Concours CCINP - independent subject practice 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.
- 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.