wpmath0000007_7

Joule-second.html

  1. k g m 2 s = J s . \frac{kg\cdot m^{2}}{s}=J\cdot s.

Jónsson_cardinal.html

  1. n \aleph_{n}
  2. n \aleph_{n}

K-edge-connected_graph.html

  1. G = ( V , E ) G=(V,E)
  2. G = ( V , E X ) G^{\prime}=(V,E\setminus X)
  3. X E X\subseteq E
  4. | X | < k |X|<k
  5. G = ( V , E ) G=(V,E)
  6. O ( n 2 ) O(n^{2})
  7. O ( n 3 ) O(n^{3})
  8. O ( n 5 ) O(n^{5})
  9. O ( n 4 ) O(n^{4})
  10. O ( n 3 ) O(n^{3})
  11. k 2 k\geq 2

K-theory_(physics).html

  1. 𝐙 N \mathbf{Z}_{N}
  2. d 3 G 3 = S q 3 G 3 + H G 3 = G 3 G 3 + H G 3 = 0 d_{3}G_{3}=Sq^{3}G_{3}+H\cup G_{3}=G_{3}\cup G_{3}+H\cup G_{3}=0
  3. \cup
  4. G 3 G 3 + H G 3 + H H = P G_{3}\cup G_{3}+H\cup G_{3}+H\cup H=P
  5. 𝐙 N \mathbf{Z}_{N}
  6. S U ( M + N ) × S U ( M ) SU(M+N)\times SU(M)

Kadomtsev–Petviashvili_equation.html

  1. x ( t u + u x u + ϵ 2 x x x u ) + λ y y u = 0 \displaystyle\partial_{x}(\partial_{t}u+u\partial_{x}u+\epsilon^{2}\partial_{% xxx}u)+\lambda\partial_{yy}u=0
  2. λ = ± 1 \lambda=\pm 1
  3. λ = + 1 \lambda=+1
  4. λ = - 1 \lambda=-1
  5. ϵ 1 \epsilon\ll 1
  6. O ( 1 / ϵ ) O(1/\epsilon)
  7. ϵ 0 \epsilon\rightarrow 0
  8. ϵ 0 \epsilon\rightarrow 0
  9. ϵ 0 \epsilon\rightarrow 0
  10. t u + u x u = 0. \displaystyle\partial_{t}u+u\partial_{x}u=0.
  11. O ( ϵ ) O(\epsilon)

Kadowaki–Woods_ratio.html

  1. R KW = A γ 2 R_{\mathrm{KW}}=\frac{A}{\gamma^{2}}

Karmarkar's_algorithm.html

  1. n n
  2. L L
  3. O ( n 3.5 L ) O(n^{3.5}L)
  4. O ( L ) O(L)
  5. O ( n 6 L ) O(n^{6}L)
  6. O ( n 3.5 L 2 log L log log L ) O(n^{3.5}L^{2}\cdot\log L\cdot\log\log L)
  7. x 0 x^{0}
  8. γ \gamma
  9. k 0 k\leftarrow 0
  10. v k b - A x k v^{k}\leftarrow b-Ax^{k}
  11. D v diag ( v 1 k , , v m k ) D_{v}\leftarrow\operatorname{diag}(v_{1}^{k},\ldots,v_{m}^{k})
  12. h x ( A T D v - 2 A ) - 1 c h_{x}\leftarrow(A^{T}D_{v}^{-2}A)^{-1}c
  13. h v - A h x h_{v}\leftarrow-Ah_{x}
  14. h v 0 h_{v}\geq 0
  15. α γ min { - v i k / ( h v ) i | ( h v ) i < 0 , i = 1 , , m } \alpha\leftarrow\gamma\cdot\min\{-v_{i}^{k}/(h_{v})_{i}\,\,|\,\,(h_{v})_{i}<0,% \,i=1,\ldots,m\}
  16. x k + 1 x k + α h x x^{k+1}\leftarrow x^{k}+\alpha h_{x}
  17. k k + 1 k\leftarrow k+1
  18. x 1 x_{1}
  19. x 2 x_{2}
  20. 2 p x 1 2px_{1}
  21. x 2 x_{2}
  22. \leq
  23. p 2 + 1 p^{2}+1
  24. p = 0.0 , 0.1 , 0.2 , , 0.9 , 1.0. p=0.0,0.1,0.2,\ldots,0.9,1.0.
  25. x 1 , x 2 x_{1},x_{2}
  26. p p

Keltner_channel.html

  1. t y p i c a l p r i c e = h i g h + l o w + c l o s e 3 typical\ price={high+low+close\over 3}

Kempe_chain.html

  1. c : V S c:V\to S
  2. c : E S . c:E\to S.

Kenneth_Davidson_(mathematician).html

  1. C C^{\ast}

Keratometer.html

  1. R = 2 d I O R=2d\frac{I}{O}

Kernel_method.html

  1. i i
  2. ( 𝐱 i , y i ) (\mathbf{x}_{i},y_{i})
  3. w i w_{i}
  4. k k
  5. 𝐱 \mathbf{x^{\prime}}
  6. 𝐱 i \mathbf{x}_{i}
  7. y ^ = sgn i = 1 n w i y i k ( 𝐱 i , 𝐱 ) \hat{y}=\operatorname{sgn}\sum_{i=1}^{n}w_{i}y_{i}k(\mathbf{x}_{i},\mathbf{x^{% \prime}})
  8. y ^ { - 1 , + 1 } \hat{y}\in\{-1,+1\}
  9. 𝐱 \mathbf{x^{\prime}}
  10. y y
  11. k : 𝒳 × 𝒳 k\colon\mathcal{X}\times\mathcal{X}\to\mathbb{R}
  12. 𝐱 , 𝐱 𝒳 \mathbf{x},\mathbf{x^{\prime}}\in\mathcal{X}
  13. n n
  14. { ( 𝐱 i , y i ) } i = 1 n \{(\mathbf{x}_{i},y_{i})\}_{i=1}^{n}
  15. y i { - 1 , + 1 } y_{i}\in\{-1,+1\}
  16. w i w_{i}\in\mathbb{R}
  17. sgn \operatorname{sgn}
  18. y ^ \hat{y}
  19. 𝐱 \mathbf{x}
  20. 𝐱 \mathbf{x^{\prime}}
  21. 𝒳 \mathcal{X}
  22. k ( 𝐱 , 𝐱 ) k(\mathbf{x},\mathbf{x^{\prime}})
  23. 𝒱 \mathcal{V}
  24. k : 𝒳 × 𝒳 k\colon\mathcal{X}\times\mathcal{X}\to\mathbb{R}
  25. φ : 𝒳 𝒱 \varphi\colon\mathcal{X}\to\mathcal{V}
  26. k ( 𝐱 , 𝐱 ) = φ ( 𝐱 ) , φ ( 𝐱 ) 𝒱 k(\mathbf{x},\mathbf{x^{\prime}})=\langle\varphi(\mathbf{x}),\varphi(\mathbf{x% ^{\prime}})\rangle_{\mathcal{V}}
  27. , 𝒱 \langle\cdot,\cdot\rangle_{\mathcal{V}}
  28. φ \varphi
  29. 𝒱 \mathcal{V}
  30. 𝒳 \mathcal{X}
  31. k k
  32. X X
  33. μ ( T ) = | T | \mu(T)=|T|
  34. T X T\subset X
  35. i = 1 n j = 1 n k ( 𝐱 i , 𝐱 j ) c i c j 0 \sum_{i=1}^{n}\sum_{j=1}^{n}k(\mathbf{x}_{i},\mathbf{x}_{j})c_{i}c_{j}\geq 0
  36. ( 𝐱 1 , , 𝐱 n ) (\mathbf{x}_{1},\ldots,\mathbf{x}_{n})
  37. 𝒳 \mathcal{X}
  38. n n
  39. ( c 1 , , c n ) (c_{1},\dots,c_{n})
  40. 𝒳 \mathcal{X}
  41. φ \varphi
  42. φ \varphi
  43. 𝐊 n × n \mathbf{K}\in\mathbb{R}^{n\times n}
  44. { 𝐱 1 , , 𝐱 n } \{\mathbf{x}_{1},\ldots,\mathbf{x}_{n}\}
  45. 𝐊 = ( k ( 𝐱 i , 𝐱 j ) ) i j \mathbf{K}=(k(\mathbf{x}_{i},\mathbf{x}_{j}))_{ij}
  46. k k
  47. k k
  48. k k
  49. k k
  50. k k
  51. 𝐊 \mathbf{K}
  52. 𝐊 \mathbf{K}
  53. 𝐊 T 𝐊 \mathbf{K}^{\mathrm{T}}\mathbf{K}

King's_graph.html

  1. n × m n\times m
  2. n × m n\times m
  3. n × m n\times m
  4. n m nm
  5. n × n n\times n
  6. n 2 n^{2}
  7. ( 2 n - 2 ) ( 2 n - 1 ) (2n-2)(2n-1)

Kirillov_character_formula.html

  1. G G
  2. j j
  3. λ \lambda
  4. 𝔱 * \mathfrak{t}^{*}
  5. ρ \rho
  6. 𝒪 λ + ρ \mathcal{O}_{\lambda+\rho}
  7. λ + ρ 𝔱 * \lambda+\rho\in\mathfrak{t}^{*}
  8. μ λ + ρ \mu_{\lambda+\rho}
  9. G G
  10. 𝒪 λ + ρ \mathcal{O}_{\lambda+\rho}
  11. dim π = d λ \dim\pi=d_{\lambda}
  12. χ π = χ λ \chi_{\pi}=\chi_{\lambda}
  13. j ( X ) χ λ ( exp X ) = 𝒪 λ + ρ e i β ( X ) d μ λ + ρ ( β ) , X 𝔤 j(X)\chi_{\lambda}(\exp X)=\int_{\mathcal{O}_{\lambda+\rho}}e^{i\beta(X)}d\mu_% {\lambda+\rho}(\beta),\;\forall\;X\in\mathfrak{g}
  14. ρ = 1 / 2 \rho=1/2
  15. λ + 1 / 2 \lambda+1/2
  16. 𝒪 λ + 1 / 2 e i β ( X ) d μ λ + 1 / 2 ( β ) = sin ( ( 2 λ + 1 ) X ) X / 2 , X 𝔤 , \int_{\mathcal{O}_{\lambda+1/2}}e^{i\beta(X)}d\mu_{\lambda+1/2}(\beta)=\frac{% \sin((2\lambda+1)X)}{X/2},\;\forall\;X\in\mathfrak{g},
  17. j ( X ) = sin X / 2 X / 2 j(X)=\frac{\sin X/2}{X/2}
  18. χ λ ( exp X ) = sin ( ( 2 λ + 1 ) X ) sin X / 2 \chi_{\lambda}(\exp X)=\frac{\sin((2\lambda+1)X)}{\sin X/2}

Kleihauer–Betke_test.html

  1. 150 m l k g 150\frac{ml}{kg}
  2. P F B = ( 3200 ) ( F C ) ( F W ) ( M C ) PFB=\frac{(3200)(FC)}{(FW)(MC)}
  3. P F B PFB
  4. F C FC
  5. M C MC
  6. M C = T C - F C MC=TC-FC
  7. T C TC
  8. F W FW
  9. T C = 5000 TC=5000
  10. F C = 200 FC=200
  11. F W = 2.0 k g FW=2.0kg
  12. P F B = ( 3200 ) ( F C ) ( F W ) ( M C ) = ( 3200 ) ( 200 ) ( 2.0 ) ( 4800 ) = 200 3 = 66.667 PFB=\frac{(3200)(FC)}{(FW)(MC)}=\frac{(3200)(200)}{(2.0)(4800)}=\frac{200}{3}=% 66.667
  13. P F B 20 PFB\geq 20
  14. P F B PFB
  15. r + 1.000 r\approx+1.000

Kleisli_category.html

  1. Obj ( 𝒞 T ) = Obj ( 𝒞 ) , Hom 𝒞 T ( X , Y ) = Hom 𝒞 ( X , T Y ) . \begin{aligned}\displaystyle\mathrm{Obj}({\mathcal{C}_{T}})&\displaystyle=% \mathrm{Obj}({\mathcal{C}}),\\ \displaystyle\mathrm{Hom}_{\mathcal{C}_{T}}(X,Y)&\displaystyle=\mathrm{Hom}_{% \mathcal{C}}(X,TY).\end{aligned}
  2. g T f = μ Z T g f g\circ_{T}f=\mu_{Z}\circ Tg\circ f
  3. id X = η X \mathrm{id}_{X}=\eta_{X}
  4. C C
  5. X X
  6. C C
  7. X T X_{T}
  8. f : X T Y f:X\to TY
  9. C C
  10. f * : X T Y T f^{*}:X_{T}\to Y_{T}
  11. C T C_{T}
  12. g * T f * = ( μ Z T g f ) * . g^{*}\circ_{T}f^{*}=(\mu_{Z}\circ Tg\circ f)^{*}.
  13. C T C_{T}
  14. id X T = ( η X ) * . \mathrm{id}_{X_{T}}=(\eta_{X})^{*}.
  15. f * = μ Y T f . f^{*}=\mu_{Y}\circ Tf.
  16. g T f = g * f . g\circ_{T}f=g^{*}\circ f.
  17. η X * \displaystyle\eta_{X}^{*}
  18. T : ob ( C ) ob ( C ) T:\mathrm{ob}(C)\to\mathrm{ob}(C)
  19. A A
  20. C C
  21. η A : A T ( A ) \eta_{A}:A\to T(A)
  22. f : A T ( B ) f:A\to T(B)
  23. C C
  24. f * : T ( A ) T ( B ) f^{*}:T(A)\to T(B)
  25. F X = X FX=X\;
  26. F ( f : X Y ) = η Y f F(f:X\to Y)=\eta_{Y}\circ f
  27. G Y = T Y GY=TY\;
  28. G ( f : X T Y ) = μ Y T f G(f:X\to TY)=\mu_{Y}\circ Tf\;
  29. ε Y = id T Y . \varepsilon_{Y}=\mathrm{id}_{TY}.

Knaster–Kuratowski–Mazurkiewicz_lemma.html

  1. Δ m \Delta_{m}
  2. C i C_{i}
  3. i I = { 1 , , m } i\in I=\{1,\dots,m\}
  4. I k I I_{k}\subset I
  5. Δ m \Delta_{m}
  6. e i e_{i}
  7. i I k i\in I_{k}
  8. C i C_{i}
  9. i I k i\in I_{k}
  10. C i C_{i}
  11. Δ 3 \Delta_{3}
  12. C 1 , C 2 , C 3 C_{1},C_{2},C_{3}
  13. C 1 C_{1}
  14. C 2 C_{2}
  15. C 3 C_{3}
  16. C 1 C_{1}
  17. C 2 C_{2}
  18. C 2 C_{2}
  19. C 3 C_{3}
  20. C 3 C_{3}
  21. C 1 C_{1}
  22. C 1 , C 2 , C 3 C_{1},C_{2},C_{3}

Knight's_graph.html

  1. n × m n\times m
  2. n × m n\times m
  3. n × m n\times m
  4. n m nm
  5. n × n n\times n
  6. n 2 n^{2}
  7. 4 ( n - 2 ) ( n - 1 ) 4(n-2)(n-1)

Knuth's_Algorithm_X.html

  1. 𝒮 \mathcal{S}
  2. 𝒮 * \mathcal{S}^{*}

Kochen–Specker_theorem.html

  1. v ( f ( 𝐀 ) ) \scriptstyle v(f({\mathbf{A}}))
  2. f ( 𝐀 ) \scriptstyle f({\mathbf{A}})
  3. v ( f ( 𝐀 ) ) = f ( v ( 𝐀 ) ) . v(f({\mathbf{A}}))=f(v({\mathbf{A}})).
  4. v ( c 1 𝐀 1 + c 2 𝐀 2 ) = c 1 v ( 𝐀 1 ) + c 2 v ( 𝐀 2 ) , v(c_{1}{\mathbf{A}}_{1}+c_{2}{\mathbf{A}}_{2})=c_{1}v({\mathbf{A}}_{1})+c_{2}v% ({\mathbf{A}}_{2}),
  5. c 1 c_{1}
  6. c 2 c_{2}
  7. v ( 𝐀 1 𝐀 2 ) = v ( 𝐀 1 ) v ( 𝐀 2 ) . v({\mathbf{A}}_{1}{\mathbf{A}}_{2})=v({\mathbf{A}}_{1})v({\mathbf{A}}_{2}).
  8. 𝐏 1 + 𝐏 2 + 𝐏 3 + 𝐏 4 = 𝐈 {\mathbf{P}}_{1}+{\mathbf{P}}_{2}+{\mathbf{P}_{3}}+{\mathbf{P}_{4}}={\mathbf{I}}
  9. v ( 𝐏 1 + 𝐏 2 + 𝐏 3 + 𝐏 4 ) = v ( 𝐈 ) = 1. v({\mathbf{P}_{1}}+{\mathbf{P}_{2}}+{\mathbf{P}}_{3}+{\mathbf{P}}_{4})=v({% \mathbf{I}})=1.
  10. v ( 𝐏 1 + 𝐏 2 + 𝐏 3 + 𝐏 4 ) = v ( 𝐏 1 ) + v ( 𝐏 2 ) + v ( 𝐏 3 ) + v ( 𝐏 4 ) v({\mathbf{P}}_{1}+{\mathbf{P}}_{2}+{\mathbf{P}}_{3}+{\mathbf{P}}_{4})=v({% \mathbf{P}}_{1})+v({\mathbf{P}}_{2})+v({\mathbf{P}}_{3})+v({\mathbf{P}}_{4})
  11. v ( 𝐏 i ) = \scriptstyle v({\mathbf{P}}_{i})=
  12. i = 1 , , 4 \scriptstyle i=1,\ldots,4
  13. v ( 𝐏 1 ) , v ( 𝐏 2 ) , v ( 𝐏 3 ) , v ( 𝐏 4 ) \scriptstyle v({\mathbf{P}}_{1}),v({\mathbf{P}}_{2}),v({\mathbf{P}}_{3}),v({% \mathbf{P}}_{4})
  14. v ( 𝐀 ) \scriptstyle v({\mathbf{A}})

Kohn–Sham_equations.html

  1. ( - 2 2 m 2 + v eff ( 𝐫 ) ) ϕ i ( 𝐫 ) = ε i ϕ i ( 𝐫 ) \left(-\frac{\hbar^{2}}{2m}\nabla^{2}+v_{\rm eff}(\mathbf{r})\right)\phi_{i}(% \mathbf{r})=\varepsilon_{i}\phi_{i}(\mathbf{r})
  2. ρ ( 𝐫 ) = i N | ϕ i ( 𝐫 ) | 2 . \rho(\mathbf{r})=\sum_{i}^{N}|\phi_{i}(\mathbf{r})|^{2}.
  3. E [ ρ ] = T s [ ρ ] + d 𝐫 v ext ( 𝐫 ) ρ ( 𝐫 ) + E H [ ρ ] + E xc [ ρ ] E[\rho]=T_{s}[\rho]+\int d\mathbf{r}\ v_{\rm ext}(\mathbf{r})\rho(\mathbf{r})+% E_{H}[\rho]+E_{\rm xc}[\rho]
  4. T s [ ρ ] = i = 1 N d 𝐫 ϕ i * ( 𝐫 ) ( - 2 2 m 2 ) ϕ i ( 𝐫 ) , T_{s}[\rho]=\sum_{i=1}^{N}\int d\mathbf{r}\ \phi_{i}^{*}(\mathbf{r})\left(-% \frac{\hbar^{2}}{2m}\nabla^{2}\right)\phi_{i}(\mathbf{r}),
  5. E H = e 2 2 d 𝐫 d 𝐫 ρ ( 𝐫 ) ρ ( 𝐫 ) | 𝐫 - 𝐫 | . E_{H}={e^{2}\over 2}\int d\mathbf{r}\int d\mathbf{r}^{\prime}\ {\rho(\mathbf{r% })\rho(\mathbf{r}^{\prime})\over|\mathbf{r}-\mathbf{r}^{\prime}|}.
  6. v eff ( 𝐫 ) = v ext ( 𝐫 ) + e 2 ρ ( 𝐫 ) | 𝐫 - 𝐫 | d 𝐫 + δ E xc [ ρ ] δ ρ ( 𝐫 ) . v_{\rm eff}(\mathbf{r})=v_{\rm ext}(\mathbf{r})+e^{2}\int{\rho(\mathbf{r}^{% \prime})\over|\mathbf{r}-\mathbf{r}^{\prime}|}d\mathbf{r}^{\prime}+{\delta E_{% \rm xc}[\rho]\over\delta\rho(\mathbf{r})}.
  7. v xc ( 𝐫 ) δ E xc [ ρ ] δ ρ ( 𝐫 ) v_{\rm xc}(\mathbf{r})\equiv{\delta E_{\rm xc}[\rho]\over\delta\rho(\mathbf{r})}
  8. E = i N ε i - V H [ ρ ] + E xc [ ρ ] - δ E xc [ ρ ] δ ρ ( 𝐫 ) ρ ( 𝐫 ) d 𝐫 E=\sum_{i}^{N}\varepsilon_{i}-V_{H}[\rho]+E_{\rm xc}[\rho]-\int{\delta E_{\rm xc% }[\rho]\over\delta\rho(\mathbf{r})}\rho(\mathbf{r})d\mathbf{r}

Kontorovich–Lebedev_transform.html

  1. g ( y ) = 0 f ( x ) K i y ( x ) d x g(y)=\int_{0}^{\infty}f(x)K_{iy}(x)\,dx
  2. f ( x ) = 2 π 2 x 0 g ( y ) K i y ( x ) sinh ( π y ) y d y . f(x)=\frac{2}{\pi^{2}x}\int_{0}^{\infty}g(y)K_{iy}(x)\sinh(\pi y)y\,dy.
  3. g ( y ) = 0 f ( x ) e - x L y ( x ) d x g(y)=\int_{0}^{\infty}f(x)e^{-x}L_{y}(x)\,dx
  4. f ( x ) = 0 g ( y ) Γ ( y ) L y ( x ) d y . f(x)=\int_{0}^{\infty}\frac{g(y)}{\Gamma(y)}L_{y}(x)\,dy.

Kozai_mechanism.html

  1. L z = ( 1 - e 2 ) cos i . L_{z}=\sqrt{(1-e^{2})}\cos i.
  2. arccos ( 3 5 ) 39.2 o \arccos\left(\sqrt{\frac{3}{5}}\right)\approx 39.2^{o}
  3. T Kozai = 2 π G M G m 2 a 2 3 a 3 / 2 ( 1 - e 2 2 ) 3 / 2 = M m 2 P 2 2 P ( 1 - e 2 2 ) 3 / 2 T_{\mathrm{Kozai}}=2\pi\frac{\sqrt{GM}}{Gm_{2}}\frac{a_{2}^{3}}{a^{3/2}}\left(% 1-e_{2}^{2}\right)^{3/2}=\frac{M}{m_{2}}\frac{P_{2}^{2}}{P}\left(1-e_{2}^{2}% \right)^{3/2}

Kramers–Wannier_duality.html

  1. Z N ( K * , L * ) = 2 e N ( K * + L * ) P Λ D ( e - 2 L * ) r ( e - 2 K * ) s Z_{N}(K^{*},L^{*})=2e^{N(K^{*}+L^{*})}\sum_{P\subset\Lambda_{D}}(e^{-2L^{*}})^% {r}(e^{-2K^{*}})^{s}
  2. tanh K = e - 2 L * , tanh L = e - 2 K * \tanh K=e^{-2L*},\ \tanh L=e^{-2K*}
  3. Z N ( K * , L * ) = 2 ( tanh K tanh L ) - N / 2 P v r w s Z_{N}(K^{*},L^{*})=2(\tanh K\;\tanh L)^{-N/2}\sum_{P}v^{r}w^{s}
  4. = 2 ( sinh 2 K sinh 2 L ) - N / 2 Z N ( K , L ) =2(\sinh 2K\;\sinh 2L)^{-N/2}Z_{N}(K,L)
  5. sinh 2 K * sinh 2 L = 1 \,\sinh 2K^{*}\sinh 2L=1
  6. sinh 2 L * sinh 2 K = 1 \,\sinh 2L^{*}\sinh 2K=1
  7. f ( K , L ) = lim N f N ( K , L ) = - k T lim N 1 N log Z N ( K , L ) f(K,L)=\lim_{N\rightarrow\infty}f_{N}(K,L)=-kT\lim_{N\rightarrow\infty}\frac{1% }{N}\log Z_{N}(K,L)
  8. f ( K * , L * ) = f ( K , L ) + 1 2 k T log ( sinh 2 K sinh 2 L ) f(K^{*},L^{*})=f(K,L)+\frac{1}{2}kT\log(\sinh 2K\sinh 2L)

Krasovskii–LaSalle_principle.html

  1. 𝐱 ˙ = f ( 𝐱 ) \dot{\mathbf{x}}=f\left(\mathbf{x}\right)
  2. 𝐱 \mathbf{x}
  3. f ( 𝟎 ) = 𝟎 f\left(\mathbf{0}\right)=\mathbf{0}
  4. C 1 C^{1}
  5. V ( 𝐱 ) V(\mathbf{x})
  6. V ( 𝐱 ) > 0 V(\mathbf{x})>0
  7. 𝐱 𝟎 \mathbf{x}\neq\mathbf{0}
  8. V ˙ ( 𝐱 ) 0 \dot{V}(\mathbf{x})\leq 0
  9. 𝐱 \mathbf{x}
  10. V ( 𝐱 ) V(\mathbf{x})\to\infty
  11. 𝐱 \mathbf{x}\to\infty
  12. V ( 𝟎 ) = V ˙ ( 𝟎 ) = 0 V(\mathbf{0})=\dot{V}(\mathbf{0})=0
  13. {\mathcal{I}}
  14. { 𝐱 : V ˙ ( 𝐱 ) = 0 } \{\mathbf{x}:\dot{V}(\mathbf{x})=0\}
  15. {\mathcal{I}}
  16. {\mathcal{I}}
  17. 𝐱 ( t ) = 𝟎 \mathbf{x}(t)=\mathbf{0}
  18. t 0 t\geq 0
  19. V ( 𝐱 ) > 0 V(\mathbf{x})>0
  20. 𝐱 𝟎 \mathbf{x}\neq\mathbf{0}
  21. V ˙ ( 𝐱 ) 0 \dot{V}(\mathbf{x})\leq 0
  22. 𝐱 \mathbf{x}
  23. D D
  24. { V ˙ ( 𝐱 ) = 0 } D \{\dot{V}(\mathbf{x})=0\}\bigcap D
  25. 𝐱 ( t ) = 𝟎 , t 0 \mathbf{x}(t)=\mathbf{0},t\geq 0
  26. V ˙ ( 𝐱 ) \dot{V}(\mathbf{x})
  27. V ˙ ( 𝐱 ) \dot{V}(\mathbf{x})
  28. m l θ ¨ = - m g sin θ - k l θ ˙ ml\ddot{\theta}=-mg\sin\theta-kl\dot{\theta}
  29. θ \theta
  30. m m
  31. l l
  32. k k
  33. x ˙ 1 = x 2 \dot{x}_{1}=x_{2}
  34. x ˙ 2 = - g l sin x 1 - k m x 2 \dot{x}_{2}=-\frac{g}{l}\sin x_{1}-\frac{k}{m}x_{2}
  35. x 1 = x 2 = 0 x_{1}=x_{2}=0
  36. V ( x 1 , x 2 ) V(x_{1},x_{2})
  37. V ( x 1 , x 2 ) = g l ( 1 - cos x 1 ) + 1 2 x 2 2 V(x_{1},x_{2})=\frac{g}{l}(1-\cos x_{1})+\frac{1}{2}x_{2}^{2}
  38. V ( x 1 , x 2 ) V(x_{1},x_{2})
  39. V ( x 1 , x 2 ) V(x_{1},x_{2})
  40. π \pi
  41. V ˙ ( x 1 , x 2 ) = g l sin x 1 x ˙ 1 + x 2 x ˙ 2 = - k m x 2 2 \dot{V}(x_{1},x_{2})=\frac{g}{l}\sin x_{1}\dot{x}_{1}+x_{2}\dot{x}_{2}=-\frac{% k}{m}x_{2}^{2}
  42. V ( 0 ) = V ˙ ( 0 ) = 0 V(0)=\dot{V}(0)=0
  43. V ˙ < 0 \dot{V}<0
  44. V ˙ 0 \dot{V}\leq 0
  45. V ˙ \dot{V}
  46. S = { ( x 1 , x 2 ) | V ˙ ( x 1 , x 2 ) = 0 } S=\{(x_{1},x_{2})|\dot{V}(x_{1},x_{2})=0\}
  47. S = { ( x 1 , x 2 ) | x 2 = 0 } S=\{(x_{1},x_{2})|x_{2}=0\}
  48. t t
  49. x 2 ( t ) = 0 x_{2}(t)=0
  50. x 1 x_{1}
  51. π \pi
  52. sin x 1 0 \sin x_{1}\neq 0
  53. x ˙ 2 ( t ) 0 \dot{x}_{2}(t)\neq 0
  54. S S
  55. t t\rightarrow\infty

Krein–Milman_theorem.html

  1. X X
  2. K K
  3. X X
  4. K K
  5. X X
  6. K . K.
  7. X X
  8. K K
  9. X X
  10. K , K,
  11. T T
  12. K K
  13. T T
  14. K K
  15. K K
  16. T . T.
  17. K K
  18. K . K.

Kronecker's_lemma.html

  1. ( x n ) n = 1 (x_{n})_{n=1}^{\infty}
  2. m = 1 x m = s \sum_{m=1}^{\infty}x_{m}=s
  3. 0 < b 1 b 2 b 3 0<b_{1}\leq b_{2}\leq b_{3}\leq\ldots
  4. b n b_{n}\to\infty
  5. lim n 1 b n k = 1 n b k x k = 0. \lim_{n\to\infty}\frac{1}{b_{n}}\sum_{k=1}^{n}b_{k}x_{k}=0.
  6. S k S_{k}
  7. 1 b n k = 1 n b k x k = S n - 1 b n k = 1 n - 1 ( b k + 1 - b k ) S k \frac{1}{b_{n}}\sum_{k=1}^{n}b_{k}x_{k}=S_{n}-\frac{1}{b_{n}}\sum_{k=1}^{n-1}(% b_{k+1}-b_{k})S_{k}
  8. S k S_{k}
  9. S k S_{k}
  10. S n - 1 b n k = 1 N - 1 ( b k + 1 - b k ) S k - 1 b n k = N n - 1 ( b k + 1 - b k ) S k S_{n}-\frac{1}{b_{n}}\sum_{k=1}^{N-1}(b_{k+1}-b_{k})S_{k}-\frac{1}{b_{n}}\sum_% {k=N}^{n-1}(b_{k+1}-b_{k})S_{k}
  11. = S n - 1 b n k = 1 N - 1 ( b k + 1 - b k ) S k - 1 b n k = N n - 1 ( b k + 1 - b k ) s - 1 b n k = N n - 1 ( b k + 1 - b k ) ( S k - s ) =S_{n}-\frac{1}{b_{n}}\sum_{k=1}^{N-1}(b_{k+1}-b_{k})S_{k}-\frac{1}{b_{n}}\sum% _{k=N}^{n-1}(b_{k+1}-b_{k})s-\frac{1}{b_{n}}\sum_{k=N}^{n-1}(b_{k+1}-b_{k})(S_% {k}-s)
  12. = S n - 1 b n k = 1 N - 1 ( b k + 1 - b k ) S k - b n - b N b n s - 1 b n k = N n - 1 ( b k + 1 - b k ) ( S k - s ) . =S_{n}-\frac{1}{b_{n}}\sum_{k=1}^{N-1}(b_{k+1}-b_{k})S_{k}-\frac{b_{n}-b_{N}}{% b_{n}}s-\frac{1}{b_{n}}\sum_{k=N}^{n-1}(b_{k+1}-b_{k})(S_{k}-s).
  13. ϵ ( b n - b N ) / b n ϵ \epsilon(b_{n}-b_{N})/b_{n}\leq\epsilon

Kröger–Vink_notation.html

  1. M S C M^{C}_{S}
  2. Al Al × \mathrm{Al}^{\times}_{\mathrm{Al}}
  3. Ni Cu × \mathrm{Ni}^{\times}_{\mathrm{Cu}}
  4. V Cl {V}^{\bullet}_{\mathrm{Cl}}
  5. Ca i \mathrm{Ca}^{\bullet\bullet}_{\mathrm{i}}
  6. Cl i \mathrm{Cl}^{{}^{\prime}}_{\mathrm{i}}
  7. O i ′′ \mathrm{O}^{{}^{\prime\prime}}_{\mathrm{i}}
  8. e \mathrm{e}^{{}^{\prime}}
  9. V Ti ′′′′ + 2 V O \Leftrightarrow V_{\mathrm{Ti}}^{\prime\prime\prime\prime}+2V_{\mathrm{O}}^{% \bullet\bullet}
  10. V Ba ′′ + V Ti ′′′′ + 3 V O \Leftrightarrow V_{\mathrm{Ba}}^{\prime\prime}+V_{\mathrm{Ti}}^{\prime\prime% \prime\prime}+3V_{\mathrm{O}}^{\bullet\bullet}
  11. Mg Mg × \mathrm{Mg}^{\times}_{\mathrm{Mg}}
  12. O O × O i ′′ \mathrm{O}^{\times}_{\mathrm{O}}\Leftrightarrow\mathrm{O}^{{}^{\prime\prime}}_% {i}
  13. V O \mathrm{V}^{\bullet\bullet}_{\mathrm{O}}
  14. Mg Mg × \mathrm{Mg}^{\times}_{\mathrm{Mg}}
  15. Mg Mg × \mathrm{Mg}^{\times}_{\mathrm{Mg}}
  16. O O × V M g ′′ \mathrm{O}^{\times}_{\mathrm{O}}\Leftrightarrow\mathrm{V}^{{}^{\prime\prime}}_% {Mg}
  17. V O \mathrm{V}^{\bullet\bullet}_{\mathrm{O}}
  18. Mg surface × \mathrm{Mg}^{\times}_{\mathrm{surface}}
  19. O surface × \mathrm{O}^{\times}_{\mathrm{surface}}
  20. V C + V A V M + V X \Leftrightarrow V^{{}^{\prime}}_{C}+V^{\bullet}_{A}\Leftrightarrow V^{{}^{% \prime}}_{M}+V^{\bullet}_{X}
  21. e + h \Leftrightarrow e^{{}^{\prime}}+h^{\bullet}
  22. V C + i C V M + i M \Leftrightarrow V^{{}^{\prime}}_{C}+i^{\bullet}_{C}\Leftrightarrow V^{{}^{% \prime}}_{M}+i^{\bullet}_{M}
  23. V A + i A V X + i X \Leftrightarrow V^{{}^{\prime}}_{A}+i^{\bullet}_{A}\Leftrightarrow V^{{}^{% \prime}}_{X}+i^{\bullet}_{X}
  24. M M × + e = M M M^{\times}_{M}+e^{{}^{\prime}}=M^{{}^{\prime}}_{M}
  25. B M + e = B M × B^{\bullet}_{M}+e^{{}^{\prime}}=B^{\times}_{M}
  26. A A × + X X × A i + 1 2 X 2 ( g ) + 2 e A^{\times}_{A}+X^{\times}_{X}\Leftrightarrow A^{\bullet\bullet}_{i}+\frac{1}{2% }X_{2}(g)+2e^{{}^{\prime}}
  27. A ( s ) A A × + V X + 2 e A(s)\Leftrightarrow A^{\times}_{A}+V^{\bullet\bullet}_{X}+2e^{{}^{\prime}}
  28. 1 2 X 2 ( g ) V A ′′ + X X × + 2 h \frac{1}{2}X_{2}(g)\Leftrightarrow V^{{}^{\prime\prime}}_{A}+X^{\times}_{X}+2h% ^{\bullet}
  29. A A × + X X × A ( s ) + X i ′′ + 2 h A^{\times}_{A}+X^{\times}_{X}\Leftrightarrow A(s)+X^{{}^{\prime\prime}}_{i}+2h% ^{\bullet}
  30. V Mg ′′ + V O \Leftrightarrow V_{\mathrm{Mg}}^{\prime\prime}+V_{\mathrm{O}}^{\bullet\bullet}
  31. [ V Mg ′′ ] [ V O ] [V_{\mathrm{Mg}}^{\prime\prime}][V_{\mathrm{O}}^{\bullet\bullet}]
  32. [ V Mg ′′ ] = [ V O ] [V_{\mathrm{Mg}}^{\prime\prime}]=[V^{\bullet\bullet}_{O}]
  33. k = e - Δ G F k B T k=e^{-\frac{\Delta G_{F}}{k_{B}T}}
  34. k B k_{B}
  35. Δ G = Δ H - T Δ S {\Delta}G={\Delta}H-T{\Delta}S
  36. e - Δ G F k B T = [ V Mg ′′ ] 2 e^{-\frac{\Delta G_{F}}{k_{B}T}}=[V_{\mathrm{Mg}}^{\prime\prime}]^{2}
  37. [ V Mg ′′ ] = e - Δ H F 2 k B T + Δ S 2 k B = A e - Δ H F 2 k B T [V_{\mathrm{Mg}}^{\prime\prime}]=e^{-\frac{\Delta H_{F}}{2k_{B}T}+\frac{\Delta S% }{2k_{B}}}=Ae^{-\frac{\Delta H_{F}}{2k_{B}T}}
  38. A A

Krull's_theorem.html

  1. R R
  2. a a
  3. R R
  4. P P
  5. a a

Kuhn_length.html

  1. N N
  2. b b
  3. n n
  4. N N
  5. L = N b L=Nb
  6. R 2 = N b 2 \langle R^{2}\rangle=Nb^{2}
  7. l l
  8. R 2 = n l 2 1 + cos ( θ ) 1 - cos ( θ ) 1 + cos ( ϕ ) 1 - cos ( ϕ ) \langle R^{2}\rangle=nl^{2}\frac{1+\cos(\theta)}{1-\cos(\theta)}\cdot\frac{1+% \langle\cos(\textstyle\phi\,\!)\rangle}{1-\langle\cos(\textstyle\phi\,\!)\rangle}
  9. cos ( ϕ ) \langle\cos(\textstyle\phi\,\!)\rangle
  10. L = n l cos ( θ / 2 ) L=nl\,\cos(\theta/2)
  11. R 2 \langle R^{2}\rangle
  12. L L
  13. N N
  14. b b

Kulkarni–Nomizu_product.html

  1. ( h k ) ( X 1 , X 2 , X 3 , X 4 ) := h ( X 1 , X 3 ) k ( X 2 , X 4 ) + h ( X 2 , X 4 ) k ( X 1 , X 3 ) - h ( X 1 , X 4 ) k ( X 2 , X 3 ) - h ( X 2 , X 3 ) k ( X 1 , X 4 ) (h{~{}\wedge\!\!\!\!\!\!\bigcirc~{}}k)(X_{1},X_{2},X_{3},X_{4}):=h(X_{1},X_{3}% )k(X_{2},X_{4})+h(X_{2},X_{4})k(X_{1},X_{3})-h(X_{1},X_{4})k(X_{2},X_{3})-h(X_% {2},X_{3})k(X_{1},X_{4})
  2. h k = k h h{~{}\wedge\!\!\!\!\!\!\bigcirc~{}}k=k{~{}\wedge\!\!\!\!\!\!\bigcirc~{}}h
  3. p = 1 n S 2 ( Ω p M ) , \bigoplus_{p=1}^{n}S^{2}(\Omega^{p}M),
  4. ( α β ) ( γ δ ) = ( α γ ) ( β δ ) (\alpha\cdot\beta){~{}\wedge\!\!\!\!\!\!\bigcirc~{}}(\gamma\cdot\delta)=(% \alpha\wedge\gamma)\cdot(\beta\wedge\delta)
  5. R = k 2 g g R=\frac{k}{2}g{~{}\wedge\!\!\!\!\!\!\bigcirc~{}}g

Kummer_ring.html

  1. [ ζ ] \mathbb{Z}[\zeta]
  2. n 0 + n 1 ζ + n 2 ζ 2 + + n m - 1 ζ m - 1 n_{0}+n_{1}\zeta+n_{2}\zeta^{2}+...+n_{m-1}\zeta^{m-1}
  3. ζ = e 2 π i / m \zeta=e^{2\pi i/m}
  4. \mathbb{Z}
  5. [ ζ ] \mathbb{Z}[\zeta]
  6. [ ζ ] \mathbb{Z}[\zeta]
  7. ϕ ( m ) \phi(m)
  8. { 1 , ζ , ζ 2 , , ζ m - 1 } \{1,\zeta,\zeta^{2},\ldots,\zeta^{m-1}\}

Kundt's_tube.html

  1. c = λ f c=\lambda f\,
  2. c m e t a l c a i r = f λ m e t a l f λ a i r = λ m e t a l λ a i r = L d \frac{c_{metal}}{c_{air}}=\frac{f\lambda_{metal}}{f\lambda_{air}}=\frac{% \lambda_{metal}}{\lambda_{air}}=\frac{L}{d}\,

Kynea_number.html

  1. 4 n + 2 n + 1 - 1 4^{n}+2^{n+1}-1
  2. ( 2 n + 1 ) 2 - 2 (2^{n}+1)^{2}-2
  3. 4 n + i = 0 n 2 i . 4^{n}+\sum_{i=0}^{n}2^{i}.

L-reduction.html

  1. OPT B ( f ( x ) ) α OPT A ( x ) \mathrm{OPT_{B}}(f(x))\leq\alpha\mathrm{OPT_{A}}(x)
  2. | OPT A ( x ) - c A ( g ( y ) ) | β | OPT B ( f ( x ) ) - c B ( y ) | |\mathrm{OPT_{A}}(x)-c_{A}(g(y^{\prime}))|\leq\beta|\mathrm{OPT_{B}}(f(x))-c_{% B}(y^{\prime})|
  3. 1 + δ 1+\delta
  4. c A ( y ) O P T A ( x ) \frac{c_{A}(y)}{OPT_{A}(x)}
  5. c A ( y ) O P T A ( x ) O P T A ( x ) + β ( c B ( y ) - O P T B ( x ) ) O P T A ( x ) \frac{c_{A}(y)}{OPT_{A}(x)}\leq\frac{OPT_{A}(x)+\beta(c_{B}(y^{\prime})-OPT_{B% }(x^{\prime}))}{OPT_{A}(x)}
  6. c A ( y ) O P T A ( x ) 1 + α β ( c B ( y ) - O P T B ( x ) O P T B ( x ) ) \frac{c_{A}(y)}{OPT_{A}(x)}\leq 1+\alpha\beta\left(\frac{c_{B}(y^{\prime})-OPT% _{B}(x^{\prime})}{OPT_{B}(x^{\prime})}\right)
  7. δ \delta
  8. 1 + α β δ 1+\alpha\beta\delta
  9. 1 1 - δ \frac{1}{1-\delta^{\prime}}
  10. c A ( y ) O P T A ( x ) \frac{c_{A}(y)}{OPT_{A}(x)}
  11. c A ( y ) O P T A ( x ) O P T A ( x ) - β ( c B ( y ) - O P T B ( x ) ) O P T A ( x ) \frac{c_{A}(y)}{OPT_{A}(x)}\geq\frac{OPT_{A}(x)-\beta(c_{B}(y^{\prime})-OPT_{B% }(x^{\prime}))}{OPT_{A}(x)}
  12. c A ( y ) O P T A ( x ) 1 - α β ( c B ( y ) - O P T B ( x ) O P T B ( x ) ) \frac{c_{A}(y)}{OPT_{A}(x)}\geq 1-\alpha\beta\left(\frac{c_{B}(y^{\prime})-OPT% _{B}(x^{\prime})}{OPT_{B}(x^{\prime})}\right)
  13. δ \delta^{\prime}
  14. 1 1 - α β δ \frac{1}{1-\alpha\beta\delta^{\prime}}
  15. 1 1 - α β δ = 1 + ϵ \frac{1}{1-\alpha\beta\delta^{\prime}}=1+\epsilon
  16. 1 1 - δ = 1 + ϵ α β ( 1 + ϵ ) - ϵ \frac{1}{1-\delta^{\prime}}=1+\frac{\epsilon}{\alpha\beta(1+\epsilon)-\epsilon}

Lagrange's_theorem_(number_theory).html

  1. f ( x ) [ x ] \textstyle f(x)\in\mathbb{Z}[x]
  2. f ( x ) p 0 f(x)\equiv_{p}0
  3. g ( x ) ( / p ) [ x ] \textstyle g(x)\in(\mathbb{Z}/p)[x]
  4. f ( x ) f(x)
  5. mod p \mod p
  6. f ( k ) f(k)
  7. p p
  8. g ( k ) = 0 g(k)=0
  9. g ( k ) g(k)
  10. g ( x ) = 0 g(x)=0
  11. f ( x ) f(x)
  12. p p
  13. g ( x ) g(x)
  14. deg g ( x ) deg f ( x ) \textstyle\deg g(x)\leq\deg f(x)
  15. g ( k ) g(k)
  16. k k
  17. / p \textstyle\mathbb{Z}/p
  18. f ( k ) mod p f(k)\mod p
  19. g ( k ) = 0 g(k)=0
  20. f ( k ) p 0 f(k)\equiv_{p}0
  21. f ( k ) f(k)
  22. p p
  23. / p \textstyle\mathbb{Z}/p
  24. p p
  25. / p \textstyle\mathbb{Z}/p
  26. f ( k 1 ) , f ( k 2 ) p 0 \textstyle f(k_{1}),f(k_{2})\equiv_{p}0
  27. k 1 p k 2 \textstyle k_{1}\not\equiv_{p}k_{2}
  28. g ( x ) g(x)
  29. deg g ( x ) \deg g(x)
  30. deg f ( x ) \deg f(x)

Lagrangian_Grassmannian.html

  1. Ω ( S p / U ) U / O \Omega(Sp/U)\simeq U/O
  2. Ω ( U / O ) Z × B O \Omega(U/O)\simeq Z\times BO
  3. U / O U/O

Lambert_azimuthal_equal-area_projection.html

  1. 3 \mathbb{R}^{3}
  2. ( x , y , z ) (x,y,z)
  3. ( X , Y ) (X,Y)
  4. ( X , Y ) = ( 2 1 - z x , 2 1 - z y ) , (X,Y)=\left(\sqrt{\frac{2}{1-z}}x,\sqrt{\frac{2}{1-z}}y\right),
  5. ( x , y , z ) = ( 1 - X 2 + Y 2 4 X , 1 - X 2 + Y 2 4 Y , - 1 + X 2 + Y 2 2 ) . (x,y,z)=\left(\sqrt{1-\frac{X^{2}+Y^{2}}{4}}X,\sqrt{1-\frac{X^{2}+Y^{2}}{4}}Y,% -1+\frac{X^{2}+Y^{2}}{2}\right).
  6. ( ϕ , θ ) (\phi,\theta)
  7. ϕ \phi
  8. θ \theta
  9. ( R , Θ ) (R,\Theta)
  10. ( R , Θ ) = ( 2 cos ( ϕ / 2 ) , θ ) , (R,\Theta)=\left(2\cos(\phi/2),\theta\right),
  11. ( ϕ , θ ) = ( 2 arccos ( R / 2 ) , Θ ) . (\phi,\theta)=\left(2\arccos(R/2),\Theta\right).
  12. ( r , θ , z ) (r,\theta,z)
  13. ( R , Θ ) (R,\Theta)
  14. ( R , Θ ) = ( 2 ( 1 + z ) , θ ) , (R,\Theta)=\left(\sqrt{2(1+z)},\theta\right),
  15. ( r , θ , z ) = ( R 1 - R 2 4 , Θ , - 1 + R 2 2 ) . (r,\theta,z)=\left(R\sqrt{1-\frac{R^{2}}{4}},\Theta,-1+\frac{R^{2}}{2}\right).
  16. 2 \sqrt{2}
  17. z < 0 z<0
  18. d A = d X d Y . dA=dX\;dY.
  19. z 0 z\leq 0
  20. z = 0 z=0
  21. 2 \sqrt{2}
  22. z 0 z\leq 0
  23. z 0 z\geq 0

Lambert_cylindrical_equal-area_projection.html

  1. x = λ - λ 0 y = sin φ \begin{aligned}\displaystyle x&\displaystyle=\lambda-\lambda_{0}\\ \displaystyle y&\displaystyle=\sin\varphi\end{aligned}
  2. φ \scriptstyle\varphi\,
  3. λ \scriptstyle\lambda\,
  4. λ 0 \scriptstyle\lambda_{0}\,

Lamm_equation.html

  1. c t = D [ ( 2 c r 2 ) + 1 r ( c r ) ] - s ω 2 [ r ( c r ) + 2 c ] \frac{\partial c}{\partial t}=D\left[\left(\frac{\partial^{2}c}{\partial r^{2}% }\right)+\frac{1}{r}\left(\frac{\partial c}{\partial r}\right)\right]-s\omega^% {2}\left[r\left(\frac{\partial c}{\partial r}\right)+2c\right]
  2. s D = m b k B T \frac{s}{D}=\frac{m_{b}}{k_{B}T}
  3. D ( c r ) - s ω 2 r c = 0 D\left(\frac{\partial c}{\partial r}\right)-s\omega^{2}rc=0

Lamport_signature.html

  1. k k
  2. P = { 0 , 1 } k P=\{0,1\}^{k}
  3. f : Y Z f:\,Y\rightarrow Z
  4. 1 i k 1\leq i\leq k
  5. j { 0 , 1 } j\in\{0,1\}
  6. y i , j Y y_{i,j}\in Y
  7. z i , j = f ( y i , j ) z_{i,j}=f(y_{i,j})
  8. K K
  9. 2 k 2k
  10. y i , j y_{i,j}
  11. 2 k 2k
  12. z i , j z_{i,j}
  13. m = m 1 m k { 0 , 1 } k m=m_{1}\ldots m_{k}\in\{0,1\}^{k}
  14. s i g ( m 1 m k ) = ( y 1 , m 1 , , y k , m k ) = ( s 1 , , s k ) sig(m_{1}\ldots m_{k})=(y_{1,m_{1}},\ldots,y_{k,m_{k}})=(s_{1},\ldots,s_{k})
  15. f ( s i ) = z i , m i f(s_{i})=z_{i,m_{i}}
  16. 1 i k 1\leq i\leq k
  17. f f
  18. 2 k 2k
  19. z i j z_{ij}
  20. ( z i j ) (z_{ij})
  21. j m i j\neq m_{i}
  22. 2 * chunk size in bits 2*\text{chunk size in bits}
  23. ( 2 chunk size in bits ) / ( chunk size in bits ) (2^{\text{chunk size in bits}})/(\text{chunk size in bits})

Landau_prime_ideal_theorem.html

  1. 2 r ( X ) + r ( X ) 2r(X)+r^{\prime}(\sqrt{X})
  2. Y 2 log Y . \frac{Y}{2\log Y}.
  3. X log X . \frac{X}{\log X}.
  4. X log X \frac{X}{\log X}
  5. Li ( X ) + O K ( X exp ( - c K log ( X ) ) , \mathrm{Li}(X)+O_{K}(X\exp(-c_{K}\sqrt{\log(X)}),\,

Laplace_expansion.html

  1. C i j = ( - 1 ) i + j M i j , C_{ij}\ =(-1)^{i+j}M_{ij}\,,
  2. | B | \displaystyle|B|
  3. B = [ 1 2 3 4 5 6 7 8 9 ] . B=\begin{bmatrix}1&2&3\\ 4&5&6\\ 7&8&9\end{bmatrix}.
  4. | B | = 1 | 5 6 8 9 | - 2 | 4 6 7 9 | + 3 | 4 5 7 8 | |B|=1\cdot\begin{vmatrix}5&6\\ 8&9\end{vmatrix}-2\cdot\begin{vmatrix}4&6\\ 7&9\end{vmatrix}+3\cdot\begin{vmatrix}4&5\\ 7&8\end{vmatrix}
  5. = 1 ( - 3 ) - 2 ( - 6 ) + 3 ( - 3 ) = 0. {}=1\cdot(-3)-2\cdot(-6)+3\cdot(-3)=0.
  6. | B | = - 2 | 4 6 7 9 | + 5 | 1 3 7 9 | - 8 | 1 3 4 6 | |B|=-2\cdot\begin{vmatrix}4&6\\ 7&9\end{vmatrix}+5\cdot\begin{vmatrix}1&3\\ 7&9\end{vmatrix}-8\cdot\begin{vmatrix}1&3\\ 4&6\end{vmatrix}
  7. = - 2 ( - 6 ) + 5 ( - 12 ) - 8 ( - 6 ) = 0. {}=-2\cdot(-6)+5\cdot(-12)-8\cdot(-6)=0.
  8. B B
  9. i , j { 1 , 2 , , n } . i,j\in\{1,2,\dots,n\}.
  10. B B
  11. i , j i,j
  12. M i j M_{ij}
  13. ( a s t ) (a_{st})
  14. 1 s , t n - 1. 1\leq s,t\leq n-1.
  15. | B | |B|
  16. b i j b_{ij}
  17. sgn τ b 1 , τ ( 1 ) b i , j b n , τ ( n ) = sgn τ b i j a 1 , σ ( 1 ) a n - 1 , σ ( n - 1 ) \operatorname{sgn}\tau\,b_{1,\tau(1)}\cdots b_{i,j}\cdots b_{n,\tau(n)}=% \operatorname{sgn}\tau\,b_{ij}a_{1,\sigma(1)}\cdots a_{n-1,\sigma(n-1)}
  18. τ ( i ) = j \tau(i)=j
  19. σ S n - 1 \sigma\in S_{n-1}
  20. τ τ
  21. σ σ
  22. τ τ
  23. σ τ \sigma\leftrightarrow\tau
  24. S n - 1 S_{n-1}
  25. { τ S n : τ ( i ) = j } . \{\tau\in S_{n}\colon\tau(i)=j\}.
  26. τ τ
  27. σ σ
  28. σ S n \sigma^{\prime}\in S_{n}
  29. σ ( k ) = σ ( k ) \sigma^{\prime}(k)=\sigma(k)
  30. 1 k n - 1 1\leq k\leq n-1
  31. σ ( n ) = n \sigma^{\prime}(n)=n
  32. sgn σ = sgn σ \operatorname{sgn}\sigma^{\prime}=\operatorname{sgn}\sigma
  33. σ = ( n , n - 1 , , j ) τ ( i , i + 1 , , n ) \sigma^{\prime}\,=(n,n-1,\ldots,j)\tau(i,i+1,\ldots,n)
  34. n - i n-i
  35. n - j n-j
  36. sgn τ = ( - 1 ) 2 n - ( i + j ) sgn σ = ( - 1 ) i + j sgn σ . \operatorname{sgn}\tau\,=(-1)^{2n-(i+j)}\operatorname{sgn}\sigma^{\prime}\,=(-% 1)^{i+j}\operatorname{sgn}\sigma.
  37. σ τ \sigma\leftrightarrow\tau
  38. τ S n : τ ( i ) = j sgn τ b 1 , τ ( 1 ) b n , τ ( n ) = σ S n - 1 ( - 1 ) i + j sgn σ b i j a 1 , σ ( 1 ) a n - 1 , σ ( n - 1 ) = b i j ( - 1 ) i + j | M i j | \begin{aligned}\displaystyle\sum_{\tau\in S_{n}:\tau(i)=j}\operatorname{sgn}% \tau\,b_{1,\tau(1)}\cdots b_{n,\tau(n)}&\displaystyle=\sum_{\sigma\in S_{n-1}}% (-1)^{i+j}\operatorname{sgn}\sigma\,b_{ij}a_{1,\sigma(1)}\cdots a_{n-1,\sigma(% n-1)}\\ &\displaystyle=b_{ij}(-1)^{i+j}\left|M_{ij}\right|\end{aligned}
  39. A = [ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ] . A=\begin{bmatrix}1&2&3&4\\ 5&6&7&8\\ 9&10&11&12\\ 13&14&15&16\end{bmatrix}.
  40. S = { { 1 , 2 } , { 1 , 3 } , { 1 , 4 } , { 2 , 3 } , { 2 , 4 } , { 3 , 4 } } S=\left\{\{1,2\},\{1,3\},\{1,4\},\{2,3\},\{2,4\},\{3,4\}\right\}
  41. b { j , k } = | a 1 j a 1 k a 2 j a 2 k | b_{\{j,k\}}=\begin{vmatrix}a_{1j}&a_{1k}\\ a_{2j}&a_{2k}\end{vmatrix}
  42. c { j , k } = | a 3 j a 3 k a 4 j a 4 k | c_{\{j,k\}}=\begin{vmatrix}a_{3j}&a_{3k}\\ a_{4j}&a_{4k}\end{vmatrix}
  43. ε { i , j } , { p , q } = sgn [ 1 2 3 4 i j p q ] \varepsilon^{\{i,j\},\{p,q\}}=\mbox{sgn}~{}\begin{bmatrix}1&2&3&4\\ i&j&p&q\end{bmatrix}
  44. | A | = H S ε H , H b H c H , |A|=\sum_{H\in S}\varepsilon^{H,H^{\prime}}b_{H}c_{H^{\prime}},
  45. H H^{\prime}
  46. H H
  47. | A | = b { 1 , 2 } c { 3 , 4 } - b { 1 , 3 } c { 2 , 4 } + b { 1 , 4 } c { 2 , 3 } + b { 2 , 3 } c { 1 , 4 } - b { 2 , 4 } c { 1 , 3 } + b { 3 , 4 } c { 1 , 2 } = | 1 2 5 6 | | 11 12 15 16 | - | 1 3 5 7 | | 10 12 14 16 | + | 1 4 5 8 | | 10 11 14 15 | + | 2 3 6 7 | | 9 12 13 16 | - | 2 4 6 8 | | 9 11 13 15 | + | 3 4 7 8 | | 9 10 13 14 | = - 4 ( - 4 ) - ( - 8 ) ( - 8 ) + ( - 12 ) ( - 4 ) + ( - 4 ) ( - 12 ) - ( - 8 ) ( - 8 ) + ( - 4 ) ( - 4 ) = 16 - 64 + 48 + 48 - 64 + 16 = 0. \begin{aligned}\displaystyle|A|&\displaystyle=b_{\{1,2\}}c_{\{3,4\}}-b_{\{1,3% \}}c_{\{2,4\}}+b_{\{1,4\}}c_{\{2,3\}}+b_{\{2,3\}}c_{\{1,4\}}-b_{\{2,4\}}c_{\{1% ,3\}}+b_{\{3,4\}}c_{\{1,2\}}\\ &\displaystyle=\begin{vmatrix}1&2\\ 5&6\end{vmatrix}\cdot\begin{vmatrix}11&12\\ 15&16\end{vmatrix}-\begin{vmatrix}1&3\\ 5&7\end{vmatrix}\cdot\begin{vmatrix}10&12\\ 14&16\end{vmatrix}+\begin{vmatrix}1&4\\ 5&8\end{vmatrix}\cdot\begin{vmatrix}10&11\\ 14&15\end{vmatrix}+\begin{vmatrix}2&3\\ 6&7\end{vmatrix}\cdot\begin{vmatrix}9&12\\ 13&16\end{vmatrix}-\begin{vmatrix}2&4\\ 6&8\end{vmatrix}\cdot\begin{vmatrix}9&11\\ 13&15\end{vmatrix}+\begin{vmatrix}3&4\\ 7&8\end{vmatrix}\cdot\begin{vmatrix}9&10\\ 13&14\end{vmatrix}\\ &\displaystyle=-4\cdot(-4)-(-8)\cdot(-8)+(-12)\cdot(-4)+(-4)\cdot(-12)-(-8)% \cdot(-8)+(-4)\cdot(-4)\\ &\displaystyle=16-64+48+48-64+16=0.\end{aligned}

Large_countable_ordinal.html

  1. ω α = α \omega^{\alpha}=\alpha
  2. ω \omega
  3. ω ω \omega^{\omega}
  4. ω ω ω \omega^{\omega^{\omega}}
  5. ε 0 + 1 , ω ε 0 + 1 = ε 0 ω , ω ω ε 0 + 1 = ( ε 0 ) ω , etc. \varepsilon_{0}+1,\qquad\omega^{\varepsilon_{0}+1}=\varepsilon_{0}\cdot\omega,% \qquad\omega^{\omega^{\varepsilon_{0}+1}}=(\varepsilon_{0})^{\omega},\qquad\,% \text{etc.}
  6. ι \iota
  7. ω α = α \omega^{\alpha}=\alpha
  8. ε ι \varepsilon_{\iota}
  9. ζ 0 \zeta_{0}
  10. ε α = α \varepsilon_{\alpha}=\alpha
  11. φ γ ( β ) \varphi_{\gamma}(\beta)
  12. φ 0 ( β ) = ω β \varphi_{0}(\beta)=\omega^{\beta}
  13. φ γ + 1 ( β ) \varphi_{\gamma+1}(\beta)
  14. β \beta
  15. φ γ \varphi_{\gamma}
  16. β \beta
  17. φ γ ( α ) = α \varphi_{\gamma}(\alpha)=\alpha
  18. φ 1 ( β ) = ε β \varphi_{1}(\beta)=\varepsilon_{\beta}
  19. δ \delta
  20. φ δ ( α ) \varphi_{\delta}(\alpha)
  21. α \alpha
  22. φ γ \varphi_{\gamma}
  23. γ < δ \gamma<\delta
  24. δ \delta
  25. φ δ ( α ) \varphi_{\delta}(\alpha)
  26. φ γ ( α ) \varphi_{\gamma}(\alpha)
  27. γ < δ \gamma<\delta
  28. φ γ \varphi_{\gamma}
  29. γ t h \gamma^{th}
  30. ω \omega
  31. φ α ( β ) < φ γ ( δ ) \varphi_{\alpha}(\beta)<\varphi_{\gamma}(\delta)
  32. α = γ \alpha=\gamma
  33. β < δ \beta<\delta
  34. α < γ \alpha<\gamma
  35. β < φ γ ( δ ) \beta<\varphi_{\gamma}(\delta)
  36. α > γ \alpha>\gamma
  37. φ α ( β ) < δ \varphi_{\alpha}(\beta)<\delta
  38. φ α ( 0 ) = α \varphi_{\alpha}(0)=\alpha
  39. Γ 0 \Gamma_{0}
  40. α Γ α \alpha\mapsto\Gamma_{\alpha}
  41. ω 1 CK \omega_{1}^{\mathrm{CK}}
  42. ω 1 CK \omega_{1}^{\mathrm{CK}}
  43. ω 1 \omega_{1}
  44. ω 1 \omega_{1}
  45. L α L_{\alpha}
  46. ω 1 CK \omega_{1}^{\mathrm{CK}}
  47. ω α CK \omega_{\alpha}^{\mathrm{CK}}
  48. α \alpha
  49. α \alpha
  50. α \alpha
  51. α \alpha
  52. α \alpha
  53. α \alpha
  54. α \alpha
  55. α \alpha
  56. α \alpha
  57. L α L_{\alpha}
  58. Σ 1 \Sigma_{1}
  59. α \alpha
  60. L α L_{\alpha}
  61. α \alpha
  62. L α L_{\alpha}
  63. ω 1 CK \omega_{1}^{\mathrm{CK}}

Large_deviations_theory.html

  1. X i X_{i}
  2. M N M_{N}
  3. N N
  4. M N = 1 N i = 1 N X i . M_{N}=\frac{1}{N}\sum_{i=1}^{N}X_{i}.
  5. M N M_{N}
  6. M N M_{N}
  7. 0.5 = E [ X 1 ] 0.5=\operatorname{E}[X_{1}]
  8. M N M_{N}
  9. N N
  10. M N M_{N}
  11. M N M_{N}
  12. P ( M N > x ) P(M_{N}>x)
  13. M N M_{N}
  14. x x
  15. N N
  16. x x
  17. E [ X 1 ] \operatorname{E}[X_{1}]
  18. N N\to\infty
  19. 0.5 < x < 1 0.5<x<1
  20. P ( M N > x ) P(M_{N}>x)
  21. I ( x ) = x ln x + ( 1 - x ) ln ( 1 - x ) + ln 2. I(x)=x\,\,\text{ln}x+(1-x)\,\,\text{ln}(1-x)+\,\text{ln}2.
  22. I ( x ) I(x)
  23. P ( M N > x ) < exp ( - N I ( x ) ) P(M_{N}>x)<\exp(-NI(x))
  24. I ( x ) I(x)
  25. N N
  26. 1 / N 1/\sqrt{N}
  27. P ( M N > x ) exp ( - N I ( x ) ) . P(M_{N}>x)\approx\exp(-NI(x)).
  28. P ( M N > x ) P(M_{N}>x)
  29. N N
  30. X , X 1 , X 2 , X,X_{1},X_{2},...
  31. lim N 1 N ln P ( M N > x ) = - I ( x ) . \lim_{N\to\infty}\frac{1}{N}\ln P(M_{N}>x)=-I(x).
  32. I ( ) I(\cdot)
  33. N N
  34. P ( M N > x ) exp [ - N I ( x ) ] , P(M_{N}>x)\approx\exp[-NI(x)],
  35. X X
  36. I ( x ) = sup θ > 0 [ θ x - λ ( θ ) ] , I(x)=\sup_{\theta>0}[\theta x-\lambda(\theta)],
  37. λ ( θ ) = ln E [ exp ( θ X ) ] \lambda(\theta)=\ln\operatorname{E}[\exp(\theta X)]
  38. E \operatorname{E}
  39. X X
  40. { X i } \{X_{i}\}
  41. 𝒳 \mathcal{X}
  42. { N } \{\mathbb{P}_{N}\}
  43. 𝒳 \mathcal{X}
  44. { a N } \{a_{N}\}
  45. lim N a N = + \lim_{N}a_{N}=+\infty
  46. I : 𝒳 [ 0 , + ] I:\mathcal{X}\to[0,+\infty]
  47. 𝒳 \mathcal{X}
  48. { N } \{\mathbb{P}_{N}\}
  49. { a n } \{a_{n}\}
  50. I I
  51. E 𝒳 E\subset\mathcal{X}
  52. - inf x E I ( x ) lim ¯ N a N - 1 log ( N ( E ) ) lim ¯ N a N - 1 log ( N ( E ) ) - inf x E ¯ I ( x ) , -\inf_{x\in E^{\circ}}I(x)\leq\underline{\lim}_{N}a_{N}^{-1}\log\big(\mathbb{P% }_{N}(E)\big)\leq\overline{\lim}_{N}a_{N}^{-1}\log\big(\mathbb{P}_{N}(E)\big)% \leq-\inf_{x\in\bar{E}}I(x),
  53. E ¯ \bar{E}
  54. E E^{\circ}
  55. E E
  56. q q
  57. N N
  58. C = Σ X i C=\Sigma X_{i}
  59. N q Nq
  60. M N M_{N}
  61. M N M_{N}

Latent_class_model.html

  1. p i 1 , i 2 , , i N t T p t n N p i n , t n , p_{i_{1},i_{2},\ldots,i_{N}}\approx\sum_{t}^{T}p_{t}\,\prod_{n}^{N}p^{n}_{i_{n% },t},
  2. p i n , t n p^{n}_{i_{n},t}
  3. p i j t T p t p i t p j t . p_{ij}\approx\sum_{t}^{T}p_{t}\,p_{it}\,p_{jt}.

Latent_Dirichlet_allocation.html

  1. θ i \theta_{i}
  2. ϕ k \phi_{k}
  3. z i j z_{ij}
  4. w i j w_{ij}
  5. w i j w_{ij}
  6. ϕ \phi
  7. D D
  8. M M
  9. N i N_{i}
  10. θ i Dir ( α ) \theta_{i}\,\sim\,\mathrm{Dir}(\alpha)
  11. i { 1 , , M } i\in\{1,\dots,M\}
  12. Dir ( α ) \mathrm{Dir}(\alpha)
  13. α \alpha
  14. ϕ k Dir ( β ) \phi_{k}\,\sim\,\mathrm{Dir}(\beta)
  15. k { 1 , , K } k\in\{1,\dots,K\}
  16. i , j i,j
  17. j { 1 , , N i } j\in\{1,\dots,N_{i}\}
  18. i { 1 , , M } i\in\{1,\dots,M\}
  19. z i , j Multinomial ( θ i ) . z_{i,j}\,\sim\,\mathrm{Multinomial}(\theta_{i}).
  20. w i , j Multinomial ( ϕ z i , j ) w_{i,j}\,\sim\,\mathrm{Multinomial}(\phi_{z_{i,j}})
  21. N i N_{i}
  22. q q
  23. z z
  24. K K
  25. V V
  26. M M
  27. N d = 1 M N_{d=1\dots M}
  28. N N
  29. N d N_{d}
  30. N = d = 1 M N d N=\sum_{d=1}^{M}N_{d}
  31. α k = 1 K \alpha_{k=1\dots K}
  32. s y m b o l α symbol\alpha
  33. α k \alpha_{k}
  34. β w = 1 V \beta_{w=1\dots V}
  35. s y m b o l β symbol\beta
  36. β w \beta_{w}
  37. ϕ k = 1 K , w = 1 V \phi_{k=1\dots K,w=1\dots V}
  38. s y m b o l ϕ k = 1 K symbol\phi_{k=1\dots K}
  39. θ d = 1 M , k = 1 K \theta_{d=1\dots M,k=1\dots K}
  40. s y m b o l θ d = 1 M symbol\theta_{d=1\dots M}
  41. z d = 1 M , w = 1 N d z_{d=1\dots M,w=1\dots N_{d}}
  42. 𝐙 \mathbf{Z}
  43. w d = 1 M , w = 1 N d w_{d=1\dots M,w=1\dots N_{d}}
  44. 𝐖 \mathbf{W}
  45. s y m b o l ϕ k = 1 K Dirichlet V ( s y m b o l β ) s y m b o l θ d = 1 M Dirichlet K ( s y m b o l α ) z d = 1 M , w = 1 N d Categorical K ( s y m b o l θ d ) w d = 1 M , w = 1 N d Categorical V ( s y m b o l ϕ z d w ) \begin{array}[]{lcl}symbol\phi_{k=1\dots K}&\sim&\operatorname{Dirichlet}_{V}(% symbol\beta)\\ symbol\theta_{d=1\dots M}&\sim&\operatorname{Dirichlet}_{K}(symbol\alpha)\\ z_{d=1\dots M,w=1\dots N_{d}}&\sim&\operatorname{Categorical}_{K}(symbol\theta% _{d})\\ w_{d=1\dots M,w=1\dots N_{d}}&\sim&\operatorname{Categorical}_{V}(symbol\phi_{% z_{dw}})\\ \end{array}
  46. φ \varphi
  47. θ \theta
  48. N N
  49. P ( s y m b o l W , s y m b o l Z , s y m b o l θ , s y m b o l φ ; α , β ) = i = 1 K P ( φ i ; β ) j = 1 M P ( θ j ; α ) t = 1 N P ( Z j , t | θ j ) P ( W j , t | φ Z j , t ) , P(symbol{W},symbol{Z},symbol{\theta},symbol{\varphi};\alpha,\beta)=\prod_{i=1}% ^{K}P(\varphi_{i};\beta)\prod_{j=1}^{M}P(\theta_{j};\alpha)\prod_{t=1}^{N}P(Z_% {j,t}|\theta_{j})P(W_{j,t}|\varphi_{Z_{j,t}}),
  50. s y m b o l φ symbol{\varphi}
  51. s y m b o l θ symbol{\theta}
  52. P ( s y m b o l Z , s y m b o l W ; α , β ) = s y m b o l θ s y m b o l φ P ( s y m b o l W , s y m b o l Z , s y m b o l θ , s y m b o l φ ; α , β ) d s y m b o l φ d s y m b o l θ = s y m b o l φ i = 1 K P ( φ i ; β ) j = 1 M t = 1 N P ( W j , t φ Z j , t ) d s y m b o l φ s y m b o l θ j = 1 M P ( θ j ; α ) t = 1 N P ( Z j , t θ j ) d s y m b o l θ . \begin{aligned}&\displaystyle P(symbol{Z},symbol{W};\alpha,\beta)=\int_{symbol% {\theta}}\int_{symbol{\varphi}}P(symbol{W},symbol{Z},symbol{\theta},symbol{% \varphi};\alpha,\beta)\,dsymbol{\varphi}\,dsymbol{\theta}\\ \displaystyle=&\displaystyle\int_{symbol{\varphi}}\prod_{i=1}^{K}P(\varphi_{i}% ;\beta)\prod_{j=1}^{M}\prod_{t=1}^{N}P(W_{j,t}\mid\varphi_{Z_{j,t}})\,dsymbol{% \varphi}\int_{symbol{\theta}}\prod_{j=1}^{M}P(\theta_{j};\alpha)\prod_{t=1}^{N% }P(Z_{j,t}\mid\theta_{j})\,dsymbol{\theta}.\end{aligned}
  53. θ \theta
  54. φ \varphi
  55. θ \theta
  56. φ \varphi
  57. θ \theta
  58. s y m b o l θ j = 1 M P ( θ j ; α ) t = 1 N P ( Z j , t θ j ) d s y m b o l θ = j = 1 M θ j P ( θ j ; α ) t = 1 N P ( Z j , t θ j ) d θ j . \int_{symbol{\theta}}\prod_{j=1}^{M}P(\theta_{j};\alpha)\prod_{t=1}^{N}P(Z_{j,% t}\mid\theta_{j})dsymbol{\theta}=\prod_{j=1}^{M}\int_{\theta_{j}}P(\theta_{j};% \alpha)\prod_{t=1}^{N}P(Z_{j,t}\mid\theta_{j})\,d\theta_{j}.
  59. θ \theta
  60. θ j P ( θ j ; α ) t = 1 N P ( Z j , t θ j ) d θ j . \int_{\theta_{j}}P(\theta_{j};\alpha)\prod_{t=1}^{N}P(Z_{j,t}\mid\theta_{j})\,% d\theta_{j}.
  61. j t h j^{th}
  62. θ j P ( θ j ; α ) t = 1 N P ( Z j , t θ j ) d θ j = θ j Γ ( i = 1 K α i ) i = 1 K Γ ( α i ) i = 1 K θ j , i α i - 1 t = 1 N P ( Z j , t θ j ) d θ j . \begin{aligned}&\displaystyle\int_{\theta_{j}}P(\theta_{j};\alpha)\prod_{t=1}^% {N}P(Z_{j,t}\mid\theta_{j})\,d\theta_{j}=&\displaystyle\int_{\theta_{j}}\frac{% \Gamma\bigl(\sum_{i=1}^{K}\alpha_{i}\bigr)}{\prod_{i=1}^{K}\Gamma(\alpha_{i})}% \prod_{i=1}^{K}\theta_{j,i}^{\alpha_{i}-1}\prod_{t=1}^{N}P(Z_{j,t}\mid\theta_{% j})\,d\theta_{j}.\end{aligned}
  63. n j , r i n_{j,r}^{i}
  64. j t h j^{th}
  65. r t h r^{th}
  66. i t h i^{th}
  67. n j , r i n_{j,r}^{i}
  68. ( ) (\cdot)
  69. n j , ( ) i n_{j,(\cdot)}^{i}
  70. j t h j^{th}
  71. i t h i^{th}
  72. t = 1 N P ( Z j , t θ j ) = i = 1 K θ j , i n j , ( ) i . \prod_{t=1}^{N}P(Z_{j,t}\mid\theta_{j})=\prod_{i=1}^{K}\theta_{j,i}^{n_{j,(% \cdot)}^{i}}.
  73. θ j \theta_{j}
  74. θ j Γ ( i = 1 K α i ) i = 1 K Γ ( α i ) i = 1 K θ j , i α i - 1 i = 1 K θ j , i n j , ( ) i d θ j \displaystyle\int_{\theta_{j}}\frac{\Gamma\bigl(\sum_{i=1}^{K}\alpha_{i}\bigr)% }{\prod_{i=1}^{K}\Gamma(\alpha_{i})}\prod_{i=1}^{K}\theta_{j,i}^{\alpha_{i}-1}% \prod_{i=1}^{K}\theta_{j,i}^{n_{j,(\cdot)}^{i}}\,d\theta_{j}
  75. θ j Γ ( i = 1 K n j , ( ) i + α i ) i = 1 K Γ ( n j , ( ) i + α i ) i = 1 K θ j , i n j , ( ) i + α i - 1 d θ j = 1. \int_{\theta_{j}}\frac{\Gamma\bigl(\sum_{i=1}^{K}n_{j,(\cdot)}^{i}+\alpha_{i}% \bigr)}{\prod_{i=1}^{K}\Gamma(n_{j,(\cdot)}^{i}+\alpha_{i})}\prod_{i=1}^{K}% \theta_{j,i}^{n_{j,(\cdot)}^{i}+\alpha_{i}-1}\,d\theta_{j}=1.
  76. θ j P ( θ j ; α ) t = 1 N P ( Z j , t θ j ) d θ j = θ j Γ ( i = 1 K α i ) i = 1 K Γ ( α i ) i = 1 K θ j , i n j , ( ) i + α i - 1 d θ j = Γ ( i = 1 K α i ) i = 1 K Γ ( α i ) i = 1 K Γ ( n j , ( ) i + α i ) Γ ( i = 1 K n j , ( ) i + α i ) θ j Γ ( i = 1 K n j , ( ) i + α i ) i = 1 K Γ ( n j , ( ) i + α i ) i = 1 K θ j , i n j , ( ) i + α i - 1 d θ j = Γ ( i = 1 K α i ) i = 1 K Γ ( α i ) i = 1 K Γ ( n j , ( ) i + α i ) Γ ( i = 1 K n j , ( ) i + α i ) . \begin{aligned}&\displaystyle\int_{\theta_{j}}P(\theta_{j};\alpha)\prod_{t=1}^% {N}P(Z_{j,t}\mid\theta_{j})\,d\theta_{j}=\int_{\theta_{j}}\frac{\Gamma\bigl(% \sum_{i=1}^{K}\alpha_{i}\bigr)}{\prod_{i=1}^{K}\Gamma(\alpha_{i})}\prod_{i=1}^% {K}\theta_{j,i}^{n_{j,(\cdot)}^{i}+\alpha_{i}-1}\,d\theta_{j}\\ \displaystyle=&\displaystyle\frac{\Gamma\bigl(\sum_{i=1}^{K}\alpha_{i}\bigr)}{% \prod_{i=1}^{K}\Gamma(\alpha_{i})}\frac{\prod_{i=1}^{K}\Gamma(n_{j,(\cdot)}^{i% }+\alpha_{i})}{\Gamma\bigl(\sum_{i=1}^{K}n_{j,(\cdot)}^{i}+\alpha_{i}\bigr)}% \int_{\theta_{j}}\frac{\Gamma\bigl(\sum_{i=1}^{K}n_{j,(\cdot)}^{i}+\alpha_{i}% \bigr)}{\prod_{i=1}^{K}\Gamma(n_{j,(\cdot)}^{i}+\alpha_{i})}\prod_{i=1}^{K}% \theta_{j,i}^{n_{j,(\cdot)}^{i}+\alpha_{i}-1}\,d\theta_{j}\\ \displaystyle=&\displaystyle\frac{\Gamma\bigl(\sum_{i=1}^{K}\alpha_{i}\bigr)}{% \prod_{i=1}^{K}\Gamma(\alpha_{i})}\frac{\prod_{i=1}^{K}\Gamma(n_{j,(\cdot)}^{i% }+\alpha_{i})}{\Gamma\bigl(\sum_{i=1}^{K}n_{j,(\cdot)}^{i}+\alpha_{i}\bigr)}.% \end{aligned}
  77. s y m b o l φ symbol{\varphi}
  78. s y m b o l φ symbol{\varphi}
  79. s y m b o l θ symbol{\theta}
  80. s y m b o l φ i = 1 K P ( φ i ; β ) j = 1 M t = 1 N P ( W j , t φ Z j , t ) d s y m b o l φ \displaystyle\int_{symbol{\varphi}}\prod_{i=1}^{K}P(\varphi_{i};\beta)\prod_{j% =1}^{M}\prod_{t=1}^{N}P(W_{j,t}\mid\varphi_{Z_{j,t}})\,dsymbol{\varphi}
  81. s y m b o l ϕ symbol{\phi}
  82. s y m b o l θ symbol{\theta}
  83. P ( s y m b o l Z , s y m b o l W ; α , β ) \displaystyle P(symbol{Z},symbol{W};\alpha,\beta)
  84. P ( s y m b o l Z \midsymbol W ; α , β ) P(symbol{Z}\midsymbol{W};\alpha,\beta)
  85. P ( s y m b o l W ; α , β ) P(symbol{W};\alpha,\beta)
  86. P ( s y m b o l Z , s y m b o l W ; α , β ) P(symbol{Z},symbol{W};\alpha,\beta)
  87. P ( Z ( m , n ) \midsymbol Z - ( m , n ) , s y m b o l W ; α , β ) = P ( Z ( m , n ) , s y m b o l Z - ( m , n ) , s y m b o l W ; α , β ) P ( s y m b o l Z - ( m , n ) , s y m b o l W ; α , β ) , P(Z_{(m,n)}\midsymbol{Z_{-(m,n)}},symbol{W};\alpha,\beta)=\frac{P(Z_{(m,n)},% symbol{Z_{-(m,n)}},symbol{W};\alpha,\beta)}{P(symbol{Z_{-(m,n)}},symbol{W};% \alpha,\beta)},
  88. Z ( m , n ) Z_{(m,n)}
  89. Z Z
  90. n t h n^{th}
  91. m t h m^{th}
  92. v t h v^{th}
  93. s y m b o l Z - ( m , n ) symbol{Z_{-(m,n)}}
  94. Z Z
  95. Z ( m , n ) Z_{(m,n)}
  96. Z ( m , n ) Z_{(m,n)}
  97. P ( Z m , n \midsymbol Z - ( m , n ) , s y m b o l W ; α , β ) P(Z_{m,n}\midsymbol{Z_{-(m,n)}},symbol{W};\alpha,\beta)
  98. Z ( m , n ) Z_{(m,n)}
  99. P ( Z ( m , n ) = k \midsymbol Z - ( m , n ) , s y m b o l W ; α , β ) \displaystyle P(Z_{(m,n)}=k\midsymbol{Z_{-(m,n)}},symbol{W};\alpha,\beta)
  100. n j , r i , - ( m , n ) n_{j,r}^{i,-(m,n)}
  101. n j , r i n_{j,r}^{i}
  102. Z ( m , n ) Z_{(m,n)}
  103. k k
  104. \displaystyle\propto
  105. K d K_{d}
  106. K w K_{w}
  107. p ( Z d , n = k ) α β C k ¬ n + V β + C k d β C k ¬ n + V β + C k w ( α + C k d ) C k ¬ n + V β p(Z_{d,n}=k)\propto\frac{\alpha\beta}{C_{k}^{\neg n}+V\beta}+\frac{C_{k}^{d}% \beta}{C_{k}^{\neg n}+V\beta}+\frac{C_{k}^{w}(\alpha+C_{k}^{d})}{C_{k}^{\neg n% }+V\beta}
  108. a , b a,b
  109. c c
  110. A = k = 1 K α β C k ¬ n + V β A=\sum_{k=1}^{K}\frac{\alpha\beta}{C_{k}^{\neg n}+V\beta}
  111. B = k = 1 K C k d β C k ¬ n + V β B=\sum_{k=1}^{K}\frac{C_{k}^{d}\beta}{C_{k}^{\neg n}+V\beta}
  112. C = k = 1 K C k w ( α + C k d ) C k ¬ n + V β C=\sum_{k=1}^{K}\frac{C_{k}^{w}(\alpha+C_{k}^{d})}{C_{k}^{\neg n}+V\beta}
  113. B B
  114. d d
  115. C C
  116. w w
  117. A A
  118. α \alpha
  119. β \beta
  120. s U ( s | A + B + C ) s\sim U(s|A+B+C)
  121. A A
  122. B B
  123. O ( K d ) O(K_{d})
  124. O ( K w + K d ) O(K_{w}+K_{d})
  125. O ( 1 ) O(1)
  126. d d
  127. Pr ( w z ) \Pr(w\mid z)
  128. Pr ( z d ) \Pr(z\mid d)
  129. d d

Lattice_constant.html

  1. V = a b c 1 + 2 cos ( α ) cos ( β ) cos ( γ ) - cos 2 ( α ) - cos 2 ( β ) - cos 2 ( γ ) . V=abc\sqrt{1+2\cos(\alpha)\cos(\beta)\cos(\gamma)-\cos^{2}(\alpha)-\cos^{2}(% \beta)-\cos^{2}(\gamma)}.
  2. V = a b c sin ( β ) . V=abc\sin(\beta).

Lattice_energy.html

  1. Δ G U = Δ G H - p Δ V m \Delta_{G}U=\Delta_{G}H-p\Delta V_{m}
  2. Δ G U \Delta_{G}U
  3. Δ G H \Delta_{G}H
  4. Δ V m \Delta V_{m}
  5. p p
  6. E = - N A M z + z - q e 2 4 π ε 0 r 0 ( 1 - 1 n ) , E=-\frac{N_{A}Mz^{+}z^{-}q_{e}^{2}}{4\pi\varepsilon_{0}r_{0}}\left(1-\frac{1}{% n}\right),

Lattice_of_subgroups.html

  1. G G
  2. G G
  3. 𝐙 / 2 𝐙 * 𝐙 / 2 𝐙 \mathbf{Z}/2\mathbf{Z}*\mathbf{Z}/2\mathbf{Z}

Lattice_reduction.html

  1. n n
  2. B B
  3. b i , i = 1 , , n b_{i},i=1,\ldots,n
  4. det ( B ) \det(B)
  5. det ( B T B ) \sqrt{\det(B^{T}B)}
  6. Λ \Lambda
  7. det ( Λ ) \det(\Lambda)
  8. d ( Λ ) d(\Lambda)
  9. δ ( B ) = Π i = 1 n || b i || det ( B T B ) = Π i = 1 n || b i || d ( Λ ) \delta(B)=\frac{\Pi_{i=1}^{n}||b_{i}||}{\sqrt{\det(B^{T}B)}}=\frac{\Pi_{i=1}^{% n}||b_{i}||}{d(\Lambda)}
  10. δ ( B ) 1 \delta(B)\geq 1
  11. δ ( B ) c \delta(B)\leq c
  12. π \pi

Lauricella_hypergeometric_series.html

  1. F A ( 3 ) ( a , b 1 , b 2 , b 3 , c 1 , c 2 , c 3 ; x 1 , x 2 , x 3 ) = i 1 , i 2 , i 3 = 0 ( a ) i 1 + i 2 + i 3 ( b 1 ) i 1 ( b 2 ) i 2 ( b 3 ) i 3 ( c 1 ) i 1 ( c 2 ) i 2 ( c 3 ) i 3 i 1 ! i 2 ! i 3 ! x 1 i 1 x 2 i 2 x 3 i 3 F_{A}^{(3)}(a,b_{1},b_{2},b_{3},c_{1},c_{2},c_{3};x_{1},x_{2},x_{3})=\sum_{i_{% 1},i_{2},i_{3}=0}^{\infty}\frac{(a)_{i_{1}+i_{2}+i_{3}}(b_{1})_{i_{1}}(b_{2})_% {i_{2}}(b_{3})_{i_{3}}}{(c_{1})_{i_{1}}(c_{2})_{i_{2}}(c_{3})_{i_{3}}\,i_{1}!% \,i_{2}!\,i_{3}!}\,x_{1}^{i_{1}}x_{2}^{i_{2}}x_{3}^{i_{3}}
  2. ( q ) i = q ( q + 1 ) ( q + i - 1 ) = Γ ( q + i ) Γ ( q ) , (q)_{i}=q\,(q+1)\cdots(q+i-1)=\frac{\Gamma(q+i)}{\Gamma(q)}~{},
  3. q q
  4. q = 0 , - 1 , - 2 , q=0,-1,-2,\ldots
  5. F A ( n ) ( a , b 1 , , b n , c 1 , , c n ; x 1 , , x n ) = i 1 , , i n = 0 ( a ) i 1 + + i n ( b 1 ) i 1 ( b n ) i n ( c 1 ) i 1 ( c n ) i n i 1 ! i n ! x 1 i 1 x n i n , F_{A}^{(n)}(a,b_{1},\ldots,b_{n},c_{1},\ldots,c_{n};x_{1},\ldots,x_{n})=\sum_{% i_{1},\ldots,i_{n}=0}^{\infty}\frac{(a)_{i_{1}+\ldots+i_{n}}(b_{1})_{i_{1}}% \cdots(b_{n})_{i_{n}}}{(c_{1})_{i_{1}}\cdots(c_{n})_{i_{n}}\,i_{1}!\cdots\,i_{% n}!}\,x_{1}^{i_{1}}\cdots x_{n}^{i_{n}}~{},
  6. F A ( 1 ) ( a , b , c ; x ) F B ( 1 ) ( a , b , c ; x ) F C ( 1 ) ( a , b , c ; x ) F D ( 1 ) ( a , b , c ; x ) F 1 2 ( a , b ; c ; x ) . F_{A}^{(1)}(a,b,c;x)\equiv F_{B}^{(1)}(a,b,c;x)\equiv F_{C}^{(1)}(a,b,c;x)% \equiv F_{D}^{(1)}(a,b,c;x)\equiv{{}_{2}}F_{1}(a,b;c;x).
  7. F D ( n ) ( a , b 1 , , b n , c ; x 1 , , x n ) = Γ ( c ) Γ ( a ) Γ ( c - a ) 0 1 t a - 1 ( 1 - t ) c - a - 1 ( 1 - x 1 t ) - b 1 ( 1 - x n t ) - b n d t , \real c > \real a > 0 . F_{D}^{(n)}(a,b_{1},\ldots,b_{n},c;x_{1},\ldots,x_{n})=\frac{\Gamma(c)}{\Gamma% (a)\Gamma(c-a)}\int_{0}^{1}t^{a-1}(1-t)^{c-a-1}(1-x_{1}t)^{-b_{1}}\cdots(1-x_{% n}t)^{-b_{n}}\,\mathrm{d}t,\quad\real\,c>\real\,a>0~{}.
  8. Π ( n , ϕ , k ) = 0 ϕ d θ ( 1 - n sin 2 θ ) 1 - k 2 sin 2 θ = sin ϕ F D ( 3 ) ( 1 2 , 1 , 1 2 , 1 2 , 3 2 ; n sin 2 ϕ , sin 2 ϕ , k 2 sin 2 ϕ ) , | \real ϕ | < π 2 . \Pi(n,\phi,k)=\int_{0}^{\phi}\frac{\mathrm{d}\theta}{(1-n\sin^{2}\theta)\sqrt{% 1-k^{2}\sin^{2}\theta}}=\sin\phi\,F_{D}^{(3)}(\tfrac{1}{2},1,\tfrac{1}{2},% \tfrac{1}{2},\tfrac{3}{2};n\sin^{2}\phi,\sin^{2}\phi,k^{2}\sin^{2}\phi),\quad|% \real\,\phi|<\frac{\pi}{2}~{}.

Law_of_supply.html

  1. ( p - p ) ( y - y ) 0 (p-p^{\prime})(y-y^{\prime})\geq 0

Law_of_the_wall.html

  1. u + = 1 κ ln y + + C + , u^{+}=\frac{1}{\kappa}\ln\,y^{+}+C^{+},
  2. y + = y u τ ν , y^{+}=\frac{y\,u_{\tau}}{\nu},
  3. u τ = τ w ρ u_{\tau}=\sqrt{\frac{\tau_{w}}{\rho}}
  4. u + = u u τ u^{+}=\frac{u}{u_{\tau}}
  5. u = u τ κ ln y y 0 {u}=\frac{u_{\tau}}{\kappa}\ln\,\frac{y}{y_{0}}
  6. k s < δ ν k_{s}<\delta_{\nu}\,
  7. k s δ ν k_{s}\approx\delta_{\nu}\,
  8. k s > δ ν k_{s}>\delta_{\nu}\,
  9. R e w = u τ k s ν . Re_{w}=\frac{u_{\tau}k_{s}}{\nu}.
  10. y 0 = ν 9 u τ y_{0}=\frac{\nu}{9u_{\tau}}
  11. y 0 = k s 30 y_{0}=\frac{k_{s}}{30}
  12. k s 3.5 D 84 , k_{s}\approx 3.5D_{84},
  13. y + < 5 y^{+}<5
  14. u + = y + u^{+}=y^{+}
  15. 5 < y + < 30 5<y^{+}<30
  16. u + y + u^{+}\neq y^{+}
  17. u + 1 κ ln y + + C + u^{+}\neq\frac{1}{\kappa}\ln\,y^{+}+C^{+}

Lazy_Eight.html

  1. \infty

Lead–lag_compensator.html

  1. Y X = s + z s + p \frac{Y}{X}=\frac{s+z}{s+p}
  2. | z | < | p | |z|<|p|
  3. | z | > | p | |z|>|p|
  4. Y X = ( s + z 1 ) ( s + z 2 ) ( s + p 1 ) ( s + p 2 ) . \frac{Y}{X}=\frac{(s+z_{1})(s+z_{2})}{(s+p_{1})(s+p_{2})}.
  5. | p 1 | > | z 1 | > | z 2 | > | p 2 | |p_{1}|>|z_{1}|>|z_{2}|>|p_{2}|
  6. Y = X - ( z 1 + z 2 ) X s + z 1 z 2 X s 2 + ( p 1 + p 2 ) Y s - p 1 p 2 Y s 2 . Y=X-(z_{1}+z_{2})\frac{X}{s}+z_{1}z_{2}\frac{X}{s^{2}}+(p_{1}+p_{2})\frac{Y}{s% }-p_{1}p_{2}\frac{Y}{s^{2}}.
  7. Y = X + 1 s ( ( p 1 + p 2 ) Y - ( z 1 + z 2 ) X + 1 s ( z 1 z 2 X - p 1 p 2 Y ) ) . Y=X+\frac{1}{s}\left((p_{1}+p_{2})Y-(z_{1}+z_{2})X+\frac{1}{s}(z_{1}z_{2}X-p_{% 1}p_{2}Y)\right).
  8. F ( x ) = A ( x ) + i B ( x ) F(x)=A(x)+iB(x)
  9. A ( x ) A(x)
  10. B ( x ) B(x)
  11. F ( x ) F(x)
  12. F ( x ) F(x)
  13. tan - 1 ( B ( x ) / A ( x ) ) \tan^{-1}(B(x)/A(x))

Leaky_integrator.html

  1. d x / d t = - A x + C dx/dt=-Ax+C\,
  2. x ( t ) = k e - A t + C x(t)=ke^{-At}+C\,

Lebedev_quadrature.html

  1. I [ f ] = 1 4 π d Ω f ( Ω ) = 1 4 π 0 π sin ( θ ) d θ 0 2 π d φ f ( θ , φ ) , I[f]=\frac{1}{4\pi}\int\mathrm{d}\Omega\ f(\Omega)=\frac{1}{4\pi}\int_{0}^{\pi% }\sin(\theta)\mathrm{d}\theta\int_{0}^{2\pi}\mathrm{d}\varphi\ f(\theta,% \varphi),
  2. I ~ [ f ] = i = 1 N w i f ( θ i , φ i ) , \tilde{I}[f]=\sum_{i=1}^{N}\ w_{i}f(\theta_{i},\varphi_{i}),
  3. I ~ 6 [ f ] = A 1 i = 1 6 f ( a i 1 ) , \tilde{I}_{6}[f]=A_{1}\sum_{i=1}^{6}f(a_{i}^{1}),
  4. a 1 a^{1}\,
  5. ( 1 , 0 , 0 ) (1,0,0)\,
  6. a 2 a^{2}\,
  7. 1 2 ( 1 , 1 , 0 ) \frac{1}{\sqrt{2}}(1,1,0)
  8. a 3 a^{3}\,
  9. 1 3 ( 1 , 1 , 1 ) \frac{1}{\sqrt{3}}(1,1,1)
  10. b k b^{k}\,
  11. ( l k , l k , m k ) (l_{k},l_{k},m_{k})\,
  12. 2 l k 2 + m k 2 = 1 2l_{k}^{2}+m_{k}^{2}=1
  13. c k c^{k}\,
  14. ( p k , q k , 0 ) (p_{k},q_{k},0)\,
  15. p k 2 + q k 2 = 1 p_{k}^{2}+q_{k}^{2}=1
  16. d k d^{k}\,
  17. ( r k , S k , W k ) (r_{k},S_{k},W_{k})\,
  18. r k 2 + S k 2 + W k 2 = 1 r_{k}^{2}+S_{k}^{2}+W_{k}^{2}=1
  19. 1 / 3 ( ± 1 , ± 1 , ± 1 ) 1/\sqrt{3}(\pm 1,\pm 1,\pm 1)
  20. 1 / 2 ( ± 1 , ± 1 , 0 ) 1/\sqrt{2}(\pm 1,\pm 1,0)
  21. I ~ 26 [ f ] = A 1 i = 1 6 f ( a i 1 ) + A 2 i = 1 12 f ( a i 2 ) + A 3 i = 1 8 f ( a i 3 ) , \tilde{I}_{26}[f]=A_{1}\sum_{i=1}^{6}f(a_{i}^{1})+A_{2}\sum_{i=1}^{12}f(a_{i}^% {2})+A_{3}\sum_{i=1}^{8}f(a_{i}^{3}),
  22. I ~ N [ f ] = A 1 i = 1 6 f ( a i 1 ) + A 2 i = 1 12 f ( a i 2 ) + A 3 i = 1 8 f ( a i 3 ) + k = 1 N 1 B k i = 1 24 f ( b i k ) + k = 1 N 2 C k i = 1 24 f ( c i k ) + k = 1 N 3 D k i = 1 48 f ( d i k ) , \begin{aligned}\displaystyle\tilde{I}_{N}[f]=A_{1}\sum_{i=1}^{6}f(a_{i}^{1})&% \displaystyle+A_{2}\sum_{i=1}^{12}f(a_{i}^{2})+A_{3}\sum_{i=1}^{8}f(a_{i}^{3})% \\ &\displaystyle+\sum_{k=1}^{N_{1}}B_{k}\sum_{i=1}^{24}f(b_{i}^{k})+\sum_{k=1}^{% N_{2}}C_{k}\sum_{i=1}^{24}f(c_{i}^{k})+\sum_{k=1}^{N_{3}}D_{k}\sum_{i=1}^{48}f% (d_{i}^{k}),\end{aligned}
  23. N = 26 + 24 ( N 1 + N 2 ) + 48 N 3 . N=26+24(N_{1}+N_{2})+48N_{3}.\,

Lebesgue's_decomposition_theorem.html

  1. μ \mu
  2. ν \nu
  3. ( Ω , Σ ) , (\Omega,\Sigma),
  4. ν 0 \nu_{0}
  5. ν 1 \nu_{1}
  6. ν = ν 0 + ν 1 \nu=\nu_{0}+\nu_{1}\,
  7. ν 0 μ \nu_{0}\ll\mu
  8. ν 0 \nu_{0}
  9. μ \mu
  10. ν 1 μ \nu_{1}\perp\mu
  11. ν 1 \nu_{1}
  12. μ \mu
  13. μ \mu
  14. ν \nu
  15. ν = ν cont + ν sing + ν pp \,\nu=\nu_{\mathrm{cont}}+\nu_{\mathrm{sing}}+\nu_{\mathrm{pp}}
  16. X = X ( 1 ) + X ( 2 ) + X ( 3 ) X=X^{(1)}+X^{(2)}+X^{(3)}
  17. X ( 1 ) X^{(1)}
  18. X ( 2 ) X^{(2)}
  19. X ( 3 ) X^{(3)}

Lebesgue's_density_theorem.html

  1. A \R n A\subset\R^{n}
  2. \R n \R^{n}
  3. d ε ( x ) = μ ( A B ε ( x ) ) μ ( B ε ( x ) ) d_{\varepsilon}(x)=\frac{\mu(A\cap B_{\varepsilon}(x))}{\mu(B_{\varepsilon}(x))}
  4. d ( x ) = lim ε 0 d ε ( x ) d(x)=\lim_{\varepsilon\to 0}d_{\varepsilon}(x)

Lecher_lines.html

  1. f = c λ f=\frac{c}{\lambda}\,
  2. c = λ f c=\lambda f\,
  3. Z 0 = 276 ln ( D / d + ( D / d ) 2 - 1 ) = ( 120 / ϵ r ) cosh - 1 ( D / d ) Z_{0}=276\ln\left(D/d+\sqrt{(D/d)^{2}-1}\right)=(120/{\sqrt{\epsilon_{r}}})% \cosh^{-1}(D/d)
  4. C = π ϵ 0 ϵ r / ln ( 2 D / d ) C=\pi\epsilon_{0}\epsilon_{r}/\ln{(2D/d)}\,
  5. Z 0 2 = L / C Z_{0}^{2}=L/C
  6. c = 1 ( L / C ) C 2 c=\frac{1}{\sqrt{(L/C)\cdot C^{2}}}
  7. = 1 Z 0 ( π ϵ 0 ϵ r ) [ ln ( 2 D / d ) ] =\frac{1}{Z_{0}\cdot\left(\pi\epsilon_{0}\epsilon_{r}\right)\cdot\left[\ln% \left({2D/d}\right)\right]}

Lee_Hwa_Chung_theorem.html

  1. α \alpha
  2. α = 0. \alpha=0.
  3. α = c × ω k 2 \alpha=c\times\omega^{\wedge\frac{k}{2}}
  4. c . c\in\mathbb{R}.

Lefschetz_hyperplane_theorem.html

  1. k n - 1 k\leq n-1
  2. k n - 1 k\leq n-1
  3. k n - 1 k\leq n-1
  4. 𝒪 X ( Y ) \mathcal{O}_{X}(Y)
  5. H q ( X , p Ω X ) H^{q}(X,\textstyle\bigwedge^{p}\Omega_{X})
  6. H q ( Y , p Ω Y ) H^{q}(Y,\textstyle\bigwedge^{p}\Omega_{Y})
  7. H q ( X , p Ω X | Y ) H^{q}(X,\textstyle\bigwedge^{p}\Omega_{X}|_{Y})
  8. \ell

Lefschetz_manifold.html

  1. ( M 2 n , ω ) (M^{2n},\omega)
  2. k = 1 n k=1\ldots n
  3. [ ω k ] : H n - k ( M , ) H n + k ( M , ) \cup[\omega^{k}]:H^{n-k}(M,\mathbb{R})\to H^{n+k}(M,\mathbb{R})
  4. M M
  5. 2 n 2n
  6. [ ω ] H D R 2 ( M ) [\omega]\in H_{DR}^{2}(M)
  7. M M
  8. L [ ω ] : H D R ( M ) H D R ( M ) , [ α ] [ ω α ] L_{[\omega]}:H_{DR}(M)\to H_{DR}(M),[\alpha]\mapsto[\omega\wedge\alpha]
  9. [ ω ] [\omega]
  10. L [ ω ] i L_{[\omega]}^{i}
  11. i i
  12. L [ ω ] L_{[\omega]}
  13. 0 i n 0\leq i\leq n
  14. L [ ω ] i : H D R n - i ( M ) H D R n + i ( M ) . L_{[\omega]}^{i}:H_{DR}^{n-i}(M)\to H_{DR}^{n+i}(M).
  15. M M
  16. H D R n - i ( M ) H_{DR}^{n-i}(M)
  17. H D R n + i ( M ) H_{DR}^{n+i}(M)
  18. L [ ω ] n - 1 : H D R 1 ( M ) H D R 2 n - 1 L_{[\omega]}^{n-1}:H_{DR}^{1}(M)\to H_{DR}^{2n-1}
  19. L [ ω ] n : H D R 0 ( M ) H D R 2 n L_{[\omega]}^{n}:H_{DR}^{0}(M)\to H_{DR}^{2n}
  20. [ ω ] [\omega]
  21. L [ ω ] i : H D R n - i ( M ) H D R n + i ( M ) L_{[\omega]}^{i}:H_{DR}^{n-i}(M)\to H_{DR}^{n+i}(M)
  22. 0 i n 0\leq i\leq n
  23. [ ω ] [\omega]
  24. ( M , ω ) (M,\omega)
  25. 2 n 2n
  26. ( M , ω ) (M,\omega)
  27. [ ω ] [\omega]
  28. ( M , ω ) (M,\omega)
  29. [ ω ] [\omega]
  30. \subset
  31. \subset
  32. \subset

Legendre's_equation.html

  1. a x 2 + b y 2 + c z 2 = 0. ax^{2}+by^{2}+cz^{2}=0.

Leibniz_operator.html

  1. 𝒮 = Fm , 𝒮 \mathcal{S}=\langle{\rm Fm},\vdash_{\mathcal{S}}\rangle
  2. T T
  3. 𝒮 \mathcal{S}
  4. T \equiv_{T}
  5. 𝒮 \mathcal{S}
  6. ϕ T ψ \phi\equiv_{T}\psi
  7. ϕ ψ T , \phi\leftrightarrow\psi\in T,
  8. \leftrightarrow
  9. T \equiv_{T}
  10. Fm / T {\rm Fm}/{\equiv_{T}}
  11. ϕ ψ T \phi\leftrightarrow\psi\in T
  12. ϕ T ψ \phi\equiv_{T}\psi
  13. T 𝒮 ϕ T\vdash_{\mathcal{S}}\phi
  14. T 𝒮 ψ T\vdash_{\mathcal{S}}\psi
  15. 𝒮 = Fm , 𝒮 , \mathcal{S}=\langle{\rm Fm},\vdash_{\mathcal{S}}\rangle,
  16. T T
  17. T T
  18. Ω ( T ) \Omega(T)
  19. ϕ , ψ Fm \phi,\psi\in{\rm Fm}
  20. ϕ Ω ( T ) ψ \phi\Omega(T)\psi
  21. α ( x , y ) \alpha(x,\vec{y})
  22. x x
  23. y \vec{y}
  24. χ \vec{\chi}
  25. y \vec{y}
  26. T 𝒮 α ( ϕ , χ ) T\vdash_{\mathcal{S}}\alpha(\phi,\vec{\chi})
  27. T 𝒮 α ( ψ , χ ) T\vdash_{\mathcal{S}}\alpha(\psi,\vec{\chi})
  28. T T
  29. ϕ Ω ( T ) ψ \phi\Omega(T)\psi
  30. ϕ T \phi\in T
  31. ψ T \psi\in T
  32. Ω \Omega
  33. T T
  34. Ω ( T ) , \Omega(T),
  35. Ω : Th 𝒮 ConFm \Omega:{\rm Th}\mathcal{S}\rightarrow{\rm Con}{\rm Fm}
  36. Th 𝒮 {\rm Th}\mathcal{S}
  37. 𝒮 \mathcal{S}
  38. ConFm {\rm Con}{\rm Fm}
  39. Fm {\rm Fm}
  40. π \pi
  41. π \pi

Lemniscatic_elliptic_function.html

  1. Γ 2 ( 1 4 ) 4 π \frac{\Gamma^{2}(\tfrac{1}{4})}{4\sqrt{\pi}}
  2. e 1 = 1 2 , e 2 = 0 , e 3 = - 1 2 . e_{1}=\tfrac{1}{2},\qquad e_{2}=0,\qquad e_{3}=-\tfrac{1}{2}.
  3. r = 0 s d t 1 - t 4 r=\int_{0}^{s}\frac{dt}{\sqrt{1-t^{4}}}
  4. cl ( r ) = c \operatorname{cl}(r)=c
  5. r = c 1 d t 1 - t 4 . r=\int_{c}^{1}\frac{dt}{\sqrt{1-t^{4}}}.
  6. G = 2 π 0 1 d t 1 - t 4 = 0.8346 . G=\frac{2}{\pi}\int_{0}^{1}\frac{dt}{\sqrt{1-t^{4}}}=0.8346\ldots.
  7. ( x 2 + y 2 ) 2 = x 2 - y 2 (x^{2}+y^{2})^{2}=x^{2}-y^{2}
  8. r = 0 s d t 1 - t 4 . r=\int_{0}^{s}\frac{dt}{\sqrt{1-t^{4}}}.

Lemoine_hexagon.html

  1. a , b , c a,b,c
  2. Δ \Delta
  3. p = a 3 + b 3 + c 3 + 3 a b c a 2 + b 2 + c 2 p=\frac{a^{3}+b^{3}+c^{3}+3abc}{a^{2}+b^{2}+c^{2}}
  4. K = a 4 + b 4 + c 4 + a 2 b 2 + b 2 c 2 + c 2 a 2 ( a 2 + b 2 + c 2 ) 2 Δ K=\frac{a^{4}+b^{4}+c^{4}+a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2}}{\left(a^{2}+b^{2}+% c^{2}\right)^{2}}\Delta
  5. p = ( a + b + c ) ( a b + b c + c a ) a 2 + b 2 + c 2 p=\frac{\left(a+b+c\right)\left(ab+bc+ca\right)}{a^{2}+b^{2}+c^{2}}
  6. K = a 2 b 2 + b 2 c 2 + c 2 a 2 ( a 2 + b 2 + c 2 ) 2 Δ K=\frac{a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2}}{\left(a^{2}+b^{2}+c^{2}\right)^{2}}\Delta

Length_constant.html

  1. λ = r m ( r i + r o ) \lambda\ =\ \sqrt{\frac{r_{m}}{(r_{i}+r_{o})}}
  2. λ = r m r i \lambda\ =\ \sqrt{\frac{r_{m}}{r_{i}}}
  3. V ( x ) = V m a x ( 1 - e - x / λ ) V(x)\ =\ V_{max}(1-e^{-x/\lambda})
  4. V ( x ) = V m a x ( e - x / λ ) V(x)\ =\ V_{max}(e^{-x/\lambda})
  5. V m a x = r m I V_{max}\ =\ r_{m}I
  6. λ = r × ρ m 2 × ρ i \lambda=\sqrt{\frac{r\times\rho_{m}}{2\times\rho_{i}}}
  7. r r
  8. ρ m = r m × 2 π r \rho_{m}=r_{m}\times 2\pi r
  9. ρ i = r i × π r 2 \rho_{i}=r_{i}\times\pi r^{2}

Lenticular_lens.html

  1. R R
  2. p p
  3. r r
  4. e e
  5. h h
  6. n n
  7. R = A - arctan ( p h ) R=A-\arctan\left({p\over h}\right)
  8. A = arcsin ( p 2 r ) A=\arcsin\left({p\over 2r}\right)
  9. h = e - f h=e-f
  10. f = r - r 2 - ( p 2 ) 2 f=r-\sqrt{r^{2}-\left({p\over 2}\right)^{2}}
  11. O O
  12. O = 2 ( A - I ) O=2(A-I)
  13. I I
  14. I = arcsin ( n sin ( R ) n a ) I=\arcsin\left({n\sin(R)\over n_{a}}\right)
  15. n a 1.003 n_{a}\approx 1.003
  16. O O
  17. F = r n - 1 F={r\over n-1}
  18. F F
  19. B F D BFD
  20. B F D = F - e n . BFD=F-{e\over n}.
  21. B F D = 0 BFD=0
  22. e = n r n - 1 . e={nr\over n-1}.
  23. e e
  24. r r

Levene's_test.html

  1. W = ( N - k ) ( k - 1 ) i = 1 k N i ( Z i - Z ) 2 i = 1 k j = 1 N i ( Z i j - Z i ) 2 , W=\frac{(N-k)}{(k-1)}\frac{\sum_{i=1}^{k}N_{i}(Z_{i\cdot}-Z_{\cdot\cdot})^{2}}% {\sum_{i=1}^{k}\sum_{j=1}^{N_{i}}(Z_{ij}-Z_{i\cdot})^{2}},
  2. W W
  3. k k
  4. N N
  5. N i N_{i}
  6. i i
  7. Y i j Y_{ij}
  8. j j
  9. i i
  10. Z i j = { | Y i j - Y ¯ i | , Y ¯ i is a mean of i-th group | Y i j - Y ~ i | , Y ~ i is a median of i-th group Z_{ij}=\left\{\begin{matrix}|Y_{ij}-\bar{Y}_{i\cdot}|,&\bar{Y}_{i\cdot}\mbox{ % is a mean of i-th group }\\ |Y_{ij}-\tilde{Y}_{i\cdot}|,&\tilde{Y}_{i\cdot}\mbox{ is a median of i-th % group }\end{matrix}\right.
  11. Z = 1 N i = 1 k j = 1 N i Z i j Z_{\cdot\cdot}=\frac{1}{N}\sum_{i=1}^{k}\sum_{j=1}^{N_{i}}Z_{ij}
  12. Z i j Z_{ij}
  13. Z i = 1 N i j = 1 N i Z i j Z_{i\cdot}=\frac{1}{N_{i}}\sum_{j=1}^{N_{i}}Z_{ij}
  14. Z i j Z_{ij}
  15. i i
  16. W W
  17. F ( α , k - 1 , N - k ) F(\alpha,k-1,N-k)
  18. F F
  19. k - 1 k-1
  20. N - k N-k
  21. α \alpha
  22. Y ¯ \bar{Y}
  23. Y ~ \tilde{Y}

Lexicographic_preferences.html

  1. x n 0 x_{n}\rightarrow 0
  2. ( x n , 0 ) > ( 0 , 1 ) (x_{n},0)>(0,1)

Leyland_number.html

  1. x y + y x x^{y}+y^{x}

Li's_criterion.html

  1. ξ ( s ) = 1 2 s ( s - 1 ) π - s / 2 Γ ( s 2 ) ζ ( s ) \xi(s)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s)
  2. λ n = 1 ( n - 1 ) ! d n d s n [ s n - 1 log ξ ( s ) ] | s = 1 . \lambda_{n}=\frac{1}{(n-1)!}\left.\frac{d^{n}}{ds^{n}}\left[s^{n-1}\log\xi(s)% \right]\right|_{s=1}.
  3. λ n > 0 \lambda_{n}>0
  4. λ n \lambda_{n}
  5. λ n = ρ [ 1 - ( 1 - 1 ρ ) n ] \lambda_{n}=\sum_{\rho}\left[1-\left(1-\frac{1}{\rho}\right)^{n}\right]
  6. ρ = lim N | ( ρ ) | N . \sum_{\rho}=\lim_{N\to\infty}\sum_{|\Im(\rho)|\leq N}.
  7. ρ 1 + | ( ρ ) | ( 1 + | ρ | ) 2 < . \sum_{\rho}\frac{1+\left|\Re(\rho)\right|}{(1+|\rho|)^{2}}<\infty.
  8. ( ρ ) 1 / 2 \Re(\rho)\leq 1/2
  9. ρ [ 1 - ( 1 - 1 ρ ) - n ] 0 \sum_{\rho}\Re\left[1-\left(1-\frac{1}{\rho}\right)^{-n}\right]\geq 0
  10. ρ ¯ \overline{\rho}
  11. 1 - ρ 1-\rho
  12. ρ [ 1 - ( 1 - 1 ρ ) n ] 0. \sum_{\rho}\left[1-\left(1-\frac{1}{\rho}\right)^{n}\right]\geq 0.

Lie_bialgebra.html

  1. 𝔤 \mathfrak{g}
  2. 𝔤 * \mathfrak{g}^{*}
  3. 𝔤 \mathfrak{g}
  4. [ , ] : 𝔤 𝔤 𝔤 [\ ,\ ]:\mathfrak{g}\otimes\mathfrak{g}\to\mathfrak{g}
  5. 𝔤 * \mathfrak{g}^{*}
  6. δ * : 𝔤 * 𝔤 * 𝔤 * \delta^{*}:\mathfrak{g}^{*}\otimes\mathfrak{g}^{*}\to\mathfrak{g}^{*}
  7. δ * \delta^{*}
  8. δ : 𝔤 𝔤 𝔤 \delta:\mathfrak{g}\to\mathfrak{g}\otimes\mathfrak{g}
  9. δ ( [ X , Y ] ) = ( ad X 1 + 1 ad X ) δ ( Y ) - ( ad Y 1 + 1 ad Y ) δ ( X ) \delta([X,Y])=\left(\operatorname{ad}_{X}\otimes 1+1\otimes\operatorname{ad}_{% X}\right)\delta(Y)-\left(\operatorname{ad}_{Y}\otimes 1+1\otimes\operatorname{% ad}_{Y}\right)\delta(X)
  10. ad X Y = [ X , Y ] \operatorname{ad}_{X}Y=[X,Y]
  11. 𝔤 * \mathfrak{g}^{*}
  12. 𝔤 \mathfrak{g}
  13. 𝔱 𝔤 \mathfrak{t}\subset\mathfrak{g}
  14. 𝔟 ± 𝔤 \mathfrak{b}_{\pm}\subset\mathfrak{g}
  15. 𝔱 = 𝔟 - 𝔟 + \mathfrak{t}=\mathfrak{b}_{-}\cap\mathfrak{b}_{+}
  16. π : 𝔟 ± 𝔱 \pi:\mathfrak{b}_{\pm}\to\mathfrak{t}
  17. 𝔤 := { ( X - , X + ) 𝔟 - × 𝔟 + | π ( X - ) + π ( X + ) = 0 } \mathfrak{g^{\prime}}:=\{(X_{-},X_{+})\in\mathfrak{b}_{-}\times\mathfrak{b}_{+% }\ \bigl|\ \pi(X_{-})+\pi(X_{+})=0\}
  18. 𝔟 - × 𝔟 + \mathfrak{b}_{-}\times\mathfrak{b}_{+}
  19. 𝔤 \mathfrak{g}
  20. 𝔤 \mathfrak{g^{\prime}}
  21. 𝔤 \mathfrak{g}
  22. ( X - , X + ) , Y := K ( X + - X - , Y ) \langle(X_{-},X_{+}),Y\rangle:=K(X_{+}-X_{-},Y)
  23. Y 𝔤 Y\in\mathfrak{g}
  24. K K
  25. 𝔤 \mathfrak{g}
  26. 𝔤 \mathfrak{g^{\prime}}
  27. 𝔤 \mathfrak{g}
  28. 𝔤 \mathfrak{g}
  29. 𝔤 \mathfrak{g}
  30. 𝔤 * \mathfrak{g^{*}}
  31. f 1 , f 2 C ( G ) f_{1},f_{2}\in C^{\infty}(G)
  32. ξ = ( d f ) e \xi=(df)_{e}
  33. ξ 𝔤 * \xi\in\mathfrak{g}^{*}
  34. 𝔤 * \mathfrak{g}^{*}
  35. [ ξ 1 , ξ 2 ] = ( d { f 1 , f 2 } ) e [\xi_{1},\xi_{2}]=(d\{f_{1},f_{2}\})_{e}\,
  36. { , } \{,\}
  37. η \eta
  38. η R \eta^{R}
  39. η R : G 𝔤 𝔤 \eta^{R}:G\to\mathfrak{g}\otimes\mathfrak{g}
  40. δ = T e η R \delta=T_{e}\eta^{R}\,
  41. [ ξ 1 , ξ 2 ] = δ * ( ξ 1 ξ 2 ) [\xi_{1},\xi_{2}]=\delta^{*}(\xi_{1}\otimes\xi_{2})

Liénard_equation.html

  1. d 2 x d t 2 + f ( x ) d x d t + g ( x ) = 0 {d^{2}x\over dt^{2}}+f(x){dx\over dt}+g(x)=0
  2. F ( x ) := 0 x f ( ξ ) d ξ F(x):=\int_{0}^{x}f(\xi)d\xi
  3. x 1 := x x_{1}:=x\,
  4. x 2 := d x d t + F ( x ) x_{2}:={dx\over dt}+F(x)
  5. [ x ˙ 1 x ˙ 2 ] = 𝐡 ( x 1 , x 2 ) := [ x 2 - F ( x 1 ) - g ( x 1 ) ] \begin{bmatrix}\dot{x}_{1}\\ \dot{x}_{2}\end{bmatrix}=\mathbf{h}(x_{1},x_{2}):=\begin{bmatrix}x_{2}-F(x_{1}% )\\ -g(x_{1})\end{bmatrix}
  6. v = d x d t v={dx\over dt}
  7. v d v d x + f ( x ) v + g ( x ) = 0 v{dv\over dx}+f(x)v+g(x)=0
  8. d 2 x d t 2 - μ ( 1 - x 2 ) d x d t + x = 0 {d^{2}x\over dt^{2}}-\mu(1-x^{2}){dx\over dt}+x=0
  9. lim x F ( x ) := lim x 0 x f ( ξ ) d ξ = ; \lim_{x\to\infty}F(x):=\lim_{x\to\infty}\int_{0}^{x}f(\xi)d\xi\ =\infty;

Life-cycle_hypothesis.html

  1. W + R Y W+RY
  2. C = W + R Y T . C=\frac{W+RY}{T}.
  3. C = 1 T W + R T Y . C=\frac{1}{T}W+\frac{R}{T}Y.
  4. C = a W + b Y , C=aW+bY,
  5. C Y = a W Y + b . \frac{C}{Y}=a\frac{W}{Y}+b.
  6. W / Y {W}/{Y}

Lifting_scheme.html

  1. ( h , g ) (h,g)
  2. s s
  3. ( h , g ) (h^{\prime},g)
  4. h ( z ) = h ( z ) + s ( z 2 ) g ( z ) h^{\prime}(z)=h(z)+s(z^{2})\cdot g(z)
  5. ( h , g ) (h,g^{\prime})
  6. g ( z ) = g ( z ) + t ( z 2 ) h ( z ) g^{\prime}(z)=g(z)+t(z^{2})\cdot h(z)
  7. ( h , g ) (h,g)
  8. ( h , g ) (h^{\prime},g)
  9. s s
  10. h ( z ) = h ( z ) + s ( z 2 ) g ( z ) h^{\prime}(z)=h(z)+s(z^{2})\cdot g(z)
  11. P P
  12. Q Q
  13. P P
  14. Q Q
  15. I I
  16. P - 1 Q P^{-1}\cdot Q

LIGA.html

  1. R a R_{a}

Lindley_equation.html

  1. F ( x ) = 0 - K ( x - y ) F ( d y ) x 0 F(x)=\int_{0^{-}}^{\infty}K(x-y)F(\,\text{d}y)\quad x\geq 0

Linear_canonical_transformation.html

  1. ( a b c d ) , \left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right),
  2. x ( t ) x(t)
  3. X ( u ) X(u)
  4. X ( a , b , c , d ) ( u ) = - i e i π d b u 2 - e - i 2 π 1 b u t e i π a b t 2 x ( t ) d t , X_{(a,b,c,d)}(u)=\sqrt{-i}\cdot e^{i\pi\frac{d}{b}u^{2}}\int_{-\infty}^{\infty% }e^{-i2\pi\frac{1}{b}ut}e^{i\pi\frac{a}{b}t^{2}}x(t)\;dt\,,
  5. X ( a , 0 , c , d ) ( u ) = d e i π c d u 2 x ( d u ) , X_{(a,0,c,d)}(u)=\sqrt{d}\cdot e^{i\pi cdu^{2}}x(du)\,,
  6. [ a b c d ] = [ 0 1 - 1 0 ] . \begin{bmatrix}a&b\\ c&d\end{bmatrix}=\begin{bmatrix}0&1\\ -1&0\end{bmatrix}.
  7. [ a b c d ] = [ cos θ sin θ - sin θ cos θ ] . \begin{bmatrix}a&b\\ c&d\end{bmatrix}=\begin{bmatrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{bmatrix}.
  8. [ a b c d ] = [ 1 λ z 0 1 ] . \begin{bmatrix}a&b\\ c&d\end{bmatrix}=\begin{bmatrix}1&\lambda z\\ 0&1\end{bmatrix}.
  9. [ a b c d ] = [ 0 i i 0 ] . \begin{bmatrix}a&b\\ c&d\end{bmatrix}=\begin{bmatrix}0&i\\ i&0\end{bmatrix}.
  10. [ a b c d ] = [ i cos θ i sin θ i sin θ - i cos θ ] . \begin{bmatrix}a&b\\ c&d\end{bmatrix}=\begin{bmatrix}i\cos\theta&i\sin\theta\\ i\sin\theta&-i\cos\theta\end{bmatrix}.
  11. X ( a , b , c , d ) ( u ) = O F ( a , b , c , d ) [ x ( t ) ] X_{(a,b,c,d)}(u)=O_{F}^{(a,b,c,d)}[x(t)]\,
  12. O F ( a 2 , b 2 , c 2 , d 2 ) { O F ( a 1 , b 1 , c 1 , d 1 ) [ x ( t ) ] } = O F ( a 3 , b 3 , c 3 , d 3 ) [ x ( t ) ] , O_{F}^{(a2,b2,c2,d2)}\left\{O_{F}^{(a1,b1,c1,d1)}[x(t)]\right\}=O_{F}^{(a3,b3,% c3,d3)}[x(t)]\,,
  13. [ a 3 b 3 c 3 d 3 ] = [ a 2 b 2 c 2 d 2 ] [ a 1 b 1 c 1 d 1 ] . \begin{bmatrix}a3&b3\\ c3&d3\end{bmatrix}=\begin{bmatrix}a2&b2\\ c2&d2\end{bmatrix}\begin{bmatrix}a1&b1\\ c1&d1\end{bmatrix}.
  14. U 0 ( x , y ) = - j λ e j k z z - - e j k 2 z [ ( x - x i ) 2 + ( y - y i ) 2 ] U i ( x i , y i ) d x i d y i , U_{0}(x,y)=-\frac{j}{\lambda}\frac{e^{jkz}}{z}\int_{-\infty}^{\infty}\int_{-% \infty}^{\infty}e^{j\frac{k}{2z}[(x-x_{i})^{2}+(y-y_{i})^{2}]}U_{i}(x_{i},y_{i% })\;dx_{i}\;dy_{i},
  15. [ a b c d ] = [ 1 λ z 0 1 ] . \begin{bmatrix}a&b\\ c&d\end{bmatrix}=\begin{bmatrix}1&\lambda z\\ 0&1\end{bmatrix}.
  16. U 0 ( x , y ) = e j k n Δ e - j k 2 f [ x 2 + y 2 ] U i ( x , y ) U_{0}(x,y)=e^{jkn\Delta}e^{-j\frac{k}{2f}[x^{2}+y^{2}]}U_{i}(x,y)
  17. [ a b c d ] = [ 1 0 - 1 λ f 1 ] . \begin{bmatrix}a&b\\ c&d\end{bmatrix}=\begin{bmatrix}1&0\\ \frac{-1}{\lambda f}&1\end{bmatrix}.
  18. [ a b c d ] = [ 1 0 - 1 λ R 1 ] . \begin{bmatrix}a&b\\ c&d\end{bmatrix}=\begin{bmatrix}1&0\\ \frac{-1}{\lambda R}&1\end{bmatrix}.
  19. [ 1 0 - 1 λ R A 1 ] . \begin{bmatrix}1&0\\ \frac{-1}{\lambda R_{A}}&1\end{bmatrix}.
  20. [ 1 0 - 1 λ R B 1 ] . \begin{bmatrix}1&0\\ \frac{-1}{\lambda R_{B}}&1\end{bmatrix}.
  21. [ 1 λ D 0 1 ] . \begin{bmatrix}1&\lambda D\\ 0&1\end{bmatrix}.
  22. [ a b c d ] = [ 1 0 - 1 λ R B 1 ] [ 1 λ D 0 1 ] [ 1 0 - 1 λ R A 1 ] = [ 1 - D R A - λ D 1 λ ( R A - 1 + R B - 1 - R A - 1 R B - 1 D ) 1 - D R B ] . \begin{bmatrix}a&b\\ c&d\end{bmatrix}=\begin{bmatrix}1&0\\ \frac{-1}{\lambda R_{B}}&1\end{bmatrix}\begin{bmatrix}1&\lambda D\\ 0&1\end{bmatrix}\begin{bmatrix}1&0\\ \frac{-1}{\lambda R_{A}}&1\end{bmatrix}=\begin{bmatrix}1-\frac{D}{R_{A}}&-% \lambda D\\ \frac{1}{\lambda}(R_{A}^{-1}+R_{B}^{-1}-R_{A}^{-1}R_{B}^{-1}D)&1-\frac{D}{R_{B% }}\end{bmatrix}\,.

Linear_dynamical_system.html

  1. N N
  2. 𝐱 \mathbf{x}
  3. 𝐀 \mathbf{A}
  4. 𝐱 \mathbf{x}
  5. 𝐱 \mathbf{x}
  6. d d t 𝐱 ( t ) = 𝐀 𝐱 ( t ) \frac{d}{dt}\mathbf{x}(t)=\mathbf{A}\cdot\mathbf{x}(t)
  7. 𝐱 \mathbf{x}
  8. 𝐱 m + 1 = 𝐀 𝐱 m \mathbf{x}_{m+1}=\mathbf{A}\cdot\mathbf{x}_{m}
  9. 𝐱 ( t ) \mathbf{x}(t)
  10. 𝐲 ( t ) \mathbf{y}(t)
  11. 𝐳 ( t ) = def α 𝐱 ( t ) + β 𝐲 ( t ) \mathbf{z}(t)\ \stackrel{\mathrm{def}}{=}\ \alpha\mathbf{x}(t)+\beta\mathbf{y}% (t)
  12. α \alpha
  13. β \beta
  14. 𝐀 \mathbf{A}
  15. 𝐱 0 = def 𝐱 ( t = 0 ) \mathbf{x}_{0}\ \stackrel{\mathrm{def}}{=}\ \mathbf{x}(t=0)
  16. 𝐫 k \mathbf{r}_{k}
  17. 𝐀 \mathbf{A}
  18. d d t 𝐱 ( t ) = 𝐀 𝐫 k = λ k 𝐫 k \frac{d}{dt}\mathbf{x}(t)=\mathbf{A}\cdot\mathbf{r}_{k}=\lambda_{k}\mathbf{r}_% {k}
  19. λ k \lambda_{k}
  20. 𝐱 ( t ) = 𝐫 k e λ k t \mathbf{x}(t)=\mathbf{r}_{k}e^{\lambda_{k}t}
  21. 𝐀 \mathbf{A}
  22. N N
  23. 𝐥 k \mathbf{l}_{k}
  24. 𝐀 \mathbf{A}
  25. 𝐱 0 = k = 1 N ( 𝐥 k 𝐱 0 ) 𝐫 k \mathbf{x}_{0}=\sum_{k=1}^{N}\left(\mathbf{l}_{k}\cdot\mathbf{x}_{0}\right)% \mathbf{r}_{k}
  26. 𝐱 ( t ) \mathbf{x}(t)
  27. 𝐱 ( t ) = k = 1 n ( 𝐥 k 𝐱 0 ) 𝐫 k e λ k t \mathbf{x}(t)=\sum_{k=1}^{n}\left(\mathbf{l}_{k}\cdot\mathbf{x}_{0}\right)% \mathbf{r}_{k}e^{\lambda_{k}t}
  28. λ n \lambda_{n}
  29. d d t 𝐱 ( t ) = 𝐀𝐱 ( t ) . \frac{d}{dt}\mathbf{x}(t)=\mathbf{A}\mathbf{x}(t).
  30. λ 2 - τ λ + Δ = 0 \lambda^{2}-\tau\lambda+\Delta=0
  31. τ \tau
  32. Δ \Delta
  33. λ 1 = τ + τ 2 - 4 Δ 2 \lambda_{1}=\frac{\tau+\sqrt{\tau^{2}-4\Delta}}{2}
  34. λ 2 = τ - τ 2 - 4 Δ 2 \lambda_{2}=\frac{\tau-\sqrt{\tau^{2}-4\Delta}}{2}
  35. Δ = λ 1 λ 2 \Delta=\lambda_{1}\lambda_{2}
  36. τ = λ 1 + λ 2 \tau=\lambda_{1}+\lambda_{2}
  37. Δ < 0 \Delta<0
  38. Δ > 0 \Delta>0
  39. τ > 0 \tau>0
  40. τ < 0 \tau<0

Linear_energy_transfer.html

  1. μ m \frac{}{μm}
  2. M e V c m \frac{MeV}{cm}
  3. L Δ = d E Δ d x L_{\Delta}=\frac{\,\text{d}E_{\Delta}}{\,\text{d}x}
  4. d E Δ \,\text{d}E_{\Delta}
  5. d x {\,\text{d}x}

Linear_stability.html

  1. d r d t = A r \frac{dr}{dt}=Ar
  2. d x d t = x - x 2 \frac{dx}{dt}=x-x^{2}
  3. d x d t = x \frac{dx}{dt}=x
  4. λ = 1 \lambda=1
  5. d r d t = ( 1 + r ) - ( 1 + r ) 2 = - r - r 2 \frac{dr}{dt}=(1+r)-(1+r)^{2}=-r-r^{2}
  6. d r d t = - r \frac{dr}{dt}=-r
  7. λ = - 1 \lambda=-1
  8. i u t = - 2 u x 2 - | u | 2 k u i\frac{\partial u}{\partial t}=-\frac{\partial^{2}u}{\partial x^{2}}-|u|^{2k}u
  9. ϕ ( x ) e - i ω t \phi(x)e^{-i\omega t}
  10. u ( x , t ) = ( ϕ ( x ) + r ( x , t ) ) e - i ω t u(x,t)=(\phi(x)+r(x,t))e^{-i\omega t}
  11. r ( x , t ) r(x,t)
  12. t [ Re r Im r ] = A [ Re r Im r ] , \frac{\partial}{\partial t}\begin{bmatrix}\,\text{Re}\,r\\ \,\text{Im}\,r\end{bmatrix}=A\begin{bmatrix}\,\text{Re}\,r\\ \,\text{Im}\,r\end{bmatrix},
  13. A = [ 0 L 0 - L 1 0 ] , A=\begin{bmatrix}0&L_{0}\\ -L_{1}&0\end{bmatrix},
  14. L 0 = - x 2 - k ϕ 2 - ω L_{0}=-\frac{\partial}{\partial x^{2}}-k\phi^{2}-\omega
  15. L 1 = - x 2 - ( 2 k + 1 ) ϕ 2 - ω L_{1}=-\frac{\partial}{\partial x^{2}}-(2k+1)\phi^{2}-\omega

Line–plane_intersection.html

  1. 𝐩 \mathbf{p}
  2. ( 𝐩 - 𝐩 𝟎 ) 𝐧 = 0 (\mathbf{p}-\mathbf{p_{0}})\cdot\mathbf{n}=0
  3. 𝐧 \mathbf{n}
  4. 𝐩 𝟎 \mathbf{p_{0}}
  5. 𝐚 𝐛 \mathbf{a}\cdot\mathbf{b}
  6. 𝐚 \mathbf{a}
  7. 𝐛 \mathbf{b}
  8. 𝐩 = d 𝐥 + 𝐥 𝟎 d \mathbf{p}=d\mathbf{l}+\mathbf{l_{0}}\quad d\in\mathbb{R}
  9. 𝐥 \mathbf{l}
  10. 𝐥 𝟎 \mathbf{l_{0}}
  11. d d
  12. ( d 𝐥 + 𝐥 𝟎 - 𝐩 𝟎 ) 𝐧 = 0 (d\mathbf{l}+\mathbf{l_{0}}-\mathbf{p_{0}})\cdot\mathbf{n}=0
  13. d 𝐥 𝐧 + ( 𝐥 𝟎 - 𝐩 𝟎 ) 𝐧 = 0 d\mathbf{l}\cdot\mathbf{n}+(\mathbf{l_{0}}-\mathbf{p_{0}})\cdot\mathbf{n}=0
  14. d d
  15. d = ( 𝐩 𝟎 - 𝐥 𝟎 ) 𝐧 𝐥 𝐧 . d={(\mathbf{p_{0}}-\mathbf{l_{0}})\cdot\mathbf{n}\over\mathbf{l}\cdot\mathbf{n% }}.
  16. 𝐥 𝐧 = 0 \mathbf{l}\cdot\mathbf{n}=0
  17. ( 𝐩 𝟎 - 𝐥 𝟎 ) 𝐧 = 0 (\mathbf{p_{0}}-\mathbf{l_{0}})\cdot\mathbf{n}=0
  18. 𝐥 𝐧 0 \mathbf{l}\cdot\mathbf{n}\neq 0
  19. d d
  20. d 𝐥 + 𝐥 𝟎 d\mathbf{l}+\mathbf{l_{0}}
  21. 𝐥 a + ( 𝐥 b - 𝐥 a ) t , t \mathbf{l}_{a}+(\mathbf{l}_{b}-\mathbf{l}_{a})t,\quad t\in\mathbb{R}
  22. 𝐥 a = ( x a , y a , z a ) \mathbf{l}_{a}=(x_{a},y_{a},z_{a})
  23. 𝐥 b = ( x b , y b , z b ) \mathbf{l}_{b}=(x_{b},y_{b},z_{b})
  24. 𝐩 0 + ( 𝐩 1 - 𝐩 0 ) u + ( 𝐩 2 - 𝐩 0 ) v , u , v \mathbf{p}_{0}+(\mathbf{p}_{1}-\mathbf{p}_{0})u+(\mathbf{p}_{2}-\mathbf{p}_{0}% )v,\quad u,v\in\mathbb{R}
  25. 𝐩 k = ( x k , y k , z k ) \mathbf{p}_{k}=(x_{k},y_{k},z_{k})
  26. k = 0 , 1 , 2 k=0,1,2
  27. 𝐥 a + ( 𝐥 b - 𝐥 a ) t = 𝐩 0 + ( 𝐩 1 - 𝐩 0 ) u + ( 𝐩 2 - 𝐩 0 ) v \mathbf{l}_{a}+(\mathbf{l}_{b}-\mathbf{l}_{a})t=\mathbf{p}_{0}+(\mathbf{p}_{1}% -\mathbf{p}_{0})u+(\mathbf{p}_{2}-\mathbf{p}_{0})v
  28. 𝐥 a - 𝐩 0 = ( 𝐥 a - 𝐥 b ) t + ( 𝐩 1 - 𝐩 0 ) u + ( 𝐩 2 - 𝐩 0 ) v , \mathbf{l}_{a}-\mathbf{p}_{0}=(\mathbf{l}_{a}-\mathbf{l}_{b})t+(\mathbf{p}_{1}% -\mathbf{p}_{0})u+(\mathbf{p}_{2}-\mathbf{p}_{0})v,
  29. [ x a - x 0 y a - y 0 z a - z 0 ] = [ x a - x b x 1 - x 0 x 2 - x 0 y a - y b y 1 - y 0 y 2 - y 0 z a - z b z 1 - z 0 z 2 - z 0 ] [ t u v ] \begin{bmatrix}x_{a}-x_{0}\\ y_{a}-y_{0}\\ z_{a}-z_{0}\end{bmatrix}=\begin{bmatrix}x_{a}-x_{b}&x_{1}-x_{0}&x_{2}-x_{0}\\ y_{a}-y_{b}&y_{1}-y_{0}&y_{2}-y_{0}\\ z_{a}-z_{b}&z_{1}-z_{0}&z_{2}-z_{0}\end{bmatrix}\begin{bmatrix}t\\ u\\ v\end{bmatrix}
  30. 𝐥 a + ( 𝐥 b - 𝐥 a ) t \mathbf{l}_{a}+(\mathbf{l}_{b}-\mathbf{l}_{a})t
  31. 𝐥 b - 𝐥 a \mathbf{l}_{b}-\mathbf{l}_{a}
  32. 𝐩 1 - 𝐩 0 \mathbf{p}_{1}-\mathbf{p}_{0}
  33. 𝐩 2 - 𝐩 0 \mathbf{p}_{2}-\mathbf{p}_{0}
  34. t [ 0 , 1 ] , t\in[0,1],
  35. 𝐥 a \mathbf{l}_{a}
  36. 𝐥 b \mathbf{l}_{b}
  37. u , v [ 0 , 1 ] , ( u + v ) 1 , u,v\in[0,1],\;\;\;(u+v)\leq 1,
  38. 𝐩 0 \mathbf{p}_{0}
  39. 𝐩 1 \mathbf{p}_{1}
  40. 𝐩 2 \mathbf{p}_{2}
  41. [ t u v ] = [ x a - x b x 1 - x 0 x 2 - x 0 y a - y b y 1 - y 0 y 2 - y 0 z a - z b z 1 - z 0 z 2 - z 0 ] - 1 [ x a - x 0 y a - y 0 z a - z 0 ] . \begin{bmatrix}t\\ u\\ v\end{bmatrix}=\begin{bmatrix}x_{a}-x_{b}&x_{1}-x_{0}&x_{2}-x_{0}\\ y_{a}-y_{b}&y_{1}-y_{0}&y_{2}-y_{0}\\ z_{a}-z_{b}&z_{1}-z_{0}&z_{2}-z_{0}\end{bmatrix}^{-1}\begin{bmatrix}x_{a}-x_{0% }\\ y_{a}-y_{0}\\ z_{a}-z_{0}\end{bmatrix}.

Link_(geometry).html

  1. X \scriptstyle X
  2. Lk ( v , X ) \scriptstyle\operatorname{Lk}(v,X)
  3. v \scriptstyle v
  4. X \scriptstyle X
  5. Lk ( v , X ) \scriptstyle\operatorname{Lk}(v,X)
  6. X \scriptstyle X
  7. v \scriptstyle v
  8. Lk ( v , X ) \scriptstyle\operatorname{Lk}(v,X)
  9. v \scriptstyle v
  10. F \scriptstyle F
  11. X \scriptstyle X
  12. Lk ( F , X ) \scriptstyle\operatorname{Lk}(F,X)
  13. G \scriptstyle G
  14. \cap
  15. \emptyset
  16. \cup
  17. \in
  18. X X
  19. Lk ( F , X ) \scriptstyle\operatorname{Lk}(F,X)
  20. X F = { G X X_{F}=\{G\in X
  21. G } \subset G\}
  22. Lk ( v , X ) \scriptstyle\operatorname{Lk}(v,X)
  23. v \scriptstyle v

Lippmann–Schwinger_equation.html

  1. + +\,
  2. - -\,
  3. | ψ ( ± ) = | ϕ + 1 E - H 0 ± i ϵ V | ψ ( ± ) . |\psi^{(\pm)}\rangle=|\phi\rangle+\frac{1}{E-H_{0}\pm i\epsilon}V|\psi^{(\pm)}% \rangle.\,
  4. | ψ ( + ) |\psi^{(+)}\rangle\,
  5. | ψ ( - ) |\psi^{(-)}\rangle\,
  6. V V\,
  7. H 0 H_{0}\,
  8. | ϕ |\phi\rangle\,
  9. E E\,
  10. i ϵ i\epsilon\,
  11. H = H 0 + V H=H_{0}+V
  12. H H
  13. H 0 = p 2 2 m H_{0}=\frac{p^{2}}{2m}
  14. V V
  15. E E
  16. H H
  17. H 0 | ϕ = E | ϕ H_{0}|\phi\rangle=E|\phi\rangle
  18. V V\,
  19. ( H 0 + V ) | ψ = E | ψ \left(H_{0}+V\right)|\psi\rangle=E|\psi\rangle
  20. | ψ | ϕ |\psi\rangle\to|\phi\rangle
  21. V 0 V\to 0
  22. | ψ = | ϕ + 1 E - H 0 V | ψ |\psi\rangle=|\phi\rangle+\frac{1}{E-H_{0}}V|\psi\rangle
  23. 1 / A 1/A
  24. A A
  25. E E
  26. | ψ ( ± ) = | ϕ + 1 E - H 0 ± i ϵ V | ψ ( ± ) |\psi^{(\pm)}\rangle=|\phi\rangle+\frac{1}{E-H_{0}\pm i\epsilon}V|\psi^{(\pm)}\rangle
  27. | ψ ( ± ) = | ϕ + d β | ϕ β E - E β ± i ϵ ϕ β | V | ψ ( ± ) , H 0 | ϕ β = E β | ϕ β |\psi^{(\pm)}\rangle=|\phi\rangle+\int d\beta\frac{|\phi_{\beta}\rangle}{E-E_{% \beta}\pm i\epsilon}\langle\phi_{\beta}|V|\psi^{(\pm)}\rangle,\quad H_{0}|\phi% _{\beta}\rangle=E_{\beta}|\phi_{\beta}\rangle
  28. ( + ) (+)
  29. ( ) (−)
  30. | ψ α ( ± ) = | ϕ + d β T β α | ϕ β E - E β ± i ϵ , T β α = ϕ β | V | ψ α ( ± ) |\psi^{(\pm)}_{\alpha}\rangle=|\phi\rangle+\int d\beta\frac{T_{\beta\alpha}|% \phi_{\beta}\rangle}{E-E_{\beta}\pm i\epsilon},\quad T_{\beta\alpha}=\langle% \phi_{\beta}|V|\psi^{(\pm)}_{\alpha}\rangle
  31. V V
  32. ψ ( ± ) \psi^{(\pm)}
  33. ϕ \phi
  34. ψ ( ± ) \psi^{(\pm)}
  35. ψ ( ± ) \psi^{(\pm)}
  36. ϕ \phi
  37. g ( E ) g(E)
  38. E E
  39. Δ E \Delta E
  40. / Δ E \hbar/\Delta E
  41. ψ g ( ± ) ( t ) = d E e - i E t g ( E ) ψ ( ± ) \psi^{(\pm)}_{g}(t)=\int dE\,e^{-iEt}g(E)\psi^{(\pm)}
  42. ϕ g ( t ) = d E e - i E t g ( E ) ϕ \phi_{g}(t)=\int dE\,e^{-iEt}g(E)\phi
  43. ψ g ( t ) \psi_{g}(t)
  44. ϕ g ( t ) \phi_{g}(t)
  45. ψ ( ± ) \psi^{(\pm)}
  46. ϕ \phi
  47. t t\rightarrow\mp\infty
  48. e - i E t e^{-iEt}
  49. ( ϕ , V ψ ± ) (\phi,V\psi^{\pm})
  50. ψ - \psi^{-}
  51. ψ - \psi^{-}
  52. ϕ \phi
  53. ψ - = ϕ \psi^{-}=\phi
  54. ψ - \psi^{-}
  55. ψ + \psi^{+}
  56. ψ + \psi^{+}
  57. ψ + \psi^{+}
  58. ϕ \phi
  59. ψ + \psi^{+}
  60. ψ \psi
  61. ± ϵ \pm\epsilon
  62. S a b = ( ψ a - , ψ b + ) S_{ab}=(\psi^{-}_{a},\psi^{+}_{b})
  63. ψ + \psi^{+}
  64. ψ - \psi^{-}
  65. ϕ \phi
  66. ψ \psi
  67. ϕ \phi
  68. S a b = δ ( a - b ) - 2 i π δ ( E a - E b ) ( ϕ a , V ψ b + ) . S_{ab}=\delta(a-b)-2i\pi\delta(E_{a}-E_{b})(\phi_{a},V\psi^{+}_{b}).
  69. ψ + \psi^{+}
  70. ϕ \phi
  71. S a b = δ ( a - b ) - 2 i π δ ( E a - E b ) ( ϕ a , V ϕ b ) S_{ab}=\delta(a-b)-2i\pi\delta(E_{a}-E_{b})(\phi_{a},V\phi_{b})\,
  72. b a b\rightarrow a
  73. | S a b - δ a b | 2 . |S_{ab}-\delta_{ab}|^{2}.\,

Liquid_mirror_telescope.html

  1. g g
  2. ω \omega
  3. m m
  4. r r
  5. h h
  6. m g mg
  7. m ω 2 r m\omega^{2}r
  8. d h d r = m ω 2 r m g \frac{dh}{dr}=\frac{m\omega^{2}r}{mg}
  9. m m
  10. h = 0 h=0
  11. r = 0 r=0
  12. h = 1 2 g ω 2 r 2 h=\frac{1}{2g}\omega^{2}r^{2}
  13. h = k r 2 h=kr^{2}
  14. k k
  15. 4 f h = r 2 4fh=r^{2}
  16. f f
  17. h h
  18. r r
  19. h h
  20. r r
  21. 2 f ω 2 = g 2f\omega^{2}=g
  22. f f
  23. ω \omega
  24. g g
  25. ω \omega
  26. g g
  27. f s 2 447 fs^{2}\approx 447
  28. f f
  29. s s

List_of_cohomology_theories.html

  1. M × I M\times I
  2. Ω * = Ω ( * ) = M S O ( * ) \Omega_{*}=\Omega(*)=MSO(*)
  3. M S O 𝐐 = H ( π * ( M S O 𝐐 ) ) MSO_{\mathbf{Q}}=H(\pi_{*}(MSO_{\mathbf{Q}}))
  4. M S O [ 2 ] = H ( π * ( M S O [ 2 ] ) ) MSO[2]=H(\pi_{*}(MSO[2]))

List_of_common_coordinate_transformations.html

  1. x = r cos θ x=r\,\cos\theta\quad
  2. y = r sin θ y=r\,\sin\theta\quad
  3. ( x , y ) ( r , θ ) = ( cos θ - r sin θ sin θ r cos θ ) \frac{\partial(x,y)}{\partial(r,\theta)}=\begin{pmatrix}\cos\theta&-r\,\sin% \theta\\ \sin\theta&r\,\cos\theta\end{pmatrix}
  4. det ( x , y ) ( r , θ ) = r \det{\frac{\partial(x,y)}{\partial(r,\theta)}}=r
  5. r = x 2 + y 2 r=\sqrt{x^{2}+y^{2}}
  6. θ = arctan | y x | \theta^{\prime}=\arctan\left|\frac{y}{x}\right|
  7. θ \theta^{\prime}
  8. 0 < θ < π 2 0<\theta<\frac{\pi}{2}
  9. θ \theta
  10. θ \theta
  11. θ \theta
  12. θ \theta^{\prime}
  13. θ = θ \theta=\theta^{\prime}
  14. θ \theta^{\prime}
  15. θ = π - θ \theta=\pi-\theta^{\prime}
  16. θ \theta^{\prime}
  17. θ = π + θ \theta=\pi+\theta^{\prime}
  18. θ \theta^{\prime}
  19. θ = 2 π - θ \theta=2\pi-\theta^{\prime}
  20. θ \theta
  21. θ \theta
  22. tan θ \tan\theta
  23. - π 2 < θ < + π 2 -\frac{\pi}{2}<\theta<+\frac{\pi}{2}
  24. π \pi
  25. r = x 2 + y 2 r=\sqrt{x^{2}+y^{2}}
  26. θ = 2 arctan y x + r \theta^{\prime}=2\arctan\frac{y}{x+r}
  27. { x = e ρ cos θ , y = e ρ sin θ . \begin{cases}x=e^{\rho}\cos\theta,\\ y=e^{\rho}\sin\theta.\end{cases}
  28. ( x , y ) = x + i y (x,y)=x+iy^{\prime}
  29. x + i y = e ρ + i θ x+iy=e^{\rho+i\theta}\,
  30. { ρ = log x 2 + y 2 , θ = arctan y x . \begin{cases}\rho=\log\sqrt{x^{2}+y^{2}},\\ \theta=\arctan\frac{y}{x}.\end{cases}
  31. x = a sinh τ cosh τ - cos σ x=a\ \frac{\sinh\tau}{\cosh\tau-\cos\sigma}
  32. y = a sin σ cosh τ - cos σ y=a\ \frac{\sin\sigma}{\cosh\tau-\cos\sigma}
  33. x = r 1 2 - r 2 2 4 c x=\frac{r_{1}^{2}-r_{2}^{2}}{4c}
  34. y = ± 1 4 c 16 c 2 r 1 2 - ( r 1 2 - r 2 2 + 4 c 2 ) 2 y=\pm\frac{1}{4c}\sqrt{16c^{2}r_{1}^{2}-(r_{1}^{2}-r_{2}^{2}+4c^{2})^{2}}
  35. r = r 1 2 + r 2 2 - 2 c 2 2 r=\sqrt{\frac{r_{1}^{2}+r_{2}^{2}-2c^{2}}{2}}
  36. θ = arctan [ 8 c 2 ( r 1 2 + r 2 2 - 2 c 2 ) r 1 2 - r 2 2 - 1 ] \theta=\arctan\left[\sqrt{\frac{8c^{2}(r_{1}^{2}+r_{2}^{2}-2c^{2})}{r_{1}^{2}-% r_{2}^{2}}-1}\right]
  37. x = cos [ κ ( s ) d s ] d s x=\int\cos\left[\int\kappa(s)\,ds\right]ds
  38. y = sin [ κ ( s ) d s ] d s y=\int\sin\left[\int\kappa(s)\,ds\right]ds
  39. κ = x y ′′ - y x ′′ ( x 2 + y 2 ) 3 / 2 \kappa=\frac{x^{\prime}y^{\prime\prime}-y^{\prime}x^{\prime\prime}}{(x^{\prime 2% }+y^{\prime 2})^{3/2}}
  40. s = a t x 2 + y 2 d t s=\int_{a}^{t}\sqrt{x^{\prime 2}+y^{\prime 2}}\,dt
  41. κ = r 2 + 2 r 2 - r r ′′ ( r 2 + r 2 ) 3 / 2 \kappa=\frac{r^{2}+2r^{\prime 2}-rr^{\prime\prime}}{(r^{2}+r^{\prime 2})^{3/2}}
  42. s = a ϕ r 2 + r 2 d ϕ s=\int_{a}^{\phi}\sqrt{r^{2}+r^{\prime 2}}\,d\phi
  43. x = ρ sin θ cos ϕ {x}=\rho\,\sin\theta\,\cos\phi\quad
  44. y = ρ sin θ sin ϕ {y}=\rho\,\sin\theta\,\sin\phi\quad
  45. z = ρ cos θ {z}=\rho\,\cos\theta\quad
  46. ( x , y , z ) ( ρ , θ , ϕ ) = ( sin θ cos ϕ ρ cos θ cos ϕ - ρ sin θ sin ϕ sin θ sin ϕ ρ cos θ sin ϕ ρ sin θ cos ϕ cos θ - ρ sin θ 0 ) \frac{\partial(x,y,z)}{\partial(\rho,\theta,\phi)}=\begin{pmatrix}\sin\theta% \cos\phi&\rho\cos\theta\cos\phi&-\rho\sin\theta\sin\phi\\ \sin\theta\sin\phi&\rho\cos\theta\sin\phi&\rho\sin\theta\cos\phi\\ \cos\theta&-\rho\sin\theta&0\end{pmatrix}
  47. d x d y d z = det ( x , y , z ) ( ρ , θ , ϕ ) d ρ d θ d ϕ = ρ 2 sin θ d ρ d θ d ϕ dx\;dy\;dz=\det{\frac{\partial(x,y,z)}{\partial(\rho,\theta,\phi)}}d\rho\;d% \theta\;d\phi=\rho^{2}\sin\theta\;d\rho\;d\theta\;d\phi\;
  48. x = r cos θ {x}={r}\,\cos\theta
  49. y = r sin θ {y}={r}\,\sin\theta
  50. z = h {z}={h}\,
  51. ( x , y , z ) ( r , θ , h ) = ( cos θ - r sin θ 0 sin θ r cos θ 0 0 0 1 ) \frac{\partial(x,y,z)}{\partial(r,\theta,h)}=\begin{pmatrix}\cos\theta&-r\sin% \theta&0\\ \sin\theta&r\cos\theta&0\\ 0&0&1\end{pmatrix}
  52. d x d y d z = det ( x , y , z ) ( r , θ , h ) d r d θ d h = r d r d θ d h dx\;dy\;dz=\det{\frac{\partial(x,y,z)}{\partial(r,\theta,h)}}dr\;d\theta\;dh={% r}\;dr\;d\theta\;dh\;
  53. ρ = x 2 + y 2 + z 2 {\rho}=\sqrt{x^{2}+y^{2}+z^{2}}
  54. ϕ = arctan ( y x ) = arccos ( x x 2 + y 2 ) = arcsin ( y x 2 + y 2 ) {\phi}=\arctan\left({\frac{y}{x}}\right)=\arccos\left(\frac{x}{\sqrt{x^{2}+y^{% 2}}}\right)=\arcsin\left(\frac{y}{\sqrt{x^{2}+y^{2}}}\right)
  55. θ = arctan ( x 2 + y 2 z ) = arccos ( z x 2 + y 2 + z 2 ) {\theta}=\arctan\left(\frac{\sqrt{x^{2}+y^{2}}}{z}\right)=\arccos\left({\frac{% z}{\sqrt{x^{2}+y^{2}+z^{2}}}}\right)
  56. ( x ρ y ρ z ρ x z ρ 2 x 2 + y 2 y z ρ 2 x 2 + y 2 - x 2 + y 2 ρ 2 - y x 2 + y 2 x x 2 + y 2 0 ) \begin{pmatrix}\frac{x}{\rho}&\frac{y}{\rho}&\frac{z}{\rho}\\ \frac{xz}{\rho^{2}\sqrt{x^{2}+y^{2}}}&\frac{yz}{\rho^{2}\sqrt{x^{2}+y^{2}}}&-% \frac{\sqrt{x^{2}+y^{2}}}{\rho^{2}}\\ \frac{-y}{x^{2}+y^{2}}&\frac{x}{x^{2}+y^{2}}&0\\ \end{pmatrix}
  57. d ρ d θ d ϕ = det ( ρ , θ , ϕ ) ( x , y , z ) d x d y d z = 1 x 2 + y 2 x 2 + y 2 + z 2 d x d y d z d\rho\ d\theta\ d\phi=\det\frac{\partial(\rho,\theta,\phi)}{\partial(x,y,z)}dx% \ dy\ dz=\frac{1}{\sqrt{x^{2}+y^{2}}\sqrt{x^{2}+y^{2}+z^{2}}}dx\ dy\ dz
  58. ρ = r 2 + h 2 {\rho}=\sqrt{r^{2}+h^{2}}
  59. ϕ = ϕ {\phi}=\phi\quad
  60. θ = arctan r h {\theta}=\arctan\frac{r}{h}
  61. ( ρ , θ , ϕ ) ( r , ϕ , h ) = ( r r 2 + h 2 0 h r 2 + h 2 h r 2 + h 2 0 - r r 2 + h 2 0 1 0 ) \frac{\partial(\rho,\theta,\phi)}{\partial(r,\phi,h)}=\begin{pmatrix}\frac{r}{% \sqrt{r^{2}+h^{2}}}&0&\frac{h}{\sqrt{r^{2}+h^{2}}}\\ \frac{h}{r^{2}+h^{2}}&0&\frac{-r}{r^{2}+h^{2}}\\ 0&1&0\end{pmatrix}
  62. det ( ρ , θ , ϕ ) ( r , ϕ , h ) = 1 r 2 + h 2 \det\frac{\partial(\rho,\theta,\phi)}{\partial(r,\phi,h)}=\frac{1}{\sqrt{r^{2}% +h^{2}}}
  63. r = x 2 + y 2 r=\sqrt{x^{2}+y^{2}}
  64. θ = { 0 if x = 0 and y = 0 arcsin ( y r ) if x 0 - arcsin ( y r ) + π if x < 0 \theta=\begin{cases}0&\mbox{if }~{}x=0\mbox{ and }~{}y=0\\ \arcsin(\frac{y}{r})&\mbox{if }~{}x\geq 0\\ -\arcsin(\frac{y}{r})+\pi&\mbox{if }~{}x<0\\ \end{cases}
  65. h = z h=z\quad
  66. θ \theta
  67. ( r , θ , h ) ( x , y , z ) = ( x x 2 + y 2 y x 2 + y 2 0 - y x 2 + y 2 x x 2 + y 2 0 0 0 1 ) \frac{\partial(r,\theta,h)}{\partial(x,y,z)}=\begin{pmatrix}\frac{x}{\sqrt{x^{% 2}+y^{2}}}&\frac{y}{\sqrt{x^{2}+y^{2}}}&0\\ \frac{-y}{x^{2}+y^{2}}&\frac{x}{x^{2}+y^{2}}&0\\ 0&0&1\end{pmatrix}
  68. r = ρ sin ϕ r=\rho\sin\phi\,
  69. θ = θ \theta=\theta\,
  70. h = ρ cos ϕ h=\rho\cos\phi\,
  71. ( r , θ , h ) ( ρ , θ , ϕ ) = ( sin ϕ 0 ρ cos ϕ 0 1 0 cos ϕ 0 - ρ sin ϕ ) \frac{\partial(r,\theta,h)}{\partial(\rho,\theta,\phi)}=\begin{pmatrix}\sin% \phi&0&\rho\cos\phi\\ 0&1&0\\ \cos\phi&0&-\rho\sin\phi\end{pmatrix}
  72. det ( r , θ , h ) ( ρ , θ , ϕ ) = - ρ \det\frac{\partial(r,\theta,h)}{\partial(\rho,\theta,\phi)}=-\rho
  73. s = 0 t x 2 + y 2 + z 2 d t s=\int_{0}^{t}\sqrt{x^{\prime 2}+y^{\prime 2}+z^{\prime 2}}\,dt
  74. κ = ( z ′′ y - y ′′ z ) 2 + ( x ′′ z - z ′′ x ) 2 + ( y ′′ x - x ′′ y ) 2 ( x 2 + y 2 + z 2 ) 3 / 2 \kappa=\frac{\sqrt{(z^{\prime\prime}y^{\prime}-y^{\prime\prime}z^{\prime})^{2}% +(x^{\prime\prime}z^{\prime}-z^{\prime\prime}x^{\prime})^{2}+(y^{\prime\prime}% x^{\prime}-x^{\prime\prime}y^{\prime})^{2}}}{(x^{\prime 2}+y^{\prime 2}+z^{% \prime 2})^{3/2}}
  75. τ = z ′′′ ( x y ′′ - y x ′′ ) + z ′′ ( x ′′′ y - x y ′′′ ) + z ( x ′′ y ′′′ - x ′′′ y ′′ ) ( x 2 + y 2 + z 2 ) ( x ′′ 2 + y ′′ 2 + z ′′ 2 ) \tau=\frac{z^{\prime\prime\prime}(x^{\prime}y^{\prime\prime}-y^{\prime}x^{% \prime\prime})+z^{\prime\prime}(x^{\prime\prime\prime}y^{\prime}-x^{\prime}y^{% \prime\prime\prime})+z^{\prime}(x^{\prime\prime}y^{\prime\prime\prime}-x^{% \prime\prime\prime}y^{\prime\prime})}{(x^{\prime 2}+y^{\prime 2}+z^{\prime 2})% (x^{\prime\prime 2}+y^{\prime\prime 2}+z^{\prime\prime 2})}

List_of_Dutch_inventions_and_discoveries.html

  1. A A
  2. B B
  3. i i
  4. F c = m v 2 r F_{c}=\frac{m\ v^{2}}{r}
  5. T = 2 π l g T=2\pi\sqrt{\frac{l}{g}}
  6. w {}_{w}
  7. 𝐅 = q ( 𝐄 + 𝐯 × 𝐁 ) \mathbf{F}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B})

List_of_exoplanet_extremes.html

  1. 3 / 2 \sqrt{3/2}

List_of_first-order_theories.html

  1. a b a b = b a \forall a\forall b\;a\vee b=b\vee a
  2. a b a b = b a \forall a\forall b\;a\wedge b=b\wedge a
  3. a b c a ( b c ) = ( a b ) c \forall a\forall b\forall c\;a\vee(b\vee c)=(a\vee b)\vee c
  4. a b c a ( b c ) = ( a b ) c \forall a\forall b\forall c\;a\wedge(b\wedge c)=(a\wedge b)\wedge c
  5. a b a ( a b ) = a \forall a\forall b\;a\vee(a\wedge b)=a
  6. a b a ( a b ) = a \forall a\forall b\;a\wedge(a\vee b)=a
  7. a b \exist c c a c b d d a d b d c \forall a\forall b\exist c\;c\leq a\wedge c\leq b\wedge\forall d\;d\leq a% \wedge d\leq b\rightarrow d\leq c
  8. a b \exist c a c b c d a d b d c d \forall a\forall b\exist c\;a\leq c\wedge b\leq c\wedge\forall d\;a\leq d% \wedge b\leq d\rightarrow c\leq d
  9. x y z x ( y z ) = ( x y ) ( x z ) \forall x\forall y\forall z\;x\vee(y\wedge z)=(x\vee y)\wedge(x\vee z)
  10. x y z x ( y ( x z ) ) = ( x y ) ( x z ) \forall x\forall y\forall z\;x\vee(y\wedge(x\vee z))=(x\vee y)\wedge(x\vee z)
  11. M A C F 0 M\models ACF_{0}
  12. a b ¬ a = b C a C and b C \forall a\forall b\;\lnot a=b\rightarrow\exists C\;aC\and bC
  13. a b C D ¬ a = b and a C and b C and a D and b D C = D \forall a\forall b\forall C\forall D\;\lnot a=b\and aC\and bC\and aD\and bD% \rightarrow C=D
  14. a b c d e G H a H and b H and e H and c G and d G and e G f I J a I and c I and f I and b J and d J and f J \forall a\forall b\forall c\forall d\forall e\forall G\forall H\;aH\and bH\and eH% \and cG\and dG\and eG\rightarrow\exists f\exists I\exists J\;aI\and cI\and fI% \and bJ\and dJ\and fJ
  15. A b c d b A and c A and d A and ¬ b = c and ¬ b = d and ¬ c = d \forall A\exists b\exists c\exists d\;bA\and cA\and dA\and\lnot b=c\and\lnot b% =d\and\lnot c=d
  16. u v ( u v ) = u v + v u \forall u\forall v\,\partial(uv)=u\,\partial v+v\,\partial u
  17. u v ( u + v ) = u + v . \forall u\forall v\,\partial(u+v)=\partial u+\partial v\ .
  18. k = { u K : ( u ) = 0 } . k=\{u\in K:\partial(u)=0\}.
  19. u ( u ) = 0 and p 1 = 0 v v p = u \forall u\,\partial(u)=0\and p1=0\rightarrow\exists v\,v^{p}=u
  20. u ( u ) = 0 and p 1 = 0 r ( u ) p = u \forall u\,\partial(u)=0\and p1=0\rightarrow r(u)^{p}=u
  21. u ¬ ( u ) = 0 r ( u ) = 0. \forall u\,\lnot\partial(u)=0\rightarrow r(u)=0.
  22. ϕ ( 0 ) ( x ϕ ( x ) ϕ ( S x ) ) ( x ϕ ( x ) ) \phi(0)\wedge(\forall x\phi(x)\rightarrow\phi(Sx))\rightarrow(\forall x\phi(x))
  23. 𝖱𝖢𝖠 0 \mathsf{RCA}_{0}
  24. 𝖶𝖪𝖫 0 \mathsf{WKL}_{0}
  25. 𝖠𝖢𝖠 0 \mathsf{ACA}_{0}
  26. 𝖠𝖳𝖱 0 \mathsf{ATR}_{0}
  27. Π 1 1 - 𝖢𝖠 0 \Pi^{1}_{1}\mbox{-}~{}\mathsf{CA}_{0}
  28. Π 1 1 \Pi^{1}_{1}

List_of_formulae_involving_π.html

  1. π \pi
  2. π \pi
  3. π = C / d \pi=C/d\!
  4. C C
  5. d d
  6. A = π r 2 A=\pi r^{2}\!
  7. A A
  8. r r
  9. V = 4 3 π r 3 V={4\over 3}\pi r^{3}\!
  10. V V
  11. r r
  12. S A = 4 π r 2 SA=4\pi r^{2}\!
  13. S A SA
  14. r r
  15. Λ = 8 π G 3 c 2 ρ \Lambda={{8\pi G}\over{3c^{2}}}\rho\!
  16. Δ x Δ p h 4 π \Delta x\,\Delta p\geq\frac{h}{4\pi}\!
  17. R μ ν - 1 2 g μ ν R + Λ g μ ν = 8 π G c 4 T μ ν R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+\Lambda g_{\mu\nu}={8\pi G\over c^{4}}T_{\mu% \nu}\!
  18. F = | q 1 q 2 | 4 π ε 0 r 2 F=\frac{\left|q_{1}q_{2}\right|}{4\pi\varepsilon_{0}r^{2}}\!
  19. μ 0 = 4 π 10 - 7 N / A 2 \mu_{0}=4\pi\cdot 10^{-7}\,\mathrm{N/A^{2}}\!
  20. T 2 π L g T\approx 2\pi\sqrt{\frac{L}{g}}\!
  21. F = π 2 E I L 2 F=\frac{\pi^{2}EI}{L^{2}}
  22. π \pi
  23. - sech ( x ) d x = π \int\limits_{-\infty}^{\infty}\,\text{sech}(x)dx=\pi\!
  24. - t e - 1 / 2 t 2 - x 2 + x t d x d t = - t e t 2 - - 1 / 2 x 2 + x t d x d t = π \int\limits_{-\infty}^{\infty}\int\limits_{t}^{\infty}e^{-1/2t^{2}-x^{2}+xt}% dxdt=\int\limits_{-\infty}^{\infty}\int\limits_{t}^{\infty}e^{{}^{-}t^{2}-1/2x% ^{2}+xt}dxdt=\pi\!
  25. - 1 1 1 - x 2 d x = π 2 \int\limits_{-1}^{1}\sqrt{1-x^{2}}\,dx=\frac{\pi}{2}\!
  26. - 1 1 d x 1 - x 2 = π \int\limits_{-1}^{1}\frac{dx}{\sqrt{1-x^{2}}}=\pi\!
  27. - d x 1 + x 2 = π \int\limits_{-\infty}^{\infty}\frac{dx}{1+x^{2}}=\pi\!
  28. - e - x 2 d x = π \int\limits_{-\infty}^{\infty}e^{-x^{2}}\,dx=\sqrt{\pi}\!
  29. d z z = 2 π i \oint\frac{dz}{z}=2\pi i\!
  30. - sin x x d x = π \int\limits_{-\infty}^{\infty}\frac{\sin x}{x}\,dx=\pi\!
  31. 0 1 x 4 ( 1 - x ) 4 1 + x 2 d x = 22 7 - π \int\limits_{0}^{1}{x^{4}(1-x)^{4}\over 1+x^{2}}\,dx={22\over 7}-\pi\!
  32. k = 0 k ! ( 2 k + 1 ) ! ! = k = 0 2 k k ! 2 ( 2 k + 1 ) ! = π 2 \sum_{k=0}^{\infty}\frac{k!}{(2k+1)!!}=\sum_{k=0}^{\infty}\frac{2^{k}k!^{2}}{(% 2k+1)!}=\frac{\pi}{2}\!
  33. 12 k = 0 ( - 1 ) k ( 6 k ) ! ( 13591409 + 545140134 k ) ( 3 k ) ! ( k ! ) 3 640320 3 k + 3 / 2 = 1 π 12\sum^{\infty}_{k=0}\frac{(-1)^{k}(6k)!(13591409+545140134k)}{(3k)!(k!)^{3}64% 0320^{3k+3/2}}=\frac{1}{\pi}\!
  34. 2 2 9801 k = 0 ( 4 k ) ! ( 1103 + 26390 k ) ( k ! ) 4 396 4 k = 1 π \frac{2\sqrt{2}}{9801}\sum^{\infty}_{k=0}\frac{(4k)!(1103+26390k)}{(k!)^{4}396% ^{4k}}=\frac{1}{\pi}\!
  35. 3 6 5 k = 0 ( ( 4 k ) ! ) 2 ( 6 k ) ! 9 k + 1 ( 12 k ) ! ( 2 k ) ! ( 127169 12 k + 1 - 1070 12 k + 5 - 131 12 k + 7 + 2 12 k + 11 ) = π \frac{\sqrt{3}}{6^{5}}\sum_{k=0}^{\infty}\frac{((4k)!)^{2}(6k)!}{9^{k+1}(12k)!% (2k)!}\left(\frac{127169}{12k+1}-\frac{1070}{12k+5}-\frac{131}{12k+7}+\frac{2}% {12k+11}\right)=\pi\!
  36. π \pi
  37. k = 0 1 16 k ( 4 8 k + 1 - 2 8 k + 4 - 1 8 k + 5 - 1 8 k + 6 ) = π \sum_{k=0}^{\infty}\frac{1}{16^{k}}\left(\frac{4}{8k+1}-\frac{2}{8k+4}-\frac{1% }{8k+5}-\frac{1}{8k+6}\right)=\pi\!
  38. 1 2 6 n = 0 ( - 1 ) n 2 10 n ( - 2 5 4 n + 1 - 1 4 n + 3 + 2 8 10 n + 1 - 2 6 10 n + 3 - 2 2 10 n + 5 - 2 2 10 n + 7 + 1 10 n + 9 ) = π \frac{1}{2^{6}}\sum_{n=0}^{\infty}\frac{{(-1)}^{n}}{2^{10n}}\left(-\frac{2^{5}% }{4n+1}-\frac{1}{4n+3}+\frac{2^{8}}{10n+1}-\frac{2^{6}}{10n+3}-\frac{2^{2}}{10% n+5}-\frac{2^{2}}{10n+7}+\frac{1}{10n+9}\right)=\pi\!
  39. ζ ( 2 ) = 1 1 2 + 1 2 2 + 1 3 2 + 1 4 2 + = π 2 6 \zeta(2)=\frac{1}{1^{2}}+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\frac{1}{4^{2}}+% \cdots=\frac{\pi^{2}}{6}\!
  40. ζ ( 4 ) = 1 1 4 + 1 2 4 + 1 3 4 + 1 4 4 + = π 4 90 \zeta(4)=\frac{1}{1^{4}}+\frac{1}{2^{4}}+\frac{1}{3^{4}}+\frac{1}{4^{4}}+% \cdots=\frac{\pi^{4}}{90}\!
  41. ζ ( 2 n ) = k = 1 1 k 2 n = 1 1 2 n + 1 2 2 n + 1 3 2 n + 1 4 2 n + = ( - 1 ) n + 1 B 2 n ( 2 π ) 2 n 2 ( 2 n ) ! \zeta(2n)=\sum_{k=1}^{\infty}\frac{1}{k^{2n}}\,=\frac{1}{1^{2n}}+\frac{1}{2^{2% n}}+\frac{1}{3^{2n}}+\frac{1}{4^{2n}}+\cdots=(-1)^{n+1}\frac{B_{2n}(2\pi)^{2n}% }{2(2n)!}\!
  42. n = 1 3 n - 1 4 n ζ ( n + 1 ) = π \sum_{n=1}^{\infty}\frac{3^{n}-1}{4^{n}}\,\zeta(n+1)=\pi\!
  43. n = 0 ( ( - 1 ) n 2 n + 1 ) 1 = 1 1 - 1 3 + 1 5 - 1 7 + 1 9 - = arctan 1 = π 4 \sum_{n=0}^{\infty}{\left(\frac{(-1)^{n}}{2n+1}\right)}^{1}=\frac{1}{1}-\frac{% 1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots=\arctan{1}=\frac{\pi}{4}\!
  44. n = 0 ( ( - 1 ) n 2 n + 1 ) 2 = 1 1 2 + 1 3 2 + 1 5 2 + 1 7 2 + = π 2 8 \sum_{n=0}^{\infty}{\left(\frac{(-1)^{n}}{2n+1}\right)}^{2}=\frac{1}{1^{2}}+% \frac{1}{3^{2}}+\frac{1}{5^{2}}+\frac{1}{7^{2}}+\cdots=\frac{\pi^{2}}{8}\!
  45. n = 0 ( ( - 1 ) n 2 n + 1 ) 3 = 1 1 3 - 1 3 3 + 1 5 3 - 1 7 3 + = π 3 32 \sum_{n=0}^{\infty}{\left(\frac{(-1)^{n}}{2n+1}\right)}^{3}=\frac{1}{1^{3}}-% \frac{1}{3^{3}}+\frac{1}{5^{3}}-\frac{1}{7^{3}}+\cdots=\frac{\pi^{3}}{32}\!
  46. n = 0 ( ( - 1 ) n 2 n + 1 ) 4 = 1 1 4 + 1 3 4 + 1 5 4 + 1 7 4 + = π 4 96 \sum_{n=0}^{\infty}{\left(\frac{(-1)^{n}}{2n+1}\right)}^{4}=\frac{1}{1^{4}}+% \frac{1}{3^{4}}+\frac{1}{5^{4}}+\frac{1}{7^{4}}+\cdots=\frac{\pi^{4}}{96}\!
  47. n = 0 ( ( - 1 ) n 2 n + 1 ) 5 = 1 1 5 - 1 3 5 + 1 5 5 - 1 7 5 + = 5 π 5 1536 \sum_{n=0}^{\infty}{\left(\frac{(-1)^{n}}{2n+1}\right)}^{5}=\frac{1}{1^{5}}-% \frac{1}{3^{5}}+\frac{1}{5^{5}}-\frac{1}{7^{5}}+\cdots=\frac{5\pi^{5}}{1536}\!
  48. n = 0 ( ( - 1 ) n 2 n + 1 ) 6 = 1 1 6 + 1 3 6 + 1 5 6 + 1 7 6 + = π 6 960 \sum_{n=0}^{\infty}{\left(\frac{(-1)^{n}}{2n+1}\right)}^{6}=\frac{1}{1^{6}}+% \frac{1}{3^{6}}+\frac{1}{5^{6}}+\frac{1}{7^{6}}+\cdots=\frac{\pi^{6}}{960}\!
  49. π = < m t p l > 1 + 1 2 + 1 3 + 1 4 - 1 5 + 1 6 + 1 7 + 1 8 + 1 9 - 1 10 + 1 11 + 1 12 - 1 13 + \pi=<mtpl>{{1}}+\frac{{1}}{{2}}+\frac{{1}}{{3}}+\frac{{1}}{{4}}-\frac{{1}}{{5}% }+\frac{{1}}{{6}}+\frac{{1}}{{7}}+\frac{{1}}{{8}}+\frac{{1}}{{9}}-\frac{{1}}{{% 10}}+\frac{{1}}{{11}}+\frac{{1}}{{12}}-\frac{{1}}{{13}}+\cdots\!
  50. π 4 = 4 arctan 1 5 - arctan 1 239 \frac{\pi}{4}=4\arctan\frac{1}{5}-\arctan\frac{1}{239}\!
  51. π 4 = arctan 1 \frac{\pi}{4}=\arctan 1
  52. π 4 = arctan 1 2 + arctan 1 3 \frac{\pi}{4}=\arctan\frac{1}{2}+\arctan\frac{1}{3}\!
  53. π 4 = 2 arctan 1 2 - arctan 1 7 \frac{\pi}{4}=2\arctan\frac{1}{2}-\arctan\frac{1}{7}\!
  54. π 4 = 2 arctan 1 3 + arctan 1 7 \frac{\pi}{4}=2\arctan\frac{1}{3}+\arctan\frac{1}{7}\!
  55. π 4 = 5 arctan 1 7 + 2 arctan 3 79 \frac{\pi}{4}=5\arctan\frac{1}{7}+2\arctan\frac{3}{79}\!
  56. π 4 = 12 arctan 1 49 + 32 arctan 1 57 - 5 arctan 1 239 + 12 arctan 1 110443 \frac{\pi}{4}=12\arctan\frac{1}{49}+32\arctan\frac{1}{57}-5\arctan\frac{1}{239% }+12\arctan\frac{1}{110443}\!
  57. π 4 = 44 arctan 1 57 + 7 arctan 1 239 - 12 arctan 1 682 + 24 arctan 1 12943 \frac{\pi}{4}=44\arctan\frac{1}{57}+7\arctan\frac{1}{239}-12\arctan\frac{1}{68% 2}+24\arctan\frac{1}{12943}\!
  58. π 2 = n = 0 arctan 1 F 2 n + 1 = arctan 1 1 + arctan 1 2 + arctan 1 5 + arctan 1 13 + \frac{\pi}{2}=\sum_{n=0}^{\infty}\arctan\frac{1}{F_{2n+1}}=\arctan\frac{1}{1}+% \arctan\frac{1}{2}+\arctan\frac{1}{5}+\arctan\frac{1}{13}+\cdots\!
  59. F n F_{n}
  60. π = 1 Z \pi=\frac{1}{Z}\!
  61. Z = n = 0 ( ( 2 n ) ! ) 3 ( 42 n + 5 ) ( n ! ) 6 16 3 n + 1 Z=\sum_{n=0}^{\infty}\frac{((2n)!)^{3}(42n+5)}{(n!)^{6}{16}^{3n+1}}\!
  62. π = 4 Z \pi=\frac{4}{Z}\!
  63. Z = n = 0 ( - 1 ) n ( 4 n ) ! ( 21460 n + 1123 ) ( n ! ) 4 441 2 n + 1 2 10 n + 1 Z=\sum_{n=0}^{\infty}\frac{(-1)^{n}(4n)!(21460n+1123)}{(n!)^{4}{441}^{2n+1}{2}% ^{10n+1}}
  64. π = 4 Z \pi=\frac{4}{Z}\!
  65. Z = n = 0 ( 6 n + 1 ) ( 1 2 ) n 3 4 n ( n ! ) 3 Z=\sum_{n=0}^{\infty}\frac{(6n+1)\left(\frac{1}{2}\right)^{3}_{n}}{{4^{n}}(n!)% ^{3}}\!
  66. π = 32 Z \pi=\frac{32}{Z}\!
  67. Z = n = 0 ( 5 - 1 2 ) 8 n ( 42 n 5 + 30 n + 5 5 - 1 ) ( 1 2 ) n 3 64 n ( n ! ) 3 Z=\sum_{n=0}^{\infty}\left(\frac{\sqrt{5}-1}{2}\right)^{8n}\frac{(42n\sqrt{5}+% 30n+5\sqrt{5}-1)\left(\frac{1}{2}\right)^{3}_{n}}{{64^{n}}(n!)^{3}}\!
  68. π = 27 4 Z \pi=\frac{27}{4Z}\!
  69. Z = n = 0 ( 2 27 ) n ( 15 n + 2 ) ( 1 2 ) n ( 1 3 ) n ( 2 3 ) n ( n ! ) 3 Z=\sum_{n=0}^{\infty}\left(\frac{2}{27}\right)^{n}\frac{(15n+2)\left(\frac{1}{% 2}\right)_{n}\left(\frac{1}{3}\right)_{n}\left(\frac{2}{3}\right)_{n}}{(n!)^{3% }}\!
  70. π = 15 3 2 Z \pi=\frac{15\sqrt{3}}{2Z}\!
  71. Z = n = 0 ( 4 125 ) n ( 33 n + 4 ) ( 1 2 ) n ( 1 3 ) n ( 2 3 ) n ( n ! ) 3 Z=\sum_{n=0}^{\infty}\left(\frac{4}{125}\right)^{n}\frac{(33n+4)\left(\frac{1}% {2}\right)_{n}\left(\frac{1}{3}\right)_{n}\left(\frac{2}{3}\right)_{n}}{(n!)^{% 3}}\!
  72. π = 85 85 18 3 Z \pi=\frac{85\sqrt{85}}{18\sqrt{3}Z}\!
  73. Z = n = 0 ( 4 85 ) n ( 133 n + 8 ) ( 1 2 ) n ( 1 6 ) n ( 5 6 ) n ( n ! ) 3 Z=\sum_{n=0}^{\infty}\left(\frac{4}{85}\right)^{n}\frac{(133n+8)\left(\frac{1}% {2}\right)_{n}\left(\frac{1}{6}\right)_{n}\left(\frac{5}{6}\right)_{n}}{(n!)^{% 3}}\!
  74. π = 5 5 2 3 Z \pi=\frac{5\sqrt{5}}{2\sqrt{3}Z}\!
  75. Z = n = 0 ( 4 125 ) n ( 11 n + 1 ) ( 1 2 ) n ( 1 6 ) n ( 5 6 ) n ( n ! ) 3 Z=\sum_{n=0}^{\infty}\left(\frac{4}{125}\right)^{n}\frac{(11n+1)\left(\frac{1}% {2}\right)_{n}\left(\frac{1}{6}\right)_{n}\left(\frac{5}{6}\right)_{n}}{(n!)^{% 3}}\!
  76. π = 2 3 Z \pi=\frac{2\sqrt{3}}{Z}\!
  77. Z = n = 0 ( 8 n + 1 ) ( 1 2 ) n ( 1 4 ) n ( 3 4 ) n ( n ! ) 3 9 n Z=\sum_{n=0}^{\infty}\frac{(8n+1)\left(\frac{1}{2}\right)_{n}\left(\frac{1}{4}% \right)_{n}\left(\frac{3}{4}\right)_{n}}{(n!)^{3}{9}^{n}}\!
  78. π = 3 9 Z \pi=\frac{\sqrt{3}}{9Z}\!
  79. Z = n = 0 ( 40 n + 3 ) ( 1 2 ) n ( 1 4 ) n ( 3 4 ) n ( n ! ) 3 49 2 n + 1 Z=\sum_{n=0}^{\infty}\frac{(40n+3)\left(\frac{1}{2}\right)_{n}\left(\frac{1}{4% }\right)_{n}\left(\frac{3}{4}\right)_{n}}{(n!)^{3}{49}^{2n+1}}\!
  80. π = 2 11 11 Z \pi=\frac{2\sqrt{11}}{11Z}\!
  81. Z = n = 0 ( 280 n + 19 ) ( 1 2 ) n ( 1 4 ) n ( 3 4 ) n ( n ! ) 3 99 2 n + 1 Z=\sum_{n=0}^{\infty}\frac{(280n+19)\left(\frac{1}{2}\right)_{n}\left(\frac{1}% {4}\right)_{n}\left(\frac{3}{4}\right)_{n}}{(n!)^{3}{99}^{2n+1}}\!
  82. π = 2 4 Z \pi=\frac{\sqrt{2}}{4Z}\!
  83. Z = n = 0 ( 10 n + 1 ) ( 1 2 ) n ( 1 4 ) n ( 3 4 ) n ( n ! ) 3 9 2 n + 1 Z=\sum_{n=0}^{\infty}\frac{(10n+1)\left(\frac{1}{2}\right)_{n}\left(\frac{1}{4% }\right)_{n}\left(\frac{3}{4}\right)_{n}}{(n!)^{3}{9}^{2n+1}}\!
  84. π = 4 5 5 Z \pi=\frac{4\sqrt{5}}{5Z}\!
  85. Z = n = 0 ( 644 n + 41 ) ( 1 2 ) n ( 1 4 ) n ( 3 4 ) n ( n ! ) 3 5 n 72 2 n + 1 Z=\sum_{n=0}^{\infty}\frac{(644n+41)\left(\frac{1}{2}\right)_{n}\left(\frac{1}% {4}\right)_{n}\left(\frac{3}{4}\right)_{n}}{(n!)^{3}5^{n}{72}^{2n+1}}\!
  86. π = 4 3 3 Z \pi=\frac{4\sqrt{3}}{3Z}\!
  87. Z = n = 0 ( - 1 ) n ( 28 n + 3 ) ( 1 2 ) n ( 1 4 ) n ( 3 4 ) n ( n ! ) 3 3 n 4 n + 1 Z=\sum_{n=0}^{\infty}\frac{(-1)^{n}(28n+3)\left(\frac{1}{2}\right)_{n}\left(% \frac{1}{4}\right)_{n}\left(\frac{3}{4}\right)_{n}}{(n!)^{3}{3^{n}}{4}^{n+1}}\!
  88. π = 4 Z \pi=\frac{4}{Z}\!
  89. Z = n = 0 ( - 1 ) n ( 20 n + 3 ) ( 1 2 ) n ( 1 4 ) n ( 3 4 ) n ( n ! ) 3 2 2 n + 1 Z=\sum_{n=0}^{\infty}\frac{(-1)^{n}(20n+3)\left(\frac{1}{2}\right)_{n}\left(% \frac{1}{4}\right)_{n}\left(\frac{3}{4}\right)_{n}}{(n!)^{3}{2}^{2n+1}}\!
  90. π = 72 Z \pi=\frac{72}{Z}\!
  91. Z = n = 0 ( - 1 ) n ( 4 n ) ! ( 260 n + 23 ) ( n ! ) 4 4 4 n 18 2 n Z=\sum_{n=0}^{\infty}\frac{(-1)^{n}(4n)!(260n+23)}{(n!)^{4}4^{4n}18^{2n}}\!
  92. π = 3528 Z \pi=\frac{3528}{Z}\!
  93. Z = n = 0 ( - 1 ) n ( 4 n ) ! ( 21460 n + 1123 ) ( n ! ) 4 4 4 n 882 2 n Z=\sum_{n=0}^{\infty}\frac{(-1)^{n}(4n)!(21460n+1123)}{(n!)^{4}4^{4n}882^{2n}}\!
  94. ( x ) n (x)_{n}\!
  95. π 4 = 3 4 5 4 7 8 11 12 13 12 17 16 19 20 23 24 29 28 31 32 \frac{\pi}{4}=\frac{3}{4}\cdot\frac{5}{4}\cdot\frac{7}{8}\cdot\frac{11}{12}% \cdot\frac{13}{12}\cdot\frac{17}{16}\cdot\frac{19}{20}\cdot\frac{23}{24}\cdot% \frac{29}{28}\cdot\frac{31}{32}\cdots\!
  96. n = 1 4 n 2 4 n 2 - 1 = 2 1 2 3 4 3 4 5 6 5 6 7 8 7 8 9 = 4 3 16 15 36 35 64 63 = π 2 \prod_{n=1}^{\infty}\frac{4n^{2}}{4n^{2}-1}=\frac{2}{1}\cdot\frac{2}{3}\cdot% \frac{4}{3}\cdot\frac{4}{5}\cdot\frac{6}{5}\cdot\frac{6}{7}\cdot\frac{8}{7}% \cdot\frac{8}{9}\cdots=\frac{4}{3}\cdot\frac{16}{15}\cdot\frac{36}{35}\cdot% \frac{64}{63}\cdots=\frac{\pi}{2}\!
  97. 2 2 2 + 2 2 2 + 2 + 2 2 = 2 π \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{2+\sqrt{2}}}{2}\cdot\frac{\sqrt{2+\sqrt{2+% \sqrt{2}}}}{2}\cdot\cdots=\frac{2}{\pi}\!
  98. π = 3 + 1 2 6 + 3 2 6 + 5 2 6 + 7 2 6 + \pi={3+\cfrac{1^{2}}{6+\cfrac{3^{2}}{6+\cfrac{5^{2}}{6+\cfrac{7^{2}}{6+\ddots% \,}}}}}
  99. π = 4 1 + 1 2 3 + 2 2 5 + 3 2 7 + 4 2 9 + \pi=\cfrac{4}{1+\cfrac{1^{2}}{3+\cfrac{2^{2}}{5+\cfrac{3^{2}}{7+\cfrac{4^{2}}{% 9+\ddots}}}}}
  100. π = 4 1 + 1 2 2 + 3 2 2 + 5 2 2 + 7 2 2 + \pi=\cfrac{4}{1+\cfrac{1^{2}}{2+\cfrac{3^{2}}{2+\cfrac{5^{2}}{2+\cfrac{7^{2}}{% 2+\ddots}}}}}\,
  101. n ! 2 π n ( n e ) n n!\sim\sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}\!
  102. e i π + 1 = 0 e^{i\pi}+1=0\!
  103. k = 1 n φ ( k ) 3 n 2 π 2 \sum_{k=1}^{n}\varphi(k)\sim\frac{3n^{2}}{\pi^{2}}\!
  104. k = 1 n φ ( k ) k 6 n π 2 \sum_{k=1}^{n}\frac{\varphi(k)}{k}\sim\frac{6n}{\pi^{2}}\!
  105. Γ ( 1 2 ) = π \Gamma\left({1\over 2}\right)=\sqrt{\pi}\!
  106. π = Γ ( 1 / 4 ) 4 / 3 agm ( 1 , 2 ) 2 / 3 2 \pi=\frac{\Gamma\left({1/4}\right)^{4/3}\mathrm{agm}(1,\sqrt{2})^{2/3}}{2}\!
  107. lim n 1 n 2 k = 1 n ( n mod k ) = 1 - π 2 12 \lim_{n\rightarrow\infty}\frac{1}{n^{2}}\sum_{k=1}^{n}(n\;\bmod\;k)=1-\frac{% \pi^{2}}{12}\!
  108. π = lim n 4 n 2 k = 1 n n 2 - k 2 \pi=\lim_{n\rightarrow\infty}\frac{4}{n^{2}}\sum_{k=1}^{n}\sqrt{n^{2}-k^{2}}
  109. π = lim n 2 4 n n ( 2 n n ) 2 \pi=\lim_{n\rightarrow\infty}\frac{2^{4n}}{n{2n\choose n}^{2}}

List_of_knapsack_problems.html

  1. 1 j n 1\leq j\leq n
  2. j = 1 n p j x j \sum_{j=1}^{n}p_{j}x_{j}
  3. j = 1 n w j x j W , \sum_{j=1}^{n}w_{j}x_{j}\leq W,
  4. x j { 0 , 1 } x_{j}\in\{0,1\}
  5. j { 1 , , n } \forall j\in\{1,\ldots,n\}
  6. j = 1 n p j x j \sum_{j=1}^{n}p_{j}x_{j}
  7. j = 1 n w j x j W , \sum_{j=1}^{n}w_{j}x_{j}\leq W,
  8. u j x j 0 , x j u_{j}\geq x_{j}\geq 0,x_{j}
  9. j = 1 n p j x j \sum_{j=1}^{n}p_{j}x_{j}
  10. j = 1 n w j x j W , \sum_{j=1}^{n}w_{j}x_{j}\leq W,
  11. x j 0 , x j x_{j}\geq 0,x_{j}
  12. N i N_{i}
  13. i = 1 k j N i p i j x i j \sum_{i=1}^{k}\sum_{j\in N_{i}}p_{ij}x_{ij}
  14. i = 1 k j N i w i j x i j W , \sum_{i=1}^{k}\sum_{j\in N_{i}}w_{ij}x_{ij}\leq W,
  15. j N i x i j = 1 , \sum_{j\in N_{i}}x_{ij}=1,
  16. 1 i k 1\leq i\leq k
  17. x i j { 0 , 1 } x_{ij}\in\{0,1\}
  18. 1 i k 1\leq i\leq k
  19. j N i j\in N_{i}
  20. j = 1 n p j x j \sum_{j=1}^{n}p_{j}x_{j}
  21. j = 1 n p j x j W , \sum_{j=1}^{n}p_{j}x_{j}\leq W,
  22. x j { 0 , 1 } x_{j}\in\{0,1\}
  23. W i W_{i}
  24. i = 1 m j = 1 n p j x i j \sum_{i=1}^{m}\sum_{j=1}^{n}p_{j}x_{ij}
  25. j = 1 n w j x i j W i , \sum_{j=1}^{n}w_{j}x_{ij}\leq W_{i},
  26. 1 i m 1\leq i\leq m
  27. i = 1 m x i j 1 , \sum_{i=1}^{m}x_{ij}\leq 1,
  28. 1 j n 1\leq j\leq n
  29. x i j { 0 , 1 } x_{ij}\in\{0,1\}
  30. 1 j n 1\leq j\leq n
  31. 1 i m 1\leq i\leq m
  32. j = 1 n p j x j + i = 1 n - 1 j = i + 1 n p i j x i x j \sum_{j=1}^{n}p_{j}x_{j}+\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}p_{ij}x_{i}x_{j}
  33. j = 1 n w j x j W , \sum_{j=1}^{n}w_{j}x_{j}\leq W,
  34. x j { 0 , 1 } x_{j}\in\{0,1\}
  35. 1 j n 1\leq j\leq n
  36. n n
  37. N = { 1 , , n } N=\{1,\ldots,n\}
  38. m m
  39. P = { 1 , , m } P=\{1,\ldots,m\}
  40. j j
  41. P j P_{j}
  42. P P
  43. j j
  44. p j p_{j}
  45. j = 1 , , n j=1,\ldots,n
  46. i i
  47. w i w_{i}
  48. i = 1 , , m i=1,\ldots,m
  49. j = 1 n p j x j \sum_{j=1}^{n}p_{j}x_{j}
  50. j = 1 n w i j x j W i , \sum_{j=1}^{n}w_{ij}x_{j}\leq W_{i},
  51. 1 i m 1\leq i\leq m
  52. x j 0 x_{j}\geq 0
  53. x j x_{j}
  54. 1 j n 1\leq j\leq n
  55. m 2 m\geq 2
  56. m 2 m\geq 2
  57. m 2 m\geq 2
  58. j = 1 n x j \sum_{j=1}^{n}x_{j}
  59. j = 1 n w j x j = W , \sum_{j=1}^{n}w_{j}x_{j}=W,
  60. x j { 0 , 1 } , x_{j}\in\{0,1\},
  61. j { 1 , , n } \forall j\in\{1,\ldots,n\}
  62. y i = 1 y_{i}=1\Leftrightarrow
  63. i = 1 n y i \sum_{i=1}^{n}y_{i}
  64. j = 1 n w j x i j W y i , \sum_{j=1}^{n}w_{j}x_{ij}\leq Wy_{i},
  65. i { 1 , , n } \forall i\in\{1,\ldots,n\}
  66. i = 1 n x i j = 1 \sum_{i=1}^{n}x_{ij}=1
  67. j { 1 , , n } \forall j\in\{1,\ldots,n\}
  68. y i { 0 , 1 } y_{i}\in\{0,1\}
  69. i { 1 , , n } \forall i\in\{1,\ldots,n\}
  70. x i j { 0 , 1 } x_{ij}\in\{0,1\}
  71. i { 1 , , n } j { 1 , , n } \forall i\in\{1,\ldots,n\}\wedge\forall j\in\{1,\ldots,n\}
  72. i = 1 m x i \sum_{i=1}^{m}x_{i}
  73. i = 1 m b i j x i B j , \sum_{i=1}^{m}b_{ij}x_{i}\leq B_{j},
  74. 1 j n 1\leq j\leq n
  75. x i { 0 , 1 , , n } x_{i}\in\{0,1,\ldots,n\}
  76. 1 i m 1\leq i\leq m
  77. i = 1 k j = 1 n p i j x i j \sum_{i=1}^{k}\sum_{j=1}^{n}p_{ij}x_{ij}
  78. i = 1 n x i j = 1 , \sum_{i=1}^{n}x_{ij}=1,
  79. 1 j n 1\leq j\leq n
  80. j = 1 n x i j = 1 , \sum_{j=1}^{n}x_{ij}=1,
  81. 1 i n 1\leq i\leq n
  82. x i j { 0 , 1 } x_{ij}\in\{0,1\}
  83. 1 i k 1\leq i\leq k
  84. j N i j\in N_{i}
  85. w 0 w_{0}
  86. j = 1 n x j p j w 0 + j = 1 n x j w j \frac{\sum_{j=1}^{n}x_{j}p_{j}}{w_{0}+\sum_{j=1}^{n}x_{j}w_{j}}
  87. j = 1 n w j x j W , \sum_{j=1}^{n}w_{j}x_{j}\leq W,
  88. x j { 0 , 1 } , x_{j}\in\{0,1\},
  89. j { 1 , , n } \forall j\in\{1,\ldots,n\}

List_of_musical_symbols.html

  1. d × ( 2 - 2 - n ) d\times(2-2^{-n})
  2. Name = 2 n + 2 th \,\text{Name}=2^{n+2}\,\text{th}
  3. n n

List_of_solution_strategies_for_differential_equations.html

  1. y + a ( x ) y = b ( x ) y^{\prime}+a(x)y=b(x)
  2. M ( x ) = exp a ( x ) d x M(x)=\exp{\int a(x)\,dx}
  3. y ( x ) = b ( x ) M ( x ) d x + C M ( x ) . y(x)=\frac{\int b(x)M(x)\,dx+C}{M(x)}.\,

List_of_triangle_inequalities.html

  1. a < b + c , b < c + a , c < a + b a<b+c,\quad b<c+a,\quad c<a+b
  2. max ( a , b , c ) < s . \,\text{max}(a,b,c)<s.
  3. a b + c + b a + c + c a + b < 2 , \frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}<2,
  4. 3 ( a b + b c + c a ) 2 ( b a + c b + a c ) + 3. 3\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)\geq 2\left(\frac{b}{a}+\frac% {c}{b}+\frac{a}{c}\right)+3.
  5. a b c ( a + b - c ) ( a - b + c ) ( - a + b + c ) . abc\geq(a+b-c)(a-b+c)(-a+b+c).\quad
  6. 1 3 a 2 + b 2 + c 2 ( a + b + c ) 2 1 2 . \frac{1}{3}\leq\frac{a^{2}+b^{2}+c^{2}}{(a+b+c)^{2}}\leq\frac{1}{2}.\quad
  7. a + b - c + a - b + c + - a + b + c a + b + c . \sqrt{a+b-c}+\sqrt{a-b+c}+\sqrt{-a+b+c}\geq\sqrt{a}+\sqrt{b}+\sqrt{c}.
  8. a 2 b ( a - b ) + b 2 c ( b - c ) + c 2 a ( c - a ) 0. a^{2}b(a-b)+b^{2}c(b-c)+c^{2}a(c-a)\geq 0.
  9. a 2 + b 2 < c 2 ; a^{2}+b^{2}<c^{2};
  10. a 2 + b 2 > c 2 . a^{2}+b^{2}>c^{2}.
  11. a 2 + b 2 > c 2 2 , a^{2}+b^{2}>\frac{c^{2}}{2},
  12. a 2 < 4 b c , b 2 < 4 a c , c 2 < 4 a b . a^{2}<4bc,\quad b^{2}<4ac,\quad c^{2}<4ab.
  13. 3 a b c a b + b c + c a a b c 3 a + b + c 3 , \frac{3abc}{ab+bc+ca}\leq\sqrt[3]{abc}\leq\frac{a+b+c}{3},
  14. cos A + c o s B + cos C 3 2 . \cos A+cosB+\cos C\leq\frac{3}{2}.
  15. ( 1 - cos A ) ( 1 - cos B ) ( 1 - cos C ) cos A cos B cos C . (1-\cos A)(1-\cos B)(1-\cos C)\geq\cos A\cdot\cos B\cdot\cos C.
  16. cos 4 A 2 + cos 4 B 2 + cos 4 C 2 s 3 2 a b c \cos^{4}\frac{A}{2}+\cos^{4}\frac{B}{2}+\cos^{4}\frac{C}{2}\leq\frac{s^{3}}{2abc}
  17. a + b + c 2 b c cos A + 2 c a cos B + 2 a b cos C . a+b+c\geq 2\sqrt{bc}\cos A+2\sqrt{ca}\cos B+2\sqrt{ab}\cos C.
  18. sin A + sin B + sin C 3 3 2 . \sin A+\sin B+\sin C\leq\frac{3\sqrt{3}}{2}.
  19. sin 2 A + sin 2 B + sin 2 C 9 4 . \sin^{2}A+\sin^{2}B+\sin^{2}C\leq\frac{9}{4}.
  20. sin A sin B sin C 3 3 8 . \sin A\cdot\sin B\cdot\sin C\leq\frac{3\sqrt{3}}{8}.
  21. sin A + sin B sin C φ \sin A+\sin B\cdot\sin C\leq\varphi
  22. φ = 1 + 5 2 , \varphi=\frac{1+\sqrt{5}}{2},
  23. sin A 2 sin B 2 sin C 2 1 8 . \sin\frac{A}{2}\cdot\sin\frac{B}{2}\cdot\sin\frac{C}{2}\leq\frac{1}{8}.
  24. tan 2 A 2 + tan 2 B 2 + tan 2 C 2 1. \tan^{2}\frac{A}{2}+\tan^{2}\frac{B}{2}+\tan^{2}\frac{C}{2}\geq 1.
  25. cot A + cot B + cot C 3 . \cot A+\cot B+\cot C\geq\sqrt{3}.
  26. sin A cos B + sin B cos C + sin C cos A 3 3 4 . \sin A\cdot\cos B+\sin B\cdot\cos C+\sin C\cdot\cos A\leq\frac{3\sqrt{3}}{4}.
  27. max ( sin A 2 , sin B 2 , sin C 2 ) 1 2 ( 1 + 1 - 2 r R ) , \max\left(\sin\frac{A}{2},\sin\frac{B}{2},\sin\frac{C}{2}\right)\leq\frac{1}{2% }\left(1+\sqrt{1-\frac{2r}{R}}\right),
  28. min ( sin A 2 , sin B 2 , sin C 2 ) 1 2 ( 1 - 1 - 2 r R ) , \min\left(\sin\frac{A}{2},\sin\frac{B}{2},\sin\frac{C}{2}\right)\geq\frac{1}{2% }\left(1-\sqrt{1-\frac{2r}{R}}\right),
  29. r R - 1 - 2 r R cos A r R + 1 - 2 r R \frac{r}{R}-\sqrt{1-\frac{2r}{R}}\leq\cos A\leq\frac{r}{R}+\sqrt{1-\frac{2r}{R}}
  30. A > B if and only if a > b , A>B\quad\,\text{if and only if}\quad a>b,
  31. 180 ° - A > max ( B , C ) . 180\,\text{°}-A>\max(B,C).
  32. B D C > A . \angle BDC>\angle A.
  33. cos 2 A + cos 2 B + cos 2 C < 1 , \cos^{2}A+\cos^{2}B+\cos^{2}C<1,
  34. p 2 12 3 T , p^{2}\geq 12\sqrt{3}\cdot T,
  35. a 2 + b 2 + c 2 4 3 T , a^{2}+b^{2}+c^{2}\geq 4\sqrt{3}\cdot T,
  36. a 2 + b 2 + c 2 ( a - b ) 2 + ( b - c ) 2 + ( c - a ) 2 + 4 3 T . a^{2}+b^{2}+c^{2}\geq(a-b)^{2}+(b-c)^{2}+(c-a)^{2}+4\sqrt{3}\cdot T.
  37. a b + b c + c a 4 3 T ab+bc+ca\geq 4\sqrt{3}\cdot T
  38. T 3 4 ( a b c ) 2 / 3 . T\leq\frac{\sqrt{3}}{4}(abc)^{2/3}.
  39. 9 a b c a + b + c 4 3 T \frac{9abc}{a+b+c}\geq 4\sqrt{3}\cdot T
  40. T a b c 2 a + b + c a 3 + b 3 + c 3 + a b c 3 4 ( a b c ) 2 / 3 ; T\leq\frac{abc}{2}\sqrt{\frac{a+b+c}{a^{3}+b^{3}+c^{3}+abc}}\leq\frac{\sqrt{3}% }{4}(abc)^{2/3};
  41. T 3 36 ( a + b + c ) 2 = 3 9 s 2 ; T\leq\frac{\sqrt{3}}{36}(a+b+c)^{2}=\frac{\sqrt{3}}{9}s^{2};
  42. 38 T 2 2 s 4 - a 4 - b 4 - c 4 38T^{2}\leq 2s^{4}-a^{4}-b^{4}-c^{4}
  43. 1 a + 1 b + 1 c < s T . \frac{1}{a}+\frac{1}{b}+\frac{1}{c}<\frac{s}{T}.
  44. 27 ( b 2 + c 2 - a 2 ) 2 ( c 2 + a 2 - b 2 ) 2 ( a 2 + b 2 - c 2 ) 2 ( 4 T ) 6 . 27(b^{2}+c^{2}-a^{2})^{2}(c^{2}+a^{2}-b^{2})^{2}(a^{2}+b^{2}-c^{2})^{2}\leq(4T% )^{6}.
  45. Area of incircle Area of triangle π 3 3 \frac{\,\text{Area of incircle}}{\,\text{Area of triangle}}\leq\frac{\pi}{3% \sqrt{3}}
  46. Area of inscribed triangle Area of reference triangle 1 4 . \frac{\,\text{Area of inscribed triangle}}{\,\text{Area of reference triangle}% }\leq\frac{1}{4}.
  47. 3 a b c 4 ( a 3 + b 3 + c 3 ) Area of triangle D E F Area of triangle A B C 1 4 . \frac{3abc}{4(a^{3}+b^{3}+c^{3})}\leq\frac{\,\text{Area of triangle}\,DEF}{\,% \text{Area of triangle}\,ABC}\leq\frac{1}{4}.
  48. m a , m b , m c m_{a},\,m_{b},\,m_{c}
  49. 3 4 ( a + b + c ) < m a + m b + m c < a + b + c . \frac{3}{4}(a+b+c)<m_{a}+m_{b}+m_{c}<a+b+c.
  50. ( m a a ) 2 + ( m b b ) 2 + ( m c c ) 2 9 4 , \left(\frac{m_{a}}{a}\right)^{2}+\left(\frac{m_{b}}{b}\right)^{2}+\left(\frac{% m_{c}}{c}\right)^{2}\geq\frac{9}{4},
  51. m a m b m c m a 2 + m b 2 + m c 2 r . \frac{m_{a}m_{b}m_{c}}{m_{a}^{2}+m_{b}^{2}+m_{c}^{2}}\geq r.
  52. M a m a + M b m b + M c m c 4. \frac{M_{a}}{m_{a}}+\frac{M_{b}}{m_{b}}+\frac{M_{c}}{m_{c}}\geq 4.
  53. G U + G V + G W A G + B G + C G GU+GV+GW\geq AG+BG+CG
  54. G U G V G W A G B G C G ; GU\cdot GV\cdot GW\geq AG\cdot BG\cdot CG;
  55. sin G B C + sin G C A + sin G A B 3 2 . \sin GBC+\sin GCA+\sin GAB\leq\frac{3}{2}.
  56. m a 2 + m b 2 + m c 2 > 6 R 2 m_{a}^{2}+m_{b}^{2}+m_{c}^{2}>6R^{2}
  57. I A 2 m a 2 + I B 2 m b 2 + I C 2 m c 2 3 4 . \frac{IA^{2}}{m_{a}^{2}}+\frac{IB^{2}}{m_{b}^{2}}+\frac{IC^{2}}{m_{c}^{2}}\leq% \frac{3}{4}.
  58. m a < m b + m c , m b < m c + m a , m c < m a + m b . m_{a}<m_{b}+m_{c},\quad m_{b}<m_{c}+m_{a},\quad m_{c}<m_{a}+m_{b}.
  59. h a + h b + h c 3 2 ( a + b + c ) h_{a}+h_{b}+h_{c}\leq\frac{\sqrt{3}}{2}(a+b+c)
  60. h a 2 + h b 2 + h c 2 3 4 ( a 2 + b 2 + c 2 ) . h_{a}^{2}+h_{b}^{2}+h_{c}^{2}\leq\frac{3}{4}(a^{2}+b^{2}+c^{2}).
  61. a b c , a\geq b\geq c,
  62. a + h a b + h b c + h c . a+h_{a}\geq b+h_{b}\geq c+h_{c}.
  63. h a 2 ( b 2 + c 2 ) h b 2 ( c 2 + a 2 ) h c 2 ( a 2 + b 2 ) ( 3 8 ) 3 . \frac{h_{a}^{2}}{(b^{2}+c^{2})}\cdot\frac{h_{b}^{2}}{(c^{2}+a^{2})}\cdot\frac{% h_{c}^{2}}{(a^{2}+b^{2})}\leq\left(\frac{3}{8}\right)^{3}.
  64. h a t a + h b t b + h c t c R + 4 r R . \frac{h_{a}}{t_{a}}+\frac{h_{b}}{t_{b}}+\frac{h_{c}}{t_{c}}\geq\frac{R+4r}{R}.
  65. 1 h a < 1 h b + 1 h c , 1 h b < 1 h c + 1 h a , 1 h c < 1 h a + 1 h b . \frac{1}{h_{a}}<\frac{1}{h_{b}}+\frac{1}{h_{c}},\quad\frac{1}{h_{b}}<\frac{1}{% h_{c}}+\frac{1}{h_{a}},\quad\frac{1}{h_{c}}<\frac{1}{h_{a}}+\frac{1}{h_{b}}.
  66. t a + t b + t c 3 2 ( a + b + c ) t_{a}+t_{b}+t_{c}\leq\frac{3}{2}(a+b+c)
  67. h a t a m a h_{a}\leq t_{a}\leq m_{a}
  68. m a + m b + m c t a + t b + t c \sqrt{m_{a}}+\sqrt{m_{b}}+\sqrt{m_{c}}\geq\sqrt{t_{a}}+\sqrt{t_{b}}+\sqrt{t_{c}}
  69. T a T b T c 8 3 9 a b c , T_{a}T_{b}T_{c}\geq\frac{8\sqrt{3}}{9}abc,
  70. T a + T b + T c 5 R + 2 r T_{a}+T_{b}+T_{c}\leq 5R+2r
  71. T a + T b + T c 4 3 ( t a + t b + t c ) . T_{a}+T_{b}+T_{c}\geq\frac{4}{3}(t_{a}+t_{b}+t_{c}).
  72. 6 r A I + B I + C I 12 ( R 2 - R r + r 2 ) . 6r\leq AI+BI+CI\leq\sqrt{12(R^{2}-Rr+r^{2})}.
  73. I L 2 + I M 2 + I N 2 r ( R + r ) . IL^{2}+IM^{2}+IN^{2}\geq r(R+r).
  74. I G < H G , IG<HG,
  75. I H < H G , IH<HG,
  76. I G < I O , IG<IO,
  77. I N < 1 2 I O ; IN<\frac{1}{2}IO;
  78. I O H < π 6 . \angle IOH<\frac{\pi}{6}.
  79. I G < 1 3 v , IG<\frac{1}{3}v,
  80. O I H \angle OIH
  81. G I H \angle GIH
  82. O G I \angle OGI
  83. O I 2 + I H 2 < O H 2 , G I 2 + I H 2 < G H 2 , O G 2 + G I 2 < O I 2 , OI^{2}+IH^{2}<OH^{2},\quad GI^{2}+IH^{2}<GH^{2},\quad OG^{2}+GI^{2}<OI^{2},
  84. O I 2 < O H 2 - 2 I H 2 < 2 O I 2 . OI^{2}<OH^{2}-2\cdot IH^{2}<2\cdot OI^{2}.
  85. If A > B then t a < t b . \,\text{If}\quad A>B\quad\,\text{then}\quad t_{a}<t_{b}.
  86. a b c , a\geq b\geq c,
  87. p a p b p_{a}\geq p_{b}
  88. p c p b . p_{c}\geq p_{b}.
  89. 2 ( P A + P B + P C ) > A B + B C + C A > P A + P B + P C 2(PA+PB+PC)>AB+BC+CA>PA+PB+PC
  90. P A + P B + P C A C + B C , P A + P B + P C A B + B C , P A + P B + P C A B + A C . PA+PB+PC\leq AC+BC,\quad PA+PB+PC\leq AB+BC,\quad PA+PB+PC\leq AB+AC.
  91. P A B C + P B C A > P C A B PA\cdot BC+PB\cdot CA>PC\cdot AB
  92. P A P B P C ( P D + P E ) ( P E + P F ) ( P F + P D ) . PA\cdot PB\cdot PC\geq(PD+PE)(PE+PF)(PF+PD).
  93. P A + P B + P C P D + P E + P F 2 \frac{PA+PB+PC}{PD+PE+PF}\geq 2
  94. P A + P B + P C P U + P V + P W 2. \frac{PA+PB+PC}{PU+PV+PW}\geq 2.
  95. P A 2 P E P F + P B 2 P F P D + P C 2 P D P E 12 ; \frac{PA^{2}}{PE\cdot PF}+\frac{PB^{2}}{PF\cdot PD}+\frac{PC^{2}}{PD\cdot PE}% \geq 12;
  96. P A P E P F + P B P F P D + P C P D P E 6 ; \frac{PA}{\sqrt{PE\cdot PF}}+\frac{PB}{\sqrt{PF\cdot PD}}+\frac{PC}{\sqrt{PD% \cdot PE}}\geq 6;
  97. P A P E + P F + P B P F + P D + P C P D + P E 3. \frac{PA}{PE+PF}+\frac{PB}{PF+PD}+\frac{PC}{PD+PE}\geq 3.
  98. ( b + c ) P A + ( c + a ) P B + ( a + b ) P C 8 T (b+c)PA+(c+a)PB+(a+b)PC\geq 8T
  99. P A a + P B b + P C c 3 . \frac{PA}{a}+\frac{PB}{b}+\frac{PC}{c}\geq\sqrt{3}.
  100. 2 ( P L + P M + P N ) 3 P G + P A + P B + P C s + 2 ( P L + P M + P N ) . 2(PL+PM+PN)\leq 3PG+PA+PB+PC\leq s+2(PL+PM+PN).
  101. k 1 ( P A ) t + k 2 ( P B ) t + k 3 ( P C ) t 2 t k 1 k 2 k 3 ( ( P D ) t k 1 + ( P E ) t k 2 + ( P F ) t k 3 ) , k_{1}\cdot(PA)^{t}+k_{2}\cdot(PB)^{t}+k_{3}\cdot(PC)^{t}\geq 2^{t}\sqrt{k_{1}k% _{2}k_{3}}\left(\frac{(PD)^{t}}{\sqrt{k_{1}}}+\frac{(PE)^{t}}{\sqrt{k_{2}}}+% \frac{(PF)^{t}}{\sqrt{k_{3}}}\right),
  102. k 1 ( P A ) t + k 2 ( P B ) t + k 3 ( P C ) t 2 k 1 k 2 k 3 ( ( P D ) t k 1 + ( P E ) t k 2 + ( P F ) t k 3 ) . k_{1}\cdot(PA)^{t}+k_{2}\cdot(PB)^{t}+k_{3}\cdot(PC)^{t}\geq 2\sqrt{k_{1}k_{2}% k_{3}}\left(\frac{(PD)^{t}}{\sqrt{k_{1}}}+\frac{(PE)^{t}}{\sqrt{k_{2}}}+\frac{% (PF)^{t}}{\sqrt{k_{3}}}\right).
  103. P A + P B + P C 6 r . PA+PB+PC\geq 6r.
  104. P A 3 + P B 3 + P C 3 + k ( P A P B P C ) 8 ( k + 3 ) r 3 PA^{3}+PB^{3}+PC^{3}+k\cdot(PA\cdot PB\cdot PC)\geq 8(k+3)r^{3}
  105. P A 2 + P B 2 + P C 2 + ( P A P B P C ) 2 / 3 16 r 2 ; PA^{2}+PB^{2}+PC^{2}+(PA\cdot PB\cdot PC)^{2/3}\geq 16r^{2};
  106. P A 2 + P B 2 + P C 2 + 2 ( P A P B P C ) 2 / 3 20 r 2 ; PA^{2}+PB^{2}+PC^{2}+2(PA\cdot PB\cdot PC)^{2/3}\geq 20r^{2};
  107. P A 4 + P B 4 + P C 4 + k ( P A P B P C ) 4 / 3 16 ( k + 3 ) r 4 PA^{4}+PB^{4}+PC^{4}+k(PA\cdot PB\cdot PC)^{4/3}\geq 16(k+3)r^{4}
  108. ( P A P B ) 3 / 2 + ( P B P C ) 3 / 2 + ( P C P A ) 3 / 2 12 R r 2 ; (PA\cdot PB)^{3/2}+(PB\cdot PC)^{3/2}+(PC\cdot PA)^{3/2}\geq 12Rr^{2};
  109. ( P A P B ) 2 + ( P B P C ) 2 + ( P C P A ) 2 8 ( R + r ) R r 2 ; (PA\cdot PB)^{2}+(PB\cdot PC)^{2}+(PC\cdot PA)^{2}\geq 8(R+r)Rr^{2};
  110. ( P A P B ) 2 + ( P B P C ) 2 + ( P C P A ) 2 48 r 4 ; (PA\cdot PB)^{2}+(PB\cdot PC)^{2}+(PC\cdot PA)^{2}\geq 48r^{4};
  111. ( P A P B ) 2 + ( P B P C ) 2 + ( P C P A ) 2 6 ( 7 R - 6 r ) r 3 . (PA\cdot PB)^{2}+(PB\cdot PC)^{2}+(PC\cdot PA)^{2}\geq 6(7R-6r)r^{3}.
  112. R r 2 , \frac{R}{r}\geq 2,
  113. R r a b c + a 3 + b 3 + c 3 2 a b c a b + b c + c a - 1 2 3 ( a b + b c + c a ) 2. \frac{R}{r}\geq\frac{abc+a^{3}+b^{3}+c^{3}}{2abc}\geq\frac{a}{b}+\frac{b}{c}+% \frac{c}{a}-1\geq\frac{2}{3}\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)% \geq 2.
  114. r R 4 a b c - a 3 - b 3 - c 3 2 a b c , \frac{r}{R}\geq\frac{4abc-a^{3}-b^{3}-c^{3}}{2abc},
  115. R r ( b + c ) 3 a + ( c + a ) 3 b + ( a + b ) 3 c 2 \frac{R}{r}\geq\frac{(b+c)}{3a}+\frac{(c+a)}{3b}+\frac{(a+b)}{3c}\geq 2
  116. ( R r ) 3 ( a b + b a ) ( b c + c b ) ( c a + a c ) 8. \left(\frac{R}{r}\right)^{3}\geq\left(\frac{a}{b}+\frac{b}{a}\right)\left(% \frac{b}{c}+\frac{c}{b}\right)\left(\frac{c}{a}+\frac{a}{c}\right)\geq 8.
  117. R r 2 ( a 2 + b 2 + c 2 ) a b + b c + c a ; \frac{R}{r}\geq\frac{2(a^{2}+b^{2}+c^{2})}{ab+bc+ca};
  118. a 3 + b 3 + c 3 8 s ( R 2 - r 2 ) a^{3}+b^{3}+c^{3}\leq 8s(R^{2}-r^{2})
  119. r ( r + 4 R ) 3 T r(r+4R)\geq\sqrt{3}\cdot T
  120. s 3 r + 4 R s\sqrt{3}\leq r+4R
  121. s 2 16 R r - 5 r 2 s^{2}\geq 16Rr-5r^{2}
  122. 2 R 2 + 10 R r - r 2 - 2 ( R - 2 r ) R 2 - 2 R r s 2 2R^{2}+10Rr-r^{2}-2(R-2r)\sqrt{R^{2}-2Rr}\leq s^{2}
  123. 2 R 2 + 10 R r - r 2 + 2 ( R - 2 r ) R 2 - 2 R r \leq 2R^{2}+10Rr-r^{2}+2(R-2r)\sqrt{R^{2}-2Rr}
  124. 9 r 2 T 1 a + 1 b + 1 c 9 R 4 T . \frac{9r}{2T}\leq\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\leq\frac{9R}{4T}.
  125. s ( 3 3 - 4 ) r + 2 R . s\leq(3\sqrt{3}-4)r+2R.
  126. A I I D + B I I E + C I I F 3. \frac{AI}{ID}+\frac{BI}{IE}+\frac{CI}{IF}\geq 3.
  127. cos A cos B cos C ( r R 2 ) 2 . \cos A\cdot\cos B\cdot\cos C\leq\left(\frac{r}{R\sqrt{2}}\right)^{2}.
  128. 18 R 3 ( a 2 + b 2 + c 2 ) R + a b c 3 18R^{3}\geq(a^{2}+b^{2}+c^{2})R+abc\sqrt{3}
  129. a 2 / 3 + b 2 / 3 + c 2 / 3 3 7 / 4 R 3 / 2 . a^{2/3}+b^{2/3}+c^{2/3}\leq 3^{7/4}R^{3/2}.
  130. a + b + c 3 3 R , a+b+c\leq 3\sqrt{3}\cdot R,
  131. 9 R 2 a 2 + b 2 + c 2 , 9R^{2}\geq a^{2}+b^{2}+c^{2},
  132. h a + h b + h c 3 3 R h_{a}+h_{b}+h_{c}\leq 3\sqrt{3}\cdot R
  133. m a 2 + m b 2 + m c 2 27 4 R 2 m_{a}^{2}+m_{b}^{2}+m_{c}^{2}\leq\frac{27}{4}R^{2}
  134. a b a + b + b c b + c + c a c + a 2 T R \frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\geq\frac{2T}{R}
  135. O U + O V + O W 3 2 R . OU+OV+OW\geq\frac{3}{2}R.
  136. O H < R , OH<R,
  137. 1 a + 1 b + 1 c 3 2 r , \frac{1}{a}+\frac{1}{b}+\frac{1}{c}\leq\frac{\sqrt{3}}{2r},
  138. 9 r h a + h b + h c 9r\leq h_{a}+h_{b}+h_{c}
  139. r a 2 + r b 2 + r c 2 6 r \sqrt{r_{a}^{2}+r_{b}^{2}+r_{c}^{2}}\geq 6r
  140. s ( a + b + c ) 2 ( r a + r b + r c ) \sqrt{s}(\sqrt{a}+\sqrt{b}+\sqrt{c})\leq\sqrt{2}(r_{a}+r_{b}+r_{c})
  141. a b c r a 3 r a + b 3 r b + c 3 r c . \frac{abc}{r}\geq\frac{a^{3}}{r_{a}}+\frac{b^{3}}{r_{b}}+\frac{c^{3}}{r_{c}}.
  142. r a r b m a m b + r b r c m b m c + r c r a m c m a 3. \frac{r_{a}r_{b}}{m_{a}m_{b}}+\frac{r_{b}r_{c}}{m_{b}m_{c}}+\frac{r_{c}r_{a}}{% m_{c}m_{a}}\geq 3.
  143. I H < r 2 , IH<r\sqrt{2},
  144. r 2 + r a 2 + r b 2 + r c 2 < 8 R 2 , r^{2}+r_{a}^{2}+r_{b}^{2}+r_{c}^{2}<8R^{2},
  145. 1 4 < A I B I C I A U B V C W 8 27 . \frac{1}{4}<\frac{AI\cdot BI\cdot CI}{AU\cdot BV\cdot CW}\leq\frac{8}{27}.
  146. 1 I X + 1 I Y + 1 I Z 3 R \frac{1}{IX}+\frac{1}{IY}+\frac{1}{IZ}\geq\frac{3}{R}
  147. 0 ( I X - I A ) + ( I Y - I B ) + ( I Z - I C ) 2 ( R - 2 r ) . 0\leq(IX-IA)+(IY-IB)+(IZ-IC)\leq 2(R-2r).
  148. E F 2 + F D 2 + D E 2 s 2 3 EF^{2}+FD^{2}+DE^{2}\leq\frac{s^{2}}{3}
  149. Perimeter of hexagon 2 3 ( Perimeter of triangle ) . \,\text{Perimeter of hexagon}\leq\frac{2}{3}(\,\text{Perimeter of triangle}).
  150. Area(DEF) min(Area(BED), Area(CFE), Area(ADF)) . \,\text{Area(DEF)}\geq\,\text{min(Area(BED), Area(CFE), Area(ADF))}.
  151. 1 x a x b 2 2 3 0.94. 1\geq\frac{x_{a}}{x_{b}}\geq\frac{2\sqrt{2}}{3}\approx 0.94.
  152. Area of triangle Area of inscribed square 2. \frac{\,\text{Area of triangle}}{\,\text{Area of inscribed square}}\geq 2.
  153. d s < d u < d v < 1 3 . \frac{d}{s}<\frac{d}{u}<\frac{d}{v}<\frac{1}{3}.
  154. a + b c 2 . a+b\leq c\sqrt{2}.
  155. 2 r c ( 2 - 1 ) , 2r\leq c(\sqrt{2}-1),
  156. h c 2 4 ( a + b ) . h_{c}\leq\frac{\sqrt{2}}{4}(a+b).
  157. 2 a c a + c > t > a c 2 a + c . \frac{2ac}{a+c}>t>\frac{ac\sqrt{2}}{a+c}.
  158. P A + P B > P C , P B + P C > P A , P C + P A > P B . PA+PB>PC,\quad PB+PC>PA,\quad PC+PA>PB.
  159. 4 ( P D 2 + P E 2 + P F 2 ) P A 2 + P B 2 + P C 2 . 4(PD^{2}+PE^{2}+PF^{2})\geq PA^{2}+PB^{2}+PC^{2}.
  160. d 2 ( b 2 + c 2 - a 2 ) + e 2 ( a 2 + c 2 - b 2 ) + f 2 ( a 2 + b 2 - c 2 ) 16 T S , d^{2}(b^{2}+c^{2}-a^{2})+e^{2}(a^{2}+c^{2}-b^{2})+f^{2}(a^{2}+b^{2}-c^{2})\geq 1% 6TS,\,
  161. c > f . c>f.
  162. cot A ( cot E + cot F ) + cot B ( cot F + cot D ) + cot C ( cot D + cot E ) 2. \cot A(\cot E+\cot F)+\cot B(\cot F+\cot D)+\cot C(\cot D+\cot E)\geq 2.

List_of_uniform_polyhedra_by_spherical_triangle.html

  1. π p π q π r {\pi\over p}\ {\pi\over q}\ {\pi\over r}
  2. π 3 π 3 π 2 {\pi\over 3}\ {\pi\over 3}\ {\pi\over 2}
  3. π 4 π 3 π 2 {\pi\over 4}\ {\pi\over 3}\ {\pi\over 2}
  4. π 5 π 3 π 2 {\pi\over 5}\ {\pi\over 3}\ {\pi\over 2}
  5. a π 3 b π 3 c π 2 {a\pi\over 3}\ {b\pi\over 3}\ {c\pi\over 2}
  6. π p π q π r {\pi\over p}\ {\pi\over q}\ {\pi\over r}
  7. π 3 π 2 2 π 3 {\pi\over 3}\ {\pi\over 2}\ {2\pi\over 3}
  8. a π 4 b π 3 c π 2 {a\pi\over 4}\ {b\pi\over 3}\ {c\pi\over 2}
  9. π p π q π r {\pi\over p}\ {\pi\over q}\ {\pi\over r}
  10. π 4 2 π 3 π 2 {\pi\over 4}\ {2\pi\over 3}\ {\pi\over 2}
  11. 3 π 4 π 3 π 2 {3\pi\over 4}\ {\pi\over 3}\ {\pi\over 2}
  12. 3 π 4 2 π 3 π 2 {3\pi\over 4}\ {2\pi\over 3}\ {\pi\over 2}
  13. a π 5 b π 3 c π 2 {a\pi\over 5}\ {b\pi\over 3}\ {c\pi\over 2}
  14. π p π q π r {\pi\over p}\ {\pi\over q}\ {\pi\over r}
  15. 2 π 5 π 3 π 2 {2\pi\over 5}\ {\pi\over 3}\ {\pi\over 2}
  16. 3 π 5 π 3 π 2 {3\pi\over 5}\ {\pi\over 3}\ {\pi\over 2}
  17. a π 5 b π 5 c π 2 {a\pi\over 5}\ {b\pi\over 5}\ {c\pi\over 2}
  18. π p π q π r {\pi\over p}\ {\pi\over q}\ {\pi\over r}
  19. π 5 2 π 5 π 2 {\pi\over 5}\ {2\pi\over 5}\ {\pi\over 2}
  20. π 5 3 π 5 π 2 {\pi\over 5}\ {3\pi\over 5}\ {\pi\over 2}
  21. a π 3 b π 3 c π 3 {a\pi\over 3}\ {b\pi\over 3}\ {c\pi\over 3}
  22. π p π q π r {\pi\over p}\ {\pi\over q}\ {\pi\over r}
  23. π 3 π 3 2 π 3 {\pi\over 3}\ {\pi\over 3}\ {2\pi\over 3}
  24. a π 4 b π 3 c π 3 {a\pi\over 4}\ {b\pi\over 3}\ {c\pi\over 3}
  25. π p π q π r {\pi\over p}\ {\pi\over q}\ {\pi\over r}
  26. a π 5 b π 3 c π 3 {a\pi\over 5}\ {b\pi\over 3}\ {c\pi\over 3}
  27. π p π q π r {\pi\over p}\ {\pi\over q}\ {\pi\over r}
  28. 3 π 5 π 3 π 3 {3\pi\over 5}\ {\pi\over 3}\ {\pi\over 3}
  29. π 5 2 π 3 π 3 {\pi\over 5}\ {2\pi\over 3}\ {\pi\over 3}
  30. a π 4 b π 4 c π 3 {a\pi\over 4}\ {b\pi\over 4}\ {c\pi\over 3}
  31. π p π q π r {\pi\over p}\ {\pi\over q}\ {\pi\over r}
  32. π 4 π 3 3 π 4 {\pi\over 4}\ {\pi\over 3}\ {3\pi\over 4}
  33. π 4 π 4 2 π 3 {\pi\over 4}\ {\pi\over 4}\ {2\pi\over 3}
  34. a π 5 b π 5 c π 3 {a\pi\over 5}\ {b\pi\over 5}\ {c\pi\over 3}
  35. π p π q π r {\pi\over p}\ {\pi\over q}\ {\pi\over r}
  36. π 3 2 π 5 3 π 5 {\pi\over 3}\ {2\pi\over 5}\ {3\pi\over 5}
  37. π 3 π 5 4 π 5 {\pi\over 3}\ {\pi\over 5}\ {4\pi\over 5}
  38. π 5 π 5 2 π 3 {\pi\over 5}\ {\pi\over 5}\ {2\pi\over 3}
  39. π 5 π 3 3 π 5 {\pi\over 5}\ {\pi\over 3}\ {3\pi\over 5}
  40. a π 5 b π 5 c π 5 {a\pi\over 5}\ {b\pi\over 5}\ {c\pi\over 5}
  41. π p π q π r {\pi\over p}\ {\pi\over q}\ {\pi\over r}
  42. 2 π 5 3 π 5 3 π 5 {2\pi\over 5}\ {3\pi\over 5}\ {3\pi\over 5}

Lithium_niobate.html

  1. n e 2 = 5.35583 + 4.629 × 10 - 7 f + 0.100473 + 3.862 × 10 - 8 f λ 2 - ( 0.20692 - 0.89 × 10 - 8 f ) 2 + 100 + 2.657 × 10 - 5 f λ 2 - ( 11.34927 ) 2 - 1.5334 × 10 - 2 λ 2 n^{2}_{e}=5.35583+4.629\times 10^{-7}f+{0.100473+3.862\times 10^{-8}f\over% \lambda^{2}-(0.20692-0.89\times 10^{-8}f)^{2}}+{100+2.657\times 10^{-5}f\over% \lambda^{2}-(11.34927)^{2}}-1.5334\times 10^{-2}\lambda^{2}
  2. n e 2 = 5.39121 + 4.968 × 10 - 7 f + 0.100473 + 3.862 × 10 - 8 f λ 2 - ( 0.20692 - 0.89 × 10 - 8 f ) 2 + 100 + 2.657 × 10 - 5 f λ 2 - ( 11.34927 ) 2 - ( 1.544 × 10 - 2 + 9.62119 × 10 - 10 λ ) λ 2 n^{2}_{e}=5.39121+4.968\times 10^{-7}f+{0.100473+3.862\times 10^{-8}f\over% \lambda^{2}-(0.20692-0.89\times 10^{-8}f)^{2}}+{100+2.657\times 10^{-5}f\over% \lambda^{2}-(11.34927)^{2}}-(1.544\times 10^{-2}+9.62119\times 10^{-10}\lambda% )\lambda^{2}
  3. n 2 = a 1 + b 1 f + a 2 + b 2 f λ 2 - ( a 3 + b 3 f ) 2 + a 4 + b 4 f λ 2 - a 5 2 - a 6 λ 2 n^{2}=a_{1}+b_{1}f+{a_{2}+b_{2}f\over\lambda^{2}-(a_{3}+b_{3}f)^{2}}+{a_{4}+b_% {4}f\over\lambda^{2}-a_{5}^{2}}-a_{6}\lambda^{2}

Live_variable_analysis.html

  1. GEN [ s ] {\mbox{GEN}~{}}[s]
  2. KILL [ s ] {\mbox{KILL}~{}}[s]
  3. LIVE i n [ s ] = GEN [ s ] ( LIVE o u t [ s ] - KILL [ s ] ) {\mbox{LIVE}~{}}_{in}[s]={\mbox{GEN}~{}}[s]\cup({\mbox{LIVE}~{}}_{out}[s]-{% \mbox{KILL}~{}}[s])
  4. LIVE o u t [ f i n a l ] = {\mbox{LIVE}~{}}_{out}[final]={\emptyset}
  5. LIVE o u t [ s ] = p s u c c [ S ] LIVE i n [ p ] {\mbox{LIVE}~{}}_{out}[s]=\bigcup_{p\in succ[S]}{\mbox{LIVE}~{}}_{in}[p]
  6. GEN [ d : y f ( x 1 , , x n ) ] = { x 1 , , x n } {\mbox{GEN}~{}}[d:y\leftarrow f(x_{1},\cdots,x_{n})]=\{x_{1},...,x_{n}\}
  7. KILL [ d : y f ( x 1 , , x n ) ] = { y } {\mbox{KILL}~{}}[d:y\leftarrow f(x_{1},\cdots,x_{n})]=\{y\}

Loading_dose.html

  1. Loading dose = C p V d F S \mbox{Loading dose}~{}=\frac{C_{p}V_{d}}{FS}

Local_consistency.html

  1. V V
  2. { 1 , 2 , 3 , 4 } \left\{1,2,3,4\right\}
  3. V 3 V\leq 3
  4. { 1 , 2 , 3 } \left\{1,2,3\right\}
  5. x i x_{i}
  6. x j x_{j}
  7. a a
  8. x i x_{i}
  9. b b
  10. x j x_{j}
  11. ( a , b ) (a,b)
  12. x i x_{i}
  13. x j x_{j}
  14. x < y x<y
  15. x x
  16. y y
  17. y y
  18. x 3 x_{3}
  19. x 2 x_{2}
  20. x 2 x_{2}
  21. x 3 x_{3}
  22. x 2 x_{2}
  23. x i x_{i}
  24. x j x_{j}
  25. x k x_{k}
  26. ( a , b ) (a,b)
  27. x i x_{i}
  28. x j x_{j}
  29. c c
  30. x k x_{k}
  31. ( a , c ) (a,c)
  32. ( b , c ) (b,c)
  33. x i x_{i}
  34. x k x_{k}
  35. x j x_{j}
  36. x k x_{k}
  37. i - 1 i-1
  38. i i
  39. i - 1 i-1
  40. i i
  41. j j
  42. j i j\leq i
  43. x 1 = 1 x_{1}=1
  44. x 2 = 1 x_{2}=1
  45. x 3 = 1 x_{3}=1
  46. n n
  47. n n
  48. alldifferent ( x 1 , , x n ) \mathop{\rm alldifferent}(x_{1},\ldots,x_{n})
  49. x i x j x_{i}\not=x_{j}
  50. i j i\not=j
  51. I I
  52. i i
  53. x 1 , , x n x_{1},\ldots,x_{n}
  54. x 1 , , x n x_{1},\ldots,x_{n}
  55. x i x_{i}
  56. x j x_{j}
  57. i < j i<j
  58. x i , x j x_{i},x_{j}
  59. x z x_{z}
  60. i , j < z i,j<z
  61. x 1 , , x n x_{1},\ldots,x_{n}
  62. x n x_{n}
  63. x 1 x_{1}
  64. x j x_{j}
  65. j j
  66. x j x_{j}
  67. x 1 = 2 x_{1}=2
  68. x 2 x_{2}
  69. x 2 = 3 x_{2}=3
  70. x 3 x_{3}
  71. x 1 x_{1}
  72. x 3 x_{3}
  73. x 3 x_{3}
  74. x 2 x_{2}
  75. x 2 = 3 x_{2}=3
  76. x 2 x_{2}
  77. x 2 = 3 x_{2}=3
  78. x 1 = 2 x_{1}=2
  79. x 1 = 3 x_{1}=3
  80. x n x_{n}
  81. x 1 x_{1}
  82. x z x_{z}
  83. x i , x j x_{i},x_{j}
  84. i , j < z i,j<z
  85. x z x_{z}
  86. x i x_{i}
  87. x z x_{z}
  88. x j x_{j}
  89. x z x_{z}
  90. x i x_{i}
  91. x j x_{j}
  92. x i x_{i}
  93. x j x_{j}
  94. i i
  95. i - 1 i-1
  96. i i
  97. i - 1 i-1
  98. i i
  99. i i
  100. i i
  101. x k x_{k}
  102. i - 1 i-1
  103. k k
  104. x k x_{k}
  105. x k x_{k}
  106. i i
  107. i i
  108. i i
  109. i i
  110. x i x_{i}
  111. x i x_{i}
  112. i i
  113. i i
  114. i i
  115. x i x_{i}
  116. x i x_{i}
  117. x i x_{i}
  118. x i x_{i}
  119. x i x_{i}
  120. x i x_{i}
  121. C C
  122. X X
  123. x x
  124. X \ { x } X\backslash\{x\}
  125. x x
  126. C C
  127. i i
  128. m m
  129. m m
  130. m m
  131. m m
  132. m m
  133. m m
  134. m m
  135. m m
  136. m m
  137. k k
  138. k < m k<m
  139. m m
  140. ( i , m ) (i,m)
  141. i i
  142. m m
  143. m m
  144. k k
  145. x 1 = a 1 , , x k = a k x_{1}=a_{1},\ldots,x_{k}=a_{k}
  146. x k + 1 x_{k+1}
  147. m m
  148. m m
  149. m m
  150. m m
  151. m m
  152. m m
  153. m m
  154. m + 1 m+1
  155. M M
  156. M i j M_{ij}
  157. i i
  158. x i x_{i}
  159. j j
  160. x j x_{j}
  161. k k
  162. k + 1 k+1
  163. k k
  164. k + 1 k+1
  165. i i
  166. i - 1 i-1

Local_nonsatiation.html

  1. x X x\in X
  2. ε > 0 \varepsilon>0
  3. y X y\in X
  4. y - x ε \|y-x\|\leq\varepsilon
  5. y y
  6. x x

Local_regression.html

  1. n n
  2. α \alpha
  3. ( λ + 1 ) / n \left(\lambda+1\right)/n
  4. λ \lambda
  5. α \alpha
  6. n α n\alpha
  7. α \alpha
  8. α \alpha
  9. α \alpha
  10. w ( x ) = ( 1 - | x | 3 ) 3 I [ | x | < 1 ] w(x)=(1-|x|^{3})^{3}\operatorname{I}\left[\left|x\right|<1\right]

Local_search_(constraint_satisfaction).html

  1. p p
  2. p p
  3. 1 - p 1-p
  4. d d
  5. e - d T e^{-d\cdot T}
  6. T T
  7. T T
  8. C o s t ( x = a ) Cost(x=a)
  9. x = a x=a
  10. y 1 , , y n y_{1},\ldots,y_{n}
  11. x x
  12. V i o l a t e s ( x = a , y i = b ) Violates(x=a,y_{i}=b)
  13. x = a , y i = b x=a,y_{i}=b
  14. x x
  15. y y
  16. C o s t ( x = a ) = i = 1 , , n min y i = b ( C o s t ( y i = b ) + V i o l a t e s ( x = a , y i = b ) ) Cost(x=a)=\sum_{i=1,\ldots,n}\min_{y_{i}=b}(Cost(y_{i}=b)+Violates(x=a,y_{i}=b))

Local_system.html

  1. γ : [ 0 , 1 ] X \gamma:[0,1]\to X
  2. γ * : E γ ( 0 ) E γ ( 1 ) \gamma_{*}:E_{\gamma(0)}\to E_{\gamma(1)}

Locally_integrable_function.html

  1. Ω Ω
  2. f : Ω f:Ω→ℂ
  3. f f
  4. Ω Ω
  5. K | f | d x < + , \int_{K}|f|\,\mathrm{d}x<+\infty,
  6. K K
  7. Ω Ω
  8. f f
  9. L 1 , loc ( Ω ) = { f : Ω measurable | f | K L 1 ( K ) K Ω , K compact } , L_{1,\mathrm{loc}}(\Omega)=\bigl\{f:\Omega\to\mathbb{C}\,\text{ measurable}\,% \big|\,f|_{K}\in L_{1}(K)\ \forall\,K\subset\Omega,\,K\,\text{ compact}\bigr\},
  10. f f
  11. K K
  12. ( X , Σ , μ ) (X, Σ,μ)
  13. Ω Ω
  14. f : Ω f:Ω→ℂ
  15. Ω | f φ | d x < + , \int_{\Omega}|f\varphi|\,\mathrm{d}x<+\infty,
  16. φ C c ( Ω ) φ∈{C}_{c}^{∞}(Ω)
  17. C c ( Ω ) {C}_{c}^{∞}(Ω)
  18. φ : Ω φ:Ω→ℝ
  19. Ω Ω
  20. f : Ω f:Ω→ℂ
  21. K | f | d x < + K Ω , K compact Ω | f φ | d x < + φ C c ( Ω ) . \int_{K}|f|\,\mathrm{d}x<+\infty\quad\forall\,K\subset\Omega,\,K\,\text{ % compact}\quad\Longleftrightarrow\quad\int_{\Omega}|f\varphi|\,\mathrm{d}x<+% \infty\quad\forall\,\varphi\in C^{\infty}_{\mathrm{c}}(\Omega).
  22. φ C c ( Ω ) φ∈{C}_{c}^{∞}(Ω)
  23. K K
  24. Ω | f φ | d x = K | f | | φ | d x φ K | f | d x < \int_{\Omega}|f\varphi|\,\mathrm{d}x=\int_{K}|f|\,|\varphi|\,\mathrm{d}x\leq\|% \varphi\|_{\infty}\int_{K}|f|\,\mathrm{d}x<\infty
  25. K K
  26. Ω Ω
  27. K K
  28. K K
  29. Ω ∂Ω
  30. Δ := d ( K , Ω ) > 0 , \Delta:=d(K,\partial\Omega)>0,
  31. δ δ
  32. Δ > 2 δ > 0 Δ>2δ>0
  33. Ω ∂Ω
  34. Δ = Δ=∞
  35. δ δ
  36. 2 δ
  37. K K
  38. K K δ K 2 δ Ω , d ( K δ , Ω ) = Δ - δ > δ > 0. K\subset K_{\delta}\subset K_{2\delta}\subset\Omega,\qquad d(K_{\delta},% \partial\Omega)=\Delta-\delta>\delta>0.
  39. φ K ( x ) = χ K δ φ δ ( x ) = n χ K δ ( y ) φ δ ( x - y ) d y , \varphi_{K}(x)={\chi_{K_{\delta}}\ast\varphi_{\delta}(x)}=\int_{\mathbb{R}^{n}% }\chi_{K_{\delta}}(y)\,\varphi_{\delta}(x-y)\,\mathrm{d}y,
  40. x K x∈K
  41. f f
  42. K | f | d x = Ω | f | χ K d x Ω | f | φ K d x < . \int_{K}|f|\,\mathrm{d}x=\int_{\Omega}|f|\chi_{K}\,\mathrm{d}x\leq\int_{\Omega% }|f|\varphi_{K}\,\mathrm{d}x<\infty.
  43. K K
  44. Ω Ω
  45. f f
  46. Ω Ω
  47. f : Ω f:Ω→
  48. p p
  49. 1 p + 1≤p≤+∞
  50. f f
  51. K | f | p d x < + , \int_{K}|f|^{p}\,\mathrm{d}x<+\infty,
  52. K K
  53. Ω Ω
  54. f f
  55. p p
  56. p p
  57. L p , loc ( Ω ) = { f : Ω measurable | f L p ( K ) , K Ω , K compact } . L_{p,\mathrm{loc}}(\Omega)=\left\{f:\Omega\to\mathbb{C}\,\text{ measurable }% \left|\ f\in L_{p}(K),\ \forall\,K\subset\Omega,K\,\text{ compact}\right.% \right\}.
  58. p p
  59. p p
  60. p p
  61. L loc p ( Ω ) , L^{p}_{\mathrm{loc}}(\Omega),
  62. L p , loc ( Ω ) , L_{p,\mathrm{loc}}(\Omega),
  63. L p ( Ω , loc ) , L_{p}(\Omega,\mathrm{loc}),
  64. d ( u , v ) = k 1 1 2 k u - v p , ω k 1 + u - v p , ω k u , v L p , loc ( Ω ) , d(u,v)=\sum_{k\geq 1}\frac{1}{2^{k}}\frac{\|u-v\|_{p,\omega_{k}}}{1+\|u-v\|_{p% ,\omega_{k}}}\qquad u,v\in L_{p,\mathrm{loc}}(\Omega),
  65. p , ω k + \scriptstyle{\|\cdot\|_{p,\omega_{k}}}\to\mathbb{R}^{+}
  66. u p , ω k = ω k | u | p d x u L p , loc ( Ω ) . {\|u\|_{p,\omega_{k}}}=\int_{\omega_{k}}|u|^{p}\,\mathrm{d}x\qquad\forall\,u% \in L_{p,\mathrm{loc}}(\Omega).
  67. f f
  68. 1 p + 1≤p≤+∞
  69. Ω Ω
  70. p = 1 p=1
  71. K K
  72. Ω Ω
  73. p + p≤+∞
  74. | Ω | χ K | q d x | 1 / q = | K d x | 1 / q = | μ ( K ) | 1 / q < + , \left|{\int_{\Omega}|\chi_{K}|^{q}\,\mathrm{d}x}\right|^{1/q}=\left|{\int_{K}% \mathrm{d}x}\right|^{1/q}=|\mu(K)|^{1/q}<+\infty,
  75. q q
  76. 1 / p + 1 / q 1/p+1/q
  77. 1 1
  78. 1 p + 1≤p≤+∞
  79. μ ( K ) μ(K)
  80. K K
  81. K | f | d x = Ω | f χ K | d x | Ω | f | p d x | 1 / p | K d x | 1 / q = f p | μ ( K ) | 1 / q < + , {\int_{K}|f|\,\mathrm{d}x}={\int_{\Omega}|f\chi_{K}|\,\mathrm{d}x}\leq\left|{% \int_{\Omega}|f|^{p}\,\mathrm{d}x}\right|^{1/p}\left|{\int_{K}\mathrm{d}x}% \right|^{1/q}=\|f\|_{p}|\mu(K)|^{1/q}<+\infty,
  82. f L 1 , loc ( Ω ) . f\in L_{1,\mathrm{loc}}(\Omega).
  83. K | f | d x = Ω | f χ K | d x | K | f | p d x | 1 / p | K d x | 1 / q = f p | μ ( K ) | 1 / q < + , {\int_{K}|f|\,\mathrm{d}x}={\int_{\Omega}|f\chi_{K}|\,\mathrm{d}x}\leq\left|{% \int_{K}|f|^{p}\,\mathrm{d}x}\right|^{1/p}\left|{\int_{K}\mathrm{d}x}\right|^{% 1/q}=\|f\|_{p}|\mu(K)|^{1/q}<+\infty,
  84. f f
  85. p p
  86. f f
  87. f f
  88. 1 1
  89. f ( x ) = { 1 / x x 0 , 0 x = 0 , f(x)=\begin{cases}1/x&x\neq 0,\\ 0&x=0,\end{cases}
  90. x = 0 x=0
  91. Ω Ω
  92. f ( x ) = { e 1 / x x 0 , 0 x = 0 , f(x)=\begin{cases}e^{1/x}&x\neq 0,\\ 0&x=0,\end{cases}
  93. f ( x ) = { k 1 e 1 / x 2 x > 0 , 0 x = 0 , k 2 e 1 / x 2 x < 0 , f(x)=\begin{cases}k_{1}e^{1/x^{2}}&x>0,\\ 0&x=0,\\ k_{2}e^{1/x^{2}}&x<0,\end{cases}
  94. x 3 d f d x + 2 f = 0. x^{3}\frac{\mathrm{d}f}{\mathrm{d}x}+2f=0.
  95. k < s u b > 1 k<sub>1
  96. K Ω K⋐Ω
  97. K Ω K⊂⊂Ω
  98. K K
  99. Ω Ω
  100. W < s u p > k , p ( Ω ) W<sup>k,p(Ω)
  101. L < s u b > 1 , l o c ( Ω ) L<sub>1,loc(Ω)
  102. E E
  103. μ μ
  104. 𝔛 𝔛
  105. E E
  106. E E
  107. E E
  108. μ μ

LOCC.html

  1. | ψ 1 = 1 2 ( | 0 A | 0 B + | 1 A | 1 B ) |\psi_{1}\rangle=\frac{1}{\sqrt{2}}\left(|0\rangle_{A}\otimes|0\rangle_{B}+|1% \rangle_{A}\otimes|1\rangle_{B}\right)
  2. | ψ 2 = 1 2 ( | 0 A | 1 B + | 1 A | 0 B ) |\psi_{2}\rangle=\frac{1}{\sqrt{2}}\left(|0\rangle_{A}\otimes|1\rangle_{B}+|1% \rangle_{A}\otimes|0\rangle_{B}\right)
  3. | 0 A | 0 B |0\rangle_{A}\otimes|0\rangle_{B}
  4. | ψ 1 |\psi_{1}\rangle
  5. 2 n \mathbb{C}^{2n}
  6. 2 n \mathbb{C}^{2}\otimes\mathbb{C}^{n}
  7. d d
  8. | ψ |\psi\rangle
  9. | ϕ |\phi\rangle
  10. | ψ = i λ i | i A | i B |\psi\rangle=\sum_{i}\sqrt{\lambda_{i}}|i_{A}\rangle\otimes|i_{B}\rangle
  11. | ϕ = i λ i | i A | i B |\phi\rangle=\sum_{i}\sqrt{\lambda_{i}^{\prime}}|i_{A}^{\prime}\rangle\otimes|% i_{B}^{\prime}\rangle
  12. λ i \sqrt{\lambda_{i}}
  13. λ 1 > λ d \lambda_{1}>\lambda_{d}
  14. | ψ |\psi\rangle
  15. | ϕ |\phi\rangle
  16. k k
  17. 1 k d 1\leq k\leq d
  18. i = 1 k λ i i = 1 k λ i \sum_{i=1}^{k}\lambda_{i}\leq\sum_{i=1}^{k}\lambda_{i}^{\prime}
  19. | ψ | ϕ iff λ λ |\psi\rangle\rightarrow|\phi\rangle\quad\,\text{iff}\quad\lambda\prec\lambda^{\prime}
  20. | ψ |\psi\rangle
  21. | ϕ |\phi\rangle
  22. d d
  23. d d

Log_probability.html

  1. [ 0 , 1 ] [0,1]
  2. [ 0 , 1 ] [0,1]
  3. x [ 0 , 1 ] x\in[0,1]
  4. x = - log ( x ) x^{\prime}=-\log(x)\in\mathbb{R}
  5. x y x\cdot y
  6. x + y x^{\prime}+y^{\prime}
  7. x + y x+y
  8. - log ( e - x + e - y ) -\log(e^{-x^{\prime}}+e^{-y^{\prime}})

Log_wind_profile.html

  1. u u
  2. z z
  3. u z = u * κ [ ln ( z - d z 0 ) + ψ ( z , z 0 , L ) ] u_{z}=\frac{u_{*}}{\kappa}\left[\ln\left(\frac{z-d}{z_{0}}\right)+\psi(z,z_{0}% ,L)\right]
  4. u * u_{*}
  5. κ \kappa
  6. d d
  7. z 0 z_{0}
  8. ψ \psi
  9. L L
  10. z / L = 0 z/L=0
  11. ψ \psi
  12. d d
  13. z 0 z_{0}
  14. z 0 z_{0}

Logarithmic_growth.html

  1. 1 + 1 2 + 1 3 + 1 4 + 1 5 + 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\cdots

Logarithmic_mean.html

  1. M lm ( x , y ) = lim ( ξ , η ) ( x , y ) η - ξ ln η - ln ξ , = { 0 if x = 0 or y = 0 , x if x = y , y - x ln y - ln x otherwise, \begin{array}[]{ll}M_{\,\text{lm}}(x,y)&=\lim_{(\xi,\eta)\to(x,y)}\frac{\eta-% \xi}{\ln\eta-\ln\xi},\\ &=\begin{cases}0&\,\text{if }x=0\,\text{ or }y=0,\\ x&\,\text{if }x=y,\\ \frac{y-x}{\ln y-\ln x}&\,\text{otherwise,}\end{cases}\end{array}
  2. x , y x,y
  3. x y M lm ( x , y ) x + y 2 for all x 0 and y 0. \sqrt{x\cdot y}\leq M_{\,\text{lm}}(x,y)\leq\frac{x+y}{2}\qquad\,\text{ for % all }x\geq 0\,\text{ and }y\geq 0.
  4. ξ [ x , y ] : f ( ξ ) = f ( x ) - f ( y ) x - y \exists\xi\in[x,y]:\ f^{\prime}(\xi)=\frac{f(x)-f(y)}{x-y}
  5. ξ \xi
  6. ln \ln
  7. f f
  8. 1 ξ = ln x - ln y x - y \frac{1}{\xi}=\frac{\ln x-\ln y}{x-y}
  9. ξ \xi
  10. ξ = x - y ln x - ln y \xi=\frac{x-y}{\ln x-\ln y}
  11. L ( x , y ) = 0 1 x 1 - t y t d t L(x,y)=\int_{0}^{1}x^{1-t}y^{t}\ \mathrm{d}t
  12. 0 1 x 1 - t y t d t = 0 1 ( y x ) t x d t = x 0 1 ( y x ) t d t = x ln y x ( y x ) t | t = 0 1 = x ln y x ( y x - 1 ) = y - x ln y - ln x \begin{array}[]{rcl}\int_{0}^{1}x^{1-t}y^{t}\ \mathrm{d}t&=&\int_{0}^{1}\left(% \frac{y}{x}\right)^{t}x\ \mathrm{d}t\\ &=&x\int_{0}^{1}\left(\frac{y}{x}\right)^{t}\mathrm{d}t\\ &=&\frac{x}{\ln\frac{y}{x}}\left(\frac{y}{x}\right)^{t}|_{t=0}^{1}\\ &=&\frac{x}{\ln\frac{y}{x}}\left(\frac{y}{x}-1\right)\\ &=&\frac{y-x}{\ln y-\ln x}\end{array}
  13. x x
  14. y y
  15. L ( c x , c y ) = c L ( x , y ) L(c\cdot x,c\cdot y)=c\cdot L(x,y)
  16. n + 1 n+1
  17. n n
  18. L MV ( x 0 , , x n ) = ( - 1 ) ( n + 1 ) n ln [ x 0 , , x n ] - n L_{\mathrm{MV}}(x_{0},\dots,x_{n})=\sqrt[-n]{(-1)^{(n+1)}\cdot n\cdot\ln[x_{0}% ,\dots,x_{n}]}
  19. ln [ x 0 , , x n ] \ln[x_{0},\dots,x_{n}]
  20. n = 2 n=2
  21. L MV ( x , y , z ) = ( x - y ) ( y - z ) ( z - x ) 2 ( ( y - z ) ln x + ( z - x ) ln y + ( x - y ) ln z ) L_{\mathrm{MV}}(x,y,z)=\sqrt{\frac{(x-y)\cdot(y-z)\cdot(z-x)}{2\cdot((y-z)% \cdot\ln x+(z-x)\cdot\ln y+(x-y)\cdot\ln z)}}
  22. S S
  23. S = { ( α 0 , , α n ) : α 0 + + α n = 1 α 0 0 α n 0 } S=\{(\alpha_{0},\dots,\alpha_{n}):\alpha_{0}+\dots+\alpha_{n}=1\ \land\ \alpha% _{0}\geq 0\ \land\ \dots\ \land\ \alpha_{n}\geq 0\}
  24. d α \mathrm{d}\alpha
  25. L I ( x 0 , , x n ) = S x 0 α 0 x n α n d α L_{\mathrm{I}}(x_{0},\dots,x_{n})=\int_{S}x_{0}^{\alpha_{0}}\cdot\dots\cdot x_% {n}^{\alpha_{n}}\ \mathrm{d}\alpha
  26. L I ( x 0 , , x n ) = n ! exp [ ln x 0 , , ln x n ] L_{\mathrm{I}}(x_{0},\dots,x_{n})=n!\cdot\exp[\ln x_{0},\dots,\ln x_{n}]
  27. n = 2 n=2
  28. L I ( x , y , z ) = - 2 x ( ln y - ln z ) + y ( ln z - ln x ) + z ( ln x - ln y ) ( ln x - ln y ) ( ln y - ln z ) ( ln z - ln x ) L_{\mathrm{I}}(x,y,z)=-2\cdot\frac{x\cdot(\ln y-\ln z)+y\cdot(\ln z-\ln x)+z% \cdot(\ln x-\ln y)}{(\ln x-\ln y)\cdot(\ln y-\ln z)\cdot(\ln z-\ln x)}
  29. L ( x 2 , y 2 ) L ( x , y ) = x + y 2 \frac{L(x^{2},y^{2})}{L(x,y)}=\frac{x+y}{2}

Logical_constant.html

  1. \mathcal{L}
  2. \mathcal{L}
  3. \Box
  4. \Diamond

Logical_matrix.html

  1. M i , j = { 1 ( x i , y j ) R 0 ( x i , y j ) R M_{i,j}=\begin{cases}1&(x_{i},y_{j})\in R\\ 0&(x_{i},y_{j})\not\in R\end{cases}
  2. ( 1 1 1 1 0 1 0 1 0 0 1 0 0 0 0 1 ) . \begin{pmatrix}1&1&1&1\\ 0&1&0&1\\ 0&0&1&0\\ 0&0&0&1\end{pmatrix}.

Loiter_(aeronautics).html

  1. E = 1 C L D ln ( W i W f ) E=\frac{1}{C}\frac{L}{D}\ln\left(\frac{W_{i}}{W_{f}}\right)
  2. E E\,\!
  3. C C\,\!
  4. L L\,\!
  5. D D\,\!
  6. W i W_{i}\,\!
  7. W f W_{f}\,\!

Lomer–Cottrell_junction.html

  1. a 2 [ 0 1 1 ] a 6 [ 1 1 2 ] + a 6 [ -1 2 1 ] \frac{a}{2}[\,\text{0 1 1}]\rightarrow\frac{a}{6}[\,\text{1 1 2}]+\frac{a}{6}[% \,\text{-1 2 1}]
  2. a 2 [ 1 0 -1 ] a 6 [ 1 1 -2 ] + a 6 [ 2 -1 -1 ] \frac{a}{2}[\,\text{1 0 -1}]\rightarrow\frac{a}{6}[\,\text{1 1 -2}]+\frac{a}{6% }[\,\text{2 -1 -1}]
  3. a 6 [ 1 1 2 ] + a 6 [ 1 1 -2 ] a 3 [ 1 1 0 ] \frac{a}{6}[\,\text{1 1 2}]+\frac{a}{6}[\,\text{1 1 -2}]\rightarrow\frac{a}{3}% [\,\text{1 1 0}]

Lomo_LC-A.html

  1. 1 / 60 {1}/{60}
  2. 1 / 500 {1}/{500}
  3. 1 / 30 {1}/{30}

London_moment.html

  1. × 10 7 \times 10^{−}7
  2. × 10 15 \times 10^{−}15
  3. B = - 2 M Q ω B=-\frac{2M}{Q}\ \omega

London_penetration_depth.html

  1. λ \lambda
  2. λ L \lambda_{L}
  3. B ( x ) = B 0 exp ( - x λ L ) , B(x)=B_{0}\exp\left(-\frac{x}{\lambda_{L}}\right),
  4. λ L \lambda_{L}
  5. e e
  6. λ L \lambda_{L}
  7. λ L = m μ 0 n q 2 \lambda_{L}=\sqrt{\frac{m}{\mu_{0}nq^{2}}}
  8. m m
  9. n n
  10. q q

Long_Josephson_junction.html

  1. λ J \lambda_{J}
  2. ϕ ( t ) \phi(t)
  3. ϕ ( x , t ) \phi(x,t)
  4. ϕ ( x , y , t ) \phi(x,y,t)
  5. ϕ \phi
  6. λ J 2 ϕ x x - ω p - 2 ϕ t t - sin ( ϕ ) = ω c - 1 ϕ t - j / j c , \lambda_{J}^{2}\phi_{xx}-\omega_{p}^{-2}\phi_{tt}-\sin(\phi)=\omega_{c}^{-1}% \phi_{t}-j/j_{c},
  7. x x
  8. t t
  9. x x
  10. t t
  11. λ J \lambda_{J}
  12. ω p \omega_{p}
  13. ω c \omega_{c}
  14. j / j c j/j_{c}
  15. j j
  16. j c j_{c}
  17. ϕ x x - ϕ t t - sin ( ϕ ) = α ϕ t - γ , \phi_{xx}-\phi_{tt}-\sin(\phi)=\alpha\phi_{t}-\gamma,
  18. λ J \lambda_{J}
  19. ω p - 1 \omega_{p}^{-1}
  20. α = 1 / β c \alpha=1/\sqrt{\beta_{c}}
  21. β c \beta_{c}
  22. γ = j / j c \gamma=j/j_{c}
  23. ϕ ( x , t ) = A exp [ i ( k x - ω t ) ] \phi(x,t)=A\exp[i(kx-\omega t)]
  24. ϕ ( x , t ) = 4 arctan exp ( ± x - u t 1 - u 2 ) \phi(x,t)=4\arctan\exp\left(\pm\frac{x-ut}{\sqrt{1-u^{2}}}\right)
  25. x x
  26. t t
  27. u = v / c 0 u=v/c_{0}
  28. v v
  29. c 0 = λ J ω p c_{0}=\lambda_{J}\omega_{p}
  30. λ J \lambda_{J}
  31. ω p - 1 \omega_{p}^{-1}

Longest_repeated_substring_problem.html

  1. k k
  2. k k

Look-ahead_(backtracking).html

  1. x k x_{k}
  2. x k x_{k}
  3. x k x_{k}
  4. x i , x j x_{i},x_{j}
  5. x i , x j x_{i},x_{j}
  6. i < j i<j

Loop_theorem.html

  1. f : ( D 2 , D 2 ) ( M , M ) f\colon(D^{2},\partial D^{2})\to(M,\partial M)\,
  2. f | D 2 f|\partial D^{2}
  3. M \partial M
  4. M M
  5. S S
  6. M \partial M
  7. N π 1 ( S ) N\subset\pi_{1}(S)
  8. ker ( π 1 ( S ) π 1 ( M ) ) - N \mathop{\mathrm{ker}}(\pi_{1}(S)\to\pi_{1}(M))-N\neq\emptyset
  9. f : D 2 M f\colon D^{2}\to M\,
  10. f ( D 2 ) S f(\partial D^{2})\subset S\,
  11. [ f | D 2 ] N . [f|\partial D^{2}]\notin N.\,
  12. g : D 2 M g\colon D^{2}\to M\,
  13. g ( D 2 ) S g(\partial D^{2})\subset S\,
  14. [ g | D 2 ] N . [g|\partial D^{2}]\notin N.\,

Lorentz–Heaviside_units.html

  1. c c
  2. 4 π
  3. 4 π
  4. 4 π \sqrt{4}{π}
  5. 4 π \sqrt{4}{π}
  6. ε = 8.854 p F / m ε=8.854pF/m
  7. L = 0.01 m L=0.01m
  8. M = 0.001 k g M=0.001kg
  9. T = 1 T=1
  10. c c
  11. 𝐃 = ρ / β , 𝐁 = 0 , κ × 𝐄 = - 𝐁 t , κ × 𝐇 = 𝐃 t + 𝐉 / β , \begin{aligned}\displaystyle\nabla\cdot\mathbf{D}&\displaystyle=\rho/\beta,\\ \displaystyle\quad\nabla\cdot\mathbf{B}&\displaystyle=0,\\ \displaystyle\quad\kappa\nabla\times\mathbf{E}&\displaystyle=-\frac{\partial% \mathbf{B}}{\partial t},\\ \displaystyle\quad\kappa\nabla\times\mathbf{H}&\displaystyle=\frac{\partial% \mathbf{D}}{\partial t}+\mathbf{J}/\beta,\end{aligned}
  12. β β
  13. κ κ
  14. β = 1 / 4 π β=1/4π
  15. κ = c κ=c
  16. β = 1 β=1
  17. κ = c κ=c
  18. β = 1 β=1
  19. κ = 1 κ=1
  20. γ γ
  21. β β
  22. γ = 4 π β γ=4πβ
  23. γ = 1 γ=1
  24. β = 1 β=1
  25. 1 1
  26. 2 π
  27. 4 π
  28. κ κ
  29. Q = I κ t Q=Iκt
  30. κ = c κ=c
  31. κ = 1 κ=1
  32. Q = I t Q=It
  33. Q = I t Q=It
  34. Q = I κ t Q=Iκt
  35. 𝐄 = ρ \nabla\cdot\mathbf{E}=\rho\,
  36. 𝐁 = 0 \nabla\cdot\mathbf{B}=0\,
  37. × 𝐄 = - 1 c 𝐁 t \nabla\times\mathbf{E}=-\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t}\,
  38. × 𝐁 = 1 c 𝐄 t + 1 c 𝐉 \nabla\times\mathbf{B}=\frac{1}{c}\frac{\partial\mathbf{E}}{\partial t}+\frac{% 1}{c}\mathbf{J}\,
  39. c c
  40. 𝐄 = 𝐃 \mathbf{E}=\mathbf{D}
  41. 𝐇 = 𝐁 \mathbf{H}=\mathbf{B}
  42. ρ ρ
  43. 𝐉 \mathbf{J}
  44. 𝐅 q = q ( 𝐄 + 𝐯 q c × 𝐁 ) \mathbf{F}_{q}=q\left(\mathbf{E}+\frac{\mathbf{v}_{q}}{c}\times\mathbf{B}% \right)\,
  45. q q
  46. ε = 1 ε=1
  47. Q Q = q q / 4 π QQ=qq/4π
  48. 4 π \sqrt{4}{π}
  49. q LH = 4 π q G q_{\mathrm{LH}}\ =\ \sqrt{4\pi}\ q_{\mathrm{G}}
  50. 𝐄 LH = 𝐄 G 4 π \mathbf{E}_{\mathrm{LH}}\ =\ {\mathbf{E}_{\mathrm{G}}\over\sqrt{4\pi}}
  51. 𝐁 LH = 𝐁 G 4 π \mathbf{B}_{\mathrm{LH}}\ =\ {\mathbf{B}_{\mathrm{G}}\over\sqrt{4\pi}}
  52. 𝐃 = ρ f \nabla\cdot\mathbf{D}=\rho\text{f}
  53. 𝐃 = 4 π ρ f \nabla\cdot\mathbf{D}=4\pi\rho\text{f}
  54. 𝐃 = ρ f \nabla\cdot\mathbf{D}=\rho\text{f}
  55. 𝐄 = ρ \nabla\cdot\mathbf{E}=\rho
  56. 𝐄 = 4 π ρ \nabla\cdot\mathbf{E}=4\pi\rho
  57. 𝐄 = ρ / ϵ 0 \nabla\cdot\mathbf{E}=\rho/\epsilon_{0}
  58. 𝐁 = 0 \nabla\cdot\mathbf{B}=0
  59. 𝐁 = 0 \nabla\cdot\mathbf{B}=0
  60. 𝐁 = 0 \nabla\cdot\mathbf{B}=0
  61. × 𝐄 = - 1 c 𝐁 t \nabla\times\mathbf{E}=-\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t}
  62. × 𝐄 = - 1 c 𝐁 t \nabla\times\mathbf{E}=-\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t}
  63. × 𝐄 = - 𝐁 t \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}
  64. × 𝐇 = 1 c 𝐉 f + 1 c 𝐃 t \nabla\times\mathbf{H}=\frac{1}{c}\mathbf{J}_{\,\text{f}}+\frac{1}{c}\frac{% \partial\mathbf{D}}{\partial t}
  65. × 𝐇 = 4 π c 𝐉 f + 1 c 𝐃 t \nabla\times\mathbf{H}=\frac{4\pi}{c}\mathbf{J}_{\,\text{f}}+\frac{1}{c}\frac{% \partial\mathbf{D}}{\partial t}
  66. × 𝐇 = 𝐉 f + 𝐃 t \nabla\times\mathbf{H}=\mathbf{J}_{\,\text{f}}+\frac{\partial\mathbf{D}}{% \partial t}
  67. × 𝐁 = 1 c 𝐉 + 1 c 𝐄 t \nabla\times\mathbf{B}=\frac{1}{c}\mathbf{J}+\frac{1}{c}\frac{\partial\mathbf{% E}}{\partial t}
  68. × 𝐁 = 4 π c 𝐉 + 1 c 𝐄 t \nabla\times\mathbf{B}=\frac{4\pi}{c}\mathbf{J}+\frac{1}{c}\frac{\partial% \mathbf{E}}{\partial t}
  69. × 𝐁 = μ 0 𝐉 + 1 c 2 𝐄 t \nabla\times\mathbf{B}=\mu_{0}\mathbf{J}+\frac{1}{c^{2}}\frac{\partial\mathbf{% E}}{\partial t}
  70. 𝐅 = q ( 𝐄 + 1 c 𝐯 × 𝐁 ) \mathbf{F}=q\left(\mathbf{E}+\frac{1}{c}\mathbf{v}\times\mathbf{B}\right)
  71. 𝐅 = q ( 𝐄 + 1 c 𝐯 × 𝐁 ) \mathbf{F}=q\left(\mathbf{E}+\frac{1}{c}\mathbf{v}\times\mathbf{B}\right)
  72. 𝐅 = q ( 𝐄 + 𝐯 × 𝐁 ) \mathbf{F}=q\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)
  73. 𝐅 = 1 4 π q 1 q 2 r 2 𝐫 ^ \mathbf{F}=\frac{1}{4\pi}\frac{q_{1}q_{2}}{r^{2}}\mathbf{\hat{r}}
  74. 𝐅 = q 1 q 2 r 2 𝐫 ^ \mathbf{F}=\frac{q_{1}q_{2}}{r^{2}}\mathbf{\hat{r}}
  75. 𝐅 = 1 4 π ϵ 0 q 1 q 2 r 2 𝐫 ^ \mathbf{F}=\frac{1}{4\pi\epsilon_{0}}\frac{q_{1}q_{2}}{r^{2}}\mathbf{\hat{r}}
  76. 𝐄 = 1 4 π q r 2 𝐫 ^ \mathbf{E}=\frac{1}{4\pi}\frac{q}{r^{2}}\mathbf{\hat{r}}
  77. 𝐄 = q r 2 𝐫 ^ \mathbf{E}=\frac{q}{r^{2}}\mathbf{\hat{r}}
  78. 𝐄 = 1 4 π ϵ 0 q r 2 𝐫 ^ \mathbf{E}=\frac{1}{4\pi\epsilon_{0}}\frac{q}{r^{2}}\mathbf{\hat{r}}
  79. 𝐁 = 1 4 π 1 c I d 𝐥 × 𝐫 ^ r 2 \mathbf{B}=\frac{1}{4\pi}\frac{1}{c}\oint\frac{Id\mathbf{l}\times\mathbf{\hat{% r}}}{r^{2}}
  80. 𝐁 = 1 c I d 𝐥 × 𝐫 ^ r 2 \mathbf{B}=\frac{1}{c}\oint\frac{Id\mathbf{l}\times\mathbf{\hat{r}}}{r^{2}}
  81. 𝐁 = μ 0 4 π I d 𝐥 × 𝐫 ^ r 2 \mathbf{B}=\frac{\mu_{0}}{4\pi}\oint\frac{Id\mathbf{l}\times\mathbf{\hat{r}}}{% r^{2}}
  82. 𝐃 = 𝐄 + 𝐏 \mathbf{D}=\mathbf{E}+\mathbf{P}
  83. 𝐃 = 𝐄 + 4 π 𝐏 \mathbf{D}=\mathbf{E}+4\pi\mathbf{P}
  84. 𝐃 = ϵ 0 𝐄 + 𝐏 \mathbf{D}=\epsilon_{0}\mathbf{E}+\mathbf{P}
  85. 𝐏 = χ e 𝐄 \mathbf{P}=\chi\text{e}\mathbf{E}
  86. 𝐏 = χ e 𝐄 \mathbf{P}=\chi\text{e}\mathbf{E}
  87. 𝐏 = χ e ϵ 0 𝐄 \mathbf{P}=\chi\text{e}\epsilon_{0}\mathbf{E}
  88. 𝐃 = ϵ 𝐄 \mathbf{D}=\epsilon\mathbf{E}
  89. 𝐃 = ϵ 𝐄 \mathbf{D}=\epsilon\mathbf{E}
  90. 𝐃 = ϵ 𝐄 \mathbf{D}=\epsilon\mathbf{E}
  91. ϵ = 1 + χ e \epsilon=1+\chi\text{e}
  92. ϵ = 1 + 4 π χ e \epsilon=1+4\pi\chi\text{e}
  93. ϵ / ϵ 0 = 1 + χ e \epsilon/\epsilon_{0}=1+\chi\text{e}
  94. ϵ \epsilon
  95. ϵ 0 \epsilon_{0}
  96. χ e \chi\text{e}
  97. ϵ \epsilon
  98. ϵ / ϵ 0 \epsilon/\epsilon_{0}
  99. χ e \chi_{e}
  100. χ eSI = χ eLH = 4 π χ eG \chi\text{e}\text{SI}=\chi\text{e}\text{LH}=4\pi\chi\text{e}\text{G}
  101. 𝐁 = 𝐇 + 𝐌 \mathbf{B}=\mathbf{H}+\mathbf{M}
  102. 𝐁 = 𝐇 + 4 π 𝐌 \mathbf{B}=\mathbf{H}+4\pi\mathbf{M}
  103. 𝐁 = μ 0 ( 𝐇 + 𝐌 ) \mathbf{B}=\mu_{0}(\mathbf{H}+\mathbf{M})
  104. 𝐌 = χ m 𝐇 \mathbf{M}=\chi\text{m}\mathbf{H}
  105. 𝐌 = χ m 𝐇 \mathbf{M}=\chi\text{m}\mathbf{H}
  106. 𝐌 = χ m 𝐇 \mathbf{M}=\chi\text{m}\mathbf{H}
  107. 𝐁 = μ 𝐇 \mathbf{B}=\mu\mathbf{H}
  108. 𝐁 = μ 𝐇 \mathbf{B}=\mu\mathbf{H}
  109. 𝐁 = μ 𝐇 \mathbf{B}=\mu\mathbf{H}
  110. μ = 1 + χ m \mu=1+\chi\text{m}
  111. μ = 1 + 4 π χ m \mu=1+4\pi\chi\text{m}
  112. μ / μ 0 = 1 + χ m \mu/\mu_{0}=1+\chi\text{m}
  113. μ \mu
  114. μ 0 \mu_{0}
  115. χ m \chi\text{m}
  116. μ \mu
  117. μ / μ 0 \mu/\mu_{0}
  118. χ m \chi\text{m}
  119. χ mSI = χ mLH = 4 π χ mG \chi\text{m}\text{SI}=\chi\text{m}\text{LH}=4\pi\chi\text{m}\text{G}
  120. 𝐄 = - ϕ \mathbf{E}=-\nabla\phi
  121. 𝐄 = - ϕ \mathbf{E}=-\nabla\phi
  122. 𝐄 = - ϕ \mathbf{E}=-\nabla\phi
  123. 𝐄 = - ϕ - 1 c 𝐀 t \mathbf{E}=-\nabla\phi-\frac{1}{c}\frac{\partial\mathbf{A}}{\partial t}
  124. 𝐄 = - ϕ - 1 c 𝐀 t \mathbf{E}=-\nabla\phi-\frac{1}{c}\frac{\partial\mathbf{A}}{\partial t}
  125. 𝐄 = - ϕ - 𝐀 t \mathbf{E}=-\nabla\phi-\frac{\partial\mathbf{A}}{\partial t}
  126. 𝐁 = × 𝐀 \mathbf{B}=\nabla\times\mathbf{A}
  127. 𝐁 = × 𝐀 \mathbf{B}=\nabla\times\mathbf{A}
  128. 𝐁 = × 𝐀 \mathbf{B}=\nabla\times\mathbf{A}
  129. μ 0 ϵ 0 = 1 / c 2 \mu_{0}\epsilon_{0}=1/c^{2}
  130. ( 𝐄 , φ ) \left(\mathbf{E},\varphi\right)
  131. 1 4 π ( 𝐄 , φ ) \frac{1}{\sqrt{4\pi}}\left(\mathbf{E},\varphi\right)
  132. ε 0 ( 𝐄 , φ ) \sqrt{\varepsilon_{0}}\left(\mathbf{E},\varphi\right)
  133. 𝐃 \mathbf{D}
  134. 1 4 π 𝐃 \frac{1}{\sqrt{4\pi}}\mathbf{D}
  135. 1 ε 0 𝐃 \frac{1}{\sqrt{\varepsilon_{0}}}\mathbf{D}
  136. ( q , ρ , I , 𝐉 , 𝐏 , 𝐩 ) \left(q,\rho,I,\mathbf{J},\mathbf{P},\mathbf{p}\right)
  137. 4 π ( q , ρ , I , 𝐉 , 𝐏 , 𝐩 ) \sqrt{4\pi}\left(q,\rho,I,\mathbf{J},\mathbf{P},\mathbf{p}\right)
  138. 1 ε 0 ( q , ρ , I , 𝐉 , 𝐏 , 𝐩 ) \frac{1}{\sqrt{\varepsilon_{0}}}\left(q,\rho,I,\mathbf{J},\mathbf{P},\mathbf{p% }\right)
  139. ( 𝐁 , Φ m , 𝐀 ) \left(\mathbf{B},\Phi\text{m},\mathbf{A}\right)
  140. 1 4 π ( 𝐁 , Φ m , 𝐀 ) \frac{1}{\sqrt{4\pi}}\left(\mathbf{B},\Phi\text{m},\mathbf{A}\right)
  141. 1 μ 0 ( 𝐁 , Φ m , 𝐀 ) \frac{1}{\sqrt{\mu_{0}}}\left(\mathbf{B},\Phi\text{m},\mathbf{A}\right)
  142. 𝐇 \mathbf{H}
  143. 1 4 π 𝐇 \frac{1}{\sqrt{4\pi}}\mathbf{H}
  144. μ 0 𝐇 \sqrt{\mu_{0}}\mathbf{H}
  145. ( 𝐦 , 𝐌 ) \left(\mathbf{m},\mathbf{M}\right)
  146. 4 π ( 𝐦 , 𝐌 ) \sqrt{4\pi}\left(\mathbf{m},\mathbf{M}\right)
  147. μ 0 ( 𝐦 , 𝐌 ) \sqrt{\mu_{0}}\left(\mathbf{m},\mathbf{M}\right)
  148. ( ε , μ ) \left(\varepsilon,\mu\right)
  149. ( ε , μ ) \left(\varepsilon,\mu\right)
  150. ( ε r , μ r ) ( ε ε 0 , μ μ 0 ) \left(\varepsilon\text{r},\mu\text{r}\right)\equiv\left(\frac{\varepsilon}{% \varepsilon_{0}},\frac{\mu}{\mu_{0}}\right)
  151. ( χ e , χ m ) \left(\chi\text{e},\chi\text{m}\right)
  152. 4 π ( χ e , χ m ) 4\pi\left(\chi\text{e},\chi\text{m}\right)
  153. ( χ e , χ m ) \left(\chi\text{e},\chi\text{m}\right)
  154. ( σ , S , C ) \left(\sigma,S,C\right)
  155. 4 π ( σ , S , C ) 4\pi\left(\sigma,S,C\right)
  156. 1 ε 0 ( σ , S , C ) \frac{1}{\varepsilon_{0}}\left(\sigma,S,C\right)
  157. ( ρ , R , L ) \left(\rho,R,L\right)
  158. 1 4 π ( ρ , R , L ) \frac{1}{4\pi}\left(\rho,R,L\right)
  159. ε 0 ( ρ , R , L ) \varepsilon_{0}\left(\rho,R,L\right)
  160. ε = μ = c = 1 ε=μ=c=1
  161. 4 π
  162. ε = 1 ε=1
  163. 4 π
  164. c = 1 c=1
  165. ε = μ = c = 1 ε=μ=c=1
  166. 𝐄 = ρ \nabla\cdot\mathbf{E}=\rho\,
  167. 𝐁 = 0 \nabla\cdot\mathbf{B}=0\,
  168. × 𝐄 = - 𝐁 t \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}\,
  169. × 𝐁 = 𝐄 t + 𝐉 \nabla\times\mathbf{B}=\frac{\partial\mathbf{E}}{\partial t}+\mathbf{J}\,
  170. 𝐅 q = q ( 𝐄 + 𝐯 q × 𝐁 ) \mathbf{F}_{q}=q(\mathbf{E}+\mathbf{v}_{q}\times\mathbf{B})\,