Concours X-ENS-ESPCI - independent subject practice flashcards

30 cards. Read the term and the definition here. The deck at the top of this page is optional.

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.

Mole

The SI amount-of-substance unit; one mole contains exactly 6.02214076 × 10^23 specified elementary entities. The type of entity must be identified when converting particle counts to amounts.

Molar mass

The mass of one mole of a substance, expressed in grams per mole and numerically equal to its relative formula mass. It converts between mass and amount in stoichiometric calculations.

Limiting reagent

The reactant that is used up first in a reaction and therefore fixes the maximum amount of product formed. Other reactants are said to be in excess.

Percentage yield

The actual mass of product obtained expressed as a percentage of the theoretical maximum calculated from stoichiometry. Losses arise from side reactions, incomplete reaction and handling.

Enthalpy change

The heat transferred at constant pressure during a reaction, given the symbol delta H. Negative values indicate exothermic processes; positive values indicate endothermic ones.

Hess's law

The principle that the total enthalpy change of a reaction is independent of the route taken, so indirect paths can be combined. It allows calculation of changes that cannot be measured directly.

Bond enthalpy

The average energy required to break one mole of a particular bond in gaseous molecules. Summing broken minus formed bond enthalpies estimates the enthalpy change of a gas-phase reaction.

Calorimetry

The experimental measurement of heat exchanged, usually by tracking a temperature change in a known mass of water or solution. The result is calculated with q = mc delta T.

Activation energy

The minimum extra energy reacting particles must possess for a collision to produce products. Raising temperature or adding a catalyst increases the fraction of collisions that succeed.

Catalyst

A substance that speeds a reaction by providing an alternative pathway of lower activation energy, without being consumed overall. It changes rate but not the position of equilibrium.