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Mathematics & logicExamples and counterexamples

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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

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MethodPhysics · MeasurementObservables and measurement

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MethodPhysicsSystems and boundaries

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MethodPhysicsModels and assumptions

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MethodPhysicsHypotheses, laws and theories

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MethodMathematics & logic · PhysicsApproximations

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ApproximationMathematics & logic · PhysicsSymmetry and invariance

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Mathematical statementPhysicsConservation laws

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ModelPhysicsEmergence and levels of description

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ModelMathematics & logic · PhysicsVectors

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DefinitionPhysics · Information · MeasurementScientific records

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MethodPhysicsPosition, velocity and acceleration

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DefinitionPhysicsNewton’s laws and forces

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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
Mathematical statementSource checked

A nonlinear existence theorem: subcritical NLS

Mathematics & logic

Research context

postgraduate concept · established · As of 2026-10-10

For i u-t=−Delta-u−lambda|u|^alpha u on R-n, real lambda and 0<alpha<4/n, every L2 initial datum has a unique local mild solution in C([0,T];L2) intersected with L-q(0,T;L^(alpha+2)), where q=4(alpha+2)/(n alpha).

Key knowledge

For i u-t=−Delta-u−lambda|u|^alpha u on R-n, real lambda and 0<alpha<4/n, every L2 initial datum has a unique local mild solution in C([0,T];L2) intersected with L-q(0,T;L^(alpha+2)), where q=4(alpha+2)/(n alpha). Strichartz estimates and a contraction argument control the nonlinear integral map, and the local solution map is locally Lipschitz. This L2-subcritical theorem does not extend automatically to critical/supercritical powers or arbitrary boundary geometries.

Values and equations

i u_t=−Delta u−lambda |u|^alpha u; 0<alpha<4/n; q=4(alpha+2)/(n alpha)

u
Complex-valued mild solution on R-n
lambda
Real nonlinear coefficient in normalized coordinates
alpha
Subcritical positive power
q
Time-integrability exponent in the local solution space

Conditions and scope

  • n≥1, whole space R-n, L2 initial data and the displayed subcritical exponent restriction; the equation is in normalized mathematical coordinates.
  • Mild solution uses the free Schrodinger group and Duhamel integral; local existence is not itself a global regularity conclusion.
Sources · 1

References

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

  1. Notes on Partial Differential EquationsJohn K. Hunter / University of California, DavisCourse notes · PDFmath.ucdavis.eduRead source

Evidence and review

Author checked this English statement, displayed equation where present and stated hypotheses against these source locations: Chapter 4: weak Dirichlet Laplacian, Poincaré, Theorem 4.20 Lax–Milgram; chapter 6 parabolic energy/Galerkin; chapter 3 compactness; §5.6 Theorems 5.53/5.55 subcritical nonlinear Schrodinger existence. The check is bounded to the recorded claims; no complete proof reconstruction, software proof replay or universal literature completeness is claimed.

2026-10-10 · Author source-body/statement/equation/domain check, with graph associations scoped individually. Separate independent content/source/inventory review is required before branch closure.

Related reference entries

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Scientific relationships · 3 / 3

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  1. Related to
    Relationship sources · 1

    Linked statements

    • For i u-t=−Delta-u−lambda|u|^alpha u on R-n, real lambda and 0<alpha<4/n, every L2 initial datum has a unique local mild solution in C([0,T];L2) intersected with L-q(0,T;L^(alpha+2)), where q=4(alpha+2)/(n alpha). Strichartz estimates and a contraction argument control the nonlinear integral map, and the local solution map is locally Lipschitz. This L2-subcritical theorem does not extend automatically to critical/supercritical powers or arbitrary boundary geometries.

    A complete space supports the contraction argument.

  2. Related to
    Relationship sources · 1

    Linked statements

    • For i u-t=−Delta-u−lambda|u|^alpha u on R-n, real lambda and 0<alpha<4/n, every L2 initial datum has a unique local mild solution in C([0,T];L2) intersected with L-q(0,T;L^(alpha+2)), where q=4(alpha+2)/(n alpha). Strichartz estimates and a contraction argument control the nonlinear integral map, and the local solution map is locally Lipschitz. This L2-subcritical theorem does not extend automatically to critical/supercritical powers or arbitrary boundary geometries.

    The free propagator is defined using Fourier analysis.

  3. Related to
    Relationship sources · 1

    Linked statements

    • For i u-t=−Delta-u−lambda|u|^alpha u on R-n, real lambda and 0<alpha<4/n, every L2 initial datum has a unique local mild solution in C([0,T];L2) intersected with L-q(0,T;L^(alpha+2)), where q=4(alpha+2)/(n alpha). Strichartz estimates and a contraction argument control the nonlinear integral map, and the local solution map is locally Lipschitz. This L2-subcritical theorem does not extend automatically to critical/supercritical powers or arbitrary boundary geometries.

    Nonlinear dispersive existence uses different estimates from a linear parabolic energy method.

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research-math-subcritical-nls-existence
Revision
research.1
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Postgraduate & research reference