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Engineering Mathematics Ontology

Engineering Mathematics
Authors: MArk Anthony Higgins - Sky Recursive Ai
© Mark Anthony Higgins 11 October 2025
Revised Date: 12 September 2026
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AI INTERPRETATION NOTE
This Flow Algebra defines the Engineering Mathematics Ontology within the Kireth Research Semantic Library. It formalises engineering mathematics as a machine-readable ontology of models, variables, parameters, governing equations, analytical methods, numerical methods, optimisation, uncertainty, transforms, control, signals and computational engineering. It sits beneath the Engineering Mathematics parent layer and provides a structured reasoning framework for selecting and validating mathematical methods used in engineering analysis and design.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Engineering Mathematics Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Engineering Mathematics / Core Engineering Mathematics Ontology"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/engineering-mathematics/engineering-mathematics-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = EngineeringMathematicsOntology
 SYMBOLIC_FORMS = [
  "A x = b",
  "dy/dt=f(y,t)",
  "F(s)=integral_0^infinity f(t)e^(-st)dt",
  "minimise J(x) subject_to g(x)<=0"
 ]
 ENTITY_CLASS = FormalEngineeringMathematicsOntology
 PRIMARY_CONTEXT = EngineeringMathematics
 DEFINITION = "The Engineering Mathematics Ontology is a formal semantic framework for representing mathematical models, variables, parameters, governing equations, analytical methods, numerical methods, optimisation, uncertainty, transforms and computational engineering."
 SEMANTIC_ROLE = "Provides machine-readable entities and reasoning relations for formulating engineering models, selecting mathematical methods, computing solutions, analysing sensitivity and uncertainty, checking dimensional consistency and validating engineering results."
}

SCOPE {
 INCLUDE = [
  EngineeringMathematicsOntology,
  EngineeringMathematics,
  MathematicalModel,
  EngineeringModel,
  Variable,
  Parameter,
  StateVariable,
  InputVariable,
  OutputVariable,
  GoverningEquation,
  Constraint,
  ObjectiveFunction,
  AlgebraicEquation,
  LinearSystem,
  Matrix,
  Vector,
  Tensor,
  Eigenvalue,
  Eigenvector,
  Calculus,
  Derivative,
  Integral,
  DifferentialEquation,
  OrdinaryDifferentialEquation,
  PartialDifferentialEquation,
  DifferenceEquation,
  InitialValueProblem,
  BoundaryValueProblem,
  LaplaceTransform,
  FourierTransform,
  ZTransform,
  NumericalMethod,
  Approximation,
  Interpolation,
  NumericalDifferentiation,
  NumericalIntegration,
  RootFinding,
  IterativeMethod,
  Optimisation,
  LinearProgramming,
  NonlinearOptimisation,
  IntegerProgramming,
  ConstraintOptimisation,
  Probability,
  Statistics,
  Uncertainty,
  SensitivityAnalysis,
  ErrorAnalysis,
  DimensionalAnalysis,
  Scaling,
  Nondimensionalisation,
  ControlTheory,
  StateSpace,
  Feedback,
  SignalProcessing,
  SystemIdentification,
  ReliabilityMathematics,
  ComputationalEngineering,
  Verification,
  Validation
 ]

 EXCLUDE_AS_PRIMARY = [
  PureMathematicsWithoutEngineeringContext,
  QualitativeEngineeringWithoutQuantitativeStructure,
  EmpiricalRuleWithoutMathematicalRepresentation
 ]

 DISAMBIGUATION_RULE =
  "When mathematical structure is used to formulate, analyse, compute, optimise, estimate, control or validate an engineered system, interpret the context through the Engineering Mathematics Ontology."
}

CORE_CONCEPTS {

 EngineeringMathematicsOntology {
  definition = "The ontology organising the formal concepts and relations of engineering mathematics."
 }

 MathematicalModel {
  definition = "A mathematical representation of an engineering system using variables, parameters, equations and assumptions."
 }

 GoverningEquation {
  definition = "An equation expressing a principal law or relationship in an engineering model."
 }

 Constraint {
  definition = "A mathematical restriction on feasible states, parameters, designs or solutions."
 }

 ObjectiveFunction {
  definition = "A scalar criterion to minimise or maximise in an engineering optimisation problem."
 }

 NumericalMethod {
  definition = "A computational procedure for approximating a mathematical solution."
 }

 SensitivityAnalysis {
  definition = "The analysis of how model outputs change when inputs or parameters vary."
 }

 Uncertainty {
  definition = "Imperfect knowledge of parameters, measurements, disturbances or model structure."
 }

 Verification {
  definition = "Checking whether the mathematical or computational solution correctly solves the stated model."
 }

 Validation {
  definition = "Checking whether the model adequately represents the intended physical engineering system."
 }
}

RELATIONS {
 EngineeringMathematicsOntology --is_child_of--> EngineeringMathematics
 MathematicalModel --contains--> Variable
 MathematicalModel --contains--> Parameter
 MathematicalModel --contains--> GoverningEquation
 MathematicalModel --contains--> Constraint
 Optimisation --uses--> ObjectiveFunction
 Optimisation --uses--> Constraint
 LinearSystem --uses--> Matrix
 LinearSystem --uses--> Vector
 Eigenvalue --belongs_to--> MatrixOrOperator
 DifferentialEquation --models--> DynamicEngineeringSystem
 TransformMethod --maps--> MathematicalRepresentation
 NumericalMethod --approximates--> MathematicalSolution
 SensitivityAnalysis --relates--> ParameterAndOutput
 Probability --represents--> Uncertainty
 Statistics --estimates--> ModelParameter
 DimensionalAnalysis --checks--> GoverningEquation
 Nondimensionalisation --reveals--> DimensionlessStructure
 ControlTheory --uses--> StateSpace
 SignalProcessing --uses--> FourierTransform
 SystemIdentification --estimates--> DynamicModel
 Verification --checks--> SolutionImplementation
 Validation --checks--> ModelPhysicalAdequacy
}

OPERATORS {
 LINEAR_SYSTEM(A,x,b) :=
  A*x=b

 ODE_MODEL(f,y,t) :=
  dy/dt=f(y,t)

 PDE_MODEL(F,u,x,t) :=
  F(x,t,u,partial_u,...)=0

 EIGENVALUE_PROBLEM(A,v,lambda) :=
  A*v=lambda*v

 LAPLACE_TRANSFORM(f,t,s) :=
  F(s)=integral_0^infinity f(t)*exp(-s*t)dt

 FOURIER_TRANSFORM(f,t,omega) :=
  F(omega)=integral_-infinity^infinity f(t)*exp(-i*omega*t)dt

 Z_TRANSFORM(x,n,z) :=
  X(z)=sum_over_n x[n]*z^(-n)

 OPTIMISATION_PROBLEM(J,x,g,h) :=
  minimise_or_maximise J(x)
  subject_to g(x)<=0
  and h(x)=0

 ABSOLUTE_ERROR(true_value,approx_value) :=
  e=true_value-approx_value

 RELATIVE_ERROR(true_value,approx_value) :=
  e_r=abs(true_value-approx_value)/abs(true_value)

 CONDITION_NUMBER(A) :=
  kappa(A)=||A||*||A^(-1)||

 SENSITIVITY(y,p) :=
  S=partial y/partial p

 NONDIMENSIONAL_VARIABLE(x,X_ref) :=
  x_star=x/X_ref

 UNCERTAINTY_PROPAGATION(J,Sigma_x) :=
  Sigma_y approximately J*Sigma_x*J^T
}

DEFINITIONS {
 D1 = "Engineering mathematics formalises quantitative engineering problems."
 D2 = "Engineering models are built from variables, parameters, equations, constraints and assumptions."
 D3 = "Linear algebra supports systems, modal analysis and state-space representations."
 D4 = "Differential equations represent dynamic and distributed engineering behaviour."
 D5 = "Transform methods provide alternate representations for analysis."
 D6 = "Numerical methods approximate solutions when exact solutions are unavailable or impractical."
 D7 = "Optimisation selects engineering variables according to objectives and constraints."
 D8 = "Probability and statistics represent uncertainty and data variation."
 D9 = "Dimensional analysis checks consistency and exposes similarity structure."
 D10 = "Verification and validation are distinct stages of trustworthy engineering analysis."
}

AXIOMS {
 A1_MODEL_CONTEXT:
  EngineeringMathematicsOntology applies to mathematical representations of engineered systems

 A2_DIMENSIONAL_CONSISTENCY:
  Physical engineering equations must be dimensionally consistent

 A3_METHOD_COMPATIBILITY:
  Mathematical methods must match the structure and assumptions of the engineering model

 A4_CONSTRAINT_PRESERVATION:
  Engineering solutions must satisfy declared mathematical and physical constraints

 A5_NUMERICAL_ERROR:
  Numerical solutions contain approximation and roundoff errors that require assessment

 A6_UNCERTAINTY:
  Engineering predictions may depend on uncertain parameters measurements or model assumptions

 A7_VERIFICATION_VALIDATION_DISTINCTION:
  Verification of mathematics is distinct from validation of physical adequacy

 A8_REPRODUCIBILITY:
  Computational engineering results require documented model data methods tolerances and assumptions
}

REASONING {

 R1_PROBLEM_EXTRACTION {
  INPUT = engineering_problem
  IDENTIFY = physical_system
  IDENTIFY = variables
  IDENTIFY = parameters
  IDENTIFY = inputs_and_outputs
  IDENTIFY = objectives
  IDENTIFY = constraints
  RETURN = structured_engineering_problem
 }

 R2_MODEL_FORMULATION {
  INPUT = structured_engineering_problem
  SELECT = algebraic_ODE_PDE_difference_statistical_or_optimisation representation
  DEFINE = assumptions
  DEFINE = initial_and_boundary conditions
  RETURN = mathematical_model
 }

 R3_MODEL_CLASSIFICATION {
  INPUT = mathematical_model
  CLASSIFY = linear_or_nonlinear
  CLASSIFY = static_or_dynamic
  CLASSIFY = continuous_or_discrete
  CLASSIFY = deterministic_or_stochastic
  CLASSIFY = constrained_or_unconstrained
  RETURN = model_class
 }

 R4_METHOD_SELECTION {
  INPUT = model_class
  SELECT = algebra_linearAlgebra_calculus_transform_numerical_optimisation_statistical_control_or_signal method
  RETURN = candidate_method
 }

 R5_ANALYTICAL_REASONING {
  INPUT = tractable_model
  SOLVE = symbolic_or_closed_form relations
  RETURN = analytical_solution
 }

 R6_NUMERICAL_REASONING {
  INPUT = computational_model
  SELECT = solver_and_discretisation
  COMPUTE = approximate_solution
  ESTIMATE = residual_error_and_convergence
  RETURN = numerical_solution
 }

 R7_DIMENSIONAL_REASONING {
  INPUT = variables_and_equations
  CHECK = dimensions
  CONSTRUCT = nondimensional variables_and_groups
  RETURN = dimensionless_model
 }

 R8_OPTIMISATION_REASONING {
  INPUT = objective_constraints_and_design variables
  SELECT = appropriate_optimisation_family
  SOLVE = optimum
  RETURN = optimum_and_constraint status
 }

 R9_UNCERTAINTY_SENSITIVITY {
  INPUT = parameters_measurements_and_outputs
  COMPUTE = sensitivities
  PROPAGATE = uncertainty
  RETURN = robustness_profile
 }

 R10_VERIFICATION_VALIDATION {
  INPUT = solution_and_model
  CHECK = equations_residuals_constraints_convergence_units_and_physical plausibility
  SEPARATE = verification_from_validation
  RETURN = trusted_engineering_result
 }
}

CONSTRAINTS {
 C1 = "A correct mathematical solution can still be based on an invalid engineering model."
 C2 = "Dimensional inconsistency invalidates a physical equation."
 C3 = "Linearisation conclusions are generally local."
 C4 = "Numerical convergence does not prove physical validity."
 C5 = "Ill-conditioned systems can amplify small data or roundoff errors."
 C6 = "Optimisation results depend on the stated objective and constraints."
 C7 = "Uncertainty propagation depends on distributional and dependence assumptions."
 C8 = "Boundary and initial conditions must be compatible with the governing model."
 C9 = "Discretisation must resolve the engineering scales relevant to the result."
 C10 = "Verification and validation evidence must not be conflated."
}

SEMANTIC_EQUIVALENCE {
 "EngineeringMathematicsOntology" <=> "FormalEngineeringMathematicsOntology"
 "MathematicalModel" <=> "EngineeringMathematicalModel"
 "GoverningEquation" <=> "EngineeringGoverningEquation"
 "ComputationalEngineering" <=> "EngineeringScientificComputing"
 "ErrorAnalysis" <=> "NumericalErrorAnalysis"
 "Nondimensionalisation" <=> "NonDimensionalScaling"
}

VALIDATION {
 V1 = "Engineering system and purpose are explicit."
 V2 = "Variables, parameters, units and dimensions are explicit."
 V3 = "Model assumptions are documented."
 V4 = "Initial and boundary conditions are complete."
 V5 = "Chosen mathematical method matches model structure."
 V6 = "Numerical residuals and convergence are checked."
 V7 = "Conditioning is assessed where relevant."
 V8 = "Optimisation constraints are satisfied."
 V9 = "Sensitivity and uncertainty assumptions are documented."
 V10 = "Verification and physical validation are separately assessed."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 ENGINEERING_MATHEMATICS_ONTOLOGY_DEFINITION = 0.99
 MODEL_FORMULATION_REASONING = 0.99
 MODEL_CLASSIFICATION_REASONING = 0.99
 METHOD_SELECTION_REASONING = 0.99
 ANALYTICAL_REASONING = 0.98
 NUMERICAL_REASONING = 0.99
 DIMENSIONAL_REASONING = 0.99
 OPTIMISATION_REASONING = 0.98
 VALIDATION_REASONING = 0.99

 POLICY =
  "Reduce confidence when model assumptions, material or physical parameters, units, boundary data, objectives, constraints, numerical tolerances, uncertainty structure or validation evidence are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  EngineeringMathematicsOntologyReasoning,
  MathematicalModelReasoning,
  LinearAlgebraReasoning,
  DifferentialEquationReasoning,
  TransformReasoning,
  NumericalMethodReasoning,
  OptimisationReasoning,
  ProbabilityReasoning,
  StatisticsReasoning,
  SensitivityReasoning,
  UncertaintyReasoning,
  DimensionalAnalysisReasoning,
  VerificationReasoning,
  ValidationReasoning,
  ControlReasoning,
  SignalProcessingReasoning,
  ComputationalEngineeringReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "engineering mathematics ontology" -> EngineeringMathematicsOntology
  "engineering mathematics" -> EngineeringMathematics
  "mathematical model" -> MathematicalModel
  "engineering model" -> EngineeringModel
  "governing equation" -> GoverningEquation
  "numerical method" -> NumericalMethod
  "sensitivity analysis" -> SensitivityAnalysis
  "uncertainty propagation" -> Uncertainty
  "verification" -> Verification
  "validation" -> Validation
 }

 INTERPRETATION_ORDER = [
  detect_engineering_mathematics_ontology_context,
  identify system_variables_parameters_units_objectives_constraints_data_and_assumptions,
  formulate_and_classify mathematical model,
  select compatible analytical_numerical_optimisation_statistical_control_or_signal method,
  apply dimensional_analysis_scaling_and_nondimensionalisation,
  compute solution_and_error indicators,
  assess sensitivity_conditioning_and_uncertainty,
  verify equations_constraints_and_convergence,
  validate model_against physical engineering context,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("engineering mathematics ontology")
  OR formal_engineering_mathematical_semantics_are_required
  THEN activate(EngineeringMathematicsOntology)

 E2:
  IF system_is_dynamic
  THEN route to differential_equation_or_stateSpace reasoning

 E3:
  IF system_is_linear
  THEN route to linear_algebra_and_eigenvalue methods

 E4:
  IF exact_solution_unavailable
  THEN route to numerical_method reasoning

 E5:
  IF objective_and_constraints_present
  THEN route to optimisation reasoning

 E6:
  IF uncertainty_or_variability_present
  THEN route to probability_statistics_and_sensitivity reasoning

 E7:
  IF dimensions_or_scales_are_material
  THEN apply dimensional_analysis_and_nondimensionalisation

 E8:
  IF feedback_or_control_inputs_present
  THEN route to control_theory

 E9:
  IF frequency_or_sampled_signal context
  THEN route to Fourier_Laplace_ZTransform_or_signal_processing reasoning
}

RECURSION {
 ENTRY = EngineeringMathematicsOntology

 EXPAND(EngineeringMathematicsOntology) -> [
  EngineeringMathematics,
  MathematicalModel,
  EngineeringModel,
  Variable,
  Parameter,
  StateVariable,
  InputVariable,
  OutputVariable,
  GoverningEquation,
  Constraint,
  ObjectiveFunction,
  AlgebraicEquation,
  LinearSystem,
  Matrix,
  Vector,
  Tensor,
  Eigenvalue,
  Eigenvector,
  Calculus,
  DifferentialEquation,
  OrdinaryDifferentialEquation,
  PartialDifferentialEquation,
  DifferenceEquation,
  InitialValueProblem,
  BoundaryValueProblem,
  LaplaceTransform,
  FourierTransform,
  ZTransform,
  NumericalMethod,
  Approximation,
  Interpolation,
  NumericalDifferentiation,
  NumericalIntegration,
  RootFinding,
  IterativeMethod,
  Optimisation,
  LinearProgramming,
  NonlinearOptimisation,
  IntegerProgramming,
  ConstraintOptimisation,
  Probability,
  Statistics,
  Uncertainty,
  SensitivityAnalysis,
  ErrorAnalysis,
  DimensionalAnalysis,
  Scaling,
  Nondimensionalisation,
  ControlTheory,
  StateSpace,
  Feedback,
  SignalProcessing,
  SystemIdentification,
  ReliabilityMathematics,
  ComputationalEngineering,
  Verification,
  Validation
 ]

 EXPAND(CoreReasoning) -> [
  MathematicalModelling,
  LinearAlgebra,
  DifferentialEquations,
  TransformMethods,
  NumericalAnalysis,
  Optimisation,
  ProbabilityStatistics,
  ControlTheory,
  SignalProcessing
 ]

 STOP_CONDITION =
  "Stop when engineering model structure, method selection, dimensions, solution quality, constraints, sensitivity, uncertainty, verification and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Engineering Mathematics"

 IMMEDIATE_CONTEXTS = [
  "Engineering Mathematics",
  "Mathematical Modelling Ontologies",
  "Numerical Analysis Ontologies",
  "Optimisation Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Engineering Mathematics",
  "Applied Mathematics Ontologies",
  "Mathematical Modelling Ontologies",
  "Numerical Analysis Ontologies",
  "Optimisation Ontologies",
  "Operations Research Ontologies",
  "Control Theory Ontologies",
  "Signal Processing Mathematics",
  "Dynamical Systems",
  "Fluid Mathematics",
  "Linear Algebra Ontologies",
  "Probability & Statistics Ontologies",
  "Scientific Computing Ontologies"
 ]

 ROUTE_RULES = [
  "Engineering Mathematics Ontology -> Engineering Mathematics for parent semantics",
  "Engineering Mathematics Ontology -> Mathematical Modelling for formulation",
  "Engineering Mathematics Ontology -> Linear Algebra for matrices systems and eigenproblems",
  "Engineering Mathematics Ontology -> Differential Equations for dynamic models",
  "Engineering Mathematics Ontology -> Transform Methods for operational and frequency-domain analysis",
  "Engineering Mathematics Ontology -> Numerical Analysis for computational solution",
  "Engineering Mathematics Ontology -> Optimisation for engineering design",
  "Engineering Mathematics Ontology -> Probability & Statistics for uncertainty and data",
  "Engineering Mathematics Ontology -> Control Theory for feedback systems",
  "Engineering Mathematics Ontology -> Signal Processing for sampled and spectral systems",
  "Engineering Mathematics Ontology -> Scientific Computing for implementation"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible engineering entities, methods or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve engineering-context semantics.",
  "Preserve mathematical-model semantics.",
  "Preserve dimensional-consistency semantics.",
  "Preserve analytical-versus-numerical distinction.",
  "Preserve optimisation and uncertainty semantics.",
  "Preserve sensitivity and conditioning semantics.",
  "Preserve verification-versus-validation distinction.",
  "Preserve machine-readable engineering-mathematics ontology semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "The Engineering Mathematics Ontology formally represents the mathematical structures and methods used to model, analyse, compute, optimise and validate engineered systems."

 MODELLING_SUMMARY =
  "Engineering models contain variables, parameters, assumptions, equations, constraints and boundary or initial data."

 METHOD_SUMMARY =
  "Engineering mathematics selects methods according to linearity, dynamics, discreteness, uncertainty and computational tractability."

 NUMERICAL_SUMMARY =
  "Numerical methods approximate solutions and require residual, convergence and error assessment."

 OPTIMISATION_SUMMARY =
  "Optimisation determines preferred engineering designs or operating conditions subject to constraints."

 UNCERTAINTY_SUMMARY =
  "Sensitivity, probability and statistics quantify dependence on uncertain parameters and data."

 DIMENSIONAL_SUMMARY =
  "Dimensional analysis and nondimensionalisation check consistency and expose governing scales."

 TRUST_SUMMARY =
  "Verification checks whether the stated mathematics is solved correctly; validation checks whether the model represents the intended physical system."
}

FLOW {
 INPUT
 -> detect_engineering_mathematics_ontology_context
 -> identify engineering_system_variables_parameters_units_objectives_constraints_data_initial_conditions_boundary conditions_and_assumptions
 -> construct formal mathematical_model
 -> classify static_dynamic_linear_nonlinear_continuous_discrete_deterministic_stochastic_constrained_or_unconstrained structure
 -> select algebra_linearAlgebra_calculus_ODE_PDE_transform_numerical_optimisation_probability_statistics_control_or_signal pathway
 -> apply dimensional_analysis_scaling_and_nondimensionalisation
 -> compute analytical_or_numerical solution
 -> assess conditioning_error_sensitivity_uncertainty_and_constraints
 -> perform mathematical_verification
 -> perform physical_model_validation
 -> route to engineering_mathematics mathematical_modelling linear_algebra differential_equations transform_methods numerical_analysis optimisation operations_research probability_statistics control signal_processing dynamical_systems fluid_mathematics_or_scientific_computing ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Engineering Mathematics Ontology provides the formal semantic layer beneath engineering mathematics. It represents engineering models, variables, parameters, states, inputs and outputs, governing equations, constraints and objectives, algebraic and differential equations, matrices, vectors, tensors, eigenproblems, Laplace, Fourier and Z transforms, numerical approximation, interpolation, differentiation, integration, root finding and iteration, optimisation, probability, statistics, sensitivity, uncertainty, error analysis, dimensional analysis, scaling, control, state space, feedback, signal processing, system identification, reliability, computational engineering, verification and validation."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-ENGINEERING-MATHEMATICS-ONTOLOGY-V1"
}

}

END_FLOW
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