Ingénieur ITPE - independent mathematics, physics and infrastructure foundations flashcards

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Pointwise convergence

Convergence at each fixed point with an index that may depend on that point, which does not alone control errors across the whole domain.

Uniform convergence

Convergence in which one sequence index controls the prescribed error simultaneously for every point in the stated domain.

Dominating function

One integrable absolute bound applying to the entire sequence in a dominated-convergence argument, rather than an individually chosen bound for each term.

Normal convergence of a function series

Convergence of the series of supremum norms in the specified setting, a sufficient condition for uniform absolute control of the series.

Radius of convergence

The distance from the center within which a power series converges absolutely, with boundary points requiring their own analysis.

Hessian

The matrix of second partial derivatives under suitable differentiability, used with its quadratic form to analyze local behavior near stationary points.

Constraint qualification

A condition on constraints that supports the applicability of an optimization theorem, so formal multiplier equations are not used without hypotheses.

Gram matrix

The matrix of pairwise inner products of basis vectors, used to compute projection coefficients when the basis is not orthonormal.

Orthogonal projection

The subspace element whose residual is orthogonal to that subspace under the stated inner product, with existence depending on the relevant setting.

Congruence transformation

A matrix transformation of a bilinear or quadratic form of the type P* A P, distinct from similarity of linear operators.

Spectral radius

The supremum or maximum modulus of spectral values in the relevant setting, distinct from the largest singular value of a general matrix.

Operator norm

A norm measuring the largest output magnitude for unit input under specified vector norms, which must not be confused with an arbitrary matrix entry.

Condition number

A measure of sensitivity of a problem’s solution to specified relative input perturbations, depending on the norm and formulation used.

Numerical stability

Control of the effects of computational perturbations and round-off by an algorithm, distinct from conditioning of the exact mathematical problem.

Residual

The discrepancy obtained by substituting an approximate solution into the original equation, which may not directly equal the solution error.

Newton iteration

A root-solving update using a function value and derivative, whose local convergence depends on root and initialization conditions.

Absolute stability

A property of a numerical evolution method controlling growth on a specified test equation and step size, separate from approximation accuracy.

Generalized coordinate

An independent coordinate describing a constrained mechanical configuration, chosen consistently with the model’s degrees of freedom.

Normal mode

A characteristic coupled oscillation pattern with a shared frequency obtained from the appropriate mass and stiffness eigenvalue problem.

Reversible adiabatic process

An idealized process with no heat transfer and no entropy production, unlike an arbitrary irreversible adiabatic process.

Effective stress

The total normal stress less pore-water pressure in a specified saturated-soil model, relevant to the skeleton response and drained strength.

Hydraulic gradient

The hydraulic-head difference divided by the stated flow-path length, with direction and the actual groundwater model identified.

Pore pressure

The fluid pressure within ground voids, distinguished from total overburden stress and from effective stress carried through the soil skeleton.

Consolidation

Time-dependent volume change associated with pore-fluid dissipation and load transfer in a specified soil drainage and compressibility model.

Undrained response

Ground response evaluated for conditions where the relevant loading duration does not allow the assumed drainage to occur.

Negative skin friction

Downward shaft action mobilized where surrounding ground settles relative to a pile, requiring compatible displacement and load assessment.

Bearing capacity

A model of ground resistance to a specified foundation load mechanism, separate from settlement acceptability and structural foundation resistance.

Active earth pressure

A limiting lateral ground-pressure condition requiring sufficient compatible outward wall movement under the stated soil and water model.

At-rest earth pressure

Lateral pressure associated with a constrained ground deformation condition, not automatically equivalent to mobilized active or passive resistance.

Limit state

A specified boundary describing an unacceptable strength, stability or serviceability condition within the actual assessment framework and loading scenario.