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H^{q}(X,\Omega _{X}^{p}\otimes E)=0
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Z_{0}={\frac {1}{\pi }}{\sqrt {\frac {\mu }{\epsilon }}}\ln \left({\frac {l}{R}}+{\sqrt {\left({\frac {l}{R}}\right)^{2}-1}}~\right)
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g:(D\times E){\to }F
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{\frac {\operatorname {d} ^{2}y}{\operatorname {d} x^{2}}}
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{\textstyle \sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}=\int _{0}^{\infty }x^{s-1}\left(\sum _{n=1}^{\infty }a_{n}e^{-nx}\right)dx={\mathcal {M}}{a_{n}}(s)}
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g=\left(1-{\frac {2M}{r}}\right)\,dt_{r}^{2}-2{\sqrt {\frac {2M}{r}}}dt_{r}dr-dr^{2}-r^{2}\,d\theta ^{2}-r^{2}\sin ^{2}\theta \,d\phi ^{2}
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\left(\sum _{i=0}^{\infty }a_{i}x^{i}\right)\cdot \left(\sum _{j=0}^{\infty }b_{j}x^{j}\right)=\sum _{k=0}^{\infty }c_{k}x^{k}
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K(x;T)=h(T)
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f_{x^{*}}(x)={\begin{cases}Ae^{-Be^{Cx}+CDx}(e^{Cx}+E)^{-F},&{\text{if}}\ d>0\\Ae^{-Bx^{2}+Cx}&{\text{if}}\ d=0\end{cases}}
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a_{\max }^{\prime }-a_{\max }\approx -1/2u
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y=f(x,u)
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\delta :L^{\times }\otimes \mathbb {Q} /\mathbb {Z} \xrightarrow {\sim } H^{1}\left(L,{\overline {K}}_{tors}\right)
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{\textstyle \left\lceil \alpha \right\rceil +z=\alpha +1}
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\{{\hat {C}}_{1}(1),{\hat {C}}_{2}(1),{\hat {I}}_{3}(1),{\hat {I}}^{2}(1),{\hat {Y}}(1),{\hat {C}}_{1}(2),{\hat {C}}_{2}(2),{\hat {I}}_{3}(2),{\hat {I}}^{2}(2),{\hat {Y}}(2)\}
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f\in \cap \operatorname {Box} _{P}=\prod B_{m_{\bullet }}
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b=d_{0}(d_{1})^{-5/4}
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\int _{0}^{\frac {b}{2}}c(y)^{2}dy
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{\mathcal {X}}^{(1)},\dots ,{\mathcal {X}}^{(k)}
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N_{s}={\frac {n{\sqrt {Q}}}{(gH)^{3/4}}}
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T:\mathbf {P} (H)\to \mathbf {P} (H)
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P_{Lo}={\frac {\sum p_{c,t_{n}}q_{c,b}}{\sum p_{c,t_{0}}q_{c,b}}}
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L_{x}^{2}=L_{y}^{2}=L_{z}^{2}\propto \mathbf {1}
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k[X][f^{-1}]
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\int _{0}^{z}{\frac {\mathrm {d} t}{\sqrt {1-t^{4}}}}=2\int _{0}^{u}{\frac {\mathrm {d} t}{\sqrt {1-t^{4}}}},\quad {\text{if }}z={\frac {2u{\sqrt {1-u^{4}}}}{1+u^{4}}}{\text{ and }}0\leq u\leq {\sqrt {{\sqrt {2}}-1}}
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{\begin{aligned}Z(X,Y)&=\log(\exp X\exp Y)\\&{}=X+Y+{\frac {1}{2}}[X,Y]+{\frac {1}{12}}\left([X,[X,Y]]+[Y,[Y,X]]\right)\\&{}\quad -{\frac {1}{24}}[Y,[X,[X,Y]]]\\&{}\quad -{\frac {1}{720}}\left([Y,[Y,[Y,[Y,X]]]]+[X,[X,[X,[X,Y]]]]\right)\\&{}\quad +{\frac {1}{360}}\left([X,[Y,[Y,[Y,X]]]]+[Y,[X,[X,[X,Y]]]]\right)\\&{}\quad +{\frac {1}{120}}\left([Y,[X,[Y,[X,Y]]]]+[X,[Y,[X,[Y,X]]]]\right)\\&{}\quad +{\frac {1}{240}}\left([X,[Y,[X,[Y,[X,Y]]]]]\right)\\&{}\quad +{\frac {1}{720}}\left([X,[Y,[X,[X,[X,Y]]]]]-[X,[X,[Y,[Y,[X,Y]]]]]\right)\\&{}\quad +{\frac {1}{1440}}\left([X,[Y,[Y,[Y,[X,Y]]]]]-[X,[X,[Y,[X,[X,Y]]]]]\right)+\cdots \end{aligned}}
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O\;\mid \;\sigma \mid \tau \;\mid \;a.\sigma
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f_{t,a,b}(z)=e^{2\pi it}z^{3}\,{\frac {1-{\overline {a}}z}{z-a}}\,{\frac {1-{\overline {b}}z}{z-b}}
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V_{\text{ads}}=V_{\text{m}}\sum _{n=1}^{\infty }n\theta _{n}=V_{\text{m}}c\theta _{0}x\sum _{n=1}^{\infty }nx^{n-1}
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\varepsilon =(1+L)\cdot \delta
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P(x{\text{ has pattern}})=\sum _{j:I(K_{j})<n}2^{-I(K_{j})}
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(x_{3}-x_{1})
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{\begin{aligned}x_{n+1}-a&=f(x_{n})-a\\&\approx f(a)+f'(a)(x_{n}-a)-a\\&=a+f'(a)(x_{n}-a)-a\end{aligned}}
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{\begin{aligned}G&=\eta D\approx (1)(1.698)=1.698\\G_{\text{dBi}}&\approx 10\,\log _{10}(1.698)\approx 2.30\,{\text{dBi}}\end{aligned}}
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V(u)=T'(u)
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Q=q_{1}^{*}+q_{2}^{*}={\frac {2(a-\chi )}{3b}}
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\operatorname {Ti} _{2}(z)={\frac {1}{2i}}\left(\operatorname {Li} _{2}(iz)-\operatorname {Li} _{2}(-iz)\right)
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\sum _{a+1\leq i\leq b}-(c_{i}/c_{a})|m+i|
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y_{n+1}=y_{n}+{\tfrac {3}{2}}hf(t_{n},y_{n})-{\tfrac {1}{2}}hf(t_{n-1},y_{n-1})
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|G|=\chi _{R}(e)=\dim(R)=\sum _{j}\dim \left((W_{j})^{\oplus (\chi _{W_{j}}|\chi _{R})}\right)=\sum _{j}(\chi _{W_{j}}|\chi _{R})\cdot \dim(W_{j})=\sum _{j}\dim(W_{j})^{2}
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L_{n>0}V_{\Delta }(z)=0\quad ,\quad L_{0}V_{\Delta }(z)=\Delta V_{\Delta }(z)\
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L=-147.56+20\ \log _{10}(d)+20\ \log _{10}(f)
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r+1>{\frac {s(s+1)(\mu +1)}{2}}
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K^{G}(T)^{2}=\beta {\begin{bmatrix}{\frac {\tau _{cg}}{T}}{\sqrt {\frac {\pi }{2}}}\mathrm {erf} ({\sqrt {2}}T/\tau _{cg})+{\frac {\tau _{cg}^{2}}{2T^{2}}}(e^{-2T^{2}/\tau _{cg}^{2}}-1)\end{bmatrix}}
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\left(p_{1}c\right)\cdot \left(p_{4}c\right)={\frac {1}{2}}\left(\left(m_{1}c^{2}\right)^{2}+\left(m_{4}c^{2}\right)^{2}-u\right)
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\psi (f)=\pi .(f-f_{0})^{2}/k
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\lim _{k\rightarrow \infty }Dq(x_{2k+1})=0
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\Phi =v_{p}^{2}-{\frac {4}{3}}v_{s}^{2}={\frac {K_{S}}{\rho }}
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a\neq 0{\text{ and }}b\neq 0
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(L\pm Q/2)^{2}-Q^{4}/4\to x_{1}^{2}+P_{1}
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pK_{a}^{\rm {ox}}<pK_{a}^{\rm {red}}
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\pi _{n}(S^{1})=0
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X=\{X_{t}:t\geq 0\}
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(p_{b}-p_{a})
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(\partial U)_{P}=-(\partial P)_{U}=C_{P}-P\left({\frac {\partial V}{\partial T}}\right)_{P}
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3,8,13,18,23,28,\ldots
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\sum _{i=1}^{n}w_{i}x_{i}\leq w'
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{\frac {1}{1-z}}\exp \left((u-1)\left({\frac {z^{n+1}}{n+1}}+{\frac {z^{n+2}}{n+2}}+\cdots \right)\right)
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{\textstyle h=-{\frac {1}{2}}A^{-1}b}
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c_{\lambda }:=a_{\lambda }b_{\lambda }=\sum _{g\in P_{\lambda },h\in Q_{\lambda }}\operatorname {sgn}(h)e_{gh}
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m_{\text{u}}=m({}^{12}{\text{C}})/12
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\mathrm {cons} _{\alpha }\ x\ l=\mathrm {roll} \ (\mathrm {inr} \ \langle x,l\rangle )
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a_{i}+b_{i}+c_{i}
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(H\otimes I)\times CNOT
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\sum _{K\subseteq N\smallsetminus I}(-1)^{|K|}|B_{K}|
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{\tfrac {1}{2}}\|\mathbf {b} \wedge \mathbf {c} \|
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{\dfrac {2mV_{0}}{\hbar ^{2}}}a^{2}=\gamma ^{2}
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\sigma _{y}=\sigma _{0}+{k_{y} \over {\sqrt {d}}}
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\operatorname {E} [\varphi (Y)-\psi (Y)]=0\implies \varphi (y)-\psi (y)=0,\theta \in \Omega
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D_{x,y}\,:\,S_{y}\rightarrow S_{x}\quad {\text{unitary}}
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\operatorname {arccsc} x=-i\,\ln \left({\frac {i}{x}}+{\sqrt {1-{\frac {1}{x^{2}}}}}\right)
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D_{0}{(cm)}\approx 0.13R^{0.14}
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P(f)=\lambda ^{-\Delta }P(\lambda f)
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g-\Omega ^{2}R\cos ^{3}(\phi )
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{\frac {d\pi }{dp}}={\frac {\partial \pi }{\partial x}}{\bigg |}_{x=x^{*}}{\frac {\partial x}{\partial p}}+{\frac {\partial \pi }{\partial p}}={\frac {\partial \pi }{\partial p}}=f(x^{*}(p))=y^{*}(p)
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\scriptstyle {\dot {M}}>10^{-3}
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0.0292487852\times 1200=35.0985422804
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{\boldsymbol {S}}_{(m)}:\delta {\boldsymbol {E}}_{(m)}
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ModD(y)\equiv -{\frac {1}{V}}\cdot {\frac {\partial V}{\partial y}}=-{\frac {\partial \ln(V)}{\partial y}}
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k_{\rm {B}}T\ll E_{\rm {F}}
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\mu (\pi )={\frac {(\mathrm {dim} \,\pi )^{2}}{|G|}}
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-log_{10}[H^{+}]_{i}=b_{0}-b_{1}E_{i^{}}
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Pr({\text{find }}H_{1}|H_{1})
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T(x)=x/c+p
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\beta _{1}={\frac {-1-i{\sqrt {3}}}{2}}
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{\frac {\partial ^{2}f}{\partial z_{i}\,\partial z_{j}}}\geq 0{\mbox{ for all }}i\neq j
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m_{\mathrm {bob} }+m_{\mathrm {rod} }
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C_{D}=\sum _{v\in V}{\frac {deg(v^{*})-deg(v)}{\mid V\mid -1}}
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(1-\delta )G\leq {\tilde {G}}\leq (1+\delta )G
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{\mathcal {B}}^{*}:={\mathcal {A}}^{*}/{\mathcal {G}};
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g_{1}(z),\ldots ,g_{m}(z)<0
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p(q_{1}+q_{2})=a-b(q_{1}+q_{2})
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{\mathcal {D}}\backslash \mathbf {sSet} \rightarrow \mathbf {sSet} ,(F\colon {\mathcal {D}}\rightarrow {\mathcal {E}})\mapsto {\mathcal {E}}/F
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q_{i}-\lfloor q_{i}\rfloor
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\ v_{\mathsf {out}}=\alpha _{1}{\Bigl (}A_{1}\cos(\omega _{1}t)+A_{2}\cos(\omega _{2}t){\Bigr )}+\alpha _{2}{\Bigl (}A_{1}^{2}\cos ^{2}(\omega _{1}t)+2A_{1}A_{2}\cos(\omega _{1}t)\ \cos(\omega _{2}t)+A_{2}^{2}\cos ^{2}(\omega _{2}t){\Bigr )}+\cdots \
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L_{\eta }=\{s\in \Sigma ^{*}\vert vP_{s}Q_{\text{accept}}>\eta \}
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Q_{-}={\frac {\text{disliked}}{\text{known}}}\times 100
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{\textstyle f_{\alpha }:f^{-1}(D_{\alpha })\to D_{\alpha }\,}
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t_{\text{threshold}}
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P_{j}(d_{j})=1-e^{-{\frac {d_{j}^{2}}{2s_{j}^{2}}}}
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{\begin{aligned}X&=a\cos \omega {\frac {\sqrt {a^{2}-b^{2}\sin ^{2}\beta -c^{2}\cos ^{2}\beta }}{\sqrt {a^{2}-c^{2}}}},\\Y&=b\cos \beta \sin \omega ,\\Z&=c\sin \beta {\frac {\sqrt {a^{2}\sin ^{2}\omega +b^{2}\cos ^{2}\omega -c^{2}}}{\sqrt {a^{2}-c^{2}}}}.\end{aligned}}
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Wikipedia LaTeX Formulas 319k Dataset
A curated collection of mathematical formulas from English Wikipedia, providing LaTeX source code paired with high-quality rendered images for training OCR and formula recognition models.
Dataset Description
Dataset Summary
This dataset represents a complete extraction of ~319k LaTeX mathematical formulas from English Wikipedia articles (October 2025 snapshot), filtered by visual complexity (score > 8) and renderability with standard LaTeX packages. Each formula is paired with its rendered PNG image at 200 DPI to support machine learning tasks in mathematical document understanding.
Key Features:
- Source: English Wikipedia (Wikimedia Enterprise API, October 2025 snapshot) - provides real-world mathematical notation as written by humans, reflecting diverse notation styles and complexity levels across scientific domains
- Content: LaTeX formula strings + rendered PNG images (200 DPI)
- Quality Filtering: Visual complexity score > 8 to filter out trivial expressions like "x", "2", "a+b" (score counts LaTeX commands, letters, digits, operators, brackets, punctuation, subscripts/superscripts). A small number of formulas requiring additional LaTeX packages beyond amsmath, amssymb, amsfonts, and mhchem were also excluded.
- Deduplication: Unique formulas only
- Format: Hugging Face Dataset with native Image feature type (auto-decoded PIL Images)
Dataset Structure
Each row contains:
formula(string): LaTeX source code (cleaned and normalized)image(Image): Rendered PNG image at 200 DPI- Automatically decoded as
PIL.Imagewhen accessed - Average size: ~2 KB per image
- Transparent background with 5pt padding
- Automatically decoded as
Usage Examples
Loading the dataset:
from datasets import load_dataset
# Load from Hugging Face Hub
ds = load_dataset("piushorn/wikipedia-latex-formulas-319k")
# Access a sample
sample = ds['train'][0]
print(sample['formula'])
# Output: r(x):=b-Ax
sample['image'].show() # Displays rendered formula as PIL Image
Example usecases for this dataset:
- Image-to-Text (LaTeX OCR): Train models to recognize LaTeX formulas from images
- Formula Understanding: Train models to parse and understand mathematical notation
- Multimodal Learning: Align visual and symbolic representations of mathematics.
Rendering Process
The images were generated from the LaTeX source code using the following pipeline:
Pipeline: LaTeX source → pdflatex → PDF → ImageMagick → PNG image
LaTeX Template:
\documentclass[preview, border=5pt]{standalone}
\usepackage{amsmath,amssymb,amsfonts}
\usepackage[version=4]{mhchem} % For chemical formulas
\usepackage{varwidth}
\begin{document}
\begin{varwidth}{25cm}
% Formula inserted here
\end{varwidth}
\end{document}
ImageMagick Conversion:
- Resolution: 200 DPI (
-density 200) - Quality: 100 (
-quality 100) - Output: PNG with transparent background
Custom Resolution & Formats: Since the dataset includes raw LaTeX source, you can regenerate images at any DPI by adjusting ImageMagick's -density parameter, or convert to different formats (JPEG, SVG, etc.) and apply other customizations (background colors, padding, etc.). This is useful for training models at different resolutions or generating outputs tailored to specific requirements.
Acknowledgments
This work has been supported by the German Federal Ministry of Research, Technology and Space (BMFTR) in the program "Forschung an Fachhochschulen in Kooperation mit Unternehmen (FH-Kooperativ)" within the joint project LLMpraxis under grant 13FH622KX2.
Licensing Information
Content License: CC BY-SA 4.0 (same as Wikipedia)
Dataset License: CC BY-SA 4.0
Wikipedia Snapshot: English Wikipedia, October 2025 Dataset Created: October 2025
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