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LaTeX (KaTeX) templates
Mathematical typesetting: equations, derivations, matrices, commutative diagrams. Renders LaTeX math in the browser.
Math — Vector Spaces
Bases, dimension and rank–nullity.
MathematicsBiology — Lotka–Volterra (Predator–Prey)
Coupled equations for predator and prey populations.
Life SciencesBiology — Competition Model
Lotka–Volterra competition and exclusion.
Life SciencesBiology — Island Biogeography
Species richness from immigration vs extinction.
Life SciencesBiology — Heritability
Narrow- and broad-sense heritability.
Life SciencesBiology — Mutation–Selection Balance
Equilibrium allele frequency under mutation and selection.
Life SciencesBiology — Biological Redox Potentials
Electron flow down the respiratory chain.
Life SciencesBiology — Receptor–Ligand Binding
Dissociation constant and fractional occupancy.
Life SciencesBiology — r- and K-Selection
Two contrasting life-history strategies.
Life SciencesPhysics — Seismic Waves
P- and S-waves and what they reveal about Earth's interior.
Chemistry & PhysicsPhysics — Earthquake Magnitude
The logarithmic Richter and moment-magnitude scales.
Chemistry & PhysicsPhysics — Isostasy
Crustal blocks floating in the mantle.
Chemistry & PhysicsPhysics — Geothermal Gradient
Heat flow through the crust.
Chemistry & PhysicsPhysics — Atmospheric Pressure
The barometric formula and lapse rate.
Chemistry & PhysicsPhysics — Radiative Forcing
Greenhouse forcing and equilibrium temperature.
Chemistry & PhysicsPhysics — Albedo & Feedback
Reflectivity and the ice-albedo feedback.
Chemistry & PhysicsPhysics — Coriolis Effect
Deflection of motion on a rotating Earth.
Chemistry & PhysicsPhysics — Ocean Waves
Deep- and shallow-water wave speed.
Chemistry & PhysicsPhysics — Tides
Tidal forces and spring/neap tides.
Chemistry & PhysicsPhysics — Stellar Nucleosynthesis
How stars forge the elements.
Chemistry & PhysicsPhysics — Jeans Instability
When a gas cloud collapses to form a star.
Chemistry & PhysicsChemistry — Ocean Acidification
How dissolved CO2 lowers ocean pH.
Chemistry & PhysicsChemistry — Chemical Weathering
Carbonate and silicate weathering as carbon sinks.
Chemistry & PhysicsChemistry — Uranium–Lead Dating
Radiometric dating of ancient rocks.
Chemistry & PhysicsChemistry — Stable Isotopes
Delta notation and isotope proxies.
Chemistry & PhysicsChemistry — Stratospheric Ozone
The Chapman cycle that maintains the ozone layer.
Chemistry & PhysicsChemistry — Water Treatment Chemistry
Coagulation and chlorination of drinking water.
Chemistry & PhysicsChemistry — Soil pH
Soil acidity and nutrient availability.
Chemistry & PhysicsChemistry — Carbonate Buffering
Calcium carbonate equilibrium and pH buffering.
Chemistry & PhysicsMath — Bisection Method
Bracketing a root and halving the interval.
MathematicsMath — Taylor Remainder
The Lagrange form of the truncation error.
MathematicsMath — Curvature
How sharply a curve bends.
MathematicsMath — Centroids & Pappus
Centroid of a region and Pappus's theorem.
MathematicsMath — Chi-square Test of Independence
Testing association in a contingency table.
MathematicsMath — Least Squares (Normal Equations)
The matrix solution to linear regression.
MathematicsBiology — Carbon Flux (GPP/NPP/NEP)
Productivity terms in the carbon budget.
Life SciencesBiology — Decomposition Kinetics
First-order decay of litter mass.
Life SciencesBiology — Nitrogen Budget
Inputs and outputs of ecosystem nitrogen.
Life SciencesBiology — Ecological Footprint
Comparing demand with biocapacity.
Life SciencesEngineering — Beam Bending
Flexural stress and deflection of a loaded beam.
Chemistry & PhysicsEngineering — Torsion of Shafts
Shear stress and twist in a circular shaft.
Chemistry & PhysicsEngineering — Euler Buckling
Critical load for a slender column.
Chemistry & PhysicsEngineering — Fracture Mechanics
Stress intensity and fracture toughness.
Chemistry & PhysicsEngineering — Fatigue
Failure under cyclic loading (S–N behaviour).
Chemistry & PhysicsEngineering — Heat Exchangers
Heat duty and the log-mean temperature difference.
Chemistry & PhysicsEngineering — Pump Power
Hydraulic power and efficiency of a pump.
Chemistry & PhysicsEngineering — Feedback Control
Closed-loop transfer function and error.
Chemistry & PhysicsEngineering — Sampling & Nyquist
The sampling theorem and aliasing.
Chemistry & PhysicsEngineering — Gain in Decibels
Power and voltage gain on the dB scale.
Chemistry & PhysicsEngineering — RC Filters
Cut-off frequency of a first-order filter.
Chemistry & PhysicsEngineering — Thermal Stress
Stress from constrained thermal expansion.
Chemistry & PhysicsChemistry — Crystal Structures
Atoms per unit cell and packing fraction.
Chemistry & PhysicsChemistry — Crystal Defects
Vacancies, interstitials and their temperature dependence.
Chemistry & PhysicsChemistry — Semiconductor Doping
n- and p-type doping and conductivity.
Chemistry & PhysicsChemistry — Polymer Molar Mass
Number- and weight-average molar mass and dispersity.
Chemistry & PhysicsChemistry — Glass Transition
The glass transition of amorphous polymers.
Chemistry & PhysicsChemistry — Galvanic Corrosion
Electrochemical corrosion of coupled metals.
Chemistry & PhysicsChemistry — Langmuir Adsorption
Monolayer adsorption isotherm.
Chemistry & PhysicsChemistry — Cracking of Hydrocarbons
Breaking long alkanes into useful products.
Chemistry & PhysicsMath — Big-O Complexity
Ordering common growth rates.
MathematicsMath — Sorting Complexity
Time complexity of sorting algorithms.
MathematicsMath — Shannon Entropy
Information content of a distribution.
MathematicsMath — Huffman Coding
Optimal prefix-free compression.
MathematicsMath — Karnaugh Maps
Minimising Boolean expressions.
MathematicsMath — Dijkstra's Algorithm
Shortest paths from a source node.
MathematicsMath — Principal Component Analysis
Dimensionality reduction by variance.
MathematicsMath — Logistic Regression
Modelling a binary outcome.
MathematicsMath — Neural Networks
Forward pass and learning by backpropagation.
MathematicsMath — Classifier Metrics
Precision, recall and the F1 score.
MathematicsMath — Monte Carlo Methods
Estimating integrals by random sampling.
MathematicsMath — Hashing
Hash tables and their performance.
MathematicsBiology — Restriction Enzymes
Cutting DNA at specific recognition sites.
Life SciencesBiology — Quantitative PCR
Real-time PCR and the threshold cycle.
Life SciencesBiology — Sanger Sequencing
Chain-termination DNA sequencing.
Life SciencesBiology — Protein Folding Thermodynamics
The free-energy basis of folding.
Life SciencesBiology — Allosteric Regulation
Cooperative, regulated binding (MWC model).
Life SciencesBiology — Signal Amplification
How signalling cascades multiply a stimulus.
Life SciencesBiology — Features of the Genetic Code
Properties of the codon code.
Life SciencesBiology — Membrane Transport Kinetics
Saturable carrier transport vs simple diffusion.
Life SciencesMedicine — Dialysis
Clearing waste solutes by diffusion across a membrane.
Life SciencesMedicine — Drug Dosing
Loading and maintenance doses.
Life SciencesMedicine — IV Infusion Kinetics
Approach to steady-state plasma concentration.
Life SciencesMedicine — Blood Flow (Poiseuille)
Why vessel radius dominates blood flow.
Life SciencesMedicine — Oxygen Delivery
How much oxygen the circulation delivers.
Life SciencesMedicine — Acid–Base Balance
Blood pH, the anion gap and disturbances.
Life SciencesMedicine — Body Fluid Compartments
Distribution of water in the body.
Life SciencesMedicine — Energy Balance
Intake, expenditure and body-mass change.
Life SciencesMedicine — Aerobic Capacity (VO2max)
Maximal oxygen uptake and METs.
Life SciencesMedicine — Lung Compliance
How easily the lungs expand.
Life SciencesMedicine — Cardiac Pressures
Ventricular pressures and ejection fraction.
Life SciencesMedicine — Ventilation/Perfusion
Matching airflow to blood flow in the lung.
Life SciencesMedical Physics — Radiotherapy
Fractionated radiation dose to treat tumours.
Chemistry & PhysicsMedical Physics — Effective Half-life
Combining physical decay and biological clearance.
Chemistry & PhysicsMedical Physics — Doppler Ultrasound
Measuring blood-flow velocity.
Chemistry & PhysicsMedical Physics — CT Imaging
Reconstructing images from X-ray projections.
Chemistry & PhysicsMedical Physics — Defibrillator
A capacitor discharge that resets the heart.
Chemistry & PhysicsMedical Physics — Sound & Hearing
The decibel scale and the audible range.
Chemistry & PhysicsMedical Physics — Radiation Shielding
Attenuating ionising radiation.
Chemistry & PhysicsMedical Physics — Dosimetry & Safety
Equivalent dose and radiation protection.
Chemistry & PhysicsMath — Survival Analysis
Kaplan–Meier estimator and hazard ratios.
Mathematics