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Formula Catalog

Explore mathematical formulas with detailed analysis and 7-Vector properties.

Showing 20 formulas

Skip-Gram Loss

Machine Learning

L=−∑t=1T∑−m≤j≤m,j≠0log⁡P(wt+j∣wt)\mathcal{L} = -\sum_{t=1}^{T}\sum_{-m\le j\le m, j\ne 0}\log P(w_{t+j}|w_t)L=−∑t=1T​∑−m≤j≤m,j=0​logP(wt+j​∣wt​)

Newton's Second Law

Physics

F⃗=ma⃗\vec{F} = m \vec{a}F=ma

Pearson Correlation Coefficient

Statistics

r=∑i=1n(xi−xˉ)(yi−yˉ)∑i=1n(xi−xˉ)2∑i=1n(yi−yˉ)2r = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n}(x_i - \bar{x})^2}\sqrt{\sum_{i=1}^{n}(y_i - \bar{y})^2}}r=∑i=1n​(xi​−xˉ)2​∑i=1n​(yi​−yˉ​)2​∑i=1n​(xi​−xˉ)(yi​−yˉ​)​

Schrödinger Equation

Quantum Mechanics

iℏ∂∂tΨ(r,t)=H^Ψ(r,t)i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t)=\hat{H}\Psi(\mathbf{r},t)iℏ∂t∂​Ψ(r,t)=H^Ψ(r,t)

Pythagorean Theorem

Geometry

a2+b2=c2a^2 + b^2 = c^2a2+b2=c2

Pythagorean Trigonometric Identity

Trigonometry

sin⁡2x+cos⁡2x=1\sin^{2}x + \cos^{2}x = 1sin2x+cos2x=1

Euler's Identity

Complex Analysis

eiπ+1=0e^{i\pi} + 1 = 0eiπ+1=0

Logistic Map

Chaos Theory

xn+1=rxn(1−xn)x_{n+1} = r x_n (1 - x_n)xn+1​=rxn​(1−xn​)

Bernoulli Distribution PMF

Probability Theory

P(X=k)=pk(1−p)1−kP(X = k) = p^{k} (1-p)^{1-k}P(X=k)=pk(1−p)1−k

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