wpmath0000002_20

Texture_mapping.html

  1. u α u_{\alpha}
  2. u 0 u_{0}
  3. u 1 u_{1}
  4. u α = ( 1 - α ) u 0 + α u 1 u_{\alpha}=(1-\alpha)u_{0}+\alpha u_{1}
  5. 0 α 1 0\leq\alpha\leq 1
  6. z z
  7. u α = ( 1 - α ) u 0 z 0 + α u 1 z 1 ( 1 - α ) 1 z 0 + α 1 z 1 u_{\alpha}=\frac{(1-\alpha)\frac{u_{0}}{z_{0}}+\alpha\frac{u_{1}}{z_{1}}}{(1-% \alpha)\frac{1}{z_{0}}+\alpha\frac{1}{z_{1}}}

Thales'_theorem.html

  1. A C ¯ \overline{AC}
  2. O A ¯ \overline{OA}
  3. O B ¯ \overline{OB}
  4. O C ¯ \overline{OC}
  5. α + ( α + β ) + β = 180 \alpha+\left(\alpha+\beta\right)+\beta=180^{\circ}
  6. 2 α + 2 β = 180 {2}\alpha+{2}\beta=180^{\circ}
  7. 2 ( α + β ) = 180 {2}(\alpha+\beta)=180^{\circ}
  8. α + β = 90 . \therefore\alpha+\beta=90^{\circ}.
  9. O = ( 0 , 0 ) O=(0,0)
  10. A = ( - 1 , 0 ) A=(-1,0)
  11. C = ( 1 , 0 ) C=(1,0)
  12. ( cos θ , sin θ ) (\cos\theta,\sin\theta)
  13. A B ¯ \overline{AB}
  14. B C ¯ \overline{BC}
  15. A B ¯ \overline{AB}
  16. B C ¯ \overline{BC}
  17. m A B = y B - y A x B - x A = sin θ cos θ + 1 m_{AB}=\frac{y_{B}-y_{A}}{x_{B}-x_{A}}=\frac{\sin\theta}{\cos\theta+1}
  18. m B C = y B - y C x B - x C = sin θ cos θ - 1 m_{BC}=\frac{y_{B}-y_{C}}{x_{B}-x_{C}}=\frac{\sin\theta}{\cos\theta-1}
  19. m A B m B C \displaystyle m_{AB}\cdot m_{BC}
  20. sin 2 θ + cos 2 θ = 1 \sin^{2}\theta+\cos^{2}\theta=1
  21. A B C ABC
  22. A B AB
  23. A B D ABD
  24. A B C ABC
  25. A B AB
  26. A B AB
  27. A C AC
  28. B D BD
  29. A D AD
  30. C B CB
  31. A B C D ABCD
  32. A B AB
  33. C D CD
  34. B C ¯ \overline{BC}
  35. A B ¯ \overline{AB}
  36. A C ¯ \overline{AC}
  37. B D ¯ \overline{BD}
  38. A C ¯ \overline{AC}
  39. A C ¯ \overline{AC}
  40. A C ¯ \overline{AC}
  41. A C ¯ \overline{AC}

The_Library_of_Babel.html

  1. 25 1 , 312 , 000 1.956 × 10 1 , 834 , 097 25^{1,312,000}\approx 1.956\times 10^{1,834,097}
  2. 10 6 10^{6}
  3. 2.18 × 10 7 2.18\times 10^{7}
  4. 1 1
  5. 24 × 1 , 312 , 000 24\times 1,312,000
  6. 24 2 ( 1 , 312 , 000 2 ) 24^{2}{\textstyle\left({{1,312,000}\atop{2}}\right)}
  7. 24 3 ( 1 , 312 , 000 3 ) 24^{3}{\textstyle\left({{1,312,000}\atop{3}}\right)}
  8. 24 4 ( 1 , 312 , 000 4 ) 24^{4}{\textstyle\left({{1,312,000}\atop{4}}\right)}
  9. 10 1 , 834 , 097 ! 10^{1,834,097}!
  10. 10 2 , 000 , 000 10^{2,000,000}

The_Market_for_Lemons.html

  1. q q
  2. q q
  3. 3 2 q \tfrac{3}{2}q
  4. q q
  5. q q
  6. q q
  7. q q
  8. 3 2 q \tfrac{3}{2}q
  9. 3 2 q avg \tfrac{3}{2}q\text{avg}
  10. q avg q\text{avg}
  11. p p
  12. p > 0 p>0
  13. p p
  14. p p
  15. p p
  16. p / 2 p/2
  17. p / 2 p/2
  18. ( 3 / 2 ) ( p / 2 ) = 3 4 p (3/2)(p/2)=\tfrac{3}{4}p
  19. p p
  20. p p

The_Method_of_Mechanical_Theorems.html

  1. 0 1 x 2 d x = 1 3 , \int_{0}^{1}x^{2}\,dx=\frac{1}{3},
  2. ρ S \rho_{S}
  3. ρ S ( x ) = x ( 2 - x ) . \rho_{S}(x)=\sqrt{x(2-x)}.\,
  4. π ρ S ( x ) 2 = 2 π x - π x 2 . \pi\rho_{S}(x)^{2}=2\pi x-\pi x^{2}.\,
  5. ρ C \rho_{C}
  6. ρ C ( x ) = x \rho_{C}(x)=x\,
  7. π ρ C 2 = π x 2 . \pi\rho_{C}^{2}=\pi x^{2}.\,
  8. M ( x ) = 2 π x . M(x)=2\pi x.\,
  9. 2 π 2\pi
  10. 2 π 2\pi
  11. 4 π 4\pi
  12. 4 π 4\pi
  13. 8 π / 3 8\pi/3
  14. V S = 4 π - 8 3 π = 4 3 π . V_{S}=4\pi-{8\over 3}\pi={4\over 3}\pi.\,
  15. S r / 3 \scriptstyle Sr/3
  16. 4 π r 3 / 3 \scriptstyle 4\pi r^{3}/3
  17. 4 π r 2 4\pi r^{2}
  18. x 2 + y 2 < 1 y 2 + z 2 < 1 x^{2}+y^{2}<1\;\;\;y^{2}+z^{2}<1
  19. x 2 + y 2 < 1     0 < z < y . x^{2}+y^{2}<1\;\;\;\;\;0<z<y.
  20. 1 - x 2 \scriptstyle\sqrt{1-x^{2}}
  21. 1 / 2 ( 1 - x 2 ) \scriptstyle 1/2(1-x^{2})
  22. - 1 1 1 2 ( 1 - x 2 ) d x \displaystyle\int_{-1}^{1}{1\over 2}(1-x^{2})\,dx
  23. x 2 / 2 \scriptstyle x^{2}/2
  24. x 2 < 1 - y 2 \scriptstyle x^{2}<1-y^{2}
  25. z 2 < 1 - y 2 \scriptstyle z^{2}<1-y^{2}
  26. 2 1 - y 2 \scriptstyle 2\sqrt{1-y^{2}}
  27. - 1 1 4 ( 1 - y 2 ) d y . \displaystyle\int_{-1}^{1}4(1-y^{2})\,dy.

The_Sand_Reckoner.html

  1. ( 10 8 ) ( 10 8 ) = 10 8 10 8 (10^{8})^{(10^{8})}=10^{8\cdot 10^{8}}
  2. ( 10 8 ) ( 10 8 ) (10^{8})^{(10^{8})}
  3. ( ( 10 8 ) ( 10 8 ) ) ( 10 8 ) = 10 8 10 16 . \left((10^{8})^{(10^{8})}\right)^{(10^{8})}=10^{8\cdot 10^{16}}.
  4. 10 a 10 b = 10 a + b 10^{a}10^{b}=10^{a+b}

Thematic_vowel.html

  1. root + suffix stem + ending word \underbrace{\underbrace{\mathrm{root+suffix}}_{\mathrm{stem}}+\mathrm{ending}}% _{\mathrm{word}}

Thermal_conduction.html

  1. k k
  2. q \overrightarrow{q}
  3. k k
  4. - T -\nabla T
  5. q = - k T \overrightarrow{q}=-k{\nabla}T
  6. q \overrightarrow{q}
  7. . k . \big.k\big.
  8. . T . \big.\nabla T\big.
  9. k k
  10. k k
  11. k k
  12. q x = - k d T d x q_{x}=-k\frac{dT}{dx}
  13. S S
  14. Q t = - k S T d A \frac{\partial Q}{\partial t}=-k\oint_{S}{{\nabla}T\cdot\,\overrightarrow{dA}}
  15. . Q t . \big.\frac{\partial Q}{\partial t}\big.
  16. d A \overrightarrow{dA}
  17. . Δ Q Δ t = - k A Δ T Δ x \big.\frac{\Delta Q}{\Delta t}=-kA\frac{\Delta T}{\Delta x}
  18. Δ T \Delta T
  19. Δ x \Delta x
  20. . U = k Δ x , \big.U=\frac{k}{\Delta x},\quad
  21. . Δ Q Δ t = U A ( - Δ T ) . \big.\frac{\Delta Q}{\Delta t}=UA\,(-\Delta T).
  22. . R = 1 U = Δ x k = A ( - Δ T ) Δ Q Δ t . \big.R=\frac{1}{U}=\frac{\Delta x}{k}=\frac{A\,(-\Delta T)}{\frac{\Delta Q}{% \Delta t}}.
  23. . 1 U = 1 U 1 + 1 U 2 + 1 U 3 + \big.\frac{1}{U}=\frac{1}{U_{1}}+\frac{1}{U_{2}}+\frac{1}{U_{3}}+\cdots
  24. . Δ Q Δ t = A ( - Δ T ) Δ x 1 k 1 + Δ x 2 k 2 + Δ x 3 k 3 + . \big.\frac{\Delta Q}{\Delta t}=\frac{A\,(-\Delta T)}{\frac{\Delta x_{1}}{k_{1}% }+\frac{\Delta x_{2}}{k_{2}}+\frac{\Delta x_{3}}{k_{3}}+\cdots}.
  25. R = V / I R=V/I\,\!
  26. G = I / V G=I/V\,\!
  27. R = ρ x / A R=\rho x/A\,\!
  28. G = k A / x G=kA/x\,\!
  29. . U = k A Δ x , \big.U=\frac{kA}{\Delta x},\quad
  30. . Q ˙ = U Δ T \big.\dot{Q}=U\,\Delta T\quad
  31. I = V / R I=V/R\,\!
  32. I = V G . I=VG.\,\!
  33. . R = Δ T Q ˙ , \big.R=\frac{\,\Delta T}{\dot{Q}},\quad
  34. R = V / I . R=V/I.\,\!
  35. r 1 r_{1}
  36. r 2 r_{2}
  37. \ell
  38. T 2 - T 1 T_{2}-T_{1}
  39. A r = 2 π r A_{r}=2\pi r\ell
  40. Q ˙ = - k A r d T d r = - 2 k π r d T d r \dot{Q}=-kA_{r}\frac{\mathrm{d}T}{\mathrm{d}r}=-2k\pi r\ell\frac{\mathrm{d}T}{% \mathrm{d}r}
  41. Q ˙ r 1 r 2 1 r d r = - 2 k π T 1 T 2 d T \dot{Q}\int_{r_{1}}^{r_{2}}\frac{1}{r}\mathrm{d}r=-2k\pi\ell\int_{T_{1}}^{T_{2% }}\mathrm{d}T
  42. Q ˙ = 2 k π T 1 - T 2 ln ( r 2 / r 1 ) \dot{Q}=2k\pi\ell\frac{T_{1}-T_{2}}{\ln(r_{2}/r_{1})}
  43. R c = Δ T Q ˙ = ln ( r 2 / r 1 ) 2 π k R_{c}=\frac{\Delta T}{\dot{Q}}=\frac{\ln(r_{2}/r_{1})}{2\pi k\ell}
  44. Q ˙ = 2 π k r m T 1 - T 2 r 2 - r 1 \dot{Q}=2\pi k\ell r_{m}\frac{T_{1}-T_{2}}{r_{2}-r_{1}}
  45. r m = r 2 - r 1 ln ( r 2 / r 1 ) r_{m}=\frac{r_{2}-r_{1}}{\ln(r_{2}/r_{1})}
  46. r 1 r_{1}
  47. r 2 r_{2}
  48. A = 4 π r 2 . \!A=4\pi r^{2}.
  49. Q ˙ = 4 k π T 1 - T 2 1 / r 1 - 1 / r 2 = 4 k π ( T 1 - T 2 ) r 1 r 2 r 2 - r 1 \dot{Q}=4k\pi\frac{T_{1}-T_{2}}{1/{r_{1}}-1/{r_{2}}}=4k\pi\frac{(T_{1}-T_{2})r% _{1}r_{2}}{r_{2}-r_{1}}
  50. 𝐵𝑖 = h L k \,\textit{Bi}=\frac{hL}{k}
  51. J m 2 s K \frac{J}{m^{2}sK}
  52. q = - h Δ T q=-h\Delta T
  53. T - T f T i - T f = exp [ - h A t ρ C p V ] \frac{T-T_{f}}{T_{i}-T_{f}}=\operatorname{exp}\left[\frac{-hAt}{\rho C_{p}V}\right]
  54. W m 2 K \mathrm{\frac{W}{m^{2}K}}
  55. 𝐹𝑜 = α t L 2 \,\textit{Fo}=\frac{\alpha t}{L^{2}}
  56. T i T_{i}
  57. t = 0 t=0
  58. x = 0 x=0
  59. T = 0 T=0
  60. x = - x=-\infty
  61. x = x=\infty
  62. t = t=\infty
  63. - x -\infty\leq x\leq\infty
  64. T ( x , t ) - T i = T i Δ X 2 π α t exp ( - x 2 4 α t ) T(x,t)-T_{i}=\frac{T_{i}\Delta X}{2\sqrt{\pi\alpha t}}\operatorname{exp}\left(% -\frac{x^{2}}{4\alpha t}\right)
  65. α \alpha
  66. α = k ρ C p \alpha=\frac{k}{\rho C_{p}}

Thermal_history_modelling.html

  1. Q = - k d T d z Q=-k\frac{dT}{dz}
  2. T z , t = T t 0 + Q t 0 z d z k z T_{z,t}=T_{t}^{0}+Q_{t}\int_{0}^{z}\frac{dz^{\prime}}{k_{z^{\prime}}}

Thermal_radiation.html

  1. α + ρ + τ = 1. \alpha+\rho+\tau=1.\,
  2. α \alpha\,
  3. ρ \rho\,
  4. τ \tau\,
  5. λ \lambda\,
  6. ϵ \epsilon\,
  7. α = ϵ = 1. \alpha=\epsilon=1.\,
  8. Q ˙ 1 2 = A 1 E b 1 F 1 2 - A 2 E b 2 F 2 1 \dot{Q}_{1\rightarrow 2}=A_{1}E_{b1}F_{1\rightarrow 2}-A_{2}E_{b2}F_{2% \rightarrow 1}
  9. A 1 F 1 2 = A 2 F 2 1 A_{1}F_{1\rightarrow 2}=A_{2}F_{2\rightarrow 1}
  10. Q ˙ 1 2 = σ A 1 F 1 2 ( T 1 4 - T 2 4 ) \dot{Q}_{1\rightarrow 2}=\sigma A_{1}F_{1\rightarrow 2}(T_{1}^{4}-T_{2}^{4})\!
  11. σ \sigma
  12. F 1 2 F_{1\rightarrow 2}
  13. Q ˙ = σ ( T 1 4 - T 2 4 ) 1 - ϵ 1 A 1 ϵ 1 + 1 A 1 F 1 2 + 1 - ϵ 2 A 2 ϵ 2 \dot{Q}=\dfrac{\sigma(T_{1}^{4}-T_{2}^{4})}{\dfrac{1-\epsilon_{1}}{A_{1}% \epsilon_{1}}+\dfrac{1}{A_{1}F_{1\rightarrow 2}}+\dfrac{1-\epsilon_{2}}{A_{2}% \epsilon_{2}}}
  14. ϵ \epsilon
  15. ν \nu
  16. u ( ν , T ) = 2 h ν 3 c 2 1 e h ν / k B T - 1 u(\nu,T)=\frac{2h\nu^{3}}{c^{2}}\cdot\frac{1}{e^{h\nu/k_{B}T}-1}
  17. u ( λ , T ) = β λ 5 1 e h c / k B T λ - 1 u(\lambda,T)=\frac{\beta}{\lambda^{5}}\cdot\frac{1}{e^{hc/k_{B}T\lambda}-1}
  18. β \beta
  19. E = h ν E=h\nu
  20. ν \nu
  21. P = σ A T 4 P=\sigma\cdot A\cdot T^{4}
  22. σ \sigma
  23. A A
  24. λ \lambda\,
  25. λ m a x = b T \lambda_{max}=\frac{b}{T}
  26. ϵ ( ν ) \epsilon(\nu)
  27. ϵ \epsilon
  28. P = ϵ σ A T 4 P=\epsilon\cdot\sigma\cdot A\cdot T^{4}
  29. h h\,
  30. b b\,
  31. k B k_{B}\,
  32. σ \sigma\,
  33. c c\,
  34. T T\,
  35. A A\,

Thermionic_emission.html

  1. J = A G T 2 e - W k T J=A_{\mathrm{G}}T^{2}\mathrm{e}^{-W\over kT}
  2. A G = λ R A 0 A_{\mathrm{G}}=\;\lambda_{\mathrm{R}}A_{0}
  3. A 0 = 4 π m k 2 e h 3 = 1.20173 × 10 6 A m - 2 K - 2 A_{0}={4\pi mk^{2}e\over h^{3}}=1.20173\times 10^{6}\,\mathrm{A\,m^{-2}\,K^{-2}}
  4. J = ( 1 - r av ) A 0 T 2 e - W k T J=(1-r_{\mathrm{av}})A_{0}T^{2}\mathrm{e}^{-W\over kT}
  5. A G = λ B ( 1 - r av ) A 0 A_{\mathrm{G}}=\lambda_{\mathrm{B}}(1-r_{\mathrm{av}})A_{0}
  6. J ( F , T , W ) = A G T 2 e - ( W - Δ W ) k T J(F,T,W)=A_{\mathrm{G}}T^{2}e^{-(W-\Delta W)\over kT}
  7. Δ W = e 3 F 4 π ϵ 0 , \Delta W=\sqrt{e^{3}F\over 4\pi\epsilon_{0}},

Thermoacoustic_heat_engine.html

  1. \Iota = T m T c r i t \Iota=\frac{\nabla T_{m}}{\nabla T_{crit}}
  2. T m = Δ T m Δ x s t a c k \nabla T_{m}=\frac{\Delta T_{m}}{\Delta x_{stack}}
  3. η = η c \Iota \eta=\frac{\eta_{c}}{\Iota}
  4. C O P = \Iota C O P c COP=\Iota\cdot COP_{c}

Thermodynamic_activity.html

  1. a i = e μ i - μ i R T a_{i}=e^{\frac{\mu_{i}-\mu^{\ominus}_{i}}{RT}}
  2. μ i = μ i + R T ln a i \mu_{i}=\mu_{i}^{\ominus}+RT\ln{a_{i}}
  3. a i = γ x , i x i = γ b , i b i b = γ w , i w i = γ c , i c i c = γ ρ , i ρ i ρ a_{i}=\gamma_{x,i}x_{i}\ =\gamma_{b,i}\frac{b_{i}}{b^{\ominus}}\,=\gamma_{w,i}% w_{i}\ =\gamma_{c,i}\frac{c_{i}}{c^{\ominus}}\,=\gamma_{\rho,i}\frac{\rho_{i}}% {\rho^{\ominus}}\,
  4. a i = f i x i a_{i}=f_{i}x_{i}\,
  5. a i = f i p = ϕ i y i p p a_{i}=\frac{f_{i}}{p^{\ominus}}=\phi_{i}y_{i}\frac{p}{p^{\ominus}}
  6. x i = n i n x_{i}=\frac{n_{i}}{n}
  7. i x i = 1 \sum_{i}x_{i}=1\,
  8. a i = f i x i a_{i}=f_{i}x_{i}\,
  9. a c , i = γ c , i c i c a_{c,i}=\gamma_{c,i}\,\frac{c_{i}}{c^{\ominus}}
  10. a b , i = γ b , i b i b a_{b,i}=\gamma_{b,i}\,\frac{b_{i}}{b^{\ominus}}
  11. b = b ( 1 + a ) b^{\prime}=b(1+a)\,
  12. μ i = μ i + R T ln a i \mu_{i}=\mu_{i}^{\ominus}+RT\ln{a_{i}}
  13. a i = c i c a_{i}=\frac{c_{i}}{c^{\ominus}}
  14. a i = p i p a_{i}=\frac{p_{i}}{p^{\ominus}}

Thermodynamic_equilibrium.html

  1. A = U - T S A=U-TS
  2. G = U - T S + P V G=U-TS+PV

Thermodynamic_potential.html

  1. U U
  2. U U
  3. T T
  4. S S
  5. p p
  6. V V
  7. F F
  8. A A
  9. i i
  10. i i
  11. U U
  12. Δ U ΔU
  13. Δ F ΔF
  14. Δ G ΔG
  15. Δ H ΔH
  16. S S
  17. U U
  18. T T
  19. F F
  20. p p
  21. H H
  22. T T
  23. p p
  24. G G
  25. T T
  26. S S
  27. P P
  28. V V
  29. U [ μ j ] = U - μ j N j U[\mu_{j}]=U-\mu_{j}N_{j}\,
  30. S , V , { N i j } , μ j ~{}~{}~{}~{}~{}S,V,\{N_{i\neq j}\},\mu_{j}\,
  31. F [ μ j ] = U - T S - μ j N j F[\mu_{j}]=U-TS-\mu_{j}N_{j}\,
  32. T , V , { N i j } , μ j ~{}~{}~{}~{}~{}T,V,\{N_{i\neq j}\},\mu_{j}\,
  33. H [ μ j ] = U + p V - μ j N j H[\mu_{j}]=U+pV-\mu_{j}N_{j}\,
  34. S , p , { N i j } , μ j ~{}~{}~{}~{}~{}S,p,\{N_{i\neq j}\},\mu_{j}\,
  35. G [ μ j ] = U + p V - T S - μ j N j G[\mu_{j}]=U+pV-TS-\mu_{j}N_{j}\,
  36. T , p , { N i j } , μ j ~{}~{}~{}~{}~{}T,p,\{N_{i\neq j}\},\mu_{j}\,
  37. U [ μ 1 , μ 2 ] = U - μ 1 N 1 - μ 2 N 2 U[\mu_{1},\mu_{2}]=U-\mu_{1}N_{1}-\mu_{2}N_{2}
  38. D D
  39. U U
  40. d U = δ Q - δ W + i μ i d N i \mathrm{d}U=\delta Q-\delta W+\sum_{i}\mu_{i}\,\mathrm{d}N_{i}
  41. δ Q δQ
  42. δ W δW
  43. i i
  44. i i
  45. δ Q δQ
  46. δ W δW
  47. d d
  48. δ Q = T d S \delta Q=T\,\mathrm{d}S\,
  49. δ W = p d V \delta W=p\,\mathrm{d}V\,
  50. T T
  51. S S
  52. p p
  53. V V
  54. d U = T d S - p d V + i μ i d N i \mathrm{d}U=T\mathrm{d}S-p\mathrm{d}V+\sum_{i}\mu_{i}\,\mathrm{d}N_{i}\,
  55. U U
  56. S S
  57. V V
  58. d U = T d S - i X i d x i + j μ j d N j dU=T\,dS-\sum_{i}X_{i}\,dx_{i}+\sum_{j}\mu_{j}\,dN_{j}\,
  59. d U \mathrm{d}U\,
  60. = \!\!=\!\!
  61. T d S T\mathrm{d}S\,
  62. - -\,
  63. p d V p\mathrm{d}V\,
  64. + i μ i d N i +\sum_{i}\mu_{i}\,\mathrm{d}N_{i}\,
  65. d F \mathrm{d}F\,
  66. = \!\!=\!\!
  67. - -\,
  68. S d T S\,\mathrm{d}T\,
  69. - -\,
  70. p d V p\mathrm{d}V\,
  71. + i μ i d N i +\sum_{i}\mu_{i}\,\mathrm{d}N_{i}\,
  72. d H \mathrm{d}H\,
  73. = \!\!=\!\!
  74. T d S T\,\mathrm{d}S\,
  75. + +\,
  76. V d p V\mathrm{d}p\,
  77. + i μ i d N i +\sum_{i}\mu_{i}\,\mathrm{d}N_{i}\,
  78. d G \mathrm{d}G\,
  79. = \!\!=\!\!
  80. - -\,
  81. S d T S\,\mathrm{d}T\,
  82. + +\,
  83. V d p V\mathrm{d}p\,
  84. + i μ i d N i +\sum_{i}\mu_{i}\,\mathrm{d}N_{i}\,
  85. d ( p V ) = d H - d U = d G - d F \mathrm{d}(pV)=\mathrm{d}H-\mathrm{d}U=\mathrm{d}G-\mathrm{d}F
  86. d ( T S ) = d U - d F = d H - d G \mathrm{d}(TS)=\mathrm{d}U-\mathrm{d}F=\mathrm{d}H-\mathrm{d}G
  87. Φ Φ
  88. d Φ = i x i d y i \mathrm{d}\Phi=\sum_{i}x_{i}\,\mathrm{d}y_{i}\,
  89. Φ Φ
  90. x j = ( Φ y j ) { y i j } x_{j}=\left(\frac{\partial\Phi}{\partial y_{j}}\right)_{\{y_{i\neq j}\}}
  91. Φ Φ
  92. U U
  93. F F
  94. H H
  95. G G
  96. + T = ( U S ) V , { N i } = ( H S ) p , { N i } +T=\left(\frac{\partial U}{\partial S}\right)_{V,\{N_{i}\}}=\left(\frac{% \partial H}{\partial S}\right)_{p,\{N_{i}\}}
  97. - p = ( U V ) S , { N i } = ( F V ) T , { N i } -p=\left(\frac{\partial U}{\partial V}\right)_{S,\{N_{i}\}}=\left(\frac{% \partial F}{\partial V}\right)_{T,\{N_{i}\}}
  98. + V = ( H p ) S , { N i } = ( G p ) T , { N i } +V=\left(\frac{\partial H}{\partial p}\right)_{S,\{N_{i}\}}=\left(\frac{% \partial G}{\partial p}\right)_{T,\{N_{i}\}}
  99. - S = ( G T ) p , { N i } = ( F T ) V , { N i } -S=\left(\frac{\partial G}{\partial T}\right)_{p,\{N_{i}\}}=\left(\frac{% \partial F}{\partial T}\right)_{V,\{N_{i}\}}
  100. μ j = ( ϕ N j ) X , Y , { N i j } ~{}\mu_{j}=\left(\frac{\partial\phi}{\partial N_{j}}\right)_{X,Y,\{N_{i\neq j}\}}
  101. ϕ ϕ
  102. U U
  103. F F
  104. H H
  105. G G
  106. X , Y , { N j i } {X,Y,\{N_{j\neq i}\}}
  107. - N j = ( U [ μ j ] μ j ) S , V , { N i j } -N_{j}=\left(\frac{\partial U[\mu_{j}]}{\partial\mu_{j}}\right)_{S,V,\{N_{i% \neq j}\}}
  108. D D
  109. D D
  110. Φ Φ
  111. ( y j ( Φ y k ) { y i k } ) { y i j } = ( y k ( Φ y j ) { y i j } ) { y i k } \left(\frac{\partial}{\partial y_{j}}\left(\frac{\partial\Phi}{\partial y_{k}}% \right)_{\{y_{i\neq k}\}}\right)_{\{y_{i\neq j}\}}=\left(\frac{\partial}{% \partial y_{k}}\left(\frac{\partial\Phi}{\partial y_{j}}\right)_{\{y_{i\neq j}% \}}\right)_{\{y_{i\neq k}\}}
  112. ( D 1 ) (\frac{D}{−1)}
  113. D ( D D\frac{(}{D}
  114. U U
  115. F F
  116. H H
  117. G G
  118. ( T V ) S , { N i } = - ( p S ) V , { N i } \left(\frac{\partial T}{\partial V}\right)_{S,\{N_{i}\}}=-\left(\frac{\partial p% }{\partial S}\right)_{V,\{N_{i}\}}
  119. ( T p ) S , { N i } = + ( V S ) p , { N i } \left(\frac{\partial T}{\partial p}\right)_{S,\{N_{i}\}}=+\left(\frac{\partial V% }{\partial S}\right)_{p,\{N_{i}\}}
  120. ( S V ) T , { N i } = + ( p T ) V , { N i } \left(\frac{\partial S}{\partial V}\right)_{T,\{N_{i}\}}=+\left(\frac{\partial p% }{\partial T}\right)_{V,\{N_{i}\}}
  121. ( S p ) T , { N i } = - ( V T ) p , { N i } \left(\frac{\partial S}{\partial p}\right)_{T,\{N_{i}\}}=-\left(\frac{\partial V% }{\partial T}\right)_{p,\{N_{i}\}}
  122. ( T N j ) V , S , { N i j } = ( μ j S ) V , { N i } \left(\frac{\partial T}{\partial N_{j}}\right)_{V,S,\{N_{i\neq j}\}}=\left(% \frac{\partial\mu_{j}}{\partial S}\right)_{V,\{N_{i}\}}
  123. ( N j V ) S , μ j , { N i j } = - ( p μ j ) S , V { N i j } \left(\frac{\partial N_{j}}{\partial V}\right)_{S,\mu_{j},\{N_{i\neq j}\}}=-% \left(\frac{\partial p}{\partial\mu_{j}}\right)_{S,V\{N_{i\neq j}\}}
  124. ( N j N k ) S , V , μ j , { N i j , k } = - ( μ k μ j ) S , V { N i j } \left(\frac{\partial N_{j}}{\partial N_{k}}\right)_{S,V,\mu_{j},\{N_{i\neq j,k% }\}}=-\left(\frac{\partial\mu_{k}}{\partial\mu_{j}}\right)_{S,V\{N_{i\neq j}\}}
  125. U U
  126. U ( { α y i } ) = α U ( { y i } ) U(\{\alpha y_{i}\})=\alpha U(\{y_{i}\})\,
  127. U ( { y i } ) = j y j ( U y j ) { y i j } U(\{y_{i}\})=\sum_{j}y_{j}\left(\frac{\partial U}{\partial y_{j}}\right)_{\{y_% {i\neq j}\}}
  128. U = T S - p V + i μ i N i U=TS-pV+\sum_{i}\mu_{i}N_{i}\,
  129. F = - p V + i μ i N i F=-pV+\sum_{i}\mu_{i}N_{i}\,
  130. H = T S + i μ i N i H=TS+\sum_{i}\mu_{i}N_{i}\,
  131. G = i μ i N i G=\sum_{i}\mu_{i}N_{i}\,
  132. U = T S - P V + i μ i N i U=TS-PV+\sum_{i}\mu_{i}N_{i}\,
  133. d U = T d S - P d V + i μ i d N i dU=TdS-PdV+\sum_{i}\mu_{i}\,dN_{i}
  134. 0 = S d T - V d P + i N i d μ i 0=SdT-VdP+\sum_{i}N_{i}d\mu_{i}\,
  135. I I
  136. I + 1 I+1
  137. Δ Δ
  138. V V
  139. p p
  140. S S
  141. Δ U ΔU
  142. Δ H ΔH
  143. T T
  144. Δ F ΔF
  145. Δ G ΔG
  146. p p
  147. T T

Thermoelectric_cooling.html

  1. W = P I t \ W=PIt

Thévenin's_theorem.html

  1. V Th = R 2 + R 3 ( R 2 + R 3 ) + R 4 V 1 V_{\mathrm{Th}}={R_{2}+R_{3}\over(R_{2}+R_{3})+R_{4}}\cdot V_{\mathrm{1}}
  2. = 1 k Ω + 1 k Ω ( 1 k Ω + 1 k Ω ) + 2 k Ω 15 V ={1\,\mathrm{k}\Omega+1\,\mathrm{k}\Omega\over(1\,\mathrm{k}\Omega+1\,\mathrm{% k}\Omega)+2\,\mathrm{k}\Omega}\cdot 15\,\mathrm{V}
  3. = 1 2 15 V = 7.5 V ={1\over 2}\cdot 15\,\mathrm{V}=7.5\,\mathrm{V}
  4. R Th = R 1 + [ ( R 2 + R 3 ) R 4 ] R_{\mathrm{Th}}=R_{1}+\left[\left(R_{2}+R_{3}\right)\|R_{4}\right]
  5. = 1 k Ω + [ ( 1 k Ω + 1 k Ω ) 2 k Ω ] =1\,\mathrm{k}\Omega+\left[\left(1\,\mathrm{k}\Omega+1\,\mathrm{k}\Omega\right% )\|2\,\mathrm{k}\Omega\right]
  6. = 1 k Ω + ( 1 ( 1 k Ω + 1 k Ω ) + 1 ( 2 k Ω ) ) - 1 = 2 k Ω . =1\,\mathrm{k}\Omega+\left({1\over(1\,\mathrm{k}\Omega+1\,\mathrm{k}\Omega)}+{% 1\over(2\,\mathrm{k}\Omega)}\right)^{-1}=2\,\mathrm{k}\Omega.
  7. R Th = R No R_{\mathrm{Th}}=R_{\mathrm{No}}\!
  8. V Th = I No R No V_{\mathrm{Th}}=I_{\mathrm{No}}R_{\mathrm{No}}\!
  9. I No = V Th / R Th . I_{\mathrm{No}}=V_{\mathrm{Th}}/R_{\mathrm{Th}}.\!
  10. V = V Eq - Z Eq I V=V_{\mathrm{Eq}}-Z_{\mathrm{Eq}}I\!
  11. I I
  12. I I
  13. I I
  14. V Eq V_{\mathrm{Eq}}
  15. Z Eq Z_{\mathrm{Eq}}

Third_law_of_thermodynamics.html

  1. T = 0 T=0
  2. S - S 0 = k B ln Ω S-S_{0}=k_{B}\ln\,\Omega
  3. Ω \Omega
  4. S - S 0 = k B ln Ω = k B ln 1 = 0 S-S_{0}=k_{B}\ln\Omega=k_{B}\ln{1}=0
  5. S - S 0 = S - 0 = 0 S-S_{0}=S-0=0
  6. S = 0 S=0
  7. Δ S = S - S 0 = k B ln Ω \Delta S=S-S_{0}=k_{B}\ln{\Omega}
  8. Δ S = S - S 0 = δ Q T \Delta S=S-S_{0}=\frac{\delta Q}{T}
  9. Δ S = S - S 0 = k B l n ( Ω ) = δ Q T \Delta S=S-S_{0}=k_{B}ln(\Omega)=\frac{\delta Q}{T}
  10. S - 0 = k B ln N = 1.38 × 10 - 23 * ln 3 × 10 22 = 70 × 10 - 23 J / K S-0=k_{B}\ln{N}=1.38\times 10^{-23}*\ln{3\times 10^{22}}=70\times 10^{-23}J/K
  11. δ Q = ϵ = h c λ = 6.62 × 10 - 34 J s * 3 × 10 8 m / s 0.01 m = 2 × 10 - 23 J \delta Q=\epsilon=\frac{hc}{\lambda}=\frac{6.62\times 10^{-34}J\cdot s*3\times 1% 0^{8}m/s}{0.01m}=2\times 10^{-23}J
  12. T = ϵ Δ S = 2 × 10 - 23 J 70 × 10 - 23 J / K = 1 35 K T=\frac{\epsilon}{\Delta S}=\frac{2\times 10^{-23}J}{70\times 10^{-23}J/K}=% \frac{1}{35}K
  13. Δ S = δ Q T . \Delta S=\frac{\delta Q}{T}.
  14. δ Q = C ( T , X ) d T . \delta Q=C(T,X)\mathrm{d}T.
  15. Δ S = C ( T , X ) d T T . \Delta S=\frac{C(T,X)\mathrm{d}T}{T}.
  16. S ( T , X ) = S ( T 0 , X ) + T 0 T C ( T , X ) T d T . S(T,X)=S(T_{0},X)+\int_{T_{0}}^{T}\frac{C(T^{\prime},X)}{T^{\prime}}\mathrm{d}% T^{\prime}.
  17. S ( T , X ) = S ( 0 , X ) + 0 T C ( T , X ) T d T . S(T,X)=S(0,X)+\int_{0}^{T}\frac{C(T^{\prime},X)}{T^{\prime}}\mathrm{d}T^{% \prime}.
  18. S ( 0 , X ) = S ( 0 ) . S(0,X)=S(0).
  19. S ( T , X ) = S ( 0 ) + 0 T C ( T , X ) T d T . S(T,X)=S(0)+\int_{0}^{T}\frac{C(T^{\prime},X)}{T^{\prime}}\mathrm{d}T^{\prime}.
  20. lim T 0 ( S ( T , X ) X ) T = 0. \lim_{T\rightarrow 0}\left(\frac{\partial S(T,X)}{\partial X}\right)_{T}=0.
  21. S ( 0 ) = 0 S(0)=0
  22. S ( T , X ) = 0 T C ( T , X ) T d T . S(T,X)=\int_{0}^{T}\frac{C(T^{\prime},X)}{T^{\prime}}\mathrm{d}T^{\prime}.
  23. T 0 T C ( T , X ) T d T = C 0 α ( T α - T 0 α ) . \int_{T_{0}}^{T}\frac{C(T^{\prime},X)}{T^{\prime}}dT^{\prime}=\frac{C_{0}}{% \alpha}(T^{\alpha}-T_{0}^{\alpha}).
  24. lim T 0 C ( T , X ) = 0. \lim_{T\rightarrow 0}C(T,X)=0.
  25. S ( T , V ) = S ( T 0 , V ) + 3 2 R ln T T 0 . S(T,V)=S(T_{0},V)+\frac{3}{2}R\ln\frac{T}{T_{0}}.
  26. C V = π 2 2 R T T F C_{V}=\frac{\pi^{2}}{2}R\frac{T}{T_{F}}
  27. T F = 1 8 π 2 N A 2 h 2 M R ( 3 π 2 N A V m ) 2 / 3 . T_{F}=\frac{1}{8\pi^{2}}\frac{N_{A}^{2}h^{2}}{MR}\left(\frac{3\pi^{2}N_{A}}{V_% {m}}\right)^{2/3}.
  28. C V = 1.93.. R ( T T B ) 3 / 2 C_{V}=1.93..R\left(\frac{T}{T_{B}}\right)^{3/2}
  29. T B = 1 11.9.. N A 2 h 2 M R ( N A V m ) 2 / 3 . T_{B}=\frac{1}{11.9..}\frac{N_{A}^{2}h^{2}}{MR}\left(\frac{N_{A}}{V_{m}}\right% )^{2/3}.
  30. L = L 0 + C p T L=L_{0}+C_{p}T
  31. S ( T , x ) = S l ( T ) + x ( L 0 T + C p ) S(T,x)=S_{l}(T)+x(\frac{L_{0}}{T}+C_{p})
  32. α V = 1 V m ( V m T ) p . \alpha_{V}=\frac{1}{V_{m}}\left(\frac{\partial V_{m}}{\partial T}\right)_{p}.
  33. ( V m T ) p = - ( S m p ) T \left(\frac{\partial V_{m}}{\partial T}\right)_{p}=-\left(\frac{\partial S_{m}% }{\partial p}\right)_{T}
  34. lim T 0 α V = 0. \lim_{T\rightarrow 0}\alpha_{V}=0.

Thomas_Johann_Seebeck.html

  1. V = a ( T h - T c ) V=a(T_{h}-T_{c})\,\!

Thomas_write_rule.html

  1. [ T 1 T 2 R e a d ( A ) R e a d ( B ) W r i t e ( C ) W r i t e ( C ) C o m m i t C o m m i t ] [ T 1 T 2 R e a d ( A ) R e a d ( B ) W r i t e ( C ) C o m m i t C o m m i t ] \begin{bmatrix}T_{1}&T_{2}\\ &Read(A)\\ Read(B)&\\ &Write(C)\\ Write(C)&\\ Commit&\\ &Commit\end{bmatrix}\Longleftrightarrow\begin{bmatrix}T_{1}&T_{2}\\ &Read(A)\\ Read(B)&\\ &Write(C)\\ &\\ Commit&\\ &Commit\\ \end{bmatrix}

Three-valued_logic.html

  1. A B = def NOT ( A ) OR B A\rightarrow B\ \overset{\underset{\mathrm{def}}{}}{=}\ \mbox{NOT}~{}(A)\ % \mbox{OR}~{}\ B
  2. K {}_{K}
  3. K {}_{K}
  4. Ł {}_{Ł}
  5. Ł {}_{Ł}

Tidal_resonance.html

  1. g h \scriptstyle\sqrt{gh}

Tilde.html

  1. x y x~{}y
  2. x x
  3. y y
  4. x y x~{}y
  5. x y x−y
  6. x y x−y
  7. x y x~{}y
  8. x x
  9. y y
  10. x x
  11. y y
  12. x y x~{}y
  13. x x
  14. y y
  15. x = y x=y
  16. f ( x ) g ( x ) f(x)~{}g(x)
  17. A B C D E F ∆ABC~{}∆DEF
  18. A B C ABC
  19. D E F DEF
  20. f ~ \tilde{f}
  21. ( x 1 , x 2 , x 3 , , x n ) = 𝐱 (x_{1},x_{2},x_{3},\ldots,x_{n})=\underset{{}^{\sim}}{\mathbf{x}}
  22. 𝐲 ~ \tilde{\mathbf{y}}
  23. 𝐲 \mathbf{y}
  24. n ~ \tilde{n}
  25. X X
  26. X x + x ~ X\to x+\tilde{x}
  27. x x
  28. x ~ \tilde{x}

Timbre.html

  1. T 1 = a 1 h = 1 H a h T1=\frac{a_{1}}{\sum_{h=1}^{H}{a_{h}}}
  2. T 2 = a 2 + a 3 + a 4 h = 1 H a h T2=\frac{a_{2}+a_{3}+a_{4}}{\sum_{h=1}^{H}{a_{h}}}
  3. T 3 = h = 5 H a h h = 1 H a h T3=\frac{\sum_{h=5}^{H}{a_{h}}}{\sum_{h=1}^{H}{a_{h}}}

Time-scale_calculus.html

  1. \mathbb{R}
  2. 𝕋 \mathbb{T}
  3. \mathbb{R}
  4. h h\mathbb{Z}
  5. t : t 𝕋 t:t\in\mathbb{T}
  6. t t
  7. σ ( t ) = inf { s 𝕋 : s > t } \sigma(t)=\inf\{s\in\mathbb{T}:s>t\}
  8. ρ ( t ) = sup { s 𝕋 : s < t } \rho(t)=\sup\{s\in\mathbb{T}:s<t\}
  9. μ \mu
  10. μ ( t ) = σ ( t ) - t . \mu(t)=\sigma(t)-t.
  11. t t
  12. σ ( t ) = t \sigma(t)=t
  13. μ ( t ) = 0 \mu(t)=0
  14. t t
  15. ρ ( t ) = t . \rho(t)=t.
  16. t 𝕋 t\in\mathbb{T}
  17. t t
  18. ρ ( t ) = t \rho(t)=t
  19. σ ( t ) = t \sigma(t)=t
  20. ρ ( t ) < t \rho(t)<t
  21. σ ( t ) > t \sigma(t)>t
  22. t 1 t_{1}
  23. t 2 t_{2}
  24. t 3 t_{3}
  25. t 4 t_{4}
  26. t t
  27. t t
  28. t t
  29. t t
  30. f : 𝕋 f:\mathbb{T}\rightarrow\mathbb{R}
  31. f Δ ( t ) f^{\Delta}(t)
  32. ϵ > 0 \epsilon>0
  33. U U
  34. t t
  35. | f ( σ ( t ) ) - f ( s ) - f Δ ( t ) ( σ ( t ) - s ) | ε | σ ( t ) - s | |f(\sigma(t))-f(s)-f^{\Delta}(t)(\sigma(t)-s)|\leq\varepsilon|\sigma(t)-s|
  36. s s
  37. U U
  38. 𝕋 = . \mathbb{T}=\mathbb{R}.
  39. σ ( t ) = t \sigma(t)=t
  40. μ ( t ) = 0 \mu(t)=0
  41. f Δ = f f^{\Delta}=f^{\prime}
  42. 𝕋 = \mathbb{T}=\mathbb{Z}
  43. σ ( t ) = t + 1 \sigma(t)=t+1
  44. μ ( t ) = 1 \mu(t)=1
  45. f Δ = Δ f f^{\Delta}=\Delta f
  46. F ( t ) F(t)
  47. f ( t ) = F Δ ( t ) f(t)=F^{\Delta}(t)
  48. r s f ( t ) Δ ( t ) = F ( s ) - F ( r ) . \int_{r}^{s}f(t)\Delta(t)=F(s)-F(r).
  49. 𝒵 { x [ z ] } = 𝒵 { x [ z + 1 ] } z + 1 \mathcal{Z}^{\prime}\{x[z]\}=\frac{\mathcal{Z}\{x[z+1]\}}{z+1}
  50. μ Δ ( A ) = λ ( ρ - 1 ( A ) ) , \mu^{\Delta}(A)=\lambda(\rho^{-1}(A)),
  51. λ \lambda
  52. ρ \rho
  53. \mathbb{R}
  54. r s f ( t ) Δ t = [ r , s ) f ( t ) d μ Δ ( t ) \int_{r}^{s}f(t)\Delta t=\int_{[r,s)}f(t)d\mu^{\Delta}(t)
  55. f Δ ( t ) = d f d μ Δ ( t ) . f^{\Delta}(t)=\frac{df}{d\mu^{\Delta}}(t).
  56. δ a ( t ) = { 1 μ ( a ) , t = a 0 , t a \delta_{a}^{\mathbb{H}}(t)=\begin{cases}\frac{1}{\mu(a)},&t=a\\ 0,&t\neq a\end{cases}

Time_(Orders_of_magnitude).html

  1. G / c 5 \sqrt{\hbar G/c^{5}}
  2. × 10 50 \times 10^{−}50
  3. {\infty}
  4. π * 10 7 \pi*10^{7}
  5. log 10 year \log_{10}\mbox{ year}~{}
  6. log 10 second \log_{10}\mbox{ second}~{}
  7. log 10 year + 7.50 \log_{10}\mbox{ year}~{}+7.50
  8. 1 year = 10 0 year = 10 0 + 7.50 seconds = 10 0.50 + 7 s = 3.16 * 10 7 s 1\mbox{ year}~{}=10^{0}\mbox{ year}~{}=10^{0+7.50}\mbox{ seconds }~{}=10^{0.50% +7}s=3.16*10^{7}s
  9. ħ / Γ {ħ}/{Γ}

Time_dilation.html

  1. A A
  2. B B
  3. L L
  4. 2 L 2L
  5. 2 L 2L
  6. Δ t = 2 L c . \Delta t=\frac{2L}{c}.
  7. v v
  8. Δ t = 2 D c . \Delta t^{\prime}=\frac{2D}{c}.
  9. D = ( 1 2 v Δ t ) 2 + L 2 . D=\sqrt{\left(\frac{1}{2}v\Delta t^{\prime}\right)^{2}+L^{2}}.
  10. D D
  11. Δ t = 2 L c 1 - v 2 c 2 \Delta t^{\prime}=\frac{\frac{2L}{c}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}
  12. Δ t Δt
  13. Δ t = Δ t 1 - v 2 c 2 \Delta t^{\prime}=\frac{\Delta t}{\sqrt{1-\frac{v^{2}}{c^{2}}}}
  14. Δ t = γ Δ t = Δ t 1 - v 2 c 2 \Delta t^{\prime}=\gamma\,\Delta t=\frac{\Delta t}{\sqrt{1-\frac{v^{2}}{c^{2}}% }}\,
  15. γ = 1 1 - v 2 c 2 . \gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\,.
  16. 1 - v 2 c 2 \scriptstyle\sqrt{1-\frac{v^{2}}{c^{2}}}
  17. d t E 2 = ( 1 - 2 G M i r i c 2 ) d t c 2 - ( 1 - 2 G M i r i c 2 ) - 1 d x 2 + d y 2 + d z 2 c 2 dt\text{E}^{2}=\left(1-\frac{2GM\text{i}}{r\text{i}c^{2}}\right)dt\text{c}^{2}% -\left(1-\frac{2GM\text{i}}{r\text{i}c^{2}}\right)^{-1}\frac{dx^{2}+dy^{2}+dz^% {2}}{c^{2}}\,
  18. v 2 = d x 2 + d y 2 + d z 2 d t c 2 . v^{2}=\frac{dx^{2}+dy^{2}+dz^{2}}{dt\text{c}^{2}}.\,
  19. d t E d t c = 1 - 2 U c 2 - v 2 c 2 - ( c 2 2 U - 1 ) - 1 v 2 c 2 \frac{dt\text{E}}{dt\text{c}}=\sqrt{1-\frac{2U}{c^{2}}-\frac{v^{2}}{c^{2}}-% \left(\frac{c^{2}}{2U}-1\right)^{-1}\frac{{v_{\shortparallel}}^{2}}{c^{2}}}\,
  20. f 0 1 - v / c and f 0 1 + v / c . \frac{f_{0}}{1-v/c}\qquad\,\text{and}\qquad\frac{f_{0}}{1+v/c}.\,
  21. 1 + v / c 1 - v / c f 0 = γ ( 1 + v / c ) f 0 and 1 - v / c 1 + v / c f 0 = γ ( 1 - v / c ) f 0 , \sqrt{\frac{1+v/c}{1-v/c}}f_{0}=\gamma\left(1+v/c\right)f_{0}\qquad\,\text{and% }\qquad\sqrt{\frac{1-v/c}{1+v/c}}f_{0}=\gamma\left(1-v/c\right)f_{0},\,
  22. f detected = f rest ( 1 - v c cos ϕ ) / 1 - v 2 / c 2 f_{\mathrm{detected}}=f_{\mathrm{rest}}{\left(1-\frac{v}{c}\cos\phi\right)/% \sqrt{1-{v^{2}}/{c^{2}}}}
  23. γ 0 = 1 1 - v 0 2 / c 2 , \gamma_{0}=\frac{1}{\sqrt{1-v_{0}^{2}/c^{2}}},
  24. x ( t ) = c 2 g ( 1 + ( g t + v 0 γ 0 ) 2 c 2 - γ 0 ) . x(t)=\frac{c^{2}}{g}\left(\sqrt{1+\frac{\left(gt+v_{0}\gamma_{0}\right)^{2}}{c% ^{2}}}-\gamma_{0}\right).
  25. v ( t ) = g t + v 0 γ 0 1 + ( g t + v 0 γ 0 ) 2 c 2 . v(t)=\frac{gt+v_{0}\gamma_{0}}{\sqrt{1+\frac{\left(gt+v_{0}\gamma_{0}\right)^{% 2}}{c^{2}}}}.
  26. τ ( t ) = τ 0 + 0 t 1 - ( v ( t ) c ) 2 d t . \tau(t)=\tau_{0}+\int_{0}^{t}\sqrt{1-\left(\frac{v(t^{\prime})}{c}\right)^{2}}% dt^{\prime}.
  27. τ ( t ) = c g ln ( g t c + 1 + ( g t c ) 2 ) = c g arsinh ( g t c ) . \tau(t)=\frac{c}{g}\ln\left(\frac{gt}{c}+\sqrt{1+\left(\frac{gt}{c}\right)^{2}% }\right)=\frac{c}{g}\operatorname{arsinh}\left(\frac{gt}{c}\right).
  28. 3 \scriptstyle\sqrt{3}

Time_hierarchy_theorem.html

  1. 𝐃𝐓𝐈𝐌𝐄 ( o ( f ( n ) log f ( n ) ) ) 𝐃𝐓𝐈𝐌𝐄 ( f ( n ) ) \mathbf{DTIME}\left(o\left(\frac{f(n)}{\log f(n)}\right)\right)\subsetneq% \mathbf{DTIME}(f(n))
  2. 𝐍𝐓𝐈𝐌𝐄 ( f ( n ) ) 𝐍𝐓𝐈𝐌𝐄 ( g ( n ) ) \mathbf{NTIME}(f(n))\subsetneq\mathbf{NTIME}(g(n))
  3. f : f:\mathbb{N}\rightarrow\mathbb{N}
  4. n n\in\mathbb{N}
  5. 𝐃𝐓𝐈𝐌𝐄 ( f ( n ) ) 𝐃𝐓𝐈𝐌𝐄 ( f ( n ) 2 ) . \mathbf{DTIME}(f(n))\subsetneq\mathbf{DTIME}\left(f(n)^{2}\right).
  6. 𝐃𝐓𝐈𝐌𝐄 ( o ( f ( n ) log f ( n ) ) ) 𝐃𝐓𝐈𝐌𝐄 ( f ( n ) ) . \mathbf{DTIME}\left(o\left(\frac{f(n)}{\log f(n)}\right)\right)\subsetneq% \mathbf{DTIME}\left(f(n)\right).
  7. o ( n 2 log n 2 ) . o\left(\frac{n^{2}}{\log{n^{2}}}\right).
  8. H f = { ( [ M ] , x ) | M accepts x in f ( | x | ) steps } . H_{f}=\left\{([M],x)\ |\ M\ \mbox{accepts}~{}\ x\ \mbox{in}~{}\ f(|x|)\ \mbox{% steps}~{}\right\}.
  9. H f 𝐓𝐈𝐌𝐄 ( f ( m ) 3 ) . H_{f}\in\mathbf{TIME}(f(m)^{3}).
  10. H f 𝐓𝐈𝐌𝐄 ( f ( m 2 ) ) H_{f}\notin\mathbf{TIME}(f(\left\lfloor\tfrac{m}{2}\right\rfloor))
  11. 𝐓𝐈𝐌𝐄 ( f ( m 2 ) ) . \mathbf{TIME}(f(\left\lfloor\tfrac{m}{2}\right\rfloor)).
  12. 𝐓𝐈𝐌𝐄 ( f ( m 2 ) ) = 𝐓𝐈𝐌𝐄 ( f ( 2 n + 1 2 ) ) = 𝐓𝐈𝐌𝐄 ( f ( n ) ) . \mathbf{TIME}(f(\left\lfloor\tfrac{m}{2}\right\rfloor))=\mathbf{TIME}(f(\left% \lfloor\tfrac{2n+1}{2}\right\rfloor))=\mathbf{TIME}(f(n)).
  13. H f 𝐓𝐈𝐌𝐄 ( f ( m 2 ) ) . H_{f}\notin\mathbf{TIME}(f(\left\lfloor\tfrac{m}{2}\right\rfloor)).
  14. H f 𝐓𝐈𝐌𝐄 ( f ( m ) 3 ) . H_{f}\in\mathbf{TIME}(f(m)^{3}).
  15. H f 𝐓𝐈𝐌𝐄 ( f ( m ) log f ( m ) ) H_{f}\in\mathbf{TIME}(f(m)\log f(m))
  16. \subsetneq
  17. \subsetneq
  18. \subsetneq
  19. \subsetneq
  20. \subsetneq
  21. \subsetneq
  22. 𝐏 𝐄𝐗𝐏𝐓𝐈𝐌𝐄 \mathbf{P}\subsetneq\mathbf{EXPTIME}
  23. 𝐏 𝐃𝐓𝐈𝐌𝐄 ( 2 n ) 𝐃𝐓𝐈𝐌𝐄 ( 2 2 n ) 𝐄𝐗𝐏𝐓𝐈𝐌𝐄 \mathbf{P}\subseteq\mathbf{DTIME}(2^{n})\subsetneq\mathbf{DTIME}(2^{2n})% \subseteq\mathbf{EXPTIME}
  24. 𝐃𝐓𝐈𝐌𝐄 ( 2 n ) 𝐃𝐓𝐈𝐌𝐄 ( o ( 2 2 n 2 n ) ) 𝐃𝐓𝐈𝐌𝐄 ( 2 2 n ) \mathbf{DTIME}\left(2^{n}\right)\subseteq\mathbf{DTIME}\left(o\left(\frac{2^{2% n}}{2n}\right)\right)\subsetneq\mathbf{DTIME}(2^{2n})

Time_value_of_money.html

  1. F V FV
  2. r r
  3. P V PV
  4. P V = F V ( 1 + r ) PV=\frac{FV}{(1+r)}
  5. F V = P V ( 1 + i ) n FV\ =\ PV\cdot(1+i)^{n}
  6. P V = F V ( 1 + i ) n PV\ =\ \frac{FV}{(1+i)^{n}}
  7. P V = t = 1 n F V t ( 1 + i ) t PV\ =\ \sum_{t=1}^{n}\frac{FV_{t}}{(1+i)^{t}}
  8. P V ( A ) = A i [ 1 - 1 ( 1 + i ) n ] PV(A)\,=\,\frac{A}{i}\cdot\left[{1-\frac{1}{\left(1+i\right)^{n}}}\right]
  9. P V = A ( i - g ) [ 1 - ( 1 + g 1 + i ) n ] PV\,=\,{A\over(i-g)}\left[1-\left({1+g\over 1+i}\right)^{n}\right]
  10. P V = A × n 1 + i PV\,=\,{A\times n\over 1+i}
  11. P V ( P ) = A i PV(P)\ =\ {A\over i}
  12. A = P ( 1 + r / n ) n t A=P(1+r/n)^{nt}
  13. = 1000 × ( 1 + .069 / 4 ) ( 5 y r s × 4 q t r s i n a y e a r ) = 1000 × ( 1 + 0.069 / 4 ) 20 1407.84 =1000\times(1+.069/4)^{({5\ yrs}\ \times\ 4\ {qtrs\ in\ a\ year})}=1000\times(% 1+0.069/4)^{20}\approx 1407.84
  14. F V ( A ) = A ( 1 + i ) n - 1 i FV(A)\,=\,A\cdot\frac{\left(1+i\right)^{n}-1}{i}
  15. F V ( A ) = A ( 1 + i ) n - ( 1 + g ) n i - g FV(A)\,=\,A\cdot\frac{\left(1+i\right)^{n}-\left(1+g\right)^{n}}{i-g}
  16. F V ( A ) = A n ( 1 + i ) n - 1 FV(A)\,=\,A\cdot n(1+i)^{n-1}
  17. F = P ( 1 + i ) n F=P\cdot(1+i)^{n}
  18. P = F ( 1 + i ) - n P=F\cdot(1+i)^{-n}
  19. A = F i ( 1 + i ) n - 1 A=F\cdot\frac{i}{(1+i)^{n}-1}
  20. A = P i ( 1 + i ) n ( 1 + i ) n - 1 A=P\cdot\frac{i(1+i)^{n}}{(1+i)^{n}-1}
  21. F = A ( 1 + i ) n - 1 i F=A\cdot\frac{(1+i)^{n}-1}{i}
  22. P = A ( 1 + i ) n - 1 i ( 1 + i ) n P=A\cdot\frac{(1+i)^{n}-1}{i(1+i)^{n}}
  23. F = G ( 1 + i ) n - i n - 1 i 2 F=G\cdot\frac{(1+i)^{n}-in-1}{i^{2}}
  24. P = G ( 1 + i ) n - i n - 1 i 2 ( 1 + i ) n P=G\cdot\frac{(1+i)^{n}-in-1}{i^{2}(1+i)^{n}}
  25. A = G [ 1 i - n ( 1 + i ) n - 1 ] A=G\cdot\left[\frac{1}{i}-\frac{n}{(1+i)^{n}-1}\right]
  26. F = D ( 1 + g ) n - ( 1 + i ) n g - i F=D\cdot\frac{(1+g)^{n}-(1+i)^{n}}{g-i}
  27. F = D n ( 1 + i ) n 1 + g F=D\cdot\frac{n(1+i)^{n}}{1+g}
  28. P = D ( 1 + g 1 + i ) n - 1 g - i P=D\cdot\frac{\left({1+g\over 1+i}\right)^{n}-1}{g-i}
  29. P = D n 1 + g P=D\cdot\frac{n}{1+g}
  30. F V = C ( 1 + i ) n - m FV\ =C(1+i)^{n-m}
  31. F V A = m = 1 n C ( 1 + i ) n - m = k = 0 n - 1 C ( 1 + i ) k FVA\ =\sum_{m=1}^{n}C(1+i)^{n-m}\ =\sum_{k=0}^{n-1}C(1+i)^{k}
  32. F V A = C ( 1 - ( 1 + i ) n ) 1 - ( 1 + i ) = C ( 1 - ( 1 + i ) n ) - i FVA\ =\frac{C(1-(1+i)^{n})}{1-(1+i)}\ =\frac{C(1-(1+i)^{n})}{-i}
  33. ( 1 + i ) n (1+i)^{n}
  34. P V A = F V A ( 1 + i ) n = C i ( 1 - 1 ( 1 + i ) n ) PVA\ =\frac{FVA}{(1+i)^{n}}=\frac{C}{i}\left(1-\frac{1}{(1+i)^{n}}\right)
  35. Principal × i = C \,\text{Principal}\times i=C
  36. Principal = C / i + g o a l \,\text{Principal}=C/i+{goal}
  37. F V = P V ( 1 + i ) n FV=PV(1+i)^{n}
  38. P V = C / i PV=C/i
  39. F V = C / i + F V A FV=C/i+FVA
  40. C i + F V A = C i ( 1 + i ) n \frac{C}{i}+FVA=\frac{C}{i}(1+i)^{n}
  41. F V A = C i [ ( 1 + i ) n - 1 ] FVA=\frac{C}{i}\left[\left(1+i\right)^{n}-1\right]
  42. ( 1 - 1 ( 1 + i ) n ) \left({1-{1\over{(1+i)^{n}}}}\right)
  43. C i {C\over i}
  44. P = F × ( P / F ) = F × 1 ( 1 + i ) n = 100 1.05 = 95.24 P=F\times(P/F)=F\times{1\over(1+i)^{n}}=\frac{100}{1.05}=95.24
  45. n = 10 years × 12 months per year = 120 months n=10{\rm\ years}\times 12{\rm\ months\ per\ year}=120{\rm\ months}
  46. i = 6 % per year 12 months per year = 0.5 % per month i={6{\rm\%\ per\ year}\over 12{\rm\ months\ per\ year}}=0.5{\rm\%\ per\ month}
  47. A = P × ( A / P ) = P × i ( 1 + i ) n ( 1 + i ) n - 1 = $ 200 , 000 × 0.005 ( 1.005 ) 120 ( 1.005 ) 120 - 1 A\ =\ P\times\left(A/P\right)\ =\ P\times{i(1+i)^{n}\over(1+i)^{n}-1}\ =\ \$20% 0,000\times{0.005(1.005)^{120}\over(1.005)^{120}-1}
  48. $ 200 , 000 × 0.01110205 $ 2 , 220.41 per month \approx\$200,000\times 0.01110205\ \approx\ \$2,220.41{\rm\ per\ month}
  49. x = b y x\ =\ b^{y}
  50. y = log ( x ) log ( b ) y\ =\ {\log(x)\over\log(b)}
  51. y = log ( F V P V ) log ( 1 + i ) = log ( 200 100 ) log ( 1.10 ) = 7.27 ( y e a r s ) y\ =\ {\log({FV\over PV})\over\log(1+i)}\ =\ {\log({200\over 100})\over\log(1.% 10)}\ =\ 7.27{(years)}
  52. i = ( F V P V ) 1 n - 1 = ( 200 100 ) 1 5 - 1 = 2 0.20 - 1 = 0.15 = 15 % i\ =\ \left({FV\over PV}\right)^{1\over n}-1\ =\ \left({200\over 100}\right)^{% 1\over 5}-1\ =\ 2^{0.20}-1\ =\ 0.15\ =\ 15\%
  53. P V A = A 1 - 1 ( 1 + i ) n i = 1000 1 - 1 ( 1 + .07 ) 20 .07 = 1000 1 - 0.258 .07 = 1000 × 10.594 $ 10 , 594 PVA\,=\,A\cdot\frac{1-\frac{1}{\left(1+i\right)^{n}}}{i}\ =\ 1000\cdot\frac{1-% \frac{1}{\left(1+.07\right)^{20}}}{.07}\ =\ 1000\cdot{1-0.258\over.07}\ =\ 100% 0\times 10.594\ \approx\ \$10,594
  54. F V = P V ( 1 + i ) n = $ 10 , 594 × ( 1 + .07 ) 20 $ 10 , 594 × 3.87 = $ 40 , 995 FV\ =\ PV(1+i)^{n}\ =\$10,594\times(1+.07)^{20}\ \approx\$10,594\times 3.87\ =% \$40,995
  55. F V = A 1 - 1 ( 1 + i ) n i ( 1 + i ) n = A ( 1 + i ) n - 1 i FV\,=\,A\cdot\frac{1-\frac{1}{\left(1+i\right)^{n}}}{i}\cdot(1+i)^{n}\,=\,A% \cdot\frac{\left(1+i\right)^{n}-1}{i}
  56. P E = 1 i = P V A {P\over E}\ =\ {1\over i}\ =\ {PV\over A}
  57. 1 P / E = i {1\over P/E}\ =\ i
  58. P E = 1 ( i - g ) {P\over E}\ =\ {1\over(i-g)}
  59. g = i - E P g\ =\ i-{E\over P}
  60. PV = FV e - r t \,\text{PV}=\,\text{FV}\cdot e^{-rt}
  61. PV = FV exp ( - 0 T r ( t ) d t ) \,\text{PV}=\,\text{FV}\cdot\exp\left(-\int_{0}^{T}r(t)\,dt\right)
  62. P V = A ( 1 - e - r t ) e r - 1 \ PV\ =\ {A(1-e^{-rt})\over e^{r}-1}
  63. P V = A e r - 1 \ PV\ =\ {A\over e^{r}-1}
  64. P V = A e - g ( 1 - e - ( r - g ) t ) e ( r - g ) - 1 \ PV\ =\ {Ae^{-g}(1-e^{-(r-g)t})\over e^{(r-g)}-1}
  65. P V = A e - g e ( r - g ) - 1 \ PV\ =\ {Ae^{-g}\over e^{(r-g)}-1}
  66. P V = 1 - e ( - r t ) r \ PV\ =\ {1-e^{(-rt)}\over r}
  67. \mathcal{L}
  68. := - t + r ( t ) . \mathcal{L}:=-\partial_{t}+r(t).
  69. f = - t f ( t ) + r ( t ) f ( t ) . \mathcal{L}f=-\partial_{t}f(t)+r(t)f(t).
  70. V = f \mathcal{L}V=f
  71. δ u ( t ) := δ ( t - u ) . \delta_{u}(t):=\delta(t-u).
  72. b ( t ; u ) := H ( u - t ) exp ( - t u r ( v ) d v ) b(t;u):=H(u-t)\cdot\exp\left(-\int_{t}^{u}r(v)\,dv\right)
  73. ; u ;u
  74. \textstyle{\int}
  75. t u \textstyle{\int_{t}^{u}}
  76. H ( u - t ) = 1 if t < u , 0 if t > u H(u-t)=1\,\text{ if }t<u,0\,\text{ if }t>u
  77. r ( v ) r , r(v)\equiv r,
  78. b ( t ; u ) = H ( u - t ) e - ( u - t ) r = { e - ( u - t ) r t < u 0 t > u , b(t;u)=H(u-t)\cdot e^{-(u-t)r}=\begin{cases}e^{-(u-t)r}&t<u\\ 0&t>u,\end{cases}
  79. ( u - t ) (u-t)
  80. T = + T=+\infty
  81. V ( t ; T ) V(t;T)
  82. V ( t ; T ) = t T f ( u ) b ( t ; u ) d u . V(t;T)=\int_{t}^{T}f(u)b(t;u)\,du.

Timeline_of_nuclear_fusion.html

  1. Q e q Q_{eq}

Timestamp-based_concurrency_control.html

  1. T i T_{i}
  2. A i x A_{ix}
  3. A i 1 A_{i1}
  4. T S ( T i ) = N O W ( ) TS(T_{i})=NOW()
  5. D E P ( T i ) = [ ] DEP(T_{i})=[]
  6. O L D ( T i ) = [ ] OLD(T_{i})=[]
  7. ( O j ) (O_{j})
  8. R T S ( O j ) RTS(O_{j})
  9. W T S ( O j ) WTS(O_{j})
  10. T i T_{i}
  11. A i x A_{ix}
  12. A i x A_{ix}
  13. O j O_{j}
  14. W T S ( O j ) > T S ( T i ) WTS(O_{j})>TS(T_{i})
  15. D E P ( T i ) . add ( W T S ( O j ) ) DEP(T_{i}).\mathrm{add}(WTS(O_{j}))
  16. R T S ( O j ) = max ( R T S ( O j ) , T S ( T i ) ) RTS(O_{j})=\max(RTS(O_{j}),TS(T_{i}))
  17. A i x A_{ix}
  18. O j O_{j}
  19. R T S ( O j ) > T S ( T i ) RTS(O_{j})>TS(T_{i})
  20. W T S ( O j ) > T S ( T i ) WTS(O_{j})>TS(T_{i})
  21. O L D ( T i ) . add ( O j , W T S ( O j ) ) OLD(T_{i}).\mathrm{add}(O_{j},WTS(O_{j}))
  22. W T S ( O j ) = T S ( T i ) WTS(O_{j})=TS(T_{i})
  23. O j O_{j}
  24. D E P ( T i ) DEP(T_{i})
  25. D E P ( T i ) DEP(T_{i})
  26. ( old O j , old W T S ( O j ) ) (\mathrm{old}O_{j},\mathrm{old}WTS(O_{j}))
  27. O L D ( T i ) OLD(T_{i})
  28. W T S ( O j ) WTS(O_{j})
  29. T S ( T i ) TS(T_{i})
  30. O j = old O j O_{j}=\mathrm{old}O_{j}
  31. W T S ( O j ) = old W T S ( O j ) WTS(O_{j})=\mathrm{old}WTS(O_{j})
  32. T 1 T_{1}
  33. T 2 T_{2}
  34. W 1 ( x ) R 2 ( x ) W 2 ( y ) C 2 R 1 ( z ) C 1 W_{1}(x)\;R_{2}(x)\;W_{2}(y)\;C_{2}\;R_{1}(z)\;C_{1}
  35. T 2 T_{2}

Tissue_expansion.html

  1. F = F e F g F=F^{e}\cdot F^{g}\,
  2. F e F^{e}
  3. F g F^{g}
  4. F g = θ g 𝕀 + [ 1 - θ g ] n 0 n 0 F^{g}=\sqrt{\theta^{g}}\mathbb{I}+[1-\sqrt{\theta^{g}}]n_{0}\otimes n_{0}\,
  5. n 0 n_{0}
  6. θ g = d e t ( F g ) = J g \theta^{g}=det(F^{g})=J^{g}

Toeplitz_matrix.html

  1. [ a b c d e f a b c d g f a b c h g f a b i h g f a ] . \begin{bmatrix}a&b&c&d&e\\ f&a&b&c&d\\ g&f&a&b&c\\ h&g&f&a&b\\ i&h&g&f&a\end{bmatrix}.
  2. A = [ a 0 a - 1 a - 2 a - n + 1 a 1 a 0 a - 1 a 2 a 1 a - 1 a - 2 a 1 a 0 a - 1 a n - 1 a 2 a 1 a 0 ] A=\begin{bmatrix}a_{0}&a_{-1}&a_{-2}&\ldots&\ldots&a_{-n+1}\\ a_{1}&a_{0}&a_{-1}&\ddots&&\vdots\\ a_{2}&a_{1}&\ddots&\ddots&\ddots&\vdots\\ \vdots&\ddots&\ddots&\ddots&a_{-1}&a_{-2}\\ \vdots&&\ddots&a_{1}&a_{0}&a_{-1}\\ a_{n-1}&\ldots&\ldots&a_{2}&a_{1}&a_{0}\end{bmatrix}
  3. A i , j = A i + 1 , j + 1 = a i - j . A_{i,j}=A_{i+1,j+1}=a_{i-j}.
  4. A x = b Ax=b
  5. n × n n\times n
  6. h h
  7. x x
  8. y = h x = [ h 1 0 0 0 h 2 h 1 h 3 h 2 0 0 h 3 h 1 0 h m - 1 h 2 h 1 h m h m - 1 h 2 0 h m h m - 2 0 0 h m - 1 h m - 2 h m h m - 1 0 0 0 h m ] [ x 1 x 2 x 3 x n ] y=h\ast x=\begin{bmatrix}h_{1}&0&\ldots&0&0\\ h_{2}&h_{1}&\ldots&\vdots&\vdots\\ h_{3}&h_{2}&\ldots&0&0\\ \vdots&h_{3}&\ldots&h_{1}&0\\ h_{m-1}&\vdots&\ldots&h_{2}&h_{1}\\ h_{m}&h_{m-1}&\vdots&\vdots&h_{2}\\ 0&h_{m}&\ldots&h_{m-2}&\vdots\\ 0&0&\ldots&h_{m-1}&h_{m-2}\\ \vdots&\vdots&\vdots&h_{m}&h_{m-1}\\ 0&0&0&\ldots&h_{m}\end{bmatrix}\begin{bmatrix}x_{1}\\ x_{2}\\ x_{3}\\ \vdots\\ x_{n}\end{bmatrix}
  9. y T = [ h 1 h 2 h 3 h m - 1 h m ] [ x 1 x 2 x 3 x n 0 0 0 0 0 x 1 x 2 x 3 x n 0 0 0 0 0 x 1 x 2 x 3 x n 0 0 0 0 0 0 x 1 x n - 2 x n - 1 x n 0 0 0 0 x 1 x n - 2 x n - 1 x n ] . y^{T}=\begin{bmatrix}h_{1}&h_{2}&h_{3}&\ldots&h_{m-1}&h_{m}\end{bmatrix}\begin% {bmatrix}x_{1}&x_{2}&x_{3}&\ldots&x_{n}&0&0&0&\ldots&0\\ 0&x_{1}&x_{2}&x_{3}&\ldots&x_{n}&0&0&\ldots&0\\ 0&0&x_{1}&x_{2}&x_{3}&\ldots&x_{n}&0&\ldots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ldots&\vdots&\vdots&\ldots&0\\ 0&\ldots&0&0&x_{1}&\ldots&x_{n-2}&x_{n-1}&x_{n}&\vdots\\ 0&\ldots&0&0&0&x_{1}&\ldots&x_{n-2}&x_{n-1}&x_{n}\end{bmatrix}.
  10. × \mathbb{Z}\times\mathbb{Z}
  11. A A
  12. 2 \ell^{2}
  13. A = [ a 0 a - 1 a - 2 a - 3 a 1 a 0 a - 1 a - 2 a 2 a 1 a 0 a - 1 a 3 a 2 a 1 a 0 ] . A=\begin{bmatrix}\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\ \ldots&a_{0}&a_{-1}&a_{-2}&a_{-3}&\ldots\\ \ldots&a_{1}&a_{0}&a_{-1}&a_{-2}&\ldots\\ \ldots&a_{2}&a_{1}&a_{0}&a_{-1}&\ldots\\ \ldots&a_{3}&a_{2}&a_{1}&a_{0}&\ldots\\ \ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\ \end{bmatrix}.
  14. A A
  15. f f
  16. f f
  17. A A
  18. A A
  19. L L^{\infty}
  20. a i = a i + n a_{i}=a_{i+n}

Tonnage.html

  1. G T = K V GT=K\cdot V
  2. K = 0.2 + 0.02 log 10 V K=0.2+0.02\cdot\log_{10}V

Top_quark.html

  1. 1 / 2 {1}/{2}
  2. 2 / 3 {2}/{3}
  3. t ¯ \overline{t}
  4. 1 / 2 {1}/{2}
  5. 2 / 3 {2}/{3}
  6. 2 / 3 {2}/{3}
  7. = y t h q u c y t v 2 ( 1 + h 0 / v ) u u c \mathcal{L}=y\text{t}hqu^{c}\rightarrow\frac{y\text{t}v}{\sqrt{2}}(1+h^{0}/v)% uu^{c}
  8. y t = 2 m t / v 1 y\text{t}=\sqrt{2}m\text{t}/v\simeq 1
  9. v = v=
  10. μ μ y t y t 16 π 2 ( 9 2 y t 2 - 8 g 3 2 - 9 4 g 2 2 - 17 20 g 1 2 ) , \mu\frac{\partial}{\partial\mu}y\text{t}\approx\frac{y\text{t}}{16\pi^{2}}% \left(\frac{9}{2}y\text{t}^{2}-8g_{3}^{2}-\frac{9}{4}g_{2}^{2}-\frac{17}{20}g_% {1}^{2}\right),
  11. μ μ
  12. μ μ y t y t 16 π 2 ( 6 y t 2 + y b 2 - 16 3 g 3 2 - 3 g 2 2 - 13 15 g 1 2 ) , \mu\frac{\partial}{\partial\mu}y\text{t}\approx\frac{y\text{t}}{16\pi^{2}}% \left(6y\text{t}^{2}+y\text{b}^{2}-\frac{16}{3}g_{3}^{2}-3g_{2}^{2}-\frac{13}{% 15}g_{1}^{2}\right),

Topologist's_sine_curve.html

  1. T = { ( x , sin 1 x ) : x ( 0 , 1 ] } { ( 0 , 0 ) } . T=\left\{\left(x,\sin\frac{1}{x}\right):x\in(0,1]\right\}\cup\{(0,0)\}.
  2. { ( 0 , y ) y [ - 1 , 1 ] } \{(0,y)\mid y\in[-1,1]\}
  3. { ( x , 1 ) x [ 0 , 1 ] } \{(x,1)\mid x\in[0,1]\}

Torsion_subgroup.html

  1. ( A , + ) (A,+)\;
  2. A T p = { g A | n , p n g = 0 } . A_{T_{p}}=\{g\in A\;|\;\exists n\in\mathbb{N}\;,p^{n}g=0\}.\;
  3. A T p P A T p . A_{T}\cong\bigoplus_{p\in P}A_{T_{p}}.\;

Torus.html

  1. x ( θ , φ ) = ( R + r cos θ ) cos φ y ( θ , φ ) = ( R + r cos θ ) sin φ z ( θ , φ ) = r sin θ \begin{aligned}\displaystyle x(\theta,\varphi)&\displaystyle=(R+r\cos\theta)% \cos{\varphi}\\ \displaystyle y(\theta,\varphi)&\displaystyle=(R+r\cos\theta)\sin{\varphi}\\ \displaystyle z(\theta,\varphi)&\displaystyle=r\sin\theta\end{aligned}
  2. ( R - x 2 + y 2 ) 2 + z 2 = r 2 , \left(R-\sqrt{x^{2}+y^{2}}\right)^{2}+z^{2}=r^{2},
  3. f ( x , y , z ) = ( R - x 2 + y 2 ) 2 + z 2 - r 2 . f(x,y,z)=\left(R-\sqrt{x^{2}+y^{2}}\right)^{2}+z^{2}-r^{2}.
  4. ( x 2 + y 2 + z 2 + R 2 - r 2 ) 2 = 4 R 2 ( x 2 + y 2 ) . (x^{2}+y^{2}+z^{2}+R^{2}-r^{2})^{2}=4R^{2}(x^{2}+y^{2}).
  5. ( R - x 2 + y 2 ) 2 + z 2 < r 2 \left(R-\sqrt{x^{2}+y^{2}}\right)^{2}+z^{2}<r^{2}
  6. A = ( 2 π r ) ( 2 π R ) = 4 π 2 R r V = ( π r 2 ) ( 2 π R ) = 2 π 2 R r 2 \begin{aligned}\displaystyle A&\displaystyle=\left(2\pi r\right)\left(2\pi R% \right)=4\pi^{2}Rr\\ \displaystyle V&\displaystyle=\left(\pi r^{2}\right)\left(2\pi R\right)=2\pi^{% 2}Rr^{2}\end{aligned}
  7. 2 \sqrt{2}
  8. π 1 ( 𝐓 2 ) = π 1 ( S 1 ) × π 1 ( S 1 ) 𝐙 × 𝐙 . \pi_{1}(\mathbf{T}^{2})=\pi_{1}(S^{1})\times\pi_{1}(S^{1})\cong\mathbf{Z}% \times\mathbf{Z}.
  9. 𝐓 n = S 1 × × S 1 n . \mathbf{T}^{n}=\underbrace{S^{1}\times\cdots\times S^{1}}_{n}.
  10. MCG ( 𝐓 n ) = Aut ( π 1 ( X ) ) = Aut ( 𝐙 n ) = GL ( n , 𝐙 ) . \operatorname{MCG}(\mathbf{T}^{n})=\operatorname{Aut}(\pi_{1}(X))=% \operatorname{Aut}(\mathbf{Z}^{n})=\operatorname{GL}(n,\mathbf{Z}).
  11. 1 Homeo 0 ( 𝐓 n ) Homeo ( 𝐓 n ) MCG ( 𝐓 n ) 1 , 1\to\operatorname{Homeo}_{0}(\mathbf{T}^{n})\to\operatorname{Homeo}(\mathbf{T}% ^{n})\to\operatorname{MCG}(\mathbf{T}^{n})\to 1,
  12. Homeo ( 𝐓 n ) Homeo 0 ( 𝐓 n ) GL ( n , 𝐙 ) . \operatorname{Homeo}(\mathbf{T}^{n})\cong\operatorname{Homeo}_{0}(\mathbf{T}^{% n})\rtimes\operatorname{GL}(n,\mathbf{Z}).
  13. 1 6 ( n 3 + 3 n 2 + 8 n ) \tfrac{1}{6}(n^{3}+3n^{2}+8n)

Totally_disconnected_space.html

  1. ω \mathbb{Q}^{\omega}
  2. X X
  3. x y x\sim y
  4. y conn ( x ) y\in\mathrm{conn}(x)
  5. conn ( x ) \mathrm{conn}(x)
  6. x x
  7. X / X/{\sim}
  8. m : x conn ( x ) m:x\mapsto\mathrm{conn}(x)
  9. X / X/{\sim}
  10. f : X Y f:X\rightarrow Y
  11. f = f ˘ m f=\breve{f}\circ m
  12. f ˘ : ( X / ) Y \breve{f}:(X/\sim)\rightarrow Y

Trace_class.html

  1. A 1 = Tr | A | := k ( A * A ) 1 / 2 e k , e k \|A\|_{1}={\rm Tr}|A|:=\sum_{k}\langle(A^{*}A)^{1/2}\,e_{k},e_{k}\rangle
  2. Tr A := k A e k , e k {\rm Tr}A:=\sum_{k}\langle Ae_{k},e_{k}\rangle
  3. k A e k , e k . \sum_{k}\langle Ae_{k},e_{k}\rangle.
  4. Tr ( a A + b B ) = a Tr ( A ) + b Tr ( B ) . \operatorname{Tr}(aA+bB)=a\,\operatorname{Tr}(A)+b\,\operatorname{Tr}(B).
  5. A , B = Tr ( A * B ) \langle A,B\rangle=\operatorname{Tr}(A^{*}B)
  6. A A
  7. B B
  8. A B AB
  9. B A BA
  10. A B 1 = Tr ( | A B | ) A B 1 , B A 1 = Tr ( | B A | ) A B 1 \|AB\|_{1}=\operatorname{Tr}(|AB|)\leq\|A\|\|B\|_{1},\qquad\|BA\|_{1}=% \operatorname{Tr}(|BA|)\leq\|A\|\|B\|_{1}
  11. Tr ( A B ) = Tr ( B A ) \operatorname{Tr}(AB)=\operatorname{Tr}(BA)
  12. A A
  13. B B
  14. A A
  15. 1 + A 1+A
  16. det ( I + A ) := n 1 [ 1 + λ n ( A ) ] {\rm det}(I+A):=\prod_{n\geq 1}[1+\lambda_{n}(A)]
  17. { λ n ( A ) } n \{\lambda_{n}(A)\}_{n}
  18. A A
  19. A A
  20. det ( I + A ) e A 1 {\rm det}(I+A)\leq e^{\|A\|_{1}}
  21. det ( I + A ) 0 {\rm det}(I+A)\neq 0
  22. ( I + A ) (I+A)
  23. A A
  24. H H
  25. { λ n ( A ) } n = 1 N , \{\lambda_{n}(A)\}_{n=1}^{N},
  26. N N\leq\infty
  27. A A
  28. λ n ( A ) \lambda_{n}(A)
  29. λ \lambda
  30. k k
  31. λ \lambda
  32. k k
  33. λ 1 ( A ) , λ 2 ( A ) , \lambda_{1}(A),\lambda_{2}(A),\dots
  34. n = 1 N λ n ( A ) = Tr ( A ) . \sum_{n=1}^{N}\lambda_{n}(A)=\operatorname{Tr}(A).
  35. n = 1 N | λ n ( A ) | m = 1 M s m ( A ) \sum_{n=1}^{N}|\lambda_{n}(A)|\leq\sum_{m=1}^{M}s_{m}(A)
  36. { λ n ( A ) } n = 1 N \{\lambda_{n}(A)\}_{n=1}^{N}
  37. { s m ( A ) } m = 1 M \{s_{m}(A)\}_{m=1}^{M}
  38. A A
  39. h H , T h = i = 1 α i h , v i u i where α i 0 and α i 0 \forall h\in H,\;Th=\sum_{i=1}\alpha_{i}\langle h,v_{i}\rangle u_{i}\quad\mbox% {where}~{}\quad\alpha_{i}\geq 0\quad\mbox{and}~{}\quad\alpha_{i}\rightarrow 0
  40. T T 2 T 1 , \|T\|\leq\|T\|_{2}\leq\|T\|_{1},
  41. T f x , y = f ( S x , y ) , \langle T_{f}x,y\rangle=f(S_{x,y}),
  42. S x , y ( h ) = h , y x . S_{x,y}(h)=\langle h,y\rangle x.
  43. i T f u i , u i = f ( I ) f , \sum_{i}\langle T_{f}u_{i},u_{i}\rangle=f(I)\leq\|f\|,
  44. I = i , u i u i . I=\sum_{i}\langle\cdot,u_{i}\rangle u_{i}.
  45. C 1 C_{1}
  46. C 1 C_{1}
  47. C 1 C_{1}

Trajectory.html

  1. ( f k ( x ) ) k (f^{k}(x))_{k\in\mathbb{N}}
  2. f f
  3. x x
  4. m m
  5. V V
  6. V V
  7. V V
  8. m d 2 x ( t ) d t 2 = - V ( x ( t ) ) m\frac{\mathrm{d}^{2}\vec{x}(t)}{\mathrm{d}t^{2}}=-\nabla V(\vec{x}(t))
  9. x = ( x , y , z ) \vec{x}=(x,y,z)
  10. V \nabla V
  11. g g
  12. v h = v cos ( θ ) v_{h}=v\cos(\theta)
  13. v v = v sin ( θ ) v_{v}=v\sin(\theta)
  14. 2 v h v v / g 2v_{h}v_{v}/g
  15. v v 2 / 2 g v_{v}^{2}/2g
  16. v v
  17. v h = v v v_{h}=v_{v}
  18. v 2 / g v^{2}/g
  19. y = x tan ( θ ) y=x\tan(\theta)
  20. y = - g t 2 / 2 y=-gt^{2}/2
  21. y = - g ( x / v h ) 2 / 2 y=-g(x/v_{h})^{2}/2
  22. y = x tan ( θ ) - g ( x / v h ) 2 / 2 y=x\tan(\theta)-g(x/v_{h})^{2}/2
  23. y = - g sec 2 θ 2 v 0 2 x 2 + x tan θ y=-{g\sec^{2}\theta\over 2v_{0}^{2}}x^{2}+x\tan\theta
  24. θ \theta
  25. R = v i 2 sin 2 θ i g R={v_{i}^{2}\sin 2\theta_{i}\over g}
  26. h = v i 2 sin 2 θ i 2 g h={v_{i}^{2}\sin^{2}\theta_{i}\over 2g}
  27. θ \theta
  28. v v
  29. v h = v cos θ , v v = v sin θ v_{h}=v\cos\theta,\quad v_{v}=v\sin\theta\;
  30. R = 2 v 2 cos ( θ ) sin ( θ ) / g = v 2 sin ( 2 θ ) / g . R=2v^{2}\cos(\theta)\sin(\theta)/g=v^{2}\sin(2\theta)/g\,.
  31. θ = 1 2 sin - 1 ( g R v 2 ) {\theta}=\frac{1}{2}\sin^{-1}\left({{gR}\over{v^{2}}}\right)
  32. θ \theta
  33. d h d_{h}
  34. θ \theta
  35. R R
  36. θ \theta
  37. d R d θ = 2 v 2 g cos ( 2 θ ) = 0 {\mathrm{d}R\over\mathrm{d}\theta}={2v^{2}\over g}\cos(2\theta)=0
  38. 2 θ = π / 2 = 90 2\theta=\pi/2=90^{\circ}
  39. θ = 45 \theta=45^{\circ}
  40. R m a x = v 2 / g R_{max}=v^{2}/g\,
  41. s i n ( π / 2 ) = 1 sin(\pi/2)=1
  42. v 2 4 g {v^{2}\over 4g}
  43. H = v 2 s i n 2 ( θ ) / ( 2 g ) H=v^{2}sin^{2}(\theta)/(2g)
  44. θ \theta
  45. d H d θ = v 2 2 cos ( θ ) sin ( θ ) / ( 2 g ) {\mathrm{d}H\over\mathrm{d}\theta}=v^{2}2\cos(\theta)\sin(\theta)/(2g)
  46. θ = π / 2 = 90 \theta=\pi/2=90^{\circ}
  47. H m a x = v 2 2 g H_{max}={v^{2}\over 2g}
  48. α \alpha
  49. θ \theta
  50. R s R_{s}
  51. R R
  52. R s R = ( 1 - cot θ tan α ) sec α \frac{R_{s}}{R}=(1-\cot\theta\tan\alpha)\sec\alpha
  53. α \alpha
  54. α \alpha
  55. tan ( - α ) = - tan α \tan(-\alpha)=-\tan\alpha
  56. sec ( - α ) = sec α \sec(-\alpha)=\sec\alpha
  57. α \alpha
  58. R s / R = ( 1 + tan θ tan α ) sec α R_{s}/R=(1+\tan\theta\tan\alpha)\sec\alpha
  59. R s / R R_{s}/R
  60. R s / R = 1 R_{s}/R=1
  61. θ c r \theta_{cr}
  62. 1 = ( 1 - tan θ tan α ) sec α 1=(1-\tan\theta\tan\alpha)\sec\alpha\quad\;
  63. θ c r = arctan ( ( 1 - csc α ) cot α ) \theta_{cr}=\arctan((1-\csc\alpha)\cot\alpha)\quad\;
  64. α \alpha
  65. θ \theta
  66. tan α \tan\alpha
  67. tan θ \tan\theta
  68. R s R = ( 1 - 0 ) sec α \frac{R_{s}}{R}=(1-0)\sec\alpha
  69. R R
  70. R = R s cos α R=R_{s}\cos\alpha
  71. m m
  72. ( x , y ) (x,y)
  73. y = m x + b y=mx+b\;
  74. y = d v y=d_{v}
  75. x = d h x=d_{h}
  76. b = 0 b=0
  77. d v = m d h d_{v}=md_{h}
  78. m x = - g 2 v 2 cos 2 θ x 2 + sin θ cos θ x mx=-\frac{g}{2v^{2}{\cos}^{2}\theta}x^{2}+\frac{\sin\theta}{\cos\theta}x
  79. x = 2 v 2 cos 2 θ g ( sin θ cos θ - m ) x=\frac{2v^{2}\cos^{2}\theta}{g}\left(\frac{\sin\theta}{\cos\theta}-m\right)
  80. y = m x = m 2 v 2 cos 2 θ g ( sin θ cos θ - m ) y=mx=m\frac{2v^{2}\cos^{2}\theta}{g}\left(\frac{\sin\theta}{\cos\theta}-m\right)
  81. R s R_{s}
  82. R s = x 2 + y 2 = ( 2 v 2 cos 2 θ g ( sin θ cos θ - m ) ) 2 + ( m 2 v 2 cos 2 θ g ( sin θ cos θ - m ) ) 2 R_{s}=\sqrt{x^{2}+y^{2}}=\sqrt{\left(\frac{2v^{2}\cos^{2}\theta}{g}\left(\frac% {\sin\theta}{\cos\theta}-m\right)\right)^{2}+\left(m\frac{2v^{2}\cos^{2}\theta% }{g}\left(\frac{\sin\theta}{\cos\theta}-m\right)\right)^{2}}
  83. = 2 v 2 cos 2 θ g ( sin θ cos θ - m ) 2 + m 2 ( sin θ cos θ - m ) 2 =\frac{2v^{2}\cos^{2}\theta}{g}\sqrt{\left(\frac{\sin\theta}{\cos\theta}-m% \right)^{2}+m^{2}\left(\frac{\sin\theta}{\cos\theta}-m\right)^{2}}
  84. = 2 v 2 cos 2 θ g ( sin θ cos θ - m ) 1 + m 2 =\frac{2v^{2}\cos^{2}\theta}{g}\left(\frac{\sin\theta}{\cos\theta}-m\right)% \sqrt{1+m^{2}}
  85. α \alpha
  86. m = tan α m=\tan\alpha
  87. R s R_{s}
  88. R s = 2 v 2 cos 2 θ g ( sin θ cos θ - tan α ) 1 + tan 2 α R_{s}=\frac{2v^{2}\cos^{2}\theta}{g}\left(\frac{\sin\theta}{\cos\theta}-\tan% \alpha\right)\sqrt{1+\tan^{2}\alpha}
  89. sec α = 1 + tan 2 α \sec\alpha=\sqrt{1+\tan^{2}\alpha}
  90. R s = 2 v 2 cos θ sin θ g ( 1 - cos θ sin θ tan α ) sec α R_{s}=\frac{2v^{2}\cos\theta\sin\theta}{g}\left(1-\frac{\cos\theta}{\sin\theta% }\tan\alpha\right)\sec\alpha
  91. R = v 2 sin 2 θ / g = 2 v 2 sin θ cos θ / g R=v^{2}\sin 2\theta/g=2v^{2}\sin\theta\cos\theta/g
  92. cos θ / sin θ = c o t a n θ \cos\theta/\sin\theta=cotan\theta
  93. R s = R ( 1 - cot θ tan α ) sec α R_{s}=R(1-\cot\theta\tan\alpha)\sec\alpha\;
  94. R s R = ( 1 - cot θ tan α ) sec α \frac{R_{s}}{R}=(1-\cot\theta\tan\alpha)\sec\alpha

Transcendental_function.html

  1. f 1 ( x ) = x π f_{1}(x)=x^{\pi}
  2. f 2 ( x ) = c x , c 0 , 1 f_{2}(x)=c^{x},\ c\neq 0,1
  3. f 3 ( x ) = x x f_{3}(x)=x^{x}
  4. f 4 ( x ) = x 1 x f_{4}(x)=x^{\frac{1}{x}}
  5. f 5 ( x ) = log c x , c 0 , 1 f_{5}(x)=\log_{c}x,\ c\neq 0,1
  6. f 6 ( x ) = sin x f_{6}(x)=\sin{x}
  7. ( f ) = { α 𝐐 ¯ : f ( α ) 𝐐 ¯ } . \mathcal{E}(f)=\{\alpha\in\overline{\mathbf{Q}}\,:\,f(\alpha)\in\overline{% \mathbf{Q}}\}.
  8. ( exp ) = { 0 } \mathcal{E}(\exp)=\{0\}
  9. ( j ) = { α 𝐇 : [ 𝐐 ( α ) : 𝐐 ] = 2 } \mathcal{E}(j)=\{\alpha\in\mathbf{H}\,:\,[\mathbf{Q}(\alpha):\mathbf{Q}]=2\}
  10. ( 2 x ) = 𝐐 \mathcal{E}(2^{x})=\mathbf{Q}
  11. ( x x ) = ( x 1 x ) = 𝐐 { 0 } . \mathcal{E}(x^{x})=\mathcal{E}(x^{\frac{1}{x}})=\mathbf{Q}\setminus\{0\}.
  12. ( e e x ) = . \mathcal{E}(e^{e^{x}})=\emptyset.

Transcranial_magnetic_stimulation.html

  1. 𝐁 = μ 0 4 π I C d 𝐥 × 𝐫 ^ r 2 \mathbf{B}=\frac{\mu_{0}}{4\pi}I\int_{C}\frac{d\mathbf{l}\times\mathbf{\hat{r}% }}{r^{2}}
  2. × 𝐄 = - 𝐁 t \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}

Transfinite_number.html

  1. 0 \scriptstyle{\aleph_{0}}
  2. 1 \scriptstyle{\aleph_{1}}
  3. 0 \scriptstyle{\aleph_{0}}
  4. 0 \scriptstyle{\aleph_{0}}

Transformation_problem.html

  1. ( B ) (B)
  2. ( D ) (D)
  3. l i l_{i}
  4. i i
  5. B B
  6. D D
  7. l i l_{i}
  8. l D l_{D}
  9. l D l B {l_{D}\over l_{B}}
  10. l D l B {l_{D}\over l_{B}}
  11. w w
  12. P D P B {P_{D}\over P_{B}}
  13. P P
  14. P i = w l i P_{i}=wl_{i}
  15. ( A ) (A)
  16. a i a_{i}
  17. a B a_{B}
  18. l B l_{B}
  19. P i = w l i + k A a i , ( i = B , D ) P_{i}=wl_{i}+k_{A}a_{i},(i=B,D)
  20. k A k_{A}
  21. P A P_{A}
  22. h 1 h\leq 1
  23. r P A rP_{A}
  24. r r
  25. h h
  26. P i = w l i + ( h + r ) P A a i P_{i}=wl_{i}+(h+r)P_{A}a_{i}
  27. P A P_{A}
  28. l A l_{A}
  29. P A = w l A P_{A}=wl_{A}
  30. h = 1 h=1
  31. P i = w l i + ( 1 + r ) w l A a i P_{i}=wl_{i}+(1+r)wl_{A}a_{i}
  32. l i l_{i}
  33. l A a i l_{A}a_{i}
  34. E i = l i + l A a i E_{i}=l_{i}+l_{A}a_{i}
  35. P D P B {P_{D}\over P_{B}}
  36. a i > 0 a_{i}>0
  37. E D E B {E_{D}\over E_{B}}
  38. P D P B {P_{D}\over P_{B}}
  39. r = 0 r=0
  40. l B l D = a B a D {l_{B}\over l_{D}}={a_{B}\over a_{D}}
  41. P D P B {P_{D}\over P_{B}}
  42. E D E B {E_{D}\over E_{B}}
  43. E D E B {E_{D}\over E_{B}}
  44. P D P B {P_{D}\over P_{B}}
  45. v v
  46. v i = l W l i v_{i}=l_{W}l_{i}
  47. c i = l A a i c_{i}=l_{A}a_{i}
  48. p i = c i + v i + s i = l A a i + l W l i + s i p_{i}=c_{i}+v_{i}+s_{i}=l_{A}a_{i}+l_{W}l_{i}+s_{i}
  49. p i p_{i}
  50. p i = l A a i + l i = E i p_{i}=l_{A}a_{i}+l_{i}=E_{i}
  51. s i s_{i}
  52. s i v i = ( 1 - l W ) l W = σ {s_{i}\over v_{i}}={(1-l_{W})\over l_{W}}=\sigma
  53. i i
  54. s i v i = σ {s_{i}\over v_{i}}=\sigma
  55. p i = c i + v i ( 1 + σ ) = l A a i + l W l i ( 1 + σ ) p_{i}=c_{i}+v_{i}(1+\sigma)=l_{A}a_{i}+l_{W}l_{i}(1+\sigma)
  56. p D p B {p_{D}\over p_{B}}
  57. P D P B {P_{D}\over P_{B}}
  58. r r
  59. Q B Q_{B}
  60. Q D Q_{D}
  61. l W = E B = l A a B + l B < 1 l_{W}=E_{B}=l_{A}a_{B}+l_{B}<1
  62. Q i c i Q_{i}c_{i}
  63. Q i v i Q_{i}v_{i}
  64. σ Q i v i \sigma Q_{i}v_{i}
  65. c i + ( 1 + σ ) v i c_{i}+(1+\sigma)v_{i}
  66. Q B l A a B Q_{B}l_{A}a_{B}
  67. Q B ( l A a B + l B ) l B Q_{B}(l_{A}a_{B}+l_{B})l_{B}
  68. σ Q B ( l A a B + l B ) l B \sigma Q_{B}(l_{A}a_{B}+l_{B})l_{B}
  69. l A a B + ( 1 + σ ) ( l A a B + l B ) l B l_{A}a_{B}+(1+\sigma)(l_{A}a_{B}+l_{B})l_{B}
  70. Q D l A a D Q_{D}l_{A}a_{D}
  71. Q D ( l A a B + l B ) l D Q_{D}(l_{A}a_{B}+l_{B})l_{D}
  72. σ Q D ( l A a B + l B ) l D \sigma Q_{D}(l_{A}a_{B}+l_{B})l_{D}
  73. l A a D + ( 1 + σ ) ( l A a B + l B ) l D l_{A}a_{D}+(1+\sigma)(l_{A}a_{B}+l_{B})l_{D}
  74. σ ( l A a B + l B ) ( Q B l B + Q D l D ) \sigma(l_{A}a_{B}+l_{B})(Q_{B}l_{B}+Q_{D}l_{D})
  75. Q i c i Q_{i}c_{i}
  76. Q i v i Q_{i}v_{i}
  77. r Q i c i rQ_{i}c_{i}
  78. v i + ( 1 + r ) c i v_{i}+(1+r)c_{i}
  79. Q B l A a B Q_{B}l_{A}a_{B}
  80. Q B ( l A a B + l B ) l B Q_{B}(l_{A}a_{B}+l_{B})l_{B}
  81. r Q B l A a B rQ_{B}l_{A}a_{B}
  82. ( l A a B + l B ) l B + ( 1 + r ) l A a B (l_{A}a_{B}+l_{B})l_{B}+(1+r)l_{A}a_{B}
  83. Q D l A a D Q_{D}l_{A}a_{D}
  84. Q D ( l A a B + l B ) l D Q_{D}(l_{A}a_{B}+l_{B})l_{D}
  85. r Q D l A a D rQ_{D}l_{A}a_{D}
  86. ( l A a B + l B ) l D + ( 1 + r ) l A a D (l_{A}a_{B}+l_{B})l_{D}+(1+r)l_{A}a_{D}
  87. r l A ( Q B A B + Q D a D ) = σ ( l A a B + l B ) ( Q B l B + Q D l D ) rl_{A}(Q_{B}A_{B}+Q_{D}a_{D})=\sigma(l_{A}a_{B}+l_{B})(Q_{B}l_{B}+Q_{D}l_{D})
  88. r r
  89. w = P B = 1 w=P_{B}=1
  90. P A = l A P_{A}=l_{A}
  91. 1 = l B + ( 1 + r ) l A a B 1=l_{B}+(1+r)l_{A}a_{B}
  92. r = ( 1 - l B ) ( l A a B ) - 1 r={(1-l_{B})\over(l_{A}a_{B})}-1
  93. P D = l D + ( 1 + r ) l A a D = l D + a D ( 1 - l B ) a B P_{D}=l_{D}+(1+r)l_{A}a_{D}=l_{D}+{a_{D}(1-l_{B})\over a_{B}}
  94. P i P_{i}
  95. i t h i^{th}
  96. P i = n = 0 l i n w ( 1 + r ) n P_{i}=\sum_{n=0}^{\infty}l_{in}w{(1+r)^{n}}
  97. n n
  98. l i n l_{in}
  99. w w
  100. r r
  101. E i = n = 0 l i n E_{i}=\sum_{n=0}^{\infty}l_{in}
  102. E i E_{i}
  103. P i P_{i}

Transitive_closure.html

  1. R + = i { 1 , 2 , 3 , } R i . R^{+}=\bigcup_{i\in\{1,2,3,\ldots\}}R^{i}.
  2. R i R^{i}
  3. R 1 = R R^{1}=R\,\!
  4. i > 0 i>0
  5. R i + 1 = R R i R^{i+1}=R\circ R^{i}
  6. \circ
  7. R R + R\subseteq R^{+}
  8. R + \displaystyle R^{+}
  9. R i \displaystyle R^{i}
  10. R + \displaystyle R^{+}
  11. R \displaystyle R
  12. R + \displaystyle R^{+}
  13. R + \displaystyle R^{+}
  14. R i \displaystyle R^{i}
  15. R + \displaystyle R^{+}
  16. ( s 1 , s 2 ) R j (s_{1},s_{2})\in R^{j}
  17. ( s 2 , s 3 ) R k (s_{2},s_{3})\in R^{k}
  18. ( s 1 , s 3 ) R j + k (s_{1},s_{3})\in R^{j+k}
  19. R + \displaystyle R^{+}
  20. R i \displaystyle R^{i}
  21. R + \displaystyle R^{+}
  22. G \displaystyle G
  23. R \displaystyle R
  24. R + G R^{+}\subseteq G
  25. i > 0 i>0
  26. R i G R^{i}\subseteq G
  27. G \displaystyle G
  28. R \displaystyle R
  29. R 1 G R^{1}\subseteq G
  30. G \displaystyle G
  31. R i G R^{i}\subseteq G
  32. R i + 1 G R^{i+1}\subseteq G
  33. R i R^{i}\,\!
  34. G \displaystyle G
  35. R i R^{i}\,\!
  36. R + \displaystyle R^{+}

Transitive_relation.html

  1. a , b , c X : ( a R b b R c ) a R c \forall a,b,c\in X:(aRb\wedge bRc)\Rightarrow aRc

Translation_(geometry).html

  1. T δ T_{\mathbf{\delta}}
  2. T δ f ( 𝐯 ) = f ( 𝐯 + δ ) . T_{\mathbf{\delta}}f(\mathbf{v})=f(\mathbf{v}+\mathbf{\delta}).
  3. T 𝐯 = [ 1 0 0 v x 0 1 0 v y 0 0 1 v z 0 0 0 1 ] T_{\mathbf{v}}=\begin{bmatrix}1&0&0&v_{x}\\ 0&1&0&v_{y}\\ 0&0&1&v_{z}\\ 0&0&0&1\end{bmatrix}
  4. T 𝐯 𝐩 = [ 1 0 0 v x 0 1 0 v y 0 0 1 v z 0 0 0 1 ] [ p x p y p z 1 ] = [ p x + v x p y + v y p z + v z 1 ] = 𝐩 + 𝐯 T_{\mathbf{v}}\mathbf{p}=\begin{bmatrix}1&0&0&v_{x}\\ 0&1&0&v_{y}\\ 0&0&1&v_{z}\\ 0&0&0&1\end{bmatrix}\begin{bmatrix}p_{x}\\ p_{y}\\ p_{z}\\ 1\end{bmatrix}=\begin{bmatrix}p_{x}+v_{x}\\ p_{y}+v_{y}\\ p_{z}+v_{z}\\ 1\end{bmatrix}=\mathbf{p}+\mathbf{v}
  5. T 𝐯 - 1 = T - 𝐯 . T^{-1}_{\mathbf{v}}=T_{-\mathbf{v}}.\!
  6. T 𝐮 T 𝐯 = T 𝐮 + 𝐯 . T_{\mathbf{u}}T_{\mathbf{v}}=T_{\mathbf{u}+\mathbf{v}}.\!
  7. ( x , y , z ) ( x + Δ x , y + Δ y , z + Δ z ) (x,y,z)\to(x+\Delta x,y+\Delta y,z+\Delta z)
  8. ( Δ x , Δ y , Δ z ) (\Delta x,\ \Delta y,\ \Delta z)
  9. ( Δ x , Δ y , Δ z ) (\Delta x,\ \Delta y,\ \Delta z)

Transmission_electron_microscopy.html

  1. d = λ 2 n sin α λ 2 NA d=\frac{\lambda}{2n\sin\alpha}\approx\frac{\lambda}{2\,\textrm{NA}}
  2. λ e h 2 m 0 E ( 1 + E 2 m 0 c 2 ) \lambda_{e}\approx\frac{h}{\sqrt{2m_{0}E\left(1+\frac{E}{2m_{0}c^{2}}\right)}}
  3. J = A T 2 exp ( - Φ k T ) , J=AT^{2}\exp\left(-\frac{\Phi}{kT}\right),
  4. I ( x ) = k t 1 - t 0 t 0 t 1 Ψ Ψ * d t I(x)=\frac{k}{t_{1}-t_{0}}\int^{t_{1}}_{t_{0}}\Psi\Psi^{\mathrm{*}}\,dt
  5. q max = 1 0.67 ( C s λ 3 ) 1 / 4 . q_{\max}=\frac{1}{0.67(C_{s}\lambda^{3})^{1/4}}.

Transmitter_power_output.html

  1. T P O × l o s s f e e d l i n e × g a i n a n t e n n a = E R P TPO\ \times\ loss_{feedline}\ \times\ gain_{antenna}\ =\ ERP

Transpose.html

  1. [ 𝐀 T ] i j = [ 𝐀 ] j i [\mathbf{A}^{\mathrm{T}}]_{ij}=[\mathbf{A}]_{ji}
  2. [ 1 2 ] T = [ 1 2 ] \begin{bmatrix}1&2\end{bmatrix}^{\mathrm{T}}=\,\begin{bmatrix}1\\ 2\end{bmatrix}
  3. [ 1 2 3 4 ] T = [ 1 3 2 4 ] \begin{bmatrix}1&2\\ 3&4\end{bmatrix}^{\mathrm{T}}=\begin{bmatrix}1&3\\ 2&4\end{bmatrix}
  4. [ 1 2 3 4 5 6 ] T = [ 1 3 5 2 4 6 ] \begin{bmatrix}1&2\\ 3&4\\ 5&6\end{bmatrix}^{\mathrm{T}}=\begin{bmatrix}1&3&5\\ 2&4&6\end{bmatrix}
  5. ( 𝐀 T ) - 1 = ( 𝐀 - 1 ) T (\mathbf{A}^{\mathrm{T}})^{-1}=(\mathbf{A}^{-1})^{\mathrm{T}}\,
  6. 𝐀 T = 𝐀 . \mathbf{A}^{\mathrm{T}}=\mathbf{A}.
  7. 𝐀 T = - 𝐀 . \mathbf{A}^{\mathrm{T}}=-\mathbf{A}.
  8. 𝐀 T = 𝐀 ¯ . \mathbf{A}^{\mathrm{T}}=\overline{\mathbf{A}}.
  9. 𝐀 T = - 𝐀 ¯ . \mathbf{A}^{\mathrm{T}}=-\overline{\mathbf{A}}.
  10. 𝐀 T = 𝐀 - 1 . \mathbf{A}^{\mathrm{T}}=\mathbf{A}^{-1}.
  11. f t ( ϕ ) = ϕ f ϕ W * . {}^{\mathrm{t}}f(\phi)=\phi\circ f\quad\forall\phi\in W^{*}.
  12. B V ( v , g ( w ) ) = B W ( f ( v ) , w ) v V , w W . B_{V}(v,g(w))=B_{W}(f(v),w)\quad\forall\ v\in V,w\in W.

Transrapid.html

  1. P = c w A Front v 3 ( density of surrounding air ) / 2 P=c_{w}\cdot A_{\rm Front}\cdot v^{3}\cdot(\mbox{density of surrounding air}~{% })/2
  2. P = 0.26 16 m 2 ( 111 m / s ) 3 1.24 kg / m 3 / 2 P = 3.53 10 6 kg m 2 / s 3 = 3.53 10 6 N m / s = 3.53 MW \begin{matrix}P&=&0{.}26\cdot 16\,\mathrm{m}^{2}\cdot(111\,\mathrm{m}/\mathrm{% s})^{3}\cdot 1{.}24\,\mathrm{kg}/\mathrm{m}^{3}/2\\ P&=&3{.}53\cdot 10^{6}\,\mathrm{kg}\cdot\mathrm{m}^{2}/\mathrm{s}^{3}=3{.}53% \cdot 10^{6}\,\mathrm{N}\cdot\mathrm{m}/\mathrm{s}=3{.}53\,\mathrm{MW}\end{matrix}

Trapdoor_function.html

  1. e e
  2. ϕ ( n ) \phi(n)
  3. f ( x ) = x e mod n f(x)=x^{e}\mod n
  4. ϕ ( n ) \phi(n)
  5. e e
  6. f ( x ) f(x)
  7. x x

Trapezoid.html

  1. K = S + T , \sqrt{K}=\sqrt{S}+\sqrt{T},
  2. sin A sin C = sin B sin D . \sin A\sin C=\sin B\sin D.
  3. p 2 + q 2 = c 2 + d 2 + 2 a b . p^{2}+q^{2}=c^{2}+d^{2}+2ab.
  4. v = | a - b | 2 . v=\frac{|a-b|}{2}.
  5. m = a + b 2 . m=\frac{a+b}{2}.
  6. h = ( - a + b + c + d ) ( a - b + c + d ) ( a - b + c - d ) ( a - b - c + d ) 2 | b - a | h=\frac{\sqrt{(-a+b+c+d)(a-b+c+d)(a-b+c-d)(a-b-c+d)}}{2|b-a|}
  7. h 2 > 0. \displaystyle h^{2}>0.
  8. K = a + b 2 h = m h K=\frac{a+b}{2}\cdot h=mh
  9. K = a + b 4 | b - a | ( - a + b + c + d ) ( a - b + c + d ) ( a - b + c - d ) ( a - b - c + d ) . K=\frac{a+b}{4|b-a|}\sqrt{(-a+b+c+d)(a-b+c+d)(a-b+c-d)(a-b-c+d)}.
  10. K = a + b | b - a | ( s - b ) ( s - a ) ( s - b - c ) ( s - b - d ) , K=\frac{a+b}{|b-a|}\sqrt{(s-b)(s-a)(s-b-c)(s-b-d)},
  11. s = 1 2 ( a + b + c + d ) s=\tfrac{1}{2}(a+b+c+d)
  12. K = ( a b 2 - a 2 b - a d 2 + b c 2 ) ( a b 2 - a 2 b - a c 2 + b d 2 ) ( 2 ( b - a ) ) 2 - ( b 2 + d 2 - a 2 - c 2 4 ) 2 . K=\sqrt{\frac{(ab^{2}-a^{2}b-ad^{2}+bc^{2})(ab^{2}-a^{2}b-ac^{2}+bd^{2})}{(2(b% -a))^{2}}-\left(\frac{b^{2}+d^{2}-a^{2}-c^{2}}{4}\right)^{2}}.
  13. p = a b 2 - a 2 b - a c 2 + b d 2 b - a , p=\sqrt{\frac{ab^{2}-a^{2}b-ac^{2}+bd^{2}}{b-a}},
  14. q = a b 2 - a 2 b - a d 2 + b c 2 b - a q=\sqrt{\frac{ab^{2}-a^{2}b-ad^{2}+bc^{2}}{b-a}}
  15. 1 F G = 1 2 ( 1 A B + 1 D C ) . \frac{1}{FG}=\frac{1}{2}\left(\frac{1}{AB}+\frac{1}{DC}\right).
  16. x = h 3 ( 2 a + b a + b ) . x=\frac{h}{3}\left(\frac{2a+b}{a+b}\right).
  17. P Q = | A D + B C - A B - C D | 2 . PQ=\frac{|AD+BC-AB-CD|}{2}.

Tree_decomposition.html

  1. X i X_{i}
  2. X j X_{j}
  3. X k X_{k}
  4. X k X_{k}
  5. X i X_{i}
  6. X j X_{j}
  7. X i X j X k X_{i}\cap X_{j}\subseteq X_{k}
  8. i I : | X i | = k + 1 i\in I:|X_{i}|=k+1
  9. ( i , j ) F : | X i X j | = k (i,j)\in F:|X_{i}\cap X_{j}|=k
  10. A ( S , i ) = | S | + j ( B ( S X j , j , i ) - | S X j | ) A(S,i)=|S|+\sum_{j}\left(B(S\cap X_{j},j,i)-|S\cap X_{j}|\right)
  11. B ( S , i , j ) = max S X i S = S X j A ( S , i ) B(S,i,j)=\max_{S^{\prime}\subset X_{i}\atop S=S^{\prime}\cap X_{j}}A(S^{\prime% },i)
  12. A ( S , i ) A(S,i)
  13. X i X_{i}

Triangular_number.html

  1. n n
  2. n n
  3. n n
  4. T n = k = 1 n k = 1 + 2 + 3 + + n = n ( n + 1 ) 2 = ( n + 1 2 ) T_{n}=\sum_{k=1}^{n}k=1+2+3+\cdots+n=\frac{n(n+1)}{2}={n+1\choose 2}
  5. ( n + 1 2 ) \textstyle{n+1\choose 2}
  6. T T
  7. L n = 3 T n - 1 = 3 ( n 2 ) ; L n = L n - 1 + 3 ( n - 1 ) , L 1 = 0. L_{n}=3T_{n-1}=3{n\choose 2};~{}~{}~{}L_{n}=L_{n-1}+3(n-1),~{}L_{1}=0.
  8. lim n T n L n = 1 3 \lim_{n\to\infty}\frac{T_{n}}{L_{n}}=\frac{1}{3}
  9. T n + T n - 1 = ( n 2 2 + n 2 ) + ( ( n - 1 ) 2 2 + n - 1 2 ) = ( n 2 2 + n 2 ) + ( n 2 2 - n 2 ) = n 2 = ( T n - T n - 1 ) 2 . T_{n}+T_{n-1}=\left(\frac{n^{2}}{2}+\frac{n}{2}\right)+\left(\frac{\left(n-1% \right)^{2}}{2}+\frac{n-1}{2}\right)=\left(\frac{n^{2}}{2}+\frac{n}{2}\right)+% \left(\frac{n^{2}}{2}-\frac{n}{2}\right)=n^{2}=(T_{n}-T_{n-1})^{2}.
  10. S n + 1 = 4 S n ( 8 S n + 1 ) S_{n+1}=4S_{n}\left(8S_{n}+1\right)
  11. S 1 = 1. S_{1}=1.
  12. S n = 34 S n - 1 - S n - 2 + 2 S_{n}=34S_{n-1}-S_{n-2}+2
  13. S 0 = 0 S_{0}=0
  14. S 1 = 1. S_{1}=1.
  15. n ( n + 1 ) ( n + 2 ) 6 . \frac{n(n+1)(n+2)}{6}.
  16. ( m + 1 ) (m+1)
  17. ( n 1 ) (n−1)
  18. C k n = k T n - 1 + 1 Ck_{n}=kT_{n-1}+1
  19. T T
  20. M p 2 p - 1 = M p ( M p + 1 ) / 2 = T M p M_{p}2^{p-1}=M_{p}(M_{p}+1)/2=T_{M_{p}}
  21. M M
  22. p p
  23. x x
  24. a x + b ax+b
  25. a a
  26. b b
  27. ( a 1 ) / 8 (a−1)/8
  28. b b
  29. b b
  30. 1 x + 0 1x+0
  31. 9 x + 1 9x+1
  32. 25 x + 3 25x+3
  33. 49 x + 6 49x+6
  34. 81 x + 10 81x+10
  35. 121 x + 15 121x+15
  36. 169 x + 21 169x+21
  37. x x
  38. n = 1 1 n 2 + n 2 = 2 n = 1 1 n 2 + n = 2. \!\ \sum_{n=1}^{\infty}{1\over{{n^{2}+n}\over 2}}=2\sum_{n=1}^{\infty}{1\over{% n^{2}+n}}=2.
  39. n = 1 1 n ( n + 1 ) = 1. \!\ \sum_{n=1}^{\infty}{1\over{n(n+1)}}=1.
  40. T a + b = T a + T b + a b T_{a+b}=T_{a}+T_{b}+ab
  41. T a b = T a T b + T a - 1 T b - 1 , T_{ab}=T_{a}T_{b}+T_{a-1}T_{b-1},
  42. n n
  43. n n
  44. b b
  45. s s
  46. n n
  47. y y
  48. n n
  49. x x
  50. x x
  51. n = 8 x + 1 - 1 2 n=\frac{\sqrt{8x+1}-1}{2}
  52. x x
  53. 8 x + 1 8x+ 1
  54. n n
  55. x x
  56. x x
  57. n n

Triangulation.html

  1. = d tan α + d tan β \ell=\frac{d}{\tan\alpha}+\frac{d}{\tan\beta}
  2. = d ( cos α sin α + cos β sin β ) \ell=d\left(\frac{\cos\alpha}{\sin\alpha}+\frac{\cos\beta}{\sin\beta}\right)
  3. = d sin ( α + β ) sin α sin β \ell=d\ \frac{\sin(\alpha+\beta)}{\sin\alpha\sin\beta}
  4. d = sin α sin β sin ( α + β ) d=\ell\ \frac{\sin\alpha\sin\beta}{\sin(\alpha+\beta)}

Trigonometric_integral.html

  1. Si ( x ) = 0 x sin t t d t \operatorname{Si}(x)=\int_{0}^{x}\frac{\sin t}{t}\,dt
  2. si ( x ) = - x sin t t d t . \operatorname{si}(x)=-\int_{x}^{\infty}\frac{\sin t}{t}\,dt~{}.
  3. S i ( x ) Si(x)
  4. s i n x / x sinx/x
  5. x = 0 x=0
  6. s i ( x ) si(x)
  7. s i n x / x sinx/x
  8. x = x=∞
  9. Si ( x ) - si ( x ) = 0 sin t t d t = π 2 . \operatorname{Si}(x)-\operatorname{si}(x)=\int_{0}^{\infty}\frac{\sin t}{t}\,% dt=\frac{\pi}{2}~{}.
  10. s i n x / x sinx/x
  11. s i n c sinc
  12. Cin ( x ) = 0 x 1 - cos t t d t , \operatorname{Cin}(x)=\int_{0}^{x}\frac{1-\cos t}{t}\,dt~{},
  13. γ γ
  14. c i ci
  15. C i Ci
  16. C i ( x ) Ci(x)
  17. c o s x / x cosx/x
  18. x = x=∞
  19. Cin ( x ) = γ + ln x - Ci ( x ) . \operatorname{Cin}(x)=\gamma+\ln x-\operatorname{Ci}(x)~{}.
  20. Shi ( x ) = 0 x sinh ( t ) t d t . \operatorname{Shi}(x)=\int_{0}^{x}\frac{\sinh(t)}{t}\,dt.
  21. Chi ( x ) = γ + ln x + 0 x cosh t - 1 t d t = chi ( x ) \operatorname{Chi}(x)=\gamma+\ln x+\int_{0}^{x}\frac{\cosh t-1}{t}\,dt=% \operatorname{chi}(x)
  22. γ \gamma
  23. Chi ( x ) = γ + ln ( x ) + 1 4 x 2 + 1 96 x 4 + 1 4320 x 6 + 1 322560 x 8 + 1 36288000 x 10 + O ( x 12 ) \operatorname{Chi}(x)=\gamma+\ln(x)+\frac{1}{4}x^{2}+\frac{1}{96}x^{4}+\frac{1% }{4320}x^{6}+\frac{1}{322560}x^{8}+\frac{1}{36288000}x^{10}+O(x^{12})
  24. f ( x ) 0 sin ( t ) t + x d t = 0 e - x t t 2 + 1 d t = Ci ( x ) sin ( x ) + [ π 2 - Si ( x ) ] cos ( x ) f(x)\equiv\int_{0}^{\infty}\frac{\sin(t)}{t+x}dt=\int_{0}^{\infty}\frac{e^{-xt% }}{t^{2}+1}dt=\operatorname{Ci}(x)\sin(x)+\left[\frac{\pi}{2}-\operatorname{Si% }(x)\right]\cos(x)
  25. g ( x ) 0 cos ( t ) t + x d t = 0 t e - x t t 2 + 1 d t = - Ci ( x ) cos ( x ) + [ π 2 - Si ( x ) ] sin ( x ) g(x)\equiv\int_{0}^{\infty}\frac{\cos(t)}{t+x}dt=\int_{0}^{\infty}\frac{te^{-% xt}}{t^{2}+1}dt=-\operatorname{Ci}(x)\cos(x)+\left[\frac{\pi}{2}-\operatorname% {Si}(x)\right]\sin(x)
  26. Si ( x ) = π 2 - f ( x ) cos ( x ) - g ( x ) sin ( x ) Ci ( x ) = f ( x ) sin ( x ) - g ( x ) cos ( x ) . \begin{array}[]{rcl}\operatorname{Si}(x)&=&\frac{\pi}{2}-f(x)\cos(x)-g(x)\sin(% x)\\ \operatorname{Ci}(x)&=&f(x)\sin(x)-g(x)\cos(x).\\ \end{array}
  27. s i , c i si,ci
  28. Si ( x ) = π 2 - cos x x ( 1 - 2 ! x 2 + 4 ! x 4 - 6 ! x 6 ) - sin x x ( 1 x - 3 ! x 3 + 5 ! x 5 - 7 ! x 7 ) \operatorname{Si}(x)=\frac{\pi}{2}-\frac{\cos x}{x}\left(1-\frac{2!}{x^{2}}+% \frac{4!}{x^{4}}-\frac{6!}{x^{6}}\cdots\right)-\frac{\sin x}{x}\left(\frac{1}{% x}-\frac{3!}{x^{3}}+\frac{5!}{x^{5}}-\frac{7!}{x^{7}}\cdots\right)
  29. Ci ( x ) = sin x x ( 1 - 2 ! x 2 + 4 ! x 4 - 6 ! x 6 ) - cos x x ( 1 x - 3 ! x 3 + 5 ! x 5 - 7 ! x 7 ) . \operatorname{Ci}(x)=\frac{\sin x}{x}\left(1-\frac{2!}{x^{2}}+\frac{4!}{x^{4}}% -\frac{6!}{x^{6}}\cdots\right)-\frac{\cos x}{x}\left(\frac{1}{x}-\frac{3!}{x^{% 3}}+\frac{5!}{x^{5}}-\frac{7!}{x^{7}}\cdots\right)~{}.
  30. ( x ) 1 ℜ(x)≫1
  31. Si ( x ) = n = 0 ( - 1 ) n x 2 n + 1 ( 2 n + 1 ) ( 2 n + 1 ) ! = x - x 3 3 ! 3 + x 5 5 ! 5 - x 7 7 ! 7 ± \operatorname{Si}(x)=\sum_{n=0}^{\infty}\frac{(-1)^{n}x^{2n+1}}{(2n+1)(2n+1)!}% =x-\frac{x^{3}}{3!\cdot 3}+\frac{x^{5}}{5!\cdot 5}-\frac{x^{7}}{7!\cdot 7}\pm\cdots
  32. Ci ( x ) = γ + ln x + n = 1 ( - 1 ) n x 2 n 2 n ( 2 n ) ! = γ + ln x - x 2 2 ! 2 + x 4 4 ! 4 \operatorname{Ci}(x)=\gamma+\ln x+\sum_{n=1}^{\infty}\frac{(-1)^{n}x^{2n}}{2n(% 2n)!}=\gamma+\ln x-\frac{x^{2}}{2!\cdot 2}+\frac{x^{4}}{4!\cdot 4}\mp\cdots
  33. x x
  34. E 1 ( z ) = 1 exp ( - z t ) t d t ( ( z ) 0 ) \operatorname{E}_{1}(z)=\int_{1}^{\infty}\frac{\exp(-zt)}{t}\,dt\qquad(\Re(z)% \geq 0)
  35. E 1 ( i x ) = i ( - π 2 + Si ( x ) ) - Ci ( x ) = i si ( x ) - ci ( x ) ( x > 0 ) . \operatorname{E}_{1}(ix)=i\left(-\frac{\pi}{2}+\operatorname{Si}(x)\right)-% \operatorname{Ci}(x)=i\operatorname{si}(x)-\operatorname{ci}(x)\qquad(x>0)~{}.
  36. ( x ) > 0 ℜ(x)>0
  37. π π
  38. 1 cos ( a x ) ln x x d x = - π 2 24 + γ ( γ 2 + ln a ) + ln 2 a 2 + n 1 ( - a 2 ) n ( 2 n ) ! ( 2 n ) 2 , \int_{1}^{\infty}\cos(ax)\frac{\ln x}{x}\,dx=-\frac{\pi^{2}}{24}+\gamma\left(% \frac{\gamma}{2}+\ln a\right)+\frac{\ln^{2}a}{2}+\sum_{n\geq 1}\frac{(-a^{2})^% {n}}{(2n)!(2n)^{2}}~{},
  39. 1 e i a x ln x x d x = - π 2 24 + γ ( γ 2 + ln a ) + ln 2 a 2 - π 2 i ( γ + ln a ) + n 1 ( i a ) n n ! n 2 . \int_{1}^{\infty}e^{iax}\frac{\ln x}{x}\,dx=-\frac{\pi^{2}}{24}+\gamma\left(% \frac{\gamma}{2}+\ln a\right)+\frac{\ln^{2}a}{2}-\frac{\pi}{2}i(\gamma+\ln a)+% \sum_{n\geq 1}\frac{(ia)^{n}}{n!n^{2}}~{}.
  40. 1 e i a x ln x x 2 d x = 1 + i a [ - π 2 24 + γ ( γ 2 + ln a - 1 ) + ln 2 a 2 - ln a + 1 - i π 2 ( γ + ln a - 1 ) ] + n 1 ( i a ) n + 1 ( n + 1 ) ! n 2 . \int_{1}^{\infty}e^{iax}\frac{\ln x}{x^{2}}dx=1+ia[-\frac{\pi^{2}}{24}+\gamma% \left(\frac{\gamma}{2}+\ln a-1\right)+\frac{\ln^{2}a}{2}-\ln a+1-\frac{i\pi}{2% }(\gamma+\ln a-1)]+\sum_{n\geq 1}\frac{(ia)^{n+1}}{(n+1)!n^{2}}~{}.
  41. 0 x 4 0≤x≤4
  42. Si ( x ) = x ( 1 - 4.54393409816329991 10 - 2 x 2 + 1.15457225751016682 10 - 3 x 4 - 1.41018536821330254 10 - 5 x 6 + 9.43280809438713025 10 - 8 x 8 - 3.53201978997168357 10 - 10 x 10 + 7.08240282274875911 10 - 13 x 12 - 6.05338212010422477 10 - 16 x 14 1 + 1.01162145739225565 10 - 2 x 2 + 4.99175116169755106 10 - 5 x 4 + 1.55654986308745614 10 - 7 x 6 + 3.28067571055789734 10 - 10 x 8 + 4.5049097575386581 10 - 13 x 10 + 3.21107051193712168 10 - 16 x 12 ) Ci ( x ) = γ + ln ( x ) + x 2 ( - 0.25 + 7.51851524438898291 10 - 3 x 2 - 1.27528342240267686 10 - 4 x 4 + 1.05297363846239184 10 - 6 x 6 - 4.68889508144848019 10 - 9 x 8 + 1.06480802891189243 10 - 11 x 10 - 9.93728488857585407 10 - 15 x 12 1 + 1.1592605689110735 10 - 2 x 2 + 6.72126800814254432 10 - 5 x 4 + 2.55533277086129636 10 - 7 x 6 + 6.97071295760958946 10 - 10 x 8 + 1.38536352772778619 10 - 12 x 10 + 1.89106054713059759 10 - 15 x 12 + 1.39759616731376855 10 - 18 x 14 ) \begin{array}[]{rcl}\operatorname{Si}(x)&=&x\cdot\left(\frac{\begin{array}[]{l% }1-4.54393409816329991\cdot 10^{-2}\cdot x^{2}+1.15457225751016682\cdot 10^{-3% }\cdot x^{4}-1.41018536821330254\cdot 10^{-5}\cdot x^{6}\\ ~{}~{}~{}+9.43280809438713025\cdot 10^{-8}\cdot x^{8}-3.53201978997168357\cdot 1% 0^{-10}\cdot x^{10}+7.08240282274875911\cdot 10^{-13}\cdot x^{12}\\ ~{}~{}~{}-6.05338212010422477\cdot 10^{-16}\cdot x^{14}\end{array}}{\begin{% array}[]{l}1+1.01162145739225565\cdot 10^{-2}\cdot x^{2}+4.99175116169755106% \cdot 10^{-5}\cdot x^{4}+1.55654986308745614\cdot 10^{-7}\cdot x^{6}\\ ~{}~{}~{}+3.28067571055789734\cdot 10^{-10}\cdot x^{8}+4.5049097575386581\cdot 1% 0^{-13}\cdot x^{10}+3.21107051193712168\cdot 10^{-16}\cdot x^{12}\end{array}}% \right)\\ &&\\ \operatorname{Ci}(x)&=&\gamma+\ln(x)+\\ &&x^{2}\cdot\left(\frac{\begin{array}[]{l}-0.25+7.51851524438898291\cdot 10^{-% 3}\cdot x^{2}-1.27528342240267686\cdot 10^{-4}\cdot x^{4}+1.05297363846239184% \cdot 10^{-6}\cdot x^{6}\\ ~{}~{}~{}-4.68889508144848019\cdot 10^{-9}\cdot x^{8}+1.06480802891189243\cdot 1% 0^{-11}\cdot x^{10}-9.93728488857585407\cdot 10^{-15}\cdot x^{12}\\ \end{array}}{\begin{array}[]{l}1+1.1592605689110735\cdot 10^{-2}\cdot x^{2}+6.% 72126800814254432\cdot 10^{-5}\cdot x^{4}+2.55533277086129636\cdot 10^{-7}% \cdot x^{6}\\ ~{}~{}~{}+6.97071295760958946\cdot 10^{-10}\cdot x^{8}+1.38536352772778619% \cdot 10^{-12}\cdot x^{10}+1.89106054713059759\cdot 10^{-15}\cdot x^{12}\\ ~{}~{}~{}+1.39759616731376855\cdot 10^{-18}\cdot x^{14}\\ \end{array}}\right)\end{array}
  43. x x
  44. f ( x ) a n d g ( x ) f(x)andg(x)
  45. 1 y f ( 1 y ) \;\;\frac{1}{\sqrt{y}}\;f\left(\frac{1}{\sqrt{y}}\right)\;\;
  46. 1 y g ( 1 y ) \;\;\frac{1}{y}\;g\left(\frac{1}{\sqrt{y}}\right)\;\;
  47. x 4 x\geq 4
  48. f ( x ) = 1 x ( 1 + 7.44437068161936700618 10 2 x - 2 + 1.96396372895146869801 10 5 x - 4 + 2.37750310125431834034 10 7 x - 6 + 1.43073403821274636888 10 9 x - 8 + 4.33736238870432522765 10 10 x - 10 + 6.40533830574022022911 10 11 x - 12 + 4.20968180571076940208 10 12 x - 14 + 1.00795182980368574617 10 13 x - 16 + 4.94816688199951963482 10 12 x - 18 - 4.94701168645415959931 10 11 x - 20 1 + 7.46437068161927678031 10 2 x - 2 + 1.97865247031583951450 10 5 x - 4 + 2.41535670165126845144 10 7 x - 6 + 1.47478952192985464958 10 9 x - 8 + 4.58595115847765779830 10 10 x - 10 + 7.08501308149515401563 10 11 x - 12 + 5.06084464593475076774 10 12 x - 14 + 1.43468549171581016479 10 13 x - 16 + 1.11535493509914254097 10 13 x - 18 ) g ( x ) = 1 x 2 ( 1 + 8.1359520115168615 10 2 x - 2 + 2.35239181626478200 10 5 x - 4 + 3.12557570795778731 10 7 x - 6 + 2.06297595146763354 10 9 x - 8 + 6.83052205423625007 10 10 x - 10 + 1.09049528450362786 10 12 x - 12 + 7.57664583257834349 10 12 x - 14 + 1.81004487464664575 10 13 x - 16 + 6.43291613143049485 10 12 x - 18 - 1.36517137670871689 10 12 x - 20 1 + 8.19595201151451564 10 2 x - 2 + 2.40036752835578777 10 5 x - 4 + 3.26026661647090822 10 7 x - 6 + 2.23355543278099360 10 9 x - 8 + 7.87465017341829930 10 10 x - 10 + 1.39866710696414565 10 12 x - 12 + 1.17164723371736605 10 13 x - 14 + 4.01839087307656620 10 13 x - 16 + 3.99653257887490811 10 13 x - 18 ) \begin{array}[]{rcl}f(x)&=&\dfrac{1}{x}\cdot\left(\frac{\begin{array}[]{l}1+7.% 44437068161936700618\cdot 10^{2}\cdot x^{-2}+1.96396372895146869801\cdot 10^{5% }\cdot x^{-4}+2.37750310125431834034\cdot 10^{7}\cdot x^{-6}\\ ~{}~{}~{}+1.43073403821274636888\cdot 10^{9}\cdot x^{-8}+4.3373623887043252276% 5\cdot 10^{10}\cdot x^{-10}+6.40533830574022022911\cdot 10^{11}\cdot x^{-12}\\ ~{}~{}~{}+4.20968180571076940208\cdot 10^{12}\cdot x^{-14}+1.00795182980368574% 617\cdot 10^{13}\cdot x^{-16}+4.94816688199951963482\cdot 10^{12}\cdot x^{-18}% \\ ~{}~{}~{}-4.94701168645415959931\cdot 10^{11}\cdot x^{-20}\end{array}}{\begin{% array}[]{l}1+7.46437068161927678031\cdot 10^{2}\cdot x^{-2}+1.9786524703158395% 1450\cdot 10^{5}\cdot x^{-4}+2.41535670165126845144\cdot 10^{7}\cdot x^{-6}\\ ~{}~{}~{}+1.47478952192985464958\cdot 10^{9}\cdot x^{-8}+4.5859511584776577983% 0\cdot 10^{10}\cdot x^{-10}+7.08501308149515401563\cdot 10^{11}\cdot x^{-12}\\ ~{}~{}~{}+5.06084464593475076774\cdot 10^{12}\cdot x^{-14}+1.43468549171581016% 479\cdot 10^{13}\cdot x^{-16}+1.11535493509914254097\cdot 10^{13}\cdot x^{-18}% \end{array}}\right)\\ &&\\ g(x)&=&\dfrac{1}{x^{2}}\cdot\left(\frac{\begin{array}[]{l}1+8.1359520115168615% \cdot 10^{2}\cdot x^{-2}+2.35239181626478200\cdot 10^{5}\cdot x^{-4}+3.1255757% 0795778731\cdot 10^{7}\cdot x^{-6}\\ ~{}~{}~{}+2.06297595146763354\cdot 10^{9}\cdot x^{-8}+6.83052205423625007\cdot 1% 0^{10}\cdot x^{-10}+1.09049528450362786\cdot 10^{12}\cdot x^{-12}\\ ~{}~{}~{}+7.57664583257834349\cdot 10^{12}\cdot x^{-14}+1.81004487464664575% \cdot 10^{13}\cdot x^{-16}+6.43291613143049485\cdot 10^{12}\cdot x^{-18}\\ ~{}~{}~{}-1.36517137670871689\cdot 10^{12}\cdot x^{-20}\end{array}}{\begin{% array}[]{l}1+8.19595201151451564\cdot 10^{2}\cdot x^{-2}+2.40036752835578777% \cdot 10^{5}\cdot x^{-4}+3.26026661647090822\cdot 10^{7}\cdot x^{-6}\\ ~{}~{}~{}+2.23355543278099360\cdot 10^{9}\cdot x^{-8}+7.87465017341829930\cdot 1% 0^{10}\cdot x^{-10}+1.39866710696414565\cdot 10^{12}\cdot x^{-12}\\ ~{}~{}~{}+1.17164723371736605\cdot 10^{13}\cdot x^{-14}+4.01839087307656620% \cdot 10^{13}\cdot x^{-16}+3.99653257887490811\cdot 10^{13}\cdot x^{-18}\end{% array}}\right)\\ \end{array}

Trigonometric_tables.html

  1. cos ( x 2 ) = ± 1 2 ( 1 + cos x ) \cos\left(\frac{x}{2}\right)=\pm\sqrt{\tfrac{1}{2}(1+\cos x)}
  2. sin ( x 2 ) = ± 1 2 ( 1 - cos x ) \sin\left(\frac{x}{2}\right)=\pm\sqrt{\tfrac{1}{2}(1-\cos x)}
  3. sin ( x ± y ) = sin ( x ) cos ( y ) ± cos ( x ) sin ( y ) \sin(x\pm y)=\sin(x)\cos(y)\pm\cos(x)\sin(y)\,
  4. cos ( x ± y ) = cos ( x ) cos ( y ) sin ( x ) sin ( y ) \cos(x\pm y)=\cos(x)\cos(y)\mp\sin(x)\sin(y)\,
  5. d s / d t = c ds/dt=c
  6. d c / d t = - s dc/dt=-s
  7. e i ( θ + Δ ) = e i θ × e i Δ θ e^{i(\theta+\Delta)}=e^{i\theta}\times e^{i\Delta\theta}

Trinification.html

  1. S U ( 3 ) C × S U ( 3 ) L × S U ( 3 ) R SU(3)_{C}\times SU(3)_{L}\times SU(3)_{R}
  2. [ S U ( 3 ) C × S U ( 3 ) L × S U ( 3 ) R ] / 3 [SU(3)_{C}\times SU(3)_{L}\times SU(3)_{R}]/\mathbb{Z}_{3}
  3. ( 3 , 3 ¯ , 1 ) (3,\bar{3},1)
  4. ( 3 ¯ , 1 , 3 ) (\bar{3},1,3)
  5. ( 1 , 3 , 3 ¯ ) (1,3,\bar{3})
  6. ( 1 , 3 , 3 ¯ ) (1,3,\bar{3})
  7. ( 1 , 3 ¯ , 3 ) (1,\bar{3},3)
  8. S U ( 3 ) L × S U ( 3 ) R SU(3)_{L}\times SU(3)_{R}
  9. [ S U ( 2 ) × U ( 1 ) ] / 2 [SU(2)\times U(1)]/\mathbb{Z}_{2}
  10. ( 3 , 3 ¯ , 1 ) ( 3 , 2 ) 1 6 ( 3 , 1 ) - 1 3 (3,\bar{3},1)\rightarrow(3,2)_{\frac{1}{6}}\oplus(3,1)_{-\frac{1}{3}}
  11. ( 3 ¯ , 1 , 3 ) 2 ( 3 ¯ , 1 ) 1 3 ( 3 ¯ , 1 ) - 2 3 (\bar{3},1,3)\rightarrow 2\,(\bar{3},1)_{\frac{1}{3}}\oplus(\bar{3},1)_{-\frac% {2}{3}}
  12. ( 1 , 3 , 3 ¯ ) 2 ( 1 , 2 ) - 1 2 ( 1 , 2 ) 1 2 2 ( 1 , 1 ) 0 ( 1 , 1 ) 1 (1,3,\bar{3})\rightarrow 2\,(1,2)_{-\frac{1}{2}}\oplus(1,2)_{\frac{1}{2}}% \oplus 2\,(1,1)_{0}\oplus(1,1)_{1}
  13. ( 8 , 1 , 1 ) ( 8 , 1 ) 0 (8,1,1)\rightarrow(8,1)_{0}
  14. ( 1 , 8 , 1 ) ( 1 , 3 ) 0 ( 1 , 2 ) 1 2 ( 1 , 2 ) - 1 2 ( 1 , 1 ) 0 (1,8,1)\rightarrow(1,3)_{0}\oplus(1,2)_{\frac{1}{2}}\oplus(1,2)_{-\frac{1}{2}}% \oplus(1,1)_{0}
  15. ( 1 , 1 , 8 ) 4 ( 1 , 1 ) 0 2 ( 1 , 1 ) 1 2 ( 1 , 1 ) - 1 (1,1,8)\rightarrow 4\,(1,1)_{0}\oplus 2\,(1,1)_{1}\oplus 2\,(1,1)_{-1}
  16. ( 3 , 1 ) - 1 3 (3,1)_{-\frac{1}{3}}
  17. ( 3 ¯ , 1 ) 1 3 (\bar{3},1)_{\frac{1}{3}}
  18. ( 1 , 2 ) 1 2 (1,2)_{\frac{1}{2}}
  19. ( 1 , 2 ) - 1 2 (1,2)_{-\frac{1}{2}}
  20. ( 1 , 3 , 3 ¯ ) H ( 3 , 3 ¯ , 1 ) ( 3 ¯ , 1 , 3 ) (1,3,\bar{3})_{H}(3,\bar{3},1)(\bar{3},1,3)
  21. ( 1 , 3 , 3 ¯ ) H ( 1 , 3 , 3 ¯ ) ( 1 , 3 , 3 ¯ ) (1,3,\bar{3})_{H}(1,3,\bar{3})(1,3,\bar{3})
  22. ( 3 , 3 ¯ , 1 ) (3,\bar{3},1)
  23. π 2 ( S U ( 3 ) × S U ( 3 ) [ S U ( 2 ) × U ( 1 ) ] / 2 ) = \pi_{2}\left(\frac{SU(3)\times SU(3)}{[SU(2)\times U(1)]/\mathbb{Z}_{2}}\right% )=\mathbb{Z}

Triviality_(mathematics).html

  1. y = y y^{\prime}=y
  2. f ′′ ( x ) = - λ f ( x ) f^{\prime\prime}(x)=-\lambda f(x)
  3. f ( 0 ) = f ( L ) = 0 f(0)=f(L)=0
  4. f ( x ) = 0 f(x)=0
  5. a n + b n = c n a^{n}+b^{n}=c^{n}
  6. a = b = c = 0 a=b=c=0
  7. 0 1 x 2 d x = 1 3 \int_{0}^{1}x^{2}\,dx=\frac{1}{3}
  8. X Y X\to Y
  9. X Y X\to Y

Trouton–Noble_experiment.html

  1. τ = - E v 2 c 2 sin 2 α \tau=-E^{\prime}\frac{v^{2}}{c^{2}}\sin 2\alpha^{\prime}
  2. τ \tau
  3. E E
  4. α \alpha
  5. f y f_{y}
  6. f x f_{x}
  7. τ = L 0 ( f x - f y ) = 0 \tau^{\prime}=L_{0}\left(f^{\prime}_{x}-f^{\prime}_{y}\right)=0
  8. τ \tau
  9. L 0 L_{0}
  10. τ = f x L 0 - f y L 0 1 - v 2 c 2 = L 0 ( f x - f y 1 - v 2 c 2 ) \tau=f_{x}\cdot L_{0}-f_{y}\cdot L_{0}\sqrt{1-\frac{v^{2}}{c^{2}}}=L_{0}\left(% f_{x}-f_{y}\sqrt{1-\frac{v^{2}}{c^{2}}}\right)
  11. f x f y = 1 - v 2 c 2 \frac{f_{x}}{f_{y}}=\sqrt{1-\frac{v^{2}}{c^{2}}}
  12. f x = f x , f y = f y 1 - v 2 c 2 f_{x}=f^{\prime}_{x},\ f_{y}=f^{\prime}_{y}\cdot\sqrt{1-\frac{v^{2}}{c^{2}}}
  13. f x f y = 1 1 - v 2 c 2 \frac{f_{x}}{f_{y}}=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}
  14. τ = - L 0 f x v 2 c 2 \tau=-L_{0}\cdot f^{\prime}_{x}\cdot\frac{v^{2}}{c^{2}}
  15. τ = E v 2 c 2 sin 2 α \tau=E^{\prime}\frac{v^{2}}{c^{2}}\sin 2\alpha^{\prime}
  16. τ = L 0 f x v 2 c 2 \tau=L_{0}\cdot f^{\prime}_{x}\cdot\frac{v^{2}}{c^{2}}
  17. tan α \tan\alpha\!
  18. f x f y = tan α \tfrac{f^{\prime}_{x}}{f^{\prime}_{y}}=\tan\alpha^{\prime}
  19. f x = f x , f y = f y 1 - v 2 c 2 , tan α = tan α 1 - v 2 c 2 f_{x}=f^{\prime}_{x},\ f_{y}=f^{\prime}_{y}\cdot\sqrt{1-\frac{v^{2}}{c^{2}}},% \ \tan\alpha=\tan\alpha^{\prime}\sqrt{1-\frac{v^{2}}{c^{2}}}
  20. f x f y = tan α 1 - v 2 c 2 \frac{f_{x}}{f_{y}}=\frac{\tan\alpha}{1-\frac{v^{2}}{c^{2}}}
  21. a x = f x m γ 3 , a y = f y m γ a_{x}=\frac{f_{x}}{m\gamma^{3}},\ a_{y}=\frac{f_{y}}{m\gamma}
  22. γ = 1 1 - v 2 c 2 \gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}
  23. a x a y = tan α \frac{a_{x}}{a_{y}}=\tan\alpha

Truncated_cube.html

  1. 2 + 2 \scriptstyle{2+\sqrt{2}}
  2. A = 2 ( 6 + 6 2 + 3 ) a 2 32.4346644 a 2 A=2\left(6+6\sqrt{2}+\sqrt{3}\right)a^{2}\approx 32.4346644a^{2}
  3. V = 1 3 ( 21 + 14 2 ) a 3 13.5996633 a 3 . V=\frac{1}{3}\left(21+14\sqrt{2}\right)a^{3}\approx 13.5996633a^{3}.
  4. 2 - 1 \scriptstyle{\sqrt{2}-1}

Truncated_cuboctahedron.html

  1. A = 12 ( 2 + 2 + 3 ) a 2 61.7551724 a 2 A=12\left(2+\sqrt{2}+\sqrt{3}\right)a^{2}\approx 61.7551724a^{2}
  2. V = ( 22 + 14 2 ) a 3 41.7989899 a 3 . V=\left(22+14\sqrt{2}\right)a^{3}\approx 41.7989899a^{3}.

Truncated_icosidodecahedron.html

  1. A = 30 a 2 ( 1 + 3 + 5 + 2 5 ) 174.2920303 a 2 . A=30a^{2}\left(1+\sqrt{3}+\sqrt{5+2\sqrt{5}}\right)\approx 174.2920303a^{2}.
  2. V = ( 95 + 50 5 ) a 3 206.803399 a 3 . V=(95+50\sqrt{5})a^{3}\approx 206.803399a^{3}.

Truncated_octahedron.html

  1. 9 8 2 \tfrac{9}{8}\scriptstyle{\sqrt{2}}
  2. 3 2 2 \tfrac{3}{2}\scriptstyle{\sqrt{2}}
  3. h = e 2 - 1 2 a 2 = 2 2 a h=\sqrt{e^{2}-\frac{1}{2}a^{2}}=\frac{\sqrt{2}}{2}a
  4. s = h 2 + 1 4 a 2 = 1 2 a 2 + 1 4 a 2 = 3 2 a s=\sqrt{h^{2}+\frac{1}{4}a^{2}}=\sqrt{\frac{1}{2}a^{2}+\frac{1}{4}a^{2}}=\frac% {\sqrt{3}}{2}a
  5. V = 1 3 a 2 h = 2 6 a 3 V=\frac{1}{3}a^{2}h=\frac{\sqrt{2}}{6}a^{3}
  6. 2 a 3 \scriptstyle{\sqrt{2}a^{3}}
  7. ( 0 , ± 1 , ± 1 ) (0,±1,±1)
  8. ( 0 , 0 , ± 1 ) (0,0,±1)
  9. ( 0 , ± 1 , 0 ) (0,±1,0)
  10. ( ± 1 , 0 , 0 ) (±1,0,0)
  11. ( ± 1 / 3 , ± 1 / 3 , ± 1 / 3 ) (±1/√3,±1/√3,±1/√3)
  12. - 1 / 3 -1/3
  13. - 1 / 3 -1/√3
  14. A = ( 6 + 12 3 ) a 2 26.7846097 a 2 A=\left(6+12\sqrt{3}\right)a^{2}\approx 26.7846097a^{2}
  15. V = 8 2 a 3 11.3137085 a 3 . V=8\sqrt{2}a^{3}\approx 11.3137085a^{3}.

Truncated_tetrahedron.html

  1. A = 7 3 a 2 12.12435565 a 2 A=7\sqrt{3}a^{2}\approx 12.12435565a^{2}
  2. V = 23 12 2 a 3 2.710575995 a 3 . V=\frac{23}{12}\sqrt{2}a^{3}\approx 2.710575995a^{3}.

Truncation.html

  1. x + x\in\mathbb{R}_{+}
  2. n 0 n\in\mathbb{N}_{0}
  3. trunc ( x , n ) = 10 n x 10 n . \operatorname{trunc}(x,n)=\frac{\lfloor 10^{n}\cdot x\rfloor}{10^{n}}.

TSU.html

  1. t s u t_{su}

Tuned_mass_damper.html

  1. m 1 m_{1}
  2. k 1 k_{1}
  3. c 1 c_{1}
  4. F 0 F_{0}
  5. m 2 m_{2}
  6. m 1 m_{1}
  7. k 2 k_{2}
  8. c 2 c_{2}
  9. F 1 F_{1}
  10. m 1 m_{1}
  11. F 0 / F 1 F_{0}/F_{1}
  12. m 2 m_{2}
  13. m 2 = 0 m_{2}=0
  14. m 2 = m 1 / 10 m_{2}=m_{1}/10
  15. m 2 m_{2}
  16. m 1 m_{1}
  17. m 1 m_{1}
  18. x 2 - x 1 x_{2}-x_{1}
  19. c 2 c_{2}

Tuple.html

  1. n n
  2. n n
  3. n n
  4. ( ) (\,\text{ })
  5. ( 2 , 7 , 4 , 1 , 7 ) (2,7,4,1,7)
  6. \langle\,\text{ }\rangle
  7. n n
  8. n n
  9. ( a 1 , a 2 , , a n ) = ( b 1 , b 2 , , b n ) (a_{1},a_{2},\ldots,a_{n})=(b_{1},b_{2},\ldots,b_{n})
  10. a 1 = b 1 , a 2 = b 2 , , a n = b n . a_{1}=b_{1},\,\text{ }a_{2}=b_{2},\,\text{ }\ldots,\,\text{ }a_{n}=b_{n}.
  11. ( 1 , 2 , 2 , 3 ) ( 1 , 2 , 3 ) (1,2,2,3)\neq(1,2,3)
  12. { 1 , 2 , 2 , 3 } = { 1 , 2 , 3 } \{1,2,2,3\}=\{1,2,3\}
  13. ( 1 , 2 , 3 ) ( 3 , 2 , 1 ) (1,2,3)\neq(3,2,1)
  14. { 1 , 2 , 3 } = { 3 , 2 , 1 } \{1,2,3\}=\{3,2,1\}
  15. n n
  16. ( a 1 , a 2 , , a n ) ( X , Y , F ) (a_{1},a_{2},\dots,a_{n})\equiv(X,Y,F)
  17. X \displaystyle X
  18. ( a 1 , a 2 , , a n ) := ( F ( 1 ) , F ( 2 ) , , F ( n ) ) . (a_{1},a_{2},\dots,a_{n}):=(F(1),F(2),\dots,F(n)).
  19. \emptyset
  20. n n
  21. n > 0 n>0
  22. ( n - 1 ) (n-1)
  23. n > 1 n>1
  24. ( a 1 , a 2 , a 3 , , a n ) = ( a 1 , ( a 2 , a 3 , , a n ) ) (a_{1},a_{2},a_{3},\ldots,a_{n})=(a_{1},(a_{2},a_{3},\ldots,a_{n}))
  25. ( n - 1 ) (n-1)
  26. ( a 1 , a 2 , a 3 , , a n ) = ( a 1 , ( a 2 , ( a 3 , ( , ( a n , ) ) ) ) ) (a_{1},a_{2},a_{3},\ldots,a_{n})=(a_{1},(a_{2},(a_{3},(\ldots,(a_{n},\emptyset% )\ldots))))
  27. ( 1 , 2 , 3 ) \displaystyle(1,2,3)
  28. \emptyset
  29. n > 0 n>0
  30. ( a 1 , a 2 , a 3 , , a n ) = ( ( a 1 , a 2 , a 3 , , a n - 1 ) , a n ) (a_{1},a_{2},a_{3},\ldots,a_{n})=((a_{1},a_{2},a_{3},\ldots,a_{n-1}),a_{n})
  31. ( a 1 , a 2 , a 3 , , a n ) = ( ( ( ( ( , a 1 ) , a 2 ) , a 3 ) , ) , a n ) (a_{1},a_{2},a_{3},\ldots,a_{n})=((\ldots(((\emptyset,a_{1}),a_{2}),a_{3}),% \ldots),a_{n})
  32. ( 1 , 2 , 3 ) \displaystyle(1,2,3)
  33. \emptyset
  34. x x
  35. n n
  36. ( a 1 , a 2 , , a n ) (a_{1},a_{2},\ldots,a_{n})
  37. x b ( a 1 , a 2 , , a n , b ) x\rightarrow b\equiv(a_{1},a_{2},\ldots,a_{n},b)
  38. x b { { x } , { x , b } } x\rightarrow b\equiv\{\{x\},\{x,b\}\}
  39. \rightarrow
  40. ( ) = ( 1 ) = ( ) 1 = { { ( ) } , { ( ) , 1 } } = { { } , { , 1 } } ( 1 , 2 ) = ( 1 ) 2 = { { ( 1 ) } , { ( 1 ) , 2 } } = { { { { } , { , 1 } } } , { { { } , { , 1 } } , 2 } } ( 1 , 2 , 3 ) = ( 1 , 2 ) 3 = { { ( 1 , 2 ) } , { ( 1 , 2 ) , 3 } } = { { { { { { } , { , 1 } } } , { { { } , { , 1 } } , 2 } } } , { { { { { } , { , 1 } } } , { { { } , { , 1 } } , 2 } } , 3 } } \begin{array}[]{lclcl}()&&&=&\emptyset\\ &&&&\\ (1)&=&()\rightarrow 1&=&\{\{()\},\{(),1\}\}\\ &&&=&\{\{\emptyset\},\{\emptyset,1\}\}\\ &&&&\\ (1,2)&=&(1)\rightarrow 2&=&\{\{(1)\},\{(1),2\}\}\\ &&&=&\{\{\{\{\emptyset\},\{\emptyset,1\}\}\},\\ &&&&\{\{\{\emptyset\},\{\emptyset,1\}\},2\}\}\\ &&&&\\ (1,2,3)&=&(1,2)\rightarrow 3&=&\{\{(1,2)\},\{(1,2),3\}\}\\ &&&=&\{\{\{\{\{\{\emptyset\},\{\emptyset,1\}\}\},\\ &&&&\{\{\{\emptyset\},\{\emptyset,1\}\},2\}\}\},\\ &&&&\{\{\{\{\{\emptyset\},\{\emptyset,1\}\}\},\\ &&&&\{\{\{\emptyset\},\{\emptyset,1\}\},2\}\},3\}\}\\ \end{array}
  41. ( x 1 , x 2 , , x n ) : 𝖳 1 × 𝖳 2 × × 𝖳 n (x_{1},x_{2},\ldots,x_{n}):\mathsf{T}_{1}\times\mathsf{T}_{2}\times\ldots% \times\mathsf{T}_{n}
  42. π 1 ( x ) : 𝖳 1 , π 2 ( x ) : 𝖳 2 , , π n ( x ) : 𝖳 n \pi_{1}(x):\mathsf{T}_{1},~{}\pi_{2}(x):\mathsf{T}_{2},~{}\ldots,~{}\pi_{n}(x)% :\mathsf{T}_{n}
  43. S 1 , S 2 , , S n S_{1},S_{2},\ldots,S_{n}
  44. [ [ 𝖳 1 ] ] = S 1 , [ [ 𝖳 2 ] ] = S 2 , , [ [ 𝖳 n ] ] = S n [\![\mathsf{T}_{1}]\!]=S_{1},~{}[\![\mathsf{T}_{2}]\!]=S_{2},~{}\ldots,~{}[\![% \mathsf{T}_{n}]\!]=S_{n}
  45. [ [ x 1 ] ] [ [ 𝖳 1 ] ] , [ [ x 2 ] ] [ [ 𝖳 2 ] ] , , [ [ x n ] ] [ [ 𝖳 n ] ] [\![x_{1}]\!]\in[\![\mathsf{T}_{1}]\!],~{}[\![x_{2}]\!]\in[\![\mathsf{T}_{2}]% \!],~{}\ldots,~{}[\![x_{n}]\!]\in[\![\mathsf{T}_{n}]\!]
  46. n n
  47. n n
  48. [ [ ( x 1 , x 2 , , x n ) ] ] = ( [ [ x 1 ] ] , [ [ x 2 ] ] , , [ [ x n ] ] ) [\![(x_{1},x_{2},\ldots,x_{n})]\!]=(\,[\![x_{1}]\!],[\![x_{2}]\!],\ldots,[\![x% _{n}]\!]\,)

Turbofan.html

  1. F n = m ( V j f e - V a ) F_{n}=m\cdot(V_{jfe}-V_{a})

Turbojet.html

  1. F N F_{N}\;
  2. F N = ( m ˙ a i r + m ˙ f ) V j - m ˙ a i r V F_{N}=(\dot{m}_{air}+\dot{m}_{f})V_{j}-\dot{m}_{air}V
  3. m ˙ a i r \dot{m}_{air}
  4. m ˙ f \dot{m}_{f}
  5. V j V_{j}\;
  6. V V\;
  7. ( m ˙ a i r + m ˙ f ) V j (\dot{m}_{air}+\dot{m}_{f})V_{j}
  8. m ˙ a i r V \dot{m}_{air}V
  9. F N = m ˙ a i r ( V j - V ) F_{N}=\dot{m}_{air}(V_{j}-V)
  10. V j V_{j}\;
  11. V V\;
  12. V j V_{j}\;

Turbulence.html

  1. v = ( v x , v y ) v=\left({{v}_{x}},{{v}_{y}}\right)
  2. v x = v x ¯ mean value + v x fluctuation , v y = v y ¯ + v y {{v}_{x}}=\underbrace{\overline{{{v}_{x}}}}_{\begin{smallmatrix}\,\text{mean}% \\ \,\text{value}\end{smallmatrix}}+\underbrace{{{{{v}^{\prime}}}_{x}}}_{\,\text{% fluctuation}}\,\text{ }\,\text{, }{{v}_{y}}=\overline{{{v}_{y}}}+{{{v}^{\prime% }}_{y}}
  3. ( T = T ¯ + T ) \left(T=\overline{T}+{T}^{\prime}\right)
  4. ( P = P ¯ + P ) \left(P=\overline{P}+{P}^{\prime}\right)
  5. τ \tau
  6. q = v y ρ c P T experimental value = - k turb T ¯ y τ = - ρ v y v x ¯ experimental value = μ turb v x ¯ y \begin{aligned}&\displaystyle q=\underbrace{{{{{v}^{\prime}}}_{y}}\rho{{c}_{P}% }{T}^{\prime}}_{\,\text{experimental value}}=-{{k}_{\,\text{turb}}}\frac{% \partial\overline{T}}{\partial y}\\ &\displaystyle\tau=\underbrace{-\rho\overline{{{{{v}^{\prime}}}_{y}}{{{{v}^{% \prime}}}_{x}}}}_{\,\text{experimental value}}={{\mu}_{\,\text{turb}}}\frac{% \partial\overline{{{v}_{x}}}}{\partial y}\\ \end{aligned}
  7. c P {{c}_{P}}
  8. ρ \rho
  9. μ turb {{\mu}_{\,\text{turb}}}
  10. k turb {{k}_{\,\text{turb}}}
  11. ν \nu
  12. ε \varepsilon
  13. η = ( ν 3 ε ) 1 / 4 \eta=\left(\frac{\nu^{3}}{\varepsilon}\right)^{1/4}
  14. η \eta
  15. η r L \eta\ll r\ll L
  16. η r L \eta\ll r\ll L
  17. ε \varepsilon
  18. E ( k ) E(k)
  19. 𝐮 ( 𝐱 ) = 3 𝐮 ^ ( 𝐤 ) e i 𝐤 𝐱 d 3 𝐤 \mathbf{u}(\mathbf{x})=\iiint_{\mathbb{R}^{3}}\widehat{\mathbf{u}}(\mathbf{k})% e^{i\mathbf{k\cdot x}}\mathrm{d}^{3}\mathbf{k}
  20. 1 / 2 u i u i 1/2\langle u_{i}u_{i}\rangle
  21. k = 2 π / r k=2\pi/r
  22. E ( k ) = C ε 2 / 3 k - 5 / 3 E(k)=C\varepsilon^{2/3}k^{-5/3}
  23. δ 𝐮 ( r ) = 𝐮 ( 𝐱 + 𝐫 ) - 𝐮 ( 𝐱 ) \delta\mathbf{u}(r)=\mathbf{u}(\mathbf{x}+\mathbf{r})-\mathbf{u}(\mathbf{x})
  24. β \beta
  25. λ \lambda
  26. δ 𝐮 ( λ r ) \delta\mathbf{u}(\lambda r)
  27. λ β δ 𝐮 ( r ) \lambda^{\beta}\delta\mathbf{u}(r)
  28. β \beta
  29. [ δ 𝐮 ( r ) ] n = C n ε n / 3 r n / 3 \langle[\delta\mathbf{u}(r)]^{n}\rangle=C_{n}\varepsilon^{n/3}r^{n/3}
  30. C n C_{n}
  31. E ( k ) k - p E(k)\propto k^{-p}
  32. 1 < p < 3 1<p<3
  33. [ δ 𝐮 ( r ) ] 2 r p - 1 \langle[\delta\mathbf{u}(r)]^{2}\rangle\propto r^{p-1}
  34. C n C_{n}

Twistor_theory.html

  1. 6 \mathbb{R}^{6}
  2. 𝐑 5 \mathbf{R}\mathbb{P}^{5}
  3. 6 \mathbb{R}^{6}
  4. 𝕄 c \mathbb{M}^{c}
  5. 𝐑 5 \mathbf{R}\mathbb{P}^{5}
  6. 𝕋 \mathbb{T}
  7. 𝕋 \mathbb{PT}
  8. 𝕋 + \mathbb{PT}^{+}
  9. 𝕋 \mathbb{PT}
  10. \mathbb{PN}
  11. 𝕋 \mathbb{PT}
  12. 𝕋 - \mathbb{PT}^{-}
  13. 𝕋 \mathbb{PT}
  14. 𝕄 c \mathbb{M}^{c}
  15. 𝕋 + \mathbb{PT}^{+}
  16. \mathbb{PN}
  17. 𝕋 - \mathbb{PT}^{-}
  18. 𝕄 c \mathbb{M}^{c}
  19. 𝕄 c \mathbb{M}^{c}
  20. \mathbb{PN}
  21. 𝕄 c \mathbb{M}^{c}
  22. 𝕋 + SU ( 2 , 2 ) / [ SU ( 2 , 1 ) × U ( 1 ) ] \mathbb{PT}^{+}\simeq\mathrm{SU}(2,2)/\left[\mathrm{SU}(2,1)\times\mathrm{U}(1% )\right]
  23. 3 | 4 \mathbb{CP}^{3|4}

Two's_complement.html

  1. N N
  2. N N
  3. N N
  4. N N
  5. N N
  6. x x
  7. x x
  8. x x
  9. w w
  10. N N
  11. a N - 1 a N - 2 a 0 a_{N-1}a_{N-2}\dots a_{0}
  12. w = - a N - 1 2 N - 1 + i = 0 N - 2 a i 2 i w=-a_{N-1}2^{N-1}+\sum_{i=0}^{N-2}a_{i}2^{i}
  13. N N
  14. N N
  15. N N
  16. x * x*
  17. x x
  18. N = 4 N=4
  19. 𝐙 / 2 k \mathbf{Z}/2^{k}
  20. N N
  21. ( 2 < s u p > N 1 1 ) (2<sup>N− 1−1)
  22. x = 0 x=0
  23. 2 < s u p > N 0 = 2 N 2<sup>N−0=2^{N}

Two-body_problem.html

  1. s y m b o l R = m 1 M s y m b o l x 1 + m 2 M s y m b o l x 2 symbol{R}=\frac{m_{1}}{M}symbol{x}_{1}+\frac{m_{2}}{M}symbol{x}_{2}
  2. s y m b o l r = s y m b o l x 1 - s y m b o l x 2 symbol{r}=symbol{x}_{1}-symbol{x}_{2}
  3. M = m 1 + m 2 M=m_{1}+m_{2}
  4. 𝐅 12 ( 𝐱 1 , 𝐱 2 ) = m 1 𝐱 ¨ 1 ( Equation 1 ) \mathbf{F}_{12}(\mathbf{x}_{1},\mathbf{x}_{2})=m_{1}\ddot{\mathbf{x}}_{1}\quad% \quad\quad(\mathrm{Equation}\ 1)
  5. 𝐅 21 ( 𝐱 1 , 𝐱 2 ) = m 2 𝐱 ¨ 2 ( Equation 2 ) \mathbf{F}_{21}(\mathbf{x}_{1},\mathbf{x}_{2})=m_{2}\ddot{\mathbf{x}}_{2}\quad% \quad\quad(\mathrm{Equation}\ 2)
  6. m 1 𝐱 ¨ 1 + m 2 𝐱 ¨ 2 = ( m 1 + m 2 ) 𝐑 ¨ = 𝐅 12 + 𝐅 21 = 0 m_{1}\ddot{\mathbf{x}}_{1}+m_{2}\ddot{\mathbf{x}}_{2}=(m_{1}+m_{2})\ddot{% \mathbf{R}}=\mathbf{F}_{12}+\mathbf{F}_{21}=0
  7. 𝐑 ¨ m 1 𝐱 ¨ 1 + m 2 𝐱 ¨ 2 m 1 + m 2 \ddot{\mathbf{R}}\equiv\frac{m_{1}\ddot{\mathbf{x}}_{1}+m_{2}\ddot{\mathbf{x}}% _{2}}{m_{1}+m_{2}}
  8. 𝐑 \mathbf{R}
  9. 𝐑 ¨ = 0 \ddot{\mathbf{R}}=0
  10. 𝐋 = 𝐫 × 𝐩 = 𝐫 × μ d 𝐫 d t \mathbf{L}=\mathbf{r}\times\mathbf{p}=\mathbf{r}\times\mu\frac{d\mathbf{r}}{dt}
  11. 𝐍 = d 𝐋 d t = 𝐫 ˙ × μ 𝐫 ˙ + 𝐫 × μ 𝐫 ¨ , \mathbf{N}=\frac{d\mathbf{L}}{dt}=\dot{\mathbf{r}}\times\mu\dot{\mathbf{r}}+% \mathbf{r}\times\mu\ddot{\mathbf{r}}\ ,
  12. 𝐍 = d 𝐋 d t = 𝐫 × 𝐅 , \mathbf{N}\ =\ \frac{d\mathbf{L}}{dt}=\mathbf{r}\times\mathbf{F}\ ,
  13. U 12 = U ( 𝐫 1 - 𝐫 2 ) ~{}U_{12}=U(\mathbf{r}_{1}-\mathbf{r}_{2})
  14. U 21 = U ( 𝐫 2 - 𝐫 1 ) ~{}U_{21}=U(\mathbf{r}_{2}-\mathbf{r}_{1})
  15. E 1 = m 1 v 1 2 2 + m 2 m 1 + m 2 U 12 = Const 1 ( t ) ~{}E_{1}=m_{1}\frac{v_{1}^{2}}{2}+\frac{m_{2}}{m_{1}+m_{2}}U_{12}=\,\text{% Const}_{1}(t)
  16. E 2 = m 2 v 2 2 2 + m 1 m 1 + m 2 U 21 = Const 2 ( t ) ~{}E_{2}=m_{2}\frac{v_{2}^{2}}{2}+\frac{m_{1}}{m_{1}+m_{2}}U_{21}=\,\text{% Const}_{2}(t)
  17. 𝐅 ( 𝐫 ) = F ( r ) 𝐫 ^ \mathbf{F}(\mathbf{r})=F(r)\hat{\mathbf{r}}
  18. μ 𝐫 ¨ = F ( r ) 𝐫 ^ , \mu\ddot{\mathbf{r}}={F}(r)\hat{\mathbf{r}}\ ,

Tychonoff's_theorem.html

  1. X = i I X i X=\prod_{i\in I}X_{i}
  2. i I A i = i I π i - 1 ( A i ) \prod_{i\in I}A_{i}=\bigcap_{i\in I}\pi_{i}^{-1}(A_{i})
  3. k = 1 N π i k - 1 ( A i k ) . \bigcap_{k=1}^{N}\pi_{i_{k}}^{-1}(A_{i_{k}})\neq\varnothing.

Type_system.html

  1. m a t r i x ( 3 , 3 ) matrix(3,3)
  2. m a t r i x m u l t i p l y : m a t r i x ( k , m ) × m a t r i x ( m , n ) m a t r i x ( k , n ) matrix_{multiply}:matrix(k,m)\times matrix(m,n)\to matrix(k,n)
  3. k k
  4. m m
  5. n n

Type_variable.html

  1. a . a a \forall a.a\to a
  2. a a

Typical_set.html

  1. 𝒳 \mathcal{X}
  2. 𝒳 \in\mathcal{X}
  3. 2 - n [ H ( X ) + ε ] p ( x 1 , x 2 , , x n ) 2 - n [ H ( X ) - ε ] 2^{-n[H(X)+\varepsilon]}\leqslant p(x_{1},x_{2},\dots,x_{n})\leqslant 2^{-n[H(% X)-\varepsilon]}
  4. H ( X ) = - y \isin 𝒳 p ( y ) log 2 p ( y ) H(X)=-\sum_{y\isin\mathcal{X}}p(y)\log_{2}p(y)
  5. ϵ > 0 \epsilon>0
  6. P r [ x ( n ) A ϵ ( n ) ] 1 - ϵ Pr[x^{(n)}\in A_{\epsilon}^{(n)}]\geq 1-\epsilon
  7. | A ε ( n ) | 2 n ( H ( X ) + ε ) \left|{A_{\varepsilon}}^{(n)}\right|\leqslant 2^{n(H(X)+\varepsilon)}
  8. | A ε ( n ) | ( 1 - ε ) 2 n ( H ( X ) - ε ) \left|{A_{\varepsilon}}^{(n)}\right|\geqslant(1-\varepsilon)2^{n(H(X)-% \varepsilon)}
  9. 𝒳 \mathcal{X}
  10. | A ϵ ( n ) | | 𝒳 ( n ) | 2 n H ( X ) 2 n log | 𝒳 | = 2 - n ( log | 𝒳 | - H ( X ) ) 0 \frac{|A_{\epsilon}^{(n)}|}{|\mathcal{X}^{(n)}|}\equiv\frac{2^{nH(X)}}{2^{n% \log|\mathcal{X}|}}=2^{-n(\log|\mathcal{X}|-H(X))}\rightarrow 0
  11. H ( X ) < log | 𝒳 | . H(X)<\log|\mathcal{X}|.
  12. - 1 n log p ( x ( n ) = ( 1 , 1 , , 1 ) ) = - 1 n log ( 0.9 n ) = 0.152 -\frac{1}{n}\log p(x^{(n)}=(1,1,\ldots,1))=-\frac{1}{n}\log(0.9^{n})=0.152
  13. 𝒳 \mathcal{X}
  14. 𝒳 \in\mathcal{X}
  15. | N ( x i ) n - p ( x i ) | < ε 𝒳 . \left|\frac{N(x_{i})}{n}-p(x_{i})\right|<\frac{\varepsilon}{\|\mathcal{X}\|}.
  16. N ( x i ) {N(x_{i})}
  17. x n x^{n}
  18. y n y^{n}
  19. ( x n , y n ) (x^{n},y^{n})
  20. p ( x n , y n ) = i = 1 n p ( x i , y i ) p(x^{n},y^{n})=\prod_{i=1}^{n}p(x_{i},y_{i})
  21. x n x^{n}
  22. y n y^{n}
  23. p ( x n ) p(x^{n})
  24. p ( y n ) p(y^{n})
  25. ( x n , y n ) (x^{n},y^{n})
  26. A ε n ( X , Y ) A_{\varepsilon}^{n}(X,Y)
  27. X ~ n \tilde{X}^{n}
  28. Y ~ n \tilde{Y}^{n}
  29. p ( x n ) p(x^{n})
  30. p ( y n ) p(y^{n})
  31. P [ ( X n , Y n ) A ε n ( X , Y ) ] 1 - ϵ P\left[(X^{n},Y^{n})\in A_{\varepsilon}^{n}(X,Y)\right]\geqslant 1-\epsilon
  32. | A ε n ( X , Y ) | 2 n ( H ( X , Y ) + ϵ ) \left|A_{\varepsilon}^{n}(X,Y)\right|\leqslant 2^{n(H(X,Y)+\epsilon)}
  33. | A ε n ( X , Y ) | ( 1 - ϵ ) 2 n ( H ( X , Y ) - ϵ ) \left|A_{\varepsilon}^{n}(X,Y)\right|\geqslant(1-\epsilon)2^{n(H(X,Y)-\epsilon)}
  34. P [ ( X ~ n , Y ~ n ) A ε n ( X , Y ) ] 2 - n ( I ( X ; Y ) - 3 ϵ ) P\left[(\tilde{X}^{n},\tilde{Y}^{n})\in A_{\varepsilon}^{n}(X,Y)\right]% \leqslant 2^{-n(I(X;Y)-3\epsilon)}
  35. P [ ( X ~ n , Y ~ n ) A ε n ( X , Y ) ] ( 1 - ϵ ) 2 - n ( I ( X ; Y ) + 3 ϵ ) P\left[(\tilde{X}^{n},\tilde{Y}^{n})\in A_{\varepsilon}^{n}(X,Y)\right]% \geqslant(1-\epsilon)2^{-n(I(X;Y)+3\epsilon)}
  36. w ^ = w ( w ) ( ( x 1 n ( w ) , y 1 n ) A ε n ( X , Y ) ) \hat{w}=w\iff(\exists w)((x_{1}^{n}(w),y_{1}^{n})\in A_{\varepsilon}^{n}(X,Y))
  37. w ^ , x 1 n ( w ) , y 1 n \hat{w},x_{1}^{n}(w),y_{1}^{n}
  38. w w
  39. A ε n ( X , Y ) A_{\varepsilon}^{n}(X,Y)
  40. p ( x 1 n ) p ( y 1 n | x 1 n ) p(x_{1}^{n})p(y_{1}^{n}|x_{1}^{n})
  41. p ( y 1 n | x 1 n ) p(y_{1}^{n}|x_{1}^{n})
  42. p ( x 1 n ) p(x_{1}^{n})

Ultimate_fate_of_the_universe.html

  1. 10 10 56 10^{10^{56}}

Ultrafinitism.html

  1. e e e 79 . e^{e^{e^{79}}}\mbox{.}~{}\!
  2. 2 6 2\uparrow\uparrow\uparrow 6
  3. 2 6 2\uparrow\uparrow\uparrow 6
  4. 2 6 2\uparrow\uparrow\uparrow 6

Ultraproduct.html

  1. i I M i \prod_{i\in I}M_{i}
  2. { i I : a i = b i } U , \left\{i\in I:a_{i}=b_{i}\right\}\in U,
  3. i I M i / U . \prod_{i\in I}M_{i}/U.
  4. R ( [ a 1 ] , , [ a n ] ) { i I : R M i ( a i 1 , , a i n ) } U , R([a^{1}],\dots,[a^{n}])\iff\left\{i\in I:R^{M_{i}}(a^{1}_{i},\dots,a^{n}_{i})% \right\}\in U,
  5. M κ / U = α < κ M / U . M^{\kappa}/U=\prod_{\alpha<\kappa}M/U.\,
  6. i I M i / U \prod_{i\in I}M_{i}/U
  7. U U
  8. I I
  9. i I i\in I
  10. M i M_{i}
  11. M M
  12. M i M_{i}
  13. U U
  14. M = i I M i / U . M=\prod_{i\in I}M_{i}/U.
  15. a 1 , , a n M i a^{1},\ldots,a^{n}\in\prod M_{i}
  16. a k = ( a i k ) i I a^{k}=(a^{k}_{i})_{i\in I}
  17. ϕ \phi
  18. M ϕ [ [ a 1 ] , , [ a n ] ] { i I : M i ϕ [ a i 1 , , a i n ] } U . M\models\phi[[a^{1}],\ldots,[a^{n}]]\iff\{i\in I:M_{i}\models\phi[a^{1}_{i},% \ldots,a^{n}_{i}]\}\in U.
  19. ϕ \phi
  20. U U
  21. S = { x M | R x } S=\{x\in M|Rx\}
  22. A n A n + 1 A_{n}\to A_{n+1}

Ultraviolet_catastrophe.html

  1. ( 1 / 2 ) k T (1/2)kT
  2. E quanta = h ν = h c λ E\text{quanta}=h\nu=h\frac{c}{\lambda}
  3. B λ ( λ , T ) = 2 h c 2 λ 5 1 e h c / ( λ k B T ) - 1 B_{\lambda}(\lambda,T)=\frac{2hc^{2}}{\lambda^{5}}\frac{1}{e^{hc/(\lambda k_{% \mathrm{B}}T)}-1}

Ultraviolet–visible_spectroscopy.html

  1. A = log 10 ( I 0 / I ) = ε c L A=\log_{10}(I_{0}/I)=\varepsilon cL
  2. I 0 I_{0}
  3. I I
  4. 1 / M * c m 1/M*cm
  5. A U / M * c m AU/M*cm
  6. I I
  7. I o I_{o}
  8. I / I o I/I_{o}
  9. A A
  10. A = - log ( % T / 100 % ) A=-\log(\%T/100\%)
  11. I I
  12. I o I_{o}
  13. I / I o I/I_{o}
  14. I o I_{o}
  15. L L

Umbral_calculus.html

  1. ( y + x ) n = k = 0 n ( n k ) y n - k x k (y+x)^{n}=\sum_{k=0}^{n}{n\choose k}y^{n-k}x^{k}
  2. B n ( y + x ) = k = 0 n ( n k ) B n - k ( y ) x k . B_{n}(y+x)=\sum_{k=0}^{n}{n\choose k}B_{n-k}(y)x^{k}.
  3. d d x x n = n x n - 1 \frac{d}{dx}x^{n}=nx^{n-1}
  4. d d x B n ( x ) = n B n - 1 ( x ) . \frac{d}{dx}B_{n}(x)=nB_{n-1}(x).
  5. B n ( x ) = k = 0 n ( n k ) b n - k x k = ( b + x ) n , B_{n}(x)=\sum_{k=0}^{n}{n\choose k}b^{n-k}x^{k}=(b+x)^{n},
  6. B n ( x ) = n ( b + x ) n - 1 = n B n - 1 ( x ) . B_{n}^{\prime}(x)=n(b+x)^{n-1}=nB_{n-1}(x).\,
  7. Δ k [ f ] \Delta^{k}[f]
  8. f ( x ) = k = 0 Δ k [ f ] ( 0 ) k ! ( x ) k f(x)=\sum_{k=0}^{\infty}\frac{\Delta^{k}[f](0)}{k!}(x)_{k}
  9. ( x ) k = x ( x - 1 ) ( x - 2 ) ( x - k + 1 ) (x)_{k}=x(x-1)(x-2)\cdots(x-k+1)
  10. L ( y n ) = B n ( 0 ) = B n . L\left(y^{n}\right)=B_{n}(0)=B_{n}.\,
  11. B n ( x ) = k = 0 n ( n k ) B n - k x k = k = 0 n ( n k ) L ( y n - k ) x k = L ( k = 0 n ( n k ) y n - k x k ) = L ( ( y + x ) n ) . B_{n}(x)=\sum_{k=0}^{n}{n\choose k}B_{n-k}x^{k}=\sum_{k=0}^{n}{n\choose k}L% \left(y^{n-k}\right)x^{k}=L\left(\sum_{k=0}^{n}{n\choose k}y^{n-k}x^{k}\right)% =L\left((y+x)^{n}\right).
  12. B n ( x ) B_{n}(x)
  13. L ( ( y + x ) n ) L((y+x)^{n})
  14. B n ( y + x ) = k = 0 n ( n k ) B n - k ( y ) x k B_{n}(y+x)=\sum_{k=0}^{n}{n\choose k}B_{n-k}(y)x^{k}
  15. k = 0 n ( n k ) B n - k ( y ) x k = k = 0 n ( n k ) L ( ( 2 y ) n - k ) x k = L ( k = 0 n ( n k ) ( 2 y ) n - k x k ) = L ( ( 2 y + x ) n ) = B n ( x + y ) . \sum_{k=0}^{n}{n\choose k}B_{n-k}(y)x^{k}=\sum_{k=0}^{n}{n\choose k}L\left((2y% )^{n-k}\right)x^{k}=L\left(\sum_{k=0}^{n}{n\choose k}(2y)^{n-k}x^{k}\right)=L% \left((2y+x)^{n}\right)=B_{n}(x+y).
  16. L 1 L 2 x n = k = 0 n ( n k ) L 1 x k L 2 x n - k . \langle L_{1}L_{2}\mid x^{n}\rangle=\sum_{k=0}^{n}{n\choose k}\langle L_{1}% \mid x^{k}\rangle\langle L_{2}\mid x^{n-k}\rangle.

Unary_coding.html

  1. P ( n ) = 2 - n \operatorname{P}(n)=2^{-n}\,
  2. n = 1 , 2 , 3 , n=1,2,3,...
  3. P ( n ) = ( k - 1 ) k - n \operatorname{P}(n)=(k-1)k^{-n}\,
  4. P ( n ) P ( n + 1 ) + P ( n + 2 ) \operatorname{P}(n)\geq\operatorname{P}(n+1)+\operatorname{P}(n+2)\,
  5. n = 1 , 2 , 3 , n=1,2,3,...

Uncorrelated_random_variables.html

  1. E [ U ] = E [ X Z ] = E [ X ] E [ Z ] = E [ X ] 0 = 0 E[U]=E[XZ]=E[X]E[Z]=E[X]\cdot 0=0
  2. cov ( U , X ) = E [ ( U - E [ U ] ) ( X - E [ X ] ) ] = E [ U ( X - 1 2 ) ] = E [ X 2 Z - 1 2 X Z ] = E [ ( X 2 - 1 2 X ) Z ] = E [ ( X 2 - 1 2 X ) ] E [ Z ] = 0 \operatorname{cov}(U,X)=E[(U-E[U])(X-E[X])]=E[U(X-\tfrac{1}{2})]=E[X^{2}Z-% \tfrac{1}{2}XZ]=E[(X^{2}-\tfrac{1}{2}X)Z]=E[(X^{2}-\tfrac{1}{2}X)]E[Z]=0
  3. Pr ( U = a X = b ) = Pr ( U = a ) \Pr(U=a\mid X=b)=\Pr(U=a)
  4. Pr ( U = 1 X = 0 ) = Pr ( X Z = 1 X = 0 ) = 0 \Pr(U=1\mid X=0)=\Pr(XZ=1\mid X=0)=0
  5. Pr ( U = 1 ) = Pr ( X Z = 1 ) = 1 / 4 \Pr(U=1)=\Pr(XZ=1)=1/4
  6. Pr ( U = 1 X = 0 ) Pr ( U = 1 ) \Pr(U=1\mid X=0)\neq\Pr(U=1)

Underclocking.html

  1. P = C V 2 f . P=CV^{2}f.

Unicity_distance.html

  1. U = H ( k ) / D U=H(k)/D
  2. U = U=\infty
  3. 26 ! = 4.0329 × 10 < s u p > 26 = 2 88.4 26!=4.0329×10<sup>26=2^{88.4}

Uniform_boundedness_principle.html

  1. sup T F T ( x ) Y < , \sup\nolimits_{T\in F}\|T(x)\|_{Y}<\infty,
  2. sup T F , x = 1 T ( x ) Y = sup T F T B ( X , Y ) < . \sup\nolimits_{T\in F,\|x\|=1}\|T(x)\|_{Y}=\sup\nolimits_{T\in F}\|T\|_{B(X,Y)% }<\infty.
  3. sup T F T ( x ) Y < . \sup\nolimits_{T\in F}\|T(x)\|_{Y}<\infty.
  4. n n\in\mathbb{N}
  5. X n = { x X : sup T F T ( x ) Y n } . X_{n}=\left\{x\in X\ :\ \sup\nolimits_{T\in F}\|T(x)\|_{Y}\leq n\right\}.
  6. X n X_{n}
  7. n 𝐍 X n = X . \bigcup\nolimits_{n\in\mathbf{N}}X_{n}=X\neq\varnothing.
  8. X m X_{m}
  9. x 0 X m x_{0}\in X_{m}
  10. B ε ( x 0 ) ¯ := { x X : x - x 0 ε } X m . \overline{B_{\varepsilon}(x_{0})}:=\{x\in X\,:\,\|x-x_{0}\|\leq\varepsilon\}% \subseteq X_{m}.
  11. T ( u ) Y \displaystyle\|T(u)\|_{Y}
  12. sup T F T B ( X , Y ) 2 ε - 1 m < . \sup\nolimits_{T\in F}\|T\|_{B(X,Y)}\leq 2\varepsilon^{-1}m<\infty.
  13. R = { x X : sup T F T x Y = } R=\left\{x\in X\ :\ \sup\nolimits_{T\in F}\|Tx\|_{Y}=\infty\right\}\neq\varnothing
  14. R = { x X : n 𝐍 : sup T F n T x Y n = } R=\left\{x\in X\ :\ \forall n\in\mathbf{N}:\sup\nolimits_{T\in F_{n}}\|Tx\|_{Y% _{n}}=\infty\right\}
  15. n , m { x X : sup T F n T x Y n m } \bigcup\nolimits_{n,m}\left\{x\in X\ :\ \sup\nolimits_{T\in F_{n}}\|Tx\|_{Y_{n% }}\leq m\right\}
  16. 𝕋 \mathbb{T}
  17. C ( 𝕋 ) C(\mathbb{T})
  18. 𝕋 \mathbb{T}
  19. C ( 𝕋 ) C(\mathbb{T})
  20. f C ( 𝕋 ) f\in C(\mathbb{T})
  21. k 𝐙 f ^ ( k ) e i k x = k 𝐙 1 2 π ( 0 2 π f ( t ) e - i k t d t ) e i k x , \sum_{k\in\mathbf{Z}}\hat{f}(k)e^{ikx}=\sum_{k\in\mathbf{Z}}\frac{1}{2\pi}% \left(\int_{0}^{2\pi}f(t)e^{-ikt}dt\right)e^{ikx},
  22. S N ( f ) ( x ) = k = - N N f ^ ( k ) e i k x = 1 2 π 0 2 π f ( t ) D N ( x - t ) d t , S_{N}(f)(x)=\sum_{k=-N}^{N}\hat{f}(k)e^{ikx}=\frac{1}{2\pi}\int_{0}^{2\pi}f(t)% D_{N}(x-t)\,dt,
  23. x 𝕋 x\in\mathbb{T}
  24. C ( 𝕋 ) C(\mathbb{T})\rightarrow\mathbb{C}
  25. φ N , x ( f ) = S N ( f ) ( x ) , f C ( 𝕋 ) , \varphi_{N,x}(f)=S_{N}(f)(x),\qquad f\in C(\mathbb{T}),
  26. C ( 𝕋 ) C(\mathbb{T})
  27. φ N , x = 1 2 π 0 2 π | D N ( x - t ) | d t = 1 2 π 0 2 π | D N ( s ) | d s = D N L 1 ( 𝕋 ) . \left\|\varphi_{N,x}\right\|=\frac{1}{2\pi}\int_{0}^{2\pi}\left|D_{N}(x-t)% \right|\,dt=\frac{1}{2\pi}\int_{0}^{2\pi}\left|D_{N}(s)\right|\,ds=\left\|D_{N% }\right\|_{L^{1}(\mathbb{T})}.
  28. 1 2 π 0 2 π | D N ( t ) | d t 0 π | sin ( ( N + 1 2 ) t ) | t d t . \frac{1}{2\pi}\int_{0}^{2\pi}|D_{N}(t)|\,dt\geq\int_{0}^{\pi}\frac{\left|\sin% \left((N+\tfrac{1}{2})t\right)\right|}{t}\,dt\to\infty.
  29. C ( 𝕋 ) C(\mathbb{T})^{\ast}
  30. C ( 𝕋 ) C(\mathbb{T})
  31. x 𝕋 x\in\mathbb{T}
  32. C ( 𝕋 ) C(\mathbb{T})
  33. 𝕋 \mathbb{T}
  34. C ( 𝕋 ) C(\mathbb{T})
  35. x 𝕋 x\in\mathbb{T}
  36. sup u H u ( x ) < , \sup\nolimits_{u\in H}\|u(x)\|<\infty,

Unijunction_transistor.html

  1. η \eta
  2. η \eta
  3. η \eta

Uniqueness_quantification.html

  1. ! n ( n - 2 = 4 ) \exists!n\in\mathbb{N}\,(n-2=4)
  2. a + 2 = 5 and b + 2 = 5. a+2=5\,\text{ and }b+2=5.\,
  3. a + 2 = b + 2. a+2=b+2.\,
  4. a = b . a=b.\,
  5. x ( P ( x ) ¬ y ( P ( y ) y x ) ) \exists x\,(P(x)\,\wedge\neg\exists y\,(P(y)\wedge y\neq x))
  6. x ( P ( x ) y ( P ( y ) y = x ) ) . \exists x\,(P(x)\wedge\forall y\,(P(y)\to y=x)).
  7. x P ( x ) y z ( ( P ( y ) P ( z ) ) y = z ) . \exists x\,P(x)\wedge\forall y\,\forall z\,((P(y)\wedge P(z))\to y=z).
  8. x y ( P ( y ) y = x ) . \exists x\,\forall y\,(P(y)\leftrightarrow y=x).

Unit_disk.html

  1. D 1 ( P ) = { Q : | P - Q | < 1 } . D_{1}(P)=\{Q:|P-Q|<1\}.\,
  2. D ¯ 1 ( P ) = { Q : | P - Q | 1 } . \bar{D}_{1}(P)=\{Q:|P-Q|\leq 1\}.\,
  3. D 1 ( 0 ) D_{1}(0)
  4. 𝔻 \mathbb{D}
  5. f ( z ) = z 1 - | z | 2 f(z)=\frac{z}{1-|z|^{2}}
  6. g ( z ) = i 1 + z 1 - z g(z)=i\frac{1+z}{1-z}

Unit_vector.html

  1. ı ^ {\hat{\imath}}
  2. 𝐮 ^ = 𝐮 𝐮 \mathbf{\hat{u}}=\frac{\mathbf{u}}{\|\mathbf{u}\|}
  3. 𝐢 ^ = [ 1 0 0 ] , 𝐣 ^ = [ 0 1 0 ] , 𝐤 ^ = [ 0 0 1 ] \mathbf{\hat{i}}=\begin{bmatrix}1\\ 0\\ 0\end{bmatrix},\,\,\mathbf{\hat{j}}=\begin{bmatrix}0\\ 1\\ 0\end{bmatrix},\,\,\mathbf{\hat{k}}=\begin{bmatrix}0\\ 0\\ 1\end{bmatrix}
  4. ı \vec{\imath}
  5. ı ^ \mathbf{\hat{\imath}}
  6. ı , \vec{\imath},
  7. ȷ , \vec{\jmath},
  8. k \vec{k}
  9. ( 𝐱 ^ , 𝐲 ^ , 𝐳 ^ ) (\mathbf{\hat{x}},\mathbf{\hat{y}},\mathbf{\hat{z}})
  10. ( 𝐱 ^ 1 , 𝐱 ^ 2 , 𝐱 ^ 3 ) (\mathbf{\hat{x}}_{1},\mathbf{\hat{x}}_{2},\mathbf{\hat{x}}_{3})
  11. ( 𝐞 ^ x , 𝐞 ^ y , 𝐞 ^ z ) (\mathbf{\hat{e}}_{x},\mathbf{\hat{e}}_{y},\mathbf{\hat{e}}_{z})
  12. ( 𝐞 ^ 1 , 𝐞 ^ 2 , 𝐞 ^ 3 ) (\mathbf{\hat{e}}_{1},\mathbf{\hat{e}}_{2},\mathbf{\hat{e}}_{3})
  13. ρ ^ \mathbf{\hat{\rho}}
  14. 𝐞 ^ \mathbf{\hat{e}}
  15. s y m b o l s ^ symbol{\hat{s}}
  16. s y m b o l φ ^ symbol{\hat{\varphi}}
  17. 𝐳 ^ \mathbf{\hat{z}}
  18. x ^ \hat{x}
  19. y ^ \hat{y}
  20. z ^ \hat{z}
  21. ρ ^ \mathbf{\hat{\rho}}
  22. cos φ 𝐱 ^ + sin φ 𝐲 ^ \cos\varphi\mathbf{\hat{x}}+\sin\varphi\mathbf{\hat{y}}
  23. s y m b o l φ ^ symbol{\hat{\varphi}}
  24. - sin φ 𝐱 ^ + cos φ 𝐲 ^ -\sin\varphi\mathbf{\hat{x}}+\cos\varphi\mathbf{\hat{y}}
  25. 𝐳 ^ = 𝐳 ^ . \mathbf{\hat{z}}=\mathbf{\hat{z}}.
  26. ρ ^ \mathbf{\hat{\rho}}
  27. s y m b o l φ ^ symbol{\hat{\varphi}}
  28. φ \varphi
  29. φ \varphi
  30. ρ ^ φ = - sin φ 𝐱 ^ + cos φ 𝐲 ^ = s y m b o l φ ^ \frac{\partial\mathbf{\hat{\rho}}}{\partial\varphi}=-\sin\varphi\mathbf{\hat{x% }}+\cos\varphi\mathbf{\hat{y}}=symbol{\hat{\varphi}}
  31. s y m b o l φ ^ φ = - cos φ 𝐱 ^ - sin φ 𝐲 ^ = - ρ ^ \frac{\partial symbol{\hat{\varphi}}}{\partial\varphi}=-\cos\varphi\mathbf{% \hat{x}}-\sin\varphi\mathbf{\hat{y}}=-\mathbf{\hat{\rho}}
  32. 𝐳 ^ φ = 0. \frac{\partial\mathbf{\hat{z}}}{\partial\varphi}=\mathbf{0}.
  33. 𝐫 ^ \mathbf{\hat{r}}
  34. s y m b o l φ ^ symbol{\hat{\varphi}}
  35. s y m b o l θ ^ symbol{\hat{\theta}}
  36. 0 θ 180 0\leq\theta\leq 180^{\circ}
  37. s y m b o l φ ^ symbol{\hat{\varphi}}
  38. s y m b o l θ ^ symbol{\hat{\theta}}
  39. φ \varphi
  40. 𝐫 ^ = sin θ cos φ 𝐱 ^ + sin θ sin φ 𝐲 ^ + cos θ 𝐳 ^ \mathbf{\hat{r}}=\sin\theta\cos\varphi\mathbf{\hat{x}}+\sin\theta\sin\varphi% \mathbf{\hat{y}}+\cos\theta\mathbf{\hat{z}}
  41. s y m b o l θ ^ = cos θ cos φ 𝐱 ^ + cos θ sin φ 𝐲 ^ - sin θ 𝐳 ^ symbol{\hat{\theta}}=\cos\theta\cos\varphi\mathbf{\hat{x}}+\cos\theta\sin% \varphi\mathbf{\hat{y}}-\sin\theta\mathbf{\hat{z}}
  42. s y m b o l φ ^ = - sin φ 𝐱 ^ + cos φ 𝐲 ^ symbol{\hat{\varphi}}=-\sin\varphi\mathbf{\hat{x}}+\cos\varphi\mathbf{\hat{y}}
  43. φ \varphi
  44. θ \theta
  45. 𝐫 ^ φ = - sin θ sin φ 𝐱 ^ + sin θ cos φ 𝐲 ^ = sin \thetasymbol φ ^ \frac{\partial\mathbf{\hat{r}}}{\partial\varphi}=-\sin\theta\sin\varphi\mathbf% {\hat{x}}+\sin\theta\cos\varphi\mathbf{\hat{y}}=\sin\thetasymbol{\hat{\varphi}}
  46. 𝐫 ^ θ = cos θ cos φ 𝐱 ^ + cos θ sin φ 𝐲 ^ - sin θ 𝐳 ^ = s y m b o l θ ^ \frac{\partial\mathbf{\hat{r}}}{\partial\theta}=\cos\theta\cos\varphi\mathbf{% \hat{x}}+\cos\theta\sin\varphi\mathbf{\hat{y}}-\sin\theta\mathbf{\hat{z}}=% symbol{\hat{\theta}}
  47. s y m b o l θ ^ φ = - cos θ sin φ 𝐱 ^ + cos θ cos φ 𝐲 ^ = cos \thetasymbol φ ^ \frac{\partial symbol{\hat{\theta}}}{\partial\varphi}=-\cos\theta\sin\varphi% \mathbf{\hat{x}}+\cos\theta\cos\varphi\mathbf{\hat{y}}=\cos\thetasymbol{\hat{% \varphi}}
  48. s y m b o l θ ^ θ = - sin θ cos φ 𝐱 ^ - sin θ sin φ 𝐲 ^ - cos θ 𝐳 ^ = - 𝐫 ^ \frac{\partial symbol{\hat{\theta}}}{\partial\theta}=-\sin\theta\cos\varphi% \mathbf{\hat{x}}-\sin\theta\sin\varphi\mathbf{\hat{y}}-\cos\theta\mathbf{\hat{% z}}=-\mathbf{\hat{r}}
  49. s y m b o l φ ^ φ = - cos φ 𝐱 ^ - sin φ 𝐲 ^ = - sin θ 𝐫 ^ - cos \thetasymbol θ ^ \frac{\partial symbol{\hat{\varphi}}}{\partial\varphi}=-\cos\varphi\mathbf{% \hat{x}}-\sin\varphi\mathbf{\hat{y}}=-\sin\theta\mathbf{\hat{r}}-\cos% \thetasymbol{\hat{\theta}}
  50. 𝐭 ^ \mathbf{\hat{t}}\,\!
  51. 𝐧 ^ \mathbf{\hat{n}}\,\!
  52. r 𝐫 ^ r\mathbf{\hat{r}}\,\!
  53. θ s y m b o l θ ^ \theta symbol{\hat{\theta}}\,\!
  54. 𝐧 ^ \mathbf{\hat{n}}\,\!
  55. 𝐧 ^ = 𝐫 ^ × s y m b o l θ ^ \mathbf{\hat{n}}=\mathbf{\hat{r}}\times symbol{\hat{\theta}}\,\!
  56. 𝐛 ^ = 𝐭 ^ × 𝐧 ^ \mathbf{\hat{b}}=\mathbf{\hat{t}}\times\mathbf{\hat{n}}\,\!
  57. 𝐞 ^ \mathbf{\hat{e}}_{\parallel}\,\!
  58. 𝐞 ^ \mathbf{\hat{e}}_{\parallel}\,\!
  59. 𝐞 ^ \mathbf{\hat{e}}_{\bot}\,\!
  60. 𝐞 ^ \mathbf{\hat{e}}_{\bot}\,\!
  61. 𝐞 ^ \mathbf{\hat{e}}_{\angle}\,\!
  62. 𝐞 ^ n \mathbf{\hat{e}}_{n}
  63. 𝐞 ^ 1 , 𝐞 ^ 2 , 𝐞 ^ 3 \mathbf{\hat{e}}_{1},\mathbf{\hat{e}}_{2},\mathbf{\hat{e}}_{3}
  64. 𝐞 ^ i 𝐞 ^ j = δ i j \mathbf{\hat{e}}_{i}\cdot\mathbf{\hat{e}}_{j}=\delta_{ij}
  65. 𝐞 ^ i ( 𝐞 ^ j × 𝐞 ^ k ) = ε i j k \mathbf{\hat{e}}_{i}\cdot(\mathbf{\hat{e}}_{j}\times\mathbf{\hat{e}}_{k})=% \varepsilon_{ijk}
  66. ε i j k \varepsilon_{ijk}

Unitary_group.html

  1. 1 1
  2. det : U ( n ) U ( 1 ) . \det\colon\operatorname{U}(n)\to\operatorname{U}(1).
  3. 1 1
  4. SU ( n ) \operatorname{SU}(n)
  5. 1 SU ( n ) U ( n ) U ( 1 ) 1. 1\to\operatorname{SU}(n)\to\operatorname{U}(n)\to\operatorname{U}(1)\to 1.
  6. U ( n ) \operatorname{U}(n)
  7. SU ( n ) \operatorname{SU}(n)
  8. U ( 1 ) \operatorname{U}(1)
  9. U ( 1 ) \operatorname{U}(1)
  10. U ( n ) \operatorname{U}(n)
  11. e i θ e^{i\theta}
  12. 1 1
  13. U ( n ) \operatorname{U}(n)
  14. n > 1 n>1
  15. U ( n ) \operatorname{U}(n)
  16. λ I \lambda I
  17. λ U ( 1 ) \lambda\in\operatorname{U}(1)
  18. U ( 1 ) \operatorname{U}(1)
  19. U ( n ) \operatorname{U}(n)
  20. 1 1
  21. U ( n ) \operatorname{U}(n)
  22. A = S diag ( e i θ 1 , , e i θ n ) S - 1 . A=S\,\mbox{diag}~{}(e^{i\theta_{1}},\dots,e^{i\theta_{n}})\,S^{-1}.
  23. t S diag ( e i t θ 1 , , e i t θ n ) S - 1 . t\mapsto S\,\mbox{diag}~{}(e^{it\theta_{1}},\dots,e^{it\theta_{n}})\,S^{-1}.
  24. π 1 ( U ( n ) ) 𝐙 . \pi_{1}(U(n))\cong\mathbf{Z}.
  25. π 1 ( U ( n ) ) π 1 ( S U ( n ) ) × π 1 ( U ( 1 ) ) . \pi_{1}(U(n))\cong\pi_{1}(SU(n))\times\pi_{1}(U(1)).
  26. diag ( e i θ 1 , , e i θ n ) diag ( e i θ σ ( 1 ) , , e i θ σ ( n ) ) \mbox{diag}~{}(e^{i\theta_{1}},\dots,e^{i\theta_{n}})\mapsto\mbox{diag}~{}(e^{% i\theta_{\sigma(1)}},\dots,e^{i\theta_{\sigma(n)}})
  27. U ( n ) = O ( 2 n ) Sp ( 2 n , 𝐑 ) GL ( n , 𝐂 ) . \operatorname{U}(n)=\operatorname{O}(2n)\cap\operatorname{Sp}(2n,\mathbf{R})% \cap\operatorname{GL}(n,\mathbf{C}).
  28. A 𝖳 J A = J A^{\mathsf{T}}JA=J
  29. A - 1 J A = J A^{-1}JA=J
  30. A 𝖳 = A - 1 A^{\mathsf{T}}=A^{-1}
  31. PSU ( n , q 2 ) PU ( n , q 2 ) \operatorname{PSU}(n,q^{2})\neq\operatorname{PU}(n,q^{2})
  32. A n 2 {}^{2}\!A_{n}
  33. D n 2 , E 6 2 , D 4 3 , {}^{2}\!D_{n},{}^{2}\!E_{6},{}^{3}\!D_{4},
  34. A n 2 {}^{2}\!A_{n}
  35. B 2 2 ( 2 2 n + 1 ) , F 4 2 ( 2 2 n + 1 ) , G 2 2 ( 3 2 n + 1 ) ; {}^{2}\!B_{2}\left(2^{2n+1}\right),{}^{2}\!F_{4}\left(2^{2n+1}\right),{}^{2}\!% G_{2}\left(3^{2n+1}\right);
  36. z Ψ 2 = z 1 2 + + z p 2 - z p + 1 2 - - z n 2 \lVert z\rVert_{\Psi}^{2}=\lVert z_{1}\rVert^{2}+\dots+\lVert z_{p}\rVert^{2}-% \lVert z_{p+1}\rVert^{2}-\dots-\lVert z_{n}\rVert^{2}
  37. Ψ ( w , z ) = w ¯ 1 z 1 + + w ¯ p z p - w ¯ p + 1 z p + 1 - - w ¯ n z n . \Psi(w,z)=\bar{w}_{1}z_{1}+\cdots+\bar{w}_{p}z_{p}-\bar{w}_{p+1}z_{p+1}-\cdots% -\bar{w}_{n}z_{n}.
  38. α : x x q \alpha\colon x\mapsto x^{q}
  39. Ψ : V × V K \Psi\colon V\times V\to K
  40. Ψ ( w , v ) = α ( Ψ ( v , w ) ) \Psi(w,v)=\alpha\left(\Psi(v,w)\right)
  41. Ψ ( w , c v ) = c Ψ ( w , v ) \Psi(w,cv)=c\Psi(w,v)
  42. Ψ ( w , v ) = w α v = i = 1 n w i q v i \Psi(w,v)=w^{\alpha}\cdot v=\sum_{i=1}^{n}w_{i}^{q}v_{i}
  43. w i , v i w_{i},v_{i}
  44. c q + 1 = 1 c^{q+1}=1
  45. a a ¯ a\mapsto\bar{a}
  46. a = a ¯ a=\bar{a}
  47. A * = A ¯ T A^{*}=\bar{A}^{\mathrm{T}}
  48. U ( n , K / k , Φ ) ( R ) := { A GL ( n , K k R ) : A * Φ A = Φ } . \operatorname{U}(n,K/k,\Phi)(R):=\left\{A\in\operatorname{GL}(n,K\otimes_{k}R)% :A^{*}\Phi A=\Phi\right\}.
  49. U ( n , 𝐂 / 𝐑 ) ( 𝐑 ) = U ( n ) \operatorname{U}(n,\mathbf{C}/\mathbf{R})(\mathbf{R})=\operatorname{U}(n)
  50. U ( n , 𝐂 / 𝐑 ) ( 𝐂 ) = GL ( n , 𝐂 ) . \operatorname{U}(n,\mathbf{C}/\mathbf{R})(\mathbf{C})=\operatorname{GL}(n,% \mathbf{C}).
  51. ε R × \varepsilon\in R^{\times}
  52. r J 2 = ε r ε - 1 r^{J^{2}}=\varepsilon r\varepsilon^{-1}
  53. ε J = ε - 1 \varepsilon^{J}=\varepsilon^{-1}
  54. Λ m i n := { r R : r - r J ε } , \Lambda_{min}:=\{r\in R\ :\ r-r^{J}\varepsilon\},
  55. Λ m a x := { r R : r J ε = - r } . \Lambda_{max}:=\{r\in R\ :\ r^{J}\varepsilon=-r\}.
  56. Λ m i n Λ Λ m a x \Lambda_{min}\subseteq\Lambda\subseteq\Lambda_{max}
  57. r J Λ r Λ r^{J}\Lambda r\subseteq\Lambda
  58. f ( x r , y s ) = r J f ( x , y ) s f(xr,ys)=r^{J}f(x,y)s
  59. x , y M x,y\in M
  60. r , s R r,s\in R
  61. h ( x , y ) := f ( x , y ) + f ( y , x ) J ε R h(x,y):=f(x,y)+f(y,x)^{J}\varepsilon\in R
  62. q ( x ) := f ( x , x ) R / Λ q(x):=f(x,x)\in R/\Lambda
  63. U ( M ) := { σ G L ( M ) : x , y M , h ( σ x , σ y ) = h ( x , y ) and q ( σ x ) = q ( x ) } . U(M):=\{\sigma\in GL(M)\ :\ \forall x,y\in M,h(\sigma x,\sigma y)=h(x,y)\,% \text{ and }q(\sigma x)=q(x)\}.
  64. J i d R , J 2 = i d R J\neq id_{R},J^{2}=id_{R}
  65. C 1 = ( u 2 + v 2 ) + ( w 2 + x 2 ) + ( y 2 + z 2 ) + C_{1}=(u^{2}+v^{2})+(w^{2}+x^{2})+(y^{2}+z^{2})+...
  66. C 2 = ( u v - v u ) + ( w x - x w ) + ( y z - z y ) + C_{2}=(uv-vu)+(wx-xw)+(yz-zy)+...
  67. Z Z ¯ Z\overline{Z}

Unitary_matrix.html

  1. U * U = U U * = I U^{*}U=UU^{*}=I\,
  2. U U = U U = I . U^{\dagger}U=UU^{\dagger}=I.\,
  3. U x , U y = x , y \langle Ux,Uy\rangle=\langle x,y\rangle
  4. U = V D V * U=VDV^{*}\;
  5. | det ( U ) | = 1 |\det(U)|=1
  6. i i
  7. i i
  8. n \mathbb{C}^{n}
  9. n \mathbb{C}^{n}
  10. U = e i φ [ a b - b * a * ] , | a | 2 + | b | 2 = 1 , U=e^{i\varphi}\begin{bmatrix}a&b\\ -b^{*}&a^{*}\\ \end{bmatrix},\qquad|a|^{2}+|b|^{2}=1,
  11. det ( U ) = e i 2 φ . \det(U)=e^{i2\varphi}.
  12. U = e i φ [ cos θ e i φ 1 sin θ e i φ 2 - sin θ e - i φ 2 cos θ e - i φ 1 ] , U=e^{i\varphi}\begin{bmatrix}\cos\theta e^{i\varphi_{1}}&\sin\theta e^{i% \varphi_{2}}\\ -\sin\theta e^{-i\varphi_{2}}&\cos\theta e^{-i\varphi_{1}}\\ \end{bmatrix},
  13. U = e i φ [ e i ψ 0 0 e - i ψ ] [ cos θ sin θ - sin θ cos θ ] [ e i Δ 0 0 e - i Δ ] . U=e^{i\varphi}\begin{bmatrix}e^{i\psi}&0\\ 0&e^{-i\psi}\end{bmatrix}\begin{bmatrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\\ \end{bmatrix}\begin{bmatrix}e^{i\Delta}&0\\ 0&e^{-i\Delta}\end{bmatrix}.
  14. U = [ 1 0 0 0 e j φ 4 0 0 0 e j φ 5 ] K [ e j φ 1 0 0 0 e j φ 2 0 0 0 e j φ 3 ] U=\begin{bmatrix}1&0&0\\ 0&e^{j\varphi_{4}}&0\\ 0&0&e^{j\varphi_{5}}\end{bmatrix}K\begin{bmatrix}e^{j\varphi_{1}}&0&0\\ 0&e^{j\varphi_{2}}&0\\ 0&0&e^{j\varphi_{3}}\end{bmatrix}

Unitary_operator.html

  1. U : H H U:H→H
  2. H H
  3. U * U = U U * = I U*U=UU*=I
  4. U * U*
  5. U U
  6. I : H H I:H→H
  7. U * U = I U*U=I
  8. U U * = I UU*=I
  9. U : H H U:H→H
  10. H H
  11. U U
  12. U U
  13. H H
  14. x x
  15. y y
  16. H H
  17. U x , U y H = x , y H . \langle Ux,Uy\rangle_{H}=\langle x,y\rangle_{H}.
  18. U : H H U:H→H
  19. H H
  20. U U
  21. H H
  22. U U
  23. H H
  24. x x
  25. y y
  26. H H
  27. U x , U y H = x , y H . \langle Ux,Uy\rangle_{H}=\langle x,y\rangle_{H}.
  28. U U
  29. U U
  30. U U
  31. H H
  32. H H
  33. H i l b ( H ) Hilb(H)
  34. U U
  35. U * U = U U * = I U*U=UU*=I
  36. I I
  37. 𝐂 \mathbf{C}
  38. 1 1
  39. θ 𝐑 θ∈\mathbf{R}
  40. θ θ
  41. θ θ
  42. 2 π
  43. 𝐂 \mathbf{C}
  44. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  45. λ U ( x ) - U ( λ x ) 2 \displaystyle\|\lambda U(x)-U(\lambda x)\|^{2}
  46. U ( x + y ) - ( U x + U y ) = 0. \|U(x+y)-(Ux+Uy)\|=0.
  47. U U
  48. λ λ
  49. | λ | = 1 |λ|=1
  50. U U
  51. f f
  52. L < s u p > 2 ( μ ) L<sup>2(μ)

Unitary_representation.html

  1. π : G U ( H ) \pi:G\rightarrow\operatorname{U}(H)

Unitary_transformation.html

  1. U : H 1 H 2 U:H_{1}\to H_{2}\,
  2. H 1 H_{1}
  3. H 2 H_{2}
  4. U x , U y H 2 = x , y H 1 \langle Ux,Uy\rangle_{H_{2}}=\langle x,y\rangle_{H_{1}}
  5. x x
  6. y y
  7. H 1 H_{1}
  8. x = y x=y
  9. H 1 H_{1}
  10. H 2 H_{2}
  11. U : H 1 H 2 U:H_{1}\to H_{2}\,
  12. U x , U y = x , y ¯ = y , x \langle Ux,Uy\rangle=\overline{\langle x,y\rangle}=\langle y,x\rangle
  13. x x
  14. y y
  15. H 1 H_{1}

Universal_joint.html

  1. γ 1 \gamma_{1}
  2. γ 2 \gamma_{2}
  3. β \beta
  4. 𝐱 ^ \hat{\mathbf{x}}
  5. 𝐲 ^ \hat{\mathbf{y}}
  6. 𝐱 ^ 1 \hat{\mathbf{x}}_{1}
  7. 𝐱 ^ 2 \hat{\mathbf{x}}_{2}
  8. 𝐱 ^ 1 \hat{\mathbf{x}}_{1}
  9. γ 1 \gamma_{1}
  10. 𝐱 ^ 2 \hat{\mathbf{x}}_{2}
  11. γ 2 \gamma_{2}
  12. 𝐱 ^ 1 \hat{\mathbf{x}}_{1}
  13. γ 1 \gamma_{1}
  14. 𝐱 ^ 1 = [ cos γ 1 , sin γ 1 , 0 ] \hat{\mathbf{x}}_{1}=[\cos\gamma_{1}\,,\,\sin\gamma_{1}\,,\,0]
  15. 𝐱 ^ 2 \hat{\mathbf{x}}_{2}
  16. x ^ = [ 1 , 0 , 0 ] \hat{x}=[1,0,0]
  17. [ π / 2 , β , 0 [\pi\!/2\,,\,\beta\,,\,0
  18. 𝐱 ^ 2 = [ - cos β sin γ 2 , cos γ 2 , sin β sin γ 2 ] \hat{\mathbf{x}}_{2}=[-\cos\beta\sin\gamma_{2}\,,\,\cos\gamma_{2}\,,\,\sin% \beta\sin\gamma_{2}]
  19. 𝐱 ^ 1 \hat{\mathbf{x}}_{1}
  20. 𝐱 ^ 2 \hat{\mathbf{x}}_{2}
  21. 𝐱 ^ 1 𝐱 ^ 2 = 0 \hat{\mathbf{x}}_{1}\cdot\hat{\mathbf{x}}_{2}=0
  22. tan γ 1 = cos β tan γ 2 \tan\gamma_{1}=\cos\beta\tan\gamma_{2}\,
  23. γ 2 \gamma_{2}
  24. γ 2 = tan - 1 [ tan γ 1 / cos β ] \gamma_{2}=\tan^{-1}[\tan\gamma_{1}/\cos\beta]\,
  25. γ 2 \gamma_{2}
  26. γ 2 \gamma_{2}
  27. - π < γ 1 < π -\pi<\gamma_{1}<\pi
  28. γ 2 = atan2 ( sin γ 1 , cos β cos γ 1 ) \gamma_{2}=\mathrm{atan2}(\sin\gamma_{1},\cos\beta\,\cos\gamma_{1})
  29. γ 1 \gamma_{1}
  30. γ 2 \gamma_{2}
  31. ω 1 = d γ 1 / d t \omega_{1}=d\gamma_{1}/dt
  32. ω 2 = d γ 2 / d t \omega_{2}=d\gamma_{2}/dt
  33. ω 2 = ω 1 cos β 1 - sin 2 β cos 2 γ 1 \omega_{2}=\frac{\omega_{1}\cos\beta}{1-\sin^{2}\beta\cos^{2}\gamma_{1}}
  34. a 1 a_{1}
  35. a 2 a_{2}
  36. a 2 = a 1 cos β 1 - sin 2 β cos 2 γ 1 - ω 1 2 cos β sin 2 β sin 2 γ 1 ( 1 - sin 2 β cos 2 γ 1 ) 2 a_{2}=\frac{a_{1}\cos\beta}{1-\sin^{2}\beta\,\cos^{2}\gamma_{1}}-\frac{\omega_% {1}^{2}\cos\beta\sin^{2}\beta\sin 2\gamma_{1}}{(1-\sin^{2}\beta\cos^{2}\gamma_% {1})^{2}}
  37. γ 1 \gamma_{1}\,
  38. γ 2 \gamma_{2}\,
  39. γ 3 \gamma_{3}\,
  40. γ 4 \gamma_{4}\,
  41. β \beta\,
  42. tan γ 2 = cos β tan γ 1 tan γ 4 = cos β tan γ 3 \tan\gamma_{2}=\cos\beta\,\tan\gamma_{1}\qquad\tan\gamma_{4}=\cos\beta\,\tan% \gamma_{3}
  43. γ 3 = γ 2 + π / 2 \gamma_{3}=\gamma_{2}+\pi/2
  44. tan ( γ + π / 2 ) = 1 / tan γ \tan(\gamma+\pi/2)=1/\tan\gamma
  45. tan γ 4 = cos β / tan γ 2 = 1 / tan γ 1 = tan ( γ 1 + π / 2 ) \tan\gamma_{4}=\cos\beta/\tan\gamma_{2}=1/\tan\gamma_{1}=\tan(\gamma_{1}+\pi/2)\,

Universal_quantification.html

  1. \forall
  2. n P ( n ) \forall n\!\in\!\mathbb{N}\;P(n)
  3. n ( Q ( n ) P ( n ) ) \forall n\!\in\!\mathbb{N}\;\bigl(Q(n)\rightarrow P(n)\bigr)
  4. n ( Q ( n ) P ( n ) ) \forall n\;\bigl(Q(n)\rightarrow P(n)\bigr)
  5. ( n ) P ( n ) (n{\in}\mathbb{N})\,P(n)
  6. ¬ \lnot
  7. x 𝐗 P ( x ) \forall{x}{\in}\mathbf{X}\,P(x)
  8. ¬ x 𝐗 P ( x ) \lnot\ \forall{x}{\in}\mathbf{X}\,P(x)
  9. x 𝐗 P ( x ) \forall{x}{\in}\mathbf{X}\,P(x)
  10. x 𝐗 ¬ P ( x ) \exists{x}{\in}\mathbf{X}\,\lnot P(x)
  11. ¬ x 𝐗 P ( x ) x 𝐗 ¬ P ( x ) \lnot\ \forall{x}{\in}\mathbf{X}\,P(x)\equiv\ \exists{x}{\in}\mathbf{X}\,\lnot P% (x)
  12. ¬ x 𝐗 P ( x ) x 𝐗 ¬ P ( x ) ¬ x 𝐗 P ( x ) x 𝐗 ¬ P ( x ) \lnot\ \exists{x}{\in}\mathbf{X}\,P(x)\equiv\ \forall{x}{\in}\mathbf{X}\,\lnot P% (x)\not\equiv\ \lnot\ \forall{x}{\in}\mathbf{X}\,P(x)\equiv\ \exists{x}{\in}% \mathbf{X}\,\lnot P(x)
  13. P ( x ) ( y 𝐘 Q ( y ) ) y 𝐘 ( P ( x ) Q ( y ) ) P(x)\land(\exists{y}{\in}\mathbf{Y}\,Q(y))\equiv\ \exists{y}{\in}\mathbf{Y}\,(% P(x)\land Q(y))
  14. P ( x ) ( y 𝐘 Q ( y ) ) y 𝐘 ( P ( x ) Q ( y ) ) , provided that 𝐘 P(x)\lor(\exists{y}{\in}\mathbf{Y}\,Q(y))\equiv\ \exists{y}{\in}\mathbf{Y}\,(P% (x)\lor Q(y)),~{}\mathrm{provided~{}that}~{}\mathbf{Y}\neq\emptyset
  15. P ( x ) ( y 𝐘 Q ( y ) ) y 𝐘 ( P ( x ) Q ( y ) ) , provided that 𝐘 P(x)\to(\exists{y}{\in}\mathbf{Y}\,Q(y))\equiv\ \exists{y}{\in}\mathbf{Y}\,(P(% x)\to Q(y)),~{}\mathrm{provided~{}that}~{}\mathbf{Y}\neq\emptyset
  16. P ( x ) ( y 𝐘 Q ( y ) ) y 𝐘 ( P ( x ) Q ( y ) ) P(x)\nleftarrow(\exists{y}{\in}\mathbf{Y}\,Q(y))\equiv\ \exists{y}{\in}\mathbf% {Y}\,(P(x)\nleftarrow Q(y))
  17. P ( x ) ( y 𝐘 Q ( y ) ) y 𝐘 ( P ( x ) Q ( y ) ) , provided that 𝐘 P(x)\land(\forall{y}{\in}\mathbf{Y}\,Q(y))\equiv\ \forall{y}{\in}\mathbf{Y}\,(% P(x)\land Q(y)),~{}\mathrm{provided~{}that}~{}\mathbf{Y}\neq\emptyset
  18. P ( x ) ( y 𝐘 Q ( y ) ) y 𝐘 ( P ( x ) Q ( y ) ) P(x)\lor(\forall{y}{\in}\mathbf{Y}\,Q(y))\equiv\ \forall{y}{\in}\mathbf{Y}\,(P% (x)\lor Q(y))
  19. P ( x ) ( y 𝐘 Q ( y ) ) y 𝐘 ( P ( x ) Q ( y ) ) P(x)\to(\forall{y}{\in}\mathbf{Y}\,Q(y))\equiv\ \forall{y}{\in}\mathbf{Y}\,(P(% x)\to Q(y))
  20. P ( x ) ( y 𝐘 Q ( y ) ) y 𝐘 ( P ( x ) Q ( y ) ) , provided that 𝐘 P(x)\nleftarrow(\forall{y}{\in}\mathbf{Y}\,Q(y))\equiv\ \forall{y}{\in}\mathbf% {Y}\,(P(x)\nleftarrow Q(y)),~{}\mathrm{provided~{}that}~{}\mathbf{Y}\neq\emptyset
  21. P ( x ) ( y 𝐘 Q ( y ) ) y 𝐘 ( P ( x ) Q ( y ) ) P(x)\uparrow(\exists{y}{\in}\mathbf{Y}\,Q(y))\equiv\ \forall{y}{\in}\mathbf{Y}% \,(P(x)\uparrow Q(y))
  22. P ( x ) ( y 𝐘 Q ( y ) ) y 𝐘 ( P ( x ) Q ( y ) ) , provided that 𝐘 P(x)\downarrow(\exists{y}{\in}\mathbf{Y}\,Q(y))\equiv\ \forall{y}{\in}\mathbf{% Y}\,(P(x)\downarrow Q(y)),~{}\mathrm{provided~{}that}~{}\mathbf{Y}\neq\emptyset
  23. P ( x ) ( y 𝐘 Q ( y ) ) y 𝐘 ( P ( x ) Q ( y ) ) , provided that 𝐘 P(x)\nrightarrow(\exists{y}{\in}\mathbf{Y}\,Q(y))\equiv\ \forall{y}{\in}% \mathbf{Y}\,(P(x)\nrightarrow Q(y)),~{}\mathrm{provided~{}that}~{}\mathbf{Y}\neq\emptyset
  24. P ( x ) ( y 𝐘 Q ( y ) ) y 𝐘 ( P ( x ) Q ( y ) ) P(x)\leftarrow(\exists{y}{\in}\mathbf{Y}\,Q(y))\equiv\ \forall{y}{\in}\mathbf{% Y}\,(P(x)\leftarrow Q(y))
  25. P ( x ) ( y 𝐘 Q ( y ) ) y 𝐘 ( P ( x ) Q ( y ) ) , provided that 𝐘 P(x)\uparrow(\forall{y}{\in}\mathbf{Y}\,Q(y))\equiv\ \exists{y}{\in}\mathbf{Y}% \,(P(x)\uparrow Q(y)),~{}\mathrm{provided~{}that}~{}\mathbf{Y}\neq\emptyset
  26. P ( x ) ( y 𝐘 Q ( y ) ) y 𝐘 ( P ( x ) Q ( y ) ) P(x)\downarrow(\forall{y}{\in}\mathbf{Y}\,Q(y))\equiv\ \exists{y}{\in}\mathbf{% Y}\,(P(x)\downarrow Q(y))
  27. P ( x ) ( y 𝐘 Q ( y ) ) y 𝐘 ( P ( x ) Q ( y ) ) P(x)\nrightarrow(\forall{y}{\in}\mathbf{Y}\,Q(y))\equiv\ \exists{y}{\in}% \mathbf{Y}\,(P(x)\nrightarrow Q(y))
  28. P ( x ) ( y 𝐘 Q ( y ) ) y 𝐘 ( P ( x ) Q ( y ) ) , provided that 𝐘 P(x)\leftarrow(\forall{y}{\in}\mathbf{Y}\,Q(y))\equiv\ \exists{y}{\in}\mathbf{% Y}\,(P(x)\leftarrow Q(y)),~{}\mathrm{provided~{}that}~{}\mathbf{Y}\neq\emptyset
  29. x 𝐗 P ( x ) P ( c ) \forall{x}{\in}\mathbf{X}\,P(x)\to\ P(c)
  30. P ( c ) x 𝐗 P ( x ) . P(c)\to\ \forall{x}{\in}\mathbf{X}\,P(x).
  31. x P ( x ) \forall{x}{\in}\emptyset\,P(x)
  32. P ( y ) x Q ( x , z ) P(y)\land\exists xQ(x,z)
  33. y z ( P ( y ) x Q ( x , z ) ) \forall y\forall z(P(y)\land\exists xQ(x,z))
  34. X X
  35. 𝒫 X \mathcal{P}X
  36. f : X Y f:X\to Y
  37. X X
  38. Y Y
  39. f * : 𝒫 Y 𝒫 X f^{*}:\mathcal{P}Y\to\mathcal{P}X
  40. f \exists_{f}
  41. f \forall_{f}
  42. f : 𝒫 X 𝒫 Y \exists_{f}\colon\mathcal{P}X\to\mathcal{P}Y
  43. S X S\subset X
  44. f S Y \exists_{f}S\subset Y
  45. f S = { y Y | there exists x S . f ( x ) = y } \exists_{f}S=\{y\in Y|\mbox{ there exists }~{}x\in S.\ f(x)=y\}
  46. f : 𝒫 X 𝒫 Y \forall_{f}\colon\mathcal{P}X\to\mathcal{P}Y
  47. f S = { y Y | for all x . f ( x ) = y x S } \forall_{f}S=\{y\in Y|\mbox{ for all }~{}x.\ f(x)=y\implies x\in S\}
  48. π : X × { T , F } { T , F } \pi:X\times\{T,F\}\to\{T,F\}
  49. { T , F } \{T,F\}
  50. S X × { T , F } S\subset X\times\{T,F\}
  51. π S = { y | x S ( x , y ) } \exists_{\pi}S=\{y\,|\,\exists x\,S(x,y)\}

Universal_set.html

  1. φ ( x ) \varphi(x)
  2. A A
  3. { x A φ ( x ) } \{x\in A\mid\varphi(x)\}
  4. x x
  5. A A
  6. φ \varphi
  7. V V
  8. { x V x x } \{x\in V\mid x\not\in x\}
  9. V V
  10. V V V\in V
  11. V V

Universally_unique_identifier.html

  1. p ( n ) 1 - e - n 2 2 x , p(n)\approx 1-e^{-\frac{n^{2}}{2x}},
  2. n 2 / 2 x n^{2}/2x
  3. p ( n ) n 2 2 x p(n)\approx\frac{n^{2}}{2x}

Universe_(mathematics).html

  1. 𝐒 X := i = 0 𝐒 i X . \mathbf{S}X:=\bigcup_{i=0}^{\infty}\mathbf{S}_{i}X\mbox{.}~{}\!
  2. V i := j < i 𝐏 V j V_{i}:=\bigcup_{j<i}\mathbf{P}V_{j}\!
  3. V := i V i V:=\bigcup_{i}V_{i}\!
  4. x u U x\in u\in U
  5. x U x\in U
  6. u U u\in U
  7. v U v\in U
  8. u × v U u\times v\in U
  9. x U x\in U
  10. 𝒫 x U \mathcal{P}x\in U
  11. x U \cup x\in U
  12. ω U \omega\in U
  13. ω = { 0 , 1 , 2 , } \omega=\{0,1,2,...\}
  14. f : a b f:a\to b
  15. a U a\in U
  16. b U b\subset U
  17. b U b\in U

Upper_half-plane.html

  1. = { x + i y | y > 0 ; x , y } . \mathbb{H}=\{x+iy\;|y>0;x,y\in\mathbb{R}\}.

Upsilon.html

  1. Υ \,\Upsilon
  2. Υ \Upsilon

Urease.html

  1. α 12 β 12 \alpha_{12}\beta_{12}

Vacuum_energy.html

  1. E = 1 2 h ν . {E}=\frac{1}{2}h\nu.

Vacuum_expectation_value.html

  1. O \langle O\rangle
  2. ψ ¯ ψ \langle\overline{\psi}\psi\rangle
  3. G μ ν G μ ν \langle G_{\mu\nu}G^{\mu\nu}\rangle

Vacuum_flask.html

  1. T d S = d H - d P TdS=dH-dP
  2. T s u r r Δ S = m c p δ T - V d p T_{surr}\Delta S=mc_{p}\delta T-Vdp
  3. T s u r r Δ S = c p ( T b - T c ) T_{surr}\Delta S=c_{p}\left(T_{b^{\prime}}-T_{c}\right)
  4. T b = T c + T s u r r Δ S c p T_{b^{\prime}}=T_{c}+\frac{T_{surr}\Delta S}{c_{p}}
  5. Q 0 = A ϵ σ ( T 4 - T 0 4 ) Q^{\prime}_{0}=A\epsilon\sigma\left(T^{4}-T_{0}^{4}\right)
  6. Q 0 = A i n ϵ s . s . σ ( T b 4 - T s u r r 4 ) Q^{\prime}_{0}=A_{in}\epsilon_{s.s.}\sigma\left(T_{b}^{\prime 4}-T_{surr}^{4}\right)
  7. Q 0 = A i n ϵ s . s . σ [ ( T c + T s u r r Δ S c p ) 4 - T s u r r 4 ] Q^{\prime}_{0}=A_{in}\epsilon_{s.s.}\sigma\left[\left(T_{c}+\frac{T_{surr}% \Delta S}{c_{p}}\right)^{4}-T_{surr}^{4}\right]
  8. α Q 0 = Q i n \alpha Q^{\prime}_{0}=Q^{\prime}_{in}
  9. Q i n = Q o u t Q^{\prime}_{in}=Q^{\prime}_{out}
  10. Q o u t = ϵ s . s . Q 0 Q^{\prime}_{out}=\epsilon_{s.s.}Q^{\prime}_{0}
  11. Q l i d = Q c o n d + Q c o n v + Q r a d Q^{\prime}_{lid}=Q^{\prime}_{cond}+Q^{\prime}_{conv}+Q^{\prime}_{rad}
  12. Q l i d = k A l i d ( T b - T s u r r Δ x ) + h A l i d ( T b - T s u r r ) + A l i d ϵ p . p . σ [ ( T c + T s u r r Δ S p . p . c p p . p . ) 4 - T s u r r 4 ] Q^{\prime}_{lid}=kA_{lid}\left(\frac{T_{b}-T_{surr}}{\Delta x}\right)+hA_{lid}% \left(T_{b}-T_{surr}\right)+A_{lid}\epsilon_{p.p.}\sigma\left[\left(T_{c}+% \frac{T_{surr}\Delta S_{p.p.}}{c_{p}^{p.p.}}\right)^{4}-T_{surr}^{4}\right]
  13. Q t o t a l = Q o u t + Q l i d Q^{\prime}_{total}=Q^{\prime}_{out}+Q^{\prime}_{lid}
  14. Δ S s y s t e m = S i n - S o u t + S g e n \Delta S_{system}=S_{in}-S_{out}+S_{gen}
  15. Δ S s y s t e m = S i n - S o u t + S g e n \Delta S^{\prime}_{system}=S^{\prime}_{in}-S^{\prime}_{out}+S^{\prime}_{gen}
  16. - d Q T s u r r + S g e n = 0 -\int\frac{dQ^{\prime}}{T_{surr}}+S^{\prime}_{gen}=0
  17. S g e n = Q 2 - Q 1 T s u r r S^{\prime}_{gen}=\frac{Q^{\prime}_{2}-Q^{\prime}_{1}}{T_{surr}}
  18. S g e n = Q t o t a l T s u r r S^{\prime}_{gen}=\frac{Q^{\prime}_{total}}{T_{surr}}

Value_at_risk.html

  1. α ( 0 , 1 ) \alpha\in(0,1)
  2. α \alpha
  3. l l
  4. L L
  5. l l
  6. ( 1 - α ) (1-\alpha)
  7. L L
  8. VaR α ( L ) \operatorname{VaR}_{\alpha}(L)
  9. α \alpha
  10. VaR α ( L ) = inf { l : P ( L > l ) 1 - α } = inf { l : F L ( l ) α } . \operatorname{VaR}_{\alpha}(L)=\inf\{l\in\mathbb{R}:P(L>l)\leq 1-\alpha\}=\inf% \{l\in\mathbb{R}:F_{L}(l)\geq\alpha\}.
  11. g ( x ) = { 0 if 0 x < 1 - α 1 if 1 - α x 1 . g(x)=\begin{cases}0&\,\text{if }0\leq x<1-\alpha\\ 1&\,\text{if }1-\alpha\leq x\leq 1\end{cases}.
  12. If X , Y 𝐋 , then ρ ( X + Y ) ρ ( X ) + ρ ( Y ) . \mathrm{If}\;X,Y\in\mathbf{L},\;\mathrm{then}\;\rho(X+Y)\leq\rho(X)+\rho(Y).
  13. X 𝐋 M + X\in\mathbf{L}_{M^{+}}
  14. 𝐋 M + \mathbf{L}_{M^{+}}
  15. VaR 1 - α ( X ) CVaR 1 - α ( X ) EVaR 1 - α ( X ) , \,\text{VaR}_{1-\alpha}(X)\leq\,\text{CVaR}_{1-\alpha}(X)\leq\,\text{EVaR}_{1-% \alpha}(X),
  16. VaR 1 - α ( X ) := inf t 𝐑 { t : Pr ( X t ) 1 - α } , CVaR 1 - α ( X ) := 1 α 0 α VaR 1 - γ ( X ) d γ , EVaR 1 - α ( X ) := inf z > 0 { z - 1 ln ( M X ( z ) / α ) } , \begin{aligned}&\displaystyle\,\text{VaR}_{1-\alpha}(X):=\inf_{t\in\mathbf{R}}% \{t:\,\text{Pr}(X\leq t)\geq 1-\alpha\},\\ &\displaystyle\,\text{CVaR}_{1-\alpha}(X):=\frac{1}{\alpha}\int_{0}^{\alpha}\,% \text{VaR}_{1-\gamma}(X)d\gamma,\\ &\displaystyle\,\text{EVaR}_{1-\alpha}(X):=\inf_{z>0}\{z^{-1}\ln(M_{X}(z)/% \alpha)\},\end{aligned}
  17. M X ( z ) M_{X}(z)
  18. X X
  19. z z
  20. X X

Van_der_Waals_equation.html

  1. ( p + a v 2 ) ( v - b ) = R T \left(p+\frac{a^{\prime}}{v^{2}}\right)\left(v-b^{\prime}\right)=RT
  2. v = V / N v=V/N
  3. N = N A n N=N_{A}n
  4. k = R / N A k=R/N_{A}
  5. ( p + n 2 a V 2 ) ( V - n b ) = n R T \left(p+\frac{n^{2}a}{V^{2}}\right)\left(V-nb\right)=nRT
  6. a = N A 2 a a=N_{A}^{2}a^{\prime}
  7. b = N A b b=N_{A}b^{\prime}
  8. T < T C T<T_{C}
  9. p = R T V m = R T v . p=\frac{RT}{V_{\mathrm{m}}}=\frac{RT}{v}.
  10. p = R T V m - b . p=\frac{RT}{V_{\mathrm{m}}-b}.
  11. b b
  12. b 2 = 4 π d 3 / 3 = 8 × ( 4 π r 3 / 3 ) b^{\prime}_{2}=4\pi d^{3}/3=8\times(4\pi r^{3}/3)
  13. b = b 2 / 2 b = 4 × ( 4 π r 3 / 3 ) b^{\prime}=b^{\prime}_{2}/2\quad\rightarrow\quad b^{\prime}=4\times(4\pi r^{3}% /3)
  14. C = N A / V m C=N_{\mathrm{A}}/V_{\mathrm{m}}
  15. a C 2 = a ( N A V m ) 2 = a V m 2 a^{\prime}C^{2}=a^{\prime}\left(\frac{N_{\mathrm{A}}}{V_{\mathrm{m}}}\right)^{% 2}=\frac{a}{V_{\mathrm{m}}^{2}}
  16. p = R T V m - b - a V m 2 ( p + a V m 2 ) ( V m - b ) = R T . p=\frac{RT}{V_{\mathrm{m}}-b}-\frac{a}{V_{\mathrm{m}}^{2}}\Rightarrow\left(p+% \frac{a}{V_{\mathrm{m}}^{2}}\right)(V_{\mathrm{m}}-b)=RT.
  17. ( p + n 2 a V 2 ) ( V - n b ) = n R T . \left(p+\frac{n^{2}a}{V^{2}}\right)(V-nb)=nRT.
  18. Q = q N N ! with q = V Λ 3 Q=\frac{q^{N}}{N!}\quad\hbox{with}\quad q=\frac{V}{\Lambda^{3}}
  19. Λ \Lambda
  20. Λ = h 2 2 π m k T \Lambda=\sqrt{\frac{h^{2}}{2\pi mkT}}
  21. u ( r ) = { when r < d , - ϵ ( d r ) 6 when r d , u(r)=\begin{cases}\infty&\hbox{when}\quad r<d,\\ -\epsilon\left(\frac{d}{r}\right)^{6}&\hbox{when}\quad r\geq d,\end{cases}
  22. ϵ \epsilon
  23. u ( r ) u(r)
  24. Q = q N / N ! Q=q^{N}/N!
  25. b = 2 π d 3 / 3 b^{\prime}=2\pi d^{3}/3
  26. q = ( V - N b ) e - ϕ / ( 2 k T ) Λ 3 . q=\frac{(V-Nb^{\prime})\,e^{-\phi/(2kT)}}{\Lambda^{3}}.
  27. ϕ = d u ( r ) N V 4 π r 2 d r , \phi=\int_{d}^{\infty}u(r)\frac{N}{V}4\pi r^{2}dr,
  28. ϕ = - 2 a N V with a = ϵ 2 π d 3 3 = ϵ b . \phi=-2a^{\prime}\frac{N}{V}\quad\hbox{with}\quad a^{\prime}=\epsilon\frac{2% \pi d^{3}}{3}=\epsilon b^{\prime}.
  29. ln Q = N ln ( V - N b ) + N 2 a V k T - N ln ( Λ 3 ) - ln N ! \ln Q=N\ln{(V-Nb^{\prime})}+\frac{N^{2}a^{\prime}}{VkT}-N\ln{(\Lambda^{3})}-% \ln{N!}
  30. p = k T ln Q V p=kT\frac{\partial\ln Q}{\partial V}
  31. p = N k T V - N b - N 2 a V 2 ( p + N 2 a V 2 ) ( V - N b ) = N k T ( p + n 2 a V 2 ) ( V - n b ) = n R T . p=\frac{NkT}{V-Nb^{\prime}}-\frac{N^{2}a^{\prime}}{V^{2}}\Rightarrow\left(p+% \frac{N^{2}a^{\prime}}{V^{2}}\right)(V-Nb^{\prime})=NkT\Rightarrow\left(p+% \frac{n^{2}a}{V^{2}}\right)(V-nb)=nRT.
  32. ( P / V ) T , N \scriptstyle\left({{\partial P}/{\partial V}}\right)_{T,N}
  33. P V ( V G - V L ) = V L V G P d V P_{V}(V_{G}-V_{L})=\int_{V_{L}}^{V_{G}}P\,dV
  34. P = ( A V ) T , N P=\left(\frac{\partial A}{\partial V}\right)_{T,N}
  35. P V = A ( V G , T , N ) - A ( V L , T , N ) V G - V L P_{V}=\frac{A(V_{G},T,N)-A(V_{L},T,N)}{V_{G}-V_{L}}
  36. A = - k T ln Q . A=-kT\ln Q.\,
  37. A ( T , V , N ) = - N k T [ 1 + ln ( ( V - N b ) T 3 / 2 N Φ ) ] - a N 2 V . A(T,V,N)=-NkT\left[1+\ln\left(\frac{(V-Nb^{\prime})T^{3/2}}{N\Phi}\right)% \right]-\frac{a^{\prime}N^{2}}{V}.
  38. Φ = T 3 / 2 Λ 3 = ( h 2 π m k ) 3 \Phi=T^{3/2}\Lambda^{3}=\left(\frac{h}{\sqrt{2\pi mk}}\right)^{3}
  39. p = - ( A V ) T = N k T V - N b - a N 2 V 2 . p=-\left(\frac{\partial A}{\partial V}\right)_{T}=\frac{NkT}{V-Nb^{\prime}}-% \frac{a^{\prime}N^{2}}{V^{2}}.
  40. S = - ( A T ) V = N k [ ln ( ( V - N b ) T 3 / 2 N Φ ) + 5 2 ] S=-\left(\frac{\partial A}{\partial T}\right)_{V}=Nk\left[\ln\left(\frac{(V-Nb% ^{\prime})T^{3/2}}{N\Phi}\right)+\frac{5}{2}\right]
  41. U = A + T S = 3 2 N k T - a N 2 V . U=A+TS=\frac{3}{2}\,NkT-\frac{a^{\prime}N^{2}}{V}.
  42. p R = p p C , v R = v v C , and T R = T T C p_{R}=\frac{p}{p_{C}},\qquad v_{R}=\frac{v}{v_{C}},\quad\hbox{and}\quad T_{R}=% \frac{T}{T_{C}}
  43. ( v - v c ) 3 = 0 {\left(v-v_{c}\right)}^{3}=0
  44. v c v_{c}
  45. v = v c v=v_{c}
  46. T c T_{c}
  47. p c p_{c}
  48. v 3 - 3 v c v 2 + 3 v c 2 v - v c 3 = 0 v^{3}-3v_{c}v^{2}+3v^{2}_{c}v-v_{c}^{3}=0
  49. v 3 - ( p b + R T p ) v 2 + a v p - a b p = 0 v^{3}-\left(\frac{pb^{\prime}+RT}{p}\right)v^{2}+\frac{a^{\prime}v}{p}-\frac{a% ^{\prime}b^{\prime}}{p}=0
  50. - 3 v c = - ( p c b + R T p c ) -3v_{c}=-\left(\frac{p_{c}b^{\prime}+RT}{p_{c}}\right)
  51. 3 v c 2 = a p c p c = a 3 v c 2 3v^{2}_{c}=\frac{a^{\prime}}{p_{c}}\rightarrow p_{c}=\frac{a^{\prime}}{3v_{c}^% {2}}
  52. - v c 3 = - a b p c v c 3 = a b a 3 v c 2 -v_{c}^{3}=-\frac{a^{\prime}b^{\prime}}{p_{c}}\rightarrow v_{c}^{3}=\frac{a^{% \prime}b^{\prime}}{\frac{a^{\prime}}{3v_{c}^{2}}}
  53. b = v c 3 b^{\prime}=\frac{v_{c}}{3}
  54. p c = a 27 b 2 p_{c}=\frac{a}{27{b^{\prime}}^{2}}
  55. 3 ( 3 b ) = a b 27 b 2 + R T c a 27 b 2 3(3b^{\prime})=\frac{\frac{a^{\prime}b^{\prime}}{27{b^{\prime}}^{2}}+RT_{c}}{% \frac{a^{\prime}}{27{b^{\prime}}^{2}}}
  56. 9 b a 27 b 2 = a 27 b + R T c \frac{9{b^{\prime}}a^{\prime}}{27{b^{\prime}}^{2}}=\frac{a^{\prime}}{27b^{% \prime}}+RT_{c}
  57. 8 a 27 b = R T c \frac{8a^{\prime}}{27b^{\prime}}=RT_{c}
  58. ( p R + 3 v R 2 ) ( v R - 1 / 3 ) = ( 8 / 3 ) T R \left(p_{R}+\frac{3}{v_{R}^{2}}\right)(v_{R}-1/3)=(8/3)T_{R}
  59. v R 3 - 1 3 ( 1 + 8 T R p R ) v R 2 + 3 p R v R - 1 p R = 0 {v_{R}^{3}}-\frac{1}{3}\left({1+\frac{8T_{R}}{p_{R}}}\right){v_{R}^{2}}+\frac{% 3}{p_{R}}v_{R}-\frac{1}{p_{R}}=0
  60. T R = p R = 1 T_{R}=p_{R}=1
  61. v R 3 - 3 v R 2 + 3 v R - 1 = ( v R - 1 ) 3 = 0 v R = 1 {v_{R}^{3}}-3v_{R}^{2}+3v_{R}-1=\left(v_{R}-1\right)^{3}=0\quad% \Longleftrightarrow\quad v_{R}=1
  62. ( p + A ) ( V - B ) = C T , (p+A)(V-B)=CT,\,

Van_der_Waals_force.html

  1. U ( z ; R 1 , R 2 ) = - A 6 ( 2 R 1 R 2 z 2 - ( R 1 + R 2 ) 2 + 2 R 1 R 2 z 2 - ( R 1 - R 2 ) 2 + ln [ z 2 - ( R 1 + R 2 ) 2 z 2 - ( R 1 - R 2 ) 2 ] ) \displaystyle U(z;R_{1},R_{2})=-\frac{A}{6}\left(\frac{2R_{1}R_{2}}{z^{2}-(R_{% 1}+R_{2})^{2}}+\frac{2R_{1}R_{2}}{z^{2}-(R_{1}-R_{2})^{2}}+\ln\left[\frac{z^{2% }-(R_{1}+R_{2})^{2}}{z^{2}-(R_{1}-R_{2})^{2}}\right]\right)
  2. z = R 1 + R 2 + r \ z=R_{1}+R_{2}+r
  3. r R 1 o r R 2 \ r\ll R_{1}orR_{2}
  4. U ( r ; R 1 , R 2 ) = - A R 1 R 2 ( R 1 + R 2 ) 6 r \ U(r;R_{1},R_{2})=-\frac{AR_{1}R_{2}}{(R_{1}+R_{2})6r}
  5. F V W ( r ) = - d d r U ( r ) \ F_{VW}(r)=-\frac{d}{dr}U(r)
  6. F V W ( r ) = - A R 1 R 2 ( R 1 + R 2 ) 6 r 2 \ F_{VW}(r)=-\frac{AR_{1}R_{2}}{(R_{1}+R_{2})6r^{2}}

Van_der_Waals_radius.html

  1. w {}_{w}
  2. w {}_{w}
  3. V w = 4 3 π r w 3 V_{\rm w}={4\over 3}\pi r_{\rm w}^{3}
  4. A {}_{A}
  5. 1 / [ u v a l , u 1000 ] {1}/{[u^{\prime}val^{\prime},u^{\prime}1000^{\prime}]}
  6. ( p + a ( n V ~ ) 2 ) ( V ~ - n b ) = n R T \left(p+a\left(\frac{n}{\tilde{V}}\right)^{2}\right)(\tilde{V}-nb)=nRT
  7. V ~ \tilde{V}
  8. a ( n V ~ ) 2 a\left(\frac{n}{\tilde{V}}\right)^{2}
  9. 3 {}^{3}
  10. 1 {}^{–1}
  11. w {}_{w}
  12. 3 {}^{3}
  13. w {}_{w}
  14. 3 {}^{3}
  15. A {}_{A}
  16. V w = b N A V_{\rm w}={b\over{N_{\rm A}}}
  17. w {}_{w}
  18. 3 {}^{3}
  19. w {}_{w}
  20. w {}_{w}
  21. V w = 4 3 π r w 3 + π r w 2 d V_{\rm w}={4\over 3}\pi r_{\rm w}^{3}+\pi r_{\rm w}^{2}d
  22. m {}_{m}
  23. V w = π V m N A 18 V_{\rm w}=\frac{\pi V_{\rm m}}{N_{\rm A}\sqrt{18}}
  24. w {}_{w}
  25. 3 {}^{3}
  26. w {}_{w}
  27. A = R T ( n 2 - 1 ) 3 p A=\frac{RT(n^{2}-1)}{3p}
  28. w {}_{w}
  29. 3 {}^{3}
  30. w {}_{w}
  31. e {}_{e}
  32. α = ϵ 0 k B T p χ e \alpha={\epsilon_{0}k_{\rm B}T\over p}\chi_{\rm e}
  33. r {}_{r}
  34. e {}_{e}
  35. r {}_{r}
  36. e {}_{e}
  37. V w = 1 4 π ϵ 0 α , V_{\rm w}={1\over{4\pi\epsilon_{0}}}\alpha,
  38. w {}_{w}
  39. 3 {}^{3}
  40. w {}_{w}
  41. 3 {}^{3}

Van_der_Waerden's_theorem.html

  1. W ( r , k ) 2 2 r 2 2 k + 9 , W(r,k)\leq 2^{2^{r^{2^{2^{k+9}}}}},
  2. W ( 2 , k ) W(2,k)
  3. ε \varepsilon
  4. W ( 2 , k ) > 2 k / k ε W(2,k)>2^{k}/k^{\varepsilon}
  5. k k
  6. W ( 3 7 ( 2 3 7 + 1 ) , 2 ) 3 7 ( 2 3 7 + 1 ) + 1. W(3^{7(2\cdot 3^{7}+1)},2)\leq 3^{7(2\cdot 3^{7}+1)}+1.
  7. a + i 1 s 1 + i 2 s 2 + i D s D a+i_{1}s_{1}+i_{2}s_{2}...+i_{D}s_{D}
  8. M = M i n N ( L , D , n ) M={\mathrm{M}inN}(L,D,n)
  9. M i n N ( L , D + 1 , n ) M * M i n N ( L , 1 , n M ) {\mathrm{M}inN}(L,D+1,n)\leq M*{\mathrm{M}inN}(L,1,n^{M})
  10. i 1 = i 2 = = i D + 1 = L i_{1}=i_{2}=...=i_{D+1}=L
  11. M i n N ( L + 1 , 1 , n ) 2 M i n N ( L , n , n ) {\mathrm{M}inN}(L+1,1,n)\leq 2{\mathrm{M}inN}(L,n,n)
  12. a + L s 1 + L s 2 + L s D - k a+Ls_{1}+Ls_{2}...+Ls_{D-k}
  13. a + L s 1 + L s 2 + L s D - p a+Ls_{1}+Ls_{2}...+Ls_{D-p}
  14. a + L * ( s 1 + s D - k ) + u * ( s D - k + 1 + s p ) a+L*(s_{1}...+s_{D-k})+u*(s_{D-k+1}...+s_{p})
  15. u = 0 , 1 , 2 , , L - 1 , L u=0,1,2,...,L-1,L

Vapnik–Chervonenkis_theory.html

  1. X 1 , , X n X_{1},\ldots,X_{n}
  2. ( 𝒳 , 𝒜 ) (\mathcal{X},\mathcal{A})
  3. Q Q
  4. Q f = f d Q Qf=\int fdQ
  5. \mathcal{F}
  6. f : 𝒳 𝐑 f:\mathcal{X}\to\mathbf{R}
  7. Q = sup { | Q f | : f } . \|Q\|_{\mathcal{F}}=\sup\{|Qf|\ :\ f\in\mathcal{F}\}.
  8. n = n - 1 i = 1 n δ X i , \mathbb{P}_{n}=n^{-1}\sum_{i=1}^{n}\delta_{X_{i}},
  9. δ δ
  10. 𝐑 \mathcal{F}\to\mathbf{R}
  11. f n f f\mapsto\mathbb{P}_{n}f
  12. P P
  13. \mathcal{F}
  14. n - P 0 , \|\mathbb{P}_{n}-P\|_{\mathcal{F}}\to 0,
  15. 𝔾 n = n ( n - P ) 𝔾 , in ( ) \mathbb{G}_{n}=\sqrt{n}(\mathbb{P}_{n}-P)\rightsquigarrow\mathbb{G},\quad\,% \text{in }\ell^{\infty}(\mathcal{F})
  16. \mathcal{F}
  17. x , sup f | f ( x ) - P f | < \forall x,\sup\nolimits_{f\in\mathcal{F}}|f(x)-Pf|<\infty
  18. \mathcal{F}
  19. P P
  20. f f
  21. f f\in\mathcal{F}
  22. \mathcal{F}
  23. \mathcal{F}
  24. \mathcal{F}
  25. N ( ε , , ) N(\varepsilon,\mathcal{F},\|\cdot\|)
  26. { g : g - f < ε } \{g:\|g-f\|<\varepsilon\}
  27. \mathcal{F}
  28. \mathcal{F}
  29. \mathcal{F}
  30. \mathcal{F}
  31. P P
  32. P P
  33. F F
  34. P F < P^{\ast}F<\infty
  35. ε > 0 sup Q N ( ε F Q , , L 1 ( Q ) ) < . \forall\varepsilon>0\quad\sup\nolimits_{Q}N(\varepsilon\|F\|_{Q},\mathcal{F},L% _{1}(Q))<\infty.
  36. \mathcal{F}
  37. 0 sup Q log N ( ε F Q , 2 , , L 2 ( Q ) ) d ε < \int_{0}^{\infty}\sup\nolimits_{Q}\sqrt{\log N\left(\varepsilon\|F\|_{Q,2},% \mathcal{F},L_{2}(Q)\right)}d\varepsilon<\infty
  38. \mathcal{F}
  39. P P
  40. P P
  41. P F 2 < P^{\ast}F^{2}<\infty
  42. f Q , 2 = ( | f | 2 d Q ) 1 2 \|f\|_{Q,2}=\left(\int|f|^{2}dQ\right)^{\frac{1}{2}}
  43. f ( n - P ) f = 1 n i = 1 n ( f ( X i ) - P f ) f\mapsto(\mathbb{P}_{n}-P)f=\dfrac{1}{n}\sum_{i=1}^{n}(f(X_{i})-Pf)
  44. f n 0 = 1 n i = 1 n ε i f ( X i ) f\mapsto\mathbb{P}^{0}_{n}=\dfrac{1}{n}\sum_{i=1}^{n}\varepsilon_{i}f(X_{i})
  45. X i X_{i}
  46. Φ : 𝐑 𝐑 Φ:\mathbf{R}→\mathbf{R}
  47. \mathcal{F}
  48. 𝔼 Φ ( n - P ) 𝔼 Φ ( 2 n 0 ) \mathbb{E}\Phi(\|\mathbb{P}_{n}-P\|_{\mathcal{F}})\leq\mathbb{E}\Phi\left(2% \left\|\mathbb{P}^{0}_{n}\right\|_{\mathcal{F}}\right)
  49. X i X_{i}
  50. Y 1 , , Y n Y_{1},\ldots,Y_{n}
  51. X 1 , , X n X_{1},\ldots,X_{n}
  52. X 1 , , X n X_{1},\ldots,X_{n}
  53. n - P = sup f 1 n | i = 1 n f ( X i ) - 𝔼 f ( Y i ) | 𝔼 Y sup f 1 n | i = 1 n f ( X i ) - f ( Y i ) | \|\mathbb{P}_{n}-P\|_{\mathcal{F}}=\sup_{f\in\mathcal{F}}\dfrac{1}{n}\left|% \sum_{i=1}^{n}f(X_{i})-\mathbb{E}f(Y_{i})\right|\leq\mathbb{E}_{Y}\sup_{f\in% \mathcal{F}}\dfrac{1}{n}\left|\sum_{i=1}^{n}f(X_{i})-f(Y_{i})\right|
  54. Φ ( n - P ) 𝔼 Y Φ ( 1 n i = 1 n f ( X i ) - f ( Y i ) ) \Phi(\|\mathbb{P}_{n}-P\|_{\mathcal{F}})\leq\mathbb{E}_{Y}\Phi\left(\left\|% \dfrac{1}{n}\sum_{i=1}^{n}f(X_{i})-f(Y_{i})\right\|_{\mathcal{F}}\right)
  55. X X
  56. 𝔼 Φ ( n - P ) 𝔼 X 𝔼 Y Φ ( 1 n i = 1 n f ( X i ) - f ( Y i ) ) \mathbb{E}\Phi(\|\mathbb{P}_{n}-P\|_{\mathcal{F}})\leq\mathbb{E}_{X}\mathbb{E}% _{Y}\Phi\left(\left\|\dfrac{1}{n}\sum_{i=1}^{n}f(X_{i})-f(Y_{i})\right\|_{% \mathcal{F}}\right)
  57. f ( X i ) - f ( Y i ) f(X_{i})-f(Y_{i})
  58. X X
  59. Y Y
  60. 𝔼 Φ ( 1 n i = 1 n e i f ( X i ) - f ( Y i ) ) \mathbb{E}\Phi\left(\left\|\dfrac{1}{n}\sum_{i=1}^{n}e_{i}f(X_{i})-f(Y_{i})% \right\|_{\mathcal{F}}\right)
  61. ( e 1 , e 2 , , e n ) { - 1 , 1 } n (e_{1},e_{2},\ldots,e_{n})\in\{-1,1\}^{n}
  62. 𝔼 Φ ( n - P ) 𝔼 ε 𝔼 Φ ( 1 n i = 1 n ε i f ( X i ) - f ( Y i ) ) \mathbb{E}\Phi(\|\mathbb{P}_{n}-P\|_{\mathcal{F}})\leq\mathbb{E}_{\varepsilon}% \mathbb{E}\Phi\left(\left\|\dfrac{1}{n}\sum_{i=1}^{n}\varepsilon_{i}f(X_{i})-f% (Y_{i})\right\|_{\mathcal{F}}\right)
  63. Φ \Phi
  64. 𝔼 Φ ( n - P ) 1 2 𝔼 ε 𝔼 Φ ( 2 1 n i = 1 n ε i f ( X i ) ) + 1 2 𝔼 ε 𝔼 Φ ( 2 1 n i = 1 n ε i f ( Y i ) ) \mathbb{E}\Phi(\|\mathbb{P}_{n}-P\|_{\mathcal{F}})\leq\dfrac{1}{2}\mathbb{E}_{% \varepsilon}\mathbb{E}\Phi\left(2\left\|\dfrac{1}{n}\sum_{i=1}^{n}\varepsilon_% {i}f(X_{i})\right\|_{\mathcal{F}}\right)+\dfrac{1}{2}\mathbb{E}_{\varepsilon}% \mathbb{E}\Phi\left(2\left\|\dfrac{1}{n}\sum_{i=1}^{n}\varepsilon_{i}f(Y_{i})% \right\|_{\mathcal{F}}\right)
  65. n 0 \mathbb{P}_{n}^{0}
  66. \mathcal{F}
  67. 𝒳 \mathcal{X}
  68. 𝒞 \mathcal{C}
  69. 𝒞 \mathcal{C}
  70. S = { x 1 , , x n } 𝒳 S=\{x_{1},\ldots,x_{n}\}\subset\mathcal{X}
  71. S = S C S=S\cap C
  72. C 𝒞 C\in\mathcal{C}
  73. 𝒞 \mathcal{C}
  74. S S
  75. V ( 𝒞 ) V(\mathcal{C})
  76. 𝒞 \mathcal{C}
  77. n n
  78. n n
  79. 𝒞 \mathcal{C}
  80. Δ n ( 𝒞 , x 1 , , x n ) \Delta_{n}(\mathcal{C},x_{1},\ldots,x_{n})
  81. 𝒞 \mathcal{C}
  82. max x 1 , , x n Δ n ( 𝒞 , x 1 , , x n ) j = 0 V ( 𝒞 ) - 1 ( n j ) ( n e V ( 𝒞 ) - 1 ) V ( 𝒞 ) - 1 \max_{x_{1},\ldots,x_{n}}\Delta_{n}(\mathcal{C},x_{1},\ldots,x_{n})\leq\sum_{j% =0}^{V(\mathcal{C})-1}{n\choose j}\leq\left(\frac{ne}{V(\mathcal{C})-1}\right)% ^{V(\mathcal{C})-1}
  83. O ( n V ( 𝒞 ) - 1 ) O(n^{V(\mathcal{C})-1})
  84. 𝒞 \mathcal{C}
  85. f : 𝒳 𝐑 f:\mathcal{X}\to\mathbf{R}
  86. 𝒳 × 𝐑 \mathcal{X}\times\mathbf{R}
  87. { ( x , t ) : t < f ( x ) } \{(x,t):t<f(x)\}
  88. \mathcal{F}
  89. 𝒞 = { 1 C : C 𝒞 } \mathcal{I}_{\mathcal{C}}=\{1_{C}:C\in\mathcal{C}\}
  90. L 1 ( Q ) L_{1}(Q)
  91. Q Q
  92. Q Q
  93. r 1 r\geq 1
  94. N ( ε , 𝒞 , L r ( Q ) ) K V ( 𝒞 ) ( 4 e ) V ( 𝒞 ) ε - r ( V ( 𝒞 ) - 1 ) N(\varepsilon,\mathcal{I}_{\mathcal{C}},L_{r}(Q))\leq KV(\mathcal{C})(4e)^{V(% \mathcal{C})}\varepsilon^{-r(V(\mathcal{C})-1)}
  95. \mathcal{F}
  96. sconv \operatorname{sconv}\mathcal{F}
  97. i = 1 m α i f i \sum_{i=1}^{m}\alpha_{i}f_{i}
  98. i = 1 m | α i | 1 \sum_{i=1}^{m}|\alpha_{i}|\leq 1
  99. N ( ε F Q , 2 , , L 2 ( Q ) ) C ε - V N\left(\varepsilon\|F\|_{Q,2},\mathcal{F},L_{2}(Q)\right)\leq C\varepsilon^{-V}
  100. \mathcal{F}
  101. log N ( ε F Q , 2 , sconv , L 2 ( Q ) ) K ε - 2 V V + 2 \log N\left(\varepsilon\|F\|_{Q,2},\operatorname{sconv}\mathcal{F},L_{2}(Q)% \right)\leq K\varepsilon^{-\frac{2V}{V+2}}
  102. 2 V V + 2 > 2 , \frac{2V}{V+2}>2,
  103. sconv \operatorname{sconv}\mathcal{F}
  104. P P
  105. \mathcal{F}
  106. f : 𝒳 𝐑 f:\mathcal{X}\to\mathbf{R}
  107. dim ( ) + 2 \dim(\mathcal{F})+2
  108. n = dim ( ) + 2 n=\dim(\mathcal{F})+2
  109. ( x 1 , t 1 ) , , ( x n , t n ) (x_{1},t_{1}),\ldots,(x_{n},t_{n})
  110. ( f ( x 1 ) , , f ( x n ) ) - ( t 1 , , t n ) (f(x_{1}),\ldots,f(x_{n}))-(t_{1},\ldots,t_{n})
  111. n 1 n−1
  112. a 0 a≠0
  113. a i > 0 a i ( f ( x i ) - t i ) = a i < 0 ( - a i ) ( f ( x i ) - t i ) , f \sum_{a_{i}>0}a_{i}(f(x_{i})-t_{i})=\sum_{a_{i}<0}(-a_{i})(f(x_{i})-t_{i}),% \quad\forall f\in\mathcal{F}
  114. S = { ( x i , t i ) : a i > 0 } S=\{(x_{i},t_{i}):a_{i}>0\}
  115. f f
  116. S = { ( x i , t i ) : f ( x i ) > t i } S=\{(x_{i},t_{i}):f(x_{i})>t_{i}\}
  117. 𝒳 \mathcal{X}
  118. 𝒴 = { 0 , 1 } \mathcal{Y}=\{0,1\}
  119. f : 𝒳 𝒴 f:\mathcal{X}\to\mathcal{Y}
  120. \mathcal{F}
  121. S ( , n ) = max x 1 , , x n | { ( f ( x 1 ) , , f ( x n ) ) , f } | S(\mathcal{F},n)=\max_{x_{1},\ldots,x_{n}}|\{(f(x_{1}),\ldots,f(x_{n})),f\in% \mathcal{F}\}|
  122. \mathcal{F}
  123. max x 1 , , x n Δ n ( 𝒞 , x 1 , , x n ) \max_{x_{1},\ldots,x_{n}}\Delta_{n}(\mathcal{C},x_{1},\ldots,x_{n})
  124. 𝒞 \mathcal{C}
  125. S ( , n ) S(\mathcal{F},n)
  126. n n
  127. \mathcal{F}
  128. 𝒞 \mathcal{C}
  129. D n = { ( X 1 , Y 1 ) , , ( X n , Y m ) } D_{n}=\{(X_{1},Y_{1}),\ldots,(X_{n},Y_{m})\}
  130. P X Y P_{XY}
  131. R ( f ) = P ( f ( X ) Y ) R(f)=P(f(X)\neq Y)
  132. P X Y P_{XY}
  133. R ( f ) R(f)
  134. R ^ n ( f ) = 1 n i = 1 n 𝕀 ( f ( X n ) Y n ) \hat{R}_{n}(f)=\dfrac{1}{n}\sum_{i=1}^{n}\mathbb{I}(f(X_{n})\neq Y_{n})
  135. P ( sup f | R ^ n ( f ) - R ( f ) | > ε ) \displaystyle P\left(\sup_{f\in\mathcal{F}}\left|\hat{R}_{n}(f)-R(f)\right|>% \varepsilon\right)
  136. \mathcal{F}
  137. S ( , n ) S(\mathcal{F},n)
  138. n n
  139. | R ^ n - R | \left|\hat{R}_{n}-R\right|_{\mathcal{F}}

Variometer.html

  1. E t o t = E p o t + E k i n E_{tot}=E_{pot}+E_{kin}
  2. E p o t E_{pot}
  3. E k i n E_{kin}
  4. Δ E t o t = Δ E p o t + Δ E k i n \Delta E_{tot}=\Delta E_{pot}+\Delta E_{kin}
  5. E p o t = m g h E_{pot}=mgh
  6. E k i n = 1 2 m V 2 E_{kin}={1\over 2}mV^{2}
  7. Δ E t o t = m g Δ h + 1 2 m Δ V 2 \Delta E_{tot}=mg\Delta h+{1\over 2}m{\Delta V}^{2}
  8. Δ E t o t m g = Δ h + Δ V 2 2 g {\Delta E_{tot}\over mg}=\Delta h+{{\Delta V}^{2}\over 2g}

VC_dimension.html

  1. f f
  2. θ \theta
  3. ( x 1 , x 2 , , x n ) (x_{1},x_{2},\ldots,x_{n})
  4. θ \theta
  5. f f
  6. f f
  7. f f
  8. h h^{\prime}
  9. h h^{\prime}
  10. h h
  11. h h
  12. f f
  13. P ( test error training error + h ( log ( 2 N / h ) + 1 ) - log ( η / 4 ) N ) = 1 - η P\left(\,\text{test error}\leq\,\text{training error}+\sqrt{h(\log(2N/h)+1)-% \log(\eta/4)\over N}\right)=1-\eta
  14. h h
  15. 0 η 1 0\leq\eta\leq 1
  16. N N
  17. h N h\ll N

Vector_bundle.html

  1. φ : U × 𝐑 k π - 1 ( U ) \varphi:U\times\mathbf{R}^{k}\to\pi^{-1}(U)
  2. ( π φ ) ( x , v ) = x (\pi\circ\varphi)(x,v)=x
  3. v < m t p l > φ ( x , v ) v<mtpl>{{\mapsto}}\varphi(x,v)
  4. \mapsto
  5. φ U : U × 𝐑 k π - 1 ( U ) , φ V : V × 𝐑 k π - 1 ( V ) \begin{aligned}\displaystyle\varphi_{U}:U\times\mathbf{R}^{k}&\displaystyle% \xrightarrow{\cong}\pi^{-1}(U),\\ \displaystyle\varphi_{V}:V\times\mathbf{R}^{k}&\displaystyle\xrightarrow{\cong% }\pi^{-1}(V)\end{aligned}
  6. φ V - 1 φ U : ( U V ) × 𝐑 k ( U V ) × 𝐑 k \varphi_{V}^{-1}\circ\varphi_{U}:(U\cap V)\times\mathbf{R}^{k}\to(U\cap V)% \times\mathbf{R}^{k}
  7. φ V - 1 φ U ( x , v ) = ( x , g U V ( x ) v ) \varphi_{V}^{-1}\circ\varphi_{U}(x,v)=\left(x,g_{UV}(x)v\right)
  8. g U V : U V GL ( k ) . g_{UV}:U\cap V\to\operatorname{GL}(k).
  9. g U U ( x ) = I , g U V ( x ) g V W ( x ) g W U ( x ) = I g_{UU}(x)=I,\quad g_{UV}(x)g_{VW}(x)g_{WU}(x)=I
  10. g U V : U V GL ( F ) g_{UV}:U\cap V\to\operatorname{GL}(F)
  11. vl v w [ f ] := d d t | t = 0 f ( v + t w ) , f C ( E x ) . \operatorname{vl}_{v}w[f]:=\frac{d}{dt}\Big|_{t=0}f(v+tw),\quad f\in C^{\infty% }(E_{x}).
  12. { Φ V : 𝐑 × ( E 0 ) ( E 0 ) ( t , v ) Φ V t ( v ) := e t v . \begin{cases}\Phi_{V}:\mathbf{R}\times(E\setminus 0)\to(E\setminus 0)\\ (t,v)\mapsto\Phi_{V}^{t}(v):=e^{t}v.\end{cases}
  13. K ( X ) K(X)
  14. E E EE
  15. 0 A B C 0 0\to A\to B\to C\to 0
  16. [ B ] = [ A ] + [ C ] [B]=[A]+[C]
  17. X X
  18. S < s u p > 2 X S<sup>2X

Very-long-baseline_interferometry.html

  1. ϵ B \epsilon_{B}
  2. ϵ C \epsilon_{C}

Vienna_Circle.html

  1. \exists\;
  2. x P ( x ) \exists\;xP(x)
  3. ¬ x \lnot\;\exists\;x
  4. ¬ x O ( x ) \lnot\;\exists\;xO(x)

Viterbi_algorithm.html

  1. S S
  2. π i \pi_{i}
  3. i i
  4. a i , j a_{i,j}
  5. i i
  6. j j
  7. y 1 , , y T y_{1},\dots,y_{T}
  8. x 1 , , x T x_{1},\dots,x_{T}
  9. V 1 , k = P ( y 1 | k ) π k V t , k = max x S ( P ( y t | k ) a x , k V t - 1 , x ) \begin{array}[]{rcl}V_{1,k}&=&\mathrm{P}\big(y_{1}\ |\ k\big)\cdot\pi_{k}\\ V_{t,k}&=&\max_{x\in S}\left(\mathrm{P}\big(y_{t}\ |\ k\big)\cdot a_{x,k}\cdot V% _{t-1,x}\right)\end{array}
  10. V t , k V_{t,k}
  11. t t
  12. k k
  13. x x
  14. Ptr ( k , t ) \mathrm{Ptr}(k,t)
  15. x x
  16. V t , k V_{t,k}
  17. t > 1 t>1
  18. k k
  19. t = 1 t=1
  20. x T = arg max x S ( V T , x ) x t - 1 = Ptr ( x t , t ) \begin{array}[]{rcl}x_{T}&=&\arg\max_{x\in S}(V_{T,x})\\ x_{t-1}&=&\mathrm{Ptr}(x_{t},t)\end{array}
  21. O ( T × | S | 2 ) O(T\times\left|{S}\right|^{2})
  22. O = { o 1 , o 2 , , o N } O=\{o_{1},o_{2},\dots,o_{N}\}
  23. S = { s 1 , s 2 , , s K } S=\{s_{1},s_{2},\dots,s_{K}\}
  24. Y = { y 1 , y 2 , , y T } Y=\{y_{1},y_{2},\ldots,y_{T}\}
  25. A A
  26. K × K K\times K
  27. A i j A_{ij}
  28. s i s_{i}
  29. s j s_{j}
  30. B B
  31. K × N K\times N
  32. B i j B_{ij}
  33. o j o_{j}
  34. s i s_{i}
  35. π \pi
  36. K K
  37. π i \pi_{i}
  38. x 1 = = s i x_{1}==s_{i}
  39. X = { x 1 , x 2 , , x T } X=\{x_{1},x_{2},\ldots,x_{T}\}
  40. Y = { y 1 , y 2 , , y T } Y=\{y_{1},y_{2},\ldots,y_{T}\}
  41. T 1 , T 2 T_{1},T_{2}
  42. K × T K\times T
  43. T 1 T_{1}
  44. T 1 [ i , j ] T_{1}[i,j]
  45. X ^ = { x ^ 1 , x ^ 2 , , x ^ j } \hat{X}=\{\hat{x}_{1},\hat{x}_{2},\ldots,\hat{x}_{j}\}
  46. x ^ j = s i \hat{x}_{j}=s_{i}
  47. Y = { y 1 , y 2 , , y j } Y=\{y_{1},y_{2},\ldots,y_{j}\}
  48. T 2 T_{2}
  49. T 2 [ i , j ] T_{2}[i,j]
  50. x ^ j - 1 \hat{x}_{j-1}
  51. X ^ = { x ^ 1 , x ^ 2 , , x ^ j - 1 , x ^ j } \hat{X}=\{\hat{x}_{1},\hat{x}_{2},\ldots,\hat{x}_{j-1},\hat{x}_{j}\}
  52. j , 2 j T \forall j,2\leq j\leq T
  53. T 1 [ i , j ] , T 2 [ i , j ] T_{1}[i,j],T_{2}[i,j]
  54. K j + i K\cdot j+i
  55. T 1 [ i , j ] = max k ( T 1 [ k , j - 1 ] A k i B i y j ) T_{1}[i,j]=\max_{k}{(T_{1}[k,j-1]\cdot A_{ki}\cdot B_{iy_{j}})}
  56. T 2 [ i , j ] = arg max k ( T 1 [ k , j - 1 ] A k i B i y j ) T_{2}[i,j]=\arg\max_{k}{(T_{1}[k,j-1]\cdot A_{ki}\cdot B_{iy_{j}})}
  57. B i y j B_{iy_{j}}
  58. i i
  59. j j
  60. O = { o 1 , o 2 , , o N } O=\{o_{1},o_{2},\dots,o_{N}\}
  61. S = { s 1 , s 2 , , s K } S=\{s_{1},s_{2},\dots,s_{K}\}
  62. Y = { y 1 , y 2 , , y T } Y=\{y_{1},y_{2},\ldots,y_{T}\}
  63. y t = = i y_{t}==i
  64. t t
  65. o i o_{i}
  66. A A
  67. K K K\cdot K
  68. A i j A_{ij}
  69. s i s_{i}
  70. s j s_{j}
  71. B B
  72. K N K\cdot N
  73. B i j B_{ij}
  74. o j o_{j}
  75. s i s_{i}
  76. π \pi
  77. K K
  78. π i \pi_{i}
  79. x 1 = = s i x_{1}==s_{i}
  80. X = { x 1 , x 2 , , x T } X=\{x_{1},x_{2},\ldots,x_{T}\}
  81. \cdot
  82. max k ( T 1 [ k , i - 1 ] A k j B j y i ) \max_{k}{(T_{1}[k,i-1]\cdot A_{kj}\cdot B_{jy_{i}})}
  83. arg max k ( T 1 [ k , i - 1 ] A k j B j y i ) \arg\max_{k}{(T_{1}[k,i-1]\cdot A_{kj}\cdot B_{jy_{i}})}
  84. arg max k ( T 1 [ k , T ] ) \arg\max_{k}{(T_{1}[k,T])}

Von_Neumann_algebra.html

  1. ( M N ) = M N , (M\otimes N)^{\prime}=M^{\prime}\otimes N^{\prime},
  2. x Tr ( 1 λ + 1 0 0 λ λ + 1 ) x . x\mapsto{\rm Tr}\begin{pmatrix}{1\over\lambda+1}&0\\ 0&{\lambda\over\lambda+1}\\ \end{pmatrix}x.

Voronoi_diagram.html

  1. X \scriptstyle X
  2. d \scriptstyle d
  3. K \scriptstyle K
  4. ( P k ) k K \scriptstyle(P_{k})_{k\in K}
  5. X \scriptstyle X
  6. R k \scriptstyle R_{k}
  7. P k \scriptstyle P_{k}
  8. X \scriptstyle X
  9. P k \scriptstyle P_{k}
  10. P j \scriptstyle P_{j}
  11. j \scriptstyle j
  12. k \scriptstyle k
  13. d ( x , A ) = inf { d ( x , a ) a A } \scriptstyle d(x,\,A)\;=\;\inf\{d(x,\,a)\mid a\,\in\,A\}
  14. x \scriptstyle x
  15. A \scriptstyle A
  16. R k = { x X d ( x , P k ) d ( x , P j ) for all j k } R_{k}=\{x\in X\mid d(x,P_{k})\leq d(x,P_{j})\;\,\text{for all}\;j\neq k\}
  17. ( R k ) k K \scriptstyle(R_{k})_{k\in K}
  18. R k \scriptstyle R_{k}
  19. P k \scriptstyle P_{k}
  20. X \scriptstyle X
  21. P 1 \scriptstyle P_{1}
  22. R 1 \scriptstyle R_{1}
  23. P 2 \scriptstyle P_{2}
  24. R 2 \scriptstyle R_{2}
  25. P 3 \scriptstyle P_{3}
  26. R 3 \scriptstyle R_{3}
  27. R k \scriptstyle R_{k}
  28. P k \scriptstyle P_{k}
  29. 2 = d [ ( a 1 , a 2 ) , ( b 1 , b 2 ) ] = ( a 1 - b 1 ) 2 + ( a 2 - b 2 ) 2 \ell_{2}=d\left[\left(a_{1},a_{2}\right),\left(b_{1},b_{2}\right)\right]=\sqrt% {\left(a_{1}-b_{1}\right)^{2}+\left(a_{2}-b_{2}\right)^{2}}
  30. d [ ( a 1 , a 2 ) , ( b 1 , b 2 ) ] = | a 1 - b 1 | + | a 2 - b 2 | d\left[\left(a_{1},a_{2}\right),\left(b_{1},b_{2}\right)\right]=\left|a_{1}-b_% {1}\right|+\left|a_{2}-b_{2}\right|
  31. O ( n 1 2 d ) \scriptstyle O\left(n^{\left\lceil\frac{1}{2}d\right\rceil}\right)