm{face}}~{}\cfrac{\mathrm{d}^{2}w_{b}}{% \mathrm{d}x^{2}}=-N^{\mathrm{botface}}_{xx}\\ \displaystyle M^{\mathrm{topface}}_{xx}&\displaystyle=-\cfrac{f^{3}~{}C_{11}^{% \mathrm{face}}}{12}\left(\cfrac{\mathrm{d}^{2}w_{b}}{\mathrm{d}x^{2}}+\cfrac{% \mathrm{d}^{2}w_{s}}{\mathrm{d}x^{2}}\right)=-\cfrac{f^{3}~{}C_{11}^{\mathrm{% face}}}{12}~{}\cfrac{\mathrm{d}^{2}w}{\mathrm{d}x^{2}}=M^{\mathrm{botface}}_{% xx}\end{aligned}
  70. M x x core := - h h z σ x x core d z = 0 M^{\mathrm{core}}_{xx}:=\int_{-h}^{h}z~{}\sigma^{\mathrm{core}}_{xx}~{}\mathrm% {d}z=0
  71. M = N x x topface ( 2 h + f ) + 2 M x x topface M=N_{xx}^{\mathrm{topface}}~{}(2h+f)+2~{}M^{\mathrm{topface}}_{xx}
  72. M = - f ( 2 h + f ) 2 2 C 11 face d 2 w b d x 2 - f 3 6 C 11 face d 2 w d x 2 M=-\cfrac{f(2h+f)^{2}}{2}~{}C_{11}^{\mathrm{face}}~{}\cfrac{\mathrm{d}^{2}w_{b% }}{\mathrm{d}x^{2}}-\cfrac{f^{3}}{6}~{}C_{11}^{\mathrm{face}}~{}\cfrac{\mathrm% {d}^{2}w}{\mathrm{d}x^{2}}
  73. Q x Q_{x}
  74. Q x core = κ - h h σ x z d z = κ ( 2 h + f ) 2 C 55 core d w s d x Q_{x}^{\mathrm{core}}=\kappa\int_{-h}^{h}\sigma_{xz}~{}dz=\tfrac{\kappa(2h+f)}% {2}~{}C_{55}^{\mathrm{core}}~{}\cfrac{\mathrm{d}w_{s}}{\mathrm{d}x}
  75. κ \kappa
  76. Q x face = d M x x face d x Q_{x}^{\mathrm{face}}=\cfrac{\mathrm{d}M_{xx}^{\mathrm{face}}}{\mathrm{d}x}
  77. Q x face = - f 3 C 11 face 12 d 3 w d x 3 Q_{x}^{\mathrm{face}}=-\cfrac{f^{3}~{}C_{11}^{\mathrm{face}}}{12}~{}\cfrac{% \mathrm{d}^{3}w}{\mathrm{d}x^{3}}
  78. D beam = - M / d 2 w d x 2 D^{\mathrm{beam}}=-M/\tfrac{\mathrm{d}^{2}w}{\mathrm{d}x^{2}}
  79. M = - f ( 2 h + f ) 2 2 C 11 face d 2 w b d x 2 - f 3 6 C 11 face d 2 w d x 2 M=-\cfrac{f(2h+f)^{2}}{2}~{}C_{11}^{\mathrm{face}}~{}\cfrac{\mathrm{d}^{2}w_{b% }}{\mathrm{d}x^{2}}-\cfrac{f^{3}}{6}~{}C_{11}^{\mathrm{face}}~{}\cfrac{\mathrm% {d}^{2}w}{\mathrm{d}x^{2}}
  80. M - f [ 3 ( 2 h + f ) 2 + f 2 ] 6 C 11 face d 2 w d x 2 M\approx-\cfrac{f[3(2h+f)^{2}+f^{2}]}{6}~{}C_{11}^{\mathrm{face}}~{}\cfrac{% \mathrm{d}^{2}w}{\mathrm{d}x^{2}}
  81. f 2 h f\ll 2h
  82. D beam f [ 3 ( 2 h + f ) 2 + f 2 ] 6 C 11 face f ( 2 h + f ) 2 2 C 11 face D^{\mathrm{beam}}\approx\cfrac{f[3(2h+f)^{2}+f^{2}]}{6}~{}C_{11}^{\mathrm{face% }}\approx\cfrac{f(2h+f)^{2}}{2}~{}C_{11}^{\mathrm{face}}
  83. D face = f 3 12 C 11 face D^{\mathrm{face}}=\cfrac{f^{3}}{12}~{}C_{11}^{\mathrm{face}}
  84. S beam = Q x / d w s d x S^{\mathrm{beam}}=Q_{x}/\tfrac{\mathrm{d}w_{s}}{\mathrm{d}x}
  85. S beam = S core = κ ( 2 h + f ) 2 C 55 core S^{\mathrm{beam}}=S^{\mathrm{core}}=\cfrac{\kappa(2h+f)}{2}~{}C_{55}^{\mathrm{% core}}
  86. n x σ x x face = n z σ z x core n_{x}~{}\sigma_{xx}^{\mathrm{face}}=n_{z}~{}\sigma_{zx}^{\mathrm{core}}
  87. n x = 1 n_{x}=1
  88. n z = 1 n_{z}=1
  89. n z = - 1 n_{z}=-1
  90. z = ± h z=\pm h
  91. 2 f h C 11 face d 2 w s d x 2 - ( 2 h + f ) C 55 core d w s d x = 4 h 2 C 11 face d 2 w b d x 2 2fh~{}C_{11}^{\mathrm{face}}~{}\cfrac{\mathrm{d}^{2}w_{s}}{\mathrm{d}x^{2}}-(2% h+f)~{}C_{55}^{\mathrm{core}}~{}\cfrac{\mathrm{d}w_{s}}{\mathrm{d}x}=4h^{2}~{}% C_{11}^{\mathrm{face}}~{}\cfrac{\mathrm{d}^{2}w_{b}}{\mathrm{d}x^{2}}
  92. n z σ z x core = d N x x face d x n_{z}~{}\sigma_{zx}^{\mathrm{core}}=\cfrac{\mathrm{d}N_{xx}^{\mathrm{face}}}{% \mathrm{d}x}
  93. d w s d x = - 2 f h ( C 11 face C 55 core ) d 3 w b d x 3 \cfrac{\mathrm{d}w_{s}}{\mathrm{d}x}=-2fh~{}\left(\cfrac{C_{11}^{\mathrm{face}% }}{C_{55}^{\mathrm{core}}}\right)~{}\cfrac{\mathrm{d}^{3}w_{b}}{\mathrm{d}x^{3}}
  94. M = D beam d 2 w s d x 2 - ( D beam + 2 D face ) d 2 w d x 2 Q = S core d w s d x - 2 D face d 3 w d x 3 \begin{aligned}\displaystyle M&\displaystyle=D^{\mathrm{beam}}~{}\cfrac{% \mathrm{d}^{2}w_{s}}{\mathrm{d}x^{2}}-\left(D^{\mathrm{beam}}+2D^{\mathrm{face% }}\right)~{}\cfrac{\mathrm{d}^{2}w}{\mathrm{d}x^{2}}\\ \displaystyle Q&\displaystyle=S^{\mathrm{core}}~{}\cfrac{\mathrm{d}w_{s}}{% \mathrm{d}x}-2D^{\mathrm{face}}~{}\cfrac{\mathrm{d}^{3}w}{\mathrm{d}x^{3}}\end% {aligned}
  95. w w
  96. w s w_{s}
  97. ( 2 D face S core ) d 4 w d x 4 - ( 1 + 2 D face D beam ) d 2 w d x 2 + ( 1 S core ) d Q d x = M D beam ( D beam S core ) d 3 w s d x 3 - ( 1 + D beam 2 D face ) d w s d x - 1 S core d M d x = - ( 1 + D beam 2 D face ) Q S core \begin{aligned}&\displaystyle\left(\frac{2D^{\mathrm{face}}}{S^{\mathrm{core}}% }\right)\cfrac{\mathrm{d}^{4}w}{\mathrm{d}x^{4}}-\left(1+\frac{2D^{\mathrm{% face}}}{D^{\mathrm{beam}}}\right)\cfrac{\mathrm{d}^{2}w}{\mathrm{d}x^{2}}+% \left(\cfrac{1}{S^{\mathrm{core}}}\right)~{}\cfrac{\mathrm{d}Q}{\mathrm{d}x}=% \frac{M}{D^{\mathrm{beam}}}\\ &\displaystyle\left(\frac{D^{\mathrm{beam}}}{S^{\mathrm{core}}}\right)\cfrac{% \mathrm{d}^{3}w_{s}}{\mathrm{d}x^{3}}-\left(1+\frac{D^{\mathrm{beam}}}{2D^{% \mathrm{face}}}\right)\cfrac{\mathrm{d}w_{s}}{\mathrm{d}x}-\cfrac{1}{S^{% \mathrm{core}}}~{}\cfrac{\mathrm{d}M}{\mathrm{d}x}=-\left(1+\cfrac{D^{\mathrm{% beam}}}{2D^{\mathrm{face}}}\right)\frac{Q}{S^{\mathrm{core}}}\end{aligned}
  98. Q d M d x ; q d Q d x Q\approx\cfrac{\mathrm{d}M}{\mathrm{d}x}~{};~{}~{}q\approx\cfrac{\mathrm{d}Q}{% \mathrm{d}x}
  99. q q
  100. ( 2 D face S core ) d 4 w d x 4 - ( 1 + 2 D face D beam ) d 2 w d x 2 = M D beam - q S core ( D beam S core ) d 3 w s d x 3 - ( 1 + D beam 2 D face ) d w s d x = - ( D beam 2 D face ) Q S core \begin{aligned}&\displaystyle\left(\frac{2D^{\mathrm{face}}}{S^{\mathrm{core}}% }\right)\cfrac{\mathrm{d}^{4}w}{\mathrm{d}x^{4}}-\left(1+\frac{2D^{\mathrm{% face}}}{D^{\mathrm{beam}}}\right)\cfrac{\mathrm{d}^{2}w}{\mathrm{d}x^{2}}=% \frac{M}{D^{\mathrm{beam}}}-\cfrac{q}{S^{\mathrm{core}}}\\ &\displaystyle\left(\frac{D^{\mathrm{beam}}}{S^{\mathrm{core}}}\right)\cfrac{% \mathrm{d}^{3}w_{s}}{\mathrm{d}x^{3}}-\left(1+\frac{D^{\mathrm{beam}}}{2D^{% \mathrm{face}}}\right)\cfrac{\mathrm{d}w_{s}}{\mathrm{d}x}=-\left(\cfrac{D^{% \mathrm{beam}}}{2D^{\mathrm{face}}}\right)\frac{Q}{S^{\mathrm{core}}}\end{aligned}
  101. M core = D beam ( d γ 2 d x + ϑ ) = D beam ( d γ d x - d 2 w d x 2 + ϑ ) M face = - D face d 2 w d x 2 Q core = S core γ Q face = - D face d 3 w d x 3 \begin{aligned}\displaystyle M^{\mathrm{core}}&\displaystyle=D^{\mathrm{beam}}% \left(\cfrac{\mathrm{d}\gamma_{2}}{\mathrm{d}x}+\vartheta\right)=D^{\mathrm{% beam}}\left(\cfrac{\mathrm{d}\gamma}{\mathrm{d}x}-\cfrac{\mathrm{d}^{2}w}{% \mathrm{d}x^{2}}+\vartheta\right)\\ \displaystyle M^{\mathrm{face}}&\displaystyle=-D^{\mathrm{face}}\cfrac{\mathrm% {d}^{2}w}{\mathrm{d}x^{2}}\\ \displaystyle Q^{\mathrm{core}}&\displaystyle=S^{\mathrm{core}}\gamma\\ \displaystyle Q^{\mathrm{face}}&\displaystyle=-D^{\mathrm{face}}\cfrac{\mathrm% {d}^{3}w}{\mathrm{d}x^{3}}\end{aligned}\,
  102. w w\,
  103. γ \gamma\,
  104. γ 1 \gamma_{1}\,
  105. γ 2 \gamma_{2}\,
  106. M core M^{\mathrm{core}}\,
  107. D beam D^{\mathrm{beam}}\,
  108. M face M^{\mathrm{face}}\,
  109. D face D^{\mathrm{face}}\,
  110. Q core Q^{\mathrm{core}}\,
  111. Q face Q^{\mathrm{face}}\,
  112. S core S^{\mathrm{core}}\,
  113. ϑ \vartheta\,
  114. α \alpha\,
  115. Q Q
  116. M M
  117. S core γ - D face d 3 w d x 3 = Q \displaystyle S^{\mathrm{core}}\gamma-D^{\mathrm{face}}\cfrac{\mathrm{d}^{3}w}% {\mathrm{d}x^{3}}=Q
  118. w w
  119. γ \gamma
  120. ( D face S core ) d 4 w d x 4 - ( 1 + D face D beam ) d 2 w d x 2 = M D beam - q S core - ϑ ( D beam S core ) d 2 γ d x 2 - ( 1 + D beam D face ) γ = - ( D beam D face ) Q S core \begin{aligned}&\displaystyle\left(\frac{D^{\mathrm{face}}}{S^{\mathrm{core}}}% \right)\cfrac{\mathrm{d}^{4}w}{\mathrm{d}x^{4}}-\left(1+\frac{D^{\mathrm{face}% }}{D^{\mathrm{beam}}}\right)\cfrac{\mathrm{d}^{2}w}{\mathrm{d}x^{2}}=\frac{M}{% D^{\mathrm{beam}}}-\cfrac{q}{S^{\mathrm{core}}}-\vartheta\\ &\displaystyle\left(\frac{D^{\mathrm{beam}}}{S^{\mathrm{core}}}\right)\cfrac{% \mathrm{d}^{2}\gamma}{\mathrm{d}x^{2}}-\left(1+\frac{D^{\mathrm{beam}}}{D^{% \mathrm{face}}}\right)\gamma=-\left(\cfrac{D^{\mathrm{beam}}}{D^{\mathrm{face}% }}\right)\frac{Q}{S^{\mathrm{core}}}\end{aligned}
  121. ( 2 D face S core ) d 4 w d x 4 - ( 1 + 2 D face D beam ) d 2 w d x 2 = M D beam - q S core \left(\frac{2D^{\mathrm{face}}}{S^{\mathrm{core}}}\right)\cfrac{\mathrm{d}^{4}% w}{\mathrm{d}x^{4}}-\left(1+\frac{2D^{\mathrm{face}}}{D^{\mathrm{beam}}}\right% )\cfrac{\mathrm{d}^{2}w}{\mathrm{d}x^{2}}=\frac{M}{D^{\mathrm{beam}}}-\cfrac{q% }{S^{\mathrm{core}}}
  122. α := 2 D face D beam ; β := 2 D face S core ; W ( x ) := d 2 w d x 2 \alpha:=\cfrac{2D^{\mathrm{face}}}{D^{\mathrm{beam}}}~{};~{}~{}\beta:=\cfrac{2% D^{\mathrm{face}}}{S^{\mathrm{core}}}~{};~{}~{}W(x):=\cfrac{\mathrm{d}^{2}w}{% \mathrm{d}x^{2}}
  123. d 2 W d x 2 - ( 1 + α β ) W = M β D beam - q D face \cfrac{\mathrm{d}^{2}W}{\mathrm{d}x^{2}}-\left(\cfrac{1+\alpha}{\beta}\right)~% {}W=\frac{M}{\beta D^{\mathrm{beam}}}-\cfrac{q}{D^{\mathrm{face}}}

Sanford-Wang_parameterisation.html

  1. π + \pi^{+}
  2. Ω \Omega
  3. d 2 σ ( p + A π + + X ) d p d Ω ( p , θ ) = \frac{d^{2}\sigma(p+A\rightarrow\pi^{+}+X)}{dpd\Omega}(p,\theta)=
  4. c 1 p c 2 ( 1 - p p b e a m ) exp [ - c 3 p c 4 p b e a m c 5 - c 6 θ ( p - c 7 p b e a m ( cos θ ) c 8 ) ] c_{1}p^{c_{2}}\left(1-\frac{p}{p_{beam}}\right)\exp\left[-c_{3}\frac{p^{c_{4}}% }{p_{beam}^{c_{5}}}-c_{6}\theta(p-c_{7}p_{beam}(\cos\theta)^{c_{8}})\right]
  5. θ \theta
  6. c 1 c 8 c_{1}\ldots c_{8}

SARG04.html

  1. a a
  2. b b
  3. n n
  4. n n
  5. | ψ = i = 1 n | ψ a i b i . |\psi\rangle=\bigotimes_{i=1}^{n}|\psi_{a_{i}b_{i}}\rangle.
  6. a i a_{i}
  7. b i b_{i}
  8. i th i^{\mathrm{th}}
  9. a a
  10. b b
  11. a i b i a_{i}b_{i}
  12. | ψ 00 = | 0 |\psi_{00}\rangle=|0\rangle
  13. | ψ 10 = | 1 |\psi_{10}\rangle=|1\rangle
  14. | ψ 01 = | + = 1 2 | 0 + 1 2 | 1 |\psi_{01}\rangle=|+\rangle=\frac{1}{\sqrt{2}}|0\rangle+\frac{1}{\sqrt{2}}|1\rangle
  15. | ψ 11 = | - = 1 2 | 0 - 1 2 | 1 . |\psi_{11}\rangle=|-\rangle=\frac{1}{\sqrt{2}}|0\rangle-\frac{1}{\sqrt{2}}|1\rangle.
  16. b i b_{i}
  17. a i a_{i}
  18. b b
  19. | ψ |\psi\rangle
  20. ε ρ = ε | ψ ψ | \varepsilon\rho=\varepsilon|\psi\rangle\langle\psi|
  21. ε \varepsilon
  22. b b
  23. b b^{\prime}
  24. b b
  25. | ψ 00 |\psi_{00}\rangle
  26. | ψ 00 |\psi_{00}\rangle
  27. | ψ 01 |\psi_{01}\rangle
  28. | ψ 00 |\psi_{00}\rangle
  29. | ψ 00 |\psi_{00}\rangle
  30. | ψ 01 |\psi_{01}\rangle
  31. | ψ 01 |\psi_{01}\rangle
  32. | ψ 11 |\psi_{11}\rangle
  33. | ψ 01 |\psi_{01}\rangle
  34. | ψ 11 |\psi_{11}\rangle
  35. | ψ 01 |\psi_{01}\rangle
  36. | ψ 11 |\psi_{11}\rangle
  37. k k
  38. k / 2 k/2

Satake_diagram.html

  1. 𝔤 \mathfrak{g}
  2. 𝔤 = 𝔨 𝔭 \mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}
  3. 𝔥 \mathfrak{h}
  4. θ ( 𝔥 ) = 𝔥 \theta(\mathfrak{h})=\mathfrak{h}
  5. 𝔥 𝔨 \mathfrak{h}\cap\mathfrak{k}
  6. 𝔥 \mathfrak{h}
  7. θ ( 𝔥 ) = 𝔥 \theta(\mathfrak{h})=\mathfrak{h}
  8. 𝔥 𝔨 \mathfrak{h}\cap\mathfrak{k}

Saxon_VIb_V.html

  1. 𝔙 \textstyle\mathfrak{V}

Saxon_X_V.html

  1. 𝔙 \textstyle\mathfrak{V}

Saxon_XI_HT.html

  1. \textstyle\mathfrak{H}

Saxon_XIV_HT.html

  1. \textstyle\mathfrak{H}

Saxon_XV_HTV.html

  1. \textstyle\mathfrak{H}
  2. 𝔙 \textstyle\mathfrak{V}

Saxon_XVIII_H.html

  1. \textstyle\mathfrak{H}

Saxon_XX_HV.html

  1. \textstyle\mathfrak{H}
  2. 𝔙 \textstyle\mathfrak{V}

SC_(complexity).html

  1. \cap
  2. \cap
  3. \cap

Scanning_SQUID_microscopy.html

  1. I I
  2. I a I_{a}
  3. I b I_{b}
  4. Φ \Phi
  5. V = R 2 I 2 - I 0 2 , = R 2 ( I 2 - ( 2 I c cos ( π Φ Φ 0 ) ) 2 ) 1 2 , \begin{aligned}\displaystyle V&\displaystyle=\frac{R}{2}\sqrt{I^{2}-I_{0}^{2}}% ,\\ &\displaystyle=\frac{R}{2}\left(I^{2}-\left(2I_{c}\cos\left(\pi\frac{\Phi}{% \Phi_{0}}\right)\right)^{2}\right)^{\frac{1}{2}},\end{aligned}

Scattered_order.html

  1. × \mathbb{Q}\times\mathbb{Z}

Schema_(genetic_algorithms).html

  1. δ ( H ) \delta(H)
  2. H H
  3. N ( H ) N(H)
  4. N ( H ) N(H)
  5. H H

Scherrer_equation.html

  1. τ = K λ β cos θ \tau=\frac{K\lambda}{\beta\cos\theta}

Schiehallion_experiment.html

  1. G G
  2. G G
  3. 17 , 804 9 , 933 \tfrac{17,804}{9,933}
  4. 9 5 \tfrac{9}{5}
  5. 9 5 \tfrac{9}{5}
  6. m m
  7. d d
  8. P P
  9. θ θ
  10. F F
  11. P P
  12. W W
  13. W W
  14. F F
  15. T T
  16. F = G m M M d 2 , W = G m M E r E 2 F=\frac{GmM_{M}}{d^{2}},\quad W=\frac{GmM_{E}}{r_{E}^{2}}
  17. G G
  18. G G
  19. m m
  20. F F
  21. W W
  22. F W = ( G m M M ) / d 2 ( G m M E ) / r E 2 = M M M E ( r E d ) 2 = ρ M ρ E V M V E ( r E d ) 2 \frac{F}{W}=\frac{(GmM_{M})/d^{2}}{(GmM_{E})/r_{E}^{2}}=\frac{M_{M}}{M_{E}}{% \left(\frac{r_{E}}{d}\right)}^{2}=\frac{\rho_{M}}{\rho_{E}}\frac{V_{M}}{V_{E}}% {\left(\frac{r_{E}}{d}\right)}^{2}
  23. T T
  24. θ θ
  25. W = T cos θ , F = T sin θ W=T\cos\theta,\quad F=T\sin\theta
  26. T T
  27. tan θ = F W = ρ M ρ E V M V E ( r E d ) 2 \tan\theta=\frac{F}{W}=\frac{\rho_{M}}{\rho_{E}}\frac{V_{M}}{V_{E}}{\left(% \frac{r_{E}}{d}\right)}^{2}
  28. d d
  29. θ θ
  30. d d
  31. ρ E ρ M = V M V E ( r E d ) 2 1 tan θ \frac{\rho_{E}}{\rho_{M}}=\frac{V_{M}}{V_{E}}{\left(\frac{r_{E}}{d}\right)}^{2% }\frac{1}{\tan\theta}
  32. 5 , 480 k g · m < s u p > 3 5,480kg·m<sup>−3

Schild's_ladder.html

  1. A 0 A_{0}
  2. A 0 X 0 A_{0}X_{0}
  3. A 0 X 0 A_{0}X_{0}
  4. A 1 X 1 A_{1}X_{1}
  5. A 1 X 1 A_{1}X_{1}
  6. A 1 . A_{1}.
  7. σ ( 0 ) = A 0 \sigma(0)=A_{0}\,
  8. σ ( 0 ) = x . \sigma^{\prime}(0)=x.\,
  9. A 0 X 0 A_{0}X_{0}
  10. A 1 A 0 X 0 , A_{1}A_{0}X_{0},

Schläfli_orthoscheme.html

  1. ( v 0 v 1 ) , ( v 1 v 2 ) , , ( v d - 1 v d ) (v_{0}v_{1}),(v_{1}v_{2}),\dots,(v_{d-1}v_{d})\,
  2. | v 0 v 1 | = | v 1 v 2 | = = | v d - 1 v d | |v_{0}v_{1}|=|v_{1}v_{2}|=\cdots=|v_{d-1}v_{d}|

Schoch_line.html

  1. ρ = 1 2 r ( 1 - r ) \rho=\frac{1}{2}r\left(1-r\right)
  2. ( r ( 1 - r ) ( 1 + r ) ( 2 - r ) , 1 2 r ( 1 + 3 r - 2 r 2 ) ) . \left(r\left(1-r\right)\sqrt{\left(1+r\right)\left(2-r\right)}~{},~{}\frac{1}{% 2}r\left(1+3r-2r^{2}\right)\right).

Schröder–Bernstein_theorem_for_measurable_spaces.html

  1. X X\,
  2. Y Y\,
  3. f : X Y , f:X\to Y,\,
  4. g : Y X , g:Y\to X,\,
  5. X X\,
  6. Y Y\,
  7. f f\,
  8. f f\,
  9. f - 1 ( B ) f^{-1}(B)\,
  10. B Y B\subset Y\,
  11. f ( A ) f(A)\,
  12. A X A\subset X\,
  13. f ( X ) f(X)\,
  14. Y , Y,\,
  15. Y . Y.\,
  16. h : X Y h:X\to Y\,
  17. f f\,
  18. g g\,
  19. h h\,
  20. f f\,
  21. g - 1 g^{-1}\,
  22. h - 1 h^{-1}\,
  23. \mathbb{R}\,
  24. 2 \mathbb{R}^{2}\,
  25. \mathbb{R}\,
  26. 2 . \mathbb{R}^{2}.\,
  27. 2 . \mathbb{R}^{2}.\,
  28. \mathbb{R}\,
  29. g : 2 g:\mathbb{R}^{2}\to\mathbb{R}\,
  30. 1 / 11 = 0.090909 1/11=0.090909\dots\,
  31. g ( x , y ) g(x,y)\,

Schulze_STV.html

  1. b b
  2. a 1 , , a M a_{1},\ldots,a_{M}
  3. b b
  4. b b

Schur's_property.html

  1. ( x n ) (x_{n})
  2. x x
  3. x x
  4. 1 \ell_{1}
  5. x x
  6. ( x n ) (x_{n})
  7. ( x n ) (x_{n})
  8. x x
  9. lim n x n - x = 0 \lim_{n\to\infty}\|x_{n}-x\|=0

Schur_complement_method.html

  1. - Δ u = f , u | Ω = 0 -\Delta u=f,\qquad u|_{\partial\Omega}=0
  2. [ A 11 0 A 1 Γ 0 A 22 A 2 Γ A Γ 1 A Γ 2 A Γ Γ ] [ U 1 U 2 U Γ ] = [ F 1 F 2 F Γ ] , \left[\begin{matrix}A_{11}&0&A_{1\Gamma}\\ 0&A_{22}&A_{2\Gamma}\\ A_{\Gamma 1}&A_{\Gamma 2}&A_{\Gamma\Gamma}\end{matrix}\right]\left[\begin{% matrix}U_{1}\\ U_{2}\\ U_{\Gamma}\end{matrix}\right]=\left[\begin{matrix}F_{1}\\ F_{2}\\ F_{\Gamma}\end{matrix}\right],
  3. Σ U Γ = F Γ - A Γ 1 A 11 - 1 F 1 - A Γ 2 A 22 - 1 F 2 , \Sigma U_{\Gamma}=F_{\Gamma}-A_{\Gamma 1}A_{11}^{-1}F_{1}-A_{\Gamma 2}A_{22}^{% -1}F_{2},
  4. Σ = A Γ Γ - A Γ 1 A 11 - 1 A 1 Γ - A Γ 2 A 22 - 1 A 2 Γ . \Sigma=A_{\Gamma\Gamma}-A_{\Gamma 1}A_{11}^{-1}A_{1\Gamma}-A_{\Gamma 2}A_{22}^% {-1}A_{2\Gamma}.
  5. A 11 - 1 A_{11}^{-1}
  6. A 22 - 1 A_{22}^{-1}
  7. A 11 U 1 = F 1 - A 1 Γ U Γ , A 22 U 2 = F 2 - A 2 Γ U Γ , A_{11}U_{1}=F_{1}-A_{1\Gamma}U_{\Gamma},\qquad A_{22}U_{2}=F_{2}-A_{2\Gamma}U_% {\Gamma},

Screw.html

  1. 1 / 2 {1}/{2}
  2. 1 / 2 {1}/{2}
  3. 1 / 2 {1}/{2}
  4. 1 / 2 {1}/{2}
  5. p = 0.9 4 \scriptstyle p=0.9^{4}
  6. 6 p 1.2 \scriptstyle 6p^{1.2}
  7. 1 / 4 {1}/{4}

Sea.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons
  4. 931 / 2 93{1}/{2}
  5. 3 / 4 {3}/{4}
  6. 473 / 4 47{3}/{4}

Secant_variety.html

  1. X n X\subset\mathbb{P}^{n}
  2. X X
  3. Σ 1 \Sigma_{1}
  4. k t h k^{th}
  5. X X
  6. Σ k \Sigma_{k}
  7. Σ k = n \Sigma_{k}=\mathbb{P}^{n}
  8. Σ k - 1 \Sigma_{k-1}
  9. X X
  10. Σ k \Sigma_{k}

Second-order_intercept_point.html

  1. V i n = A cos ( w t ) V_{in}=A\cos(wt)
  2. V o u t , 2 n d o r d e r = k 2 A 2 cos 2 ( w t ) V_{out,2ndorder}=k_{2}A^{2}\cos^{2}(wt)
  3. V o u t , 2 n d o r d e r = k 2 A 2 2 + k 2 A 2 2 cos ( 2 w t ) V_{out,2ndorder}=\frac{k_{2}A^{2}}{2}+\frac{k_{2}A^{2}}{2}\cos(2wt)
  4. V i n = A 1 cos ( w 1 t ) + A 2 cos ( w 2 t ) V_{in}=A_{1}\cos(w_{1}t)+A_{2}\cos(w_{2}t)
  5. V o u t , 2 n d o r d e r = k 2 [ A 1 cos ( w 1 t ) + A 2 cos ( w 2 t ) ] 2 V_{out,2ndorder}=k_{2}[A_{1}\cos(w_{1}t)+A_{2}\cos(w_{2}t)]^{2}
  6. V o u t , 2 n d o r d e r = k 2 { A 1 2 + A 2 2 2 + A 1 2 cos ( 2 w 1 t ) 2 + A 2 2 cos ( 2 w 2 t ) 2 + A 1 A 2 cos [ ( w 1 + w 2 ) t ] + A 1 A 2 cos [ ( w 1 - w 2 ) t ] } V_{out,2ndorder}=k_{2}\{\frac{A_{1}^{2}+A_{2}^{2}}{2}+\frac{A_{1}^{2}\cos(2w_{% 1}t)}{2}+\frac{A_{2}^{2}\cos(2w_{2}t)}{2}+A_{1}A_{2}\cos[(w_{1}+w_{2})t]+A_{1}% A_{2}\cos[(w_{1}-w_{2})t]\}
  7. 1 I S O I c a s = 1 I S O I 1 + 1 I S O I 2 / G p , 1 + + 1 I S O I n / G p , 1 G p , 2 G p , 3 G p , n - 1 \frac{1}{\sqrt{ISOI_{cas}}}=\frac{1}{\sqrt{ISOI_{1}}}+\frac{1}{\sqrt{ISOI_{2}/% G_{p,1}}}+...+\frac{1}{\sqrt{ISOI_{n}/G_{p,1}G_{p,2}G_{p,3}...G_{p,n-1}}}
  8. 1 O S O I c a s = 1 G p , 2 G p , 3 G p , 4 G p , n O S O I 1 + 1 G p , 3 G p , 4 G p , n O S O I 2 + + 1 O S O I n \frac{1}{\sqrt{OSOI_{cas}}}=\frac{1}{\sqrt{G_{p,2}G_{p,3}G_{p,4}...G_{p,n}OSOI% _{1}}}+\frac{1}{\sqrt{G_{p,3}G_{p,4}...G_{p,n}OSOI_{2}}}+...+\frac{1}{\sqrt{% OSOI_{n}}}
  9. 1 I S O I c a s = 1 I S O I 1 + 1 I S O I 2 / G p , 1 + + 1 I S O I n / G p , 1 G p , 2 G p , 3 G p , n - 1 \frac{1}{ISOI_{cas}}=\frac{1}{ISOI_{1}}+\frac{1}{ISOI_{2}/G_{p,1}}+...+\frac{1% }{ISOI_{n}/G_{p,1}G_{p,2}G_{p,3}...G_{p,n-1}}
  10. 1 O S O I c a s = 1 G p , 2 G p , 3 G p , 4 G p , n O S O I 1 + 1 G p , 3 G p , 4 G p , n O S O I 2 + + 1 O S O I n \frac{1}{OSOI_{cas}}=\frac{1}{G_{p,2}G_{p,3}G_{p,4}...G_{p,n}OSOI_{1}}+\frac{1% }{G_{p,3}G_{p,4}...G_{p,n}OSOI_{2}}+...+\frac{1}{OSOI_{n}}
  11. S O , d B \bigtriangleup_{SO,dB}
  12. S O , d B = P o u t , f , d B m - P o u t , 2 f , d B m \bigtriangleup_{SO,dB}=P_{out,f,dBm}-P_{out,2f,dBm}
  13. S O , d B = I S O I d B m - P i n , f , d B m \bigtriangleup_{SO,dB}=ISOI_{dBm}-P_{in,f,dBm}
  14. S O , d B = O S O I d B m - P o u t , f , d B m \bigtriangleup_{SO,dB}=OSOI_{dBm}-P_{out,f,dBm}

Second_moment_method.html

  1. P ( X > 0 ) ( E [ X ] ) 2 E [ X 2 ] . \operatorname{P}(X>0)\geq\frac{(\operatorname{E}[X])^{2}}{\operatorname{E}[X^{% 2}]}.
  2. E [ X ] = E [ X 1 { X > 0 } ] E [ X 2 ] 1 / 2 P ( X > 0 ) 1 / 2 . \operatorname{E}[X]=\operatorname{E}[X\,\mathbf{1}_{\{X>0\}}]\leq\operatorname% {E}[X^{2}]^{1/2}\operatorname{P}(X>0)^{1/2}.
  3. P ( X > 0 ) \operatorname{P}(X>0)
  4. E [ X n 2 ] c 1 E [ X n ] 2 E\left[X_{n}^{2}\right]\leq c_{1}E[X_{n}]^{2}
  5. E [ X n ] c 2 E\left[X_{n}\right]\geq c_{2}
  6. P ( X n c 2 θ ) ( 1 - θ ) 2 c 1 . P(X_{n}\geq c_{2}\theta)\geq\frac{(1-\theta)^{2}}{c_{1}}.
  7. lim sup n 1 X n > 0 > 0 \limsup_{n\to\infty}1_{X_{n}>0}>0
  8. inf n P ( X n > 0 ) > 0 \inf_{n\to\infty}P(X_{n}>0)>0
  9. E [ X n ] 2 E [ X n 2 ] E [ ( 1 X n > 0 ) 2 ] = E [ X n 2 ] P ( X n > 0 ) . E[X_{n}]^{2}\leq E[X_{n}^{2}]\,E\left[(1_{X_{n}>0})^{2}\right]=E[X_{n}^{2}]\,P% (X_{n}>0).
  10. inf n E [ X n ] 2 E [ X n 2 ] > 0 , \inf_{n}\frac{E\left[X_{n}\right]^{2}}{E\left[X_{n}^{2}\right]}>0\,,
  11. P ( v K ) = p n . P(v\in K)=p^{n}.\,
  12. | T n | = 2 n |T_{n}|=2^{n}
  13. E [ X n ] = 2 n p n E[X_{n}]=2^{n}\,p^{n}
  14. E [ X n 2 ] = E [ v T n u T n 1 v K 1 u K ] = v T n u T n P ( v , u K ) . E\!\left[X_{n}^{2}\right]=E\!\left[\sum_{v\in T_{n}}\sum_{u\in T_{n}}1_{v\in K% }\,1_{u\in K}\right]=\sum_{v\in T_{n}}\sum_{u\in T_{n}}P(v,u\in K).
  15. P ( v , u K ) = p 2 n - k ( v , u ) . P(v,u\in K)=p^{2n-k(v,u)}.
  16. 2 s 2 n - s 2 n - s - 1 = 2 2 n - s - 1 2^{s}\,2^{n-s}\,2^{n-s-1}=2^{2n-s-1}
  17. E [ X n 2 ] = s = 0 n 2 2 n - s - 1 p 2 n - s = 1 2 ( 2 p ) n s = 0 n ( 2 p ) s = 1 2 ( 2 p ) n ( 2 p ) n + 1 - 1 2 p - 1 p 2 p - 1 E [ X n ] 2 , E\!\left[X_{n}^{2}\right]=\sum_{s=0}^{n}2^{2n-s-1}p^{2n-s}=\frac{1}{2}\,(2p)^{% n}\sum_{s=0}^{n}(2p)^{s}=\frac{1}{2}\,(2p)^{n}\,\frac{(2p)^{n+1}-1}{2p-1}\leq% \frac{p}{2p-1}\,E[X_{n}]^{2},
  18. P ( v , u K ) = P ( v K ) P ( u K ) P(v,u\in K)=P(v\in K)\,P(u\in K)
  19. X n = v T n 1 v K . X_{n}=\sum_{v\in T_{n}}1_{v\in K}.
  20. X n = f n ( t ) d μ ( t ) , X_{n}=\int f_{n}(t)\,d\mu(t),
  21. E [ X n 2 ] \displaystyle E\left[X_{n}^{2}\right]

Secret_sharing_using_the_Chinese_remainder_theorem.html

  1. 𝐙 / n 𝐙 \mathbf{Z}/n\mathbf{Z}
  2. n > 0 n>0
  3. k 2 , m 1 , , m k 2 k\geqslant 2,m_{1},...,m_{k}\geqslant 2
  4. b 1 , , b k 𝐙 b_{1},...,b_{k}\in\mathbf{Z}
  5. { x b 1 mod m 1 x b k mod m k \begin{cases}x\equiv&b_{1}\ \bmod\ m_{1}\\ &\vdots\\ x\equiv&b_{k}\ \bmod\ m_{k}\\ \end{cases}
  6. 𝐙 \mathbf{Z}
  7. b i b j mod ( m i , m j ) b_{i}\equiv b_{j}\bmod(m_{i},m_{j})
  8. 1 i , j k 1\leqslant i,j\leqslant k
  9. ( m i , m j ) (m_{i},m_{j})
  10. 𝐙 / n 𝐙 \mathbf{Z}/n\mathbf{Z}
  11. n = [ m 1 , , m k ] n=[m_{1},...,m_{k}]
  12. m 1 , , m k m_{1},...,m_{k}
  13. m 1 , m 2 , , m k m_{1},m_{2},...,m_{k}
  14. S < i = 1 k m i S<\prod_{i=1}^{k}m_{i}
  15. m 1 < m 2 < < m n m_{1}<m_{2}<...<m_{n}
  16. s 1 , s 2 , , s n s_{1},s_{2},...,s_{n}
  17. s i = S mod m i s_{i}=S\bmod\ m_{i}
  18. i = 1 , 2 , , n i=1,2,...,n
  19. i = n - k + 2 n m i < S < i = 1 k m i . \prod_{i=n-k+2}^{n}m_{i}<S<\prod_{i=1}^{k}m_{i}.
  20. k 1 k−1
  21. k 1 k−1
  22. k 1 k−1
  23. 2 k n 2≤k≤n
  24. m 1 < < m n m_{1}<\cdots<m_{n}
  25. ( m i , m j ) = 1 (m_{i},m_{j})=1
  26. m n - k + 2 m n < m 1 m k m_{n-k+2}\cdots m_{n}<m_{1}\cdots m_{k}
  27. 1 i n 1≤i≤n
  28. s i 1 , , s i k s_{i_{1}},\ldots,s_{i_{k}}
  29. { x s i 1 mod m i 1 x s i k mod m i k \begin{cases}x\equiv&s_{i_{1}}\ \bmod\ m_{i_{1}}\\ &\vdots\\ x\equiv&s_{i_{k}}\ \bmod\ m_{i_{k}}\\ \end{cases}
  30. m i 1 , , m i k m_{i_{1}},\ldots,m_{i_{k}}
  31. m i 1 m i k m_{i_{1}}\cdots m_{i_{k}}
  32. 2 k n 2≤k≤n
  33. m 0 < < m n m_{0}<...<m_{n}
  34. m 0 . m n - k + 2 m n < m 1 m k m_{0}.m_{n-k+2}...m_{n}<m_{1}...m_{k}
  35. α α
  36. S + α . m 0 < m 1 m k S+\alpha.m_{0}<m_{1}...m_{k}
  37. S + α . m 0 S+\alpha.m_{0}
  38. 1 i n 1≤i≤n
  39. I i = ( s i , m i ) I_{i}=(s_{i},m_{i})
  40. I i 1 , , I i k I_{i_{1}},...,I_{i_{k}}
  41. { x s i 1 mod m i 1 x s i k mod m i k \begin{cases}x\equiv&s_{i_{1}}\ \bmod\ m_{i_{1}}\\ &\vdots\\ x\equiv&s_{i_{k}}\ \bmod\ m_{i_{k}}\\ \end{cases}
  42. m i 1 , , m i k m_{i_{1}},...,m_{i_{k}}
  43. m i 1 m i k m_{i_{1}}...m_{i_{k}}
  44. α α
  45. S S
  46. i = n - k + 2 n m i α } < i = 1 k m i m 0 \left.\begin{array}[]{r}\prod_{i=n-k+2}^{n}m_{i}\\ \alpha\end{array}\right\}<\frac{\prod_{i=1}^{k}m_{i}}{m_{0}}
  47. α α
  48. i = n - k + 2 n m i . \prod_{i=n-k+2}^{n}m_{i}.
  49. k 1 k−1
  50. k 1 k−1
  51. k 1 k−1
  52. S S
  53. m 0 = 3 , m 1 = 11 , m 2 = 13 , m 3 = 17 m_{0}=3,m_{1}=11,m_{2}=13,m_{3}=17
  54. m 4 = 19 m_{4}=19
  55. 3 17 19 < 11 13 17 3\cdot 17\cdot 19<11\cdot 13\cdot 17
  56. S S
  57. α = 51 \alpha=51
  58. 2 + 51 3 = 155 2+51\cdot 3=155
  59. { x 1 mod 11 x 12 mod 13 x 2 mod 17 \begin{cases}x\equiv&1\ \bmod\ 11\\ x\equiv&12\ \bmod\ 13\\ x\equiv&2\ \bmod\ 17\\ \end{cases}
  60. M = 11 13 17 M=11\cdot 13\cdot 17
  61. x 0 = 1 e 1 + 12 e 2 + 2 e 3 x_{0}=1\cdot e_{1}+12\cdot e_{2}+2\cdot e_{3}
  62. ( m i , M / m i ) = 1 (m_{i},M/m_{i})=1
  63. r i . m i + s i . M / m i = 1 r_{i}.m_{i}+s_{i}.M/m_{i}=1
  64. e i = s i M i / m i e_{i}=s_{i}\cdot M_{i}/m_{i}
  65. 1 = 1 221 - 20 11 = ( - 5 ) 187 + 72 13 = 5 143 + ( - 42 ) 17 1=1\cdot 221-20\cdot 11=(-5)\cdot 187+72\cdot 13=5\cdot 143+(-42)\cdot 17
  66. e 1 = 221 , e 2 = - 935 , e 3 = 715 e_{1}=221,e_{2}=-935,e_{3}=715
  67. 11 13 17 11\cdot 13\cdot 17
  68. S = 155 2 mod 3 S=155\equiv 2\mod 3

Section_modulus.html

  1. S = b h 2 6 S=\cfrac{bh^{2}}{6}
  2. S = B H 2 6 - b h 3 6 H S=\cfrac{BH^{2}}{6}-\cfrac{bh^{3}}{6H}
  3. S = B 2 ( H - h ) 6 + ( B - b ) 3 h 6 B S=\cfrac{B^{2}(H-h)}{6}+\cfrac{(B-b)^{3}h}{6B}
  4. S = π r 3 4 = π d 3 32 S=\cfrac{\pi r^{3}}{4}=\cfrac{\pi d^{3}}{32}
  5. S = π ( r 2 4 - r 1 4 ) 4 r 2 = π ( d 2 4 - d 1 4 ) 32 d 2 S=\cfrac{\pi\left(r_{2}^{4}-r_{1}^{4}\right)}{4r_{2}}=\cfrac{\pi(d_{2}^{4}-d_{% 1}^{4})}{32d_{2}}
  6. S = B H 2 6 - b h 3 6 H S=\cfrac{BH^{2}}{6}-\cfrac{bh^{3}}{6H}
  7. S = B H 2 24 S=\cfrac{BH^{2}}{24}
  8. S = B H 2 6 - b h 3 6 H S=\cfrac{BH^{2}}{6}-\cfrac{bh^{3}}{6H}
  9. Z P = A C y C + A T y T Z_{P}=A_{C}y_{C}+A_{T}y_{T}
  10. Z P = b h 2 4 Z_{P}=\cfrac{bh^{2}}{4}
  11. Z P = b h 2 4 - ( b - 2 t ) ( h 2 - t ) 2 Z_{P}=\cfrac{bh^{2}}{4}-(b-2t)(\cfrac{h}{2}-t)^{2}
  12. Z P = b 1 t 1 y 1 + b 2 t 2 y 2 Z_{P}=b_{1}t_{1}y_{1}+b_{2}t_{2}y_{2}\,
  13. b 1 , b 2 b_{1},b_{2}
  14. t 1 , t 2 t_{1},t_{2}
  15. y 1 , y 2 y_{1},y_{2}
  16. Z P = b t f ( d - t f ) + 0.25 t w ( d - 2 t f ) 2 Z_{P}=bt_{f}(d-t_{f})+0.25t_{w}(d-2t_{f})^{2}
  17. Z P = ( b 2 t f ) / 2 + 0.25 t w 2 ( d - 2 t f ) Z_{P}=(b^{2}t_{f})/2+0.25t_{w}^{2}(d-2t_{f})
  18. Z P = d 3 6 Z_{P}=\cfrac{d^{3}}{6}
  19. Z P = d 2 3 - d 1 3 6 Z_{P}=\cfrac{d_{2}^{3}-d_{1}^{3}}{6}
  20. k = Z S k=\cfrac{Z}{S}

Sectional_density.html

  1. S D = M A SD=\frac{M}{A}
  2. S D = M lb d in 2 = M gr 7000 d in 2 SD=\frac{M_{\mathrm{lb}}}{{d_{\mathrm{in}}}^{2}}=\frac{M_{\mathrm{gr}}}{7000\,% {d_{\mathrm{in}}}^{2}}

Sectrix_of_Maclaurin.html

  1. P P
  2. P 1 P_{1}
  3. P = ( 0 , 0 ) P=(0,0)
  4. P 1 = ( a , 0 ) P_{1}=(a,0)
  5. t t
  6. P P
  7. θ = κ t + α \theta=\kappa t+\alpha
  8. P 1 P_{1}
  9. θ 1 = κ 1 t + α 1 \theta_{1}=\kappa_{1}t+\alpha_{1}
  10. κ \kappa
  11. α \alpha
  12. κ 1 \kappa_{1}
  13. α 1 \alpha_{1}
  14. t t
  15. θ 1 = q θ + θ 0 \theta_{1}=q\theta+\theta_{0}
  16. q = κ 1 / κ q=\kappa_{1}/\kappa
  17. θ 0 = α 1 - q α \theta_{0}=\alpha_{1}-q\alpha
  18. q q
  19. Q Q
  20. ψ \psi
  21. Q Q
  22. ψ = θ 1 - θ \psi=\theta_{1}-\theta
  23. r r
  24. P P
  25. Q Q
  26. r sin θ 1 = a sin ψ {r\over\sin\theta_{1}}={a\over\sin\psi}\!
  27. r = a sin θ 1 sin ψ = a sin [ q θ + θ 0 ] sin [ ( q - 1 ) θ + θ 0 ] r=a\frac{\sin\theta_{1}}{\sin\psi}=a\frac{\sin[q\theta+\theta_{0}]}{\sin[(q-1)% \theta+\theta_{0}]}\!
  28. θ 0 = 0 \theta_{0}=0
  29. q = n q=n
  30. n n
  31. r = a sin n θ sin ( n - 1 ) θ r=a\frac{\sin n\theta}{\sin(n-1)\theta}\!
  32. θ 0 = 0 \theta_{0}=0
  33. q = - n q=-n
  34. n n
  35. r = a sin n θ sin ( n + 1 ) θ r=a\frac{\sin n\theta}{\sin(n+1)\theta}\!
  36. r 1 = ( - a ) sin [ ( 1 / q ) θ 1 - θ 0 / q ] sin [ ( 1 / q - 1 ) θ 1 - θ 0 / q ] r_{1}=(-a)\frac{\sin[(1/q)\theta_{1}-\theta_{0}/q]}{\sin[(1/q-1)\theta_{1}-% \theta_{0}/q]}\!
  37. r 1 r_{1}
  38. θ 1 \theta_{1}
  39. a a
  40. q = m / n q=m/n
  41. m m
  42. n n
  43. θ 1 = q θ + θ 0 \theta_{1}=q\theta+\theta_{0}
  44. n θ 1 = m θ + n θ 0 n\theta_{1}=m\theta+n\theta_{0}
  45. z = x + i y z=x+iy
  46. θ = arg ( z ) , θ 1 = arg ( z - a ) \theta=\arg(z),\ \theta_{1}=\arg(z-a)
  47. n arg ( z - a ) = m arg ( z ) + n θ 0 n\ \arg(z-a)=m\ \arg(z)+n\ \theta_{0}
  48. m arg ( z ) - n arg ( z - a ) = arg ( z m ( z - a ) - n ) = c o n s t m\ \arg(z)-n\ \arg(z-a)=\arg(z^{m}(z-a)^{-n})=const
  49. Re ( z m ( z - a ) - n ) Im ( z m ( z - a ) - n ) = c o n s t . \frac{\operatorname{Re}(z^{m}(z-a)^{-n})}{\operatorname{Im}(z^{m}(z-a)^{-n})}=const.
  50. w = z m ( z - a ) - n w=z^{m}(z-a)^{-n}
  51. a r g ( w ) = c o n s t . arg(w)=const.
  52. | w | = c o n s t |w|=const
  53. | z | m | z - a | n = c o n s t . \frac{|z|^{m}}{|z-a|^{n}}=const.
  54. q = m / n q=m/n
  55. m m
  56. n n
  57. θ = n p \theta=np
  58. p p
  59. x = a sin [ m p + θ 0 ] cos n p sin [ ( m - n ) p + θ 0 ] , y = a sin [ m p + θ 0 ] sin n p sin [ ( m - n ) p + θ 0 ] x=a\frac{\sin[mp+\theta_{0}]\cos np}{\sin[(m-n)p+\theta_{0}]},y=a\frac{\sin[mp% +\theta_{0}]\sin np}{\sin[(m-n)p+\theta_{0}]}\!
  60. x = a sin [ m p + θ 0 ] cos n p sin [ ( m - n ) p + θ 0 ] = a + a cos [ m p + θ 0 ] sin n p sin [ ( m - n ) p + θ 0 ] = a 2 + a 2 sin [ ( m + n ) p + θ 0 ] sin [ ( m - n ) p + θ 0 ] x=a\frac{\sin[mp+\theta_{0}]\cos np}{\sin[(m-n)p+\theta_{0}]}=a+a\frac{\cos[mp% +\theta_{0}]\sin np}{\sin[(m-n)p+\theta_{0}]}={a\over 2}+{a\over 2}\frac{\sin[% (m+n)p+\theta_{0}]}{\sin[(m-n)p+\theta_{0}]}\!
  61. x = a 2 sin [ ( m + n ) p + θ 0 ] sin [ ( m - n ) p + θ 0 ] , y = a sin [ m p + θ 0 ] sin n p sin [ ( m - n ) p + θ 0 ] x={a\over 2}\cdot\frac{\sin[(m+n)p+\theta_{0}]}{\sin[(m-n)p+\theta_{0}]},y=a% \frac{\sin[mp+\theta_{0}]\sin np}{\sin[(m-n)p+\theta_{0}]}\!
  62. θ 0 = 0 \theta_{0}=0
  63. x = a 2 sin ( m + n ) p sin ( m - n ) p , y = a sin m p sin n p sin ( m - n ) p x={a\over 2}\frac{\sin(m+n)p}{\sin(m-n)p},y=a\frac{\sin mp\sin np}{\sin(m-n)p}\!
  64. r = a sin [ q θ + θ 0 ] sin [ ( q - 1 ) θ + θ 0 ] r=a\frac{\sin[q\theta+\theta_{0}]}{\sin[(q-1)\theta+\theta_{0}]}
  65. r = a sin [ ( q - 1 ) θ + θ 0 ] sin [ q θ + θ 0 ] = a sin [ ( 1 - q ) θ - θ 0 ] sin [ ( ( 1 - q ) - 1 ) θ - θ 0 ] r=a\frac{\sin[(q-1)\theta+\theta_{0}]}{\sin[q\theta+\theta_{0}]}=a\frac{\sin[(% 1-q)\theta-\theta_{0}]}{\sin[((1-q)-1)\theta-\theta_{0}]}
  66. q , 1 q , 1 - q , 1 1 - q , q - 1 q , q q - 1 q,\ \frac{1}{q},\ 1-q,\frac{1}{1-q},\ \frac{q-1}{q},\ \frac{q}{q-1}
  67. q = m / n q=m/n
  68. m m
  69. n n
  70. θ 0 \theta_{0}
  71. θ 0 \theta_{0}
  72. φ \varphi
  73. P P
  74. P 1 P_{1}
  75. P 1 P_{1}
  76. φ + θ 0 \varphi+\theta_{0}
  77. Q Q
  78. P Q PQ
  79. θ \theta
  80. φ + θ 0 = θ 1 = q θ + θ 0 \varphi+\theta_{0}=\theta_{1}=q\theta+\theta_{0}
  81. θ = n φ m \theta=\frac{n\varphi}{m}
  82. θ \theta
  83. φ \varphi
  84. φ / m \varphi/m
  85. φ \varphi
  86. P P
  87. Q Q^{\prime}
  88. P Q P^{\prime}Q^{\prime}
  89. θ 1 = q θ + θ 0 = q φ + θ 0 \theta_{1}=q\theta+\theta_{0}=q\varphi+\theta_{0}
  90. θ 0 \theta_{0}
  91. q φ = m φ n q\varphi=\frac{m\varphi}{n}
  92. φ / n \varphi/n
  93. P P
  94. π / 2 - φ - θ 0 \pi/2-\varphi-\theta_{0}
  95. P P^{\prime}
  96. π / 2 + φ + θ 0 \pi/2+\varphi+\theta_{0}
  97. C C
  98. P P PP^{\prime}
  99. C C
  100. P P
  101. P P^{\prime}
  102. P C P = 2 ( φ + θ 0 ) \angle PCP^{\prime}=2(\varphi+\theta_{0})
  103. φ + θ 0 \varphi+\theta_{0}
  104. P P
  105. P P^{\prime}
  106. Q ′′ Q^{\prime\prime}
  107. φ + θ 0 = P Q ′′ P = ψ = θ 1 - θ = ( q - 1 ) θ + θ 0 \varphi+\theta_{0}=\angle PQ^{\prime\prime}P^{\prime}=\psi=\theta_{1}-\theta=(% q-1)\theta+\theta_{0}
  108. φ = ( m - n ) θ n , θ = n θ m - n \varphi=\frac{(m-n)\theta}{n},\ \theta=\frac{n\theta}{m-n}
  109. φ / ( m - n ) \varphi/(m-n)
  110. r = a sin θ 0 sin ( - θ + θ 0 ) r=a\frac{\sin\theta_{0}}{\sin(-\theta+\theta_{0})}
  111. ( a , 0 ) (a,0)
  112. ( a , 0 ) (a,0)
  113. r = a sin ( θ + θ 0 ) sin θ 0 r=a\frac{\sin(\theta+\theta_{0})}{\sin\theta_{0}}
  114. | z - a | | z | = c o n s t . \frac{|z-a|}{|z|}=const.
  115. ( 0 , 0 ) (0,0)
  116. ( a , 0 ) (a,0)
  117. r = a sin ( - θ + θ 0 ) sin ( - 2 θ + θ 0 ) r=a\frac{\sin(-\theta+\theta_{0})}{\sin(-2\theta+\theta_{0})}
  118. a r g ( z ( z - a ) ) = c o n s t . arg(z(z-a))=const.
  119. x 2 - y 2 - x = c ( 2 x y - y ) x^{2}-y^{2}-x=c(2xy-y)
  120. θ = θ 0 / 2 \theta=\theta_{0}/2
  121. θ 0 / 2 + π / 2 \theta_{0}/2+\pi/2
  122. ( a / 2 , 0 ) (a/2,0)
  123. | z | | z - a | = c |z||z-a|=c
  124. ( 0 , 0 ) (0,0)
  125. ( a , 0 ) (a,0)
  126. q = 3 q=3
  127. q = 1 / 3 q=1/3
  128. θ 0 = 0 \theta_{0}=0
  129. r = a sin 3 θ sin 2 θ = a 2 4 cos 2 θ - 1 cos θ = a 2 ( 4 cos θ - sec θ ) r=a\frac{\sin 3\theta}{\sin 2\theta}={a\over 2}\frac{4\cos^{2}\theta-1}{\cos% \theta}={a\over 2}(4\cos\theta-\sec\theta)
  130. q = 3 / 2 q=3/2
  131. q = 2 / 3 q=2/3
  132. θ 0 = 0 \theta_{0}=0
  133. r = a sin 3 2 θ sin 1 2 θ = a ( 3 cos 2 1 2 θ - sin 2 1 2 θ ) = a ( 1 + 2 cos θ ) r=a\frac{\sin\tfrac{3}{2}\theta}{\sin\tfrac{1}{2}\theta}=a(3\cos^{2}\tfrac{1}{% 2}\theta-\sin^{2}\tfrac{1}{2}\theta)=a(1+2\cos\theta)
  134. r = - a sin 2 3 θ sin - 1 3 θ = 2 a cos 1 3 θ r=-a\frac{\sin\tfrac{2}{3}\theta}{\sin-\tfrac{1}{3}\theta}=2a\cos\tfrac{1}{3}\theta

Seed_production_and_gene_diversity.html

  1. i = 1 N j = 1 N \sum_{i=1}^{N}\sum_{j=1}^{N}
  2. i = 1 N \sum_{i=1}^{N}
  3. i = 1 N \sum_{i=1}^{N}
  4. j = 1 N \sum_{j=1}^{N}
  5. i = 1 N \sum_{i=1}^{N}

Segment_addition_postulate.html

  1. \geq

Selberg_integral.html

  1. S n ( α , β , γ ) = 0 1 0 1 i = 1 n t i α - 1 ( 1 - t i ) β - 1 1 i < j n | t i - t j | 2 γ d t 1 d t n = j = 0 n - 1 Γ ( α + j γ ) Γ ( β + j γ ) Γ ( 1 + ( j + 1 ) γ ) Γ ( α + β + ( n + j - 1 ) γ ) Γ ( 1 + γ ) \begin{aligned}\displaystyle S_{n}(\alpha,\beta,\gamma)&\displaystyle=\int_{0}% ^{1}\cdots\int_{0}^{1}\prod_{i=1}^{n}t_{i}^{\alpha-1}(1-t_{i})^{\beta-1}\prod_% {1\leq i<j\leq n}|t_{i}-t_{j}|^{2\gamma}\,dt_{1}\cdots dt_{n}\\ &\displaystyle=\prod_{j=0}^{n-1}\frac{\Gamma(\alpha+j\gamma)\Gamma(\beta+j% \gamma)\Gamma(1+(j+1)\gamma)}{\Gamma(\alpha+\beta+(n+j-1)\gamma)\Gamma(1+% \gamma)}\end{aligned}
  2. 0 1 0 1 ( i = 1 k t i ) i = 1 n t i α - 1 ( 1 - t i ) β - 1 1 i < j n | t i - t j | 2 γ d t 1 d t n \int_{0}^{1}\cdots\int_{0}^{1}\left(\prod_{i=1}^{k}t_{i}\right)\prod_{i=1}^{n}% t_{i}^{\alpha-1}(1-t_{i})^{\beta-1}\prod_{1\leq i<j\leq n}|t_{i}-t_{j}|^{2% \gamma}\,dt_{1}\cdots dt_{n}
  3. = S n ( α , β , γ ) j = 1 k α + ( n - j ) γ α + β + ( 2 n - j - 1 ) γ . =S_{n}(\alpha,\beta,\gamma)\prod_{j=1}^{k}\frac{\alpha+(n-j)\gamma}{\alpha+% \beta+(2n-j-1)\gamma}.
  4. 1 ( 2 π ) n / 2 - - i = 1 n e - t i 2 / 2 1 i < j n | t i - t j | 2 γ d t 1 d t n . \frac{1}{(2\pi)^{n/2}}\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}% \prod_{i=1}^{n}e^{-t_{i}^{2}/2}\prod_{1\leq i<j\leq n}|t_{i}-t_{j}|^{2\gamma}% \,dt_{1}\cdots dt_{n}.
  5. j = 1 n Γ ( 1 + j γ ) Γ ( 1 + γ ) . \prod_{j=1}^{n}\frac{\Gamma(1+j\gamma)}{\Gamma(1+\gamma)}.
  6. 1 ( 2 π ) n / 2 | r 2 ( x , r ) ( r , r ) | γ e - ( x 1 2 + + x n 2 ) / 2 d x 1 d x n = j = 1 n Γ ( 1 + d j γ ) Γ ( 1 + γ ) \frac{1}{(2\pi)^{n/2}}\int\cdots\int\left|\prod_{r}\frac{2(x,r)}{(r,r)}\right|% ^{\gamma}e^{-(x_{1}^{2}+\cdots+x_{n}^{2})/2}dx_{1}\cdots dx_{n}=\prod_{j=1}^{n% }\frac{\Gamma(1+d_{j}\gamma)}{\Gamma(1+\gamma)}

Selberg_sieve.html

  1. S ( A , P , z ) = | A p P ( z ) A p | . S(A,P,z)=\left|A\setminus\bigcup_{p\mid P(z)}A_{p}\right|.
  2. | A d | = 1 f ( d ) X + R d . \left|A_{d}\right|=\frac{1}{f(d)}X+R_{d}.
  3. g ( n ) = d n μ ( d ) f ( n / d ) g(n)=\sum_{d\mid n}\mu(d)f(n/d)
  4. f ( n ) = d n g ( d ) f(n)=\sum_{d\mid n}g(d)
  5. V ( z ) = d < z d P ( z ) μ 2 ( d ) g ( d ) . V(z)=\sum_{\begin{smallmatrix}d<z\\ d\mid P(z)\end{smallmatrix}}\frac{\mu^{2}(d)}{g(d)}.
  6. S ( A , P , z ) X V ( z ) + O ( d 1 , d 2 < z d 1 , d 2 P ( z ) | R [ d 1 , d 2 ] | ) . S(A,P,z)\leq\frac{X}{V(z)}+O\left({\sum_{\begin{smallmatrix}d_{1},d_{2}<z\\ d_{1},d_{2}\mid P(z)\end{smallmatrix}}\left|R_{[d_{1},d_{2}]}\right|}\right).
  7. V ( z ) d z 1 f ( d ) . V(z)\geq\sum_{d\leq z}\frac{1}{f(d)}.\,

Semantic_parameterization.html

  1. G G
  2. M M
  3. G G
  4. M M
  5. p 1 p_{1}
  6. \sqcap
  7. p 2 p_{2}
  8. \sqcap
  9. p 3 p_{3}
  10. p 1 p_{1}
  11. \sqcap
  12. p 4 p_{4}
  13. \sqcap
  14. p 4 p_{4}
  15. \sqcap
  16. p 5 p_{5}
  17. p 1 p_{1}
  18. \sqcap
  19. p 5 p_{5}
  20. p 2 p_{2}
  21. \sqcap
  22. p 5 p_{5}
  23. p 3 p_{3}
  24. \sqcap
  25. p 5 p_{5}
  26. p 4 p_{4}
  27. \sqcap
  28. p 5 p_{5}
  29. p 1 p_{1}

Semi-infinite_programming.html

  1. min x X f ( x ) \min_{x\in X}\;\;f(x)
  2. subject to: \,\text{subject to: }
  3. g ( x , y ) 0 , y Y g(x,y)\leq 0,\;\;\forall y\in Y
  4. f : R n R f:R^{n}\to R
  5. g : R n × R m R g:R^{n}\times R^{m}\to R
  6. X R n X\subseteq R^{n}
  7. Y R m . Y\subseteq R^{m}.

Semialgebraic_space.html

  1. 𝒪 𝐑 n \mathcal{O}_{\mathbf{R}^{n}}
  2. ( X , 𝒪 X ) (X,\mathcal{O}_{X})

Semidiameter.html

  1. ( x a ) 2 + ( y b ) 2 = 1 ; \left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2}=1;\,\!
  2. ( x a ) 2 + ( y b ) 2 + ( z c ) 2 = 1 ; \left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2}+\left(\frac{z}{c}% \right)^{2}=1;\,\!

Separoid.html

  1. S S
  2. 2 S × 2 S \mid\ \subseteq 2^{S}\times 2^{S}
  3. A , B S A,B\subseteq S
  4. A B B A , A\mid B\Leftrightarrow B\mid A,
  5. A B A B = , A\mid B\Rightarrow A\cap B=\varnothing,
  6. A B and A A A B . A\mid B\hbox{ and }A^{\prime}\subset A\Rightarrow A^{\prime}\mid B.
  7. A B A\mid B
  8. φ : S T \varphi\colon S\to T
  9. A , B T A,B\subseteq T
  10. A B φ - 1 ( A ) φ - 1 ( B ) . A\mid B\Rightarrow\varphi^{-1}(A)\mid\varphi^{-1}(B).
  11. A B a A and b B : a b E . A\mid B\Leftrightarrow\forall a\in A\hbox{ and }b\in B\colon ab\not\in E.

Sequential_dynamical_system.html

  1. F i ( x ) = ( x 1 , x 2 , , x i - 1 , f i ( x [ i ] ) , x i + 1 , , x n ) . F_{i}(x)=(x_{1},x_{2},\ldots,x_{i-1},f_{i}(x[i]),x_{i+1},\ldots,x_{n})\;.
  2. [ F Y , w ] = F w ( m ) F w ( m - 1 ) F w ( 2 ) F w ( 1 ) . [F_{Y},w]=F_{w(m)}\circ F_{w(m-1)}\circ\cdots\circ F_{w(2)}\circ F_{w(1)}\;.

Sequential_minimal_optimization.html

  1. max α i = 1 n α i - 1 2 i = 1 n j = 1 n y i y j K ( x i , x j ) α i α j , \max_{\alpha}\sum_{i=1}^{n}\alpha_{i}-\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}y% _{i}y_{j}K(x_{i},x_{j})\alpha_{i}\alpha_{j},
  2. 0 α i C , for i = 1 , 2 , , n , 0\leq\alpha_{i}\leq C,\quad\mbox{ for }~{}i=1,2,\ldots,n,
  3. i = 1 n y i α i = 0 \sum_{i=1}^{n}y_{i}\alpha_{i}=0
  4. α i \alpha_{i}
  5. α i \alpha_{i}
  6. α 1 \alpha_{1}
  7. α 2 \alpha_{2}
  8. 0 α 1 , α 2 C , 0\leq\alpha_{1},\alpha_{2}\leq C,
  9. y 1 α 1 + y 2 α 2 = k , y_{1}\alpha_{1}+y_{2}\alpha_{2}=k,
  10. k k
  11. α 1 \alpha_{1}
  12. α 2 \alpha_{2}
  13. ( α 1 , α 2 ) (\alpha_{1},\alpha_{2})

Sersic_profile.html

  1. I I
  2. R R
  3. n n
  4. ln I ( R ) = ln I 0 - k R 1 / n , \ln\ I(R)=\ln\ I_{0}-kR^{1/n},
  5. I 0 I_{0}
  6. R = 0 R=0
  7. n n
  8. n n
  9. d ln I d ln R = - ( k / n ) R 1 / n . \frac{\mathrm{d}\ln I}{\mathrm{d}\ln R}=-(k/n)\ R^{1/n}.
  10. I ( R ) e - k R 1 / 4 I(R)\propto e^{-kR^{1/4}}
  11. I ( R ) e - k R I(R)\propto e^{-kR}
  12. I I
  13. ρ \rho
  14. R R
  15. r r

Sextupole_magnet.html

  1. 𝐅 = q ( 𝐄 + 𝐯 × 𝐁 ) , \mathbf{F}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B}),

SHA-3.html

  1. ( i j ) = ( 3 2 1 0 ) t ( 0 1 ) \begin{pmatrix}i\\ j\end{pmatrix}=\begin{pmatrix}3&2\\ 1&0\end{pmatrix}^{t}\begin{pmatrix}0\\ 1\end{pmatrix}

Shape_factor_(image_analysis_and_microscopy).html

  1. A R = d min d max A_{R}=\frac{d_{\min}}{d_{\max}}
  2. f circ = 4 π A P 2 f\text{circ}=\frac{4\pi A}{P^{2}}
  3. f elong = i 2 i 1 f\text{elong}=\sqrt{\frac{i_{2}}{i_{1}}}
  4. f comp = A 2 2 π i 1 2 + i 2 2 f\text{comp}=\frac{A^{2}}{2\pi\sqrt{{i_{1}}^{2}+{i_{2}}^{2}}}
  5. f wav = P cvx P f\text{wav}=\frac{P\text{cvx}}{P}
  6. A R = 1290 2670 = 0.483 A_{R}=\frac{1290}{2670}=0.483
  7. f circ = 4 π ( 2166086 ) 39330 2 = 0.0176. f\text{circ}=\frac{4\pi(2166086)}{39330^{2}}=0.0176.

Shapiro's_lemma.html

  1. Ext R n ( N , M R ) Ext S n ( S R N , M ) \operatorname{Ext}^{n}_{R}(N,{}_{R}M)\cong\operatorname{Ext}^{n}_{S}(S\otimes_% {R}N,M)
  2. Ext R n ( M R , N ) Ext S n ( M , Hom R ( S , N ) ) \operatorname{Ext}^{n}_{R}({}_{R}M,N)\cong\operatorname{Ext}^{n}_{S}(M,% \operatorname{Hom}_{R}(S,N))
  3. Ext G n ( M , N H G ) Ext H n ( M H G , N ) \operatorname{Ext}^{n}_{G}(M,N\uparrow_{H}^{G})\cong\operatorname{Ext}^{n}_{H}% (M\downarrow_{H}^{G},N)

Sharp-P-completeness_of_01-permanent.html

  1. A = ( a i j ) A=(a_{ij})
  2. a i j a_{ij}
  3. G ϕ G_{\phi}
  4. G ϕ G_{\phi}
  5. ϕ \phi
  6. G ϕ G_{\phi}
  7. ϕ \phi
  8. G ϕ G_{\phi}
  9. 12 m 12^{m}
  10. G ϕ G_{\phi}
  11. ( # ϕ ) (\#\phi)
  12. ϕ \phi
  13. 12 m ( # ϕ ) 12^{m}\cdot(\#\phi)
  14. { - 1 , 0 , 1 , 2 , 3 } \{-1,0,1,2,3\}
  15. 4 t ( ϕ ) ( # ϕ ) 4^{t(\phi)}\cdot(\#\phi)
  16. t ( ϕ ) t(\phi)
  17. ϕ \phi
  18. m m
  19. o > j o>j
  20. x i x_{i}
  21. ϕ \phi
  22. G ϕ G_{\phi}
  23. x i x_{i}
  24. C j C_{j}
  25. x i x_{i}
  26. x i x_{i}
  27. ϕ \phi
  28. C j C_{j}
  29. C j C_{j}
  30. ϕ \phi
  31. x i x_{i}
  32. C j C_{j}
  33. x i x_{i}
  34. x i x_{i}
  35. x i x_{i}
  36. x i x_{i}
  37. x i x_{i}
  38. G ϕ G_{\phi}
  39. | ϕ | |\phi|
  40. G ϕ G_{\phi}
  41. ϕ \phi
  42. G ϕ G_{\phi}
  43. ϕ \phi
  44. x i x_{i}
  45. x i x_{i}
  46. x i x_{i}
  47. x i x_{i}
  48. ϕ \phi
  49. C j C_{j}
  50. C j C_{j}
  51. ϕ \phi
  52. x i x_{i}
  53. ϕ \phi
  54. x i x_{i}
  55. 12 m 12^{m}
  56. 0
  57. C j C_{j}
  58. G ϕ G_{\phi}
  59. M L M\cup L
  60. G ϕ G_{\phi}
  61. L M L^{M}
  62. G ϕ G_{\phi}
  63. Z M Z^{M}
  64. G ϕ G_{\phi}
  65. Z M Z^{M}
  66. Z M Z^{M}
  67. Z M Z^{M}
  68. 12 m 12^{m}
  69. C j C_{j}
  70. Z M Z^{M}
  71. 0
  72. { - 1 , 0 , 1 , 2 , 3 } \{-1,0,1,2,3\}
  73. G ϕ G_{\phi}
  74. A A^{\prime}
  75. A A
  76. A A^{\prime}
  77. A A
  78. n × n n\times n
  79. μ \mu
  80. Q = 2 n ! μ n + 1 Q=2n!\cdot\mu^{n}+1
  81. | Perm ( A ) | n ! μ n |\operatorname{Perm}(A)|\leq n!\cdot\mu^{n}
  82. A = A mod Q A^{\prime}=A\,\bmod\,Q
  83. P = Perm ( A ) mod Q P=\operatorname{Perm}(A^{\prime})\,\bmod\,Q
  84. P < Q / 2 P<Q/2
  85. Perm ( A ) = P - Q \operatorname{Perm}(A)=P-Q
  86. A A
  87. A A^{\prime}
  88. n n
  89. log ( μ ) \log(\mu)
  90. Q Q
  91. n n
  92. log ( μ ) \log(\mu)
  93. A = [ 2 - 2 - 2 1 ] A=\begin{bmatrix}2&-2\\ -2&1\end{bmatrix}
  94. Perm ( A ) = 2 1 + ( - 2 ) ( - 2 ) = 6 \operatorname{Perm}(A)=2\cdot 1+(-2)\cdot(-2)=6
  95. n = 2 n=2
  96. μ = 2 \mu=2
  97. μ n = 4 \mu^{n}=4
  98. Q = 17 Q=17
  99. A = A mod 17 = [ 2 15 15 1 ] . A^{\prime}=A\,\bmod\,17=\begin{bmatrix}2&15\\ 15&1\end{bmatrix}.
  100. Perm ( A ) = 2 1 + 15 15 = 227 \operatorname{Perm}(A^{\prime})=2\cdot 1+15\cdot 15=227
  101. P = 227 mod 17 = 6 P=227\,\bmod\,17=6
  102. P < Q / 2 P<Q/2
  103. Perm ( A ) = P = 6. \operatorname{Perm}(A)=P=6.
  104. 13 = 2 3 + 2 2 + 2 0 13=2^{3}+2^{2}+2^{0}
  105. G G
  106. n n
  107. W W
  108. e e
  109. w w
  110. w = 2 x 1 + 2 x 2 + + 2 x r w=2^{x_{1}}+2^{x_{2}}+\cdots+2^{x_{r}}
  111. 0 x 1 x 2 x r log ( w ) 0\leq x_{1}\leq x_{2}\leq\cdots\leq x_{r}\leq\log(w)
  112. r r
  113. 3 r 3r
  114. G G^{\prime}
  115. G G
  116. G G^{\prime}
  117. R R
  118. G G
  119. e e
  120. R R
  121. R R
  122. R R^{\prime}
  123. e e
  124. R R
  125. G G^{\prime}
  126. u u
  127. v v
  128. r r
  129. G G
  130. G G
  131. G G^{\prime}
  132. G G^{\prime}
  133. n n
  134. log W \log W
  135. n n
  136. G G^{\prime}
  137. n n
  138. p p
  139. 2 p 2^{p}
  140. e = ( u , v ) e=(u,v)
  141. w = 2 r > 1 w=2^{r}>1
  142. J e J_{e}
  143. 2 r 2r
  144. 6 r 6r
  145. J e J_{e}
  146. R R
  147. G G
  148. e e
  149. R R
  150. J e J_{e}
  151. J e J_{e}
  152. G G^{\prime}
  153. u u
  154. v v
  155. r r
  156. 2 r 2^{r}
  157. u u
  158. v v
  159. 2 r 2^{r}
  160. n 5 / 2 n^{5/2}

Sharp_map.html

  1. M M
  2. Γ ( T M ) \,\Gamma(TM)
  3. g Γ ( T * M 2 ) g\in\Gamma(T^{*}M^{\otimes 2})
  4. X := i X g = g ( X , . ) X^{\flat}:=i_{X}g=g(X,.)
  5. : Γ ( T M ) Γ ( T * M ) \flat:\Gamma(TM)\to\Gamma(T^{*}M)
  6. g g
  7. := - 1 : Γ ( T * M ) Γ ( T M ) \sharp:=\flat^{-1}:\Gamma(T^{*}M)\to\Gamma(TM)

Shear_and_moment_diagram.html

  1. F = 0 , M A = 0 . \sum F=0~{},~{}~{}\sum M_{A}=0\,.
  2. - 10 - ( 1 ) ( 15 ) + R a + R b + R c = 0 -10-(1)(15)+R_{a}+R_{b}+R_{c}=0
  3. ( R a ) ( 10 ) + ( R b ) ( 25 ) + ( R c ) ( 50 ) - ( 1 ) ( 15 ) ( 17.5 ) - 50 + M c = 0 . (R_{a})(10)+(R_{b})(25)+(R_{c})(50)-(1)(15)(17.5)-50+M_{c}=0\,.
  4. R b = 37.5 - 1.6 R a + 0.04 M c R_{b}=37.5-1.6R_{a}+0.04M_{c}
  5. R c = - 12.5 + 0.6 R a - 0.04 M c . R_{c}=-12.5+0.6R_{a}-0.04M_{c}\,.
  6. ( 10 ) ( 10 ) - ( 1 ) ( 15 ) ( 7.5 ) + ( R b ) ( 15 ) + ( R c ) ( 40 ) - 50 + M c = 0 . (10)(10)-(1)(15)(7.5)+(R_{b})(15)+(R_{c})(40)-50+M_{c}=0\,.
  7. ( 10 ) ( 25 ) - ( R a ) ( 15 ) + ( 1 ) ( 15 ) ( 7.5 ) + ( R c ) ( 25 ) - 50 + M c = 0 . (10)(25)-(R_{a})(15)+(1)(15)(7.5)+(R_{c})(25)-50+M_{c}=0\,.
  8. ( 10 ) ( 50 ) - ( R a ) ( 40 ) - ( R b ) ( 25 ) + ( 1 ) ( 15 ) ( 32.5 ) - 50 + M c = 0 . (10)(50)-(R_{a})(40)-(R_{b})(25)+(1)(15)(32.5)-50+M_{c}=0\,.
  9. F = - 10 - V 1 = 0 \sum F=-10-V_{1}=0
  10. M A = - V 1 x + M 1 = 0 . \sum M_{A}=-V_{1}x+M_{1}=0\,.
  11. V 1 = - 10 and M 1 = - 10 x . V_{1}=-10\quad\,\text{and}\quad M_{1}=-10x\,.
  12. F = - 10 + R a - ( 1 ) ( x - 10 ) - V 2 = 0 \sum F=-10+R_{a}-(1)(x-10)-V_{2}=0
  13. M A = R a ( 10 ) - ( 1 ) ( x - 10 ) ( x + 10 ) 2 - V 2 x + M 2 = 0 . \sum M_{A}=R_{a}(10)-(1)(x-10)\frac{(x+10)}{2}-V_{2}x+M_{2}=0\,.
  14. V 2 = R a - x and M 2 = - 50 + R a ( x - 10 ) - x 2 2 . V_{2}=R_{a}-x\quad\,\text{and}\quad M_{2}=-50+R_{a}(x-10)-\frac{x^{2}}{2}\,.
  15. M A = 10 x - R a ( x - 10 ) + ( 1 ) ( x - 10 ) ( x - 10 ) 2 + M 2 = 0 . \sum M_{A}=10x-R_{a}(x-10)+(1)(x-10)\frac{(x-10)}{2}+M_{2}=0\,.
  16. M 2 = - 50 + R a ( x - 10 ) - x 2 2 . M_{2}=-50+R_{a}(x-10)-\frac{x^{2}}{2}\,.
  17. - 10 + R a + R b - ( 1 ) ( 15 ) - V 3 = 0 -10+R_{a}+R_{b}-(1)(15)-V_{3}=0
  18. ( 10 ) ( x ) - R a ( x - 10 ) - R b ( x - 25 ) + ( 1 ) ( 15 ) ( x - 17.5 ) + M 3 = 0 . (10)(x)-R_{a}(x-10)-R_{b}(x-25)+(1)(15)(x-17.5)+M_{3}=0\,.
  19. V 3 = 25 - R a - R b = R c V_{3}=25-R_{a}-R_{b}=R_{c}
  20. M 3 = 262.5 + R a ( x - 10 ) + R b ( x - 25 ) - 25 x = - 675 + R a ( 30 - 0.6 x ) - M c ( 1 - 0.04 x ) + 12.5 x . M_{3}=262.5+R_{a}(x-10)+R_{b}(x-25)-25x=-675+R_{a}(30-0.6x)-M_{c}(1-0.04x)+12.% 5x\,.
  21. - 10 + R a + R b - ( 1 ) ( 15 ) - V 4 = 0 -10+R_{a}+R_{b}-(1)(15)-V_{4}=0
  22. ( 10 ) ( x ) - R a ( x - 10 ) - R b ( x - 25 ) + ( 1 ) ( 15 ) ( x - 17.5 ) - 50 + M 4 = 0 . (10)(x)-R_{a}(x-10)-R_{b}(x-25)+(1)(15)(x-17.5)-50+M_{4}=0\,.
  23. V 4 = 25 - R a - R b = R c V_{4}=25-R_{a}-R_{b}=R_{c}
  24. M 4 = 312.5 + R a ( x - 10 ) + R b ( x - 25 ) - 25 x = - 625 + R a ( 30 - 0.6 x ) + M c ( 0.04 x - 1 ) + 12.5 x . M_{4}=312.5+R_{a}(x-10)+R_{b}(x-25)-25x=-625+R_{a}(30-0.6x)+M_{c}(0.04x-1)+12.% 5x\,.
  25. M 4 = M c = - 937.5 + 40 R a + 25 R b . M_{4}=M_{c}=-937.5+40R_{a}+25R_{b}\,.
  26. d 2 w d x 2 = - M E I \frac{d^{2}w}{dx^{2}}=-\frac{M}{EI}
  27. w 1 = 5 3 E I x 3 + C 1 + C 2 x w 2 = 1 24 E I x 2 [ x 2 + 600 - 4 R a ( x - 30 ) ] + C 3 + C 4 x w 3 = 1 100 E I [ x 3 3 ( - 625 + 30 R a - 2 M c ) - 50 x 2 ( - 675 + 30 R a - M c ) ] + C 5 + C 6 x w 4 = 1 100 E I [ x 3 3 ( - 625 + 30 R a - 2 M c ) - 50 x 2 ( - 625 + 30 R a - M c ) ] + C 7 + C 8 x \begin{aligned}\displaystyle w_{1}&\displaystyle=\frac{5}{3EI}\,x^{3}+C_{1}+C_% {2}\,x\\ \displaystyle w_{2}&\displaystyle=\frac{1}{24EI}\,x^{2}\,\left[x^{2}+600-4R_{a% }(x-30)\right]+C_{3}+C_{4}\,x\\ \displaystyle w_{3}&\displaystyle=\frac{1}{100EI}\left[\frac{x^{3}}{3}(-625+30% R_{a}-2M_{c})-50x^{2}(-675+30R_{a}-M_{c})\right]+C_{5}+C_{6}\,x\\ \displaystyle w_{4}&\displaystyle=\frac{1}{100EI}\left[\frac{x^{3}}{3}(-625+30% R_{a}-2M_{c})-50x^{2}(-625+30R_{a}-M_{c})\right]+C_{7}+C_{8}\,x\end{aligned}
  28. C 7 = - 1250 3 E I ( - 625 + M c + 30 R a ) and C 8 = 125 E I ( - 125 + 6 R a ) . C_{7}=-\frac{1250}{3EI}(-625+M_{c}+30R_{a})\quad\,\text{and}\quad C_{8}=\frac{% 125}{EI}(-125+6R_{a})\,.
  29. w 4 = - 1 300 E I ( x - 50 ) 2 [ - 5 ( 6 R a - 125 ) ( x - 50 ) + 2 M c ( x + 25 ) ] . w_{4}=-\frac{1}{300EI}(x-50)^{2}\left[-5(6R_{a}-125)(x-50)+2M_{c}(x+25)\right]\,.
  30. C 5 = - 625 12 E I ( - 5675 + 8 M c + 240 R a ) and C 6 = 250 E I ( 3 R a - 70 ) . C_{5}=-\frac{625}{12EI}(-5675+8M_{c}+240R_{a})\quad\,\text{and}\quad C_{6}=% \frac{250}{EI}\left(3R_{a}-70\right)\,.
  31. w 3 = 1 300 E I [ 30 R a ( - 50 + x ) 3 - 2 M c ( - 50 + x ) 2 ( 25 + x ) - 625 ( - 141875 + x ( 8400 + ( - 162 + x ) x ) ) ] . \begin{aligned}\displaystyle w_{3}=\frac{1}{300EI}\Bigl[&\displaystyle 30R_{a}% (-50+x)^{3}-2M_{c}(-50+x)^{2}(25+x)-\\ &\displaystyle 625(-141875+x(8400+(-162+x)x))\Bigr]\,.\end{aligned}
  32. C 3 = - 3125 24 E I ( - 1645 + 4 M c + 64 R a ) and C 4 = 25 12 E I ( - 40325 + 6 M c + 120 R a ) . C_{3}=-\frac{3125}{24EI}(-1645+4M_{c}+64R_{a})\quad\,\text{and}\quad C_{4}=% \frac{25}{12EI}\left(-40325+6M_{c}+120R_{a}\right)\,.
  33. w 2 = 1 24 E I [ - 3125 ( - 1645 + 4 M c + 64 R a ) + 50 ( - 4025 + 6 M c + 120 R a ) x + 120 ( 5 + R a ) x 2 - 4 R a x 3 + x 4 ] . \begin{aligned}\displaystyle w_{2}=\frac{1}{24EI}\Bigl[&\displaystyle-3125(-16% 45+4M_{c}+64R_{a})+\\ &\displaystyle 50(-4025+6M_{c}+120R_{a})x+120(5+R_{a})x^{2}-4R_{a}x^{3}+x^{4}% \Bigr]\,.\end{aligned}
  34. C 1 = - 125 24 E I ( - 40145 + 100 M c + 1632 R a ) and C 2 = 25 4 E I ( - 1315 + 2 M c + 48 R a ) . C_{1}=-\frac{125}{24EI}(-40145+100M_{c}+1632R_{a})\quad\,\text{and}\quad C_{2}% =\frac{25}{4EI}(-1315+2M_{c}+48R_{a})\,.
  35. w 1 = 5 24 E I [ 1026125 - 39450 x + 8 x 3 + 20 M c ( - 125 + 3 x ) + 480 R a ( - 85 + 3 x ) ] . w_{1}=\frac{5}{24EI}\left[1026125-39450x+8x^{3}+20M_{c}(-125+3x)+480R_{a}(-85+% 3x)\right]\,.
  36. M c = 175 - 7.5 R a . M_{c}=175-7.5R_{a}\,.
  37. R a = 25.278 M c = - 14.585 . R_{a}=25.278\quad\implies\quad M_{c}=-14.585\,.
  38. d Q d x = - q \frac{dQ}{dx}=-q
  39. d M d x = Q \frac{dM}{dx}=Q

Shear_velocity.html

  1. u = τ ρ u_{\star}=\sqrt{\frac{\tau}{\rho}}
  2. τ \tau
  3. ρ \rho
  4. u = τ b ρ u_{\star}=\sqrt{\frac{\tau_{b}}{\rho}}
  5. τ b \tau_{b}
  6. 0 = ν 2 u ¯ y 2 - y ( u v ¯ ) 0={\nu}{\partial^{2}\overline{u}\over\partial y^{2}}-\frac{\partial}{\partial y% }(\overline{u^{\prime}v^{\prime}})
  7. u u_{\star}
  8. ν u \frac{\nu}{u_{\star}}
  9. τ w ρ = ν u y - u v ¯ \frac{\tau_{w}}{\rho}=\nu\frac{\partial u}{\partial y}-\overline{u^{\prime}v^{% \prime}}
  10. τ w ρ u 2 = u + y + + τ T + ¯ \frac{\tau_{w}}{\rho u_{\star}^{2}}=\frac{\partial u^{+}}{\partial y^{+}}+% \overline{\tau_{T}^{+}}
  11. u = τ w ρ u_{\star}=\sqrt{\frac{\tau_{w}}{\rho}}
  12. τ w \tau_{w}

Shewanella_algae.html

  1. C 6 H 12 O 6 + 12 U 6 + + 6 H 2 O + 24 e - 6 C O 2 + 12 U 4 + + 24 H + W a t e r S o l u b l e W a t e r I n s o l u b l e \begin{array}[]{lcr}C_{6}H_{12}O_{6}+12U^{6+}+6H_{2}O+24e^{-}\longrightarrow&6% CO_{2}+12U^{4+}+24H^{+}\\ \qquad Water\ Soluble&Water\ Insoluble\end{array}

Shifted_log-logistic_distribution.html

  1. z = ( x - μ ) / σ z=(x-\mu)/\sigma\,
  2. ( 1 + ( 1 + ξ z ) - 1 / ξ ) - 1 \left(1+(1+\xi z)^{-1/\xi}\right)^{-1}\,
  3. z = ( x - μ ) / σ z=(x-\mu)/\sigma\,
  4. μ + σ ξ ( α csc ( α ) - 1 ) \mu+\frac{\sigma}{\xi}(\alpha\csc(\alpha)-1)
  5. α = π ξ \alpha=\pi\xi\,
  6. μ \mu\,
  7. μ + σ ξ [ ( 1 - ξ 1 + ξ ) ξ - 1 ] \mu+\frac{\sigma}{\xi}\left[\left(\frac{1-\xi}{1+\xi}\right)^{\xi}-1\right]
  8. σ 2 ξ 2 [ 2 α csc ( 2 α ) - ( α csc ( α ) ) 2 ] \frac{\sigma^{2}}{\xi^{2}}[2\alpha\csc(2\alpha)-(\alpha\csc(\alpha))^{2}]
  9. α = π ξ \alpha=\pi\xi\,
  10. δ \delta
  11. X X
  12. X + δ X+\delta
  13. Y Y
  14. log ( Y - δ ) \log(Y-\delta)
  15. F ( x ; μ , σ , ξ ) = 1 1 + ( 1 + ξ ( x - μ ) σ ) - 1 / ξ F(x;\mu,\sigma,\xi)=\frac{1}{1+\left(1+\frac{\xi(x-\mu)}{\sigma}\right)^{-1/% \xi}}
  16. 1 + ξ ( x - μ ) / σ 0 1+\xi(x-\mu)/\sigma\geqslant 0
  17. μ \mu\in\mathbb{R}
  18. σ > 0 \sigma>0\,
  19. ξ \xi\in\mathbb{R}
  20. κ = - ξ \kappa=-\xi\,\!
  21. f ( x ; μ , σ , ξ ) = ( 1 + ξ ( x - μ ) σ ) - ( 1 / ξ + 1 ) σ [ 1 + ( 1 + ξ ( x - μ ) σ ) - 1 / ξ ] 2 , f(x;\mu,\sigma,\xi)=\frac{\left(1+\frac{\xi(x-\mu)}{\sigma}\right)^{-(1/\xi+1)% }}{\sigma\left[1+\left(1+\frac{\xi(x-\mu)}{\sigma}\right)^{-1/\xi}\right]^{2}},
  22. 1 + ξ ( x - μ ) / σ 0. 1+\xi(x-\mu)/\sigma\geqslant 0.
  23. ξ \xi
  24. | ξ | > 1 |\xi|>1
  25. x = μ - σ / ξ x=\mu-\sigma/\xi
  26. ξ \xi
  27. x = μ . x=\mu.
  28. μ = σ / ξ , \mu=\sigma/\xi,
  29. ξ \xi
  30. ξ = 1 \xi=1
  31. ξ = 1. \xi=1.

Shockley–Queisser_limit.html

  1. η = q V ( ϕ s - ϕ r ) σ T s u n 4 \eta=\frac{qV(\phi_{s}-\phi_{r})}{\sigma T_{sun}^{4}}
  2. ϕ s \phi_{s}
  3. ϕ r \phi_{r}
  4. σ \sigma
  5. T s u n T_{sun}

Shoelace_formula.html

  1. 𝐀 \displaystyle\mathbf{A}
  2. 𝐀 = 1 2 | i = 1 n x i ( y i + 1 - y i - 1 ) | = 1 2 | i = 1 n y i ( x i + 1 - x i - 1 ) | = 1 2 | i = 1 n x i y i + 1 - x i + 1 y i | = 1 2 | i = 1 n det ( x i x i + 1 y i y i + 1 ) | \mathbf{A}={1\over 2}\Big|\sum_{i=1}^{n}x_{i}(y_{i+1}-y_{i-1})\Big|={1\over 2}% \Big|\sum_{i=1}^{n}y_{i}(x_{i+1}-x_{i-1})\Big|={1\over 2}\Big|\sum_{i=1}^{n}x_% {i}y_{i+1}-x_{i+1}y_{i}\Big|={1\over 2}\Big|\sum_{i=1}^{n}\det\begin{pmatrix}x% _{i}&x_{i+1}\\ y_{i}&y_{i+1}\end{pmatrix}\Big|
  3. 𝐀 tri. = 1 2 | x 1 y 2 + x 2 y 3 + x 3 y 1 - x 2 y 1 - x 3 y 2 - x 1 y 3 | \mathbf{A}\text{tri.}={1\over 2}|x_{1}y_{2}+x_{2}y_{3}+x_{3}y_{1}-x_{2}y_{1}-x% _{3}y_{2}-x_{1}y_{3}|
  4. 𝐀 pent. = 1 2 | x 1 y 2 + x 2 y 3 + x 3 y 4 + x 4 y 5 + x 5 y 1 - x 2 y 1 - x 3 y 2 - x 4 y 3 - x 5 y 4 - x 1 y 5 | \mathbf{A}\text{pent.}={1\over 2}|x_{1}y_{2}+x_{2}y_{3}+x_{3}y_{4}+x_{4}y_{5}+% x_{5}y_{1}-x_{2}y_{1}-x_{3}y_{2}-x_{4}y_{3}-x_{5}y_{4}-x_{1}y_{5}|
  5. 𝐀 quad. = 1 2 | x 1 y 2 + x 2 y 3 + x 3 y 4 + x 4 y 1 - x 2 y 1 - x 3 y 2 - x 4 y 3 - x 1 y 4 | \mathbf{A}\text{quad.}={1\over 2}|x_{1}y_{2}+x_{2}y_{3}+x_{3}y_{4}+x_{4}y_{1}-% x_{2}y_{1}-x_{3}y_{2}-x_{4}y_{3}-x_{1}y_{4}|
  6. 𝐀 \displaystyle\mathbf{A}
  7. [ 2 4 3 - 8 1 2 2 4 ] \begin{bmatrix}2&4\\ 3&-8\\ 1&2\\ 2&4\end{bmatrix}

Shortcut_model.html

  1. 𝐙 d \textstyle\mathbf{Z}^{d}
  2. j \textstyle j
  3. r ( i , j ) \textstyle r(i,j)
  4. i \textstyle i
  5. r ( i , j ) d \textstyle r(i,j)^{d}
  6. r ( i , j ) \textstyle r(i,j)
  7. i \textstyle i
  8. j \textstyle j
  9. j \textstyle j
  10. r ( i , j ) \textstyle r(i,j)
  11. i \textstyle i
  12. i \textstyle i
  13. n = ( n 1 , , n d ) 𝐙 d \textstyle\vec{n}=(n_{1},\dots,n_{d})\in\mathbf{Z}^{d}
  14. d \textstyle d
  15. n \textstyle\|\vec{n}\|
  16. n \textstyle\vec{n}
  17. n = n 1 2 + + n d 2 . \|\vec{n}\|=\sqrt{n_{1}^{2}+\cdots+n_{d}^{2}}.
  18. L 1 \textstyle L^{1}
  19. n 1 = n 1 + + n d . \|\vec{n}\|_{1}=\|n_{1}\|+\cdots+\|n_{d}\|.
  20. L 1 \textstyle L^{1}
  21. V ( r ) = k r d , V(r)=kr^{d},
  22. k \textstyle k
  23. S ( r ) \textstyle S(r)
  24. r \textstyle r
  25. S ( r ) \textstyle S(r)
  26. S ( r ) = k d r d - 1 . S(r)=kdr^{d-1}.
  27. N \textstyle N
  28. N \textstyle N
  29. p \textstyle p
  30. m \textstyle m
  31. p = 0 \textstyle p=0
  32. N \textstyle N
  33. p = 1 \textstyle p=1
  34. m \textstyle m
  35. N / m \textstyle N/m
  36. p \textstyle p
  37. 0 \textstyle 0
  38. 1 \textstyle 1
  39. size = N , \,\text{size}=N,\,
  40. shortcut distance = m , \,\text{shortcut distance}=m,\,
  41. rewiring probability = p . \,\text{rewiring probability}=p.\,
  42. r \textstyle r
  43. 1 / r α \textstyle 1/r^{\alpha}
  44. 0 α 1 \textstyle 0\leq\alpha\leq 1
  45. N \textstyle N\rightarrow\infty
  46. H = - 1 2 i , j J ( r ( i , j ) ) s i s j H=-\frac{1}{2}\sum_{i,j}J(r(i,j))s_{i}s_{j}
  47. s i \textstyle s_{i}
  48. r ( i , j ) \textstyle r(i,j)
  49. i \textstyle i
  50. j \textstyle j
  51. J ( r ( i , j ) ) \textstyle J(r(i,j))
  52. J ( r ( i , j ) ) \textstyle J(r(i,j))
  53. 1 / r α \textstyle 1/r^{\alpha}
  54. α \textstyle\alpha
  55. ρ = i , j J ( r ( i , j ) ) , \rho=\sum_{i,j}J(r(i,j)),
  56. ρ \textstyle\rho
  57. ρ \textstyle\rho
  58. ζ ( α - d + 1 ) \textstyle\zeta(\alpha-d+1)
  59. α > d . \alpha>d.\,
  60. p \textstyle p

Shunt_equation.html

  1. Q s / Q t Q_{s}/Q_{t}
  2. Q s / Q t = ( C c O 2 - C a O 2 ) / ( C c O 2 - C v O 2 ) Q_{s}/Q_{t}=(Cc_{O_{2}}-Ca_{O_{2}})/(Cc_{O_{2}}-Cv_{O_{2}})
  3. Q t C v O 2 Q_{t}\cdot Cv_{O_{2}}
  4. C v O 2 Cv_{O_{2}}
  5. Q t Q_{t}
  6. Q t C a O 2 Q_{t}\cdot Ca_{O_{2}}
  7. C a O 2 Ca_{O_{2}}
  8. Q s Q_{s}
  9. Q c Q_{c}
  10. Q t = Q s + Q c Q_{t}=Q_{s}+Q_{c}
  11. Q c Q_{c}
  12. Q c = Q t - Q s Q_{c}=Q_{t}-Q_{s}
  13. Q t C a O 2 = Q s C v O 2 + ( Q t - Q s ) C c O 2 Q_{t}\cdot Ca_{O_{2}}=Q_{s}\cdot Cv_{O_{2}}+(Q_{t}-Q_{s})\cdot Cc_{O_{2}}
  14. Q t C a O 2 = Q s C v O 2 + Q t C c O 2 - Q s C c O 2 Q_{t}\cdot Ca_{O_{2}}=Qs\cdot Cv_{O_{2}}+Q_{t}\cdot Cc_{O_{2}}-Qs\cdot Cc_{O_{% 2}}
  15. Q s C c O 2 - Q s C v O 2 = Q t C c O 2 - Q t C a O 2 Q_{s}\cdot Cc_{O_{2}}-Qs\cdot Cv_{O_{2}}=Q_{t}\cdot Cc_{O_{2}}-Qt\cdot Ca_{O_{% 2}}
  16. Q s ( C c O 2 - C v O 2 ) = Q t ( C c O 2 - C a O 2 ) Q_{s}\cdot(Cc_{O_{2}}-Cv_{O_{2}})=Q_{t}\cdot(Cc_{O_{2}}-Ca_{O_{2}})
  17. Q s Q t = C c O 2 - C a O 2 C c O 2 - C v O 2 \dfrac{Q_{s}}{Q_{t}}=\dfrac{Cc_{O_{2}}-Ca_{O_{2}}}{Cc_{O_{2}}-Cv_{O_{2}}}

SIC-POVM.html

  1. d d
  2. \mathcal{H}
  3. M M
  4. { F i } \{F_{i}\}
  5. i = 1 M F i = I . \sum_{i=1}^{M}F_{i}=I.
  6. d 2 d^{2}
  7. F i , F j F_{i},F_{j}
  8. Tr ( F i F j ) = Tr ( Π i Π j ) d 2 = | ψ i | ψ j | 2 d 2 = 1 d 2 ( d + 1 ) i j . \mathrm{Tr}\left(F_{i}F_{j}\right)=\frac{\mathrm{Tr}\left(\Pi_{i}\Pi_{j}\right% )}{d^{2}}=\frac{\left|\langle\psi_{i}|\psi_{j}\rangle\right|^{2}}{d^{2}}=\frac% {1}{d^{2}(d+1)}\quad i\neq j.
  9. M M
  10. F i = 1 d Π i , F_{i}=\frac{1}{d}\Pi_{i},
  11. Π i \Pi_{i}
  12. 1 d α Π α = I \frac{1}{d}\sum_{\alpha}\Pi_{\alpha}=I
  13. Tr ( Π α Π β ) = μ 2 \mathrm{Tr}(\Pi_{\alpha}\Pi_{\beta})=\mu^{2}\;
  14. d \displaystyle d
  15. Tr ( Π i Π j ) = | ψ i | ψ j | 2 = 1 d + 1 i j \mathrm{Tr}\left(\Pi_{i}\Pi_{j}\right)=\left|\langle\psi_{i}|\psi_{j}\rangle% \right|^{2}=\frac{1}{d+1}\quad i\neq j
  16. ( ) ( ) \mathcal{B}(\mathcal{H})\rightarrow\mathcal{B}(\mathcal{H})
  17. 𝒢 : ( ) ( ) A α | ψ α ψ α | A | ψ α ψ α | \begin{aligned}\displaystyle\mathcal{G}:\mathcal{B}(\mathcal{H})&\displaystyle% \rightarrow\mathcal{B}(\mathcal{H})\\ \displaystyle A&\displaystyle\mapsto\displaystyle\sum_{\alpha}|\psi_{\alpha}% \rangle\langle\psi_{\alpha}|A|\psi_{\alpha}\rangle\langle\psi_{\alpha}|\end{aligned}
  18. 𝒢 ( Π β ) = α Π α | ψ α | ψ β | 2 = Π β + 1 d + 1 α β Π α = d d + 1 Π β + 1 d + 1 Π β + 1 d + 1 α β Π α = d d + 1 Π β + d d + 1 α 1 d Π α = d d + 1 ( Π β + I ) \begin{aligned}\displaystyle\mathcal{G}(\Pi_{\beta})&\displaystyle=% \displaystyle\sum_{\alpha}\Pi_{\alpha}\left|\langle\psi_{\alpha}|\psi_{\beta}% \rangle\right|^{2}\\ &\displaystyle=\displaystyle\Pi_{\beta}+\frac{1}{d+1}\sum_{\alpha\neq\beta}\Pi% _{\alpha}\\ &\displaystyle=\displaystyle\frac{d}{d+1}\Pi_{\beta}+\frac{1}{d+1}\Pi_{\beta}+% \frac{1}{d+1}\sum_{\alpha\neq\beta}\Pi_{\alpha}\\ &\displaystyle=\displaystyle\frac{d}{d+1}\Pi_{\beta}+\frac{d}{d+1}\sum_{\alpha% }\frac{1}{d}\Pi_{\alpha}\\ &\displaystyle=\displaystyle\frac{d}{d+1}\left(\Pi_{\beta}+I\right)\end{aligned}
  19. G = d d + 1 ( + I ) G=\frac{d}{d+1}\left(\mathcal{I}+I\right)
  20. ( A ) = A and I ( A ) = Tr ( A ) I \mathcal{I}(A)=A\,\text{ and }I(A)=\mathrm{Tr}(A)I
  21. G - 1 = 1 d [ ( d + 1 ) I - ] G^{-1}=\frac{1}{d}\left[\left(d+1\right)I-\mathcal{I}\right]
  22. I = G - 1 G = 1 d α [ ( d + 1 ) Π α Π α - I Π α ] I=G^{-1}G=\frac{1}{d}\sum_{\alpha}\left[(d+1)\Pi_{\alpha}\odot\Pi_{\alpha}-I% \odot\Pi_{\alpha}\right]
  23. ρ \rho
  24. ρ = I | ρ ) \displaystyle\rho=I|\rho)
  25. | ρ ) |\rho)
  26. ( ) \mathcal{B}(\mathcal{H})
  27. ρ \rho
  28. ( d + 1 ) p α - 1 d (d+1)p_{\alpha}-\frac{1}{d}
  29. P P
  30. G G
  31. d 2 d^{2}
  32. | ψ ψ | P , U g G , U g | ψ P \forall|\psi\rangle\langle\psi|\in P,\quad\forall U_{g}\in G,\quad U_{g}|\psi% \rangle\in P
  33. | ψ ψ | , | ϕ ϕ | P , U g G , U g | ϕ = | ψ \forall|\psi\rangle\langle\psi|,|\phi\rangle\langle\phi|\in P,\quad\exists U_{% g}\in G,\quad U_{g}|\phi\rangle=|\psi\rangle
  34. | ϕ |\phi\rangle
  35. | ϕ | U g | ϕ | 2 = 1 d + 1 g i d |\langle\phi|U_{g}|\phi\rangle|^{2}=\frac{1}{d+1}\ \forall g\neq id
  36. U g U_{g}
  37. | ϕ |\phi\rangle
  38. d × d \mathbb{Z}_{d}\times\mathbb{Z}_{d}
  39. d × d \mathbb{Z}_{d}\times\mathbb{Z}_{d}
  40. U ( d ) U(d)
  41. | e i |e_{i}\rangle
  42. \mathcal{H}
  43. T | e i = ω i | e i T|e_{i}\rangle=\omega^{i}|e_{i}\rangle
  44. ω = e 2 π i d \omega=e^{\frac{2\pi i}{d}}
  45. S | e i = | e i + 1 mod d S|e_{i}\rangle=|e_{i+1\mod{d}}\rangle
  46. W ( p , q ) = S p T q W(p,q)=S^{p}T^{q}
  47. W ( p , q ) W ( p , q ) = S p T q T - q S - p = I d \begin{aligned}\displaystyle W(p,q)W^{\dagger}(p,q)&\displaystyle=S^{p}T^{q}T^% {-q}S^{-p}\\ &\displaystyle=Id\end{aligned}
  48. ( p , q ) d × d W ( p , q ) (p,q)\in\mathbb{Z}_{d}\times\mathbb{Z}_{d}\rightarrow W(p,q)
  49. d 2 d\geq 2
  50. E 0 E_{0}
  51. H d H_{d}
  52. E 0 E_{0}
  53. J d = H d S L ( 2 , d ) J_{d}=H_{d}\rtimes SL(2,\mathbb{Z}_{d})
  54. H d H_{d}
  55. d × d \mathbb{Z}_{d}\times\mathbb{Z}_{d}
  56. d d\in\mathbb{N}
  57. { k } k = 0 d - 1 \left\{k\right\}_{k=0}^{d-1}
  58. d \mathbb{C}^{d}
  59. ω = e 2 π i d , D j , k = ω j k 2 m = 0 d - 1 ω j m | k + m mod d m | \displaystyle\omega=e^{\frac{2\pi i}{d}},\quad\quad D_{j,k}=\omega^{\frac{jk}{% 2}}\sum_{m=0}^{d-1}\omega^{jm}|k+m\mod{d}\rangle\langle m|
  60. | ϕ d \exists|\phi\rangle\in\mathbb{C}^{d}
  61. { D j , k | ϕ } j , k = 1 d \left\{D_{j,k}|\phi\rangle\right\}_{j,k=1}^{d}
  62. d = 1 , , 15 , 19 , 24 , 35 , 48 d=1,\dots,15,19,24,35,48
  63. d × d \mathbb{Z}_{d}\times\mathbb{Z}_{d}
  64. d = 67 d=67
  65. S = { | ϕ k : | ϕ k 𝕊 d } S=\left\{|\phi_{k}\rangle:|\phi_{k}\rangle\in\mathbb{S}^{d}\right\}
  66. t t h t^{th}
  67. f t ( ψ ) f_{t}(\psi)
  68. S S
  69. f t ( ψ ) f_{t}(\psi)
  70. | ψ |\psi\rangle
  71. t = i = 1 t \mathcal{H}_{t}=\displaystyle\bigotimes_{i=1}^{t}\mathcal{H}
  72. S t = k = 1 n | Φ k t Φ k t | , | Φ k t = | ϕ k t S_{t}=\displaystyle\sum_{k=1}^{n}|\Phi_{k}^{t}\rangle\langle\Phi_{k}^{t}|,% \quad|\Phi_{k}^{t}\rangle=|\phi_{k}\rangle^{\otimes t}
  73. { | ϕ k 𝕊 d } k = 1 n \left\{|\phi_{k}\rangle\in\mathbb{S}^{d}\right\}_{k=1}^{n}
  74. n ( t + d - 1 d - 1 ) n\geq{t+d-1\choose d-1}
  75. Tr [ S t 2 ] = j , k | ϕ j | ϕ k | 2 t = n 2 t ! ( d - 1 ) ! ( t + d - 1 ) ! \displaystyle\mathrm{Tr}\left[S_{t}^{2}\right]=\sum_{j,k}\left|\langle\phi_{j}% |\phi_{k}\rangle\right|^{2t}=\frac{n^{2}t!(d-1)!}{(t+d-1)!}
  76. Tr ( S 2 2 ) = j , k | ϕ j | ϕ k | 4 = 2 d 3 d + 1 \mathrm{Tr}(S^{2}_{2})=\displaystyle\sum_{j,k}|\langle\phi_{j}|\phi_{k}\rangle% |^{4}=\frac{2d^{3}}{d+1}
  77. { | ψ i } , { | ϕ j } \left\{|\psi_{i}\rangle\right\},\left\{|\phi_{j}\rangle\right\}
  78. | ψ i | ϕ j | 2 = 1 d , i , j \displaystyle|\langle\psi_{i}|\phi_{j}\rangle|^{2}=\frac{1}{d},\quad\forall i,j
  79. d \mathbb{C}^{d}
  80. N + 1 N+1

Siddhaṃ_script.html

  1. \cdots
  2. \cdots
  3. \vdots
  4. \cdots
  5. \cdots

Sidon_sequence.html

  1. x + O ( x 4 ) \sqrt{x}+O(\sqrt[4]{x})
  2. x ( 1 - o ( 1 ) ) \sqrt{x}(1-o(1))
  3. lim inf x A ( x ) log x x 1 \liminf_{x\to\infty}\frac{A(x)\sqrt{\log x}}{\sqrt{x}}\leq 1
  4. A ( x ) > c x 3 A(x)>c\sqrt[3]{x}
  5. A ( x ) > x log x 3 . A(x)>\sqrt[3]{x\log x}.
  6. A ( x ) > x 2 - 1 - o ( 1 ) A(x)>x^{\sqrt{2}-1-o(1)}
  7. A ( x ) > x 1 / 2 - o ( 1 ) A(x)>x^{1/2-o(1)}
  8. a i - a j = a k - a l a_{i}-a_{j}=a_{k}-a_{l}
  9. a i + a l = a k + a j a_{i}+a_{l}=a_{k}+a_{j}

Signal-to-noise_ratio_(imaging).html

  1. μ sig \mu_{\mathrm{sig}}
  2. σ bg \sigma_{\mathrm{bg}}
  3. SNR = μ sig σ bg \mathrm{SNR}=\frac{\mu_{\mathrm{sig}}}{\sigma_{\mathrm{bg}}}
  4. σ bg \sigma_{\mathrm{bg}}
  5. μ sig \mu_{\mathrm{sig}}
  6. σ sig \sigma_{\mathrm{sig}}
  7. SNR = μ sig σ sig \mathrm{SNR}=\frac{\mu_{\mathrm{sig}}}{\sigma_{\mathrm{sig}}}
  8. f i = j = 0 m i = 1 n a j x i j f_{i}=\sum_{j=0}^{m}\sum_{i=1}^{n}a_{j}x_{i}^{j}
  9. f f\,
  10. m m\,
  11. x x\,
  12. n n\,
  13. a j a_{j}\,
  14. [ 1 x 1 x 1 2 1 x 2 x 2 2 1 x n x n 2 ] [ a 2 a 1 a 0 ] = [ f 1 f 2 f n ] \begin{bmatrix}1&x_{1}&x_{1}^{2}\\ 1&x_{2}&x_{2}^{2}\\ \vdots&\vdots&\vdots\\ 1&x_{n}&x_{n}^{2}\end{bmatrix}\begin{bmatrix}a_{2}\\ a_{1}\\ a_{0}\\ \end{bmatrix}=\begin{bmatrix}f_{1}\\ f_{2}\\ \vdots\\ f_{n}\end{bmatrix}
  15. [ n x i x i 2 x i x i 2 x i 3 x i 2 x i 3 x i 4 ] [ a 2 a 1 a 0 ] = [ f i f i x i f i x i 2 ] \begin{bmatrix}n&\sum x_{i}&\sum x_{i}^{2}\\ \sum x_{i}&\sum x_{i}^{2}&\sum x_{i}^{3}\\ \sum x_{i}^{2}&\sum x_{i}^{3}&\sum x_{i}^{4}\end{bmatrix}\begin{bmatrix}a_{2}% \\ a_{1}\\ a_{0}\end{bmatrix}=\begin{bmatrix}\sum f_{i}\\ \sum f_{i}x_{i}\\ \sum f_{i}x_{i}^{2}\end{bmatrix}
  16. μ sig = i = 1 n ( X i - f i ) n μ bkg = i = 1 n ( X i - f i ) n \mu\text{sig}=\frac{\sum_{i=1}^{n}(X_{i}-f_{i})}{n}\qquad\qquad\mu\text{bkg}=% \frac{\sum_{i=1}^{n}(X_{i}-f_{i})}{n}
  17. μ sig \mu\text{sig}
  18. μ bkg \mu\text{bkg}
  19. n n\,
  20. X i X_{i}\,
  21. f i f_{i}\,
  22. signal = μ sig - μ bkg \,\text{signal}=\mu\text{sig}-\mu\text{bkg}
  23. RMS noise = i = 1 n ( X i - i = 1 n X i n ) 2 n \,\text{RMS noise}=\sqrt{\frac{\sum_{i=1}^{n}(X_{i}-\frac{\sum_{i=1}^{n}X_{i}}% {n})^{2}}{n}}
  24. SNR = signal RMS noise \,\text{SNR}=\frac{\,\text{signal}}{\,\text{RMS noise}}
  25. SNR = 20 log 10 signal RMS noise dB \,\text{SNR}=20\log_{10}\frac{\,\text{signal}}{\,\text{RMS noise}}\,\mbox{dB}~{}

Signal_transfer_function.html

  1. n n\,
  2. ( x i , y i ) (x_{i}\,,y_{i}\,)
  3. y = m x + b y=mx+b\,
  4. m = x i y i n - x i n y i n x i 2 n - ( x i n ) 2 b = y i n - m x i n m=\frac{\frac{\sum x_{i}y_{i}}{n}-\frac{\sum x_{i}}{n}\frac{\sum y_{i}}{n}}{% \frac{\sum x_{i}^{2}}{n}-(\frac{\sum x_{i}}{n})^{2}}\qquad\qquad b=\frac{\sum y% _{i}}{n}-m\frac{\sum x_{i}}{n}

Signature_operator.html

  1. M M
  2. 2 l 2l
  3. d : Ω p ( M ) Ω p + 1 ( M ) d:\Omega^{p}(M)\rightarrow\Omega^{p+1}(M)
  4. i i
  5. M M
  6. M M
  7. \star
  8. ω , η = M ω η \langle\omega,\eta\rangle=\int_{M}\omega\wedge\star\eta
  9. d * : Ω p + 1 ( M ) Ω p ( M ) d^{*}:\Omega^{p+1}(M)\rightarrow\Omega^{p}(M)
  10. d d
  11. d * = ( - 1 ) 2 l ( p + 1 ) + 2 l + 1 d = - d d^{*}=(-1)^{2l(p+1)+2l+1}\star d\star=-\star d\star
  12. d + d * d+d^{*}
  13. Ω ( M ) = p = 0 2 l Ω p ( M ) \Omega(M)=\bigoplus_{p=0}^{2l}\Omega^{p}(M)
  14. τ \tau
  15. τ ( ω ) = i p ( p - 1 ) + l ω , ω Ω p ( M ) \tau(\omega)=i^{p(p-1)+l}\star\omega\quad,\quad\omega\in\Omega^{p}(M)
  16. d + d * d+d^{*}
  17. τ \tau
  18. ( ± 1 ) (\pm 1)
  19. Ω ± ( M ) \Omega_{\pm}(M)
  20. τ \tau
  21. d + d * = ( 0 D D * 0 ) d+d^{*}=\begin{pmatrix}0&D\\ D^{*}&0\end{pmatrix}
  22. d + d * d+d^{*}
  23. D : Ω + ( M ) Ω - ( M ) D:\Omega_{+}(M)\rightarrow\Omega_{-}(M)
  24. M M
  25. i ( d + d * ) τ i(d+d^{*})\tau
  26. M M
  27. l = 2 k l=2k
  28. M M
  29. index ( D ) = sign ( M ) \mathrm{index}(D)=\mathrm{sign}(M)
  30. H 2 k ( M ) H^{2k}(M)
  31. sign ( M ) = M L ( p 1 , , p l ) \mathrm{sign}(M)=\int_{M}L(p_{1},\ldots,p_{l})
  32. L L
  33. p i p_{i}
  34. M M

Significance_analysis_of_microarrays.html

  1. False discovery rate ( FDR ) = Median ( or 90 th percentile ) of # of falsely called genes Number of genes called significant \mathrm{False\ discovery\ rate\ (FDR)=\frac{Median\ (or\ 90^{th}\ percentile)% \ of\ \#\ of\ falsely\ called\ genes}{Number\ of\ genes\ called\ significant}}

Signomial.html

  1. X X
  2. X = ( x 1 , x 2 , x 3 , , x n ) T X=(x_{1},x_{2},x_{3},\dots,x_{n})^{T}
  3. f ( x 1 , x 2 , , x n ) = i = 1 M ( c i j = 1 n x j a i j ) f(x_{1},x_{2},\dots,x_{n})=\sum_{i=1}^{M}\left(c_{i}\prod_{j=1}^{n}x_{j}^{a_{% ij}}\right)
  4. c k c_{k}
  5. a i j a_{ij}
  6. c i c_{i}
  7. a i j a_{ij}
  8. f ( x 1 , x 2 , x 3 ) = 2.7 x 1 2 x 2 - 1 / 3 x 3 0.7 - 2 x 1 - 4 x 3 2 / 5 f(x_{1},x_{2},x_{3})=2.7x_{1}^{2}x_{2}^{-1/3}x_{3}^{0.7}-2x_{1}^{-4}x_{3}^{2/5}

Silverman's_game.html

  1. T T
  2. ν \nu
  3. 0 < ν < 0<\nu<\infty
  4. ν \nu

Simple-homotopy_equivalence.html

  1. τ ( f ) . \tau(f).

SIMPLE_algorithm.html

  1. p k + 1 = p k + urf p p^{k+1}=p^{k}+\,\text{urf}\cdot p^{{}^{\prime}}
  2. p b p_{b}^{{}^{\prime}}
  3. m ˙ f k + 1 = m ˙ f * + m ˙ f \dot{m}_{f}^{k+1}=\dot{m}_{f}^{*}+\dot{m}_{f}^{{}^{\prime}}
  4. v k + 1 = v * - Vol p < m t p l > a P v \vec{v}^{k+1}=\vec{v}^{*}-\frac{{\,\text{Vol}\ \nabla p^{{}^{\prime}}}}{<}mtpl% >{{\vec{a}_{P}^{v}}}
  5. p {\nabla p^{{}^{\prime}}}
  6. a P v {\vec{a}_{P}^{v}}

Simply_connected_at_infinity.html

  1. π 1 ( X - D ) π 1 ( X - C ) \pi_{1}(X-D)\to\pi_{1}(X-C)\,
  2. n 5 n\geq 5

SimRank.html

  1. a a
  2. b b
  3. c c
  4. d d
  5. c c
  6. d d
  7. v v
  8. I ( v ) I(v)
  9. O ( v ) O(v)
  10. v v
  11. I i ( v ) I_{i}(v)
  12. 1 i | I ( v ) | 1\leq i\leq\left|I(v)\right|
  13. O i ( v ) O_{i}(v)
  14. 1 i | O ( v ) | 1\leq i\leq\left|O(v)\right|
  15. a a
  16. b b
  17. s ( a , b ) [ 0 , 1 ] s(a,b)\in[0,1]
  18. s ( a , b ) s(a,b)
  19. a = b a=b
  20. s ( a , b ) s(a,b)
  21. 1 1
  22. s ( a , b ) = C | I ( a ) | | I ( b ) | i = 1 | I ( a ) | j = 1 | I ( b ) | s ( I i ( a ) , I j ( b ) ) s(a,b)=\frac{C}{\left|I(a)\right|\left|I(b)\right|}\sum_{i=1}^{\left|I(a)% \right|}\sum_{j=1}^{\left|I(b)\right|}s(I_{i}(a),I_{j}(b))
  23. C C
  24. 0
  25. 1 1
  26. a a
  27. b b
  28. a a
  29. b b
  30. s ( a , b ) = 0 s(a,b)=0
  31. 0
  32. I ( a ) = I(a)=\emptyset
  33. I ( b ) = I(b)=\emptyset
  34. 𝐒 \mathbf{S}
  35. [ 𝐒 ] a , b [\mathbf{S}]_{a,b}
  36. s ( a , b ) s(a,b)
  37. 𝐀 \mathbf{A}
  38. [ 𝐀 ] a , b = 1 | ( b ) | [\mathbf{A}]_{a,b}=\tfrac{1}{|\mathcal{I}(b)|}
  39. a a
  40. b b
  41. 𝐒 = max { C ( 𝐀 T 𝐒 𝐀 ) , 𝐈 } , {{\mathbf{S}}}=\max\{C\cdot(\mathbf{A}^{T}\cdot{{\mathbf{S}}}\cdot{{\mathbf{A}% }}),{{\mathbf{I}}}\},
  42. 𝐈 \mathbf{I}
  43. G G
  44. n n
  45. G G
  46. k k
  47. n 2 n^{2}
  48. s k ( * , * ) s_{k}(*,*)
  49. s k ( a , b ) s_{k}(a,b)
  50. a a
  51. b b
  52. k k
  53. s k + 1 ( * , * ) s_{k+1}(*,*)
  54. s k ( * , * ) s_{k}(*,*)
  55. s 0 ( * , * ) s_{0}(*,*)
  56. s 0 ( a , b ) s_{0}(a,b)
  57. s ( a , b ) s(a,b)
  58. s 0 ( a , b ) = { 1 , if a = b , 0 , if a b . s_{0}(a,b)=\begin{cases}1\mbox{ }~{},\mbox{ }~{}\mbox{if }~{}a=b\mbox{ }~{},\\ 0\mbox{ }~{},\mbox{ }~{}\mbox{if }~{}a\neq b\mbox{ }~{}.\end{cases}
  59. s k + 1 ( a , b ) s_{k+1}(a,b)
  60. s k ( * , * ) s_{k}(*,*)
  61. s k + 1 ( a , b ) = C | I ( a ) | | I ( b ) | i = 1 | I ( a ) | j = 1 | I ( b ) | s k ( I i ( a ) , I j ( b ) ) s_{k+1}(a,b)=\frac{C}{\left|I(a)\right|\left|I(b)\right|}\sum_{i=1}^{\left|I(a% )\right|}\sum_{j=1}^{\left|I(b)\right|}s_{k}(I_{i}(a),I_{j}(b))
  62. a b a\neq b
  63. s k + 1 ( a , b ) = 1 s_{k+1}(a,b)=1
  64. a = b a=b
  65. k + 1 k+1
  66. ( a , b ) (a,b)
  67. ( a , b ) (a,b)
  68. k k
  69. s k ( * , * ) s_{k}(*,*)
  70. k k
  71. s ( * , * ) s(*,*)
  72. a , b V a,b\in V
  73. lim k s k ( a , b ) = s ( a , b ) \lim_{k\to\infty}s_{k}(a,b)=s(a,b)
  74. C = 0.8 C=0.8
  75. K = 5 K=5
  76. C C
  77. K K
  78. C = 0.6 C=0.6
  79. 𝐒 \mathbf{S}
  80. [ 𝐒 ] a , b [\mathbf{S}]_{a,b}
  81. s ( a , b ) s(a,b)
  82. 𝐀 \mathbf{A}
  83. 𝐒 = C ( 𝐀 T 𝐒 𝐀 ) + 𝐈 , {{\mathbf{S}}}=C\cdot(\mathbf{A}^{T}\cdot{{\mathbf{S}}}\cdot{{\mathbf{A}}})+{{% \mathbf{I}}},
  84. 𝐈 \mathbf{I}
  85. p ( 0 ) ( i ) = e i p^{(0)}(i)=e_{i}
  86. e i e_{i}
  87. i i
  88. p ( k ) = A p ( k - 1 ) p^{(k)}=Ap^{(k-1)}
  89. s ( i , j ) = k = 0 C k p ( k ) ( i ) , p ( k ) ( j ) s(i,j)=\sum_{k=0}^{\infty}C^{k}\langle p^{(k)}(i),p^{(k)}(j)\rangle
  90. p ( 0 ) ( i ) p^{(0)}(i)
  91. 𝒪 ( K d 2 n 2 ) \mathcal{O}(Kd^{2}n^{2})
  92. 𝒪 ( K d n 2 ) \mathcal{O}(Kdn^{2})
  93. K K
  94. d d
  95. n n
  96. I ( a ) I(a)
  97. Partial I ( a ) s k ( j ) = i I ( a ) s k ( i , j ) , ( j I ( b ) ) \,\text{Partial}_{I(a)}^{s_{k}}(j)=\sum_{i\in I(a)}s_{k}(i,j),\qquad(\forall j% \in I(b))
  98. s k + 1 ( a , b ) s_{k+1}(a,b)
  99. Partial I ( a ) s k ( j ) \,\text{Partial}_{I(a)}^{s_{k}}(j)
  100. s k + 1 ( a , b ) = C | I ( a ) | | I ( b ) | j I ( b ) Partial I ( a ) s k ( j ) . s_{k+1}(a,b)=\frac{C}{|I(a)||I(b)|}\sum_{j\in I(b)}\,\text{Partial}_{I(a)}^{s_% {k}}(j).
  101. Partial I ( a ) s k ( j ) \,\text{Partial}_{I(a)}^{s_{k}}(j)
  102. j I ( b ) \forall j\in I(b)
  103. s k + 1 ( a , * ) s_{k+1}(a,*)
  104. a a