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3364Shadow Price InterpretationWhy is the optimal multiplier often interpreted as the marginal value of relaxing or tightening a constraint?数学中等essay未尝试面试订阅3365Why Slater-Type Regularity MattersWhy do regularity conditions such as Slater's condition matter when applying KKT?数学中等essay未尝试面试订阅3399Balancing Truncation and Rounding ErrorSuppose a numerical derivative has total error model E(h)=3h 2 + 10 -4 /h. What h minimizes this approximation?数学中等derivation未尝试面试订阅3400Minimum Error at the Balanced MeshUsing E(h)=3h 2 + 10 -4 /h and the optimal h from first-order balancing, what is the minimum modeled error?数学中等derivation未尝试面试订阅3401Explicit Heat Stability Cap 1For an explicit heat-equation update, diffusion coefficient kappa = 0.5, and grid spacing Delta x = 0.1, what is the largest stable time step under the standard Delta t <= Delta x 2 / (2 kappa) rule?数学中等derivation未尝试面试订阅3402Explicit Heat Stability Cap 2For an explicit heat update with kappa = 1.2 and Delta x = 0.05, what is the largest stable Delta t under the same rule?数学中等derivation未尝试面试订阅3403Upwind Advection CFL LimitAn explicit upwind advection scheme has speed u = 2 and grid spacing Delta x = 0.03. What is the largest stable Delta t under the CFL rule Delta t <= Delta x / |u|?数学中等derivation未尝试面试订阅3404Advection-Diffusion Explicit Time-Step Rule 1Use the simple sufficient rule Delta t <= 1 / (|u|/Delta x + 2 kappa / Delta x 2) for an explicit advection-diffusion update. If u = 1, kappa = 0.2, and Delta x = 0.1, what is the resulting Delta t cap?数学中等derivation未尝试面试订阅3405Advection-Diffusion Explicit Time-Step Rule 2Using the same sufficient explicit rule Delta t <= 1 / (|u|/Delta x + 2 kappa / Delta x 2), what is the Delta t cap when u = 0.5, kappa = 0.1, and Delta x = 0.05?数学中等derivation未尝试面试订阅3406Zero-Flux Ghost PointAt a left boundary, a zero-flux Neumann condition is approximated by (u 1 - u -1 )/(2h)=0. What relation does this impose on the ghost point u -1 ?数学中等derivation未尝试面试订阅3407Ghost Point Under a Positive Boundary SlopeAt x=0, approximate u x(0)=3 using (u 1-u -1 )/(2h) with h=0.1 and u 1=1.4. What is the ghost point value u -1 ?数学中等derivation未尝试面试订阅3408Boundary Curvature From a Nonzero Slope ConditionSuppose u(0)=4, u 1=5, h=0.25, and the left-boundary slope satisfies u x(0)=-2 approximated by (u 1-u -1 )/(2h). What second derivative estimate do you get from (u -1 - 2u 0 + u 1)/h 2?数学中等derivation未尝试面试订阅3409Boundary Curvature Under Zero FluxSuppose u(0)=7, u 1=6, h=0.5, and the left boundary has zero flux. What second derivative estimate follows from the ghost-point stencil (u -1 -2u 0+u 1)/h 2?数学中等derivation未尝试面试订阅3411Why Richardson Extrapolation HelpsWhy does Richardson extrapolation often improve an estimate even when you never see the exact derivative value?数学中等essay未尝试面试订阅3412Why Smaller h Is Not Always BetterWhy can making h smaller eventually worsen a finite-difference estimate?数学中等essay未尝试面试订阅3413Why Diffusion Stability Scales With Delta x SquaredWhy does the explicit heat-equation time-step restriction shrink like Delta x squared rather than just like Delta x?数学中等essay未尝试面试订阅3414Why Boundary Conditions Change the StencilWhy can the same derivative operator require a different finite-difference stencil near a boundary than in the interior?数学中等essay未尝试面试订阅3415Why Central Differences Are Usually More Accurate Than One-Sided OnesWhy do central differences often achieve a higher order of accuracy than one-sided differences using similar nearby points?数学中等essay未尝试面试订阅3481Expected Questions in a Hot-State-First Binary TreeA binary decision tree asks first whether the state is S1. If not, it asks whether it is S2. If not, it asks whether it is S3, and otherwise concludes S4. If prior probabilities are [0.5, 0.25, 0.15, 0.10], what is the expected number of questions?数学中等derivation未尝试面试订阅3482Expected Questions for a Balanced Three-State TriageA binary tree first asks whether the state is in S1, S2 . If yes, one more question separates S1 from S2. If no, the answer is S3. If prior probabilities are [0.5, 0.25, 0.25], what is the expected number of questions?数学中等derivation未尝试面试订阅