equation stringlengths 7 1.51k | spoken_English stringlengths 0 4.53k |
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G_{0}( \tau , \tau ^{ \prime })= \frac {( \beta - \tau ^{ \prime }) \, \tau }{ \beta } \, \theta ( \tau ^{ \prime }- \tau )+ \frac {( \beta - \tau ) \, \tau ^{ \prime }}{ \beta } \, \theta ( \tau - \tau ^{ \prime }), | G sub 0 of open paren tau comma tau prime close paren equals open paren beta minus tau prime close paren times tau over beta times theta times open paren tau prime minus tau close paren plus open paren beta minus tau close paren times tau prime over beta times theta times open paren tau minus tau prime close paren comm... |
{ \cal L}(M_{m+1 \,j})= \lambda M_{m+1 \,j} \, | script L of open paren M sub m plus 1 j close paren equals lamda M sub m plus 1 j |
T( \omega ) \gamma ^{A}[T( \omega )]^{-1}= \Lambda _{ \cdot \,B}^{A \, \cdot }( \omega ) \gamma ^{B} | T of omega gamma to the A power times open bracket T of omega close bracket to the negative 1 power equals normal Lamda sub times B raised to the A times power of omega gamma to the B power |
V(- \vec {x},-t) \longrightarrow -V( \vec {x},t) | V of open paren negative x right arrow comma negative t close paren long right arrow negative V of open paren x right arrow comma t close paren |
\Phi _{i}(x)= \left( \begin{matrix}{ \varphi _{1}(x)} \\ {0} \\ \end{matrix} \right), \quad \Phi _{f}(x)= \left( \begin{matrix}{0} \\ { \varphi _{2}(x)} \\ \end{matrix} \right), | normal Phi sub i of x equals the 2 by 1 column matrix phi sub 1 of x 0 comma normal Phi sub f of x equals the 2 by 1 column matrix 0 phi sub 2 of x comma |
\Lambda \equiv \: \{A: \: \parallel A \parallel ^{2} \leq \: \parallel \,^{g} \!A \parallel ^{2} \mathrm {forall} \;g \}. | normal Lamda is identical to the set of all A such that the metric of A squared is less than or equal to the metric of raised to the g power times A squared forall g period |
\hat {w}_{ \mu }= \frac {1}{2} \, \epsilon _{ \mu \nu \rho \sigma } \hat {P}^{ \nu }J^{ \rho \sigma }=w_{ \mu }M^{-1} \,, | w hat sub mu equals one half times epsilon sub mu nu rho sigma P hat to the nu-th power J raised to the rho sigma power equals w sub mu M to the negative 1 power comma |
a(k,t) \,= \,H_{n}(t \beta ) \,= \,(-1)^{n} \, \mathrm {e}^{(t \beta )^{2}}{ \frac {d^{n} \mathrm {e}^{-(t \beta )^{2}}}{d(t \beta )^{n}}}, | a of open paren k comma t close paren equals H sub n of t beta equals open paren negative 1 close paren to the n-th power times normal e raised to the exponent open paren t beta close paren squared end exponent d to the n-th power normal e raised to the exponent negative open paren t beta close paren squared end expone... |
{ \cal M}^{a \mu \nu }= \frac {1}{2} \; \left[{ \cal P}^{a \mu }X^{ \nu }-2 \sqrt {- \gamma }K_{i}n^{ \mu \,i}e^{ \nu \,a}-( \mu \leftrightarrow \nu ) \right] \,. | script M raised to the a mu nu power equals one half times open bracket script P raised to the a mu power X to the nu-th power minus 2 the square root of negative gamma K sub i n raised to the mu i power e raised to the nu a power minus open paren mu left right arrow nu close paren close bracket period |
\left[ \hat {x}^{i}, \hat {x}^{j} \right]= \frac {i \hbar }{eB} \epsilon ^{ij}. | open bracket x hat to the i-th power comma x hat to the j-th power close bracket equals i italic h over two pi over e B epsilon raised to the i j power period |
{ \cal L}= \frac {1}{4} \int d^{4} \theta { \cal K}+ \frac {1}{2} \left[ \int d^{2} \theta { \cal W}+ \mathrm {h.c.} \right] \; \;, | script L equals one fourth the integral of d to the fourth power theta script K plus one half times open bracket the integral of d squared theta script W plus normal h period normal c period close bracket comma |
\begin{array}{c}{{ \cal H}(z)= \left( \begin{array}{cc}{ \tilde { \hat {w}}(z)- \textstyle { \frac {1}{2}} \hat { \lambda }(z)+i \textstyle { \frac {1}{2}} \hat { \psi }(z)}&{2 \hat { \rho }_{b}(z)} \\ {0}&{ \hat {t}_{~b}^{a}(z)+i \textstyle { \frac {2}{N}} \hat { \psi }(z) \delta _{~b}^{a}} \\ \end{array} \right)} \\ ... | 3 lines Line 1: script H of z equals the 2 by 2 matrix Row 1: Column 1, w hat tilde of z minus one half lamda hat of z plus i one half psi hat of z Column 2, 2 rho hat sub b of z Row 2: Column 1, 0 Column 2, t hat sub b to the a-th power of z plus i 2 over N psi hat of z delta sub b to the a-th power Line 2: blank Line... |
\sigma _{1}= \left( \begin{array}{cc}{0}&{1} \\ {1}&{0} \\ \end{array} \right) \,, \sigma _{2}= \left( \begin{array}{cc}{0}&{-i} \\ {i}&{0} \\ \end{array} \right) \,, \sigma _{3}= \left( \begin{array}{cc}{1}&{0} \\ {0}&{-1} \\ \end{array} \right) \,. | sigma sub 1 equals the 2 by 2 matrix Row 1: 0 1 Row 2: 1 0 comma sigma sub 2 equals the 2 by 2 matrix Row 1: 0 negative i Row 2: i 0 comma sigma sub 3 equals the 2 by 2 matrix Row 1: 1 0 Row 2: 0 negative 1 period |
\Phi \rightarrow \Phi ^{ \prime }= \Phi , | normal Phi right arrow normal Phi prime equals normal Phi comma |
{ \cal L}_{W}(t, \partial _{x})=P_{W}(t, \partial _{x}) \partial _{x}P_{W}^{-1}(t, \partial _{x}). | script L sub W of open paren t comma partial differential sub x close paren equals P sub W of open paren t comma partial differential sub x close paren partial differential sub x P sub W inverse of open paren t comma partial differential sub x close paren period |
M= \frac {1}{2} \left[e^{-2 \phi } \sqrt { \frac {g_{2}}{g_{1}}} \frac {dg_{1}}{dr} \right]_{ \infty } \ , \S =2 \pi \left[e^{-2 \phi } \right]_{horizon}. | M equals one half times open bracket e raised to the negative 2 phi power the square root of the fraction with numerator g sub 2 and denominator g sub 1 d g sub 1 over d r close bracket sub normal infinity comma section sign equals 2 pi times open bracket e raised to the negative 2 phi power close bracket sub h of o r ... |
\partial _{ \mu } \langle j_{ \mu }(x_{1})j_{ \nu }(x_{2}) \rangle =C_{j} \partial _{ \nu } \Box ^{ \frac {D-2}{2}} \delta (x_{12}), | partial differential sub mu left angle bracket j sub mu of open paren x sub 1 close paren j sub nu of open paren x sub 2 close paren right angle bracket equals C sub j partial differential sub nu white medium square raised to the fraction with numerator D minus 2 and denominator 2 power delta of open paren x sub 12 clo... |
\vec { \mathrm { \bf ~f}}_{1}=(1,{ \frac {1}{ \sqrt {3}}}) \qquad \vec { \mathrm { \bf ~f}}_{2}=(0,{ \frac {2}{ \sqrt {3}}}) | bold f right arrow sub 1 equals open paren 1 comma the fraction with numerator 1 and denominator the square root of 3 close paren bold f right arrow sub 2 equals open paren 0 comma the fraction with numerator 2 and denominator the square root of 3 close paren |
W= \left( \zeta +a_{0} \frac {x_{8}^{4-n}z^{8-n}}{ \zeta } \right)+x^{2}+z^{5}-zy^{2}+c_{5}z^{4}+c_{4}z^{3}+c_{3}z^{2}+c_{2}z+c_{1}+ \cdots , | W equals open paren zeta plus a sub 0 x sub 8 raised to the 4 minus n power z raised to the 8 minus n power over zeta close paren plus x squared plus z to the fifth power minus z y squared plus c sub 5 z to the fourth power plus c sub 4 z cubed plus c sub 3 z squared plus c sub 2 z plus c sub 1 plus dot dot dot comma |
G( \omega )= \frac {1}{ \omega -H_{ \gamma }}= \frac {1}{T^{*}}. | G of omega equals the fraction with numerator 1 and denominator omega minus H sub gamma equals the fraction with numerator 1 and denominator T raised to the asterisk power period |
\hat {q}^{ \mu } \psi (q)= \int dq^{ \prime } \sqrt {g(q^{ \prime })} \left\langle q \right|q^{ \mu } \left|q^{ \prime } \right\rangle \left\langle q^{ \prime } \right. \left| \psi \right\rangle =q^{ \mu } \psi (q). | q hat to the mu-th power psi of q equals the integral of d q prime the square root of g of open paren q prime close paren left angle bracket q divides q to the mu-th power divides q prime right angle bracket left angle bracket q prime divides psi right angle bracket equals q to the mu-th power psi of q period |
\lambda ( \gamma _{1}) \lambda ( \gamma _{2}) \lambda ( \gamma _{1} \gamma _{2})^{-1}= \sigma ( \gamma _{1}, \gamma _{2}), \qquad \forall \gamma _{1}, \gamma _{2} \in \pi _{1}(X), | lamda of open paren gamma sub 1 close paren lamda of open paren gamma sub 2 close paren lamda of open paren gamma sub 1 gamma sub 2 close paren to the negative 1 power equals sigma of open paren gamma sub 1 comma gamma sub 2 close paren comma for all gamma sub 1 comma gamma sub 2 is a member of pi sub 1 of X comma |
\left[K_{ab} \right] \stackrel { \Sigma _{j}}{=}0 \quad \quad \quad \quad \quad \quad (j=1,2), | open bracket K sub a b close bracket equals sign with normal Sigma sub j over it 0 open paren j equals 1 comma 2 close paren comma |
\prod _{{v \in Vert( \Gamma )} \atop {val(v)=2} \quad {{}^{ \sharp }S_{v}}=0} \frac {1}{w_{F_{1}(v)}+w_{F_{2}(v)}} | the product over 2 lines Line 1: v is a member of V e r t of normal Gamma Line 2: v a l of v equals 2 raised to the normal ♯ power S sub v equals 0 of the fraction with numerator 1 and denominator w sub F sub 1 of v plus w sub F sub 2 of v |
C_{R} \delta _{ab}= \frac {Tr(T^{a}T^{b})(N^{2}-1)}{D_{R}}. | C sub R delta sub a b equals the fraction with numerator T r of open paren T to the a-th power T to the b-th power close paren times open paren N squared minus 1 close paren and denominator D sub R period |
<r|Q|s>= \int d^{d}x~ \Psi _{r}^{B \dagger }Q \Psi _{s}^{B} | is less than r times the absolute value of Q times s is greater than equals the integral of d to the d-th power x normal Psi sub r raised to the B dagger power Q normal Psi sub s to the B power |
\Omega _{ \cal Z}^{(c)}: \tilde {o}(z)~=~ \tilde {o}(T(z_{1}),z_{2}/T^{ \prime }(z_{1})) | normal Omega sub script Z raised to the c power colon o tilde of z equals o tilde of open paren T of open paren z sub 1 close paren comma z sub 2 divided by T prime of open paren z sub 1 close paren close paren |
ds^{2}= \alpha ^{ \prime } \left\{ \frac {U^{2}}{R^{2}} \left((1-U_{T}^{4}/U^{4})dt^{2}+ \sum _{i}dx_{i}^{2} \right)+ \frac {R^{2}}{U^{2}} \frac {dU^{2}}{1-U_{T}^{4}/U^{4}}+R^{2}d \Omega _{5}^{2} \right\}. | d s squared equals alpha prime times open brace the fraction with numerator U squared and denominator R squared times open paren open paren 1 minus U sub T to the fourth power divided by U to the fourth power close paren times d t squared plus the sum over i of d x sub i squared close paren plus the fraction with numer... |
B^{n} \equiv (S,S)^{n}= \delta _{KT}S^{n+1}+D^{n}(S^{0},...,S^{n}) \ , | B to the n-th power is identical to open paren S comma S close paren to the n-th power equals delta sub K T S raised to the n plus 1 power plus the n-th power of D of open paren S to the 0 power comma period period period comma S to the n-th power close paren comma |
V_{mass}= \frac {1}{2 \alpha ^{ \prime }e^{2}} \Phi ^{2}+ \frac {1}{2 \alpha ^{ \prime }} \left(A- \overline {{A}} \right)_{i} \left(A- \overline {{A}} \right)^{i} | V sub m a s s equals 1 over 2 alpha prime e squared normal Phi squared plus 1 over 2 alpha prime times open paren A minus A horizontal bar close paren sub i times open paren A minus A horizontal bar close paren to the i-th power |
\xi _{j}= \sqrt { \kappa }z_{j} \ , | xi sub j equals the square root of kappa z sub j comma |
Tr(L^{2})=2I_{ \Lambda }{ \cal H}, | T r of open paren L squared close paren equals 2 I sub normal Lamda script H comma |
V^{ \prime }(x)=x^{3}-4c_{1}x^{2}+2c_{2}x+8c_{1} \,, \quad V(0)=0 \,. | V prime of x equals x cubed minus 4 c sub 1 x squared plus 2 c sub 2 x plus 8 c sub 1 comma V of 0 equals 0 period |
\vec {b}_{DES}^{ \, \prime }=R_{z}( \phi ^{ \prime }-{ \frac { \pi }{2}})R_{x}({ \frac { \pi }{2}}- \theta ^{ \prime })R_{z}(2v)R_{z}(- \theta _{2}^{ \prime })R_{x}( \phi _{2}^{ \prime })R_{z}( \theta _{2}^{ \prime })R_{z}(v) \hat {x}, | b right arrow sub D E S prime equals R sub z of open paren phi prime minus pi over 2 close paren R sub x of open paren pi over 2 minus theta prime close paren R sub z of 2 v R sub z of open paren negative theta sub 2 prime close paren R sub x of open paren phi sub 2 prime close paren R sub z of open paren theta sub 2 p... |
S= \oint _{ \Sigma } \theta ^{a}n^{a}=- \oint _{ \Sigma } \varepsilon ^{abc}n^{a}F^{bc} | S equals the contour integral over normal Sigma of theta to the a-th power n to the a-th power equals negative the contour integral over normal Sigma of epsilon raised to the a b c power n to the a-th power the b-th c-th power of F |
\Omega _{ \phi ^{ \prime }}=-i \lambda ^{ \prime } \sqrt {2} \int d \tau d \theta \theta (-{ \frac {1}{4}}D^{2}X_{ \mu }+{ \frac {1}{4}} \bar {X}D \bar {X})X^{ \prime } | normal Omega sub phi prime equals negative i lamda prime the square root of 2 the integral of d tau d theta theta times open paren negative one fourth D squared X sub mu plus one fourth X bar D X bar close paren times X prime |
\epsilon _{1}+( \epsilon _{+})^{ \prime }=0. | epsilon sub 1 plus epsilon sub plus prime equals 0 period |
g= \left( \begin{array}{cc}{( \alpha ^{t})^{-1}}&{0} \\ {0}&{ \alpha } \\ \end{array} \right) | g equals the 2 by 2 matrix Row 1: Column 1, alpha to the t-th power to the negative 1 power Column 2, 0 Row 2: Column 1, 0 Column 2, alpha |
\partial _{ \chi }{ \cal H}=0,~~~~~~~~ \partial _{ \theta }{ \cal H}=0,~~~~~~~~~ \partial _{ \phi }{ \cal H}=0. | partial differential sub chi script H equals 0 comma partial differential sub theta script H equals 0 comma partial differential sub phi script H equals 0 period |
J_{0}=-1+ \frac {V^{ \prime } \left(- \left| \lambda \right|^{2} \right)}{ \left| \lambda \right|^{2}} \varphi \star \varphi ^{ \dagger }+ \left(1- \frac {1}{2}J_{ij} \star J^{ij} \right)^{1/2}. | J sub 0 equals negative 1 plus the fraction with numerator V prime of open paren negative the absolute value of lamda squared close paren and denominator the absolute value of lamda squared phi star phi raised to the dagger power plus open paren 1 minus one half J sub i j star J raised to the i j power close paren rais... |
2m^{I}- \widehat A_{IJ} \,m^{J}= \widehat C_{IJ} \,m^{J}~, | 2 m to the I power minus A hat sub I J times m to the J power equals C hat sub I J times m to the J power comma |
r_{n}= \sqrt {( \alpha _{n-1}+ \alpha _{n})/2}. | r sub n equals the square root of open paren alpha sub n minus 1 plus alpha sub n close paren divided by 2 period |
T^{ \pm }= \frac {1}{2}(a^{ \pm })^{2} | T raised to the plus or minus power equals one half times a raised to the plus or minus power squared |
\beta _{ \bar {g}}= \bar {g} \left( \gamma _{ \sigma }- \tilde { \gamma }_{ \sigma } \right) \equiv \bar {g} \left[ \gamma _{ \sigma }( \bar {g},M;N)- \gamma _{ \sigma }(1/ \bar {g},N;M) \right]. | beta sub g bar equals g bar of open paren gamma sub sigma minus gamma tilde sub sigma close paren is identical to g bar of open bracket gamma sub sigma of open paren g bar comma M semicolon N close paren minus gamma sub sigma of open paren 1 divided by g bar comma N semicolon M close paren close bracket period |
\frac {n}{T^{3}} \approx 4 \times 10^{-13} \,( \, \frac {g}{G_{F}m_{N}^{2}} \,)^{2} \, \frac {T}{M} \,. | the fraction with numerator n and denominator T cubed almost equals 4 times 10 to the negative 13 power times open paren the fraction with numerator g and denominator G sub F of m sub N squared close paren squared times T over M period |
\begin{array}{ll}{ \lambda _{ \theta }=-p_{n}i \sigma ^{n} \bar { \psi },}&{ \qquad \lambda _{ \bar { \theta }}=i \psi \sigma ^{n}p_{n},} \\ { \lambda _{ \rho }=-2p_{n}i \sigma ^{n} \lambda _{ \bar { \theta }} \approx 0,}&{ \qquad \lambda _{ \bar { \rho }}=2i \lambda _{ \theta } \sigma ^{n}p_{n} \approx 0.} \\ \end{arr... | 2 lines Line 1: lamda sub theta equals negative p sub n i sigma to the n-th power psi bar comma lamda sub theta bar equals i psi sigma to the n-th power p sub n comma Line 2: lamda sub rho equals negative 2 p sub n i sigma to the n-th power lamda sub theta bar almost equals 0 comma lamda sub rho bar equals 2 i lamda su... |
\mathrm {d}s^{2}={{ \hat {g}}_{ \alpha \beta }(r)} \mathrm {d}x^{ \alpha } \mathrm {d}x^{ \beta }+{ \mathrm {e}^{B(r)}} \mathrm {d} \vec {x} \cdot \mathrm {d} \vec {x}, | normal d s squared equals g hat sub alpha beta of r normal d x to the alpha-th power normal d x to the beta-th power plus normal e raised to the B of r power normal d x right arrow times normal d x right arrow comma |
S=- \frac {1}{4e^{2}} \int _{ \cal M}F_{ \mu \nu }^{I}F_{I}^{ \mu \nu }+ \frac {K_{IJ}}{8 \pi } \int _{ \cal M} \epsilon ^{ \mu \nu \lambda }A_{ \mu }^{I} \partial _{ \nu }A_{ \lambda }^{J} \,, | S equals negative 1 over 4 e squared the integral over script M of F sub mu nu to the I power of F sub I raised to the mu nu power plus the fraction with numerator K sub I J and denominator 8 pi the integral over script M of epsilon raised to the mu nu lamda power A sub mu to the I power partial differential sub nu A s... |
\alpha _{2}=- \frac { \alpha ^{2}}{m+ \alpha N},~~ \alpha _{3}=- \frac { \alpha m}{m+ \alpha N},~~~ \alpha \neq - \frac {m}{N},~~m= \pm 2. | alpha sub 2 equals negative the fraction with numerator alpha squared and denominator m plus alpha N comma alpha sub 3 equals negative the fraction with numerator alpha m and denominator m plus alpha N comma alpha is not equal to negative m over N comma m equals plus or minus 2 period |
H_{c}= \frac {1}{2} \int _{- \pi }^{ \pi }dx[ \{ \Pi _{i}(x,t)-T_{ij} \partial _{x} \phi _{i}(x,t) \}^{2}+( \partial _{x} \phi _{i}(x,t))^{2}] | H sub c equals one half the integral from negative pi to pi of d x times open bracket open brace normal Pi sub i of open paren x comma t close paren minus T sub i j partial differential sub x phi sub i of open paren x comma t close paren close brace squared plus open paren partial differential sub x phi sub i of open p... |
\mu \, \, \ll \, \,{ \frac {1}{R}} \, \, \ll \, \,{ \frac {1}{ \sqrt { \alpha ^{ \prime }}}} \,. | mu is much less than 1 over R is much less than the fraction with numerator 1 and denominator the square root of alpha prime period |
{ \frac { \partial ^{2} \Phi }{ \partial \sigma ^{2}}}+{ \frac { \partial ^{2} \Phi }{ \partial \tau ^{2}}}= \sqrt { \gamma }e^{ \displaystyle 2 \sqrt { \gamma } \Phi }. | the fraction with numerator partial differential squared normal Phi and denominator partial differential sigma squared plus the fraction with numerator partial differential squared normal Phi and denominator partial differential tau squared equals the square root of gamma e raised to the 2 the square root of gamma norm... |
\Gamma ^{(0,2)} \left(0;m,g \right)=m^{2} | the open paren 0 comma second close paren power of normal Gamma of open paren 0 semicolon m comma g close paren equals m squared |
\frac {1}{1-(F/E)^{-2c}}< \frac {-1}{2c} \left[ \frac {1}{1-(F/E)} \right] \,. | the fraction with numerator 1 and denominator 1 minus open paren F of slash E close paren raised to the negative 2 c power is less than negative 1 over 2 c times open bracket the fraction with numerator 1 and denominator 1 minus open paren F of slash E close paren close bracket period |
\left( \partial _{z}^{2}+{ \frac {1}{2}}T^{F}(z) \right) \partial _{ \bar {z}}^{l}e^{- \varphi /2}=0, \qquad l=0,1, \ldots . | open paren partial differential sub z squared positive one half the F power of T of z close paren partial differential sub z bar to the l-th power e raised to the negative phi divided by 2 power equals 0 comma l equals 0 comma 1 comma dot dot dot period |
2^{r+3} \mathrm {Im} \left[ \frac {(z_{1}z_{2})^{r}}{r!r!} \int _{0}^{ \infty }e^{i \beta \mu \xi }(1-e^{-r \xi }) \sum _{n=0}^{ \infty }<n| \lambda ^{r}|n+r>^{2}e^{-(n+1/2) \xi } \frac {r}{p^{2}-r^{2}} \right]. | 2 raised to the r plus 3 power the imaginary of open bracket the fraction with numerator open paren z sub 1 z sub 2 close paren to the r-th power and denominator r factorial r factorial the integral from 0 to normal infinity of e raised to the i beta mu xi power times open paren 1 minus e raised to the negative r xi po... |
\partial _{ \nu }L^{ \mu \nu }(x)=0 \quad \mathrm {if~x~ \ne ~0~}. | partial differential sub nu the mu-th nu-th power of L of x equals 0 if normal x is not equal to 0 period |
\delta G_{ \mu \nu }=G_{ \mu \nu } \star a_{h}-a_{h} \star G_{ \mu \nu }. | delta G sub mu nu equals G sub mu nu star a sub h minus a sub h star G sub mu nu period |
A(z) \,B(w)= \,:A(z) \,B(w):+<A(z) \,B(w)> \, \,, | A of z B of w equals colon A of z B of w colon plus is less than A of z B of w is greater than comma |
S={ \frac {a^{2}}{4 \pi }} \int _{0}^{ \infty }s(p)dp^{2}={ \frac { \cal A}{4G}}~, | S equals the fraction with numerator a squared and denominator 4 pi the integral from 0 to normal infinity of s of p d p squared equals script A over 4 G comma |
H_{ \mathrm {LC}}^{ \mathrm {eff}}( \omega ):= \hat {P}H_{ \mathrm {LC}} \hat {P}+ \hat {P}H_{ \mathrm {LC}} \hat {Q}( \omega -H_{ \mathrm {LC}})^{-1} \hat {Q}H_{ \mathrm {LC}} \hat {P} | H sub LC raised to the eff power of omega colon equals P hat H sub LC of P hat plus P hat H sub LC of Q hat times open paren omega minus H sub LC close paren to the negative 1 power times Q hat H sub LC of P hat |
g_{0}-g_{ \theta }= \int _{0}^{ \infty }M^{4}{ \cal L}({ \cal G}_{0}-{ \cal G}_{ \theta })dx \equiv 12 \int _{0}^{ \infty } \biggl (M^{4}{ \cal L}m^{2}( \ell -m) \biggl )^{ \prime }dx, | g sub 0 minus g sub theta equals the integral from 0 to normal infinity of M to the fourth power script L times open paren script G sub 0 minus script G sub theta close paren d x is identical to 12 the integral from 0 to normal infinity of open paren M to the fourth power script L m squared times open paren script l mi... |
ds_{string}^{2}=ds_{r}^{2}+| \lambda |^{2} \tilde { \lambda } \otimes \tilde { \lambda }. | d s sub s t r i n g squared equals d s sub r squared plus the absolute value of lamda squared lamda tilde circled times lamda tilde period |
0=2 \,V_{[ \check {E}, \check {F}]}( \tau , \vec { \sigma }) \,U^{ \check {F}}( \tau , \vec { \sigma })+T( \tau , \vec { \sigma }) \,s_{, \check {E}}( \tau , \vec { \sigma }), | 0 equals 2 V sub open bracket E ˇ comma F ˇ close bracket of open paren tau comma sigma right arrow close paren the F ˇ power of U of open paren tau comma sigma right arrow close paren plus T of open paren tau comma sigma right arrow close paren s sub comma E ˇ of open paren tau comma sigma right arrow close paren comm... |
e^{ik_{1} \cdot X}e^{ik_{2} \cdot X} \sim \left( \tau - \tau ^{ \prime } \right)^{2 \alpha ^{ \prime }g^{ \mu \nu }k_{1 \mu } \cdot k_{2 \nu }} \times e^{i \left(k_{1}+k_{2} \right) \cdot X}+ \cdots . | e raised to the i k sub 1 times X power e raised to the i k sub 2 times X power tilde open paren tau minus tau prime close paren raised to the exponent 2 alpha prime the mu-th nu-th power of g of k sub 1 mu times k sub 2 nu end exponent times e raised to the i times open paren k sub 1 plus k sub 2 close paren times X p... |
\tilde {u}_{ \tilde { \omega }km}= \frac {1}{ \sqrt {8 \pi ^{2}}}e^{-i \tilde { \omega }t_{+}+im \varphi _{+}+ikz}J_{m} \left( \sqrt {( \tilde { \omega }+m \Omega )^{2}-k^{2}}R \right), | u tilde sub omega tilde k m equals the fraction with numerator 1 and denominator the square root of 8 pi squared e raised to the negative i omega tilde t sub plus plus i m phi sub plus plus i k z power J sub m of open paren the square root of open paren omega tilde plus m normal Omega close paren squared minus k square... |
{ \cal L}=-{ \frac {1}{2}}Tr \big (F_{ \mu \nu }F^{ \mu \nu } \big )-2Tr \big ( \lambda nA \big ). | script L equals negative one half T r of F sub mu nu of the mu-th nu-th power of F minus 2 T r of open paren lamda n A close paren period |
F=-f( \lambda ,N) \frac { \pi ^{2}}{6}N^{2}T^{4}V | F equals negative f of open paren lamda comma N close paren the fraction with numerator pi squared and denominator 6 N squared T to the fourth power V |
P_{ \alpha \beta }= \lambda _{ \alpha } \lambda _{ \beta } | P sub alpha beta equals lamda sub alpha lamda sub beta |
\alpha ( \tilde { \beta }^{b} \tilde { \gamma }) \alpha ^{-1}=( \tilde { \beta }^{b+ \frac {2k^{ \prime }}{ \delta }} \tilde { \gamma }), | alpha of open paren beta tilde to the b-th power gamma tilde close paren alpha inverse equals open paren beta tilde raised to the exponent b plus 2 k prime over delta end exponent gamma tilde close paren comma |
\xi _{I}+ \eta _{I}=2 \xi _{II}=2 \eta _{II}, | xi sub I plus eta sub I equals 2 xi sub I I equals 2 eta sub I I comma |
\Upsilon (x)-D_{ \Box }^{2} \left( \Psi ^{2} \right)(x)=0, | normal Upsilon of x minus D sub white medium square squared of open paren normal Psi squared close paren times x equals 0 comma |
ds^{2}= \frac {1}{ \eta ^{2 \alpha }} \left(-d \eta ^{2}+ \sum _{i=1}^{D-1}(dx^{i})^{2} \right) \ , | d s squared equals the fraction with numerator 1 and denominator eta raised to the 2 alpha power times open paren negative d eta squared plus the sum from i equals 1 to D minus 1 of open paren d x to the i-th power close paren squared close paren comma |
\left(f(x)+f(x+a) \right)_{x}+f^{2}(x)-f^{2}(x+a)= \mu . | open paren f of x plus f of open paren x plus a close paren close paren sub x plus f squared of x minus f squared of open paren x plus a close paren equals mu period |
\mathrm {{Const}}= \int _{r}^{ \infty }{ \frac {ds}{ \sqrt {s^{4}+a^{4}}}}+ \int _{0}^{r}{ \frac {ds}{ \sqrt {s^{4}+a^{4}}}}= \frac {1}{a} \mathrm {K}( \frac {1}{2}) | Const equals the integral from r to normal infinity of the fraction with numerator d s and denominator the square root of s to the fourth power plus a to the fourth power plus the integral from 0 to r of the fraction with numerator d s and denominator the square root of s to the fourth power plus a to the fourth power ... |
(A_{1},A_{2},A_{3},A_{0})=(A_{1},A_{2},0,0)+ \lambda ^{ \prime }(0,0, \omega , \omega ), | open paren A sub 1 comma A sub 2 comma A sub 3 comma A sub 0 close paren equals open paren A sub 1 comma A sub 2 comma 0 comma 0 close paren plus lamda prime of open paren 0 comma 0 comma omega comma omega close paren comma |
\{{ \Psi }_{ai}^{(n+1)},H_{C} \}=(-1)^{n+1}{ \epsilon }_{ij}{ \epsilon }_{abc} \partial _{b}{ \Psi }_{cj}^{(n+1)} \approx 0 \ , | the set normal Psi sub a i raised to the open paren n plus 1 close paren power comma H sub C equals open paren negative 1 close paren raised to the n plus 1 power times epsilon sub i j epsilon sub a b c partial differential sub b normal Psi sub c j raised to the open paren n plus 1 close paren power almost equals 0 com... |
x(u, \xi )x^{(1/2)}(-u, \xi )+x(-u, \xi )x^{(1/2)}(u, \xi )=-2 \wp (u)+g( \xi ), | x of open paren u comma xi close paren the open paren first divided by second close paren power of x of open paren negative u comma xi close paren plus x of open paren negative u comma xi close paren the open paren first divided by second close paren power of x of open paren u comma xi close paren equals negative 2 nor... |
\Pi _{ \mu \nu }^{(1) \,AB}(p)= \Pi _{ \mu \nu }^{(1) \,AB(ghost)}(p)+ \Pi _{ \mu \nu }^{(1) \,(gauge)}(p)=0 | normal Pi sub mu nu raised to the 1 times A B power of p equals normal Pi sub mu nu raised to the 1 times A B of g of h of o s t power of p plus normal Pi sub mu nu raised to the 1 times open paren g of a u g of e close paren power of p equals 0 |
\hat {Q}_{1}= \int _{S^{2}}*dV \ ; | Q hat sub 1 equals the integral over S squared of asterisk d V semicolon |
\partial P= \partial _{i}Pdz_{i}, \quad \bar { \partial }P= \bar { \partial }_{i}Pd \bar {z}_{i}. | partial differential P equals partial differential sub i P d z sub i comma partial differential sign with bar over it P equals partial differential sign with bar over it sub i P d z bar sub i period |
u= \gamma (z^{ \frac {2}{ \gamma -2}} \psi -1) | u equals gamma times open paren z raised to the fraction with numerator 2 and denominator gamma minus 2 power psi minus 1 close paren |
x(2u, \xi )x_{d}(-u, \xi )+x(-2u, \xi )x_{d}(u, \xi )=- \wp (u)+ \wp ( \xi ), | x of open paren 2 u comma xi close paren x sub d of open paren negative u comma xi close paren plus x of open paren negative 2 u comma xi close paren x sub d of open paren u comma xi close paren equals negative normal script cap P of u plus normal script cap P of xi comma |
S_{ij}( \theta )S_{ji}(- \theta )=1 \quad , | S sub i j of theta S sub j i of negative theta equals 1 comma |
V_{int}= \int \frac { \lambda (a)}{4!} \phi _{a}^{4}(b)d \mu (a,b), \quad \lambda (a) \sim a^{ \nu }. | V sub i n t equals the integral of the fraction with numerator lamda of a and denominator 4 factorial phi sub a to the fourth power of b d mu of open paren a comma b close paren comma lamda of a tilde a to the nu-th power period |
\frac { \delta S_{eff}}{ \delta A^{ \mu }}= \langle J_{ \mu } \rangle , | delta S sub e f of f over delta A to the mu-th power equals left angle bracket J sub mu right angle bracket comma |
4 \pi G_{4} \delta \rho =-3 \dot { \alpha }_{0}^{2} \Phi _{0}, | 4 pi G sub 4 of delta rho equals negative 3 alpha dot above sub 0 squared normal Phi sub 0 comma |
X(u)= \frac {x_{1}}{2((b,c))}(be+B(u) \bar {e}b). | X of u equals the fraction with numerator x sub 1 and denominator 2 times open paren open paren b comma c close paren close paren times open paren b e plus B of u e bar b close paren period |
V(x)= \sum _{m=0}^{ \infty } \frac {V^{(m)}(q)}{m!}(x-q)^{m}= \sum _{m=0}^{ \infty }C_{m}(q)(x-q)^{m} | V of x equals the sum from m equals 0 to normal infinity of the fraction with numerator the m-th power of V of q and denominator m factorial times open paren x minus q close paren to the m-th power equals the sum from m equals 0 to normal infinity of C sub m of q times open paren x minus q close paren to the m-th power |
S= \int d^{2}{ \xi } \sqrt {g} \ g^{ab}{ \cal D}_{a}x_{ \mu } \! \left(T-s{ \cal D}^{2}+{ \frac {1}{M^{2}}}{ \cal D}^{4} \right) \ { \cal D}_{b}x_{ \mu } \ , | S equals the integral of d squared xi the square root of g the a-th b-th power of g of script D sub a x sub mu times open paren T minus s script D squared plus the fraction with numerator 1 and denominator M squared script D to the fourth power close paren script D sub b x sub mu comma |
\overline {{V_{A}^{a}}}=V_{ \dot {A}a}, \overline {{V^{Aa}}}=-V_{a}^{ \dot {A}}, \overline {{V_{Aa}}}=-V_{ \dot {A}}^{a}, \overline {{V_{a}^{A}}}=V^{ \dot {A}a}, \overline {{ \varepsilon _{ab}}}= \varepsilon ^{ab}, \overline {{ \chi ^{Aa} \psi ^{Bb}}}=- \overline {{ \chi ^{Aa}}} \ \overline {{ \psi ^{Bb}}} | V sub A to the a-th power horizontal bar equals V sub A dot above a comma V raised to the A a power horizontal bar equals negative V sub a raised to the A dot above power comma V sub A a horizontal bar equals negative V sub A dot above to the a-th power comma V sub a to the A power horizontal bar equals V raised to the... |
ds_{ \mathrm {inv}}^{2}= \sum _{i,j}g_{ij}g_{ij}dg^{ii}dg^{jj}+ \frac {1}{2} \sum _{i<j;k<l} \left(g_{ik}g_{jl}+g_{il}g_{jk} \right)dg^{ij}dg^{kl}+2 \sum _{i;k<l}g_{ik}g_{il}dg^{ii}dg^{kl} | d s sub inv squared equals the sum over i comma j of g sub i j of g sub i j of d the i-th i-th power of g of d the j-th j-th power of g plus one half the sum over i is less than j semicolon k is less than l of open paren g sub i k of g sub j l plus g sub i l of g sub j k close paren times d the i-th j-th power of g of ... |
\delta [Z_{D}^{ \prime }(0)]= \gamma [ \delta c_{0})]+ \mathrm {Finite}_{ \epsilon \to 0}[ \int d^{2}x \,2 \delta \sigma (x)K_{D}(x,x; \epsilon )- \sum _{ \nu : \, \lambda _{ \nu }=0} \int _{D}d^{2}x \,2 \delta \sigma (x)| \Psi _{ \nu }(x)|^{2}] \,. | delta of Z sub D prime of 0 equals gamma of delta of c sub 0 close bracket positive Finite sub epsilon right arrow 0 times open bracket the integral of d squared x times 2 delta of sigma of x K sub D of open paren x comma x semicolon epsilon close paren minus the sum over nu colon lamda sub nu equals 0 of the integral ... |
\Phi _{n}( \lambda ;l)= \Phi _{n}( \lambda ;l+1)+V_{n-1}(l) \Phi _{n-1}( \lambda ;l+1), | normal Phi sub n of open paren lamda semicolon l close paren equals normal Phi sub n of open paren lamda semicolon l plus 1 close paren plus V sub n minus 1 of l normal Phi sub n minus 1 of open paren lamda semicolon l plus 1 close paren comma |
\psi ^{-}= \Omega ^{-} \psi ^{out} | psi raised to the minus power equals normal Omega raised to the minus power psi raised to the o u t power |
i \Pi (q)=-4 \int \frac {d^{4}k}{(2 \pi )^{4}} \frac {k^{2}+k^{ \mu }q_{ \mu }-m^{2}}{[(k+q)^{2}-m^{2}][k^{2}-m^{2}]}. | i normal Pi of q equals negative 4 the integral of the fraction with numerator d to the fourth power k and denominator open paren 2 pi close paren to the fourth power the fraction with numerator k squared plus k to the mu-th power q sub mu minus m squared and denominator open bracket open paren k plus q close paren squ... |
H(N)= \widehat {H}(N) | H of N equals H hat of N |
\frac {d \lambda }{d \tau }= \left( \sum _{ \mu , \nu =0}^{3}g_{ \mu \nu } \frac {dx^{ \mu }}{d \lambda } \frac {dx^{ \nu }}{d \lambda } \right)^{-1/2}. | d lamda over d tau equals open paren the sum from mu comma nu equals 0 to 3 of g sub mu nu of d x to the mu-th power over d lamda d x to the nu-th power over d lamda close paren raised to the exponent negative 1 divided by 2 end exponent period |
\delta \varphi (x)~=~ \varphi ^{ \prime }(x)- \varphi (x) \quad , | delta phi of x equals phi prime of x minus phi of x comma |
\varphi ^{ \alpha }(t) \rightarrow \varphi ^{ \alpha }(t)+2 \pi n^{ \alpha } | the alpha-th power of phi of t right arrow the alpha-th power of phi of t plus 2 pi n to the alpha-th power |
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