Equation Database

EQUATION LIST

economics

• Allocative Efficiency Condition

  #Economics #Microeconomics

  $$P = MC \\ \text{Marginal Social Benefit (MSB)} = \text{Marginal Social Cost (MSC)}$$

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• Annuities Due

  #Economics

  $$S = R \times \frac{(1+i)^(n+1) - 1}{i} - R$$

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• Average Fixed Cost

  #Economics #Microeconomics

  $$AFC = \frac{Total Fixed Cost (TFC)}{Quantity of Output (Q)}$$

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• Average Product

  #Economics #Microeconomics

  $$\text{AP} = \frac{\text{Total Product}}{\text{Quantity of Input}}$$

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• Average Revenue

  #Economics #Microeconomics

  $$\text{Average Revenue} = \frac{\text{Total Revenue}}{\text{Quantity}}$$

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• Average Total Cost

  #Economics #Microeconomics

  $$\text{Average Total Cost(ATC)} = \frac{\text{Total Cost (TC)}}{\text{Quantity of Output (Q)}}$$

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• Average Variable Cost

  #Economics #Microeconomics

  $$\text{Average Variable Cost(AVC)} = \frac{\text{Total Variable Cost (TVC)}}{\text{Quantity of Output (Q)}}$$

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• Compound Interest

  #Economics

  $$A = P(1+\frac{r}{m})^{mt} \\ A = Pe^{rt}$$

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• Cross-Price Elasticity of Demand

  #Economics #Microeconomics

  $$\text{Elasticity of Demand} = \frac{\text{Percentage Change in Quantity Demanded of Good X}}{\text{Percentage Change in Price of Good Y}}$$

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• Effective Rate

  #Economics

  $$r_{e} = (1 + \frac{r}{m})^{m} - 1 \\ r_{e} = e^{r} - 1$$

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• Elasticity of Supply

  #Economics #Microeconomics

  $$\text{Elasticity of Supply} = \frac{\text{Percentage Change in Quantity Supplied}}{\text{Percentage Change in Price}}$$

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• Factor of Production Hiring Rule

  #Economics #Microeconomics

  $$\text{MRP} = \text{MFC}$$

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• Future Value of Ordinary Annuities

  #Economics

  $$S = R \times \frac{(1+i)^n - 1}{i} \\ R = S \times \frac{i}{(1+i)^n - 1}$$

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• Marginal Cost

  #Economics #Microeconomics

  $$\text{MC} = \frac{\Delta \text{TC}}{\Delta \text{Q}} = \frac{\Delta \text{TVC}}{\Delta \text{Q}}$$

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• Marginal Revenue Product

  #Economics #Microeconomics

  $$\text{MRP} = \text{MP} \times \text{MR}$$

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• Present Value of Ordinary Annuities

  #Economics

  $$P = R \frac{1 - (1+r)^{-n}}{i} \\ R = P \frac{i}{1 - (1+r)^{-n}}$$

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• Simple Interest

  #Economics

  $$I=Prt \\ A=P(1+rt)$$

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• Distributive Efficiency Condition

  #Economics #Microeconomics

  $$\frac{MU_{F}}{P_{F}} = \frac{MU_{C}}{P_{C}}$$

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• Gini Coefficient

  #Economics #Microeconomics

  $$G = \frac{S_{A}}{S_{B}} \\ G = 1 - \sum^{n}_{i=1} P_{i} \times (2 Q_{i} - W_{i}) \\ Q_{i} = \sum^{i}_{k = 1} W_{k}$$

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• Investment

  #Economics #MacroEconomics

  $$I=I_{P}+I_{U}$$

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• Marginal Factor Cost MFC

  #Economics #Microeconomics

  $$\text{MFC} = \frac{\Delta \text{TC}}{\Delta \text{f}}$$

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• Marginal Product of Labor

  #Economics #Microeconomics

  $$\text{MPL} = \frac{\Delta \text{TP}}{\Delta \text{L}}$$

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• Marginal Revenue

  #Economics #Microeconomics

  $$\text{MR} = \frac{\Delta \text{TR}}{\Delta \text{Q}}$$

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• Marginal Revenue Product of Labor MRPL

  #Economics #Microeconomics

  $$\text{MRP}_{L} = \text{MP}_{L} \times \text{P}$$

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• Optimal Combination of Resources Condition

  #Economics #Microeconomics

  $$\text{MRP}_{L} = \text{MP}_{L} \times \text{P}$$

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• Optimal Consumption Rule

  #Economics #Microeconomics

  $$\frac{MU_{x}}{P_{x}} = \frac{MU_{Y}}{P_{Y}}$$

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• Price Elasticity of Demand

  #Economics #Microeconomics

  $$\text{Price Elasticity of Demand} = \frac{% \Delta Q_{d}}{% \Delta P} = \frac{\frac{\Delta Q_{d}}{Q}}{\frac{\Delta P}{P}}$$

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• Price for a Competitive Firm

  #Economics #Microeconomics

  $$P = MR$$

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• Production Efficiency Condition

  #Economics #Microeconomics

  $$\frac{w}{r} = \frac{MP_{L}}{MP_{K}}$$

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• Profit

  #Economics #Microeconomics

  $$\text{Profit} = \text{TR} – \text{TC}$$

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• Profit-Maximizing Output Level

  #Economics #Microeconomics

  $$MR = MC$$

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• Slope of the Total Product Curve

  #Economics #Microeconomics

  $$\text{Marginal Product} = \frac{\Delta P}{\Delta L} = \frac{\text{Rise}}{\text{Run}} = \frac{\text{Change in Total Product}}{\text{Change in the Number of Units of an Input}}$$

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• Socially Optimal Level of Output

  #Economics #Microeconomics

  $$\text{MSB} = \text{MSC}$$

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• Total Costs

  #Economics #Microeconomics

  $$\text{TC} = \text{TFC} + \text{TVC}$$

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• Autonomous Spending Multiplier

  #Economics #MacroEconomics

  $$\text{Multiplier} = \frac{1}{1-MPC} = \frac{1}{MPS}$$

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• Balanced Budget Multiplier

  #Economics #MacroEconomics

  $$\text{Balanced Budget Multiplier} = \frac{1}{1-MPC} + \frac{-MPC}{1-MPC} = 1$$

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• Banks Reserve Ratio

  #Economics #MacroEconomics

  $$\text{Reserve Ratio} = \frac{\text{Bank Reserves}}{\text{Total Deposits}}$$

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• Calls and Puts with Different Strikes

  #Financial #Economics

  $$K_{1} < K_{2} \\ 0 \le c(K_{1}) - c(K_{2}) \le (K_{2} - K_{1})e^{-rT} \\ 0 \le p(K_{2}) - p(K_{1}) \le (K_{2}) - K_{1})e^{-rT} \\ \frac{c(K_{1}) - c(K_{2})}{K_{2} - K_{1}} \ge \frac{c(K_{2}) - c(K_{3})}{K_{3} - K_{2}} \\ \frac{p(K_{1}) - p(K_{2})}{K_{2} - K_{1}} \le \frac{p(K_{3}) - p(K_{2})}{K_{3} - K_{2}}$$

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• Consumer Price Index CPI

  #Economics #MacroEconomics

  $$\text{CPI} = \frac{\text{Base Year Quantities} \times \text{Current Year Prices}}{\text{Base Year Quantities} \times \text{Base Year Prices}} \times 100$$

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• Consumption Function

  #Economics #MacroEconomics

  $$C = C_{a} + \text{MPC}(Y)$$

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• Equality of Leakages and Injections

  #Economics #MacroEconomics

  $$\text{S} + \text{T} + \text{M} = \text{I} + \text{G} + \text{X}$$

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• Equation of Exchange

  #Economics #MacroEconomics

  $$\text{MV} = \text{PQ}$$

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• Forwards

  #Financial #Economics

  $$F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\text{Dividends}) = S_{t}e^{(r-\delta)(T-t)}$$

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• Gross Domestic Product Deflator

  #Economics #MacroEconomics

  $$\text{GDP Deflator}= \frac{\text{Current Year Quantities} \times \text{Current Year Prices}}{\text{Current Year Quantities} \times \text{Base Year Prices}} \times 100$$

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• Merchandise Trade Balance

  #Economics #MacroEconomics

  $$\text{Merchandise Trade Balance}=\text{Value of Merchandise Exports} - \text{Value of Merchandise Imports}$$

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• Nominal Interest Rate

  #Economics #MacroEconomics

  $$\text{Nominal Interest Rate}=\text{Real Interest Rate} + \text{Anticipated Inflation}$$

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• Put-Call Parity

  #Financial #Economics

  $$c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}$$

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• Real GDP

  #Economics #MacroEconomics

  $$\text{Real GDP}=\frac{Nominal GDP}{CPI for the same year as the nominal figure} \times 100$$

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• Real Interest Rate

  #Economics #MacroEconomics

  $$\text{Real Interest Rate} = \text{Nominal Interest Rate} – \text{Anticipated Inflation}$$

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• Tax Multiplier

  #Economics #MacroEconomics

  $$\text{Tax Multiplier} = -\frac{MPC}{MPS}$$

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• Unemployment Rate

  #Economics #MacroEconomics

  $$\text{Unemployment Rate} = \frac{\text{Unemployed}}{\text{Labor Force}}$$

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• Asian Options

  #Financial #Economics

  $$A(T) = \frac{1}{n} \sum S(ih) \\ G(T) = [\prod S(ih)]^{\frac{1}{n}}$$

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• Black-Derman-Toy BDT

  #Financial #Economics

  $$\text{First Node: 1-year bond price} \\ P_{0} = \frac{1}{1 + R_{0}} \\ \text{Second Node} \\ P_{1} = \frac{1}{1+R_{0}} [\frac{1}{2} P(1,2,r_{u}) + \frac{1}{2} P(1,2,r_{d})] \\ = \frac{1}{1+R_{0}} [\frac{1}{2(1 + R_{1}e^{2\sigma_{1}})} + \frac{1}{2(1 + R_{1})}] \\ R_{0} = \frac{1}{2} \ln (\frac{R_{1} e^{2\sigma_{1}} }{R_{1}})$$

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• Bonds and Interest Rates

  #Financial #Economics

  $$P(0, S) = \frac{1}{[1 + r(0, s)]^{s}} \text{or} e^{-r(0,s)s} \\ \text{Forward bond price} \\ F_{t,T}[P(T, T+s)] = \frac{P(t, T+s)}{P(t, T)} \\ P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)$$

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• Call and Put Price Bounds

  #Financial #Economics

  $$(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \le c(S_{t},K,t,T) \le F^{P}_{t,T}(S) \\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \le p(S_{t},K,t,T) \le Ke^{-r(T-t)} \\ c(S_{t},K,t,T) \le C(S_{t},K,t,T) \le S_{t} \\ p(S_{t},K,t,T) \le P(S_{t},K,t,T) \le K$$

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• Calls and Puts Arbitrage

  #Financial #Economics

  $$K_{1} < K_{2} < K_{3} \\ K_{2} = \lambda K_{1} + (1 - \lambda) K_{3} \\ \lambda = \frac{K_{3} - K_{2}}{K_{3} - K_{1}}$$

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• Cox-Ingersoll-Ross CIR

  #Financial #Economics

  $$\mathrm{d} r(t) = a[b - r(t)] \mathrm{d} t + \sigma \sqrt{r(t)} \mathrm{d} Z(t) \\ P(r, t, T) = A(T-t)e^{-rB(T-t)} \\ \gamma = \sqrt{(a-\bar{\phi})^{2} + 2 \sigma^{2}} \\ q(r, t, T) = \sigma \sqrt{r} B(T-t) \\ \text{yield to maturity} \\ \frac{2ab}{ a - \bar{\phi} + \gamma}$$

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• Early Exercise for American Options

  #Financial #Economics

  $$P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)$$

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• Geometric Brownian Motion

  #Financial #Economics

  $$Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\mu t + \sigma Z(t)]} \\ E(e^{kU}) = e^{kE(U) + \frac{1}{2}k^{2}\text{Var}(U)} \\ E[Y^{k}(t)] = Y^{k}(0) e^{(k\mu + \frac{1}{2}k^{2}\sigma^{2})t} \\ \ln Y(t) \sim N(\ln Y(0) + \mu t, \sigma^{2} t)$$

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• Geometric Brownian Motion SDEs

  #Financial #Economics

  $$\mathrm{d}Y(t) = \mu Y(t)dt + \sigma Y(t) \mathrm{d}Z(t) \\ \mathrm{d}[\ln Y(t)] = (\mu - \frac{\sigma^2}{2}) \mathrm{d}t + \sigma \mathrm{d}Z(t) \\ Y(t) = T(0) e^{(\mu - \frac{\sigma^2}{2})t + \sigma Z(t)}$$

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• Ito Lemma

  #Financial #Economics

  $$\mathrm{d}X(t) = a(t, X(t)) \mathrm{d}t + b(t, X(t))\mathrm{d} Z(t) \\ Y(t) = f(t, X(t)) \mathrm{d}t \\ \mathrm{d} Y(t) = f_{t}(t, X(t)) + f_{x}(t, X(t))\mathrm{d} X(t) + \frac{1}{2} f_{xx}(t, X(t))[\mathrm{d}X(t)]^{2} \\ [\mathrm{d} X(t)]^{2} = b^{2}(t, X(t))\mathrm{d} t$$

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• Monte-Carlo Simulations

  #Financial #Economics

  $$S(T) = S(0) e^{(a - \delta - \frac{\sigma^2}{2})T + \sigma \sqrt{T} z} \\ S(T) = S(t) e^{(a - \delta - \frac{\sigma^2}{2})(T-t) + \sigma (Z(T) - Z(t))} \\ \text{Variance} \\ e^{-2rT} \times \frac{s^{2}}{n} \\ s^{2} = \frac{1}{n-1} \sum [(g(S_{i}) - \bar{g})]^{2}$$

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• Risk-Neutral Valuation and Power Contracts

  #Financial #Economics

  $$\frac{\mathrm{d}S(t)}{S(t)} = (r - \delta) \mathrm{d}t + \sigma \mathrm{d} \tilt{Z}(t) \\ \tilt{Z}(t) = Z(t) + \phi t \\ V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)] \\ F^{p}_{t, T}(S^{a}) = S^{a}(t) e ^{ (-r + a(r-\delta) + \frac{1}{2} a(a-1)\sigma^{2})(T-t)}$$

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• Sharpe Ratio

  #Financial #Economics

  $$\frac{\mathrm{d}X(t)}{X(t)} = m \mathrm{d}t + s \mathrm{d}Z(t) \\ \phi = \frac{m + \delta -r }{s} \\ \phi = \frac{a - r}{\sigma}$$

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• Standard Brownian Motion

  #Financial #Economics

  $$Z(t) \sim N(0, t) \\ Z(t+s) - Z(t) \sim N(0, s) \\ Z(t+s) \sim N(Z(t), s)$$

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• Stock Prices as Geometric Brownian Motion

  #Financial #Economics

  $$\frac{\mathrm{d}S(t)}{S(t)} = (a - \delta) \mathrm{d}t + \sigma \mathrm{d}Z(t) \\ S(t) = S(0) e^{(a - \delta - \frac{\sigma^{2}}{2})t + \sigma Z(t)} \\ \mathrm{d}[\ln S(t)] = (a - \delta - \frac{\sigma^{2}}{2}) \mathrm{d}t + \sigma \sigma \mathrm{d} Z(t) \\ S(t) \sim \ln( \ln S(0) + (a - \delta - \frac{\sigma^2}{2})t, \sigma^{2}t)$$

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• Varying Times to Expiration

  #Financial #Economics

  $$T_{2} \ge T_{1} \\ C(S_{t},K,t,T_{2}) \ge C(S_{t},K,t,T_{1}) \le S_{t} \\ P(S_{t},K,t,T_{2}) \ge P(S_{t},K,t,T_{1}) \le S_{t}$$

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nlp

• BLEU Bilingual Evaluation Understudy

  #nlp #BLEU #evaluation

  $$ \text{BLEU}_{w}(\hat{S},S)=BP(\hat{S};S) \times \exp{\sum^{\infty}_{n=1}w_{n} \ln p_{n}(\hat{S};S)}, p_{n}(\hat{S};S)=\frac{\sum^{M}_{i=1}\sum_{s \in G_{n}(\hat{y})} \min(C(s,\hat{y}),\max_{y \in S_{i}} C(s,y))}{\sum^{M}_{i=1}\sum_{s \in G_{n}(\hat{y})}C(s, \hat{y})}, p_{n}(\hat{y};y)=\frac{\sum_{s \in G_{n}(\hat{y})} \min(C(s,\hat{y}), C(s,y))}{\sum_{s \in G_{n}(\hat{y})}C(s, \hat{y})}, BP(\hat{S};S) = e^{-(r/c-1)^{+}}$$

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• Conditional Random Field CRF

  #machine learning #nlp

  $$P(y|x)=\frac{1}{Z(x)}\exp(\sum_{i,k}\lambda_{k}t_{k}(y_{i-1},y_{i},x,i))+\sum_{i,l}\mu_{l}s_{l}(y_{i},x,i)) \\ Z(x)=\sum_{y}\exp(\sum_{i,k}\lambda_{k}t_{k}(y_{i-1},y_{i},x,i))+\sum_{i,l}\mu_{l}s_{l}(y_{i},x,i))$$

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• Direct Policy Optimization DPO

  #nlp #llm #RLHF

  $$\pi_{r} (y|x) = \frac{1}{Z(x)} \pi_{ref} (y|x) \exp(\frac{1}{\beta} r(x,y) ) , r(x,y) = \beta \log \frac{\pi_{r} (y|x)}{\pi_{ref} (y|x)} + \beta \log Z(x) , p^{*}(y_{1} > y_{2} |x) = \frac{1}{1+\exp{(\beta \frac{\pi^{*} (y_{2}|x)}{\pi_{ref} (y_{2}|x)} - \beta \frac{\pi^{*} (y_{1}|x)}{\pi_{ref} (y_{1}|x)} )}} , \mathcal{L}_{DPO}(\pi_{\theta};\pi_{ref}) = -\mathbb{E}_{(x, y_{w},y_{l}) \sim D } [\log \sigma (\beta \frac{\pi_{\theta} (y_{w}|x)}{\pi_{ref} (y_{w}|x)} - \beta \frac{\pi_{\theta} (y_{l}|x)}{\pi_{ref} (y_{l}|x)} )] , \nabla \mathcal{L}_{DPO}(\pi_{\theta};\pi_{ref}) = - \beta \mathbb{E}_{(x, y_{w},y_{l}) \sim D } [ \sigma ( \hat{r}_{\theta} (x, y_{l}) - \hat{r}_{\theta} (x, y_{w})) [\nabla_{\theta} \log \pi (y_{w}|x) - \nabla_{\theta} \log \pi (y_{l}|x) ] ] , \hat{r}_{\theta} (x, y) = \beta \log (\frac{\pi_{\theta} (y|x)}{\pi_{ref} (y|x)})$$

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• Hidden Markov Model

  #machine learning #nlp

  $$Q=\{q_{1},q_{2},...,q_{N}\}, V=\{v_{1},v_{2},...,v_{M}\} \\ I=\{i_{1},i_{2},...,i_{T}\},O=\{o_{1},o_{2},...,o_{T}\} \\ A=[a_{ij}]_{N \times N}, a_{ij}=P(i_{t+1}=q_{j}|i_{t}=q_{i}) \\ B=[b_{j}(k)]_{N \times M},b_{j}(k)=P(o_{t}=v_{k}|i_{t}=q_{j})$$

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• KTO Kahneman-Tversky Optimisation Equation

  #nlp #llm #AI

  $$f(\pi_\theta, \pi_\text{ref}) = \mathbb{E}_{x,y\sim\mathcal{D}}[ a_{x,y} v(r_\theta(x,y) - \mathbb{E}_{Q}[r_\theta(x, y')])] + C_\mathcal{D}$$

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• LOW RANK ADAPTATION LORA

  #nlp #llm #RLHF

  $$W_{0} + \Delta W_{0} = W_{0} + BA, h=W_{0}x + \Delta W_{0}x = W_{0}x + BAx, \text{Initialization:} A \sim N(0, \sigma^{2}), B = 0$$

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• Perplexity of Language Model

  #nlp #LLM #metric

  $$\text{PPL}(X) = \exp \{- \frac{1}{t} \sum^{t}_{i} \log p_{\theta} (x_{i} | x_{ \lt i}) \}$$

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• RLHF Reinforcement Learning from Human Feedback

  #nlp #LLM #equation

  $$p^*(y_w \succ y_l|x) = \sigma(r^*(x,y_w) - r^*(x,y_l)) $$ $$ \mathcal{L}_R(r_\phi) = \mathbb{E}_{x,y_w,y_l \sim D}[- \log \sigma(r_\phi(x, y_w) - r_\phi(x, y_l))] $$ $$ \mathbb{E}_{x \in D, y \in \pi_\theta} [r_\phi(x,y)] - \beta D_{\text{KL}}(\pi_\theta(y|x) \| \pi_{\text{ref}}(y|x)) $$

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• Transformer

  #machine learning #nlp #gpt

  $$\text{Attention}(Q, K, V) = \text{softmax}(\frac{QK^T}{\sqrt{d_k}})V$$

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physics

• Amplitude of a driven oscillation

  #physics #amplitude #oscillation

  $$A = \frac{{F_0 }}{{\sqrt {m^2 \left( {\omega _0^2 - \omega ^2 } \right)^2 + b^2 \omega ^2 } }}$$

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• Angular frequency for a damped oscillation

  #physics #angular #damped oscillation #oscillation

  $$\omega ' = \omega _0 \sqrt {1 - \left( {\frac{b}{{2m\omega _0 }}} \right)^2 } = \omega _0 \sqrt {1 - \frac{1}{{4Q^2 }}}$$

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• Coupled Conductors and Transformers

  #physics #coupled #conductors #transformers

  $$M_{12}=M_{21}:=M=k\sqrt{L_1L_2}=\frac{N_1\Phi_1}{I_2}=\frac{N_2\Phi_2}{I_1}\sim N_1N_2 \\ \frac{V_1}{V_2}=\frac{I_2}{I_1}=-\frac{i\omega M}{i\omega L_2+R_{\rm load}}\approx-\sqrt{\frac{L_1}{L_2}}=-\frac{N_1}{N_2} \\ \Phi_{12}=M_{12}I_2 \\ \Phi_{21}=M_{21}I_1$$

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• Cylindrical Waves

  #physics #cylindrical waves

  $$\frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}-\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial u}{\partial r}\right)=0 \\ u(r,t)=\frac{\hat{u}}{\sqrt{r}}\cos(k(r\pm vt))$$

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• Einstein Field Equations

  #physics

  $$G^{\alpha\beta}:=R^{\alpha\beta}-\frac{1}{2}g^{\alpha\beta}R \\ G_{\alpha\beta}=\frac{8\pi \kappa}{c^{2}}T_{\alpha\beta}$$

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• Electric Oscillations

  #physics #oscillations

  $$\text{Impedance} \\ Z=R+ix \\ \text{Series connection} \\ V=IZ, Z_{\rm tot}=\sum_i Z_i~,~~L_{\rm tot}=\sum_i L_i~,~~ \frac{1}{C_{\rm tot}}=\sum_i\frac{1}{C_i}~,~~Q=\frac{Z_0}{R}~,~~ Z=R(1+iQ\delta) \\ \text{Parallel connection} \\ \frac{1}{Z_{\rm tot}}=\sum_i\frac{1}{Z_i}~,~~ \frac{1}{L_{\rm tot}}=\sum_i\frac{1}{L_i}~,~~ C_{\rm tot}=\sum_i C_i~,~~Q=\frac{R}{Z_0}~,~~ Z=\frac{R}{1+iQ\delta}$$

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• General Relativity

  #physics #relativity

  $$\frac{\mathrm{d}^{2} x^{\alpha}}{\mathrm{d} s^{2}}+\Gamma^{\alpha}_{\beta\gamma}\frac{\mathrm{d} x^{\beta}}{\mathrm{d} s}\frac{\mathrm{d} x^{\gamma}}{\mathrm{d} s}=0$$

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• Harmonic Oscillations

  #physics #harmonic #oscillations

  $$\Psi(t)=\hat{\Psi}(t)e^{i(\omega t \pm \phi)}=\hat{\Psi}(t)\cos (\omega t \pm \phi) \\ \sum_{i} \hat{\Psi_{i}}\cos(\alpha_{i} \pm \omega t) =\hat{\Phi}\cos (\beta \pm \omega t) \\ \tan (\beta)=\frac{\sum_{i} \hat{\Psi_{i}} \sin (\alpha_{i})}{\sum_{i} \hat{\Psi_{i}} \cos (\alpha_{i})} \\ \hat{\Phi}^{2} = \sum_{i} \hat{\Psi_{i}^{2}} + 2 \sum_{j > i} \sum_{i} \hat{\Psi_{i}} \hat{\Psi_{j}} \cos (\alpha_{i} - \alpha_{j}) \\ \int x(t) dt=\frac{x(t)}{i \omega} \\ \frac{d^{n}(x(t))}{d t^{n}}=(i \omega)^{n} x(t)$$

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• Lorentz Transformation Latex

  #physics #relativity #lorentz #transformation

  $$(\vec{x}^{'}, t^{'}) = (\vec{x}^{'}(\vec{x},t), t^{'}(\vec{x},t)) \\ \vec{x}^{'} = \vec{x} +\frac{(\gamma-1)(\vec{x}\vec{v}) \vec{v}}{|v|^{2}} - \gamma \vec{v}t \\ t^{'} = \frac{\gamma(t-\vec{x}\vec{v})}{c^{2}} \\ \gamma = \frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}} \\ \frac{\partial^{2}}{\partial{x^{2}}} + \frac{\partial^{2}}{\partial{y^{2}}} + \frac{\partial^{2}}{\partial{z^{2}}} - \frac{1}{c^{2}} \frac{\partial^{2}}{\partial{t^{2}}} = \frac{\partial^{2}}{\partial{{x^{'}}^{2}}} + \frac{\partial^{2}}{\partial{{y^{'}}^{2}}} + \frac{\partial^{2}}{\partial{{z^{'}}^{2}}} - \frac{1}{c^{2}} \frac{\partial^{2}}{\partial{{t^{'}}^{2}}}$$

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• Maxwell Equations Differential

  #physics #maxwell #electricity #magnetism

  $$\nabla \cdot \vec{D}=\rho_{free} \\ \nabla \cdot \vec{B}=0 \\ \nabla \times \vec{E}=-\frac{\partial{\vec{B}}}{\partial{t}} \\ \nabla \times \vec{H}=\vec{J}_{free}+\frac{\partial{\vec{D}}}{\partial{t}}$$

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• Maxwell Equations Integral

  #physics #maxwell #electricity #magnetism

  $$\oiint (\vec{D}\cdot \vec{n}) \mathrm{d}^{2}A=Q_{\text{free,included}}\\ \oiint (\vec{B}\cdot \vec{n}) \mathrm{d}^{2}A=0 \\ \oint \vec{E} \mathrm{d}\vec{s}=-\frac{\mathrm{d}\Phi}{\mathrm{d}t}\\ \oint \vec{H} \mathrm{d}\vec{s}=I_{\text{free,included}}+\frac{\mathrm{d}\Psi }{\mathrm{d}t}$$

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• Mechanic Oscillations

  #physics #mechanic #oscillations

  $$m\ddot{x}=F(t)-k\dot{x}-Cx \\ F(t)=\hat{F}\cos(\omega t) \\ -m\omega^2 x=F-Cx-ik\omega x \\ \omega_0^2=C/m \\ x=\frac{F}{m(\omega_0^2-\omega^2)+ik\omega} \\ \dot{x}=\frac{F}{i\sqrt{Cm}\delta+k} \\ \delta=\frac{\omega}{\omega_0}-\frac{\omega_0}{\omega} \\ Z=F/\dot{x} \\ Q=\frac{\sqrt{Cm}}{k}$$

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• Pendulums

  #physics #pendulums

  $$T=1/f \\ T=2\pi\sqrt{m/C} \\ T=2\pi\sqrt{I/\tau} \\ T=2\pi\sqrt{I/\kappa} \\ T=2\pi\sqrt{l/g}$$

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• Plane Waves

  #physics #plane waves

  $$u(\vec{x},t)=2^n\hat{u}\cos(\omega t)\sum_{i=1}^n\sin(k_ix_i) \\ u(\vec{x},t)=\hat{u}\cos(\vec{k}\cdot\vec{x}\pm\omega t+\varphi) \\ \frac{f}{f_0}=\frac{v_{\rm f}-v_{\rm obs}}{v_{\rm f}}$$

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• Red and Blue Shift

  #physics #red and blue shift

  $$\vec{e_{v}}\vec{e_{r}}=\cos(\Phi) \\ \frac{f^{'}}{f}=\gamma(1-\frac{v\cos(\Phi)}{c}) \\ \frac{\Delta f}{f}=\frac{\kappa M}{rc^{2}} \\ \frac{\lambda_{0}}{\lambda_{1}}=\frac{R_{0}}{R_{1}}$$

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• Riemannian Tensor

  #physics #riemannian rensor

  $$R^{\mu}_{v \alpha \beta}T^{v}=\triangledown_{\alpha}\triangledown_{\beta}T^{\mu}-\triangledown_{\beta}\triangledown_{\alpha}T^{\mu} \\ \triangledown_{j}a^{i}=\partial_{j}a^{i}+\Gamma^{i}_{jk}a^{k} \\ \triangledown_{j}a_{i}=\partial_{j}a_{i}-\Gamma^{k}_{ij}a_{k} \\ \Gamma^{i}_{jk}=\frac{\partial^{2} \bar{x}^{l}}{\partial{x^{j}}\partial{x^{k}}}\frac{\partial{x^{i}}}{\partial{\bar{x}^{l}}}$$

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• Spherical Waves

  #physics #spherical waves

  $$\frac{1}{v^2}\frac{\partial^2 (ru)}{\partial t^2}-\frac{\partial^2 (ru)}{\partial r^2}=0 \\ u(r,t)=C_1\frac{f(r-vt)}{r}+C_2\frac{g(r+vt)}{r}$$

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• Stress-energy Tensor

  #physics #stress-energy tensor

  $$T_{\mu v}=(\varrho c^{2}+p)u_{p}u_{v}+pg_{\mu v}+\frac{1}{c^{2}}(F^{\mu}_{\alpha}F^{\alpha v} + \frac{1}{4}g^{\mu v}F^{\alpha\beta}F_{\alpha\beta}) \\ \triangledown_{v} T_{\mu v}=0 \\ F_{\alpha\beta}=\frac{\partial{A_{\beta}}}{\partial{x^{\alpha}}} - \frac{\partial{A_{\alpha}}}{\partial{x^{\beta}}} \\ \frac{d p_{\alpha}}{d \tau}=qF_{\alpha\beta}u^{\beta}$$

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• The general solution of Wave Equation

  #physics #general solution #waves

  $$\frac{\partial^2 u(x,t)}{\partial t^2}=\sum_{m=0}^{N}\left(b_m\frac{\partial ^m}{\partial x^m}\right)u(x,t) \\ u(x,t)=\int\limits_{-\infty}^{\infty}\left(a(k){\rm e}^{i(kx-\omega_1(k)t)}+ b(k){\rm e}^{i(kx-\omega_2(k)t)}\right)dk \\ u(x,t)=A{\rm e}^{i(kx-\omega t)} \\ \omega_j=\omega_j(k)$$

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• Wave Equation

  #physics #wave

  $$\nabla^2u-\frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+\frac{\partial^2 u}{\partial z^2}-\frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}=0 \\ v=f\lambda \\ k\lambda=2\pi \\ \omega=2\pi f \\ v_{\rm g}=\frac{d\omega}{dk}=v_{\rm ph}+k\frac{dv_{\rm ph}}{dk}= v_{\rm ph}\left(1-\frac{k}{n}\frac{dn}{dk}\right) \\ v=\sqrt{\kappa/\varrho}$$

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• Waves In Long Conductors

  #physics #waves #oscillations

  $$Z_0=\sqrt{\frac{dL}{dx}\frac{dx}{dC}} \\ v=\sqrt{\frac{dx}{dL}\frac{dx}{dC}}$$

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• Angular frequency for a damped oscillation

  #Physics #Angular frequency

  $$\omega ' = \omega _0 \sqrt {1 - \left( {\frac{b}{{2m\omega _0 }}} \right)^2 } = \omega _0 \sqrt {1 - \frac{1}{{4Q^2 }}}$$

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• Bending of light Fermat's principle

  #physics #optics #Fermat's principle

  $$\delta\int\limits_1^2 dt=\delta\int\limits_1^2\frac{n(s)}{c}ds=0\Rightarrow \delta\int\limits_1^2 n(s)ds=0$$

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• Bending of light Snell's law

  #physics #optics #Snell's law

  $$n_i\sin(\theta_i)=n_t\sin(\theta_t) \\ \frac{n_2}{n_1}=\frac{\lambda_1}{\lambda_2}=\frac{v_1}{v_2} \\ n^2=1+\frac{n_{\rm e}e^2}{\varepsilon_0m}\sum_j\frac{f_j}{\omega_{0,j}^2-\omega^2-i\delta\omega} \\ v_{\rm g}=c/(1+(n_{\rm e}e^2/2\varepsilon_0m\omega^2)) $$

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• Birefringence and Dichroism

  #physics #optics #girefringence #dichroism

  $$\text{NA}$$

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• Diffraction

  #physics #optics #diffraction

  $$\frac{I(\theta)}{I_0}=\left(\frac{\sin(u)}{u}\right)^2\cdot \left(\frac{\sin(Nv)}{\sin(v)}\right)^2 \\ u=\pi b\sin(\theta)/\lambda \\ v=\pi d\sin(\theta)/\lambda$$

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• Displacement of a driven oscillator

  #physics #displacement

  $$x = A\cos \left( {\omega t + \delta } \right)$$

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• Fabry Perot Interferometer

  #physics #optics #fabry perot

  $$T+R+A=1$$

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• Fresnel Equations

  #physics #optics #fresnel

  $$r_\parallel=\frac{\tan(\theta_i-\theta_t)}{\tan(\theta_i+\theta_t)} \\ r_\perp =\frac{\sin(\theta_t-\theta_i)}{\sin(\theta_t+\theta_i)} \\ t_\parallel=\frac{2\sin(\theta_t)\cos(\theta_i)}{\sin(\theta_t+\theta_i)\cos(\theta_t-\theta_i)} \\ t_\perp =\frac{2\sin(\theta_t)\cos(\theta_i)}{\sin(\theta_t+\theta_i)}$$

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• Green functions for the initial-value problem

  #physics #green functions

  $$u(x,t)=\int\limits_{-\infty}^\infty f(x')Q(x,x',t)dx'+ \int\limits_{-\infty}^\infty g(x')P(x,x',t)dx'$$

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• Mirrors

  #physics #optics #mirrors

  $$\frac{1}{f}=\frac{1}{v}+\frac{1}{b}=\frac{2}{R}+\frac{h^2}{2}\left(\frac{1}{R}-\frac{1}{v}\right)^2$$

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• Non-linear Wave Equation

  #physics #non-linear wave

  $$\frac{d^2x}{dt^2}-\varepsilon\omega_0(1-\beta x^2)\frac{dx}{dt}+\omega_0^2x=0 \\ \frac{d}{dt}\left\{\frac{1}{2}\left(\frac{dx}{dt}\right)^2+\frac{1}{2} \omega_0^2x^2\right\}= \varepsilon\omega_0(1-\beta x^2)\left(\frac{dx}{dt}\right)^2 \\ \frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}-\underbrace{au\frac{\partial u}{\partial x}}_{\rm non-lin}+ \underbrace{b^2\frac{\partial ^3u}{\partial x^3}}_{\rm dispersive}=0 \\ u(x-ct)=\frac{-d}{\cosh^2(e(x-ct))}$$

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• Optical System Matrix Methods

  #physics #optics #Matrix Methods

  $$\left(\begin{array}{c}n_2\alpha_2\\y_2\end{array}\right)=M \left(\begin{array}{c}n_1\alpha_1\\y_1\end{array}\right) \\ {\rm Tr}(M)=1$$

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• Optics Magnification

  #physics #optics #magnification

  $$N=-\frac{b}{v} \\ N_{\alpha}=-\frac{\alpha_{\rm syst}}{\alpha_{\rm none}}$$

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• Optics Polarization

  #physics #optics #polarization

  $$P=\frac{I_{\rm p}}{I_{\rm p}+I_{\rm u}}=\frac{I_{\rm max}-I_{\rm min}}{I_{\rm max}+I_{\rm min}} \\ I(\theta)=I(0)\cos^2(\theta)$$

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• Optics Principal Planes

  #physics #optics #principal planes

  $$h_1=n\frac{m_{11}-1}{m_{12}} \\ h_2=n\frac{m_{22}-1}{m_{12}}$$

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• Prisms and Dispersion

  #physics #optics #prisms #dispersion

  $$\delta=\theta_i+\theta_{i'}+\alpha \\ n=\frac{\sin(\frac{1}{2}(\delta_{\rm min}+\alpha))}{\sin(\frac{1}{2}\alpha)} \\ D=\frac{d\delta}{d\lambda}=\frac{d\delta}{dn}\frac{dn}{d\lambda} \\ \frac{d\delta}{dn}=\frac{2\sin(\frac{1}{2}\alpha)}{\cos(\frac{1}{2}(\delta_{\rm min}+\alpha))}$$

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• Reflection and Transmission

  #physics #optics #reflection #transmission

  $$r_\parallel\equiv\left(\frac{E_{0r}}{E_{0i}}\right)_\parallel \\ r_\perp\equiv\left(\frac{E_{0r}}{E_{0i}}\right)_\perp \\ t_\parallel\equiv\left(\frac{E_{0t}}{E_{0i}}\right)_\parallel \\ t_\perp\equiv\left(\frac{E_{0t}}{E_{0i}}\right)_\perp$$

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• The Stationary Phase Method Wave Equation

  #physics #stationary phase

  $$\int\limits_{-\infty}^\infty a(k){\rm e}^{i(kx-\omega(k)t)}dk\approx \sum_{i=1}^{N}\sqrt{\frac{2\pi}{\frac{d^2\omega(k_i)}{dk_i^2}}} \exp\left[-i\mbox{$\frac{1}{4}$}\pi+i(k_ix-\omega(k_i)t)\right]$$

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• Thermodynamics Ideal Mixtures

  #physics #thermodynamics #ideal mixtures

  $$U_{\rm mixture}=\sum_i n_i U^0_i \\ H_{\rm mixture}=\sum_i n_i H^0_i \\ S_{\rm mixture}=n\sum_i x_i S^0_i+\Delta S_{\rm mix} \\ \Delta S_{\rm mix}=-nR\sum\limits_i x_i\ln(x_i)$$

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• Thermodynamics Statistical Basis

  #physics #thermodynamics

  $$P=N!\prod_i\frac{g_i^{n_i}}{n_i!} \\ n_i=\frac{N}{Z}g_i\exp\left(-\frac{W_i}{kT}\right) \\ Z=\sum\limits_ig_i\exp(-W_i/kT) \\ Z=\frac{V(2\pi mkT)^{3/2}}{h^3} \\ \text{Entropy in Thermodynamic Equilibrium} \\ S=\frac{U}{T}+kN\ln\left(\frac{Z}{N}\right)+kN\approx\frac{U}{T}+k\ln\left(\frac{Z^N}{N!}\right) \\ \text{Ideal gas} \\ S=kN+kN\ln\left(\frac{V(2\pi mkT)^{3/2}}{Nh^3}\right)$$

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• Waveguides and resonating cavities

  #physics #waveguides

  $$\begin{array}{ll} \vec{n}\cdot(\vec{D}_2-\vec{D}_1)=\sigma~~&~~\vec{n}\times(\vec{E}_2-\vec{E}_1)=0\\ \vec{n}\cdot(\vec{B}_2-\vec{B}_1)=0~~&~~\vec{n}\times(\vec{H}_2-\vec{H}_1)=\vec{K} \end{array} \\ \vec{E}(\vec{x},t)=\vec{\cal E}(x,y){e}^{i(kz-\omega t)} \\ \vec{B}(\vec{x},t)=\vec{\cal{B}}(x,y){e}^{i(kz-\omega t)}$$

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• Angular Momentum

  #physics #mechanics

  $$M = I\omega$$

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• Average Acceleration

  #physics #mechanics #Average Acceleration

  $$a_{av} = \frac{{\Delta \upsilon }}{{\Delta t}}$$

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• Bulk Modulus

  #physics #mechanics #bulk modulus

  $$B = - \frac{P}{{\Delta V}/V}$$

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• Centripetal Acceleration

  #physics #mechanics

  $$a=\frac{v^{2}}{r}$$

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• Displacement of a slightly damped oscillator

  #physics #oscillations

  $$x = A_0 \exp \left( { - \frac{b}{{2m}}t} \right)\cos \left( {\omega 't + \delta } \right)$$

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• Energy change in a damped oscillation

  #physics #oscillations

  $$\frac{{\Delta E}}{E} = - \frac{b}{m}T \\ E = E_0 \exp \left( { - \frac{b}{m}t} \right) = E_0 \exp \left( { - \frac{t}{\tau }} \right)$$

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• Energy transmitted by a harmonic wave

  #physics #oscillations

  $$\Delta E = \frac{1}{2}\mu \omega ^2 A^2 \Delta x = \frac{1}{2}\mu \omega ^2 A^2 \upsilon \Delta t$$

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• Harmonic Wave Function

  #physics #wave

  $$y(x,t) = A\sin \left[ {2\pi \left( {\frac{x}{\lambda } - \frac{t}{T}} \right)} \right] = A\sin \left[ {k(x - \upsilon t)} \right]$$

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• Isobaric Processes

  #physics #thermodynamics #isobaric processes

  $$\text{Isobaric Processes} \\ H_2-H_1=\int_1^2 C_pdT \\ \text{Reversible isobaric process} \\ H_2-H_1=Q_{\rm rev}$$

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• Mechanics Compressibility

  #physics #mechanics #compressibility

  $$k = \frac{1}{B} = - \frac{{\Delta V}/V}{P}$$

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• Phase Transitions

  #physics #thermodynamics #phase transitions

  $$\Delta S_m=S_m^\alpha - S_m^\beta=\frac{r_{\beta\alpha}}{T_0} \\ S_m=\left(\frac{\partial G_m}{\partial T}\right)_{p} \\ \frac{dp}{dT}=\frac{S_m^\alpha-S_m^\beta}{V_m^\alpha-V_m^\beta}= \frac{r_{\beta\alpha}}{(V_m^\alpha-V_m^\beta)T} \\ p=p_0{\rm e}^{-r_{\beta\alpha/RT}} \\ r_{\beta\alpha}=r_{\alpha\beta} \\ r_{\beta\alpha}=r_{\gamma\alpha}-r_{\gamma\beta} \\ \frac{dp}{dT}=\frac{S_m^\alpha-S_m^\beta}{V_m^\alpha-V_m^\beta}= \frac{r_{\beta\alpha}}{(V_m^\alpha-V_m^\beta)T} \\ p=p_0{\rm e}^{-r_{\beta\alpha/RT}}$$

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• Phase constant of a driven oscillation

  #physics #oscillation

  $$\tan \delta = \frac{{b\omega }}{{m\left( {\omega _0^2 - \omega ^2 } \right)}}$$

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• Reversible Adiabatic Processes

  #physics #thermodynamics

  $$\text{adiabatic processes} \\ W=U_1-U_2 \\ \text{reversible adiabatic processes} \\ \gamma=C_p/C_V$$

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• State functions Maxwell Relations

  #physics #thermodynamics

  $$\left(\frac{\partial T}{\partial V}\right)_{S}=-\left(\frac{\partial p}{\partial S}\right)_{V}~,~~\left(\frac{\partial T}{\partial p}\right)_{S}=\left(\frac{\partial V}{\partial S}\right)_{p}~,~~ \left(\frac{\partial p}{\partial T}\right)_{V}=\left(\frac{\partial S}{\partial V}\right)_{T}~,~~\left(\frac{\partial V}{\partial T}\right)_{p}=-\left(\frac{\partial S}{\partial p}\right)_{T} \\ TdS=C_VdT+T\left(\frac{\partial p}{\partial T}\right)_{V}dV~~\mbox{and}~~TdS=C_pdT-T\left(\frac{\partial V}{\partial T}\right)_{p}dp$$

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• The Carnot Cycle

  #physics #thermodynamics #the carnot cycle

  $$\eta=1-\frac{|Q_2|}{|Q_1|}=1-\frac{T_2}{T_1}:=\eta_{\rm C} \\ \xi=\frac{|Q_2|}{W}=\frac{|Q_2|}{|Q_1|-|Q_2|}=\frac{T_2}{T_1-T_2}$$

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• Thermal Heat Capacity

  #physics #thermodynamics #heat capacity

  $$C_p-C_V=T\left(\frac{\partial p}{\partial T}\right)_{V}\cdot\left(\frac{\partial V}{\partial T}\right)_{p}=-T\left(\frac{\partial V}{\partial T}\right)_{p}^2\left(\frac{\partial p}{\partial V}\right)_{T}\geq0 \\ \displaystyle C_X=T\left(\frac{\partial S}{\partial T}\right)_{X} \\ \displaystyle C_p=\left(\frac{\partial H}{\partial T}\right)_{p} \\ \displaystyle C_V=\left(\frac{\partial U}{\partial T}\right)_{V} \\ C_{mp}-C_{mV}=R$$

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• Thermodynamic Potential

  #physics #thermodynamics

  $$dG=-SdT+Vdp+\sum_i\mu_idn_i \\ \displaystyle\mu=\left(\frac{\partial G}{\partial n_i}\right)_{p,T,n_j} \\ V=\sum_{i=1}^c n_i\left(\frac{\partial V}{\partial n_i}\right)_{n_j,p,T}:=\sum_{i=1}^c n_i V_i \\ \begin{aligned} V_m&=&\sum_i x_i V_i\\ 0&=&\sum_i x_i dV_i\end{aligned}$$

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• Thermodynamics Conservation of Energy

  #physics #thermodynamics

  $$Q=\Delta U+W \\ d\hspace{-1ex}\rule[1.25ex]{2mm}{0.4pt} Q=dU+d\hspace{-1ex}\rule[1.25ex]{2mm}{0.4pt}W \\ d\hspace{-1ex}\rule[1.25ex]{2mm}{0.4pt}W=pdV \\ Q=\Delta H+W_{\rm i}+\Delta E_{\rm kin}+\Delta E_{\rm pot}$$

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• Thermodynamics Definitions

  #physics #thermodynamics

  $$f(x,y,z)=0 \\ dz=\left(\frac{\partial z}{\partial x}\right)_{y}dx+\left(\frac{\partial z}{\partial y}\right)_{x}dy \\ \left(\frac{\partial x}{\partial y}\right)_{z}\cdot\left(\frac{\partial y}{\partial z}\right)_{x}\cdot\left(\frac{\partial z}{\partial x}\right)_{y}=-1 \\ \varepsilon^m F(x,y,z)=F(\varepsilon x,\varepsilon y,\varepsilon z) \\ mF(x,y,z)=x\frac{\partial F}{\partial x}+y\frac{\partial F}{\partial y}+z\frac{\partial F}{\partial z}$$

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• Thermodynamics Nernst Law

  #physics #thermodynamics

  $$\lim_{T\rightarrow0}\left(\frac{\partial S}{\partial X}\right)_{T}=0$$

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• Throttle Processes

  #physics #thermodynamics #throttle processes

  $$\text{Isobaric Processes} \\ H_2-H_1=\int_1^2 C_pdT \\ \text{Reversible Isobaric Process} \\ H_2-H_1=Q_{\rm rev}$$

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• Black Body Radiation

  #physics #quantum #Black Body Radiation

  $$w(f)=\frac{8\pi hf^3}{c^3}\frac{1}{{\rm e}^{hf/kT}-1} \\ w(\lambda)=\frac{8\pi hc}{\lambda^5}\frac{1}{{\rm e}^{hc/\lambda kT}-1} \\ P=A\sigma T^4 \\ T\lambda_{\rm max}=k_{\rm W}$$

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• Harmonic Oscillator Quantum

  #physics #quantum

  $$H=\frac{p^2}{2m}+\frac{1}{2} m\omega^2 x^2= \frac{1}{2} \hbar\omega+\omega A^\dagger A \\ A=\sqrt{\mbox{$\frac{1}{2}$}m\omega}x+\frac{ip}{\sqrt{2m\omega}} \\ A^\dagger=\sqrt{\mbox{$\frac{1}{2}$}m\omega}x-\frac{ip}{\sqrt{2m\omega}} \\ HAu_E=(E-\hbar\omega)Au_E \\ u_n=\frac{1}{\sqrt{n!}}\left(\frac{A^\dagger}{\sqrt{\hbar}}\right)^nu_0 \\ u_0=\sqrt[4]{\frac{m\omega}{\pi\hbar}}\exp\left(-\frac{m\omega x^2}{2\hbar}\right) \\ E_n=( \frac{1}{2} +n)\hbar\omega$$

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• Mechanics Continuity Equation

  #physics #mechanics #continuity equation

  $$I_{V} = \upsilon A$$

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• Mechanics Displacement

  #physics #mechanics #displacement

  $${\Delta x}=x-x_{0}=v_{0}t+\frac{1}{2}at^{2}$$

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• Mechanics Stress

  #physics #mechanics #stress

  $${\text{Stress}} = \frac{F}{A}$$

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• Newton Second Law Force

  #physics #mechanics #newton

  $$F = ma$$

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• Operators in Quantum Physics

  #physics #quantum

  $$\int\psi_1^*A\psi_2d^3V=\int\psi_2(A\psi_1)^*d^3V \\ A\Psi=a\Psi \\ \Psi=\sum\limits_nc_nu_n \\ \frac{dA}{dt}=\frac{\partial A}{\partial t}+\frac{[A,H]}{i\hbar}$$

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• Parity

  #physics #quantum

  $${\cal P}\psi(x)=\psi(-x) \\ \psi(x)= \underbrace{\frac{1}{2} (\psi(x)+\psi(-x))}_{\rm even:~\hbox{$\psi^+$}}+ \underbrace{\frac{1}{2} (\psi(x)-\psi(-x))}_{\rm odd:~\hbox{$\psi^-$}} \\ \psi^+= \frac{1}{2} (1+{\cal P})\psi(x,t) \\ \psi^-= \frac{1}{2} (1-{\cal P})\psi(x,t)$$

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• Quantum Wave Functions

  #physics #quantum #wave

  $$\Phi(k,t)=\frac{1}{\sqrt{h}}\int\Psi(x,t){\rm e}^{-ikx}dx \\ \Psi(x,t)=\frac{1}{\sqrt{h}}\int\Phi(k,t){\rm e}^{ikx}dk \\ v_{\rm g}=p/m \\ E=\hbar\omega \\ \left\langle f(t) \right\rangle=\int\hspace{-1.5ex}\int\hspace{-1.5ex}\int\Psi^* f\Psi d^3V \\ \left\langle f_p(t) \right\rangle=\int\hspace{-1.5ex}\int\hspace{-1.5ex}\int\Phi^*f\Phi d^3V_p \\ \left\langle f(t) \right\rangle=\left\langle \Phi|f|\Phi \right\rangle \\ \left\langle \Phi|\Phi \right\rangle=\left\langle \Psi|\Psi \right\rangle=1$$

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• Shear Modulus

  #physics #mechanics #shear modulus

  $$M_{s} = \frac{F_{s} / A}{ {\Delta x} /L}=\frac{F_{s}/A}{\tan \theta}M_{s} = \frac{F_{s}/A}{\Delta x/L} = \frac{F_{s}/A}{\tan \theta}$$

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• Simple Harmonic Motion Acceleration

  #physics #mechanics

  $$a = - \omega ^2 x = - \omega ^2 r\sin (\omega t)$$

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• Speed of Sound Waves in a Fluid

  #physics #mechanics

  $$\upsilon = \sqrt {\frac{B}{\rho }}$$

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• The Compton Effect

  #physics #quantum #Compton Effect

  $$\lambda'=\lambda+\frac{h}{mc}(1-\cos\theta)=\lambda+\lambda_{\rm C}(1-\cos\theta)$$

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• The Schrödinger Equation

  #physics #quantum

  $$-\dfrac{\hbar^{2}}{2m}\bigtriangledown ^{2} \psi +U\psi=E\psi = i\hbar \dfrac{\partial \psi}{\partial t} \\ H=p^2/2m+U, H\psi=E\psi \\ \psi(x,t)=\left(\sum+\int dE\right)c(E)u_E(x)\exp\left(-\frac{iEt}{\hbar}\right) \\ \displaystyle J=\frac{\hbar}{2im}(\psi^*\nabla\psi-\psi\nabla\psi^*) \\ \displaystyle\frac{\partial P(x,t)}{\partial t}=-\nabla J(x,t)$$

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• The Tunnel Effect

  #physics #quantum

  $$\psi(x)=a^{-1/2}\sin(kx) \\ E_n=n^2h^2/8a^2m \\ \psi_1=A{\rm e}^{ikx}+B{\rm e}^{-ikx} \\ \psi_2=C{\rm e}^{ik'x}+D{\rm e}^{-ik'x} \\ \psi_3=A'{\rm e}^{ikx} \\ k'^2=2m(W-W_0)/\hbar^2 \ k^2=2mW \\ T=|A'|^2/|A|^2$$

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• The Uncertainty Principle

  #physics #quantum

  $$(\Delta A)^2=\left\langle \psi|A_{\rm op}-\left\langle A \right\rangle|^2\psi \right\rangle=\left\langle A^2 \right\rangle-\left\langle A \right\rangle^2 \\ \Delta A\cdot\Delta B\geq \frac{1}{2} |\left\langle \psi|[A,B]|\psi \right\rangle| \\ \Delta E\cdot\Delta t\geq\hbar \\ \Delta p_x\cdot\Delta x\geq \frac{1}{2} \hbar \\ \Delta L_x\cdot\Delta L_y\geq \frac{1}{2} \hbar L_z$$

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• Velocity

  #physics #mechanics #velocity

  $$\upsilon = \upsilon _0 + at$$

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• Viscous Flow

  #physics #mechanics #viscous flow

  $$F = \eta \frac{{\upsilon A}}{z}$$

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• Youngs Modulus

  #physics #mechanics #youngs modulus

  $$\Upsilon = \frac{F/A}{\Delta L/L} =\frac{\text{Stress}}{\text{Strain}}$$

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math

• Arithmetic and Geometric Progressions

  #math #arithmetic #geometric #progressions

  $$S_{n}=a+(a+d)+(a+2d)+...+[a+(n-1)d]=\frac{n}{2}[2a+(n-1)d] \\ S_{n}=a+ar+ar^{2}+...+ar^{n-1}=a\frac{1-r^{n}}{1-r}$$

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• Bessel Equation

  #math #bessel equation

  $$x^{2}\frac{\mathrm{d}^{2} y}{\mathrm{d} x^{2}}+x\frac{\mathrm{d} y}{\mathrm{d} x}+(x^{2}-m^{2})y=0 \\ J_{m}(x)=\sum^{\infty}_{k=0}\frac{(-1)^{k} (x/2)^{m+2k}}{k!(m+k)!}$$

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• Binomial Expansion

  #math #binomial #expansion

  $$(1+x)^{n}=1+nx+\frac{n(n-1)}{2!}x^{2}+...+ C^{r}_{n}x^{r} +...+x^{n} \\ C^{r}_{n}=\frac{n!}{r!(n-r)!}$$

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• Convergence of Series

  #math #convergence #series

  $$S_{n}=u_{1}+u_{2}+...+u_{n} \\ \text{Converge AS } n \rightarrow \infty \text{, If} \lim_{n \rightarrow \infty} |\frac{u_{n+1}}{u_{n}}| < 1$$

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• Determinants of a Matrix

  #math #determinants

  $$|A|=\sum_{i,j,k,...}\epsilon_{ijk}A_{1i}A_{2j}A_{3k}...$$

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• Diffusion Conduction Equation

  #math #diffusion #conduction

  $$\kappa u_{xx} = u_{t} \\ \frac{\partial \Psi}{\partial t}=\kappa \triangledown^{2} \Psi$$

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• Eigenvalues and Eigenvectors

  #math #eigenvalues #eigenvectors

  $$A\mathbf{u}=\lambda\mathbf{u} \\ P_{n}(\lambda)=|A-\lambda I|$$

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• Fourier Series

  #math #fourier series

  $$y(x)=c_{0}+\sum^{M}_{m=1}c_{m}\cos mx+\sum^{M^{'}}_{m=1}s_{m}\sin mx \\ c_{0}=\frac{1}{2\pi}\int^{\pi}_{-\pi}y(x) \mathrm{d} x \\ c_{m}=\frac{1}{\pi}\int^{\pi}_{-\pi}y(x) \cos mx \mathrm{d} x \\ s_{m}=\frac{1}{\pi}\int^{\pi}_{-\pi}y(x) \sin mx \mathrm{d} x$$

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• Fourier Transforms

  #math #fourier transforms

  $$y(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty} \hat{y}(\omega) e^{i\omega t} \mathrm{d} \omega \\ \hat{y}(\omega)=\int_{-\infty}^{\infty} y(t) e^{-i\omega t} \mathrm{d} t$$

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• Heat Equation

  #math #heat equation

  $$u_{t}={\alpha}^{2}u_{xx}$$

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• Laplace Equation

  #math #laplace equation

  $$u_{xx} = 0$$

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• Laplace Transform

  #math #laplace transform

  $$\bar{y}(s)=\mathcal{L}(y(t))=\int^{\infty}_{0} e^{-st} y(t) \mathrm{d}t$$

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• Legendre Equation

  #math #legendre equation

  $$(1-x^{2})\frac{\mathrm{d}^{2} y}{\mathrm{d} x^{2}}-2x\frac{\mathrm{d} y}{\mathrm{d} x}+l(l+1)y=0 \\ P_{l}(x)=\frac{1}{2^{l}l!}(\frac{\mathrm{d}}{\mathrm{d} x})^{l}(x^2-1)^{l}\\ P_{l}(x)=\frac{1}{l}[(2l-1)xP_{l-1}(x)-(l-1)P_{l-2}(x)]$$

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• Maclaurin Series

  #math #maclaurin series

  $$y(x)=y(a+u)=y(0)+x\frac{\mathrm{d} y}{\mathrm{d} x}+\frac{1}{2!}x^{2}\frac{\mathrm{d}^2 y}{\mathrm{d} x^{2}}+\frac{1}{3!}x^{3}\frac{\mathrm{d}^3 y}{\mathrm{d} x^{3}}+...$$

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• Plane Wave Expansion

  #math #plane wave expansion

  $$\exp(ikz)=\exp(ikr\cos \theta=\sum^{\infty}_{l=0}(2l+1)i^{l}j_{l}(kr)P_{l}(\cos \theta)$$

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• Power Series with Real Variables

  #math #power #series

  $$e^{x}=1+x+\frac{x^{2}}{2!}+...+\frac{x^{n}}{n!}+... \\ \ln(1+x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} + ... + (-1)^{n+1}\frac{x^{n}}{n!} +... \\ \cos(x) = \frac{e^{ix}+e^{-ix}}{2}=1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\frac{x^{6}}{6!}+...\\ \sin(x) = \frac{e^{ix}-e^{-ix}}{2i}=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}+...$$

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• Regression Least Squares Fitting

  #math #regression

  $$y_{i}=\alpha + \beta (x_{i} - \bar{x}) + \epsilon_{i} \\ \hat{\alpha}=\bar{y} \\ \hat{\beta}=\frac{s^{2}_{xy}}{s^{2}_{x}}$$

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• Spherical Harmonics Equation

  #math #spherical harmonics

  $$[\frac{1}{\sin \theta} \frac{\partial}{\partial \theta}(\sin \theta \frac{\partial}{\partial \theta}) + \frac{1}{\sin^{2} \theta} \frac{\partial^{2}}{\partial \phi^{2}}) ] Y^{m}_{l} + l(l+1) Y^{m}_{l}=0 \\ Y^{m}_{l}(\theta,\phi)=\sqrt{\frac{2l+1}{4 \pi} \frac{(l-|m|)!}{(l+|m|)!}}P^{m}_{l}(\cos \theta) e^{im \phi} \times \begin{cases}(-1)^{m} & m\ge 0 \\ 1 & m <0 \end{cases}$$

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• Taylor Series

  #math #taylor series

  $$y(x)=y(a+u)=y(a)+u\frac{\mathrm{d} y}{\mathrm{d} x}+\frac{1}{2!}u^{2}\frac{\mathrm{d}^2 y}{\mathrm{d} x^{2}}+\frac{1}{3!}u^{3}\frac{\mathrm{d}^3 y}{\mathrm{d} x^{3}}+...$$

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• Wave Equation

  #math #wave equation

  $$u_{tt}=c^{2}u_{xx}$$

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• Area of a Circle

  #Math #Geometry

  $$A = \pi r^2$$

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• Area of a Ellipse

  #Math #Geometry

  $$A = \pi r_1 r_2$$

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• Area of a Parallelogram

  #Math #Geometry

  $$A = bh$$

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• Area of a Rectangle

  #Math #Geometry

  $$A = lw$$

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• Area of a Regular Polygon

  #Math #Geometry

  $$A = \frac{{nsr}}{2} = \frac{{pr}}{2}$$

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• Area of a Rhombus

  #Math #Geometry

  $$A = \frac{{x_1 x_2 }}{2}$$

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• Area of a Sector

  #Math #Geometry

  $$A = \frac{{\theta r^2 }}{2}$$

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• Area of a Square

  #Math #Geometry

  $$A = x^2$$

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• Area of a Trapezoid

  #Math #Geometry

  $$A = \frac{1}{2}(x_1 + x_2 )h$$

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• Area of a Triangle

  #Math #Geometry

  $$A = \frac{1}{2}(x_1 + x_2 )h$$

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• Binomial Distribution

  #Math #Statistics

  $$X \sim B(n,p) \\f(x)=\begin{pmatrix}n\\ x\end{pmatrix}p^{x}q^{n-x}=C^{k}_{n}p^{x}q^{n-x},q=1-p\\\text{Binominal Mean}\ \mu=np\\\text{Binominal Variance}\ \sigma^2=npq$$

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• Complex Numbers

  #math #complex numbers

  $$z=x+iy=r(\cos \theta + i \sin \theta)=re^{i(\theta+2n\pi)} \\ z^{*}=x-iy=r(\cos \theta - i \sin \theta)=re^{-i\theta} \\ zz^{*}=|z|^{2}=x^{2}+y^{2}$$

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• De Moivre's Theorem

  #math #complex number #De Moivre

  $$(\cos \theta + i \sin \theta)^{n}=e^{in\theta}=\cos n\theta + i \sin n\theta$$

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• Exponential Distribution

  #Math #Statistics

  $$f(x, \lambda)=\lambda e^{-\lambda x} \\ F(x, \lambda)=1 - e^{-\lambda x}$$

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• Fisher Skewness

  #Math #Statistics

  $$\gamma_1 = \frac{{\mu_3 }}{{\mu_2 ^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} }} = \frac{{\mu_3 }}{{\sigma ^3 }}$$

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• Gamma Distribution

  #Math #Statistics

  $$\Gamma \left( a \right) = \int\limits_0^\infty {s^{a - 1} } e^{ - s} ds \\ P(x) = \frac{x^{\alpha-1} e^{-frac{x}{\theta}}}{\Gamma(\alpha) \theta^{\alpha}} \\ \mu = \alpha \theta \\ \sigma^{2} = \alpha \theta^{2} \\ \gamma_{1} = \frac{2}{\sqrt{\alpha}} \\ \gamma_{2} = \frac{6}{\alpha}$$

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• Normal Gaussian Distribution

  #Math #Statistics

  $$X \sim \mathcal{N}(\mu,\sigma^2) \\ f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp{[-\frac{(x-\mu)^{2}}{2\sigma^{2}}]}$$

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• Poisson Distribution

  #Math #Statistics

  $$X \sim \pi(\mu) \\f(x)=\frac{\mu^{x}}{x!}e^{-\mu}\\ \text{Poisson Mean} \mu \\ \text{Poisson Variance}\sigma^2=\mu$$

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• Power Series for Complex Variables

  #math #complex variables

  $$e^{z}=1+z+\frac{z^{2}}{2!}+\frac{z^{3}}{3!}+...+\frac{z^{n}}{n!}+...\\ \sin z=z-\frac{z^{3}}{3!}+\frac{z^{5}}{5!}-...\\ \cos z=1-\frac{z^{2}}{2!}+\frac{z^{4}}{4!}-...\\ \ln (1+z)=1-\frac{z^{2}}{2!}+\frac{z^{3}}{3!}-...\\ (1+z)^{n}=1+nz+\frac{n(n-1)}{2!}z^{2}+\frac{n(n-1)(n-2)}{3!}z^{3}+...$$

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• gaussian process

  #math #gaussian process

  $$\log p(y|X) \propto -[y^{T}(K + \sigma^{2}I)^{-1}y+\log|K + \sigma^{2}I|] \\ f(X)=\[f(x_{1}),f(x_{2}),...,f(x_{N}))\]^{T} \sim \mathcal{N}(\mu, K_{X,X}) \\ f_{*}|X_{*},X,y \sim \mathcal{N}(\mathbb{E}(f_{*}),\text{cov}(f_{*})) \\ \text{cov}(f_{*})=K_{X_{*},X_{*}}-K_{X_{*},X}[K_{X,X}+\sigma^{2}I]^{-1}K_{X,X_{*}}$$

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• Area of a Equilateral Triangle

  #Math #Geometry

  $$A = \frac{{h^2 \sqrt 3 }}{3}$$

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• Cartesian to Cylindrical Coordinates

  #Math #Geometry

  $$\begin{array}{*{20}c} {x = r\cos \theta } & {r = \sqrt {x^2 + y^2 } } & {} \\ {y = r\sin \theta } & {\theta = \tan ^{ - 1} \left( {\frac{y}{x}} \right)} & {} \\ {z = z} & {z = z} & {} \\ \end{array}$$

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• Cartesian to Polar Coordinates

  #Math #Geometry

  $$\begin{array}{*{20}c} {x = r\cos \theta } & {r = \sqrt {x^2 + y^2 } } \\ {y = r\sin \theta } & {\theta = \tan ^{ - 1} \left( {\frac{y}{x}} \right)} \\ \end{array}$$

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• Cartesian to Spherical Coordinates

  #Math #Geometry

  $$\begin{array}{*{20}c} {x = R\sin \theta \cos \phi } & {R = \sqrt {x^2 + y^2 + z^2 } } & {} \\ {y = R\sin \theta \sin \phi } & {\phi = \tan ^{ - 1} \left( {\frac{y}{x}} \right)} & {} \\ {z = R\cos \theta } & {\theta = \cos ^{ - 1} \left( {\frac{z}{{\sqrt {x^2 + y^2 + z^2 } }}} \right)} & {} \\ \end{array}$$

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• Circle

  #Math #Geometry

  $$\left( {x - x_0 } \right)^2 + \left( {y - y_0 } \right)^2 = R^2$$

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• Cylindrical to Spherical Coordinates

  #Math #Geometry

  $$\begin{array}{*{20}c} {r = R\sin \theta } & {R = \sqrt {r^2 + z^2 } } & {} \\ {z = R\sin \theta } & {\phi = \theta } & {} \\ {\theta = \phi } & {\theta = \tan ^{ - 1} \left( {\frac{r}{z}} \right)} & {} \\ \end{array}$$

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• Distance Between Two Points 2D

  #Math #Geometry

  $$d = \sqrt {\left( {x_1 - x_2 } \right)^2 + \left( {y_1 - y_2 } \right)^2 }$$

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• Distance Between Two Points 3D

  #Math #Geometry

  $$d = \sqrt {\left( {x_1 - x_2 } \right)^2 + \left( {y_1 - y_2 } \right)^2 + \left( {z_1 - z_2 } \right)^2 }$$

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• Eccentricity of a Hyperbola

  #Math #Geometry

  $$\varepsilon = \frac{{\sqrt {a^2 + b^2 } }}{a}$$

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• Eccentricity of an Ellipse

  #Math #Geometry

  $$\varepsilon = \frac{{\sqrt {a^2 - b^2 } }}{a}$$

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• Ellipse

  #Math #Geometry

  $$\frac{{\left( {x - x_0 } \right)^2 }}{{a^2 }} + \frac{{\left( {y - y_0 } \right)^2 }}{{b^2 }} = 1 $$

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• Hyperbola

  #Math #Geometry

  $$\frac{{\left( {x - x_0 } \right)^2 }}{{a^2 }} - \frac{{\left( {y - y_0 } \right)^2 }}{{b^2 }} = 1 $$

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• Hyperbolic Paraboloid

  #Math #Geometry

  $$\frac{{\left( {x - x_0 } \right)^2 }}{{a^2 }} - \frac{{\left( {y - y_0 } \right)^2 }}{{b^2 }} = \frac{{\left( {z - z_0 } \right)}}{c} $$

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• Parabola

  #Math #Geometry

  $$\left( {y - y_0 } \right)^2 = 4a\left( {x - x_0 } \right)$$

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• Plane

  #Math #Geometry

  $$Ax + By + Cz + D = 0$$

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• Sphere

  #Math #Geometry

  $$\left( {x - x_0 } \right)^2 + \left( {y - y_0 } \right)^2 + \left( {z - z_0 } \right)^2 = R^2 $$

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• Surface area of a Cuboid

  #Math #Geometry

  $$S = 2lw + 2lh + 2wh$$

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• Surface area of a Sphere

  #Math #Geometry

  $$S = 4\pi r^2$$

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• Surface area of a cube

  #Math #Geometry

  $$S = 6x^2$$

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• Surface area of a cylinder

  #Math #Geometry

  $$S = 2\pi r^2 + 2\pi rh$$

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• Bernoulli Distribution

  #Math #Statistics

  $$Pr(X=1) = p = 1- Pr(X=0) = 1 - q, \\ f(x)=p \text{ if } k = 1 \text{ else } q \text{ if } k = 0$$

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• Beta Binomial Distribution

  #Math #Statistics

  $$x \sim f(x | n, \alpha, \beta), \\ f(x | n, \alpha, \beta) = \int_{0}^{1} \text{Bin}(x|n,p)\text{Beta}(p|\alpha,\beta) dp , \\ f(x | n, \alpha, \beta) = C^{x}_{n} \frac{B(x+\alpha, n -x + \beta)}{B(\alpha,\beta)}$$

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• Beta Distribution

  #Math #Statistics

  $$X \sim Beta(\alpha,\beta), \\ f(x)=\frac{ x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}, \\ B(\alpha,\beta)=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}, \\ F(x)=I_{x}(\alpha+\beta)$$

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• Chi-Squared Distribution

  #Math #Statistics

  $$Q = \sum^{k}_{i=1} Z^{2}_{i} \sim \chi^{2}(k), \\ f(k) = \frac{1}{2^{k/2} \Gamma (k/2)} x^{k/2-1} e^{-x/2}, \\ F(k) = \frac{1}{\Gamma (k/2)} \gamma (\frac{k}{2},\frac{x}{2})$$

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• Ellipsoid

  #Math #Geometry

  $$\frac{{\left( {x - x_0 } \right)^2 }}{{a^2 }} + \frac{{\left( {y - y_0 } \right)^2 }}{{b^2 }} + \frac{{\left( {z - z_0 } \right)^2 }}{{c^2 }} = 1 $$

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• Elliptic Cone

  #Math #Geometry

  $$\frac{{\left( {x - x_0 } \right)^2 }}{{a^2 }} + \frac{{\left( {y - y_0 } \right)^2 }}{{b^2 }} = \frac{{\left( {z - z_0 } \right)^2 }}{{c^2 }}$$

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• Elliptic Cylinder

  #Math #Geometry

  $$\frac{{\left( {x - x_0 } \right)^2 }}{{a^2 }} + \frac{{\left( {y - y_0 } \right)^2 }}{{b^2 }} = 1 $$

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• Elliptic Paraboloid

  #Math #Geometry

  $$\frac{{\left( {x - x_0 } \right)^2 }}{{a^2 }} + \frac{{\left( {y - y_0 } \right)^2 }}{{b^2 }} = \frac{{\left( {z - z_0 } \right)}}{c} $$

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• Euler identity

  #math #euler

  $$e^{i\pi} + 1 = 0$$

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• Gaussian Integral

  #math #calculus

  $${\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}}\,dx={\sqrt {\pi }}.} $$

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• Geometric Distribution

  #Math #Statistics

  $$Pr(X=k) = (1-p)^{k-1}q, \\ f(x)=(1-p)^{k-1}q, \\ F(x)=1 - (1-p)^{[x]}$$

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• Gumbel Distribution

  #Math #Statistics

  $$x \sim \text{Gumbel}(\mu,\beta), \\ \frac{1}{\beta} e^{-(z + e^{-z})}, z=\frac{x - \mu}{\beta}, \\ e^{-e^{-(x-\mu)/\beta}}$$

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• Laplace Distribution

  #Math #Statistics

  $$x \sim \text{Laplace}(\mu,b), \\ f(x | \mu,b) =\frac{1}{2b} \exp (-\frac{|x-\mu|}{b}), \\ F(x | \mu,b) = \frac{1}{2} \exp (\frac{x - \mu}{b}) \text{ if } x \le \mu, 1 - \frac{1}{2} \exp (-\frac{x - \mu}{b}) \text{ if } x \ge \mu$$

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• Poisson Binomial Distribution

  #Math #Statistics

  $$Pr(K = k) = \sum_{A \in F_{k}} \prod_{i \in A} p_{i} \prod_{j \in A_{c}} (1-p_{j})$$

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• Pythagorean Theorem

  #math #geometry

  $$a^{2}+b^{2}=c^{2}$$

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• Riemann Hypothesis

  #math

  $$\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots $$ $$ \zeta (s)=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}}={\frac {1}{1-2^{-s}}}\cdot {\frac {1}{1-3^{-s}}}\cdot {\frac {1}{1-5^{-s}}}\cdot {\frac {1}{1-7^{-s}}}\cdots$$

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• Spiral of Archimedes

  #Math #Geometry

  $$r = a \theta$$

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• Student t-Distribution

  #Math #Statistics

  $$f(t) = \frac{\Gamma(\frac{v + 1}{2})}{\sqrt{v\pi}\Gamma(v/2)} (1+\frac{t^{2}}{v})^{-(v+1)/2}, \\ F(t) = \frac{1}{2} + x \Gamma(\frac{v+1}{2}) \times \frac{2^{F_{1}}(1/2,\frac{v+1}{2};3/2;-\frac{x^{2}}{v})}{\sqrt{\pi v} \Gamma(\frac{v}{2})}, \\ F(t) = 1 - \frac{1}{2} I_{x(t)} (\frac{v}{2}, \frac{1}{2}), x(t)=\frac{v}{t^{2} + v}$$

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• Uniform Distribution

  #Math #Statistics

  $$X \sim U(a,b) \\ f(x)=\frac{1}{b-a} \text{for} a \le x \le b \text{else} 0 \\ F(x)=\frac{x-a}{b-a} \text{for} a \le x \le b, 1 \text{for} x > b, 0 \text{for} x < a$$

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machine learning

• Area Under Uplift Curve AUUC

  #machine learning #causual inference

  $$f(t)=(\frac{Y^{T}_{t}}{N^{T}_{t}} - \frac{Y^{C}_{t}}{N^{C}_{t}})(N^{T}_{t}+N^{C}_{t})$$

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• Average Treatment Effect ATE

  #machine learning #causual inference

  $$\text{ATE}:=\mathbb{E}[Y(1)-Y(0)]$$

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• Bellman Equation

  #machine learning

  $$v_{\pi}(s)=\sum_{a}\pi(a|s)\sum_{s^{'},r}p(s^{'},r|s,a)[r+\gamma v_{\pi}(s^{'})]$$

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• Conditional Average Treatment Effect CATE

  #machine learning #causual inference

  $$\tau(x):=\mathbb{E}[Y(1)-Y(0)|X=x]$$

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• Deep Kernel Learning

  #machine learning #Deep Kernel Learning

  $$k(x_{i},x_{j}|\phi)=k(h(x_i,w_k),h(x_j,w_k)|w_k,\phi)$$

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• Diffusion Model Forward Process

  #machine learning #diffusion

  $$q(x_{t}|x_{t-1})=\mathcal{N}(x_{t};\sqrt{1-\beta_{t}}x_{t-1},\beta_{t}I) \\q(x_{1:T}|x_{0})=\prod_{t=1}^{T}q(x_{t}|x_{t-1})$$

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• Diffusion Model Forward Process Reparameterization

  #machine learning #diffusion

  $$x_{t}=\sqrt{\alpha_{t}}x_{t-1}+\sqrt{1-\alpha_{t}} \epsilon_{t-1}\\=\sqrt{\alpha_{t}\alpha_{t-1}}x_{t-2} + \sqrt{1-\alpha_{t}\alpha_{t-1}} \bar{\epsilon}_{t-2}\\=\text{...}\\=\sqrt{\bar{\alpha}_{t}}x_{0}+\sqrt{1-\bar{\alpha}_{t}}\epsilon \\\alpha_{t}=1-\beta_{t}, \bar{\alpha}_{t}=\prod_{t=1}^{T}\alpha_{t}$$

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• Diffusion Model Reverse Process

  #machine learning #diffusion

  $$p_\theta(\mathbf{x}_{0:T}) = p(\mathbf{x}_T) \prod^T_{t=1} p_\theta(\mathbf{x}_{t-1} \vert \mathbf{x}_t) \\ p_\theta(\mathbf{x}_{t-1} \vert \mathbf{x}_t) = \mathcal{N}(\mathbf{x}_{t-1}; \boldsymbol{\mu}_\theta(\mathbf{x}_t, t), \boldsymbol{\Sigma}_\theta(\mathbf{x}_t, t))$$

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• Diffusion Model Variational Lower Bound

  #machine learning #diffusion

  $$\begin{aligned} - \log p_\theta(\mathbf{x}_0) &\leq - \log p_\theta(\mathbf{x}_0) + D_\text{KL}(q(\mathbf{x}_{1:T}\vert\mathbf{x}_0) \| p_\theta(\mathbf{x}_{1:T}\vert\mathbf{x}_0) ) \\ &= -\log p_\theta(\mathbf{x}_0) + \mathbb{E}_{\mathbf{x}_{1:T}\sim q(\mathbf{x}_{1:T} \vert \mathbf{x}_0)} \Big[ \log\frac{q(\mathbf{x}_{1:T}\vert\mathbf{x}_0)}{p_\theta(\mathbf{x}_{0:T}) / p_\theta(\mathbf{x}_0)} \Big] \\ &= -\log p_\theta(\mathbf{x}_0) + \mathbb{E}_q \Big[ \log\frac{q(\mathbf{x}_{1:T}\vert\mathbf{x}_0)}{p_\theta(\mathbf{x}_{0:T})} + \log p_\theta(\mathbf{x}_0) \Big] \\ &= \mathbb{E}_q \Big[ \log \frac{q(\mathbf{x}_{1:T}\vert\mathbf{x}_0)}{p_\theta(\mathbf{x}_{0:T})} \Big] \\ \text{Let }L_\text{VLB} &= \mathbb{E}_{q(\mathbf{x}_{0:T})} \Big[ \log \frac{q(\mathbf{x}_{1:T}\vert\mathbf{x}_0)}{p_\theta(\mathbf{x}_{0:T})} \Big] \geq - \mathbb{E}_{q(\mathbf{x}_0)} \log p_\theta(\mathbf{x}_0) \end{aligned}$$

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• Diffusion Model Variational Lower Bound Loss

  #machine learning #diffusion

  $$\begin{aligned} L_\text{VLB} &= L_T + L_{T-1} + \dots + L_0 \\ \text{where } L_T &= D_\text{KL}(q(\mathbf{x}_T \vert \mathbf{x}_0) \parallel p_\theta(\mathbf{x}_T)) \\ L_t &= D_\text{KL}(q(\mathbf{x}_t \vert \mathbf{x}_{t+1}, \mathbf{x}_0) \parallel p_\theta(\mathbf{x}_t \vert\mathbf{x}_{t+1})) \text{ for }1 \leq t \leq T-1 \\ L_0 &= - \log p_\theta(\mathbf{x}_0 \vert \mathbf{x}_1) \end{aligned}$$

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• Generative Adversarial Networks GAN

  #machine learning #gan

  $$\min_{G} \max_{D} V(D,G)=\mathbb{E}_{x \sim p_{data}(x)}[\log D(x)]+\mathbb{E}_{z \sim p_{z}(z)}[\log(1-D(G(z)))]$$

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• Individual Treatment Effect ITE

  #machine learning #causual inference

  $$\text{ITE}_{i}:=Y_{i}(1)-Y_{i}(0)$$

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• Jensen-Shannon Divergence JS-Divergence

  #machine learning

  $$JS(P||Q)=\frac{1}{2}KL(P||\frac{(P+Q)}{2})+\frac{1}{2}KL(Q||\frac{(P+Q)}{2})$$

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• KL-Divergence

  #machine learning

  $$KL(P||Q)=\sum_{x}P(x)\log(\frac{P(x)}{Q(x)})$$

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• Kullback-Leibler Divergence

  #machine learning #kl divergence

  $$KL(P||Q)=\sum_{x}P(x)\log(\frac{P(x)}{Q(x)})$$

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• Mahalanobis Distance

  #machine learning #mahalanobis

  $$D_{M}(x,y)=\sqrt{(x-y)^{T}\Sigma^{-1}(x-y)}$$

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• Maximum Mean Discrepancy MMD

  #machine learning #mmd

  $$\textup{MMD}(\mathbb{F},X,Y):=\sup_{f \in\mathbb{F}}(\frac{1}{m}\sum_{i=1}^{m}f(x_{i}) -\frac{1}{n}\sum_{j=1}^{n}f(y_{j}))$$

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• Propensity Score

  #machine learning #causual inference

  $$e := p(W=1|X=x)$$

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• QINI

  #machine learning #causual inference

  $$g(t)=Y^{T}_{t}-\frac{Y^{C}_{t}N^{T}_{t}}{N^{C}_{t}},\\ f(t)=g(t) \times \frac{N^{T}_{t}+N^{C}_{t}}{N^{T}_{t}}$$

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• S-Learner

  #machine learning #causual inference

  $$\mu(x,w)=\mathbb{E}[Y_{i}|X=x_{i},W=w] \\ \hat{\tau}(x)=\hat{\mu}(x,1)-\hat{\mu}(x,0)$$

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• Support Vector Machine SVM

  #machine learning #svm

  $$\max_{w,b} \frac{2}{||w||} \\ s.t.\ y_{i}(w^{T}x_{i} + b) \geq 1, i=1,2,...,m \\ L(w,b,\alpha)=\frac{1}{2}||w||^2 + \sum^{m}_{i=1}a_{i}(1-y_{i}(w^{T}x_{i} + b))$$

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• T-Learner

  #machine learning #causual inference

  $$\mu_{0}(x)=\mathbb{E}[Y(0)|X=x],\mu_{1}(x)=\mathbb{E}[Y(1)|X=x],\\ \hat{\tau}(x)=\hat{\mu}_{1}(x)-\hat{\mu}_{0}(x)$$

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• TransE

  #machine learning

  $$\mathcal{L}=\sum_{(h,r,t) \in S} \sum_{(h^{'},r^{'},t^{'}) \in S^{'}_{(h,r,t)}} \[ \gamma + d(h + r, t) - d(h^{'} + r^{'}, t^{'}) \]_{+} \\ S^{'}_{(h,r,t)}=\{(h^{'},r,t)|h^{'} \in E\} \cup \{(h,r,t^{'})|t^{'} \in E\} \\ d(h + r, t)=||h + r - t||^{2}_{2}$$

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• TransH

  #machine learning #KG

  $$f_{r}(h,t) =||h_{\perp} + d_{r} - t_{\perp} ||^{2}_{2}=||(h - w_{r}hw_{r}) + d_{r} - (t - w_{r}tw_{r}) ||^{2}_{2}$$

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• Unconfoundedness Assumption

  #machine learning #causual inference

  $$\{Y_{i}(0),Y_{i}(1)\}\perp W_{i}|X_{i}$$

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• Variational AutoEncoder VAE

  #machine learning #VAE

  $$\log p_{\theta}(x)=\mathbb{E}_{q_{\phi}(z|x)}[\log p_{\theta}(x)] \\ =\mathbb{E}_{q_{\phi}(z|x)}[\log \frac{p_{\theta}(x,z)}{p_{\theta}(z|x)}] \\ =\mathbb{E}_{q_{\phi}(z|x)}[\log [\frac{p_{\theta}(x,z)}{q_{\phi}(z|x)} \times \frac{q_{\phi}(z|x)}{p_{\theta}(z|x)}]] \\ =\mathbb{E}_{q_{\phi}(z|x)}[\log [\frac{p_{\theta}(x,z)}{q_{\phi}(z|x)} ]] +D_{KL}(q_{\phi}(z|x) || p_{\theta}(z|x))\\$$

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• Wasserstein Distance Optimal Transport

  #machine learning #wasserstein

  $$W_{p}(P,Q)=(\inf_{J \in J(P,Q)} \int{||x-y||^{p}dJ(X,Y)})^\frac{1}{p}$$

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• X-Learner

  #machine learning #causual inference

  $$\tilde{D}^{1}_{i}:=Y^{1}_{i}-\hat{\mu}_{0}(X^{1}_{i}),\tilde{D}^{0}_{i}:=\hat{\mu}_{1}(X^{0}_{i})-Y^{0}_{i}\\ \hat{\tau}(x)=g(x)\hat{\tau}_{0}(x) + (1-g(x))\hat{\tau}_{1}(x)$$

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• Bound on Target Domain Error

  #machine learning #transfer learning

  $$\epsilon_{T}(h) \le \hat{\epsilon}_{S}(h) + \sqrt{\frac{4}{m}(d \log \frac{2em}{d} + \log \frac{4}{\delta })} + d_{\mathcal{H}}(\tilde{\mathcal{D}}_{S}, \tilde{\mathcal{D}}_{T}) + \lambda \\ \lambda = \lambda_{S} + \lambda_{T}$$

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• Bregman Divergences

  #machine learning

  $$d_{\phi}(z,z^{'})=\phi(z) - \phi(z^{'})-(z-z^{'})^{T} \nabla \phi(z^{'})$$

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• Coefficient of Determination

  #machine learning #metric

  $$SS_{res} = \sum_{i} (y_{i} - f_{i})^{2} = \sum_{i} e_{i}^{2}, SS_{total} = \sum_{i} (y_{i} - \bar{y} )^{2}, y=\frac{1}{n} \sum^{n}_{i=1} y_{i}, R^{2} = 1 - \frac{SS_{res}}{SS_{total}}$$

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• Conditional Random Field CRF

  #machine learning #nlp

  $$P(y|x)=\frac{1}{Z(x)}\exp(\sum_{i,k}\lambda_{k}t_{k}(y_{i-1},y_{i},x,i))+\sum_{i,l}\mu_{l}s_{l}(y_{i},x,i)) \\ Z(x)=\sum_{y}\exp(\sum_{i,k}\lambda_{k}t_{k}(y_{i-1},y_{i},x,i))+\sum_{i,l}\mu_{l}s_{l}(y_{i},x,i))$$

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• Cross-Stitch Network

  #machine learning #multi task

  $$\begin{bmatrix} \tilde{x}^{ij}_{A}\\\tilde{x}^{ij}_{B}\end{bmatrix}=\begin{bmatrix} a_{AA} & a_{AB}\\ a_{BA} & a_{BB} \end{bmatrix}\begin{bmatrix} x^{ij}_{A}\\ x^{ij}_{B} \end{bmatrix}$$

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• Domain Adaptation H-Divergence

  #machine learning #transfer learning

  $$d_{\mathcal{H}}(\mathcal{D},\mathcal{D}^{'})=2\sup_{h \in \mathcal{H}}|\Pr_{\mathcal{D}}[I(h)]-\Pr_{\mathcal{D}^{'}}[I(h)]|$$

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• Domain-Adversarial Neural Networks DANN

  #machine learning #transfer learning

  $$\min [\frac{1}{m}\sum^{m}_{1}\mathcal{L}(f(\textbf{x}^{s}_{i}),y_{i})+\lambda \max(-\frac{1}{m}\sum^{m}_{i=1}\mathcal{L}^{d}(o(\textbf{x}^{s}_{i}),1)-\frac{1}{m^{'}}\sum^{m^{'}}_{i=1}\mathcal{L}^{d}(o(\textbf{x}^{t}_{i}),0))]$$

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• Entire Space Multi-Task Model ESSM

  #machine learning #multi task

  $$L(\theta_{cvr},\theta_{ctr})=\sum^{N}_{i=1}l(y_{i},f(x_{i};\theta_{ctr}))+\sum^{N}_{i=1}l(y_{i}\&z_{i},f(x_{i};\theta_{ctr}) \times f(x_{i};\theta_{cvr})) $$

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• Graph Attention Network GAT

  #machine learning #graph #GNN

  $$h=\{\vec{h_{1}},\vec{h_{2}},...,\vec{h_{N}}\}, \\ \vec{h_{i}} \in \mathbb{R}^{F} \\ W \in \mathbb{R}^{F \times F^{'}} \\ e_{ij}=a(Wh_{i},Wh_{j}) \\ k \in \mathcal{N}_{i},\text{ neighbourhood nodes}\\ a_{ij}=\text{softmax}_{j}(e_{ij})=\frac{\exp(e_{ij})}{\sum_{k \in \mathcal{N}_{i}} \exp(e_{ik})}$$

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• Graph Convolutional Networks GCN

  #machine learning #graph #GNN

  $$H^{(l+1)}=\sigma(\tilde{D}^{-\frac{1}{2}}\tilde{A}\tilde{D}^{-\frac{1}{2}}H^{l}W^{l})\\ \tilde{A}=A+I_{N}\\ \tilde{D}_{ii}=\sum_{j}\tilde{A}_{ij} \\ H^{0}=X \\ \mathcal{L}=-\sum_{l \in Y}\sum^{F}_{f=1} Y_{lf} \ln Z_{lf}$$

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• Graph Laplacian

  #machine learning #graph #GNN

  $$L=I_{N}-D^{-\frac{1}{2}}AD^{-\frac{1}{2}} \\ L=U\Lambda U^{T}$$

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• GraphSage

  #machine learning #graph #GNN

  $$h^{0}_{v} \leftarrow x_{v} \\ \textbf{for} k \in \{1,2,...,K\} \text{do}\\ \textbf{for} v \in V \text{do} \\ h^{k}_{N_{v}} \leftarrow \textbf{AGGREGATE}_{k}(h^{k-1}_{u}, u \in N(v)); \\ h^{k}_{v} \leftarrow \sigma (W^{k} \textbf{concat}(h^{k-1}_{v},h^{k}_{N_{v}})) \\ \textbf{end} \\ h^{k}_{v}=h^{k}_{v}/||h^{k}_{v}||_{2},\forall v \in V \\ \textbf{end} \\ z_{v} \leftarrow h^{k}_{v} \\ J_{\textbf{z}_{u}}=-\log (\sigma (\textbf{z}_{u}^{T}\textbf{z}_{v})) - Q \mathbb{E}_{v_{n} \sim p_n(v)} \log(\sigma (-\textbf{z}_{u}^{T}\textbf{z}_{v_{n}}))$$

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• Hidden Markov Model

  #machine learning #nlp

  $$Q=\{q_{1},q_{2},...,q_{N}\}, V=\{v_{1},v_{2},...,v_{M}\} \\ I=\{i_{1},i_{2},...,i_{T}\},O=\{o_{1},o_{2},...,o_{T}\} \\ A=[a_{ij}]_{N \times N}, a_{ij}=P(i_{t+1}=q_{j}|i_{t}=q_{i}) \\ B=[b_{j}(k)]_{N \times M},b_{j}(k)=P(o_{t}=v_{k}|i_{t}=q_{j})$$

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• Huber Loss

  #machine learning

  $$L_{\delta}(y,f(x)) = \left \{ \begin{aligned} & \frac{1}{2}(y-f(x))^{2}, \text{for} |y-f(x)| \le \delta \cr & \delta \times (|y-f(x)| - \frac{1}{2}\delta) \cr \end{aligned} \right. $$

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• Language Modelling

  #machine learning

  $$p(x)=\prod^{n}_{i=1} p(s_{n}|s_{1},...,s_{n-1})$$

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• Mean Absolute Error MAE

  #machine learning #metric

  $$\text{MAE} = \frac{1}{n} \sum^{n}_{i=1} |Y_{i} - \hat{Y}_{i}| = \frac{1}{n} \sum^{n}_{i=1} |e_{i}|$$

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• Mean Squared Error MSE

  #machine learning #metric

  $$\text{MSE} = \frac{1}{n} \sum^{n}_{i=1} (Y_{i} - \hat{Y}_{i}) ^ {2}$$

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• Model-Agnostic Meta-Learning MAML

  #machine learning #meta learning

  $$\min_{\theta} \sum_{\mathcal{T}_{i} \sim p(\mathcal{T})} \mathcal{L}_{\mathcal{T}_{i}}(f_{\theta^{'}_{i}}) = \sum_{\mathcal{T}_{i} \sim p(\mathcal{T})} \mathcal{L}_{\mathcal{T}_{i}}(f_{\theta_{i} - \alpha \nabla_{\theta} \mathcal{L}_{\mathcal{T}_{i}} (f_{\theta}) })$$

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• Multi-Gate Mixture of Experts MMoE

  #machine learning #multi task

  $$g^{k}(x)=\text{softmax}(W_{gk}x) \\ f^{k}(x)=\sum^{n}_{i=1}g^{k}(x)_{i}f_{i}(x) \\ y_{k}=h^{k}(f^{k}(x))$$

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• Progressive Layered Extraction PLE

  #machine learning #multi task

  $$g^{k}(x)=w^{k}(x)S^{k}(x) \\ w^{k}(x)=\text{softmax}(W^{k}_{g}x) \\ S^{k}(x)=\[E^{T}_{(k,1)},E^{T}_{(k,2)},...,E^{T}_{(k,m_{k})},E^{T}_{(s,1)},E^{T}_{(s,2)},...,E^{T}_{(s,m_{s})}\]^{T} \\ y^{k}(x)=t^{k}(g^{k}(x)) \\ g^{k,j}(x)=w^{k,j}(g^{k,j-1}(x))S^{k,j}(x) $$

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• Prototypical Networks Protonets

  #machine learning #meta learning

  $$c_{k}=\frac{1}{|S_{k}|}\sum_{(x_{i},y_{i}) \in S_{k}} f_{\phi}(x) \\ p_{\phi}(y=k|x)=\frac{\exp(-d(f_{\phi}(x), c_{k}))}{\sum_{k^{'}} \exp(-d(f_{\phi}(x), c_{k^{'}})} \\\min J(\phi)=-\log p_{\phi}(y=k|x)$$

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• Proximal Policy Optimization

  #machine learning #AI #LLM

  $$\arg\max\limits_{\pi}{ E_{p \sim D,g \sim \pi} [R(g|p)] }, R(g|p) = \tilde{R}_{c}(g|p) - \beta D_{KL}( \pi_{\theta} (g|p) || \pi_{0} (g|p))$$

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• Proximal Policy Optimization PPO

  #machine learning

  $$L^{CLIP}(\theta)=E_{t}[\min(r_{t}(\theta))A_{t}, \text{clip}(r_{t}(\theta), 1-\epsilon,1+\epsilon)A_{t}]$$

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• RotatE

  #machine learning #KG

  $$f_{r}(h, t) = ||h \circ r - t||^{2}_{2}$$

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• SME Linear

  #machine learning #KG

  $$\epsilon(lhs,rel,rhs)=E_{lhs(rel)}^{T}E_{rhs(rel)} \\=(W_{l1}E_{lhs}^{T} + W_{l2}E_{rel}^{T} + b_{l})^{T}(W_{r1}E_{rhs}^{T} + W_{r2}E_{rel}^{T} + b_{r})$$

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• SimplE

  #machine learning #KG

  $$s(e_{i}, r, e_{j}) = \frac{1}{2}( + )$$

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• TransR

  #machine learning #KG

  $$h_{r}=hM_{r}, t_{r}=tM_{r} \\f_{r}(h, t) = ||h_{r} + r - t_{r}||^{2}_{2}=||hM_{r}+r-tM_{r}||^{2}_{2}$$

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• Transformer

  #machine learning #nlp #gpt

  $$\text{Attention}(Q, K, V) = \text{softmax}(\frac{QK^T}{\sqrt{d_k}})V$$

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statistics

• Bernoulli Distribution

  #Math #Statistics

  $$Pr(X=1) = p = 1- Pr(X=0) = 1 - q, \\ f(x)=p \text{ if } k = 1 \text{ else } q \text{ if } k = 0$$

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• Beta Binomial Distribution

  #Math #Statistics

  $$x \sim f(x | n, \alpha, \beta), \\ f(x | n, \alpha, \beta) = \int_{0}^{1} \text{Bin}(x|n,p)\text{Beta}(p|\alpha,\beta) dp , \\ f(x | n, \alpha, \beta) = C^{x}_{n} \frac{B(x+\alpha, n -x + \beta)}{B(\alpha,\beta)}$$

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• Beta Distribution

  #Math #Statistics

  $$X \sim Beta(\alpha,\beta), \\ f(x)=\frac{ x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}, \\ B(\alpha,\beta)=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}, \\ F(x)=I_{x}(\alpha+\beta)$$

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• Binomial Distribution

  #Math #Statistics

  $$X \sim B(n,p) \\f(x)=\begin{pmatrix}n\\ x\end{pmatrix}p^{x}q^{n-x}=C^{k}_{n}p^{x}q^{n-x},q=1-p\\\text{Binominal Mean}\ \mu=np\\\text{Binominal Variance}\ \sigma^2=npq$$

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• Chi-Squared Distribution

  #Math #Statistics

  $$Q = \sum^{k}_{i=1} Z^{2}_{i} \sim \chi^{2}(k), \\ f(k) = \frac{1}{2^{k/2} \Gamma (k/2)} x^{k/2-1} e^{-x/2}, \\ F(k) = \frac{1}{\Gamma (k/2)} \gamma (\frac{k}{2},\frac{x}{2})$$

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• Exponential Distribution

  #Math #Statistics

  $$f(x, \lambda)=\lambda e^{-\lambda x} \\ F(x, \lambda)=1 - e^{-\lambda x}$$

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• Fisher Skewness

  #Math #Statistics

  $$\gamma_1 = \frac{{\mu_3 }}{{\mu_2 ^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} }} = \frac{{\mu_3 }}{{\sigma ^3 }}$$

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• Gamma Distribution

  #Math #Statistics

  $$\Gamma \left( a \right) = \int\limits_0^\infty {s^{a - 1} } e^{ - s} ds \\ P(x) = \frac{x^{\alpha-1} e^{-frac{x}{\theta}}}{\Gamma(\alpha) \theta^{\alpha}} \\ \mu = \alpha \theta \\ \sigma^{2} = \alpha \theta^{2} \\ \gamma_{1} = \frac{2}{\sqrt{\alpha}} \\ \gamma_{2} = \frac{6}{\alpha}$$

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• Geometric Distribution

  #Math #Statistics

  $$Pr(X=k) = (1-p)^{k-1}q, \\ f(x)=(1-p)^{k-1}q, \\ F(x)=1 - (1-p)^{[x]}$$

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• Gumbel Distribution

  #Math #Statistics

  $$x \sim \text{Gumbel}(\mu,\beta), \\ \frac{1}{\beta} e^{-(z + e^{-z})}, z=\frac{x - \mu}{\beta}, \\ e^{-e^{-(x-\mu)/\beta}}$$

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• Laplace Distribution

  #Math #Statistics

  $$x \sim \text{Laplace}(\mu,b), \\ f(x | \mu,b) =\frac{1}{2b} \exp (-\frac{|x-\mu|}{b}), \\ F(x | \mu,b) = \frac{1}{2} \exp (\frac{x - \mu}{b}) \text{ if } x \le \mu, 1 - \frac{1}{2} \exp (-\frac{x - \mu}{b}) \text{ if } x \ge \mu$$

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• Normal Gaussian Distribution

  #Math #Statistics

  $$X \sim \mathcal{N}(\mu,\sigma^2) \\ f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp{[-\frac{(x-\mu)^{2}}{2\sigma^{2}}]}$$

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• Poisson Binomial Distribution

  #Math #Statistics

  $$Pr(K = k) = \sum_{A \in F_{k}} \prod_{i \in A} p_{i} \prod_{j \in A_{c}} (1-p_{j})$$

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• Poisson Distribution

  #Math #Statistics

  $$X \sim \pi(\mu) \\f(x)=\frac{\mu^{x}}{x!}e^{-\mu}\\ \text{Poisson Mean} \mu \\ \text{Poisson Variance}\sigma^2=\mu$$

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• Student t-Distribution

  #Math #Statistics

  $$f(t) = \frac{\Gamma(\frac{v + 1}{2})}{\sqrt{v\pi}\Gamma(v/2)} (1+\frac{t^{2}}{v})^{-(v+1)/2}, \\ F(t) = \frac{1}{2} + x \Gamma(\frac{v+1}{2}) \times \frac{2^{F_{1}}(1/2,\frac{v+1}{2};3/2;-\frac{x^{2}}{v})}{\sqrt{\pi v} \Gamma(\frac{v}{2})}, \\ F(t) = 1 - \frac{1}{2} I_{x(t)} (\frac{v}{2}, \frac{1}{2}), x(t)=\frac{v}{t^{2} + v}$$

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• Uniform Distribution

  #Math #Statistics

  $$X \sim U(a,b) \\ f(x)=\frac{1}{b-a} \text{for} a \le x \le b \text{else} 0 \\ F(x)=\frac{x-a}{b-a} \text{for} a \le x \le b, 1 \text{for} x > b, 0 \text{for} x < a$$

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