png imagewidth (px) 10 1.78k ⌀ | json dict | __key__ stringlengths 4 22 | __url__ stringclasses 204
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"tex": "$$ J\\leq \\frac{E(u_0,u_1)-E(v_0,v_1)}{E(W,0)}.$$",
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"width": 502
} | 2103_01293_13261 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"tex": "$$ \\limsup_{n\\to\\infty}\\|\\vec{u}(t_n)\\|_{\\mathcal{H}}\\leq C_0.$$",
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"width": 424
} | 2103_01293_13313 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"id": "2103.01293_13316",
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"tex": "$$\\liminf_{t\\to\\infty} \\|\\vec{u}(t)\\|_{\\mathcal{H}}<\\infty$$",
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"width": 327
} | 2103_01293_13316 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"id": "2103.01293_13324",
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"tex": "$$\\limsup_{t\\to T_+}\\|\\vec{u}(t)\\|_{\\mathcal{H}}\\leq C_0.$$",
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} | 2103_01293_13324 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
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"tex": "\\begin{equation} \\lim_{t\\to T_+}\\|\\partial_tu(t)-v_1\\|_{L^2}=0. \\end{equation}",
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"width": 543
} | 2103_01293_13330 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"tex": "\\begin{equation} \\lambda_{1,n}\\ll \\ldots \\ll \\lambda_{J,n}\\ll T_+-t_n, \\end{equation}",
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"width": 109
} | 2103_01293_13362 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"tex": "\\begin{equation} E(u_0,u_1)=E(v_0,v_1)+JE(W,0). \\end{equation}",
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"width": 272
} | 2103_01293_13407 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"id": "2103.01293_13444",
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"tex": "$$E(u_0,u_1)=E(v_0,v_1)+J'E(W,0)+\\frac{1}{2}\\varepsilon_0^2.$$",
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"width": 396
} | 2103_01293_13444 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"id": "2103.01293_13451",
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"tex": "$$\\frac{1}{2}\\varepsilon_0^2=(J-J')E(W,0),$$",
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"width": 442
} | 2103_01293_13451 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"id": "2103.01293_13498",
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"tex": "$$ J=\\frac{E(u_0,u_1)-E(v_0,v_1)}{E(W,0)}$$",
"tokens": 23,
"width": 207
} | 2103_01293_13498 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"id": "2103.01293_13747",
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"tex": "$$ \\|(\\psi_0-\\ell\\pi,\\psi_1)\\|_{\\mathbf{H}}^2\\approx E_M(\\psi_0,\\psi_1).$$",
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"width": 390
} | 2103_01293_13747 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
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"id": "2103.01293_14449",
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"tex": "$$(\\phi_0,\\phi_1)(r)=(\\psi(T,r),\\partial_t\\psi(T,r)), r\\geq T,$$",
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"width": 492
} | 2103_01293_14449 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
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"id": "2103.01293_14695",
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"tex": "\\begin{equation} \\sup_nE_M(\\psi_{0,n},\\psi_{1,n})<\\infty. \\end{equation}",
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"width": 351
} | 2103_01293_14695 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"id": "2103.01293_14874",
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"tex": "$$\\sum_{j\\in \\mathcal{J}_C}E_M(\\Psi_{0}^j,\\Psi_{1}^j)<\\infty.$$",
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"width": 498
} | 2103_01293_14874 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
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"id": "2103.01293_15295",
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"tex": "\\begin{equation} \\lim_{J\\to\\infty} \\limsup_{n\\to\\infty}\\varepsilon_n^J=0, \\end{equation}",
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"width": 212
} | 2103_01293_15295 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
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"id": "2103.01293_15815",
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"tex": "\\begin{equation*} \\lim_{n\\to\\infty} t_n=T_+, \\forall n,\\; 0\\leq t_n<T_+. \\end{equation*}",
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"width": 318
} | 2103_01293_15815 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
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"id": "2103.01293_15839",
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"tex": "$$\\lambda_{1,n}\\ll \\ldots \\ll \\lambda_{J,n}\\ll T_+-t_n,$$",
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"width": 331
} | 2103_01293_15839 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"id": "2103.01293_15917",
"lines": 1,
"tex": "$$\\operatorname{supp} (\\varphi-\\psi)\\subset \\{r\\leq 1-t\\}.$$",
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"width": 118
} | 2103_01293_15917 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"id": "2103.01293_16488",
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"tex": "\\begin{equation} \\lim_{n\\to\\infty} \\frac{\\lambda_{J_0,n}}{1-t_n}=0. \\end{equation}",
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"width": 167
} | 2103_01293_16488 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"id": "2103.01293_16562",
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"tex": "$$\\iota_{J_0}\\int_{0}^{\\infty}(Q-\\pi)(r)\\chi(r)dr=0, $$",
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"width": 292
} | 2103_01293_16562 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"id": "2103.01293_16573",
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"tex": "\\begin{equation} \\lim_{n\\to\\infty} \\|\\omega_{0,n}\\|_{H}=0 \\end{equation}",
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} | 2103_01293_16573 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"id": "2103.01293_16850",
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"tex": "\\begin{equation} E_M(\\psi_0,\\psi_1)=E_M(\\varphi_0,\\varphi_1)+JE_M(Q,0). \\end{equation}",
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"width": 360
} | 2103_01293_16850 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"id": "2103.01293_16968",
"lines": 1,
"tex": "$$0<\\lambda_{1,n}\\ll \\ldots \\ll \\lambda_{J,n}\\ll 1-t_n,$$",
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"width": 655
} | 2103_01293_16968 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
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"id": "2103.01293_17259",
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"tex": "$$\\lim_{t\\to\\infty} \\int_0^1 |G(r-t)|^2dr=0,$$",
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"width": 546
} | 2103_01293_17259 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"id": "2103.01293_17274",
"lines": 1,
"tex": "$$\\eta\\mapsto (\\eta+\\tau)^{\\frac{N-1}{2}}\\chi(\\eta+\\tau)(\\partial_t u)(\\tau,\\eta+\\tau)$$",
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"width": 658
} | 2103_01293_17274 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"id": "2103.01293_18229",
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"tex": "$$ (Aw)(t,r)=\\frac{1}{r^2}\\int_{0}^t w(\\tau,r)d\\tau.$$",
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"width": 551
} | 2103_01293_18229 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"id": "2103.01293_18233",
"lines": 1,
"tex": "$$ \\left\\|Aw\\right\\|_{(L^2L^{8/3})(R)} \\lesssim \\|w\\|_{(L^2L^8)(R)},$$",
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"width": 447
} | 2103_01293_18233 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"id": "2103.01293_18346",
"lines": 1,
"tex": "$$ f(t,r)=|w(t,r)|1\\! 1_{\\{r>t+R\\}}.$$",
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"width": 295
} | 2103_01293_18346 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
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"tex": "$$\\left|A(w)(t,r)\\right|\\lesssim \\frac{|t|}{r^2}M(f)(t,r).$$",
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"width": 262
} | 2103_01293_18365 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"tex": "\\begin{equation} (\\kappa \\, *\\, k )(t) = \\{1 \\},\\ t>0. \\end{equation}",
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} | 2103_15000_432 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
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"tex": "\\begin{equation} (I_{0+}^0\\, f)(t) := f(t). \\end{equation}",
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} | 2103_15000_2906 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
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"tex": "\\begin{equation} a_{ij}=-2\\nu_T\\bar{\\mathbf{S}}, \\end{equation}",
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} | 2103_16963_242 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
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"tex": "\\begin{equation} b_{ij}=B_{ij}(\\widehat{\\mathbf{S}},\\widehat{\\mathbf{\\Omega}}), \\end{equation}",
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"width": 63
} | 2103_16963_379 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
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"tex": "\\begin{equation} \\nu_{T}=f_{\\mu}C_{\\mu}\\frac{k^2}{\\varepsilon}, \\end{equation}",
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} | 2103_16963_588 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
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"tex": "\\begin{equation} f_{\\mu}= \\mbox{exp}(\\frac{-2.5}{1+Re_{L}/50}). \\end{equation}",
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"width": 523
} | 2103_16963_598 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
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"tex": "\\begin{equation*} 0 = t_0 < t_1 < t_2 < \\dots < t_{N_t -1} < t_{N_t} = T \\end{equation*}",
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} | 2103_04324_1083 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"tex": "\\begin{equation*} \\mathcal T_\\nu := \\{\\omega_i \\subset \\mathbb{R}^{d} \\colon i=1,\\dots,N_x\\} \\end{equation*}",
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} | 2103_04324_1122 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
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"tex": "\\begin{equation} K_h \\underline{u} = \\underline{\\widetilde f} \\end{equation}",
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} | 2103_04324_1291 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
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"tex": "\\begin{equation} Q:= \\Omega \\times(0,T) := (0,1) \\times (0,10). \\end{equation}",
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} | 2103_04324_1504 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"tex": "\\begin{equation} x_0 = 0, x_1 = 1/4, x_2 = 1, \\end{equation}",
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} | 2103_04324_1524 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
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"tex": "\\begin{equation} t_0 = 0, t_1 = T/8, t_2 = T/4, t_3 = T, \\end{equation}",
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"width": 356
} | 2103_04324_1538 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"tex": "\\[ |zf''(z)+\\alpha(f'(z)-1)|<\\beta, z\\in\\mathbb{D},\\]",
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"width": 283
} | 2103_10140_740 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"tex": "$$zF''_{\\lambda}(z)+\\alpha F'_{\\lambda}(z) \\prec \\alpha +\\beta z, z \\in \\mathbb{D}.$$",
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} | 2103_10140_1189 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"tex": "$$F'_{\\lambda}(z) \\prec 1+\\frac{\\beta}{1+\\alpha} z, z \\in \\mathbb{D}.$$",
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"width": 255
} | 2103_10140_1195 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"tex": "\\begin{equation*} |h'(z)| +|g'(z)| \\leq 1+ \\frac{\\beta}{1+\\alpha}|z| \\end{equation*}",
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"width": 684
} | 2103_10140_1205 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
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"tex": "\\begin{equation*} |h'(z)| -|g'(z)| \\geq 1-\\frac{\\beta}{1+\\alpha}|z|. \\end{equation*}",
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} | 2103_10140_1207 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
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"tex": "\\begin{equation} \\sum_{n=2}^{\\infty} n (n+\\alpha-1)(|a_n|+|b_n|)\\leq \\beta, \\end{equation}",
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} | 2103_10140_1356 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"tex": "$$\\sum_{n=2}^{\\infty}n(n+\\alpha-1)\\left( |a_n|+|b_n|\\right) =\\beta.$$",
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} | 2103_10140_1478 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
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"tex": "\\begin{equation} \\ h(z_1)-h(z_2) = \\overline{g(z_2)-g(z_1).} \\end{equation}",
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} | 2103_10140_1690 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"tex": "\\[h(z_1)-h(z_2)=e^{2i\\theta} \\left( g(z_2)-g(z_1)\\right), \\]",
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} | 2103_10140_1733 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"tex": "\\[ f(z)=\\sum_{k=1}^{n} t_kf_k(z)=h(z)+\\overline{g(z)},\\]",
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} | 2103_10140_1863 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
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"tex": "\\[ f\\ast\\left(\\phi+\\lambda \\overline{\\phi} \\right) = h\\ast \\phi +\\overline{\\overline{\\lambda}(g\\ast \\phi)}. \\]",
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"width": 696
} | 2103_10140_1891 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"id": "2103.10140_1928",
"lines": 1,
"tex": "\\[ \\left | zF_\\lambda ''(z)+\\alpha (F_\\lambda'(z)-1)\\right | \\leq \\beta, z\\in\\mathbb{D}. \\]",
"tokens": 30,
"width": 616
} | 2103_10140_1928 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 159,
"id": "2103.10140_2029",
"lines": 1,
"tex": "\\[\\Lambda=\\frac{\\Gamma(c)\\Gamma(c-a-b)}{\\Gamma(c-a)\\Gamma(c-b)}. \\]",
"tokens": 30,
"width": 430
} | 2103_10140_2029 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 31,
"id": "2103.10140_2238",
"lines": 1,
"tex": "\\[ (-m)_n =\\frac{(-1)^nm!}{(m-n)!}.\\]",
"tokens": 17,
"width": 83
} | 2103_10140_2238 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 99,
"id": "2103.10588_536",
"lines": 1,
"tex": "$$\\operatorname{index}(\\Sigma) \\le \\operatorname{index}(\\check{\\Sigma}) \\le 2\\operatorname{index}(\\Sigma) -1.$$",
"tokens": 27,
"width": 635
} | 2103_10588_536 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"id": "2103.10588_554",
"lines": 1,
"tex": "$$\\operatorname{index}(\\Sigma) + 1\\le \\operatorname{index}(\\check{\\Sigma}) \\le 2\\operatorname{index}(\\Sigma) -2.$$",
"tokens": 29,
"width": 235
} | 2103_10588_554 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 92,
"id": "2103.10588_591",
"lines": 1,
"tex": "$$2 = \\operatorname{index}(\\Sigma) \\le \\operatorname{index}(\\check{\\Sigma}) \\le 2\\operatorname{index}(\\Sigma) -1 = 3.$$",
"tokens": 31,
"width": 737
} | 2103_10588_591 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 139,
"id": "2103.10588_1015",
"lines": 1,
"tex": "$$Q_{X}(f,f) := \\int_X(|\\nabla f|^2 - |\\nabla N|^2f^2)dA,$$",
"tokens": 28,
"width": 722
} | 2103_10588_1015 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 97,
"id": "2103.10588_1285",
"lines": 1,
"tex": "$$\\operatorname{index}(X) = \\operatorname{index}_-(X) + \\operatorname{index}_+(X),$$",
"tokens": 25,
"width": 594
} | 2103_10588_1285 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 116,
"id": "2103.10588_1291",
"lines": 1,
"tex": "$$\\operatorname{nul}(X) = \\operatorname{nul}_-(X) + \\operatorname{nul}_+(X).$$",
"tokens": 28,
"width": 634
} | 2103_10588_1291 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 81,
"id": "2103.10588_2167",
"lines": 1,
"tex": "$$\\operatorname{index}(\\check{\\Sigma}) = \\operatorname{index}(\\Sigma) +\\operatorname{index}_\\bullet(\\Sigma)$$",
"tokens": 26,
"width": 470
} | 2103_10588_2167 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 121,
"id": "2103.10588_2171",
"lines": 1,
"tex": "$$\\operatorname{index}(\\Sigma) \\ge \\operatorname{index}_\\bullet(\\Sigma) + \\operatorname{nul}_\\bullet(\\Sigma)$$",
"tokens": 28,
"width": 643
} | 2103_10588_2171 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 28,
"id": "2103.10588_2648",
"lines": 1,
"tex": "$$\\operatorname{index}_{NN}(\\Lambda) + \\operatorname{index}_{DN}(\\Lambda) = \\operatorname{index}(\\Sigma),$$",
"tokens": 29,
"width": 186
} | 2103_10588_2648 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 86,
"id": "2103.10588_2649",
"lines": 1,
"tex": "$$\\operatorname{index}_{ND}(\\Lambda) + \\operatorname{index}_{DD}(\\Lambda) = \\operatorname{index}_\\bullet(\\Sigma),$$",
"tokens": 32,
"width": 566
} | 2103_10588_2649 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 120,
"id": "2103.10588_2714",
"lines": 1,
"tex": "$$\\operatorname{index}_{NN}(\\Lambda) + \\operatorname{index}_{DN}(\\Lambda) = \\operatorname{index}(\\Sigma).$$",
"tokens": 29,
"width": 775
} | 2103_10588_2714 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 58,
"id": "2103.10588_2814",
"lines": 1,
"tex": "$$\\operatorname{index}_{++}(X) \\ge \\operatorname{index}_{+-}(X) + \\operatorname{nul}_{+-}(X),$$",
"tokens": 31,
"width": 378
} | 2103_10588_2814 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 49,
"id": "2103.10588_2815",
"lines": 1,
"tex": "$$\\operatorname{index}_{++}(X) \\ge \\operatorname{index}_{-+}(X) + \\operatorname{nul}_{-+}(X),$$",
"tokens": 31,
"width": 354
} | 2103_10588_2815 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 82,
"id": "2103.10588_2816",
"lines": 1,
"tex": "$$\\operatorname{index}_{+-}(X) \\ge \\operatorname{index}_{--}(X) + \\operatorname{nul}_{--}(X),$$",
"tokens": 31,
"width": 553
} | 2103_10588_2816 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 120,
"id": "2103.10588_2817",
"lines": 1,
"tex": "$$\\operatorname{index}_{-+}(X) \\ge \\operatorname{index}_{--}(X) + \\operatorname{nul}_{--}(X).$$",
"tokens": 31,
"width": 789
} | 2103_10588_2817 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 52,
"id": "2103.09367_11220",
"lines": 1,
"tex": "\\begin{equation} Gini > -0.14\\times M20 + 0.33 \\end{equation}",
"tokens": 22,
"width": 274
} | 2103_09367_11220 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"id": "2103.14443_1071",
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"tex": "\\begin{equation*} \\left\\{\\mathbf{e}_i \\right\\} = \\operatorname{KGE} (\\mathcal{G}), \\end{equation*}",
"tokens": 28,
"width": 390
} | 2103_14443_1071 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"id": "2103.10559_2219",
"lines": 1,
"tex": "\\begin{align} \\min_{\\theta}\\ f(\\theta | M_0) + \\lambda \\|\\theta\\|_1, \\end{align}",
"tokens": 26,
"width": 678
} | 2103_10559_2219 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"id": "2103.06811_920",
"lines": 1,
"tex": "\\begin{equation} f(t, D) \\propto \\int_0^D n_e( t , z) dz \\end{equation}",
"tokens": 26,
"width": 292
} | 2103_06811_920 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
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"id": "2103.06811_1133",
"lines": 1,
"tex": "\\begin{equation} \\eta= k_z D/2 \\ \\ \\ \\ \\ \\ \\ ; \\ \\ \\ \\mu= k_x D/2 \\end{equation}",
"tokens": 31,
"width": 422
} | 2103_06811_1133 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 91,
"id": "2103.06811_1381",
"lines": 1,
"tex": "\\begin{equation} D= (5.5\\pm 1.8) \\times 10^8 cm \\end{equation}",
"tokens": 24,
"width": 411
} | 2103_06811_1381 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 164,
"id": "2103.01763_2271",
"lines": 1,
"tex": "$$\\hat V_{\\rm Coul}(r)=-\\frac{4}{3}\\alpha_s \\frac{F_1(r)F_2(r)}{r}$$",
"tokens": 29,
"width": 641
} | 2103_01763_2271 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 22,
"id": "2103.01763_2320",
"lines": 1,
"tex": "\\begin{equation} F(r)=1-e^{-\\xi r -\\zeta r^2}. \\end{equation}",
"tokens": 24,
"width": 73
} | 2103_01763_2320 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 49,
"id": "2103.01763_2809",
"lines": 1,
"tex": "$$X(2^{++})=(Qq)_{S=1}(\\bar Q\\bar q')_{S=1}.$$",
"tokens": 26,
"width": 252
} | 2103_01763_2809 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 58,
"id": "2103.08439_2282",
"lines": 1,
"tex": "\\begin{equation} \\mathbf{A}_i = \\mathbf{V}_i \\times \\mathbf{E}_i^{\\prime}, \\end{equation}",
"tokens": 28,
"width": 144
} | 2103_08439_2282 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 98,
"id": "2103.14149_4081",
"lines": 1,
"tex": "\\begin{equation} \\gamma = \\frac{1.44q}{1.23 + 0.67q -2 q^2} \\end{equation}",
"tokens": 31,
"width": 282
} | 2103_14149_4081 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 48,
"id": "2103.10899_1447",
"lines": 1,
"tex": "\\begin{equation} w_x(d/2)=w_y(d/2). \\end{equation}",
"tokens": 20,
"width": 155
} | 2103_10899_1447 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 100,
"id": "2103.10899_1467",
"lines": 1,
"tex": "\\begin{equation} R_{x,z}(d/2)=R_{y,z}(d/2). \\end{equation}",
"tokens": 28,
"width": 367
} | 2103_10899_1467 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 162,
"id": "2103.10899_1507",
"lines": 1,
"tex": "\\begin{equation} w_i = \\sqrt{\\frac{2 f_i}{k}}, \\end{equation}",
"tokens": 19,
"width": 270
} | 2103_10899_1507 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 26,
"id": "2103.10899_1508",
"lines": 1,
"tex": "\\begin{equation} f_i \\cdot f_o = 2\\cdot d^2, \\end{equation}",
"tokens": 18,
"width": 68
} | 2103_10899_1508 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 89,
"id": "2103.07446_3520",
"lines": 1,
"tex": "$$d^*(x,y)=1\\Leftrightarrow (x-\\bar{x})(y-\\bar{y}(x))\\geqslant 0.$$",
"tokens": 27,
"width": 594
} | 2103_07446_3520 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 60,
"id": "2103.07446_8278",
"lines": 1,
"tex": "$$d^*(x,y)=1\\Leftrightarrow (x-\\bar{x})y\\geqslant 0,$$",
"tokens": 20,
"width": 292
} | 2103_07446_8278 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 40,
"id": "2103.07446_13001",
"lines": 1,
"tex": "$$\\bar{\\Pi}_\\tau(\\theta)=\\tau\\bar{\\Pi}_1(\\theta)+(1-\\tau)\\bar{\\Pi}_0(\\theta),$$",
"tokens": 31,
"width": 220
} | 2103_07446_13001 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 122,
"id": "2103.07446_13872",
"lines": 1,
"tex": "$$\\frac{\\hat{x}}{\\hat{y}}=\\frac{1-\\hat{x}}{1-\\hat{y}},$$",
"tokens": 25,
"width": 207
} | 2103_07446_13872 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 110,
"id": "2103.07446_13998",
"lines": 1,
"tex": "$$\\hat{x}=\\mathbb{E}\\left[x|(x-\\hat{x})y<0\\right],$$",
"tokens": 24,
"width": 475
} | 2103_07446_13998 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 72,
"id": "2103.07446_14031",
"lines": 1,
"tex": "$$\\hat{x}=\\mathbb{E}\\left[x|(x-\\hat{x})y<0\\right].$$",
"tokens": 24,
"width": 291
} | 2103_07446_14031 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 94,
"id": "2103.00282_2275",
"lines": 1,
"tex": "\\[ \\underset{y,k}{\\sup} g_{p}(y,k)<C(p) \\]",
"tokens": 22,
"width": 242
} | 2103_00282_2275 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 168,
"id": "2103.00282_2339",
"lines": 1,
"tex": "\\[ \\underset{k\\in\\mathbb{\\mathbb{Z}}_{\\geq1}}{\\sup}G_{p}(k) \\]",
"tokens": 25,
"width": 284
} | 2103_00282_2339 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 111,
"id": "2103.00282_3310",
"lines": 1,
"tex": "\\[ J_{k}(\\varphi):J_{k}(X/S)\\rightarrow J_{k}(Y/S), \\]",
"tokens": 25,
"width": 559
} | 2103_00282_3310 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 85,
"id": "2103.00282_5008",
"lines": 1,
"tex": "\\[ \\left|H_{F}(x)\\right|\\leq\\left|G_{F}(x)\\right|, \\]",
"tokens": 23,
"width": 309
} | 2103_00282_5008 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 63,
"id": "2103.00282_5015",
"lines": 1,
"tex": "\\[ \\left|H_{F'}(x)\\right|\\leq\\left|G_{F'}(x)\\right|, \\]",
"tokens": 23,
"width": 230
} | 2103_00282_5015 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 59,
"id": "2103.00282_7924",
"lines": 1,
"tex": "\\[ h_{F}(y',k)<C_{1}q_{F}^{-1}. \\]",
"tokens": 20,
"width": 184
} | 2103_00282_7924 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 71,
"id": "2103.00282_9154",
"lines": 1,
"tex": "\\[ h_{F}(y',k)<C_{1}q_{F}^{-E}. \\]",
"tokens": 20,
"width": 246
} | 2103_00282_9154 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 44,
"id": "2103.00282_9523",
"lines": 1,
"tex": "\\[ \\lim_{q_F \\to \\infty} q_F^{-\\delta} g_F(k)=0. \\]",
"tokens": 19,
"width": 121
} | 2103_00282_9523 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar | |
{
"height": 62,
"id": "2103.00282_9764",
"lines": 1,
"tex": "\\[ g_F(k) = H_F(k)+ E_F(k) \\leq | H_F(k)|+ |E_F(k)|. \\]",
"tokens": 26,
"width": 477
} | 2103_00282_9764 | hf://datasets/topdu/MER-17M@0a8d647367b1ea9da0d7b9a4b212d7cd2968437a/tokens_17_32/b-part-000097.tar |
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