wpmath0000006_13

Tanner_graph.html

  1. R R
  2. m m
  3. n n
  4. R 1 - ( 1 - r ) m R\geq 1-(1-r)m\,

Tarski's_axioms.html

  1. u v a b . \forall u\forall v\ldots\exists a\exists b\dots.
  2. x y y x . xy\equiv yx\,.
  3. x y z z x = y . xy\equiv zz\rightarrow x=y.
  4. ( x y z u and x y v w ) z u v w . (xy\equiv zu\and xy\equiv vw)\rightarrow zu\equiv vw.
  5. x y z w xy\equiv zw
  6. x y xy
  7. z w zw
  8. x y x y xy\equiv xy\,
  9. x y z w z w x y xy\equiv zw\rightarrow zw\equiv xy\,
  10. ( x y z u and z u v w ) x y v w (xy\equiv zu\and zu\equiv vw)\rightarrow xy\equiv vw\,
  11. x y z w x y w z xy\equiv zw\rightarrow xy\equiv wz\,
  12. x y z w y x z w xy\equiv zw\rightarrow yx\equiv zw\,
  13. x y z w y x w z xy\equiv zw\rightarrow yx\equiv wz\,
  14. B x y x x = y . Bxyx\rightarrow x=y.
  15. x x xx
  16. x x
  17. ( B x u z and B y v z ) a ( B u a y and B v a x ) . (Bxuz\and Byvz)\rightarrow\exists a\,(Buay\and Bvax).
  18. x u v y xuvy
  19. a x y [ ( ϕ ( x ) and ψ ( y ) ) B a x y ] b x y [ ( ϕ ( x ) and ψ ( y ) ) B x b y ] . \exists a\,\forall x\,\forall y\,[(\phi(x)\and\psi(y))\rightarrow Baxy]% \rightarrow\exists b\,\forall x\,\forall y\,[(\phi(x)\and\psi(y))\rightarrow Bxby].
  20. a b c [ ¬ B a b c and ¬ B b c a and ¬ B c a b ] . \exists a\,\exists b\,\exists c\,[\neg Babc\and\neg Bbca\and\neg Bcab].
  21. ( x u x v and y u y v and z u z v and u v ) ( B x y z B y z x B z x y ) . (xu\equiv xv\and yu\equiv yv\and zu\equiv zv\and u\neq v)\rightarrow(% BxyzByzxBzxy).
  22. ( ( B x y w and x y y w ) and ( B x u v and x u u v ) and ( B y u z and y u z u ) ) y z v w . ((Bxyw\and xy\equiv yw)\and(Bxuv\and xu\equiv uv)\and(Byuz\and yu\equiv zu))% \rightarrow yz\equiv vw.
  23. B x y z B y z x B z x y a ( x a y a and x a z a ) . BxyzByzxBzxy\exists a\,(xa\equiv ya\and xa\equiv za).
  24. ( B x u v and B y u z and x u ) a b ( B x y a and B x z b and B a v b ) . (Bxuv\and Byuz\and x\neq u)\rightarrow\exists a\,\exists b\,(Bxya\and Bxzb\and Bavb).
  25. ( x y and B x y z and B x y z and x y x y and y z y z and x u x u and y u y u ) z u z u . {(x\neq y\and Bxyz\and Bx^{\prime}y^{\prime}z^{\prime}\and xy\equiv x^{\prime}% y^{\prime}\and yz\equiv y^{\prime}z^{\prime}\and xu\equiv x^{\prime}u^{\prime}% \and yu\equiv y^{\prime}u^{\prime})}\rightarrow zu\equiv z^{\prime}u^{\prime}.
  26. z [ B x y z and y z a b ] . \exists z\,[Bxyz\and yz\equiv ab].
  27. a b c d ab\leq cd
  28. a b ab
  29. c d cd
  30. x y z u v ( z v u v w ( x w y w and y w u v ) ) . xy\leq zu\leftrightarrow\forall v(zv\equiv uv\rightarrow\exists w(xw\equiv yw% \and yw\equiv uv)).
  31. B x y z u ( ( u x x y and u z z y ) u = y ) . Bxyz\leftrightarrow\forall u((ux\leq xy\and uz\leq zy)\rightarrow u=y).

Tarski–Kuratowski_algorithm.html

  1. Σ 0 0 \Sigma^{0}_{0}
  2. Π 0 0 \Pi^{0}_{0}
  3. Σ k + 1 0 \Sigma^{0}_{k+1}
  4. Π k + 1 0 \Pi^{0}_{k+1}

Tate_conjecture.html

  1. H 2 i ( V k s , 𝐐 l ( i ) ) = W H^{2i}(V_{k_{s}},\mathbf{Q}_{l}(i))=W
  2. Hom ( A , B ) 𝐙 𝐐 l Hom G ( H 1 ( A k s , 𝐐 l ) , H 1 ( B k s , 𝐐 l ) ) \,\text{Hom}(A,B)\otimes_{\mathbf{Z}}\mathbf{Q}_{l}\rightarrow\,\text{Hom}_{G}% (H_{1}(A_{k_{s}},\mathbf{Q}_{l}),H_{1}(B_{k_{s}},\mathbf{Q}_{l}))

Tate_module.html

  1. T p ( A ) = lim A [ p n ] T_{p}(A)=\underset{\longleftarrow}{\lim}A[p^{n}]
  2. z ( a n ) n = ( ( z mod p n ) a n ) n . z(a_{n})_{n}=((z\,\text{ mod }p^{n})a_{n})_{n}.
  3. H et 1 ( G × K K s , 𝐙 p ) H^{1}_{\,\text{et}}(G\times_{K}K^{s},\mathbf{Z}_{p})
  4. Hom K ( A , B ) 𝐙 p Hom G K ( T p ( A ) , T p ( B ) ) \mathrm{Hom}_{K}(A,B)\otimes\mathbf{Z}_{p}\cong\mathrm{Hom}_{G_{K}}(T_{p}(A),T% _{p}(B))
  5. k ( C ) k ^ ( C ) A ( p ) k(C)\subset\hat{k}(C)\subset A^{(p)}
  6. k ^ \hat{k}
  7. k ^ ( C ) \hat{k}(C)
  8. K ^ \hat{K}
  9. K ^ \hat{K}
  10. T p ( K ) = Gal ( A ( p ) / K ^ ) . T_{p}(K)=\mathrm{Gal}(A^{(p)}/\hat{K})\ .
  11. λ m + μ p m + κ . \lambda m+\mu p^{m}+\kappa\ .

Tate_twist.html

  1. V 𝐐 p ( - 1 ) m . V\otimes\mathbf{Q}_{p}(-1)^{\otimes m}.

Tautological_bundle.html

  1. 𝒪 n ( - 1 ) \mathcal{O}_{\mathbb{P}^{n}}(-1)
  2. 𝒪 n ( 1 ) \mathcal{O}_{\mathbb{P}^{n}}(1)
  3. G n ( n + k ) End ( n + k ) , V p V , G_{n}(\mathbb{R}^{n+k})\to\operatorname{End}(\mathbb{R}^{n+k}),\,V\mapsto p_{V},
  4. p V p_{V}
  5. ϕ : π - 1 ( U ) G n ( n + k ) × X \phi:\pi^{-1}(U)\to G_{n}(\mathbb{R}^{n+k})\times X
  6. [ X , G n ] Vect n ( X ) , f f * ( γ n ) [X,G_{n}]\to\operatorname{Vect}^{\mathbb{R}}_{n}(X),\,f\mapsto f^{*}(\gamma_{n})
  7. E X × n + k E\hookrightarrow X\times\mathbb{R}^{n+k}
  8. f E : X G n , x E x f_{E}:X\to G_{n},\,x\mapsto E_{x}
  9. [ X , G n ] = Vect n ( X ) [X,G_{n}]=\operatorname{Vect}^{\mathbb{R}}_{n}(X)
  10. H = n - 1 \sub n H=\mathbb{P}^{n-1}\sub\mathbb{P}^{n}
  11. Γ ( U , O ( D ) ) = { f K | ( f ) + D 0 on U } \Gamma(U,O(D))=\{f\in K|(f)+D\geq 0\,\text{ on }U\}
  12. O ( H ) O ( 1 ) , f f x 0 O(H)\simeq O(1),f\mapsto fx_{0}
  13. A = k [ y 0 , , y n ] A=k[y_{0},\dots,y_{n}]
  14. n = Proj A \mathbb{P}^{n}=\operatorname{Proj}A
  15. 𝐒𝐩𝐞𝐜 ( 𝒪 n [ x 0 , , x n ] ) = 𝔸 n n + 1 = 𝔸 n + 1 × k n \mathbf{Spec}(\mathcal{O}_{\mathbb{P}^{n}}[x_{0},\dots,x_{n}])=\mathbb{A}^{n+1% }_{\mathbb{P}^{n}}=\mathbb{A}^{n+1}\times_{k}{\mathbb{P}^{n}}
  16. L = 𝐒𝐩𝐞𝐜 ( 𝒪 n [ x 0 , , x n ] / I ) L=\mathbf{Spec}(\mathcal{O}_{\mathbb{P}^{n}}[x_{0},\dots,x_{n}]/I)
  17. x i y j - x j y i x_{i}y_{j}-x_{j}y_{i}
  18. 𝔸 n n + 1 \mathbb{A}^{n+1}_{\mathbb{P}^{n}}
  19. n \mathbb{P}^{n}
  20. 𝔸 n + 1 × k n \mathbb{A}^{n+1}\times_{k}{\mathbb{P}^{n}}
  21. n \mathbb{P}^{n}
  22. 𝔸 n + 1 \mathbb{A}^{n+1}
  23. 𝐒𝐩𝐞𝐜 ( Sym E ˇ ) \mathbf{Spec}(\operatorname{Sym}\check{E})
  24. 0 I 𝒪 n [ x 0 , , x n ] x i y i Sym 𝒪 n ( 1 ) 0 , 0\to I\to\mathcal{O}_{\mathbb{P}^{n}}[x_{0},\dots,x_{n}]\overset{x_{i}\mapsto y% _{i}}{\to}\operatorname{Sym}\mathcal{O}_{\mathbb{P}^{n}}(1)\to 0,
  25. 𝒪 n ( - 1 ) \mathcal{O}_{\mathbb{P}^{n}}(-1)
  26. ( V ) \mathbb{P}(V)
  27. 𝒪 ( - 1 ) \mathcal{O}(-1)
  28. 𝒪 ( 1 ) \mathcal{O}(1)

Taylor_cone.html

  1. π - θ 0 \pi-\theta_{0}\,
  2. θ 0 \theta_{0}\,
  3. P 1 / 2 ( cos θ 0 ) P_{1/2}(\cos\theta_{0})\,
  4. R \sqrt{R}\,
  5. V = V 0 + A R 1 / 2 P 1 / 2 ( cos θ 0 ) V=V_{0}+AR^{1/2}P_{1/2}(\cos\theta_{0})\,
  6. V = V 0 V=V_{0}\,
  7. θ 0 \theta_{0}
  8. V = V 0 V=V_{0}\,
  9. P 1 / 2 ( cos θ 0 ) P_{1/2}(\cos\theta_{0})\,
  10. π \pi\,

Taylor_number.html

  1. Ta = 4 Ω 2 R 4 ν 2 \mathrm{Ta}=\frac{4\Omega^{2}R^{4}}{\nu^{2}}
  2. Ω \Omega
  3. ν \nu
  4. Ta = Ω 2 R 1 ( R 2 - R 1 ) 3 ν 2 \mathrm{Ta}=\frac{\Omega^{2}R_{1}(R_{2}-R_{1})^{3}}{\nu^{2}}

Taylor–Couette_flow.html

  1. Ta \mathrm{Ta}
  2. Ta c \mathrm{Ta_{c}}
  3. Ta < Ta c , \mathrm{Ta}<\mathrm{Ta_{c}},
  4. Ta \mathrm{Ta}
  5. Ta c \mathrm{Ta_{c}}
  6. Ta > Ta c \mathrm{Ta}>\mathrm{Ta_{c}}
  7. μ = Ω 2 / Ω 1 \mu=\Omega_{2}/\Omega_{1}
  8. η = R 1 / R 2 \eta=R_{1}/R_{2}
  9. μ \mu
  10. η \eta
  11. Ta \mathrm{Ta}
  12. η 1 \eta\rightarrow 1
  13. μ 1 \mu\rightarrow 1
  14. Ta c 1708 \mathrm{Ta_{c}}\simeq 1708

TCP_tuning.html

  1. Throughput RWIN RTT \mathrm{Throughput}\leq\frac{\mathrm{RWIN}}{\mathrm{RTT}}\,\!
  2. Throughput MSS RTT P loss \mathrm{Throughput}\leq\frac{\mathrm{MSS}}{\mathrm{RTT}\sqrt{P_{\mathrm{loss}}}}

Temperature_dependence_of_liquid_viscosity.html

  1. μ ( T ) = μ 0 exp ( - b T ) \mu(T)\,=\,\mu_{0}\exp(-bT)
  2. μ 0 \mu_{0}
  3. b b
  4. μ ( T ) = μ 0 exp ( E R T ) \mu(T)\,=\,\mu_{0}\exp\left(\frac{E}{RT}\right)
  5. μ 0 \mu_{0}
  6. μ ( T ) = μ 0 exp ( - C 1 ( T - T r ) C 2 + T - T r ) \mu(T)\,=\,\mu_{0}\exp\left(\frac{-C_{1}(T-T_{r})}{C_{2}+T-T_{r}}\right)
  7. C 1 C_{1}
  8. C 2 C_{2}
  9. T r T_{r}
  10. μ 0 \mu_{0}
  11. T r T_{r}
  12. C 1 C_{1}
  13. C 2 C_{2}
  14. T r T_{r}
  15. T g T_{g}
  16. C 1 C_{1}\approx
  17. C 2 51.6 C_{2}\approx 51.6
  18. T r = T g + 43 T_{r}\,=\,T_{g}+43
  19. C 1 8.86 C_{1}\approx 8.86
  20. C 2 C_{2}\approx
  21. log ( η / η g ) = A [ exp { B ( T g - T ) T } - 1 ] \log(\eta/\eta_{g})=A\left[\exp\left\{\frac{B(T_{g}-T)}{T}\right\}-1\right]
  22. μ ( T ) = 2.414 × 10 - 5 × 10 247.8 / ( T - 140 ) \mu(T)=2.414\times 10^{-5}\times 10^{247.8/(T-140)}
  23. log 10 [ log 10 ( ν + λ ) ] = A - B log 10 ( T ) \log_{10}[\log_{10}(\nu+\lambda)]=A-B\,\log_{10}(T)
  24. log 10 [ log 10 [ ν + λ + f ( ν ) ] ] = A - B log 10 ( T ) \log_{10}[\log_{10}[\nu+\lambda+f(\nu)]]=A-B\,\log_{10}(T)
  25. ln ( ln ( ν + 0.7 + e - ν K 0 ( ν + 1.244067 ) ) ) = A - B * ln ( T ) \ln\left({\ln\left({\nu+0.7+e^{-\nu}K_{0}\left({\nu+1.244067}\right)}\right)}% \right)=A-B*\ln\left(T\right)
  26. ν \nu
  27. K 0 K_{0}
  28. ln ( ln ( ν + 0.7 + e - ν K 0 ( ν + 1.244067 ) ) ) = A - B T \ln\left({\ln\left({\nu+0.7+e^{-\nu}K_{0}\left({\nu+1.244067}\right)}\right)}% \right)=A-{B\over T}

Temperley–Lieb_algebra.html

  1. R R
  2. δ R \delta\in R
  3. T L n ( δ ) TL_{n}(\delta)
  4. R R
  5. U 1 , U 2 , , U n - 1 U_{1},U_{2},\ldots,U_{n-1}
  6. U i 2 = δ U i U_{i}^{2}=\delta U_{i}
  7. 1 i n - 1 1\leq i\leq n-1
  8. U i U i + 1 U i = U i U_{i}U_{i+1}U_{i}=U_{i}
  9. 1 i n - 2 1\leq i\leq n-2
  10. U i U i - 1 U i = U i U_{i}U_{i-1}U_{i}=U_{i}
  11. 2 i n - 1 2\leq i\leq n-1
  12. U i U j = U j U i U_{i}U_{j}=U_{j}U_{i}
  13. 1 i , j n - 1 1\leq i,j\leq n-1
  14. | i - j | 1 |i-j|\neq 1
  15. T L n ( δ ) TL_{n}(\delta)
  16. T L 3 ( δ ) TL_{3}(\delta)
  17. U i U_{i}
  18. T L 5 ( δ ) TL_{5}(\delta)
  19. T L 5 ( δ ) TL_{5}(\delta)
  20. L L
  21. = j = 1 L - 1 ( 1 - e j ) \mathcal{H}=\sum_{j=1}^{L-1}(1-e_{j})
  22. e j = U ( λ ) / sin λ e_{j}=U(\lambda)/\sin\lambda
  23. λ R \lambda\in R
  24. L = 3 L=3
  25. = 2 - e 1 - e 2 \mathcal{H}=2-e_{1}-e_{2}
  26. \mathcal{H}
  27. \mathcal{H}
  28. \mathcal{H}
  29. \mathcal{H}
  30. \mathcal{H}
  31. = ( 1 - 1 - 1 1 ) \mathcal{H}=\left(\begin{array}[]{rr}1&-1\\ -1&1\end{array}\right)
  32. \mathcal{H}
  33. λ 0 \lambda_{0}
  34. \mathcal{H}
  35. λ 0 = 0 \lambda_{0}=0
  36. ψ 0 = ( 1 , 1 ) \psi_{0}=(1,1)
  37. L L
  38. L L
  39. ψ 0 \psi_{0}
  40. L L
  41. ψ 0 \psi_{0}
  42. ( 3 3 , 1 2 ) (3_{3},1_{2})
  43. ( 11 , 5 2 , 4 , 1 ) (11,5_{2},4,1)
  44. ( 26 4 , 10 2 , 9 2 , 8 2 , 5 2 , 1 2 ) (26_{4},10_{2},9_{2},8_{2},5_{2},1_{2})
  45. ( 170 , 75 2 , 71 , 56 2 , 50 , 30 , 14 4 , 6 , 1 ) (170,75_{2},71,56_{2},50,30,14_{4},6,1)
  46. ( 646 , ) (646,\ldots)
  47. \vdots
  48. \vdots
  49. \vdots
  50. \vdots
  51. m n = ( m , , m ) m_{n}=(m,\ldots,m)
  52. n n
  53. 5 2 = ( 5 , 5 ) 5_{2}=(5,5)
  54. \mathcal{H}
  55. 1 , 2 , 11 , 170 , = j = 0 n - 1 ( 3 j + 1 ) ( 2 j ) ! ( 6 j ) ! ( 4 j ) ! ( 4 j + 1 ) ! 1,2,11,170,\ldots=\prod_{j=0}^{n-1}\left(3j+1\right)\frac{(2j)!(6j)!}{(4j)!(4j% +1)!}
  56. 1 , 3 , 26 , 646 , = j = 0 n - 1 ( 3 j + 2 ) ( 2 j + 2 ) ! ( 6 j + 3 ) ! ( 4 j + 2 ) ! ( 4 j + 3 ) ! . 1,3,26,646,\ldots=\prod_{j=0}^{n-1}(3j+2)\frac{(2j+2)!(6j+3)!}{(4j+2)!(4j+3)!}.
  57. L L
  58. L L
  59. ( 2 n + 1 ) × ( 2 n + 1 ) (2n+1)\times(2n+1)
  60. X X Z XXZ
  61. Δ = - 1 / 2 \Delta=-1/2
  62. X X Z XXZ
  63. Δ = - 1 / 2 \Delta=-1/2

Template:Elementbox_heatvaporiz_kjpmol.html

  1. Δ v a p H \Delta_{vap}{H}^{\ominus}

Temporal_logic_of_actions.html

  1. [ A ] t [A]_{t}
  2. x + x * y = y x+x^{\prime}*y=y^{\prime}
  3. [ A ] t [A]_{t}

Tensor_product_of_Hilbert_spaces.html

  1. , 1 \langle\cdot,\cdot\rangle_{1}
  2. , 2 \langle\cdot,\cdot\rangle_{2}
  3. ϕ 1 ϕ 2 , ψ 1 ψ 2 = ϕ 1 , ψ 1 1 ϕ 2 , ψ 2 2 for all ϕ 1 , ψ 1 H 1 and ϕ 2 , ψ 2 H 2 \langle\phi_{1}\otimes\phi_{2},\psi_{1}\otimes\psi_{2}\rangle=\langle\phi_{1},% \psi_{1}\rangle_{1}\,\langle\phi_{2},\psi_{2}\rangle_{2}\quad\mbox{for all }~{% }\phi_{1},\psi_{1}\in H_{1}\mbox{ and }~{}\phi_{2},\psi_{2}\in H_{2}
  4. x 1 x 2 x_{1}\otimes x_{2}
  5. x * H 1 * x^{*}\in H^{*}_{1}
  6. x * x * ( x 1 ) x 2 x^{*}\mapsto x^{*}(x_{1})\,x_{2}
  7. H 1 H 2 H_{1}\otimes H_{2}
  8. T 1 , T 2 = n T 1 e n * , T 2 e n * , \langle T_{1},T_{2}\rangle=\sum_{n}\langle T_{1}e_{n}^{*},T_{2}e_{n}^{*}\rangle,
  9. ( e n * ) (e_{n}^{*})
  10. H = H 1 H 2 H=H_{1}\otimes H_{2}
  11. i , j = 1 | L ( e i , f j ) , u | 2 d 2 || u || 2 \sum\nolimits_{i,j=1}^{\infty}\big|\langle\,L(e_{i},f_{j}),u\rangle\big|^{2}% \leq d^{2}\,||u||^{2}
  12. \in
  13. H n H_{n}
  14. ξ n \xi_{n}
  15. L 2 L^{2}
  16. n = 1 ψ n \otimes_{n=1}^{\infty}\psi_{n}
  17. ψ n \psi_{n}
  18. ξ n \xi_{n}
  19. 𝔄 i \mathfrak{A}_{i}
  20. H i H_{i}
  21. i = 1 , 2 i=1,2
  22. A 1 A 2 A_{1}\otimes A_{2}
  23. A i 𝔄 i A_{i}\in\mathfrak{A}_{i}
  24. i = 1 , 2 i=1,2
  25. H 1 H 2 H_{1}\otimes H_{2}

Tensor_product_of_modules.html

  1. M R N M\otimes_{R}N
  2. : M × N M R N \otimes:M\times N\to M\otimes_{R}N
  3. f : M × N G f:M\times N\to G\,
  4. f ~ : M R N G \tilde{f}:M\otimes_{R}N\to G
  5. f ~ = f . \tilde{f}\circ\otimes=f.
  6. Hom ( M R N , G ) L R ( M , N ; G ) , g g . \operatorname{Hom}_{\mathbb{Z}}(M\otimes_{R}N,G)\simeq\operatorname{L}_{R}(M,N% ;G),\,g\mapsto g\circ\otimes.
  7. \otimes
  8. G = M R N G=M\otimes_{R}N
  9. L R ( M , N ; G ) = Hom R ( M , Hom ( N , G ) ) \operatorname{L}_{R}(M,N;G)=\operatorname{Hom}_{R}(M,\operatorname{Hom}_{% \mathbb{Z}}(N,G))
  10. Hom ( M R N , G ) Hom R ( M , Hom ( N , G ) ) \operatorname{Hom}_{\mathbb{Z}}(M\otimes_{R}N,G)\simeq\operatorname{Hom}_{R}(M% ,\operatorname{Hom}_{\mathbb{Z}}(N,G))
  11. : M × N M R N \otimes:M\times N\to M\otimes_{R}N
  12. M R N M\otimes_{R}N
  13. Q = ( M R N ) / L Q=(M\otimes_{R}N)/L
  14. 0 = q 0=q\circ\otimes
  15. 0 = 0 0=0\circ\otimes
  16. \square
  17. M R N M\otimes_{R}N
  18. r ( x y ) := r x y = x r y r\cdot(x\otimes y):=rx\otimes y=x\otimes ry
  19. M R N M\otimes_{R}N
  20. M R N M\otimes_{R}N
  21. Hom R ( M R N , G ) { \operatorname{Hom}_{R}(M\otimes_{R}N,G)\simeq\{
  22. M × N M\times N
  23. G } , g g G\},\,g\mapsto g\circ\otimes
  24. M R N M\otimes_{R}N
  25. s ( x y ) := s x y s\cdot(x\otimes y):=sx\otimes y
  26. M R N M\otimes_{R}N
  27. f : M M f:M\to M^{\prime}
  28. g : N N g:N\to N^{\prime}
  29. f g : M R N M R N f\otimes g:M\otimes_{R}N\to M^{\prime}\otimes_{R}N^{\prime}
  30. ( f g ) ( x y ) = f ( x ) g ( y ) (f\otimes g)(x\otimes y)=f(x)\otimes g(y)
  31. M R - : R - 𝐌𝐨𝐝 𝐀𝐛 M\otimes_{R}-:R-\mathbf{Mod}\to\mathbf{Ab}
  32. f : R S f:R\to S
  33. M R N M S N M\otimes_{R}N\to M\otimes_{S}N
  34. M × N S M S N M\times N\overset{\otimes_{S}}{\to}M\otimes_{S}N
  35. M 12 R 2 M 20 M_{12}\otimes_{R_{2}}M_{20}
  36. M 02 R 2 M 23 M_{02}\otimes_{R_{2}}M_{23}
  37. ( M 01 R 1 M 12 ) R 2 M 20 = M 01 R 1 ( M 12 R 2 M 20 ) (M_{01}\otimes_{R_{1}}M_{12})\otimes_{R_{2}}M_{20}=M_{01}\otimes_{R_{1}}(M_{12% }\otimes_{R_{2}}M_{20})
  38. R R R = R R\otimes_{R}R=R
  39. m n = : m R n mn=:m\otimes_{R}n
  40. R R M = M R\otimes_{R}M=M
  41. ( M R N ) R P = M R ( N R P ) (M\otimes_{R}N)\otimes_{R}P=M\otimes_{R}(N\otimes_{R}P)
  42. M R N R P := M R ( N R P ) M\otimes_{R}N\otimes_{R}P:=M\otimes_{R}(N\otimes_{R}P)
  43. M R N = N R M M\otimes_{R}N=N\otimes_{R}M
  44. M 1 R R M n M σ ( 1 ) R R M σ ( n ) M_{1}\otimes_{R}\cdots\otimes_{R}M_{n}\to M_{\sigma(1)}\otimes_{R}\cdots% \otimes_{R}M_{\sigma(n)}
  45. x 1 x n x_{1}\otimes\cdots\otimes x_{n}
  46. x σ ( 1 ) x σ ( n ) . x_{\sigma(1)}\otimes\cdots\otimes x_{\sigma(n)}.
  47. M R ( N P ) = ( M R N ) ( M R P ) M\otimes_{R}(N\oplus P)=(M\otimes_{R}N)\oplus(M\otimes_{R}P)
  48. M R ( i I N i ) = i I M R N i M\otimes_{R}(\bigoplus_{i\in I}N_{i})=\bigoplus_{i\in I}M\otimes_{R}N_{i}
  49. N i N_{i}
  50. M R i = 1 n N i = i = 1 n M R N i M\otimes_{R}\prod_{i=1}^{n}N_{i}=\prod_{i=1}^{n}M\otimes_{R}N_{i}
  51. S - 1 ( M R N ) = S - 1 M S - 1 R S - 1 N S^{-1}(M\otimes_{R}N)=S^{-1}M\otimes_{S^{-1}R}S^{-1}N
  52. S - 1 R S^{-1}R
  53. S - 1 R S^{-1}R
  54. S - 1 - = S - 1 R R - S^{-1}-=S^{-1}R\otimes_{R}-
  55. - S = S R - -_{S}=S\otimes_{R}-
  56. ( M R N ) S = M S S N S ; (M\otimes_{R}N)_{S}=M_{S}\otimes_{S}N_{S};
  57. ( lim M i ) R N = lim ( M i R N ) . (\underrightarrow{\lim}M_{i})\otimes_{R}N=\underrightarrow{\lim}(M_{i}\otimes_% {R}N).
  58. 0 N 𝑓 N 𝑔 N ′′ 0 0\to N^{\prime}\overset{f}{\to}N\overset{g}{\to}N^{\prime\prime}\to 0
  59. M R N 1 f M R N 1 g M R N ′′ 0 M\otimes_{R}N^{\prime}\overset{1\otimes f}{\to}M\otimes_{R}N\overset{1\otimes g% }{\to}M\otimes_{R}N^{\prime\prime}\to 0
  60. ( 1 f ) ( x y ) = x f ( y ) . (1\otimes f)(x\otimes y)=x\otimes f(y).
  61. Hom R ( M R N , P ) = Hom R ( M , Hom R ( N , P ) ) \operatorname{Hom}_{R}(M\otimes_{R}N,P)=\operatorname{Hom}_{R}(M,\operatorname% {Hom}_{R}(N,P))
  62. Hom R ( M , N ) P Hom R ( M , N P ) , \operatorname{Hom}_{R}(M,N)\otimes P\to\operatorname{Hom}_{R}(M,N\otimes P),
  63. Hom R ( M , N ) Hom R ( M , N ) Hom R ( M M , N N ) \operatorname{Hom}_{R}(M,N)\otimes\operatorname{Hom}_{R}(M^{\prime},N^{\prime}% )\to\operatorname{Hom}_{R}(M\otimes M^{\prime},N\otimes N^{\prime})
  64. ( M , N ) (M,N)
  65. ( M , M ) (M,M^{\prime})
  66. e i , i I e_{i},i\in I
  67. f j , j J f_{j},j\in J
  68. M = i I R e i M=\bigoplus_{i\in I}Re_{i}
  69. M R N = i , j R ( e i f j ) M\otimes_{R}N=\bigoplus_{i,j}R(e_{i}\otimes f_{j})
  70. e i f j , i I , j J e_{i}\otimes f_{j},\,i\in I,j\in J
  71. M R N M\otimes_{R}N
  72. / p n = 0 \mathbb{Q}\otimes_{\mathbb{Z}}\mathbb{Z}/p^{n}=0
  73. ( lim / p n ) = p = p [ p - 1 ] = p (\underleftarrow{\lim}\mathbb{Z}/p^{n})\otimes_{\mathbb{Z}}\mathbb{Q}=\mathbb{% Z}_{p}\otimes_{\mathbb{Z}}\mathbb{Q}=\mathbb{Z}_{p}[p^{-1}]=\mathbb{Q}_{p}
  74. p , p \mathbb{Z}_{p},\mathbb{Q}_{p}
  75. M R N M\otimes_{R}N
  76. N R M N\otimes_{R}M
  77. M R N M\otimes_{R}N
  78. N R M N\otimes_{R}M
  79. ( M R N ) S P = M R ( N S P ) (M\otimes_{R}N)\otimes_{S}P=M\otimes_{R}(N\otimes_{S}P)
  80. Hom S ( M R N , P ) = Hom R ( M , Hom S ( N , P ) ) , f f \operatorname{Hom}_{S}(M\otimes_{R}N,P)=\operatorname{Hom}_{R}(M,\operatorname% {Hom}_{S}(N,P)),\,f\mapsto f^{\prime}
  81. f f^{\prime}
  82. f ( x ) ( y ) = f ( x y ) . f^{\prime}(x)(y)=f(x\otimes y).
  83. Hom S ( S , - ) = - \operatorname{Hom}_{S}(S,-)=-
  84. Hom S ( M R S , P ) = Hom R ( M , Res R ( P ) ) . \operatorname{Hom}_{S}(M\otimes_{R}S,P)=\operatorname{Hom}_{R}(M,\operatorname% {Res}_{R}(P)).
  85. - R S -\otimes_{R}S
  86. Res R \operatorname{Res}_{R}
  87. - R S -\otimes_{R}S
  88. R n R S = S n R^{n}\otimes_{R}S=S^{n}
  89. R R
  90. S R R [ x 1 , , x n ] = S [ x 1 , , x n ] S\otimes_{R}R[x_{1},\dots,x_{n}]=S[x_{1},\dots,x_{n}]
  91. S R ( R [ x 1 , , x n ] / I ) = S [ x 1 , , x n ] / I S [ x 1 , , x n ] , I S\otimes_{R}(R[x_{1},\dots,x_{n}]/I)=S[x_{1},\dots,x_{n}]/IS[x_{1},\dots,x_{n}% ],\,I
  92. = [ x ] / ( x 2 + 1 ) \mathbb{C}=\mathbb{R}[x]/(x^{2}+1)
  93. = [ x ] / ( x 2 + 1 ) = [ x ] / ( x + i ) × [ x ] / ( x - i ) = 2 \mathbb{C}\otimes_{\mathbb{R}}\mathbb{C}=\mathbb{C}[x]/(x^{2}+1)=\mathbb{C}[x]% /(x+\mathrm{i})\times\mathbb{C}[x]/(x-\mathrm{i})=\mathbb{C}^{2}
  94. [ i ] = [ i ] = \mathbb{R}\otimes_{\mathbb{Z}}\mathbb{Z}[\mathrm{i}]=\mathbb{C}[\mathrm{i}]=% \mathbb{C}
  95. G = 0 \mathbb{Q}\otimes_{\mathbb{Z}}G=0
  96. G \mathbb{Q}\otimes_{\mathbb{Z}}G
  97. x = i r i g i x=\sum_{i}r_{i}\otimes g_{i}
  98. r i , g i G r_{i}\in\mathbb{Q},g_{i}\in G
  99. n i n_{i}
  100. g i g_{i}
  101. x = ( r i / n i ) n i g i = r i / n i n i g i = 0. x=\sum(r_{i}/n_{i})n_{i}\otimes g_{i}=\sum r_{i}/n_{i}\otimes n_{i}g_{i}=0.
  102. / / = 0. \mathbb{Q}/\mathbb{Z}\otimes_{\mathbb{Z}}\mathbb{Q}/\mathbb{Z}=0.
  103. R / I R M = M / I M R/I\otimes_{R}M=M/IM
  104. I M = I R M IM=I\otimes_{R}M
  105. M / I M R / I N / I N = M R N R R / I . M/IM\otimes_{R/I}N/IN=M\otimes_{R}N\otimes_{R}R/I.
  106. R / I R R / J = R / ( I + J ) R/I\otimes_{R}R/J=R/(I+J)
  107. 0 I R R / I 0 0\to I\to R\to R/I\to 0
  108. I R M 𝑓 R R M = M R / I R M 0 I\otimes_{R}M\overset{f}{\to}R\otimes_{R}M=M\to R/I\otimes_{R}M\to 0
  109. i x i x i\otimes x\mapsto ix
  110. R / I R R / J = R / J I ( R / J ) = R / J ( I + J ) / J = R / ( I + J ) R/I\otimes_{R}R/J={R/J\over I(R/J)}={R/J\over(I+J)/J}=R/(I+J)
  111. \square
  112. G / n = G / n G G\otimes_{\mathbb{Z}}\mathbb{Z}/n=G/nG
  113. / n / m = / gcd ( n , m ) \mathbb{Z}/n\otimes_{\mathbb{Z}}\mathbb{Z}/m=\mathbb{Z}/{\operatorname{gcd}(n,% m)}
  114. μ n \mu_{n}
  115. μ n / n \mu_{n}\approx\mathbb{Z}/n
  116. μ n μ m μ g . \mu_{n}\otimes_{\mathbb{Z}}\mu_{m}\approx\mu_{g}.
  117. \mathbb{Q}\otimes_{\mathbb{Z}}\mathbb{Q}
  118. \mathbb{Q}\otimes_{\mathbb{Q}}\mathbb{Q}
  119. \mathbb{Q}\otimes_{\mathbb{Z}}\mathbb{Q}
  120. \mathbb{Q}
  121. \mathbb{Q}\otimes_{\mathbb{Z}}\mathbb{Q}\to\mathbb{Q}\otimes_{\mathbb{Q}}% \mathbb{Q}
  122. r s x y - x r s y {r\over s}x\otimes y-x\otimes{r\over s}y
  123. r s x y = r s x s s y = x r s y , {r\over s}x\otimes y={r\over s}x\otimes{s\over s}y=x\otimes{r\over s}y,
  124. = = . \mathbb{Q}\otimes_{\mathbb{Z}}\mathbb{Q}=\mathbb{Q}\otimes_{\mathbb{Q}}\mathbb% {Q}=\mathbb{Q}.
  125. \mathbb{R}\otimes_{\mathbb{Z}}\mathbb{R}
  126. \mathbb{R}\otimes_{\mathbb{R}}\mathbb{R}
  127. = \mathbb{R}\otimes_{\mathbb{Z}}\mathbb{R}=\mathbb{R}\otimes_{\mathbb{Q}}\mathbb% {R}
  128. \mathbb{R}
  129. \mathbb{R}\otimes_{\mathbb{Q}}\mathbb{R}
  130. \mathbb{R}\otimes_{\mathbb{Z}}\mathbb{R}\approx\mathbb{R}\otimes_{\mathbb{R}}% \mathbb{R}
  131. : M × N M R N , ( m , n ) [ m * n ] \otimes:M\times N\to M\otimes_{R}N,\,(m,n)\mapsto[m*n]
  132. M × R × N σ × 1 1 × τ M × N M R N , M\times R\times N{{{}\atop\overset{\sigma\times 1}{\rightarrow}}\atop{% \underset{1\times\tau}{\rightarrow}\atop{}}}M\times N\overset{\otimes}{\to}M% \otimes_{R}N,
  133. m ( n + n ) = m n + m n , m\otimes(n+n^{\prime})=m\otimes n+m\otimes n^{\prime},
  134. ( m + m ) n = m n + m n . (m+m^{\prime})\otimes n=m\otimes n+m^{\prime}\otimes n.
  135. M R N M\otimes_{R}N
  136. M S N M\otimes_{S}N
  137. x r S y - x S r y , r R , x M , y N xr\otimes_{S}y-x\otimes_{S}ry,\,r\in R,x\in M,y\in N
  138. x S y x\otimes_{S}y
  139. ( x , y ) (x,y)
  140. : M × N M S N . \otimes:M\times N\to M\otimes_{S}N.
  141. ( ϕ + ψ ) ( u ) = ϕ ( u ) + ψ ( u ) , ϕ , ψ E * , u E (\phi+\psi)(u)=\phi(u)+\psi(u),\quad\phi,\psi\in E^{*},u\in E
  142. ( r ϕ ) ( u ) = r ϕ ( u ) , ϕ E * , u E , r R , (r\cdot\phi)(u)=r\cdot\phi(u),\quad\phi\in E^{*},u\in E,r\in R,
  143. , : E * × E R : ( e , e ) e , e = e ( e ) \langle\cdot,\cdot\rangle:E^{*}\times E\to R:(e^{\prime},e)\mapsto\langle e^{% \prime},e\rangle=e^{\prime}(e)
  144. , : F × F * R : ( f , f ) f , f = f ( f ) . \langle\cdot,\cdot\rangle:F\times F^{*}\to R:(f,f^{\prime})\mapsto\langle f,f^% {\prime}\rangle=f^{\prime}(f).
  145. r g , h s = r g , h s , r , s R . \langle r\cdot g,h\cdot s\rangle=r\cdot\langle g,h\rangle\cdot s,\quad r,s\in R.
  146. E * R E R E^{*}\otimes_{R}E\to R
  147. ϕ x ϕ ( x ) \phi\otimes x\mapsto\phi(x)
  148. E * R E = End R ( E ) E^{*}\otimes_{R}E=\operatorname{End}_{R}(E)
  149. tr : End R ( E ) R . \operatorname{tr}:\operatorname{End}_{R}(E)\to R.
  150. 𝔗 q p = Γ ( M , T M ) p R Γ ( M , T * M ) q \mathfrak{T}^{p}_{q}=\Gamma(M,TM)^{\otimes p}\otimes_{R}\Gamma(M,T^{*}M)^{% \otimes q}
  151. p \otimes p
  152. 𝔗 q p \mathfrak{T}^{p}_{q}
  153. 𝔗 p q \mathfrak{T}^{q}_{p}
  154. 𝔗 q p . \mathfrak{T}^{p}_{q}.
  155. E = Γ ( M , T M ) E=\Gamma(M,TM)
  156. E * = Γ ( M , T * M ) E^{*}=\Gamma(M,T^{*}M)
  157. E p × E * q E p - 1 × E * q - 1 , ( X 1 , , X p , ω 1 , , ω q ) X k , ω l ( X 1 , , X l ^ , , X p , ω 1 , , ω l ^ , , ω q ) E^{p}\times{E^{*}}^{q}\to E^{p-1}\times{E^{*}}^{q-1},\,(X_{1},\dots,X_{p},% \omega_{1},\dots,\omega_{q})\mapsto\langle X_{k},\omega_{l}\rangle(X_{1},\dots% ,\widehat{X_{l}},\dots,X_{p},\omega_{1},\dots,\widehat{\omega_{l}},\dots,% \omega_{q})
  158. E p E^{p}
  159. 1 p E \prod_{1}^{p}E
  160. C l k : 𝔗 q p 𝔗 q - 1 p - 1 . C^{k}_{l}:\mathfrak{T}^{p}_{q}\to\mathfrak{T}^{p-1}_{q-1}.
  161. C l k ( X 1 X p ω 1 ω q ) = X k , ω l X 1 X l ^ X p ω 1 ω l ^ ω q . C^{k}_{l}(X_{1}\otimes\cdots\otimes X_{p}\otimes\omega_{1}\otimes\cdots\otimes% \omega_{q})=\langle X_{k},\omega_{l}\rangle X_{1}\otimes\cdots\widehat{X_{l}}% \cdots\otimes X_{p}\otimes\omega_{1}\otimes\cdots\widehat{\omega_{l}}\cdots% \otimes\omega_{q}.
  162. - R - : Mod R × R Mod Ab -\otimes_{R}-:\mathrm{Mod}\mbox{--}~{}R\times R\mbox{--}~{}\mathrm{Mod}% \rightarrow\mathrm{Ab}
  163. M R - : R Mod Ab M\otimes_{R}-:R\mbox{--}~{}\mathrm{Mod}\rightarrow\mathrm{Ab}
  164. - R N : Mod R Ab -\otimes_{R}N:\mathrm{Mod}\mbox{--}~{}R\rightarrow\mathrm{Ab}
  165. Hom R ( - , - ) \mathrm{Hom}_{R}(-,-)
  166. { m i n j i I , j J } \{m_{i}\otimes n_{j}\mid i\in I,j\in J\}
  167. { m i n j i I , j J } \{m_{i}\otimes n_{j}\mid i\in I,j\in J\}
  168. ( X R Y ) n = i + j = n X i R Y i . (X\otimes_{R}Y)_{n}=\sum_{i+j=n}X_{i}\otimes_{R}Y_{i}.
  169. C G C\otimes_{\mathbb{Z}}G
  170. ( T M ) p O ( T * M ) q (TM)^{\otimes p}\otimes_{O}(T^{*}M)^{\otimes q}
  171. T M , T * M TM,T^{*}M
  172. f f f\mapsto f^{\prime}
  173. f ( x ) ( y ) = f ( x , y ) f^{\prime}(x)(y)=f(x,y)
  174. Hom ( N , G ) \operatorname{Hom}_{\mathbb{Z}}(N,G)
  175. ( g r ) ( y ) = g ( r y ) (g\cdot r)(y)=g(ry)
  176. f ( x r ) = f ( x ) r f ( x r , y ) = f ( x , r y ) . \Leftrightarrow f^{\prime}(xr)=f^{\prime}(x)\cdot r\Leftrightarrow f(xr,y)=f(x% ,ry).
  177. ( M R N ) S = ( S S M ) R N = M S R N = M S S S R N = M S S N S (M\otimes_{R}N)_{S}=(S\otimes_{S}M)\otimes_{R}N=M_{S}\otimes_{R}N=M_{S}\otimes% _{S}S\otimes_{R}N=M_{S}\otimes_{S}N_{S}
  178. T * M T^{*}M

Tensor–vector–scalar_gravity.html

  1. M M
  2. r r
  3. a = - G M r 2 , a=-\frac{GM}{r^{2}},
  4. G G
  5. m m
  6. F = m a . F=ma.
  7. F = μ ( a / a 0 ) m a , F=\mu(a/a_{0})ma,
  8. μ ( x ) \mu(x)
  9. μ ( x ) = 1 if | x | 1 , \mu(x)=1~{}\mathrm{if}~{}|x|\gg 1,
  10. μ ( x ) = x if | x | 1. \mu(x)=x~{}\mathrm{if}~{}|x|\ll 1.
  11. = - a 0 2 8 π G f ( | Φ | 2 a 0 2 ) - ρ Φ , {\mathcal{L}}=-\frac{a_{0}^{2}}{8\pi G}f\left(\frac{|\nabla\Phi|^{2}}{a_{0}^{2% }}\right)-\rho\Phi,
  12. Φ \Phi
  13. ρ \rho
  14. f ( y ) f(y)
  15. a = - Φ a=-\nabla\Phi
  16. μ ( y ) = d f ( y ) / d y \mu(\sqrt{y})=df(y)/dy
  17. g μ ν g_{\mu\nu}
  18. u α u^{\alpha}
  19. σ \sigma
  20. ϕ \phi
  21. ϕ \phi
  22. S TeVeS = ( g + s + v ) d 4 x . S_{\mathrm{TeVeS}}=\int\left({\mathcal{L}}_{g}+{\mathcal{L}}_{s}+{\mathcal{L}}% _{v}\right)d^{4}x.
  23. [ + , - , - , - ] [+,-,-,-]
  24. c = 1 c=1
  25. g = - 1 16 π G R - g , {\mathcal{L}}_{g}=-\frac{1}{16\pi G}R\sqrt{-g},
  26. R R
  27. g g
  28. s = - 1 2 [ σ 2 h α β α ϕ β ϕ + 1 2 G l 2 σ 4 F ( k G σ 2 ) ] - g , {\mathcal{L}}_{s}=-\frac{1}{2}\left[\sigma^{2}h^{\alpha\beta}\partial_{\alpha}% \phi\partial_{\beta}\phi+\frac{1}{2}\frac{G}{l^{2}}\sigma^{4}F(kG\sigma^{2})% \right]\sqrt{-g},
  29. h α β = g α β - u α u β h^{\alpha\beta}=g^{\alpha\beta}-u^{\alpha}u^{\beta}
  30. l l
  31. k k
  32. F F
  33. v = - K 32 π G [ g α β g μ ν ( B α μ B β ν ) + 2 λ K ( g μ ν u μ u ν - 1 ) ] - g {\mathcal{L}}_{v}=-\frac{K}{32\pi G}\left[g^{\alpha\beta}g^{\mu\nu}(B_{\alpha% \mu}B_{\beta\nu})+2\frac{\lambda}{K}(g^{\mu\nu}u_{\mu}u_{\nu}-1)\right]\sqrt{-g}
  34. B α β = α u β - β u α B_{\alpha\beta}=\partial_{\alpha}u_{\beta}-\partial_{\beta}u_{\alpha}
  35. K K
  36. k k
  37. K K
  38. K = k 2 π K=\frac{k}{2\pi}
  39. v {\mathcal{L}}_{v}
  40. F F
  41. g ^ μ ν = e 2 ϕ g μ ν - 2 u α u β sinh ( 2 ϕ ) . {\hat{g}}^{\mu\nu}=e^{2\phi}g^{\mu\nu}-2u^{\alpha}u^{\beta}\sinh(2\phi).
  42. S m = ( g ^ μ ν , f α , f | μ α , ) - g ^ d 4 x , S_{m}=\int{\mathcal{L}}({\hat{g}}_{\mu\nu},f^{\alpha},f^{\alpha}_{|\mu},...)% \sqrt{-{\hat{g}}}d^{4}x,
  43. g ^ μ ν {\hat{g}}_{\mu\nu}
  44. | |
  45. E G E_{G}
  46. E G = 0.392 ± 0.065 E_{G}=0.392\pm{0.065}
  47. f ( R ) f(R)
  48. E G = 0.22 E_{G}=0.22

Tent_map.html

  1. f μ := μ min { x , 1 - x } , f_{\mu}:=\mu\min\{x,\,1-x\},
  2. x n x_{n}
  3. x n + 1 = f μ ( x n ) = { μ x n for x n < 1 2 μ ( 1 - x n ) for 1 2 x n x_{n+1}=f_{\mu}(x_{n})=\begin{cases}\mu x_{n}&\mathrm{for}~{}~{}x_{n}<\frac{1}% {2}\\ \\ \mu(1-x_{n})&\mathrm{for}~{}~{}\frac{1}{2}\leq x_{n}\end{cases}
  4. μ = 2 \mu=2
  5. 0.61 0.585 0.6225 0.56625 0.650625 0.61\to 0.585\to 0.6225\to 0.56625\to 0.650625\ldots
  6. μ μ 2 + 1 μ 2 μ 2 + 1 μ μ 2 + 1 appears at μ = 1 \frac{\mu}{\mu^{2}+1}\to\frac{\mu^{2}}{\mu^{2}+1}\to\frac{\mu}{\mu^{2}+1}\mbox% { appears at }~{}\mu=1
  7. μ μ 3 + 1 μ 2 μ 3 + 1 μ 3 μ 3 + 1 μ μ 3 + 1 appears at μ = 1 + 5 2 \frac{\mu}{\mu^{3}+1}\to\frac{\mu^{2}}{\mu^{3}+1}\to\frac{\mu^{3}}{\mu^{3}+1}% \to\frac{\mu}{\mu^{3}+1}\mbox{ appears at }~{}\mu=\frac{1+\sqrt{5}}{2}
  8. μ μ 4 + 1 μ 2 μ 4 + 1 μ 3 μ 4 + 1 μ 4 μ 4 + 1 μ μ 4 + 1 appears at μ 1.8393 \frac{\mu}{\mu^{4}+1}\to\frac{\mu^{2}}{\mu^{4}+1}\to\frac{\mu^{3}}{\mu^{4}+1}% \to\frac{\mu^{4}}{\mu^{4}+1}\to\frac{\mu}{\mu^{4}+1}\mbox{ appears at }~{}\mu% \approx 1.8393
  9. x 0 x_{0}
  10. x n x_{n}
  11. x n x_{n}
  12. x n {x_{n}}
  13. μ = 2 \mu=2
  14. y n y_{n}
  15. x n = 2 π sin - 1 ( y n 1 / 2 ) . x_{n}=\tfrac{2}{\pi}\sin^{-1}(y_{n}^{1/2}).
  16. μ = 2 \mu=2
  17. v n + 1 = { v n / a for v n [ 0 , a ) ( 1 - v n ) / ( 1 - a ) for v n [ a , 1 ] v_{n+1}=\begin{cases}v_{n}/a&\mathrm{for}~{}~{}v_{n}\in[0,a)\\ \\ (1-v_{n})/(1-a)&\mathrm{for}~{}~{}v_{n}\in[a,1]\end{cases}
  18. a [ 0 , 1 ] a\in[0,1]
  19. μ = 2 \mu=2
  20. a = 1 2 a=\tfrac{1}{2}
  21. v n v_{n}
  22. w n + 1 = ( 2 a - 1 ) w n + u n + 1 w_{n+1}=(2a-1)w_{n}+u_{n+1}
  23. u n u_{n}

Tephigram.html

  1. ϕ \phi
  2. ϕ \phi

Term_indexing.html

  1. S S
  2. q q
  3. S S
  4. t t
  5. q q
  6. t t
  7. q q
  8. t t
  9. θ \theta
  10. q θ q\theta
  11. t θ t\theta
  12. t t
  13. q q
  14. θ \theta
  15. q θ q\theta
  16. t t
  17. t t
  18. q q
  19. θ \theta
  20. q q
  21. t θ t\theta
  22. q q
  23. t t
  24. θ \theta
  25. q θ q\theta
  26. t t
  27. q q
  28. t t
  29. θ \theta
  30. t θ t\theta
  31. q q
  32. t t
  33. S S
  34. S S

Ternary_Golay_code.html

  1. [ 11 , 6 , 5 ] 3 [11,6,5]_{3}
  2. [ 1 1 1 2 2 0 1 0 0 0 0 1 1 2 1 0 2 0 1 0 0 0 1 2 1 0 1 2 0 0 1 0 0 1 2 0 1 2 1 0 0 0 1 0 1 0 2 2 1 1 0 0 0 0 1 ] . \left[\begin{array}[]{ccccccccccc}1&1&1&2&2&0&1&0&0&0&0\\ 1&1&2&1&0&2&0&1&0&0&0\\ 1&2&1&0&1&2&0&0&1&0&0\\ 1&2&0&1&2&1&0&0&0&1&0\\ 1&0&2&2&1&1&0&0&0&0&1\end{array}\right].
  3. x 12 + y 12 + z 12 + 22 ( x 6 y 6 + y 6 z 6 + z 6 x 6 ) + 220 ( x 6 y 3 z 3 + y 6 z 3 x 3 + z 6 x 3 y 3 ) . x^{12}+y^{12}+z^{12}+22\left(x^{6}y^{6}+y^{6}z^{6}+z^{6}x^{6}\right)+220\left(% x^{6}y^{3}z^{3}+y^{6}z^{3}x^{3}+z^{6}x^{3}y^{3}\right).

Test_particle.html

  1. m 1 m_{1}
  2. m 2 m_{2}
  3. F ( r ) = - G m 1 m 2 ( r 1 - r 2 ) 2 F(r)=-G\frac{m_{1}m_{2}}{(r_{1}-r_{2})^{2}}
  4. r 1 r_{1}
  5. r 2 r_{2}
  6. R = m 1 r 1 + m 2 r 2 m 1 + m 2 R=\frac{m_{1}r_{1}+m_{2}r_{2}}{m_{1}+m_{2}}
  7. m 1 m 2 m_{1}>>m_{2}
  8. g ( r ) = G m 1 r 2 g(r)=\frac{Gm_{1}}{r^{2}}
  9. r r
  10. a ( r ) = F ( r ) m 2 = - g ( r ) a(r)=\frac{F(r)}{m_{2}}=-g(r)
  11. 𝐄 = k q r 2 r ^ \,\textbf{E}=k\frac{q}{r^{2}}\hat{r}
  12. q test q_{\textrm{test}}

TEX86.html

  1. G D G T r a t i o - 2 = ( [ G D G T - 2 ] + [ G D G T - 3 ] + [ G D G T - 4 ] [ G D G T - 1 ] + [ G D G T - 2 ] + [ G D G T - 3 ] + [ G D G T - 4 ] ) GDGTratio-2=\left(\tfrac{[GDGT-2]+[GDGT-3]+[GDGT-4^{\prime}]}{[GDGT-1]+[GDGT-2% ]+[GDGT-3]+[GDGT-4^{\prime}]}\right)
  2. G D G T r a t i o - 1 = ( [ G D G T - 2 ] [ G D G T - 1 ] + [ G D G T - 2 ] + [ G D G T - 3 ] ) GDGTratio-1=\left(\tfrac{[GDGT-2]}{[GDGT-1]+[GDGT-2]+[GDGT-3]}\right)

Tf–idf.html

  1. f t , d f_{t,d}
  2. log ( 1 + f t , d ) \log(1+f_{t,d})
  3. 0.5 + 0.5 f t , d max f t , d 0.5+0.5\frac{f_{t,d}}{\max{f_{t,d}}}
  4. K + ( 1 - K ) f t , d max f t , d K+(1-K)\frac{f_{t,d}}{\max{f_{t,d}}}
  5. tf ( t , d ) = 0.5 + 0.5 × f ( t , d ) max { f ( t , d ) : t d } \mathrm{tf}(t,d)=0.5+\frac{0.5\times\mathrm{f}(t,d)}{\max\{\mathrm{f}(t,d):t% \in d\}}
  6. log N n t \log\frac{N}{n_{t}}
  7. log ( 1 + N n t ) \log(1+\frac{N}{n_{t}})
  8. log ( 1 + max t n t n t ) \log\left(1+\frac{\max_{t}n_{t}}{n_{t}}\right)
  9. log N - n t n t \log\frac{N-n_{t}}{n_{t}}
  10. idf ( t , D ) = log N | { d D : t d } | \mathrm{idf}(t,D)=\log\frac{N}{|\{d\in D:t\in d\}|}
  11. N N
  12. | { d D : t d } | |\{d\in D:t\in d\}|
  13. t t
  14. tf ( t , d ) 0 \mathrm{tf}(t,d)\neq 0
  15. 1 + | { d D : t d } | 1+|\{d\in D:t\in d\}|
  16. tfidf ( t , d , D ) = tf ( t , d ) × idf ( t , D ) \mathrm{tfidf}(t,d,D)=\mathrm{tf}(t,d)\times\mathrm{idf}(t,D)
  17. f t , d × log N n t f_{t,d}\times\log\frac{N}{n_{t}}
  18. ( 0.5 + 0.5 f t , q max t f t , q ) × log N n t \left(0.5+0.5\frac{f_{t,q}}{\max_{t}f_{t,q}}\right)\times\log\frac{N}{n_{t}}
  19. 1 + log f t , d 1+\log f_{t,d}
  20. log ( 1 + N n t ) \log(1+\frac{N}{n_{t}})
  21. ( 1 + log f t , d ) × log N n t (1+\log f_{t,d})\times\log\frac{N}{n_{t}}
  22. ( 1 + log f t , q ) × log N n t (1+\log f_{t,q})\times\log\frac{N}{n_{t}}
  23. d d
  24. t t
  25. P ( t | d ) = | { d D : t d } | N P(t|d)=\frac{|\{d\in D:t\in d\}|}{N}
  26. idf = - log P ( t | d ) = log 1 P ( t | d ) = log N | { d D : t d } | \begin{aligned}\displaystyle\mathrm{idf}&\displaystyle=-\log P(t|d)\\ &\displaystyle=\log\frac{1}{P(t|d)}\\ &\displaystyle=\log\frac{N}{|\{d\in D:t\in d\}|}\end{aligned}
  27. idf ( 𝗍𝗁𝗂𝗌 , D ) = log N | { d D : t d } | \mathrm{idf}(\mathsf{this},D)=\log\frac{N}{|\{d\in D:t\in d\}|}
  28. idf ( 𝗍𝗁𝗂𝗌 , D ) = log 2 2 = 0 \mathrm{idf}(\mathsf{this},D)=\log\frac{2}{2}=0
  29. tf ( 𝖾𝗑𝖺𝗆𝗉𝗅𝖾 , d 2 ) = 3 \mathrm{tf}(\mathsf{example},d_{2})=3
  30. idf ( 𝖾𝗑𝖺𝗆𝗉𝗅𝖾 , D ) = log 2 1 0.3010 \mathrm{idf}(\mathsf{example},D)=\log\frac{2}{1}\approx 0.3010
  31. tfidf ( 𝖾𝗑𝖺𝗆𝗉𝗅𝖾 , d 2 ) = tf ( 𝖾𝗑𝖺𝗆𝗉𝗅𝖾 , d 2 ) × idf ( 𝖾𝗑𝖺𝗆𝗉𝗅𝖾 , D ) = 3 × 0.3010 0.9030 \mathrm{tfidf}(\mathsf{example},d_{2})=\mathrm{tf}(\mathsf{example},d_{2})% \times\mathrm{idf}(\mathsf{example},D)=3\times 0.3010\approx 0.9030

The_Monkey_and_the_Hunter.html

  1. Y dart = V Y 0 t - 1 2 g t 2 Y_{\rm dart}=V_{Y0}t-\frac{1}{2}gt^{2}
  2. Y monkey = h - 1 2 g t 2 . Y_{\rm monkey}=h-\frac{1}{2}gt^{2}.
  3. V Y 0 t - 1 2 g t 2 = h - 1 2 g t 2 . V_{Y0}t-\frac{1}{2}gt^{2}=h-\frac{1}{2}gt^{2}.
  4. V Y 0 t = h . V_{Y0}t=h.
  5. V Y 0 , V_{Y0},
  6. t = h V Y 0 . t=\frac{h}{V_{Y0}}.
  7. V Y 0 , V_{Y0},

Theil_index.html

  1. T T = T α = 1 = 1 N i = 1 N ( x i x ¯ ln x i x ¯ ) T_{T}=T_{\alpha=1}=\frac{1}{N}\sum_{i=1}^{N}\left(\frac{x_{i}}{\overline{x}}% \cdot\ln{\frac{x_{i}}{\overline{x}}}\right)
  2. x ¯ {\overline{x}}
  3. x x
  4. ln N \ln N
  5. ln N \ln N
  6. S S
  7. S = k i = 1 N ( p i log 1 p i ) = - k i = 1 N ( p i log p i ) S=k\sum_{i=1}^{N}\left(p_{i}\log{\frac{1}{p_{i}}}\right)=-k\sum_{i=1}^{N}\left% (p_{i}\log{p_{i}}\right)
  8. p i p_{i}
  9. i i
  10. k k
  11. k = 1 k=1
  12. p i p_{i}
  13. x i x_{i}
  14. N x ¯ N\overline{x}
  15. S Theil S\text{Theil}
  16. S Theil = i = 1 N ( x i N x ¯ ln N x ¯ x i ) S\text{Theil}=\sum_{i=1}^{N}\left(\frac{x_{i}}{N\overline{x}}\ln{\frac{N% \overline{x}}{x_{i}}}\right)
  17. T T = S max - S Theil T_{T}=S\text{max}-S\text{Theil}
  18. S max S\text{max}
  19. x i = x ¯ x_{i}=\overline{x}
  20. i i
  21. S Theil S\text{Theil}
  22. S max = ln N S\text{max}=\ln N
  23. S Theil S\text{Theil}
  24. S max S\text{max}
  25. x x
  26. S Theil S\text{Theil}
  27. N N
  28. i i
  29. x i x_{i}
  30. N x ¯ N\overline{x}
  31. T T T_{T}
  32. m m
  33. s i s_{i}
  34. i i
  35. T T i T_{Ti}
  36. x ¯ i \overline{x}_{i}
  37. i i
  38. T T = i = 1 m s i T T i + i = 1 m s i ln x ¯ i x ¯ T_{T}=\sum_{i=1}^{m}s_{i}T_{T_{i}}+\sum_{i=1}^{m}s_{i}\ln{\frac{\overline{x}_{% i}}{\overline{x}}}

Theorem_on_friends_and_strangers.html

  1. n n\,
  2. K n K_{n}\,
  3. K 6 K_{6}\,
  4. K 6 K_{6}\,
  5. R ( 3 , 3 : 2 ) = 6. R(3,3:2)=6.\,

Theoretical_astronomy.html

  1. τ d y n a m i c a l R v = R 3 2 G M 1 / G ρ \tau_{dynamical}\simeq\frac{R}{v}=\sqrt{\frac{R^{3}}{2GM}}\sim 1/\sqrt{G\rho}

Theoretical_production_ecology.html

  1. G T = T S k = 1 N ( T - T crit ) GT=\frac{TS}{\sum_{k=1}^{N}(T-T\text{crit})}

Thermal_velocity.html

  1. k B k_{B}
  2. T T
  3. m m
  4. v t h v_{th}
  5. v t h = k B T m v_{th}=\sqrt{\frac{k_{B}T}{m}}
  6. v t h v_{th}
  7. v t h = 2 k B T π m v_{th}=\sqrt{\frac{2k_{B}T}{\pi m}}
  8. v t h v_{th}
  9. 1 / e 1/e
  10. v t h v_{th}
  11. k B T k_{B}T
  12. v t h = 2 k B T m v_{th}=\sqrt{\frac{2k_{B}T}{m}}
  13. v t h v_{th}
  14. v t h = ( 1.1 ± 0.3 ) k B T m v_{th}=(1.1\pm 0.3)\sqrt{\frac{k_{B}T}{m}}
  15. v t h v_{th}
  16. v t h = 2 k B T m v_{th}=\sqrt{\frac{2k_{B}T}{m}}
  17. v t h v_{th}
  18. v t h = 3 k B T m v_{th}=\sqrt{\frac{3k_{B}T}{m}}
  19. v t h v_{th}
  20. v t h = 8 k B T m π v_{th}=\sqrt{\frac{8k_{B}T}{m\pi}}
  21. v t h v_{th}
  22. v t h = ( 1.6 ± 0.2 ) k B T m v_{th}=(1.6\pm 0.2)\sqrt{\frac{k_{B}T}{m}}

Thermodynamic_beta.html

  1. β 1 k B ( S E ) V , N = 1 k B T , \beta\triangleq\frac{1}{k_{B}}\left(\frac{\partial S}{\partial E}\right)_{V,N}% =\frac{1}{k_{B}T}\,,
  2. Ω = Ω 1 ( E 1 ) Ω 2 ( E 2 ) = Ω 1 ( E 1 ) Ω 2 ( E - E 1 ) . \Omega=\Omega_{1}(E_{1})\Omega_{2}(E_{2})=\Omega_{1}(E_{1})\Omega_{2}(E-E_{1}).\,
  3. d d E 1 Ω = Ω 2 ( E 2 ) d d E 1 Ω 1 ( E 1 ) + Ω 1 ( E 1 ) d d E 2 Ω 2 ( E 2 ) d E 2 d E 1 = 0. \frac{d}{dE_{1}}\Omega=\Omega_{2}(E_{2})\frac{d}{dE_{1}}\Omega_{1}(E_{1})+% \Omega_{1}(E_{1})\frac{d}{dE_{2}}\Omega_{2}(E_{2})\cdot\frac{dE_{2}}{dE_{1}}=0.
  4. d E 2 d E 1 = - 1. \frac{dE_{2}}{dE_{1}}=-1.
  5. Ω 2 ( E 2 ) d d E 1 Ω 1 ( E 1 ) - Ω 1 ( E 1 ) d d E 2 Ω 2 ( E 2 ) = 0 \Omega_{2}(E_{2})\frac{d}{dE_{1}}\Omega_{1}(E_{1})-\Omega_{1}(E_{1})\frac{d}{% dE_{2}}\Omega_{2}(E_{2})=0
  6. d d E 1 ln Ω 1 = d d E 2 ln Ω 2 at equilibrium. \frac{d}{dE_{1}}\ln\Omega_{1}=\frac{d}{dE_{2}}\ln\Omega_{2}\quad\mbox{at % equilibrium.}~{}
  7. β = d ln Ω d E . \beta=\frac{d\ln\Omega}{dE}.
  8. S = k B ln Ω , S=k_{B}\ln\Omega,\,
  9. d ln Ω = 1 k B d S . d\ln\Omega=\frac{1}{k_{B}}dS.
  10. β = 1 k B d S d E . \beta=\frac{1}{k_{B}}\frac{dS}{dE}.
  11. d S d E = 1 T , \frac{dS}{dE}=\frac{1}{T},
  12. β = 1 k B T = 1 τ \beta=\frac{1}{k_{B}T}=\frac{1}{\tau}
  13. τ \tau

Thermodynamic_limit.html

  1. N , V , N / V = const N\to\infty,V\to\infty,N/V=\,\text{const}

Thermophotovoltaic.html

  1. Δ G = E p h o t o n - E g \Delta G=E_{photon}-E_{g}
  2. η = 1 - T c e l l T e m i t \eta=1-\frac{T_{cell}}{T_{emit}}
  3. I ( λ , T ) = 2 h c 2 λ 5 1 e h c λ k T - 1 I^{\prime}(\lambda,T)=\frac{2hc^{2}}{\lambda^{5}}\frac{1}{e^{\frac{hc}{\lambda kT% }}-1}
  4. λ max = b T \lambda_{\mathrm{max}}=\frac{b}{T}

Theta_divisor.html

  1. ( g - k + r r ) . {g-k+r\choose r}.

Theta_graph.html

  1. Θ \Theta
  2. Θ \Theta
  3. Θ \Theta
  4. Θ \Theta
  5. Θ \Theta
  6. p p
  7. l l
  8. Θ \Theta
  9. k k
  10. p p
  11. p p
  12. θ = 2 π / k \theta=2\pi/k
  13. p p
  14. C 1 C_{1}
  15. C k C_{k}
  16. C 1 C_{1}
  17. V 1 V_{1}
  18. V k V_{k}
  19. p p
  20. p p
  21. l l
  22. V i V_{i}
  23. v V i v\in V_{i}
  24. l l
  25. r r
  26. { p , r } \{p,r\}
  27. { p , q } \{p,q\}
  28. Θ \Theta
  29. O ( n log n ) O(n\log{n})
  30. Θ \Theta
  31. k k
  32. Θ \Theta
  33. k n = O ( n ) k\cdot n=O(n)
  34. θ = 2 π / k \theta=2\pi/k
  35. k 9 k\geq 9
  36. Θ \Theta
  37. 1 / ( cos θ - sin θ ) 1/(\cos\theta-\sin\theta)
  38. l l
  39. k 7 k\geq 7
  40. 1 / ( 1 - 2 sin ( π / k ) ) 1/(1-2\sin(\pi/k))
  41. k = 1 k=1
  42. Θ \Theta
  43. k = 2 k=2
  44. k = 4 k=4
  45. 5 5
  46. 6 6
  47. 7 \geq 7
  48. Θ \Theta
  49. Θ \Theta
  50. k = 3 k=3
  51. k k
  52. Θ k \Theta_{k}
  53. Θ k \Theta_{k}
  54. Θ k \Theta_{k}
  55. Θ 6 \Theta_{6}
  56. Θ 6 \Theta_{6}
  57. Θ 6 \Theta_{6}

Thin_set_(Serre).html

  1. O ( N 1 / 2 ) O\left({N^{1/2}}\right)
  2. N ( H ) = O ( H n - 1 / 2 log H ) . N(H)=O\left({H^{n-1/2}\log H}\right).

Thomson_problem.html

  1. r = 1 r=1
  2. U ( N ) U(N)
  3. e i = e j = e e_{i}=e_{j}=e
  4. e e
  5. U i j ( N ) = k e e i e j r i j U_{ij}(N)=k_{e}{e_{i}e_{j}\over r_{ij}}
  6. k e k_{e}
  7. r i j = | 𝐫 i - 𝐫 j | r_{ij}=|\mathbf{r}_{i}-\mathbf{r}_{j}|
  8. 𝐫 i \mathbf{r}_{i}
  9. 𝐫 j \mathbf{r}_{j}
  10. e = 1 e=1
  11. k e = 1 k_{e}=1
  12. U i j ( N ) = 1 r i j U_{ij}(N)={1\over r_{ij}}
  13. U ( N ) = i < j 1 r i j U(N)=\sum_{i<j}\frac{1}{r_{ij}}
  14. U ( N ) U(N)
  15. r i j = 2 r = 2 r_{ij}=2r=2
  16. U ( 2 ) = 1 2 U(2)={1\over 2}
  17. i < j f ( | x i - x j | ) \sum_{i<j}f(|x_{i}-x_{j}|)
  18. f ( x ) = x - α f(x)=x^{-\alpha}
  19. N N
  20. E 1 E_{1}
  21. r i r_{i}
  22. v i v_{i}
  23. e e
  24. f 3 f_{3}
  25. f 4 f_{4}
  26. θ 1 \theta_{1}
  27. E 1 E_{1}
  28. | 𝐫 i | \left|\sum\mathbf{r}_{i}\right|
  29. v 3 v_{3}
  30. v 4 v_{4}
  31. v 5 v_{5}
  32. v 6 v_{6}
  33. v 7 v_{7}
  34. v 8 v_{8}
  35. e e
  36. f 3 f_{3}
  37. f 4 f_{4}
  38. θ 1 \theta_{1}
  39. D h D_{\infty h}
  40. D 3 h D_{3h}
  41. T d T_{d}
  42. D 3 h D_{3h}
  43. O h O_{h}
  44. D 5 h D_{5h}
  45. D 4 d D_{4d}
  46. D 3 h D_{3h}
  47. D 4 d D_{4d}
  48. C 2 v C_{2v}
  49. I h I_{h}
  50. C 2 v C_{2v}
  51. D 6 d D_{6d}
  52. D 3 D_{3}
  53. T T
  54. D 5 h D_{5h}
  55. D 4 d D_{4d}
  56. C 2 v C_{2v}
  57. D 3 h D_{3h}
  58. C 2 v C_{2v}
  59. T d T_{d}
  60. D 3 D_{3}
  61. O O
  62. C s C_{s}
  63. C 2 C_{2}
  64. D 5 h D_{5h}
  65. T T
  66. D 3 D_{3}
  67. D 2 D_{2}
  68. C 3 v C_{3v}
  69. I h I_{h}
  70. C s C_{s}
  71. D 2 D_{2}
  72. C 2 C_{2}
  73. D 2 D_{2}
  74. D 5 h D_{5h}
  75. D 6 d D_{6d}
  76. D 3 h D_{3h}
  77. T d T_{d}
  78. D 3 h D_{3h}
  79. D 5 h D_{5h}
  80. C 2 v C_{2v}
  81. O h O_{h}
  82. D 3 D_{3}
  83. T T
  84. C s C_{s}
  85. O O
  86. C 3 C_{3}
  87. D 6 d D_{6d}
  88. D 3 D_{3}
  89. C 3 C_{3}
  90. C 2 v C_{2v}
  91. C 2 C_{2}
  92. C 2 C_{2}
  93. D 2 D_{2}
  94. D 3 D_{3}
  95. D 2 D_{2}
  96. C 2 C_{2}
  97. D 3 D_{3}
  98. C 1 C_{1}
  99. D 5 D_{5}
  100. D 3 D_{3}
  101. D 2 D_{2}
  102. C 2 C_{2}
  103. C 2 C_{2}
  104. D 5 D_{5}
  105. D 2 D_{2}
  106. D 3 D_{3}
  107. D 2 d D_{2d}
  108. C 2 C_{2}
  109. I I
  110. C 2 C_{2}
  111. C 2 C_{2}
  112. D 3 D_{3}
  113. C 2 C_{2}
  114. D 5 D_{5}
  115. T h T_{h}
  116. C s C_{s}
  117. D 4 d D_{4d}
  118. C 2 C_{2}
  119. D 2 D_{2}
  120. C 2 C_{2}
  121. C 2 C_{2}
  122. C 2 C_{2}
  123. C 2 C_{2}
  124. C 2 C_{2}
  125. D 2 D_{2}
  126. C 2 C_{2}
  127. D 3 D_{3}
  128. C 2 C_{2}
  129. D 2 D_{2}
  130. C 2 C_{2}
  131. D 2 D_{2}
  132. C 2 C_{2}
  133. C 2 C_{2}
  134. C 2 C_{2}
  135. C 2 C_{2}
  136. C 2 C_{2}
  137. T T
  138. D 3 D_{3}
  139. D 3 D_{3}
  140. C 2 C_{2}
  141. D 6 D_{6}
  142. D 3 D_{3}
  143. D 2 D_{2}
  144. C 2 C_{2}
  145. C 2 C_{2}
  146. C 2 C_{2}
  147. D 6 D_{6}
  148. D 3 D_{3}
  149. D 5 D_{5}
  150. D 3 D_{3}
  151. C 2 C_{2}
  152. C 3 C_{3}
  153. C 2 C_{2}
  154. C 2 C_{2}
  155. C 2 C_{2}
  156. C 2 C_{2}
  157. C s C_{s}
  158. C 3 C_{3}
  159. I h I_{h}
  160. C 2 v C_{2v}
  161. D 2 D_{2}
  162. C 2 C_{2}
  163. D 4 D_{4}
  164. D 5 D_{5}
  165. C 2 C_{2}
  166. C 2 C_{2}
  167. C 2 C_{2}
  168. C 2 C_{2}
  169. I I
  170. C 3 C_{3}
  171. C 2 C_{2}
  172. D 3 D_{3}
  173. T T
  174. D 5 D_{5}
  175. C 2 C_{2}
  176. C 2 C_{2}
  177. C 1 C_{1}
  178. C 2 v C_{2v}
  179. C 2 C_{2}
  180. C 2 C_{2}
  181. D 2 D_{2}
  182. C s C_{s}
  183. D 2 D_{2}
  184. C 2 C_{2}
  185. C 2 C_{2}
  186. C 1 C_{1}
  187. T T
  188. C 2 C_{2}
  189. D 2 D_{2}
  190. D 3 D_{3}
  191. C 2 C_{2}
  192. C 2 C_{2}
  193. C 2 C_{2}
  194. C 2 C_{2}
  195. C 2 C_{2}
  196. C 2 C_{2}
  197. D 2 D_{2}
  198. C 2 C_{2}
  199. D 3 D_{3}
  200. C 2 C_{2}
  201. D 2 D_{2}
  202. C 2 C_{2}
  203. D 2 d D_{2d}
  204. C 2 C_{2}
  205. D 3 D_{3}
  206. C s C_{s}
  207. D 2 d D_{2d}
  208. D 3 D_{3}
  209. C 2 v C_{2v}
  210. C s C_{s}
  211. D 2 D_{2}
  212. C 2 C_{2}
  213. C 1 C_{1}
  214. D 5 D_{5}
  215. D 2 D_{2}
  216. C 1 C_{1}
  217. D 2 D_{2}
  218. C 2 C_{2}
  219. D 5 D_{5}
  220. C 1 C_{1}
  221. T T
  222. C 2 C_{2}
  223. C 1 C_{1}
  224. D 5 D_{5}
  225. D 2 D_{2}
  226. C 2 C_{2}
  227. C 3 C_{3}
  228. C 1 C_{1}
  229. I I
  230. C 2 C_{2}
  231. C 1 C_{1}
  232. D 3 D_{3}
  233. C 2 C_{2}
  234. D 5 D_{5}
  235. C 2 C_{2}
  236. C 1 C_{1}
  237. D 2 D_{2}
  238. C 1 C_{1}
  239. D 5 D_{5}
  240. C s C_{s}
  241. T h T_{h}
  242. I I
  243. T T
  244. D 3 D_{3}
  245. D 5 D_{5}
  246. T T
  247. D 3 D_{3}
  248. T T
  249. D 5 D_{5}
  250. I h I_{h}
  251. I I
  252. D 5 D_{5}
  253. T h T_{h}
  254. C 2 C_{2}
  255. D 3 D_{3}
  256. D 5 D_{5}
  257. D 3 D_{3}
  258. T T
  259. T h T_{h}
  260. D 5 D_{5}
  261. T T
  262. I I
  263. D 5 D_{5}
  264. T h T_{h}
  265. I I
  266. T T
  267. D 5 D_{5}
  268. D 3 D_{3}
  269. T T
  270. T T
  271. S 6 S_{6}
  272. S 6 S_{6}
  273. N N
  274. ( 2 N 112 ) (2\leq N\leq 112)

Thouless_energy.html

  1. E T = D / L 2 E_{T}=\hbar D/L^{2}
  2. t D = L 2 / D t_{D}=L^{2}/D

Three-dimensional_space_(mathematics).html

  1. 3 \scriptstyle{\mathbb{R}}^{3}
  2. V = 4 3 π r 3 V=\frac{4}{3}\pi r^{3}
  3. 4 \mathbb{R}^{4}
  4. P = ( x , y , z , t ) P=(x,y,z,t)
  5. x 2 + y 2 + z 2 + t 2 = 1 x^{2}+y^{2}+z^{2}+t^{2}=1
  6. 𝐀 𝐁 = A 1 B 1 + A 2 B 2 + A 3 B 3 \mathbf{A}\cdot\mathbf{B}=A_{1}B_{1}+A_{2}B_{2}+A_{3}B_{3}
  7. 𝐀 \|\mathbf{A}\|
  8. 𝐀 𝐁 = 𝐀 𝐁 cos θ , \mathbf{A}\cdot\mathbf{B}=\|\mathbf{A}\|\,\|\mathbf{B}\|\cos\theta,
  9. 𝐀 𝐀 = 𝐀 2 , \mathbf{A}\cdot\mathbf{A}=\|\mathbf{A}\|^{2},
  10. 𝐀 = 𝐀 𝐀 , \|\mathbf{A}\|=\sqrt{\mathbf{A}\cdot\mathbf{A}},
  11. f = f x 𝐢 + f y 𝐣 + f z 𝐤 \nabla f=\frac{\partial f}{\partial x}\mathbf{i}+\frac{\partial f}{\partial y}% \mathbf{j}+\frac{\partial f}{\partial z}\mathbf{k}
  12. div 𝐅 = 𝐅 = U x + V y + W z . \operatorname{div}\,\mathbf{F}=\nabla\cdot\mathbf{F}=\frac{\partial U}{% \partial x}+\frac{\partial V}{\partial y}+\frac{\partial W}{\partial z}.
  13. | 𝐢 𝐣 𝐤 x y z F x F y F z | \begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\ \\ {\frac{\partial}{\partial x}}&{\frac{\partial}{\partial y}}&{\frac{\partial}{% \partial z}}\\ \\ F_{x}&F_{y}&F_{z}\end{vmatrix}
  14. ( F z y - F y z ) 𝐢 + ( F x z - F z x ) 𝐣 + ( F y x - F x y ) 𝐤 \left(\frac{\partial F_{z}}{\partial y}-\frac{\partial F_{y}}{\partial z}% \right)\mathbf{i}+\left(\frac{\partial F_{x}}{\partial z}-\frac{\partial F_{z}% }{\partial x}\right)\mathbf{j}+\left(\frac{\partial F_{y}}{\partial x}-\frac{% \partial F_{x}}{\partial y}\right)\mathbf{k}
  15. C f d s = a b f ( 𝐫 ( t ) ) | 𝐫 ( t ) | d t . \int\limits_{C}f\,ds=\int_{a}^{b}f(\mathbf{r}(t))|\mathbf{r}^{\prime}(t)|\,dt.
  16. a < b a<b
  17. C 𝐅 ( 𝐫 ) d 𝐫 = a b 𝐅 ( 𝐫 ( t ) ) 𝐫 ( t ) d t . \int\limits_{C}\mathbf{F}(\mathbf{r})\cdot\,d\mathbf{r}=\int_{a}^{b}\mathbf{F}% (\mathbf{r}(t))\cdot\mathbf{r}^{\prime}(t)\,dt.
  18. S f d S = T f ( 𝐱 ( s , t ) ) 𝐱 s × 𝐱 t d s d t \iint_{S}f\,\mathrm{d}S=\iint_{T}f(\mathbf{x}(s,t))\left\|{\partial\mathbf{x}% \over\partial s}\times{\partial\mathbf{x}\over\partial t}\right\|\mathrm{d}s\,% \mathrm{d}t
  19. f ( x , y , z ) , f(x,y,z),
  20. D f ( x , y , z ) d x d y d z . \iiint\limits_{D}f(x,y,z)\,dx\,dy\,dz.
  21. φ : U n \varphi:U\subseteq\mathbb{R}^{n}\to\mathbb{R}
  22. φ ( 𝐪 ) - φ ( 𝐩 ) = γ [ 𝐩 , 𝐪 ] φ ( 𝐫 ) d 𝐫 . \varphi\left(\mathbf{q}\right)-\varphi\left(\mathbf{p}\right)=\int_{\gamma[% \mathbf{p},\,\mathbf{q}]}\nabla\varphi(\mathbf{r})\cdot d\mathbf{r}.
  23. Σ × 𝐅 d 𝚺 = Σ 𝐅 d 𝐫 . \iint_{\Sigma}\nabla\times\mathbf{F}\cdot\mathrm{d}\mathbf{\Sigma}=\oint_{% \partial\Sigma}\mathbf{F}\cdot\mathrm{d}\mathbf{r}.
  24. V V
  25. n \mathbb{R}^{n}
  26. n = 3 , V n=3,V
  27. S S
  28. V = S ∂V=S
  29. 𝐅 \mathbf{F}
  30. V V
  31. V V
  32. V V
  33. V ∂V
  34. V V
  35. 𝐧 \mathbf{n}
  36. V ∂V
  37. d 𝐒 d\mathbf{S}
  38. 𝐧 d S \mathbf{n}dS
  39. 3 \scriptstyle{\mathbb{R}}^{3}

Three-torus.html

  1. 𝕋 3 = S 1 × S 1 × S 1 . \mathbb{T}^{3}=S^{1}\times S^{1}\times S^{1}.

Throughput_(business).html

  1. I = R * T I=R*T
  2. R = I / T R=I/T

Tidal_tensor.html

  1. U U
  2. Δ U = 4 π μ \Delta U=4\pi\,\mu
  3. μ \mu
  4. Φ a b = J a b - 1 3 J m m η a b \Phi_{ab}=J_{ab}-\frac{1}{3}\,{J^{m}}_{m}\,\eta_{ab}
  5. J a b = 2 U x a x b J_{ab}=\frac{\partial^{2}U}{\partial x^{a}\,\partial x^{b}}
  6. d s 2 = d x 2 + d y 2 + d z 2 , - < x , y , z < ds^{2}=dx^{2}+dy^{2}+dz^{2},\;-\infty<x,y,z<\infty
  7. d s 2 = d ρ 2 + ρ 2 ( d θ 2 + sin ( θ ) 2 d ϕ 2 ) ds^{2}=d\rho^{2}+\rho^{2}\,\left(d\theta^{2}+\sin(\theta)^{2}\,d\phi^{2}\right)
  8. 0 < ρ < , 0 < θ < π , - π < ϕ < π 0<\rho<\infty,\;0<\theta<\pi,\;-\pi<\phi<\pi
  9. ϵ 1 = r , ϵ 2 = 1 r θ , ϵ 3 = 1 r sin θ ϕ \vec{\epsilon}_{1}=\partial_{r},\;\vec{\epsilon}_{2}=\frac{1}{r}\,\partial_{% \theta},\;\vec{\epsilon}_{3}=\frac{1}{r\sin\theta}\,\partial_{\phi}
  10. m / ( r + h ) 2 - m / r 2 = - 2 m / r 3 h + 3 m / r 4 h 2 + O ( h 3 ) m/(r+h)^{2}-m/r^{2}=-2m/r^{3}\,h+3m/r^{4}\,h^{2}+O(h^{3})
  11. Φ 11 = - 2 m / r 3 \Phi_{11}=-2m/r^{3}
  12. r = r 0 r=r_{0}
  13. m r 0 2 sin ( θ ) m r 0 2 h r 0 = m r 0 3 h \frac{m}{r_{0}^{2}}\,\sin(\theta)\approx\frac{m}{r_{0}^{2}}\,\frac{h}{r_{0}}=% \frac{m}{r_{0}^{3}}\,h
  14. O ( h 2 ) O(h^{2})
  15. Φ 22 = Φ 33 = m / r 3 \Phi_{22}=\Phi_{33}=m/r^{3}
  16. Φ a ^ b ^ = m r 3 diag ( - 2 , 1 , 1 ) \Phi_{\hat{a}\hat{b}}=\frac{m}{r^{3}}\operatorname{diag}(-2,1,1)
  17. U = - m / ρ U=-m/\rho
  18. U = - m / ( x 2 + y 2 + z 2 ) U=-m/\sqrt{(}x^{2}+y^{2}+z^{2})
  19. Φ a b = m ( x 2 + y 2 + z 2 ) 5 / 2 [ y 2 + z 2 - 2 x 2 - 3 x y - 3 x z - 3 x y x 2 + z 2 - 2 y 2 - 3 y z - 3 x z - 3 y z x 2 + y 2 - 2 z 2 ] \Phi_{ab}=\frac{m}{(x^{2}+y^{2}+z^{2})^{5/2}}\,\left[\begin{matrix}y^{2}+z^{2}% -2x^{2}&-3xy&-3xz\\ -3xy&x^{2}+z^{2}-2y^{2}&-3yz\\ -3xz&-3yz&x^{2}+y^{2}-2z^{2}\end{matrix}\right]
  20. ρ = x \rho=x

Tight_binding.html

  1. φ m ( s y m b o l r ) \varphi_{m}(symbol{r})
  2. H a t H_{at}
  3. Δ U \Delta U
  4. H H
  5. H ( s y m b o l r ) = s y m b o l R n H at ( s y m b o l r - R n ) + Δ U ( s y m b o l r ) . H(symbol{r})=\sum_{symbol{R_{n}}}H_{\mathrm{at}}(symbol{r-R_{n}})+\Delta U(% symbol{r})\ .
  6. ψ r \psi_{r}
  7. φ m ( s y m b o l r - R n ) \varphi_{m}(symbol{r-R_{n}})
  8. ψ ( s y m b o l r ) = m , s y m b o l R n b m ( s y m b o l R n ) φ m ( s y m b o l r - R n ) \psi(symbol{r})=\sum_{m,symbol{R_{n}}}b_{m}(symbol{R_{n}})\ \varphi_{m}(symbol% {r-R_{n}})
  9. m m
  10. R n R_{n}
  11. ψ ( s y m b o l r + R ) = e i s y m b o l k R ψ ( s y m b o l r ) , \psi(symbol{r+R_{\ell}})=e^{isymbol{k\cdot R_{\ell}}}\psi(symbol{r})\ ,
  12. k {k}
  13. m , s y m b o l R n b m ( s y m b o l R n ) φ m ( s y m b o l r - R n + R ) = e i s y m b o l k R m , s y m b o l R n b m ( s y m b o l R n ) φ m ( s y m b o l r - R n ) . \sum_{m,symbol{R_{n}}}b_{m}(symbol{R_{n}})\ \varphi_{m}(symbol{r-R_{n}+R_{\ell% }})=e^{isymbol{k\cdot R_{\ell}}}\sum_{m,symbol{R_{n}}}b_{m}(symbol{R_{n}})\ % \varphi_{m}(symbol{r-R_{n}})\ .
  14. s y m b o l R p = s y m b o l R n - s y m b o l R symbol{R_{p}}=symbol{R_{n}}-symbol{R_{\ell}}
  15. b m ( s y m b o l R p + R ) = e i s y m b o l k R b m ( s y m b o l R p ) , b_{m}(symbol{R_{p}+R_{\ell}})=e^{isymbol{k\cdot R_{\ell}}}b_{m}(symbol{R_{p}})\ ,
  16. s y m b o l R n symbol{R_{n}}
  17. s y m b o l R p symbol{R_{p}}
  18. b m ( s y m b o l R p ) = e i s y m b o l k R p b m ( s y m b o l 0 ) . b_{m}(symbol{R_{p}})=e^{isymbol{k\cdot R_{p}}}b_{m}(symbol{0})\ .
  19. d 3 r ψ * ( s y m b o l r ) ψ ( s y m b o l r ) = 1 \int d^{3}r\ \psi^{*}(symbol{r})\psi(symbol{r})=1
  20. = s y m b o l R n b * ( s y m b o l R n ) s y m b o l R b ( s y m b o l R ) d 3 r φ * ( s y m b o l r - R n ) φ ( s y m b o l r - R ) =\sum_{symbol{R_{n}}}b^{*}(symbol{R_{n}})\sum_{symbol{R_{\ell}}}b(symbol{R_{% \ell}})\int d^{3}r\ \varphi^{*}(symbol{r-R_{n}})\varphi(symbol{r-R_{\ell}})
  21. = b * ( 0 ) b ( 0 ) s y m b o l R n e - i s y m b o l k R n s y m b o l R e i s y m b o l k R d 3 r φ * ( s y m b o l r - R n ) φ ( s y m b o l r - R ) =b^{*}(0)b(0)\sum_{symbol{R_{n}}}e^{-isymbol{k\cdot R_{n}}}\sum_{symbol{R_{% \ell}}}e^{isymbol{k\cdot R_{\ell}}}\ \int d^{3}r\ \varphi^{*}(symbol{r-R_{n}})% \varphi(symbol{r-R_{\ell}})
  22. = N b * ( 0 ) b ( 0 ) s y m b o l R p e - i s y m b o l k R p d 3 r φ * ( s y m b o l r - R p ) φ ( s y m b o l r ) =Nb^{*}(0)b(0)\sum_{symbol{R_{p}}}e^{-isymbol{k\cdot R_{p}}}\ \int d^{3}r\ % \varphi^{*}(symbol{r-R_{p}})\varphi(symbol{r})
  23. = N b * ( 0 ) b ( 0 ) s y m b o l R p e i s y m b o l k R p d 3 r φ * ( s y m b o l r ) φ ( s y m b o l r - R p ) , =Nb^{*}(0)b(0)\sum_{symbol{R_{p}}}e^{isymbol{k\cdot R_{p}}}\ \int d^{3}r\ % \varphi^{*}(symbol{r})\varphi(symbol{r-R_{p}})\ ,
  24. b * ( 0 ) b ( 0 ) = 1 N 1 1 + s y m b o l R p 0 e i s y m b o l k R p α ( s y m b o l R p ) , b^{*}(0)b(0)=\frac{1}{N}\ \cdot\ \frac{1}{1+\sum_{symbol{R_{p}\neq 0}}e^{% isymbol{k\cdot R_{p}}}\alpha(symbol{R_{p}})}\ ,
  25. b n ( 0 ) 1 N , b_{n}(0)\approx\frac{1}{\sqrt{N}}\ ,
  26. ψ ( s y m b o l r ) 1 N m , s y m b o l R n e i s y m b o l k R n φ m ( s y m b o l r - R n ) . \psi(symbol{r})\approx\frac{1}{\sqrt{N}}\sum_{m,symbol{R_{n}}}e^{isymbol{k% \cdot R_{n}}}\ \varphi_{m}(symbol{r-R_{n}})\ .
  27. ε m \varepsilon_{m}
  28. ε m = d 3 r ψ * ( s y m b o l r ) H ( s y m b o l r ) ψ ( s y m b o l r ) \varepsilon_{m}=\int d^{3}r\ \psi^{*}(symbol{r})H(symbol{r})\psi(symbol{r})
  29. = s y m b o l R n b * ( s y m b o l R n ) d 3 r φ * ( s y m b o l r - R n ) H ( s y m b o l r ) ψ ( s y m b o l r ) =\sum_{symbol{R_{n}}}b^{*}(symbol{R_{n}})\ \int d^{3}r\ \varphi^{*}(symbol{r-R% _{n}})H(symbol{r})\psi(symbol{r})
  30. = s y m b o l R s y m b o l R n b * ( s y m b o l R n ) d 3 r φ * ( s y m b o l r - R n ) H at ( s y m b o l r - R ) ψ ( s y m b o l r ) + s y m b o l R n b * ( s y m b o l R n ) d 3 r φ * ( s y m b o l r - R n ) Δ U ( s y m b o l r ) ψ ( s y m b o l r ) . =\sum_{symbol{R_{\ell}}}\ \sum_{symbol{R_{n}}}b^{*}(symbol{R_{n}})\ \int d^{3}% r\ \varphi^{*}(symbol{r-R_{n}})H_{\mathrm{at}}(symbol{r-R_{\ell}})\psi(symbol{% r})\ +\sum_{symbol{R_{n}}}b^{*}(symbol{R_{n}})\ \int d^{3}r\ \varphi^{*}(% symbol{r-R_{n}})\Delta U(symbol{r})\psi(symbol{r})\ .
  31. E m + b * ( 0 ) s y m b o l R n e - i s y m b o l k R n d 3 r φ * ( s y m b o l r - R n ) Δ U ( s y m b o l r ) ψ ( s y m b o l r ) . \approx E_{m}+b^{*}(0)\sum_{symbol{R_{n}}}e^{-isymbol{k\cdot R_{n}}}\ \int d^{% 3}r\ \varphi^{*}(symbol{r-R_{n}})\Delta U(symbol{r})\psi(symbol{r})\ .
  32. ε m ( s y m b o l k ) = E m - N | b ( 0 ) | 2 ( β m + s y m b o l R n 0 l γ m , l ( s y m b o l R n ) e i s y m b o l k s y m b o l R n ) , \varepsilon_{m}(symbol{k})=E_{m}-N\ |b(0)|^{2}\left(\beta_{m}+\sum_{symbol{R_{% n}}\neq 0}\sum_{l}\gamma_{m,l}(symbol{R_{n}})e^{isymbol{k}\cdot symbol{R_{n}}}% \right)\ ,
  33. = E m - β m + s y m b o l R n 0 l e i s y m b o l k s y m b o l R n γ m , l ( s y m b o l R n ) 1 + s y m b o l R n 0 l e i s y m b o l k R n α m , l ( s y m b o l R n ) , =E_{m}-\ \frac{\beta_{m}+\sum_{symbol{R_{n}}\neq 0}\sum_{l}e^{isymbol{k}\cdot symbol% {R_{n}}}\gamma_{m,l}(symbol{R_{n}})}{\ \ 1+\sum_{symbol{R_{n}\neq 0}}\sum_{l}e% ^{isymbol{k\cdot R_{n}}}\alpha_{m,l}(symbol{R_{n}})}\ ,
  34. α m , l \alpha_{m,l}
  35. β m \beta_{m}
  36. γ m , l \gamma_{m,l}
  37. β m = - φ m * ( s y m b o l r ) Δ U ( s y m b o l r ) φ m ( s y m b o l r ) d 3 r \beta_{m}=-\int\varphi_{m}^{*}(symbol{r})\Delta U(symbol{r})\varphi_{m}(symbol% {r})\,d^{3}r
  38. γ m , l ( s y m b o l R n ) = - φ m * ( s y m b o l r ) Δ U ( s y m b o l r ) φ l ( s y m b o l r - R n ) d 3 r , \gamma_{m,l}(symbol{R_{n}})=-\int\varphi_{m}^{*}(symbol{r})\Delta U(symbol{r})% \varphi_{l}(symbol{r-R_{n}})\,d^{3}r\ ,
  39. α m , l ( s y m b o l R n ) = φ m * ( s y m b o l r ) φ l ( s y m b o l r - R n ) d 3 r \alpha_{m,l}(symbol{R_{n}})=\int\varphi_{m}^{*}(symbol{r})\varphi_{l}(symbol{r% -R_{n}})\,d^{3}r
  40. β m \beta_{m}
  41. β m \beta_{m}
  42. γ m , l \gamma_{m,l}
  43. α m , l \alpha_{m,l}
  44. ψ m ( 𝐤 , 𝐫 ) = 1 N n a m ( 𝐑 𝐧 , 𝐫 ) e 𝐢𝐤 𝐑 𝐧 , \psi_{m}\mathbf{(k,r)}=\frac{1}{\sqrt{N}}\sum_{n}{a_{m}\mathbf{(R_{n},r)}}e^{% \mathbf{ik\cdot R_{n}}}\ ,
  45. a m ( 𝐑 𝐧 , 𝐫 ) = 1 N 𝐤 e - 𝐢𝐤 𝐑 𝐧 ψ m ( 𝐤 , 𝐫 ) = 1 N 𝐤 e 𝐢𝐤 ( 𝐫 - 𝐑 𝐧 ) u m ( 𝐤 , 𝐫 ) . a_{m}\mathbf{(R_{n},r)}=\frac{1}{\sqrt{N}}\sum_{\mathbf{k}}{e^{\mathbf{-ik% \cdot R_{n}}}\psi_{m}\mathbf{(k,r)}}=\frac{1}{\sqrt{N}}\sum_{\mathbf{k}}{e^{% \mathbf{ik\cdot(r-R_{n})}}u_{m}\mathbf{(k,r)}}.
  46. a m ( 𝐑 𝐧 , 𝐫 ) {a_{m}\mathbf{(R_{n},r)}}
  47. H = - t i , j , σ ( c i , σ c j , σ + h . c . ) H=-t\sum_{\langle i,j\rangle,\sigma}(c^{\dagger}_{i,\sigma}c_{j,\sigma}+h.c.)
  48. c i σ , c j σ c^{\dagger}_{i\sigma},c_{j\sigma}
  49. σ \displaystyle\sigma
  50. t \displaystyle t
  51. i , j \displaystyle\langle i,j\rangle
  52. t \displaystyle t
  53. γ \displaystyle\gamma
  54. t 0 t\rightarrow 0
  55. t > 0 \displaystyle t>0
  56. H e e = 1 2 n , m , σ n 1 m 1 , n 2 m 2 | e 2 | r 1 - r 2 | | n 3 m 3 , n 4 m 4 c n 1 m 1 σ 1 c n 2 m 2 σ 2 c n 4 m 4 σ 2 c n 3 m 3 σ 1 \displaystyle H_{ee}=\frac{1}{2}\sum_{n,m,\sigma}\langle n_{1}m_{1},n_{2}m_{2}% |\frac{e^{2}}{|r_{1}-r_{2}|}|n_{3}m_{3},n_{4}m_{4}\rangle c^{\dagger}_{n_{1}m_% {1}\sigma_{1}}c^{\dagger}_{n_{2}m_{2}\sigma_{2}}c_{n_{4}m_{4}\sigma_{2}}c_{n_{% 3}m_{3}\sigma_{1}}
  57. | k = 1 N n = 1 N e i n k a | n |k\rangle=\frac{1}{\sqrt{N}}\sum_{n=1}^{N}e^{inka}|n\rangle
  58. k k
  59. - π a k π a -\frac{\pi}{a}\leqq k\leqq\frac{\pi}{a}
  60. n | H | n = E 0 = E i - U . \langle n|H|n\rangle=E_{0}=E_{i}-U\ .
  61. n ± 1 | H | n = - Δ \langle n\pm 1|H|n\rangle=-\Delta
  62. n | n = 1 ; \langle n|n\rangle=1\ ;
  63. n ± 1 | n = S . \langle n\pm 1|n\rangle=S\ .
  64. n ± 1 | H | n = - Δ \langle n\pm 1|H|n\rangle=-\Delta
  65. E i , j E_{i,j}
  66. σ \sigma
  67. E s , s = V s s σ E_{s,s}=V_{ss\sigma}
  68. | k |k\rangle
  69. H | k = 1 N n e i n k a H | n H|k\rangle=\frac{1}{\sqrt{N}}\sum_{n}e^{inka}H|n\rangle
  70. k | H | k = 1 N n , m e i ( n - m ) k a m | H | n \langle k|H|k\rangle=\frac{1}{N}\sum_{n,\ m}e^{i(n-m)ka}\langle m|H|n\rangle
  71. = 1 N n n | H | n + 1 N n n - 1 | H | n e + i k a + 1 N n n + 1 | H | n e - i k a =\frac{1}{N}\sum_{n}\langle n|H|n\rangle+\frac{1}{N}\sum_{n}\langle n-1|H|n% \rangle e^{+ika}+\frac{1}{N}\sum_{n}\langle n+1|H|n\rangle e^{-ika}
  72. = E 0 - 2 Δ cos ( k a ) , =E_{0}-2\Delta\,\cos(ka)\ ,
  73. 1 N n n | H | n = E 0 1 N n 1 = E 0 , \frac{1}{N}\sum_{n}\langle n|H|n\rangle=E_{0}\frac{1}{N}\sum_{n}1=E_{0}\ ,
  74. 1 N n n - 1 | H | n e + i k a = - Δ e i k a 1 N n 1 = - Δ e i k a . \frac{1}{N}\sum_{n}\langle n-1|H|n\rangle e^{+ika}=-\Delta e^{ika}\frac{1}{N}% \sum_{n}1=-\Delta e^{ika}\ .
  75. 1 N n n - 1 | n e + i k a = S e i k a 1 N n 1 = S e i k a . \frac{1}{N}\sum_{n}\langle n-1|n\rangle e^{+ika}=Se^{ika}\frac{1}{N}\sum_{n}1=% Se^{ika}\ .
  76. | k |k\rangle
  77. E ( k ) = E 0 - 2 Δ cos ( k a ) 1 + 2 S cos ( k a ) E(k)=\frac{E_{0}-2\Delta\,\cos(ka)}{1+2S\,\cos(ka)}
  78. k = 0 k=0
  79. E = ( E 0 - 2 Δ ) / ( 1 + 2 S ) E=(E_{0}-2\Delta)/(1+2S)
  80. k = π / ( 2 a ) k=\pi/(2a)
  81. E = E 0 E=E_{0}
  82. e i π / 2 e^{i\pi/2}
  83. k = π / a k=\pi/a
  84. E = ( E 0 + 2 Δ ) / ( 1 - 2 S ) E=(E_{0}+2\Delta)/(1-2S)
  85. E i , j ( r n , n ) = n , i | H | n , j E_{i,j}(\vec{{r}}_{n,n^{\prime}})=\langle n,i|H|n^{\prime},j\rangle
  86. V s s σ V_{ss\sigma}
  87. V p p π V_{pp\pi}
  88. V d d δ V_{dd\delta}
  89. r n , n = ( r x , r y , r z ) = d ( l , m , n ) \vec{{r}}_{n,n^{\prime}}=(r_{x},r_{y},r_{z})=d(l,m,n)
  90. E s , s = V s s σ E_{s,s}=V_{ss\sigma}
  91. E s , x = l V s p σ E_{s,x}=lV_{sp\sigma}
  92. E x , x = l 2 V p p σ + ( 1 - l 2 ) V p p π E_{x,x}=l^{2}V_{pp\sigma}+(1-l^{2})V_{pp\pi}
  93. E x , y = l m V p p σ - l m V p p π E_{x,y}=lmV_{pp\sigma}-lmV_{pp\pi}
  94. E x , z = l n V p p σ - l n V p p π E_{x,z}=lnV_{pp\sigma}-lnV_{pp\pi}
  95. E s , x y = 3 l m V s d σ E_{s,xy}=\sqrt{3}lmV_{sd\sigma}
  96. E s , x 2 - y 2 = 3 2 ( l 2 - m 2 ) V s d σ E_{s,x^{2}-y^{2}}=\frac{\sqrt{3}}{2}(l^{2}-m^{2})V_{sd\sigma}
  97. E s , 3 z 2 - r 2 = [ n 2 - ( l 2 + m 2 ) / 2 ] V s d σ E_{s,3z^{2}-r^{2}}=[n^{2}-(l^{2}+m^{2})/2]V_{sd\sigma}
  98. E x , x y = 3 l 2 m V p d σ + m ( 1 - 2 l 2 ) V p d π E_{x,xy}=\sqrt{3}l^{2}mV_{pd\sigma}+m(1-2l^{2})V_{pd\pi}
  99. E x , y z = 3 l m n V p d σ - 2 l m n V p d π E_{x,yz}=\sqrt{3}lmnV_{pd\sigma}-2lmnV_{pd\pi}
  100. E x , z x = 3 l 2 n V p d σ + n ( 1 - 2 l 2 ) V p d π E_{x,zx}=\sqrt{3}l^{2}nV_{pd\sigma}+n(1-2l^{2})V_{pd\pi}
  101. E x , x 2 - y 2 = 3 2 l ( l 2 - m 2 ) V p d σ + l ( 1 - l 2 + m 2 ) V p d π E_{x,x^{2}-y^{2}}=\frac{\sqrt{3}}{2}l(l^{2}-m^{2})V_{pd\sigma}+l(1-l^{2}+m^{2}% )V_{pd\pi}
  102. E y , x 2 - y 2 = 3 2 m ( l 2 - m 2 ) V p d σ - m ( 1 + l 2 - m 2 ) V p d π E_{y,x^{2}-y^{2}}=\frac{\sqrt{3}}{2}m(l^{2}-m^{2})V_{pd\sigma}-m(1+l^{2}-m^{2}% )V_{pd\pi}
  103. E z , x 2 - y 2 = 3 2 n ( l 2 - m 2 ) V p d σ - n ( l 2 - m 2 ) V p d π E_{z,x^{2}-y^{2}}=\frac{\sqrt{3}}{2}n(l^{2}-m^{2})V_{pd\sigma}-n(l^{2}-m^{2})V% _{pd\pi}
  104. E x , 3 z 2 - r 2 = l [ n 2 - ( l 2 + m 2 ) / 2 ] V p d σ - 3 l n 2 V p d π E_{x,3z^{2}-r^{2}}=l[n^{2}-(l^{2}+m^{2})/2]V_{pd\sigma}-\sqrt{3}ln^{2}V_{pd\pi}
  105. E y , 3 z 2 - r 2 = m [ n 2 - ( l 2 + m 2 ) / 2 ] V p d σ - 3 m n 2 V p d π E_{y,3z^{2}-r^{2}}=m[n^{2}-(l^{2}+m^{2})/2]V_{pd\sigma}-\sqrt{3}mn^{2}V_{pd\pi}
  106. E z , 3 z 2 - r 2 = n [ n 2 - ( l 2 + m 2 ) / 2 ] V p d σ + 3 n ( l 2 + m 2 ) V p d π E_{z,3z^{2}-r^{2}}=n[n^{2}-(l^{2}+m^{2})/2]V_{pd\sigma}+\sqrt{3}n(l^{2}+m^{2})% V_{pd\pi}
  107. E x y , x y = 3 l 2 m 2 V d d σ + ( l 2 + m 2 - 4 l 2 m 2 ) V d d π + ( n 2 + l 2 m 2 ) V d d δ E_{xy,xy}=3l^{2}m^{2}V_{dd\sigma}+(l^{2}+m^{2}-4l^{2}m^{2})V_{dd\pi}+(n^{2}+l^% {2}m^{2})V_{dd\delta}
  108. E x y , y z = 3 l m 2 n V d d σ + l n ( 1 - 4 m 2 ) V d d π + l n ( m 2 - 1 ) V d d δ E_{xy,yz}=3lm^{2}nV_{dd\sigma}+ln(1-4m^{2})V_{dd\pi}+ln(m^{2}-1)V_{dd\delta}
  109. E x y , z x = 3 l 2 m n V d d σ + m n ( 1 - 4 l 2 ) V d d π + m n ( l 2 - 1 ) V d d δ E_{xy,zx}=3l^{2}mnV_{dd\sigma}+mn(1-4l^{2})V_{dd\pi}+mn(l^{2}-1)V_{dd\delta}
  110. E x y , x 2 - y 2 = 3 2 l m ( l 2 - m 2 ) V d d σ + 2 l m ( m 2 - l 2 ) V d d π + l m ( l 2 - m 2 ) / 2 V d d δ E_{xy,x^{2}-y^{2}}=\frac{3}{2}lm(l^{2}-m^{2})V_{dd\sigma}+2lm(m^{2}-l^{2})V_{% dd\pi}+lm(l^{2}-m^{2})/2V_{dd\delta}
  111. E y z , x 2 - y 2 = 3 2 m n ( l 2 - m 2 ) V d d σ - m n [ 1 + 2 ( l 2 - m 2 ) ] V d d π + m n [ 1 + ( l 2 - m 2 ) / 2 ] V d d δ E_{yz,x^{2}-y^{2}}=\frac{3}{2}mn(l^{2}-m^{2})V_{dd\sigma}-mn[1+2(l^{2}-m^{2})]% V_{dd\pi}+mn[1+(l^{2}-m^{2})/2]V_{dd\delta}
  112. E z x , x 2 - y 2 = 3 2 n l ( l 2 - m 2 ) V d d σ + n l [ 1 - 2 ( l 2 - m 2 ) ] V d d π - n l [ 1 - ( l 2 - m 2 ) / 2 ] V d d δ E_{zx,x^{2}-y^{2}}=\frac{3}{2}nl(l^{2}-m^{2})V_{dd\sigma}+nl[1-2(l^{2}-m^{2})]% V_{dd\pi}-nl[1-(l^{2}-m^{2})/2]V_{dd\delta}
  113. E x y , 3 z 2 - r 2 = 3 [ l m ( n 2 - ( l 2 + m 2 ) / 2 ) V d d σ - 2 l m n 2 V d d π + l m ( 1 + n 2 ) / 2 V d d δ ] E_{xy,3z^{2}-r^{2}}=\sqrt{3}\left[lm(n^{2}-(l^{2}+m^{2})/2)V_{dd\sigma}-2lmn^{% 2}V_{dd\pi}+lm(1+n^{2})/2V_{dd\delta}\right]
  114. E y z , 3 z 2 - r 2 = 3 [ m n ( n 2 - ( l 2 + m 2 ) / 2 ) V d d σ + m n ( l 2 + m 2 - n 2 ) V d d π - m n ( l 2 + m 2 ) / 2 V d d δ ] E_{yz,3z^{2}-r^{2}}=\sqrt{3}\left[mn(n^{2}-(l^{2}+m^{2})/2)V_{dd\sigma}+mn(l^{% 2}+m^{2}-n^{2})V_{dd\pi}-mn(l^{2}+m^{2})/2V_{dd\delta}\right]
  115. E z x , 3 z 2 - r 2 = 3 [ l n ( n 2 - ( l 2 + m 2 ) / 2 ) V d d σ + l n ( l 2 + m 2 - n 2 ) V d d π - l n ( l 2 + m 2 ) / 2 V d d δ ] E_{zx,3z^{2}-r^{2}}=\sqrt{3}\left[ln(n^{2}-(l^{2}+m^{2})/2)V_{dd\sigma}+ln(l^{% 2}+m^{2}-n^{2})V_{dd\pi}-ln(l^{2}+m^{2})/2V_{dd\delta}\right]
  116. E x 2 - y 2 , x 2 - y 2 = 3 4 ( l 2 - m 2 ) 2 V d d σ + [ l 2 + m 2 - ( l 2 - m 2 ) 2 ] V d d π + [ n 2 + ( l 2 - m 2 ) 2 / 4 ] V d d δ E_{x^{2}-y^{2},x^{2}-y^{2}}=\frac{3}{4}(l^{2}-m^{2})^{2}V_{dd\sigma}+[l^{2}+m^% {2}-(l^{2}-m^{2})^{2}]V_{dd\pi}+[n^{2}+(l^{2}-m^{2})^{2}/4]V_{dd\delta}
  117. E x 2 - y 2 , 3 z 2 - r 2 = 3 [ ( l 2 - m 2 ) [ n 2 - ( l 2 + m 2 ) / 2 ] V d d σ / 2 + n 2 ( m 2 - l 2 ) V d d π + ( 1 + n 2 ) ( l 2 - m 2 ) / 4 V d d δ ] E_{x^{2}-y^{2},3z^{2}-r^{2}}=\sqrt{3}\left[(l^{2}-m^{2})[n^{2}-(l^{2}+m^{2})/2% ]V_{dd\sigma}/2+n^{2}(m^{2}-l^{2})V_{dd\pi}+(1+n^{2})(l^{2}-m^{2})/4V_{dd% \delta}\right]
  118. E 3 z 2 - r 2 , 3 z 2 - r 2 = [ n 2 - ( l 2 + m 2 ) / 2 ] 2 V d d σ + 3 n 2 ( l 2 + m 2 ) V d d π + 3 4 ( l 2 + m 2 ) 2 V d d δ E_{3z^{2}-r^{2},3z^{2}-r^{2}}=[n^{2}-(l^{2}+m^{2})/2]^{2}V_{dd\sigma}+3n^{2}(l% ^{2}+m^{2})V_{dd\pi}+\frac{3}{4}(l^{2}+m^{2})^{2}V_{dd\delta}

Time-dependent_density_functional_theory.html

  1. H ^ ( t ) = T ^ + V ^ ext ( t ) + W ^ , \hat{H}(t)=\hat{T}+\hat{V}_{\mathrm{ext}}(t)+\hat{W},
  2. H ^ ( t ) | Ψ ( t ) = i t | Ψ ( t ) , | Ψ ( 0 ) = | Ψ . \hat{H}(t)|\Psi(t)\rangle=i\hbar\frac{\partial}{\partial t}|\Psi(t)\rangle,\ % \ \ |\Psi(0)\rangle=|\Psi\rangle.
  3. H ^ s ( t ) = T ^ + V ^ s ( t ) , \hat{H}_{s}(t)=\hat{T}+\hat{V}_{s}(t),
  4. H ^ s ( t ) | Φ ( t ) = i t | Φ ( t ) , | Φ ( 0 ) = | Φ , \hat{H}_{s}(t)|\Phi(t)\rangle=i\frac{\partial}{\partial t}|\Phi(t)\rangle,\ \ % \ |\Phi(0)\rangle=|\Phi\rangle,
  5. ( - 1 2 2 + v s ( 𝐫 , t ) ) ϕ i ( 𝐫 , t ) = i t ϕ i ( 𝐫 , t ) ϕ i ( 𝐫 , 0 ) = ϕ i ( 𝐫 ) , \left(-\frac{1}{2}\nabla^{2}+v_{s}(\mathbf{r},t)\right)\phi_{i}(\mathbf{r},t)=% i\frac{\partial}{\partial t}\phi_{i}(\mathbf{r},t)\ \ \ \phi_{i}(\mathbf{r},0)% =\phi_{i}(\mathbf{r}),
  6. ρ s ( 𝐫 , t ) = i = 1 N | ϕ i ( 𝐫 , t ) | 2 , \rho_{s}(\mathbf{r},t)=\sum_{i=1}^{N}|\phi_{i}(\mathbf{r},t)|^{2},
  7. ρ s ( 𝐫 , t ) = ρ ( 𝐫 , t ) . \rho_{s}(\mathbf{r},t)=\rho(\mathbf{r},t).
  8. v s ( 𝐫 , t ) = v ext ( 𝐫 , t ) + v J ( 𝐫 , t ) + v xc ( 𝐫 , t ) . v_{s}(\mathbf{r},t)=v_{\rm ext}(\mathbf{r},t)+v_{J}(\mathbf{r},t)+v_{\rm xc}(% \mathbf{r},t).\,
  9. A [ Ψ ] = d t Ψ ( t ) | H - i t | Ψ ( t ) . A[\Psi]=\int\mathrm{d}t\ \langle\Psi(t)|H-i\frac{\partial}{\partial t}|\Psi(t)\rangle.
  10. A [ ρ ] = A [ Ψ [ ρ ] ] , A[\rho]=A[\Psi[\rho]],\,
  11. δ V e x t ( t ) \delta V^{ext}(t)
  12. H ( t ) = H + δ V e x t ( t ) H^{\prime}(t)=H+\delta V^{ext}(t)
  13. H K S [ ρ ] ( t ) = H K S [ ρ ] + δ V H [ ρ ] ( t ) + δ V x c [ ρ ] ( t ) + δ V e x t ( t ) H^{\prime}_{KS}[\rho](t)=H_{KS}[\rho]+\delta V_{H}[\rho](t)+\delta V_{xc}[\rho% ](t)+\delta V^{ext}(t)
  14. δ ρ ( 𝐫 t ) = χ ( 𝐫 t , 𝐫 t ) δ V e x t ( 𝐫 t ) \delta\rho(\mathbf{r}t)=\chi(\mathbf{r}t,\mathbf{r^{\prime}}t^{\prime})\delta V% ^{ext}(\mathbf{r^{\prime}}t^{\prime})
  15. δ ρ ( 𝐫 t ) = χ K S ( 𝐫 t , 𝐫 t ) δ V e f f [ ρ ] ( 𝐫 t ) \delta\rho(\mathbf{r}t)=\chi_{KS}(\mathbf{r}t,\mathbf{r^{\prime}}t^{\prime})% \delta V^{eff}[\rho](\mathbf{r^{\prime}}t^{\prime})
  16. δ V e f f [ ρ ] ( t ) = δ V e x t ( t ) + δ V H [ ρ ] ( t ) + δ V x c [ ρ ] ( t ) \delta V^{eff}[\rho](t)=\delta V^{ext}(t)+\delta V_{H}[\rho](t)+\delta V_{xc}[% \rho](t)
  17. δ V H [ ρ ] ( 𝐫 ) = δ V H [ ρ ] δ ρ δ ρ = 1 | 𝐫 - 𝐫 | δ ρ ( 𝐫 ) \delta V_{H}[\rho](\mathbf{r})=\frac{\delta V_{H}[\rho]}{\delta\rho}\delta\rho% =\frac{1}{|\mathbf{r}-\mathbf{r^{\prime}}|}\delta\rho(\mathbf{r^{\prime}})
  18. δ V x c [ ρ ] ( 𝐫 ) = δ V x c [ ρ ] δ ρ δ ρ = f x c ( 𝐫 t , 𝐫 t ) δ ρ ( 𝐫 ) \delta V_{xc}[\rho](\mathbf{r})=\frac{\delta V_{xc}[\rho]}{\delta\rho}\delta% \rho=f_{xc}(\mathbf{r}t,\mathbf{r^{\prime}}t^{\prime})\delta\rho(\mathbf{r^{% \prime}})
  19. χ ( 𝐫 1 t 1 , 𝐫 2 t 2 ) = χ K S ( 𝐫 𝟏 t 1 , 𝐫 2 t 2 ) + χ K S ( 𝐫 𝟏 t 1 , 𝐫 2 t 2 ) ( 1 | 𝐫 2 - 𝐫 1 | + f x c ( 𝐫 2 t 2 , 𝐫 1 t 1 ) ) χ ( 𝐫 1 t 1 , 𝐫 2 t 2 ) \chi(\mathbf{r}_{1}t_{1},\mathbf{r}_{2}t_{2})=\chi_{KS}(\mathbf{r_{1}}t_{1},% \mathbf{r}_{2}t_{2})+\chi_{KS}(\mathbf{r_{1}}t_{1},\mathbf{r}_{2}^{\prime}t_{2% }^{\prime})\left(\frac{1}{|\mathbf{r}_{2}^{\prime}-\mathbf{r}_{1}^{\prime}|}+f% _{xc}(\mathbf{r}_{2}^{\prime}t_{2}^{\prime},\mathbf{r}_{1}^{\prime}t_{1}^{% \prime})\right)\chi(\mathbf{r}_{1}^{\prime}t_{1}^{\prime},\mathbf{r}_{2}t_{2})

Time_dilation_of_moving_particles.html

  1. L = L 0 / γ L=L_{0}/\gamma
  2. T = T 0 γ T=T_{0}\cdot\gamma
  3. M Newton M_{\mathrm{Newton}}
  4. M SR M_{\mathrm{SR}}
  5. N N
  6. M M
  7. Z Z
  8. T 0 T_{0}
  9. M Newton = N exp [ - Z / T 0 ] M SR = N exp [ - Z / ( γ T 0 ) ] \begin{aligned}\displaystyle M_{\mathrm{Newton}}&\displaystyle=N\exp\left[-Z/T% _{0}\right]\\ \displaystyle M_{\mathrm{SR}}&\displaystyle=N\exp\left[-Z/\left(\gamma T_{0}% \right)\right]\end{aligned}

Tinkerbell_map.html

  1. x n + 1 = x n 2 - y n 2 + a x n + b y n x_{n+1}=x_{n}^{2}-y_{n}^{2}+ax_{n}+by_{n}\,
  2. y n + 1 = 2 x n y n + c x n + d y n y_{n+1}=2x_{n}y_{n}+cx_{n}+dy_{n}\,
  3. a = 0.9 , b = - 0.6013 , c = 2.0 , d = 0.50 a=0.9,b=-0.6013,c=2.0,d=0.50
  4. a = 0.3 , b = 0.6000 , c = 2.0 , d = 0.27 a=0.3,b=0.6000,c=2.0,d=0.27

Tomita–Takesaki_theory.html

  1. ϕ ( x ) = ( x Ω , Ω ) \phi(x)=(x\Omega,\Omega)
  2. S 0 ( m Ω ) = m * Ω S_{0}(m\Omega)=m^{*}\Omega
  3. F 0 ( m Ω ) = m * Ω F_{0}(m\Omega)=m^{*}\Omega
  4. S = J | S | = J Δ 1 / 2 = Δ - 1 / 2 J S=J|S|=J\Delta^{1/2}=\Delta^{-1/2}J
  5. F = J | F | = J Δ - 1 / 2 = Δ 1 / 2 J F=J|F|=J\Delta^{-1/2}=\Delta^{1/2}J
  6. J = J - 1 = J * J=J^{-1}=J^{*}
  7. Δ = S * S = F S \Delta=S^{*}S=FS
  8. Δ i t M Δ - i t = M \Delta^{it}M\Delta^{-it}=M
  9. J M J = M , JMJ=M^{\prime},
  10. σ ϕ t ( x ) = Δ i t x Δ - i t \sigma^{\phi_{t}}(x)=\Delta^{it}x\Delta^{-it}
  11. σ ψ t ( x ) = u t σ ϕ t ( x ) u t - 1 \sigma^{\psi_{t}}(x)=u_{t}\sigma^{\phi_{t}}(x)u_{t}^{-1}
  12. u s + t = u s σ ϕ s ( u t ) u_{s+t}=u_{s}\sigma^{\phi_{s}}(u_{t})
  13. F ( t ) = ϕ ( A α t ( B ) ) , F ( t + i ) = ϕ ( a t ( B ) A ) F(t)=\phi(A\alpha_{t}(B)),F(t+i)=\phi(a_{t}(B)A)

Tone_mapping.html

  1. V out = V in V in + 1 , V_{\,\text{out}}=\frac{V_{\,\text{in}}}{V_{\,\text{in}}+1},
  2. [ 0 , ) [0,\infty)
  3. [ 0 , 1 ) . [0,1).
  4. V out = A V in γ , V_{\,\text{out}}=A\,V_{\,\text{in}}^{\gamma},
  5. A > 0 A>0
  6. [ 0 , 1 / A 1 / γ ] [0,1/A^{1/\gamma}]
  7. [ 0 , 1 ] . [0,1].
  8. A < 1 A<1

Topological_algebra.html

  1. : A × A A \cdot:A\times A\longrightarrow A
  2. ( a , b ) a b (a,b)\longmapsto a\cdot b

Topological_divisor_of_zero.html

  1. X X
  2. X X

Topological_K-theory.html

  1. K K
  2. K K
  3. X X
  4. k = 𝐑 , 𝐂 k=\mathbf{R},\mathbf{C}
  5. k k
  6. X X
  7. K K
  8. K ( X ) K(X)
  9. K K
  10. K K
  11. K O ( X ) KO(X)
  12. K K
  13. K K
  14. K K
  15. K ~ ( X ) \widetilde{K}(X)
  16. X X
  17. K ( X ) K(X)
  18. E E
  19. F F
  20. K ~ ( X ) \widetilde{K}(X)
  21. X X
  22. K K
  23. ( X , A ) (X,A)
  24. K ~ ( X / A ) K ~ ( X ) K ~ ( A ) \widetilde{K}(X/A)\to\widetilde{K}(X)\to\widetilde{K}(A)
  25. K ~ ( S X ) K ~ ( S A ) K ~ ( X / A ) K ~ ( X ) K ~ ( A ) . \cdots\to\widetilde{K}(SX)\to\widetilde{K}(SA)\to\widetilde{K}(X/A)\to% \widetilde{K}(X)\to\widetilde{K}(A).
  26. n n
  27. K ~ - n ( X ) := K ~ ( S n X ) , n 0. \widetilde{K}^{-n}(X):=\widetilde{K}(S^{n}X),\qquad n\geq 0.
  28. K - n ( X ) = K ~ - n ( X + ) . K^{-n}(X)=\widetilde{K}^{-n}(X_{+}).
  29. K ~ n \widetilde{K}^{n}
  30. K K
  31. 𝐙 \mathbf{Z}
  32. K K
  33. B U × 𝐙 BU×\mathbf{Z}
  34. 𝐙 \mathbf{Z}
  35. , ,
  36. B U BU
  37. K ~ ( X ) [ X , 𝐙 × B U ] . \widetilde{K}(X)\cong[X,\mathbf{Z}\times BU].
  38. K K
  39. B O BO
  40. K K
  41. K K
  42. K K
  43. K K
  44. K ( X ) K ~ ( T ( E ) ) , K(X)\cong\widetilde{K}(T(E)),
  45. T ( E ) T(E)
  46. E E
  47. X X
  48. E E
  49. K K
  50. K K
  51. K ~ n + 2 ( X ) = K ~ n ( X ) . \widetilde{K}^{n+2}(X)=\widetilde{K}^{n}(X).
  52. Ω < s u p > 2 B U B U × 𝐙 Ω<sup>2BU≅BU×\mathbf{Z}

Topological_manifold.html

  1. + n = { ( x 1 , , x n ) n : x n 0 } . \mathbb{R}^{n}_{+}=\{(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}:x_{n}\geq 0\}.

Topological_quantum_number.html

  1. S 3 S_{3}
  2. S 3 S_{3}
  3. π 3 ( S 3 ) = \pi_{3}(S^{3})=\mathbb{Z}
  4. π 1 ( S 1 ) = \pi_{1}(S^{1})=\mathbb{Z}
  5. \mathbb{Z}
  6. \mathbb{Z}

Topological_tensor_product.html

  1. A B A\otimes B
  2. A B A\otimes B
  3. x = i = 1 n a i b i x=\sum_{i=1}^{n}a_{i}\otimes b_{i}
  4. n n
  5. x x
  6. a i A a_{i}\in A
  7. b i B b_{i}\in B
  8. i = 1 , , n i=1,\ldots,n
  9. A B A\otimes B
  10. p ( a b ) = a b p(a\otimes b)=\|a\|\|b\|
  11. p ( a b ) = a b . p^{\prime}(a^{\prime}\otimes b^{\prime})=\|a^{\prime}\|\|b^{\prime}\|.
  12. π \pi
  13. π ( x ) = inf { i = 1 n a i b i : x = i a i b i } \pi(x)=\inf\left\{\sum_{i=1}^{n}\|a_{i}\|\|b_{i}\|:x=\sum_{i}a_{i}\otimes b_{i% }\right\}
  14. x A B x\in A\otimes B
  15. ε \varepsilon
  16. ε ( x ) = sup { | ( a b ) ( x ) | : a A , b B , a = b = 1 } \varepsilon(x)=\sup\{|(a^{\prime}\otimes b^{\prime})(x)|:a^{\prime}\in A^{% \prime},b^{\prime}\in B^{\prime},\|a^{\prime}\|=\|b^{\prime}\|=1\}
  17. x A B x\in A\otimes B
  18. A ^ π B A\hat{\otimes}_{\pi}B
  19. A ^ ε B A\hat{\otimes}_{\varepsilon}B
  20. A ^ σ B A\hat{\otimes}_{\sigma}B
  21. ( X , Y ) (X,Y)
  22. X Y X\otimes Y
  23. X X
  24. W W
  25. Y Y
  26. Z Z
  27. S : X W S:X\to W
  28. T : Y Z T:Y\to Z
  29. S T : X α Y W α Z S\otimes T:X\otimes_{\alpha}Y\to W\otimes_{\alpha}Z
  30. S T S T \|S\otimes T\|\leq\|S\|\|T\|
  31. A B A\otimes B
  32. A B A\otimes B
  33. A α B A\otimes_{\alpha}B
  34. A α B A\otimes_{\alpha}B
  35. A ^ α B A\hat{\otimes}_{\alpha}B
  36. A B A\otimes B
  37. A ^ α B A\hat{\otimes}_{\alpha}B
  38. A ^ α B A\hat{\otimes}_{\alpha}B
  39. A α B A\otimes_{\alpha}B
  40. α A , B ( x ) \alpha_{A,B}(x)
  41. α ( x ) \alpha(x)
  42. α \alpha
  43. ( X , Y ) (X,Y)
  44. u X Y u\in X\otimes Y
  45. α ( u ; X Y ) = inf { α ( u ; M N ) : dim M , dim N < } . \alpha(u;X\otimes Y)=\inf\{\alpha(u;M\otimes N):\dim M,\dim N<\infty\}.
  46. α \alpha
  47. ( X , Y ) (X,Y)
  48. u X Y u\in X\otimes Y
  49. α ( u ) = sup { α ( ( Q E Q F ) u ; ( X / E ) ( Y / F ) ) : dim X / E , dim Y / F < } . \alpha(u)=\sup\{\alpha((Q_{E}\otimes Q_{F})u;(X/E)\otimes(Y/F)):\dim X/E,\dim Y% /F<\infty\}.
  50. π \pi
  51. ε \varepsilon
  52. ε A , B ( x ) α A , B ( x ) π A , B ( x ) . \varepsilon_{A,B}(x)\leq\alpha_{A,B}(x)\leq\pi_{A,B}(x).

Torelli_theorem.html

  1. 2 \geq 2

Torsion_(algebra).html

  1. M F T ( M ) , M\simeq F\oplus T(M),
  2. M Q = M R Q , M_{Q}=M\otimes_{R}Q,
  3. M S = M R R S , M_{S}=M\otimes_{R}R_{S},

Torsion_(mechanics).html

  1. T = J T r τ = J T G θ T=\frac{J_{T}}{r}\tau=\frac{J_{T}}{\ell}G\theta
  2. τ \tau
  3. τ φ z = T r J T \tau_{\varphi_{z}}={Tr\over J_{T}}
  4. φ = T G J T . \varphi=\frac{T\ell}{GJ_{T}}.
  5. ω = 2 π f \omega=2\pi f
  6. P = T ω P=T\omega
  7. T max = τ max J z z r T_{\max}=\frac{{\tau}_{\max}J_{zz}}{r}
  8. D = ( 16 T max π τ max ) 1 / 3 D=(\frac{16T_{\max}}{\pi{\tau}_{\max}})^{1/3}

Torsion_of_a_curve.html

  1. s s
  2. κ \kappa
  3. 𝐧 = 𝐭 κ , 𝐛 = 𝐭 × 𝐧 , \mathbf{n}=\frac{\mathbf{t}^{\prime}}{\kappa},\quad\mathbf{b}=\mathbf{t}\times% \mathbf{n},
  4. s s
  5. τ \tau
  6. 𝐛 = - τ 𝐧 . \mathbf{b}^{\prime}=-\tau\mathbf{n}.
  7. τ = - 𝐧 𝐛 . \tau=-\mathbf{n}\cdot\mathbf{b}^{\prime}.
  8. σ = 1 τ . \sigma=\frac{1}{\tau}.
  9. τ ( s ) \displaystyle\tau(s)
  10. 𝐫 ( t ) , 𝐫 ′′ ( t ) \mathbf{r^{\prime}}(t),\mathbf{r^{\prime\prime}}(t)
  11. τ = det ( r , r ′′ , r ′′′ ) r × r ′′ 2 = ( r × r ′′ ) r ′′′ r × r ′′ 2 . \tau={{\det\left({r^{\prime},r^{\prime\prime},r^{\prime\prime\prime}}\right)}% \over{\left\|{r^{\prime}\times r^{\prime\prime}}\right\|^{2}}}={{\left({r^{% \prime}\times r^{\prime\prime}}\right)\cdot r^{\prime\prime\prime}}\over{\left% \|{r^{\prime}\times r^{\prime\prime}}\right\|^{2}}}.
  12. τ = x ′′′ ( y z ′′ - y ′′ z ) + y ′′′ ( x ′′ z - x z ′′ ) + z ′′′ ( x y ′′ - x ′′ y ) ( y z ′′ - y ′′ z ) 2 + ( x ′′ z - x z ′′ ) 2 + ( x y ′′ - x ′′ y ) 2 . \tau=\frac{x^{\prime\prime\prime}(y^{\prime}z^{\prime\prime}-y^{\prime\prime}z% ^{\prime})+y^{\prime\prime\prime}(x^{\prime\prime}z^{\prime}-x^{\prime}z^{% \prime\prime})+z^{\prime\prime\prime}(x^{\prime}y^{\prime\prime}-x^{\prime% \prime}y^{\prime})}{(y^{\prime}z^{\prime\prime}-y^{\prime\prime}z^{\prime})^{2% }+(x^{\prime\prime}z^{\prime}-x^{\prime}z^{\prime\prime})^{2}+(x^{\prime}y^{% \prime\prime}-x^{\prime\prime}y^{\prime})^{2}}.

Total_revenue_test.html

  1. E d = - ( ( Q 2 - Q 1 ) / ( P 2 - P 1 ) ) ( P 1 / Q 1 ) E_{d}=-\left(\left(Q_{2}-Q_{1}\right)/\left(P_{2}-P_{1}\right)\right)\cdot% \left(P_{1}/Q_{1}\right)
  2. P P
  3. Q Q
  4. ( Q 2 - Q 1 ) \left(Q_{2}-Q_{1}\right)
  5. ( P 2 - P 1 ) \left(P_{2}-P_{1}\right)
  6. E d = - d Q d P P Q E_{d}=-\frac{dQ}{dP}\cdot\frac{P}{Q}
  7. T R = P Q TR=P\cdot Q
  8. Q = f ( P ) , Q=f(P),
  9. T R = P f ( P ) . TR=P\cdot f(P).
  10. d T R d P = 1 f ( P ) + P f ( P ) \frac{dTR}{dP}=1\cdot f(P)+P\cdot f^{\prime}(P)
  11. Q = f ( P ) Q=f(P)
  12. d T R d P = f ( P ) P + Q \frac{dTR}{dP}=f^{\prime}(P)\cdot P+Q
  13. Q Q
  14. d T R d P = Q ( f ( P ) P Q + 1 ) . \frac{dTR}{dP}=Q\left(f^{\prime}(P)\cdot\frac{P}{Q}+1\right).
  15. - f ( P ) P Q -f^{\prime}(P)\cdot\frac{P}{Q}
  16. d T R d P = Q ( - E d + 1 ) = Q ( 1 - E d ) \frac{dTR}{dP}=Q(-E_{d}+1)=Q(1-E_{d})
  17. E d > 1 E_{d}>1\!
  18. d R d P < 0 \dfrac{dR}{dP}<0\!
  19. E d < 1 E_{d}<1\!
  20. d R d P > 0 \dfrac{dR}{dP}>0\!
  21. E d = 1 E_{d}=1
  22. d R d P = 0 \frac{dR}{dP}=0
  23. P 1 P_{1}
  24. P 2 P_{2}
  25. P 1 P 2 C A P_{1}P_{2}CA
  26. Q 1 Q 2 B C Q_{1}Q_{2}BC
  27. E d < 1 E_{d}<1\!
  28. E d > 1 E_{d}>1\!

Total_sum_of_squares.html

  1. TSS = i = 1 n ( y i - y ¯ ) 2 \mathrm{TSS}=\sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)^{2}
  2. y ¯ \bar{y}
  3. 𝐓 = 𝐖 + 𝐁 , \mathbf{T}=\mathbf{W}+\mathbf{B},

Totally_positive_matrix.html

  1. A = ( A i j ) {A}=(A_{ij})
  2. A [ p ] = ( A i k j ) \displaystyle{A}_{[p]}=(A_{i_{k}j_{\ell}})
  3. det ( A [ p ] ) 0 \det({A}_{[p]})\geq 0

Traffic_flow.html

  1. v t = ( 1 / m ) i = 1 m v i v_{t}=(1/m)\sum_{i=1}^{m}v_{i}
  2. v s = n ( i = 1 n ( 1 / v i ) ) - 1 v_{s}=n\left(\sum_{i=1}^{n}(1/v_{i})\right)^{-1}
  3. v t = v s + σ s 2 v s v_{t}=v_{s}+\frac{\sigma_{s}^{2}}{v_{s}}
  4. σ s 2 \sigma_{s}^{2}
  5. k = 1 s k=\frac{1}{s}
  6. K ( L , t 1 ) = n L = 1 s ¯ ( t 1 ) K(L,t_{1})=\frac{n}{L}=\frac{1}{\bar{s}(t_{1})}
  7. k ( A ) = n L = n d t L d t = t t | A | k(A)=\frac{n}{L}=\frac{n\,dt}{L\,dt}=\frac{tt}{\left|A\right|}
  8. q = k v q=kv\,
  9. q = 1 / h q=1/h\,
  10. q ( T , x 1 ) = m T = 1 h ¯ ( x 1 ) q(T,x_{1})=\frac{m}{T}=\frac{1}{\bar{h}(x_{1})}
  11. q ( B ) = m T = m d x T d x = t d | B | q(B)=\frac{m}{T}=\frac{m\,dx}{T\,dx}=\frac{td}{\left|B\right|}
  12. q ( C ) = t d | C | q(C)=\frac{td}{\left|C\right|}
  13. k ( C ) = t t | C | k(C)=\frac{tt}{\left|C\right|}
  14. t d = i = 1 m d x i td=\sum_{i=1}^{m}\,dx_{i}
  15. t t = i = 1 n d t i tt=\sum_{i=1}^{n}\,dt_{i}
  16. f ( t , x , V ) f(t,x,V)
  17. t t
  18. x x
  19. V V
  20. average delay ( w avg ) = total delay experienced by m vehicles total number of delayed vehicles = T D m \,\text{average delay (}w\text{avg}\,\text{)}=\frac{\,\text{total delay % experienced by }m\,\text{ vehicles}}{\,\text{total number of delayed vehicles}% }=\frac{TD}{m}
  21. average queue ( Q avg ) = total delay experienced by m vehicles duration of congestion = T D ( t 2 - t 1 ) \,\text{average queue (}Q\text{avg}\,\text{)}=\frac{\,\text{total delay % experienced by }m\,\text{ vehicles}}{\,\text{duration of congestion}}=\frac{TD% }{(t_{2}-t_{1})}
  22. k t + q x = 0 , \frac{\partial k}{\partial t}+\frac{\partial q}{\partial x}=0,
  23. q = F ( k ) {q}={F(k)}
  24. k ( t , x ) {k(t,x)}
  25. t = 0 {t}={0}
  26. k ( 0 , x ) = g ( x ) {k(0,x)}={g(x)}
  27. g ( x ) {g(x)}
  28. g ( t ) {g(t)}
  29. x = 0 {x=0}
  30. k ( t , 0 ) = g ( t ) {k(t,0)}={g(t)}
  31. C m i n C_{min}

Trajectory_optimization.html

  1. H ( u ) = ϕ ( x , λ , t ) u + H(u)=\phi(x,\lambda,t)u+\cdots
  2. a u ( t ) b a\leq u(t)\leq b
  3. H ( u ) H(u)
  4. u u
  5. ϕ ( x , λ , t ) \phi(x,\lambda,t)
  6. u ( t ) = { b , ϕ ( x , λ , t ) < 0 ? , ϕ ( x , λ , t ) = 0 a , ϕ ( x , λ , t ) > 0. u(t)=\begin{cases}b,&\phi(x,\lambda,t)<0\\ ?,&\phi(x,\lambda,t)=0\\ a,&\phi(x,\lambda,t)>0.\end{cases}
  7. ϕ \phi
  8. b b
  9. a a
  10. ϕ \phi
  11. ϕ \phi
  12. t 1 t t 2 t_{1}\leq t\leq t_{2}

Trans-tubular_potassium_gradient.html

  1. T T K G = u r i n e K p l a s m a K ÷ u r i n e o s m p l a s m a o s m TTKG={\frac{urine_{K}}{plasma_{K}}}\div{\frac{urine_{osm}}{plasma_{osm}}}

Transient_recovery_voltage.html

  1. d u d t = Z d i d t \frac{du}{dt}=Z{di\over dt}\,
  2. ( l / c ) \sqrt{(}l/c)\,

Transitive_set.html

  1. X X \bigcup X\subseteq X
  2. X \bigcup X
  3. X = { y ( x X ) y x } \bigcup X=\{y\mid(\exists x\in X)y\in x\}
  4. X \bigcup X
  5. X X X\cup\bigcup X
  6. X 𝒫 ( X ) . X\subset\mathcal{P}(X).
  7. { X , X , X , X , X , } . \bigcup\{X,\bigcup X,\bigcup\bigcup X,\bigcup\bigcup\bigcup X,\bigcup\bigcup% \bigcup\bigcup X,\ldots\}.

Translation_plane.html

  1. P P
  2. l l
  3. P P
  4. l l
  5. l l
  6. P P
  7. P P
  8. l l
  9. P P
  10. l l
  11. l l
  12. Π Π
  13. l l
  14. l l
  15. Π Π
  16. Π Π
  17. P G ( 3 , q ) PG(3,q)
  18. P G ( 3 , q ) PG(3,q)
  19. S S
  20. P G ( 3 , q ) PG(3,q)
  21. π ( S ) π(S)
  22. P G ( 3 , q ) PG(3,q)
  23. P G ( 4 , q ) PG(4,q)
  24. A ( S ) A(S)
  25. P G ( 4 , q ) PG(4,q)
  26. P G ( 3 , q ) PG(3,q)
  27. P G ( 4 , q ) PG(4,q)
  28. P G ( 3 , q ) PG(3,q)
  29. S S
  30. A ( S ) A(S)
  31. π ( S ) π(S)
  32. A ( S ) A(S)
  33. P G ( 3 , q ) PG(3,q)
  34. R R
  35. q + 1 q+1
  36. R R
  37. R R
  38. R R
  39. R R
  40. P G ( 3 , q ) PG(3,q)
  41. S S
  42. P G ( 3 , q ) PG(3,q)
  43. S S
  44. S S
  45. q > 2 q>2
  46. S S
  47. P G ( 3 , q ) PG(3,q)
  48. P G ( 3 , 2 ) PG(3,2)
  49. V V
  50. 2 n 2n
  51. F F
  52. V V
  53. S S
  54. n n
  55. V V
  56. V V
  57. S S
  58. A A
  59. V V
  60. v + U v+U
  61. v v
  62. V V
  63. U U
  64. S S
  65. A A
  66. x x + w x→x+w
  67. w w
  68. V V
  69. q q
  70. V V
  71. 2 n 2n
  72. F F
  73. V = { ( x , y ) : x , y F n } . V=\{(x,y)\colon x,y\in F^{n}\}.
  74. n × n n×n
  75. F F
  76. i j i≠j
  77. V i = { ( x , x M i ) : x F n } , V_{i}=\{(x,xM_{i})\colon x\in F^{n}\},
  78. V q n = { ( 0 , y ) : y F n } , V_{q^{n}}=\{(0,y)\colon y\in F^{n}\},
  79. x = 0 x=0
  80. V V
  81. F F
  82. E E
  83. n n
  84. F F
  85. 2 n 2n
  86. F F
  87. V V
  88. E E
  89. n n
  90. F F
  91. V V
  92. n × n n×n
  93. F = G F ( q ) F=GF(q)
  94. E E
  95. F F
  96. x α x x→αx
  97. α α
  98. E E
  99. F F
  100. n × n n×n
  101. F F
  102. G F ( 9 ) GF(9)
  103. G F ( 3 ) GF(3)
  104. A G ( 2 , 9 ) AG(2,9)
  105. [ 0 0 0 0 ] , [ 1 0 0 1 ] , [ 2 0 0 2 ] , [ 0 1 2 0 ] , [ 1 1 2 1 ] , [ 2 1 2 2 ] , [ 0 2 1 0 ] , [ 1 2 1 1 ] , [ 2 1 2 2 ] . \left[\begin{matrix}0&0\\ 0&0\end{matrix}\right],\left[\begin{matrix}1&0\\ 0&1\end{matrix}\right],\left[\begin{matrix}2&0\\ 0&2\end{matrix}\right],\left[\begin{matrix}0&1\\ 2&0\end{matrix}\right],\left[\begin{matrix}1&1\\ 2&1\end{matrix}\right],\left[\begin{matrix}2&1\\ 2&2\end{matrix}\right],\left[\begin{matrix}0&2\\ 1&0\end{matrix}\right],\left[\begin{matrix}1&2\\ 1&1\end{matrix}\right],\left[\begin{matrix}2&1\\ 2&2\end{matrix}\right].
  106. R R
  107. R R
  108. S S
  109. P G ( 3 , q ) PG(3,q)
  110. R R
  111. R R
  112. S < s u p > S<sup>∗

Transposition_(logic).html

  1. ( P Q ) ( ¬ Q ¬ P ) (P\to Q)\Leftrightarrow(\neg Q\to\neg P)
  2. \Leftrightarrow
  3. ( P Q ) ( ¬ Q ¬ P ) (P\to Q)\vdash(\neg Q\to\neg P)
  4. \vdash
  5. ( ¬ Q ¬ P ) (\neg Q\to\neg P)
  6. ( P Q ) (P\to Q)
  7. P Q ¬ Q ¬ P \frac{P\to Q}{\therefore\neg Q\to\neg P}
  8. P Q P\to Q
  9. ¬ Q ¬ P \neg Q\to\neg P
  10. ( P Q ) ( ¬ Q ¬ P ) (P\to Q)\to(\neg Q\to\neg P)
  11. P P
  12. Q Q
  13. P Q P\rightarrow Q
  14. ¬ P Q \neg PQ
  15. Q ¬ P Q\neg P
  16. ¬ Q ¬ P \neg Q\rightarrow\neg P

Tree_(descriptive_set_theory).html

  1. X X
  2. X X
  3. X X
  4. X < ω X^{<\omega}
  5. T T
  6. X < ω X^{<\omega}
  7. x 0 , x 1 , , x n - 1 \langle x_{0},x_{1},\ldots,x_{n-1}\rangle
  8. n n
  9. T T
  10. 0 m < n 0\leq m<n
  11. x 0 , x 1 , , x m - 1 \langle x_{0},x_{1},\ldots,x_{m-1}\rangle
  12. T T
  13. m = 0 m=0
  14. T T
  15. X X
  16. T T
  17. T T
  18. [ T ] [T]
  19. T T
  20. T T
  21. T T
  22. x 0 , x 1 , , x n - 1 T \langle x_{0},x_{1},\ldots,x_{n-1}\rangle\in T
  23. x x
  24. X X
  25. x 0 , x 1 , , x n - 1 , x T \langle x_{0},x_{1},\ldots,x_{n-1},x\rangle\in T
  26. T T
  27. T T
  28. T T
  29. U U
  30. T < U T<U
  31. T T
  32. U U
  33. X X
  34. X ω X^{\omega}
  35. C C
  36. X ω X^{\omega}
  37. [ T ] [T]
  38. T T
  39. T T
  40. C C
  41. [ T ] [T]
  42. T T
  43. X × Y X\times Y
  44. ( X × Y ) < ω (X\times Y)^{<\omega}
  45. X < ω × Y < ω X^{<\omega}\times Y^{<\omega}
  46. [ T ] [T]
  47. X < ω × Y < ω X^{<\omega}\times Y^{<\omega}
  48. [ T ] [T]
  49. p [ T ] = { x X ω | ( y Y ω ) x , y [ T ] } p[T]=\{\vec{x}\in X^{\omega}|(\exists\vec{y}\in Y^{\omega})\langle\vec{x},\vec% {y}\rangle\in[T]\}

Trembling_hand_perfect_equilibrium.html

  1. Up , Left \langle\,\text{Up},\,\text{Left}\rangle
  2. Down , Right \langle\,\text{Down},\,\text{Right}\rangle
  3. U , L \langle\,\text{U},\,\text{L}\rangle
  4. ( 1 - ε , ε ) (1-\varepsilon,\varepsilon)
  5. 0 < ε < 1 0<\varepsilon<1
  6. 1 ( 1 - ε ) + 2 ε = 1 + ε 1(1-\varepsilon)+2\varepsilon=1+\varepsilon
  7. 0 ( 1 - ε ) + 2 ε = 2 ε 0(1-\varepsilon)+2\varepsilon=2\varepsilon
  8. ε \varepsilon
  9. ( 1 - ε , ε ) (1-\varepsilon,\varepsilon)
  10. U , L \langle\,\text{U},\,\text{L}\rangle
  11. D , R \langle\,\text{D},\,\text{R}\rangle
  12. ( ε , 1 - ε ) (\varepsilon,1-\varepsilon)
  13. 1 ε + 2 ( 1 - ε ) = 2 - ε 1\varepsilon+2(1-\varepsilon)=2-\varepsilon
  14. 0 ( ε ) + 2 ( 1 - ε ) = 2 - 2 ε 0(\varepsilon)+2(1-\varepsilon)=2-2\varepsilon
  15. ε \varepsilon
  16. D , R \langle\,\text{D},\,\text{R}\rangle

Triangle_group.html

  1. a 2 = b 2 = c 2 = 1 a^{2}=b^{2}=c^{2}=1
  2. ( a b ) l = ( b c ) n = ( c a ) m = 1. (ab)^{l}=(bc)^{n}=(ca)^{m}=1.
  3. Δ ( l , m , n ) = a , b , c a 2 = b 2 = c 2 = ( a b ) l = ( b c ) n = ( c a ) m = 1 . \Delta(l,m,n)=\langle a,b,c\mid a^{2}=b^{2}=c^{2}=(ab)^{l}=(bc)^{n}=(ca)^{m}=1\rangle.
  4. 1 l + 1 m + 1 n = 1. \frac{1}{l}+\frac{1}{m}+\frac{1}{n}=1.
  5. 1 l + 1 m + 1 n > 1. \frac{1}{l}+\frac{1}{m}+\frac{1}{n}>1.
  6. 1 l + 1 m + 1 n < 1. \frac{1}{l}+\frac{1}{m}+\frac{1}{n}<1.
  7. D ( l , m , n ) = x , y x l , y m , ( x y ) n . D(l,m,n)=\langle x,y\mid x^{l},y^{m},(xy)^{n}\rangle.

Triangular_prism.html

  1. V = 1 2 b h l V=\frac{1}{2}bhl

Triangular_tiling.html

  1. 2 * {}^{*}_{2}
  2. 2 3 {}^{3}_{2}
  3. π 12 \frac{\pi}{\sqrt{12}}
  4. 2 * {}^{*}_{2}

Tribometer.html

  1. F = μ 𝒩 F=\mu\mathcal{N}
  2. μ = m H / m T \mu\ =m_{H}/m_{T}

Trident_curve.html

  1. x y + a x 3 + b x 2 + c x = d xy+ax^{3}+bx^{2}+cx=d\,
  2. a x 3 + b x 2 z + c x z 2 + x z = d z 3 , ax^{3}+bx^{2}z+cxz^{2}+xz=dz^{3},\,

Trigamma_function.html

  1. ψ 1 ( z ) \psi_{1}(z)
  2. ψ 1 ( z ) \psi_{1}(z)
  3. ψ 1 ( z ) = d 2 d z 2 ln Γ ( z ) \psi_{1}(z)=\frac{d^{2}}{dz^{2}}\ln\Gamma(z)
  4. ψ 1 ( z ) = d d z ψ ( z ) \psi_{1}(z)=\frac{d}{dz}\psi(z)
  5. ψ ( z ) \psi(z)
  6. ψ 1 ( z ) = n = 0 1 ( z + n ) 2 , \psi_{1}(z)=\sum_{n=0}^{\infty}\frac{1}{(z+n)^{2}},
  7. ψ 1 ( z ) = ζ ( 2 , z ) . \psi_{1}(z)=\zeta(2,z).\frac{}{}
  8. 1 - z 1-z
  9. ψ 1 ( z ) = 0 1 0 y x z - 1 y 1 - x d x d y \psi_{1}(z)=\int_{0}^{1}\int_{0}^{y}\frac{x^{z-1}y}{1-x}\,dx\,dy
  10. ψ 1 ( z ) = - 0 1 x z - 1 ln x 1 - x d x \psi_{1}(z)=-\int_{0}^{1}\frac{x^{z-1}\ln{x}}{1-x}\,dx
  11. ψ 1 ( z ) = 1 z + 1 2 z 2 + k = 1 B 2 k z 2 k + 1 = k = 0 B k z k + 1 \psi_{1}(z)=\frac{1}{z}+\frac{1}{2z^{2}}+\sum_{k=1}^{\infty}\frac{B_{2k}}{z^{2% k+1}}=\sum_{k=0}^{\infty}\frac{B_{k}}{z^{k+1}}
  12. B 1 = 1 / 2 B_{1}=1/2
  13. ψ 1 ( z + 1 ) = ψ 1 ( z ) - 1 z 2 \psi_{1}(z+1)=\psi_{1}(z)-\frac{1}{z^{2}}
  14. ψ 1 ( 1 - z ) + ψ 1 ( z ) = π 2 sin 2 ( π z ) \psi_{1}(1-z)+\psi_{1}(z)=\frac{\pi^{2}}{\sin^{2}(\pi z)}\,
  15. ψ 1 ( 1 4 ) = π 2 + 8 K \psi_{1}\left(\frac{1}{4}\right)=\pi^{2}+8K
  16. ψ 1 ( 1 2 ) = π 2 2 \psi_{1}\left(\frac{1}{2}\right)=\frac{\pi^{2}}{2}
  17. ψ 1 ( 1 ) = π 2 6 \psi_{1}(1)=\frac{\pi^{2}}{6}
  18. ψ 1 ( 3 2 ) = π 2 2 - 4 \psi_{1}\left(\frac{3}{2}\right)=\frac{\pi^{2}}{2}-4
  19. ψ 1 ( 2 ) = π 2 6 - 1 \psi_{1}(2)=\frac{\pi^{2}}{6}-1
  20. ψ 1 \psi_{1}
  21. z n , z n ¯ z_{n},\overline{z_{n}}
  22. ( z ) < 0 \Re(z)<0
  23. ( z n ) = - n + 1 / 2 \Re(z_{n})=-n+1/2
  24. z 1 = - 0.4121345 + i 0.5978119 z_{1}=-0.4121345\ldots+i0.5978119\ldots
  25. z 2 = - 1.4455692 + i 0.6992608 z_{2}=-1.4455692\ldots+i0.6992608\ldots
  26. ( z ) > 0 \Im(z)>0
  27. n = 1 n 2 - 1 2 ( n 2 + 1 2 ) 2 [ ψ 1 ( n - i 2 ) + ψ 1 ( n + i 2 ) ] = - 1 + 2 4 π coth ( π 2 ) - 3 π 2 4 sinh 2 ( π 2 ) + π 4 12 sinh 4 ( π 2 ) ( 5 + cosh ( π 2 ) ) . \sum_{n=1}^{\infty}\frac{n^{2}-\frac{1}{2}}{\left(n^{2}+\frac{1}{2}\right)^{2}% }\left[\psi_{1}\left(n-\frac{i}{\sqrt{2}}\right)+\psi_{1}\left(n+\frac{i}{% \sqrt{2}}\right)\right]=-1+\frac{\sqrt{2}}{4}\pi\coth\left(\frac{\pi}{\sqrt{2}% }\right)-\frac{3\pi^{2}}{4\sinh^{2}\left(\frac{\pi}{\sqrt{2}}\right)}+\frac{% \pi^{4}}{12\sinh^{4}\left(\frac{\pi}{\sqrt{2}}\right)}\left(5+\cosh\left(\pi% \sqrt{2}\right)\right).

Trinomial.html

  1. 3 x + 5 y + 8 z 3x+5y+8z
  2. x , y , z x,y,z
  3. 3 t + 9 s 2 + 3 y 3 3t+9s^{2}+3y^{3}
  4. t , s , y t,s,y
  5. 3 t s + 9 t + 5 s 3ts+9t+5s
  6. t , s t,s
  7. A x a y b z c + B t + C s Ax^{a}y^{b}z^{c}+Bt+Cs
  8. x , y , z , t , s x,y,z,t,s
  9. a , b , c a,b,c
  10. A , B , C A,B,C
  11. P x a + Q x b + R x c Px^{a}+Qx^{b}+Rx^{c}
  12. x x
  13. a , b , c a,b,c
  14. P , Q , R P,Q,R
  15. x = q + x m x=q+x^{m}

Trisectrix_of_Maclaurin.html

  1. P = ( 0 , 0 ) P=(0,0)
  2. P 1 = ( a , 0 ) P_{1}=(a,0)
  3. P P
  4. θ \theta
  5. P 1 P_{1}
  6. 3 θ 3\theta
  7. Q Q
  8. Q Q
  9. 2 θ 2\theta
  10. r sin 3 θ = a sin 2 θ {r\over\sin 3\theta}={a\over\sin 2\theta}\!
  11. r = a sin 3 θ sin 2 θ = a 2 4 cos 2 θ - 1 cos θ = a 2 ( 4 cos θ - sec θ ) r=a\frac{\sin 3\theta}{\sin 2\theta}={a\over 2}\frac{4\cos^{2}\theta-1}{\cos% \theta}={a\over 2}(4\cos\theta-\sec\theta)\!
  12. 2 x ( x 2 + y 2 ) = a ( 3 x 2 - y 2 ) 2x(x^{2}+y^{2})=a(3x^{2}-y^{2})\!
  13. r = a 2 cos θ 3 r=\frac{a}{2\cos{\theta\over 3}}\!
  14. ϕ \phi
  15. ( a , 0 ) (a,0)
  16. x x
  17. ϕ \phi
  18. x x
  19. ϕ / 3 \phi/3
  20. 3 a 2 3a\over 2
  21. x = - a 2 x={-{a\over 2}}
  22. ( a , ± 1 3 a ) (a,{\pm{1\over\sqrt{3}}a})
  23. 2 x = a ( 3 x 2 - y 2 ) 2x=a(3x^{2}-y^{2})
  24. ( x + a ) 2 + y 2 = a 2 (x+a)^{2}+y^{2}=a^{2}
  25. x = a 2 x={a\over 2}
  26. y 2 = 2 a ( x - 3 2 a ) y^{2}=2a(x-\tfrac{3}{2}a)
  27. ( a , 0 ) (a,0)

Trix_(technical_analysis).html

  1. p 0 p_{0}
  2. p 1 p_{1}
  3. f = 1 - 2 N + 1 = N - 1 N + 1 f=1-{2\over N+1}={N-1\over N+1}
  4. T r i p l e E M A 0 = ( 1 - f ) 3 ( p 0 + 3 f p 1 + 6 f 2 p 2 + 10 f 3 p 3 + ) TripleEMA_{0}=(1-f)^{3}(p_{0}+3fp_{1}+6f^{2}p_{2}+10f^{3}p_{3}+\dots)
  5. f n f^{n}

Trochoid.html

  1. x = a θ - b sin ( θ ) x=a\theta-b\sin(\theta)\,
  2. y = a - b cos ( θ ) y=a-b\cos(\theta)\,
  3. ( x , y ) (x,y)
  4. ( x , y ) (x^{\prime},y^{\prime})
  5. x = x + r 1 cos ( ω 1 t + ϕ 1 ) , y = y + r 1 sin ( ω 1 t + ϕ 1 ) , r 1 > 0 , x=x^{\prime}+r_{1}\cos(\omega_{1}t+\phi_{1}),\ y=y^{\prime}+r_{1}\sin(\omega_{% 1}t+\phi_{1}),\ r_{1}>0,
  6. x = x 0 + v 2 x t , y = y 0 + v 2 y t x = x 0 + r 1 cos ( ω 1 t + ϕ 1 ) + v 2 x t , y = y 0 + r 1 sin ( ω 1 t + ϕ 1 ) + v 2 y t , \begin{array}[]{lcl}x^{\prime}=x_{0}+v_{2x}t,\ y^{\prime}=y_{0}+v_{2y}t\\ \therefore x=x_{0}+r_{1}\cos(\omega_{1}t+\phi_{1})+v_{2x}t,\ y=y_{0}+r_{1}\sin% (\omega_{1}t+\phi_{1})+v_{2y}t,\\ \end{array}
  7. ( x 0 , y 0 ) (x_{0},y_{0})
  8. x = x 0 + r 2 cos ( ω 2 t + ϕ 2 ) , y = y 0 + r 2 sin ( ω 2 t + ϕ 2 ) , r 2 0 x = x 0 + r 1 cos ( ω 1 t + ϕ 1 ) + r 2 cos ( ω 2 t + ϕ 2 ) , y = y 0 + r 1 sin ( ω 1 t + ϕ 1 ) + r 2 sin ( ω 2 t + ϕ 2 ) , \begin{array}[]{lcl}x^{\prime}=x_{0}+r_{2}\cos(\omega_{2}t+\phi_{2}),\ y^{% \prime}=y_{0}+r_{2}\sin(\omega_{2}t+\phi_{2}),\ r_{2}\geq 0\\ \therefore x=x_{0}+r_{1}\cos(\omega_{1}t+\phi_{1})+r_{2}\cos(\omega_{2}t+\phi_% {2}),\ y=y_{0}+r_{1}\sin(\omega_{1}t+\phi_{1})+r_{2}\sin(\omega_{2}t+\phi_{2})% ,\\ \end{array}
  9. ω 1 / ω 2 \omega_{1}/\omega_{2}
  10. p / q p/q
  11. p p
  12. q q
  13. p p
  14. q q
  15. ( x 0 , y 0 ) (x_{0},y_{0})
  16. r 1 r_{1}
  17. R R
  18. epicycloid: ω 1 / ω 2 = p / q = r 2 / r 1 = R / r 1 + 1 , | p - q | cusps hypocycloid: ω 1 / ω 2 = p / q = - r 2 / r 1 = - ( R / r 1 - 1 ) , | p - q | = | p | + | q | cusps \begin{array}[]{lcl}\,\text{epicycloid: }&\omega_{1}/\omega_{2}&=p/q=r_{2}/r_{% 1}=R/r_{1}+1,\ |p-q|\,\text{ cusps}\\ \,\text{hypocycloid: }&\omega_{1}/\omega_{2}&=p/q=-r_{2}/r_{1}=-(R/r_{1}-1),\ % |p-q|=|p|+|q|\,\text{ cusps}\end{array}
  19. r 2 r_{2}

Trudinger's_theorem.html

  1. Ω \Omega
  2. n \mathbb{R}^{n}
  3. m p = n mp=n
  4. p > 1 p>1
  5. A ( t ) = exp ( t n / ( n - m ) ) - 1. A(t)=\exp\left(t^{n/(n-m)}\right)-1.
  6. W m , p ( Ω ) L A ( Ω ) W^{m,p}(\Omega)\hookrightarrow L_{A}(\Omega)
  7. L A ( Ω ) = { u M f ( Ω ) : u A , Ω = inf { k > 0 : Ω A ( | u ( x ) | k ) d x 1 } < } . L_{A}(\Omega)=\left\{u\in M_{f}(\Omega):\|u\|_{A,\Omega}=\inf\{k>0:\int_{% \Omega}A\left(\frac{|u(x)|}{k}\right)~{}dx\leq 1\}<\infty\right\}.
  8. L A ( Ω ) L_{A}(\Omega)

Truel.html

  1. p > q > r p>q>r\,
  2. p < q ( 1 + q ) 1 - q + q 2 p<\frac{q(1+q)}{1-q+q^{2}}
  3. ( 1 - q ) ( 1 - q + q 2 ) p 2 - q ( 1 - q ) ( 1 + 2 q ) p - q 3 > 0 (1-q)(1-q+q^{2})p^{2}-q(1-q)(1+2q)p-q^{3}>0\,
  4. r > p ( p - q - p q - q 2 + p q 2 ) p 2 ( 1 - q ) + q 2 ( 1 - p ) 2 r>\frac{p(p-q-pq-q^{2}+pq^{2})}{p^{2}(1-q)+q^{2}(1-p)^{2}}

Trust_region.html

  1. A Δ x = b A\Delta x=b
  2. Δ x \Delta x
  3. ( A + λ diag ( A ) ) Δ x = b (A+\lambda\operatorname{diag}(A))\Delta x=b
  4. diag ( A ) \operatorname{diag}(A)
  5. Δ x = 0 \Delta x=0
  6. Δ f pred \Delta f_{\mathrm{pred}}
  7. Δ x \Delta x
  8. Δ f actual = f ( x + Δ x ) - f ( x ) \Delta f_{\mathrm{actual}}=f(x+\Delta x)-f(x)
  9. Δ f pred / Δ f actual \Delta f_{\mathrm{pred}}/\Delta f_{\mathrm{actual}}
  10. Δ f pred \Delta f_{\mathrm{pred}}
  11. Δ f actual \Delta f_{\mathrm{actual}}

Truth-table_reduction.html

  1. A 1 B A m B A t t B A w t t B A T B A\leq_{1}B\Rightarrow A\leq_{m}B\Rightarrow A\leq_{tt}B\Rightarrow A\leq_{wtt}% B\Rightarrow A\leq_{T}B

TSL_color_space.html

  1. T = { 1 2 π arctan r g + 1 4 , if g > 0 1 2 π arctan r g + 3 4 , if g < 0 0 , if g = 0 T=\begin{cases}\frac{1}{2\pi}\arctan{\frac{r^{\prime}}{g^{\prime}}}+\frac{1}{4% },&\mbox{if}~{}~{}g^{\prime}>0\\ \frac{1}{2\pi}\arctan{\frac{r^{\prime}}{g^{\prime}}}+\frac{3}{4},&\mbox{if}~{}% ~{}g^{\prime}<0\\ 0,&\mbox{if}~{}~{}g^{\prime}=0\\ \end{cases}
  2. S = 9 5 ( r 2 + g 2 ) S=\sqrt{\frac{9}{5}\left(r^{\prime 2}+g^{\prime 2}\right)}
  3. L = 0.299 R + 0.587 G + 0.114 B L=0.299R+0.587G+0.114B
  4. r = r - 1 3 r^{\prime}=r-\tfrac{1}{3}
  5. g = g - 1 3 g^{\prime}=g-\tfrac{1}{3}
  6. r = R R + G + B r=\tfrac{R}{R+G+B}
  7. g = G R + G + B g=\tfrac{G}{R+G+B}
  8. R = k r R=k\cdot r
  9. G = k g G=k\cdot g
  10. B = k ( 1 - r - g ) B=k\cdot(1-r-g)
  11. r = { 5 3 S , if T = 0 x g + 1 3 , if T 0 r=\begin{cases}\frac{\sqrt{5}}{3}S,&\mbox{if}~{}~{}T=0\\ x\cdot g+\frac{1}{3},&\mbox{if}~{}~{}T\neq 0\\ \end{cases}
  12. g = { - 5 9 ( x 2 + 1 ) S , if T > 1 2 5 9 ( x 2 + 1 ) S , if T < 1 2 0 , if T = 0 g=\begin{cases}-\sqrt{\frac{5}{9(x^{2}+1)}}\cdot S,&\mbox{if}~{}~{}T>\frac{1}{% 2}\\ \sqrt{\frac{5}{9(x^{2}+1)}}\cdot S,&\mbox{if}~{}~{}T<\frac{1}{2}\\ 0,&\mbox{if}~{}~{}T=0\\ \end{cases}
  13. k = L 0.185 r + 0.473 g + 0.114 k=\frac{L}{0.185r+0.473g+0.114}
  14. x = - cot ( 2 π T ) x=-\cot({2\pi\cdot T})

Turbulence_modeling.html

  1. - ρ υ i υ j ¯ -\rho\overline{\upsilon_{i}^{\prime}\upsilon_{j}^{\prime}}
  2. R i j = - ρ υ i υ j ¯ R_{ij}=-\rho\overline{\upsilon_{i}^{\prime}\upsilon_{j}^{\prime}}
  3. R i j R_{ij}
  4. ν t > 0 \nu_{t}>0
  5. - υ i υ j ¯ = ν t ( υ ¯ i x j + υ ¯ j x i ) - 2 3 K δ i j -\overline{\upsilon_{i}^{\prime}\upsilon_{j}^{\prime}}=\nu_{t}\left(\frac{% \partial\bar{\upsilon}_{i}}{\partial x_{j}}+\frac{\partial\bar{\upsilon}_{j}}{% \partial x_{i}}\right)-\frac{2}{3}K\delta_{ij}
  6. - υ i υ j ¯ = 2 ν t S i j - 2 3 K δ i j -\overline{\upsilon_{i}^{\prime}\upsilon_{j}^{\prime}}=2\nu_{t}S_{ij}-\frac{2}% {3}K\delta_{ij}
  7. S i j S_{ij}
  8. ν t \nu_{t}
  9. K = 1 2 υ i υ i ¯ K=\frac{1}{2}\overline{\upsilon_{i}^{\prime}\upsilon_{i}^{\prime}}
  10. δ i j \delta_{ij}
  11. ν t = | u y | l m 2 \nu_{t}=\left|\frac{\partial u}{\partial y}\right|l_{m}^{2}
  12. u y \frac{\partial u}{\partial y}
  13. l m l_{m}
  14. ν t = Δ x Δ y ( u x ) 2 + ( v y ) 2 + 1 2 ( u y + v x ) 2 \nu_{t}=\Delta x\Delta y\sqrt{\left(\frac{\partial u}{\partial x}\right)^{2}+% \left(\frac{\partial v}{\partial y}\right)^{2}+\frac{1}{2}\left(\frac{\partial u% }{\partial y}+\frac{\partial v}{\partial x}\right)^{2}}
  15. ν t \nu_{t}

Turnover_number.html

  1. [ E ] T [E]_{T}
  2. V max V_{\max}
  3. [ E ] T [E]_{T}
  4. k cat = V max [ E ] T k_{\mathrm{cat}}=\frac{V_{\max}}{[E]_{T}}

Tverberg's_theorem.html

  1. ( d + 1 ) ( r - 1 ) + 1 (d+1)(r-1)+1

TW_Hydrae.html

  1. ± \pm

Twin_deficits_hypothesis.html

  1. Y = C + I + G + ( X - M ) Y=C+I+G+(X-M)
  2. Y = C + S + T Y=C+S+T
  3. Y = C + I + G + N X Y=C+I+G+NX
  4. Y - C - T = S Y-C-T=S
  5. S = G - T + N X + I S=G-T+NX+I
  6. ( S - I ) + ( T - G ) = ( N X ) (S-I)+(T-G)=(NX)
  7. C A ( N X ) = ( p r i v a t e ) S - I + ( T - G ) CA(NX)=(private)S-I+(T-G)
  8. C u r r e n t A c c o u n t = ( P r i v a t e S a v i n g s - I n v e s t m e n t ) + ( T a x - G o v e r n m e n t E x p e n d i t u r e ) CurrentAccount=(PrivateSavings-Investment)+(Tax-GovernmentExpenditure)
  9. S a v i n g s + T r a d e D e f i c i t = I n v e s t m e n t + B u d g e t D e f i c i t . Savings+TradeDeficit=Investment+BudgetDeficit.
  10. B u d g e t D e f i c i t = S a v i n g s + T r a d e D e f i c i t - I n v e s t m e n t . BudgetDeficit=Savings+TradeDeficit-Investment.

Twist_per_inch.html

  1. T P I = T M × c o u n t TPI=TM\times\sqrt{count}
  2. T M TM
  3. K K

Twistor_space.html

  1. A ( A Ω B ) = 0 \nabla_{A^{\prime}}^{(A}\Omega^{B)}=0
  2. 𝕄 \mathbb{M}
  3. Ω A ( x ) = ω A - i x A A π A \Omega^{A}(x)=\omega^{A}-ix^{AA^{\prime}}\pi_{A^{\prime}}
  4. ω A \omega^{A}
  5. π A \pi_{A^{\prime}}
  6. x A A = σ μ A A x μ x^{AA^{\prime}}=\sigma^{AA^{\prime}}_{\mu}x^{\mu}
  7. Z α = ( ω A , π A ) Z^{\alpha}=(\omega^{A},\pi_{A^{\prime}})
  8. Σ ( Z ) = ω A π ¯ A + ω ¯ A π A \Sigma(Z)=\omega^{A}\bar{\pi}_{A}+\bar{\omega}^{A^{\prime}}\pi_{A^{\prime}}
  9. ω A = i x A A π A . \omega^{A}=ix^{AA^{\prime}}\pi_{A^{\prime}}.
  10. 3 \mathbb{CP}^{3}
  11. x M x\in M
  12. 1 \mathbb{CP}^{1}
  13. π A \pi_{A^{\prime}}

Two-element_Boolean_algebra.html

  1. B , + , . , . . ¯ , 1 , 0 \langle B,+,.,\overline{..},1,0\rangle
  2. 2 , 2 , 1 , 0 , 0 \langle 2,2,1,0,0\rangle
  3. 1 + 1 = 1 + 0 = 0 + 1 = 1 \displaystyle 1+1=1+0=0+1=1
  4. A + A = A \displaystyle A+A=A
  5. A ( B + C ) = A B + A C ; \ A\cdot(B+C)=A\cdot B+A\cdot C;
  6. A + ( B C ) = ( A + B ) ( A + C ) . \ A+(B\cdot C)=(A+B)\cdot(A+C).
  7. A B = A ¯ + B ¯ ¯ A\cdot B=\overline{\overline{A}+\overline{B}}
  8. A + B = A ¯ B ¯ ¯ . A+B=\overline{\overline{A}\cdot\overline{B}}.
  9. A B C = B C A \ ABC=BCA
  10. A ¯ A = 1 \overline{A}A=1
  11. A 0 = A \ A0=A
  12. A A B ¯ = A B ¯ A\overline{AB}=A\overline{B}
  13. A A = A , A ¯ ¯ = A , 1 + A = 1 AA=A,\overline{\overline{A}}=A,1+A=1

Two-graph.html

  1. ρ \sqrt{\rho}

Two-stream_instability.html

  1. 𝐕 0 \mathbf{V}_{0}
  2. 𝐄 1 = ξ 1 exp [ i ( k x - ω t ) ] 𝐱 ^ \mathbf{E}_{1}=\xi_{1}\exp[i(kx-\omega t)]\mathbf{\hat{x}}
  3. t - i ω \partial_{t}\rightarrow-i\omega
  4. i k \nabla\rightarrow ik
  5. 1 = ω p e 2 [ m e / m i ω 2 + 1 ( ω - k v 0 ) 2 ] , 1=\omega_{pe}^{2}\left[\frac{m_{e}/m_{i}}{\omega^{2}}+\frac{1}{(\omega-kv_{0})% ^{2}}\right],
  6. ω \omega
  7. ω j = ω j R + i γ j \omega_{j}=\omega_{j}^{R}+i\gamma_{j}
  8. I m ( ω j ) Im(\omega_{j})
  9. 𝐄 = ξ exp [ i ( k x - ω t ) ] 𝐱 ^ \mathbf{E}=\xi\exp[i(kx-\omega t)]\mathbf{\hat{x}}
  10. I m ( ω j ) 0 Im(\omega_{j})\neq 0
  11. 𝐄 = ξ exp [ i ( k x - ω j R t ) ] exp [ γ t ] 𝐱 ^ \mathbf{E}=\xi\exp[i(kx-\omega_{j}^{R}t)]\exp[\gamma t]\mathbf{\hat{x}}
  12. γ \gamma
  13. γ < 0 \gamma<0
  14. γ > 0 \gamma>0
  15. v p h v_{ph}
  16. v > v p h v>v_{ph}

Two-vector.html

  1. 𝐟 = f α β e α e β \mathbf{f}=f^{\alpha\beta}\,\vec{e}_{\alpha}\otimes\vec{e}_{\beta}
  2. M : V V M:V\rightarrow V
  3. 𝐟 : V ~ V \mathbf{f}:\tilde{V}\rightarrow V

Two_envelopes_problem.html

  1. E ( B ) \displaystyle\operatorname{E}(B)
  2. E ( B ) = 1 2 2 x + 1 2 x = 3 2 x \operatorname{E}(B)=\frac{1}{2}2x+\frac{1}{2}x=\frac{3}{2}x
  3. E ( B ) = E ( A A < B ) + 1 4 E ( A A > B ) = x + 1 4 2 x = 3 2 x \operatorname{E}(B)=\operatorname{E}(A\mid A<B)+\frac{1}{4}\operatorname{E}(A% \mid A>B)=x+\frac{1}{4}2x=\frac{3}{2}x
  4. E = 1 2 + A 3 A / 2 + 1 2 - A / 2 3 A / 4 = 0 E=\frac{1}{2}\cdot\frac{+A}{3A/2}+\frac{1}{2}\cdot\frac{-A/2}{3A/4}=0
  5. E ( B | A = a ) > a E(B|A=a)>a
  6. P ( { 1 , 2 } 2 ) = P ( { 1 , 2 } ) / 2 P ( { 1 , 2 } ) / 2 + P ( { 2 , 4 } ) / 2 = P ( { 1 , 2 } ) P ( { 1 , 2 } ) + P ( { 2 , 4 } ) = 1 / 3 1 / 3 + 2 / 9 = 3 / 5 , \begin{aligned}\displaystyle P(\{1,2\}\mid 2)&\displaystyle=\frac{P(\{1,2\})/2% }{P(\{1,2\})/2+P(\{2,4\})/2}\\ &\displaystyle=\frac{P(\{1,2\})}{P(\{1,2\})+P(\{2,4\})}\\ &\displaystyle=\frac{1/3}{1/3+2/9}=3/5,\end{aligned}
  7. P ( { 1 , 2 } ) / 2 P(\{1,2\})/2
  8. P ( { 2 , 4 } ) / 2 P(\{2,4\})/2
  9. 3 5 a 2 + 2 5 2 a = 11 10 a \frac{3}{5}\frac{a}{2}+\frac{2}{5}2a=\frac{11}{10}a
  10. E ( B | A = a ) > a E(B|A=a)>a
  11. E ( X ) = E(X)=\infty
  12. E ( B | A = a ) < a E(B|A=a)<a
  13. E ( X ) = E(X)=\infty
  14. E ( B ) = E ( A ) = E(B)=E(A)=\infty
  15. A = a A=a
  16. E ( B | A = a ) > a E(B|A=a)>a
  17. E ( B ) = E ( A ) = E(B)=E(A)=\infty
  18. E ( B ) = E ( A ) = E(B)=E(A)=\infty
  19. u ( w ) u(w)
  20. E ( u ( B ) | A = a ) < u ( a ) E(u(B)|A=a)<u(a)
  21. A = a A=a
  22. u ( w ) u(w)
  23. u ( w ) u(w)
  24. u ( A ) u(A)
  25. u ( B ) u(B)
  26. u ( A ) u ( M ) u(A)\leq u(M)
  27. u ( B ) u ( M ) u(B)\leq u(M)
  28. E ( X ) = E(X)=\infty
  29. p ( x ) p(x)
  30. p ( a ) / ( p ( a / 2 ) + p ( a ) ) p(a)/(p(a/2)+p(a))
  31. p ( a / 2 ) / ( p ( a / 2 ) + p ( a ) ) p(a/2)/(p(a/2)+p(a))
  32. p ( a / 2 ) < 2 p ( a ) p(a/2)<2p(a)
  33. p ( a / 2 ) > 2 p ( a ) p(a/2)>2p(a)
  34. f ( x ) f(x)
  35. 2 f ( a ) / ( f ( a / 2 ) + 2 f ( a ) ) 2f(a)/(f(a/2)+2f(a))
  36. f ( a / 2 ) / ( f ( a / 2 ) + 2 f ( a ) ) f(a/2)/(f(a/2)+2f(a))
  37. f ( a / 2 ) < 4 f ( a ) f(a/2)<4f(a)
  38. f ( a ) h ( 1 / 2 ) f(a)\cdot h\cdot(1/2)
  39. f ( a / 2 ) ( h / 2 ) ( 1 / 2 ) f(a/2)\cdot(h/2)\cdot(1/2)
  40. f ( a ) h ( 1 / 2 ) / ( f ( a ) h ( 1 / 2 ) + f ( a / 2 ) ( h / 2 ) ( 1 / 2 ) ) = 2 f ( a ) / ( 2 f ( a ) + f ( a / 2 ) ) f(a)\cdot h\cdot(1/2)/(f(a)\cdot h\cdot(1/2)+f(a/2)\cdot(h/2)\cdot(1/2))=2f(a)% /(2f(a)+f(a/2))
  41. p ( a / 2 ) < p ( a ) p(a/2)<p(a)
  42. f ( a / 2 ) < 2 f ( a ) f(a/2)<2f(a)
  43. q ( a ) q(a)
  44. E ( B | A = a ) > a E(B|A=a)>a

Tympanometry.html

  1. S {}_{S}
  2. D {}_{D}

Types_of_capacitor.html

  1. C = ε A d C=\frac{\varepsilon A}{d}
  2. Z \scriptstyle Z
  3. Z = | Z | e j θ \ Z=|Z|e^{j\theta}
  4. | Z | \scriptstyle|Z|
  5. j \scriptstyle j
  6. θ \scriptstyle\theta
  7. Z = R + j X \ Z=R+jX
  8. R \scriptstyle R
  9. E S R \scriptstyle ESR
  10. X \scriptstyle X
  11. C C
  12. L ( E S L ) L(ESL)
  13. R ( E S R ) R(ESR)
  14. ω \omega
  15. X C = - 1 ω C X_{C}=-\frac{1}{\omega C}
  16. X L = ω L ESL X_{L}=\omega L_{\mathrm{ESL}}
  17. Z \scriptstyle Z
  18. Z Z
  19. Z = E S R 2 + ( X C + ( - X L ) ) 2 Z=\sqrt{{ESR}^{2}+(X_{\mathrm{C}}+(-X_{\mathrm{L}}))^{2}}
  20. Z = u ^ ı ^ = U eff I eff . Z=\frac{\hat{u}}{\hat{\imath}}=\frac{U_{\mathrm{eff}}}{I_{\mathrm{eff}}}.
  21. X C = - 1 ω C X_{C}=-\frac{1}{\omega C}
  22. X L = ω L ESL X_{L}=\omega L_{\mathrm{ESL}}
  23. X C = X L X_{C}=X_{L}
  24. E S R {ESR}
  25. X C X_{C}
  26. X L X_{L}
  27. I R I_{R}
  28. X C - X L X_{C}-X_{L}
  29. E S L ESL
  30. tan δ = E S R ω C \tan\delta=ESR\cdot\omega C
  31. B B
  32. f 0 f_{0}
  33. Q = 1 t a n δ = f 0 B Q=\frac{1}{tan\delta}=\frac{f_{0}}{B}
  34. P P
  35. E S R ESR
  36. I I
  37. P = I 2 E S R P=I^{2}\cdot ESR
  38. t a n δ tan\delta
  39. P = U 2 t a n δ 2 π f C P=\frac{U^{2}\cdot tan\delta}{2\pi f\cdot C}
  40. d v / d t dv/dt
  41. I p I_{p}
  42. I p = C d v / d t I_{p}=C\cdot dv/dt
  43. I p I_{p}
  44. C C
  45. d v / d t dv/dt
  46. u ( t ) = U 0 e - t / τ s , u(t)=U_{0}\cdot\mathrm{e}^{-t/\tau_{\mathrm{s}}},
  47. U 0 U_{0}
  48. τ s = R ins C \tau_{\mathrm{s}}=R_{\mathrm{ins}}\cdot C
  49. τ s \tau_{\mathrm{s}}\,
  50. U 0 U_{0}
  51. C ε A d C\approx\frac{\varepsilon A}{d}
  52. E stored = 1 2 C V 2 , E_{\mathrm{stored}}=\frac{1}{2}CV^{2},

Ultrahydrophobicity.html

  1. γ S G = γ S L + γ L G cos θ \gamma_{SG}\ =\gamma_{SL}+\gamma_{LG}\cos{\theta}
  2. γ S G \gamma_{SG}
  3. γ S L \gamma_{SL}
  4. γ L G \gamma_{LG}
  5. θ W * \theta_{W}*
  6. cos θ W * = r cos θ \cos{\theta}_{W}*=r\cos{\theta}
  7. θ C B * \theta_{CB}*
  8. cos θ C B * \cos{\theta}_{CB}*
  9. Λ C = - ρ g V 1 / 3 ( ( 1 - c o s ( θ a ) s i n ( θ a ) ) ( 3 + ( 1 - c o s ( θ a ) s i n ( θ a ) ) 2 ) ) 2 / 3 ( 36 π ) 1 / 3 γ c o s ( θ a , 0 + w - 90 ) \Lambda_{C}=\frac{-\rho{g}{V^{1/3}}((\frac{1-cos(\theta_{a})}{sin(\theta_{a})}% )(3+(\frac{1-cos(\theta_{a})}{sin(\theta_{a})})^{2}))^{2/3}}{(36\pi)^{1/3}% \gamma cos(\theta_{a,0}+w-90)}
  10. θ a = λ p ( θ a , 0 + w ) + ( 1 - λ p ) θ a i r \theta_{a}=\lambda_{p}(\theta_{a,0}+w)+(1-\lambda_{p})\theta_{air}
  11. θ r = λ p θ r , 0 + ( 1 - λ p ) θ a i r \theta_{r}=\lambda_{p}\theta_{r,0}+(1-\lambda_{p})\theta_{air}
  12. θ a = λ p ( θ a , 0 + w ) + ( 1 - λ p ) θ a , 0 \theta_{a}=\lambda_{p}(\theta_{a,0}+w)+(1-\lambda_{p})\theta_{a,0}
  13. θ r = λ p ( θ r , 0 - w ) + ( 1 - λ p ) θ r , 0 \theta_{r}=\lambda_{p}(\theta_{r,0}-w)+(1-\lambda_{p})\theta_{r,0}

Ultrasonic_flow_meter.html

  1. t u p t_{up}
  2. t d o w n t_{down}
  3. L L
  4. α \alpha
  5. v = L < m t p l > 2 sin ( α ) t u p - t d o w n t u p t d o w n v=\frac{L}{<}mtpl>{{2\;\sin\left(\alpha\right)}}\;\frac{{t_{up}-t_{down}}}{{t_% {up}\;t_{down}}}
  6. c = L 2 t u p + t d o w n t u p t d o w n c=\frac{L}{2}\;\frac{{t_{up}+t_{down}}}{{t_{up}\;t_{down}}}
  7. v v
  8. c c

Ultrastrong_topology.html

  1. p ω ( x ) = ω ( x * x ) 1 / 2 p_{\omega}(x)=\omega(x^{*}x)^{1/2}
  2. ω \omega
  3. L * ( H ) L_{*}(H)
  4. x ( n = 1 || x ξ n || 2 ) 1 / 2 , x\mapsto\left(\sum_{n=1}^{\infty}||x\xi_{n}||^{2}\right)^{1/2},
  5. n = 1 || ξ n || 2 < \sum_{n=1}^{\infty}||\xi_{n}||^{2}<\infty

Umbelliferone.html

  1. P A L \xrightarrow{PAL}
  2. C 4 H \xrightarrow{C4H}
  3. C 2 H \xrightarrow{C2H}
  4. \longrightarrow

Umov_effect.html

  1. P = I - I I + I , P=\frac{I_{\perp}-I_{\|}}{I_{\perp}+I_{\|}}\ ,
  2. I I_{\perp}
  3. I I_{\|}
  4. P 1 α , P\propto\frac{1}{\alpha}\ ,

Unconditional_convergence.html

  1. X X
  2. I I
  3. x i X x_{i}\in X
  4. i I i\in I
  5. i I x i \textstyle\sum_{i\in I}x_{i}
  6. x X x\in X
  7. I 0 := { i I : x i 0 } I_{0}:=\{i\in I:x_{i}\neq 0\}
  8. I 0 := { i I : x i 0 } I_{0}:=\{i\in I:x_{i}\neq 0\}
  9. i = 1 x i = x \sum_{i=1}^{\infty}x_{i}=x
  10. ( ε n ) n = 1 (\varepsilon_{n})_{n=1}^{\infty}
  11. ε n { - 1 , + 1 } \varepsilon_{n}\in\{-1,+1\}
  12. n = 1 ε n x n \sum_{n=1}^{\infty}\varepsilon_{n}x_{n}
  13. x n \sum x_{n}

Uniform_boundedness.html

  1. = { f i : X K , i I } \mathcal{F}=\{f_{i}:X\to K,i\in I\}
  2. I I
  3. X X
  4. K K
  5. \mathcal{F}
  6. M M
  7. | f i ( x ) | M i I x X . |f_{i}(x)|\leq M\qquad\forall i\in I\quad\forall x\in X.
  8. Y Y
  9. d d
  10. = { f i : X Y , i I } \mathcal{F}=\{f_{i}:X\to Y,i\in I\}
  11. a a
  12. Y Y
  13. M M
  14. d ( f i ( x ) , a ) M i I x X . d(f_{i}(x),a)\leq M\qquad\forall i\in I\quad\forall x\in X.
  15. f n ( x ) = sin n x f_{n}(x)=\sin nx\,
  16. x x
  17. n n
  18. f n ( x ) = n cos n x , f^{\prime}_{n}(x)=n\,\cos nx,
  19. f n f^{\prime}_{n}\,
  20. | n | , |n|,\,
  21. M M
  22. | n | M |n|\leq M
  23. n . n.

Uniform_polyhedron.html

  1. { p , q } \begin{Bmatrix}p,q\end{Bmatrix}
  2. t { p , q } t\begin{Bmatrix}p,q\end{Bmatrix}
  3. { p q } \begin{Bmatrix}p\\ q\end{Bmatrix}
  4. t { q , p } t\begin{Bmatrix}q,p\end{Bmatrix}
  5. { q , p } \begin{Bmatrix}q,p\end{Bmatrix}
  6. r { p q } r\begin{Bmatrix}p\\ q\end{Bmatrix}
  7. t { p q } t\begin{Bmatrix}p\\ q\end{Bmatrix}
  8. s { p q } s\begin{Bmatrix}p\\ q\end{Bmatrix}
  9. { p , 2 } \begin{Bmatrix}p,2\end{Bmatrix}
  10. t { p , 2 } t\begin{Bmatrix}p,2\end{Bmatrix}
  11. { p 2 } \begin{Bmatrix}p\\ 2\end{Bmatrix}
  12. t { 2 , p } t\begin{Bmatrix}2,p\end{Bmatrix}
  13. { 2 , p } \begin{Bmatrix}2,p\end{Bmatrix}
  14. r { p 2 } r\begin{Bmatrix}p\\ 2\end{Bmatrix}
  15. t { p 2 } t\begin{Bmatrix}p\\ 2\end{Bmatrix}
  16. s { p 2 } s\begin{Bmatrix}p\\ 2\end{Bmatrix}

Uniformization_(set_theory).html

  1. R R
  2. X × Y X\times Y
  3. X X
  4. Y Y
  5. f f
  6. R R
  7. X X
  8. Y Y
  9. x x
  10. f ( x ) f(x)
  11. { x X | y Y ( x , y ) R } \{x\in X|\exists y\in Y(x,y)\in R\}\,
  12. R R
  13. R R
  14. R R
  15. X X
  16. Y Y
  17. R R
  18. s y m b o l Γ symbol{\Gamma}
  19. R R
  20. s y m b o l Γ symbol{\Gamma}
  21. s y m b o l Γ symbol{\Gamma}
  22. s y m b o l Π 1 1 symbol{\Pi}^{1}_{1}
  23. s y m b o l Σ 2 1 symbol{\Sigma}^{1}_{2}
  24. s y m b o l Π 2 n + 1 1 symbol{\Pi}^{1}_{2n+1}
  25. s y m b o l Σ 2 n + 2 1 symbol{\Sigma}^{1}_{2n+2}
  26. n n

Unifying_Theories_of_Programming.html

  1. P 1 P_{1}
  2. P 2 P_{2}
  3. P 2 P_{2}
  4. P 1 P_{1}
  5. P 1 P 2 if and only if [ P 2 P 1 ] P_{1}\sqsubseteq P_{2}\quad\,\text{if and only if}\quad\left[P_{2}\Rightarrow P% _{1}\right]
  6. [ X ] \left[X\right]
  7. v v
  8. v v^{\prime}
  9. 𝐬𝐤𝐢𝐩 v = v \mathbf{skip}\equiv v^{\prime}=v
  10. E E
  11. a a
  12. a a^{\prime}
  13. E E
  14. u u
  15. a := E a = E u = u a:=E\equiv a^{\prime}=E\land u^{\prime}=u
  16. P 1 ; P 2 v 0 P 1 [ v 0 / v ] P 2 [ v 0 / v ] P_{1};P_{2}\equiv\exists v_{0}\bullet P_{1}[v_{0}/v^{\prime}]\land P_{2}[v_{0}% /v]
  17. P 1 P 2 P 1 P 2 P_{1}\sqcap P_{2}\equiv P_{1}\lor P_{2}
  18. P 1 C P 2 ( C P 1 ) ( ¬ C P 2 ) P_{1}\triangleleft C\triangleright P_{2}\equiv(C\land P_{1})\lor(\lnot C\land P% _{2})
  19. μ 𝐅 \mu\mathbf{F}
  20. 𝐅 \mathbf{F}
  21. μ X 𝐅 ( X ) { X 𝐅 ( X ) X } \mu X\bullet\mathbf{F}(X)\equiv\sqcap\left\{X\mid\mathbf{F}(X)\sqsubseteq X\right\}

Unique_games_conjecture.html

  1. x 1 \displaystyle x_{1}

Unit_function.html

  1. ε ( n ) = { 1 , if n = 1 0 , if n 1 \varepsilon(n)=\begin{cases}1,&\mbox{if }~{}n=1\\ 0,&\mbox{if }~{}n\neq 1\end{cases}

Unit_propagation.html

  1. l l
  2. l l
  3. l l
  4. ¬ l \neg l
  5. l l
  6. a a
  7. { a b , ¬ a c , ¬ c d , a } \{a\vee b,\neg a\vee c,\neg c\vee d,a\}
  8. a b a\vee b
  9. a a
  10. ¬ a c \neg a\vee c
  11. a a
  12. { \{
  13. a b , a\vee b,
  14. ¬ a c , \neg a\vee c,
  15. ¬ c d , \neg c\vee d,
  16. a a
  17. } \}
  18. ¬ a \neg a
  19. { \{
  20. c , c,
  21. ¬ c d , \neg c\vee d,
  22. a a
  23. } \}
  24. { c , ¬ c d , a } \{c,\neg c\vee d,a\}
  25. c c
  26. ¬ c d \neg c\vee d
  27. d d
  28. a a
  29. a a
  30. { 1 : a b , 2 : ¬ a c , 3 : ¬ c d , 4 : a } \{1:a\vee b,2:\neg a\vee c,3:\neg c\vee d,4:a\}
  31. a : 1 2 4 a:1\ 2\ 4
  32. b : 1 b:1\,
  33. c : 2 3 c:2\ 3
  34. d : 3 d:3\,

Unit_root.html

  1. { y t , t = 1 , , } \{y_{t},t=1,\ldots,\infty\}
  2. y t = a 1 y t - 1 + a 2 y t - 2 + + a p y t - p + ε t . y_{t}=a_{1}y_{t-1}+a_{2}y_{t-2}+\cdots+a_{p}y_{t-p}+\varepsilon_{t}.
  3. { ε t , t = 0 , } \{\varepsilon_{t},t=0,\infty\}
  4. σ 2 \sigma^{2}
  5. y 0 = 0 y_{0}=0
  6. m = 1 m=1
  7. m p - m p - 1 a 1 - m p - 2 a 2 - - a p = 0 m^{p}-m^{p-1}a_{1}-m^{p-2}a_{2}-\cdots-a_{p}=0
  8. I ( 1 ) I(1)
  9. y t = a 1 y t - 1 + ε t y_{t}=a_{1}y_{t-1}+\varepsilon_{t}
  10. a 1 = 1 a_{1}=1
  11. m - a 1 = 0 m-a_{1}=0
  12. m = 1 m=1
  13. t t
  14. y t = y t - 1 + ε t . y_{t}=y_{t-1}+\varepsilon_{t}.
  15. y t = y 0 + j = 1 t ε j y_{t}=y_{0}+\sum_{j=1}^{t}\varepsilon_{j}
  16. y t y_{t}
  17. Var ( y t ) = j = 1 t σ 2 = t σ 2 . \operatorname{Var}(y_{t})=\sum_{j=1}^{t}\sigma^{2}=t\sigma^{2}.
  18. Var ( y 1 ) = σ 2 \operatorname{Var}(y_{1})=\sigma^{2}
  19. Var ( y 2 ) = 2 σ 2 \operatorname{Var}(y_{2})=2\sigma^{2}
  20. Δ y t = y t - y t - 1 = ε t \Delta y_{t}=y_{t}-y_{t-1}=\varepsilon_{t}
  21. y t = a 1 y t - 1 + a 2 y t - 2 + ε t y_{t}=a_{1}y_{t-1}+a_{2}y_{t-2}+\varepsilon_{t}
  22. ( 1 - λ 1 L ) ( 1 - λ 2 L ) y t = ε t (1-\lambda_{1}L)(1-\lambda_{2}L)y_{t}=\varepsilon_{t}
  23. L y t = y t - 1 Ly_{t}=y_{t-1}
  24. λ 2 = 1 \lambda_{2}=1
  25. z t = Δ y t z_{t}=\Delta y_{t}
  26. z t = λ 1 z t - 1 + ε t z_{t}=\lambda_{1}z_{t-1}+\varepsilon_{t}
  27. | λ 1 | < 1 |\lambda_{1}|<1
  28. λ 1 \lambda_{1}
  29. Y t Y_{t}
  30. Δ Y t = Y t - Y t - 1 \Delta Y_{t}=Y_{t}-Y_{t-1}
  31. y t = y t - 1 + c + e t y_{t}=y_{t-1}+c+e_{t}
  32. e t e_{t}
  33. y t y_{t}
  34. y t = a + c t y_{t}=a+ct
  35. y t = k t + u t y_{t}=k\cdot t+u_{t}
  36. u t u_{t}
  37. y t y_{t}

UNITY_(programming_language).html

  1. Θ ( n ) \Theta(n)
  2. Θ ( n ) \Theta(n)
  3. Θ ( n 2 ) \Theta(n^{2})
  4. Θ ( n ) \Theta(n)
  5. { 0 , 1 } \{0,1\}
  6. Θ ( log n ) \Theta(\log n)
  7. Θ ( n 2 ) \Theta(n^{2})
  8. Θ ( n 2 ) \Theta(n^{2})
  9. Θ ( n ) \Theta(n)
  10. Θ ( n 2 ) \Theta(n^{2})
  11. Θ ( n 3 ) \Theta(n^{3})
  12. Θ ( n 3 ) \Theta(n^{3})
  13. Θ ( n 3 log n ) \Theta(n^{3}\log n)
  14. r r
  15. i i
  16. j j
  17. 0 r 0\dots r
  18. 0 2 r 0\dots 2r

Universal_bundle.html

  1. G G
  2. B G BG
  3. G G
  4. M M
  5. M B G M→BG
  6. G G
  7. E G EG
  8. G G
  9. E G B G EG→BG
  10. G G
  11. G G
  12. U ( n ) U(n)
  13. n n
  14. E U ( n ) EU(n)
  15. E G EG
  16. E U ( n ) EU(n)
  17. E U ( n ) EU(n)
  18. U ( n ) U(n)
  19. M M
  20. P M P→M
  21. G G
  22. f : M B G f:M→BG
  23. P P
  24. G G
  25. E G B G EG→BG
  26. f f
  27. π : E G B G π:EG→BG
  28. P × E G P×EG
  29. G G
  30. P M P→M
  31. P × E G P×EG
  32. P P × E G E G π M s P × G E G B G \begin{array}[]{rcccl}P&\to&P\times EG&\to&EG\\ \downarrow&&\downarrow&&\downarrow\pi\\ M&\to_{\!\!\!\!\!\!\!s}&P\times_{G}EG&\to&BG\end{array}
  33. p p
  34. E G EG
  35. p p
  36. s s
  37. f f
  38. f : M B G f:M→BG
  39. P M P→M
  40. p p
  41. f f
  42. f f
  43. Φ : { ( x , u ) M × E G : f ( x ) = π ( u ) } P \Phi:\left\{(x,u)\in M\times EG\ :\ f(x)=\pi(u)\right\}\to P
  44. { M P × G E G x [ Φ ( x , u ) , u ] \begin{cases}M\to P\times_{G}EG\\ x\mapsto[\Phi(x,u),u]\end{cases}
  45. p p
  46. f f
  47. E G EG
  48. G G
  49. G G
  50. X X
  51. Y = X × E G Y=X×EG
  52. E G EG
  53. X X
  54. Y Y
  55. Y Y
  56. G G
  57. X X
  58. E G EG
  59. X X

Universal_code_(data_compression).html

  1. E ( l ) = 6 π 2 l = 1 1 l = . E(l)=\frac{6}{\pi^{2}}\sum_{l=1}^{\infty}\frac{1}{l}=\infty.\,
  2. 1 / n 2 1/n^{2}
  3. 1 / n q 1/n^{q}
  4. q = 1 / log 2 ( φ ) 1.44 , q=1/\log_{2}(\varphi)\simeq 1.44,
  5. φ \varphi
  6. q = 1 + log 3 ( 4 / 3 ) 1.26 q=1+\log_{3}(4/3)\simeq 1.26

Universal_Rule.html

  1. R = 0.2 L S D 3 R=\frac{0.2\cdot L\cdot\sqrt{S}}{\sqrt[3]{D}}
  2. R R
  3. R R
  4. 𝐑𝐚𝐭𝐢𝐧𝐠 = 0.182 𝐋 S a i l A r e a [ 3 ] D i s p l a c e m e n t \,\textbf{Rating}=\frac{0.182\cdot\,\textbf{L}\cdot\sqrt{SailArea}}{\sqrt{}}[3% ]{Displacement}
  5. 𝐋 = L . W . L . + .5 ( q . b . l . - 100 - L . W . L . 100 L . W . L . ) \,\textbf{L}=L.W.L.+.5(q.b.l.-\frac{100-\sqrt{L.W.L.}}{100}\cdot L.W.L.)

Universal_Transverse_Mercator_coordinate_system.html

  1. a = 6378.137 a=6378.137
  2. 1 / f = 298.257 223 563 1/f=298.257\,223\,563
  3. φ \,\varphi
  4. λ \,\lambda
  5. k k\,\!
  6. γ \gamma\,\!
  7. λ 0 \lambda_{0}
  8. N 0 = 0 N_{0}=0
  9. N 0 = 10000 N_{0}=10000
  10. k 0 = 0.9996 k_{0}=0.9996
  11. E 0 = 500 E_{0}=500
  12. n = f 2 - f , A = a 1 + n ( 1 + n 2 4 + n 4 64 + ) , n=\frac{f}{2-f},\quad A=\frac{a}{1+n}\left(1+\frac{n^{2}}{4}+\frac{n^{4}}{64}+% \cdots\right),
  13. α 1 = 1 2 n - 2 3 n 2 + 5 16 n 3 , α 2 = 13 48 n 2 - 3 5 n 3 , α 3 = 61 240 n 3 , \alpha_{1}=\frac{1}{2}n-\frac{2}{3}n^{2}+\frac{5}{16}n^{3},\,\,\,\alpha_{2}=% \frac{13}{48}n^{2}-\frac{3}{5}n^{3},\,\,\,\alpha_{3}=\frac{61}{240}n^{3},
  14. β 1 = 1 2 n - 2 3 n 2 + 37 96 n 3 , β 2 = 1 48 n 2 + 1 15 n 3 , β 3 = 17 480 n 3 , \beta_{1}=\frac{1}{2}n-\frac{2}{3}n^{2}+\frac{37}{96}n^{3},\,\,\,\beta_{2}=% \frac{1}{48}n^{2}+\frac{1}{15}n^{3},\,\,\,\beta_{3}=\frac{17}{480}n^{3},
  15. δ 1 = 2 n - 2 3 n 2 - 2 n 3 , δ 2 = 7 3 n 2 - 8 5 n 3 , δ 3 = 56 15 n 3 . \delta_{1}=2n-\frac{2}{3}n^{2}-2n^{3},\,\,\,\delta_{2}=\frac{7}{3}n^{2}-\frac{% 8}{5}n^{3},\,\,\,\delta_{3}=\frac{56}{15}n^{3}.
  16. t = sinh ( tanh - 1 sin φ - 2 n 1 + n tanh - 1 ( 2 n 1 + n sin φ ) ) , t=\sinh\left(\tanh^{-1}\sin\varphi-\frac{2\sqrt{n}}{1+n}\tanh^{-1}\left(\frac{% 2\sqrt{n}}{1+n}\sin\varphi\right)\right),
  17. ξ = tan - 1 ( t cos ( λ - λ 0 ) ) , η = tanh - 1 ( sin ( λ - λ 0 ) 1 + t 2 ) , \xi^{\prime}=\tan^{-1}\left(\frac{t}{\cos(\lambda-\lambda_{0})}\right),\,\,\,% \eta^{\prime}=\tanh^{-1}\left(\frac{\sin(\lambda-\lambda_{0})}{\sqrt{1+t^{2}}}% \right),
  18. σ = 1 + j = 1 3 2 j α j cos ( 2 j ξ ) cosh ( 2 j η ) , τ = j = 1 3 2 j α j sin ( 2 j ξ ) sinh ( 2 j η ) . \sigma=1+\sum_{j=1}^{3}2j\alpha_{j}\cos\left(2j\xi^{\prime}\right)\cosh\left(2% j\eta^{\prime}\right),\,\,\,\tau=\sum_{j=1}^{3}2j\alpha_{j}\sin\left(2j\xi^{% \prime}\right)\sinh\left(2j\eta^{\prime}\right).
  19. E = E 0 + k 0 A ( η + j = 1 3 α j cos ( 2 j ξ ) sinh ( 2 j η ) ) , E=E_{0}+k_{0}A\left(\eta^{\prime}+\sum_{j=1}^{3}\alpha_{j}\cos\left(2j\xi^{% \prime}\right)\sinh\left(2j\eta^{\prime}\right)\right),
  20. N = N 0 + k 0 A ( ξ + j = 1 3 α j sin ( 2 j ξ ) cosh ( 2 j η ) ) , N=N_{0}+k_{0}A\left(\xi^{\prime}+\sum_{j=1}^{3}\alpha_{j}\sin\left(2j\xi^{% \prime}\right)\cosh\left(2j\eta^{\prime}\right)\right),
  21. k = k 0 A a { 1 + ( 1 - n 1 + n tan φ ) 2 } σ 2 + τ 2 t 2 + cos 2 ( λ - λ 0 ) , k=\frac{k_{0}A}{a}\sqrt{\left\{1+\left(\frac{1-n}{1+n}\tan\varphi\right)^{2}% \right\}\frac{\sigma^{2}+\tau^{2}}{t^{2}+\cos^{2}(\lambda-\lambda_{0})}},
  22. γ = tan - 1 ( τ 1 + t 2 + σ t tan ( λ - λ 0 ) σ 1 + t 2 - τ t tan ( λ - λ 0 ) ) . \gamma=\tan^{-1}\left(\frac{\tau\sqrt{1+t^{2}}+\sigma t\tan(\lambda-\lambda_{0% })}{\sigma\sqrt{1+t^{2}}-\tau t\tan(\lambda-\lambda_{0})}\right).
  23. ξ = N - N 0 k 0 A , η = E - E 0 k 0 A , \xi=\frac{N-N_{0}}{k_{0}A},\,\,\,\eta=\frac{E-E_{0}}{k_{0}A},
  24. ξ = ξ - j = 1 3 β j sin ( 2 j ξ ) cosh ( 2 j η ) , η = η - j = 1 3 β j cos ( 2 j ξ ) sinh ( 2 j η ) , \xi^{\prime}=\xi-\sum_{j=1}^{3}\beta_{j}\sin\left(2j\xi\right)\cosh\left(2j% \eta\right),\,\,\,\eta^{\prime}=\eta-\sum_{j=1}^{3}\beta_{j}\cos\left(2j\xi% \right)\sinh\left(2j\eta\right),
  25. σ = 1 - j = 1 3 2 j β j cos ( 2 j ξ ) cosh ( 2 j η ) , τ = j = 1 3 2 j β j sin ( 2 j ξ ) sinh ( 2 j η ) , \sigma^{\prime}=1-\sum_{j=1}^{3}2j\beta_{j}\cos\left(2j\xi\right)\cosh\left(2j% \eta\right),\,\,\,\tau^{\prime}=\sum_{j=1}^{3}2j\beta_{j}\sin\left(2j\xi\right% )\sinh\left(2j\eta\right),
  26. χ = sin - 1 ( sin ξ cosh η ) . \chi=\sin^{-1}\left(\frac{\sin\xi^{\prime}}{\cosh\eta^{\prime}}\right).
  27. φ = χ + j = 1 3 δ j sin ( 2 j χ ) , \varphi=\chi+\sum_{j=1}^{3}\delta_{j}\sin\left(2j\chi\right),
  28. λ 0 = Zone × 6 - 183 \lambda_{0}=\mathrm{Z}\mathrm{o}\mathrm{n}\mathrm{e}\times 6^{\circ}-183^{% \circ}\,
  29. λ = λ 0 + tan - 1 ( sinh η cos ξ ) , \lambda=\lambda_{0}+\tan^{-1}\left(\frac{\sinh\eta^{\prime}}{\cos\xi^{\prime}}% \right),
  30. k = k 0 A a { 1 + ( 1 - n 1 + n tan φ ) 2 } cos 2 ξ + sinh 2 η σ 2 + τ 2 , k=\frac{k_{0}A}{a}\sqrt{\left\{1+\left(\frac{1-n}{1+n}\tan\varphi\right)^{2}% \right\}\frac{\cos^{2}\xi^{\prime}+\sinh^{2}\eta^{\prime}}{\sigma^{\prime 2}+% \tau^{\prime 2}}},
  31. γ = Hemi × tan - 1 ( τ + σ tan ξ tanh η σ - τ tan ξ tanh η ) . \gamma=\mathrm{H}\mathrm{e}\mathrm{m}\mathrm{i}\times\tan^{-1}\left(\frac{\tau% ^{\prime}+\sigma^{\prime}\tan\xi^{\prime}\tanh\eta^{\prime}}{\sigma^{\prime}-% \tau^{\prime}\tan\xi^{\prime}\tanh\eta^{\prime}}\right).

Universally_measurable_set.html

  1. A A
  2. X X
  3. X X
  4. X X
  5. X X
  6. ν ( A ) = i = 0 1 2 n + 1 μ ( A N i ) \nu(A)=\sum_{i=0}^{\infty}\frac{1}{2^{n+1}}\mu(A\cap N_{i})
  7. A A
  8. 2 ω 2^{\omega}
  9. A A
  10. A A
  11. A A
  12. A A
  13. A A
  14. A A
  15. A A
  16. A A
  17. A A
  18. A A
  19. A A^{\prime}
  20. A A
  21. A A^{\prime}
  22. A A
  23. A A^{\prime}
  24. A A^{\prime}
  25. A A^{\prime}

Unscrupulous_diner's_dilemma.html

  1. h - l > g - b h-l>g-b
  2. g - 1 n h > b - 1 n l g-\frac{1}{n}h>b-\frac{1}{n}l
  3. 1 n x + 1 n l \frac{1}{n}x+\frac{1}{n}l
  4. 1 n x + 1 n h \frac{1}{n}x+\frac{1}{n}h
  5. g - 1 n x - 1 n h g-\frac{1}{n}x-\frac{1}{n}h
  6. b - 1 n x - 1 n l b-\frac{1}{n}x-\frac{1}{n}l
  7. g - h g-h
  8. b - l b-l
  9. b - l > g - h b-l>g-h

Unusual_number.html

  1. n \sqrt{n}
  2. n \sqrt{n}
  3. lim n u ( n ) n = ln ( 2 ) = 0.693147 . \lim_{n\rightarrow\infty}\frac{u(n)}{n}=\ln(2)=0.693147\dots\,.

Upper_and_lower_probabilities.html

  1. P ( S ) P(S)\,\!
  2. m : P ( S ) R m:P(S)\rightarrow R
  3. m ( ) = 0 ; A P ( X ) m ( A ) = 1. m(\varnothing)=0\,\,\,\,\,\,\!;\,\,\,\,\,\,\sum_{A\in P(X)}m(A)=1.\,\!
  4. bel ( A ) = B B A m ( B ) ; pl ( A ) = B B A m ( B ) \operatorname{bel}(A)=\sum_{B\mid B\subseteq A}m(B)\,\,\,\,;\,\,\,\,% \operatorname{pl}(A)=\sum_{B\mid B\cap A\neq\varnothing}m(B)
  5. S S
  6. bel \operatorname{bel}
  7. env 1 ( A ) = inf p C p ( A ) ; env 2 ( A ) = sup p C p ( A ) \operatorname{env_{1}}(A)=\inf_{p\in C}p(A)\,\,\,\,;\,\,\,\,\operatorname{env_% {2}}(A)=\sup_{p\in C}p(A)

Uranium-lead_dating.html

  1. N Now = N Orig e - λ t N_{\mathrm{Now}}=N_{\mathrm{Orig}}e^{-\lambda t}\,
  2. N Now N_{\mathrm{Now}}
  3. N Orig N_{\mathrm{Orig}}
  4. λ \lambda
  5. t t
  6. N U = ( N U + N Pb ) e - λ U t N_{\mathrm{U}}=\left(N_{\mathrm{U}}+N_{\mathrm{Pb}}\right)e^{-\lambda_{\mathrm% {U}}t}\,
  7. N U N U + N Pb = e - λ U t {{N_{\mathrm{U}}}\over{N_{\mathrm{U}}+N_{\mathrm{Pb}}}}=e^{-\lambda_{\mathrm{U% }}t}\,
  8. 1 + N Pb N U = e λ U t 1+{{N_{\mathrm{Pb}}}\over{N_{\mathrm{U}}}}=e^{\lambda_{\mathrm{U}}t}\,
  9. 206 Pb * 238 U = e λ 238 t - 1 {{\text{206}\,\!\,\text{Pb}^{*}}\over{\text{238}\,\!\,\text{U}}}=e^{\lambda_{2% 38}t}-1
  10. 207 Pb * 235 U = e λ 235 t - 1 {{\text{207}\,\!\,\text{Pb}^{*}}\over{\text{235}\,\!\,\text{U}}}=e^{\lambda_{2% 35}t}-1

Urea_reduction_ratio.html

  1. U R R = U p r e - U p o s t U p r e × 100 % URR=\frac{U_{pre}-U_{post}}{U_{pre}}\times 100\%
  2. V V\,
  3. V V\,
  4. m l m i n \frac{ml}{min}
  5. L h r \frac{L}{hr}
  6. t t\,
  7. K t K\cdot t
  8. m l m i n m i n = m l \frac{ml}{min}\cdot min=ml
  9. L h r h r = L \frac{L}{hr}\cdot hr=L
  10. V V\,
  11. K t V \frac{K\cdot t}{V}
  12. m l m l \frac{ml}{ml}
  13. L L \frac{L}{L}
  14. K t V \frac{K\cdot t}{V}
  15. U R R URR\,
  16. K t V = - l n ( 1 - U R R ) \frac{K\cdot t}{V}=-ln(1-URR)
  17. K t V = - l n ( ( 1 - U R R ) - 0.008 t ) + ( 4 - 3.5 ( 1 - U R R ) ) 0.55 U F V \frac{K\cdot t}{V}=-ln((1-URR)-0.008\cdot t)+(4-3.5(1-URR))\cdot\frac{0.55% \cdot UF}{V}
  18. ( 0.008 t ) (0.008\cdot t)
  19. ( 4 - 3.5 ( 1 - U R R ) ) 0.55 U F V (4-3.5(1-URR))\cdot\frac{0.55\cdot UF}{V}
  20. 0.55 U F V \frac{0.55\cdot UF}{V}
  21. U F W \frac{UF}{W}
  22. K t V = - l n ( ( 1 - U R R ) - 0.03 ) + ( 4 - 3.5 ( 1 - U R R ) ) U F W \frac{K\cdot t}{V}=-ln((1-URR)-0.03)+(4-3.5(1-URR))\cdot\frac{UF}{W}

Utm_theorem.html

  1. φ 1 , φ 2 , φ 3 , \varphi_{1},\varphi_{2},\varphi_{3},...
  2. u : 2 u:\mathbb{N}^{2}\to\mathbb{N}
  3. u ( i , x ) := φ i ( x ) i , x u(i,x):=\varphi_{i}(x)\qquad i,x\in\mathbb{N}
  4. u u

Vacuum_permittivity.html

  1. F C = 1 4 π ε 0 q 1 q 2 r 2 \ F_{C}=\frac{1}{4\pi\varepsilon_{0}}\frac{q_{1}q_{2}}{r^{2}}
  2. ε 0 = 1 μ 0 c 2 \varepsilon_{0}=\frac{1}{\mu_{0}c^{2}}
  3. ε 0 = 1 μ 0 c 2 = e 2 2 α h c , \varepsilon_{0}=\frac{1}{\mu_{0}c^{2}}=\frac{e^{2}}{2\alpha hc}\ ,
  4. α = μ 0 c e 2 2 h . \alpha=\frac{\mu_{0}ce^{2}}{2h}\ .
  5. ε 0 \varepsilon_{0}\,
  6. ϵ 0 \epsilon_{0}\,
  7. F = k e Q 2 r 2 , F=\;k_{\mathrm{e}}\frac{Q^{2}}{r^{2}},
  8. F = q s 2 r 2 . F=\frac{{q_{\,\text{s}}}^{2}}{r^{2}}.
  9. F = k e q s 2 4 π r 2 . F=\;k^{\prime}_{\mathrm{e}}\frac{{q^{\prime}_{\,\text{s}}}^{2}}{4\pi r^{2}}.
  10. F = 1 4 π ε 0 q 2 r 2 . \ F=\frac{1}{4\pi\varepsilon_{0}}\frac{q^{2}}{r^{2}}.
  11. q s = q 4 π ε 0 . \ q_{\,\text{s}}=\frac{q}{\sqrt{4\pi\varepsilon_{0}}}.
  12. 𝐃 = ε 0 𝐄 + 𝐏 . \mathbf{D}=\varepsilon_{0}\mathbf{E}+\mathbf{P}.
  13. 𝐃 ( 𝐫 , t ) = - t d t d 3 𝐫 ε ( 𝐫 , t ; 𝐫 , t ) 𝐄 ( 𝐫 , t ) . \mathbf{D}(\mathbf{r},\ t)=\int_{-\infty}^{t}dt^{\prime}\int d^{3}\mathbf{r}^{% \prime}\ \varepsilon(\mathbf{r},\ t;\mathbf{r}^{\prime},\ t^{\prime})\mathbf{E% }(\mathbf{r}^{\prime},\ t^{\prime}).
  14. 𝐃 = ε 𝐄 = ε r ε 0 𝐄 \mathbf{D}=\varepsilon\mathbf{E}=\varepsilon_{\,\text{r}}\varepsilon_{0}% \mathbf{E}

Valiant–Vazirani_theorem.html

  1. Ω ( 1 / n ) \Omega(1/n)
  2. F ( x 1 , , x n ) F(x_{1},\dots,x_{n})
  3. n n
  4. x 1 , , x n x_{1},\dots,x_{n}
  5. F ( x 1 , , x n ) F^{\prime}(x_{1},\dots,x_{n})
  6. F F^{\prime}
  7. F F
  8. F F
  9. Ω ( 1 / n ) \Omega(1/n)
  10. F F^{\prime}
  11. ( a 1 , , a n ) (a_{1},\dots,a_{n})
  12. t t
  13. F 1 , , F t F^{\prime}_{1},\dots,F^{\prime}_{t}
  14. t = O ( n ) t=O(n)
  15. F i F^{\prime}_{i}
  16. 1 / 2 1/2
  17. F F
  18. F i F^{\prime}_{i}
  19. F F
  20. k k
  21. GF ( 2 ) n \,\text{GF}(2)^{n}
  22. k { 1 , , n } k\in\{1,\dots,n\}
  23. Ω ( 1 / n 8 ) \Omega(1/n^{8})

Value_product.html

  1. = V =V
  2. = C c =C_{c}
  3. = C f =C_{f}
  4. = S =S
  5. V P n = C c + C f + V + S = VP_{n}=C_{c}+C_{f}+V+S=
  6. V P = V + S = VP=V+S=

Van_Deemter_equation.html

  1. H E T P = A + B u + ( C s + C m ) u HETP=A+\frac{B}{u}+(C_{s}+C_{m})\cdot u
  2. u = B C u=\sqrt{\frac{B}{C}}
  3. H = L N H=\frac{L}{N}\,
  4. L L\,
  5. N N\,
  6. t R t_{R}\,
  7. σ \sigma\,
  8. N = ( t R σ ) 2 N=\left(\frac{t_{R}}{\sigma}\right)^{2}\,
  9. W 1 / 2 W_{1/2}\,
  10. N = 8 ln ( 2 ) ( t R W 1 / 2 ) 2 N=8\ln(2)\cdot\left(\frac{t_{R}}{W_{1/2}}\right)^{2}\,
  11. N = 16 ( t R W b a s e ) 2 N=16\cdot\left(\frac{t_{R}}{W_{base}}\right)^{2}\,
  12. H = 2 λ d p + 2 γ D m u + ω ( d p or d c ) 2 u D m + R d f 2 u D s H=2\lambda d_{p}+{2\gamma D_{m}\over u}+{\omega(d_{p}\mbox{ or }~{}d_{c})^{2}u% \over D_{m}}+{Rd_{f}^{2}u\over D_{s}}

Van_Stockum_dust.html

  1. r r
  2. e 0 = t , e 1 = f ( r ) z , e 2 = f ( r ) r , e 3 = 1 r ϕ - h ( r ) t \vec{e}_{0}=\partial_{t},\;\vec{e}_{1}=f(r)\,\partial_{z},\;\vec{e}_{2}=f(r)\,% \partial_{r},\;\vec{e}_{3}=\frac{1}{r}\,\partial_{\phi}-h(r)\,\partial_{t}
  3. σ 0 = - d t + h ( r ) r d ϕ , σ 1 = 1 f ( r ) d z , σ 2 = 1 f ( r ) d r , σ 3 = r d ϕ \sigma^{0}=-dt+h(r)r\,d\phi,\;\sigma^{1}=\frac{1}{f(r)}\,dz,\;\sigma^{2}=\frac% {1}{f(r)}\,dr,\;\sigma^{3}=rd\phi
  4. g = - σ 0 σ 0 + σ 1 σ 1 + σ 2 σ 2 + σ 3 σ 3 g=-\sigma^{0}\otimes\sigma^{0}+\sigma^{1}\otimes\sigma^{1}+\sigma^{2}\otimes% \sigma^{2}+\sigma^{3}\otimes\sigma^{3}
  5. d s 2 = - d t 2 - 2 h ( r ) r d t d ϕ + ( 1 - h ( r ) 2 ) r 2 d ϕ 2 + d z 2 + d r 2 f ( r ) 2 ds^{2}=-dt^{2}-2h(r)r\,dt\,d\phi+(1-h(r)^{2})r^{2}\,d\phi^{2}+\frac{dz^{2}+dr^% {2}}{f(r)^{2}}
  6. - < t , z < , 0 < r < , - π < ϕ < π -\infty<t,z<\infty,\;0<r<\infty,\;-\pi<\phi<\pi
  7. e 0 \vec{e}_{0}
  8. G m ^ n ^ = 8 π μ diag ( 1 , 0 , 0 , 0 ) + 8 π p diag ( 0 , 1 , 1 , 1 ) G^{\hat{m}\hat{n}}=8\pi\mu\,\operatorname{diag}(1,0,0,0)+8\pi p\,\operatorname% {diag}(0,1,1,1)
  9. f ′′ = ( f ) 2 f + f r , ( h ) 2 + 2 h h r + h 2 r 2 = 4 f r f f^{\prime\prime}=\frac{(f^{\prime})^{2}}{f}+\frac{f^{\prime}}{r},\;(h^{\prime}% )^{2}+\frac{2h^{\prime}h}{r}+\frac{h^{2}}{r^{2}}=\frac{4f^{\prime}}{r\,f}
  10. f f
  11. h h
  12. e 0 = t , e 1 = exp ( a 2 r 2 / 2 ) z , e 2 = exp ( a 2 r 2 / 2 ) r , e 3 = 1 r ϕ - a r t \vec{e}_{0}=\partial_{t},\;\vec{e}_{1}=\exp(a^{2}r^{2}/2)\,\partial_{z},\;\vec% {e}_{2}=\exp(a^{2}r^{2}/2)\,\partial_{r},\;\vec{e}_{3}=\frac{1}{r}\,\partial_{% \phi}-ar\,\partial_{t}
  13. r > 0 r>0
  14. μ = a 2 2 π exp ( a 2 r 2 ) \mu=\frac{a^{2}}{2\pi}\,\exp(a^{2}r^{2})
  15. r = 0 r=0
  16. ξ 1 = t , ξ 2 = z , ξ 3 = ϕ \vec{\xi}_{1}=\partial_{t},\;\vec{\xi}_{2}=\partial_{z},\;\vec{\xi}_{3}=% \partial_{\phi}
  17. ξ 1 \vec{\xi}_{1}
  18. e 0 \vec{e}_{0}
  19. Ω = - a exp ( a 2 r 2 / 2 ) e 1 \vec{\Omega}=-a\,\exp(a^{2}r^{2}/2)\,\vec{e}_{1}
  20. r = 0 r=0
  21. a a
  22. E m ^ n ^ = a 2 exp ( a 2 r 2 ) diag ( 0 , 1 , 1 ) E_{\hat{m}\hat{n}}=a^{2}\,\exp(a^{2}r^{2})\,\operatorname{diag}(0,1,1)
  23. B m ^ n ^ = - a 3 exp ( a 2 r 2 ) [ 0 1 0 1 0 0 0 0 0 ] B_{\hat{m}\hat{n}}=-a^{3}\,\exp(a^{2}r^{2})\,\left[\begin{matrix}0&1&0\\ 1&0&0\\ 0&0&0\end{matrix}\right]
  24. z z
  25. e 0 e 1 , e 0 e 2 , e 0 e 3 \nabla_{\vec{e}_{0}}\vec{e}_{1},\;\nabla_{\vec{e}_{0}}\vec{e}_{2},\;\nabla_{% \vec{e}_{0}}\vec{e}_{3}
  26. e 1 \vec{e}_{1}
  27. f 0 = e 0 , f 1 = e 1 , f 2 = cos ( θ ) e 2 + sin ( θ ) e 3 , f 3 = - sin ( θ ) e 2 + cos ( θ ) e 3 \vec{f}_{0}=\vec{e}_{0},\;\vec{f}_{1}=\vec{e}_{1},\;\vec{f}_{2}=\cos(\theta)\,% \vec{e}_{2}+\sin(\theta)\,\vec{e}_{3},\;\vec{f}_{3}=-\sin(\theta)\,\vec{e}_{2}% +\cos(\theta)\,\vec{e}_{3}
  28. θ = t q ( r ) \theta=t\,q(r)
  29. θ = a t exp ( a 2 r 2 / 2 ) \theta=a\,t\,\exp(a^{2}r^{2}/2)
  30. r = a - 1 r=a^{-1}
  31. t = t 0 t=t_{0}
  32. r = a - 1 r=a^{-1}

Varghese_Mathai.html

  1. L 2 L^{2}

Variational_message_passing.html

  1. H H
  2. V V
  3. V V
  4. Q Q
  5. ln P ( V ) = H Q ( H ) ln P ( H , V ) P ( H | V ) = H Q ( H ) [ ln P ( H , V ) Q ( H ) - ln P ( H | V ) Q ( H ) ] \ln P(V)=\sum_{H}Q(H)\ln\frac{P(H,V)}{P(H|V)}=\sum_{H}Q(H)\Bigg[\ln\frac{P(H,V% )}{Q(H)}-\ln\frac{P(H|V)}{Q(H)}\Bigg]
  6. L ( Q ) = H Q ( H ) ln P ( H , V ) Q ( H ) L(Q)=\sum_{H}Q(H)\ln\frac{P(H,V)}{Q(H)}
  7. P P
  8. Q Q
  9. L L
  10. V V
  11. Q Q
  12. P P
  13. P P
  14. Q Q
  15. Q ( H ) = i Q i ( H i ) , Q(H)=\prod_{i}Q_{i}(H_{i}),
  16. H i H_{i}
  17. log P \log P
  18. Q Q
  19. L ( Q ) L(Q)
  20. H i H_{i}
  21. Q j Q_{j}
  22. Q j * Q_{j}^{*}
  23. Q j Q_{j}
  24. Q j * Q_{j}^{*}
  25. Q j * ( H j ) = 1 Z e 𝔼 - j { ln P ( H , V ) } Q_{j}^{*}(H_{j})=\frac{1}{Z}e^{\mathbb{E}_{-j}\{\ln P(H,V)\}}
  26. 𝔼 - j { ln P ( H , V ) } \mathbb{E}_{-j}\{\ln P(H,V)\}
  27. Q i Q_{i}
  28. Q j Q_{j}
  29. Q j Q_{j}
  30. Q j * Q_{j}^{*}
  31. L L
  32. σ \sigma

Vector_area.html

  1. S S
  2. n ^ \hat{n}
  3. 𝐒 \mathbf{S}
  4. 𝐒 = 𝐧 ^ S \mathbf{S}=\mathbf{\hat{n}}S
  5. S i S_{i}
  6. 𝐒 = i 𝐧 ^ i S i \mathbf{S}=\sum_{i}\mathbf{\hat{n}}_{i}S_{i}
  7. 𝐧 ^ i \mathbf{\hat{n}}_{i}
  8. S i S_{i}
  9. d 𝐒 = 𝐧 ^ d S d\mathbf{S}=\mathbf{\hat{n}}dS
  10. 𝐧 ^ \mathbf{\hat{n}}
  11. d S dS
  12. 𝐒 = d 𝐒 \mathbf{S}=\int d\mathbf{S}
  13. 𝐒 𝐳 = | 𝐒 | cos θ \mathbf{S_{z}}=\left|\mathbf{S}\right|\cos\theta
  14. θ \theta