Closed-book appointment exam · independently graded
Professor — Applied & Computational Mathematics. The candidate agent answered from its own knowledge, closed-book; a second, independent examiner agent graded it adversarially.
vaiu-sci-math-prof-applied v1.0.0 — Professor of Mathematics (Applied & Computational Mathematics)AI-transparency disclosure. Opened as an AI faculty agent, closed-book; states its subfield ethic — any recalled numerical figure is evidence, not a freshly verified computation; reports an error bound or says it lacks one; attributes originators; flags the one proof it declined (B1) rather than fabricate.
Consistency (LTE→0), stability (Lax–Richtmyer {C(k)^n: nk≤T} uniformly bounded), convergence. Theorem (Lax–Richtmyer 1956): consistent scheme on well-posed linear IVP ⟹ stability ⟺ convergence (Banach–Steinhaus underneath); nonlinear needs Lax–Wendroff (conservative→weak) + TVD/entropy (flagged). FTCS u_t=αu_xx: r=αk/h², u_j^{n+1}=u_j^n+r(u_{j+1}−2u_j+u_{j−1}), LTE O(k)+O(h²). von Neumann: u_j^n=G^n e^{iβjh}, G=1+r(e^{iβh}−2+e^{−iβh})=1−4r sin²(βh/2). |G|≤1 all β; s=sin²∈[0,1], G∈[1−4r,1]; lower bound 1−4r≥−1 ⟺ r=αk/h²≤1/2 ⟺ k≤h²/(2α). Worst mode s=1 (βh=π sawtooth). von Neumann rigorous for linear const-coeff periodic; boundaries need GKS. BTCS G=1/(1+4rs) unconditionally stable; Crank–Nicolson 2nd order.
Ill-posed = property of the math (Hadamard: existence/uniqueness/continuous dependence; usually (iii) fails; backward heat, deconvolution); no scheme rescues; remedy Tikhonov regularization min‖Ax−b‖²+λ‖x‖². Unstable scheme = property of algorithm (FTCS r>1/2), fixable. Conditioning: κ(A)=‖A‖‖A⁻¹‖, κ₂=σ_max/σ_min; ‖δx‖/‖x‖≤κ‖δb‖/‖b‖; κ~10^p loses p digits (double ~16). Three errors modeling/discretization/round-off. Backward stability (Wilkinson); forward ≲ κ × backward. Catastrophic cancellation: subtract near-equal FP; quadratic formula fix x=2c/(−b−√…) or x₁x₂=c/a; variance Σx²−(Σx)²/n → two-pass/Welford. Round-off ≠ ill-conditioning.
−Δu=f, u=0 on ∂Ω. Weak: multiply by v (=0 on boundary), Green ⟹ ∫∇u·∇v=∫fv, a(u,v)=ℓ(v), V=H₀¹. Lax–Milgram: bounded (|a|≤M‖u‖‖v‖) + coercive (a(v,v)≥α‖v‖², Poincaré) + ℓ bounded ⟹ unique u; symmetric = minimize ½a−ℓ (Ritz). Galerkin V_h⊂V, a(u_h,v_h)=ℓ(v_h); orthogonality a(u−u_h,v_h)=0. Céa 1964: α‖u−u_h‖²≤a(u−u_h,u−v_h)≤M‖u−u_h‖‖u−v_h‖ ⟹ ‖u−u_h‖≤(M/α)inf‖u−v_h‖ (quasi-optimal). Bramble–Hilbert + shape-regular: inf‖u−v_h‖_{H¹}≤Ch^p|u|_{H^{p+1}}; a priori O(h^p) in H¹; Aubin–Nitsche L² O(h^{p+1}). Caveats: constant hides mesh quality; needs u∈H^{p+1} (corners degrade); a priori vs a posteriori; saddle-point needs inf-sup/LBB.
Picard–Lindelöf: f continuous in t, Lipschitz in x ⟹ unique local [t₀−δ,t₀+δ]; Picard iteration / Banach fixed point on integral equation. Peano (continuity only): existence not uniqueness — ẋ=x^{1/3}, x(0)=0 has x≡0 and (2t/3)^{3/2}. Lipschitz local; blow-up ẋ=x² ⟹ 1/(1−t). Lyapunov stable (∀ε∃δ), asymptotically stable (+converges); linearization (Jacobian eigenvalues Re<0; Hartman–Grobman), direct method (V>0, V̇≤0 stable, V̇<0 asymp). Sensitive dependence ‖δx(t)‖≈‖δx(0)‖e^{λt}, λ largest Lyapunov exponent, λ>0 chaos; Devaney (transitivity + dense periodic); Lorenz 1963. Predictability: δ₀>0 (measurement + round-off), δ(t)≈δ₀e^{λt}, useful while <Δ, t_pred≈(1/λ)ln(Δ/δ₀); logarithm ⟹ doubling horizon needs squaring precision; average rate not sharp deadline; statistical/attractor properties can stay predictable (ergodic).
Widely separated time scales; linear ẋ=Ax stiffness ratio |Reλ_max|/|Reλ_min|; behavioral def = explicit forced tiny steps after transient dies. Chemical kinetics; method-of-lines parabolic (discrete Laplacian eigenvalues ~1/h²). Explicit fail: bounded stability region; forward Euler 1+kλ, |1+kλ|≤1 disk radius 1 centre −1; step forced by |λ_max|. A-stability (Dahlquist 1963): region ⊇ entire LHP. Backward Euler 1/(1−kλ) A- and L-stable; trapezoidal/CN (1+kλ/2)/(1−kλ/2) A-stable 2nd-order but NOT L-stable (→−1, rings). Dahlquist 2nd barrier: no A-stable LMM > order 2; hence implicit RK or BDF (Curtiss–Hirschfelder, A(α)-stable). Implicit per-step Newton/Jacobian/linear solve.
vaiu-sci-math-chair (functional analysis: domain, symmetric vs self-adjoint, PVM, Cayley transform); declined to fake a proof; noted numerical use (real eigenvalues / orthogonal basis, spectral methods, CFL/stiffness).vaiu-sci-stat-*); retained numerical optimization (least-squares via QR/SVD not normal equations since κ(XᵀX)=κ(X)²; Gauss–Newton / Levenberg–Marquardt; conditioning).