\usepackage or this document’s own macros. Corpus-wide, \bm occurs 327 times, \Perp is defined by no package and spans 10 documents, and 11,088 of 11,624 undefined occurrences are the source document’s own macros. A row that looks wrong here may be correct, and one that looks right here may not compile. The LaTeX report renders through the document’s own preamble and is the surface to judge from.2970 rows
The page, the confidence and the picture of an inline formula are its HOST LINE's --- a formula has none of its own. A line's confidence is not a formula's.
| Identifier | Page | Conf. | LaTeX source | Rendered | Image |
|---|---|---|---|---|---|
| gilmore-lie-groups_FO0002 | 63 | 0.978 | S O(3) | | |
| gilmore-lie-groups_FO0003 | 63 | 1.000 | S O(4) | | |
| gilmore-lie-groups_FO0004 | 224 | 0.997 | S O(4,1) | | |
| gilmore-lie-groups_FO0005 | 215 | 0.943 | S O(4,2) | | |
| gilmore-lie-groups_FO0006 | 15 | 0.764 | (+,-, \times, \div) | | |
| gilmore-lie-groups_FO0007 | 15 | 0.764 | n | | |
| gilmore-lie-groups_FO0008 | 17 | 1.000 | G | | |
| gilmore-lie-groups_FO0009 | 17 | 1.000 | G=G_{0} \supset G_{1} \supset | | |
| gilmore-lie-groups_FO0010 | 17 | 1.000 | \cdots \supset G_{\omega}=I | | |
| gilmore-lie-groups_FO0011 | 17 | 1.000 | G_{i+1} | | |
| gilmore-lie-groups_FO0012 | 17 | 1.000 | G_{i} | | |
| gilmore-lie-groups_FO0013 | 17 | 1.000 | G_{i} / G_{i+1} | | |
| gilmore-lie-groups_FO0014 | 17 | 0.998 | G=\left\{g_{1}, g_{2}, \ldots\right\} | | |
| gilmore-lie-groups_FO0015 | 18 | 1.000 | g_{i} \in G, g_{j} \in G | | |
| gilmore-lie-groups_FO0016 | 18 | 1.000 | g_{i} \cdot g_{j} \in G | | |
| gilmore-lie-groups_FO0017 | 18 | 1.000 | g_{i} \in G, g_{j} \in G, g_{k} \in G | | |
| gilmore-lie-groups_FO0018 | 18 | 0.835 | I | | |
| gilmore-lie-groups_FO0019 | 18 | 1.000 | g_{i} | | |
| gilmore-lie-groups_FO0020 | 18 | 1.000 | g_{i}^{-1} | | |
| gilmore-lie-groups_FO0021 | 18 | 1.000 | z_{1}, z_{2}, \ldots, z_{n} | | |
| gilmore-lie-groups_FO0022 | 18 | 0.658 | P_{n} | | |
| gilmore-lie-groups_FO0023 | 18 | 0.658 | S_{n} | | |
| gilmore-lie-groups_FO0024 | 18 | 0.658 | n! | | |
| gilmore-lie-groups_FO0025 | 18 | 0.996 | n \times n | | |
| gilmore-lie-groups_FO0026 | 18 | 1.000 | 6=3!3 \times 3 | | |
| gilmore-lie-groups_FO0027 | 18 | 1.000 | S_{3} | | |
| gilmore-lie-groups_FO0028 | 19 | 0.998 | z_{1} | | |
| gilmore-lie-groups_FO0029 | 19 | 0.998 | z_{2}, z_{2} | | |
| gilmore-lie-groups_FO0030 | 19 | 0.999 | z_{3} | | |
| gilmore-lie-groups_FO0031 | 19 | 0.768 | \times | | |
| gilmore-lie-groups_FO0032 | 19 | 1.000 | G \rightarrow \Gamma(G) | | |
| gilmore-lie-groups_FO0033 | 19 | 0.988 | \left(\Gamma\left(g_{i}\right) \times \Gamma\left(g_{j}\right)\right) | | |
| gilmore-lie-groups_FO0034 | 19 | 0.999 | \left(\Gamma\left(g_{i} \cdot g_{j}\right)\right) | | |
| gilmore-lie-groups_FO0035 | 19 | 0.999 | g_{i}, g_{j} \in G | | |
| gilmore-lie-groups_FO0036 | 19 | 0.874 | 3 \times 3 | | |
| gilmore-lie-groups_FO0037 | 19 | 1.000 | H | | |
| gilmore-lie-groups_FO0038 | 19 | 1.000 | A_{3} | | |
| gilmore-lie-groups_FO0039 | 19 | 1.000 | A_{3} \subset S_{3} | | |
| gilmore-lie-groups_FO0040 | 19 | 1.000 | S_{2}(i j) | | |
| gilmore-lie-groups_FO0041 | 20 | 0.706 | (123) \cdot(123)=(321) | | |
| gilmore-lie-groups_FO0042 | 20 | 1.000 | G, H_{1} \subset G | | |
| gilmore-lie-groups_FO0043 | 20 | 1.000 | H_{2} \subset G | | |
| gilmore-lie-groups_FO0044 | 20 | 1.000 | g \in G | | |
| gilmore-lie-groups_FO0045 | 20 | 1.000 | S_{2}(12) | | |
| gilmore-lie-groups_FO0046 | 20 | 1.000 | S_{2}(13) | | |
| gilmore-lie-groups_FO0047 | 20 | 1.000 | G=S_{3} | | |
| gilmore-lie-groups_FO0048 | 20 | 1.000 | H \subset G | | |
| gilmore-lie-groups_FO0049 | 21 | 1.000 | f | | |
| gilmore-lie-groups_FO0050 | 21 | 1.000 | g_{1}, g_{2}, \ldots | | |
| gilmore-lie-groups_FO0051 | 21 | 0.999 | h_{1}, h_{2}, \ldots | | |
| gilmore-lie-groups_FO0052 | 21 | 1.000 | h_{i} \in H | | |
| gilmore-lie-groups_FO0053 | 21 | 1.000 | g_{j} \in G | | |
| gilmore-lie-groups_FO0054 | 21 | 1.000 | h \in H | | |
| gilmore-lie-groups_FO0055 | 21 | 0.933 | g \in G\left(h_{1}=f\left(g_{1}\right)\right. | | |
| gilmore-lie-groups_FO0056 | 21 | 0.933 | \left.h_{2}=f\left(g_{2}\right), h_{1}=h_{2} \Rightarrow g_{1}=g_{2}\right) | | |
| gilmore-lie-groups_FO0057 | 21 | 1.000 | S_{2} | | |
| gilmore-lie-groups_FO0058 | 22 | 0.987 | h | | |
| gilmore-lie-groups_FO0059 | 22 | 1.000 | G / H | | |
| gilmore-lie-groups_FO0060 | 22 | 1.000 | |G|\left(S_{3}\right. | | |
| gilmore-lie-groups_FO0061 | 22 | 0.637 | 3!=6 | | |
| gilmore-lie-groups_FO0062 | 22 | 0.866 | |G / H|=|G| /|H| | | |
| gilmore-lie-groups_FO0063 | 22 | 0.866 | H=A_{3}=\{I | | |
| gilmore-lie-groups_FO0064 | 22 | 0.866 | \} | | |
| gilmore-lie-groups_FO0065 | 22 | 0.992 | H=S_{2}(23)=\{I,(23)\} | | |
| gilmore-lie-groups_FO0066 | 22 | 1.000 | |G| /|H| | | |
| gilmore-lie-groups_FO0067 | 22 | 0.692 | G / H=S_{3} / A_{3}=\{I | | |
| gilmore-lie-groups_FO0068 | 22 | 0.997 | \{I,(23)\} | | |
| gilmore-lie-groups_FO0069 | 22 | 0.997 | S_{2}(23) | | |
| gilmore-lie-groups_FO0070 | 22 | 0.997 | G / H=S_{3} / S_{2}(23)= | | |
| gilmore-lie-groups_FO0071 | 22 | 0.556 | \{I,(123),(321)\} | | |
| gilmore-lie-groups_FO0072 | 22 | 1.000 | S_{3} / A_{3} | | |
| gilmore-lie-groups_FO0073 | 22 | 0.630 | \operatorname{group} G / H: G=G / H \times H | | |
| gilmore-lie-groups_FO0074 | 22 | 1.000 | k | | |
| gilmore-lie-groups_FO0075 | 22 | 1.000 | I_{k} | | |
| gilmore-lie-groups_FO0076 | 22 | 1.000 | z_{i} | | |
| gilmore-lie-groups_FO0077 | 23 | 0.992 | I_{k}(k=1,2, \ldots, n) | | |
| gilmore-lie-groups_FO0078 | 23 | 0.992 | \left(z_{1}, z_{2}, \ldots, z_{n}\right) | | |
| gilmore-lie-groups_FO0079 | 23 | 1.000 | f\left(z_{1}, z_{2}, \ldots, z_{n}\right) | | |
| gilmore-lie-groups_FO0080 | 23 | 1.000 | I_{1}, I_{2}, \ldots, I_{n} | | |
| gilmore-lie-groups_FO0081 | 23 | 1.000 | G_{\omega-1} \supset G_{\omega}=I | | |
| gilmore-lie-groups_FO0082 | 23 | 1.000 | G_{\omega-1} / G_{\omega}=G_{\omega-1} | | |
| gilmore-lie-groups_FO0083 | 23 | 1.000 | \left|G_{\omega-1}\right| /\left|G_{\omega}\right| | | |
| gilmore-lie-groups_FO0084 | 23 | 1.000 | G_{\omega-1} | | |
| gilmore-lie-groups_FO0085 | 23 | 1.000 | G_{\omega}=I | | |
| gilmore-lie-groups_FO0086 | 23 | 1.000 | G_{\omega-2} \supset G_{\omega-1} | | |
| gilmore-lie-groups_FO0087 | 23 | 1.000 | G_{\omega-2} | | |
| gilmore-lie-groups_FO0088 | 23 | 1.000 | G=G_{0} \supset | | |
| gilmore-lie-groups_FO0089 | 23 | 1.000 | G_{1} | | |
| gilmore-lie-groups_FO0090 | 24 | 1.000 | \Gamma^{1}, \Gamma^{2} | | |
| gilmore-lie-groups_FO0091 | 24 | 1.000 | r_{1}-r_{2} | | |
| gilmore-lie-groups_FO0092 | 24 | 0.899 | \left(r_{1}-r_{2}\right) | | |
| gilmore-lie-groups_FO0093 | 24 | 1.000 | \Gamma^{2} | | |
| gilmore-lie-groups_FO0094 | 24 | 1.000 | \Gamma^{1} | | |
| gilmore-lie-groups_FO0095 | 24 | 1.000 | \left(r_{1}-r_{2}\right)^{2} | | |
| gilmore-lie-groups_FO0096 | 24 | 0.999 | I_{1}, I_{2} | | |
| gilmore-lie-groups_FO0097 | 25 | 1.000 | D | | |
| gilmore-lie-groups_FO0098 | 25 | 1.000 | \left(r_{1}-r_{2}\right)= \pm \sqrt{D} | | |
| gilmore-lie-groups_FO0099 | 25 | 1.000 | x | | |
| gilmore-lie-groups_FO0100 | 25 | 1.000 | z | | |
| gilmore-lie-groups_FO0101 | 26 | 1.000 | S_{3} / A_{3}=S_{2} | | |
| gilmore-lie-groups_FO0102 | 26 | 1.000 | A_{3} / I=A_{3} | | |
| gilmore-lie-groups_FO0103 | 26 | 1.000 | A_{3} \supset I | | |
| gilmore-lie-groups_FO0105 | 35 | 1.000 | \omega | | |
| gilmore-lie-groups_FO0110 | 26 | 1.000 | (123)^{-1} | | |
| gilmore-lie-groups_FO0111 | 26 | 1.000 | v_{2} | | |
| gilmore-lie-groups_FO0112 | 27 | 1.000 | v_{1} | | |
| gilmore-lie-groups_FO0113 | 27 | 1.000 | v_{3} | | |
| gilmore-lie-groups_FO0114 | 27 | 1.000 | S_{3} \supset A_{3} | | |
| gilmore-lie-groups_FO0115 | 27 | 1.000 | S_{2}=S_{3} / A_{3} | | |
| gilmore-lie-groups_FO0116 | 27 | 1.000 | v_{2}^{3} | | |
| gilmore-lie-groups_FO0117 | 27 | 1.000 | v_{3}^{3} | | |
| gilmore-lie-groups_FO0118 | 27 | 1.000 | J_{1}, J_{2} | | |
| gilmore-lie-groups_FO0119 | 27 | 0.995 | I_{1}, I_{2}, I_{3} | | |
| gilmore-lie-groups_FO0120 | 27 | 0.995 | J_{1} | | |
| gilmore-lie-groups_FO0121 | 27 | 0.995 | J_{2} | | |
| gilmore-lie-groups_FO0122 | 27 | 1.000 | I_{1}=s_{1}+s_{2}+ | | |
| gilmore-lie-groups_FO0123 | 27 | 1.000 | s_{3}=0 | | |
| gilmore-lie-groups_FO0124 | 28 | 1.000 | J_{1}=v_{2}^{3}+v_{3}^{3} | | |
| gilmore-lie-groups_FO0125 | 28 | 1.000 | J_{2}=v_{2}^{3} v_{3}^{3} | | |
| gilmore-lie-groups_FO0126 | 28 | 1.000 | I_{2}^{\prime}, I_{3}^{\prime} | | |
| gilmore-lie-groups_FO0127 | 28 | 1.000 | v_{2}^{3}, v_{3}^{3} | | |
| gilmore-lie-groups_FO0128 | 28 | 1.000 | s_{1}, s_{2}, s_{3} | | |
| gilmore-lie-groups_FO0129 | 28 | 1.000 | v_{1}, v_{2}, v_{3} | | |
| gilmore-lie-groups_FO0130 | 29 | 1.000 | z=z^{\prime}+\frac{1}{4} I_{1} | | |
| gilmore-lie-groups_FO0131 | 29 | 1.000 | S_{4} | | |
| gilmore-lie-groups_FO0132 | 29 | 1.000 | A_{4} | | |
| gilmore-lie-groups_FO0133 | 29 | 0.957 | V_{4} | | |
| gilmore-lie-groups_FO0134 | 29 | 0.957 | \{I,(12)(34),(13)(24),(14)(23)\} | | |
| gilmore-lie-groups_FO0135 | 30 | 1.000 | S_{4} / A_{4}=S_{2} | | |
| gilmore-lie-groups_FO0136 | 30 | 0.987 | A_{4} / V_{4}=C_{3}=\{I | | |
| gilmore-lie-groups_FO0137 | 30 | 0.987 | V_{4} / I=V_{4}=\{I | | |
| gilmore-lie-groups_FO0143 | 30 | 0.971 | w_{2}, w_{3}, w_{4} | | |
| gilmore-lie-groups_FO0144 | 30 | 1.000 | A_{4} \supset V_{4} | | |
| gilmore-lie-groups_FO0145 | 30 | 1.000 | A_{4} / V_{4} | | |
| gilmore-lie-groups_FO0146 | 30 | 0.999 | w_{1}=I_{1} | | |
| gilmore-lie-groups_FO0147 | 30 | 0.999 | w_{2}^{2}, w_{3}^{2}, w_{4}^{2} | | |
| gilmore-lie-groups_FO0148 | 30 | 1.000 | w_{j}^{2}(j=2,3,4) | | |
| gilmore-lie-groups_FO0149 | 30 | 1.000 | C_{3}=A_{4} / V_{4} | | |
| gilmore-lie-groups_FO0150 | 31 | 1.000 | J_{k} | | |
| gilmore-lie-groups_FO0151 | 31 | 1.000 | C_{3} | | |
| gilmore-lie-groups_FO0152 | 31 | 1.000 | S_{4} \supset A_{4} | | |
| gilmore-lie-groups_FO0153 | 31 | 1.000 | y_{2}, y_{3}, y_{4} | | |
| gilmore-lie-groups_FO0154 | 31 | 1.000 | w_{2} w_{3} w_{4}=8 I_{3}^{\prime} | | |
| gilmore-lie-groups_FO0155 | 31 | 1.000 | \pm \sqrt{y_{j}} | | |
| gilmore-lie-groups_FO0156 | 31 | 1.000 | 8 I_{3}^{\prime} | | |
| gilmore-lie-groups_FO0157 | 31 | 1.000 | t_{i} | | |
| gilmore-lie-groups_FO0158 | 31 | 1.000 | I_{1} | | |
| gilmore-lie-groups_FO0159 | 31 | 1.000 | w_{j}\left(I^{\prime}\right) | | |
| gilmore-lie-groups_FO0160 | 31 | 1.000 | w_{j} | | |
| gilmore-lie-groups_FO0161 | 31 | 1.000 | S_{5} | | |
| gilmore-lie-groups_FO0162 | 32 | 1.000 | A_{5} | | |
| gilmore-lie-groups_FO0163 | 32 | 1.000 | S_{5} / A_{5}=S_{2} | | |
| gilmore-lie-groups_FO0164 | 32 | 1.000 | A_{5} / I=A_{5} | | |
| gilmore-lie-groups_FO0165 | 33 | 1.000 | t_{1}^{\prime}=-3, t_{2}^{\prime}=-2, t_{3}^{\prime}= | | |
| gilmore-lie-groups_FO0166 | 33 | 0.961 | 1, t_{4}^{\prime}=4 | | |
| gilmore-lie-groups_FO0167 | 33 | 1.000 | y=y^{\prime}+\frac{1}{3} J_{1}=y^{\prime}+\frac{1}{3}(4+16+100)=y^{\prime}+40 | | |
| gilmore-lie-groups_FO0168 | 34 | 0.990 | v_{2}^{3}+v_{3}^{3}, v_{2}^{3} v_{3}^{3} | | |
| gilmore-lie-groups_FO0169 | 34 | 0.990 | J_{2}^{\prime}, J_{3}^{\prime} | | |
| gilmore-lie-groups_FO0170 | 34 | 1.000 | x=x^{\prime}+\frac{1}{2} K_{1} | | |
| gilmore-lie-groups_FO0171 | 35 | 1.000 | y_{1}, y_{2}, y_{3} | | |
| gilmore-lie-groups_FO0172 | 35 | 0.982 | v_{2}, v_{3} | | |
| gilmore-lie-groups_FO0173 | 35 | 1.000 | w_{2} w_{3} w_{4}=8 I_{3}^{\prime}=80 | | |
| gilmore-lie-groups_FO0174 | 36 | 1.000 | S_{4} / A_{4}, A_{4} / V_{4}, V_{4} | | |
| gilmore-lie-groups_FO0175 | 36 | 0.986 | V_{8} | | |
| gilmore-lie-groups_FO0176 | 36 | 0.986 | S_{4} \supset V_{8} \supset V_{4} | | |
| gilmore-lie-groups_FO0177 | 36 | 1.000 | z^{3}-7 z+6=0((z-1)(z-2)(z+3)=0) | | |
| gilmore-lie-groups_FO0178 | 36 | 1.000 | \left(x-v_{2}^{3}\right)\left(x-v_{3}^{3}\right)=x^{2}-162 x+ | | |
| gilmore-lie-groups_FO0179 | 36 | 0.996 | 9261=0 | | |
| gilmore-lie-groups_FO0180 | 36 | 0.996 | v_{2}^{3}, v_{3}^{3}=81 \pm i 30 \sqrt{3} | | |
| gilmore-lie-groups_FO0181 | 36 | 0.996 | v_{2}, v_{3}=\frac{1}{2}(3 \pm | | |
| gilmore-lie-groups_FO0182 | 36 | 1.000 | i 5 \sqrt{3} | | |
| gilmore-lie-groups_FO0183 | 36 | 0.543 | x+y, x-y, x y | | |
| gilmore-lie-groups_FO0184 | 36 | 0.543 | x / y | | |
| gilmore-lie-groups_FO0185 | 36 | 1.000 | \sqrt{x} | | |
| gilmore-lie-groups_FO0186 | 36 | 1.000 | x+i y=(x, y) | | |
| gilmore-lie-groups_FO0187 | 36 | 1.000 | K | | |
| gilmore-lie-groups_FO0188 | 36 | 1.000 | K=2^{n} | | |
| gilmore-lie-groups_FO0189 | 36 | 1.000 | \pi | | |
| gilmore-lie-groups_FO0190 | 36 | 1.000 | x^{2}-\pi=0 | | |
| gilmore-lie-groups_FO0191 | 36 | 1.000 | 1^{3}=1 | | |
| gilmore-lie-groups_FO0192 | 36 | 1.000 | x^{3}-2=0 | | |
| gilmore-lie-groups_FO0193 | 36 | 1.000 | 3 \neq 2^{n} | | |
| gilmore-lie-groups_FO0194 | 36 | 1.000 | 3 \theta | | |
| gilmore-lie-groups_FO0195 | 36 | 1.000 | \frac{1}{3}(3 \theta)=\theta | | |
| gilmore-lie-groups_FO0196 | 37 | 1.000 | \cos (3 \theta) | | |
| gilmore-lie-groups_FO0197 | 37 | 1.000 | \cos (\theta) | | |
| gilmore-lie-groups_FO0198 | 37 | 1.000 | \left(x^{2}+a x+b\right)(x+c)=0 | | |
| gilmore-lie-groups_FO0199 | 37 | 1.000 | a, b, c | | |
| gilmore-lie-groups_FO0200 | 37 | 1.000 | \cos (3 \theta)=0, c=0 | | |
| gilmore-lie-groups_FO0201 | 37 | 1.000 | a=0 | | |
| gilmore-lie-groups_FO0202 | 37 | 1.000 | b=-3 / 4 | | |
| gilmore-lie-groups_FO0203 | 37 | 1.000 | \cos (\theta)=0 | | |
| gilmore-lie-groups_FO0204 | 37 | 0.986 | \pm \sqrt{3} / 2 | | |
| gilmore-lie-groups_FO0205 | 37 | 0.986 | 3 \theta=\pi / 2(+), 3 \pi / 2(0) | | |
| gilmore-lie-groups_FO0206 | 37 | 0.986 | 5 \pi / 2(-) | | |
| gilmore-lie-groups_FO0207 | 38 | 1.000 | g_{i}, g_{j}, g_{k}, \ldots | | |
| gilmore-lie-groups_FO0208 | 38 | 1.000 | g_{i} \circ g_{j} \in G | | |
| gilmore-lie-groups_FO0209 | 38 | 1.000 | e | | |
| gilmore-lie-groups_FO0210 | 38 | 1.000 | g_{i} \in G | | |
| gilmore-lie-groups_FO0211 | 39 | 1.000 | 2 \times 2 | | |
| gilmore-lie-groups_FO0212 | 39 | 1.000 | S L(2 ; \mathbb{R}) | | |
| gilmore-lie-groups_FO0213 | 39 | 1.000 | \alpha, \beta, \gamma, \delta | | |
| gilmore-lie-groups_FO0214 | 39 | 0.797 | A | | |
| gilmore-lie-groups_FO0215 | 39 | 0.797 | B | | |
| gilmore-lie-groups_FO0216 | 39 | 0.797 | A \circ B=C | | |
| gilmore-lie-groups_FO0217 | 39 | 0.797 | \circ | | |
| gilmore-lie-groups_FO0218 | 39 | 0.922 | C | | |
| gilmore-lie-groups_FO0219 | 39 | 0.922 | \operatorname{det}(A)=+1 | | |
| gilmore-lie-groups_FO0220 | 39 | 0.922 | \operatorname{det}(B)=+1 | | |
| gilmore-lie-groups_FO0221 | 39 | 1.000 | \operatorname{det}(C)=\operatorname{det}(A) \operatorname{det}(B)=+1 | | |
| gilmore-lie-groups_FO0222 | 39 | 0.998 | (A \circ B) \circ C | | |
| gilmore-lie-groups_FO0223 | 39 | 0.998 | A \circ(B \circ C) | | |
| gilmore-lie-groups_FO0224 | 39 | 1.000 | g_{i} \rightarrow g(x) | | |
| gilmore-lie-groups_FO0225 | 39 | 1.000 | i | | |
| gilmore-lie-groups_FO0226 | 39 | 1.000 | S^{2} \subset R^{3}: x^{2}+y^{2}+z^{2}=1 | | |
| gilmore-lie-groups_FO0227 | 39 | 1.000 | R^{2} | | |
| gilmore-lie-groups_FO0228 | 39 | 1.000 | S^{2} | | |
| gilmore-lie-groups_FO0229 | 39 | 1.000 | M^{n} | | |
| gilmore-lie-groups_FO0230 | 39 | 1.000 | T | | |
| gilmore-lie-groups_FO0231 | 39 | 1.000 | U_{\alpha} | | |
| gilmore-lie-groups_FO0232 | 39 | 0.896 | T: \cup_{\alpha} U_{\alpha}=T | | |
| gilmore-lie-groups_FO0233 | 40 | 1.000 | p | | |
| gilmore-lie-groups_FO0234 | 40 | 1.000 | \phi_{\alpha} | | |
| gilmore-lie-groups_FO0235 | 40 | 1.000 | \phi_{\alpha}\left(U_{\alpha}\right)=V_{\alpha} \subset R^{n} | | |
| gilmore-lie-groups_FO0236 | 40 | 1.000 | V_{\alpha} | | |
| gilmore-lie-groups_FO0237 | 40 | 1.000 | \phi_{\alpha} \circ \phi_{\beta}^{-1}: \phi_{\beta}\left(U_{\alpha} \cap U_{\beta}\right) \rightarrow \phi_{\alpha}\left(U_{\alpha} \cap\right. | | |
| gilmore-lie-groups_FO0238 | 40 | 0.988 | U_{\beta} | | |
| gilmore-lie-groups_FO0239 | 40 | 0.988 | R^{n} | | |
| gilmore-lie-groups_FO0240 | 40 | 1.000 | \phi_{\alpha} \circ \phi_{\beta}^{-1} | | |
| gilmore-lie-groups_FO0241 | 40 | 1.000 | R^{n} \rightarrow R^{n} | | |
| gilmore-lie-groups_FO0242 | 40 | 0.697 | C^{k} | | |
| gilmore-lie-groups_FO0243 | 40 | 1.000 | x_{1} \rightarrow 0 | | |
| gilmore-lie-groups_FO0244 | 40 | 1.000 | \times S^{1} | | |
| gilmore-lie-groups_FO0245 | 41 | 0.999 | z^{2}-x^{2}-y^{2}=1 | | |
| gilmore-lie-groups_FO0247 | 41 | 0.698 | \left.U_{\alpha}\right) | | |
| gilmore-lie-groups_FO0248 | 41 | 0.698 | \cup_{\alpha}^{\text {finite }} T \subset U_{\alpha} | | |
| gilmore-lie-groups_FO0249 | 41 | 1.000 | \left(\left|x-x^{\prime}\right|^{2}=\left|x_{1}-x_{1}^{\prime}\right|^{2}+\cdots+\right. | | |
| gilmore-lie-groups_FO0250 | 41 | 1.000 | \left|x_{n}-x_{n}^{\prime}\right|^{2} | | |
| gilmore-lie-groups_FO0251 | 42 | 1.000 | \left(g(x), x \in M^{n}\right) | | |
| gilmore-lie-groups_FO0252 | 42 | 1.000 | g(x) \circ g(y)= | | |
| gilmore-lie-groups_FO0253 | 42 | 1.000 | g(z) | | |
| gilmore-lie-groups_FO0254 | 42 | 1.000 | z \in M^{n} | | |
| gilmore-lie-groups_FO0255 | 42 | 1.000 | x \in M^{n} | | |
| gilmore-lie-groups_FO0256 | 42 | 1.000 | y \in M^{n} | | |
| gilmore-lie-groups_FO0257 | 42 | 1.000 | z=\phi(x, y) | | |
| gilmore-lie-groups_FO0258 | 42 | 1.000 | g(x) \circ g(y)=g(z) | | |
| gilmore-lie-groups_FO0259 | 42 | 1.000 | y=\psi(x) | | |
| gilmore-lie-groups_FO0260 | 42 | 1.000 | g(x)^{-1}=g(y) | | |
| gilmore-lie-groups_FO0261 | 42 | 1.000 | g(\phi(x, y)) | | |
| gilmore-lie-groups_FO0262 | 42 | 1.000 | \phi | | |
| gilmore-lie-groups_FO0263 | 42 | 1.000 | x_{1} | | |
| gilmore-lie-groups_FO0264 | 42 | 1.000 | y_{1} | | |
| gilmore-lie-groups_FO0265 | 42 | 1.000 | x_{2}-x_{3} | | |
| gilmore-lie-groups_FO0266 | 42 | 1.000 | x_{1}=0 | | |
| gilmore-lie-groups_FO0267 | 42 | 1.000 | y_{2}-y_{3} | | |
| gilmore-lie-groups_FO0268 | 42 | 1.000 | y_{1}=0 | | |
| gilmore-lie-groups_FO0269 | 43 | 0.983 | \left(y_{1}, y_{2}, y_{3}\right) | | |
| gilmore-lie-groups_FO0270 | 43 | 0.983 | \left[g\left(x_{1}, x_{2}, x_{3}\right)\right]^{-1} | | |
| gilmore-lie-groups_FO0271 | 43 | 0.998 | \left(z_{1}, z_{2}, z_{3}\right)=(1,0,0) | | |
| gilmore-lie-groups_FO0272 | 43 | 0.998 | \left(x_{1}, x_{2}, x_{3}\right) | | |
| gilmore-lie-groups_FO0273 | 43 | 1.000 | [g(x)]^{-1}=g(y)=g(\psi(x)) | | |
| gilmore-lie-groups_FO0274 | 43 | 0.964 | (x, y,|z|, \theta) | | |
| gilmore-lie-groups_FO0275 | 43 | 0.882 | (-x,-y,-|z|, \theta+\pi) | | |
| gilmore-lie-groups_FO0276 | 43 | 1.000 | H^{2+} \times S^{1} | | |
| gilmore-lie-groups_FO0277 | 43 | 1.000 | H^{2+} | | |
| gilmore-lie-groups_FO0278 | 43 | 1.000 | \phi(x, y) | | |
| gilmore-lie-groups_FO0279 | 43 | 1.000 | g(x) \circ g(y)=g(z)=g(\phi(x, y)) | | |
| gilmore-lie-groups_FO0280 | 44 | 1.000 | [g(x)]^{-1}= | | |
| gilmore-lie-groups_FO0281 | 44 | 1.000 | g(y)=g(\psi(x)) | | |
| gilmore-lie-groups_FO0282 | 44 | 0.863 | \operatorname{SL}(2 ; \mathbb{R}) | | |
| gilmore-lie-groups_FO0283 | 44 | 1.000 | M | | |
| gilmore-lie-groups_FO0284 | 44 | 1.000 | S L(n ; \mathbb{R}) | | |
| gilmore-lie-groups_FO0285 | 44 | 1.000 | S | | |
| gilmore-lie-groups_FO0286 | 44 | 0.755 | O | | |
| gilmore-lie-groups_FO0287 | 44 | 0.755 | S O(n): M=S O | | |
| gilmore-lie-groups_FO0288 | 44 | 1.000 | S=\left(M M^{t}\right)^{1 / 2} | | |
| gilmore-lie-groups_FO0289 | 44 | 1.000 | O=S^{-1} M | | |
| gilmore-lie-groups_FO0290 | 44 | 1.000 | (x, y, z) \rightarrow\left(x^{\prime}, y^{\prime}, z^{\prime}\right) | | |
| gilmore-lie-groups_FO0291 | 44 | 1.000 | M_{1} | | |
| gilmore-lie-groups_FO0292 | 44 | 1.000 | M_{2} | | |
| gilmore-lie-groups_FO0293 | 44 | 0.995 | [S O(1,1)] | | |
| gilmore-lie-groups_FO0294 | 44 | 0.972 | (c t)^{2}-x^{2} | | |
| gilmore-lie-groups_FO0295 | 44 | 0.972 | S O(2) | | |
| gilmore-lie-groups_FO0296 | 44 | 0.999 | x^{2}+y^{2} | | |
| gilmore-lie-groups_FO0297 | 45 | 1.000 | (c t)^{2}-x^{2}-y^{2}-z^{2} | | |
| gilmore-lie-groups_FO0298 | 45 | 1.000 | O(3,1) | | |
| gilmore-lie-groups_FO0299 | 45 | 0.535 | S L(2 ; \mathbb{C}) | | |
| gilmore-lie-groups_FO0300 | 45 | 0.788 | \left[\begin{array}{ll}\alpha & \beta \\ \gamma & \delta\end{array}\right] | | |
| gilmore-lie-groups_FO0301 | 45 | 0.987 | \alpha \delta-\beta \gamma=1 | | |
| gilmore-lie-groups_FO0302 | 45 | 0.987 | X | | |
| gilmore-lie-groups_FO0303 | 45 | 1.000 | \mathbf{x} | | |
| gilmore-lie-groups_FO0304 | 45 | 1.000 | \mathbf{x}=(x, y, z) | | |
| gilmore-lie-groups_FO0305 | 45 | 1.000 | \sigma=\left(\sigma_{1}, \sigma_{2}, \sigma_{3}\right)=\left(\sigma_{x}, \sigma_{y}, \sigma_{z}\right) | | |
| gilmore-lie-groups_FO0306 | 45 | 1.000 | X^{\dagger} \equiv\left(X^{t}\right)^{*}=X | | |
| gilmore-lie-groups_FO0307 | 45 | 1.000 | g \in S L(2 ; \mathbb{C}) | | |
| gilmore-lie-groups_FO0308 | 45 | 1.000 | g^{\dagger} X g=X^{\prime}=H\left(x^{\prime}, y^{\prime}, z^{\prime}, c t^{\prime}\right) | | |
| gilmore-lie-groups_FO0309 | 45 | 0.964 | \left(x^{\prime}, y^{\prime}, z^{\prime}, c t^{\prime}\right) | | |
| gilmore-lie-groups_FO0310 | 45 | 0.918 | (x, y, z, c t) | | |
| gilmore-lie-groups_FO0311 | 45 | 1.000 | g | | |
| gilmore-lie-groups_FO0312 | 45 | 1.000 | \alpha^{*}, \beta^{*}, \gamma^{*}, \delta^{*} | | |
| gilmore-lie-groups_FO0313 | 45 | 1.000 | g^{\dagger} | | |
| gilmore-lie-groups_FO0314 | 45 | 0.990 | t^{\prime}=t | | |
| gilmore-lie-groups_FO0315 | 45 | 0.990 | S U(2) \subset S L(2 ; \mathbb{C}) | | |
| gilmore-lie-groups_FO0316 | 45 | 0.952 | g=k h | | |
| gilmore-lie-groups_FO0317 | 45 | 0.952 | h \in S U(2), h^{\dagger}=h^{-1}, h | | |
| gilmore-lie-groups_FO0318 | 45 | 0.461 | h=\operatorname{EXP}\left(\frac{i}{2} \sigma \cdot \theta\right) | | |
| gilmore-lie-groups_FO0319 | 45 | 0.461 | k \in \operatorname{SL}(2 ; \mathbb{C}) / \operatorname{SU}(2), k^{\dagger}=k^{+1}, k | | |
| gilmore-lie-groups_FO0320 | 45 | 1.000 | k=\operatorname{EXP}\left(\frac{1}{2} \sigma \cdot \mathbf{b}\right) | | |
| gilmore-lie-groups_FO0321 | 45 | 1.000 | \mathbf{b} | | |
| gilmore-lie-groups_FO0322 | 45 | 1.000 | \theta | | |
| gilmore-lie-groups_FO0323 | 45 | 0.899 | k^{\dagger} H(x, y, z, c t) k=H\left(x^{\prime}, y^{\prime}, z^{\prime}, c t^{\prime}\right) | | |
| gilmore-lie-groups_FO0324 | 45 | 0.893 | \mathbf{b}=(0,0, b) | | |
| gilmore-lie-groups_FO0325 | 45 | 0.458 | k\left(b^{\prime}\right) | | |
| gilmore-lie-groups_FO0326 | 45 | 0.458 | k(b) | | |
| gilmore-lie-groups_FO0327 | 45 | 0.458 | (a) k\left(b^{\prime}+b\right) | | |
| gilmore-lie-groups_FO0328 | 45 | 0.981 | \mathbf{b}^{\prime} | | |
| gilmore-lie-groups_FO0329 | 45 | 0.981 | k\left(\mathbf{b}^{\prime}\right) k(\mathbf{b})=k\left(\mathbf{b}^{\prime \prime}\right) h(\theta) | | |
| gilmore-lie-groups_FO0330 | 45 | 0.981 | \mathbf{b}^{\prime \prime}, \theta | | |
| gilmore-lie-groups_FO0331 | 45 | 1.000 | \theta \rightarrow \theta^{\prime}=\theta+k+f(\theta) | | |
| gilmore-lie-groups_FO0332 | 45 | 1.000 | 0 \leq k<2 \pi | | |
| gilmore-lie-groups_FO0333 | 45 | 1.000 | f(\theta) | | |
| gilmore-lie-groups_FO0334 | 45 | 0.991 | f(\theta+2 \pi)=f(\theta) | | |
| gilmore-lie-groups_FO0335 | 45 | 0.991 | 1: 1 | | |
| gilmore-lie-groups_FO0336 | 45 | 1.000 | f(\theta): d f(\theta) / d \theta>-1 | | |
| gilmore-lie-groups_FO0337 | 46 | 0.769 | (a, b, c, d) | | |
| gilmore-lie-groups_FO0338 | 46 | 1.000 | R P^{1} | | |
| gilmore-lie-groups_FO0339 | 46 | 1.000 | (\lambda a, \lambda b, \lambda c, \lambda d)=\lambda(a, b, c, d)(\lambda \neq 0) | | |
| gilmore-lie-groups_FO0340 | 46 | 1.000 | A, B, C, D | | |
| gilmore-lie-groups_FO0341 | 46 | 0.668 | (\lambda, 0,0, \lambda) | | |
| gilmore-lie-groups_FO0342 | 46 | 0.999 | x^{\prime} \rightarrow x | | |
| gilmore-lie-groups_FO0343 | 46 | 0.999 | \lambda(d,-b,-c, a) | | |
| gilmore-lie-groups_FO0344 | 46 | 1.000 | \lambda \neq 0 | | |
| gilmore-lie-groups_FO0345 | 46 | 1.000 | D=a d-b c \neq 0 | | |
| gilmore-lie-groups_FO0346 | 46 | 1.000 | x^{\prime}=(a, b, c, d) x=\lambda(a, b, c, d) x | | |
| gilmore-lie-groups_FO0347 | 46 | 1.000 | a, b, c, d | | |
| gilmore-lie-groups_FO0348 | 46 | 1.000 | D=a d-b c=1 | | |
| gilmore-lie-groups_FO0349 | 46 | 0.980 | (y, z) | | |
| gilmore-lie-groups_FO0350 | 46 | 1.000 | x=y / z | | |
| gilmore-lie-groups_FO0351 | 46 | 0.477 | x_{1}, x_{2}, x_{3} | | |
| gilmore-lie-groups_FO0352 | 46 | 0.477 | \infty | | |
| gilmore-lie-groups_FO0353 | 46 | 0.996 | D=1 | | |
| gilmore-lie-groups_FO0354 | 46 | 1.000 | \left(x_{1}^{\prime}, x_{2}^{\prime}, x_{3}^{\prime}\right) | | |
| gilmore-lie-groups_FO0355 | 46 | 1.000 | R P^{n} | | |
| gilmore-lie-groups_FO0356 | 46 | 0.987 | R^{n+1} | | |
| gilmore-lie-groups_FO0357 | 46 | 0.987 | S L(n+1 ; \mathbb{R}) \operatorname{maps} x=\left(x_{1}, x_{2}, \ldots, x_{n+1}\right) \in R^{n+1} | | |
| gilmore-lie-groups_FO0358 | 46 | 0.987 | x^{\prime} \in R^{n+1} | | |
| gilmore-lie-groups_FO0359 | 46 | 1.000 | x^{\prime} \neq 0 \leftrightarrow x \neq 0 | | |
| gilmore-lie-groups_FO0360 | 46 | 1.000 | x^{\prime}=0 \leftrightarrow x=0 | | |
| gilmore-lie-groups_FO0361 | 46 | 1.000 | x \neq 0 | | |
| gilmore-lie-groups_FO0362 | 46 | 1.000 | y \neq 0 | | |
| gilmore-lie-groups_FO0363 | 46 | 1.000 | y=\lambda x | | |
| gilmore-lie-groups_FO0364 | 46 | 1.000 | y | | |
| gilmore-lie-groups_FO0365 | 46 | 1.000 | R^{n+1}: y \in S^{n} \subset R^{n+1} | | |
| gilmore-lie-groups_FO0366 | 46 | 1.000 | \lambda | | |
| gilmore-lie-groups_FO0367 | 46 | 1.000 | \lambda= \pm 1 /\left(\sum_{i=1}^{n+1} x_{i}^{2}\right)^{1 / 2} | | |
| gilmore-lie-groups_FO0368 | 46 | 1.000 | R^{3} | | |
| gilmore-lie-groups_FO0369 | 46 | 0.983 | (x, y, z) | | |
| gilmore-lie-groups_FO0370 | 46 | 0.983 | (X, Y)= | | |
| gilmore-lie-groups_FO0371 | 46 | 1.000 | (x / z, y / z) | | |
| gilmore-lie-groups_FO0372 | 46 | 1.000 | z \neq 0 | | |
| gilmore-lie-groups_FO0373 | 46 | 1.000 | x \rightarrow x^{\prime}=M x, M \in S L(3 ; \mathbb{R}) | | |
| gilmore-lie-groups_FO0374 | 47 | 1.000 | x^{\prime} | | |
| gilmore-lie-groups_FO0375 | 47 | 1.000 | R P^{n} \rightarrow R P^{n} | | |
| gilmore-lie-groups_FO0376 | 47 | 0.989 | S L(2 ; \mathbb{R}) / S O(2) \simeq\left[\begin{array}{cc}z+x & y \\ y & z-x\end{array}\right] | | |
| gilmore-lie-groups_FO0377 | 47 | 1.000 | x=r \cos \phi, y=r \sin \phi | | |
| gilmore-lie-groups_FO0378 | 47 | 0.633 | S O(3) / S O(2) | | |
| gilmore-lie-groups_FO0379 | 47 | 0.633 | S^{2} \subset R^{3} | | |
| gilmore-lie-groups_FO0380 | 47 | 1.000 | z^{2}+\left(x^{2}+y^{2}\right)=1 | | |
| gilmore-lie-groups_FO0381 | 47 | 1.000 | d s^{2}=d z^{2}+\left(d x^{2}+d y^{2}\right) | | |
| gilmore-lie-groups_FO0382 | 47 | 1.000 | H^{2} | | |
| gilmore-lie-groups_FO0383 | 47 | 1.000 | 1+r^{2} \rightarrow 1-r^{2} | | |
| gilmore-lie-groups_FO0384 | 47 | 1.000 | 0 \leq r \leq 1,0 \leq \phi \leq 2 \pi | | |
| gilmore-lie-groups_FO0385 | 47 | 1.000 | r=0 | | |
| gilmore-lie-groups_FO0386 | 47 | 1.000 | r=1 | | |
| gilmore-lie-groups_FO0387 | 47 | 1.000 | \phi=0 | | |
| gilmore-lie-groups_FO0388 | 47 | 1.000 | s=\int_{0}^{1} d r / \sqrt{1-r^{2}}=\pi / 2 | | |
| gilmore-lie-groups_FO0389 | 47 | 1.000 | V=\int_{r=0}^{r=1} \int_{\phi=0}^{\phi=2 \pi} d V(r, \phi)=\int_{0}^{1} r d r / \sqrt{1-r^{2}} \int_{0}^{2 \pi} d \phi= | | |
| gilmore-lie-groups_FO0390 | 47 | 1.000 | 2 \pi | | |
| gilmore-lie-groups_FO0391 | 48 | 0.969 | G L(n ; \mathbb{F}) | | |
| gilmore-lie-groups_FO0392 | 48 | 1.000 | \mathbb{F} | | |
| gilmore-lie-groups_FO0393 | 48 | 1.000 | H_{1} \subset G | | |
| gilmore-lie-groups_FO0394 | 48 | 1.000 | H_{12}=H_{1} \cap H_{2} | | |
| gilmore-lie-groups_FO0395 | 48 | 1.000 | H_{1} \cap H_{2} | | |
| gilmore-lie-groups_FO0396 | 48 | 0.950 | H_{1} | | |
| gilmore-lie-groups_FO0397 | 48 | 0.817 | H_{2} | | |
| gilmore-lie-groups_FO0398 | 49 | 1.000 | \pm I_{2} | | |
| gilmore-lie-groups_FO0399 | 49 | 0.950 | (\mathbb{F}=\mathbb{R}) | | |
| gilmore-lie-groups_FO0400 | 49 | 0.950 | \mathbb{F}=\mathbb{C} | | |
| gilmore-lie-groups_FO0401 | 49 | 0.950 | \mathbb{F}=\mathbb{Q} | | |
| gilmore-lie-groups_FO0402 | 49 | 1.000 | 1, \mathcal{I}, J, K | | |
| gilmore-lie-groups_FO0403 | 49 | 0.900 | [\operatorname{det}(\mathrm{M})=+1] | | |
| gilmore-lie-groups_FO0404 | 49 | 0.999 | G L(1 ; \mathbb{Q}) | | |
| gilmore-lie-groups_FO0405 | 49 | 0.999 | 1 \times 1 | | |
| gilmore-lie-groups_FO0406 | 50 | 1.000 | A_{i}^{j} | | |
| gilmore-lie-groups_FO0407 | 50 | 1.000 | \epsilon^{i_{1} i_{2} \cdots i_{n}} | | |
| gilmore-lie-groups_FO0408 | 50 | 0.719 | 1,2, \ldots, n ;-1 | | |
| gilmore-lie-groups_FO0409 | 50 | 1.000 | i_{*} | | |
| gilmore-lie-groups_FO0410 | 50 | 1.000 | U T(p, q) | | |
| gilmore-lie-groups_FO0411 | 50 | 1.000 | n \times n(n=p+q) | | |
| gilmore-lie-groups_FO0412 | 50 | 1.000 | U T(1,1) | | |
| gilmore-lie-groups_FO0413 | 50 | 0.991 | y=0 | | |
| gilmore-lie-groups_FO0414 | 50 | 0.991 | y=0 \rightarrow y^{\prime}=0 | | |
| gilmore-lie-groups_FO0415 | 50 | 1.000 | x=0 | | |
| gilmore-lie-groups_FO0416 | 50 | 1.000 | V_{p} \oplus V_{q} | | |
| gilmore-lie-groups_FO0417 | 50 | 1.000 | V_{q} | | |
| gilmore-lie-groups_FO0418 | 50 | 1.000 | V_{p} | | |
| gilmore-lie-groups_FO0419 | 50 | 1.000 | q | | |
| gilmore-lie-groups_FO0420 | 51 | 1.000 | H T(p, q) | | |
| gilmore-lie-groups_FO0421 | 51 | 1.000 | H T(1,1)\left(m_{22}=1\right) | | |
| gilmore-lie-groups_FO0422 | 51 | 1.000 | x \rightarrow | | |
| gilmore-lie-groups_FO0423 | 51 | 0.997 | x^{\prime}=a x+b | | |
| gilmore-lie-groups_FO0424 | 51 | 0.999 | U T(p, q, r) | | |
| gilmore-lie-groups_FO0425 | 51 | 1.000 | U T(p, q+r) \cap U T(p+q, r) | | |
| gilmore-lie-groups_FO0426 | 52 | 0.460 | S U(1,1) | | |
| gilmore-lie-groups_FO0427 | 52 | 1.000 | \operatorname{Sol}(n)=U T(1,1,1, \ldots, 1) | | |
| gilmore-lie-groups_FO0428 | 52 | 0.971 | U T(1,1,1) | | |
| gilmore-lie-groups_FO0429 | 52 | 1.000 | \hat{n}=a^{\dagger} a | | |
| gilmore-lie-groups_FO0430 | 52 | 1.000 | a^{\dagger} | | |
| gilmore-lie-groups_FO0431 | 52 | 1.000 | a | | |
| gilmore-lie-groups_FO0432 | 52 | 0.994 | I=a a^{\dagger}-a^{\dagger} a=\left[a, a^{\dagger}\right] | | |
| gilmore-lie-groups_FO0433 | 52 | 1.000 | \operatorname{Nil}(n) | | |
| gilmore-lie-groups_FO0434 | 52 | 1.000 | \operatorname{Sol}(n) | | |
| gilmore-lie-groups_FO0435 | 52 | 0.998 | \operatorname{Nil}(3) | | |
| gilmore-lie-groups_FO0436 | 52 | 0.976 | a^{\dagger}, a, I | | |
| gilmore-lie-groups_FO0437 | 52 | 0.716 | (p | | |
| gilmore-lie-groups_FO0438 | 52 | 0.716 | q) | | |
| gilmore-lie-groups_FO0439 | 52 | 0.716 | [p, q]=\hbar / i | | |
| gilmore-lie-groups_FO0440 | 52 | 1.000 | \langle p \mid q\rangle=\frac{1}{\sqrt{2}} e^{2 \pi i p q / h} | | |
| gilmore-lie-groups_FO0441 | 52 | 0.995 | A(p, q) | | |
| gilmore-lie-groups_FO0442 | 52 | 0.998 | (p, q) | | |
| gilmore-lie-groups_FO0443 | 53 | 1.000 | A B=B A | | |
| gilmore-lie-groups_FO0444 | 53 | 1.000 | A(1,1) | | |
| gilmore-lie-groups_FO0445 | 53 | 0.798 | x \rightarrow x^{\prime}=x+a | | |
| gilmore-lie-groups_FO0446 | 53 | 1.000 | M^{\dagger} G M=G | | |
| gilmore-lie-groups_FO0447 | 53 | 0.679 | G=I_{n} | | |
| gilmore-lie-groups_FO0448 | 53 | 0.573 | G=I_{p, q} | | |
| gilmore-lie-groups_FO0449 | 53 | 1.000 | I_{n} | | |
| gilmore-lie-groups_FO0450 | 54 | 0.936 | G L(n ; \mathbb{F}), \mathbb{F}=\mathbb{R}, \mathbb{C}, \mathbb{Q} | | |
| gilmore-lie-groups_FO0451 | 54 | 1.000 | C^{2} | | |
| gilmore-lie-groups_FO0452 | 54 | 1.000 | \mathbb{Q} | | |
| gilmore-lie-groups_FO0453 | 54 | 0.808 | S U(1 ; \mathbb{Q}) | | |
| gilmore-lie-groups_FO0454 | 54 | 1.000 | R^{4} | | |
| gilmore-lie-groups_FO0455 | 54 | 0.992 | G=I_{p, q}, p+q=n | | |
| gilmore-lie-groups_FO0456 | 54 | 1.000 | p \neq 0, q \neq 0 | | |
| gilmore-lie-groups_FO0457 | 54 | 1.000 | x^{2}+y^{2}+z^{2}-(c t)^{2} | | |
| gilmore-lie-groups_FO0458 | 55 | 0.999 | N \times N | | |
| gilmore-lie-groups_FO0459 | 55 | 1.000 | \operatorname{det}(G)=\operatorname{det}\left(G^{t}\right)=\operatorname{det}(-G)=(-)^{N} \operatorname{det}(G), N | | |
| gilmore-lie-groups_FO0460 | 55 | 1.000 | N=2 n | | |
| gilmore-lie-groups_FO0461 | 55 | 1.000 | i \sigma_{y}=\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right] | | |
| gilmore-lie-groups_FO0462 | 55 | 0.990 | \operatorname{Sp}(2 n ; \mathbb{R}) | | |
| gilmore-lie-groups_FO0463 | 55 | 0.935 | \operatorname{Sp}(2 ; \mathbb{R}) \subset G L(2 ; \mathbb{R}) | | |
| gilmore-lie-groups_FO0464 | 55 | 0.510 | a d-b c=+1 | | |
| gilmore-lie-groups_FO0465 | 55 | 0.510 | \operatorname{Sp}(2 ; \mathbb{R})=S L(2 ; \mathbb{R}) | | |
| gilmore-lie-groups_FO0466 | 55 | 1.000 | M G M^{t}=G | | |
| gilmore-lie-groups_FO0467 | 56 | 0.999 | E(3) | | |
| gilmore-lie-groups_FO0468 | 56 | 0.607 | \mathbf{t} | | |
| gilmore-lie-groups_FO0469 | 56 | 0.792 | A \in S O(3,1), A I_{3,1} A^{t}=I_{3,1} | | |
| gilmore-lie-groups_FO0470 | 56 | 0.608 | \mathbf{v} | | |
| gilmore-lie-groups_FO0471 | 56 | 0.608 | (\mathbf{t}) | | |
| gilmore-lie-groups_FO0472 | 56 | 0.608 | \left(t_{4}\right) | | |
| gilmore-lie-groups_FO0473 | 57 | 0.991 | \left[\begin{array}{cc}a & b \\ c & d\end{array}\right] \in S L(2 ; \mathbb{R}) \subset G L(2 ; \mathbb{R}) | | |
| gilmore-lie-groups_FO0474 | 57 | 0.998 | U(1,1) | | |
| gilmore-lie-groups_FO0475 | 57 | 1.000 | a^{*} a-b^{*} b=+1 | | |
| gilmore-lie-groups_FO0476 | 57 | 1.000 | U(n) | | |
| gilmore-lie-groups_FO0477 | 57 | 1.000 | U^{\dagger} U=I_{n} | | |
| gilmore-lie-groups_FO0478 | 57 | 1.000 | 2 n \times 2 n | | |
| gilmore-lie-groups_FO0479 | 57 | 1.000 | M^{t} M=I_{2 n} | | |
| gilmore-lie-groups_FO0480 | 57 | 1.000 | { }^{\dagger} | | |
| gilmore-lie-groups_FO0481 | 57 | 1.000 | { }^{t} | | |
| gilmore-lie-groups_FO0482 | 57 | 1.000 | I_{2 n} | | |
| gilmore-lie-groups_FO0483 | 57 | 0.712 | S O(2 n) | | |
| gilmore-lie-groups_FO0484 | 57 | 1.000 | O U(2 n) | | |
| gilmore-lie-groups_FO0485 | 58 | 0.999 | U(n ; \mathbb{Q})=S p(n) | | |
| gilmore-lie-groups_FO0486 | 58 | 1.000 | M^{\dagger} M=I_{2 n} | | |
| gilmore-lie-groups_FO0487 | 58 | 1.000 | \mathbb{C} | | |
| gilmore-lie-groups_FO0488 | 58 | 0.945 | \operatorname{SU}(2 n) | | |
| gilmore-lie-groups_FO0489 | 58 | 1.000 | U \operatorname{Sp}(2 n) | | |
| gilmore-lie-groups_FO0490 | 58 | 1.000 | \mathbb{R} | | |
| gilmore-lie-groups_FO0491 | 58 | 1.000 | G L(n ; \mathbb{Z}) | | |
| gilmore-lie-groups_FO0492 | 58 | 1.000 | m \in G L(n ; \mathbb{Z}), \operatorname{det}(m)= \pm 1 | | |
| gilmore-lie-groups_FO0493 | 58 | 1.000 | S L(n ; \mathbb{Z}) | | |
| gilmore-lie-groups_FO0494 | 58 | 1.000 | m \in S L(n ; \mathbb{Z}), \operatorname{det}(m)=+1 | | |
| gilmore-lie-groups_FO0495 | 58 | 1.000 | P S L(n ; \mathbb{Z}), n | | |
| gilmore-lie-groups_FO0496 | 58 | 1.000 | P S L(n ; \mathbb{Z})=S L(n ; \mathbb{Z}) /\left\{I_{n},-I_{n}\right\} | | |
| gilmore-lie-groups_FO0497 | 58 | 0.999 | n=2 | | |
| gilmore-lie-groups_FO0498 | 58 | 0.999 | \left[\begin{array}{ll}a & b \\ c & d\end{array}\right] | | |
| gilmore-lie-groups_FO0499 | 58 | 1.000 | \operatorname{det}(m)=n | | |
| gilmore-lie-groups_FO0500 | 58 | 1.000 | \operatorname{det}\left(m^{-1}\right)=1 / n | | |
| gilmore-lie-groups_FO0501 | 58 | 1.000 | \operatorname{det}(m)= | | |
| gilmore-lie-groups_FO0502 | 58 | 0.714 | \pm 1 | | |
| gilmore-lie-groups_FO0503 | 58 | 0.714 | G L(2 ; \mathbb{Z}) | | |
| gilmore-lie-groups_FO0504 | 58 | 0.714 | S L(2 ; \mathbb{Z}) \subset | | |
| gilmore-lie-groups_FO0505 | 58 | 1.000 | P S L(2 ; \mathbb{Z}) | | |
| gilmore-lie-groups_FO0506 | 58 | 0.906 | S L(2 ; \mathbb{Z}) | | |
| gilmore-lie-groups_FO0507 | 58 | 0.906 | \left[\begin{array}{ll}-a & -b \\ -c & -d\end{array}\right] \simeq\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] | | |
| gilmore-lie-groups_FO0508 | 59 | 1.000 | F(n) | | |
| gilmore-lie-groups_FO0509 | 59 | 1.000 | n=1(F(0)=0, F(1)=1) | | |
| gilmore-lie-groups_FO0510 | 59 | 0.998 | G L(2 ; \mathbb{Z}), G L(3 ; \mathbb{Z}) | | |
| gilmore-lie-groups_FO0511 | 59 | 0.998 | G L(n ; \mathbb{R}) | | |
| gilmore-lie-groups_FO0512 | 59 | 1.000 | M^{t} I_{n} M=I_{n} | | |
| gilmore-lie-groups_FO0513 | 59 | 1.000 | O(n ; \mathbb{Z}) | | |
| gilmore-lie-groups_FO0514 | 59 | 1.000 | \operatorname{det}(m)=+1 | | |
| gilmore-lie-groups_FO0515 | 60 | 1.000 | A_{n} | | |
| gilmore-lie-groups_FO0516 | 60 | 0.932 | O(2 ; \mathbb{Z}) | | |
| gilmore-lie-groups_FO0517 | 60 | 0.932 | 8=2^{2} \times 2! | | |
| gilmore-lie-groups_FO0518 | 60 | 0.974 | O(3 ; \mathbb{Z}) | | |
| gilmore-lie-groups_FO0519 | 60 | 0.974 | 2^{3} \times 3!=48 | | |
| gilmore-lie-groups_FO0520 | 60 | 0.974 | 6=3! | | |
| gilmore-lie-groups_FO0521 | 60 | 1.000 | A_{3} \subset S_{3} \subset O(3 ; \mathbb{Z}) | | |
| gilmore-lie-groups_FO0522 | 60 | 0.997 | G_{2}, F_{4}, E_{6}, E_{7}, E_{8} | | |
| gilmore-lie-groups_FO0523 | 60 | 1.000 | G L(n ; \mathbb{Z}), S L(n ; \mathbb{Z}) | | |
| gilmore-lie-groups_FO0524 | 60 | 1.000 | P S L(n ; \mathbb{Z}) | | |
| gilmore-lie-groups_FO0525 | 61 | 1.000 | \mathcal{I} J=-\mathcal{K} | | |
| gilmore-lie-groups_FO0526 | 61 | 0.997 | \sigma_{x}, \sigma_{y}, \sigma_{z} | | |
| gilmore-lie-groups_FO0527 | 61 | 0.924 | \{\mathcal{I}, \mathcal{J}\}=\mathcal{I} \mathcal{J}+\mathcal{J} \mathcal{I}=0 | | |
| gilmore-lie-groups_FO0528 | 61 | 1.000 | \mathcal{I}, J, K | | |
| gilmore-lie-groups_FO0529 | 61 | 0.464 | S U(1 ; \mathbb{Q}) \sim S U(2 ; \mathbb{C}) | | |
| gilmore-lie-groups_FO0530 | 61 | 1.000 | p+q=n | | |
| gilmore-lie-groups_FO0531 | 61 | 0.996 | S L_{i}(n ; \mathbb{C}) | | |
| gilmore-lie-groups_FO0532 | 61 | 0.996 | G L(n ; \mathbb{C}) | | |
| gilmore-lie-groups_FO0533 | 61 | 1.000 | \phi, \lambda, r | | |
| gilmore-lie-groups_FO0534 | 61 | 1.000 | r \neq 0 | | |
| gilmore-lie-groups_FO0535 | 61 | 1.000 | 2 n^{2}-1 | | |
| gilmore-lie-groups_FO0536 | 61 | 1.000 | S L_{3}(n ; \mathbb{C}) | | |
| gilmore-lie-groups_FO0537 | 61 | 1.000 | S L(n ; \mathbb{C})=S L_{1}(n ; \mathbb{C}) \cap S L_{2}(n ; \mathbb{C}) | | |
| gilmore-lie-groups_FO0538 | 61 | 1.000 | \mathbb{C} \rightarrow \mathbb{R} | | |
| gilmore-lie-groups_FO0539 | 61 | 1.000 | \mathbb{C} \rightarrow \mathbb{Q} | | |
| gilmore-lie-groups_FO0540 | 61 | 0.936 | \left[\begin{array}{cc}-1 & a \\ 0 & 1\end{array}\right], a \in R | | |
| gilmore-lie-groups_FO0541 | 61 | 1.000 | R^{1} | | |
| gilmore-lie-groups_FO0542 | 61 | 1.000 | \left[\begin{array}{cc}1 & a \\ 0 & -1\end{array}\right] | | |
| gilmore-lie-groups_FO0547 | 84 | 1.000 | n(n-1) / 2 | | |
| gilmore-lie-groups_FO0549 | 62 | 1.000 | \mathbf{F}=d \mathbf{p} / d t | | |
| gilmore-lie-groups_FO0551 | 62 | 0.814 | S U(2) | | |
| gilmore-lie-groups_FO0552 | 62 | 1.000 | c_{1}=a_{1}+i b_{1} | | |
| gilmore-lie-groups_FO0553 | 62 | 1.000 | c_{2}=a_{2}+i b_{2} | | |
| gilmore-lie-groups_FO0554 | 62 | 1.000 | a_{1}^{2}+b_{1}^{2}+a_{2}^{2}+b_{2}^{2}=1 | | |
| gilmore-lie-groups_FO0555 | 62 | 0.991 | S^{3} \subset R^{4} | | |
| gilmore-lie-groups_FO0556 | 62 | 1.000 | a_{1}^{2}+b_{1}^{2}-a_{2}^{2}-b_{2}^{2}=1 | | |
| gilmore-lie-groups_FO0557 | 62 | 1.000 | \left[\begin{array}{cc}m_{11} & x \\ m_{21} & m_{22}\end{array}\right] | | |
| gilmore-lie-groups_FO0558 | 62 | 1.000 | m_{11}^{2}+x^{2}=1 | | |
| gilmore-lie-groups_FO0559 | 62 | 1.000 | m_{11}= \pm \sqrt{1-x^{2}} | | |
| gilmore-lie-groups_FO0560 | 63 | 1.000 | m_{21} m_{11}+m_{22} x=0 | | |
| gilmore-lie-groups_FO0561 | 63 | 0.994 | m_{i j} i \geq j | | |
| gilmore-lie-groups_FO0562 | 63 | 0.495 | Z_{2} | | |
| gilmore-lie-groups_FO0563 | 63 | 0.495 | Z_{1} | | |
| gilmore-lie-groups_FO0564 | 63 | 0.495 | (x, y) | | |
| gilmore-lie-groups_FO0565 | 63 | 0.854 | M \in G L(n ; \mathbb{Z}) | | |
| gilmore-lie-groups_FO0566 | 63 | 0.854 | \operatorname{det}(M) | | |
| gilmore-lie-groups_FO0567 | 63 | 1.000 | O(n ; \mathbb{Z}) \supset S_{n} \supset A_{n} | | |
| gilmore-lie-groups_FO0568 | 63 | 1.000 | 2^{n} \times n!, n!, \frac{1}{2} n! | | |
| gilmore-lie-groups_FO0569 | 63 | 1.000 | \lambda_{ \pm}=\frac{1}{2}(1 \pm \sqrt{5}) | | |
| gilmore-lie-groups_FO0570 | 63 | 0.999 | F(0)=0, F(1)=1 | | |
| gilmore-lie-groups_FO0571 | 63 | 1.000 | \left[\begin{array}{cc}2 & -1 \\ -1 & 1\end{array}\right] | | |
| gilmore-lie-groups_FO0572 | 63 | 1.000 | |n l m\rangle | | |
| gilmore-lie-groups_FO0573 | 63 | 1.000 | n^{2} | | |
| gilmore-lie-groups_FO0574 | 63 | 1.000 | E(n l m)=-E_{0} / n^{2}\left(E_{0}=13.6 \mathrm{eV}\right) | | |
| gilmore-lie-groups_FO0575 | 63 | 1.000 | S O(4) \downarrow S O(3) | | |
| gilmore-lie-groups_FO0576 | 63 | 0.993 | n(n=1,2,3, \ldots) | | |
| gilmore-lie-groups_FO0577 | 63 | 0.980 | l, l=0,1,2 \ldots, n-1 | | |
| gilmore-lie-groups_FO0578 | 63 | 0.980 | \sum_{l=0}^{l=n-1}(2 l+1)=n^{2} | | |
| gilmore-lie-groups_FO0579 | 63 | 1.000 | l | | |
| gilmore-lie-groups_FO0580 | 63 | 1.000 | 2 l+1 | | |
| gilmore-lie-groups_FO0581 | 63 | 1.000 | +Z e | | |
| gilmore-lie-groups_FO0582 | 64 | 1.000 | \delta=0.28 | | |
| gilmore-lie-groups_FO0583 | 64 | 0.608 | (n, l) \rightarrow 1 s | | |
| gilmore-lie-groups_FO0584 | 64 | 0.608 | 2 s | | |
| gilmore-lie-groups_FO0585 | 64 | 0.608 | 2 p | | |
| gilmore-lie-groups_FO0586 | 64 | 0.608 | 3 s | | |
| gilmore-lie-groups_FO0587 | 64 | 0.608 | 3 p | | |
| gilmore-lie-groups_FO0588 | 64 | 0.608 | 4 s | | |
| gilmore-lie-groups_FO0589 | 64 | 0.608 | 3 d | | |
| gilmore-lie-groups_FO0590 | 64 | 0.608 | 4 p | | |
| gilmore-lie-groups_FO0591 | 64 | 0.957 | 5 s, 4 d, 5 p ; 6 s, 4 f, 5 d, 6 p ; 7 s, 5 f, 6 d, 7 p ; 8 s, 6 f, 7 d, 8 p ; \ldots | | |
| gilmore-lie-groups_FO0592 | 64 | 1.000 | G_{2} | | |
| gilmore-lie-groups_FO0593 | 64 | 1.000 | 2 n | | |
| gilmore-lie-groups_FO0594 | 64 | 1.000 | M^{t} G_{i} M=G_{i} | | |
| gilmore-lie-groups_FO0595 | 64 | 1.000 | y^{j}=x^{i} M_{i}{ }^{j} | | |
| gilmore-lie-groups_FO0596 | 64 | 0.990 | f(q, p) | | |
| gilmore-lie-groups_FO0597 | 64 | 0.990 | g(q, p) | | |
| gilmore-lie-groups_FO0598 | 65 | 0.666 | (Q, P) | | |
| gilmore-lie-groups_FO0599 | 65 | 1.000 | A^{t} C | | |
| gilmore-lie-groups_FO0600 | 65 | 1.000 | B^{t} D | | |
| gilmore-lie-groups_FO0601 | 65 | 1.000 | A^{t} D-B^{t} C=I_{n} | | |
| gilmore-lie-groups_FO0602 | 65 | 0.999 | \left[q_{j}, q_{k}\right]=\left[p_{j}, p_{k}\right]=0 | | |
| gilmore-lie-groups_FO0603 | 65 | 0.999 | \left[q_{j}, p_{k}\right]=i \hbar \delta_{j k} | | |
| gilmore-lie-groups_FO0604 | 65 | 1.000 | e^{+i k x} | | |
| gilmore-lie-groups_FO0605 | 65 | 0.644 | m | | |
| gilmore-lie-groups_FO0606 | 65 | 0.644 | (+) | | |
| gilmore-lie-groups_FO0607 | 65 | 0.644 | \hbar k | | |
| gilmore-lie-groups_FO0608 | 65 | 1.000 | E=(\hbar k)^{2} / 2 m | | |
| gilmore-lie-groups_FO0609 | 65 | 1.000 | A_{L} | | |
| gilmore-lie-groups_FO0610 | 65 | 1.000 | \langle\hat{p}\rangle=\left(\left|A_{L}\right|^{2}-\left|B_{L}\right|^{2}\right) \hbar k | | |
| gilmore-lie-groups_FO0611 | 65 | 1.000 | \hat{p}=\frac{\hbar}{i} \frac{d}{d x} | | |
| gilmore-lie-groups_FO0615 | 66 | 1.000 | A_{L}, A_{R} | | |
| gilmore-lie-groups_FO0616 | 66 | 1.000 | B_{L}, B_{R} | | |
| gilmore-lie-groups_FO0617 | 66 | 1.000 | E | | |
| gilmore-lie-groups_FO0618 | 66 | 1.000 | T(E) \in U(1,1) | | |
| gilmore-lie-groups_FO0619 | 66 | 1.000 | T \in S U(1,1) | | |
| gilmore-lie-groups_FO0620 | 66 | 1.000 | S \in U(2) | | |
| gilmore-lie-groups_FO0621 | 66 | 1.000 | S(E) | | |
| gilmore-lie-groups_FO0622 | 66 | 1.000 | T(E) | | |
| gilmore-lie-groups_FO0623 | 66 | 1.000 | t_{11}(E) | | |
| gilmore-lie-groups_FO0624 | 66 | 1.000 | r_{j} /\left[\left(E-E_{j}\right)+i\left(\Gamma_{j} / 2\right)\right] | | |
| gilmore-lie-groups_FO0625 | 66 | 1.000 | E_{j} | | |
| gilmore-lie-groups_FO0626 | 66 | 1.000 | \Gamma_{j} / \hbar | | |
| gilmore-lie-groups_FO0627 | 66 | 1.000 | V_{1} | | |
| gilmore-lie-groups_FO0628 | 66 | 1.000 | V_{2} | | |
| gilmore-lie-groups_FO0629 | 66 | 1.000 | T_{1} | | |
| gilmore-lie-groups_FO0630 | 66 | 1.000 | T_{2} | | |
| gilmore-lie-groups_FO0631 | 66 | 1.000 | S_{1} | | |
| gilmore-lie-groups_FO0632 | 67 | 1.000 | S_{\text {Tot }} | | |
| gilmore-lie-groups_FO0633 | 67 | 1.000 | 1 /\left(1-s_{12} s_{43}\right) | | |
| gilmore-lie-groups_FO0634 | 67 | 1.000 | V_{1}{ }^{\prime} | | |
| gilmore-lie-groups_FO0635 | 67 | 1.000 | V_{2}{ }^{\prime} | | |
| gilmore-lie-groups_FO0636 | 67 | 1.000 | T_{i}(E) \rightarrow T_{i}^{\prime}(E) | | |
| gilmore-lie-groups_FO0637 | 67 | 1.000 | S_{i}(E) \rightarrow S_{i}^{\prime}(E) | | |
| gilmore-lie-groups_FO0638 | 67 | 1.000 | i=1,2 | | |
| gilmore-lie-groups_FO0639 | 67 | 1.000 | E, S_{\text {Tot }}^{\prime}(E)=S_{\text {Tot }}(E) | | |
| gilmore-lie-groups_FO0640 | 67 | 1.000 | S_{1}^{\prime}(E) | | |
| gilmore-lie-groups_FO0641 | 67 | 1.000 | S_{2}^{\prime}(E) | | |
| gilmore-lie-groups_FO0642 | 67 | 0.998 | S_{\text {Tot }}(E) | | |
| gilmore-lie-groups_FO0643 | 67 | 0.999 | U(4) \supset U(2) \otimes U(2) \downarrow U(2) | | |
| gilmore-lie-groups_FO0644 | 67 | 0.997 | T_{1}(E) T_{2}(E)=T_{\text {Tot }}(E)=T_{1}^{\prime}(E) T_{2}^{\prime}(E) | | |
| gilmore-lie-groups_FO0645 | 67 | 0.997 | T_{1}^{\prime}(E)=T_{1}(E) R | | |
| gilmore-lie-groups_FO0646 | 67 | 0.997 | T_{2}^{\prime}(E)= | | |
| gilmore-lie-groups_FO0647 | 67 | 1.000 | R^{-1} T_{2}(E) | | |
| gilmore-lie-groups_FO0648 | 67 | 1.000 | R \in U(1,1) | | |
| gilmore-lie-groups_FO0649 | 67 | 1.000 | U(2,2) \supset U(1,1) \otimes U(1,1) \downarrow U(1,1) | | |
| gilmore-lie-groups_FO0650 | 67 | 0.932 | T_{\text {Tot }}(E) | | |
| gilmore-lie-groups_FO0651 | 67 | 0.932 | \left(T_{1}(E) R, R^{-1} T_{2}(E)\right. | | |
| gilmore-lie-groups_FO0652 | 67 | 0.850 | \left(S_{1}^{\prime}(E), S_{2}^{\prime}(E)\right) | | |
| gilmore-lie-groups_FO0653 | 68 | 0.987 | n_{1}, n_{2}, \ldots, n_{k} | | |
| gilmore-lie-groups_FO0654 | 68 | 0.718 | k S | | |
| gilmore-lie-groups_FO0655 | 68 | 0.718 | n_{j} \times n_{j}(j=1,2, \ldots, k) | | |
| gilmore-lie-groups_FO0656 | 68 | 1.000 | \Gamma | | |
| gilmore-lie-groups_FO0657 | 68 | 0.999 | \left[o_{e}\right]=S_{\text {Network }}\left[i_{e}\right] | | |
| gilmore-lie-groups_FO0658 | 68 | 1.000 | S_{\text {Newtork }} | | |
| gilmore-lie-groups_FO0659 | 68 | 1.000 | S_{\text {Newtork }}^{\dagger}=S_{\text {Newtork }}, S_{\text {Newtork }} \subset U(d) | | |
| gilmore-lie-groups_FO0660 | 68 | 1.000 | U\left(\sum_{j=1}^{k} n_{j}\right) \supset \Pi_{j=1}^{k} \otimes U\left(n_{j}\right) | | |
| gilmore-lie-groups_FO0661 | 68 | 1.000 | U(d) | | |
| gilmore-lie-groups_FO0662 | 68 | 1.000 | d | | |
| gilmore-lie-groups_FO0663 | 68 | 0.917 | \Pi_{j=1}^{k} \otimes U\left(n_{j}\right) | | |
| gilmore-lie-groups_FO0664 | 68 | 1.000 | U | | |
| gilmore-lie-groups_FO0665 | 70 | 1.000 | b | | |
| gilmore-lie-groups_FO0666 | 70 | 1.000 | b a^{-1} | | |
| gilmore-lie-groups_FO0667 | 70 | 1.000 | a^{-1} b | | |
| gilmore-lie-groups_FO0668 | 71 | 0.953 | (a, b, c) | | |
| gilmore-lie-groups_FO0669 | 71 | 1.000 | \phi(x, y) \rightarrow \phi(x, 0+\delta y) | | |
| gilmore-lie-groups_FO0670 | 72 | 1.000 | \epsilon | | |
| gilmore-lie-groups_FO0671 | 72 | 1.000 | I+\epsilon X | | |
| gilmore-lie-groups_FO0672 | 72 | 0.744 | X_{a} | | |
| gilmore-lie-groups_FO0673 | 72 | 0.744 | X_{b} | | |
| gilmore-lie-groups_FO0674 | 72 | 0.744 | X_{c} | | |
| gilmore-lie-groups_FO0675 | 72 | 1.000 | X^{n} | | |
| gilmore-lie-groups_FO0676 | 72 | 1.000 | X^{0}=I_{2}, X^{1}=X | | |
| gilmore-lie-groups_FO0677 | 72 | 1.000 | X^{2}=\theta^{2} I_{2} | | |
| gilmore-lie-groups_FO0678 | 72 | 1.000 | X^{2} | | |
| gilmore-lie-groups_FO0679 | 72 | 1.000 | X^{3}=X^{2} X^{1} | | |
| gilmore-lie-groups_FO0680 | 72 | 0.991 | X\left(=\theta^{2} X\right), X^{4} | | |
| gilmore-lie-groups_FO0681 | 72 | 0.995 | X^{0}=I_{2} | | |
| gilmore-lie-groups_FO0682 | 72 | 0.995 | X^{1}=X | | |
| gilmore-lie-groups_FO0683 | 72 | 1.000 | X^{3} | | |
| gilmore-lie-groups_FO0684 | 72 | 0.759 | I_{2} | | |
| gilmore-lie-groups_FO0685 | 73 | 1.000 | f_{0}, f_{1} | | |
| gilmore-lie-groups_FO0686 | 73 | 1.000 | \theta^{2}=a^{2}+b c | | |
| gilmore-lie-groups_FO0687 | 73 | 1.000 | f_{0}\left(\theta^{2}\right)=1+\theta^{2} / 2!+\theta^{4} / 4!+\theta^{6} / 6!+\cdots=\cosh \theta | | |
| gilmore-lie-groups_FO0688 | 73 | 1.000 | f_{1}\left(\theta^{2}\right)=1+\theta^{2} / 3!+\theta^{4} / 5!+\theta^{6} / 7!+\cdots=\sinh (\theta) / \theta | | |
| gilmore-lie-groups_FO0689 | 73 | 1.000 | Y | | |
| gilmore-lie-groups_FO0690 | 73 | 1.000 | g_{1}=I+\epsilon X | | |
| gilmore-lie-groups_FO0691 | 73 | 1.000 | I+\epsilon \alpha X | | |
| gilmore-lie-groups_FO0692 | 74 | 1.000 | \alpha | | |
| gilmore-lie-groups_FO0693 | 74 | 1.000 | \epsilon, \delta | | |
| gilmore-lie-groups_FO0694 | 74 | 1.000 | g_{1}(\epsilon)= | | |
| gilmore-lie-groups_FO0695 | 74 | 1.000 | \operatorname{EXP}(\epsilon X) | | |
| gilmore-lie-groups_FO0696 | 74 | 1.000 | g_{1}(\epsilon)^{-1}=\operatorname{EXP}(-\epsilon X) | | |
| gilmore-lie-groups_FO0697 | 74 | 1.000 | g_{2}(\delta)^{ \pm 1}=\operatorname{EXP}( \pm \delta Y) | | |
| gilmore-lie-groups_FO0698 | 74 | 0.996 | g_{1}(\epsilon)=\operatorname{EXP}(\epsilon X) | | |
| gilmore-lie-groups_FO0699 | 74 | 0.996 | g_{2}(\delta)= | | |
| gilmore-lie-groups_FO0700 | 74 | 1.000 | \operatorname{EXP}(\delta Y) | | |
| gilmore-lie-groups_FO0701 | 74 | 1.000 | Y,[X, Y]=(X Y-Y X) | | |
| gilmore-lie-groups_FO0702 | 74 | 1.000 | \mathfrak{g} | | |
| gilmore-lie-groups_FO0703 | 74 | 1.000 | g_{1} H g_{1}^{-1} \subset H | | |
| gilmore-lie-groups_FO0704 | 74 | 1.000 | X, Y, Z | | |
| gilmore-lie-groups_FO0705 | 74 | 0.996 | ([X, Y]=X Y-Y X) | | |
| gilmore-lie-groups_FO0706 | 74 | 1.000 | X Y | | |
| gilmore-lie-groups_FO0707 | 75 | 1.000 | X \in \mathfrak{g} | | |
| gilmore-lie-groups_FO0708 | 75 | 1.000 | Y \in \mathfrak{g} | | |
| gilmore-lie-groups_FO0709 | 75 | 1.000 | X \in \mathfrak{g}, Y \in \mathfrak{g} | | |
| gilmore-lie-groups_FO0710 | 75 | 1.000 | Z \in \mathfrak{g} | | |
| gilmore-lie-groups_FO0711 | 75 | 0.996 | X_{1}, X_{2}, \ldots, X_{n} | | |
| gilmore-lie-groups_FO0712 | 75 | 1.000 | \left(X_{i}, X_{j}\right) | | |
| gilmore-lie-groups_FO0713 | 76 | 1.000 | C_{i j}{ }^{k} | | |
| gilmore-lie-groups_FO0714 | 76 | 0.865 | X_{a}, X_{b}, X_{c} | | |
| gilmore-lie-groups_FO0715 | 76 | 0.865 | \mathfrak{s l}(2 ; R) | | |
| gilmore-lie-groups_FO0716 | 76 | 1.000 | Z | | |
| gilmore-lie-groups_FO0717 | 76 | 1.000 | R(Z) | | |
| gilmore-lie-groups_FO0718 | 77 | 1.000 | \mathfrak{s l}(2 ; \mathbb{R}) | | |
| gilmore-lie-groups_FO0719 | 77 | 0.844 | a, b | | |
| gilmore-lie-groups_FO0720 | 77 | 0.844 | c | | |
| gilmore-lie-groups_FO0721 | 78 | 0.899 | X_{a}, X_{b} | | |
| gilmore-lie-groups_FO0722 | 78 | 0.997 | E(2) | | |
| gilmore-lie-groups_FO0723 | 78 | 1.000 | P_{x}, P_{y} | | |
| gilmore-lie-groups_FO0724 | 78 | 1.000 | L_{z} | | |
| gilmore-lie-groups_FO0725 | 78 | 0.985 | A, B | | |
| gilmore-lie-groups_FO0726 | 78 | 0.985 | p \times q | | |
| gilmore-lie-groups_FO0727 | 79 | 0.998 | X_{i} | | |
| gilmore-lie-groups_FO0728 | 79 | 0.998 | C_{i \star}{ }^{*}=0 | | |
| gilmore-lie-groups_FO0729 | 79 | 1.000 | C_{* \star}^{i}=0 | | |
| gilmore-lie-groups_FO0730 | 79 | 1.000 | V_{0} | | |
| gilmore-lie-groups_FO0731 | 79 | 1.000 | V_{-} | | |
| gilmore-lie-groups_FO0732 | 79 | 1.000 | V_{+} | | |
| gilmore-lie-groups_FO0733 | 80 | 1.000 | X_{ \pm}=X_{b} \pm X_{c} | | |
| gilmore-lie-groups_FO0734 | 80 | 1.000 | X_{-} | | |
| gilmore-lie-groups_FO0735 | 80 | 1.000 | X_{a}, X_{+} | | |
| gilmore-lie-groups_FO0736 | 81 | 0.644 | \alpha^{1}, \alpha^{2}, \ldots, \alpha^{n} | | |
| gilmore-lie-groups_FO0737 | 81 | 0.840 | \alpha^{1}+d \alpha^{1}, \alpha^{2}+d \alpha^{2}, \ldots, \alpha^{n}+ | | |
| gilmore-lie-groups_FO0738 | 81 | 0.866 | d \alpha^{n} | | |
| gilmore-lie-groups_FO0739 | 81 | 0.866 | \left(x^{1}, x^{2}, \ldots, x^{n}\right) | | |
| gilmore-lie-groups_FO0740 | 81 | 0.831 | \alpha^{1}+d \alpha^{1}, \alpha^{2}+d \alpha^{2}, \ldots, \alpha^{n}+d \alpha^{n} | | |
| gilmore-lie-groups_FO0741 | 81 | 0.831 | x^{1}+d x^{1}, x^{2}+ | | |
| gilmore-lie-groups_FO0742 | 81 | 1.000 | d x^{2}, \ldots, x^{n}+d x^{n} | | |
| gilmore-lie-groups_FO0743 | 81 | 0.997 | d x | | |
| gilmore-lie-groups_FO0744 | 81 | 0.997 | d \alpha | | |
| gilmore-lie-groups_FO0745 | 81 | 1.000 | d s | | |
| gilmore-lie-groups_FO0746 | 81 | 1.000 | \alpha^{i}+d \alpha^{i} | | |
| gilmore-lie-groups_FO0747 | 81 | 0.671 | x, g_{r s}(x) | | |
| gilmore-lie-groups_FO0748 | 81 | 1.000 | x, g(x) | | |
| gilmore-lie-groups_FO0749 | 82 | 0.976 | \left(d \alpha^{1}, d \alpha^{2}, d \alpha^{3}\right) | | |
| gilmore-lie-groups_FO0750 | 82 | 0.999 | (d x, d y, d z) | | |
| gilmore-lie-groups_FO0751 | 83 | 1.000 | [X, Y]=(X Y-Y X) | | |
| gilmore-lie-groups_FO0752 | 83 | 1.000 | g_{1}=(I+\epsilon X), g_{1}^{-1}=(I+\epsilon X)^{-1}=I- | | |
| gilmore-lie-groups_FO0753 | 83 | 1.000 | \epsilon X+\epsilon^{2} X^{2}-\cdots | | |
| gilmore-lie-groups_FO0754 | 83 | 1.000 | g_{2} | | |
| gilmore-lie-groups_FO0755 | 83 | 1.000 | X^{\prime} | | |
| gilmore-lie-groups_FO0756 | 83 | 1.000 | (X, X)=2\left(a^{2}+b c\right) | | |
| gilmore-lie-groups_FO0757 | 83 | 1.000 | Y=e^{X} | | |
| gilmore-lie-groups_FO0758 | 83 | 0.570 | \lambda_{i} | | |
| gilmore-lie-groups_FO0759 | 83 | 0.570 | 0<\lambda_{i}<+2 | | |
| gilmore-lie-groups_FO0760 | 83 | 0.570 | Y \in \operatorname{SL}(2 ; \mathbb{R}) | | |
| gilmore-lie-groups_FO0761 | 83 | 0.570 | \operatorname{tr} Y<-2 | | |
| gilmore-lie-groups_FO0762 | 84 | 1.000 | X \in \mathfrak{s l}(2 ; \mathbb{R}) | | |
| gilmore-lie-groups_FO0763 | 84 | 0.637 | \operatorname{tr} e^{X}<-2 | | |
| gilmore-lie-groups_FO0764 | 84 | 0.952 | \mathbf{L}= | | |
| gilmore-lie-groups_FO0765 | 84 | 1.000 | \left(L_{1}, L_{2}, L_{3}\right)=\left(X_{23}, X_{31}, X_{12}\right) | | |
| gilmore-lie-groups_FO0766 | 84 | 1.000 | \mathbf{X}=\theta \cdot \mathbf{L} | | |
| gilmore-lie-groups_FO0767 | 84 | 1.000 | \theta^{2}=\theta_{1}^{2}+\theta_{2}^{2}+\theta_{3}^{2} | | |
| gilmore-lie-groups_FO0768 | 84 | 1.000 | S O(n) | | |
| gilmore-lie-groups_FO0769 | 84 | 1.000 | X_{i j}=-X_{j i}(1 \leq i \neq j \leq n) | | |
| gilmore-lie-groups_FO0770 | 84 | 1.000 | \left(X_{i j}\right)_{\alpha \beta}=\delta_{i \alpha} \delta_{j \beta}-\delta_{i \beta} \delta_{j \alpha} | | |
| gilmore-lie-groups_FO0771 | 84 | 1.000 | \mathcal{X}_{i j}=x^{i} \partial_{j}-x^{j} \partial_{i} | | |
| gilmore-lie-groups_FO0772 | 84 | 1.000 | \mathcal{B}_{i j}= | | |
| gilmore-lie-groups_FO0773 | 84 | 1.000 | b_{i}^{\dagger} b_{j}-b_{j}^{\dagger} b_{i}(1 \leq i \neq j \leq n) | | |
| gilmore-lie-groups_FO0774 | 84 | 0.963 | \mathcal{F}_{i j}= | | |
| gilmore-lie-groups_FO0775 | 84 | 1.000 | f_{i}^{\dagger} f_{j}-f_{j}^{\dagger} f_{i}(1 \leq i \neq j \leq n) | | |
| gilmore-lie-groups_FO0776 | 84 | 1.000 | D, Y, Z | | |
| gilmore-lie-groups_FO0777 | 84 | 0.998 | \mathfrak{s o}(4) | | |
| gilmore-lie-groups_FO0778 | 84 | 0.998 | 4 \times 4 | | |
| gilmore-lie-groups_FO0779 | 85 | 1.000 | a_{i j} X_{i j} | | |
| gilmore-lie-groups_FO0780 | 85 | 1.000 | S O(p, q) | | |
| gilmore-lie-groups_FO0781 | 85 | 1.000 | \operatorname{EXP}\left(V_{-}\right) | | |
| gilmore-lie-groups_FO0782 | 85 | 0.546 | S O(3,1) | | |
| gilmore-lie-groups_FO0783 | 85 | 0.546 | S O(2,2) | | |
| gilmore-lie-groups_FO0784 | 85 | 0.982 | T^{2} | | |
| gilmore-lie-groups_FO0785 | 85 | 0.752 | e^{\theta \cdot \mathbf{L}} \in S O(3) | | |
| gilmore-lie-groups_FO0786 | 85 | 0.752 | \rho(\theta) | | |
| gilmore-lie-groups_FO0787 | 85 | 1.000 | g_{\mu \nu}(\theta) | | |
| gilmore-lie-groups_FO0788 | 85 | 1.000 | d V(x)=\|g(x)\|^{1 / 2} d^{n} x | | |
| gilmore-lie-groups_FO0789 | 85 | 0.623 | (\mathbf{x}, \mathbf{x}) | | |
| gilmore-lie-groups_FO0790 | 85 | 1.000 | g_{r s}=\left(\mathbf{e}_{r}, \mathbf{e}_{s}\right) | | |
| gilmore-lie-groups_FO0791 | 85 | 1.000 | \mathbf{x}=\mathbf{e}_{i} x^{i} | | |
| gilmore-lie-groups_FO0792 | 85 | 1.000 | g^{r s} | | |
| gilmore-lie-groups_FO0793 | 85 | 1.000 | \mathbf{y}=G \mathbf{x},(\mathbf{y}, \mathbf{y})=(\mathbf{x}, \mathbf{x}) | | |
| gilmore-lie-groups_FO0794 | 85 | 1.000 | G^{t} g G=g | | |
| gilmore-lie-groups_FO0795 | 85 | 1.000 | H^{t} g+g H=0 | | |
| gilmore-lie-groups_FO0796 | 85 | 1.000 | X_{r s}=g_{r t} x^{t} \partial_{s}-g_{s t} x^{t} \partial_{r} | | |
| gilmore-lie-groups_FO0797 | 86 | 1.000 | X_{r s} | | |
| gilmore-lie-groups_FO0798 | 86 | 0.887 | M=S O | | |
| gilmore-lie-groups_FO0808 | 98 | 0.986 | A^{t}=-A | | |
| gilmore-lie-groups_FO0809 | 86 | 0.999 | M M^{t}=S^{2}=e^{2 \Sigma} | | |
| gilmore-lie-groups_FO0810 | 86 | 1.000 | O=S^{-1} M=e^{-\Sigma} M | | |
| gilmore-lie-groups_FO0811 | 86 | 1.000 | \Sigma | | |
| gilmore-lie-groups_FO0812 | 86 | 1.000 | S-I_{2} | | |
| gilmore-lie-groups_FO0813 | 86 | 1.000 | M=H U | | |
| gilmore-lie-groups_FO0814 | 86 | 1.000 | H^{\dagger}=H^{+1} | | |
| gilmore-lie-groups_FO0815 | 86 | 1.000 | U^{\dagger}=U^{-1} | | |
| gilmore-lie-groups_FO0816 | 86 | 1.000 | \hbar k_{L} | | |
| gilmore-lie-groups_FO0817 | 86 | 1.000 | -\hbar k_{R} | | |
| gilmore-lie-groups_FO0818 | 86 | 1.000 | m_{i j} | | |
| gilmore-lie-groups_FO0819 | 86 | 1.000 | \hbar k_{L}=\hbar k_{R} | | |
| gilmore-lie-groups_FO0820 | 86 | 0.679 | \operatorname{SU}(1,1) | | |
| gilmore-lie-groups_FO0821 | 87 | 1.000 | \kappa_{R} | | |
| gilmore-lie-groups_FO0822 | 87 | 1.000 | \kappa_{L} | | |
| gilmore-lie-groups_FO0823 | 87 | 1.000 | \kappa_{L}=\kappa_{R} | | |
| gilmore-lie-groups_FO0824 | 87 | 1.000 | k_{*}=\sqrt{2 m\left(E-V_{*}\right) / \hbar^{2}} | | |
| gilmore-lie-groups_FO0825 | 87 | 1.000 | E>V_{*} | | |
| gilmore-lie-groups_FO0826 | 87 | 1.000 | \kappa_{*}=\sqrt{2 m\left(V_{*}-E\right) / \hbar^{2}} | | |
| gilmore-lie-groups_FO0827 | 87 | 1.000 | E<V_{*} | | |
| gilmore-lie-groups_FO0828 | 87 | 0.978 | *=L, R | | |
| gilmore-lie-groups_FO0829 | 88 | 0.998 | \operatorname{det}(M) \neq 0 | | |
| gilmore-lie-groups_FO0830 | 88 | 1.000 | I+\epsilon A | | |
| gilmore-lie-groups_FO0831 | 88 | 1.000 | \mathfrak{g l}(n ; \mathbb{F}) | | |
| gilmore-lie-groups_FO0832 | 89 | 1.000 | \mathfrak{u t}(p, q) | | |
| gilmore-lie-groups_FO0833 | 89 | 0.998 | \mathfrak{h t}(p, q) | | |
| gilmore-lie-groups_FO0834 | 90 | 1.000 | (a, b)=(1,0) | | |
| gilmore-lie-groups_FO0835 | 90 | 0.999 | \left[X_{a}, X_{b}\right]=X_{b} | | |
| gilmore-lie-groups_FO0836 | 90 | 1.000 | \mathfrak{u t}(p, q, r) | | |
| gilmore-lie-groups_FO0837 | 90 | 0.999 | \mathfrak{u t}(1,2,1) | | |
| gilmore-lie-groups_FO0838 | 90 | 1.000 | j | | |
| gilmore-lie-groups_FO0839 | 90 | 1.000 | \left[X_{i}, X_{j}\right] | | |
| gilmore-lie-groups_FO0845 | 91 | 1.000 | X_{\delta} | | |
| gilmore-lie-groups_FO0854 | 91 | 1.000 | a^{\dagger} a^{\dagger} | | |
| gilmore-lie-groups_FO0855 | 91 | 1.000 | a a | | |
| gilmore-lie-groups_FO0863 | 90 | 1.000 | \hat{n}=a^{\dagger} a+\frac{1}{2} | | |
| gilmore-lie-groups_FO0864 | 91 | 0.996 | I=\left[a, a^{\dagger}\right] | | |
| gilmore-lie-groups_FO0865 | 91 | 0.887 | \mathfrak{s o l}(n)=\mathfrak{u t}(1,1,1, \ldots, 1) | | |
| gilmore-lie-groups_FO0866 | 91 | 1.000 | \mathfrak{u t}(1,1,1) | | |
| gilmore-lie-groups_FO0867 | 91 | 0.936 | \mathfrak{n i l}(n) | | |
| gilmore-lie-groups_FO0868 | 91 | 1.000 | \eta=0 | | |
| gilmore-lie-groups_FO0869 | 91 | 1.000 | \mathfrak{a}(p, q) | | |
| gilmore-lie-groups_FO0870 | 92 | 0.998 | O(3) | | |
| gilmore-lie-groups_FO0871 | 92 | 0.998 | U(2) | | |
| gilmore-lie-groups_FO0872 | 92 | 0.728 | \sigma_{\mu} | | |
| gilmore-lie-groups_FO0873 | 92 | 1.000 | \mathfrak{s o}(2,1) | | |
| gilmore-lie-groups_FO0874 | 93 | 1.000 | \mathfrak{s o}(3,1) | | |
| gilmore-lie-groups_FO0875 | 93 | 1.000 | M=-G^{-1} M^{\dagger} G | | |
| gilmore-lie-groups_FO0876 | 93 | 1.000 | (p+q) \times(p+q) | | |
| gilmore-lie-groups_FO0877 | 93 | 0.983 | \left[\begin{array}{ll}g & 0 \\ 0 & 0\end{array}\right] | | |
| gilmore-lie-groups_FO0878 | 93 | 0.983 | \operatorname{det}(g) \neq 0 | | |
| gilmore-lie-groups_FO0879 | 93 | 1.000 | \operatorname{diag}(1,1,1,0) | | |
| gilmore-lie-groups_FO0880 | 94 | 1.000 | \theta_{i} | | |
| gilmore-lie-groups_FO0881 | 94 | 1.000 | s_{4} | | |
| gilmore-lie-groups_FO0882 | 94 | 1.000 | t^{\prime}=e^{s_{4}} t | | |
| gilmore-lie-groups_FO0883 | 94 | 1.000 | E(3)=I S O(3) | | |
| gilmore-lie-groups_FO0884 | 94 | 0.981 | I S O(3,1)(3.26) | | |
| gilmore-lie-groups_FO0885 | 94 | 0.837 | 5 \times 5 | | |
| gilmore-lie-groups_FO0886 | 94 | 0.837 | G=\operatorname{diag}(1,1,1,-1,0) | | |
| gilmore-lie-groups_FO0887 | 94 | 0.988 | I S O(3,1) | | |
| gilmore-lie-groups_FO0888 | 94 | 1.000 | \mathfrak{s u}(n) | | |
| gilmore-lie-groups_FO0889 | 94 | 1.000 | \mathfrak{u}(n) | | |
| gilmore-lie-groups_FO0890 | 94 | 1.000 | \mathfrak{s l}(n ; \mathbb{C}) | | |
| gilmore-lie-groups_FO0891 | 95 | 1.000 | M_{i j}(i<j)(M \in \mathfrak{u}(n)) | | |
| gilmore-lie-groups_FO0892 | 95 | 0.588 | M_{i i} | | |
| gilmore-lie-groups_FO0893 | 95 | 0.761 | M_{i j} | | |
| gilmore-lie-groups_FO0894 | 95 | 0.761 | (i>j) | | |
| gilmore-lie-groups_FO0895 | 95 | 0.999 | \mathfrak{u}(n) \rightarrow \mathfrak{o u}(2 n) \subset \mathfrak{o}(2 n) | | |
| gilmore-lie-groups_FO0896 | 95 | 0.999 | \mathfrak{o} \mathfrak{u}(2 n) | | |
| gilmore-lie-groups_FO0897 | 95 | 0.640 | \mathcal{u}(n): 2 \times n(n-1) / 2+1 \times n=n^{2} | | |
| gilmore-lie-groups_FO0898 | 95 | 0.914 | S p(n) | | |
| gilmore-lie-groups_FO0899 | 95 | 0.791 | \operatorname{Sp}(n) | | |
| gilmore-lie-groups_FO0900 | 95 | 0.712 | M_{i j}(i<j)(M \in \mathfrak{s p}(n)) | | |
| gilmore-lie-groups_FO0901 | 95 | 0.998 | q_{0}=0, q_{i} | | |
| gilmore-lie-groups_FO0902 | 95 | 0.615 | i>j | | |
| gilmore-lie-groups_FO0903 | 95 | 1.000 | \mathfrak{s p}(n) \rightarrow \mathfrak{u s p}(2 n) \subset \mathfrak{s u}(2 n) | | |
| gilmore-lie-groups_FO0904 | 95 | 1.000 | \mathfrak{u s p}(2 n) | | |
| gilmore-lie-groups_FO0905 | 95 | 1.000 | \mathfrak{s p}(n) | | |
| gilmore-lie-groups_FO0906 | 95 | 1.000 | 4 \times n(n-1) / 2+n \times 3=2 n(2 n+1) / 2 | | |
| gilmore-lie-groups_FO0907 | 96 | 0.999 | (X, X)=0 \Rightarrow X= | | |
| gilmore-lie-groups_FO0908 | 96 | 0.949 | X, X^{\prime} | | |
| gilmore-lie-groups_FO0909 | 96 | 0.811 | (X, Y) | | |
| gilmore-lie-groups_FO0910 | 97 | 1.000 | \exp (\theta X) | | |
| gilmore-lie-groups_FO0911 | 97 | 1.000 | -\pi \leq \theta \leq+\pi | | |
| gilmore-lie-groups_FO0912 | 97 | 1.000 | -\pi | | |
| gilmore-lie-groups_FO0913 | 97 | 1.000 | +\pi | | |
| gilmore-lie-groups_FO0914 | 97 | 1.000 | -\infty<\theta<+\infty | | |
| gilmore-lie-groups_FO0915 | 97 | 1.000 | S^{1} | | |
| gilmore-lie-groups_FO0916 | 97 | 1.000 | R^{k} | | |
| gilmore-lie-groups_FO0917 | 97 | 1.000 | G=I_{n}, I_{p, q} | | |
| gilmore-lie-groups_FO0918 | 97 | 0.989 | (A, A)= | | |
| gilmore-lie-groups_FO0919 | 97 | 1.000 | \operatorname{tr}(A)^{2}=a^{2}+c^{2} | | |
| gilmore-lie-groups_FO0920 | 97 | 1.000 | \mathfrak{u t}(1,1) | | |
| gilmore-lie-groups_FO0921 | 98 | 1.000 | (A, A)=\operatorname{tr} R(A)^{2}=(a-c)^{2} | | |
| gilmore-lie-groups_FO0922 | 98 | 0.998 | X_{a}+X_{c} | | |
| gilmore-lie-groups_FO0923 | 98 | 1.000 | \mathfrak{g l}(1 ; \mathbb{R}) \subset \mathfrak{g l}(1 ; \mathbb{C}) \subset \mathfrak{g l}(1 ; \mathbb{Q}) | | |
| gilmore-lie-groups_FO0924 | 98 | 1.000 | X_{\eta}, X_{r}, X_{l}, X_{\delta} | | |
| gilmore-lie-groups_FO0925 | 98 | 1.000 | \hat{E}=\left(a^{\dagger} a+\frac{1}{2}\right) \hbar \omega \rightarrow a^{\dagger} a+\frac{1}{2} | | |
| gilmore-lie-groups_FO0926 | 98 | 1.000 | \hbar \omega=1 | | |
| gilmore-lie-groups_FO0927 | 98 | 1.000 | X, X^{\prime} \in \mathfrak{h} | | |
| gilmore-lie-groups_FO0928 | 98 | 1.000 | Y, Y^{\prime} \in \mathfrak{p} | | |
| gilmore-lie-groups_FO0929 | 98 | 1.000 | \left[X, X^{\prime}\right] \in \mathfrak{h},[X, Y] \in \mathfrak{p},\left[Y, Y^{\prime}\right] \in \mathfrak{h} | | |
| gilmore-lie-groups_FO0930 | 98 | 1.000 | \mathfrak{h} | | |
| gilmore-lie-groups_FO0931 | 98 | 1.000 | \mathfrak{p} | | |
| gilmore-lie-groups_FO0932 | 98 | 0.995 | X=\left[\begin{array}{cc}A & 0 \\ 0 & D\end{array}\right] \in \mathfrak{h} | | |
| gilmore-lie-groups_FO0933 | 98 | 0.995 | Y=\left[\begin{array}{cc}0 & B \\ C & 0\end{array}\right] \in \mathfrak{p} | | |
| gilmore-lie-groups_FO0934 | 98 | 1.000 | (X, Y)=0 | | |
| gilmore-lie-groups_FO0935 | 98 | 0.970 | \mathfrak{s o}(p, q) | | |
| gilmore-lie-groups_FO0936 | 98 | 0.970 | \left[\begin{array}{cc}A & B \\ B^{t} & C\end{array}\right] | | |
| gilmore-lie-groups_FO0937 | 98 | 0.970 | p \times p | | |
| gilmore-lie-groups_FO0938 | 98 | 0.970 | q \times q | | |
| gilmore-lie-groups_FO0939 | 98 | 0.986 | C^{t}=-C | | |
| gilmore-lie-groups_FO0940 | 98 | 0.986 | X=\left[\begin{array}{cc}A & 0 \\ 0 & C\end{array}\right] \in \mathfrak{h} | | |
| gilmore-lie-groups_FO0941 | 98 | 0.986 | Y=\left[\begin{array}{cc}0 & B \\ B^{t} & 0\end{array}\right] \in \mathfrak{p} | | |
| gilmore-lie-groups_FO0942 | 98 | 0.988 | (X, X) \leq 0, \quad(X, X)=0 \Rightarrow X=0 | | |
| gilmore-lie-groups_FO0943 | 98 | 1.000 | (Y, Y) \geq 0, \quad(Y, Y)=0 \Rightarrow Y=0 | | |
| gilmore-lie-groups_FO0944 | 99 | 0.818 | \mathfrak{s u}(p, q) | | |
| gilmore-lie-groups_FO0945 | 99 | 0.818 | \left[\begin{array}{c}A \\ B^{\dagger} \\ C\end{array}\right] | | |
| gilmore-lie-groups_FO0946 | 99 | 0.999 | A^{\dagger}=-A, C^{\dagger}=-C | | |
| gilmore-lie-groups_FO0947 | 99 | 0.999 | \operatorname{tr}(A+C)=0 | | |
| gilmore-lie-groups_FO0948 | 99 | 1.000 | Y=\left[\begin{array}{cc}0 & B \\ B^{\dagger} & 0\end{array}\right] \in \mathfrak{p} | | |
| gilmore-lie-groups_FO0949 | 99 | 1.000 | (X, X)=0 \Rightarrow X=0 | | |
| gilmore-lie-groups_FO0950 | 99 | 1.000 | (Y, Y)=0 \Rightarrow Y=0 | | |
| gilmore-lie-groups_FO0951 | 99 | 1.000 | \mathfrak{s l}(n ; \mathbb{R}) | | |
| gilmore-lie-groups_FO0952 | 99 | 1.000 | B^{t}=B | | |
| gilmore-lie-groups_FO0953 | 99 | 1.000 | \operatorname{tr} B=0 | | |
| gilmore-lie-groups_FO0954 | 99 | 1.000 | (A, A)=0 \Rightarrow A=0 | | |
| gilmore-lie-groups_FO0955 | 99 | 1.000 | (B, B)=0 \Rightarrow B=0 | | |
| gilmore-lie-groups_FO0956 | 99 | 1.000 | A^{\dagger}=-A | | |
| gilmore-lie-groups_FO0957 | 99 | 1.000 | H^{\dagger}=H | | |
| gilmore-lie-groups_FO0958 | 99 | 1.000 | (H, H)=0 \Rightarrow H=0 | | |
| gilmore-lie-groups_FO0959 | 99 | 1.000 | \mathfrak{g}=\mathfrak{h}+\mathfrak{p} | | |
| gilmore-lie-groups_FO0960 | 100 | 1.000 | \mathfrak{g}^{\prime}=\mathfrak{h}+i \mathfrak{p}=\mathfrak{h}+\mathfrak{p}^{\prime} | | |
| gilmore-lie-groups_FO0961 | 100 | 1.000 | \mathfrak{g}=\mathfrak{s l}(n ; \mathbb{Q}) | | |
| gilmore-lie-groups_FO0962 | 100 | 1.000 | \mathfrak{h}=\mathfrak{s l}(n ; \mathbb{C}) | | |
| gilmore-lie-groups_FO0963 | 210 | 1.000 | \mu | | |
| gilmore-lie-groups_FO0964 | 126 | 1.000 | \sigma | | |
| gilmore-lie-groups_FO0969 | 100 | 1.000 | \mathfrak{g}=A | | |
| gilmore-lie-groups_FO0970 | 100 | 1.000 | \operatorname{tr} A^{2} \leq 0,=0 \Rightarrow A=0 | | |
| gilmore-lie-groups_FO0971 | 100 | 1.000 | \operatorname{EXP}(t A) | | |
| gilmore-lie-groups_FO0972 | 100 | 1.000 | t | | |
| gilmore-lie-groups_FO0973 | 100 | 1.000 | \lambda_{i}=\gamma n_{i}, n_{i} | | |
| gilmore-lie-groups_FO0974 | 100 | 1.000 | \gamma \neq 0 | | |
| gilmore-lie-groups_FO0975 | 100 | 1.000 | t_{0} | | |
| gilmore-lie-groups_FO0976 | 100 | 0.979 | \mathfrak{s o}(3) | | |
| gilmore-lie-groups_FO0977 | 100 | 1.000 | (\delta x, \delta y, \delta z) | | |
| gilmore-lie-groups_FO0978 | 100 | 1.000 | g(x, y, z) | | |
| gilmore-lie-groups_FO0979 | 100 | 1.000 | d \mu(x, y, z) | | |
| gilmore-lie-groups_FO0980 | 101 | 1.000 | (\mathfrak{g} \cap \mathfrak{h}=\mathfrak{h}, \mathfrak{g} \cap \mathfrak{p}=\mathfrak{p}) | | |
| gilmore-lie-groups_FO0981 | 102 | 0.987 | b_{i}^{\dagger} | | |
| gilmore-lie-groups_FO0982 | 102 | 0.987 | b_{j} | | |
| gilmore-lie-groups_FO0983 | 102 | 0.971 | \left[b_{i}, b_{j}\right],\left[b_{i}^{\dagger}, b_{j}^{\dagger}\right],\left[b_{i}, I\right],\left[b_{j}^{\dagger}, I\right] | | |
| gilmore-lie-groups_FO0984 | 102 | 1.000 | \mathcal{A} | | |
| gilmore-lie-groups_FO0985 | 103 | 1.000 | [A, B]=C | | |
| gilmore-lie-groups_FO0986 | 103 | 0.999 | 2 n+1 | | |
| gilmore-lie-groups_FO0987 | 103 | 0.999 | b_{i}, b_{j}^{\dagger}, I(1 \leq i, j \leq n) | | |
| gilmore-lie-groups_FO0988 | 103 | 1.000 | X_{i j} | | |
| gilmore-lie-groups_FO0989 | 103 | 1.000 | b_{i}^{\dagger} b_{j} | | |
| gilmore-lie-groups_FO0990 | 103 | 0.970 | f_{i}^{\dagger} | | |
| gilmore-lie-groups_FO0991 | 103 | 0.970 | f_{j} | | |
| gilmore-lie-groups_FO0992 | 103 | 1.000 | \left\{f_{i}, f_{j}\right\},\left\{f_{i}^{\dagger}, f_{j}^{\dagger}\right\} | | |
| gilmore-lie-groups_FO0993 | 104 | 1.000 | S U(n) | | |
| gilmore-lie-groups_FO0994 | 104 | 1.000 | \mathcal{L}_{1}=x_{2} \partial_{3}-x_{3} \partial_{2}, \mathcal{L}_{2}=x_{3} \partial_{1}-x_{1} \partial_{3}, \mathcal{L}_{3}=x_{1} \partial_{2}-x_{2} \partial_{1} | | |
| gilmore-lie-groups_FO0995 | 104 | 0.705 | L_{i} | | |
| gilmore-lie-groups_FO0996 | 104 | 0.705 | \mathcal{L}_{i} | | |
| gilmore-lie-groups_FO0997 | 105 | 1.000 | E(2)=I S O(2) | | |
| gilmore-lie-groups_FO0998 | 105 | 1.000 | x-y | | |
| gilmore-lie-groups_FO0999 | 105 | 1.000 | t_{1} | | |
| gilmore-lie-groups_FO1000 | 105 | 1.000 | t_{2} | | |
| gilmore-lie-groups_FO1001 | 105 | 1.000 | \mathcal{L}_{z}=x_{1} \partial_{2}-x_{2} \partial_{1} | | |
| gilmore-lie-groups_FO1002 | 105 | 1.000 | \mathcal{T}_{i}=\partial_{i} | | |
| gilmore-lie-groups_FO1003 | 105 | 0.867 | (a, b) | | |
| gilmore-lie-groups_FO1004 | 105 | 1.000 | x(p) | | |
| gilmore-lie-groups_FO1005 | 105 | 1.000 | S^{\prime} | | |
| gilmore-lie-groups_FO1006 | 105 | 1.000 | p, x^{\prime}(p) | | |
| gilmore-lie-groups_FO1007 | 105 | 1.000 | f_{S} | | |
| gilmore-lie-groups_FO1008 | 105 | 1.000 | f_{S^{\prime}} | | |
| gilmore-lie-groups_FO1009 | 105 | 1.000 | x^{\prime}(p) | | |
| gilmore-lie-groups_FO1010 | 106 | 1.000 | G(x, y) | | |
| gilmore-lie-groups_FO1011 | 106 | 1.000 | \left(x^{\prime}, y^{\prime}\right) | | |
| gilmore-lie-groups_FO1012 | 107 | 1.000 | \mathcal{A}=\sum_{i} \sum_{j} A_{i j} X_{i j} | | |
| gilmore-lie-groups_FO1013 | 107 | 1.000 | [A, B]=C \Leftrightarrow[\mathcal{A}, \mathcal{B}]=\mathcal{C} | | |
| gilmore-lie-groups_FO1014 | 107 | 1.000 | e^{A} e^{B}=e^{D} \Leftrightarrow e^{\mathcal{A}} e^{\mathcal{B}}=e^{\mathcal{D}} | | |
| gilmore-lie-groups_FO1015 | 107 | 1.000 | a_{i}^{\dagger} a_{j}, 1 \leq i, j \leq 2 | | |
| gilmore-lie-groups_FO1016 | 107 | 1.000 | \hat{n}=a_{1}^{\dagger} a_{1}+a_{2}^{\dagger} a_{2} | | |
| gilmore-lie-groups_FO1017 | 107 | 1.000 | \hat{n} | | |
| gilmore-lie-groups_FO1018 | 107 | 1.000 | a_{1}^{\dagger} a_{1}-a_{2}^{\dagger} a_{2}, a_{1}^{\dagger} a_{2} | | |
| gilmore-lie-groups_FO1019 | 107 | 1.000 | a_{2}^{\dagger} a_{1} | | |
| gilmore-lie-groups_FO1020 | 108 | 0.999 | \frac{1}{2}\left(a_{1}^{\dagger} a_{1}-a_{2}^{\dagger} a_{2}\right), a_{1}^{\dagger} a_{2} | | |
| gilmore-lie-groups_FO1021 | 108 | 1.000 | J_{z}, J_{ \pm} | | |
| gilmore-lie-groups_FO1022 | 108 | 1.000 | J^{2}=J_{x}^{2}+J_{y}^{2}+J_{z}^{2} | | |
| gilmore-lie-groups_FO1023 | 108 | 1.000 | i(i=1,2) | | |
| gilmore-lie-groups_FO1024 | 108 | 1.000 | \left|n_{i}\right\rangle | | |
| gilmore-lie-groups_FO1025 | 108 | 1.000 | n_{i} | | |
| gilmore-lie-groups_FO1026 | 108 | 1.000 | a_{i}^{\dagger} | | |
| gilmore-lie-groups_FO1027 | 108 | 1.000 | a_{i} | | |
| gilmore-lie-groups_FO1028 | 108 | 1.000 | \left|n_{1}\right\rangle \otimes\left|n_{2}\right\rangle=\left|n_{1}, n_{2}\right\rangle | | |
| gilmore-lie-groups_FO1029 | 108 | 1.000 | \left|n_{1}, n_{2}\right\rangle=|j m\rangle | | |
| gilmore-lie-groups_FO1030 | 108 | 0.994 | \frac{1}{2}\left(a_{1}^{\dagger} a_{1}-a_{2}^{\dagger} a_{2}\right) | | |
| gilmore-lie-groups_FO1031 | 108 | 0.994 | J_{z} | | |
| gilmore-lie-groups_FO1032 | 108 | 0.994 | a_{1}^{\dagger} a_{2}, a_{2}^{\dagger} a_{1} | | |
| gilmore-lie-groups_FO1033 | 108 | 1.000 | J_{+} | | |
| gilmore-lie-groups_FO1034 | 108 | 1.000 | J_{-} | | |
| gilmore-lie-groups_FO1035 | 108 | 1.000 | a_{i}^{\dagger} a_{j} | | |
| gilmore-lie-groups_FO1036 | 108 | 1.000 | n_{1}+n_{2} | | |
| gilmore-lie-groups_FO1037 | 108 | 1.000 | J^{2}|j m\rangle=j(j+1)|j m\rangle | | |
| gilmore-lie-groups_FO1038 | 108 | 1.000 | J_{z}|j m\rangle=m|j m\rangle | | |
| gilmore-lie-groups_FO1039 | 108 | 1.000 | J_{+}|j m\rangle=a_{1}^{\dagger} a_{2}\left|n_{1}, n_{2}\right\rangle=\sqrt{n_{1}+1} \sqrt{n_{2}}\left|n_{1}+1, n_{2}-1\right\rangle= | | |
| gilmore-lie-groups_FO1040 | 108 | 0.999 | |j, m+1\rangle \sqrt{j+m+1} \sqrt{j-m} | | |
| gilmore-lie-groups_FO1041 | 108 | 1.000 | J_{-}|j m\rangle=a_{2}^{\dagger} a_{1}\left|n_{1}, n_{2}\right\rangle=\sqrt{n_{1}} \sqrt{n_{2}+1}\left|n_{1}-1, n_{2}+1\right\rangle= | | |
| gilmore-lie-groups_FO1042 | 108 | 0.999 | |j, m-1\rangle \sqrt{j+m} \sqrt{j-m+1} | | |
| gilmore-lie-groups_FO1043 | 108 | 1.000 | J_{ \pm}|j m\rangle=|j, m \pm 1\rangle \sqrt{(j \pm m+1)(j \mp m)} | | |
| gilmore-lie-groups_FO1044 | 108 | 1.000 | J_{+}|j, j\rangle=0, J_{-}|j,-j\rangle= | | |
| gilmore-lie-groups_FO1045 | 108 | 1.000 | \left\langle j^{\prime} m^{\prime}\right| J_{ \pm}|j m\rangle=\sqrt{\left(j^{\prime} \pm m^{\prime}\right)(j \mp m)} \delta_{j^{\prime} j} \delta_{m^{\prime}, m \pm 1} | | |
| gilmore-lie-groups_FO1046 | 108 | 0.999 | \mathfrak{u}(3) | | |
| gilmore-lie-groups_FO1047 | 108 | 0.999 | U(3) | | |
| gilmore-lie-groups_FO1048 | 108 | 1.000 | 1 \leq i, j \leq 3 | | |
| gilmore-lie-groups_FO1049 | 108 | 1.000 | \left|n_{1}, n_{2}, n_{3}\right\rangle=\left|n_{1}\right\rangle \otimes\left|n_{2}\right\rangle \otimes\left|n_{3}\right\rangle | | |
| gilmore-lie-groups_FO1050 | 108 | 1.000 | b_{i}\left|n_{i}\right\rangle=\left|n_{i}-1\right\rangle \sqrt{n_{i}} | | |
| gilmore-lie-groups_FO1051 | 109 | 1.000 | N=\sum_{i=1}^{3} n_{i} | | |
| gilmore-lie-groups_FO1052 | 109 | 1.000 | D=(N+3-1)!/ N!(3- | | |
| gilmore-lie-groups_FO1053 | 109 | 0.953 | N | | |
| gilmore-lie-groups_FO1054 | 109 | 0.953 | (n) | | |
| gilmore-lie-groups_FO1055 | 109 | 0.852 | n . D | | |
| gilmore-lie-groups_FO1056 | 109 | 0.999 | f\left(x_{1}, x_{2}, \ldots, x_{n}\right) | | |
| gilmore-lie-groups_FO1057 | 109 | 1.000 | I_{D} | | |
| gilmore-lie-groups_FO1058 | 109 | 1.000 | \mathcal{O} | | |
| gilmore-lie-groups_FO1059 | 109 | 1.000 | f_{i}^{\dagger} f_{j} | | |
| gilmore-lie-groups_FO1060 | 109 | 1.000 | d, d^{\dagger} | | |
| gilmore-lie-groups_FO1061 | 109 | 0.996 | a, a^{\dagger} | | |
| gilmore-lie-groups_FO1062 | 110 | 0.996 | \left[d, d^{\dagger}\right]=1 | | |
| gilmore-lie-groups_FO1063 | 110 | 0.996 | [d, d]=\left[d^{\dagger}, d^{\dagger}\right]=0 | | |
| gilmore-lie-groups_FO1064 | 110 | 0.996 | A D | | |
| gilmore-lie-groups_FO1065 | 110 | 1.000 | B C=1 | | |
| gilmore-lie-groups_FO1066 | 110 | 0.998 | \operatorname{Sp}(2 ; \mathbb{R})= | | |
| gilmore-lie-groups_FO1067 | 110 | 0.982 | \operatorname{Sp}(2 ; \mathbb{R}) | | |
| gilmore-lie-groups_FO1068 | 110 | 1.000 | x, \partial | | |
| gilmore-lie-groups_FO1069 | 110 | 0.946 | a_{1}, a_{2}, \ldots, a_{n} | | |
| gilmore-lie-groups_FO1070 | 110 | 1.000 | \left|n_{1}, n_{2}, \ldots, n_{N}\right\rangle | | |
| gilmore-lie-groups_FO1071 | 110 | 1.000 | \hbar \omega(n+ | | |
| gilmore-lie-groups_FO1072 | 110 | 0.531 | \frac{N}{2} | | |
| gilmore-lie-groups_FO1073 | 110 | 0.531 | \operatorname{deg}(N, n)=(n+N-1)!/ n!(N-1)! | | |
| gilmore-lie-groups_FO1074 | 110 | 0.999 | f\left(x_{1}, x_{2}, \ldots, x_{N}\right) | | |
| gilmore-lie-groups_FO1075 | 110 | 1.000 | N^{2} | | |
| gilmore-lie-groups_FO1076 | 110 | 1.000 | \mathfrak{u}(N) | | |
| gilmore-lie-groups_FO1077 | 110 | 1.000 | \left[\mathcal{H}, a_{i}^{\dagger} a_{j}\right]=0 | | |
| gilmore-lie-groups_FO1078 | 110 | 1.000 | \mathfrak{s u}(N) | | |
| gilmore-lie-groups_FO1079 | 110 | 1.000 | \mathcal{H} | | |
| gilmore-lie-groups_FO1080 | 110 | 1.000 | a_{j} | | |
| gilmore-lie-groups_FO1081 | 110 | 1.000 | (N+1)^{2} | | |
| gilmore-lie-groups_FO1082 | 110 | 1.000 | n^{\prime} | | |
| gilmore-lie-groups_FO1083 | 110 | 1.000 | R, S, T, U, \ldots | | |
| gilmore-lie-groups_FO1084 | 110 | 1.000 | a^{\dagger}= | | |
| gilmore-lie-groups_FO1085 | 110 | 0.999 | a_{1}^{\dagger}, a_{2}^{\dagger}, \ldots, a_{n}^{\dagger} | | |
| gilmore-lie-groups_FO1086 | 111 | 1.000 | \mathcal{R}=a^{\dagger} R a=a_{i}^{\dagger} R_{i j} a_{j} | | |
| gilmore-lie-groups_FO1087 | 111 | 1.000 | \mathcal{S}, T, U, \ldots | | |
| gilmore-lie-groups_FO1088 | 111 | 1.000 | [R, S]=T \Leftrightarrow[\mathcal{R}, S]=\mathcal{T} | | |
| gilmore-lie-groups_FO1089 | 111 | 1.000 | e^{R} e^{S}=e^{U} \Leftrightarrow e^{\mathcal{R}} e^{\mathcal{S}}=e^{\mathcal{U}} | | |
| gilmore-lie-groups_FO1090 | 111 | 0.994 | \left[\frac{d}{d x}, e^{-x^{2} / 2}\right]=-x e^{-x^{2} / 2} | | |
| gilmore-lie-groups_FO1091 | 111 | 1.000 | a=\frac{1}{\sqrt{2}}\left(x+\frac{d}{d x}\right) | | |
| gilmore-lie-groups_FO1092 | 111 | 1.000 | \langle x \mid 0\rangle | | |
| gilmore-lie-groups_FO1093 | 111 | 1.000 | a\langle x \mid 0\rangle=0 | | |
| gilmore-lie-groups_FO1094 | 111 | 0.990 | \langle x \mid 0\rangle=e^{-x^{2} / 2} / \sqrt{1 \sqrt{\pi}} | | |
| gilmore-lie-groups_FO1095 | 111 | 1.000 | a^{\dagger}=\frac{1}{\sqrt{2}}\left(x-\frac{d}{d x}\right) | | |
| gilmore-lie-groups_FO1096 | 111 | 1.000 | \psi_{n}(x) | | |
| gilmore-lie-groups_FO1097 | 111 | 1.000 | \langle x \mid n\rangle= | | |
| gilmore-lie-groups_FO1098 | 111 | 0.999 | \frac{\left(a^{\dagger}\right)^{n}}{\sqrt{n!}}\langle x \mid 0\rangle | | |
| gilmore-lie-groups_FO1099 | 111 | 0.749 | L_{i j}=a_{i}^{\dagger} a_{j}-a_{j}^{\dagger} a_{i} | | |
| gilmore-lie-groups_FO1100 | 111 | 0.749 | Q_{i j}=a_{i}^{\dagger} a_{j}+a_{j}^{\dagger} a_{i} | | |
| gilmore-lie-groups_FO1101 | 111 | 1.000 | i \leq j | | |
| gilmore-lie-groups_FO1102 | 111 | 0.999 | \Gamma_{j i}^{*}=\Gamma_{i j} | | |
| gilmore-lie-groups_FO1103 | 112 | 0.992 | b_{i}=m_{i j} a_{j}: H=\sum_{i=1}^{n} \hbar \omega_{i}^{\prime}\left(b_{i}^{\dagger} b_{i}+\frac{1}{2}\right)+ | | |
| gilmore-lie-groups_FO1104 | 112 | 1.000 | \Gamma(H) | | |
| gilmore-lie-groups_FO1105 | 114 | 1.000 | \operatorname{EXP}(X) | | |
| gilmore-lie-groups_FO1106 | 114 | 1.000 | \operatorname{Tr} X=0 | | |
| gilmore-lie-groups_FO1107 | 114 | 1.000 | \pm \theta | | |
| gilmore-lie-groups_FO1108 | 114 | 1.000 | \pm i \theta | | |
| gilmore-lie-groups_FO1109 | 116 | 0.997 | a=b=0 | | |
| gilmore-lie-groups_FO1110 | 116 | 1.000 | 2 \pi n | | |
| gilmore-lie-groups_FO1111 | 116 | 0.651 | S O(2) \subset S L(2 ; R) | | |
| gilmore-lie-groups_FO1112 | 116 | 0.651 | a, b, 0 | | |
| gilmore-lie-groups_FO1113 | 116 | 1.000 | z=\cosh r \geq 1 | | |
| gilmore-lie-groups_FO1114 | 116 | 1.000 | (x, y)=(b, a) \sinh (r) / r, r^{2}=a^{2}+ | | |
| gilmore-lie-groups_FO1115 | 116 | 1.000 | b^{2} | | |
| gilmore-lie-groups_FO1116 | 116 | 1.000 | H_{2+}^{2} | | |
| gilmore-lie-groups_FO1119 | 117 | 1.000 | H_{1}^{2} | | |
| gilmore-lie-groups_FO1120 | 118 | 1.000 | c, 0 \leq c<2 \pi | | |
| gilmore-lie-groups_FO1121 | 118 | 0.999 | S L(2 ; \mathbb{R}) / S O(2) | | |
| gilmore-lie-groups_FO1122 | 118 | 0.996 | R^{2} \times S^{1} | | |
| gilmore-lie-groups_FO1123 | 118 | 1.000 | [S L(2 ; \mathbb{R}) / S O(1,1)] \times S O(1,1)\left(S O(1,1) \simeq R^{1}\right) | | |
| gilmore-lie-groups_FO1124 | 118 | 1.000 | R^{1} \times S^{1} | | |
| gilmore-lie-groups_FO1125 | 118 | 1.000 | R^{N}, N=n^{2} | | |
| gilmore-lie-groups_FO1126 | 119 | 1.000 | R^{m} | | |
| gilmore-lie-groups_FO1127 | 119 | 0.943 | S O(2,1) | | |
| gilmore-lie-groups_FO1128 | 119 | 0.993 | S O(2,1) / S O(2) | | |
| gilmore-lie-groups_FO1129 | 119 | 0.993 | S U(1,1) / U(1) | | |
| gilmore-lie-groups_FO1130 | 119 | 0.969 | U(1) | | |
| gilmore-lie-groups_FO1131 | 119 | 0.969 | 2: 1 | | |
| gilmore-lie-groups_FO1132 | 119 | 0.936 | b_{3} | | |
| gilmore-lie-groups_FO1133 | 119 | 1.000 | U\left(b_{3}+2 \pi\right)=-U\left(b_{3}\right) | | |
| gilmore-lie-groups_FO1134 | 119 | 1.000 | a_{3} | | |
| gilmore-lie-groups_FO1135 | 120 | 0.984 | U(1) \subset S U(1,1) | | |
| gilmore-lie-groups_FO1136 | 120 | 0.984 | 4 \pi | | |
| gilmore-lie-groups_FO1137 | 120 | 0.984 | S O(2) \subset S O(2,1) | | |
| gilmore-lie-groups_FO1138 | 120 | 0.811 | \operatorname{SO}(2,1) | | |
| gilmore-lie-groups_FO1139 | 120 | 0.992 | 2 \rightarrow 1 | | |
| gilmore-lie-groups_FO1140 | 120 | 1.000 | \sqrt{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}} \leq | | |
| gilmore-lie-groups_FO1141 | 120 | 0.994 | |\mathbf{a}|=\pi | | |
| gilmore-lie-groups_FO1142 | 120 | 0.999 | \operatorname{SU}(2) | | |
| gilmore-lie-groups_FO1143 | 120 | 0.998 | 2 \pi\left(\sqrt{b_{1}^{2}+b_{2}^{2}+b_{3}^{2}}<2 \pi\right) | | |
| gilmore-lie-groups_FO1144 | 120 | 0.986 | -I_{2} | | |
| gilmore-lie-groups_FO1145 | 121 | 0.652 | x^{2}+y^{2}+z^{2}=1 | | |
| gilmore-lie-groups_FO1146 | 121 | 0.652 | S U(2) / U(1) | | |
| gilmore-lie-groups_FO1147 | 121 | 0.989 | x^{\prime}, y^{\prime}, z^{\prime} | | |
| gilmore-lie-groups_FO1148 | 121 | 0.989 | x^{\prime 2}+y^{\prime 2}+z^{\prime 2}=1 | | |
| gilmore-lie-groups_FO1149 | 121 | 0.604 | S U(2) \rightarrow S O(3) | | |
| gilmore-lie-groups_FO1150 | 121 | 1.000 | \bar{G} | | |
| gilmore-lie-groups_FO1151 | 121 | 1.000 | \bar{G} / D | | |
| gilmore-lie-groups_FO1152 | 121 | 1.000 | \bar{G}: g d_{i}=d_{i} g | | |
| gilmore-lie-groups_FO1153 | 121 | 1.000 | d_{i} \in D | | |
| gilmore-lie-groups_FO1154 | 121 | 0.863 | D_{\mathrm{MAX}} | | |
| gilmore-lie-groups_FO1155 | 121 | 0.979 | D_{\text {MAX }} | | |
| gilmore-lie-groups_FO1157 | 121 | 1.000 | G_{1}=\bar{G} / D_{1} | | |
| gilmore-lie-groups_FO1158 | 121 | 0.848 | \bar{G} / D_{\mathrm{MAX}} | | |
| gilmore-lie-groups_FO1161 | 122 | 1.000 | \lambda I_{2} | | |
| gilmore-lie-groups_FO1162 | 122 | 1.000 | \lambda^{*} \lambda=1 | | |
| gilmore-lie-groups_FO1163 | 122 | 1.000 | \operatorname{det}\left(\lambda I_{2}\right)=+1 | | |
| gilmore-lie-groups_FO1164 | 122 | 1.000 | \lambda= \pm 1 . D | | |
| gilmore-lie-groups_FO1165 | 122 | 1.000 | D=\left\{I_{2},-I_{2}\right\} | | |
| gilmore-lie-groups_FO1166 | 122 | 1.000 | S O(3), D=\lambda I_{3} | | |
| gilmore-lie-groups_FO1167 | 122 | 0.874 | \lambda=+1 | | |
| gilmore-lie-groups_FO1168 | 122 | 0.874 | S U(2) /\left\{I_{2},-I_{2}\right\}=S O(3) / I_{3}=S O(3) | | |
| gilmore-lie-groups_FO1169 | 122 | 1.000 | \overline{S O(2,1)}= | | |
| gilmore-lie-groups_FO1170 | 122 | 1.000 | \overline{S U(1,1)} | | |
| gilmore-lie-groups_FO1171 | 122 | 1.000 | \overline{S O(2,1) / S O(2)} \times \overline{S O(2)}=\overline{S U(1,1) / U(1)} \times \overline{U(1)}=[S O(2,1) / S O(2)] \times | | |
| gilmore-lie-groups_FO1172 | 122 | 1.000 | \overline{S O(2)}=S U(1,1) / U(1) \times \overline{U(1)}=R^{2} \times R^{1} | | |
| gilmore-lie-groups_FO1173 | 122 | 1.000 | \overline{S O(2,1)}=\overline{S U(1,1)} | | |
| gilmore-lie-groups_FO1174 | 123 | 1.000 | R_{+}^{2} | | |
| gilmore-lie-groups_FO1175 | 123 | 0.974 | (w, z) | | |
| gilmore-lie-groups_FO1176 | 123 | 1.000 | R_{+}^{2}(x>0, y) | | |
| gilmore-lie-groups_FO1177 | 123 | 1.000 | R^{2}(w, z) | | |
| gilmore-lie-groups_FO1178 | 123 | 0.995 | x>0 | | |
| gilmore-lie-groups_FO1179 | 124 | 0.981 | a, a^{\dagger}, I | | |
| gilmore-lie-groups_FO1180 | 124 | 1.000 | l=l^{\prime}=l^{\prime \prime}, r=r^{\prime}=r^{\prime \prime}, \delta^{\prime}=\delta+\frac{1}{2} l r=\delta^{\prime \prime}+l^{\prime \prime} r^{\prime \prime} | | |
| gilmore-lie-groups_FO1181 | 124 | 1.000 | \delta=0 | | |
| gilmore-lie-groups_FO1182 | 125 | 1.000 | \hat{n}=a^{\dagger} a, a, a^{\dagger}, I | | |
| gilmore-lie-groups_FO1183 | 125 | 1.000 | \operatorname{EXP}\left(\eta a^{\dagger} a+r a^{\dagger}+l a\right) | | |
| gilmore-lie-groups_FO1184 | 125 | 1.000 | \operatorname{EXP}\left(r^{\prime} a^{\dagger}\right) \operatorname{EXP}\left(\eta^{\prime} a^{\dagger} a+\delta^{\prime} I\right) \operatorname{EXP}\left(l^{\prime} a\right) | | |
| gilmore-lie-groups_FO1185 | 125 | 0.948 | e^{l^{\prime} a}|0\rangle=|0\rangle,\langle 0| e^{r^{\prime} a^{\dagger}}=\langle 0| | | |
| gilmore-lie-groups_FO1186 | 125 | 0.948 | e^{\eta^{\prime} a^{\dagger} a}|0\rangle=|0\rangle | | |
| gilmore-lie-groups_FO1187 | 125 | 1.000 | \mathfrak{s u}(2) | | |
| gilmore-lie-groups_FO1188 | 125 | 1.000 | (|j,-j\rangle) | | |
| gilmore-lie-groups_FO1189 | 126 | 1.000 | \theta_{z}^{\prime} | | |
| gilmore-lie-groups_FO1190 | 126 | 1.000 | \mathbf{J} \rightarrow \frac{1}{2} \sigma | | |
| gilmore-lie-groups_FO1191 | 126 | 1.000 | \theta_{ \pm}=\theta_{1} \pm i \theta_{2} | | |
| gilmore-lie-groups_FO1192 | 127 | 1.000 | e^{\alpha J_{+}} e^{\beta J_{-}} | | |
| gilmore-lie-groups_FO1193 | 127 | 1.000 | \operatorname{EXP}\left(\beta^{\prime} J_{-}\right) | | |
| gilmore-lie-groups_FO1194 | 127 | 0.999 | \operatorname{EXP}\left(n^{\prime} J_{z}\right) \operatorname{EXP}\left(\alpha^{\prime} J_{+}\right) | | |
| gilmore-lie-groups_FO1195 | 127 | 0.999 | \alpha^{\prime}, \beta^{\prime}, n^{\prime} | | |
| gilmore-lie-groups_FO1196 | 127 | 1.000 | (1+\alpha \beta)^{2 j} | | |
| gilmore-lie-groups_FO1197 | 127 | 1.000 | \mathcal{A}, \mathcal{B} | | |
| gilmore-lie-groups_FO1198 | 127 | 0.995 | e^{\mathcal{A}} e^{\mathcal{B}} | | |
| gilmore-lie-groups_FO1199 | 127 | 0.999 | e^{\mathcal{B}^{\prime}} e^{\mathcal{A}^{\prime}}\left(\mathcal{A}^{\prime}, \mathcal{B}^{\prime}\right. | | |
| gilmore-lie-groups_FO1200 | 127 | 0.999 | \left.\mathcal{A}, \mathcal{B}\right) | | |
| gilmore-lie-groups_FO1201 | 127 | 1.000 | e^{A} e^{B} | | |
| gilmore-lie-groups_FO1202 | 127 | 1.000 | e^{B^{\prime}} e^{A^{\prime}} | | |
| gilmore-lie-groups_FO1203 | 127 | 1.000 | A^{\prime}, B^{\prime} | | |
| gilmore-lie-groups_FO1204 | 127 | 1.000 | A^{\prime} \leftrightarrow \mathcal{A}^{\prime} B^{\prime} \leftrightarrow \mathcal{B}^{\prime} | | |
| gilmore-lie-groups_FO1205 | 127 | 1.000 | (\mathcal{A}, \mathcal{B}) \leftrightarrow | | |
| gilmore-lie-groups_FO1206 | 127 | 0.998 | \left(\mathcal{A}^{\prime}, \mathcal{B}^{\prime}\right) | | |
| gilmore-lie-groups_FO1207 | 127 | 1.000 | \mathcal{A}, \mathcal{B}, \ldots | | |
| gilmore-lie-groups_FO1208 | 128 | 1.000 | A, B, \ldots | | |
| gilmore-lie-groups_FO1209 | 128 | 1.000 | \mathcal{A} \leftrightarrow A | | |
| gilmore-lie-groups_FO1210 | 128 | 1.000 | t+\delta t | | |
| gilmore-lie-groups_FO1211 | 128 | 1.000 | \left|\psi\left(t_{f}\right)\right\rangle | | |
| gilmore-lie-groups_FO1212 | 128 | 1.000 | t_{f} | | |
| gilmore-lie-groups_FO1213 | 128 | 1.000 | \left|\psi\left(t_{f}\right)\right\rangle= | | |
| gilmore-lie-groups_FO1214 | 128 | 0.996 | U\left(t_{f}, t_{i}\right)\left|\psi\left(t_{i}\right)\right\rangle | | |
| gilmore-lie-groups_FO1215 | 128 | 1.000 | H\left(t^{\prime}\right) | | |
| gilmore-lie-groups_FO1216 | 128 | 1.000 | H(t), t^{\prime} \neq t | | |
| gilmore-lie-groups_FO1217 | 129 | 1.000 | 2 j+1 | | |
| gilmore-lie-groups_FO1218 | 129 | 1.000 | \left|j, m_{j}\right\rangle | | |
| gilmore-lie-groups_FO1219 | 129 | 1.000 | g(t) \in S U(2) | | |
| gilmore-lie-groups_FO1220 | 129 | 1.000 | (2 j+1) \times(2 j+1) | | |
| gilmore-lie-groups_FO1221 | 129 | 1.000 | a(t) | | |
| gilmore-lie-groups_FO1222 | 129 | 1.000 | b(t) | | |
| gilmore-lie-groups_FO1223 | 129 | 1.000 | \sigma_{1}, \sigma_{2}, \sigma_{3} | | |
| gilmore-lie-groups_FO1224 | 129 | 1.000 | a\left(t_{i}\right)= | | |
| gilmore-lie-groups_FO1225 | 129 | 0.999 | 1, b\left(t_{i}\right)=0 | | |
| gilmore-lie-groups_FO1226 | 129 | 0.999 | a\left(t_{f}\right), b\left(t_{f}\right) | | |
| gilmore-lie-groups_FO1227 | 129 | 0.971 | (2 j+1) \times | | |
| gilmore-lie-groups_FO1228 | 129 | 1.000 | (2 j+1) | | |
| gilmore-lie-groups_FO1229 | 129 | 1.000 | \left|\psi\left(t_{i}\right)\right\rangle | | |
| gilmore-lie-groups_FO1230 | 130 | 1.000 | |n\rangle, n=0,1,2, \ldots | | |
| gilmore-lie-groups_FO1231 | 130 | 1.000 | U= | | |
| gilmore-lie-groups_FO1232 | 130 | 1.000 | \operatorname{EXP}\left(i\left[n(t) a^{\dagger} a+r(t) a^{\dagger}+r^{*}(t) a+d(t) I\right]\right) | | |
| gilmore-lie-groups_FO1233 | 130 | 0.982 | \left(n, r, r^{*}, d\right) | | |
| gilmore-lie-groups_FO1234 | 130 | 1.000 | d \omega(t) / d t=0 | | |
| gilmore-lie-groups_FO1235 | 130 | 1.000 | \rho | | |
| gilmore-lie-groups_FO1236 | 130 | 1.000 | \rho=e^{-\beta H} / Z | | |
| gilmore-lie-groups_FO1237 | 130 | 1.000 | Z=\operatorname{tr} e^{-\beta H} | | |
| gilmore-lie-groups_FO1238 | 130 | 1.000 | \beta=1 / k_{B} T, k_{B} | | |
| gilmore-lie-groups_FO1239 | 131 | 1.000 | \left\langle e^{\Lambda}\right\rangle | | |
| gilmore-lie-groups_FO1240 | 131 | 1.000 | \Lambda=\lambda \cdot \mathbf{J} | | |
| gilmore-lie-groups_FO1241 | 131 | 1.000 | H \cdot \Lambda=(H, \Lambda)=\frac{1}{2} \operatorname{tr} H \Lambda | | |
| gilmore-lie-groups_FO1242 | 131 | 1.000 | |H|=\sqrt{(H, H)} | | |
| gilmore-lie-groups_FO1243 | 131 | 1.000 | |\Lambda|=\sqrt{(\Lambda, \Lambda)} | | |
| gilmore-lie-groups_FO1244 | 131 | 1.000 | 2^{N} | | |
| gilmore-lie-groups_FO1245 | 131 | 1.000 | 2 J+1 | | |
| gilmore-lie-groups_FO1246 | 131 | 1.000 | N=2 J | | |
| gilmore-lie-groups_FO1247 | 131 | 1.000 | \mu(H, \Lambda, T) | | |
| gilmore-lie-groups_FO1248 | 131 | 1.000 | \left\langle J_{-}\right\rangle | | |
| gilmore-lie-groups_FO1249 | 131 | 1.000 | \frac{\partial}{\partial \lambda^{*}}\left\langle e^{\Lambda}\right\rangle /\left\langle e^{0}\right\rangle | | |
| gilmore-lie-groups_FO1250 | 131 | 1.000 | \Lambda=0 | | |
| gilmore-lie-groups_FO1251 | 131 | 1.000 | \left.\frac{\partial}{\partial \lambda^{*}} \log \left(\left\langle e^{\Lambda}\right\rangle\right)\right|_{\Lambda=0} | | |
| gilmore-lie-groups_FO1252 | 132 | 0.999 | \rho=e^{-\beta\left(\hbar \omega a^{\dagger} a+\alpha a^{\dagger}+\alpha^{*} a+\delta I\right)} / Z | | |
| gilmore-lie-groups_FO1253 | 132 | 1.000 | \chi(H, \Lambda, T)=\operatorname{tr} e^{-\beta H} e^{\lambda_{n} a^{\dagger} a+\lambda a^{\dagger}+\lambda^{*} a+d I} / Z=\left\langle e^{\Lambda}\right\rangle | | |
| gilmore-lie-groups_FO1254 | 132 | 0.967 | |0\rangle,|1\rangle,|2\rangle, \ldots | | |
| gilmore-lie-groups_FO1255 | 132 | 0.989 | H, \Lambda | | |
| gilmore-lie-groups_FO1256 | 132 | 0.999 | Z_{l} | | |
| gilmore-lie-groups_FO1257 | 132 | 0.999 | Z_{r} | | |
| gilmore-lie-groups_FO1258 | 133 | 1.000 | A<0 | | |
| gilmore-lie-groups_FO1259 | 133 | 1.000 | \lambda_{n}= | | |
| gilmore-lie-groups_FO1260 | 133 | 1.000 | d=0 | | |
| gilmore-lie-groups_FO1261 | 133 | 1.000 | D=\mathrm{Id}, G | | |
| gilmore-lie-groups_FO1262 | 133 | 1.000 | \lambda I_{n} | | |
| gilmore-lie-groups_FO1263 | 134 | 0.940 | i t / \hbar \leftrightarrow 1 / k_{B} T | | |
| gilmore-lie-groups_FO1264 | 134 | 1.000 | \phi\left(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\right) | | |
| gilmore-lie-groups_FO1265 | 134 | 1.000 | \phi\left(\left(w_{1}, z_{1}\right),\left(w_{2}, z_{2}\right)\right) | | |
| gilmore-lie-groups_FO1266 | 134 | 1.000 | a^{2}+b^{2}-c^{2}<0 | | |
| gilmore-lie-groups_FO1267 | 134 | 1.000 | S O(2) \subset | | |
| gilmore-lie-groups_FO1268 | 134 | 0.999 | c_{T} | | |
| gilmore-lie-groups_FO1269 | 134 | 0.999 | a^{2}+b^{2}>0\left(\sqrt{a^{2}+b^{2}}=\beta \times c,|\beta|<1\right) | | |
| gilmore-lie-groups_FO1270 | 134 | 0.999 | 2 \pi \gamma | | |
| gilmore-lie-groups_FO1271 | 134 | 0.999 | \gamma=1 / \sqrt{1-\beta^{2}} | | |
| gilmore-lie-groups_FO1272 | 134 | 1.000 | \beta^{2}=\left(a^{2}+b^{2}\right) / c^{2} | | |
| gilmore-lie-groups_FO1273 | 134 | 0.947 | S U(3) | | |
| gilmore-lie-groups_FO1274 | 134 | 0.954 | \left\{I_{3}, \lambda I_{3}, \lambda^{2} I_{3}\right\} | | |
| gilmore-lie-groups_FO1275 | 134 | 0.954 | \lambda=e^{2 \pi i / 3} | | |
| gilmore-lie-groups_FO1276 | 134 | 0.954 | \operatorname{SU}(3) / D_{\text {MAX }} | | |
| gilmore-lie-groups_FO1277 | 134 | 0.229 | [\Re e g(\mathfrak{s u}(3))] | | |
| gilmore-lie-groups_FO1278 | 134 | 1.000 | \epsilon I_{n}, \epsilon=e^{2 \pi i / n} | | |
| gilmore-lie-groups_FO1279 | 134 | 0.988 | S U(n) / D_{\text {MAX }} | | |
| gilmore-lie-groups_FO1280 | 135 | 1.000 | a^{\dagger}, a | | |
| gilmore-lie-groups_FO1281 | 135 | 1.000 | \left\langle n^{\prime}\right| x^{k}|n\rangle | | |
| gilmore-lie-groups_FO1282 | 135 | 1.000 | e^{\lambda x} | | |
| gilmore-lie-groups_FO1283 | 135 | 1.000 | x^{k} | | |
| gilmore-lie-groups_FO1284 | 135 | 1.000 | \left[a, a^{\dagger}\right]=I | | |
| gilmore-lie-groups_FO1285 | 135 | 0.955 | e^{r a^{\dagger}} e^{\delta I} e^{l a} | | |
| gilmore-lie-groups_FO1286 | 135 | 0.999 | \left\langle n^{\prime}\right| x^{4}|n\rangle | | |
| gilmore-lie-groups_FO1287 | 135 | 1.000 | \left\langle e^{i k x}\right\rangle | | |
| gilmore-lie-groups_FO1288 | 136 | 1.000 | P_{n} \simeq e^{-n \beta \hbar \omega} | | |
| gilmore-lie-groups_FO1289 | 136 | 1.000 | k \rightarrow 0 | | |
| gilmore-lie-groups_FO1290 | 136 | 1.000 | \left\langle e^{i k x}\right\rangle=e^{\delta} | | |
| gilmore-lie-groups_FO1291 | 136 | 0.976 | \delta | | |
| gilmore-lie-groups_FO1292 | 136 | 1.000 | \infty \times \infty | | |
| gilmore-lie-groups_FO1293 | 136 | 0.999 | \omega^{\prime}=\omega | | |
| gilmore-lie-groups_FO1294 | 136 | 0.999 | \alpha, \beta | | |
| gilmore-lie-groups_FO1295 | 136 | 0.999 | \gamma | | |
| gilmore-lie-groups_FO1296 | 136 | 1.000 | \left[X_{i}, X_{j}\right]=\sum_{k=1}^{N} C_{i j}{ }^{k} X_{k} | | |
| gilmore-lie-groups_FO1297 | 136 | 1.000 | R | | |
| gilmore-lie-groups_FO1298 | 136 | 1.000 | \left(R\left(a^{i} X_{i}\right)\right)^{\dagger}=\left(a^{i} R\left(X_{i}\right)\right)^{\dagger}=\left(a^{i}\right)^{*} R^{\dagger}\left(X_{i}\right) | | |
| gilmore-lie-groups_FO1299 | 136 | 1.000 | S\left(a^{i} X_{i}\right)=0 \Rightarrow a^{i}=0 | | |
| gilmore-lie-groups_FO1300 | 136 | 1.000 | \mathcal{H}=R\left(a^{i} X_{i}\right) | | |
| gilmore-lie-groups_FO1301 | 136 | 0.999 | H_{1}, H_{2}, \ldots, H_{r} \in \mathfrak{g} | | |
| gilmore-lie-groups_FO1302 | 136 | 1.000 | \left[H_{i}, H_{j}\right]=0,1 \leq i, j \leq r | | |
| gilmore-lie-groups_FO1303 | 136 | 1.000 | R\left(H_{i}\right) | | |
| gilmore-lie-groups_FO1304 | 136 | 1.000 | \left[R\left(H_{i}\right)\right]_{\alpha \beta}=r_{\alpha}(i) \delta_{\alpha \beta} | | |
| gilmore-lie-groups_FO1305 | 137 | 1.000 | \left[S\left(H_{i}\right), S\left(H_{j}\right)\right]=0 | | |
| gilmore-lie-groups_FO1306 | 137 | 1.000 | r | | |
| gilmore-lie-groups_FO1307 | 137 | 1.000 | S\left(H_{i}\right) | | |
| gilmore-lie-groups_FO1308 | 137 | 1.000 | U(t)=R\left(e^{-\frac{i}{\hbar} \mathcal{H} t}\right)=e^{-\frac{i}{\hbar} R(\mathcal{H}) t} | | |
| gilmore-lie-groups_FO1309 | 137 | 1.000 | \rho(T)=e^{-\beta \mathcal{H}} / Z= | | |
| gilmore-lie-groups_FO1310 | 137 | 0.975 | R\left(e^{-\beta \mathcal{H}}\right) / Z=e^{-\beta R(\mathcal{H})} / Z | | |
| gilmore-lie-groups_FO1311 | 137 | 1.000 | U(t) | | |
| gilmore-lie-groups_FO1312 | 137 | 1.000 | \rho(T) | | |
| gilmore-lie-groups_FO1313 | 137 | 1.000 | i t / \hbar \leftrightarrow \beta=1 / k_{B} T | | |
| gilmore-lie-groups_FO1314 | 137 | 1.000 | G=e^{\mathfrak{g}} | | |
| gilmore-lie-groups_FO1315 | 137 | 1.000 | H_{i} | | |
| gilmore-lie-groups_FO1316 | 137 | 1.000 | x^{i} | | |
| gilmore-lie-groups_FO1317 | 137 | 1.000 | S=e^{y^{k} X_{k}} | | |
| gilmore-lie-groups_FO1318 | 137 | 1.000 | e^{x^{i} X_{i}} | | |
| gilmore-lie-groups_FO1319 | 137 | 1.000 | U(\alpha)=e^{\left(\alpha a^{\dagger}-\alpha^{*} a\right)} | | |
| gilmore-lie-groups_FO1320 | 138 | 0.896 | a^{\dagger}|n\rangle= | | |
| gilmore-lie-groups_FO1321 | 138 | 1.000 | |n+1\rangle \sqrt{n+1} | | |
| gilmore-lie-groups_FO1322 | 138 | 1.000 | \langle\beta \mid \alpha\rangle | | |
| gilmore-lie-groups_FO1323 | 138 | 1.000 | \langle\alpha \mid \alpha\rangle=1 | | |
| gilmore-lie-groups_FO1324 | 138 | 1.000 | a|\alpha\rangle=\alpha|\alpha\rangle | | |
| gilmore-lie-groups_FO1325 | 138 | 1.000 | \langle\alpha| x|\alpha\rangle=\left(\alpha^{*}+\alpha\right) / \sqrt{2} | | |
| gilmore-lie-groups_FO1326 | 138 | 0.966 | e^{i \phi J_{z}}\left|{ }_{-j}^{j}\right\rangle=\left|{ }_{-j}^{j}\right\rangle e^{-i j \phi} | | |
| gilmore-lie-groups_FO1327 | 138 | 1.000 | e^{i\left(\theta_{x} J_{x}+\theta_{y} J_{y}\right)} | | |
| gilmore-lie-groups_FO1328 | 138 | 1.000 | e^{i \alpha_{+} J_{+}} e^{i \alpha_{z} J_{z}} e^{i \alpha_{-} J_{-}} | | |
| gilmore-lie-groups_FO1329 | 138 | 0.859 | e^{i \alpha_{-} J_{-}}\left|{ }_{-j}^{j}\right\rangle=\left|{ }_{-j}^{j}\right\rangle | | |
| gilmore-lie-groups_FO1330 | 138 | 0.952 | e^{i \alpha_{z} J_{z}}\left|{ }_{-j}^{j}\right\rangle=\left|{ }_{-j}^{j}\right\rangle e^{-i j \alpha_{z}} | | |
| gilmore-lie-groups_FO1331 | 139 | 0.815 | \left.\left.J_{-}\right|_{\theta_{x} \theta_{y}} ^{j}\right\rangle | | |
| gilmore-lie-groups_FO1332 | 139 | 0.815 | \left|\begin{array}{c}j \\ \theta_{x} \theta_{y}\end{array}\right\rangle | | |
| gilmore-lie-groups_FO1333 | 139 | 0.815 | \left|\begin{array}{c}j \\ +j\end{array}\right\rangle | | |
| gilmore-lie-groups_FO1334 | 139 | 1.000 | H_{4} | | |
| gilmore-lie-groups_FO1335 | 139 | 1.000 | e^{-i \phi}=\left(\theta_{x}-i \theta_{y}\right) / \theta | | |
| gilmore-lie-groups_FO1336 | 139 | 1.000 | \theta^{\prime} | | |
| gilmore-lie-groups_FO1337 | 139 | 1.000 | |u\rangle | | |
| gilmore-lie-groups_FO1338 | 139 | 1.000 | J_{3} | | |
| gilmore-lie-groups_FO1339 | 139 | 1.000 | \left(E^{\prime} J_{3}-Z\right)|v\rangle=0 | | |
| gilmore-lie-groups_FO1340 | 139 | 1.000 | E^{\prime}=\sqrt{E^{2}+p^{2}} | | |
| gilmore-lie-groups_FO1341 | 139 | 1.000 | |v\rangle=U|u\rangle | | |
| gilmore-lie-groups_FO1342 | 139 | 1.000 | E= \pm \sqrt{(Z / m)^{2}-p^{2}} | | |
| gilmore-lie-groups_FO1343 | 139 | 0.871 | (p \rightarrow 0) | | |
| gilmore-lie-groups_FO1344 | 139 | 1.000 | j, p, Z | | |
| gilmore-lie-groups_FO1345 | 139 | 0.986 | \left.|u\rangle=\left.e^{i \theta J_{2}}\right|_{m} ^{j}\right\rangle | | |
| gilmore-lie-groups_FO1346 | 139 | 1.000 | z=\cos (\beta) | | |
| gilmore-lie-groups_FO1347 | 139 | 1.000 | D_{m n}^{j} | | |
| gilmore-lie-groups_FO1348 | 139 | 1.000 | j=l | | |
| gilmore-lie-groups_FO1349 | 140 | 1.000 | I_{2} \in S U(2) | | |
| gilmore-lie-groups_FO1350 | 140 | 1.000 | I_{3} \in S O(3) | | |
| gilmore-lie-groups_FO1351 | 140 | 0.835 | I_{3} | | |
| gilmore-lie-groups_FO1352 | 140 | 0.999 | S U(2) / U(1)(z=1, x=y=0) | | |
| gilmore-lie-groups_FO1353 | 140 | 0.793 | z=-1, x=y=0 | | |
| gilmore-lie-groups_FO1354 | 140 | 0.701 | S O(3) / S O(2)(z=1, x=y=0) | | |
| gilmore-lie-groups_FO1355 | 140 | 0.701 | (z=-1, x=y=0) | | |
| gilmore-lie-groups_FO1356 | 140 | 1.000 | S U(2) / U(1)=S^{2}=S O(3) / S O(2) | | |
| gilmore-lie-groups_FO1357 | 140 | 0.986 | S U(2) \downarrow S O(3) | | |
| gilmore-lie-groups_FO1358 | 140 | 0.918 | U(1) \downarrow S O(2) | | |
| gilmore-lie-groups_FO1359 | 140 | 1.000 | e^{2 \pi i k / r} I_{n} | | |
| gilmore-lie-groups_FO1360 | 140 | 1.000 | n / r | | |
| gilmore-lie-groups_FO1361 | 140 | 1.000 | e^{2 \pi i k / n} I_{n} | | |
| gilmore-lie-groups_FO1362 | 140 | 0.999 | \left[\begin{array}{cc}-\lambda & 0 \\ 0 & -1 / \lambda\end{array}\right] | | |
| gilmore-lie-groups_FO1363 | 140 | 1.000 | \lambda>1 | | |
| gilmore-lie-groups_FO1364 | 140 | 1.000 | \hbar \omega | | |
| gilmore-lie-groups_FO1365 | 140 | 1.000 | \sigma_{z}^{(i)} | | |
| gilmore-lie-groups_FO1366 | 140 | 1.000 | a^{\dagger} a | | |
| gilmore-lie-groups_FO1367 | 140 | 0.999 | \sigma_{+}^{(j)}\left(\sigma_{ \pm}^{(j)}=\frac{1}{2}\left(\sigma_{x}^{(j)} \pm i \sigma_{y}^{(j)}\right)\right) | | |
| gilmore-lie-groups_FO1368 | 140 | 1.000 | \sigma_{-}^{(j)} a^{\dagger} | | |
| gilmore-lie-groups_FO1369 | 140 | 1.000 | \sum_{i=1}^{N} \frac{1}{2} \sigma_{z}^{(i)} \rightarrow J_{z} | | |
| gilmore-lie-groups_FO1370 | 140 | 0.999 | \sum_{i=1}^{N} \sigma_{ \pm}^{(i)} \rightarrow J_{ \pm} | | |
| gilmore-lie-groups_FO1371 | 140 | 0.999 | \mathfrak{s} \mathfrak{u}(2) | | |
| gilmore-lie-groups_FO1372 | 141 | 1.000 | J_{z} \rightarrow\left\langle J_{z}(t)\right\rangle, J_{+} \rightarrow\left\langle J_{+}(t)\right\rangle, J_{-} \rightarrow\left\langle J_{-}(t)\right\rangle=\left\langle J_{+}(t)\right\rangle^{*} | | |
| gilmore-lie-groups_FO1373 | 141 | 1.000 | |\alpha(t)\rangle=U(\alpha(t))|0\rangle=e^{\alpha a^{\dagger}-\alpha^{*} a}|0\rangle | | |
| gilmore-lie-groups_FO1374 | 141 | 1.000 | \alpha(t) | | |
| gilmore-lie-groups_FO1375 | 141 | 1.000 | \left\langle J_{z}(t)\right\rangle | | |
| gilmore-lie-groups_FO1376 | 141 | 1.000 | \left\langle J_{+}(t)\right\rangle=\left\langle J_{-}(t)\right\rangle^{*} | | |
| gilmore-lie-groups_FO1377 | 141 | 1.000 | |\beta\rangle | | |
| gilmore-lie-groups_FO1378 | 141 | 1.000 | \beta | | |
| gilmore-lie-groups_FO1379 | 141 | 1.000 | a^{\dagger} \rightarrow\langle a(t)\rangle^{*} | | |
| gilmore-lie-groups_FO1380 | 141 | 1.000 | a \rightarrow\langle a(t)\rangle | | |
| gilmore-lie-groups_FO1381 | 141 | 1.000 | a^{\dagger} a \rightarrow\langle a(t)\rangle^{*}\langle a(t)\rangle | | |
| gilmore-lie-groups_FO1382 | 141 | 1.000 | m=-\frac{1}{2} | | |
| gilmore-lie-groups_FO1383 | 141 | 1.000 | M=-J, J=N / 2 | | |
| gilmore-lie-groups_FO1384 | 141 | 0.997 | |\theta(t)\rangle=e^{i \theta(t) \cdot \mathbf{J}}|J,-J\rangle | | |
| gilmore-lie-groups_FO1385 | 141 | 0.997 | J=N / 2 | | |
| gilmore-lie-groups_FO1386 | 141 | 1.000 | \theta(t) | | |
| gilmore-lie-groups_FO1387 | 141 | 1.000 | \langle a(t)\rangle | | |
| gilmore-lie-groups_FO1388 | 141 | 1.000 | \langle a(t)\rangle^{*} | | |
| gilmore-lie-groups_FO1389 | 141 | 1.000 | \langle a\rangle | | |
| gilmore-lie-groups_FO1390 | 141 | 1.000 | \langle a\rangle^{*} | | |
| gilmore-lie-groups_FO1391 | 141 | 1.000 | \left\langle J_{+}\right\rangle=\left\langle J_{-}\right\rangle^{*} | | |
| gilmore-lie-groups_FO1392 | 141 | 1.000 | T\left(\beta=1 / k_{B} T\right) | | |
| gilmore-lie-groups_FO1393 | 141 | 1.000 | \left\langle\sigma_{+}^{(i)}\right\rangle_{T} | | |
| gilmore-lie-groups_FO1394 | 141 | 1.000 | a^{\dagger}, a, a^{\dagger} a | | |
| gilmore-lie-groups_FO1395 | 142 | 1.000 | \sigma_{z}, \sigma_{+}, \sigma_{-} | | |
| gilmore-lie-groups_FO1396 | 142 | 1.000 | \epsilon, \hbar \omega, \lambda, N | | |
| gilmore-lie-groups_FO1397 | 142 | 1.000 | \left\langle J_{+}\right\rangle_{T} \neq 0 | | |
| gilmore-lie-groups_FO1398 | 142 | 1.000 | \lambda^{2} / \epsilon \hbar \omega | | |
| gilmore-lie-groups_FO1399 | 142 | 1.000 | a(t), b(t) | | |
| gilmore-lie-groups_FO1400 | 142 | 1.000 | a_{r}^{2}+a_{i}^{2}+b_{r}^{2}+b_{i}^{2}=1 | | |
| gilmore-lie-groups_FO1401 | 142 | 0.542 | \left(a_{r}, a_{i}, b_{r}, b_{i}\right) | | |
| gilmore-lie-groups_FO1402 | 142 | 1.000 | \lambda_{3} \rightarrow 0 | | |
| gilmore-lie-groups_FO1403 | 142 | 1.000 | \lambda_{n} \rightarrow 0 | | |
| gilmore-lie-groups_FO1404 | 142 | 1.000 | d=\delta=0 | | |
| gilmore-lie-groups_FO1405 | 142 | 1.000 | \left\langle J_{+} J_{-}+J_{-} J_{+}\right\rangle | | |
| gilmore-lie-groups_FO1406 | 142 | 1.000 | \left\langle a a^{\dagger}+a^{\dagger} a\right\rangle | | |
| gilmore-lie-groups_FO1407 | 142 | 1.000 | \left\langle J_{+} J_{-}\right\rangle | | |
| gilmore-lie-groups_FO1408 | 142 | 1.000 | \left\langle a^{\dagger} a\right\rangle | | |
| gilmore-lie-groups_FO1409 | 143 | 0.749 | X_{j}=A_{j}^{s} Y_{s} | | |
| gilmore-lie-groups_FO1410 | 144 | 1.000 | N=p q | | |
| gilmore-lie-groups_FO1411 | 144 | 1.000 | \mathfrak{s o l}(n) | | |
| gilmore-lie-groups_FO1412 | 145 | 1.000 | \hat{n}= | | |
| gilmore-lie-groups_FO1413 | 145 | 0.993 | \frac{1}{2}\left\{a, a^{\dagger}\right\}=a^{\dagger} a+\frac{1}{2}, a^{\dagger 2}, a^{2}, a^{\dagger}, a, I=\left[a, a^{\dagger}\right] | | |
| gilmore-lie-groups_FO1414 | 145 | 1.000 | X_{i j}=-X_{j i}, 1 \leq i, j \leq 4 | | |
| gilmore-lie-groups_FO1415 | 146 | 1.000 | \mathfrak{s o}(n), n>4 | | |
| gilmore-lie-groups_FO1416 | 146 | 0.999 | \mathfrak{s u}(n)(n \geq 2), \mathfrak{s o}(n)(n>4) | | |
| gilmore-lie-groups_FO1417 | 146 | 0.999 | \mathfrak{s p}(n)(n \geq 1) | | |
| gilmore-lie-groups_FO1418 | 147 | 0.999 | \left[Z, X_{i}\right] | | |
| gilmore-lie-groups_FO1419 | 147 | 1.000 | X_{j} | | |
| gilmore-lie-groups_FO1420 | 147 | 1.000 | j \geq i | | |
| gilmore-lie-groups_FO1421 | 147 | 1.000 | X_{i}, X_{i+1}, \ldots, X_{n} | | |
| gilmore-lie-groups_FO1422 | 147 | 0.593 | i=1,2, \ldots, n | | |
| gilmore-lie-groups_FO1423 | 147 | 1.000 | V_{i} | | |
| gilmore-lie-groups_FO1424 | 147 | 1.000 | V_{j}, i>j | | |
| gilmore-lie-groups_FO1425 | 148 | 1.000 | \left[V_{2}, V_{2}\right] \subseteq V_{2}, V_{2} | | |
| gilmore-lie-groups_FO1426 | 148 | 1.000 | V_{2}, V_{2} | | |
| gilmore-lie-groups_FO1427 | 149 | 1.000 | \mathfrak{g}^{\prime}=\mathfrak{g}-V_{0} | | |
| gilmore-lie-groups_FO1428 | 149 | 1.000 | \mathfrak{g}^{\prime} | | |
| gilmore-lie-groups_FO1429 | 150 | 1.000 | V_{0}^{\prime}=0 | | |
| gilmore-lie-groups_FO1430 | 150 | 1.000 | a^{\dagger 2} | | |
| gilmore-lie-groups_FO1431 | 150 | 1.000 | a^{2} | | |
| gilmore-lie-groups_FO1432 | 150 | 1.000 | a^{\dagger 2}+a^{2} | | |
| gilmore-lie-groups_FO1433 | 150 | 1.000 | a^{\dagger} a+\frac{1}{2} | | |
| gilmore-lie-groups_FO1434 | 150 | 1.000 | a^{\dagger 2}-a^{2} | | |
| gilmore-lie-groups_FO1435 | 151 | 1.000 | \hat{n}, a^{\dagger}, a, I | | |
| gilmore-lie-groups_FO1436 | 151 | 0.786 | \left[b_{i}, b_{j}^{\dagger}\right]=I \delta_{i j}, 1 \leq i, j \leq n | | |
| gilmore-lie-groups_FO1437 | 151 | 1.000 | b_{i}, b_{j}^{\dagger}, I | | |
| gilmore-lie-groups_FO1438 | 151 | 1.000 | b_{i}^{\dagger} b_{j}, b_{i}, b_{j}^{\dagger}, I | | |
| gilmore-lie-groups_FO1439 | 151 | 1.000 | b_{i}^{\dagger} b_{j}^{\dagger}, b_{i}^{\dagger} b_{j}+\frac{1}{2} \delta_{i j}, b_{i} b_{j} | | |
| gilmore-lie-groups_FO1440 | 151 | 0.998 | b_{i}^{\dagger} b_{j}^{\dagger}, b_{i}^{\dagger} b_{j}+\frac{1}{2} \delta_{i j}, b_{i} b_{j}, b_{i}, b_{j}^{\dagger}, I | | |
| gilmore-lie-groups_FO1441 | 151 | 1.000 | b, b^{\dagger} b, b^{\dagger} b^{\dagger} b | | |
| gilmore-lie-groups_FO1442 | 151 | 0.788 | b^{\dagger} | | |
| gilmore-lie-groups_FO1443 | 151 | 0.788 | b^{\dagger} b, b^{\dagger} b b | | |
| gilmore-lie-groups_FO1444 | 151 | 1.000 | \left\{f_{i}, f_{j}^{\dagger}\right\}=\delta_{i j} | | |
| gilmore-lie-groups_FO1445 | 151 | 0.985 | f_{i}^{\dagger} f_{j}^{\dagger} | | |
| gilmore-lie-groups_FO1446 | 151 | 0.985 | f_{i}^{\dagger} f_{j}+\frac{1}{2} \delta_{i j} | | |
| gilmore-lie-groups_FO1447 | 151 | 0.985 | f_{i} f_{j} | | |
| gilmore-lie-groups_FO1448 | 151 | 1.000 | \left(x_{i}\right) | | |
| gilmore-lie-groups_FO1449 | 151 | 1.000 | \left(\partial_{j}\right) | | |
| gilmore-lie-groups_FO1450 | 151 | 1.000 | x_{i}, \partial_{j}, I | | |
| gilmore-lie-groups_FO1451 | 151 | 1.000 | x_{i} \partial_{j} | | |
| gilmore-lie-groups_FO1452 | 151 | 1.000 | x_{i} \partial_{j}, x_{i}, \partial_{j}, I | | |
| gilmore-lie-groups_FO1453 | 151 | 0.991 | x_{i} x_{j}, x_{i} \partial_{j}+\frac{1}{2} I \delta_{i j}, \partial_{i} \partial_{j} | | |
| gilmore-lie-groups_FO1454 | 151 | 1.000 | x_{i} x_{j}, x_{i} \partial_{j}, \partial_{i} \partial_{j}, x_{i}, \partial_{j}, I | | |
| gilmore-lie-groups_FO1455 | 151 | 1.000 | \frac{d}{d x}, x \frac{d}{d x}, x^{2} \frac{d}{d x} | | |
| gilmore-lie-groups_FO1456 | 151 | 1.000 | x, x \frac{d}{d x}, x \frac{d^{2}}{d x^{2}} | | |
| gilmore-lie-groups_FO1457 | 151 | 0.799 | 2+1 | | |
| gilmore-lie-groups_FO1458 | 151 | 0.799 | (x, y | | |
| gilmore-lie-groups_FO1459 | 151 | 0.799 | t) | | |
| gilmore-lie-groups_FO1463 | 152 | 1.000 | x_{i}, \partial_{j} | | |
| gilmore-lie-groups_FO1464 | 152 | 0.781 | b_{i}^{\dagger}, b_{j} | | |
| gilmore-lie-groups_FO1465 | 152 | 1.000 | g_{i j}=C_{i r}{ }^{s} C_{j s}{ }^{r} | | |
| gilmore-lie-groups_FO1466 | 152 | 1.000 | g^{i j} | | |
| gilmore-lie-groups_FO1467 | 152 | 0.940 | \mathcal{C}^{2}=g^{i j} X_{i} X_{j} | | |
| gilmore-lie-groups_FO1468 | 152 | 0.940 | \left[\mathcal{C}^{2}, X_{k}\right]=0 | | |
| gilmore-lie-groups_FO1469 | 152 | 1.000 | \mathcal{C}^{2} | | |
| gilmore-lie-groups_FO1470 | 152 | 1.000 | C_{i j k}=C_{i j}{ }^{r} g_{r k} | | |
| gilmore-lie-groups_FO1471 | 152 | 1.000 | C_{i j k}=C_{j k i}=C_{k i j}= | | |
| gilmore-lie-groups_FO1472 | 152 | 1.000 | -C_{k j i}=-C_{j i k}=-C_{i k j} | | |
| gilmore-lie-groups_FO1473 | 154 | 0.989 | \mathfrak{s u}(1,1) | | |
| gilmore-lie-groups_FO1474 | 154 | 1.000 | Z, X | | |
| gilmore-lie-groups_FO1475 | 154 | 1.000 | X=\sum_{i=1}^{N} a^{i} X_{i} | | |
| gilmore-lie-groups_FO1476 | 154 | 1.000 | a^{i} | | |
| gilmore-lie-groups_FO1477 | 154 | 1.000 | \phi_{j}(Z) | | |
| gilmore-lie-groups_FO1478 | 154 | 0.973 | z^{i}\left(Z=\sum z^{i} X_{i}\right) | | |
| gilmore-lie-groups_FO1479 | 155 | 1.000 | a^{\dagger}, a, I=\left[a, a^{\dagger}\right] | | |
| gilmore-lie-groups_FO1480 | 155 | 1.000 | (-\lambda)^{N}=0 | | |
| gilmore-lie-groups_FO1481 | 155 | 0.852 | X=\sum a_{i} X_{i} \in \mathfrak{s u}(2) | | |
| gilmore-lie-groups_FO1482 | 155 | 0.953 | \mathfrak{d e f}(X) | | |
| gilmore-lie-groups_FO1483 | 155 | 0.953 | \mathfrak{R e g}(X) | | |
| gilmore-lie-groups_FO1484 | 155 | 1.000 | \phi_{2}(\mathbf{a}) \geq 0 | | |
| gilmore-lie-groups_FO1485 | 155 | 0.760 | \mathfrak{s} \mathfrak{u}(2)) | | |
| gilmore-lie-groups_FO1486 | 155 | 0.760 | \lambda=0, \lambda= \pm i a | | |
| gilmore-lie-groups_FO1487 | 155 | 0.999 | a^{2}=+a_{1}^{2}+a_{2}^{2}+a_{3}^{2} | | |
| gilmore-lie-groups_FO1488 | 156 | 0.861 | Y=\sum b_{i} Y_{i} \in \mathfrak{s u}(1,1) | | |
| gilmore-lie-groups_FO1489 | 156 | 0.976 | \mathfrak{d e f}(Y) | | |
| gilmore-lie-groups_FO1490 | 156 | 0.976 | \mathfrak{R e g}(Y) | | |
| gilmore-lie-groups_FO1491 | 156 | 0.999 | \left(a_{1}, a_{2}, a_{3}\right) \rightarrow\left(i b_{1}, i b_{2}, b_{3}\right) | | |
| gilmore-lie-groups_FO1492 | 156 | 0.999 | \phi_{2}(\mathbf{a})=a_{1}^{2}+a_{2}^{2}+a_{3}^{2} | | |
| gilmore-lie-groups_FO1493 | 156 | 0.953 | \phi_{2}(\mathbf{b})=-b_{1}^{2}-b_{2}^{2}+b_{3}^{2} | | |
| gilmore-lie-groups_FO1494 | 156 | 1.000 | \phi_{2} | | |
| gilmore-lie-groups_FO1495 | 157 | 0.515 | \Re e g | | |
| gilmore-lie-groups_FO1496 | 157 | 1.000 | \phi_{j}\left(z^{i}\right) | | |
| gilmore-lie-groups_FO1497 | 157 | 1.000 | X=Z | | |
| gilmore-lie-groups_FO1498 | 157 | 1.000 | l^{2} \sim N | | |
| gilmore-lie-groups_FO1499 | 157 | 1.000 | \mathfrak{m} | | |
| gilmore-lie-groups_FO1503 | 158 | 0.955 | \phi_{j} | | |
| gilmore-lie-groups_FO1505 | 158 | 0.751 | R^{3},=\epsilon_{i j k}=+1 | | |
| gilmore-lie-groups_FO1506 | 158 | 0.751 | (i j k) | | |
| gilmore-lie-groups_FO1507 | 159 | 1.000 | \phi_{j}\left(X^{r}{ }_{s}\right) | | |
| gilmore-lie-groups_FO1508 | 159 | 1.000 | O(n) | | |
| gilmore-lie-groups_FO1509 | 159 | 0.651 | d \times d | | |
| gilmore-lie-groups_FO1510 | 159 | 0.651 | \mathfrak{s o}(n) | | |
| gilmore-lie-groups_FO1511 | 159 | 0.651 | d=n(n-1) / 2) | | |
| gilmore-lie-groups_FO1512 | 159 | 1.000 | \mathfrak{d e f}(X)^{t}=-\mathfrak{d e f}(X) | | |
| gilmore-lie-groups_FO1513 | 159 | 0.986 | [n / 2] | | |
| gilmore-lie-groups_FO1514 | 159 | 0.996 | j=2 | | |
| gilmore-lie-groups_FO1515 | 159 | 0.996 | X_{i j}=-X_{j i} | | |
| gilmore-lie-groups_FO1516 | 159 | 1.000 | \mathfrak{s o}(5) | | |
| gilmore-lie-groups_FO1518 | 160 | 0.940 | v^{m}=\epsilon^{i j k l m} X_{i j} X_{k l} | | |
| gilmore-lie-groups_FO1519 | 160 | 1.000 | n / 2 | | |
| gilmore-lie-groups_FO1520 | 160 | 1.000 | Z=0 | | |
| gilmore-lie-groups_FO1521 | 160 | 1.000 | [Z, X]=0 X | | |
| gilmore-lie-groups_FO1522 | 160 | 1.000 | \hat{n}+\frac{1}{2}=\frac{1}{2}\left\{a, a^{\dagger}\right\} | | |
| gilmore-lie-groups_FO1523 | 160 | 1.000 | a^{\dagger 2}, a^{\dagger}, I=\left[a, a^{\dagger}\right], a, a^{2} | | |
| gilmore-lie-groups_FO1524 | 161 | 1.000 | \pm 2 z_{1}, \pm z_{1}, 0,0 | | |
| gilmore-lie-groups_FO1525 | 161 | 0.999 | X_{(2,0)}=a^{\dagger 2}, X_{(1,0)}=a^{\dagger}, X_{(0,0)}=\hat{n}+\frac{1}{2} I, I, X_{(-1,0)}=a, X_{(-2,0)}=a^{2} | | |
| gilmore-lie-groups_FO1526 | 161 | 1.000 | k, l \in\{-2,-1,0,+1,+2\} | | |
| gilmore-lie-groups_FO1527 | 161 | 1.000 | k+l | | |
| gilmore-lie-groups_FO1528 | 161 | 1.000 | \{-2,-1,0,+1,+2\} | | |
| gilmore-lie-groups_FO1529 | 161 | 1.000 | k+l=0 | | |
| gilmore-lie-groups_FO1530 | 161 | 0.968 | (0,0): \hat{n}+\frac{1}{2} | | |
| gilmore-lie-groups_FO1531 | 162 | 1.000 | \alpha_{1}, \alpha_{2}, \ldots, \alpha_{l} | | |
| gilmore-lie-groups_FO1532 | 162 | 0.570 | \left(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{l}\right) | | |
| gilmore-lie-groups_FO1533 | 162 | 1.000 | V_{\beta} | | |
| gilmore-lie-groups_FO1534 | 162 | 1.000 | V_{\alpha+\beta} | | |
| gilmore-lie-groups_FO1535 | 162 | 1.000 | \alpha+\beta | | |
| gilmore-lie-groups_FO1536 | 162 | 1.000 | H_{1}, H_{2}, \ldots, H_{l} | | |
| gilmore-lie-groups_FO1537 | 162 | 1.000 | V_{\alpha}(\alpha \neq 0) | | |
| gilmore-lie-groups_FO1538 | 162 | 0.995 | E_{\alpha} | | |
| gilmore-lie-groups_FO1539 | 162 | 0.995 | \left[V_{0}, V_{\alpha}\right] \subset V_{\alpha} | | |
| gilmore-lie-groups_FO1540 | 162 | 1.000 | -\alpha | | |
| gilmore-lie-groups_FO1541 | 162 | 1.000 | c \alpha | | |
| gilmore-lie-groups_FO1542 | 162 | 1.000 | |c|=1 | | |
| gilmore-lie-groups_FO1543 | 162 | 1.000 | E_{-\alpha} | | |
| gilmore-lie-groups_FO1544 | 162 | 1.000 | \alpha^{i} | | |
| gilmore-lie-groups_FO1545 | 162 | 1.000 | \alpha_{j} | | |
| gilmore-lie-groups_FO1546 | 162 | 1.000 | \alpha^{i}=h^{i j} \alpha_{j} | | |
| gilmore-lie-groups_FO1547 | 163 | 1.000 | h^{i j}=\delta^{i j} | | |
| gilmore-lie-groups_FO1548 | 163 | 1.000 | \alpha_{i}: \alpha^{i}=\alpha_{i} | | |
| gilmore-lie-groups_FO1549 | 163 | 1.000 | N_{\alpha, \beta} | | |
| gilmore-lie-groups_FO1550 | 163 | 1.000 | E_{\alpha}, E_{\beta}, E_{\gamma} | | |
| gilmore-lie-groups_FO1551 | 163 | 1.000 | \alpha, \beta+k \alpha | | |
| gilmore-lie-groups_FO1553 | 164 | 1.000 | N_{\alpha, \beta+n \alpha}=0 | | |
| gilmore-lie-groups_FO1554 | 164 | 1.000 | N_{-\alpha, \beta-m \alpha}^{2}=N_{\alpha, \beta-(m+1) \alpha}^{2}=0 | | |
| gilmore-lie-groups_FO1555 | 164 | 1.000 | N_{\alpha, \beta+k \alpha}^{2}=(n-k)(m+k+1)(\alpha \cdot \alpha) / 2 \geq 0 | | |
| gilmore-lie-groups_FO1556 | 164 | 1.000 | \alpha \cdot \beta>0 | | |
| gilmore-lie-groups_FO1557 | 164 | 1.000 | m- | | |
| gilmore-lie-groups_FO1558 | 164 | 1.000 | n>0 | | |
| gilmore-lie-groups_FO1559 | 164 | 1.000 | \alpha \cdot \beta<0 | | |
| gilmore-lie-groups_FO1560 | 164 | 1.000 | m-n<0 | | |
| gilmore-lie-groups_FO1563 | 165 | 0.456 | l(l | | |
| gilmore-lie-groups_FO1564 | 165 | 0.977 | \mathbf{H}=\left(H_{1}, H_{2}, \ldots, H_{l}\right) | | |
| gilmore-lie-groups_FO1565 | 165 | 0.994 | \alpha=\left(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{l}\right) | | |
| gilmore-lie-groups_FO1566 | 166 | 1.000 | C_{2} | | |
| gilmore-lie-groups_FO1569 | 166 | 1.000 | \mathbf{e}_{1} | | |
| gilmore-lie-groups_FO1570 | 166 | 1.000 | \mathbf{e}_{2} | | |
| gilmore-lie-groups_FO1571 | 166 | 1.000 | \pm 2 \mathbf{e}_{1}, \pm 2 \mathbf{e}_{2}, \pm \mathbf{e}_{1} \pm \mathbf{e}_{2} | | |
| gilmore-lie-groups_FO1572 | 166 | 0.986 | H_{i}, i=1,2 | | |
| gilmore-lie-groups_FO1573 | 166 | 1.000 | \sum \alpha \cdot \alpha=2 | | |
| gilmore-lie-groups_FO1609 | 168 | 1.000 | \mathfrak{s l}(2 ; \mathbb{C})) | | |
| gilmore-lie-groups_FO1610 | 168 | 1.000 | \alpha_{i} | | |
| gilmore-lie-groups_FO1611 | 168 | 1.000 | \frac{1}{2}\left\{a^{\dagger}, a\right\} | | |
| gilmore-lie-groups_FO1612 | 168 | 1.000 | a^{\dagger^{2}}, a^{\dagger}, I, a, a^{2} | | |
| gilmore-lie-groups_FO1613 | 168 | 1.000 | X_{i j}=x^{i} \partial_{j}-x^{j} \partial_{i}(1 \leq i<j \leq 4) | | |
| gilmore-lie-groups_FO1614 | 168 | 1.000 | \mathcal{C}_{2}=\sum_{i<j} X_{i j}^{2} | | |
| gilmore-lie-groups_FO1615 | 168 | 0.980 | \mathcal{C}_{2}^{\prime}=X_{12} X_{34}-X_{13} X_{24}+X_{14} X_{23} | | |
| gilmore-lie-groups_FO1616 | 168 | 0.994 | \mathfrak{s u}(4) | | |
| gilmore-lie-groups_FO1617 | 168 | 1.000 | \mathfrak{s o}(2 n+1) | | |
| gilmore-lie-groups_FO1618 | 168 | 1.000 | 2,4, \ldots, 2 n | | |
| gilmore-lie-groups_FO1619 | 168 | 1.000 | \mathfrak{s o}(2 n) | | |
| gilmore-lie-groups_FO1620 | 168 | 0.600 | \mathcal{C}_{n}^{\prime}=\epsilon^{i_{1} i_{2} \cdots i_{2 n}} X_{i_{1} i_{2}} X_{i_{3} i_{4}} \cdots X_{i_{2 n-1}, i_{2 n}} | | |
| gilmore-lie-groups_FO1621 | 168 | 0.600 | \mathcal{C}_{2}^{\prime} | | |
| gilmore-lie-groups_FO1622 | 168 | 0.600 | \mathcal{C}_{3}^{\prime} | | |
| gilmore-lie-groups_FO1623 | 168 | 0.600 | \mathfrak{s o}(6) | | |
| gilmore-lie-groups_FO1624 | 168 | 1.000 | \mathfrak{s u}(4)=\mathfrak{s o}(6) | | |
| gilmore-lie-groups_FO1625 | 168 | 1.000 | a_{i}^{\dagger} a_{j}+\frac{1}{2} \delta_{i j} | | |
| gilmore-lie-groups_FO1626 | 169 | 0.999 | (1 \leq i, j \leq 2) | | |
| gilmore-lie-groups_FO1627 | 169 | 0.999 | a_{i}^{\dagger} a_{j}^{\dagger} | | |
| gilmore-lie-groups_FO1628 | 169 | 0.999 | a_{i} a_{j}\left(a_{i} a_{j}=a_{j} a_{i}\right) | | |
| gilmore-lie-groups_FO1629 | 169 | 1.000 | a_{i}^{\dagger} a_{i}+\frac{1}{2} \leftrightarrow H_{i}, a_{i}^{\dagger} a_{j}^{\dagger} \leftrightarrow E_{+\mathbf{e}_{i}+\mathbf{e}_{j}}(i \neq j), a_{i}^{\dagger} a_{j} \leftrightarrow E_{+\mathbf{e}_{i}-\mathbf{e}_{j}}(i \neq j) | | |
| gilmore-lie-groups_FO1630 | 169 | 1.000 | a_{i} a_{j} \leftrightarrow E_{-\mathbf{e}_{i}-\mathbf{e}_{j}}(i \neq j), a_{i}^{\dagger} a_{i}^{\dagger} \leftrightarrow E_{+2 \mathbf{e}_{i}}, a_{i} a_{i} \leftrightarrow E_{-2 \mathbf{e}_{i}} | | |
| gilmore-lie-groups_FO1631 | 169 | 1.000 | D_{2} | | |
| gilmore-lie-groups_FO1632 | 169 | 0.995 | f_{i}^{\dagger} f_{i}^{\dagger}=0 | | |
| gilmore-lie-groups_FO1633 | 169 | 1.000 | X, Y \in \mathfrak{s u}(2) | | |
| gilmore-lie-groups_FO1634 | 169 | 0.577 | \operatorname{tr}[\mathfrak{d e f}(X) \mathfrak{d e f}(Y)] | | |
| gilmore-lie-groups_FO1635 | 169 | 0.569 | \mathfrak{s u}(2):(X, Y)=\operatorname{tr}[\Re \mathfrak{e g}(X) \Re \mathfrak{e g}(Y)] | | |
| gilmore-lie-groups_FO1636 | 169 | 0.995 | \left(n^{2}-1\right) \times\left(n^{2}-1\right) | | |
| gilmore-lie-groups_FO1637 | 169 | 1.000 | Y=X | | |
| gilmore-lie-groups_FO1638 | 169 | 1.000 | X=H_{1} | | |
| gilmore-lie-groups_FO1639 | 169 | 0.999 | 1 \leq i, j \leq N | | |
| gilmore-lie-groups_FO1640 | 169 | 1.000 | \mathbf{R}=\frac{1}{2} \sum_{\alpha>0} \alpha | | |
| gilmore-lie-groups_FO1641 | 169 | 1.000 | B_{n}, R_{i}=\frac{1}{2}(2 n+1)-i | | |
| gilmore-lie-groups_FO1642 | 169 | 0.996 | \mathbf{M} | | |
| gilmore-lie-groups_FO1643 | 169 | 1.000 | \mathfrak{s o}(3), \mathbf{M}=j, \mathbf{R}=R_{1}=\frac{1}{2} | | |
| gilmore-lie-groups_FO1644 | 169 | 1.000 | \mathcal{C}^{2}(j)=\left(j+\frac{1}{2}\right)^{2}-\left(0+\frac{1}{2}\right)^{2}= | | |
| gilmore-lie-groups_FO1645 | 169 | 1.000 | j(j+1) | | |
| gilmore-lie-groups_FO1646 | 169 | 1.000 | A_{n-1} | | |
| gilmore-lie-groups_FO1647 | 169 | 1.000 | D_{n} | | |
| gilmore-lie-groups_FO1648 | 169 | 0.919 | 2^{n-1} n! | | |
| gilmore-lie-groups_FO1649 | 169 | 0.919 | B_{n} | | |
| gilmore-lie-groups_FO1650 | 169 | 0.919 | C_{n} | | |
| gilmore-lie-groups_FO1651 | 169 | 0.919 | 2^{n} n! | | |
| gilmore-lie-groups_FO1652 | 170 | 1.000 | f_{r}(\Gamma) | | |
| gilmore-lie-groups_FO1653 | 170 | 1.000 | \mathcal{C}^{r} | | |
| gilmore-lie-groups_FO1654 | 170 | 1.000 | \mathfrak{s o}(2 n), \mathfrak{s o}(2 n+1), \mathfrak{s p}(2 n) | | |
| gilmore-lie-groups_FO1655 | 170 | 1.000 | \sum_{i j} a^{i j} M_{i j} | | |
| gilmore-lie-groups_FO1656 | 170 | 1.000 | \mathfrak{s o}(2 n), \mathfrak{s p}(2 n) | | |
| gilmore-lie-groups_FO1657 | 170 | 1.000 | (2 n+1) \times(2 n+1) | | |
| gilmore-lie-groups_FO1658 | 170 | 1.000 | a^{i j} | | |
| gilmore-lie-groups_FO1659 | 170 | 1.000 | \phi_{r}\left(a^{i j}\right) | | |
| gilmore-lie-groups_FO1660 | 170 | 0.999 | \epsilon_{i_{1} i_{2} \cdots i_{l}} | | |
| gilmore-lie-groups_FO1661 | 170 | 1.000 | (l=2 n, 2 n+1,2 n) | | |
| gilmore-lie-groups_FO1662 | 170 | 1.000 | \phi_{2 n}\left(a^{i j}\right) | | |
| gilmore-lie-groups_FO1663 | 171 | 0.999 | \operatorname{tr} M^{2}=-2 \theta \cdot \theta | | |
| gilmore-lie-groups_FO1664 | 171 | 0.987 | \operatorname{tr} \mathcal{M}^{2}=-2 \mathbf{J} \cdot \mathbf{J} | | |
| gilmore-lie-groups_FO1665 | 171 | 1.000 | \left[\mathbf{J}, \operatorname{tr} \mathcal{M}^{2}\right]=0 | | |
| gilmore-lie-groups_FO1666 | 171 | 0.999 | \operatorname{tr} \mathcal{M}^{2 n+1}=0 | | |
| gilmore-lie-groups_FO1667 | 171 | 0.999 | \operatorname{tr} \mathcal{M}^{2 n}=(-2)^{n}(\mathbf{J} \cdot \mathbf{J})^{n} | | |
| gilmore-lie-groups_FO1668 | 171 | 1.000 | \left[X_{i}, X_{j}\right]=C_{i j}{ }^{k} X_{k} | | |
| gilmore-lie-groups_FO1669 | 171 | 1.000 | V^{(1)} | | |
| gilmore-lie-groups_FO1670 | 171 | 1.000 | V^{(2)} | | |
| gilmore-lie-groups_FO1671 | 171 | 1.000 | X_{i} \rightarrow \Gamma^{(1)}\left(X_{i}\right)= | | |
| gilmore-lie-groups_FO1672 | 171 | 1.000 | Y_{i} | | |
| gilmore-lie-groups_FO1673 | 171 | 1.000 | X_{i} \rightarrow \Gamma^{(2)}\left(X_{i}\right)=Z_{i} | | |
| gilmore-lie-groups_FO1674 | 171 | 1.000 | g^{i j} Y_{i} Z_{j} | | |
| gilmore-lie-groups_FO1675 | 171 | 0.857 | (Y+Z)_{k} | | |
| gilmore-lie-groups_FO1676 | 171 | 0.857 | \Gamma^{(1)}\left(X_{i}\right) \otimes I_{\operatorname{dim} V^{(2)}}+I_{\operatorname{dim} V^{(1)}} \otimes \Gamma^{(2)}\left(X_{i}\right) | | |
| gilmore-lie-groups_FO1677 | 171 | 1.000 | X^{0}=I_{n} | | |
| gilmore-lie-groups_FO1678 | 171 | 1.000 | \lambda_{i}(X) | | |
| gilmore-lie-groups_FO1679 | 171 | 1.000 | \phi_{i}(X) | | |
| gilmore-lie-groups_FO1680 | 171 | 0.967 | e^{i \phi J_{z}} | | |
| gilmore-lie-groups_FO1681 | 171 | 0.999 | j=\frac{1}{2}, 1, \frac{3}{2}, 2 | | |
| gilmore-lie-groups_FO1682 | 172 | 1.000 | f_{j}(\phi) | | |
| gilmore-lie-groups_FO1683 | 172 | 1.000 | e^{X} | | |
| gilmore-lie-groups_FO1684 | 172 | 1.000 | X \in \mathfrak{s u}(2) | | |
| gilmore-lie-groups_FO1686 | 257 | 1.000 | \frac{1}{2} | | |
| gilmore-lie-groups_FO1705 | 172 | 1.000 | j=\frac{1}{2} | | |
| gilmore-lie-groups_FO1706 | 172 | 1.000 | l=1 | | |
| gilmore-lie-groups_FO1707 | 173 | 1.000 | l+1 | | |
| gilmore-lie-groups_FO1708 | 174 | 1.000 | m, n | | |
| gilmore-lie-groups_FO1709 | 174 | 1.000 | \beta+k \alpha | | |
| gilmore-lie-groups_FO1710 | 174 | 1.000 | k=-m, \ldots,+n | | |
| gilmore-lie-groups_FO1711 | 174 | 0.999 | \beta^{\prime} | | |
| gilmore-lie-groups_FO1712 | 174 | 1.000 | 2(\alpha \cdot \beta) /(\alpha \cdot \alpha)=n | | |
| gilmore-lie-groups_FO1713 | 174 | 1.000 | 2(\alpha \cdot \beta) /(\beta \cdot \beta)=n^{\prime} | | |
| gilmore-lie-groups_FO1714 | 174 | 0.992 | \pm \mathbf{e}_{1} | | |
| gilmore-lie-groups_FO1715 | 174 | 1.000 | A_{2}, B_{2}=C_{2}, D_{2} | | |
| gilmore-lie-groups_FO1734 | 175 | 0.998 | (l-1) | | |
| gilmore-lie-groups_FO1735 | 175 | 0.998 | D_{l} | | |
| gilmore-lie-groups_FO1736 | 175 | 0.998 | \pm\left(\mathbf{e}_{i}+\mathbf{e}_{j}\right) | | |
| gilmore-lie-groups_FO1737 | 175 | 0.998 | A_{l-1} | | |
| gilmore-lie-groups_FO1738 | 175 | 0.998 | B_{l}, C_{l} | | |
| gilmore-lie-groups_FO1739 | 175 | 1.000 | \pm \mathbf{e}_{i}, \pm 2 \mathbf{e}_{i} | | |
| gilmore-lie-groups_FO1740 | 175 | 1.000 | A_{l-1}, D_{l}, B_{l}, C_{l} | | |
| gilmore-lie-groups_FO1743 | 178 | 1.000 | A_{1} | | |
| gilmore-lie-groups_FO1744 | 178 | 1.000 | S L(l ; C) | | |
| gilmore-lie-groups_FO1745 | 178 | 0.997 | \pm\left(\mathbf{e}_{1}-\mathbf{e}_{2}\right) | | |
| gilmore-lie-groups_FO1746 | 178 | 1.000 | \pm\left(\mathbf{e}_{1}+\mathbf{e}_{2}\right) | | |
| gilmore-lie-groups_FO1747 | 178 | 0.964 | B_{2} | | |
| gilmore-lie-groups_FO1748 | 178 | 0.513 | S O(5) | | |
| gilmore-lie-groups_FO1749 | 178 | 0.513 | S p(2)=U(2 ; \mathbb{Q}) | | |
| gilmore-lie-groups_FO1750 | 178 | 1.000 | U(2 ; \mathbb{Q}) | | |
| gilmore-lie-groups_FO1751 | 178 | 0.975 | S O(3)\left(B_{1}\right) | | |
| gilmore-lie-groups_FO1752 | 178 | 0.975 | U(1 ; Q) | | |
| gilmore-lie-groups_FO1753 | 178 | 0.995 | \operatorname{SU}(2)\left(A_{1}\right) | | |
| gilmore-lie-groups_FO1771 | 184 | 1.000 | C_{l} | | |
| gilmore-lie-groups_FO1773 | 180 | 1.000 | n_{i j} | | |
| gilmore-lie-groups_FO1774 | 180 | 1.000 | 0,1,2 | | |
| gilmore-lie-groups_FO1775 | 180 | 1.000 | G_{2}+B_{3} | | |
| gilmore-lie-groups_FO1776 | 180 | 0.665 | \mathbf{v}_{i} | | |
| gilmore-lie-groups_FO1777 | 180 | 0.665 | \mathbf{u} | | |
| gilmore-lie-groups_FO1778 | 180 | 1.000 | \mathbf{u} \cdot \mathbf{v}_{i} | | |
| gilmore-lie-groups_FO1779 | 181 | 1.000 | \mathbf{u}_{i}=\alpha_{i} /\left|\alpha_{i}\right| | | |
| gilmore-lie-groups_FO1780 | 181 | 1.000 | 2 \mathbf{u}_{i} \cdot \mathbf{u}_{j} \leq-1 | | |
| gilmore-lie-groups_FO1781 | 181 | 0.979 | (B, C, F) | | |
| gilmore-lie-groups_FO1782 | 182 | 1.000 | \mathbf{u}_{i}, \mathbf{v}_{j} | | |
| gilmore-lie-groups_FO1783 | 182 | 1.000 | \mathbf{v}_{i}=\alpha_{j} /\left|\alpha_{j}\right| | | |
| gilmore-lie-groups_FO1784 | 182 | 0.985 | p \geq q | | |
| gilmore-lie-groups_FO1785 | 182 | 0.657 | (D, E) | | |
| gilmore-lie-groups_FO1786 | 182 | 0.997 | p \geq q \geq r \geq 2 | | |
| gilmore-lie-groups_FO1788 | 327 | 0.991 | E_{6} | | |
| gilmore-lie-groups_FO1789 | 327 | 0.546 | E_{7} | | |
| gilmore-lie-groups_FO1790 | 327 | 0.731 | E_{8} | | |
| gilmore-lie-groups_FO1791 | 182 | 1.000 | l-1 | | |
| gilmore-lie-groups_FO1793 | 184 | 1.000 | U(l) | | |
| gilmore-lie-groups_FO1814 | 185 | 0.852 | D, E | | |
| gilmore-lie-groups_FO1815 | 185 | 1.000 | 1 \leq i, j \leq n | | |
| gilmore-lie-groups_FO1816 | 185 | 1.000 | n=3 | | |
| gilmore-lie-groups_FO1817 | 185 | 1.000 | \hat{\mathbf{Q}} | | |
| gilmore-lie-groups_FO1818 | 185 | 1.000 | \hat{N} | | |
| gilmore-lie-groups_FO1819 | 185 | 0.998 | \mathbf{L}+\hat{\mathbf{Q}} | | |
| gilmore-lie-groups_FO1820 | 185 | 0.623 | \mathbf{L}(3 \rightarrow n(n-1) / 2) | | |
| gilmore-lie-groups_FO1821 | 185 | 0.623 | \mathbf{Q} | | |
| gilmore-lie-groups_FO1822 | 185 | 1.000 | (6 \rightarrow n(n+1) / 2) | | |
| gilmore-lie-groups_FO1823 | 185 | 1.000 | \hat{N}(3 \rightarrow n) | | |
| gilmore-lie-groups_FO1824 | 186 | 1.000 | Z=r^{i} X_{i} | | |
| gilmore-lie-groups_FO1825 | 186 | 1.000 | r^{i} | | |
| gilmore-lie-groups_FO1826 | 186 | 1.000 | \sum \sum r^{i}\left[R_{i}{ }^{j}(Z)-\lambda \delta_{i}{ }^{j}\right] X_{j}=0 | | |
| gilmore-lie-groups_FO1827 | 186 | 1.000 | h^{i} | | |
| gilmore-lie-groups_FO1828 | 186 | 1.000 | e^{\alpha} | | |
| gilmore-lie-groups_FO1829 | 187 | 1.000 | A_{n-1}, D_{n}, B_{n}, C_{n} | | |
| gilmore-lie-groups_FO1830 | 187 | 1.000 | a_{1}, a_{2}, a_{3} | | |
| gilmore-lie-groups_FO1831 | 187 | 1.000 | \left(a_{1}, a_{2}, a_{3}\right) | | |
| gilmore-lie-groups_FO1832 | 187 | 1.000 | \left(b_{1}, b_{2}, b_{3}\right) | | |
| gilmore-lie-groups_FO1833 | 188 | 0.998 | H_{i}, E_{\alpha} | | |
| gilmore-lie-groups_FO1834 | 188 | 1.000 | \frac{1}{\sqrt{2}}\left(E_{\alpha} \pm E_{-\alpha}\right) | | |
| gilmore-lie-groups_FO1835 | 189 | 0.826 | H_{i}, \frac{1}{\sqrt{2}}\left(E_{\alpha}+E_{-\alpha}\right) | | |
| gilmore-lie-groups_FO1836 | 189 | 0.826 | \frac{1}{\sqrt{2}}\left(E_{\alpha}-E_{-\alpha}\right) | | |
| gilmore-lie-groups_FO1837 | 189 | 0.826 | \alpha \neq 0 | | |
| gilmore-lie-groups_FO1838 | 189 | 1.000 | \frac{1}{\sqrt{2}}\left(E_{\alpha}+E_{-\alpha}\right) | | |
| gilmore-lie-groups_FO1839 | 189 | 0.999 | A_{1}, \mathfrak{s l}(2 ; \mathbb{R}) | | |
| gilmore-lie-groups_FO1840 | 189 | 1.000 | h_{r}, h_{i} ; a_{r}, a_{i} ; b_{r}, b_{i} | | |
| gilmore-lie-groups_FO1841 | 190 | 0.850 | n_{+}, n_{-} | | |
| gilmore-lie-groups_FO1842 | 190 | 1.000 | n_{+} | | |
| gilmore-lie-groups_FO1843 | 190 | 1.000 | n_{-} | | |
| gilmore-lie-groups_FO1844 | 190 | 1.000 | \chi=n_{+}-n_{-} | | |
| gilmore-lie-groups_FO1845 | 190 | 1.000 | i H_{i}, i \frac{1}{\sqrt{2}}\left(E_{\alpha}+E_{-\alpha}\right) | | |
| gilmore-lie-groups_FO1846 | 190 | 0.917 | g_{\mu, \nu}=\operatorname{diag}(1,1,1,-1) | | |
| gilmore-lie-groups_FO1847 | 190 | 0.870 | x, y, z, i c t | | |
| gilmore-lie-groups_FO1848 | 190 | 0.870 | g_{\mu, \nu}= | | |
| gilmore-lie-groups_FO1849 | 190 | 0.672 | \operatorname{diag}(1,1,1,1) | | |
| gilmore-lie-groups_FO1850 | 190 | 1.000 | \mathfrak{p} \rightarrow \mathfrak{p}^{\prime}=i \mathfrak{p} | | |
| gilmore-lie-groups_FO1851 | 191 | 1.000 | \mathfrak{g}^{\prime}, \mathfrak{h} | | |
| gilmore-lie-groups_FO1852 | 191 | 1.000 | \mathfrak{p}^{\prime} | | |
| gilmore-lie-groups_FO1853 | 191 | 1.000 | T^{2}=I | | |
| gilmore-lie-groups_FO1854 | 192 | 1.000 | \mathfrak{u}(n, \mathbb{F}) | | |
| gilmore-lie-groups_FO1855 | 192 | 0.949 | A_{p}=-A_{p}^{\dagger}, A_{q}=-A_{q}^{\dagger} | | |
| gilmore-lie-groups_FO1856 | 192 | 0.999 | I_{p+q} | | |
| gilmore-lie-groups_FO1857 | 192 | 0.999 | \mathfrak{u}(n ; \mathbb{F}) | | |
| gilmore-lie-groups_FO1858 | 192 | 1.000 | I_{p, q} | | |
| gilmore-lie-groups_FO1859 | 192 | 1.000 | \mathfrak{u}(p, q ; \mathbb{F}) | | |
| gilmore-lie-groups_FO1860 | 192 | 0.995 | \mathbb{F}=\mathbb{R}, \mathbb{C}, \mathbb{Q} | | |
| gilmore-lie-groups_FO1861 | 192 | 0.995 | (D, B), A, C | | |
| gilmore-lie-groups_FO1862 | 192 | 0.998 | \mathfrak{s p}(2 n ; \mathbb{R}) | | |
| gilmore-lie-groups_FO1863 | 193 | 1.000 | \mathfrak{s p}(n)=\mathfrak{u}(n ; \mathbb{Q}) | | |
| gilmore-lie-groups_FO1864 | 193 | 1.000 | \alpha, \beta, \gamma | | |
| gilmore-lie-groups_FO1865 | 193 | 0.999 | \mathfrak{o u}(2 n) | | |
| gilmore-lie-groups_FO1866 | 193 | 1.000 | \underline{2 n} \times 2 n | | |
| gilmore-lie-groups_FO1867 | 194 | 1.000 | R^{2 n} | | |
| gilmore-lie-groups_FO1868 | 194 | 1.000 | p_{1}, q_{1}, p_{2}, q_{2}, \ldots, p_{n}, q_{n} | | |
| gilmore-lie-groups_FO1869 | 194 | 1.000 | v_{i}^{\prime} G_{i j} v_{j} | | |
| gilmore-lie-groups_FO1870 | 194 | 0.967 | M \in \operatorname{Sp}(2 n ; \mathbb{R}) | | |
| gilmore-lie-groups_FO1871 | 194 | 1.000 | M^{t} G M=G | | |
| gilmore-lie-groups_FO1872 | 194 | 1.000 | \mathfrak{p} \rightarrow i \mathfrak{p} | | |
| gilmore-lie-groups_FO1873 | 194 | 1.000 | \mathfrak{s} \mathfrak{o}^{*}(2 n) | | |
| gilmore-lie-groups_FO1874 | 194 | 1.000 | \mathfrak{s} \mathfrak{u}^{*}(2 n) | | |
| gilmore-lie-groups_FO1875 | 194 | 0.991 | A_{2 n-1} | | |
| gilmore-lie-groups_FO1876 | 195 | 1.000 | \mathfrak{s u}(2)), B_{1}(\mathfrak{s o}(3)) | | |
| gilmore-lie-groups_FO1877 | 195 | 1.000 | C_{1}(\mathfrak{s p}(1)) | | |
| gilmore-lie-groups_FO1878 | 195 | 1.000 | (\mathfrak{s o}(5)) | | |
| gilmore-lie-groups_FO1879 | 195 | 1.000 | C_{2}(\mathfrak{s p}(2)) | | |
| gilmore-lie-groups_FO1880 | 195 | 1.000 | A_{3}(\mathfrak{s u}(4)) | | |
| gilmore-lie-groups_FO1881 | 195 | 1.000 | D_{3}(\mathfrak{s o}(6)) | | |
| gilmore-lie-groups_FO1903 | 234 | 0.679 | D_{3} | | |
| gilmore-lie-groups_FO1918 | 327 | 0.999 | F_{4} | | |
| gilmore-lie-groups_FO1944 | 196 | 1.000 | T(\mathfrak{p})=-\mathfrak{p} | | |
| gilmore-lie-groups_FO1945 | 196 | 1.000 | T(\mathfrak{h})=+\mathfrak{h} | | |
| gilmore-lie-groups_FO1946 | 196 | 1.000 | \mathfrak{g}^{\prime}: \mathfrak{g}=\mathfrak{h}+\mathfrak{p} \rightarrow \mathfrak{g}^{\prime}=\mathfrak{h}+i \mathfrak{p}^{\prime} | | |
| gilmore-lie-groups_FO1947 | 197 | 1.000 | 1 \leq i | | |
| gilmore-lie-groups_FO1948 | 197 | 1.000 | j \leq 2 | | |
| gilmore-lie-groups_FO1949 | 197 | 1.000 | a_{1}^{\dagger} a_{1}+a_{2}^{\dagger} a_{2} | | |
| gilmore-lie-groups_FO1950 | 197 | 1.000 | a_{1}^{\dagger} a_{2}- | | |
| gilmore-lie-groups_FO1951 | 197 | 1.000 | a_{1}^{\dagger} a_{1}-a_{2}^{\dagger} a_{2} | | |
| gilmore-lie-groups_FO1952 | 197 | 1.000 | a_{1}^{\dagger} a_{2}+a_{2}^{\dagger} a_{1} | | |
| gilmore-lie-groups_FO1953 | 197 | 1.000 | i \sigma_{j} | | |
| gilmore-lie-groups_FO1954 | 197 | 1.000 | \sigma_{j} | | |
| gilmore-lie-groups_FO1955 | 197 | 0.982 | k \rightarrow 1 | | |
| gilmore-lie-groups_FO1956 | 197 | 0.844 | \overline{S O(n)}=S p i n(n) | | |
| gilmore-lie-groups_FO1957 | 197 | 1.000 | \operatorname{Spin}(n) | | |
| gilmore-lie-groups_FO1958 | 197 | 1.000 | n=3,4,5,6 | | |
| gilmore-lie-groups_FO1959 | 197 | 1.000 | n>6 | | |
| gilmore-lie-groups_FO1960 | 198 | 0.968 | H_{i}:\left|n_{1}, n_{2}, \ldots, n_{r}\right\rangle | | |
| gilmore-lie-groups_FO1961 | 198 | 1.000 | H_{j} \rightarrow | | |
| gilmore-lie-groups_FO1962 | 198 | 1.000 | i H_{j} | | |
| gilmore-lie-groups_FO1963 | 198 | 1.000 | |l, 0, \ldots, 0\rangle | | |
| gilmore-lie-groups_FO1964 | 198 | 0.989 | -m^{2} | | |
| gilmore-lie-groups_FO1965 | 198 | 0.989 | e^{i m \phi} | | |
| gilmore-lie-groups_FO1966 | 198 | 0.989 | -l(l+1) | | |
| gilmore-lie-groups_FO1967 | 198 | 0.989 | Y_{m}^{l}(\theta, \phi) | | |
| gilmore-lie-groups_FO1968 | 198 | 1.000 | J_{i}, i=1,2,3 | | |
| gilmore-lie-groups_FO1969 | 198 | 0.926 | \operatorname{EXP}\left(i r^{k} J_{k}\right) | | |
| gilmore-lie-groups_FO1970 | 198 | 0.926 | r^{k} | | |
| gilmore-lie-groups_FO1971 | 198 | 1.000 | \mathfrak{s} \mathfrak{u}(1,1) | | |
| gilmore-lie-groups_FO1972 | 199 | 0.826 | K_{3} | | |
| gilmore-lie-groups_FO1973 | 199 | 1.000 | e^{i m \phi} \delta_{m^{\prime} m} | | |
| gilmore-lie-groups_FO1974 | 199 | 1.000 | \phi \rightarrow \phi+4 \pi | | |
| gilmore-lie-groups_FO1975 | 199 | 1.000 | m=\frac{1}{2}\left(n_{1}-n_{2}\right) | | |
| gilmore-lie-groups_FO1976 | 199 | 1.000 | J_{ \pm} | | |
| gilmore-lie-groups_FO1977 | 199 | 1.000 | K_{ \pm} | | |
| gilmore-lie-groups_FO1978 | 199 | 1.000 | n_{1}, n_{2} | | |
| gilmore-lie-groups_FO1979 | 199 | 1.000 | \left|n_{1}, n_{2}\right\rangle=\left|n_{1}\right\rangle \otimes | | |
| gilmore-lie-groups_FO1980 | 199 | 1.000 | \left|n_{2}\right\rangle | | |
| gilmore-lie-groups_FO1981 | 199 | 0.999 | J_{3}, J_{ \pm}, K_{3}, K_{ \pm} | | |
| gilmore-lie-groups_FO1982 | 199 | 1.000 | J_{x} | | |
| gilmore-lie-groups_FO1983 | 199 | 1.000 | J_{y} | | |
| gilmore-lie-groups_FO1984 | 199 | 0.749 | (0,0) | | |
| gilmore-lie-groups_FO1985 | 199 | 0.975 | \left|{ }_{m}^{j}\right\rangle=\left|n_{1}, n_{2}\right\rangle, j=\frac{1}{2}\left(n_{1}+n_{2}\right), m=\frac{1}{2}\left(n_{1}-n_{2}\right) | | |
| gilmore-lie-groups_FO1986 | 199 | 1.000 | n_{1} | | |
| gilmore-lie-groups_FO1987 | 199 | 1.000 | n_{2} | | |
| gilmore-lie-groups_FO1996 | 200 | 0.998 | \left|n_{1}+k, n_{2}-k\right\rangle | | |
| gilmore-lie-groups_FO1997 | 200 | 1.000 | \left|n_{1}, n_{2}=-1\right\rangle | | |
| gilmore-lie-groups_FO1998 | 200 | 1.000 | n_{1}=0,1,2, \ldots | | |
| gilmore-lie-groups_FO1999 | 200 | 0.995 | \left|{ }_{m}^{j}\right\rangle=\left|n_{1}, n_{2}\right\rangle | | |
| gilmore-lie-groups_FO2000 | 200 | 0.995 | j+\frac{1}{2}= | | |
| gilmore-lie-groups_FO2001 | 200 | 0.993 | 0, \frac{1}{2}, 1, \frac{3}{2}, \ldots | | |
| gilmore-lie-groups_FO2002 | 200 | 0.993 | 2 j+1=0,1,2,3, \ldots | | |
| gilmore-lie-groups_FO2003 | 200 | 0.993 | m=j+1, j+2, \ldots | | |
| gilmore-lie-groups_FO2004 | 200 | 1.000 | \mathcal{D}_{+}^{j} | | |
| gilmore-lie-groups_FO2005 | 200 | 0.831 | \mathbf{j} | | |
| gilmore-lie-groups_FO2006 | 200 | 1.000 | \left|n_{1}=-1, n_{2}\right\rangle | | |
| gilmore-lie-groups_FO2007 | 200 | 1.000 | n_{2}=0,1,2, \ldots | | |
| gilmore-lie-groups_FO2008 | 200 | 1.000 | m=-j-1,-j-2, \ldots | | |
| gilmore-lie-groups_FO2009 | 200 | 0.999 | \mathcal{D}_{-}^{j} | | |
| gilmore-lie-groups_FO2010 | 201 | 0.998 | \left(\mathcal{D}_{+}^{j}\right) | | |
| gilmore-lie-groups_FO2011 | 201 | 0.998 | \left(\mathcal{D}_{-}^{j}\right) | | |
| gilmore-lie-groups_FO2012 | 201 | 0.999 | \mid n_{1}+k, n_{2}- | | |
| gilmore-lie-groups_FO2013 | 201 | 1.000 | k\rangle(k=\ldots,-2,-1,0,+1,+2, \ldots) | | |
| gilmore-lie-groups_FO2014 | 201 | 1.000 | |p, q\rangle | | |
| gilmore-lie-groups_FO2015 | 201 | 1.000 | -1 \leq p, q \leq 0 | | |
| gilmore-lie-groups_FO2016 | 201 | 1.000 | \frac{1}{2}(p-q)= | | |
| gilmore-lie-groups_FO2017 | 201 | 0.891 | p-q=0 | | |
| gilmore-lie-groups_FO2018 | 201 | 0.891 | -1 \leq p=q \leq 0 | | |
| gilmore-lie-groups_FO2019 | 201 | 0.679 | \left.\left.\right|_{m} ^{j}\right\rangle | | |
| gilmore-lie-groups_FO2020 | 201 | 0.679 | -\frac{1}{2} \leq j+\frac{1}{2} \leq+\frac{1}{2} | | |
| gilmore-lie-groups_FO2021 | 201 | 0.679 | \mathcal{D}^{p} | | |
| gilmore-lie-groups_FO2022 | 201 | 1.000 | j+\frac{1}{2}=i \beta | | |
| gilmore-lie-groups_FO2023 | 201 | 1.000 | j^{\prime}<-\frac{1}{2} | | |
| gilmore-lie-groups_FO2024 | 201 | 0.693 | \left(n_{1}, n_{2}\right)=\left(-\frac{1}{2},-\frac{1}{2}\right) | | |
| gilmore-lie-groups_FO2025 | 201 | 1.000 | j>-\frac{1}{2} | | |
| gilmore-lie-groups_FO2026 | 201 | 0.311 | j+\frac{1}{2}=-\left(j^{\prime}+\frac{1}{2}\right) | | |
| gilmore-lie-groups_FO2027 | 201 | 0.311 | \left|k_{m^{\prime}}^{j^{\prime}}\right\rangle \simeq\left|\underset{m=m^{\prime}}{j}\right\rangle | | |
| gilmore-lie-groups_FO2028 | 201 | 0.996 | X=h^{i} H_{i}+e^{\alpha} E_{\alpha} | | |
| gilmore-lie-groups_FO2029 | 201 | 0.996 | h^{i}, e^{\alpha} | | |
| gilmore-lie-groups_FO2030 | 201 | 1.000 | \frac{1}{2}(n-l) | | |
| gilmore-lie-groups_FO2031 | 201 | 1.000 | \left(E_{\alpha}-E_{-\alpha}\right) / \sqrt{2} | | |
| gilmore-lie-groups_FO2032 | 202 | 1.000 | \mathfrak{s p}(p, q) | | |
| gilmore-lie-groups_FO2033 | 202 | 1.000 | \mathfrak{s p}(p+q) | | |
| gilmore-lie-groups_FO2034 | 203 | 0.999 | \mathfrak{p}, i \mathfrak{p} | | |
| gilmore-lie-groups_FO2035 | 203 | 1.000 | i \mathfrak{p} | | |
| gilmore-lie-groups_FO2036 | 203 | 1.000 | g_{i j} | | |
| gilmore-lie-groups_FO2037 | 203 | 1.000 | \operatorname{EXP}(\mathfrak{p}) | | |
| gilmore-lie-groups_FO2038 | 203 | 1.000 | \operatorname{EXP}(i \mathfrak{p}) | | |
| gilmore-lie-groups_FO2039 | 203 | 0.997 | S^{2} \sim S U(2) / U(1) | | |
| gilmore-lie-groups_FO2040 | 203 | 0.997 | H_{2+}^{2}=S L(2 ; \mathbb{R}) / | | |
| gilmore-lie-groups_FO2041 | 203 | 1.000 | S O(2)=S U(1,1) / U(1) | | |
| gilmore-lie-groups_FO2042 | 203 | 1.000 | H_{1}^{2}=S L(2 ; \mathbb{R}) / S O(1,1) | | |
| gilmore-lie-groups_FO2043 | 203 | 0.943 | \mathfrak{s u}(2)-\mathfrak{u}(1) | | |
| gilmore-lie-groups_FO2044 | 203 | 0.999 | S U(2) / U(1) \sim S^{2} | | |
| gilmore-lie-groups_FO2045 | 203 | 0.991 | \mathfrak{s u}(1,1)-\mathfrak{u}(1) \simeq \mathfrak{s l}(2 ; \mathbb{R})-\mathfrak{s o}(2) | | |
| gilmore-lie-groups_FO2046 | 203 | 1.000 | H_{2+}^{2}=S U(1,1) / S O(2) | | |
| gilmore-lie-groups_FO2049 | 204 | 0.998 | \mathfrak{s u}(1,1)-\mathfrak{s o}(1,1) | | |
| gilmore-lie-groups_FO2050 | 204 | 0.966 | \operatorname{EXP}[\mathfrak{s u}(1,1)-\mathfrak{s o}(1,1)]=S U(1,1) / S O(1,1) | | |
| gilmore-lie-groups_FO2051 | 204 | 0.952 | \mathfrak{s o}(n), \mathfrak{s u}(n), \mathfrak{s p}(n) | | |
| gilmore-lie-groups_FO2052 | 205 | 1.000 | \mathfrak{p}(i \mathfrak{p}) | | |
| gilmore-lie-groups_FO2053 | 205 | 1.000 | P=G / H=\operatorname{EXP}(\mathfrak{p}) | | |
| gilmore-lie-groups_FO2054 | 205 | 1.000 | P | | |
| gilmore-lie-groups_FO2055 | 205 | 0.999 | P^{\prime}=G^{\prime} / H=\operatorname{EXP}(i \mathfrak{p}) | | |
| gilmore-lie-groups_FO2056 | 205 | 1.000 | P^{\prime}=\operatorname{EXP}(i \mathfrak{p}) | | |
| gilmore-lie-groups_FO2057 | 205 | 1.000 | n=\operatorname{dim} i \mathfrak{p} | | |
| gilmore-lie-groups_FO2058 | 205 | 1.000 | P=\operatorname{EXP}(\mathfrak{p}) | | |
| gilmore-lie-groups_FO2059 | 205 | 1.000 | \mathfrak{g}=\mathfrak{k}+\mathfrak{p} | | |
| gilmore-lie-groups_FO2060 | 205 | 1.000 | [\mathfrak{k}, \mathfrak{k}] \subseteq \mathfrak{k},[\mathfrak{k}, \mathfrak{p}] \subseteq \mathfrak{p} | | |
| gilmore-lie-groups_FO2061 | 205 | 1.000 | [\mathfrak{p}, \mathfrak{p}] \subseteq \mathfrak{k} | | |
| gilmore-lie-groups_FO2062 | 205 | 1.000 | P= | | |
| gilmore-lie-groups_FO2063 | 205 | 1.000 | G / K | | |
| gilmore-lie-groups_FO2064 | 205 | 1.000 | P=G / H | | |
| gilmore-lie-groups_FO2065 | 205 | 1.000 | P^{\prime}=G^{\prime} / H | | |
| gilmore-lie-groups_FO2066 | 205 | 0.993 | S O(p, q) / S O(p) \times S O(q) | | |
| gilmore-lie-groups_FO2067 | 206 | 1.000 | B^{\dagger} B | | |
| gilmore-lie-groups_FO2068 | 206 | 1.000 | B B^{\dagger} | | |
| gilmore-lie-groups_FO2069 | 206 | 1.000 | \min (p, q) | | |
| gilmore-lie-groups_FO2070 | 206 | 1.000 | P\left(P^{\prime}\right) | | |
| gilmore-lie-groups_FO2072 | 206 | 0.999 | G^{\prime} | | |
| gilmore-lie-groups_FO2078 | 313 | 0.994 | n-1 | | |
| gilmore-lie-groups_FO2082 | 207 | 1.000 | S O(p, q) / S O(p) \otimes S O(q) | | |
| gilmore-lie-groups_FO2106 | 207 | 1.000 | G^{\prime} / H \rightarrow G / H | | |
| gilmore-lie-groups_FO2107 | 207 | 1.000 | S O(p+q) / S O(p) \otimes S O(q) | | |
| gilmore-lie-groups_FO2108 | 207 | 1.000 | P, P^{\prime} | | |
| gilmore-lie-groups_FO2109 | 207 | 0.987 | d x(\mathrm{Id}) | | |
| gilmore-lie-groups_FO2110 | 207 | 1.000 | d x(p) | | |
| gilmore-lie-groups_FO2111 | 208 | 1.000 | M^{i}{ }_{\mu}(p) | | |
| gilmore-lie-groups_FO2112 | 208 | 0.998 | S O(n, 1) / S O(n), S U(n, 1) / U(n), S p(n, 1) / S p(n) \times S p(1) | | |
| gilmore-lie-groups_FO2113 | 208 | 1.000 | M^{i}{ }_{\mu}(X) | | |
| gilmore-lie-groups_FO2114 | 208 | 1.000 | W^{-1} | | |
| gilmore-lie-groups_FO2115 | 208 | 1.000 | H_{2}^{2}=S O(2,1) / S O(2) | | |
| gilmore-lie-groups_FO2116 | 209 | 0.999 | S^{2}=S O(3) / S O(2) | | |
| gilmore-lie-groups_FO2117 | 209 | 1.000 | S O(2+1) / S O(2) | | |
| gilmore-lie-groups_FO2118 | 209 | 0.999 | Y^{2}=1-\left(x^{2}+y^{2}\right) \geq 0 | | |
| gilmore-lie-groups_FO2119 | 209 | 1.000 | x, y | | |
| gilmore-lie-groups_FO2120 | 209 | 1.000 | x^{i}, i=1,2, \ldots, N | | |
| gilmore-lie-groups_FO2121 | 209 | 1.000 | N=2 | | |
| gilmore-lie-groups_FO2122 | 210 | 1.000 | g^{i j} \simeq \delta^{i j}-x^{i} x^{j} | | |
| gilmore-lie-groups_FO2123 | 210 | 0.900 | g^{i j} \rightarrow \delta^{i j} | | |
| gilmore-lie-groups_FO2124 | 210 | 1.000 | R_{\mu \sigma, \alpha \beta}= | | |
| gilmore-lie-groups_FO2125 | 210 | 1.000 | \delta_{\alpha \mu} \delta_{\beta \sigma}-\delta_{\alpha \sigma} \delta_{\beta \mu} | | |
| gilmore-lie-groups_FO2126 | 210 | 0.874 | R=2 | | |
| gilmore-lie-groups_FO2127 | 210 | 1.000 | H_{2}^{2} | | |
| gilmore-lie-groups_FO2128 | 210 | 0.848 | \Gamma_{\mu \nu}^{\sigma} \rightarrow-\delta_{\mu \nu} x^{\sigma} | | |
| gilmore-lie-groups_FO2129 | 210 | 0.848 | R=-2 | | |
| gilmore-lie-groups_FO2130 | 211 | 0.998 | d s^{2}=g_{\mu \nu} d x^{\mu} d x^{\nu}>0 | | |
| gilmore-lie-groups_FO2131 | 211 | 0.998 | \Rightarrow d x=0 | | |
| gilmore-lie-groups_FO2132 | 211 | 0.958 | \left(d s^{2}<0\right) | | |
| gilmore-lie-groups_FO2133 | 211 | 0.958 | \|g\| \neq 0 | | |
| gilmore-lie-groups_FO2134 | 211 | 1.000 | \mathfrak{g}^{\prime \prime} | | |
| gilmore-lie-groups_FO2135 | 211 | 1.000 | \mathfrak{h}^{\prime \prime} | | |
| gilmore-lie-groups_FO2136 | 211 | 1.000 | \mathfrak{p}^{\prime \prime} | | |
| gilmore-lie-groups_FO2137 | 211 | 1.000 | H^{\prime \prime}= | | |
| gilmore-lie-groups_FO2138 | 211 | 1.000 | \operatorname{EXP}\left(\mathfrak{h}^{\prime \prime}\right) | | |
| gilmore-lie-groups_FO2139 | 211 | 1.000 | T_{1}, T_{2} | | |
| gilmore-lie-groups_FO2140 | 211 | 0.819 | T_{1}^{2}=I | | |
| gilmore-lie-groups_FO2141 | 211 | 0.819 | T_{2}^{2}=I | | |
| gilmore-lie-groups_FO2142 | 211 | 1.000 | T_{1} \neq T_{2} | | |
| gilmore-lie-groups_FO2143 | 211 | 1.000 | \mathfrak{g}_{ \pm, \pm} | | |
| gilmore-lie-groups_FO2144 | 212 | 1.000 | \mathfrak{h}^{\prime \prime}, \mathfrak{p}^{\prime \prime} | | |
| gilmore-lie-groups_FO2145 | 212 | 0.996 | \mathrm{p}^{\prime \prime} | | |
| gilmore-lie-groups_FO2146 | 212 | 0.999 | T_{1}= | | |
| gilmore-lie-groups_FO2147 | 212 | 1.000 | T_{2}= | | |
| gilmore-lie-groups_FO2151 | 213 | 1.000 | T_{3}=T_{1} T_{2} | | |
| gilmore-lie-groups_FO2152 | 212 | 1.000 | \mathfrak{g}_{+,+}=0, \mathfrak{g}_{+,-}=i \sigma_{3}, \mathfrak{g}_{-,+}=i \sigma_{2}, \mathfrak{g}_{-,-}=i \sigma_{1} | | |
| gilmore-lie-groups_FO2153 | 212 | 1.000 | T_{i} | | |
| gilmore-lie-groups_FO2154 | 212 | 0.999 | \mathfrak{h}^{\prime} | | |
| gilmore-lie-groups_FO2155 | 212 | 0.984 | P=G / K | | |
| gilmore-lie-groups_FO2156 | 212 | 1.000 | \phi_{j}\left(B, B^{\dagger}\right) | | |
| gilmore-lie-groups_FO2157 | 212 | 0.585 | (p-q) | | |
| gilmore-lie-groups_FO2158 | 212 | 0.585 | (q-p) | | |
| gilmore-lie-groups_FO2159 | 213 | 1.000 | \Delta^{2} | | |
| gilmore-lie-groups_FO2160 | 213 | 1.000 | G / H=P | | |
| gilmore-lie-groups_FO2161 | 213 | 1.000 | \Delta^{2}=g^{i j}\left(\partial_{i} \partial_{j}-\Gamma_{i j}{ }^{k} \partial_{k}\right) | | |
| gilmore-lie-groups_FO2162 | 213 | 1.000 | x^{2}+y^{2} \leq 1 | | |
| gilmore-lie-groups_FO2163 | 213 | 1.000 | S^{n} | | |
| gilmore-lie-groups_FO2164 | 213 | 1.000 | H^{n}, n>2 | | |
| gilmore-lie-groups_FO2165 | 213 | 1.000 | T_{1}^{2}=T_{2}^{2}=I | | |
| gilmore-lie-groups_FO2166 | 213 | 1.000 | T_{1} T_{2}=T_{2} T_{1} | | |
| gilmore-lie-groups_FO2167 | 213 | 1.000 | T_{3} \neq I | | |
| gilmore-lie-groups_FO2168 | 213 | 1.000 | \mathfrak{g}=\mathfrak{g}_{+,+}+\mathfrak{g}_{+,-}+\mathfrak{g}_{-,+}+\mathfrak{g}_{-,-} | | |
| gilmore-lie-groups_FO2169 | 213 | 1.000 | 3!/ 1!=6 | | |
| gilmore-lie-groups_FO2170 | 213 | 1.000 | b_{3}=0 | | |
| gilmore-lie-groups_FO2171 | 213 | 1.000 | \phi_{j}(\mathfrak{p}) | | |
| gilmore-lie-groups_FO2172 | 213 | 0.992 | \phi_{j}(\mathfrak{h}, \mathfrak{p}) | | |
| gilmore-lie-groups_FO2173 | 213 | 0.998 | \mathfrak{h}=0 | | |
| gilmore-lie-groups_FO2174 | 213 | 0.891 | x_{0}^{2}-x_{1}^{2}-x_{2}^{2}=1 | | |
| gilmore-lie-groups_FO2175 | 213 | 1.000 | -d s^{2}=d x_{0}^{2}-d x_{1}^{2}- | | |
| gilmore-lie-groups_FO2176 | 213 | 0.973 | d x_{2}^{2} | | |
| gilmore-lie-groups_FO2177 | 214 | 1.000 | x_{1}, x_{2} | | |
| gilmore-lie-groups_FO2178 | 214 | 1.000 | (r, \theta), x_{1}=r \cos (\theta), x_{2}=r \sin (\theta) | | |
| gilmore-lie-groups_FO2179 | 214 | 1.000 | S O(1,2) | | |
| gilmore-lie-groups_FO2180 | 214 | 1.000 | \left(x_{1}, x_{2}\right) \in H_{2}^{2} | | |
| gilmore-lie-groups_FO2181 | 214 | 1.000 | g_{i j}(x) | | |
| gilmore-lie-groups_FO2182 | 214 | 1.000 | X_{r s}=g_{r t} x^{t} \partial_{s}- | | |
| gilmore-lie-groups_FO2183 | 214 | 1.000 | g_{s t} x^{t} \partial_{r} | | |
| gilmore-lie-groups_FO2184 | 214 | 1.000 | \left[X_{a b}, \Delta\right]=0 | | |
| gilmore-lie-groups_FO2185 | 214 | 1.000 | \Delta=G^{a b ; r s} X_{a b} X_{r s} | | |
| gilmore-lie-groups_FO2186 | 214 | 0.546 | G_{a b ; r s}=\operatorname{tr}\left\{\mathfrak{d e f}\left(X_{a b}\right) \mathfrak{d e f}\left(X_{r s}\right)\right\} | | |
| gilmore-lie-groups_FO2187 | 214 | 0.546 | G^{a b ; r s} | | |
| gilmore-lie-groups_FO2188 | 214 | 0.546 | G_{a b ; r s} | | |
| gilmore-lie-groups_FO2189 | 214 | 0.989 | \Delta | | |
| gilmore-lie-groups_FO2190 | 214 | 1.000 | \partial_{r} | | |
| gilmore-lie-groups_FO2191 | 214 | 0.765 | \Gamma_{r}{ }_{s}{ }^{t} | | |
| gilmore-lie-groups_FO2192 | 214 | 1.000 | \left(r, \phi_{2}, \phi_{3}, \ldots, \phi_{n}\right) | | |
| gilmore-lie-groups_FO2193 | 214 | 1.000 | S^{n} \subset R^{n+1} | | |
| gilmore-lie-groups_FO2194 | 215 | 0.995 | S O(n+1) | | |
| gilmore-lie-groups_FO2195 | 215 | 1.000 | f_{1}(\phi) | | |
| gilmore-lie-groups_FO2196 | 215 | 1.000 | f_{2}(\phi) | | |
| gilmore-lie-groups_FO2197 | 215 | 0.994 | a_{i}^{\dagger} a_{j}\left(\mathcal{H}=h_{i j}(t) a_{i}^{\dagger} a_{j}, 1 \leq i, j \leq\right. | | |
| gilmore-lie-groups_FO2198 | 215 | 0.985 | i \mathcal{H} | | |
| gilmore-lie-groups_FO2199 | 215 | 1.000 | S U(n) / U(n-1) | | |
| gilmore-lie-groups_FO2200 | 215 | 1.000 | V^{(n)} | | |
| gilmore-lie-groups_FO2201 | 215 | 0.999 | (x, x)_{m}=m_{i j} x^{i} x^{j} | | |
| gilmore-lie-groups_FO2202 | 215 | 0.999 | n+2 | | |
| gilmore-lie-groups_FO2203 | 215 | 1.000 | W^{(n+2)} | | |
| gilmore-lie-groups_FO2204 | 215 | 1.000 | (y, y)_{M}=M_{\mu \nu} y^{\mu} y^{\nu}=(s x, s x)_{m}-\frac{2}{2} s\left[s(x, x)_{m}\right]=0 | | |
| gilmore-lie-groups_FO2205 | 215 | 1.000 | G=O(n) | | |
| gilmore-lie-groups_FO2206 | 215 | 1.000 | O(n+1,1) | | |
| gilmore-lie-groups_FO2207 | 215 | 1.000 | n_{1}, n_{2}\left(n_{1}+n_{2}=n\right) | | |
| gilmore-lie-groups_FO2208 | 215 | 1.000 | G=O\left(n_{1}, n_{2}\right) | | |
| gilmore-lie-groups_FO2209 | 215 | 1.000 | H=O\left(n_{1}+1, n_{2}+1\right) | | |
| gilmore-lie-groups_FO2210 | 215 | 1.000 | S O\left(n_{1}+1, n_{2}+1\right) / S O\left(n_{1}, n_{2}\right) | | |
| gilmore-lie-groups_FO2211 | 215 | 0.996 | y \rightarrow y^{\prime} | | |
| gilmore-lie-groups_FO2212 | 215 | 0.996 | x \rightarrow x^{\prime} | | |
| gilmore-lie-groups_FO2213 | 215 | 0.996 | x^{\prime i}=y^{\prime i} / y^{\prime n+1} | | |
| gilmore-lie-groups_FO2214 | 215 | 0.997 | (+1,-1,-1,-1) | | |
| gilmore-lie-groups_FO2215 | 216 | 0.988 | L_{\mu \nu} | | |
| gilmore-lie-groups_FO2216 | 216 | 1.000 | P_{\mu} | | |
| gilmore-lie-groups_FO2217 | 216 | 1.000 | K_{\mu} | | |
| gilmore-lie-groups_FO2218 | 216 | 1.000 | x_{\mu}=g_{\mu \nu} x^{\nu} | | |
| gilmore-lie-groups_FO2219 | 216 | 0.989 | e^{c^{\mu} K_{\mu}}\left(x^{\nu}\right)=x^{\prime \nu}=\frac{x^{\nu}+c^{\nu}(x, x)}{1+2(c, x)+(c, c)(x, x)} | | |
| gilmore-lie-groups_FO2220 | 216 | 0.999 | P_{\mu} \rightarrow P_{\mu}^{\prime}=x_{\mu} | | |
| gilmore-lie-groups_FO2221 | 216 | 0.999 | K_{\mu} \rightarrow K_{\mu}^{\prime}=2(x, \partial) \partial_{\mu}- | | |
| gilmore-lie-groups_FO2222 | 216 | 0.719 | (\partial, \partial) x_{\mu} | | |
| gilmore-lie-groups_FO2223 | 216 | 1.000 | z=x+i y | | |
| gilmore-lie-groups_FO2224 | 216 | 1.000 | P S L(2, \mathbb{R}) | | |
| gilmore-lie-groups_FO2225 | 216 | 1.000 | M,-M \in S L(2 ; \mathbb{R}) | | |
| gilmore-lie-groups_FO2226 | 216 | 1.000 | y^{\prime}>0 | | |
| gilmore-lie-groups_FO2227 | 216 | 1.000 | y>0 | | |
| gilmore-lie-groups_FO2228 | 216 | 1.000 | y^{\prime}=0 | | |
| gilmore-lie-groups_FO2229 | 216 | 0.997 | (y=0) | | |
| gilmore-lie-groups_FO2230 | 217 | 1.000 | d z^{\prime}=d z /|c z+d|^{2} | | |
| gilmore-lie-groups_FO2231 | 217 | 1.000 | d \mu=d x d y / y^{2} | | |
| gilmore-lie-groups_FO2232 | 217 | 1.000 | z_{2} | | |
| gilmore-lie-groups_FO2233 | 217 | 1.000 | w=x+i y | | |
| gilmore-lie-groups_FO2234 | 217 | 1.000 | \bar{w} w=x^{2}+y^{2} \leq 1 | | |
| gilmore-lie-groups_FO2235 | 217 | 1.000 | M,-M \in S U(1,1) | | |
| gilmore-lie-groups_FO2236 | 217 | 1.000 | w=e^{i \phi} \rightarrow w^{\prime}=e^{i \psi} | | |
| gilmore-lie-groups_FO2237 | 217 | 1.000 | \psi(\phi) | | |
| gilmore-lie-groups_FO2238 | 217 | 1.000 | w_{1} | | |
| gilmore-lie-groups_FO2239 | 217 | 1.000 | w_{2} | | |
| gilmore-lie-groups_FO2240 | 217 | 1.000 | w | | |
| gilmore-lie-groups_FO2241 | 217 | 1.000 | z_{0} | | |
| gilmore-lie-groups_FO2242 | 218 | 1.000 | z_{0}=i | | |
| gilmore-lie-groups_FO2243 | 218 | 1.000 | e^{i \phi}=i | | |
| gilmore-lie-groups_FO2244 | 219 | 1.000 | Y_{r} | | |
| gilmore-lie-groups_FO2245 | 219 | 1.000 | Y_{r}= | | |
| gilmore-lie-groups_FO2246 | 219 | 1.000 | M_{r}{ }^{i}(\epsilon) X_{i} | | |
| gilmore-lie-groups_FO2247 | 219 | 0.986 | C_{r s}{ }^{t}(\epsilon)=M_{r}{ }^{i}(\epsilon) M_{s}{ }^{j}(\epsilon) C_{i j}{ }^{k}\left(M(\epsilon)^{-1}\right)_{k}{ }^{t} | | |
| gilmore-lie-groups_FO2248 | 219 | 1.000 | C_{r s}{ }^{t}(\epsilon) | | |
| gilmore-lie-groups_FO2249 | 220 | 0.998 | \mathfrak{g} \rightarrow \mathfrak{g}^{\prime} | | |
| gilmore-lie-groups_FO2250 | 220 | 1.000 | \operatorname{dim}(\mathfrak{h}) | | |
| gilmore-lie-groups_FO2251 | 220 | 1.000 | \epsilon \rightarrow 0 | | |
| gilmore-lie-groups_FO2252 | 220 | 1.000 | \left[\mathfrak{h}, \mathfrak{p}^{\prime}\right] \subseteq \mathfrak{p}^{\prime} | | |
| gilmore-lie-groups_FO2254 | 220 | 0.998 | L_{1}=X_{23}= | | |
| gilmore-lie-groups_FO2255 | 220 | 1.000 | x_{2} \partial_{3}-x_{3} \partial_{2}=\epsilon_{1 j k} x_{j} \partial_{k} | | |
| gilmore-lie-groups_FO2256 | 220 | 1.000 | L_{2} | | |
| gilmore-lie-groups_FO2257 | 220 | 1.000 | L_{3} | | |
| gilmore-lie-groups_FO2258 | 221 | 1.000 | L_{1} | | |
| gilmore-lie-groups_FO2259 | 221 | 1.000 | I S O(2)=E(2) | | |
| gilmore-lie-groups_FO2260 | 221 | 1.000 | L_{3}, P_{1}, P_{2} | | |
| gilmore-lie-groups_FO2261 | 221 | 0.983 | R^{2}, I S O(2) | | |
| gilmore-lie-groups_FO2262 | 221 | 1.000 | P_{1}=\partial_{1} | | |
| gilmore-lie-groups_FO2263 | 221 | 1.000 | P_{2}=\partial_{2} | | |
| gilmore-lie-groups_FO2264 | 221 | 0.698 | x^{2}+y^{2}+z^{2}=R^{2} | | |
| gilmore-lie-groups_FO2265 | 221 | 0.698 | 0,0, R | | |
| gilmore-lie-groups_FO2266 | 221 | 1.000 | R \rightarrow \infty | | |
| gilmore-lie-groups_FO2267 | 221 | 0.992 | -P_{2},+P_{1} | | |
| gilmore-lie-groups_FO2268 | 221 | 0.992 | -y | | |
| gilmore-lie-groups_FO2269 | 221 | 0.992 | +x | | |
| gilmore-lie-groups_FO2270 | 221 | 1.000 | \theta_{1}, \theta_{2} | | |
| gilmore-lie-groups_FO2271 | 221 | 1.000 | d_{1}, d_{2} | | |
| gilmore-lie-groups_FO2272 | 221 | 1.000 | R \theta_{i}(i=1,2) | | |
| gilmore-lie-groups_FO2273 | 221 | 1.000 | \theta_{2}=d_{1} / R | | |
| gilmore-lie-groups_FO2274 | 221 | 1.000 | d_{1} | | |
| gilmore-lie-groups_FO2275 | 221 | 1.000 | \theta_{1}=d_{2} / R | | |
| gilmore-lie-groups_FO2276 | 221 | 1.000 | -d_{2} | | |
| gilmore-lie-groups_FO2278 | 223 | 0.418 | P_{i} | | |
| gilmore-lie-groups_FO2279 | 223 | 0.418 | i=1,2,3 | | |
| gilmore-lie-groups_FO2280 | 223 | 1.000 | I S O(3) | | |
| gilmore-lie-groups_FO2281 | 223 | 1.000 | S O(3) \rightarrow I S O(2) | | |
| gilmore-lie-groups_FO2282 | 223 | 1.000 | \mathbf{P} \cdot \mathbf{P}=\nabla^{2} | | |
| gilmore-lie-groups_FO2283 | 223 | 1.000 | \mathbf{L} \cdot \mathbf{P}=-\mathbf{L} \cdot \nabla | | |
| gilmore-lie-groups_FO2285 | 224 | 0.592 | [S O(3,1)] | | |
| gilmore-lie-groups_FO2286 | 224 | 0.997 | S O(3,2) | | |
| gilmore-lie-groups_FO2287 | 224 | 0.998 | \epsilon^{\alpha \beta \gamma \mu \nu} | | |
| gilmore-lie-groups_FO2288 | 224 | 1.000 | W_{\alpha} | | |
| gilmore-lie-groups_FO2289 | 224 | 1.000 | W^{\alpha} | | |
| gilmore-lie-groups_FO2290 | 224 | 0.615 | v | | |
| gilmore-lie-groups_FO2291 | 225 | 1.000 | \epsilon^{\alpha \beta \gamma \mu 5} X_{\beta \gamma}\left(\partial / \partial x^{\mu}\right) | | |
| gilmore-lie-groups_FO2292 | 225 | 1.000 | W^{\alpha} W_{\alpha} | | |
| gilmore-lie-groups_FO2293 | 225 | 1.000 | \mathbf{P} \cdot \mathbf{P}=\sum P_{\mu} P^{\mu}=-(m c)^{2} | | |
| gilmore-lie-groups_FO2294 | 225 | 0.862 | \mathfrak{u}(2) | | |
| gilmore-lie-groups_FO2295 | 225 | 0.862 | J_{3}, J_{ \pm}, J_{0} | | |
| gilmore-lie-groups_FO2296 | 225 | 0.996 | h_{3}, h_{ \pm}, h_{0} | | |
| gilmore-lie-groups_FO2297 | 226 | 1.000 | c \rightarrow 0 | | |
| gilmore-lie-groups_FO2298 | 226 | 1.000 | J_{0} \rightarrow h_{0} | | |
| gilmore-lie-groups_FO2299 | 226 | 1.000 | \left(h_{0} / 2 c\right)^{2} | | |
| gilmore-lie-groups_FO2300 | 226 | 1.000 | c \rightarrow 0,\left(c h_{3}\right)^{2} \rightarrow 0 | | |
| gilmore-lie-groups_FO2301 | 227 | 1.000 | \hat{n}+\frac{1}{2} I, a^{\dagger}, a | | |
| gilmore-lie-groups_FO2302 | 227 | 1.000 | \mathfrak{h}_{4} | | |
| gilmore-lie-groups_FO2303 | 227 | 1.000 | \theta_{0} | | |
| gilmore-lie-groups_FO2304 | 227 | 1.000 | \theta_{0}-\theta_{3} / 2 c^{2} | | |
| gilmore-lie-groups_FO2305 | 227 | 0.999 | h_{3} | | |
| gilmore-lie-groups_FO2306 | 227 | 0.999 | |J, M\rangle | | |
| gilmore-lie-groups_FO2307 | 227 | 1.000 | |J,-J\rangle | | |
| gilmore-lie-groups_FO2308 | 227 | 1.000 | M=-J | | |
| gilmore-lie-groups_FO2309 | 227 | 0.985 | J | | |
| gilmore-lie-groups_FO2310 | 227 | 0.985 | (2 J+1) | | |
| gilmore-lie-groups_FO2311 | 228 | 1.000 | n=J+M | | |
| gilmore-lie-groups_FO2312 | 229 | 1.000 | \lim _{c \rightarrow 0} \zeta / c \rightarrow \alpha | | |
| gilmore-lie-groups_FO2313 | 229 | 1.000 | P_{m}^{l}(\cos \theta) | | |
| gilmore-lie-groups_FO2314 | 229 | 1.000 | u \rightarrow | | |
| gilmore-lie-groups_FO2315 | 229 | 1.000 | x / \sqrt{l} | | |
| gilmore-lie-groups_FO2316 | 229 | 1.000 | l+m=n | | |
| gilmore-lie-groups_FO2317 | 230 | 1.000 | c \rightarrow 0, l \rightarrow \infty, l+m=n, 2 l c^{2}=1 | | |
| gilmore-lie-groups_FO2318 | 230 | 1.000 | 1 / \sqrt{\pi} | | |
| gilmore-lie-groups_FO2319 | 230 | 1.000 | \left(2^{n} n!\right)^{-1} | | |
| gilmore-lie-groups_FO2320 | 230 | 1.000 | H_{n}(x) | | |
| gilmore-lie-groups_FO2321 | 231 | 1.000 | I S O(2) | | |
| gilmore-lie-groups_FO2322 | 231 | 1.000 | P_{ \pm} | | |
| gilmore-lie-groups_FO2323 | 231 | 1.000 | \mathfrak{i s o}(2) | | |
| gilmore-lie-groups_FO2324 | 231 | 1.000 | J_{k}(x) | | |
| gilmore-lie-groups_FO2325 | 231 | 0.992 | A_{k}^{l} | | |
| gilmore-lie-groups_FO2326 | 231 | 0.992 | \theta^{\prime}, \phi^{\prime} | | |
| gilmore-lie-groups_FO2327 | 231 | 0.998 | \mathbf{L} \cdot \mathbf{L} | | |
| gilmore-lie-groups_FO2328 | 231 | 0.998 | \nabla^{2} | | |
| gilmore-lie-groups_FO2329 | 231 | 1.000 | \mathfrak{u}(2) \rightarrow \mathfrak{h}_{4} | | |
| gilmore-lie-groups_FO2330 | 232 | 1.000 | N_{n} | | |
| gilmore-lie-groups_FO2331 | 232 | 0.993 | N_{n}=1 / \sqrt{2^{n} n!\sqrt{\pi}} | | |
| gilmore-lie-groups_FO2332 | 232 | 0.984 | J_{3}, J_{ \pm}\left(\left[J_{3}, J_{ \pm}\right]= \pm J_{ \pm},\left[J_{+}, J_{-}\right]=2 J_{3}\right) | | |
| gilmore-lie-groups_FO2333 | 232 | 0.993 | \left(P^{\prime}, T^{\prime}, V^{\prime}\right) | | |
| gilmore-lie-groups_FO2334 | 232 | 0.991 | \partial_{x}, \partial_{t}, t \partial_{x} | | |
| gilmore-lie-groups_FO2335 | 232 | 1.000 | \mathfrak{a}_{1} | | |
| gilmore-lie-groups_FO2336 | 232 | 0.985 | \mathfrak{g a l}(1) | | |
| gilmore-lie-groups_FO2337 | 232 | 1.000 | S^{n}=S O(n+1) / S O(n) | | |
| gilmore-lie-groups_FO2338 | 232 | 1.000 | R^{n}=I S O(n) / S O(n) | | |
| gilmore-lie-groups_FO2339 | 233 | 1.000 | \tau=(\zeta /|\zeta|) \tan (|\zeta|) | | |
| gilmore-lie-groups_FO2340 | 233 | 1.000 | \alpha=\lim _{c \rightarrow 0} \zeta / c | | |
| gilmore-lie-groups_FO2341 | 233 | 1.000 | \langle X\rangle=\operatorname{tr} X e^{-\beta \mathcal{H}} / \operatorname{tr} e^{-\beta \mathcal{H}} | | |
| gilmore-lie-groups_FO2342 | 233 | 1.000 | \left\langle e^{\alpha X}\right\rangle= | | |
| gilmore-lie-groups_FO2343 | 233 | 1.000 | \operatorname{tr} e^{\alpha X} e^{-\beta \mathcal{H}} / \operatorname{tr} e^{-\beta \mathcal{H}} | | |
| gilmore-lie-groups_FO2344 | 233 | 0.998 | \mathcal{H}=\epsilon J_{3} | | |
| gilmore-lie-groups_FO2345 | 233 | 1.000 | 2 j+1=2 | | |
| gilmore-lie-groups_FO2346 | 234 | 0.974 | \mathcal{C}^{2}=\sum_{i j} X_{i j}^{2}, \mathcal{C}^{3}=\epsilon^{\text {abcdef }} X_{a b} X_{c d} X_{e f} | | |
| gilmore-lie-groups_FO2347 | 234 | 0.974 | \mathcal{C}^{4}= | | |
| gilmore-lie-groups_FO2348 | 234 | 0.812 | \sum_{i j} Y_{i j}^{2} | | |
| gilmore-lie-groups_FO2349 | 234 | 0.812 | Y_{i j}=\epsilon^{i j c d e f} X_{c d} X_{e f} | | |
| gilmore-lie-groups_FO2350 | 234 | 0.812 | S O(6) | | |
| gilmore-lie-groups_FO2351 | 234 | 0.995 | S O(4) \otimes S O(2) | | |
| gilmore-lie-groups_FO2352 | 234 | 0.933 | A_{i}=\lim _{\epsilon \rightarrow 0} \epsilon X_{i 5} | | |
| gilmore-lie-groups_FO2353 | 234 | 0.933 | B_{i}=\lim _{\epsilon \rightarrow 0} \epsilon X_{i 6} | | |
| gilmore-lie-groups_FO2354 | 234 | 1.000 | S O(4,2) /[S O(4) \otimes S O(2)] | | |
| gilmore-lie-groups_FO2355 | 234 | 1.000 | I[S O(4) \otimes S O(2)] /[S O(4) \otimes S O(2)] | | |
| gilmore-lie-groups_FO2356 | 234 | 1.000 | X_{\alpha} | | |
| gilmore-lie-groups_FO2357 | 234 | 0.987 | Y_{\alpha}=\lim _{\epsilon \rightarrow 0} \in X_{\alpha} | | |
| gilmore-lie-groups_FO2358 | 236 | 1.000 | |\psi\rangle | | |
| gilmore-lie-groups_FO2359 | 236 | 1.000 | \langle S \mid \psi\rangle | | |
| gilmore-lie-groups_FO2360 | 236 | 1.000 | \left\langle S^{\prime} \mid \psi\right\rangle | | |
| gilmore-lie-groups_FO2361 | 236 | 0.998 | \left\langle S^{\prime} \mid S\right\rangle | | |
| gilmore-lie-groups_FO2362 | 236 | 1.000 | \left\langle S \mid S^{\prime}\right\rangle | | |
| gilmore-lie-groups_FO2363 | 237 | 1.000 | \left|\psi^{\prime}\right\rangle | | |
| gilmore-lie-groups_FO2364 | 237 | 0.997 | S O(3)) | | |
| gilmore-lie-groups_FO2365 | 237 | 0.997 | \left\langle S^{\prime}\right| H\left|S^{\prime}\right\rangle=\left\langle S^{\prime} \mid S\right\rangle\langle S| H|S\rangle\left\langle S \mid S^{\prime}\right\rangle | | |
| gilmore-lie-groups_FO2366 | 237 | 1.000 | \left\langle S^{\prime}\right| H\left|S^{\prime}\right\rangle= | | |
| gilmore-lie-groups_FO2367 | 237 | 1.000 | \langle S| H|S\rangle | | |
| gilmore-lie-groups_FO2368 | 237 | 1.000 | 2 p_{z} | | |
| gilmore-lie-groups_FO2369 | 237 | 1.000 | 2 p_{x} | | |
| gilmore-lie-groups_FO2370 | 237 | 1.000 | 2 p_{y} | | |
| gilmore-lie-groups_FO2371 | 237 | 0.996 | p^{\mu} p_{\mu}=g_{\mu \nu} p^{\mu} p^{\nu}=(m c)^{2} | | |
| gilmore-lie-groups_FO2372 | 237 | 1.000 | \mathbf{p} | | |
| gilmore-lie-groups_FO2373 | 237 | 1.000 | p_{\mu} \rightarrow \pi_{\mu}=p_{\mu}-(q / c) A_{\mu} | | |
| gilmore-lie-groups_FO2374 | 237 | 0.974 | \Phi | | |
| gilmore-lie-groups_FO2375 | 237 | 0.997 | \mathbf{B}=\nabla \times \mathbf{A} | | |
| gilmore-lie-groups_FO2376 | 237 | 0.997 | \mathbf{E}=-\nabla \Phi-(1 / c)(\partial \mathbf{A} / \partial t) | | |
| gilmore-lie-groups_FO2377 | 237 | 0.997 | q= | | |
| gilmore-lie-groups_FO2378 | 237 | 0.928 | -e | | |
| gilmore-lie-groups_FO2379 | 237 | 1.000 | \Phi=e / r | | |
| gilmore-lie-groups_FO2380 | 237 | 1.000 | \mathbf{A}=\mathbf{0} | | |
| gilmore-lie-groups_FO2381 | 237 | 1.000 | E \rightarrow E+e^{2} / r | | |
| gilmore-lie-groups_FO2382 | 237 | 1.000 | \mathbf{p} \rightarrow(\hbar / i) \nabla | | |
| gilmore-lie-groups_FO2383 | 237 | 1.000 | \psi(\mathbf{x}) | | |
| gilmore-lie-groups_FO2384 | 238 | 0.999 | \left\langle S^{\prime}\right| H\left|S^{\prime}\right\rangle=\langle S| H|S\rangle | | |
| gilmore-lie-groups_FO2385 | 238 | 0.999 | \left\langle S^{\prime} \mid S\right\rangle \in S O(3) | | |
| gilmore-lie-groups_FO2386 | 238 | 0.623 | m c^{2} | | |
| gilmore-lie-groups_FO2387 | 238 | 0.906 | E=m c^{2}+W | | |
| gilmore-lie-groups_FO2388 | 238 | 1.000 | W | | |
| gilmore-lie-groups_FO2389 | 238 | 0.884 | (\simeq 0.0025 \%) | | |
| gilmore-lie-groups_FO2390 | 238 | 1.000 | \left(W+e^{2} / r\right)^{2} / m c^{2} | | |
| gilmore-lie-groups_FO2391 | 238 | 0.727 | [A, B] / i \hbar=\{A, B\} | | |
| gilmore-lie-groups_FO2392 | 238 | 0.860 | (r, \theta, \phi) | | |
| gilmore-lie-groups_FO2393 | 239 | 1.000 | \mathcal{L}^{2}\left(S^{2}\right) | | |
| gilmore-lie-groups_FO2394 | 239 | 0.980 | \mathcal{L}^{2}\left(S^{2}\right) Y_{m}^{l}(\theta, \phi)=-l(l+1) Y_{m}^{l}(\theta, \phi) | | |
| gilmore-lie-groups_FO2395 | 239 | 0.980 | (l, m) | | |
| gilmore-lie-groups_FO2396 | 239 | 0.980 | l=0,1,2, \ldots | | |
| gilmore-lie-groups_FO2397 | 239 | 0.999 | -l \leq m \leq+l | | |
| gilmore-lie-groups_FO2398 | 239 | 1.000 | A, B, C | | |
| gilmore-lie-groups_FO2404 | 239 | 1.000 | n= | | |
| gilmore-lie-groups_FO2405 | 239 | 1.000 | 0,1,2, \ldots | | |
| gilmore-lie-groups_FO2406 | 239 | 1.000 | m_{\text {red }}^{-1}=m_{e}^{-1}+M_{p}^{-1} | | |
| gilmore-lie-groups_FO2409 | 271 | 1.000 | \mathcal{L}^{2} | | |
| gilmore-lie-groups_FO2412 | 240 | 1.000 | f(r) | | |
| gilmore-lie-groups_FO2418 | 240 | 1.000 | \hbar | | |
| gilmore-lie-groups_FO2420 | 241 | 1.000 | \alpha \rightarrow Z \alpha | | |
| gilmore-lie-groups_FO2421 | 241 | 1.000 | \left|W_{1}\right|=\frac{1}{2} m c^{2} \alpha^{2} | | |
| gilmore-lie-groups_FO2422 | 241 | 0.999 | 1 / N^{2} | | |
| gilmore-lie-groups_FO2423 | 241 | 0.999 | N=n+l+1 | | |
| gilmore-lie-groups_FO2424 | 241 | 1.000 | N^{\prime} | | |
| gilmore-lie-groups_FO2425 | 241 | 1.000 | \frac{1}{2} m c^{2} \alpha^{2} | | |
| gilmore-lie-groups_FO2426 | 241 | 1.000 | \frac{1}{2} m c^{2}(Z \alpha)^{2} | | |
| gilmore-lie-groups_FO2427 | 241 | 1.000 | 1 s | | |
| gilmore-lie-groups_FO2428 | 242 | 1.000 | g_{i}|\psi\rangle | | |
| gilmore-lie-groups_FO2429 | 242 | 1.000 | |\psi\rangle=\psi_{2 p_{z}}(\mathbf{x}) | | |
| gilmore-lie-groups_FO2430 | 242 | 1.000 | \pi / 2 | | |
| gilmore-lie-groups_FO2431 | 242 | 1.000 | \psi_{2 p_{x}}(\mathbf{x}) | | |
| gilmore-lie-groups_FO2432 | 242 | 1.000 | -\psi_{2 p_{y}}(\mathbf{x}) | | |
| gilmore-lie-groups_FO2433 | 242 | 1.000 | H=\mathbf{p} \cdot \mathbf{p} / 2 m-e^{2} / r | | |
| gilmore-lie-groups_FO2434 | 242 | 1.000 | \mathbf{p} \cdot \mathbf{p}=-\hbar^{2} \nabla^{2} | | |
| gilmore-lie-groups_FO2435 | 242 | 1.000 | -e^{2} / r | | |
| gilmore-lie-groups_FO2436 | 242 | 1.000 | i: \epsilon_{i j k} x_{j} \partial_{k} | | |
| gilmore-lie-groups_FO2437 | 242 | 1.000 | \mathbf{L}_{i}=(\mathbf{r} \times \mathbf{p})_{i}=(\hbar / i) \epsilon_{i j k} x_{j} \partial_{k} | | |
| gilmore-lie-groups_FO2438 | 242 | 0.998 | \mathbf{L}=\mathbf{r} \times \mathbf{p} | | |
| gilmore-lie-groups_FO2439 | 242 | 0.992 | \mathbf{r} \times \nabla | | |
| gilmore-lie-groups_FO2440 | 242 | 1.000 | \hbar / i | | |
| gilmore-lie-groups_FO2441 | 242 | 1.000 | \left(L_{+}\right) | | |
| gilmore-lie-groups_FO2442 | 242 | 1.000 | \left(L_{-}\right) | | |
| gilmore-lie-groups_FO2443 | 242 | 1.000 | L_{ \pm}=L_{x} \pm i L_{y} | | |
| gilmore-lie-groups_FO2444 | 242 | 1.000 | L_{z}=\hbar \frac{1}{2}\left(a_{1}^{\dagger} a_{1}-a_{2}^{\dagger} a_{2}\right), L_{+}=\hbar a_{1}^{\dagger} a_{2}, L_{-}=\hbar a_{2}^{\dagger} a_{1} | | |
| gilmore-lie-groups_FO2445 | 242 | 1.000 | \left|n_{1} n_{2}\right\rangle | | |
| gilmore-lie-groups_FO2446 | 242 | 0.838 | \left.=\left.\right|_{m} ^{j}\right\rangle | | |
| gilmore-lie-groups_FO2447 | 242 | 0.838 | n_{1}=0,1,2, \ldots, n_{2}=0,1,2, \ldots, n_{1}+n_{2}=2 j, n_{1}-n_{2}=2 m | | |
| gilmore-lie-groups_FO2448 | 243 | 1.000 | -j \leq m \leq+j | | |
| gilmore-lie-groups_FO2449 | 243 | 0.994 | ((x, y, z) \rightarrow(r, \theta, \phi) | | |
| gilmore-lie-groups_FO2450 | 243 | 0.994 | x=r \sin \theta \cos \phi) | | |
| gilmore-lie-groups_FO2451 | 243 | 1.000 | Y_{-l}^{l}(\theta, \phi) | | |
| gilmore-lie-groups_FO2452 | 243 | 1.000 | L_{-} Y_{-l}^{l}(\theta, \phi)=0 | | |
| gilmore-lie-groups_FO2453 | 266 | 1.000 | l=0 | | |
| gilmore-lie-groups_FO2466 | 244 | 1.000 | L_{+} | | |
| gilmore-lie-groups_FO2467 | 244 | 0.815 | (l=0,1,2,3) | | |
| gilmore-lie-groups_FO2468 | 244 | 0.662 | \theta, \phi | | |
| gilmore-lie-groups_FO2469 | 244 | 1.000 | \hbar^{2} l(l+1) | | |
| gilmore-lie-groups_FO2470 | 244 | 1.000 | E_{N}=-\frac{1}{2} m c^{2} \alpha^{2} \frac{1}{N^{2}} | | |
| gilmore-lie-groups_FO2471 | 244 | 1.000 | \sum_{l=0}^{l=N-1}(2 l+1)=N^{2} | | |
| gilmore-lie-groups_FO2472 | 244 | 1.000 | 1 / r^{2} | | |
| gilmore-lie-groups_FO2473 | 244 | 1.000 | d \mathbf{p} / d t=-K \mathbf{r} / r^{3} | | |
| gilmore-lie-groups_FO2474 | 245 | 1.000 | K=G M m, G | | |
| gilmore-lie-groups_FO2475 | 245 | 1.000 | \mathbf{r}=x \hat{\mathbf{i}}+y \hat{\mathbf{j}}+z \hat{\mathbf{k}} | | |
| gilmore-lie-groups_FO2476 | 245 | 1.000 | \mathbf{p} \times \mathbf{L} | | |
| gilmore-lie-groups_FO2477 | 245 | 0.999 | \mathbf{L} | | |
| gilmore-lie-groups_FO2478 | 245 | 1.000 | 1 / r | | |
| gilmore-lie-groups_FO2479 | 245 | 0.967 | \mathbf{r} \times \mathbf{L} | | |
| gilmore-lie-groups_FO2480 | 245 | 0.967 | \dot{\mathbf{r}} | | |
| gilmore-lie-groups_FO2481 | 245 | 1.000 | (d / d t)(\mathbf{r} / r)=\dot{\mathbf{r}} / \mathbf{r}-(\dot{\mathbf{r}} \cdot \mathbf{r}) \mathbf{r} / r^{3} | | |
| gilmore-lie-groups_FO2482 | 245 | 0.985 | d \mathbf{M} / d t=0 | | |
| gilmore-lie-groups_FO2483 | 245 | 1.000 | [H, \mathbf{M}]=0 | | |
| gilmore-lie-groups_FO2484 | 245 | 1.000 | L_{i}, M_{j} | | |
| gilmore-lie-groups_FO2485 | 245 | 0.993 | (E<0) | | |
| gilmore-lie-groups_FO2486 | 245 | 0.993 | (E>0) | | |
| gilmore-lie-groups_FO2487 | 246 | 1.000 | \mathbf{M}^{\prime}=(-m / 2 H)^{1 / 2} \mathbf{M} | | |
| gilmore-lie-groups_FO2488 | 246 | 1.000 | E>0 | | |
| gilmore-lie-groups_FO2489 | 246 | 1.000 | -\rightarrow+ | | |
| gilmore-lie-groups_FO2490 | 246 | 1.000 | S O(4) \rightarrow S O(3,1) | | |
| gilmore-lie-groups_FO2491 | 246 | 0.834 | \mathbf{A} | | |
| gilmore-lie-groups_FO2492 | 246 | 0.834 | \mathbf{B} | | |
| gilmore-lie-groups_FO2493 | 246 | 0.996 | A_{3}=\frac{1}{2}\left(a_{1}^{\dagger} a_{1}-\right. | | |
| gilmore-lie-groups_FO2494 | 246 | 0.910 | a_{2}^{\dagger} a_{2} | | |
| gilmore-lie-groups_FO2495 | 246 | 0.910 | A_{+}=a_{1}^{\dagger} a_{2}, A_{-}=a_{2}^{\dagger} a_{1} | | |
| gilmore-lie-groups_FO2496 | 246 | 0.910 | \hbar \rightarrow 1 | | |
| gilmore-lie-groups_FO2497 | 246 | 1.000 | b_{1}, b_{2} | | |
| gilmore-lie-groups_FO2498 | 246 | 1.000 | \left|p_{1}, p_{2}\right\rangle | | |
| gilmore-lie-groups_FO2499 | 246 | 1.000 | p_{1}+p_{2}=2 j_{a} | | |
| gilmore-lie-groups_FO2500 | 246 | 1.000 | p_{1}-p_{2}=m_{a} | | |
| gilmore-lie-groups_FO2501 | 246 | 1.000 | 2 j_{a}+1 | | |
| gilmore-lie-groups_FO2502 | 246 | 0.968 | p_{1}=2 j_{a}, p_{2}=0 ; p_{1}=2 j_{a}-1, p_{2}=1 | | |
| gilmore-lie-groups_FO2503 | 246 | 1.000 | \left|q_{1}, q_{2}\right\rangle | | |
| gilmore-lie-groups_FO2504 | 246 | 1.000 | q_{1}+q_{2}=2 j_{b} | | |
| gilmore-lie-groups_FO2505 | 246 | 1.000 | q_{1}-q_{2}=m_{b} | | |
| gilmore-lie-groups_FO2506 | 246 | 1.000 | \mathbf{A} \cdot \mathbf{A}=j_{a}\left(j_{a}+1\right) | | |
| gilmore-lie-groups_FO2507 | 246 | 1.000 | \mathbf{B} \cdot \mathbf{B}=j_{b}\left(j_{b}+1\right) | | |
| gilmore-lie-groups_FO2508 | 246 | 1.000 | \mathbf{A} \cdot \mathbf{A}=\mathbf{B} \cdot \mathbf{B} | | |
| gilmore-lie-groups_FO2509 | 246 | 1.000 | j_{a}=j_{b} | | |
| gilmore-lie-groups_FO2510 | 246 | 1.000 | (2 j+ | | |
| gilmore-lie-groups_FO2511 | 246 | 1.000 | 2 j+1=N=n+l+1 | | |
| gilmore-lie-groups_FO2512 | 246 | 0.996 | \operatorname{good} l | | |
| gilmore-lie-groups_FO2513 | 247 | 1.000 | M_{+}^{\prime}= | | |
| gilmore-lie-groups_FO2514 | 247 | 1.000 | A_{+}-B_{+}=a_{1}^{\dagger} a_{2}-b_{1}^{\dagger} b_{2} | | |
| gilmore-lie-groups_FO2515 | 247 | 0.998 | \mathbf{M}^{\prime} | | |
| gilmore-lie-groups_FO2516 | 247 | 0.608 | \mathbf{W}=\mathbf{L} \times \mathbf{M} | | |
| gilmore-lie-groups_FO2517 | 247 | 1.000 | \mathbf{r} \cdot \mathbf{L}=0 | | |
| gilmore-lie-groups_FO2518 | 247 | 1.000 | \mathbf{p} \cdot \mathbf{L}=0 | | |
| gilmore-lie-groups_FO2519 | 248 | 0.986 | \mathbf{W} | | |
| gilmore-lie-groups_FO2520 | 248 | 0.986 | p_{z}=0, p_{x}=\mathbf{p} \cdot \mathbf{M} / \sqrt{\mathbf{M} \cdot \mathbf{M}} | | |
| gilmore-lie-groups_FO2521 | 248 | 0.986 | p_{y}= | | |
| gilmore-lie-groups_FO2522 | 248 | 1.000 | \mathbf{p} \cdot \mathbf{W} / \sqrt{\mathbf{W} \cdot \mathbf{W}} | | |
| gilmore-lie-groups_FO2523 | 248 | 1.000 | m K / L | | |
| gilmore-lie-groups_FO2524 | 248 | 1.000 | m M / L | | |
| gilmore-lie-groups_FO2525 | 248 | 1.000 | p_{0}=\sqrt{-2 E / m} | | |
| gilmore-lie-groups_FO2526 | 248 | 1.000 | \hat{\mathbf{u}} \in S^{3} \subset R^{4} | | |
| gilmore-lie-groups_FO2527 | 248 | 1.000 | \hat{\mathbf{w}} | | |
| gilmore-lie-groups_FO2528 | 248 | 0.995 | \hat{\mathbf{u}} | | |
| gilmore-lie-groups_FO2529 | 248 | 1.000 | S^{3} | | |
| gilmore-lie-groups_FO2530 | 248 | 0.803 | S O(4) / S O(3) | | |
| gilmore-lie-groups_FO2531 | 248 | 1.000 | p_{0} | | |
| gilmore-lie-groups_FO2532 | 249 | 1.000 | N^{2}=(n+l+1)^{2} | | |
| gilmore-lie-groups_FO2533 | 249 | 1.000 | -N^{3} | | |
| gilmore-lie-groups_FO2534 | 249 | 1.000 | \mathbf{x}^{\prime}=M \mathbf{x} | | |
| gilmore-lie-groups_FO2535 | 249 | 1.000 | (\mathbf{x}, \mathbf{x})_{N}=\mathbf{x}^{t} \mathbf{g x}=x_{i} g_{i j} x_{j} | | |
| gilmore-lie-groups_FO2536 | 249 | 0.749 | \mathbf{y} | | |
| gilmore-lie-groups_FO2537 | 249 | 1.000 | \mathbf{y}=\lambda \mathbf{x} | | |
| gilmore-lie-groups_FO2538 | 249 | 1.000 | z_{1}=\lambda | | |
| gilmore-lie-groups_FO2539 | 249 | 1.000 | z_{2}= | | |
| gilmore-lie-groups_FO2540 | 249 | 1.000 | \lambda(\mathbf{x}, \mathbf{x})_{N} | | |
| gilmore-lie-groups_FO2541 | 249 | 1.000 | N+2 | | |
| gilmore-lie-groups_FO2542 | 249 | 0.998 | \mathbf{y}, z_{1}, z_{2} | | |
| gilmore-lie-groups_FO2543 | 249 | 0.986 | \mathbf{y}, y_{N+1}, y_{N+2} | | |
| gilmore-lie-groups_FO2544 | 249 | 0.986 | y_{N+1}=\frac{1}{2}\left(z_{1}+z_{2}\right) | | |
| gilmore-lie-groups_FO2545 | 249 | 0.986 | y_{N+2}=\frac{1}{2}\left(z_{1}-z_{2}\right) | | |
| gilmore-lie-groups_FO2546 | 249 | 1.000 | R^{N} | | |
| gilmore-lie-groups_FO2547 | 250 | 0.999 | R^{N+2} | | |
| gilmore-lie-groups_FO2548 | 250 | 0.999 | S O(p+1, q+1) | | |
| gilmore-lie-groups_FO2549 | 250 | 1.000 | S^{3} \subset R^{4}(E<0) | | |
| gilmore-lie-groups_FO2550 | 250 | 1.000 | H^{3} \subset R^{4}(E>0) | | |
| gilmore-lie-groups_FO2551 | 250 | 1.000 | \mathbf{u}^{t} G \mathbf{u}=1 | | |
| gilmore-lie-groups_FO2552 | 251 | 0.999 | \operatorname{diag}\left(1, \pm I_{3},-1,+1\right) | | |
| gilmore-lie-groups_FO2553 | 251 | 0.984 | S O(5,1) | | |
| gilmore-lie-groups_FO2554 | 251 | 0.984 | E<0 | | |
| gilmore-lie-groups_FO2555 | 251 | 0.984 | S O(2,4) | | |
| gilmore-lie-groups_FO2556 | 251 | 1.000 | z_{1}=z_{2}, y_{4}=\lambda | | |
| gilmore-lie-groups_FO2557 | 251 | 1.000 | y_{5}=0 | | |
| gilmore-lie-groups_FO2558 | 251 | 0.934 | \left(y_{0}, \mathbf{y}=\lambda \mathbf{u}, y_{4}=\frac{1}{2}\left(z_{1}+z_{2}\right), y_{5}=\frac{1}{2}\left(z_{1}-z_{2}\right)\right) | | |
| gilmore-lie-groups_FO2559 | 251 | 0.908 | \left[\begin{array}{cc}M & 0 \\ 0 & 1\end{array}\right] | | |
| gilmore-lie-groups_FO2560 | 251 | 0.997 | \operatorname{diag}\left(1, \pm I_{3},-1\right) | | |
| gilmore-lie-groups_FO2561 | 251 | 0.997 | R^{5} | | |
| gilmore-lie-groups_FO2562 | 251 | 0.997 | S O(1,4) | | |
| gilmore-lie-groups_FO2563 | 251 | 0.761 | \mathbf{u}^{\prime} | | |
| gilmore-lie-groups_FO2564 | 251 | 1.000 | A^{t} G A-C^{t} C=G | | |
| gilmore-lie-groups_FO2565 | 251 | 1.000 | u | | |
| gilmore-lie-groups_FO2566 | 251 | 1.000 | \left(u_{0} \rightarrow u_{4}\right) | | |
| gilmore-lie-groups_FO2567 | 252 | 0.967 | 1 / \sqrt{2 m|E|} | | |
| gilmore-lie-groups_FO2568 | 252 | 1.000 | L_{i}, M_{i}^{\prime} | | |
| gilmore-lie-groups_FO2569 | 252 | 1.000 | B_{\mu} | | |
| gilmore-lie-groups_FO2570 | 252 | 0.993 | a_{i}, a_{j}^{\dagger} | | |
| gilmore-lie-groups_FO2571 | 252 | 1.000 | b_{i}, b_{j}^{\dagger} | | |
| gilmore-lie-groups_FO2572 | 252 | 1.000 | \left|m_{1}, m_{2} ; n_{1}, n_{2}\right\rangle | | |
| gilmore-lie-groups_FO2573 | 252 | 1.000 | j_{a}=\frac{1}{2}\left(m_{1}+m_{2}\right) | | |
| gilmore-lie-groups_FO2574 | 252 | 1.000 | j_{b}=\frac{1}{2}\left(n_{1}+n_{2}\right) | | |
| gilmore-lie-groups_FO2575 | 253 | 1.000 | N=2 j_{a}+1=2 j_{b}+1=\left(j_{a}+j_{b}\right)+1 | | |
| gilmore-lie-groups_FO2576 | 253 | 1.000 | j_{a} | | |
| gilmore-lie-groups_FO2577 | 253 | 1.000 | j_{b} | | |
| gilmore-lie-groups_FO2578 | 253 | 0.991 | \left(H_{1}, H_{2}, H_{3}, H_{4}\right)=\left(a_{1}^{\dagger} a_{1}, a_{2}^{\dagger} a_{2}, b_{1}^{\dagger} b_{1}\right. | | |
| gilmore-lie-groups_FO2579 | 253 | 0.829 | b_{2}^{\dagger} b_{2} | | |
| gilmore-lie-groups_FO2580 | 253 | 1.000 | \pm \frac{1}{2} | | |
| gilmore-lie-groups_FO2581 | 253 | 1.000 | \pi / 4 | | |
| gilmore-lie-groups_FO2582 | 253 | 1.000 | 3 \pi / 4 | | |
| gilmore-lie-groups_FO2583 | 253 | 1.000 | A_{3}=D_{3} | | |
| gilmore-lie-groups_FO2584 | 254 | 1.000 | a_{1}^{\dagger} a_{1}+a_{2}^{\dagger} a_{2}+b_{1}^{\dagger} b_{1}+b_{2}^{\dagger} b_{2} | | |
| gilmore-lie-groups_FO2585 | 254 | 1.000 | \mathfrak{s o}(4)+\mathfrak{s o}(2) | | |
| gilmore-lie-groups_FO2586 | 254 | 0.890 | \mathfrak{s} \mathfrak{o}(4,2)=\mathfrak{s} \mathfrak{u}(2,2) | | |
| gilmore-lie-groups_FO2587 | 254 | 1.000 | \left(q_{1}, q_{2}, q_{3}, q_{4}\right) | | |
| gilmore-lie-groups_FO2588 | 254 | 0.559 | \left(Q_{1}, Q_{2}, Q_{3}\right) | | |
| gilmore-lie-groups_FO2589 | 254 | 1.000 | Q_{4} | | |
| gilmore-lie-groups_FO2590 | 254 | 1.000 | q_{1}^{2}+q_{2}^{2}+q_{3}^{2}+q_{4}^{2} \neq 0 | | |
| gilmore-lie-groups_FO2591 | 254 | 1.000 | R=\sqrt{Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}} | | |
| gilmore-lie-groups_FO2592 | 254 | 1.000 | q=\sqrt{q_{1}^{2}+q_{2}^{2}+q_{3}^{2}+q_{4}^{2}} | | |
| gilmore-lie-groups_FO2593 | 254 | 1.000 | R=q^{2} | | |
| gilmore-lie-groups_FO2594 | 254 | 1.000 | P_{4}=0 | | |
| gilmore-lie-groups_FO2595 | 254 | 1.000 | P^{2}=P_{1}^{2}+P_{2}^{2}+P_{3}^{2}=\left(1 / 4 R p^{2}\right)-\left(\zeta^{2} / 4 R^{2}\right) \rightarrow | | |
| gilmore-lie-groups_FO2596 | 254 | 0.980 | (1 / 4 R)\left(p_{1}^{2}+p_{2}^{2}+p_{3}^{2}+p_{4}^{2}\right) | | |
| gilmore-lie-groups_FO2597 | 255 | 1.000 | \left(q_{1}, q_{2}, q_{3}, q_{4} ; p_{1}, p_{2}, p_{3}, p_{4}\right) | | |
| gilmore-lie-groups_FO2598 | 255 | 1.000 | \zeta=0 | | |
| gilmore-lie-groups_FO2599 | 255 | 1.000 | M^{t} G_{1} M=G_{1} | | |
| gilmore-lie-groups_FO2600 | 255 | 1.000 | M^{t} G_{2} M=G_{2} | | |
| gilmore-lie-groups_FO2601 | 256 | 0.988 | \operatorname{Sp}(8 ; \mathbb{R}) | | |
| gilmore-lie-groups_FO2602 | 256 | 1.000 | S O(4,4) | | |
| gilmore-lie-groups_FO2603 | 256 | 0.998 | (q, p) | | |
| gilmore-lie-groups_FO2604 | 256 | 0.998 | (s, r) | | |
| gilmore-lie-groups_FO2605 | 256 | 1.000 | U(2,2) | | |
| gilmore-lie-groups_FO2606 | 256 | 0.825 | \left(V(r)=-e^{2} / r\right) | | |
| gilmore-lie-groups_FO2607 | 257 | 1.000 | \mathfrak{s o}(4,1) | | |
| gilmore-lie-groups_FO2608 | 257 | 1.000 | \mathfrak{s o}(4,2) | | |
| gilmore-lie-groups_FO2609 | 257 | 1.000 | J_{i}, M_{i}, A_{i}, \Gamma_{i}(i=1,2,3) | | |
| gilmore-lie-groups_FO2610 | 257 | 0.996 | A_{4}, \Gamma_{4}, \Gamma_{5} | | |
| gilmore-lie-groups_FO2611 | 257 | 1.000 | \Phi(r)=e / r | | |
| gilmore-lie-groups_FO2612 | 257 | 1.000 | A_{4}, \Gamma_{4} | | |
| gilmore-lie-groups_FO2613 | 257 | 1.000 | \Gamma_{5} | | |
| gilmore-lie-groups_FO2614 | 257 | 0.860 | (\mathbf{E}, \mathbf{B}) | | |
| gilmore-lie-groups_FO2615 | 257 | 1.000 | A_{\mu}=(\phi, \mathbf{A}) | | |
| gilmore-lie-groups_FO2616 | 257 | 1.000 | q=-e | | |
| gilmore-lie-groups_FO2617 | 257 | 1.000 | H_{D} \psi=E \psi | | |
| gilmore-lie-groups_FO2621 | 260 | 1.000 | \Gamma_{4} | | |
| gilmore-lie-groups_FO2634 | 258 | 1.000 | \pi=\mathbf{p}-\frac{q}{c} \mathbf{A} | | |
| gilmore-lie-groups_FO2635 | 258 | 0.971 | \gamma_{i} | | |
| gilmore-lie-groups_FO2636 | 258 | 0.982 | \sigma_{i} | | |
| gilmore-lie-groups_FO2637 | 259 | 1.000 | \mathbf{p} \rightarrow \pi=\mathbf{p}-\frac{q}{c} \mathbf{A} | | |
| gilmore-lie-groups_FO2638 | 259 | 1.000 | J_{i}, M_{i}, A_{i}, \Gamma_{i} | | |
| gilmore-lie-groups_FO2639 | 259 | 0.990 | \Gamma_{4}, \Gamma_{5} | | |
| gilmore-lie-groups_FO2640 | 259 | 1.000 | D=d / d r | | |
| gilmore-lie-groups_FO2641 | 259 | 1.000 | r D^{2}+r | | |
| gilmore-lie-groups_FO2642 | 259 | 1.000 | r D^{2}-r | | |
| gilmore-lie-groups_FO2643 | 260 | 1.000 | C^{2}=\Gamma_{5}^{2}-\Gamma_{4}^{2}-M_{4}^{2}=-a | | |
| gilmore-lie-groups_FO2644 | 260 | 0.978 | a \rightarrow A | | |
| gilmore-lie-groups_FO2645 | 260 | 1.000 | C<0 | | |
| gilmore-lie-groups_FO2646 | 260 | 1.000 | e^{-\theta}+C e^{\theta}=0 | | |
| gilmore-lie-groups_FO2647 | 260 | 1.000 | u(r)=e^{\theta M_{4}} R(r) | | |
| gilmore-lie-groups_FO2648 | 260 | 0.798 | N=-\frac{1}{2}+ | | |
| gilmore-lie-groups_FO2649 | 260 | 1.000 | \sqrt{\left(\frac{1}{2}\right)^{2}-A}+1+n, n=0,1,2, \ldots | | |
| gilmore-lie-groups_FO2650 | 261 | 1.000 | W=-\frac{1}{2} m c^{2} \alpha^{2}\left(1 / N^{2}\right) | | |
| gilmore-lie-groups_FO2651 | 261 | 1.000 | N=l+1+k, k=0,1,2, \ldots | | |
| gilmore-lie-groups_FO2652 | 261 | 1.000 | m_{l} | | |
| gilmore-lie-groups_FO2653 | 261 | 0.449 | |N, \ln \rangle \leftrightarrow|N \pm 1, l m\rangle | | |
| gilmore-lie-groups_FO2660 | 261 | 1.000 | L_{-} Y_{m=-l}^{l}(\theta, \phi)=0 | | |
| gilmore-lie-groups_FO2662 | 261 | 1.000 | C>0 | | |
| gilmore-lie-groups_FO2663 | 261 | 1.000 | e^{-\theta}-C e^{\theta}=0 | | |
| gilmore-lie-groups_FO2664 | 261 | 1.000 | k \simeq e^{-\theta} | | |
| gilmore-lie-groups_FO2665 | 261 | 1.000 | E=\hbar^{2} k^{2} / 2 m | | |
| gilmore-lie-groups_FO2666 | 262 | 1.000 | \delta(j)=\arg [\Gamma(j+1-i(\alpha / k)] | | |
| gilmore-lie-groups_FO2667 | 262 | 1.000 | j=-\frac{1}{2}+\sqrt{\left(\frac{1}{2}\right)^{2}-A} | | |
| gilmore-lie-groups_FO2668 | 262 | 1.000 | -1 / r | | |
| gilmore-lie-groups_FO2669 | 262 | 1.000 | -1 / r^{2} | | |
| gilmore-lie-groups_FO2670 | 262 | 1.000 | V(r)=-e^{2} / r \rightarrow-e^{2} / r-\mu_{l}\left(\hbar^{2} / 2 m\right) / r^{2} | | |
| gilmore-lie-groups_FO2671 | 262 | 1.000 | j \rightarrow j^{\prime}=j+\Delta j | | |
| gilmore-lie-groups_FO2672 | 262 | 1.000 | \Delta j= | | |
| gilmore-lie-groups_FO2673 | 262 | 1.000 | -\mu_{l} /(2 l+1) | | |
| gilmore-lie-groups_FO2674 | 262 | 1.000 | \Delta j | | |
| gilmore-lie-groups_FO2675 | 263 | 1.000 | G \supset S O(3) | | |
| gilmore-lie-groups_FO2676 | 264 | 0.987 | 3^{\text {deg }} | | |
| gilmore-lie-groups_FO2677 | 264 | 1.000 | \mathbf{F}=m \mathbf{g} | | |
| gilmore-lie-groups_FO2678 | 264 | 1.000 | \mathbf{A}=\frac{1}{2} \mathbf{B} \times \mathbf{r} | | |
| gilmore-lie-groups_FO2679 | 264 | 0.996 | (r, \theta, \phi)=\left(\theta_{3}, \theta_{2}, \theta_{1}\right) | | |
| gilmore-lie-groups_FO2680 | 264 | 1.000 | \mathcal{L}^{2}\left(S^{1}\right)=\partial^{2} / \partial \theta_{1}^{2} | | |
| gilmore-lie-groups_FO2681 | 264 | 1.000 | \mathcal{L}^{2}\left(S^{3}\right) | | |
| gilmore-lie-groups_FO2682 | 264 | 1.000 | \left(\partial / \partial \cos \theta_{3}\right)^{2} | | |
| gilmore-lie-groups_FO2683 | 264 | 1.000 | \mathcal{L}^{2}\left(S^{n}\right) | | |
| gilmore-lie-groups_FO2684 | 265 | 1.000 | r \rightarrow \infty | | |
| gilmore-lie-groups_FO2685 | 265 | 1.000 | \gamma=\frac{1}{2} \pm \sqrt{\left(\frac{1}{2}\right)^{2}-A} | | |
| gilmore-lie-groups_FO2686 | 265 | 1.000 | \lambda= \pm \sqrt{-C} | | |
| gilmore-lie-groups_FO2687 | 265 | 1.000 | \sqrt{\left(\frac{1}{2}\right)^{2}-A} | | |
| gilmore-lie-groups_FO2688 | 265 | 1.000 | \pm \sqrt{-C} | | |
| gilmore-lie-groups_FO2689 | 265 | 1.000 | R(r)=r^{\gamma} e^{\lambda r} f(r) | | |
| gilmore-lie-groups_FO2690 | 265 | 1.000 | f(r)=\sum_{j=0}^{\infty} f_{j} r^{j} | | |
| gilmore-lie-groups_FO2691 | 265 | 1.000 | j \rightarrow \infty | | |
| gilmore-lie-groups_FO2692 | 265 | 1.000 | f(r) \rightarrow e^{-2 \lambda r} | | |
| gilmore-lie-groups_FO2693 | 265 | 1.000 | \lambda<0 | | |
| gilmore-lie-groups_FO2694 | 265 | 0.963 | r^{n} | | |
| gilmore-lie-groups_FO2695 | 265 | 0.963 | f_{n} \neq 0 | | |
| gilmore-lie-groups_FO2696 | 265 | 0.963 | f_{n+1}=0\left(\Rightarrow f_{n+2}=\right. | | |
| gilmore-lie-groups_FO2697 | 265 | 1.000 | f_{n+3}=\cdots=0 | | |
| gilmore-lie-groups_FO2698 | 265 | 1.000 | 2 \lambda(n+\gamma)+B=0 | | |
| gilmore-lie-groups_FO2699 | 266 | 1.000 | \frac{1}{r} r^{\gamma} f(r) e^{\lambda r} | | |
| gilmore-lie-groups_FO2700 | 266 | 0.907 | (0, \infty) | | |
| gilmore-lie-groups_FO2701 | 266 | 1.000 | Z e / r | | |
| gilmore-lie-groups_FO2702 | 266 | 1.000 | N^{\prime}=n+\frac{1}{2}+\sqrt{\left(l+\frac{1}{2}\right)^{2}-\alpha^{2}} | | |
| gilmore-lie-groups_FO2703 | 266 | 1.000 | e^{\lambda r} | | |
| gilmore-lie-groups_FO2704 | 266 | 1.000 | \lambda^{-1} | | |
| gilmore-lie-groups_FO2705 | 266 | 1.000 | a \simeq 1 /|\lambda| | | |
| gilmore-lie-groups_FO2706 | 266 | 0.910 | (n, l) | | |
| gilmore-lie-groups_FO2717 | 267 | 1.000 | a_{B}=\hbar^{2} / m e^{2}=0.529 \times 10^{-8} \mathrm{~cm} | | |
| gilmore-lie-groups_FO2718 | 267 | 0.906 | n, l | | |
| gilmore-lie-groups_FO2719 | 267 | 1.000 | W=-\frac{1}{2} m c^{2} \alpha^{2} / N^{2} | | |
| gilmore-lie-groups_FO2720 | 267 | 1.000 | 1 / m=1 / m_{1}+1 / m_{2} | | |
| gilmore-lie-groups_FO2721 | 267 | 1.000 | (E=0) | | |
| gilmore-lie-groups_FO2722 | 267 | 1.000 | 1 / r^{3} | | |
| gilmore-lie-groups_FO2723 | 267 | 1.000 | \alpha=\sqrt{1-\eta}, \eta=C / K a | | |
| gilmore-lie-groups_FO2724 | 267 | 1.000 | \alpha \simeq 1 | | |
| gilmore-lie-groups_FO2725 | 267 | 1.000 | \eta | | |
| gilmore-lie-groups_FO2726 | 268 | 0.651 | E=\sqrt{\left(m c^{2}\right)^{2}+(\mathbf{p} c)^{2}}-K / r | | |
| gilmore-lie-groups_FO2727 | 268 | 1.000 | E=\left(m c^{2}\right)+\left(p^{2} / 2 m\right)-\left(p^{2} / 2 m\right)^{2} /\left(2 m c^{2}\right)-K / r=m c^{2}+W | | |
| gilmore-lie-groups_FO2728 | 268 | 1.000 | -\left(p^{2} / 2 m\right)^{2} /\left(2 m c^{2}\right) | | |
| gilmore-lie-groups_FO2729 | 268 | 1.000 | -(W+K / r)^{2} /\left(2 m c^{2}\right) | | |
| gilmore-lie-groups_FO2730 | 268 | 1.000 | K^{\prime} | | |
| gilmore-lie-groups_FO2731 | 268 | 1.000 | C^{\prime} | | |
| gilmore-lie-groups_FO2732 | 268 | 1.000 | K^{\prime}=K\left(1+W / m c^{2}\right) | | |
| gilmore-lie-groups_FO2733 | 268 | 1.000 | C^{\prime}=-K^{2} /\left(2 m c^{2}\right) | | |
| gilmore-lie-groups_FO2734 | 268 | 1.000 | K \rightarrow K^{\prime} | | |
| gilmore-lie-groups_FO2735 | 268 | 1.000 | C=2 C^{\prime} | | |
| gilmore-lie-groups_FO2736 | 268 | 1.000 | \delta \theta \simeq \eta / 2 | | |
| gilmore-lie-groups_FO2737 | 268 | 1.000 | \epsilon=0.206 | | |
| gilmore-lie-groups_FO2738 | 268 | 1.000 | T=0.24 | | |
| gilmore-lie-groups_FO2739 | 268 | 1.000 | m= | | |
| gilmore-lie-groups_FO2740 | 268 | 1.000 | m_{0} / \sqrt{1-(v / c)^{2}} | | |
| gilmore-lie-groups_FO2741 | 268 | 1.000 | \mathbf{F}(r)=\left(-K / r^{2}+p(r)\right) \hat{\mathbf{r}} | | |
| gilmore-lie-groups_FO2742 | 268 | 1.000 | p(r) | | |
| gilmore-lie-groups_FO2743 | 268 | 1.000 | 1 / r=\left(m K / L^{2}\right)(1+(M / m K) \cos \theta) | | |
| gilmore-lie-groups_FO2744 | 268 | 1.000 | L | | |
| gilmore-lie-groups_FO2745 | 268 | 1.000 | C / r^{3} | | |
| gilmore-lie-groups_FO2746 | 268 | 1.000 | C \times 2 \pi \frac{m K}{L^{2}} | | |
| gilmore-lie-groups_FO2747 | 268 | 1.000 | K=G M m | | |
| gilmore-lie-groups_FO2748 | 268 | 1.000 | M \gg m, \omega | | |
| gilmore-lie-groups_FO2749 | 269 | 1.000 | \omega=42^{\prime \prime} | | |
| gilmore-lie-groups_FO2750 | 269 | 1.000 | L_{-} | | |
| gilmore-lie-groups_FO2751 | 269 | 1.000 | Y_{m}^{l}(\theta, \phi)=P_{-l}^{l}(\theta) e^{-i l \phi} | | |
| gilmore-lie-groups_FO2752 | 269 | 1.000 | P_{-l}^{l}=(\sin \theta)^{l} | | |
| gilmore-lie-groups_FO2753 | 269 | 1.000 | N_{l} | | |
| gilmore-lie-groups_FO2754 | 269 | 1.000 | Y_{ \pm l}^{l}(\theta, \phi) | | |
| gilmore-lie-groups_FO2755 | 269 | 0.999 | N_{0}=\sqrt{1 / 4 \pi} | | |
| gilmore-lie-groups_FO2756 | 269 | 1.000 | N_{3} | | |
| gilmore-lie-groups_FO2757 | 269 | 0.999 | \left\langle{ }_{m^{\prime}}^{l}\right| L_{+}\left|{ }_{m}^{l}\right\rangle=\sqrt{\left(l+m^{\prime}\right)(l-m)} | | |
| gilmore-lie-groups_FO2758 | 269 | 1.000 | \delta_{m^{\prime}, m+1} | | |
| gilmore-lie-groups_FO2759 | 269 | 1.000 | l=N-1 | | |
| gilmore-lie-groups_FO2760 | 269 | 0.980 | n=0 | | |
| gilmore-lie-groups_FO2761 | 269 | 0.848 | \mathbf{r} | | |
| gilmore-lie-groups_FO2762 | 270 | 1.000 | \mathbf{M} \cdot \mathbf{L}=\mathbf{0} | | |
| gilmore-lie-groups_FO2763 | 270 | 1.000 | \mathbf{M} \cdot \mathbf{M}=(2 \mathbf{L} \cdot \mathbf{L} / m)(\mathbf{p} \cdot \mathbf{p} / 2 m-K / r)+K^{2} | | |
| gilmore-lie-groups_FO2764 | 270 | 1.000 | \mathbf{M} \cdot \mathbf{r}=\mathbf{L} \cdot \mathbf{L} / m-K r | | |
| gilmore-lie-groups_FO2765 | 270 | 0.999 | \mathbf{M} \cdot \mathbf{r}=M r \cos \theta | | |
| gilmore-lie-groups_FO2766 | 270 | 1.000 | r=(\mathbf{L} \cdot \mathbf{L} / m K / 1+(M / K) \cos \theta) | | |
| gilmore-lie-groups_FO2767 | 270 | 1.000 | L^{2} / m K | | |
| gilmore-lie-groups_FO2768 | 270 | 1.000 | \epsilon=M / K | | |
| gilmore-lie-groups_FO2769 | 270 | 1.000 | \mathbf{A} \cdot \mathbf{A}=\left(-1 / 4 \hbar^{2}\right)\left(\mathbf{L} \cdot \mathbf{L}+\mathbf{M}^{\prime} \cdot \mathbf{M}^{\prime}+\mathbf{L} \cdot \mathbf{M}^{\prime}+\mathbf{M}^{\prime} \cdot \mathbf{L}\right) | | |
| gilmore-lie-groups_FO2770 | 270 | 1.000 | \mathbf{B} \cdot \mathbf{B} | | |
| gilmore-lie-groups_FO2771 | 270 | 0.880 | \mathbf{L} \cdot \mathbf{M}=\mathbf{M} \cdot \mathbf{L}=\mathbf{0} | | |
| gilmore-lie-groups_FO2772 | 270 | 1.000 | p_{x}=\mathbf{p} \cdot \mathbf{M} / M | | |
| gilmore-lie-groups_FO2773 | 270 | 1.000 | p_{y}=\mathbf{p} \cdot \mathbf{W} / W | | |
| gilmore-lie-groups_FO2774 | 270 | 1.000 | p_{x}^{2}+\left(p_{y}-a\right)^{2}=r^{2} | | |
| gilmore-lie-groups_FO2775 | 270 | 0.900 | 0, a | | |
| gilmore-lie-groups_FO2776 | 270 | 1.000 | x^{a} y^{b} z^{c} | | |
| gilmore-lie-groups_FO2777 | 270 | 0.996 | a+b+c=l | | |
| gilmore-lie-groups_FO2778 | 270 | 0.996 | N(l, 3)=(l+3-1) / l!(3-1)! | | |
| gilmore-lie-groups_FO2779 | 270 | 0.997 | x_{1}, x_{2}, \ldots, x_{N} | | |
| gilmore-lie-groups_FO2780 | 270 | 0.997 | 3 \rightarrow N | | |
| gilmore-lie-groups_FO2781 | 270 | 1.000 | r^{l} Y_{m}^{l}(\theta, \phi) | | |
| gilmore-lie-groups_FO2782 | 270 | 1.000 | x, y, z | | |
| gilmore-lie-groups_FO2783 | 270 | 1.000 | l-2 | | |
| gilmore-lie-groups_FO2784 | 270 | 1.000 | \operatorname{dim}\left\{Y_{m}^{l}\right\}=N(l, 3)-N(l-2,3)=2 l+1 | | |
| gilmore-lie-groups_FO2785 | 270 | 0.999 | \operatorname{dim}\left\{\mathcal{Y}_{l m}^{n}\right\}=N(n, 4)-N(n-2,4)=(n+1)^{2} | | |
| gilmore-lie-groups_FO2786 | 270 | 0.995 | \psi(\mathbf{x})_{n l m} | | |
| gilmore-lie-groups_FO2787 | 270 | 0.995 | n=0,1,2 ; l=0, \ldots, n-1 | | |
| gilmore-lie-groups_FO2788 | 271 | 1.000 | \frac{1}{2}, \frac{3}{2}, \frac{5}{2}, \ldots | | |
| gilmore-lie-groups_FO2789 | 271 | 1.000 | N(l, d)= | | |
| gilmore-lie-groups_FO2790 | 271 | 1.000 | N(l, d-1)+N(l-1, d) | | |
| gilmore-lie-groups_FO2791 | 271 | 1.000 | S^{n-1} | | |
| gilmore-lie-groups_FO2792 | 271 | 1.000 | (l+1)^{2}=l^{2}+(2 l+1) | | |
| gilmore-lie-groups_FO2793 | 271 | 1.000 | \mathcal{Y}^{0}\left(S^{n}\right)=1=\operatorname{dim} \mathcal{Y}^{0}\left(S^{n}\right) | | |
| gilmore-lie-groups_FO2794 | 271 | 1.000 | \operatorname{dim} \mathcal{Y}^{l}\left(S^{n}\right)=\frac{(l+n-2)!}{l!(n-1)!}(2 l+n-1) | | |
| gilmore-lie-groups_FO2795 | 271 | 1.000 | S^{D-1} | | |
| gilmore-lie-groups_FO2796 | 271 | 1.000 | \mathcal{Y}^{l}\left(S^{D-1}\right) | | |
| gilmore-lie-groups_FO2797 | 271 | 1.000 | -\left[(l+\alpha)^{2}-\alpha^{2}\right] | | |
| gilmore-lie-groups_FO2798 | 271 | 0.709 | S O(D) | | |
| gilmore-lie-groups_FO2799 | 271 | 1.000 | \alpha=D-2 | | |
| gilmore-lie-groups_FO2800 | 271 | 1.000 | \psi(\mathbf{x})=\left(1 / r^{(D-1) / 2}\right) \mathcal{Y}^{l} | | |
| gilmore-lie-groups_FO2801 | 272 | 1.000 | \mathrm{a}-1 / r^{2} | | |
| gilmore-lie-groups_FO2802 | 272 | 0.999 | l(l+1) \rightarrow l(l+1)-\mu_{l} | | |
| gilmore-lie-groups_FO2803 | 272 | 0.999 | s | | |
| gilmore-lie-groups_FO2804 | 272 | 0.953 | \mu_{0} \gg \mu_{1}>\cdots | | |
| gilmore-lie-groups_FO2805 | 272 | 1.000 | L_{i j}=a_{i}^{\dagger} a_{j}-a_{j}^{\dagger} a_{i}=-L_{j i} | | |
| gilmore-lie-groups_FO2806 | 272 | 1.000 | Q_{i j}= | | |
| gilmore-lie-groups_FO2807 | 272 | 1.000 | a_{i}^{\dagger} a_{j}+a_{j}^{\dagger} a_{i}=+Q_{j i} | | |
| gilmore-lie-groups_FO2808 | 272 | 1.000 | \left[a_{i}, a_{j}^{\dagger}\right]=1 | | |
| gilmore-lie-groups_FO2809 | 272 | 1.000 | a_{i} a_{j} | | |
| gilmore-lie-groups_FO2810 | 273 | 0.999 | \mathbf{E}(\mathbf{x}, t), \mathbf{B}(\mathbf{x}, t) | | |
| gilmore-lie-groups_FO2811 | 273 | 0.935 | \mathbf{E}(k), \mathbf{B}(k) | | |
| gilmore-lie-groups_FO2812 | 274 | 1.000 | \mathbf{E}(\mathbf{x}, t) | | |
| gilmore-lie-groups_FO2813 | 274 | 1.000 | \mathbf{B}(\mathbf{x}, t) | | |
| gilmore-lie-groups_FO2814 | 274 | 1.000 | \mathbf{E}(k) | | |
| gilmore-lie-groups_FO2815 | 274 | 1.000 | \mathbf{B}(k) | | |
| gilmore-lie-groups_FO2816 | 274 | 1.000 | k \cdot k=\mathbf{k} \cdot \mathbf{k}-k_{4} k_{4}=0 | | |
| gilmore-lie-groups_FO2817 | 274 | 0.798 | k_{4} | | |
| gilmore-lie-groups_FO2818 | 274 | 1.000 | j=1 | | |
| gilmore-lie-groups_FO2819 | 274 | 1.000 | k \cdot k=0 | | |
| gilmore-lie-groups_FO2820 | 275 | 0.999 | (x, y, z, i c t) | | |
| gilmore-lie-groups_FO2821 | 275 | 0.999 | (x, y, z, i c t)^{\prime} | | |
| gilmore-lie-groups_FO2822 | 275 | 0.865 | \Lambda | | |
| gilmore-lie-groups_FO2823 | 275 | 1.000 | k \cdot a=\Lambda k \cdot | | |
| gilmore-lie-groups_FO2824 | 275 | 1.000 | \Lambda a | | |
| gilmore-lie-groups_FO2825 | 275 | 1.000 | \Lambda \in O(3,1) | | |
| gilmore-lie-groups_FO2826 | 275 | 0.936 | \mathbf{J}, \mathbf{K} | | |
| gilmore-lie-groups_FO2827 | 276 | 0.988 | a=(x, y, z, c t) | | |
| gilmore-lie-groups_FO2828 | 276 | 0.995 | (\partial / \partial x, \partial / \partial y, \partial / \partial z, i \partial / \partial(c t)) | | |
| gilmore-lie-groups_FO2829 | 276 | 1.000 | \{\Lambda, 0\} | | |
| gilmore-lie-groups_FO2830 | 276 | 1.000 | \{I, a\} | | |
| gilmore-lie-groups_FO2831 | 276 | 1.000 | \Gamma^{k} | | |
| gilmore-lie-groups_FO2832 | 276 | 1.000 | |k\rangle | | |
| gilmore-lie-groups_FO2833 | 277 | 1.000 | D_{2}=A_{1}+A_{1} | | |
| gilmore-lie-groups_FO2834 | 277 | 1.000 | \mathbf{J}^{(1)} | | |
| gilmore-lie-groups_FO2835 | 277 | 1.000 | D^{j} | | |
| gilmore-lie-groups_FO2836 | 277 | 1.000 | \mathbf{J}^{(2)} | | |
| gilmore-lie-groups_FO2837 | 277 | 0.970 | 2 j^{\prime}+1 | | |
| gilmore-lie-groups_FO2838 | 277 | 0.970 | D^{j^{\prime}} | | |
| gilmore-lie-groups_FO2839 | 277 | 1.000 | (2 j+1)\left(2 j^{\prime}+1\right) | | |
| gilmore-lie-groups_FO2840 | 277 | 1.000 | D^{j j^{\prime}} | | |
| gilmore-lie-groups_FO2841 | 278 | 1.000 | \mathbf{J}^{(j)} | | |
| gilmore-lie-groups_FO2842 | 278 | 0.990 | D^{j j^{\prime}}(\Lambda) | | |
| gilmore-lie-groups_FO2843 | 278 | 1.000 | S O(3) \subset S O(3,1) | | |
| gilmore-lie-groups_FO2844 | 278 | 1.000 | j^{\prime}=0 | | |
| gilmore-lie-groups_FO2845 | 278 | 1.000 | j=0 | | |
| gilmore-lie-groups_FO2846 | 278 | 1.000 | T_{\mu \nu}(x) | | |
| gilmore-lie-groups_FO2847 | 278 | 1.000 | \Lambda \in S O(3,1) | | |
| gilmore-lie-groups_FO2848 | 278 | 1.000 | x^{\prime}=x \Lambda^{-1} | | |
| gilmore-lie-groups_FO2849 | 279 | 0.774 | \left|\begin{array}{cc}j & j^{\prime} \\ \mu & \mu^{\prime}\end{array}\right\rangle | | |
| gilmore-lie-groups_FO2855 | 280 | 1.000 | \{\Lambda, a\} | | |
| gilmore-lie-groups_FO2856 | 280 | 1.000 | \Gamma^{k}(\{I, a\}) | | |
| gilmore-lie-groups_FO2857 | 280 | 1.000 | |k ; \xi\rangle | | |
| gilmore-lie-groups_FO2858 | 280 | 1.000 | \xi | | |
| gilmore-lie-groups_FO2859 | 280 | 0.999 | k^{\prime}=\Lambda k | | |
| gilmore-lie-groups_FO2860 | 280 | 1.000 | M_{\xi^{\prime} \xi}(\Lambda) | | |
| gilmore-lie-groups_FO2861 | 280 | 1.000 | k^{\prime} | | |
| gilmore-lie-groups_FO2862 | 281 | 1.000 | M(\Lambda) | | |
| gilmore-lie-groups_FO2863 | 281 | 1.000 | k^{0} | | |
| gilmore-lie-groups_FO2864 | 281 | 1.000 | \left|k^{0} ; \xi\right\rangle | | |
| gilmore-lie-groups_FO2865 | 281 | 0.822 | H_{k^{0}} | | |
| gilmore-lie-groups_FO2866 | 281 | 0.822 | C_{k} | | |
| gilmore-lie-groups_FO2867 | 282 | 1.000 | \theta_{1}=\theta_{2}=b_{3}=0, \theta_{3}, b_{1}, b_{2} | | |
| gilmore-lie-groups_FO2868 | 282 | 1.000 | (0,0,1,-i)=T(0,0,1,+i) | | |
| gilmore-lie-groups_FO2869 | 282 | 1.000 | Y_{1}=J_{1}-K_{2}, Y_{2}= | | |
| gilmore-lie-groups_FO2870 | 282 | 1.000 | J_{2}+K_{1}, Y_{3}=J_{3} | | |
| gilmore-lie-groups_FO2871 | 282 | 1.000 | \mathbf{b}=0 | | |
| gilmore-lie-groups_FO2872 | 283 | 1.000 | D_{\xi^{\prime} \xi}\left(H_{k^{0}}\right) | | |
| gilmore-lie-groups_FO2873 | 283 | 0.999 | k \cdot k>0 | | |
| gilmore-lie-groups_FO2874 | 283 | 1.000 | j=0, \frac{1}{2}, 1, \frac{3}{2}, \ldots | | |
| gilmore-lie-groups_FO2875 | 283 | 1.000 | \kappa=\left(\kappa_{1}, \kappa_{2}\right) | | |
| gilmore-lie-groups_FO2876 | 283 | 1.000 | \kappa \in R^{2}, \kappa \cdot \kappa \geq 0 | | |
| gilmore-lie-groups_FO2877 | 283 | 1.000 | |\kappa\rangle | | |
| gilmore-lie-groups_FO2878 | 283 | 0.999 | \left|\kappa^{\prime}\right\rangle | | |
| gilmore-lie-groups_FO2879 | 283 | 0.999 | \kappa^{\prime} \cdot \kappa^{\prime}=\kappa \cdot \kappa | | |
| gilmore-lie-groups_FO2880 | 283 | 0.999 | \kappa^{\prime}=\left(\kappa_{1}^{\prime}, \kappa_{2}^{\prime}\right) | | |
| gilmore-lie-groups_FO2881 | 283 | 1.000 | \kappa^{\prime}=R(\theta) \kappa | | |
| gilmore-lie-groups_FO2882 | 283 | 1.000 | \kappa \cdot \kappa | | |
| gilmore-lie-groups_FO2883 | 283 | 0.999 | \kappa \cdot \kappa>0 \quad | | |
| gilmore-lie-groups_FO2884 | 283 | 0.996 | \kappa \cdot \kappa=0 \quad | | |
| gilmore-lie-groups_FO2885 | 283 | 0.996 | =I S O(2) | | |
| gilmore-lie-groups_FO2886 | 284 | 1.000 | \kappa^{2} | | |
| gilmore-lie-groups_FO2887 | 284 | 1.000 | \kappa^{2}>0 | | |
| gilmore-lie-groups_FO2888 | 284 | 1.000 | \kappa=0 | | |
| gilmore-lie-groups_FO2889 | 284 | 0.961 | \left(Y_{1} \rightarrow 0, Y_{2} \rightarrow 0\right) | | |
| gilmore-lie-groups_FO2890 | 284 | 1.000 | \left(j, j^{\prime}\right) | | |
| gilmore-lie-groups_FO2891 | 285 | 1.000 | \left\{H_{k^{0}}, 0\right\} | | |
| gilmore-lie-groups_FO2892 | 285 | 1.000 | H_{k^{0}}=\operatorname{EXP}\left(\Theta J_{3}+\theta_{1} Y_{1}+\theta_{2} Y_{2}\right) | | |
| gilmore-lie-groups_FO2893 | 285 | 0.494 | \left.\left.\left|k^{0}\right\rangle\right|_{\mu} ^{j} \underset{\mu^{\prime}}{j^{\prime}}\right\rangle | | |
| gilmore-lie-groups_FO2894 | 285 | 0.119 | \left|k^{0}\right\rangle\left|\begin{array}{ll}j_{0}^{0} & 0 \\ j^{0}\end{array}\right\rangle | | |
| gilmore-lie-groups_FO2895 | 285 | 0.119 | \xi>0 | | |
| gilmore-lie-groups_FO2896 | 285 | 0.119 | j=+\xi | | |
| gilmore-lie-groups_FO2897 | 285 | 0.983 | \left|k^{0}\right\rangle\left|\begin{array}{cc}0 & j^{\prime} \\ 0 & -j^{\prime}\end{array}\right\rangle | | |
| gilmore-lie-groups_FO2898 | 285 | 0.983 | \xi<0 | | |
| gilmore-lie-groups_FO2899 | 285 | 0.983 | j^{\prime}=-\xi | | |
| gilmore-lie-groups_FO2900 | 286 | 0.560 | |k\rangle\left|\begin{array}{cc}j & j^{\prime} \\ \mu & \mu^{\prime}\end{array}\right\rangle | | |
| gilmore-lie-groups_FO2901 | 286 | 1.000 | \langle k ; \xi \mid \psi\rangle | | |
| gilmore-lie-groups_FO2902 | 286 | 0.741 | \left\langle k ;{ }_{\mu}^{j} j^{\prime} \mid \psi\right\rangle | | |
| gilmore-lie-groups_FO2903 | 286 | 1.000 | \Lambda k \cdot \Lambda k=0, k \neq 0 | | |
| gilmore-lie-groups_FO2904 | 286 | 1.000 | \xi= \pm 1 | | |
| gilmore-lie-groups_FO2905 | 286 | 0.712 | \mu, \mu^{\prime}:-j \leq \mu \leq+j,-j^{\prime} \leq \mu^{\prime} \leq+j^{\prime} | | |
| gilmore-lie-groups_FO2906 | 286 | 1.000 | \xi=j>0 | | |
| gilmore-lie-groups_FO2907 | 286 | 1.000 | \left\langle k^{0} ; j \mid \psi\right\rangle | | |
| gilmore-lie-groups_FO2908 | 286 | 1.000 | \left|k^{0} ; j\right\rangle | | |
| gilmore-lie-groups_FO2909 | 286 | 0.993 | \left\langle k^{0} ;{ }_{j}^{j}{ }_{0}^{0} \mid \psi\right\rangle | | |
| gilmore-lie-groups_FO2910 | 286 | 0.999 | \left\langle k^{0} ;{ }_{m}^{j}{ }_{0}^{0} \mid \psi\right\rangle, m \neq j | | |
| gilmore-lie-groups_FO2911 | 286 | 0.719 | \{\cdot\} | | |
| gilmore-lie-groups_FO2912 | 286 | 0.719 | (j-j) k_{3}^{0}=0 | | |
| gilmore-lie-groups_FO2913 | 286 | 0.999 | (m-j) k_{3}^{0} | | |
| gilmore-lie-groups_FO2914 | 286 | 1.000 | \left\langle k^{0} ;{ }_{m}^{j}{ }_{0}^{0} \mid \psi\right\rangle | | |
| gilmore-lie-groups_FO2915 | 286 | 1.000 | (m-j) k_{3}^{0} \neq 0 | | |
| gilmore-lie-groups_FO2916 | 286 | 0.961 | m \neq j | | |
| gilmore-lie-groups_FO2917 | 286 | 1.000 | \xi=-j | | |
| gilmore-lie-groups_FO2918 | 286 | 0.556 | \left|k^{0}\right\rangle\left|\begin{array}{cc}j & j^{\prime} \\ \mu & \mu^{\prime}\end{array}\right\rangle | | |
| gilmore-lie-groups_FO2919 | 286 | 0.556 | \left.\left.|k\rangle\right|_{v} ^{j} \begin{array}{cc}j^{\prime} \\ v^{\prime}\end{array}\right\rangle | | |
| gilmore-lie-groups_FO2920 | 287 | 0.847 | \left\langle k ;{ }_{\mu}^{j}{ }_{\mu^{\prime}}^{j^{\prime}} \mid \psi\right\rangle | | |
| gilmore-lie-groups_FO2921 | 287 | 1.000 | \left|k^{0}\right\rangle | | |
| gilmore-lie-groups_FO2922 | 287 | 1.000 | \xi=j | | |
| gilmore-lie-groups_FO2923 | 287 | 1.000 | M^{j j^{\prime}}\left(k^{0}\right)=M^{j 0}\left(k^{0}\right) | | |
| gilmore-lie-groups_FO2924 | 287 | 0.943 | |k\rangle\left|\begin{array}{ll}1 & 0 \\ \mu & 0\end{array}\right\rangle | | |
| gilmore-lie-groups_FO2925 | 287 | 1.000 | \langle x \mid k\rangle\left\langle k ;{ }_{m}^{j}{ }_{0}^{0} \mid \psi\right\rangle | | |
| gilmore-lie-groups_FO2926 | 287 | 1.000 | \psi_{j m}(x),(j=1, m=+1,0,-1 | | |
| gilmore-lie-groups_FO2927 | 288 | 1.000 | D^{01} | | |
| gilmore-lie-groups_FO2928 | 289 | 0.999 | k^{0}=(0,0,1, i) | | |
| gilmore-lie-groups_FO2929 | 289 | 0.999 | \left\langle k ;{ }_{m}^{j=1}{ }_{0}^{0} \mid \psi\right\rangle | | |
| gilmore-lie-groups_FO2930 | 289 | 1.000 | m=+1 | | |
| gilmore-lie-groups_FO2931 | 289 | 1.000 | -\left(v_{x}+i v_{y}\right) | | |
| gilmore-lie-groups_FO2932 | 289 | 1.000 | \mathbf{v}=\left(v_{x}, v_{y}, 0\right) | | |
| gilmore-lie-groups_FO2933 | 289 | 1.000 | \mathbf{k}^{0}=(0,0,1): \mathbf{k}^{0} \cdot \mathbf{v}=0 | | |
| gilmore-lie-groups_FO2934 | 289 | 1.000 | B_{z} | | |
| gilmore-lie-groups_FO2935 | 289 | 0.976 | \mathbf{k} | | |
| gilmore-lie-groups_FO2936 | 289 | 1.000 | \mathbf{k} \cdot \mathbf{v}(\mathbf{k})=0 | | |
| gilmore-lie-groups_FO2937 | 290 | 0.999 | \nabla \cdot \mathbf{E}=0 | | |
| gilmore-lie-groups_FO2938 | 290 | 0.999 | \nabla \cdot \mathbf{B}=0 | | |
| gilmore-lie-groups_FO2939 | 290 | 1.000 | \mathbf{x}(j) | | |
| gilmore-lie-groups_FO2940 | 290 | 1.000 | e_{j} | | |
| gilmore-lie-groups_FO2941 | 290 | 1.000 | m_{j} | | |
| gilmore-lie-groups_FO2948 | 290 | 0.997 | T, P, T P | | |
| gilmore-lie-groups_FO2949 | 290 | 1.000 | 1 / c | | |
| gilmore-lie-groups_FO2950 | 291 | 0.867 | \phi=\pi / 2 | | |
| gilmore-lie-groups_FO2951 | 291 | 0.867 | (\mathbf{B}, \mathbf{E}) \rightarrow | | |
| gilmore-lie-groups_FO2952 | 291 | 1.000 | (*)=0 | | |
| gilmore-lie-groups_FO2953 | 291 | 1.000 | *= | | |
| gilmore-lie-groups_FO2954 | 291 | 0.998 | D^{j j^{\prime}+j^{\prime} j}(\Lambda) | | |
| gilmore-lie-groups_FO2955 | 291 | 0.998 | j-j^{\prime}= \pm 2 | | |
| gilmore-lie-groups_FO2956 | 291 | 0.998 | \left(j, j^{\prime}\right)=(2,0) | | |
| gilmore-lie-groups_FO2957 | 291 | 0.990 | \mathbf{G}_{\mathbf{e}} | | |
| gilmore-lie-groups_FO2958 | 291 | 0.990 | \mathbf{G}_{\mathbf{m}} | | |
| gilmore-lie-groups_FO2959 | 291 | 0.866 | \mathbf{J} \cdot \nabla | | |
| gilmore-lie-groups_FO2960 | 292 | 1.000 | F_{i j}=F_{j i}, \sum_{i} F_{i i}=0 | | |
| gilmore-lie-groups_FO2961 | 292 | 1.000 | \partial^{i} F_{i j}=0 | | |
| gilmore-lie-groups_FO2962 | 292 | 0.999 | U_{i j}=\sum_{k} m_{k}\left(\mathbf{x}_{k}(t) \mathbf{x}_{k}(t)\right)_{i j} | | |
| gilmore-lie-groups_FO2963 | 292 | 0.999 | \mathbf{G}_{\mathbf{m}}+i \mathbf{G}_{\mathbf{e}} \rightarrow e^{i \phi}\left(\mathbf{G}_{\mathbf{m}}+i \mathbf{G}_{\mathbf{e}}\right) | | |
| gilmore-lie-groups_FO2964 | 292 | 0.999 | \mathbf{J}_{\mathbf{m}}+i \mathbf{J}_{\mathbf{e}} \rightarrow e^{i \phi}\left(\mathbf{J}_{\mathbf{m}}+i \mathbf{J}_{\mathbf{e}}\right) | | |
| gilmore-lie-groups_FO2965 | 292 | 1.000 | D^{j j}(\Lambda) | | |
| gilmore-lie-groups_FO2966 | 292 | 0.871 | (J, M) | | |
| gilmore-lie-groups_FO2967 | 292 | 1.000 | (J, M)=(0,0),(1,0),(1, \pm 1),(2,0),(2 \pm 1) | | |
| gilmore-lie-groups_FO2968 | 293 | 1.000 | N \lambda | | |
| gilmore-lie-groups_FO2969 | 293 | 1.000 | N \lambda^{\prime} | | |
| gilmore-lie-groups_FO2970 | 293 | 1.000 | \rho(\mathbf{x}, t)=\rho(t) | | |
| gilmore-lie-groups_FO2971 | 293 | 1.000 | m_{e}(t) / M_{p}(t) | | |
| gilmore-lie-groups_FO2972 | 293 | 0.981 | v=\frac{1}{2}\left(m c^{2} / \hbar\right) \times\left|\left(1 / n_{1}^{2}-1 / n_{2}^{2}\right)\right| | | |
| gilmore-lie-groups_FO2973 | 293 | 1.000 | H_{\alpha} | | |
| gilmore-lie-groups_FO2974 | 294 | 1.000 | \Phi(x) | | |
| gilmore-lie-groups_FO2975 | 294 | 1.000 | S^{t} g_{\text {grav }} S=g_{\text {flat }} | | |
| gilmore-lie-groups_FO2976 | 294 | 1.000 | V | | |
| gilmore-lie-groups_FO2977 | 294 | 0.867 | \mathbf{E} | | |
| gilmore-lie-groups_FO2978 | 294 | 0.523 | a, p(x)=q \delta(x) | | |
| gilmore-lie-groups_FO2979 | 294 | 1.000 | |\mathbf{E}(a)|=q /|\mathbf{a}| | | |
| gilmore-lie-groups_FO2980 | 294 | 0.808 | \left(\oint \mathbf{E} \cdot d \mathbf{S}=\int 2 \pi \rho d A\right) | | |
| gilmore-lie-groups_FO2981 | 294 | 1.000 | a=R \theta | | |
| gilmore-lie-groups_FO2982 | 295 | 1.000 | 2 \pi R \sin \theta | | |
| gilmore-lie-groups_FO2983 | 295 | 1.000 | q / a | | |
| gilmore-lie-groups_FO2984 | 295 | 1.000 | a(a=R \theta) | | |
| gilmore-lie-groups_FO2985 | 295 | 1.000 | a=c t | | |
| gilmore-lie-groups_FO2986 | 295 | 1.000 | H^{n}=S O(n, 1) / S O(n), R^{n}= | | |
| gilmore-lie-groups_FO2987 | 295 | 0.999 | I S O(n) / S O(n) | | |
| gilmore-lie-groups_FO2988 | 295 | 1.000 | k=(-1,0,+1) | | |
| gilmore-lie-groups_FO2989 | 295 | 1.000 | H^{n}, R^{n}, S^{n} | | |
| gilmore-lie-groups_FO2990 | 295 | 1.000 | H^{n}, S^{n} | | |
| gilmore-lie-groups_FO2991 | 295 | 1.000 | \Omega | | |
| gilmore-lie-groups_FO2992 | 295 | 1.000 | S^{n-1} \subset H^{n}, R^{n} | | |
| gilmore-lie-groups_FO2993 | 295 | 1.000 | \int e^{-x^{2}} d x=\sqrt{\pi} | | |
| gilmore-lie-groups_FO2994 | 295 | 1.000 | \Omega=2 \pi^{n / 2} / \Gamma(n / 2) | | |
| gilmore-lie-groups_FO2995 | 296 | 1.000 | H^{n} | | |
| gilmore-lie-groups_FO2996 | 296 | 1.000 | R, c, t | | |
| gilmore-lie-groups_FO2997 | 296 | 0.839 | (l, b)=\left(263^{\circ}, 48^{\circ}\right) | | |
| gilmore-lie-groups_FO2998 | 296 | 1.000 | T(\theta, \phi ; t)= | | |
| gilmore-lie-groups_FO2999 | 296 | 1.000 | \sum_{l, m} A_{m}^{l}(t) Y_{m}^{l}(\theta, \phi) | | |
| gilmore-lie-groups_FO3000 | 296 | 0.998 | \Delta U \Delta(1 / T) \geq k | | |
| gilmore-lie-groups_FO3001 | 297 | 1.000 | j=\frac{1}{2}, 1,2 | | |
| gilmore-lie-groups_FO3002 | 297 | 1.000 | v(t) | | |
| gilmore-lie-groups_FO3003 | 297 | 1.000 | v(t) \simeq v\left(t_{0}\right) e^{-\left(t-t_{0}\right) / \tau} | | |
| gilmore-lie-groups_FO3004 | 297 | 1.000 | \tau | | |
| gilmore-lie-groups_FO3005 | 297 | 0.990 | \tau / T_{p} | | |
| gilmore-lie-groups_FO3006 | 297 | 0.990 | T_{p} | | |
| gilmore-lie-groups_FO3007 | 297 | 0.990 | T_{p} \simeq 13.7 \mathrm{BY} | | |
| gilmore-lie-groups_FO3008 | 297 | 1.000 | T_{B H} | | |
| gilmore-lie-groups_FO3009 | 297 | 1.000 | R=2 G M / c^{2} | | |
| gilmore-lie-groups_FO3010 | 297 | 1.000 | T_{B H}=\hbar c^{3} / 8 \pi k G M | | |
| gilmore-lie-groups_FO3011 | 297 | 1.000 | \gamma \pi R^{2} | | |
| gilmore-lie-groups_FO3012 | 297 | 1.000 | \gamma=3^{3} / 2^{2} | | |
| gilmore-lie-groups_FO3013 | 298 | 1.000 | d y / d x=g(x) | | |
| gilmore-lie-groups_FO3014 | 298 | 1.000 | y: d y / d x=g(x) | | |
| gilmore-lie-groups_FO3015 | 298 | 1.000 | d y / d x=p | | |
| gilmore-lie-groups_FO3016 | 299 | 1.000 | F(x, y, p)=0 | | |
| gilmore-lie-groups_FO3017 | 299 | 1.000 | R, S, T | | |
| gilmore-lie-groups_FO3018 | 299 | 1.000 | R(x, y) | | |
| gilmore-lie-groups_FO3019 | 299 | 1.000 | S(x, y) | | |
| gilmore-lie-groups_FO3020 | 299 | 1.000 | T(x, y, p) | | |
| gilmore-lie-groups_FO3021 | 299 | 1.000 | d y / d x | | |
| gilmore-lie-groups_FO3022 | 299 | 0.999 | F(R,-, T)=0 | | |
| gilmore-lie-groups_FO3023 | 299 | 0.999 | d S / d R=f(R,-, T) | | |
| gilmore-lie-groups_FO3024 | 299 | 1.000 | y-G(x)=0 | | |
| gilmore-lie-groups_FO3025 | 299 | 1.000 | y+c-G(x)= | | |
| gilmore-lie-groups_FO3026 | 299 | 1.000 | e^{c \partial / \partial y} | | |
| gilmore-lie-groups_FO3027 | 300 | 1.000 | F(x, y, p)=p-g(x) | | |
| gilmore-lie-groups_FO3028 | 300 | 1.000 | x, y, p | | |
| gilmore-lie-groups_FO3029 | 300 | 1.000 | \partial / \partial x, \partial / \partial y, \partial / \partial p | | |
| gilmore-lie-groups_FO3030 | 300 | 1.000 | F(x, y, p) | | |
| gilmore-lie-groups_FO3031 | 300 | 0.798 | \left(d^{n} y / d x^{n}, n=1\right) | | |
| gilmore-lie-groups_FO3032 | 300 | 1.000 | p^{m}=(d y / d x)^{m}, m=1 | | |
| gilmore-lie-groups_FO3033 | 300 | 1.000 | p-g(x) | | |
| gilmore-lie-groups_FO3034 | 300 | 1.000 | \frac{\partial}{\partial y} F(x, y, p) \neq 0 | | |
| gilmore-lie-groups_FO3035 | 300 | 0.999 | (R, S, T) | | |
| gilmore-lie-groups_FO3036 | 300 | 0.999 | R=R(x, y) | | |
| gilmore-lie-groups_FO3037 | 300 | 1.000 | S=S(x, y) | | |
| gilmore-lie-groups_FO3038 | 300 | 1.000 | T=T(x, y, p) | | |
| gilmore-lie-groups_FO3039 | 301 | 1.000 | x \rightarrow x | | |
| gilmore-lie-groups_FO3040 | 301 | 1.000 | y \rightarrow y+\epsilon | | |
| gilmore-lie-groups_FO3041 | 301 | 1.000 | \xi=0, \eta=1 | | |
| gilmore-lie-groups_FO3042 | 301 | 1.000 | \zeta(x, y, p) | | |
| gilmore-lie-groups_FO3043 | 301 | 1.000 | \xi(x, y) | | |
| gilmore-lie-groups_FO3044 | 301 | 1.000 | \eta(x, y) | | |
| gilmore-lie-groups_FO3045 | 301 | 1.000 | \eta_{x}=\partial \eta / \partial x | | |
| gilmore-lie-groups_FO3046 | 302 | 1.000 | \xi(x, y), \eta(x, y) | | |
| gilmore-lie-groups_FO3047 | 302 | 1.000 | y: p=p(x, y) | | |
| gilmore-lie-groups_FO3048 | 302 | 1.000 | X F(x, y, p(x, y))=0 | | |
| gilmore-lie-groups_FO3049 | 302 | 1.000 | d_{\xi}, d_{\eta} | | |
| gilmore-lie-groups_FO3050 | 302 | 0.999 | X F=0 | | |
| gilmore-lie-groups_FO3051 | 302 | 0.999 | \sum C_{i j} x^{i} y^{j}=0 | | |
| gilmore-lie-groups_FO3052 | 302 | 1.000 | C_{i j} | | |
| gilmore-lie-groups_FO3053 | 302 | 0.999 | \xi_{i j}, \eta_{i j} | | |
| gilmore-lie-groups_FO3054 | 303 | 1.000 | X(x, y, p) S(x, y)=1 | | |
| gilmore-lie-groups_FO3055 | 303 | 1.000 | X S=+1 | | |
| gilmore-lie-groups_FO3056 | 303 | 1.000 | X S=-1 | | |
| gilmore-lie-groups_FO3057 | 303 | 1.000 | X S=k \neq 0 | | |
| gilmore-lie-groups_FO3058 | 303 | 1.000 | X R=0 | | |
| gilmore-lie-groups_FO3059 | 303 | 1.000 | X T=0 | | |
| gilmore-lie-groups_FO3060 | 304 | 1.000 | R: T=T(R) | | |
| gilmore-lie-groups_FO3061 | 304 | 1.000 | x=x(R, S), y=y(R, S) | | |
| gilmore-lie-groups_FO3065 | 305 | 1.000 | y \rightarrow \alpha y | | |
| gilmore-lie-groups_FO3066 | 305 | 1.000 | x \rightarrow \beta x | | |
| gilmore-lie-groups_FO3067 | 305 | 1.000 | \alpha\left(x p+y-(\alpha \beta) x y^{2}\right)=0 | | |
| gilmore-lie-groups_FO3068 | 305 | 1.000 | \alpha \beta=1 | | |
| gilmore-lie-groups_FO3069 | 305 | 1.000 | x \rightarrow \lambda x, y \rightarrow \lambda^{-1} y, p \rightarrow \lambda^{-2} p | | |
| gilmore-lie-groups_FO3070 | 305 | 1.000 | X(x, y, p) | | |
| gilmore-lie-groups_FO3071 | 305 | 1.000 | p=y^{2}-y / x | | |
| gilmore-lie-groups_FO3072 | 306 | 1.000 | p: p(x, y)= | | |
| gilmore-lie-groups_FO3073 | 306 | 1.000 | -y / x+y^{2} | | |
| gilmore-lie-groups_FO3074 | 306 | 1.000 | X F(x, y, p)=0 | | |
| gilmore-lie-groups_FO3075 | 306 | 1.000 | \eta: \xi=\xi_{00}, \eta=\eta_{00} | | |
| gilmore-lie-groups_FO3076 | 306 | 0.999 | y / x, 1 | | |
| gilmore-lie-groups_FO3077 | 306 | 0.999 | x y | | |
| gilmore-lie-groups_FO3078 | 306 | 1.000 | \xi_{00}, \eta_{00} | | |
| gilmore-lie-groups_FO3079 | 307 | 1.000 | \xi_{01}=\xi_{00}=\eta_{00}=\eta_{10}=0 | | |
| gilmore-lie-groups_FO3080 | 307 | 1.000 | x y^{2} | | |
| gilmore-lie-groups_FO3081 | 307 | 1.000 | -\xi_{10}-\eta_{01}=0 | | |
| gilmore-lie-groups_FO3082 | 307 | 1.000 | \xi(x, y)=x | | |
| gilmore-lie-groups_FO3083 | 307 | 1.000 | \eta(x, y)=-y | | |
| gilmore-lie-groups_FO3084 | 307 | 1.000 | \zeta=-2 p | | |
| gilmore-lie-groups_FO3085 | 307 | 1.000 | -y d S(y) / d y=1 | | |
| gilmore-lie-groups_FO3086 | 307 | 1.000 | -\ln (y) | | |
| gilmore-lie-groups_FO3087 | 307 | 1.000 | S(x, y)=\ln (y) | | |
| gilmore-lie-groups_FO3088 | 307 | 1.000 | y d x=-x d y | | |
| gilmore-lie-groups_FO3089 | 307 | 1.000 | d(x y)=0 | | |
| gilmore-lie-groups_FO3090 | 307 | 1.000 | R(x, y)=x y | | |
| gilmore-lie-groups_FO3091 | 307 | 1.000 | -d p / 2 p | | |
| gilmore-lie-groups_FO3092 | 307 | 1.000 | d x / x | | |
| gilmore-lie-groups_FO3093 | 307 | 1.000 | d x / x=-d p / 2 p | | |
| gilmore-lie-groups_FO3094 | 307 | 1.000 | (1 / x) d\left(x^{2} p\right)=0 | | |
| gilmore-lie-groups_FO3095 | 307 | 1.000 | T(x, y, p)=x^{2} p | | |
| gilmore-lie-groups_FO3101 | 308 | 1.000 | T=T(R, S) | | |
| gilmore-lie-groups_FO3102 | 308 | 1.000 | R: T(R)=R^{2}-R | | |
| gilmore-lie-groups_FO3103 | 309 | 0.999 | (R, S) | | |
| gilmore-lie-groups_FO3104 | 309 | 1.000 | x d / d x | | |
| gilmore-lie-groups_FO3105 | 309 | 1.000 | e^{\lambda x d / d x} x=e^{\lambda} x | | |
| gilmore-lie-groups_FO3106 | 309 | 0.766 | (x, y, p) | | |
| gilmore-lie-groups_FO3107 | 309 | 1.000 | \ln (y) | | |
| gilmore-lie-groups_FO3108 | 309 | 1.000 | \ln \left(e^{-\lambda} y\right)=\ln (y)-\lambda | | |
| gilmore-lie-groups_FO3109 | 309 | 1.000 | x^{2} p | | |
| gilmore-lie-groups_FO3110 | 309 | 1.000 | \ln \left(x y^{2}\right) | | |
| gilmore-lie-groups_FO3111 | 309 | 1.000 | x^{3} y p | | |
| gilmore-lie-groups_FO3112 | 309 | 1.000 | F\left(x y, x^{2} p\right)=0 | | |
| gilmore-lie-groups_FO3113 | 309 | 1.000 | x^{2} p=h(x y) | | |
| gilmore-lie-groups_FO3114 | 309 | 1.000 | d y / d x=x^{-2} h(x y) | | |
| gilmore-lie-groups_FO3115 | 309 | 1.000 | h(z)=-z+z^{2} | | |
| gilmore-lie-groups_FO3116 | 309 | 1.000 | d y / d x+y^{2}-2 / x^{2}=0, h(z)=z^{2}-2 | | |
| gilmore-lie-groups_FO3117 | 309 | 1.000 | y^{\prime 2}+y^{4}-x^{-4}=0 | | |
| gilmore-lie-groups_FO3118 | 309 | 1.000 | R^{4}+T^{2}=1 | | |
| gilmore-lie-groups_FO3119 | 310 | 1.000 | p= \pm \sqrt{x^{-4}-y^{4}} | | |
| gilmore-lie-groups_FO3120 | 310 | 1.000 | T= \pm \sqrt{1-R^{4}} | | |
| gilmore-lie-groups_FO3130 | 310 | 0.998 | y^{(2)} | | |
| gilmore-lie-groups_FO3234 | 312 | 1.000 | F\left(x, y, y^{\prime}, y^{\prime \prime}\right)=0 | | |
| gilmore-lie-groups_FO3235 | 312 | 1.000 | S=\ln y | | |
| gilmore-lie-groups_FO3236 | 312 | 1.000 | F(R,-, T, U)=0 | | |
| gilmore-lie-groups_FO3237 | 312 | 1.000 | y, T=T\left(x, y, y^{\prime}\right) | | |
| gilmore-lie-groups_FO3238 | 312 | 1.000 | U=U\left(x, y, y^{\prime}, y^{\prime \prime}\right) | | |
| gilmore-lie-groups_FO3239 | 312 | 1.000 | d T / d R | | |
| gilmore-lie-groups_FO3240 | 312 | 1.000 | y^{\prime \prime} | | |
| gilmore-lie-groups_FO3241 | 313 | 1.000 | F\left(x, y, \ldots, y^{(n)}\right)=0 | | |
| gilmore-lie-groups_FO3242 | 313 | 1.000 | T\left(x, y, y^{(1)}\right) | | |
| gilmore-lie-groups_FO3243 | 313 | 1.000 | d T^{(j)} / d R^{(j)}, j=0 | | |
| gilmore-lie-groups_FO3244 | 313 | 1.000 | j=1,2, \ldots, n-1 | | |
| gilmore-lie-groups_FO3245 | 313 | 1.000 | y^{(j+1)} | | |
| gilmore-lie-groups_FO3246 | 314 | 1.000 | x^{i} \rightarrow \lambda x^{i}, u \rightarrow \alpha u | | |
| gilmore-lie-groups_FO3247 | 314 | 1.000 | \delta(x) \rightarrow \delta(\lambda x)=\lambda^{-n} \delta(x) | | |
| gilmore-lie-groups_FO3248 | 314 | 1.000 | \alpha=\lambda^{2-n} | | |
| gilmore-lie-groups_FO3249 | 314 | 1.000 | R=R(x, u) | | |
| gilmore-lie-groups_FO3250 | 314 | 1.000 | R \sim u|x|^{n-2} | | |
| gilmore-lie-groups_FO3251 | 314 | 1.000 | u \sim|x|^{2-n}=k|x|^{2-n} | | |
| gilmore-lie-groups_FO3252 | 314 | 1.000 | V\left(S^{n-1}\right)=2 \pi^{n / 2} / \Gamma\left(\frac{n}{2}\right) | | |
| gilmore-lie-groups_FO3253 | 314 | 1.000 | R^{n}(n \neq 2) | | |
| gilmore-lie-groups_FO3254 | 314 | 1.000 | u(x, t) | | |
| gilmore-lie-groups_FO3255 | 314 | 1.000 | u \rightarrow \alpha u, t \rightarrow \beta t | | |
| gilmore-lie-groups_FO3256 | 314 | 1.000 | x^{i} \rightarrow \lambda x^{i} | | |
| gilmore-lie-groups_FO3257 | 314 | 1.000 | \alpha \lambda^{n}=1 | | |
| gilmore-lie-groups_FO3258 | 314 | 1.000 | \beta / \lambda^{2}=1 | | |
| gilmore-lie-groups_FO3259 | 315 | 1.000 | n+1 | | |
| gilmore-lie-groups_FO3260 | 315 | 1.000 | x^{i}, t | | |
| gilmore-lie-groups_FO3261 | 315 | 1.000 | R=u t^{n / 2} e^{|x|^{2} / 4 t} | | |
| gilmore-lie-groups_FO3262 | 315 | 1.000 | F(R,-, T)= | | |
| gilmore-lie-groups_FO3263 | 315 | 0.980 | \int f(R,-, T(R)) d R | | |
| gilmore-lie-groups_FO3265 | 316 | 1.000 | \zeta(x, y, p)=\eta^{(1)}\left(x, y, y^{(1)}\right) | | |
| gilmore-lie-groups_FO3266 | 316 | 1.000 | X=\xi \partial / \partial x+\eta \partial / \partial y+\zeta \partial / \partial p | | |
| gilmore-lie-groups_FO3267 | 316 | 1.000 | F=0 | | |
| gilmore-lie-groups_FO3268 | 316 | 1.000 | X R=0, X S=1, X T=0 | | |
| gilmore-lie-groups_FO3269 | 316 | 1.000 | F \rightarrow F(R,-, T)=0 | | |
| gilmore-lie-groups_FO3270 | 317 | 1.000 | S: S=\int f(R,-, T(R))+c | | |
| gilmore-lie-groups_FO3271 | 317 | 1.000 | x=x(R, S) | | |
| gilmore-lie-groups_FO3272 | 317 | 1.000 | y=y(R, S) | | |
| gilmore-lie-groups_FO3273 | 317 | 1.000 | n=0,1 | | |
| gilmore-lie-groups_FO3274 | 317 | 1.000 | V(\mathbf{x}) | | |
| gilmore-lie-groups_FO3275 | 317 | 1.000 | m \rightarrow \alpha m), \mathbf{x} \rightarrow \beta \mathbf{x}, t \rightarrow \gamma t | | |
| gilmore-lie-groups_FO3276 | 317 | 1.000 | k: V(\beta \mathbf{x}) \rightarrow \beta^{k} V(\mathbf{x}) | | |
| gilmore-lie-groups_FO3277 | 317 | 1.000 | \alpha^{1} \beta^{2-k} | | |
| gilmore-lie-groups_FO3278 | 317 | 1.000 | \gamma^{-2}=1 | | |
| gilmore-lie-groups_FO3279 | 317 | 1.000 | \alpha=1 | | |
| gilmore-lie-groups_FO3280 | 317 | 1.000 | \gamma^{2}=\beta^{2-k} | | |
| gilmore-lie-groups_FO3281 | 317 | 1.000 | k=-1, k=0, k,=+1, k=+2 | | |
| gilmore-lie-groups_FO3286 | 318 | 0.785 | \gamma^{2} | | |
| gilmore-lie-groups_FO3287 | 318 | 0.785 | \beta^{3} | | |
| gilmore-lie-groups_FO3288 | 318 | 0.785 | R^{\prime} | | |
| gilmore-lie-groups_FO3289 | 318 | 1.000 | T^{\prime} | | |
| gilmore-lie-groups_FO3290 | 318 | 1.000 | P^{\prime} | | |
| gilmore-lie-groups_FO3291 | 318 | 1.000 | \beta^{3} \rightarrow\left(R^{\prime} / R\right)^{3}=\left(T^{\prime} / T\right)^{2} \leftarrow \gamma^{2} | | |
| gilmore-lie-groups_FO3292 | 318 | 0.860 | (\beta) | | |
| gilmore-lie-groups_FO3293 | 318 | 0.860 | (\gamma) | | |
| gilmore-lie-groups_FO3294 | 318 | 1.000 | V=m g z | | |
| gilmore-lie-groups_FO3295 | 318 | 1.000 | \gamma=1 | | |
| gilmore-lie-groups_FO3296 | 318 | 1.000 | \beta=1 | | |
| gilmore-lie-groups_FO3297 | 318 | 1.000 | \sqrt{M} | | |
| gilmore-lie-groups_FO3298 | 318 | 0.994 | \alpha \beta^{2} \gamma^{-2}=\beta^{k} | | |
| gilmore-lie-groups_FO3299 | 318 | 1.000 | \delta \int \mathcal{L}(\mathbf{x}, \dot{\mathbf{x}}) d \mathbf{x}=0 | | |
| gilmore-lie-groups_FO3300 | 318 | 1.000 | v_{i} | | |
| gilmore-lie-groups_FO3301 | 318 | 1.000 | e^{\epsilon v_{i}} f(x, t) | | |
| gilmore-lie-groups_FO3307 | 319 | 1.000 | u \partial_{u} | | |
| gilmore-lie-groups_FO3317 | 319 | 1.000 | \lambda e^{-\epsilon \lambda^{2} x^{2}} f\left(\lambda^{2} x, \lambda^{2} t\right) | | |
| gilmore-lie-groups_FO3318 | 319 | 1.000 | \lambda^{2}=1 /(1+4 \epsilon t) | | |
| gilmore-lie-groups_FO3319 | 319 | 1.000 | D_{2}=u \partial_{u} | | |
| gilmore-lie-groups_FO3320 | 319 | 1.000 | S O(2+1,1+1)=S O(3,2) | | |
| gilmore-lie-groups_FO3321 | 319 | 1.000 | S O(3+1,1+1)=S O(4,2) | | |
| gilmore-lie-groups_FO3322 | 319 | 1.000 | u_{x x}-u_{t}=0 | | |
| gilmore-lie-groups_FO3323 | 319 | 1.000 | X=\xi^{i} \frac{\partial}{\partial x^{i}}+\eta \frac{\partial}{\partial u}+\cdots=\xi^{1} \frac{\partial}{\partial x}+\xi^{2} \frac{\partial}{\partial t}+\eta \frac{\partial}{\partial u}+\cdots | | |
| gilmore-lie-groups_FO3324 | 320 | 1.000 | h(x, t) | | |
| gilmore-lie-groups_FO3325 | 320 | 0.986 | \left(t \rightarrow t^{\prime}=T(t, x, \epsilon)=t+\right. | | |
| gilmore-lie-groups_FO3326 | 320 | 1.000 | \left.\epsilon \xi(t, x), x_{i} \rightarrow x_{i}^{\prime}=X_{i}(t, x, \epsilon)=x_{i}+\epsilon \eta_{i}(t, x)\right) | | |
| gilmore-lie-groups_FO3327 | 320 | 1.000 | d t^{\prime} / d t=\partial T / \partial t+\left(\partial T / \partial x_{i}\right) d x_{i} / d t | | |
| gilmore-lie-groups_FO3328 | 320 | 1.000 | \epsilon=0 | | |
| gilmore-lie-groups_FO3329 | 321 | 1.000 | L[u]= | | |
| gilmore-lie-groups_FO3330 | 321 | 1.000 | \int \mathcal{L}(x, u) d x, x \in R^{p}, u \in R^{q} | | |
| gilmore-lie-groups_FO3331 | 321 | 1.000 | d \rho(g) | | |
| gilmore-lie-groups_FO3332 | 321 | 0.995 | \operatorname{Vol}(G)=\int d \rho(g), \Gamma_{\mu \nu}^{\lambda}(g) | | |
| gilmore-lie-groups_FO3333 | 321 | 1.000 | \phi(g), \psi(g) | | |
| gilmore-lie-groups_FO3334 | 322 | 1.000 | \psi(g)=\langle g \mid \psi\rangle | | |
| gilmore-lie-groups_FO3335 | 322 | 1.000 | \left\langle{ }_{\mu \nu}^{\lambda} \mid \psi\right\rangle= | | |
| gilmore-lie-groups_FO3336 | 322 | 1.000 | \int d \rho(g)\left\langle{ }_{\mu \nu}^{\lambda} \mid g\right\rangle\langle g \mid \psi\rangle | | |
| gilmore-lie-groups_FO3337 | 322 | 1.000 | \phi(g)=\langle g \mid \phi\rangle | | |
| gilmore-lie-groups_FO3338 | 322 | 1.000 | \int \phi^{*}(g) \psi(g) d \rho(g) | | |
| gilmore-lie-groups_FO3339 | 323 | 1.000 | \mathrm{He}^{+} | | |
| gilmore-lie-groups_FO3340 | 324 | 0.868 | A_{n} ; D_{n}, B_{n}, C_{n}, J | | |
| gilmore-lie-groups_FO3341 | 325 | 0.866 | \mathrm{mx} /{ }^{\sim} | | |
| gilmore-lie-groups_FO3342 | 327 | 0.995 | A(p q), 39,48 | | |
| gilmore-lie-groups_FO3343 | 327 | 0.978 | A_{1}, 161 | | |
| gilmore-lie-groups_FO3344 | 327 | 0.991 | A_{2} | | |
| gilmore-lie-groups_FO3345 | 327 | 0.865 | A_{3}, 46,161,162 | | |
| gilmore-lie-groups_FO3346 | 327 | 0.946 | A_{n}, 46,49,161,166 | | |
| gilmore-lie-groups_FO3347 | 327 | 0.992 | B_{1}, 161 | | |
| gilmore-lie-groups_FO3348 | 327 | 0.749 | B_{2}, 151,160,161,164 | | |
| gilmore-lie-groups_FO3349 | 327 | 0.761 | B_{3}, 162 | | |
| gilmore-lie-groups_FO3350 | 327 | 0.729 | \boldsymbol{B}_{n}, 161,166,168 | | |
| gilmore-lie-groups_FO3351 | 327 | 1.000 | C_{1}, 161 | | |
| gilmore-lie-groups_FO3352 | 327 | 0.523 | C_{2}, 151,160,161,164 | | |
| gilmore-lie-groups_FO3353 | 327 | 0.712 | C_{3}, 162 | | |
| gilmore-lie-groups_FO3354 | 327 | 0.997 | C_{n}, 161,166,168 | | |
| gilmore-lie-groups_FO3355 | 327 | 0.840 | D_{2}, 151,160,162 | | |
| gilmore-lie-groups_FO3356 | 327 | 0.435 | G L(1 ; \mathbb{Q}), 40,47 | | |
| gilmore-lie-groups_FO3357 | 327 | 0.979 | G L(2 ; \mathbb{C}), 47 | | |
| gilmore-lie-groups_FO3358 | 327 | 0.788 | G L(2 ; \mathbb{R}) | | |
| gilmore-lie-groups_FO3359 | 327 | 0.731 | G L(2 ; \mathbb{Z}), 45,49 | | |
| gilmore-lie-groups_FO3360 | 327 | 0.903 | G L(3 ; \mathbb{Z}), 45 | | |
| gilmore-lie-groups_FO3361 | 327 | 0.471 | G L(n ; \mathbb{F}), 34,36,74 | | |
| gilmore-lie-groups_FO3362 | 327 | 0.698 | G L(n ; \mathbb{Q}) | | |
| gilmore-lie-groups_FO3363 | 327 | 0.973 | G_{2}, 151,160,162,165 | | |
| gilmore-lie-groups_FO3364 | 327 | 0.933 | \operatorname{HT}(p, q), 37,48 | | |
| gilmore-lie-groups_FO3365 | 327 | 0.634 | \operatorname{Nil}(n), 38,48 | | |
| gilmore-lie-groups_FO3366 | 327 | 0.788 | O(31), 261 | | |
| gilmore-lie-groups_FO3367 | 327 | 0.988 | O(3), 40,78 | | |
| gilmore-lie-groups_FO3368 | 327 | 1.000 | O(3 ; \mathbb{Z}), 46 | | |
| gilmore-lie-groups_FO3369 | 327 | 0.991 | O(n), 40,43,145 | | |
| gilmore-lie-groups_FO3370 | 327 | 1.000 | O(n ; \mathbb{G}), 41 | | |
| gilmore-lie-groups_FO3371 | 327 | 1.000 | O(n ; \mathbb{Z}), 45,49 | | |
| gilmore-lie-groups_FO3372 | 327 | 0.737 | O(p, q) | | |
| gilmore-lie-groups_FO3373 | 327 | 0.939 | P_{n}, 45 | | |
| gilmore-lie-groups_FO3374 | 327 | 0.901 | S L(2 ; \mathbb{C}), 43 | | |
| gilmore-lie-groups_FO3375 | 327 | 0.985 | S L(2 ; \mathbb{R}), 26,28,29,30,41,43,56,58,62,100,102 | | |
| gilmore-lie-groups_FO3376 | 327 | 0.956 | S L(n ; \mathbb{R}), 30 | | |
| gilmore-lie-groups_FO3377 | 327 | 0.600 | S L(n ; \mathbb{C}), 43,47 | | |
| gilmore-lie-groups_FO3378 | 327 | 0.519 | \operatorname{SL}(n ; \mathbb{Q}) | | |
| gilmore-lie-groups_FO3379 | 327 | 0.747 | \operatorname{SL}(n ; \mathbb{R}) | | |
| gilmore-lie-groups_FO3380 | 327 | 0.880 | S L(n ; \mathbb{Z}), 45 | | |
| gilmore-lie-groups_FO3381 | 327 | 0.703 | S O(2,1), 105 | | |
| gilmore-lie-groups_FO3382 | 327 | 0.943 | S O(2,1) / S O(2), 106 | | |
| gilmore-lie-groups_FO3383 | 327 | 0.820 | \operatorname{SO}(2), 48,164 | | |
| gilmore-lie-groups_FO3384 | 327 | 0.621 | S O(2 n), 164 | | |
| gilmore-lie-groups_FO3385 | 327 | 1.000 | S O(2 n+1), 164 | | |
| gilmore-lie-groups_FO3386 | 327 | 0.607 | S O(3,1), 263 | | |
| gilmore-lie-groups_FO3387 | 327 | 0.979 | \operatorname{SO}(3), 49,90,106 | | |
| gilmore-lie-groups_FO3388 | 327 | 0.875 | S O(3) / S O(2), 107 | | |
| gilmore-lie-groups_FO3389 | 327 | 0.498 | \operatorname{SO}(4,1), 210 | | |
| gilmore-lie-groups_FO3390 | 327 | 0.463 | \operatorname{SO}(n) | | |
| gilmore-lie-groups_FO3391 | 327 | 0.637 | \operatorname{SU}(1 ; \mathbb{Q}), 40,48 | | |
| gilmore-lie-groups_FO3392 | 327 | 0.998 | \operatorname{SU}(1,1), 38,43,48,105 | | |
| gilmore-lie-groups_FO3393 | 327 | 1.000 | S U(1,1) / U(1), 106 | | |
| gilmore-lie-groups_FO3394 | 327 | 0.606 | \operatorname{SU}(2), 48,106 | | |
| gilmore-lie-groups_FO3395 | 327 | 0.999 | S U(2) / U(1), 107 | | |
| gilmore-lie-groups_FO3396 | 327 | 0.983 | \operatorname{SU}(n), 43,90,164 | | |
| gilmore-lie-groups_FO3397 | 327 | 0.962 | \operatorname{SU}(p, q), 43,164 | | |
| gilmore-lie-groups_FO3398 | 327 | 0.994 | S_{3}, 5,46 | | |
| gilmore-lie-groups_FO3399 | 328 | 0.886 | S_{n}, 45,49 | | |
| gilmore-lie-groups_FO3400 | 328 | 0.925 | \operatorname{Sol}(n), 38,48 | | |
| gilmore-lie-groups_FO3401 | 328 | 0.986 | \operatorname{Sp}(1), 40 | | |
| gilmore-lie-groups_FO3402 | 328 | 0.995 | \operatorname{Sp}(2 ; \mathbb{R}), 41 | | |
| gilmore-lie-groups_FO3403 | 328 | 0.883 | \operatorname{Sp}(2 n ; \mathbb{R}), 41 | | |
| gilmore-lie-groups_FO3404 | 328 | 0.739 | \operatorname{Sp}(n), 40,164 | | |
| gilmore-lie-groups_FO3405 | 328 | 0.999 | \operatorname{Sp}(n ; \mathbb{C}), 41 | | |
| gilmore-lie-groups_FO3406 | 328 | 0.944 | \operatorname{Sp}(n ; \mathbb{G}), 41 | | |
| gilmore-lie-groups_FO3407 | 328 | 0.621 | \operatorname{Sp}(n ; \mathbb{R}, 41 | | |
| gilmore-lie-groups_FO3408 | 328 | 0.903 | \operatorname{Sp}(p, q), 164 | | |
| gilmore-lie-groups_FO3409 | 328 | 0.970 | U(1,1), 43 | | |
| gilmore-lie-groups_FO3410 | 328 | 0.619 | U(2), 40,78 | | |
| gilmore-lie-groups_FO3411 | 328 | 1.000 | U(2 ; \mathbb{Q}), 164 | | |
| gilmore-lie-groups_FO3412 | 328 | 0.804 | U(n), 40,43,90 | | |
| gilmore-lie-groups_FO3413 | 328 | 1.000 | U(n ; \mathbb{G}), 41 | | |
| gilmore-lie-groups_FO3414 | 328 | 1.000 | U(p, q), 43 | | |
| gilmore-lie-groups_FO3415 | 328 | 0.713 | U \operatorname{Sp}(2 n), 44 | | |
| gilmore-lie-groups_FO3416 | 328 | 0.397 | \operatorname{UT}(p, q), 48 | | |
| gilmore-lie-groups_FO3417 | 328 | 1.000 | \mathbb{Z} | | |
| gilmore-lie-groups_FO3418 | 328 | 0.999 | S_{2}, 10 | | |
| gilmore-lie-groups_FO3419 | 328 | 0.743 | S_{3}, 12 | | |
| gilmore-lie-groups_FO3420 | 328 | 0.712 | C_{2}, 153 | | |
| gilmore-lie-groups_FO3421 | 329 | 1.000 | \overline{S O(2,1) / S O(2)}, 108 | | |
| gilmore-lie-groups_FO3422 | 329 | 0.560 | \overline{S U(1,1) / U(1)}, 108 | | |
| gilmore-lie-groups_FO3423 | 330 | 1.000 | \mathfrak{a}(p, q), 77,129 | | |
| gilmore-lie-groups_FO3424 | 330 | 0.874 | \mathfrak{g} \mathfrak{l}(n ; \mathbb{F}), 74,83 | | |
| gilmore-lie-groups_FO3425 | 330 | 0.808 | \mathfrak{h t}(p, q), 75 | | |
| gilmore-lie-groups_FO3426 | 330 | 0.891 | \operatorname{nil}(n), 77,130 | | |
| gilmore-lie-groups_FO3427 | 330 | 1.000 | \mathfrak{o u}(2 n), 179 | | |
| gilmore-lie-groups_FO3428 | 330 | 1.000 | \mathfrak{o}(n ; G), 79 | | |
| gilmore-lie-groups_FO3429 | 330 | 0.997 | \mathfrak{o}(p, q), 78 | | |
| gilmore-lie-groups_FO3430 | 330 | 0.630 | \mathfrak{s} \mathfrak{l}(2 ; \mathbb{R}), 100,102,154,173 | | |
| gilmore-lie-groups_FO3431 | 330 | 0.811 | \mathfrak{s l}(n), 80 | | |
| gilmore-lie-groups_FO3432 | 330 | 0.637 | \mathfrak{s} \mathfrak{l}(n ; \mathbb{C}), 80,85,86 | | |
| gilmore-lie-groups_FO3433 | 330 | 0.940 | \mathfrak{s l}(n ; \mathbb{Q}), 80,86 | | |
| gilmore-lie-groups_FO3434 | 330 | 0.743 | \mathfrak{s} \mathfrak{l}(n ; \mathbb{R}), 80,85,178,180 | | |
| gilmore-lie-groups_FO3435 | 330 | 0.808 | \mathfrak{s} \mathfrak{o} \mathfrak{l}(n), 77,130 | | |
| gilmore-lie-groups_FO3436 | 330 | 0.932 | \mathfrak{s} \mathfrak{o}(2,1), 78 | | |
| gilmore-lie-groups_FO3437 | 330 | 0.608 | \mathfrak{s o}(2 n), 180 | | |
| gilmore-lie-groups_FO3438 | 330 | 0.802 | \mathfrak{s} \mathfrak{o}(3,1), 79 | | |
| gilmore-lie-groups_FO3439 | 330 | 0.887 | \mathfrak{s} \mathfrak{o}(3,2), 86 | | |
| gilmore-lie-groups_FO3440 | 330 | 1.000 | \mathfrak{s} \mathfrak{o}(3), 86,90,154 | | |
| gilmore-lie-groups_FO3441 | 330 | 0.666 | \mathfrak{s} \mathfrak{o}(4,1), 86 | | |
| gilmore-lie-groups_FO3442 | 330 | 0.930 | \mathfrak{s o}(4), 132 | | |
| gilmore-lie-groups_FO3443 | 330 | 0.562 | \mathfrak{s} \mathfrak{o}(5), 86,146 | | |
| gilmore-lie-groups_FO3444 | 330 | 0.651 | \mathfrak{s o}(n), 132,145,178 | | |
| gilmore-lie-groups_FO3445 | 330 | 0.986 | \mathfrak{s} \mathfrak{o}(p, q), 84,178 | | |
| gilmore-lie-groups_FO3446 | 330 | 0.972 | \mathfrak{s} \mathfrak{o}^{*}(2 n), 180 | | |
| gilmore-lie-groups_FO3447 | 330 | 0.986 | \mathfrak{s} \mathfrak{p}(2 n ; \mathbb{R}), 178,179,180 | | |
| gilmore-lie-groups_FO3448 | 330 | 0.994 | \mathfrak{s p}(G ; \mathbb{C}), 79 | | |
| gilmore-lie-groups_FO3449 | 330 | 0.788 | \mathfrak{s} \mathfrak{p}(G ; \mathbb{R}), 79 | | |
| gilmore-lie-groups_FO3450 | 330 | 0.986 | \mathfrak{s p}(n), 132,178 | | |
| gilmore-lie-groups_FO3451 | 330 | 0.791 | \mathfrak{s p}(n ; G), 79 | | |
| gilmore-lie-groups_FO3452 | 330 | 0.767 | \mathfrak{s p}(p, q), 78,178 | | |
| gilmore-lie-groups_FO3453 | 331 | 0.990 | \mathfrak{s} \mathfrak{u}(1,1), 140,141,143,173 | | |
| gilmore-lie-groups_FO3454 | 331 | 0.623 | \mathfrak{s u}(2), 111,140,141,143,173 | | |
| gilmore-lie-groups_FO3455 | 331 | 0.943 | \mathfrak{s} \mathfrak{u}(2 n), 180 | | |
| gilmore-lie-groups_FO3456 | 331 | 0.857 | \mathfrak{s} \mathfrak{u}(n), 80,132,178 | | |
| gilmore-lie-groups_FO3457 | 331 | 1.000 | \mathfrak{s} \mathfrak{u}(p, q), 85,178 | | |
| gilmore-lie-groups_FO3458 | 331 | 0.882 | \mathfrak{s} \mathfrak{u}^{*}(2 n), 180 | | |
| gilmore-lie-groups_FO3459 | 331 | 0.999 | \mathfrak{u s p}(2 n), 179 | | |
| gilmore-lie-groups_FO3460 | 331 | 1.000 | \mathfrak{u t}(1,1), 83 | | |
| gilmore-lie-groups_FO3461 | 331 | 0.993 | \mathfrak{u t}(p, q, r), 76 | | |
| gilmore-lie-groups_FO3462 | 331 | 1.000 | \mathfrak{u t}(p, q), 75,131 | | |
| gilmore-lie-groups_FO3463 | 331 | 0.973 | \mathfrak{u}(n), 80 | | |
| gilmore-lie-groups_FO3464 | 331 | 1.000 | \mathfrak{u}(n ; G), 79 | | |
| gilmore-lie-groups_FO3465 | 331 | 0.858 | \mathfrak{u}(p, q), 78 | | |
| gilmore-lie-groups_FO3466 | 331 | 0.996 | \mathfrak{u}(p, q ; \mathbb{F}), 178 | | |
| gilmore-lie-groups_FO3467 | 331 | 0.982 | \mathfrak{s l}(2 ; \mathbb{C}), 154 | | |
| gilmore-lie-groups_FO3468 | 332 | 0.676 | S U(2), 187 | | |
| gilmore-lie-groups_FO3469 | 332 | 0.679 | \operatorname{SU}(1,1), 187 | | |
| gilmore-lie-groups_FO3470 | 332 | 0.977 | c, 282 | | |
| gilmore-lie-groups_FO3471 | 333 | 0.644 | S O(n), 183 | | |
| gilmore-lie-groups_FO3472 | 333 | 0.933 | S O(3), 164 | | |
| gilmore-lie-groups_FO3473 | 333 | 0.662 | S O(5), 164 | |