Operators

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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.