Editing Operators

== Arithmetic Operators ==
* Addition (+)
** Adds two quantities together.
*** Basic arithmetic, summing values.

* Subtraction (-)
** Subtracts one quantity from another.
*** Basic arithmetic, finding the difference between values.

* Multiplication (*)
** Multiplies two quantities together.
*** Repeated addition, scaling, area calculation.

* Division (/)
** Divides one quantity by another.
*** Sharing equally, finding rates, calculating proportions.

* Exponentiation (^ or **)
** Raises a base to a power.
*** Compound interest, growth/decay, geometric progression.

* Modulus (%)
** Finds the remainder of a division operation.
*** Determining divisibility, cyclic patterns, hashing algorithms.

* Floor Division (//)
** Divides one quantity by another, rounding down to the nearest integer.
*** Integer division, partitioning, array indexing in programming languages.

== Relational Operators ==
* Equality (==)
** Checks if two quantities are equal.
*** Comparison, conditional statements, database queries.

* Inequality (!=)
** Checks if two quantities are not equal.
*** Comparison, conditional statements, filtering data.

* Greater Than (>)
** Determines if one quantity is greater than another.
*** Ranking, sorting, conditional statements.

* Less Than (<)
** Determines if one quantity is less than another.
*** Ranking, sorting, conditional statements.

* Greater Than or Equal To (>=)
** Determines if one quantity is greater than or equal to another.
*** Ranking, sorting, conditional statements.

* Less Than or Equal To (<=)
** Determines if one quantity is less than or equal to another.
*** Ranking, sorting, conditional statements.

== Logical Operators ==
* AND (&&)
** Returns true if both conditions are true.
*** Conditional statements, filtering data.

* OR (||)
** Returns true if at least one condition is true.
*** Conditional statements, filtering data.

* NOT (!)
** Negates the value of a condition.
*** Conditional statements, filtering data.

== Set Theory Operators ==
* Union (∪)
** Combines two sets into one, containing all unique elements.
*** Set operations, combining data sets.

* Intersection (∩)
** Finds the common elements between two sets.
*** Set operations, filtering data.

* Complement (')
** Contains all elements not in a given set.
*** Set operations, filtering data.

* Subset (⊆)
** Checks if all elements of one set are in another set.
*** Set operations, subset testing.

* Superset (⊇)
** Checks if one set contains all elements of another set.
*** Set operations, superset testing.

* Empty Set (∅)
** A set containing no elements.
*** Placeholder, indicating absence of data.

* Set Difference (-)
** Contains elements in one set but not in another.
*** Set operations, data comparison.

== [[Vector and Matrix Operators]] ==
* Dot Product (· or ⋅)
** Multiplies corresponding components of two vectors and sums the results.
*** Finding angles between vectors, projection calculations.

* Cross Product (× or ⨯)
** Produces a vector perpendicular to two given vectors.
*** Determining orientations, calculating torque.

* Matrix Multiplication (*)
** Multiplies corresponding elements of two matrices and sums the results.
*** Transformations, solving systems of equations.

* Transpose (T or ')
** Swaps the rows and columns of a matrix.
*** Matrix manipulation, solving linear equations.

* Determinant (det)
** Scalar value that can be computed from the elements of a square matrix.
*** Volume scaling factor in linear transformations.

* Inverse (⁻¹)
** Matrix that, when multiplied by the original matrix, results in the identity matrix.
*** Solving systems of linear equations, finding inverses.

* Trace (Tr)
** Sum of the elements on the main diagonal of a square matrix.
*** Characteristic polynomial, calculating exponentials of matrices.

* Adjoint (adj)
** Matrix obtained by taking the transpose of the cofactor matrix of a given square matrix.
*** Finding inverses of matrices, solving systems of linear equations.

* Kronecker Product (⊗)
** Produces a block matrix from two given matrices.
*** Quantum mechanics, signal processing, and image processing.

== Calculus Operators ==
* Derivative (d/dx or ∂/∂x)
** Measures the rate at which a quantity changes with respect to another.
*** Rate of change, optimization, slope of curves.

* Integral (∫)
** Computes the area under a curve.
*** Accumulation of quantities, probability density functions.

* Gradient (grad or ∇)
** Vector that points in the direction of the greatest rate of increase of a scalar field.
*** Optimization, direction of steepest ascent.

* Laplacian (∇²)
** Scalar operator that measures the divergence of the gradient of a scalar field.
*** Heat flow, diffusion equations.

* Divergence (div)
** Measures the extent to which a vector field diverges from or converges towards a point.
*** Fluid flow, flux of a vector field.

* Curl (curl)
** Measures the rotation of a vector field.
*** Fluid dynamics, electromagnetic fields.

* Partial Differential (∂²/∂x²)
** Derivative with respect to multiple independent variables.
*** Heat conduction, wave equations.

* Limit (lim)
** Describes the behavior of a function as its input approaches a certain value.
*** Continuity, asymptotic behavior.

* Taylor Series Expansion
** Representation of a function as an infinite sum of terms.
*** Approximation of functions, numerical analysis.

== Trigonometric Operators ==
* Sine (sin)
** Ratio of the length of the side opposite an angle to the length of the hypotenuse.
*** Modeling periodic phenomena, waveforms.

* Cosine (cos)
** Ratio of the length of the adjacent side to the length of the hypotenuse.
*** Modeling periodic phenomena, oscillations.

* Tangent (tan)
** Ratio of the length of the side opposite an angle to the length of the side adjacent to the angle.
*** Slope of lines, angular velocity.

* Secant (sec)
** Reciprocal of the cosine function.
*** Light rays, sound waves.

* Cosecant (csc)
** Reciprocal of the sine function.
*** Antennas, vibrations.

* Cotangent (cot)
** Reciprocal of the tangent function.
*** Electrical circuits, mechanical systems.

* Arcsine (arcsin)
** Inverse of the sine function.
*** Solving trigonometric equations, calculating angles.

* Arccosine (arccos)
** Inverse of the cosine function.
*** Solving trigonometric equations, calculating angles.

* Arctangent (arctan)
** Inverse of the tangent function.
*** Solving trigonometric equations, calculating angles.

* Hyperbolic Sine (sinh)
** Analog of the trigonometric sine function for hyperbolic functions.
*** Modeling exponential growth, catenary curves.

* Hyperbolic Cosine (cosh)
** Analog of the trigonometric cosine function for hyperbolic functions.
*** Modeling exponential growth, catenary curves.

* Hyperbolic Tangent (tanh)
** Analog of the trigonometric tangent function for hyperbolic functions.
*** Modeling logistic growth, saturation phenomena.

* Inverse Hyperbolic Sine (arsinh)
** Inverse of the hyperbolic sine function.
*** Solving hyperbolic equations, calculating areas.

* Inverse Hyperbolic Cosine (arcosh)
** Inverse of the hyperbolic cosine function.
*** Solving hyperbolic equations, calculating areas.

* Inverse Hyperbolic Tangent (artanh)
** Inverse of the hyperbolic tangent function.
*** Solving hyperbolic equations, calculating areas.

== Probability and Statistics Operators ==
* Expected Value (E)
** Mean value of a random variable.
*** Estimating outcomes, decision-making.

* Variance (Var)
** Measure of how spread out the values in a data set are around the mean.
*** Assessing risk, quantifying variability.

* Standard Deviation (σ or SD)
** Square root of the variance.
*** Measuring dispersion, error bars in plots.

* Probability (P)
** Measure of the likelihood of an event occurring.
*** Assessing uncertainty, predicting outcomes.

* Conditional Probability (P(A|B))
** Probability of event A occurring given that event B has already occurred.
*** Bayesian inference, probability trees.

* Joint Probability (P(A ∩ B))
** Probability of both events A and B occurring simultaneously.
*** Probability distribution functions, contingency tables.

* Marginal Probability (P(A))
** Probability of a single event occurring regardless of other events.
*** Summing joint probabilities, independence testing.

* Cumulative Distribution Function (CDF)
** Function that maps the probability of a random variable being less than or equal to a certain value.
*** Probability distribution functions, hypothesis testing.

* Probability Density Function (PDF)
** Function that describes the likelihood of a random variable taking on a particular value.
*** Modeling continuous random variables, integration.

* Covariance
** Measure of the joint variability of two random variables.
*** Assessing correlation, portfolio management.

* Correlation
** Measure of the strength and direction of the relationship between two random variables.
*** Analyzing relationships, predictive modeling.

== Number Theory Operators ==
* Greatest Common Divisor (gcd)
** Largest positive integer that divides each of the integers.
*** Simplifying fractions, reducing fractions to lowest terms.

* Least Common Multiple (lcm)
** Smallest positive integer that is divisible by each of the integers.
*** Finding common multiples, adding fractions with unlike denominators.

* Prime Counting Function (π)
** Count of the number of prime numbers less than or equal to a given number.
*** Analyzing distribution of primes, cryptography.

* Euler's Totient Function (φ)
** Count of the positive integers less than n that are coprime to n.
*** Cryptography, number theory algorithms.

* Mobius Function (μ)
** Arithmetic function defined on the positive integers.
*** Analyzing number theory functions, combinatorics.

== Special Functions ==
* Gamma Function (Γ)
** Generalization of the factorial function to complex numbers.
*** Analyzing divergent series, statistical distributions.

* Beta Function (B)
** Function defined by an integral.
*** Statistical distributions, probability density functions.

* Riemann Zeta Function (ζ)
** Analytic function that generalizes the sum of the infinite series.
*** Analyzing prime numbers, number theory.

* Bessel Functions (J, Y, I, K)
** Solutions to Bessel's differential equation.
*** Modeling oscillations, heat conduction.

* Legendre Polynomials (P)
** Orthogonal polynomials that arise in the solution of Laplace's equation.
*** Quantum mechanics, celestial mechanics.

* Hermite Polynomials (H)
** Orthogonal polynomials that arise in the solution of the quantum harmonic oscillator.
*** Quantum mechanics, statistical mechanics.

* Laguerre Polynomials (L)
** Orthogonal polynomials that arise in the solution of the quantum hydrogen atom.
*** Quantum mechanics, atomic physics.

== Matrix Decomposition ==
* Singular Value Decomposition (SVD)
** Decomposes a matrix into singular vectors and singular values.
*** Dimensionality reduction, image compression.

* Eigenvalue Decomposition
** Decomposes a matrix into eigenvectors and eigenvalues.
*** Stability analysis, systems of ordinary differential equations.

* QR Decomposition
** Decomposes a matrix into an orthogonal matrix and an upper triangular matrix.
*** Solving linear least squares problems, numerical stability.

* Cholesky Decomposition
** Decomposes a symmetric positive definite matrix into the product of a lower triangular matrix and its conjugate transpose.
*** Solving linear systems, generating correlated random variables.

* LU Decomposition
** Decomposes a matrix into the product of a lower triangular matrix and an upper triangular matrix.
*** Solving linear systems, matrix inversion.

* Schur Decomposition
** Decomposes a matrix into a unitary matrix and an upper triangular matrix.
*** Solving linear systems, stability analysis.

* Jordan Normal Form
** Canonical form of a matrix under a similarity transformation.
*** Stability analysis, solving systems of linear differential equations.

== Other Mathematical Operators ==
* Summation (∑)
** Adds up a sequence of numbers.
*** Finding totals, calculating areas under curves.

* Product (∏)
** Multiplies a sequence of numbers.
*** Finding products, calculating volumes of irregular shapes.

* Integral with Limits (∫ₐᵇ)
** Computes the area under a curve between two points.
*** Finding areas, computing volumes, calculating work.

* Partial Derivative (∂)
** Derivative of a function of several variables with respect to one variable, holding others constant.
*** Multivariable calculus, optimization.

* Infimum (inf)
** Greatest lower bound of a set.
*** Optimization, analysis of functions.

* Supremum (sup)
** Least upper bound of a set.
*** Optimization, analysis of functions.

* Absolute Value (|x|)
** Distance of a number from zero.
*** Distance calculations, finding magnitudes.

* Factorial (!)
** Product of all positive integers less than or equal to a given number.
*** Counting permutations, combinations.

* Round (round)
** Rounds a number to the nearest integer.
*** Approximation, formatting numbers.

* Logarithm (log)
** Inverse operation to exponentiation.
*** Scaling, orders of magnitude, solving exponential equations.

* Greatest Integer Function (⌊x⌋)
** Greatest integer less than or equal to a given number.
*** Floor function, truncating decimals.

* Ceiling Function (⌈x⌉)
** Smallest integer greater than or equal to a given number.
*** Ceiling function, rounding up.

* Signum Function (sgn)
** Returns the sign of a number.
*** Determining direction, characterizing solutions.