AlFrauch/im2latex · Datasets at Hugging Face

image imagewidth (px) 20 14k	text stringlengths 1 18.8k
	\displaystyle\tilde{\Delta}_{n}(\lambda)
	\displaystyle F_{3,l}
	\displaystyle x^{d}\eta_{0}x^{-d}=\eta_{0},\ \ x^{d}\xi_{0}x^{-d}=\xi_{0},
	\gamma^{u}_{k}[m^{u},n^{u}]=g^{u}_{k}[m^{u}]p^{u}_{k}[m^{u},n^{u}],
	\displaystyle\mathrm{C3:}s_{k}^{u}[m^{u},\tau+o]+s^{d}_{k}[m^{d},n^{d}]\leq 1,
	\displaystyle\mathrm{C4:}s^{d}_{k}[m^{d},n^{d}]=0,\forall n^{d}\geq D_{k}-\tau.
	\displaystyle\mbox{C15}:\bar{p}_{k}^{u}[m^{u},n^{u}]\geq p_{k}^{u}[m^{u},n^{u}]
	\displaystyle\mbox{C19}:\bar{p}_{k}^{d}[m^{d},n^{d}]\geq p_{k}^{d}[m^{d},n^{d}]
	\displaystyle\quad\;\;\mathrm{C12,C13-C20.}
	\displaystyle\mathrm{C6b:}E^{u}(\mathbf{s}^{{u}})-H^{u}(\mathbf{s}^{{u}})\leq 0,
	\displaystyle\;\qquad\mathrm{C3-C5,C6a,C7-C9,C10a,C11-C20}.
	\displaystyle-\frac{1}{\pi}\partial_{\lambda}\delta_{n}(\lambda)
	\mathcal{L}_{\textrm{int}}=a\sum_{\psi}\bar{\psi}\left(g_{\psi}^{s}+ig_{\psi}^{p}\gamma_{5}\right)\psi\ .
	\delta E=g_{e}^{s}g_{e}^{p}\Omega W_{\textrm{ax}}^{(ee)}(m_{a}),
	\delta E=\bar{g}_{N}^{s}g_{e}^{p}\Omega W_{\textrm{ax}}^{(eN)}(m_{a}).
	\Psi=e^{\hat{T}}\Phi_{0}.
	\hat{T}=\hat{T}_{1}+\hat{T}_{2}+\hat{T}_{3}+\dots,
	\displaystyle a_{n}(\lambda)
	F_{m}(T)=\int_{0}^{1}dtt^{2m}e^{-Tt^{2}}.
	G_{m}(T,U)=\int_{0}^{1}dtt^{2m}e^{-Tt^{2}+U(1-\frac{1}{t^{2}})}.
	G_{-1}=\frac{e^{-T}}{4}\sqrt{\frac{\pi}{U}}\left[e^{k^{2}}\textnormal{erfc}(k)+e^{\lambda^{2}}\textnormal{erfc}(\lambda)\right],
	G_{0}=\frac{e^{-T}}{4}\sqrt{\frac{\pi}{T}}\left[e^{k^{2}}\textnormal{erfc}(k)-e^{\lambda^{2}}\textnormal{erfc}(\lambda)\right],
	k=-\sqrt{T}+\sqrt{U},
	\lambda=\sqrt{T}+\sqrt{U},
	G_{m}=\frac{1}{2T}[(2m-1)G_{m-1}+2UG_{m-2}-e^{-T}].
	G_{0}=1-e^{U}\sqrt{\pi U}\textnormal{erfc}(\sqrt{U}).
	G_{m}(0,U)=\frac{1}{2m+1}[1-2UG_{m-1}(0,U)].
	G_{m}(T,U)=\sum_{k=0}^{\infty}\frac{(-T)^{k}}{k!}G_{m+k}(0,U),
	\displaystyle\frac{2n}{\pi}\frac{1}{(n^{2}+4\lambda^{2})};
	\delta E=d_{\mathrm{e}}\Omega W_{d},
	\delta E\approx 23~{}\mu{\rm Hz}
	\delta E\approx\frac{\bar{g}_{N}^{s}g_{e}^{p}}{m_{a}^{2}}\Omega\widetilde{W}^{eN},
	\widetilde{W}^{eN}=\lim_{m_{a}\rightarrow+\infty}m_{a}^{2}W_{\textrm{ax}}^{(eN)}(m_{a}).
	\|\widetilde{W}^{eN}\|\approx 5.87\times 10^{-11}\ \textrm{GeV}^{2}\times\frac{m_{e}c}{\hbar}.
	\widetilde{W}^{ee}=\lim_{m_{a}\rightarrow+\infty}m_{a}^{2}W_{\textrm{ax}}^{(ee)}(m_{a}).
	\|\widetilde{W}^{ee}\|\approx 1.1\times 10^{-14}\ \textrm{GeV}^{2}\times\frac{m_{e}c}{\hbar}.
	\hat{f}\ni f\hat{\psi}_{16}\hat{\psi}_{16}\hat{\phi}_{10}+\cdots
	m_{16},\ m_{10},\ M_{D}^{2},\ m_{1/2},\ A_{0},\ \tan\beta,\ sign(\mu)
	\displaystyle A_{nm}(\lambda)
	R=\frac{max(f_{t},f_{b},f_{\tau})}{min(f_{t},f_{b},f_{\tau})},
	A_{0}\sim-2\,m_{16},\ m_{10}\sim 1.2\,m_{16},
	\Omega_{\rm DM}h^{2}=0.111^{+0.011}_{-0.015}\ \ (2\sigma).
	4\,m_{16}^{2}=2\,m_{10}^{2}=A_{0}^{2}
	P(X^{t+1}=x\|X^{t}=x^{t},...,X^{1}=x^{1})=P(X^{t+1}=x\|X^{t}=x_{t}).
	p=\frac{P(x^{c})Q(x^{t};x^{c})}{P(x^{t})Q(x^{c};x^{t})},
	\chi^{2}_{R}=\left(\frac{R(x)-R_{unification}}{\sigma_{R}}\right)^{2}
	m_{16},\ m_{1/2},\ A_{0},\ \tan\beta,\ m_{A},\ \mu
	\displaystyle P_{n+1}(x)
	\displaystyle=P_{n}(x)+x^{2^{n}}Q_{n}(x),
	\displaystyle\delta_{nm}\delta(\lambda)+a_{\|n-m\|}(\lambda)
	\displaystyle Q_{n+1}(x)
	\displaystyle=P_{n}(x)-x^{2^{n}}Q_{n}(x).
	r(2n)=r(n),\qquad r(2n+1)=(-1)^{n}r(n).
	\displaystyle P_{n+2s}(\omega)-C_{s}(\omega)P_{n+s}(\omega)+(-2)^{s}P_{n}(\omega)
	\displaystyle Q_{n+2s}(\omega)-C_{s}(\omega)Q_{n+s}(\omega)+(-2)^{s}Q_{n}(\omega)
	t(2n)=t(n),\qquad t(2n+1)=-t(n).
	T(n;x)=\sum_{m=0}^{n-1}t(m)x^{m}
	A(n;x)=\sum_{m=0}^{n-1}a(m)x^{m},
	t(n)=(-1)^{s_{2}(n)},
	T(2^{s}n;\omega)=T(2^{s};\omega)T(n;\omega).
	\rho(q)=\rho_{p}(q)+\rho_{h}(q).
	T(2^{ms_{0}};\omega)=(T(2^{s_{0}};\omega))^{m}.
	\displaystyle M(k^{t};x)
	\displaystyle M^{R}(k^{t};x)
	\displaystyle m_{ij}(k^{t};x)
	\displaystyle m^{R}_{ij}(k^{t};x)
	\displaystyle M(k^{t};x)=\sum_{w\in\Sigma_{k}^{t}}M_{w}x^{[w]_{k}},
	\displaystyle M^{R}(k^{t};x)=\sum_{w\in\Sigma_{k}^{t}}M_{w}x^{[w^{R}]_{k}}.
	\displaystyle\rho_{p}(q)
	f_{p}=\sum_{j=0}^{c}\alpha_{pj}f_{j},
	\widehat{m}_{ij}(x)=m_{ij}(x)+\sum_{p=c+1}^{d-1}\alpha_{pj}m_{ip}(x).
	\displaystyle\widehat{M}(k^{t};x)
	\displaystyle\widehat{M}^{R}(k^{t};x)
	\widehat{M}(k^{t};x)=\sum_{m=0}^{k^{t}-1}\widehat{M}_{m}x^{m},
	\widehat{M}(n;x)=\sum_{m=0}^{n-1}\widehat{M}_{m}x^{m}.
	\sum_{m=0}^{l}C_{m}(\omega)A(k^{ms}n;\omega)=0,
	\displaystyle F_{i}(n;x)
	\displaystyle F_{i}^{R}(n;x)
	\displaystyle\widehat{F}(n;x)
	\displaystyle\Theta(B-q)\rho(q);
	\displaystyle\widehat{F}^{R}(n;x)
	\displaystyle\widehat{F}(k^{u}n;x)
	\displaystyle=\widehat{M}(n;x^{k^{u}})\widehat{F}(k^{u};x),
	\displaystyle\widehat{F}^{R}(k^{u}n;x)
	\displaystyle=\widehat{M}^{R}(k^{u};x)\widehat{F}^{R}(n;x^{k^{u}}).
	C(x,y)=\sum_{m=0}^{l}C_{m}(x)y^{m}.
	\sum_{m=0}^{l}C_{m}(\omega)A(k^{ms}n;\omega)=0.
	\displaystyle C(x,\lambda)=\lambda^{2}-(x^{3}+x^{2}+x+1)\lambda+4x^{3},
	\displaystyle R(2^{4}n;1)-4R(2^{2}n;1)+4R(n;1)
	\displaystyle\rho_{h}(q)
	\displaystyle R(2^{4}n;\omega)-R(2^{2}n;\omega)+4R(n;\omega)
	R(2^{2s}n;\omega)-C_{s}(\omega)R(2^{s}n;\omega)+(-2)^{s}R(n;\omega)=0,
	B(2^{2s}n;\omega)-C_{s}(\omega)B(2^{s}n;\omega)+(-1)^{s}B(n;\omega)=0,
	\widetilde{t}(n)=s_{2}(n)\bmod{2},
	\widetilde{T}(n;x)=\sum_{m=0}^{n-1}\widetilde{t}(m)x^{m}.
	2\widetilde{T}(n;x)=\frac{x^{n}-1}{x-1}-T(n;x).
	\displaystyle\widetilde{T}(2^{s}n;\omega)
	\displaystyle=C\widetilde{T}(n;\omega)+\frac{(1-C)(\omega^{n}-1)}{2(\omega-1)}.
	\widetilde{T}(2^{s}n;\omega)=\widetilde{C}\widetilde{T}(n;\omega),
	\displaystyle\Theta(q-B)\rho(q).

\displaystyle\tilde{\Delta}_{n}(\lambda)

\displaystyle F_{3,l}

\displaystyle x^{d}\eta_{0}x^{-d}=\eta_{0},\ \ x^{d}\xi_{0}x^{-d}=\xi_{0},

\gamma^{u}_{k}[m^{u},n^{u}]=g^{u}_{k}[m^{u}]p^{u}_{k}[m^{u},n^{u}],

\displaystyle\mathrm{C3:}s_{k}^{u}[m^{u},\tau+o]+s^{d}_{k}[m^{d},n^{d}]\leq 1,

\displaystyle\mathrm{C4:}s^{d}_{k}[m^{d},n^{d}]=0,\forall n^{d}\geq D_{k}-\tau.

\displaystyle\mbox{C15}:\bar{p}_{k}^{u}[m^{u},n^{u}]\geq p_{k}^{u}[m^{u},n^{u}]

\displaystyle\mbox{C19}:\bar{p}_{k}^{d}[m^{d},n^{d}]\geq p_{k}^{d}[m^{d},n^{d}]

\displaystyle\quad\;\;\mathrm{C12,C13-C20.}

\displaystyle\mathrm{C6b:}E^{u}(\mathbf{s}^{{u}})-H^{u}(\mathbf{s}^{{u}})\leq 0,

\displaystyle\;\qquad\mathrm{C3-C5,C6a,C7-C9,C10a,C11-C20}.

\displaystyle-\frac{1}{\pi}\partial_{\lambda}\delta_{n}(\lambda)

\mathcal{L}_{\textrm{int}}=a\sum_{\psi}\bar{\psi}\left(g_{\psi}^{s}+ig_{\psi}^{p}\gamma_{5}\right)\psi\ .

\delta E=g_{e}^{s}g_{e}^{p}\Omega W_{\textrm{ax}}^{(ee)}(m_{a}),

\delta E=\bar{g}_{N}^{s}g_{e}^{p}\Omega W_{\textrm{ax}}^{(eN)}(m_{a}).

\Psi=e^{\hat{T}}\Phi_{0}.

\hat{T}=\hat{T}_{1}+\hat{T}_{2}+\hat{T}_{3}+\dots,

\displaystyle a_{n}(\lambda)

F_{m}(T)=\int_{0}^{1}dtt^{2m}e^{-Tt^{2}}.

G_{m}(T,U)=\int_{0}^{1}dtt^{2m}e^{-Tt^{2}+U(1-\frac{1}{t^{2}})}.

G_{-1}=\frac{e^{-T}}{4}\sqrt{\frac{\pi}{U}}\left[e^{k^{2}}\textnormal{erfc}(k)+e^{\lambda^{2}}\textnormal{erfc}(\lambda)\right],

G_{0}=\frac{e^{-T}}{4}\sqrt{\frac{\pi}{T}}\left[e^{k^{2}}\textnormal{erfc}(k)-e^{\lambda^{2}}\textnormal{erfc}(\lambda)\right],

k=-\sqrt{T}+\sqrt{U},

\lambda=\sqrt{T}+\sqrt{U},

G_{m}=\frac{1}{2T}[(2m-1)G_{m-1}+2UG_{m-2}-e^{-T}].

G_{0}=1-e^{U}\sqrt{\pi U}\textnormal{erfc}(\sqrt{U}).

G_{m}(0,U)=\frac{1}{2m+1}[1-2UG_{m-1}(0,U)].

G_{m}(T,U)=\sum_{k=0}^{\infty}\frac{(-T)^{k}}{k!}G_{m+k}(0,U),

\displaystyle\frac{2n}{\pi}\frac{1}{(n^{2}+4\lambda^{2})};

\delta E=d_{\mathrm{e}}\Omega W_{d},

\delta E\approx 23~{}\mu{\rm Hz}

\delta E\approx\frac{\bar{g}_{N}^{s}g_{e}^{p}}{m_{a}^{2}}\Omega\widetilde{W}^{eN},

\widetilde{W}^{eN}=\lim_{m_{a}\rightarrow+\infty}m_{a}^{2}W_{\textrm{ax}}^{(eN)}(m_{a}).

|\widetilde{W}^{eN}|\approx 5.87\times 10^{-11}\ \textrm{GeV}^{2}\times\frac{m_{e}c}{\hbar}.

\widetilde{W}^{ee}=\lim_{m_{a}\rightarrow+\infty}m_{a}^{2}W_{\textrm{ax}}^{(ee)}(m_{a}).

|\widetilde{W}^{ee}|\approx 1.1\times 10^{-14}\ \textrm{GeV}^{2}\times\frac{m_{e}c}{\hbar}.

\hat{f}\ni f\hat{\psi}_{16}\hat{\psi}_{16}\hat{\phi}_{10}+\cdots

m_{16},\ m_{10},\ M_{D}^{2},\ m_{1/2},\ A_{0},\ \tan\beta,\ sign(\mu)

\displaystyle A_{nm}(\lambda)

R=\frac{max(f_{t},f_{b},f_{\tau})}{min(f_{t},f_{b},f_{\tau})},

A_{0}\sim-2\,m_{16},\ m_{10}\sim 1.2\,m_{16},

\Omega_{\rm DM}h^{2}=0.111^{+0.011}_{-0.015}\ \ (2\sigma).

4\,m_{16}^{2}=2\,m_{10}^{2}=A_{0}^{2}

P(X^{t+1}=x|X^{t}=x^{t},...,X^{1}=x^{1})=P(X^{t+1}=x|X^{t}=x_{t}).

p=\frac{P(x^{c})Q(x^{t};x^{c})}{P(x^{t})Q(x^{c};x^{t})},

\chi^{2}_{R}=\left(\frac{R(x)-R_{unification}}{\sigma_{R}}\right)^{2}

m_{16},\ m_{1/2},\ A_{0},\ \tan\beta,\ m_{A},\ \mu

\displaystyle P_{n+1}(x)

\displaystyle=P_{n}(x)+x^{2^{n}}Q_{n}(x),

\displaystyle\delta_{nm}\delta(\lambda)+a_{|n-m|}(\lambda)

\displaystyle Q_{n+1}(x)

\displaystyle=P_{n}(x)-x^{2^{n}}Q_{n}(x).

r(2n)=r(n),\qquad r(2n+1)=(-1)^{n}r(n).

\displaystyle P_{n+2s}(\omega)-C_{s}(\omega)P_{n+s}(\omega)+(-2)^{s}P_{n}(\omega)

\displaystyle Q_{n+2s}(\omega)-C_{s}(\omega)Q_{n+s}(\omega)+(-2)^{s}Q_{n}(\omega)

t(2n)=t(n),\qquad t(2n+1)=-t(n).

T(n;x)=\sum_{m=0}^{n-1}t(m)x^{m}

A(n;x)=\sum_{m=0}^{n-1}a(m)x^{m},

t(n)=(-1)^{s_{2}(n)},

T(2^{s}n;\omega)=T(2^{s};\omega)T(n;\omega).

\rho(q)=\rho_{p}(q)+\rho_{h}(q).

T(2^{ms_{0}};\omega)=(T(2^{s_{0}};\omega))^{m}.

\displaystyle M(k^{t};x)

\displaystyle M^{R}(k^{t};x)

\displaystyle m_{ij}(k^{t};x)

\displaystyle m^{R}_{ij}(k^{t};x)

\displaystyle M(k^{t};x)=\sum_{w\in\Sigma_{k}^{t}}M_{w}x^{[w]_{k}},

\displaystyle M^{R}(k^{t};x)=\sum_{w\in\Sigma_{k}^{t}}M_{w}x^{[w^{R}]_{k}}.

\displaystyle\rho_{p}(q)

f_{p}=\sum_{j=0}^{c}\alpha_{pj}f_{j},

\widehat{m}_{ij}(x)=m_{ij}(x)+\sum_{p=c+1}^{d-1}\alpha_{pj}m_{ip}(x).

\displaystyle\widehat{M}(k^{t};x)

\displaystyle\widehat{M}^{R}(k^{t};x)

\widehat{M}(k^{t};x)=\sum_{m=0}^{k^{t}-1}\widehat{M}_{m}x^{m},

\widehat{M}(n;x)=\sum_{m=0}^{n-1}\widehat{M}_{m}x^{m}.

\sum_{m=0}^{l}C_{m}(\omega)A(k^{ms}n;\omega)=0,

\displaystyle F_{i}(n;x)

\displaystyle F_{i}^{R}(n;x)

\displaystyle\widehat{F}(n;x)

\displaystyle\Theta(B-q)\rho(q);

\displaystyle\widehat{F}^{R}(n;x)

\displaystyle\widehat{F}(k^{u}n;x)

\displaystyle=\widehat{M}(n;x^{k^{u}})\widehat{F}(k^{u};x),

\displaystyle\widehat{F}^{R}(k^{u}n;x)

\displaystyle=\widehat{M}^{R}(k^{u};x)\widehat{F}^{R}(n;x^{k^{u}}).

C(x,y)=\sum_{m=0}^{l}C_{m}(x)y^{m}.

\sum_{m=0}^{l}C_{m}(\omega)A(k^{ms}n;\omega)=0.

\displaystyle C(x,\lambda)=\lambda^{2}-(x^{3}+x^{2}+x+1)\lambda+4x^{3},

\displaystyle R(2^{4}n;1)-4R(2^{2}n;1)+4R(n;1)

\displaystyle\rho_{h}(q)

\displaystyle R(2^{4}n;\omega)-R(2^{2}n;\omega)+4R(n;\omega)

R(2^{2s}n;\omega)-C_{s}(\omega)R(2^{s}n;\omega)+(-2)^{s}R(n;\omega)=0,

B(2^{2s}n;\omega)-C_{s}(\omega)B(2^{s}n;\omega)+(-1)^{s}B(n;\omega)=0,

\widetilde{t}(n)=s_{2}(n)\bmod{2},

\widetilde{T}(n;x)=\sum_{m=0}^{n-1}\widetilde{t}(m)x^{m}.

2\widetilde{T}(n;x)=\frac{x^{n}-1}{x-1}-T(n;x).

\displaystyle\widetilde{T}(2^{s}n;\omega)

\displaystyle=C\widetilde{T}(n;\omega)+\frac{(1-C)(\omega^{n}-1)}{2(\omega-1)}.

\widetilde{T}(2^{s}n;\omega)=\widetilde{C}\widetilde{T}(n;\omega),

\displaystyle\Theta(q-B)\rho(q).