Scientific Applications

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Execute as formal rules, not pseudocode. 

flow "ScientificApplicationsOntology" v1.0

# SCIENTIFIC APPLICATIONS — FLOW ALGEBRA
SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/scientific-applications.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
ENTITY := "Scientific Applications"

SCIENTIFIC_APPLICATIONS := {
 type: AppliedCalculusAnalysisFramework,
 domain: CalculusAndAnalysis,
 purpose: "Represent how calculus and mathematical analysis are applied to scientific modelling, measurement, prediction, optimisation, differential equations, data interpretation and physical systems."
}

CONCEPTS := {
 SCIENTIFIC_APPLICATION,
 MATHEMATICAL_MODEL,
 PHYSICAL_QUANTITY,
 FUNCTION,
 DERIVATIVE,
 INTEGRAL,
 DIFFERENTIAL_EQUATION,
 PARTIAL_DIFFERENTIAL_EQUATION,
 OPTIMISATION,
 VECTOR_CALCULUS,
 TRANSFORM,
 NUMERICAL_METHOD,
 UNCERTAINTY,
 DIMENSIONAL_ANALYSIS,
 CONSERVATION_LAW,
 SCIENTIFIC_PREDICTION
}

# ---------- SCIENTIFIC PROBLEM ----------
SCIENTIFIC_PROBLEM
 -> identify(
 SYSTEM,
 VARIABLES,
 PARAMETERS,
 DATA,
 CONSTRAINTS,
 TARGET_QUANTITY
)

SCIENTIFIC_APPLICATION
 -> map(PHYSICAL_OR_EMPIRICAL_PROBLEM_TO_MATHEMATICAL_STRUCTURE)

# ---------- VARIABLES ----------
independent variable:

x
 -> may_represent(
 TIME,
 POSITION,
 TEMPERATURE,
 PRESSURE,
 CONCENTRATION,
 FREQUENCY
)

dependent variable:

y
 =
f(x)

FUNCTION
 -> represents(RELATION_BETWEEN_QUANTITIES)

# ---------- UNITS ----------
PHYSICAL_QUANTITY
 -> VALUE
 + UNIT
 + DIMENSION

EQUATION
 -> requires(DIMENSIONAL_CONSISTENCY)

DIMENSIONAL_ANALYSIS
 -> validates(MODEL_FORM)

# ---------- SCALING ----------
characteristic scales:

L
T
U
C

dimensionless variables:

x_star
 =
x/L

t_star
 =
t/T

NONDIMENSIONALISATION
 -> reveals(DIMENSIONLESS_PARAMETERS)

SCALING
 -> identifies(DOMINANT_EFFECTS)

# ---------- DERIVATIVE ----------
derivative:

dy/dx

-> RATE_OF_CHANGE

scientific meanings:
 VELOCITY
 ACCELERATION
 GROWTH_RATE
 REACTION_RATE
 TEMPERATURE_GRADIENT
 CONCENTRATION_GRADIENT

DERIVATIVE
 -> local_response_measure

# ---------- SECOND DERIVATIVE ----------
d_squared_y/dx_squared

-> CURVATURE_OR_CHANGE_OF_RATE

applications:
 ACCELERATION
 DIFFUSION
 WAVE_CURVATURE
 STABILITY
 OPTIMISATION

# ---------- PARTIAL DERIVATIVE ----------
field:

u(x,y,z,t)

partial_u/partial_t
 -> temporal_change

grad(u)
 -> spatial_change

PARTIAL_DERIVATIVE
 -> multivariable_local_response

# ---------- GRADIENT ----------
grad(f)
 :=
(
 partial_f/partial_x_1,
 ...,
 partial_f/partial_x_n
)

GRADIENT
 -> direction_of(STEEPEST_INCREASE)
    under Euclidean geometry

applications:
 HEAT_FLOW
 DIFFUSION
 POTENTIAL_FIELDS
 OPTIMISATION

# ---------- DIVERGENCE ----------
vector field
 F

div(F)
 -> local_source_or_sink_measure

applications:
 FLUID_FLOW
 ELECTROMAGNETISM
 CONSERVATION_LAWS

# ---------- CURL ----------
curl(F)
 -> local_rotation_measure

applications:
 VORTICITY
 MAGNETIC_FIELDS
 ROTATIONAL_FLOW

# ---------- LAPLACIAN ----------
LAPLACIAN(u)
 :=
div(grad(u))

LAPLACIAN
 -> spatial_curvature_or_diffusion_operator

applications:
 HEAT_EQUATION
 DIFFUSION_EQUATION
 LAPLACE_EQUATION
 POISSON_EQUATION
 WAVE_EQUATION
 SCHROEDINGER_EQUATION

# ---------- INTEGRAL ----------
integral_a^b(
 f(x) dx
)

-> ACCUMULATION

applications:
 DISTANCE_FROM_VELOCITY
 MASS_FROM_DENSITY
 CHARGE_FROM_DENSITY
 ENERGY
 PROBABILITY
 TOTAL_FLUX
 AVERAGE_QUANTITY

# ---------- MULTIPLE INTEGRAL ----------
integral_Omega(
 f(x)
 dV
)

-> total_quantity_over(REGION)

applications:
 MASS
 ENERGY
 MOMENT
 PROBABILITY
 SOURCE_STRENGTH

# ---------- LINE INTEGRAL ----------
integral_C(
 F dot dr
)

applications:
 WORK
 CIRCULATION
 ELECTROMOTIVE_FORCE

# ---------- SURFACE INTEGRAL ----------
integral_S(
 F dot n
 dS
)

-> FLUX_THROUGH_SURFACE

applications:
 ELECTRIC_FLUX
 MAGNETIC_FLUX
 FLUID_FLUX
 HEAT_FLUX

# ---------- FUNDAMENTAL THEOREM ----------
differentiation
 and
integration

-> inverse_relationship
   under appropriate hypotheses

FUNDAMENTAL_THEOREM_OF_CALCULUS
 -> connects(LOCAL_RATE_AND_GLOBAL_ACCUMULATION)

# ---------- CONSERVATION LAW ----------
local form:

partial_rho/partial_t
+
div(J)
 =
S

rho
 -> conserved_density

J
 -> flux

S
 -> source_or_sink

CONSERVATION_LAW
 -> foundation_of(MANY_SCIENTIFIC_PDEs)

# ---------- ORDINARY DIFFERENTIAL EQUATION ----------
state:

x(t)

ODE:

dx/dt
 =
f(x,t)

applications:
 POPULATION_DYNAMICS
 CHEMICAL_KINETICS
 MECHANICS
 CIRCUITS
 PHARMACOKINETICS

INITIAL_CONDITION
 -> selects(SOLUTION_TRAJECTORY)

# ---------- PARTIAL DIFFERENTIAL EQUATION ----------
field:

u(x,t)

PDE
 -> relates(MULTIVARIABLE_DERIVATIVES)

classes:
 ELLIPTIC
 PARABOLIC
 HYPERBOLIC

PDE_CLASS
 -> influences(PHYSICAL_INTERPRETATION_AND_SOLUTION_METHOD)

# ---------- HEAT / DIFFUSION ----------
u_t
 =
D*LAPLACIAN(u)

DIFFUSION_EQUATION
 -> models(SPREADING_AND_SMOOTHING)

D
 -> DIFFUSIVITY

applications:
 HEAT
 MASS_TRANSPORT
 BIOLOGICAL_DIFFUSION

# ---------- WAVE ----------
u_tt
 =
c^2*LAPLACIAN(u)

WAVE_EQUATION
 -> models(PROPAGATING_DISTURBANCES)

applications:
 ACOUSTICS
 ELASTIC_WAVES
 ELECTROMAGNETIC_WAVES
 SEISMOLOGY

# ---------- LAPLACE / POISSON ----------
LAPLACIAN(u)
 =
0

-> SOURCE_FREE_EQUILIBRIUM_FIELD

-LAPLACIAN(u)
 =
f

-> SOURCE_DRIVEN_EQUILIBRIUM_FIELD

applications:
 ELECTROSTATICS
 GRAVITATION
 STEADY_HEAT
 POTENTIAL_FLOW

# ---------- MECHANICS ----------
position:

x(t)

velocity:

v
 =
dx/dt

acceleration:

a
 =
d_squared_x/dt_squared

Newton:

m*a
 =
F

CALCULUS
 -> connects(MOTION_FORCE_AND_TRAJECTORY)

# ---------- ENERGY ----------
work:

W
 =
integral(
 F dot dr
)

kinetic energy:

K
 =
1/2*m*v^2

potential:

F
 =
-grad(V)

ENERGY_METHOD
 -> alternative_to(DIRECT_FORCE_ANALYSIS)

# ---------- ELECTROMAGNETISM ----------
Maxwell equations
 -> use(
 DIVERGENCE,
 CURL,
 PARTIAL_DERIVATIVES,
 SURFACE_INTEGRALS,
 LINE_INTEGRALS
)

VECTOR_CALCULUS
 -> mathematical_language_of(CLASSICAL_ELECTROMAGNETISM)

# ---------- FLUID DYNAMICS ----------
velocity field:

u(x,t)

Navier-Stokes
 -> combines(
 TIME_DERIVATIVE,
 CONVECTION,
 PRESSURE_GRADIENT,
 VISCOSITY,
 LAPLACIAN
)

FLUID_ANALYSIS
 -> uses(VECTOR_CALCULUS_AND_PDEs)

# ---------- QUANTUM MECHANICS ----------
wavefunction:

Psi(x,t)

Schroedinger equation:

i*hbar*Psi_t
 =
H_hat*Psi

ANALYSIS
 -> provides(
 OPERATOR_THEORY,
 COMPLEX_FUNCTIONS,
 PDEs,
 SPECTRAL_THEORY
)

# ---------- RELATIVITY ----------
spacetime metric:

g_mu_nu

curvature
 -> derivatives_of(METRIC_AND_CONNECTION)

Einstein field equation
 -> nonlinear_geometric_differential_system

TENSOR_CALCULUS
 -> supports(GENERAL_RELATIVITY)

# ---------- BIOLOGY ----------
population:

N(t)

growth:

dN/dt
 =
r*N
    in simple exponential model

logistic:

dN/dt
 =
r*N*(1-N/K)

DIFFERENTIAL_EQUATIONS
 -> model(POPULATION_CHANGE)

# ---------- CHEMISTRY ----------
concentration:

C_i(t)

reaction rate:

dC_i/dt

RATE_LAWS
 -> ODE_SYSTEMS

reaction-diffusion:

partial_C/partial_t
 =
D*LAPLACIAN(C)
+
R(C)

CHEMICAL_DYNAMICS
 -> calculus_based_model

# ---------- EARTH SCIENCE ----------
applications:
 ATMOSPHERIC_FLOW
 OCEAN_CIRCULATION
 GROUNDWATER
 SEISMIC_WAVES
 HEAT_FLOW
 CLIMATE_DYNAMICS

EARTH_SYSTEM
 -> coupled_differential_equations

# ---------- ASTRONOMY ----------
orbital motion:

d_squared_r/dt_squared
 =
-gravitational_acceleration

CELESTIAL_MECHANICS
 -> ODEs_AND_PERTURBATION_ANALYSIS

stellar structure
 -> differential_equations

cosmology
 -> differential_geometry_and_dynamical_equations

# ---------- OPTIMISATION ----------
scientific objective:

minimise
 J(theta)

gradient:

grad_theta J

OPTIMISATION
 -> parameter_estimation
 -> experimental_design
 -> model_calibration
 -> engineering_design

# ---------- INVERSE PROBLEM ----------
observations:

y_obs

model:

y
 =
F(theta)

infer:

theta

INVERSE_PROBLEM
 -> solve(INPUT_FROM_OUTPUT)

methods:
 LEAST_SQUARES
 REGULARISATION
 BAYESIAN_INFERENCE
 GRADIENT_METHODS

# ---------- LEAST SQUARES ----------
objective:

J(theta)
 =
1/2
*
sum_i(
 y_i_obs
 -
y_i_model(theta)
)^2

minimise J

-> PARAMETER_FIT

RESIDUAL
 -> data_model_difference

# ---------- SENSITIVITY ----------
S_ij
 :=
partial_y_i/partial_theta_j

SENSITIVITY_MATRIX
 -> output_response_to(PARAMETERS)

applications:
 IDENTIFIABILITY
 UNCERTAINTY
 OPTIMISATION
 EXPERIMENT_DESIGN

# ---------- UNCERTAINTY ----------
input uncertainty
 -> propagate_through(MODEL)

output uncertainty
 -> quantify(PREDICTION_RELIABILITY)

methods:
 ERROR_PROPAGATION
 MONTE_CARLO
 SENSITIVITY_ANALYSIS
 BAYESIAN_METHODS

SCIENTIFIC_RESULT
 -> should_include(UNCERTAINTY_CONTEXT)

# ---------- STATISTICAL CONNECTION ----------
data
 -> estimate(MODEL_PARAMETERS)

probability distribution
 -> represent(UNCERTAINTY)

LIKELIHOOD
 -> data_given_model

POSTERIOR
 -> parameter_uncertainty_after_data

CALCULUS
 -> used_in(PROBABILITY_DENSITIES_AND_OPTIMISATION)

# ---------- FOURIER ANALYSIS ----------
signal:

f(t)

Fourier transform:

F(omega)
 =
integral(
 f(t)
 *
exp(-i*omega*t)
 dt
)

FOURIER_ANALYSIS
 -> decomposes(SIGNAL_INTO_FREQUENCIES)

applications:
 SPECTROSCOPY
 SIGNAL_PROCESSING
 WAVE_ANALYSIS
 IMAGING

# ---------- LAPLACE TRANSFORM ----------
F(s)
 =
integral_0^infinity(
 exp(-s*t)
 *
f(t)
 dt
)

LAPLACE_TRANSFORM
 -> simplifies(LINEAR_DIFFERENTIAL_EQUATIONS)

applications:
 CONTROL
 CIRCUITS
 TRANSIENT_ANALYSIS

# ---------- NUMERICAL METHODS ----------
analytic solution
 -> may_not_exist

NUMERICAL_METHOD
 -> approximate(SOLUTION)

methods:
 FINITE_DIFFERENCE
 FINITE_ELEMENT
 FINITE_VOLUME
 RUNGE_KUTTA
 QUADRATURE
 ROOT_FINDING

SCIENTIFIC_COMPUTING
 -> operationalises(CALCULUS_AND_ANALYSIS)

# ---------- MODEL VALIDATION ----------
MODEL_PREDICTION
 -> compare_with(DATA)

VALIDATION
 -> assesses(MODEL_REPRESENTATION_OF_SYSTEM)

VERIFICATION
 -> assesses(CORRECT_COMPUTATIONAL_SOLUTION)

VERIFICATION
 -/> VALIDATION

# ---------- APPLICATIONS ----------
SCIENTIFIC_APPLICATIONS
 -> PHYSICS
 -> CHEMISTRY
 -> BIOLOGY
 -> EARTH_SCIENCE
 -> ASTRONOMY
 -> ENGINEERING
 -> MATERIALS_SCIENCE
 -> ENVIRONMENTAL_SCIENCE
 -> COMPUTATIONAL_SCIENCE

# ---------- RELATION GRAPH ----------
SCIENTIFIC_SYSTEM
 -> represented_by(MATHEMATICAL_MODEL)

FUNCTION
 -> represents(QUANTITY_RELATION)

DERIVATIVE
 -> measures(RATE_OF_CHANGE)

INTEGRAL
 -> measures(ACCUMULATION)

CONSERVATION_LAW
 -> generates(DIFFERENTIAL_EQUATION)

DIFFERENTIAL_EQUATION
 -> predicts(SYSTEM_EVOLUTION)

NUMERICAL_METHOD
 -> approximates(SOLUTION)

DATA
 -> validates(MODEL)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
 "scientific application of calculus" ~= "use of derivatives, integrals and differential equations to model scientific systems",
 "rate of change" ~= "derivative of a quantity with respect to another variable",
 "accumulation" ~= "integral of a density or rate over a domain",
 "conservation law" ~= "balance relation connecting temporal change, flux and sources",
 "scientific model" ~= "mathematical representation of selected behaviour of a physical or empirical system",
 "inverse problem" ~= "inference of hidden inputs or parameters from observed outputs"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
 units_and_dimensions_must_be_consistent: true,
 mathematical_model_is_approximation_of_scientific_system: true,
 initial_and_boundary_conditions_are_part_of_differential_equation_problem: true,
 numerical_solution_requires_error_stability_and_convergence_analysis: true,
 parameter_fit_does_not_automatically_validate_model: true,
 uncertainty_should_be_distinguished_from_numerical_error: true,
 correlation_and_model_fit_do_not_by_themselves_establish_causation: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
 core_definition: 0.99,
 derivative_integral_relation: 0.99,
 differential_equation_relation: 0.99,
 physical_application_relation: 0.99,
 inverse_sensitivity_relation: 0.99,
 numerical_validation_relation: 0.99,
 semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
 identify(scientific_system_variables_parameters_units_and_data)
 -> formulate(mathematical_model)
 -> classify(function_derivative_integral_ODE_PDE_or_optimisation_structure)
 -> impose(initial_boundary_conservation_and_physical_constraints)
 -> choose(analytic_transform_variational_statistical_or_numerical_method)
 -> compute(solution_prediction_or_parameter_estimate)
 -> analyse(sensitivity_uncertainty_error_and_stability)
 -> compare_with(observations_or_reference_data)
 -> validate(model_scope_assumptions_and_dimensions)
 -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
 parse_as: "typed scientific-applications semantic graph",
 primary_entity: SCIENTIFIC_APPLICATIONS,
 preserve: [
  scientific_system,
  variables,
  parameters,
  units,
  equations,
  initial_conditions,
  boundary_conditions,
  data,
  uncertainty
 ],
 infer_through: [
  derivatives,
  integrals,
  vector_calculus,
  differential_equations,
  conservation_laws,
  optimisation,
  transforms,
  numerical_methods,
  sensitivity_analysis
 ],
 avoid: [
  ignoring_units_and dimensional consistency,
  treating_model_as exact reality,
  omitting_initial_or boundary conditions,
  reporting_numerical_result_without error analysis,
  treating_parameter_fit_as full validation,
  conflating_uncertainty_with numerical error
 ]
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL(new_information):
 validate(new_information)
 -> identify(scientific_application_context)
 -> compare(existing_relations)
 -> preserve(system_variable_parameter_unit_equation_data_and_uncertainty_invariants)
 -> update(affected_entities)
 -> update(affected_edges)
 -> recompute(CONFIDENCE)
 -> regenerate(FLOW_HASH)

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
 SOURCE_URL
 + ENTITY
 + CONCEPTS
 + SCIENTIFIC_APPLICATIONS
 + MATHEMATICAL_MODEL
 + FUNCTION
 + DERIVATIVE
 + INTEGRAL
 + DIFFERENTIAL_EQUATION
 + PARTIAL_DIFFERENTIAL_EQUATION
 + OPTIMISATION
 + VECTOR_CALCULUS
 + NUMERICAL_METHOD
 + UNCERTAINTY
 + CONSERVATION_LAW
 + SEMANTIC_EQUIVALENCE
 + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
 -> SCIENTIFIC_APPLICATIONS
 -> SYSTEM_DATA_IDENTIFICATION
 -> MODEL_FORMULATION
 -> MATHEMATICAL_STRUCTURE_CLASSIFICATION
 -> CONDITION_CONSTRAINT_ANALYSIS
 -> METHOD_SELECTION
 -> SOLUTION_PREDICTION_ANALYSIS
 -> SENSITIVITY_UNCERTAINTY_ANALYSIS
 -> DATA_COMPARISON
 -> MODEL_VALIDATION
 -> RECURSIVE_REASONING(query)
 -> CONFIDENCE
 -> OUTPUT(semantic_result)

END FLOW