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Showing 1–30 of 1709 reference entries

Mathematics & logicExamples and counterexamples

Learn what a claim covers, explain why it works, and recognise an allowed case that shows it needs changing.

MethodMathematics & logicCounting and zero

Count each object once, explain the final count, and distinguish quantity, position and identifying labels. Make zero’s role explicit.

DefinitionMathematics & logicNumber lines and order

Use number-line positions to compare whole numbers. Read the scale, recognise equality, and distinguish numerical distance from drawing size.

DefinitionMathematics & logicAddition and subtraction

Combine separate counts, remove a part, and find differences or missing parts. Connect objects, equations and number-line moves, then use tens and ones to regroup without changing value.

DefinitionMathematics & logicMultiplication and division

Count equal groups, distinguish sharing from making groups, and check with multiplication. Split calculations into parts; interpret zero and remainders in context.

DefinitionMathematics & logicFractions

Choose a whole, count equal fractional units, and connect sharing with division. Place fractions on the number line, including values beyond one, and explain what their numerator and denominator mean.

DefinitionMathematics & logic · Physics · MeasurementRatios and rates

Distinguish part-to-part ratios, fractions of a whole and rates with units. Scale quantities together, compare unit rates, and explain when a relationship is proportional. Check zero cases, starting amounts and whether a result fits its context.

DefinitionMathematics & logic · Physics · MeasurementQuantities and units

Describe a quantity as a numerical value with a unit, and distinguish its physical kind from its dimension. Build length, area, volume and rate units, check meaningful combinations, and separate exact unit relations from measurement uncertainty.

DefinitionMathematics & logic · Physics · MeasurementUnit conversions

Convert quantities with exact unit ratios, track cancellation through compound units, and apply the correct powers to area and volume. Distinguish temperature readings from differences, check dimensions and context, and retain measurement uncertainty when reporting.

MethodMathematics & logic · Physics · MeasurementEstimation and uncertainty

Distinguish exact model values, rounded approximations and estimates from measurements. Declare the rounding rule, preserve information through calculations and unit changes, and interpret resolution, precision, accuracy and the stated uncertainty type without turning a report’s digits into guaranteed bounds.

DefinitionMathematics & logicVariables and expressions

Use symbols to name numbers or quantities, describe their allowed values, and evaluate expressions by replacing every occurrence consistently. Preserve grouping, signs, fractions and units, and distinguish a calculated value from solving an equation or proving two rules equivalent.

DefinitionMathematics & logicEquality and equations

Read equality as a relationship, find the permitted values that make an equation true, and justify one-step solutions by reversible operations on both sides. Work exactly with negative numbers and fractions, recognize all-solution and no-solution cases, and check the original equation, units and context.

DefinitionMathematics & logic · PhysicsCoordinates and tables

Locate a point from an ordered pair, read named axes and their scales, and connect the same values in words, tables and plots. Keep signs, fractions and units exact, distinguish an axis point from a quadrant point, and separate selected observations from a stated rule and its domain before drawing connections.

DefinitionMathematics & logic · PhysicsSlope and straight lines

Read a straight line as a stated relationship, calculate its slope from consistent signed coordinate changes, and interpret its intersections with the axes. Keep horizontal, vertical and coincident-point cases distinct, attach units and domain conditions to context models, and use numerical scales rather than appearance alone to judge a line’s change.

DefinitionMathematics & logic · Physics · MeasurementReading graphs

Read a graph’s quantities, units, ticks and context before judging its visual impression. Calculate differences, reference-based percentages and interval rates from the represented values; distinguish legitimate zoom from distorted spacing; and keep samples, interpolation, missing information, uncertainty and causal explanations separate.

MethodPhysics · MeasurementObservables and measurement

Choose the physical quantity you want to investigate, specify the point, event or location involved, and explain how a reading relates to that quantity.

MethodPhysicsSystems and boundaries

Choose what belongs to an investigation, describe its condition at a stated time, and identify interactions with what lies outside its boundary.

MethodPhysicsModels and assumptions

Choose what a model represents, make its simplifications explicit, and check whether its predictions answer the physical question you actually asked.

MethodPhysicsHypotheses, laws and theories

Identify what a scientific statement does, examine its evidence and conditions, and avoid treating its name as a score for certainty.

MethodMathematics & logic · PhysicsApproximations

Build a useful estimate, compare a correction with a stated reference scale, and check what is lost when you leave out a term.

ApproximationMathematics & logic · PhysicsSymmetry and invariance

Specify a transformation, identify what stays unchanged, and distinguish symmetry of an object from invariance of a physical model.

Mathematical statementPhysicsConservation laws

Use a defined system and a named quantity to track transfers, distinguish changing parts from conserved totals, and check the conditions of energy and momentum balances.

ModelPhysicsEmergence and levels of description

Choose a descriptive level, connect collective quantities with component behavior, and check when a simpler physical description answers a specified question.

ModelMathematics & logic · PhysicsVectors

Describe displacement with fixed-axis components, combine compatible vectors, and use small exact examples to introduce span, independence, basis and real vector spaces.

DefinitionPhysics · Information · MeasurementScientific records

Build an inspectable physics record, test and revise a bounded cart model, preserve processing history, and plan findable data with appropriate sharing conditions.

MethodPhysicsPosition, velocity and acceleration

Describe a point's motion on a chosen straight axis, then connect signed changes, instantaneous rates and one exact set of position, velocity and acceleration graphs.

DefinitionPhysicsNewton’s laws and forces

Choose one body, identify the forces acting on it, and use their vector sum to distinguish acceleration, balanced motion and mutual interactions.

ModelPhysicsWork, energy and power

Connect force and displacement to changes in motion, follow a spring’s stored energy, and distinguish an energy transfer from its rate.

ModelPhysics · MeasurementSpeed of light in vacuum

c denotes the speed of light in vacuum. Its numerical value is fixed exactly in SI units.

DefinitionPhysics · MeasurementPlanck constant

h denotes the Planck constant. Its numerical value is fixed exactly in SI units.

Definition
DefinitionSource checked

Conditioning of a numerical problem

Mathematics & logic

Conditioning measures sensitivity of the exact answer to data perturbations.

Key knowledge

Conditioning measures sensitivity of the exact answer to data perturbations. For differentiable scalar f at nonzero x with f(x)≠0, its relative condition number is |xf′(x)/f(x)|.

Values and equations

κ_f(x)=|xf′(x)/f(x)|

f
Differentiable scalar problem
x
Nonzero input with f(x)≠0
κ_f
Local relative condition number

Conditions and scope

  • Relative conditioning is ill-defined at a zero scale; use absolute or other scales there.
  • Conditioning concerns the problem, not the chosen algorithm.
Sources · 1

References

Sources retain their recorded scope. A source link alone does not establish verification of every statement.

  1. Fundamentals of Numerical Computation: Problems and conditioningTobin A. Driscoll and Richard J. BraunTextbook chapterfncbook.comRead source

Evidence and review

Definition 1.2.1 and equation 1.2.6.

2026-10-10 · Independent AI editorial review: All241 compact claims, hypotheses, displayed equations/conventions and all596 origin semantic links read; entire241-entry author exact-source inspection matrix and declared limitations reviewed. Prior105 actual inherited Mathematics/Information records read for whole-inventory breadth. Targeted independent primary sections are listed below; this does NOT claim every cited source body or every proof was independently re-read. This is AI reference-content review, not human disciplinary certification, exhaustive textbook reproduction or experimental replication. Receipt: mathematics-information-independent-content-review.1.

Related reference entries

Explore relationships within this area of study and its connections to other areas.

Showing 1–17 of 17 relationships

Scientific relationships · 17 / 17

The relationship type describes the connection. Its linked statements, sources and conditions define the scope.

  1. Uses definition
    Relationship sources · 1

    Linked statements

    • Conditioning measures sensitivity of the exact answer to data perturbations. For differentiable scalar f at nonzero x with f(x)≠0, its relative condition number is |xf′(x)/f(x)|.
  2. Uses definition
    Relationship sources · 1

    Linked statements

    • Forward error compares a computed answer with the exact answer. Backward error asks how much the input must change to make the computed answer exact. A backward-stable algorithm solves a nearby problem with a small data perturbation.
  3. Uses definition
    Relationship sources · 1

    Linked statements

    • For invertible A and an induced matrix norm, κ(A)=‖A‖‖A⁻¹‖. If A(x+Δx)=b+Δb with b≠0, then ‖Δx‖/‖x‖≤κ(A)‖Δb‖/‖b‖.
  4. Uses definition
    Relationship sources · 1

    Linked statements

    • Interpolation chooses p with p(tᵢ)=yᵢ at specified distinct nodes. Global high-degree polynomials can behave poorly between nodes; piecewise low-degree interpolation limits degree locally, and node choice affects conditioning.
  5. Uses definition
    Relationship sources · 1

    Linked statements

    • For continuous real f with f(a)f(b)<0, test the midpoint and retain a half interval with opposite endpoint signs, unless the midpoint is a root. After k halvings the retained width is (b−a)/2ᵏ, and its midpoint is within half that width of a root.
  6. Uses definition
    Relationship sources · 1

    Linked statements

    • For uᵢ₊₁=uᵢ+hφ(tᵢ,uᵢ,h), if the exact-value one-step defect is bounded by Chᵖ⁺¹ and |∂ᵤφ|≤L, L>0, then exact initialization gives |u(tᵢ)−uᵢ|≤(Chᵖ/L)(eᴸ⁽ᵗⁱ⁻ᵃ⁾−1). On a fixed interval this is O(hᵖ).
  7. Related to
    Relationship sources · 1

    Linked statements

    • For constant pumping from an ideal confined aquifer, drawdown at radius r and time t is s=QW(u)/(4πT), where u=r²S/(4Tt) and W(u)=∫ᵤ∞e^(−ξ)dξ/ξ. The response depends jointly on transmissivity T and storativity S. Fitting an observed curve estimates model parameters only if the aquifer and observation process satisfy the assumed model.

    Parameter sensitivity and conditioning affect what a fitted drawdown can constrain.

  8. Related to
    Relationship sources · 1

    Linked statements

    • Different subsurface density structures can yield the same external gravitational field. For example, concentric spherically symmetric bodies of different radii/densities but the same total excess mass produce the same exterior −GΔM/r potential wherever the observer is outside them. Finite noisy anomaly observations add further resolution limits; geological constraints narrow admissible models rather than guarantee uniqueness.

    Noise amplification and mathematical nonuniqueness are distinct concerns.

  9. Related to
    Relationship sources · 2

    Linked statements

    • A weak-field linear isotropic rock model separates magnetizationM=M_r+χH into retained remanence and an induced response to localH. A scalar survey's anomaly relative to vector backgroundB₀ isΔF=|B₀+b|−|B₀|, approximatelyB̂₀·b when|b|≪|B₀|. The localized geological fieldb depends on magnetization, geometry and observation position; scalar anomalies alone do not uniquely recover those properties.

    Limited noisy scalar data constrain interpretation.

  10. Related to
    Relationship sources · 1

    Linked statements

    • Regularized empirical minimization adds lambda times a complexity penalty to sample loss. An L2 penalty controls coefficient size and an L1 penalty can favor sparse solutions, but correlated predictors and tuning affect interpretation. Convexity can support global optimization guarantees; a nonconvex neural objective generally offers weaker local guarantees. A smaller optimization error does not by itself reduce estimation error or prove a causal relation.

    Optimization stability depends on parameterization and conditioning.

  11. Related to
    Relationship sources · 1

    Linked statements

    • For separable H=pᵀM⁻¹p/2+V(q), fixed-step kick–drift–kick Verlet is second order, reversible and symplectic. It generally preserves neither the exact Hamiltonian nor every observable exactly. Backward-error analysis explains near-conservation of a modified energy on suitable long intervals when smoothness, step-size and bounded-trajectory hypotheses hold.

    Round-off, stiffness and finite step size remain separate errors.

  12. Related to
    Relationship sources · 1

    Linked statements

    • Single-site dynamical mean-field theory maps an interacting lattice problem to a self-consistent impurity problem with a frequency-dependent local self-energy. It retains local temporal correlations that static mean field misses. Its locality becomes exact in a suitably scaled infinite-coordination limit; finite-dimensional nonlocal correlations and material double-counting remain approximation issues.

    Inverse continuation from imaginary-time data is an ill-conditioned inference.

  13. Related to
    Relationship sources · 4

    Linked statements

    • DNS aims to resolve the dynamically relevant viscous scales of a specified flow; LES filters resolved scales and models subfilter stress. Changing resolution changes an LES filter/model problem as well as numerical error. Verification checks conservation, discretization and sampling; validation compares selected observables with independent uncertain measurements for the declared flow regime.

    Stable code and converged residuals do not prove continuum/model accuracy.

  14. Related to
    Relationship sources · 3

    Linked statements

    • Yee FDTD staggers electric/magnetic fields in space and time to approximate Maxwell curl evolution. In a uniform isotropic nondispersive medium, the three-dimensional explicit scheme requires c_mΔt√(Δx⁻²+Δy⁻²+Δz⁻²)≤1. Stability alone does not guarantee correct phase, interfaces or absorption; finite PML and numerical dispersion need convergence tests.

    Convergence, round-off and physical model error remain distinct.

  15. Related to
    Relationship sources · 2

    Linked statements

    • A normalized frequency-domain Tikhonov inverse G=H*/(|H|²+λ²|L|²) suppresses ill-conditioned frequencies instead of dividing by an arbitrarily small H. The estimate depends on penalty and prior assumptions. Include calibration-parameter, noise, timing, interpolation and regularization-error contributions; filtering cannot recover absent bandwidth without additional information.

    Small transfer response creates unstable inversion.

  16. Related to
    Relationship sources · 2

    Linked statements

    • Calibration parameters are identifiable only when distinct parameter values lead to distinguishable predictions for the available design and reference information. Shared offset, scale and model discrepancy can be confounded. Reference anchors, excitation across the range and independent validation constrain this ambiguity; a densely sampled curve cannot resolve an unobserved degree of freedom.

    Nearly indistinguishable predictions create unstable parameter estimation.

  17. Related to
    Relationship sources · 1

    Linked statements

    • Molecular-dynamics convergence requires checking integration step, constraints and conserved quantities for the chosen ensemble. A thermostat can hide integration-related energy drift; apparent temperature control is therefore insufficient evidence for accurate trajectories.

    Solver errors and sensitivity must be separated from the physical approximation.

Entry detailsPermanent link to this entry ↗
Record identity
math-numerical-conditioning
Revision
university-core.1
Structured reference record
University core expansion