wpmath0000016_13

Serre's_theorem_on_affineness.html

  1. H 1 ( X , I ) = 0 H^{1}(X,I)=0

Sesquithujene_synthase.html

  1. \rightleftharpoons

Set_intersection_oracle.html

  1. O ( | S i | + | S j | ) = O ( N ) O(|S_{i}|+|S_{j}|)=O(N)
  2. O ( 1 ) O(1)
  3. O ( n 2 ) O(n^{2})
  4. N \sqrt{N}
  5. N \sqrt{N}
  6. O ( N ) O(N)
  7. O ( N 3 / 2 ) O(N^{3/2})
  8. O ( 1 ) O(1)
  9. O ( N ) O(\sqrt{N})
  10. O ( N ) O(\sqrt{N})
  11. N c N^{c}
  12. N 1 - c N^{1-c}
  13. O ( N 2 - c ) O(N^{2-c})
  14. O ( N c ) O(N^{c})
  15. Ω ( n 2 ) \Omega(n^{2})
  16. O ( 1 ) O(1)

Seventh-order_Korteweg–de_Vries_equation.html

  1. u t + 6 u u x + u x x x - u x x x x x + α u x x x x x x x = 0. u_{t}+6uu_{x}+u_{xxx}-u_{xxxxx}+\alpha u_{xxxxxxx}=0.
  2. x x
  3. t t
  4. α \alpha

Severity_factor.html

  1. 𝐅𝐃𝐒𝐅 ( ω ) = < 𝐦𝐭𝐩𝐥 > 𝐄𝐒𝐃 𝐧𝐨𝐒𝐭𝐝 ( ω ) 𝐄𝐒𝐃 𝐞𝐧𝐯𝐨𝐥 ( ω ) \mathbf{FDSF(\omega)}=\frac{\mathbf{<mtpl>{{ESD}}_{noStd}}(\omega)}{\mathbf{{{% ESD}}_{envol}}(\omega)}
  2. 𝐓𝐃𝐒𝐅 ( 𝐢 ) = < 𝐦𝐭𝐩𝐥 > 𝚫 𝐕 𝐧𝐨𝐒𝐭𝐝 ( i ) 𝚫 𝐕 𝐞𝐧𝐯𝐨𝐥 ( i ) \mathbf{TDSF(i)}=\frac{\mathbf{<mtpl>{{\Delta V}}_{noStd}}(i)}{\mathbf{{{% \Delta V}}_{envol}}(i)}

Shadow_heap.html

  1. A A
  2. B B
  3. O ( | A | + min { log | B | log log | B | , log | A | log | B | } ) O(|A|+\min\{\log|B|\log\log|B|,\log|A|\log|B|\})
  4. A A
  5. B B
  6. A A
  7. B B
  8. C C
  9. A A
  10. C C
  11. | A | |A|
  12. C C
  13. C C
  14. P P
  15. C C
  16. T T
  17. | P | |P|
  18. S S
  19. S S
  20. | S | > | C | |S|>|C|
  21. S S
  22. C C
  23. C C
  24. | S | | C | |S|\leq|C|
  25. | P | |P|
  26. C C
  27. S S
  28. S S
  29. C C
  30. T T
  31. P P
  32. T T
  33. C C
  34. P P
  35. C C
  36. O ( log | B | ) O(\log|B|)
  37. T T
  38. d d
  39. d + 1 d+1
  40. O ( | A | ) O(|A|)
  41. P P
  42. | P | |P|
  43. S S
  44. O ( log | B | ) O(\log|B|)
  45. | S | > | C | |S|>|C|
  46. P P
  47. C C
  48. O ( log | A | log | B | ) O(\log|A|\log|B|)
  49. O ( | A | + log | B | log log | B | ) O(|A|+\log|B|\log\log|B|)
  50. T T
  51. O ( | A | ) O(|A|)
  52. O ( | A | + min { log | A | log | B | , log | B | log log | B | } ) O(|A|+\min\{\log|A|\log|B|,\log|B|\log\log|B|\})
  53. H H
  54. f ( H ) f(H)
  55. B B
  56. A A
  57. A A
  58. A A
  59. B B
  60. H = ( B , A ) H=(B,A)
  61. log | H | f ( H ) log | H | log log | H | \log|H|\leq f(H)\leq\log|H|\log\log|H|
  62. | B | |B|
  63. f ( H ) f(H)
  64. Ψ ( H ) \Psi(H)
  65. Ψ ( H ) = | A | ( 1 + min { log | B | log log | B | , log | B | log | A | } / f ( H ) ) \Psi(H)=|A|(1+\min\{\log|B|\log\log|B|,\log|B|\log|A|\}/f(H))
  66. H H
  67. Ψ \Psi
  68. O ( 1 ) O(1)
  69. x x
  70. H H
  71. B B
  72. A A
  73. O ( 1 ) O(1)
  74. O ( 1 + min { log | B | log log | B | , log | B | log | A | } / f ( H ) ) O(1+\min\{\log|B|\log\log|B|,\log|B|\log|A|\}/f(H))
  75. O ( log | H | log log | H | / f ( H ) ) O(\log|H|\log\log|H|/f(H))
  76. H H
  77. O ( | A | + log | B | ) O(|A|+\log|B|)
  78. f ( H ) f(H)
  79. f f
  80. A A
  81. B B
  82. O ( | B | ) O(|B|)
  83. A A
  84. B B
  85. O ( | A | log | B | ) O(|A|\log|B|)
  86. O ( | A | + log | A | log | B | ) O(|A|+\log|A|\log|B|)
  87. | A | > log | B | |A|>\log|B|
  88. O ( 1 ) O(1)
  89. Ω ( | A | ) \Omega(|A|)

Shannon_capacity_of_a_graph.html

  1. Θ ( G ) = sup k α ( G k ) k = lim k α ( G k ) k , \Theta(G)=\sup_{k}\sqrt[k]{\alpha(G_{k})}=\lim_{k\rightarrow\infty}\sqrt[k]{% \alpha(G_{k})},
  2. 5 \sqrt{5}
  3. Θ ( G ) R ( G ) = min B rank ( B ) , \Theta(G)\leq R(G)=\min_{B}\operatorname{rank}(B),

Shannon_coding.html

  1. l i = - log p i ( x ) l_{i}=\left\lceil{-\log}p_{i}(x)\right\rceil
  2. i = k i - 1 p k ( x ) \sum\limits_{i=k}^{i-1}p_{k}(x)
  3. x \left\lceil x\right\rceil
  4. x x

Shannon_Criteria.html

  1. log D = k - log Q \log D=k-\log Q
  2. 10 k = Q * D 10^{k}=Q*D

Shapiro-Stiglitz_theory.html

  1. u ( w , e ) = w - e u(w,e)=w-e
  2. V u V_{u}
  3. V e = w T + e - r T [ b T V u + ( 1 - b T ) V e ] , V_{e}=wT+e^{-rT}[bTV_{u}+(1-bT)V_{e}]\;,
  4. V e = w T + ( 1 - r T ) [ b T V u + ( 1 - b T ) V e ] , V_{e}=wT+(1-rT)[bTV_{u}+(1-bT)V_{e}]\;,
  5. V e = w T + b T V u - r b T 2 V u r T + b T - r b T 2 , V_{e}=\frac{wT+bTV_{u}-rbT^{2}V_{u}}{rT+bT-rbT^{2}}\;,
  6. lim t 0 V e = w + b V u r + b . \lim_{t\rightarrow 0}V_{e}=\frac{w+bV_{u}}{r+b}\;.
  7. r V e = w + b ( V u - V e ) . rV_{e}=w+b(V_{u}-V_{e})\;.
  8. r V e , N = w - e + b ( V u - V e , N ) , rV_{e,N}=w-e+b(V_{u}-V_{e,N})\;,
  9. r V e , S = w + ( b + q ) ( V u - V e , S ) , rV_{e,S}=w+(b+q)(V_{u}-V_{e,S})\;,
  10. V e , N = w - e + b V u r + b , V_{e,N}=\frac{w-e+bV_{u}}{r+b}\;,
  11. V e , S = w + ( b + q ) V u r + b + q . V_{e,S}=\frac{w+(b+q)V_{u}}{r+b+q}\;.
  12. V e , S < V e , N V_{e,S}<V_{e,N}
  13. w ^ = r V u + e ( r + b + q ) q < w , \hat{w}=rV_{u}+\frac{e(r+b+q)}{q}<w\;,
  14. w ^ \hat{w}
  15. a a
  16. a = b L N - L , a=\frac{bL}{N-L}\;,
  17. L L
  18. N N
  19. w ¯ \overline{w}
  20. w ¯ + e + e ( a + b + r ) q = w ^ < w , \overline{w}+e+\frac{e(a+b+r)}{q}=\hat{w}<w\;,
  21. e + w ¯ + e q ( b u + r ) < w , e+\overline{w}+\frac{e}{q}(\frac{b}{u}+r)<w\;,
  22. u = N - L N u=\frac{N-L}{N}
  23. F ( L ) F(L)
  24. w ¯ = 0 \overline{w}=0
  25. w ^ = d F ( L ) d L . \hat{w}=\frac{dF(L)}{dL}\quad.
  26. F ( L ) = w ^ = w * F^{\prime}(L)=\hat{w}=w^{*}
  27. w * w^{*}
  28. F ( L ) = w ^ = e + e q ( b u + r ) = e ( 1 + r + b + a q ) . F^{\prime}(L)=\hat{w}=e+\frac{e}{q}(\frac{b}{u}+r)=e\left(1+\frac{r+b+a}{q}% \right)\;\;.
  29. w * w^{*}
  30. w * w^{*}
  31. W = L p W=\frac{L}{p}
  32. π = g ( N ) - N L p - N M , \pi=g(N)-\frac{NL}{p}-NM,
  33. f ( t ) = 1 - e - t . f(t)=1-e^{-t}.
  34. π N = g N - L p . \frac{\partial\pi}{\partial N}=\frac{\partial g}{\partial N}-\frac{L}{p}.
  35. 2 π N 2 = 2 g ( N ) N 2 . \frac{\partial^{2}\pi}{\partial N^{2}}=\frac{\partial^{2}g(N)}{\partial N^{2}}.
  36. π N = 0 \frac{\partial\pi}{\partial N}=0
  37. g ( N ) N = L p + M . \frac{\partial g(N)}{\partial N}=\frac{L}{p}+M.
  38. 2 g ( N ) N 2 N R = - L p p R . \frac{\partial^{2}g(N)}{\partial N^{2}}\frac{\partial N}{\partial R}=-\frac{L}% {p}\frac{\partial p}{\partial R}.
  39. N R \frac{\partial N}{\partial R}

Shared_snapshot_objects.html

  1. \in
  2. d ¯ \overline{d}
  3. P i , P j \langle P_{i},P_{j}\rangle
  4. O ( n 2 ) O(n^{2})
  5. O ( 2 n ) O(2^{n})
  6. O ( n log 2 n ) O(n\log^{2}n)
  7. O ( n 3 / 2 log 2 n ) O(n^{3/2}\log^{2}n)
  8. Ω ( min { w , r } ) \Omega(\min\{w,r\})
  9. O ( n log n ) O(n\log n)
  10. O ( n log n ) O(n\log n)
  11. O ( n ) O(n)

Sharp_series.html

  1. v = R [ 2 + p ] 2 - R [ m + s ] 2 with m = 2 , 3 , 4 , 5 , 6 , v=\frac{R}{\left[2+p\right]^{2}}-\frac{R}{\left[m+s\right]^{2}}\,\text{ with }% m=2,3,4,5,6,...
  2. 2 P 3 2 2P_{\frac{3}{2}}
  3. 2 P 1 2 2P_{\frac{1}{2}}
  4. ν s = R ( Z 3 p 2 3 2 - Z n s 2 n 2 ) n = 4 , 5 , 6 , \nu_{s}=R\left(\frac{Z_{3p}^{2}}{3^{2}}-\frac{Z_{ns}^{2}}{n^{2}}\right)n=4,5,6% ,...
  5. ν d = R ( Z 3 p 2 3 2 - Z n d 2 n 2 ) n = 4 , 5 , 6 , \nu_{d}=R\left(\frac{Z_{3p}^{2}}{3^{2}}-\frac{Z_{nd}^{2}}{n^{2}}\right)n=4,5,6% ,...

Shc_the_shell_script_compiler.html

  1. [ - f f i l e n a m e ] [-ffilename]
  2. f i l e n a m e . x . c filename.x.c
  3. f i l e n a m e filename
  4. c c - $ C F L A G S f i l e n a m e . x . c cc-\$CFLAGSfilename.x.c

Sheaf_of_logarithmic_differential_forms.html

  1. Ω X p ( log D ) \Omega^{p}_{X}(\log D)
  2. D = D j D=\sum D_{j}
  3. 0 Ω X p Ω X p ( log D ) 𝛽 j i j * Ω D j p - 1 0 , p 1 0\to\Omega^{p}_{X}\to\Omega^{p}_{X}(\log D)\overset{\beta}{\to}\oplus_{j}{i_{j% }}_{*}\Omega^{p-1}_{D_{j}}\to 0,\,p\geq 1
  4. i j : D j X i_{j}:D_{j}\to X
  5. D j , 1 j k D_{j},1\leq j\leq k
  6. D j , j > k D_{j},j>k
  7. d u 1 u 1 , , d u k u k , d u k , , d u n {du_{1}\over u_{1}},\dots,{du_{k}\over u_{k}},\,du_{k},\dots,du_{n}
  8. Ω X 1 ( log D ) \Omega^{1}_{X}(\log D)
  9. u j u_{j}
  10. u j , 1 j k u_{j},1\leq j\leq k
  11. D j , 1 j k D_{j},1\leq j\leq k

Sheaf_of_modules.html

  1. 𝐙 ¯ \underline{\mathbf{Z}}
  2. H i ( X , - ) \operatorname{H}^{i}(X,-)
  3. Γ ( X , - ) \Gamma(X,-)
  4. Ω X \Omega_{X}
  5. ω X \omega_{X}
  6. Ω X \Omega_{X}
  7. F O G F\otimes_{O}G
  8. F G F\otimes G
  9. U F ( U ) O ( U ) G ( U ) . U\mapsto F(U)\otimes_{O(U)}G(U).
  10. O ( 1 ) O ( - 1 ) = O O(1)\otimes O(-1)=O
  11. o m O ( F , G ) \mathcal{H}om_{O}(F,G)
  12. U Hom O | U ( F | U , G | U ) U\mapsto\operatorname{Hom}_{O|_{U}}(F|_{U},G|_{U})
  13. o m O ( F , O ) \mathcal{H}om_{O}(F,O)
  14. F ˇ \check{F}
  15. E ˇ F o m O ( E , F ) \check{E}\otimes F\to\mathcal{H}om_{O}(E,F)
  16. L ˇ L O , \check{L}\otimes L\simeq O,
  17. H 1 ( X , 𝒪 * ) \operatorname{H}^{1}(X,\mathcal{O}^{*})
  18. E ˇ E End O ( E ) O \check{E}\otimes E\simeq\operatorname{End}_{O}(E)\to O
  19. k F \wedge^{k}F
  20. U O ( U ) k F ( U ) U\mapsto\wedge^{k}_{O(U)}F(U)
  21. n F \wedge^{n}F
  22. r F n - r F det ( F ) . \wedge^{r}F\otimes\wedge^{n-r}F\to\operatorname{det}(F).
  23. f * F f_{*}F
  24. f * G f^{*}G
  25. f - 1 G f - 1 O O f^{-1}G\otimes_{f^{-1}O^{\prime}}O
  26. f - 1 G f^{-1}G
  27. f - 1 O O f^{-1}O^{\prime}\to O
  28. O f * O O^{\prime}\to f_{*}O
  29. f * f_{*}
  30. f * f^{*}
  31. Hom O ( f * G , F ) Hom O ( G , f * F ) \operatorname{Hom}_{O}(f^{*}G,F)\simeq\operatorname{Hom}_{O^{\prime}}(G,f_{*}F)
  32. f * ( F f * E ) f * F E . f_{*}(F\otimes f^{*}E)\simeq f_{*}F\otimes E.
  33. i I O F 0 \oplus_{i\in I}O\to F\to 0
  34. Γ ( X , - ) \Gamma(X,-)
  35. X = Spec ( R ) X=\operatorname{Spec}(R)
  36. 𝒪 X \mathcal{O}_{X}
  37. M ~ \widetilde{M}
  38. M ~ p \widetilde{M}_{p}
  39. Γ ( X , M ~ ) = M \Gamma(X,\widetilde{M})=M
  40. M M ~ M\mapsto\widetilde{M}
  41. 𝒪 X \mathcal{O}_{X}
  42. X = Spec R X=\operatorname{Spec}R
  43. F Γ ( X , F ) F\mapsto\Gamma(X,F)
  44. M M ~ M\mapsto\widetilde{M}
  45. 𝒪 X \mathcal{O}_{X}
  46. M ~ \widetilde{M}
  47. M ~ | { f 0 } ( M [ f - 1 ] 0 ) \widetilde{M}|_{\{f\neq 0\}}\simeq(M[f^{-1}]_{0})^{\sim}
  48. { f 0 } = Spec ( R [ f - 1 ] 0 ) \{f\neq 0\}=\operatorname{Spec}(R[f^{-1}]_{0})
  49. M ~ \widetilde{M}
  50. O ( 1 ) = R ( 1 ) ~ O(1)=\widetilde{R(1)}
  51. F ( n ) = F O ( n ) F(n)=F\otimes O(n)
  52. ( n 0 Γ ( X , F ( n ) ) ) F (\oplus_{n\geq 0}\Gamma(X,F(n)))^{\sim}\to F
  53. H i ( X , F ( n ) ) = 0 , i 1 , n n 0 \operatorname{H}^{i}(X,F(n))=0,\,i\geq 1,n\geq n_{0}
  54. 0 F G H 0. 0\rightarrow F\rightarrow G\rightarrow H\rightarrow 0.
  55. Ext O 1 ( H , F ) \mathrm{Ext}_{O}^{1}(H,F)
  56. Ext O 1 ( H , F ) \mathrm{Ext}_{O}^{1}(H,F)
  57. H i ( X , F ) = Ext O i ( O , F ) , \operatorname{H}^{i}(X,F)=\mathrm{Ext}_{O}^{i}(O,F),
  58. Γ ( X , - ) = Hom O ( O , - ) . \Gamma(X,-)=\operatorname{Hom}_{O}(O,-).
  59. Ext O i ( F , G ( n ) ) = Γ ( X , 𝓍 𝓉 O i ( F , G ( n ) ) ) , n n 0 \operatorname{Ext}_{O}^{i}(F,G(n))=\Gamma(X,\mathcal{Ext}_{O}^{i}(F,G(n))),\,n% \geq n_{0}
  60. o m O ( F , O ) x Hom O x ( F x , O x ) , \mathcal{H}om_{O}(F,O)_{x}\to\operatorname{Hom}_{O_{x}}(F_{x},O_{x}),
  61. F G O F\otimes G\simeq O

Sheaf_of_spectra.html

  1. π * F π * G \pi_{*}F\to\pi_{*}G

Shearlet.html

  1. f L 2 ( \R 2 ) f\in L^{2}(\R^{2})
  2. L 2 ( \R 2 ) L^{2}(\R^{2})
  3. f L 2 ( \R 2 ) f\in L^{2}(\R^{2})
  4. f f
  5. [ 0 , 1 ] 2 [0,1]^{2}
  6. C 2 C^{2}
  7. C 2 C^{2}
  8. L 2 L^{2}
  9. N N
  10. N N
  11. f - f N L 2 2 C N - 2 ( log N ) 3 , N , \|f-f_{N}\|_{L^{2}}^{2}\leq CN^{-2}(\log N)^{3},\quad N\to\infty,
  12. C C
  13. f f
  14. f f^{{}^{\prime}}
  15. f ′′ f^{{}^{\prime\prime}}
  16. N N
  17. O ( N - 1 ) O(N^{-1})
  18. L 2 ( \R d ) , d 2 L^{2}(\R^{d}),d\geq 2
  19. A a = [ a 0 0 a 1 / 2 ] , a > 0 A_{a}=\begin{bmatrix}a&0\\ 0&a^{1/2}\end{bmatrix},\quad a>0
  20. S s = [ 1 s 0 1 ] , s \R S_{s}=\begin{bmatrix}1&s\\ 0&1\end{bmatrix},\quad s\in\R
  21. S s S_{s}
  22. s \Z s\in\Z
  23. S s \Z 2 \Z 2 . S_{s}\Z^{2}\subseteq\Z^{2}.
  24. ψ L 2 ( \R 2 ) \psi\in L^{2}(\R^{2})
  25. ψ \psi
  26. SH cont ( ψ ) = { ψ a , s , t = a 3 / 4 ψ ( S s A a ( - t ) ) a > 0 , s \R , t \R 2 } , \operatorname{SH}_{\mathrm{cont}}(\psi)=\{\psi_{a,s,t}=a^{3/4}\psi(S_{s}A_{a}(% \cdot-t))\mid a>0,s\in\R,t\in\R^{2}\},
  27. f 𝒮 ψ f ( a , s , t ) = f , ψ a , s , t , f L 2 ( \R 2 ) , ( a , s , t ) \R > 0 × \R × \R 2 . f\mapsto\mathcal{SH}_{\psi}f(a,s,t)=\langle f,\psi_{a,s,t}\rangle,\quad f\in L% ^{2}(\R^{2}),\quad(a,s,t)\in\R_{>0}\times\R\times\R^{2}.
  28. SH cont ( ψ ) \operatorname{SH}_{\mathrm{cont}}(\psi)
  29. \R > 0 × \R × \R 2 . \R_{>0}\times\R\times\R^{2}.
  30. { ( 2 j , k , A 2 j - 1 S k - 1 m ) j \Z , k \Z , m \Z 2 } \R > 0 × \R × \R 2 . \{(2^{j},k,A_{2^{j}}^{-1}S_{k}^{-1}m)\mid j\in\Z,k\in\Z,m\in\Z^{2}\}\subseteq% \R_{>0}\times\R\times\R^{2}.
  31. ψ \psi
  32. SH ( ψ ) = { ψ j , k , m = 2 3 j / 4 ψ ( S k A 2 j - m ) j \Z , k \Z , m \Z 2 } , \operatorname{SH}(\psi)=\{\psi_{j,k,m}=2^{3j/4}\psi(S_{k}A_{2^{j}}\cdot{}-m)% \mid j\in\Z,k\in\Z,m\in\Z^{2}\},
  33. f 𝒮 ψ f ( j , k , m ) = f , ψ j , k , m , f L 2 ( \R 2 ) , ( j , k , m ) \Z × \Z × \Z 2 . f\mapsto\mathcal{SH}_{\psi}f(j,k,m)=\langle f,\psi_{j,k,m}\rangle,\quad f\in L% ^{2}(\R^{2}),\quad(j,k,m)\in\Z\times\Z\times\Z^{2}.
  34. ψ 1 L 2 ( \R ) \psi_{1}\in L^{2}(\R)
  35. j \Z | ψ ^ 1 ( 2 - j ξ ) | 2 = 1 , ξ \R , \sum_{j\in\Z}|\hat{\psi}_{1}(2^{-j}\xi)|^{2}=1,\quad\xi\in\R,
  36. ψ ^ 1 C ( \R ) \hat{\psi}_{1}\in C^{\infty}(\R)
  37. supp ψ ^ 1 [ - 1 2 , - 1 16 ] [ 1 16 , 1 2 ] , \operatorname{supp}\hat{\psi}_{1}\subseteq[-\tfrac{1}{2},-\tfrac{1}{16}]\cup[% \tfrac{1}{16},\tfrac{1}{2}],
  38. ψ ^ 1 \hat{\psi}_{1}
  39. ψ 1 . \psi_{1}.
  40. ψ 1 \psi_{1}
  41. ψ 2 L 2 ( \R ) \psi_{2}\in L^{2}(\R)
  42. ψ ^ 2 C ( \R ) , \hat{\psi}_{2}\in C^{\infty}(\R),
  43. supp ψ ^ 2 [ - 1 , 1 ] \operatorname{supp}\hat{\psi}_{2}\subseteq[-1,1]
  44. k Z | ψ ^ 2 ( ξ + k ) | 2 = 1 , ξ \R . \sum_{k\in Z}|\hat{\psi}_{2}(\xi+k)|^{2}=1,\quad\xi\in\R.
  45. ψ ^ 2 \hat{\psi}_{2}
  46. ψ L 2 ( \R 2 ) \psi\in L^{2}(\R^{2})
  47. ψ ^ ( ξ ) = ψ ^ 1 ( ξ 1 ) ψ ^ 2 ( ξ 2 ξ 1 ) , ξ = ( ξ 1 , ξ 2 ) \R 2 , \hat{\psi}(\xi)=\hat{\psi}_{1}(\xi_{1})\hat{\psi}_{2}\left(\tfrac{\xi_{2}}{\xi% _{1}}\right),\quad\xi=(\xi_{1},\xi_{2})\in\R^{2},
  48. SH ( ψ ) \operatorname{SH}(\psi)
  49. L 2 ( \R 2 ) L^{2}(\R^{2})
  50. ψ L 2 ( \R 2 ) \psi\in L^{2}(\R^{2})
  51. SH ( ψ ) \operatorname{SH}(\psi)
  52. L 2 ( \R 2 ) L^{2}(\R^{2})
  53. SH ( ψ ) \operatorname{SH}(\psi)
  54. f L 2 ( \R 2 ) f\in L^{2}(\R^{2})
  55. ξ 2 \xi_{2}
  56. s s
  57. ξ 2 \xi_{2}
  58. \displaystyle\mathcal{R}
  59. ϕ , ψ , ψ ~ L 2 ( \R 2 ) \phi,\psi,\tilde{\psi}\in L^{2}(\R^{2})
  60. c = ( c 1 , c 2 ) ( \R > 0 ) 2 : c=(c_{1},c_{2})\in(\R_{>0})^{2}:
  61. SH ( ϕ , ψ , ψ ~ ; c ) = Φ ( ϕ ; c 1 ) Ψ ( ψ ; c ) Ψ ~ ( ψ ~ ; c ) , \operatorname{SH}(\phi,\psi,\tilde{\psi};c)=\Phi(\phi;c_{1})\cup\Psi(\psi;c)% \cup\tilde{\Psi}(\tilde{\psi};c),
  62. Φ ( ϕ ; c 1 ) = { ϕ m = ϕ ( - c 1 m ) m \Z 2 } , Ψ ( ψ ; c ) = { ψ j , k , m = 2 3 j / 4 ψ ( S k A 2 j - M c m ) j 0 , | k | 2 j / 2 , m \Z 2 } , Ψ ~ ( ψ ~ ; c ) = { ψ ~ j , k , m = 2 3 j / 4 ψ ( S ~ k A ~ 2 j - M ~ c m ) j 0 , | k | 2 j / 2 , m \Z 2 } , \begin{aligned}\displaystyle\Phi(\phi;c_{1})&\displaystyle=\{\phi_{m}=\phi(% \cdot{}-c_{1}m)\mid m\in\Z^{2}\},\\ \displaystyle\Psi(\psi;c)&\displaystyle=\{\psi_{j,k,m}=2^{3j/4}\psi(S_{k}A_{2^% {j}}\cdot{}-M_{c}m)\mid j\geq 0,|k|\leq\lceil 2^{j/2}\rceil,m\in\Z^{2}\},\\ \displaystyle\tilde{\Psi}(\tilde{\psi};c)&\displaystyle=\{\tilde{\psi}_{j,k,m}% =2^{3j/4}\psi(\tilde{S}_{k}\tilde{A}_{2^{j}}\cdot{}-\tilde{M}_{c}m)\mid j\geq 0% ,|k|\leq\lceil 2^{j/2}\rceil,m\in\Z^{2}\},\end{aligned}
  63. A ~ a = [ a 1 / 2 0 0 a ] , a > 0 , S ~ s = [ 1 0 s 1 ] , s \R , M c = [ c 1 0 0 c 2 ] , and M ~ c = [ c 2 0 0 c 1 ] . \begin{aligned}&\displaystyle\tilde{A}_{a}=\begin{bmatrix}a^{1/2}&0\\ 0&a\end{bmatrix},\;a>0,\quad\tilde{S}_{s}=\begin{bmatrix}1&0\\ s&1\end{bmatrix},\;s\in\R,\quad M_{c}=\begin{bmatrix}c_{1}&0\\ 0&c_{2}\end{bmatrix},\quad\,\text{and}\quad\tilde{M}_{c}=\begin{bmatrix}c_{2}&% 0\\ 0&c_{1}\end{bmatrix}.\end{aligned}
  64. Ψ ( ψ ) \Psi(\psi)
  65. Ψ ~ ( ψ ~ ) \tilde{\Psi}(\tilde{\psi})
  66. x 1 x_{1}
  67. x 2 x_{2}
  68. 𝒞 h \mathcal{C}_{\mathrm{h}}
  69. 𝒞 v \mathcal{C}_{\mathrm{v}}
  70. ϕ \phi
  71. \mathcal{R}
  72. α \alpha

Sheppard's_correction.html

  1. m k m_{k}
  2. μ ^ k \hat{\mu}_{k}
  3. c c
  4. μ ^ 2 \displaystyle\hat{\mu}_{2}

Shifted_force_method.html

  1. i i
  2. 𝐅 ( 𝐫 ) = j i F ( r ) 𝐫 ^ ; F ( r ) = q i q j 4 π ε 0 r 2 \mathbf{F}(\mathbf{r})=\sum_{j\neq i}F(r)\mathbf{\hat{r}}\,\,\,;\,\,F(r)=\frac% {q_{i}q_{j}}{4\pi\varepsilon_{0}r^{2}}
  3. 𝐫 \mathbf{r}
  4. j j
  5. r r
  6. i i
  7. j j
  8. 𝐫 ^ \mathbf{\hat{r}}
  9. j j
  10. i i
  11. F ( r ) F(r)
  12. q i q_{i}
  13. q j q_{j}
  14. i i
  15. j j
  16. r - 2 r^{-2}
  17. F C U T ( r ) = { q i q j 4 π ε 0 r 2 for r r c 0 for r > r c . \displaystyle F_{CUT}(r)=\begin{cases}\frac{q_{i}q_{j}}{4\pi\varepsilon_{0}r^{% 2}}&\,\text{for }r\leq r_{c}\\ 0&\,\text{for }r>r_{c}.\end{cases}
  18. r c r_{c}
  19. r c r_{c}
  20. r c r_{c}
  21. F S F ( r ) = { q i q j 4 π ε 0 r 2 - q i q j 4 π ε 0 r c 2 for r r c 0 for r > r c . \displaystyle F_{SF}(r)=\begin{cases}\frac{q_{i}q_{j}}{4\pi\varepsilon_{0}r^{2% }}-\frac{q_{i}q_{j}}{4\pi\varepsilon_{0}r_{c}^{2}}&\,\text{for }r\leq r_{c}\\ 0&\,\text{for }r>r_{c}.\end{cases}

Shionone_synthase.html

  1. \rightleftharpoons

Shoolery's_rule.html

  1. δ = 0.23 ppm + S A + S B \delta=0.23\,\mathrm{ppm}+S_{\mathrm{A}}+S_{\mathrm{B}}

Short_fiber_thermoplastics.html

  1. s = l d s=\frac{l}{d}

Siegel_Eisenstein_series.html

  1. C , D 1 det ( C Z + D ) k \sum_{C,D}\frac{1}{\det(CZ+D)^{k}}

Siegel_theta_series.html

  1. Θ L g ( T ) = λ L g exp ( π i T r ( λ T λ t ) ) \Theta_{L}^{g}(T)=\sum_{\lambda\in L^{g}}\exp(\pi iTr(\lambda T\lambda^{t}))

Sight_reduction.html

  1. s i n H c = s i n B * s i n D e c + c o s B * c o s D e c * c o s L H A sinHc=sinB*sinDec+cosB*cosDec*cosLHA\,
  2. c o s Z = ( s i n D e c - s i n H c * s i n B ) / ( c o s H c * c o s B ) cosZ=(sinDec-sinHc*sinB)/(cosHc*cosB)\,

Signal-to-interference-plus-noise_ratio.html

  1. SINR ( x ) < m t p l P I + N \mathrm{SINR}(x)<mtpl>{{=}}\frac{P}{I+N}
  2. ( | x - y | ) = | x - y | α \ell(|x-y|)=|x-y|^{\alpha}
  3. SINR ( x i ) < m t p l ( | x i | ) F i j n [ ( | x j | ) F j ] - ( | x i | ) F i + N \mathrm{SINR}(x_{i})<mtpl>{{=}}\frac{\ell(|x_{i}|)F_{i}}{\sum_{j}^{n}[\ell(|x_% {j}|)F_{j}]-\ell(|x_{i}|)F_{i}+N}
  4. SINR ( x i ) < m t p l ( | x i | ) F i j i [ ( | x j | ) F j ] + N \mathrm{SINR}(x_{i})<mtpl>{{=}}\frac{\ell(|x_{i}|)F_{i}}{\sum_{j\neq i}[\ell(|% x_{j}|)F_{j}]+N}
  5. SINR ( x i ) < m t p l | x i | α F i j i | x j | α F j + N \mathrm{SINR}(x_{i})<mtpl>{{=}}\frac{|x_{i}|^{\alpha}F_{i}}{\sum_{j\neq i}|x_{% j}|^{\alpha}F_{j}+N}

Signal_magnitude_area.html

  1. { x 1 , x 2 , , x n } \{x_{1},x_{2},\dots,x_{n}\}
  2. x sma = i = 1 n x i x\text{sma}=\sum_{i=1}^{n}x_{i}
  3. f sma = 1 T 0 T | x ( t ) - a x | + | y ( t ) - a y | + | z ( t ) - a z | d t f\text{sma}={1\over T}\int_{0}^{T}|x(t)-a_{x}|+|y(t)-a_{y}|+|z(t)-a_{z}|\,dt

Similarity_measure.html

  1. s ( x , y ) = - || x - y || 2 2 s(x,y)=-||x-y||_{2}^{2}
  2. || x - y || 2 2 ||x-y||_{2}^{2}

Simmons–Su_protocols.html

  1. Θ ( 1 ϵ n ) \Theta(\frac{1}{\epsilon^{n}})

Simple_matching_coefficient.html

  1. S M C = Number of Matching Attributes Number of Attributes = M 00 + M 11 M 00 + M 01 + M 10 + M 11 SMC={\,\text{Number of Matching Attributes}\over\,\text{Number of Attributes}}% ={{M_{00}+M_{11}}\over{M_{00}+M_{01}+M_{10}+M_{11}}}
  2. M 11 M_{11}
  3. M 01 M_{01}
  4. M 10 M_{10}
  5. M 00 M_{00}

Simplicial_commutative_ring.html

  1. π 0 A \pi_{0}A
  2. π i A \pi_{i}A
  3. π * A \pi_{*}A
  4. π 0 A \pi_{0}A
  5. π * A = i 0 π i A \pi_{*}A=\oplus_{i\geq 0}\pi_{i}A
  6. π * A \pi_{*}A
  7. S 1 S^{1}
  8. x : ( S 1 ) i A , y : ( S 1 ) j A x:(S^{1})^{\wedge i}\to A,\,\,y:(S^{1})^{\wedge j}\to A
  9. ( S 1 ) i × ( S 1 ) j A × A A (S^{1})^{\wedge i}\times(S^{1})^{\wedge j}\to A\times A\to A
  10. ( S 1 ) i ( S 1 ) j A (S^{1})^{\wedge i}\wedge(S^{1})^{\wedge j}\to A
  11. π i + j A \pi_{i+j}A
  12. π i A × π j A π i + j A \pi_{i}A\times\pi_{j}A\to\pi_{i+j}A
  13. x y = ( - 1 ) | x | | y | y x xy=(-1)^{|x||y|}yx
  14. S 1 S 1 S 1 S 1 S^{1}\wedge S^{1}\to S^{1}\wedge S^{1}
  15. Spec A \operatorname{Spec}A

Simplicial_homotopy.html

  1. f , g : X Y f,g:X\to Y
  2. h : X × 1 Y h:X\times\triangle^{1}\to Y
  3. f ( x ) = h ( x , 0 ) f(x)=h(x,0)
  4. g ( x ) = h ( x , 1 ) g(x)=h(x,1)

Simplicial_localization.html

  1. π 0 \pi_{0}
  2. C [ W - 1 ] C[W^{-1}]
  3. π 0 L C ( x , y ) = C [ W - 1 ] ( x , y ) \pi_{0}LC(x,y)=C[W^{-1}](x,y)

Simplicial_presheaf.html

  1. Hom ( - , U ) \operatorname{Hom}(-,U)
  2. B G BG
  3. B GL = lim B GL n B\operatorname{GL}=\underrightarrow{\lim}B\operatorname{GL_{n}}
  4. f : X Y f:X\to Y
  5. f : X Y \mathbb{Z}f:\mathbb{Z}X\to\mathbb{Z}Y
  6. π * F \pi_{*}F
  7. f : X Y f:X\to Y
  8. ( π 0 pr F ) ( X ) = π 0 ( F ( X ) ) (\pi_{0}\text{pr}F)(X)=\pi_{0}(F(X))
  9. ( π i pr ( F , s ) ) ( f ) = π i ( F ( Y ) , f * ( s ) ) (\pi_{i}\text{pr}(F,s))(f)=\pi_{i}(F(Y),f^{*}(s))
  10. π i F \pi_{i}F
  11. π i pr F \pi_{i}\text{pr}F
  12. S o p Δ o p S e t s S^{op}\to\Delta^{op}Sets
  13. 𝒢 \mathcal{F}\to\mathcal{G}
  14. ( U ) 𝒢 ( U ) \mathcal{F}(U)\to\mathcal{G}(U)
  15. F ( X ) holim F ( H n ) F(X)\to\operatorname{holim}F(H_{n})
  16. [ n ] = { 0 , 1 , , n } F ( H n ) [n]=\{0,1,\dots,n\}\mapsto F(H_{n})
  17. F ( X ) F(X)
  18. F π 0 F F\mapsto\pi_{0}F
  19. K ( A , 1 ) K(A,1)
  20. K ( A , i ) = K ( K ( A , i - 1 ) , 1 ) K(A,i)=K(K(A,i-1),1)
  21. H i ( X ; A ) = [ X , K ( A , i ) ] \operatorname{H}^{i}(X;A)=[X,K(A,i)]

Simpson's_Rules_(Ship_Stability).html

  1. A r e a = h 3 ( a + 4 b + 2 c + 4 d + 2 e + 4 f + g ) Area=\frac{h}{3}(a+4b+2c+4d+2e+4f+g)
  2. A r e a = 3 8 ( s u m . . o f . . p r o d u c t s ) Area=\frac{3}{8}(sum..of..products)
  3. A r e a = h 12 ( 5 a + 8 b - c ) Area=\frac{h}{12}(5a+8b-c)

Simpson_correspondence.html

  1. π 1 ( X ) \pi_{1}(X)

Singular_trace.html

  1. Tr ( A ) = n = 0 μ ( n , A ) = ( μ ( A ) ) {\rm Tr}(A)=\sum_{n=0}^{\infty}\mu(n,A)=\sum\left(\mu(A)\right)
  2. Tr ( A ) = n = 0 λ ( n , A ) = ( λ ( A ) ) . {\rm Tr}(A)=\sum_{n=0}^{\infty}\lambda(n,A)=\sum(\lambda(A)).
  3. Tr ( A ) = n = 0 A e n , e n = ( { A e n , e n } n = 0 ) {\rm Tr}(A)=\sum_{n=0}^{\infty}\langle Ae_{n},e_{n}\rangle=\sum(\{\langle Ae_{% n},e_{n}\rangle\}_{n=0}^{\infty})
  4. φ ( A ) = f ( { A e n , e n } n = 0 ) \varphi(A)={\rm f}(\{\langle Ae_{n},e_{n}\rangle\}_{n=0}^{\infty})
  5. S = Tr ( S T ) = n = 0 S e n , e n λ ( n , T ) = v T ( { S e n , e n } n = 0 ) \langle S\rangle={\rm Tr}(ST)=\sum_{n=0}^{\infty}\langle Se_{n},e_{n}\rangle% \lambda(n,T)=v_{T}(\{\langle Se_{n},e_{n}\rangle\}_{n=0}^{\infty})
  6. = =
  7. φ ( S T ) = f ( { S e n , e n λ ( n , T ) } n = 0 ) = v φ , T ( { S e n , e n } n = 0 ) \varphi(ST)={\rm f}(\{\langle Se_{n},e_{n}\rangle\lambda(n,T)\}_{n=0}^{\infty}% )=v_{\varphi,T}(\{\langle Se_{n},e_{n}\rangle\}_{n=0}^{\infty})
  8. = =
  9. = =
  10. = =
  11. = =
  12. = =
  13. S = “limit at infinity” S e n , e n \langle S\rangle=\,\text{``limit at infinity''}\langle Se_{n},e_{n}\rangle
  14. = =
  15. M f = X f ( x ) d x . \int M_{f}=\int_{X}f(x)\,dx.
  16. S = lim n k = 0 n 1 1 + k S e k , e k k = 0 n 1 1 + k \int S=\lim_{n\to\infty}\frac{\sum_{k=0}^{n}\frac{1}{1+k}\langle Se_{k},e_{k}% \rangle}{\sum_{k=0}^{n}\frac{1}{1+k}}
  17. = =
  18. = =
  19. = =
  20. L p = { A K ( H ) : ( n = 0 μ ( n , A ) p ) 1 p < } , L_{p}=\{A\in K(H):\left(\sum_{n=0}^{\infty}\mu(n,A)^{p}\right)^{\frac{1}{p}}<% \infty\},
  21. = =
  22. L p , = { A K ( H ) : μ ( n , A ) = O ( n - 1 p ) } L_{p,\infty}=\{A\in K(H):\mu(n,A)=O(n^{-\frac{1}{p}})\}
  23. = =
  24. = =
  25. Tr ω ( A ) = ω ( { 1 log ( 1 + n ) k = 0 n λ ( k , A ) } n = 0 ) , A L 1 , . {\rm Tr}_{\omega}(A)=\omega\left(\left\{\frac{1}{\log(1+n)}\sum_{k=0}^{n}% \lambda(k,A)\right\}_{n=0}^{\infty}\right),\quad A\in L_{1,\infty}.
  26. E k = { A K ( H ) : μ ( n , A ) = O ( log k - 1 ( n ) / n ) } E_{\otimes k}=\{A\in K(H):\mu(n,A)=O(\log^{k-1}(n)/n)\}
  27. Tr ω k ( A ) = ω ( { 1 log k ( 1 + n ) j = 0 n λ ( j , A ) } n = 0 ) , A E k . {\rm Tr}^{k}_{\omega}(A)=\omega\left(\left\{\frac{1}{\log^{k}(1+n)}\sum_{j=0}^% {n}\lambda(j,A)\right\}_{n=0}^{\infty}\right),\quad A\in E_{\otimes k}.
  28. L ψ = { A K ( H ) : 1 ψ ( 1 + n ) j = 0 n μ ( n , A ) < } . L_{\psi}=\{A\in K(H):\frac{1}{\psi(1+n)}\sum_{j=0}^{n}\mu(n,A)<\infty\}.
  29. Tr ω ψ ( A ) = ω ( { 1 ψ ( 1 + n ) j = 0 n λ ( j , A ) } n = 0 ) , A L ψ {\rm Tr}^{\psi}_{\omega}(A)=\omega\left(\left\{\frac{1}{\psi(1+n)}\sum_{j=0}^{% n}\lambda(j,A)\right\}_{n=0}^{\infty}\right),\quad A\in L_{\psi}
  30. lim inf n ψ ( 2 n ) ψ ( n ) = 1. \liminf_{n\to\infty}\frac{\psi(2n)}{\psi(n)}=1.
  31. = =

Sinhc_function.html

  1. Sinhc ( z ) = sinh ( z ) z \operatorname{Sinhc}(z)=\frac{\sinh(z)}{z}
  2. w ( z ) z - 2 d d z w ( z ) - z d 2 d z 2 w ( z ) = 0 w(z)z-2\,\frac{d}{dz}w(z)-z\frac{d^{2}}{dz^{2}}w(z)=0
  3. Im ( sinh ( x + i y ) x + i y ) \operatorname{Im}\left(\frac{\sinh(x+iy)}{x+iy}\right)
  4. Re ( sinh ( x + i y ) x + i y ) \operatorname{Re}\left(\frac{\sinh(x+iy)}{x+iy}\right)
  5. | sinh ( x + i y ) x + i y | \left|\frac{\sinh(x+iy)}{x+iy}\right|
  6. 1 - sinh ( z ) ) 2 z - sinh ( z ) z 2 \frac{1-\sinh(z))^{2}}{z}-\frac{\sinh(z)}{z^{2}}
  7. - Re ( - 1 - ( sinh ( x + i y ) ) 2 x + i y + sinh ( x + i y ) ( x + i y ) 2 ) -\operatorname{Re}\left(-\frac{1-(\sinh(x+iy))^{2}}{x+iy}+\frac{\sinh(x+iy)}{(% x+iy)^{2}}\right)
  8. - Im ( - 1 - ( sinh ( x + i y ) ) 2 x + i y + sinh ( x + i y ) ( x + i y ) 2 ) -\operatorname{Im}\left(-\frac{1-(\sinh(x+iy))^{2}}{x+iy}+\frac{\sinh(x+iy)}{(% x+iy)^{2}}\right)
  9. | - 1 - ( sinh ( x + i y ) ) 2 x + i y + sinh ( x + i y ) ( x + i y ) 2 | \left|-\frac{1-(\sinh(x+iy))^{2}}{x+iy}+\frac{\sinh(x+iy)}{(x+iy)^{2}}\right|
  10. Sinhc ( z ) = KummerM ( 1 , 2 , 2 z ) e z \operatorname{Sinhc}(z)=\frac{{\rm KummerM}(1,\,2,\,2\,z)}{{\rm e}^{z}}
  11. Sinhc ( z ) = HeunB ( 2 , 0 , 0 , 0 , 2 z ) e z \operatorname{Sinhc}(z)=\frac{\operatorname{HeunB}\left(2,0,0,0,\sqrt{2}\sqrt{% z}\right)}{{\rm e}^{z}}
  12. Sinhc ( z ) = 1 / 2 WhittakerM ( 0 , 1 / 2 , 2 z ) z \operatorname{Sinhc}(z)=1/2\,\frac{{{\rm WhittakerM}(0,\,1/2,\,2\,z)}}{z}
  13. Sinhc z ( 1 + 1 3 z 2 + 2 15 z 4 + 17 315 z 6 + 62 2835 z 8 + 1382 155925 z 10 + 21844 6081075 z 12 + 929569 638512875 z 14 + O ( z 16 ) ) \operatorname{Sinhc}z\approx\left(1+\frac{1}{3}z^{2}+\frac{2}{15}z^{4}+\frac{1% 7}{315}z^{6}+\frac{62}{2835}z^{8}+\frac{1382}{155925}z^{10}+\frac{21844}{60810% 75}z^{12}+\frac{929569}{638512875}z^{14}+O(z^{16})\right)
  14. 𝑆𝑖𝑛ℎ𝑐 ( z ) = ( 1 + 53272705 360869676 z 2 + 38518909 7217393520 z 4 + 269197963 3940696861920 z 6 + 4585922449 15605159573203200 z 8 ) ( 1 - 2290747 120289892 z 2 + 1281433 7217393520 z 4 - 560401 562956694560 z 6 + 1029037 346781323848960 z 8 ) - 1 {\it Sinhc}\left(z\right)=\left(1+{\frac{53272705}{360869676}}\,{z}^{2}+{\frac% {38518909}{7217393520}}\,{z}^{4}+{\frac{269197963}{3940696861920}}\,{z}^{6}+{% \frac{4585922449}{15605159573203200}}\,{z}^{8}\right)\left(1-{\frac{2290747}{1% 20289892}}\,{z}^{2}+{\frac{1281433}{7217393520}}\,{z}^{4}-{\frac{560401}{56295% 6694560}}\,{z}^{6}+{\frac{1029037}{346781323848960}}\,{z}^{8}\right)^{-1}

Sitnikov_problem.html

  1. ( m 1 = m 2 = m 2 ) \left(m_{1}=m_{2}=\tfrac{m}{2}\right)
  2. ( m 3 = 0 ) (m_{3}=0)
  3. ( m = 1 ) (m=1)
  4. ( 2 π ) (2\pi)
  5. ( a = 1 ) (a=1)
  6. E \,E
  7. E = 1 2 ( d z d t ) 2 - 1 r E=\frac{1}{2}\left(\frac{dz}{dt}\right)^{2}-\frac{1}{r}
  8. d 2 z d t 2 = - z r 3 \frac{d^{2}z}{dt^{2}}=-\frac{z}{r^{3}}
  9. r 2 = a 2 + z 2 = 1 + z 2 r^{2}=a^{2}+z^{2}=1+z^{2}
  10. d 2 z d t 2 = - z ( 1 + z 2 ) 3 \frac{d^{2}z}{dt^{2}}=-\frac{z}{\left(\sqrt{1+z^{2}}\right)^{3}}

Six_operations.html

  1. f * f^{*}
  2. f * f_{*}
  3. f ! f_{!}
  4. f ! f^{!}
  5. f * f^{*}
  6. f * f_{*}
  7. f ! f_{!}
  8. f ! f^{!}
  9. \ell
  10. \ell
  11. L g * R f ! R f ! L g * , Lg^{*}\circ Rf_{!}\to Rf^{\prime}_{!}\circ Lg^{\prime*},
  12. R g * f ! f ! R g * . Rg^{\prime}_{*}\circ f^{\prime!}\to f^{!}\circ Rg_{*}.
  13. ( R f ! M ) Y N R f ! ( M X L f * N ) , (Rf_{!}M)\otimes_{Y}N\to Rf_{!}(M\otimes_{X}Lf^{*}N),
  14. RHom Y ( R f ! M , N ) R f * RHom X ( M , f ! N ) , \operatorname{RHom}_{Y}(Rf_{!}M,N)\to Rf_{*}\operatorname{RHom}_{X}(M,f^{!}N),
  15. f ! RHom Y ( M , N ) RHom X ( L f * M , f ! N ) . f^{!}\operatorname{RHom}_{Y}(M,N)\to\operatorname{RHom}_{X}(Lf^{*}M,f^{!}N).
  16. R j ! j ! 1 R i * i * R j ! j ! [ 1 ] , Rj_{!}j^{!}\to 1\to Ri_{*}i^{*}\to Rj_{!}j^{!}[1],
  17. 1 Z ( - c ) [ - 2 c ] i ! 1 S , 1_{Z}(-c)[-2c]\to i^{!}1_{S},
  18. \ell
  19. M D X ( D X ( M ) ) , M\to D_{X}(D_{X}(M)),
  20. D X ( M D X ( M ) ) RHom ( M , M ) , D_{X}(M\otimes D_{X}(M^{\prime}))\to\operatorname{RHom}(M,M^{\prime}),
  21. D X ( f * N ) f ! ( D Y ( N ) ) , D_{X}(f^{*}N)\cong f^{!}(D_{Y}(N)),
  22. D X ( f ! N ) f * ( D Y ( N ) ) , D_{X}(f^{!}N)\cong f^{*}(D_{Y}(N)),
  23. D Y ( f ! M ) f * ( D X ( M ) ) , D_{Y}(f_{!}M)\cong f_{*}(D_{X}(M)),
  24. D Y ( f * M ) f ! ( D X ( M ) ) . D_{Y}(f_{*}M)\cong f_{!}(D_{X}(M)).

Skin_friction_drag.html

  1. Re = 𝐕𝐋 < m t p l > ν \mathrm{Re}=\frac{{\mathbf{V}}{\mathbf{L}}}{<}mtpl>{{\nu}}
  2. 𝐕 {\mathbf{V}}
  3. 𝐋 {\mathbf{L}}
  4. ν {\nu}
  5. C f = 1.328 Re C_{f}=\frac{1.328}{\sqrt{\mathrm{Re}}}
  6. C f l = .664 Re l C_{fl}=\frac{.664}{\sqrt{\mathrm{Re}_{l}}}
  7. C f = .455 l o g ( Re ) 2.58 C_{f}=\frac{.455}{log(\mathrm{Re})^{2.58}}
  8. F = C f ρ V 2 2 S w e t t e d F=C_{f}\frac{\rho V^{2}}{2}\ S_{wetted}
  9. 𝐒 w e t t e d {\mathbf{S}_{wetted}}

SL2.html

  1. 𝔰 𝔩 2 \mathfrak{sl}_{2}

Slice_theorem_(differential_geometry).html

  1. G / G x M , [ g ] g x G/G_{x}\to M,\,[g]\mapsto g\cdot x
  2. G / G x G/G_{x}
  3. G × G x T x M / T x ( G x ) G\times_{G_{x}}T_{x}M/T_{x}(G\cdot x)
  4. M / G M/G

Smith_space.html

  1. X X
  2. K K
  3. T X T\subseteq X
  4. T λ K T\subseteq\lambda\cdot K
  5. λ > 0 \lambda>0
  6. X X
  7. X X^{\star}
  8. X X
  9. X X^{\star}
  10. X X
  11. X X^{\star}
  12. f : X f:X\to\mathbb{C}
  13. X X

Smooth_maximum.html

  1. { x 1 , , x n } max { x 1 , , x n } , \{x_{1},\ldots,x_{n}\}\mapsto\max\{x_{1},\ldots,x_{n}\},\,
  2. α > 0 \alpha>0
  3. 𝒮 α ( { x i } i = 1 n ) = i = 1 n x i e α x i i = 1 n e α x i \mathcal{S}_{\alpha}(\{x_{i}\}_{i=1}^{n})=\frac{\sum_{i=1}^{n}x_{i}e^{\alpha x% _{i}}}{\sum_{i=1}^{n}e^{\alpha x_{i}}}
  4. 𝒮 α \mathcal{S}_{\alpha}
  5. 𝒮 α max \mathcal{S}_{\alpha}\to\max
  6. α \alpha\to\infty
  7. 𝒮 0 \mathcal{S}_{0}
  8. 𝒮 α min \mathcal{S}_{\alpha}\to\min
  9. α - \alpha\to-\infty
  10. 𝒮 α \mathcal{S}_{\alpha}
  11. x i 𝒮 α ( { x i } i = 1 n ) = e α x i j = 1 n e α x j [ 1 + α ( x i - 𝒮 α ( { x i } i = 1 n ) ) ] . \nabla_{x_{i}}\mathcal{S}_{\alpha}(\{x_{i}\}_{i=1}^{n})=\frac{e^{\alpha x_{i}}% }{\sum_{j=1}^{n}e^{\alpha x_{j}}}[1+\alpha(x_{i}-\mathcal{S}_{\alpha}(\{x_{i}% \}_{i=1}^{n}))].
  12. g ( x 1 , , x n ) = log ( exp ( x 1 ) + + exp ( x n ) ) g(x_{1},\ldots,x_{n})=\log(\exp(x_{1})+\ldots+\exp(x_{n}))

Smooth_topology.html

  1. l \mathbb{Q}_{l}
  2. B 𝔾 m B\mathbb{G}_{m}
  3. Spec 𝐅 q \operatorname{Spec}\mathbf{F}_{q}
  4. B 𝔾 m = Spec 𝐅 q B\mathbb{G}_{m}=\operatorname{Spec}\mathbf{F}_{q}
  5. B 𝔾 m B\mathbb{G}_{m}
  6. P \mathbb{C}P^{\infty}
  7. B 𝔾 m B\mathbb{G}_{m}

Snaith's_theorem.html

  1. P \mathbb{C}P^{\infty}

Snub_rhombicuboctahedron.html

  1. s r { 4 3 } sr\begin{Bmatrix}4\\ 3\end{Bmatrix}

SNV_calling_from_NGS_data.html

  1. P ( G | D ) \displaystyle P(G|D)
  2. D D
  3. G G
  4. G i G_{i}
  5. P ( G ) P(G)
  6. P ( D | G ) P(D|G)
  7. P ( D | G ) P(D|G)
  8. P ( D | G ) P(D|G)

Social_cognitive_optimization.html

  1. f ( x ) f(x)
  2. x x
  3. S S
  4. f f
  5. N c N_{c}
  6. N L N_{L}
  7. x i x_{i}
  8. i i
  9. X X
  10. S S
  11. t t
  12. ( t = 1 , , T ) (t=1,\ldots,T)
  13. i i
  14. ( i = 1 , , N c ) (i=1,\ldots,N_{c})
  15. x M x_{M}
  16. X ( t ) X(t)
  17. τ B \tau_{B}
  18. x i ( t ) x_{i}(t)
  19. x M x_{M}
  20. x B a s e x_{Base}
  21. x R e f x_{Ref}
  22. x B a s e x_{Base}
  23. x R e f x_{Ref}
  24. x i ( t + 1 ) x_{i}(t+1)
  25. x i ( t + 1 ) x_{i}(t+1)
  26. x B a s e x_{Base}
  27. x B a s e x_{Base}
  28. x R e f x_{Ref}
  29. x B a s e x_{Base}
  30. x i ( t + 1 ) S x_{i}(t+1)\in S
  31. x i ( t + 1 ) x_{i}(t+1)
  32. X X
  33. i i
  34. x i ( t ) x_{i}(t)
  35. x i ( t + 1 ) x_{i}(t+1)
  36. X ( t ) X(t)
  37. X ( t + 1 ) X(t+1)
  38. τ W \tau_{W}
  39. X ( t ) X(t)
  40. N c N_{c}
  41. N L N_{L}
  42. T T
  43. N L + N c * ( T + 1 ) N_{L}+N_{c}*(T+1)
  44. N L N_{L}
  45. T T
  46. N c N_{c}
  47. N c = 1 N_{c}=1
  48. N c N_{c}
  49. N L N_{L}

Sodium_bifluoride.html

  1. \overrightarrow{\leftarrow}

Soil-Adjusted_Vegetation_Index.html

  1. SAVI = ( 1 + L ) ( NIR - Red ) ( NIR + Red + L ) \mbox{SAVI}~{}=\frac{(\mbox{1}~{}+\mbox{L}~{})(\mbox{NIR}~{}-\mbox{Red}~{})}{(% \mbox{NIR}~{}+\mbox{Red}~{}+\mbox{L}~{})}

Solid_bowl_centrifuge.html

  1. P e r c e n t a g e o f S o l i d R e c o v e r y = s o l i d i n f e e d - s o l i d i n e f f l u e n t s o l i d i n f e e d × 100 % {Percentage\ of\ Solid\ Recovery}=\frac{solid\ in\ feed-solid\ in\ effluent}{% solid\ in\ feed}\times{100\%}

Solid_partition.html

  1. n n
  2. n i , j , k n_{i,j,k}
  3. i , j , k 1 i,\ j,\ k\geq 1
  4. i , j , k n i , j , k = n \sum_{i,j,k}n_{i,j,k}=n
  5. n i + 1 , j , k n i , j , k , n i , j + 1 , k n i , j , k and n i , j , k + 1 n i , j , k , i , j and k . n_{i+1,j,k}\leq n_{i,j,k}\quad,\quad n_{i,j+1,k}\leq n_{i,j,k}\quad\,\text{and% }\quad n_{i,j,k+1}\leq n_{i,j,k}\quad,\quad\forall\ i,\ j\,\text{ and }k\ .
  6. p 3 ( n ) p_{3}(n)
  7. n n
  8. n n
  9. n n
  10. λ = ( 𝐲 1 , 𝐲 2 , , 𝐲 n ) \lambda=(\mathbf{y}_{1},\mathbf{y}_{2},\ldots,\mathbf{y}_{n})
  11. 𝐲 i 0 4 \mathbf{y}_{i}\in\mathbb{Z}_{\geq 0}^{4}
  12. 𝐚 = ( a 1 , a 2 , a 3 , a 4 ) λ \mathbf{a}=(a_{1},a_{2},a_{3},a_{4})\in\lambda
  13. 𝐲 = ( y 1 , y 2 , y 3 , y 4 ) \mathbf{y}=(y_{1},y_{2},y_{3},y_{4})
  14. 0 y i a i 0\leq y_{i}\leq a_{i}
  15. i = 1 , 2 , 3 , 4 i=1,2,3,4
  16. ( 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1 0 0 ) , \left(\begin{smallmatrix}0\\ 0\\ 0\\ 0\end{smallmatrix}\begin{smallmatrix}0\\ 0\\ 1\\ 0\end{smallmatrix}\begin{smallmatrix}0\\ 1\\ 0\\ 0\end{smallmatrix}\begin{smallmatrix}1\\ 0\\ 0\\ 0\end{smallmatrix}\begin{smallmatrix}1\\ 1\\ 0\\ 0\end{smallmatrix}\right)\ ,
  17. 5 5
  18. S 4 S_{4}
  19. n i , j , k n_{i,j,k}
  20. ( i - 1 , j - 1 , k - 1 , * ) (i-1,j-1,k-1,*)
  21. * *
  22. n i , j , k n_{i,j,k}
  23. n i , j , k n_{i,j,k}
  24. n i , j , k n_{i,j,k}
  25. n i , j , k n_{i,j,k}
  26. ( i - 1 , j - 1 , k - 1 , y 4 ) (i-1,j-1,k-1,y_{4})
  27. 0 y 4 < n i , j , k 0\leq y_{4}<n_{i,j,k}
  28. 5 5
  29. n 1 , 1 , 1 = n 2 , 1 , 1 = n 1 , 2 , 1 = n 1 , 1 , 2 = n 2 , 2 , 1 = 1 n_{1,1,1}=n_{2,1,1}=n_{1,2,1}=n_{1,1,2}=n_{2,2,1}=1
  30. n i , j , k n_{i,j,k}
  31. p 3 ( 0 ) 1 p_{3}(0)\equiv 1
  32. P 3 ( q ) P_{3}(q)
  33. P 3 ( q ) := n = 0 p 3 ( n ) q n = 1 + q + 4 q 2 + 10 q 3 + 26 q 4 + 59 q 5 + 140 q 6 + P_{3}(q):=\sum_{n=0}^{\infty}p_{3}(n)\ q^{n}=1+q+4\ q^{2}+10\ q^{3}+26\ q^{4}+% 59\ q^{5}+140\ q^{6}+\cdots
  34. n 28 n\leq 28
  35. n 50 n\leq 50
  36. n 72 n\leq 72
  37. p 3 ( 72 ) = 3464274974065172792 , p_{3}(72)=3464274974065172792\ ,
  38. lim n n - 3 / 4 ln p 3 ( n ) a constant . \lim_{n\rightarrow\infty}n^{-3/4}\ln p_{3}(n)\rightarrow\,\text{a constant}.
  39. 1.79 ± 0.01 1.79\pm 0.01

Solution_algorithms_for_pressure-velocity_coupling_in_steady_flows.html

  1. ρ u x + ρ v y = 0 \frac{\partial\rho u}{\partial x}+\frac{\partial\rho v}{\partial y}=0
  2. ρ u u x + ρ v u y = ν u x x + ν u y y - p x + S u \frac{\partial\rho uu}{\partial x}+\frac{\partial\rho vu}{\partial y}=\frac{% \partial\frac{\nu\partial u}{\partial x}}{\partial x}+\frac{\partial\frac{\nu% \partial u}{\partial y}}{\partial y}-\frac{\partial p}{\partial x}+S_{u}
  3. ρ u v x + ρ v v y = ν v x x + ν v y y - p y + S v \frac{\partial\rho uv}{\partial x}+\frac{\partial\rho vv}{\partial y}=\frac{% \partial\frac{\nu\partial v}{\partial x}}{\partial x}+\frac{\partial\frac{\nu% \partial v}{\partial y}}{\partial y}-\frac{\partial p}{\partial y}+S_{v}
  4. ρ \rho
  5. p x = P p - P w x u \frac{\partial p}{\partial x}=\frac{P_{p}-P_{w}}{\partial x_{u}}
  6. p y = P p - P s y v \frac{\partial p}{\partial y}=\frac{P_{p}-P_{s}}{\partial y_{v}}
  7. a i , J u i , J = a n b u n b - Δ V u P I , J - P I - 1 , J x u + S Δ V u a_{i},_{J}u_{i},_{J}=\sum a_{n}bu_{n}b-\Delta V_{u}\frac{P_{I},_{J}-P_{I-1},_{% J}}{\partial x_{u}}+S\Delta V_{u}
  8. a n b u n b \sum a_{nb}u^{\prime}_{nb}
  9. a n b v n b \sum a_{nb}v^{\prime}_{nb}

Solvent_models.html

  1. H ^ t o t a l ( r m ) = H ^ m o l e c u l e ( r m ) + V ^ m o l e c u l e + s o l v e n t ( r m ) \hat{H}^{total}(r_{m})=\hat{H}^{molecule}(r_{m})+\hat{V}^{molecule+solvent}(r_% {m})
  2. ( r m ) (r_{m})
  3. V ^ m o l e c u l e s + s o l v e n t \hat{V}^{molecules+solvent}
  4. Q ( m ) = Q c a v i t y + Q e l e c t r o s t a t i c + Q d i s p e r s i o n + Q r e p u l s i o n Q(m)=Q_{cavity}+Q_{electrostatic}+Q_{dispersion}+Q_{repulsion}
  5. G = G c a v i t y + G e l e c t r o s t a t i c + G d i s p e r s i o n + G r e p u l s i o n + G t h e r m a l m o t i o n G=G_{cavity}+G_{electrostatic}+G_{dispersion}+G_{repulsion}+G_{thermal\;motion}
  6. h ( r - r ; Θ - Θ ) = g ( r - r ; Θ - Θ ) - 1 h(r-r^{{}^{\prime}};\Theta-\Theta^{{}^{\prime}})=g(r-r^{{}^{\prime}};\Theta-% \Theta^{{}^{\prime}})-1
  7. h ( r ; Θ ) h(r;\Theta)
  8. g ( r ; Θ ) g(r;\Theta)
  9. c ( r 1 , 3 ) c(r_{1,3})
  10. h ( r 2 , 3 ) h(r_{2,3})
  11. h ( r ) = c ( r 1 , 2 ) + d r 3 c ( r 1 , 3 ) ρ ( r 3 ) h ( r 2 , 3 ) h(r)=c(r_{1,2})+\int dr_{3}c(r_{1,3})\rho(r_{3})h(r_{2,3})
  12. h α ( r ) = { e - β U ( r ) + T ( r ) - 1 ( when - β υ a ( r ) + h a ( r ) - c a ( r ) 0 ) - β U ( r ) + T ( r ) ( when - β υ a ( r ) + h a ( r ) - c a ( r ) > 0 ) h_{\alpha}(r)=\begin{cases}e^{-\beta U(r)+T(r)}-1&(\,\text{when}-\beta\upsilon% _{a}(r)+h_{a}(r)-c_{a}(r)\leq 0)\\ -\beta U(r)+T(r)&(\,\text{when}-\beta\upsilon_{a}(r)+h_{a}(r)-c_{a}(r)>0)\end{cases}
  13. β = 1 k B T \beta=\frac{1}{k_{B}T}
  14. U ( r ) U(r)
  15. U ( r ) = 4 ϵ [ ( σ 1 r 12 ) 12 - ( σ 2 r 12 ) 6 ] + Q 1 Q 2 r 12 U(r)=4\epsilon\left[\left(\frac{\sigma_{1}}{r_{12}}\right)^{12}-\left(\frac{% \sigma_{2}}{r_{12}}\right)^{6}\right]+\frac{Q_{1}Q_{2}}{r_{12}}

Sommerfeld_parameter.html

  1. η η
  2. η = Z 1 Z 2 e 2 4 π ϵ 0 v \eta=\frac{Z_{1}Z_{2}e^{2}}{4\pi\epsilon_{0}\hbar v}
  3. e e
  4. v v
  5. P P
  6. P = exp ( - 2 π η ) P=\exp(-2\pi\eta)

Sonar_signal_processing.html

  1. s ( t , r ) s(t,\vec{r})
  2. t t
  3. r = ( x , y , z ) {}\vec{r}=(x,y,z)
  4. s ( w , k ) = s ( t , r ) e - j ( w t - k r ) d x d t , {}s(w,\vec{k})=\iiiint\limits\,s(t,\vec{r})\cdot e^{-j(wt-\vec{k}\vec{r})}d% \vec{x}dt,
  5. k = ( k x , k y , k z ) , \vec{k}=(k_{x},k_{y},k_{z}),
  6. s ( t , r ) = s ( w , k ) e j ( w t - k r ) d k d w , {}s(t,\vec{r})=\iiiint\limits\,s(w,\vec{k})\cdot e^{j(wt-\vec{k}\vec{r})}d\vec% {k}dw,
  7. w w
  8. k \vec{k}
  9. s ( t , r ) = e - j ( w t - k r ) , s(t,\vec{r})=e^{-j(wt-\vec{k}\vec{r})},
  10. k \vec{k}
  11. v = w | k | v=\frac{w}{|\vec{k}|}
  12. λ = 2 π | k | \lambda=\frac{2\pi}{|\vec{k}|}
  13. s ( t , r ) s(t,\vec{r})
  14. r ( t ) = r ( n T ) = s ( r , n T ) r(t)=r(nT)=s(\vec{r},nT)
  15. r i ( t ) = s ( x i , t ) r_{i}(t)=s(\vec{x}_{i},t)
  16. x i = ( x i , 0 , 0 ) = ( i D , 0 , 0 ) \vec{x}_{i}=(x_{i},0,0)=(iD,0,0)
  17. k = k 0 \vec{k}=\vec{k}_{0}
  18. t i = k 0 w x i t_{i}=\frac{\vec{k}_{0}}{w}\vec{x}_{i}
  19. i i
  20. x i ( i = 0 , 1 , , M - 1 ) x_{i}(i=0,1,...,M-1)
  21. r i ( t ) r_{i}(t)
  22. b ( t ) = 1 M i = 0 i = M - 1 w i r i ( t - t i ) b(t)=\frac{1}{M}\sum_{i=0}^{i=M-1}{w_{i}r_{i}(t-t_{i})}
  23. y ( n ) = k = 0 N - 1 h ( k ) x ( n - k ) y(n)=\sum_{k=0}^{N-1}{h(k)x(n-k)}
  24. y ( n 1 , n 2 ) = k 1 = 0 M 1 - 1 k 2 = 0 M 2 - 1 h ( k 1 , k 2 ) x ( n 1 - k 1 , n 2 - k 2 ) y(n1,n2)=\sum_{k_{1}=0}^{M_{1}-1}\sum_{k_{2}=0}^{M_{2}-1}{h(k_{1},k_{2})x(n_{1% }-k_{1},n_{2}-k_{2})}
  25. y ( n ) = k = 0 N - 1 a k x ( n - k ) + k = 0 M - 1 b k y ( n - k ) y(n)=\sum_{k=0}^{N-1}{a_{k}x(n-k)}+\sum_{k=0}^{M-1}{b_{k}y(n-k)}
  26. y ( n 1 , n 2 ) = r 1 = 0 N 1 - 1 r 2 = 0 N 2 - 1 a ( r 1 , r 2 ) x ( n 1 - r 1 , n 2 - r 2 ) - l 1 = 0 M 1 - 1 l 2 = 0 M 2 - 1 b ( l 1 , l 2 ) y ( l 1 , l 2 ) y(n_{1},n_{2})=\sum_{r_{1}=0}^{N_{1}-1}\sum_{r_{2}=0}^{N_{2}-1}{a(r_{1},r_{2})% x(n_{1}-r_{1},n_{2}-r_{2})}-\sum_{l_{1}=0}^{M_{1}-1}\sum_{l_{2}=0}^{M_{2}-1}{b% (l_{1},l_{2})y(l_{1},l_{2})}

Sound_exposure.html

  1. E = t 0 t 1 p ( t ) 2 d t , E=\int_{t_{0}}^{t_{1}}p(t)^{2}\,\mathrm{d}t,
  2. L E = 1 2 ln ( E E 0 ) Np = log 10 ( E E 0 ) B = 10 log 10 ( E E 0 ) dB , L_{E}=\frac{1}{2}\ln\!\left(\frac{E}{E_{0}}\right)\!~{}\mathrm{Np}=\log_{10}\!% \left(\frac{E}{E_{0}}\right)\!~{}\mathrm{B}=10\log_{10}\!\left(\frac{E}{E_{0}}% \right)\!~{}\mathrm{dB},
  3. E 0 = 400 μ Pa 2 s . E_{0}=400~{}\mathrm{\mu Pa^{2}\cdot s}.

Space_cloth.html

  1. R = r 1 r 2 η 2 π ρ d ρ = η 2 π ln r 2 r 1 R=\int_{r_{1}}^{r_{2}}\frac{\eta}{2\pi\rho}d\rho=\frac{\eta}{2\pi}\ln\frac{r_{% 2}}{r_{1}}
  2. Z 0 = η π ln 2 D d Z_{0}=\frac{\eta}{\pi}\ln\frac{2D}{d}
  3. R = R s π ln 2 D d R=\frac{R_{s}}{\pi}\ln\frac{2D}{d}
  4. R I = η h 1 w R_{I}=\eta\frac{h_{1}}{w}
  5. R I I = R I I I = η h 2 w R_{II}=R_{III}=\eta\frac{h_{2}}{w}
  6. Z C M = R I I 2 = η h 2 2 w Z_{CM}=\frac{R_{II}}{2}=\frac{\eta h_{2}}{2w}
  7. Z D M = ( R I ) ( 2 R I I ) R I + 2 R I I = η w 2 h 2 h 1 h 1 + 2 h 2 Z_{DM}=\frac{(R_{I})(2R_{II})}{R_{I}+2R_{II}}=\frac{\eta}{w}\frac{2h_{2}h_{1}}% {h_{1}+2h_{2}}

Sparse_matrix-vector_multiplication.html

  1. y = A x y=Ax
  2. A A
  3. x x
  4. y y
  5. y = A x y=Ax
  6. A A
  7. A A

Sparse_network.html

  1. L = N 2 L=N^{2}
  2. L < L m a x L<L_{max}
  3. L = N ( N - 1 ) / 2 L=N(N-1)/2

Spatial_Poisson_process.html

  1. d d
  2. d \mathbb{R}^{d}
  3. d 1 d\geq 1
  4. d \mathbb{R}^{d}
  5. Π \Pi
  6. d \mathbb{R}^{d}
  7. Π \Pi
  8. Π \Pi
  9. | Π A | |\Pi\cap A|
  10. A d A\subset\mathbb{R}^{d}
  11. A A
  12. A d A\in\mathcal{B}^{d}
  13. d \mathcal{B}^{d}
  14. σ \sigma
  15. | A | |A|
  16. A A
  17. S d S\not\subset\mathbb{R}^{d}
  18. S S
  19. σ \sigma
  20. μ \mu
  21. | Π A | |\Pi\cap A|
  22. μ ( Π A ) \mu(\Pi\cap A)
  23. λ \lambda
  24. A A
  25. Π \Pi
  26. d \mathbb{R}^{d}
  27. λ \lambda
  28. A d A\in\mathcal{B}^{d}
  29. N ( A ) := | Π A | N(A):=|\Pi\cap A|
  30. N ( A ) N(A)
  31. λ | A | \lambda|A|
  32. A 1 , A 2 , , A n A_{1},A_{2},\ldots,A_{n}
  33. d \mathcal{B}^{d}
  34. N ( A 1 ) , N ( A 2 ) , , N ( A n ) N(A_{1}),N(A_{2}),\ldots,N(A_{n})
  35. N N
  36. λ > 0 \lambda>0
  37. | A | = |A|=\infty
  38. ( | Π A | = ) = 1 \mathbb{P}(|\Pi\cap A|=\infty)=1
  39. λ \lambda
  40. Λ ( A ) \Lambda(A)
  41. | A | |A|
  42. λ d 𝐱 \lambda d\,\textbf{x}
  43. λ ( 𝐱 ) d 𝐱 \lambda(\,\textbf{x})d\,\textbf{x}
  44. Λ ( A ) := A λ ( 𝐱 ) d 𝐱 , A d . \Lambda(A):=\int_{A}\lambda(\,\textbf{x})\,d\,\textbf{x},\quad A\in\mathcal{B}% ^{d}.
  45. λ : d \lambda:\mathbb{R}^{d}\to\mathbb{R}
  46. Λ ( A ) < \Lambda(A)<\infty
  47. A A
  48. Π d \Pi\subset\mathbb{R}^{d}
  49. λ \lambda
  50. d \mathcal{B}^{d}
  51. N ( A ) = | Π A | N(A)=|\Pi\cap A|
  52. N ( A ) N(A)
  53. Λ ( A ) \Lambda(A)
  54. A 1 , A 2 , , A n A_{1},A_{2},\ldots,A_{n}
  55. d \mathcal{B}^{d}
  56. N ( A 1 ) , N ( A 2 ) , , N ( A n ) N(A_{1}),N(A_{2}),\ldots,N(A_{n})
  57. Λ ( A ) , A d \Lambda(A),A\in\mathcal{B}^{d}
  58. Π \Pi
  59. λ \lambda
  60. λ \lambda
  61. ( N ( A i ) = k i , 1 i n ) = ( λ A 1 ) k 1 k 1 ! e - λ | A 1 | ( λ A n ) k n k n ! e - λ | A n | , \mathbb{P}(N(A_{i})=k_{i},1\leq i\leq n)=\dfrac{(\lambda A_{1})^{k_{1}}}{k_{1}% !}\cdot e^{-\lambda|A_{1}|}\cdots\dfrac{(\lambda A_{n})^{k_{n}}}{k_{n}!}\cdot e% ^{-\lambda|A_{n}|},
  62. A 1 , , A n A_{1},...,A_{n}
  63. k 1 , , k n k_{1},\ldots,k_{n}
  64. N ( A ) N(A)
  65. N ( A ) N(A)
  66. λ \lambda
  67. N ( A ) N(A)
  68. { 0 , 1 , 2 , } \{0,1,2,\ldots\}
  69. 0 < ( N ( A ) = 0 ) < 1 0<\mathbb{P}(N(A)=0)<1
  70. 0 < | A | < 0<|A|<\infty
  71. N ( A ) N(A)
  72. A A
  73. | A | |A|
  74. 0 < ( N ( A ) 1 ) = λ | A | + o ( | A | ) 0<\mathbb{P}(N(A)\geq 1)=\lambda|A|+o(|A|)
  75. | A | 0. |A|\downarrow 0.
  76. m 2 , m\geq 2,
  77. A 1 , , A m A_{1},\ldots,A_{m}
  78. N ( A 1 ) , N ( A 2 ) , , N ( A m ) N(A_{1}),N(A_{2}),\ldots,N(A_{m})
  79. N ( A 1 A 2 A m ) = N ( A 1 ) + N ( A 2 ) + + N ( A m ) . N(A_{1}\cup A_{2}\cup\cdots\cup A_{m})=N(A_{1})+N(A_{2})+\cdots+N(A_{m}).
  80. lim | A | 0 ( N ( A ) 1 ) ( N ( A ) = 1 ) = 1. \lim\limits_{|A|\to 0}\dfrac{\mathbb{P}(N(A)\geq 1)}{\mathbb{P}(N(A)=1)}=1.
  81. N ( A ) N(A)
  82. A A
  83. A A
  84. | A | > 0 |A|>0
  85. N ( A ) = 1 N(A)=1
  86. A A
  87. ( N ( B ) = 1 N ( A ) = 1 ) = | B | | A | \mathbb{P}(N(B)=1\mid N(A)=1)=\dfrac{|B|}{|A|}
  88. B A B\subset A
  89. | A | > 0 |A|>0
  90. A A
  91. n n
  92. A A
  93. A 1 , , A m A_{1},\ldots,A_{m}
  94. A A
  95. k i k_{i}
  96. k 1 + + k m = n k_{1}+\cdots+k_{m}=n
  97. ( N ( A 1 ) = k 1 , , N ( A m ) = k m N ( A ) = n ) = n ! k 1 ! k m ! ( | A 1 | | A | ) k 1 ( | A m | | A | ) k m . \mathbb{P}(N(A_{1})=k_{1},\ldots,N(A_{m})=k_{m}\mid N(A)=n)=\dfrac{n!}{k_{1}!% \cdots k_{m}!}\left(\dfrac{|A_{1}|}{|A|}\right)^{k_{1}}\cdots\left(\dfrac{|A_{% m}|}{|A|}\right)^{k_{m}}.
  98. Π \Pi^{\prime}
  99. Π ′′ \Pi^{\prime\prime}
  100. d \mathbb{R}^{d}
  101. λ \lambda^{\prime}
  102. λ ′′ \lambda^{\prime\prime}
  103. Π = Π Π ′′ \Pi=\Pi^{\prime}\cup\Pi^{\prime\prime}
  104. λ = λ + λ ′′ \lambda=\lambda^{\prime}+\lambda^{\prime\prime}
  105. Π \Pi
  106. d \mathbb{R}^{d}
  107. λ ( 𝐱 ) \lambda(\,\textbf{x})
  108. Π \Pi
  109. Π \Pi
  110. γ ( 𝐱 ) \gamma(\,\textbf{x})
  111. σ ( 𝐱 ) = 1 - γ ( 𝐱 ) \sigma(\,\textbf{x})=1-\gamma(\,\textbf{x})
  112. Γ \Gamma
  113. Σ \Sigma
  114. Γ \Gamma
  115. Σ \Sigma
  116. γ ( 𝐱 ) λ ( 𝐱 ) \gamma(\,\textbf{x})\lambda(\,\textbf{x})
  117. σ ( 𝐱 ) λ ( 𝐱 ) \sigma(\,\textbf{x})\lambda(\,\textbf{x})

SPEARpesticides.html

  1. S P E A R p e s t i c i d e s = i = 1 n log ( x i + 1 ) y i = 1 n log ( x i + 1 ) SPEAR_{pesticides}=\frac{\sum_{i=1}^{n}\log(x_{i}+1)y}{\sum_{i=1}^{n}\log(x_{i% }+1)}

Specific_mechanical_energy.html

  1. e k + e u e_{k}+e_{u}

Specific_potential_energy.html

  1. e u = g h e_{u}=gh

Spectral_estimation_of_multidimensional_signals.html

  1. P ( K x , w ) = - - φ s s ( x , t ) e - j ( w t - k x ) d x d t P\left(K_{x},w\right)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\varphi_{% ss}\left(x,t\right)\ e^{-j\left(wt-k^{\prime}x\right)}\,dx\,dt
  2. φ s s ( x , t ) = s [ ( ξ , τ ) s * ( ξ - x , τ - t ) ] \varphi_{ss}\left(x,t\right)=s\left[\left(\xi,\tau\right)s*\left(\xi-x,\tau-t% \right)\right]
  3. P B ( w ) = 1 d e t N l | n x ( n + M I ) e - j ( w n ) | 2 P_{B}\left(w\right)=\frac{1}{detN}\sum_{l}|\sum_{n}\ x\left(n+MI\right)\ e^{-j% \left(w^{\prime}n\right)}|^{2}
  4. P M ( w ) = 1 d e t N | n g ( n ) x ( n ) e - j ( w n ) | 2 P_{M}\left(w\right)=\frac{1}{detN}|\sum_{n}\ g\left(n\right)\ x\left(n\right)% \ e^{-j\left(w^{\prime}n\right)}|^{2}
  5. P W ( w ) = 1 d e t N l | n g ( n ) x ( n + M I ) e - j ( w n ) | 2 P_{W}\left(w\right)=\frac{1}{detN}\sum_{l}|\sum_{n}\ g\left(n\right)\ x\left(n% +MI\right)\ e^{-j\left(w^{\prime}n\right)}|^{2}
  6. P C ( K o x , w o ) = E [ | y ( i , n ) | 2 ] = 1 α = 0 N - 1 β = 0 M - 1 l = 0 N - 1 m = 0 M - 1 ψ e ( l , α ; m , β ) P_{C}\left(K_{o}x,w_{o}\right)=E\left[|y\left(i,n\right)|^{2}\right]=\frac{1}{% \sum_{\alpha=0}^{N-1}\sum_{\beta=0}^{M-1}\sum_{l=0}^{N-1}\sum_{m=0}^{M-1}\psi_% {e}\left(l,\alpha;m,\beta\right)}
  7. P A ( k x , w ) P_{A}(k_{x},w)
  8. r ( i , n ) r(i,n)
  9. P A ( k x , w ) = P e ( k x , w ) | 1 1 - A ( k x , w ) | 2 P_{A}\left(k_{x},w\right)=P_{e}\left(k_{x},w\right)|\frac{1}{1-A\left(k_{x},w% \right)}|^{2}
  10. P e ( k x , w ) P_{e}\left(k_{x},w\right)
  11. e ( i , n ) e(i,n)
  12. | 1 1 - A ( k x , w ) | |\frac{1}{1-A\left(k_{x},w\right)}|
  13. r ( i , n ) r(i,n)
  14. A ( k x , w ) = p = o N - 1 q = 0 M - 1 a ( p , q ) e x p ( j k x p - j w q ) A\left(k_{x},w\right)=\sum_{p=o}^{N-1}\sum_{q=0}^{M-1}a(p,q)exp(jk_{x}p-jwq)
  15. a ( p , q ) a\left(p,q\right)
  16. φ ( l , m ) \varphi\left(l,m\right)
  17. H = 1 4 π 2 - π π - π π l o g P ( k x , w ) d k x d w H=\frac{1}{4\pi^{2}}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}logP\left(k_{x},w\right)% dk_{x}dw
  18. P ( k , w ) P\left(k,w\right)
  19. P M E = 1 l m λ ( l , m ) e x p ( j k x l - j w m ) P_{ME}=\frac{1}{\sum_{l}\sum_{m}\lambda\left(l,m\right)exp\left(jk_{x}l-jwm% \right)}
  20. I M L M ( k : A , B ) = 1 1 M L M ( k : A ) - 1 M L M ( k : B ) IMLM\left(k:A,B\right)=\frac{1}{\frac{1}{MLM\left(k:A\right)}-\frac{1}{MLM% \left(k:B\right)}}
  21. I M L M ( k : A , B ) = M L M ( k : A ) M L M ( k : B ) M L M ( k : B ) - M L M ( k : A ) IMLM\left(k:A,B\right)=\frac{MLM\left(k:A\right)MLM\left(k:B\right)}{MLM\left(% k:B\right)-MLM\left(k:A\right)}
  22. I M L M ( k : A , B ) = M L M ( k : A ) W A B ( k ) IMLM\left(k:A,B\right)=MLM\left(k:A\right)W_{AB}\left(k\right)
  23. W A B ( k ) W_{AB}\left(k\right)
  24. W A B ( k ) = M L M ( k : B ) M L M ( k : B ) - M L M ( k : A ) W_{AB}\left(k\right)=\frac{MLM\left(k:B\right)}{MLM\left(k:B\right)-MLM\left(k% :A\right)}

Spectral_invariants.html

  1. α H = a 1 x 1 + a 2 x 2 + \alpha_{H}=a_{1}x_{1}+a_{2}x_{2}+\cdots
  2. c H ( α ) = min max { A H ( x i ) : a i 0 } . c_{H}(\alpha)=\min\max\{A_{H}(x_{i})\ :\ a_{i}\neq 0\}.

Spectral_line_ratios.html

  1. e r g / c m 3 s erg/cm^{3}s
  2. P u l = N u ω u l A u l P_{u\rightarrow l}=N_{u}\ \hbar\omega_{u\rightarrow l}\ A_{u\rightarrow l}
  3. N u N_{u}
  4. ω u l \hbar\omega_{u\rightarrow l}
  5. A u l A_{u\rightarrow l}
  6. P u 1 l 1 P u 2 l 2 = N u 1 ω u 1 l 1 A u 1 l 1 N u 2 ω u 2 l 2 A u 2 l 2 \frac{P_{u_{1}\rightarrow l_{1}}}{P_{u_{2}\rightarrow l_{2}}}=\frac{N_{u_{1}}% \omega_{u_{1}\rightarrow l_{1}}A_{u_{1}\rightarrow l_{1}}}{N_{u_{2}}\omega_{u_% {2}\rightarrow l_{2}}A_{u_{2}\rightarrow l_{2}}}
  7. N u 1 N_{u_{1}}
  8. N u 2 N_{u_{2}}

Spectrometer.html

  1. p = m v = q B r p=mv=qBr

Spherical_basis.html

  1. 𝐀 = A x 𝐞 x + A y 𝐞 y + A z 𝐞 z \mathbf{A}=A_{x}\mathbf{e}_{x}+A_{y}\mathbf{e}_{y}+A_{z}\mathbf{e}_{z}
  2. 𝐀 = A + 𝐞 + + A - 𝐞 - + A 0 𝐞 0 \mathbf{A}=A_{+}\mathbf{e}_{+}+A_{-}\mathbf{e}_{-}+A_{0}\mathbf{e}_{0}
  3. 𝐞 + = - 1 2 𝐞 x - i 2 𝐞 y 𝐞 - = + 1 2 𝐞 x - i 2 𝐞 y 𝐞 ± = 1 2 ( 𝐞 x ± i 𝐞 y ) \begin{aligned}\displaystyle\mathbf{e}_{+}&\displaystyle=-\frac{1}{\sqrt{2}}% \mathbf{e}_{x}-\frac{i}{\sqrt{2}}\mathbf{e}_{y}\\ \displaystyle\mathbf{e}_{-}&\displaystyle=+\frac{1}{\sqrt{2}}\mathbf{e}_{x}-% \frac{i}{\sqrt{2}}\mathbf{e}_{y}\\ \end{aligned}\quad\rightleftharpoons\quad\mathbf{e}_{\pm}=\mp\frac{1}{\sqrt{2}% }\left(\mathbf{e}_{x}\pm i\mathbf{e}_{y}\right)\,
  4. 𝐞 0 = 𝐞 z \mathbf{e}_{0}=\mathbf{e}_{z}
  5. 𝐞 x \displaystyle\mathbf{e}_{x}
  6. k k
  7. T q ( k ) T_{q}^{(k)}
  8. [ J ± , T q ( k ) ] = ( k q ) ( k ± q + 1 ) T q ± 1 ( k ) [J_{\pm},T_{q}^{(k)}]=\hbar\sqrt{(k\mp q)(k\pm q+1)}T_{q\pm 1}^{(k)}
  9. [ J z , T q ( k ) ] = q T q ( k ) [J_{z},T_{q}^{(k)}]=\hbar qT_{q}^{(k)}
  10. 𝒟 ( R ) \mathcal{D}(R)
  11. R R
  12. 𝒟 ( R ) T q ( k ) 𝒟 ( R ) = q = - k k 𝒟 q q ( k ) T q ( k ) \mathcal{D}^{\dagger}(R)T_{q}^{(k)}\mathcal{D}(R)=\sum_{q^{\prime}=-k}^{k}% \mathcal{D}_{qq^{\prime}}^{(k)}T_{q^{\prime}}^{(k)}
  13. \langle
  14. \rangle
  15. A + = 𝐞 + , 𝐀 = - A x 2 + i A y 2 A - = 𝐞 - , 𝐀 = + A x 2 + i A y 2 A ± = 𝐞 ± , 𝐀 = 1 2 ( A x + i A y ) \begin{aligned}\displaystyle A_{+}&\displaystyle=\left\langle\mathbf{e}_{+},% \mathbf{A}\right\rangle=-\frac{A_{x}}{\sqrt{2}}+\frac{iA_{y}}{\sqrt{2}}\\ \displaystyle A_{-}&\displaystyle=\left\langle\mathbf{e}_{-},\mathbf{A}\right% \rangle=+\frac{A_{x}}{\sqrt{2}}+\frac{iA_{y}}{\sqrt{2}}\\ \end{aligned}\quad\rightleftharpoons\quad A_{\pm}=\left\langle\mathbf{e}_{\pm}% ,\mathbf{A}\right\rangle=\frac{1}{\sqrt{2}}\left(\mp A_{x}+iA_{y}\right)
  16. A 0 = 𝐞 0 , 𝐀 = 𝐞 z , 𝐀 = A z A_{0}=\left\langle\mathbf{e}_{0},\mathbf{A}\right\rangle=\left\langle\mathbf{e% }_{z},\mathbf{A}\right\rangle=A_{z}
  17. A x \displaystyle A_{x}
  18. 𝐚 , 𝐛 = 𝐚 𝐛 = j a j b j \left\langle\mathbf{a},\mathbf{b}\right\rangle=\mathbf{a}\cdot\mathbf{b}^{% \star}=\sum_{j}a_{j}b_{j}^{\star}
  19. \langle
  20. \rangle
  21. 𝐞 + , 𝐞 - = 𝐞 - , 𝐞 0 = 𝐞 0 , 𝐞 + = 0 \left\langle\mathbf{e}_{+},\mathbf{e}_{-}\right\rangle=\left\langle\mathbf{e}_% {-},\mathbf{e}_{0}\right\rangle=\left\langle\mathbf{e}_{0},\mathbf{e}_{+}% \right\rangle=0
  22. 𝐞 + , 𝐞 + = 𝐞 - , 𝐞 - = 𝐞 0 , 𝐞 0 = 1 \left\langle\mathbf{e}_{+},\mathbf{e}_{+}\right\rangle=\left\langle\mathbf{e}_% {-},\mathbf{e}_{-}\right\rangle=\left\langle\mathbf{e}_{0},\mathbf{e}_{0}% \right\rangle=1
  23. 2 \sqrt{2}
  24. ( 𝐞 + 𝐞 - 𝐞 0 ) = 𝐔 ( 𝐞 x 𝐞 y 𝐞 z ) , 𝐔 = ( - 1 2 - i 2 0 + 1 2 - i 2 0 0 0 1 ) , \begin{pmatrix}\mathbf{e}_{+}\\ \mathbf{e}_{-}\\ \mathbf{e}_{0}\end{pmatrix}=\mathbf{U}\begin{pmatrix}\mathbf{e}_{x}\\ \mathbf{e}_{y}\\ \mathbf{e}_{z}\end{pmatrix}\,,\quad\mathbf{U}=\begin{pmatrix}-\frac{1}{\sqrt{2% }}&-\frac{i}{\sqrt{2}}&0\\ +\frac{1}{\sqrt{2}}&-\frac{i}{\sqrt{2}}&0\\ 0&0&1\end{pmatrix}\,,
  25. ( 𝐞 x 𝐞 y 𝐞 z ) = 𝐔 - 1 ( 𝐞 + 𝐞 - 𝐞 0 ) , 𝐔 - 1 = ( - 1 2 + 1 2 0 + i 2 + i 2 0 0 0 1 ) . \begin{pmatrix}\mathbf{e}_{x}\\ \mathbf{e}_{y}\\ \mathbf{e}_{z}\end{pmatrix}=\mathbf{U}^{-1}\begin{pmatrix}\mathbf{e}_{+}\\ \mathbf{e}_{-}\\ \mathbf{e}_{0}\end{pmatrix}\,,\quad\mathbf{U}^{-1}=\begin{pmatrix}-\frac{1}{% \sqrt{2}}&+\frac{1}{\sqrt{2}}&0\\ +\frac{i}{\sqrt{2}}&+\frac{i}{\sqrt{2}}&0\\ 0&0&1\end{pmatrix}\,.
  26. ( A + A - A 0 ) = 𝐔 * ( A x A y A z ) , 𝐔 * = ( - 1 2 + i 2 0 + 1 2 + i 2 0 0 0 1 ) , \begin{pmatrix}A_{+}\\ A_{-}\\ A_{0}\end{pmatrix}=\mathbf{U}^{\mathrm{*}}\begin{pmatrix}A_{x}\\ A_{y}\\ A_{z}\end{pmatrix}\,,\quad\mathbf{U}^{\mathrm{*}}=\begin{pmatrix}-\frac{1}{% \sqrt{2}}&+\frac{i}{\sqrt{2}}&0\\ +\frac{1}{\sqrt{2}}&+\frac{i}{\sqrt{2}}&0\\ 0&0&1\end{pmatrix}\,,
  27. ( A x A y A z ) = ( 𝐔 * ) - 1 ( A + A - A 0 ) , ( 𝐔 * ) - 1 = ( - 1 2 + 1 2 0 - i 2 - i 2 0 0 0 1 ) . \begin{pmatrix}A_{x}\\ A_{y}\\ A_{z}\end{pmatrix}=(\mathbf{U}^{\mathrm{*}})^{-1}\begin{pmatrix}A_{+}\\ A_{-}\\ A_{0}\end{pmatrix}\,,\quad(\mathbf{U}^{\mathrm{*}})^{-1}=\begin{pmatrix}-\frac% {1}{\sqrt{2}}&+\frac{1}{\sqrt{2}}&0\\ -\frac{i}{\sqrt{2}}&-\frac{i}{\sqrt{2}}&0\\ 0&0&1\end{pmatrix}\,.
  28. 𝐞 q × 𝐞 q = s y m b o l 0 \mathbf{e}_{q}\times\mathbf{e}_{q}=symbol{0}
  29. 𝐞 ± × 𝐞 = ± i 𝐞 0 \mathbf{e}_{\pm}\times\mathbf{e}_{\mp}=\pm i\mathbf{e}_{0}
  30. 𝐞 ± × 𝐞 0 = ± i 𝐞 ± \mathbf{e}_{\pm}\times\mathbf{e}_{0}=\pm i\mathbf{e}_{\pm}
  31. 𝐀 , 𝐁 = A + B + + A - B - + A 0 B 0 \left\langle\mathbf{A},\mathbf{B}\right\rangle=A_{+}B_{+}^{\star}+A_{-}B_{-}^{% \star}+A_{0}B_{0}^{\star}

Spherical_contact_distribution_function.html

  1. 𝐑 d \textstyle\,\textbf{R}^{d}
  2. x \textstyle x
  3. N \textstyle{N}
  4. x N , \textstyle x\in{N},
  5. N \textstyle{N}
  6. B \textstyle B
  7. N ( B ) , \textstyle{N}(B),
  8. H s ( r ) = P ( N ( b ( o , r ) ) = 0 ) . H_{s}(r)=P({N}(b(o,r))=0).
  9. 𝐑 d \textstyle\,\textbf{R}^{d}
  10. B \textstyle B
  11. B \textstyle B
  12. r 0 \textstyle r\geq 0
  13. H B ( r ) = P ( N ( r B ) = 0 ) . H_{B}(r)=P({N}(rB)=0).
  14. N \textstyle{N}
  15. 𝐑 d \textstyle\,\textbf{R}^{d}
  16. Λ \textstyle\Lambda
  17. H s ( r ) = 1 - e - Λ ( b ( o , r ) ) , H_{s}(r)=1-e^{-\Lambda(b(o,r))},
  18. H s ( r ) = 1 - e - λ | b ( o , r ) | , H_{s}(r)=1-e^{-\lambda|b(o,r)|},
  19. | b ( o , r ) | \textstyle|b(o,r)|
  20. r \textstyle r
  21. 𝐑 2 \textstyle\,\textbf{R}^{2}
  22. H s ( r ) = 1 - e - λ π r 2 . H_{s}(r)=1-e^{-\lambda\pi r^{2}}.
  23. J J
  24. J J
  25. r r
  26. J ( r ) = 1 - D o ( r ) 1 - H s ( r ) J(r)=\frac{1-D_{o}(r)}{1-H_{s}(r)}
  27. J J
  28. J ( r ) J(r)
  29. J ( r ) J(r)
  30. J J

Spherical_wave_transformation.html

  1. k k
  2. x = k 2 x x 2 + y 2 , y = k 2 y x 2 + y 2 x^{\prime}=\frac{k^{2}x}{x^{2}+y^{2}},\quad y^{\prime}=\frac{k^{2}y}{x^{2}+y^{% 2}}
  3. x , y , z x,y,z
  4. δ x 2 + δ y 2 + δ z 2 = λ ( δ x 2 + δ y 2 + δ z 2 ) \delta x^{\prime 2}+\delta y^{\prime 2}+\delta z^{\prime 2}=\lambda\left(% \delta x^{2}+\delta y^{2}+\delta z^{2}\right)
  5. λ = 1 \lambda=1
  6. λ 1 \lambda\neq 1
  7. λ \sqrt{\lambda}
  8. λ = k 4 / ( x 2 + y 2 + z 2 ) 2 \lambda=k^{4}/\left(x^{2}+y^{2}+z^{2}\right)^{2}
  9. x = k 2 x x 2 + y 2 + z 2 , y = k 2 y x 2 + y 2 + z 2 , z = k 2 z x 2 + y 2 + z 2 x^{\prime}=\frac{k^{2}x}{x^{2}+y^{2}+z^{2}},\quad y^{\prime}=\frac{k^{2}y}{x^{% 2}+y^{2}+z^{2}},\quad z^{\prime}=\frac{k^{2}z}{x^{2}+y^{2}+z^{2}}
  10. n n
  11. δ x 1 2 + + δ x n 2 = λ ( δ x 1 2 + + δ x n 2 ) \delta x_{1}^{\prime 2}+\dots+\delta x_{n}^{\prime 2}=\lambda\left(\delta x_{1% }^{2}+\dots+\delta x_{n}^{2}\right)
  12. x = k 2 x x 2 + y 2 + z 2 - r 2 , z = k 2 z x 2 + y 2 + z 2 - r 2 , y = k 2 y x 2 + y 2 + z 2 - r 2 , r = ± k 2 r x 2 + y 2 + z 2 - r 2 . \begin{aligned}\displaystyle x^{\prime}&\displaystyle=\frac{k^{2}x}{x^{2}+y^{2% }+z^{2}-r^{2}},&\displaystyle z^{\prime}&\displaystyle=\frac{k^{2}z}{x^{2}+y^{% 2}+z^{2}-r^{2}},\\ \displaystyle y^{\prime}&\displaystyle=\frac{k^{2}y}{x^{2}+y^{2}+z^{2}-r^{2}},% &\displaystyle r^{\prime}&\displaystyle=\frac{\pm k^{2}r}{x^{2}+y^{2}+z^{2}-r^% {2}}.\end{aligned}
  13. x , y x,y
  14. r r
  15. x , y , z x,y,z
  16. z = i r z=ir
  17. z = r z=r
  18. x , y , z , r x,y,z,r
  19. x , y , z , r x^{\prime},y^{\prime},z^{\prime},r^{\prime}
  20. ( x - x ) 2 + ( y - y ) 2 + ( z - z ) 2 - ( r - r ) 2 = 0 (x-x^{\prime})^{2}+(y-y^{\prime})^{2}+(z-z^{\prime})^{2}-(r-r^{\prime})^{2}=0
  21. t = i r t=ir
  22. ( x - x ) 2 + ( y - y ) 2 + ( z - z ) 2 + ( t - t ) 2 = 0 (x-x^{\prime})^{2}+(y-y^{\prime})^{2}+(z-z^{\prime})^{2}+(t-t^{\prime})^{2}=0
  23. G 15 G_{15}
  24. δ x 2 + δ y 2 + δ z 2 + δ u 2 = λ ( δ x 2 + δ y 2 + δ z 2 + δ u 2 ) \delta x^{\prime 2}+\delta y^{\prime 2}+\delta z^{\prime 2}+\delta u^{\prime 2% }=\lambda\left(\delta x^{2}+\delta y^{2}+\delta z^{2}+\delta u^{2}\right)
  25. u = i c t u=ict
  26. t t
  27. c c
  28. r r
  29. c t ct
  30. c c
  31. λ \lambda
  32. λ = 1 \lambda=1
  33. G 10 G_{10}
  34. λ 1 \lambda\neq 1
  35. λ \lambda
  36. l = λ l=\sqrt{\lambda}
  37. x = γ l ( x - v t ) , y = l y , z = l z , t = γ l ( t - x v c 2 ) x^{\prime}=\gamma l\left(x-vt\right),\quad y^{\prime}=ly,\quad z^{\prime}=lz,% \quad t^{\prime}=\gamma l\left(t-x\frac{v}{c^{2}}\right)
  38. l = 1 l=1
  39. λ = r 4 / ( x 2 + y 2 + z 2 + u 2 ) 2 \lambda=r^{4}/\left(x^{2}+y^{2}+z^{2}+u^{2}\right)^{2}
  40. x = k 2 x x 2 + y 2 + z 2 + u 2 , z = k 2 z x 2 + y 2 + z 2 + u 2 , y = k 2 y x 2 + y 2 + z 2 + u 2 , u = k 2 u x 2 + y 2 + z 2 + u 2 , \begin{aligned}\displaystyle x^{\prime}&\displaystyle=\frac{k^{2}x}{x^{2}+y^{2% }+z^{2}+u^{2}},&\displaystyle z^{\prime}&\displaystyle=\frac{k^{2}z}{x^{2}+y^{% 2}+z^{2}+u^{2}},\\ \displaystyle y^{\prime}&\displaystyle=\frac{k^{2}y}{x^{2}+y^{2}+z^{2}+u^{2}},% &\displaystyle u^{\prime}&\displaystyle=\frac{k^{2}u}{x^{2}+y^{2}+z^{2}+u^{2}}% ,\end{aligned}
  41. c t ct
  42. u = i c t u=ict
  43. x 2 + y 2 + z 2 - c 2 t 2 x^{2}+y^{2}+z^{2}-c^{2}t^{2}
  44. R R
  45. R R^{\prime}
  46. D D
  47. D D^{\prime}
  48. D 2 - D 2 = R 2 - R 2 , D - D = α ( R - R ) , D + D = 1 α ( R + R ) , D^{2}-D^{\prime 2}=R^{2}-R^{\prime 2},\quad D-D^{\prime}=\alpha(R-R^{\prime}),% \quad D+D^{\prime}=\frac{1}{\alpha}(R+R^{\prime}),
  49. D = D ( 1 + α 2 ) - 2 α R 1 - α 2 , R = 2 α D - R ( 1 + α 2 ) 1 - α 2 . D^{\prime}=\frac{D\left(1+\alpha^{2}\right)-2\alpha R}{1-\alpha^{2}},\quad R^{% \prime}=\frac{2\alpha D-R\left(1+\alpha^{2}\right)}{1-\alpha^{2}}.
  50. z = D z=D
  51. k = α k=\alpha
  52. x x
  53. y y
  54. x = x , z = 1 + k 2 1 - k 2 z - 2 k R 1 - k 2 , y = y , R = 2 k z 1 - k 2 - 1 + k 2 1 - k 2 R , \begin{aligned}\displaystyle x^{\prime}&\displaystyle=x,&\displaystyle z^{% \prime}&\displaystyle=\frac{1+k^{2}}{1-k^{2}}z-\frac{2kR}{1-k^{2}},\\ \displaystyle y^{\prime}&\displaystyle=y,&\displaystyle R^{\prime}&% \displaystyle=\frac{2kz}{1-k^{2}}-\frac{1+k^{2}}{1-k^{2}}R,\end{aligned}
  55. z + R = 1 + k 1 - k ( z - R ) , z - R = 1 - k 1 + k ( z + R ) , z^{\prime}+R^{\prime}=\frac{1+k}{1-k}(z-R),\quad z^{\prime}-R^{\prime}=\frac{1% -k}{1+k}(z+R),
  56. x 2 + y 2 + z 2 - R 2 = x 2 + y 2 + z 2 - R 2 x^{\prime 2}+y^{\prime 2}+z^{\prime 2}-R^{\prime 2}=x^{2}+y^{2}+z^{2}-R^{2}
  57. p = κ 2 + 1 κ 2 - 1 p - 2 κ κ 2 - 1 R , R = 2 κ κ 2 - 1 p - κ 2 + 1 κ 2 - 1 R p^{\prime}=\frac{\kappa^{2}+1}{\kappa^{2}-1}p-\frac{2\kappa}{\kappa^{2}-1}R,% \quad R^{\prime}=\frac{2\kappa}{\kappa^{2}-1}p-\frac{\kappa^{2}+1}{\kappa^{2}-% 1}R
  58. κ = R - R p - p , p 2 - p 2 = R 2 - R 2 . \kappa=\frac{R^{\prime}-R}{p^{\prime}-p},\quad p^{\prime 2}-p^{2}=R^{\prime 2}% -R^{2}.
  59. c = 1 c=1
  60. x = x - v t 1 - v 2 , y = y , z = z , t = t - v x 1 - v 2 x^{\prime}=\frac{x-vt}{\sqrt{1-v^{2}}},\quad y^{\prime}=y,\quad z^{\prime}=z,% \quad t^{\prime}=\frac{t-vx}{\sqrt{1-v^{2}}}
  61. x 2 + y 2 + z 2 - t 2 x^{2}+y^{2}+z^{2}-t^{2}
  62. x 2 + y 2 + z 2 - R 2 x^{2}+y^{2}+z^{2}-R^{2}
  63. z z
  64. R = c t R=ct
  65. R = c t R^{\prime}=ct^{\prime}
  66. k k
  67. D = D ( 1 + α 2 ) - 2 α R 1 - α 2 , R = 2 α D - R ( 1 + α 2 ) 1 - α 2 . D^{\prime}=\frac{D\left(1+\alpha^{2}\right)-2\alpha R}{1-\alpha^{2}},\quad R^{% \prime}=\frac{2\alpha D-R\left(1+\alpha^{2}\right)}{1-\alpha^{2}}.
  68. 2 α 1 + α 2 = w \frac{2\alpha}{1+\alpha^{2}}=w
  69. 1 - α 2 1 + α 2 = 1 - w 2 , 2 α 1 - α 2 = w 1 - w 2 , \frac{1-\alpha^{2}}{1+\alpha^{2}}=\sqrt{1-w^{2}},\quad\frac{2\alpha}{1-\alpha^% {2}}=\frac{w}{\sqrt{1-w^{2}}},
  70. D = x , D = x , R = t , R = t D=x,D^{\prime}=x^{\prime},R=t,R^{\prime}=t^{\prime}
  71. t - v x t-vx
  72. w x - t wx-t
  73. x = x - w t 1 - w 2 , t = w x - t 1 - w 2 x^{\prime}=\frac{x-wt}{\sqrt{1-w^{2}}},\quad t^{\prime}=\frac{wx-t}{\sqrt{1-w^% {2}}}
  74. π 1 \pi_{1}
  75. π \pi
  76. π \pi
  77. r r
  78. x x
  79. x + r = 1 + λ 2 1 - λ 2 ( x + r ) , x - r x + r = 1 - λ 1 + λ x - r x + r , x^{\prime}+r^{\prime}=\sqrt{\frac{1+\lambda^{2}}{1-\lambda^{2}}}(x+r),\quad% \frac{x^{\prime}-r^{\prime}}{x^{\prime}+r^{\prime}}=\frac{1-\lambda}{1+\lambda% }\cdot\frac{x-r}{x+r},
  80. 1 - λ 2 x = x - λ r , 1 - λ 2 r = r - λ x . \sqrt{1-\lambda^{2}}\cdot x^{\prime}=x-\lambda r,\quad\sqrt{1-\lambda^{2}}% \cdot r^{\prime}=r-\lambda x.
  81. λ = v / c \lambda=v/c
  82. r = c t r=ct
  83. α = α 1 1 - λ 2 - R λ 1 - λ 2 , β = β , γ = γ , R = α - λ 1 - λ 2 + R 1 1 - λ 2 \alpha^{\prime}=\alpha\frac{1}{\sqrt{1-\lambda^{2}}}-R\frac{\lambda}{\sqrt{1-% \lambda^{2}}},\quad\beta^{\prime}=\beta,\quad\gamma^{\prime}=\gamma,\quad R^{% \prime}=\alpha\frac{-\lambda}{\sqrt{1-\lambda^{2}}}+R\frac{1}{\sqrt{1-\lambda^% {2}}}
  84. x = α , y = β , z = γ , R = c t , x = α , y = β , z = γ , R = c t , \begin{aligned}\displaystyle x&\displaystyle=\alpha,&\displaystyle y&% \displaystyle=\beta,&\displaystyle z&\displaystyle=\gamma,&\displaystyle R&% \displaystyle=ct,\\ \displaystyle x^{\prime}&\displaystyle=\alpha^{\prime},&\displaystyle y^{% \prime}&\displaystyle=\beta^{\prime},&\displaystyle z^{\prime}&\displaystyle=% \gamma^{\prime},&\displaystyle R^{\prime}&\displaystyle=ct^{\prime},\end{aligned}
  85. λ = v c \lambda=\frac{v}{c}
  86. d x 2 + d y 2 - d r 2 dx^{2}+dy^{2}-dr^{2}
  87. d x 2 + d y 2 + d z 2 - d r 2 dx^{2}+dy^{2}+dz^{2}-dr^{2}
  88. d x 2 + d y 2 + d z 2 + d r 2 dx^{2}+dy^{2}+dz^{2}+dr^{2}
  89. d x 1 2 + d x 2 2 + d x 3 2 - d x 4 2 dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-dx_{4}^{2}
  90. P P
  91. E 3 E^{3}
  92. P P
  93. P P
  94. 4 \mathbb{R}^{4}
  95. 4 \mathbb{R}^{4}
  96. x 1 2 + x 2 2 + x 3 2 - x 4 2 = 0 x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}=0
  97. ( x - x ) 2 + ( y - y ) 2 + ( z - z ) 2 - ( t - t ) 2 \scriptstyle(x-x^{\prime})^{2}+(y-y^{\prime})^{2}+(z-z^{\prime})^{2}-(t-t^{% \prime})^{2}
  98. ( x - x ) 2 + ( y - y ) 2 + ( z - z ) 2 + ( r - r ) 2 \scriptstyle\sqrt{(x-x^{\prime})^{2}+(y-y^{\prime})^{2}+(z-z^{\prime})^{2}+(r-% r^{\prime})^{2}}

Spherinder.html

  1. D = { ( x , y , z , w ) | x 2 + y 2 + z 2 r 1 2 , w 2 r 2 2 } D=\{(x,y,z,w)|x^{2}+y^{2}+z^{2}\leq r_{1}^{2},\ w^{2}\leq r_{2}^{2}\}

Spitzer_resistivity.html

  1. η \eta
  2. η = π Z e 2 m 1 / 2 ln Λ ( 4 π ε 0 ) 2 ( k B T ) 3 / 2 \eta=\dfrac{\pi Ze^{2}m^{1/2}\ln\Lambda}{\left(4\pi\varepsilon_{0}\right)^{2}% \left(k_{B}T\right)^{3/2}}

Splaysort.html

  1. x log d x \sum_{x}\log d_{x}
  2. x log i x \sum_{x}\log i_{x}

Spline_wavelet.html

  1. N 1 ( x ) = { 1 0 x < 1 0 otherwise N_{1}(x)=\begin{cases}1&0\leq x<1\\ 0&\,\text{otherwise}\end{cases}
  2. N m ( x ) = 0 1 N m - 1 ( x - t ) d t N_{m}(x)=\int_{0}^{1}N_{m-1}(x-t)dt
  3. m > 1 m>1
  4. N m ( x ) N_{m}(x)
  5. [ 0 , m ] [0,m]
  6. N m ( x ) N_{m}(x)
  7. N m ( x ) > 0 N_{m}(x)>0
  8. 0 < x < m 0<x<m
  9. k = - N m ( x - k ) = 1 \sum_{k=-\infty}^{\infty}N_{m}(x-k)=1
  10. x x
  11. N m ( x ) = x m - 1 N m - 1 ( x ) + m - x m - 1 N m - 1 ( x - 1 ) N_{m}(x)=\frac{x}{m-1}N_{m-1}(x)+\frac{m-x}{m-1}N_{m-1}(x-1)
  12. N m ( x ) N_{m}(x)
  13. x = m 2 x=\frac{m}{2}
  14. N m ( m 2 - x ) = N m ( m 2 + x ) N_{m}\left(\frac{m}{2}-x\right)=N_{m}\left(\frac{m}{2}+x\right)
  15. N m ( x ) N_{m}(x)
  16. N m ( x ) = N m - 1 ( x ) - N m - 1 ( x - 1 ) N_{m}^{\prime}(x)=N_{m-1}(x)-N_{m-1}(x-1)
  17. - N m ( x ) d x = 1 \int_{-\infty}^{\infty}N_{m}(x)\,dx=1
  18. N m ( x ) = k = 0 m 2 - m + 1 ( m k ) N m ( 2 x - k ) N_{m}(x)=\sum_{k=0}^{m}2^{-m+1}{m\choose k}N_{m}(2x-k)
  19. A A
  20. B B
  21. { c k } k = - \{c_{k}\}_{k=-\infty}^{\infty}
  22. A { c k } 2 k = - c k N m ( x - k ) 2 B { c k } 2 A\left\|\{c_{k}\}\right\|^{2}\leq\left\|\sum_{k=-\infty}^{\infty}c_{k}N_{m}(x-% k)\right\|^{2}\leq B\left\|\{c_{k}\}\right\|^{2}
  23. \|\cdot\|
  24. N 1 ( x ) N_{1}(x)
  25. N 1 ( x ) N_{1}(x)
  26. N 1 ( x ) = { 1 0 x < 1 0 otherwise N_{1}(x)=\begin{cases}1&0\leq x<1\\ 0&\,\text{otherwise}\end{cases}
  27. N 1 ( x ) = N 1 ( 2 x ) + N 1 ( 2 x - 1 ) N_{1}(x)=N_{1}(2x)+N_{1}(2x-1)
  28. N 1 ( x ) N_{1}(x)
  29. N 2 ( x ) N_{2}(x)
  30. N 2 ( x ) = { x 0 x < 1 - x + 2 1 x < 2 0 otherwise N_{2}(x)=\begin{cases}x&0\leq x<1\\ -x+2&1\leq x<2\\ 0&\,\text{otherwise}\end{cases}
  31. N 2 ( x ) = 1 2 N 2 ( 2 x ) + N 2 ( 2 x - 1 ) + 1 2 N 2 ( 2 x - 2 ) N_{2}(x)=\frac{1}{2}N_{2}(2x)+N_{2}(2x-1)+\frac{1}{2}N_{2}(2x-2)
  32. N 2 ( x ) N_{2}(x)
  33. N 3 ( x ) N_{3}(x)
  34. N 3 ( x ) = { 1 2 x 2 0 x < 1 - x 2 + 3 x - 3 2 1 x < 2 1 2 x 2 - 3 x + 9 2 2 x < 3 0 otherwise N_{3}(x)=\begin{cases}\frac{1}{2}x^{2}&0\leq x<1\\ -x^{2}+3x-\frac{3}{2}&1\leq x<2\\ \frac{1}{2}x^{2}-3x+\frac{9}{2}&2\leq x<3\\ 0&\,\text{otherwise}\end{cases}
  35. N 3 ( x ) = 1 4 N 3 ( 2 x ) + 3 4 N 3 ( 2 x - 1 ) + 3 4 N 3 ( 2 x - 2 ) + 1 4 N 3 ( 2 x - 3 ) N_{3}(x)=\frac{1}{4}N_{3}(2x)+\frac{3}{4}N_{3}(2x-1)+\frac{3}{4}N_{3}(2x-2)+% \frac{1}{4}N_{3}(2x-3)
  36. N 3 ( x ) N_{3}(x)
  37. N 4 ( x ) N_{4}(x)
  38. N 4 ( x ) = { 1 6 x 3 0 x < 1 - 1 2 x 3 + 2 x 2 - 2 x + 2 3 1 x < 2 1 2 x 3 - 4 x 2 + 10 x - 22 3 2 x < 3 - 1 6 x 3 + 2 x 2 - 8 x + 32 3 3 x < 4 0 otherwise N_{4}(x)=\begin{cases}\frac{1}{6}x^{3}&0\leq x<1\\ -\frac{1}{2}x^{3}+2x^{2}-2x+\frac{2}{3}&1\leq x<2\\ \frac{1}{2}x^{3}-4x^{2}+10x-\frac{22}{3}&2\leq x<3\\ -\frac{1}{6}x^{3}+2x^{2}-8x+\frac{32}{3}&3\leq x<4\\ 0&\,\text{otherwise}\end{cases}
  39. N 4 ( x ) = 1 8 N ( 2 x ) + 1 2 N ( 2 x - 1 ) + 3 4 N ( 2 x - 2 ) + 1 2 N 4 ( 2 x - 3 ) + 1 8 N ( 2 x - 4 ) N_{4}(x)=\frac{1}{8}N(2x)+\frac{1}{2}N(2x-1)+\frac{3}{4}N(2x-2)+\frac{1}{2}N_{% 4}(2x-3)+\frac{1}{8}N(2x-4)
  40. N 4 ( x ) N_{4}(x)
  41. N 5 ( x ) N_{5}(x)
  42. N 5 ( x ) = { 1 24 x 4 0 x < 1 - 1 6 x 4 + 5 6 x 3 - 5 4 x 2 + 5 6 x - 5 24 1 x < 2 1 4 x 4 - 5 2 x 3 + 35 4 x 2 - 25 2 x + 155 24 2 x < 3 - 1 6 x 4 + 5 2 x 3 - 55 4 x 2 + 65 2 x - 655 24 3 x < 4 1 24 x 4 - 5 6 x 3 + 25 4 x 2 - 125 6 x + 625 24 4 x < 5 0 otherwise N_{5}(x)=\begin{cases}\frac{1}{24}x^{4}&0\leq x<1\\ -\frac{1}{6}x^{4}+\frac{5}{6}x^{3}-\frac{5}{4}x^{2}+\frac{5}{6}x-\frac{5}{24}&% 1\leq x<2\\ \frac{1}{4}x^{4}-\frac{5}{2}x^{3}+\frac{35}{4}x^{2}-\frac{25}{2}x+\frac{155}{2% 4}&2\leq x<3\\ -\frac{1}{6}x^{4}+\frac{5}{2}x^{3}-\frac{55}{4}x^{2}+\frac{65}{2}x-\frac{655}{% 24}&3\leq x<4\\ \frac{1}{24}x^{4}-\frac{5}{6}x^{3}+\frac{25}{4}x^{2}-\frac{125}{6}x+\frac{625}% {24}&4\leq x<5\\ 0&\,\text{otherwise}\end{cases}
  43. N 5 ( x ) = 1 16 N 5 ( 2 x ) + 5 16 N 5 ( 2 x - 1 ) + 10 16 N 5 ( 2 x - 2 ) + 10 16 N 5 ( 2 x - 3 ) + 5 16 N 5 ( 2 x - 4 ) + 1 16 N 5 ( 2 x - 5 ) N_{5}(x)=\frac{1}{16}N_{5}(2x)+\frac{5}{16}N_{5}(2x-1)+\frac{10}{16}N_{5}(2x-2% )+\frac{10}{16}N_{5}(2x-3)+\frac{5}{16}N_{5}(2x-4)+\frac{1}{16}N_{5}(2x-5)
  44. N 6 ( x ) N_{6}(x)
  45. N 6 ( x ) = { 1 120 x 5 0 x < 1 - 1 24 x 5 + 1 4 x 4 - 1 2 x 3 + 1 2 x 2 - 1 4 x + 1 20 1 x < 2 1 12 x 5 - x 4 + 9 2 x 3 - 19 2 x 2 + 39 4 x - 79 20 2 x < 3 - 1 12 x 5 + 3 2 x 4 - 21 2 x 3 + 71 2 x 2 - 231 4 x + 731 20 3 x < 4 1 24 x 5 - x 4 + 19 2 x 3 - 89 2 x 2 + 409 4 x - 1829 20 4 x < 5 - 1 120 x 5 + 1 4 x 4 - 3 x 3 + 18 x 2 - 54 x + 324 5 5 x < 6 0 otherwise N_{6}(x)=\begin{cases}\frac{1}{120}x^{5}&0\leq x<1\\ -\frac{1}{24}x^{5}+\frac{1}{4}x^{4}-\frac{1}{2}x^{3}+\frac{1}{2}x^{2}-\frac{1}% {4}x+\frac{1}{20}&1\leq x<2\\ \frac{1}{12}x^{5}-x^{4}+\frac{9}{2}x^{3}-\frac{19}{2}x^{2}+\frac{39}{4}x-\frac% {79}{20}&2\leq x<3\\ -\frac{1}{12}x^{5}+\frac{3}{2}x^{4}-\frac{21}{2}x^{3}+\frac{71}{2}x^{2}-\frac{% 231}{4}x+\frac{731}{20}&3\leq x<4\\ \frac{1}{24}x^{5}-x^{4}+\frac{19}{2}x^{3}-\frac{89}{2}x^{2}+\frac{409}{4}x-% \frac{1829}{20}&4\leq x<5\\ -\frac{1}{120}x^{5}+\frac{1}{4}x^{4}-3x^{3}+18x^{2}-54x+\frac{324}{5}&5\leq x<% 6\\ 0&\,\text{otherwise}\end{cases}
  46. N m ( x ) N_{m}(x)
  47. N m ( x ) N_{m}(x)
  48. L 2 ( R ) L^{2}(R)
  49. k , j k,j
  50. N m , k j ( x ) = N m ( 2 k x - j ) N_{m,kj}(x)=N_{m}(2^{k}x-j)
  51. k k
  52. V k V_{k}
  53. L 2 ( R ) L^{2}(R)
  54. { N m , k j ( x ) : j = , - 2 , - 1 , 0 , 1 , 2 , } \{N_{m,kj}(x):j=\cdots,-2,-1,0,1,2,\cdots\}
  55. V k V_{k}
  56. V - 2 V - 1 V 0 V 1 V 2 \cdots\subset V_{-2}\subset V_{-1}\subset V_{0}\subset V_{1}\subset V_{2}\subset\cdots
  57. L 2 ( R ) L^{2}(R)
  58. V k V_{k}
  59. L 2 ( R ) L^{2}(R)
  60. V k V_{k}
  61. k k
  62. { N m , k j ( x ) : j = , - 2 , - 1 , 0 , 1 , 2 , } \{N_{m,kj}(x):j=\cdots,-2,-1,0,1,2,\cdots\}
  63. V k V_{k}
  64. N m ( x ) N_{m}(x)
  65. ψ m ( x ) \psi_{m}(x)
  66. L 2 ( R ) L^{2}(R)
  67. N m ( x ) N_{m}(x)
  68. L 2 ( R ) L^{2}(R)
  69. { ψ m ( x - j ) : j = , - 2 , - 1 , 0 , 1 , 2 , } \{\psi_{m}(x-j):j=\cdots,-2,-1,0,1,2,\cdots\}
  70. W 0 W_{0}
  71. V 0 V_{0}
  72. V 1 V_{1}
  73. ψ m ( x ) \psi_{m}(x)
  74. ψ m ( x ) \psi_{m}(x)
  75. ψ m ( x ) \psi_{m}(x)
  76. N m ( x ) N_{m}(x)
  77. N m ( x ) N_{m}(x)
  78. { f j : j = , - 2 , - 1 , 0 , 1 , 2 , } \{f_{j}:j=\cdots,-2,-1,0,1,2,\cdots\}
  79. { c m , k : k = , - 2 , - 1 , 0 , 1 , 2 , } \{c_{m,k}:k=\cdots,-2,-1,0,1,2,\cdots\}
  80. k = - c m , k N m ( j + m 2 - k ) = f j \sum_{k=-\infty}^{\infty}c_{m,k}N_{m}\left(j+\frac{m}{2}-k\right)=f_{j}
  81. j j
  82. { f j } \{f_{j}\}
  83. δ 0 j \delta_{0j}
  84. δ i j \delta_{ij}
  85. δ i j \delta_{ij}
  86. δ i j = { 1 , if i = j 0 , if i j \delta_{ij}=\begin{cases}1,&\,\text{ if }i=j\\ 0,&\,\text{ if }i\neq j\end{cases}
  87. L m ( x ) L_{m}(x)
  88. L m ( x ) = k = - c m , k N m ( x + m 2 - k ) L_{m}(x)=\sum_{k=-\infty}^{\infty}c_{m,k}N_{m}\left(x+\frac{m}{2}-k\right)
  89. { c m , k } \{c_{m,k}\}
  90. k = - c m , k N m ( j + m 2 - k ) = δ 0 j \sum_{k=-\infty}^{\infty}c_{m,k}N_{m}\left(j+\frac{m}{2}-k\right)=\delta_{0j}
  91. L m ( x ) L_{m}(x)
  92. A ( z ) = k = - δ k 0 z k = 1 , A(z)=\sum_{k=-\infty}^{\infty}\delta_{k0}z^{k}=1,
  93. B m ( z ) = k = - N m ( k + m 2 ) z k , B_{m}(z)=\sum_{k=-\infty}^{\infty}N_{m}\left(k+\frac{m}{2}\right)z^{k},
  94. C m ( z ) = k = - c m , k z k , C_{m}(z)=\sum_{k=-\infty}^{\infty}c_{m,k}z^{k},
  95. c m , k c_{m,k}
  96. B m ( z ) C m ( z ) = A ( z ) B_{m}(z)C_{m}(z)=A(z)
  97. C m ( z ) = 1 B m ( z ) C_{m}(z)=\frac{1}{B_{m}(z)}
  98. c m , k c_{m,k}
  99. L 4 ( x ) L_{4}(x)
  100. B m ( z ) B_{m}(z)
  101. B 4 ( x ) = k = - N 4 ( 2 + k ) z k B_{4}(x)=\sum_{k=-\infty}^{\infty}N_{4}(2+k)z^{k}
  102. N 4 ( k + 2 ) N_{4}(k+2)
  103. k = - 1 , 0 , 1 k=-1,0,1
  104. N 4 ( 1 ) = 1 6 , N 4 ( 2 ) = 4 6 , N 4 ( 3 ) = 1 6 . N_{4}(1)=\frac{1}{6},N_{4}(2)=\frac{4}{6},N_{4}(3)=\frac{1}{6}.
  105. B 4 ( z ) B_{4}(z)
  106. B 4 ( z ) = 1 6 z - 1 + 4 6 z 0 + 1 6 z 1 = 1 + 4 z + z 2 6 z B_{4}(z)=\frac{1}{6}z^{-1}+\frac{4}{6}z^{0}+\frac{1}{6}z^{1}=\frac{1+4z+z^{2}}% {6z}
  107. C 4 ( z ) C_{4}(z)
  108. C 4 ( z ) = 6 z 1 + 4 z + z 2 C_{4}(z)=\frac{6z}{1+4z+z^{2}}
  109. c 4 , k c_{4,k}
  110. L 4 ( x ) L_{4}(x)
  111. L 4 ( x ) = k = - ( - 1 ) k 3 ( 2 - 3 ) | k | N 4 ( x + 2 - k ) L_{4}(x)=\sum_{k=-\infty}^{\infty}(-1)^{k}\sqrt{3}(2-\sqrt{3})^{|k|}N_{4}(x+2-k)
  112. ψ m ( x ) \psi_{m}(x)
  113. ψ I , m ( x ) = d m d x m L 2 m ( 2 x - 1 ) \psi_{I,m}(x)=\frac{d^{m}}{dx^{m}}L_{2m}(2x-1)
  114. N m ( x ) N_{m}(x)
  115. ψ I , m \psi_{I,m}
  116. ψ I , 2 ( x ) = d 2 d x 2 L 4 ( 2 x - 1 ) \psi_{I,2}(x)=\frac{d^{2}}{dx^{2}}L_{4}(2x-1)
  117. L 4 ( x ) L_{4}(x)
  118. ψ I , 2 ( x ) = d 2 d x 2 k = - ( - 1 ) k 3 ( 2 - 3 ) | k | N 4 ( 2 x + 1 - k ) \psi_{I,2}(x)=\frac{d^{2}}{dx^{2}}\sum_{k=-\infty}^{\infty}(-1)^{k}\sqrt{3}(2-% \sqrt{3})^{|k|}N_{4}(2x+1-k)
  119. N m ( x ) N_{m}(x)
  120. N m - 1 ( x ) N_{m-1}(x)
  121. ψ 2 ( x ) \psi_{2}(x)
  122. ψ I , 2 ( x ) = k = - ( - 1 ) k 4 3 ( 2 - 3 ) | k | ( ( N 2 ( 2 x + k - 1 ) - 2 N 2 ( 2 x + k - 2 ) + N 2 ( 2 x + k - 3 ) ) \psi_{I,2}(x)=\sum_{k=-\infty}^{\infty}(-1)^{k}4\sqrt{3}(2-\sqrt{3})^{|k|}\Big% ((N_{2}(2x+k-1)-2N_{2}(2x+k-2)+N_{2}(2x+k-3)\Big)
  123. ψ 2 ( x ) \psi_{2}(x)
  124. k = - 3 , , 3 k=-3,\ldots,3
  125. ψ 2 ( x ) \psi_{2}(x)
  126. ψ I , 2 ( x ) { 0.07142668 x + 0.17856670 - 2.5 < x - 2 - 0.48084803 x - 0.92598272 - 2 < x - 1.5 2.0088293 x + 2.8085333 - 1.5 < x - 1 - 7.5684795 x - 6.7687755 - 1 < x - 0.5 28.245949 x + 11.138439 - 0.5 < x 0 - 57.415316 x + 11.138439 0 < x 0.5 57.415316 x - 46.276878 0.5 < x 1 - 28.245949 x + 39.384388 1 < x 1.5 7.5684795 x - 14.337255 1.5 < x 2 - 2.0088293 x + 4.8173625 2 < x 2.5 0.48084803 x - 1.4068308 2.5 < x 3 - 0.07142668 x + 0.24999338 3 < x 3.5 0 o t h e r w i s e \psi_{I,2}(x)\approx\begin{cases}0.07142668x+0.17856670&-2.5<x\leq-2\\ -0.48084803x-0.92598272&-2<x\leq-1.5\\ 2.0088293x+2.8085333&-1.5<x\leq-1\\ -7.5684795x-6.7687755&-1<x\leq-0.5\\ 28.245949x+11.138439&-0.5<x\leq 0\\ -57.415316x+11.138439&0<x\leq 0.5\\ 57.415316x-46.276878&0.5<x\leq 1\\ -28.245949x+39.384388&1<x\leq 1.5\\ 7.5684795x-14.337255&1.5<x\leq 2\\ -2.0088293x+4.8173625&2<x\leq 2.5\\ 0.48084803x-1.4068308&2.5<x\leq 3\\ -0.07142668x+0.24999338&3<x\leq 3.5\\ 0&{otherwise}\end{cases}
  127. ψ m ( x ) \psi_{m}(x)
  128. ψ I , m ( x ) = - q n N m ( 2 x - n ) \psi_{I,m}(x)=\sum_{-\infty}^{\infty}q_{n}N_{m}(2x-n)
  129. q n = j = 0 m ( - 1 ) j ( m j ) c m + n - j - 1 . q_{n}=\sum_{j=0}^{m}(-1)^{j}{m\choose j}c_{m+n-j-1}.
  130. N m ( x ) N_{m}(x)
  131. ψ C , m ( x ) \psi_{C,m}(x)
  132. [ 0 , 2 m - 1 ] [0,2m-1]
  133. ψ C , m ( x ) = 1 2 2 m - 1 j = 0 2 m - 2 ( - 1 ) j N 2 m ( j + 1 ) d m d x m N 2 m ( 2 x - j ) \psi_{C,m}(x)=\frac{1}{2^{2m-1}}\sum_{j=0}^{2m-2}(-1)^{j}N_{2m}(j+1)\frac{d^{m% }}{dx^{m}}N_{2m}(2x-j)
  134. ψ C , 1 ( x ) = 1 2 N 2 ( 1 ) d d x N 2 ( 2 x ) = { 1 0 x < 1 2 - 1 1 2 x < 1 0 otherwise \psi_{C,1}(x)=\frac{1}{2}N_{2}(1)\frac{d}{dx}N_{2}(2x)=\begin{cases}1&0\leq x<% \frac{1}{2}\\ -1&\frac{1}{2}\leq x<1\\ 0&\,\text{otherwise}\end{cases}
  135. ψ C , m ( x ) \psi_{C,m}(x)
  136. [ 0 , 2 m - 1 ] [0,2m-1]
  137. ψ C , m ( x ) \psi_{C,m}(x)
  138. η ( x ) W 0 \eta(x)\in W_{0}
  139. W 0 W_{0}
  140. 2 m - 1 2m-1
  141. η ( x ) = c 0 ψ C , m ( x - n 0 ) \eta(x)=c_{0}\psi_{C,m}(x-n_{0})
  142. c 0 c_{0}
  143. n 0 n_{0}
  144. ψ C , m ( x ) \psi_{C,m}(x)
  145. ψ m ( x ) \psi_{m}(x)
  146. ψ C , m ( x ) = n = 0 3 m - 2 q n N m ( 2 x - n ) \psi_{C,m}(x)=\sum_{n=0}^{3m-2}q_{n}N_{m}(2x-n)
  147. q n = ( - 1 ) n 2 m - 1 j = 0 m ( m j ) N 2 m ( n - j + 1 ) q_{n}=\frac{(-1)^{n}}{2^{m-1}}\sum_{j=0}^{m}{m\choose j}N_{2m}(n-j+1)
  148. N m ( 2 x - l ) = k = - [ a m , l - 2 k N m ( x - k ) + b m , l - 2 k ψ C , m ( x - k ) ] N_{m}(2x-l)=\sum_{k=-\infty}^{\infty}\left[a_{m,l-2k}N_{m}(x-k)+b_{m,l-2k}\psi% _{C,m}(x-k)\right]
  149. a m , j a_{m,j}
  150. b m , j b_{m,j}
  151. a m , j = - ( - 1 ) j 2 l = - q - j + 2 m - 2 l + 1 c 2 m , l , a_{m,j}=-\frac{(-1)^{j}}{2}\sum_{l=-\infty}^{\infty}q_{-j+2m-2l+1}c_{2m,l},
  152. b m , j = ( - 1 ) j 2 l = - p - j + 2 m - 2 l + 1 c 2 m , l . b_{m,j}=\frac{(-1)^{j}}{2}\sum_{l=-\infty}^{\infty}p_{-j+2m-2l+1}c_{2m,l}.
  153. c 2 m , l c_{2m,l}
  154. ψ C , 1 ( x ) = N 1 ( 2 x ) - N 1 ( 2 x - 1 ) \psi_{C,1}(x)=N_{1}(2x)-N_{1}(2x-1)
  155. ψ C , 1 ( x ) = { 1 0 x < 1 2 - 1 1 2 x < 1 0 otherwise \psi_{C,1}(x)=\begin{cases}1&0\leq x<\frac{1}{2}\\ -1&\frac{1}{2}\leq x<1\\ 0&\,\text{otherwise}\end{cases}
  156. ψ C , 2 ( x ) = 1 12 ( N 2 ( 2 x ) - 6 N 2 ( 2 x - 1 ) + 10 N 2 ( 2 x - 2 ) - 6 N 2 ( 2 x - 3 ) + N 2 ( 2 x - 4 ) ) \psi_{C,2}(x)=\frac{1}{12}\left(N_{2}(2x)-6N_{2}(2x-1)+10N_{2}(2x-2)-6N_{2}(2x% -3)+N_{2}(2x-4)\right)
  157. ψ C , 2 ( x ) = { 1 6 x 0 x < 1 2 - 7 6 x + 2 3 1 2 x < 1 8 3 x - 19 6 1 x < 3 2 - 8 3 x + 29 6 3 2 x < 2 7 6 x - 17 6 2 x < 5 2 - 1 6 x + 1 2 5 2 x < 3 0 otherwise \psi_{C,2}(x)=\begin{cases}\frac{1}{6}x&0\leq x<\frac{1}{2}\\ -\frac{7}{6}x+\frac{2}{3}&\frac{1}{2}\leq x<1\\ \frac{8}{3}x-\frac{19}{6}&1\leq x<\frac{3}{2}\\ -\frac{8}{3}x+\frac{29}{6}&\frac{3}{2}\leq x<2\\ \frac{7}{6}x-\frac{17}{6}&2\leq x<\frac{5}{2}\\ -\frac{1}{6}x+\frac{1}{2}&\frac{5}{2}\leq x<3\\ 0&\,\text{otherwise}\end{cases}
  158. ψ C , 3 ( x ) = 1 480 [ ( N 3 ( 2 x ) - 29 N 3 ( 2 x - 1 ) + 147 N 3 ( 2 x - 2 ) - 303 N 3 ( 2 x - 3 ) + \psi_{C,3}(x)=\frac{1}{480}\Big[(N_{3}(2x)-29N_{3}(2x-1)+147N_{3}(2x-2)-303N_{% 3}(2x-3)+
  159. 303 N 3 ( 2 x - 4 ) - 147 N 3 ( 2 x - 5 ) + 29 N 3 ( 2 x - 6 ) - N 3 ( 2 x - 7 ) ] 303N_{3}(2x-4)-147N_{3}(2x-5)+29N_{3}(2x-6)-N_{3}(2x-7)\Big]
  160. ψ C , 3 ( x ) = { 1 240 x 2 0 x < 1 2 - 31 240 x 2 + 2 15 x - 1 30 1 2 x < 1 103 120 x 2 - 221 120 x + 229 240 1 x < 3 2 - 313 120 x 2 + 1027 120 x - 1643 240 3 2 x < 2 22 5 x 2 - 779 40 x + 339 16 2 x < 5 2 - 22 5 x 2 + 981 40 x - 541 16 5 2 x < 3 313 120 x 2 - 701 40 x + 2341 80 3 x < 7 2 - 103 120 x 2 + 809 120 x - 3169 240 7 2 x < 4 31 240 x 2 - 139 120 x + 623 240 4 x < 9 2 - 1 240 x 2 + 1 24 x - 5 48 9 2 x < 5 0 otherwise \psi_{C,3}(x)=\begin{cases}\frac{1}{240}x^{2}&0\leq x<\frac{1}{2}\\ -\frac{31}{240}x^{2}+\frac{2}{15}x-\frac{1}{30}&\frac{1}{2}\leq x<1\\ \frac{103}{120}x^{2}-\frac{221}{120}x+\frac{229}{240}&1\leq x<\frac{3}{2}\\ -\frac{313}{120}x^{2}+\frac{1027}{120}x-\frac{1643}{240}&\frac{3}{2}\leq x<2\\ \frac{22}{5}x^{2}-\frac{779}{40}x+\frac{339}{16}&2\leq x<\frac{5}{2}\\ -\frac{22}{5}x^{2}+\frac{981}{40}x-\frac{541}{16}&\frac{5}{2}\leq x<3\\ \frac{313}{120}x^{2}-\frac{701}{40}x+\frac{2341}{80}&3\leq x<\frac{7}{2}\\ -\frac{103}{120}x^{2}+\frac{809}{120}x-\frac{3169}{240}&\frac{7}{2}\leq x<4\\ \frac{31}{240}x^{2}-\frac{139}{120}x+\frac{623}{240}&4\leq x<\frac{9}{2}\\ -\frac{1}{240}x^{2}+\frac{1}{24}x-\frac{5}{48}&\frac{9}{2}\leq x<5\\ 0&\,\text{otherwise}\end{cases}
  161. ψ C , 4 ( x ) = 1 40320 [ N 4 ( 2 x ) - 124 N 4 ( 2 x - 1 ) + 1677 N 4 ( 2 x - 2 ) - 7904 N 4 ( 2 x - 3 ) + 18482 N 4 ( 2 x - 4 ) - \psi_{C,4}(x)=\frac{1}{40320}\Big[N_{4}(2x)-124N_{4}(2x-1)+1677N_{4}(2x-2)-790% 4N_{4}(2x-3)+18482N_{4}(2x-4)-
  162. 24264 N 4 ( 2 x - 5 ) + 18482 N 4 ( 2 x - 6 ) - 7904 N 4 ( 2 x - 7 ) + 1677 N 4 ( 2 x - 8 ) - 124 N 4 ( 2 x - 9 ) + N 4 ( 2 x - 10 ) ] 24264N_{4}(2x-5)+18482N_{4}(2x-6)-7904N_{4}(2x-7)+1677N_{4}(2x-8)-124N_{4}(2x-% 9)+N_{4}(2x-10)\Big]
  163. ψ C , 4 ( x ) = { 1 30240 x 3 0 x < 1 2 - 127 30240 x 3 + 2 315 x 2 - 1 315 x + 1 1890 1 2 x < 1 19 280 x 3 - 47 224 x 2 + 2147 10080 x - 103 1440 1 x < 3 2 - 1109 2520 x 3 + 465 224 x 2 - 32413 10080 x + 16559 10080 3 2 x < 2 5261 3360 x 3 - 33463 3360 x 2 + 42043 2016 x - 145193 10080 2 x < 5 2 - 35033 10080 x 3 + 93577 3360 x 2 - 148517 2016 x + 216269 3360 5 2 x < 3 4832 945 x 3 - 27691 560 x 2 + 113923 720 x - 28145 168 3 x < 7 2 - 4832 945 x 3 + 58393 1008 x 2 - 52223 240 x + 2048227 7560 7 2 x < 4 35033 10080 x 3 - 75827 1680 x 2 + 981101 5040 x - 234149 840 4 x < 9 2 - 5261 3360 x 3 + 38509 1680 x 2 - 112487 1008 x + 30347 168 9 2 x < 5 1109 2520 x 3 - 24077 3360 x 2 + 78311 2016 x - 141311 2016 5 x < 11 2 - 19 280 x 3 + 1361 1120 x 2 - 14617 2016 x + 4151 288 11 2 x < 6 127 30240 x 3 - 55 672 x 2 + 5359 10080 x - 11603 10080 6 x < 13 2 - 1 30240 x 3 + 1 1440 x 2 - 7 1440 x + 49 4320 13 2 x < 7 0 otherwise \psi_{C,4}(x)=\begin{cases}\frac{1}{30240}x^{3}&0\leq x<\frac{1}{2}\\ -\frac{127}{30240}x^{3}+\frac{2}{315}x^{2}-\frac{1}{315}x+\frac{1}{1890}&\frac% {1}{2}\leq x<1\\ \frac{19}{280}x^{3}-\frac{47}{224}x^{2}+\frac{2147}{10080}x-\frac{103}{1440}&1% \leq x<\frac{3}{2}\\ -\frac{1109}{2520}x^{3}+\frac{465}{224}x^{2}-\frac{32413}{10080}x+\frac{16559}% {10080}&\frac{3}{2}\leq x<2\\ \frac{5261}{3360}x^{3}-\frac{33463}{3360}x^{2}+\frac{42043}{2016}x-\frac{14519% 3}{10080}&2\leq x<\frac{5}{2}\\ -\frac{35033}{10080}x^{3}+\frac{93577}{3360}x^{2}-\frac{148517}{2016}x+\frac{2% 16269}{3360}&\frac{5}{2}\leq x<3\\ \frac{4832}{945}x^{3}-\frac{27691}{560}x^{2}+\frac{113923}{720}x-\frac{28145}{% 168}&3\leq x<\frac{7}{2}\\ -\frac{4832}{945}x^{3}+\frac{58393}{1008}x^{2}-\frac{52223}{240}x+\frac{204822% 7}{7560}&\frac{7}{2}\leq x<4\\ \frac{35033}{10080}x^{3}-\frac{75827}{1680}x^{2}+\frac{981101}{5040}x-\frac{23% 4149}{840}&4\leq x<\frac{9}{2}\\ -\frac{5261}{3360}x^{3}+\frac{38509}{1680}x^{2}-\frac{112487}{1008}x+\frac{303% 47}{168}&\frac{9}{2}\leq x<5\\ \frac{1109}{2520}x^{3}-\frac{24077}{3360}x^{2}+\frac{78311}{2016}x-\frac{14131% 1}{2016}&5\leq x<\frac{11}{2}\\ -\frac{19}{280}x^{3}+\frac{1361}{1120}x^{2}-\frac{14617}{2016}x+\frac{4151}{28% 8}&\frac{11}{2}\leq x<6\\ \frac{127}{30240}x^{3}-\frac{55}{672}x^{2}+\frac{5359}{10080}x-\frac{11603}{10% 080}&6\leq x<\frac{13}{2}\\ -\frac{1}{30240}x^{3}+\frac{1}{1440}x^{2}-\frac{7}{1440}x+\frac{49}{4320}&% \frac{13}{2}\leq x<7\\ 0&\,\text{otherwise}\end{cases}
  164. ψ C , 5 ( x ) = 1 5806080 [ N 5 ( 2 x ) - 507 N 5 ( 2 x - 1 ) + 17128 N 5 ( 2 x - 2 ) - 166304 N 5 ( 2 x - 3 ) + 748465 N 5 ( 2 x - 4 ) \psi_{C,5}(x)=\frac{1}{5806080}\Big[N_{5}(2x)-507N_{5}(2x-1)+17128N_{5}(2x-2)-% 166304N_{5}(2x-3)+748465N_{5}(2x-4)
  165. - 1900115 N 5 ( 2 x - 5 ) + 2973560 N 5 ( 2 x - 6 ) - 2973560 N 5 ( 2 x - 7 ) + 1900115 N 5 ( 2 x - 8 ) -1900115N_{5}(2x-5)+2973560N_{5}(2x-6)-2973560N_{5}(2x-7)+1900115N_{5}(2x-8)
  166. - 748465 N 5 ( 2 x - 9 ) + 166304 N 5 ( 2 x - 10 ) - 17128 N 5 ( 2 x - 11 ) + 507 N 5 ( 2 x - 12 ) - N 5 ( 2 x - 13 ) ] -748465N_{5}(2x-9)+166304N_{5}(2x-10)-17128N_{5}(2x-11)+507N_{5}(2x-12)-N_{5}(% 2x-13)\Big]
  167. ψ C , 5 ( x ) = { 1 8709120 x 4 0 x < 1 2 - 73 1244160 x 4 + 1 8505 x 3 - 1 11340 x 2 + 1 34020 x - 1 272160 1 2 x < 1 9581 4354560 x 4 - 19417 2177280 x 3 + 1303 96768 x 2 - 19609 2177280 x + 6547 2903040 1 x < 3 2 - 118931 4354560 x 4 + 366119 2177280 x 3 - 186253 483840 x 2 + 121121 311040 x - 427181 2903040 3 2 x < 2 759239 4354560 x 4 - 3146561 2177280 x 3 + 6466601 1451520 x 2 - 13202873 2177280 x + 26819897 8709120 2 x < 5 2 - 2980409 4354560 x 4 + 5183893 725760 x 3 - 13426333 483840 x 2 + 426589 8960 x - 12635243 414720 5 2 x < 3 7873577 4354560 x 4 - 16524079 725760 x 3 + 7385369 69120 x 2 - 17868671 80640 x + 497668543 290304 3 x < 7 2 - 14714327 4354560 x 4 + 108543091 2177280 x 3 - 56901557 207360 x 2 + 1454458651 2177280 x - 5286189059 8709120 7 2 x < 4 15619 3402 x 4 - 33822017 435456 x 3 + 15828929 32256 x 2 - 597598433 435456 x + 277413649 193536 4 x < 9 2 - 15619 3402 x 4 + 38150335 435456 x 3 - 20157247 32256 x 2 + 859841695 435456 x - 64472345 27648 9 2 x < 5 14714327 4354560 x 4 - 4466137 62208 x 3 + 165651247 290304 x 2 - 875490655 435456 x + 4614904015 1741824 5 x < 11 2 - 7873577 4354560 x 4 + 30717383 725760 x 3 - 179437319 483840 x 2 + 16606729 11520 x - 869722273 414720 11 2 x < 6 2980409 4354560 x 4 - 12698561 725760 x 3 + 16211669 96768 x 2 - 19138891 26880 x + 3289787993 2903040 6 x < 13 2 - 759239 4354560 x 4 + 10519741 2177280 x 3 - 10403603 207360 x 2 + 71964499 311040 x - 3481646837 8709120 13 2 x < 7 118931 4354560 x 4 - 1774639 2177280 x 3 + 630259 69120 x 2 - 14096161 311040 x + 245108501 2903040 7 x < 15 2 - 9581 4354560 x 4 + 21863 311040 x 3 - 407387 483840 x 2 + 9758873 2177280 x - 25971499 2903040 15 2 x < 8 73 1244160 x 4 - 4343 2177280 x 3 + 5273 207360 x 2 - 313703 2177280 x + 380873 1244160 8 x < 17 2 - 1 8709120 x 4 + 1 241920 x 3 - 1 17920 x 2 + 3 8960 x - 27 35840 17 2 x < 9 0 otherwise \psi_{C,5}(x)=\begin{cases}\frac{1}{8709120}x^{4}&0\leq x<\frac{1}{2}\\ -\frac{73}{1244160}x^{4}+\frac{1}{8505}x^{3}-\frac{1}{11340}x^{2}+\frac{1}{340% 20}x-\frac{1}{272160}&\frac{1}{2}\leq x<1\\ \frac{9581}{4354560}x^{4}-\frac{19417}{2177280}x^{3}+\frac{1303}{96768}x^{2}-% \frac{19609}{2177280}x+\frac{6547}{2903040}&1\leq x<\frac{3}{2}\\ -\frac{118931}{4354560}x^{4}+\frac{366119}{2177280}x^{3}-\frac{186253}{483840}% x^{2}+\frac{121121}{311040}x-\frac{427181}{2903040}&\frac{3}{2}\leq x<2\\ \frac{759239}{4354560}x^{4}-\frac{3146561}{2177280}x^{3}+\frac{6466601}{145152% 0}x^{2}-\frac{13202873}{2177280}x+\frac{26819897}{8709120}&2\leq x<\frac{5}{2}% \\ -\frac{2980409}{4354560}x^{4}+\frac{5183893}{725760}x^{3}-\frac{13426333}{4838% 40}x^{2}+\frac{426589}{8960}x-\frac{12635243}{414720}&\frac{5}{2}\leq x<3\\ \frac{7873577}{4354560}x^{4}-\frac{16524079}{725760}x^{3}+\frac{7385369}{69120% }x^{2}-\frac{17868671}{80640}x+\frac{497668543}{290304}&3\leq x<\frac{7}{2}\\ -\frac{14714327}{4354560}x^{4}+\frac{108543091}{2177280}x^{3}-\frac{56901557}{% 207360}x^{2}+\frac{1454458651}{2177280}x-\frac{5286189059}{8709120}&\frac{7}{2% }\leq x<4\\ \frac{15619}{3402}x^{4}-\frac{33822017}{435456}x^{3}+\frac{15828929}{32256}x^{% 2}-\frac{597598433}{435456}x+\frac{277413649}{193536}&4\leq x<\frac{9}{2}\\ -\frac{15619}{3402}x^{4}+\frac{38150335}{435456}x^{3}-\frac{20157247}{32256}x^% {2}+\frac{859841695}{435456}x-\frac{64472345}{27648}&\frac{9}{2}\leq x<5\\ \frac{14714327}{4354560}x^{4}-\frac{4466137}{62208}x^{3}+\frac{165651247}{2903% 04}x^{2}-\frac{875490655}{435456}x+\frac{4614904015}{1741824}&5\leq x<\frac{11% }{2}\\ -\frac{7873577}{4354560}x^{4}+\frac{30717383}{725760}x^{3}-\frac{179437319}{48% 3840}x^{2}+\frac{16606729}{11520}x-\frac{869722273}{414720}&\frac{11}{2}\leq x% <6\\ \frac{2980409}{4354560}x^{4}-\frac{12698561}{725760}x^{3}+\frac{16211669}{9676% 8}x^{2}-\frac{19138891}{26880}x+\frac{3289787993}{2903040}&6\leq x<\frac{13}{2% }\\ -\frac{759239}{4354560}x^{4}+\frac{10519741}{2177280}x^{3}-\frac{10403603}{207% 360}x^{2}+\frac{71964499}{311040}x-\frac{3481646837}{8709120}&\frac{13}{2}\leq x% <7\\ \frac{118931}{4354560}x^{4}-\frac{1774639}{2177280}x^{3}+\frac{630259}{69120}x% ^{2}-\frac{14096161}{311040}x+\frac{245108501}{2903040}&7\leq x<\frac{15}{2}\\ -\frac{9581}{4354560}x^{4}+\frac{21863}{311040}x^{3}-\frac{407387}{483840}x^{2% }+\frac{9758873}{2177280}x-\frac{25971499}{2903040}&\frac{15}{2}\leq x<8\\ \frac{73}{1244160}x^{4}-\frac{4343}{2177280}x^{3}+\frac{5273}{207360}x^{2}-% \frac{313703}{2177280}x+\frac{380873}{1244160}&8\leq x<\frac{17}{2}\\ -\frac{1}{8709120}x^{4}+\frac{1}{241920}x^{3}-\frac{1}{17920}x^{2}+\frac{3}{89% 60}x-\frac{27}{35840}&\frac{17}{2}\leq x<9\\ 0&\,\text{otherwise}\end{cases}
  168. F ( t ) F(t)
  169. F ^ ( ω ) \hat{F}(\omega)
  170. N m ( x ) N_{m}(x)
  171. N m ( x ) N_{m}(x)
  172. N ^ m ( ω ) \hat{N}_{m}(\omega)
  173. ϕ m ( t ) \phi_{m}(t)
  174. ϕ ^ m ( ω ) = N ^ m ( ω ) ( k = - | N ^ m ( ω + 2 π k ) | 2 ) 1 / 2 . \hat{\phi}_{m}(\omega)=\frac{\hat{N}_{m}(\omega)}{\left(\sum_{k=-\infty}^{% \infty}|\hat{N}_{m}(\omega+2\pi k)|^{2}\right)^{1/2}}.
  175. ψ B L , m ( t ) \psi_{BL,m}(t)
  176. ψ ^ B L , m ( ω ) = - e - i ω / 2 ϕ ^ m ( ω + 2 π ) ¯ ϕ ^ m ( ω 2 ) ϕ ^ m ( ω 2 + π ) ¯ \hat{\psi}_{BL,m}(\omega)=-\frac{e^{-i\omega/2}\,\,\overline{\hat{\phi}_{m}(% \omega+2\pi)}\,\,\hat{\phi}_{m}\left(\frac{\omega}{2}\right)}{\overline{\hat{% \phi}_{m}\left(\frac{\omega}{2}+\pi\right)}}

Sporulenol_synthase.html

  1. \rightleftharpoons

Squalene—-hopanol_cyclase.html

  1. \rightleftharpoons

Squalene—hopene_cyclase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Squirmer.html

  1. R R
  2. u r ( r , θ ) = 2 3 ( R 3 r 3 - 1 ) B 1 P 1 ( cos θ ) + n = 2 ( R n + 2 r n + 2 - R n r n ) B n P n ( cos θ ) , u_{r}(r,\theta)=\frac{2}{3}\left(\frac{R^{3}}{r^{3}}-1\right)B_{1}P_{1}(\cos% \theta)+\sum_{n=2}^{\infty}\left(\frac{R^{n+2}}{r^{n+2}}-\frac{R^{n}}{r^{n}}% \right)B_{n}P_{n}(\cos\theta)\;,
  3. u θ ( r , θ ) = 2 3 ( R 3 2 r 3 + 1 ) B 1 V 1 ( cos θ ) + n = 2 1 2 ( n R n + 2 r n + 2 + ( 2 - n ) R n r n ) B n V n ( cos θ ) . u_{\theta}(r,\theta)=\frac{2}{3}\left(\frac{R^{3}}{2r^{3}}+1\right)B_{1}V_{1}(% \cos\theta)+\sum_{n=2}^{\infty}\frac{1}{2}\left(n\frac{R^{n+2}}{r^{n+2}}+(2-n)% \frac{R^{n}}{r^{n}}\right)B_{n}V_{n}(\cos\theta)\;.
  4. B n B_{n}
  5. P n ( cos θ ) P_{n}(\cos\theta)
  6. V n ( cos θ ) = - 2 n ( n + 1 ) θ P n ( cos θ ) V_{n}(\cos\theta)=\frac{-2}{n(n+1)}\partial_{\theta}P_{n}(\cos\theta)
  7. P 1 ( cos θ ) = cos θ , P 2 ( cos θ ) = 1 2 ( 3 cos 2 θ - 1 ) , , V 1 ( cos θ ) = sin θ , V 2 ( cos θ ) = 1 2 sin 2 θ , P_{1}(\cos\theta)=\cos\theta,P_{2}(\cos\theta)=\tfrac{1}{2}(3\cos^{2}\theta-1)% ,\dots,V_{1}(\cos\theta)=\sin\theta,V_{2}(\cos\theta)=\tfrac{1}{2}\sin 2\theta,\dots
  8. u θ ( R , θ ) = n = 1 B n V n u_{\theta}(R,\theta)=\sum_{n=1}^{\infty}B_{n}V_{n}
  9. u r ( R , θ ) = 0 u_{r}(R,\theta)=0
  10. 𝐔 = - 1 2 𝐮 ( R , θ ) sin θ d θ = 2 3 B 1 𝐞 z \mathbf{U}=-\tfrac{1}{2}\int\mathbf{u}(R,\theta)\sin\theta\mathrm{d}\theta=% \tfrac{2}{3}B_{1}\mathbf{e}_{z}
  11. 𝐮 L = 𝐮 + 𝐔 \mathbf{u}^{L}=\mathbf{u}+\mathbf{U}
  12. u r L ( r , θ ) = R 3 r 3 U P 1 ( cos θ ) + n = 2 ( R n + 2 r n + 2 - R n r n ) B n P n ( cos θ ) , u_{r}^{L}(r,\theta)=\frac{R^{3}}{r^{3}}UP_{1}(\cos\theta)+\sum_{n=2}^{\infty}% \left(\frac{R^{n+2}}{r^{n+2}}-\frac{R^{n}}{r^{n}}\right)B_{n}P_{n}(\cos\theta)\;,
  13. u θ L ( r , θ ) = R 3 2 r 3 U V 1 ( cos θ ) + n = 2 1 2 ( n R n + 2 r n + 2 + ( 2 - n ) R n r n ) B n V n ( cos θ ) . u_{\theta}^{L}(r,\theta)=\frac{R^{3}}{2r^{3}}UV_{1}(\cos\theta)+\sum_{n=2}^{% \infty}\frac{1}{2}\left(n\frac{R^{n+2}}{r^{n+2}}+(2-n)\frac{R^{n}}{r^{n}}% \right)B_{n}V_{n}(\cos\theta)\;.
  14. U = | 𝐔 | U=|\mathbf{U}|
  15. lim r 𝐮 L = 0 \lim_{r\rightarrow\infty}\mathbf{u}^{L}=0
  16. u r L ( R , θ ) 0 u^{L}_{r}(R,\theta)\neq 0
  17. n = 2 n=2
  18. r R r\gg R
  19. u θ ( R , θ ) = B 1 sin θ + 1 2 B 2 sin 2 θ u_{\theta}(R,\theta)=B_{1}\sin\theta+\tfrac{1}{2}B_{2}\sin 2\theta
  20. β = B 2 / | B 1 | \beta=B_{2}/|B_{1}|
  21. n = 1 n=1
  22. 1 / r 3 \propto 1/r^{3}
  23. U U
  24. n = 2 n=2
  25. 1 / r 2 \propto 1/r^{2}
  26. β \beta
  27. β \beta
  28. β < 0 \beta<0
  29. β = 0 \beta=0
  30. β > 0 \beta>0
  31. β = ± \beta=\pm\infty
  32. 𝐮 1 / r 2 \mathbf{u}\propto 1/r^{2}
  33. 𝐮 1 / r 3 \mathbf{u}\propto 1/r^{3}
  34. 𝐮 1 / r 2 \mathbf{u}\propto 1/r^{2}
  35. 𝐮 1 / r 2 \mathbf{u}\propto 1/r^{2}
  36. 𝐮 1 / r \mathbf{u}\propto 1/r
  37. B 1 B_{1}
  38. β = 0 \beta=0
  39. B 2 B_{2}
  40. β 0 \beta\neq 0

Sriramachakra.html

  1. M = [ a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 a 41 a 42 a 43 a 44 ] M=\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\ a_{21}&a_{22}&a_{23}&a_{24}\\ a_{31}&a_{32}&a_{33}&a_{34}\\ a_{41}&a_{42}&a_{43}&a_{44}\\ \end{bmatrix}
  2. a m , n + a m , n + 1 + a m + 1 , n + a m + 1 , n + 1 = 34 a_{m,n}+a_{m,n+1}+a_{m+1,n}+a_{m+1,n+1}=34
  3. a 23 + a 24 + a 33 + a 34 = 34 a_{23}+a_{24}+a_{33}+a_{34}=34
  4. a 24 + a 21 + a 34 + a 31 = 34 a_{24}+a_{21}+a_{34}+a_{31}=34
  5. a 44 + a 41 + a 14 + a 11 = 34 a_{44}+a_{41}+a_{14}+a_{11}=34

Stable_range_condition.html

  1. v 0 , , v n v_{0},\dots,v_{n}
  2. t 1 , , t n t_{1},\dots,t_{n}
  3. v i - v 0 t i , 1 i n v_{i}-v_{0}t_{i},\,1\leq i\leq n
  4. d + 1 d+1

Stable_∞-category.html

  1. X ( i ) , i X(i),i\in\mathbb{Z}
  2. E r p , q E^{p,q}_{r}
  3. π p + q colim X ( i ) . \pi_{p+q}\operatorname{colim}X(i).

Staggered_tuning.html

  1. B = m - 1 Q B={\sqrt{m-1}\over Q}
  2. m - 1 \sqrt{m-1}
  3. m = 2 1 / n m=2^{1/n}
  4. A ( s ) = g m s L s 2 L C + s L G + 1 A(s)=\frac{g_{\mathrm{m}}sL}{s^{2}LC+sLG+1}
  5. ω 0 = 1 L C \omega_{0}={1\over\sqrt{LC}}
  6. A 0 := A ( ω 0 ) = g m G A_{0}:=A(\omega_{0})=\frac{g_{\mathrm{m}}}{G}
  7. Q = 1 ω 0 L G Q={1\over\omega_{0}LG}
  8. A ( s ) = A 0 s ω 0 s 2 Q + s ω 0 + ω 0 2 Q A(s)=A_{0}\frac{s\omega_{0}}{s^{2}Q+s\omega_{0}+\omega_{0}^{2}Q}
  9. A ( ω ) = A 0 i ω ω 0 i ω ω 0 + ω 0 2 Q - ω 2 Q A(\omega)=A_{0}\frac{i\omega\omega_{0}}{i\omega\omega_{0}+\omega_{0}^{2}Q-% \omega^{2}Q}
  10. | A ( ω c ) | = A 0 m |A(\omega_{c})|=\frac{A_{0}}{\sqrt{m}}
  11. Δ ω c = ω c1 - ω c2 = ω 0 ( m - 1 ) Q \Delta\omega_{\mathrm{c}}=\omega_{\mathrm{c}1}-\omega_{\mathrm{c}2}=\frac{% \omega_{0}\sqrt{(m-1)}}{Q}
  12. B := Δ ω c ω 0 = m - 1 Q B:=\frac{\Delta\omega_{\mathrm{c}}}{\omega_{0}}=\frac{\sqrt{m-1}}{Q}
  13. A T = A 1 A 2 A 3 A_{\mathrm{T}}=A_{1}A_{2}A_{3}\cdots
  14. A ( s ) = A 0 1 + Q ( s ω 0 + ω 0 s ) A(s)=\frac{A_{0}}{1+Q\left(\frac{s}{\omega_{0}}+\frac{\omega_{0}}{s}\right)}
  15. Q ( s ω 0 + ω 0 s ) s ω c Q\left(\frac{s}{\omega_{0}}+\frac{\omega_{0}}{s}\right)\to\frac{s}{\omega_{c}^% {\prime}}
  16. A ( s ) = A 0 Q s ω 0 ( s - p ) ( s - p * ) A(s)=\frac{A_{0}}{Q}\frac{s\omega_{0}}{(s-p)(s-p^{*})}
  17. A T = s a 1 ( s - p 1 ) ( s - p 1 * ) s a 2 ( s - p 2 ) ( s - p 2 * ) s a 3 ( s - p 3 ) ( s - p 3 * ) A_{\mathrm{T}}=\frac{sa_{1}}{(s-p_{1})(s-p_{1}^{*})}\cdot\frac{sa_{2}}{(s-p_{2% })(s-p_{2}^{*})}\cdot\frac{sa_{3}}{(s-p_{3})(s-p_{3}^{*})}\cdot\cdots
  18. p k , p k * = 1 2 ( q k ω 0 B ω c Q eff ± ( q k ω 0 B ω c Q eff ) 2 - 4 ω 0 B 2 ) p_{k},p^{*}_{k}={1\over 2}\left(\frac{q_{k}\omega_{0\mathrm{B}}}{\omega^{% \prime}_{\mathrm{c}}Q_{\mathrm{eff}}}\pm\sqrt{\left(\frac{q_{k}\omega_{0% \mathrm{B}}}{\omega^{\prime}_{\mathrm{c}}Q_{\mathrm{eff}}}\right)^{2}-4{\omega% _{0\mathrm{B}}}^{2}}\right)
  19. p k , p k * = 1 2 ( q k ± q k 2 - 4 ω 0 B 2 ) p_{k},p^{*}_{k}={1\over 2}\left(q_{k}\pm\sqrt{q_{k}^{2}-4{\omega_{0\mathrm{B}}% }^{2}}\right)
  20. p k , p k * q k 2 ± i ω 0 B p_{k},p^{*}_{k}\approx{q_{k}\over 2}\pm i\omega_{0\mathrm{B}}

State-transition_equation.html

  1. d x ( t ) d t = A x ( t ) + B u ( t ) + E w ( t ) , \frac{dx(t)}{dt}=Ax(t)+Bu(t)+Ew(t),
  2. s X ( s ) - x ( 0 ) = A X ( s ) + B U ( s ) + E W ( s ) sX(s)-x(0)=AX(s)+BU(s)+EW(s)
  3. t = 0 t=0
  4. X ( s ) X(s)
  5. X ( s ) = ( s I - A ) - 1 x ( 0 ) + ( s I - A ) - 1 [ B U ( s ) + E W ( s ) ] . X(s)=(sI-A)^{-1}x(0)+(sI-A)^{-1}[BU(s)+EW(s)].
  6. x ( t ) = L - 1 [ ( s I - A ) - 1 ] x ( 0 ) + L - 1 ( s I - A ) - 1 [ B U ( s ) + E W ( s ) ] = ϕ ( t ) x ( 0 ) + 0 t ϕ ( t - τ ) [ B u ( τ ) + E w ( τ ) ] d t . x(t)=L^{-1}[(sI-A)^{-1}]x(0)+L^{-1}{(sI-A)^{-1}[BU(s)+EW(s)]}=\phi(t)x(0)+\int% _{0}^{t}\phi(t-\tau)[Bu(\tau)+Ew(\tau)]dt.
  7. t = 0 t=0
  8. t 0 t_{0}
  9. x ( t 0 ) x(t_{0})
  10. u ( t ) u(t)
  11. w ( t ) w(t)
  12. t = t 0 , t=t_{0},
  13. x ( 0 ) x(0)
  14. x ( 0 ) = ϕ ( - t 0 ) x ( t 0 ) - ϕ ( - t 0 ) 0 t 0 ϕ ( t 0 - τ ) [ B u ( τ ) + E w ( τ ) ] d τ . x(0)=\phi(-t_{0})x(t_{0})-\phi(-t_{0})\int_{0}^{t_{0}}\phi(t_{0}-\tau)[Bu(\tau% )+Ew(\tau)]d\tau.

Stationary_or_bubbling_fluidized_bed.html

  1. d p d_{p}
  2. u m f = [ ( ρ p - ρ ) g d p ε 3 1.75 ρ ] 1 2 u_{mf}=\left[\frac{(\rho_{p}-\rho)gd_{p}\varepsilon^{3}}{1.75\rho}\right]^{% \frac{1}{2}}
  3. ρ p \rho_{p}
  4. Δ P L = 150 μ ( 1 - ε ) 2 u 0 ε 3 d p 2 + 1.75 ( 1 - ε ) ρ u 0 2 ε 3 d p \frac{\Delta P}{L}=\frac{150\mu(1-\varepsilon)^{2}u_{0}}{\varepsilon^{3}d_{p}^% {2}}+\frac{1.75(1-\varepsilon)\rho u_{0}^{2}}{\varepsilon^{3}d_{p}}
  5. u m b u m f = 4.125 × 10 4 μ 0.9 ρ 0.1 ( ρ p - ρ ) g d p \frac{u_{mb}}{u_{mf}}=\frac{4.125\times 10^{4}\mu^{0.9}\rho^{0.1}}{(\rho_{p}-% \rho)gd_{p}}

Steady_flight.html

  1. T cos α cos β - W sin γ - D = 0 ( x w -axis ) T\cos{\alpha}\cos{\beta}-W\sin{\gamma}-D=0\quad(x_{w}\,\text{-axis})
  2. C cos μ + L sin μ + T ( sin α sin μ + cos α cos μ sin β ) = W g ( V cos γ ) 2 R ( x E - y E plane radial direction ) C\cos{\mu}+L\sin{\mu}+T(\sin{\alpha}\sin{\mu}+\cos{\alpha}\cos{\mu}\sin{\beta}% )=\frac{W}{g}\frac{(V\cos{\gamma})^{2}}{R}\quad(x_{E}\,\text{-}y_{E}\,\text{ % plane radial direction})
  3. W cos γ + C sin μ - L cos μ - T sin α cos μ = 0 ( axis perpendicular to x w in the x w - z E plane ) W\cos{\gamma}+C\sin{\mu}-L\cos{\mu}-T\sin{\alpha}\cos{\mu}=0\quad(\,\text{axis% perpendicular to }x_{w}\,\text{ in the }x_{w}\,\text{-}z_{E}\,\text{ plane})
  4. T = W γ + D T=W\gamma+D
  5. L sin μ = W g V 2 R L\sin{\mu}=\frac{W}{g}\frac{V^{2}}{R}
  6. L cos μ = W L\cos{\mu}=W
  7. R = V 2 g tan μ R=\frac{V^{2}}{g\tan{\mu}}

Steiner_conic.html

  1. C h a r 2 Char\neq 2
  2. B ( U ) , B ( V ) B(U),B(V)
  3. U , V U,V
  4. U U
  5. V V
  6. π \pi
  7. B ( U ) B(U)
  8. B ( V ) B(V)
  9. π \pi
  10. B ( U ) B(U)
  11. B ( V ) B(V)
  12. a a
  13. π \pi
  14. \R \R
  15. \Q \Q
  16. \C \C
  17. U , V U,V
  18. a a
  19. Z Z
  20. b b
  21. a , u , w a,u,w
  22. π ( a ) = b , π ( u ) = w , π ( w ) = v \pi(a)=b,\pi(u)=w,\pi(w)=v
  23. π \pi
  24. π b , π a \pi_{b},\pi_{a}
  25. π b \pi_{b}
  26. U U
  27. O O
  28. b b
  29. π a \pi_{a}
  30. O O
  31. V V
  32. a a
  33. π = π a π b \pi=\pi_{a}\pi_{b}
  34. π ( a ) = b , π ( u ) = w , π ( w ) = v \pi(a)=b,\pi(u)=w,\pi(w)=v
  35. g g
  36. π ( g ) = π a π b ( g ) \pi(g)=\pi_{a}\pi_{b}(g)
  37. u u
  38. v v
  39. U U
  40. V V
  41. u u
  42. v v
  43. w w
  44. O O
  45. U , V U,V
  46. E = ( 1 , 1 ) E=(1,1)
  47. y = 1 / x y=1/x
  48. u , v u,v
  49. π \pi
  50. u u
  51. v v
  52. π \pi
  53. u u
  54. v v
  55. Z Z
  56. π \pi
  57. C h a r = 2 Char=2
  58. C h a r 2 Char\neq 2
  59. A , U , W A,U,W
  60. π ( A ) = B , π ( U ) = W , π ( W ) = V \pi(A)=B,\,\pi(U)=W,\,\pi(W)=V
  61. π \pi
  62. π B , π A \pi_{B},\pi_{A}
  63. π B \pi_{B}
  64. u u
  65. o o
  66. B B
  67. π A \pi_{A}
  68. o o
  69. v v
  70. A A
  71. π = π A π B \pi=\pi_{A}\pi_{B}
  72. π ( A ) = B , π ( U ) = W , π ( W ) = V \pi(A)=B,\,\pi(U)=W,\,\pi(W)=V
  73. G G
  74. π ( G ) = π A π B ( G ) \pi(G)=\pi_{A}\pi_{B}(G)
  75. G π ( G ) ¯ \overline{G\pi(G)}
  76. U U
  77. V V
  78. u u
  79. v v
  80. U U
  81. V V
  82. u , v u,v
  83. U , V U,V

Steinmetz's_equation.html

  1. Q = η * B 1.6 Q=\eta*B^{1.6}
  2. Q = k * η a * B b Q=k*\eta^{a}*B^{b}
  3. P = 1 T 0 T k i | d B d t | a ( Δ B b - a ) d t P=\frac{1}{T}\int_{0}^{T}k_{i}{\left|\frac{dB}{dt}\right|}^{a}(\Delta B^{b-a})dt
  4. Δ B \Delta B
  5. k i k_{i}
  6. k i = k ( 2 π ) a - 1 0 2 π | c o s θ | a * 2 b - a d θ k_{i}=\frac{k}{{(2\pi)}^{a-1}\int_{0}^{2\pi}{\left|cos\theta\right|}^{a}*2^{b-% a}d\theta}

Stemar-13-ene_synthase.html

  1. \rightleftharpoons

Stemod-13(17)-ene_synthase.html

  1. \rightleftharpoons

Steric_tesseractic_honeycomb.html

  1. B ~ 4 {\tilde{B}}_{4}

Stericantellated_tesseractic_honeycomb.html

  1. C ~ 4 {\tilde{C}}_{4}

Stericantic_tesseractic_honeycomb.html

  1. B ~ 4 {\tilde{B}}_{4}

Stericantitruncated_16-cell_honeycomb.html

  1. F ~ 4 {\tilde{F}}_{4}

Stericantitruncated_tesseractic_honeycomb.html

  1. C ~ 4 {\tilde{C}}_{4}

Stericated_24-cell_honeycomb.html

  1. F ~ 4 {\tilde{F}}_{4}

Steriruncic_tesseractic_honeycomb.html

  1. B ~ 4 {\tilde{B}}_{4}

Steriruncicantic_tesseractic_honeycomb.html

  1. B ~ 4 {\tilde{B}}_{4}

Steriruncitruncated_tesseractic_honeycomb.html

  1. C ~ 4 {\tilde{C}}_{4}

Steritruncated_16-cell_honeycomb.html

  1. F ~ 4 {\tilde{F}}_{4}

Steritruncated_tesseractic_honeycomb.html

  1. C ~ 4 {\tilde{C}}_{4}
  2. B ~ 4 {\tilde{B}}_{4}

Stimulated_raman_adiabatic_passage.html

  1. | g 1 |g_{1}\rangle
  2. | g 2 |g_{2}\rangle
  3. | e |e\rangle
  4. | g 1 |g_{1}\rangle
  5. | g 1 |g_{1}\rangle
  6. | g 2 |g_{2}\rangle
  7. | g 2 |g_{2}\rangle
  8. | e |e\rangle
  9. | g 2 |g_{2}\rangle
  10. | e |e\rangle
  11. | g 1 |g_{1}\rangle
  12. | g 2 |g_{2}\rangle
  13. | e |e\rangle

Stochastic_geometry_models_of_wireless_networks.html

  1. SINR = S I + N \mathrm{SINR}=\frac{S}{I+N}
  2. p out = P ( SINR t ) p_{\mathrm{out}}=P(\mathrm{SINR}\leq t)
  3. p c = P ( SINR > t ) = 1 - p out p_{\mathrm{c}}=P(\mathrm{SINR}>t)=1-p_{\mathrm{out}}
  4. C = B log 2 ( 1 + SINR ) C=B\log_{2}(1+\mathrm{SINR})
  5. ( | x - y | ) < m t p l | x - y | α \ell(|x-y|)<mtpl>{{=}}|x-y|^{\alpha}
  6. P ( n ) < m t p l λ | B | n n ! e - λ | B | , P(n)<mtpl>{{=}}\frac{\lambda|B|^{n}}{n!}e^{-\lambda|B|},
  7. n n
  8. λ < m t p l i = 1 n λ i . \lambda<mtpl>{{=}}\sum_{i=1}^{n}\lambda_{i}.

Stochastic_portfolio_theory.html

  1. X i , X_{i},
  2. i = 1 , , n , i=1,\dots,n,
  3. d log X i ( t ) = γ i ( t ) d t + ν = 1 d ξ i ν ( t ) d W ν ( t ) d\log X_{i}(t)=\gamma_{i}(t)\,dt+\sum_{\nu=1}^{d}\xi_{i\nu}(t)\,dW_{\nu}(t)
  4. W := ( W 1 , , W d ) W:=(W_{1},\dots,W_{d})
  5. n n
  6. d n d\geq n
  7. γ i \gamma_{i}
  8. ξ i ν \xi_{i\nu}
  9. { t } = { t W } \{\mathcal{F}_{t}\}=\{\mathcal{F}^{W}_{t}\}
  10. γ i ( t ) \gamma_{i}(t)
  11. X i , X_{i},
  12. log X i \log X_{i}
  13. log X j \log X_{j}
  14. σ i j ( t ) = ν = 1 d ξ i ν ( t ) ξ j ν ( t ) . \sigma_{ij}(t)=\sum_{\nu=1}^{d}\xi_{i\nu}(t)\xi_{j\nu}(t).
  15. i , i,
  16. ξ i , 1 2 ( t ) + + ξ i d 2 ( t ) \xi_{i,1}^{2}(t)+\cdots+\xi_{id}^{2}(t)
  17. t . t\rightarrow\infty.
  18. α i ( t ) , \alpha_{i}(t),
  19. α i ( t ) = γ i ( t ) + σ i i ( t ) 2 \alpha_{i}(t)=\gamma_{i}(t)+\frac{\sigma_{ii}(t)}{2}
  20. X i ( t ) X_{i}(t)
  21. i i
  22. t , t,
  23. X ( t ) = X 1 ( t ) + + X n ( t ) X(t)=X_{1}(t)+\cdots+X_{n}(t)
  24. π = ( π 1 , , π n ) \pi=(\pi_{1},\cdots,\pi_{n})
  25. π i ( t ) \pi_{i}(t)
  26. i i
  27. t t
  28. π 0 ( t ) := 1 - i = 1 n π i ( t ) \pi_{0}(t):=1-\sum_{i=1}^{n}\pi_{i}(t)
  29. κ 0 ( κ 0 1 ) \kappa\equiv 0(\kappa_{0}\equiv 1)
  30. π \pi
  31. π 1 ( t ) + + π n ( t ) = 1 \pi_{1}(t)+\cdots+\pi_{n}(t)=1
  32. Z π Z_{\pi}
  33. π \pi
  34. d log Z π ( t ) = i = 1 n π i ( t ) d log X i ( t ) + γ π * ( t ) d t d\log Z_{\pi}(t)=\sum_{i=1}^{n}\pi_{i}(t)\,d\log X_{i}(t)+\gamma_{\pi}^{*}(t)% \,dt
  35. γ π * \gamma_{\pi}^{*}
  36. γ π * ( t ) := 1 2 i = 1 n π i ( t ) σ i i ( t ) - 1 2 i , j = 1 n π i ( t ) π j ( t ) σ i j ( t ) \gamma_{\pi}^{*}(t):=\frac{1}{2}\sum_{i=1}^{n}\pi_{i}(t)\sigma_{ii}(t)-\frac{1% }{2}\sum_{i,j=1}^{n}\pi_{i}(t)\pi_{j}(t)\sigma_{ij}(t)
  37. π i ( t ) \pi_{i}(t)
  38. μ i ( t ) := X i ( t ) X 1 ( t ) + + X n ( t ) \mu_{i}(t):=\frac{X_{i}(t)}{X_{1}(t)+\cdots+X_{n}(t)}
  39. i = 1 , , n i=1,\dots,n
  40. μ \mu
  41. Z μ ( 0 ) = X ( 0 ) , Z_{\mu}(0)=X(0),
  42. Z μ ( t ) = X ( t ) Z_{\mu}(t)=X(t)
  43. t . t.
  44. ( σ i j ( t ) ) 1 i , j n (\sigma_{ij}(t))_{1\leq i,j\leq n}
  45. lim t t - 1 log ( μ i ( t ) ) = 0 \operatorname{lim}_{t\rightarrow\infty}t^{-1}\log(\mu_{i}(t))=0
  46. i = 1 , , n . i=1,\dots,n.
  47. [ 0 , T ] [0,T]
  48. ε > 0 \varepsilon>0
  49. μ max ( t ) 1 - ε \mu_{\max}(t)\leq 1-\varepsilon
  50. t [ 0 , T ] . t\in[0,T].
  51. [ 0 , T ] [0,T]
  52. ε > 0 \varepsilon>0
  53. 1 T 0 T μ max ( t ) d t 1 - ε \frac{1}{T}\int_{0}^{T}\mu_{\max}(t)\,dt\leq 1-\varepsilon
  54. S ( μ ( t ) ) = - i = 1 n μ i ( t ) log ( μ i ( t ) ) . S(\mu(t))=-\sum_{i=1}^{n}\mu_{i}(t)\log(\mu_{i}(t)).
  55. ( μ ( 1 ) ( t ) , , μ ( n ) ( t ) ) , (\mu_{(1)}(t),\dots,\mu_{(n)}(t)),
  56. 0 t < 0\leq t<\infty
  57. max 1 i n μ i ( t ) = : μ ( 1 ) ( t ) μ ( 2 ) ( t ) μ ( n ) ( t ) := min 1 i n μ i ( t ) \max_{1\leq i\leq n}\mu_{i}(t)=:\mu_{(1)}(t)\geq\mu_{(2)}(t)\geq\cdots\mu_{(n)% }(t):=\min_{1\leq i\leq n}\mu_{i}(t)
  58. G ( k , k + 1 ) ( t ) := log ( μ ( k ) ( t ) / μ ( k + 1 ) ( t ) ) , G^{(k,k+1)}(t):=\log(\mu_{(k)}(t)/\mu_{(k+1)}(t)),
  59. 0 t < 0\leq t<\infty
  60. k = 1 , , n - 1 k=1,\dots,n-1
  61. Λ ( k , k + 1 ) ( t ) = L G ( k , k + 1 ) ( t ; 0 ) \Lambda^{(k,k+1)}(t)=L^{G^{(k,k+1)}}(t;0)
  62. k k
  63. k + 1 k+1
  64. [ 0 , t ] [0,t]
  65. ( μ ( 1 ) ( t ) , , μ ( n ) ( t ) ) (\mu_{(1)}(t),\cdots,\mu_{(n)}(t))
  66. t t\rightarrow\infty
  67. ( M ( 1 ) , , M ( n ) ) (M_{(1)},\cdots,M_{(n)})
  68. { ( x 1 , , x n ) x 1 > x 2 > > x n and i = 1 n x i = 1 } \{(x_{1},\dots,x_{n})\mid x_{1}>x_{2}>\dots>x_{n}\,\text{ and }\sum_{i=1}^{n}x% _{i}=1\}
  69. lim t Λ ( k , k + 1 ) ( t ) t = λ ( k , k + 1 ) > 0 \lim_{t\rightarrow\infty}\frac{\Lambda^{(k,k+1)}(t)}{t}=\lambda^{(k,k+1)}>0
  70. λ ( 1 , 2 ) , , λ ( n - 1 , n ) . \lambda^{(1,2)},\dots,\lambda^{(n-1,n)}.
  71. π , ρ \pi,\rho
  72. T > 0 T>0
  73. π \pi
  74. ρ \rho
  75. [ 0 , T ] [0,T]
  76. ( Z π ( T ) Z ρ ( T ) ) 1 \mathbb{P}(Z_{\pi}(T)\geq Z_{\rho}(T))\geq 1
  77. ( Z π ( T ) > Z ρ ( T ) ) > 0 \mathbb{P}(Z_{\pi}(T)>Z_{\rho}(T))>0
  78. ( Z π ( T ) > Z ρ ( T ) ) = 1. \mathbb{P}(Z_{\pi}(T)>Z_{\rho}(T))=1.
  79. ρ \rho
  80. κ 0 , \kappa\equiv 0,
  81. ν \nu
  82. π \pi
  83. Z π / Z ν Z_{\pi}/Z_{\nu}
  84. \mathbb{P}
  85. 1 / Z ν 1/Z_{\nu}
  86. ν \nu
  87. \mathbb{P}
  88. \mathbb{P}
  89. ν \nu
  90. lim sup T 1 T log ( Z π ( T ) Z ν ( T ) ) 0 \limsup_{T\rightarrow\infty}\frac{1}{T}\log\left(\frac{Z_{\pi}(T)}{Z_{\nu}(T)}% \right)\leq 0
  91. π \pi
  92. π \pi
  93. T > 0 T>0
  94. 𝔼 [ log ( Z π ( T ) ] 𝔼 [ log ( Z ν ( T ) ) ] . \mathbb{E}[\log(Z_{\pi}(T)]\leq\mathbb{E}[\log(Z_{\nu}(T))].
  95. α ( t ) = ( α 1 ( t ) , , α n ( t ) ) \alpha(t)=(\alpha_{1}(t),\cdots,\alpha_{n}(t))^{\prime}
  96. σ ( t ) = ( σ ( t ) ) 1 i , j n \sigma(t)=(\sigma(t))_{1\leq i,j\leq n}
  97. ν ( t ) = arg max p n ( p α ( t ) - 1 2 p α ( t ) p ) for all 0 t < \nu(t)=\arg\max_{p\in\mathbb{R}^{n}}(p^{\prime}\alpha(t)-\tfrac{1}{2}p^{\prime% }\alpha(t)p)\qquad\,\text{ for all }0\leq t<\infty
  98. \mathbb{Q}
  99. [ 0 , T ] [0,T]
  100. \mathbb{P}
  101. T \mathcal{F}_{T}
  102. X 1 ( t ) , , X n ( t ) X_{1}(t),\dots,X_{n}(t)
  103. 0 t T 0\leq t\leq T
  104. \mathbb{Q}
  105. [ 0 , T ] [0,T]
  106. κ \kappa
  107. μ \mu
  108. ρ \rho
  109. Z ρ Z_{\rho}
  110. π , ρ \pi,\rho
  111. [ 0 , T ] [0,T]
  112. G : U ( 0 , ) G:U\rightarrow(0,\infty)
  113. U U
  114. n \mathbb{R}^{n}
  115. π i 𝔾 ( t ) := μ i ( t ) ( D i log ( 𝔾 ( μ ( t ) ) ) + 1 - j = 1 n μ j ( t ) D j log ( 𝔾 ( μ ( t ) ) ) ) for 1 i n \pi_{i}^{\mathbb{G}}(t):=\mu_{i}(t)\left(D_{i}\log(\mathbb{G}(\mu(t)))+1-\sum_% {j=1}^{n}\mu_{j}(t)D_{j}\log(\mathbb{G}(\mu(t)))\right)\qquad\,\text{ for }1% \leq i\leq n
  116. 𝔾 \mathbb{G}
  117. 𝔾 \mathbb{G}
  118. π 𝔾 \pi_{\mathbb{G}}
  119. μ \mu
  120. log ( Z π 𝔾 ( T ) Z μ ( T ) ) = log ( 𝔾 ( μ ( T ) ) 𝔾 ( μ ( 0 ) ) ) + 0 T g ( t ) d t \log\left(\frac{Z_{\pi^{\mathbb{G}}}(T)}{Z_{\mu}(T)}\right)=\log\left(\frac{% \mathbb{G}(\mu(T))}{\mathbb{G}(\mu(0))}\right)+\int_{0}^{T}g(t)\,dt
  121. g ( t ) := - 1 2 𝔾 ( μ ( t ) ) i = 1 n j = 1 n D i j 2 𝔾 ( μ ( t ) ) μ i ( t ) μ j ( t ) τ i j μ ( t ) g(t):=\frac{-1}{2\mathbb{G}(\mu(t))}\sum_{i=1}^{n}\sum_{j=1}^{n}D_{ij}^{2}% \mathbb{G}(\mu(t))\mu_{i}(t)\mu_{j}(t)\tau_{ij}^{\mu}(t)
  122. 𝔾 \mathbb{G}
  123. τ i j μ ( t ) := ν = 1 n ( ξ i ν ( t ) - ξ ν μ ( t ) ) ( ξ j ν ( t ) - ξ ν μ ( t ) ) , ξ i ν ( t ) := i = 1 n μ i ( t ) ξ i ν ( t ) \tau_{ij}^{\mu}(t):=\sum_{\nu=1}^{n}(\xi_{i\nu}(t)-\xi_{\nu}^{\mu}(t))(\xi_{j% \nu}(t)-\xi_{\nu}^{\mu}(t)),\qquad\xi_{i\nu}(t):=\sum_{i=1}^{n}\mu_{i}(t)\xi_{% i\nu}(t)
  124. 1 i , j n 1\leq i,j\leq n
  125. log ( X i ) \log(X_{i})
  126. log ( X j ) \log(X_{j})
  127. 𝔾 := w > 0 \mathbb{G}:=w>0
  128. μ \mu
  129. ( x ) := ( x 1 x n ) 1 n \mathbb{H}(x):=(x_{1}\cdots x_{n})^{\frac{1}{n}}
  130. φ i ( n ) = 1 n \varphi_{i}(n)=\frac{1}{n}
  131. 1 i n 1\leq i\leq n
  132. 𝕊 c ( x ) = c - i = 1 n x i log ( x i ) \mathbb{S}^{c}(x)=c-\sum_{i=1}^{n}x_{i}\cdot\log(x_{i})
  133. c > 0 c>0
  134. 𝔻 ( p ) ( x ) := ( i = 1 n x i p ) 1 p \mathbb{D}^{(p)}(x):=(\sum_{i=1}^{n}x_{i}^{p})^{\frac{1}{p}}
  135. 0 < p < 1 0<p<1
  136. δ i ( p ) ( t ) = ( μ i ( t ) ) p i = 1 n ( μ i ( t ) ) p \delta_{i}^{(p)}(t)=\frac{(\mu_{i}(t))^{p}}{\sum_{i=1}^{n}(\mu_{i}(t))^{p}}
  137. ( 1 - p ) γ δ ( p ) * ( t ) (1-p)\gamma_{\delta^{(p)}}^{*}(t)
  138. 2 γ μ * ( t ) = i = 1 n μ i ( t ) τ i i μ ( t ) 2\gamma_{\mu}^{*}(t)=\sum_{i=1}^{n}\mu_{i}(t)\tau_{ii}^{\mu}(t)
  139. γ μ * ( t ) = 1 2 i = 1 n μ i ( t ) τ i i μ ( t ) h > 0 , \gamma_{\mu}^{*}(t)=\frac{1}{2}\sum_{i=1}^{n}\mu_{i}(t)\tau_{ii}^{\mu}(t)\geq h% >0,
  140. 0 t < 0\leq t<\infty
  141. h h
  142. T > 𝕊 ( μ ( 0 ) ) / h , T>\mathbb{S}(\mu(0))/h,
  143. c > 0 c>0
  144. Θ ( c ) \Theta^{(c)}
  145. μ \mu
  146. [ 0 , T ] [0,T]
  147. ( σ i j ( t ) ) 1 i , j n (\sigma_{ij}(t))_{1\leq i,j\leq n}
  148. γ μ * h > 0 \gamma_{\mu}^{*}\geq h>0
  149. μ max 1 - ε \mu_{\max}\leq 1-\varepsilon
  150. ε ( 0 , 1 ) . \varepsilon\in(0,1).
  151. δ ( p ) \delta^{(p)}
  152. d log ( X i ( t ) ) = α 2 μ i ( t ) d t + σ μ i ( t ) d W i ( t ) d\log(X_{i}(t))=\frac{\alpha}{2\mu_{i}(t)}\,dt+\frac{\sigma}{\mu_{i}(t)}\,dW_{% i}(t)
  153. 1 i n 1\leq i\leq n
  154. α 0 \alpha\geq 0
  155. n n
  156. ( W 1 , , W n ) . (W_{1},\dots,W_{n}).
  157. X X
  158. μ i \mu_{i}
  159. 1 i n 1\leq i\leq n
  160. μ \mu
  161. m { 2 , , n - 1 } m\in\{2,\dots,n-1\}
  162. m m
  163. ζ \zeta
  164. n - m n-m
  165. η \eta
  166. ζ i ( t ) = k = 1 m μ ( k ) ( t ) 𝟏 { μ i ( t ) = μ ( k ) ( t ) } l = 1 m μ ( l ) ( t ) and η i ( t ) = k = m + 1 n μ ( k ) ( t ) 𝟏 { μ i ( t ) = μ ( k ) ( t ) } l = m + 1 n μ ( l ) ( t ) \zeta_{i}(t)=\frac{\sum_{k=1}^{m}\mu_{(k)}(t)\mathbf{1}_{\{\mu_{i}(t)=\mu_{(k)% }(t)\}}}{\sum_{l=1}^{m}\mu_{(l)}(t)}\qquad\,\text{ and }\eta_{i}(t)=\frac{\sum% _{k=m+1}^{n}\mu_{(k)}(t)\mathbf{1}_{\{\mu_{i}(t)=\mu_{(k)}(t)\}}}{\sum_{l=m+1}% ^{n}\mu_{(l)}(t)}
  167. 1 i n . 1\leq i\leq n.
  168. log ( Z ζ ( t ) Z μ ( t ) ) = log ( μ ( 1 ) ( T ) + + μ ( m ) ( T ) μ ( 1 ) ( 0 ) + + μ ( m ) ( 0 ) ) - 1 2 0 T μ ( m ) ( t ) μ ( 1 ) ( t ) + + μ ( m ) ( t ) d Λ ( m , m + 1 ) ( t ) . \log\left(\frac{Z_{\zeta}(t)}{Z_{\mu}(t)}\right)=\log\left(\frac{\mu_{(1)}(T)+% \cdots+\mu_{(m)}(T)}{\mu_{(1)}(0)+\cdots+\mu_{(m)}(0)}\right)-\frac{1}{2}\int_% {0}^{T}\frac{\mu_{(m)}(t)}{\mu_{(1)}(t)+\cdots+\mu_{(m)}(t)}\,d\Lambda^{(m,m+1% )}(t).
  169. [ 0 , T ] [0,T]
  170. ζ \zeta
  171. m m
  172. T T
  173. m m
  174. ζ \zeta
  175. Λ ( m , m + 1 ) \Lambda^{(m,m+1)}
  176. ζ \zeta
  177. η \eta
  178. log ( Z η ( t ) Z μ ( t ) ) = log ( μ ( m + 1 ) ( T ) + + μ ( n ) ( T ) μ ( m + 1 ) ( 0 ) + + μ ( n ) ( 0 ) ) + 1 2 0 T μ ( m + 1 ) ( t ) μ ( m + 1 ) ( t ) + + μ ( n ) ( t ) . \log\left(\frac{Z_{\eta}(t)}{Z_{\mu}(t)}\right)=\log\left(\frac{\mu_{(m+1)}(T)% +\cdots+\mu_{(n)}(T)}{\mu_{(m+1)}(0)+\cdots+\mu_{(n)}(0)}\right)+\frac{1}{2}% \int_{0}^{T}\frac{\mu_{(m+1)}(t)}{\mu_{(m+1)}(t)+\cdots+\mu_{(n)}(t)}.
  179. ζ \zeta
  180. η \eta
  181. lim T 1 T log ( Z η ( t ) Z μ ( t ) ) = λ ( m , m + 1 ) 𝔼 ( M ( 1 ) M ( 1 ) + + M ( m ) + M ( m + 1 ) M ( m + 1 ) + + M ( n ) ) > 0. \lim_{T\rightarrow\infty}\frac{1}{T}\log\left(\frac{Z_{\eta}(t)}{Z_{\mu}(t)}% \right)=\lambda^{(m,m+1)}\mathbb{E}\left(\frac{M_{(1)}}{M_{(1)}+\cdots+M_{(m)}% }+\frac{M_{(m+1)}}{M_{(m+1)}+\cdots+M_{(n)}}\right)>0.
  182. X 1 , , X n X_{1},\ldots,X_{n}
  183. σ k 2 = lim t t - 1 log μ ( k ) ( t ) \mathbf{\sigma}_{k}^{2}=\lim_{t\to\infty}t^{-1}\langle\log\mu_{(k)}\rangle(t)
  184. 𝐠 k = lim T 1 T 0 T i = 1 n 𝟏 { r t ( i ) = k } d log μ i ( t ) \mathbf{g}_{k}=\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}\sum_{i=1}^{n}\mathbf{1% }_{\{r_{t}(i)=k\}}\,d\log\mu_{i}(t)
  185. k = 1 , , n k=1,\ldots,n\,
  186. r t ( i ) r_{t}(i)
  187. X i ( t ) X_{i}(t)
  188. X ^ 1 , , X ^ n {\widehat{X}}_{1},\ldots,{\widehat{X}}_{n}
  189. d log X ^ i ( t ) = k = 1 n 𝐠 k 1 { r ^ t ( i ) = k } d t + k = 1 n σ k 𝟏 { r ^ t ( i ) = k } d W i ( t ) , d\log{\widehat{X}}_{i}(t)=\sum_{k=1}^{n}\mathbf{g}_{k}\,\mathbf{1}_{\{{\hat{r}% }_{t}(i)=k\}}\,dt+\sum_{k=1}^{n}\mathbf{\sigma}_{k}\mathbf{1}_{\{{\hat{r}}_{t}% (i)=k\}}\,dW_{i}(t),
  190. r ^ t ( i ) {\hat{r}}_{t}(i)
  191. X ^ i ( t ) \widehat{X}_{i}(t)
  192. ( W 1 , , W n ) (W_{1},\ldots,W_{n})
  193. n n
  194. X 1 , , X n X_{1},\ldots,X_{n}
  195. ( 1 / n ) (1/n)

Stokes'_paradox.html

  1. u u
  2. ψ \psi
  3. 𝐮 = ( ψ y - ψ x ) \mathbf{u}=\begin{pmatrix}{\partial\psi\over\partial y}&-{\partial\psi\over% \partial x}\end{pmatrix}
  4. ψ \psi
  5. ψ \psi
  6. z ¯ f ( z ) + g ( z ) \bar{z}f(z)+g(z)
  7. z = x + i y z=x+iy
  8. i i
  9. z ¯ = x - i y \bar{z}=x-iy
  10. f ( z ) , g ( z ) f(z),g(z)
  11. u u
  12. u = u x + i u y u=u_{x}+iu_{y}
  13. u u
  14. u = - 2 i ψ z ¯ u=-2i\frac{\partial\psi}{\partial\bar{z}}
  15. 1 2 i u = ψ z ¯ \frac{1}{2}iu=\frac{\partial\psi}{\partial\bar{z}}
  16. 1 2 i u = f ( z ) + z f ¯ ( z ) + g ¯ ( z ) \frac{1}{2}iu=f(z)+z\bar{f\prime}(z)+\bar{g\prime}(z)
  17. lim z u = 1 \lim_{z\to\infty}u=1
  18. u = 0 u=0
  19. | z | = 1 |z|=1
  20. f , g f,g
  21. f ( z ) = n = - f n z n , g ( z ) = n = - g n z n f(z)=\sum_{n=-\infty}^{\infty}f_{n}z^{n},g(z)=\sum_{n=-\infty}^{\infty}g_{n}z^% {n}
  22. f n = 0 , g n = 0 f_{n}=0,g_{n}=0
  23. n 2 n\geq 2
  24. z z
  25. z n = r n e i n θ , z ¯ n = r n e - i n θ z^{n}=r^{n}e^{in\theta},\bar{z}^{n}=r^{n}e^{-in\theta}
  26. r = 1 r=1
  27. n = - e i n θ ( f n + ( 2 - n ) f ¯ 2 - n + ( 1 - n ) g ¯ 1 - n ) = 0 \sum_{n=-\infty}^{\infty}e^{in\theta}\left(f_{n}+(2-n)\bar{f}_{2-n}+(1-n)\bar{% g}_{1-n}\right)=0
  28. e i n θ e^{in\theta}
  29. n n
  30. f f
  31. g g
  32. f ( z ) = a z + b , g ( z ) = - b z + c f(z)=az+b,g(z)=-bz+c
  33. a a
  34. b b
  35. c c
  36. u u
  37. u = 0 u=0
  38. u x u_{x}
  39. u y u_{y}
  40. r r

Stokes_approximation_and_artificial_time.html

  1. ρ ( 𝐯 t + 𝐯 𝐯 ) = - p + μ 2 𝐯 + 𝐟 . \rho\left(\frac{\partial\mathbf{v}}{\partial t}+\mathbf{v}\cdot\nabla\mathbf{v% }\right)=-\nabla p+\mu\nabla^{2}\mathbf{v}+\mathbf{f}.
  2. R e . t u ~ + R e . u ~ , ~ u ~ + R e . p ~ - Δ ~ u ~ = 0 R_{e}.\partial_{t}\tilde{u}+R_{e}.\left\langle\tilde{u},\tilde{\nabla}\right% \rangle\tilde{u}+R_{e}.\nabla\tilde{p}-\tilde{\Delta}\tilde{u}=0
  3. R e . p - Δ u = 0 R_{e}.\nabla p-\Delta u=0
  4. t ψ - L [ ψ ] = 0 w h e r e ψ = ( u 1 , u 2 , u 3 , p ) T \partial_{t}\psi-L[\psi]=0where\psi=(u_{1},u_{2},u_{3},p)^{T}

Strang_splitting.html

  1. d 𝐲 d x = L 1 ( 𝐲 ) + L 2 ( 𝐲 ) \frac{d\mathbf{y}}{dx}=L_{1}(\mathbf{y})+L_{2}(\mathbf{y})
  2. L 1 L_{1}
  3. L 2 L_{2}
  4. d 𝐲 d x = L 1 , 2 ( 𝐲 ) \frac{d\mathbf{y}}{dx}=L_{1,2}(\mathbf{y})
  5. x 0 x_{0}
  6. 𝐲 ( x 0 ) \mathbf{y}\left(x_{0}\right)
  7. x 0 + Δ x x_{0}+\Delta x
  8. 𝐲 ( x 0 + Δ x ) = exp ( L 1 , 2 ( 𝐲 ( x 0 ) ) ) 𝐲 ( x 0 ) . \mathbf{y}(x_{0}+\Delta x)=\exp(L_{1,2}(\mathbf{y}(x_{0})))\mathbf{y}(x_{0}).
  9. 𝐲 ( x 0 + Δ x ) = exp ( L 1 ( 𝐲 ( x 0 ) ) ) exp ( L 2 ( 𝐲 ( x 0 ) ) ) 𝐲 ( x 0 ) . \mathbf{y}(x_{0}+\Delta x)=\exp(L_{1}(\mathbf{y}(x_{0})))\exp(L_{2}(\mathbf{y}% (x_{0})))\mathbf{y}(x_{0}).

Strategic_Air_Forces_Command.html

  1. C E P 90 {CEP}_{90}

Stratified_flow.html

  1. j * = 0.5 α 3 / 2 j^{*}=0.5\alpha^{3/2}
  2. α = ( h G H ) \alpha={\left(\frac{h_{G}}{H}\right)}
  3. j * = [ U G α ρ G g H ( ρ L - ρ G ) ] j^{*}=\left[\frac{U_{G}\alpha{\sqrt{\rho_{G}}}}{\sqrt{gH(\rho_{L}-\rho_{G})}}% \right]\quad
  4. h L h_{L}
  5. h L h_{L}
  6. j * > ( 1 - h L H ) α 3 / 2 j^{*}>{\left(1-\frac{h_{L}}{H}\right)\alpha^{3/2}}
  7. U G > ( 1 - h L H ) ( ( ρ L - ρ G ) g A G ρ G d A L / d h L ) 1 / 2 U_{G}>{\left(1-\frac{h_{L}}{H}\right)}{\left(\frac{(\rho_{L}-\rho_{G})gA_{G}}{% \rho_{G}dA_{L}/dh_{L}}\right)^{1/2}}
  8. ( 1 - h L H ) = α {\left(1-\frac{h_{L}}{H}\right)}=\alpha
  9. ( h L H ) = 0.5 {\left(\frac{h_{L}}{H}\right)}=0.5
  10. ( 1 - h L H ) = 0.5 {\left(1-\frac{h_{L}}{H}\right)}=0.5
  11. h L h_{L}

Stratified_flows.html

  1. j * = 0.5 α 3 / 2 j^{*}=0.5\alpha^{3/2}
  2. α = ( h G H ) \alpha={\left(\frac{h_{G}}{H}\right)}
  3. j * = [ U G α ρ G g H ( ρ L - ρ G ) ] j^{*}=\left[\frac{U_{G}\alpha{\sqrt{\rho_{G}}}}{\sqrt{gH(\rho_{L}-\rho_{G})}}% \right]\quad
  4. h L h_{L}
  5. h L h_{L}
  6. j * > ( 1 - h L H ) α 3 / 2 j^{*}>{\left(1-\frac{h_{L}}{H}\right)\alpha^{3/2}}
  7. U G > ( 1 - h L H ) ( ( ρ L - ρ G ) g A G ρ G d A L / d h L ) 1 / 2 U_{G}>{\left(1-\frac{h_{L}}{H}\right)}{\left(\frac{(\rho_{L}-\rho_{G})gA_{G}}{% \rho_{G}dA_{L}/dh_{L}}\right)^{1/2}}
  8. ( 1 - h L H ) = α {\left(1-\frac{h_{L}}{H}\right)}=\alpha
  9. ( h L H ) = 0.5 {\left(\frac{h_{L}}{H}\right)}=0.5
  10. ( 1 - h L H ) = 0.5 {\left(1-\frac{h_{L}}{H}\right)}=0.5
  11. h L h_{L}

Strichartz_estimate.html

  1. u 0 u_{0}
  2. e i t Δ / 2 u 0 e^{it\Delta/2}u_{0}
  3. d \mathbb{R}^{d}
  4. 2 q , r 2\leq q,r\leq\infty
  5. 2 q + d r = d 2 \frac{2}{q}+\frac{d}{r}=\frac{d}{2}
  6. ( q , r , d ) ( 2 , , 2 ) (q,r,d)\neq(2,\infty,2)
  7. e i t Δ / 2 u 0 L t q L x r C d , q , r u 0 2 . \|e^{it\Delta/2}u_{0}\|_{L^{q}_{t}L^{r}_{x}}\leq C_{d,q,r}\|u_{0}\|_{2}.
  8. q ~ , r ~ \tilde{q},\tilde{r}
  9. q , r q,r
  10. q ~ , r ~ \tilde{q}^{\prime},\tilde{r}^{\prime}
  11. e - i s Δ / 2 F ( s ) d s L x 2 C d , q ~ , r ~ F L t q ~ L x r ~ . \left\|\int_{\mathbb{R}}e^{-is\Delta/2}F(s)\,ds\right\|_{L^{2}_{x}}\leq C_{d,% \tilde{q},\tilde{r}}\|F\|_{L^{\tilde{q}^{\prime}}_{t}L^{\tilde{r}^{\prime}}_{x% }}.
  12. s < t e - i ( t - s ) Δ / 2 F ( s ) d s L t q L x r C d , q , r , q ~ , r ~ F L t q ~ L x r ~ . \left\|\int_{s<t}e^{-i(t-s)\Delta/2}F(s)\,ds\right\|_{L^{q}_{t}L^{r}_{x}}\leq C% _{d,q,r,\tilde{q},\tilde{r}}\|F\|_{L^{\tilde{q}^{\prime}}_{t}L^{\tilde{r}^{% \prime}}_{x}}.

Strömberg_wavelet.html

  1. S m ( t ) P ( m ) ( A 1 ) . S^{m}(t)\in P^{(m)}(A_{1}).
  2. S m ( t ) = 1 \|S^{m}(t)\|=1
  3. R | S m ( t ) | 2 d t = 1. \int_{R}|S^{m}(t)|^{2}\,dt=1.
  4. S m ( t ) S^{m}(t)
  5. P ( m ) ( A 0 ) P^{(m)}(A_{0})
  6. R S m ( t ) f ( t ) d t = 0 \int_{R}S^{m}(t)\,f(t)\,dt=0
  7. f ( t ) P ( m ) ( A 0 ) . f(t)\in P^{(m)}(A_{0}).
  8. { 2 j / 2 S m ( 2 j t - k ) : j , k integers } \left\{2^{j/2}S^{m}(2^{j}t-k):j,k\,\text{ integers }\right\}
  9. S 0 ( k ) = S 0 ( 1 ) ( 3 - 2 ) k - 1 S^{0}(k)=S^{0}(1)(\sqrt{3}-2)^{k-1}
  10. k = 1 , 2 , 3 , k=1,2,3,\ldots
  11. S 0 ( 1 2 ) = - S 0 ( 1 ) ( 3 + 1 2 ) S^{0}(\tfrac{1}{2})=-S^{0}(1)\left(\sqrt{3}+\tfrac{1}{2}\right)
  12. S 0 ( 0 ) = S 0 ( 1 ) ( 2 3 - 2 ) S^{0}(0)=S^{0}(1)(2\sqrt{3}-2)
  13. S 0 ( - k 2 ) = S 0 ( 1 ) ( 2 3 - 2 ) ( 3 - 2 ) k S^{0}(-\tfrac{k}{2})=S^{0}(1)(2\sqrt{3}-2)(\sqrt{3}-2)^{k}
  14. k = 1 , 2 , 3 , k=1,2,3,\ldots
  15. S 0 ( - k / 2 ) = ( 10 - 6 3 ) S 0 ( k ) S^{0}(-k/2)=(10-6\sqrt{3})S^{0}(k)
  16. k = 1 , 2 , 3 , . k=1,2,3,\ldots\,.

Structurable_algebra.html

  1. x x ¯ x\mapsto\bar{x}
  2. V x , y z := ( x y ¯ ) z + ( z y ¯ ) x - ( z x ¯ ) y V_{x,y}z:=(x\bar{y})z+(z\bar{y})x-(z\bar{x})y
  3. [ x , y ] = x y - y x [x,y]=xy-yx
  4. [ V x , y , V z , w ] = V V x , y z , w - V z , V y , x w . [V_{x,y},V_{z,w}]=V_{V_{x,y}z,w}-V_{z,V_{y,x}w}.

Structural_cut-off.html

  1. E k k E_{kk^{\prime}}
  2. k k
  3. k k^{\prime}
  4. k k k\neq k^{\prime}
  5. k = k k=k^{\prime}
  6. E k k E_{kk^{\prime}}
  7. m k k m_{kk^{\prime}}
  8. r k k E k k m k k = k P ( k , k ) min { k P ( k ) , k P ( k ) , N P ( k ) P ( k ) } r_{kk^{\prime}}\equiv\frac{E_{kk^{\prime}}}{m_{kk^{\prime}}}=\frac{\langle k% \rangle P(k,k^{\prime})}{\min\{kP(k),k^{\prime}P(k^{\prime}),NP(k)P(k^{\prime}% )\}}
  9. k \langle k\rangle
  10. N N
  11. P ( k ) P(k)
  12. k k
  13. P ( k , k ) = E k k / k N P(k,k^{\prime})=E_{kk^{\prime}}/\langle k\rangle N
  14. k k
  15. k k^{\prime}
  16. r k k 1 r_{kk^{\prime}}\leq 1
  17. k s k_{s}
  18. r k s k s = 1 r_{k_{s}k_{s}}=1
  19. k s ( k N ) 1 / 2 k_{s}\sim(\langle k\rangle N)^{1/2}
  20. k k s k\geq k_{s}
  21. γ \gamma
  22. P ( k ) k - γ P(k)\sim k^{-\gamma}
  23. k max N 1 γ - 1 k_{\,\text{max}}\sim N^{\frac{1}{\gamma-1}}
  24. γ < 3 \gamma<3
  25. k max k\text{max}
  26. k s N 1 / 2 k_{s}\sim N^{1/2}
  27. k max > k s k\text{max}>k_{s}
  28. k > k s k>k_{s}

Structural_holes.html

  1. Redundancy = p i q m j q \,\text{Redundancy}=p_{iq}m_{jq}
  2. Effective size of i’s network = j [ 1 - q p i q m j q ] , q i , j , \,\text{Effective size of i's network}=\sum_{j}\left[1-\sum_{q}p_{iq}m_{jq}% \right],\quad q\neq i,j,
  3. c i j = ( p i j + q p i q p q j ) 2 , i q j c_{ij}=(p_{ij}+\sum_{q}p_{iq}p_{qj})^{2},\quad i\neq q\neq j

Studies_for_Player_Piano_(Nancarrow).html

  1. 2 \sqrt{2}
  2. 5 / 7 {5}/{7}
  3. 3 / 4 {3}/{4}
  4. 1 / 2 {1}/{2}
  5. 1 / 2 {1}/{2}
  6. 1 / 4 {1}/{4}
  7. 9 4 / 5 / 6 \tfrac{9}{4/5/6}
  8. 10 4 / 5 / 6 \tfrac{10}{4/5/6}
  9. 11 4 / 5 / 6 \tfrac{11}{4/5/6}
  10. e / π e / π \tfrac{e/\pi}{e/\pi}
  11. 1 π / 2 / 3 \tfrac{1}{\sqrt{\pi}}/\sqrt{2/3}
  12. 1 3 π / 13 / 16 3 \tfrac{1}{3\sqrt{\pi}}/\sqrt[3]{13/16}
  13. 1 3 π / 13 / 16 3 1 π / 2 / 3 \tfrac{\tfrac{1}{3\sqrt{\pi}}/\sqrt[3]{13/16}}{\tfrac{1}{\sqrt{\pi}}/\sqrt{2/3}}
  14. 2 \sqrt{2}
  15. 9 4 / 5 / 6 \tfrac{9}{4/5/6}
  16. 10 4 / 5 / 6 \tfrac{10}{4/5/6}
  17. 11 4 / 5 / 6 \tfrac{11}{4/5/6}
  18. 5 / 7 {5}/{7}
  19. 3 / 4 {3}/{4}
  20. 1 / 2 {1}/{2}
  21. 1 / 2 {1}/{2}
  22. 1 / 4 {1}/{4}
  23. e / π e / π \tfrac{e/\pi}{e/\pi}
  24. 1 3 π / 13 / 16 3 1 π / 2 / 3 \tfrac{\tfrac{1}{3\sqrt{\pi}}/\sqrt[3]{13/16}}{\tfrac{1}{\sqrt{\pi}}/\sqrt{2/3}}
  25. 1 π / 2 / 3 \tfrac{1}{\sqrt{\pi}}/\sqrt{2/3}
  26. 1 3 π / 13 / 16 3 \tfrac{1}{3\sqrt{\pi}}/\sqrt[3]{13/16}

Sub-Gaussian_random_variable.html

  1. X X
  2. C , v C,v
  3. t > 0 t>0
  4. P ( | X | > t ) C e - v t 2 . P(|X|>t)\leq Ce^{-vt^{2}}.
  5. X ψ 2 = inf { s > 0 E e ( X / s ) 2 - 1 1 } . \|X\|_{\psi_{2}}=\inf\{s>0\mid Ee^{(X/s)^{2}}-1\leq 1\}.
  6. X X
  7. ψ 2 \psi_{2}
  8. a > 0 E e a X 2 < + \exists a>0\ Ee^{aX^{2}}<+\infty
  9. B , b > 0 λ E e λ X B e λ 2 b \exists B,b>0\ \forall\lambda\in\mathbb{R}\ \ Ee^{\lambda X}\leq Be^{\lambda^{% 2}b}
  10. K > 0 p 1 ( E | X | p ) 1 / p K p \exists K>0\ \forall p\geq 1\ \left(E|X|^{p}\right)^{1/p}\leq K\sqrt{p}

Sum_of_two_squares_theorem.html

  1. n > 1 n>1
  2. n = a 2 + b 2 n=a^{2}+b^{2}
  3. a , b a,b

Summability_kernel.html

  1. 𝕋 := / \mathbb{T}:=\mathbb{R}/\mathbb{Z}
  2. ( k n ) (k_{n})
  3. L 1 ( 𝕋 ) L^{1}(\mathbb{T})
  4. 𝕋 k n ( t ) d t = 1 \int_{\mathbb{T}}k_{n}(t)\,dt=1
  5. 𝕋 | k n ( t ) | d t M \int_{\mathbb{T}}|k_{n}(t)|\,dt\leq M
  6. δ | t | 1 2 | k n ( t ) | d t 0 \int_{\delta\leq|t|\leq\frac{1}{2}}|k_{n}(t)|\,dt\to 0
  7. n n\to\infty
  8. δ > 0 \delta>0
  9. k n 0 k_{n}\geq 0
  10. n n
  11. ( k n ) (k_{n})
  12. 𝕋 = / 2 π \mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}
  13. 1 2 π 𝕋 k n ( t ) d t = 1 \frac{1}{2\pi}\int_{\mathbb{T}}k_{n}(t)\,dt=1
  14. π \pi
  15. \mathbb{R}
  16. 𝕋 \mathbb{T}
  17. \mathbb{R}
  18. | t | > δ |t|>\delta
  19. ( k n ) (k_{n})
  20. * *
  21. ( k n ) , f 𝒞 ( 𝕋 ) (k_{n}),f\in\mathcal{C}(\mathbb{T})
  22. 𝕋 \mathbb{T}
  23. k n * f f k_{n}*f\to f
  24. 𝒞 ( 𝕋 ) \mathcal{C}(\mathbb{T})
  25. n n\to\infty
  26. ( k n ) , f L 1 ( 𝕋 ) (k_{n}),f\in L^{1}(\mathbb{T})
  27. k n * f f k_{n}*f\to f
  28. L 1 ( 𝕋 ) L^{1}(\mathbb{T})
  29. n n\to\infty
  30. ( k n ) (k_{n})
  31. f L 1 ( 𝕋 ) f\in L^{1}(\mathbb{T})
  32. k n * f f k_{n}*f\to f
  33. n n\to\infty
  34. ( k n ) (k_{n})
  35. k ~ n ( x ) := sup | y | | x | k n ( y ) \widetilde{k}_{n}(x):=\sup_{|y|\geq|x|}k_{n}(y)
  36. sup n k n ~ 1 < \sup_{n\in\mathbb{N}}\|\widetilde{k_{n}}\|_{1}<\infty

Super-proportional_division.html

  1. V > 1 / n V>1/n
  2. V 1 / n V\geq 1/n
  3. d > 1 ( i - 1 ) i ( q ( i - 1 ) - 1 ) d>\frac{1}{(i-1)i(q(i-1)-1)}
  4. q i - 1 qi-1
  5. q q
  6. ( q i - 1 ) - q q i - 1 = q ( i - 1 ) - 1 q i - 1 \frac{(qi-1)-q}{qi-1}=\frac{q(i-1)-1}{qi-1}
  7. 1 i - 1 + d \frac{1}{i-1}+d
  8. q ( i - 1 ) - 1 ( i - 1 ) ( q i - 1 ) \frac{q(i-1)-1}{(i-1)(qi-1)}
  9. 1 i ( i - 1 ) ( q i - 1 ) \frac{1}{i(i-1)(qi-1)}
  10. ( q i - 1 ) ( i - 1 ) ( i - 1 ) i ( q i - 1 ) = 1 i \frac{(qi-1)(i-1)}{(i-1)i(qi-1)}=\frac{1}{i}
  11. q q i - 1 > 1 i \frac{q}{qi-1}>\frac{1}{i}

Super-resolution_optical_fluctuation_imaging.html

  1. k = 1 N δ ( r - r k ) ε k s k ( t ) \sum_{k=1}^{N}\delta(\vec{r}-\vec{r}_{k})\cdot\varepsilon_{k}\cdot s_{k}(t)
  2. r \vec{r}
  3. F ( r , t ) = k = 1 N U ( r - r k ) ε k s k ( t ) . F(\vec{r},t)=\sum_{k=1}^{N}U(\vec{r}-\vec{r}_{k})\cdot\varepsilon_{k}\cdot s_{% k}(t).
  4. r k \vec{r}_{k}
  5. ε k s k \varepsilon_{k}\cdot s_{k}
  6. ε k \varepsilon_{k}
  7. s k ( t ) s_{k}(t)
  8. δ F ( r , t ) = F ( r , t ) - F ( r , t ) t \delta F(\vec{r},t)=F(\vec{r},t)-\langle F(\vec{r},t)\rangle_{t}
  9. t \langle\cdots\rangle_{t}
  10. τ \tau
  11. δ F ( r , t ) = δ F ( r , t + τ ) δ F ( r , t ) t \delta F(\vec{r},t)=\langle\delta F(\vec{r},t+\tau)\cdot\delta F(\vec{r},t)% \rangle_{t}
  12. 2 \sqrt{2}
  13. A C n ( r , τ 1 n - 1 = 0 ) = k = 1 N U n ( r - r k ) ε k n w k ( 0 ) AC_{n}(\vec{r},\tau_{1\ldots n-1}=0)=\sum_{k=1}^{N}U^{n}(\vec{r}-\vec{r}_{k})% \varepsilon^{n}_{k}w_{k}(0)
  14. w k w_{k}
  15. K n ( r ) = μ n ( r ) - i = 1 n - 1 ( n - 1 i ) K n - i ( r ) μ i ( r ) K_{n}(\vec{r})=\mu_{n}(\vec{r})-\sum_{i=1}^{n-1}\begin{pmatrix}n-1\\ i\end{pmatrix}K_{n-i}(\vec{r})\mu_{i}(\vec{r})
  16. μ \mu
  17. C C n ( r , τ 1 n - 1 = 0 ) = j < l n U ( r j - r l n ) i = 1 N U n ( r i - k n r k n ) ε i n w i ( 0 ) CC_{n}(\vec{r},\tau_{1\ldots n-1}=0)=\prod_{j<l}^{n}U\Bigg(\frac{\vec{r}_{j}-% \vec{r}_{l}}{\sqrt{n}}\Bigg)\cdot\sum_{i=1}^{N}U^{n}\Bigg(\vec{r}_{i}-\frac{% \sum_{k}^{n}\vec{r}_{k}}{n}\Bigg)\varepsilon^{n}_{i}w_{i}(0)
  18. U ( r j - r l / n ) U(r_{j}-r_{l}/\sqrt{n})
  19. K n ( r = 1 n i = 1 n r i ) = P ( - 1 ) | P | - 1 ( | P | - 1 ) ! p P i p F ( r ) i t K_{n}\Bigg(\vec{r}=\frac{1}{n}\sum_{i=1}^{n}\vec{r_{i}}\Bigg)=\sum_{P}(-1)^{|P% |-1}(|P|-1)!\prod_{p\in P}\Big\langle\prod_{i\in p}F(\vec{r})_{i}\Big\rangle_{t}

Supercritical_liquid–gas_boundaries.html

  1. G ( r ) G(\vec{r})
  2. 3 k B 3k_{B}
  3. k B k_{B}
  4. 1 k B 1k_{B}
  5. c V = 2 k B c_{V}=2k_{B}

Supermembranes.html

  1. ( σ , θ ) (\sigma,\theta)
  2. X μ k ( σ , θ ) = x μ k + O ( σ , θ ) X^{k}_{\mu}(\sigma,\theta)=x^{k}_{\mu}+O(\sigma,\theta)
  3. Φ [ X ] = Φ [ x , a , . . ] = ϕ ( x ) + a ( y ) ϕ ( x , y ) d y + a ( y ) a ( z ) ϕ ( x , y , z ) d y d z + \Phi[X]=\Phi[x,{a},..]=\phi(x)+\int{a}(y)\phi(x,y)dy+\int{a}(y){a}(z)\phi(x,y,% z)dydz+...
  4. S = Φ [ X ] Q Φ [ X ] D [ X ] S=\int\Phi[X]{Q}\Phi[X]D[X]
  5. G ( X , Y ) = Q - 1 ( X , Y ) G(X,Y)={Q^{-1}}(X,Y)

Supersingular_Isogeny_Key_Exchange.html

  1. j ( E ) = 1728 4 a 3 4 a 3 + 27 b 2 . j(E)=1728\frac{4a^{3}}{4a^{3}+27b^{2}}.
  2. p = w A e A w B e B f ± 1. p=w_{A}^{e_{A}}\cdot w_{B}^{e_{B}}\cdot f\pm 1.
  3. 𝔽 p 2 \mathbb{F}_{p^{2}}
  4. P A , Q A , P B , Q B on E . P_{A},Q_{A},P_{B},Q_{B}\,\text{ on }E.
  5. P A P_{A}
  6. Q A Q_{A}
  7. ( w A ) e A . (w_{A})^{e_{A}}.
  8. P B P_{B}
  9. Q B Q_{B}
  10. ( w B ) e B (w_{B})^{e_{B}}
  11. P A P_{A}
  12. Q A Q_{A}
  13. P B P_{B}
  14. Q B Q_{B}
  15. R A R_{A}
  16. R B R_{B}
  17. P A P_{A}
  18. Q A Q_{A}
  19. P B P_{B}
  20. Q B Q_{B}
  21. R A R_{A}
  22. R B R_{B}
  23. ϕ A \phi_{A}
  24. ϕ B \phi_{B}
  25. P A P_{A}
  26. Q A Q_{A}
  27. P B P_{B}
  28. Q B Q_{B}
  29. ϕ A \phi_{A}
  30. ϕ B \phi_{B}
  31. ϕ B ( P A ) \phi_{B}(P_{A})
  32. ϕ B ( Q A ) \phi_{B}(Q_{A})
  33. ϕ A ( P B ) \phi_{A}(P_{B})
  34. ϕ A ( Q B ) \phi_{A}(Q_{B})
  35. S B A S_{BA}
  36. S A B S_{AB}
  37. R A := m A ( P A ) + n A ( Q A ) . R_{A}:=m_{A}\cdot(P_{A})+n_{A}\cdot(Q_{A}).
  38. R A R_{A}
  39. ϕ A : E E A \phi_{A}:E\rightarrow E_{A}
  40. E A E_{A}
  41. E . E.
  42. ϕ A \phi_{A}
  43. P B P_{B}
  44. Q B Q_{B}
  45. E A : ϕ A ( P B ) E_{A}:\phi_{A}(P_{B})
  46. ϕ A ( Q B ) . \phi_{A}(Q_{B}).
  47. E A , ϕ A ( P B ) E_{A},\phi_{A}(P_{B})
  48. ϕ A ( Q B ) . \phi_{A}(Q_{B}).
  49. E B , ϕ B ( P A ) E_{B},\phi_{B}(P_{A})
  50. ϕ B ( Q A ) . \phi_{B}(Q_{A}).
  51. m A , n A , ϕ B ( P A ) m_{A},n_{A},\phi_{B}(P_{A})
  52. ϕ B ( Q A ) \phi_{B}(Q_{A})
  53. S B A := m A ( ϕ B ( P A ) ) + n A ( ϕ B ( Q A ) ) . S_{BA}:=m_{A}(\phi_{B}(P_{A}))+n_{A}(\phi_{B}(Q_{A})).
  54. S B A S_{BA}
  55. ψ B A \psi_{BA}
  56. ψ B A \psi_{BA}
  57. E B A E_{BA}
  58. K := j-invariant ( j B A ) K:=\,\text{ j-invariant }(j_{BA})
  59. E B A E_{BA}
  60. m B , n B , ϕ A ( P B ) m_{B},n_{B},\phi_{A}(P_{B})
  61. ϕ A ( Q B ) \phi_{A}(Q_{B})
  62. S A B = m B ( ϕ A ( P B ) ) + n B ( ϕ A ( Q B ) ) S_{AB}=m_{B}(\phi_{A}(P_{B}))+n_{B}(\phi_{A}(Q_{B}))
  63. S A B S_{AB}
  64. ψ A B \psi_{AB}
  65. ψ A B \psi_{AB}
  66. E A B E_{AB}
  67. j A B ) j_{AB})
  68. E A B E_{AB}
  69. E A B E_{AB}
  70. E B A E_{BA}

Supertransitive_class.html

  1. ( x ) ( x A 𝒫 ( x ) A ) . (\forall x)(x\in A\to\mathcal{P}(x)\subseteq A).

SUPS.html

  1. N N
  2. c c
  3. S U P S = c × N SUPS=c\times N
  4. υ \upsilon
  5. υ c N \upsilon cN
  6. Δ t \Delta t
  7. c N Δ t \frac{cN}{\Delta t}
  8. Δ t \Delta t
  9. Δ t < 1 υ N \Delta t<\frac{1}{\upsilon N}
  10. υ c N 2 \upsilon cN^{2}

Surface_chemistry_of_neural_implants.html

  1. Z = 1 i ω C b + 1 / R b Z=\frac{1}{{i}\omega C_{b}+1/R_{b}}
  2. γ \gamma
  3. γ = ( E s - E b ) 2 A \gamma=\frac{(E_{s}-E_{b})}{2A}
  4. γ \gamma
  5. Δ a d s G = Δ a d s H - T Δ a d s S \Delta_{ads}G=\Delta_{ads}H-T\Delta_{ads}S

Surface_chemistry_of_paper.html

  1. p I = p K a + p K b 2 pI={{pKa}+{pKb}\over 2}
  2. γ SL + γ LG cos θ c = γ SG \gamma_{\mathrm{SL}}+\gamma_{\mathrm{LG}}\cos{\theta_{\mathrm{c}}}=\gamma_{% \mathrm{SG}}\,
  3. γ SL \gamma_{\mathrm{SL}}
  4. γ LG \gamma_{\mathrm{LG}}
  5. γ SG \gamma_{\mathrm{SG}}

Surface_energy_transfer.html

  1. k S E T = ( 1 / τ D ) ( d / d 0 ) 4 k_{SET}=({{1}/{\tau_{D}})({d}/{d_{0}})^{4}}
  2. τ D \tau_{D}
  3. d d
  4. d 0 d_{0}
  5. ϕ S E T = 1 1 + ( d / d 0 ) 4 \phi_{SET}=\frac{1}{1+({d}/{d_{0}})^{4}}

Surface_hopping.html

  1. 𝐪 \mathbf{q}
  2. 𝐑 \mathbf{R}
  3. m i m_{i}
  4. H = i - 2 2 m i q i 2 + V ( 𝐪 , 𝐑 ) H=\sum_{i}-\frac{\hbar^{2}}{2m_{i}}\nabla_{q_{i}}^{2}+V(\mathbf{q},\mathbf{R})
  5. V V
  6. H H
  7. 𝐑 \mathbf{R}
  8. ϕ i ( 𝐪 ; 𝐑 ) \phi_{i}(\mathbf{q};\mathbf{R})
  9. 𝐪 \mathbf{q}
  10. 𝐅 𝐑 \displaystyle\mathbf{F}_{\mathbf{R}}
  11. 𝐑 \mathbf{R}
  12. ψ ( 𝐪 ; 𝐑 , t ) = n c n ( t ) ϕ n ( 𝐪 ; 𝐑 ) \psi(\mathbf{q};\mathbf{R},t)=\sum_{n}c_{n}(t)\phi_{n}(\mathbf{q};\mathbf{R})
  13. c n ( t ) c_{n}(t)
  14. i c j ˙ = n c n ( V j n - i 𝐑 ˙ . 𝐝 j n ) i\hbar\dot{c_{j}}=\sum_{n}c_{n}\left(V_{jn}-i\hbar\dot{\mathbf{R}}.\mathbf{d}_% {jn}\right)
  15. V j n V_{jn}
  16. 𝐝 j n \mathbf{d}_{jn}
  17. V j n = ϕ j | H | ϕ n = ϕ j | H | ϕ j δ j n 𝐝 j n = ϕ j | 𝐑 ϕ n > \begin{aligned}\displaystyle V_{jn}&\displaystyle=\langle\phi_{j}|H|\phi_{n}% \rangle=\langle\phi_{j}|H|\phi_{j}\rangle\delta_{jn}\\ \displaystyle\mathbf{d}_{jn}&\displaystyle=\langle\phi_{j}|\nabla_{\mathbf{R}}% \phi_{n}>\end{aligned}
  18. | c j ( t ) | 2 |c_{j}(t)|^{2}
  19. | c j ( t ) | 2 |c_{j}(t)|^{2}
  20. | c j ( t ) | 2 ˙ = n 2 I m ( a n j V j n ) - 2 R e ( a n j 𝐑 ˙ . 𝐝 j n ) \dot{|c_{j}(t)|^{2}}=\sum_{n}\frac{2}{\hbar}Im(a_{nj}V_{jn})-2Re(a_{nj}\dot{% \mathbf{R}}.\mathbf{d}_{jn})
  21. a n j = c n c j * a_{nj}=c_{n}c_{j}^{*}
  22. | c j ( t ) | 2 |c_{j}(t)|^{2}
  23. | c j ( t + d t ) | 2 - | c j ( t ) | 2 | c j ( t ) | 2 d t a j j n 2 I m ( a n j V j n ) - 2 R e ( a n j 𝐑 ˙ . 𝐝 j n ) \frac{|c_{j}(t+dt)|^{2}-|c_{j}(t)|^{2}}{|c_{j}(t)|^{2}}\approx\frac{dt}{a_{jj}% }\sum_{n}\frac{2}{\hbar}Im(a_{nj}V_{jn})-2Re(a_{nj}\dot{\mathbf{R}}.\mathbf{d}% _{jn})
  24. j j
  25. P j n = d t a j j ( 2 I m ( a n j V j n ) - 2 R e ( a n j 𝐑 ˙ . 𝐝 j n ) ) P_{j\to n}=\frac{dt}{a_{jj}}\left(\frac{2}{\hbar}Im(a_{nj}V_{jn})-2Re(a_{nj}% \dot{\mathbf{R}}.\mathbf{d}_{jn})\right)
  26. < ϕ j | 𝐑 H | ϕ n > = 𝐑 < ϕ j | H | ϕ n > - < 𝐑 ϕ j | H | ϕ n > - < ϕ j | H | 𝐑 ϕ n > = 𝐑 E j δ j n + ( E j - E n ) 𝐝 j n , \begin{aligned}\displaystyle<\phi_{j}|\nabla_{\mathbf{R}}H|\phi_{n}>&% \displaystyle=\nabla_{\mathbf{R}}<\phi_{j}|H|\phi_{n}>-<\nabla_{\mathbf{R}}% \phi_{j}|H|\phi_{n}>-<\phi_{j}|H|\nabla_{\mathbf{R}}\phi_{n}>\\ &\displaystyle=\nabla_{\mathbf{R}}E_{j}\delta_{jn}+(E_{j}-E_{n})\mathbf{d}_{jn% },\end{aligned}
  27. E j = < ϕ j | H | ϕ j > E_{j}=<\phi_{j}|H|\phi_{j}>
  28. d j n = - d n j d_{jn}=-d_{nj}
  29. 𝐝 j n \mathbf{d}_{jn}
  30. 𝐝 j n \mathbf{d}_{jn}
  31. δ t = / 2 Δ E \delta t=\hbar/2\Delta E
  32. Δ E \Delta E
  33. t t
  34. Δ t \Delta t
  35. t + Δ t t+\Delta t
  36. t t
  37. t + Δ t t+\Delta t
  38. δ t \delta t
  39. δ t \delta t
  40. Δ t \Delta t

Surface_triangulation.html

  1. x 4 + y 4 + z 4 = 1 x^{4}+y^{4}+z^{4}=1
  2. x 4 + y 4 + z 4 = 1 x^{4}+y^{4}+z^{4}=1

Survey_camp.html

  1. cos θ \cos\theta\!
  2. sin θ \sin\theta
  3. θ \theta

Suslin_operation.html

  1. 𝒜 \mathcal{A}

Suspension_of_a_ring.html

  1. Σ R \Sigma R
  2. Σ ( R ) = C ( R ) / M ( R ) \Sigma(R)=C(R)/M(R)
  3. C ( R ) C(R)
  4. M ( R ) M(R)
  5. K i ( R ) K i + 1 ( Σ R ) K_{i}(R)\simeq K_{i+1}(\Sigma R)

Swirl_function.html

  1. S ( k , n , r , θ ) = sin ( k cos ( r ) - n θ ) S(k,n,r,\theta)=\sin(k\cos(r)-n\theta)
  2. f ( - k , n , r , θ ) = - f ( k , n , r , - θ ) f(-k,n,r,\theta)=-f(k,n,r,-\theta)
  3. f ( - k , n , r , θ ) = - f ( k , - n , r , θ ) f(-k,n,r,\theta)=-f(k,-n,r,\theta)
  4. f ( - k , - n , r , θ ) = - f ( k , n , r , θ ) f(-k,-n,r,\theta)=-f(k,n,r,\theta)
  5. f ( - k , n , r , - θ ) = - f ( k , n , r , θ ) f(-k,n,r,-\theta)=-f(k,n,r,\theta)
  6. f ( - k , n , r , θ ) = - f ( k , n , - r , - θ ) f(-k,n,r,\theta)=-f(k,n,-r,-\theta)
  7. f ( - k , n , - r , - θ ) = - f ( k , n , r , θ ) f(-k,n,-r,-\theta)=-f(k,n,r,\theta)
  8. f ( - k , - n , - r , θ ) = - f ( k , n , r , θ ) f(-k,-n,-r,\theta)=-f(k,n,r,\theta)
  9. f ( - k , n , - r , - θ ) = - f ( k , n , r , θ ) f(-k,n,-r,-\theta)=-f(k,n,r,\theta)
  10. f ( k , - n , r , θ ) = f ( k , n , r , - θ ) f(k,-n,r,\theta)=f(k,n,r,-\theta)
  11. f ( k , - n , r , - θ ) = f ( k , n , r , θ ) f(k,-n,r,-\theta)=f(k,n,r,\theta)
  12. f ( k , n , - r , θ ) = f ( k , n , r , θ ) f(k,n,-r,\theta)=f(k,n,r,\theta)
  13. f ( k , n , - r , θ ) = f ( k , n , r , θ ) f(k,n,-r,\theta)=f(k,n,r,\theta)
  14. f ( k , n , - r , θ ) = f ( k , - n , r , - θ ) f(k,n,-r,\theta)=f(k,-n,r,-\theta)
  15. f ( k , - n , - r , - θ ) = f ( k , n , r , θ ) f(k,-n,-r,-\theta)=f(k,n,r,\theta)
  16. f ( k , n , - r , θ ) - f ( k , n , r , θ ) f(k,n,-r,\theta)-f(k,n,r,\theta)
  17. S ( k , n , r , θ + 2 π n ) = S ( k , n , r , θ ) S\left(k,n,r,\theta+\frac{2\pi}{n}\right)=S(k,n,r,\theta)

Sylvester_graph.html

  1. { 5 , 4 , 2 ; 1 , 1 , 4 } \{5,4,2;1,1,4\}

Symbol_(number_theory).html

  1. ( a p ) \left(\frac{a}{p}\right)
  2. ( a b ) \left(\frac{a}{b}\right)
  3. ( a b ) \left(\frac{a}{b}\right)
  4. ( a b ) = ( a b ) m \left(\frac{a}{b}\right)=\left(\frac{a}{b}\right)_{m}
  5. ( a , b ) p = ( a , b p ) = ( a , b p ) m (a,b)_{p}=\left(\frac{a,b}{p}\right)=\left(\frac{a,b}{p}\right)_{m}
  6. θ L / K ( α ) = ( α , L / K ) = ( L / K α ) \theta_{L/K}(\alpha)=(\alpha,L/K)=\left(\frac{L/K}{\alpha}\right)
  7. ψ L / K ( α ) = ( α , L / K ) = ( L / K α ) = ( ( L / K ) / α ) \psi_{L/K}(\alpha)=(\alpha,L/K)=\left(\frac{L/K}{\alpha}\right)=((L/K)/\alpha)
  8. [ ( L / K ) / P ] = [ L / K P ] [(L/K)/P]=\left[\frac{L/K}{P}\right]
  9. ( a , χ p ) \left(\frac{a,\chi}{p}\right)
  10. ( ( α , L / K ) / p ) = ( α , L / K p ) ((\alpha,L/K)/p)=\left(\frac{\alpha,L/K}{p}\right)

Symbolic_power_of_a_prime_ideal.html

  1. P ( n ) = P n R P R = { f R s f P n for some s R - P } . P^{(n)}=P^{n}R_{P}\cap R=\{f\in R\mid sf\in P^{n}\,\text{ for some }s\in R-P\}.
  2. P ( 1 ) = P P^{(1)}=P
  3. P ( n ) = P n P^{(n)}=P^{n}
  4. I R I\cap R
  5. R R P R\to R_{P}

Symmetric_spectrum.html

  1. Σ n \Sigma_{n}
  2. X n X_{n}
  3. S 1 S 1 X n S 1 S 1 X n + 1 S 1 X n + p - 1 X n + p S^{1}\wedge\dots\wedge S^{1}\wedge X_{n}\to S^{1}\wedge\dots\wedge S^{1}\wedge X% _{n+1}\to\dots\to S^{1}\wedge X_{n+p-1}\to X_{n+p}
  4. Σ p × Σ n \Sigma_{p}\times\Sigma_{n}
  5. 𝒮 p Σ \mathcal{S}p^{\Sigma}
  6. 𝒮 p Σ \mathcal{S}p^{\Sigma}

Symmetry_(geometry).html

  1. m m
  2. ( m k ) (m−k)
  3. k k
  4. m m
  5. m m
  6. k k
  7. m m
  8. m m
  9. m m
  10. m m
  11. m m
  12. m m
  13. m × m m×m
  14. m m
  15. m m
  16. T a ( p ) = p + a \scriptstyle T_{a}(p)\;=\;p\,+\,a
  17. m θ \scriptstyle m\theta
  18. A A
  19. A A

Symmetry_in_quantum_mechanics.html

  1. Ω ^ ψ ( 𝐫 , t ) = ψ ( 𝐫 , t ) \widehat{\Omega}\psi(\mathbf{r},t)=\psi(\mathbf{r}^{\prime},t^{\prime})
  2. Ω ^ \widehat{\Omega}
  3. Ω ^ - 1 = Ω ^ \widehat{\Omega}^{-1}=\widehat{\Omega}^{\dagger}
  4. Ω ^ | 𝐫 ( t ) = | 𝐫 ( t ) \widehat{\Omega}\left|\mathbf{r}(t)\right\rangle=\left|\mathbf{r}^{\prime}(t^{% \prime})\right\rangle
  5. Ω ^ \widehat{\Omega}
  6. Ω ^ = Ω ^ \widehat{\Omega}=\widehat{\Omega}^{\dagger}
  7. A ^ \widehat{A}
  8. Ω ^ \widehat{\Omega}
  9. A ^ ψ = Ω ^ A ^ Ω ^ ψ Ω ^ A ^ ψ = A ^ Ω ^ ψ \widehat{A}\psi=\widehat{\Omega}^{\dagger}\widehat{A}\widehat{\Omega}\psi\quad% \Rightarrow\quad\widehat{\Omega}\widehat{A}\psi=\widehat{A}\widehat{\Omega}\psi
  10. [ Ω ^ , A ^ ] ψ = 0 [\widehat{\Omega},\widehat{A}]\psi=0
  11. A ^ = A ^ \widehat{A}=\widehat{A}^{\dagger}
  12. g = G ( ξ 1 , ξ 2 , ) g=G(\xi_{1},\xi_{2},\cdots)
  13. I = G ( 0 , 0 ) I=G(0,0\cdots)
  14. X j = g ξ j | ξ j = 0 X_{j}=\left.\frac{\partial g}{\partial\xi_{j}}\right|_{\xi_{j}=0}
  15. X j = i g ξ j | ξ j = 0 X_{j}=i\left.\frac{\partial g}{\partial\xi_{j}}\right|_{\xi_{j}=0}
  16. [ X a , X b ] = i f a b c X c \left[X_{a},X_{b}\right]=if_{abc}X_{c}
  17. D [ g ( ξ j ) ] D ( ξ j ) = e i ξ j D ( X j ) D[g(\xi_{j})]\equiv D(\xi_{j})=e^{i\xi_{j}D(X_{j})}
  18. D ( ξ a ) D ( ξ b ) = D ( ξ a ξ b ) . D(\xi_{a})D(\xi_{b})=D(\xi_{a}\xi_{b}).
  19. T ^ ( Δ 𝐫 ) \widehat{T}(\Delta\mathbf{r})
  20. T ^ \widehat{T}
  21. 𝐩 ^ \widehat{\mathbf{p}}
  22. E ^ \widehat{E}
  23. T ^ \widehat{T}
  24. U ^ \widehat{U}
  25. T ^ ( Δ 𝐫 ) ψ ( 𝐫 , t ) = ψ ( 𝐫 + Δ 𝐫 , t ) \widehat{T}(\Delta\mathbf{r})\psi(\mathbf{r},t)=\psi(\mathbf{r}+\Delta\mathbf{% r},t)
  26. U ^ ( Δ t ) ψ ( 𝐫 , t ) = ψ ( 𝐫 , t + Δ t ) \widehat{U}(\Delta t)\psi(\mathbf{r},t)=\psi(\mathbf{r},t+\Delta t)
  27. T ^ ( Δ 𝐫 ) = I + i Δ 𝐫 𝐩 ^ \widehat{T}(\Delta\mathbf{r})=I+\frac{i}{\hbar}\Delta\mathbf{r}\cdot\widehat{% \mathbf{p}}
  28. U ^ ( Δ t ) = I - i Δ t E ^ \widehat{U}(\Delta t)=I-\frac{i}{\hbar}\Delta t\widehat{E}
  29. lim N ( I + i Δ 𝐫 N 𝐩 ^ ) N = exp ( i Δ 𝐫 𝐩 ^ ) = T ^ ( Δ 𝐫 ) \lim_{N\rightarrow\infty}\left(I+\frac{i}{\hbar}\frac{\Delta\mathbf{r}}{N}% \cdot\widehat{\mathbf{p}}\right)^{N}=\exp\left(\frac{i}{\hbar}\Delta\mathbf{r}% \cdot\widehat{\mathbf{p}}\right)=\widehat{T}(\Delta\mathbf{r})
  30. lim N ( I - i Δ t N E ^ ) N = exp ( - i Δ t E ^ ) = U ^ ( Δ t ) \lim_{N\rightarrow\infty}\left(I-\frac{i}{\hbar}\frac{\Delta t}{N}\cdot% \widehat{E}\right)^{N}=\exp\left(-\frac{i}{\hbar}\Delta t\widehat{E}\right)=% \widehat{U}(\Delta t)
  31. 𝐩 ^ = - i \widehat{\mathbf{p}}=-i\hbar\nabla
  32. E ^ = i t \widehat{E}=i\hbar\frac{\partial}{\partial t}
  33. [ T ^ ( 𝐫 1 ) , T ^ ( 𝐫 2 ) ] ψ ( 𝐫 , t ) = 0 \left[\widehat{T}(\mathbf{r}_{1}),\widehat{T}(\mathbf{r}_{2})\right]\psi(% \mathbf{r},t)=0
  34. [ U ^ ( t 1 ) , U ^ ( t 2 ) ] ψ ( 𝐫 , t ) = 0 \left[\widehat{U}(t_{1}),\widehat{U}(t_{2})\right]\psi(\mathbf{r},t)=0
  35. [ p ^ i , p ^ j , ] ψ ( 𝐫 , t ) = 0 \left[\widehat{p}_{i},\widehat{p}_{j},\right]\psi(\mathbf{r},t)=0
  36. [ E ^ , p ^ i , ] ψ ( 𝐫 , t ) = 0 \left[\widehat{E},\widehat{p}_{i},\right]\psi(\mathbf{r},t)=0
  37. U ^ ( t ) = exp ( - i Δ t E ) \widehat{U}(t)=\exp\left(-\frac{i\Delta tE}{\hbar}\right)
  38. ψ ( 𝐫 , t + t 0 ) = U ^ ( t - t 0 ) ψ ( 𝐫 , t 0 ) \psi(\mathbf{r},t+t_{0})=\widehat{U}(t-t_{0})\psi(\mathbf{r},t_{0})
  39. U ^ ( t - t 0 ) U ( t , t 0 ) \widehat{U}(t-t_{0})\equiv U(t,t_{0})
  40. R ( Δ θ , 𝐧 ^ ) ψ ( 𝐫 , t ) = ψ ( 𝐫 , t ) {R}(\Delta\theta,\hat{\mathbf{n}})\psi(\mathbf{r},t)=\psi(\mathbf{r}^{\prime},t)
  41. 𝐧 ^ = ( n 1 , n 2 , n 3 ) \hat{\mathbf{n}}=(n_{1},n_{2},n_{3})
  42. 𝐫 = R ^ ( Δ θ , 𝐧 ^ ) 𝐫 . \mathbf{r}^{\prime}=\widehat{R}(\Delta\theta,\hat{\mathbf{n}})\mathbf{r}\,.
  43. R ^ ( Δ θ , 𝐧 ^ ) \widehat{R}(\Delta\theta,\hat{\mathbf{n}})
  44. Δ θ 𝐧 ^ = Δ θ ( n 1 , n 2 , n 3 ) \Delta\theta\hat{\mathbf{n}}=\Delta\theta(n_{1},n_{2},n_{3})
  45. 𝐞 ^ x , 𝐞 ^ y , 𝐞 ^ z \hat{\mathbf{e}}_{x},\hat{\mathbf{e}}_{y},\hat{\mathbf{e}}_{z}
  46. R ^ x R ^ ( Δ θ , 𝐞 ^ x ) = ( 1 0 0 0 cos Δ θ - sin Δ θ 0 sin Δ θ cos Δ θ ) , \widehat{R}_{x}\equiv\widehat{R}(\Delta\theta,\hat{\mathbf{e}}_{x})=\begin{% pmatrix}1&0&0\\ 0&\cos\Delta\theta&-\sin\Delta\theta\\ 0&\sin\Delta\theta&\cos\Delta\theta\\ \end{pmatrix}\,,
  47. J x J 1 = i R ^ ( θ , 𝐞 ^ x ) θ | θ = 0 = i ( 0 0 0 0 0 - 1 0 1 0 ) , J_{x}\equiv J_{1}=i\left.\frac{\partial\widehat{R}(\theta,\hat{\mathbf{e}}_{x}% )}{\partial\theta}\right|_{\theta=0}=i\begin{pmatrix}0&0&0\\ 0&0&-1\\ 0&1&0\\ \end{pmatrix}\,,
  48. R ^ y R ^ ( Δ θ , 𝐞 ^ y ) = ( cos Δ θ 0 sin Δ θ 0 1 0 - sin Δ θ 0 cos Δ θ ) , \widehat{R}_{y}\equiv\widehat{R}(\Delta\theta,\hat{\mathbf{e}}_{y})=\begin{% pmatrix}\cos\Delta\theta&0&\sin\Delta\theta\\ 0&1&0\\ -\sin\Delta\theta&0&\cos\Delta\theta\\ \end{pmatrix}\,,
  49. J y J 2 = i R ^ ( θ , 𝐞 ^ y ) θ | θ = 0 = i ( 0 0 1 0 0 0 - 1 0 0 ) , J_{y}\equiv J_{2}=i\left.\frac{\partial\widehat{R}(\theta,\hat{\mathbf{e}}_{y}% )}{\partial\theta}\right|_{\theta=0}=i\begin{pmatrix}0&0&1\\ 0&0&0\\ -1&0&0\\ \end{pmatrix}\,,
  50. R ^ z R ^ ( Δ θ , 𝐞 ^ z ) = ( cos Δ θ - sin Δ θ 0 sin Δ θ cos Δ θ 0 0 0 1 ) , \widehat{R}_{z}\equiv\widehat{R}(\Delta\theta,\hat{\mathbf{e}}_{z})=\begin{% pmatrix}\cos\Delta\theta&-\sin\Delta\theta&0\\ \sin\Delta\theta&\cos\Delta\theta&0\\ 0&0&1\\ \end{pmatrix}\,,
  51. J z J 3 = i R ^ ( θ , 𝐞 ^ z ) θ | θ = 0 = i ( 0 - 1 0 1 0 0 0 0 0 ) . J_{z}\equiv J_{3}=i\left.\frac{\partial\widehat{R}(\theta,\hat{\mathbf{e}}_{z}% )}{\partial\theta}\right|_{\theta=0}=i\begin{pmatrix}0&-1&0\\ 1&0&0\\ 0&0&0\\ \end{pmatrix}\,.
  52. 𝐧 ^ \hat{\mathbf{n}}
  53. [ R ^ ( θ , 𝐧 ^ ) ] i j = ( δ i j - n i n j ) cos θ - ε i j k n k sin θ + n i n j [\widehat{R}(\theta,\hat{\mathbf{n}})]_{ij}=(\delta_{ij}-n_{i}n_{j})\cos\theta% -\varepsilon_{ijk}n_{k}\sin\theta+n_{i}n_{j}
  54. 𝐞 ^ z \hat{\mathbf{e}}_{z}
  55. 𝐧 ^ \hat{\mathbf{n}}
  56. R ^ ( Δ θ , 𝐞 ^ z ) ψ ( x , y , z , t ) = ψ ( x - Δ θ y , Δ θ x + y , z , t ) \widehat{R}(\Delta\theta,\hat{\mathbf{e}}_{z})\psi(x,y,z,t)=\psi(x-\Delta% \theta y,\Delta\theta x+y,z,t)
  57. R ^ ( Δ θ , 𝐧 ^ ) ψ ( r i , t ) = ψ ( R i j r j , t ) = ψ ( r i - ε i j k n k Δ θ r j , t ) \widehat{R}(\Delta\theta,\hat{\mathbf{n}})\psi(r_{i},t)=\psi(R_{ij}r_{j},t)=% \psi(r_{i}-\varepsilon_{ijk}n_{k}\Delta\theta r_{j},t)
  58. R ^ ( Δ θ , 𝐞 ^ z ) = I - i Δ θ L ^ z \widehat{R}(\Delta\theta,\hat{\mathbf{e}}_{z})=I-\frac{i}{\hbar}\Delta\theta% \widehat{L}_{z}
  59. R ^ ( Δ θ , 𝐧 ^ ) = I - ( - Δ θ n k ε k i j r j ) r i = I - ( Δ θ n k ε k j i r j ) r i = I - Δ θ 𝐧 ^ ( 𝐫 × ) = I - i Δ θ 𝐧 ^ 𝐋 ^ \begin{aligned}\displaystyle\widehat{R}(\Delta\theta,\hat{\mathbf{n}})&% \displaystyle=I-(-\Delta\theta n_{k}\varepsilon_{kij}r_{j})\frac{\partial}{% \partial r_{i}}\\ &\displaystyle=I-(\Delta\theta n_{k}\varepsilon_{kji}r_{j})\frac{\partial}{% \partial r_{i}}\\ &\displaystyle=I-\Delta\theta\hat{\mathbf{n}}\cdot(\mathbf{r}\times\nabla)\\ &\displaystyle=I-\frac{i\Delta\theta}{\hbar}\hat{\mathbf{n}}\cdot\widehat{% \mathbf{L}}\\ \end{aligned}
  60. R ^ = 1 - i Δ θ 𝐧 ^ 𝐋 ^ , 𝐋 ^ = i 𝐧 ^ θ \widehat{R}=1-\frac{i}{\hbar}\Delta\theta\hat{\mathbf{n}}\cdot\widehat{\mathbf% {L}}\,,\quad\widehat{\mathbf{L}}=i\hbar\hat{\mathbf{n}}\frac{\partial}{% \partial\theta}
  61. lim N ( 1 - i Δ θ N 𝐧 ^ 𝐋 ^ ) N = exp ( - i Δ θ 𝐧 ^ 𝐋 ^ ) = R ^ \lim_{N\rightarrow\infty}\left(1-\frac{i}{\hbar}\frac{\Delta\theta}{N}\hat{% \mathbf{n}}\cdot\widehat{\mathbf{L}}\right)^{N}=\exp\left(-\frac{i}{\hbar}% \Delta\theta\hat{\mathbf{n}}\cdot\widehat{\mathbf{L}}\right)=\widehat{R}
  62. L ^ z = i θ \widehat{L}_{z}=i\hbar\frac{\partial}{\partial\theta}
  63. 𝐋 ^ \widehat{\mathbf{L}}
  64. 𝐧 ^ \hat{\mathbf{n}}
  65. 𝐧 ^ 𝐋 ^ \hat{\mathbf{n}}\cdot\widehat{\mathbf{L}}
  66. R ( θ 1 + θ 2 , 𝐞 i ) = R ( θ 1 𝐞 i ) R ( θ 2 𝐞 i ) , [ R ( θ 1 𝐞 i ) , R ( θ 2 𝐞 i ) ] = 0 . R(\theta_{1}+\theta_{2},\mathbf{e}_{i})=R(\theta_{1}\mathbf{e}_{i})R(\theta_{2% }\mathbf{e}_{i})\,,\quad[R(\theta_{1}\mathbf{e}_{i}),R(\theta_{2}\mathbf{e}_{i% })]=0\,.
  67. [ L i , L j ] = i ε i j k L k . [L_{i},L_{j}]=i\hbar\varepsilon_{ijk}L_{k}.
  68. 𝐒 ^ = ( S x ^ , S y ^ , S z ^ ) \widehat{\mathbf{S}}=(\widehat{S_{x}},\widehat{S_{y}},\widehat{S_{z}})
  69. \hbar
  70. 𝐧 ^ \hat{\mathbf{n}}
  71. n ^ \hat{n}
  72. 𝐒 ^ = 2 s y m b o l σ \widehat{\mathbf{S}}=\frac{\hbar}{2}symbol{\sigma}
  73. σ 1 = σ x = ( 0 1 1 0 ) , σ 2 = σ y = ( 0 - i i 0 ) , σ 3 = σ z = ( 1 0 0 - 1 ) \sigma_{1}=\sigma_{x}=\begin{pmatrix}0&1\\ 1&0\end{pmatrix}\,,\quad\sigma_{2}=\sigma_{y}=\begin{pmatrix}0&-i\\ i&0\end{pmatrix}\,,\quad\sigma_{3}=\sigma_{z}=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}
  74. 𝐉 ^ = 𝐋 ^ + 𝐒 ^ \widehat{\mathbf{J}}=\widehat{\mathbf{L}}+\widehat{\mathbf{S}}
  75. J ^ ( θ , 𝐧 ^ ) = exp ( - i θ 𝐧 ^ 𝐉 ^ ) \widehat{J}(\theta,\hat{\mathbf{n}})=\exp\left(-\frac{i}{\hbar}\theta\hat{% \mathbf{n}}\cdot\widehat{\mathbf{J}}\right)
  76. 𝐚 ^ = ( a 1 , a 2 , a 3 ) \hat{\mathbf{a}}=(a_{1},a_{2},a_{3})
  77. 𝐧 ^ = ( n 1 , n 2 , n 3 ) \hat{\mathbf{n}}=(n_{1},n_{2},n_{3})
  78. φ 𝐚 ^ = φ ( a 1 , a 2 , a 3 ) \varphi\hat{\mathbf{a}}=\varphi(a_{1},a_{2},a_{3})
  79. θ 𝐧 ^ = θ ( n 1 , n 2 , n 3 ) \theta\hat{\mathbf{n}}=\theta(n_{1},n_{2},n_{3})
  80. R ^ x , R ^ y , R ^ z \widehat{R}_{x},\widehat{R}_{y},\widehat{R}_{z}
  81. R ^ ( Δ θ , 𝐞 ^ x ) = ( 0 0 0 0 0 1 0 0 0 0 cos Δ θ - sin Δ θ 0 0 sin Δ θ cos Δ θ ) , \widehat{R}(\Delta\theta,\hat{\mathbf{e}}_{x})=\begin{pmatrix}0&0&0&0\\ 0&1&0&0\\ 0&0&\cos\Delta\theta&-\sin\Delta\theta\\ 0&0&\sin\Delta\theta&\cos\Delta\theta\\ \end{pmatrix}\,,
  82. J x = J 1 = i ( 0 0 0 0 0 0 0 0 0 0 0 - 1 0 0 1 0 ) , J_{x}=J_{1}=i\begin{pmatrix}0&0&0&0\\ 0&0&0&0\\ 0&0&0&-1\\ 0&0&1&0\\ \end{pmatrix}\,,
  83. R ^ ( Δ θ , 𝐞 ^ y ) = ( 0 0 0 0 0 cos Δ θ 0 sin Δ θ 0 0 1 0 0 - sin Δ θ 0 cos Δ θ ) , \widehat{R}(\Delta\theta,\hat{\mathbf{e}}_{y})=\begin{pmatrix}0&0&0&0\\ 0&\cos\Delta\theta&0&\sin\Delta\theta\\ 0&0&1&0\\ 0&-\sin\Delta\theta&0&\cos\Delta\theta\\ \end{pmatrix}\,,
  84. J y = J 2 = i ( 0 0 0 0 0 0 0 1 0 0 0 0 0 - 1 0 0 ) , J_{y}=J_{2}=i\begin{pmatrix}0&0&0&0\\ 0&0&0&1\\ 0&0&0&0\\ 0&-1&0&0\\ \end{pmatrix}\,,
  85. R ^ ( Δ θ , 𝐞 ^ z ) = ( 0 0 0 0 0 cos Δ θ - sin Δ θ 0 0 sin Δ θ cos Δ θ 0 0 0 0 1 ) , \widehat{R}(\Delta\theta,\hat{\mathbf{e}}_{z})=\begin{pmatrix}0&0&0&0\\ 0&\cos\Delta\theta&-\sin\Delta\theta&0\\ 0&\sin\Delta\theta&\cos\Delta\theta&0\\ 0&0&0&1\\ \end{pmatrix}\,,
  86. J z = J 3 = i ( 0 0 0 0 0 0 - 1 0 0 1 0 0 0 0 0 0 ) . J_{z}=J_{3}=i\begin{pmatrix}0&0&0&0\\ 0&0&-1&0\\ 0&1&0&0\\ 0&0&0&0\\ \end{pmatrix}\,.
  87. 𝐀 = R ^ ( Δ θ , 𝐞 ^ x ) 𝐀 \mathbf{A}^{\prime}=\widehat{R}(\Delta\theta,\hat{\mathbf{e}}_{x})\mathbf{A}
  88. 𝐞 ^ x , 𝐞 ^ y , 𝐞 ^ z \hat{\mathbf{e}}_{x},\hat{\mathbf{e}}_{y},\hat{\mathbf{e}}_{z}
  89. B ^ x , B ^ y , B ^ z \widehat{B}_{x},\widehat{B}_{y},\widehat{B}_{z}
  90. B ^ x B ^ ( φ , 𝐞 ^ x ) = ( cosh φ sinh φ 0 0 sinh φ cosh φ 0 0 0 0 1 0 0 0 0 1 ) , \widehat{B}_{x}\equiv\widehat{B}(\varphi,\hat{\mathbf{e}}_{x})=\begin{pmatrix}% \cosh\varphi&\sinh\varphi&0&0\\ \sinh\varphi&\cosh\varphi&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{pmatrix}\,,
  91. K x = K 1 = i B ^ ( φ , 𝐞 ^ x ) φ | φ = 0 = i ( 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ) , K_{x}=K_{1}=i\left.\frac{\partial\widehat{B}(\varphi,\hat{\mathbf{e}}_{x})}{% \partial\varphi}\right|_{\varphi=0}=i\begin{pmatrix}0&1&0&0\\ 1&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ \end{pmatrix}\,,
  92. B ^ y B ^ ( φ , 𝐞 ^ y ) = ( cosh φ 0 sinh φ 0 0 1 0 0 sinh φ 0 cosh φ 0 0 0 0 1 ) , \widehat{B}_{y}\equiv\widehat{B}(\varphi,\hat{\mathbf{e}}_{y})=\begin{pmatrix}% \cosh\varphi&0&\sinh\varphi&0\\ 0&1&0&0\\ \sinh\varphi&0&\cosh\varphi&0\\ 0&0&0&1\\ \end{pmatrix}\,,
  93. K y = K 2 = i B ^ ( φ , 𝐞 ^ y ) φ | φ = 0 = i ( 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 ) , K_{y}=K_{2}=i\left.\frac{\partial\widehat{B}(\varphi,\hat{\mathbf{e}}_{y})}{% \partial\varphi}\right|_{\varphi=0}=i\begin{pmatrix}0&0&1&0\\ 0&0&0&0\\ 1&0&0&0\\ 0&0&0&0\\ \end{pmatrix}\,,
  94. B ^ z B ^ ( φ , 𝐞 ^ z ) = ( cosh φ 0 0 sinh φ 0 1 0 0 0 0 1 0 sinh φ 0 0 cosh φ ) , \widehat{B}_{z}\equiv\widehat{B}(\varphi,\hat{\mathbf{e}}_{z})=\begin{pmatrix}% \cosh\varphi&0&0&\sinh\varphi\\ 0&1&0&0\\ 0&0&1&0\\ \sinh\varphi&0&0&\cosh\varphi\\ \end{pmatrix}\,,
  95. K z = K 3 = i B ^ ( φ , 𝐞 ^ z ) φ | φ = 0 = i ( 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 ) . K_{z}=K_{3}=i\left.\frac{\partial\widehat{B}(\varphi,\hat{\mathbf{e}}_{z})}{% \partial\varphi}\right|_{\varphi=0}=i\begin{pmatrix}0&0&0&1\\ 0&0&0&0\\ 0&0&0&0\\ 1&0&0&0\\ \end{pmatrix}\,.
  96. 𝐀 = B ^ ( φ , 𝐞 ^ x ) 𝐀 \mathbf{A}^{\prime}=\widehat{B}(\varphi,\hat{\mathbf{e}}_{x})\mathbf{A}
  97. [ J a , J b ] = i ε a b c J c \left[J_{a},J_{b}\right]=i\varepsilon_{abc}J_{c}
  98. [ K a , K b ] = - i ε a b c J c \left[K_{a},K_{b}\right]=-i\varepsilon_{abc}J_{c}
  99. [ J a , K b ] = i ε a b c K c \left[J_{a},K_{b}\right]=i\varepsilon_{abc}K_{c}
  100. [ D ( J a ) , D ( J b ) ] = i ε a b c D ( J c ) \left[{D(J_{a})},{D(J_{b})}\right]=i\varepsilon_{abc}{D(J_{c})}
  101. [ D ( K a ) , D ( K b ) ] = - i ε a b c D ( J c ) \left[{D(K_{a})},{D(K_{b})}\right]=-i\varepsilon_{abc}{D(J_{c})}
  102. [ D ( J a ) , D ( K b ) ] = i ε a b c D ( K c ) \left[{D(J_{a})},{D(K_{b})}\right]=i\varepsilon_{abc}{D(K_{c})}
  103. B ^ ( φ , 𝐚 ^ ) = exp ( - i φ 𝐚 ^ 𝐊 ) \widehat{B}(\varphi,\hat{\mathbf{a}})=\exp\left(-\frac{i}{\hbar}\varphi\hat{% \mathbf{a}}\cdot\mathbf{K}\right)
  104. R ^ ( θ , 𝐧 ^ ) = exp ( - i θ 𝐧 ^ 𝐉 ) \widehat{R}(\theta,\hat{\mathbf{n}})=\exp\left(-\frac{i}{\hbar}\theta\hat{% \mathbf{n}}\cdot\mathbf{J}\right)
  105. Λ ( φ , 𝐚 ^ , θ , 𝐧 ^ ) = exp [ - i ( φ 𝐚 ^ 𝐊 + θ 𝐧 ^ 𝐉 ) ] \Lambda(\varphi,\hat{\mathbf{a}},\theta,\hat{\mathbf{n}})=\exp\left[-\frac{i}{% \hbar}\left(\varphi\hat{\mathbf{a}}\cdot\mathbf{K}+\theta\hat{\mathbf{n}}\cdot% \mathbf{J}\right)\right]
  106. D [ B ^ ( φ , 𝐚 ^ ) ] = exp ( - i φ 𝐚 ^ D ( 𝐊 ) ) D[\widehat{B}(\varphi,\hat{\mathbf{a}})]=\exp\left(-\frac{i}{\hbar}\varphi\hat% {\mathbf{a}}\cdot D(\mathbf{K})\right)
  107. D [ R ^ ( θ , 𝐧 ^ ) ] = exp ( - i θ 𝐧 ^ D ( 𝐉 ) ) D[\widehat{R}(\theta,\hat{\mathbf{n}})]=\exp\left(-\frac{i}{\hbar}\theta\hat{% \mathbf{n}}\cdot D(\mathbf{J})\right)
  108. D [ Λ ( θ , 𝐧 ^ , φ , 𝐚 ^ ) ] = exp [ - i ( φ 𝐚 ^ D ( 𝐊 ) + θ 𝐧 ^ D ( 𝐉 ) ) ] D[\Lambda(\theta,\hat{\mathbf{n}},\varphi,\hat{\mathbf{a}})]=\exp\left[-\frac{% i}{\hbar}\left(\varphi\hat{\mathbf{a}}\cdot D(\mathbf{K})+\theta\hat{\mathbf{n% }}\cdot D(\mathbf{J})\right)\right]
  109. M 0 a = - M a 0 = K a , M a b = ε a b c J c . M^{0a}=-M^{a0}=K_{a}\,,\quad M^{ab}=\varepsilon_{abc}J_{c}\,.
  110. ω 0 a = - ω a 0 = φ a a , ω a b = θ ε a b c n c , \omega_{0a}=-\omega_{a0}=\varphi a_{a}\,,\quad\omega_{ab}=\theta\varepsilon_{% abc}n_{c}\,,
  111. Λ ( φ , 𝐚 ^ , θ , 𝐧 ^ ) = exp ( - i 2 ω α β M α β ) = exp [ - i 2 ( φ 𝐚 ^ 𝐊 + θ 𝐧 ^ 𝐉 ) ] \Lambda(\varphi,\hat{\mathbf{a}},\theta,\hat{\mathbf{n}})=\exp\left(-\frac{i}{% 2}\omega_{\alpha\beta}M^{\alpha\beta}\right)=\exp\left[-\frac{i}{2}\left(% \varphi\hat{\mathbf{a}}\cdot\mathbf{K}+\theta\hat{\mathbf{n}}\cdot\mathbf{J}% \right)\right]
  112. 𝐀 = Λ ( φ , 𝐚 ^ , θ , 𝐧 ^ ) 𝐀 \mathbf{A}^{\prime}=\Lambda(\varphi,\hat{\mathbf{a}},\theta,\hat{\mathbf{n}})% \mathbf{A}
  113. ψ σ ( 𝐫 , t ) D ( Λ ) ψ σ ( Λ - 1 ( 𝐫 , t ) ) \psi_{\sigma}(\mathbf{r},t)\rightarrow D(\Lambda)\psi_{\sigma}(\Lambda^{-1}(% \mathbf{r},t))
  114. ψ ( 𝐫 , t ) = [ ψ σ = s ( 𝐫 , t ) ψ σ = s - 1 ( 𝐫 , t ) ψ σ = - s + 1 ( 𝐫 , t ) ψ σ = - s ( 𝐫 , t ) ] ψ ( 𝐫 , t ) = [ ψ σ = s ( 𝐫 , t ) ψ σ = s - 1 ( 𝐫 , t ) ψ σ = - s + 1 ( 𝐫 , t ) ψ σ = - s ( 𝐫 , t ) ] \psi(\mathbf{r},t)=\begin{bmatrix}\psi_{\sigma=s}(\mathbf{r},t)\\ \psi_{\sigma=s-1}(\mathbf{r},t)\\ \vdots\\ \psi_{\sigma=-s+1}(\mathbf{r},t)\\ \psi_{\sigma=-s}(\mathbf{r},t)\end{bmatrix}\quad\rightleftharpoons\quad{\psi(% \mathbf{r},t)}^{\dagger}=\begin{bmatrix}{\psi_{\sigma=s}(\mathbf{r},t)}^{\star% }&{\psi_{\sigma=s-1}(\mathbf{r},t)}^{\star}&\cdots&{\psi_{\sigma=-s+1}(\mathbf% {r},t)}^{\star}&{\psi_{\sigma=-s}(\mathbf{r},t)}^{\star}\end{bmatrix}
  115. 𝐀 = 𝐉 + i 𝐊 2 , 𝐁 = 𝐉 - i 𝐊 2 , \mathbf{A}=\frac{\mathbf{J}+i\mathbf{K}}{2}\,,\quad\mathbf{B}=\frac{\mathbf{J}% -i\mathbf{K}}{2}\,,
  116. [ A i , A j ] = ε i j k A k , [ B i , B j ] = ε i j k B k , [ A i , B j ] = 0 , \left[A_{i},A_{j}\right]=\varepsilon_{ijk}A_{k}\,,\quad\left[B_{i},B_{j}\right% ]=\varepsilon_{ijk}B_{k}\,,\quad\left[A_{i},B_{j}\right]=0\,,
  117. ( A x ) m n , m n = δ n n ( J x ( m ) ) m m ( B x ) m n , m n = δ m m ( J x ( n ) ) n n \left(A_{x}\right)_{m^{\prime}n^{\prime},mn}=\delta_{n^{\prime}n}\left(J_{x}^{% (m)}\right)_{m^{\prime}m}\,\quad\left(B_{x}\right)_{m^{\prime}n^{\prime},mn}=% \delta_{m^{\prime}m}\left(J_{x}^{(n)}\right)_{n^{\prime}n}
  118. ( A y ) m n , m n = δ n n ( J y ( m ) ) m m ( B y ) m n , m n = δ m m ( J y ( n ) ) n n \left(A_{y}\right)_{m^{\prime}n^{\prime},mn}=\delta_{n^{\prime}n}\left(J_{y}^{% (m)}\right)_{m^{\prime}m}\,\quad\left(B_{y}\right)_{m^{\prime}n^{\prime},mn}=% \delta_{m^{\prime}m}\left(J_{y}^{(n)}\right)_{n^{\prime}n}
  119. ( A z ) m n , m n = δ n n ( J z ( m ) ) m m ( B z ) m n , m n = δ m m ( J z ( n ) ) n n \left(A_{z}\right)_{m^{\prime}n^{\prime},mn}=\delta_{n^{\prime}n}\left(J_{z}^{% (m)}\right)_{m^{\prime}m}\,\quad\left(B_{z}\right)_{m^{\prime}n^{\prime},mn}=% \delta_{m^{\prime}m}\left(J_{z}^{(n)}\right)_{n^{\prime}n}
  120. ( J z ( m ) ) m m = m δ m m ( J x ( m ) ± i J y ( m ) ) m m = m δ a , a ± 1 ( a m ) ( a ± m + 1 ) \left(J_{z}^{(m)}\right)_{m^{\prime}m}=m\delta_{m^{\prime}m}\,\quad\left(J_{x}% ^{(m)}\pm iJ_{y}^{(m)}\right)_{m^{\prime}m}=m\delta_{a^{\prime},a\pm 1}\sqrt{(% a\mp m)(a\pm m+1)}
  121. ( 𝐀 ) m n , m n [ ( A x ) m n , m n , ( A y ) m n , m n , ( A z ) m n , m n ] \left(\mathbf{A}\right)_{m^{\prime}n^{\prime},mn}\equiv\left[\left(A_{x}\right% )_{m^{\prime}n^{\prime},mn},\left(A_{y}\right)_{m^{\prime}n^{\prime},mn},\left% (A_{z}\right)_{m^{\prime}n^{\prime},mn}\right]
  122. ( 𝐁 ) m n , m n [ ( B x ) m n , m n , ( B y ) m n , m n , ( B z ) m n , m n ] \left(\mathbf{B}\right)_{m^{\prime}n^{\prime},mn}\equiv\left[\left(B_{x}\right% )_{m^{\prime}n^{\prime},mn},\left(B_{y}\right)_{m^{\prime}n^{\prime},mn},\left% (B_{z}\right)_{m^{\prime}n^{\prime},mn}\right]
  123. ( 𝐉 ( m ) ) m m [ ( J x ( m ) ) m m , ( J y ( m ) ) m m , ( J z ( m ) ) m m ] \left(\mathbf{J}^{(m)}\right)_{m^{\prime}m}\equiv\left[\left(J_{x}^{(m)}\right% )_{m^{\prime}m},\left(J_{y}^{(m)}\right)_{m^{\prime}m},\left(J_{z}^{(m)}\right% )_{m^{\prime}m}\right]
  124. 𝐏 ^ = ( E ^ c , - 𝐩 ^ ) = i ( 1 c t , ) , \widehat{\mathbf{P}}=\left(\frac{\widehat{E}}{c},-\widehat{\mathbf{p}}\right)=% i\hbar\left(\frac{1}{c}\frac{\partial}{\partial t},\nabla\right)\,,
  125. X ^ ( Δ 𝐗 ) = exp ( - i Δ 𝐗 𝐏 ^ ) = exp [ - i ( Δ t E ^ + Δ 𝐫 𝐩 ^ ) ] . \widehat{X}(\Delta\mathbf{X})=\exp\left(-\frac{i}{\hbar}\Delta\mathbf{X}\cdot% \widehat{\mathbf{P}}\right)=\exp\left[-\frac{i}{\hbar}\left(\Delta t\widehat{E% }+\Delta\mathbf{r}\cdot\widehat{\mathbf{p}}\right)\right]\,.
  126. [ P μ , P ν ] = 0 [P_{\mu},P_{\nu}]=0\,
  127. 1 i [ M μ ν , P ρ ] = η μ ρ P ν - η ν ρ P μ \frac{1}{i}[M_{\mu\nu},P_{\rho}]=\eta_{\mu\rho}P_{\nu}-\eta_{\nu\rho}P_{\mu}\,
  128. 1 i [ M μ ν , M ρ σ ] = η μ ρ M ν σ - η μ σ M ν ρ - η ν ρ M μ σ + η ν σ M μ ρ \frac{1}{i}[M_{\mu\nu},M_{\rho\sigma}]=\eta_{\mu\rho}M_{\nu\sigma}-\eta_{\mu% \sigma}M_{\nu\rho}-\eta_{\nu\rho}M_{\mu\sigma}+\eta_{\nu\sigma}M_{\mu\rho}\,
  129. W μ = 1 2 ε μ ν ρ σ J ν ρ P σ , W_{\mu}=\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}J^{\nu\rho}P^{\sigma},
  130. [ P μ , W ν ] = 0 , \left[P^{\mu},W^{\nu}\right]=0\,,
  131. [ J μ ν , W ρ ] = i ( η ρ ν W μ - η ρ μ W ν ) , \left[J^{\mu\nu},W^{\rho}\right]=i\left(\eta^{\rho\nu}W^{\mu}-\eta^{\rho\mu}W^% {\nu}\right)\,,
  132. [ W μ , W ν ] = - i ϵ μ ν ρ σ W ρ P σ . \left[W_{\mu},W_{\nu}\right]=-i\epsilon_{\mu\nu\rho\sigma}W^{\rho}P^{\sigma}\,.
  133. U ^ \widehat{U}
  134. U ^ = U ^ \widehat{U}=\widehat{U}^{\dagger}
  135. [ U ^ , H ^ ] = 0 \left[\widehat{U},\widehat{H}\right]=0
  136. U ^ \widehat{U}
  137. U ^ \widehat{U}
  138. U = e - i θ U=e^{-i\theta}
  139. U = e - i a θ U=e^{-ia\theta}
  140. U = ( a b - b a ) U=\begin{pmatrix}a&b\\ -b^{\star}&a^{\star}\\ \end{pmatrix}
  141. det ( U ) = a a + b b = | a | 2 + | b | 2 = 1 \det(U)=aa^{\star}+bb^{\star}={|a|}^{2}+{|b|}^{2}=1
  142. [ σ a , σ b ] = 2 i ε a b c σ c [\sigma_{a},\sigma_{b}]=2i\hbar\varepsilon_{abc}\sigma_{c}
  143. U ( θ , 𝐞 ^ j ) = e i θ σ j / 2 U(\theta,\hat{\mathbf{e}}_{j})=e^{i\theta\sigma_{j}/2}
  144. U ( θ , 𝐞 ^ j ) = exp ( - i 2 n = 1 8 θ n λ n ) U(\theta,\hat{\mathbf{e}}_{j})=\exp\left(-\frac{i}{2}\sum_{n=1}^{8}\theta_{n}% \lambda_{n}\right)
  145. [ λ a , λ b ] = 2 i f a b c λ c \left[\lambda_{a},\lambda_{b}\right]=2if_{abc}\lambda_{c}
  146. | r = ( 1 0 0 ) , | g = ( 0 1 0 ) , | b = ( 0 0 1 ) |r\rangle=\begin{pmatrix}1\\ 0\\ 0\end{pmatrix}\,,\quad|g\rangle=\begin{pmatrix}0\\ 1\\ 0\end{pmatrix}\,,\quad|b\rangle=\begin{pmatrix}0\\ 0\\ 1\end{pmatrix}
  147. S U ( 3 ) SU(3)
  148. s u ( 3 ) su(3)
  149. S U ( 3 ) SU(3)
  150. F ^ j = 1 2 λ j \hat{F}_{j}=\frac{1}{2}\lambda_{j}
  151. [ F ^ j , H ^ ] = 0 \left[\hat{F}_{j},\hat{H}\right]=0
  152. | ψ | 2 \left|\psi\right|^{2}
  153. ψ \psi

Symmetry_of_diatomic_molecules.html

  1. * {}^{\prime}*^{\prime}
  2. x , y G x,y\in G
  3. x * y G x*y\in G
  4. ( x * y ) * z = x * ( y * z ) x , y , z G (x*y)*z=x*(y*z)\forall x,y,z\in G
  5. x * e = e * x = x ; x G x*e=e*x=x;\forall x\in G
  6. x G x\in G
  7. y G \,\text{ }\exists\,\text{ }y\in G
  8. x * y = y * x = e x*y=y*x=e
  9. x , y G \forall x,y\in G
  10. x * y = y * x x*y=y*x
  11. U U
  12. H H
  13. [ H , U ] = 0 [H,U]=0
  14. H = U H U = H H U = U H [ H , U ] = 0 ; U G \begin{aligned}&\displaystyle{H}^{\prime}={{U}^{\dagger}}HU=H\\ &\displaystyle\Rightarrow HU=UH\\ &\displaystyle\Rightarrow[H,U]=0;\forall U\in G\\ \end{aligned}
  15. T T
  16. d T d t = d Ψ | T | Ψ d t = Ψ t | T | Ψ + Ψ | T t | Ψ + Ψ | T | Ψ t \frac{d\left\langle T\right\rangle}{dt}=\frac{d\left\langle\Psi|T|\Psi\right% \rangle}{dt}=\left\langle\frac{\partial\Psi}{\partial t}|T|\Psi\right\rangle+% \left\langle\Psi|\frac{\partial T}{\partial t}|\Psi\right\rangle+\left\langle% \Psi|T|\frac{\partial\Psi}{\partial t}\right\rangle
  17. i | Ψ t = H | Ψ i\hbar\frac{\partial\left|\Psi\right\rangle}{\partial t}=H\left|\Psi\right\rangle
  18. d T d t = - 1 i Ψ | H T | Ψ + 1 i Ψ | T H | Ψ + T t \frac{d\left\langle T\right\rangle}{dt}=-\frac{1}{i\hbar}\left\langle\Psi|HT|% \Psi\right\rangle+\frac{1}{i\hbar}\left\langle\Psi|TH|\Psi\right\rangle+\left% \langle\frac{\partial T}{\partial t}\right\rangle
  19. d T d t = 1 i [ H , T ] + T t \frac{d\left\langle T\right\rangle}{dt}=\frac{1}{i\hbar}\left\langle[H,T]% \right\rangle+\left\langle\frac{\partial T}{\partial t}\right\rangle
  20. [ H , T ] = 0 [H,T]=0
  21. d T d t = 0 \frac{d\left\langle T\right\rangle}{dt}=0
  22. T \Rightarrow\left\langle T\right\rangle
  23. | Ψ \left|\Psi\right\rangle
  24. < m t p l > L z <mtpl>{{L}}_{{z}}
  25. R ( α ) = e - i α L z R(\alpha)={{e}^{\frac{-i\alpha{{L}_{z}}}{\hbar}}}
  26. < m t p l > L z <mtpl>{{L}}_{{z}}
  27. H H
  28. c n {{c}_{n}}
  29. 360 0 n \frac{{{360}^{0}}}{n}
  30. c n {{c}_{n}}
  31. 360 0 n \frac{{{360}^{0}}}{n}
  32. n = n=\infty
  33. C v {{C}_{\infty v}}
  34. D h {{D}_{\infty h}}
  35. C v {{C}_{\infty v}}
  36. C v {{C}_{\infty v}}
  37. C ( ϕ ) C(\phi)
  38. ϕ \phi
  39. C n v {{C}_{nv}}
  40. n n\to\infty
  41. σ v {{\sigma}_{v}}
  42. C v {{C}_{\infty v}}
  43. σ v {{\sigma}_{v}}
  44. C ( ± ϕ ) C(\pm\phi)
  45. \infty
  46. σ v \infty{{\sigma}_{v}}
  47. 2 cos ( ϕ ) 2\cos(\phi)
  48. 2 cos ( 2 ϕ ) 2\cos(2\phi)
  49. 2 cos ( 3 ϕ ) 2\cos(3\phi)
  50. D h {{D}_{\infty h}}
  51. D h {{D}_{\infty h}}
  52. C v {{C}_{\infty v}}
  53. D h {{D}_{\infty h}}
  54. C v {{C}_{\infty v}}
  55. D h = C v × C i {{D}_{\infty h}}={{C}_{\infty v}}\times{{C}_{i}}
  56. C v {{C}_{\infty v}}
  57. D h {{D}_{\infty h}}
  58. \infty
  59. σ h \infty{{\sigma}_{h}}
  60. \infty
  61. c 2 c_{2}^{{}^{\prime}}
  62. 2 cos ( ϕ ) 2\cos(\phi)
  63. - 2 cos ( ϕ ) -2\cos(\phi)
  64. 2 cos ( 2 ϕ ) 2\cos(2\phi)
  65. 2 cos ( 2 ϕ ) 2\cos(2\phi)
  66. 2 cos ( 3 ϕ ) 2\cos(3\phi)
  67. - 2 cos ( 3 ϕ ) -2\cos(3\phi)
  68. 2 cos ( ϕ ) 2\cos(\phi)
  69. 2 cos ( ϕ ) 2\cos(\phi)
  70. 2 cos ( 2 ϕ ) 2\cos(2\phi)
  71. - 2 cos ( 2 ϕ ) -2\cos(2\phi)
  72. 2 cos ( 3 ϕ ) 2\cos(3\phi)
  73. 2 cos ( 3 ϕ ) 2\cos(3\phi)
  74. C v {{C}_{\infty v}}
  75. c ϕ c_{{}_{\infty}}^{\phi}
  76. 2 c 2c_{{}_{\infty}}
  77. σ v \infty{{\sigma}_{v}}
  78. \infty
  79. \infty
  80. D h {{D}_{\infty h}}
  81. c ϕ c_{{}_{\infty}}^{\phi}
  82. c 2 c_{2}^{{}^{\prime}}
  83. 2 c 2c_{{}_{\infty}}
  84. σ h \infty{{\sigma}_{h}}
  85. \infty
  86. c 2 c_{2}^{{}^{\prime}}
  87. \infty
  88. \infty
  89. L 2 {{L}^{2}}
  90. l l
  91. L z {{L}_{z}}
  92. J z {{J}_{z}}
  93. H H
  94. L x , L y {{L}_{x}},{{L}_{y}}
  95. H H
  96. S 2 {{S}^{2}}
  97. S z {{S}_{z}}
  98. H H
  99. y i - y i {{y}_{i}}\to-{{y}_{i}}
  100. A y {{A}_{y}}
  101. [ A y , H ] = 0 [{{A}_{y}},H]=0
  102. { H , J z , L z , S 2 , S z , A } \{H,\,\text{ }{{J}_{z}},{{L}_{z}},{{S}^{2}},{{S}_{z}},A\}
  103. A A
  104. O 16 {{O}^{16}}
  105. O 18 {{O}^{18}}
  106. r i - r i {{\vec{r}}_{i}}\to-{{\vec{r}}_{i}}
  107. Π \Pi
  108. { H , J z , L z , S 2 , S z , A , Π } \left\{H,\,\text{ }{{J}_{z}},{{L}_{z}},{{S}^{2}},{{S}_{z}},A,\,\text{ }\Pi\right\}
  109. [ H , L 2 ] 0 [H,{{L}^{2}}]\neq 0
  110. l l
  111. L 2 {{L}^{2}}
  112. L J 2 S + 1 {}^{2S+1}{{L}_{J}}
  113. [ H , L z ] = 0 [H,{{L}_{z}}]=0
  114. L z {{L}_{z}}
  115. L z | Ψ = M L | Ψ ; M L = 0 , ± 1 , ± 2 , . L z | Ψ = ± Λ | Ψ ; Λ = 0 , 1 , 2 , . . \begin{aligned}&\displaystyle{{L}_{z}}\left|\Psi\right\rangle={{M}_{L}}\hbar% \left|\Psi\right\rangle;{{M}_{L}}=0,\pm 1,\pm 2,..........\\ &\displaystyle\Rightarrow{{L}_{z}}\left|\Psi\right\rangle=\pm\Lambda\hbar\left% |\Psi\right\rangle;\Lambda=0,1,2,...........\\ \end{aligned}
  116. Λ = | M L | \Lambda=\left|{{M}_{L}}\right|
  117. Λ \Lambda
  118. Λ \Lambda
  119. λ = | m l | \lambda=\left|{{m}_{l}}\right|
  120. [ A y , H ] = 0 [{{A}_{y}},H]=0
  121. A y L z = - L z A y {{A}_{y}}{{L}_{z}}=-{{L}_{z}}{{A}_{y}}
  122. L z = - i ( x y - y x ) {{L}_{z}}=-i\hbar(x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x})
  123. Λ 0 \Lambda\neq 0
  124. A y {{A}_{y}}
  125. Λ \Lambda\hbar
  126. < m t p l > L z <mtpl>{{L}}_{{z}}
  127. - Λ -\Lambda\hbar
  128. Λ 0 \Lambda\neq 0
  129. Π , Δ , Φ , . \Pi,\Delta,\Phi,................
  130. Λ 0 \Lambda\neq 0
  131. Λ \Lambda
  132. Λ = 0 \Lambda=0
  133. Σ \Sigma
  134. Σ \Sigma
  135. Λ = 0 \Lambda=0
  136. H H
  137. < m t p l > L z <mtpl>{{L}}_{{z}}
  138. < m t p l > A y <mtpl>{{A}}_{{y}}
  139. A y 2 = 1 A_{y}^{2}=1
  140. < m t p l > A y <mtpl>{{A}}_{{y}}
  141. ± 1 \pm 1
  142. Σ \Sigma
  143. Σ + {{\Sigma}^{+}}
  144. Σ - {{\Sigma}^{-}}
  145. H 2 , N 2 , O 2 , {{\,\text{H}}_{2}},{{N}_{2}},{{\,\text{O}}_{2,...............}}
  146. O 16 {{O}^{16}}
  147. O 18 {{O}^{18}}
  148. r i - r i {{\vec{r}}_{i}}\to-{{\vec{r}}_{i}}
  149. Π \Pi
  150. < m t p l > L z <mtpl>{{L}}_{{z}}
  151. Λ \Lambda
  152. r i - r i {{\vec{r}}_{i}}\to-{{\vec{r}}_{i}}
  153. Σ g , Σ u , Π g , Π u , {{\Sigma}_{g}},{{\Sigma}_{u}},{{\Pi}_{g}},{{\Pi}_{u}},......
  154. Σ \Sigma
  155. Σ g - , Σ g + , Σ u - , Σ g + \Sigma_{g}^{-},\Sigma_{g}^{+},\Sigma_{u}^{-},\Sigma_{g}^{+}
  156. s ( s + 1 ) 2 s(s+1){{\hbar}^{2}}
  157. 2 s + 1 2s+1
  158. s s
  159. Λ \Lambda
  160. 2 s + 1 2s+1
  161. Λ 2 s + 1 {}^{2s+1}\Lambda
  162. Π 3 {}^{3}\Pi
  163. Λ = 2 \Lambda=2
  164. s = 1 s=1
  165. X X
  166. s = 0 s=0
  167. Σ + 1 {}^{1}{{\Sigma}^{+}}
  168. X Σ + 1 X{}^{1}{{\Sigma}^{+}}
  169. A , B , C , A,B,C,...
  170. Σ g + 1 {}^{1}\Sigma_{g}^{+}
  171. X Σ g + 1 X{}^{1}\Sigma_{g}^{+}
  172. M J {{M}_{J}}
  173. M J = M S + M L {{M}_{J}}={{M}_{S}}+{{M}_{L}}
  174. M J {{M}_{J}}
  175. Ω \Omega
  176. Ω = Λ + M S \Omega=\Lambda+{{M}_{S}}
  177. Λ Ω , ( g / u ) ( + / - ) 2 S + 1 {}^{2S+1}\!\Lambda^{(+/-)}_{\Omega,(g/u)}
  178. Λ \Lambda
  179. Ω \Omega
  180. E S ( R ) {{E}_{S}}(R)
  181. R R
  182. E 1 ( R ) {{E}_{1}}(R)
  183. E 2 ( R ) {{E}_{2}}(R)
  184. E 1 ( R ) {{E}_{1}}(R)
  185. E 2 ( R ) {{E}_{2}}(R)
  186. R c {{R}}_{{c}}
  187. E 1 ( R C ) {{E}_{1}}({{R}_{C}})
  188. E 2 ( R C ) {{E}_{2}}({{R}_{C}})
  189. E 1 ( R ) {{E}_{1}}(R)
  190. E 2 ( R ) {{E}_{2}}(R)
  191. R C R C + Δ R {{R}_{C}}\to{{R}_{C}}+\Delta R
  192. E 1 ( 0 ) = E 1 ( R C ) E_{1}^{(0)}={{E}_{1}}({{R}_{C}})
  193. E 2 ( 0 ) = E 2 ( R C ) E_{2}^{(0)}={{E}_{2}}({{R}_{C}})
  194. H 0 = H ( R C ) {{H}_{0}}=H({{R}_{C}})
  195. | Φ 1 ( 0 ) \left|\Phi_{1}^{(0)}\right\rangle
  196. | Φ 2 ( 0 ) \left|\Phi_{2}^{(0)}\right\rangle
  197. H H ( R C + Δ R ) = H 0 + H H\equiv H({{R}_{C}}+\Delta R)={{H}_{0}}+H^{\prime}
  198. H = H 0 R C Δ R H^{\prime}=\frac{\partial{{H}_{0}}}{\partial{{R}_{C}}}\Delta R
  199. H i j = Φ i ( 0 ) | H | Φ j ( 0 ) ; i , j = 1 , 2 H_{ij}^{{}^{\prime}}=\left\langle\Phi_{i}^{(0)}|H^{\prime}|\Phi_{j}^{(0)}% \right\rangle;i,j=1,2
  200. E 1 ( R ) {{E}_{1}}(R)
  201. E 2 ( R ) {{E}_{2}}(R)
  202. R C + Δ R {{R}_{C}}+\Delta R
  203. E 1 ( 0 ) - E 2 ( 0 ) + H 11 - H 22 = 0 E_{1}^{(0)}-E_{2}^{(0)}+H_{11}^{{}^{\prime}}-H_{22}^{{}^{\prime}}=0
  204. H 12 = 0 H_{12}^{{}^{\prime}}=0
  205. | Φ 1 ( 0 ) \left|\Phi_{1}^{(0)}\right\rangle
  206. | Φ 2 ( 0 ) \left|\Phi_{2}^{(0)}\right\rangle
  207. | Φ ( 0 ) = c 1 | Φ 1 ( 0 ) + c 2 | Φ 2 ( 0 ) \left|{{\Phi}^{(0)}}\right\rangle={{c}_{1}}\left|\Phi_{1}^{(0)}\right\rangle+{% {c}_{2}}\left|\Phi_{2}^{(0)}\right\rangle
  208. H H
  209. c 1 {{c}_{1}}
  210. c 2 {{c}_{2}}
  211. ( H 0 + H ) | Φ ( 0 ) = E | Φ ( 0 ) ({{H}_{0}}+H^{\prime})\left|{{\Phi}^{(0)}}\right\rangle=E\left|{{\Phi}^{(0)}}\right\rangle
  212. c 1 ( E 1 ( 0 ) + H - E ) | Φ 1 ( 0 ) + c 2 ( E 2 ( 0 ) + H - E ) | Φ 2 ( 0 ) = 0 {{c}_{1}}(E_{1}^{(0)}+H^{\prime}-E)\left|\Phi_{1}^{(0)}\right\rangle+{{c}_{2}}% (E_{2}^{(0)}+H^{\prime}-E)\left|\Phi_{2}^{(0)}\right\rangle=0
  213. c 1 ( E 1 ( 0 ) + H - E ) Φ 1 ( 0 ) | Φ 1 ( 0 ) + c 2 ( E 2 ( 0 ) + H - E ) Φ 1 ( 0 ) | Φ 2 ( 0 ) = 0 {{c}_{1}}(E_{1}^{(0)}+H^{\prime}-E)\left\langle\Phi_{1}^{(0)}|\Phi_{1}^{(0)}% \right\rangle+{{c}_{2}}(E_{2}^{(0)}+H^{\prime}-E)\left\langle\Phi_{1}^{(0)}|% \Phi_{2}^{(0)}\right\rangle=0
  214. c 1 ( E 1 ( 0 ) + H - E ) Φ 2 ( 0 ) | Φ 1 ( 0 ) + c 2 ( E 2 ( 0 ) + H - E ) Φ 2 ( 0 ) | Φ 2 ( 0 ) = 0 {{c}_{1}}(E_{1}^{(0)}+H^{\prime}-E)\left\langle\Phi_{2}^{(0)}|\Phi_{1}^{(0)}% \right\rangle+{{c}_{2}}(E_{2}^{(0)}+H^{\prime}-E)\left\langle\Phi_{2}^{(0)}|% \Phi_{2}^{(0)}\right\rangle=0
  215. | Φ 1 ( 0 ) \left|\Phi_{1}^{(0)}\right\rangle
  216. | Φ 2 ( 0 ) \left|\Phi_{2}^{(0)}\right\rangle
  217. H 0 {{H}_{0}}
  218. H 0 {{H}_{0}}
  219. Φ i ( 0 ) | Φ j ( 0 ) = δ i j \left\langle\Phi_{i}^{(0)}|\Phi_{j}^{(0)}\right\rangle={{\delta}_{ij}}
  220. c 1 ( E 1 ( 0 ) + H 11 - E ) + c 2 H 12 = 0 {{c}_{1}}(E_{1}^{(0)}+H_{11}^{{}^{\prime}}-E)+{{c}_{2}}H_{12}^{{}^{\prime}}=0
  221. c 1 H 21 + c 2 ( E 2 ( 0 ) + H 22 - E ) = 0 {{c}_{1}}H_{21}^{{}^{\prime}}+{{c}_{2}}(E_{2}^{(0)}+H_{22}^{{}^{\prime}}-E)=0
  222. H H^{\prime}
  223. H 11 H_{11}^{{}^{\prime}}
  224. H 22 H_{22}^{{}^{\prime}}
  225. H 12 = H 21 * H_{12}^{{}^{\prime}}=H_{21}^{{}^{\prime}*}
  226. c 1 {{c}_{1}}
  227. c 2 {{c}_{2}}
  228. | E 1 ( 0 ) + H 11 - E H 12 H 21 E 2 ( 0 ) + H 22 - E | = 0 \left|\begin{matrix}E_{1}^{(0)}+H_{11}^{{}^{\prime}}-E&H_{12}^{{}^{\prime}}\\ H_{21}^{{}^{\prime}}&E_{2}^{(0)}+H_{22}^{{}^{\prime}}-E\\ \end{matrix}\right|=0
  229. E = 1 2 ( E 1 ( 0 ) + E 2 ( 0 ) + H 11 + H 22 ) ± 1 4 ( E 1 ( 0 ) - E 2 ( 0 ) + H 11 - H 22 ) 2 + | H 12 | 2 E=\frac{1}{2}(E_{1}^{(0)}+E_{2}^{(0)}+H_{11}^{{}^{\prime}}+H_{22}^{{}^{\prime}% })\pm\sqrt{\frac{1}{4}{{(E_{1}^{(0)}-E_{2}^{(0)}+H_{11}^{{}^{\prime}}-H_{22}^{% {}^{\prime}})}^{2}}+{{\left|H_{12}^{{}^{\prime}}\right|}^{2}}}
  230. ( R C + Δ R ) ({{R}_{C}}+\Delta R)
  231. E E
  232. E 1 ( 0 ) - E 2 ( 0 ) + H 11 - H 22 = 0 E_{1}^{(0)}-E_{2}^{(0)}+H_{11}^{{}^{\prime}}-H_{22}^{{}^{\prime}}=0
  233. H 12 = 0 H_{12}^{{}^{\prime}}=0
  234. Δ R \Delta R
  235. H H^{\prime}
  236. | Φ 1 ( 0 ) \left|\Phi_{1}^{(0)}\right\rangle
  237. | Φ 2 ( 0 ) \left|\Phi_{2}^{(0)}\right\rangle
  238. H 12 H_{12}^{{}^{\prime}}
  239. H 12 H_{12}^{{}^{\prime}}
  240. Δ R \Delta R
  241. R R
  242. H H^{\prime}
  243. H H
  244. | Φ 1 ( 0 ) \left|\Phi_{1}^{(0)}\right\rangle
  245. | Φ 2 ( 0 ) \left|\Phi_{2}^{(0)}\right\rangle
  246. Λ \Lambda
  247. Σ + {{\Sigma}^{+}}
  248. Σ - {{\Sigma}^{-}}
  249. | Φ 1 ( 0 ) \left|\Phi_{1}^{(0)}\right\rangle
  250. | Φ 2 ( 0 ) \left|\Phi_{2}^{(0)}\right\rangle
  251. H 12 H_{12}^{{}^{\prime}}
  252. R R
  253. Δ R \Delta R
  254. R R
  255. λ \lambda
  256. λ \lambda
  257. | m | \left|m\right|
  258. R R
  259. 0
  260. \infty
  261. R R
  262. 0
  263. \infty
  264. D D
  265. | Ψ α \left|{{\Psi}_{\alpha}}\right\rangle
  266. α \alpha
  267. | Φ s \left|{{\Phi}_{s}}\right\rangle
  268. | v \left|v\right\rangle
  269. | ϕ , M , Λ \left|{{\phi}_{\Im,{{M}_{\Im}},\Lambda}}\right\rangle
  270. J 2 {{J}^{2}}
  271. < m t p l > J z <mtpl>{{J}}_{{z}}
  272. \Im
  273. M {{M}_{\Im}}
  274. D D
  275. D α α = Ψ α | D | Ψ α ; α ( s , v , , M , Λ ) {{D}_{\alpha\alpha}}=\left\langle{{\Psi}_{\alpha}}|D|{{\Psi}_{\alpha}}\right% \rangle;\,\text{ }\alpha\equiv(s,v,\Im,{{M}_{\Im}},\Lambda)
  276. α \alpha
  277. D α α {{D}_{\alpha\alpha}}
  278. D D
  279. H C l HCl
  280. Λ = 0 \Lambda=0
  281. Δ = ± 1 \displaystyle\Delta\Im=\pm 1
  282. Λ 0 \Lambda\neq 0
  283. Δ = 0 , ± 1 \displaystyle\Delta\Im=0,\pm 1
  284. \Im
  285. Δ Λ = 0 \Delta\Lambda=0
  286. ω + 1 , = E r ( + 1 ) - E r ( ) = 2 B ( + 1 ) \hbar{{\omega}_{\Im+1,\Im}}={{E}_{r}}(\Im+1)-{{E}_{r}}(\Im)=2B(\Im+1)
  287. B = 2 2 μ R 0 2 B=\frac{{{\hbar}^{2}}}{2\mu R_{0}^{2}}
  288. Λ \Im\geq\Lambda
  289. μ v , v = v | μ | v {{\mu}_{v,v^{\prime}}}=\left\langle v^{\prime}|\mu|v\right\rangle
  290. μ \mu
  291. α \alpha
  292. R R
  293. μ = μ 0 + ( d μ d x ) 0 x + 1 2 ( d 2 μ d x 2 ) 0 x 2 + . \mu={{\mu}_{0}}+{{(\frac{d\mu}{dx})}_{0}}x+\frac{1}{2}{{(\frac{{{d}^{2}}\mu}{d% {{x}^{2}}})}_{0}}{{x}^{2}}+.......
  294. μ 0 {{\mu}_{0}}
  295. v | μ | v = μ 0 v | v + ( d μ d x ) 0 v | x | v + 1 2 ( d 2 μ d x 2 ) 0 v | x 2 | v + . = ( d μ d x ) 0 v | x | v + 1 2 ( d 2 μ d x 2 ) 0 v | x 2 | v + . \left\langle v^{\prime}|\mu|v\right\rangle={{\mu}_{0}}\left\langle v^{\prime}|% v\right\rangle+{{(\frac{d\mu}{dx})}_{0}}\left\langle v^{\prime}|x|v\right% \rangle+\frac{1}{2}{{(\frac{{{d}^{2}}\mu}{d{{x}^{2}}})}_{0}}\left\langle v^{% \prime}|{{x}^{2}}|v\right\rangle+.......={{(\frac{d\mu}{dx})}_{0}}\left\langle v% ^{\prime}|x|v\right\rangle+\frac{1}{2}{{(\frac{{{d}^{2}}\mu}{d{{x}^{2}}})}_{0}% }\left\langle v^{\prime}|{{x}^{2}}|v\right\rangle+.......
  296. μ \mu
  297. μ v , v = v | μ | v = ( d μ d x ) 0 v | x | v {{\mu}_{v,v^{\prime}}}=\left\langle v^{\prime}|\mu|v\right\rangle={{(\frac{d% \mu}{dx})}_{0}}\left\langle v^{\prime}|x|v\right\rangle
  298. 2 ( α x ) H v ( α x ) = 2 v H v - 1 ( α x ) + H v + 1 ( α x ) 2(\alpha x){{H}_{v}}(\alpha x)=2v{{H}_{v-1}}(\alpha x)+{{H}_{v+1}}(\alpha x)
  299. x | v x\left|v\right\rangle
  300. x H v ( α x ) x{{H}_{v}}(\alpha x)
  301. | v + 1 \left|v+1\right\rangle
  302. | v - 1 \left|v-1\right\rangle
  303. μ v , v {{\mu}_{v,v^{\prime}}}
  304. v = v ± 1 v^{\prime}=v\pm 1
  305. Δ v = ± 1 \Delta v=\pm 1
  306. Σ \Sigma
  307. ( v , ) (v,\Im)
  308. ( v , ) (v^{\prime},\Im^{\prime})
  309. v v
  310. v = v + 1 v^{\prime}=v+1
  311. Δ = + 1 \Delta\Im=+1
  312. Δ = - 1 \Delta\Im=-1
  313. Δ = + 1 \Delta\Im=+1
  314. ω R = E ( v + 1 , + 1 ) - E ( v , ) = 2 B ( + 1 ) + ω 0 ; = 0 , 1 , 2 , \hbar{{\omega}^{R}}=E(v+1,\Im+1)-E(v,\Im)=2B(\Im+1)+\hbar{{\omega}_{0}};\,% \text{ }\Im=0,1,2,......
  315. Δ = - 1 \Delta\Im=-1
  316. ω P = E ( v + 1 , - 1 ) - E ( v , ) = - 2 B + ω 0 ; = 1 , 2 , 3 , \hbar{{\omega}^{P}}=E(v+1,\Im-1)-E(v,\Im)=-2B\Im+\hbar{{\omega}_{0}};\,\text{ % }\Im=1,2,3,......
  317. Σ \Sigma
  318. Λ 0 \Lambda\neq 0
  319. Δ = 0 \Delta\Im=0
  320. ω Q {{\omega}^{Q}}
  321. \Im
  322. B v {{B}_{v}}
  323. B v + 1 {{B}_{v+1}}
  324. ω Q = E ( v + 1 , ) - E ( v , ) = ω 0 \hbar{{\omega}^{Q}}=E(v+1,\Im)-E(v,\Im)=\hbar{{\omega}_{0}}
  325. B v + 1 = B v {{B}_{v+1}}={{B}_{v}}
  326. H 2 + \,\text{H}_{2}^{+}
  327. H 2 + \,\text{H}_{2}^{+}
  328. 𝐫 | 𝟏 = 1 π a 0 3 e - | 𝐫 - 𝐑 2 | a 0 \left\langle\mathbf{r}|\mathbf{1}\right\rangle=\frac{1}{\sqrt{\pi a_{0}^{3}}}{% {e}^{-\frac{\left|\mathbf{r}-\frac{\mathbf{R}}{2}\right|}{{{a}_{0}}}}}
  329. 𝐫 | 𝟐 = 1 π a 0 3 e - | 𝐫 + 𝐑 2 | a 0 \left\langle\mathbf{r}|\mathbf{2}\right\rangle=\frac{1}{\sqrt{\pi a_{0}^{3}}}{% {e}^{-\frac{\left|\mathbf{r}+\frac{\mathbf{R}}{2}\right|}{{{a}_{0}}}}}
  330. H 2 + \,\text{H}_{2}^{+}
  331. H = 𝐩 2 2 m e - e 2 | 𝐫 - 𝐑 / 2 | - e 2 | 𝐫 + 𝐑 / 2 | + e 2 R H=\frac{{{\mathbf{p}}^{2}}}{2{{m}_{e}}}-\frac{{{e}^{2}}}{\left|\mathbf{r}-% \mathbf{R}/2\right|}-\frac{{{e}^{2}}}{\left|\mathbf{r}+\mathbf{R}/2\right|}+% \frac{{{e}^{2}}}{R}
  332. | 1 \left|1\right\rangle
  333. | 2 \left|2\right\rangle
  334. H 2 + \,\text{H}_{2}^{+}
  335. H 2 + \,\text{H}_{2}^{+}
  336. H 11 = H 22 and H 12 = H 21 {{H}_{11}}={{H}_{22}}\,\text{ and }{{H}_{12}}={{H}_{21}}
  337. H 11 = 1 | 𝐩 2 2 m e - e 2 | 𝐫 - 𝐑 / 2 | | 1 - 1 | e 2 | 𝐫 + 𝐑 / 2 | | 1 + e 2 R 1 | 1 {{H}_{11}}=\left\langle 1|\frac{{{\mathbf{p}}^{2}}}{2{{m}_{e}}}-\frac{{{e}^{2}% }}{\left|\mathbf{r}-\mathbf{R}/2\right|}|1\right\rangle-\left\langle 1|\frac{{% {e}^{2}}}{\left|\mathbf{r}+\mathbf{R}/2\right|}|1\right\rangle+\frac{{{e}^{2}}% }{R}\left\langle 1|1\right\rangle
  338. H 11 = E 1 - d 3 r e 2 | 𝐫 + 𝐑 / 2 | | 𝐫 | 1 | 2 + e 2 R \Rightarrow{{H}_{11}}={{E}_{1}}-\int{{{d}^{3}}r}\frac{{{e}^{2}}}{\left|\mathbf% {r}+\mathbf{R}/2\right|}{{\left|\left\langle\mathbf{r}|1\right\rangle\right|}^% {2}}+\frac{{{e}^{2}}}{R}
  339. H 11 = E 1 - d 3 r e 2 | 𝐫 + 𝐑 / 2 | | 𝐫 | 1 | 2 + e 2 R \Rightarrow{{H}_{11}}={{E}_{1}}-\int{{{d}^{3}}r}\frac{{{e}^{2}}}{\left|\mathbf% {r}+\mathbf{R}/2\right|}{{\left|\left\langle\mathbf{r}|1\right\rangle\right|}^% {2}}+\frac{{{e}^{2}}}{R}
  340. E 1 {{E}_{1}}
  341. H 22 = 2 | 𝐩 2 2 m e - e 2 | 𝐫 + 𝐑 / 2 | | 2 - 2 | e 2 | 𝐫 - 𝐑 / 2 | | 2 + e 2 R 2 | 2 {{H}_{22}}=\left\langle 2|\frac{{{\mathbf{p}}^{2}}}{2{{m}_{e}}}-\frac{{{e}^{2}% }}{\left|\mathbf{r}+\mathbf{R}/2\right|}|2\right\rangle-\left\langle 2|\frac{{% {e}^{2}}}{\left|\mathbf{r}-\mathbf{R}/2\right|}|2\right\rangle+\frac{{{e}^{2}}% }{R}\left\langle 2|2\right\rangle
  342. H 22 = E 1 - d 3 r e 2 | 𝐫 - 𝐑 / 2 | | 𝐫 | 2 | 2 + e 2 R = H 11 \Rightarrow{{H}_{22}}={{E}_{1}}-\int{{{d}^{3}}r}\frac{{{e}^{2}}}{\left|\mathbf% {r}-\mathbf{R}/2\right|}{{\left|\left\langle\mathbf{r}|2\right\rangle\right|}^% {2}}+\frac{{{e}^{2}}}{R}={{H}_{11}}
  343. | 𝐫 | 2 | 2 = | 𝐫 | 1 | 2 = 1 π a 0 3 {{\left|\left\langle\mathbf{r}|2\right\rangle\right|}^{2}}={{\left|\left% \langle\mathbf{r}|1\right\rangle\right|}^{2}}=\frac{1}{\pi a_{0}^{3}}
  344. H 12 = 1 | 𝐩 2 2 m e - e 2 | 𝐫 + 𝐑 / 2 | | 2 - 1 | e 2 | 𝐫 - 𝐑 / 2 | | 2 + e 2 R 1 | 2 {{H}_{12}}=\left\langle 1|\frac{{{\mathbf{p}}^{2}}}{2{{m}_{e}}}-\frac{{{e}^{2}% }}{\left|\mathbf{r}+\mathbf{R}/2\right|}|2\right\rangle-\left\langle 1|\frac{{% {e}^{2}}}{\left|\mathbf{r}-\mathbf{R}/2\right|}|2\right\rangle+\frac{{{e}^{2}}% }{R}\left\langle 1|2\right\rangle
  345. H 12 = ( E 1 + e 2 R ) 1 | 2 - d 3 r e 2 | 𝐫 - 𝐑 / 2 | 1 | 𝐫 𝐫 | 2 \Rightarrow{{H}_{12}}=({{E}_{1}}+\frac{{{e}^{2}}}{R})\left\langle 1|2\right% \rangle-\int{{{d}^{3}}r}\frac{{{e}^{2}}}{\left|\mathbf{r}-\mathbf{R}/2\right|}% \left\langle 1\left|\mathbf{r}\right\rangle\left\langle\mathbf{r}\right|2\right\rangle
  346. d 3 r | 𝐫 𝐫 | \int{{{d}^{3}}r}\left|\mathbf{r}\right\rangle\left\langle\mathbf{r}\right|
  347. 1 | 2 = d 3 r 1 | 𝐫 𝐫 | 2 \left\langle 1|2\right\rangle=\int{{{d}^{3}}r}\left\langle 1\left|\mathbf{r}% \right\rangle\left\langle\mathbf{r}\right|2\right\rangle
  348. H 21 = 2 | 𝐩 2 2 m e - e 2 | 𝐫 - 𝐑 / 2 | | 1 - 2 | e 2 | 𝐫 + 𝐑 / 2 | | 1 + e 2 R 2 | 1 {{H}_{21}}=\left\langle 2|\frac{{{\mathbf{p}}^{2}}}{2{{m}_{e}}}-\frac{{{e}^{2}% }}{\left|\mathbf{r}-\mathbf{R}/2\right|}|1\right\rangle-\left\langle 2|\frac{{% {e}^{2}}}{\left|\mathbf{r}+\mathbf{R}/2\right|}|1\right\rangle+\frac{{{e}^{2}}% }{R}\left\langle 2|1\right\rangle
  349. H 21 = ( E 1 + e 2 R ) 2 | 1 - d 3 r e 2 | 𝐫 + 𝐑 / 2 | 2 | 𝐫 𝐫 | 1 = H 12 \Rightarrow{{H}_{21}}=({{E}_{1}}+\frac{{{e}^{2}}}{R})\left\langle 2|1\right% \rangle-\int{{{d}^{3}}r}\frac{{{e}^{2}}}{\left|\mathbf{r}+\mathbf{R}/2\right|}% \left\langle 2\left|\mathbf{r}\right\rangle\left\langle\mathbf{r}\right|1% \right\rangle={{H}_{12}}
  350. H 11 = H 22 and H 12 = H 21 {{H}_{11}}={{H}_{22}}\,\text{ and }{{H}_{12}}={{H}_{21}}
  351. H 11 = H 22 {{H}_{11}}={{H}_{22}}
  352. H 12 = H 21 {{H}_{12}}={{H}_{21}}
  353. | 1 \left|1\right\rangle
  354. | 2 \left|2\right\rangle
  355. | ± = 1 2 ± 2 1 | 2 ( | 1 ± | 2 ) \left|\pm\right\rangle=\frac{1}{\sqrt{2\pm 2\left\langle 1|2\right\rangle}}(% \left|1\right\rangle\pm\left|2\right\rangle)
  356. [ H , Π ] = 0 [H,\Pi]=0
  357. H 2 + \,\text{H}_{2}^{+}
  358. | ± \left|\pm\right\rangle
  359. Π \Pi
  360. | + \left|+\right\rangle
  361. | - \left|-\right\rangle
  362. Π \Pi
  363. 𝐫 | + \left\langle\mathbf{r}|+\right\rangle
  364. 𝐫 | - \left\langle\mathbf{r}|-\right\rangle
  365. E ± = 1 1 ± 1 | 2 ( H 11 ± H 12 ) {{E}_{\pm}}=\frac{1}{1\pm\left\langle 1|2\right\rangle}({{H}_{11}}\pm{{H}_{12}})
  366. E + {{E}_{+}}
  367. 1.3 A 0 1.3\overset{0}{\mathop{\,\text{A}}}\,
  368. E + = - 15.4 e V {{E}_{+}}=-15.4eV
  369. - 13.6 e V -13.6eV
  370. 1.8 e V 1.8eV
  371. H 2 + \,\text{H}_{2}^{+}
  372. X 2 Σ g + {{X}^{2}}\Sigma_{g}^{+}
  373. | + \left|+\right\rangle
  374. H 2 + \,\text{H}_{2}^{+}
  375. C {{C}_{\infty}}
  376. C h {{C}_{\infty h}}
  377. D {{D}_{\infty}}