wpmath0000010_3

Reprojection_error.html

  1. 𝐗 ^ \hat{\mathbf{X}}
  2. 𝐱 \mathbf{x}
  3. 𝐏 \mathbf{P}
  4. 𝐱 ^ \hat{\mathbf{x}}
  5. 𝐗 ^ \hat{\mathbf{X}}
  6. 𝐱 ^ = 𝐏 𝐗 ^ \hat{\mathbf{x}}=\mathbf{P}\,\hat{\mathbf{X}}
  7. 𝐗 ^ \hat{\mathbf{X}}
  8. d ( 𝐱 , 𝐱 ^ ) d(\mathbf{x},\,\hat{\mathbf{x}})
  9. d ( 𝐱 , 𝐱 ^ ) d(\mathbf{x},\,\hat{\mathbf{x}})
  10. 𝐱 \mathbf{x}
  11. 𝐱 ^ \hat{\mathbf{x}}
  12. { 𝐱 𝐢 𝐱 𝐢 } \{\mathbf{x_{i}}\leftrightarrow\mathbf{x_{i}}^{\prime}\}
  13. 𝐇 ^ \hat{\mathbf{H}}
  14. 𝐱 𝐢 ^ \hat{\mathbf{x_{i}}}
  15. 𝐱 ^ i \hat{\mathbf{x}}_{i}^{\prime}
  16. 𝐱 𝐢 ^ = H ^ 𝐱 ^ 𝐢 \hat{\mathbf{x_{i}}}^{\prime}=\hat{H}\mathbf{\hat{x}_{i}}
  17. i d ( 𝐱 𝐢 , 𝐱 𝐢 ^ ) 2 + d ( 𝐱 𝐢 , 𝐱 𝐢 ^ ) 2 \sum_{i}d(\mathbf{x_{i}},\hat{\mathbf{x_{i}}})^{2}+d(\mathbf{x_{i}}^{\prime},% \hat{\mathbf{x_{i}}}^{\prime})^{2}
  18. 𝐱 𝐢 ^ , 𝐱 𝐢 ^ \hat{\mathbf{x_{i}}},\hat{\mathbf{x_{i}}}^{\prime}

Resistive_ballooning_mode.html

  1. γ \gamma
  2. γ 2 = - g e f f p p \gamma^{2}=-\vec{g_{eff}}\cdot\frac{\nabla p}{p}
  3. | p | p L p |\nabla p|\sim\frac{p}{L_{p}}
  4. g e f f = c s | B B | 1 / R 0 g_{eff}=c_{s}|\frac{\nabla B}{B}|\sim 1/R_{0}
  5. g \vec{g}
  6. g e f f \vec{g}_{eff}
  7. g \vec{g}
  8. ρ \rho
  9. g e f f \vec{g}_{eff}
  10. p p

Resistive_skin_time.html

  1. τ R = μ 0 a 2 η \tau_{R}=\frac{\mu_{0}a^{2}}{\eta}
  2. η \eta
  3. a a
  4. μ 0 \mu_{0}

Retardation_factor.html

  1. R = quantity of substance in the mobile phase total quantity of substance in the system \ R=\frac{\mbox{quantity of substance in the mobile phase}~{}}{\mbox{total % quantity of substance in the system}~{}}
  2. R f = migration distance of substance migration distance of solvent front \ R_{f}=\frac{\mbox{migration distance of substance}~{}}{\mbox{migration % distance of solvent front}~{}}
  3. R = 1 k + 1 \ R=\frac{1}{k+1}
  4. k = 1 - R R \ k=\frac{1-R}{R}

Reticuline_oxidase.html

  1. \rightleftharpoons

Retinal_dehydrogenase.html

  1. \rightleftharpoons

Retinal_isomerase.html

  1. \rightleftharpoons

Retinal_oxidase.html

  1. \rightleftharpoons

Retinol_dehydrogenase.html

  1. \rightleftharpoons

Retinol_isomerase.html

  1. \rightleftharpoons

Reversible_diffusion.html

  1. d X t = b ( X t ) d t + d B t \mathrm{d}X_{t}=b(X_{t})\,\mathrm{d}t+\mathrm{d}B_{t}
  2. d μ ( x ) d x = exp ( - 2 Φ ( x ) ) \frac{\mathrm{d}\mu(x)}{\mathrm{d}x}=\exp\left(-2\Phi(x)\right)
  3. 𝐑 d exp ( - 2 Φ ( x ) ) d x = 1. \int_{\mathbf{R}^{d}}\exp\left(-2\Phi(x)\right)\,\mathrm{d}x=1.

Ribitol-5-phosphate_2-dehydrogenase.html

  1. \rightleftharpoons

Ribitol_2-dehydrogenase.html

  1. \rightleftharpoons

Ribose_1-dehydrogenase_(NADP+).html

  1. \rightleftharpoons

Ribose_isomerase.html

  1. \rightleftharpoons

Ribosyldihydronicotinamide_dehydrogenase_(quinone).html

  1. \rightleftharpoons

Richardson's_theorem.html

  1. e a x 2 e^{ax^{2}}
  2. a = 0 a=0

Rifamycin-B_oxidase.html

  1. \rightleftharpoons

Right_half-plane.html

  1. { z : Re ( z ) > 0 } \{z\in\mathbb{C}\quad:\quad\mbox{Re}~{}(z)>0\}

Robbins_lemma.html

  1. E ( X f ( X - 1 ) ) = λ E ( f ( X ) ) . \operatorname{E}(Xf(X-1))=\lambda\operatorname{E}(f(X)).\,

Robinson's_joint_consistency_theorem.html

  1. T 1 T_{1}
  2. T 2 T_{2}
  3. T 1 T_{1}
  4. T 2 T_{2}
  5. T 1 T 2 T_{1}\cap T_{2}
  6. T 1 T_{1}
  7. T 2 T_{2}
  8. T 1 T 2 T_{1}\cup T_{2}
  9. T φ T\vdash\varphi
  10. T ¬ φ T\vdash\neg\varphi
  11. T 1 T_{1}
  12. T 2 T_{2}
  13. T 1 T_{1}
  14. T 2 T_{2}
  15. φ \varphi
  16. T 1 T_{1}
  17. T 2 T_{2}
  18. T 1 φ T_{1}\vdash\varphi
  19. T 2 ¬ φ T_{2}\vdash\neg\varphi
  20. T 1 T 2 T_{1}\cup T_{2}

Rook_polynomial.html

  1. R B ( x ) = k = 0 r k ( B ) x k R_{B}(x)=\sum_{k=0}^{\infty}r_{k}(B)x^{k}
  2. R 1 ( x ) \displaystyle R_{1}(x)
  3. R m , n ( x ) = n ! x n L n ( m - n ) ( - x - 1 ) . R_{m,n}(x)=n!x^{n}L_{n}^{(m-n)}(-x^{-1}).
  4. ( 64 8 ) = 64 ! 8 ! ( 64 - 8 ) ! = 4 , 426 , 165 , 368. {64\choose 8}=\frac{64!}{8!(64-8)!}=4,426,165,368.
  5. ( m k ) {\left({{m}\atop{k}}\right)}
  6. ( n k ) {\left({{n}\atop{k}}\right)}
  7. ( m k ) ( n k ) {\left({{m}\atop{k}}\right)}{\left({{n}\atop{k}}\right)}
  8. r k = ( m k ) ( n k ) k ! = n ! m ! k ! ( n - k ) ! ( m - k ) ! . r_{k}={\left({{m}\atop{k}}\right)}{\left({{n}\atop{k}}\right)}k!=\frac{n!m!}{k% !(n-k)!(m-k)!}.
  9. 8 ! 8 ! 3 ! 5 ! 5 ! = 18 , 816 \textstyle{\frac{8!8!}{3!5!5!}}=18,816
  10. R m , n ( x ) = k = 0 min ( m , n ) ( m k ) ( n k ) k ! x k = k = 0 min ( m , n ) n ! m ! k ! ( n - k ) ! ( m - k ) ! x k . R_{m,n}(x)=\sum_{k=0}^{\min(m,n)}{\left({{m}\atop{k}}\right)}{\left({{n}\atop{% k}}\right)}k!x^{k}=\sum_{k=0}^{\min(m,n)}\frac{n!m!}{k!(n-k)!(m-k)!}x^{k}.
  11. ( m n k ) = ( m n ) ! k ! ( m n - k ) ! {\left({{mn}\atop{k}}\right)}=\frac{(mn)!}{k!(mn-k)!}
  12. Q n = 1 + ( n 2 ) + 1 1 × 2 ( n 2 ) ( n - 2 2 ) + 1 1 × 2 × 3 ( n 2 ) ( n - 2 2 ) ( n - 4 2 ) + . Q_{n}=1+{\left({{n}\atop{2}}\right)}+\frac{1}{1\times 2}{\left({{n}\atop{2}}% \right)}{\left({{n-2}\atop{2}}\right)}+\frac{1}{1\times 2\times 3}{\left({{n}% \atop{2}}\right)}{\left({{n-2}\atop{2}}\right)}{\left({{n-4}\atop{2}}\right)}+\cdots.

RRNA_(adenine-N6-)-methyltransferase.html

  1. \rightleftharpoons

RRNA_(adenosine-2'-O-)-methyltransferase.html

  1. \rightleftharpoons

RRNA_(guanine-N1-)-methyltransferase.html

  1. \rightleftharpoons

RRNA_(guanine-N2-)-methyltransferase.html

  1. \rightleftharpoons

Rubredoxin—NAD(+)_reductase.html

  1. \rightleftharpoons

Rubredoxin—NAD(P)(+)_reductase.html

  1. \rightleftharpoons

Rutherford_backscattering_spectrometry.html

  1. E 1 = k E 0 , E_{1}=k\cdot E_{0},
  2. k = ( m 1 cos θ 1 ± m 2 2 - m 1 2 ( sin θ 1 ) 2 m 1 + m 2 ) 2 , k=\left(\frac{m_{1}\cos{\theta_{1}}\pm\sqrt{m_{2}^{2}-m_{1}^{2}(\sin{\theta_{1% }})^{2}}}{m_{1}+m_{2}}\right)^{2},
  3. θ 1 \theta_{1}
  4. d ω d Ω = ( Z 1 Z 2 e 2 4 E 0 ) 2 1 ( sin θ / 2 ) 4 , \frac{d\omega}{d\Omega}=\left(\frac{Z_{1}Z_{2}e^{2}}{4E_{0}}\right)^{2}\frac{1% }{\left(\sin{\theta/2}\right)^{4}},
  5. Z 1 Z_{1}
  6. Z 2 Z_{2}
  7. θ 1 \theta_{1}
  8. θ \theta
  9. S ( E ) = - d E d x . S(E)=-{dE\over dx}.
  10. Z 2 E \frac{Z_{2}}{E}
  11. R = 2 Z 1 Z 2 e 2 L E 0 R=2\sqrt{\frac{Z_{1}Z_{2}e^{2}L}{E_{0}}}
  12. Z 1 Z 2 E 0 d \sqrt{\frac{Z_{1}Z_{2}}{E_{0}d}}

Rydberg_matter.html

  1. d = 2.9 n 2 a 0 d=2.9n^{2}a_{0}\,

S-(hydroxymethyl)glutathione_dehydrogenase.html

  1. \rightleftharpoons

S-(hydroxymethyl)glutathione_synthase.html

  1. \rightleftharpoons

S-alkylcysteine_lyase.html

  1. \rightleftharpoons

S-carboxymethylcysteine_synthase.html

  1. \rightleftharpoons

S-matrix_theory.html

  1. S S = 1 SS^{\dagger}=1

S-methyl-5-thioribose-1-phosphate_isomerase.html

  1. \rightleftharpoons

S-ribosylhomocysteine_lyase.html

  1. \rightleftharpoons

S5_(modal_logic).html

  1. \Box
  2. \Diamond
  3. ( A B ) ( A B ) \Box(A\to B)\to(\Box A\to\Box B)
  4. A A \Box A\to A
  5. A A \Diamond A\to\Box\Diamond A
  6. A A \Box A\to\Box\Box A
  7. A A A\to\Box\Diamond A

Saccharopine_dehydrogenase_(NAD+,_L-glutamate-forming).html

  1. \rightleftharpoons

Saccharopine_dehydrogenase_(NAD+,_L-lysine-forming).html

  1. \rightleftharpoons

Saccharopine_dehydrogenase_(NADP+,_L-glutamate-forming).html

  1. \rightleftharpoons

Saccharopine_dehydrogenase_(NADP+,_L-lysine-forming).html

  1. \rightleftharpoons

Saint-Venant's_theorem.html

  1. ρ \rho
  2. σ \sigma
  3. P = 4 sup f ( D f d x d y ) 2 D f x 2 + f y 2 d x d y . P=4\sup_{f}\frac{\left(\int\int\limits_{D}f\,dx\,dy\right)^{2}}{\int\int% \limits_{D}f_{x}^{2}+f_{y}^{2}\,dx\,dy}.
  4. P P circle A 2 2 π . P\leq P_{\,\text{circle}}\leq\frac{A^{2}}{2\pi}.
  5. P < 4 ρ 2 A P<4\rho^{2}A

Salicylaldehyde_dehydrogenase.html

  1. \rightleftharpoons

Salicylate_1-monooxygenase.html

  1. \rightleftharpoons

Salutaridine_reductase_(NADPH).html

  1. \rightleftharpoons

Salutaridine_synthase.html

  1. \rightleftharpoons

Sammon_mapping.html

  1. d i j * \scriptstyle d^{*}_{ij}
  2. d i j \scriptstyle d_{ij}
  3. E = 1 i < j d i j * i < j ( d i j * - d i j ) 2 d i j * . E=\frac{1}{\sum\limits_{i<j}d^{*}_{ij}}\sum_{i<j}\frac{(d^{*}_{ij}-d_{ij})^{2}% }{d^{*}_{ij}}.

Sarcosine_dehydrogenase.html

  1. \rightleftharpoons

Sarcosine_reductase.html

  1. \rightleftharpoons

Scaling_pattern_of_occupancy.html

  1. P a = P 0 a D / 2 - 1 P_{a}=P_{0}a^{D/2-1}\,
  2. P a j = f 0 f 1 f j . P_{a_{j}}=f_{0}f_{1}\cdots f_{j}.\,
  3. p ( 4 a ) + = 1 - Ω 4 p\,{(4a)_{+}}=1-\frac{{\Omega}^{4}}{\mho}
  4. q ( 4 a ) + / + = Ω 10 - 2 Ω 4 2 + 3 2 ( - Ω 4 + ) q\,{(4a)_{+/+}}=\frac{{\Omega}^{10}-2\,{\Omega}^{4}\,{\mho}^{2}+{\mho}^{3}}{{% \mho}^{2}\,\left(-{\Omega}^{4}+\mho\right)}
  5. \mho
  6. P a = 1 - b c 2 a 1 / 2 h a P_{a}=1-bc^{2a^{1/2}}h^{a}\,

Scattering-matrix_method.html

  1. E t o t = E i n c + E s c a t t E_{tot}=E_{inc}+E_{scatt}

Schilder's_theorem.html

  1. f : [ 0 , T ] 𝐑 d f:[0,T]\longrightarrow\mathbf{R}^{d}
  2. f ( 0 ) = 0 f(0)=0
  3. I ( ω ) = 1 2 0 T | ω ˙ ( t ) | 2 d t I(\omega)=\frac{1}{2}\int_{0}^{T}|\dot{\omega}(t)|^{2}\,\mathrm{d}t
  4. lim sup ε 0 ε log 𝐖 ε ( F ) - inf ω F I ( ω ) \limsup_{\varepsilon\downarrow 0}\varepsilon\log\mathbf{W}_{\varepsilon}(F)% \leq-\inf_{\omega\in F}I(\omega)
  5. lim inf ε 0 ε log 𝐖 ε ( G ) - inf ω G I ( ω ) . \liminf_{\varepsilon\downarrow 0}\varepsilon\log\mathbf{W}_{\varepsilon}(G)% \geq-\inf_{\omega\in G}I(\omega).
  6. 𝐖 ( C 0 𝐁 c ( 0 ; ) ) 𝐏 [ B > c ] , \mathbf{W}(C_{0}\setminus\mathbf{B}_{c}(0;\|\cdot\|_{\infty}))\equiv\mathbf{P}% \big[\|B\|_{\infty}>c\big],
  7. B > c ε B A := { ω C 0 | | ω ( t ) | > 1 for some t [ 0 , T ] } . \|B\|_{\infty}>c\iff\sqrt{\varepsilon}B\in A:=\big\{\omega\in C_{0}\big||% \omega(t)|>1\mbox{ for some }~{}t\in[0,T]\big\}.
  8. lim c 1 c 2 log 𝐏 [ B > c ] \lim_{c\to\infty}\frac{1}{c^{2}}\log\mathbf{P}\big[\|B\|_{\infty}>c\big]
  9. = lim ε 0 ϵ 𝐏 [ ε B A ] =\lim_{\varepsilon\to 0}\epsilon\mathbf{P}\big[\sqrt{\varepsilon}B\in A\big]
  10. = - inf { 1 2 0 T | ω ˙ ( t ) | 2 d t | ω A } =-\inf\left\{\left.\frac{1}{2}\int_{0}^{T}|\dot{\omega}(t)|^{2}\,\mathrm{d}t% \right|\omega\in A\right\}
  11. = - 1 2 0 T 1 T 2 d t =-\frac{1}{2}\int_{0}^{T}\frac{1}{T^{2}}\,\mathrm{d}t
  12. = - 1 2 T , =-\frac{1}{2T},
  13. 1 c 2 log 𝐏 [ B > c ] - 1 2 T , \frac{1}{c^{2}}\log\mathbf{P}\big[\|B\|_{\infty}>c\big]\approx-\frac{1}{2T},
  14. 𝐏 [ B > c ] exp ( - c 2 2 T ) . \mathbf{P}\big[\|B\|_{\infty}>c\big]\approx\exp\left(-\frac{c^{2}}{2T}\right).
  15. 𝐏 [ sup 0 t T | ε B t | c ] 4 n exp ( - c 2 2 n T ε ) . \mathbf{P}\left[\sup_{0\leq t\leq T}\big|\sqrt{\varepsilon}B_{t}\big|\geq c% \right]\leq 4n\exp\left(-\frac{c^{2}}{2nT\varepsilon}\right).

Schrödinger_field.html

  1. L = ψ ( i t + 2 2 m ) ψ . L=\psi^{\dagger}\left(i{\partial\over\partial t}+{\nabla^{2}\over 2m}\right)\psi.
  2. ψ \psi
  3. ψ \psi
  4. V ( x ) V(x)
  5. S = x t ψ ( i t + 2 2 m ) ψ - ψ ( x ) ψ ( x ) V ( x ) . S=\int_{xt}\psi^{\dagger}\left(i{\partial\over\partial t}+{\nabla^{2}\over 2m}% \right)\psi-\psi^{\dagger}(x)\psi(x)V(x).
  6. ϕ i ( x ) \phi_{i}(x)
  7. E i E_{i}
  8. ψ ( x ) = i ψ i ϕ i ( x ) . \psi(x)=\sum_{i}\psi_{i}\phi_{i}(x).\,
  9. S = t i ψ i ( i t - E i ) ψ i S=\int_{t}\sum_{i}\psi_{i}^{\dagger}\left(i{\partial\over\partial t}-E_{i}% \right)\psi_{i}\,
  10. S = t i 2 ψ r d ψ i d t - E i ( ψ r 2 + ψ i 2 ) S=\int_{t}\sum_{i}2\psi_{r}{d\psi_{i}\over dt}-E_{i}(\psi_{r}^{2}+\psi_{i}^{2})
  11. ψ r \psi_{r}
  12. S = t i 1 E i ( d ψ i d t ) 2 - E i ψ i 2 S=\int_{t}\sum_{i}{1\over E_{i}}\left({d\psi_{i}\over dt}\right)^{2}-E_{i}\psi% _{i}^{2}
  13. ψ i \scriptstyle\psi_{i}
  14. E i E_{i}
  15. V ( x 1 , x 2 ) V(x_{1},x_{2})
  16. S = x t ψ ( i t + 2 2 m ) ψ - x y ψ ( x ) ψ ( x ) V ( x , y ) ψ ( y ) ψ ( y ) . S=\int_{xt}\psi^{\dagger}\left(i{\partial\over\partial t}+{\nabla^{2}\over 2m}% \right)\psi-\int_{xy}\psi^{\dagger}(x)\psi(x)V(x,y)\psi^{\dagger}(y)\psi(y).
  17. V ( x , y ) = q 2 | x - y | . V(x,y)={q^{2}\over|x-y|}.
  18. V ( x 1 , x 2 ) V(x_{1},x_{2})
  19. S = x ψ ( i t + 2 2 m ) ψ + λ ( ψ ψ ) 2 . S=\int_{x}\psi^{\dagger}\left(i{\partial\over\partial t}+{\nabla^{2}\over 2m}% \right)\psi+\lambda(\psi^{\dagger}\psi)^{2}.
  20. λ \lambda
  21. L i = x ( ψ ψ ) ( x 1 ) ( ψ ψ ) ( x 2 ) ( ψ ψ ) ( x n ) V ( x 1 , x 2 , , x n ) . L_{i}=\int_{x}(\psi^{\dagger}\psi)(x_{1})(\psi^{\dagger}\psi)(x_{2})\cdots(% \psi^{\dagger}\psi)(x_{n})V(x_{1},x_{2},\dots,x_{n}).\,
  22. ψ \psi
  23. Π ( x ) = i ψ . \Pi(x)=i\psi^{\dagger}.\,
  24. [ ψ ( x ) , ψ ( y ) ] = δ ( x - y ) . [\psi(x),\psi^{\dagger}(y)]=\delta(x-y).
  25. H = S - Π ( x ) d d t ψ = | ψ | 2 2 m + x y V ( x , y ) ψ ( x ) ψ ( x ) ψ ( y ) ψ ( y ) H=S-\int\Pi(x){d\over dt}\psi=\int{|\nabla\psi|^{2}\over 2m}+\int_{xy}V(x,y)% \psi^{\dagger}(x)\psi(x)\psi^{\dagger}(y)\psi(y)\,
  26. i t ψ = - 2 2 m ψ + ( y V ( x , y ) ψ ( y ) ψ ( y ) ) ψ ( x ) . i{\partial\over\partial t}\psi=-{\nabla^{2}\over 2m}\psi+\left(\int_{y}V(x,y)% \psi^{\dagger}(y)\psi(y)\right)\psi(x).\,
  27. G ( k ) = 1 i ω - k 2 2 m . G(k)={1\over i\omega-{k^{2}\over 2m}}.\,
  28. i d d t ψ = ( 1 2 2 m + 2 2 2 m + + N 2 2 m + V ( x 1 , x 2 , , x N ) ) ψ i{d\over dt}\psi=\left(\frac{\nabla_{1}^{2}}{2m}+\frac{\nabla_{2}^{2}}{2m}+% \cdots+\frac{\nabla_{N}^{2}}{2m}+V(x_{1},x_{2},\dots,x_{N})\right)\psi\,
  29. H = p 1 2 2 m + p 2 2 2 m + + p N 2 2 m + V ( x 1 , , x N ) . H=\frac{p_{1}^{2}}{2m}+\frac{p_{2}^{2}}{2m}+\cdots+\frac{p_{N}^{2}}{2m}+V(x_{1% },\dots,x_{N}).\,
  30. ψ ( x 1 , x 2 , ) = ψ ( x 2 , x 1 , ) for bosons \psi(x_{1},x_{2},\dots)=\psi(x_{2},x_{1},\dots)\qquad\quad\,\text{for bosons}
  31. ψ ( x 1 , x 2 , ) = - ψ ( x 2 , x 1 , ) for fermions \psi(x_{1},x_{2},\dots)=-\psi(x_{2},x_{1},\dots)\qquad\,\text{for fermions}
  32. V ( x 1 , , x N ) = V 1 ( x 1 ) + V 2 ( x 2 ) + + V N ( x N ) V(x_{1},\dots,x_{N})=V_{1}(x_{1})+V_{2}(x_{2})+\cdots+V_{N}(x_{N})\,
  33. V 1 = V 2 = = V N V_{1}=V_{2}=\cdots=V_{N}
  34. V ( x 1 , x N ) = V 1 , 2 ( x 1 , x 2 ) + V 1 , 3 ( x 2 , x 3 ) + V 2 , 3 ( x 1 , x 2 ) V(x_{1}...,x_{N})=V_{1,2}(x_{1},x_{2})+V_{1,3}(x_{2},x_{3})+V_{2,3}(x_{1},x_{2% })\,
  35. V 1 , 2 = V 1 , 3 = V 2 , 3 V_{1,2}=V_{1,3}=V_{2,3}
  36. | N ; x 1 , , x N |N;x_{1},\ldots,x_{N}\rangle\,
  37. ψ 0 | 0 + x ψ 1 ( x ) | 1 ; x + x 1 x 2 ψ 2 ( x 1 , x 2 ) | 2 ; x 1 x 2 + \psi_{0}|0\rangle+\int_{x}\psi_{1}(x)|1;x\rangle+\int_{x_{1}x_{2}}\psi_{2}(x_{% 1},x_{2})|2;x_{1}x_{2}\rangle+\ldots\,
  38. ψ 0 \psi_{0}
  39. 1 ; x 1 | 1 ; y 1 = δ ( x 1 - y 1 ) \langle 1;x_{1}|1;y_{1}\rangle=\delta(x_{1}-y_{1})\,
  40. 2 ; x 1 x 2 | 2 ; y 1 y 2 = δ ( x 1 - y 1 ) δ ( x 2 - y 2 ) ± δ ( x 1 - y 2 ) δ ( x 2 - y 1 ) \langle 2;x_{1}x_{2}|2;y_{1}y_{2}\rangle=\delta(x_{1}-y_{1})\delta(x_{2}-y_{2}% )\pm\delta(x_{1}-y_{2})\delta(x_{2}-y_{1})\,
  41. ψ ( x ) \scriptstyle\psi^{\dagger}(x)
  42. ψ ( x ) | N ; x 1 x n = | N + 1 ; x 1 , , x n , x \psi^{\dagger}(x)|N;x_{1}...x_{n}\rangle=|N+1;x_{1},...,x_{n},x\rangle\,
  43. ψ \psi
  44. ψ \psi^{\dagger}
  45. ψ \scriptstyle\psi^{\dagger}
  46. ψ \psi
  47. ψ ( x ) | N ; x 1 , x N = δ ( x - x 1 ) | N - 1 ; x 2 , x N + δ ( x - x 2 ) | N - 1 ; x 1 , x 3 , x N + \psi(x)|N;x_{1}...,x_{N}\rangle=\delta(x-x_{1})|N-1;x_{2}...,x_{N}\rangle+% \delta(x-x_{2})|N-1;x_{1},x_{3}...,x_{N}\rangle+\ldots\,
  48. ψ ( k ) = x e - i k x ψ ( x ) \psi^{\dagger}(k)=\int_{x}e^{-ikx}\psi^{\dagger}(x)\,
  49. ψ ( k ) = x e i k x ψ ( x ) \psi(k)=\int_{x}e^{ikx}\psi(x)\,
  50. ψ ( k ) ψ ( k ) - ψ ( k ) ψ ( k ) = 0 \psi^{\dagger}(k)\psi^{\dagger}(k^{\prime})-\psi^{\dagger}(k^{\prime})\psi^{% \dagger}(k)=0\,
  51. ψ ( k ) ψ ( k ) - ψ ( k ) ψ ( k ) = 0 \psi(k)\psi(k^{\prime})-\psi(k^{\prime})\psi(k)=0\,
  52. ψ ( k ) ψ ( k ) - ψ ( k ) ψ ( k ) = δ ( k - k ) \psi(k)\psi^{\dagger}(k^{\prime})-\psi(k^{\prime})\psi^{\dagger}(k)=\delta(k-k% ^{\prime})\,
  53. | n 1 , n 2 , n k |n_{1},n_{2},...n_{k}\rangle\,
  54. ψ k \scriptstyle\psi_{k}
  55. ψ ( k ) | . . , n k , = n k + 1 | , n k + 1 , \psi^{\dagger}(k)|..,n_{k},\ldots\rangle=\sqrt{n_{k}+1}\,|...,n_{k}+1,\ldots\rangle
  56. ψ ( k ) | , n k , = n k | , n k - 1 , \psi(k)|...,n_{k},\ldots\rangle=\sqrt{n_{k}}\,|...,n_{k}-1,\ldots\rangle
  57. k ψ ( k ) ψ ( k ) = x ψ ( x ) ψ ( x ) \sum_{k}\psi^{\dagger}(k)\psi(k)=\int_{x}\psi^{\dagger}(x)\psi(x)
  58. ψ ( x ) \scriptstyle\psi(x)
  59. ψ ( x ) \scriptstyle\psi^{\dagger}(x)
  60. [ ψ ( x ) , ψ ( y ) ] = [ ψ ( x ) , ψ ( y ) ] = 0 [\psi(x),\psi(y)]=[\psi^{\dagger}(x),\psi^{\dagger}(y)]=0
  61. [ ψ ( x ) , ψ ( y ) ] = δ ( x - y ) [\psi(x),\psi^{\dagger}(y)]=\delta(x-y)
  62. ψ ( x ) ψ ( x ) \scriptstyle\psi^{\dagger}(x)\psi(x)
  63. ψ ψ \scriptstyle\psi^{\dagger}\nabla\psi
  64. H = x ψ ( x ) 2 2 m ψ ( x ) H=\int_{x}\psi^{\dagger}(x){\nabla^{2}\over 2m}\psi(x)\,
  65. ψ i d d t ψ = ψ - 2 2 m ψ \psi^{\dagger}i{d\over dt}\psi=\psi^{\dagger}{-\nabla^{2}\over 2m}\psi\,
  66. ψ \scriptstyle\psi^{\dagger}
  67. i t ψ = - 2 2 m ψ i{\partial\over\partial t}\psi={-\nabla^{2}\over 2m}\psi\,
  68. H = ψ ψ 2 m H={\nabla\psi^{\dagger}\nabla\psi\over 2m}
  69. L = ψ ( i t + 2 2 m ) ψ L=\psi^{\dagger}\left(i{\partial\over\partial t}+{\nabla^{2}\over 2m}\right)\psi\,

Schuette–Nesbitt_formula.html

  1. Ω Ω
  2. N ( ω ) = n = 1 m 1 A n ( ω ) , ω Ω , N(\omega)=\sum_{n=1}^{m}1_{A_{n}}(\omega),\qquad\omega\in\Omega,
  3. ω Ω ω∈Ω
  4. k 0 , 1 , , m k∈{0,1,...,m}
  5. N k ( ω ) = J { 1 , , m } | J | = k 1 j J A j ( ω ) , ω Ω , N_{k}(\omega)=\sum_{\scriptstyle J\subset\{1,\ldots,m\}\atop\scriptstyle|J|=k}% 1_{\cap_{j\in J}A_{j}}(\omega),\qquad\omega\in\Omega,
  6. k k
  7. ω ω
  8. Ω Ω
  9. V V
  10. R R
  11. R R
  12. n = 0 m 1 { N = n } c n = k = 0 m N k l = 0 k ( - 1 ) k - l ( k l ) c l , \sum_{n=0}^{m}1_{\{N=n\}}c_{n}=\sum_{k=0}^{m}N_{k}\sum_{l=0}^{k}(-1)^{k-l}{% \left({{k}\atop{l}}\right)}c_{l},
  13. ω Ω ω∈Ω
  14. N ( ω ) = n N(ω)=n
  15. ( k l ) \textstyle{\left({{k}\atop{l}}\right)}
  16. V V
  17. Ω Ω
  18. ω Ω ω∈Ω
  19. n = N ( ω ) n=N(ω)
  20. I I
  21. i 1 , , m i∈{1,...,m}
  22. I I
  23. n n
  24. J 1 , , m J⊂{1,...,m}
  25. k k
  26. ω ω
  27. J J
  28. I I
  29. ( n k ) \textstyle{\left({{n}\atop{k}}\right)}
  30. k > n k>n
  31. ω ω
  32. k = 0 m ( n k ) l = 0 k ( - 1 ) k - l ( k l ) c l = l = 0 m k = l n ( - 1 ) k - l ( n k ) ( k l ) = : ( * ) c l , \sum_{k=0}^{m}{\left({{n}\atop{k}}\right)}\sum_{l=0}^{k}(-1)^{k-l}{\left({{k}% \atop{l}}\right)}c_{l}=\sum_{l=0}^{m}\underbrace{\sum_{k=l}^{n}(-1)^{k-l}{% \left({{n}\atop{k}}\right)}{\left({{k}\atop{l}}\right)}}_{=:\,(*)}c_{l},
  33. k > n k>n
  34. ( * ) = k = l n ( - 1 ) k - l n ! k ! ( n - k ) ! k ! l ! ( k - l ) ! = n ! l ! ( n - l ) ! = ( n l ) k = l n ( - 1 ) k - l ( n - l ) ! ( n - k ) ! ( k - l ) ! = : ( * * ) \begin{aligned}\displaystyle(*)&\displaystyle=\sum_{k=l}^{n}(-1)^{k-l}\frac{n!% }{k!\,(n-k)!}\,\frac{k!}{l!\,(k-l)!}\\ &\displaystyle=\underbrace{\frac{n!}{l!\,(n-l)!}}_{={\left({{n}\atop{l}}\right% )}}\underbrace{\sum_{k=l}^{n}(-1)^{k-l}\frac{(n-l)!}{(n-k)!\,(k-l)!}}_{=:\,(**% )}\\ \end{aligned}
  35. j = k l j=k−l
  36. ( * * ) = j = 0 n - l ( - 1 ) j ( n - l ) ! ( n - l - j ) ! j ! = j = 0 n - l ( - 1 ) j ( n - l j ) = ( 1 - 1 ) n - l = δ l n , \begin{aligned}\displaystyle(**)&\displaystyle=\sum_{j=0}^{n-l}(-1)^{j}\frac{(% n-l)!}{(n-l-j)!\,j!}\\ &\displaystyle=\sum_{j=0}^{n-l}(-1)^{j}{\left({{n-l}\atop{j}}\right)}=(1-1)^{n% -l}=\delta_{ln},\end{aligned}
  37. n n
  38. l l
  39. 1 1
  40. l = n l=n
  41. ω ω
  42. c < s u b > n c<sub>n
  43. V V
  44. R x x Rxx
  45. x x
  46. n = 0 m 1 { N = n } x n = k = 0 m N k ( x - 1 ) k . \sum_{n=0}^{m}1_{\{N=n\}}x^{n}=\sum_{k=0}^{m}N_{k}(x-1)^{k}.
  47. ω ω
  48. n 0 , , m n∈{0,...,m}
  49. n = 0 m 1 { N = n } x n = k = 0 m N k l = 0 k ( k l ) ( - 1 ) k - l x l = ( x - 1 ) k , \sum_{n=0}^{m}1_{\{N=n\}}x^{n}=\sum_{k=0}^{m}N_{k}\underbrace{\sum_{l=0}^{k}{% \left({{k}\atop{l}}\right)}(-1)^{k-l}x^{l}}_{=\,(x-1)^{k}},
  50. E E
  51. Δ Δ
  52. V V
  53. E : V 0 V 0 , E ( c 0 , c 1 , c 2 , c 3 , ) ( c 1 , c 2 , c 3 , ) , \begin{aligned}\displaystyle E:V^{\mathbb{N}_{0}}&\displaystyle\to V^{\mathbb{% N}_{0}},\\ \displaystyle E(c_{0},c_{1},c_{2},c_{3},\ldots)&\displaystyle\mapsto(c_{1},c_{% 2},c_{3},\ldots),\\ \end{aligned}
  54. Δ : V 0 V 0 , Δ ( c 0 , c 1 , c 2 , c 3 ) ( c 1 - c 0 , c 2 - c 1 , c 3 - c 2 , ) . \begin{aligned}\displaystyle\Delta:V^{\mathbb{N}_{0}}&\displaystyle\to V^{% \mathbb{N}_{0}},\\ \displaystyle\Delta(c_{0},c_{1},c_{2},c_{3}\ldots)&\displaystyle\mapsto(c_{1}-% c_{0},c_{2}-c_{1},c_{3}-c_{2},\ldots).\\ \end{aligned}
  55. x = E x=E
  56. n = 0 m 1 { N = n } E n = k = 0 m N k Δ k , \sum_{n=0}^{m}1_{\{N=n\}}E^{n}=\sum_{k=0}^{m}N_{k}\Delta^{k},
  57. Δ = E I Δ=E–I
  58. I I
  59. I I
  60. k k
  61. n = 0 m 1 { N = n } E n = j = 1 m ( 1 A j c I + 1 A j E ) \sum_{n=0}^{m}1_{\{N=n\}}E^{n}=\prod_{j=1}^{m}(1_{A_{j}^{\mathrm{c}}}I+1_{A_{j% }}E)
  62. Ω Ω
  63. ω ω
  64. Ω Ω
  65. k k
  66. k 0 , , m k∈{0,...,m}
  67. ω ω
  68. k k
  69. E E
  70. I I
  71. 1 A j c I + 1 A j E = I - 1 A j I + 1 A j E = I + 1 A j ( E - I ) = I + 1 A j Δ , j { 0 , , m } . \begin{aligned}\displaystyle 1_{A_{j}^{\mathrm{c}}}I+1_{A_{j}}E&\displaystyle=% I-1_{A_{j}}I+1_{A_{j}}E\\ &\displaystyle=I+1_{A_{j}}(E-I)=I+1_{A_{j}}\Delta,\qquad j\in\{0,\ldots,m\}.% \end{aligned}
  72. n = 0 m 1 { N = n } E n = k = 0 m J { 1 , , m } | J | = k 1 j J A j Δ k , \sum_{n=0}^{m}1_{\{N=n\}}E^{n}=\sum_{k=0}^{m}\sum_{\scriptstyle J\subset\{1,% \ldots,m\}\atop\scriptstyle|J|=k}1_{\cap_{j\in J}A_{j}}\Delta^{k},
  73. k k
  74. n = 0 m 1 { N = n } c n = k = 0 m N k ( Δ k c ) 0 . \sum_{n=0}^{m}1_{\{N=n\}}c_{n}=\sum_{k=0}^{m}N_{k}(\Delta^{k}c)_{0}.
  75. ( Ω , , ) (Ω,\mathcal{F}, ℙ)
  76. 𝐄 \mathbf{E}
  77. N N
  78. S k = 𝔼 [ N k ] = J { 1 , , m } | J | = k ( j J A j ) , k { 0 , , m } , S_{k}=\mathbb{E}[N_{k}]=\sum_{\scriptstyle J\subset\{1,\ldots,m\}\atop% \scriptstyle|J|=k}\mathbb{P}\biggl(\bigcap_{j\in J}A_{j}\biggr),\qquad k\in\{0% ,\ldots,m\},
  79. Ω Ω
  80. R R
  81. n = 0 m ( N = n ) x n = k = 0 m S k ( x - 1 ) k \sum_{n=0}^{m}\mathbb{P}(N=n)x^{n}=\sum_{k=0}^{m}S_{k}(x-1)^{k}
  82. R x x Rxx
  83. R R
  84. N N
  85. n = 0 m ( N = n ) E n = k = 0 m S k Δ k \sum_{n=0}^{m}\mathbb{P}(N=n)E^{n}=\sum_{k=0}^{m}S_{k}\Delta^{k}
  86. n = 0 m ( N = n ) c n = k = 0 m S k ( Δ k c ) 0 . \sum_{n=0}^{m}\mathbb{P}(N=n)\,c_{n}=\sum_{k=0}^{m}S_{k}\,(\Delta^{k}c)_{0}.
  87. V V
  88. E E
  89. c = ( 0 , 1 , 1 , ) c=(0,1,1,...)
  90. N 1 {N≥1}
  91. k 1 , , m k∈{1,...,m}
  92. m + 1 m+1
  93. E E
  94. Δ Δ
  95. ( m + 1 ) (m+1)
  96. ( m + 1 ) × ( m + 1 ) (m+1)×(m+1)
  97. E = ( 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 ) , Δ = ( - 1 1 0 0 0 - 1 1 0 0 0 - 1 1 0 0 0 - 1 ) , E=\begin{pmatrix}0&1&0&\cdots&0\\ 0&0&1&\ddots&\vdots\\ \vdots&\ddots&\ddots&\ddots&0\\ 0&\cdots&0&0&1\\ 0&\cdots&0&0&0\end{pmatrix},\qquad\Delta=\begin{pmatrix}-1&1&0&\cdots&0\\ 0&-1&1&\ddots&\vdots\\ \vdots&\ddots&\ddots&\ddots&0\\ 0&\cdots&0&-1&1\\ 0&\cdots&0&0&-1\end{pmatrix},
  98. I I
  99. ( m + 1 ) (m+1)
  100. ( m + 1 ) (m+1)
  101. T T
  102. c c
  103. E E
  104. Δ Δ
  105. E E
  106. I I
  107. N = n {N=n}
  108. N n {N≥n}
  109. ( k n ) \textstyle{\left({{k}\atop{n}}\right)}
  110. ( N = n ) = k = n m ( - 1 ) k - n ( k n ) S k , n { 0 , , m } , \mathbb{P}(N=n)=\sum_{k=n}^{m}(-1)^{k-n}{\left({{k}\atop{n}}\right)}S_{k},% \qquad n\in\{0,\ldots,m\},
  111. ( N n ) = k = n m ( - 1 ) k - n ( k - 1 n - 1 ) S k , n { 1 , , m } , \mathbb{P}(N\geq n)=\sum_{k=n}^{m}(-1)^{k-n}{\left({{k-1}\atop{n-1}}\right)}S_% {k},\qquad n\in\{1,\ldots,m\},
  112. Δ k = ( E - I ) k = j = 0 k ( k j ) ( - 1 ) k - j E j , k 0 . \Delta^{k}=(E-I)^{k}=\sum_{j=0}^{k}{\left({{k}\atop{j}}\right)}(-1)^{k-j}E^{j}% ,\qquad k\in\mathbb{N}_{0}.
  113. c = ( 0 , , 0 , 1 , 0 , 0 , ) c=(0,...,0,1,0,0,...)
  114. n n
  115. j = n j=n
  116. N = n {N=n}
  117. c = ( 0 , , 0 , 1 , 1 , 1 , ) c=(0,...,0,1,1,1,...)
  118. n n
  119. j n j≥n
  120. ( N n ) = k = n m S k j = n k ( k j ) ( - 1 ) k - j . \mathbb{P}(N\geq n)=\sum_{k=n}^{m}S_{k}\sum_{j=n}^{k}{\left({{k}\atop{j}}% \right)}(-1)^{k-j}.
  121. j = n k ( k j ) ( - 1 ) k - j = - j = 0 n - 1 ( k j ) ( - 1 ) k - j = ( - 1 ) k - n ( k - 1 n - 1 ) . \sum_{j=n}^{k}{\left({{k}\atop{j}}\right)}(-1)^{k-j}=-\sum_{j=0}^{n-1}{\left({% {k}\atop{j}}\right)}(-1)^{k-j}=(-1)^{k-n}{\left({{k-1}\atop{n-1}}\right)}.
  122. N n {N≥n}
  123. m m
  124. t t
  125. n n
  126. m m
  127. t t
  128. t t
  129. j j
  130. t t
  131. σ σ
  132. 1 , , m {1,...,m}
  133. j j
  134. σ σ
  135. J J
  136. 1 , , m {1,...,m}
  137. ( m | J | ) ! (m–|J|)!
  138. m | J | m–|J|
  139. ( j J A j ) = ( m - | J | ) ! m ! . \mathbb{P}\biggl(\bigcap_{j\in J}A_{j}\biggr)=\frac{(m-|J|)!}{m!}.
  140. ( m k ) \textstyle{\left({{m}\atop{k}}\right)}
  141. J J
  142. 1 , , m {1,...,m}
  143. k k
  144. S k = ( m k ) ( m - k ) ! m ! = 1 k ! . S_{k}={\left({{m}\atop{k}}\right)}\frac{(m-k)!}{m!}=\frac{1}{k!}.
  145. N N
  146. 𝔼 [ x N ] = k = 0 m ( x - 1 ) k k ! , x . \mathbb{E}[x^{N}]=\sum_{k=0}^{m}\frac{(x-1)^{k}}{k!},\qquad x\in\mathbb{R}.
  147. x 1 x–1
  148. 1 1
  149. m m
  150. N N
  151. 1 1

Scyllo-inosamine-4-phosphate_amidinotransferase.html

  1. \rightleftharpoons

Scymnol_sulfotransferase.html

  1. \rightleftharpoons

Seashell_surface.html

  1. x \displaystyle x
  2. 0 u < 2 π 0\leq u<2\pi
  3. - 2 π v < 2 π -2\pi\leq v<2\pi
  4. F ( θ , φ ) = e α φ ( cos ( φ ) , - sin ( φ ) , 0 sin ( φ ) , cos ( φ ) , 0 0 , 0 , 1 ) F ( θ , 0 ) \vec{F}\left({\theta,\varphi}\right)=e^{\alpha\varphi}\left({\begin{array}[]{*% {20}c}{\cos\left(\varphi\right),}&{-\sin(\varphi),}&{\rm{0}}\\ {\sin(\varphi),}&{\cos\left(\varphi\right),}&0\\ {0,}&{{\rm{0,}}}&1\\ \end{array}}\right)\vec{F}\left({\theta,0}\right)
  5. F ( θ , 0 ) \vec{F}\left({\theta,0}\right)

Secologanin_synthase.html

  1. \rightleftharpoons

Secondary-alcohol_oxidase.html

  1. \rightleftharpoons

Sedimentation_potential.html

  1. E s = - ε ζ ( ρ - ρ 0 ) ϕ p g σ η E_{s}=-\frac{\varepsilon\zeta(\rho-\rho_{0})\phi_{p}g}{\sigma^{\infty}\eta}
  2. σ = e 2 k B T z i 2 D i n i \sigma^{\infty}=\frac{e^{2}}{k_{B}T}\sum z_{i}^{2}D_{i}n_{i\infty}
  3. E s = - ε ζ ( ρ - ρ 0 ) ϕ p σ η g H ( κ α ) + ϑ ( ζ 2 ) E_{s}=-\frac{\varepsilon\zeta(\rho-\rho_{0})\phi_{p}}{\sigma^{\infty}\eta}gH(% \kappa\alpha)+\vartheta(\zeta^{2})
  4. ζ = η λ E s ε r ε 0 ( ρ - ρ 0 ) g \zeta=\frac{\eta\lambda E_{s}}{\varepsilon_{r}\varepsilon_{0}(\rho-\rho_{0})g}
  5. E s E_{s}
  6. η \eta
  7. λ \lambda
  8. ε r \varepsilon_{r}
  9. ε 0 \varepsilon_{0}
  10. ρ \rho
  11. ρ 0 \rho_{0}
  12. g g

Segment_tree.html

  1. ( - , p 1 ) , [ p 1 , p 1 ] , ( p 1 , p 2 ) , [ p 2 , p 2 ] , , ( p m - 1 , p m ) , [ p m , p m ] , ( p m , + ) (-\infty,p_{1}),[p_{1},p_{1}],(p_{1},p_{2}),[p_{2},p_{2}],...,(p_{m-1},p_{m}),% [p_{m},p_{m}],(p_{m},+\infty)

Selenate_reductase.html

  1. \rightleftharpoons

Selenocysteine_lyase.html

  1. \rightleftharpoons

Semialgebraic_set.html

  1. P ( x 1 , , x n ) = 0 P(x_{1},...,x_{n})=0
  2. Q ( x 1 , , x n ) > 0 Q(x_{1},...,x_{n})>0

Semicircular_potential_well.html

  1. 0
  2. π \pi
  3. π \pi
  4. 2 π 2\pi
  5. 0
  6. π \pi
  7. S 1 S^{1}
  8. - 2 2 m 2 ψ = E ψ ( 1 ) -\frac{\hbar^{2}}{2m}\nabla^{2}\psi=E\psi\quad(1)
  9. 2 = 1 s 2 2 ϕ 2 ( 2 ) \nabla^{2}=\frac{1}{s^{2}}\frac{\partial^{2}}{\partial\phi^{2}}\quad(2)
  10. - 2 2 m s 2 d 2 ψ d ϕ 2 = E ψ ( 3 ) -\frac{\hbar^{2}}{2ms^{2}}\frac{d^{2}\psi}{d\phi^{2}}=E\psi\quad(3)
  11. I = def V r 2 ρ ( r , ϕ , z ) r d r d ϕ d z I\ \stackrel{\mathrm{def}}{=}\ \iiint_{V}r^{2}\,\rho(r,\phi,z)\,rdr\,d\phi\,dz\!
  12. I = m s 2 I=ms^{2}
  13. - 2 2 I d 2 ψ d ϕ 2 = E ψ -\frac{\hbar^{2}}{2I}\frac{d^{2}\psi}{d\phi^{2}}=E\psi
  14. 0
  15. π \pi
  16. ψ ( ϕ ) = A cos ( m ϕ ) + B sin ( m ϕ ) ( 4 ) \ \psi(\phi)=A\cos(m\phi)+B\sin(m\phi)\quad(4)
  17. m = 2 I E 2 m=\sqrt{\frac{2IE}{\hbar^{2}}}
  18. E = m 2 2 2 I E=\frac{m^{2}\hbar^{2}}{2I}
  19. ψ \psi
  20. d ψ d ϕ \frac{d\psi}{d\phi}
  21. 0 π | ψ ( ϕ ) | 2 d ϕ = 1 ( 5 ) \int_{0}^{\pi}\left|\psi(\phi)\right|^{2}\,d\phi=1\ \quad(5)
  22. ϕ = 0 \phi=0
  23. ϕ = π \phi=\pi
  24. ψ ( 0 ) = ψ ( π ) = 0 ( 6 ) \ \psi(0)=\psi(\pi)=0\quad(6)
  25. ψ ( 0 ) = 0 \ \psi(0)=0
  26. cos ( 0 ) = 1 \ \cos(0)=1
  27. ϕ = π \phi=\pi
  28. ψ ( π ) = 0 = B sin ( m π ) \ \psi(\pi)=0=B\sin(m\pi)
  29. sin ( n π ) = 0 \sin(n\pi)=0
  30. E = m 2 2 2 I E=\frac{m^{2}\hbar^{2}}{2I}
  31. ψ = 0 \psi=0
  32. B = 2 π B=\sqrt{\frac{2}{\pi}}
  33. ψ ( ϕ ) = 2 π sin ( m ϕ ) ( 7 ) \ \psi(\phi)=\sqrt{\frac{2}{\pi}}\sin(m\phi)\quad(7)
  34. E = 2 2 I E=\frac{\hbar^{2}}{2I}
  35. ψ ( ϕ ) \ \psi(\phi)
  36. ψ ( ϕ ) 2 \ \psi(\phi)^{2}
  37. ϕ \phi
  38. ϕ \phi
  39. L z L_{z}
  40. ϕ \phi
  41. L z L_{z}
  42. ϕ \phi
  43. - i d d ϕ -i\hbar\frac{d}{d\phi}
  44. [ ϕ , L z ] = i ψ ( ϕ ) ( 8 ) [\phi,L_{z}]=i\hbar\ \psi(\phi)\quad(8)
  45. ( Δ ϕ ) ( Δ L z ) 2 (\Delta\phi)(\Delta L_{z})\geq\frac{\hbar}{2}
  46. Δ ψ ϕ = ϕ 2 ψ - ϕ ψ 2 \Delta_{\psi}\phi=\sqrt{\langle{\phi}^{2}\rangle_{\psi}-\langle{\phi}\rangle_{% \psi}^{2}}
  47. Δ ψ L z = L z 2 ψ - L z ψ 2 ( 9 ) \Delta_{\psi}L_{z}=\sqrt{\langle{L_{z}}^{2}\rangle_{\psi}-\langle{L_{z}}% \rangle_{\psi}^{2}}\quad(9)
  48. 2 π 2\pi
  49. - π 2 \frac{-\pi}{2}
  50. π 2 \frac{\pi}{2}
  51. ψ o ( ϕ ) = 2 π cos ( m ϕ ) ( 10 ) \ \psi_{o}(\phi)=\sqrt{\frac{2}{\pi}}\cos(m\phi)\quad(10)
  52. ψ e ( ϕ ) = 2 π sin ( m ϕ ) ( 11 ) \ \psi_{e}(\phi)=\sqrt{\frac{2}{\pi}}\sin(m\phi)\quad(11)
  53. ψ o ( ϕ ) \ \psi_{o}(\phi)
  54. ψ e ( ϕ ) \ \psi_{e}(\phi)
  55. ϕ \phi
  56. L z L_{z}

Senecionine_N-oxygenase.html

  1. \rightleftharpoons

Separable_algebra.html

  1. L / K \scriptstyle L/K
  2. A K L \scriptstyle A\otimes_{K}L
  3. p = i = 1 n x i y i p=\sum_{i=1}^{n}x_{i}\otimes y_{i}
  4. A e = A K A op A^{e}=A\otimes_{K}A^{\rm op}
  5. i = 1 n x i y i = 1 A \sum_{i=1}^{n}x_{i}y_{i}=1_{A}
  6. p 2 = p p^{2}=p
  7. A e A^{e}
  8. A e A^{e}
  9. e = i = 1 n x i y i = i = 1 n y i x i e=\sum_{i=1}^{n}x_{i}\otimes y_{i}=\sum_{i=1}^{n}y_{i}\otimes x_{i}
  10. L / K \scriptstyle L/K
  11. a a
  12. p ( x ) = ( x - a ) i = 0 n - 1 b i x i p(x)=(x-a)\sum_{i=0}^{n-1}b_{i}x^{i}
  13. i = 0 n - 1 a i K b i p ( a ) \sum_{i=0}^{n-1}a^{i}\otimes_{K}\frac{b_{i}}{p^{\prime}(a)}
  14. σ 1 , , σ n \sigma_{1},\ldots,\sigma_{n}
  15. i = 1 n σ i ( x ) \sum_{i=1}^{n}\sigma_{i}(x)
  16. 1 o ( G ) g G g K g - 1 \frac{1}{o(G)}\sum_{g\in G}g\otimes_{K}g^{-1}
  17. R S R R R\otimes_{S}R\rightarrow R
  18. m ( i r i S t i ) = i r i t i m(\sum_{i}r_{i}\otimes_{S}t_{i})=\sum_{i}r_{i}t_{i}
  19. R R S R R\rightarrow R\otimes_{S}R
  20. r r 1 r\mapsto r\otimes 1
  21. H n ( R , S ; M ) H^{n}(R,S;M)

Sepiapterin_reductase.html

  1. \rightleftharpoons

Sequoyitol_dehydrogenase.html

  1. \rightleftharpoons

Serine-sulfate_ammonia-lyase.html

  1. \rightleftharpoons

Serine_2-dehydrogenase.html

  1. \rightleftharpoons

Serine_3-dehydrogenase.html

  1. \rightleftharpoons

Shell_balance.html

  1. V x V_{x}

Shift_theorem.html

  1. P ( D ) ( e a x y ) e a x P ( D + a ) y . P(D)(e^{ax}y)\equiv e^{ax}P(D+a)y.\,
  2. P ( D ) = D n P(D)=D^{n}\,
  3. D ( e a x y ) = e a x ( D + a ) y . D(e^{ax}y)=e^{ax}(D+a)y.\,
  4. D k ( e a x y ) = e a x ( D + a ) k y . D^{k}(e^{ax}y)=e^{ax}(D+a)^{k}y.\,
  5. D k + 1 ( e a x y ) d d x { e a x ( D + a ) k y } = e a x d d x { ( D + a ) k y } + a e a x { ( D + a ) k y } = e a x { ( d d x + a ) ( D + a ) k y } = e a x ( D + a ) k + 1 y . \begin{aligned}\displaystyle D^{k+1}(e^{ax}y)&\displaystyle\equiv\frac{d}{dx}% \{e^{ax}(D+a)^{k}y\}\\ &\displaystyle{}=e^{ax}\frac{d}{dx}\{(D+a)^{k}y\}+ae^{ax}\{(D+a)^{k}y\}\\ &\displaystyle{}=e^{ax}\left\{\left(\frac{d}{dx}+a\right)(D+a)^{k}y\right\}\\ &\displaystyle{}=e^{ax}(D+a)^{k+1}y.\end{aligned}
  6. 1 P ( D ) ( e a x y ) = e a x 1 P ( D + a ) y . \frac{1}{P(D)}(e^{ax}y)=e^{ax}\frac{1}{P(D+a)}y.\,
  7. t < a t<a
  8. ( e a t f ( t ) ) = ( f ( t - a ) ) . \scriptstyle\mathcal{L}(e^{at}f(t))=\scriptstyle\mathcal{L}(f(t-a)).\,

Shikimate_dehydrogenase.html

  1. \rightleftharpoons

Shilov_boundary.html

  1. 𝒜 \mathcal{A}
  2. Δ 𝒜 \Delta\mathcal{A}
  3. 𝒜 * {\mathcal{A}}^{*}
  4. F F
  5. Δ 𝒜 \Delta{\mathcal{A}}
  6. 𝒜 {\mathcal{A}}
  7. max f Δ 𝒜 | x ( f ) | = max f F | x ( f ) | \max_{f\in\Delta{\mathcal{A}}}|x(f)|=\max_{f\in F}|x(f)|
  8. x 𝒜 x\in\mathcal{A}
  9. S = { F : F is a boundary of 𝒜 } S=\bigcap\{F:F\,\text{ is a boundary of }{\mathcal{A}}\}
  10. S S
  11. 𝒜 {\mathcal{A}}
  12. S Δ 𝒜 S\subset\Delta\mathcal{A}
  13. S S
  14. 𝒜 \mathcal{A}
  15. F F
  16. 𝒜 \mathcal{A}
  17. S F S\subset F
  18. 𝔻 = { z : | z | < 1 } \mathbb{D}=\{z\in\mathbb{C}:|z|<1\}
  19. 𝒜 = ( 𝔻 ) 𝒞 ( 𝔻 ¯ ) {\mathcal{A}}={\mathcal{H}}(\mathbb{D})\cap{\mathcal{C}}(\bar{\mathbb{D}})
  20. 𝔻 \mathbb{D}
  21. 𝔻 \mathbb{D}
  22. Δ 𝒜 = 𝔻 ¯ \Delta{\mathcal{A}}=\bar{\mathbb{D}}
  23. S = { | z | = 1 } S=\{|z|=1\}

Shrikhande_graph.html

  1. 4 × 4 \mathbb{Z}_{4}\times\mathbb{Z}_{4}
  2. { ± ( 1 , 0 ) , ± ( 0 , 1 ) , ± ( 1 , 1 ) } \{\pm(1,0),\pm(0,1),\pm(1,1)\}
  3. λ = μ = 2 \lambda=\mu=2
  4. ( x - 6 ) ( x - 2 ) 6 ( x + 2 ) 9 (x-6)(x-2)^{6}(x+2)^{9}

Siegel–Tukey_test.html

  1. Pr [ X min ( U A , U B ) ] \Pr\left[X\leq\min(U_{A},U_{B})\right]\!
  2. X Wilcoxon ( m , n ) X\sim\,\text{Wilcoxon}(m,n)\,\!
  3. Pr [ x 16 ] = 0.2669. \Pr\left[x\leq 16\right]=0.2669.\,\!

Simple_magic_cube.html

  1. S = m ( m 3 + 1 ) 2 . S=\frac{m(m^{3}+1)}{2}.

Single-linkage_clustering.html

  1. D ( X , Y ) = min x X , y Y d ( x , y ) , D(X,Y)=\min_{x\in X,y\in Y}d(x,y),
  2. N × N N\times N
  3. 𝒪 ( n 3 ) \mathcal{O}(n^{3})
  4. O ( n 2 ) O(n^{2})
  5. O ( n ) O(n)
  6. n n
  7. C C
  8. i i
  9. π \pi
  10. i i
  11. C C
  12. λ \lambda
  13. i i
  14. C C
  15. O ( n ) O(n)
  16. O ( n ) O(n)
  17. O ( n 2 ) O(n^{2})
  18. O ( n ) O(n)
  19. O ( n log n ) O(n\log n)
  20. O ( n ) O(n)

Singular_integral.html

  1. T ( f ) ( x ) = K ( x , y ) f ( y ) d y , T(f)(x)=\int K(x,y)f(y)\,dy,
  2. H ( f ) ( x ) = 1 π lim ε 0 | x - y | > ε 1 x - y f ( y ) d y . H(f)(x)=\frac{1}{\pi}\lim_{\varepsilon\to 0}\int_{|x-y|>\varepsilon}\frac{1}{x% -y}f(y)\,dy.
  3. K i ( x ) = x i | x | n + 1 K_{i}(x)=\frac{x_{i}}{|x|^{n+1}}
  4. x i x_{i}
  5. T ( f ) ( x ) = lim ε 0 | y - x | > ε K ( x - y ) f ( y ) d y . T(f)(x)=\lim_{\varepsilon\to 0}\int_{|y-x|>\varepsilon}K(x-y)f(y)\,dy.
  6. K ^ L ( 𝐑 n ) \hat{K}\in L^{\infty}(\mathbf{R}^{n})
  7. sup y 0 | x | > 2 | y | | K ( x - y ) - K ( x ) | d x C . \sup_{y\neq 0}\int_{|x|>2|y|}|K(x-y)-K(x)|\,dx\leq C.
  8. p . v . K [ ϕ ] = lim ϵ 0 + | x | > ϵ ϕ ( x ) K ( x ) d x \operatorname{p.v.}\,\,K[\phi]=\lim_{\epsilon\to 0^{+}}\int_{|x|>\epsilon}\phi% (x)K(x)\,dx
  9. R 1 < | x | < R 2 K ( x ) d x = 0 , R 1 , R 2 > 0 \int_{R_{1}<|x|<R_{2}}K(x)\,dx=0,\ \forall R_{1},R_{2}>0
  10. sup R > 0 R < | x | < 2 R | K ( x ) | d x C , \sup_{R>0}\int_{R<|x|<2R}|K(x)|\,dx\leq C,
  11. K C 1 ( 𝐑 n { 0 } ) K\in C^{1}(\mathbf{R}^{n}\setminus\{0\})
  12. | K ( x ) | C | x | n + 1 |\nabla K(x)|\leq\frac{C}{|x|^{n+1}}
  13. ( a ) | K ( x , y ) | C | x - y | n (a)\qquad|K(x,y)|\leq\frac{C}{|x-y|^{n}}
  14. ( b ) | K ( x , y ) - K ( x , y ) | C | x - x | δ ( | x - y | + | x - y | ) n + δ whenever | x - x | 1 2 max ( | x - y | , | x - y | ) (b)\qquad|K(x,y)-K(x^{\prime},y)|\leq\frac{C|x-x^{\prime}|^{\delta}}{\bigl(|x-% y|+|x^{\prime}-y|\bigr)^{n+\delta}}\,\text{ whenever }|x-x^{\prime}|\leq\frac{% 1}{2}\max\bigl(|x-y|,|x^{\prime}-y|\bigr)
  15. ( c ) | K ( x , y ) - K ( x , y ) | C | y - y | δ ( | x - y | + | x - y | ) n + δ whenever | y - y | 1 2 max ( | x - y | , | x - y | ) (c)\qquad|K(x,y)-K(x,y^{\prime})|\leq\frac{C|y-y^{\prime}|^{\delta}}{\bigl(|x-% y|+|x-y^{\prime}|\bigr)^{n+\delta}}\,\text{ whenever }|y-y^{\prime}|\leq\frac{% 1}{2}\max\bigl(|x-y^{\prime}|,|x-y|\bigr)
  16. g ( x ) T ( f ) ( x ) d x = g ( x ) K ( x , y ) f ( y ) d y d x , \int g(x)T(f)(x)\,dx=\iint g(x)K(x,y)f(y)\,dy\,dx,
  17. T ( f ) L 2 C f L 2 , \|T(f)\|_{L^{2}}\leq C\|f\|_{L^{2}},
  18. | T ( τ x ( φ r ) ) ( y ) τ x ( ψ r ) ( y ) d y | C r - n \left|\int T\bigl(\tau^{x}(\varphi_{r})\bigr)(y)\tau^{x}(\psi_{r})(y)\,dy% \right|\leq Cr^{-n}
  19. M b 2 T M b 1 M_{b_{2}}TM_{b_{1}}
  20. T ( b 1 ) T(b_{1})
  21. T t ( b 2 ) , T^{t}(b_{2}),

Sirohydrochlorin_cobaltochelatase.html

  1. \rightleftharpoons

Sirohydrochlorin_ferrochelatase.html

  1. \rightleftharpoons

Slope_deflection_method.html

  1. K = I a b L a b K=\frac{I_{ab}}{L_{ab}}
  2. ψ = Δ L a b \psi=\frac{\Delta}{L_{ab}}
  3. L a b L_{ab}
  4. E a b I a b E_{ab}I_{ab}
  5. M a b M_{ab}
  6. M b a M_{ba}
  7. θ a - Δ L a b = L a b 3 E a b I a b M a b - L a b 6 E a b I a b M b a \theta_{a}-\frac{\Delta}{L_{ab}}=\frac{L_{ab}}{3E_{ab}I_{ab}}M_{ab}-\frac{L_{% ab}}{6E_{ab}I_{ab}}M_{ba}
  8. θ b - Δ L a b = - L a b 6 E a b I a b M a b + L a b 3 E a b I a b M b a \theta_{b}-\frac{\Delta}{L_{ab}}=-\frac{L_{ab}}{6E_{ab}I_{ab}}M_{ab}+\frac{L_{% ab}}{3E_{ab}I_{ab}}M_{ba}
  9. Σ ( M f + M m e m b e r ) = Σ M j o i n t \Sigma\left(M^{f}+M_{member}\right)=\Sigma M_{joint}
  10. M m e m b e r M_{member}
  11. M f M^{f}
  12. M j o i n t M_{joint}
  13. L = 10 m L=10\ m
  14. P = 10 k N P=10\ kN
  15. a = 3 m a=3\ m
  16. q = 1 k N / m q=1\ kN/m
  17. P = 10 k N P=10\ kN
  18. θ A \theta_{A}
  19. θ B \theta_{B}
  20. θ C \theta_{C}
  21. M A B f = - P a b 2 L 2 = - 10 × 3 × 7 2 10 2 = - 14.7 kN m M_{AB}^{f}=-\frac{Pab^{2}}{L^{2}}=-\frac{10\times 3\times 7^{2}}{10^{2}}=-14.7% \mathrm{\,kN\,m}
  22. M B A f = P a 2 b L 2 = 10 × 3 2 × 7 10 2 = 6.3 kN m M_{BA}^{f}=\frac{Pa^{2}b}{L^{2}}=\frac{10\times 3^{2}\times 7}{10^{2}}=6.3% \mathrm{\,kN\,m}
  23. M B C f = - q L 2 12 = - 1 × 10 2 12 = - 8.333 kN m M_{BC}^{f}=-\frac{qL^{2}}{12}=-\frac{1\times 10^{2}}{12}=-8.333\mathrm{\,kN\,m}
  24. M C B f = q L 2 12 = 1 × 10 2 12 = 8.333 kN m M_{CB}^{f}=\frac{qL^{2}}{12}=\frac{1\times 10^{2}}{12}=8.333\mathrm{\,kN\,m}
  25. M C D f = - P L 8 = - 10 × 10 8 = - 12.5 kN m M_{CD}^{f}=-\frac{PL}{8}=-\frac{10\times 10}{8}=-12.5\mathrm{\,kN\,m}
  26. M D C f = P L 8 = 10 × 10 8 = 12.5 kN m M_{DC}^{f}=\frac{PL}{8}=\frac{10\times 10}{8}=12.5\mathrm{\,kN\,m}
  27. M A B = E I L ( 4 θ A + 2 θ B ) = 0.4 E I θ A + 0.2 E I θ B M_{AB}=\frac{EI}{L}\left(4\theta_{A}+2\theta_{B}\right)=0.4EI\theta_{A}+0.2EI% \theta_{B}
  28. M B A = E I L ( 2 θ A + 4 θ B ) = 0.2 E I θ A + 0.4 E I θ B M_{BA}=\frac{EI}{L}\left(2\theta_{A}+4\theta_{B}\right)=0.2EI\theta_{A}+0.4EI% \theta_{B}
  29. M B C = 2 E I L ( 4 θ B + 2 θ C ) = 0.8 E I θ B + 0.4 E I θ C M_{BC}=\frac{2EI}{L}\left(4\theta_{B}+2\theta_{C}\right)=0.8EI\theta_{B}+0.4EI% \theta_{C}
  30. M C B = 2 E I L ( 2 θ B + 4 θ C ) = 0.4 E I θ B + 0.8 E I θ C M_{CB}=\frac{2EI}{L}\left(2\theta_{B}+4\theta_{C}\right)=0.4EI\theta_{B}+0.8EI% \theta_{C}
  31. M C D = E I L ( 4 θ C ) = 0.4 E I θ C M_{CD}=\frac{EI}{L}\left(4\theta_{C}\right)=0.4EI\theta_{C}
  32. M D C = E I L ( 2 θ C ) = 0.2 E I θ C M_{DC}=\frac{EI}{L}\left(2\theta_{C}\right)=0.2EI\theta_{C}
  33. Σ M A = M A B + M A B f = 0.4 E I θ A + 0.2 E I θ B - 14.7 = 0 \Sigma M_{A}=M_{AB}+M_{AB}^{f}=0.4EI\theta_{A}+0.2EI\theta_{B}-14.7=0
  34. Σ M B = M B A + M B A f + M B C + M B C f = 0.2 E I θ A + 1.2 E I θ B + 0.4 E I θ C - 2.033 = 0 \Sigma M_{B}=M_{BA}+M_{BA}^{f}+M_{BC}+M_{BC}^{f}=0.2EI\theta_{A}+1.2EI\theta_{% B}+0.4EI\theta_{C}-2.033=0
  35. Σ M C = M C B + M C B f + M C D + M C D f = 0.4 E I θ B + 1.2 E I θ C - 4.167 = 0 \Sigma M_{C}=M_{CB}+M_{CB}^{f}+M_{CD}+M_{CD}^{f}=0.4EI\theta_{B}+1.2EI\theta_{% C}-4.167=0
  36. θ A = 40.219 E I \theta_{A}=\frac{40.219}{EI}
  37. θ B = - 6.937 E I \theta_{B}=\frac{-6.937}{EI}
  38. θ C = 5.785 E I \theta_{C}=\frac{5.785}{EI}
  39. M A B = 0.4 × 40.219 + 0.2 × ( - 6.937 ) - 14.7 = 0 M_{AB}=0.4\times 40.219+0.2\times\left(-6.937\right)-14.7=0
  40. M B A = 0.2 × 40.219 + 0.4 × ( - 6.937 ) + 6.3 = 11.57 M_{BA}=0.2\times 40.219+0.4\times\left(-6.937\right)+6.3=11.57
  41. M B C = 0.8 × ( - 6.937 ) + 0.4 × 5.785 - 8.333 = - 11.57 M_{BC}=0.8\times\left(-6.937\right)+0.4\times 5.785-8.333=-11.57
  42. M C B = 0.4 × ( - 6.937 ) + 0.8 × 5.785 + 8.333 = 10.19 M_{CB}=0.4\times\left(-6.937\right)+0.8\times 5.785+8.333=10.19
  43. M C D = 0.4 × 5.785 - 12.5 = - 10.19 M_{CD}=0.4\times 5.785-12.5=-10.19
  44. M D C = 0.2 × 5.785 + 12.5 = 13.66 M_{DC}=0.2\times 5.785+12.5=13.66

Smart_ligand.html

  1. K d K_{d}
  2. k o f f k_{off}
  3. k o n k_{on}

Smith–Minkowski–Siegel_mass_formula.html

  1. m ( f ) = Λ 1 | Aut ( Λ ) | m(f)=\sum_{\Lambda}{1\over|\operatorname{Aut}(\Lambda)|}
  2. m ( f ) = 2 π - n ( n + 1 ) / 4 j = 1 n Γ ( j / 2 ) p prime 2 m p ( f ) m(f)=2\pi^{-n(n+1)/4}\prod_{j=1}^{n}\Gamma(j/2)\prod_{p\,\text{ prime}}2m_{p}(f)
  3. m p ( f ) = p ( r n ( n - 1 ) + s ( n + 1 ) ) / 2 N ( p r ) m_{p}(f)={p^{(rn(n-1)+s(n+1))/2}\over N(p^{r})}
  4. X tr A X A mod p r X\text{tr}AX\equiv A\ \bmod\ p^{r}
  5. α p ( f ) = N ( p r ) p r n ( n - 1 ) / 2 = p s ( n + 1 ) / 2 m p ( f ) \alpha_{p}(f)={N(p^{r})\over p^{rn(n-1)/2}}={p^{s(n+1)/2}\over m_{p}(f)}
  6. std p ( f ) = 1 2 ( 1 - p - 2 ) ( 1 - p - 4 ) ( 1 - p 2 - n ) ( 1 - ( ( - 1 ) n / 2 det ( f ) p ) p - n / 2 ) \operatorname{std}_{p}(f)={1\over 2(1-p^{-2})(1-p^{-4})\cdots(1-p^{2-n})(1-{(-% 1)^{n/2}\det(f)\choose p}p^{-n/2})}\quad
  7. std p ( f ) = 1 2 ( 1 - p - 2 ) ( 1 - p - 4 ) ( 1 - p 1 - n ) \operatorname{std}_{p}(f)={1\over 2(1-p^{-2})(1-p^{-4})\cdots(1-p^{1-n})}
  8. std ( f ) = 2 π - n ( n + 1 ) / 4 ( j = 1 n Γ ( j / 2 ) ) ζ ( 2 ) ζ ( 4 ) ζ ( n - 1 ) \operatorname{std}(f)=2\pi^{-n(n+1)/4}\left(\prod_{j=1}^{n}\Gamma(j/2)\right)% \zeta(2)\zeta(4)\cdots\zeta(n-1)
  9. std ( f ) = 2 π - n ( n + 1 ) / 4 ( j = 1 n Γ ( j / 2 ) ) ζ ( 2 ) ζ ( 4 ) ζ ( n - 2 ) ζ D ( n / 2 ) \operatorname{std}(f)=2\pi^{-n(n+1)/4}\left(\prod_{j=1}^{n}\Gamma(j/2)\right)% \zeta(2)\zeta(4)\cdots\zeta(n-2)\zeta_{D}(n/2)
  10. ζ D ( s ) = p 1 1 - ( D p ) p - s \zeta_{D}(s)=\prod_{p}{1\over 1-{D\choose p}p^{-s}}
  11. ζ ( s ) = ( 2 π ) s 2 × s ! | B s | . \zeta(s)={(2\pi)^{s}\over 2\times s!}|B_{s}|.
  12. m ( f ) = std ( f ) p | 2 det ( f ) m p ( f ) std p ( f ) . m(f)=\operatorname{std}(f)\prod_{p|2\det(f)}{m_{p}(f)\over\operatorname{std}_{% p}(f)}.
  13. f = q f q f=\sum qf_{q}
  14. m p ( f ) = q M p ( f q ) × q < q ( q / q ) n ( q ) n ( q ) / 2 × 2 n ( I , I ) - n ( I I ) m_{p}(f)=\prod_{q}M_{p}(f_{q})\times\prod_{q<q^{\prime}}(q^{\prime}/q)^{n(q)n(% q^{\prime})/2}\times 2^{n(I,I)-n(II)}
  15. 1 2 ( 1 - p - 2 ) ( 1 - p - 4 ) ( 1 - p 1 - n ) {1\over 2(1-p^{-2})(1-p^{-4})\cdots(1-p^{1-n})}
  16. 1 2 ( 1 - p - 2 ) ( 1 - p - 4 ) ( 1 - p 2 - n ) ( 1 - p - n / 2 ) {1\over 2(1-p^{-2})(1-p^{-4})\cdots(1-p^{2-n})(1-p^{-n/2})}
  17. 1 2 ( 1 - p - 2 ) ( 1 - p - 4 ) ( 1 - p 2 - n ) ( 1 + p - n / 2 ) {1\over 2(1-p^{-2})(1-p^{-4})\cdots(1-p^{2-n})(1+p^{-n/2})}
  18. 1 2 ( 1 - p - 2 ) ( 1 - p - 4 ) ( 1 - p - 2 t ) {1\over 2(1-p^{-2})(1-p^{-4})\cdots(1-p^{-2t})}
  19. 1 2 ( 1 - p - 2 ) ( 1 - p - 4 ) ( 1 - p 2 - 2 t ) ( 1 - p - t ) {1\over 2(1-p^{-2})(1-p^{-4})\cdots(1-p^{2-2t})(1-p^{-t})}
  20. 1 2 ( 1 - p - 2 ) ( 1 - p - 4 ) ( 1 - p 2 - 2 t ) ( 1 + p - t ) {1\over 2(1-p^{-2})(1-p^{-4})\cdots(1-p^{2-2t})(1+p^{-t})}
  21. ( D m ) {D\choose m}
  22. ζ D ( s ) = L ( s , χ ) = L ( s , ψ ) p | k ( 1 - ψ ( p ) p s ) \zeta_{D}(s)=L(s,\chi)=L(s,\psi)\prod_{p|k}\left(1-{\psi(p)\over p^{s}}\right)
  23. L ( 1 - s , ψ ) = k 1 s - 1 Γ ( s ) ( 2 π ) s ( i - s + ψ ( - 1 ) i s ) G ( ψ ) L ( s , ψ ) L(1-s,\psi)={k_{1}^{s-1}\Gamma(s)\over(2\pi)^{s}}(i^{-s}+\psi(-1)i^{s})G(\psi)% L(s,\psi)
  24. G ( ψ ) = r = 1 k 1 ψ ( r ) e 2 π i r / k 1 . G(\psi)=\sum_{r=1}^{k_{1}}\psi(r)e^{2\pi ir/k_{1}}.
  25. L ( 1 - s , ψ ) = - k 1 s - 1 s r = 1 k 1 ψ ( r ) B s ( r / k 1 ) L(1-s,\psi)=-{k_{1}^{s-1}\over s}\sum_{r=1}^{k_{1}}\psi(r)B_{s}(r/k_{1})
  26. Λ 1 | Aut ( Λ ) | = | B n / 2 | n 1 j < n / 2 | B 2 j | 4 j \sum_{\Lambda}{1\over|\operatorname{Aut}(\Lambda)|}={|B_{n/2}|\over n}\prod_{1% \leq j<n/2}{|B_{2j}|\over 4j}
  27. | B 4 | 8 | B 2 | 4 | B 4 | 8 | B 6 | 12 = 1 / 30 8 1 / 6 4 1 / 30 8 1 / 42 12 = 1 696729600 . {|B_{4}|\over 8}{|B_{2}|\over 4}{|B_{4}|\over 8}{|B_{6}|\over 12}={1/30\over 8% }\;{1/6\over 4}\;{1/30\over 8}\;{1/42\over 12}={1\over 696729600}.
  28. | B 8 | 16 | B 2 | 4 | B 4 | 8 | B 6 | 12 | B 8 | 16 | B 10 | 20 | B 12 | 24 | B 14 | 28 = 691 277667181515243520000 . {|B_{8}|\over 16}{|B_{2}|\over 4}{|B_{4}|\over 8}{|B_{6}|\over 12}{|B_{8}|% \over 16}{|B_{10}|\over 20}{|B_{12}|\over 24}{|B_{14}|\over 28}={691\over 2776% 67181515243520000}.
  29. 1 970864271032320000 + 1 685597979049984000 = 691 277667181515243520000 . {1\over 970864271032320000}+{1\over 685597979049984000}={691\over 277667181515% 243520000}.

Smooth_coarea_formula.html

  1. M , N \scriptstyle M,\,N
  2. m n \scriptstyle m\,\geq\,n
  3. F : M N \scriptstyle F:M\,\longrightarrow\,N
  4. F \scriptstyle F
  5. φ : M [ 0 , ] \scriptstyle\varphi:M\,\longrightarrow\,[0,\infty]
  6. x M φ ( x ) d M = y N x F - 1 ( y ) φ ( x ) 1 N J F ( x ) d F - 1 ( y ) d N \int_{x\in M}\varphi(x)\,dM=\int_{y\in N}\int_{x\in F^{-1}(y)}\varphi(x)\frac{% 1}{N\!J\;F(x)}\,dF^{-1}(y)\,dN
  7. x M φ ( x ) N J F ( x ) d M = y N x F - 1 ( y ) φ ( x ) d F - 1 ( y ) d N \int_{x\in M}\varphi(x)N\!J\;F(x)\,dM=\int_{y\in N}\int_{x\in F^{-1}(y)}% \varphi(x)\,dF^{-1}(y)\,dN
  8. N J F ( x ) \scriptstyle N\!J\;F(x)
  9. F \scriptstyle F
  10. y N \scriptstyle y\,\in\,N
  11. F \scriptstyle F
  12. F - 1 ( y ) \scriptstyle F^{-1}(y)
  13. M \scriptstyle M

Sn-glycerol-1-phosphate_dehydrogenase.html

  1. \rightleftharpoons

Sn1CB_mechanism.html

  1. r = k [ Co ( NH 3 ) 5 Cl 2 + ] [ OH - ] r=k[\mathrm{Co(NH_{3})_{5}Cl^{2+}}][\mathrm{OH}^{-}]

Sod_shock_tube.html

  1. ( ρ L P L v L ) = ( 1.0 1.0 0.0 ) \left(\begin{array}[]{c}\rho_{L}\\ P_{L}\\ v_{L}\end{array}\right)=\left(\begin{array}[]{c}1.0\\ 1.0\\ 0.0\end{array}\right)
  2. ( ρ R P R v R ) = ( 0.125 0.1 0.0 ) \left(\begin{array}[]{c}\rho_{R}\\ P_{R}\\ v_{R}\end{array}\right)=\left(\begin{array}[]{c}0.125\\ 0.1\\ 0.0\end{array}\right)
  3. ρ \rho
  4. c s 1 = γ P L ρ L cs_{1}=\sqrt{\gamma\frac{P_{L}}{\rho_{L}}}
  5. c s 5 = γ P R ρ R cs_{5}=\sqrt{\gamma\frac{P_{R}}{\rho_{R}}}
  6. Γ = γ - 1 γ + 1 \Gamma=\frac{\gamma-1}{\gamma+1}
  7. β = γ - 1 2 γ \beta=\frac{\gamma-1}{2\gamma}
  8. ρ 4 = ρ 5 P 4 + Γ P 5 P 5 + Γ P 4 \rho_{4}=\rho_{5}\frac{P_{4}+\Gamma P_{5}}{P_{5}+\Gamma P_{4}}
  9. P 4 = P 3 P_{4}=P_{3}
  10. u 2 u_{2}
  11. u 4 u_{4}
  12. u 4 = ( P 3 - P 5 ) 1 - Γ ρ R ( P 3 + Γ P 5 ) u_{4}=\left(P_{3}^{\prime}-P_{5}\right)\sqrt{\frac{1-\Gamma}{\rho_{R}(P_{3}^{% \prime}+\Gamma P_{5})}}
  13. u 2 = ( P 1 β - P 3 β ) ( 1 - Γ 2 ) P 1 1 / γ Γ 2 ρ L u_{2}=\left(P_{1}^{\beta}-P_{3}^{\prime\beta}\right)\sqrt{\frac{(1-\Gamma^{2})% P_{1}^{1/\gamma}}{\Gamma^{2}\rho_{L}}}
  14. u 2 - u 4 = 0 u_{2}-u_{4}=0
  15. P 3 = calculate ( P 3 , s , s , , ) P_{3}=\operatorname{calculate}(P_{3},s,s,,)
  16. u 3 = u 5 + ( P 3 - P 5 ) ρ 5 2 ( ( γ + 1 ) P 3 + ( γ - 1 ) P 5 ) u_{3}=u_{5}+\frac{(P_{3}-P_{5})}{\sqrt{\frac{\rho_{5}}{2}((\gamma+1)P_{3}+(% \gamma-1)P_{5})}}
  17. u 4 = u 3 u_{4}=u_{3}
  18. ρ 3 \rho_{3}
  19. ρ 3 = ρ 1 ( P 3 P 1 ) 1 / γ \rho_{3}=\rho_{1}\left(\frac{P_{3}}{P_{1}}\right)^{1/\gamma}

Software_equation.html

  1. E = [ L O C . B .333 P ] 3 ( 1 t 4 ) E=\left[\frac{LOC.B^{.333}}{P}\right]^{3}\left(\frac{1}{t^{4}}\right)

Soil_respiration.html

  1. F = b V / A F=bV/A

Sorbitol-6-phosphate_2-dehydrogenase.html

  1. \rightleftharpoons

Sorbose_5-dehydrogenase_(NADP+).html

  1. \rightleftharpoons

Sorbose_dehydrogenase.html

  1. \rightleftharpoons

Sorbose_reductase.html

  1. \rightleftharpoons

Souders–Brown_equation.html

  1. V = ( k ) ρ L - ρ V ρ V V=\left(k\right)\sqrt{\frac{\rho_{L}-\rho_{V}}{\rho_{V}}}
  2. V V
  3. ρ L \rho_{L}
  4. ρ V \rho_{V}
  5. k k

SP-DEVS.html

  1. M = < X , Y , S , s 0 , τ , δ x , δ y Align g t ; M=<X,Y,S,s_{0},\tau,\delta_{x},\delta_{y}&gt;
  2. X X
  3. Y Y
  4. S S
  5. s 0 S s_{0}\in S
  6. τ : S [ 0 , ] \tau:S\rightarrow\mathbb{Q}_{[0,\infty]}
  7. [ 0 , ] \mathbb{Q}_{[0,\infty]}
  8. δ x : S × X S \delta_{x}:S\times X\rightarrow S
  9. δ y : S Y ϕ × S \delta_{y}:S\rightarrow Y^{\phi}\times S
  10. Y ϕ = Y { ϕ } Y^{\phi}=Y\cup\{\phi\}
  11. ϕ Y \phi\notin Y
  12. M = < X , Y , S , s 0 , τ , δ x , δ y > M=<X,Y,S,s_{0},\tau,\delta_{x},\delta_{y}>
  13. X X
  14. Y Y
  15. S S
  16. s 0 s_{0}
  17. τ \tau
  18. τ \tau
  19. τ \tau
  20. τ \tau
  21. τ \tau
  22. τ \tau
  23. τ \tau
  24. δ x \delta_{x}
  25. δ x \delta_{x}
  26. \neq
  27. δ y \delta_{y}
  28. δ y \delta_{y}
  29. δ y \delta_{y}
  30. ϕ \phi
  31. δ y \delta_{y}
  32. δ y \delta_{y}
  33. δ y \delta_{y}
  34. δ y \delta_{y}
  35. t s [ 0 , ] t_{s}\in\mathbb{Q}_{[0,\infty]}
  36. t e [ 0 , ] t_{e}\in[0,\infty]
  37. s S s\in S
  38. ( s , t s , t e ) (s,t_{s},t_{e})
  39. t e t_{e}
  40. t e t_{e}
  41. M = < X , Y , S , s 0 , τ , δ x , δ y > M=<X,Y,S,s_{0},\tau,\delta_{x},\delta_{y}>
  42. = < X , Y , S , s 0 , t a , δ e x t , δ i n t , λ Align g t ; \mathcal{M}=<X,Y,S^{\prime},s_{0}^{\prime},ta,\delta_{ext},\delta_{int},% \lambda&gt;
  43. X , Y X,Y
  44. \mathcal{M}
  45. M M
  46. S = { ( s , t s ) : s S , t s 𝕋 } S^{\prime}=\{(s,t_{s}):s\in S,t_{s}\in\mathbb{T}^{\infty}\}
  47. s 0 = ( s 0 , τ ( s 0 ) ) s_{0}^{\prime}=(s_{0},\tau(s_{0}))
  48. ( s , t s ) S (s,t_{s})\in S^{\prime}
  49. t a ( s , t s ) = t s . ta(s,t_{s})=t_{s}.
  50. ( s , t s ) S (s,t_{s})\in S^{\prime}
  51. x X , δ e x t ( s , t s , t e , x ) = ( s , t s - t e ) if δ x ( s , x ) = s . x\in X,\delta_{ext}(s,t_{s},t_{e},x)=(s^{\prime},t_{s}-t_{e})\,\text{ if }% \delta_{x}(s,x)=s^{\prime}.
  52. ( s , t s ) S (s,t_{s})\in S^{\prime}
  53. δ i n t ( s , t s ) = ( s , τ ( s ) \delta_{int}(s,t_{s})=(s^{\prime},\tau(s^{\prime})
  54. δ y ( s ) = ( y , s ) . \delta_{y}(s)=(y,s^{\prime}).
  55. ( s , t s ) S (s,t_{s})\in S^{\prime}
  56. λ ( s , t s ) = y \lambda(s,t_{s})=y
  57. δ y ( s ) = ( y , s ) . \delta_{y}(s)=(y,s^{\prime}).
  58. ( s , t s , t e ) (s,t_{s},t_{e})
  59. t s = t_{s}=\infty
  60. δ y \delta_{y}
  61. λ : S Y ϕ \lambda:S\rightarrow Y^{\phi}
  62. δ i n t : S S \delta_{int}:S\rightarrow S

Spacetime_topology.html

  1. M M
  2. 4 \mathbb{R}^{4}
  3. ρ \rho
  4. E M E\subset M
  5. c c
  6. O O
  7. E c = O c E\cap c=O\cap c
  8. M M
  9. I + ( p , U ) I - ( p , U ) p I^{+}(p,U)\cup I^{-}(p,U)\cup p
  10. p M p\in M
  11. U M U\subset M
  12. I ± I^{\pm}
  13. I + ( E ) I^{+}(E)
  14. I - ( E ) I^{-}(E)
  15. E M E\subset M
  16. I + ( x ) I - ( y ) I^{+}(x)\cap I^{-}(y)
  17. x , y M \,x,y\in M
  18. I + ( E ) I^{+}(E)

Spache_readability_formula.html

  1. Grade Level = ( 0.141 × Average sentence length ) + ( 0.086 × Number of unique unfamiliar words ) + 0.839 \mbox{Grade Level}~{}=\left(0.141\times\mbox{Average sentence length}~{}\right% )+\left(0.086\times\mbox{Number of unique unfamiliar words}~{}\right)+0.839
  2. Grade Level = ( 0.121 × Average sentence length ) + ( 0.082 × Number of unique unfamiliar words ) + 0.659 \mbox{Grade Level}~{}=\left(0.121\times\mbox{Average sentence length}~{}\right% )+\left(0.082\times\mbox{Number of unique unfamiliar words}~{}\right)+0.659

Sparse_language.html

  1. ( n k ) {\left({{n}\atop{k}}\right)}
  2. 𝐍𝐏 𝐏 / poly \,\textbf{NP}\subseteq\,\textbf{P}/\,\text{poly}

Spectral_clustering.html

  1. A A
  2. A i j 0 A_{ij}\geq 0
  3. i i
  4. j j
  5. ( B 1 , B 2 ) (B_{1},B_{2})
  6. v v
  7. L n o r m := I - D - 1 / 2 A D - 1 / 2 L^{norm}:=I-D^{-1/2}AD^{-1/2}
  8. D D
  9. D i i = j A i j . D_{ii}=\sum_{j}A_{ij}.
  10. P = D - 1 A P=D^{-1}A
  11. L := D - A L:=D-A
  12. m m
  13. v v
  14. v v
  15. m m
  16. B 1 B_{1}
  17. B 2 B_{2}
  18. k ( x i , x j ) = ϕ T ( x i ) ϕ ( x j ) k(x_{i},x_{j})=\phi^{T}(x_{i})\phi(x_{j})
  19. w r w_{r}
  20. max { C s } r = 1 k w r x i , x j C r k ( x i , x j ) . \max_{\{C_{s}\}}\sum_{r=1}^{k}w_{r}\sum_{x_{i},x_{j}\in C_{r}}k(x_{i},x_{j}).
  21. F F
  22. F i j = w r F_{ij}=w_{r}
  23. i , j C r i,j\in C_{r}
  24. K K
  25. max F trace ( K F ) \max_{F}\operatorname{trace}\left(KF\right)
  26. F = G n × k G n × k T F=G_{n\times k}G_{n\times k}^{T}
  27. G T G = I G^{T}G=I
  28. rank ( G ) = k \,\text{rank}(G)=k
  29. F F
  30. F 𝕀 = 𝕀 F\cdot\mathbb{I}=\mathbb{I}
  31. 𝕀 \mathbb{I}
  32. F T 𝕀 = 𝕀 F^{T}\mathbb{I}=\mathbb{I}
  33. max G trace ( G T G ) . \max_{G}\,\text{ trace }\left(G^{T}G\right).
  34. F F
  35. F F

Spectral_density_estimation.html

  1. s i n ( t ) sin(t)
  2. N N
  3. Δ t \Delta t
  4. x n x_{n}
  5. S ( f ) = Δ t N | n = 0 N - 1 x n e - i 2 π n f | 2 , - 1 2 Δ t < f 1 2 Δ t S(f)=\frac{\Delta t}{N}\left|\sum_{n=0}^{N-1}x_{n}e^{-i2\pi nf}\right|^{2},% \qquad-\frac{1}{2\Delta t}<f\leq\frac{1}{2\Delta t}
  6. 1 / ( 2 Δ t ) 1/(2\Delta t)
  7. N N\rightarrow\infty
  8. x n x_{n}
  9. S ( f ; a 1 , , a p ) S(f;a_{1},\ldots,a_{p})
  10. f f
  11. p p
  12. a 1 , , a p a_{1},\ldots,a_{p}
  13. A R ( p ) AR(p)
  14. p p
  15. { Y t } \{Y_{t}\}
  16. A R ( p ) AR(p)
  17. Y t = ϕ 1 Y t - 1 + ϕ 2 Y t - 2 + + ϕ p Y t - p + ϵ t , Y_{t}=\phi_{1}Y_{t-1}+\phi_{2}Y_{t-2}+\cdots+\phi_{p}Y_{t-p}+\epsilon_{t},
  18. ϕ 1 , , ϕ p \phi_{1},\ldots,\phi_{p}
  19. ϵ t \epsilon_{t}
  20. σ p 2 \sigma^{2}_{p}
  21. S ( f ; ϕ 1 , , ϕ p , σ p 2 ) = σ p 2 Δ t | 1 - k = 1 p ϕ k e - 2 i π f k Δ t | 2 | f | < f N , S(f;\phi_{1},\ldots,\phi_{p},\sigma^{2}_{p})=\frac{\sigma^{2}_{p}\Delta t}{% \left|1-\sum_{k=1}^{p}\phi_{k}e^{-2i\pi fk\Delta t}\right|^{2}}\qquad|f|<f_{N},
  22. Δ t \Delta t
  23. f N f_{N}
  24. ϕ 1 , , ϕ p , σ p 2 \phi_{1},\ldots,\phi_{p},\sigma^{2}_{p}
  25. A R ( p ) AR(p)
  26. A R ( p ) AR(p)
  27. A R ( p ) AR(p)
  28. x ( n ) x(n)
  29. p p
  30. w ( n ) w(n)
  31. x ( n ) = i = 1 p A i e j n ω i + w ( n ) x(n)=\sum_{i=1}^{p}A_{i}e^{jn\omega_{i}}+w(n)
  32. x ( n ) x(n)
  33. p p
  34. P ^ P H D ( e j ω ) = 1 | 𝐞 H 𝐯 m i n | 2 \hat{P}_{PHD}(e^{j\omega})=\frac{1}{|\mathbf{e}^{H}\mathbf{v}_{min}|^{2}}
  35. P ^ M U ( e j ω ) = 1 i = p + 1 M | 𝐞 H 𝐯 i | 2 \hat{P}_{MU}(e^{j\omega})=\frac{1}{\sum_{i=p+1}^{M}|\mathbf{e}^{H}\mathbf{v}_{% i}|^{2}}
  36. P ^ E V ( e j ω ) = 1 i = p + 1 M 1 λ i | 𝐞 H 𝐯 i | 2 \hat{P}_{EV}(e^{j\omega})=\frac{1}{\sum_{i=p+1}^{M}\frac{1}{\lambda_{i}}|% \mathbf{e}^{H}\mathbf{v}_{i}|^{2}}
  37. P ^ M N ( e j ω ) = 1 | 𝐞 H 𝐚 | 2 ; 𝐚 = λ 𝐏 n 𝐮 1 \hat{P}_{MN}(e^{j\omega})=\frac{1}{|\mathbf{e}^{H}\mathbf{a}|^{2}};\mathbf{a}=% \lambda\mathbf{P}_{n}\mathbf{u}_{1}

Spermidine_dehydrogenase.html

  1. \rightleftharpoons

Spermine_synthase.html

  1. \rightleftharpoons

Spinodal_decomposition.html

  1. \nabla
  2. \nabla
  3. = 𝐱 ^ x + 𝐲 ^ y + 𝐳 ^ z \nabla=\mathbf{\hat{x}}{\partial\over\partial x}+\mathbf{\hat{y}}{\partial% \over\partial y}+\mathbf{\hat{z}}{\partial\over\partial z}
  4. { 𝐱 ^ , 𝐲 ^ , 𝐳 ^ } \{\mathbf{\hat{x}},\mathbf{\hat{y}},\mathbf{\hat{z}}\}
  5. grad c = c x 𝐱 ^ + c y 𝐲 ^ + c z 𝐳 ^ = c . \mbox{grad}~{}\,c={\partial c\over\partial x}\mathbf{\hat{x}}+{\partial c\over% \partial y}\mathbf{\hat{y}}+{\partial c\over\partial z}\mathbf{\hat{z}}=\nabla c.
  6. d F = N v [ F 0 + k ( c ) 2 ] d V dF=N_{v}[F_{0}+k(\nabla c)^{2}]~{}dV
  7. F = v [ F 0 + k ( c ) 2 ] d V F=\int_{v}[F_{0}+k(\nabla c)^{2}]~{}dV
  8. N v N_{v}
  9. F 0 F_{0}
  10. \nabla
  11. \nabla
  12. f ( c ) = f ( c o ) + ( c - c o ) ( f c ) c = c o + 1 2 ( c - c o ) 2 ( 2 f c 2 ) c = c o . f(c)=f(c_{o})+\left(c-c_{o}\right)\left(\frac{\partial f}{\partial c}\right)_{% c\,=\,c_{o}}+\frac{1}{2}\,\left(c-c_{o}\right)^{2}\left(\frac{\partial^{2}f}{% \partial c^{2}}\right)_{c\,=\,c_{o}}.
  13. ( c - c o ) = A cos β x \left(c-c_{o}\right)=A\;\cos\,\beta x\,
  14. Δ F V = ( A 2 4 ) [ ( 2 f c 2 ) + 2 κ β 2 ] . \frac{\Delta F}{V}=\left(\frac{A^{2}}{4}\right)\left[\left(\frac{\partial^{2}f% }{\partial c^{2}}\right)+2\,\kappa\,\beta^{2}\right].
  15. ( c - c o ) 2 ( 2 f c 2 ) > 2 κ ( c ) 2 . \left(c-c_{o}\right)^{2}\,\left(\frac{\partial^{2}f}{\partial c^{2}}\right)>2% \kappa\left(\nabla c\right)^{2}.
  16. \nabla
  17. ( c - c o ) = A cos β x \left(c-c_{o}\right)=A\,\cos\,\beta x
  18. Δ F V = ( A 2 4 ) ( f ′′ + 2 κ β 2 ) . \frac{\Delta F}{V}=\left(\frac{A^{2}}{4}\right)\left(f^{\prime\prime}+2\,% \kappa\,\beta^{2}\right).
  19. β c = f ′′ 2 κ \beta_{c}=\sqrt{\frac{f^{\prime\prime}}{2\kappa}}
  20. λ c = 8 π 2 κ f ′′ . \lambda_{c}=\sqrt{\frac{8\pi^{2}\kappa}{f^{\prime\prime}}}.
  21. - J ~ = M ( μ a - μ b ) -\tilde{J}=M\nabla(\mu_{a}-\mu_{b})
  22. μ a - μ b = f c \mu_{a}-\mu_{b}=\frac{\partial f}{\partial c}
  23. - J a = M 2 f c 2 c -J_{a}=M\frac{\partial^{2}f}{\partial c^{2}}\nabla c
  24. - J ~ = - J a = D c -\tilde{J}=-J_{a}=D\nabla_{c}
  25. D = M 2 f c 2 D=M\frac{\partial^{2}f}{\partial c^{2}}
  26. ( μ a - μ b ) = f c - 2 K 2 c (\mu_{a}-\mu_{b})=\frac{\partial f}{\partial c}-2K\nabla^{2}c
  27. - J ~ = M 2 f c 2 c - 2 M K 3 c -\tilde{J}=M\frac{\partial^{2}f}{\partial c^{2}}\nabla c-2MK\nabla^{3}c
  28. c t = M 2 f c 2 2 c - 2 M k 4 c \frac{\partial c}{\partial t}=M\frac{\partial^{2}f}{\partial c^{2}}\nabla^{2}c% -2Mk\nabla^{4}c
  29. N v ( μ 2 - μ 1 ) = d f d c N_{v}(\mu_{2}-\mu_{1})=\frac{df}{dc}
  30. J = - M ( d d x ) d f d c J=-M\left(\frac{d}{dx}\right)\frac{df}{dc}
  31. c t = - ( 1 N v ) ( J x ) = ( m N v ) f ′′ 2 f x 2 \frac{\partial c}{\partial t}=-\left(\frac{1}{N_{v}}\right)\left(\frac{% \partial J}{\partial x}\right)=\left(\frac{m}{N_{v}}\right)f^{\prime\prime}% \frac{\partial^{2}f}{\partial x^{2}}
  32. c t = D ( 2 c x 2 ) \frac{\partial c}{\partial t}=D\left(\frac{\partial^{2}c}{\partial x^{2}}\right)
  33. M = D N v f ′′ M=\frac{DN_{v}}{f^{\prime\prime}}
  34. c - c 0 = A ( β , t ) exp [ i β x ] c-c_{0}=A(\beta,t)\exp\left[i\beta x\right]
  35. A ( β , t ) = A ( β , 0 ) exp [ R ( β ) t ] A(\beta,t)=A(\beta,0)\exp\left[R(\beta)t\right]
  36. R ( β ) = - M N β 2 f ′′ R(\beta)=-\frac{M}{N}\beta^{2}f^{\prime\prime}
  37. ϵ = a - a 0 a 0 \epsilon=\frac{a-a_{0}}{a_{0}}
  38. σ x = σ y = σ z \sigma_{x^{\prime}}=\sigma_{y^{\prime}}=\sigma_{z^{\prime}}
  39. W E = 1 2 i σ i ϵ i W_{E}=\frac{1}{2}\displaystyle\sum_{i}\sigma_{i}\epsilon_{i}
  40. W E ( 1 ) = 3 2 ( c 11 + 2 c 12 ) σ 2 W_{E}(1)=\frac{3}{2}(c_{11}+2c_{12})\sigma^{2}
  41. W E ( 2 ) = σ 2 ( c 11 + 2 c 22 ) 2 c 11 W_{E}(2)=\frac{\sigma^{2}(c_{11}+2c_{22})}{2c_{11}}
  42. W E = W E ( 1 ) - W E ( 2 ) W_{E}=W_{E}(1)-W_{E}(2)
  43. W E = ( σ 2 2 ) ( c 11 + 2 c 12 ) ( 3 - [ c 11 - 2 c 12 c 1 1 ] ) W_{E}=\left(\frac{\sigma^{2}}{2}\right)(c_{11}+2c_{12})\left(3-\left[\frac{c_{% 11}-2c_{12}}{c_{1^{\prime}1^{\prime}}}\right]\right)
  44. c 1 1 = c 11 + 2 ( 2 c 44 - c 11 + c 12 ) ( l 2 m 2 + m 2 n 2 + l 2 n 2 ) c_{1^{\prime}1^{\prime}}=c_{11}+2(2c_{44}-c_{11}+c_{12})(l^{2}m^{2}+m^{2}n^{2}% +l^{2}n^{2})
  45. W E = Y σ 2 W_{E}=Y\sigma^{2}
  46. Y = 1 2 ( c 11 + 2 c 12 ) [ 3 - c 11 + 2 c 12 c 11 + 2 ( 2 c 44 - c 11 + c 12 ) ( l 2 m 2 + m 2 n 2 + l 2 n 2 ) ] Y=\frac{1}{2}(c_{11}+2c_{12})\left[3-\frac{c_{11}+2c_{12}}{c_{11}+2(2c_{44}-c_% {11}+c_{12}})(l^{2}m^{2}+m^{2}n^{2}+l^{2}n^{2})\right]
  47. W E = 4 Y σ 2 d x W_{E}=4\int Y\sigma^{2}~{}dx
  48. a = a 0 [ 1 + η [ c - c 0 ] + ] a=a_{0}[1+\eta[c-c_{0}]+\cdots]
  49. η = ( 1 a 0 ) ( d a d c ) + d ln a d c \eta=\left(\frac{1}{a_{0}}\right)\left(\frac{da}{dc}\right)+\frac{d\ln a}{dc}
  50. σ = a - a 0 a 0 = η ( c - c 0 ) \sigma=\frac{a-a_{0}}{a_{0}}=\eta(c-c_{0})
  51. W E = A η 2 Y ( c - c 0 ) 2 d x W_{E}=A\int\eta^{2}Y(c-c_{0})^{2}~{}dx
  52. 2 c 44 - c 11 + c 12 2c_{44}-c_{11}+c_{12}
  53. Y [ iso ] = c 11 + c 12 - 2 ( c 12 2 c 11 ) Y[\mathrm{iso}]=c_{11}+c_{12}-2(\frac{c_{12}^{2}}{c_{11}})
  54. c 11 = E ( 1 - ν ) ( 1 - 2 ν ) ( 1 + ν ) c_{11}=\frac{E(1-\nu)}{(1-2_{\nu})(1+\nu)}
  55. c 12 = E ν ( 1 - 2 ν ) ( 1 + ν ) c_{12}=\frac{E_{\nu}}{(1-2_{\nu})(1+\nu)}
  56. Y [ iso ] = E 1 - ν Y[\mathrm{iso}]=\frac{E}{1-\nu}
  57. 2 c 44 - c 11 + c 12 2c_{44}-c_{11}+c_{12}
  58. Y [ 100 ] = c 11 + c 12 - 2 ( c 12 2 c 11 ) Y[\mathrm{100}]=c_{11}+c_{12}-2\left(\frac{c_{12}^{2}}{c_{11}}\right)
  59. Y [ 111 ] = 6 c 44 ( c 11 + 2 c 12 ) c 11 + 2 c 12 + 4 c 44 Y[\mathrm{111}]=\frac{6c_{44}(c_{11}+2c_{12})}{c_{11}+2c_{12}+4c_{44}}
  60. F t = A f ( c ) + η Y ( c - c 0 ) 2 + K ( d c d x ) 2 d x F_{t}=A\int f(c)+\eta Y(c-c_{0})^{2}+K\left(\frac{dc}{dx}\right)^{2}~{}dx
  61. c t = ( M N ν ) ( [ f ′′ + 2 η Y ] ( d 2 c d x 2 ) - 2 K ( d 4 c d x 4 ) ) \frac{\partial c}{\partial t}=\left(\frac{M}{N_{\nu}}\right)\left([f^{\prime% \prime}+2\eta Y]\left(\frac{d^{2}c}{dx^{2}}\right)-2K\left(\frac{d^{4}c}{dx^{4% }}\right)\right)
  62. c t = ( [ 1 + 2 η Y f ′′ ] d 2 c d x 2 - 2 K F f ′′ d 4 c d x 4 ) \frac{\partial c}{\partial t}=\left(\left[1+\frac{2\eta Y}{f^{\prime\prime}}% \right]\frac{d^{2}c}{dx^{2}}-\frac{2KF}{f^{\prime\prime}}\frac{d^{4}c}{dx^{4}}\right)
  63. c - c 0 = A ( β ) exp ( i β x ) d β c-c_{0}=\int A(\beta)\exp(i\beta x)~{}d\beta
  64. A ( β ) = 1 2 π ( c - c 0 ) exp ( - i β x ) d x A(\beta)=\frac{1}{2\pi}\int(c-c_{0})\exp(-i\beta x)~{}dx
  65. d A ( β ) d t = - M N ν [ f ′′ + 2 η 2 Y + 2 Y β 2 ] β 2 A ( β ) \frac{dA(\beta)}{dt}=-\frac{M}{N_{\nu}}[f^{\prime\prime}+2\eta^{2}Y+2Y\beta^{2% }]\beta^{2}A(\beta)
  66. A ( β , t ) = A ( β , 0 ) exp [ R ( β ) t ] A(\beta,t)=A(\beta,0)\exp[R(\beta)t]
  67. R ( β ) = - M N ν ( f ′′ + 2 η Y + 2 k β 2 ) β 2 R(\beta)=-\frac{M}{N_{\nu}}(f^{\prime\prime}+2\eta Y+2k\beta^{2})\beta^{2}
  68. R ( β ) = - D ~ ( 1 + 2 η 2 Y f ′′ + 2 K f ′′ β 2 ) β 2 R(\beta)=-\tilde{D}\left(1+\frac{2\eta^{2}Y}{f^{\prime\prime}}+\frac{2K}{f^{% \prime\prime}}\beta^{2}\right)\beta^{2}
  69. c t = M 2 f c 2 2 c - 2 M K 4 c ) \frac{\partial c}{\partial t}=M\frac{\partial^{2}f}{\partial c^{2}}\nabla^{2}c% -2MK\nabla^{4}c)
  70. c - c 0 = e x p [ R β ¯ t ] c o s β r c-c_{0}=exp[R\bar{\beta}t]cos\beta\cdot r
  71. R ( β ¯ ) - M β 2 ( 2 f c 2 + 2 K β 2 ) R(\bar{\beta})-M\beta^{2}\left(\frac{\partial^{2}f}{\partial c^{2}}+2K\beta^{2% }\right)
  72. R ( β ¯ ) = - M β 2 ( 2 f c 2 + 2 η 2 Y + 2 K β 2 ) R(\bar{\beta})=-M\beta^{2}\left(\frac{\partial^{2}f}{\partial c^{2}}+2\eta^{2}% Y+2K\beta^{2}\right)
  73. Y = E 1 - ν Y=\frac{E}{1-\nu}
  74. 2 \sqrt{2}
  75. 2 \sqrt{2}
  76. A A
  77. B B
  78. F = ( A 2 ϕ 2 + B 4 ϕ 4 + κ 2 ( ϕ ) 2 ) d x . F=\int\!\left(\frac{A}{2}\phi^{2}+\frac{B}{4}\phi^{4}+\frac{\kappa}{2}\left(% \nabla\phi\right)^{2}\right)~{}dx\;.
  79. ϕ = ρ A - ρ B \phi=\rho_{A}-\rho_{B}
  80. A A
  81. B B
  82. κ \kappa
  83. t ϕ = ( m μ + ξ ( x ) ) , \partial_{t}\phi=\nabla(m\nabla\mu+\xi(x))\;,
  84. m m
  85. ξ ( x ) \xi(x)
  86. ξ ( x ) = 0 \langle\xi(x)\rangle=0
  87. μ \mu
  88. μ = δ F δ ϕ = A ϕ + B ϕ 3 - κ 2 ϕ . \mu=\frac{\delta F}{\delta\phi}=A\phi+B\phi^{3}-\kappa\nabla^{2}\phi\;.
  89. A < 0 A<0
  90. ϕ = 0 \phi=0
  91. ξ \xi
  92. ϕ = ϕ i n \phi=\phi_{in}
  93. k k
  94. t ϕ ~ ( k , t ) = - m ( ( A + 3 B ϕ i n 2 ) k 2 + κ k 4 ) ϕ ~ ( k , t ) = R ( k ) ϕ ~ ( k , t ) , \partial_{t}\tilde{\phi}(k,t)=-m((A+3B\phi_{in}^{2})k^{2}+\kappa k^{4})\tilde{% \phi}(k,t)=R(k)\tilde{\phi}(k,t)\;,
  95. ϕ ~ ( k , t ) = exp ( R ( k ) t ) . \tilde{\phi}(k,t)=\exp(R(k)t)\;.
  96. R ( k ) R(k)
  97. k s p = - ( A + 3 B ϕ i n 2 ) 2 κ , k_{sp}=\sqrt{\frac{-(A+3B\phi_{in}^{2})}{2\kappa}}\;,
  98. λ s p = 2 π k s p = 2 π 2 κ - ( A + 3 B ϕ i n 2 ) . \lambda_{sp}=\frac{2\pi}{k_{sp}}=2\pi\sqrt{\frac{2\kappa}{-(A+3B\phi_{in}^{2})% }}\;.
  99. R ( k s p ) = - m ( ( A + 3 B ϕ i n 2 ) k s p 2 + κ k s p 4 ) = m ( A + 3 B ϕ i n 2 ) 2 4 κ = 1 t s p R(k_{sp})=-m((A+3B\phi_{in}^{2})k_{sp}^{2}+\kappa k_{sp}^{4})=\frac{m(A+3B\phi% _{in}^{2})^{2}}{4\kappa}=\frac{1}{t_{sp}}
  100. t s p t_{sp}

Squarewave_voltammetry.html

  1. Δ i p = n F A D 0 1 / 2 C 0 * ( π t p ) 1 / 2 Δ Ψ p \Delta i_{p}=\frac{nFAD_{0}^{1/2}C_{0}^{*}}{(\pi t_{p})^{1/2}}\Delta\Psi_{p}

SSS*.html

  1. ( J , s , h ) (J,s,h)
  2. J J
  3. ϵ \epsilon
  4. s { L , S } s\in\{L,S\}
  5. J J
  6. h ( - , ) h\in(-\infty,\infty)
  7. h h
  8. h h
  9. Γ \Gamma
  10. p = ( J , s , h ) p=(J,s,h)

Stack_(mathematics).html

  1. \rightarrow
  2. \rightarrow
  3. Bun G ( X ) \operatorname{Bun}_{G}(X)

Statistically_close.html

  1. X X
  2. Y Y
  3. D D
  4. Δ ( X , Y ) = 1 2 α D | Pr [ X = α ] - Pr [ Y = α ] | \Delta(X,Y)=\frac{1}{2}\sum_{\alpha\in D}|\Pr[X=\alpha]-\Pr[Y=\alpha]|
  5. { X k } k 𝒩 \{X_{k}\}_{k\in\mathcal{N}}
  6. { Y k } k 𝒩 \{Y_{k}\}_{k\in\mathcal{N}}
  7. Δ ( X k , Y k ) \Delta(X_{k},Y_{k})
  8. k k

Stearoyl-CoA_9-desaturase.html

  1. \rightleftharpoons

Steffensen's_inequality.html

  1. b - k b f ( x ) d x a b f ( x ) g ( x ) d x a a + k f ( x ) d x , \int_{b-k}^{b}f(x)\,dx\leq\int_{a}^{b}f(x)g(x)\,dx\leq\int_{a}^{a+k}f(x)\,dx,
  2. k = a b g ( x ) d x . k=\int_{a}^{b}g(x)\,dx.

Stein's_method.html

  1. m m
  2. ( 1.1 ) d ( P , Q ) = sup h | h d P - h d Q | = sup h | E h ( W ) - E h ( Y ) | (1.1)\quad d(P,Q)=\sup_{h\in\mathcal{H}}\left|\int hdP-\int hdQ\right|=\sup_{h% \in\mathcal{H}}\left|Eh(W)-Eh(Y)\right|
  3. P P
  4. Q Q
  5. 𝒳 \mathcal{X}
  6. W W
  7. Y Y
  8. P P
  9. Q Q
  10. E E
  11. \mathcal{H}
  12. 𝒳 \mathcal{X}
  13. \mathcal{H}
  14. \mathcal{H}
  15. \mathcal{H}
  16. P P
  17. Q Q
  18. Q Q
  19. Q Q
  20. 𝒜 \mathcal{A}
  21. f f
  22. 𝒳 \mathcal{X}
  23. Q Q
  24. ( 2.1 ) E ( 𝒜 f ) ( Y ) = 0 for all f Y has distribution Q . (2.1)\quad E(\mathcal{A}f)(Y)=0\,\text{ for all }f\quad\iff\quad Y\,\text{ has% distribution }Q.
  25. ( 2.2 ) E ( f ( Y ) - Y f ( Y ) ) = 0 for all f C b 1 Y has standard normal distribution. (2.2)\quad E\left(f^{\prime}(Y)-Yf(Y)\right)=0\,\text{ for all }f\in C_{b}^{1}% \quad\iff\quad Y\,\text{ has standard normal distribution.}
  26. ( 2.3 ) ( 𝒜 f ) ( x ) = f ( x ) - x f ( x ) . (2.3)\quad(\mathcal{A}f)(x)=f^{\prime}(x)-xf(x).
  27. P P
  28. Q Q
  29. d d
  30. 𝒜 \mathcal{A}
  31. P = Q P=Q
  32. E ( 𝒜 f ) ( W ) = 0 E(\mathcal{A}f)(W)=0
  33. P Q P\approx Q
  34. E ( 𝒜 f ) ( W ) 0 E(\mathcal{A}f)(W)\approx 0
  35. f = f h f=f_{h}
  36. ( 3.1 ) ( 𝒜 f ) ( x ) = h ( x ) - E [ h ( Y ) ] for all x . (3.1)\quad(\mathcal{A}f)(x)=h(x)-E[h(Y)]\qquad\,\text{ for all }x.
  37. x x
  38. W W
  39. W W
  40. ( 3.2 ) E ( 𝒜 f ) ( W ) = E [ h ( W ) ] - E [ h ( Y ) ] . (3.2)\quad E(\mathcal{A}f)(W)=E[h(W)]-E[h(Y)].
  41. Q Q
  42. ( 3.3 ) f ( x ) - x f ( x ) = h ( x ) - E [ h ( Y ) ] for all x . (3.3)\quad f^{\prime}(x)-xf(x)=h(x)-E[h(Y)]\qquad\,\text{for all }x.
  43. ( 3.4 ) ( 𝒜 f ) ( x ) = f ( x ) + f ( x ) q ( x ) / q ( x ) . (3.4)\quad(\mathcal{A}f)(x)=f^{\prime}(x)+f(x)q^{\prime}(x)/q(x).
  44. ( 4.1 ) f ( x ) = e x 2 / 2 - x [ h ( s ) - E h ( Y ) ] e - s 2 / 2 d s . (4.1)\quad f(x)=e^{x^{2}/2}\int_{-\infty}^{x}[h(s)-Eh(Y)]e^{-s^{2}/2}ds.
  45. 𝒜 \mathcal{A}
  46. ( Z t ) t 0 (Z_{t})_{t\geq 0}
  47. ( 4.2 ) f ( x ) = - 0 [ E x h ( Z t ) - E h ( Y ) ] d t , (4.2)\quad f(x)=-\int_{0}^{\infty}[E^{x}h(Z_{t})-Eh(Y)]dt,
  48. E x E^{x}
  49. Z Z
  50. x x
  51. h h\in\mathcal{H}
  52. f f
  53. h h
  54. ( 5.1 ) || D k f || C k , l || D l h || , (5.1)\quad||D^{k}f||\leq C_{k,l}||D^{l}h||,
  55. k , l = 0 , 1 , 2 , k,l=0,1,2,\dots
  56. k l k\geq l
  57. k l - 1 k\geq l-1
  58. | | | | ||\cdot||
  59. D k D^{k}
  60. C k , l C_{k,l}
  61. Q Q
  62. ( 5.2 ) || f || min { π / 2 || h || , 2 || h || } , || f || min { 2 || h || , 4 || h || } , || f ′′ || 2 || h || , (5.2)\quad||f||_{\infty}\leq\min\{\sqrt{\pi/2}||h||_{\infty},2||h^{\prime}||_{% \infty}\},\quad||f^{\prime}||_{\infty}\leq\min\{2||h||_{\infty},4||h^{\prime}|% |_{\infty}\},\quad||f^{\prime\prime}||_{\infty}\leq 2||h^{\prime}||_{\infty},
  63. h h
  64. W W
  65. W = i = 1 n X i W=\sum_{i=1}^{n}X_{i}
  66. E [ W ] = 0 E[W]=0
  67. var [ W ] = 1 \operatorname{var}[W]=1
  68. i = 1 , , n i=1,\dots,n
  69. A i { 1 , 2 , , n } A_{i}\subset\{1,2,\dots,n\}
  70. X i X_{i}
  71. X j X_{j}
  72. j A i j\not\in A_{i}
  73. X i X_{i}
  74. B i { 1 , 2 , , n } B_{i}\subset\{1,2,\dots,n\}
  75. X j X_{j}
  76. j A i j\in A_{i}
  77. X k X_{k}
  78. k B i k\not\in B_{i}
  79. B i B_{i}
  80. X i X_{i}
  81. A { 1 , 2 , , n } A\subset\{1,2,\dots,n\}
  82. X A := j A X j X_{A}:=\sum_{j\in A}X_{j}
  83. ( 6.1 ) | E ( f ( W ) - W f ( W ) ) | || f ′′ || i = 1 n ( 1 2 E | X i X A i 2 | + E | X i X A i X B i A i | + E | X i X A i | E | X B i | ) (6.1)\quad\left|E(f^{\prime}(W)-Wf(W))\right|\leq||f^{\prime\prime}||_{\infty}% \sum_{i=1}^{n}\left(\frac{1}{2}E|X_{i}X_{A_{i}}^{2}|+E|X_{i}X_{A_{i}}X_{B_{i}% \setminus A_{i}}|+E|X_{i}X_{A_{i}}|E|X_{B_{i}}|\right)
  84. || h || ||h^{\prime}||_{\infty}
  85. h h
  86. f ′′ f^{\prime\prime}
  87. || f || ||f||_{\infty}
  88. || f || ||f^{\prime}||_{\infty}
  89. W W
  90. d W d_{W}
  91. ( 6.2 ) d W ( ( W ) , N ( 0 , 1 ) ) 2 i = 1 n ( 1 2 E | X i X A i 2 | + E | X i X A i X B i A i | + E | X i X A i | E | X B i | ) . (6.2)\quad d_{W}(\mathcal{L}(W),N(0,1))\leq 2\sum_{i=1}^{n}\left(\frac{1}{2}E|% X_{i}X_{A_{i}}^{2}|+E|X_{i}X_{A_{i}}X_{B_{i}\setminus A_{i}}|+E|X_{i}X_{A_{i}}% |E|X_{B_{i}}|\right).
  92. h h
  93. || h || 1 ||h^{\prime}||\leq 1
  94. W W
  95. X i X_{i}
  96. A i A_{i}
  97. B i B_{i}
  98. E X i = 0 EX_{i}=0
  99. v a r X i = 1 varX_{i}=1
  100. W = n - 1 / 2 X i W=n^{-1/2}\sum X_{i}
  101. A i = B i = { i } A_{i}=B_{i}=\{i\}
  102. ( 7.1 ) d W ( ( W ) , N ( 0 , 1 ) ) 5 E | X 1 | 3 n 1 / 2 . (7.1)\quad d_{W}(\mathcal{L}(W),N(0,1))\leq\frac{5E|X_{1}|^{3}}{n^{1/2}}.
  103. E h ( X 1 + + X n ) - E h ( Y 1 + + Y n ) Eh(X_{1}+...+X_{n})-Eh(Y_{1}+...+Y_{n})
  104. ψ ( t ) \psi(t)
  105. ψ ( t ) + t ψ ( t ) = 0 \psi^{\prime}(t)+t\psi(t)=0
  106. t t
  107. ψ W ( t ) \psi_{W}(t)
  108. W W
  109. ψ W ( t ) + t ψ W ( t ) 0 \psi^{\prime}_{W}(t)+t\psi_{W}(t)\approx 0
  110. ψ W ( t ) ψ ( t ) \psi_{W}(t)\approx\psi(t)
  111. W W

Stein–Strömberg_theorem.html

  1. M f ( x ) = sup r > 0 1 λ n ( B r ( x ) ) B r ( x ) | f ( y ) | d λ n ( y ) , Mf(x)=\sup_{r>0}\frac{1}{\lambda^{n}\big(B_{r}(x)\big)}\int_{B_{r}(x)}|f(y)|\,% \mathrm{d}\lambda^{n}(y),
  2. M f L p C p f L p . \|Mf\|_{L^{p}}\leq C_{p}\|f\|_{L^{p}}.
  3. M f L p C p , n f L p \|Mf\|_{L^{p}}\leq C_{p,n}\|f\|_{L^{p}}

Sten_scores.html

  1. ( s - 23.5 ) 4.2 2 + 5.5 \frac{(s-23.5)}{4.2}2+5.5

Stengle's_Positivstellensatz.html

  1. W = { x R n f F f ( x ) 0 } , W=\{x\in R^{n}\mid\forall f\in F\,f(x)\geq 0\},
  2. x W p ( x ) > 0 \forall x\in W\;p(x)>0
  3. f 1 , f 2 C p f 1 = 1 + f 2 \exists f_{1},f_{2}\in C\;pf_{1}=1+f_{2}
  4. { x R n f F f ( x ) 0 g G g ( x ) = 0 h H h ( x ) 0 } = \{x\in R^{n}\mid\forall f\in F\,f(x)\geq 0\land\forall g\in G\,g(x)=0\land% \forall h\in H\,h(x)\neq 0\}=\emptyset
  5. f C , g I , n f + g + ( H ) 2 n = 0. \exists f\in C,g\in I,n\in\mathbb{N}\;f+g+\left(\prod H\right)^{2n}=0.

Sterigmatocystin_8-O-methyltransferase.html

  1. \rightleftharpoons

Steroid_11beta-monooxygenase.html

  1. \rightleftharpoons

Steroid_17alpha-monooxygenase.html

  1. \rightleftharpoons

Steroid_9alpha-monooxygenase.html

  1. \rightleftharpoons

Steroid_Delta-isomerase.html

  1. \rightleftharpoons

Steroid_sulfotransferase.html

  1. \rightleftharpoons

Sterol-4alpha-carboxylate_3-dehydrogenase_(decarboxylating).html

  1. \rightleftharpoons

Sterol_14-demethylase.html

  1. \rightleftharpoons

Sterol_24-C-methyltransferase.html

  1. \rightleftharpoons

Sticking_coefficient.html

  1. P a P_{a}
  2. P m P_{m}
  3. P d P_{d}
  4. P a + P m + P d = 1 P_{a}+P_{m}+P_{d}=1
  5. P d + P m = 1 P_{d}^{\prime}+P_{m}^{\prime}=1
  6. P m 1 = P m ( 1 - θ ) + P m ( θ ) P_{m1}=P_{m}(1-\theta)+P_{m}^{\prime}(\theta)
  7. P d 1 = P d ( 1 - θ ) + P d ( θ ) P_{d1}=P_{d}(1-\theta)+P_{d}^{\prime}(\theta)
  8. P a 1 = P m ( 1 - θ ) P_{a1}=P_{m}(1-\theta)
  9. P m 2 = P m 1 × P m 1 = P m 1 2 P_{m2}=P_{m1}\times P_{m1}=P_{m1}^{2}
  10. s c s_{c}
  11. s = P a ( 1 - θ ) + P m 1 P a ( 1 - θ ) + P m 1 2 P a ( 1 - θ ) s=P_{a}(1-\theta)+P_{m1}P_{a}(1-\theta)+P_{m1}^{2}P_{a}(1-\theta)...
  12. s = P a ( 1 - θ ) n = 0 P m 1 n s=P_{a}(1-\theta)\sum_{n=0}^{\infty}P_{m1}^{n}
  13. n = 0 x n = 1 1 - x x < 1 \sum_{n=0}^{\infty}x^{n}=\frac{1}{1-x}\forall x<1
  14. s = P a ( 1 - θ ) 1 1 - P m 1 \therefore s=P_{a}(1-\theta)\frac{1}{1-P_{m1}}
  15. s 0 s_{0}
  16. θ = 0 \theta=0
  17. 1 - P m 1 = P a + P d 1-P_{m1}=P_{a}+P_{d}
  18. s 0 = P a P a + P d s_{0}=\frac{P_{a}}{P_{a}+P_{d}}
  19. s s 0 = P a ( 1 - θ ) 1 - P m 1 P a + P d P a \frac{s}{s_{0}}=\frac{P_{a}(1-\theta)}{1-P_{m1}}\frac{P_{a}+P_{d}}{P_{a}}
  20. P m 1 = 1 - P d ( 1 - θ ) - P d ( θ ) - P a ( 1 - θ ) P_{m}1=1-P_{d}(1-\theta)-P_{d}^{\prime}(\theta)-P_{a}(1-\theta)
  21. s s 0 = [ 1 + P d θ ( P a + P d ) ( 1 - θ ) ] - 1 \frac{s}{s_{0}}=[1+\frac{P_{d}^{\prime}\theta}{(P_{a}+P_{d})(1-\theta)}]^{-1}
  22. s s 0 = [ 1 + K θ 1 - θ ] - 1 \frac{s}{s_{0}}=[1+\frac{K\theta}{1-\theta}]^{-1}
  23. K = def P d P a + P d K\overset{\underset{\mathrm{def}}{}}{=}\frac{P_{d}^{\prime}}{P_{a}+P_{d}}

Stizolobate_synthase.html

  1. \rightleftharpoons

Stizolobinate_synthase.html

  1. \rightleftharpoons

Stokes'_law_of_sound_attenuation.html

  1. α \alpha
  2. α = 2 η ω 2 3 ρ V 3 \alpha=\frac{2\eta\omega^{2}}{3\rho V^{3}}
  3. η \eta
  4. ω \omega
  5. ρ \rho
  6. V V
  7. A 0 A_{0}
  8. d d
  9. A ( d ) A(d)
  10. A ( d ) = A 0 e - α d A(d)=A_{0}e^{-\alpha d}
  11. α \alpha
  12. m - 1 \mathrm{m}^{-1}
  13. α = 1 m - 1 \alpha=1\mathrm{m}^{-1}
  14. 1 / e 1/e
  15. η v \eta^{\mathrm{v}}
  16. α = 2 ( η + 3 η v / 4 ) ω 2 3 ρ V 3 \alpha=\frac{2(\eta+3\eta^{\mathrm{v}}/4)\omega^{2}}{3\rho V^{3}}
  17. 2 ( α V ω ) 2 = 1 1 + ω 2 τ 2 - 1 1 + ω 2 τ 2 2\left(\frac{\alpha V}{\omega}\right)^{2}=\frac{1}{\sqrt{1+\omega^{2}\tau^{2}}% }-\frac{1}{1+\omega^{2}\tau^{2}}
  18. τ \tau
  19. τ = 4 η / 3 + η v ρ V 2 \tau=\frac{4\eta/3+\eta^{\mathrm{v}}}{\rho V^{2}}
  20. 10 - 12 s 10^{-12}\mathrm{s}

Strain_rate.html

  1. ϵ \epsilon
  2. ϵ ( t ) = L ( t ) - L 0 L 0 \epsilon(t)=\frac{L(t)-L_{0}}{L_{0}}
  3. L 0 L_{0}
  4. L ( t ) L(t)
  5. t t
  6. ϵ ˙ ( t ) = d ϵ d t = d d t ( L ( t ) - L 0 L 0 ) = 1 L 0 d L d t ( t ) = v ( t ) L 0 \dot{\epsilon}(t)=\frac{d\epsilon}{dt}=\frac{d}{dt}\left(\frac{L(t)-L_{0}}{L_{% 0}}\right)=\frac{1}{L_{0}}\frac{dL}{dt}(t)=\frac{v(t)}{L_{0}}
  7. v ( t ) v(t)
  8. t t
  9. X ( y , t ) X(y,t)
  10. y y
  11. X ( y + d , t ) - X ( y , t ) X(y+d,t)-X(y,t)
  12. d d
  13. ϵ ( y , t ) = lim d 0 X ( y + d , t ) - X ( y , t ) d = X y ( y , t ) \epsilon(y,t)=\lim_{d\rightarrow 0}\frac{X(y+d,t)-X(y,t)}{d}=\frac{\partial X}% {\partial y}(y,t)
  14. ϵ ˙ ( y , t ) = ( t X y ) ( y , t ) = ( y X t ) ( y , t ) = V y ( y , t ) \dot{\epsilon}(y,t)=\left(\frac{\partial}{\partial t}\frac{\partial X}{% \partial y}\right)(y,t)=\left(\frac{\partial}{\partial y}\frac{\partial X}{% \partial t}\right)(y,t)=\frac{\partial V}{\partial y}(y,t)
  15. V ( y , t ) V(y,t)
  16. y y

Strictosidine_synthase.html

  1. \rightleftharpoons

Strombine_dehydrogenase.html

  1. \rightleftharpoons

Studentized_range.html

  1. q n , ν = max { x 1 , x n } - min { x 1 , x n } s = max i , j = 1 , , n { x i - x j s } q_{n,\nu}=\frac{\max\{\,x_{1},\ \dots\ x_{n}\,\}-\min\{\,x_{1},\ \dots\ x_{n}% \}}{s}=\max_{i,j=1,\dots,n}\left\{\frac{x_{i}-x_{j}}{s}\right\}
  2. s 2 = 1 n - 1 i = 1 n ( x i - x ¯ ) 2 , s^{2}=\frac{1}{n-1}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2},
  3. x ¯ = x 1 + + x n n \overline{x}=\frac{x_{1}+\cdots+x_{n}}{n}

Styrene-oxide_isomerase.html

  1. \rightleftharpoons

Subscript_and_superscript.html

  1. 67 68 \tfrac{67}{68}
  2. 34 35 \tfrac{34}{35}
  3. X a b X_{ab}
  4. X a b X^{ab}

Succinate-semialdehyde_dehydrogenase.html

  1. \rightleftharpoons

Succinate-semialdehyde_dehydrogenase_(NAD(P)+).html

  1. \rightleftharpoons

Succinate—citramalate_CoA-transferase.html

  1. \rightleftharpoons

Succinate—hydroxymethylglutarate_CoA-transferase.html

  1. \rightleftharpoons

Succinyl-CoA:(R)-benzylsuccinate_CoA-transferase.html

  1. \rightleftharpoons

Succinylglutamate-semialdehyde_dehydrogenase.html

  1. \rightleftharpoons

Sulcatone_reductase.html

  1. \rightleftharpoons

Sulfiredoxin.html

  1. \rightleftharpoons

Sulfite_dehydrogenase.html

  1. \rightleftharpoons

Sulfite_reductase.html

  1. \rightleftharpoons

Sulfite_reductase_(NADPH).html

  1. \rightleftharpoons

Sulfolactate_sulfo-lyase.html

  1. \rightleftharpoons

Sulfur_oxygenase::reductase.html

  1. \rightleftharpoons

Sullivan_conjecture.html

  1. G G
  2. B G BG
  3. B G BG
  4. X X
  5. B G BG
  6. X X
  7. X X
  8. F ( B G , X ) F(BG,X)
  9. B G BG
  10. X X
  11. x x
  12. X X
  13. x x
  14. F ( B G , X ) F(BG,X)
  15. F ( B G , X ) F(BG,X)
  16. G G
  17. X X
  18. G G
  19. X X
  20. F ( E G , X ) G F(EG,X)^{G}
  21. F ( E G , X ) F(EG,X)
  22. E G EG
  23. B G BG
  24. X X
  25. G G
  26. F ( E G , X ) F(EG,X)
  27. g g
  28. G G
  29. f f
  30. F ( E G , X ) F(EG,X)
  31. g f g - 1 gfg^{-1}
  32. G G
  33. E G EG
  34. * *
  35. X G = F ( * , X ) G X^{G}=F(*,X)^{G}
  36. F ( E G , X ) G F(EG,X)^{G}
  37. G G
  38. X X
  39. G G
  40. B Z / 2 BZ/2
  41. X X
  42. G = Z / 2 G=Z/2
  43. ( X G ) p (X^{G})_{p}
  44. F ( E G , ( X ) p ) G F(EG,(X)_{p})^{G}
  45. G G
  46. ( X ) p (X)_{p}
  47. X X
  48. F ( E G , X ) G F(EG,X)^{G}

Sulochrin_oxidase_((+)-bisdechlorogeodin-forming).html

  1. \rightleftharpoons

Sulochrin_oxidase_((-)-bisdechlorogeodin-forming).html

  1. \rightleftharpoons

Super-prime.html

  1. x ( log x ) 2 + O ( x log log x ( log x ) 3 ) \frac{x}{(\log x)^{2}}+O\left(\frac{x\log\log x}{(\log x)^{3}}\right)

Superoxide_reductase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Surface_conductivity.html

  1. κ σ = 4 F 2 C z 2 D ( 1 + 3 m / z 2 ) R T κ ( cosh z F ζ 2 R T - 1 ) {\kappa}^{\sigma}=\frac{4F^{2}Cz^{2}D(1+3m/z^{2})}{RT\kappa}\left(\cosh\frac{% zF\zeta}{2RT}-1\right)
  2. m = 2 ε 0 ε m R 2 T 2 3 η F 2 D m=\frac{2\varepsilon_{0}\varepsilon_{m}R^{2}T^{2}}{3\eta F^{2}D}

Surface_gradient.html

  1. S S
  2. u u
  3. S u = u - 𝐧 ^ ( 𝐧 ^ u ) \nabla_{S}u=\nabla u-\mathbf{\hat{n}}(\mathbf{\hat{n}}\cdot\nabla u)
  4. 𝐧 ^ \mathbf{\hat{n}}

Surface_second_harmonic_generation.html

  1. I s 2 ω ( γ ) = C | s 1 sin 2 γ χ x x z | 2 ( I ω ) 2 \mathrm{I}_{s}^{2\omega}(\gamma)=C|s_{1}\sin{2\gamma}\ \chi_{xxz}|^{2}(I^{% \omega})^{2}
  2. I p 2 ω ( γ ) = C | s 5 χ z x x + cos 2 γ ( s 2 χ x x z + s 3 χ z x x + s 4 χ z z z - s 5 χ z x x ) | 2 ( I ω ) 2 \mathrm{I}_{p}^{2\omega}(\gamma)=C|s_{5}\chi_{zxx}+\cos^{2}{\gamma}\ {(s_{2}% \chi_{xxz}+s_{3}\chi_{zxx}+s_{4}\chi_{zzz}-s_{5}\chi{zxx})}|^{2}(I^{\omega})^{2}
  3. χ Z Z Z = N s [ cos 3 θ β Z Z Z + cos θ sin 2 θ sin 2 Ψ ( β Z X X + β X X Z ) ] \chi_{ZZZ}=N_{s}[\langle\cos^{3}\theta\rangle\beta_{Z^{\prime}Z^{\prime}Z^{% \prime}}+\langle\cos\theta\sin^{2}\theta\sin^{2}\Psi\rangle(\beta_{Z^{\prime}X% ^{\prime}X^{\prime}}+\beta_{X^{\prime}X^{\prime}Z^{\prime}})]
  4. χ Z X X = 1 2 N s [ cos θ sin 2 θ β Z Z X + cos θ β Z X X - cos θ sin 2 θ sin 2 Ψ ( β Z X X + β X X Z ) ] \chi_{ZXX}=\frac{1}{2}N_{s}[\langle\cos\theta\sin^{2}\theta\rangle\beta_{Z^{% \prime}Z^{\prime}X^{\prime}}+\langle\cos\theta\rangle\beta_{Z^{\prime}X^{% \prime}X^{\prime}}-\langle\cos\theta\sin^{2}\theta\sin^{2}\Psi\rangle(\beta_{Z% ^{\prime}X^{\prime}X^{\prime}}+\beta_{X^{\prime}X^{\prime}Z^{\prime}})]
  5. χ X X Z = 1 2 N s [ cos θ sin 2 θ β Z Z X + cos θ β X X Z - cos θ sin 2 θ sin 2 Ψ ( β Z X X + β X X Z ) ] \chi_{XXZ}=\frac{1}{2}N_{s}[\langle\cos\theta\sin^{2}\theta\rangle\beta_{Z^{% \prime}Z^{\prime}X^{\prime}}+\langle\cos\theta\rangle\beta_{X^{\prime}X^{% \prime}Z^{\prime}}-\langle\cos\theta\sin^{2}\theta\sin^{2}\Psi\rangle(\beta_{Z% ^{\prime}X^{\prime}X^{\prime}}+\beta_{X^{\prime}X^{\prime}Z^{\prime}})]

Surveyor's_wheel.html

  1. 3 4 π \frac{3}{4}\pi
  2. 3 8 \frac{3}{8}

Swastika_curve.html

  1. y 4 - x 4 = x y , y^{4}-x^{4}=xy,\,
  2. r 2 = - tan ( 2 θ ) / 2. r^{2}=-\tan(2\theta)/2.\,
  3. x 4 - y 4 = x y . x^{4}-y^{4}=xy.\,

Symplectization.html

  1. ( V , ξ ) (V,\xi)
  2. x V x\in V
  3. S x V = { β T x * V - { 0 } ker β = ξ x } T x * V S_{x}V=\{\beta\in T^{*}_{x}V-\{0\}\mid\ker\beta=\xi_{x}\}\subset T^{*}_{x}V
  4. x x
  5. ξ x \xi_{x}
  6. S V = x V S x V T * V SV=\bigcup_{x\in V}S_{x}V\subset T^{*}V
  7. V V
  8. π : S V V \pi:SV\to V
  9. V V
  10. \R * \R - { 0 } \R^{*}\equiv\R-\{0\}
  11. ξ \xi
  12. α \alpha
  13. ξ \xi
  14. α \alpha
  15. S x + V = { β T x * V - { 0 } | β = λ α , λ > 0 } T x * V , S^{+}_{x}V=\{\beta\in T^{*}_{x}V-\{0\}\,|\,\beta=\lambda\alpha,\,\lambda>0\}% \subset T^{*}_{x}V,
  16. S + V = x V S x + V T * V . S^{+}V=\bigcup_{x\in V}S^{+}_{x}V\subset T^{*}V.
  17. ξ \xi
  18. π : S V V \pi:SV\to V

Synaptic_augmentation.html

  1. A ( t ) = [ TransmitterRelease ( t ) / TransmitterRelease ( 0 ) ] - 1 , A(t)=[{\rm TransmitterRelease}(t)/{\rm TransmitterRelease}(0)]-1,
  2. A A
  3. t t
  4. A * A^{*}
  5. d A * d t = J ( t ) a * - k A * A * \frac{dA^{*}}{dt}=J(t)a^{*}-k_{A^{*}}A^{*}
  6. J ( t ) J(t)
  7. a * a^{*}
  8. A * A^{*}
  9. k A * k_{A^{*}}
  10. A * A^{*}
  11. a * = a 0 * Z S T a^{*}=a^{*}_{0}Z^{ST}
  12. a 0 * a^{*}_{0}
  13. Z Z
  14. a * a^{*}
  15. S S
  16. T T

T-norm_fuzzy_logics.html

  1. * : [ 0 , 1 ] 2 [ 0 , 1 ] *\colon[0,1]^{2}\to[0,1]
  2. x * y = y * x x*y=y*x
  3. ( x * y ) * z = x * ( y * z ) (x*y)*z=x*(y*z)
  4. x y x\leq y
  5. x * z y * z x*z\leq y*z
  6. 1 * x = x 1*x=x
  7. 0 * x = 0 0*x=0
  8. * *
  9. * *
  10. * *
  11. * *
  12. \Rightarrow
  13. x * y z x*y\leq z
  14. x y z . x\leq y\Rightarrow z.
  15. ( x y ) = sup { z z * x y } . (x\Rightarrow y)=\sup\{z\mid z*x\leq y\}.
  16. x * ( x y ) y . x*(x\Rightarrow y)\leq y.
  17. ¬ x = ( x 0 ) \neg x=(x\Rightarrow 0)
  18. x y = ( x y ) * ( y x ) . x\Leftrightarrow y=(x\Rightarrow y)*(y\Rightarrow x).
  19. Δ x = 1 \Delta x=1
  20. x = 1 x=1
  21. Δ x = 0 \Delta x=0
  22. * , *,
  23. * - *\mbox{-}~{}
  24. * - *\mbox{-}~{}
  25. * , *,
  26. x * y = max ( x + y - 1 , 0 ) x*y=\max(x+y-1,0)
  27. x * y = min ( x , y ) x*y=\min(x,y)
  28. x * y = x y x*y=x\cdot y
  29. \rightarrow
  30. & \And
  31. \otimes
  32. \wedge
  33. A B A & ( A B ) . A\wedge B\equiv A\mathbin{\And}(A\rightarrow B).
  34. \bot
  35. 0
  36. 0 ¯ \overline{0}
  37. \bot
  38. ¬ \neg
  39. ¬ A A \neg A\equiv A\rightarrow\bot
  40. \leftrightarrow
  41. A B ( A B ) ( B A ) A\leftrightarrow B\equiv(A\rightarrow B)\wedge(B\rightarrow A)
  42. ( A B ) & ( B A ) . (A\rightarrow B)\mathbin{\And}(B\rightarrow A).
  43. \vee
  44. A B ( ( A B ) B ) ( ( B A ) A ) A\vee B\equiv((A\rightarrow B)\rightarrow B)\wedge((B\rightarrow A)\rightarrow A)
  45. \top
  46. 1 1
  47. 1 ¯ \overline{1}
  48. \top
  49. . \top\equiv\bot\rightarrow\bot.
  50. \triangle
  51. A \triangle A
  52. L L
  53. L . L_{\triangle}.
  54. r r
  55. r ¯ . \overline{r}.
  56. & r , s ¯ ( & r ¯ , s ¯ ) , \overline{r\mathbin{\And}s}\leftrightarrow(\overline{r}\mathbin{\And}\overline% {s}),
  57. r s ¯ ( r ¯ , s ¯ ) , \overline{r\rightarrow s}\leftrightarrow(\overline{r}\mathbin{\rightarrow}% \overline{s}),
  58. \sim
  59. ¬ ¬ A A \neg\neg A\leftrightarrow A
  60. L L
  61. L L_{\sim}
  62. L L
  63. \oplus
  64. A B ( A & B ) . A\oplus B\equiv\mathrm{\sim}(\mathrm{\sim}A\mathbin{\And}\mathrm{\sim}B).
  65. \forall
  66. \exists
  67. L L
  68. L . L\forall.
  69. L L
  70. L L
  71. L L
  72. L L
  73. L L
  74. L L
  75. L L
  76. L L
  77. L L
  78. L L_{\sim}
  79. L L
  80. \sim
  81. f ( x ) = 1 - x , f_{\sim}(x)=1-x,
  82. \sim
  83. L L_{\sim}

Tabersonine_16-hydroxylase.html

  1. \rightleftharpoons

Tabersonine_16-O-methyltransferase.html

  1. \rightleftharpoons

Tagaturonate_reductase.html

  1. \rightleftharpoons

Takebe_Kenko.html

  1. ( arcsin ( x ) ) 2 (\arcsin(x))^{2}
  2. π \pi

Tartrate_dehydrogenase.html

  1. \rightleftharpoons

Tartrate_epimerase.html

  1. \rightleftharpoons

Taurine_dehydrogenase.html

  1. \rightleftharpoons

Taurine_dioxygenase.html

  1. \rightleftharpoons

Tauropine_dehydrogenase.html

  1. \rightleftharpoons

Taxadiene_5alpha-hydroxylase.html

  1. \rightleftharpoons

Taxane_10beta-hydroxylase.html

  1. \rightleftharpoons

Taxane_13alpha-hydroxylase.html

  1. \rightleftharpoons

Taxifolin_8-monooxygenase.html

  1. \rightleftharpoons

Technetium_(99mTc)_albumin_aggregated.html

  1. % R i g h t - t o - l e f t s h u n t = ( ( T o t a l b o d y c o u n t s - T o t a l l u n g c o u n t s ) T o t a l b o d y c o u n t s ) × 100 % \%Right-to-leftshunt=\left(\frac{(Total\ body\ counts\ -\ Total\ lung\ counts)% }{Total\ body\ counts}\right)\times 100\%

Template:All.html

  1. \forall

Template:Eqv.html

  1. \leftrightarrow

Template:Exist.html

  1. \exists

Template:HAZMAT_Class_6_Acute_LD50.html

  1. C A T A + C B T B + C Z T Z = 100 T M \tfrac{C_{A}}{T_{A}}+\tfrac{C_{B}}{T_{B}}+\tfrac{C_{Z}}{T_{Z}}=\tfrac{100}{T_{% M}}
  2. C C
  3. T T
  4. T M T_{M}

Template:HAZMAT_Class_6_Packing_Group_Hazard_Zone_Inhaling_Vapors.html

  1. V 500 L C 50 V\geq 500LC_{50}
  2. L C 50 200 m L m 3 LC_{50}\leq 200\tfrac{mL}{m^{3}}
  3. V 10 L C 50 V\geq 10LC_{50}
  4. L C 50 1000 m L m 3 LC_{50}\leq 1000\tfrac{mL}{m^{3}}
  5. V L C 50 V\geq LC_{50}
  6. L C 50 3000 m L m 3 LC_{50}\leq 3000\tfrac{mL}{m^{3}}
  7. V .2 L C 50 V\geq.2LC_{50}
  8. L C 50 5000 m L m 3 LC_{50}\leq 5000\tfrac{mL}{m^{3}}
  9. m L m 3 \tfrac{mL}{m^{3}}

Template:HAZMAT_Class_6_Packing_Group_Hazard_Zone_Not_Inhaling_Vapors.html

  1. m g k g \tfrac{mg}{kg}
  2. m g k g \tfrac{mg}{kg}
  3. m g L \tfrac{mg}{L}
  4. 5 \leq 5
  5. 40 \leq 40
  6. 0.5 \leq 0.5
  7. 5 \geq 5
  8. 50 \leq 50
  9. 40 \geq 40
  10. 200 \leq 200
  11. 0.5 \geq 0.5
  12. 2 \leq 2
  13. 50 \geq 50
  14. 200 \leq 200
  15. 50 \geq 50
  16. 500 \leq 500
  17. 200 \geq 200
  18. 1000 \leq 1000
  19. 2 \geq 2
  20. 10 \leq 10

Template:HAZMAT_Class_6_Packing_Group_LC50_and_R.html

  1. R 500 R\geq 500
  2. L C 50 ( m i x t u r e ) 200 m L m 3 LC_{50}(mixture)\leq 200\tfrac{mL}{m^{3}}
  3. R 10 R\geq 10
  4. L C 50 ( m i x t u r e ) 1000 m L m 3 LC_{50}(mixture)\leq 1000\tfrac{mL}{m^{3}}
  5. R 1 R\geq 1
  6. L C 50 ( m i x t u r e ) 3000 m L m 3 LC_{50}(mixture)\leq 3000\tfrac{mL}{m^{3}}
  7. R 1 5 R\geq\tfrac{1}{5}
  8. L C 50 ( m i x t u r e ) 5000 m L m 3 LC_{50}(mixture)\leq 5000\tfrac{mL}{m^{3}}

Template:HAZMAT_Class_6_Packing_Groups.html

  1. L C 50 LC_{50}
  2. L C 50 LC_{50}
  3. L C 50 ( m i x t u r e ) = 1 i = 1 n f i L C 50 i LC_{50}(mixture)=\cfrac{1}{\textstyle\sum_{i=1}^{n}\cfrac{f_{i}}{LC_{50i}}}
  4. f i f_{i}
  5. i t h i^{th}
  6. L C 50 i LC_{50i}
  7. i t h i^{th}
  8. m L m 3 \tfrac{mL}{m^{3}}
  9. V i = P i × 10 6 101.3 [ m L m 3 ] V_{i}=P_{i}\times\tfrac{10^{6}}{101.3}[\tfrac{mL}{m^{3}}]
  10. P i P_{i}
  11. i t h i^{th}
  12. P i P_{i}
  13. R = i = 1 n f i L C 50 i R=\textstyle\sum_{i=1}^{n}\tfrac{f_{i}}{LC_{50i}}

Template:Or-.html

  1. \lor

Template:Tee.html

  1. \vdash

Template:Xor.html

  1. \nleftrightarrow

Tennenbaum's_theorem.html

  1. M \scriptstyle M
  2. N × N \scriptstyle N\times N
  3. N \scriptstyle N
  4. n 0 , n 1 \scriptstyle n_{0},n_{1}
  5. ( N , + , × , < , n 0 , n 1 ) M , (N,+,\times,<,n_{0},n_{1})\equiv M,\,
  6. \scriptstyle\equiv
  7. N \scriptstyle N

Terephthalate_1,2-cis-dihydrodiol_dehydrogenase.html

  1. \rightleftharpoons

Terephthalate_1,2-dioxygenase.html

  1. \rightleftharpoons

Terzaghi's_principle.html

  1. σ \sigma^{\prime}
  2. σ \sigma
  3. u u
  4. σ = σ + u \sigma=\sigma^{\prime}+u

Testosterone_17beta-dehydrogenase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Tetrachloroethene_reductive_dehalogenase.html

  1. \rightleftharpoons

Tetrahydroberberine_oxidase.html

  1. \rightleftharpoons

Tetrahydrocolumbamine_2-O-methyltransferase.html

  1. \rightleftharpoons

Tetrahydromethanopterin_S-methyltransferase.html

  1. \rightleftharpoons

Tetrahydroxynaphthalene_reductase.html

  1. \rightleftharpoons

Tetrahydroxypteridine_cycloisomerase.html

  1. \rightleftharpoons

The_Quadrature_of_the_Parabola.html

  1. Area = T + 2 ( T 8 ) + 4 ( T 8 2 ) + 8 ( T 8 3 ) + . \mbox{Area}~{}\;=\;T\,+\,2\left(\frac{T}{8}\right)\,+\,4\left(\frac{T}{8^{2}}% \right)\,+\,8\left(\frac{T}{8^{3}}\right)\,+\,\cdots.
  2. Area = ( 1 + 1 4 + 1 16 + 1 64 + ) T . \mbox{Area}~{}\;=\;\left(1\,+\,\frac{1}{4}\,+\,\frac{1}{16}\,+\,\frac{1}{64}\,% +\,\cdots\right)T.
  3. 1 + 1 4 + 1 16 + 1 64 + = 4 3 . 1\,+\,\frac{1}{4}\,+\,\frac{1}{16}\,+\,\frac{1}{64}\,+\,\cdots\;=\;\frac{4}{3}.
  4. 1 4 + 1 16 + 1 64 + . \frac{1}{4}\,+\,\frac{1}{16}\,+\,\frac{1}{64}\,+\,\cdots.

Theobromine_synthase.html

  1. \rightleftharpoons

Theodore_Motzkin.html

  1. [ 1 + - 19 2 ] \mathbb{Z}\left[\frac{1+\sqrt{-19}}{2}\right]

Thetin—homocysteine_S-methyltransferase.html

  1. \rightleftharpoons

Thiamine_oxidase.html

  1. \rightleftharpoons

Thiocyanate_isomerase.html

  1. \rightleftharpoons

Thioether_S-methyltransferase.html

  1. \rightleftharpoons

Thiol_oxidase.html

  1. \rightleftharpoons

Thiol_S-methyltransferase.html

  1. \rightleftharpoons

Thiol_sulfotransferase.html

  1. \rightleftharpoons

Thiomorpholine-carboxylate_dehydrogenase.html

  1. \rightleftharpoons

Thiophene-2-carbonyl-CoA_monooxygenase.html

  1. \rightleftharpoons

Thiosulfate_dehydrogenase.html

  1. \rightleftharpoons

Thiosulfate_dehydrogenase_(quinone).html

  1. \rightleftharpoons

Thiosulfate_sulfurtransferase.html

  1. \rightleftharpoons

Thiosulfate—dithiol_sulfurtransferase.html

  1. \rightleftharpoons

Thiosulfate—thiol_sulfurtransferase.html

  1. \rightleftharpoons

Threo-3-hydroxyaspartate_ammonia-lyase.html

  1. \rightleftharpoons

Threonine_racemase.html

  1. \rightleftharpoons

Thymaridas.html

  1. x = ( m 1 + m 2 + + m n - 1 ) - s n - 2 . x=\frac{(m_{1}+m_{2}+...+m_{n-1})-s}{n-2}.

Thymidylate_synthase_(FAD).html

  1. \rightleftharpoons

Thymine_dioxygenase.html

  1. \rightleftharpoons

Tidewater_glacier_cycle.html

  1. V C = C H w + D V_{C}=CH_{w}+D
  2. V C V_{C}
  3. C C
  4. H w H_{w}
  5. D D

Tilted_large_deviation_principle.html

  1. J ε ( S ) = S e - F ( x ) / ε d μ ε ( x ) J_{\varepsilon}(S)=\int_{S}e^{-F(x)/\varepsilon}\,\mathrm{d}\mu_{\varepsilon}(x)
  2. ν ε ( S ) = J ε ( S ) J ε ( X ) . \nu_{\varepsilon}(S)=\frac{J_{\varepsilon}(S)}{J_{\varepsilon}(X)}.
  3. I F ( x ) = sup y X [ F ( y ) - I ( y ) ] - [ F ( x ) - I ( x ) ] . I^{F}(x)=\sup_{y\in X}\big[F(y)-I(y)\big]-\big[F(x)-I(x)\big].

Time-of-flight_mass_spectrometry.html

  1. E p = q U E_{\mathrm{p}}=qU\,
  2. E k = 1 2 m v 2 E_{\mathrm{k}}=\frac{1}{2}mv^{2}
  3. E p = E k E_{\mathrm{p}}=E_{\mathrm{k}}\,
  4. q U = 1 2 m v 2 qU=\frac{1}{2}mv^{2}\,
  5. v = d t v=\frac{d}{t}\,
  6. q U = 1 2 m ( d t ) 2 qU=\frac{1}{2}m\left(\frac{d}{t}\right)^{2}\,
  7. t 2 = d 2 2 U m q t^{2}=\frac{d^{2}}{2U}\frac{m}{q}\,
  8. t = d 2 U m q t=\frac{d}{\sqrt{2U}}\sqrt{\frac{m}{q}}\,
  9. d 2 U \frac{d}{\sqrt{2U}}
  10. t = k m q t=k\sqrt{\frac{m}{q}}\,
  11. t = 1.5 m 2 ( 15000 V ) ( 1000 Da ) ( 1.660538921 × 10 - 27 kg Da - 1 ) + 1.602 × 10 - 19 C t=\frac{1.5\;\mathrm{m}}{\sqrt{2(15000\;\mathrm{V})}}\sqrt{\frac{(1000\;% \mathrm{Da})(1.660538921\times 10^{-27}\;\mathrm{kg\;Da}^{-1})}{+1.602\times 1% 0^{-19}\;\mathrm{C}}}
  12. t = 2.792 × 10 - 5 s t=2.792\times 10^{-5}\;\mathrm{s}

Timoshenko_beam_theory.html

  1. u x ( x , y , z ) = - z φ ( x ) ; u y ( x , y , z ) = 0 ; u z ( x , y ) = w ( x ) u_{x}(x,y,z)=-z~{}\varphi(x)~{};~{}~{}u_{y}(x,y,z)=0~{};~{}~{}u_{z}(x,y)=w(x)
  2. ( x , y , z ) (x,y,z)
  3. u x , u y , u z u_{x},u_{y},u_{z}
  4. φ \varphi
  5. w w
  6. z z
  7. d 2 d x 2 ( E I d φ d x ) = q ( x , t ) \displaystyle\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}\left(EI\frac{\mathrm{d}% \varphi}{\mathrm{d}x}\right)=q(x,t)
  8. E I κ L 2 A G 1 \frac{EI}{\kappa L^{2}AG}\ll 1
  9. L L
  10. A A
  11. E E
  12. G G
  13. I I
  14. κ \kappa
  15. κ = 5 / 6 \kappa=5/6
  16. E I d 4 w d x 4 = q ( x ) - E I κ A G d 2 q d x 2 EI~{}\cfrac{\mathrm{d}^{4}w}{\mathrm{d}x^{4}}=q(x)-\cfrac{EI}{\kappa AG}~{}% \cfrac{\mathrm{d}^{2}q}{\mathrm{d}x^{2}}
  17. M x x M_{xx}
  18. Q x Q_{x}
  19. w w
  20. φ \varphi
  21. M x x = - E I φ x and Q x = κ A G ( - φ + w x ) . M_{xx}=-EI~{}\frac{\partial\varphi}{\partial x}\quad\,\text{and}\quad Q_{x}=% \kappa~{}AG~{}\left(-\varphi+\frac{\partial w}{\partial x}\right)\,.
  22. u x ( x , y , z , t ) = - z φ ( x , t ) ; u y ( x , y , z , t ) = 0 ; u z ( x , y , z ) = w ( x , t ) u_{x}(x,y,z,t)=-z~{}\varphi(x,t)~{};~{}~{}u_{y}(x,y,z,t)=0~{};~{}~{}u_{z}(x,y,% z)=w(x,t)
  23. ε x x = u x x = - z φ x ; ε x z = 1 2 ( u x z + u z x ) = 1 2 ( - φ + w x ) \varepsilon_{xx}=\frac{\partial u_{x}}{\partial x}=-z~{}\frac{\partial\varphi}% {\partial x}~{};~{}~{}\varepsilon_{xz}=\frac{1}{2}\left(\frac{\partial u_{x}}{% \partial z}+\frac{\partial u_{z}}{\partial x}\right)=\frac{1}{2}\left(-\varphi% +\frac{\partial w}{\partial x}\right)
  24. κ \kappa
  25. ε x z = 1 2 κ ( - φ + w x ) \varepsilon_{xz}=\frac{1}{2}~{}\kappa~{}\left(-\varphi+\frac{\partial w}{% \partial x}\right)
  26. δ U = L A ( σ x x δ ε x x + 2 σ x z δ ε x z ) d A d L = L A [ - z σ x x ( δ φ ) x + σ x z κ ( - δ φ + ( δ w ) x ) ] d A d L \delta U=\int_{L}\int_{A}(\sigma_{xx}\delta\varepsilon_{xx}+2\sigma_{xz}\delta% \varepsilon_{xz})~{}\mathrm{d}A~{}\mathrm{d}L=\int_{L}\int_{A}\left[-z~{}% \sigma_{xx}\frac{\partial(\delta\varphi)}{\partial x}+\sigma_{xz}~{}\kappa% \left(-\delta\varphi+\frac{\partial(\delta w)}{\partial x}\right)\right]~{}% \mathrm{d}A~{}\mathrm{d}L
  27. M x x := A z σ x x d A ; Q x := κ A σ x z d A M_{xx}:=\int_{A}z~{}\sigma_{xx}~{}\mathrm{d}A~{};~{}~{}Q_{x}:=\kappa~{}\int_{A% }\sigma_{xz}~{}\mathrm{d}A
  28. δ U = L [ - M x x ( δ φ ) x + Q x ( - δ φ + ( δ w ) x ) ] d L \delta U=\int_{L}\left[-M_{xx}\frac{\partial(\delta\varphi)}{\partial x}+Q_{x}% \left(-\delta\varphi+\frac{\partial(\delta w)}{\partial x}\right)\right]~{}% \mathrm{d}L
  29. δ U = L [ ( M x x x - Q x ) δ φ - Q x x δ w ] d L \delta U=\int_{L}\left[\left(\frac{\partial M_{xx}}{\partial x}-Q_{x}\right)~{% }\delta\varphi-\frac{\partial Q_{x}}{\partial x}~{}\delta w\right]~{}\mathrm{d}L
  30. q ( x , t ) q(x,t)
  31. δ W = L q δ w d L \delta W=\int_{L}q~{}\delta w~{}\mathrm{d}L
  32. δ U = δ W L [ ( M x x x - Q x ) δ φ - ( Q x x + q ) δ w ] d L = 0 \delta U=\delta W\implies\int_{L}\left[\left(\frac{\partial M_{xx}}{\partial x% }-Q_{x}\right)~{}\delta\varphi-\left(\frac{\partial Q_{x}}{\partial x}+q\right% )~{}\delta w\right]~{}\mathrm{d}L=0
  33. M x x x - Q x = 0 ; Q x x + q = 0 \frac{\partial M_{xx}}{\partial x}-Q_{x}=0~{};~{}~{}\frac{\partial Q_{x}}{% \partial x}+q=0
  34. M x x = A z σ x x d A = A z E ε x x d A = - A z 2 E φ x d A = - E I φ x Q x = A σ x z d A = A 2 G ε x z d A = A κ G ( - φ + w x ) d A = κ A G ( - φ + w x ) \begin{aligned}\displaystyle M_{xx}&\displaystyle=\int_{A}z~{}\sigma_{xx}~{}% \mathrm{d}A=\int_{A}z~{}E~{}\varepsilon_{xx}~{}\mathrm{d}A=-\int_{A}z^{2}~{}E~% {}\frac{\partial\varphi}{\partial x}~{}\mathrm{d}A=-EI~{}\frac{\partial\varphi% }{\partial x}\\ \displaystyle Q_{x}&\displaystyle=\int_{A}\sigma_{xz}~{}\mathrm{d}A=\int_{A}2G% ~{}\varepsilon_{xz}~{}\mathrm{d}A=\int_{A}\kappa~{}G~{}\left(-\varphi+\frac{% \partial w}{\partial x}\right)~{}\mathrm{d}A=\kappa~{}AG~{}\left(-\varphi+% \frac{\partial w}{\partial x}\right)\end{aligned}
  35. x ( E I φ x ) + κ A G ( w x - φ ) = 0 x [ κ A G ( w x - φ ) ] + q = 0 \begin{aligned}\displaystyle\frac{\partial}{\partial x}\left(EI\frac{\partial% \varphi}{\partial x}\right)+\kappa AG~{}\left(\frac{\partial w}{\partial x}-% \varphi\right)&\displaystyle=0\\ \displaystyle\frac{\partial}{\partial x}\left[\kappa AG\left(\frac{\partial w}% {\partial x}-\varphi\right)\right]+q&\displaystyle=0\end{aligned}
  36. 2 x 2 ( E I φ x ) = q w x = φ - 1 κ A G x ( E I φ x ) \begin{aligned}&\displaystyle\frac{\partial^{2}}{\partial x^{2}}\left(EI\frac{% \partial\varphi}{\partial x}\right)=q\\ &\displaystyle\frac{\partial w}{\partial x}=\varphi-\cfrac{1}{\kappa AG}~{}% \frac{\partial}{\partial x}\left(EI\frac{\partial\varphi}{\partial x}\right)% \end{aligned}
  37. w w
  38. M x x M_{xx}
  39. φ \varphi
  40. Q x Q_{x}
  41. w w
  42. φ \varphi
  43. Q x Q_{x}
  44. M x x M_{xx}
  45. x x
  46. z z
  47. x x
  48. z z
  49. M x x M_{xx}
  50. Q x Q_{x}
  51. z z
  52. x = L x=L
  53. x = 0 x=0
  54. P P
  55. z z
  56. - P x - M x x = 0 M x x = - P x -Px-M_{xx}=0\implies M_{xx}=-Px
  57. P + Q x = 0 Q x = - P . P+Q_{x}=0\implies Q_{x}=-P\,.
  58. P x = E I d φ d x and - P = κ A G ( - φ + d w d x ) . Px=EI\,\frac{d\varphi}{dx}\qquad\,\text{and}\qquad-P=\kappa AG\left(-\varphi+% \frac{dw}{dx}\right)\,.
  59. φ = 0 \varphi=0
  60. x = L x=L
  61. φ ( x ) = - P 2 E I ( L 2 - x 2 ) . \varphi(x)=-\frac{P}{2EI}\,(L^{2}-x^{2})\,.
  62. d w d x = - P κ A G - P 2 E I ( L 2 - x 2 ) . \frac{dw}{dx}=-\frac{P}{\kappa AG}-\frac{P}{2EI}\,(L^{2}-x^{2})\,.
  63. w = 0 w=0
  64. x = L x=L
  65. w ( x ) = P ( L - x ) κ A G - P x 2 E I ( L 2 - x 2 3 ) + P L 3 3 E I . w(x)=\frac{P(L-x)}{\kappa AG}-\frac{Px}{2EI}\,\left(L^{2}-\frac{x^{2}}{3}% \right)+\frac{PL^{3}}{3EI}\,.
  66. σ x x ( x , z ) = E ε x x = - E z d φ d x = - P x z I = M x x z I . \sigma_{xx}(x,z)=E\,\varepsilon_{xx}=-E\,z\,\frac{d\varphi}{dx}=-\frac{Pxz}{I}% =\frac{M_{xx}z}{I}\,.
  67. u x ( x , y , z , t ) = - z φ ( x , t ) ; u y ( x , y , z , t ) = 0 ; u z ( x , y , z , t ) = w ( x , t ) u_{x}(x,y,z,t)=-z~{}\varphi(x,t)~{};~{}~{}u_{y}(x,y,z,t)=0~{};~{}~{}u_{z}(x,y,% z,t)=w(x,t)
  68. ( x , y , z ) (x,y,z)
  69. u x , u y , u z u_{x},u_{y},u_{z}
  70. φ \varphi
  71. w w
  72. z z
  73. ρ A 2 w t 2 - q ( x , t ) = x [ κ A G ( w x - φ ) ] \rho A\frac{\partial^{2}w}{\partial t^{2}}-q(x,t)=\frac{\partial}{\partial x}% \left[\kappa AG\left(\frac{\partial w}{\partial x}-\varphi\right)\right]
  74. ρ I 2 φ t 2 = x ( E I φ x ) + κ A G ( w x - φ ) \rho I\frac{\partial^{2}\varphi}{\partial t^{2}}=\frac{\partial}{\partial x}% \left(EI\frac{\partial\varphi}{\partial x}\right)+\kappa AG\left(\frac{% \partial w}{\partial x}-\varphi\right)
  75. w ( x , t ) w(x,t)
  76. φ ( x , t ) \varphi(x,t)
  77. ρ \rho
  78. A A
  79. E E
  80. G G
  81. I I
  82. κ \kappa
  83. κ = 5 / 6 \kappa=5/6
  84. q ( x , t ) q(x,t)
  85. m := ρ A m:=\rho A
  86. J := ρ I J:=\rho I
  87. E I 4 w x 4 + m 2 w t 2 - ( J + E I m k A G ) 4 w x 2 t 2 + m J k A G 4 w t 4 = q ( x , t ) + J k A G 2 q t 2 - E I k A G 2 q x 2 EI~{}\cfrac{\partial^{4}w}{\partial x^{4}}+m~{}\cfrac{\partial^{2}w}{\partial t% ^{2}}-\left(J+\cfrac{EIm}{kAG}\right)\cfrac{\partial^{4}w}{\partial x^{2}~{}% \partial t^{2}}+\cfrac{mJ}{kAG}~{}\cfrac{\partial^{4}w}{\partial t^{4}}=q(x,t)% +\cfrac{J}{kAG}~{}\cfrac{\partial^{2}q}{\partial t^{2}}-\cfrac{EI}{kAG}~{}% \cfrac{\partial^{2}q}{\partial x^{2}}
  88. ( 1 ) m 2 w t 2 = κ A G ( 2 w x 2 - φ x ) + q ( x , t ) ; m := ρ A ( 2 ) J 2 φ t 2 = E I 2 φ x 2 + κ A G ( w x - φ ) ; J := ρ I \begin{aligned}\displaystyle(1)&&\displaystyle\quad m~{}\frac{\partial^{2}w}{% \partial t^{2}}&\displaystyle=\kappa AG~{}\left(\frac{\partial^{2}w}{\partial x% ^{2}}-\frac{\partial\varphi}{\partial x}\right)+q(x,t)~{};~{}~{}m:=\rho A\\ \displaystyle(2)&&\displaystyle\quad J~{}\frac{\partial^{2}\varphi}{\partial t% ^{2}}&\displaystyle=EI~{}\frac{\partial^{2}\varphi}{\partial x^{2}}+\kappa AG~% {}\left(\frac{\partial w}{\partial x}-\varphi\right)~{};~{}~{}J:=\rho I\end{aligned}
  89. ( 3 ) φ x = - m κ A G 2 w t 2 + 2 w x 2 + q κ A G ( 4 ) 2 q t 2 = m 4 w t 4 - κ A G ( 4 w x 2 t 2 - 3 φ x t 2 ) \begin{aligned}\displaystyle(3)&&\displaystyle\quad\frac{\partial\varphi}{% \partial x}&\displaystyle=-\cfrac{m}{\kappa AG}~{}\frac{\partial^{2}w}{% \partial t^{2}}+\frac{\partial^{2}w}{\partial x^{2}}+\cfrac{q}{\kappa AG}\\ \displaystyle(4)&&\displaystyle\quad\frac{\partial^{2}q}{\partial t^{2}}&% \displaystyle=m~{}\cfrac{\partial^{4}w}{\partial t^{4}}-\kappa AG~{}\left(% \cfrac{\partial^{4}w}{\partial x^{2}\partial t^{2}}-\cfrac{\partial^{3}\varphi% }{\partial x\partial t^{2}}\right)\end{aligned}
  90. ( 5 ) 3 φ x 3 = - m κ A G 4 w x 2 t 2 + 4 w x 4 + 1 κ A G 2 q x 2 (5)\qquad\cfrac{\partial^{3}\varphi}{\partial x^{3}}=-\cfrac{m}{\kappa AG}~{}% \cfrac{\partial^{4}w}{\partial x^{2}\partial t^{2}}+\cfrac{\partial^{4}w}{% \partial x^{4}}+\cfrac{1}{\kappa AG}~{}\frac{\partial^{2}q}{\partial x^{2}}
  91. ( 6 ) 3 φ x t 2 = E I J 3 φ x 3 + κ A G J ( 2 w x 2 - φ x ) (6)\qquad\cfrac{\partial^{3}\varphi}{\partial x\partial t^{2}}=\cfrac{EI}{J}~{% }\cfrac{\partial^{3}\varphi}{\partial x^{3}}+\cfrac{\kappa AG}{J}~{}\left(% \frac{\partial^{2}w}{\partial x^{2}}-\frac{\partial\varphi}{\partial x}\right)
  92. ( 7 ) 1 κ A G 2 q t 2 - m κ A G 4 w t 4 + 4 w x 2 t 2 = E I J 3 φ x 3 + κ A G J ( 2 w x 2 - φ x ) (7)\qquad\cfrac{1}{\kappa AG}~{}\frac{\partial^{2}q}{\partial t^{2}}-\cfrac{m}% {\kappa AG}~{}\cfrac{\partial^{4}w}{\partial t^{4}}+\cfrac{\partial^{4}w}{% \partial x^{2}\partial t^{2}}=\cfrac{EI}{J}~{}\cfrac{\partial^{3}\varphi}{% \partial x^{3}}+\cfrac{\kappa AG}{J}~{}\left(\frac{\partial^{2}w}{\partial x^{% 2}}-\frac{\partial\varphi}{\partial x}\right)
  93. ( 8 ) 1 κ A G 2 q t 2 - m κ A G 4 w t 4 + 4 w x 2 t 2 = E I J 3 φ x 3 + m J 2 w t 2 - q J (8)\qquad\cfrac{1}{\kappa AG}~{}\frac{\partial^{2}q}{\partial t^{2}}-\cfrac{m}% {\kappa AG}~{}\cfrac{\partial^{4}w}{\partial t^{4}}+\cfrac{\partial^{4}w}{% \partial x^{2}\partial t^{2}}=\cfrac{EI}{J}~{}\cfrac{\partial^{3}\varphi}{% \partial x^{3}}+\cfrac{m}{J}~{}\frac{\partial^{2}w}{\partial t^{2}}-\cfrac{q}{J}
  94. ( 9 ) J κ A G 2 q t 2 - m J κ A G 4 w t 4 + J 4 w x 2 t 2 = - m E I κ A G 4 w x 2 t 2 + E I 4 w x 4 + E I κ A G 2 q x 2 + m 2 w t 2 - q (9)\qquad\cfrac{J}{\kappa AG}~{}\frac{\partial^{2}q}{\partial t^{2}}-\cfrac{mJ% }{\kappa AG}~{}\cfrac{\partial^{4}w}{\partial t^{4}}+J~{}\cfrac{\partial^{4}w}% {\partial x^{2}\partial t^{2}}=-\cfrac{mEI}{\kappa AG}~{}\cfrac{\partial^{4}w}% {\partial x^{2}\partial t^{2}}+EI~{}\cfrac{\partial^{4}w}{\partial x^{4}}+% \cfrac{EI}{\kappa AG}~{}\frac{\partial^{2}q}{\partial x^{2}}+m~{}\frac{% \partial^{2}w}{\partial t^{2}}-q
  95. E I 4 w x 4 + m 2 w t 2 - ( J + m E I κ A G ) 4 w x 2 t 2 + m J κ A G 4 w t 4 = q + J κ A G 2 q t 2 - E I κ A G 2 q x 2 EI~{}\cfrac{\partial^{4}w}{\partial x^{4}}+m~{}\frac{\partial^{2}w}{\partial t% ^{2}}-\left(J+\cfrac{mEI}{\kappa AG}\right)~{}\cfrac{\partial^{4}w}{\partial x% ^{2}\partial t^{2}}+\cfrac{mJ}{\kappa AG}~{}\cfrac{\partial^{4}w}{\partial t^{% 4}}=q+\cfrac{J}{\kappa AG}~{}\frac{\partial^{2}q}{\partial t^{2}}-\cfrac{EI}{% \kappa AG}~{}\frac{\partial^{2}q}{\partial x^{2}}\quad\square
  96. u x ( x , y , z , t ) = u 0 ( x , t ) - z φ ( x , t ) ; u y ( x , y , z , t ) = 0 ; u z ( x , y , z ) = w ( x , t ) u_{x}(x,y,z,t)=u_{0}(x,t)-z~{}\varphi(x,t)~{};~{}~{}u_{y}(x,y,z,t)=0~{};~{}~{}% u_{z}(x,y,z)=w(x,t)
  97. u 0 u_{0}
  98. x x
  99. m 2 w t 2 = x [ κ A G ( w x - φ ) ] + q ( x , t ) J 2 φ t 2 = N ( x , t ) w x + x ( E I φ x ) + κ A G ( w x - φ ) \begin{aligned}\displaystyle m\frac{\partial^{2}w}{\partial t^{2}}&% \displaystyle=\frac{\partial}{\partial x}\left[\kappa AG\left(\frac{\partial w% }{\partial x}-\varphi\right)\right]+q(x,t)\\ \displaystyle J\frac{\partial^{2}\varphi}{\partial t^{2}}&\displaystyle=N(x,t)% ~{}\frac{\partial w}{\partial x}+\frac{\partial}{\partial x}\left(EI\frac{% \partial\varphi}{\partial x}\right)+\kappa AG\left(\frac{\partial w}{\partial x% }-\varphi\right)\end{aligned}
  100. J = ρ I J=\rho I
  101. N ( x , t ) N(x,t)
  102. N x x ( x , t ) = - h h σ x x d z N_{xx}(x,t)=\int_{-h}^{h}\sigma_{xx}~{}dz
  103. σ x x \sigma_{xx}
  104. 2 h 2h
  105. E I 4 w x 4 + N 2 w x 2 + m 2 w t 2 - ( J + m E I κ A G ) 4 w x 2 t 2 + m J κ A G 4 w t 4 = q + J κ A G 2 q t 2 - E I κ A G 2 q x 2 EI~{}\cfrac{\partial^{4}w}{\partial x^{4}}+N~{}\cfrac{\partial^{2}w}{\partial x% ^{2}}+m~{}\frac{\partial^{2}w}{\partial t^{2}}-\left(J+\cfrac{mEI}{\kappa AG}% \right)~{}\cfrac{\partial^{4}w}{\partial x^{2}\partial t^{2}}+\cfrac{mJ}{% \kappa AG}~{}\cfrac{\partial^{4}w}{\partial t^{4}}=q+\cfrac{J}{\kappa AG}~{}% \frac{\partial^{2}q}{\partial t^{2}}-\cfrac{EI}{\kappa AG}~{}\frac{\partial^{2% }q}{\partial x^{2}}
  106. η ( x ) w t \eta(x)~{}\cfrac{\partial w}{\partial t}
  107. m 2 w t 2 + η ( x ) w t = x [ κ A G ( w x - φ ) ] + q ( x , t ) m\frac{\partial^{2}w}{\partial t^{2}}+\eta(x)~{}\cfrac{\partial w}{\partial t}% =\frac{\partial}{\partial x}\left[\kappa AG\left(\frac{\partial w}{\partial x}% -\varphi\right)\right]+q(x,t)
  108. J 2 φ t 2 = N w x + x ( E I φ x ) + κ A G ( w x - φ ) J\frac{\partial^{2}\varphi}{\partial t^{2}}=N\frac{\partial w}{\partial x}+% \frac{\partial}{\partial x}\left(EI\frac{\partial\varphi}{\partial x}\right)+% \kappa AG\left(\frac{\partial w}{\partial x}-\varphi\right)
  109. E I 4 w x 4 + N 2 w x 2 + m 2 w t 2 - ( J + m E I κ A G ) 4 w x 2 t 2 + m J κ A G 4 w t 4 + J η ( x ) κ A G 3 w t 3 - E I κ A G 2 x 2 ( η ( x ) w t ) + η ( x ) w t = q + J κ A G 2 q t 2 - E I κ A G 2 q x 2 \begin{aligned}\displaystyle EI~{}\cfrac{\partial^{4}w}{\partial x^{4}}&% \displaystyle+N~{}\cfrac{\partial^{2}w}{\partial x^{2}}+m~{}\frac{\partial^{2}% w}{\partial t^{2}}-\left(J+\cfrac{mEI}{\kappa AG}\right)~{}\cfrac{\partial^{4}% w}{\partial x^{2}\partial t^{2}}+\cfrac{mJ}{\kappa AG}~{}\cfrac{\partial^{4}w}% {\partial t^{4}}+\cfrac{J\eta(x)}{\kappa AG}~{}\cfrac{\partial^{3}w}{\partial t% ^{3}}\\ &\displaystyle-\cfrac{EI}{\kappa AG}~{}\cfrac{\partial^{2}}{\partial x^{2}}% \left(\eta(x)\cfrac{\partial w}{\partial t}\right)+\eta(x)\cfrac{\partial w}{% \partial t}=q+\cfrac{J}{\kappa AG}~{}\frac{\partial^{2}q}{\partial t^{2}}-% \cfrac{EI}{\kappa AG}~{}\frac{\partial^{2}q}{\partial x^{2}}\end{aligned}
  110. A τ d A = κ A G φ \int_{A}\tau dA=\kappa AG\varphi\,
  111. κ = 10 ( 1 + ν ) 12 + 11 ν \kappa=\cfrac{10(1+\nu)}{12+11\nu}
  112. κ = 6 ( 1 + ν ) 7 + 6 ν \kappa=\cfrac{6(1+\nu)}{7+6\nu}

Tocopherol_O-methyltransferase.html

  1. \rightleftharpoons

Toda_bracket.html

  1. W f X g Y h Z W\stackrel{f}{\ \to\ }X\stackrel{g}{\ \to\ }Y\stackrel{h}{\ \to\ }Z
  2. π S = k 0 π k S \pi_{\ast}^{S}=\bigoplus_{k\geq 0}\pi_{k}^{S}

Toluene_dioxygenase.html

  1. \rightleftharpoons

Tomlinson_model.html

  1. η = 4 π 2 V 0 k a 2 , \eta=\frac{4\pi^{2}V_{0}}{ka^{2}},

Toric_lens.html

  1. S = n - 1 R + r S=\frac{n-1}{R+r}
  2. s = n - 1 r s=\frac{n-1}{r}

Toroidal_and_poloidal.html

  1. ζ \zeta
  2. ϕ \phi
  3. θ \theta
  4. ψ \psi
  5. χ \chi
  6. ρ \rho
  7. r r
  8. ζ \zeta
  9. θ \theta
  10. x = ( R 0 + r cos θ ) cos ζ x=(R_{0}+r\cos\theta)\cos\zeta\,
  11. y = s ζ ( R 0 + r cos θ ) sin ζ y=s_{\zeta}(R_{0}+r\cos\theta)\sin\zeta\,
  12. z = s θ r sin θ . z=s_{\theta}r\sin\theta.\,
  13. s θ = ± 1 , s ζ = ± 1 s_{\theta}=\pm 1,s_{\zeta}=\pm 1
  14. s θ = s ζ = + 1 s_{\theta}=s_{\zeta}=+1
  15. r , θ , ζ r,\theta,\zeta
  16. r , θ , ζ r,\theta,\zeta
  17. r θ × ζ > 0 \nabla r\cdot\nabla\theta\times\nabla\zeta>0
  18. s θ = - 1 , s ζ = + 1 s_{\theta}=-1,s_{\zeta}=+1
  19. s θ = + 1 , s ζ = - 1 s_{\theta}=+1,s_{\zeta}=-1
  20. s θ = s ζ = + 1 s_{\theta}=s_{\zeta}=+1
  21. ( r , θ , ζ ) \left(r,\theta,\zeta\right)
  22. 𝐞 r = ( cos θ cos ζ cos θ sin ζ sin θ ) 𝐞 θ = ( - sin θ cos ζ - sin θ sin ζ cos θ ) 𝐞 ζ = ( - sin ζ cos ζ 0 ) \mathbf{e}_{r}=\begin{pmatrix}\cos\theta\cos\zeta\\ \cos\theta\sin\zeta\\ \sin\theta\end{pmatrix}\quad\mathbf{e}_{\theta}=\begin{pmatrix}-\sin\theta\cos% \zeta\\ -\sin\theta\sin\zeta\\ \cos\theta\end{pmatrix}\quad\mathbf{e}_{\zeta}=\begin{pmatrix}-\sin\zeta\\ \cos\zeta\\ 0\end{pmatrix}
  23. 𝐫 = ( r + R 0 cos θ ) 𝐞 r - R 0 sin θ 𝐞 θ \mathbf{r}=\left(r+R_{0}\cos\theta\right)\mathbf{e}_{r}-R_{0}\sin\theta\mathbf% {e}_{\theta}
  24. 𝐫 ˙ = r ˙ 𝐞 r + r θ ˙ 𝐞 θ + ζ ˙ ( R 0 + r cos θ ) 𝐞 ζ \mathbf{\dot{r}}=\dot{r}\mathbf{e}_{r}+r\dot{\theta}\mathbf{e}_{\theta}+\dot{% \zeta}\left(R_{0}+r\cos\theta\right)\mathbf{e}_{\zeta}
  25. 𝐫 ¨ = ( r ¨ - r θ ˙ 2 - r ζ ˙ 2 cos 2 θ - R 0 ζ ˙ 2 cos θ ) 𝐞 r \mathbf{\ddot{r}}=\left(\ddot{r}-r\dot{\theta}^{2}-r\dot{\zeta}^{2}\cos^{2}% \theta-R_{0}\dot{\zeta}^{2}\cos\theta\right)\mathbf{e}_{r}
  26. + ( 2 r ˙ θ ˙ + r θ ¨ + r ζ ˙ 2 cos θ sin θ + R 0 ζ ˙ 2 sin θ ) 𝐞 θ +\left(2\dot{r}\dot{\theta}+r\ddot{\theta}+r\dot{\zeta}^{2}\cos\theta\sin% \theta+R_{0}\dot{\zeta}^{2}\sin\theta\right)\mathbf{e}_{\theta}
  27. + ( 2 r ˙ ζ ˙ cos θ - 2 r θ ˙ ζ ˙ sin θ + ζ ¨ ( R 0 + r cos θ ) ) 𝐞 ζ +\left(2\dot{r}\dot{\zeta}\cos\theta-2r\dot{\theta}\dot{\zeta}\sin\theta+\ddot% {\zeta}\left(R_{0}+r\cos\theta\right)\right)\mathbf{e}_{\zeta}

Trans-1,2-dihydrobenzene-1,2-diol_dehydrogenase.html

  1. \rightleftharpoons

Trans-2-decenoyl-(acyl-carrier_protein)_isomerase.html

  1. \rightleftharpoons

Trans-2-enoyl-CoA_reductase_(NAD+).html

  1. \rightleftharpoons

Trans-2-enoyl-CoA_reductase_(NADPH).html

  1. \rightleftharpoons

Trans-acenaphthene-1,2-diol_dehydrogenase.html

  1. \rightleftharpoons

Trans-aconitate_2-methyltransferase.html

  1. \rightleftharpoons

Trans-aconitate_3-methyltransferase.html

  1. \rightleftharpoons

Trans-cinnamate_2-monooxygenase.html

  1. \rightleftharpoons

Trans-cinnamate_4-monooxygenase.html

  1. \rightleftharpoons

Trans-epoxysuccinate_hydrolase.html

  1. \rightleftharpoons

Transmissibility_(structural_dynamics).html

  1. f m a x f_{max}
  2. P 0 P_{0}
  3. T R = f m a x P 0 = R d 1 + ( 2 ζ β ) 2 TR=\frac{f_{max}}{P_{0}}=R_{d}\sqrt{1+(2\zeta\beta)^{2}}
  4. ζ \zeta
  5. β \beta
  6. R d R_{d}

Transmission_delay.html

  1. D T = N / R D_{T}=N/R
  2. D T D_{T}

Transversal_(geometry).html

  1. α \alpha
  2. γ \gamma

Triangulation_(computer_vision).html

  1. 𝐎 1 \mathbf{O}_{1}
  2. 𝐎 2 \mathbf{O}_{2}
  3. 𝐲 1 \mathbf{y}_{1}
  4. 𝐲 2 \mathbf{y}_{2}
  5. 𝐲 1 \mathbf{y}_{1}
  6. 𝐲 2 \mathbf{y}_{2}
  7. 𝐲 1 \mathbf{y}_{1}
  8. 𝐲 2 \mathbf{y}_{2}
  9. 𝐲 1 \mathbf{y}^{\prime}_{1}
  10. 𝐲 2 \mathbf{y}^{\prime}_{2}
  11. 𝐲 1 \mathbf{y}_{1}
  12. 𝐲 2 \mathbf{y}_{2}
  13. 𝐲 1 \mathbf{y}^{\prime}_{1}
  14. 𝐲 2 \mathbf{y}^{\prime}_{2}
  15. 𝐲 1 \mathbf{y}^{\prime}_{1}
  16. 𝐲 2 \mathbf{y}^{\prime}_{2}
  17. 𝐲 1 \mathbf{y}^{\prime}_{1}
  18. 𝐲 2 \mathbf{y}^{\prime}_{2}
  19. 𝐲 1 = 𝐲 1 \mathbf{y}_{1}=\mathbf{y}^{\prime}_{1}
  20. 𝐲 2 = 𝐲 2 \mathbf{y}_{2}=\mathbf{y}^{\prime}_{2}
  21. τ \tau\,
  22. 𝐱 τ ( 𝐲 1 , 𝐲 2 , 𝐂 1 , 𝐂 2 ) \mathbf{x}\sim\tau(\mathbf{y}^{\prime}_{1},\mathbf{y}^{\prime}_{2},\mathbf{C}_% {1},\mathbf{C}_{2})
  23. 𝐲 1 , 𝐲 2 \mathbf{y}^{\prime}_{1},\mathbf{y}^{\prime}_{2}
  24. 𝐂 1 , 𝐂 2 \mathbf{C}_{1},\mathbf{C}_{2}
  25. \sim\,
  26. τ \tau\,
  27. τ \tau\,
  28. 𝐲 1 , 𝐲 2 , 𝐂 1 , 𝐂 2 \mathbf{y}^{\prime}_{1},\mathbf{y}^{\prime}_{2},\mathbf{C}_{1},\mathbf{C}_{2}
  29. 4 × 4 4\times 4
  30. 𝐓 \mathbf{T}
  31. 𝐱 ¯ 𝐓 𝐱 \mathbf{\bar{x}}\sim\mathbf{T}\,\mathbf{x}
  32. 𝐂 ¯ k 𝐂 k 𝐓 - 1 \mathbf{\bar{C}}_{k}\sim\mathbf{C}_{k}\,\mathbf{T}^{-1}
  33. 𝐲 k 𝐂 ¯ k 𝐱 ¯ = 𝐂 k 𝐱 \mathbf{y}_{k}\sim\mathbf{\bar{C}}_{k}\,\mathbf{\bar{x}}=\mathbf{C}_{k}\,% \mathbf{x}
  34. τ \tau
  35. 𝐓 \mathbf{T}
  36. 𝐱 ¯ est 𝐓 𝐱 est \mathbf{\bar{x}}_{\rm est}\sim\mathbf{T}\,\mathbf{x}_{\rm est}
  37. τ ( 𝐲 1 , 𝐲 2 , 𝐂 1 , 𝐂 2 ) 𝐓 - 1 τ ( 𝐲 1 , 𝐲 2 , 𝐂 1 𝐓 - 1 , 𝐂 2 𝐓 - 1 ) , \tau(\mathbf{y}^{\prime}_{1},\mathbf{y}^{\prime}_{2},\mathbf{C}_{1},\mathbf{C}% _{2})\sim\mathbf{T}^{-1}\,\tau(\mathbf{y}^{\prime}_{1},\mathbf{y}^{\prime}_{2}% ,\mathbf{C}_{1}\,\mathbf{T}^{-1},\mathbf{C}_{2}\,\mathbf{T}^{-1}),
  38. 𝐲 1 , 𝐲 2 \mathbf{y}^{\prime}_{1},\mathbf{y}^{\prime}_{2}
  39. τ \tau
  40. τ \tau
  41. τ \tau
  42. 𝐲 1 \mathbf{y}^{\prime}_{1}
  43. 𝐲 2 \mathbf{y}^{\prime}_{2}
  44. 𝐋 1 \mathbf{L}^{\prime}_{1}
  45. 𝐋 2 \mathbf{L}^{\prime}_{2}
  46. 𝐂 1 , 𝐂 2 \mathbf{C}_{1},\mathbf{C}_{2}
  47. d d\,
  48. d ( 𝐋 , 𝐱 ) = d(\mathbf{L},\mathbf{x})=
  49. 𝐋 \mathbf{L}
  50. 𝐱 \mathbf{x}
  51. d ( 𝐋 1 , 𝐱 ) 2 + d ( 𝐋 2 , 𝐱 ) 2 d(\mathbf{L}^{\prime}_{1},\mathbf{x})^{2}+d(\mathbf{L}^{\prime}_{2},\mathbf{x}% )^{2}

Triglucosylalkylacylglycerol_sulfotransferase.html

  1. \rightleftharpoons

Trimethylamine-N-oxide_reductase.html

  1. \rightleftharpoons

Trimethylamine-N-oxide_reductase_(cytochrome_c).html

  1. \rightleftharpoons

Trimethylamine_dehydrogenase.html

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Trimethyllysine_dioxygenase.html

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Trimethylsulfonium—tetrahydrofolate_N-methyltransferase.html

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Trithionate_hydrolase.html

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TRNA-pseudouridine_synthase_I.html

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TRNA_(5-methylaminomethyl-2-thiouridylate)-methyltransferase.html

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TRNA_(adenine-N1-)-methyltransferase.html

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TRNA_(adenine-N6-)-methyltransferase.html

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TRNA_(cytosine-5-)-methyltransferase.html

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TRNA_(guanine-N1-)-methyltransferase.html

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TRNA_(guanine-N2-)-methyltransferase.html

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TRNA_(guanine-N7-)-methyltransferase.html

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TRNA_(uracil-5-)-methyltransferase.html

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TRNA_guanosine-2'-O-methyltransferase.html

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TRNA_sulfurtransferase.html

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Tropinone_reductase_I.html

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Tropinone_reductase_II.html

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Trypanothione-disulfide_reductase.html

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Tryptophan_2'-dioxygenase.html

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Tryptophan_2,3-dioxygenase.html

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Tryptophan_2-C-methyltransferase.html

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Tryptophan_2-monooxygenase.html

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Tryptophan_alpha,beta-oxidase.html

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Tryptophan_dehydrogenase.html

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Tutte–Berge_formula.html

  1. 1 2 min U V ( | U | - odd ( G - U ) + | V | ) , \frac{1}{2}\min_{U\subseteq V}\left(|U|-\,\text{odd}(G-U)+|V|\right),

Tyramine_N-methyltransferase.html

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Tyrosine-ester_sulfotransferase.html

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Tyrosine_2,3-aminomutase.html

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U-quadratic_distribution.html

  1. f ( x | a , b , α , β ) = α ( x - β ) 2 , for x [ a , b ] . f(x|a,b,\alpha,\beta)=\alpha\left(x-\beta\right)^{2},\quad\,\text{for }x\in[a,% b].
  2. β = b + a 2 \beta={b+a\over 2}
  3. α = 12 ( b - a ) 3 \alpha={12\over\left(b-a\right)^{3}}
  4. { ( - a - b + 2 x ) f ( x ) - 4 f ( x ) = 0 , f ( 0 ) = - 3 ( a + b ) 2 ( a - b ) 3 } \left\{(-a-b+2x)f^{\prime}(x)-4f(x)=0,f(0)=-\frac{3(a+b)^{2}}{(a-b)^{3}}\right\}
  5. { ( x - β ) f ( x ) - 2 f ( x ) = 0 , f ( 0 ) = α β 2 } \left\{(x-\beta)f^{\prime}(x)-2f(x)=0,f(0)=\alpha\beta^{2}\right\}
  6. \cap
  7. M X ( t ) = - 3 ( e a t ( 4 + ( a 2 + 2 a ( - 2 + b ) + b 2 ) t ) - e b t ( 4 + ( - 4 b + ( a + b ) 2 ) t ) ) ( a - b ) 3 t 2 M_{X}(t)={-3\left(e^{at}(4+(a^{2}+2a(-2+b)+b^{2})t)-e^{bt}(4+(-4b+(a+b)^{2})t)% \right)\over(a-b)^{3}t^{2}}
  8. ϕ X ( t ) = 3 i ( e i a t e i b t ( 4 i - ( - 4 b + ( a + b ) 2 ) t ) ) ( a - b ) 3 t 2 \phi_{X}(t)={3i\left(e^{iate^{ibt}}(4i-(-4b+(a+b)^{2})t)\right)\over(a-b)^{3}t% ^{2}}

UDP-arabinose_4-epimerase.html

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UDP-galactopyranose_mutase.html

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UDP-glucosamine_4-epimerase.html

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UDP-glucose_6-dehydrogenase.html

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UDP-glucuronate_4-epimerase.html

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UDP-glucuronate_5'-epimerase.html

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UDP-N-acetylgalactosamine-4-sulfate_sulfotransferase.html

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UDP-N-acetylglucosamine_2-epimerase.html

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UDP-N-acetylglucosamine_4-epimerase.html

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UDP-N-acetylglucosamine_6-dehydrogenase.html

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UDP-N-acetylmuramate_dehydrogenase.html

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UDP-sulfoquinovose_synthase.html

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UNIFAC.html

  1. a i a_{i}
  2. γ c \gamma^{c}
  3. γ r \gamma^{r}
  4. i th i^{\mathrm{th}}
  5. ln γ i = ln γ i c + ln γ i r \ln\gamma_{i}=\ln\gamma_{i}^{c}+\ln\gamma_{i}^{r}
  6. R R
  7. Q Q
  8. τ i j \tau_{ij}
  9. U i U_{i}
  10. ln γ i c = ln ϕ i x i + z 2 q i ln θ i ϕ i + L i - ϕ i x i j = 1 n x j L j \ln\gamma_{i}^{c}=\ln\frac{\phi_{i}}{x_{i}}+\frac{z}{2}q_{i}\ln\frac{\theta_{i% }}{\phi_{i}}+L_{i}-\frac{\phi_{i}}{x_{i}}\displaystyle\sum_{j=1}^{n}x_{j}L_{j}
  11. θ i \theta_{i}
  12. ϕ i \phi_{i}
  13. i th i^{\mathrm{th}}
  14. L i L_{i}
  15. r r
  16. z z
  17. q q
  18. z z
  19. θ i = x i q i j = 1 n x j q j ; ϕ i = x i r i j = 1 n x j r j ; L i = z 2 ( r i - q i ) - ( r i - 1 ) ; z = 10 \theta_{i}=\frac{x_{i}q_{i}}{\displaystyle\sum_{j=1}^{n}x_{j}q_{j}}\mathrm{\,% \,;\,\,}\phi_{i}=\frac{x_{i}r_{i}}{\displaystyle\sum_{j=1}^{n}x_{j}r_{j}}% \mathrm{\,\,;\,\,}L_{i}=\frac{z}{2}(r_{i}-q_{i})-(r_{i}-1)\mathrm{\,\,;\,\,}z=10
  20. q i q_{i}
  21. r i r_{i}
  22. Q Q
  23. R R
  24. ν k \nu_{k}
  25. r i = k = 1 n ν k R k ; q i = k = 1 n ν k Q k r_{i}=\displaystyle\sum_{k=1}^{n}\nu_{k}R_{k}\mathrm{\,\,;\,\,}q_{i}=% \displaystyle\sum_{k=1}^{n}\nu_{k}Q_{k}
  26. γ r \gamma^{r}
  27. i t h i^{th}
  28. n n
  29. ln γ i r = k n ν k ( i ) [ ln Γ k - ln Γ k ( i ) ] \ln\gamma_{i}^{r}=\displaystyle\sum_{k}^{n}\nu_{k}^{(i)}\left[\ln\Gamma_{k}-% \ln\Gamma_{k}^{(i)}\right]
  30. Γ k ( i ) \Gamma_{k}^{(i)}
  31. i i
  32. Γ k ( i ) \Gamma_{k}^{(i)}
  33. ln Γ k - ln Γ k ( i ) \ln\Gamma_{k}-\ln\Gamma_{k}^{(i)}
  34. Γ k \Gamma_{k}
  35. Γ k ( i ) \Gamma_{k}^{(i)}
  36. ln Γ k = Q k [ 1 - ln m Θ m Ψ m k - m Θ m Ψ k m n Θ n Ψ n m ] \ln\Gamma_{k}=Q_{k}\left[1-\ln\displaystyle\sum_{m}\Theta_{m}\Psi_{mk}-% \displaystyle\sum_{m}\frac{\Theta_{m}\Psi_{km}}{\displaystyle\sum_{n}\Theta_{n% }\Psi_{nm}}\right]
  37. Θ m \Theta_{m}
  38. m m
  39. θ i \theta_{i}
  40. Ψ m n \Psi_{mn}
  41. X n X_{n}
  42. n n
  43. Θ m = Q m X m n Q n X n \Theta_{m}=\frac{Q_{m}X_{m}}{\displaystyle\sum_{n}Q_{n}X_{n}}
  44. Ψ m n = exp [ - U m n - U n m R T ] ; \Psi_{mn}=\mathrm{exp}\left[-\frac{U_{mn}-U_{nm}}{RT}\right]\mathrm{\,;\,\,}
  45. X m = j ν m j x j j n ν n j x j X_{m}=\frac{\displaystyle\sum_{j}\nu^{j}_{m}x_{j}}{\displaystyle\sum_{j}% \displaystyle\sum_{n}\nu_{n}^{j}x_{j}}
  46. U m n U_{mn}
  47. U m n = U n m U_{mn}=U_{nm}
  48. Ψ m n = exp - a m n T \Psi_{mn}=\mathrm{exp}\frac{-a_{mn}}{T}
  49. a m n a_{mn}
  50. m m
  51. n n

Unistochastic_matrix.html

  1. B i j = | U i j | 2 for i , j = 1 , , n . B_{ij}=|U_{ij}|^{2}\,\text{ for }i,j=1,\dots,n.\,
  2. n = 3 n=3
  3. B = 1 2 [ 1 1 0 0 1 1 1 0 1 ] B=\frac{1}{2}\begin{bmatrix}1&1&0\\ 0&1&1\\ 1&0&1\end{bmatrix}
  4. n 3 n\geq 3
  5. n = 3 n=3
  6. n = 3 n=3
  7. 8 π 2 / 105 75.2 % 8\pi^{2}/105\approx 75.2\%

Universal_conductance_fluctuations.html

  1. l ϕ \textstyle l_{\phi}
  2. l m \textstyle l_{m}
  3. l ϕ < l c \textstyle l_{\phi}<l_{c}
  4. l c = M × l m l_{c}=M\times l_{m}
  5. M \textstyle M
  6. l m \textstyle l_{m}
  7. G o = 2 e 2 / h \textstyle G_{o}=2e^{2}/h

Universal_parabolic_constant.html

  1. P = ln ( 1 + 2 ) + 2 = 2.29558714939 P=\ln(1+\sqrt{2})+\sqrt{2}=2.29558714939\dots
  2. y = x 2 4 f y=\frac{x^{2}}{4f}
  3. p = 2 f p=2f
  4. = 2 f \ell=2f
  5. P : = 1 p - 1 + ( d y d x ) 2 d x = 1 2 f - 2 f 2 f 1 + x 2 4 f 2 d x = - 1 1 1 + t 2 d t ( x = 2 f t ) = arcsinh ( 1 ) + 2 = ln ( 1 + 2 ) + 2 . \begin{aligned}\displaystyle P&\displaystyle:=\frac{1}{p}\int_{-\ell}^{\ell}% \sqrt{1+\left(\frac{dy}{dx}\right)^{2}}\,dx\\ &\displaystyle=\frac{1}{2f}\int_{-2f}^{2f}\sqrt{1+\frac{x^{2}}{4f^{2}}}\,dx\\ &\displaystyle=\int_{-1}^{1}\sqrt{1+t^{2}}\,dt\quad(x=2ft)\\ &\displaystyle=\operatorname{arcsinh}(1)+\sqrt{2}\\ &\displaystyle=\ln(1+\sqrt{2})+\sqrt{2}.\end{aligned}
  6. P - 2 = ln ( 1 + 2 ) \!\ P-\sqrt{2}=\ln(1+\sqrt{2})
  7. e ln ( 1 + 2 ) = 1 + 2 \!\ e^{\ln(1+\sqrt{2})}=1+\sqrt{2}
  8. d avg = P 6 . d\text{avg}={P\over 6}.
  9. d avg : = 8 0 1 2 0 x x 2 + y 2 d y d x = 8 0 1 2 1 2 x 2 ( ln ( 1 + 2 ) + 2 ) d x = 4 P 0 1 2 x 2 d x = P 6 . \begin{aligned}\displaystyle d\text{avg}&\displaystyle:=8\int_{0}^{1\over 2}% \int_{0}^{x}\sqrt{x^{2}+y^{2}}\,dy\,dx\\ &\displaystyle=8\int_{0}^{1\over 2}{1\over 2}x^{2}(\ln(1+\sqrt{2})+\sqrt{2})\,% dx\\ &\displaystyle=4P\int_{0}^{1\over 2}x^{2}\,dx\\ &\displaystyle={P\over 6}.\end{aligned}

Unspecific_monooxygenase.html

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Ureidoglycolate_dehydrogenase.html

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Ureidoglycolate_lyase.html

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Uronate_dehydrogenase.html

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Vackář_oscillator.html

  1. C v C 0 , C_{v}\ll C_{0},
  2. C g C v , C_{g}\gg C_{v},
  3. C a C 0 , C_{a}\gg C_{0},

Valine_dehydrogenase_(NADP+).html

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Van_der_Waals_surface.html

  1. A w = 4 π r w 2 A_{\rm w}=4\pi r_{\rm w}^{2}
  2. V w = 4 3 π r w 3 V_{\rm w}={4\over 3}\pi r_{\rm w}^{3}

Vanillate_monooxygenase.html

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Vanillin_dehydrogenase.html

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Vanillyl-alcohol_oxidase.html

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Vantieghems_theorem.html

  1. 1 k n - 1 ( 2 k - 1 ) n mod ( 2 n - 1 ) . \prod_{1\leq k\leq n-1}\left(2^{k}-1\right)\equiv n\mod\left(2^{n}-1\right).
  2. 1 k n - 1 ( X k - 1 ) n - ( X n - 1 ) / ( X - 1 ) mod ( X n - 1 ) \prod_{1\leq k\leq n-1}\left(X^{k}-1\right)\equiv n-\left(X^{n}-1\right)/\left% (X-1\right)\mod\left(X^{n}-1\right)
  3. 1 k n - 1 ( X k - 1 ) n mod ( X n - 1 ) / ( X - 1 ) . \prod_{1\leq k\leq n-1}\left(X^{k}-1\right)\equiv n\mod\left(X^{n}-1\right)/% \left(X-1\right).

Varadhan's_lemma.html

  1. lim M lim sup ε 0 ε log 𝐄 [ exp ( ϕ ( Z ε ) / ε ) 𝟏 ( ϕ ( Z ε ) M ) ] = - , \lim_{M\to\infty}\limsup_{\varepsilon\to 0}\varepsilon\log\mathbf{E}\big[\exp% \big(\phi(Z_{\varepsilon})/\varepsilon\big)\mathbf{1}\big(\phi(Z_{\varepsilon}% )\geq M\big)\big]=-\infty,
  2. lim sup ε 0 ε log 𝐄 [ exp ( γ ϕ ( Z ε ) / ε ) ] < + . \limsup_{\varepsilon\to 0}\varepsilon\log\mathbf{E}\big[\exp\big(\gamma\phi(Z_% {\varepsilon})/\varepsilon\big)\big]<+\infty.
  3. lim ε 0 ε log 𝐄 [ exp ( ϕ ( Z ε ) / ε ) ] = sup x X ( ϕ ( x ) - I ( x ) ) . \lim_{\varepsilon\to 0}\varepsilon\log\mathbf{E}\big[\exp\big(\phi(Z_{% \varepsilon})/\varepsilon\big)\big]=\sup_{x\in X}\big(\phi(x)-I(x)\big).

Variance_inflation_factor.html

  1. σ 2 \sigma^{2}
  2. var ^ ( β ^ j ) = s 2 ( n - 1 ) var ^ ( X j ) 1 1 - R j 2 , {\rm\widehat{var}}(\hat{\beta}_{j})=\frac{s^{2}}{(n-1)\widehat{\rm var}(X_{j})% }\cdot\frac{1}{1-R_{j}^{2}},
  3. var ^ ( X j ) \widehat{\rm var}(X_{j})
  4. X 1 = α 2 X 2 + α 3 X 3 + + α k X k + c 0 + e X_{1}=\alpha_{2}X_{2}+\alpha_{3}X_{3}+\cdots+\alpha_{k}X_{k}+c_{0}+e
  5. β ^ i \hat{\beta}_{i}
  6. VIF = 1 1 - R i 2 \mathrm{VIF}=\frac{1}{1-R^{2}_{i}}
  7. X i X_{i}
  8. VIF ( β ^ i ) \operatorname{VIF}(\hat{\beta}_{i})
  9. VIF ( β ^ i ) > 10 \operatorname{VIF}(\hat{\beta}_{i})>10

Vaught_conjecture.html

  1. T T
  2. I ( T , α ) I(T,\alpha)
  3. α \alpha
  4. T T
  5. 0 < I ( T , 0 ) < 2 0 \aleph_{0}<I(T,\aleph_{0})<2^{\aleph_{0}}

Vellosimine_dehydrogenase.html

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Vinorine_hydroxylase.html

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Vinylacetyl-CoA_Delta-isomerase.html

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Virtual_black_hole.html

  1. Δ R μ Δ x μ P 2 = G c 3 \Delta R_{\mu}\Delta x_{\mu}\geq\ell^{2}_{P}=\frac{\hbar G}{c^{3}}
  2. R μ R_{\mu}
  3. x μ x_{\mu}
  4. P \ell_{P}
  5. \hbar
  6. G G
  7. c c
  8. R ^ μ \hat{R}_{\mu}
  9. x ^ μ \hat{x}_{\mu}
  10. [ R ^ μ , x ^ μ ] = - 2 i P 2 [\hat{R}_{\mu},\hat{x}_{\mu}]=-2i\ell^{2}_{P}
  11. R μ = 2 G c 3 m c U μ R_{\mu}=\frac{2G}{c^{3}}m\,c\,U_{\mu}
  12. P 2 = G c 3 \ell^{2}_{P}=\frac{\hbar\,G}{c^{3}}
  13. Δ P μ Δ x μ = Δ ( m c U μ ) Δ x μ 2 \Delta P_{\mu}\Delta x_{\mu}=\Delta(mc\,U_{\mu})\Delta x_{\mu}\geq\frac{\hbar}% {2}
  14. U 0 = 1 , U i = 0 ( i = 1 , 2 , 3 ) U_{0}=1,U_{i}=0\,(i=1,2,3)
  15. Δ R 0 Δ x 0 = Δ r s Δ r P 2 \Delta R_{0}\Delta x_{0}=\Delta r_{s}\Delta r\geq\ell^{2}_{P}
  16. r s r_{s}
  17. r r
  18. d S 2 dS^{2}
  19. d S 2 = ( 1 - r s r ) c 2 d t 2 - d r 2 1 - r s / r - r 2 ( d Ω 2 + sin 2 Ω d φ 2 ) dS^{2}=\left(1-\frac{r_{s}}{r}\right)c^{2}dt^{2}-\frac{dr^{2}}{1-{r_{s}}/{r}}-% r^{2}(d\Omega^{2}+\sin^{2}\Omega d\varphi^{2})
  20. r s P 2 / r r_{s}\approx\ell^{2}_{P}/r
  21. d S 2 = ( 1 - P 2 r 2 ) c 2 d t 2 - d r 2 1 - P 2 / r 2 - r 2 ( d Ω 2 + sin 2 Ω d φ 2 ) dS^{2}=\left(1-\frac{\ell^{2}_{P}}{r^{2}}\right)c^{2}dt^{2}-\frac{dr^{2}}{1-{% \ell^{2}_{P}}/{r^{2}}}-r^{2}(d\Omega^{2}+\sin^{2}\Omega d\varphi^{2})
  22. r = P r=\ell_{P}

Vitali_convergence_theorem.html

  1. ( X , , μ ) (X,\mathcal{F},\mu)
  2. μ ( X ) < \mu(X)<\infty
  3. { f n } \{f_{n}\}
  4. f n ( x ) f ( x ) f_{n}(x)\to f(x)
  5. n n\to\infty
  6. | f ( x ) | < |f(x)|<\infty
  7. f 1 ( μ ) f\in\mathcal{L}^{1}(\mu)
  8. lim n X | f n - f | d μ = 0 \lim_{n\to\infty}\int_{X}|f_{n}-f|d\mu=0
  9. X | f | d μ lim inf n X | f n | d μ \int_{X}|f|d\mu\leq\liminf_{n\to\infty}\int_{X}|f_{n}|d\mu
  10. E | f n | d μ < 1 \int_{E}|f_{n}|d\mu<1
  11. E E
  12. μ ( E ) < δ \mu(E)<\delta
  13. f n {f_{n}}
  14. E C E^{C}
  15. E C | f n - f p | d μ < 1 \int_{E^{C}}|f_{n}-f_{p}|d\mu<1
  16. p p
  17. n > p \forall n>p
  18. E C | f n | d μ E C | f p | d μ + 1 = M \int_{E^{C}}|f_{n}|d\mu\leq\int_{E^{C}}|f_{p}|d\mu+1=M
  19. X | f - f n | d μ E | f | d μ + E | f n | d μ + E C | f - f n | d μ \int_{X}|f-f_{n}|d\mu\leq\int_{E}|f|d\mu+\int_{E}|f_{n}|d\mu+\int_{E^{C}}|f-f_% {n}|d\mu
  20. E X E\in X
  21. μ ( E ) < δ \mu(E)<\delta
  22. f n f_{n}
  23. n > N n>N
  24. ( X , , μ ) (X,\mathcal{F},\mu)
  25. μ ( X ) < \mu(X)<\infty
  26. f n 1 ( μ ) f_{n}\in\mathcal{L}^{1}(\mu)
  27. lim n E f n d μ \lim_{n\to\infty}\int_{E}f_{n}d\mu
  28. E E\in\mathcal{F}
  29. { f n } \{f_{n}\}

Vitamin-K-epoxide_reductase_(warfarin-insensitive).html

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Vitamin-K-epoxide_reductase_(warfarin-sensitive).html

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Volume_hologram.html

  1. Δ λ \Delta\lambda
  2. Δ λ / λ Λ / L \Delta\lambda/\lambda\approx\Lambda/L
  3. λ \lambda
  4. Λ \Lambda
  5. L L
  6. Λ = λ / ( 2 n ) \Lambda=\lambda/(2n)
  7. n n
  8. λ = 500 nm , L = 1 cm , n = 1.5 \lambda=500\mbox{ nm}~{},\,L=1\mbox{ cm}~{},\,n=1.5
  9. Δ λ / λ 10 - 5 \Delta\lambda/\lambda\approx 10^{-5}
  10. Δ Θ \Delta\Theta
  11. Δ Θ Λ / d \Delta\Theta\approx\Lambda/d
  12. d d
  13. Λ \Lambda
  14. Λ = ( λ / 2 sin Θ \Lambda=(\lambda/2\sin\Theta
  15. λ = 500 nm , d = 1 cm , Θ = 45 \lambda=500\mbox{ nm}~{},\,d=1\mbox{ cm}~{},\,\Theta=45^{\circ}
  16. Δ Θ 4 × 10 - 5 rad = 0.002 \Delta\Theta\approx 4\times 10^{-5}\mbox{ rad}~{}=0.002^{\circ}

Volume_viscosity.html

  1. - 1 3 T a a = p , -{1\over 3}T_{a}^{a}=p,
  2. ρ ( 𝐯 t + 𝐯 𝐯 ) = - p + μ 2 𝐯 + 𝐟 + ( 1 3 μ + ζ ) ( 𝐯 ) \rho\left(\frac{\partial\mathbf{v}}{\partial t}+\mathbf{v}\cdot\nabla\mathbf{v% }\right)=-\nabla p+\mu\nabla^{2}\mathbf{v}+\mathbf{f}+(\frac{1}{3}\mu+\zeta)% \nabla(\nabla\cdot\mathbf{v})
  3. ζ \zeta

Vomifoliol_dehydrogenase.html

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Vomilenine_reductase.html

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Waldhausen_category.html

  1. 0 A \scriptstyle 0\,\rightarrowtail\,A
  2. A B \scriptstyle A\,\rightarrowtail\,B
  3. A C \scriptstyle A\,\to\,C
  4. B A C \scriptstyle B\,\cup_{A}\,C
  5. C B A C \scriptstyle C\,\rightarrowtail\,B\,\cup_{A}\,C
  6. X X
  7. 0 X 0\to X
  8. C b ( 𝒜 ) \scriptstyle C^{b}(\mathcal{A})
  9. 𝒜 \scriptstyle\mathcal{A}
  10. S n 𝒞 \scriptstyle S_{n}\mathcal{C}
  11. A r ( Δ n ) 𝒞 \scriptstyle Ar(\Delta^{n})\,\to\,\mathcal{C}
  12. 𝒞 \scriptstyle\mathcal{C}
  13. I \scriptstyle I
  14. 𝒞 I \scriptstyle\mathcal{C}^{I}
  15. 𝒞 \scriptstyle\mathcal{C}

Ward_Leonard_control.html

  1. W i W_{i}
  2. K K
  3. t t
  4. J J
  5. D D
  6. G G
  7. s s
  8. V g f = R g f I g f + L g f I g f V_{g}^{f}=R_{g}^{f}I_{g}^{f}+L_{g}^{f}I_{g}^{f}
  9. - G g f a I g f W g r + ( R g a + R m a ) I a + ( L g a + L m a ) I a + G m f a I m f W m r = 0 -G_{g}^{f}aI_{g}^{f}W_{g}^{r}+(R_{g}^{a}+R_{m}^{a})I^{a}+(L_{g}^{a}+L_{m}^{a})% I^{a}+G_{m}^{f}aI_{m}^{f}W_{m}^{r}=0
  10. - T L = J m W m r + D m W m r -T_{L}=J_{m}W_{m}^{r}+D_{m}W_{m}^{r}
  11. L g a + L m a L_{g}^{a}+L_{m}^{a}
  12. T L = 0 T_{L}=0
  13. W m r ( S ) V g f ( S ) = K B K v / D m ( t g f s + 1 ) ( t m s + K m D m ) \frac{W_{m}^{r}(S)}{V_{g}^{f}(S)}=\cfrac{K_{B}K_{v}/D_{m}}{\left(t_{g}^{f}s+1% \right)\left(t_{m}s+\frac{K_{m}}{D_{m}}\right)}
  14. K B = G m f a V m f R m f ( R g a + R m a ) K_{B}=\frac{G_{m}^{f}aV_{m}^{f}}{R_{m}^{f}(R_{g}^{a}+R_{m}^{a})}
  15. K v = G g f a W g r R g f K_{v}=\frac{G_{g}^{f}aW_{g}^{r}}{R_{g}^{f}}
  16. t m = J m D m t_{m}=\frac{J_{m}}{D_{m}}
  17. t g f = L g f R g f t_{g}^{f}=\frac{L_{g}^{f}}{R_{g}^{f}}
  18. K m = D m + K B 2 ( R g a + R m a ) K_{m}=D_{m}+K_{B}^{2}(R_{g}^{a}+R_{m}^{a})

Watasenia-luciferin_2-monooxygenase.html

  1. \rightleftharpoons

Weierstrass_p.html

  1. \wp

Werner_Kuhn.html

  1. S = k log W S=k\log W\!

Whipple_formulae.html

  1. P - μ - 1 2 - ν - 1 2 ( z z 2 - 1 ) = ( z 2 - 1 ) 1 / 4 e - i μ π Q ν μ ( z ) ( π / 2 ) 1 / 2 Γ ( ν + μ + 1 ) P_{-\mu-\frac{1}{2}}^{-\nu-\frac{1}{2}}\biggl(\frac{z}{\sqrt{z^{2}-1}}\biggr)=% \frac{(z^{2}-1)^{1/4}e^{-i\mu\pi}Q_{\nu}^{\mu}(z)}{(\pi/2)^{1/2}\Gamma(\nu+\mu% +1)}
  2. Q - μ - 1 2 - ν - 1 2 ( z z 2 - 1 ) = - i ( π / 2 ) 1 / 2 Γ ( - ν - μ ) ( z 2 - 1 ) 1 / 4 e - i ν π P ν μ ( z ) . Q_{-\mu-\frac{1}{2}}^{-\nu-\frac{1}{2}}\biggl(\frac{z}{\sqrt{z^{2}-1}}\biggr)=% -i(\pi/2)^{1/2}\Gamma(-\nu-\mu)(z^{2}-1)^{1/4}e^{-i\nu\pi}P_{\nu}^{\mu}(z).
  3. ν , μ , \nu,\mu,
  4. z z
  5. P ν - 1 2 μ ( z ) = 2 Γ ( μ - ν + 1 2 ) π 3 / 2 ( z 2 - 1 ) 1 / 4 [ π sin μ π P μ - 1 2 ν ( z z 2 - 1 ) + cos π ( ν + μ ) e - i ν π Q μ - 1 2 ν ( z z 2 - 1 ) ] P_{\nu-\frac{1}{2}}^{\mu}(z)=\frac{\sqrt{2}\Gamma(\mu-\nu+\frac{1}{2})}{\pi^{3% /2}(z^{2}-1)^{1/4}}\biggl[\pi\sin\mu\pi P_{\mu-\frac{1}{2}}^{\nu}\biggl(\frac{% z}{\sqrt{z^{2}-1}}\biggr)+\cos\pi(\nu+\mu)e^{-i\nu\pi}Q_{\mu-\frac{1}{2}}^{\nu% }\biggl(\frac{z}{\sqrt{z^{2}-1}}\biggr)\biggr]
  6. Q ν - 1 2 μ ( z ) = e i μ π Γ ( μ - ν + 1 2 ) ( π / 2 ) 1 / 2 ( z 2 - 1 ) 1 / 4 [ P μ - 1 2 ν ( z z 2 - 1 ) - 2 π e - i ν π sin ν π Q μ - 1 2 ν ( z z 2 - 1 ) ] . Q_{\nu-\frac{1}{2}}^{\mu}(z)=\frac{e^{i\mu\pi}\Gamma(\mu-\nu+\frac{1}{2})(\pi/% 2)^{1/2}}{(z^{2}-1)^{1/4}}\biggl[P_{\mu-\frac{1}{2}}^{\nu}\biggl(\frac{z}{% \sqrt{z^{2}-1}}\biggr)-\frac{2}{\pi}e^{-i\nu\pi}\sin\nu\pi Q_{\mu-\frac{1}{2}}% ^{\nu}\biggl(\frac{z}{\sqrt{z^{2}-1}}\biggr)\biggr].
  7. P m - 1 2 n ( cosh η ) = ( - 1 ) m Γ ( m - n + 1 2 ) 2 π sinh η Q n - 1 2 m ( coth η ) P_{m-\frac{1}{2}}^{n}(\cosh\eta)=\frac{(-1)^{m}}{\Gamma(m-n+\frac{1}{2})}\sqrt% {\frac{2}{\pi\sinh\eta}}Q_{n-\frac{1}{2}}^{m}(\coth\eta)
  8. Q m - 1 2 n ( cosh η ) = ( - 1 ) m π Γ ( m - n + 1 2 ) π 2 sinh η P n - 1 2 m ( coth η ) Q_{m-\frac{1}{2}}^{n}(\cosh\eta)=\frac{(-1)^{m}\pi}{\Gamma(m-n+\frac{1}{2})}% \sqrt{\frac{\pi}{2\sinh\eta}}P_{n-\frac{1}{2}}^{m}(\coth\eta)

Whole_body_counting.html

  1. M D A = 2.707 + 4.65 N E T MDA=\frac{2.707+4.65\sqrt{N}}{ET}

Wiener_index.html

  1. 3 × 1 + 2 × 2 + 1 × 3 = 10. 3\times 1+2\times 2+1\times 3=10.
  2. 3 × 1 + 3 × 2 = 9. 3\times 1+3\times 2=9.
  3. ( n 3 - n ) / 6 (n^{3}-n)/6
  4. n n
  5. ( n - 1 ) 2 (n-1)^{2}
  6. n n

Wigner–Seitz_radius.html

  1. r s r_{s}
  2. N N
  3. V V
  4. 4 3 π r s 3 = V N . \frac{4}{3}\pi r_{s}^{3}=\frac{V}{N}.
  5. r s r_{s}
  6. r s = ( 3 4 π n ) 1 / 3 , r_{s}=\left(\frac{3}{4\pi n}\right)^{1/3}\,,
  7. n n
  8. 2 r s 2r_{s}
  9. r s = ( 3 M 4 π ρ N A ) 1 3 , r_{s}=\left(\frac{3M}{4\pi\rho N_{A}}\right)^{\frac{1}{3}}\,,
  10. M M
  11. ρ \rho
  12. N A N_{A}
  13. r s r_{s}
  14. r s / a 0 r_{s}/a_{0}

Wijsman_convergence.html

  1. d ( x , A ) = inf a A d ( x , a ) . d(x,A)=\inf_{a\in A}d(x,a).
  2. d ( x , A i ) d ( x , A ) . d(x,A_{i})\to d(x,A).
  3. d H ( A , B ) = sup x X | d ( x , A ) - d ( x , B ) | . d_{\mathrm{H}}(A,B)=\sup_{x\in X}\big|d(x,A)-d(x,B)\big|.

Willmore_conjecture.html

  1. W ( M ) = M H 2 d A . W(M)=\int_{M}H^{2}\,dA.

Wilson–Cowan_model.html

  1. E ( t ) E(t)
  2. I ( t ) I(t)
  3. r r
  4. t - r t E ( t ) d t \int_{t-r}^{t}E(t^{\prime})dt^{\prime}
  5. 1 - t - r t E ( t ) d t 1-\int_{t-r}^{t}E(t^{\prime})dt^{\prime}
  6. D ( θ ) D(\theta)
  7. S ( x ) = 0 x ( t ) D ( θ ) d θ S(x)=\int_{0}^{x(t)}D(\theta)d\theta
  8. S ( x ) = θ x ( t ) C ( w ) d w S(x)=\int_{\frac{\theta}{x(t)}}^{\infty}C(w)dw
  9. - t α ( t - t ) [ c 1 E ( t ) - c 2 I ( t ) + P ( t ) ] d t \int_{-\infty}^{t}\alpha(t-t^{\prime})[c_{1}E(t^{\prime})-c_{2}I(t^{\prime})+P% (t^{\prime})]dt^{\prime}
  10. α ( t ) \alpha(t)
  11. c 1 c_{1}
  12. c 2 c_{2}
  13. E ( t ) = [ 1 - t - r t E ( t ) d t ] S ( x ) d t E(t)=[1-\int_{t-r}^{t}E(t^{\prime})dt^{\prime}]S(x)dt
  14. E ( t + τ ) = [ 1 - t - r t E ( t ) d t ] S e { - t α ( t - t ) [ c 1 E ( t ) - c 2 I ( t ) + P ( t ) ] d t } E(t+\tau)=[1-\int_{t-r}^{t}E(t^{\prime})dt^{\prime}]S_{e}\left\{\int_{-\infty}% ^{t}\alpha(t-t^{\prime})[c_{1}E(t^{\prime})-c_{2}I(t^{\prime})+P(t^{\prime})]% dt^{\prime}\right\}
  15. I ( t + τ ) = [ 1 - t - r t I ( t ) d t ] S i { - t α ( t - t ) [ c 3 E ( t ) - c 4 I ( t ) + Q ( t ) ] d t } I(t+\tau)=[1-\int_{t-r}^{t}I(t^{\prime})dt^{\prime}]S_{i}\left\{\int_{-\infty}% ^{t}\alpha(t-t^{\prime})[c_{3}E(t^{\prime})-c_{4}I(t^{\prime})+Q(t^{\prime})]% dt^{\prime}\right\}
  16. τ d E ¯ d t = - E ¯ + ( 1 - r E ¯ ) S e [ k c 1 E ¯ ( t ) + k P ( t ) ] \tau\frac{d\bar{E}}{dt}=-\bar{E}+(1-r\bar{E})S_{e}[kc_{1}\bar{E}(t)+kP(t)]
  17. c 2 I = c 1 E - S e - 1 ( E k e - r e E ) + P c_{2}I=c_{1}E-S_{e}^{-1}\left(\frac{E}{k_{e}-r_{e}E}\right)+P
  18. S ( x ) = 1 1 + exp [ - a ( x - θ ) ] - 1 1 + exp ( a θ ) S(x)=\frac{1}{1+\exp[-a(x-\theta)]}-\frac{1}{1+\exp(a\theta)}

Winnow_(algorithm).html

  1. X = { 0 , 1 } n X=\{0,1\}^{n}
  2. w i w_{i}
  3. i { 1... n } i\in\{1...n\}
  4. ( x 1 , x n ) (x_{1},...x_{n})
  5. i = 1 n w i x i > Θ \sum_{i=1}^{n}w_{i}x_{i}>\Theta
  6. Θ \Theta
  7. Θ = n / 2 \Theta=n/2
  8. α \alpha
  9. α \alpha
  10. α \alpha
  11. α > 1 \alpha>1
  12. Θ 1 / α \Theta\geq 1/\alpha
  13. k k
  14. f ( x 1 , x n ) = x i 1 x i k f(x_{1},...x_{n})=x_{i_{1}}\cup...\cup x_{i_{k}}
  15. α k ( log α Θ + 1 ) + n Θ \alpha k(\log_{\alpha}\Theta+1)+\frac{n}{\Theta}

Word_(group_theory).html

  1. s 1 ε 1 s 2 ε 2 s n ε n s_{1}^{\varepsilon_{1}}s_{2}^{\varepsilon_{2}}\cdots s_{n}^{\varepsilon_{n}}
  2. x x y - 1 z y z z z x - 1 x - 1 xxy^{-1}zyzzzx^{-1}x^{-1}\,
  3. x 2 y - 1 z y z 3 x - 2 . x^{2}y^{-1}zyz^{3}x^{-2}.\,
  4. x 2 y ¯ z y z 3 x ¯ 2 . x^{2}\overline{y}zyz^{3}\overline{x}^{2}.\,
  5. x - 1 y x = y 2 . x^{-1}yx=y^{2}.\,
  6. \mathcal{R}
  7. \mathcal{R}
  8. S \langle S\mid\mathcal{R}\rangle
  9. \mathcal{R}
  10. i , j i 2 = 1 , j 2 = 1 , i j = j i . \langle i,j\mid i^{2}=1,\,j^{2}=1,\,ij=ji\rangle.
  11. S \langle S\rangle
  12. y - 1 z x x - 1 y y - 1 z y . y^{-1}zxx^{-1}y\;\;\longrightarrow\;\;y^{-1}zy.
  13. x z y - 1 x x - 1 y z - 1 z z - 1 y z x y z . xzy^{-1}xx^{-1}yz^{-1}zz^{-1}yz\;\;\longrightarrow\;\;xyz.
  14. S \langle S\mid\;\rangle
  15. ( x z y z - 1 ) ( z y - 1 x - 1 y ) = x z y z - 1 z y - 1 x - 1 y . \left(xzyz^{-1}\right)\left(zy^{-1}x^{-1}y\right)=xzyz^{-1}zy^{-1}x^{-1}y.
  16. ( z y - 1 x - 1 y ) - 1 = y - 1 x y z - 1 . \left(zy^{-1}x^{-1}y\right)^{-1}=y^{-1}xyz^{-1}.
  17. z y - 1 x - 1 y y - 1 x y z - 1 = 1. zy^{-1}x^{-1}y\;y^{-1}xyz^{-1}=1.
  18. x - 1 ( x y - 1 z - 1 y z ) x = y - 1 z - 1 y z x . x^{-1}\left(xy^{-1}z^{-1}yz\right)x=y^{-1}z^{-1}yzx.
  19. S \langle S\mid\mathcal{R}\rangle

Wu's_method_of_characteristic_set.html

  1. x , y , z , I ( x , y , z , ) f ( x , y , z , ) \forall x,y,z,\dots I(x,y,z,\dots)\implies f(x,y,z,\dots)\,
  2. V ( F ) = W ( T 1 ) W ( T e ) , V(F)=W(T_{1})\cup\cdots\cup W(T_{e}),\,

Xanthine_dehydrogenase.html

  1. \rightleftharpoons

Xanthommatin_reductase.html

  1. \rightleftharpoons

Xanthotoxol_O-methyltransferase.html

  1. \rightleftharpoons

Xanthoxin_dehydrogenase.html

  1. \rightleftharpoons

Xylitol_oxidase.html

  1. \rightleftharpoons

Yamabe_invariant.html

  1. M M
  2. n 2 n\geq 2
  3. \mathcal{E}
  4. g g
  5. M M
  6. ( g ) = M R g d V g ( M d V g ) n - 2 n , \mathcal{E}(g)=\frac{\int_{M}R_{g}\,dV_{g}}{\left(\int_{M}\,dV_{g}\right)^{% \frac{n-2}{n}}},
  7. R g R_{g}
  8. g g
  9. d V g dV_{g}
  10. g g
  11. c c
  12. ( c g ) = ( g ) \mathcal{E}(cg)=\mathcal{E}(g)
  13. ( g ) \mathcal{E}(g)
  14. g g
  15. M M
  16. ( g ) \mathcal{E}(g)
  17. Y ( g ) = inf f ( e 2 f g ) , Y(g)=\inf_{f}\mathcal{E}(e^{2f}g),
  18. f f
  19. M M
  20. - -\infty
  21. Y ( g ) - ( M | R g | n / 2 d V g ) 2 / n Y(g)\geq-\left(\textstyle\int_{M}|R_{g}|^{n/2}\,dV_{g}\right)^{2/n}
  22. Y ( g ) Y(g)
  23. g g
  24. g g
  25. Y ( g ) Y(g)
  26. ( g 0 ) \mathcal{E}(g_{0})
  27. g 0 g_{0}
  28. n n
  29. S n S^{n}
  30. σ ( M ) = sup g Y ( g ) , \sigma(M)=\sup_{g}Y(g),
  31. M M
  32. σ ( M ) ( g 0 ) \sigma(M)\leq\mathcal{E}(g_{0})
  33. σ ( M ) \sigma(M)
  34. M M
  35. n = 2 n=2
  36. ( g ) = M R g d V g = M 2 K g d V g , \mathcal{E}(g)=\int_{M}R_{g}\,dV_{g}=\int_{M}2K_{g}\,dV_{g},
  37. K g K_{g}
  38. 2 π χ ( M ) 2\pi\chi(M)
  39. χ ( M ) \chi(M)
  40. σ ( M ) = 4 π χ ( M ) . \sigma(M)=4\pi\chi(M).
  41. 8 π 8\pi
  42. C P 2 CP^{2}
  43. M M
  44. n 5 n\geq 5
  45. σ ( M ) 0 \sigma(M)\geq 0
  46. n n
  47. n = 4 n=4
  48. 4 4
  49. ( g 0 ) \mathcal{E}(g_{0})
  50. 6 ( 2 π 2 ) 2 / 3 6(2\pi^{2})^{2/3}
  51. σ 1 \sigma_{1}
  52. M M
  53. σ ( M ) \sigma(M)
  54. S 3 S^{3}
  55. σ 1 \sigma_{1}
  56. S 2 × S 1 S^{2}\times S^{1}
  57. σ 1 \sigma_{1}
  58. S 1 S^{1}
  59. S 2 × S 1 S^{2}\stackrel{\sim}{\times}S^{1}
  60. σ 1 \sigma_{1}
  61. S 1 S^{1}
  62. \R 3 \R\mathbb{P}^{3}
  63. σ 1 / 2 2 / 3 \sigma_{1}/2^{2/3}
  64. \R 2 × S 1 \R\mathbb{P}^{2}\times S^{1}
  65. σ 1 / 2 2 / 3 \sigma_{1}/2^{2/3}
  66. T 3 T^{3}
  67. 0
  68. M M
  69. σ ( M ) \sigma(M)
  70. M M
  71. 3 \mathbb{RP}^{3}
  72. S p i n c Spin^{c}

Yuri_Valentinovich_Nesterenko.html

  1. e π 3 e^{\pi\sqrt{3}}
  2. e π n e^{\pi\sqrt{n}}

Zariski's_main_theorem.html

  1. Γ V × W \Gamma\subset V\times W
  2. p 1 p_{1}
  3. U V U\subset V
  4. p 1 - 1 ( U ) p_{1}^{-1}(U)
  5. p 2 p 1 - 1 p_{2}\circ p_{1}^{-1}
  6. p 2 p 1 - 1 p_{2}\circ p_{1}^{-1}
  7. f : X Y f:X\to Y
  8. X Z Y X\to Z\to Y

Zeatin_reductase.html

  1. \rightleftharpoons

ZOOMQ3D.html

  1. x [ K x x ϕ x ] + y [ K y y ϕ y ] + z [ K z z ϕ z ] = S S ϕ t - q \frac{\partial}{\partial x}\left[K_{xx}\frac{\partial\phi}{\partial x}\right]+% \frac{\partial}{\partial y}\left[K_{yy}\frac{\partial\phi}{\partial y}\right]+% \frac{\partial}{\partial z}\left[K_{zz}\frac{\partial\phi}{\partial z}\right]=% S_{S}\frac{\partial\phi}{\partial t}-q
  2. ϕ ( x , y , z , t ) \phi(x,y,z,t)
  3. ( x , y , z ) (x,y,z)
  4. ( t ) (t)
  5. K x x K_{xx}
  6. K y y K_{yy}
  7. K z z K_{zz}
  8. q q
  9. S S S_{S}
  10. x [ T x x h x ] + y [ T y y h y ] = S c h t - q - L above + L below \frac{\partial}{\partial x}\left[T_{xx}\frac{\partial h}{\partial x}\right]+% \frac{\partial}{\partial y}\left[T_{yy}\frac{\partial h}{\partial y}\right]=S_% {c}\frac{\partial h}{\partial t}-q-L_{\mathrm{above}}+L_{\mathrm{below}}
  11. h h
  12. t , t,
  13. T x x T_{xx}
  14. T y y T_{yy}
  15. q q
  16. S S
  17. L L
  18. L L

Ε-quadratic_form.html

  1. ( - ) n (-)^{n}
  2. 0 Q ϵ ( M ) B ( M ) 1 - ϵ T B ( M ) Q ϵ ( M ) 0 0\to Q^{\epsilon}(M)\to B(M)\stackrel{1-\epsilon T}{\longrightarrow}B(M)\to Q_% {\epsilon}(M)\to 0
  3. Q ϵ ( M ) := ker ( 1 - ϵ T ) Q^{\epsilon}(M):=\mbox{ker}~{}\,(1-\epsilon T)
  4. Q ϵ ( M ) := coker ( 1 - ϵ T ) Q_{\epsilon}(M):=\mbox{coker}~{}\,(1-\epsilon T)
  5. Q ϵ ( M ) B ( M ) Q ϵ ( M ) Q^{\epsilon}(M)\to B(M)\to Q_{\epsilon}(M)
  6. ( 1 + ϵ T ) B ( M ) < Q ϵ ( M ) (1+\epsilon T)B(M)<Q^{\epsilon}(M)
  7. R = 𝐙 [ 1 + i 2 ] R=\mathbf{Z}\left[\textstyle{\frac{1+i}{2}}\right]
  8. λ = 1 ± i 2 \lambda=\textstyle{\frac{1\pm i}{2}}
  9. ( a b c d ) \begin{pmatrix}a&b\\ c&d\end{pmatrix}
  10. ( a b b c ) \begin{pmatrix}a&b\\ b&c\end{pmatrix}
  11. ( 0 b - b 0 ) \begin{pmatrix}0&b\\ -b&0\end{pmatrix}
  12. ( a b c d ) \begin{pmatrix}a&b\\ c&d\end{pmatrix}
  13. a x 2 + b x y + c y x + d y 2 = a x 2 + ( b + c ) x y + d y 2 ax^{2}+bxy+cyx+dy^{2}=ax^{2}+(b+c)xy+dy^{2}\,
  14. e x 2 + f x y + g y 2 ex^{2}+fxy+gy^{2}
  15. ( 2 e f f 2 g ) \begin{pmatrix}2e&f\\ f&2g\end{pmatrix}
  16. ( e f 0 g ) \begin{pmatrix}e&f\\ 0&g\end{pmatrix}
  17. 2 e x 2 + 2 f x y + 2 g y 2 = 2 ( e x 2 + f x y + g y 2 ) 2ex^{2}+2fxy+2gy^{2}=2(ex^{2}+fxy+gy^{2})
  18. ( a z z ¯ c ) \begin{pmatrix}a&z\\ \bar{z}&c\end{pmatrix}
  19. ( b i z - z d i ) \begin{pmatrix}bi&z\\ -z&di\end{pmatrix}
  20. v 2 = Q ( v ) v^{2}=Q(v)
  21. H ϵ ( R ) Q ϵ ( R R * ) H_{\epsilon}(R)\in Q_{\epsilon}(R\oplus R^{*})
  22. ( ( v 1 , f 1 ) , ( v 2 , f 2 ) ) f 2 ( v 1 ) ((v_{1},f_{1}),(v_{2},f_{2}))\mapsto f_{2}(v_{1})
  23. 4 k + 2 , 4k+2,
  24. 4 k , 4k,
  25. S 2 k × S 2 k S^{2k}\times S^{2k}
  26. S 2 k + 1 × S 2 k + 1 S^{2k+1}\times S^{2k+1}
  27. ( 0 1 1 0 ) \left(\begin{smallmatrix}0&1\\ 1&0\end{smallmatrix}\right)
  28. ( 0 1 - 1 0 ) . \left(\begin{smallmatrix}0&1\\ -1&0\end{smallmatrix}\right).
  29. π 1 s \pi^{s}_{1}
  30. ( 0 1 - 1 0 ) \begin{pmatrix}0&1\\ -1&0\end{pmatrix}

Ω-logic.html

  1. H 2 H_{\aleph_{2}}
  2. H 1 H_{\aleph_{1}}
  3. H 2 H_{\aleph_{2}}
  4. 1 \aleph_{1}
  5. Π 2 \Pi_{2}
  6. H 2 H_{\aleph_{2}}
  7. 2 \aleph_{2}
  8. Σ 2 \Sigma_{2}
  9. P ( α ) P(\alpha)
  10. V α 𝔹 V^{\mathbb{B}}_{\alpha}
  11. α \alpha
  12. 𝔹 \mathbb{B}
  13. Π 2 \Pi_{2}