allenai/olmOCR-bench · Datasets at Hugging Face (original) (raw)
arxiv_math/2503.04048_pg46.pdf
https://arxiv.org/pdf/2503.04048
{\mathcal{V}}(\psi_m)\rightarrow +\infty
arxiv_math/2503.07228_pg12.pdf
https://arxiv.org/pdf/2503.07228
arxiv_math/2503.04993_pg18.pdf
https://arxiv.org/pdf/2503.04993
\frac{d}{d\epsilon} J_1(u_1 + \epsilon v_1, u_2) \Big|_{\epsilon=0}
arxiv_math/2503.04993_pg18.pdf
https://arxiv.org/pdf/2503.04993
\frac{d}{d\epsilon} J_1(u_1 + \epsilon v_1, u_2) \Big|_{\epsilon=0}=I_1 + I_2
arxiv_math/2503.06865_pg3.pdf
https://arxiv.org/pdf/2503.06865
F(q,r)=\gamma _{J\nu (q)}(r)
arxiv_math/2503.06865_pg3.pdf
https://arxiv.org/pdf/2503.06865
F:N\times \mathbb{R}\rightarrow \mathbb{C}H^{2}
arxiv_math/2503.06865_pg3.pdf
https://arxiv.org/pdf/2503.06865
arxiv_math/2503.06865_pg3.pdf
https://arxiv.org/pdf/2503.06865
B(\gamma _{J\nu (p)}(-t))=\gamma _{J\nu (p)}(t)% \text{.}
arxiv_math/2503.09550_pg2.pdf
https://arxiv.org/pdf/2503.09550
\{f_j:X_n \rightarrow \mathbb{R}\}
arxiv_math/2503.09550_pg2.pdf
https://arxiv.org/pdf/2503.09550
-1<\beta_{\vert X \vert} \leq \ldots \leq \beta_2< \beta_1=1
arxiv_math/2503.08077_pg42.pdf
https://arxiv.org/pdf/2503.08077
\{ \tfrac{-1}{2} , 0 , \tfrac{+1}{2} \}
arxiv_math/2503.04045_pg2.pdf
https://arxiv.org/pdf/2503.04045
arxiv_math/2503.05614_pg12.pdf
https://arxiv.org/pdf/2503.05614
R\Gamma_{\text{global}}(E, \mathcal{D}) \simeq R\text{Hom}(R\Gamma_{\text{global}}(E, \mathcal{D}), \mathbb{Q}/\mathbb{Z}(1))[1]
arxiv_math/2503.05614_pg12.pdf
https://arxiv.org/pdf/2503.05614
C^\bullet(E) \simeq R\text{Hom}(C^\bullet(E), \mathbb{Q}/\mathbb{Z}(1))[1]
arxiv_math/2503.05614_pg12.pdf
https://arxiv.org/pdf/2503.05614
\Lambda(E, s) = \varepsilon_E \cdot \Lambda(E, 2 - s)
arxiv_math/2503.05614_pg12.pdf
https://arxiv.org/pdf/2503.05614
arxiv_math/2503.05614_pg12.pdf
https://arxiv.org/pdf/2503.05614
E(\mathbb{Q})/E(\mathbb{Q})_{\text{tors}}
arxiv_math/2503.05614_pg12.pdf
https://arxiv.org/pdf/2503.05614
R\Gamma(\mathbb{Q}_v, \mathcal{D}_v) \simeq R\text{Hom}(R\Gamma(\mathbb{Q}_v, \mathcal{D}_v), \mathbb{Q}/\mathbb{Z}(1))[1]
arxiv_math/2503.05360_pg2.pdf
https://arxiv.org/pdf/2503.05360
\vdash \phi \quad \text{iff} \quad M \vdash g \tag{\text{$\dagger$}}
arxiv_math/2503.05360_pg2.pdf
https://arxiv.org/pdf/2503.05360
\chi = \chi_1 \land \chi_2
arxiv_math/2503.08031_pg36.pdf
https://arxiv.org/pdf/2503.08031
h(x,x') = \sum_{k=1}^{K} \phi_k(x)\phi_k(x')
arxiv_math/2503.08031_pg36.pdf
https://arxiv.org/pdf/2503.08031
\Sigma\in \mathbb{R}^{K\times K}
arxiv_math/2503.08031_pg36.pdf
https://arxiv.org/pdf/2503.08031
h: \mathbb{R}^p \times \mathbb{R}^p \to \mathbb{R}
arxiv_math/2503.08031_pg36.pdf
https://arxiv.org/pdf/2503.08031
\phi_1,\dots,\phi_K:\mathbb{R}^p\to\mathbb{R}
arxiv_math/2503.08031_pg36.pdf
https://arxiv.org/pdf/2503.08031
\phi(X_1) = (\phi_1(X_1),\ldots,\phi_{K}(X_1))
arxiv_math/2503.05717_pg14.pdf
https://arxiv.org/pdf/2503.05717
\beta=0, \,-10, \,-20, \,-30
arxiv_math/2503.05717_pg14.pdf
https://arxiv.org/pdf/2503.05717
arxiv_math/2503.05717_pg14.pdf
https://arxiv.org/pdf/2503.05717
10^4\text{mm}^{1/2}\text{Pa}
arxiv_math/2503.04108_pg45.pdf
https://arxiv.org/pdf/2503.04108
\mathfrak{g}\supset \mathfrak{g}^{\prime}
arxiv_math/2503.04108_pg45.pdf
https://arxiv.org/pdf/2503.04108
\mathfrak{su}(4) \supset \mathfrak{su}(2) \times \mathfrak{su}(2)
arxiv_math/2503.08522_pg23.pdf
https://arxiv.org/pdf/2503.08522
arxiv_math/2503.08675_pg30.pdf
https://arxiv.org/pdf/2503.08675
arxiv_math/2503.08675_pg30.pdf
https://arxiv.org/pdf/2503.08675
\liminf_{i\to\infty}d(i)\geq R
arxiv_math/2503.08675_pg30.pdf
https://arxiv.org/pdf/2503.08675
arxiv_math/2503.05752_pg4.pdf
https://arxiv.org/pdf/2503.05752
arxiv_math/2503.04620_pg35.pdf
https://arxiv.org/pdf/2503.04620
\log(z) = \log|z| + i(\arg(z) + 2\pi m), \forall m \in \mathbb{Z}
arxiv_math/2503.04620_pg35.pdf
https://arxiv.org/pdf/2503.04620
B_{kj} = \begin{cases} \delta_{k, j-1}, & \text{for } k = 1, 2, \ldots, 2r-1 ,\\ -\dfrac{h_{j-1}}{\tilde{a}_r - i \tilde{b}_r}, & \text{for } k = 2r, \end{cases}
arxiv_math/2503.04620_pg35.pdf
https://arxiv.org/pdf/2503.04620
x_t = (\arg(z_t) + 2\pi m) - i \log|z_t|, \quad t = 1, 2, \ldots, 2r, \quad \forall m \in \mathbb{Z}
arxiv_math/2503.04620_pg35.pdf
https://arxiv.org/pdf/2503.04620
\mathbf{B} = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -\frac{\tilde{a}_2 + i \tilde{b}_2}{\tilde{a}_2 - i \tilde{b}_2} & -\frac{\tilde{a}_1 + i \tilde{b}_1}{\tilde{a}_2 - i \tilde{b}_2} & 0 & - \frac{\tilde{a}_1 - i \tilde{b}_1}{\tilde{a}_2 - i \tilde{b}_2} \end{bmatrix}
arxiv_math/2503.04620_pg35.pdf
https://arxiv.org/pdf/2503.04620
h_j = \begin{cases} \tilde{a}_{r-j} + i \tilde{b}_{r-j}, & j = 0, 1, \ldots, r-1 ,\\ 2 \tilde{a}_0, & j = r, \\ \tilde{a}_{j-r} - i \tilde{b}_{j-r}, & j = r+1, r+2, \ldots, 2r. \end{cases}
arxiv_math/2503.04620_pg35.pdf
https://arxiv.org/pdf/2503.04620
\tilde{a}_k = \begin{cases} 0, & \text{for } k = 0\\ b_k k, & \text{for } k = 1, 2, \ldots, r \end{cases} \quad \text{and} \quad \tilde{b}_k = -a_k k, \text{for } k = 1, 2, \ldots, r
arxiv_math/2503.04620_pg35.pdf
https://arxiv.org/pdf/2503.04620
x_t = \arg(z_t) + 2\pi m, \quad \text{when} \quad |z_t| = 1
arxiv_math/2503.05276_pg14.pdf
https://arxiv.org/pdf/2503.05276
V(s) \approx \hat{v}_w(s) \coloneqq w^\intercal \psi(s)
arxiv_math/2503.08351_pg5.pdf
https://arxiv.org/pdf/2503.08351
arxiv_math/2503.08351_pg5.pdf
https://arxiv.org/pdf/2503.08351
S_F: U(b)\subset L^q([0,b], H_p)\multimap L^q([0,b],L^p(D)^3)
arxiv_math/2503.08351_pg5.pdf
https://arxiv.org/pdf/2503.08351
F(t,\cdot): H_p\multimap L^p(D)^3
arxiv_math/2503.08351_pg5.pdf
https://arxiv.org/pdf/2503.08351
F: [0,a]\times H_p \multimap L^p(D)^3
arxiv_math/2503.08351_pg5.pdf
https://arxiv.org/pdf/2503.08351
\eta_F:[0,\infty)\to [0,\infty)
arxiv_math/2503.08351_pg5.pdf
https://arxiv.org/pdf/2503.08351
S_F(u)=\{f\in L^q([0,b],L^p(D)^3): f(t)\in F(t,u(t)),~\text{a.a.}~t\in [0,b]\}
arxiv_math/2503.08351_pg5.pdf
https://arxiv.org/pdf/2503.08351
p\in L^q([0,b], W^{1,p}(D)^3)
arxiv_math/2503.08351_pg5.pdf
https://arxiv.org/pdf/2503.08351
(D,f_1,f_2)\in \Sigma\times K\times K
arxiv_math/2503.08351_pg5.pdf
https://arxiv.org/pdf/2503.08351
arxiv_math/2503.08351_pg5.pdf
https://arxiv.org/pdf/2503.08351
f\in L^q([0,b], L^p(D)^3)
arxiv_math/2503.09178_pg16.pdf
https://arxiv.org/pdf/2503.09178
\left\{ \begin{aligned} \varphi(0,\mu) &= \mu^{2} + const, \quad \text{if } \mu > 0, \\ \varphi(1,\mu) &= \mu^{2} + const, \quad \text{if } \mu < 0. \end{aligned} \right.
arxiv_math/2503.09178_pg16.pdf
https://arxiv.org/pdf/2503.09178
\varphi(x,\mu) = \mu^{2}\cos^{4}\pi x + const
arxiv_math/2503.09178_pg16.pdf
https://arxiv.org/pdf/2503.09178
s(x,\mu) = -4 \pi \mu^{3} \cos^{3}\pi x \sin\pi x + \Sigma_{t}(\mu^{2}\cos^{4}\pi x + const) - \Sigma_{s}(const + \frac{\cos^{4}\pi x}{3})
arxiv_math/2503.09178_pg16.pdf
https://arxiv.org/pdf/2503.09178
\Sigma_{t} = 22000, \quad \Sigma_{s} = 1
arxiv_math/2503.09178_pg16.pdf
https://arxiv.org/pdf/2503.09178
\Vert u-u_{N}^{M} \Vert_{L^{2}}
arxiv_math/2503.06379_pg4.pdf
https://arxiv.org/pdf/2503.06379
f:\mathcal{P}\rightarrow\mathcal{Q}
arxiv_math/2503.06379_pg4.pdf
https://arxiv.org/pdf/2503.06379
f : (\mathcal{P},\leq_\mathcal{P})\rightarrow(\mathcal{Q},\leq_\mathcal{Q})
arxiv_math/2503.06379_pg4.pdf
https://arxiv.org/pdf/2503.06379
|f|, |g|: |\mathcal{P}| \to |\mathcal{Q}|
arxiv_math/2503.06379_pg4.pdf
https://arxiv.org/pdf/2503.06379
|f| : |\mathcal{P}|\rightarrow|\mathcal{Q}|
arxiv_math/2503.06379_pg4.pdf
https://arxiv.org/pdf/2503.06379
f \circ g \simeq \text{Id}_Y
arxiv_math/2503.06379_pg4.pdf
https://arxiv.org/pdf/2503.06379
g \circ f \simeq \text{Id}_X
arxiv_math/2503.06379_pg4.pdf
https://arxiv.org/pdf/2503.06379
f(x) \leq_\mathcal{Q} f(y)
arxiv_math/2503.07251_pg28.pdf
https://arxiv.org/pdf/2503.07251
\tau_{1},\ldots, \tau_{L^{G}}
arxiv_math/2503.07251_pg28.pdf
https://arxiv.org/pdf/2503.07251
\Delta \tau=\tau_{i+1}-\tau_{i}
arxiv_math/2503.07251_pg28.pdf
https://arxiv.org/pdf/2503.07251
arxiv_math/2503.03827_pg10.pdf
https://arxiv.org/pdf/2503.03827
I = \langle f(x,y),~g(x,y) \rangle
arxiv_math/2503.03827_pg10.pdf
https://arxiv.org/pdf/2503.03827
\vec{a}_1 = (0, \frac{n}{2}), \quad \vec{a}_2 = (1, \gamma), \quad \text{with } 0 \leq \gamma < \frac{n}{2}
arxiv_math/2503.03827_pg10.pdf
https://arxiv.org/pdf/2503.03827
[[254, 28, 14 \leq d \leq 20]]
arxiv_math/2503.08770_pg37.pdf
https://arxiv.org/pdf/2503.08770
M_{z+w}\otimes N_w\otimes P_0
arxiv_math/2503.08770_pg37.pdf
https://arxiv.org/pdf/2503.08770
M_{z+w}\otimes (N_w\otimes P_0)
arxiv_math/2503.08770_pg37.pdf
https://arxiv.org/pdf/2503.08770
(1\otimes \Delta_w) \Delta_{z+w}=(\Delta_z\otimes 1)\Delta_w
arxiv_math/2503.08770_pg37.pdf
https://arxiv.org/pdf/2503.08770
(M_z\otimes N)_w\otimes P_0
arxiv_math/2503.08646_pg17.pdf
https://arxiv.org/pdf/2503.08646
\mu_{u^\perp}+\mu_{v^\perp}=0
arxiv_math/2503.08646_pg17.pdf
https://arxiv.org/pdf/2503.08646
w\in \mathrm{T}_p\mu^{-1}(0)/\mathrm{T}_pL
arxiv_math/2503.08646_pg17.pdf
https://arxiv.org/pdf/2503.08646
arxiv_math/2503.08646_pg17.pdf
https://arxiv.org/pdf/2503.08646
{\mathrm{Span}\left(\Bigl\{ u_i \Bigr\}_{i=2}^n\right)=\mathrm{Span}\left(\Bigl\{ v_i^\ast \Bigr\}_{i=2}^n\right)}
arxiv_math/2503.08646_pg17.pdf
https://arxiv.org/pdf/2503.08646
\mathrm{T}_p\mu^{-1}(0)\simeq \mathrm{T}_pL
arxiv_math/2503.08646_pg17.pdf
https://arxiv.org/pdf/2503.08646
d\mu[v]=\omega(\bullet, v)
arxiv_math/2503.08646_pg17.pdf
https://arxiv.org/pdf/2503.08646
\mu_u^\perp=\sum_{i=2}^n\,\left(u_i\otimes u_i^\dagger-u_i^\ast \otimes u_i^t\right)
arxiv_math/2503.08646_pg17.pdf
https://arxiv.org/pdf/2503.08646
(w, u^\perp)\mapsto \Bigl(u, v, w\Bigr)
arxiv_math/2503.08646_pg17.pdf
https://arxiv.org/pdf/2503.08646
\mathrm{Span}(\mathrm{T}_pL, w)
arxiv_math/2503.09588_pg25.pdf
https://arxiv.org/pdf/2503.09588
\overline{\alpha}\colon S_0\to S_1
arxiv_math/2503.09588_pg25.pdf
https://arxiv.org/pdf/2503.09588
\iota_1\circ c' = \iota_0\circ c\circ \alpha^{-1}
arxiv_math/2503.09588_pg25.pdf
https://arxiv.org/pdf/2503.09588
arxiv_math/2503.09588_pg25.pdf
https://arxiv.org/pdf/2503.09588
arxiv_math/2503.09588_pg25.pdf
https://arxiv.org/pdf/2503.09588
\{H_1,\dots, H_{k-1}, K_k\}
arxiv_math/2503.09588_pg25.pdf
https://arxiv.org/pdf/2503.09588
arxiv_math/2503.09588_pg25.pdf
https://arxiv.org/pdf/2503.09588
c'\circ\alpha = \overline{\alpha} \circ c
arxiv_math/2503.09588_pg25.pdf
https://arxiv.org/pdf/2503.09588
arxiv_math/2503.04448_pg10.pdf
https://arxiv.org/pdf/2503.04448
d(X_1,X_2) + d(X_2,X_1) = 1
arxiv_math/2503.04448_pg10.pdf
https://arxiv.org/pdf/2503.04448
\lambda_i/(\sum_j \lambda_j)
arxiv_math/2503.05358_pg8.pdf
https://arxiv.org/pdf/2503.05358
v_{1,{\max}} = \sqrt{C3} + v^E = \sqrt{C3} + \sqrt {\frac{\mu^S}{r^E}}
arxiv_math/2503.05358_pg8.pdf
https://arxiv.org/pdf/2503.05358
\cos(\theta_{12}) = \frac{1}{e} \left( \frac{p}{r^F} - 1 \right)
arxiv_math/2503.05358_pg8.pdf
https://arxiv.org/pdf/2503.05358
l^F_2 = \Omega^F + \omega^F + n^F (t_2 - t^F_p)
arxiv_math/2503.05358_pg8.pdf
https://arxiv.org/pdf/2503.05358
E_{\max} = \frac{v^2_{1,{\max}}}{2} - \frac{\mu^S}{r^E} \to a_{\max} = \frac{-\mu^S}{2E_{\max}} \to e_{max} = 1 - \frac{r^E}{a_{\max}}
arxiv_math/2503.05358_pg8.pdf
https://arxiv.org/pdf/2503.05358
a_{\min} = \frac{r^E + r^F}{2} \to e_{\min} = 1 - \frac{r^E}{a_{\min}}
arxiv_math/2503.06716_pg12.pdf
https://arxiv.org/pdf/2503.06716
d(u_k,\mathcal{M})\to L\in\left[0,\sqrt{\mu_{s,t}}\right]